STP 1387
Multiaxial Fatigue and Deformation: Testing and Prediction Sreeramesh Kalluri and Peter J. Bonacuse, editors
ASTM Stock Number: STP1387
ASTM 100 Barr Harbor Drive P.O. Box C700 West Conshohocken, PA 19428-2959 Printed in the U.S.A.
Library of Congress Cataloging-in-Publication Data Multiaxial fatigue and deformation: testing and prediction/Sreeramesh Kalluri and Peter J. Bonacuse, editors. p. cm.--(STP; 1387) "ASTM stock number: STP 1387." Includes bibliographical references and index. ISBN 0-803-2865-7 1. Materials-Fatigue. 2. Axial loads. 3. Materials-Dynamic testing. 4. Deformations (Mechanics) I. Kalluri, Sreeramesh. I1. Bonacuse, Peter J., 1960TA418.38.M86 2000 620.11126-dc21 00-059407
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Foreword
This publication, Multiaxial Fatigue and Deformation: Testing and Prediction, contains papers presented at the Symposium on Multiaxial Fatigue and Deformation: Testing and Prediction, which was held in Seattle, Washington during 19-20 May 1999. The Symposium was sponsored by the ASTM Committee E-8 on Fatigue and Fracture and its Subcommittee E08.05 on Cyclic Deformation and Fatigue Crack Formation. Sreeramesh Kalluri, Ohio Aerospace Institute, NASA Glenn Research Center at Lewis Field, and Peter J. Bonacuse, Vehicle Technology Directorate, U.S. Army Research Laboratory, NASA Glenn Research Center at Lewis Field, presided as symposium co-chairmen and both were editors of this publication.
Contents Overview
vii MULTIAXIAL STRENGTH OF MATERIALS
Keynote Paper: Strength of a G-10 Composite Laminate Tube Under Multiaxial Loading--D. SOCrEANDJ. WANG Biaxial Strength Testing of Isotropic and Anisotropic Monoliths--J. A. SALE~AND
1 13
M. G. JENKINS
In-Plane Biaxial Failure Surface of Cold-Rolled 304 Stainless Steel Sheets--s. J. COVEY AND P. A. BARTOLOTrA
26
MULTIAXIAL DEFORMATION OF MATERIALS
Analysis of Characterization Methods for Inelastic Composite Material Deformation Under Multiaxial Stresses--J. AHMAD, G. M. NEWAZ, AND T. NICHOLAS Deformation and Fracture of a Particulate MMC Under Nonradial Combined Loadings--D. w. A. REESAND Y. H. J. AU M u l t i a x i a l S t r e s s - S t r a i n N o t c h Analysis--A. BUCZYNSKI AND G. GLINKA Axial-Torsional Load Effects of Haynes 188 at 650 ~ C----c. J. LlSSENDEN,M. a. WALKER, AND B. A. LERCH
A Newton Algorithm for Solving Non-Linear Problems in Mechanics of Structures Under Complex Loading Histories--M. ARZT,W. BROCKS,ANDR. MOHR
41 54 82
99 126
FATIGUE LIFE PREDICTION UNDER GENERIC MULTIAXIAL LOADS
A Numerical Approach for High-Cycle Fatigue Life Prediction with Multiaxial Loading--M. DE FREITAS, B. LI, AND J. L. T. SANTOS Experiences with Lifetime Prediction Under Multlaxial Random Loading--K. POTTER,F. YOUSEFI, AND H. ZENNER
Generalization of Energy-Based Multiaxial Fatigue Criteria to Random Loading--T. LAGODA AND E. MACHA Fatigue Strength of Welded Joints Under Multiaxial Loading: Comparison Between Experiments and Calculations--M. WITT,F. YOUSEFLANDH. ZENNER
139 157 173 191
FATIGUE LIFE PREDICTION UNDER SPECIFIC MULT1AXIAL LOADS
The Effect of Periodic Overloads on Biaxial Fatigue of Normalized SAE 1045 Steel--J. J. F. BONNEN AND T. H. TOPPER Fatigue of the Quenched and Tempered Steel 42CrMo4 (SAE 4140) Under Combined In- and Out-of-Phase Tension and Torsion---a. LOVaSCH,~. BOMAS,AND P. MAYR
In-Phase and Out-of-Phase Combined Bendlng-Torsion Fatigue of a Notched Specimen--J. PARKANDD. V. NELSON
213
232 246
vi
CONTENTS
The Application of a Biaxial Isothermal Fatigue Model to Thermomechanical Loading for Austenitic Stainless Steel--s. v. ZAMRIKANDM.L. RENAULD Cumulative Axial and Torsional Fatigue: An Investigation of Load-Type Sequencing Effects--s. KALLURI AND P. J. BONACUSE
266 281
MULTIAXIAL FATIGUE LIFE AND CRACK GROWTH ESTIMATION
A New Multiaxial Fatigue Life and Crack Growth Rate Model for Various In-Phase and Out-of-Phase Strain Paths--A. VARVANI-FARAHANIANDT. H. TOPPER Modeling of Short Crack Growth Under Biaxial Fatigue: Comparison Between Simulation and Experiment--H.A. SUHARTONO, K. POTTER, A. SCHRAM, AND H. ZENNER
305
323
Micro-Crack Growth Modes and Their Propagation Rate Under Multiaxial Low-Cycle Fatigue at High Temperature--N. ISOBEANDS. SAKURAI
340
MULTIAXIAL EXPERIMENTAL TECHNIQUES
Keynote Paper: System Design for Multiaxial High-Strain Fatigue Testing--R. D. LOHR An In-Plane Biaxial Contact Extensometer--o. L. KRAUSEANDP. A. BARTOLOTTA Design of Specimens and Reusable Fixturing for Testing Advanced Aeropropulsion Materials Under In-Plane Biaxial Loading--J. R. ELLIS,G. S. SANDLASS,AND M. BAYYARI
Cruciform Specimens for In-Plane Biaxiai Fracture, Deformation, and Fatigue Testing----c. DALLE DONNE, K.-H. TRAUTMANN, AND H. AMSTUTZ Development of a True Trlaxlal Testing Facility for Composite Materials--J. s. WELSH AND D. F. ADAMS
Indexes
355 369
382 405 423 439
Overview Engineering materials are subjected to multiaxial loading conditions routinely in aeronautical, astronautical, automotive, chemical, power generation, petroleum, and transportation industries. The extensive use of engineering materials over such a wide range of applications has generated extraordinary interest in the deformation behavior and fatigue durability of these materials under multiaxial loading conditions. Specifically, the technical areas of interest include strength of the materials under multiaxial loading conditions, multiaxial deformation and fatigue of materials, and development of multiaxial experimental capabilities to test materials under controlled prototypical loading conditions. During the last 18 years, the American Society for Testing and Materials (ASTM) has sponsored four symposia to address these technical areas and to disseminate the technical knowledge to the scientific community. Three previously sponsored symposia have yielded the following Special Technical Publications (STPs): (1) Multiaxial Fatigue, ASTM STP 853, (2) Advances in Multiaxial Fatigue, ASTM STP 1191, and (3) Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280. This STP is the result of the fourth ASTM symposium on the multiaxial fatigue and deformation aspects of engineering materials. A symposium entitled "Multiaxial Fatigue and Deformation: Testing and Prediction" was sponsored by ASTM Committee E-8 on Fatigue and Fracture and its Subcommittee E08.05 on Cyclic Deformation and Fatigue Crack Formation. The symposium was held during 19-20 May 1999 in Seattle, Washington. The symposium's focus was primarily on state-of-the-art multiaxial testing techniques and analytical methods for characterizing the fatigue and deformation behaviors of engineering materials. The objectives of the symposium were to foster interaction in the areas of multiaxial fatigue and deformation among researchers from academic institutions, industrial research and development establishments, and government laboratories and to disseminate recent developments in analytical modeling and experimental techniques. All except one of the 25 papers in this publication were presented at the symposium. Technical papers in this publication are broadly classified into the following six groups: (1) Multiaxial Strength of Materials, (2) Multiaxial Deformation of Materials, (3) Fatigue Life Prediction under Generic Multiaxial Loads, (4) Fatigue Life Prediction under Specific Multiaxial Loads, (5) Multiaxial Fatigue Life and Crack Growth Estimation, and (6) Multiaxial Experimental Techniques. This classification is intended to be neither exclusive nor all encompassing for the papers published in this publication. In fact, a few papers overlap two or more of the categories. A brief outline of the papers for each of the six groups is provided in the following sections. Multiaxial Strength of Materials
Multiaxial strengths of metallic and composite materials are commonly investigated with either tubular or cruciform specimens. Three papers in this section address multiaxial strength characterization of materials. The first, and one of the two keynote papers in this publication, describes an experimental study on the strength and failure modes of woven glass fiber/epoxy matrix, laminated composite tubes under several combinations of tensile, compressive, torsional, internal pressure, and external pressure loads. This investigation illustrated the importance of failure modes in addition to the states of stress for determining the failure envelopes for tubular composite materials. The second paper describes a test rig for biaxial flexure strength testing of isotropic and anisotropic materials with the pressure-on-ring approach. The tangential and radial stresses generated in the disk specimens and the strains measured at failure in the experiments are compared with the theoretical predictions. The vii
viii
OVERVIEW
third paper deals with in-plane biaxial testing of cruciform specimens manufactured from thin, coldrolled, 304 stainless steel sheets. In particular the influence of texture, which occurs in the material from the rolling operation, on the effective failure stress is illustrated and some guidelines are proposed to minimize the rejection rates while forming the thin, cold-rolled, stainless steel into components.
Multiaxial Deformation of Materials Constitutive relationships and deformation behavior of materials under multiaxial loading conditions are the subjects of investigation f6r the five papers in this section. The first paper documents detailed analyses of tests performed on off-axis tensile specimens and biaxially loaded cruciform specimens of unidirectional,fiber reinforced, metal matrix composites. The simplicity associated with the off-axis tensile tests to characterize the nonlinear stress-strain behavior of a unidirectional composite under biaxial stress states is illustrated. In addition, the role of theoretical models and biaxial cruciform tests for determining the nonlinear deformation behavior of composites under multiaxial stress states is discussed. Deformation and fracture behaviors of a particulate reinforced metal matrix alloy subjected to non-radial, axial-torsional, cyclic loading paths are described in the second paper. Even though the composite's flow behavior was qualitatively predicted with the application of classical kinematic hardening models to the matrix material, it is pointed out that additional refinements to the model are required to properly characterize the experimentally observed deformation behavior of the composite material. The third paper describes a methodology for calculating the notch tip stresses and strains in materials subjected to cyclic multiaxial loading paths. The Mroz-Garud cyclic plasticity model is used to simulate the stress-strain response of the material and a formulation based on the total distortional strain energy density is employed to estimate the elasto-plastic notch tip stresses and strains. The fourth paper contains experimental results on the elevated temperature flow behavior of a cobalt-base superalloy under both proportional and nonproportional axial and torsional loading paths. The database generated could eventually be used to validate viscoplastic models for predicting the multiaxial deformation behavior of the superalloy. Deformation behavior of a rotating turbine disk is analyzed with an internal variable model and a Newton algorithm in conjunction with a commercial finite element package in the fifth paper. Specifically, the inelastic stress-strain responses at the bore and the neck of the turbine disk and contour plots depicting the variation of hoop stress with the number of cycles are discussed.
Fatigue Life Prediction under Generic Multiaxial Loads Estimation of fatigue life under general multiaxial loads has been a challenging task for many researchers over the last several decades. Four papers in this section address this topic. The first paper proposes a minimum circumscribed ellipse approach to calculate the effective shear stress amplitude and mean value for a complex multiaxial loading cycle. Multiaxial fatigue data with different waveforms, frequencies, out-of-phase conditions, and mean stresses are used to validate the proposed approach. Multiaxial fatigue life predictive capabilities of the integral and critical plane approaches are compared in the second paper for variable amplitude tests conducted under bending and torsion on smooth and notched specimens. Fatigue life predictions by the two approaches are compared with the experimental results for different types of multiaxial tests (pure bending with superimposed mean shear stress; pure torsion with superimposed mean tensile stress; and in-phase, 90 ~ out-of-phase, and noncorrelated bending and torsional loads) and the integral approach has been determined to be better than the critical plane approach. In the third paper, a generalized energy-based criterion that considers both the shear and normal strain energy densities is presented for predicting fatigue life under multiaxial random loading. A successful application of the energy method to estimate the fatigue lives under uniaxial and biaxial nonproportional random loads is illustrated. Estimation of the fatigue lives of welded joints subjected to multiaxial loads is the subject of the fourth paper. Experimental results on flange-tube type welded joints subjected to cyclic bending and torsion are reported and a
OVERVIEW
ix
fatigue lifetime prediction software is used to calculate the fatigue lives under various multiaxial loading conditions.
Fatigue Life Prediction under Specific Multiaxial Loads Biaxial and multiaxial fatigue and life estimation under combinations of cyclic loading conditions such as axial tension/compression, bending, and torsion are routinely investigated to address specific loading conditions. Five papers in this publication address such unique issues and evaluate appropriate life prediction methodologies. The effects of overloads on the fatigue lives of tubular specimens manufactured from normalized SAE 1045 steel are established in the first paper by performing a series of biaxial, in-phase, tension-torsion experiments at five different shear strain to axial strain ratios. The influence of periodic overloads on the endurance limit of the steel, variation of the crack initiation and propagation planes due to changes in the strain amplitudes and strain ratios, and evaluation of commonly used multiaxial damage parameters with the experimental data are reported. Combined in- and out-of-phase tension and torsion fatigue behavior of quenched and tempered SAE 4140 steel is the topic of investigation for the second paper. Cyclic softening of the material, orientation of cracks, and fatigue life estimation under in- and out-of-phase loading conditions, and calculation of fatigue limits in the normal stress and shear stress plane both with and without the consideration of residual stress state are reported. High cycle fatigue behavior of notched 1%Cr-Mo-V steel specimens tested under cyclic bending, torsion, and combined in- and out-of-phase bending and torsion is discussed in the third paper. Three multiaxial fatigue life prediction methods (a von Mises approach, a critical plane method, and an energy-based approach) are evaluated with the experimental data and surface crack growth behavior under the investigated loading conditions is reported. The fourth paper illustrates the development and application of a biaxial, thermomechanical, fatigue life prediction model to 316 stainless steel. The proposed life prediction model extends an isothermal biaxial fatigue model by introducingfrequency and phase factors to address time dependent effects such as creep and oxidation and the effects of cycling under in- and out-of-phase thermomechanical conditions, respectively. Cumulative fatigue behavior of a wrought superalloy subjected to various single step sequences of axial and torsional loading conditions is investigated in the fifth paper. Both high/low load ordering and load-type sequencing effects are investigated and fatigue life predictive capabilities of Miner's linear damage rule and the nonlinear damage curve approach are discussed.
Multiaxial Fatigue Life and Crack Growth Estimation Monitoring crack growth under cyclic rnultiaxial loading conditions and determination of fatigue life can be cumbersome. In general, crack growth monitoring is only possible for certain specimen geometries and test setups. The first paper proposes a multiaxial fatigue parameter that is based on the normal and shear energies on the critical plane and discusses its application to several materials tested under various in- and out-of-phase axial and torsional strain paths. The parameter is also used to derive the range of an effective stress intensity factor that is subsequently used to successfully correlate the closure free crack growth rates under multiple biaxial loading conditions. The second paper on modelling of short crack growth behavior under biaxial fatigue received the Best Presented Paper Award at the symposium. The surface of a polycrystalline material is modeled as hexagonal grains with different crystallographic orientations and both shear (stage I) and normal (stage II) crack growth phases are simulated to determine crack propagation. Distributions of microcracks estimated with the model are compared with experimental results obtained for a ferritic steel and an aluminum alloy subjected various axial and torsional loads. Initiation of fatigue cracks and propagation rates of cracks developed under cyclic axial, torsional, and combined axial-torsional loading conditions are investigated for 316 stainless steel, 1Cr-Mo-V steel, and Hastelloy-X in the third paper. For each material, fatigue microcrack initiation mechanisms are identified and appropriate strain parameters to correlate the fatigue crack growth rates are discussed.
X
OVERVIEW
Multiaxial Experimental Techniques State-of-the-art experimental methods and novel apparati are necessary to generate multiaxial deformation and fatigue data that are necessary to develop and verify both constitutive models for describing the flow behavior of materials and fatigue life estimation models. Five papers in this publication address test systems, extensometers, and design of test specimens and fixtures to facilitate multiaxial testing of engineering materials. The second of the two keynote papers reviews progress made in the design of multiaxial fatigue testing systems over the past five decades. Different types of loading schemes for tubular and planar specimens and the advantages and disadvantages associated with each of those schemes are summarized in the paper. Development of an extensometer system for conducting in-plane biaxial tests at elevated temperatures is described in the second paper. Details on the calibration and verification of the biaxial extensometer system and its operation under cyclic loading conditions at room temperature and static and cyclic loading conditions at elevated temperatures are discussed. Designing reusable fixtures and cruciform specimens for in-plane biaxial testing of advanced aerospace materials is the topic of investigation for the third paper. Feasibility of a fixture arrangement with slots and fingers to load the specimens and optimal specimen designs are established with finite element analyses. Details on three types of cruciform specimens used for biaxial studies involving fracture mechanics, yield surfaces, and fatigue of riveted joints are described in the fourth paper. Methods used for resolving potentially conflicting specimen design requirements such as uniform stress distribution within the test section and low cost of fabrication are discussed for the three types of specimens. The final paper describes the development and evaluation of a computer-controlled, electromechanical test system for characterizing mechanical behavior of composite materials under biaxial and triaxial loading conditions. Verification of the test system with uniaxial and biaxial tests on 6061-T6 aluminum, biaxial and triaxial test results generated on a carbon/epoxy cross-ply laminate, and proposed modifications to the test facility and specimen design to improve the consistency and accuracy of the experimental data are discussed. The papers published in this book provide glimpses into the technical achievements in the areas of multiaxial fatigue and deformation behaviors of engineering materials. It is our sincere belief that the information contained in this book describes state-of-the-art advances in the field and will serve as an invaluable reference material. We would like to thank all the authors for their significant contributions and the reviewers for their critical reviews and constructive suggestions for the papers in this publication. We are grateful to the excellent support received from the staff at ASTM. In particular, we would like to express our gratitude to the following individuals: Ms. Dorothy Fitzpatrick, Ms. Hannah Sparks, and Ms. Helen Mahy for coordinating the symposium in Seattle, Washington; Ms. Monica Siperko for efficiently managing the reviews and revisions for all the papers; and Ms. Susan Sandler and Mr. David Jones for coordinating the compilation and publication of the STP.
Sreeramesh Kalluri Ohio Aerospace Institute NASA Glenn Research Center at Lewis Field Cleveland, Ohio Symposium Co-Chairman and Editor
Peter J. Bonacuse Vehicle Technology Directorate US. Army Research Laboratory NASA Glenn Research Center at Lewis Field Cleveland, Ohio Symposium Co-Chairman and Editor
Multiaxial Strength of Materials
Darell Socie a and Jerry Wang 2
Strength of a G-IO Composite Laminate Tube Under Multiaxial Loading REFERENCE: Socie, D. and Wang, J., "Strength of a G-10 Composite Laminate Tube Under Multiaxial Loading," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 138Z S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 3-12.
ABSTRACT: An experimental study of the strength and failure behavior of an orthotropic G-10 glass fiber-reinforced epoxy laminate has been conducted. Tubular specimens were loaded in combinations of tension, compression, torsion, internal pressure, and external pressure to produce a variety of stress states. Previous work involved the loading of two simultaneously applied in-plane stresses. This investigation furthers the previous work by simultaneously applying three in-plane stresses. One interesting observation from this work is that combined axial compression and torsion loading results in a much lower failure strength than combined hoop compression and torsion even though the stress state is identical. For the same torsion stress, axial compression is more damaging than hoop compression because torsion loading rotates fibers aligned in the axial direction to accommodate the shear strains. Hoop fibers do not rotate and remain aligned in the compressive loading direction. A simple failure mode dependent maximum stress theory that considers low-energy compressive failure modes such as delamination and fiber buckling provides a reasonable fit to the experimental data. KEYWORDS: composite strength, multiaxial loading, failure theories
High specific strength and stiffness of composite materials make them attractive candidates for replacing metals in many weight-critical applications. Many of these applications involved complicated stress states. Although the behavior of composite materials has been studied for many years, much of the work on multiaxial stress states has been limited to theoretical studies and off-axis testing. Failure of composite materials is more complicated than monolithic materials because: (1) Failure modes of composite materials under a particular stress state are determined not only by their internal properties such as constituent properties and microstructural parameters, but also by geometric variables, loading type, and boundary conditions. (2) Stress caused by applied external loads does not distribute homogeneously between the fiber and matrix because of large differences between their elastic properties. From a strength viewpoint, composite materials cannot be considered as homogeneous anisotropic materials. Failure of composite materials is controlled by either the fiber, matrix, or interface between them, depending on the geometry and external loading. (3) Identical laminae have different behavior in various angle-ply laminates. Laminate failure is difficult to predict with only the lamina properties. Failure of composite laminates can be studied from many different levels: micromechanics, lamina, and laminates. Failure behavior of composite laminates is expected to be predicted by the properties of individual lamina which might be obtained from basic properties of the resin and matrix. Mechanical Engineering Department, University of Illinois at Urbana-Champaign, Urbana, IL. 2 Ford Motor Company, Dearborn, MI.
Copyright9
by ASTM International
3 www.astm.org
4
MULTIAXIALFATIGUE AND DEFORMATION
However, the failure behavior of composite laminates in a structure is much more complicated. This complication is demonstrated in that failure behavior among constituents, lamina, and laminate are quite different. Lamina properties, particularly those involving in-plane shear, are not easily obtained from the properties of the constituents. Interaction between the fiber and resin cannot be predicted from properties of the constituents. In composite structural analysis, laminate properties are frequently obtained from laminate theory with properties of the lamina obtained from experiments. To model complicated behavior of a structure, various anisotropic strength criteria have been developed for both the lamina and laminate level. Anisotropic strength theories may be classified broadly into one of three categories. In the first category, anisotropic strength theories are failure mode dependent. Failure will occur if any or all of the longitudinal, transverse, or shear stresses or strains exceed the limits determined by unidirectional tests. The simplest forms include maximum stress and maximum strain theories. These simple estimates have been shown to overestimate the strength in the comer regions of the failure envelope [1]. Many extensions to these simple ideas have been made to accommodate different failure modes. For example, Hart-Smith [2] advocates cutting off the corners of the failure envelope to account for shear failure modes caused by in-plane principal stresses of opposite signs. In the second category, anisotropic strength theories are failure mode independent and a gradual transition from one failure mode to another is assumed. Although they have been developed many years ago, the Tsai [3] and Tsai-Wu [4] failure theories are still widely used failure criteria. Almost all failure mode independent strength criteria are in the form Fz + (Fijo-i~) '~ = 1, with or without nonlinear terms. For an in-plane loading ~x, ~y and ~-this criterion becomes FIt"x + F2o'y + F6T + (Fll O'2 + F22o-y2 + 2F12o'xO'y +
F66~'2) ~ =
1
(1)
The term F12o'x oy represents the interaction among stress components and is negative to account for shear produced by in-plane loadings of opposite signs. Jiang and Tennyson [5] have added cubic terms to Eq 1 in the form Fi + F i j ~ + Fijko-i~o'k = 1. These types of failure theories contain enough adjustable constants, Fi, to include many failure modes. In the absence of an applied shear stress, these criteria predict that composite laminates are stronger in biaxial tension or biaxial compression loading and are weaker under biaxial tension-compression loading. Although the parameters can be adjusted to fit different sets of test data, physical meaning of the parameters and resulting failure envelope described by these criteria are not very clear and can lead to unrealistic results when extrapolated outside the range of test data. The third group of models includes micromechanical theories where stresses and strains in the matrix and fibers are computed. Ardic et al. [6] use strains computed from classical lamination plate theory for the laminate as input to calculate the strains in each layer using a three-dimensional elasticity approach. Layer strains are then used to compute fiber and matrix stresses and strains. Failure surfaces are then constructed based on the allowable stresses and strains for the fiber, matrix, and lamina. Sun and T a t [7] have computed failure envelopes with linear laminated plate theory using a failure criterion that seperates fiber and matrix failure modes. Many lamina failure criteria and laminate failure analysis methods have been proposed [8]. Soden et al. [9] provides a good review of the predictive capabilities of failure theories for composite laminates. They reported that predicted failure loads for a quasi-isotropic carbon/epoxy laminate varied by as much as 1900% for the various failure theories considered. This paper presents new test results that explore the failure envelope for combined loading experiments utilizing glass fiber-reinforced epoxy G-10 laminate tubes. These results are combined with previous test results [10,11] on the same composite to evaluate the failure envelope for three simultaneously applied in-plane stresses.
SOCIE AND WANG ON MULTIAXIAL LOADING
5
TABLE 1--Loading combinations. 1
Tension Compression Internal pressure External pressure Torsion
2
3
4
5
X
6
7
X
X
X X
X X
X X
8
9
X X
X
10
11
12
13
X
14
X X X
X X
15
X
X
X X
X
X X
X
X
Experiments
In this study, a N E M A / A S T M G-10 epoxy resin reinforced laminate with E-glass plain woven fabric was used. This industrial composite was selected because it is commercially available in both sheet and tubular forms. E-glass plain woven fabric consists of fill and warp yarns crossing alternatively above and below the adjacent yarns along the entire length and width. Fiber volume fractions in the two perpendicular directions are slightly different such that the nominal fiber volume in the fill direction is about 75% of that in the warp direction. This results in a laminate with nearly equal tensile strengths in both directions. The laminates are stacked in plies with fill fibers in the same direction. Crimp angles, a measure of the waviness of the fibers, for both fill and warp yarns were less than l0 ~ Tubular specimens with an inside diameter of 45 mm and length of 300 mm were employed in this study with the fill fibers running along the axis of the tube. Specimens were mechanically ground to reduce the wall thickness from 5 to 3 mm to form a reduced gage section with a length of 100 mm. Specially designed test fixtures were used to achieve tensile or compressive stresses in the warp (hoop) direction. A mandrel was used for internal pressure tests to generate hoop tension. Hoop compression was obtained with an external pressure vessel that used high pressure seals on the grip diameter of the specimens. These fixtures were placed in a conventional tension-torsion servohydraulic testing system to generate the various combinations of in-plane loads given in Table 1. Fifteen different combinations of loading were used in the study. Failure is determined in the pressure loading experiments by a sudden loss of pressure. This corresponds to a longitudinal split in the tube. In torsion, failure is determined by excessive angular deformation which corresponds to a spiral crack around the circumference of the tube. Tension and compression failures are determined by a sudden drop in load. Additional details of the specimen and test system can be found in Ref 10. Results and D i s c u s s i o n
The failure envelope for combinations of biaxial tension and compression loading is shown in Fig. 1. These test results are shown by the open square symbols. The X symbols are the results of threeaxis loading and will be discussed later in the paper. Failure modes were determined by scanning electron microscope (SEM) observations of failed specimens and are indicated in the figure. The tensile strength in both the fill and warp directions is similar. Compressive strength in the warp direction is much lower than that in the fill direction for the tubular specimen. This is caused by a change in failure mode. In the fill direction, the failure mechanism is out-of-plane kinking of the fibers. Figure 2 illustrates the difference between in-plane and out-of-plane shear stresses for the composite laminate. Under this loading condition, the in-plane shear failure stress is twice as large as the outof-plane shear failure stress. Both sets of fill and warp fibers need to be broken for an in-plane failure while only one set of either fill or warp fibers needs to be broken for an out-of-plane or tensile failure. Delamination failures occurred during compressive loading in the warp direction. This is a
6
MULTIAXlAL FATIGUE AND DEFORMATION 400 -
Tensile fiber fracture
Out-of-plane kinking
q
.
j
EL -400 |
T
I
-200
200
I
I I
400
I I
_.~____~I Delamination / ~ -40(1 afu ~ , M P a
FIG. l--Biaxial tension-compression failure envelope.
common failure mode in hoop compression of a tubular specimen. In contrast to tube tests, small coupon specimens cut from fiat plates show the same compressive strength in both fill and warp directions and fail in a mode known as kink buckling. The shear cutoff predicted by many theories for tension-compressionloading is not observed in this material. For a stiff fiber and soft matrix, the interaction between fill and warp fibers will be small. External loads are carried by fibers parallel to the applied loads. Although each fiber is in an in-plane biaxial stress state, the transverse stress on a fiber is small because the more compliant matrix accommodates the transverse strain. A simple rule of mixtures approach based on fiber modulus, matrix modulus, and volume fraction shows that the transverse stresses in the fibers are less than 15% of the longitudinal fiber stress during equibiaxial tensile loading so that little interaction is expected between ten-
.4-----
1 In-plane shear
Out-of-plane shear
FIG. 2--1n-plane and out-of-plane shear stress.
SOCIE AND WANG ON MULTIAXIAL LOADING
7
sile loads in the fill and warp directions. The net result is that there is little interaction between loads in the axial and hoop direction and final composite failure is dictated by the lowest energy failure mode in either direction for all combinations of biaxial tension and compression loading along the fill and warp directions. Compressive loading in the hoop direction is expected to generate delamination failures between the plies. Kachanov [12] employed a simple energy analysis to model the delarnination buckling of composite tubes under external pressure. The critical compressive stress, o'er, of tubular specimens is given by
F(hol '
(R, 1"2 Kr~ho)J
+
~cr-- 0.916Ew I_\Ri)
(2) Kr = 4.77yEwRi where Ew is the composite elastic modulus in the warp direction, ho is the thickness of the buckled layer, Ri is the inner radius, and y is the specific fracture energy according to Griffith. The weak layer can be found by differentiating the above expression with the result ho = (Kr/2) 1/3 Ri. For the epoxy resin, y is about 700 J/m2 [13,14] and the critical failure stress is computed to be 200 MPa. This is about 18% higher than the experimental data. Experimental evidence of delaminationis shown in Fig. 8 of Ref 10. It is worth noting that compression and tension-compression tests of a flat plate G-10 laminate specimen did not show evidence of delamination [15]. For these tests, the compressive strengths in the fill and warp directions were the same and the failure envelope shown in Fig. 1 was a square. This shows the importance of considering the specimen design when evaluating failure criteria for any particular application. Two types of in-plane shear can be applied to a tubular specimen: (1) tension and compression along the fiber directions, or (2) with torsion applied along the tube axis. Under torsion loading, shear stresses act in the direction of the fibers. During torsion loading, most of the in-plane shear stress is first taken by the soft matrix as both fill and warp yarns rotate. Interaction between fill and warp yarns under in-plane shear loading influence the failure strength even though the shear strength is predominantly controlled by the weaker matrix. The failure envelope for hoop stress and torsion is given in Fig. 3 and in Fig. 4 for axial stress and torsion. These test results are shown by the open square sym-
Matrixcracking Interfacedebonding \ Fiber pun-out ~
Tensileliberfracture
,ooX-- "E~I/ / U
Delamination~ [ ~ - - ~ I I I I
!
-400
.
.
.
.
.
.
.
I I I I I
I-~
IITI'I -200
.
I
0
200
IT~
warp I
400
o.. m , MPa FIG. 3--Failure envelope for combined tension~compression and torsion in warp direction.
8
MULTIAXIALFATIGUE AND DEFORMATION
Matrix cracking Interface debonding Fiber puU-out X
,oo \
Fiber buckling ~
I -400
El"
x D, E~Jee"
I -200
Tensile fiber fracture
~
0
ll'-I
I I'lTn I 200 400
~r~, MPa
FIG. 4--Failure
envelope for combined tension~compression and torsion in fill direction.
bols. The X symbols are the results of three axis loading and will be discussed later in the paper. Two distinct types of behavior are observed. Even though the stress states are identical, there is an interaction between axial compression and torsion shown in Fig. 4. No interaction between hoop compression and torsion was observed in the test results shown in Fig. 3. No interaction was observed between tension and torsion loads in either direction. For a combination of tensile stress in the axial direction and in-plane shear stress, the tensile stress is carried by the fill fibers and the in-plane shear stress is carried by the matrix. This laminate should not be affected by the direction of the in-plane shear stress and the failure envelope should be symmetric about the O'fill-O'warp plane. In tension the macroscopic failure surface is perpendicular or 90 ~ to tube axis. Fracture surfaces in torsion are oriented 60 ~ with respect to the tube axis. The combined loading experiments failed on one of these planes. Two different failure modes are found on the fracture surface but there is no observed interaction between the failure mechanisms. Under SEM examination, the failure surface oriented at 90 ~ shows a typical tensile failure mode of fiber fracture while the 60 ~ planes show evidence of matrix cracking, interface debonding, and fiber pull-out typical of the torsion tests. Combined tension in the hoop direction and shear loading resulted in the same failure mechanisms that were observed in the axial direction. The maximum in-plane shear strain is about 20% which corresponds to a 10 ~ rotation of the fill fibers from the axial direction. While these strains may be considered unreasonably high for a high-performance composite, they could easily occur in a composite pressure vessel and piping system during an overload condition. The combined action of the applied tensile and shear stresses increases the fiber stress about 9% compared to that of uniaxial tension so that a small reduction in the strength may be expected. Scatter in the data was such that this small difference could not be observed and the addition of an in-plane shear stress did not reduce the tensile strength of the laminate. Hoop or warp compression and shear loading results in failures that are caused by either delamination followed by out-of-plane kinking as a result of the hoop compressive stress or by matrix cracking and interface debonding followed by fiber pullout. It might be anticipated that the interface debonding from the torsion stresses would lead to premature delamination from the compressive loads and result in a lower strength. This was not observed in either the experiments or the SEM observations of the fracture surfaces and we conclude that the applied in-plane shear stress does not change the failure mode or strength in combined loading in this direction. Figure 4 shows a substantial interaction between the shear and compressive stress. Fiber buckling is the dominant compression failure mechanism. Fracture surfaces for torsion and combined torsion
SOClE AND WANG ON MULTIAXIALLOADING
9
FIG. 5--Failure surfaces. and compression are compared in Fig. 5. In torsion the final fracture plane was oriented about 30~ to the axial direction and perpendicular to the specimen surface. Evidence of matrix cracking, interface debonding and final fracture by fiber pullout is shown in the SEM micrograph for torsion. Since Gl0 is a woven fabric laminate, fibers pull out in bundles. The addition of axial compression changed the failure mode to in-plane fiber buckling shown in Fig. 5 followed by out-of-plane kinking. The macroscopic fracture surface was oriented 45 ~ to the specimen surface. The difference in behavior of the hoop and fill fibers is illustrated in Fig. 6. Rotated fill fibers lose compression load-carrying ability. These fiber rotations from the in-plane shear loading lead to much lower compressive strengths because it activates a low energy fiber buckling mechanism followed by a kink band failure. Fiber rotations did not affect the tensile load-carrying capability. Hoop fibers do not rotate and the compressive load-carrying capacity is not reduced by an additional in-plane shear loading. Budiansky and Fleck [16] have shown that remote shear stresses activate yielding within a microbuckle band and greatly reduce the compressive strength of unidirectional composites with a remotely applied shear stress, r. The critical compressive stress, ~rcr,is found to be
1.2ry-r ~
FIG. 6--Rotation of fill fibers.
(3)
10
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 7--Failure envelope.
where ~-yis the shear yield strength of the matrix and th is the initial misalignmentangle of the fibers. Equation 3 suggests that the shear stress will have a large influence on the compressive strength. This work was extended by Jelf and Fleck [17] to include the effects of fiber rotations under combined compression and torsion loading. Their data for unidirectional carbon fiber epoxy tubes follows the same linear reduction in compressive strength with applied torsion that is shown in Fig. 4. The addition of an in-plane shear stress does not affect the delamination failure mode because delamination is not controlled by shear yielding of the matrix. Since no degradation of compressive strength was observed in the hoop direction, we conclude that fiber rotations are more important than shear yielding of the matrix. This failure mode would only be identified by compression-torsion testing of a tubular specimen. Test data from Figs. 1, 3, and 4 are combined into a single failure envelope in Fig. 7. The failure envelope can be described by five material properties: in-plane shear strength, fill tensile strength, fill compressive strength, warp tensile strength, warp compressive strength, and the knowledge of the interaction between compressive and in-plane shear loading in the fill direction. A series of experiments was conducted to probe the extremes of the three-dimensional failure envelope. Five combinations of loading shown in Fig. 8 were selected for testing. Table 2 gives the expected failure stresses normalized with the static strength for each direction. A negative ratio indicates compression. Specimens were loaded in load, torque, and internal pressure control with a ratio between them determined by the expected failure strength. A common command signal was used to control the three loads in the tests and no attempt was made to control the exact phasing between the channels. All of the loads should be in-phase; however, each test took several minutes and that is well
FIG. 8--Experimental load points.
SOCIE AND WANG ON MULTIAXIALLOADING
11
TABLE 2--Experimental results for combined loading. Expected
A-1 A-2 A-3 B-1 C-1 C-2 D-1 E-2 E-2
Observed
O'filI
O'war p
T
O'fil 1
O'wax p
7
1.0 1.0 1.0 -0.3 -0.7 -0.7 -0.3 -0.7 -0.7
1.0 1.0 1.0 1.0 1.0 1.0 0.7 0.7 0.7
1.0 1.0 1.0 0.7 0.3 0.3 0.7 0.3 0.3
0.98 0.88 0.70 -0.32 -0.48 -0.67 -0.40 -0.96 -0.79
0.83 0.88 0.76 0.96 0.71 1.01 0.78 0.95 0.91
0.82 0.90 0.60 0.65 0.24 0.34 0.79 0.48 0.40
within the control capabilities of the servohydraulic system. Failure is expected when any one of the stress components reaches the expected strength. The first series of tests designated A in Fig. 8 was designed so that all three stress components reached a maximum at the same time, There were three repetitions of this test. Macroscopic fracture surfaces were examined and compared to those under uniaxial loading. Specimen A-1 had a fracture surface that closely resembled that of a uniaxial tensile test. Specimens A-2 and A-3 fractured from the hoop tension loading. When a specimen contains a vertical split along the specimen axis, we conclude that internal pressure was the first failure mode. If tension or shear fractures occurred first, the specimen would leak oil and the internal pressure would decrease and not be able to split open the tube. Once a large tensile or hoop crack forms, the specimen loses torsional stiffness and the shear loads lead to final fracture. None of these tests reached the expected failure strengths and one of the tests failed at loads much lower than the other tests. Results of these three tests are plotted in Fig. 1 with the X symbols. These data fall in line with the other data shown in Fig. 1 that do not have shear loading. The dashed lines are drawn through the uniaxial strengths rather than as a best fit to all the data to form the expected failure envelope. For high stresses, the data for tension-tension loading falls inside the failure envelope indicating some interaction between the two stress systems at high loads. Similarly, the test data for compression-compression loading in Fig. l also falls inside the failure envelope. The remainder of the tests, B-E, were conducted in a region where there is interaction between the in-plane shear and normal stresses. The loading was chosen so that none of the specimens would be expected to fail from the torsion loading. Rather, the torsion loads were expected to reduce the compressive load-carrying capacity in the fill direction. All of these tests had fracture surfaces that were similar to uniaxial compression tests in the fill direction. The failure plane was perpendicular to the axial direction and oriented 45 ~ to the specimen surface indicating that the failures were due to outof-plane shear stresses. Results of these three tests are plotted in Fig. 4 with the X symbols. The failure envelope was constructed by drawing a straight line between the shear and compressive strength rather than a fit to the experimental data. All of the data scatter around this line.
Summary Longitudinal and transverse fiber stresses are decoupled in a composite laminate with stiff fibers and a compliant matrix such as the G-IO woven fabric laminate used in this study. As a result, a simple maximum stress theory provides a reasonable fit to the experimental data for combined tensiontension multiaxial loading when low-energy failure mode cutoffs are employed. The in-plane shear cutoff predicted by many of the anisotropic strength criteria for composite laminates under a biaxial
12
MULTIAXIAL FATIGUE AND DEFORMATION
tension-compression loading was not observed. More important, tubular specimens have low-energy compressive failure modes such as delamination and fiber buckling that must be considered. Delamination results in a lower compressive strength in the hoop direction when compared to the axial direction, and a delamination cutoff must be added to the maximum stress criterion for hoop compression tests of tubular specimens. The state of stress for axial compression and torsion is identical to that of hoop compression and torsion. The failure modes and resulting strengths are quite different. Fiber rotations in the axial direction lead to fiber buckling and a strong interaction is observed between torsional shear and axial compressive loads. These interactions are not predicted by any of the anisotropic strength theories. Failure mode-dependent theories are required to obtain the failure envelope of this material.
Acknowledgment The three-dimensional loading tests were conducted by Mr. David Waller for a course entitled "Laboratory Investigations in Mechanical Engineering" at the University of Illinois.
References [1] Abu-Farsakh, G. A. and Abdel-Jawad, Y. A., "A New Failure Criterion for Nonlinear Composite Materials," Journal of Composites Technology and Research, JCTRER, Vol. 16, No. 2, 1994, pp. 138-145. [2] Hart-Smith, L. J., "Predictions of a Generalized Maximum Shear Stress Criterion for Certain Fiberous Composite Laminates," Composites Science and Technology, Vol. 58, 1998, pp. 1179-1208. [3] Tsai, S. W., "Strength Characteristics of Composite Materials," NASA CR-224, April, 1965. [4] Tsai, S. W. and Wu, E. M., "A General Theory of Strength for Anisotropic Materials," Journal of Composite Materials, Vol. 5, 1971, pp. 58-80. [5] Jiang, Z. and Tennyson, R. C., "Closure of the Cubic Tensor Polynomial Failure Surface," Journal of Composite Materials, Vol. 23, 1989, pp. 208-231. [6] Ardic, E. S., Anlas, G., and Eraslanoglu, G., "Failure Prediction for Laminated Composites Under Multiaxial Loading," Journal of Reinforced Plastics and Composites, Vol. 18, No. 2, 1999, pp. 138-150. [7] Sun, C. T. and Tao, J., "Prediction of Failure Envelops and Stress/Strain Behavior of Composite Laminates," Composites Science and Technology, Vol. 58, 1998, pp. 1125-1136. [8] Nahas, M. N., "Survey of Failure and Post-Failure Theories of Laminated Fiber-Reinforced Composites," Journal of Composite Technology Research, Vol. 8, 1986, pp. 1138-1153. [9] Soden, P. O., Hinton, M. J., and Kaddour, A. S., "A Comparison of the Predictive Capabilities of Current Failure Theories for Composite Laminates," Composites Science and Technology, Vol. 58, 1998, pp. 1225-1254. [10] Wang, J. Z. and Socie, D. F., "Biaxial Testing and Failure Mechanisms in Tubular G-10 Composite Laminates" ASTMSTP 1206, 1993, pp. 136-149. [11] Socie, D. F. and Wang, Z. Q., "Failure Strength and Mechanisms of a Woven Composite Laminate Under Multiaxial In-Plane Loading," Durability and Damage Tolerance, ASME AD-Vol. 43, 1994, pp. 149-164. [12] Kachanov, L. M., Delamination Buckling of Composite Materials, Kluer Academic Publishers, 1988. [13] Sih, G. C., Hilton, P. D., Badaliance, R., Shenberger, P. S., and Villarreal, G., "Fracture Mechanics for Fibrous Composites," ASTM STP 521, 1973, pp. 98-132. [14] Browning, C. E. and Schwartz, H. S., "Delamination Resistance Composite Concepts," ASTM STP 893, 1986, pp. 256-265. [15] Wang, Z. Q. and Socie, D. F., "A Biaxial Tension-Compression Test Method for Composite Laminates," Journal of Composites Technology and Research, JCTRER, Vol. 16, No. 4, 1994, pp. 336-342. [16] Budiansky, B. and Fleck, N. A., "Compressive Failure of Fiber Composites," Journal of Mechanics and Physics of Solids, Vol. 41, No. 1, 1993, p. 183. [17] Jelf, P. M. and Fleck, N. A., "The Failure of Composite Tubes Due to Combined Compression and Torsion," Journal of Materials Science, Vol. 29, 1994, pp. 3080-3084.
J. A. S a l e m I a n d M. G. Jenkins 2
Biaxial Strength Testing of Isotropic and Anisotropic Monoliths REFERENCE: Salem, J. A. and Jenkins, M. G., "Biaxial Strength Testing of Isotropic and Anisotropic Monoliths," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kallnri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 13-25. ABSTRACT: A test apparatus for measuring the multiaxial strength of circular plates was developed and experimentally verified. Contact and frictional stresses were avoided in the highly stressed regions of the test specimen by using fluid pressurization to load the specimen. Both isotropic plates and singlecrystal NiA1 plates were considered, and the necessary strain functions for anisotropic plates were formulated. For isotropic plates and single-crystal NiA1 plates, the maximum stresses generated in the test rig were within 2% of those calculated by plate theory when the support ring was lubricated. KEYWORDS: anisotropy, single crystals, ceramics, composites, multiaxial strength, nickel aluminide, tungsten carbide, displacement, strain, stress
Nomenclature An Bn bq Cz Do D* ep eq ~p ~z ki k* P r Rs RD So
SDxz t v o-q O-p O-q
Series constant in anisotropic, plate displacement solution Series constant in anisotropic, plate displacement solution Reduced elastic stiffness Constants in the anisotropic displacement stress solution Flexural rigidities Effective flexural rigidity of an anisotropic plate Measured major principal strain component Measured minor principal strain component Measured principal strain uncorrected for transverse sensitivity Measured strain uncorrected for transverse sensitivity; i = 1,2,3 Reduced flexnral rigidity Effective, reduced flexural rigidity of an anisotropic plate Pressure Radius Radius of support ring Radius of disk test specimen Elastic compliance Standard deviation of x~ variable Disk test specimen thickness Poisson's ratio Stress component Measured major principal stress Measured minor principal stress
1 NASA Glenn Research Center, MS 49-7, 21000 Brookpark Rd., Cleveland, OH 44135. University of Washington, Box 352600, Seattle, WA 98195.
Copyright9
by ASTM International
13 www.astm.org
14
MULTIAXIAL FATIGUE AND DEFORMATION
(a)
~\\\\\\\\\\\\\\\\\~
(b)
KN\\\\\\\\\\\\\\\\\~
fT"
P
(c)
[~\\\\\\\\\\\\\\\\\"~1
J_ t
f FIG. 1--Schematic of typical testing configurations used to generate biaxial tensile stresses in plate specimens: (a) ball-on-ring, (b) ring-on-ring, and (c) pressure-on-ring.
O'rr Radial stress o00
Tangential stress
~rO Shear stress o's w x y z
Correction term for effect of lateral stresses on plate deflection Plate deflection in the z-direction Abscissa as measured from plate center Ordinate as measured from plate center Distance from midsurface of plate ranging over +_t/2
The strength of brittle materials such as ceramics, glasses, and semiconductors is a function of the test specimen size and the state of applied stress [1]. Engineering applications of such materials (e.g., ceramics as heat engine components, glasses as insulators, silicon and germanium as semiconductors) involve components with volumes, shapes, and stresses substantially different from those of standard test specimens used to generate design data. Although a variety of models [2] exist that can use conventional test specimen data to estimate the strength of large test specimens or components subjected to multiaxial stresses, it is frequently necessary to measure the strength of a brittle material under multiaxial stresses. Such strength data can be used to verify the applicability of various design models to a particular material or to mimic the multiaxial stress state generated in a component during service. Further, these materials tend to be brittle, and machining and handling of test specimens can lead to spurious chips at the specimen edges which in turn can induce failure not representative of the flaw population distributed through the materials' bulk. In the case of a plate subjected to lateral pressure, the stresses developed are lower at the edges, thereby minimizing spurious failure from damage at the edges.
SALEM
AND
JENKINS
ON
BIAXIAL
STRENGTH
TESTING
15
For components that are subjected to multiaxial bending, three different loading assemblies, shown schematically in Fig. 1, can be used to mimic component conditions by flexing circular or square plates: ball-on-ring, ring-on-ring (R-O-R), or pressure-on-ring (P-O-R). The R-O-R and the P-O-R are preferred because more of the test specimen volume is subjected to larger stresses. However, significant frictional or wedging stresses associated with the loading ring can be developed in the highly stressed regions of the R-O-R specimen [3,4]. These stresses are not generated in the P-O-R configuration. Rickerby [5] developed a P-O-R system that used a neoprene membrane to transmit pressure to a disk test specimen (diameter to thickness ratio of 2Ro/t ~ 17). The reported stresses were in excellent agreement with plate theory at the disk center (< 0.5% difference). At 0.43Rs the differences in radial and tangential stresses were -3.6 and -2.5%, respectively, and at 0.85Rs the differences were ~27 and -2.4%, respectively, where Rs is the support ring radius. The biaxial test rig used by Shetty et al. [6] included a 0.25 mm spring steel membrane between the disk test specimen (2Ro/t ~ 13) compressive surface and the pressurization medium. Despite the steel membrane, the rig was reported to produce stresses in reasonable agreement with plate theory. The measured stresses at the disk center were -3.5% greater than theoretical predictions. The radial and tangential stresses were -1.5 and -1.9% greater, respectively, at 0.25Rs, and at 0.8Rs the radial error was -10%. Reliability calculations are strongly dependent on the peak stress regions, and thus the differences need to be small in the central region of the disk. Although the overall differences are not large, -10% toward the disk edge, they are somewhat greater than Rickerby's at the highly stressed central region. This may be due to the clamped edge of the steel membrane. The objective of this work was to design, build, and experimentally verify a P-O-R biaxial flexure test rig for strength and fatigue testing of both isotropic and anisotropic materials. One goal was to eliminate the membrane between the pressurization medium and the test specimen, thereby eliminating interaction between the test specimen and membrane.
Biaxial Test Apparatus The rigs consist of a pressurization chamber, reaction ring and cap, extensometer, and oil inlet and drain ports, as shown in Fig. 2. The desired pressurization cycle is supplied to the test chamber and
TFRT
c- v ?- E- i~i ~T,r7 iwIF T ['I D
FIG. 2--Schematic of pressure-on-ring assembly and test specimen.
16
MULTIAXIAL FATIGUE AND DEFORMATION
specimen via a servohydraulic actuator connected to a closed loop controller. The feedback to the controller is supplied by a commercial pressure transducer connected to the oil inlet line. The test chamber and cap are 304 stainless steel, and the reaction ring is cold rolled, half-hard copper or steel depending on the strength of the material tested. For low strength specimens, minor misalignments or specimen curvatures can be accommodated via the copper support ring. The hydraulic oil is contained on the compressive face of the specimen by a nitrile O-ring retained in a groove. A cross section of the test rig, which can accommodate 38.1 or 50.8 mm diameter disks by using different seals and cap/reaction ring assemblies, is shown in Fig. 2. A similar rig for testing specimens with 25.4 mm diameters was also developed.
Stress Analysis of the P-O-R Test Specimen
Isotropic Materials The radial and tangential stresses generated in a circular, isotropic plate of radius RD and thickness t that is supported on a ring of radius Rs and subjected to a lateral pressure P within the support ring are [7]
O'rr
-
-
o00 = ~
8t 2
(1 - v) R~ + 2(1 + u) - (3 + v)
E
(1 - v) R--~o+ 2(1 + v) - (1 + 3v)
+ ~rs
+ os
(1)
e(3 + v) O ' s - 4(1 - u) where r is the radius of interest. The term O's is a small correction factor to the simple plate theory for the effects of the sheafing stresses and lateral pressure on the plate deflection [8]. Equation 1 is based on small-deflection theory and thus assumes that the plate is thin and deflects little relative to the plate thickness.
Anisotropic Materials The displacement solution for a circular, orthotropic plate of unit radius and thickness subjected to a unit lateral pressure was determined by Okubu [9] in the form of a series solution and as an empirical approximation. Such a solution is useful in the testing and analysis of composite plates and plates made from single crystals such as silicon, germanium, or nickel aluminide. The analysis was based on small-deflection theory and thus assumes that the plate is thin and deflects little relative to the plate thickness (i.e., less than 10%).
Approximate Solution The approximate displacement solution given by Okubu for a plate of unit radius is w ~ ( 1 - rP2 ) ( k
* - r 2)
(2)
where
1
D* = ~ (3Dlt + 2D12 + 4D66 + 3022)
k* =
7Dll + 10D12 + 12D66 + 7D22 2(Dll + 2D12 + D22)
(3)
SALEM AND JENKINS ON BIAXlAL STRENGTH TESTING
17
and Oil
t3
S22
t3
12 SHS12 - $22
Sl 1
D22 = 1-~ $11S12 - $22 (4)
--t 3 S12 D12 = 12 S l l S 1 2 - $22
t3 1 12 $66
D66
where the S o terms are the material compliances or single crystal elastic constants. The plate rigidity terms, Dii, and associated functions are written in the more standard notation used by Hearmon [10] instead of that used by Okubu [9]. For the general case of a plate of variable support radius the displacement becomes w -~ ~
P
(R~ - ? ) ( k * 1 ~ -
,~)
(5)
As the symmetry of cubic crystals and orthotropic composites is orthogonal, the elastic constants are in Cartesian form and the stress and strain solutions are determined in Cartesian coordinates: OZw
02W
ax2
~ 02W Z Oy2 , e66 = - - A Z O ~ y
__ 02__..~ W .
811 = --Z-0-'~-; e22 =
- P [2R2(k * + 1) - 12x2 - 4y 2] 64D* (6)
02W
Oy2
- P [2R~(k* + 1) - 12y 2 - 4x2] 64D* O2w OxOy
P
8D* [xy]
where z is the distance from the midsurface of the plate. The stresses are determined from the strains by [10]
0.11 = --Z~Oll " - ~
L
-I- b12 OY2 /
~w
o2.,]
(7)
where bla = $221(SalS22 - Sa2), b22 = Snl(Sa~S22 - $12), b ~ = 11Sa6, and b12 -- -S121($11S22 - S~2). As the plate is cylindrical, a description of the stresses in polar coordinates is more intuitive, and the Cartesian values at any point in the plate can be converted to polar coordinates with O'rr = Orll COS2 0 q- 0"22 sin 2 0 + 0"a2 sin 20 or00 = 0"22 COS2 0 + O"11sin 2 0 -- 012 sin 20
O'ro
=
(0"22 -- O"11) sin 0 cos 0 + o'12 cos 20
where 0 is the angle counterclockwise from the x-axis.
(8)
18
MULTIAXIAL FATIGUE AND DEFORMATION
The Series Solution If the series displacement solution given by Okubu is redetermined for the case of variable radius, thickness, and pressure, the following displacement function results
2(1 - ~) ~ A n n=2
cosh(2n + 2)a' ( 2 - n ~ ( 2 - n ~ ) i~
1 ] cosh2nc~' cos2n/3" + 2)/3' - [ (2n + 1)2n + 2n(2n1)
cosh(2n - 2)a +(~n_--l)-~--_~cos(Zn - 2)/3
P~
+ 2(1 - ~) ~ B.
w = ~-~-
n=2
cosh(2n + 2)a" os(2n i~-(~--~(~-n---~-~) c /
( 1 2 ) i f ' - _ (2n+ 1)2n
2n(2n - 1)-] cosh2nd' cos2n/3"
cosh(2n - 2)a' l + (2nn-- 1)-~----2) cos(2n - 2)if' + {(Cx - C2 + C3)(cos4fl + 3) + 4(Ct
~+
4{(C4
s
C5) cos 2/3 + C4 + C5}
-
r~
C3) COS2/3 + 4C2}
+ 8C6
R~ (9)
The curvatures are 02w _ ax 2
P~J- [~=2 (A. cosh 2ncd cos 2nfl' + B~ cosh 2na" cos2nff') re + 2C4] + (6C1 + C2 + (6C1 - C2) cos 2/3) R--~
02w OY2 - PR~ t3 [ -- .~=2(A.k 2 cosh 2nct' cos 2n13' + B.k~ cosh 2nd' cos 2nff') (10) rEz + 2C5] + (6C3 + C2 - (6C3 - C2)cos 2fl) Rs
02W OxlOx2
ta
-
(Ankl sinh 2ncg sin 2n/3' q .2 + B~kz sinh 2ha" sin 2nil') + 2Cz ~ sin 213/ Rs J
SALEM AND JENKINS ON BIAXIAL STRENGTH
TESTING
19
TABLE 1--Constants ( XlO -6 rn2/MN) for NiAI and graphite/epoxy plates of unit thickness and radius subjected to a unit lateral pressure. NiAI: $22 = SI1 = 1.0428, Sa2 = -0.421, $66 = 0.892 (• 10 -5 m2/MN) [11] Cl
C2
C3
C4
C5
C6
A2
B2
A3
03
1.392
2.009
1.392
-7.253
-7.253
5.958
0.474
-0.105
10 -15
10 -16
Graphite Epoxy: Sll = 0.6667, $22 = 11.11, Sl2 = -0.2000, $66 = 14.08 (• 10-5 m2/MN) [12] C1 2.741
C2 9.046
(73 4.080
C4 -15.52
C5 -16.34
C6 12.24
A2 0.385
B2 0.385
A3 0.079
B3 0.079
where
DI/z
D~a kl = (D2 +
D4
-'F
{(D2
+ D4) 2 -
D1D3}lt2) 112
k2 = (D2 + D4
--
{(D2
+ 04) 2 -
D1D3}1/2) 1/2 (11)
and theAn, Bn, and Ci terms are constants determined from the boundary conditions, and the/3,/3' and /3" terms are functions describing the angular position of interest. The solution converges rapidly for a plate of cubic material in the "standard" orientation and only the constants A2, Bz, and Ci are needed, as shown in Table 1. For an orthotropic material such as graphite-epoxy, the higher order constants are small but significant. The stresses generated in a NiA1 (nickel aluminide) plate of {001 } crystal orientation are shown in polar coordinates in Fig. 3. The stresses are a function of both radius and angle, with the peak stresses being tangential components occurring at the (110) crystal directions. The effect of anisotropy is most
/k
Tangential Stress, r/R= 0.2 Radial Stress, r / R ==0~2 . 9 Tangential Stress, r / R = 0.8 -- Radial Stress, r/R, = 0.8
.....
Q V
---
1.2 1.0 0.8 0.6 0.4 0.2 <010>
0.0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
Stress/Pressure
FIG. 3--Stresses generated in a NiA1 single crystal plate of unit radius and thickness subjected to a unit pressure.
20
MULTIAXlAL FATIGUE AND DEFORMATION
apparent at the plate edges where the stresses vary with angular position by -45% for r/Rs = 0.8. At r/Rs = 0, the stresses become equibiaxial as in the isotropic case.
Test Rig Verification Isotropic Materials Ideally a test rig will generate stresses described by simple plate theory. A comparison was made between Eq 1 and the stresses measured with stacked, rectangular strain gage rosettes placed at eight radial positions on the tensile surfaces of two 4340 steel disk test specimens and at seven positions on two WC (tungsten carbide) disk test specimens. The strain-gaged specimens were inserted, pressurized, and removed repeatedly while the strain was recorded as a function of pressure. Three supporting conditions were considered: (1) unlubricated, (2) lubricated with hydraulic oil, and (3) lubricated with an anti-seizing compound. The average of at least three slopes, as determined by linear regression of strain as function of pressure, was used to calculate the mean strains and stresses in the usual manner [13,14] at the pressure level of interest. As the calculation of stress from strain via constitutive equations requires the elastic modulus and Poisson's ratio, measurements of the steel were made with biaxial strain gages mounted on tension test specimens, and by ASTM Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio for Advanced Ceramics by Impulse Excitation of Vibration (C 1259-94) on beams fabricated from the same plate of material as the disk test specimens. The elastic modulus as estimated from the strain gage measurements was 209.3 _+ 0.9 GPa and the Poisson's ratio was 0.29, in good agreement with handbook values [15]. The elastic modulus as estimated from ASTM C 1259-94 was 209.9 • 0.5 GPa. The elastic modulus and Poisson's ratio of the WC material were measured by using ASTM C 1259-94 on ten 50.8 mm diameter disk specimens. The elastic modulus was 607 + 3 GPa and Poisson's ratio was 0.22. The stresses generated in the steel specimens with the lubricant on the copper reaction ring were consistently greater than those generated without lubricant. However, the differences were small (4.2 MPa at -400 MPa equibiaxial stress) and approximately one standard deviation of the measurements. For an applied pressure of 3.45 MPa, agreement between plate theory and the measurements on the steel specimens without lubrication on the boundary were within 1% at the disk center, within 2% at 0.49Rs, and within 7 and 8%, respectively, for the radial and tangential components at 0.75Rs. In general, the differences increase with increasing radial position, particularly for the tangential component. In contrast, the WC specimens, which were tested on a steel support due to the large strength, exhibited a substantial effect of friction. The maximum stresses decreased by - 5 % when the specimens were tested without anti-seizing lubricant, and the use of hydraulic oil on the support ring did little to reduce friction. For the WC specimens and anti-seizing lubricant on the boundary, agreement between plate theory and the measurements at a pressure of 8.3 MPa was within 2% at the disk center, within 2% at 0.49Rs, and within 6 and 9%, respectively, for the radial and tangential components at 0.75Rs. The significance of the differences between the plate theory and the measured stresses can be assessed by estimating the standard deviations and confidence bands of the measurements. The standard deviations of the strains and stresses were calculated from the apparent strain variances by applying a truncated Taylor series approximation [16] to the transverse sensitivity correction equations, the strain transformation equations, and the stress-strain relations. For a rectangular strain rosette, the standard deviations of principal stress, principal strain, and principal strain uncorrected for transverse strain errors are
SD~p
_
E N / S D ~ + ~2SD2~ 1 - v2
SALEM AND JENKINS ON BIAXIAL STRENGTH TESTING
SD,~ -
SD~o =
21
E 1 - - v 2 X'/ ~SD2" + SD2~ 1 - vokt 1 -- -s 2 %/SD}" + ~ SD2~ (12) 1 - vokt - ~
SD~q = -f-el_e_2
~,
2 S D 2 + (2~2
.
2
Ve, SDr + SD}q ~1 - ~3)2 SD22 +
+
SD~3
v5 =
(gl
-- ~2) 2 +
(2~2
- - ~1 - - g 3 ) 2
where E and v are the elastic modulus and Poisson' s ratio of the test material, vo is Poisson' s ratio of the strain gage manufacturer's calibration material, kt is the transverse sensitivity of the strain gages, ~1, ~2, ~3 are the apparent strains, and the SDxi terms are the standard deviations of the following xi variables: ~p and ~q being the principal strains uncorrected for transverse effects, ep and eq being the corrected principal strains, and o-p and O'q being the corrected principal stresses. The elastic constants in Eq 12 are assumed to be exact for a single test specimen. The results along with 95% confidence bands are summarized in Tables 2 and 3 and shown in Fig. 4 for the condition of a lubricated boundary. Because the 95% confidence bands of the tangential stress measurements on the 4340 steel specimens do not overlap the theory for radii greater than 0.5Rs, the differences are significant. The radial stresses are in good agreement for all radii. For the WC specimens, overall agreement between theory and the experiment is better than for the steel specimen.
TABLE 2--Measured stresses, standard deviations, and theoretical stresses for a 2.3-mm-thick, 51-mm-diameter 4340 steel plate supported on a 45.6-mm-diameter copper ring and subjected to 3.45 MPa uniform pressure. Radial Position Tangential Stress, MPa
Radial Stress, MPa Percent of Support Radius and Lubrication 0, Dry 17, Dry 33, Dry 49, Dry 61, Dry 74, Dry 75, Dry 76, Dry 0, Anti-seizing I 49, Anti-seizing 1 0, Clamped only 49, Clamped only
Plate Theory 2 403.8 391.5 357.4 302.1 247.9 173.5 168.0 164.1 403.8 302.1
Measured 3 401.3 +- 2.7 389.8 + 2.3 351.6 + 1.0 297.8 -+ 1.3 236.4 - 0.9 164.9 --- 3.6 156.1 - 1.5 ' 165.1 + 3.3 403.7 + 2.9 300.0 + 1.0 5.6 + 2.4 8.0 + 2.4
Percent Difference
Plate Theory2
-0.6 -0.4 -1.6 -1.4 -4.6 -5.0 -7.1 0.6 0.0 -0.7
403.8 396.8 377.6 346.4 315.7 273.8 270.6 268.5 403.8 346.4
Never-Seez, Never-Seez Compound Corp., Broadview, IL. 2 See Ref 7. 3 Mean _+ one standard deviation.
Measured 3 405.4 _+ 4.1 396.8 -+ 1.3 369.7 -+ 2.9 338.2 -+ 1.4 301.8 -+ 1.1 252.4 + 1.1 249.8 _+ 0.5 248.7 _+ 2.2 409.6 _+ 4.1 339.5 _+ 2.1 -2.2 + 1.3 -2.0 _+ 1.3
Percent Difference 0.4 0.0 -2.1 -2.4 -4.4 -7.8 -7.7 -7.4 1.4 -2.0
22
M U L T I A X l A L FATIGUE A N D D E F O R M A T I O N
TABLE 3--Measured stresses, standard deviations, and theoretical stresses for a 2.2-ram-thick, 51-rnm diameter WC plate supported on a 45.4-ram-diameter steel ring and subjected to 8.3 MPa uniform pressure. Radial Position Tangential Stress, MPa
Radial Stress, MPa Percent of Support Radius and Lubrication
Plate Theory 2
0, Dry 16, Dry 32, Dry 49, Dry 72, Dry 73, Dry 83, Dry 0, Hydraulic oil 49, Hydraulic oil 72, Hydraulic oil 73, Hydraulic oil 0, Anti-seizing] 16, Anti-seizing] 32, Anti-seizing ] 49, Anti-seizing ] 72, Anti-seizing] 73, Anti-seizing] 83, Anti-seizing ]
1005.5 977.8 899.1 755.7 466.8 449.7 288.1 1005.5 755.7 466.8 449.7 1005.5 977.8 899.1 755.7 466.8 449.7 288.1
Measured 3 939.0 950.7 878.1 701.4 403.7 349.8 262.4 942.6 709.1 410.3 377.1 983.1 972.7 892.6 756.4 458.5 423.3 271.5
Percent Difference
Plate Theory 2
-6.6 -2.8 -2.3 -7.2 -13.5 -22.2 -8.9 -6.3 -6.2 - 12.1 -16.2 -2.2 -0.5 -0.7 0.1 -1.8 -5.9 -5.8
1005.5 991.2 950.7 876.7 727.8 719.0 635.7 1005.5 876.7 727.8 719.0 1005.5 991.2 950.7 876.7 727.8 719.0 635.7
• 1.9 • 6.3 • 5.1 +_ 4.5 • 5.6 • 8.8 • 2.5 • 17.2 • 9.6 + 6.1 • • 27.5 • 2.7 • 5.3 • 18.3 • 22.9 • 20.4 • 2.6
Percent Difference
Measured 3 968.3 969.4 942.9 829.3 668.0 616.0 777.9 975.9 822.1 671.0 618.5 1019.0 992.2 962.1 866.8 712.7 655.8 789.8
• 3.0 • 8.7 • 8.6 ___6.9 + 8.2 • 11.7 • 9.2 • 10.4 • 15.3 • 3.4 • 15.6 • 19.2 • 5.3 --- 7.3 • 35.2 • 15.3 • 33.4 • 7.2
-3.7 -2.2 -0.8 -5.4 -8.2 -14.3 22.4 -2.9 -6.2 -7.8 -14.0 1.3 0.1 1.2 -1.1 -2.1 -8.8 24.2
1Never-Seez, Never-Seez Compound Corp., Broadview, IL. 2 See Ref 7. 3 Mean • one standard deviation.
T h e forces exerted by the O-ring on the test s p e c i m e n resulted in stresses on the s p e c i m e n surfaces. T h e level a n d c o n s i s t e n c y o f t h e s e stresses were m e a s u r e d at the disk center a n d at 0.49Rs b y repeatedly inserting a n d r e m o v i n g an unlubricated, steel s t r a i n - g a g e d test s p e c i m e n into a n d f r o m the fixture. T h e stresses generated b y c l a m p i n g varied with orientation a n d radial position. 125
125 ungsten Carbide
340 Steel 100
~
100
b-
~
75
n
75
\
ffl
r o9
50
09
25
o o 9 9 ----
Disk 2 Radial Disk 2, Tangential Disk 1, Radial Disk 1, Tangential Theory, Tangential Theory, Radial
\
~
~ \
\
\
N
\
0
so 25
o o ----
&
Measured, Radial Y'\ Measured, Tangential ~ Theory, Radial \ Theory, Tangential
0 0.0
0.2
0.4
0.6
0.8
1.0
Radial Position/Support Radius, r / R s
0.0
0.2
0.4
0.6
0.8
1.0
Radial P o s i t i o n / S u p p o r t radius, r/R s
FIG. 4---Measured and theoretical stresses as a function o f normalized radial position. Error bars indicate the 95% confidence bands: (left) steel disk on a copper support, and (right) tungsten carbide disk on a steel support.
23
S A L E M A N D J E N K I N S ON BIAXIAL S T R E N G T H T E S T I N G
70
70 60
~
5O
~
40
~
30
~
I
{001} <100>
6=
t
50 40
\ ,~
20 10
9
----
Radial, Measured Tangential, Measured
\\
r r
0.2
0.4
0.6
\\ ~9
\
20 10
\ 0.8
\
z,
0
0.0
\
\
30
,,
Radial, Theory Tangential, Theory
{001} <110>
9 ----
\
\
Measured, Radial
Measured, Tangential Theory, Radial Theory, Tangential
olo
1.0
\
o14
\
\
\\
\
o18
1.o
Radial Position/Support Radius, r/Rs
Radial P o s i t i o n / S u p p o r t Radius, r/R s
FIG. 5---Measured and theoretical stresses for a [001} NiA1 disk as a function o f normalized radial position. Error bars indicate the 95% confidence bands." (left) (100) direction and (right) (110) direction.
During seven clampings, the mean principal stresses ( + one standard deviation) were 5.6 -+ 2.4 and - 2 . 2 _+ 1.3 MPa, respectively, at the disk center, and 8.0 + 2.4 and - 2 . 0 + 1.3 MPa, respectively, at 0.49Rs. The maximum principal stresses observed during a clamping were 9.5 and 3.8 MPa at the disk center. Anisotropic Materials To compare the stresses generated in the test rig with the solutions of Okubu, single crystal NiA1 disk test specimens were machined with face of the disk corresponding to the {001 }. One specimen was strain gaged at four locations and pressurized to 4.8 MPa in the rig with anti-seizing lubricant on the steel support. The resulting stresses are shown in Fig. 5 and summarized in Table 4. The stresses calculated with the series solution are within 2% of the measured stresses at the plate center and within 7% at radii less than 50% of the support radius.
TABLE ~-Measured stresses, standard deviations, and theoretical stresses for a 1.55-ram-thick, 25.4-mm-diameter [001} NiAl single crystal plate supported on a 23.1-mm-diameter lubricated steel ring and subjected to a 4.8 MPa uniform pressure. Radial Position Tangenital Stress, MPa
Radial Stress, MPa Percent of Support Radius and Angular Position
Plate Theory 1
2, center 44, < 100 > 51, < 500> 50, < 110 >
305.7 259.8 234.2 239.8
] See Ref 9. 2 Mean • one standard deviation.
Measured 2 300.1 251.3 232.9 223.7
• 1.0 _+ 3.1 • 1.0 __+1.0
Percent Difference
Plate Theory 1
-1.8 -3.3 -5.6 -6.7
305.7 272.2 274.8 275.6
Measured 2 311.2 264.4 262.8 288.8
• • + •
1.2 1.7 1.0 1.0
Percent Difference +1.8 -2.9 -4.4 +4.8
24
MULTIAXIAL FATIGUE AND DEFORMATION
1.10
1.05
| O
!
n,, 1.00
9
| 0.95
9
|
0.90
Measured
Approximate
Measured
Exact
FIG. 6--Measured and theoretical strains at failure for {001} NiA1 disk test specimens. The measured strains are normalized with Okubu's approximate and series solutions [9].
Additionally, disk test specimens were strain gaged at the center and pressurized to failure. The strain at failure is compared to those calculated with Eqs 6 and 10 in Fig. 6. The strains generated in the rig lie between those of the solutions, with the approximate solution overestimating the strains by - 5 % and the series solution underestimating the rig data by -3%. However, neither the approximate or series solutions consider the effect of lateral pressure and shear on the strains and stresses. If the isotropic correction term, os, in Eq 1 is used with the Poisson's ratio of polycrystalline NiA1 (~0.31 [17]) to approximate the error, an additional strain of -1.7% is expected, implying that the bending stresses generated by the test rig closely approximate the series solution.
Summary A test apparatus for measuring the multiaxial strength of brittle materials was developed and experimentally verified. Contact and frictional stresses were avoided in the highly stressed regions of the test specimen by using fluid pressurization to load the specimen. For isotropic plates, the experimental differences relative to plate theory were functions of radial position with the maximum differences occurring toward the seal where the stresses are the least. The maximum stresses generated in the test rig were within 2% of those calculated by plate theory when the support ring was lubricated. The effects of friction and the clamping forces due to the seal were typically less than 2% of the equibiaxial (maximum) applied stress when an unlubricated copper support ring was used. When an unlubricated steel ring was used, the effect of friction on lapped tungsten carbide was approximately 5% of the maximum stress. Application of a lubricant to the support eliminated the detectable effects of friction. For a single-crystal NiA1 plate, the maximum stresses generated in the test rig were within 2% of those calculated by plate theory when the support ring was lubricated. For radial positions of less than 50% of the support radius, the calculated and measured stresses were within 7%. The stress distribution in a single-crystal plate of cubic symmetry is a function of both radial position and orientation. The maximum stresses at any radius are tangential and occur at (110) orientations.
References [1] Weibull, W., "A Statistical Theory of the Strength of Materials," Ingeniors Vetenskaps Akademien Handlinger, No. 151, 1939. [2] Batdorf, S. B. and Crose, J. G., "A Statistical Theory for the Fracture of Brittle Structures Subjected to Nonuniform Polyaxial Stresses," Journal of Applied Mechanics, Vol. 41, No. 2, June 1974, pp. 459~-64.
SALEM AND JENKINS ON BIAXIAL STRENGTH TESTING
25
[3] Adler, W. F. and Mihora, D. J., "Biaxial Flexure Testing: Analysis and Experimental Results," Fracture Mechanics of Ceramics, Vol. 10, R. C. Bradt, D. P. H. Hasselman, D. Munz, M. Sakai, and V. Shevchenko, Eds., Plenum Press, New York, 1991, pp. 227-246. [4] Fessler, H. and Fricker, D. C., "A Theoretical Analysis of the Ring-On-Ring Loading Disk Tests," Journal American Ceramic Society, Vol. 67, No. 9, 1984, pp. 582-588. [5] Rickerby, D. G., "Weibull Statistics for Biaxial Strength Testing," Fracture 1977, Vol. 2, ICF4, Waterloo, Canada, 19-24 June 1977, pp. 1133-1141. [6] Shetty, D. K., Rosenfield, A. R., Duckworth, W. H., and Held, P. R., "A Biaxial Test for Evaluating Ceramic Strengths," Journal of the American Ceramic Society, Vol. 66, No. 1, Jan. 1983, pp. 36-42. [7] Szilard, R., Theory and Analysis of Plates, Classical and Numerical Methods, Prentice-Hall, Englewood Cliffs, NJ, 1974, p. 628. [8] Timoshenko, S. P. and Woinowsky-Krieger, S., Theory of Plates and Shells, 2nd ed., McGraw-Hill, New York, NY, 1959, p. 72. [9] Okubu, H., "Bending of a Thin Circular Plate of an Aeolotropic Material Under Uniform Lateral Load (Supported Edge)," Journal of Applied Physics, Vol. 20, Dec., 1949, pp. 1151-1154. [10] Hearmon, R. F. S., An Introduction to Applied Anisotropic Elasticity, Oxford University Press, 1961. [11] Wasilewski, R. J., "Elastic Constants and Young's Modulus of NiAI," Transactions of the Metallurgical Society ofAIME, Vol. 236, 1966, pp. 455-456. [12] Lee, H. J. and Saravanos, D. A., "Generalized Finite Element Formulation for Smart Multilayered Thermal Piezoelectric Composite Plates," International Journal of Solids Structures, Vol. 34, No. 26, 1997, pp. 3355-3371. [13] "Errors Due to Transverse Sensitivity in Strain Gages," Measurements Group Tech Note TN-509, Measurements Group, Raleigh, NC. [14] "Strain Gage Rosettes--Selection, Application and Data Reduction," Measurements Group Tech Note TN515, Measurements Group, Raleigh, NC. [15] Aerospace Structural Metals Handbook, CINDAS/USAF CRDA Handbook Operations, West Lafayette, IN, Vol. 1, 1997, p. 41. [16] Hangen, E. B., Probabilistic Mechanical Design, Wiley, New York, 1980. [17] Noebe, R. D, Bowman, R. R., and Nathal, M. V., "Physical and Mechanical Properties of the B2 Compound NiAI," International Materials Reviews, Vol. 38, No. 4, 1993, pp. 193-232.
Steven J. Covey I and PauI A. Bartolotta 2
In-Plane Biaxial Failure Surface of Cold-Rolled 304 Stainless Steel Sheets REFERENCE: Covey, S. J. and Bartolotta, P. A., "In-Plane Biaxial Failure Surface of ColdRolled 304 Stainless Steel Sheets," Multiaxial Fatigue and Deformation: Testing and Prediction,
ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 2000, pp. 26--37. ABSTRACT: Cold forming of thin metallic plates and sheets is a common inexpensive manufacturing
process for many thin lightweight components. Unfortunately, part rejection rates of cold (or warm) rolled sheet metals are high. This is especially true for materials that have a texture (i.e., cold-rolled stainless steel sheets) and are being cold-formed into geometrically complex parts. To obtain an understanding on how cold forming affects behavior and subsequent high rejection rates, a series of in-plane biaxial tests was conducted on thin 0.l-ram (0.004-in.) fully cold-rolled 304 stainless steel sheets. The sheets were tested using an in-plane biaxial test system with acoustic emission. A failure surface was mapped out for the 304 stainless steel sheet. Results from this study indicated that an angle of 72 ~ from the transverse orientation for the peak strain direction during forming should be avoided. This was microstructurally related to the length-to-width ratio of the elongated 304 stainless steel grains. Thus on rejected parts, it is expected that a high number of cracks will be located in the plastic deformation regions of cold-formed details with the same orientation. KEYWORDS: in-plane biaxial failure surfaces, stainless steel, texture, cold forming, equivalent stress,
failure loads
Metals are among the most common manufacturing materials in the world. Unless cast to shape, metals are typically solidified in large billets and then subsequently processed via cold (or warm) working into near final shape. This cold working of a material into the final shape changes the material's microstructure and associated properties. In fact, the metal's grains take on a preferred orientation (or texturing) which aligns the crystal structure differently in the direction of rolling (longitudinal) than in the direction perpendicular to rolling (transverse). Texturing can transform a material with similar properties in all directions (isotropic) to one with substantial variations in material properties with direction (nonisotropic). In most cases, yield strength is higher in the rolling direction while strain-to-failure is higher in the transverse direction. Tensile strength, strength coefficient (K) and strain hardening exponent (n) values (as defined in ASTM E 646) and other mechanical properties can also he affected. For manufacturing facilities which utilize many rolling or forming operations, it is important to understand how the material properties may be evolving in each direction from one forming process to the next. During the forming of sheet metal components, a biaxial stress state is encountered by the material. Biaxial stress states can result in a much different stress-strain behavior than observed under uniaxial loading conditions. Generally, the strength, and associated forming forces, can increase by up to 30% depending on the biaxiality of the stress state as discussed by Shiratori and Ikegami [1] and Kreibig and Schindler [2]. Strain-to-failure also depends on states of stress. Another point of interest is how subsequent material behavior is affected by a substantial inelastic strain. For example, a sheet 1 St. Cloud State University, St. Cloud, MN. 2 NASA Glenn Research Center, Cleveland, OH.
Copyright9
by ASTM International
26 www.astm.org
COVEY AND BARTOLO-I-I'AON STAINLESSSTEEL
27
metal may be plastically deformed in one manufacturing process and then subsequently deformed in another operation. It is hypothesized that these types of complex processing are typically the cause for high part rejection rates in sheet metal components. Consequently, an understanding of material behavior under complex stress states is essential for detailed tool and process design. To investigate the intricacies of the sheet metal forming process, a series of in-plane biaxial tests was conducted on thin 0.1 rnm (0.004 in.) fully cold-rolled 304 stainless steel sheets. This paper discusses the results of the study describes briefly the unique capabilities of the biaxial test system that was used to generate the failure surface data.
Material Details The material used in this study was a fully cold-rolled 304 stainless steel sheet 0.1 mm (0.004 in.) thick. Using a standard etching solution (10 mL HNO3, 10 mL acetic acid, 15 mL HCL, and 5 mL glycerol), the textured microstructure of the 304 stainless steel is clearly visible (Fig. 1). The grain length is three times longer than its width indicating the rolling direction of the material. Initial uniaxial static tests were conducted on coupon samples. These samples were cut from the same lot of 304 stainless steel as used in the subsequent biaxial tests. The test specimens were machined in two orientations: longitudinal (parallel with the rolling direction) and transverse (perpendicular with the rolling direction). The specimens were 12 mm wide by 0.1 mm thick with a 114.3mm-long test section. The extensometer gage length was 50.9 mm. The specimens were tested in displacement control at a rate of 0.5 mm/min up to 0.75 mm displacement and then at a faster displacement rate of 5 mm/s until failure.
FIG. 1--Photomicrograph of the 304 stainless steel grain structure showing that the rolling direction grain size is three times that of the transverse direction (original magnification m400, electropolished).
28
MULTIAXIAL FATIGUE AND DEFORMATION TABLE 1--Uniaxial tensile properties of 304 stainless steel sheet.
Orientation
Modulus, GPa
0.2% Yield Stress, MPa
Ultimate Tensile Strength, MPa
Failure Strain, %
n
K, MPa
Longitudinal Transverse
160 183
1225 1181
1343 1409
2.48 4.73
0.285 0.137
4437 2477
Uniaxial longitudinal and transverse properties are summarized in Table 1. The data are averages from 12 tests for each direction. Standard deviations on stress and elastic modulus values are less than 0.5%. Note that the elastic modulus values differ by almost 15% and the strain-to-failure by nearly a factor of two for this "homogeneous" material. Experimental Details
Specimen Geometry The specimens were machined from 300 m m (12-in.) square plates with geometry based on the work of Shiratori and Ikegami [1] and Kreibig and Schindler [2]. These specimens had a reduced width gage section with a double reduction of radius of curvature from about 11 m m (0.43 in.) to about half that at the comer root (Fig. 2). The intent of the specimen geometry was to induce a true uniform biaxial stress state over as much of the gage section as possible, without a large stress concentration within the comer root. Shiratori and Ikegami [1] and Kreibig and Schindler [2] report a fairly uniform stress distribution as defined by numerical, strain gage, photoelastic, and failure results. The specimen geometry used here should provide useful results even though fabrication of these thin specimens required some minor changes from those in the references. Generally, verification of stress state quality in the gage section of cruciform test specimens requires extensive finite-element analysis and utilizes a reduced thickness for optimization among the relevant parameters.
5.5 mm
f~
11 mm "-4 ~ / - - F ~ d i u s
150 m m
r
<
300 m m FIG. 2--Cruciform specimen geometry.
IL
COVEY AND BARTOLOTTA ON STAINLESS STEEL
29
Demmerle and Boehler [3] give an excellent discussion of these methods and the resulting optimized geometry. These reduced section specimens are very expensive and are not applicable for testing of thin sheets. However, a uniform stress state free of stress risers is still of concern and is discussed later.
Equipment and Test Details Most biaxial material tests are performed on tubes in an axial/torsion test rig. However, actual sheet metal geometries (i.e., thin plates) prohibit such testing. NASA Glenn Research Center, in Cleveland, Ohio, has two in-plane biaxial test rigs for this type of testing. These rigs are computercontrolled servohydraulic test frames with hydraulic grips. Figure 3 shows the grip configuration with a strain-gaged sample used to verify alignment. Even though they have a large force capacity of 500 kN (110 kip), testing of these thin plates was successfully performed. Since the grip wedges would not allow testing of such thin sheets, hardened steel shims were glued on each side of the cruciform's arms. Then, cardboard was glued onto the shims to provide enough lateral stiffness to allow mounting the specimen into the test machine. Alignment of the load frame and grips was performed using a precision steel specimen carefully equipped with 44 strain gages: eleven in each direction on each side. Alignment was considered adequate when the strain levels on both sides and at each arm were within 5% of the nominal applied strains for equal X and Y loading with no indications of significant localized bending strains. Due to the thinness and the relative geometry of the specimen, all tests were performed in load control. In displacement control, the risk for off-axis and unequal loading is significantly high thereby compromising the stress uniformity in the specimen test section. Furthermore, since the 304 stainless steel sheets have a relatively low ductility (<5%), the affects of load control mode on the materials yielding behavior is minimized. It is the authors' opinion that the benefits of load control for these tests outweigh its disadvantages. The X and Y loads were controlled to provide a constant effective
FIG. 3--Biaxial test system with strain gaged alignment specimen.
30
MULTIAXIAL FATIGUE AND DEFORMATION
stress rate of 3.44 MPa/s (0.5 ksi/s) until the sample failed. Only tension-tension (i.e., first quadrant of biaxial stress space) testing was performed because of this thin material's inability to support compressive stresses. Fourteen tests were planned with two each at seven different 0-angles (0 ~ (uniaxial X, transverse), 18, 36, 45, 54, 72, and 90 ~ (uniaxial Y, longitudinal)). For this study, "0-angle" refers to the orientation of the maximum principal stress plane with respect to the transverse rolling direction. Test control software and sample damage evaluation techniques are discussed below.
Control Software Data acquisition and waveform generation were performed via a program written using an objectoriented control software (Labview). This software allows design of virtual instruments using graphical icons that can be wired together. Each icon serves a unique purpose not unlike a subroutine of a lower level programming language. The control software ramped up load on both the X- and Y-axis at any selected 0-angle at any effective stress rate to any maximum effective stress. The control software requires as input: full-scale load and strain levels, the specimen's cross-sectional area, and the expected elastic modulus and Poisson's ratio. During the test, the control software provides a realtime effective stress versus effective strain curve and large digital readouts of effective stress and X and Y load. Stress levels in the X and Y directions were determined independently from load via load cells and strain via strain gages. The material elastic modulus was determined in the X and Y directions using stress via load and strain via strain gage. End of test could be determined via change of elastic modulus by a certain percentage, maximum effective stress, magnitude of inelastic strain, or increase in acoustic emission signal. At any time during the test, the user could stop, hold/pause, or return to zero load using three large buttons on the front panel. Once the software received the maximum effective stress or other test-end signal, the program returned the load to zero in ten seconds. All time, load, strain, and displacement data are written in a spreadsheet-usable ASCII format file name selected by the user. The effective stress (and hence effective stress rate) and effective strain are defined by a von Mises equivalent approach as used by Shiratori and Ikegami [1] with O'eq = ~r
-- O"x O'y -~- OV~y
(1)
( 8 x2 Jr- 8 x 8y q- 8 2)
(2)
and eeq :
where ex and e r were measured using strain gages. The stresses, o'x and o-r, were calculated using a specimen area defined by Kreibig and Schindler's work. The areas were calibrated for each specimen by using strain measurements (at a load level well within the material's elastic region), calculating the stress using equations of elasticity and compare them to the calculated stresses using Kreibig and Schindler solution. Results and Discussion
Failure Surface The first quadrant (tension-tension) failure surface was successfully obtained and is given in Fig. 4. Note that failure was defined as complete specimen fracture. The data are also given in Table 2. The surface is essentially elliptical with a major axis in the transverse direction. The cruciform specimen geometry was justified because the specimen failed at an applied stress within 10% of the uniaxial strength data when loaded along that axis only. However, when loaded only in the longitu-
COVEY AND BARTOLO'I-[A ON STAINLESS STEEL 1750 t
Uniaxial Rollinq Direction Pronerties: UTS = 1350 MPa Strain-to-Failure 2.5%
31
. . . . . . . . .
140QP
• =Z._~
r
1050, O
~.--= 700 350
i t i t J i
,
,
350
,
i
i
i
I,,,, i ,
t
1750
1050
Xr:nt~Ss (MPa) .. se uirectlon
~ ~
Uniaxlal Transverse Direction Properties: UTS = 1410 MPa Strain-to-Failure = 4.7%
FIG. 4--In-plane biaxial failure surface of fully cold-rolled 304 stainless steel sheets 0,1 mm (0.004 in,) thick.
dinal direction, the cruciform specimen failed at an applied stress 20% lower than that obtained from uniaxial coupons. It is likely that the higher uniaxial strain-to-failure in the transverse direction (4.73%) allowed the specimen comer stress concentration factors to reduce to one via plastic deformation while the more brittle rolling direction (strain-to-failure = 2.48%) maintained a stress concentration factor greater than one until failure. Failed samples were clearly indicative of their 0-angle. Figure 5 shows failed specimens for 18 ~ and 45 ~ 0-angle tests. Note that the failure planes match the loading 0-angle (i.e., tests with an 18 ~ [from the transverse axis] 0-angle had an 18 ~ [from the transverse axis] failure). Figure 6 shows micrographs of the same two specimens at two magnifications. The fracture planes are transgranular (i.e., through the grains) with the plane coinciding with the 0-angle. The observations shown in
TABLE 2--O-Angle and associated effective stress at failure. 0-Angle, deg
Effective Stress at Failure, MPa (ksi)
0 (transverse) 0 (transve~e) 18 18 36 36 45 45 54 72 72 90(long~udinai)
1316q 191) 1337q 194) 1137~ 165) 1068~ 155) 1075~ 156) 992~ 144) 896~ 130) 896~ 130) 861, 125) 827, 120) 813, 118) 1020, 148)
FIG. 5 - - F a i l e d s p e c i m e n s e c t i o n s f o r 1 8 ~ a n d 4 5 ~ l o a d i n g c o n d i t i o n s .
z
>
c m > z o o rn 71 0 ~J
1-71
x
c 1----t
po
FIG. 6 - - M i c r o s t r u c t u r e o f fracture plane f o r 18 ~ and 45 ~ loaded specimens.
O~ r
-..t m m r-
m
r-
z
-.t
z
0
tO
>
w
< m -< > z
0 0
FIG. 7--Typical microstructure of fracture plane for 72 ~ loaded specimens showing intergranular failure.
z
0
"11
z ~D o m
c m
W" "11
x
I-"
c
COVEY AND BARTOLOTTA ON STAINLESS STEEL
35
both Figs. 5 and 6 are typical for all of the biaxial specimens for each loading 0-angle with the exception of the 72 ~ 0-angle tests. In the 72 ~ 0-angle tests, the fracture planes were oriented at approximately 72 ~ from the transverse rolling direction. However, the fracture planes of the 72 ~ 0angle tests exhibited mostly intergranular fracture (i.e., along the grain boundary), which is indicative to the lower failure stress (Fig. 7). These findings further support that the cruciform geometry used here provided at least a reasonably uniform stress state without gross stress concentration factors. From these results there appear to be two useful pieces of information for optimization of manufacturing processes with this 304 stainless steel material. Industries attempting to form this material with significant cutouts or discontinuities should be able to decrease part rejects by orienting the peak strain axis along an angle less than 36 ~ from the transverse direction. Even though a biaxial test may not have been necessary to know forming would be best along the transverse direction, the magnitude of benefit for discontinuous geometries (25%) and limiting angle (32 ~ from transverse) would not have been available from uniaxial data. In general, a stress concentration factor greater than one implies a strain localization which is undesirable for most forming operations. The second point worth noting is that 72 ~ from the transverse direction shows a global minimum peak strength to less than 40% of the uniaxial data. Figure 8 shows this global minimum effective stress at the 72 ~ angle more clearly. The cause for this sharp decrease in strength is likely due to the ratio of the grain size in the roiling direction to that in the transverse direction. Since the grains are about three times longer in the rolling/longitudinal direction, the angle between the average longitudinal grain boundary length and the average transverse grain boundary length is exactly 72 ~ (see Fig. 9). Consequently, lower strength at this 0-angle may indicate increased grain boundary failure over the other orientations. Forming operations should be set up to avoid peak strains in the direction 72 ~
1400
8
1300
~"
1200
1100
0
0
1000
900
800
700 - 18
I
I
0
18
I
I
I
I
36
54
72
90
e-angle (degrees)
FIG. 8--Plot of effective stress vs. O-angle showing minimum at 72 ~
36
MULTIAXIAL FATIGUE AND DEFORMATION
Transverse Direction
c FIG. 9--Sketch of the 304 stainless steel sheet grain structure illustrating the 72 ~plane of minimum strength.
from transverse. A careful study of failed parts would likely show a high percentage of cracks at a 72 ~ angle from the transverse direction. Conclusions and Recommendations
In-plane biaxial testing of thin 0.1 mm (0.004 in.) fully cold-rolled 304 stainless sheet steel prodnced the following conclusions. Consideration of these items during process design could reduce the number of rejected parts during subsequent forming operations. Careful study of the failure surface obtained during the course of this work provides the following recommendations: (1) Forming operations with cutouts or discontinuities should align the maximum strain orientation to an angle less than 36 ~ from the transverse direction to reduce stress (and hence strain) concentration effects and therefore reduce rejected parts. (2) An angle of 72 ~ from the transverse direction for the peak strain direction during forming should be avoided, if at all possible, because a significant strain localization occurs there. It is likely that this localization is due to the ratio of grain boundary length in the rolling direction to that in the transverse direction, which makes a 72 ~ angle to the transverse direction. In general forming, one would expect an unusually high number of cracks to be found on rejected parts at that orientation. Acknowledgments S. Covey was supported by the American Society of Engineering Educators/NASA Summer Faculty Fellowship program. The help of S. Smith with sample preparation was greatly appreciated.
COVEY AND BARTOLOI-FA ON STAINLESS STEEL
37
References [1 ]
Shiratori, E. and Ikegami, K., "Experimental Study of the Subsequent Yield Surface by Using Cross-Shaped Specimens," Journal of the Mechanics and Physics of Solids. Vol. 41, 1993, pp. 143-181. [2] Kreibig, R. and Schindler, J., "Some Experimental Results on Yield Condition in Plane Stress State," Acta Mechanica, Vol. 65, 1986, pp. 169-179. [3] Demmerle, S. and Boehler, J., "Optimal Design of Biaxial Tensile Cruciform Specimens," Journal of the Mechanics and Physics of Solids. Vol. 41, 1993, pp. 143-181. [4] LABV1EW User Manual, National Instruments Corporation, 1994. [5] Miller, R. and McIntire, P., Eds., Nondestructive Testing Handbook Volume 5: Acoustic Emission Testing," American Society of Nondestructive Testing, 2nd ed., 1985. [6] Winstone, M., "Influence of Prestress on the Yield Surface of the Cast Nickel Superalloy MAR-M002 at Elevated Temperature," Proceedings, Presented at the Mechanical Behaviour of Materials--IV, Stockholm, Sweden, August 1983, pp. 199-205.
Multiaxial Deformation of Materials
Jalees Ahmad, 1 G o l a m M. Newaz, 2 a n d Theodore Nicholas 3
Analysis of Characterization Methods for Inelastic Composite Material Deformation Under Multiaxial Stresses REFERENCE: Ahmad, J., Newaz, G. M., and Nicholas, T., "Analysis of Characterization Methods for Inelastic Composite Material Deformation Under Multiaxial Stresses," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 41-53. ABSTRACT: Off-axis tension (OAT) tests and biaxially loaded cruciform-shaped specimen (BC) tests on unidirectional fiber-reinforced metal matrix composites are subjected to detailed analyses using a newly developed model. The results indicate that the off-axis tension test is a viable method for biaxial stress-strain characterization of composites in the nonlinear deformation range. It is found that, besides being much more expensive to conduct, the specific cruciform-shaped specimen test considered in the study is less amenable to unambiguous interpretation.
KEYWORDS: multiaxial, analysis, testing, metal matrix composite, modeling Nomenclature
g Es-hEm h (V~- 1)/Vr T Ey Em E/j /~j Ipp ./2 jz Nij Rij SU Sa Tp Vf de 0 de~ ds~
Temperature Young's modulus of fiber material Young's modulus of matrix material Elastic moduli of composite Interfacial resistance to sliding Debond strength in direction p Second invariant of deviatoric stress tensor Analog of J2 Dimensionless parameters Residual stress components Deviatoric stress components Analog of S o Stress-free (processing) temperature Fiber volume fraction Strain increment Elastic strain increment Inelastic strain increment
1 Research Applications, Inc., 11772 Sorrento Valley Road, Suite 145, San Diego, CA 92121-1085. 2 Wayne State University, Mechanical Engineering Department, 5050 Anthony Wayne Drive, Detroit, MI 48202. 3 Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson AFB, OH 454337817.
Copyright9
by ASTM lntcrnational
41 www.astm.org
42
MULTIAXIALFATIGUE AND DEFORMATION af o/m ~s ~m e0 ~0
Coefficient of thermal expansion of fiber material Coefficient of thermal expansion of matrix material Average a s over temperature range Average c~mover temperature range Strain components Strain-rate components ~ij'm'i Components of inelastic strain-rate in matrix s~ Elastic strain components Kij Components of dimensionless sliding damage parameter A Constitutive equation parameter for matrix material )tij Constitutive equation parameters for composite us Poisson's ratio of the fiber material Um Poisson's ratio of the matrix material O'e Effective stress ~ j Stress components ~ro Yield strength of matrix material Anisotropy of composite materials often necessitates their characterization under multiaxial stress states. There are, of course, established test methods for multiaxial characterization of composites in the linear-elastic deformation range. For example, under biaxial loading in the linear range, one can determine the Young's modulus values (Eli, i = 1,2) of a continuous fiber composite by conducting two individual tests in which a uniaxial load is applied consecutively in directions 1 and 2. The modulus in any other direction can then be reliably predicted using well-known stress and strain transformation relations. However, in the nonlinear range, one would need a number of tests, each requiring simultaneous application of several combinations of loads in the two directions. This is necessitated due to the lack of an established inelastic deformation theory (and stress-strain relations) for composites. In the context of metal matrix composites (MMCs), there have been some attempts to develop characterization methods under multiaxial stress states. For example, test data on the deformation behavior of MMC tubes subjected to combined tension and torsion loads have been reported [1]. Recently, Kirpatrick [2] has reported test data using off-axis tension (OAT) and biaxially loaded cruciformshaped (BC) test specimens of unidirectional MMC panels (Fig. 1). An especially valuable contribution to the study of MMC deformation under multiaxial stress states is due to Sun et al. [3]. They seem to be among the first to recognize that OAT tests provide an effective way of developing test data on MMCs under a wide range of biaxial stress states. Recently, OAT data have been reported by Ahmad
FIG. 1 - - A n off-axis tensile specimen (a) and a biaxial cruciform-shaped specimen (b). The darker areas indicated gripped regions. Typical dimensions f o r eight-ply thick (t ~ 1.5 mm) composite specimens are: L = 100 to 150 mm, W = 12 to 20 m m and g = 25 to 30 mm.
AHMAD ET AL. ON CHARACTERIZATION METHODS
43
et al. [4]. On the analytical side, variations of the Hill's theory of anisotropic plasticity [5] have been attempted [2,6]. Titanium-basedMMCs with ceramic fibers are particularly prone to fiber-matrix interface damage. Both the type and extent of damage changes with micromechanical stress state, which, in part, is affected by applied stresses. Depending on the stress state, micromechanical damage can significantly affect the macroscopic (global) deformation behavior of these composites. Additionally, residual stresses induced by the composite consolidation process have a significant effect on the global deformation behavior. A Hill's type theory does not explicitly account for these important factors. Therefore, its use causes difficulties in correlating test data. Recently, Abroad and Nicholas [7] and Abroad et al. [4] have presented a somewhat different theory (called jA) for predicting nonlinear deformation of composites. The theory explicitly includes the effects of matrix material inelasticity (in the form of plasticity, creep, and viscoplasticity), micromechanical residual stresses, and interfacial damage on global deformation. It has been found [8] that, under plane stress, the theory provides reasonably accurate predictions of global inelastic response of titanium-based MMCs. Implementation of the theory requires knowledge of the deformation and damage characteristics of only the constituent (fiber and matrix) materials and the fiber-matrix interface, and not of the composite. Laboratory testing of the composite is needed only for characterizing the interface. Once the interface is characterized, the theory enables one to predict, rather than only correlate, nonlinear global deformation characteristics of a composite under multiaxial stress states. In the present work, the jA theory is used in investigating the efficacy of the OAT and BC test methods for characterizing nonlinear composite deformation behavior under multiaxial stress states. Based on comparisons of theoretical predictions and test data, conclusions are drawn regarding relative merits of the test methods. Of course, the conclusions are provisional in that the theoretical predictions are assumed reasonably accurate. Description of Theory In plane stress, designating 1 and 2 as the orthogonal principal material directions, the potential function in the J~ theory can be expressed as follows [4]: 2 1
1
jA = i~-1 y " "~ (Nii~ q- Rii)2 A- (N120r12 -t- R12) 2 - -~ [(NnoM + Rll)(N220"22 A- R22)]
(1)
in which No are material specific dimensionless parameters. The parameters R 0 are the components of the average residual stress in the matrix material. For a given composite system, values ofN 0 and Rii can be found using micromechanics considerations [4]. ~rUare components of the average stress imposed on the composite by external sources, such as mechanical or thermal straining. The effective stress is defined as: o-~ = N/~2A
(2)
If elastic-plastic deformation behavior is assumed, the condition for initial yield of the composite is: O'e = O'o
(3)
in which o-0 is the yield strength of the matrix material, assumed isotropic. In composites, the deviation from its initial linear elastic stress-strain behavior can occur by mechanisms other than matrix inelasticity. For example, in MMCs, fiber-matrix interfacial damage in the form of separation ("debond") or sliding can occur. Therefore, certain damage conditions need to be brought into consideration. For example, if it is assumed that interfacial debond and sliding consti-
44
MULTIAXIAL FATIGUE AND DEFORMATION
tute the dominant damage mechanisms affecting the global deformation response of MMCs, the following damage conditions can be used:
o'pp - lpp -- 0 (p = 2,3)
(4)
fordebond, and
I0"01 -
10. ~ 0, (i ~j,
O-pp - Ipp < 0)
(5)
for sliding. In the above equations, Ipp represents the debond strength and 10.(i ~ j) represents resistance to mutual sliding between a fiber and the surrounding matrix. Both Ipp and 10. (i 4=j) can depend on the internal stress state (Rij) as well as on externally imposed stresses, ~i. Also,//j (i v~j) need not be independent of lpp. Generally, determination of these parameters would require laboratory testing of the composite. In composites with a "weak" fiber matrix bond, it can be assumed that I22 = -R22 and 112 = I 0"22
- - K12122 I
(6)
where, tq2 is a positive constant. Equations 1 through 6 describe the initial yield/damage surface of a unidirectional composite. The elastic strain (e~) prior to the occurrence of yield or damage is found using the Generalized Hooke's law for orthotropic solids and elastic properties of the composite. Beyond initial yield/damage, incremental stress-incremental strain relations must be prescribed. The total strain increment (de0) due to an external stress increment is composed of elastic strain increment (de~), inelastic strain increment (delj) caused by matrix material inelasticity, and strain increment due to the occurrence of damage. Prior to occurrence of any damage, the elastic strain increment de~ can be found by using the Generalized Hooke's law for orthotropic solids. The strain increment due to damage is found by assuming that the load (in the damaged region only) is supported entirely (or, largely) by the matrix. Thus, in the damaged region, the average stress components in the matrix are found by considering the volume initially occupied by the fibers to be voids, and redistributing local residual stresses. Then, within the elastic range, the strain increment after the occurrence of damage is found by using the Hooke' s law and the matrix material' s elastic constants. To determine the inelastic strain increments delj, the rate form of the Prandtl-Reuss relations is used. For the matrix material (m), these relations can be expressed as: ~,i=
}~Sij
(7)
in which A is a scalar whose value may depend on one or more state variables. One can use one of a number of constitutive models available in the literature to establish A for the matrix material. For the composite, the deviatoric stress (SA) is defined as: 3
S A = Nii0-ii .-}- Rii - ~1- ~_
(Niio.ii _]_ Rii)
and Sa = 2(N0.~j + Rij) for i :~ j
(8)
45
AHMAD ET AL. ON CHARACTERIZATION METHODS
Then, the components of the inelastic strain rate in the composite are: ~0 = A,JSA
(9)
The parameters A0 are scalars analogous to A for the isotropic material case. To find inelastic strain rate after the occurrence of damage, one needs to estimate the stress redistribution caused by damage, assume that the stress is largely carried by the matrix material, then use Eq 7 to find the inelastic strain rate components. The parameters Aij corresponding to each inelastic strain rate component are found by using micromechanics considerations. Estimation of the various parameters involved in Eqs 1 to 9 is discussed in detail in Refs 4 and 8. Briefly, equilibrium of forces under purely axial loading gives: N1, = [ 1
VfEfell]](1 O'11
]
--
VI)
(10a)
In the linear range, Eq 10 gives: (10b)
N1 l(e) = E,,,/E11
Assuming, under purely transverse and under purely shear loading, that the average matrix stress is the same as the applied stress, the parameter N22 = N12 = N33 = N23 = N31 ~- 1.0
(11)
where Em is the Young's modulus of the matrix material and E~I is the elastic modulus of the composite in the fiber direction, given by: E l l = VIEy + (1 - Vy)E,n
(12)
In the above equation, E s represents the Young's modulus of the fiber material, and Vf is the volume fraction of the fiber in the composite. Using an elastic cylinder within a cylinder model of a unit composite cell, estimates for the residual stress components are: R12 = 0 . 0
(13)
EmEyVI
EI~
Rll
(~s - ~m)(T-
Te)
and R22 = Rll
E11(1 - Vy) V~
(14)
where g + vmEf - h ufEm = hEr[g(1 + Vm) + Em(U,, -- Uf)] -- [v.fhEm - ~',nEf - g][(2Um -- DE s - 2hvfEm - g] g= El-
hEmandh=(V
I-
1)/Vf
46
MULTIAXIAL FATIGUE AND DEFORMATION
In the above, T is the temperature and T e is the (stress free) composite consolidation temperature. The parameters ~y and ~m are temperature-averaged values of the coefficient of thermal expansion (CTE) of the fiber and the matrix materials, respectively. For titanium matrix composites having weak fiber-matrix interfaces 122 :
--R22
(15)
and 112 = I 0"22 - -
K12h2 I
(16)
In plane stress, the relevant parameters in Eq 9 are All, )t22, and A12. References 4 and 8 provide the following estimate for these parameters: A0 = AN0(e) (1 - Vy)
(17)
in which A is the parameter associated with the constitutive model used in describing the matrix material behavior. Equations 9 and 17 are used in predicting a composite body's global deformation response in the inelastic range. As discussed earlier, if one or both the damage conditions are met, N o, Rij, and Aii are modified to model post damage behavior. The nonlinear problem represented by the above equations requires numerical solution. In the present work, the solutions were obtained by the finite-element method. The details of the method relevant to the present study can be found in Ref 8.
Analysis Results The experimental data selected for the present study are on SCS-6/Ti-6-4 [0113 and TIMETAL SCS-6/TIMETAL 21S [0]6 composites. OAT test data on the SCS-6/Ti-6-4 composite are available in Refs 3 and 8. OAT and BC test data on the SCS-6/TIMETAL 21S [0]6 composite were found in Ref 2. The estimated values of the material specific parameters for these material systems are given in the Appendix. The Bodner-Partom viscoplastic model [9] was selected to represent the inelastic deformation response of the matrix materials. The material constants associated with this model are also given in the Appendix. The only parameter that cannot be determined using the matrix and fiber material properties is/(12, which is an assumed characteristic of the fiber-matrix interface. As discussed in Refs 4 and 8, this parameter can be determined using an Iosipescu shear test [10] on the composite, losipescu test data were not available for the SCS-6/Ti-6-4 (Vs = 0.4) and the SCS-6/TIMETAL 21S (Vs = 0.29) materials. As an expedient, the /(12 = 2.2 value found in Ref4 (for SCS-6/Ti-6-4 [0Is with Vf = 0.27) was used in the pi:esent study. This assumption would be of significance only in analyses of those cases in which fiber-matrix sliding damage might occur. The analyses involving the J~ theory were performed using the IDAC finite-elementanalysis code [S]. Off-Axis Tensile Tests
Previous studies [11] have shown that, with judiciously selected specimen dimensions and loading arrangement, the stress state in the gage section of an OAT test specimen is sufficiently uniform. As-
AHMAD ET AL. ON CHARACTERIZATION METHODS
47
TABLE 1--Stress states in off-axis tests corresponding to a unit applied stress.
Off-Axis Angle, degrees
0-11
0.0 15.0 22.5 30.0 45.0 60.0 90.0
0.933 0.854 0.750 0.500 0.250 0.000
0"22
0"12
0.000 0.067 0.146 0.250 0.500 0.750 1.000
1.000
0.000 0.250 0.354 0.433 0.500 0.433 0.000
suming a uniform stress state, Table 1 gives the relative magnitude of the three relevant stress components for several off-axis angles. Figure 2 shows the numerical predictions together with the test data of Sun et al. [3] for various off-axis angles. The test data correspond to SCS-6/Ti-6-4 [0113 composite (Vs = 0.40) at room temperature. In the analyses, only the gage section was included in the finite-element model. The analyses were performed assuming a state of plane-stress. It is seen that, overall, analysis predictions and test data are consistent. In all cases, the analysis results show deviation from linear stress-strain behavior. This deviation is due to matrix material yielding and interfacial damage. Table 2 is a summary of the predicted sequence of inelastic deformation/damage mechanisms corresponding to each OAT test shown in Fig. 2. The predicted sequence is in agreement with reported experimental observations for the zero and 90-degree cases; for example, see Ref 12. For the remaining off-axis angles, direct experimental validation of the predicted damage sequence is not possible because of lack of corresponding experimental observations.
1400
~.
1300
. . . . . . . . . . . . -
1200 1100 1000
ft. v 09
09
900
800
-
, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Predictions
o O "
0~ Test D a t a 15~ Test Data 300 Test Data
v O O
45~ Test Data 60~ Test Data 90~ Test Data
Ti-6-4/SCS-6 (Vf=0.40 h Room Temperature 0o / o o O/ ~
_ _ _
15 ~
yfj_-
700
600
45~
~
5oo
400
/'~^
300
~
'~ ~
~,,f~"(...,..,,~-#"~_..o,.--~
~ _ I
~
_I_. ~-._.~---;.--r o
; o ~ 9o o
loo 0
~
0.000
. . . .
0.001
0.002
i . . . . . . . . .
0.003
J . . . . . . . . .
0.004
J . . . . . . . . .
0.005
~_
I
0.006
Strain
FIG. 2--Off-axis test data [3] and predictions f o r Ti-6-4/SCS-6
[0113
composite.
48
MULTIAXIAL FATIGUE AND DEFORMATION
TABLE 2--Predicted damage sequence in off-axis tests [3]. Inelasticity/Damage Sequence Analysis of Ref 3 Data
Analysis of Ref 2 Data
Off-Axis Angle, degrees
1
2
1
2
0.0 15.0 22.5 30.0 45.0 60.0 90.0
yielding yielding . . . sliding debonding* debonding* debonding
fiber fracture sliding
yielding yielding sliding sliding debonding* debonding* debonding
fiber fracture sliding yielding yielding yielding yielding yielding
.
.
. yielding yielding yielding yielding
* In these cases, it is assumed that sliding also occurs if the debonding criterion is satisfied.
Figure 3 shows the test data [2] and predictions for OAT tests on SCS-6/TIMETAL 21S [0]6 (gf = 0.29) at room temperature. The procedure used in these analyses was the same as in the analysis of Ref 3 data. The sequence of inelastic deformation/damage corresponding to each test is shown in Table 2. There are considerably more pronounced discrepancies among test data and predictions in Fig. 3 than in Fig. 2. Especially, the test data corresponding to off-axis angles larger than thirty degrees show stress plateaus (saturation levels) to be lower than predicted. Plausible reasons for these discrepancies include overestimation of one or both interracial strength parameters and overestimation of strain-hardening in the constitutive model used for describing the stress-strain behavior of the matrix material. Nevertheless, predictions and test data are not entirely inconsistent. Additional comparisons of predictions and test data on OAT tests on an SCS-6/Ti-6-4 composite can be found in Refs 4 and 8. Overall, it can be concluded that the OAT test does provide deformation response data on MMCs that is consistent with the stress states shown in Table h
2600
.
.
.
.
.
.
.
.
,
.
.
.
.
.
.
.
.
.
.
t
.
.
.
.
.
.
.
.
.
2400 TIMETAL
21S/SCS-6,
Vt=0.29 , Room
Temperature
2200
- -
2000
0 data 15 data 22 data ~7 30 data O 45 data 0 60 data 9 90 data
0o 1800
~ ~
,--. 1600
/ 1400 r r 1200 1,1.1 I..,
0 v
/o
/
[]
/ O n
/.,~'~
1000
15
9
fy
400
V~I~.
2OO
__22 ~
,~ .
~
o
~
800
i
v
30 ~
~,.
\ ~
Predictions
O [] A
oo
.O
~
90 ~
f
0.00
0.01
0.02
0.03
STRAIN
FIG. 3--Off-axis data [2] and predictions for SCS-6/TIMETAL 21S [0]6 composite.
AHMAD ET AL. ON CHARACTERIZATION METHODS
49
Biaxial-Load Tests Figure 4 is a schematic drawing of the cruciform-shaped test specimen used in Ref 2 for testing a unidirectional SCS-6/TIMETAL 21S [0]6 (Vf = 0.29) composite under biaxial loading. Tests were performed at room temperature. The fibers were aligned with one of the load directions. Tests were performed for several fiber direction load (PI) to transverse load (P2) ratios. The loading in all cases was monotonic under applied load control conditions. Even though the specimen itself is essentially in a state of plane-stress, the BC specimen test setup (Fig. 4) requires a three-dimensional analysis to determine the stress state in the gage section. In its present form the jA theory can be applied only to plane-stress problems. To circumvent this difficulty, first a three-dimensional analysis was performed for each load case to detemfine the stress state within the linear deformation range. Next, a two-dimensional nonlinear analysis was started in which the specimen was assumed to be in plane-stress, and stresses at the boundary of the gage section were assigned based on the results of the linear three-dimensional analysis. Subsequently, all stress components were proportionally increased to simulate specimen loading in the inelastic deformation regime. Three-dimensional linear finite-element analyses were performed using the ABAQUS generalpurpose finite-element code. The MMC was assumed to be a homogeneous, linearly elastic, orthotropic solid ( E ] I = 197 GPa, E22 = E33 = 163 GPa, G I 2 = G13 = 60.8 GPa, G23 -~ 60.1 GPa, v12 = v13 = 0.311, v23 = 0.353). The titanium alloy was assumed to be linearly elastic and isotropic (E = 117 GPa, v = 0.34). For P2/P1 = 0.25, Fig. 5 shows the computed stress distribution in the MMC along the X-axis (Fig. 4). It is seen that, despite Pz being positive (tensile), ~r= in the gage section is
FIG. 4--The BC test specimen and grip configuration used in Ref 2.
50
MULTIAXIAL FATIGUE AND DEFORMATION 2.00 ~ . . . . . . . . .
~.........
~.
. . . . . . . .
~,,
. . . . . . .
3-D Linear :EM Analysis o a
~1.50 LL t-- 1.40
\ ~H/'Z
(/)
//
O9
022/~-,
/
"0
.--- 0.0o r
if)
Remote appl ed stress in fil: er direction=T. Remote appl ed stress in tr~ nsverse directi~ )n=YJ4
-0.05 3
Distance From Center Along X-Axis (mm)
FIG. 5--Stress distribution across the centerline of the BC specimen gage section.
compressive. For the same P2 and P2/P1 <0.25, the 0"22component was found to be even more compressive than shown in Fig. 5. Thus, the analyses give the unexpected result that, for P2/P] -<0.25, the test data in Ref 2 correspond to tension-compression stress-strain behavior of the composite even though both P1 and P2 are tensile. Analysis in the nonlinear deformation range was performed using the IDAC finite-element analysis code [8] that incorporates the jA theory for analysis of two-dimensionalproblems. Figure 6 shows
2000
1500
1000 12. u~ U)
500
,
.
t
.
.
.
.
i
i
i
i
i
i
I
i
i
i
r
h
=
,
I
I
.,
TIM ETAL 21S/SCS-6 (V{=0.29, 23~ Under Biaxial Loading
1: Axial Specimen (0 ~ load) 2"9 P2/Pl=0.125 3:P2/P1--0.000 4:P2/PI=-0.200 (negative P2) 5:Pz/Pt=-0.330 (negative Pz)
Ell VS.
(Y] 1
0.010
0.015
O9
0 3
-500
4~ 522
VS. (~
22
-1000
-o.;IO
-ooo5
o.ooo
0.005
0.020
Strain
FIG. 6--Analysis results for BC test specimen of SCS-6/TIMETAL 21S [0]6 composite.
AHMAD ET AL. ON CHARACTERIZATION METHODS
80000
O [3 ,', O
Z v "O tO O ._1
TIMETAL 21S/SCS-6
1 : P2/PI=0.125 2:Pz/PI=0.000 3:P=/PI=-0.200 (negative P2) 4:P=/PI=-0.330 (negative P2)
60000
51
- Predictions Test Data for1 Test Data for 2 Test Data for 3 Test Data for 4
40000
20000
4 3 2 1
[ e22 Vs. P2 -
0
-0.010
-0.005
0.000
0.005
0.010
Strain
FIG. 7--BC test data [2] and analysis predictions for SCS-6/TIMETAL 21S composite.
the results of the two-dimensional analyses for P2/PI = 0.125, 0.0, - 0 . 2 , and -0.33. The negative load ratios correspond to compressive transverse load (P2). Also shown, for reference, is the predicted uniaxial tensile (0 ~ stress-strain curve, previously shown in Fig. 3. It is seen that the zero-degree curve is not the same as the curve corresponding to P2/P1 = 0.0. This is because the latter corresponds to a tensile ( ~ 1) compressive (~r22) stress state. In addition, the results show decreasing stiffness with increasing compression in the transverse direction. Figure 7 shows the data on BC specimen tests reported in Ref 2. The data from an additional test (with a higher positive load ratio) reported in Ref 2 have been omitted because, during the particular test, the load ratio changed by an uncertain amount due to test machine limitations. For direct comparison with the test data, the analysis results of Fig. 6 have been plotted as load-strain curves. As shown, the predictions are generally consistent with the test data. The analysis results indicate that the stress states in all the tests reported in Ref 2 were such that the nonlinearity in the stress-strain response of the material is only due to matrix inelasticity. Much larger load ratios (P2/P0 would be required to induce interfacial damage. Conclusions
The results of the present study demonstrate the viability of the off-axis tension test as a method for measuring nonlinear stress-strain response of a unidirectional composite under biaxial stress states. Both the conduct and the interpretation of the OAT test are considerably simpler than the BC specimen test. The specific BC specimen design considered in the present work is also limited in its ability to induce (and, therefore, characterize) interfacial damage mechanisms in the composite. It may be possible to overcome this difficulty by using alternative specimen designs and loading arrangements. However, there appears to be no significant advantage in developing a BC specimen test that would provide the same information as a much simpler OAT test. The only advantage may be that the BC specimen test would provide the means for generating test data over a wider range of 0"n/022 ratios in the absence of shear stress (0-12). However, this is not a
52
MULTIAXIALFATIGUE AND DEFORMATION
significant advantage if theoretical models can provide a reasonably accurate representation of nonlinear deformation behavior of composites under multiaxial stress states. In that case, one would need a limited number of OAT tests on a specific composite to validate the theoretical model, and then use the model to predict composite response under any other biaxial stress states of interest. After further validation under a broader set of test conditions, the theoretical model used in the present study may be found useful for this purpose. Acknowledgment
The authors gratefully acknowledge the support of the Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson AFB, OH. The help of Mr. Swamy Chandu, of Research Applications, Inc., in three-dimensional finite-element analyses is appreciated.
APPENDIX If inelastic strain rate in the matrix material can be expressed as: e~l = ASu
the Bodner-Partom model used in the present work is described by the following equation: A = ~2_2D0exp[_ 1 (.(Z~ +3j_ezD)2/N].] J
in which ZI=ml(ZI-Zt)Wi"-A1ZI~,
{ Z 1 -- Z21rl , OZ2 .j~/ Z1 -- ZI ~ Z,
]
+-~I~Z~-~}
W i . = o-~l~l, z ~ =/3klu,z
_
[ ~ x ~
r2
[3ij = m : (Z3uq - flo)W i" - A2Z1 ~ - - - 1 )
"ij ~- ~ ,
,/~ij 023 vO + -~3 0 T
I
]Vij-- .Vl[~kl[3kl
m, = tomb + (m,~ -- m i d exp[--ml~ (Z1 - Zo)]
and A1 = Alb + (AI~ - Alb) expI-Alc (Z 1 -- Zz)]
The values of the constants involved in the Bodner-Partom model are included in Table A-1. These values were obtained using test data on Ti-6-4 and TIMETAL 21S matrix materials in "fiber-less" composite form.
53
AHMAD ET AL. ON CHARACTERIZATION METHODS TABLE A- 1--Constituent material properties (SI units). E, MPa
v
a * 10-6/~
E, MPa
Ti-6-4 TIMETAL 21S
120000 170000
0.31 0.34
7.5039 6.3440
mlb
Za
rt
r2
N
Z2
Z3
Ti-6-4 TIMETAL 21S
1.24 2.00
250.0 350.0
2.54 3.50
2.54 3.50
1.7922 1.5000
2225.77 3000.00
414.00 180.00
m2
Alb
A2
mla
mlc
Ala
Aac
Ti-6-4 TIMETAL 21S
0.00085 4.00000
0.0 0.0
0.0 0.0
1.6939 20.000
0.00015565 0.00100000
0.0 0.0
0.0030 0.005
SCS-6 393000 . . . . . . . . .
v
a * 10 6/~
0.25
3.4149
TABLE A-2--J A Theory constants. R12 = 0.0, 1 b = I ~ = 0.0, K12 = 2.2, Tp :: 950~ N22 : N12 : 1.0
Rll ,
R22 ,
NII(s )
MPa
MPa
SCS-6/Ti-6-4 (Vf = 0.40) SCS-6/TIMETAL 21S (Vf = 0.29)
0.52112 0.59480
427.21 200.71
- 115.60 -70.31
References [1]
Lissenden, C. J., Pindera, M-J., and Herakovich, C. T., "Stiffness Degradation of SiC/Ti Tubes Subjected to Biaxial Loading," Composites Science and Technology, Vol. 50, 1994, pp. 23-36. [2] Kirpatrick, S. W., "Damage and Failure Behavior of Metal Matrix Composites Under Biaxial Loads," SRI International Report to the Air Force Office of Scientific Research (No. F49620-96-C-0051) and a Ph.D. Dissertation submitted to the Department of Mechanical Engineering, Stanford University, September 1998. [3] Sun, C. T., Chen, J. L., Sha, G. T., and Koop, W. E., "Mechanical Characterization of SCS-6/Ti-6-4 Metal Matrix Composite," Journal of Composite Materials, Vol. 24, 1990. [4] Ahmad, J., Newaz, G. M., and Nicholas, T., "Prediction of Metal Matrix Composite Response to Multiaxial Stress," Recent Advances in Mechanics of Aerospace Structures and Materials--1998, B. V. Sankar, Ed., ASME, New York, AD-Vol. 56, 1998, pp. 43-59. [5] Hill, R., The Mathematical Theory Plasticity, Oxford Univ. Press, 1950. [6] Arnold, S. M. and Wilt, T. E. J., "A Deformation and Life Prediction of a Circumferentially Reinforced SiC/Ti-15-3 Ring," Reliability, Stress Analysis, and Failure Prevention, R. J. Schaller, Ed., DE-Vol. 55, 1993, pp. 231-238. [7] Ahmad, J. and Nicholas, T., "Modeling of Inelastic Metal Matrix Composite Response Under Multiaxial Loading," Failure Mechanisms and Mechanism-Based Modeling in High Temperature Composites: Proceedings of the ASMEAerospace and Materials Divisions, W. S. Chan, M. L. Dunn, W. F. Jones, G. M. Newaz, P. V. D. McLaughlin, and R. C. Wetherhold, Eds., ASME, New York, 1996, pp. 311-323. [8] Ahmad, J., Chandu, S., Santhosh, U., and Newaz, G. M., "Nonlinear Multiaxial Stress Analysis of Composites," Research Applications, Inc., Final Report to the Air Force Research Laboratory, Materials and Manufacturing Directorate, Contract F33615-96-C-5261, Wright-Patterson AFB, OH, 1999. [9] Nicholas, T. and Kroupa, J. L., "Micromechanics Analysis and Life Prediction of Metal Matrix Composites," Journal of Composites Technology & Research, JCTRER, Vol. 20, 1998, pp. 79-88. [10] Zhang, K., Newaz, G. M., Ahmad, J., and Chandu, S., "Analysis of Quasi-Isotropic MMC Using Iosipescu Shear Test," to appear in Journal of Composite Materials, 1997. [11] Chandu, S., Ahmad, J., and Newaz, G. M., "Detailed Interpretation of Off-Axis and Iosipescu Test Data on Metal Matrix Composites," Journal of Reinforced Plastics and Composites, Vol. 16, No. 13, 1997, pp. 1156-1167. [12] Majumdar, B. S., Newaz, G. M., and Ellis, R., "Evolution of Damage and Plasticity in Titanium-Based Composites," Metallurgical Transactions, Vol. 24A, 1993, p. 1597.
David W. A. Rees I and Y. H. Joc A u 1
Deformation and Fracture of a Particulate MMC Under Nonradial Combined Loadings REFERENCE: Rees, D. W. A. and Au, Y. H. J., "Deformation and Fracture of a Particulate M M C Under Nonradial Combined Loadings." Multiaxial Fatigue and Deformation: Testing and Prediction, ASTMSTP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 54-81.
ABSTRACT: Nonradial, elastic-plastic cyclic loading paths have been applied to a 17% SiC particulate/2124 aluminum alloy metal matrix composite at room temperature. By combining axial loading with torsion these cycles involve stepped loading and reversals through zero with successively increasing peak stresses through to failure. Strain data taken from a three-element rosette strain gage, bonded to the outer diameter of the tubular test piece, are processed to enable an examination of the axial and shear surface strain response to the loading paths. The component stress-strain plots and the total strain paths are given for the elastic-plastic deformation preceding fracture in this composite. The behavior is complex although it permits an assessment of the existence of a yield surface and the appropriateness of the concepts of the normality rule, isotropic and kinematic hardening. A qualitative judgment is offered in the form of rotation and distortion accompanying a translating yield surface. It is shown how a rigid translation can admit reversed softening and ratchet strain. The particulate arrangement offers greater resistance to tensile plasticity so that strain ratcheting occurs predominantly under compressive stress. This alloy offers no preference to shear flow from forward and reversed torsion other than the usual appearance of a Banschinger effect within the metal matrix when one torsional mode follows the other. However, cyclic torsional ratcheting appears with simultaneous axial tension. The rotation and distortion features are implied from an examination of the strain paths. Their gradients are a guide to the direction of a plastic strain increment vector lying normal to the yield surface. Further plots reveal the degree to which the principal axes of stress rotate and deviate from the principal strain axes. This composite is light and strong but the SiC particulates sacrifice the available tensile ductility of the matrix alloy. Under cyclic loading the likelihood of an early brittle tensile failure appears to be offset by the prior compressive flow that occurs under a comparable stress magnitude. The compressive residual strain can lessen the severity of a subsequent tensile strain, thus contributing to survival when cycling to combined stress levels of -+500 MPa in torsion and _+200 MPa in tension/compression. KEYWORDS: metal matrix composite, SiC particulate, 2124 matrix, nonradial loading, cyclic plasticity, kinematic hardening, ratcheting Nomenclature
A E G J ri, ro T W Y, k e
Cross-sectional area, l n l n 2 Tensile modulus, GPa Shear modulus, GPa Polar second m o m e n t o f area, m m 4 Inner and outer radii, m m Applied torque, N m Axial force, N Tensile and shear yield stresses Direct strain, %
1 Senior lecturer and lecturer, respectively, Department of Systems Engineering, Brunel University, Uxbridge, Middlesex, UB8 3 PH, UK.
Copyright9
by ASTM lntcrnational
54 www.astm.org
REES AND AU ON DEFORMATION AND FRACTURE
y tr r v 0o~ G 0o x, y
55
Shear strain, % Axial stress, MPa Shear stress, MPa Poisson's ratio Inclination of principal stress and strain axes (o) Gage misalignment Reference coordinates
This paper examines the behavior of a metal matrix composite (MMC) when subjected to complex cyclic loading paths of the type shown in Fig. la-f. All tests combine tension and compression with forward and reversed torsion at room temperature for a particulate MMC consisting of 17% SiC, 5 /~m particles embedded in a 2124 aluminum alloy matrix. The literature reveals similar studies for wrought metallic alloys [1-3]. A little work on the combined loading of MMCs [4,5] has been done in addition to that reported from simple loadings [6, 7]. The purpose of the investigations [4,5] was to appraise the predictions from classical plasticity theory. Thus, for a radial outward path, for which stress components increased in proportion, a linear plastic strain path was produced. According to the theory, at the point where the stress vector pierces the initial yield locus the increments of plastic
3
1 901
.r
a
t '
_--
4
(c)
(b)
-r
r
l
4
(a)
9.-
z
0"
1
5 qt
!
."-
.i
dr
4
3 "-
I
--
t
'i
2
3
"-
(a)
(e) FIG. 1--Nonproportional cyclic stress paths.
6
O"
56
MULTIAXIAL FATIGUE AND DEFORMATION
strain are components to a vector that lies in the position of the exterior normal to this locus. Thereafter, as a subsequent yield locus contains the stress components under a radial stress path, the plastic strain path will remain linear if the current normal direction remains constant. This normality rule is employed with a uniformly expanding yield locus by the isotropic hardening rule or, for a locus that translates rigidly, by the kinematic hardening rule. When the similar rules of normality and hardening are applied to outward, nonradial stepped paths they predict a nonlinear plastic strain path. The subsequent yield locus becomes the plastic potential in a flow rule of plasticity which allows strain increments to be found under the current stress state. Experiment [4,8,9] has confirmed that the normality applies to both radial and nonradial outward loadings for wrought metallic alloys and particulate MMCs. The simpler isotropic hardening rule has become established formally within the Levy-Mises theory of plasticity and the Prandtl-Reuss theory of elastic-plasticity. They provide reliable predictions to plasticity arising from outward loadings. Kinematic hardening has been recognized as the essential model of plasticity when a stress reversal occurs as with all the present load paths (see Fig. la-f). The Banschinger effect can be identified with a reduced reversed yield stress when an initial yield locus, centered at the origin, translates in a rigid manner to any stress point in the elastic-plastic region. The predictive capability of the kinematic hardening is extended within multi-surface models. These apply translations to initially concentric surfaces of constant equivalent plastic strain or tangent moduli [3]. They have been found useful for predicting the strain response to proportional cyclic loadings [10] in which the material may harden, soften, or attain a stable state. The reliability of these models, when applied to more complex nonproportional cyclic loading paths, is less certain though advances have been made for single-step stress paths of the type shown in Figs. lb and c [11]. Recent work [12-16], aligned with the theme of the present symposium, has considered cyclic plasticity and fatigue behavior under multi-axial stress states including elevated temperatures. In the present test program torsional cycling is performed in the presence of tensile and compressive, initially elastic, stresses. There is a need to establish reliable data for cyclic combined stress paths of the type shown in Fig. l d - f where the behavior of a MMC is less well understood than for wrought alloys. This is the purpose of the present work. The cycles in Fig. 1 are made progressively more complex from (a)-(f) in order to build a picture of the multiplicity of effects and associated trends when cyclically loading a MMC in a complex manner beyond its yield point. The tests reveal unusual ratcheting phenomena that place a demand upon the predictive capability of a mathematical model. Model Predictions
It will be seen that the measured strain response of this composite to each load path can be complicated by irregular flow curves and unpredictable failures. However, certain trends do appear which allow a qualitative appraisal of plasticity theory. In particular, the validity of the normality and kinematic hardening rules will be investigated. It will be assumed by Prager's rule [17] that the initial yield surface translates rigidly in a direction parallel to the outward normal at the current stress point. This rule assumes a linear hardening material such that the path traced by the center of the locus is proportional to the path of plastic strain. Thus, for loadings within a stress space ~, z, the normals will provide the gradient 63,P/See to a center path of yP versus e e as shown in Figs. 2a-e. In its simplest form this rule describes the Bauschinger effect, which is a consistent feature of cyclic plasticity. Also revealed are axial plastic ratchet strains when an elastic-plastic torsion is superimposed upon a steady axial tension or a compression. This refers to the increase in axial strain/cycle that occurs with the repeated application of shear stress within the plastic range even though the axial stress is initially elastic and remains constant. The present material consistently ratchets under a steady compressive stress but not under a steady tensile stress. Allied to this is the shift in the hysteresis loop during torsional cycling, showing that an increment of plastic shear strain is a necessary requirement for the axial plastic ratchet strain.
b..
W ft.
I-C) u_ 0 Z Z
o_ er 0 iii D Z 0
O Z U.I W rr
r..
21.
(q)
f3
"f-v I ~tllvd g~a.lly a!]ada xopun ~t]ivd u?vd~s oI ~uo~.la!poxd 8tquopdv~] a.~laluou!)l-- E "Did
I ,,"Io :L
(P)
a3QI
58
MULTIAXIALFATIGUE AND DEFORMATION
Many refinements have been made to match the observed behavior of engineering alloys more closely as with nonlinear hardening, temperature, and rate effects [12-16]. In addition, the restriction of a rigid translation may be removed consistent with observations on rotation, cross-hardening, and distortion of the subsequent yield surface [18,19]. Examined here is the extent to which the results from an experimental investigation on a particulate MMC appear within a basic model of kinematic hardening.
Experimental Material The particulate MMC under test was manufactured by Aerospace Metal Composites Ltd., U.K. It consists of 17% by volume of 2 - 5 / x m SiC particles embedded in a 2124 aluminum alloy matrix. A Keller's reagent was used to expose the angular appearance of the particles and their distribution within the matrix (see Fig. 3). To achieve this microstructure atomized aluminum powder is first mixed with the particles and then blended. There follows solid state compaction by hot isostatic pressing to produce a billet. Thereafter, a conventional extrusion was used to produce the 25 mm bar. The mechanical properties were later optimized at an intermediate stage within specimen machining with a solution treatment at 505~ for 1 h, followed by a water quench and aging at room temperature for one week. Table 1 compares the uniaxial and shear properties of this MMC with those of its matrix material. Poisson's ratio v was calculated assuming the elastic constants for MMC conformed to the relationship: E = 2G (1 + v). The gain in strength and stiffness offered by the particulate clearly oc-
FIG. 3--Fine SiC particulate MMC structure.
REES AND AU ON DEFORMATION AND FRACTURE
59
TABLE 1--Mechanicalpropertiesfor MMC and 2124 aluminum alloy. Elastic Moduli MatedN 2124A1 17%MMC
Fracture Strain
Yield Stress
Ultimate Strength
E (GPa)
G (GPa)
v
• (%)
Tf (%)
Y (MP~
k (MPa)
Tension (MPa)
Shear (MPa)
74 105
28 40
0.32 0.31
25 4, -20
20 4
300 460
170 270
475 650
275 375
curs at the expense of ductility. Both the tensile and torsional strains in MMC are restricted to a few percent. However, this material can sustain far greater compression strain to failure. Pure compression studies [4] of short solid cylinders revealed that strains of this order (20%) accompany a shear sliding mechanism and failure along 45 ~ planes. In contrast, a pure tensile failure followed a brittle path normal to the stress axis. As with the 2124 alloy, the yield and ultimate strengths of an MMC obey a Mises relationship (e.g., k "-~ Y/~J3) but the fracture strains do not (i.e., Ys ~: ~/3es). Instead, YS and e s are of comparable magnitude but with a reduction factor of 5 compared with the 2124 alloy.
Machining and Specimen Design Carbide-tipped tools were used, with frequent regrinds, for producing the blank for heat treatment. For finish boring and turning of the test piece gage area, fine cuts were taken using diamond tools. The short tubular specimen (see Fig. 4) had a 5 mm parallel gage length with inner and outer diameters of 12.00 -+o.02mm and 14.25 -+0.02mm, respectively. At the ends of the 5 mm gage length the ends were enlarged through 1 mm radii to 25 mm diameter ends with 18 mm fiats to fit within close-fitting grips. The design was chosen to offset shear stress gradients, bending due to misalignment, and torsional buckling. One disadvantage of this design is the possibility of a restraining influence of the grips upon the deformation within the short gage length. However, there was not a direct connection between the grips and the test section. Loads were first transmitted from the grips to the enlarged
Sec tion XX
1R
7 I 4D
27
14.25 25 FIG. 4
(ram)
67
Test piece showing test section and enlarged ends.
60
MULTIAXIALFATIGUE AND DEFORMATION
ends, in the manner described below, and then to the integral gage section. The stress concentration at the fillet radius was not of an order to influence the elastic shear moduli. This consistently lays in the range 38 to 40 GPa for all tests.
Test Machine Full details of the combined tension-torsion test rig (see Fig. 5) have been cited elsewhere [4,5]. The grips were designed to transmit torque on the flats, tension to the shoulders, and compression to the ends of the test piece. One grip was connected in-line to a load cell and an air cylinder. The grip and cell were free to move back and forth in the axial direction by the air pressure in the double acting cylinder. The motion of the load cell was guided by a keyway in its housing that also served to prevent rotation. The other grip was free to rotate but prevented from axial movement. The shaft to which this grip was connected was seated in a thrust bearing that was supported in the aforementioned housing. At the point where it emerged from the housing the shaft was connected to a pulley-wire system that allowed a torque to be applied to the specimen. Weights were applied tangentially at two points along the grooved periphery of the pulley in either direction. The rig allowed an alternation between tension/compression and forward/reversed torsion for the purpose of investigating complex loading paths. For example, to achieve the path shown in Fig. lb an elastic tensile stress was first applied to the specimen by raising pressure in the air cylinder. Thereafter, a superimposed shear stress to point 1 was attained by applying weights to the pulley to give a positive torque. The weights were then removed and reapplied in an opposing direction along the pulley to give negative torque until the shear stress at point 2 was reached. This torque was then removed and was followed by removal of the pressure so it returned to the stress-free origin of the path in the o-, ~"coordinates shown (cycle
FIG. 5--Test machine showing air cylinder, test block, and torsion pulley.
REES AND AU ON DEFORMATION AND FRACTURE
61
1). The loading order was then repeated to raise the shear stress to point 3 and then 4 before retuming to the stress origin (cycle 2). Cycling was continued by incrementing __+ ~'maxunder the constant o-until failure eventually occurred. For all paths shown a cycle is designated 0120, 0340, 0560, etc., by starting and finishing at the origin. For example, the first two cycles are shown in Fig. I a - e and the first three cycles in Fig. I f The bar length supplied and the cost of specimen manufacture precluded more than one test per load path. Not all test pieces attained the fracture strains expected of them. MMC is prone to a number of inherent defects that initiate premature failure. In particular, an embrittling defect can already exist within a section of extruded bar or be imparted during the machining process. The load cell provided the magnitudes of the axial load and torque carried by the test piece. The maximum loads to be measured were --_10 kN and +_140 Nm. For this the cell was bonded with two separate four-arm Wheatstone bridges. One was capable of recording the axial load with a sensitivity to 135.6 N//~e and the other torque with a sensitivity of 0.876 Nm//ze. The torque calibration was achieved within the rig since torque becomes the product of the tangential force and the pulley radius. The axial load calibration was conducted by removing the cell and compressing it between its ends in a 500 kN hydraulic test machine. These loads were converted to stress using o- = - W/A and "r = +-Tro/J where A is the gage section area, ro is the outer radius and J = J27r(ro 4 - r 4) is the polar second moment of the gage section. There was a small amount of inertia under axial tensile loading associated with the weight of the load cell and the friction within its guides. The loss in a load cell reading under low tensile loads resuited in a false modulus initially but the true value recovered with increasing tension to equal the E value for compression 100 to 110 GPa. Strain M e a s u r e m e n t
The reliable measurement of strain within the specimen gage length is essential for the data analyses proposal. Initially, tranducers were used to measure displacement and angular twist between the ends from which the axial and shear strains could be calculated. Because of space restriction and some through-zero slack their sitings at remote locations led to inaccuracies in elastic strain when compared to directly bonded strain gage readings. The strain gage method was therefore preferred for the room temperature elastic-plastic strain measurements made here. Strains were not large. They lay within the measurable range of the 2 mm gage length, 5 nun diameter polymer backed, post-yield rosette and this ensured that they remained bonded through to failure. A single, three-element, selftemperature compensated rosette was bonded to the outer gage diameter with one element being aligned with the axial direction and the other two elements lying in orientations at ---45~ to this axis. The orientations of each 2 mm superimposed grid ensured they lay in a region of uniform strain central to the 5 mm test length. Each 1201) strain gage was connected as a separate 88bridge using a commercial strain meter connected to a bridge completion channel selector unit. A second rosette was not employed to obtain an average strain for each direction as this would complicate the gage misalignment problem. The elastic moduli in torsion and compression, as found from a single rosette, agreed with quoted values and thus the influence of bending was judged to be negligible. During each test the load paths were followed incrementally by combining deadweights with pressure loading. This allowed time to record the three strain gage readings and the two load cell readings, following each load step. During elastic loadings the specimen strain readings were stable but under higher stress levels within the elastic-plastic range the strains continue to increase with time [19]. Normally, primary creep behavior is associated with a changing strain rate that diminishes to a steady state. The material appeared to behave in a similar manner, although the 15 rain maximum time allowed between final load increments was often insufficient to attain completely steady strain readings. For successive cycling of stress and strain, creep in a positive (or negative) direction resumed suddenly at its previous peak stress levels in a similar manner.
62
MULTIAXIALFATIGUE AND DEFORMATION
Data Analysis The conversion of the rosette's three strain readings into the specimen axial and shear surface strains requires an allowance for misalignments arising from bonding and the rotation that occurs in each gage when torque is applied to the test piece in either direction. The problem is to determine the true axial and shear specimen strains, ~ and y, from the three strains CA, es, and 0oc recorded by the rosette's arms, given that there may be slight misalignment 00 between gage B and the specimen axis following bonding of the strain gage (see Fig. 6). It is possible to calculate 0o from the initial loading phase within each cycle. For initial tension or compression, Sa and 0oc should be equal when there is true alignment. For initial torsion, if arms A and C lie true at - 4 5 % then 0oa = --eC. The analysis must also admit a rotation in each arm when the specimen is placed under torque. W h e n the elastic limit is exceeded the initial misalignment is modified by the twist that remains in limb B. The modified misalignment from cycle to cycle is used in the calculation of 0oand 3/. Let x be aligned with arm B and y be perpendicular to B (see Fig. 6). In the axes x and y there are two normal strains ex and ey and a shear strain Yxy which may be transformed to give the normal and shear strain for any direction inclined at positive 0 to x as: e = ex cos 2 0 + Sy sin z 0 + ~/2y~ysin 20
(la)
y = - ( e x - ey) sin 20 + T~y cos 20
(lb)
When 0 = 0 ~ and 0 = ---45 ~ Eq l a gives the solutions to ex, ey, and Yxy as: 0ox = 0oB,0oy = 0OA+ 0OC-- 0OS,Yxy = 0OA-- 0OC
(2a,b,c)
The principal strains and their orientations, relative to x, are found from Eqs 2a-c as:
~176 : 1/2 (0ox + 0Or) + I/2~V'/[(0Ox-- 0oy)2 _1_ ,~2y]
(3a)
tan 20x = Yxy/(ex - 0oy)
(3b)
Now let the initial misalignment between the test piece axis and the gage B axis be 00 as shown in Fig. 6. It follows that 0~ = 0x - 00 is the true inclination between the test piece axis and the princi-
62 FIG. 6 - - S t r a i n gage rosette showing initial misalignment with specimen axis.
REES AND AU ON DEFORMATION AND FRACTURE
63
pal planes. The axial and shear strains referred to the axis of the test piece follow from setting 0 = 0o in Eqs 1a,b using the x, y strain components from Eqs 2a,b,c. Note that IAYxr gives the rotation in direction B under load. The residual value 'A3,xyremaining upon completion of a cycle was used to modify the gage misalignment 0o in the calculation of e, y for the cycle that followed. The plastic shear strain modified the misalignment only by a small amount so that the 0o values rarely exceeded +3 ~ and - 4 ~ but, as will be seen, this is not insignificant when there are contrasting magnitudes between e and YThe principal stresses 0.1, 0"2 and their orientations O,~to the test piece axis are found from the loading independently. Once the loads have been converted to axial and shear stresses 0" and r it follows that: 0"1, 2 = 1/20" + 1/2 "V/(o "2 -[- 4 r 2)
(4a)
tan 20,~ = 2r/0.
(4b)
The calculations in Eqs 1-4 were programmed for each increment in each cycle. Particular care was taken with the signs of strains and their orientations in relation to the initial misalignment (usually -< + 2 ~ and the directions of the applied loading. A manual check on the correctness of the strain calculations was made from a Mohr's strain circle construction from at least one set of data within each cycle. The derived quantities within each test were then presented using a graphical software package in the following forms: (i) r versus y, (ii) 0. versus e, (iii) y versus e, (iv) 0~ versus 0~, (v) r versus 0o~ (vi) y versus 0~, (vii) el/e2 versus 0,, (viii) 0.1/0.2 versus 0,, and (ix) 0o versus 3,. In addition, certain cross plots: r versus e, and 0. versus y, and the plastic strain trajectory, y e versus e P, were found to assist with the interpretation of flow behavior. There follows a description of these selected plots for each test.
Results and Discussion Test 1 Forward and reversed elastic-plastic torsional cycling (see Fig. la) revealed the hysteresis loop in axes of r(MPa) versus % 3/given in Fig. 7. The loop was marked by a clear division between linear elasticity and a flattened elastic-plastic regime for the forward torque. Following a torque reversal at -~-"/'max,positive elastic strains recovered and a Bauschinger effect appeared within a well-rounded softening region characteristic of the matrix alloy. This is consistent with the kinematic translation shown in Fig. 2a. The material responded elastically to a further torque reversal at - r m ~ , but with some loss in stiffness (G < 40 GPa), to mirror the previous trace within the first loop. The loop asymmetry is partly due to a slight difference between + rm~ but is mostly indicative of a likely ratcheting mechanism in shear, in which the loop shifts along the 7 axis; an effect that appears in the combined stress tests. Examples of these can be seen in Figs. 9a and l l a . The specimen sufferered a sudden failure at 3' = 3.2% on the forward side of this loop before the second "/'maxlimit was reached. Consequently, it was not possible to confirm the ratcheting phenomenon under pure shear and whether cyclic axial strains were present. In the combined stress tests the material endured many torsional cycles from imposing peak shear strains less than 3%. In this regard plastic flow in an MMC is severely limited by the damage it creates, this being previously attributed to particle debonding, cracking, and void formation in the adjoining matrix material [4,5]. Table 1 shows that the fracture strains in tension and torsion are of comparable magnitudes. Even though these are small when compared to those for the matrix material, the present work shows that the material is capable of sustaining many load cycles provided the peak cyclic strains in a given direction remain within these fracture strains. Under this condition cycling serves to increase peak strains so that the permissible cyclic strain range becomes twice the static fracture value.
64
MULTIAXIALFATIGUEAND DEFORMATION _.--x failure
400 F
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-2
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HG. 7--Hysteresis behavior for MMC under pure torsion.
Test 2 In Fig. lb forward and reversed torsion is superimposed upon an elastic tensile stress o- = 250 MPa. The test piece endured four complete half cycles before failing at a peak shear stress of 245 MPa during the next half cycle. At failure the tensile strain was limited to 0,38% in combination with a maximum shear strain of 0.5%. The limited ductility under this load combination appears within the component flow plots in Figs. 8a, b. Shear straining remained predominantly linear elastic, with a modulus of 40 GPa, although a limited amount of hysteresis appears (Fig. 8a). There is some evidence within Fig. 8b that tensile flow becomes elastic-plastic when accommodating a limited amount of ratcheting in the positive e-direction. The steepest elastic gradients for loading and unloading confirm a tensile elastic modulus E ~- 100 GPa. The initial shallow gradient is believed to be a machine and not a material effect since it did not appear in compression (e.g., see Fig. 9b where E = 100 GPa). The strain path (Fig. 8c) shows how e is altered by 3'. This is due entirely to the axial plastic strain component 6e p of the strain vector 3yP/8e p, which lies in the normal position as the yield locus is carried by the stress vector. Taking a kinematic translation parallel to the direction of 3TP/8e e implies that the motion of center traces the path of plastic strain. The path shown in Fig. 2b reveals that in following the path 012, 8e P adds to the elastic axial strain under the tensile branch. Reversal from 2 produces 8y e without adding further 8e P. The model partly agrees with the observed strain path in
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(b)
65
66
MULTIAXIAL FATIGUE AND DEFORMATION
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67
REES AND AU ON DEFORMATION AND FRACTURE
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MULTIAXIAL FATIGUE AND DEFORMATION
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REES AND AU O N D E F O R M A T I O N A N D F R A C T U R E
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70
MULTIAXIALFATIGUE AND DEFORMATION
Fig. 8c. If it is to remain consistent within the positive strain quadrant in Fig. 8c, then a rotation or a local distortion of the subsequent yield locus should occur. The cross-plot in Fig. 8d shows the dependence of y upon o. The near verticals apply to tension and the horizontals to torsion. This shows that the air cylinder is capable of sustaining tension while the test piece undergoes torsion but there is some axial stress variation between cycles.
Test 3 In Fig. lc, forward and reversed torsion is superimposed upon an elastic compressive stress o- = - 2 5 0 MPa. The specimen failed in cycle 10 and Figs. 9a, b show that far greater stress and strain levels (~- = 350 MPa, y = -L-_2%,6 = - 1 . 2 % ) were achieved than in Test 2 (see Figs. 8a, b). The elastic moduli in compression were E = 100 MPa and in shear G = 38 GPa. The effect of increasing --+~'m~x within each cycle is shown in the stress-strain plots of Figs. 9a, b. These show that the width of the hysteresis loop increases approximately symmetrically by 26y p about its origin while the axial plastic compressive strain, 6e e, increments by between 0.1 and 0.2% per cycle. In a model of kinematic hardening (mirrored about the z-axis in Fig. 2b) the yield locus is raised and lowered by -+ rmax. The inelastic strains 6e P and gyp are the components of the plastic strain increment vector and the plastic strain path, 3,e versus eP, is proportional to the path traced by the center of the translating locus as shown. In this, as with all cycles involving a stress reversal, the model will describe the Bauschinger and ratcheting behavior. The deviations and irregularities observed may be attributed to asymmetries in yield locus motion arising from progressive strain damage to the material. The plot between the two total strains (Fig. 9c) shows the growth in strain within each cycle. The horizontal limbs show the elastic strain arising and recovered from the compression. By taking a gradient to this plot we see how the direction 6y/re gradually changes as the yield locus is carried to the stress point with the progress of plasticity. Most axial strain is accumulated from forward torsion which is broadly consistent with the model's prediction that path 012 produces e p and where only yP is found following the reversal at point 2. A similar observation was made for Test 2 but now the extent of compressive ratchet strain is greater than its tensile counterpart (compare Fig. 9b with Fig. 8b). In all the present tests the principal axes of stress and strain rotate. When shear stress is absent no such rotation occurs. This is the fundamental difference between conducting nonradial load tests with and without shear stress. The plot in Fig. 9d reveals how the principal direction of stress and total strain (Eqs 3b and 4b) alternate to either side of the specimen axis with increasing stress and strain in these cycles. Figures 9e a n d f show that as these axes rotate they do not remain coincident. The principal strain and stress ratios are calculated from Eqs 3a and 4a. If we were to subtract the elastic component of strain from Fig. 9e then the principal planes of plastic strain become more nearly aligned with the principal stress planes, which is an assumption made in the classical theory of isotropic plasticity.
Test 4 The loading sequence Fig. l d was preceded by two cycles of "elastic" loading. Shown in Fig. 10a is the complete strain history, which was apparently important to the integrity of this specimen. A precompression OA and superimposed forward torsion AB are sensibly elastic since their strains are recovered when these loads are removed. A further compression-tension cycle OCDO and a forwardreversed torsion cycle OEOFO also appear elastic since very little plastic strain remains with their removal. Then the first cycle 012 (see Fig. ld) was applied for which the material failed at point 2. The cross-plot Fig. 10b shows that there clearly had been some plastic shear strain from the branch 01. If the strains were wholly elastic then Figs. lOa, b would appear geometrically similar, with G the multiplying factor between their ordinates. It is suspected that a defect existed in this test piece that rendered it unable to sustain the reversal through tension to point 2.
REES A N D AU ON D E F O R M A T I O N A N D F R A C T U R E
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(b)
71
72
MULTIAXIALFATIGUE AND DEFORMATION
The path EF shows the manner in which axial strain can arise under pure torsion. It may be tensile or compressive depending upon the sense of the torque and appears with a hysteresis accompanying load-unload. This is not attributed to gage misalignment or rotation since these influences have been removed. The literature has attributed other examples of this phenomenon to anisotropy in wrought metals [20]. Here it is believed to arise from the mismatch in principal strains for a damaged composite with alternating tension and compression along its • ~ principal planes. If torsion produces axial plastic strain then there would be an increase in the axial yield stress given that a material hardens. Consequently, following a torsional prestrain path, the subsequent yield locus in or, ~"axes would show a cross-effect, i.e., a widening in a direction parallel to the ~ axis.
Test 5 The stress-strain plots shown in Figs. 1 la, b reveal further information on how a rotation might accompany the translation in the yield locus. The asymmetry in the hysteresis loops (Fig. 1 la) is due to a greater amount of plastic shear strain arising from reversed torsion. Axial strains arise with the translations to points 1 and 3, etc. (Fig. le) but are more dominant on the compressive side (2,4, etc.) which lead to a ratcheting along the negative c-axis (see Fig. 1 lb). We may interpret these results when the kinematic motion within each cycle is as shown in Fig. 2d. Greater amounts of each component strain would appear consistent within the observed strain path if we were to allow successive rotations as the yield locus translates into its subsequent positions 3 and 4, etc. These rotations can serve to accommodate either a steady or an accelerating axial ratchet strain within the normal vector as the locus is dragged back along the shear stress axis. It appears from Fig. l i b that the tensile branches to points 2, 4, etc., lie within this locus and remain essentially elastic. In returning to point 3, 5, etc., the unloading is elastic within this locus before carrying it back in the +T-direction. The rotation should only admit further plastic strain within each compressive limb to account for the ratcheting seen in Fig. 1 lb. The total strain trajectory in Fig. 1 lc again shows the inelastic strain increasing in the compressive direction while attaining a fixed strain on the tensile side. Increases in the negative shear strain account for the progressive widening of the loop in Fig. 1 la. Thus, while axial compressive ratchet strains apply to points 1, 3, etc., shear ratchet strain applies to points 2, 4, etc. This is consistent with the change to the normal gradient that should accompany a translation and rotation of the yield locus at these points. Figure 1 ld shows the alternation in the principal planes of strain following the cyclic application of shear strain.
Test 6 The specimen endured 23 cycles of the type shown in Fig. lf. Within a cycle the shear stress arising from forward torsion was increased to a given value and then the axial stress was alternated between fixed limits: from compression 1, 3, 5, etc., to tension 2, 4, 6, etc., before unloading. Figures 12a and b show the component stress-strain plots in which failure occurred when y and e reached 3.5% and - 0 . 4 % respectively at corresponding stress levels ~-= 525 MPa and o- = - 2 0 0 MPa. Figure 12a is typical of the 2124 alloy matrix elastic-plastic response to incremental torsional loading despite a slight irregularity at a peak shear stress arising from the application of compression/tension. The hysteresis is narrow, thus preserving the elasticity in shear between loading and unloading. The axial stress-strain response (see Fig. 12b) reveals that the particulates are more effective in inhibiting tensile flow, where c < 0.1%. With advanced cycling the compression branch leads to a net compressive strain with progressive ratcheting. Inelastic axial strain behavior shows that the loops widen from the origin and cross over on the compressive side thus incrementing the plastic strain. The total strain path (Fig. 12c) reveals the manner in which c accumulates with + 3,. Compressive strain dominates and is only partly recovered by the application of tension. The gradient, 6Tire, is an approximate indicator of how a rotation in the yield surface should modify the kinematic translation shown
FIG. 11--Flow curves and strain path showing compressive ratcheting under stress path le. Ca~
..-t c m
71
z
z
m 71 0
c 0 z
z
m m (~
74
MULTIAXlAL FATIGUE AND DEFORMATION
C I d
REES AND AU ON DEFORMATION AND FRACTURE
75
in Fig. I f Here, if the center traces the plastic strain path, then, with the addition of elastic strains: e e = ~/E, ,/e = T/G, the total strain path may be predicted. The cross plots in Figs. 12d and e show how e and y depend, respectively, upon an alternating ~" and or. They reveal again the bias for negative axial strain ratcheting. Cycling between points l and 2 (see Fig. le) remains elastic. This is confirmed from Fig. 12b if we ignore the through-zero machine irregularity. Thus the current yield locus at point 1 will contain point 2 within its interior. Most inelastic shear strain arises with the loading from the origin to point 1 and in unloading to the origin from point 2, since here the stress path crosses the boundary. Figure 12fshows the rotation in the principal axes of stress for this test as calculated from Eq 4b. The rotations lie to either side of the 45 ~ orientations and decrease with increasing shear stress. Ratch-
FIG. 12--Flow curves, strain path, and orientations showing ratcheting for stress path I f
76
MULTIAXIALFATIGUE AND DEFORMATION
FIG.
12----(Continued.)
eting increases as the rotation decreases. With different rotations per cycle a new plane in the section is placed under maximum tension, compression, and shear. The damage arising from both modes would thus be spread over a wider area of material than under torsion alone and may serve to prolong the strain to failure. Plastic Strain Trajectories
The elastic strain components are removed from the total strains by applying: 7P = T -
~'IG a n d ~ e = ~ -
tr/E
(5a,b)
REES AND AU ON DEFORMATION AND FRACTURE
77
FIG. 12--(Continued.)
where E and G are the elastic moduli given in Table 1. The plot of 3,p versus e e defines the plastic strain trajectory in a given test. Figure 13 gives an example of the trajectory derived from applying Eqs 5a,b to the corresponding total strain plots in Fig. 9c (Test 3) for which plasticity was significant. The bias for compressive ratcheting is clearly evident as is the shift and widening of the torsional hysteresis loop. Shift of the hysteresis loop occurs negatively with accompanying widening. We have seen that according to Prager's kinematic hardening rule [17] the trajectory: (1) is directly proportional to the motion of the center, and (2) ties normal to the boundary of the current yield locus. In (1) a single work hardening constant c must connect the plastic strains (e p, TP) to the center coordinates (a,/3) of the current yield locus in an incremental manner for a nonradial path. That is: da = c de e
78
MULTIAXIALFATIGUE AND DEFORMATION
1.0
0.5
-0.0
-0.6
-1.0
-1.5
-2.0
'
'
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l
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-0.8
-0.6
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FIG. 13--Plastic strain trajectory for Test 3.
and dfl = c d'gP. The present tests show that linear hardening cannot be assumed. Thus c is not constant; this violates the assumption of a rigid translation made within Prager's rule. An appropriate function for c for this material will be examined in a future paper. Gage Misalignment and Rotation The short gage length of 2 mm resulted in a slight misalignment 0o with the test piece axis during bonding (see Fig. 6). Once known, 0o was used to correct rosette strains for the true axial and shear strains, i.e., e and 3', aligned with the test piece axis. However, 0o is continually altered with the application of shear strain and it becomes necessary to upgrade 0o for the calculation of axis strains at each load step. For this, it is necessary to determine the shear strain lying in the direction defined by 0o. Assuming pure shear we can ignore rigid body rotation and take one half the shear strain increment d3'~ to upgrade 0o by addition or subtraction depending upon the sense of the torque. This procedure was programmed so that the dependence of 0o upon 3' could be monitored throughout each test. Figure 14 shows the worst case of Test 6, where shear strain was not reversed (see Fig. 12a). Here 0o grows l!nearly with 3' within each cycle but is disrupted by the alternating axial stress imposed at peak shear stress. The result is that 0o varies from near zero initially to - 4 ~ at fracture ( - v e means thatx lies on the opposite side of the x-axis in Fig. 6). Correspondingly, the rosette strains were eA = 1.873%, en = --0.139% and ec = -1.816%. From Eqs 2a-c the x, y coordinate strain values are ex = -0.139%, ey = 0.196%, and 3'xy = 3.689%. When the rotation effect is ignored the axial and shear strains are the ex and 3'xyvalues given above. Compare these to the test piece axis strain components: -0.392% and 3.606%, as calculated from Eqs la,b. Clearly an unacceptable error arises in the estimation of the small axial strain value despite a relatively insignificant 2% error in a far greater shear strain.
Fracture Finally, Fig. 15 shows that failure surfaces were aligned with the planes of maximum shear. As the shear stress in each test increases so these planes become more closely aligned with the axial and
REES AND AU ON DEFORMATION AND FRACTURE I
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FIG. 14--Effect o f shear strain upon gage misalignment f o r Test 6.
FIG. 15--MMC specimens showing shear failures.
79
80
MULTIAXIALFATIGUE AND DEFORMATION
transverse directions of the specimen. With the exception of an explosive fracture under compression, which fragmented the test section, crack paths lay in these directions. The specimen ends were kept in line by the grips and so torsion alone permits a relative sliding between the adjacent faces of the shear planes. Some shear cracks formed within the lead-in to the fillet radii and were accompanied by secondary cracks that ran helically into the gage section. Unlike the case of pure compression where sliding along 45 ~ planes can accommodate an axial displacement, here the axial ratcheting is due to compressive plasticity of matrix material on transverse sections. Broadly, the results of the present tests show that the flow behavior of this MMC may be understood from a knowledge of the plasticity behavior of its aluminum matrix under similar load paths. The concept of a hardening rule involving a translating yield locus has long been applied to metals and is particularly useful here to provide a qualitative description of the results obtained. This may not be surprising since the SiC particles remain brittle and do not themselves contribute to plastic strain. However, these particles impede the flow to promote a semi-brittle behavior. This is seen in the low ductility of a MMC composite compared to its matrix metal/alloy. Despite its limited strain range, the MMC composite permits cyclic applications of nonradial loads to high stress levels. Initially, the usual features of cyclic elasto-plasticity for metals also appear in the composite. These are linear elasticity, strain hardening, the Bauschinger effect, creep, ratcheting, and elastic recovery. With continued cycling some of these features become less clear, this being most likely due to an accumulation of damage under tensile stressing, where the material remains essentially brittle. That is, it does not flow plastically when tension is applied either monotonically or in a repeated manner. In contrast, compression induces a plastic ratcheting mechanism than enables it to sustain cyclic loading. It is believed that an advantageous interplay between compressive ratchet strain, residual stress, and bond strength permits repeated tensile cycling in the absence of plasticity. A possible description of the complex behavior observed may follow from assuming that each particulate acts as a metallurgical notch around which a stress concentration exists. This should be combined, say, from using the rule of mixtures, with the features of traditional matrix plasticity reported here. Conclusion The various responses of a particulate MMC to combined cyclic loading paths appear complex but are not wholly unpredictable. It has been shown how classical kinematic hardening model predictions, as applied to the matrix material, are in qualitative agreement with the composite flow behavior. These experimental results show, however, that certain refinements would be necessary to model some of the more unusual features of this material. In particular is its capability to sustain a greater degree of compressive flow in combination with essentially brittle tensile behavior. Crucial to maintaining integrity of this composite are the plastic strains from compression and the damage from tension branches of a given cycle. The material will undergo an axial compressive ratchet strain for many cycles and this appears to prolong the ability to bear tension. Microscopically, there appears to be an advantageous interplay between existing residual stress in the matrix and the subsequent strain from load cycling. Macroscopically, a continually changing internal stress can be identified with the center coordinates of a translating yield locus. The preference for matrix compressive flow suggests, within the rule of normality, that a rotation and possibly a distortion will accompany the translation. References [1] Ikegami, K., "An Historical Perspective of the Experimental Study of Subsequent Yield Surfaces for Metals," Parts 1 and 2, Brit. Ind. & Sci. Int Trans Ser., BISITS 14420, Sept 1976, The Metals Society, London. [2] Ikegami, K., "Experimental Plasticity on the Anisotropy of Metals," Proceedings, Euromech Col1115, Mechanical Behaviour of Anisotropic Solids, J-P. Boehler, Ed., No. 295, CNRS 1982, pp. 201-242. [3] Rees, D. W. A., "A Survey of Hardening in Metallic Materials," Failure Criteria of Structured Media, JP. Boehler, Ed., Balkema, 1993, pp. 69-97.
REES AND AU ON DEFORMATION AND FRACTURE [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17]
[18] [19] [20]
81
Rees, D. W. A., "Deformation and Fracture of Metal Matrix Particulate Composites Under Combined Loadings," Composites PartA, Vol. 29A, 1998, pp. 171-182. Rees, D. W. A. and Liddiard, M., "Elasticity and Flow Behaviour of a Metal Matrix Composite," Key Engineering Materials, Trans Tech., Vol. 118-119, 1996, pp. 179-185. Majumdar, B. S., Yegneswaran, A. H., and Rohatgi, P. K., "Strength and Fracture Behaviour of Metal Matrix Particulate Composites," Mat Sci and Engng, Vol. 68, 1984, pp. 85-95. Everett, R. K. and Arsenault, R. J., Metal Matrix Composites: Mechanisms and Properties, Academic Press Ltd, 1991. Rees, D. W. A., "Applications of Classical Plasticity Theory to Non-Radial Loading Paths," Proceedings, Royal Society, Vol. A410, 1987, pp. 443--475. Rees, D. W. A., "A Review of Stress-Strain Relations and Constitutive Relations in the Plastic Range," Journal of Strain Analysis, Vol. 16, No. 4, 1981, pp. 235-249. Chaboche, J. L., "Time-Independent Constitutive Theories for Cyclic Plasticity, International Journal of Plasticity, Vol. 2, No. 2, 1986, pp. 149-188. Lamba, H. S. and Sidebottom, O. M., "Cyclic Plasticity for Non-Proportional Load Paths, Parts 1 and 2, J1 Engng Mat Tech, Vol. 100, 1978, pp. 96-103, pp. 104--111. McDowell, D. L., "Experimental Study on Structure of Constitutive Equations for Non-Proportional Cyclic Plasticity," Jl Engng Mater Technol., Vol. 107, 1985, pp. 307-315. Abdul-Latif, A., Clavel, M., Feruey, V., and Saanouni, K., "Modelling of Non-Proportional Cyclic Plasticity of Waspalloy," Jl Engng Mat Tech, VoL 116, No. 1, 1994, pp. 35-44. Beruallal, A., Cailletaud, G., Chaboche, J. L., Marquis, D., Nouailhas, D., and Rousser, M., "Description and Modelling of Non-Proportional Effects in Cyclic Plasticity," Proceedings: Biaxial andMulti-Axial Fatigue, M. W. Brown and K. J. Miller, Eds., EG3 Pub. 3, 1989, pp. 107-129. Armstrong, P. J. and Frederick, C. O., "A Mathematical Representation of the Multi-Axial Bauschinger Effect," CEGB Rpt. RD/B/N/731. Chaboche, J. L., "Constitutive Equations for Cyclic Plasticity and Cyclic Visco-Plasticity," International Journal of Plasticity, Vol. 5, No. 3, 1989, pp. 247-302. Prager, W., "A New Method of Analyzing Stresses and Strains in Work Hardening Plastic Solids," Journal of Applied Mechanics, Vol. 23, 1956, pp. 483-496. Phillips, A., "A Review of Quasistatic Experimental Plasticity and Viscoplasticity," International Journal of Plasticity, Vol. 4, No. 2, 1986, pp. 315-328. Wu, H. C. and Yeh, W. C., "Experimental Determination of Yield Surfaces and Some Results of Annealed Stainless Steel," International Journal of Plasticity, Vol. 7, No. 8, 1991, pp. 803-826. Billington, E. W., "Non-Linear Response of Various Metals: Permanent Length Changes in Twisted Tubes," Jl Phys D: Appl Physics, Vol. 10, 1977, pp. 533-545.
A. Buczynski I and G. Glinka 2
Multiaxial Stress-Strain Notch Analysis REFERENCE: Buczynski, A. and Glinka, G., "Multiaxial Stress-Strain Notch Analysis," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 82-98.
ABSTRACT: Fatigue and durability analyses require the use of analytical and/or numerical methods for calculating elastic-plastic notch tip stresses and strains in bodies subjected to nonproportional loading sequences. The method discussed in the paper is based on the incremental relationships, which relate the elastic and elastic-plastic strain energy densities at the notch tip and the material stress-strain behavior, simulated according to the Mroz-Garud cyclic plasticity model. The formulation described below is based on the equivalence of the total distortional strain energy density, which appears to give the upper-bound estimations for the elastic-plastic notch tip strains and stresses. The formulation consists of a set of algebraic incremental equations that can easily be solved for elasticplastic stress and strain increments, based on the increments of the hypothetical elastic notch tip stress history and the material stress-strain curve. The validation of the proposed model against the experimental and numerical data includes several nonproportional loading histories. The basic equations involving the equivalence of the strain energy density are carefully examined and discussed. Finally, the numerical procedure for solving the two sets of equations is briefly described. The method is particularly suitable for fatigue life analyses of notched bodies subjected to cyclic multiaxial loading paths. KEYWORDS: notches, multiaxial stress state, elastic-plastic strain analysis
Nomenclature Coordinates of center of mth (fro) plasticity surface Modulus of elasticity Equivalent strain energy density e~ Actual elastic-plastic strains at notch tip e~j Hypothetical elastic strains at notch tip G Shear modulus of elasticity K' Cyclic strength coefficient Kr Stress concentration factor due to axial load Kr Stress concentration factor due to torsional load k,n Load increment n u m b e r 11p Cyclic strain hardening exponent 8ii Kronecker delta, 6ij 1 for i = j and 6ij = 0 for i :~ j Plastic strain increments Elastic strain increments Actual elastic-plastic strain increments aSe~ Equivalent plastic strain increment Pseudo-elastic stress increments
E ESED
=
i Institute of Heavy Machinery Engineering, Warsaw University of Technology, ul. Narbutta 85, 02-524 Warsaw, Poland. 2 Department of Mechanical Engineering, University of Waterloo, Ontario N2L 3G1, Canada.
Copyright9
by ASTM International
82 www.astm.org
BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS
83
A,~ Actual stress increments Ao-eaq Actual equivalent stress increment s~ Deviatoric stresses of elastic input s~ Actual deviatoric stresses pa Actual equivalent plastic strain ~'eq Actual elasto-plastic notch-tip strains Elastic notch tip strain components Nominal strain F.n Poisson' s ratio O-~q Actual equivalent stress at notch tip a Size parameter of the mth (fm) plasticity surface O'eq, m Actual stress tensor components in notch tip Notch tip stress tensor components of elastic input o-o Parameter of the material stress-strain curve P Axial load T Torque R Radius of the cylindrical specimen Notches and other geometrical irregularities cause significant stress concentration. Such an increase of stresses results often in localized plastic deformation, leading to premature initiation of fatigue cracks. Therefore, the fatigue strength and durability estimations of notched components require detailed knowledge of stresses and strains in such regions. The stress state in the notch tip region is in most cases multiaxial in nature. Axles and shafts may experience, for example, combined outof-phase torsion and bending loads. Although modem finite-element commercial software packages make it possible to determine notch tip stresses in elastic and elastic plastic bodies with a high accuracy for short loading histories, such methods are still impractical in the case of long loading histories experienced by machines in service. A representative cyclic loading history may contain from a few thousands to a few millions of cycles. Therefore, incremental elastic-plastic finite-element analysis of such a history would require prohibitively long computing time. For this reason more efficient methods of elastic-plastic stress analysis are necessary in the case of fatigue life estimations of notched bodies subjected to lengthy cyclic stress histories. One such method, suitable for calculating multiaxial elastic-plastic stresses and strains in notched bodies subjected to proportional and nonproportional loading histories, is discussed in the paper.
Loading Histories The notch tip stresses and strains are dependent on the notch geometry, material properties and the loading history applied to the body. If all components of a stress tensor change proportionally, the loading is called proportional. When the applied load causes the directions of the principal stresses and the ratio of the principal stress magnitudes to change after each load increment, the loading is termed nonproportional. If plastic yielding takes place at the notch tip then almost always the stress path in the notch tip region is nonproportional regardless of whether the remote loading is proportional or not. The nonproportional loading/stress paths are usually defined by successive increments of load/stress parameters and all calculations have to be carried out incrementally. In addition the material stress-strain response to nonproportional cyclic loading paths has to be simulated, including the material memory effects.
Stress State at the Notch Tip For the case of general multiaxial loading applied to a notched body, the state of stress near the notch tip is triaxial. However, the stress state at the notch tip is biaxial because of the notch-tip stress
84
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 1--Stress state at a notch tip (notation).
free surface (Fig. 1). Since equilibrium of the element at the notch tip must be maintained, i.e., ~r23 = 0"32 and 823 = 832, there are three nonzero stress components and four nonzero strain components. Therefore, there are seven unknowns all together and a set of seven independent equations is required for the determination of all stress and strain components at the notch tip
O'i=
0"~2 0"23
<.=/0
and
0-~2 O'~3J
8~2 823 8~2
(1)
8ff3J
The material constitutive relationships provide four equations, leaving three additional equations to be established. Material Constitutive Model In the case of proportional or nearly proportional notch tip stress path, the Hencky total deformation equations of plasticity can be used in the analysis E
-~ o'~k 8o
2 o'aeq ~lJ
where 1
Sa = ~ a - -5 ~
a
a~J
(2)
BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS
85
The most frequently used model of incremental plasticity is the Prandtl-Reuss flow rule. For an isotropic body, the Prandtl-Reuss strain-stress relationships can be expressed as Ae~ = 1 + v A o - ~ - v E
.
3 Aeerq
~ Ao'kk 6 0 + -2- 0-" a- q
(3)
The multiaxial incremental stress-strain relation (3) is obtained from the uniaxial stress-strain curve by relating the equivalent plastic strain increment to the equivalent stress increment such that
A~.Pe~- f(~ d d
/k0-aq
(4)
0"eaq
The function, ePq = f(O'eq), is identical to the plastic strain-stress relationship obtained experimentally from the uniaxial tension test.
Load-Notch Tip Stress-Strain Relations The load or the load parameter, in the case of notched bodies, is usually represented by the nominal or reference stress being proportional to the remote applied load. In the case of notched bodies in plane stress or plane strain state the relationship between the load and the elastic-plastic notch tip strains and stresses in the localized plastic zone is often approximated by the Neuber rule [1] or the equivalent strain energy density (ESED) equation [2]. It was shown later [3,4] that both methods can be extended for multiaxial proportional and nonproportional modes of loading. Similar methods were also proposed by Hoffman and Seeger [5] and Barkey et al. [6]. All methods consist of two parts, namely, the constitutive equations and the relationships linking the fictitious linear elastic stressstrain state (0"~, e~) at the notch tip with the actual elastic-plastic stress-strain response (0"9, ~ ) as shown in Fig. 2. The Neuber and the ESED rule [2,3] for proportional loading, where the Hencky stress-strain relationships are applicable, can be written for the uniaxial and multiaxial stress state in the form of Eqs. 5a and 5b, respectively. e e = a a O'22~22 0"22~22
(5a)
4 ~ = 0"680 a a
(Sb)
The Neuber rule (5a) represents the equality of the total strain energy (the strain energy and the complementary strain energy density) at the notch tip, represented by the rectangles A and B in Fig. 3a. The ESED method (6a) is based on the equivalence of the strain energy density, which can be interpreted as the equality between the strain energy density at the notch tip of a linear elastic body (Fig. 2) and the notch tip strain energy density of a geometrically identical elastic-plastic body subjected to the same load. '~2
e
e
0"22ds22 =
a
f2
a
a
0-22d/322
(6a)
This relationship is shown graphically in Fig. 3b, and represents the equality of the area under the linear-elastic curve and the area under the actual elastic-plastic 0 - ~ 2 - - ~ 2 material curve. In the case
86
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 2--Stress states in geometrically identical elastic and elastic-plastic bodies subjected to identical boundary conditions.
FIG. 3--Graphical interpretation of (a) Neuber's rule, and (b) equivalent strain energy density (ESED) method.
BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS
87
of multiaxially stressed notches the strain energy density equations can be written as: eFj
f~
e
e
o'~jdeij
:
eg
f~
a
a
oijdeij
(6b)
The overall strain energy density equivalence, Eq 5b or 6b, relating the pseudo-elastic and the actual elastic-plastic notch tip strains and stresses at the notch tip has been generally accepted as a good approximation rule, but the additional conditions, necessary for the complete formulation of a multiaxial stress state problem, are the subject of controversy. Hoffman and Seeger [5] assumed that the ratio of the actual principal strains at the notch tip is to be equal to the ratio of the fictitious elastic principal strain components, while Barkey et al. [6] suggested to use the ratio of principal stresses. The data presented by Moftakhar [7] found that the accuracy of the stress or strain ratio based analysis depended on the degree of constraint at the notch tip. Therefore, Moftakhar et al. proposed [7] to use the ratios of strain energy density contributed by each pair of corresponding stress and strain components. It was confirmed later by Singh et al. [4] that the accuracy of the additional energy equations was also good when used in an incremental form. Because the ratios of strain energy density increments seem to be less dependent on the geometry and constraint conditions at the notch tip than the ratios of stresses or strains, the analyst is not forced to make any arbitrary decisions about the constraint while using these equations. However, the additional strain energy density equations [4, 7] have a theoretical drawback indicated by Chu [8], namely, the estimated elastic-plastic notch tip strains and stresses may depend on the selected system of coordinates. Fortunately, the dependence is not very strong and with suitably chosen system of axis it could be sufficiently accurate for a variety of engineering applications. It was also found that the set of seven equations involving the strain, stress, and the strain energy density increments can be singular at some specific ratios of stress components, which is due to the conflict between the plasticity model (normality rule) and strain energy density equations. Such a conflict can be avoided if the principal idea of Neuber is implemented in the incremental form. Namely, it should be noted that the original Neuber rule (5a) was derived for bodies in pure shear stress state. It means that the Neuber equation states the equivalence of only distortional strain energies. Therefore, in order to formulate the set of necessary equations for a multiaxial nonproportional analysis of elastic-plastic stresses and strains at the notch tip, the equality of increments of the total distortional strain energy density can be used. Thus all equations should be written in terms of deviatoric stresses and strains. Deviatoric
Stress-Strain
Relationships
The notch tip deviatoric stresses of the hypothetical linear-elastic input are determined as e
e
1
Sij = o"0 - ~ o'~,rij
(7)
The elastic deviatoric strains and strain increments can be calculated from the Hooke law
AN
Ae~ = 2G
(8)
The actual deviatoric stress components in the notch tip can analogously be defined as a
1
a
= ~'J - 3- c r ~ 8,j
(9)
88
MULTIAXIAL FATIGUE AND DEFORMATION
The incremental deviatoric stress-strain relations based on the associated Prandtl-Reuss flow rule can be subsequently written as
a~ Ae~ = - ~ - + ~ dA
(10)
where pa 3 AF'eq. dA = ~- o.eaq , Ae~e~=
3 ~ ~//j; (oreaq)2= 2-
do~q Ao'~q
The form and specific parameters of the stress-strain function, e~q imentally from an uniaxial cyclic test.
=
f(O'eq), must be obtained exper-
Equivalence of Increments of the Total Distortional Strain Energy Density It is proposed, analogously to the original Neuber rule, to use the equivalence of increments of the total distortional strain energy density contributed by each pair of associated stress and strain components.
FIG.
4--Graphical representation of incremental Neuber rule.
BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS
89
S~2Ae~2 + e~2AS~2 = S~2Ae~2 + e~2AS~22 e
e
e
e
a
a
$33Ae33 + e33AS33 = $33Ae33 + e~3AS~33
(1t)
e e a a $23Ae23 + e~3A~23 = ~223Ae~3+ e23AS23
The equalities of strain energy increments for each set of corresponding hypothetical elastic and actual elastic-plastic strains and stress increments at the notch tip can be shown graphically (Fig. 4) as the equality of surface areas of the two pairs of rectangular elements representing the increments of strain energy density. The area of dotted rectangles represents the total strain energy increment of the hypothetical elastic notch tip input stress while the area of the hatched rectangles represents the total strain energy density of the actual elastic-plastic material response at the notch tip. Equations 10 and 1 1 form a set of seven simultaneous equations from which all deviatofic strain and stress increments can be determined, based on the linear hypothetical elastic notch tip stress path data, i.e., increments A~j, obtained from the linear-elastic analysis and the constitutive stress-strain curve (3).
A~I
Ae?l = ~
3 A~3aq a
+ ~---@-e~qSll
AS~22
3
A~ea_~qS~22
AN3
3
ACq
AS'~3
3 Ae'~q 20-aq 23
(12)
S~2Ae~2 + e~2ASq~2 = S~2Ae~2 + e~2AS~22 S~3Ae~3 + e~3AS~3 = ~333Ae~3+ e~3AS~33 S~3Ae~3 + e~3ASg 3 = S~23Ae~3+ e~3AS~23
For each increment of the external load, represented by the increments of pseudo-elastic deviatoric stresses, AS~j, the deviatoric elastic-plastic notch tip strain and stress increments, Aeg. and AS~, are computed from the Eq set 12. With the help of Eq 9 the calculated deviatoric stress increments, AS~., can subsequently be converted into the actual stress increments,
I
AS~22 = Ao-~2 -- "~ (Ao'~2 -1- AO..~3)
AS~3 = Ao-~3 --
1
"~
(Ao'~2 + Ao'~3)
AS~3 = AO'~3
(13)
90
MULTIAXIALFATIGUE AND DEFORMATION
The deviatoric and the actual stress components 5,q and o-~ at the end of a given load increment are determined from Eqs 14 and 15. n--I
S'//jn = S~0.~ + ~
AS0k + AS~/jn
(14)
k=l
n--1 k=l
where n denotes the number of the load increment. The actual strain increments, Ae~, can finally be determined from the constitutive Eq 3.
Comparison of Calculated Elastic-Plastic Notch Tip Strains and Stresses with Finite-Element Data Monotonic Nonproportional Loading Path In the case of monotonic (no unloading) nonproportional stress path the qualitative correctness and accuracy of the method was demonstrated by comparing the calculated notch tip stress-strain histories to those obtained from the finite-element method. The elastic-plastic finite-element stress results of Ref 4 were obtained using the ABAQUS finite-element package. The isotropic sWain-hardening plasticity model was used for calculations, The geometry of the notched element was that of the circunfferentially notched bar shown in Fig. 5. The basic proportions of the cylindrical component were p/t = 0.3 and R/t = 7 resulting in the torsional and tensile stress concentration factor Kr = 3.31 and KF = 1.94, respectively. The ratio of the notch tip hoop to axial stress under tensile loading was cr~31cr~z= 0.284. The stress concentration factors for the axial and torsion loads were defined as:
KF =
0"~2 o-,~
and
Kr -
0"~2 ~'nr
(16)
FIG. 5--Geometry and dimensions of notched bar tested under nonproportional tension and torsion loading.
BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS
91
200180 m I1.
160
~
140'
initiation of notch tip yielding
d 12o 0 ~ 100 Q. "
80
0 '-
60
0
~
40 20 0
i
0
50
100
f
150
200
250
300
350
400
Axial notch tip stress, ~22e (MPa) FIG. 6---Monotonic torsion-tension stress path.
while the nominal stresses in the net cross section were determined as: F trn - ~r(R - t) 2
and
r,
2T 7r(R -- t ) 3
(17)
The loads applied to the bar were increasing torsion in the first phase and then increasing tension in the second phase with the torsion load being kept constant as shown in Fig. 6. The torque Tinduced the "linear elastic" shear stress 0% at the notch tip and the axial load Finduced the normal stress components trY2 and o~3. The increments of the hypothetical "elastic" stress components try3, try2, and o% and associated strains were used as the input into the equation set 12. The pseudo-elastic equivalent stress of the input at the notch tip was increasing throughout the entire loading process to ensure a monotonic loading path. The material for the notched bar was SAE 1045 steel with a cyclic stress-strain curve approximated by the Ramberg-Osgood relation. The material properties were: E = 202 GPa, v = 0.3, Sr = 202 MPa, n = 0.208, and K = 1258 MPa. o-
(18)
The maximum applied load levels were chosen to be 50% higher than would be required to induce yielding at the notch tip if each load was applied separately. and e23, and the stress components, The calculated and the FEM determined strain components, e22 a a
92
MULTIAXlAL FATIGUE AND DEFORMATION
~r~z and ~r~3, are shown in Figs. 7 and 8. Note that the calculated stresses and strains and the results of the finite-element analysis are identical in the elastic range. This is expected since the model converges to the elastic solution in the elastic range. Just beyond the onset of yielding at the notch tip, the strain results that were predicted using the proposed model and the finite-element data begin gradually to diverge. It can be concluded that the method based on the equivalence of the total strain energy increments overestimates the actual notch tip strains, but the predicted strains are reasonably close to the numerical FEM data.
Cyclic Plasticity Model In order to predict the notch tip stress-strain response of a notched component subjected to multiaxial cyclic loading, the incremental equations discussed above have to be linked with the cyclic plasticity model. Several plasticity models are available in the literature. The most popular is the model proposed by Mroz [9]. According to Mroz [9] the uniaxial stress-strain material curve can be represented in a multiaxial stress space by a set of work-hardening surfaces. O.eq, m
=
(Sij _
m a au)(Su
-
(19)
ol~j )
In the case of a two-dimensional stress state, such as that one at a notch tip, the work-hardening surfaces can be represented by ellipses on the coordinate plane for which the axes are defined by the directions of principal stress components (Fig. 9). The equation of each work-hardening ellipse in the
0.002
m 0.0015 r e,,,
~ e,,
0.001
,9,0 O
E
0.0005
0.0005
0.001
Axial notch tip stress,
0.0015
0.002
~ 22 a
FIG. 7 Comparison of calculated and FEM determined strain paths for monotonic torsion-tension input.
93
BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS 160
140
~
120
m 13 100 e3
8
"W'
80
Q. ,.C
N
6o
9
40
0
Calc.
20
i "-'-" FEM
0
50
100
150
200
250
300
350
Axial notch tip stress, (~2za (MPa)
FIG. 8--Comparison of calculated and FEM determined stress paths for monotonic torsion-tension input.
~l
~a
0
~a
s a
FIG. 9--Piecewise linearization of material ~r--e curve and corresponding work-hardening sur-
faces.
94
MULTIAXlAL FATIGUE AND DEFORMATION
principal stress space is: O'~q,m = V ' ( o ' ~
-
o~) 2 -
(o'~ -
ol~")(o-~ -
ol~)
+ (o'~ -
(20)
a~) 2
The essential elements of the plasticity model can be presented in such a case graphically in a twodimensional stress space. The load path dependency effects are modeled by prescribing a translation rule for the translation of ellipses in the o-~ - o-~ plane. The translation of these ellipses is assumed to be caused by the sought stress increment, which can be represented in the principal stress space as a vector. The ellipses can be translated with respect to each other over distances dependent on magnitude of the stress/load increment. The ellipses move within the boundaries of each other, but they do not intersect. If an ellipse comes in contact with another, they move together as one rigid body. However, it has been found that the ellipses in the original Mroz model may sometimes intersect each other, which is not permitted. Therefore, Garud [10] proposed an improved translation rule that prevents any intersections of plasticity surfaces. The principle idea of the Garud translation rule is illustrated in Fig. 10. (a) The line of action of the stress increment, Ao-a, is extended to intersect the next larger nonactive surface, f2, at point Bz. (b) Point B 2 is connected to the center, O2, of the surface f2. (c) A line is extended through the center of the smaller active surface, O1, parallel to the line 02B2 to find point B1 on surface fl. (d) The conjugate points B1 and BE are connected by the line B1B2. (e) Surface f l is translated from point O1 to point O{ such that vector O10~ is parallel to line B1B2. The translation is complete when the end of the vector defined by the stress increment, Atr, lies on the translated surface f~.
t~2a
12
t
i if
01
0
,"
(~la
FIG. l O--Geometrical illustration of translation rule in Garud incremental plasticity model.
BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS
95
The mathematics reflecting these operations can be found in the original paper of Mroz [9] or Garud [10] or in any recent textbook on the theory of plasticity. The Mroz and Garud models are relatively simple but they are not very efficient numerically, especially in the case of long load histories with a large number of small increments. If the computation time is of some concern the model based on infinite number of plasticity surfaces proposed by Chu [11] can be used in lengthy fatigue analyses. The cyclic plasticity models enable the AePeq - Atre~qrelationship to be established providing the actual plastic modulus for given stress/load increment, Atri. In other words, the plasticity model determines which piece of the stress-strain curve (Fig. 9) has to be utilized during given stress/load increment. Two or more tangent ellipses translate together as rigid bodies and the largest moving ellipse indicates which linear piece of the constitutive relationship should be used for a given stress increment. The slope of the actual element of the stress-strain curve defines the plastic modulus, AO-eq/A~3Pq,necessary for the determination of parameter, dA, in the constitutive Eq 10. The plasticity models are described in most publications as algorithms for calculating strain increments that result from a given series of stress increments or vice versa. This is called the stress or strain controlled input. In the case of the notch analysis neither stresses nor strains are directly inputted into the plasticity model. The input is given in the form of the total deviatoric strain energy density increments and both the deviatoric strain and stress increments are to be found simultaneously by solving the Eq set (12). Therefore, the plasticity model is needed only to indicate which work-hardening surface will be active during the current load increment, which subsequently determines the instantaneous value of the parameter dA. In order to find the elastic-plastic deviatoric stress and strain increment A~r~ and Ae~ from the Eq set (12) the value of parameter dA is determined first based on the current configuration of plasticity surfaces. After calculating the stress increments, Ao-~, and the resultant stress increment, A~ a, the plasticity surfaces are translated as shown in Fig. 10. The process is repeated for each subsequent increment of the "elastic" input, Ao-~. The Mroz and Garud models were chosen here as an illustration. Obviously, any other plasticity model can be associated with the incremental stress-strain notch analysis proposed above.
Multiaxial Cyclic Loading Paths The experimental data concerning measured notch tip strains induced by nonproportional cyclic loading histories were obtained by Barkey [12] who used a cylindrical bar with a circumferential notch similar to that one shown in Fig. 5. The basic proportions of the cylindrical specimen were p/t = 1 and R/t = 2 resulting in the tensile and torsion stress concentration factor KF = 1.41 and Kr = 1.15, respectively. The ratio of the notch tip hoop stress to the axial stress under tensile axial loading w a s O'~3/O'~2 = 0.184. The actual radius of the cylindrical specimen was R = 25.4 mm. The material for the notched bar was SAE 1070 steel with a cyclic stress-strain curve approximated by the Ramberg-Osgood relation, Eq 18. The material properties were: E = 210 GPa, ~, = 0.3, S~, = 242 MPa, n' = 0.199, and K' = 1736 MPa. The first cyclic stress path of the pseudo-elastic notch tip stresses, O'~2 - - O'~3 , is shown in Fig. 11. The rectangular path was repeated counterclockwise more than a hundred times while recording the strains in the notch tip. The notch tip strain components were measured using electric resistance strain gages mounted at the notch tip. The maximum nominal tensile stress and the nominal torsion stresses were trn = 296 MPa and ~'n = 193 MPa, respectively. The corresponding notch tip pseudo-elastic input stresses were o'~2 = 417.3 MPa and O'~3 ~ 221.9 MPa, respectively. Comparison of the measured and calculated notch tip strain paths are shown in Fig. 12. The pseudo-elastic input strain path has been also included as a reference. It can be noted that the agreement between the calculated and measured strain paths is qualitatively and quantitatively good. The experimental solid lines represent notch tip strains measured during the 1st and the 50th loading cycle. The remaining experimental data are not shown in order to preserve the clarity of the diagram. The measured strain path is not sym-
96
MULTIAXlAL FATIGUE AND DEFORMATION
Rectangular Stress Path 250 200 150
a. 3E
100 50
r
b
u)
0
i
-500
400
-300
-200
<.w U)
- - 7
-100
100
200
300
40(
500
-50 -100
tU)
-150
L
-200 A
-250
A x i a l s t r e s s , 022 e ( M P a )
FIG. 11--Rectangular (box) tension-torsion cyclic stress~loading path.
m e~ 03 t,.m
E
I
r/i ,m .J~
, r
-
-
-0.003
- * - Calculated
J I Exp. 1st cyc. i Exp. 50th cyc. j
)03 ~ -o-Input
o
1
r-
ct~
Axial notch tip strain,
~23
a
FIG. 12--Measured and calculated strain paths in notch tip induced by rectangular input loading path.
BUCZYNSKI AND GLINKA ON NOTCH ANALYSIS
97
Unequal-Frequency Stress Path
250
.~~ "
~ 2 0 0
. -" "
150
9
. . . .
9
"~176
b
~9 -500
-400 J0
-200 -100-50 ~ 3 ; 0 -100 -150
400 500
-200 -250 Axial Stress, (~22 e (MPa)
FIG. 13--Unequal frequency tension-torsion stress/loading path.
0.0025 I
ff
~-~176176 ~-~176176 ~ ~ Q.
_o.oo2 ~ I
~176176 o,oo~ i-~176
1
L - - Exper'/
-0.0025 j Axial notch tip strain, E;22 a
FIG. 14--Measured and calculated strain paths in notch tip induced by unequalfrequency tensiontorsion input loading path.
98
MULTIAXIAL FATIGUE AND DEFORMATION
metric with respect to the center of coordinates and the yielding during the first make up cycle, which is always slightly different from the subsequent cyclically stabilized material response, might cause this shift. This might be the reason for the offset of one part of the measured strain path with respect to the calculated symmetric strain path, which was obtained from the stabilized cyclic stress-strain curve. The second cyclic stress path of Barkey [12] is shown in Fig. 13. The resulting elastic-plastic notch tip strain paths are shown in Fig. 14. The maximum nominal stresses were again tr, = 296 MPa and ~', = 193 MPa. Again the qualitative and quantitative agreement between the measured and calculated strain histories was good as in the case of the box stress path.
Conclusions A method for calculating elastic-plastic notch tip strains and stresses induced by multiaxial loading paths has been proposed. The method has been formulated using the equivalence of the total distortional strain energy density. The generalized equations of the total equivalent strain energy density yield a conservative solution for the notch tip strains and stresses in the case of monotonic nonproportional loading. The method has been verified by comparison with finite-element and experimental data obtained for nonproportional loading paths and nonlinear stress-strain material behavior. The notch tip strains and stresses calculated for cyclic load paths can be used for the estimation of fatigue lives for multiaxial cyclic loading histories.
References [1] Neuber, H., "Theory of Stress Concentration of Shear Strained Prismatic Bodies with Arbitrary Non Linear Stress-Strain Law," ASME Journal of Applied Mechanics, Vol. 28, 1961, pp. 544-550. [2] Molski, K. and Glinka, G., "A Method of Elastic-Plastic Stress and Strain Calculation at a Notch Root," Material Science and Engineering, Vol. 50, 1981, pp. 93-100. [3] Moftakhar, A., Buczynski, A., and Glinka, G., "Calculation of Elasto-Plastic Strains and Stresses in Notches under Multiaxial Loading," International Journal of Fracture, Vol. 70, 1995, pp. 357-373. [4] Singh, M. N. K., "Notch Tip Stress-Strain Analysis in Bodies Subjected to Non-Proportional Cyclic Loads," Ph.D. Dissertation, Department of Mechanical Engineering, University of Waterloo, Ontario, Canada, 1998. [5] Seeger, T. and Hoffman, M., "The Use of Hencky's Equations for the Estimation of Multiaxial Elastic-Plastic Notch Stresses and Strains," Report No. FB-3/1986, Technische Hochschule Darmstadt, Darmstadt, 1986. [6] Barkey, M. E., Socie, D. F., and Hsia, K. J., "A Yield Surface Approach to the Estimation of Notch Strains for Proportional and Non-proportional Cyclic Loading," ASME Journal of Engineering Materials and Technology, Vol. 116, 1994, pp. 173-180. [7] Moftakhar, A. A., "Calculation of Time Independent and Time-Dependent Strains and Stresses in Notches," Ph.D. Dissertation, University of Waterloo, Department of Mechanical Engineering, Waterloo, Ontario, Canada, 1994. [8] Chu, C.-C., "Incremental Multiaxial Neuber Correction for Fatigue Analysis," International Congress and Exposition, Detroit, 1995, SAE Technical Paper No. 950705, Warrendaie, PA, 1995. [9] Mroz, Z., "On the Description of Anisotropic Workhardening," Journal of Mechanics and Physics of Solids, Vol. 15, 1967, pp. 163-175. [10] Garud, Y. S., "A New Approach to the Evaluation of Fatigue under Multiaxial Loading," Journal of Engineering Materials and Technology, ASME, Vol. 103, 1981, pp. 118-125. [11] Chu, C.-C., "A Three-Dimensional Model of Anisotropic Hardening in Metals and Its Application to the Analysis of Sheet Metal Forming," Journal of Mechanics and Physics of Solids, Vol. 32, 1984, pp. 197-212. [12] Barkey, M. E., "Calculation of Notch Strains under Multiaxial Nominal Loading," Ph. D. Dissertation, Department of Theoretical and Applied Mechanics, University of Illinois at Urbana-Champaign, Urbana, IL, 1993.
Cliff J. Lissenden, 1 Mark A. Walker, 2 and Bradley A. Lerch 3
Axial-Torsional Load Effects of Haynes 188 at 650~ REFERENCE: Lissenden, C. J., Walker, M. A., and Lerch, B. A., "Axial-Torsional Load Effects of Haynes 188 at 650~ '' Multiaxial Fatigue and Deformation: Testing and Prediction, A S T M STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 99-125. ABSTRACT: Threshold and flow loci were constructed experimentally at 650~ in the axial-shear stress plane for the cobalt-based alloy Haynes 188. These loci suggest that the stress-dependence of the dissipation potential is on the second deviatoric stress invariant, -/2, alone. The functional form of the dissipation is critical to associative potential-based viscoplasticity models because it dictates the direction of the inelastic strain rate vector through normality. The directions of the inelastic strain rate vectors for proportional loading were in generally good agreement with the normality rule. Data suitable for characterizing the material parameters of a viscoplasticity model are presented. Serrated yielding was often present for axial, torsional, and combined axial-torsional loading. By applying proportional and nonproportional load histories at identical equivalent strain rates, -/2 was found to generally reduce the data to a single curve.
KEYWORDS: superalloy, viscoplasticity, multiaxial loading, equivalent stress, path-dependence, stress relaxation Nomenclature
13eq :
~ij ~ij
'
~eq --
'F'ij
O'eq = ~ 2 2
1
912 = ~ S g j S i j sij = ~rij - o-kk ~ij
/~J =
for i r j
Equivalent strain
Equivalent inelastic strain Equivalent stress Second invariant o f deviatoric stress Deviatofic stress tensor Kronecker delta
o-11 and o'12 Axial stress and shear stress c o m p o n e n t s ~311 and 1312 Axial and (tensorial) shear strain c o m p o n e n t s ~/1 and e{2 Axial and (tensorial) shear inelastic strain c o m p o n e n t s 1 Assistant professor, Department of Engineering Science and Mechanics, Penn State University, University Park, PA. 2 Undergraduate student, Department of Engineering Science and Mechanics, Penn State University, University Park, PA 16802. 3 Research engineer, Life Prediction Branch, NASA Glenn Research Center, Cleveland OH, 44135.
Copyright9
by ASTM lntcrnational
99 www.astm.org
100
MULTIAXIAL FATIGUE AND DEFORMATION
1P: ++4
Inelastic power (density) Dissipation Conjugate force-like and displacement-like internal state variables O'y Tensile yield strength E, G, and v Elastic engineering properties: Axial and shear moduli, Poisson's ratio O'p, Tp Linear elastic fit parameters O-o, %, a, b Ellipse fit parameters
D = oijei"Ij - p ~ : Pa and ~::
Structural elements in aeronautics applications often experience complex multiaxial mechanical and thermal loads in service. Elastic-viscoplastic models, for example [1-5], are commonly used to predict time-dependent deformation behavior of the metallic materials commonly used in these applications. In addition, these models are used to predict the inelastic strain range necessary to determine low cycle fatigue life using the Coffin-Manson relation. Whether these models are able to accurately predict material behavior in the presence of multiaxial stress states under these conditions is not known due to a lack of experimental results. Herein, we present the results of axial-torsional experiments conducted at 650~ on the cobalt-based alloy Haynes 188, which is commonly used for transition ducts and combustor liners in jet engines. All of the tests were aimed at viscoplastic model verification. Two general types of tests were conducted: flow surface determination tests and path-dependence tests. Some of the test results are currently being used to characterize the material through determination of model parameters. Flow Surfaces
Flow surfaces provide a measure of multiaxial inelastic flow; they are contours of constant flow. In addition, associative models that use the thermodynamically admissible concept of dissipation potential require the inelastic strain rate vector to be normal to the surface of constant dissipation (SCD). Herein, families of surfaces of constant inelastic power (SCIPs) in the axial-shear stress plane are determined for target values ranging from 0-28 000 Pa/sec. It is important to keep in mind that inelastic power (density) and dissipation defined by, IP = +j ~
(1) D = o'/j ~/~ - p : ~: respectively, are different quantities due to the evolution of the material state with inelastic deformation, except in the vicinity of the threshold where they are essentially the same. Thus, the inelastic strain rate vector is not required by the model to be normal to the SCIPs unless SCDs and SCIPs are proportional. The differences between the various definitions for inelastic flow are discussed by Iyer et al. [6]. As part of the test program, near-threshold yield loci are determined by multiprobe tests on single specimens using the procedure pioneered by Phillips et al. [7] and Liu [8] and used more recently by Gil et al. [9]. Additionally, a family of SCIPs is determined in the axial-shear stress plane by following one proportional loading path on each of the six specimens. These tests parallel those in references [10,11] for rate-independent definitions of inelasticity. Normality of the inelastic strain rate vector with respect to SCIPs is found to be reasonably well satisfied. Path-Dependence
Arnold et at. [12] asserts that nonproportional loading accentuates the difference between a complete, thermodynamically based viscoplastic model and other less complete models. Therefore,
LISSENDEN ETAL. ON HAYNES 188
101
strain-controlled loadings (one proportional and two nonproportional) having the same start and end points, but different paths were applied. At the end of each load path, the strain was held constant and the stress allowed to relax. Similar tests were conducted with stress-controlled loadings followed by a creep period.
Experimental Procedures Test Equipment Tests were conducted on a computer-controlled axial-torsional servohydraulic test machine (222 000 N axial capacity, 2260 N-m torque capacity). Each specimen was held in a vertical position by water-cooled hydraulic grips. The top grip remains stationary, while the bottom grip can translate vertically and rotate during a test. These movements are independently controlled by analog controllers, which are interfaced with a personal computer for control and data acquisition. The specimen was heated by a 15 kW radiofrequency induction heating system having three adjustable coils that surround the gage section [13]. Three therrnocouples spaced approximately 12 mm apart were spot-welded along a vertical line at the mid-length of the specimen. The center thermocouple was used to control the specimen temperature and the other two were used to ensure that the thermal gradient in the gage section was less than 1% of the test temperature. An off-the-shelf high-temperature biaxial extensometer that is spring-loaded against small indents on the surface of the specimen was used to measure axial and shear strains. The maximum strain ranges are 10% axial and 2.3% for shear (tensorial shear strain, given a 26 mm diameter gage section). Yield surface testing demands high resolution strain measurement and the suitability of this extensometer for this purpose has been shown by Lissenden et al. [14].
Specimen Details The Haynes 188 specimens were machined from the same heat (188061714) of wrought bar stock. Results presented by Kalhiri and Bonacuse [15] are also from this heat. Many results have been published from an earlier heat (188081742) of material, but differences in the flow behavior between the two heats were observed for uniaxial tension. Specifically, the earlier heat exhibited a lower proportional limit, a much larger transition region, and much less serrated yielding. The chemical composition (weight percent) provided by the manufacturer is 22.66Ni, 22.11Cr, 14.06W, 1.17Fe, 0.09C, 0.80Mn, 0.35Si, 0.005P, 0.002S, 0.052La, 0.003B, and balance Co. After machining to the specimen dimensions shown in Fig. la, each specimen was solution annealed (1175~ for 1 h in vacuum then quick cooled in argon to room temperature). Haynes 188 is a solid solution alloy and contains second phase carbide particles, which provide an additional strengthening component. The Rockwell hardness was found to be approximately 60 on the A scale and the average grain size was an ASTM 6 (-35 /xm grain diameter) using the lineal intercept technique.
Test Control Equivalent strain or equivalent stress-controlled proportional and nonproportional load paths were followed, where ~eq =
~2-~ (1 + 2/.'2)s21 + 4 d2 (2)
94.0
L
p ~
I ,~.6--------.P
~
bhEa~
~ ' C lloU JUOwN
J
~,....j~_.....,~
f
-
26.01 dill
I F I G . 1--Axial-shear loading, (a) thin-walled Haynes 188 tubular specimen with all dimensions in mm, (b) proportional load paths for multiple probes of a yield locus on a single specimen, (c) proportional load paths for determination of families of SC1Ps, (d) proportional and nonproportional load paths used for the path-dependence study.
BothEndB t ~-~
0,76x45"
c,..,,~./]
(a)
z
5
m "n 0
z
c ITI
---I
I-" "1"1
x
c I"" ---I
8r~
LISSENDEN ET AL. ON HAYNES 188
103
for axial-torsional loading. Poisson's ratio was not measured, but assumed to follow the rule of mixtures (as in [16]),
v
ell + e/1
(3)
where the elastic and inelastic Poisson's ratios were taken to be 0.32 and 0.50, respectively. The validity of Eq 3 has been verified for the nickel-based alloy Inconel 718, where for an axial strain of 8000/xm/m, the error in effective Poisson's ratio, Eq 3, relative to strain gage readings was only 6% (and this was the maximum error). Initial yield surface tests in the axial-shear stress plane were controlled using a technique described previously [9]. Each locus was probed using a single specimen and an equivalent strain rate of 100 /xm/m/sec in 16 unique directions as shown in Fig. lb. Each probe of the locus was unloaded when the equivalent inelastic strain (calculated during the test by the control software) reached 20/zm/m. With this method a small offset yield surface could be constructed using one specimen. However, for determining surfaces at much larger offsets where large inelastic strains would significantlyinfluence subsequent loadings, multiple specimens were required. Hence, the equivalent strain-controlledproportional load paths shown in Fig. lc were also followed. Here, the same strain rate as above was used, but each load path was applied to a different specimen and the unload criterion was a total equivalent strain of 20 000/zm/m. The path-dependence of the material behavior was assessed by following load paths having the same start and end points, but that were otherwise completely different as shown in Fig. ld. Path 1, having a constant ratio of shear to axial load, is proportional loading. Paths 2 and 3 are nonproportional load paths consisting of tension followed by shear and shear followed tension, respectively. Following these nonproportional load paths with a constant equivalent strain rate required that the second leg of the path have a continuously changing rate. Consider, for example, a strain-controlled Path 2 test. The increment in shear strain in the second leg, while the tensile strain is held constant,
(b)
~2 s
7
i
15
1
11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
............
~
,~
12
/ e
-
,,,, I
i
~
10 i 4 |
14
FIG. 1--(Continued)
104
MULTIAXIALFATIGUE AND DEFORMATION
(C)
2
.~-6|2 72*
! i
i
~t
540
i
.
i///j~,'~ . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
! ! t i
(d)
c
End point D
v
A
B
axial
FIG. l~-( Continued)
is given by
d~12 =
(Seq + d ~ e q ) 2 - - ~ ( 1 + 2 v 2 ) 8 ~ 1
-
~12
(4)
After attaining the maximum strain at point D in Fig. ld, both strain components were held constant and the stress allowed to relax for an extended period (-12 h). The stress-controlled tests were different in that the three paths did not have a common endpoint D, but rather a final equivalent stress of 303 MPa. The limited material hardening within the measurable strain range precluded attaining a common endpoint well beyond the threshold using stress-control. That is, even a very small increase in stress resulted in a large increase in strain.
LISSENDEN ET AL. ON HAYNES 188
105
T A B L E 1 --Test matrix.
Specimen
Control Mode
Loading Type
Load Path
HY-42 HY-48 HY-46 HY-36 HY-54 HY-31 HY-47 HY-41 HY-35 HY-37 HY-39 HY-40 HY-38 HY-33
Strain Strain Strain Strain Strain Strain Strain Stram Strain Strain Strain Stress Stress Stress
P P P P
0 ~ (tension) to 1% 90~ (torsion) to 1% 180~ (compression) to 1% 270~ (torsion) to 1% 18~ to 1.9% 36~ to 1.9% 54~ to 1.9% 72~ to 1.9% 225 ~ to 2% Path 3 to 2% Path 2 to 2% 45 ~ to 303 MPa Path 3 to 303 MPa Path 2 to 303 MPa
P
P P P P NP NP P NP NP
Hold (h) 12 12 12 12 0
12 12 12 18 18 3.3 6.6 0 12
Test Matrix All tests were conducted at 650~ with strain-controlled tests loaded at an equivalent strain rate of 100/zm/m/sec and stress-controlled tests loaded at an equivalent stress rate of 69 MPa/s (that is, an equivalent elastic strain rate of 381/zm/m/s). At least one initial yield surface test was conducted on each specimen in the pristine condition. Table 1 shows the details of the tests conducted on each of the 14 specimens. Specimen designations were actually HYII-xx, but have been shortened to HYxx for convenience. Results
Flow Surface Data First, the initial yield locus for each specimen was mapped out in the normal stress-shear stress plane. The offset strain target value of 2 0 / z m / m used here is small enough that there is minimal difference between this locus and the actual threshold locus. The yield loci for four specimens are shown in a modified stress plane in Fig. 2. In this stress plane the familiar Mises ellipse is a circle. The symbols in Fig. 2 denote individual yield points and the lines represent the best fit ellipse, (O'11 -- 0"0) 2 a2
(~r
-- ~/3~'o) 2 (X/~b) 2
-
1
(5)
where a and N/~b are the axes of the ellipse and (0-0, X/3Zo) is the center of the ellipse in the modified stress plane. The regressed values for the axes and center of the ellipse are also shown in Fig. 2. While there is some scatter in the yield points that comprise each locus, as well as between the loci for different specimens, the Mises yield criterion, X / ~ z = 0-r
(6)
represents initial yielding very well. These yield loci were helpful for minimizing specimen to specimen scatter for the load path dependence study where equivalent stress-strain data from one specim e n are compared to that obtained from a different load history applied to another specimen. In essence, the specimens used in this program were screened for uniformity by their initial yield surfaces.
216 204 219 206
HY-42 HY-48 HY-46 HY-36
223 207 209 218
b, MPa
(11,4) (6,12) (17,9) (-8,-1)
Center, MPa
F I G . 2--1nitial yield loci from four different specimens using a 20 txm/m offset strain definition. Ellipses have been fit to yield point data and major and minor axes as well as the center of ellipse are shown.
a, MPa
Specimen
z
.-I
0
"11
11"i
c m > z
I-" "11
x
I-"
c -q
o
LISSENDEN ET AL. ON HAYNES 188
107
Loading excursions along the coordinate axes in the axial-shear stress plane were performed, using strain control, to assess whether the subsequent flow response was symmetric like that of initial yielding. The direction of loading did not have a significant effect on the equivalent stress-strain responses, as shown in Fig. 3a. Serrated yielding (sharp fluctuations in stress) is evident in all but the compression data. Serrated yielding is known to occur in Haynes 188 at some temperatures and strain rates [17,18], and is generally associated with dynamic strain aging, commonly termed the PortevinLe Chatelier effect. The occurrence of dynamic strain aging in Haynes 188 has been associated with pronounced planar slip, an abundance of stacking faults, and very high dislocation densities [17]. It is unclear why serrated yielding did not occur in the compression test, because it did occur in other compression tests under the same conditions. The equivalent stress-strain response in compression (specimen HY-46) may be slightly higher than that in the other loading directions due to the absence of serrated yielding. Stress relaxation data from these four loading excursions are shown in Fig. 3b. Unfortunately, the strain signal was not as constant as we would have liked; it had a tendency to decrease in magnitude as shown in Fig. 4a, where each strain signal has been translated to start at zero at the beginning of the hold time. In the best case (specimen HY-48), the deviation in strain was less than 20/xm/m (0.2% of the nominal constant equivalent strain), while in the worst case (specimen HY-46), the strain variation was approximately 90/zm/m (0.9% of the nominal constant equivalent strain). Both tensile and compressive relaxation tests had a relatively large drop in strain between one and four h. This poor control was remedied by engaging the Reset Integrator, and this resulted in much better control as shown in Figs. 4b and 4c. We believe that the large initial stress drop for specimen HY-46 shown in Fig. 3b is due in part to the decrease in strain. Other contributors to the relatively large stress relaxation of specimen HY-46 are that it had the highest initial stress and did not exhibit serrated yielding. Notice that when the strain signal did not drift (heavy lines in Fig. 3b), the stress relaxation rate was approximately the same for all four loading directions. A priori expectations that the initial yield surface is an ellipse in the axial-shear stress plane and the results of the loading excursions along the axes suggest that it is sufficient to describe flow in just one quadrant of the axial-shear stress plane. Thus, proportional loading excursions in the tension-positive torsion quadrant (I) of the axial-shear stress plane were conducted at angles of 18, 36, 54, and 72 ~ (Fig. lc). The equivalent stress-strain and relaxation responses are shown in Figs. 3c and 3d, respectively. Stress relaxation data from specimen HY-54 was not obtained because the control thermocouple lost contact with the specimen, ending the test. The stress-strain data from the loading portion of these probes are analyzed in the data analysis section. The strain control was good during the hold period, leading to more tightly grouped relaxation curves in Fig. 3d, where the equivalent stress relaxed to between 65 and 75% of its initial value in 12 h.
Path-Dependence Data As shown in Fig. 5a, the constant equivalent strain rate was accurately controlled for all three load paths (Fig. ld). The applied loads were compression and negative shear for this strain-controlled sequence of tests. Details of the load paths are provided in Table 2. The three load paths result in essentially the same equivalent stress-strain response (Fig. 5b) until the equivalent strain reaches approximately 15 000/xm/m, at which point the equivalent stress increases significantly for the nonproportional toad paths, but not for the proportional load path. This point corresponds to a change in the loading direction for the nonproportional load paths (points B and C in Fig. ld). The maximum strain (point D in Fig. ld) was held constant to allow the stress to relax, with the hope that the stress relaxation would provide a rigorous test for a viscoplastic model. Normalized equivalent stress relaxation data are shown in Fig. 5c. Again, the equivalent stress and strain definitions do a good job of collapsing the data to a single curve. The hold period after following Path 2 was cut short due to a loss of hydraulic pressure.
FIG. 3--Proportional load paths for tension, positive shear, compression, and negative shear: (a) stress-strain response, (b) stress relaxation. Four proportional load paths in quadrant 1: (c) stressstrain response, (d) stress relaxation.
LISSENDEN ET AL. ON HAYNES 188
FIG. 3--(Continued)
109
110
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 4--Controlled constant strain signal as a function of time with the Reset Integrator (a) turned off, (b) turned on--shear strain, (c) turned on--axial strain.
The axial and shear component response shown in Fig. 6 is more informative than the equivalent response in terms of describing the path dependence. Figures 6a and 6b show the axial and shear stress-strain diagrams, respectively. The axial response for proportional loading (Path 1) is softer than that for the first leg of axial-then-shear nonproportional loading (Path 2). Likewise, the shear response for proportional loading is softer than that for the first leg of shear-then-axial nonproportional loadTABLE 2--Load path specification. Key Point Coordinates: (ell, e12) in/xm/m, [Oql, o12] in MPa Path
Control Mode
A
B
1
Strain
.
2
Strain
3
Strain
1
Stress
2
Stress
3
Stress
(0,0) [0, 0] (0, 0) [0, 0] (0, 0) [0, 0] (0, 0) [0, 0] (0, 0) [0, 0] (0, 0) [0, 0]
C .
.
.
(-14350, - 7 ) [-310, 3] ... .
.
.
(8110, -393) [260, 0] ...
.
. ...
.
( - 10, - 12260) [-9, - 173] . . ... (-217, 5576) [0, 150]
D ( - 14450, - 12120) [-242, - 120] (-14330, -12260) [-4, -216] ( - 14390, - 12260) [-366, - 11] (11980, 9766) [222, 124] (16580, 2999) [266, 88] (4894, 16450) [153, 155]
FIG. 5--Strain-controlled Paths 1, 2, and 3,
(a) controlled equivalent strain versus time, (b) equivalent stress-strain, and (c) stress relaxation.
W"
-< z m
0 z -r"
.r-
CD m z o m z m
112
MULTIAXIALFATIGUE AND DEFORMATION
(C) 1o -~ I
0.9-
t~ "13
!
~
Patti l (HY.35, 3 | 4 5 MPa)
!
O
Pllth 2 (HY.39. 370.5 MPa)
I
,~
Path 3 (HY.37. 356.3 MPa)
0.8
o 0.7
0.6 10600
21600
32400
43200
54000
6480
Time (sec)
FIG. 5--(Continued)
ing (Path 3). However, it is surprising that the axial response for Path 3 reaches higher axial stresses than Paths 1 and 2 (Fig. 6a). Apparently, this happens due to relaxation of the shear stress to almost zero between points C and D for Path 3 (Fig. 6b). Most of this relaxation happens almost immediately such that the stress state in segment CD of Path 3 is nearly uniaxial compression. Thus, the higher axial stresses attained for Path 3 appear to be due to the hardening of the material during the shear loading segment AC. Similarly, the shear stresses for Path 2 are higher than their counterparts for Paths 1 and 3 and the axial stress for Path 2 relaxed to almost zero during segment BD. The relaxation of the axial and shear stress components under constant strain, that is, after reaching point D, is shown in Fig. 6c and 6d, respectively. Stress components have been normalized with respect to their values at point D. Axial stress relaxation for Path 2 and shear stress relaxation for Path 3 are not shown because their stress magnitudes at point D were very small. For the stress-controlled tests the rapid loading rate made accurate control difficult, but as shown in Fig. 7a the control was excellent until the equivalent stress reached approximately 230 MPa when large strains accompanied an increase in stress. The applied stresses were tension and positive shear. The rapid loading rate was used to not give inelastic strain time to develop until the end point of the load path was reached. The end point was attained for all three load paths, with the equivalent stressstrain curves shown in Fig. 7b. These three curves are similar, but the proportional load path (Path 1) resulted in a slightly higher yield point and subsequent stresses, perhaps due to the better control (Fig. 7a). The accumulation of normalized equivalent strain with the equivalent stress held constant is shown in Fig. 7c for Path i and Path 2. The Path 3 test (shear-then-axial load) was stopped before adequate strain accumulation data was collected due to the shear strain exceeding the present limit. The Path 2 test also stopped prematurely due to a problem with the hydraulics system. Clearly, there is a difference between the accumulated (creep) equivalent strains after Paths 1 and 2. More data is required to address whether this difference is due to path-dependence or other causes. Figure 8 shows the axial and shear component stress-strain-time responses to the load Paths 1, 2, and 3. The stress-strain response for proportional loading (Path 1) is, as expected, in between the responses to the nonproportional load paths. The axial response to Path 3 and the shear response to Path
LISSENDEN ET AL. ON HAYNES 188
1 13
FIG. 6 - - A x i a l and shear stress component response to strain-controlled Paths 1, 2, and 3, (a) axial stress-strain, (b) shear stress-strain, (c) axial stress relaxation, and (d) shear stress relaxation.
114
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 6--(Continued)
accumulation.
FIG. 7--Stress-controlled Paths 1, 2, and 3, (a) controlled equivalent stress versus time, (b) equivalent stress-strain, and (c) strain
01
.< Z m Oo
Z "1-
0
m z o m z m
W"
116
MULTIAXlALFATIGUE AND DEFORMATION
FIG. 7--(Continued)
2 exhibited no elastic region due to the presence of the other stress component (Figs. 8a and 8b). The accumulation of axial and shear strain components under constant load is shown in Figs. 8c and 8d, respectively.
Data Analysis Data from proportional loading strain-controlled probes at angles of 0, 18, 36, 54, 72, and 90 ~ were analyzed to construct contours of constant inelastic flow (SCIPs). The test data that were post-processed to construct SCIPs are: time, axial and shear stress, and axial and shear strain. First, the equation of the linear elastic loading curve (coefficients E, G, ~rp,and rp) were determined using linear regression, which enabled the inelastic strain components to be calculated from, 0"11 --O'p /3/1 = e l l
E (7)
~12 = /312
O"12 - -
?p
2G
where the stress axis intercepts, O-p,and ~-p,were typically less than 10 MPa. The typical stress-strain diagrams in Fig. 9a (for a proportional load path at 54 ~ indicate the magnitude of the stress fluctuations due to serrated yielding. In order to obtain reasonably smooth inelastic strains, these fluctuations were removed from the data by only using the peaks, as indicated by the heavy lines in Fig. 9a. The inelastic strain components were then calculated using Eq 7, and are shown as a function of the applied (total) strain components in Fig. 9b. These curves are representative of the proportional load path data. The material response is divisible into three regions: linear elastic, transition, and linear hardening. Based on the strains applied in this program, the linear hardening region is the largest, and the linear elastic region is the smallest. In the linear hardening region, the rate of change of the
LISSENDEN ET AL. ON HAYNES 188
117
FIG. 8 - - A x i a l and shear stress component response to stress-controlled Paths I, 2, and 3, (a) axial stress-strain, (b) shear stress-strain, (c) axial strain accumulation, and (d) shear strain accumulation.
118
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 8---(Continued)
FIG. 9--Data processing, (a) stress-strain diagrams showing that only peaks of yield serrations are used, (b) inelastic strain components with respect to applied strain components, (c) enlarged view of elastic and transition regions of axial inelastic strain-total strain diagram.
-t. co
O Z "I" > -< Z m
.t-
m z ~3 m z l'n -4
i--
120
MULTIAXIAL FATIGUE AND DEFORMATION 1200
(c)
.c
9
e~
U3 .g
!
,
! !ii
.I
.~
4oo
0
500
1000 Axial Strain (Prn/m)
1500
20O
FIG. 9--(Continued)
inelastic strain with respect to the total strain was nearly the same for axial and shear strain components for all proportional load paths. Figure 9c shows the linear elastic and transition regions at a larger scale and identifies the threshold. The inelastic strain rates are needed to calculate the inelastic power. Curves were fit to the inelastic strain-total strain data to permit the rates to be calculated by differentiation of polynomials,
~1
dS~l d811 dell dt
(8)
d8112 d~:12 ,~12 = d,~12 dt where representative polynomials are shown in Table 3 for a proportional load path at 54 ~ A quartic polynomial was the lowest order polynomial that gave a good fit to the inelastic strain-total strain data in the transition region. The inelastic power was then calculated using
I P = o'11 g{1 +2o"12 g{2
(9)
The rate-dependent, inelastic power definition of flow will be compared with the rate-independent, equivalent inelastic strain definition of flow, because while the former definition is more important than the latter for elevated temperature environments, the latter is more traditional. The results of using these two definitions are shown in Fig. 10. The accumulation of equivalent inelastic strain with time is shown in Fig. 10a for proportional loading paths in quadrant I of the axial-shear stress plane. Contours of constant equivalent inelastic strain, Sleq, are constructed in Fig. 10b, which also shows stress paths associated with the strain-controlled tests. Regression is used to fit ellipses to the data points. The threshold, which delimits the elastic region, agrees well with the initial yield loci shown in Fig. 2.
LISSENDEN ET AL. ON HAYNES 188
121
TABLE 3--Curve fits for specimen HY-47 Axial Strain-Time
611 = 60.55t + 40.66/xm/m
Stress-Strain:
O-ll = 0.1812611 + 8.684 MPa
Inelastic Strain-
for 611 < 600/zm/m
r
Total Strain: 6~1 =
215.9 -- 1.153611 + 1.540 • 10 -3 621 -4.695 • 10-7 6131+ 5.097 • 10 -11 641
for 600 4611 <3000/zm/m
-857.2 + 0.9793611
for ooll > 3000/xm/m Shear
Strain-Time:
612 = 70.08t + 8.952/zm/m
Stress-Strain:
o12 = 0.1388612 + 1.311 MPa
Inelastic Strain-
0
for e12 < 800/xm/m
2494 - 8.297612 + 9.368 • 10 _3 o~2 --4.159 • 10-6 632 + 5.879 • 10 10 642
for 800 < e12 < 2000/xm/m
-873.9 + 0.9823612
for $12 > 2000/xm/m
Total Strain: 6~2 =
The related accumulation of inelastic power with time and contours of constant inelastic power (SCIPs) are shown in Figs. 10c and 10d, respectively. The shapes of the inelastic power-time curves (Fig. 10c) resemble those of the stress-strain curves. The range of times at which the inelastic power attained a value of 28 000 Pa/s seems to suggest a meandering SCIP in the axial-shear stress plane. But, as shown in Fig. 10d, this is not the case, because the stress rate is very small in the linear hardening region. The IP values shown in the legend of Fig. 10d are nolninal values. The actual IP values were as much as 19% above the minimum due to the stress data being discontinuous and having a scatter band. Although not done here, this could be addressed in the linear hardening range by determining the stress rate from the total strain rate and inelastic strain rate. In the linear hardening range where the inelastic strain rate is constant and the stress rate is small there is only a minimal increase in IP. Thus, SCIPs constructed for increasing values of IP beyond approximately 25 000 Pals are not expected to increase substantially in size. Due to the uniformity of the inelastic response in different loading directions shown in Figs. 2 and 3, symmetry can be used to extrapolate the inelastic power contours found in quadrant I (Fig. 10d) to the entire axial-shear stress plane. The directions of the inelastic strain rate vector with respect to the 28 000 Pals SCIP for proportional load paths of 18, 36, 54, and 72 ~ are also shown in Fig. 10d. Although shown with respect to the 28 000 Pals SCIP, these directions did not change substantially in the linear hardening region. Based on visual inspection, these directions appear to be very close to the direction of the outward
122
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. l O---Processed data, (a) equivalent inelastic strain-time showing levels at which contours are constructed, (b) contours of equivalent inelastic strain in the axial-shear stress plane, stress paths are also shown, (c) inelastic power-time showing levels at which contours are constructed, (d) SCIPs in the axial-shear stress plane with directions of inelastic strain rate vectors also shown.
LISSENDEN ETAL. ON HAYNES 188
FIG. l O---(Continued)
123
124
MULTIAXlALFATIGUE AND DEFORMATION
normal to the SCIP. However, as pointed out above, the applicability of the normality criterion can not be assessed yet, because the relationship between the SC1P and SCD is unknown at this point. The direction of the inelastic strain rate vector at point D of the strain-controlled load paths 1, 2, and 3 (see Fig. ld) was determined. For both proportional and nonproportional load paths it was found to be oriented within 3 ~ of the direction of the current strain increment. Additionally, the inelastic power at point D was found to be approximately 40 300 Pa/s for nonproportional load paths 2 and 3 and 31 700 Pa/s for proportional load path 1. The inelastic power for the nonproportional load paths was actually decreasing through segments BD (path 2) and CD (path 3) owing to the decrease in normal stress for path 2 and the decrease in shear stress for path 3. This is in contrast to the inelastic power for proportional loading, which was nearly constant (slightly increasing) due to the linear hardening response.
Discussion Initial yielding under various ratios of axial-torsional loading was found to closely follow the Mises criterion as shown in Fig. 2. The equivalent stress and strain definitions used in this work are based on this yield criterion. The equivalent stress-strain curves shown in Figs. 3a, 3c, and 5b suggest that these definitions of equivalent stress and strain are effective in reducing proportional loading data to a single curve. However, this is not the case for nonproportional load paths as shown in Fig. 5b where the equivalent stress rate increases at the change in load direction. The objective of this work was to generate experimental data that can be used to assess the ability of viscoplastic models to predict multiaxial material response. An important theoretical issue is whether the concept of generalized normality [19] is applicable. The directions of inelastic strain rate vectors are shown with respect to SCIPs in Fig. 10d. However, due to the evolution of internal state SCIPs and SCDs are different. In order to assess the normality criterion a model will have to be used to relate SCIPs and SCDs. Implicit in this comparison will be the definition of internal state variables and their evolution. To facilitate this comparison the material is currently being characterized. Once characterized, the path-dependence tests will be used for validation.
Summary and Conclusion Combined axial-torsional loads have been applied to the cobalt-based alloy Haynes 188 tubular specimens at 650~ Both proportional and nonproportional load paths were followed in strain control to equivalent strains of up to 2%. These strains were then held constant to measure the stress relaxation. Additionally, stress controlled tests were conducted to an equivalent stress of 303 MPa, after which the stress was held constant and the accumulated strain measured. These tests are intended to be validation experiments for the multiaxial form of elastic-viscoplastic models. Key experimental results are as follows: 9 The Mises yield criterion fits the initial yield data quite well (20/xm/m offset strain definition, which is essentially the threshold). 9 The Mises equivalent stress definition and associated equivalent strain definition adequately correlate the stress-strain response for proportional loading. However, for nonproportional strain-controlled loading there is a considerable increase in the equivalent stress rate associated with a change in the loading direction. This increase is not present for proportional loading. 9 For a strain-controlled axial-then-shear nonproportional load path, the axial stress was reduced to almost zero very quickly with the axial strain held constant during shear loading. Likewise, the shear stress reduced to almost zero with the shear strain held constant during axial loading of a shear-then-axial load path.
LISSENDEN ET AL. ON HAYNES 188
125
9 Families of surfaces of constant inelastic power (SCIP) have been constructed in the tensionshear stress plane for inelastic power values of zero (the threshold) to 28 000 Pa/s. The direction of the inelastic strain rate vector is approximately normal to these surfaces.
Acknowledgments CJL and M A W gratefully acknowledge the financial support of the NASA Glenn Research Center (grant NCC3-597).
References [1] Bodner, S. R. and Partom, Y., "Constitutive Equations for Elastic-Viscoplastic Strain-Hardening Materials," Journal of Applied Mechanics, Vol. 42, 1975, pp. 385-389. [2] Robinson, D. N., "A Unified Creep-Plasticity Model for Structural Metals at High Temperature," ORNL TM-5969, Oak Ridge National Laboratory, 1978. [3] Walker, K. P., "Research and Development Program for Nonlinear Structural Modeling with Advanced Time-Temperature Dependent Constitutive Relationships," NASA CR-165533, 1981. [4] Freed, A. D. and Walker, K. P., "Viscoplasticity with Creep and Plasticity Bounds," International Journal of Plasticity, Vol. 9, 1993, pp. 213-242. [5] Arnold, S. M., Saleeb, A. F., and Castelli, M. G., "A Fully Associative, Nonlinear Kinematic, Unified Viscoplastic Model for Titanium-Based Matrices," Life Prediction Methodology for Titanium Matrix Composites, ASTM STP 1253, W. S. Johnson, J. M. Larsen, and B. N. Cox, Eds., American Society for Testing and Materials, 1996, pp. 231-256. [6] Iyer, S. K., Lissenden, C. J., and Arnold, S. M., "Local and Overall Flow Surfaces in Composites Predicted by Micromechanics," Composites Part B: Engineering, in press. [7] Phillips, A., Liu, C. S., and Justusson, J. W., "An Experimental Investigation of Yield Surfaces at Elevated Temperatures," Acta Mechanica, Vol. 14, 1972, pp. 119-146. [8] Liu, K. C., "Room Temperature Elastic-Plastic Response of Thin-Walled Tubes Subjected to Nonradial Combinations of Axial and Torsional Loadings," Verification and Qualification of Inelastic Analysis ComputerPrograms, ASME STP, 1975, pp. 1-12. [9] Gil, C. M., Lissenden, C. J., and Lerch, B. A., "Determination of Yield in Inconel 718 for Axial-Torsional Loading at Temperatttres up to 649~ '' NASA TM- 1998-208658. [10] Williams, J. ]F. and Svensson, N. L., "Effect of Torsional Prestrain on the Yield Locus of 1100-F Aluminum," Journal of Strain Analysis, Vol. 6, 1971, pp. 263-272. [11] Khan, A. S. and Wang, X., "An Experimental Study on Subsequent Yield Surface After Finite Shear Prestraining," International Journal of Plasticity, Vol. 9, 1993, pp. 889-905. [12] Arnold, S. M., Saleeb, A. F., and Wilt, T. E., "A Modeling Investigation of Thermal and Strain Induced Recovery and Nonlinear Hardening in Potential Based Viscoplasticity," Journal of Engineering Materials and Technology, Vol. 117, 1995, pp. 157-167. [13] Ellis, J. R. and Bartolotta, P. A., "Adjustable Work Coil Fixture Facilitating the Use of Induction Heating in Mechanical Testing," Multiaxial Fatigue and Deformation Testing Techniques. ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 43-62. [14] Lissenden, C. J., Lerch, B. A., Ellis, J. R., and Robinson, D. N., "Experimental Determination of Yield and Flow Surfaces under Axial-Torsional Loading," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 92-112. [15] Kalluri, S. and Bonacuse, P. J. "Cumulative Axial and Torsional Fatigue: An Investigation of Load-Type Sequencing Effects," ASTM Symposium on Multiaxial Fatigue and Deformation: Testing and Prediction, 19-20 May 1999, Seattle, WA. [16] Bakis, C. E., Castelli, M. G., and Ellis, J. R., "Thermomechanical Loading in Pure Torsion: Test Control and Deformation Behavior," Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and J. R. Ellis, Eds., American Society for Testing and Materials, West Conshohocken, PA, 1993, pp. 223-243. [17] Bhanu Sankara Rao, K., Castelli, M. G., Allen, G. P., and Ellis, J. R., "A Critical Assessment of the Mechanistic Aspects in HAYNES 188 during Low-Cycle Fatigue in the Range 25~ to 1000~ '' Metallurgical and Materials Transactions A, Vol. 28A, 1997, pp. 347-361. [18] Bhanu Sankara Rao, K., Castelli, M. G., and Ellis, J. R., "On the Low Cycle Fatigue Deformation of Haynes 188 Superalloy in the Dynamic Strain Aging Regime," Scripta Metallurgica Et Materialia, Vol. 33, 1995, pp. 1005-1012. [19] Lubliner, J., Plasticity Theory, Macmillan, New York, 1990.
Markus Arzt, 1 Wolfgang Brocks, 1 and Tainer Mohr I
A Newton Algorithm for Solving Non-Linear Problems in Mechanics of Structures Under Complex Loading Histories REFERENCE: Arzt, M., Brocks, W., and Mohr, T., "A Newton Algorithm for Solving Non-Linear Problems in Mechanics of Structures Under Complex Loading Histories," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 126-135.
ABSTRACT: A Newton algorithm is discussed. This algorithm allows for studying nonlinear material behavior, such as plasticity and viscoplasticity. The material behavior is described by so-called internal variables. The model applied in this paper was developed by Lemaitre and Chaboche. The Newton algorithm is implemented as a user-subroutine in the commercial finite-element method package ABAQUS. The results of a numerical analysis of the rotation of an aircraft turbine disk are presented. Cyclic loading conditions are studied, and the results are compared with numerical results in the literature. KEYWORDS: cyclic loading, constitutive equations, elasto-viscoplastic material, finite elements, numerical algorithms Nomenclature
a A a A
Scalar Scalar Vector Matrix
Structures under cyclic loading conditions are studied. Numerical simulation of inelastic behavior under cyclic loading is an interesting task since the failure of structures is often caused by repeated cycles of loading, e.g., an aircraft turbine disk [1-3]. To simulate the material behavior in the case of cyclic loading, multiaxial and out-of-phase loading conditions, nonlinear constitutive relations have to be applied [4,5]. At present, constitutive equations described by internal variables are the common approach to simulate the phenomena of isotropic and kinematic hardening [6, 7]. The viscoplastic model applied is based on the works of Chaboche [8,9] and Lemaitre and Chaboche [10]. These constitutive relations are developed to simulate cyclic loading. The model of Chaboche, implemented in the commercial finite-element model (FEM) package ABAQUS is rateindependent, to simulate plasticity behavior [11]. The model applied in this paper is rate-dependent, to simulate viscoplastic behavior. This model is implemented as a user-subroutine in the FEM package ABAQUS. The unknowns of the problem are defined in the second paragraph. The constitutive relations are strongly nonlinear. Therefore, the numerical analysis of structures is complex and the integration methods are difficult to develop. A Newton algorithm is adopted to determine the equivalent plastic strain increment and the flow 1 GKSS Forschungszentrum Geesthacht GmbH, Institut fur Werkstofforschung, Max-Planck-Slrafle, 21502 Geesthacth, Germany. 126
Copyright9
by ASTM lntcrnational
www.astm.org
ARZT ET AL. ON A NEWTON ALGORITHM
127
direction as well. One procedure is implemented to calculate the equivalent plastic strain increment, and a second to calculate the flow direction. First, the equivalent plastic strain increment is calculated for fixed values of the flow direction, and then the flow direction is determined for the previous, fixed value of the equivalent plastic strain increment. Both are part of an iteration procedure to determine the solution. The theory is developed in Chaboche and Cailletaud [12]. Once the solution is known, a Euler scheme is applied to integrate the unknowns. The consistent material tangent is determined in order to reduce the number of iterations required, and to increase the time increment size [11]. The required processing time and memory for storage to solve these problems is still important. Even today there are few complex computations carried out in industry, although the performances of computers are increasing and strategies are developed to carry out these computations [13-15]. These methods allow the reduction of the processing time and decrease the storage capacities required [16-18].
Material Behavior The unknowns of the problem are defined: stress, back stress, isotropic hardening, and the inelastic strain [8-10]. Mechanical behavior is described by so-called internal variables, the equivalent plastic strain, p, and the kinematic hardening variable, a. in order to simulate cyclic phenomena. The expansion or contraction, and translation of the yield surface are represented by the corresponding variables, the isotropic hardening function, R, and the back stress, x. respectively. Isotropic and kinematic hardening are combined, and kinematic hardening is nonlinear. The potential of the free energy, gt, where the density is p, is taken as p ~ = ~-D1
eel Eel _[_3-1caaT
a + h(p)
(1)
where e"el is the elastic strain. The elastic stiffness or Hooke matrix is D. Kinematic hardening constants are c and a. For the isotropic hardening, the function h(p) is chosen as the potential: Q h(p) = Qp - f f (1 - e -bp)
(2)
Asymptotic value of isotropic hardening function is Q, Eq 5, and b is an exponent. The state equations of elasticity and kinematic hardening are differentials of the free energy potential, Eq 1:
~el
(3)
x = ~ ca ot
(4)
o" = D 2
Differentiating the isotropic hardening potential, Eq 2, using the potential of the free energy, Eq 1, the state equation of the isotropic hardening is obtained: R = Q (1 - e -bp)
(5)
The constitutive equations are derived from the dissipation potential, ~O*:
g
[_(f(o', X_,e)).] n+l
~0* (g, x_, R) = n + 1 [
K
where n stands for the hardening exponent, and K represents the coefficient of resistance.
(6)
128
MULTIAXIALFATIGUE AND DEFORMATION
The yield function f(o-, x_,R) is defined as: (7)
f(ff, x_, R) = J2(o" - x_) - R - k and the second invariant or the equivalent von Mises stress is
/ 3 [_o, dev - x)T (crdov - x) J2(_~ - x) = ~/~where _o-devis the deviatoric part of the stress ~. The yield strength at zero plastic strain is k. The MacCauley bracket, a function, is defined as (y) = y i f y > 0 and (y) = 0 i f y < 0 Derivation of dissipation potential, ~b*, leads to constitutive equations [10]. T h e rates of inelastic strain, ~in, back stress, __f,and isotropic hardening,/~, are given: ~i,
-
3 -~
- x
2 J2(ff - x)/~
x_' = c [ J2(ff - x)
where
/~ =
[(f(ffK
qb(p)x /~ -- d
J~x)
'
R))]"
(8)
and
(9)
(10)
k = b(Q - R)[~
The second term, the relaxation term, of the kinematic hardening is introduced to decrease the effect of the plastic strain. The function of the relaxation term, r is 9 (p) : ~
+ (1 - ~ ) e -wp
The constants qb~ and oJ are material parameters. The third term describes the static recovery, the parameters are the constant d and the exponent r. This version of the model is a rather simple one. More kinematic hardening variables could be introduced, in order to describe the phenomenon of hysteresis and the so-called Bauschinger effect more realistically.
Newton Algorithm A Newton algorithm is applied to determine the equivalent plastic strain increment and the flow direction. Once the solution is determined, the unknowns are integrated using a Euler scheme. The advantage of this algorithm is that plasticity problems can be studied. The basic theory is given in Chaboche and Cailletaud [12]. The consistent material tangent is used in order to reduce the number of iterations and to increase the length of the time increments [11,19]. First, the equivalent plastic strain increment is calculated for fixed values of the flow direction. And then the flow direction is calculated for the previous, fixed value of the equivalent plastic strain increment. The initial values of the two schemes are zero. The set of the two Eqs 11 and 12 allows the determination of the equivalent plastic strain and the flow direction, respectively. Taking Eq 7, the first equation is written [Apqa/n ~'= J z ( o ' o - X_o) - K : O
where
K = Ro + k + K|~-{IL
j
(11)
ARZT ET AL. ON A NEWTON ALGORITHM
129
the thirdterm of the function r represents the viscoplastic overstress, this term is equal to zero in case of plasticity. The second equation gives a relation to determine the flow direction, and is written as: w = L ( o 0 - x-0) - Kn0 = 0
L = 3 idev
and
(12)
where the matrix L is an abbreviation, and la~V stands for the deviatoric unit matrix. Since a Euler scheme is used to integrate the unknowns, an integration parameter, 0, is introduced. The bounds of 0 are stated: 0 < 0 --- 1. Where the midpoint rule, 0 = ~/2,and the implicit integration method, 0 = 1, are included, but not the explicit integration method, 0 = 0. For the stress, the expression is: fro=fit+
0 D Ae- ODnoAp
(13)
The equation for the back stress is the integrated expression of Eq 9:
(14)
x_o = ~oX~ + ~o0 2 ca n_oAp where 1 rl~ = 1 + 0 (c crp(po)Ap + s(xo)At)
[ J2(x-o) l r 1 , s (x-0) = - d [ - - Y - ]
with
,
and s(x_o) stands for the static recovery term. For the isotropic hardening, Eq 10, and the equivalent plastic strain are taken Ro = Q (1 - e -bp~
and
Po = pt + OAp
(15)
The first equation of the system, Eq 11, is solved, using Eqs 13 and 14, and Eq 15, for fixed values of the flow direction, no. The scalar Newton algorithm is applied, and a value for Ap at iteration m + 1 is determined:
Ap (re+l)=mp(m)
J)(mp(m)
l/ ( mp (m))
with
v'(Ap(m)) 4:0
(16)
The second equation of the system, Eq 12, is solved for a fixed plastic strain increment, Ap. The Newton algorithm is adopted and a value for no is calculated at iteration m + 1: (17) The consistent material tangent is chosen in order to decrease the number of iterations and to increase the length of the time increments. The theory is explained in Chaboche and Cailletaud [12] and is an extension of Doghri [I1 ]. The variation of the stress and back stress at the midpoint, t + OAt, and at the end of the increment, t + At, are used to determine the variation of the plastic strain increment, the two relations follow: bf0 = 0~tr
and
bx-0 = 00x-
130
MULTIAXIAL FATIGUE AND DEFORMATION
Applying the variation of the yield function, Eq 11, and the variation of the function K, using the abbreviation H, OK = n.n_( Oo" -- 3x)
and
OK = - H O A p
a relation for the variation of the plastic flow direction to the variation of the plastic strain increment is established. Taking the variation of the plastic flow direction, On = 1__N(0_o" - 0x)
where
N= L -
n nT
(18)
K - -
the variation of the plastic strain increment is deduced. Finally, the following equation is obtained 3 A p = 1g n T _D O e + 1g r~o[C Ocrp(po) A p + Os(xo)At]n TX_0
(19)
where and
g = 3IX + H + rloh
h = ca - c45(po) n_Txo
In order to determine the consistent tangent matrix, the variations of the stress and back stress are taken and a linear system is obtained. Therefore, the variation of the plastic flow direction, Eq 18, and the variation of the plastic strain increment, Eq 19, are applied. The following linear system of equations has to be solved:
[,)A ,;A
[:]
The solution of the system leads to the definition of the consistent tangent matrix: Oo" = K O e
where
K
=
F-1H__,
(21)
and the expressions: __F= [__/+ A ] - A [ / + B ]
-1 _
and
H = U +A
[__/}-n]
- 1 V
(22)
The components of the matrix are: A =AN, A=2/z ~ --
--
and
B=B
K
--
N . B = 70ca A p --
(23)
K
For the variation of the strains, the components are the following:
U
=
D
-
1__(2/x)2n g
nT
and
V__V= 1 2 t z r l o ca n n_T - 1 2p, rlocei~(po) n sTo
--
g
g
The inverse of matrix, _,F is: F - 1 = / + F_~N --
--
--
where
F
A 1 +~(A
+B)
(24)
ARZT ET AL. ON A NEWTON ALGORITHM
FIG.
131
1--Mesh of aircraft turbine disk.
T h e c o n s i s t e n t t a n g e n t m a t r i x is obtained: K = D - 2/x3F/dev
+[21xF-l(21~,2+121x3F~loCqb(po) n_Wx_o]n..__~n T
(25)
1 21~Frl~ cCrp(po)xonT g Numerical Simulation T h e rotation o f an aircraft turbine d i s k is carried o u t as a n u m e r i c a l simulation. T h e p r o b l e m is quasi-static; i s o t h e r m a l conditions a n d s m a l l strains are a s s u m e d . T h e m e c h a n i c a l loading is the centrifugal force. T h e p r o b l e m is a x i s y m m e t r i c . T h e diameter o f the disk is 300 m m , and the d i a m e t e r o f the hole at the center is 75 m m . T h e t h i c k n e s s o f the disk at the bore is 50 m m , a n d at the n e c k the t h i c k n e s s is 14 m m . A t the outer r i m the t h i c k n e s s is 40 m m . In Fig. 1 the m e s h o f the turbine disk is s h o w n . T h e n u m b e r o f three-node e l e m e n t s is 571, a n d the n u m b e r o f n o d e s is 335. T h e plane o f s y m m e t r y is the plane 1-3. T h e total n u m b e r o f degrees o f f r e e d o m is about 629. T h e temperature field is h o m o g e n e o u s and the t e m p e r a t u r e is 550~ T h e material is a superalloy called I N C O N E L 718; the data are g i v e n in Table 1 [4,20]. The data allow for s t u d y i n g cyclic m e a n stress relaxation, in contrast to the data g i v e n in C h a b o c h e and Cailletaud [1].
TABLE E = 169 400 MPa Q = - 1 8 5 MPa c = 500 d = 0 1/s n = 4
l--Parameters of Chaboche model.
v = 0.3 b = 60 a = 340 MPa r = 0 K = 622 MPa s 1/"
k = 646 MPa qb= = 1
o~ = 0 p = 8190 kg/m 3
132
MULTIAXlALFATIGUE AND DEFORMATION
FIG. 2--First three cycles of loading.
The loading of 1000 rotation cycles is simulated. The shape of the first three loading cycles is presented in Fig. 2. The minimum rotation speed is 1500 rpm, and the maximum rotation speed is 27 700
rpm [1,18]. The processing time on an IBM Risc/System 6000 computer is 13 h, CPU. The integration parameter is chosen equal to 5. The results are compared with results obtained by Lesne and Savalle [18]. In Fig. 3 the hoop stress versus inelastic hoop strain for a node at the bore and at the neck is presented. An important variation of the stress is noted. Although the inelastic strain magnitudes of 0.6% and 0.1% at the bore and at the neck, respectively, are lower, compared to Lesne and Savalle [18], these differences are due to the data allowing the study of cyclic mean-stress relaxation. The hoop stress at the beginning of maximum rotation speed for the first and tenth cycle and for cycles 100 and 1000 are given in Figs. 4, 5, 6 and 7, respectively. The stress distribution versus the number of cycles is observed. The minimum level of the hoop stress decreases and the maximum level of the hoop stress increases. The distributions give an idea of the zones where plastic flow occurs. Note a serious level of the stress at the bore and at the neck. Important stress gradients are observed at the bore and at the neck. At cycle 100, the stable stress distribution is nearly reached.
FIG. 3--Hoop stress versus inelastic hoop strain at bore and neck.
ARZT ET AL. ON A NEWTON ALGORITHM
FIG. 4---Hoop stress at beginning of maximum rotation speed for Cycle 1.
FIG. 5--Hoop stress at beginning of maximum rotation speed for Cycle 10.
FIG. 6-~Hoop stress at beginning of maximum rotation speed for Cycle 100.
FIG. 7--Hoop stress at beginning of maximum rotation speed for Cycle 1000.
133
134
MULTIAXIALFATIGUEAND DEFORMATION
Conclusions Performance of the method developed by Chaboche and Cailletaud [12] is shown. A numerical simulation is carried out of the rotation of an aircraft turbine disk [2,3,18]. Numerical results correspond to the results obtained by Lesne and Savalle [18], although the material data of INCONEL 718 at a temperature of 550~ are not the data applied in the literature mentioned above. Since the data are not given, the data of Benallal and Ben Cheikh [4] are used. These data allow for studying mean-stress relaxation. The Newton algorithm is implemented as a user-subroutine in the commercial finite-element method package ABAQUS. This user-subroutine will be extended. The aim is to take into account anisotropy, anisothermal conditions, and damage as well. The increment size will be controlled by procedures in order to accelerate convergence and decrease processing time.
Acknowledgments The authors would like to thank Jtirgen Olschewski and Rainer Sievert at Bundesanstalt ftir Materialforschung und-prtifung, Berlin, for providing experimental data. These data enabled the verification of the subroutine applied to carry out the numerical simulation.
References [1] Chaboche, J.-L. and Cailletaud, G., "Influence of Material Behaviour on Stress Redistribution in Cyclic Plasticity," Proceedings of the International Conference on Numerical Methods in Engineering: Theory and Applications, N. Pande and J. Midleton, Eds., 1985, pp. 401-409. [2] Dambrine, B. and Mascarell, J. P., "About the Interest of Using Unified Viscoplastic Models in Engine Hot Components Life Prediction," Proceedings of the International Seminar on High Temperature Fracture Mechanisms and Mechanics, P. Bensussan and J. P. Mascarell, Eds., Mechanical Engineering Publications, London, 1990, pp. 1-15. [3] Dhondt, G. and Krhl, M., "The Effect of the Geometry and the Load Level on the Dynamic Fatigue of Rotating Disks," International Journal of Solids and Structures, Vol. 36, No. 6, 1999, pp. 789-812. [4] Benallal, A. and Ben Cheikh, A., "Constitutive Equations for Anisothermal Elasto-Viscoplasticity," Constitutive Laws for Engineering Materials: Theory and Applications, C. S. Desai, et al., Eds., Elsevier Science Publishing Co. Inc., 1987, pp. 667-674. [5] Ohno, N., "Constitutive Modelling of Cyclic Plasticity with Emphasis on Ratchetting," International Journal of Mechanical Science, Vol. 40, Nos. 2-3, Elsevier Science Ltd., 1997, pp. 251-261. [6] Nouailhas, D., "Unified Modelling of Cyclic Viscoplasticity: Application to Austenitic Stainless Steels," International Journal of Plasticity, Vol. 5, Pergamon Press Plc., 1989, pp. 501-520. [7] Wang, J.-D. and Ohno, N., "Two Equivalent Forms of Non-Linear Kinematic Hardening: Application to Non-isothermal Plasticity," International Journal of Plasticity, Vol. 7, Pergamon Press Plc., 1991, pp. 637-650. [8] Chaboche, J.-L., "Cyclic Viscoplastic Constitutive Equations, Part I: A Thermodynamically Consistent Formulation," Journal of Applied Mechanics, Vol. 60, Transactions of the ASME, New York, December 1993, pp. 813-821. [9] Chaboche, J.-L., "Cyclic Viscoplastic Constitutive Equations, Part II: Stored Energy~omparison Between Models and Experiments," Journal of Applied Mechanics, Vol. 60, Transactions of the ASME, New York, December 1993, pp. 822-828. [10] Lemaitre, J. and Chaboche, J.-L., Mechanics of Solid Materials, Cambridge University Press, UK, 1994. [11] Doghri, I., "Fully Implicit Integration and Consistent Tangent Modulus in Elasto-Plasticity," International Journal for Numerical Methods in Engineering, Vol. 36, Wiley & Sons, Ltd., 1993, pp. 3915-3932. [12] Chaboche, J.-L. and Cailletaud, G., "Integration Methods for Complex Plastic Constitutive Equations," Computational Methods in Applied Mechanical Engineering, Vol. 133, Elsevier Science S. A., 1996, pp. 125-155. [13] Ladev~ze, P. and Rougre, P., "Plasticit6 et viscoplasticit6 sous chargement cyclique: proprirtrs et caleul du cycle limite," Comptes Rendus de l'Acad~mie des Sciences, tome 301, srrie II, no. 13, Gauthier-Villars, Paris, 1985, pp. 891-894. [14] Cognard, J.-Y. and Ladev~ze, P., "A Parallel Computer Implementation for Elastoplastic Calculations with the Large Time Increment Method," Non-Linear Engineering Computations, Swansea, 1991, pp. 1-10. [15] Arzt, M., Cognard, J.-Y., and Ladev~ze, P., "A Large Time Increment Strategy for the Analysis of Vis-
ARZT ET AL. ON A NEWTON ALGORITHM
[16] [17] [18] [19] [20]
135
coplastic Structures Under Complex Loading Histories," Proceedings of the International Seminar on Multiaxial Plasticity, A. Benallal, R. BiUardon, and D. Marquis, Eds., Laboratoire de MEcanique et Technologic, Ecole Normale SupErieure de Cachan, France, 1992, pp. 434-460. Chaboche, J.-L., "A Review of Computational Methods for Cyclic Plasticity and Viscoplasticity," Proceedings of the First International Conference on Computational Plasticity, 1987, pp. 379-411. Ladev~ze, P., "La mEthode ~t grand increment de temps pour l'analyse de structures ~ comportement non linEaire dEcrit par variables intermes," Comptes Rendus de l'Acad~mie des Sciences, tome 309, sErie II, Gauthier-Villars, Paris, 1989, pp. 1095-1099. Lesne, M. P. and Savalle, S., "An Efficient Cycles Jump Technique for Viscoplastic Structure Calculations Involving Large Number of Cycles," Proceedings of the 2nd International Conference on Computational Plasticity, D. R. J. Owen, E. Ofiate, and E. Hinton, Eds., Pineridge Press, Ltd., Swansea, 1989, pp. 591-602. Peirce, D., Shih, C. F., and Needelman, A., "A Tangent Modulus Method for Rate Dependent Solids," Computers and Structures, Vol. 18, Pergamon Press, Ltd., 1984, pp. 875-887. Wanhill, R. J. H., "Significance of Dwell Cracking for IN718 Turbine Discs," Proceedings of the 4th International Conference on Low Cycle Fatigue and Elasto-Plastic Behaviour of Materials, K.-T. Rie and P. D. Portella, Eds., Elsevier Science, Ltd., 1998, pp. 801-806.
Fatigue Life Prediction Under Generic Multiaxial Loads
M. de Freitas, 1 B. Li, 1 a n d J. L. T. Santos 1
A Numerical Approach for High-Cycle Fatigue Life Prediction with Multiaxial Loading REFERENCE: de Freitas, M., Li, B., and Santos, J. L. T., "A Numerical Approach for High-Cycle Fatigue Life Prediction with Muitiaxiai Loading," Multiaxial Fatigueand Deformation: Testingand Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 139-156. ABSTRACT: This paper presents an efficient approach for predicting high-cycle fatigue crack initiation life under general multiaxial fatigue loadings. A minimum circumscribed ellipse approach is proposed for evaluating the effective shear stress amplitude and mean value throughout a complex loading cycle. The idea of this approach is to construct a minimum circumscribed ellipse enclosing the loading path in the transformed deviatoric stress space. The new definition of the effective shear stress amplitude is the root mean square of the major semi-axis and the minor semi-axis of the minimum circumscribed ellipse. In this way, the out-of-phase loading effects are taken into account and improvement is made over the previous approaches such as the longest projection, the longest chord, and the minimum circumscribed circle methods. By using mathematical programming techniques, an efficient numerical algorithm is proposed for solving the min-max problem of finding the minimum circumscribed ellipse that can enclose the whole loading path. This new approach allows extension of the Sines or Crossland fatigue criteria to fatigue life prediction under general multiaxial loading with arbitrary stress-time histories. Multiaxial fatigue test results collected from literature, which include complex stress histories with different waveforms, frequencies, out-of-phase angles and mean stresses, were used to validate the approach here proposed. KEYWORDS: multiaxial fatigue, nonproportional loadings, fatigue life prediction, high-cycle fatigue, numerical method Nomenclature Phase angle Material plane Length of the chord joining any two points of the loading path 5-dimensional Euclidean space Fully reversed bending endurance limit Fully reversed bending strength at N cycles Repeated bending endurance limit fo(N) Repeated bending strength at N cycles I Error index measuring the deviation from the predicted to the experimentally determined fatigue strength I The second order unit tensor k Material constant parameter k(N) Material constant parameter at N cycles A Material constant parameter A(N) Material constant parameter at N cycles
A D E5 f-t f - t(N) f0
Professor, research assistant, and associate professor, respectively. Department of Mechanical Engineering, Instituto Superior Tdcnico, Technical University of Lisbon, Av. Rovisco Pais 1, 1049-001 Lisbon, Portugal. 139
Copyright9
by ASTM lntcrnational
www.astm.org
140
MULTIAXIALFATIGUE AND DEFORMATION
N N n
Pn PHmean PHmax t l
t ~(N) Ra
Rb S
Si (Jr
(ro O'ij, a
~rij.m tr((r) T
%, Ta,max
"rm t W* Wi
Finite fatigue life cycles Normal stress vector Unit normal stress vector Hydrostatic stress Mean value of the hydrostatic stress during a loading cycle Maximum value of the hydrostatic stress during a loading cycle Fully reversed torsion limit Fully reversed torsion strength at N cycles Major semi-axis radius of the minimum circumscribed ellipse Minor semi-axis radius of the minimum circumscribed ellipse Transformed deviatoric stress vector Components of the transformed deviatoric stress vector S Stress vector Components of the stress vector ~r Amplitude of the stress component o'ij Mean value of the stress component ~j First invariant of the stress vector ~r Shear stress vector Shear stress amplitude throughout a loading cycle Shear stress amplitude at the critical plane Mean value of shear stress throughout a loading cycle Time instant Center point of the minimum circumscribed ellipse Center coordinates of the minimum circumscribed ellipsecenter point w*
Square root of the amplitude of the second invariant of the stress deviator throughout a loading cycle. Equivalent shear stress amplitude ~p Spherical angle of the material plane orientation 0 Spherical angle of the material plane orientation Trace of the stress vector a(t) Trace of the shear stress vector ~(t) Trace of the second invariant of the stress deviator X/~a Most engineering components/structures in service are subjected to a two- or three-dimensional stress state with irregular stress-time histories. The origin of multiaxiality comes from various factors such as multiaxial external loading, complex geometry of the component/structure, residual stresses, etc. Such multiaxial stresses are often non-proportional, that is, the corresponding principal directions and/or principal stress ratios vary with time. Advanced engineering design methodologies such as computer aided design and structural optimization require efficient, accurate and easy-of-use methods of fatigue life prediction for components/structures under general complex multiaxial fatigue loadings. Current industrial mechanical design is heavily concerned with the high-cycle fatigue resistance of the designed components. A usual procedure in fatigue design starts by computing the local stress-time histories at critical locations by the finite element method. Then, it proceeds by evaluating whether or not these critical locations will undergo the specified finite number of loading cycles without initiating a fatigue crack, by applying an appropriate multiaxial fatigue criterion. The multiaxial fatigue criteria proposed in the literature may be categorized in three groups: stressbased, strain-based and energy-based methods [1,2]. For high-cycle fatigue, most of the fatigue criteria are stress-based [3-13]. Although there are numerous multiaxial fatigue criteria in the literature, design engineers are often faced with difficulties in applying these criteria to modern engineering design. One difficulty is that most of the existing multiaxial fatigue criteria can only provide good pre-
DE FREITAS ET AL. ON HIGH-CYCLE FATIGUE LIFE PREDICTION
141
dictions for proportional (in-phase) loadings, where the principal direction remains fixed throughout the loading history. Another difficulty involves in their implementation for general complex multiaxial fatigue loadings. Among current multiaxial fatigue criteria, the Sines [5] and the Crossland [6] criteria are very popular and easy to use for engineering design. They can provide good predictions for proportional loads with mean stress effects [24]. Unfortunately, they are not applicable for general complex multiaxial fatigue loadings. In this paper, the nonproportional loading effects are studied and a new approach is proposed for evaluating the effective shear stress amplitude and mean value throughout a complex loading cycle. An efficient numerical algorithm is developed for implementing the proposed approach. Then, the Sines and Crossland criteria are extended to fatigue life prediction under general complex multiaxial loadings. Finally, multiaxial fatigue test results collected from the literature, which include complex stress histories with different waveforms, frequencies, out-of-phase angles and mean stresses, are used for validating the developed approach.
Nonproportional Loading Effects Trace o f the Shear Stress Vector on a Material Plane
As shown in Fig. 1, consider a material plane, denoted as A, passing through the point under consideration. The plane A is located by its unit normal vector n. This unit vector in turn is described by its spherical angles (~b, 0). The stress vector o- acting on the plane A is decomposed into two vectors: the normal stress vector N and the shear stress vector r. During a complex cyclic loading, the tip of the stress vector o'(t) describes a closed space curve ~0, see Fig. 2. It is clear that the direction of the normal stress N(0 remains invariant, showing that during a cycle of a complex periodic loading the normal stress vector, N(t), changes in magnitude but not in direction. Therefore, the definition of the amplitude and mean value of the normal stress can be based on its algebraic values, which is a scalar periodic function of time. The semi-difference between the maximum and minimum values that the function achieves during a load cycle provides the amplitude of the normal stress, whereas the semisum yields the mean value of the normal stress.
Z
"''""...
S
"'',
n
../..,7""
. .......y,,....~ ,...,.
~
.....,,..,,............
.. O"
~"N..
0
FIG. 1--Stresses on a material plane A.
Y
142
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 2 - - T h e trace t) o f a stress vector o" (t) and the trace ~ ' o f the shear stress vector r(t) acting on a material plane A , during a complex cyclic loading.
However, the shear stress vector 7It) changes both in magnitude and in direction during a complex loading cycle. So, the definition of the amplitude and mean value of the shear stress is much more complex than that for normal stress. As shown in Fig. 2, the tip of the shear stress vector ~(t) describes a closed plane curve qs', which is the projection on A of the space curve ~ described by the tip of the stress vector oIt). The plane curve ga describes the trace of the shear stress vector r(t) acting on the material plane A as shown in Fig. 2. In the case of proportional (in-phase) loadings, the trace ga will become a rectilinear line. The plane curve qa described by the tip of the shear stress vector ~t) is different on different planes passing through the point under consideration. Therefore, the shear stress amplitude ~'a depends on the orientation of the plane on which it acts, which means that ra is a function of the plane orientation angles ~band 0, i.e. ~-a (qS, 0). To find the maximum shear stress amplitude ~-~..... it is necessary to search the maximum of ~'a (~b, 0) over the plane orientation angles q5 and 0. Trace o f the Shear Stress in the Deviatoric Hyperplane
Another way to evaluate the shear stress during a loading period is to use the deviatoric part o-'(t) 1
of the stress vector o-(t), defined by cr'(t) = o(t) - -~ tr(cr(t))I. The general loading path can be described by the trace of the orthogonal projection of the stress vector o(t) onto the deviatoric hyperplane. Stress invariant based multiaxial fatigue criteria, such as the Sines [5] and Crossland [6] criteria, are expressed by the amplitude of the equivalent shear stress (octahedral shear stress) and the mean (Sines) or maximum (Crossland) value of the hydrostatic stress (octahedral normal stress) throughout a loading cycle. The amplitude of the equivalent shear stress can be expressed as the square root of the second invariant of the deviatoric stresses N/~2,~. Thus, stress invariant-based multiaxial fatigue criteria do not need to search the orientation of the critical material plane. To make it easier to compute the equivalent shear stress amplitude N/~2,~, the following transfor-
DE FREITAS ET AL. ON HIGH-CYCLE FATIGUE LIFE PREDICTION
143
mation rules can be used [3]:
x/g,
S 1 = ~ - - - - Oxx ,
1 t t --~(O'yy - - O-zz),
S2
t S 3 = O'xy ,
t S 4 -~ Orxz,
t S 5 = O'yz
(1)
where o% o-~y,o% O-~y,o% O'yz, are the six components of the deviatoric stress vector o-', and S1, 52, $3, $4, $5, are the five components of the transformed deviatoric stress vector S. With the above transformation, the deviatoric stress vector ~'(t) is mapped onto a vector S(t) in a 5-dimensional Euclidean space Es. In this way, the stress deviator is fully described by a smaller number of components in the transformed space. During a periodic loading, the tip of the vector S(t) describes a closed curve qb' in the transformed deviatoric stress space. Current Approaches for Evaluating the Amplitude and Mean Value of the Shear Stress For a general solution of the problem of evaluating the shear-stress amplitude ~'a and the mean value ~'mduring a complex multiaxial loading cycle, three proposals have been formulated in the literature as described in [3,22]. The earliest method, known as the longest projection method, is illustrated in Fig. 3. This method starts by considering all the lines that lie on the material plane, A, and pass through the origin, O, and proceeds by projecting the loading path curve, a/a, on all these lines. The shear stress amplitude is defined to be equal to half the length of the longest projection of a/a, denoted by ~'al in Fig. 3. The mean
Minimum circumscribed circle Ta2
Longest j WClord
Projection of~' ~
/
/
'
Longes-'~ "" / / ~'/
/
i
/
/
i
!f /t
0 / .t
f
FIG. 3--Definitions of the shear stress amplitude and mean value by current approaches.
144
MULTIAXIALFATIGUE AND DEFORMATION
value of the shear stress is defined to be equal to the length of the segment joining the origin, O, with the midpoint of the longest projection of a/a, denoted as 7ml in Fig. 3. The second method, known as the longest chord method, is also illustrated in Fig. 3. This method considers all the chords joining any two points of the loading path curve ~ and finds the maximum length chord. The amplitude of the shear stress acting on A is defined to be equal to half the length of the longest chord, denoted as Ta2 in Fig. 3. The mean value of the shear stress is defined to be equal to the length of the segment joining the origin O with the midpoint of the longest chord of q~, denoted as ~'m2in Fig. 3. The longest chord method is formulated as:
Ta2 =
1 max/ {maxj II r(tj) - "r(ti)II}
(2)
Where maxj II ~(tj) - ~-(t,) II means to find the longest line segment which can be drawn inside the shear stress trace curve with a time instant t/as a reference point, then repeat this process relative to a different reference time point ti and find the overall longest chord as expressed by max/{ }. The half of the longest chord length is defined to be the shear stress amplitude ~'a2. These two methods have some drawbacks, since they may lead to inconsistent results for some complex loading cases [3,22]. To overcome these drawbacks, a third method was proposed based on the concept of the minimum circumscribed circle by Dang Van and Papadopoulos [13,22]. The idea of the third method is to find the minimum circumscribed circle that can enclose the whole loading path qa inside it. The amplitude of the shear stress is defined to be equal to the radius Ra of the minimum circumscribed circle, denoted as ~'a3in Fig. 3. The mean value of the shear stress is the length of the vector w that points from the origin O to the center of the minimum circumscribed circle, denoted as ~'m3in Fig. 3. The formulation of the minimum circumscribed circle method is illustrated in Fig. 3, The center w* and the radius R = Ra of the minimum circumscribed circle are computed according to [3,22]: w*: rain {max II ~-(t) - w II] W
(3)
t
R = max II r(t) - w*ll
(4)
t
Where maxt II ~-(t) - w II means to find the longest line segment which can be drawn between any point of the shear stress trace curve and any reference point w, then repeat this process relative to different reference point w and find the point w* which minimizes the longest line segment as expressed as minw { }. The radius R of the minimum circumscribed circle is equal to the length of the longest line segment relative to the point w*.
Current Approaches to Account for Nonproportional Loading Effects Consider two loading paths illustrated in Fig. 4. Load path 1 is a closed curve representing a general non-proportional loading case, and load path 2 is a rectilinear path representing a proportional loading case. The current approaches, such as the longest projection, the longest chord, or the minimum circumscribed circle approaches, lead to the same shear stress amplitude for both loading paths, which can not take into account the nonproportional loading effects. That means the current approaches do not make any difference between linear and complex loading paths (Fig. 4), whereas a difference is demonstrated by experiments [29,30]. This explains the larger errors associated with the fatigue damage evaluation under non-proportional loading, made by the current methods. The effects of non-proportional loadings on the fatigue resistance have been a very active research topic during the last two decades [14-21]. Grubisic and Simburger [14] proposed an effective strain-
DE FREITAS ET AL. ON HIGH-CYCLE FATIGUE LIFE PREDICTION
Load Path 1 ~
~
M
(General)
i
n
i
m
u
m
145
circumscribed
circle
//
i'"h
i
--
,------M-- Load Path 2
o FIG. 4~Definition of the shear stress amplitude ~-, and mean value "rmby current approach.
ing approach (root mean square expression) which involves examining all planes in a body, for the most unfavorable combinations of mean and alternating shear stress on each. Liu and Zenner [17] formulated their criterion through a double integral, over the spherical coordinates 0 and r of a material orientation, based on the combination of the amplitudes and mean values of the shear stress and the normal stress acting on a material plane. This criterion requires four material parameters that can be obtained from the results of two sets of experiments. The disadvantage of these integral approaches is the difficulty in implementation for general complex loading situations. Deperrois [18] proposed a criterion that is based on the representation of the loading path qb in the transformed deviatoric stress space Es. Deperrois' criterion appears appealing because it attempts to provide a detailed characterization of the loading path in the deviatoric space. However, this approach leads to inconsistent results for some complex loading cases due to the weakness of the longest chord method used in the approach. Papadopoulos [19] further developed the mesoscopic scale approach and proposed a criterion based on the average measure of the accumulated plastic strain within an elementary volume V. Starting from the study at the scale of the grains of a metal, Papadopoulos arrived at a fatigue criterion that depends upon the usual macroscopic stress fields. For the particular case of synchronous sinusoidal multiaxial loads, Papadopoulos' mesoscopic criterion can be evaluated analytically, and improved prediction results were provided for synchronous sinusoidal out-of-phase loadings. However, for non-synchronous muttiaxial loading, the analytical evaluation is difficult to carry out. More recently, Duprat [20] suggested that the equivalent shear stress amplitude be defined as the half-perimeter instead of the half-longest-chord of the loading path ellipse of synchronous sinusoidal out-of-phase stresses. Morel et al. [21] also proposed a method to account for the openness of the loading path ellipse of synchronous sinusoidal out-of-phase stresses. However, for general non-proportional loading, such as non-sinusoidal or different frequency stress histories, these methods are difficult to apply. In this study, a general and easy-to-use approach is proposed for considering the non-proportional loading effects in evaluating the effective shear stress amplitude under general complex multiaxial loadings with arbitrary stress-time histories. This new approach is presented in the next section.
146
MULTIAXIAL FATIGUE AND DEFORMATION
The Minimum Circumscribed Ellipse Approach A new approach, called the minimum circumscribed ellipse approach, is developed to account for the non-proportional loading effect [23]. The idea is to construct a minimum circumscribed ellipse that can enclose the whole loading path throughout a loading block. The graphical representation of the new method and the relation with the minimum circumscribed circle approach [3,22] is illustrated in Fig. 5. Rather than defining ~-~= Ra by the minimum circumscribed circle method, a new definition of "/'a = ~v/R] + Rbz, is proposed. The important advantage of this new approach is that it can take into account the non-proportional loading effects in an easy way. As shown in Fig. 5, for the general out-ofphase loading path 1, the shear stress amplitude is defined as ~'a = ~ + R~,.For the rectilinear loading path 2 (in-phase-loading),it is defined as % = Ra, since Rb is equal to zero in this in-phase loading case.
Numerical Implementation of the Minimum Circumscribed Ellipse Approach How to find the minimum circumscribed ellipse, which can enclose the whole loading path ~rt,, is a very difficult min-max problem, if solved by usual analytical methods. In this research, a numerical mathematical programming method is efficiently used to solve this problem [23,28]. This problem is stated as follows: Find the center point coordinates, the major semi-axis, Ra, and the minor semi-axis, Rb, of the minimum circumscribed ellipse, which can enclose the whole loading path ~/a of a cyclic loading period. This min-max problem can be solved efficiently by using a sequential linear programming optimizer in conjunction with the simplex method [25]. To facilitate the solution of this problem, it is formulated as a two-step problem in this study. It means that the minimum circumscribed circle is searched in the first step. With the identified circle radius as the major semi-axis, then a search for the minimum minor semi-axis is conducted in the second step. The first step is to find the radius R~ and the center coordinates w* (Wl, w2, w3, w4, ws) of the min-
The minimum Load Path 1 (General)
~
Circle
.i. ~ R
Load Path 2 b
~
(Rectilinear)
/
The Minimum Circumscribed Ellipse o
FIG. 5--The minimum circumscribed circle and ellipse approaches.
DE FREITAS ET AL. ON HIGH-CYCLE FATIGUE LIFE PREDICTION
147
imum circumscribed circle, in the 5-dimensional Euclidean space of the transformed deviatoric stresses $1, $2, $3, $4 and $5. The bound formulation [27] of the min-max problem Eqs 3 and 4 is used, and a sequential linear programming optimizer in conjunction with the simplex method are used for the efficient solution of the problem. The second step is to find the minor semi-axis Rb of the minimum circumscribed ellipse, with the same center coordinates w* (wl, we, w3, w4, ws) and major semi-axis Ra as obtained in the solution of the above first step. The mathematical statement of this problem can be expressed as: Minimize Rb
(5)
Subject to
( Ro]
r~ (ti)12+ ('r'2(ti)12<~ l i = 1,2,
\R~J
.,m
(6)
" "
where r~(ti) and "r~(ti) are the two components of each discretized point of the loading path ~O',in the new coordinate system, with the center w* (wl, w2, w3, w4, ws) of the minimum circle as the origin O', and the major semi-axis R a a s one of the coordinate axes. The algorithm used for determining the minimum minor semi-axis R9 of the ellipse, solution of the Eqs 5 and 6, is based on an efficient bisection searching method.
A Numerical Procedure for Multiaxial Fatigue Life Prediction The numerical approach for evaluating the effective shear stress amplitude throughout a loading cycle makes it possible to extend the Sines and the Crossland multiaxial fatigue criteria for finite fatigue life prediction under general multiaxial loadings. The Sines criterion [5] is formulated as: ~'a + k P~/. . . .
=
A
(7)
The material parameters k and A can be obtained from the results of two sets of experiments, known as the repeated bending limit f0 and the fully reversed torsion limit t_ 1: /3t-i] _ V3
k=\ fo J
A = t-1
(8)
Failure occurs when the left-hand side of Eq 7 is equal or greater than its right-hand side. The Crossland criterion [6] differs from the Sines criterion only on how to account for the influence of the hydrostatic stress. According to Crossland, the maximum hydrostatic stress should be considered rather than the mean hydrostatic stress in the fatigue formula:
"ca + k PHmax = A
(9)
The material parameters k and A can be obtained from the results of two sets of experiments, the fully reversed bending limit, f _ l, and the fully reversed torsion limit, t_ 1 (3t-1~ k=\f_,)
_ "V~
A = t-1
Failure occurs when the left-hand side of Eq 9 is equal or greater than its right-hand side.
(10)
148
MULTIAXIALFATIGUEAND DEFORMATION
If the two S - N curves (the stress amplitude corresponding to failure at N cycles), i.e., the uniaxial repeated bending fatigue strength f o ( N ) at N cycles and the reversed torsion fatigue strength t_ ~(N) at N cycles are available, then the Sines formulation Eq 7 becomes: ra + k(N) PH . . . . .
=
A(N)
(11)
A(N) = t-1 (N)
(12)
where { 3 t - 1 (N) I k(N) = k f o (N) ]
_ V3
Equation 11 is an extension of the Sines fatigue limit formulation to high-cycle fatigue. Using the Newton-Raphson algorithm, the finite fatigue life N can be obtained from the iterative solution of Eq 11. If the two S - N curves, the uniaxial reversed bending fatigue strength f_ I(N) at N cycles and the reversed torsion fatigue strength t_ I(N) at N cycles are available, then the Crossland formulation, Eq 9 becomes: (13)
ra + k(N) PH .... = A(N)
where
(3t_, (N) I - N/~
k(N) = \ ~ ]
A(N) = t-l(N)
(14)
Equation 13 is an extension of the Crossland fatigue limit formulation to high-cycle fatigue. Using the Newton-Raphson algorithm, the fatigue life N can be obtained from the iterative solution of Eq 13. For Eqs 11 and 13, the key step is the evaluation of the equivalent shear stress amplitude ra and hydrostatic stresses PH, throughout a general multiaxial cyclic loading block. After computing the local stress-time histories at the critical locations of the component/structure by the finite element method, the numerical procedure described next can be followed. Compute the Hydrostatic a n d Deviatoric Stresses
Split the stress tensor or(t) into its deviatoric and spherical parts (15)
~(t) = ~'(t) + 1 tr(~r(t))I
where tr(cr(t)) is the first stress invariant given by (16)
tr(~r(t)) = (~rxx(t) + Oyy(t) + crzz(t))
The stress deviator oJ is expressed as: .2~ Od ~
t
- ~ y y - ~= 3 ~yx
2~
-- ~ -- ~= 3 ~Zy
~z 2~rzz- ~ x x - O'yy 3
(17)
DE FREITAS ET AL. ON HIGH-CYCLE FATIGUE LIFE PREDICTION
149
The hydrostatic stress Pl4(t) is calculated as: PH(t) =
1 tr(tr(t)) = "~ [O'xx(t ) + O'yy(t) -b trzz(t)
(18)
For a cyclic loading, PH(t) is a periodic scalar function within the time period. The hydrostatic stress amplitude is computed as: 1 (tr(o'(t)) 3
P H,a = "2 ~ m a x
_ min t~o{t))~
3 ]
(19)
The mean value of the hydrostatic stress is computed as: 1 ( max t~a(t)) + man ~ PH. . . . . = ~3
)
(20)
The maximum value of the hydrostatic stress is equal to: PH, max = PH, . . . . q- PH, a
(21)
Transform the Stress Deviator ~r' to Euclidean Space Using Eq 1, the deviatoric stress vector o"(t) is mapped onto a vector S(t) in a 5-dimensional Euclidean space Es. Computation of the Equivalent Shear Stress Amplitude In the transformed deviatoric stress space, the loading path is a closed curve, ~', in the deviatoric hyperplane. Using a sequential linear programming optimizer in conjunction with the simplex method, the minimum circumscribed ellipse, which can enclose the whole loading path 4 ' of a cyclic loading period, can be found by solving the Eqs 3-6. The major semi-axis radius Ra and the minor semi-axis radius Rb of the minimum circumscribed ellipse are used to calculate the equivalent shear stress amplitude ra = ~ R~ Application Examples and Validation of the Proposed Approach The capability of the proposed approach to predict finite fatigue life under general multiaxial stress histories will be considered for cases where experimental data are available. Fatigue Evaluation under Complex Multiaxial Loading Histories A group of biaxial (O'xxand O'yy) fatigue loading cases, with the stress-time histories during a loading block, as shown in Fig. 6, is analyzed by the approach proposed and implemented in this paper. The loading cases cover a wide range including the out-of-phase effects, different waveforms and frequencies. The capability of the developed approach for handling complex fatigue load cases is shown, and the predicted results are compared with the experimental results by McDiarmid [16], a series of fatigue tests were carried out on thin-wall tubular specimens of EN24T steel. According to the numerical procedure described earlier, the loading stresses are transformed to the deviatoric stress space, where all the loading cases have the same fluctuating stress amplitude, O'x~,a = ~yy,a = 326 MPa. The loading paths, in the transformed deviatoric stress space, are shown in Fig. 7. It may be noticed that the loading path curves are strongly influenced by the out-of-phase angles, waveforms and frequency ratios.
150
MULTIAXIAL FATIGUE AND DEFORMATION
Load Case 1 400 300 200 ~" 100 ~
Load Case 2
_
svl
400 300 ~" 200 100 "~ 0
-200
-300 -400
-300 -40O
Time(s)
Time(s)
oo%
~
-loo
~-100
Load Case 3
Load Case 4
400
A
200
Syy
== -loo
-200
400 -.
20O 100
0
-200
~ -200
-300 -400
-300 -400
Time(s)
~
Time(s)
A
FIG. 6--Biaxial stress histories during a loading block, with different waveform, frequency, and out-of-phase angle.
Table 1 shows the importance of considering the non-proportional loading effects in the definition of the effective shear stress amplitude. The last two rows in Table 1 display the shear stress amplitude values obtained with the definition used by the minimum circumscribed circle approach and the definition proposed in the minimum circumscribed ellipse approach. The new definition can characterize the non-proportional loading effects. Table 2 shows the predicted results, using the Crossland criterion with the new definition of the equivalent shear stress amplitude, compared with the experimental results by McDiarmid [16]. The uniaxial longitudinal fatigue strength reported in [16] are used, since these values are closer to the test data from the solid specimen of the similar material. At 106 cycles, O'Ais the uniaxial longitudinal fatigue strength. For each load case at 106 cycles, ~ta is the allowable maximum principal stress amplitude. The reduction of fatigue strength due to the bi-axial loading effects is represented by ~xJtra. Agreement between predicted and observed fatigue strength is satisfactory, except for load case No. 2, where predicted strength is about 25% higher than actual strength.
Fatigue Evaluation for Specified Finite Fatigue Life The proposed approach will be used to predict the fatigue strength under combined loading for a specified number of cycles. Two series of test results, according to Simburger [26], are considered.
151
DE FREITAS ETAL. ON HIGH-CYCLE FATIGUE LIFE PREDICTION
Load Case 5
Load Case 6
400 . . . . . . . . . . . . . . . . . . 300 ~" 200 100
400 300 200 =. 100 0 == -100 -200 -300 -400 A r
}o
p -lOO
-200 -300 -400 ~. . . . . .
I
-.SS
Time(s)
1
Time(s)
Load Case 7
Load Case 8
400 300 ~" 200
300 200
100
-- '
_,oo
~ -100 -200 -300
-200
-300 -4O0
-400 E
Time(s)
Time(s)
B FIG. 6---(Continued)
The tested material is a heat-treated steel CK45. Specimens are thin-walled cylinders. Two test series were performed: tests with combined tension and internal pressure, and tests with combined tension and torsion. Test data are given as fatigue strength for N = 100 0130 cycles and 50% crack initiation probability, as summarized in Table 3. The last column of Table 3 gives the prediction error index/, which measures the deviation of the experimentally observed results to those predicted by the developed approach. In this example, the Crossland criterion with the new definition of equivalent shear stress amplitude, formulation Eq 13, was used. The fatigue data of material CK45 at N = 1 x 105 cycles are given as: f - 1 = 423 MPa, t_ 1 = 287 MPa. The error index I is expressed as the relative difference between the left and right hand sides of Eq 13:
I -
left hand side - right hand side right hand side (%)
(22)
If the error index I is equal to zero, it means that the prediction agrees exactly with experimental observation. Negative or positive value of I means the prediction is underestimated or overestimated of the fatigue damage, respectively. Table 3 shows reasonably good correlation results for the multiaxial fatigue strength evaluation at N = 1 • 105 cycles.
152
MULTIAXIAL FATIGUE A N D D E F O R M A T I O N
Load case 2
LoadCase 1
jo-,ooI
200
l
200
O
i~
0
t
y
-100
-200 -100
-100 -50
0
50
100
-200 -300-200-100
Sl(MPa)
Load case 3
100 200 300
Load Case 4
20O A
0
SI(MPa)
200
100
-100 -200 -300-200-100
0
100 200 300
-300
-100
100
300
St(MPa)
SI(MPa)
! Load Case 6
Load Case 5 200 T
i -200 ~ -300-200-100
0
- ~100 200 300
SI(MPa)
Load Case 7
-300-200-100
0
100 200 300
SI(MPa)
Load Case 8
20O
A 210 i | ": : r =-~k~--
-300-200-100
0
100 200 300
Sl(MPa)
-300-200-100 0
100 200 300
St(MPa)
FIG. 7--Loading paths in the transformed deviatoric stress space.
153
DE FREITAS ET AL. ON HIGH-CYCLE FATIGUE LIFE PREDICTION
T A B L E l--Comparison of the results obtained with the minimum circumscribed circle and ellipse approaches. Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Case 8
188.2 0 188.2 188.2
325.9 0 325.9 325.9
286.5 164.7 286.5 330.5
256.5 233.1 256.5 346.6
326.0 148.0 326.0 358.0
306.2 177.0 306.2 353.6
188.0 187.0 188.0 265.4
326.0 188.0 326.0 376.5
Major semi-axis R~ Minor semi-axis Rb ~'a (Old definition), MPa za (New definition), MPa
T A B L E 2--Comparison between numerical predictions and experimental results [16]. O'la/ O"A
Load Case
Predicted
Experimental
1 2 3 4 5 6
0.81 0.79 0.59 0.57 0.58 0.56
0.82 0.63 0.61 0.63 0.57 0.63
T A B L E 3--Accuracy of multiaxial fatigue strength evaluations. Stresses, MPa O'xxa
O'xxm
215 234 300 275 183 250 300 336 308 300 327 250 288 292 285 304 400 277 0
-215 0 -300 -275 183 250 0 0 308 300 0 250 0 0 0 0 0 277 417
NOTE: o's(t)
=
236 256 330 302 367 275 330 368 339 330 0 0 0 0 0 0 0 0 0 O ' x x m -}- o ' a x a
236 256 330 302 367 275 330 368 339 330 0 0 0 0 0 0 0 0 0 sin(tot).
O'yy(t) = O'yym ~- O'yya sin(cot + 6y ). o's(t) = O'xym+ O'xyasin(tot + 6xy ).
Phase Angle, deg
0 0 0 0 0 0 0 0 0 0 188 144 165 167 163 174 0 159 241
0 0 0 0 0 0 0 0 0 0 0 0 165 0 163 0 200 0 0
180 180 0 90 0 180 90 0 0 90 ... ... ... ... ... ... . . . . . . ... . . . . . .
... ... ... ... ... ... ... ... ... ... 0 90 90 60 0 90 0
Validity (%)
-19.5 -4.5 -13.2 -2.2 3.4 10.8 17.1 9.2 10.9 25.7 4.4 - 11.5 -8.4 -6.8 -9.3 -3.3 -4.6 -1.9 -0.5
154
MULTiAXIALFATIGUEAND DEFORMATION 3,0E+05
,
,"
. Test65A 9 Test65B
/" / s s
.-"
"6 2,0E+05
.-"
~, Test66A A
9 Test66B
.-"
o
+ Test67A 9 Test67B -Test68A
/ j"
.""
~K
9 13/ J IV'~<
1,0E+05 I--
/" 9
[] Test68B O Test69A
0
oTest69B
~6 9 0,0E+00 ." 0,0E+00
i
1,0E+05
9 Test70A /~ Test70B
i
2,0E+05
3,0E+05
X Test71A
Predicted life, cycles
• Test71B
FIG. 8--Comparison between predicted results and experimental observations.
Finite Fatigue Life Prediction A c o m p a r i s o n o f the c o m b i n e d t e n s i o n / t o r s i o n t e s t r e s u l t s , as g e n e r a t e d w i t h the h e a t - t r e a t e d s t e e l C K 4 5 s p e c i m e n s b y S i m b u r g e r [26], a n d p r e d i c t i o n s b y the p r o p o s e d m e t h o d , u s i n g E q 13, are p r e s e n t e d i n F i g . 8, w h e r e the t e s t c o d e s r e p r e s e n t t h e t e s t c o n d i t i o n s o f s t r e s s r a t i o R s t r e s s a m p l i t u d e r a t i o s = ~rxy,a/~rx,a, a n d o u t - o f - p h a s e a n g l e 6 as l i s t e d i n T a b l e 4.
=
O'ij,min/O'ij,max,
I n Fig. 8, the p o i n t s are l o c a t e d i n t h e v i c i n i t y o f t h e 4 5 ~ line. T h e m u l t i a x i a l c a s e s are p r e d i c t e d reasonably well.
T A B L E 4--Test conditions for bi-axial loading cases [26].
Test 65A Test 65B Test 66A Test 66B Test 67A Test 67B Test 68A Test 68B Test 69A Test 69B Test 70A Test 70B Test 71A Test 71B
R+,~
R~y
-
-
-
l 1 1 1 1 1 0 0 0 0 1 1 1 1
NOTE: try(t) = trxxm + O r x x a sin(tot). tr~y(t) = trxy,. + tr~y~sin(o~t + ~).
1 1 1 1 1 1 1 1 1 1 0 0 0 0
o'~x,a (MPa)
O'xy, alO'xx, a
6 (deg)
344 314 344 314 344 314 304 275 304 275 304 275 304 275
0.575 0.575 0.575 0.575 0.575 0.575 0.575 0.575 0.575 0.575 0.575 0.575 0.575 0.575
0 0 60 60 90 90 0 0 90 90 0 0 90 90
DE FREITAS ET AL. ON HIGH-CYCLE FATIGUE LIFE PREDICTION
155
Conclusions The minimum circumscribed ellipse approach provides an efficient and easy-to-use approach to fully characterize the non-proportional loading effects. With this new approach for evaluating the effective shear stress amplitude, the Sines and Crossland multiaxial fatigue criteria can be extended for finite fatigue life prediction. Multiaxial fatigue test results collected from the literature, including complex stress histories with different waveforms, frequencies, out-of-phase angles and mean stresses, are used for validating the developed approach in this paper. The correlation is satisfactory. The numerical algorithm for computation of the major and minor ellipse semi-axes, required to evaluate the shear stress amplitude and mean value, is general and efficient. It provides a unified approach for fatigue design that is particularly suitable for integration with computer aided design.
References [1] Garud, Y. S., "Multiaxial Fatigue: A Survey of the State of the Art," Journal of Testing and Evaluation, JTEVA, Vol. 9, No. 3, 1981, pp. 165-178. [2] Lagoda, T. and Macha, E., "A Review of High-cycle Fatigue Models under Non-proportional Loading," Proceedings of Fracture from Defects, ECF12, Sheffield, September 14-18, 1998, pp. 73-79. [3] Papadopoulos, I. V. "A Review of Multiaxial Fatigue Limit Criteria," Advanced Course on High-Cycle Metal Fatigue, International Center of Mechanical Sciences, Udine, Italy, 1997. [4] Gough, H. J., Pollard, H. V. and Clenshaw, W. J., "Some Experiments on the Resistance of Metals to Fatigue under Combined Stress," Memo 2522, Aeronautical Research Council, HMSO, London, 1951. [5] Sines, G., "Behavior of Metals under Complex Static and Alternating Stresses," Metal Fatigue, by G. Sines and J. L. Waisman, Eds. McGraw Hill, New York, 1959, pp. 145-169. [6] Crossland, B., "Effect of Large Hydrostatic Pressures on the Torsional Fatigue Strength of an Alloy Steel," Proceedings of the International Conference on Fatigue of Metals, Institution of Mechanical Engineers, London, 1956, pp. 138-149. [7] Stulen, F. B. and Cummings, H. N., "A Failure Criterion for Multiaxial Fatigue Stresses," Proceedings, ASTM, Vol. 54, 1954, pp. 822-835. [8] Findley, W. N., "A Theory for the Effect of Mean Stress on Fatigue of Metals under Combined Torsion and Axial Load or Bending," Journal of Engineering for Industry, 1959, pp. 301-306. [9] Kakuno, H. and Kawada, Y., "A New Criterion of Fatigue Strength of a Round Bar Subjected to Combined Static and Repeated Bending and Torsion," Fatigue and Fracture of Engineering Materials and Structures, No. 2, 1979, pp. 229-236. [10] McDiarmid, D. L., "A General Criterion for High Cycle Multiaxial Fatigue Failure," Fatigue and Fracture of Engineering Materials and Structures, Vol. 14, No. 4, 1991, pp. 429-453. [11] Kenmeugne, B., Weber, B., Carmet, A. and Robert, J. L., "A Stress-Based Approach for Fatigue Assessment under Multiaxial Variable Amplitude Loading," 5th International Conference on Biaxial/Multiaxial Fatigue and Fracture, Cracow '97, Poland, 1997, pp. 557-573. [12] Dang Van, K. and Papadopoulos, Y. V., "Multiaxial Fatigue Failure Criterion: A New Approach," Proceedings of the Third International Conference on Fatigue and Fatigue Thresholds, Fatigue 87, University of Virginia, Charlottesville, Virginia, 1987, pp. 997-1008. [13] Dang Van, K., Griveau, B. and Message, O., "On a New Multiaxial Fatigue Limit Criterion: Theory and Application," Biaxial and Multiaxial Fatigue, EGF 3 M. W. Brown and K. J. Miller, Eds., Mechanical Engineering Publications, London, 1989, pp. 479-496. [14] Grubisic, V. and Simburger, A., "Fatigue under Combined Out-of-phase Multiaxial Stresses," Proceedings of the International Conference on Fatigue Testing and Design, Society of Environmental Engineers, London, 1976, pp. 27.1-27.8. [15] Lee, S. B., "A Criterion for Fully Reversed Out-of-Phase Torsion and Bending," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 553-568. [16] McDiarmid, D. L., "The Effect of Mean Stress on Biaxial Fatigue Where the Stresses are Out-of-phase and at Different Frequencies," Biaxial and Multiaxial Fatigue, EGF3, M. W. Brown and K. J. Miller, Eds., Mechanical Engineering Publications, London, 1989, pp. 557-573. [17] Liu, J. and Zenner, H., "Berechnung der Dauerschwingfestigkeit bei Mehrachsinger Beanspruchung," Mat.-wiss. u. Werkstofftech, Vol. 24, 1993, pp. 240-249. [18] Ballard, P., Dang Van, K., Deperrois, A. and Papadopoulos, Y. V., "High Cycle Fatigue and A Finite Element Analysis," Fatigue and Fracture of Engineering Materials and Structures, Vol. 18, No. 3, 1995, pp. 397-411.
156
MULTIAXlALFATIGUE AND DEFORMATION
[19] Papadopoulos, I. V., "Exploring the High-cycle Fatigue Behavior of Metals from the Mesoscopic Scale," Advanced Course on High Cycle Metal Fatigue, International Center of Mechanical Sciences, 1997, Udine, Italy. [20] Duprat, D., "A Model to Predict Fatigue Life of Aeronautical Structures with Out-of-Phase Multiaxial Stress Condition," 5th International Conference on Biaxial/Multiaxial Fatigue and Fracture, Cracow '97, Poland, 1997, pp. 111-123. [21] Morel, F., Ranganathan, N., Petit, J. and Bignonnet, A., "A Mesoscopic Approach for Fatigue Life Prediction under Multiaxial Loading," 5th International Conference on Biaxial,,3,1uhiaxial Fatigue and Fracture, Cracow '97, Poland, 1997, pp. 155-171. [22] Papadopoulos, I. V., "Critical Plane Approaches in High-cycle Fatigue: On the Definition of the Amplitude and Mean Value of the Shear Stress Acting on the Critical Plane," Fatigue & Fracture of Engineering Materials & Structures, Vol. 21, 1998, pp. 269-285. [23] Li, B., "Numerical Optimization of Structural Fatigue Design under Multiaxial Loading," Ph.D. thesis, Department of Mechanical Engineering, Instituto Superior Tdcnico, Technical University of Lisbon, Lisbon, Portugal, 1999. [24] Fuchs, H. O. and Stephens, R. I., Metal Fatigue in Engineering, Addison-Wesley, 1980. [25] Arora, J. S., Introduction to Optimum Design, McGraw-Hill, New York, 1989. [26] Simbtirger, A., "Festigkeitsverhalten z~iher Werstoffe bei einer Mehrachsiger Phasenverschobenen Schwingbeanspruchung mit korperfesten und veranderlichen Hauptspannungsrichtungen," LBF Darmstadt, Bericht, Nr-FB-121, 1975. [27] Olhoff, N., "Multicriterion Structural Optimization via Bound Formulation and Mathematical Programming," Structural Optimization, Vol. 1, 1989, pp. 11-17. [28] Li, B., Santos, J. L. T. and Freitas, M., "A Unified Numerical Approach for Multiaxial Fatigue Limit Evaluation," International Journal of Mechanics of Structures and Machines, accepted. [29] Liu, J., "Weakest Link Theory and Multiaxial Criteria," Proceedings of the 5th International Conference on Biaxial/Multiaxial Fatigue and Fracture, Cracow, Poland, 1997, pp. 45-62. [30] Heidenreich, R., Zenner H., and Richter I., "Dauerschwingfestigkeit bei mehrachsiger Beanspruchung," Forschungshefte FKM, Heft 105, 1983.
K u r t POtter, s F a r h a d Yousefi, 1 a n d H a r o l d Z e n n e r I
Experiences with Lifetime Prediction Under Multiaxial Random Loading REFERENCE: P6tter, K., Yousefi, F., and Zenner, H., "Experiences with Lifetime Prediction Under Multiaxial Random Loading," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 157-172.
ABSTRACT: A number of multiaxial fatigue criteria have been developed during the past decades. They can be subdivided into hypotheses of the critical plane approach, hypothesis of integral approach, as well as empirical criteria. In the case of the integral approach the equivalent stress is calculated as an integral of the stresses over all intersection planes, whereas in the case of the critical plane approach only the intersection plane with the critical stress combination is considered. The difference between the two approaches is insignificant with proportional in-phase loading, but becomes clearly visible in the case of noncorrelated or phase-shifted loading. In the present paper the two approaches are compared. For the model verification, variable-amplitude tests under combined bending and torsion with smooth and notched specimens are applied. The integral approach was found to be superior to the critical plane approach under varying principal stress directions.
KEYWORDS: multiaxial fatigue criteria, influencing parameters, random load, phase difference, superimposed mean stress, test results Nomenclature
As, Z
Elongation at fracture, area reduction at fracture
CA C~- Normal- and shear-stress coefficient for equivalent stress calculation D Damage sum according to Miner's Rule 6 Phase difference between normal and shear stress H
K, k L R era
O'~oa o's
O'eq,~oa "ra
'T~a 7S
Magnitude of load spectrum Direction of intersecting plane Stress concentration factor for notched specimens Exponent (slope) of S-N curve Calculated lifetime Stress ratio, R = trmiJ~rm~x Normal stress amplitude Normal stress amplitude of intersection plane Fatigue strength at normal stress Equivalent stress amplitude of intersection plane Shear stress amplitude Shear stress amplitude of intersection plane q~ Fatigue strength at shear stress
1 Institute for Plant Engineering and Fatigue Analysis (1MAB), Technical University of Clausthal, Leibnizstral3e 32, 38678 Clausthal-Zellerfeld, Germany. 157
Copyright9
by ASTM lntcrnational
www.astm.org
158
MULTIAXIAL FATIGUE AND DEFORMATION
Subscripts a, m Amplitude, mean value B, T Bending, torsion N = 106 Corresponding to 106 load cycles Many engineering structures are subjected to multiaxial fatigue loading. Often the multiaxial stress state is of a very complex nature with mutually independent stress components of changing frequency and direction. In general, the classical multiaxial criteria, such as the von Mises criterion or the maximum shear stress criterion, are not applicable in the case of complex variable multiaxial fatigue due to the fact that they are suitable only for proportional loading. Moreover, for more complex situations, with changing ratios between the principal stresses as well as changing principal stress directions during one load cycle, the classical criteria are inadequate. For assessing the fatigue behavior of components under multiaxial loading a number of strength criteria have been developed during the past decades, Most of the concepts have been reviewed, for example, by Garud [1] or have been discussed concerning their applicability for special practical problems [2-4]. Nevertheless, a comparative assessment of different concepts is hardly possible, since different parameters were used for different applications. The concepts are stress-based [5-8], or strain-based [9-11], developed for the determination of the multiaxial fatigue limit, lifetime prediction under multiaxial random loading, and for low-cycle fatigue problems. Some concepts are generated from the extension of classical static multiaxial criteria; other approaches use a correlation between plastic work and fatigue [12]. But although many investigations on multiaxial fatigue have been carried out, for example Refs 13 and 14, there is still a lack of test results for all influencingload cases, which leads to an inadequate verification of the concepts. The multiaxial fatigue assessments are conducted as a post-processing of finite-element analyses [15]. Hence, it is important to know whether the concepts differ with special stress states and how they influence the calculated lifetime. In the present paper, two hypotheses are compared, namely, the critical plane criteria and the criteria of integral exertion. These hypotheses have been developed for fatigue lifetime assessments under variable amplitude loading in the high-cycle regime and were based on nominal or local elastic stresses. Multiaxial Criteria Multiaxial fatigue criteria must comply with two basic requirements. The hypotheses should consider the physical damaging process, which is difficult to assess due to the fact that we do not know how multiaxial loading affects the basic mechanisms of fatigue. Furthermore, a formal requirement of invariance has to be satisfied because of the temporal variation of the stress tensor during complex multiaxial loading. As a result, the application of the classical strength hypotheses, like the maximum shear stress criterion or the distortion energy criterion, is limited to situations in which the direction of the principal stresses do not vary during a load cycle, e.g., proportional loading. Generally, two types of fatigue strength hypotheses satisfy the postulation of invariance. They can be characterized as hypotheses of the critical intersection plane and hypotheses of an integral exertion [7]. In the case of the critical plane approach, failure is the result of the damage summation in the "most damaging plane." The integral approach, on the other hand, gathers all damaged planes of a specific critical volume. The applicability of both approaches will be discussed subsequently. The following algorithms are used.
Application of Critical Plane Approach and Integral Approach Based upon the real-time load sequence, a transformation of the stress state into all intersection planes ~pof an infinite volume at the highest stressed point is executed. After that, an equivalent stress
POTTER ET AL. ON LIFETIME PREDICTION
159
amplitude treq,~ can be determined, for example, by superimposing the plane stress components r ~ and o-~ [6] O'eq,q~a = Cr'Tr&wa + r
(1)
In this case, the empirical constants c, and c,, depend on the fatigue resistance for bending and torsion, respectively [16]. By utilizing the subsequently described damage accumulation rules of the critical plane or integral exertion, the coefficients are determined by compliance with the boundary conditions of uniaxial constant amplitude fatigue life. Therefore, the coefficients depend on the strength hypotheses used. Alternative definitions of the "equivalent stresses," for example, the normal stress hypotheses, can be applied. To determine the plane damage sum D~, a linear damage accumulation for each intersection plane follows. In the case of the critical plane approach, the intersection plane with the maximum damage sum has to be evaluated. While with the integral approach, all accumulated plane damages are assumed to be lifetime decisive D = D,p,rnax
Critical Plane
(2) D =~
D~dq~
Integral Exertion
A distribution of the plane damage sum D~ versus the direction of the intersecting plane q~in the case of proportional loading is shown in Fig. 1. Besides the linear integration of the plane damage sums, also a calculation of the root mean square can be proposed.
FIG. l--Distribution of plane damage sum D~ versus the intersecting plane ~o.
160
MULTIAXIALFATIGUE AND DEFORMATION
According to the Miner Rule, the calculated lifetime L will be estimated by the summation of the damages per load cycle. Whereas the definition of load cycles at multiaxial stress time functions is very difficult to describe and has not been developed satisfactorily. 1
L = ~.H
(4)
Subsequently the critical plane approach (CP) and the integral approach (Integral Multiaxial Damage Hypothesis (IMDH) [16]), using the above given equivalent stress, Eq 1, are considered. In addition, a critical plane approach based on the normal stress hypothesis (NH) and an integral approach utilizing the root mean square of the plane damages (IQDH), are developed.
Comparison of Critical Plane and Integral Approach In the following paragraph a comparison of the calculated fatigue lives under constant amplitude loading will be performed with no respect to the experiment. For this purpose, two S-N curves are defined for bending and torsion, respectively.
The calculation is based on fictitious S-N curves. This gives the opportunity for a parameter study of the influencing factors like the type of multiaxiality as well as the influence of the uniaxial fatigue properties. Basically, both hypotheses have to comply with the boundary conditions of uniaxial loading, pure bending, and pure torsion. As mentioned before, classical strength hypotheses are applicable at proportional loading. Based on the equivalent stress hypothesis, Gough et al. [17] proposed the formulation of an ellipse arc and an ellipse quadrant in a ~-a/O'adiagram for brittle and ductile materials, respectively. For adoption of the uniaxial fatigue limit the formulation utilizes the fatigue strength ratio (~-s/o's), with the shear stress fatigue limit ~'s (torsion) and the normal stress fatigue limit ~rs using test results for tensioncompression or bending. This proposal is known to be in good agreement with experimental results. The equations can also be adopted for assessing finite fatigue life. In this case the uniaxial fatigue limits are replaced by the endurable stress amplitudes for a given lifetime, i.e., o'a,u-105 and ~'a.U-105. Assuming two fictitious S-N curves for bending and torsion, Fig. 2, two proposals are made. In the diagram the torsion shear stress amplitude % is plotted versus the normal stress amplitude o'a for bending. The two lines characterize combinations of ~-aand o'~ causing the same calculated lifetime of L = 105 cycles under constant-amplitude loading. Additionally, lifetime calculations with the CP and the IMDH have been performed for different ra/O'a ratios and plotted in Fig. 2. It can be concluded that both hypotheses comply with the uniaxial boundary conditions and that their difference is basically insignificant at proportional loading. Furthermore, the calculated results are in good agreement with the formulation of Gough et al. [17]. While calculated lifetimes with critical plane and integral approaches are comparable in the case of proportional loading, their difference becomes clearly visible in the case of phase shifted bending and torsion. This is due to the fact that with a phase difference of 90 ~ between bending and torsion, the principal stress direction rotates during one load cycle and therefore causes damage in all intersecting planes, Fig. 3. As a result, the critical plane approach predicted a higher lifetime with rising phase difference, whereas only a slight extension was found with the integral approach, Fig. 4. The lifetime ratio was highest at a phase difference of 90 ~ see Fig. 4. The effect of the phase difference on the lifetime estimation depends on additional parameters. Figure 5, for example, indicates the maximum lifetime extension that can be found with an amplitude ratio of 'Ta/O" a 0.5 between bending and torsion. In addition, the fatigue strength ratio (~-y/o-s)exhibits =
POTTERETAL. ONLIFETIMEPREDICTION 500
I
I
500
600
161
400
-~ 300 ,-&
IMDH
9 ellipse quadrant ~
-
-Gough et il !
~"
--S-N curves:
200
bending
torsion
Of = 400 MPa xf = 230 MPa --slope k = -5 slope k = -5 point of deflection of S-N curve: cy?les, i '
100
,Nf? !06
,Nf? il06cycl.e~ . . . .
0 0
100
200
300
400
700
normal stress amplitude o a [MPa] FIG. 2--Calculated finite life (N = 105 cycles) amplitude for combined bending and torsion.
a clear influence on the predicted life, as shown in Fig. 6. The fatigue strength ratio usually indicates the type of failure of materials (brittle or ductile). Besides the phase shifted loading, superimposed mean stresses can also cause a temporal variation of the principal stress direction. Figures 7 and 8 show the influence of superimposed tension and shear stresses, respectively. A comparison of integral and critical plane approaches indicates no remarkable discrepancy between the approaches. Even with uniaxial alternating tension-compression loading a case of multiaxial fatigue may appear, if for example, static shear stresses are superimposed. Figures 9 and 10 show the influence of static shear and static tension stresses of the fatigue life prediction un-
D-
q
"~xy
lo-x
cyx
= Cxa sin(0~t)
~xy
= Xxya sin(~
"~xya = 0 . 5 . O x a ~xy
tension- and shear-stress
~ 0 1l~[/ /V',c~xy O? - 1
~
=90~
principal stress
(Yl 1 ~ C~20 ~ ~ / - 1
/
principal stress direction
*
90o~~~ 0
90~
nCOt
"~
FIG. 3--Stress state with revolving principal stress direction.
162
MULTIAXIAL FATIGUE AND DEFORMATION
r
P
NH lifetime ratio
N =o
/
/
stress ratio: R = -1 s l o p e S-N curve: k = -5. "~a/r = 0,5
N5
Critical Plane
j/j
"~f/Of= 43
Integral Exertion
IQDH--
1:
0
i
0o
15 ~
30 ~
45 ~
60 ~
75 ~
90 ~
phase difference 8 [degree] between bending and torsion FIG.
4--Influence of phase difference on calculated lifetime under CA combined bending and tor-
sion.
der uniaxial alternating tension and torsion, respectively. With an increase in shear mean stress, the predicted life decreased significantly in the case of the critical plane approach, while this effect with the integral approach was less pronounced. Fig. 9. With superimposed tension stress there was a minor difference between both the hypotheses, Fig. 10. The integral approach even leads to a greater lifetime reduction.
r
lifetime ratio N8
4
I'
I
stress ratio: R = -1 - - s l o p e S-N curve: k = -5 9f/of = 43 __phase shift 8 = 90 ~ m
'
I
. ~ 8 . , = 90 ~
'/~'
.:
I
3
/f
NS=0
IMDII 0
i
0
i
i
i
0.2
i
0.4
*
i
0.6
,
0.8
amplitude ratio "ca/cya FIG.
5--Normalized calculated lifetimes at different amplitude ratios with 90 ~phase shift.
Ps
ET AL. ON LIFETIME PREDICTION
i
stress ratio: R = -I slope S-N curve: k =
ii -5
.
,,
.90o ,,
V.. V.r
%1~a = 0.5 o __~ise shift 8 = 90
lifetime ratio
163
N8
~CP~ - - - - ' - - - - "IMDH 0
,
,
i
0.5
,
,
i
0.6
I
,
,
0.7
J
J
i
i
.
.
.
0.8
.
,
,
,
0.9
1.0
fatigue strength ratio xf/Of FIG. 6---Normalized calculated lifetimes with different fatigue strength ratios and 90 ~phase shift.
Comparison with Experimental Results The aim of several research projects performed at the Institute of Fatigue Analysis at the Technical University of Clausthal was to examine the influence of load sequences and different specimen shapes on the multiaxial fatigue process [I&19]. More details about the experimental setup and procedure are given in Ref 20. The tests were performed on a machine capable of applying bending and
1.2
,
1
1 ) ~ a ~.-"Ca
1.0 ' k ~ lifetime ratio
N,=O
I'
: i~)' t,.,J
0.8
t
00164
".~J ~m t,,)-
t
slope S-N curve: k = -5 "ITf/Of = •3
~rn = Oa
Ra=0
0.2 0 0
0.2
0.4
0.6
0.8
1.0
amplitude ratio % / o a FIG. 7--Calculated lifetime under combined bending~torsion with superimposed tension stress.
164
MULTIAXlALFATIGUE AND DEFORMATION
1.2
'
''1
''
~ [ d a [ ~ .-'Ca "r - t - - - t c : / - ~ - 7 --Sin
1.01
U
lifetime ratio
d--
I
N,r
0.4
--
I
slope S-N curve: k = -5"r = "43 Om = 0 P~ = -1 m='~a R, = 0
0.6 N~=0
r
C P ~
0.2 0
. . . . .
0
0.2
0.4
0.6
0.8
1.0
amplitude ratio x a/~a FIG. 8--Calculated lifetime under combined bending/torsion with superimposed shear stress.
torsion simultaneously. The load sequences either had a constant amplitude or were standardized variable amplitude load time histories, Fig. 11. The smooth cylindrical specimen of 30 CrNiMo 8 with a notch factor Kt = 1 and the notched specimen of 42 CrMo 4 with a notch factor for bending, KtB = 2.0, and torsion, Ktr = 1.6, are shown in Fig. 12. The static material properties are given in Table 1. The S-N curves of the constant-amplitude fatigue tests are presented in Table 2, with the use
1.2 1.0 lifetime ratio
~
IMDH
~.,._._.,..__
0.8 CP 0.6
N,~m
N-~m=0
0.4 --slope - 5 ,=S-N r"43 curve: k = 0.2
t~f~a- { ~
"~m lira
--Om:0 ,,
0
0.2
I
0.4
0.6
-
_
I,,
0.8
1.0
amplitude ratio "~m/r~a
FIG. 9--Calculated lifetime under fully reversed bending with superimposed static shear stress.
POTTER ET AL. ON LIFETIME PREDICTION
165
1.2
lifetime ratio
1.0
CP
0.8
IMDH
0.6 No-m
N•rn=0
0.4
_ s l o p e S-N curve: k = - 5 _
~fmf = q3 0.2
z
m=0
R x = -1
....
0 0.5
0
1.0
1.5
L .... 2.0
L .... 2.5
3.0
amplitude ratio cym / % FIG. l O---Calculated lifetime under fully reversed torsion with superimposed static tension stress. 1.0 normalised loadamplitude Sa m
Sa
0.5
x = 5.25652 - ( S a l S a ) 5.25652
- Crestfactor
Ho = 106 - block sequence size a
- m a x i m u m load I
100
I
I
I
103
106
cumulative frequency H section o f load-time history
time t lrregularityfactor I= 0.99 (narrow b a n d )
FIG. 11--Standardized random load-time history.
166
MULTIAXIAL FATIGUE AND DEFORMATION
Q
_
4o
_]
-
-'
180 I
t
f r /70
FIG. 1 2 - - S m o o t h and notched specimens f o r combined bending and torsion fatigue tests.
TABLE 1--Material tensile properties.
30 CrNiMo 8 42 CrMo 4
Ultimate Strength Rm, MPa
Yield Strength Re 02 MPa
A5 %
Z %
1014 920
912 743
7.6 21.0
68 69
TABLE 2--S-N curves used for lifetime calculation.
Load Case
Fatigue Limit oaf, MPa
Point of Deflection,
Specimen
Slope, k
Kt = 1.0
bending torsion bending torsion
550 370 330 225
3.0" 105 2.2-106 2.5' 105 2.0. 106
-8.0 -24.6 -4.4 -10.8
K , = 1.0 Kt8 = 2.0 KtT = 1.6
POI-I'ER ET AL. ON LIFETIME PREDICTION
167
TABLE 3--Test results with smooth specimens under variable amplitude. Test Series
~r~.N= 10 6, MPa
Proportional load, %/tr~ = 0.5 Phase difference 90~ za/o'a = 0.5 Noncorrelated, ~',JO'a= 0.5 Pure alternating bending, superimposed ~'m= 500 [MPa] Pure alternating torsion, superimposed tr,, = 550 [MPa]
799 747 837 996 = 663
Slope, k -9.5 -15.5 -8.7 -11.6 -18.1
of Eq 5 utilizing the fatigue limit ~rs and the point of deflection N s as a reference point on the S - N curve. The S - N curves for variable-amplitude loading of the smooth and notched specimens are given in Table 3 and 4, respectively. For this, the normal stress amplitude o-a, representing a fatigue life of N = 106 load cycles, was taken as a reference point. In the case of notched specimens crack initiation was detected by direct current potential drop measurements. The detectable crack length was about 1 mm. Tests with smooth specimens under proportional and phase-shifted bending and torsion have been performed with fully reversed loading and with an amplitude ratio ~'a/O'a 0.5. AS a result the phase difference of 90 ~ causes a decreased lifetime compared with proportional loading, Fig. 13. The test results are plotted as number of load cycles to failure versus the maximum value of load spectra. Additional tests were carried out with load sequences for bending and torsion without any correlation of phase, amplitude, or frequency. The maximum values of the load spectra for bending and torsion have a ratio of Za/O'a = 0.5, but do not occur at the same time. This "noncorrelated" loading causes a slight increase in fatigue life compared to proportional loading, Fig. 14. For fatigue life prediction the critical plane approach (CP) and the integral approach (IMDH) were used. The damage accumulation, using Miner's linear damage summation method [21] was based upon experimentally determined S - N curves for constant amplitude uniaxial bending and torsion conditions, Table 2. A damage sum o f D = 1 was considered to be the failure criterion [21]. Besides the experimentally determined fatigue life, the predicted results are given in Figs. 13 and 14. For proportional loading the predicted fatigue life by critical plane and integral approaches are similar. Compared to the test results, the calculation tends to be nonconservative. To examine the total prediction error, a distinction has to be made between the error caused by the damage accumulation (Miner's Rule) and the strength hypothesis. While it is well known that the Miner Rule tends to overestimate the fatigue life under variable amplitude loading [22], a nonconservative prediction is not unusual. As a result, to compare both strength hypotheses with respect to their accuracy, their ability to predict the tendency of the test results has to be regarded. In the case of 90 ~ phase difference the critical plane approach indicates an increased fatigue life compared to the proportional loading, which is in contrast to test data. The integral approach, however, shows a reduced lifetime and therefore complies with the tendency of the test results, =
TABLE 4--Test results for crak initiation with notched specimens under variable amplitude. Test Series
o'a,N = 10 6, MPa
Slope, k
Proportional load, "ralo',~= 0.5 Noncorrelated, zJoo = 0.5 Noncorrelated, "ra/o',~= 0.75
544 529 468
-8.2 - 11.1 -8.5
168
MULTIAXIAL FATIGUE AND DEFORMATION
I
proportional: test IMDH CP
I000 ,
900
~ ~.
k = -10
700 600
II -~
IMDH CP 90 ~ phase shift
multiaxial 90 ~ phase shift 500 . "~a = 0.5 9 (Ya Rc~ = -1
400
300
t
30 CrNiMo 8 ~
. . . . . . . . . . . . . . . . . 10 6 107
105
cycles to failure N (log) FIG. 13--Variable-amplitude tests with smooth specimens with 90 ~ phase shift. Compared with proportional loading, experiment, and calculation.
Fig. 13. In the case of noncorrelated loading both approaches come up with an increased fatigue life which corresponds to the experiments. But a lifetime increase of 10 times in the case of the critical plane approach is much too high compared with test results. In the case of the integral approach a minor effect can be observed.
PM r p22i~
1:
1000 900 ~
800
tz ~
700
.~
600
-test k = - 1 0 _proportional _ _
II
500
--
~ 400 e..,
IMDH CP non-correlated
non-correlated ,Ca=0.5.a a -
Ed
R
~
a
105
-1
30 CrNiMo 8
t , lk-
R x = -1 I
300
=
I
I
I
I
I
10 6
i
i
i
i i
i
i
i
10 v
cycles to failure N (log)
FIG. 14--Variable-amplitude tests with smooth specimens with noncorrelated loading. Compared with proportional loading, experiment, and calculation.
POI-FER ET AL. ON LIFETIME PREDICTION
k = -12--.._ " "--._
1000 900 ~
800 / 7.-.. "">'CP
"700
= T~
169
IMDH
600 500 400
superimposed shear stress
~
R~=-I
30 CrNiMo 8
300 10 5
........
[ 10 6
t
~Kt
9 "% = 500 [MPa]
e-,
i
i
i
i
i
i
i
i
= l'0~i i
i
i
lO 7
cycles to failure N (log) FIG. 15--Variable-amplitude tests with smooth specimens under vibrating bending with superimposed static shear stress. Experiment and calculation.
Figures 15 and 16 present test results at pure bending and pure torsion with superimposed mean shear stress and mean tension stress, respectively. In this case, both approaches are in adequate agreement with the test results, whereas the critical plane approach overestimates the experimentally determined life compared to the integral approach, especially in Fig. 16. According to Figs. 9 and 10,
1000 900 C P ~
800
IMDH~....
700 "~
600
"~c~ 500 "~
400
. superimposed tension stress
l ~
300
'- (3m = 550 [MPa] 30 CrNiMo 8 R~=-I i
........ 105
10 6
i
,
,ill
107
cycles to failure N (log) FIG. 16--Variable-amplitude tests with smooth specimens under vibrating torsion with superimposed static tension stress. Experiment and calculation.
170
MULTIAXIALFATIGUE AND DEFORMATION
.~
800
o
700
r
600 ~
500
b
400
r H
300
k
---8
multiaxial proportional -
"~a = 0 . 5
~
200 - -
42 CrMo 4 V KtB = 2.0
R o = -1
~ i
9 ~a
r,- R~ = - I i
i
i
iii
104
= 1.6
KtT i
i
i
i
ilml~
i
i
i
i roll
10 6
l0 s
107
cycles to crack initiation N i (log) FIG. 17--Variable-amplitude tests with notched specimens and proportional loading. Experiment and calculation. the difference between both approaches in the case of superimposed mean stresses should be marginal, because of the relatively small changes in the principal stress direction. With notched specimens subjected to proportional bending and torsion load with variable amplitudes, a comparison between predicted and experimental fatigue lives is presented in Fig. 17. As ex-
,~
800
O
700
IMDH k = -12
600
~
~
500
tz
400
c5 H
300
\
multiaxial non-correlated "ca = 0.5 90 a
42 CrMo 4 V -
R o = -1
-
"
KtB = 2.0
200 - I
104
I
I
105
I
I
I
I I I I~
I
I
10 6
I
I
I I I
107
cycles to crack initiation N i (log) FIG. 18--Variable-amplitude tests with notched specimens and noncorrelated loading, with maximum of load spectra ~'a/~a = 0.5. Experiment and calculation.
POTTER ET AL. ON LIFETIME PREDICTION
800
b~
IMDH\
700 k
60O
=
171
/ /CP
-9
500
400 multiaxial non-correlated II
300
"ca = 0.75 9~a
--42
CrMo 4 V -
R a = -1
-
'
"
KtB = 2.0
2O0 = -1 i
104
i~ll
KtT = 1.6 t
105
i
i
:
~llll
i
i
106
]
: :ll
107
cycles to crack initiation N i (log) FIG. 19--Variable-amplitude tests with notched specimens and noncon'elated loading, with maximum of load spectra "ra/~ra= O.75. Experiment and calculation.
pected in the case of proportional loading, there is little difference between critical plane and integral approach. The agreement with the experimental results is satisfying. In addition to the proportional loading, tests with noncorrelated bending and torsion and a ratio between the maximum values of the load spectra of ~-a/~ = 0.5 and 0.75 were performed. In this case both concepts overestimate the experimentally determined fatigue life, Figs. 18 :,_nd 19. However, the integral approach was found to be a favorable approach.
Conclusion The fatigue process under a multiaxial cyclic stress state is strongly influenced by the temporal correlation of the stress components. The different evaluation of the spatial damage state causes the discrepancy in the calculated fatigue life between critical plane and integral approach. For an assessment of the strength hypotheses, their ability to describe the tendency of the test results, depending on different load cases, has to be considered. The results of the variable-amplitude tests indicate that a 90 ~ phase shift between bending and torsion causes a reduced fatigue life compared to the proportional loading, while a load case with no correlation between bending and torsion leads to an increased lifetime. The integral approach agrees with the tendency of the test results, whereas the critical plane approach predicts an increased fatigue life in both cases. For variable-amplitude bending or torsion with superimposed shear or tension stresses, both approaches are in adequate agreement with the test results. Because of the small changes of the principal stress direction, the difference between the critical plane and the integral approach is marginal.
Acknowledgment The authors wish to thank the Deutsche Forschungsgemeinschaft DFG and the Arbeitsgemeinschaft industrieller Forschung AiF for financial support of this work
172
MULTIAXIAL FATIGUE AND DEFORMATION
References [1] Garud, Y. S., "Multiaxial Fatigue: A Survey of the State of the Art," Journal of Testing and Evaluation, JTEVA, Vol. 9, No. 3, May 1981, pp. 165-178. [2] Balthazar, J. C. and Araujo, J. A., "Biaxial Fatigue: An Analysis of the Combined Bending/Torsion Loading Case," Proceedings of the 5th International Conference on Bio.rial/Multiaxial Fatigue & Fracture, Cracow, Poland, 8-12 September 1997, pp. 9-23. [3] Papadopoulos, I. V., Davoli, P., Grola, C., Filippini, M., and Bemasconi, A., "A Comparative Study on Multiaxial High-Cycle Fatigue Criteria for Metals," International Journal of Fatigue, Vol. 19, No. 3, 1997, pp. 219-235. [4] Sonsino, C. M., "Overview of the State of the Art on Multiaxial Fatigue of Welds," Proceedings of the 5th International Conference on Biaxial/Multiaxial Fatigue & Fracture, Cracow, Poland, 8-12 September 1997, pp. 395-419. [5] McDiarmid, D. L., "The Effects of Mean Stress and Stress Concentration on Fatigue under Combined Bending and Torsion," Fatigue Fract. Eng. Mater. Struct., Vol. 8, No. 1, pp. 1-12, 1985. [6] Nckleby, J. O., "Fatigue under Multiaxial Stress Conditions," Report MD-81 001, Div. Masch. Elem., The Norw. Institute of Technology, Trondheim/Norwegen, 1981. [7] Liu, J., "Weakest Link Theory and Multiaxial Criteria," Proceedings of the 5th International Conference on Biaxial/Multiaxial Fatigue & Fracture, Cracow, Poland, 8-12 September 1997, pp. 45--62. [8] Sines, G., "Fatigue of Materials under Combined Repeated Stresses with Superimposed Static Stresses," Technical Note 2495, NACA, Washington, DC, 1955. [9] Lohr, R. D. and Ellison, E. G., "A Simple Theory for Low Cycle Multiaxial Fatigue," Fatigue of Engineering Materials and Structures, Vol. 3, 1980, pp. 1-17. [10] Kandil, F. A., Brown, M. W., and Miller, K. J., "Biaxial Low Cycle Fatigue Fracture of 316 Stainless Steel at Elevated Temperatures," The Metals Society, Book 280, London, 1982, pp. 203-210. [11] Socie, D., Waill, L., and Dittmer, D., "Biaxial Fatigue of Inconel 718 Including Mean Stress Effects," Multiaxial Fatigue, ASTM Technical Publ. 853, 1985, pp. 461-481. [12] Ott, W., Baumgart, O., Trautmann, K. H., and Nowack, H., "A New Crack Initiation Life Prediction Method for Arbitrary Multiaxial Loading Considering Mean Stress Effect," Proceedings of the 6th International Fatigue Congress, Vol. 2, 1996, pp. 1007-1012. [13] Bannantine, J. W. and Socie, D. F., A Multiaxial Fatigue Life Estimation Technique, ASTM STP 1122, Philadelphia, 1992, pp. 249-275. [14] Bonnen, J., Conle, F., and Chu, C., "Biaxial Torsion-Bending Fatigue of SAE Axle Shafts," SAE Paper No. 910164, Society of Automotive Engineers, Warrendale, PA, 1991. [15] Jung, L., Ktttgen, V. B., Mascher, G., Reissel, M., and Zhang, G., "Numerische Betriebsfestigkeitsanalyse eines Pkw-Schwenklagers im Rahmen des FEM-Postprocessing," DVM-Bericht 123, Betriebsfestigkeit und Entwicklungszeitverkiirzung, 22-23. October 1997, pp. 151-162. [16] Esderts, A., "Betriebsfestigkeit bei mehrachsiger Biege- und Torsionsbeanspruchung," Ph.D. Thesis, TU Clausthal, 1995. [17] Gough, H. J., Pollard, H. V., and Clenshaw, W. J., "Some Experiments on the Resistance of Metals to Fatigue under Combined Stress," Mem. 2522, Aeronautical Research Council, His Majesty's Stationary Ofrice, London, 1951. [18] Sanetra, C., Zenner, H., Amstutz, H., and Seeger, T., "Betriebsfestigkeit bei mehrachsiger Beanspruchung I, FKM-Report, Vol. 153, 1990. [19] Esderts, A., Zenner, H., Amstutz, H., and Seeger, T. "Betriebsfestigkeit bei mehrachsiger Beanspruchung II," FKM-Report, Vol. 188, 1994. [20] Esderts, A., Pttter, K., and Zenner, H., "Fatigue of Smooth and Notched Specimens Under Multiaxial Random LoadingIExperimental Results and Prediction," Proceedings of the 5th International Conference on Biaxial/Multiaxial Fatigue & Fracture, Cracow, Poland, 8-12 September 1997. pp. 609--620. [21] Miner, M. A., "Cumulative Damage in Fatigue," Trans. ASME, Journal of Applied Mechanics, Vol. 12, 1945, pp. A159-A169. [22] Kotte, K. L. and Zenner, H., "Lifetime Prediction---Comparison Between Calculation and Experiment on a Large Database," Proceedings of the 4th International Conference on Low Cycle Fatigue and ElastoPlastic Behaviour of Materials, Garmisch-Partenkirchen, Germany, 7-11 September 1998, pp. 721-728.
Tadeusz Lagoda 1 and Ewald Macha I
Generalization of Energy-Based Multiaxial Fatigue Criteria to Random Loading REFERENCE: Lagoda, T. and Macha, E., "Generalization of Energy-Based Multiaxial Fatigue Criteria to Random Loading," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 173-190. ABSTRACT: Results from a review of energy-based criteria of multiaxial fatigue indicate that they may be divided into three groups, depending on the kind of strain energy density per cycle which is assumed as a damage parameter. They are: criteria based on elastic strain energy for high-cycle fatigue, criteria based on plastic strain energy for low-cycle fatigue, and criteria based on the sum of plastic and elastic strain energies for both low- and high-cycle fatigue. Lately, special attention has been paid to criteria taking into account strain energy density on the critical plane. However, in the energy approach to multiaxial fatigue there is an important unsolved problem, namely, the evaluation of energy, especially plastic strain energy density, from the closed stressstrain hysteresis loops under random loading. In this paper, energy parameters defined for random loading are introduced. Under uniaxial loading we distinguish between the strain energy density for tension (positive) and the strain energy density for compression (negative). As a consequence, if there is no mean component in the random loading, we obtain a random history of strain (elastic and plastic) energy density with a mean of zero. Next, some known energy criteria of multiaxial cyclic fatigue were generalized for the random loading. The new criterion has been successfully used for fatigue life calculation under uniaxial and biaxial random tension-compression of plane specimens made of 10HNAP steel. For this material the equivalent strain energy density on the critical plane, Weq (t), seems to be an efficient parameter under nonproportional random loading in the range of a great number of cycles. The equivalent strain energy density is derived from the new criterion. KEYWORDS: multiaxial fatigue, nonproportional random loading, generalized strain energy density criterion, lifetime, critical plane approach
Service loading in elements o f machines and structures usually causes random stress states in the materials. U n d e r such loading the assumed criteria o f multiaxial random fatigue play a very important role in the algorithm o f fatigue life prediction. The first criteria o f multiaxial random fatigue can be found in R e f 1. They are a result o f generalization o f the k n o w n stress-based criteria for cyclic loading. However, in R e f 2 some theoretical limitations precluding generalization o f some criteria to the range o f random loading are discussed. As a c o n s e q u e n c e o f more detailed considerations, the criteflon o f m a x i m u m shear and normal stress in the fracture plane (critical plane) has been formulated max{Bzns(t) + Ktrn(t)} = F
(1)
where Zns(t) and trn(t) are the shear stress in direction ~ and the normal stress in the critical plane with normal ~, respectively. B, K and F are constants for a particular version o f the criterion. This crite-
Assistant professor and professor, respectively, Department of Mechanics and Machine Design, Technical University of Opole, ul.Mikolajczyka 5, 45-271 Opole, Poland.
Copyright9
by ASTM International
173 www.astm.org
174
MULTIAXIAL FATIGUE AND DEFORMATION
rion is applied for reduction of the multiaxial random stress state to the uniaxial one and determination of an equivalent stress. Then the equivalent stress history is analyzed in the same way as in the case of uniaxial random loading and high-cycle fatigue region. In a similar way the strain criterion of multiaxial random fatigue [3] can be defined (2)
max{bens(t) + ken(t)} = q
This criterion can be applied for both long- and short-lasting fatigue strength. There are also two other theories based on the critical plane concepts: one, proposed by Bennantine and Socie [4], is an extension of the local strain approach for uniaxial variable amplitude loading, and the other, proposed by Wang and Brown, employs a counting algorithm for multiaxis random loading [5]. New mathematical models of the criteria are still being searched [6]. At present, special attention is paid to the energy criteria [7-18]. Results from some tests performed under uniaxial and mnltiaxial cyclic fatigue suggest, that the energy approach to fatigue of structural materials seems to be promising. Thus, an attempt of generalization of the energy-based criteria of multiaxial cyclic fatigue to the range of random loading is one of the most important aims of this paper. It has been assumed that strain and stress tensors are stationary and ergodic. Moreover, let us assume that histories of the components of both tensors have low-band frequency spectrums in the case of which the influence of loading frequency on the fatigue life can be neglected.
Energy Parameter Energy Parameter in Uniaxial Stress State
The first attempts of energy approach to fatigue under random loading have been presented in Ref 15. In the case of random loading, application of the energy fatigue models is quite difficult. The difficulties occur while calculating plastic strain energy density with the use of the closed stress-strain hysteresis loops. We can, however, apply a fatigue parameter, namely, elastic and plastic strain energy density, which takes into account the difference between energies under tension and compression. Simulation tests were conducted in order to observe energy change versus time. Here we present a simulation for uniaxial tension-compression and low-cycle fatigue (with plastic strain) where the stress was controlled. At first, a stress history with o'~ = 400 MPa and w = 1/s was generated (see Fig. 1) o'(t) = -era cos~ot
0 --< t --< T
(3)
Then, we assumed that the cyclic stress-strain curve was of the Ramberg-Osgood type
{
(4)
Ae = ---if- + 2 ~2K, } and we determined the strain history (Fig. 1) according to the relation (see the Appendix)
(
- t r a coswt E
e(t) =
--o"a coswt E
( O'a~lint
2..OSOOt.)l/n' __
2 (~)l/n' [( 1 +20soot)1In' --(1)lforZ]2~t<
where E = 215 GPa, K' = 853 MPa, n' = 0.156.
T
(5)
175
LAGODA AND MACHA ON GENERALIZATION OF ENERGY
400 -
. _.
~(t)
0.010-
"~W"~
[Ml'a]
i
E
0.005-
200 -
t. . . . . .
'
*"I'"
1
/
2/
~ ......
'
W,m,]
l
I
4
'",
-200-
)05
-J-2
-400
FIG. 1--Stress, strain, and strain energy density histories during one cycle. Let us note that under uniaxial tension-compression the material grains are subjected to displacements and damage resulting from specific work of stresses on different paths of strains in the first, (0 - TI2), and second, (7"12 - T), half-period of the fatigue cycle. Strain energy density is equal to that work and in the first half-cycle is represented by the area A Wo-r/2 in Fig. 2, in the next half-cycle-by the area AW772-T, and in the full cycle--by AWo-r. The considered energies can be determined from
AWo-r/2 = Ioa~ Ao-d(Ae)
(6)
&d le,'=lo~
.::::::::::.I
,",",".",""X.:.:-2'.,:4
/'::;
..:.'.:.:.:.:.~
. . . . .
. . . . . . .
B == Wll WUM=
===================== e.=o* I**e Io ,e,o,~ 9
Q
*
9
9
~
9
9
=
9
- - " 9
9
=
M
9"-:l
;I::'-.Z..::':::I ."
o'pel~176
=~mmm " o ' , ' .
o
9
,
el 9
.
|
__
0
As
Aw _t
0
As
Aw0_t=Aw0_ +Aw _t
FIG. 2--Strain energy density in first, AWo-T/2, and second AWT/2-T half-cycle and in full loading cycle, AWo-T = Ao-Ae.
176
MULTIAXIAL FATIGUE AND DEFORMATION
AWr/2-r =
Ao'd(As) =
A W o _ T = A W o _ T / 2 -t- A W T / 2
T =
C
Aed(Ao-)
Ao'd(As) +
Asd(Ao')
(7)
= Ao-Ae
(8)
From Eq 8 it appears that strain energy density in the full loading cycle AW0 r is equal to the product of the stress and strain ranges. Thus, the energy differential d(AWo-r) = d(Ao'Ae) = Ao-d(Ae) + Aed(Ao')
(9)
is the total differential of the product of stress and strain ranges in a loading half-cycle (see Fig. 3). If the energy differential d(AW0 r) according to Eq 9 is integrated over the time interval including a loading half-cycle and an unloading half-cycle, we obtain double energy expressed by Eq 8, i.e.,
fo
d[Wo_r(t)] = 2AcrAs
(10)
Thus, in the case of integration over this time domain we should use a half of the total differential of the energy density, i.e., d[Wo-r(t)] = 1 {Acr(t)d[Ae(t)] + Ae(t)d[A~r(t)] }
(11)
From Eq 11 we obtain the differential of strain energy density for any moment of time dW(t) = 1 { ~r(t)d[e(t)] + e(t)d[~r(t)] }
(12)
)" Aa
If 0
dI,cl
I
At
FIG. 3--Differential of strain energy density as full differential of a product of stress and strain ranges in a loading half-cycle.
~LAGODA AND MACHA ON GENERALIZATION OF ENERGY
177
and strain energy density
W(t) = l~r(t)e(t)
(13)
For distinguishing positive and negative specific works in a fatigue cycle we introduce functions sgn[e(t)] and sgn[tr(t)] to Eq 13 in the following way (see Fig. 1)
W(t) = lo'(t)e(t)sgn[e(t)] + lo'(t)e(t) sgn[o'(t)] (14)
= ltr(t)e(t){sgn[tr(t)] + sgn[e(t)]} = l~r(t)e(t) /.
sgn[tr(t)] + sgn[e(t)]
2
In such a case, during a loading half-period a sign of specific work 0.25tr(t) e(t)sgn[e(t)] will be influenced by the stress sign, and during an unloading half-cycle a sign of specific work 0.25tr(t)e(t)sgn[~r(t)] will be influenced by the strain sign. Let us introduce the two-argument logical function sgn(x,y), sensitive to signs of variables x and y, defined as 1 when sgn(x) = sgn(y) = 1 0.5when x = O a n d sgn(y) = 1 o r y = 0 and sgn(x) = 1 sgn(x,y) =
sgn(x) + sgn(y) =
O when sgn(x) = - s g n ( y )
(15)
- 0 . 5 w h e n x = O a n d sgn(y) = - 1 o r y = 0 and sgn(x) = - 1 - l w h e n sgn(x) = sgn(y) = - 1 Then Eq 14 takes a form
W(t) = l ~ ( t ) e ( t ) sgn[~r(t),e(t)]
(16)
Let us see that Eq 16 expresses positive and negative strain energy densities in a fatigue cycle and it allows us to distinguish energy (specific work) for tension and energy (specific work) for compression. The advantage of Eq 16 is that strain energy density history has zero mean value, when cyclic stresses and strains change symmetrically in relation to the zero levels. If the stress and strain reach their maximum values, tra and e,, then the maximum energy density value, according to Eq 16, i.e., its amplitude is
Wa = 0.5craea
(17)
Assuming W(t) as the fatigue damage parameter according to Eq 16, we can rescale the standard characteristics of cyclic fatigue (~a - NS) and (ea - N s) and obtain a new one, (Wa - NS). In the case of high-cycle fatigue, when the characteristic (04 - ivy) is used, the axis o-a should be replaced by W~, where ~ a~
Wa = ~
(18)
178
MULTIAXlALFATIGUE AND DEFORMATION
In the case of low- and high-cycle fatigue, when the characteristic (ea -- Ny) is used, we can do similar rescaling. From the Manson-Coffin-Basquin equation and Eq 17 we obtain e~ = e~ + ePa = Tty] (2Ns)b + e}(2Nfy
(19)
r ToJ (2N,) b + e] (2N,) c] W~ = To'a /L
(20)
oa = o}(2Ny) b
(21)
Let us assume that
then Eq 20 becomes a new fatigue characteristic (W~ - NS) Wa = ~(~' * s s
~2b + 0.5e} o'~(2Us)b+c
(22)
same as SWT parameter.
Energy Parameter in Multiaxial Stress State The random strain and stress tensors are analyzed. Each of them is a six-dimensional stationary and ergodic Gaussian process with the wide-band frequency spectrum and zero expected values. The proposed generalized energy criterion is based on the selected components of specific work of stress on the strains in the critical plane and it has a mathematical form similar to criteria (1) and (2).
Generalized Criterion of Maximum Shear and Normal Strain Energy Density on the Critical Plane--Let us make the following assumptions: (1) Fatigue fracture is caused by that part of strain energy density which corresponds to the specific work of normal stress trn(t) on normal strain %(t), i.e., Wn(t) and specific work of shear stress %,(t) on shear strain ens(t) acting in the ~ direction, on the plane with a normal ~, i.e.,
W~s(t). (2) The direction ~ on the critical plane (the expected fracture plane) coincides with the mean direction along which the maximum shear strain energy density W~m~(t) occurs. (3) In the limit state that conforms to the fatigue strength the maximum value of combined Wn(t) and Wns(t) energies satisfies the following equation
max{flWn,(t) + KWh(t)} = Q
(23)
where fl is constant for a particular form of Eq 23, and K and Q are material constants determined from sinusoidal fatigue tests. The left side of Eq 23 can be written as maxt {W(t) } and should be interpreted as the 100% quantile of the random variable W. If the maximum value of W(t) exceeds the value of Q, then damage wilt accumulate, resulting in fracture. The random process W(t) can be interpreted as a stochastic process of the fatigue strength of a material. The positions of the unit vectors ~ and ~ are determined with the use of one of the following procedures: weight functions method, variance method, or damage accumulation method [19].
LAGODA AND MACHA ON GENERALIZATIONOF ENERGY
179
A choice of constants/3, K and Q in Eq 23, together with the assumed position of the critical plane, leads to particular cases of the generalized criterion. Three special cases are considered here.
Criterion of Maximum Normal Strain Energy Density on the Critical Plane--If/3 = 0, tr = 1 and Q = Way (fatigue limit under tension compression expressed by normal strain energy density) and if we assume that the unit vector ~ coincides with the mean direction along which the maximum norreal strain energy density Wnmax(t) occurs, i.e.,
= [ni + rhnj + hnk
(24)
max { Wn(t) } = W~y
(25)
then criterion (23) becomes
where fn,rhn,fi n = mean direction cosines of ~ in relation to the constant system of axes Oxyz, and a i ,j,k = versors of the axes 0xyz. The equivalent strain energy density derived from criterion (25) is
Weq(t) =- Wn(t) = ltrn(t)en(t) sgn[~n(t),en(t)]
(26)
where c%(t) and en(t) are normal stress and strain on the critical plane, respectively, i.e.: On(t ) = [20"xx(t) + rn 20"yy(t) + nn^2trzz(t) (27)
+ 2[nrfin~xy(t) + 2[nfintrxz(t) + 2rhnhno'yz(t) en(t) = [~:.~(t) + m^ 2n eyy(t) + ~2 ezz(t) (28) + 21nrhnexy(t) + 21nil n exz(t) + 2thrift n eyz(t) If under proportional mulfiaxial sinusoidal loading we assume that normal stress and normal strain having the maximum amplitudes act along the axis x, i.e. tr=(t) = ~
(29)
sinoJt; e~x(t) = e ~ sinwt
and further if we assume in = 1, then according to Eqs 25-28, we obtain
1 max {~tr~,(t)e~,(t)sgn[trx~ (t),
ex~(t)]} -- --21 r
-- War
(30)
This result leads to the energy parameter used by Socie [16], based on the idea of Smith et al. [17] and to the criterion proposed by Nitta et al. [14] for Mode I. Thus criterion (25) is a generalization of the mentioned energy criteria, applied under cyclic loading.
Criterion of Maximum Shear Strain Energy Density on Critical Plane--For/3 = 4/(1 + 1,), (v = Poisson ratio), K = 0 and Q = W~fwe assume that the critical plane with normal ~ is determined by the mean position of one of two planes on which the maximum shear strain energy acts. On this plane
180
MULTIAXIAL FATIGUE AND DEFORMATION
we choose a direction ~ coincident with the mean position along which the energy W n . . . . (t) occurs, i.e. = l~i + rh~j + t~sk
(31)
where l'~, the, ri~ are mean direction cosines of ~ in relation to the axes 0xyz. Under the above assumptions, the criterion (23) becomes { 4 max ~
} Wns(t) = Way
(32)
The equivalent strain energy density derived from criterion (32) is as follows
W~q(t) = ~
4
2
Wn~(t ) = T-~v'rn~(t)ens(t) sgn[~'n~(t), end(t)]
(33)
where ~'ns(t) =ijs~rxx(t) + rh,rhsCryy(t) + finfis~zz(t)
(34)
+ 21"nrhsO'xy(t) + 21n~s~rxz(t) + 2rhnt~sO-yz(t)
ens(t ) = ~n~se~.(t) + rhnrhseyy(t) + ~nPtsezz(t)
(35) + 21nrhs~xy(t) + 21nfis~xz(t) + 2rhnPtseyz(t) In the case of proportional multiaxial sinusoidal loading when the normal stress and strain having the maximum amplitudes act along the x-axis and the normal stress and strain with minimum amplitudes act along the z-axis, i.e. O'x~(t) = cro~ sin tot; exx(t) = e ~ sin oJt (36) O-zz(t) = o-~zzsin oJt; ezz(t) = e~zz sin oJt and when
~ =__~1 rh =0, fin=
~ = ~ 2 ,,hs = 0 , ~ -
1
x /1~
(37)
then according to Eqs 32-37, we obtain
(38) 1+ v
2
2
1 q- 1-' Tamax "Yamax ~
Waf
Strain energy density expressed by Eq 38 is also applied in the criterion proposed by Nitta et al. for
4_AGODA AND MACHA ON GENERALIZATION OF ENERGY
181
Mode II under cyclic loading [14]. Thus, criterion (32) is a generalization of the next energy criterion applied under cyclic loading.
Criterion of the Maximum Shear and Normal Strain Energy Density on the Critical Plane--Case / - - F o r / 3 = 2/(1 + v), K = 2/(1 - v) and Q = WafWe assume, as in the above section, that the critical plane with normal ~ is determined by the mean position of one of two planes on which the maximum shear strain energy acts. On this plane the direction ~ coincident with the mean position along which the energy Wn. . . . (t) occurs--see Eq 31. The general criterion (Eq 23) now has the following form W~s(t) +
max ~
W,(t) = Way
(39)
From criterion (Eq 39) we can derive the equivalent strain energy density as
Weq(t) = ~
2
Wns(t) +
~
Wn(t) = ~
• sgn[%s(t), e,s(t)] + ~
1
Zns(t)ens 1
o-,(t)en(t)sgn[trn(t), en(t)]
(40)
where ~'ns(t), e,Ts(t), trn(t), and en(t) are expressed by Eqs 34, 35, 27, and 28, respectively. Under multiaxial sinusoidal in-phase loading and on the assumption as in the previous section-see Eqs 36 and 37, from Eqs 39 and 40 it follows that 2
1 (41)
+
6%(t)en(t)sgn[gn(t), e,(t)] = 2(1 + v) z"max %max + ~
o'onean = Way
where o'an and ean are amplitudes of normal stress and strain on the plane of the maximum shear stress and strain amplitudes Zamaxand Yama~,respectively. The strain energy density in Eq 41 is also assumed by Liu [18] in his virtual strain-energy parameter (VSE) under in-phase cyclic loading for Mode II fracture (for/3 = K = 1). Thus, the criterion (39) is a generalization of the energy criterion formulated by Liu [18] to the range of random loading. Moreover, it is possible to prove that for/3 = 2 and K = 1 the criterion (Eq 39) is a generalization of the energy criterion proposed by Glinka et al. [11] for cyclic loading. Further cases of this criterion may be defined by choosing other positions of critical plane. Fatigue Life Determination Under Multiaxial Random Stress State Figure 4 shows an algorithm for determination of fatigue life with use of the general criterion of maximum shear and normal strain energy density on a critical plane. At first (Stage 1) we measure the strain state components and next we calculate stress histories. Having tensors of strain and stress histories we can determine courses of normal and shear strain energy density (Stage 2) in all the planes with the distinguished direction r/, s with use of one of the particular forms of the generalized energy criterion. In the algorithm for the fatigue life evaluation it is important to determine the expected critical (fatigue fracture) plane position (Stage 3). The strain and stress state existing in the material belongs to the most important factors determining the plane position. The position is determined by the given values of the direction cosines l.,
182
MULTIAXIALFATIGUE AND DEFORMATION
1
I
2
Measurement and calculation of eij(t), 6ij(t), (ij = x, y, z)
I
J Calculation of Wn(t) andWns(t )
I
Determination of the critical plane and directions of the unit vectors and~
4
Calculation of the equivalent strain energy density histories
I
5
[ Cycle and half-cycle amplitudes counting on the critical plane
6
[ Fatigue damage aecamulation
7
] Fatigue life determination
I
I
FIG. 4---Algorithm for fatigue life calculation under multiaxial random loading.
rh,, ~in (n = ~/, s) of the unit vectors ~ and ~ occurring in the fatigue criteria. The following three methods of determination of the expected critical plane position are proposed: (1) The method of weight function, presented in Ref 19, insists in averaging the random values of angles O/n(t), fin(t), ~/n(t), determining instantaneous positions of the principal strain/stress axes position in relation to the constant system of 0xyz axes with use of special weight functions. (2) The method of damage accumulation, presented in Ref 19, has the fatigue damage accumulated on all possible planes. Next, the plane on which damage is maximum is selected. Thus, we obtain not only direction of the expected critical plane, but the fatigue life as well. (3) The method of variance maximum. It is the method most often applied so far. It gives good results when the stress and strain criteria are applied [19-21]. Here, in the method of variance, it is assumed that the planes in which equivalent strain energy density variance according to the chosen criterion reaches its maximum are critical for the material. When the energy density history at the given critical plane (Stage 5) is determined, the energy cycles are counted with the rain flow algorithm. Fatigue damage accumulation (Stage 6) is accomplished according to the Palmgren-Miner hypothesis
LAGODA AND MACHA ON GENERALIZATION OF ENERGY
[+
ni
183
for Waeqi-->aWaf
(42)
S(To)=[i~=l0 Ny~ forWaeqi
S(To) = fatigue damage degree at time of observation To, Way= fatigue limit expressed in strain energy density, a = coefficient allowing the taking into account of amplitude W~.qi, less than W~f, in the damage accumulation process (a = 0.25), Nfi = number of cycles corresponding to the strain energy density W,eqi, determined from Eq 22 ni a number of cycles with amplitude Wa~i (two of the same half-cycles form one cycle). =
When degree of damage at observation time To is determined, we calculated the fatigue life (Stage 6) To Tc~a = S(To)
(43)
Experimental Verification of the Energy Parameter
Fatigue Tests Specimens made of 10HNAP steel were used. The steel was delivered as hot-rolled sheets; its chemical composition and static parameters are shown in Tables 1 and 2. For uniaxial (tension-compression) fatigue tests, fiat specimens were used. After the fatigue tests under symmetric uniaxial cyclic loading [22] the regression model was determined according to ASTM Standard E 739-80. Under random uniaxial tension-compression the tests were conducted for the zero expected value and several loading levels in the range from medium to long-life. The random histories were generated via minicomputer and the matrix method (only the extreme values were generated). Cruciform specimens of a shape used by B~dkowski [20,21] were tested under the biaxial stress state. Central parts of the specimens had spherical cutouts with the radius of curvature 250 mm. The specimens were 1 mm in thickness in the central region. The strain histories ex~(t) and eyy(t) in the central part of the specimens were measured by means of two strain gages. The stress components are calculated according to the following equations
o's(t) = E ,e=(t) + very(t) 1 -
ery(t) + ve,=(t) O'yy(t) = E
1~2
1 -
v2
(44)
TABLE 1--Chemical composition of lOHNAPsteel (in [%]). C
Mn
0.115
0.71
Si
P
0 . 4 1 0.082
S 0.028
Cr
Cu
0.81 0.30
Ni
Fe
0.50
the rest
184
MULTIAXIAL FATIGUE AND DEFORMATION
TABLE 2--Static parameters of IOHNAP steel. Rolling Direction
R0.2 MPa
R,, MPa
Alo %
Z %
E GPa
v
Longitudinal Transverse
414 382
566 565
32 30
60 43
215
0.29
Under biaxial nonproportional tension-compression the cruciform specimens were tested for the stress correlation coefficients from the range ( - 1, + 1) [20,21]. The coefficient was determined from
ro.~o33,
i
~o'xxo'yy ~VF.~_
y
(45)
where/Zo~,/Zoyy, ]s are components of the stress covariance matrix. A few specimens were tested under cyclic biaxial tension-compression, with the remaining under random loading. Particular specimens were loaded by force histories with the stress correlation coefficients near 1, O, + 1 and different loading described by the following stress standard deviation ratio -
Ux/y
=
~ / W ~ y
(46)
Experimental Verification For calculations of fatigue life according to the algorithm shown in Fig. 4, we used ~r~ = 994 MPa, b = -0.095, ~} = 0.244, c = -0.464, and the fatigue limit Waf = 0.155 MJ/m 3 for the tested material. First of all we determined fatigue lives according to the proposed model for uniaxial random stress state. The calculation and experimental results are shown in Fig. 5. All the calculation results are inside the scatterband of a factor of 3, the same as for cyclic tests [22]. The same calculation procedure was repeated for tests under biaxial random and cyclic stress states using criterion of the maximum normal strain energy density on the critical plane (26) Weq(t) = Wn(t) = 0.25cr,(t)e,(t)[sgncrn(t) + sgnen(t)]
(47)
on(t) = [/~xx(t) + rh~ryy(t)]
(48)
e,(t) = [/~e~(t) + rh~eyy(t) + ~Z~ezz(t)]
(49)
where
In the plane stress state the normal vector orientation to the critical plane ~ may be described with the use of one angle in relation to the x-axis. Thus, the direction cosines of ~ are is = cos a,
rh n = sin a,
rl n = 0
(50)
The vector position ~ was determined with the use of the damage accumulation method. The calculation results of fatigue life for cruciform specimens and the experimental results are shown in Fig. 5. Most of the calculation results are inside the scatterband of a factor of 3. From Fig. 5 it appears that
~LAGODAAND MACHA ON GENERALIZATIONOF ENERGY
10 4
10 5
185
10 6
FIG. 5--Comparison of calculated and experimental fatigue lives of lOHNAP steel under uniaxial and biaxial nonproportional random loading.
the results of fatigue life calculations, Teal, for uniaxial tension are a little overestimated and they are lowered under biaxial tension in relation to the experimental results, T~xp. This results from assuming the model of a perfectly elastic body. Thus, in the case of uniaxial tension where strains are calculated from stresses, we obtain lowered strains. Then also energy is lowered, and the calculated life is overestimated. In the case of biaxial tension (the inverse situation) stresses are calculated from strains, so they are overestimated. Thus, energy is overestimated and the calculated life is lowered. Consequently, the assumption of a more realistic elastic-plastic model might decrease the differences between Tca~and Texp, and the normal strain energy density on the critical plane may be accepted as an efficient parameter for fatigue life prediction under random uniaxial and biaxial nonproportional tension-compression in the range from medium to long-life. Conclusions
1. Suitable introduction of strain and stress signs into the strain energy density parameter allows us to distinguish positive strain energy density in tension and negative strain energy density in compression paths. Under random loadings with zero expected values it leads to the centered stochastic energy density process. 2. With the use of the energy parameter the new generalized criterion of maximum shear and normal strain energy density on the critical plane for multiaxial random loading was formulated. Also shown was the generalization of some known energy criteria of multiaxial cyclic fatigue.
186
MULTIAXlALFATIGUE AND DEFORMATION
3. Application of the energy parameter for calculation of fatigue life of plane specimens under uniaxial random stress state and cruciform specimens subjected to random biaxial tension-compression shows that the calculation results for 10HNAP steel are included in the scatterband of a factor of 3, the same as for cyclic tests. 4. The criterion of maximum normal strain energy density on the critical plane, which is a particular case of generalized criterion, seems to be an efficient parameter for high-cycle fatigue calculation of 10HNAP steel under biaxial random nonproportional tension-compression.
APPENDIX Derivation of the Equation for Elastic-Plastic Strain Under the Controlled Stress and Uniaxial Tension-Compression When the stress (see Fig. A.1) tr(t) = - t r a cos wt
(A.1)
is controlled, assuming that the cyclic stress-strain curve is of the Ramberg-Osgood type
Ae
~1In' Ao" + 2 (\ - Ao~]
= g~
(A.2)
400 -a(t)
D
[MPal 200 -
C B
0
E
-200 -
J A -400
A t [s] '
0
1
2
~
I
4
'
t
6
'
I
8
FIG. A. 1--Stress cycle course causing elastic-plastic strain of material.
,LAGODA AND MACHA ON GENERALIZATION OF ENERGY
187
the stress range course can be expressed as A~r(t) = -~r~ coscot + t r a = ~.(1 - costot)
(A.3)
The strain range course takes the form o'~(1 - cosoJt) + 2 [/ ~r~(1 E \
Ae(t) -
COSO)t)~lint
Tl~
(A.4)
]
From (A.4) we obtain for t = 0 --->Ae(0) = 0
(item A in Fig. A.2)
O"a t = T/4 --) Ae(T/4) = ~
{O'a]l/n'(_~) l/n'
(A.5)
+ 2 \~7]
(item B in Fig. A.2)
Thus, the strain course in the first half-period of the cycle (0 - TI2) (the interval AD of the curve in Fig. A.2) can be described with
e(t) = Ae(t) - Ae(O)
(A.6)
Substituting Eqs A.4 and A.5 to A.6 we obtain
E
+ 2 ~-r]
2
AG
(~
[ [MPa]
/ [MPa]
600-
, for 0 < t < TI2
-
D
300 l
,oo- l 400-
~ /
300
'
C
"
~/"
- 0.004
200 --
/
'
o
? '
0.004
-100
/ 1O0
/
-200 -
/ CA
0
AE
I
0.007
0.014
FIG. A,2--Relation between stress and strain in first half-period (0 - T/2).
(A.7)
188
MULTIAXIAL FATIGUE AND DEFORMATION
Ao
I
[MPa] 0,014 A8
0,007
,
600--
I
E
I
500
-200-
400
-100
300 -
t;
,
I
0.004
'
200100
~
0
200
I
'
/ /
,
300
-0.004
- / , ~ --
'
0
/ o - 100
~
0
0
'E/fl
-
100 ,
D
---- 400
F
500
I
L
0.007
I
At;
-
600
0.014A~ [MPa]
FIG. A . 3 - - R e l a t i o n b e t w e e n strain a n d stress in inversed system in s e c o n d half-period o f cycle (T/2 - T).
Similarly, after inversion of the coordinate system shown in Fig. A.3 and taking Eq A. 1 into account we have Ao- = o'. cosog + ~r~ = o'~(1 + coscot)
(A.8)
Thus for t = T/2 ---) Ao'(T/2) = 0
(item D in Fig. A.3)
t = T---) Ao-(T) = 2cr~
(item A in Fig. A.3)
substituting Eqs A.8 into A.4 we obtain
As(t) --
O'a(1 + costot) + 2 [/ ffa(1 + E \
COSC~Ot)~lint
(A.9)
From Eq A.9 we have (item D in Fig. A.3)
t = TI2 --~ A e ( T / 2 ) = 0
O"a t = 3/4 T---> ae(3/4T) = T
+
{O'a~l/nt (1)l/nt 2 ~K']
(item A in Fig. A.3)
(A.IO)
LAGODA AND MACHA ON GENERALIZATION OF ENERGY
0,0081
189
D
s(t)
0,004--]
E
C/
\F
0,000
-0,004 ~
B "~,
\A t [s] -0,00a
r
0
I
r
]
2
~
4
~
~ --L
6
8
FIG. A.4--Course of elastic-plastic strain cycle according to Eq A.13.
Thus, the strain course in the second half-period of the cycle (T/2 - T) (the interval DA in Fig. A.3) can be described with
e(t) = Ae(t) - Ae(1/2T)
(A.11)
Substituting Eqs A.9 and A.10 into A.11 and after inversion of the coordinate system, we obtain
coso, (oo)l,n[(,+to< (1)] E
2 ~-
.
.
.
.
, for T/2 < t < r
(A.12)
Finally, the equation for the strain course can be written in form of Eqs A.7 and A. 12, i.e.
--~aCOS~t +2(---~,)l/n'[(1--~OSCOt) 1/# - - ( 1 ) ] f o r O < _ t < T / 2 E
e(t) =
--~aCOS~t
E
(5)
2 (tTa)l/n['(l+2oscot)l/'~7-
-- ( 1 ) l forT~2 < - t < T
The strain course according to Eq A.13 is shown in Fig. A.4. References
[1] Macha,E., "Simulation of Fatigue Process in Material Subjected to Random Complex State of Stress," Simulation of Systems, L. Dekker, Ed., North-HollandPublishing Company, Amsterdam, 1976, pp. 1033-1041.
190
MULTIAXIAL FATIGUE AND DEFORMATION
[2] Macha, E., "Generalization of Fatigue Fracture Criteria for Multiaxial Sinusoidal Loadings in the Range of Random Loadings," Biaxial and Multiaxial Fatigue, EGF 3, M. Brown and K. J. Miller, Eds., Mech. Engineering Publication, London, 1989, pp. 425-436. [3] Macha, E., "Generalization of Strain Criteria of Multiaxial Cyclic Fatigue to Random Loadings," Fortschr.-Ber. VDI, Reiche 18, Nr 52, VDI-Verlag, Dusseldorf, 1988, p. 102. [4] Bannantine, J. A. and Socie, D. F., "A Variable Amplitude Multiaxial Fatigue Life Prediction Method," Fatigue Under Biaxial and Multiaxial Loading, ESIS 10, K. F. Kussmaul, D. L. McDiarmid and D. F. Socie, Eds., MEP, London, 1999, pp. 35-51. [5] Wang, C. H. and Brown, M. W., "Life Prediction Techniques for Variable Amplitude Multiaxial Fatigue-Part 1: Theories; Part 2: Comparison with Experimental Results," Trans. ASME JEMT, Vol. 118, 1996, pp. 367-370; 371-374. [6] Kenmeugne, B., Weber, B., Carmet, A., and Robert, J.-L., "A Stress-Based Approach for Fatigue Assessment Under Multiaxial Variable Amplitude Loading," Proceedings, 5th International Conference on Biaxial/Multiaxial Fatigue and Fracture, E. Macha and Z. Mrtz, Eds., TU Opole, Poland, 1997, Vol. 1, pp. 557-573. [7] Andrews, R. M. and Brown, M. W., "Elevated Temperature Out-of-Phase Fatigue Behaviour of a Stainless Steel," Biaxial and Multiaxial Fatigue EGF3, M. W. Brown and K. J. Miller, Eds., MEP, London, 1989, pp. 641-658. [8] Curioni, S. and Freddi, A., "Energy-Based Torsional Low-Cycle Fatigue Analysis," Fatigue Under Biaxial andMultiaxial Loading, ESIS 10, K. F. Kussmaul, D. L. McDiarmid, and D. F. Socie, Eds., MEP, London, 1991, pp. 23-33. [9] Ellyin, F. and Golog, K., "Multiaxial Fatigue Damage Criterion," Trans. ASME JEMT, Vol. 110, 1988, pp. 63-68. [10] Garud, Y. S., "A New Approach to the Evaluation of Fatigue Under Multiaxial Loadings," Trans. ASME JEMT, Vol. 103, 1981, pp. 118-125. [11] Glinka, G., Shen, G., and Plumtree, A., "A Multiaxial Fatigue Strain Energy Density Parameter Related to the Critical Fracture Plane," Fatigue Fract. Engng Mater. Struct., Vol. 18, No. 1, 1995, pp. 37-64. [12] Golog, K., "An Energy Based Multiaxial Fatigue Criterion," Engineering Transactions, Vo]. 36, No. l, 1988, pp. 55-63. [13] Lefebvre, D., Neale, K. W., and Ellyin, F., "A Criterion for Low-Cycle Fatigue Failure Under Biaxial States of Stress," Trans. ASME JEMT, Vol. 103, 1988, pp. 1-6. [14] Nitta, A., Ogata, T., and Kuwabara, K., "Fracture Mechanisms and Life Assessment Under High-Strain Biaxial Cyclic Loading of Type 304 Stainless Steel," Fatigue Fract. Engng. Mater. Struct. Vol. 12, No. 2, 1989, pp. 77-92. [15] Lagoda, T. and Macha, E., "Energy-Based Approach to Damage Cumulation in Random Fatigue," Reliability Assessment of Cyclically Loaded Engineering Structures, R. A. Smith, Ed., Kluwer Academic Publishers, 1997, pp. 435-442. [16] Socie, D. F., "Multiaxial Fatigue Damage Models," Trans. ASME JEMT, Vol. 109, 1987, pp. 293-298. [17] Smith, K. N., Watson, P., and Topper, T. H., "A Stress-Strain Function for the Fatigue of Metals," Journal of Materials, Vol. 5, No. 4, 1970, pp. 767-776. [18] Liu, K. C., "A Method Based on Virtual Strain-Energy Parameters for Multiaxial Fatigue Life Prediction," Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and R. Ellis, Eds., American Society for Testing and Materials, Philadelphia, 1993, pp. 78-84. [19] Macha, E., "Simulation Investigations of the Position of Fatigue Fracture Plane in Materials with Biaxial Loads," Mat.-wiss.u. Werkstofftech. VCH Verlagsgesellschaft mbH, D-6940 Weinheim (Germany) 1989, 20, Heft 4/89, pp. 132-136; Heft 5/89, pp. 159-164. [20] Be~lkowski,W., "Determination of the Critical Plane and Effort Criterion in Fatigue Life Evaluation for Materials Under Multiaxial Random Loading. Experimental Verification Based on Fatigue Tests of Cruciform Specimens," Proceedings, 4th International Conference on Biaxiab'Multiaxial Fatigue, St Germain en Laye-France, 1994, VoL l, pp. 435-447. [21] Lagoda, T., Macha, E., Dragon, A., and Petit, J., "Influence of Correlations Between Stresses on Calculated Fatigue Life on Machine Elements," International Journal of Fatigue, Vol. 18, No. 8, 1996, pp. 547-555. [22] Lachowicz, C., Lagoda, T., Macha, E., Dragon, A., and Petit, J., "Selections of Algorithms for Fatigue Life Calculation of Elements Made of 10HNAP Steel Under Uniaxial Random Loadings," Studia Geotechnica etMechanica, Vol. XVIII, No. 1-2, 1996, pp. 19-43.
M a r i o Witt, 1 Farhad Yousefi, 1 and H a r o l d Zenner 1
Fatigue Strength of Welded Joints Under Multiaxial Loading" Comparison Between Experiments and Calculations REFERENCE: Witt, M., Yousefi, F., and Zenner, H., "Fatigue Strength of Welded Joints Under Multiaxial Loading: Comparison Between Experiments and Calculations," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 191-210. ABSTRACT: In reality most welded components are loaded with a combination of different variable forces and moments which often cause a state of multiaxial stress in the fatigue critical areas of the weldment. If the multiaxial loading is nonproportional, conventional hypotheses are not able to give a satisfying lifetime prediction. This investigation is a cooperation of three German research institutes to build an experimental database for the verification of different concepts of lifetime prediction. In accordance with former investigations, a flange-tube connection made of steel P460 is used. The test program is divided into constant amplitude and variable amplitude tests. The ratio between the nominal bending and shear stress is 1. The ratio between the local shear and normal stress at the critical point is 0.6. For the variable amplitude tests, a Gaussian standard is used. A lifetime prediction software for multiaxial state of cyclic stress was developed. The software has a modular structure and allows calculations with different hypotheses and methods. The calculations are based on the local elastic stresses. This is an acceptable method for high-cycle fatigue. The failure criteria "critical plane approach" and "integral damage approach" are used in the software. Lifetime predictions for the flange-tube connection were performed using various hypotheses and concepts. In this work, one type of calculation, the integral approach with elementary Miner's Rule and damage sum D = 1, is illustrated. The calculations correspond well with the experimental results. KEYWORDS: combined loading, welded joints, experimental database, lifetime prediction, prediction software, critical plane approach, integral approach Nomenclature R S T
Mb M~
Wb Wt
Fb Ft 1 Da di tr
Stress ratio N o m i n a l stress N o m i n a l shear Bending m o m e n t Torsional m o m e n t Section modulus (bending) Section modulus (torsion) Bending force Torsional force Length Outside diameter Inside diameter Local stress
1 Technical University of Clausthal, Institute for Plant Engineering and Fatigue Analysis, Leibnizstr. 32, D368678 Clausthal-Zellerfeld, Germany. 191
Copyright9
by ASTM International
www.astm.org
192
MULTIAXIALFATIGUE AND DEFORMATION
~- Local shear
Ktb Notch factor bending Ktt D ~v~a cr~ r~ c~ c, 6
Notch factor torsion Damage sum Local equivalent stress amplitude in an intersecting plane Local stress amplitude in an intersecting plane Local shear amplitude in an intersecting plane Factor for bending Factor for torsion Phase shift
Under operational conditions, welded components are loaded with a combination of different variable forces and moments which often cause a state of multiaxial stress in the fatigue-critical areas of the weldment. The current information on fatigue life calculation for welded components under multiaxial stresses is very limited. Besides insufficiently developed hypotheses that consider multiaxiality and damage accumulation, there is also a lack of experimental results by which calculation concepts and procedures could be developed and checked. This investigation is a cooperation of three German institutes (LBF, Fraunhofer Institut fiir Betriebsfestigkeit, Darmstadt, Institut fiir Stahlbau und Werkstoffmechanik, TH Darmstadt, IMAB, Institut fur Maschinelle Anlagentechnikund Betriebsfestigkeit, TU Clausthal) and aims to build an experimental database for the verification of different concepts of lifetime prediction methodologies. New methods (modifications) for fatigue life calculation are developed and checked. The presented test results and calculations are only a small part of the whole investigation. General Section
Material and Specimen A flange-tube joint of fine grained steel P 460 with a yield strength of 520 MPa and an ultimate tensile strength of 670 MPa was used for the experiment. This material is often used in pipe, tank, and pressure vessel construction and allows comparisons to be made with existing test results [1]. Customary rolled panel material with a thickness of 30 mm and rolled pipe material with a diameter of 88.9 mm and a wall thickness of 10 mm were chosen for the semi-finished products. The specimen geometry and the joint preparation are shown in Figs. 1 and la. The seams were welded in three steps (Fig. lb). Before welding the flanges and tubes have been heated to 150 to
240
~ vn
FIG.
25 U~
1--Geometry of the specimen; all dimensions are in ram.
WIFE ET AL. ON WELDED JOINTS
193
~
60
FIG. l a - - D e t a i l o f the seam preparation; all dimensions are in mm.
160~ The welding parameters for the individual layers were adjusted according to the table below. The following filler material has been used: Oerlikon Carbofil NiMo 1, AWS/ASME SFA-5.28 ER 90 S-G with a diameter of 1 mm and shielding gas: EN 439-M21 (82% Ar, 18% CO2). Layer Root Hot pass filler weld Cover layer
Us
Is
vs
Type of Arc
2 0 . . . 21 V 2 7 . . . 28 V 30 V
160 A 180... 190 A 240 A
32 cm/nfin 44 cm/min 30 cm/min
Short arc Pulsed arc Spray arc
All specimens were stress-relief annealed for 150 min at 540~ before testing. Experimental Equipment
The test bench principle was developed by the LBF for the fatigue strength experiments on the flange-tube specimens (Fig. 2). The force for bending and torsion is provided by a separate hydraulic actuator and a lever. A 100 kN hydraulic actuator for the bending and a 250 kN hydraulic actuator for the torsion are fitted. The actuators are each kept in two beatings. The geometry of the test setup ensures that through the introduction of torsion, there is practically no bending in the critical area [2]. The tests are carried out under force control. Controlling the two hydraulic cylinders was a problem in these tests. A variable-amplitude test should also have a test frequency as high as possible, and an exact control of the phase angle of the power signal. The Technical University of Clausthal developed a software that allowed a constant phase difference between the two power signals to be adjusted and controlled [3].
FIG. t b - - D e t a i l o f seam.
194
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 2 - - T e s t bench.
The criterion for lifetime (cycles to failure) is determined by internal pressure leakage of the specimen which occurs when the crack has grown through the wall of the tube. When starting the test, the interior of the specimen is set below atmospheric pressure. The pressure balance that occurs during the leak of the specimen serves as a signal for the end of the test. Beside the number of cycles to detect a leak, the technical crack initiation (cracks with a depth of 0.5 n u n and a length of 10 mm) were determined using a DC potential probe [2].
Test Results Results o f the Constant-Amplitude Tests For the constant-amplitude tests, the test program shown in Table 1 was carried out. The results of the constant-amplitude tests are shown in Figs. 3-5. For the tests the nominal stresses for bending and torsion are determined according to the following formulas and Fig. 6: S
Mb _
Wb
Fb" l Wb
and
Wb
T = Mt = Ft" l W~ W,
~(D~ - d:) 32 D a
Da = 84.9 m m
9
and
di = 68.9 m m
with
Wt -
~r(D~ - d:) 16 9 Da
l = 350 m m
Wl-I-r ET AL. ON WELDED JOINTS
195
T A B L E 1--Program of the constant amplitude tests.
No. W1 W2 W3 W4 W5 W6 W7 W8 W9 Wl0 Wll W12
Type of Loading Bending
Stress Ratio R
Type of Loading Torsion
+ ...
-1
. +
+
"~'i
+ + + + S m = T~ + ... + +
- l - 1 - 1 -1 S m = const. 0
Ta = S. Ta = Sa Ta = Sa
.
"0"
T~ = Tm= + . + T~ =
0
Ta = S,,
W i t h t h e n o t c h f a c t o r s Ktb = 2 . 2 a n d K , proportion:
=
Stress Ratio R
.
.
.
. . - 1
.
.
-1
6 ""0 ~
- 1
6 =
- 1 - 1
Sa
Tm= const . .
.
.
90 ~
frlf~ = 1/5 frlf~ = 5/1
S~
.
Phase Difference, Frequency Ratio
.
.
.
-1
...
0 0 0
"0" 6 = 0~ ~ = 90 ~
.
S,
1.32 the local stresses result in the following
'r =
0,6
O"
P u r e t o r s i o n at t h e s a m e n o m i n a l s t r e s s a m p l i t u d e as p u r e b e n d i n g l e a d s to a s h o r t e r f a t i g u e life. C o m b i n e d i n - p h a s e b e n d i n g a n d t o r s i o n l e a d s to a r e d u c t i o n o f t h e f a t i g u e l i f e b y a f a c t o r o f 10, c o m -
MPa
I
t
..........{
i
{ i
~
Lltlt
', . ,.~.ii.......
I
i
i
i i~1
i
i
r
..............~. ~
bending
i
IIIF
!
i
t
i
tit
,~..~ ..............~......~...~ ~ .~..~
i i i iii
I
i ! i iii
3OO
l~" E
-,~
80
E
60
t--
rsion
~..bending ........ii ......
100
0
i a!-.-i-..i-i.i-} ~............. ~ i
El............. : ii i-i!i ..
Q-- i .+....i...i. ~ ~-.i. iiI................... !iFi "{ i -.i-.-.i.@ ii il i
ii i l ii ii!!il i i III ii :iii ZZ ii Z
iii"-iiii
........... - ,,"
.
,~
- " I~ T !"
~i
ii
1 03
1 04
. . .i.i..........ii
Z~Z~ZIZ
i
!
iiiii............................
i i ,i .......... f""'"]"'"~"~i "'~'~
i6=90~ i
ZZIII ii
i
i
i i
1 05 cycles to failure
106
i
i i ii!i
i
i
i
i
i
107
N
F I G . 3 - - E x p e r i m e n t a l results o f constant-amplitude tests: R = - 1 ; bending, torsion, combined in and out o f phase.
196
MULTIAXlAL FATIGUE AND DEFORMATION
M Pa -o
3 00
= .'~i -E
c-
i
~, ~N( J t.,-,.,. Y] '~ . ~ ~
/
iii
i,
!
i
i, / i ii ii )i )i 'alternating berld'ng" 9
i))/i
i!~
~ ~ , ~ .... ~. - . - . ~ y C d .~. . - - . : ! - - % ~ 4 ~ - . - : ) . ~ - ~ - ( ~ l ~ i i i iilii / ] -"~ , i ~ , ,~ bendin g and tors on with ' "L~.~4). -IU` d,fferent f r e q u e n ~ ; , ; " ~ ' ~ ~ / - - . ~
,
................ t ...-'--+-'. f-.-F-.'}--i..~..i-
,
"" r -~ .c:
.....................~...-...)...................rt-i--i--i-i ....... ii)..........T--i--ri-i--t-~l ...... ...........i.-.-.-..-.-.t-----i -- ~--i--ti ) ~ ~ ) M bending i I i i il ........................ .~.--..~-.4-..~..~ .......... ~.........L-I-..F-LI..H.I .........-i-.........i.-...-~...-.L.~d4 ....... r ..........~-..-.....L....'-..L..I-.~..pL ........
100 80
and constant Iormon
------
. . . . . . . . . . . . .
, .4., , ,, . I . .~ . . ,,,, ~ i'i~i! i,-
9
, , , , i,,
i' 'gn-ado s'n 'th J.. i L ~.~LJ4-]-'~-Z]......... L.]. J..i..i..k i i i idifferentfrequency(f=5*f)i i-i i(.l.~l~._...~ i i iili ~ .............. TTTI t b F--T-TT/"iT~'iT........!..T..T T T M I Fi ...........................'........... ......EJ-t-f!ft~i? ......... .........i--f---~--i-ialtemating ....... torsion I-.I ....... I;L .+i.I. ...............f--i.....i..-iand ......... constant bending R--I lil ~ ~ i !!~
_ _L_..i...i i
0ell
!iIi
i! i! ............. ' ' ''""I .......it'!, .....,...,,,,,,ill .. .......i+d-f, .....,...... i
.
40 i
i i iiiii
103
i
.
.
.
i i iiiii
104
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.
Ti
.
.
i
.
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.
i
i
i
i ' fill
105
i
i
i i i!i '
. . . .
i iiii
10 6
cycles t o f a i l u r e
10 7
N
FIG. 4--Experimental results of constant-amplitude tests: R = - l ; bending, torsion, combined with different frequencies and different mean stresses.
m
MPa
,-i................ i~~~' ~i " "~..~..' "' ' ~~~ i .................... ~..........;......~.....i i i ..... } ~....i~bending .4..H .................... [ ' ~..........,.....~.....i (R ' =LI~) '''"i[.!ii .4..;..iu...~.
300
I-.............. L...-U~~.....~i-Xr
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.
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k .............. _L_....!. "r"
.
.
.
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.
.
.
.
i !i
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=
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i i i ii'"
.............. : ................ L.+.-~.§ i
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-.
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.
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=
I l,l,ll
.
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9
.
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.
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~_s
.t .................. f---+---t-+-H-MF
I
, , ,i,,,
i
.+
--
cycles t o f a i l u r e
) ) ))))J L.L~.L'
.--~
........... ~-t--t-i+t+H
, , II,,,
10s
i
10 6
,
, ,
,,~,
107
N
FIG. 5--Experimental results of constant-amplitude tests: R = O; bending, torsion, combined in and out of phase.
WI-I"T ET AL. ON WELDED JOINTS
197
FIG. 6--Geometric dam for calculating nominal stresses.
pared with the S-N curve for bending. The combination of bending and torsion with a phase difference of 90 ~ causes, as other experiments on notched specimens show [1], a reduction of the fatigue life compared with the in-phase load. This applies to the stress ratio R = - 1 as well as R = 0. The combination of bending and torsion with the following frequency proportions (Eq 1) leads to a reduction of the fatigue life (given in the number of stress cycles of the higher frequency load) by a factor of 5 compared with the S-N curve for pure bending. fbend
ftorsion
__
1 5
and
fb~.d _ 5 ftorsio~ 1
(1)
The combination of an alternating bending stress and a constant torsion stress leads to a small increase in the fatigue life (Fig. 4) compared with pure alternating bending stress. Up to now it is not possible to explain this experimental result and the fact that the comparison of the cycles to crack initiation shows that a tendency for a longer fatigue life, especially at low stress, is discernable. The combination of alternating torsion stress and a constant bending stress leads to the same S-N curve as pure torsion (R = - 1). A factor which might reduce the fatigue life is not disceruable (Fig. 4). With pulsating load (R = 0) there is only a reduction of the fatigue life referred to the values of an alternating load (R = - 1 ) in the case of bending. With torsion and combined bending and torsion, there is no significant influence of the stress ratio. The S-N curves for R = 0 and R = - 1 fall together (Fig. 5).
Results of the Fatigue Tests with Variable-Amplitude Loading In the fatigue tests the following test program shown in Table 2 was carried out. A Gauss-standard load spectrum with a capacity of 50 000 load cycles is used [10]. Two different modifications were used, one with a stress ratio of the maximum amplitudes of R = - 1, the other with R = 0 (Figs. 7, 8).
198
MULTIAXIAL FATIGUE AND DEFORMATION TABLE
No. BF1 BF2 BF3 BF4 BF5 BF6 BF7 BF8 BF9 BF10
2--Program of the variable amplitude tests.
Type of Loading Bending
Stress Ratio R
Type of Loading Torsion
+ ... + + + + + ...
-1
. . + Ta = S,, T,, = S,~
+
"0"
+
0
-
1 1 1 1 0
FIG.
Stress Ratio R
.
.
.
. . - 1 - 1 - 1
Ta = Sa + . . +
- 1 - 1 .
.
.
.
. 6 = 0~ 6 = 90 ~ fr/fs = 5/1 uncorrelated
.
T~ = S a Ta = Sa
7 - - P a r t o f load history: R = - 1 .
F1
/-'x t
FIG.
Phase Difference, Frequency Ratio
8 ~ P a r t o f load history: R = O.
0
"0"
0
6 = 0~
0
6 = 90 ~
199
WlTT ET AL. ON WELDED JOINTS
The results of the variable amplitude tests are shown in Figs. 9-1 l. The fatigue life is at its greatest with pure bending (R = - 1 ) ; torsion (R = - 1 ) leads to a shorter fatigue life than bending stress. The combination of bending and torsion in phase (R = - 1) results in a reduction of the fatigue life by a factor of 10 compared with pure bending. Bending and torsion with a phase difference of 90 ~ (R = - 1) results in a reduction of the fatigue life compared with the tests without a phase difference, the same as the S-N tests. The effect is more distinct in the variable amplitude tests than in the constant amplitude tests (Fig. 9). The combination of bending and torsion (R = - 1 ) with the following frequency proportions (Eq 2) leads to a fatigue life (given in the number of stress cycles of the high-frequency load) similar to the combination of bending and torsion with the same frequency (Fig. 10).
fbend ftorsion
1
5
(2)
The uncorrelated combination of torsion and bending (Fig. 12) meaning the load history for bending and torsion are generated in a different way, the experiment leads to a reduction of the fatigue life (for leak) compared to the combined in-phase loading. This effect is quite distinct (factor ~ 3) especially in low stress areas (Fig. 10). Pulsating loading (R = 0) results in a reduction of the fatigue life by a factor of 2 to 3 relative to alternating loading (R = - 1) (Fig. 11).
Calculation of Fatigue Life Lifetime Prediction with Design Standards For the prediction of fatigue life for the multiaxial state of cyclic stress, no valid general method exists up to now. Therefore, different strength hypotheses are used depending on the type of material
t
I
I
li
I
I
I
I
9
MPa
.................i'--"i--i'"'i'"'i''i"id"
"!'--"...bending
__'+'-~
300
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and lorsmon
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i/ii~l .rll
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). . . . . n
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: 7/)-..... - ........... ~-~, .... ~'"
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I
V.b+.~'..,................ ~- bending)d..~
-~i ,
bending
~)i/
/
and torsion
i. I
I
i
il ........ )-t-)-i
ilii
i
I
to failure
i i iiiii
......... ~ ........ i-i....i.
I l i i ,
105 cycles
i
.... i .............. t ........... til
.l.....-..i .'..~
illil
104
! I
,
10 6
i
i
i
i
iiii
10 7
N
FIG. 9--Experimental results of variable-amplitude tests: R = - 1 ; bending, torsion, combined in and out of phase.
MULTIAXIAL FATIGUE AND DEFORMATION
200
(1)
'''''
MPa
'
.
'
'
','''
!
!
!
"O
300 Q.
E o~ o~
i !iii,, bending a~ndtorsi;n~ , ~ .......
09
~ 8 = 90 ~
100
E
~
i i i i i!i,. i ! .~ i i ioenainging a,d andlors , o ~ I ~ ~ ~ uncorrelated
~ - " / ~
80
E O r-
60
E
40 103
104
105 cycles
106
107
to failure N
FIG. l O--Experimental results of variable-amplitude tests: R = -1; bending, torsion, combined with different frequencies and uncorrelated.
= =
.....
I
!
I I I ,!~l
!
i i iiii~ending
............ ~..---f---.~-i.--i i-i-il ...............-...-----b--..i-id-.i.. /..................i..-..-.-i-.-f-i-1 i.1-.1 bending
aoo
...........
o.
i
i iiiii}/
/
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. . . . . . . . .
.
i
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.
.
.
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.
.
.
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.
~
l
Ill
J'i
i'1
i { i t',i', i ', i i~,';iilbendingandtorsion ', i i i~;;: ...............1--YiiTri ......... --t-i-i-i-~-t?i:i_ ...." ..... ]-,.,~ ""~ ~'................................... ...... ~-FTT~H "~--t'""i'"i"'F1 .............. g"'i""f"i"-~U'i'~ ................ , ,, , , , , ,, r
.......................... i-.i..-....q...-.'..............r
o
.
! I ,!!r]
(R =-1)~
~ i i!!! i i i ii!i . . . . . . . . J ............. f ........ O-,.:...~d.4.. t .............................. i ........ iiiiit i i i !!i ~ :../;: ! ~.~? i i ! iiii~ l iijibendingandtorsion ~ i i i~.i -A i i i i ijii J i i i o ,t :,j i i ~i -" i i i ill i ~ i 5=0 i i~ ~[ ~< i i i i;
~1)
,"
. . . . . . . .
M Pa
6 0
~
.......... ~-.F-i--H-..f.4 ................f--..-.-..~-.-H-i--~
~
~
~ i i i i !1
.........................................................
40
' , /
10 a
i
....
,'
i;i:
~'
l
i
i i lii
i
104
!
I
l__L
T,, ................ r-r-!
i
i
ili.i
i i iJ[ii
I ~
-
, , , ,,
_2 .
i:i i iii ::::i ~ ! i
~,
i
i
i i iiti
105
i
106
L -L
,
_._L.k
i
! i
i
i i
,
';
107
cycles to failure N FIG. l l--Experimental results of variable-amplitude tests: R = O; bending, torsion, combined in and out of phase.
wl-n- ET AL. ON WELDED JOINTS
201
torsion bending
FIG. 12--Part of load history of uncorrelated combination of bending and torsion.
and failure mechanisms. There are many standards for the correct design of welded joints. Most standards take into consideration the multiaxial effect by using equivalent stress. The "distortion energy hypothesis" and the "maximum shear stress hypothesis" are used in Refs 4-7, and "maximum normal stress hypothesis" in Ref 8. The fatigue life of welded joints predicted by the standards is valid in the case of proportional stress with constant direction of principal stress. However, these methods are not satisfactory in the case of nonproportional stress with changing direction of principal stress which, as experiments show, results in a shortening of fatigue life compared to proportional. For this reason additional hypotheses are being developed and checked.
Calculation Software "maxi pro" In the past two years we have developed at TU-Clausthal a lifetime prediction software for multiaxial cyclic loading named maxi pro. With this software it is possible to perform extensive calculations for complex combined cyclic stress (e.g., for loads with phase shift and/or different frequencies and uncorrelated). The software is based on a modem object-orientated programming language as a graphical user interface (GUI) application and will run on a standard personal computer. Due to the modular program structure it is possible to perform a variety of multiaxial calculations with different concepts and hypotheses. New calculation methods and hypotheses can be easily integrated and proved. The program structure is shown in Fig. 13. The software allows fatigue life prediction for two kinds of material according to the failure mechanisms. It distinguishes between: 9 Group A: ductile and defect-free 9 Group B: brittle or ductile and defective (micro cracks) The calculations are based on the local elastic stress which is acceptable for high-cycle fatigue. According to the type of material and load, different failure criterion will be used:
9 Critical Plane Approach--Failure is caused in the intersecting plane with maximum damage Omax. 9 Integral Damage Approach--Failure is determined by the integral of the damages in all intersecting planes Dim.
202
MULTIAXIAL FATIGUE AND DEFORMATION
Different fatigue hypotheses are used for the calculation of equivalent stress based on the material type and parameters. For damage accumulation the software contains the different modifications of Miner's Rule. Data input for the calculation is (Fig. 13): - - L o c a l stresses at the critical location of the part. These stresses can be taken from an FEM (finite-element method) analysis or directly from a measurement at the critical location with a strain gage.
FIG. 13--Structure of software.
WIFE ET AL. ON WELDED JOINTS
203
FIG. 14--Steps offatigue life prediction.
--Material data--fatigue strength under reversed bending (~rw) fatigue strength under reversed torsion (zw) pulsating fatigue strength (~rsch) --S-N curve data. The fatigue life is predicted according to the following steps (Fig. 14): --Calculation of normal stress (~r~) and shear stress (~-r in each intersecting plane q~from 0 ~ to 180 ~. --Calculation of equivalent stress ~rv~in each intersecting plane. --Damage accumulation (Miner's Rule) D~ in each intersecting plane. --Determination of Dmax for critical plane approach or predicting Dim for integral damage approach. --Calculation of fatigue life.
Prediction of Fatiguefor the Flange-Tube Connection The calculation of the flange-tube connection was performed using various concepts and hypotheses. Due to the vast amount of data involved, only one type of calculation will be shown below: The following conditions are chosen: Type of material: Hypothesis: Damage accumulation: Damage sum:
Ductile (group A) Square integral damage hypothesis (QIDH) Elementary Miner (no endurance limit) D=I
The QIDH is based on a suggestion by Esderts [9] to obtain an equivalent stress by a linear combination of the shear stress and normal stress amplitude.
204
MULTIAXIAL FATIGUE AND DEFORMATION
(rv~a = c~ ~'~a + c,~ o'r
(3)
The constants c, and c~ are determined for the special case of pure bending and pure torsion with constant amplitude at 105 cycles. This takes into account the fatigue strength behavior of the basic S-N curves in the finite life fatigue strength zone when calculating the fatigue life. The calculation for the damage accumulation in each intersecting plane will be performed by using the equivalent stress calculated with Eq 3. For the QIDH the fatigue life will be determined by using the integral of squared damages (square averages of the damage sums) in all intersecting planes. Calculation Results Calculation with Constant Amplitude~The calculated S-N curves are shown in Figs. 15 to 17. The results of the fatigue life estimation at R = - 1 for pure bending, pure torsion, and combined bending and torsion with 6 = 0 ~ (proportional load) are similar to the results in the experiments. Shifting the phase to 6 = 90 ~ causes the results for the fatigue life calculation to increase by a factor of 2 (Fig. 15). Calculations with a variable slope of the S-N curve based on the stress ratio and on data from pure bending and pure torsion S-N curves correspond welt with the experimental results (Fig. 18). Calculations for combined load with different frequencies show, similar to the experiments, a shortened fatigue life (Fig. 16). Calculations for alternating bending and constant torsion show no change in fatigue life compared with pure alternating bending (Fig. 16). Calculations for alternating torsion and constant bending show, similar to the experiments, no influence on fatigue life compared to alternating torsion (Fig. 16). The prediction for bending at R = 0 compared to R = - 1 shows no real change for the fatigue life (Fig. 17). A reason might be the small value of c~ = 0.0128 (in comparison to c~ = 2.533). The calculated S-N curves of torsion, bending, and combined torsion and bending at 6 = 0 ~ and = 90 ~ at R = 0 match the S-N curves at R = - 1.
MPa 9o
. . . . . :___:__. , , . 4 :
. . . . -___:__:_v:.,.
9 ! .!!!!L!
300
L. . . . . . . .
E
"-
...........
' "{
iiii'l~176
bend./tor. 9 /
tJ} ,,~
. . . . . ~__!_L ~ U_L,
"
':
100 .
.
.
.
.
r
E
40
. . . . . . . . . . . . . . . . . . . I : ', : : : : : : 1 : : : :::;:1 I
10 3
I
I
I IIII
I
I
10 4
l
Illll
:
:6=90~ .......
I
I
I
10 5 cycles
to
failure
.......... I : ', : : : : : : I
I Ill
I
10 6
I
I
IIIII
10 7
N
FIG. 15--Calculated results of constant-amplitude tests." R = -1; bending, torsion, combined in and out of phase.
wl-r-] ET AL. ON WELDED JOINTS
MPa "1~
.....
300
-
- ....
!__.!..'~,';.,!/
....
!._.!._:__v.!::L
:
:~--i~,bending
"
"
" : """"
ii::i
'
"
......
i_.i..:_i_i_!_-i_
;_.!..:_!_:_,.!LI .....
!..!_!_:r
:
:
: : :::::
: : :::::
" " " ~""::~ " : " "":"" -:-r:: . . . . . alternating bending and :T
:2
':: .....
b e n ' d . / i ~
~
....
205
T
~
!...;
._i_i~i
....
"--+_Li_ii!-
"
::ill
100
.E
40
.... ::--r-:-:'ii
I
.....
:"I
10 3
.... i---i--i--i-~,., frequency, !if_5, fb)
.....
,
10 4
",
10 s
, -ii!',"ii
...... -1
10 6
10 7
c y c l e s to failure N FIG. 16~Calculated results of constant-amplitude tests: R = - 1 ; bending, torsion, combined with different frequencies and different mean stresses.
Calculation with Variable-Amplitude (Operating Load)~The calculated fatigue life curves are shown in Figs. 19 and 20. The calculations for pure bending, pure torsion, and combined bending and torsion 6 = 0 ~ show results similar to the experiment results (Fig. 19).
MPa
---iiiii i- i-iiiilI
9
t
, , . . . . . . . . . . .
~-.... i-300
I
.... ..............
~
/
6o
"
I 40 L
,
,
.....
,
,
,[,,,
......... i--ii-iiiil,._,_,_
i i
or si
L.... i
oE
. . . . . . .
e n d i n g ( R = - I ) i-i-[[i] ..... [ - - i i i i - i - i
-
-
~
"
:-_~-i+ii.~
_---i--!~i?Ti-::-ii::~
103
i i i!iiiil : : ::',:::~
i i i i!iill : : :::::~
i ibend./tor.! i iiiiiil : : :~3=90 ~ : ', ::::::1
I
I
I
I
I
IIIII
104
I
I
I IIII
I
I
105
IIIlil
I
106
I
I
IIII
10 7
cycles to failure N FIG. 17--Calculated results of constant-amplitude tests: R and out-of-phase.
=
O; bending, torsion, combined in-
206
MULTIAXIAL
FATIGUE
MPa -o
300
.~
~,.
W
,, =
i
i
!
',
'. : ', ',',',:1
i
i
Illl
'-
. . . . . . . . . . . . . . . . . .
"
: : ::::::"
,,
,. . . . , . . , . . ,
I
i i i iiiiir
. . . . . . .
E1~
|
AND DEFORMATION
100
.
.
.
.
.
.
:
.: , . . . , , , , i
.
.
.
:
.................
:
,
:
....
:_:_:, , , , , ,
.. .. . . . . . . . . .
,
i i~iiiil,,,4Li .
.
.
.
.
test prediction
9
: : ::::::" ., . . . , . . , . . ,
b
.
', ', ',:',',',1
.
w
;
,,,,
,
,. . . . , . . ,. . ,. , , , , ,
i iiiiil
i i i iiiiil
. . . . . .
;
;
;
. . . . .
80 0o
e0
,' ,
,, ,. . . , . ,. ,. ,, . . . . . . . . I
I
I
10 3
I
, , i
b b
Illl
., . . . , . . , . . , . . . . . . . I
I
I
,,,,, . .
, .
.
.
I l l l l
10 4
.
.
.
.. .. . . . . . . . . . . . . . . . . i
I
I
105
.
,,,i . .
, . .
[ I L l
,. . . . , . . ,. . , . , , , , ,
I
I
I
10 6
I I I I
10 7
c y c l e s to failure N FIG. 18--Comparison of calculated and test result; variable slope of S - N curve: R = - 1 ; combined bending and torsion out-of-phase.
B y s h i f t i n g the p h a s e to 6 = 90 ~ a n e x t e n s i o n in f a t i g u e life w a s c a l c u l a t e d c o m p a r e d w i t h 6 = 0 ~ (Fig. 20). A n a d d i t i o n a l f a t i g u e life e x t e n s i o n is p r e d i c t e d for the u n c o r r e l a t e d case. T h e c a l c u l a t i o n s at R = 0 for p u r e b e n d i n g a n d c o m b i n e d b e n d i n g a n d t o r s i o n at t~ = 0 ~ c o m p a r e d to R = - 1 s h o w a s h o r t e n e d f a t i g u e life (Fig. 20).
MPa "ID
.=_
I
300
.
E
:ii
r
i
c
, m
E
100 80 60
O
t,-
I
/:::;: ~"~ii"i~::~: ..... 'bencl'ng' ::i::.... t0rsion~~LL ii
a.
.=
I
.
.
.
.
.
.
.
.
::/ii:
bend.~o;.~
:: ii
-:...... :--" bend./tor.i---i--i- iii ::!..... [ i!&=90~ '::::::!::[: ]ii
, - ~--:, - ~--:. . . . . ~,...:..:.
. . . . ::'" i'*::.
.
.
.
40 i
10 3
;;
I ;
10 4
I
; [
10 s
I[;
10 6
I
; ,
!i,~
10 7
c y c l e s to failure N FIG. 19--Calculated results of variable-amplitude tests: R = - 1 ; bending, torsion, combined inand out-of-phase.
WlTT
MPa ",=r o
300
I
t
Q.
I
ET AL. ON WELDED
I
. . . . .
,,I
I ,
1
]
:: i i ::::::::::L bending:;(R:'~....i "'-?--?'-,'?'iiii l .... iibe~dii~iii~
207
JOINTS
, '
, I It ' I , , ,
:::: ::::::i~
E
t~
.= w I
100 80
t-.i
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. . . . . . . .
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. . . .
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. . . . . . . .
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,
,
,
,
,,,,,
. . . . . . . .
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,
40 I
10 3
I
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I
I
I
Jill
10 3 cycles to failure N
10 6
I
I
IIII
10 7
FIG. 20---Calculated results of variable-amplitude tests: R = O; bending, torsion, combined inand out-of-phase. Comparison of Experiment and Calculation--The experimentally determined fatigue life and the calculated results are compared in Figs. 21 to 24. It is revealed that the calculations with a damage sum o f D = 1 are on the unsafe side as compared with the experimental results either in the case of a phase shift of 6 = 90 ~ or in the case of an uncorrelated loading. Fatigue life calculations for combined loading cases with different frequencies of the individual loads and for loading cases with an additional mean stress show results comparable to the experimentally achieved data. Summary The test results accumulated a large database which allows one to check and improve concepts for fatigue life calculation. Because of the geometry and the size of the specimen, it is possible to transfer the results of these tests to fatigue life problems of real welded structures. The constant-amplitude tests show that torsion and combined bending and torsion lead to reductions of lifetime by a factor of 5 to 10 compared with pure bending. A significant influence of the stress ratio (R = - 1 and R = 0) was not observed. The variable-amplitude tests show the same effects as the constant-amplitude tests, but in these tests there is an influence of the stress ratio. A stress ratio of R = 0 leads to a reduction of fatigue life by a factor of 2 to 3 relative to R = - 1. The calculations with constant amplitude show that torsion and combined bending and torsion lead to a reduction of lifetime compared with pure bending. An influence of the stress ratio (R = - 1 and R = 0) was not observed. This analytical result matches well with the test results. The calculation of phase shift at 90 ~ shows an increase of fatigue life in this case, although the test results indicate a decrease of fatigue life. The calculations with variable-amplitude loading show the same effects as the constant amplitude. But a stress ratio of R = 0 leads to a reduction of fatigue life relative to R = - 1, as was observed by experiment.
208
MULTIAXIAL
FATIGUE
AND
DEFORMATION
1 ,E+07
11 u n s a f e r
Ira" o
--*
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1,E+05
1,E+06
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c y c l e s to failure; e x p e r i m e n t FIG. 21--Nc~z~. versus N~p.for constant-amplitude tests: R = - 1 ; bending, torsion, combined.
1,E+07 Jl
.0 9~
__ ~ c - -~ F--I
1,E+06
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, ....... 9 . . . . bend. andtors, wlthdiffersnt f r e q u e n c y (ft=5~b) ( W 6 )
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cycles
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1,E+05 to failure;
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experiment
FIG. 22--Ncaz~. versus N~xp.for constant-amplitude tests: R = - 1 ; bending, torsion, combined.
WITT
ET AL, ON WELDED
209
JOINTS
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cycles to failure; experiment FIG. 23--N~az~. versus N ~ . for constant-amplitude tests: R = O; bending, torsion, combined.
1,E+07 ~
. . . . . . . . , . . . .
9 bending
(BF1)
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:::-::::]:-
o
~-r.,
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. . . . 7 - - -p- - q - -,- -, r -,- - - | b e n d i n g R = - I ( B F 1 ) Y -,~-,~ - F : bendJtor. ~: ~ - ,~ ~ - ~B . . . . . . . . . . ",~ . . . . . :.,, . . . . . .
~
bendJt~176 R=-I IBF3,
:
",
l~:~ I, , ,
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~
:
1,E+06 C y c l e s to failure; e x p e r i m e n t
FIG. 24--'Ncazc. versus Nexp.for variable-amplitude tests: bending, torsion, combined.
1 ,E+07
210
MULTIAXIAL FATIGUE AND DEFORMATION
Acknowledgment The authors would like to thank the German Research Foundation (DFG) for the financial support of this research program (contract ZE 248/8-1 and ZE 248/8-2).
References [1] Sonsino, C. M., "Fatigue Life to Crack Initiation of Welded Components under Complex Elasto-Plastic Multiaxial Deformation," Forschungsvorhaben der Europ~tischen Gemeinschaft fiir Kohle und Stahl (EGKS), EUR-Report No. 16024, Luxemburg, 1997. [2] Witt, M. and Zenner, H., "Multiaxial Fatigue Behavior of Welded Flange-Tube Connections under Combined Loading. Experiments and Lifetime Prediction," Proceedings, 5th International Conference on Biaxial/Multiaxial Fatigue & Fracture, Vol. 1, Cracow, Poland, Sept. 1997, pp. 421-434. [3] Pfeiffer, J. and Witt, M., "Model Based Control of Hydraulic Test Benches," Proceedings, 8th Congreso Latinoamericano de ControlAutomatico, Vol. 1, Vina del Mar, Chile, Nov. 1998, pp. 189-193. [4] EUROCODE Nr. 3, "Gemeinsame einheitliche Regeln ftir Stahlbanten," Kommission der Europiiischen Gemeinschaft, Bericht Nr. EUR 8849, DE, EN, FR, Stahlbau-Verlagsgesellschaft mbH, Cologne, 1984. [5] AD-Merkblatt $2, "Berechnung gegen Schwingbeansprnchung," Beuth-Verlag, Berlin / Cologne, 1982. [6] ASME Boiler and Pressure Vessel Code, Section III, Division 1, Subsection NA, Article XIV.1212, Subsection NB-3352.4, New York, 1984. [7] Hobbacher, A., "Recommendations on Fatigue of Welded Joints and Components," International Institute of Welding, doc. XIII-1539-96/XV-845-96, Paris, 1996. [8] British Standard BS 5400, "Steel, Concrete and Composite Bridges," Part 10 Code of Practice for Fatigue, British Standard Institution, 1980. [9] Esderts, A., "Betriebsfestigkeit bei mehrachsiger Biege- und Torsionsbeanspruchung," Dissertation TUClausthal, 1995. [10] Amstutz, H., Seeger, T., Ktippers, M., Sonsino, C. M., Witt, M., Yousefi, F., and Zenner, H., "Schwingfestigkeit yon Schweiflverbindangen bei mehrachsiger Beanspruchung," Abschlu[3bericht DFG_Forschungsvorhaben, Darmstadt, Clausthal, (anticipated date of publication Sept. 2000).
Fatigue Life Prediction Under Specific Multiaxial Loads
John J. F. Bonnen I and T. H. Topper 2
The Effect of Periodic Overloads on Biaxial Fatigue of Normalized SAE 1045 Steel REFERENCE: Bonnen, J. J. F. and Topper, T. H., "Effect of Periodic Overloads on Biaxiai Fatigue of Normalized SAE 1045 Steel," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 213-231. ABSTRACT: During the past decade it has been observed that periodically applied overloads of yield stress magnitude can significantly reduce or eliminate crack closure under uniaxial or Mode I loading. This paper reports the results of a series of biaxial in-phase tension-torsion experiments that were performed to evaluate the effects of overloads on the fatigue life of smooth tubes constructed of normalized SAE 1045 steel. Five strain ratios were investigated, including uniaxial ()t = e ~ / e = = 0), pure torsion (A = co), and three intermediate ratios (A = 3/4, 3/2, and 3). Periodically applied overloads of yield stress magnitude allowed cracks to grow under crack face interference-free conditions. Strain-life curves were developed by computationally removing the overload cycle damage from test results and calculating equivalent fatigue lives. A factor of two reduction in the fatigue limit was found at all ratios when these results were compared with constant-amplitude results. Cracking behavior was observed and it was noted that for strain ratios greater than one, cracks initiated along the rolling direction (longitudinally); otherwise, the cracks initiated on maximum shear planes. This observation was used to help explain the similarity in fatigue life results for all strain ratios for both constant-amplitude and overload data. Parameter-life curves were developed using the equivalent fatigue life data and several common multiaxial damage parameters, and the damage parameters were evaluated. It was found that the simple maximum shear strain criterion together with uniaxial overload data provided a good estimate of the fatigue behavior for all strain ratios. KEYWORDS: multiaxial fatigue, biaxial fatigue, fatigue (materials), fracture (materials), steels, overloads, sequence effects, tension-torsion loading, axial torsion loading, in-phase loading, proportional loading, testing, crack closure, crack face interference, mean stresses, cracking behavior Nomenclature
D e e~y y r/ A ni N NI N/ P
D a m a g e to c o m p o n e n t subjected to fatigue Nominal strain Local strain Tensorial shear strain Engineering shear strain N u m b e r o f small cycles b e t w e e n overloads Biaxial strain ratio, e ~ / e = N u m b e r o f cycles applied at amplitude i N u m b e r o f cycles applied N u m b e r o f cycles to failure N u m b e r o f cycles to failure at amplitude i A damage parameter
1 Research staff, Manufacturing Systems Dept., Ford Motor Co., 3135 MD3135 SRL, P. O. Box 2053, Dearborn MI 48121-2053. 2 Professor, Civil Engineering Department, University of Waterloo, Ontario, Canada, N2L3G 1.
Copyright9
by ASTM International
213 www.astm.org
214
MULTIAXIALFATIGUE AND DEFORMATION
R~ Strain ratio, emin/emax S Nominal stress Sop Crack opening stress o" Local stress ~y Cyclic yield stress ~" Local shear stress
Subscripts, Superscripts and Operators x,y,z 1,2,3 a max min rn
op ol sc n A
Orthogonal specimen coordinates Orthogonal principal stress/strain coordinates Amplitude of variable during load cycle Maximum value of variable in load cycle Minimum value of variable in load cycle Mean value of variable during load cycle Value of variable when crack is just fully open Value of variable for overload cycle Value of variable for small cycle Normal to crack plane Range of variable during loading cycle
Introduction The fatigue analysis of typical vehicle structures is greatly complicated by the variable-amplitude loads to which most of these structures are subjected. The analysis is further complicated by multiaxial loading. Recent investigations indicate that, while uniaxial fatigue analysis is satisfactory in most cases, it is estimated that it produces seriously nonconservative life estimates 5 to 10% of the time because of multiaxial loading [1]. A large analytical infrastructure has been built up around variable-amplitude loading, but even with these tools a life estimation error of as much as a factor of 20 [2] is possible. One of the problems with estimating the damage resulting from variable-amplitude loading is that the mean stress rules used are typically developed using constant-amplitudefatigue data for which crack closure levels are higher than for variable-amplitude loading. In the past two decades, several researchers have linked mean stress effects to crack closure levels [3-5]. It has long been known that overloads can alter the crack closure level and radically change experimental fatigue life. Overload studies, usually performed with compact tension specimens, focus on a stress range limited to much less than half of the net section yield stress. Unfortunately, the combination of severe service loading and notched engineering components can often lead to stresses well beyond the range normally investigated in conventionalfatigue crack growth studies. Only in the past decade [5, 6] was it determined that yield stress level overloads, whether compressive or tensile, substantially accelerated short fatigue crack growth rates by reducing closure levels. For the purposes of this paper the term overload will be used to refer to these large yield stress level excursions. DuQuesnay [5] determined that overloads which are of the order of the yield stress significantly reduce crack closure. Compressive overloads drastically reduce closure by flattening asperities and eliminating the interaction between crack faces. Tension overloads stretch the crack mouth open-separating the crack faces and eliminating closure. He was able to determine, with the aid of acetate replicas and with strain gages laid across the crack that overloads of sufficient amplitude and frequency can maintain a fully open crack throughout the life of a specimen. Later researchers [7,8] were able to corroborate these observations via direct optical measurements. Varvani-Farahani et al. [9] obtained similar results for Stage I cracks using a confocal scanning laser microscope.
BONNEN AND TOPPER ON PERIODICOVERLOADS
215
The concept of closure does not work well for cases involvingmultiaxial loading. Instead, the term "crack face interference" is used to extend the closure analogy into combined Mode I/II/III loading systems in which a crack may not "open" or "close." Various researchers [10-14], have found that crack growth rates could be strongly influenced by secondary stresses. Tschegg [15] observed that as crack lengths increased in AISI C1018 steel, crack face interference under Mode III loading consumed an increasing part of the crack driving force, especially at low crack driving force levels. Crack-face interference is also reduced by applying Mode III overloads during Mode III crack growth. Ritchie et al. [16] applied Mode III overloads during a Mode III crack growth test and found that overloads accelerated the smaller cycle crack growth. One group of researchers [17] applied bending overloads to a notched shaft during torsion cycling and found that the overloads not only increased smaller cycle damage but substantially lowered the torsional fatigue limit. Varvani-Farahani [18] removed crack face interference by applying large compressive (Mode I) overloads across shear cracks growing under various strain ratios and obtained accelerated crack growth rates. The current work explores the interactions of overloads with smaller cycles over a range of five different in-phase biaxial strain ratios that varied from pure uniaxial loading to pure torsional loading. Crack face interference-free strain life diagrams were developed for each ratio, and observations were made of the cracking behavior in each strain ratio and used to help explain the fatigue behavior. Finally, the effectiveness of different multiaxial parameters in consolidating test data for different biaxial strain ratios was examined. Materials and Procedures The material used in this study was an SAE 1045 steel in the normalized condition. The steel was hot rolled into a 63.5-ram-diameter bar and normalized to produce a Brinell hardness of 203 BHN. A synopsis of the history of the material, monotonic and cyclic material properties, and the chemical composition may be found in Ref 17. Figure 1 presents the microstructure of the steel in the longitudinal-transverse (L-T) orientations. It has a pearlitic/ferritic microstructure and, because of the nor-
FIG. 1--Microstructure of normalized SAE 1045 steel ( X400), L-T orientation.
216
MULTIAXIAL FATIGUE AND DEFORMATION
malizatiou procedure, has equiaxed grains of roughly 25/xm. Also observable in Fig. 1 is the banding characteristic in this lot of material, but this banding is not evident in the short-transverse orientation. MnS stringers are present in this steel and are aligned with the rolling direction. They range in length from 0.1 to 2 mm.
Testing Techniques Uniaxial Tests--The techniques, methods, and specimen used in uniaxial testing are discussed in Ref 17. Uniaxial Periodic Overload Tests--Overload tests were conducted in strain control. The histories used in the uniaxial overload tests, shown in Fig. 2, consisted of a single compression-tension overload cycle followed by a number of smaller cycles whose peak tensile strains were the same as the peak tensile strain of the overload cycle. The amplitude of the overload cycle itself was selected to be 0.48% strain, which corresponded roughly to 10 000 cycles to failure under conventional constantamplitude testing. A number of smaller constant-amplitude cycles (7) followed the overload cycle, and the smaller cycle amplitude was set, depending on the test, at a value between 0.2% strain and 0.06% strain. Finally, the number of smaller cycles, r/, placed between the overload cycles was chosen such that the overload cycles constituted no more than approximately 25% of the total damage. In this paper the term "small cycles" (sc) is used when referring to overload tests to indicate the smaller cycles in the overload history. Axial-Torsion Tests--A tension-torsion machine performed multiaxial fatigue life tests on the tubular specimens shown in Fig. 3. Rough machining of the specimen consisted of lathe turning on the outer surface and boring the inside. Finish machining consisted of low-stress grinding and sanding on the outer surface and honing the inner surface with successively finer stones. The outer surface was polished to 1 /xm for the purpose of observing cracking behavior. The inner diameter received a 5/zm finish.
1]
caLi ~min
-0
FIG. 2--Uniaxial overload history.
BONNEN ANDTOPPER ON PERIODIC OVERLOADS =_
- 52+0.1/-0.00
217
200+1.0/-0.1
, 33.0-+03,30.0+0~I-0.00
~45+0.0/-1.O R75_+1
\
\
FIG. 3--Axial-torsion specimen. All dimensions in mm.
The axial-torsional tests were conducted using an axial-torsion load frame capable of exerting a 250 kN axial force and a 2250 Nm torque on test specimens. The specimens were mounted with hydraulic grips developed at the University of Illinois [19]. Strains were measured with an axialtorsion extensometer that had the advantage of sensing directly the shear strain at the specimen surface rather than measuring specimen twist. Tests were conducted in strain control, and strains were controlled to an accuracy of 1% by the adaptive parametric control program described in Ref 20. A maximum test frequency of 40 Hz was employed for some high cycle tests while slower frequencies were used for shorter lives. Frequencies above 8 Hz were used only when specimen stress-strain response was elastic and load control was then used on both axes. Specimen failure was defined as the first discernible compliance change. This technique resulted in an estimated failure crack length of 1 to 3 ram. Five strain ratios (A) were used in this research: ~, 3, 3/2, 3/4, and 0 (uniaxial). For tests performed under A = ~ loading the axial actuator is left in load control at 0 load for the duration of the test. Biaxial Periodic Overload Tests--Just as in the uniaxial periodic overload tests a large, fully reversed, overload cycle was applied and followed by 7/smaller cycles, as in Fig. 2. The overload cycle was set such that it alone would cause specimen failure in 10 000 cycles, and it was applied inphase with the subsequent smaller cycles, while the smaller cycles were set such that they shared the same peak strain as the overload cycle.
Biaxial Fatigue Results Uniaxial Fatigue Behavior, A = 0 As discussed in the introduction, the presumption that the mean stress effect observed in fatigue is a direct result of fatigue crack closure implies that if a cycle is free of closure, then the mean stress of that cycle will not affect the crack growth rate--except where the maximum stress intensity approaches the critical stress intensity. The Palmgren-Miner rule [21,22] was used to calculate the equivalent number of cycles to failure for the small cycles in the overload histories. This equivalent life was then paired with the small cycle amplitude, and the crack-closure-free result was plotted (along with conventional fully reversed
218
MULTIAXIAL FATIGUE AND DEFORMATION
fatigue data). For the history shown in Fig. 2, the equivalent number of cycles to failure for the small cycles, N~, is given [17] by ni
Z~//=
1
(1)
= 1
(2)
i
nol
nsc
No--~t+ ~
N~c-
1 nsc
1
l
(3)
~lNol
where ni is the number of cycles at amplitude i, N~ is the expected constant-amplitude life at amplitude i, "oF' indicates overload cycle, "sc" indicates the small cycle, and 7/is the ratio of the total number of small cycles applied to the total number of overload cycles applied (~/= n~c/nol). The life for the overload cycle, No~, is taken from the constant amplitude strain life curve. Figure 4 depicts the constant-amplitude (R~ = - 1, plotted as open circles) and overload data (plotted as triangles). The fatigue limit was reduced from a strain amplitude of 0.0017 to 0.00072--a reduction to two-fifths of its original value by the overloads. A curve which approximates fully open fatigue crack growth is also presented in Fig. 4 and labeled as Ae~g/2. The underlying theory and the development of this curve is described in Refs 17 and 23. Crack Closure-Free Fatigue Life Testing--In the tests in this study the method used to determine whether the cracks were growing under fully open conditions involved performing an overload test
10 -1 "'E
~ &
EE 102
~,'0.
9 Constant amplitude Equivalent. small cycles (z~~ = 0.0048)
i"~/~eff/2
= (~i +/~*)/2
O "O
< C 9
lo
.3
..........
_0_:9_0_
O3 '"~,
~~
= 0.0065
~1;'12 = 0.50(2N~)"~
10 4
........ , ........ , ........ , ....... . ....... . ....... . ........ , ........ , 104 102 103 104 10 s 106 107 108 109 Reversals to Failure (2Nf)
FIG. 4---Uniaxial (~ = g~ ~" = O) periodic overload and effective strain curves for normalized SAE 1045 steel.
219
BONNEN AND TOPPER ON PERIODIC OVERLOADS
1 0 -2 '
X=oo
E
_.E E E
"--" 10-3
Periodic Overload
(~ty). = 0.0035 11
10 -4
' ' .,,,,.| 10 ~
10 2
,,
,,,,,,i
Constant
...,1,,,|
10 3
10 4
..
.,,.,! 10 5
. ,,,,,l|
,
Amplitude
. ...,i,
10 6
I
10 7
,
. IH...
I
I
,
.
10 e
Equivalent Cycles to Failure FIG. 5 - - F a t i g u e response o f normalized S A E 1045 under torsional loading, )t -~ ~ = o~ Circled datapoint indicates special overload level (exy)a ol = 0.00425, (exy)a sc = 0.00125, and ~ = 50.
with an overload amplitude considerably larger than the one used in the standard overload tests. This type of test is termed a "companion test." The circled data point in Fig. 4 gives the life of the companion test for )t = 0. This specimen was subjected to an overload strain amplitude e ~ 0.0065, a small cycle amplitude e sc = 0.00125, and r / = 250. With the exception of the overload amplitude, the adjacent point and the the circled data point share the same test conditions. Since a larger overload is expected to reduce the crack closure stress, the failure of the larger overload to further reduce the fatigue life suggests that, at the lower overload level, the crack was already fully open, i.e., the opening stress was below the minimum stress (Sop <- Stain)- Hence, the overload amplitude of e ~ = 0.0048 produced fully open crack growth. It follows that in tests with the small cycle strain amplitudes below this level (e~c<- 0.00125) cracks are also growing under crack closure-free conditions. Torsion Tests, A = co
The torsional periodic overload tests were conducted using the shear strain history shown in the inset in Fig. 5. The torsional overload cycle, maintained at (exy)~ ot = 0.0035 in all tests, was followed by torsional small cycles. The results of the torsional experiments are also presented in Fig. 5. This and subsequent fatigue life data sets are presented in tensorial shear strain, exy. Tensorial shear strain is denoted by exy while %y is used to indicate engineering shear strain (Yxy = 2e~y). Calculations to determine the equivalent cycles to failure were made in the same way as for uniaxial loading (A = 0), described in "Uniaxial Fatigue Behavior" above. The introduction of the overloads caused a reduction of the constant amplitude endurance limit ca sc from (exy)a = 0.0016 to (exy)a = 0.0007 at 107 cycles--a little less than one-half of its original value. The superscript sc indicates small cycle and the superscript ol indicates overload cycle.
MULTtAXIALFATIGUE AND DEFORMATION
220
Crack Face Interference-Free Crack Growth Under A = o~---A test was performed in order to determine whether the cracks faces under in-phase A = 0 loading were growing free of interference with each other. The test is indicated in Fig. 5 by a circle around the data point. This test has an overload strain level of (exy)~ ol = 0.00425 (constant-amplitude life of 4985 cycles to failure), (eS~y)a = 0.00125, and r / = 50. The rest of the overload data points were subjected to (exr)a~ = 0.0035, including the point adjacent to the circled data point. This data point shared the same small cycle amplitude, (e~)~ = 0.00125, but had ~1 = 20. The two tests have similar lives and it is presumed that since the larger overload, which would further reduce crack face interference, did not reduce the fatigue life, the (e~ = 0.0035 overload cycle has eliminated crack face interference. Intermediate Fatigue Results ()t = 3/4, 3/2, 3) The results of testing under the intermediate strain ratios ()t = 3/4, 3/2, and 3) can be seen in Figs. 6-8, and, like the torsional fatigue data, are plotted as tensorial shear strain amplitude versus equivalent cycles to failure. In the fatigue strain-life diagram for each strain ratio an example history is presented. The overload cycle was applied simultaneously in both the axial and torsional channels so that the peak negative axial strain occurred at the same instant as the peak "negative" torsion strain. The overload cycle ends at the peak positive strain on both axes, and it is followed by the small cycles. The application of the small cycles follows the same ratio: the positive peaks and negative peaks occur at the same time in both axes, and the peak values of these cycles match that of the peak from the overload cycle. The ratio of the torsional to the axial strain is maintained continuously throughout all tests. For each strain ratio Table 1 summarizes both the standard and companion test overload levels, the constant amplitude life at the overload strain level, and the fatigue limit strain for both the constant-
1 0
"2
'
~=3
~=~ a,x
(E~'x ),=0.0015 o (%)a=0.0045 E E
10.3
g~
~
Periodic Overload (s)a=0.001167, (sxy)a=0.0035 Ot
(s~) =o.ool 1s7 {s~ )a=0.0035 = _-
f
""'~e~__.~
ol
Constant Amplitude
' x x [ ~
1 0 .4
'
101
'
' " ' " I
102
'
'
' " ' " I
,
103
'
' " ' " i
104
.
.
. ' " " i
105
.
.
,,.'"i
,
106
.
..,.,.i
=
10~
'
'"'-i
108
Equivalent Cycles to Failure (Nf) FIG. 6--Periodic overload and constant-amplitude fatigue response for )t = ~ E ~ = 3. Circled datapoint indicates special overload level.
BONNEN AND TOPPER ON PERIODICOVERLOADS 10
221
-2 "
~,=3/2
Constant Amplitude
E E "~ 10-3
.-Periodic
g~
Overload
(,~'),=ooo233, (e,~),=0.0035 o,
I0
-4
'
'
''''"I
101
'
'
''''"I
10 2
'
'
'''"'I
10 3
'
'''"'I
10 4
'
'
''''"I
10 ~
'
'
'''"'I
10 6
'
'
''''"I
10 7
10 s
Equivalent Cycles to Failure (Nf) FIG. 7--Periodic overload and constant-amplitude fatigue response for A datapoint indicates special overload level.
10
"2 '
Constant Amplitude
,xxl
~=3/4
E E ,~E10.3 gr
Overload
Periodic
~
(S~', )~0.003, (e~),=0.00225
"~,,~ ~ ,
"
%I ~ -4
'
101
04
(
9
10
E~ - 3/2. Circled e~
'
'''"'I
(~:),=o.oo3 '
102
'
'''"'I
'
103
'
='''"I
'
104
''''"I
'
10s
'
''''"I
'
106
'
'
'''H'I
107
'
' '''"1
108
EquivalentCyclesto Failure(Nf) FIG. 8--Periodic overload and constant-amplitude fatigue response for A = ~ = 3/4. Circled datapoint indicates special overload level.
222
MULTIAXlAL FATIGUE AND DEFORMATION
TABLE l--Summary offatigue datafor normalized SAE 1045 steel. h 8xy
-ex~ -
Standard Overload o/
(e?3)~
(e~)~,
mm[mm 0 3/4 3/2 3 cr
0.0048 0.0030 0.0023 0.0012 0
0 0.0023 0.0035 0.0035 0.0035
Companion Overload
C A Life
Nf 12500 16300 9900 10000 10400
(e~)~
(e~)a ~u
mm/mm 0.0065 0.0040 0.0027 0.0015 0
0 0.0030 0.0040 0.0045 0.0043
C A Life
Fatigue Limit @ Nf = POL
CA
Nf
(e'sx~)a
(6~)a
5000 6500 6900 6500 5000
0.0017" 0.0080 0.0011 0.0014 0.0016
0.00085* 0.00040 0.00057 0.00068 0.00075
10 7
POL/CA
0.50 0.50 0.52 0.49 0.47
CA = Constant Amplitude test, POL = Periodic Overload test, and "*" indicates (e~)a.
amplitude and periodic overload tests. As can be seen from the table, the periodic overload fatigue limit strain at 107 cycles is about one-half that of the constant-amplitude curves for all biaxial strain ratios (h). In the uniaxial tests (h = 0) where periodic overload data were taken out to 10 s cycles the ratio (POL/CA) drops further to 0.42. In a fashion similar to that discussed in "Crack Closure-Free Fatigue Life Testing" a companion test was used in each test series to demonstrate that the fatigue cracks were growing under crackface interference-free conditions. Each graph in Figs. 6-8 contains a circled data point. This data point represents a companion test with an overload strain level which was larger than the standard overload used for that series of tests. As stated previously, since the lives of the companion tests did not significantly depart from those of the standard tests in Figs. 6-8, the standard overloads used in these strain ratios may be presumed to have eliminated crack-face interference for the smaller cycles.
Cracking Behavior The development of cracks in the constant amplitude and the periodic overload tests was similar. As can be seen in Fig. 9, this material showed a very strong tendency towards longitudinal cracking. The microstructure shows strong longitudinal banding into alternating regions of dense pearlite and dense ferrite grains (see Figs. 1 and 9), and cracks tended to initiate and grow in the ferrite-rich regions or along the ferrite-pearlite grain boundaries. This behavior was observed clearly in tests with biaxial strain ratios greater than 1 (A = 0% 3 and 3/2). The cracking behavior observed here is very similar to that described by Socie [24] for another SAE 1045 steel. The cracking behavior for A > 1 is illustrated in Fig. 10 which shows typical initiation cracks for a specimen subjected to A = 3/2. Initiation predominantly takes place on maximum shear strain planes (as marked by solid lines in the figure) and along the longitudinal direction. These longitudinal cracks became dominant and grew to failure. At the lowest strain levels cracks would eventually switch over to growth on maximum tensile strain planes regardless of strain ratio. As the strain level was increased for A > 1, the length of the initial longitudinal shear crack increased until the shear crack would grow the length of the specimen. Cracking for specimens subjected to A < 1 can be seen in Fig. 11 where a specimen subjected to A = 3/4 loading is presented. In these tests cracks also initiated in ferrite-rich regions, but after growing a short distance (-50/xm) they switched over to growth on maximum tensile strain planes and grew to failure. As in A > 1 tests the initial shear crack length would lengthen with increasing strain amplitude, but as A increased the strength of this effect was reduced.
FIG. 9--Preferential cracking through ferrite grains and along the ferrite/pearlite grain boundaries. Specimen subjected to (exy)a ox = 0.00425, (/3xy)a sc = 0.00125, and r1 = 50, and photo taken at Nf = 108 540.
FIG. l O ~ T y p i c a l small crack growth under ~ = ~-~ = 3/2 loading o f normalized SAE 1045 steel. Specimen subjected to (e~ax)a = 0.0027 and (s~xay)a = 0.004. Photo taken at Nf = 6930. 223
224
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 11--Small cracks developed under in-phase A = 3/4 loading. Specimen subjected to (eca)a = 0.004 and (e ~x~y)a= 0.003. Photo taken at Nf = 6470.
Crack Face Interactions It was observed that, for all values of A, shear cracks initiated on shear planes. Under A < 1 conditions cracks soon moved onto maximum tensile strain planes, while for A > 1, cracks mostly grew along the longitudinal shear plane. Shear Crack Growth--At crack lengths of 0.5 mm and greater, shear-crack faces are observed to slide back and forth across each other, and it is believed that this is also true for shorter crack lengths. Under this motion (Modes II and III crack growth) the energy available at the crack tip is reduced by the interaction of crack-face asperities. After being smeared by the overload the asperities offer much less resistance to crack-face motion and the crack driving force is increased at the crack tip. Lateral smearing of asperities consistent with this model was observed on fracture surfaces of specimens subjected to A > 1 straining. Tensile Crack Growth--Under tensile (Mode I) loading asperities contribute significantly to crack face interference (closure) [5]. Compressive overloads applied across the crack face have been shown to flatten crack face asperities [18]. The crack is open during more of the loading cycle and the effective stress intensity range is increased. The difference between the tensile and shear mode overloads is that the asperities are crushed rather than smeared. Crushed asperities were visible in the ten-
225
BONNEN AND TOPPER ON PERIODIC OVERLOADS
sile growth regions of fracture surfaces when A < 1 (and on those for A > 1 which also demonstrated tensile growth).
Consolidation of Fatigue Life Results Constant-Amplitude Tests Cracking in the constant amplitude tests as in the periodic overload tests was dominated by longitudinal anisotropy and cracks were aligned with the longitudinal axis. Presumably, these cracks are not greatly influenced by tension loading and it follows that the life of these tests was controlled by the applied shear strain. Figure 12 shows all of the constant-amplitude data from tube tests and some additional fatigue life data for this material from notched shafts [17] plotted against the applied shear strain (ex~y)a. The figure clearly demonstrates that, in all regions except for data at the endurance limit, the applied shear strain as a fatigue parameter reduces data for A > 1 into a single characteristic curve. The data for A = 3/4 falls on a separate but parallel curve below that of the other data. At the fatigue limit the A = ~ (open squares), A = 3 (dot-center squares), and notched shaft (filled squares) data fall into a single narrow band. The notched shaft data fall in the gap between the runout and regular fatigue data, and, while A = 3/2 data (bar-center squares) initially follows the trend of the other data, it then departs from the trend at the A = ~ endurance limit and proceeds along an extension of the trend curve to a lower level.
10 -2
-~'~,.O'~.
Initiation T y p e ~
-
E
E
E 10 .3
E
k---~
=
[] 9
Tension-Torsion (I) Notched Shaft (i)
;L=3/2 = ;L=3/4
Tension-Torsion
10 .4
[]
X=3
(I)
[] []
~,=3/2 X=3/4
(I) (/)
'
02
'
' ' ' ' " !
'
10 3
'
' ' ' ' " 1
10 4
Initiation Key (I) = longitudinal ( / ) = inclined
'
'
' ' ' ' " 1
'
'
' ' ' ' " 1
10 8
10 8
'
'
' ' ' ' " 1
10 7
'
'
''
....
I
10 8
Cycles to Failure
FIG. 12--Constant-amplitude curves for tension-torsion and notched shaft tests [ 17] plotted on the basis of applied shear strain amplitude ((eC~,)a).
226
MULTIAXIAL FATIGUE AND DEFORMATION
Also shown in the figure is the initiation angle observed for each strain ratio. For biaxial strain ratios of 0% 3, and 3/2 initiation is aligned with the longitudinal axis as discussed in "Cracking Behavior." A single failure at A = 3/2 lies below the A = oo and 3 endurance limit, and this failure did not initiate longitudinally; its initiation was along the plane of maximum tensile strain. It is assumed that this change in initiation type is responsible for the endurance limit shift in the ,~ = 3/2 data. The A = 3/4 data lie below all of the rest of the data in this figure. This is presumably because the initiation plane is aligned with the maximum shear planes. It seems that the influence of strains applied on the tension axis was small since, as seen in Fig. 12, there was essentially no difference in the fatigue response between ~ = 0% 3 and 3/2 loading--the life is the same for a given applied torsional strain. Under A = 3/4 loading an influence of tension loading is seen in the fatigue response of the material--for a given applied torsional strain amplitude the )t = 3/4 life is shorter than that of the other strain ratios. Since the bulk of the life in A = 3/4 and ~ = 0 loading is spent initiating and growing a crack along planes of maximum shear strain these data are plotted in Fig. 13 against the maximum shear strain amplitude, (e 12)a, while the rest of the data is plotted against the applied torsional shear strain amplitude (resolved shear strain on along the x-axis), (exy)a. When the data of Fig. 13 are presented in terms of maximum shear strain amplitude, (e 12)a, in Fig. 14 there is only a small shift in the h = oo and 3. = 3 strain ratios (increases of 3 and 10%, respectively). The correlation of the data based on maximum shear strain amplitude is almost as good as that given by using the shear strain on the initiation plane. As a predictive tool its advantage is that it does not require tests to determine the initiation angle.
10 2
E E
Torsion, ;L=oo (~=~12) [] Tension-Torsion " ~ 1 0 -3. 9 Notched Shaft #
I~
=
Tension-Torsion []
X=3
[]
X=3/2 ;L=3/4
[]
(~,~) (Zxy) (s12)
Uniaxial, ;L=O 104
........ 10 2
i 10 3
........
i 10 4
........
!
........
10 6
i 10 6
........
i 10 z
........
i 10 8
Cycles to Failure FIG. 13--Constant-amplitude tension-torsion and notched shaft tests [ 17] considered on the basis of initiation plane.
BONNEN AND TOPPER ON PERIODIC OVERLOADS
227
1 0 .2
E
Torsion, X=oo
E
E 104 ,~E
[] 9
,~~1~[3----~
Tension-Torsion Notched Shaft
N
Tension-Torsion []
;L=3
[]
;L=3/2
[]
X=3/4
Uniaxial, X=O 10.4 10 2
10 3
10 4
10 5
10 6
10 7
10 8
Cycles to Failure FIG. 14---Constant-amplitude tension-torsion and notched shaft tests [17] plotted using maximum shear strain amplitude.
Overload Tests The mean stress corrections found in most damage criteria are corrections for the presence of closure/crack face interference. When an increasing tensile mean stress is applied to a growing crack, the actual stress range which reaches the crack tip increases because the crack opening stress increases less rapidly than the maximum stress. In other words, Smax - Sop increases because Smax increases more than Sop. The increased effective stress range causes the crack to grow faster. A graph of fatigue life versus maximum shear strain for the periodic overload data is presented in Fig. 15. As observed for the constant-amplitude data, there were no large differences between the resuits when viewed in terms of the resolved shear on the initiation plane, as shown in Fig. 16, and the results when viewed in terms of the maximum shear. In Fig. 15 about two thirds of the data fall within the 2x bands, and most of the data are contained by the 5x bands, while in Fig. 16 the result is quite similar. The Brown and Miller parameter [25], which can be expressed [26,27] as PsM = (exy)~ + 0.5(exx)a when applied to the experimental data results in the plot in Fig. 17. This parameter provides a good condensation of the overload data. Most of the data points fall within the 2x bands, and the 5x bands contain all of the rest of the tension-torsion data. The parameter-life plot for the Fatemi-Socie-Kurath parameter [28,29], as expressed [26,27] by PF = Ya (1 + KF ~ ) , where KF is taken to be 0.3, is presented in Fig. 18. Under this parameter the data fall in much the same fashion as in Brown and Miller's parameter, with somewhat more of the data falling outside the 2x bands. The parameter-life diagram which demonstrates the least scatter is the
1 0 -2 "
5x
2x
2x
"',,
5x
\
E
",,
,
E l 0 "3
V
~.= O, u n i a x i a l
0
Z=3/4
~ > V
..
".
E
,
\\
~
x\
x
@,
.
~ v ~ ....
Z=3/2
10 . 4
'
<>
~.=3
[]
;~=oo. t o r s i o n
'''""I
'
102
10 ~
- -
'''""I
'
'''""I
103
L=O (uniaxial) e f f e c t i v e c u r v e
' ''""I
104
'
105
'''""I
'
'''""I
10 s
'
107
' ''""I
'
' ''""I
10 s
109
E q u i v a l e n t C y c l e s to Failure
FIG. 15--Maximum shear strain amplitude (e ]2)a curves for periodic overload tests.
10
"2 '
5x 2x
2x
5x
E E vE r 10 .3
v
10 .4
[]
~=~
(%)
0
;~=3
(%)
z~ o
k=3/2 Z=3/4
(%) (~12)
- -
v X=O (~12) (uniaxial) . . . . . . . . i . .......i . . .... ..i . . . . . . . . i 101 102 103 104 10 s
X=O (uniaxial) e f f e c t i v e c u r v e
.......i . ....... i . . . . . ...i . . ...... i 106 10 z 108 109
E q u i v a l e n t C y c l e s to Failure
FIG. 16---Periodic overload data considered on the basis of initiation plane. 228
10 .2
5x
2x
2x
5x
"'.. \Q
.''.
o_==10-s
1
0 -4
'
v
3,=0, uniaxial
o
3,=3/4
A
3,=3/2
O []
3,=3 3,=00, torsion
''''"'l
101
'
' ''""I
1 02
'
- -
''"~"I
1 03
'
Z=0 (uniaxial) effective c u r v e
'''"'q
1 04
'
'''""l
10 s
'
'''""I
1 06
'
''''"'l
107
'
~ ''""I
1 08
1 09
E q u i v a l e n t C y c l e s to Failure
FIG. 17--Brown and Miller [25] parameter-life plot for periodic overload tests. 1 0 -2 -
5x
2•
2x
5x
"',
u. D_
"..
10 -3
'.
10 "4 10 ~
v
3,=0, uniaxiat
o
3.=3/4
A 0
X=3/2 3,=3
[]
3,=00, torsion
........ | 1 02
,
......
- -
,i
1 03
,
",Q
,
"..
k--o (uniaxial) effective curve
,, ..... ! . . . . . . . . i 10" 10s
,
,
......
i
. . . . . .
10 s
,,i
1 07
,
,, ..... i 1 08
,
,
......
I
1 09
E q u i v a l e n t C y c l e s to Failure
FIG. t 8--Fatemi-Socie-Kurath [28,29] parameter-life plot for periodic overload tests. 229
230
MULTIAXIAL FATIGUE AND DEFORMATION
Brown and Miller parameter. However, the simpler maximum shear parameter provides very nearly as good a consolidation of the data.
Summary and Conclusions The purpose of this investigation was to observe the effect of overloads in multiaxial fatigue. Inphase strain controlled constant-amplitude and periodic overload tests were conducted on tubular specimens, and the tension-torsion strain ratios (A = ~ x y / ~ ) were selected to be % 3, 3/2, 3/4, and 0. Both the overloads and small cycles shared the same strain ratio. 9 Periodic overloads reduced the 107 cycle endurance limit of normalized SAE 1045 steel to about one-half the constant-amplitude value for all strain ratios. Experiments in which cycling was continued to l0 s cycles exhibited a further endurance limit reduction to 2/5 of the constant-amplitude value. 9 It was found that for h = E~y > 1 cracks initiated and grew along the specimen longitudinal ~ axis. For h < 1 cracks tended to initiate on planes of maximum shear strain, and eventually move onto planes of maximum tensile strain. Another trend was also noted where, at low strain amplitudes, long cracks would tend to grow on maximum tensile planes but, at high strain amplitudes, long shear cracks dominated. 9 Companion tests with overloads higher than those used in the standard test series were performed on one specimen at each strain ratio in order to determine whether the overload level used in the regular tests was large enough to produce crack-face interference-free conditions. These tests indicated that the overloads used did produce a maximum fatigue life reduction, and it should follow that for small cycle amplitudes below that employed in the companion test, the fatigue cracks would grow under crack-face interference-free conditions. Simple models, supported by fractographic evidence, are used to describe the nature of crack-face interference and explain how it was reduced by overloads. 9 A series of multiaxial damage parameters that correlate fatigue data for different strain ratios was examined. For constant-amplitude data it was found that plotting the resolved shear strain on the initiation plane against fatigue life provided good data consolidation. Maximum shear strain also gave good consolidation of the constant-amplitude data. For the periodic overload fatigue data the Brown and Miller parameter gave the best consolidation. However, the maximum shear strain parameter also provided good consolidation of the data and is simpler to implement.
References [1] Chu, C.-C. and Htibner, A., Personal communication, 1997. [2] Bannantine, J., Comer, J., and Handrock, J., Fundamentals of Metal Fatigue Analysis, Prentice-Hall, 1990. [3] Vosikovski, O., "The Effect of Stress Ratio on Fatigue Crack Growth Rates in Steels," Engineering Fracture Mechanics, Vol. 11, 1979, pp. 595-602. [4] Yu, M. and Topper, T., "The Effects of Material Strength, Stress Ratio and Compressive Overloads on the Threshold Behavior of a SAE 1045 Steel," Journal of Engineering Materials and Technology, Vol. 107, 1985, pp. 19-25. [5] DuQuesnay, D., "Fatigue Damage Accumulation in Metals Subjected to High Mean Stress and Overload Cycles," Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1991. [6] Jurcevic, R., DuQuesnay, D., Topper, T., and Pompetzki, M., "Fatigue Damage Accumulation in 2024T351 Aluminium Subjected to Periodic Reversed Overloads," International Journal of Fatigue, Vol. 12, No. 4, 1990, pp. 259-266. [7] Vormvald, M. and Seeger, T., "The Consequences of Short Crack Closure on Fatigue Crack Growth under Variable Amplitude Loading," Fatigue and Fracture of Engineering Materials and Structures, Vol. 14, No. 2/3, 1991, pp. 205-225. [8] Dabayeh, A. and Topper, T., "Changes in Crack Opening Stress After Overloads in 2024-T351 Aluminum Alloy," International Journal of Fatigue, Vol. 17, No. 4, 1995, pp. 261-269.
BONNEN AND TOPPER ON PERIODIC OVERLOADS
231
[9] Varvani-Farahani, A., Topper, T., and Plumtree, A., "Confocal Scanning Laser Microscopy Measurements of the Growth and Morphology of Microstructurally Short Fatigue Cracks in A12024-T351 Alloy," Fatigue and Fracture of Engineering Materials and Structures, Vol. 19, No. 9, 1996, pp. 1153-1159. [10] Brown, M. and Miller, K., "Mode I Fatigue Crack Growth Under Biaxial Stress at Room and Elevated Temperature," Multiaxial Fatigue, ASTM STP 853, American Society for Testing and Materials, pp. 135-152. [11] Youshi, H., Brown, M., and Miller, K., "Fatigue Crack Growth from a Circular Notch under High Levels of Biaxial Stress," Fatigue and Fracture of Engineering Materials and Structures, Vol. 15, No. 12, 1992, pp. 1185-1197. [12] Hourlier, F. and Pineau, A., "Fatigue Crack Propagation Behavior Under Complex Mode Loading," Advances in Fracture Research (Fracture 81), D. Francois, Ed., Vol. 4, Oxford, Pergamon Press, 1982. Held in Cannes, March 29-April 3, 1981, pp. 1833-1840. [13] Brown, M., Hay, E., and Miller, K., "Fatigue at Notches Subjected to Reversed Torsion and Static Axial Loads," Fatigue and Fracture of Engineering Materials and Structures, Vol. 8, No. 3, 1985, pp. 243-258. [14] Fatemi, A. and Socie, D., "A Critical Plane Approach to Multiaxial Fatigue Damage Including Out-ofPhase Loading," Fatigue and Fracture of Engineering Materials and Structures, Vol. 11, No. 3, 1988, pp. 149-169. [15] Tschegg, E., "Sliding Mode Crack Closure and Mode III Fatigue Crack Growth in Mild Steel," Acta Metallurgica, Vol. 31, No. 9, 1983, pp. 1323-1330. [16] Ritchie, R., McClintock, F., Tschegg, E., and Nayeb-Hashemi, H., "Mode III Fatigue Crack Growth under Combined Torsional and Axial Loading," Multiaxial Fatigue, ASTM STP 853, American Society for Testing and Materials, pp. 203-227. [17] Bonnen, J. and Topper, T., "The Effect of Bending Overloads on Torsional Fatigue in Normalized SAE 1045 Steel," International Journal of Fatigue, Vol. 21, No. 1, January 1999, pp. 23-33. [18] Varvani-Farahani, A., "Biaxial Fatigue Crack Growth and Crack Closure under Constant Amplitude and Periodic Compressive Overload Histories in 1045 Steel," Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1998. [19] Kurath, P., Personal communication, 1994. [20] Bonnen, J. and Conle, F., "An Adaptable, Multichannel, Multiaxial Control System," Technical Paper #950703, Society of Automotive Engineers, 1995. Also in Recent Developments in Fatigue Technology, SAE PT-67, Warrendale, PA, 1997. [21] Palmgren, A., "Die Lebensdauer yon Kugellagern (Fatigue Life of Ball Bearings)," Zeitschrift Verein Deutscherlngenieure, Vol. 68, No. 14, 1924, pp. 339-34l. In German. [22] Miner, M., "Cumulative Damage in Fatigue," Journal of Applied Mechanics, Vol. 67, September 1945, pp. A159-A164. [23] Topper, T. and Lam, T., "Effective-Strain Fatigue Life Data for Variable Amplitude Fatigue," International Journal of Fatigue, Vol. 19, Supplement No. 1, 1997, pp. S137-S143. [24] Socie, D., "Critical Plane Approaches for Multiaxial Fatigue Damage Assessment," Advances in Multiaxial Fatigue, ASTMSTP 1191, American Society for Testing and Materials, 1993, pp. 7-36. [25] Brown, M. and Miller, K., "A Theory for Fatigue Failure under Multiaxial Stress-Strain Conditions," The Institution of Mechanical Engineers Proceedings, Vol. 187, No. 65, 1973, pp. 745-755. [26] Chu, C.-C., "Fatigue Damage Calculation Using the Critical Plane Approach," Journal of Engineering Materials and Technology, Vol. 117, 1995, pp. 41-49. [27] Chu, C.-C., "Critical Plane Fatigue Analysis of Various Constant Amplitude Tests for SAE 1045 Steels," Technical Paper #940246, Society of Automotive Engineers, 1994. [28] Fatemi, A. and Kurath, P., "Multiaxial Fatigue Life Predictions Under the Influence of Mean-Stresses," Journal of Engineering Materials and Technology, Vol. 110, October 1988, pp. 380-388. [29] Socie, D., Waill, L., and Dittmer, D., "Biaxial Fatigue of Inconel 718 Including Mean Stress Effects," in Multiaxial Fatigue, STP 853, American Society for Testing and Materials, pp. 463-481. [30] American Society for Testing and Materials, Multiaxial Fatigue, ASTM STP 853, 1985.
Giinther Lgwisch, 1 Hubert Bomas, 1 and Peter Mayr 1
Fatigue of the Quenched and Tempered Steel 42CrMo4 (SAE 4140) under Combined In- and Out-of-Phase Tension and Torsion REFERENCE: L6wisch, G., Bomas, H., and Mayr, P., "Fatigue of the Quenched and Tempered Steel 42CrMo4 (SAE 4140) under Combined In- and Out-of Phase Tension and Torsion," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 232-245. ABSTRACT: Two types of specimens of the quenched and hardened steel 42CrMo4 (similar to SAE 4140) that differed in their residual stress state were tested by combined tension-torsion in- and out-ofphase loading. Under cyclic, stress controlled loading an elastic behavior is registered until 50% of the lifetime. Then a continuous softening occurs, the velocity of which correlates with the von-Mises equivalent stress in the case of in-phase loading. The residual stresses have no influence on the lifetime when cyclic softening occurs. Analytically, the lifetime is best described by the fatigue criterion of Zenner which considers the integral average of the stress state in every plane. This stress state is described by a function of the shear stress amplitude and the normal stress amplitude. Below the cyclic yield strength, the residual stresses must be taken into account as static stresses. The comparison of the local residual stress distributions is possible by using the weakest link model of Heckel.
KEYWORDS: quenched and hardened steel, multiaxial fatigue, residual stresses, fatigue criteria, weakest-link model
Nomenclature A A0 M ma my N No Nf PE V Vo
Surface o f a specimen, m m 2 Reference surface o f a specimen, I n . l I l 2 M e a n stress sensivity Weibull exponent for surface Weibull e x p o n e n t for volume N u m b e r o f cycles N u m b e r o f cycles at the beginning o f plastic softening N u m b e r o f cycles to failure Probability for endurance o f a specimen V o l u m e o f a specimen, m m 3 Reference volume o f a specimen, m m 3 Normal strain ~'pa Equivalent plastic strain amplitude 3' Shear strain A z~/~ra; loading ratio for c o m b i n e d loading O'a Normal stress amplitude, N / m m 2 cra,~q Equivalent stress amplitude, N / m m 2
1 Research engineer, senior research engineer, and managing director, respectively, Stiftung Institut fuer Werkstofflechnik, Badgasteiner Stra[3e 3, D-29359 Bremen, Germany. 232
Copyright9
by ASTM lntcrnational
www.astm.org
LOWlSCH ET AL. ON QUENCHED AND TEMPERED STEEL O'a,eq,A O'a,eq, V O'WAO O'WV0
,ra
233
Equivalent stress amplitude at the surface, N/mm 2 Equivalent stress amplitude in the volume, N/ram 2 50% endurance limit of the reference surface At N/mm 2 50% endurance limit of the reference volume Vo, N/mm 2 Shear stress amplitude, N/mm 2
High Cycle Fatigue Criteria The influence of multiaxial load and mean stresses on high cycle fatigue lifetime and endurance limit is usually described by high cycle fatigue criteria, which define a scalar equivalent value that allows a comparison of the actual cyclic load with a uniaxial mean stress free push-pull load. The equivalent stress amplitude is deduced from the equivalent value as that push-pull stress amplitude that generates the same equivalent value as the actual load. Many high cycle fatigue criteria can be assigned to four classes which are characterized by different methods of building the equivalent value: 9 The equivalent value is a linear combination of the maximum shear stress amplitude that arises in a critical plane and a normal stress in this plane, 9 The equivalent value is a linear combination of the maximum shear stress amplitude that arises in a critical plane and a hydrostatic stress. 9 The equivalent value is a linear combination oftbe octahedral shear stress amplitude and a hydrostatic stress. 9 The equivalent value is a mean over all planes of a function of the shear stresses and normal stresses in these planes Table 1 shows the fatigue criteria that were examined in this work.
Inhomogeneous Stress States In components, the local stresses that are originated by an outside load usually are inhomogeneously distributed at the surface and in the volume. This may be due to the component geometry or the kind of load. Residual stresses are also inhomogeneously distributed, which is due to the balance of the residual forces. For higher strength materials, the influence of such stress states on the fatigue limit is well described by the weakest-link model. This model was developed by Weibull to describe the scattering static strength of brittle materials [10]. Later, it was transferred by Heckel and his group to cyclic loads [11-13]. Different authors have made observations that are in good agreement with the weakest link concept [14-16]. The basis of the model is the assumption that surface or volume defects are equally distributed, and that the worst defect initiates a fatigue crack which leads to failure. In the opinion of the Heckel group, there exists a fracture mechanics relation between the defect size and a threshold stress amplitude which is identical with the fatigue limit. Since crack propagation is not considered, the application of this model is restricted to materials or conditions where crack propagation is of less importance. With respect to different crack initiation mechanisms at the surface and in the volume of a material, a distinction has to be made between surface crack initiation and volume crack initiation. In the first step, crack initiation in the volume shall be considered. Proceeding from a reference volume V0, the probability for endurance of this volume is described as a function of the stress amplitude:
=
"v
Present value
Mean value and amplitude
Maximum value
Ta,ma x ~- O/ On,ma x
McDiarmid [2]:
Ta,ma x Jr- O/ O"n
Findley [1 ]:
Type of Extensions
Mean value
Extensions of Tresca Criterion with Normal Stresses
~'a,max+ CrPm + /3 pa
Dang Van [3]: ~a,max -I- a Prnax Bomas, Linkewitz and Mayr [4]:
Extensions of Tresca Criterion with Hydrostatic Stresses
l ~ a ~ + b,~) 9 (1 + d,~..)2d,p
~,oct + a Pm + ~ Pa
S. = f ('ra, o'a, ~'m, or,.)
~s~dV
Simburger [9]:
Zenner [8]:
Criteria with Averaged Stress Functions
Kakuno and Kawada [7]:
7a,oc t + Ot Pmax
Crossland [6]:
Ta oct + Ot Pm
Sines [5]:
Extensions of von-Mises Criterion with Hydrostatic Stresses
TABLE l--Critical values of some high cycle fatigue criteria.
z
5
m -I1 O
z
c m
--rl --4
v-
x
r'-
C
4~
fad
LOWlSCH ET AL. ON QUENCHED AND TEMPERED STEEL
235
Regarding the more general case of an inhomogeneous stress distribution in a part or specimen, the whole volume has to be divided into volume elements with the following probability of endurance derived from Eq 1 by the rules of probability calculation:
dr( ~o )"v
e~(dV) = 2-v~ ~
(2)
The endurance probability of the total stressed volume can be calculated by multiplying the probabilities of all volume elements; this means an integration in the exponent:
Pe(V) = 2-f~ (~@o) "v d-Evv~
(3)
In the case of multiaxial stresses or mean stresses the stress amplitude o-~ has to be replaced by an equivalent stress amplitude. The volume considerations are only sufficient if subsurface crack initiation is the failure process. If surface crack initiation is the origin of failure analogous derivations have to be made with replacement of the volume V by the surface A. The total endurance probability of a part with both, surface and subsurface crack initiation, is the product of the probabilities for surface and volume:
Pe(V)=2
[ [ ....... Ite~AdA_[ ( ...... ImVdw JA\r At Jv~O'wvo] Vo
(4)
For this general case, two reference fatigue limits, trwvo and OrWAO,tWO Weibull exponents m A and my, as well as two fatigue criteria for surface and volume have to be distinguished.
Experimental The high-cycle fatigue behavior of the steel, 42CrMo4 (SAE 4140), under combined tension and torsion was examined at tubular specimens. The material came from one batch which was continuously cast under air exclusion and magnetic stirring. The chemical composition which fulfills the demands of EN 10083-1 is shown in Table 2. The specimen fabrication and its geometry is shown in Fig. 1. The ultimate tensile strength of the specimen was 950 N/mm 2. In order to remove the distortion caused by heat treatment, the specimens of a first series, which will be called "A," have been ground outside and honed inside. Due to the honing, compressive residual stresses were introduced in the inner surface. A part of the specimens was annealed after honing so that the residual stresses were reduced. This series will be called "B". Figure 2 shows the residual stresses which were measured across the wall of the specimens. The load was exercised force- and moment-controlled with a PTT-type machine built by SCHENCK which has both a longitudinally and torsionally working cylinder with servohydraulic
TABLE 2--Chemical composition of the material. Element Mass, % Element Mass-%
C 0.43 Mo 0.22
Si 0.21 Ni 0.23
Mn 0.77 Cu 0.13
P 0.010 A1 0.029
S 0.030 N 0.008
Cr 103 Ti 0.004
236
MULTIAXIAL FATIGUE AND DEFORMATION
Heat treatment Drilling, |850~ 30 min / ~I~1~ Tuming ..,~ Oil, 60~ + . Cutting | 650~ 2h / | Water
Honing Grinding Polishing I ~
2'~ identical
Heat treatment
olishing
FIG. 1--Specimen geometry and manufacturing procedure, dimensions in ram.
drives. The tube form of the specimens avoids stress gradients under torsion. The ratio between shear stresses and normal stresses, A = ~-~/o-~, varied from 0.5 to 1. The phase shift, ~, between shear stresses and normal stresses varied between 0 and 90 ~ Table 4 shows the load variants which were free of mean stresses. The load frequency was 10 Hz at the fatigue limit and 1 Hz at a lifetime N s = 10 000. Between these loads, it was interpolated linearly.
\~-~-%'~\\'~\\~- 9.~\~\'~,\\~.\%%\'~ t ~ ~,~
E E
-loo
~~
u
Z o~
r
-200 (3
(D
-g_
-300
u)
-400 I
,
-200
0
200
1
1400
Distance to the outer surface [~n] FIG. 2--Residual stresses across the specimen wall.
1600
LOWlSCH
ET AL. O N Q U E N C H E D
AND TEMPERED
STEEL
237
TABLE 3--Load variants. Short Description
Normal Stress Amplitude o-a
Shear Stress Amplitude r,
Phase Shift 6
TC AT 0.5, 0~
oa o o-~
0
...
I-.
o;
% = o-J2
0.5, 90 ~ 1, 0~
o-. o-.
~'. =
o-~,
90 ~ 0~
1, 90~
o-~
r~ = or.
90 ~
r = o-,,/2
The strain measurement was achieved by a clip device that measured axial shear, 1/, and axial strain, e. The strain amplitude was determined by drawing the y - e pairs of one cycle in a e - X / ~ y coordinate system and taking the radius of the circumscribing circle as the total strain amplitude. An analogous procedure was exercised for the determination of the von-Mises equivalent stress amplitude and the von-Mises equivalent plastic strain amplitude. The actual plastic strains were calculated with Hooke's law.
Cyclic Strain Response Under loads in the finite lifetime region first, the material deforms elastically, but later exhibits a progressive plastic behavior (Fig. 3). The plastic strain amplitude is a linear function of the logarithm of the number of cycles and can be described as
I?,pa :
rap"
log
N)
(5)
The number of cycles, No, at the beginning of plastic softening is about half of the number of cycles to failure. Figure 4 shows an analysis of this relation. The increase, mp, can be described as a function of the von-Mises stress amplitude. All proportional loads have the same function, whereas the non-proportional loads have extra functions (Fig. 5). In the latter cases the softening is more pronounced, especially in the case of A = 0.5, where the highest softening rate is observed. It is assumed that this is due to the constant maximum shear stress which rotates over all planes perpendicular to the specimen surface, and allows the movement of dislocations in many gliding systems.
Crack Initiation The specimens were loaded until a crack of at least lO-mm surface length occurred. Usually, this crack changed its direction after a surface length of some mm. Only the cracks that were initiated un-
TABLE 4--Model parameters for fatigue limit calculation. O'WAON/mm~
Criterion Findley Zenner
Ao = 1 mma
mA
~rwvoN/mm2 Vo = 1 mm 3
mv
M
a a
b
d
619 625
13 13
980 972
10 10
0.4 0.4
0.33 0.33
0.22
1.22 10 3
200
400
C O >
R
600
9~
1
_'%'
i
s
l
i
I
I
Illlll
100
I
III1||
1000
|
I
I
10000
I I l | l l
cycles to failure Nf
l
/ ~ 8 I
I
|
100000
IIII1|
=0";Nt =92.730
/ % = 230 N/mm2; x = 230 N/mm2;
FIG. 3--Cyclic deformation behavior of two specimens.
10
I l l l l l
both specimen: eq(von Mises)..._. ~ = 508 N/mm 2_
z
0 3o
-11
m
z
cm
/
C) ,~ 800 % = 450 N/mm2;~, = 225 N/mm2; 8=90";Nf=8.515
-n
,..~1000
x
cI-"
r~ Go co
L(SWISCH ET AL. ON QUENCHED AND TEMPERED STEEL
239
100000 o
Z
~o
10000
o0 ~ / /
[]
-~
O
go
1000
9 9
13
J~
N, = 2,15 NoI'~ 100 1000
.
.
,
,
,
,
,.J
,
10000
,
t
i
o []
tension torsion
~x
;~=0,5; ~5=0"
o
~=0,5; 5=90*
9
~=1;(5=0"
9
7~=1; 5 = 9 0 *
11111
,
,
=
=
,
100000
,
||1
1000000
cycles to failure Nf FIG. 4--Correlation between lifetime NF and numbers of cycles No to the beginning of cyclic soft-
ening.
540
D
Q
~176176 I
A
520
E
500
..
.z. ~- 480 b [] *-'...... LU
0
O
.--'=
; proportional:
9
9
o
tension
.O, . . . . . . . .
n n
torsion ~.=0,5; 5=0 ~
9
;~=1; 5=0 ~
"
,'"
~o-~" I
O____O..
440 420
0
~. mp
400 0
d~pa/dIOgN with N>N o
I 2000
o .
.....
L=0,5; 8=90 ~
........ L = I ; 5=90 ~ I 40O0
slope m r [10 "6] FIG. 5--Slope of the plastic strain amplitude versus the von-Mises equivalent stress.
240
MULTIAXIALFATIGUE AND DEFORMATION
der push-pull conditions did not change their direction of propagation. In all cases, the crack that led to failure was the only one that could be detected in the specimen. The crack orientations before the change of direction are shown in Fig. 6 and Fig. 7 for the loads with the stress ratio, A = 1. The vertical axis has no meaning for the experimental points. Their shift on this axis is just to make the individual points visible. In case of the proportional load, most of the crack orientations lie between - 15 and + 30 ~ In case of the non-proportional load, most crack orientations lie at 0 ~ The crack orientations were observed only at the specimens surfaces. This means that it is the orientation of the axis of intersection between surface and crack face. A true crack face orientation measurement is very expensive and could not be realized within the project. However, a relation was searched between the frequency of crack initiation at a certain intersection axis, and the maximum shear stress and normal stress amplitude that can be found in all planes that have this intersection axis common. These stresses are also shown in the figures. The relation between the experimental frequency and the curves of the stresses supports the idea that both shear stresses and normal stresses are enhancing crack initiation, which is in accordance to the fatigue criteria presented in Table 1.
Lifetimeand FatigueLimit Lifetime and fatigue limit were predicted with all fatigue criteria shown in Table 1. The best predictions were achieved with the Zenner criterion. The following two examples, the Findley prediction, the Zenner prediction and their results are described in detail. Since in the region of limited lifetime the residual stresses had little influence, the lifetime was predicted without taking them into account. The specific parameters, c~, a and b, of the criteria were determined as functions of the number of cycles to failure by using push-pull and the alternating torsion results as reference. The comparison of the S-N curves is shown in Fig. 8 and Fig. 9.
1,2
1,0
Z=I
; 8=90
~
-.
..
"~0,8 ~ 0,6
3 e" 0,4 0,2 ..... 0,0 "90 ~
=
I
I
-60 ~
I -30 ~
i
I 0~
shear =
I
stress I
30 ~
amplitude I = 60 ~
I 90 ~
apearance of the crack at the surface FIG. 6---Crack orientation and stresses on the crack planes under combined in-phase loading with A=I.
241
L(~WISCH ET AL. ON QUENCHED AND TEMPERED STEEL
1,8
Z = I ; 8=0 ~
1,6 1,4 i~| 1 , 2 1,0
~ 0,8 ~
0,6 0,4 0,2
0,0 ,
I
.90 ~
,
I
_60 ~
,
I
.30 ~
I
,
0~
,
I
30 ~
,
60 ~
I
90 ~
a p p e a r a n c e of the crack at the surface FIG. 7 - - C r a c k orientation and stresses on the crack planes under combined out-of-phase loading vith A = 1.
600
A con
~
~
o
o []
9
A 0 []
O
9
0
A
9
9
0
9 o@ 9 O O O
[]
O O
O 0
400
,
O 9
Findley: "~am.x + ~ a . ,
i
i
,=1
10000
i
=
9
0
0
,
A
0
[]
m
]
torsion ~,=0,5; 8=0 ~ ~,=0,5; 6=90 ~ ~=1; 6 = 0 ~ ~,=1; 6=90 ~
o
A
ooooo 500
tension
[] A
9
E E Z
o
=
,
=
i , i i
i =
~
9
o o i
100000
,
""
,
,
,
=,1
1000000
cycles to failure Nf FIG. 8--Lifetime as a function o f the Findley equivalent stress.
"
242
MULTIAXIAL FATIGUE AND DEFORMATION O 600
22
o ~~
Zenner: l(a% +b% ) dm
I
9 Z~O~ A
9
9
O
500
"o :h
E Z
t)"
400
300
,
i
o
tension
[] zx
torsion ;~=0,5; 5=0 ~
0
~.=0,5; 5=90 ~
9 9
L=I; 5=0 ~ L=I; 5=90 ~
1 1 , 1 , 1
A@
O
o
,
,
10000
|
i
I
,
,11
I
i
i
1 , 1 , , 1
100000
1000000
cycles to failure Nf
FIG. 9~Lifetime as a function of the Zenner equivalent stress.
The residual stresses can not be neglected at near fatigue limit loads. Figure 10 shows this for the push-pull S-N curves of the specimen series A and B which are different in residual stress state. At stress amplitudes near the fatigue limit, the lifetimes differ from series to series, whereas at higher stress amplitudes where plastic deformation is observed no difference
regression ' cyclic
5O0
"E E Z
series A
O
550
Nf = 440 184 (ca/424 N/mm2) ls'a
0o
softening elastic
O O~.
()
behaviour 450
o=I..%
9
o"2
400
9
series B
........ regression 350
Nf = 785 772 (cr=/357 N/mm 2) -8,e I
'
'
'
'
I I I
10000
I
I
I
I
'
j
' ' l
=
100000
t
I
|
i
|
i
.l
1000000
cycles to failure Nf FIG. I O---S-N diagram of the series A and B under tension-compression load.
L(gWlSCH ET AL. ON QUENCHED AND TEMPERED STEEL
243
can be detected. This can be explained by the declining of the residual stresses by plastic deformation. At the fatigue limit a comparison of the specimen series could only be achieved by applying the weakest-link model where the local residual stresses were introduced as mean stresses in the local equivalent stress amplitudes. According to Eq 2, 3 and 4 the total endurance probability was calculated by variation of the stress amplitude until the total endurance probability is 0.5 the fatigue limit is gained. The model parameters were determined on the base of reference variants. These were the specimens of series A and B under tension compression and the alternating torsion variant. The surface reference fatigue limit, trwao, and the Weibull exponent, mA, were taken from the specimens of series B under push-pull. The volume reference fatigue limit, O-wvo, and the Weibull exponent mv were taken from the specimens of series A under push-pull. A mean stress sensitivity defined by Schuetz [17] of M = 0.4 was assumed, according to results from literature [18]. The parameters a, a and b which describe the damaging effect of the normal stress against the shear stress were gained with the help of the torsion fatigue limit. The Zenner parameter, d, can be calculated from the mean stress sensitivity M [8]. Table 4 gives a survey over the calculation parameters. Figures 11 and 12 show the results of the calculations in a o-~-~-~diagram. The open squares indicate the reference variants. The filled symbols show the experimental fatigue limits of the combined loads. The lines show the calculations without considering the residual stresses, and the open symbols show the predictions which include the residual stress influence. Generally, the Findley criterion predicts large differences between in- and out-of-phase loading, whereas the Zenner criterion predicts no differences for these loads.
3O0
i
i
Findley:
(NP'--I
E E z
q
1;a,max "k 0,33 % @
i
200
13
. O
~
E ~
t-
100
O
[] reference ecperiment 9 in-phase 9 out-pf-phase .calculation O in-phase O out-pf-phase calculation without residual stresses . . . . . in-phase out-pf-phase I
0
I
100
i
I
200
~
~ %
I
In
300
400
normal stress amplitude a [N/mm 2] FIG. l l--Measured fatigue limits in a t r a - ra plane in comparison with the prediction using the Findley criterion.
244
MULTIAXIALFATIGUE AND DEFORMATION 300
i
z
Zenner:
E E
(a%2+bo" 2)(1 +d~m )2
d~
Z
i
~ 200 "o Q..
E 100 I,.,.
t~ (lJ t,-
--
[] reference experiment 9 in-phase 9 out-of-phase calculation O in-phaseand out-of-phase calculationwithout residualstresses in-phase and out-of-phase
0
i
I
1O0
v
I
200
300
400
normal stress amplitude % [N/mm 2] FIG. 12--Measured fatigue limits in a ~a - "l'aplane in comparison with the prediction using the Zenner criterion.
Conclusion The presented experiments on tubular specimens of the steel 42CrMo4 (SAE 4140) show that under combined constant amplitude stress-controlled fatigue loads with longitudinal forces and torsion moments the plastic strain response under proportional loading can be described by the von-Mises equivalent stress. The cyclic softening that starts at about 50% of the lifetime increases when the loads are non-proportional. The observed crack orientations support the idea that initiation of the cracks is controlled by shear stresses and normal stresses. This is reflected in the fatigue criteria that were tested. A satisfying prediction of the lifetime and fatigue limit is possible with the Zenner criterion. At non-proportional loading, the equivalent stress amplitude is close to the maximum value for a large numbers of planes. Therefore, the probability of crack initiation is higher than under an in-phase loading with the same equivalent stress in the critical plane. The Zenner criterion considers the stresses in the material in every plane, which is obviously a better approach to fatigue damage than restriction to a critical plane. For fatigue limit prediction it is necessary to take account of the local residual stresses. This was achieved with the weakest-link concept where the residual stresses were handled like local mean stresses. Acknowledgement The work that is presented here was supported by the German Bundesministerium fuer Wirtschaft and the Arbeitsgemeinschaft Industrieller Forschungsvereinigungen under contract number AiF 10 058. The authors are grateful for this.
LOWlSCH ET AL. ON QUENCHED AND TEMPERED STEEL
245
References [1] Findley, W. N., "Effect of Stress on Fatigue of 76S-T61 Aluminium Alloy under Combined Stresses Which Produce Yielding," Journal of Applied Mechanics, Vol. 75, 1953, pp. 365-374. [2] McDiarrnid, D. L., "A General Criterion for High Cycle Multiaxial Fatigue Failure," Fatigue and Fracture of Engineering Materials and Structures, Vol. 14, 1990, pp. 429--454. [3] Dang Van, K., Griveau, B., Message, O., "On a New Multiaxial Fatigue Criterion: Theory and Application," Brown, M. W. and Miller, K. J., Eds. Biaxial andMultiaxial Fatigue, EGF 3, 1989, p. 459. [4] Bomas, H., Linkewitz, T., Mayr, P., "Application of a Weakest-Link Concept to the Fatigue Limit of the Beating Steel SAE 52 100 in a Bainitic Condition," Fatigue and Fracture of Engineering Materials and Structures, Vol. 22, 1999, pp. 738-741. [5] Sines, G, "Behaviour of Metals under Complex Static and Alternating Stresses," Metal Fatigue, Herausgegeben von Sines, G., und Waismann, J. L., Eds., McGraw Hill, New York, 1959. [6] Crossland, B., "Effect of Large Hydrostatic Pressure on the Torsional Fatigue Strength of an Alloy Steel," Proceedings of the International Conference on the Fatigue of Metals, Institute of Mechanical Engineers, London, 1956, pp. 138-149. [7] Kakuno, K. and Kawada, Y., "A New Criterion of Fatigue Strength of a Round Bar Subjected to Combined Static and Repeated Bending and Torsion," Fatigue of Engineering Materials and Structures, Vol. 2, 1979, pp. 229-236. [8] Zenner, H., Heidenreich, R., and Richter, I., "Schubspannungsintensit~itshypothese-Erweiterung und experimentelle Absttitzung einer neuen Festigkeitshypothese fiir schwingende Beanspruchung," Konstruktion, Vol. 32, 1980, pp. 143-152. [9] Simbiirger, A., "Festigkeitsverhalten ZSher Werkstoffe bei Einer Mehrachsigen, Phasenverschobenen Schwingbeanspruchung mit K~Srperfestenund Verg.nderlichen," Hauptspannungsrichtungen. LBF Darmstadt. Bericht Nr. FB-121, 1975. [10] Weibull, W., "Zur Abh~ingigkeit der Festigkeit vonder Probengrrge," Ingenieur-Archiv, Vol. 28, 1959, pp. 360-362. [i1] Brhm, J. and Heckel, K., "Experimentelle Dauerschwingfestigkeit unter Be~cksichtigung des Statistischen GrSl3eneinflusses," Zeitschrififiir Werkstofftechnik, Vol. 13, 1982, pp. 120-128. [12] Heckel, K. and KOhler, J., "Experimentelle Untersuchung des Statistischen Grrfleneinflusses irn Dauerschwingversuch an Ungekerbten Stahlproben. Zeitschrift ftir Werkstofftechnick" Vol. 6, 1975, pp. 52-54. [13] Kra, C., "Beschreibung des Lebensdauerverhaltens Gekerbter Proben unter Betriebsbelastung anf der Basis des Statistischen Grrfleneinflusses," Dissertation, M~inchen, 1988. [14] Kuguel, R., "A Relation between Theoretical Stress Concentration Factor and Fatigue Notch Factor Deduced from the Concept of Highly Stressed Volume," ASTM Proceedings 61, 1961, pp. 732-744. [15] Liu, J. and Zenner, H., "Berechnung von BauteilwShlerlinien unter Berticksichtigung der Statistischen und Spannungsmechanischen Sttitzziffer," Materialwissenschaft und Werkstofftechnik, Vol. 26, 1995, 25-33. [16] Sonsino, C. M., "Zur Bewertung des Schwingfestigkeitsverhaltens von Bauteilen mit Hilfe 13rtlicher Beansprnchungen," Konstruktion, Vol. 45, 1993, pp. 25-33. [17] Schtitz, W., Ober eine Beziehung Zwischen der Lebensdaner bei Konstanter und bei Veranderlicher Beanspruchungsamplitude und Ober Ihre Anwendbarkeit anf die Bemessung von Flugzeugbauteilen," Zeitschriftfiir Flugwissenschafien, Vol. 15, 1967, pp. 407--419. [18] Macherauch, E. and Wohlfahrt, H., "Eigenspannung und Ermiidung, Ermtidungsverhalten MetaUischer Werkstoffe," DGM-lnformationsgesellschafi, D. Munz (Hrsg.), 1985, pp. 237-283.
Jinsoo P a r k 1 and D r e w V. Nelson 2
In-Phase and Out-of-Phase Combined Bending-Torsion Fatigue of a Notched Specimen REFERENCES: Park, J. and Nelson, D. V., "In-Phase and Out-of-Phase Combined Bending-Torsion Fatigue of a Notched Specimen," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. KaUuri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 246-265. ABSTRACT: An experimental study of the high-cycle biaxial fatigue behavior of notched specimens is reported. Solid round bars of I%Cr-Mo-V steel having a circumferential semicircular groove were tested under fully reversed constant-amplitude bending, torsion, and combined bending and torsion, inphase, and 90 ~ out-of-phase. Smooth specimens of the same material were also tested under bending as well as torsion to provide baseline data. Observed fatigue life data are used to evaluate several multiaxial fatigue life prediction models, including a critical plane method, a yon Mises approach, and an energy-based approach. Crack growth behavior as observed on the surface in smooth and notched specimens is presented and discussed.
KEYWORDS: fatigue, biaxial, multiaxial, crack growth, stress concentration, notch, crack initiation Nomenclature
a, A b B d D E Ae,~ ~eq e,,om eiz el,lit G "Yeq Y, om k Ktb Ktt h MSE
Slope and intercept in fatigue life relation Shear fatigue strength exponent Bending moment Diameter of notched cross section Diameter of unnotched cross section Modulus of elasticity Deviatoric elastic strain ranges Equivalent nominal bending strain amplitude Nominal bending strain amplitude Strain measured along specimen axis Strains measured at • ~ to specimen axis Shear modulus Equivalent nominal torsional shear strain amplitude Nominal torsional shear strain amplitude Constant used to merge bending and torsional data Elastic stress concentration factor for bending Elastic stress concentration factor for torsion Ratio of 'Yeq to F.eq Mean squared error
1 Senior researcher, Hyundai Heavy Industries Co., Ltd., 1 Cheonha-Dong, Dong-Gu, Ulsan, Korea 682-792. : Professor, Mechanical Engineering Department, Stanford University, Stanford, CA 94305-4021. 246
Copyright9
by ASTM International
www.astm.org
PARK AND NELSON ON BENDING-TORSION FATIGUE n
2N
Ni No P tO
4, SeQA As o O-b Crx,~ry,~rz Orxa, O'ya or1 On,max
O'eq A Oemax t T ";a
# % Txya A'Tmax
0
we
247
Number of data points Number of reversals Cycles to crack initiation (1.0 mm crack) Cycles to 10% load drop Observed life Predicted life Poisson's ratio Frequency Phase angle between normal and shear stresses Angle relative to specimen axis Equivalent stress amplitude Deviatoric stress ranges Bending stress amplitude Normal stress components Amplitudes of o'x, Cry Maximum normal stress amplitude Maximum normal stress on plane of % yon Mises equivalent stress amplitude Maximum principal stress range Time Torque Maximum amplitude of shear stress Shear fatigue strength coefficient Torsional shear stress amplitude Amplitude of ~'~ Shear stress components Maximum range of shear stress Phase angle between normal stress components Elastic distortion energy fatigue damage parameter
Despite its importance in mechanical design, experimental research on the topic of multiaxial fatigue of specimens with stress concentrations (notches) has been relatively sparse. In the forties, Gough [1] investigated fatigue limits of specimens with four types of notches (e.g., V-type groove, oil hole, transition fillet, and longitudinal spline) subjected to in-phase bending and torsion. Test resuits were correlated by the so-called ellipse arc and quadrant based on nominal stress amplitudes. Several decades later, steel shaft specimens with a shoulder fillet were investigated by Simburger [2] and in an SAE biaxial fatigue test program [3]. Combined bending and torsion was applied, both inphase and out-of-phase, producing lives over the range of 103 to 106 cycles. Correlations by various life prediction models for those tests are presented in Refs 2, 4, and 5. During the late eighties, studies of the low-cycle, proportional, axial-torsional, elevated temperature fatigue behavior of 304 stainless steel specimens with a hole or V-notch were reported [6, 7]. More recently, Yip and Jen [8] reported studies of crack initiation at the edge of a hole in 1045 steel solid round bars for low-cycle, proportional, axial-torsional loadings. Subsequently, those authors used nonproportional, axial-torsional loadings in low-cycle tests of AIS1 316 stainless steel round bars with a semicircular circumferential notch [9]. Many structural members and machine parts contain various kinds of stress concentrations and operate in the high-cycle regime, experiencing macroscopically elastic or small elastic-plastic deformation at the stress concentrations. As indicated by the review above, published research in this cycle range has been limited, especially for nonproportional stresses. A main objective of this experimental program was to investigate effects of a stress concentration on high-cycle biaxial fa-
248
MULTIAXIAL FATIGUE AND DEFORMATION
tigue behavior in the range of approximately 105 to 2 • 106 cycles. A circumferential semicircular notch in a solid round bar specimen was chosen as a stress concentration and specimens tested under combined bending and torsion, in-phase, and 90 ~ out-of-phase. The selection of 90 ~ out-of-phase loading was motivated by the desire to investigate the effect of nonproportional cyclic stresses. Also, 90 ~ out-of-phase cyclic bending and torsion simulates the stresses felt by a surface element on a rotating shaft experiencing a steady torque and bending moment at a given cross section, a situation commonly encountered in turbines and other machinery. A further objective of these tests was to make observations of crack initiation and small crack growth in the notch, rather than just recording life to fracture, as was the practice in earlier studies such as that of Gough. In addition, results of the test program will be used to evaluate three life prediction approaches: a critical plane method, a version of the von Mises criterion, and a new energy-based method.
Life Prediction Approaches Critical Plane Method The following critical plane approach suggested by Socie [10] for high-cycle multiaxial fatigue will be considered in this paper
ra + kon,max = r}(2N) b
(1)
where ra is the maximum value of shear stress amplitude, On.max is the maximum normal stress on the plane of ra, and r} and b are the shear fatigue strength coefficient and exponent, respectively. The value of k may be determined from two different sets of test data, for example, from axial (or bending) and torsional test data, as a value merging the two sets of data into a line on a plot of ra + ko'n.max versus 2N.
Von Mises Criterion The von Mises criterion has been widely used for correlating high-cycle multiaxial fatigue life under proportional cyclic stresses, when ratios of principal stresses and their directions remain fixed during cycling. For nonproportional cases, a stress-based version of the ASME Boiler and Pressure Vessel Code Case N-47-23 [11] may be used as an extension of the von Mises criterion, in which an equivalent stress amplitude parameter, SEQA, is defined from stress ranges &rx, Aoy, Aoz, Ar~y, Aryz, Arzx in the form
SEQA = ~
[(Am
-
AO'y)2
-{- ( A O ' y
-- aO'z) 2
- - 2 1112 + (Ao. - A~z)2 + 6Ar 2 + 6Areyz + ~ozarLd
(2)
where Aox = o'x(t0 - ~x(t2), A% = ~y(tl) -- Oy(t2), etc. SEQA is maximized with respect to two arbitrary instants, tl and t2, during a fatigue loading cycle. For constant amplitude bending and torsional stresses such as ox = ob sin(ogt) and z~ = rtsin(~ot - q~), where 4' is the phase angle between ~rx and r.y, Eq 2 becomes
SEQA where K = 2rt/o'b.
=o[ -~
I+~-K
+
1
1-~K 4
]1,2
(3)
PARK AND NELSON ON BENDING-TORSION FATIGUE
249
Elastic Distortion Strain Energy The SEQA approach can be considered a form of a distortion energy criterion. A different distortion energy approach can be derived based on the ranges of deviatoric stress and strain, Asij, and Ae~, for a load cycle. An elastic distortion energy fatigue damage parameter, We, can be defined as
From the relations si/= 2Ge~j = Ee~jl(1 + v) in the elastic range, Eq 4 can be expressed in terms of deviatoric stresses
We = 1 + v (AsL + AsZy + Ask + 2As 2 + 2Adz + 2As~)
4E
(5)
For biaxial fatigue with two normal stress components and a shear stress component, ~rx = O'xa sin(wt), % = O'yasin(wt - 0), rxy = rxya sin(oJt - 05), where 0 is the phase angle between crx and ~y, and 05 is the phase angle between Cxyand o-x, Eq 5 becomes
We= 2(1 q- lJ) [~
~176176176 3
2
1
+ rZxya
(6)
For in-phase stress, Eq 6 reduces to
We =
2(1 + 3E
O-e2q
(7)
where OTqis the yon Mises equivalent stress amplitude = ((3/2)$2).1/2It is of interest to note that Eq 6 has the same mathematical form as the average resolved shear stress amplitude for all of the planes in stress space as derived by Papadapolous [12], who also showed that this approach successfully correlated high-cycle biaxial fatigue data generated with (a) out-of-phase combined axial-torsional loading, or (b) out-of-phase normal stress components.
Experimental Program Material The test material used for this investigation was a hot-rolled I%Cr-Mo-V steel, which is used for bolts, nuts, and pins in turbines and many other machine parts. The steel was quenched in oil after a solution heat treatment at 930~ for 2 h and then tempered at 680~ for 3 h. Prior to being machined into specimens, solid round bars with a diameter of 40 mm were stress-relieved at 650~ for 3 h and cooled in air. The chemical composition and mechanical properties of the material are summarized in Tables 1 and 2, respectively.
TABLE 1--Chemical composition of 1% Cr-Mo-V steel (weight %). C 0.42~0.50
Si 0.20-0.35
Mn 0.45~0.70
P Max. 0.025
S Max. 0.025
Ni Max. 0.25
Cr 0.80-1.15
Mo 0.45-0.65
V 0.25~0.35
A1 Max. 0.015
250
MULTIAXIAL FATIGUE AND DEFORMATION
TABLE 2--Mechanical properties of l % Cr-Mo-V steel. Modulus of elasticity, E Poisson's ratio, v Ultimate strength, o-. Yield strength (0.2% offset), try Total elongation, ef Reduction in area, RA Brinell hardness, HB Cyclic strength coefficient, K' Cyclic yield strength (0.2% offset), try
211000 MPa 0.29 828 MPa (min) 725 MPa (min) 18% (min) 50% (rain) 302 (max) 1442 MPa 515 MPa
Specimens
The geometries of the smooth and notched specimens used in the tests are shown in Fig. 1. Smooth solid round bar specimens had an hourglass test section with a minimum diameter of 16.5 mm. Notched specimens had a circumferential semicircular groove of 1.5 mm radius with an inner diameter of 14 mm at the section of the notch. Theoretical elastic stress concentration factors (SCF) for the notch are 1.95 for bending and 1.49 for torsion from Peterson's charts [13], 1.97 for bending and 1.48 for torsion from a closed-form analysis [14], and 1.98 for bending and 1.50 for torsion from a finite-element analysis (FEA) of the specimen. The values of SCFs obtained from the FEA were used in this investigation. Both ends of the specimens were designed to have fiats for gripping. Surfaces in the notch root and the smooth specimen test section were polished with diamond paste, ending with paste of approximately 1/xm particle size. (Attempts to electropolish the specimens led to rapid formarion of ferric oxides.) Test Procedure
All of the tests were conducted under fully reversed, constant-amplitude bending and torsion using a machine described in Ref 15 that applied desired angles of twist and/or bending deflections to specimens. Smooth specimens were first tested under bending and then torsion to obtain baseline data. Strain amplitudes were chosen to result in fatigue lives ranging between about 105 and 2 x 106 cycles. Strain amplitudes applied in the smooth specimen tests are listed in Table 3, as determined from strain gage rosettes attached on the top and bottom of the test section of a sample specimen (Fig. 2a). Use of the sample specimen allowed angle of twist and bending deflection applied by the test machine to be adjusted prior to each test to produce the desired strains in the specimens used for fatigue testing. Measurements of specimen diameters showed that they differed by less than 0.06% from the diameter of the sample specimen, so that strains in the specimens used for fatigue testing would be expected to be within 0.2% of those in the sample specimen (with strain varying as the cube of diameter). Fully reversed cycling was confirmed using the top and bottom rosettes of the sample specimen. For the notched specimens, four fixed ratios A of equivalent nominal torsional strain amplitude ')teq to equivalent nominal bending strain amplitude eeq, A 0 (bending), 1, 2, and ~ (torsion), were used for in-phase and 90 ~ out-of-phase bending and torsion =
')teq a
-
eeq
~//3~tno m -
2(I + v)e,,om
(8)
where 3'eq = N/3T,,oml2(l + v) and eeq = e,,om based on the von Mises criteria, and e.om and 3'no,~ are, respectively, nominal bending and torsional strain amplitudes, and v is elastic Poisson's ratio.
I_
105
---0.5
1.6/
t
105 •176
1,6/ ~,,
I 28,5(~ ='~
I
i 28,5(~
"-.05
Prior to
+--I
I_ '-
-, (b)
375
•
0.~//(IJ.m] Priorto 20~1/1.5R=.o2 polishing 25R-+1
(a)
375
t
0'4X//[I-tm] polishing
105 •176
105 •176
-1
-I
_1
21
TESTSECTION
" ~J~28.,5
FIG. 1--Geometries of(a) smooth and (b) notched specimens (dimensions in ram).
--
-r-"
=!-
+02/+
%
r •176
O"l
c m
z
0
4
z 0
0 Z 0 z W m z
Z m i-"
Z
~D
"o
252
MULTIAXIAL FATIGUE AND DEFORMATION TABLE 3--Smooth specimen bending and torsionalfatigue data. Strain Amplitude
Fatigue Life
Test No.
e (%)
,~1 (%)
N~ (cycles)
N~(cycles)
S1 $2 $3 $4 $5 $6 $7 $8 $9 $10 Sll S12 S13 S14 $15
0.262 0.257 0.246 0.236 0.224 0.210 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0.423 0.415 0.409 0.409 0.396 0.381 0.371 0.365 0.351
84 200 138 000 394 000 595 000 1 198 000 >3 400 000 72 300 89 000 140 0013 306 000 232 000 >1 500 000 371 000 >1 582 000 >2 912 000
110 000 161 000 416 000 633 000 1 249 000 102"000-123 000 181 000 389 000 275 000 ---449"000 .....
1Engineering shear strain. 2 Number of cycles to 1.0 mm surface crack length. 3 Number of cycles to 10% load drop.
(a)
Rosette
MovingClamp
Fixed
Clamp Displacement
Rosette
(b) Fixed Clamp
lo.51 =1=2o.I ~
MovingClamp
(dimensions : mm)
!
o0,e of
I TW's' Displacement
Rosette
Top View of Rosette
q,y/~4 E;[I[ 5o
SpecimenAxis
45
8i FIG. 2--Schematic showing deflection-controlled loadings and arrangement of strain rosettes in (a) smooth and (b) notched specimens.
PARK AND NELSON ON BENDING-TORSION FATIGUE
253
The enom and "Ynomvalues were calculated from the bending and torsional moments, B and T, at the notch cross section
enom
32B ETrd 3
and
16T "Y,,o,,, - Gird 3
(9)
where E and G are, respectively, the elastic and shear moduli, and d is the diameter of the notch cross section. B was assumed to vary linearly between the two strain gage rosettes installed on the top of a sample notched specimen (Fig. 2b), which was used in the same manner as the sample smooth specimen to set up desired strains. The values of B at the positions of the rosettes were computed as et1E~rD3/32, where D is the diameter of the unnotched section and Su is the strain value of the second element (in direction of specimen axis) of a rosette. The value of B at the notch cross section was determined using the two values o r B computed from the two rosettes based on the aforementioned assumption of a linear distribution of B between the two rosettes. This assumption was confirmed from the linear distribution of bending stress along the direction of the specimen axis (except near the notch) obtained from the finite element analysis. Twas assumed to be constant along the direction of the specimen axis and computed as (el - elzl)GcrD3/16 where ~:1and si11(~I > F'Ill) are the strain values of the first and third elements (45 ~ with respect to the specimen axis) of the rosette. The value of T at the notch cross section was taken as the average of the two values of T (which were very close) determined from the two top rosettes. The bottom rosette was used to confirm fully-reversed cycling. FEA of the notched specimen showed that stresses at the positions of the rosettes were not affected by the notch geometry; in other words, the rosettes were located sufficiently far from the shoulder of the notch so that measured strains were nominal strains. The small size of the rosettes (2 mm gage length) minimized effects of the curvature of the specimens on strain measurements. The ability of the test machine to produce desired in-phase and 90 ~ out-of-phase loadings was verified by examining bending and torsional strain histories measured from the rosettes for different loading (in-phase and 90 ~ out-of-phase) conditions. All of the strain amplitudes applied to the notched specimens were selected to result in approximately the same range of fatigue lives as the smooth specimen tests. The nominal strain amplitudes and phase differences applied are listed in Table 4. Crack initiation was defined as a 1.0-ram-long surface crack and final failure as a 10% drop in load monitored by a load cell in the test machine fixture holding the clamped end of a specimen. The drop corresponded to a comparable decrease in bending moment or torque for the bending or torsion tests, respectively. For combined bending and torsion, the drop was created by a loss of both bending stiffness and torsional rigidity. Crack formation and growth behavior were observed through a microscope installed vertically over the specimen. (Attempts to take surface replicas of notches were unsuccessful due to the small radii and double curvature of the notches.) All of the tests were performed at a frequency of about 2.5 ~ 3.5 Hz in air at room temperature. Crack initiation and final failure are further discussed in a later section. Results
Smooth Specimens In bending tests of smooth specimens, all of the cracks that initiated propagated on the plane of maximum principal stress range Atrm~x, i.e., along the direction perpendicular to the specimen axis (Fig. 3). At the higher strain amplitudes, it appeared that cracks initiated on the plane of maximum shear stress range A~-. . . . grew to a length of approximately 10 - 50/xm, then proceeded in a zigzag manner perpendicular to Ao'r~a~. At the lower strain amplitudes, initiation by shear was not clear. Only one crack occurred on the top or bottom of a specimen and propagated to final failure. In the torsion tests, cracks always formed and grew on the plane of A~'maxalong the specimen axis to a length of approximately 50 - 400/zm; then the cracks changed direction onto the plane of Ao'max
254
MULTIAXlAL FATIGUE AND DEFORMATION TABLE 4 ~ N o t c h e d specimen bending, torsional, and combined bending-torsional fatigue data. Strain Amplitude
Test No. N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 Nil N12 N13 N14 N15 N16 N17 N18 N19 N20 N21 N22 N23 N24 N25 N26 N27 N28
8no ml
(%)
0.133 0.128 0.124 0.116 0 0 0 0 0 0 0.110 0.106 0.103 0.100 0.079 0.078 0.077 0.075 0.073 0.125 0.122 0.116 0.112 0.089 0.089 0.089 0.083 0.079
Phase
Fatigue Life 3
3~o,~ (%)
~0(deg.)
N i (cycles)
0 0 0 0 0.304 0.304 0.293 0.290 0.275 0.252 0.161 0.158 0.153 0.149 0.235 0.231 0.227 0.222 0.218 0.184 0.179 0.171 0.166 0.265 0.265 0.265 0.246 0.235
... ... ... ... ... ... ... ... ... ...
130 000 208 000 >1 700 000 >1 700 000 65 000 142 000 97 000 700 000 N.M. 5 260 000 232 000 233 000 165 000 >1 300 000 157 000 385 000 165 000 > 1 300 000 >1 300 000 197 000 151 000 353 000 >1 300 000 125 000 130 000 289 000 N.M. > 1 300 000
0 0 0 0 0 0 0 0 0 90 90 90 90 90 90 90 90 90
Nominal bending strain amplitude at the notch section, e,,om --
4
N f (cycles)
262 000 380 000 ... .H
110 000 420 000 164 000 1 230 000 184 000 870 000 453 000 524 000 334 000 315"000-725 000 309 000 ... 236"000-185 000 440 000 160000 213 000 440 000 246 000 ...
32B E ~rd~ "
16T 2 Nominal torsional strain amplitude at the notch section, Y, om= Gird 3. 3 Number of cycles to 1.0 m m surface crack length. 4 Number of cycles to 10% load drop. 5 Not measured.
(Fig. 4). In t h o s e tests, f r o m one to a few cracks occurred a r o u n d the c i r c u m f e r e n c e o f a s p e c i m e n at the m i n i m u m section, o n e or two o f t h o s e cracks leading to final failure. Elastically calculated strain a m p l i t u d e s applied to the s m o o t h s p e c i m e n s a n d r e s u l t i n g fatigue lives, N / ( 1 . 0 m m surface crack) a n d N s (10% load drop), are listed in T a b l e 3. M o s t o f the life w a s s p e n t in the crack initiation stage, f o r m i n g m i l l i m e t e r - s i z e d cracks; for instance, N i / N f w a s 75 ~ 9 5 % for b e n d i n g a n d 70 - 85% for torsion. Figure 5 represents the b e n d i n g a n d torsional fatigue data u s i n g the s h e a r stress-based critical p l a n e p a r a m e t e r o f Eq 1, w h e r e the solid line w a s d r a w n f r o m the 10% load drop data u s i n g the relation: ~'a + 0.23o',,m~x = 7 2 3 N ] -~176
(mPa)
(10)
T h e c o n s t a n t s in Eq 10 were d e t e r m i n e d b y c o m p u t i n g ~'a for the torsion data points a n d ~'a a n d On,max for the b e n d i n g points. T h e n a least-squares r e g r e s s i o n w a s p e r f o r m e d for the data set o f the torsion
PARK AND NELSON ON BENDING-TORSION FATIGUE
255
FIG. 3--Crack growth in a smooth specimen tested in bending. A~mox and A'rmaxdenote ranges of maximum principal stress and maximum shear stress, respectively. plus bending points using log (i-~ + ko-. . . . . ) = logA + a log Nf
(11)
with k = 0 for the torsion points and k a variable for the bending points. The value of k was varied from 0 to 1 in steps of 0.01 to see which value maximized the sample correlation coefficient, also al-
FIG. 4---Crack growth in a smooth specimen tested in torsion. A~r,~ and A'Cmaxdenote ranges of maximum principal stress and maximum shear stress, respectively.
256
MULTIAXlAL FATIGUE AND DEFORMATION
500
1%Cr-Mo-V Steel Smooth Specimen []
1.0 mm crack Bending Torsion
|
Bending
[]
Torsion
0 (2.
10% load drop
400 x E C
b 03
350
....
C)
I-I --i.--
"'"-..
300 0"--~
25O i
i
i
i
i
ii
i
i
i
~
10 s
i
i
i
i]
i
106
I
i
i
i
i
i
i~
107
N (cycles) FIG. 5--Correlation of smooth specimen bending and torsion fatigue data by the critical plane approach. The dashed line is based on the 1.0 mm crack data and the solid line on the 10% load drop data.
lowing determination of the values of A and a. The same procedure was applied separately for the lives to 1.0 mm cracking. The k value again turned out to be 0.23, but the right-hand side of Eq 10 became 661Ni -~176 as given by the dashed line in Fig. 5. The elastic distortion strain energy parameter We of Eq 6 was also used to represent the smooth specimen data, as shown in Fig. 6, where the solid line is
We = 5.50Ni ~
(MJ/m 3)
(12)
from the bending test data for final failure and the dashed line is We = 4.72Ni-~ from the bending data for crack initiation. Note that lives in torsion were somewhat greater than those in bending for the same value of We since We does not use a parameter such as k to merge bending and torsional data. Figure 7 shows correlations by the SEQA parameter of Eq 3, where the solid line is from the bending data for final failure
SEQA = 1159 N f 0"063
(MPa)
(13)
and the dashed line is SEQA = 1074N~ -~176 from the bending data for crack initiation. The above baseline equations will be used in the next section for comparison with the notched specimen fatigue test results.
Notched Specimens In bending tests, cracks initiated on the top or bottom of a notch and grew along the notch root to final failure. Characteristics of cracking behavior were almost the same as that in smooth specimens. In torsion tests, cracks initiated on the plane of maximum shear stress range Aq'max either along the
PARK AND NELSON ON BENDING-TORSION FATIGUE
1%Cr-Mo-V Steel Smooth S p e c i m e n
E
2 1.5
O []
1.0 mm crack Bending Torsion
| []
10% load d r o p Bending Torsion
257
[]
0--'~
.6
,
,
i
,
, , i
i
i
i
i
10 2
i
[
i
i [
[
i
i
i
i
10 8
N
i
i
ii
10 7
(cycles)
FIG. 6--Correlation of smooth specimen bending and torsion fatigue data by the elastic distortion energy parameter We. The dashed line is based on the 1.0 mm crack data and the solid line on the 10% load drop data.
1000
1%Cr-Mo-V Steel Smooth S p e c i m e n O []
1.0 mm crack Bending Torsion
| []
10% load d r o p Bending Torsion
800 t~ (2_
< (2 UJ 09
[]
600
...
E]
"-'11~
500 0---~ 400
......
I
,
,
, ,,
105
,,,I
106 N
I
I
I
I
, ,,,I
107
(cycles)
FIG. 7--Correlation of smooth specimen bending and torsion fatigue data by the equivalent stress
parameter SEQA. The dashed line is based on the 1.0 mm crack data and the solid line on the 10% load drop data.
258
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 8--Crack growth in a notched specimen under torsion. A0-,no_~and A.r,~, denote ranges of maximum principal stress and maximum shear stress, respectively.
direction of the specimen axis or along the circumferential notch root. The surface lengths over which this shear mode cracking occurred, however, were less than that in the smooth specimens, for instance, approximately 50-200/xm, which may be due to the effect of geometry of the notch. Then, the cracks turned into the tensile mode (Fig. 8) and propagated on the plane of maximum principal stress range A0"maxto final failure. Under in-phase bending and torsion, cracks formed on the plane of ATmax and grew to a length of about 10~50/~m and then turned onto the A0-maxplane for both cases of A = 1 and 2 (Fig. 9). Under 90 ~ out-of-phase combined bending and torsion, the two cases of A = 1 and 2 showed different cracking behavior. For A = 1, cracks initiated on the A~'m~xplane but propagated on the AO'maxplane (Fig. 10) as with the in-phase cases. For A = 2, cracks initiated on the A ~'maxplane as for other load cases; however, the cracks did not follow the Ao'max plane exactly (Fig. 11) but rather grew in a direction between the planes of Arm,x and AO'max. The difference in crack growth behavior for the case of A = 2 with a phase difference of 90 ~ might be explained by examining the magnitude of maximum normal stress amplitude o"1 (= A0-m,x/2) on planes whose normals are at angle 0 with respect to the specimen axis 0-1 = maxt [0-~ COS2 ~/ -~ Txy sin(20)]
(14)
where o'x is the notch root normal stress given as gtbEeno m sin(tot), ~'xyis the notch root shear stress given as gttG'Ynom sin(tot - th), Ktb and Kit are, respectively, theoretical stress concentration factors for bending and torsion, and th is the phase difference between bending and torsion. The value of o1 in Eq 14 is maximized with respect to the time t. Figure 12 shows the variation of 0-1 versus the angle of plane 0, where it can be seen that for the case of 90 ~ out-of-phase loading with A = 2, the maximum value of 0-1 changes little (less than about 5%) in the range of 0 = 0~176 and 140~ ~ Therefore, cracks initiated on the Armax plane seemed to have followed a plane in that range. On the other hand, the 0-1 values for the other loading cases drop abruptly from the maximum point and all of the cracks propagated on the trl planes.
PARK AND NELSON ON BENDING-TORSION FATIGUE
259
FIG. 9--Crack growth in a notched specimen under in-phase bending and torsion. A~mo~and A'cm~x denote ranges of maximum principal stress and maximum shear stress, respectively.
FIG. lO--Crack growth in a notched specimen under 90 ~ out-of-phase bending and torsion ( A = 1). A~m~ and A'r,,~ denote ranges of maximum principal stress and maximum shear stress, respectively.
260
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 11--Crack growth in a notched specimen under 90 ~ out-of-phase bending and torsion ( A = 2). AO'ma x and A ~'m~xdenote ranges of maximum principal stress and maximum shear stress, respectively.
-
Bending - In-Phase (X=I)
-
---------. . . . . . . . .
.... 1.0
0.8
0.6
0.4
7
In-Phase (~.=2) 90* (~,=1) 90* (~=2) Torsion
,,,, .~~'~.'~\-.~..
/...c;'~.. i:~,-~"f-
/"'~\,Q.\X'\"\ ,?.;',.
/./I"" ..i/.\.X'~.I
'_ / \"\.\'Q
,......,.,i-\/ ..
-/ _!
/ / 7 /I7.,"\
\ -~,;..!,/>-/,
0.2 9
0.0 0
.
,
,
30
,
,
,
,
60
."-C'k/~,./,
90
T
,",,,/, ',~/. 120
150
,
'
180
(degree)
FIG. 12--Maximum normal stress amplitude 0 1 vs. angle of plane ~ with respect to the specimen axis under combined bending and torsion.
PARK A N D NELSON ON B E N D I N G - T O R S I O N FATIGUE
261
To evaluate results of the tests with notched specimens, nominal elastic stresses at the notch were multiplied by Ktb and K , to obtain notch stresses. Multiaxial cyclic elastic-plastic analyses of notch strains were not attempted because of the significant computational uncertainties that would be involved, especially for nonproportional stresses, and because notch plastic strains were relatively small in any case. The nominal strain amplitudes applied and corresponding fatigue lives are listed in Table 4. Figures 13 to 15 show correlations by the parameters ~'a + ktrn,m,x, We, and SEQA, respectively, based on elastically calculated notch stresses. Solid and dashed lines in the figures were obtained based on the smooth bending and/or torsional test data, as described in the previous section. The parameters ~-a + ko'~,m~• and We resulted in conservative correlations. The SEQA parameter correlated the test data conservatively except for 90 ~ out-of-phase loadings with h = 1. As a measure of the relative performance of the different life prediction approaches, mean squared errors (MSE) were computed from
MSE =
_1 _s ~.~ (log Np -
(15)
log No)2i
n i=l
where n is the number of data points, Np is predicted life, and No is observed life. MSE for the parameters ~'a + k~ We, and SEQA in Figs. 13 to 15 were computed to be 0.72, 1.23, and 0.52 for the 1.0 mm surface crack data, and 0.76, 1.29, and 0.64 for the 10% load drop data, respectively.
Crack Initiation and Final Failure In order to compare lives spent in the crack initiation and propagation stages, numbers of cycles to a 0.1 mm surface crack, to a 1.0 mm surface crack, and to a 10% load drop are compared in
800 1%Cr-Mo-V Steel Notched Specimen ~3
600
O [] A
empty symbol : 1.0 mm crack marked symbol: 10% load drop
x
E cO co oq o 4-
~|174
400
Bending Torsion In-Phase (;L=I)
v
In-Phase (X=2)
0
90- (~=1)
0
90 ~ (;L=2)
|
300 250
......
r
. . . . . . . .
105
r
106
N
,
. . . . . . .
107
(cycles)
FIG. 13--Correlation of notched specimen bending and torsion fatigue data by the critical plane parameter. The dashed line is based on the smooth specimen 1.0 mm crack data and the solid line on the 10% load drop data.
262
MULTIAXlAL FATIGUE AND DEFORMATION
1%Cr-Mo-V Steel Notched Specimen
2
O
90 o (x=2)
[]
Z~ V
E v
O
Bending Torsion In-Phase (2~=1) In-Phase (X=2) 90 ~ (Z=l)
O
empty symbol : 1.0 mm crack marked symbol: 10% load drop
~ | 1 7 4
|
1.5
0 . 7
r
r
i
i
i i i
i
I
I
i
r
i
105
,
,r
i
,
,
,
,
108 N
,
,
,
107
(cycles)
FIG. 14---Correlation of notched specimen bending and torsion fatigue data by the elastic distortion energy parameter We. The dashed line is based on the smooth specimen 1.0 mm crack data and the solid line on the 10% load drop data.
1000
1%Cr-Mo-V Steel Notched Specimen empty symbol : 1.0 mm crack marked symbol: 10% load drop
O [] L& v O O
n
Bending Torsion In-Phase (;L=I) In-Phase (;L=2) 90 ~ (~=1) 90 ~ (X=2)
v
< O
700 []
[] [] []
LU 03
[] []
i~
[]
500
400
......
, 105
. . . . . . . . N
i 106
....... 107
(cycles)
FIG. 15--Correlation of notched specimen bending and torsion fatigue data by the equivalent stress parameter SEQA. The dashed line is based on the smooth specimen 1.0 mm crack data and the solid line on the 10% load drop data.
PARK AND NELSON ON BENDING-TORSIONFATIGUE
263
Fig. 16, where it can be seen that for smooth specimens most of the life was spent in forming a crack of 1 mm length, and slightly more so for bending. On the other hand, notched specimens spent more of their lives after cracks reached 1 ram, for instance, about 40-50% of total lives for bending, torsion, and in-phase bending and torsion. Under 90 ~ out-of-phase loadings, remaining lives were reduced to about 20~30%, especially for the case of A = 1. It is also of interest to note that most of the life was spent in forming a 0.1 mm surface crack under 90 ~ out-of-phase loadings.
Discussion The scatter in fatigue lives of notched specimens as correlated by the critical plane, We and SEQA parameters in Figs. 13-15 is likely associated with the stress levels used in testing being close to a fatigue limit, a situation that tends to increase scatter [16]. The visual appearance of the scatter is also somewhat exaggerated in those figures by the difference in log coordinates used for the abscissae and
FIG. 16---Comparison of fatigue lives to crack initiation and final failure.
264
MULTIAXIALFATIGUE AND DEFORMATION
ordinates. The correlations by the SEQA approach follow the trends observed in other studies [5,15] of being nonconservative for out-of-phase combined bending and torsion, but the somewhat lower mean squared error of the correlations by that parameter compared to the We or critical plane parameters was unexpected. The three fatigue damage parameters considered in this paper were evaluated with elastically calculated stresses. Over the years, elastic stresses have often been used to correlate high-cycle fatigue data from notched specimens, even when there was some plastic straining at notch roots. In future work, it might be of interest to evaluate the test results in this paper using notch strains estimated by various multiaxial cyclic elastic-plastic notch analyses that have been under development in recent years [17]. In high-cycle fatigue of lab specimens, it is generally assumed based on numerous empirical observations that most of the fatigue life (to 10% load drop) is spent in forming millimeter-sized cracks [18]. Such is the case in tests of smooth specimens reported here. However, in most of the tests of the notched specimens, roughly half of the life was spent in initiating cracks of that size and the remainder in crack propagation, The role of crack growth in high-cycle multiaxial fatigue of notched specimens may be even more significant for other types of notches where the geometry of the notch offers greater resistance to crack growth than the one used in these tests. Such geometries might include splines, keyways, or circumferential V-grooves. Thus, the multiaxial fatigue life of notched specimens or components should depend not only on surface stresses in the notch but also on notch geometry. The test results reported here also showed that, in many cases, small cracks initiated by shear grew a small distance (<0.5 mm) and then turned to grow by normal stress to failure. That behavior suggests that perhaps the high-cycle multiaxial fatigue lives of notched specimens should be predicted using a shear stress based parameter for initiation and growth of small cracks to the transition in crack direction and a normal stress based parameter for their subsequent propagation to failure. However, implementing such an approach would require a way of determining when to switch from one parameter to the other, knowledge not currently available in advance of testing. Further studies of specimens containing different types of notches and subject to a variety of multiaxial fatigue loadings are needed, including observations of the formation of small cracks in the notches and their growth behavior. Also needing further study are effects of residual stresses and changes in fatigue behavior arising from manufacturing processes (e.g., shot peening and heat treatments). Conclusions (1) In high-cycle bending tests of smooth and notched specimens made of 1% Cr-Mo-V steel, cracks formed on planes of maximum shear stress amplitude in some cases but not as clearly in others. In all cases, cracks propagated to failure on planes of maximum principal stress amplitude. In torsion and combined bending-torsiontests, cracks formed on planes of maximum amplitude of shear stress, and then propagated on planes of maximum principal stress amplitude. An exception to that trend was for 90 ~ out-of-phase tests of notched specimens with a ratio of equivalent nominal bending strain to equivalent shear strain of approximately two; cracking proceeded on planes between those of maximum shear and maximum principal stress amplitude. (2) In bending and torsion tests of smooth specimens, nearly all of the life to failure (10% load drop) was spent in forming millimeter-sized cracks. However, in bending, torsion, and inphase bending and torsion tests of notched specimens, approximately half of the life was spent in initiating cracks of that size, with the remainder spent in growth to failure. In the out-ofphase tests, somewhat more of the life was spent in crack initiation. (3) A shear stress based critical plane and an elastic distortion energy parameter both correlated notched specimen fatigue lives conservatively. An extension of the traditional von Mises approach was nonconservative for certain 90 ~ out-of-phase loadings but otherwise provided reasonable correlations of lives.
PARK AND NELSON ON BENDING-TORSION FATIGUE
265
References [1] Gough, H. J., "Engineering Steels Under Combined Cyclic and Static Stresses," Proceedings, Institution of Mechanical Engineers, Vol. 160, 1949, pp. 417-440. [2] Simburger, A., "Festigkeitverhalten zaher Werkstoffe bei einer mehrachsigen, phasenverschobenen Schwingbeanspruchung mit Korperfesten und verandedichen Hauptspannungsrichtungen," Bericht Nr. BR-121, Laboratorium fur Betriebsfestigkeit, Darmstadt, Germany, 1975. [3] Kurath, P., Downing, S. D., and Galliart, D. R., "Summary of Non-Hardened Notched Shaft Round Robin Program," Multiaxial Fatigue: Analysis and Experiments, SAEAE-14, D. Socie and G. Leese, Eds., Society of Automotive Engineers, 1989, pp. 13-31. [4] Fash, J. W., Socie, D. F., and McDowell, D. L., "Fatigue Life Estimates for a Simple Notched Component under Biaxial Loading," Multiaxial Fatigue, STP 853, K. Miller and M. Brown, Eds., American Society for Testing and Materials, 1985, pp. 497-513. [5] Tipton, S. M. and Nelson D. V., "Fatigue Life Predictions for a Notched Shaft in Combined Bending and Torsion," Multiaxial Fatigue, STP 853, K. Miller and M. Brown, Eds., American Society for Testing and Materials, 1985, pp. 514-550. [6] Sakane, M., Ohnami, M., and Hamada, N., "Biaxial Low Cycle Fatigue for Notched, Cracked and Smooth Specimens at High Temperatures," Journal of Engineering Materials and Technology, Vol. 110, 1988, pp. 48-54. [7] Umeda, H., Sakane, M., and Ohnami, M., "Notch Effect in Biaxial Low Cycle Fatigue at Elevated Temperatures," Journal of Engineering Materials and Technology, Vol. 111, 1989, pp. 286-293. [8] Yip, M.-C. and Jen, Y.-M., "Biaxial Fatigue Crack Initiation Life Prediction of Solid Cylindrical Specimens with Transverse Circular Holes," International Journal of Fatigue, Vol. 18, No. 2, 1996, pp. 111-117. [9] Yip, M.-C. and Jen, Y.-M., "Mean Strain Effect on Crack Initiation Lives for Notched Specimens under Biaxial Nonproportional Loading Paths," Journal of Engineering Materials and Technology, Vol. 119, 1997, pp. 104-112. [10] Socie, D., "Critical Plane Approaches for Multiaxial Fatigue Damage Assessment," Advances in Multiaxial Fatigue, STP 1191, D. McDowell and R. Ellis, Eds., American Society for Testing and Materials, 1993, pp. 7-36. [11] "Class 1 Components in Elevated Temperature Service, App. T," Cases of ASME Boiler and Pressure Vessel Code, Sec. III, Div. 1, Code Case N-47-23, American Society of Mechanical Engineers, 1988. [12] Papadopoulos, I. V., "A High-Cycle Fatigue Criterion Applied in Biaxial and Triaxial Out-of-Phase Stress Conditions," Fatigue and Fracture of Engineering Materials and Structures, Vol. 18, No. 1, 1995, pp. 79-91. [13] Peterson, R. E., Stress Concentration Factors, Wiley, New York, 1974. [14] Nisitani, H. and Noda, N.-A., "Stress Concentration of a Cylindrical Bar with a V-Shaped Circumferential Groove Under Torsion, Tension or Bending," Engineering Fracture Mechanics, Vol. 20, No. 5/6, 1984, pp. 743-766. [15] Nelson, D. V. and Rostami, A., "Biaxial Fatigue of A533B Pressure Vessel Steel," Journal of Pressure Vessel Technology, Vol. 119, 1997, pp. 325-331. [16] Forest, P. G., Fatigue of Metals, Pergamon Press, Oxford, 1962. [17] Tipton, S. M. ancl Nelson, D. V., "Advances in Multiaxial Fatigue Life Prediction for Components with Stress Concentrations," International Journal of Fatigue, Vol. 19, No. 6, 1997, pp. 503-515. [18] Nelson, D. V. and Socie, D. F., "Crack Initiation and Propagation Approaches to Fatigue Analysis," Design of Fatigue and Fracture Resistant Structures, STP 761, P. R. Abelkis and C. M. Hudson, Eds., American Society for Testing and Materials, 1982, pp. 110-132.
S a m Y. Z a m r i k 1 a n d M a r k L. R e n a u l d 2
The Application of a Biaxial Isothermal Fatigue Model to Thermomechanical Loading for Austenitic Stainless Steel REFERENCE: Zamrik, S.Y. and Renauld, M.L., "The Application of a Biaxial Isothermal Fatigue Model to Thermomechanical Loading for Austenitie Stainless Steel," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 266-280. ABSTRACT: A biaxial thermomechanical fatigue (TMF) model has been developed by extending a biaxial fatigue model for isothermal condition. The proposed model assesses the in-phase and out-ofphase type cycle incorporating the effect of oxidation and creep. The isothermal fatigue model utilizes the concept of triaxiality factor (TF), which accounts for the state of stress effect on the material's fracture ductility. The TMF biaxial strain ratios varied from 0 to 3.65 at cyclic temperatures of 399 to 621 ~ (750 to 1150~ All tests were strain controlled using tubular specimens. Heating was by induction and the cooling was by natural convection. KEYWORDS: biaxial fatigue, thermomechanical, triaxiality factor, phase factors, Z-parameter, 316 stainless steel
Notation Aeteq Ae i" Ae p ej~. eyo y A ~b ~blp ~b~ cr} o-a o'2 ~3 b, c f Nf N90
Mises equivalent strain range Inelastic strain range Plastic strain range Fracture ductility coefficient Effective ductility Tensile ductility Shear strain Biaxial strain ratio Phase factor In-phase factor Out-of-phase factor Fatigue strength coefficient Principal stress Principal stress Principal stress Material constants Frequency Cycles to failure 10% drop in load
1 Professor emeritus, Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA 16802. 2 Research engineer, United Technology Research Center, (Pratt & Whitney), Materials Structures Technology Department, East Hartford, CT 06108. 266
Copyright9
by ASTM lntcrnational
www.astm.org
ZAMRIK AND RENAULD ON AUSTENITIC STAINLESS STEEL
T TF Z E, G ~,~ A
267
Temperature Triaxiality factor Z-parameter Elastic module Elastic Poisson's ratio Ductility constant
Introduction
Thermomechanical fatigue (TMF) has recently surfaced as a major contributor to industrial component failure. The TMF process is a result of thermal cycling superimposed on mechanical load. If the thermal strain is added to the mechanical strain, the cycle is known as in phase (IP); however, if the thermal strain is not additive, then the TMF cycle is out of phase (OP). The life of structural components is greatly affected by these two types of cycles. For example, for material such as Inconel 738LC, which is a high-strength but low-ductility material used as gas turbine blades, the fatigue life is considerably reduced when the strain and temperature are out of phase with each other; however, for material such as 316 stainless steel, a low-strength and high-ductility material, the in-phase condition is more damaging to fatigue life. Also, the life of the component is affected, in addition to the phasing of the TMF cycle, by the state of stress. Since most components experience a multiaxial/biaxial state of stress, basic life prediction models developed for isothermal conditions were found inadequate to address the effect of TMF. In searching the literature, one finds a limited amount of published results that address TMF and the state of stress. Yamauchi et al. [1] established a variable temperature distribution along a pipe specimen by induction heating of the outer surface and cooling the inner surface with water. This testing condition generated an equal biaxial stress state under out-of-phase loading. The loss in life was attributed to a reduction in the material's ductility. Castelli et al. [2] investigated the TMF deformation of Hastelloy X in torsion at three temperature ranges. The emphasis was not on fatigue life prediction but rather on the applicability of the Jz yield criterion under TMF. The correlation was good at the highest temperature range, but decreased with decreasing temperature. Meerman et al. [3] published biaxial TMF results on IN-738LC where various combinations of stress and strain parameters were used to correlate all fatigue data, resulting in limited success. Bonacuse and Kalluri [4] investigated the axial, torsion, and axial-torsion TMF response of wrought cobalt-base superalloy, Haynes 188, at a temperature range of 316 to 760~ (600 to 1400~ and axial and shear strain ranges of 0.8
t9,05
e/ IE.7 D
q. 52.96 [.
.
.
.
158,75
ALL DIHEN$IDN$ IN MH UNLESS OTHERWISE NOTED, TDLERANCs ARE + / - 0.05 HH FIG. l--Biaxial TMF test specimens.
+0.05 -0,00
, I
AXIAL ISTRAIN ---
l~S'mMm
LaA~ TDRBL~
Tr~srrtter
i Isot~ted
I
~_~ i,.
l~,,,.a,.ek ~n,,t:t
IT,,en,m~ :,~ir~, mm.ll
I~,~,t ~r,,t
----iStr.Qln E,ndl,e v t i s
FIG. 2--Schematic o f TMF testing system.
He~r
| ~:ih-,.d I a-tO,,~ ~O*"I
o-~ rot.i: ileal
Vest 1~o75 Tev,pero,'tu~
I
I
~, L O O P
I I:~omed
Controller
Teb-kS~l~r
|solttid ]--~Tenper~tu~ co.and Tr~nsrd*t~r I 10-5 Volt slOno.! _
,,.i
=m';
z
z o o m -n 0 7J
c m
~>
i'-n
x
I'-
c
ZAMRIK AND RENAULD ON AUSTENITIC STAINLESS STEEL
269
and 1.4%. They investigated the applicability of four life prediction models: the von Mises equivalent strain range, the Modified Multiaxiality Factor (MF) [5], the modified Smith-Watson [6], and the Fatemi-Socie-Karath model [7]. The Mises and the Modified Multiaxiality factor methods provided the best data fit. Material and Test Procedure Type 316 stainless steel material, used in this research program, was provided by Oak Ridge National Laboratory in the form of 25.4-mm (1 in.) diameter rod hot rolled at 1180~ Tubular fatigue specimens were machined from the rolled bars by a low stress grinding process to the dimensions shown in Fig. 1. Machined specimens were solution annealed at 1065~ in argon for 30 min. The TMF testing equipment contains three closed-loop control loops, one for temperature and two for mechanical actuator (axial and torsional), as shown schematically in Fig. 2. Figure 3 outlines the TMF
TEMP
I IN-PHASE TMF I
/•
MAX
MEAN
/,~,\
~ + MAX
TEMP
~TRAIN MIN
//
I I
TEMP I
RAMP
START TIME THERMAL CYCLE
A TEHP
4> START TMF CYCLE
L,nUT-OF-PHA~E
TIME
TMF ]
MAX ~. MEAN
TEMP
'
-- --
\/
',l ; :J ."
0 STRAIN HECH
HIN
/
i r _v RAMP
B
START TIME THERHAL CYCLE
v START TMF CYCLE
TIME
FIG. 3--(a) Relationship between thermal and mechanical strains at start-up of lP TMF test. (b) Relationship between thermal and mechanical strains at start-up of OP TMF test.
270
MULTIAXIAL FATIGUE AND DEFORMATION
testing procedure. An MTS high-temperature biaxial extensometer was used to control the axial displacement and an angle of rotation, which in turn translated to strains. The temperature command procedure for the TMF cycle is shown in Fig. 4, which also shows a "hump" in the exact temperature path. This is due to the cut off procedure in the command signal (upper decoupling point), which gives the temperature enough time to reach the maximum value and then to drop following the command path.
Biaxial Experimental Results Axial-torsional IP TMF experiments were conducted with a 0.3 to 0.5% mechanical axial strain range superimposed on a 0.3 to 1.45% torsional strain range producing biaxiality ratios, A = Ay/ Asmech, which varied from 1 to 3.625. The temperature cycle was 399 to 621~ (750 to 1150~ Typical biaxial IP TMF mid-life hysteresis loops are shown in Fig. 5. The total loop is larger than the mechanical loop since the TMF test is IP where thermal and mechanical strains are additive. Biaxial strain test results for IP TMF are plotted in Fig. 6 in terms of yon Mises yield criterion: {A~fq = [(A~P) 2 + 1/3(AyP)2] m } along with uniaxial IP and OP TMF and biaxial isothermal fatigue data previously generated [8]. Serrated behavior was noticed in the axial strain loop that was absent in the torsional loop, which can be contributed to smoother deformation characteristics in torsion. Failure, for all tests, was determined by a 10% load drop. In general, the IP TMF data generally shows shorter lives than the isothermal data, indicating that this type of cycle is more damaging to the life of the 316 SS material. However, most of the damage is attributed to the axial strain component and not the torsional strain component.
TEMP ~
(621) [B713
~
ACTUAL SPECIHEN TEMPERATURE
.-(613) [857] T \\
TEMPERATURE COMMAND SIGNAL
(510) C677] MEAN
,.,(407) [496]
(399) C482]
I UPPER DECOUPLE PAINT
LOVER DECOUPLE PAINT
TIME
FIG. 4~Temperature command signal and resulting specimen response during TMF experiment without hold time.
ZAMRIK ANO RENAULD ON AUSTENITIC STAINLESS STEEL
1-5340N (1200 Lb)
I
I
150 in,tb)
I
I
N=2719 ' : ~
total
mech
~~i_(17N .0,m 0.0087 ~n/In
0.005 in/in N=2718 . . . . . V,
271
N=2718 -
|
'
.L I
t..-t I~ ' i
J II • II
II I1:
roman
It,:).:.:!:
I |11 I I |
a
'" I/
2 ~ Ji'i i iii,i , ITI
i
i
!11 J | !
_fi!, P I1
I: I I:
,~,L,/I] /!'i ~l,[,/
i ,
~
l; I/: ', '1~ I I
~'.,,,1:~
,,,:!7~7~,,, , , ,,I,
(a)
I
Ill
I
~ 7:1 V
J/J
] :
/~',.
':I ii
(c)
,LII;
(b)
FIG. 5--Mid-life hysteresis loops from biaxial lP TMF testing of Type 316SS: (a) mechanical, (b) total, and (c) torsional A~rnec h = 0.4%, A y = 0.4%, AT = 399 to 621~
Development of Biaxial TMF Life Prediction Model
Biaxial fatigue TMF, plotted in Fig. 6 on the basis of the von Mises criterion, shows considerable scatter in the data that highlights the effect of thermal/mechanical strain cycling. To address this problem, a model has been developed to assess biaxial TMF. The proposed biaxial TMF model is based on the isothermal biaxial fatigue model developed by Zamrik et al. [8]. The model had three basic components: a triaxiality factor (TF), a modified equivalent strain, and a transition cycle Z-parameter. The triaxiality factor concept, developed by Davis and Connelly [9], is used to account for the effect of state of stress on ductility. This is due to cavities that require tensile triaxiality to growth. The coalescence of the cavities during their growth stage considerably reduces the macroscopic strain to fracture, although the intrinsic ductility of the matrix is unchanged. They proposed that an effective ductility, ~s, under multiaxial loading condition may be obtained by dividing the tensile ductility (tensile elongation), ego, by a tri-
MULTIAXIAL FATIGUE A N D D E F O R M A T I O N
272
10 ~
I ~n.As
I !S0riI~MAk T=ePI*CI 0
1.=0 1=3 6~5 ).=2 5 ).=2 0
A
D
v
o x
9 9 * e 9
).=10 ~oo
AY.s,~2~e I oP. IP. IP. IP, IP, IP,
L-O ~0 1-3 625 1,=2.0 1,=1.36 l,=l.O
o" g=,
, ,,.,t
W +
1o ~
IO
&~
A
9
o" q) q~ to
x
~N( Q e tOs '
X
y~ x
o~
x
a
t~ Q.) ~
o 10-1
~
. . . . . .
102
!
.
.
.
.
.
.
.
.
10 s
1
104
C y c l e s to Fa ilu re,
. . . . . . . 10 s
Ng0
FIG. 6--Correlation of Type 316 SS fatigue data using the von Mises equivalent strain range cri-
terion.
axiality factor: efo
(1)
ES-TF
where:
TF =
( 0 " 1 -~ 0" 2 -.]- 0"3)
1
(2) [(0-~ - 0-2) 2 + (0"2 - 0-3) 2 + (o-3 - 0"02] ]/2
The triaxiality factor (TF) is equal to one for uniaxial case, zero for pure torsion case, and two for equal biaxial case. Their view was that the effective ductility reduces as TF increases. Manjoine [10] has applied the concept of triaxiality factor to explain the effect of stress state on ductility at elevated temperatures under monotonic loading. The measured reduction of ductility for a number of materials versus triaxiality factor is shown in Fig. 7. Manjoine's experimental results indicated that for a given temperature and strain rate, ductility decreases as triaxiality factor increases. Zamrik and Mirdamadi [8] showed that the triaxiality factor can improve the Mises equivalent strain predictive capability when expressed as: A~teq =
A e e e q -[-
ACeq
(3)
ZAMRIK AND RENAULD ON AUSTENITIC STAINLESS STEEL
L4
273
oI ~'0 lYD< 0
Cede l. 0.3~: Steel
_4
~-o
I
2. O.k~C5l~J
3. Ann. Brass 4. 20"L4-T3 AI 5, 7075-T651 A1 ..:1 =.
~''"
Strain Rate
1.2
Totwndature
n
2
'9 o.s Q
Constant
3.4
-I
0.4 q " q.1, q. b
!
!
v
v
t
v
-!
O
l
2
3
4
~ " l o | § 2 4TRI~IIALIIY
FJ~'It::~
031
u
~01 _oZi2 §
+1o3..~112 ]
l/Z
FIG. 7--Effective ductility as a function o f stress state as calculated by the triaxiality f a c t o r [10].
in terms of fatigue cycles: = B (2 Ny)b + C(2Ns) c
(4)
constants B and C are replaced by: t
B = (~ 1-TF]&f \--
(5)
i E
and C = ( A 1 TF) F~}
(6)
Therefore the equivalent strain can now be expressed as: A ~ t q __ - ,71-L TF ~0"~ - .~. .' .S. ) b + A I TF F.~ ( 2 N / ) c
(7)
The Z parameter is taken as the ratio of the fatigue transition cycles between the axial and the torsion
274
MULTIAXIAL FATIGUE A N D D E F O R M A T I O N
strain where the elastic strain is equated to the plastic strain and defined as:
Z - 2(1 + ~,e~ o-y/ET;
(8)
r~ and 7~ are strength and ductility coefficients. Experimental results using the model are shown in Fig. 8. A is a ductility coefficient that was found to be 1.85 and was rounded to 2.0 [8]. However, when Eq 7 was applied to IP TMF biaxial fatigue data, considerable scatter was observed and the isothermal approach was not adequate as shown in Fig. 9. One difference between the isothermal data and the TMF data is the cycle frequency. Under TMF, the cycle is much slower than under isothermal cycling, and by cycling at a slower rate, creep or environmental damage per cycle can be induced. Therefore, a frequency term, f, has to be included mainly in the inelastic strain term of Eq 7 rather than in the elastic term. Figure 9 shows that the time of the cycle had a dominant effect in the low-cycle regime. Hence, the modified term is expressed as
A.e*eq=(ZTF-l) Aeeq+(f)m(ATF-])AePeq
(9)
where Z = 1.35, A = 2, andre = 0.10. Using Eq 9, isothermal and TMF data are plotted as shown in Fig. 10. Two observations can be made from the results: one is that the TMF data has shifted, relative to the isothermal data, towards
101 0 A El
09
0 x
.w..q
l,=O, T=fi2l~ ).=3.62. T=62 Iq~ 1=25, T=62 I~ ),.=2 O, T=62 I~ 1.=1 o, T=621oC ~=~l T=62 IOC k*r 9 =0 9878(N90 )-0"i~ 14 -I- 33 .O(N90 }-'0594n ~ j -
v
C::Y"
10 o Q)
,~
..-,
o
E-"-.
I >
L"q
Z=l 35
A--2 o
10 2
10 3
Cycles to Failure,
I0 4
I0 s
N90
FIG. 8--Analysis of isothermal axial, torsional, and axial-torsional fatigue data on Type 316 SS using "Z-parameter" and triaxiality factor [8].
275
ZAMRIK AND RENAULD ON AUSTENITIC STAINLESS STEEL
10'
I larm~.RuAt, c~
IT"~"*L-"~"A"!.CAL AT~0O-a~n~ I oP, 1.=o
).=0
~, []
L=3.625 ).=2.5
9 9
IP. ~ 0 IP. 1.=3.625
1.=2.0 ).=i.o 1.=~
*
IP. ),=20
0 •
9
IP, I.=i.3G
9
IP, ),=1.0
.~..q
w
T4~t*q
(3
1 (1)
X=A't/AE
•
v
10 ~
c~
9 Q)
~1~i ~
Q.)
t QJ
A
oo
I 0
Z=1.35
v
A=2 0
i0-I
~
10 2
. . . . . . .
I
lO s
......
I
10 4
..... lO s
Cycles to Failure, N90 FIG. 9--Analysis of uniaxial and biaxial isothermal and thermal-mechanical fatigue data on Type 316 SS using "Z-parameter" and triaxiality factor.
larger equivalent strain ranges. This is because the IP and OP TMF cycle time is much longer (180 s) than the isothermal cycle time (6 s). The second observation is that the OP data lie above the isothermal data while the IP data falls slightly below, suggesting a thermal/mechanical phase effect. Neu and Sehitoglu [11] proposed specific mathematical relations for thermal/mechanical strain phasing to account for the effect of oxidation and creep damage due to TMF loading. The phasing factor, q~, was written in the form of two exponential functions in terms of thermal/mechanical strain rates. One function is an oxidation effect under OP TMF and the other for the creep effect encountered in the IP TMF cycle. In their study, they showed that the phase factor for oxidation decreases from a maximum value of + 1 where OP TMF conditions are most detrimental to zero where no oxidation damage is predicted. Similarly, for creep damage, the phase factor also decreases from a maximum value of + 1. The value of + 1 is obtained when thermal strain rate is equal to the mechanical strain rate resulting in a ratio of + 1. On the other hand, if the thermal strain rate is much greater than the mechanical strain rate, the phase factor approaches zero. Their oxidation damage formulation is based on repeated oxidation formation at the crack tip with high-temperature exposure followed by oxide rupture at the low temperature. Also, the creep damage term is based on void formation at grain boundaries and grain boundary intersections. These voids grow simply in tension in such a way that creep-induced cracking occurs along the grain boundaries. However, in this investigation, the frequency term was incorporated in the analysis for possible creep-like damage. Definitely, the failure mode in IP was intergranular with grain boundary cracking. This can be attributed to higher strain accumulation, thermal and mechanical, with longer exposure time where cracking took place at the
276
M U L T I A X I A L FATIGUE A N D D E F O R M A T I O N l0 t
IIS0rH.~aUALr=62!*Cl
v C:r"
9o..,e
l:~
tO
I
rJ~
Cz.. [_,
[ 111~UAL-IIECltAHICAkAT~99--62!~ J oP, 1.=o
O
}.=0
A
1,=3825
9
CI
~.=25
9
O X
).=1.0 ~.=ao
9
IP, IP. IP, IP,
$
IP. ).=1.0
bO 1.=.'1.825 1.=20 ).=1 38
v q.)
•
%
10 ~ f..=3 c/3
v
co q) e'-, 0
q,~
x
t~
tO
Z=135 "C7
A=2
,,~ o
6--,
m=0.10 iO-I
.
10 z
.
.
.
.
.
I
,
,
,
, . , I
I0 ~
104
9
,
,
. . ,
l0 s
Cycles to Failure, N90 FIG. lO---Analysis of uniaxial and biaxial isothermal and thermal-mechanical fatigue data on Type 316 SS using "Z-parameter" triaxiality factor, and frequency modification.
grain boundaries, inducing damage similar to creep. Oxidation, on the other hand, was observed to be at a minimum. Since the frequency term was included in Eq 9, data showed some improvement in its life assessment, but still considerable scatter was observed in the low-cycle region. This discrepancy was attributed to phase effect where large strain had accumulated in the IP mode under biaxial loading. Since phase effect was questioned, a simple method was developed to estimate the magnitude of the phase factors for oxidation and creep relative to isothermal fatigue and to see if they are within Neu et al. [11] suggested maximum ranges of + 1 for creep and oxidation. However, since the frequency term (an inducing time-dependent damage mechanism) was used, the frequency modified inelastic strain range was plotted against life for isothermal, IP, and OP TMF data sets as shown in Fig. 11. The IP and OP phase factors, (1)1,~ and q~op, are then computed from the equivalent strain ratios with the isothermal equivalent strain as the common denominator. For example, at a given life of 2000 cycles in Fig. 11, the following ratios were calculated:
(Dip : [Aei(1/f)~ [Aei'(1/f)~
_ 0.47 p
= = 1.15 0.40
(10)
and (Ti)oP_ [Ae'in(l[f)O'l]iso -- 0.47 [Aein(l[f)~ p 0.66
0.70
(11)
ZAMRIK
|
AND RENAULD
i~176
ON AUSTENITIC
0
~--~i
. . . . 0.:6.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
STEEL
277
Isothermal, 6210C (I 1500P)
I-]
IP TklF'. 399-621~
(750-1150~
A
0P Tlff. 399-621~
(750-1150~
0.47
C/3 ~
O
.~
STAINLESS
""
0 40
. . . . . . . . . . . . . . . . . . .
_~
~
<3
0
E7
(D
10-1
, ,
102
,
,
.....
I
~
.
.
Cycles to
.
.
.
104
i05
.
.
.
.
|05
Failure, N90
FIG. 11--Identification of IP and OP phase factors for thermal-mechanical strain cycling of Type
316 SS from 399 to 621~ Using the phase factors as determined, the von Mises modified strain range, Eq 9, was modified to:
1)m
m-~eq ~- (zTF--1) mF"eq-J'- ~ 7
(ATF 1) AePeq
(12)
where
[O]~blP = 1 for isothermal condition
(13)
Equation 12 was used to analyze all type 316 SS isothermal and TMF data generated under uniaxial and biaxial loading. The data is replotted in Fig. 12. As seen in Fig. 12, the phase modification in Eq 12 shows excellent ability to predict the fatigue life for the high-ductility 316 SS under a variety of uniaxial and biaxial isothermal TMF conditions. Arguments can always be made to the validity of the assumptions made, but results definitely showed reasonable account for the assumptions used in the analysis. S u m m a r y and Conclusions The TMF biaxial fatigue analysis is based on an established isothermal biaxial fatigue model to relate tmiaxial, torsional, and axial-torsional fatigue data [8]. Two parameters, namely frequency and phase factors, are introduced in the biaxial TMF analysis to account for the effect of IP or OP cycling. The effect of frequency on fatigue life at elevated temperature is an important factor since oxidation or creep could become major damaging mechanisms. Also, the frequency term provides time-dependent effects since TMF cycle times are significantly longer than isothermal fatigue cycle times.
278
MULTIAXlAL FATIGUE AND DEFORMATION
1Ot
l TH~MAL-MgL'IIAIIIeAklT='qgg"821q~I
o" a~
(3 A
t~
[3 0
•
l=O ~ 3 625 ~2 5 1.,=2.0
u 9
I.=I0 I.=oo
9 9
x:a~/A~ IOo t::r"
w :~
Z=1.35
9
~
OP. 1.=0 IP. L=O IP. 1.=3.R25 IP, l.=:~.0 IP. 1=1.36 IP, 1.=1.0
3{N9o)-o.~26 .,,..~9.7(~9.q!-O.S",e
h=2 m=O.lO
§ or'
o
I.15
in-pha~
l oo
Isothermal
0.70
0ul-ol-phase
qJ
= o
6~ |O~
,
102
,.
. . .. .
,I .
IO s
. ,
.
,
. .,
.
,,I
.
10 4
,
l0 s
Cycles to Failure, N90 FIG. 12--Analysis of uniaxial and biaxial isothermal and thermal-mechanical fatigue data on Type 316 SS using "Z-parameter" triaxiality factor, frequency modification, and phase factor. The phase factor approach was introduced by Neu and Sehitoglu [11] to assess oxidation and creep damage arising from thermal/mechanical strain cycling interaction in a thermally controlled environment. However, The IP TMF phase factor for biaxial loading was + 1.15, which is more than + 1 by 15%. The 15% increase also reflects the observed increase in the effective strain range, which is a combination of plastic and creep like strain producing the intergranular failure characteristics. In
FIG. 13--SEM fractograph showing intergranular crack in semi-circular initiation region. = 0.3%, A T = 0.3%, IP TMF, AT = 399 to 621~ Ngo = 23229.
A~mec
h
ZAMRIK AND RENAULDON AUSTENITICSTAINLESSSTEEL
FIG. 14a--SEM micrograph showing mixed-mode crack propagation in 316 SS. A y = 0.68%, IP TMF, AT = 399 to 621~ N9o = 2144.
A~mech =
279
0.5%,
addition, this increase may account for the effect of biaxiality that was not observed in the Neu's uniaxial test. On the other hand, the OP TMF phase factor was 0.70, an indication that the oxidation was not a factor. In that case, the failure mode was transgranular and the equivalent strain range was reduced by 30% with an increase in life. In a previous uniaxial IP TMF study on type 316 SS by Zamrik et al. [12], failure by intergranular cracking was the dominant failure mechanism, whereas, for the OP tests, crack growth was transgranular. Similarly, the microscopic analysis in this study also shows that biaxial IP thermal-mechanical strain cycling with time exposure produced primarily intergranular crack growth with severe intergranular secondary cracking as shown in Figs. 13 and t4. However, the exact location of crack
FIG. 14b--SEM fractograph showing severe intergranular secondary cracking and severe rubbing on the fracture surface due to a high torsional strain range. Aemech = 0.4%, A T = 1.45%, 1P TMF, A T = 399 to 621~ Ngo -~ 1253.
280
MULTIAXIALFATIGUEAND DEFORMATION
initiation was often difficult to identify, especially under higher torsional loading and, in some cases, small regions of striated transgranular growth were observed. The use of the Z-parameter, triaxiality factor, and frequency and phase factor for biaxial IP and OP cycling has shown a good life predictive capability when applied to 316 stainless steel. Acknowledgment This research program has been supported by the National Science Foundation under Grant No. MSS-9215694 and the Pressure Vessel Research Council (PVRC). The 316 stainless material was provided by Mr. Robert Swindeman of Oak Ridge Laboratory. The support received from NSF and PVRC is greatly appreciated.
References [1] Yamauchi, M., Ohtani, T., and Takahashi, Y., "Thermal Fatigue Behavior of a SU304 Pipe Under Longitudinal Cyclic Movement of Axial Temperature Distribution," Thermomechanical Fatigue Behavior of Materials: Second Volume, ASTM STP 1263, M. J. Verrilli and M. G. Castelli, Eds., American Society for Testing and Materials, West Conshohocken, PA, 1996. [2] Castelli, M. G., Bakis, C. E., and Ellis, J. R., "Experimental Investigation of Cyclic Thermomechanical Deformation in Torsion," NASA TM-105938, 1992. [3] Meerman, J., Frenz, H., Zeibs, J., Kuhn, H. J., and Forest, S., "Thermo-Mechanical Fatigue Behavior of IN738LC and SC16," Thermal Mechanical Fatigue of Aircraft Engine Materials, A GARD-CP- 569, 81st Meeting of the AGARD Structures and Materials Panel, Bnaff, Canada, 1995, pp. 19.1-19.11. [4] Bonacuse, P. J. and Kalluri, S., "Axial-Torsional Thermomechanical Fatigue Behavior of Haynes 188 Superalloy," Thermal Mechanical Fatigue of Aircraft Engine Materials, AGARD-CP-569, 81 st Meeting of the AGARD Structures and Materials Panel, Bnaff, Canada, 1995, pp. 15.1-15.10. [5] Bonacuse, P. J. and Kalluri, S., "Elevated Temperature and Axial Torsional Fatigue Behavior of Haynes 188," Journal of Engineering Materials and Technology, 1995, pp. 191-199. [6] Socie, D. F., "Multiaxial Fatigue Damage Models," Journal of Engineering Materials and Technology, Vol. 109, No. 4, 1987, pp. 293-298. [7] Fatemi, A. and Socie, D. F., "A Critical Plane Approach to Multiaxial Fatigue Damage Including Out-OfPhase Loading," Fatigue and Fracture of Engineering Materials and Structures, u 11, No. 3, 1988, pp. 149-165. [8] Zamrik, S. Y., Mirdamadi, M., and Davis, D. C., "A Proposed Model for Biaxial Fatigue Analysis Using the Triaxiality Factor Concept," Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and R. Ellis, Eds., American Society for Testing and Materials, West Conshohocken, PA, 1993, pp. 85-106. [9] Davis, E. A. and Connelly, F. M., "Stress Distribution and Plastic Deformation in Rotating Cylinders of Strain Hardening Materials," Journal of Applied Mechanics, 1959, pp. 25-30. [10] Manjoine, M. J., "Damage and Failure at Elevated Temperature," Journal of Pressure Vessel Technology, Vol. 105, 1983, pp. 58~52. [11] Neu, R. W. and Sehitoglu, H., "Therrnomechanical Fatigue, Oxidation and Creep: Part II. Life Prediction," Metallurgical Transactions A, Vol. 20A, 1989, pp. 1769-1783. [12] Zamrik, S. Y., Davis, D. C., and Firth, L. C. in Thermo-Mechanical Fatigue Behavior of Materials, ASTM STP 1263, M. J. Verrilli, and M. G. Castelli, Eds., American Society for Testing and Materials, West Conshohocken, PA, 1994, pp. 96-116. [13] ASM Handbook, Fatigue and Fracture, Vol. 19, 1996, pp. 527-556.
Sreeramesh Kalluri 1 and Peter J. Bonacuse 2
Cumulative Axial and Torsional Fatigue: An Investigation of Load-Type Sequencing Effects REFERENCE: Kalluri, S. and Bonacuse, P. J., "Cumulative Axial and Torsional Fatigue: An Investigation of Load-Type Sequencing Effects," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 281-301. ABSTRACT: Cumulative fatigue behavior of a wrought cobalt-base superalloy, Haynes 188, was investigated at 538~ under various single-step sequences of axial and torsional loading conditions. Initially, fully-reversed, axial and torsional fatigue tests were conducted under strain control at 538~ on thin-walled tubular specimens to establish baseline fatigue life relationships. Subsequently, four sequences (axial/axial, torsional/torsional, axial/torsional, and torsional/axial) of two load-level fatigue tests were conducted to characterize both the load-order (high/low) and load-type sequencing effects. For the two load-level tests, summations of life fractions and the remaining fatigue lives at the second load level were computed by the Miner's linear damage rule (LDR) and a nonlinear damage curve approach (DCA). In general, for all four cases predictions by LDR were unconservative. Predictions by the DCA were within a factor of two of the experimentally observed fatigue lives for a majority of the cumulative axial and torsional fatigue tests. KEYWORDS: axial fatigue, cumulative fatigue, cyclic hardening, damage curve approach, life prediction, linear damage rule, load-type sequencing, torsional fatigue
Nomenclature b, c
Exponents o f elastic and inelastic strain range-life relations N u m b e r o f applied cycles at a load level in a cumulative fatigue test Coefficients o f elastic and inelastic strain range-life relations M F Multiaxiality factor N N u m b e r o f cycles TF Triaxiality factor e Engineering axial strain y Engineering shear strain u Frequency o f w a v e f o m l in a fatigue test A Denotes range o f a variable tr Axial stress ~- Shear stress
n B, C
Subscripts 1 2
First load level in a two-load-level cumulative fatigue test Second load level in a two-load-level cumulative fatigue test
1 Senior research engineer, Ohio Aerospace Institute, NASA Glenn Research Center, Cleveland, OH. 2 Materials research engineer, Vehicle Technology Directorate, U.S. Army Research Laboratory, NASA Glenn Research Center, Cleveland, OH.
Copyright9
by ASTM International
281 www.astm.org
282
MULTIAXIALFATIGUE AND DEFORMATION
el Elastic in Inelastic m f A T I II III
Mean value failure Axial Torsional First principal Second principal Third principal
Accumulation of damage in materials subjected to fatigue under multiple load levels and estimation of cyclic life under cumulative fatigue have been the subjects of investigation for the past 75 years [1-13]. In these cumulative fatigue investigations materials have been typically subjected to the same load type (for example, axial tension/compression [3,13], torsion [4,8,9], or rotating bending [5,6]), albeit to different magnitudes, during the multiple loading levels. In engineering design, fatigue life under cumulative fatigue loading conditions is commonly estimated with a linear damage rule (LDR) [1-3], primarily due to its simplicity, associated ease of implementation, and lack of proven applicability of alternative rules. However, inadequacy of the LDR to properly account for the load order effects (either high/low or low/high for a given load type) has been well documented in the literature [6-13]. Within a specified load type, the high/low load ordering typically generates a sum of life fractions less than unity, whereas the low/high load ordering typically generates a sum of life fractions greater than unity. Several nonlinear damage accumulation models [6-12] have been developed to overcome the disadvantages of the LDR for predicting fatigue lives of materials subjected to multiple load levels. Most of the nonlinear damage accumulation models capture the well-known load order effects adequately for a given load type under cumulative cyclic loading conditions. The cumulative fatigue behavior of materials under dissimilar load types could potentially be different from that under a single load type due to either a lack of interaction or a potential synergistic interaction between the deformation and damage modes and their orientation associated with the two load types. Investigationsinvolving cumulative fatigue of materials with dissimilar load types are relatively recent in comparison to those involving the same load type [14-22]. During the past 15 to 20 years, researchers have investigated accumulation of fatigue damage in materials under dissimilar load types such as: (1) tension/compression,torsion, and proportional and nonproportionalcombined axial-torsional loads [14-17,19-22] and (2) torsion and bending [18]. For cumulative fatigue involving axial and torsional loading conditions most of the previous studies have been conducted with: (1) the same equivalent strain range [14,17,19,20], (2) the same fatigue lives [16], or (3) with the same equivalent damage [21]. Equivalency in terms of strain range, fatigue life, or damage is selected primarily to separate the load order effects from the load-type sequencing effects. In general, under equivalent loading conditions, cyclic tension/compression followed by cyclic torsion-type load sequencing has been found to be more benign than that predicted by LDR (for example, with a sum of cycle fractions greater than unity), whereas the load-type sequencing of cyclic torsion followed by cyclic tension/compressionhas been reported to be more damaging than that estimated by LDR (with a sum of cycle fractions less than or equal to unity) [16,17,19,20]. However, in a few investigations [19-21] it has been reported that the load-type sequencing of cyclic tension/compressionfollowed by cyclic torsion is more detrimental than cyclic torsion followed by cyclic tension/compression. This reversal in load-type sequencing effects has been attributed to differences in the cracking patterns of materials that are caused by temperature dependent environmental effects (for example, oxidation) and inherent differences in microstructures. Cumulative fatigue investigations that considered load order as well as load-type sequencing effects under cyclic axial and torsional loads are rather limited in number [15,22]. As far as fatigue life estimation is concerned, noticeable deviations from the LDR have been reported in both studies.
KALLURI AND BONACUSEON LOAD-TYPE SEQUENCING EFFECTS
283
The objective of the present study was to evaluate the effects of both load-type sequencing and high/low load ordering under cumulative axial and torsional loading conditions. A test program was designed to investigate the cumulative fatigue behavior of a representative high temperature superalloy under various sequences of axial and torsional loading conditions. The wrought cobalt-base superalloy, Haynes 188, was selected for this purpose. Examples of the many applications of this superalloy include the cryogenic oxygen carrying tubes in the main injector of the reusable space shuttle main engine and the combustor liner in the T-800 turboshaft engine for the RAH-66 Comanche helicopter. Axial, torsional, and combined axial-torsional fatigue behavior of Haynes 188 under isothermal (316 and 760~ and thermomechanical (316 to 760~ loading conditions on a single heat of the superalloy was previously documented by the authors [23-27]. In the current investigation, axial and torsional fatigue tests were conducted at 538~ on material from another heat of Haynes 188 to establish baseline fatigue lives. Subsequently four sequences (axial/axial, torsional/torsional, axial/torsional, and torsional/axial) of two load-level (single-step) fatigue tests were conducted (same heat as that used for the baseline tests) at 538~ to characterize the cumulative fatigue behavior of the superalloy. For the two load-level tests, summations of life fractions and the remaining fatigue lives at the second load level were estimated with two models, LDR [1-3] and the nonlinear damage curve approach (DCA) [7,10]. This paper summarizes details of the test program, results from the axial and torsional cumulative fatigue tests, and predictive capabilities of the models.
Material and Specimens Solution annealed, hot rolled, cobalt-base superalloy, Haynes 188, was supplied by the manufacturer in the form of round bars with a diameter of 50.8 mm (heat number: 1-1880-6-1714). The composition of the superalloy in weight percent was as follows: <0.002 S, 0.003 B, <0.005 P, 0.09 C, 0.35 Si, 0.052 La, 0.8 Mn, 1.17 Fe, 14.06 W, 22.11 Cr, 22.66 Ni, balance Co. Thin-walled tubular specimens with nominal inner and outer diameters of 22 and 26 mm, respectively, in the straight section (41 mm) and an overall length of 229 mm were machined from the bar stock. Bores of the tubular specimens were finished with a honing operation and the external surfaces of the specimens were polished. Additional details on machining of tubular specimens are available in Ref 28. In the middle of the straight section of the tubular specimen, two indentations (25 mm apart and 80/xm deep) were pressed with a fixture to define the gage section and to positively locate the extensometer probes. Average values of the elastic modulus, shear modulus, and Poisson's ratio for Haynes 188 at 538~ were 190 GPa, 73 GPa, and 0.3, respectively.
Experimental Details All the tests were performed in an axial-torsional fatigue test system [27] equipped with a personal computer and a data acquisition system. Tubular specimens were heated to the test temperature of 538~ in air with a three-coil fixture [29] connected to a 15 kW induction heating unit. Specimen temperature in the gage section was measured with a noncontacting optical temperature measurement device. Thermocouples spot-welded in the shoulder regions of the specimens were used to control and monitor the temperature during fatigue tests. Axial and engineering shear strains within the gage section of each specimen were measured with a water-cooled, axial-torsional extensometer. Test control software written in C language was used to generate triangular, axial, and torsional command waveforms at the appropriate frequencies for the strain-controlled fatigue tests. For each axial and torsional fatigue test, test control software increased the strain to the full amplitude by increasing the strain increments linearly over 10 cycles. For axial strain-controlled fatigue tests, the torsional servocontroller was in load-control at zero torque and for torsional strain-controlled fatigue tests, the axial servocontroller was in load-control at zero load. In the case of cumulative fatigue tests with two load levels, after completing the required number of
284
MULTIAXIAL FATIGUE AND DEFORMATION TABLE 1--Axial fatigue data of Haynes 188 at 538~
Specimen Number
v (Hz)
Ae
Air (MPa)
trm (MPa)
Aeet
Aei,,
Ns (Cycles)
Crack Orientationa
HYII5 HYII6 HYII78 HYII77 HYII10 HYII11 HYII14 HYII15 HYII13
0.1 0.1 0.1 0.1 0.5 0.5 0.5 0.5 0.5
0.0202 0.0202 0.0143 0.0102 0.0071 0.0071 0.0067 0.0067 0.0060
1 274 1 266 1 137 1 011 937 980 955 962 1 056
- 14 -12 -12 -11 -16 -20 - 18 - 19 -22
0.0067 0.0067 0.0060 0.0053 0.0049 0.0052 0.0050 0.0051 0.0056
0.0135 0.0135 0.0083 0.0049 0.0022 0.0019 0.0017 0.0016 0.0004
787 864 1 639 4 755 24 105 24 030 35 889 42 937 252 3510
90 ~ 90 ~ 90 ~ 90 ~ 90 ~ 90 ~ 80 ~ 80 ~ ...
a Measured with specimen's axis as the reference (0% o Runout (no cracks were observed).
cycles at the first load level, test control software decreased the strain amplitude to zero by reducing the strain increments linearly over 10 cycles. This procedure w a s necessary to return the material to an approximately zero stress and zero strain state in a carefully controlled manner. Software was also used to acquire axial and torsional load, strain, and stroke data at logarithmic intervals in cycles and to s h u t d o w n each test in a controlled manner. For axial and torsional fatigue tests, failure w a s defined as a 10% load drop f r o m a previously recorded cycle. If a specimen did not fail after 250 000 cycles, then that test was declared a runout.
Results Baseline Axial and Torsional Fatigue Tests Fully reversed, strain-controlled, axial and torsional fatigue tests were conducted at 538~ to establish baseline fatigue data for the subsequent cumulative fatigue tests. Axial and torsional fatigue data obtained f r o m near half-life cycles are listed in Tables 1 and 2, respectively. For both axial and torsional tests at higher strain ranges a frequency o f 0.1 Hz was used, whereas at lower strain ranges
TABLE 2--Torsional fatigue data of Haynes 188 at 538~C. Specimen Number
v (Hz)
AT
Ar (MPa)
r,, (MPa)
A~lel
A~/in
Ns (Cycles)
Crack Orientationa
HYII1 HYII2 HYII79 HYII17 HYII21 HYII20 HYII23 HYII7 HYII16 HYII4 HYII16
0.1 0.1 0.1 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
0.0346 0.0345 0.0217 0.0139 0.0130 0.0130 0.0122 0.0121 0.0101 0.0095 0.0073
737 727 625 579 571 607 610 603 624 629 527
2 1 0 0 0 0 1 0 0 0 -2
0.0101 0.0100 0.0086 0.0079 0.0078 0.0083 0.0084 0.0083 0.0085 0.0086 0.0072
0.0245 0.0245 0.0131 0.0060 0.0052 0.0047 0.0038 0.0038 0.00t6 0.0009 0.0001
1 630 1 882 4 023 19 080 20 762 43 678 52 280 65 612 250 475 b 367 4470 550 183 b
0~ 0~ 0~ 0~ 0~ 0~ 0~ 0~ ... ... ...
Measured with specimen's axis as the reference (0~ b Runout (no cracks were observed).
a
KALLURI AND BONACUSE ON LOAD-TYPE SEQUENCING EFFECTS
285
a frequency of 0.5 Hz was used. Lowering of the frequency at higher strain ranges was necessary to achieve adequate control of the fatigue test in the presence of "serrated yielding" [30] exhibited by Haynes 188 at 538~ [31]. Slight compressive mean stresses were observed in all the axial fatigue tests, whereas no appreciable mean stresses were noticed in the torsional fatigue tests. In a majority of the torsional fatigue tests, axial strain ratcheting in the positive direction was observed. Such axial strain ratcheting in materials subjected to torsional loading was reported by other investigators [32-35] and was also observed in Haynes 188 at 760~ [36]. However, for the tests conducted in the present study, magnitudes of the mean axial ratcheting strains near half-lives were either less than or of the same order as the equivalent strains calculated from the imposed engineering shear strains. In a separate study [37] also conducted on Haynes 188, no significant influence of mean axial strain, either tensile or compressive, was observed on the axial fatigue life of the superalloy. In the baseline tests, orientation of the crack(s) that led to specimen failure was nearly perpendicular to the maximum normal stress in the axial figure tests, whereas in the torsional fatigue tests the crack orientation was always parallel to one of the two maximum shear directions (Tables 1 and 2). Axial and torsional fatigue life relations (Eqs 1 and 2) were computed by separating the total strain range for each test into elastic and inelastic components (Tables 1 and 2), and subsequently performing a regression between logarithms of the strain range components and the fatigue lives. Fatigue data from the mnout tests were omitted while computing the life relationships. Constants for the axial and torsional life relationships are shown in Table 3. A~ = B(Ns) ~ + C ( N f ) ~
(t)
a 7 = BT(N:) b~ + C r ( N : ) c~
(2)
Axial and torsional fatigue data and the corresponding life relationships are plotted in Figs. 1 and 2, respectively. Note that for Haynes 188 at 538~ the slopes of the elastic and inelastic life relations for the axial and torsional loading conditions are very similar. Axial and torsional fatigue data are compared by using von Mises equivalent strain range (Aeeq = Ay/N/3) in Fig. 3. Most of the torsional fatigue data fall near the axial fatigue curve. The torsional fatigue life relationship was estimated from the axial fatigue life relationship by the modified multiaxiality factor (MMF) approach (Eqs 3 and 4).
(3) where
TABLE 3--Constants for axial and torsional fatigue life relationships. Constants
Axial Life Relation
Constants
Torsional Life Relation
b c B C
-0.08 -0.544 0.0113 0.501
bTCT Br Cr
-0.082 --0.534 0.0187 1.24
286
MULTIAXlAL FATIGUE AND DEFORMATION 10-1
:
,
i
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I
.
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10-4
I -Total . . . . . Elastic ---
10-5
,
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10 2
i
i
10 3
i iiiill
i
,
i iiiill
10 4
i i iiii1
10 s
t0 s
Cyclic Life, Nf
FIG. l--Axial fatigue life relationships for Haynes 188 at 538~ M F = TF; TF >-- 1 (4)
TF=
X/(o-i - o'11)2 + (o'ii - o'1.) 2 + (o'm - o'i)2
v~ This approach was previously used to estimate torsional fatigue behavior from axial fatigue life relationships o f Haynes 188 at 316 and 760~ [24,25]. For torsion (o"i = -O'ni and oil = 0; TF = 0; and ,10-1
<~
C
9~ u)
10-2
...........
10"3
.C
10-4
-..... ---
Total Elastic Inelastic
10.5 , , ,.,...I . , ,,,,,,I , , .,,,,,I , , ,,,,, 10 2 10 s 10 4 lO s 10 e Cyclic Life, Nf
FIG. 2--Torsional fatigue life relationships for Haynes 188 at 538~
KALLURI AND BONACUSE ON LOAD-TYPE SEQUENCING EFFECTS 10-1
'
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,
,
,i,,,,
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287
,i,,,i
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~
10 .2
.
I
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9
Axial Fatigue Data
I
9
Torsional Fatigue Data
I
- MMF Approach, MF = 0.5 Axial Fatigue Life Relation, MF = 1.0 0 >
104 102
=
,
,,,,=,1
,
,
,,,,l,I
,
103
,
,,,,,,I
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104
,
,,,,
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Cyclic Life, Nf FIG. 3--Estimation of torsional fatigue lives with modified multiaxiality factor approach.
MF = 0.5), the estimated torsional fatigue life relation from Eqs 3 and 4 forms an upper bound to the experimentally observed torsional fatigue data at 538~ (Fig. 3). Four nominal strain ranges, two each for axial (Ae 1 = 0.02 and Ae2 = 0.0067) and torsional (A3'1 = 0.035 and AT2 = 0.012) loading conditions, were selected for the subsequent cumulative fatigue tests. Duplicate tests were conducted in the baseline test program to evaluate repeatability of the cyclic deformation behavior and to provide a more accurate estimate of the fatigue life for each test condition. The evolution of cyclic axial and shear stresses are plotted in Fig. 4 for the baseline axial and torsional tests. In each of these tests, Haynes 188 exhibited cyclic hardening for a majority of the life with a slight softening towards the end of the test. No significant differences were observed be-
1400 D. b <1 o~
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400 r
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200/01- . . . . .... .
100
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,
,,,,.,I
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10=
,
.,llp,I
.
103
0.02 (HYII5) 0.02 (HYII6) 0.067 (HYII14) 0.067 (HYII15) .
,,11..I
,
104
.
,,,,
105
Number of Cycles, N (a) FIG. 4--Cyclic stress evolution in baseline fatigue tests (a) axial tests, (b) torsional tests.
288
MULTIAXlAL FATIGUE AND DEFORMATION
800
'
'
J~="l
'
~ ~'''"1
~
~ ~lJ,,
I
,
,
L~Hr
I
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,
,,,H~
600
soo~
n~ 400,, ~ ' - ~ " U)
300 t JZ
200 J/ 100 r / 100 O "
--0- ~--'V~ '
,
,,,,,,I
,
101
,
,,*,,,I
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102
,
t~, = 0.035 (HYII1) ~,'/= 0.035 (HYll2) Ay= 0.012 (HYII7) Ay = 0.012 (HYII23)
,i,,,,I
,
.......
103
I
104
,
,
,,,,
105
Number of Cycles, N (b)
FIG. 4---(Continued)
tween the cyclic hardening behaviors of the duplicate tests. For a given cyclic loading condition, scatter in fatigue life typically exhibits a log-normal distribution. Therefore, geometric mean lives (arithmetic means of the logarithms of fatigue lives) of the duplicated axial and torsional baseline fatigue tests were used to design the cumulative fatigue test matrices. The calculated geometric mean lives for the four axial and torsional fatigue loading conditions were as follows: N1A = 825 cycles, N2A = 39 255 cycles, Nix = 1 751 cycles, and N2x = 58 568 cycles.
Cumulative Fatigue Tests Four types of two load-level, axial and torsional cumulative fatigue tests were conducted to quantify the load order as well as load-type sequencing effects. The cumulative fatigue tests involving load-order effects (high/low) without load-type sequencing effects were as follows: (1) axial/axial, and (2) torsional/torsional. Cumulative fatigue tests involving both load-order (high/low) and loadtype (axial and torsional) sequencing effects were as follows: (1) axial/torsional, and (2) torsional/axial. For each type of two load-level test, four different life fractions (nl/N1 = 0.1, 0.2, 0.4, and 0.6) at the first load level were investigated. All the cumulative fatigue tests were started at the first load level and after completing the required number of cycles the remainder of the test was carried out to failure at the second load level. Axial and torsional interaction fatigue data obtained from the near middle cycle for each load segment are listed in Table 4. In all cumulative fatigue tests, after the completion of the first load segment any existing mean strains (for example, axial ratcheting strain due to cyclic torsional loading) were re-zeroed before starting the second load segment. In the axial/axial and torsional/axial cumulative fatigue tests orientation of the crack(s) leading to failure of the specimens were nearly perpendicular to the maximum normal stress direction induced by the axial loads (Table 4). In the case of torsional/torsional fatigue tests cracks were oriented along both of the maximum shear stress planes. Axial/torsional cumulative fatigue testing resulted in crack orientations both along planes of maximum shear stress and perpendicular to the maximum normal stress direction. The cyclic hardening behavior exhibited by Haynes 188 during the different types of axial and torsional cumulative fatigue tests is shown in Figs. 5a to d. The corresponding baseline cyclic hardening behaviors are also included for comparison. Note that the cyclic hardening behavior from the sec-
Atrl (MPa)
0.0205 1 138 0.0202 1 230 0.0201 1 276 0.0200 1 256 . . . . . . . . . . . . . . . . . . . . . _0.0202 1"154 0.0205 1 255 0.0202 1 256 0.0203 1 252 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
As1
. . . . .
. . . . .
. . . . . .
-12 - 14 - 12 - 13
-10 - 12 - 11 -13
trml (MPa)
0.03"49 0.0349 0.0346 0.0345 0.0345
0.03"47 0.0352 0.0349 0.0345 ... ... ...
... ... ...
AT1
731 751 749 760
739 738 753 . . . . . . . . . . . . . . . . . .
i -1
-1
0
1
1 1 3
'i
~'ml (MPa)
. . . . . . . . . . . . . . . . . .
A'/'I (MPa)
M e a s u r e d w i t h s p e c i m e n ' s axis as the r e f e r e n c e ( 0 ~ b Initial orientation; c r a c k e v e n t u a l l y c h a n g e d to 90 ~ orientation.
HYII75 HYII64 HYII65 HYII70 HYII76 HYII66 HYII67 HYII71 HYII73 HYII8 HYII61 HYII68 HYII74 HYII62 HYII63 HYII72 HYII69
Specimen Number
First L o a d L e v e l ; v = 0.1 H z
350 700 1 051 1 151
175
350 700 1 051 83 165 330 495
175
83 165 330 495
nl (Cycles)
0.0062 0.0064 0.0064 0.0063 0.0062
0.0065 0.0065 0.0065 0.0064
Ae2 015 035 055 054
1"i52 1 162 1 179 1 143 1 162
1 1 1 1
Ao"2 (MPa)
-4 6
0 2 5
-5 -6 -7 -5 ... ... ... ... ... ... ...
O'm2 (MPa)
AT2 (MPa)
0.0i'22 0.0122 0.0122 0.0122 0.0121 0.0122 0.0122 0.0122 . . . . . . . . . . . . . . . . . . . .
646 650 652 646 652 679 689 683 . . . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Ay 2
S e c o n d L o a d L e v e l ; v -*- 0.5 H z
T A B L E 4--Axial and torsional interaction fatigue data of Haynes 188 at 538~
"'6 0 0 -1 1 2 2 3
Tin2 (MPa)
14 17 9 11 28 18 8 4 2
17 10 8 4 59 29 4
923 540 306 857 143 547 156 951 732 033 919 700 980 380 799 642 824
n2 (Cycles)
90 ~ 75 ~ 75 ~ 60 ~ 0~ 0~ 85 ~ 90 ~ 75 ~ 0~ 75 ~ 90 ~ 75 ~ 80 ~ 90 ~ 90 ~ 80 ~
Crack Orientation a
290
MULTIAXIAL FATIGUE AND DEFORMATION
1400 I1.
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Baseline, AE = 0.02 ] Baseline, At = 0.0067 Cumulative, na= 83 Cumulative, nl= 165 Cumulative, nl= 330 Cumulative, nl= 405
101
10=
103
104
10s
Number of Cycles, N (a) 800,
........
,
........
,
........
,
- , o o ~ l ,oot
,
......
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/1
---
-
500 i~
........
~ ~
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~" u)
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Baseline, AT = 0.035 - - - Baseline,/',? = 0.012 --B-- Cumulative, nl= 175 Cumulative, nl= 350 Cumulative, nl= 700 Cumulative, nl= 1 051 i
10o
i ill,HI
,
101
, ,IL,,,I
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102
= ,l,,,,l
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, ,,,,,,[
103
,
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,,,=
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Number of Cycles, N (b)
FIG. 5--Cyclic stress evolution at the second load level in cumulative fatigue tests, (a) axial~axial tests, (b) torsional/torsional tests, (c) axial~torsional tests, (d) torsional~axial tests.
KALLURI AND BONACUSE ON LOAD-TYPE SEQUENCING EFFECTS
800 ~'
,,, I
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100 0 100
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400 x
<
--I-
200 0 100
Cumulative,nl= 700 Cumulative, nl= 1 051 Cumulative,n~= 1 151
rl I
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Number of Cycles, N
FIG.
5---(Continued)
I
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'IIII
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105
291
292
MULTIAXIALFATIGUEANDDEFORMATION 1.0
'
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r
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tto
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m tr
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x ~ 5 NI~A=825 5
0.4
\
0.2
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0.4
0.6
0.8
1.0
Applied Life Fraction, nllN 1 (a)
1.2
o~ 1,0 :~ .2
L
L\
'
I
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LDR
--DCA
0.8
~
~
'
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N1T
---" 1 7 5 " 1
o.e
.J
:_E
E
oc
0.4 0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Applied Life Fraction, nl/N 1
(b) FIG. 6---Fatigue life estimation of two load-level cumulative fatigue tests, (a) axial~axial tests, (b) torsionalltorsional tests, (c) axial~torsional tests, (d) torsionallaxial tests. ond load level alone is included in each of these plots. In all the cumulative fatigue tests, Haynes 188 cyclically softened during the latter segments of the two load-level tests. This was to be expected because the material was cyclically hardened during the first load segments, which were applied at much higher strain ranges compared to the lower strain ranges in the second load segments. In axial/axial and torsional/axial cumulative fatigue tests, hardening of the material during the first load segments resulted in the second segment axial strain range levels to be slightly lower than the intended nominal value of Ae2 = 0.0067 (Table 4).
KALLURI AND BONACUSE ON LOAD-TYPE SEQUENCING EFFECTS
293
Applied and remaining life fractions in the two load-level axial and torsional cumulative tests are shown in Figs. 6a to d. In all the figures, corresponding predictions from the LDR (Eq 5) and the nonlinear DCA (Eq 6) are also included.
1.0
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~
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,
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,
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Applied Life Fraction, nl/N 1
(e) 1.0
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I
'
I
'
r
z O4 r
V
0.8
_
;c:
.
0 .m 0
0.6
U.
\
\ \
.--I
_
Haynes188 [
I
.,.--,7,, \N~=43855
0.4
._= ._= m
E
n,,
0.2
0.0 0.0
0.2
0.4
0.6
0.8
Applied Life Fraction, nllN 1
(d)
FIG. 6---(Continued)
1.0
294
MULTIAXIAL FATIGUE AND DEFORMATION
n2 = 1 -
(6)
Applied and summations of the life fractions observed in the two load-level axial and torsional cumulative fatigue tests and the corresponding predictions from the LDR and the nonlinear DCA are listed in Table 5 and are also shown in Figs. 7a to d. The baseline fatigue lives used for the LDR and the DCA computations are indicated in the Figs. 6 and 7. In most instances, the baseline fatigue lives (NIA, N1T, N2A, and NaT) were obtained directly from the geometric mean fatigue lives as mentioned earlier (Figs. 6a to c and Figs. 7a to c). In the case of torsional/axial cumulative tests, however, N2A value corresponding to the lower than nominal axial strain range was calculated from Eq 1 (Figs. 6d and 7d). For axial/axial cumulative fatigue tests, which primarily involve load-order (high/low) interaction effect within axial loading, the curve from the DCA predicted the trend in the data very well, whereas the predictions by the LDR were unconservative (Fig. 7a). In the case of torsional/torsional cumulative fatigue tests, which involves load-order interaction effect within torsional loading, predictions by the LDR were unconservative in three out of four instances (Fig. 7b) and the DCA was able to only approximate the general trend of the data. In fact, for two data points the predictions by DCA were conservative and for the remaining data points they were unconservative. The cumulative fatigue behavior under torsional loading will be discussed further in the following section. In the axial/torsional and torsional/axial cumulative fatigue tests, which consider both the high/low load-ordering and loadtype sequencing effects, the DCA was able to closely predict the trends of the data and the LDR was unconservative in both instances (Figs. 7c and d). Experimentally observed remaining fatigue lives at the second load level and corresponding predictions by the LDR and the DCA are compared in Figs. 8a and b, respectively, for the cumulative axial and torsional fatigue tests. For a majority of the tests, predictions by the LDR were unconservative by more than a factor of two. Predicted remaining fatigue lives by the DCA were generally within a factor of two of the observed fatigue lives with a few exceptions. Discussion The baseline torsional fatigue data, when compared on the basis of yon Mises equivalent strain range, fall slightly above the axial fatigue data (Fig. 3). This observation indicates that for Haynes 188 at 538~ torsion is a slightly more benign type of loading compared to axial loading. The predictions by the MMF approach, as indicated earlier, form a close upper bound to the torsional fatigue data. In the absence of any torsional fatigue data, the MMF approach can be used for Haynes 188 to estimate the torsional fatigue lives from the axial fatigue life relationship at the appropriate temperature. Axial/axial and torsional/torsional cumulative fatigue tests were performed mainly to investigate the high/low load ordering effects without load-type sequencing. The axial/axial cumulative fatigue behavior of Haynes 188 was predicted accurately by the DCA (Fig. 7a and Table 5), whereas the torsional/torsional behavior did not closely confirm to the predictions by either the LDR or the DCA (Fig. 7b and Table 5). The orientation of crack(s) did not vary significantly in the axial/axial tests (Table 4). In the torsional/torsional tests, the orientation of the crack(s) leading to the failure of the specimens shifted from one maximum shear plane (0 ~ or parallel to the specimen's axis) to another (90 ~ or perpendicular to the specimen's axis) as the life fraction of the first load segment was increased from 0.1 to 0.6 (Table 4). It is also interesting to note that the predictions by the DCA are conservative for nl/N1 = 0.1 and 0.2 and unconservative for nffN1 = 0.4 and 0.6. Cracking pattern appears to influence the cumulative fatigue behavior of Haynes 188 under torsion. Wood and Reimann [4] also observed some unusual cumulative fatigue behavior while testing brass and copper under torsion. In their study, the unusual behavior was attributed to a change in the damage mechanisms of the
HYII75 HYII64 HYII65 HYII70 HYII76 HYII66 HYII67 HYII71 HYII73 HYII8 HYII61 HYII68 HYII74 HYII62 HYII63 HYII72 HYII69
Specimen Number
Axial/Axial Axial/Axial Axial/Axial Axial/Axial Torsional/Torsional Torsional/Torsional Torsional/Torsional Torsional/Torsional Axial/Torsional Axial/Torsional Axial/Torsional Axial/Torsional Torsional/Axial Torsional/Axial Torsional/Axial Torsional/Axial Torsional/Axial
Test Type 0.101 0.200 0.400 0.600 0.100 0.200 0.400 0.600 0.101 0.200 0.400 0.600 0.100 0.200 0.400 0.600 0.657
Applied Life Fraction, (nllN1) 0.899 0.800 0.600 0.400 0.900 0.800 0.600 0.400 0.899 0.800 0.600 0.400 0.900 0.800 0.600 0.400 0.343
LDR 0.387 0.291 0.178 0.103 0.432 0.327 0.202 0.118 0.341 0.254 0.153 0.089 0.470 0.358 0.223 0.131 0.109
DCA 0.457 0.269 0.212 0.124 1.010 0.504 0.071 0.016 0.252 0.291 0.169 0.200 0.661 0.419 0.201 0.106 0.064
Experiment
Remaining Life Fraction, (n2/Nz)
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
LDR 0.488 0.491 0.578 0.703 0.532 0.527 0.602 0.718 0.442 0.454 0.553 0.689 0.570 0.558 0.623 0.731 0.766
DCA
0.558 0.469 0.612 0.724 1.110 0.704 0.471 0.616 0.353 0.491 0.569 0.800 0.761 0.619 0.601 0.706 0.721
Experiment
Sum of Life Fractions, (nllN1) + (n2/N2)
TABLE 5--Comparison of life fractions for axial and torsional cumulative fatigue tests.
o-!
r~ s
Co
m o
m '-m "1"I
z if)
_(2
m 0 c m z
"U m
,u
0
0 z > o c G0 m 0 z
> z o
i
C 33
I-" r"
296
MULTIAXlAL FATIGUE AND DEFORMATION
2.0
1.8
Z t+
N2A = 39 255
----
1.4 6 eo
Haynes 188 LDR DCA
N1A = 825
1.6
1.2 1.0
o LI.
0.8
--I
0.6
E or)
/
/
/
/
Y
0.4 0.2 I
0,0 0.0
i
I
0.2
i
I
0.4
f
I
0.6
1.0
0.8
A p p l i e d Life Fraction, n l / N 1 (a) 2.0 z re~ -
1.8 []
NiT = 1 751
1.6
+
N2T -- 5 8 5 6 8
----
1.4
E 6 0
1.2
Haynes 188 LDR DCA
[]
1.0
0
.m I.L -.I
E O~
0.8 0.6
[] ---- --~ - ~ I " []
~
/
/
/
/
/
/
/
[]
0.4 0.2 0.0 0.0
,
I 0.2
,
I 0.4
,
I
0.6
,
I
0.8
i
1.0
Applied Life Fraction, nl/N 1 (b)
FIG. 7--Comparison of the summation of life fractions in two load-level cumulative fatigue tests, (a) axial~axial tests, (b) torsional~torsional tests, (c) axial~torsional tests, (d) torsional~axial tests.
KALLURI AND BONACUSE ON LOAD-TYPE SEQUENCING EFFECTS
2.0 z C 4-
i
i
t
[
1.8 NIA = 825 N2T= 58 568
1.6
.A
Haynes 188 LDR DCA
1.4
E 6 O
1.2 1.0 /
O
A
U.
0.8
..J
0,6 \
E o0
0.4
~11
/
~i
J
/
/
/
A
0.2 I
0.0 0.0
,
I
0.2
,
I
0.4
,
I
0.6
0.8
1.0
Applied Life Fraction, nl/N 1 (c) 2.0 Z 4-
1.8 1.6 1.4
E 6 O
V
N1T = 1 751 Nz~ = 43 855
Haynes 188 I LDR DCA
----
1.2 1.0
/
O
U. ..J
E O0
i
0.8 ~
0.8 0.4
0.2
V
l i
I\-___v~I ~ l j ~ v
I
0.0
,
0.0
I
0.2
,
I
,
0.4
I
0.6
,
I
0.8
Applied Life Fraction, n l l N 1 (d) FIG.
7---(Continued)
1.0
297
298
MULTIAXIAL FATIGUE AND DEFORMATION
10 5
0
v ~/~//"
104
..J
.2 m
/
(J
o ._= r.
/'
10 3
, / // //
E G) n,
/
] r-I Torsional/Torsional
/ ~ .x,.,~o~,on.,
..... ,
10 2 10 2
,, !, L,,
10 3
10 4
10 5
Remaining Cyclic Life, (n2)OBSERVED (a) 10 5
,
,
,
, , i , I
,
i
,
/
,,,
/
-
r
A
~.
10 4
,.J
~ac,orso,~
U
on Life
/ /
//~,"
0
=~ ._= c E n~
10 3
/'//
0
/~ / ~ , / i 10 2 10 2
,
~ ,,,,,,
i
t0 3
,
Axial/Axial Torsional/Torsional Axial/Torsional Torsional/Axial
,,,,,,I
,
,
,,,,,,I
10 4
10 5
Remaining Cyclic Life, (n2)OBSERVED (b) FIG. 8--Comparison of remaining cyclic lives in the axial and torsional cumulative fatigue tests, (a) linear damage rule, (b) damage curve approach.
KALLURI AND BONACUSEON LOAD-TYPE SEQUENCING EFFECTS
299
materials [4]. Additional torsional/torsional cumulative fatigue tests on Haynes 188 at closer nl[N 1 intervals are required to characterize this behavior in a systematic manner. Comprehensive microstructural examination of the material subjected to interrupted tests could determine whether different damage mechanisms are involved. Axial/torsional and torsional/axial tests were conducted to understand the influences of both high/low load ordering and load-type sequencing on the cumulative fatigue behavior of Haynes 188. The predictions by the DCA for both types of cumulative fatigue tests closely followed the experimental results even though there were some variations in the failure crack orientations (Table 4). Typically, under equivalent loading conditions, load-type sequencing effect results in a total life fraction of greater than unity for either the axial/torsional or the torsional/axial cumulative fatigue tests [16,17,19-21]. In the present study total life fractions from both the axial/torsional and the torsional/axial cumulative fatigue tests were less than unity (Figs. 7c and d and Table 5). This result clearly indicated that for Haynes 188 at 538~ and the test conditions investigated in this study, the high/low load ordering effect was much stronger than the load-type sequencing effect. Hua and Socie [15] and Hua and Femando [22] also observed similar dominance of load-ordering effects on the cumulative fatigue lives of other materials. However, load-type sequencing can significantly influence the cumulative fatigue behavior of a material when load-ordering effect is either eliminated by careful design of the cumulative fatigue experiments (equivalent strain range equal cyclic lives, or equivalence in damage) or minimized in comparison to the load-type sequencing effect
[14,16,17,19-21].
Summary Cumulative fatigue behavior of a wrought cobalt-base superalloy, Haynes 188, was investigated under axial and torsional loading conditions at 538~ Four different types of two load-level (singlestep), high/low load-ordering, cumulative fatigue tests were performed with (axial/torsional and torsional/axial) and without load-type sequencing (axial/axial and torsional/tosional). The cyclic lives in the cumulative fatigue tests were estimated with the LDR and the DCA. Important issues identified from this study are summarized as follows: (1) In baseline fatigue tests, orientation of dominant crack(s) was nearly perpendicular to the maximum normal stress direction under axial loading, whereas under torsional loading the orientation was always parallel to the maximum shear stress planes. (2) For the axial/axial and torsional/torsional cumulative fatigue tests, which involve only high/low ordering effects and no load-type sequencing effects, the summations of life fractions were less than unity in all except one torsional/torsional test. This confirmed the presence of a high/low order effect in Haynes 188 superalloy. Even in the case of axial/torsional and torsional/axial cumulative fatigue tests, which potentially contain both the high/low ordering and load-type sequencing effects, the summations of life fractions were less than unity in all the tests. This essentially indicated that for the test conditions evaluated in this study high/low load ordering effect was more predominant than the load-type sequencing effects. (3) In the cumulative fatigue tests, predicted summations of the life fractions by the LDR were unconservative for all except one torsional/torsional test, whereas those predicted by the DCA closely matched the experimental data for axial/axial, axial/torsional, and torsional/axial tests. In the case of torsional/torsionaltests the DCA was only able to predict the general trend in the data. (4) Remaining cyclic life predictions by the LDR were unconservative by more than a factor of two for a majority of the cumulative axial and torsional fatigue tests, and corresponding predictions by the DCA were generally within a factor of two of the experimental data with few exceptions.
300
MULTIAXlALFATIGUE AND DEFORMATION
Acknowledgment Valuable technical discussions with Dr. Gary R. Halford (NASA Glenn Research Center) and the diligent efforts of Mr. Christopher S. Burke (Dynacs Engineering Company, Inc.) in the High Temperature Fatigue and Structures Laboratory are gratefully acknowledged.
References [1] Palmgren, A., "Die Lebensdaner von Kugellagern," Zeitschrifi des Vereinesdeutscher Ingenierure, Vol. 68, No. 14, April 1924, (The Service Life of Ball Bearings, NASA Technical Translation of German Text, NASA TT 1-13460, 1971), pp. 339-341. [2] Langer, B. F., "Fatigue Failure from Stress Cycles of Varying Amplitude," Journal of Applied Mechanics, Vol. 4, No. 3, September 1937, (Transactions of the American Society of Mechanical Engineers, Vol. 59, 1937), pp. A160-A162. [3] Miner, M. A., "Cumulative Damage in Fatigue," Journal of Applied Mechanics, Vol. 12, No. 3, September 1945, (Transactions of the American Society of Mechanical Engineers, Vol. 67, 1945), pp. A159-A164. [4] Wood, W. A. and Reimann, W. H., "Observations on Fatigue Damage Produced by Combinations of Am. plitudes in Copper and Brass," Journal of the Institute of Metals, Vol. 94, 1966, pp. 66-70. [5] Manson, S. S., Nachtigall, A. J., Ensign, C. R., and Freche, J. C., "Further Investigation of a Relation for Cumulative Fatigue Damage in Bending," Journal of Engineering for Industry, 1965, pp. 25-35. [6] Manson, S. S., Freche, J. C., and Ensign, C. R. "Application of a Double Linear Damage Rule to Cumulative Fatigue," Fatigue Crack Propagation, ASTM STP 415, American Society for Testing and Materials, 1967, pp. 384-412. [7] Manson, S. S. and Halford, G. R., "Practical Implementation of the Double Linear Damage Rule and Damage Curve Approach for Treating Cumulative Fatigue Damage," International Journal of Fracture, Vol. 17, No. 2, 198t, pp. 169-192. [8] Bui-Quoc, T., "Cumulative Damage with Interaction Effect Due to Fatigue Under Torsion Loading," Experimental Mechanics, 1982, pp. 180-187. [9] Miller, K. J. and Ibrahim, M. F. E., "Damage Accumulation During Initiation and Short Crack Growth Regimes," Fatigue of Engineering Materials and Structures, Vol. 4, No. 3, 1981, pp. 263-277. [10] Manson, S. S. and Halford, G. R., "Re-examination of Cumulative Fatigue Damage Analysis--An Engineering Perspective," Engineering Fracture Mechanics, Vol. 25, Nos. 5/6, 1986, pp. 539-571. [1l] Golos, K. and Ellyin, F., "Generalization of Cumulative Damage Criterion to Multilevel Cyclic Loading," Theoretical and Applied Fracture Mechanics, Vol. 7, 1987, pp. 169-176. [12] Chaboche, J. L. and Lesne, P. M., "A Non-Linear Continuous Fatigue Damage Model," Fatigue and Fracture of Engineering Materials and Structures, Vol. 11, No. 1, 1988, pp. 1-17. [13] McGaw, M. A., Kalluri, S., Moore, D., and Heine, J., "The Cumulative Fatigue Damage Behavior of MarM 247 in Air and High Pressure Hydrogen," Advances in Fatigue Lifetime Predictive Techniques: Second Volume, ASTMSTP 1211, M. R. Mitchell and R. W. Landgraf, Eds., American Society for Testing and Materials, 1993, pp. 117-131. [14] Miller, K. J. and Brown, M. W., "Multiaxial Fatigue: A Brief Review," Fracture 84, Proceedings of the 6th International Conference on Fracture, 1984, New Delhi, India, Pergamon Press, pp. 31-56. [15] Hua, C. T. and Socie, D. F., "Fatigue Damage in 1045 Steel Under Variable Amplitude Biaxial Loading," Fatigue and Fracture of Engineering Materials and Structures, Vol. 8, No. 2, 1985, pp. 101-114. [16] Miller, K. J., "Metal Fatigue--Past, Current, and Future," Proceedings of the Institution of Mechanical Engineers, Vol. 205, 1991, pp. 1-14. [17] Robillard, M. and Cailletand, G. " 'Directionally Defined Damage' in Multiaxial Low-Cycle Fatigue: Experimental Evidence and Tentative Modelling," Fatigue Under Biaxial and Multiaxial Loading, ESIS10, K. Kussmaul, D. McDiarmid, and D. Socie, Eds. 1991, Mechanical Engineering Publications, London, pp. 103-130. [18] Harada, S. and Endo, T. "On the Validity of Miner's Rule under Sequential Loading of Rotating Bending and Cyclic Torsion," Fatigue Under Biaxial and Multiaxial Loading, ESISI0, K. Kussmaul, D. McDiarmid, and D. Socie, Eds., 1991, Mechanical Engineering Publications, London, pp. 161-178. [19] Weiss, J. and Pineau, A., "Continuous and Sequential Multiaxial Low-Cycle Fatigue Damage in 316 Stainless Steel," Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and R. Ellis, Eds., American Society for Testing and Materials, 1993, pp. 183-203. [20] Weiss, J. and Pineau, A., "Fatigue and Creep-Fatigue Damage of Austenitic Stainless Steels under Multiaxial Loading," Metallurgical Transactions A, Vol. 24A, 1993, pp. 2247-2261. [2l] Lin, H., Nayeb-Hashemi, H., and Berg, C. A., "Cumulative Damage Behavior of Anisotropic A1-6061-T6 as a Function of Axial-Torsional Loading Mode Sequence," Journal of Engineering Materials and Technology, Vol. 116, 1994, pp. 27-34.
KALLURI AND BONACUSEON LOAD-TYPESEQUENCING EFFECTS
301
[22] Hua, G. and Fernando, U. S., "Effect of Non-Proportional Overloading on Fatigue Life," Fatigue and Fracture of Engineering Materials and Structures, Vol. 19, No. 10, 1996, pp. 1197-1206. [23] Kalluri, S. and Bonacuse, P. J., "In-Phase and Out-of-Phase Axial-Torsional Fatigue Behavior of Haynes 188 Superalloy at 760~ '' Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and J. R. Ellis, Eds., American Society for Testing and Materials, 1993, pp. 133-150. [24] Kalluri, S. and Bonacuse, P. J., "Estimation of Fatigue Life under Axial-Torsional Loading," Material Durability/Life Prediction Modeling: Materials for the 21st Century., PVP-Vol. 290, S. Y. Zamrik and G. R. Halford, Eds., American Society of Mechanical Engineers, 1994, pp. 17-33. [25] Bonacuse, P. J. and Kalluri, S., "Elevated Temperature Axial and Torsional Fatigue Behavior of Haynes 188," Journal of Engineering Materials and Technology, Vol. 117, April 1995, pp. 191-199. [26] Bonacuse, P. J. and Kalluri, S., "Axial-Torsional, Thermomechanical Fatigue Behavior of Haynes 188 Superalloy," Thermal Mechanical Fatigue of Aircraft Engine Materials, AGARD-CP-569, Advisory Group for Aerospace Research & Development, Neuilly-sur-Seine, France, 1996, pp. 15-1-15-10. [27] Kalluri, S. and Bonacuse, P. J., "An Axial-Torsional, Thermomechanical Fatigue Testing Technique," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 184-207. [28] Bonacuse, P. J. and Kalluri, S., "Axial-Torsional Fatigue: A Study of Tubular Specimen Thickness Effects," Journal of Testing and Evaluation, JTEVA, Vol. 21, No. 3, 1993, pp. 160-167. [29] Ellis, J. R. and Bartolotta, P. A., "Adjustable Work Coil Fixture Facilitating the Use of Induction Heating in Mechanical Testing," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 43~52. [30] Bhanu Sankara Rao, K., Kalluri, S., Halford, G. R., McGaw, M. A., "Serrated Flow and Deformation Substructure at Room Temperature in Inconel 718 Superalloy during Strain Controlled Fatigue," Scripta Metallurgica et Materialia, Vol. 32, No. 4, 1995, pp. 493-498. [31] Bhanu Sankara Rao, K., Castelli, M. G., Allen, G. P., and Ellis, J. R., "A Critical Assessment of the Mechanistic Aspects in Haynes 188 during Low-Cycle Fatigue in the Range 25~ to 1000~ '' Metallurgical and Material Transactions A, Vol. 28A, 1997, pp. 347-361. [32] Poynting, J. H., "On Pressure Perpendicular to the Shear Planes in Finite Shears, and on the Lengthening of Loaded Wires when Twisted," Proceedings of the Royal Society, London, Series A, Vol. 82, 1909, pp. 546-559. [33] Poynting, J. H., "Changes in Dimensions of Steel Wire when Twisted and Pressure of Distortional Waves in Steel," Proceedings of the Royal Society, London, Series A, Vol. 86, 1912, pp. 543-561. [34] Swift, H. W., "Length Changes in Metals under Torsional Overstrain," Engineering, Vol. 163, 1947, pp. 253-257. [35] Wack, B., "The Torsion of Tube (or a Rod): General Cylindrical Kinematics and Some Axial Deformation and Ratchetting Measurements," Acta Mechanica, Vol. 80, 1989, pp. 39-59. [36] Bonacnse, P, J. and Kalluri, S., "Cyclic Axial-Torsional Deformation Behavior of a Cobalt-Base Superalloy," Cyclic Deformation, Fracture, and Nondestructive Evaluation of Advanced Materials: Second Volume, ASTM STP 1184, M. R. Mitchell and O. Buck, Eds., American Society for Testing and Materials, 1994, pp. 204-229. [37] Kalluri, S., McGaw, M. A., and Halford, G. R., "Fatigue Life Estimation under Cumulative Cyclic Loading Conditions," accepted for publication in Fatigue and Fracture Mechanics: 31st Volume, STP 1389, G. R. Halford and J. P. Gallagher, Eds., American Society for Testing and Materials, 2000.
Multiaxial Fatigue Life And Crack Growth Estimation
A. Varvani-Farahani 1 and T. H. Topper 2
A New Multiaxial Fatigue Life and Crack Growth Rate Model for Various In-Phase and Out-of-Phase Strain Paths REFERENCE: Varvani-Farahani, A. and Topper, T. H., "A New Multiaxial Fatigue Life and Crack Growth Rate Model for Various In-Phase and Out-of.Phase Strain Paths," Multiaxial Fatigue Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 305-322. ABSTRACT: A new multiaxial fatigue parameter for in-phase and out-of-phase straining is proposed. The parameter proposed is the sum of the normal energy range and the shear energy range calculated for the critical plane on which the stress and strain Mohr's circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies used in this parameter are divided by the tensile and shear fatigue properties, respectively. The proposed parameter, unlike many other parameters, does not use an empirical fitting factor. The proposed parameter successfully correlates multiaxial fatigue lives for: (a) various in-phase and out-of-phase multiaxial fatigue straining conditions, (b) tests in which a mean stress was applied normal to the maximum shear plane, and (c) out-of-phase tests in which there was additional hardening. An effective (closure free) intensity factor range, AKeff, was derived based on the proposed parameter. This effective intensity factor successfully correlated the closure-free crack growth rates for straining of various biaxial strain ratios.
KEYWORDS: multiaxial fatigue model, crack growth rate, in-phase and out-of-phase strain paths, shear and normal energies, critical plane, mean stress effect, strain hardening, effective (closure-free) fatigue data Nomenclature
Shear and axial strain ranges, respectively
(da/dN) a
A e o, Atri: AKb AK3 ~e~AVM~S, A VBM A ~I~m . yeS t.aLXeff , z.al~-eff
, ~aO, e f f
Maximum shear strain range and normal strain range acting on critical plane, respectively Closure-free crack growth rate due to both small cycles and overstrain cycles Crack depth Applied tensorial strain range, stress range, and shear stress range, respectively Strain and stress tensor ranges (where i a n d j = 1, 2, 3) Strain intensity factor range for opening mode and shear mode, respectively Effective strain intensity factor ranges based on the shear strain parameter, the Brown-Miller parameter, the Kandil-Miller-Brown parameter, and the Fatemi-Socie parameter, respectively
i Assistant professor, Department of Mechanical Engineering-Ryerson Polytechnic University, Toronto, Ontario, M5B 2K3, Canada. 2 Full professor, Civil Engineering Department, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada. 305
Copyright9
by ASTM lntcrnational
www.astm.org
306
MULTIAXIAL FATIGUE AND DEFORMATION
a e7 Ar
Actn
E,G 81, /32, 83 /3e, ~p
4, Ei, Fm A N v~, Vp, v4f
01, 02 O"1, 0-2, 0"3
0-~ 0-max 0-nrain
0-~,6
Effective intensity factor range based on the Varvani-Topper parameter proposed in this study Maximum shear stress range and normal stress range, respectively Elastic and shear moduli, respectively Principal strains (el > e2 > e3) Elastic strain and plastic strain, respectively Phase delay between strains on the axial and torsional axes Shape factor for a semielliptical opening mode crack, and for a semielliptical shear crack, respectively Biaxial strain ratio (A = e3/el) Equivalent number of small cycles to failure Elastic, plastic, and effective Poisson's ratios, respectively Angles during loading and unloading parts of a cycle, respectively, at which the Mohr's circles are the largest Principal stresses (0-1 > 0"2 > 0"3) Mean normal stress Maximum normal stress Minimum normal stress Axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively Shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively
Many engineering components that undergo fatigue loading experience multiaxial stresses in which two or three principal stresses fluctuate with time; i.e., the corresponding principal stresses are out-of-phase or the principal directions change during a cycle of loading. Extensive reviews of multiaxial fatigue life prediction methods are presented by Garud [1], Brown and Miller [2], and You and Lee [3]. Methods for predicting multiaxial fatigue life will be briefly reviewed. Equivalent Stress/Strain Approach The most commonly used method of correlating uniaxial and multiaxial fatigue life transforms cyclic multiaxial stresses into an equivalent uniaxial stress amplitude thought to produce the same fatigue life as the multiaxial stresses. The most popular methods for making the transformation are extensions of the von Mises yield criterion in which static values of principal stress are replaced by amplitudes and the yield strength is replaced by the uniaxial fatigue strength. Jordan [4] and Garud [5] showed that the von Mises criterion successfully correlates multiaxial life data only under proportional loading in the high-cycle fatigue regime. Energy Approach Fatigue is generally believed to involve cyclic plastic deformations which are dependent on the stress-strain path. Garud [6] applied this approach in conjunction with incremental plasticity theory to predict fatigue crack initiation life under complex nonproportional multiaxial loading conditions. Ellyin et al. [7,8] tried to correlate uniaxial and torsional data using the total strain energy density. They proposed that the durability of components should be characterized by the quantity of energy that a material could absorb. Critical Plane Approach Fatigue analysis using the concept of a critical plane of maximum shear strain is very effective because the critical plane concept is based on the fracture mode or the initiation mechanism of cracks.
VARVANI-FARAHANIAND TOPPER ON STRAIN PATHS
307
In the critical plane concept, after determining the maximum shear strain plane, many researchers define fatigue parameters as combinations of the maximum shear strain (or stress) and normal strain (or stress) on that plane to explain multiaxial fatigue behavior [5,9-11]. Strain terms are used in the region of low-cycle fatigue (LCF) and stress terms are used in the high-cycle fatigue (HCF) region in these critical plane approaches to multiaxial fatigue analysis. Brown and Miller [9] tried to analyze multiaxial fatigue in the low-cycle fatigue region by using the state of strain on the plane where the maximum shear strain occurred, while Findley [10] and Stulen and Commings [12] used stress terms in the high-cycle fatigue region. Combined Energy~Critical Plane Approach
Critical plane parameters have been criticized for lack of adherence to rigorous continuum mechanics fundamentals. To compensate for this lack, Liu [13], Chu et al. [14], and Glinka et al. [15] used the energy criterion in conjunction with the critical plane approach. Liu [13] calculated the virtual strain energy (VSE) in the critical plane by the use of crack initiation modes and stress-strain Mohr's circles. In the calculation of VSE, Liu included both elastic energy and plastic energy while the elastic energy was not considered in Garud's model [6]. Chu et al. [14] formulated normal and shear energy components based on the Smith-Watson-Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. Glinka et al. [15] proposed a multiaxial life parameter based on the summation of the products of normal and shear strains and stresses on the critical shear plane. In the present study, a multiaxial fatigue parameter for various in-phase and out-of-phase strain paths is proposed. The parameter is given by the sum of the normal energy range and the shear energy range calculated for the critical plane at which the stress and strain Mohr's circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies in this parameter have been weighted by the tensile and shear fatigue properties, respectively, and the parameter requires no empirical fitting factor. This parameter takes into account the effect of the mean stress applied normal to the maximum shear plane. The proposed parameter also increases when there is additional hardening caused by out-of-phase straining, while strain-based parameters fail to take into account this effect. The proposed parameter gives a good correlation of multiaxial fatigue lives and crack growth rates for various in-phase and out-of-phase straining conditions.
Materials and Multiaxial Fatigue Data Table 1 lists the references for in-phase and out-of-phase multiaxial fatigue data used in this study and tabulates the fatigue properties of the materials used. Fatigue coefficients tr} and e~ are the axial
TABLE 1--Fatigue properties of materials used in this study. Materials and Fatigue Data
E, GPa
~
o-j MPa
G, GPa
yj
~'~MPa
Ni-Cr-Mo-V steel* [16] 1 1045 steel [17-19] 1 Incone1718 [20]1 Haynes 188 [21]1 Waspaloy [22, 23] 2 Mild steel [24]2 Stainless steel [25]1
200 206 208.5 170.2 362 210 185
1.14 0.26 2.67 0.489 0.381 0.1516 0.171
680 948 1640 823 2610 1009 1000
77 79.2 80.2 65.5 139.2 80.8 71
1.69 0.413 3.62 1.78 0.516 0.322 0.413
444 505 1030 635 1640 431 709
* Ni-Cr-Mo-V steel is known as rotor steel. 1Fatigue properties are given by referenced papers. 2 Fatigue properties are calculated from uniaxial and torsional fatigue life-strain data.
308
MULTIAXIAL FATIGUE AND DEFORMATION
" [ ~axial
l
(a)
-- ~
fatigue curve
~,.
Log (fatigue life-cycles) 107
'I•\
(b)
i
'atigue
1 Log (fatigue life-cycles) 107
FIG. 1--Schematic presentation of fatigue life-strain curves for (a) uniaxial loading, and (b) torsional loading.
fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and ~-}and y~ are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively. These coefficients are illustrated in Fig. 1 for: (a) uniaxial, and (b) torsional fatigue loading life-strain curves. In-Phase and Out-of-Phase Strain Paths
In this study, for convenience of presentation, in-phase and out-of-phase strain paths have been categorized into three kinds: (a) in-phase strain paths (strain histories A1, A2, A3, A4, A5, and A6), (b) linear out-of-phase strain paths (strain histories B1, B2,), and finally (c) nonlinear out-of-phase strain paths (C1, C2, C3, C4, and C5). In-Phase Strain Paths In in-phase straining, both axial and shear strain cycles are alternating with no phase difference. Strain paths in in-phase straining are linear. For the in-phase straining data used in this study, the strain histories, strain paths, and strain and stress Mohr's circles are presented in Fig. 2a and Fig. 2b. The largest stress and strain Mohr's circles during the loading part (at 0a) and unloading part (at 02) of a cycle for which the maximum shear stress and strain and corresponding normal stress and normal strain values are calculated are illustrated in Fig. 2. In this figure, the light Mohr's circles are the largest during the loading part, and the dark Mohr's circles are the largest during the unloading part of a cycle. The strain histories A1, A2, and A3 correspond to uniaxial straining, torsional straining, and inphase combined axial and torsional straining, respectively. The linear in-phase strain paths shown in Fig. 2b have mean values. Strain history A4 is a combined axial and torsional strain path with an axial mean strain. Strain history A5 has a torsional mean strain, and finally, strain history A6 has both axial and torsional mean strains. Linear Out-of-Phase Strain Paths In out-of-phase alternating straining there is a phase difference between the axial and shear strain cycles. Strain history B 1 (Box) and strain history B2 (Two-Box) shown in Fig. 2c, are linear out-ofphase strain histories. Nonlinear Out-of-Phase Strain Paths In the nonlinear out-of-phase strain histories examined there is a phase delay between the axial strain and torsional strain. Strain paths are elliptical and as the phase difference increases the ellipti-
VARVANI-FARAHANIANDTOPPERONSTRAINPATHS Strain Path
~/~History A 1
02=270~
v
I
,I
4 90~
0o at 270 ~
q
,
-! 02=270~
'
~
~'i
'
F)o
7~.
f
,
at 270
I x ~,,rz/
k_LJi ' ' '"Lo
270o~
I 7/q3 ~
r o1~0o~Q/ .-
Ii
Stress Mohr's Circle ~
t90~
.._
270~
.~
/~
17 / "/'~
Strain Mohr's Circle ~ 7/21" , / 7 ~ / 2 i at90 ~
309
f at 25o ~
t ACt .
I
~'m:
Y)r r
Q'
A^o
p,
~Ae,,~ Ao n
FIG. 2a--Strain history, strain path, and Mohr's circle presentation for in-phase strain paths.
~istoryA4
Strain Path
Strain Mohr's Circle
270~ ~
Q ' Ae,
Q
!
Stress Molar's Circle
270~ t~al. ~Q'
l
/r,,r--X..b/.
. ~
~
,
~
I
-
-
,.~., 7/21 ~./90~
x! ,D/90o
/ 2 7 0 ~
90~
I
Ao- u
FIG. 2b----In-phase strain histories, paths, and Mohr's circle presentations for in-phase paths containing mean strain values.
310
MULTIAXIAL FATIGUE AND DEFORMATION
I 4Strai~a History B 1 02
Strain Path
Strain Mohr's Circle
Stress Mohr's Circle
t
Ae~
AO"n
AE n
AtT~
01
\ T
lil
I I
i/
FIG. 2c--Linear out-of-phase strain history, path, and strain-stress Mohr' s circles.
cal path becomes larger in its minor diameter, and finally, at a 90 ~ phase difference the strain path becomes circular. Strain histories C1, C2, C3, and C4 present out-of-phase axial and torsional straining with phase delays of 30, 45, 60, and 90 ~ respectively. Strain history C5 corresponds to a 90 ~ out-ofphase strain path containing an axial mean strain value. Figure 2d presents nonlinear out-of-phase strain paths, strain histories, and strain and stress Molar's circles. The maximum shear strains for in-phase and out-of-phase strain paths were numerically calculated at 10~ increments through a cycle and are presented in Figs. 3a and b, respectively. Proposed Parameter and Analysis
Figure 4 illustrates a thin-walled tubular specimen subjected to combined axial and torsional fatigue. The strain and stress tensors for a thin-walled tubular specimen subjected to axial and torsional fatigue are given by Eq I and Eq 2, respectively f
- - ~,effA~ap
Aeap
(1)
0
(L
(2) 0
where axial and shear strain ranges Aeap, A(Tap[2), respectively, are given by Eq 3 and Eq 4 as Aeap = Aea sinO
(3)
311
VARVANI-FARAHANI AND TOPPER ON STRAIN PATHS
Strain History C 1 I,/~ 02~300~
strain Path
Strain Mohr's Circle yI2[ y,.=/2 at 120~
e
Stress Mohr's Circle v~ ~ at 120~
AT
Am~
at 300~ al~-e. I
300" I/'~
310; 9
Y[ 1 2 0 o ~
\
~..~'vrg'
i
Va~3()0~ ] ~ a .
A~e.[
3~-A~Ja "
Y/2~ ~.-/13(P
"r I
3l O ~ [ - ~ e J
~
I
I
Ig ~11
1 II
~/130 ~
31( W ' 4 " ~
~.3a:r~
m
27 o 0 ~ , , ' ~
l / r'-,,
n
FIG. 2d--Strain histories, paths, and Mohr' s circle presentations for nonlinear out-of-phase strain paths without and with mean strain values.
9~ 1.2
--~
o~!
1.0 - m- HistoryAS]
i.,--,,,'~
(a)
B
'~
i
~0.8 E.. 8--g~--~...~. "~0.6 ~0.4 " i~9 . . . . . --~ , "~ 0.2 r"..~--" ! "'l ~o.o 20 40 60 80 100 120 140 160 0 (Degrees)
~'0.40
"go.35 0.30 0.25 .~ 0.20 ~0.15 ~0.10 "~ 0.05 0.00
.. ~ . i ~ - - ~ , ~ . o x "~,
z-v.. 9
---'l'--Hi~o~
C1
(~=30,
el~1201
-,~-aistory c2 (~-~51m=12o~I --A--History
C3
0~'60,
01ffi130~
--o--Historye4 (~--9o.ol--9o)| , i . , , 1 1 . ,
60
80
i,**
i,
,.i
ii
ii
i i I
I00 120 140 160 180 0 (Degrees) FIG. 3--Maximum shear strain through loading part of a cycle for various (a) in-phase loading, and (b) out-of-phase loading conditions.
312
MULTIAXIALFATIGUE AND DEFORMATION
I.
x
zr (a)
(b)
(c)
FIG. 4 ~ ( a ) Thin-walled tubular specimen subjected to combined axial and torsional fatigue, (b) 3-D presentation of strain state, and (c) stress state.
(4) where "sa and ya/2 are the applied axial and shear amplitude strains, respectively. The angle 0 is the angle during a cycle of straining at which the Mohr's circle is the largest and has the maximum value of shear strain. Angle th corresponds to the phase delay between strains on the axial and torsional axes. In Eq 2 AO'aand hra are the ranges of axial and shear stresses, respectively. In Eq 1 veff is the effective Poisson's ratio which is given by
l)eff =
(5)
llee e + 1.'pSp "se -~- "sp
where Ve = 0.3 is the elastic Poisson's ratio and Vp = 0.5 is the plastic Poisson's ratio. The axial elastic and plastic strains are given by Eq 6a and Eq 6b, respectively o'a
"sp z ,Sap
(6a)
oa E
(6b)
The range of maximum shear strain and the corresponding normal strain range on the critical plane at which both strain and stress Mohr's circles are the largest during loading (at the angle 01) and unloading (at the angle 02) of a cycle (see Fig. 2) are calculated as
\2}
\
2
= ('$1 +'s,) A's.
\
2
,]Ol
(7a)
\~]o2
('s1 +'$3) J01 - \
2
,/ee
(7b)
where el, e2, and "$3are the principal strain values @1 > "$2> "$3)which are calculated from the strain Mohr's circle (see Fig. 5a) as: 81 = ( 1 - -
1-'eft)
+ ~
"sap(1 + b'eff) 2 +
(8a)
VARVANI-FARAHANI AND TOPPER ON STRAIN PATHS
~max
313
max
(a)
(b)
FIG. 5--(a) Strain M o h r ' s circle, a n d (b) stress M o h r ' s circle.
e3 = (1 - Ve,~)
8"2 = -- Peffgap
(8b)
- - ~1 eap 2
(8c)
(1 + veff)2 +
Similarly, the range of maximum shear stress and the corresponding normal stress range are calculated from the largest stress Mohr's circle during loading (at the angle 01) and unloading (at the angle 02) of a cycle as:
(0-1 -- 0-3~
( 0-1 -- 0-3~
Agmax = \ ~ ] 0 1
-- \
(0"1 +
A0"n = \
2
2
(0"1
i]01 --
t~
7o2
(ha)
]oz
(9b)
where 0-1, (re, and 0-3 are the principal stress values (0-1 > 0-2 > 0-3) and they are calculated from the stress Molar's circle (see Fig. 5b) as: OVa 0-1 = ~-+ ~1 [(~,2 + 4,/.211/2
(lOa)
0"2 = 0
(lOb)
03 - 0-a 2
21 [o-.2 + 4r 2] 1/2
(10c)
In strain paths with no mean strain, the largest strain and stress Mohr's circles, obtained during loading (at 01) and unloading (at 02) in a cycle, have equal diameters. In these strain paths, to achieve the plane of maximum shear strain, the plane P (obtained at 01) and plane Q (obtained at 02) should rotate counterclockwise with the angle of c~ = tan-
1
[A~alA~#a~
on the Mohr's circles (see Fig.
2a--history A3). For strain paths having a mean strain, the largest Molar's circles obtained at 01 and 02 do not have equal diameters (see Fig. 2b). To achieve the same critical plane, both planes P and Q on Mohr's circles have to rotate through an angle a (see Fig. 2b--histories A4 and A5). The ranges of shear strain and normal strain for strain histories containing axial and shear mean strains are shown in Fig. 2b. For the strain history A4 which has an axial mean strain, the ranges of shear strains and
314
MULTIAXIAL FATIGUE AND DEFORMATION
stresses are calculated by multiplying the second terms of Eqs 7a and 9a by cosc~ and the ranges of normal strains and stresses are calculated by multiplying the second terms of Eqs 7b and 9b by sina. For strain history A5 which has a mean shear strain, the ranges of shear strains and stresses are calculated by multiplying the second term of Eqs 7a and 9a by sin~ which, in calculating the ranges of normal strains and stresses, the second terms of Eq 7b and 9b are multiplied by 1 + cosa. For strain history A5, containing both axial and shear mean strains, the second terms of Eqs 7 and 9 become zero. The range of maximum shear stress A~'ma~and shear strain A (~-~-~) obtained from the largest stress and strain Mohr's circles at angles 0j and 02 during the loading and unloading parts of a cycle and the corresponding normal stress range A~r, and the normal strain range mE n o n that plane are the components of the proposed parameter. Both the normal and shear strain energies are weighted by the axial and shear fatigue properties, respectively:
(o-~ ~)
(~s ~'~) \
where o-} and ~} are the axial fatigue strength coefficient and axial fatigue ductility coefficient, respectively, and ~-~and ~ are the shear fatigue strength coefficient and shear fatigue ductility coefficient, respectively. Multiaxial fatigue energy based models have been long discussed in terms of normal and shear energy weights. In Garud's approach [6] he found that an empirical weighting factor of C = 0.5 in the shear energy part of his model (Eq 12) gave a good correlation of multiaxial fatigue results for 1% Cr-Mo-V steel for both in-phase and out-of-phase loading conditions. AeA~r + CA~,AT = I(Ns)
(12)
Tipton [34] found that a good multiaxial fatigue life correlation was obtained for 1045 steel with a scaling factor C of 0.90. Andrews [35] found that a C factor of 0.30 yielded the best correlation of multiaxial life data for AISI 316 stainless steel. Chu et al. [14] weighted the shear energy part of their formulation by a factor of C = 2 to obtain a good correlation of fatigue results. Liu's [13] and Glinka et al.'s [15] formulations provided an equal weight of normal and shear energies. The empirical factors (C) suggested by each of the above authors gave a good fatigue life correlation for a specific material which suggests that the empirical weighting factor C is material dependent. In the present study, the proposed model correlates rnultiaxial fatigue lives by normalizing the normal and shear energies using the axial and shear material fatigue properties, respectively, and hence the parameter uses no empirical weighting factor.
Out-of-Phase Strain Hardening Under out-of-phase loading, the principal stress and strain axes rotate during fatigue loading often causing additional cyclic hardening of materials. A change of loading direction allows more grains to undergo their most favorable orientation for slip, and leads to more active slip systems in producing dislocation interactions and dislocation tangles to form dislocation cells. Interactions strongly affect the hardening behavior and as the degree of out-of-phase increases, the number of active slip systems increases. Socie et al. [25] performed in-phase and 90 ~ out-of-phase fatigue tests with the same shear strain range on 304 stainless steel. Even though both loading histories had the same shear strain range, cyclic stabilized stress-strain hysteresis loops in the 90 ~ out-of-phase tests had stress ranges twice as large as those of the in-phase tests. They concluded that the higher magnitude of strain and stress ranges in the out-of-phase tests was due to the effect of an additional strain hardening in the material [26]. During out-of-phase straining, the magnitude of the normal strain and stress ranges is larger than that for in-phase straining with the same applied shear strain ranges per cycle. The proposed param-
VARVANI-FARAHANI AND TOPPER ON STRAIN PATHS
315
eter via its stress range term increases with the additional hardening caused by out-of-phase tests, whereas critical plane models that include only strain terms do not change when there is strain path dependent hardening. To calculate the additional hardening for out-of-phase fatigue tests, these approaches may be modified by a proportionality factor like the one proposed by Kanazawa et al. [27].
Mean Stress Correction Under multiaxial fatigue loading mean tensile and compressive stresses have a substantial effect on fatigue life. Sines [28] showed compressive mean stresses are beneficial to the fatigue life while tensile mean stresses are detrimental. He also showed that a mean axial tensile stress superimposed on torsional loading has a significant effect on the fatigue life. In 1942 Smith [29] reported experimental results for 27 different materials from which it was concluded that mean shear stresses have very little effect on fatigue life and endurance limit. Sines [28] reported his findings and Smith's results by plotting mean stress normalized by monotonic yield stress versus the amplitude of alternating stress normalized by fatigue limit (R = - 1) values (see Fig. 6). Figures 6a and 6b show the ef-
1
i .... ' .... ' .....
1.5
'''' ';+cli'c'str" i'a)
...a
I
9
O [] <> 9
0.5
<
0.41% C steel I A1 2024 [ 0.65% C steel[ 0.44% C steel I
-1.5
-1
-0.5
--
0
0.5
1
1.5
Mean stress/yield stress
2
Cyclic str~
(b)
1.5 "Z-~
1 0.5
0
9 Mild steel ] [] A1 6061-T6 [ O NI-Cr-Mo steelJ ,
-1.5
,
,
,
l
-I
,
,
,
,
I
. . . .
I
,
,
,
,
I
,
,
-0.5 0 0.5 M e a n stress/yield stress
,
,
I
1
,
,
l
,
1.5
FIG. 6---The effect of axial mean stress on (a)pull-push fatigue loading, and (b) torsional fatigue
loading.
316
MULTIAXIAL FATIGUE AND DEFORMATION
fect of a static tensile and compressive stress for various materials on axial and torsional fatigue, respectively. The relation is linear as long as the maximum stress during a cycle does not exceed the yield stress of the material [26]. Concerning the effect of mean strain on fatigue life, Bergmann et al. [30] found almost no effect in the low-cycle fatigue region and very little effect in the high-cycle fatigue region. Mean stress effects are included into fatigue parameters in different ways [26]. One approach was applied earlier by Fatemi and Socie [31] to incorporate mean stress using the maximum value of normal stress during a cycle to modify the damage parameter. Considering the effect of axial mean stress, a similar mean stress correction factor t +
m Eq 11 showed a good correlation of multiaxial
fatigue data containing mean stress values for both in-phase and out-of-phase straining conditions. This correction is based on the mean normal stress applied to the critical plane. To take into account the effect of mean axial stress on the proposed parameter, Eq 11 is rewritten as: O-m -I- v n
1
(O.tf Fvf) (AO'nA~n)~-
/ /.v_..~ \\ (Ttf ~ltf) ~ m . ' r r n a x A ~ - } ) : f ( g f )
(13)
where the normal mean stress o-m acting on the critical plane is given by: o'nm _- i1
(o.max
+
o.mi.)
(14)
In Eq 14, o-max and O'n ~in are the maximum and minimum normal stresses which are calculated from the stress Mohr's circles.
Parameter for the Correlation of Multiaxial Fatigue Crack Growth Rates In a previous study [18], strain intensity factor range values were calculated for a semielliptical surface crack under tensile mode straining and shear mode straining, respectively, using AK1 = Ft E A~, ~
(15a)
AK3 = FixI GA "}/max~
(15b)
where the AK1 and AK3 are the strain intensity factor range for opening mode and shear straining, FI and Fni are the shape factors for the deepest point of an opening mode crack, and a shear crack, respectively (solutions for geometry factors FI and Fin are given in Refs 18, 19, and 32), and a is the crack depth on the plane of maximum shear strain. DuQuesnay et al. [33] found that a mean stress has no effect when the crack opening stresses are below the minimum stresses for 1045 steel and 2024-T351 aluminum alloys. The present authors also performed two series of biaxial crack growth and life test series [18,19], one in which constant amplitude fully-reversed strains were applied and another in which large periodic compressive strain cycles causing strains normal to the crack plane were inserted in a constant amplitude history of smaller strain cycles. Ratios of hoop strain (e3) to axial strain (el) of A = - 1 (pure shear), A = 0.625, A = u (uniaxial straining), and A = + 1 (equibiaxial straining) were used in each test series. The magnitude and frequency of application of the periodic compressive overstrain cycles in the second test series was chosen to reduce the crack opening stress to a level below the minimum stress level of the constant amplitude small cycles so that they experienced closure-free crack growth. The compressive underloads caused a large decrease in the small cycle fatigue resistance. Crack size, crack growth mechanisms, closure-free crack growth rates and life data for SAE 1045 steel under various biaxial fatigue straining were extensively reported in a previous paper [18]. -
VARVANI-FARAHANIAND TOPPER ON STRAIN PATHS
317
In the present study, the effective intensity factor range, AKeff, is formulated based on the proposed parameter (Eq 11). The components of this formulation consist of the shear energy range and the corresponding normal energy range acting on the critical plane obtained from the largest Mohr' s circles for a cycle, the square root of the crack depth, and the geometry factors F~ and Fm
AKeff= (AK2 + AK2)1/2
(16)
where AKn is the normal part of intensity factor range, and AKs is the shear part of intensity factor range. These components are given as
AK.=
FE(kA- 0- --"~A] ~ N / - ~a
(17a)
( (m'rmaxm( ~ax)) t Substituting Eqs 17a and 17b into Eq 16, and G = E/2(1 + re) = EI2.6, Eq 16 can be rewritten as
(r 2{A0-nAI3n~2 (FItI)2{A'l"max~p('~lmax[2)~2~l~-a~a AKe~=E k I k 0"}8} } + [ 2 . 6 ] [ #YS ]1
(18)
Correlations of Fatigue Data Using the Proposed Parameter In order to assess the capability of the proposed parameter to correlate multiaxial fatigue lives and crack growth rates for both in-phase and out-of-phase loading conditions, fatigue data for different materials and various in-phase and out-of-phase strain paths available in the literature were used. Figures 7a-7g present multiaxial fatigue life correlations based on the proposed parameter (Eq 13) for seven different materials subjected to the various in-phase and out-of-phase strain paths and strain histories which are shown in Fig. 2. A very good correlation of multiaxial fatigue lives is obtained for Ni-Cr-Mo-V steel (Fig. 7a), Incone1718 (Fig. 7c), Haynes 188 (Fig. 7d), and Waspaloy (Fig. 7e) for the various in-phase and out-of-phase conditions within a factor of 1.5 for both low-cycle and highcycle fatigue lives. Fatigue life correlation for 1045 steel (Fig. 7b), stainless steel (Fig. 7f), and mild steel (Fig. 7g) fell within factors of 2, 2.5, and 2, respectively. In the previous studies, the effective (closure-free) strain intensity factor range, AK~ff,was modeled [18,19,36] based on critical plane approaches of: (a) the maximum shear strain (MSS) parameter, (b) the Brown-Miller (BM) parameter [37], (c) the Kandil-Miller-Brown (KMB) parameter [38], and (d) the Fatemi-Socie (FS) parameter [31]. Critical plane approaches postulate that cracks initiate and propagate on the maximum shear strain plane and that the normal strain on this plane assists in the fatigue crack growth process. The components of these models consist of the maximum shear strain range and the normal strain range acting on the maximum shear strain plane. Figure 8 presents and compares the correlation of effective biaxial fatigue crack growth and life data for SAE 1045 steel tubular specimens based on: (a) the maximum shear strain parameter, (b) the Brown-Miller parameter [37], (c) the Kandil-Miller-Brown parameter [38], (d) the Fatemi-Socie parameter [31], and finally (e) the new parameter proposed in this study. The results of fatigue life predictions obtained from the various AKeff-da/dN curves presented in Figs. 8a-h show that the strain-based critical plane parameters [31,37-38] correlate the effective fatigue life data within a factor of -+2 for the low-cycle fatigue regime, 103 < N --< 105, and a factor of -+3 for the high-cycle fatigue regime, N > 105. Figure 8i-j shows that the proposed parameter suc-
318
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 7 Multiaxial fatigue life correlation for various in-phase and out-of-phase strain histories and seven different materials: (a) Ni-Cr-Mo-V steel [16], (b) 1045 steel [17-19], (c) Incone1718 [20], (d) Haynes 188 [21], (e) Waspaloy [22,23], (f) Stainless steel [25], and (g) Mild steel [24].
VARVANI-FARAHANI AND TOPPER ON STRAIN PATHS
319
FIG. 8--Effective fatigue life correlations and crack growth rate correlations of SAE 1045 steel based on: (a, b) the maximum shear strain (MSS) parameter, (c, d) the Brown-Miller (BM) parameter [37], (e, f) the Kandil-Miller-Brown (KMB) parameter [38], (g, h) the Fatemi-Socie (FS) parameter [31], and (i, j) the Varvani-Topper (VT) parameter proposed in this paper.
320
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 8--(Confinued)
cessfully correlates the biaxial effective fatigue lives and crack growth rates within a factor of + 1.5 for both low- and high-cycle regimes. Discussion Energy-critical plane parameters [13-15], including the parameter proposed in the present paper, are defined on specific planes and account for states of stress through combinations of the normal and shear strain and stress ranges. These parameters depend upon the choice of the critical plane and the stress and strain ranges acting on that plane. For the proposed parameter, the critical plane is defined by the largest shear strain and stress Mohr's circles during the loading and unloading parts of a cycle and the parameter consists of tensorial stress and strain range components acting on this critical plane. The critical plane in Liu's parameter [13], on the other hand, is associated with two different physical modes of failure and the parameter consists of Mode I and Mode II energy components. Liu's parameter does not account for the effect of mean stress. Chu et al. [14] formulated normal and shear energy components based on the Smith-Watson-Topper parameter. They determined the critical plane and the largest damage parameter from the transformation of strains and stresses onto planes spaced at equal increments using a generalized Mroz model. This parameter is based on the maximum value of the damage parameter rather than being defined on planes of maximum stress or strain. Glinka et al. [15] proposed a multiaxial fatigue life parameter based on the summation of the products of normal and shear strains and stresses on the critical plane which is assumed to be the plane of maximum shear strain. In their papers, Liu, Chu et al., and Glinka et al. reported that their parameters were capable of correlating multiaxial fatigue life results for both in-phase and out-of-phase loading paths. The proposed parameter successfully correlated multiaxial fatigue lives within a factor that varied with materials from 1.5 to 2.5 for both low- and high-cycle fatigue regimes for various in-phase and out-of phase multiaxial fatigue straining conditions. The poorest correlation, a factor of 2.5 in fatigue life in stainless steel, may be due to crack growth mechanism in this material. Observations of crack formation and early crack growth for this material reported by Socie [25] showed that in tensile loading, Mode I failures were observed at all strain amplitudes. In torsion, Mode II shear failures were observed at high-strain amplitude and Mode I failures at low-strain amplitudes. However, for other materials studied in the present paper, in the early stage of crack growth a Mode II crack was dominant independent of stress state. The proposed parameter successfully correlated multiaxial fatigue lives for tests in which a mean stress was applied normal to the maximum shear plane. The proposed parameter via its stress range term takes into account the effect of additional hardening in out-of-phase fatigue tests. This parame-
VARVANI-FARAHANI AND TOPPER ON STRAIN PATHS
321
ter successfully correlated closure-free crack growth rates for various biaxial strain ratios in SAE 1045 steel. The correlations shown in Fig. 8 are for fully effective (closure-free) fatigue data, the only data of this type presently available.
Conclusions A multiaxial fatigue parameter is proposed by the sum of the normal energy range and the shear energy range calculated for the critical plane on which the stress and strain M o h r ' s circles are the largest during the loading and unloading parts of a cycle. The normal and shear energies in this parameter have been weighted by the tensile and shear fatigue properties, respectively. The proposed parameter successfully correlated multiaxial fatigue lives and crack growth rates by taking into account: (a) various in-phase and out-of phase multiaxial fatigue straining conditions, (b) the effect of a mean stress applied normal to the maximum shear plane, and (c) the proposed parameter via its stress range term increases when there is an additional hardening caused by out-of-phase tests, whereas critical plane approaches that include only strain terms do not change when there is strain path-dependent hardening. The proposed parameter has shown a very good correlation of multiaxial low-cycle and high-cycle fatigue lives for various in-phase and out-of-phase straining conditions for different material fatigue data reported in the literature. An effective intensity factor based on the proposed parameter successfully correlated the biaxial effective crack growth rates for various biaxial strain ratios.
References [1] Garud, Y. S., "Multiaxial Fatigue: A Survey of the State-of-the-Art," Journal of Testing and Evaluation, Vol. 9, No. 3, 1981, pp. 165-178. [2] Brown, M. W. and Miller, K. J., "Two Decades of Progress in the Assessment of Multiaxial Low-Cycle Fatigue Life," Low-Cycle Fatigue and Life Prediction, ASTM STP 770, C. Amzallag, B. Leis, and P. Rabbe, Eds., American Society for Testing and Materials, 1982, pp. 482--499. [3] You, B. R. and Lee, S. B., "A Critical Review on Multiaxial Fatigue Assessments of Metals," International Journal of Fatigue, Vol. 18, No. 4, 1996, pp. 235-244. [4] Jordan, E. H., "Fatigue-Multiaxial Aspects," Pressure Vessel and Piping Design Technology--A Decade of Progress, ASME, American Society of Mechanical Engineers, 1982, pp. 507-518. [5] Garud, Y. S., "A New Approach to the Evaluation of Fatigue Under Multiaxial Loadings," Proceedings, Symposium on Methods for Predicting Materials Life in Fatigue, W. J. Ostergren and J. R. Whitehead, Eds., ASME, American Society of Mechanical Engineers, 1979, pp. 247-263. [6] Garud, Y. S., "A New Approach to the Evaluation of Fatigue Under Multiaxial Loadings," Transaction of theASME, Vol. 103, 1981, pp. 118-125. [7] Ellyin, F. and Kujawski, F., "A Multiaxial Fatigue Criterion Including Mean Stress Effect," Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and R. Ellis, Eds., American Society for Testing and Materials, Philadelphia, 1993, pp. 55-66. [8] Ellyin, F. and Xia, Z., "A General Fatigue Theory and Its Application to Out-of-Phase Cyclic Loading," ASME, J. Engng Mater. Tech., American Society of Mechanical Engineers, Vol. 115, 1993, pp. 411-416. [9] Brown, M. W. and Miller, K. J., "A Theory for Fatigue Under Multiaxial Stress-Strain Conditions," Proceedings of the Institution Mechanical Engineering, Vol. 187, 1973, pp. 745-755. [10] Findley, W. N., "A Theory for the Effect of Mean Stress on Fatigue of Metals Under Combined Torsion and Axial Load or Bending," Journal of the Engineering Industry, Vol. 81, 1959, pp. 301-306. [11] Flavenot, J. F. and Skalli, N., "A Critical Depth Criterion for Evaluation of Long Life Fatigue Strength Under Multiaxial Loadings and Stress Gradient," Biaxial and Multiaxial Fatigue, M. W. Brown and K. J. Miller, Eds., ESIS Publication No. EGF3, London, 1989, pp. 355-365. [12] Stulen, F. B. and Cummings, H. N., "A Failure Criterion for Multiaxial Fatigue Stresses," Proceedings, ASTM, Vol. 54, 1954, pp. 822-835. [13] Liu, K. C., "A Method Based on Virtual Strain-Energy Parameters for Multiaxial Fatigue," Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and R. Ellis Eds., American Society for Testing and Materials, Philadelphia, 1993, pp. 67-84. [14] Chu, C. C., Conle, F. A., and Bonnen, J. F., "Multiaxiai Stress-Strain Modeling and Fatigue Life Prediction of SAE Axle Shafts," Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and R. Ellis, Eds., American Society for Testing and Materials, Philadelphia, 1993, pp. 37-54.
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[15] Glinka, G., Shen, G., and Plumtree, A., "A Multiaxial Fatigue Strain Energy Density Parameter Related to the Critical Plane," Fatigue and Fracture of Engineering Materials and Structure, Vol. 18, 1995, pp. 37-46.
[16] Williams, R. A., Placek, R. J., Khifas, O., Adams, S. L., and Gonyea, D. C., "Biaxial/Torsional Fatigue Turbine Generator Rotor Steel," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 440-462.
[17] Kurath, P., Downing, S. D., and Galliart, D. R., "Summary of Non-Hardened Notched Shaft Round Robin Program," Multiaxial Fatigue: Analysis and Experiments, G. E. Leese and D. Socie, Eds., Society of Automotive Engineers, 1989, pp. 13-31.
[18] Varvani-Farahani, A. and Topper, T. H., "Closure-Free Biaxial Fatigue Crack Growth Rate and Life Prediction Under Various Biaxiality Ratios in 1045 Steel," Fatigue and Fracture of Engineering Materials and Structures, Vol. 22, 1999, pp. 697-710. [19] Varvani-Farahani, A. and Topper, T. H., "The Effect of Biaxial Strain Ratio and Periodic Compressive Overstrains on Fatigue Crack Growth Mode and Crack Growth Rate," ASTM STP 1360, K. L. Jerina and P. C. Paris, Eds., American Society for Testing and Materials, 2000, pp. 299-312.
[20] Koch, J. L., "Proportional and Non-Proportional Biaxial Fatigue of Incone1718," Report No. 121, Department of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, 1985.
[21] Kalluri, S. and Bonacuse, P. J., "In-Phase and Out-of-Phase Axial-Torsional Fatigue Behaviour of Haynes 188 Superalloy at 760~ '' Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell and R. Ellis Eds., American Society for Testing and Materials, Philadelphia, 1993, pp. 133-150.
[22] Lerch, B. A. and Jayaraman, N., "A Study of Fatigue Damage Mechanisms in Waspaloy from 25 to 800~ '' Materials Science and Engineering, Vol. 66, 1984, pp. 151-166. [23] Jayaraman, N. and Ditmars, M. M., "Torsional and Biaxail (Tension-Torsion) Fatigue Damage Mechanisms in Waspaloy at Room Temperature," International Journal of Fatigue, Vol. 11, 1989, pp. 309-318. [24] Doquet, V. and Pineau, A., Multiaxial Low-Cycle Fatigue Behaviour of a Mild Steel," Fatigue under Biaxial and Multiaxial Loading, ESIS 10, K. Kussmaul, D. McDiarmid, and D. Socie, Eds., Mechanical Engineering Publication, London, 1991, pp. 81-101.
[25] Socie, D., "Multiaxial Fatigue Damage Models," Journal of Engineering Materials and Technology, Vol. 109, 1987, pp. 293-298.
[26] Multiaxial Fatigue, D. Socie and G. Marquis, Eds., Society of Automotive Engineers (SAE) International, under publication, Fall- 1999.
[27] Kanazawa, K., Miller, K. J., and Brown, M. W., "Cyclic Deformation of 1% Cr-Mo-V Steel Under Out-ofPhase Loads," Fatigue and Fracture of Engineering Materials and Structures, Vol. 2, 1979, pp. 217-228. [28] Sines, G., "The Prediction of Fatigue Fracture Under Combined Stresses at Stress Concentrations," Bulletin of the Japan Society for Mechanical Engineers, Vol. 4, No. 15, 1961, pp. 443--453. [29] Smith, J. O., "Effect of Range of Stress on Fatigue Strength of Metals," University of Illinois, Engineering Experiment Station, Bulletin No. 334, Vol. 39, No. 26, 1942.
[30] Bergmann, J., Klee, S., and Seeger, T., "Effect of Mean Strain and Mean Stress on the Cyclic Stress-Strain and Fracture Behaviour of Steel StE70," Materialpruefung, Vol. 19, No. 1, 1977, pp. 10-17. [31] Fatemi, A. and Socie, D. F., "A Critical Plane Approach to Multiaxial Fatigue Damage Including Out-ofPhase Loading," Fatigue and Fracture of Engineering Materials and Structures, Vol. 11, 1988, pp. 149-165.
[32] Socie, D. F., Hua, C. T., and Worthem, D. W., "Mixed Mode Small Crack Growth," Fatigue and Fracture of Engineering Materials and Structures, Vol. 10, 1987, pp. 1-16. [33] DuQuesnay, D. L, Topper, T. H., Yu, M. T., and Pompetzki, M. A., "The Effective Stress Range as a Mean Stress Parameter," International Journal of Fatigue, Vol. 14, No. 1, 1992, pp. 45-50. [34] Tipton, S. M., "Fatigue Behaviour Under Multiaxial Loading in the Presence of a Notch: Methodologies [35] [36] [37] [38]
for the Prediction of Life to Crack Initiation and Life Spent in Crack Propagation," Ph.D. Thesis, Mechanical Engineering Department, Stanford University, Stanford, CA, 1984. Andrews, R. M., "High Temperature Fatigue of AISI 316 Stainless Steel Under Complex Biaxial Loading," Ph.D. Thesis, University of Sheffield, UK, 1986. Varvani-Farahani, A. "Biaxial Fatigue Crack Growth and Crack Closure Under Constant Amplitude and Periodic Compressive Overload Histories in 1045 Steel," Ph.D. Thesis, University of Waterloo, Canada, 1998. Brown, M. W. and Miller, K. J., "A Theory for Fatigue Failure Under Multiaxial Stress-Strain Conditions," Proceedings, Institution of Mechanical Engineering, u 187, 1973, pp. 745-755. Kandil, F. A., Miller, K. J., and Brown, M. W., "Creep and Aging Interactions in Biaxial Fatigue of Type 316 Stainless Steel," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 651~568.
H. A. Suhartono, a K. POtter, 1 A. Schram, 2 and H. Zenner 1
Modeling of Short Crack Growth Under Biaxial Fatigue: Comparison Between Simulation and Experiment* REFERENCE: Suhartono, H. A., Prtter, K., Schram, A., and Zenner, H., "Modeling of Short Crack Growth Under Biaxial Fatigue: Comparison Between Simulation and Experiment," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 323-339. ABSTRACT: Grain boundaries, microstructural barriers and differences in stress/strain state play a dominant role in the early stages of the fatigue crack growth of metals. Many studies on the growth of short cracks have revealed anomalies in the behavior predicted by LEFM analysis. A simulation of fatigue crack growth is presented. The polycrystalline metal was modeled as an aggregate of hexagonal grains with a different crystallographic orientation of each grain. The effect of grain boundaries on Stage I crack growth is considered in the model. The mode of shear crack growth is used to compute the crack growth. This mode is analyzed on the basis of microstructural crack growth within the first few grains, where the crack growth decelerates as the crack tip gets closer to the grain boundary. Normal crack growth has been considered for those cracks which are longer than microstructurai cracking (physically short cracks). The transition from Stage I to Stage II growth is considered. The model is applied for thinwailed tubular specimens of the ferritic steel AISI 1015 and the aluminum alloy AIMgSil subjected to tension and torsion loading as well as in-phase and out-of-phase combined tension-torsion loading, sequential tension and torsion loading. The microstructural crack pattern and crack distribution can be successfully simulated with the model, and the simulated microstructural crack growth rate is presented. KEYWORDS: modeling, microcrack growth, AISI 1015, A1MgSil
Nomenclature a A B d D N r rc s a /3 '~max
Crack length Crack growth coefficient (Stage I) Crack growth coefficient (Stage II) Grain diameter Crack growth threshold N u m b e r o f cycles Distance b e t w e e n crack tips Critical distance b e t w e e n crack tips Distance b e t w e e n crack tip and barrier Crack growth exponent (Stage I) Crack growth exponent (Stage II) M a x i m u m shear strain
i Institut fiir Maschinelle Anlagentechnik und Betriebsfestigkeit, TU Clausthal, Leibnizstr. 32, 38678 Clausthal-Zelleffeld, Germany. 2 Institut fur Schweisstechnik und Trennende Fertigungsverfahren, TU Clausthal, Agricolastr. 2, 38678 Clausthal-Zellerfeld, Germany. * This paper received the Best Presented Pager award at the ASTM Symposium on Multiaxial Fatigue and Deformation: Testing and Prediction.
Copyright9
by ASTM International
323 www.astm.org
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MULTIAXIAL FATIGUE AND DEFORMATION
Maximum principal normal strain Intermediate principal normal strain Minimum principal normal strain $3 o-x, O'y Normal stress in the Cartesian axis direction ~-~y Shear stress in the Cartesian axis direction 7,o Shear stress on the slip plane O) Orientation of the slip system The accuracy of life prediction under repetitive loading is still unsatisfactory. Inadequate predictions appear particularly when the loading sequence involves effects with high-low/low-high changes, overloads or mixture and changing of mean stresses. If these loading effects are combined with multiaxial stress states, most life prediction concepts fail. One reason for the disagreement between calculated and experimental results is the fact that it fails to generate a substantial model of the microstructural damage process and to make it accessible for calculations. Different groups of scientists and engineers have explored this area [1-13] and have described the damage process under different aspects. All of the research results recognize that the propagation of microcracks is strongly influenced by the microstructure of the material. For the examinations, the damage process is subdivided into microcrack nucleation, microcrack growth, microcrack coalescence, and macrocrack propagation. In order to take into account the influence of the microstructure on the damage process, a simulation model that considers the local stress state and the random nature of the material structure in the form of grain boundary and slip systems is proposed. The first suggestions for fatigue simulation models were submitted by Hoshide and Stele [14], followed by other publications in [9,11,13,15,16]. In the following paper a simulation model that describes the microcrack nucleation and microcrack growth as simplified as possible is introduced.
Modeling of Mieroeraek Formation and Growth The polycrystalline material is modeled as a two-dimensional hexagonal network of grains with specific sizes of diameter, d = 60/xm. Multiple slip systems are active in each grain with a randomized crystallographic orientation 0). The stress state in the slip plane of each grain is dependent on its orientation and the applied loading. Only the material surface with its plane stress state is currently considered. The simulation does not yet consider the crack growth in the depth direction of the material. Furthermore, the deformation behavior of microstructure, the cyclic hardening and softening of the material, the crack-opening effects, as well as the texture and anisotropy of the material are not considered. Figure 1 shows the simulated microstructure, the stress state, and crack growth. The location of the microcrack nucleation is given by a random generator. The shape of the microcrack seed is a point with no spatial extension, denoting an initial crack length of zero. It is assumed that the points of crack nucleation are given at the beginning of the simulation and that the crack growth starts with the first load cycle. The driving force for the crack growth is the applied loading. The shear stress in individual slip plane directions, 0), is calculated from Eq 1. To = - ( ~ ) s i n
(2w)+ Txycos (20))
(1)
The crack propagation of microstructural short cracks (Stage I) is calculated with an equation proposed by Hobson et al. [1]. The equation describes the development of fatigue cracks within the first few grains of a polycrystalline material. With several experimental investigations taken into account, the cracks are driven by the cyclic shear stress on the slip planes. The crack growth rate depends on the shear stress and on the distances between the crack tips and the dominant microstructural barri-
SUHARTONO ET AL. ON SHORT CRACK GROWTH
325
Microstructure and stress-state Txy
Stage I crack growth
Crack coalescence
FIG. 1--Simulated microstructure, stress state, and crack growth.
ers, in this case, the grain boundary. The microcrack growth equation has the form da
,A
oL
~-n = ,azaro)'s
(2)
where s is the crack tip distance to the next barrier, and A and a are material parameters. At first the crack growth is fast, but when the crack approaches the barrier (s ~ 0) the crack growth rate tends toward zero. In the current model the grain boundary is regarded to be the dominant material barrier. When the crack is sufficiently long to permit an opening of the crack front, the development of Stage II (tensile) crack occurs. At this point, the influence of the microstructure is limited, and crack growth can be described by continuum mechanics. The equation of Stage II crack growth proposed by Hobson et al. [1] is used in the model: d__q_.a= BAtr~a _ D dn
(3)
where Atr represents the tensile stress perpendicular to the crack plane, and r, B, and D are experimentally determined material parameters. The material parameters used in the simulation are taken
326
MULTIAXlAL FATIGUE AND DEFORMATION
from Hobson et al. [1]. Further experimental investigations have to be performed to validate these assumptions. The crack length at the transition from Stage I to Stage II can be easily introduced by assigning the number of the grains. Taylor and Knott [17 ] suggest a value of about three grain diameters for the transition. In the transition zone the crack growth is calculated by using the higher value between Eq 2 and Eq 3. Besides the cyclic growth of microcracks, a rapid spread of the crack length can be observed during the experiment by the linking of cracks. The crack coalescence is described by assuming that the linking of cracks appears when the length of the cracks reaches 75% of the grain size, and the distance r between their tips is less than a critical distance re. In the simulation of Socie and Furman [13], the critical distance is 25% of the grain diameter. The simulation ends when the predetermined number of load cycles is reached or the microcrack reaches the predetermined crack length. The crack length is defined by the direct line between both crack tips. If a crack was formed by linking of several microcracks, the crack length is always represented by the crack tips with the longest distance. A flow chart of the simulation model is given in Fig. 2. In Figs. 3 and 4 the material structure is
3LOADING S E L E C T I O N : 1) Tension, torsion or synchrone 2) O~ iofphasr loading ) Consecutive loading
i Input : Grain diameter crack density stress or strain state crack growth-Parameter
l
graphic presentation of the E L ] grains ~ l ]
random generator : angles of slip planes
|
J
random generator : determination of microcrack seeds I Calculate resolved shear ( 1 stress on each slip plane
~ A'~ = I/2(Ao y -Acsx) sin2t0 + &xy cos2m
Calculate miZ-ocrack growth i.e. length < 3 grain size da/dn = A(A'~o) a (d - a)
microcrack stop
~ c
micr~ oales~
short crack growth i.e. length > 3 grain size dMdn = B(&~) I~a - C
~
crack or the end of simulation
FIG. 2--Flow chart of the modeling.
SUHARTONO ET AL. ON SHORT CRACK GROWTH
327
FIG. 3--Simulated cracking behavior under tension-compression loading.
represented with the randomized slip systems and the simulated microcrack growth under tensioncompression loading (Fig. 3) and in-phase tension-compression and torsion loading (Fig. 4). The figures show the crack pattern at different numbers of load cycles. The initial cracks grow within the first few grains in the direction of the maximum shear stress. After they have reached a critical length, possibly by crack coalescence, the transition from Stage I to Stage II takes place. Finally, the macrocrack growth is presented. The crack length is plotted versus the number of cycles in Figs. 5 and 6, respectively, for the load cases considered. If damage is assumed to be represented by the maximum crack length, it can be seen that the accumulation of damage is unsteady, and changes rapidly in the case of crack coalescence. As determined by the interaction between the crack, microstructure, and stress state, different cracks appear to be the most critical or damaging during the simulation,
Experimental Procedure The materials tested during the experimental analyses were AISI 1015 steel [9] and A1MgSil aluminum. The hollow cylindrical specimens, Fig. 7, of AISI 1015 and AIMgSil were taken from the material in the orientation parallel with the rolling direction, machined, and polished.
328
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 4--Simulated cracking behavior under proportional loading.
The experiments were carried out on a multiaxial servo hydraulic test device capable of imposing torsion- and tension/compression-loads; the device comprises a 10 (kN-m) hydraulic actuator for torsion and a 100 kN actuator for axial loading. The tests are performed under controlled strain with the use of a multiaxial extensiometer. The different loading cases are presented in Table 1. Microscopic surface investigations were performed to determine the nucleation and propagation of the microcracks. The microscopic examinations were conducted at regular intervals during the test at 12 areas (three areas in the longitudinal direction, and four areas in the tangential direction), see Fig. 8.
Comparison Between Simulation and Experimental Results Experimental Results with Steel AIS11015 The orientation, length, and density of microcracks depend on the magnitude and type of loading. The density and orientation of microcracks taken from experimental results with steel AISI 1015 [9] and simulation are shown in Fig. 9 for different loading cases (see Table 1). The tests were performed with a thin-walled tubular specimen under multiaxial loading.
S U H A R T O N O ET AL. ON S H O R T C R A C K G R O W T H
329
7~176 l 600 +
500
"~ 4 0 0
Cra l y
~300 O 200
100
' ~ O, 1
4 i
10
i 1000
100
10000
N (Cycles)
FIG. 5--Propagation of crack growth versus number of cycles of tension-compression load.
600
6+7 5OO
9+3
crack 6 and cra~ 7
-
8
O
crack 9 and crack3 ~
200
1
6e ~ 100
9
10
~
~1"~
-
100
1000
10000
N (Cycles)
FIG. 6~Propagation of crack growth versus number of cycles of in-phase tension-compression and torsion loading.
330
MULTIAXIAL FATIGUE AND DEFORMATION
1.5x45~
..... f
......
t--t
100 130 160
F I G . 7--Smooth hollow specimen geometry used in the experimental analyses.
TABLE 1--Experimental programs with different loading cases [9].
Loading
L o a d ratio
2
(R )
rla
Stress ( M P a )
Torsion
-1
o~
~'a = 200
Tension -compression
-1
0
or. = 300
In-phase
-1
89
era = 275 ~'~ = 137.5
Out-of-phase
-1
V2
Time-function
'hnnn lUUUUt
tra= 275 ~'a = 137.5 6=90
Interchanging
-1
1/2
o'a = 275
~'a = 137.5
Consecutive
-1
2/3
tra= 300 Va = 2 0 0
VL~iJiiIIv~VVVV cr~A~A
x
t
SUHARTONO ET AL. ON SHORT CRACK GROWTH
331
FIG. 8--Areas on specimen examined with an optical microscope.
Figure 9 shows the number of Stage I cracks, counted with respect to their direction. The angle of orientation is defined in relation to the specimen axis. An angle of 0 ~ represents a plane perpendicular to the specimen axis. In addition to the crack density, the variation of normal and shear stress amplitudes acting on the crack plane are also plotted with solid and dashed lines, respectively. The directions of the microcracks are distributed over all orientations, but the maximum crack density appears in the direction of maximum shear planes in all loading cases, except for out-of-phase loading. From the experiment [9], it can be observed that under torsion loading most microcracks are oriented in the 0 ~ and 90 ~ directions. Under tension loading the microcrack distribution is represented by a sinusoidal function with a maximum density under 45 ~ In the case of multiaxial in-phase loading, the microcrack orientation likewise shows an accumulation of cracks near the direction of the plane of maximum shear. A multiaxial load case with a 90 ~ phase shift between tension-compression and torsion generates a continuously revolving vector of the maximum shear stress. As a result, the same density of microcracks in all directions can be found in the simulation. In contrast, the experiment yields an irregular distribution of microcracks. It is assumed that this effect is caused by
332
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 9--Comparison of microcrack density and orientation between experiment (AISI 1015) [9] and simulation.
SUHARTONO ET AL. ON SHORT CRACK GROWTH
333
FIG. lO--Surface microcracks of torsion loading (Steel A1S11015, N: 3500) [20] and microcrack simulation.
anisotropy of the material structure. The anisotropy might be caused by the rolling process. As a result, there are preferred directions of microcrack initiation and growth. This effect becomes clear with phase-shifted loading, where an equally distributed crack orientation is expected. Besides the tests with simultaneous tension and torsion loading, tests with sequentially applied loading were performed. With interchanging tension-torsion loading, the tension loading generates a maximum shear stress at angular intervals of -+45~ with respect to the specimen axis, whereas with torsion loading the maximum shear stress occurs in the 0 ~ and 90 ~ directions. The simulation indicates a microcrack orientation that is uniformly distributed, for the most part, whereas the experiment yields similar results for the crack distribution, as for the tests with phase-shifted loading. Examples of the microcrack simulation and the experimentally determined cracking behavior of steel AISI 1015 [9] are presented in Figs. l0 to 15. In Figs. l0 and l l , the crack pattern is shown for uniaxial tension-compression and torsion load, respectively. The cracks grow in the direction of the maximum shear stress. Crack growth with multiaxial loading is shown in Figs. 12 and 13. In comparison with in-phase loading, Fig. 12, a much higher crack density can be found with out-of-phase loading, Fig. 13, because of the revolving shear stress direction. Otherwise, the maximum crack length with in-phase loading exceeds the maximum crack length with out-of-phase loading.
FIG. 11--Surface microcracks of tension-compression loading (Steel AISI 1015, N: 1000) [20] and microcrack simulation.
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MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 12--Surface microcracks of in-phase loading (Steel AISI 1015, N: 1000) [20] and microcrack simulation.
FIG. 13--Surface microcracks of out-of-phase loading (Steel AISI 1015, N: 1000) [20] and microcrack simulation.
FIG. 14--Surface microcracks of interchanging tension-compression and torsion loading (Steel AIS11015, N: 3000) [20] and microcrack simulation.
SUHARTONO ET AL. ON SHORT CRACK GROWTH
335
FIG. 15--Surface microcracks of consecutive tension-compression and torsion loading (Steel AISl 1015, N: 3000) [20] and microcrack simulation.
Experimental Results with Aluminum In Figs. 16 and 17, the experimentally determined density and orientation of microcracks are shown for aluminum A1MgSil. The tests were performed with tension-compressionand torsion loading. Distinctions were found in the microcrack behavior of steel and A1MgSil under tension-compression and torsion loading. In accordance with Miller [18], there are two basic types of crack extension in ductile metals. The cracks occur and grow on the plane of maximum shear strain. As dictated by the ratio of the principal strains, the maximum shear strain can follow two directions. The maximum shear is parallel to the surface if the intermediate principal strain 62 appears perpendicular to the surface, whereas the orientation of the maximum shear strain is in the depth direction in the case of an intermediate principal strain e2 parallel to the surface. In correspondence with the stress state and the material properties, two types of crack systems can be defined: surface cracks (Type A) and cracks that grow away from the surface into the material (Type B). The microcracks in the steel AISI 1015 are shear cracks
FIG. 16---Relation between density and orientation of microcracks of AlSiMg l under tension-compression loading (N = 30).
336
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 17--Relation between density and orientation of microcracks of AlSiMg l under torsion loading (N = 900).
whose direction is parallel to the surface plane (Case A type of crack [18], while the microcracks observed in the A1MgSil specimens are either parallel to the surface plane (Case A) or propagate away from the surface into the interior plane (Case B type of crack [18]). In addition to the optical micrographs, the surface of some fatigue specimens have been examined with the scanning electron microscope in the area of the failure on the specimen surface. From the experimental results, the fatigue crack nucleation after torsion loading was found to be situated around the Si particle (Fig. 18a). The hole where the crack nucleation takes place in the scanning electron micrograph is the site of the Si particle that was removed during the specimen preparation. It has been pointed out by some researchers that retardation of the growth rate was observed when the crack approaches microstructural barriers [1,19]; the same phenomena after in-phase tension compressiontorsion loading tension are shown in Fig. 18b.
FIG. 18--(a) Scanning electron microscope picture of A1MgSi l fatigue specimen surface after torsion loading showing the site of crack initiation; (b) after in-phase tension compression-torsion loading showing the arresting of microcrack at grain boundary.
SUHARTONO ET AL. ON SHORT CRACK GROWTH
337
FIG. 19--The behavior of microcrack under tension-compression loading.
In Fig. 19, the cracking behavior is shown for A1MgSil under tension-compression loading as observed in the experiment. Figure 19 shows that the microcrack can grow within the grain until the crack tip is close to the first microstructural barrier, for example, a grain boundary. If the orientation of the next grain is slightly different from that of the first grain, the microcrack will be deflected, or it will follow the easiest path, for example, along the grain boundary. However, if the orientation of the next grain is very different from that of the first grain, the microcrack will be arrested. The direction of the microcrack growth which is parallel with the shear stress is indicated in Fig. 20. The crack coalescence can be seen in Fig. 21. The phenomena demonstrated in the experiment can be successfully simulated in the model (see Figs. 3 and 4).
FIG. 20--Microcrack growth under tension-compression loading (the microcrack grows in the slip plane which is parallel to the shear stress).
338
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 21--Behavior of microcracks under tension-compression loading showing coalescence among the microcracks.
Conclusions A simple microcrack growth simulation model is presented. It is based on fundamental assumptions and takes into account the rate and direction of microcrack growth, the interaction between the crack and the material barrier, as well as the crack coalescence. 1. The few simple assumptions lead to qualitatively good agreement with experimentally determined results on steel AISI 1015 and aluminum A1MgSi 1. 2. Material anisotropy, which is not taken into account in the model, causes a discrepancy between simulation and experiment under multiaxial out-of-phase loading. 3. The simulation indicates a higher crack density under out-of-phase loading, while the maximum crack length is greater under in-phase loading. 4. A two-dimensional model appears to be insufficient for materials with pronounced Type B crack behavior like A1MgSil. The development of the program is focused on a simulation with different grain sizes and the consideration of the material texture. Furthermore, a 3-dimensional simulation could be suggested.
Acknowledgments The authors would like to thank the Deutsche Forschung Gesellschaft for the financial support of the research program, and Mr. Suhartono wishes to thank the Deutsche Akademische Austauschdienst for a scholarship.
References [1] Hobson, P. D., Brown, M. W., de los Rios, E. R., "Two Phases of Short Crack Growth in Medium Carbon Steel," The Behaviour of Short Fatigue Cracks, EGF Pub. 1, K. J. Miller and E. R. de los Rios, Eds., London, 1986, pp. 441-459. [2] Bannantine, J. A., "Observation of Tension and Torsion Fatigue Cracking Behaviour and the Effect on Multiaxial Damage Correlation," University of Illinois UILN-Eng. 86-3605, Rep. No. 128, 1986. [3] Doquet, V. and Pineau, A., "Multiaxial Low-Cycle Fatigue Behaviour of a Mild Steel," Fatigue UnderBiaxial and Multiaxial Loading, ESIS 10, K. Kussmaul, D. McDiarmid, and D. Socie, Eds., London, 1991, pp. 81-101. [4] Prochotta, J., "Verhalten kurzer Risse in Stiihlen bei biaxialen Betriebsbelastungen," Dissertation RWTH Aachen, 1991.
SUHARTONO ET AL. ON SHORT CRACK GROWTH
339
[5] Vormwald, M., "Anrif~lebensdauervorhersage auf der Basis der Schwingbruclamechanik ftir kurze Risse," Publication of Instituts flit Stahlbau und Werkstoffmechanik, Book 47, TH Darmstadt, Darmstadt, 1989. [6] Miller, K. J., "Metal Fatigue--Past, Current and Future," Proceedings of the Institution of Mechanical Engineers, Preprint No. 3, 1991. [7] Bomas H., Lohrmann, M., Ltwisch, G., and Mayr, P., "Riflbildung und -ausbreitung im Stahl 1.3964 unter mehrachsiger Schwingbeanspruchung," Report for 25. Symposium of DVM-AK Bruchvorgange, 1993, pp. 75-84. [8] Brown, M. W., Miller, K. J., Fernando, U. S., Yates, J. R., and Suker, D. K., "Aspect of Multiaxial Fatigue Crack Propagation," Proceedings, 4th International Conference on Biaxial/Multiaxial Fatigue, ESIS, Paris, France, Vol. I, 1994, pp. 3-16. [9] Hug, J., Einflul3 der Mehrachsigeit auf die Sch~idigung bei schwingender Beanspruchung. Dissertation Technische Universitaet Clausthal, Germany, 1994. [10] Ltiwisch, G., Bomas, H., and Mayr, P., "Fatigue Crack Initiation and Propagation in Ductile Steels Under Multiaxial Loading," Proceedings, 4th International Conference on Biaxial/Multiaxial Fatigue, ESIS, Paris, France, Vol. II, 1994, pp. 27-42. [11] McDowell, D. L. and Poindexter, V., "Multiaxial Fatigue Modeling Based on Microcrack Propagation: Stress State and Amplitude Effects," Proceedings, 4th International Conference on Biaxial/Multiaxial Fatigue, ESIS, Paris, France, Vol. I, 1994, pp. 115-130. [12] James, M. N. and de los Rios, E. R., "Variable Amplitude Loading of Small Fatigue Cracks in 626-T6 Aluminum Alloy," Fatigue Fracture Engineering Materials Structures Vol. 19, No. 4, 1994, pp. 413-426. [13] Socie, D. and Furman, S., "Fatigue Damage Simulation Models for Multiaxial Loading," Fatigue 96, 6th International Fatigue Congress, G. Ltitjering and H. Nowack, Eds., Berlin, Germany, 1996, pp. 967-976. [14] Hoshide T. and Socie, D. F., "Crack Nucleation and Growth Modeling in Biaxial Fatigue," Engineering Fracture Mechanics, Vol. 29, No. 3, 1988, pp. 287-299. [15] Argence, D., Weiss, J. and Pineu, A., "Observation and Modeling of Transgranular and Intergranular Multiaxial Low Cycle Fatigue Damage of Austenitic Stainless Steels," Proceedings, 4th International Conference on Biaxial/Multiaxial Fatigue, ESIS, Paris, France, Vol. I, 1994, pp. 309-322. [16] Hoshide, T. and Kusuura, K., "Life Prediction by Simulation of Crack Growth in Notched Components with Different Microstructures and under Multiaxial Fatigue," Fatigue Fracture Engineering Materials Structures, Vol. 21, 1998, pp. 201-213. [17] Taylor, D. and Knott, J. F., "Fatigue Crack Propagation Behaviour of Short Cracks: The Effect of Microstructure," Fatigue Fracture Engineering Materials Structures, Vol. 4, 1981, p. 147. [18] Miller, K. J., "Multiaxial Fatigue: A Review," DVM-Final Colloqium: Sch~idigungsfriihererkennung und Schadensablauf bei metallischen Bauteilen, Darmstadt, 1993. [19] Miller, K. J., "Initiation and Growth Rates of Short Fatigue Cracks," Fundamentals of Deformation and Fracture, B. A. Bilby, K. J. Miller, and J. R. Willis, Eds., Sheffield, 1984. [20] Schram, A. and Liu, J., "Einflul3 der Mehrachsigkeit auf die Sch~idigung bei Schwingender Beanspruchung," Final Report Vol II, DFG - Forschungsvorhaben Ze 248/4, 1993.
Nobuhiro Isobe 1 and Shigeo Sakurai 2
Micro-Crack Growth Modes and Their Propagation Rate under Multiaxial LowCycle Fatigue at High Temperature REFERENCE: Isobe, N. and Sakurai, S., "Micro-Crack Growth Modes and Their Propagation Rate under Multiaxial Low-Cycle Fatigue at High Temperature," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTMSTP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 340--352. ABSTRACT: Crack growth behavior and propagation rates in SUS316L steel, 1CrMoV steel, and Hastelloy-X in multiaxial low-cycle fatigue tests at high temperature were investigated. Crack growth mode in pure torsion tests changed from the principal strain plane to the maximum shear plane with increasing temperature or strain range. The dominant mechanism of micro-crack initiation, however, differed among the three materials. The dominant process of micro-cracking was considered to be slip in grains for SUS316L steel, whereas oxidation film cracking and grain boundary cracking were the mechanisms for 1CrMoV steel and Hastelloy-X, respectively. Tile appropriate parameter for evaluating the crack growth rate under multiaxial conditions is discussed in relation to the micro-crack growth mechanisms. Good correlation was obtained between the crack growth rate and the strain parameter based on the micro-cracking mechanism of each material. KEYWORDS: multiaxial low-cycle fatigue, micro-crack, crack propagation, principal strain, shear plane, crack growth rate
Most high-temperature components are subjected to multiaxial loading rather than uniaxial loading. Equivalent stresses or strains with safety margins are used to design components. For example, von Mises' equivalent strain range has been used to evaluate the fatigue damage in the ASME Code Section I I I - - N H [I]. Some parameters based on a strength model or a physical theory have been proposed [2~/], but most of these parameters need experimental constants. These constants are usually derived from failure data for specimens, not necessarily based on the m e c h a n i s m of damage. In high-temperature low-cycle fatigue, micro-cracks initiate in the early stage of life and their propagation dominates the life [5]. Therefore, it is important for the life assessment of hot components to estimate the effect of multiaxial states on micro-crack growth. In this study, we investigated micro-crack growth behaviors in an austenitic stainless steel, a lowalloy steel, and a nickel-base superalloy under combined tensile and torsional loading at high temperature. Crack growth modes and the evaluation of crack propagation rates are discussed based on the micro-crack growth mechanism of each material in order to improve the life assessment of hightemperature components.
1 Researcher, Hitachi, Ltd., Mechanical Engineering Research Laboratory, 3-1-1, Saiwai, Hitachi Ibaraki, 3178511 Japan. 2 Senior researcher, Hitachi, Ltd., Mechanical Engineering Research Laboratory, 3-1-1, Saiwai, Hitachi Ibaraki, 317-8511 Japan.
Copyright9
by ASTM International
34O www.astm.org
ISOBE AND SAKURAI ON MICRO-CRACKGROWTH MODES
341
TABLE 1--Chemical compositions of tested materials (wt%). (a) 316L Stainless Steel C
Si
Mn
P
S
Ni
Cr
Mo
0.020
0.58
0.80
0.028
0.006
12.23
17.47
2.23
(b) 1CrMoV Steel C
Si
Mn
P
S
Ni
Cr
Mo
V
Sb
Sn
0.28
0.24
0.77
0.010
0.007
0.38
1.08
1.23
0.25
0.0016
0.0013
(c) Hastelloy-X C
Si
Mn
P
S
Ni
Cr
Co
Mo
W
Fe
0.06
0.40
0.69
0.013
0.001
Bal.
21.7
1.0
8.9
0.50
17.6
Experimental Procedure Three structural alloys, 316L stainless steel, 1CrMoV steel, and Hastelloy-X, were tested. Their chemical compositions are listed in Table 1. 1CrMoV steel was obtained from the lower temperature zone of an actual steam turbine rotor after 80,000 h of service [6]. SUS316L steel and Hastelloy-X were solution-treated at 1100 and 1150~ respectively. Their mechanical properties are listed in Table 2. Figure 1 shows the shape and dimensions of the specimens. Hollow cylindrical specimens with an outer diameter of 22 m m and an inner diameter of 18 m m in the gage portion were employed. A servo-hydraulic, axial-torsional machine, of 245-kN axial load capacity and 2.8 kN.m torque capacity, was used for the strain-controlled multiaxial fatigue tests. The extensometer was an axial type with conical-point quartz extension rods that can control and measure both the axial and shear strain independently. The gage length of the extensometer was 25 ram. Strain-controlled multiaxial fatigue tests with combined axial and torsional loading were carried out at several yon Mises strain ranges. Strain waves were in-phase with fully reversed triangular waves at a strain rate of 0.1%/s. The prin cipal strain ratios ~b( = e3/el) employed were - 1 (pure torsion), . - 0 64 (combined, e = i v y ) , and - 0 . 5 0 (axial only). Specimens were heated by induction heating. Tests were performed at room temperature, 550 and 650~ for SUS316L steel, 550~ for 1CrMoV steel, and 700~ for Hastelloy-X. The failure of the specimen was defined as the number of cycles at 25% reduction of the stress range from the saturated value. Cellulose acetate film replicas were employed to observe fatigue crack
TABLE 2--Mechanicalproperties of tested materials.
SUS316L 1CrMoV Hastelloy-X
0"0.2,
O'B,
Elongation,
Temperature
MPa ('0"o.o2)
MPa
%
RT 550~ 550~ RT
243 133 355" 382
562 339 521 764
61.7 45.8 20.0 52.1
342
MULTIAXIAL FATIGUE AND DEFORMATION
.Y f
85 ~1~
60 230
85
I
b
A
FIG. 1--Shape and dimensions of specimen (mm).
growth with occasional interruptions of the test. Surface replication had been conducted without removing the specimen and extensometer from test apparatus. The surface of the specimen was not polished before replication. Test Results and Discussion
Crack Growth Mode in Multiaxial States 316L Stainless Steel--Combined tensile and torsional fatigue tests of 316L were performed with a von Mises strain range of 0.7% at three temperature conditions. Figure 2 shows the cracks observed in the tests. These photographs were taken when the cracks were about 2 mm long. The angles between the principal strain plane and the specimen axis were 90.0, 65.4, and +45.0 for conditions of d? = 0.50, -0.64, and - 1.00, respectively. Cracks almost all propagated along the principal plane in the room temperature test. In the 4) = -0.50 and -0.64 tests, cracks propagated in tangential direction to the specimen and their paths became wavy with increasing temperature. In the pure torsion test (~b = - 1.00), crack growth behaviors changed distinctively with temperature. The crack growth path changed from the principal strain plane to the maximum shear plane with increasing temperature. In the room temperature test, many micro-cracks were observed and the main crack was formed by the linking of these micro-cracks. Macroscopically, the cracks propagated on the principal plane. At 550~ cracks initiated in the axial direction that corresponded to the maximum shear plane and branched when they grew to about 0.2 mm. Cracks propagated on the principal plane after branching and X-like cracks were observed when specimens failed. On the other hand, shear plane cracking was observed in the pure torsion test at 650~ Figure 3 shows the relationship between test temperature and failure life in the pure torsion test. In the pure torsion fatigue test of 304 and 316 stainless steels at room temperature, a variation of the crack growth path from the principal plane to the shear plane with increasing strain range was described by Bannantine and Socie [7]. Temperature rise has the same effect on crack growth path as increase in the strain range. Even for the same total strain range, the inelastic strain range usually increases with increasing temperature. The variation of crack growth path is affected by the amount of inelastic strain. Furthermore, the shear plane cracking in the pure torsion test is caused by the normal strain not acting on the maximum shear plane. The main crack in the pure torsion test at room temperature propagated along the principal strain plane, but it was formed by coalescence of micro-cracks that initiated in various directions. The orientations of individual grains may affect the directions of the micro-cracks. Therefore, slip is considered to be the dominant mechanism of micro-crack growth in SUS316L. The effect of normal strain on the maximum shear plane is also important in conditions other than ~b = -1.00. The appropriate parameter for evaluating crack growth will be one based on the maximum shear strain and the normal strain acting on the shear plane.
343
ISOBE AND SAKURAI ON MICRO-CRACK GROW-rH MODES
%
.2
344
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 3--Three regions of cracking behavior observed in pure torsion tests of 316L steel.
1CrMoV Steel--Figure 4 shows the crack growth behavior observed in the pure torsion tests of 1CrMoV steel. A small hole approximately 0.2 mm in diameter was made in the center of the gage section to facilitate crack observation in the tests of 1CrMoV steel. The same tendency as for SUS316L steel, in which the crack growth path changed from the principal plane to the shear plane with increasing strain range, was observed. In the medium strain range (Aeeq = 0.7%), however, cracks initiated on the principal plane and changed direction to the shear plane as they propagated. In the combined axial and torsional loading tests, cracks propagated perpendicular to the specimen axis in the higher strain range, whereas a change in crack path from the principal plane to the direction of tangential to the specimen was observed in the lower strain range, as shown in Fig. 5. We also observed micro-cracks distributed in bands running along both the specimen axis and its transverse direction, especially in the higher strain range. These micro-cracks had X-like shapes and the main cracks propagated among these cracks except in the case of the lower strain range tests. These micro-cracks were caused by the surface oxide film cracking. At high strains, a large plastic deformation area will be formed along the maximum shear plane, so the oxide film cracks should be distributed along the specimen axis and its transverse direction. Hastelloy-X--The crack growth path in the pure torsion test of Hastelloy-X also changed with strain range, as shown in Fig. 6. The main crack in the pure torsion test of 1.0% von Mises strain range propagated along the specimen axis. Grain boundary cracking, however, was observed microscopically and many micro-cracks were initiated on the grain boundary with an angle of 45 ~ to the specimen axis. The normal strain acting on the grain boundary produces these micro-cracks. At higher strains, a large deformation area will be formed along the axial direction and the coalescence of the micro-cracks initiated in this area formed the macroscopic shear cracking, as in 1CrMoV steel. The dominant mechanism of surface micro-crack initiation for Hastelloy-X is grain boundary cracking. Grain boundary sliding, which is a distinctive phenomenon in creep, did not occur in these
ISOBE AND SAKURAI ON MICRO-CRACK GROWTH MODES
345
FIG. 4--Cracking behavior in pure torsion test of 1CrMoV steel
tests since the strain rate was not so slow and grain boundary cracks were observed only around the surface as shown in Fig. 7. The embrittlement of grain boundary by oxidation affected this grain boundary cracking at the surface, since the coalescence of grain boundary micro-cracks produced macro-cracks. Therefore, crack growth rates in Hastelloy-X will show good correlation with the principal strain even in tests in which macroscopic shear cracking was observed.
346
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 5--Cracking behavior in combined tension and torsion test of 1 CrMoV steel.
Crack Growth Rate in Multiaxial Tests Figure 8 shows the relationship between the maximum surface crack length and life fraction. Closed symbols indicate that the cracks propagated along the shear plane macroscopically. Crack lengths were almost proportional to the life fraction on a semi-logarithm graph except for the result for SUS316L at 650~ and ~b = -1.00, in which crack growths on the maximum shear plane were observed. In that test, crack growth rates were reduced when the crack grew to about 3 mm. In these conditions, cracks were observed to propagate on the tangential direction as well as the axial direction, as shown in Fig. 9, and their interference reduced the crack growth rate. For 1CrMoV steel and Hastelloy-X, cracks propagated almost in proportion to the life fraction even in the test in which the crack grew on the maximum shear plane. The mechanisms of micro-crack initiation were different in the three materials. We consider them to be slip for SUS316L steel, oxidation film cracking for 1CrMoV, and grain boundary cracking for Hastelloy-X. These differences in micro-cracking affect the crack growth behaviors. Micro-crack growths are observed from the early stage of life in a high-temperature low-cycle fatigue test, and it is important to identify the dominant mechanism of micro-cracking in order to predict the life of hot components. The appropriate parameter for evaluating crack growth
ISOBE AND SAKURAI ON MICRO-CRACK GROWTH MODES
347
FIG. 6~Cracking behavior in pure torsion test of Hastelloy-X.
for SUS316L steel is one based on the maximum shear strain and the normal stress acting on the shear plane because the slip is considered to be the mechanism of micro-cracking. On the other hand, the principal strain is the appropriate parameter for 1CrMoV steel and Hastelloy-X. Next, the crack growth rate evaluation is discussed based on the observation results mentioned above. The principal strain or stress is often used to evaluate crack growth rates because the crack opening mode is considered to dominate the crack growth. Figure 10 shows the relationship between the normalized crack growth rate and the principal strain range for three materials. It can be seen that the principal strain is not the appropriate parameter for estimating the crack growth rate for SUS316L steel, whereas good correlation was obtained for 1CrMoV steel and Hastelloy-X.
348
MULTIAXIALFATIGUE AND DEFORMATION
FIG. 7--Cracking behavior of Hastelloy-X in cross section of specimen. Lohr and Ellison proposed a strain parameter called the/"*-plane parameter to correlate the multiaxial fatigue life [3]. It is described using the maximum shear strain and the normal strain on the maximum shear plane that intersects the free surface at an angle of 45 ~ as follows, A ~-+Ke
=C
(1)
ISOBE AND SAKURAI ON MICRO-CRACK GROWTH MODES
'
I
SUS316L, {=-
1s
d
I
.50
I
I
E E
O-
04
....
1
" Y
O 0 . 1 [3t3
..-''""
..-"
0
0 RT,0.7~
zx 550~ [] 650~
...--" I.
-"
0.0
=
02
I
=
I
1.0
SUS316L, OF-0.50
I
.~
I
SUS316L, ~= .64
E 1(
=
0.4 0.6 0.8 Life fraction N/Nf
(a)-I '
I
I
I
E 0 Oq
f-
_m 0
,m 0 O, II~~
.Z>~'_ v
[] 650~C,0.7%
.-'6" " v
~..-6;,
~,
r,-- YV
'[
0.0
v 55~'c,1.0%
.
i
I
0.2
.
i
0.4
I
0
65o~c,l.o~
=
I
0.6
0.8
1.0
Life fraction N/Nf
(a)-2 SUS316L, d~=-0.64 '
E
"~ I-
c
F
9
.~-'v
~
'-' n tl. - ~
~
...-'"
~" S / ~ o q . . - -
I t , / ~
'
0.0
'
&
0.2
.-i
.-'"
O
~
."
v 550~
./-:'" I
'
,~ 550oc,0.7O/o 9 659~ , e5o~1.o%
?--'~'"
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~-'"~
'
_v A. v
9
I-.--""-~/o ,'x
'
I
0.4
~
I
RT,0.7%
~
0.6
t
0.8
i
1.0
Life fraction N/N f
(a)-3
SUS316L, d~=-l.00
FIG. 8--Relationship between crack length and life fraction.
349
350
MULTIAXIAL FATIGUE AND DEFORMATION '
I
I
I
I
1( I CrMoV,550*C
,
~ ~.v~Z~
E E
y .v-""y
d t-
0.
i~,~ 7
O
.-""
9 .... e"" _.~-"
.--"
0.'
0.05 0.0
,
,
l
~---0.50,0.4%
0 o & v
0.2
I
,
I
0.4
$=-0.50,0.7% $=-0.64,0.4% Jt=-0.64,0.7% ,
I
0.6
0.8
Life fraction N/Nf (b)-! i
'
]CrMoV, ~=-0.50,-0.64 I
I
I
I
lCrMoV,55s176
E E r
O
t-"
0.
L)
0,"
[] 9 9
0.01 0.0
,
I
,
0.2
I
0.4
,
I
,
0.6
I
0.4% 0.7% 1.0% ,
0.8
1.0
Life fraction N/Nf (b)-2 1CrMoV,q~-1.00 1C 9 H a s t e l l o y - X , 7 0 0 ~
s
z~
1 0.~
0.1 0.0~
0.2
0.4
0.6
Life fraction N/Nf
(c) Hastelloy-X FIG. 8---(Continued)
0.8
1.0
ISOBEAND SAKURAION MICRO-CRACKGROWTHMODES
351
FIG. 9--Surface cracks observed in pure torsion fatigue test of 316L steel at 650~ where y* = maximum shear strain on the maximum shear plane that intersects the free surface at an angle of 45~ e* = normal strain on the maximum shear plane that intersects the free surface at an angle of 45 ~ K = constant having a value of 0.2 for stainless steel and CrMoV steel. The appropriate parameter for SUS316L steel is one based on the maximum shear strain and the normal stress acting on the shear plane. The relationship between the normalized crack growth rate and t h e / ' * - p l a n e parameter for SUS316L steel is shown in Fig. 11. Good correlation was obtained for S US316L steel at three temperatures. Therefore, strain parameters for evaluating the crack growth rate of materials under multiaxial states should be selected based on the mechanism of microcracking.
-~
10
SUS316L' eRT 9 550~
i ....
'
ICrMoV
,,/~
o.c
9
' 7
I
,,'~,,,
i I
# t,, 99
it
i
ii I
e ~~
s~
10" et
r
10"~
tt
9
10" Factorof ~
"r0
"
/ ~ /"
? 9
o z
'
10~
it/, ,~
epI
9
b ~ 1~
E
I
9
2O~
|
'
H~l~,oy-•
' A II~V~V
i I
10
I . . . . I 10 0.5 1.0 0.5 1.0 Maximum principal strain range ~ 1 . % 9
I
. . . .
i
FIG. lO---Relationship between normalized crack growth rate and principal strain range.
352
MULTIAXlAL FATIGUE AND DEFORMATION
" • ~
lff
I
SUS316L
, iS
,," ,,"
.~Z
~
l(T
,'" ,," ,,',
E..~.
O,c-
:~ :~0 10
e RT 9 550~ o 650~ 015 .... 11,0
F*-plane parameter A(~'*t2+0.2~Zn*) FIG. 11--Relationship between normalized crack growth rate and I'*-plane parameter in 316L
steel. Conclusions Multiaxial fatigue tests on 316L stainless steel, 1CrMoV steel, and Hastelloy-X at high temperature were carried out and the mechanism and growth rate of micro-cracks were discussed to improve the life assessment of high-temperature components. As a result of the present investigation, the following conclusions were obtained: 1. The crack growth mode in pure torsion tests changed from the principal strain plane to the maximum shear plane with increasing temperature or strain range. 2. The dominant mechanism of micro-crack initiation differed among the three materials. The dominant process of micro-cracking was slip in grains for SUS316L steel, oxidation film cracking for 1CrMoV steel, and grain boundary cracking for Hastelloy-X. 3. The crack growth rates in multiaxial states were correlated with the strain parameter which is related to the mechanism of micro-cracking. T h e / ' * - p l a n e parameter is appropriate for SUS316L steel and the principal strain is appropriate for ICrMoV steel and Hastelloy-X.
References [1] ASME, "Boiler and Pressure Vessel Code, Section IIl, Division 1--Subsection NH," 1995. [2] Kandil, F. A., Brown, M. W., and Miller, K. J., "Biaxial Low-Cycle Fatigue Fracture of 316 Stainless Steel at Elevated Temperatures," Mech. Behav. Nucl. Appl. Stainless Steel Elevated Temperature, 1982, pp. 203-209. [3] Lohr, R. D. and Ellison, E. G., "A Simple Theory for Low Cycle Multiaxial Fatigue," Fatigue Fract. Eng. Mater. Struct., Vol. 3, No. 1, 1980, pp. 1-17. [4] Hamada, N., Sakane, M., and Ohnami, M., "Creep-Fatigue Studies Under a Biaxial Stress State at Elevated Temperature," Fatigue Fract. Eng. Mater. Struct., Vol. 7, No. 2, 1984, pp. 85-96. [5] Sakurai, S., Usami, S., and Miyata, H., "Micro-Crack Initiation and Growth Behavior Under Creep-Fatigue in a Plain Specimen of Degraded CrMoV Cast Steel," JSME (Japan Society of Mechanical Engineers), Int. J., Vol. 30, 1987, pp. 1732-1740. [6] Sakurai, S., Fukuda, Y., Isobe, N., and Kaneko, R., "Micro-Crack Growth and Life Prediction of a 1CrMoV Steel Under Axial-Torsional Low Cycle Fatigue at 550~ '' Fatigue Fract. Eng. Mater. Struct., Vol. 17, No. 11, 1994, pp. 1271-1279. [7] Bannantine, J. A. and Socie, D. F., "Observations of Cracking Behavior in Tension and Torsion Low Cycle Fatigue," Low Cycle Fatigue, ASTM STP 942, 1988, pp. 899-921.
Multiaxial Experimental Techniques
Raymond D. Lohr 1
System Design for Multiaxial High-Strain Fatigue Testing REFERENCE: Lohr, R. D., "System Design for Multiaxial High-Strain Fatigue Testing" Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kallud and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 355-368.
ABSTRACT: System design starts with the specimen and the forces that need to be applied to simulate service conditions by creating specified combinations of multiaxial stress or strain. Based on this premise a semigraphical tabular presentation is used to review the technical and cost-related advantages, and disadvantages, of alternative design schemes. References to real-world multiaxial test systems illustrate progress to the present day. This paper sets out to review salient features that have marked out progress in the design of multiaxial systems over the past 50 years. It also aims to provide foresight in the specification of future equipment and to improve the understanding of those factors, other than intrinsic material behavior, which influence the characterization of high-strain multiaxial fatigue life. KEYWORDS: multiaxial, torsion, internal pressure, external pressure, biaxial strain ratio, cruciform, thin-walled tube, modal control, low-cycle fatigue, LCF, crack propagation, thermomechanical fatigue, TMF
Nomenclature q5 Biaxial surface principal strain ratio = e2/el where el > e2 v Poisson's ratio The development of high-strain multiaxial fatigue testing has been driven by the inability of uniaxial data plus classical yield criteria to adequately predict in-service fatigue failures. Many engineering components are subject to rotational and pressure loading and the high biaxial strain fields induced by thermal transients during startup and rapid shutdown of a steam and gas turbine power plant. The design of materials testing systems for multiaxial high-strain fatigue investigations can be traced back to 1950. Early work revealed the need to test single-geometry specimens over the full range of strain ratios. Two strands of machine designs developed: those reflecting the need for testing sheet or plate material, and those where material form was not a restriction. The latter enabled the design of systems capable of controlling and monitoring all biaxial stresses, strains, and their plastic components. The need for better simulation of components in-service has led to the provision of outof-phase loading, modal control, and thermomechanical fatigue. Digital control and signal processing techniques now supply high-resolution closed-loop control, data acquisition, analysis, and archiving of results. This paper provides a chronological review of multiaxial system design, noting key developments, and presents a semi-analytical tool for the specification of future equipment. It also addresses the choice of failure criteria, which are often predicated by the design of the testing machine and specimen, and can significantly skew the analysis of results. 1 Corporate research director, Instron Corporation, High Wycombe, HPI2 3SY, UK. 355
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Design Review--1 Recognizing the danger of cyclic loading in highly stressed elements of pressure vessels, the Pressure Vessel Research Committee in 1950 instigated a program of low-cycle biaxial fatigue studies for steels. The 2:1 stress ratio encountered in thin-walled pressure vessels supporting their own end load and the equibiaxial stress state generated due to thermal stressing no doubt influenced the development of test facilities capable of achieving positive biaxial stress ratios.
Bending Cantilever bending of beams with width:thickness >5 approaches a nominal plane strain condition, th = 0, and a stress ratio approaching 2:1 for the fully plastic condition. At Lehigh University, Gross and Stout [1] adopted this approach for several pressure vessel steels using constant deflection and constant load cycling. Later, Sachs et al. [2] varied the width-to-thickness ratio achieving strain ratios - v -< ~b < O. Repeated pressurization of rectangular plate specimens was developed by Blaser et al. [3] at Babcock and Wilcox, achieving a stress ratio of 2:1. The technique was improved considerably in the 1960s by Ives et al. [4] through adopting a circular specimen, simply supported at its edges and alternately pressurizing each face to provide a fully reversed equibiaxial stress and strain field, ~b = + 1, at the center. Zamrik at Pennsylvania State University [5] introduced different specimen shapes from oval to circular providing a range of multiaxiality +0.5 --< ~ - 1. Anticlastic bending of rhombic plates, by applying equal and opposite point loads at adjacent corners, generates a relatively large region over which a given strain field is produced. By varying the ratio of the diagonals, strain ratios - 1 ~ ~b -< 0.5 can be generated. First reported by Zamrik [6] in the late 1960s, an improved system was described more recently by Zamrik and Davis [7]. The above bending techniques are all relevant to plate material and there are numerous references in the literature for studies of steels and light alloys. Table 1, which provides the format used throughout the paper to discuss multiaxial systems, shows that each bending technique is generally limited to a relatively small range of biaxiality. Perhaps, more importantly, the specimen geometries have to be varied in order to change the strain ratio. Benefits are the rig simplicity, specimen resistance to buckling, and in the case of anticlastic bending, an essentially constant strain field over a significant area which, together with ease of observation, still make it a useful technique for crack growth studies.
Torsion Pure torsion provides a simple technique for generating shear strain, 4) = - 1, albeit with the principal axes inclined at 45 ~ to the specimen axes. However the analysis of data from solid specimens is complicated by the reduction in strain with decreasing radius, especially in plastic cycling when the surface layers yield first. In the early 1960s, Halford and Morrow [8] published LCF data for torsion of thin-walled tubular specimens. The removal of core material and a mean diameter to wall thickness ratio of 10 or higher enable a much simpler stress/strain system to be realized. Subsequently, Miller and Chandler [9] demonstrated a progressive reduction in torsional fatigue life with reducing wall thickness which they attributed to the removal of elastic constraint. This emphasizes the need to maintain constant geometry in multiaxial studies if the understanding of constitutive laws is to be achieved.
Axial + Torsion At Tohoku, in 1965, Yokobori et al. [10] reported torsional and uniaxial LCF data, derived from separate machines but with identical gage length geometry. At Kyoto, Taira et al. [11] performed
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combined axial + torsional fatigue at ambient and elevated temperature for in-phase, and subsequently, out-of phase cycling. From the 1970s onwards the axial + torsion system provided by closed loop servohydraulic testing machines has been widely used. As Table 2 shows, while there are numerous benefits, the achievable range of surface strains is restricted to - 1 -< ~b -< - ~,, i.e., no positive strain ratios or even plane strain can be realized. In addition, buckling, a problem in many LCF specimens, can be present both axially and torsionally. It is also important to consider the effect of rotation of principal strain axes (as the applied strains change from pure axial to pure torsion) when testing anisotropic materials. Axial 4- Torsion 4- Internal Pressure Results at elevated temperature for creep fatigue tests employing cyclic axial + constant internal pressure and separately cyclic torsion were reported in 1963 by Kennedy [12] at Oak Ridge. Subsequently Crosby et al. [13] utilized combined cyclic axial and internal pressure, although plastic buckling instability was a reported problem. Commercial biaxial fatigue systems providing axial + internal pressure at elevated temperature and axial + torsion + constant internal pressure at ambient were reported by the author previously [14]. Reference to Table 2 shows that the full range of strain ratios is achievable with this triple loading configuration, however, because it is not possible to apply negative internal pressure, tests are not possible in the lower left quadrant. The consequence of repeated internal pressure cycling, above the yield stress, is ratcheting on the circumferential axis. The cycle may become elastic after the first cycle as a result of monotonic strain hardening or somewhat later due to cyclic hardening. The conclusion is that fully reversed biaxial tests are not possible without the addition of external pressure,
Design Review--2 Earlier work had shown that a single geometry specimen capable of being loaded in two orthogonal directions, under fully reversed strain-controlled cycling, was the key to progress in high-strain multiaxial fatigue. Two distinct approaches developed, which to this day have distinct advantages and disadvantages. Thin-Walled Tube Tubular specimens have the advantage that all axial forces, pressures, and torques are fully carried by the gage length. Stresses and plastic strains can be determined during the test unambiguously to the clear benefit of studies aimed at modeling material behavior. Axial 4- Differential Pressure Previous studies had shown that adding external pressure to axial force + internal pressure would permit testing in all four principal strain quadrants. At Waterloo, Havard [15] reported in 1968 on a rig which potentially provided these facilities; however, the use of a single hydraulic supply for force and pressure required the specimen design to be varied to change the strain ratio. Closed Loop Control In Bristol, during the 1970s, Andrews and Ellison [16], followed by Lohr and Ellison [17], developed a system which achieved the design goals of a single specimen capable of fully reversed cycles for all - 1 -< 4>--< + 1 with the ability to monitor stress, strain, and plastic strain continuously on both axes (Fig. 1). By using one closed loop actuator for axial force ( • 90 kN), a second to drive an intensifier for internal pressure (• 110 MPa), and adopting a fixed external pressure adjustable up to 55 MPa so that a controllable differential pressure could be achieved, the solution was also economic.
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FIG. 1--Bristol biaxial specimen and pressure vessel.
Finite-element analysis and experimental optimization led to specimen dimensions of 25.4 mm bore, 0.8 mm wall thickness, 9.5 mm parallel length with 25.4 mm fillet radii for tests on RR58 A1 alloy and 1Cr-Mo-V steel. The wall thickness was subsequently increased to 1 mm by Shatil et al. [18] to enable testing at higher plastic strain ranges without buckling on EN15R steel. Averaging axial and diametral capacitive extensometry enabled strain control throughout all tests. The advantages of this approach are identified in Table 2 and include the potential for an elevated temperature version using inert gas as the pressurizing medium since the internal volume is small and the external volume, at constant pressure, could readily be maintained at constant temperature. However, the pressurizing system does result in the hydrostatic component varying cyclicly which may have a second-order influence on fatigue life. Axial + Differential Pressure + Torsion The final evolution in mechanical design took place during the 1980s at Sheffield by Found et al. [19] and Fernando et al. [20] where four independent closed loops provide control of axial force (-+400 kN), internal and external pressure (160 MPa) plus torsion (-+ 1 kNm). The high pressures are required to be able to plastically deform steel specimens of 16 mm bore, 2 mm wall thickness, and 20 mm parallel length with 25 mm fillet radii. In such a system, torsion provides the ability to rotate the principal applied strain axes with respect to the specimen and investigate anisotropic effects. The system is large and complex, but fully comprehensive in its ability to command not only strain ratio but also the direction of the principal strain axes. However, this approach is not suitable for high temperatures since pumping and compressing large gas volumes for external pressure would make it impossible to stabilize temperature.
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Composite Specimens At Alberta, Ellyin and Wolodko [21] recently reported on a new system for testing tubular samples with capabilities similar to the Sheffield machine. Axial forces of (_+260 kN) plus independent internal (82 MPa) and external (41 MPa) pressure plus torsion (_+2.7 kNm) are achievable. Structurally, with the actuator assembly directly bolted to the pressure vessel, the system is similar in concept to the Bristol machine. Modern PC computer control and software provide improved data acquisition and test flexibility. Of particular interest are the specimens of 38.2 mm bore, 1.4 mm wall thickness, and 102 mm parallel length manufactured from glass fiber/epoxy composite with bonded segmented aluminum tab ends. Initial hiaxial results were given for monotonic tensile behavior under stress control.
Cruciform Systems A specimen lending itself directly to biaxial testing is a cross-shaped plate, or cruciform, loaded in-plane by four orthogonal actuators. A higher tendency to buckle in compression than a tube, and a gage area that does not react all the load (some is shunted around its periphery) means that, in fatigue studies, stresses and plastic strains cannot be directly measured.
Tension In the early 1960s, at the Chance Vought Corporation, a rig was developed capable of applying tensile loads to a cruciform specimen. Initially, biaxial monotonic, and subsequently,biaxial fatigue tests were reported by McClaren and Terry [22] for equibiaxial and 2: I stress ratios on plate specimens with no reduced central section.
Open to Closed Loop Control At Cambridge, the development and application of cruciform testing spanned a period from the early 1960s to the mid-1980s. Pascoe and de Villiers [23] reported on the first practical rig based on a stiff octagonal frame carrying four 200 kN double acting actuators. Specimen development resulted in a design with central spherical recesses of 76 mm radius and a minimum thickness of 1 mm; a flat bottom was rejected despite favorable finite-element analysis because of premature failure at the fillet. Pressure limit cycling enabled tension-compression fatigue tests under 1:1 (equibiaxial), 1:-1 (pure shear), and uniaxial conditions to be performed. Strain gages enabled the total strain ranges to be measured; however, because of the ring reinforcement around the gage area, stresses and plastic strains remained indeterminate. Considerable development resulting in full servo-control of the actuators, and the ability to perform fully reversed biaxial fatigue tests at any strain ratio, was reported in 1975 by Parsons and Pascoe [24]. They found it necessary [25] to modify the specimen geometry by halving the spherical radius and doubling the wall thickness to obviate buckling at shorter lives on QT 35 ferritic and AISI 304 austenitic steels.
Specimen Development From the early 1980s, further development of the cruciform testing technique took place at Sheffield. In 1985, Brown and Miller [26] described a new specimen featuring a recessed flat-bottomed square gage area (100 mm X 100 mm • 4 ram) connected to the loading arms by sets of fingers created by slotting the arms (Fig. 2). This geometry effectively decouples the adjacent loading arms and means that the majority of the force on either axis is carried by the gage area over which the strain field is substantially uniform. The only restrictions are that high compressive forces, generating plastic specimen deformation, will result in buckling, while fatigue tests with an unnotched spec-
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FIG. 2--Sheffield cruciform specimen. imen may result in first cracks initiating from the slot roots. For crack propagation studies, however, where plasticity is essentially a crack tip phenomenon, this new geometry represented a major step forward and has been influential in subsequent research worldwide.
Center Control A particular problem with the operation of cruciform systems has been controlling to a minimum the movement of the specimen center. Such unwanted motion generates specimen side forces, and hence bending, and is problematic for dynamic crack observation. The principal cause is the 100% cross coupling between opposing actuators which results in a "fight" if each actuator is separately controlled both for deformation and center position. The complete solution was developed by McA1lister for JUTEM [27] when, for each axis, the deformation and center position control loops were made independent of each other (Fig. 3). Considering one axis in displacement control, deformation is provided by the sum of the position transducer signals, while the center position is given by their semi-difference. The same principles enable load control and strain control to be realized (Fig. 4). A further useful consequence is the ability to simultaneously apply strain control for deformation and load control for zero side force. These are examples of "modal control," when two or more actuators are each driven by more than one control loop. The end result was that for cycling in strain control at 1 Hz the center position could be held stationary to circa _+1/xm.
New Materials The JUTEM system described by Masumoto and Tanaka [28], utilizes radio frequency (RF) heating plus susceptor in vacuum to enable temperatures up to 1800~ for testing structural composites at up to - 100 kN over the full range of biaxiality under strain control with crack observation by laser scanning microscope. At NASA Lewis, Bartolotta et al. [29] reported on a + 500 kN system designed
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FIG. 3--Cruciform: modal control of deformation and center position.
DEFORMATION CONTROL
TRANSLATION CONTROL
9 Sum of LVDT readings
9 Half-Difference of LVDT readings
9 Average of LoadceU readings
9 Difference of Loadcell readings
9 Extensometer FIG. 4---Cruciform: multiple control modes.
for testing CMCs, intermetallics, and other advanced aerospace materials. Modified Sheffield type specimens, with central flat gage area 95 turn square x 2mm thick, can be heated up to 1500~ using an advanced quartz lamp radiant furnace. These, and similar systems, demonstrate the advantages of the cruciform solution for biaxial fatigue and crack growth studies of materials whose received form is sheet or thin plate (see Table 1).
Thermomechanical Fatigue High-strain thermomechanical fatigue of uniaxial specimens can trace its history back to at least to the mid-1970s, Taira [30] and Hopkins [31], However, the problems associated with the test under multiaxial conditions have only been addressed in the 1990s following the development of multiaxis digital closed loop controllers with high-speed data acquisition and software enabling flexibility in test design and data analysis.
Axial-Torsion The system initially selected for TMF studies has been the thin-walled tube under axial + torsional loading. This provides deterministic stress/strain relationships along with the practical benefits of "relatively simple to mechanically load and heat" and the possibility of"blowdown cooling" through the center. However, the range of biaxiality is a limitation.
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System Description Kalluri and Bonacuse [32] reported on the development, at NASA Lewis, of four basic TMF test sequences derived from the traditional cases of 0 ~ and 90 ~ phasing between axial and torsional mechanical cycles and 0 ~ and 180 ~ phasing between mechanical and thermal cycles. Using a commercial machine providing axial force (-+220 kN) and torque (-+2.2 kNm), they developed a specimen of 22 mm bore with 2 mm wall thickness, a parallel length of 41 mm and fillet radii of 86 mm. Biaxial extensometry locates in a pair of dimples impressed within the parallel length of the specimen 25 mm apart. Heating is provided by audio frequency induction (50 kW, 10 kHz) with three independently adjustable coils. During TMF testing, real time thermal strain compensation is provided by "learned" polynomial relations for heating and cooling. Hysteresis loops and fatigue endurance data were reported for Haynes 188 superalloy for strain ranges of _+0.4% axial and _+0.7% shear over the temperature range 316~ to 760~ In order to prevent local buckling, temperature deviations in the parallel length were held to -+-I~ which resulted in a cycle time of 10 min and hence heating and cooling rates of 1.5~ s -I.
Complex Cycles At BAM, Bedim using similar servohydraulic hardware, Meersman et al. [33] reported an extended program of tests for nickel-based superalloys IN 738 LC and SC 16. Simple TMF implied linear, diamond, and sinusoidal cycles, whereas complex TMF referred to simulation of a strain-time history representative of the leading edge of a first stage "bucket" in service. Tests were performed within the range 450~ to 950~ at rates up to 4.2~ s -~ for equivalent strain ranges between 0.6% and 1.24% and at strain rates 10 5s 1 and 10-4s -1.
Failure Criteria The definition of failure in uniaxial low-cycle fatigue is often characterized by a specified reduction in tensile stress amplitude measured in relation to the current half-life value or to the stabilized trend value after the end of cyclic hardening or softening. The percentage drop can vary between 2 and 50% in different studies. Another technique compares the unloading moduli from the peaks of the hysteresis loop, the lower value being associated with unloading from tension. Both approaches reflect cumulative cracking damage which, because of crack closure, is not registered in compression. However, in multiaxial fatigue studies the specimen geometry and the method of loading can often make other criteria more relevant or even unavoidable. Cruciform fatigue specimens, which generally employ dished center sections, may show little loss of peak tensile force, even with quite large cracks, because load is shunted around the gage area by thicker material. As a result, specified surface crack length has often been used as the endof-test criterion. In contrast, thin-walled tubes, loaded by axial force and differential pressure, will invariably require the test to be terminated soon after the fatigue crack has penetrated through the wall resulting in a coupling between the internal and external pressure systems and subsequent loss of control. Crack morphology and rate of crack growth has been shown to vary with strain ratio and with the plane in which the maximum shear strain lies [18,34]. Although the various theories and microstructural mechanisms governing crack propagation are outside the scope of this paper, it is instructive to look at the three-dimensional Mohr's strain circles (Fig. 5) in which four biaxial strain ratios (~b = - 1, - u, 0, + 1) are depicted for el = constant and u = 0.5 (fully plastic). At q5 = - 1, maximum shear strain YzO,given by the diameter of the largest circle (ez - e0), lies in the surface plane and long shallow surface cracks are observed for all test geometries and loading systems. As a result, in a cruciform, the crack grows more quickly to a specified surface length, whereas, in a pressurized thin-walled tube, penetration of the thickness is delayed because of the shallow ha-
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do=-1 ~__..[~___~ Uniax 1 3 3 ~ 1 3 t 13o
13z
~13
133
131
8r
13z
do=01~ 1
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131
z
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ture of the crack. At ~b = + 1, maximum shear strains Yzr and "YOrlie in the planes which intersect the surface plane and short deep surface cracks are observed for both cruciforms and thin-walled tubes. This time in a cruciform the growth of surface crack length is retarded, while in the case of the thinwalled tube, penetration is accelerated. These observations can help to explain why, for pure shear, cruciforms (with surface crack length failure criteria) may give relatively shorter lives than pressurized thin-walled tubes (with penetration failure criteria) and why, for equibiaxial straining, cruciform lives may be relatively extended. Similar arguments can be applied in the case of plane strain, ~b = 0, where the Mohr's circle is geometrically identical with pure shear but the maximum shear strain is Tzr, not ~tzO. In summary, different failure criteria can be expected to modify the relative fatigue lives measured. It would be interesting to establish the effect in a cruciform test series when crack penetration of the specimen thickness is the failure criterion rather than surface crack length.
System Selection In this final section the author offers a process for system specification. In conjunction, Table 1 provides information for systems required to test plate and sheet materials, while Table 2 addresses tubular specimen test systems for which thicker material must be available. Research Purpose
It is important to first decide whether the system is for fundamental materials properties determination (e.g., inputs to constitutive equations), crack growth studies, or component simulation. These considerations should decide the range of biaxiality to be provided by the system. The proposed environment (ambient, elevated temperature, or TMF) should then identify a particular scheme. The capital budget is an issue here since generally speaking the cost of a system is related to the number of actuators and the complexity of the environment.
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Specimen Geometry Specimen definition forms the cornerstone of the subsequent design process. Cruciform optimization has been discussed in some detail in Design Review - 2; however, optimization of the gage length for thin-walled tubular specimens is worthy of a summary. Tube mean diameter and wall thickness, together with material strength, determine the axial force, differential pressure, and torque requirements. In uniaxial LCF (solid samples) it is normal to have a parallel length of at least twice the gage diameter and a large fillet radius to achieve a low-strain concentration at the fillet runout. For axial-torsion (thin-walled tubes) only a modest reduction in parallel length may be necessary to maintain geometric stability. However, specimens subject to the full biaxial range of - 1 --< th --< + 1 may need parallel length and fillet radius reduced to the mean diameter or less in order to avoid buckling under plastic equibiaxial conditions. The ratio of mean diameter to wall thickness should approach 20:1 to minimize through-thickness strain gradients under pressure and .torsion. However, buckling considerations often reduce this ratio nearer to 10:1. Finite-element analysis is recommended for specimen geometry optimization.
Loadstring With the specimen defined, loadcells, axial and torsional actuators, and pressure intensifiers should be sized to at least 110% of the maximum required to break, in each mode, the strongest material envisaged for testing. Grips or pullrods should be specified in terms of capacity, operating temperature, and hydraulic or manual clamping.
Environment and Extensometry Sizing of specimen and grips together with maximum temperature and heating rates enables the furnace type and power rating to be defined along with any requirements for an enclosure such as a vacuum chamber. Extensometry can now be specified for operating environment, number of axes, averaging or not, strain ranges, and performance class.
Reaction Frame With force capacities and working space (crosshead to table, and between columns) defined, the frame can be specified, All low-cycle fatigue frames and loadstrings require high lateral stiffness as well as axial stiffness to ensure minimum specimen bending during plastic deformation. Under multiaxial conditions, when torsion may also be applied, it is advisable to specify larger diameter columns to ensure adequate lateral and torsional frame stiffness.
Controller and Software A modem digital closed loop multiaxis controller should provide the ability to generate and control complex mechanical and thermal waveforms. The latest graphical user interface (GUI) based software running in an open architecture system should provide flexibility and forward compatibility. Conclusions
1. Techniques for multiaxial testing derived from half a century of research work have been reviewed with special focus on those systems that broke new ground. 2. Thin-walled tubes under axial + differential pressure permit a two actuator economic design achieving full biaxiality with the unambiguous determination of all stresses and strains, essential for materials modeling. Adding torsion enables rotation of the principal axes.
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3. Cruciform systems have been confirmed as the only approach for testing materials in plate and sheet form over the full biaxial range - 1 <-- ~b<- + 1, and the overall method of choice for crack propagation studies especially at elevated temperature. 4. Multiaxial TMF has been facilitated by the combination of axial + torsion loading applied to thin-walled tubes and the advent of digital multiaxis controllers and software. The simulation of complex thermomechanical component cycles is now possible. 5. Failure criteria have been examined and an explanation offered for some of the differences in multiaxial fatigue endurance seen by cruciforms and thin-walled tubes under axial + differential pressure. 6. A procedure for specifying multiaxial high-strain fatigue systems has been presented with a tabular presentation of the advantages and disadvantages of alternative design schemes. Future Opportunities 1. At the time of writing, no TMF cruciform systems have been reported; however, with the full range of biaxiality beckoning, this can only be a matter of time. The ability to digitally control two mechanical straining axes and a temperature controller already exists as do cruciform test machines featuring biaxial extensometry and induction or radiant heating systems capable of temperatures in excess of 1500~ and heating rates up to 50~ s -~. 2. Digital modeling of materials behavior continues to advance and in the multiaxial field there already exists the ability to simulate hysteresis loops based on general strain-time demands. The concept of a high-capability multiaxial testing system able to reliably generate the input parameters to the constitutive equations for a given material creates the prospect of fewer but better tests per material and faster information to the designer. References [1] Gross, J. H. and Stout, R. D., "Plastic Fatigue Properties of High Strength Pressure Vessel Steels," The Welding Journal Vol. 34, Research Supplement, 1955, pp. 161-166. [2] Sachs, G., Gerberich, W. W., Weiss, V., and Latorre, J. V., "Low Cycle Fatigue of Pressure Vessel Materials," Proceedings of the American Society for Testing and Materials, Vol. 60, 1960, pp. 512-529. [3] Blaser, R. U., Tucker, J. T., and Kooistra, L. F., "Biaxial Fatigue Tests on Flat Plate Specimens," The Welding Journal, Vol. 31, Research Supplement, 1952, pp. 161-168. [4] Ives, K. D., Kooistra, L. F., and Tucker, J. T., "Equibiaxial Low-Cycle Fatigue Properties of Typical Pressure Vessel Steels," Journal of Basic Engineering, Transactions of the ASME, Vol. 88, 1966, pp. 745-754. [5] Zamrik, S. T., "An Investigation of Strain Cycling Behavior of 7075-T6 Aluminum Under Combined States of Strain," Progress Report No. 5, Pennsylvania State University, October 1968. [6] Zamrik, S. Y., "An Investigation of Strain Cycling Behavior of 7075-T6 Aluminum Under Combined States of Strain," Third Annual Progress Report, Pennsylvania State University, October 1967. [7] Zamrik, S. Y. and Davis, D. C., "A Simple Test Method and Apparatus for Biaxial Fatigue and Crack Growth Studies," Advances in Multiaxial Fatigue, ASTM STP 1191, D. L. McDowell, and J. R. Ellis, Eds., American Society for Testing and Materials, Philadelphia, 1993, pp. 204-219. [8] Halford, G. R. and Morrow, J., "Low Cycle Fatigue in Torsion," Proceedings of the American Society for Testing and Materials, Vol. 62, 1962, pp. 695-709. [9] Miller, K. J. and Chandler, D. C., "High Strain Torsion Fatigue of Solid and Tubular Specimens," Proceedings of the Institution of Mechanical Engineers, Vol. 184, 1970, pp. 433-448. [10] Yokobori, T., Yamanouchi, H., and Yamamoto, S., "Low Cycle Fatigue of Thin-Walled Hollow Cylindrical Specimens of Mild Steel in Uni-Axial and Torsional Tests at Constant Strain Amplitude," International Journal of Fracture Mechanics, Vol. 1, 1965, pp. 3-13. [11] Taira, S., Inoue, T., and Takahashi, M., "Low Cycle Fatigue Under Multiaxial Stresses (in the Case of Combined Cyclic Tension-Compression and Cyclic Torsion in the Same Phase at Elevated Temperature)," Proceedings, lOth Japan Congress on Testing Materials, 1967, pp. 18-23. [12] Kennedy, C. R., "Effect of Stress State on Low-Cycle Fatigue," Fatigue of Aircraft Structures, ASTM STP 338, American Society for Testing and Materials, Philadelphia, 1962, pp. 92-104. [13] Crosby, I. R., Bums, D. J., and Benham, P. P., "Effect of Stress Biaxiality on the High Strain Fatigue Behavior of an Aluminum Copper Alloy," Experimental Mechanics, Vol. 9, 1969, pp. 305-312.
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[14] Lohr, R. D., "Comparative Techniques for Multiaxial Testing," Low Cycle Fatigue and Elasto-Plastic Behavior of Materials, K-T. Rie and P. D. Portella, Eds., Elsevier, 1998, pp. 235-240. [15] Havard, D. G., "New Equipment for Biaxial-Fatigue Testing of Pressure Vessel Materials," Ontario Hydro Research Quarterly, 1st Quarter, 1968, pp. 14-19. [16] Andrews, J. M. H. and Ellison, E. G., "A Testing Rig for Cycling at High Biaxial Strains," Journal of Strain Analysis, Vol. 8, No. 3, 1973, pp. 168-175. [17] Lohr, R. D. and Ellison, E. G., "Biaxial High Strain Fatigue of 1Cr-Mo-V Steel," Fatigue of Engineering Materials and Structures, Vol. 3, No. 1, 1980, pp. 19-37. [18] Shatil, G., Smith, D., and Ellison, E. G., "High Strain Biaxial Fatigue of a Structural Steel," Fatigue and Fracture of Engineering of Materials and Structures, Vol. 13, No. 2, 1994, pp. 159-170. [19] Found, M. S., Fernando, U. S., and Miller, K. J., "Requirements of a New Multiaxial Fatigue Testing Facility," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 11-23. [20] Feruando, U. S., Miller, K. J., and Brown, M. W., "Computer Aided Multiaxial Fatigue Testing," Fatigue and Fracture of Engineering Materials and Structures, Vol. 13, No. 4, 1990, pp. 387-398. [21] Ellyin, F. and Wolodko, J. D., "Testing Facilities for Multiaxial Loading of Tubular Specimens," Multiaxial Fatigue and Deformation Testing Techniques. ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 7-24. [22] McLaren, S. W. and Terry, E. L., "Characteristics of Aerospace Materials Subjected to Biaxial Static and Fatigue Loading Conditions," ASME Publication 63-WA-315, 1963. [23] Pascoe, K. J. and de Villiers, J. W. R., "Low Cycle Fatigue of Steels Under Biaxial Straining," Journal of Strain Analysis, Vol. 2, No. 2, 1967, pp. 117-126. [24] Parsons, M. W. and Pascoe, K. J., "Development of a Biaxial Fatigue Testing Rig," Journal of Strain Analysis, VoL 10, 1975, pp. 1-9. [25] Parsons, M. W. and Pascoe, K. J., "Low-Cycle Fatigue Under Biaxial Stress," Proceedings of the Institution of Mechanical Engineers, Vol. 188, 1974, pp. 657-671. [26] Brown, M. W. and Miller, K. J., "Mode 1 Fatigue Crack Growth Under Biaxial Stress at Room and Elevated Temperature," Multiaxial Fatigue, ASTM 8TP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 135-152. [27] McAllister, J., "The Control of Cruciform Testing Systems Using Opposed Pairs of Servohydranlic Actuators," Proceedings, 7th Bath International Fluid Power Workshop, Bath, UK, September 1994, pp. 311-320. [28] Masumoto, H. and Tanaka, M., "Ultra High Temperature In-Plane Biaxial Fatigue Testing System with InSitu Observation," Ultra High Temperature Mechanical Testing, R. D. Lohr and M. Steen, Eds., Woodhead, Cambridge, UK, pp. 193-207. [29] Bartolotta, P. A. and Abdul-Aziz, A., "A Structural Test Facility for In-Plane Testing of Advanced Materials," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 25-42. [30] Taira, S., "Relationship Between Thermal Fatigue and Low Cycle Fatigue at Elevated Temperature," Fatigue at Elevated Temperature, ASTM STP 520, American Society for Testing and Materials, Philadelphia, 1973, pp. 80-101. [31] Hopkins, S. W., "Low-Cycle Thermal Mechanical Fatigue Testing," Thermal Fatigue of Materials and Components, ASTM STP 612, D. A. Spera and D. F. Mowbray, Eds., American Society for Testing and Materials, Philadelphia, 1976, pp. 157-169. [32] Kalluri, S. and Bonacuse, P. J., "An Axial-Torsional, Thermomechanical Fatigue Testing Technique," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 184-207. [33] Meersman, J., Frenz, H., Ziebs, J., Klingelhoeffer, H., and Kuehn, H.-J., "Thermomechanical Deformation Behaviour of IN 738 LC and SC 16," Proceedings, 5th International Conference on Biaxial/Multiaxial Fatigue and Fracture, E. Macha and Z. Mroz, Eds., Cracow, Poland, 1997, Vol. 1, pp. 303-322. [34] Parsons, M. W. and Pascoe, K. J., "Observations of Surface Deformation, Crack Initiation and Crack Growth in Low-Cycle Fatigue under Biaxial Stress," Materials Science and Engineering, Vol. 22, 1976, pp. 31-50.
D a v i d L. K r a u s e I a n d Paul A. Bartolotta 1
An In-Plane Biaxial Contact Extensometer REFERENCE: Krause, D. L. and Bartolotta, P. A., "An In-Plane Biaxial Contact Extensometer," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 369-381. ABSTRACT: A new extensometer system was developed to measure strains in cruciform specimens under in-plane biaxial loading at elevated temperature. This system incorporates standard axial contact extensometers to provide a cost-effective high-precision instrument. Capabilities include a minimum strain range of 25% and a maximum sensitivity of eight microstrain (txm/m). Allowable environments include elevated temperature, presently tested to 600~ and pressures from atmospheric to high vacuum. Descriptions of the test facility, modifications, mounting components, and installation procedures are reported. Calibration of the extensometers is described, including setup, error determination, and classification. Test results are presented for a stainless steel specimen. Strain gage and extensometer outputs are compared for static load cases and for combinations of waveform shape, strain range, frequency range, and test duration for cyclic room temperature tests. Elevated temperature results are presented for free thermal growth conditions and for instrument stability during hot cyclic operation. The results are summarized by the following: the new extensometer system calibrated with a maximum error of 0.8%; room temperature correlation with strain gage data yielded an average variation of 58/xm/m; operation under cyclic conditions resulted in tracking errors less than 3%; elevated temperature results compared accurately with theoretical predictions; and, long duration testing proved to be stable. KEYWORDS: extensometers, in-plane biaxial testing, strain measurement, experimental techniques,
cruciform specimen, cyclic testing, elevated temperature, multiaxial fatigue, deformation
Components of mechanical equipment under load are routinely subjected to multiaxial states of stress. These conditions result from both multiaxial loadings and component geometry. In aircraft engines and many other applications, components are subjected to these complex stress states at elevated temperatures. For these conditions, biaxial testing is crucial for deriving basic material characteristics and mechanical properties used in design and life prediction. In addition, many materials exhibit anisotropic properties; for these, biaxial testing provides a meaningful bridge between testing of uniaxial coupons and that of higher-level structures. In-plane biaxial testing of cruciform specimens is one arrangement used for obtaining biaxial stress states. Accurate strain measurement during in-plane biaxial testing is important for several reasons. One reason is that strain measurement permits calculating specimen test area stresses under the various loading conditions. Strain measurement is also important in verifying the specimen design by correlating mechanical and thermal loads to the strains in the test area. Additionally, real-time strain measurement permits observation of deformation behavior under biaxial loading conditions. Finally, electronic continuous measurement is used in closed-loop test control for strain-controlled experiments and in all test types for sensing test-termination strain limits. In this paper, a contact extensometer system is described that measures test area strains along two orthogonal axes in flat cruciform specimens. The device was validated for use by extensive testing of a stainless steel specimen, with specimen temperatures ranging from room temperature to 600~ In1 Materials research engineer, NASA Glenn Research Center, Mail Stop 51-1, 21000 Brookpark Road, Cleveland, OH 44135-3191. 369
Copyright9
by ASTM International
www.astm.org
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MULTIAXlAL FATIGUE AND DEFORMATION
plane loading conditions included several static biaxial load ratios, plus cyclic loadings of various waveform shapes, frequencies, magnitudes, and durations. The extensometer system measurements are benchmarked to strain gage data at room temperature and to calculated strain values for elevated temperature measurements. This strain measurement system was developed to test advanced materials for the Advanced High Temperature Engine Materials Technology Program (HITEMP) program [1] and the High Speed Civil Transport propulsion system. The candidate materials would be used in turbine engine components that are under highly multiaxial states of stress. Although monolithic, these cast materials also exhibit directionality of mechanical properties due to large grain sizes. Future uses of the extensometer system are expected to include testing of other high-temperature materials, including polymeric and ceramic matrix composite materials.
Background Biaxial stress states in material specimens are frequently produced in the laboratory by two common test systems. The first is a versatile configuration that employs thin tubular specimens with simultaneously applied axial load and torsional load [2, 3]. In addition, some of these systems also have internal or external pressure capabilities. This highly specialized arrangement produces every possible biaxial stress state by independently imposing the shear stress and the two normal stresses on the test section. The axial-torsional system is limited to specimen geometry where thin-wall theory applies and to loadings where buckling instability does not occur. In addition, this method is not suitable for testing material product forms not readily fabricated into the hollow-cylinder specimen form.
In-Plane Biaxial Testing The second common test system is the in-plane biaxial test configuration [4~5]. With this method, force actuators apply axial loads along two orthogonal axes to a cruciform-shaped flat specimen. This system overcomes some of the limitations inherent in other test machines. A uniform through-thickness stress condition in the specimen test area is achieved regardless of dimension, assuming proper specimen design and test machine capacity. Both sides of the entire test area are visible, and the specimen shape is well suited for plate materials, sheet materials, and composite structures fabricated in plate form. However, other difficulties arise with this method. Because external shear can not be applied to the specimen, principal stress axes are always coincident with the loading axes. Accordingly, specimen geometry must be pre-aligned with material microstructure when anisotropic material effects are important. For unnotched biaxial testing, another complexity is designing a specimen that produces maximum stress in the test area. Local stresses at features in the cruciform legs and around the test section often exceed the test area stresses. Lastly, difficulty arises in actually determining the test area stress states under the multitude of loading combinations possible. These values can be derived analytically by such means as the finite element method. Experimentally, the elastic stress state can be determined from test area strain measurements when the material's elastic modulus and Poisson's ratio are known. Elementary mechanics of materials texts provide the methodology for the derivation of this plane stress condition [7].
Strain Measurement Techniques For a well-designed cruciform specimen under accurately aligned in-plane biaxial load, the central test area principal strains are coincident with the two orthogonal load axes. By definition, the shear strains perpendicular to these axes are nonexistent. Therefore, these conditions permit determination of the complete state of strain (and consequently, the state of stress) by measuring the extensional strains only along the two load axes.
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371
Experimentally, various measurement techniques are used to determine the test area strains. Optical methods are a recent development [8]. Laser interferometry uses optical interference fringe patterns to obtain a global depiction of strain contours. The white light speckle pattern correlation method compares optically obtained displacements of applied or naturally occurring surface features to derive the global strain state. These methods possess some limitations. For example, integration of strain control for closed-loop cyclic testing is not readily available. Also, visual access to the test area is required. For room temperature testing, adhesively bonded electrical resistance strain gages are commonly used [9]. This method is accurate and relatively inexpensive; however, due to the sensing element being directly mounted to the specimen, it is limited by the element's strain range capacity and maximum temperature. Extensometers are used to overcome restrictions of the other methods. These devices remotely locate the strain-sensing electrical element away from the specimen. Using the mechanical advantage of probe rods, strains in the sensor can be much smaller than strains of the measured surface. The method of coupling to the specimen varies depending on the design of the device. Probes can be mechanically fastened or adhesively bonded. In contact extensometers, probes are friction-coupled with the normal force applied through a spring arrangement. Similarly, probes can be affixed with the spring arrangement pressing specially shaped probe ends into matching machined notches or dimples in the specimen. For elevated temperature testing, the extensometer probe rods are fabricated of low thermal conductance material, heat shields are installed, and water-cooling passages are added, all to insulate the strain-sensing element from heat. No commercial sources were available for a high-temperature extensometer system designed for in-plane biaxial cruciform specimens; several sources prepared unexpectedly high cost estimates for development of an all-new system. A small number of experimental configurations were developed for specific test systems [10]. None of these could be applied to the test system at hand due to the elevated temperature requirement and the existing radiant furnace arrangement. Therefore, it was decided to adapt standard contact extensometers in a novel configuration to provide a solution.
Biaxial Extensometer Apparatus
In-Plane Biaxial Test System The extensometer system is installed at the Benchmark Test Facility at NASA Glenn Research Center, Cleveland, Ohio [11]. This facility was developed to characterize advanced materials under complex loading conditions. At the facility the extensometer system is mounted on a large in-plane biaxial load frame that places the fiat plate specimen in a vertical orientation (Fig. 1). This versatile load frame is presently configured for a cruciform specimen of established design, approximately 305 by 305 mm with a 100-ram-square test area (Fig. 2). Loads are applied through four hydraulic actuators; each one has a capacity of 490 kN in both tension and compression. The specimen is mounted to the load train with water-cooled hydraulic wedge grips of equal capacity. The grips are aligned with a strain-gaged specimen to produce bending strain of less than 10% of the axial strain at 100 /.trn/m. The hydraulic actuation system operates under closed-loop control with a state-of-the-art digital controller. The controller allows each actuator to operate independently or to operate under centroid control. Centroid control maintains the specimen center at a fixed point in space; it is accomplished by the digital controller, using cross-compensation between opposing actuators to force the actuators' position difference equal to zero. A water-cooled quartz lamp radiant furnace maintains specimen temperature. This versatile system produces rapid heat transfer rates and is divided into eight controllable zones to provide a uniform (or non-uniform, if desired) temperature field. The furnace has heated a ceramic specimen's 150-ram-square test section to 1500~ the maximum system capacity. The furnace is constructed of two hinged "clam-shell" halves that enclose the cruciform specimen when in operation. Through the
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MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 1--In-plane biaxial load frame at NASA Glenn Research Center.
center of each half and perpendicular to the specimen test area is a 90 by 55-mm open passageway, originally designed as a viewport for optical microscopy. For vacuum and inert environment testing, a large water-cooled vacuum chamber surrounds the test area, grips, and furnace. A roughing pump and diffusion pump provide a quick transition to absolute vacuum levels less than 13/zPa (10 -7 torr). The chamber can be backfilled with purified argon for inert environment testing. Large front and rear doors are equipped with small sight ports to view the specimen center and swing open on vertical hinges for easy access.
2--/
12.5---t
FIG. 2--Cruciform specimen (dimensions in mm).
KRAUSE AND BARTOLOTTA ON EXTENSOMETERS
373
In-Plane Biaxial Contact Extensometer Description The strain-sensing elements of the in-plane biaxial contact extensometer are commercially available high-temperature axial contact extensometers [12]. They are designed to measure and control specimen strain in static and fatigue-type uniaxial testing for compressive and tensile loading. These vacuum-capable instruments incorporate water cooling to provide acceptable and constant instrument temperature and to assure accurate measurements. Specimen strain is transferred through contact probe rods and mechanical linkages to precision resistance strain gages attached to an internal bending metallic element. The strain gages produce a voltage output proportional to the displacement of the probe rods and consequently to the specimen strain. The gage length between centerlines of the probes is 12.7 mm for one extensometer and 25.4 mm for the other. These axial extensometers are nestled in a configuration that places the two gage length dimensions perpendicular to each other with coincident central axes (Fig. 3). The gage lengths are aligned to the cruciform specimen's principal axes at the specimen center. The shorter gage-length extensometer measures strain in the vertical direction, while the longer measures strain in the horizontal direction. The vertical extensometer is mounted inboard of (closer to the specimen than) the horizontal one, which straddles the inner extensometer transducer body. The axial extensometers are mounted on their original water-cooled radiation heat shields. The heat shields contain integral spring systems that press the contact probes to the cruciform. The heat shields also serve as the attachment mechanism to the load frame. Special mounting blocks and shims are used to rigidly attach the heat shields to the back support structure of the radiant furnace rear half. The blocks and shims are fabricated with oversized fastener holes to allow precise alignment of each extensometer to the cruciform specimen. The furnace support structure to which the heat shields are attached is itself fixed to the test system vacuum chamber, which in turn is rigidly attached to the load frame. This provides a high
FIG. 3--Biaxial extensometer installation.
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MULTIAXlALFATIGUE AND DEFORMATION
stiffness path from the extensometers to structural "ground." Extended-length conical-tipped alumina probes (193 and 274 mm long by 3.55 mm diameter) pass through the furnace central viewport to contact the cruciform specimen at its center (Fig. 4). A quartz window thermally protects the viewport opening and is machined with a small opening to allow passage of the probes. This arrangement impairs visual observation of the rear face of the cruciform specimen, although full access to the front surface is maintained. Electrical connection of the extensometers is customary. A commercial portable digital controller with two d-c transducer conditioners is used to provide excitation voltage, variable gain (for improved linearity), scaling, and offset for each extensometer [13]. The portable controller provides a display of each transducer voltage output; the readout can also be programmed to directly show strain values and other data such as minimum and maximum strains. Digital and analog signal outputs are available that can be used for plotting, data acquisition, strain limits, or test control. All components are mounted outside the test system vacuum chamber except for the extensometers, heat shields, and a short length of cable that ties through the chamber via a vacuum-rated connector.
Mounting Considerations Installation of the extensometer system is straightforward, albeit lengthier and requiring more care than a typical uniaxial setup. Prior to specimen installation, sight lines are drawn on the cruciform specimen test area indicating vertical and horizontal centerlines along with gage spacings. Next, the two contact extensometers and transducer conditioners are "bench-calibrated" with a standard extensometer calibrator. The calibrator is temporarily attached to the vacuum chamber near the mounting site so that all electronic components and cables to be used during the test are also used during calibration. The final step of calibration assures that output is set to zero when strain is at zero. Following this, the rear furnace half is hard-fastened to the vacuum chamber. After specimen installation, the inner vertical extensometer is mounted into its heat shield, and alumina probes are roughly positioned onto the marked gage lines. If required, the heat shield is repositioned in its slotted mounted holes to align to the extensometer body. Similarly, the outer horizontal extensometer is slid into position over the inner one, mounted, roughly positioned, and checked for heat shield alignment. Next, cooling water lines are connected for elevated temperature testing. Now, observing the conditioner digital display, the probe tips are nudged into final position. This is performed by lightly tapping the tips with a stiff bent metal wire that fits in the narrow space between the specimen and the furnace.
FIG. 4---Alumina probes at rear furnace face.
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KRAUSE AND BARTOLOTTA ON EXTENSOMETERS Strain R a n g e = + 0.8% Max. Error = 0.80% of range
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When the conditioner display readouts approach zero voltage, the correct gage lengths are achieved. Final zeroing is accomplished electronically with the conditioner zero-offset function. Several practical considerations unique to the in-plane biaxial extensometer configuration are worth noting. First, the alumina probes must be very accurately installed in the extensometer bodies. Any inaccuracy in setting the probe depth dimension is amplified greatly by the long probes and resuits in the extensometer body being misaligned in the heat shield. Another concern is due to the tight clearance between the inner extensometer body and the outer's probes. The configuration makes it easy to decouple the inner extensometer from the specimen while installing the outer extensometer. Thermal expansion is responsible for a final consideration that results in a special cruciform specimen-mountingprocedure for some elevated temperature testing. The procedure allows the specimen center to be maintained at the load frame center (which is required for acceptable extensometer operation) and minimizes unwanted gripping-induced thermal stress in the specimen. Hydraulic grip pressure is set to a low value, and at room temperature one leg of the specimen is clamped. The extensometer system is installed per this section, followed by gradual specimen heating with the radiant furnace. The specimen center is maintained manually by gradually moving the gripped actuator out. Once equilibrium is reached at the test temperature, the other three grips are clamped onto the specimen, and the original gripped leg is released and then reclamped. The final step is to raise the grip hydraulic pressure to that required for the test loads.
Extensometer Specifications and Calibration The commercial contact extensometers, modified with long probes and used with the portable digital controller, yield an allowable strain range of + 175% (tension) to - 3 5 % (compression) in the vertical direction, and + 125% to - 2 5 % in the horizontal direction. When electrically configured for highest gain (and smallest range), the strain sensitivity is 4/zm/m vertically and 8/xm/m horizontally. Final calibration analysis (Fig. 5) gave maximum values of fixed error equal to 64 tzrn/m (< 1000/.din allowed), relative error equal to 0.8% (< 1%), and relative error of gage length equal to 0.008% (<1%). This resulted in a Class C designation in accordance with ASTM Standard Practice for Verification and Classification of Extensometers (E 83).
Validation Testing and Experimental Results To verify the in-plane biaxial extensometer operation under a multitude of conditions, a series of tests were performed on an AISI Type 304 stainless steel cruciform specimen. The specimen was in-
376
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 6---Specimen strain gage locations. strumented with six, two-element 90 ~ strain gage rosettes for the room temperature tests (Fig. 6). Extensometer outputs were manually documented as well as recorded with the load signal on a fourchannel thermal plotter. Note that the vacuum capability of the system was not verified.
Room Temperature Testing Seven strain states were investigated at room temperature by applying monotonic ramped waveforms with varying peak loads along the two orthogonal load axes. These included equiaxial loading (the vertical and horizontal loads were equal), uniaxial loadings (the non-load axis was set to zero), and maximum shear loadings (the vertical and horizontal loads were equal but opposite in sign). Elastic strain values were maintained by limiting strains to less than 1000 ~m/m. Extensometer values were compared to: (1) the "center average" of the four strain gages oriented in the direction of the extensometer gage and lying on the specimen centerline, and (2) an average of all six strain gages oriented in the direction of the extensometer gage (Fig. 7, for example). For all strain states, the vertical direction yielded an average maximum difference of 5 8 / ~ m / m between extensometer reading and strain gage "center average" over the 1000/xm/m range. Likewise, the horizontal direction had a difference of 38/~m/m. These results indicated good agreement with strain gage data and showed minimal "cross-talk" between the two measured directions. Cyclic room temperature testing that simulated low-cycle fatigue conditions was performed next. Peak extensometer readings were monitored for 1500 cycles under an equiaxial sinusoidal load
Vertical Extensometer 70 60 ~" 50
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FIG. 7--Extensometer/strain gage correlation for equiaxial loading.
377
KRAUSE AND BARTOLO-I-FA ON EXTENSOMETERS
700 !
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FIG. 8--Cyclic operation for equiaxial loading at room temperature.
(Fig. 8). The sequence was repeated for six useful frequencies from 0.5 to 3.0 Hz. The average difference for all six frequencies in peak reading from the mean peak strain of 400 to 700/~rn/m over the 1500 cycles was 2.1% in the vertical direction and 2.9% in the horizontal direction. A second cyclic test was carried out to validate use of the extensometer signal for strain-controlled testing; time plots of strain and load signals were examined to perform this analysis (Figs. 9 and 10). Offsets in arrival time between the strain and load waveform maxima were measured for sinusoidal equiaxial loads for the same six frequencies. In all cases this tracking error was less than 3% of the waveform period. A like study was made for triangular and square load waveforms at 0.5 Hz. The maximum resuiting tracking error was 1.5%. This testing proved good coupling between the extensometer probes and the specimen, and excellent tracking for cyclic tests.
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378
MULTIAXIAL FATIGUE AND DEFORMATION
40 kN p-p equiaxial load (9 kN p-p for square w a v e )
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Elevated Temperature Testing Acceptable use at elevated temperatures was confirmed by additional testing. In one test the cruciform specimen was held in centroid control at zero load and gradually heated from room temperature to 600~ producing thermal strains greater than 9000/~m/m (Fig. 11). The recorded extensometer outputs were compared with a simplified linear growth model rather than strain gage data, as conventional adhesively applied strain gages burn off at temperatures above approximately 150~ The average difference between the measured and predicted strain values was 2.2% in the vertical direction and 3.1% in the horizontal direction. The expected effect of the water-cooled grips on constraining free thermal expansion of the test area was not observable. Another elevated temperature test was performed similar to the first, except an initial equiaxiat toad was imposed at room temperature to a moderate strain level (Fig. 12). After heating the specimen to 500~ the load was released and the specimen was allowed to cool back to room temperature. Again, measured strain values closely matched the linear model. These static hot tests demonstrated accurate strain measurement and good extensometer coupling at elevated temperatures. Fatigue-type loading at elevated temperature was evaluated next. For the first test, the specimen was heated to a steady state at 480~ and the extensometer conditioners were re-zeroed. Next, a small equiaxial sinusoidal loading (22 kN) was applied for short durations at 0.25, 0.5, and 1.0 Hz, followed by a moderate loading (45 kN) at 0.25, 0.5, 1, 2, and 3 Hz. After the load was returned to zero, residual strain readings were taken. The signal "walk" was 14% ( - 8 0 / ~ m / m ) of the strain range in the vertical direction and 7% ( - 4 0 / x m / m ) in the horizontal. This "walk" indicated possible slippage of the probe arms under hot cyclic conditions, which could be minimized by use of either a dimpled specimen or chisel-edged (instead of conical-tipped) probe rods. A second test evaluated performance over a longer period of time. The specimen was heated to 480~ and loaded with a moderate equiaxial load (33 kN at 1.0 Hz). The extensometers' output and the load signals were recorded on the plotter for 350 000 cycles. An evaluation of the traces over time showed the strain signals "wandered" about the mean value by + 16% of the strain range (_+49/zm/m) in the vertical direction and -+20% (-+ 67/xm/m)
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KRAUSE AND BARTOLOTTA ON EXTENSOMETERS
381
in the horizontal. This strain "wander" may be explained by specimen thermal drift, as the two strain oscillations were roughly in phase and had a long period (approximately 10 h). Strains were very sensitive to temperature; for example, a 4~ change produced 68 p,m/m thermal strain in this material.
Conclusions A mechanism to measure strains in flat cruciform specimens at room and elevated temperatures has been developed from substantially off-the-shelf components to produce a reliable, accurate, low-cost in-plane biaxial contact extensometer. The device attaches two standard axial extensometers to the test system support structure using adjustable alignment fixtures. Final calibration produced a maximum relative error of 0.8%, resulting in the extensometer designation of Class C according to ASTM Practice E 83. An extensive series of tests were performed to validate operation of the biaxial extensometer. Static and dynamic tests at room and elevated temperatures to 600~ established accurate strain measurement, good specimen coupling, and excellent load signal tracking. Using a different probe tip configuration could alleviate a small amount of offset discovered upon unloading in hot cyclic testing. Strain "wander" in long duration testing is thought to be a real measurement, possibly related to slight specimen temperature fluctuations. Acknowledgments The valuable technical assistance and mechanical skills provided during extensometer development by Stephen J. Smith (Gilcrest Electric and Supply Company), and insights provided into the fine points of extensometry by Christopher S. Burke (Dynacs Engineering Inc.) are gratefully acknowledged.
References [1] Ginty, C. A., "Overview of NASA's Advanced High Temperature Engine Materials Technology Program," HITEMP Review 1997: Advanced High Temperature Engine Materials Technology Program Volume L NASA Conference Publication 10192, 1997, pp. 2-1 to 2-19. [2] Ellyin, F. and Wolodko, J. D, "Testing Facilities for Multiaxial Loading of Tubular Specimens," Multiaxial Fatigue and Deformation Techniques, STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohochen, PA, 1997, pp. 7-24. [3] Lefebvre, D. F., Ameziane-Hassani, H., and Neale, K. W., "Accuracy of Multiaxial Fatigue Testing with Thin-Walled Tubular Specimens," Factors that Affect the Precision of Mechanical Tests, STP 1025, R. Papimo and H. C. Weiss, Eds., American Society for Testing and Materials, West Conshohochen, PA, 1989, pp. 103-114. [4] Makinde, A., Thibodeau, L., and Neale, K. W., "Development of an Apparatus for Biaxial Testing Using Cruciform Specimens," ExperimentalMechanics, Vol. 32, No. 2, 1992, pp. 138-144. [5] Boehler, J. P., Demmerle, S., and Koss, S., "A New Direct Biaxial Testing Machine for Anisotropic Materials," Experimental Mechanics, Vol. 34, No. 1, 1994, pp. 1-9. [6] Demmerle, S. and Boekler, J. P., "Optimal Design of Biaxial Tensile Cruciform Specimens," Journal of the Mechanics and Physics of Solids, Vol. 41, No. 1, 1993, pp. 143-181. [7] Popov, E. P., Mechanics of Materials, 2nded., Prentice-Hall Inc., Englewood Cliffs, NJ, 1976, pp. 235-265. [8] Bhat, G. K., "Electronic Speckle Pattern Interferometry Applied to the Characterization of Materials at Elevated Temperature," NDT Solution, American Society for Nondestructive Testing Inc., January 1998. [9] Vishay Measurements Group, "Strain Gage Selection: Criteria, Procedures, Recommendations, TN-5054," Measurements Group Tech Note, Measurements Group Inc., 1989, pp. 1-15. [10] Makinde, A., Thibodeau, L., Neale, K. W., and Lefebvre, D., "Design of a Biaxial Extensometer for Measuring Strains in Cruciform Specimens," ExperimentalMechanics, Vol. 32, No. 2, 1992, pp. 132-137. [11] Bartolotta, P. A., Ellis, J. R., and Abdul-Aziz, A., "A Structural Test Facility for In-Plane Biaxial Testing of Advanced Materials," Multiaxial Fatigue and Deformation Techniques, STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 25-42. [12] MTS Systems Corporation, "Series 632 High Temperature Extensometers," Product Manual, MTS Systems Corporation, 1976. [13] MTS Systems Corporation, "Model 407 Controller, Version 3.0," Product Manual, MTS Systems Corporation, 1995.
J. R. Ellis, l G. S. Sandlass, 2 and M. Bayyari 3
Design of Specimens and Reusable Fixturing for Testing Advanced Aeropropulsion Materials Under In-Plane Biaxial Loading REFERENCE: Ellis, J. R., Sandlass, G. S., and Bayyari, M., "Design of Specimens and Reusable Fixturing for Testing Advanced Aeropropulsion Materials Under In-Plane Biaxial Loading," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 382-404. ABSTRACT: A design study was undertaken to investigate the feasibility of using simple specimen designs and reusable fixturing for in-plane biaxial tests planned for advanced aeropropulsion materials. Materials of interest in this work include: advanced metallics, polymeric matrix composites, metal and intermetallic matrix composites, and ceramic matrix composites. Early experience with advanced metallics showed that the cruciform specimen design typically used in this type of testing was impractical for these materials, primarily because of concerns regarding complexity and cost. The objective of this research was to develop specimen designs, fixturing, and procedures that would allow in-plane biaxial tests to be conducted on a wide range of aeropropulsion materials while at the same time keeping costs within acceptable limits. With this goal in mind, a conceptual design was developed centered on a specimen incorporating a relatively simple arrangement of slots and fingers for attachment and loading purposes. The ANSYS finite-element code was used to demonstrate the feasibility of the approach and also to develop a number of optimized specimen designs. The same computer code was used to develop the reusable fixturing needed to position and grip the specimens in the load frame. The design adopted uses an assembly of slotted fingers which can be reconfigured as necessary to obtain optimum biaxial stress states in the specimen gage area. Most recently, prototype fixturing was manufactured and is being evaluated over a range of uniaxial and biaxial loading conditions. KEYWORDS: in-plane biaxial testing, advanced aeropropulsion materials, cruciform specimen design, reusable fixturing, finite-element analysis, optimization techniques, attachment methods, prototype fixturing
One technique for investigating material behavior under complex stress states is to use in-plane biaxial loading. Using this approach, cruciform specimens fabricated from plate or sheet material are gripped at four locations and loaded along two orthogonal axes. Servo-hydraulic loading systems are used in this application which are similar to those used for uniaxial testing. Thus, the technique has the advantage that the loading arrangement is relatively straightforward and uses equipment which has seen extensive development over the past 30 years. Also, the test method allows a wide range of biaxial stress states to be investigated with minimum complication from the load application viewpoint. For these reasons, the test method has been used to generate a sizable body of biaxial test data for both monolithic and composite materials [1-29]. l Senior research engineer, NASA Glenn Research Center, Cleveland, OH 44135. 2 Structural analyst, MTS Systems Corp., Eden Prairie, MN 55344. 3 Principal research engineer, Research Applications, Inc., San Diego, CA 92121. 382
Copyright9
by ASTM lntcrnational
www.astm.org
ELLIS ET AL. ON AEROPROPULSIONMATERIALS
383
One difficulty facing these investigations has been the selection and/or development of the most suitable specimen design for the particular program. It should be noted that consensus standards do not exist for this method of testing, and so the experimentalist is faced with a wide range of possibilities. A major complication here is that use of the cruciform specimen configuration and associated gripping fixtures results in "coupling" between the two loading directions. In the present research, specimens are positioned in the load frame using four hydraulic grips which rigidly constrain the specimen over the gripped regions. It follows that loading applied in one direction is partially reacted by the specimen and partially by the grips associated with loading in the second direction. One method of minimizing this effect is to use specimen designs which incorporate fairly complicated arrangements of flexures as illustrated in Fig. 1. It has been demonstrated that flexures with low bending stiffness in the plane of loading can be used to minimize the constraint imposed by specimen gripping. Also, it has been shown that the geometry of the flexures can be optimized and tailored to give near-uniform stress/strain conditions in the gage area for specific biaxial loading conditions. One obvious disadvantage of using flexures is that regions of high stress concentration can be introduced into specimens in close proximity to the gage area. Of particular concern are stress concentrations at the ends and intersection points of the flexures. This raises the possibility that failure can be initiated outside of the gage area in regions where stress/strain conditions are ill-defined. Traditionally, this problem has been addressed by incorporating a gage area within which specimen thickness is reduced significantly from the value in the gripped regions. In the case of plate specimens incorporating flexures, experience has shown that thickness reduction factors as high as ten are needed to achieve acceptable performance. That is failure initiating within the gage area where stress/strain conditions are both relatively uniform and relatively well defined. Although the above approach has been used effectively in the case of conventional structural alloys, it has proved impractical for the materials of interest in this work including: advanced metallics, polymeric matrix composites, metal and intermetallic matrix composites, and ceramic matrix composites. Problems include the unavailability of material in "large" product forms and also the diffi-
FIG. 1--Current NASA specimen design.
384
MULTIAXIALFATIGUEAND DEFORMATION
culties associated with machining complex three-dimensional geometries in complex multi-phase materials. The aim of the present work was to develop an alternative approach involving use of a simplified specimen design and use of reusable fixtures incorporating the design features needed to decouple the applied biaxial loading. A further goal was to develop fixturing and procedures which would allow in-plane biaxial tests to be conducted on a wide range of advanced materials while at the same time keeping costs within acceptable limits.
Specimen Design and Analysis The current cruciform specimen design (Fig. 1) developed at NASA for testing conventional structural alloys served as a starting point for this work. In the new approach, the gripped regions of the current design are replaced by four individual fixtures which incorporate slotted fingers to decouple the biaxial loading. The gage section of the current design is replaced by a specimen incorporating a reduced gage area and four sets of slots and fingers for attachment purposes. The focus of the preliminary design and analysis work was on determining whether a specimen design with this configuration would meet two straightforward design requirements. These were: (1) that the maximum stress in the part should occur within the gage area, and (2) that the stress/strain distribution in the gage area should be reasonably uniform, say, within +-5% of the mean. Details of the conceptual design and the results of specimen design and analysis work are described in the following.
Conceptual Design A conceptual design for the new approach is shown in Fig. 2. One constraint on the overall size of the assembly was that a 432 X 432 mm envelope is available within the load frame for installation and gripping purposes. Four slotted finger fixtures are shown attached to a specimen fabricated from a 229 X 229 • 6 mm plate. These dimensions were selected to give a relatively large gage area, 76 mm outside diameter in the case of specimens with circular gage areas. This approach was adopted primarily with instrumentation requirements in mind. The attachment method is not shown in Fig. 2 for simplicity of drawing. Details of a slotted finger attachment are given in Fig. 3. The design shown is idealized in that no stress relieving blend radii were included so as to simplify finite-element analysis of the complete assembly. This approach was acceptable because the focus of initial work was on the performance of the specimen rather than on the performance of the fixturing. The slotted finger attachments are assumed to be gripped over 152 • 38 mm 2 areas on both top and bottom surfaces. Earlier experiments using the current NASA cruciform specimen design had shown that this arrangement met the loading requirements of planned test programs. Experience gained in these earlier experiments was also used to size the finger and flexure configuration shown in Fig. 3. The four finger arrangement was an obvious choice given the need to locate a slot on the fixtures centerline and also given the need to maintain symmetry about the fixtures centerline. Details of the initial specimen design are given in Fig. 4. As noted earlier, it is assumed to be fabricated from a 229 x 229 X 6 mm plate. Perhaps the most important design feature is the arrangement of slots and fingers used for attachment and loading purposes. It was recognized at the outset that the slot configuration would play a key role in obtaining an optimized specimen design. As indicated in Fig. 4, a slot width of 10 mm was selected for initial feasibility studies. Other dimensions shown in symbolic form were treated as variables in subsequent optimization analyses.
Stress Analysis Details The ANSYS finite-element code, version 5.4, was selected for this work, primarily because it features an optimization package. The plan was to model '/8of the complete assembly with the six specimen dimensions shown in Fig. 4 expressed as variables and to perform fully automated analyses un-
385
ELLIS ET AL. ON AEROPROPULSION MATERIALS 432 mm square
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MULTIAXlALFATIGUE AND DEFORMATION
_1 v I
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FIG. 4--Conceptual specimen design with circular gage area.
til an optimum set of specimen dimensions had been obtained. To facilitate this process, a command input file was created using the ANSYS Parametric Design Language (APDL). This provides the means to create finite-element models in terms of variables, which in turn allows for easy and rapid design changes. Approximately 1000 lines of ANSYS commands were used to define parameters, generate the finite-element model, solve, evaluate results, and begin optimization looping. As indicated earlier, external loading was applied to the assembly over 152 • 38 mm 2 areas. An equibiaxial stress state was introduced into the finite-element model by constraining all surface nodes within the gripped regions to displace 0.127 mm in the positive sense in the two loading directions. Similarly, clamping within the specimen grips was simulated by constraining all surface nodes within the gripped region to displace 0.005 mm in the thickness sense. Regarding the attachment method, the specimen and the slotted finger fixtures were modeled as a single unit in the early analyses which focused on specimen performance. Note that the material properties used in this work were handbook values for Inconel 718 and that the results of stress analyses are expressed in the form of von Mises equivalent stress throughout. Stress Analysis Results
The results shown in Figs. 5 and 6 were obtained in one of a large number of stress analyses performed during the initial stages of the research. One interesting feature of the overall deformation behavior of the assembly shown in Fig. 5 is the ability of the finger and flexure arrangement to accommodate large overall displacements and rotations without becoming overstressed. This flexibility is, of course, the mechanism by which loading in the two directions is partially decoupled. As might be
ELLIS ET AL. ON AEROPROPULSION MATERIALS
387
expected, analysis of the results showed that stress concentrations occurred at three locations outside of the gage area. As indicated in Fig. 6, these locations were the center slot, the outer slot, and the fillet region. Simply stated, the optimization process involved varying specimen dimensions in a systematic manner until the maximum stress values at these locations were less than the average stress in the gage area, say, with a 20% margin of conservatism. Analysis of the data shown in Figs. 5 and 6 indicated that this condition had partially been achieved with the particular specimen design shown. The average and maximum stresses in the gage area were 392 MPa and 397 MPa, respectively. The maximum stress levels in the center and the outer slots were 367 MPa and 352 MPa, and the maximum stress in the fillet region was 382 MPa. Thus, this design met the design requirement that the maximum stress should occur within the gage area. However, it was apparent that additional work was needed to achieve the 20% margin of conservatism. The second design requirement was that the stress distribution in the gage area should be uniform within -+5% of the mean. Analysis of the data shown in Fig. 6 showed that the stress distribution in the gage area fell within ---2.4% of the mean, easily meeting the target value.
Optimization Method The combination of specimen dimensions used in obtaining the above result was as follows: 9 9 9 9
Gage section radius (R1) = 41.28 m m Center slot length (SLA) = 35.50 m m Outer slot length (SLB) = 37.00 m m Fillet radius (FR) = 27.94 m m
FIG. 5--Stress distribution in assembly.
388
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 6---Stress distribution in specimen.
9 Gage area thickness (T1) = 1.25 m m 9 Thickness transition radius (TR1) = 12.70 m m The optimization process followed in obtaining these results was not straightforward and proved to be extremely time-consuming. Evaluation of the various ANSYS optimization routines showed that in this application, the routines had difficulty converging on optimum sets of values. The factorial routine was found to be most useful for the present work as it allowed predetermined combinations of specimen dimensions to be investigated in a straightforward manner. As described in the following, the "automatic" process was supplemented by presenting the results of stress analysis in graphical form and by analyzing the results by hand. This approach effectively narrowed the design space and allowed optimum data sets to be determined more efficiently. As a first step, fully automated stress analyses were conducted for up to 36 combinations of center and outer slot lengths and gage section thicknesses. As illustrated in Fig. 7, these data were plotted to establish the optimum combination of center and outer slot length. This was defined as the intersection point giving the minimum stress condition at the two slot locations. It can be seen in Fig. 7 that the optimum values of center and outer slot length established in this manner were 35.50 m m and 37.00 mm. One important result was that the curves representing stress conditions at the two slot locations and at the fillet region were found to be unaffected for the most part by changing the gage area thickness. The most important effect of changing this variable was to shift the position of the gage area curve
ELLIS ET AL. ON A E R O P R O P U L S I O N M A T E R I A L S
389
in the vertical sense relative to the stress axis. Thus, selecting the optimum gage area thickness simply involved identifying the curve that fell above the intersection point with some reasonable level of conservatism. The optimum value of gage area thickness selected in this manner was 1.25 mm. Additional stress analyses were then conducted to determine the optimum value of fillet radius. This approach was possible because changing this variable did not have a major effect on the stress states at the slots and within the gage area. As illustrated in Fig. 8, these results were also plotted to determine the minimum feasible value of fillet radius which was 26.9 mm for the case shown. A value of 27.94 mm was selected as being optimum for the particular specimen design as it provided some margin of conservatism.
Final Specimen Designs Given this encouraging result, attention was shifted to the design of gripping and attachment methods and to modifying the initial specimen design as found to be necessary. By way of background, it was anticipated that load levels as high as _+222 kN would be needed in tests planned for aeropropulsion materials of interest. Relatively simple attachment methods using bolts, for example, as the primary means of load transfer proved inadequate, primarily because of the previously noted size constraints. The approach adopted to resolve this difficulty was to incorporate tapers on the specimen fingers to allow load transfer by means of shear. This change was made reluctantly as it was viewed as introducing a major element of complexity into the specimen design. A further change was that the width of the slots was increased to reduce stress levels at the root locations and to increase margins of conservatism. Details of the final specimen design featuring a circular gage area are given in Figs. 9 and 10. One change from the initial design is that the width of the fingers is reduced from 19 mm to 16 mm. This
O Center slot A Outer slot [] Fillet radius
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390
ELLIS ET AL. ON AEROPROPULSIONMATERIALS
391
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allowed the slot width to be increased from 10 m m to 14 mm. Further, an 8~ ' --- 15' taper was incorporated on the gripped section to facilitate load transfer into the specimen (Fig. 10). Optimization exercises similar to those described above were conducted for two values of overall plate thickness, 19 nun and 25.4 mm. The optimized set of dimensions for the two thicknesses are summarized in Table 1, and the results of stress analyses are summarized in Table 2. The relative merits of the designs will be discussed later in the paper. TABLE 1--Summary of specimen types and optimized specimen designs. Optimized Specimen Dimensions (mm) Specimen Type
TI
T2
SLA
SLB
FR
R1
Design with 19 mm overall thickness, and 14 mm slot width
2.1 2.2 2.3
1.524 1.524 2.032
6.35 6.35 6.35
35.56 30.48 30.48
36.45 30.78 30.78
40.64 50.80 50.80
44.45 44.45 44.45
Design with 25.4 mm overall thickness, and 14 mm slot width
3.1 3.2 3.3
2.032 2.032 2.540
12.70 12.70 12.70
35.56 30.48 30.48
36.45 30.78 30.78
40.64 45.72 45.72
48.46 48.46 48.46
Specimen Details
NOTE: L1 = 80.65 mm and TR1 = 25.4 mm throughout.
3.3
3.2
3.1
2.3
3l 6 (_+2.6%) 305 (_+2.4%) 283 (_+2.8%)
361 (_+2.7%) 352 (_+ 1.9%) 317 (_+2.0%)
MPa
1
1
1
1
1
1
Normalized
241
253
293
268
288
326
MPa
0.85
0.83
0.93
0.85
0.82
0.90
Normalized
Center Slot (maximum)
248
248
293
270
271
330
MPa
0.88
0.81
0.93
0.85
0.77
0.92
Normalized
Outer Slot (maximum)
248
249
272
297
299
346
MPa
0.88
0.82
0.86
0.94
0.85
0.96
Normalized
Fillet Radius (maximum)
NOTES: Values in brackets are % deviations about the mean. Normalization of von Mises equivalent stress obtained using average gage section stress.
Design with 25.4 mm overall thickness, and 14 mm slot width
2.1
Design with 19 mm overall thickness, and 14 mm slot width
2.2
Specimen Type
Specimen Details
Gage Section (average)
Von Mises Equivalent Stress at Location Shown
TABLE 2--Summary of stress results at critical locations for optimized specimen designs.
Z
rrl -n O
c m > z c3
w" -n
X
r--
03 r PO
ELLIS ET AL. ON AEROPROPULSIONMATERIALS
393
Reusable Fixturing Design and Analysis The first objective of the fixturing design work was to identify the optimum attachment method for specimens with the slot and finger configuration described above. A number of attachment methods were considered during initial design studies and all but two were rejected, primarily because of concerns regarding complexity and cost. Work on the two most promising concepts was continued through detailed design and in one case through manufacture. The preferred approach uses a yoke arrangement which has the important advantage of not requiring any further modifications to the specimen designs described earlier. The design and analysis process followed in achieving a final design is outlined as follows.
Design Details As a first step, a number of changes were made to the slotted finger attachment (Fig. 3) which served as a starting point for this work. It was recognized that the design in its original form was going to be difficult and expensive to manufacture and, with this in mind, the original unit design was broken down into nine subcomponents. These were eight fingers and a single mounting plate for assembly purposes. Details of this assembly are shown in Figs. 11 and 12. Here it can be seen that the mounting plate incorporates eight slots to accommodate the fingers and that setscrews are used for
152
TI
/ / /
Mounting plate ~
Slots are 2.50 N wide with 1.25 ///i radii at each end ~ " tl i
UIIIU•l
152
i
i. . . .
i-
--
Slotted
finger ~ . . . .
]ii
..... L
~ 14
,
~
I
14, typical ~ . ~ 3 places 106
Note: All dimensions in millimeters. FIG. 11--Slottedfinger attachment: plan view.
- ~ 1 6 , typical
v l
4 places
394
MULTIAXIAL FATIGUE AND DEFORMATION
Mounting p l a t e ~ . . ..... -.
33-~....... - ~
~-
9.5
~+o--~+-r..r " ~ " ' ~
-L.
f-L ~J
Slotted - f i n g e r s - ~ ....
7o30'+0o15,
,
typical 2
f r
152
......
places~.._~/I "
Flat and undercut,
typical 2 places
Preload
i
bracket
Note: All dimensions in m i l l i m e t e r s . FIG. 12--Slottedfinger attachment: side view.
assembly purposes. The finger design includes tapers on two surfaces to match the corresponding tapers on the specimen and the yoke. A further design feature worth noting is the fiats which were provided to ensure positive location between the end of the specimen and the fingers. An undercut was included at the same location to ensure proper specimen seating and also to provide some flexibility for specimen gripping. The important features of the yoke gripping arrangement are shown in Figs. 13, 14, and 15 where, for simplicity, a single pair of fingers is shown attached to a rectangular block. The plan was to first subject the fixturing to detailed evaluation under uniaxial loading. The prototype fixturing shown in these figures was manufactured specifically for this purpose. The primary function of the yoke fixture is to prevent the fingers from separating under load and to prevent relative motion between the specimen and the fingers. Stated differently, the yoke's function is to ensure that the end of the specimen stays in full contact with the mating surfaces of the fingers under both tensile and compressive loadings. This condition was to be achieved by effectively clamping the ends of the specimen between finger pairs by applying suitable preload to the assembly. This preloading was to be achieved using the two preload bolts in conjunction with the tapers on the mating surfaces of the fingers and the yoke. The first goal of analysis was to determine whether effective clamping could be obtained without overstressing either the bolts or the yoke. Also, the stiffness and load transfer characteristics of the attachment method were investigated to establish the useful loading range available with the design.
ELLIS ET A L ON AEROPROPULSION MATERIALS
395
Top view
r-- Uniaxial specimen I I I I
T . _ ~
__ _' ~.~ mm
o __
I I T F" --'-----L "----"=-= ~ P'J aTl--.~-r------~.,,/~-7,7",7
ILII.~q - ~.~-JL~---L;.I
~
~ "tl
/
I~ I~
56 mm
|
|
I
I
_L-,,"I- -~-r--r .... -~-I Preload bolt - J
' I
~ I
r
I
~-- Yoke
Side view
,~- ~-'--~ , . . ~ q - - - ~ ' - - ,~,:,!
I I
I I I
',I I
,
f/
'~-~ '' .'~ I I II
'~-~1 I| I ', I iI 9
\\
~-- Water cooling Note: FIG.
Taper angle on yoke = 7 ~ 00'.
13--Yoke gripping arrangement and setup for evaluation under uniaxial loading.
FIG. 14--Prototype fixturing in partially disassembled form.
396
MULTIAXlALFATIGUE AND DEFORMATION
FIG. 15--Prototype fixturing in assembled form.
Stress Analysis Details The ANSYS finite-element code, version 5.4, was again used for this work. The plan was to model one half of the assembly shown in Fig. 13 and to investigate the characteristics of the attachment method over a range of uniaxial loading conditions. The effect of simply preloading the assembly was the first loading case considered. Bolt preloading in these analyses was obtained by applying known displacements to the underside of the bolt head. Further, it was of interest to investigate how the stress state resulting from preloading was modified by the superimposition of external loading. As in earlier analyses, external loading was applied by constraining surface nodes in the gripped region to displace predetermined amounts in the loading direction. The magnitudes of these displacements were varied to simulate strength tests under both tensile and compressive loading. As with most general purpose finite-element codes, contact elements are available within ANSYS for analyzing joints and attachments. Line contact elements were used in the present work to model the contact surfaces between the mating components. One complication with this approach is that a value of contact stiffness has to be specified up front for these nonlinear analyses. Also, it is well known that the correct choice of contact stiffness is critical in regard to the accuracy of the solution and also in regard to the time taken to converge on a solution. To address this issue, a series of preliminary analyses was conducted investigating the effect of varying contact stiffness over the range 1.776 x 101 to 1.776 • 106 kN/mm. The highest value was selected for the final analyses as it was shown to provide reasonably accurate solutions within acceptable time periods. As would be expected, a coefficient of friction value has to be specified which is judged typical for the surface condition of the mating parts under consideration. In the absence of experimental data, a value of 0.1 was assumed for the majority of analyses and a limited number of spot checks were conducted using a value of 0.2. As in the case of the earlier work, elastic constants used in these analyses were handbook values for Inconel 718.
Stress Analysis Results The results shown in Figs. 16 and 17 were obtained in one of a large number of stress analyses performed on the yoke assembly. These data were obtained using a contact stiffness of 1.776 X l06 kN/mm, a coefficient of friction of 0.2, and a bolt preload of 31.6 kN. Analysis of the results showed that the maximum stress in the yoke, 692 MPa, occurred at location (2) in Fig. 16. The maximum stress in the finger, 1792 MPa, occurred at location (4) in Fig. 17. Clearly, the preload assumed in this
ELLIS ET AL. ON AEROPROPULSION MATERIALS
397
FIG. 16~Stress distribution in central section of yoke for 31.6 kN bolt preload.
analysis was less than ideal since it resulted in unacceptably high stresses in both the yoke and the finger. This situation was corrected in subsequent analyses by using more realistic values of bolt preload. One result of changing this variable was to change the location of maximum stress. For example, in analyses using a bolt preload of 5.816 kN and a coefficient of friction of 0.1, the maximum stress condition occurred at location (1) in Fig. 16 and at location (3) in Fig. 17. The results of all analyses performed using a contact stiffness of 1.776 • l 0 6 kN/mm are summarized in Table 3.
Stiffness and Load Transfer Characteristics The efficiency of the yoke attachment method was investigated by generating plots of applied grip displacement versus the corresponding specimen load. Such a curve is shown in Fig. 18 for a bolt preload of 5.816 kN and a coefficient of friction of 0.1. As would be expected, the performance of the fixture under tensile loading is significantly different from that under compressive loading. This is because the load path between the specimen and the fingers is completely different for the two loading cases. Under tension, a minor change in slope occurs at point (A) and a major change occurs at point (B) where slippage apparently occurs. Under compression, behavior is more straightforward with a minor change in slope occurring at point (C). The above data are summarized in more quantitative form in Table 4 along with results obtained for other combinations of preload and coefficient of friction.
398
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 17--Stress distribution in tapered section of finger for 31.6 kN bolt preload.
TABLE 3--Stress analysis resultsfor the yoke gripping arrangement. Coefficient of Friction
Bolt Preload (kN)
0.1
5.816 (0.381m m) 5
0.2
31.60 (0.762 ram)
Grip Displacement (ram)
Bolt Load (kN)
Specimen Force (kN)
0
5.816
0
0.152
5.790
12.725
0.254
6.016
17.298
-0.152
6.966
-14.140
-0.254
6.962
-23.000
0
31.600
0
0.152
31.620
12.721
0.254
31.620
21.760
Maximum Yoke Stress (MPa)
Maximum Finger Stress (MPa)
450 (1) 450 (1) 519 (1) 411 (1) 394 (1) 692 (2) 704 (2) 709 (2)
718 (3) 763 (3) 830 (3) 826 (3) 909 (3) 1792 (4) 1772 (4) 1765 (4)
NOTE: (1) and (2) see Fig. 15 for maximum stress locations in yoke; (3) and (4) see Fig. 16 for maximum stress locations in finger; and (5) corresponding bolt head displacement in mm.
ELLIS ET AL. ON AEROPROPULSION MATERIALS
399
22.5 18.0 - -
13.5 -z -~
9.0 --
4.5O '0 9 E -4.5 "~. r -9.0 --
_
/
13.29~ // i 0.1 50 // / / - - Secant i ~p ~" modulus Secant ~ , = 88.6 modulus /,, "~- Initial slope = 90.33 --~TZ I", = 91.8 yC '"~ Initial slope 0.150/ = 92.8 /
-13.5
-18.0 ~-- /
~ ] 1/.55 3 I
J
I
I
I
-22.5
-25 -20 - 1 5 - 1 0 -5 0 5 10 15 20 25 Grip displacement, m m x l 0 -2
Note: Assumed coefficient of friction = 0.1 and bolt preload = 5.816 kN. FIG. 18--Stiffness and load transfer characteristics under tensile and compressive loading.
An attempt was made to investigate the cause of the slope changes using the contact status option available with ANSYS. Using this option, the status of contacting surfaces is given in three categories: 1. 2. 3.
Gap closed, no sliding. Gap closed, sliding. Gap open.
TABLE 4---Stiffness and load transfer characteristics of the yoke gripping arrangement.
Coefficient of Friction
Bolt Preload (kN)
0.1
5.816 (0.381/rnm) 2 11.62 (0.762 mm) 31.64 (0.762 mm)
0.2
Maximum Grip Displacement (mm) 0.254 -0.254 0.381 -0.351 0.254
Initial Slope (kN/mm)
0.150 Secant Modulus (kN/mm)
Limit of Proportionality Load (kN)/Displacement (nun)
91.8 92.8 96.4 91.44 93.93
88.6 90.3 91.26 (I) 92.15
5.19/0.0559 5.36/0.0584 12.36/0.130 (1) 11.04/0.119
NOTE: (1) Discontinuities in the load versus displacement data prevented determination of these values. (2) Corresponding bolt head displacements in ram.
400
MULTIAXIALFATIGUE AND DEFORMATION
Data of this type were determined at frequent intervals during simulated loading for both the finger/yoke interface and the specimen/finger interface. The approach adopted in analyzing the data was simply to identify any significant changes in contact status that occurred during the various stages of loading. Overall, the data did not appear reliable and proved difficult to analyze. For example, there were no obvious changes in contact status at load/displacement combinations corresponding to points (A), (B), and (C) in Fig. 18. One expected result was that the status at both interfaces was predominantly category (1) for the case of preload only. Also, as expected, the status at the specimen/finger interface changed to mixed (2) and (3) almost immediately on load application as a result of specimen straining. Beyond this, the data were judged to have little value and will not be discussed further.
Discussion As noted at the beginning of this paper, the aim of this research was to develop specimen designs and fixturing which would allow in-plane biaxial tests to be conducted on a wide range of aeropropulsion materials including advanced metallics and composites. The plan was to develop optimized specimen designs with relatively simple geometries to facilitate manufacture and to keep costs within reasonable limits. Further, reusable fixturing was to be developed for specimen gripping and loading purposes that would incorporate the flexures needed to decouple the applied biaxial loading. The fixturing was to be manufactured using conventional structural alloys, again with the goal of keeping costs within acceptable limits. Inspection of Figs. 9 and 10 and Tables 1 and 2 shows that the goal of developing a relatively simple specimen design was met as a result of this research. The design feature which enabled this result was the arrangement of slots and fingers used for attachment and loading purposes. As expected, the geometry of the slots played a key role in obtaining optimized specimen designs. Over the course of the design process, the width of the slots was progressively increased to a final value of 14 mm with the aim of reducing stress levels at the ends of the slots. Thus far, the slots have been configured using a simple circular detail at the root location with ease of manufacture in mind. Clearly, the possibility exists that noncircular geometries could be used to give improved results. Slot length also played an important role in the optimization process. It was shown that best results in terms of stress distribution at the root locations were obtained when the shorter slot was located on the specimen's centerline. It was also shown that increasing slot length caused increased stress levels at the ends of the slots and reduced stress levels at the fillet radius. Best results were obtained for slot lengths in the range 30.48 to 38.10 mm and fillet radii in the range 40.64 to 50.80 ram. Interestingly, these results applied for all the various specimen thicknesses considered. The importance of these results is that it allows some flexibility in the choice of gage area thickness (T1). For example, increased values of this variable might be preferred in test programs investigating the effects of preexisting notches or defects. One advantage offered by the slotted finger attachment is that the flexures are located some distance from the specimen gage area. This allows the use of generous fillet radii at the specimen corners to reduce the stress concentrations at this location. As indicated in Table 1, fillet radii as high as 50.80 mm were used to obtain feasible specimen designs. Such an approach is not possible with the current NASA cruciform specimen design shown in Fig. 1. In this case, use of large fillet radii increases the stiffness of the outermost flexures resulting in unacceptably high stress concentrations at the corner locations. The results summarized in Table 1 show that it is possible to develop feasible specimen designs for a range of specimen dimensions. Six designs are shown in this table with overall thicknesses of 19 mm and 25.4 mm and with gage area thicknesses ranging from 1.524 mm to 2.540 ram. One method of evaluating the various designs is to normalize the maximum stresses at the slot and fillet radius locations using the average gage area stress. This procedure was followed with the results shown in
ELLIS ET AL. ON AEROPROPULSION MATERIALS
401
Table 2. The normalized values give an indication of "margin of conservatism" and can be used to rank the various designs. Adopting this approach, review of the data showed that most favorable results were obtained for center and outer slot lengths of 30.48 mm and 30.78 mm. With one exception, the normalized values were 0.88 or less, giving about a 10% design margin. The ranking process was carried one stage further by evaluating the results for various gage area thicknesses. Best results were obtained for a 1.524 mm gage area thickness in the case of the 19 mm specimen, and for a gage area thickness of 2.032 mm in the case of the 25.4 nun specimen. The margin of conservatism for these particular designs was about 20%. The above result illustrates that the subject specimen design offers some flexibility in meeting the particular size requirements of test programs investigating the behavior of advanced materials. Turning to the design of reusable fixturing, the slotted finger attachment in its final form is shown in Figs. 11 and 12. One important modification was that the original unit design was broken down into nine subcomponents with the primary aim of simplifying manufacture. Subsequent to the design study, eight slotted fingers were fabricated from Inconel 718 using the electrical discharge machining (EDM) method. The wire EDM method was used to machine the somewhat complicated finger profile shown in Fig. 12 in a single part about 32 mm wide. This part was then cut into two 16 mm widths to form a matched pair of fingers. This approach ensured the symmetry of the finger pair about the central plane of the fixture. The obvious concern here was load alignment and the need to minimize the effects of variability in machining. A similar issue addressed during the manufacturing exercise concerned the tolerances specified for the various tapered surfaces. As noted earlier, one important goal of the fixturing design was to ensure that the end of the specimen stays in full contact with the fingers under both tensile and compressive loading. For this condition to be achieved, analysis showed that careful attention had to be given to the taper angles specified for the mating parts. Assuming a less than perfect machining job, it was shown that the taper angle on the finger should be less than that on the specimen and greater than that on the yoke. For simplicity, an angle of 7030 ' --_ 15' was selected for both taper angles on the slotted fingers. The taper angle selected for the specimen was 8000 ' -+ 15' and that selected for the yoke was 7000 ' • 15'. It remains to experimentally verify that this approach will provide the required results in terms of gripping efficiency and effective load transfer under both tensile and compressive loading. Turning to the yoke attachment, the final design is shown in Figs. 13, 14, and 15 and the results of stress analysis are shown in Figs. 16 and 17 and in Table 3. It should be noted that a single value of contact stiffness, 1.776 • 106 kN/mm, was used throughout this work. Initial analysis focused on investigating the effect of varying bolt preload and coefficient of friction on the stress distribution in the assembly. Stress analysis results are summarized in Table 3 for two combinations of bolt preload and coefficient of friction (/x). Here, it can be seen that for/z = 0.2 and bolt preload = 31.6 kN, the maximum stress in the yoke is 692 MPa and that in the finger is 1792 MPa. Since the ultimate tensile strength for Inconel 718 in an aged condition is about 1448 MPa, the latter stress was known to be unrealistic. However, data for this particular combination were retained as it provided useful insight regarding the load transfer characteristics of the assembly for higher values of coefficient of friction and bolt preload. More reasonable results were obtained for/z = 0.1 and bolt preload = 5.816 kN. For this combination, the maximum stress in the yoke was 450 MPa and that in the finger was 718 MPa. These values were judged acceptable as the 0.2% yield strength for Inconel 718 in an aged condition is about 1172 MPa. The analyses were carried one stage further by simulating tile effect of specimen loading. This was done by applying a range of grip displacements in both the tensile and compressive senses. One interesting result was that superimposition of tensile loading had little effect on the bolt preload, whereas compressive loading caused the bolt preload to increase by about 20%. Stresses in the yoke were found to increase by about 15% under tensile loading and to decrease by about 13% under compressive loading. In contrast, stress levels in the finger increased by about 22% for both tensile and
402
MULTIAXIALFATIGUE AND DEFORMATION
compressive loading. The important result here is that the major component of stress in the assembly resulted from bolt preloading and that subsequent specimen loading had a relatively minor effect. Data regarding the stiffness and load transfer characteristics of the assembly are shown in Fig. 18 and in Table 4. Note that the data shown in this figure were determined for a coefficient of friction of 0.1 and for a bolt preload of 5.816 kN. It can be seen in Fig. 18 that at relatively low load levels, the initial slope for tensile loading is 91.8 kN/mm and that for compressive loading is 92.8 kN/mm. Relatively small changes in slope occurred at load (kN)/displacement (mm) combinations of 5.19/0.0559 in the case of tensile loading and 5.36/0.0584 in the case of compressive loading. The 0.150 secant modulii for the two loading directions were 88.6 kN/mm and 90.33 kN/mm. Thus, the magnitude of the slope changes at points (A) and (C) are relatively small, about 5% on average. Under tensile loading, a major change in slope occurred at point (B) which corresponds to a load level of about 14.20 kN. Load-carrying capability was lost at this point giving an upper limit on the useful range of the fixture for the particular combination of bolt preload and coefficient of friction considered. Under compressive loading, behavior was better behaved with near-linear response extending to at least 22.50 kN. Turning to the results summarized in Table 4, similar behavior as that described above was observed for higher values of coefficient of friction and preload. More specifically, for a coefficient of friction --- 0.1 and a bolt preload = 11.62 kN, the initial slope in tension was 96.41 kN/mm and that in compression was 91.44 kN/mm. Further, the 0.150 secant modulus for tensile loading was 91.26 kN/mm. The magnitude of these values was very close to those determined for a bolt preload = 5.816 kN indicating that the value of initial slope is not a function of this variable. Under tensile loading, the change in slope occurred at a load (kN)/displacement (mm) combination of 12.36/0.130. Apparently, increasing bolt preload by a factor of two effectively doubled the initial linear range of the fixture. Inspection of Table 4 shows that this trend was not continued when bolt preload was increased to 31.64 kN for a coefficient of friction of 0.2. In this case, increasing preload did not result in any increase in the linear range. One important result here is that the stiffness characteristics of the fixturing are not a function of coefficient of friction or bolt preload. Had this been the case, the design would not have been useful for in-plane biaxial testing. Conclusions The following conclusions were drawn from this design study aimed at developing improved specimen designs and fixturing for in-plane biaxial testing: 1. The feasibility of using specimen designs incorporating relatively simple arrangements of slots and fingers for loading purposes was demonstrated by analysis for conventional structural alloys. 2. A number of optimized specimen designs were developed with gage area thicknesses ranging from 1.524 to 2.032 ram. These designs were suitable for investigating material behavior under equibiaxial stress states. 3. Reusable fixturing was developed incorporating an assembly of slotted fingers which provide the flexibility needed to decouple the applied biaxial loading. This assembly can be reconfigured as necessary to obtain optimum biaxial stress states in the specimen's gage area. 4. A yoke gripping arrangement was developed which facilitates specimen loading while avoiding the need for holes or other forms of discontinuity in the specimen. Future Work The slotted finger and yoke fixtures will be subjected to detailed experimental evaluation under uniaxial loading with the focus on stiffness and load transfer characteristics. Given a positive result, a second series of experiments will investigate the performance of the fixturing under in-plane biaxial loading.
ELLIS ET AL. ON AEROPROPULSION MATERIALS
403
References [1] Johnson, A. E., "Creep Under Complex Stress Systems at Elevated Temperatures," Proceedings, Institution of Mechanical Engineers, Vol. 164, No. 4, 1951, pp. 432---447. [2] Mtnch, E. and Galster, D., "A Method for Producing a Defined Uniform Biaxial Tensile Stress Field," British Journal of Applied Physics, Vol. 14, 1963, pp. 810-812. [3] Pascoe, K. J. and DeVillers, J. W. R., "Low-Cycle Fatigue of Steels Under Biaxial Straining," Journal of Strain Analysis, Vol. 2, No. 2, 1967, pp. 117-126. [4] Wilson, I. H. and White, D. J., "Cruciform Specimens for Biaxial Fatigue Tests: An Investigation Using Finite Element Analysis and Photoelastic Coating Techniques," Journal of Strain Analysis, Vol. 6, 1971, pp. 27-37. [5] Pascoe, K. J., "Low Cycle Biaxial Fatigue Testing at Elevated Temperatures," Proceedings, 3rd International Conference on Fracture, Munich, Verein Deutscher Eisenhuttenleiite, Dusseldorf, 1973, Vol. 6, paper V-524/A. [6] Hayhurst, D. R., "A Biaxial-tension Creep Rupture Testing Machine," Journal of Strain Analysis, Vol. 8, No. 2, 1973, pp. 119-123. [7] Parsons, M. W. and Pascoe, K. J., "Low Cycle Fatigue Under Biaxial Stress," Proceedings, Institution of Mechanical Engineers, Vol. 188, 1974, pp. 657~571. [8] Mon-ison, C. J., "Development of a High Temperature Biaxial Testing Machine," Leicester University report, Vol. 71, No. 13, 1974. [9] Odqvist, F. K. G., Mathematical Theory of Creep Rupture, Oxford Mathematical Monographs, 2nd ed., Clarendon Press, Oxford, 1974. [10] Weerasooriya, T., "Fatigue Under Biaxial Loading at 565~ and Deformation Characteristics of 2 1/4% Cr1% Mo Steel," Ph.D. Thesis, University of Cambridge, Jan. 1978. [11] Duggan, M. F., "An Experimental Evaluation of the Slotted-Tension Shear Test for Composite Materials," ExperimentalMechanics, 1980, pp. 233-239. [12] Charvat, I. M. H. and Garrett, G. G., "The Development of Closed Loop Servo-Hydraulic Test System for Direct Stress Monotonic and Cyclic Crack Propagation Studies under Biaxial Loading," Journal of Testing and Evaluation, Vol. 8, 1980, pp. 9-17. [13] Brown, M. W., "Low Cycle Fatigue Testing Under Multiaxial Stresses at Elevated Temperature," Measurement of High Temperature Properties of Materials, M. S. Loveday, M. F. Day, and B. F. Dyson, Eds., HMSO, 1982, pp. 185-203. [14] Henderson, J. and Dyson, B. F., "Multiaxial Creep Testing," Measurement of High Temperature Properties of Materials, M. S. Loveday, M. F. Day, and B. F. Dyson, Eds., HMSO, 1982, pp. 171-184. [15] Jones, D. L., Poulose, P. K., and Liebowitz, H., "Effect of Biaxial Loads on the Static and Fatigue Properties of Composite Materials," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 413-427. [16] Radon, J. C. and Wachnicki, C. R., "Biaxial Fatigue of Glass Fiber Reinforced Polyester Resin," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 396--412. [17] Found, M. S., "A Review of the Multiaxial Fatigue Testing of Fiber Reinforced Plastics," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 381-395. [18] Brown, M. W. and Miller, K. J., "Mode I Fatigue Crack Growth Under Biaxial Stress at Room and Elevated Temperature," Multiaxial Fatigue, ASTM STP 853, K. J. Miller and M. W. Brown, Eds., American Society for Testing and Materials, Philadelphia, 1985, pp. 135-152. [19] Sakane, M. and Ohnami, M., "Creep-Fatigue in Biaxial Stress Using Cruciform Specimens," Third International Conference on Biaxial/Multiaxial Fatigue, Vol. 2, University Stuttgart, Paper No. 46, 1989, pp. 1-18. [20] Susuki, I., "Fatigue Damage of Composite Laminate under Biaxial Loads," Mechanical Behavior of Materials--W, Vol. 2 (ICM 6), Pergamon Press, Oxford, 1991, pp. 543-548. [21] Trautman, K.-H., Dtker, H., and Nowack, H., "Biaxial Testing," Materials Research and Engineering, H. Buhl, Ed., Springer Verlag, Berlin, 1992, pp. 308-319. [22] Makinde, A., Thibodeau, L., and Neale, K. W., "Development of an Apparatus for Biaxial Testing Using Cruciform Specimens," Experimental Mechanics, Vol. 32, 1992, pp. 138-144. [23] Demmerle, J. and Boehler, J. P., "Optimal Design of Biaxial Tensile Cruciform Specimens," Journal of the Mechanics and Physics of Solids, Vol. 41, No. 1, 1993, pp. 143-181. [24] Boehler, J. P., Demmerle, S., and Koss, S., "A New Direct Biaxial Testing Machine for Anisotropic Materials," ExperimentalMechanics, Vol. 34, 1994, pp. 1-9. [25] Wang, J. Z. and Socie, D. F., "A Biaxial Tension-Compression Test Method for Composite Laminates," Journal of Composites Technology & Research, Vol. 16, No. 4, Oct. 1994, pp. 336-342.
404
MULTIAXIAL FATIGUE AND DEFORMATION
[26] Masumoto, H. and Tanaka, M., "Ultra High Temperature In-Plane Biaxial Fatigue Testing System with InSitu Observation," Ultra High Temperature Mechanical Testing, R. F. Lohr, and M. Steen, Eds., Woodhead Publishing Limited, Cambridge, 1995, pp. 193-207. [27] Bartolotta, P. A., Ellis, J. R., and Abdul-Aziz, A., "A Structural Test Facility for In-Plane Biaxial Testing of Advanced Materials," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 25-42. [28] Trautmann, K.-H., Maldfeld, E., and Nowack, H., "Crack Propagation in Cruciform 1MI 834 Specimens Under Variable Biaxial Loading," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 290-309. [29] Dalle Donne, C. and D6ker, H., "Plane Stress Crack Resistance Curves of an Inclined Crack Under Biaxial Loading," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, 1997, pp. 243-263.
Claudio Dalle Donne, t Karl-Heinz Trautmann I and Hans Amstutz 2
Cruciform Specimens for In-Plane Biaxial Fracture, Deformation, and Fatigue Testing REFERENCE: Dalle Donne, C., Trautmann, K.-H., and Amstutz, H., "Crueiform Specimens for InPlane Biaxial Fracture, Deformation, and Fatigue Testing," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 405-422. ABSTRACT: Three different types of cruciform specimens, which have been used successfully on the DLR biaxial test rig for investigations in the fields of fracture mechanics, yield surface evaluation, and fatigue of riveted joints are presented in detail. The following three characteristics were required from these specimens: a uniform stress and strain distribution in the central testing regions, low manufacturing costs, and easy mounting in the DLR biaxial test rig. It is shown that the first and the second requirement are in conflict. Because of the uneven stress distribution in the simple and inexpensive fracture mechanics specimen, additional finite-element calculations were needed for a full fracture mechanical characterization. On the other hand, impelling a very uniform stress distribution in the riveted cruciform specimens increased the manufacturing costs considerably. The specimen which satisfies the demands best is the yield locus evaluation specimen of deep drawing steel. It has low manufacturing costs and a very uniform strain distribution in the central testing section. KEYWORDS: cruciform specimen, biaxial loading, stress intensity factor, plastic limit load. yield surface, deep drawing, compression testing, riveted joint, fiber metal laminate
Nomenclature a
B E F1, F2 Fl,y J
KI N W Y1, Y2 A O'1, 0"2 O'1 ,lig, O-2,1ig O'vM3ig O-y
.y
Half-crack length in center cracked cruciform specimen Thickness Young' s modulus Forces in main and secondary loading axis respectively, F1 --> F2 Plastic limit load Mode I J-integral Mode I stress intensity factor Ramberg-Osgood hardening exponent, N > 1 Half-width of center cracked cruciform specimen Finite width correction factors for KI induced by F1 and F2, respectively Biaxiality ratio, F2/F1 Gross stresses in main and secondary loading axis, respectively, o"1 ~ 0-2 Ligament stresses in main and secondary loading axis, respectively von Mises equivalent ligament stress Yield strength Ligament width normalized by specimen width
1 Research engineer and senior engineer, respectively, Institute of Materials Research, German Aerospace Center DLR, Linder Hthe, D-51147 Cologne, Germany. 2 Senior research engineer, Department of Mechanics of Materials, University of Darmstadt, Petersenstr, 13, D64287 Darmstadt, Germany. 4O5
Copyright9
by ASTM lntcrnational
www.astm.org
406
MULTIAXlALFATIGUE AND DEFORMATION
Introduction The importance of studying the effects of biaxial loading on fracture, fatigue, and deformation behavior of materials and structures has been recognized for many years. The objectives of such biaxial investigations were either the better understanding of biaxial loading effects on the failure or deformation behavior of materials, or a more realistic simulation of the complex loading situation of the structure to be assessed. The most realistic experimental test method to create in-plane biaxial loading in flat sheets is the in-plane biaxial test with cruciform specimens (see review in Ref 1). A uniform stress and strain distribution in the central rectangular or circular testing region of the crossshaped specimens is usually achieved through an array of slots in the loading arms, Fig. 1. The applied load is distributed by the material's "fingers" along the edge of the testing region. Moreover, the slots minimize the specimen strain cross-sensitivity between the two loading axes, since as the load is applied on one axis, the individual fingers on the other loading axis are able to flex relatively freely. As shown in Fig. 1, the thickness in the gage section is sometimes reduced to ensure maximum stresses to occur in this region. In this paper, three different types of cruciform specimens, which have been used successfully on the DLR biaxial test rig [2] for investigations in the fields of fracture mechanics, fatigue, and plastic deformation, are presented in detail. They were optimized for uniform stress and strain distributions in the central testing regions, for easy mounting in the DLR biaxial test rig and for low manufacturing costs.
Fracture Mechanics Cruciform Specimen The fracture mechanics cruciform specimen was used in a basic research program on biaxial load effects on stable crack growth in ductile steel and aluminum alloy sheets [3-7]. To keep manufacturing costs low, the possibility of a thickness reduction of the cruciform specimens was rejected at the beginning of the program. Therefore, only the influence of the slots in the loading arms on the stress intensity factor was investigated in a preliminary finite-element study. The mesh of a Mode I cracked cruciform specimen employed in the plane stress finite-element (FE) calculations is displayed in Fig. 2. Due to symmetry, a quarter mesh was used with the relevant boundary conditions applied to the specimens edges. The mesh consisted of 560 eight-node isoparametric elements with four integration points. These elements were employed through the whole specimen, also in the refined area around the crack tip. The FE program delivered J-integral values ob-
slotted
F-
FIG. 1--Schematic of a typical cruciform specimen for in-plane biaxial testing.
DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS
407
=;LF
FIG. 2--Finite-element model of the cruciform fracture mechanics specimen used for stress intensity factor evaluation and plastic limit load calculations.
tained through the virtual crack extension method of Parks and De Lorenzi [8,9]. Since in linear elasticity the J-integral corresponds to the energy release rate, the stress intensity factor could be calculated from the well known relationship between K~ and the energy release rate [10] K, = x / ~
O)
The notation E indicates the Young's modulus. The high accuracy of this approach (within 1% of the analytical solution of a simple cracked configuration) was proven in a previous study [ll]. The loading arms contained 11 slots, which were modeled by uncoupled knots on both sides of each slot. Figure 3 shows the influence of the slot length on the stress intensity factor KI of a cruciform specimen with a crack to width ratio of a/W = 0.25 at different applied load biaxiality ratios h. Here h is defined by the ratio of the crack parallel load F2 to the load perpendicular to the crack F1. The stress intensity factor is normalized by the KI= value
El
of an infinite plate under the same remote biaxial loading. In this particular case, remote crack parallel loading (/;2) has no influence on KI,= [12]. Very long, open ended slots introduced uniform stress and strain distribution in the specimens. The K1 values are therefore very close to the infinite plate solution for any biaxiality ratio A. Decreasing slot length increased the stiffness of the loading arms. As shown in Fig. 4, the load trajectories are deviated in the adjacent loading arms causing severe notch stresses at the comer fillets of the specimens.
408
MULTIAXlAFATI L GUEANDDEFORMATION 1.1 ~ -I
1
21q 18o /
.
0.8
0
~
~
W=150mm ao/W=0"25
0.7
-1.o
~
0
-
slot length [ram]
I
-o.5
o'.o
;L
1.o
FIG. 3--1nfluence of the slot length on stress intensity factor of a cruciform specimen.
Especially in the case of tensile crack parallel stresses (A > 0) these notch stresses led to an unloading of the cracked region and therefore to K~ values which are considerably lower than the idealized Kr,= values. It should be noted here that the independence of the infinite plate K~,~has often led researchers to the misinterpretation that also in the case of finite specimens the load biaxiality ratio has no influence on Kr. It is clear from Fig. 3 that this assumption is only true in connection with very long slots in the loading arms. The often contradictory experimental results of biaxial load effects on fracture toughness and fatigue crack growth can therefore be explained with a misinterpretation of the data caused by the wrong K1 calibration [13-16]. To prevent buckling problems at negative A, an intermediate value of slot length of 45 mm was chosen for the fracture mechanics specimen. The final shape is shown in Fig. 5. A cutout in the less load carrying area of the loading arms simplified the gripping procedure, since the servohydraulic actuators of the biaxial test rig could remain almost in their final testing position.
FIG. 4---Deviation of force trajectories in adjacent loading arms in cruciform specimens.
DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS
409
FIG. 5--Final shape offracture mechanics cruciform specimen (all dimensions in ram). The gripping procedure was simplified by a cutout in the loading arms.
The stress intensity factor KI is presented as the sum of the components perpendicular (F1) and parallel (F2) to the crack K1 = 2__B~ V~-aa~y1 (_~)
F2
2-gff
a
where Y1 and Y2 are the finite width correction factors for Kr induced by F1 and F2, respectively. The Y1 and Y2values obtained by the FE calculations are displayed in Fig. 6. Since the crack parallel component (Y2) is negative, dominant crack parallel loading could lead to crack face contact and interaction. The FE calculations were performed with and without the central 8-mm-wide hole. It is evident from Fig. 6 that the hole affects the crack parallel correction factor for cracks shorter than a/W = 0.4. The main loading axis correction factor is fitted by the following equation Y~ = 0 . 9 9 1 + 0.786 ( W ) 2532
(4)
The secondary loading axis correction factor for the specimen with the central hole is approximated by Y2 = -0.1 + 0.0544 W
(5)
410
MULTIAXlAL FATIGUE AND DEFORMATION 1.5"
O 1.4-
Y1 with central 8 mm hole Y
1.3> Z 1.21.11.0-
g
w
0.9 0.0 ' 0'.1 ' 012
017
0.8
-0.10 0.0 ' 0'.1 ' 012 ' 0'.3 ' 0'.4 ' 015 ' 0'.6 ' 0:7 a/W
0.8
-0.05|
ix
0.3 ' " 0:4 a/W
0.5 . . . 0.6 .
Y2 with central 8 mm hole
/
-0.061 -0.071 >_~ -0.08-
-0.09-
FIG. 6---Correction factors for stress intensity factor of cruciform fracture mechanics specimen.
whereas without the central hole Y2 is given by
Y2 = -0.04966 - 0.03476 cos 1.1126 ~ ~-
(6)
Taking into account the uncertainties of the FE calculations, these fits should be accurate within 2.5% for 0.2 --< a/W <--0.8. In fracture mechanics, the extreme opposed to linear elastic fracture is the fully plastic fracture, which is often described by the ligament yielding load Ft,r. A simple limit load estimation for the cracked cruciform specimen is obtained by assuming constant von Mises equivalent stresses (rvM,lig in the ligament. These equivalent stresses are equaled to the material's yield strength (rr: CraM,rig= X/~r~jig + ~Jig -- ~x,lig~ZJig= ~ r
(7)
DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS
411
Knowing that or I O'l,lig = - -
3'
O'2,1ig = AO" 1
y=l
(8)
a W
one obtains the main loading axis limit load F1,y by inserting Eq 8 into Eq 7 and solving for 0"1:
2BWo'r
(9)
1
Fly = /
,/
A 2 + T-"~- - - -"~
To check the validity of the limit load estimation, two-dimensional plane stress FE analyses were performed using conventional deformation plasticity with a Ramberg-Osgood stress-strain law and a very low hardening exponent of N = 100. The analyses were conducted using the FE mesh of Fig. 2. The plastic limit load was obtained from the intersection of the elastic and fully plastic branches of the double logarithmic J versus F1 curve as shown in the insert of Fig. 7. In linear elasticity the relationship between J and F1 is quadratic (Eq 1). In the fully plastic regime, J scales with the (N+ 1) power of FI [10]. Therefore the change in slope of the double-logarithmic J-F1 curve can be used as an indication that the plastic limit load of the cracked structure is reached. In Fig. 7 the von Mises limit load estimation is compared with the results of the FE calculations. Equation 3 qualitatively predicts the trends of increasing limit load with increasing A up to a value of 1 and then again lower limit loads for A = +2, but especially for long cracks (a/W > 0.5) and 0 --< A -< + 1 the accuracy is very low. As discussed in the linear elastic case, the discrepancies between the semi-infinite limit load solution and the FE calculations of the real specimen are due to the unloading of the central region by the load-carrying crack parallel specimen arms. To account for this effect
1
/
J3
10
F evaluationfrom FE calc.
+1
9
0
~
.-j
~o.8 Od
! "
~->.0.6,,~
~
"
+2 _ - - - - - ~ .
-, ~
0.4-
2
0.2-
sym
"
~
-=
~
~
.
~
~
-
~
1,Y
Ig
1
~ ' ~
9
v o n Mises eqn. (9)
0.0
0.0
0.2 '
0.4 ' a/W
0
16
o'.8
FIG. 7--Comparison of the simple limit load estimation based on the yon Mises criterion to the FE calculations.
412
MULTIAXIAL FATIGUE AND DEFORMATION
1.2 1.0-
+1 ,~
.....
0a0.6-
0.2-
-
+2.,~= . . . . . . . . ~ . . . . . . . .
~
~
symbols FE results . . . . . yon Mises eqn. (9) with y*=0.7(0.5+ ~,)
0.0
0.0
0:2
0]4 a/W
0:6
0[8
FIG. 8--Comparison of the limit load estimation based on the von Mises criterion with fictitiously lengthened ligaments to the FE calculations.
the ligament length 3' in Eq 9 is fictitiously lengthened by replacing 3' with 3'* = 0.7(3' + 0.5)
(lO)
This linear relationship was obtained by fitting Eq 9 to the FE results of A = 0. Even though only the uniaxial loading case was considered, a general improvement of the limit load prediction of Eq 9 can be observed, Fig. 8. A simple experimental J-integral estimation formula based on the area under the load displacement curve and Eq 9 is found in Ref 3. It was shown in this paragraph that the lack of agreement between the actual and the ideal KI and F1,y formulas is connected with a nonuniform stress distribution in the testing area of the cruciform specimen. The uniformity of the stress distribution could be improved by a thickness reduction in the central part of the specimen. As stated at the beginning of this paragraph, milling or grinding in thickness direction was, however, rejected to keep the production costs low. In fact, the great advantage of the cruciform fracture mechanics specimens are the very low manufacturing costs, since a stack of five specimens was machined at the same time by laser (steel) or waterjet cutting (aluminum alloy).
Specimens for Yield Locus Evaluation Initial yield surfaces, the variation of these surfaces with biaxial straining, and the dependencies of the yield loci from given plastic predeformations were required as input parameters for numerical deep drawing simulations of automotive body parts. In collaboration with the company3 performing the simulations for German carmakers, a cruciform specimen able to sustain up to 10% uniaxial strain had to be developed. Starting from a very simple configuration, Fig. 9, the fillet comers of the specimens were successively sharpened and slots in the loading arms were introduced to achieve the wanted deformation level in the test section. Besides heterogeneous strain distributions in the testing area, the main problem was premature failure of the loading arms. The final optimized shape of the specimen is shown in Fig. 10. Extremely long and compliant loading arms of a higher strength steel were 31NPRO GmbH,Berlin, Germany.
DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS
413
680
R150
FIG. 9--Simple cruciform specimen for yield locus evaluation of deep drawing sheets used as the starting point for the optimization process (dimensions in mm).
FIG. lO---Optimized cruciform specimen for yield locus evaluation of O.8- to 1.5-mm-thick deepdrawing sheets (dimensions in ram).
414
MULTIAXIALFATIGUEAND DEFORMATION
electron beam welded to the quadratic testing piece of 0.8 to 1.5 mm thick deep drawing steels [17,18]. In the test program two deep-drawing steel qualities, which covered most of the automotive applications, were tested: the low strength steel Stl4 (or = 140 MPa) and the microalloyed high strength steel ZStE 340 (Or = 340 MPa). Higher strength loading arms were electron beam welded to testing sections: ZStE 340 to St14 and ZStE 690 (~y = 690 MPa) to ZStE 340. Additionally, some biaxial tests with the deep-drawing aluminum alloy A1Mg5Mn (similar to 5182, o'y = 140 MPa) were performed [19]. Here the loading arms had to be glued to the quadratic testing region with a standard aerospace adhesive. Therefore, each loading arm of the Al-specimens consisted of two pieces. The final shape of the specimens and the slots in the loading arms were always cut after the joining procedure by laser jet cutting in the case of the steel and by waterjet cutting in the case of the Al-alloy specimens. The uniformity of the strain distribution in the gage section of the specimens was controlled by FE calculations, by strain gage measurements and by post-test evaluations of the plastic displacements of grids scribed on the specimen surfaces, Fig. 11. All examinations yielded a very uniform strain distribution (within _ 12% of the average strain) in the central 80 mm times 80 mm part of the quadratic gage. An example of the plastic strain distribution in the Fl-direction of an equibiaxial (A = FE/F1 = l) test with an aluminum alloy specimen is given in Fig. 12. The strains were calculated from the displacements of the intersection points of the distorted grid of Fig. 11. The highest strains were found at the measurement points close to the specimen comers (recall Fig. 4). During yield locus experiments, the displacements in both loading axes were measured by two standard clip gages over a span of 10 mm. The two clip gages were fixed in the center of the specimens on either side of the sheet by rubber bands and magnetic or glued pads, Fig. 13. Equibiaxial straining up to 9% equivalent strain in the steel specimens and up to 4% equivalent strain in the aluminum alloy specimens was accomplished without failure of the specimens. At higher loads the spec-
F2
I
120
~.~
FIG. 11--Grid scribed on the gage section of a yield locus evaluation specimen (dimensions in ram).
DALLE DONNE ET AL. ON CRUCIFORMSPECIMENS
415
strain values in Fl-direction [%] ~,,,3.90-,~ ~,,-3.50-~ ~M3.40-,~ II~.3.80,.~-4
~..3.95.~ ~..3.60-~ ~-3.4s-~ ~.-3.7s-~ -3 ~.3.90-~ ~.-s.4s-~ ~-3.60.~ ~.-3.75.~ -2 ~-3.8s,~ ~.s.50..), ~-3.70-), ~-3.7s..),-1 ~.4.1 o-~ ~ - s . ~
.~.3.4o.~ ~s.8o~ -0
4 3 2 1 0 FIG. 12--Plastic strain distribution in F1 direction after an equibiaxial AlMg5Mn specimen obtained from the distorted grid of Fig. 11.
(,h = F 2 / F 1 =
1) test of an
imens usually failed in the welded or glued joint. Sometimes crack growth from the fillet comers towards the center of the specimens was observed. Obviously the very compliant cruciform specimens cannot be loaded with compressive stresses. Since some points in the negative sectors of the yield surfaces were also needed for the numerical deep drawing simulations, special load frames for compression testing in standard testing machines
FIG. 13--Clip gages for displacement measurements in F1 and F2 direction on the yield surface evaluation specimens.
416
MULTIAXIAL FATIGUE AND DEFORMATION
were developed, Figs. 14 and 15. Both frames were designed for small, quadratic sheet samples, which could be cut out from previously strained cruciform specimens. With the very simple uniaxial load frame of Fig. 14 additional points on the negative 0-~ and 02 axes of the yield locus diagram were gained, whereas the equibiaxial compression apparatus shown in Fig. 15 delivered a further point in the negative 0"]-0"2sector. Theoretically the alignment of the hinged joints with the lateral edge of the specimen should enforce a pure biaxial compression on the quadratic samples. Under these circumstances also the magnitude of the compressive loading should be easily calculated from the testing machine load output. In reality, however, small misalignments caused an overall tilting of the load frame, which altered slightly the load biaxiality ratio at the specimen. Therefore, load ceils were mounted between the pressure plates and the specimen to monitor the real loading conditions. Currently some design changes to the testing device are planned that should ensure defined biaxial loading conditions. Out-of-plane buckling was prevented through Teflon(trifluoroethylene)-coatedplates in both compression devices (not shown in Figs. 14 and 15). These plates were slightly smaller than the quadratic samples and slightly pressed on the samples by a screw. The displacement was again measured with standard clip gages with a span of 5 mm. An example for the yield locus evaluation is given in Fig. 16 where the influence of uniaxial and equibiaxial prestraining on the 0.05% yield locus of a deep drawing steel is investigated. In each plot the yield ellipse of the virgin material is compared to the yield surface after a defined predeformation. As may be expected, the yield surfaces of the predeformed specimens were increased in the relative prestraining direction. Equibiaxial prestraining expands and translates the yield ellipse towards the positive 0]-0-2 sector. Just as most of the structural metals, the steel therefore shows a combination of isotropic (expansion of yield ellipse) and kinematic (translation of yield ellipse) hardening.
testing machine grips
/
~20~
a screw extensometer
FIG. 14---Cut through the uniaxial compression device for small, quadratic sheet samples (all dimensions in mm).
DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS
417
FIG. 15--Apparatus for biaxial compression testing of small quadratic sheet samples in a conventional testing machine (all dimensions in mm).
Cruciform Specimens with Riveted Joints Two types of cruciform specimens with riveted central testing parts, Fig. 17, were developed in cooperation with Fokker and NLR 4 for a comparison study between a standard fuselage aluminum alloy (2024-T3) and a modern metal fiber laminate called GLARE [20,21]. This material is built up from thin aluminum alloy sheets and intermediate adhesive layers which are reinforced with crossply glass fibers [22]. The configuration and loading conditions of the specimens simulated the conditions encountered in fatigue critical areas of an airplane fuselage. The longitudinal lap joint specimen represented a longitudinal lap joint of the crown section of a fuselage. The shear lap joint specimens characterized a riveted fuselage part close to the wing attachment where additional shear stresses are present. In the
4 Fokd~erAerostrucnlres, The Netherlands, NLR: National Aerospace Laboratory, The Netherlands.
MULTIAXIALFATIGUE AND DEFORMATION
418
l
500 300 Q.
~1oo -100
j/
-300
/1
f-/-
\.J
-100
100
ooii
F,{F27oo ' 300 500
-300
I'1\
9
-100
0% pre-def. / 6% pre-deI.
-300
~f ~ __...~."~-
300
/7" [
-500 -500
f
,,,,..,,-
500
Iw~.~
~
~ v
-500 -500 -300 -100
"
i ,
,
100
500
300
0.1 [Mpa] 50o
0.2, F2
300 100 0"1, F 1
f
./"0% )re-def.
t
j
/Z
-100 -300
I
/
.//
/
~i
_,~ I 4% pre-def. -F1/F2=1 -500 ' ' ' -500 -300 -100 100 300 500 FIG. 16--Influence of a uniaxial or biaxial predeformation on the 0.05% yield locus of the ZstE 360 deep drawing steel (the lines are best fit ellipses).
context of a realistic simulation of the fuselage loading, two different biaxial fatigue load sequences were applied to the specimens. More details on riveting, load spectra, and the carrying out of the experiments are found in Refs 20 and 21. The cruciform riveted specimens can be regarded as an intermediate test between a very simple uniaxial fatigue test with limited information value and a very expensive full-scale fatigue test of a curved fuselage panel or even complete pressurized section (so-called "barrel") of the airplane body. Biaxial loading of the specimens is of crucial importance for the simulation of single lap joints in the biaxially stressed upper fuselage, since the tensile load parallel to the joint (axial direction in Fig. 17) straightens the specimen and prevents tilting. This specimen tilting, which is present under uniaxial loading of a rivet joint, is due to secondary bending stresses which are induced by the eccentric nature of a single lap joint. The riveted rectangular parts of 1.27-mm thick 2024-T3 and 0.85 to 1.1-mm thick fiber metal laminate constituted the central part of the specimens. All parts of the specimen arms were made of 1- to 2-mm-thick 2024-T3. Similarly to fracture mechanics specimens, specimen handling during the gripping operation was facilitated by the shorter loading arm in the axial direction. Each individual arm consisted of several parts to compensate the out-of-plane eccentricity of the single lap joint and to provide symmetrical loading conditions. The blanks of these arms where machined by milling and
DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS
419
FIG. 17--Configuration of the riveted cruciform specimens (all dimensions in mm).
16
0
E d)
E
AI2024-T3 rivet: . - ~ 15 right
12-
21 right 21 left 24 right
8-
O r
longitudinal lap joint max. axial stress 191 MPa max. tang. stress 130 MPa fuselage spectrum loading
G L A R E 3 3/2 rivet: --O-, 6 left --II-- 12 right ---A-, 15 right --.~. 18 left A,. ~-,.~l
t--
L. 0
9
o
i
loo
260
number of flights [xl000 ]
400
FIG. 18--Crack propagation rates in a longitudinal lap joint specimen. The numbers in the legend indicate individual rivets [20,21].
FIG. 19--Crack formation in the A l layers o f a f i b e r laminate longitudinal lap j o i n t which was dismantled after 360 000 simulated flights.
z
5
0 m -n 0 ~J
z (D
c m
x w"1"1
cw"
o
4~
DALLE DONNE ET AL. ON CRUCIFORM SPECIMENS
421
assembled with bonding layers and centering pins in a mounting device. The whole assembly was then pressed between the heating plates of a press to cure the glue. The final shape including the slots in the specimen arms were again machined by a numerically controlled water jet cutting device. The high expenditure was primarily needed to achieve a very uniform stress distribution in the central part of the riveted specimens. The uniform stress distribution is essential for this type of specimen since it prevents preferential loading of single rivets close to the loading arm attachments which may cause premature cracking and specimen failure. A typical result of the investigations is shown in Fig. 18 where crack length at the individually numbered rivets is plotted as a function of simulated flights. The crack lengths were measured optically through a microscope on the top sheet of the lap joint, Fig. 19. The fatigue cracks grew much faster in the standard aluminum alloy. The very slow crack growth in the fiber laminate GLARE is attributed to the crack bridging of intact fibers in the wake of the crack. While the crack growth remains limited to the aluminum alloy layers, the crack bridges the fibrous restrain crack opening and therefore reduces the local driving force for metal crack advance. Some GLARE specimens where dismantled after three or four times the design life was reached. A qualitative hint on the uniformity of the stress distribution was given by the damage distribution in the single aluminum layers. The cracks were randomly arranged around the rivet holes of the load-exposed third rivet row of the top sheet and first rivet row of the bottom sheet, Fig. 19. This suggests that no particular load concentrations or preferential loading paths were present in the testing area.
Concluding Remarks The following characteristics were required for the cruciform specimens presented here: a uniform stress and strain distribution in the test area, low manufacturing costs, and easy mounting in the DLR test rig. While the last demand was rather easy to meet, the requirements of low manufacturing costs and a uniform strain distribution in the gage section were in conflict. The inexpensive fracture mechanics specimen has the disadvantage of an uneven stress and strain distribution in the central testing part. Therefore, finite-element calculations were needed to fully characterize this specimen. On the other hand, the very uniform stress distribution in the riveted specimens was attained through an expensive assembly process. The specimen which satisfies the demands best is the yield locus evaluation specimen of deep-drawing steel. It has low manufacturing costs and a very uniform strain distribution in the central testing section. Acknowledgments
The authors thank Mr. C. Sick and Mr. H. Frauenrath for their experimental assistance.
References [1] Demmerle, S. and Boehler, J. P., "Optimal Design of Biaxial Tensile Cruciform Specimens," Journal of the Mechanics and Physics of Solids, Vol. 41, No. 1, 1993, pp. 143-181. [2] Trautmann, K.-H., Maldfeld, E., and Nowack, H., "Small Crack and Macrocrack Propagation in Cruciform IMI834 Specimens under Variable Biaxial Loading," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., ASTM, Philadelphia, t997, pp. 290-309. [3] Dalle Donne, C. and Drker, H., "Biaxial Load Effects on Plane Stress J-Aa and 85-Aa-Curves," Proceeding of the lOth European Conference on Fracture ECF 10, Vol. 2, K.-H. Schwalbe, and C. Berger, Eds., Engineering Materials Advisory Services Ltd, 1994, pp. 891-900. [4] Dalle Donne, C. and Drker, H., "Plane Stress Crack Resistance Curves of an Inclined Crack under Biaxial Loading," Multiaxial Fatigue and Deformation Testing Techniques, ASTM STP 1280, S. Kalluri and P. J. Bonacuse, Eds., ASTM, Philadelphia, 1997, pp. 243-263. [5] Dalle Donne, C., Obertragbarkeit yon Ri/3widerstandskurven yon Standardproben auf biaxial belastete, bauteilfihnliche Kreuzproben, VDI Verlag, DUsseldorf, 1997.
422
MULTIAXIALFATIGUE AND DEFORMATION
[6] Dalle Donne, C., "A CTOD Approach to Assess Stable Tearing under Complex Loading Conditions," The Proceedings, The 2nd Joint NASAIFAA/DoD Conference on Aging Aircraft, NASA/CP-1999208982/PART 2, C. E. Harris, Ed., 1999, pp. 555-564. [7] Dalle Donne, C., "The Crack Tip Displacement Vector Approach to Mixed-Mode Fracture," Mixed-Mode Crack Behavior, ASTM STP 1359, K. J. Miller and D. L. McDowell, Eds., ASTM, Philadelphia, 1999, pp. 21-40. [8] Parks, D. M., "The Virtual Crack Extension Method for Nonlinear Material Behaviour," Computer Methods in Applied Mechanics and Engineering, Vol. 12, 1977, pp. 353-364. [9] De Lorenzi, H. G., "Energy Release Rate Calculations by the Finite Element Method," Engineering Fracture Mechanics, Vol. 21, 1985, pp. 129-143. [10] Anderson, T. L., Fracture Mechanics--Fundamentals and Applications, 2nd ed., CRC Press, Boca Raton, FL, 1995. [11] Amstutz, H. and Seeger, T., Untersuchungen zur Rissnahfeldapproximation mit Finiten Elementen, Internal Report, University of Darmstadt, Germany, 1991. [12] Williams, M. L., "On the Stress Distribution on the Base of a Stationary Crack," Journal of Applied Mechanics, Vol. 24, 1957, pp. 109-114. [13] Moyer, E. T. and Liebowitz, H., "Biaxial Load Effects in the Mechanics of Fracture," Journal of the Aereonautical Society oflndia, Vol. 36, 1984, pp. 221-236. [14] Eftis, J. and Jones, D. L., "Influence of Load Biaxiality on the Fracture Load of Center Cracked Sheets," International Journal of Fracture, Vol. 20, 1982, pp. 267-289. [15] Kibler, J. J. and Roberts, R., '~'lae Effect of Biaxial Stresses on Fatigue and Fracture," Journal of Engineeringforlndustry, Trans. ASME, Vol. 92, 1970, pp. 727-734. [16] Hunt, R. T., "Crack Propagation and Residual Strength of Stiffened and Unstiffened Sheet," Proceedings of the Symposium on Current Aeronautical Fatigue Problems, J. Schijve, J. R. Heath-Smith, and E. R. Welbourne, Eds., Pergamon Press, 1963, pp. 287-324. [17] Mathiak, F. U., Krawietz, H., Nowack, H., and Trautmann, K.-H., German Patent DE 3914966 C1, 1989. [18] Mathiak, F. U., Fuchs, F. A., Trautmann, K.-H., and Frauenrath, H., "Biaxpriifung von Tiefziehst~len des Karosseriebaus," Werkstoffpriifung '92, DVM, Berlin, 1992, pp. 139-147. [19] Trautmann, K.-H. and Fedelich, B., "Bestimmung von Flie/3ortkurven einer Aluminiumlegierung bei biaxialer Beanspruchung fiir die numerische Simulation des Tiefziehens," DLR Werkstoff-Kolloquium 1993, Metallische Leichtbauwerkstoffe, W. A. Kaysser, Ed., DLR, 1993, pp. 17-19. [20] Trautmann, K.-H., Dalle Donne, C., and Schendera, C., "Biaxial Fatigue of Riveted Lap Joints in Fiber Metal Laminates, Mis-Matching of Interfaces and Welds," M. Koqak and K.-H. Schwalbe, Eds., GKSS Research Center Publications, Geesthacht, 1997, pp. 489-500. [21] Trautmann, K.-H., Dalle Donne, C., and Schendera, C., "Riveted Lap Joints of the Fiber Laminate GLARE under Biaxial Loading," Proceedings of the 19th Symposium of the International Committee on Aeronautical Fatigue ICAF 97, Vol. 2, P. Poole, Ed., Engineering Materials Advisory Services Ltd., 1997, pp. 1001-1014. [22] Roebroeks, G. H. J. J., "Fibre-Metal Laminates, Recent Developments and Applications," International Journal of Fatigue, Vol. 16, No. 1, 1994, pp. 33-42.
Jeffry S.
Welsh I
and Donald F. A d a m s 2
Development of a True Triaxial Testing Facility for Composite Materials REFERENCE: Welsh, J. S. and Adams, D. F., "Development of a True Triaxlal Testing Facility for Composite Materials," Multiaxial Fatigue and Deformation: Testing and Prediction, ASTM STP 1387, S. Kalluri and P. J. Bonacuse, Eds., American Society for Testing and Materials, West Conshohocken, PA, 2000, pp. 423-437.
ABSTRACT: An electromechanical testing facility capable of applying any combination of tensile and/or compressive forces to three mutually orthogonal axes of a thickness-tapered composite cruciform specimen was designed, fabricated, assembled, and evaluated. Any stress ratio in biaxial (crx - r or triaxial (o1 - 02 - 0"3)stress space can be explored using this computer-controlled test facility. A brief description of the testing machine and its capabilities as well as the present test specimen design is included. Once fully assembled, uniaxial and biaxial tests were performed on 606 l-T6 aluminum using this facility. The excellent agreement between the uniaxial and biaxial results obtained in the present study for this material, with accepted handbook values and applicable failure theories, confirmed the performance of several aspects of the testing facility. These aspects included the intra-axis alignmerit, machine compliance, specimen fabrication and testing procedures, automated computer testing algorithms, data acquisition algorithms, and calibration values. In addition, biaxial and triaxial tests were performed on an AS4/3501-6 carbon/epoxy cross-ply laminate. While most of these tests are considered valid, they revealed aspects of the present thickness-tapered cruciform specimen design that could be improved. More specifically, an undesirable failure mode was encountered in some biaxial tension tests, and triaxial tension tests were not performed successfully. Nevertheless, the overall acceptable performance of the triaxial testing facility is believed to have been demonstrated.
KEYWORDS: composite materials, biaxial and triaxial testing, thickness-tapered cruciform specimen, failure envelope, failure surface.
Major advances in the development of new classes of composite materials have been made by the composite materials research and development community during the past two decades. While test methods used to evaluate the in-plane shear, axial and transverse tension, and axial and transverse compression have made strides to meet the demand for experimental axial data [1,2], test methods for characterizing these materials under multiaxial stress states have not kept pace [3-5]. Considerable effort has been expended in developing suitable testing techniques for determining the multiaxial response of engineering metals, but only in limited areas of study such as developing testing machines and test methods, evaluating fracture and fatigue performance under multiaxial stress fields, and evaluating failure criteria [6]. While this has apparently satisfied the metallic materials community, the composites community has not been as fortunate. In an attempt to extend the knowledge gained from metals testing to composite materials, many new complications have been identified. Undoubtedly, the nonhomogeneous and anisotropic response of composite materials has been the source of most of the difficulty in making significant ad-
Assistant professor, Mechanical Engineering Department, South Dakota State University, Brookings, SD 57007. 2 Director, Composite Materials Research Group and professor, Department of Mechanical Engineering, University of Wyoming, Laramie, WY 82071-3295. 423
Copyright9
by ASTM International
www.astm.org
424
MULTIAXIALFATIGUEAND DEFORMATION
vances in this area. More specifically, there is very little existing capability in the composites community to characterize the full three-dimensional properties of composite materials, even though there is a large demonstrated demand for such information [7-10]. Obtaining the multiaxial (biaxial or triaxial) response of composite materials is particularly difficult because test methods have not been developed to the point where reliable data are being produced. Many test methods and specimen configurations have been proposed over the past two decades in an attempt to further advance the knowledge associated with multiaxial testing. While this initial research has produced a certain amount of useful experimental data, a single biaxial or triaxial test method that is capable of consistently producing acceptable results over the entire range of biaxial and triaxial stress space has not emerged [11]. The test methods used to evaluate the multiaxial response of composite materials simply have not been pursued as rigorously as those used to evaluate the uniaxial material response. Thus, the composite materials community is currently in a confused state as to how to accurately determine the multiaxial response of composite materials. An important consequence of this lack of reliable experimental data is that existing failure theories have not been fully evaluated. Numerous failure theories have been proposed in the past in an effort to accurately predict the response of composite materials. The use of these failure theories is a critical component in the design process as they represent the only means to predict failure in composite structures without actually testing each structural component. An acceptable level of confidence in these proposed failure theories has not been developed because the existing failure theories have not been shown to perform satisfactorily even in carefully controlled laboratory tests. The lack of reliable experimental biaxial and triaxial data is directly responsible for preventing this critical step in the evolution of acceptable failure theories. It is this lack of multiaxial experimental data that led to the present study. A triaxial test facility capable of applying any combination of tensile or compressive forces to three mutually orthogonal axes of a composite specimen was fabricated in the present study. This test facility utilizes a thickness-tapered cruciform specimen to evaluate the in-plane (biaxial) material response. The same specimen configuration is used to perform triaxial tests by applying through-thethickness tensile or compressive forces to the gage section of the thickness-tapered specimen. A discussion of the primary features of the triaxial test facility and thickness-tapered cruciform specimen is included in this paper. Experimental results obtained in the present study from testing 6061T6 aluminum and an AS4/3501-6 carbon/epoxy cross-ply lanainate are also presented.
Machine Description Show in Fig. 1, the triaxial testing facility contains three primary subsystems referred to as the reaction frame, the test fixture, and the computer control system.
Reaction Frame The reaction frame represents the main structural component of the test facility and is used to provide precise alignment of all six force actuators, precise alignment of the test fixture with respect to these force actuators, and a working surface to support the test fixture. As shown in Fig. 1, two octagon-shaped reaction frames were assembled together to form the full triaxial test facility. Each separate reaction frame utilizes a sandwich frame design and was constructed from 51-mm-thick (2 in.) steel that was ground flat and parallel to within __+0.004mm (• in.) per linear meter. In addition, all of the alignment dowel pin and bolt holes used to assemble the triaxial test frame were located within • 0.013 and _+0.13 mm (_+0.0005 and ---0.005 in.) with respect to the center of the test frame, respectively. This level of tolerance allowed the precise alignment of all six force actuators to be achieved without using any post-assembly alignment techniques [11]. The horizontal frame represents the primary reaction frame as it contains the four actuators used to apply the in-plane forces for both biaxial and triaxial tests. The two axes in the horizontal frame have
WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY
425
FIG. 1--Photograph of the triaxial test facility.
been designated the X- and Y-axes. The horizontal flame measures 188 cm (74 in.) across any two parallel edges and stands 174 cm (68 in.) above the surface of the floor. While the bottom horizontal flame plate was designed to be essentially solid to provide the necessary working surface for the triaxial testing facility, the top horizontal flame plate measures 25.4-cm-wide (10 in.) around the perimeter. The top vertical reaction flame, which houses the Zl-actuator, measures 188 cm (74 in.) wide, 25.4 cm (10 in.) deep, and stands 94 cm (37 in.) from the top surface of the horizontal flame. The Z2-actuator is contained within the bottom vertical reaction flame that measures 188 cm (74 in.) wide, 25.4 cm (10 in.) deep, and hangs 77 cm (30.3 in.) from the bottom surface of the horizontal flame. Under the most demanding loading conditions, the deflection of the horizontal flame, top vertical frame, and bottom vertical frame is conservatively estimated to be 0.1, 1.5, and 1.0 mm (0.004, 0.06, and 0.04 in.), respectively [11]. Although the reaction frame was designed for maximum forces of _+133 kN (---30 kip), the six identical force actuators (two per loading axis) can only generate maximum axial forces of approximately ---94 kN (+21 kip) because of undersized drive motors. Utilizing two force actuators for each loading axis allows the position of the specimen centroid to be precisely controlled with respect to the center of the test frame during testing. Each force actuator uses a 2.25-in-diameter, 0.25-in-pitch lead screw enclosed by a zero-backlash ball beating screw assembly to provide the required translation of each axis. Each ball bearing screw is rigidly captured between the sandwich flame design of the three reaction frames and is powered by a stepper motor and accompanying gear train. This configuration results in a lead screw translation of approximately 0.0005 mm (0.00002 in.) per motor step when operated in full-step mode. In addition, each actuator is equipped with a load cell (mounted to the end of the lead screw), position transducer, and multiple limit switches to monitor and control the performance of each actuator during a multiaxial test. A complete description of the instrumentation used for this facility is included in Ref 12.
426
MULTIAXIAL FATIGUE AND DEFORMATION
FIG. 2--Photograph of the triaxial test fixture.
Test Fixture Because the ball beating screws used in the present triaxial testing facility exhibit a small amount of undesirable runout, the triaxial test fixture was designed as the part of the facility to overcome any misalignments induced by the actuators. Shown installed in the center of the test frame in Fig. 2, the test fixture is used to provide accurate alignment between the test specimen and the applied forces from the test frame for both biaxial and triaxial tests. The test fixture consists of two guide plates equipped with linear bearings that position the in-plane (X- and Y-axes) tension and compression housings as well as the Z-axis loading assemblies. This configuration provides a nearly frictionless interface between the test frame and the cruciform test specimen in all three loading directions. The critical dimensions of each of the test fixture components were maintained within at least --_0.05 mm (---0.002 in.), to help ensure that precise alignment was achieved between the test frame and the test specimen. Because the mass of the fully assembled test fixture exceeds 136 kg (300 lb), a fixture loading crane was also included as part of the test frame to facilitate installation of the fixture into the test frame. This fixture loading crane is shown in the upper right hand corner of Fig. 1. Each tension and compression housing serves as an interface between the force actuators and the in-plane loading arms of the thickness-tapered cruciform specimen. As a result, a total of eight housings (four tension and four compression) are required to perform all combinations of biaxial (T/T, T/C, and C/C) loading configurations. The tension and compression housings are also used to apply in-plane forces during a triaxial test. Each tension and compression housing utilizes conventional wedge grips to transfer the force generated by an actuator to the loading arms of the cruciform specimen. Two sets of interchangeable, reversible wedge grips (12 ~ taper angle) are used with the tension and compression housings to generate the necessary loading. Each set of wedge grips contains tungsten-particle gripping surfaces to enhance the coefficient of friction between the wedge grips and
WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY
427
untabbed specimen loading arms, and utilizes two alignment dowel pins to provide precise alignment of the grips relative to the tension and/or compression housings. In turn, the precise alignment of each tension or compression housing is maintained with respect to the test fixture framework through the use of four precision linear bearing assemblies per housing. The test fixture components described above represent the portion of the test fixture referred to as the biaxial test fixture. To apply the through-the-thickness forces necessary to perform triaxial tests, several additional components must be added to the biaxial test fixture. The primary components added to perform triaxial tests are referred to as Z-axis assemblies. One is shown in Fig. 2 positioned on top of the top guide plate; the other is mounted on the bottom guide plate. Each assembly houses two additional linear bearings that provide precise alignment of the Z-axis load trains relative to the center of the test specimen. However, because an essentially two-dimensional composite cruciform specimen is used in a triaxial test, conventional wedge grips could not be used on the Z-axes to transfer the force generated in the through-the-thickness direction to the test specimen. This obstacle was overcome by using two steel Z-axis attachments that are bonded to the gage section of the thicknesstapered cruciform specimen, as shown in Fig. 3. It should be noted that it is not necessary to adhesively bond the Z-axis attachments to the specimen to generate compressive stresses in the throughthe-thickness direction of the test specimen. The performance of this system in successfully generating through-the-thickness stresses is described in Ref 13. Once the Z-axis attachments are positioned with respect to the test specimen, additional load train components are used to form a rigid connection between the Z-axis attachments and the Z-axis force actuators. These components include two concentric loading shafts and a loading shaft end cap. Again, only precision linear bearings and shafts were used in the Z-axis load train, to ensure that the relative alignment between the Z-axis load train and the test specimen was accurate.
FIG. 3 ~ P h o t o g r a p h o f a triaxial tension test specimen with Z-axis attachments bonded to the gage section.
428
MULTIAXIALFATIGUE AND DEFORMATION
Computer Control Because performing a triaxial test involves actively controlling six force actuators, monitoring and recording data generated fi-om six force transducers, and recording numerous strain gage data simultaneously, it is almost mandatory that a computer be used to control the facility. For the present study, a 266 MHz Pentium II computer was selected to perform all of the tasks associated with biaxial and triaxial tests. These tasks include initializing the stepper motors, individually moving any actuator, pre-loading a test specimen, performing a biaxial or triaxial test, and analyzing subsequent experimental data. While many of these tasks do not require sophisticated computer programs, performing an accurate multiaxial test involves many automated algorithms. The burden this creates on the controlling computer's CPU was eased by utilizing two stepper motor controller boards and a single data acquisition board that were all equipped with onboard microprocessors. While this configuration added minor complexities, the controlling computer was operated in an essentially closed-loop configuration using force transducer feedback signals. The present triaxial testing facility is only configured to perform multiaxial static tests in a load control mode. However, this configuration was not found to be limiting as the primary function of the present test facility is to investigate the response of composite materials subjected to a predetermined ratio of biaxial or triaxial normal stresses. This is most easily achieved by controlling from force transducer feedback signals. Once a biaxial or triaxial test is initiated using the triaxial testing facility, the drive axes (X-axes) are moved at a constant crosshead rate in the appropriate direction. The slave axes (Y- and Z-axes) are automatically monitored and adjusted to maintain the desired stress ratio until ultimate specimen failure. Automated algorithms are included to monitor and correct for side forces (a difference between the force transducers of two opposing loading arms) and to acquire data. The current configuration allows the user to select (and vary during a test) any three-dimensional stress ratio, data acquisition rate, maximum allowable side force, maximum slave-axis crosshead rate, and maximum allowable stress ratio error through the use of real-time feedback controls. Using this configuration for the AS4/3501-6 carbon/epoxy cross-ply specimens tested in the present study, typical side force and stress ratio errors were found to be approximately 4% and 2%, respectively [11]. These data indicate that both the desired stress ratio and stress state were achieved in the gage section of the cruciform test specimen [13].
Specimen Description Because the new facility was designed to test composite materials subjected to either a biaxial or a triaxial stress state, several unique design considerations had to be imposed on the test specimen to ensure that this goal was achieved. While many studies have been performed to investigate the biaxial response of composite materials, few have been extended to include stresses in the third direction. The majority of previous studies have utilized either thin-walled tubular specimens or cruciformshaped specimens to investigate only the biaxial response of composite materials [6]. The extension of biaxial testing knowledge to tfiaxial testing is not an intuitive process. The addition of a third loading axis to a two-dimensional test specimen presents several unique challenges. Nevertheless, it has been demonstrated in the present study that a thickness-tapered cruciform specimen can be used to perform both biaxial and triaxial tests, by applying through-the-thickness forces perpendicular to the gage section. This general specimen design was developed after a literature review of existing biaxial and triaxial test methods. A thickness-tapered cruciform specimen was chosen since it is the only configuration capable of performing triaxial tests in any trl - trz - tr3 stress ratio using a single specimen configuration [11]. Thickness-tapered cruciform geometries have been previously investigated for biaxial test specimens only [14-16]. The specific thickness-tapered cruciform specimen developed and used for all tests in the present study is shown in Fig. 4. It is 161 mm (6.35 in.) long across op-
WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY
429
@
(8 places)
@
@
(~
24.7 mm (0.975
in)
1
@
~
4.o6
I
I I
mm
(o.~6 in)
[ t
9
~- 2.03 ..~i
1
mm
~
161.4 mm (6.354 ~n)
FIG. 4
(o.oB in) I I
I t
I 9
Schematic drawing of the biaxial/triaxial test specimen.
posing 24.8-mm-wide (0.98 in.) loading arms, each of which contain two 4.76-mm-diameter (0.188 in.) holes used to align each set of wedge grips. The gage section detail of this cruciform specimen is shown in Fig. 5. All dimensions shown in Fig. 5 assume a specimen thickness of 4.06 mm (0.160 in.) and a gage section thickness-taper fillet radius of 12.7 mm (0.50 in.). In this configuration, the gage section consists of a 21.6-mm-square (0.850 in.) test region with 1.59-mm-radius (0.063 in.) loading arm fillets used to merge adjacent loading arms at the corners of the gage section. The gage section detail shown in Fig. 5 was machined into both surfaces of all test specimens used in the present study to produce the desired symmetric thickness-tapered geometry. The specific specimen design shown in Figs. 4 and 5 was selected after performing an anisotropic, linear-elastic, finite-element analysis. Specific concerns included the location and magnitude of stress concentrations, the interaction of the three normal stress components, and the effect of laminate configuration. The finite-element analysis demonstrated the ability of this specimen to produce a nearly homogeneous triaxial stress state in the gage section [11]. In addition, the three normal stresses were found to be reasonably independent [11]. That is, appreciable normal stresses in one loading direc-
430
MULTIAXIAL FATIGUE AND DEFORMATION
15.5 mm(ty~)I / :
0.612 in (typ)
"]
R3.18 mm (typ) R1/8 in (typ)
R1.59 mm (typ) R1/16 in (typ)
lg,2z:It;;i
4.98 mm 0.196 in
R0.256 R6.50 mm
\ i
i
FIG. 5--Detail of the biaxial/triaxial test specimen gage section.
tion did not extend a significant distance into adjacent loading arms. This is critical in minimizing undesirable load sharing effects in cruciform-shaped test specimens. One of the primary concerns associated with testing cruciform specimens is the location and magnitude of stress concentrations. More specifically, the stress concentration generated at the intersection of two adjacent loading arms is a frequent concern [14-16]. The present finite-element analysis indicated that the magnitude of this in-plane stress concentration was approximately 2.5 for a quasiisotropic AS4/3501-6 carbon-epoxy laminate. This value increased to approximately 3.0 for a unidirectional laminate of the same material. Although not specifically modeled in the present study, it is assumed that similar stress concentrations would be present in a cross-ply laminate. One stress concentration that is desired when using thickness-tapered cruciform specimens is a direct result of thinning the gage section. Perhaps more accurately described as a stress riser, this geometric feature is exploited in an attempt to increase the stresses, and produce subsequent failures, in the gage section. The present finite-element analysis indicated that the specimen geometry shown in Figs. 4 and 5 produced a stress riser of approximately 1.25 [11]. Unfortunately, this value limits the laminate configurations that can successfully be tested, due to biaxial strengthening effects. That is, any degree of biaxial loading results in an increase in the ultimate strength of some laminates compared to uniaxial loading, meaning that the ultimate strength of a biaxially loaded laminate should be higher than a uniaxially loaded specimen. This is obviously an undesirable situation for a cruciform specimen that is loaded biaxially in the gage section and uniaxially in each of the four loading arms, as unacceptable failures will occur in the arms of the specimen rather than the gage section. As a resuit, it is believed that only cross-ply laminates can overcome this affect and be successfully tested
WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY
431
TABLE 1--Average uniaxia16061-T6 aluminum tensile and compressive yield strength, ultimate strength, and
modulus results. Average Yield Strength
Ultimate Strength
Average Modulus
Axis
Test Mode
MPa
ksi
MPa
ksi
CVa %
GPa
Msi
X Y Z X Y Z
T T T C C C
2871 2901 2922 -272 a -2761 -2782
41.611 42.11 42.32 -39.51 -40.01 -40.32
323 324 322 -279 -278 -339
46.8 47.0 46.7 -40.5 -40.4 -49.13
0.1 0.1 0.3 1.1 0.9 0.3
68.21 68.9 a 68.92 -68.21 -68.91 NA
9.91 10.01 10.02 -9.91 -I0.01 NA
CV 4
%
0.0 1.0 1.0 0.3 0.4 NA
NA - Not Available. 1 Average of 2 specimens. 2 Average of 3 specimens. 3 Specimen buckling. 4 Coefficient of variation.
using a thickness-tapered cruciform test specimen [8,11]. For this reason, only cross-ply laminates were tested in the present study.
Experimental Results 6061-T6 Aluminum As an initial evaluation of the performance of the triaxial testing facility, numerous uniaxial and biaxial tests were performed using 6061-T6 aluminum. Specifically, these tests were used to investigate the repeatability of, the accuracy of, and the intra-axis performance of the test machine under uniaxial and biaxial testing conditions. Experimental data generated were compared to established handbook values and applicable failure theories. A total of 30 uniaxial and 27 biaxial 6061-T6 aluminum specimens were tested. Tables 1 and 2 present the average uniaxial and biaxial results, re-
TABLE 2--Average biaxia16061-T6 aluminum yield strength results. Average X-Direction Yield Strength
Average Y-Direction Yield Strength
Specimen Group I.D.
Stress Ratio
MPa
ksi
CV 1%
MPa
ksi
CV 1%
JAHll0 JAH320 JAH310 JAH260 JAH160 JAH170 JAH680 JAH780 JAH660
1/1/0 3/2/0 3/1/0 2 / - 1/0 1/- 1/0 1/-2/0 -1/-3/0 -2/-3/0 -1/-1/0
306 333 317 213 172 109 -141 -261 -318
44.4 48.3 46.0 30.9 24.9 15.8 -20.5 -37.8 -46.1
1.6 1.8 0.8 1.8 0.0 2.5 3.2 6.7 0.9
303 227 112 - 112 - 168 -212 -347 -348 -316
44.0 32.9 16.2 - 16.2 -24.3 -30.7 -50.4 -50.5 -45.9
1.7 1.8 1.l 0.3 0.0 2.5 1.1 3.8 1.1
1 Coefficient of variation.
432
MULTIAXIALFATIGUE AND DEFORMATION
spectively. Each average value shown in Tables 1 and 2 was obtained from testing five and three individual specimens, respectively, unless otherwise noted. Details of specimen fabrication and testing procedures are presented in Ref 11.
Discussion of 6061-T6 Aluminum Results One of the most notable features of the data presented in Table 1 is that the average strength and modulus values were very consistent. Inspection of Table 1 reveals that the average ultimate tensile strength values for all three loading axes were within 0.4% of the mean value of 323 MPa (46.8 ksi), while the average modulus values were within 1.0% of the mean value of 68.9 GPa (10.0 Msi). Because all of the uniaxial compression specimens failed by gross (Euler) column buckling after initial specimen yielding, an analysis of the ultimate strength average values is of limited use. However, the average yield strength for the compression specimens were within 2.5% of the mean value of -275 MPa (-39.9 ksi), and the average compressive modulus values were within 0.7% of the mean value of -68.9 GPa ( - 10.0 Msi), These data indicate that the triaxial testing facility is capable of generating very consistent experimental data. Perhaps more important than the consistency of the data presented in Table 1 is the fact that the yield strength, ultimate strength, and modulus values obtained in the present study compare very favorably with accepted handbook values for this material. Values of 276 MPa, 310 MPa, and 68.9 GPa (40 ksi, 45 ksi, and 10.0 Msi) for the yield strength, tensile strength, and modulus of elasticity, respectively, are generally accepted handbook values for 6061-T6 aluminum [17]. The fact that comparable data were generated in the present study was interpreted as a verification of the numerous fabrication, testing, calibration, and data acquisition and reduction procedures that were developed specifically for the present study. In addition, numerous aspects of the triaxial testing facility itself were verified by obtaining consistent, accurate data from each of the three loading axes. These aspects included the reaction frame compliance, force actuator alignment, and the assumption that the various linear bearings provide a nearly frictionless interface between the test fixture framework and the tension and compression housings. While a comparison of the biaxial yield strength data presented in Table 2 for the various stress ratios is of limited use in form, these data can readily be compared to existing failure theories when plotted in ~rl - ~2 stress space. Figure 6 presents such a plot of the data, along with Tresca and von Mises yield criteria predictions. One of the most significant features of the data presented in Fig. 6 is that they are in close agreement with the von Mises yield criterion in all quadrants. The largest difference occurred for the three stress ratios in the C/C quadrant, in which the experimental data exceeded the von Mises prediction by approximately 13%. This discrepancy is believed to be primarily a result of the technique used to account for the biaxial load sharing between adjacent loading arms for the present specimen design. Very briefly, a biaxial tensile test specimen loaded uniaxially was used to experimentally determine the amount of load sharing between adjacent arms of the cruciform specimen [11]. It is believed that a more accurate representation of the tests performed in the C/C quadrant could be achieved by using a uniaxial compression specimen to establish the level of load sharing for specimens tested in this quadrant [11]. Another significant feature of Fig. 6 is that each group of three specimens tested at a particular stress ratio were very consistent. As with the uniaxial test specimen results, this was viewed as a confirmation of the biaxial specimen fabrication and testing procedures developed specifically for the present study. In addition, inspection of the individual specimen test results presented in Ref 11 indicates that the desired biaxial stress ratio was achieved at failure for nearly every biaxial test. This fact, coupled with the close agreement of the uniaxial and biaxial yield strengths obtained in the present study with established values generated reasonable confidence in the triaxial testing facility.
WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY
433
(1~) Cry (MPa)
Yield Surfaces . . . . -"3!-
60
Tresea | yon i~4iscs [ Experimental Data.,/
~_ 400 Axis o f
Symmetry
/ 200 /
/
/ /
/ //
-40O
/
/
/
/-200
200
400 fgP~)
r -60
-30
/
l
3o/
J
/ Uniaxial Results
60 (ksi)
/ / :,
/
,,M
-30 / 9~
~/-'~
Uniaxial Results
/
++-------~ _6o~- -400 ++
FIG. 6---Biaxial yield envelope for 6061-T6 aluminum.
AS4/3501-6 Carbon~Epoxy The triaxial test facility had been extensively evaluated using 6061-T6 aluminum uniaxial and biaxial tests. The primary objective of these tests was to characterize the biaxial and triaxial response of a composite material. It was anticipated that the orthotropic response of such a material would provide an additional challenge to the triaxial testing facility. A total of 27 biaxial [0/9019s Hercules AS4/3501-6 carbon/epoxy specimens were tested in o1 - 0"2 stress space. More specifically, the same stress ratios used to evaluate the response of the 6061-T6 aluminum specimens were used to evaluate the AS4/3501-6 carbon/epoxy material system in the T/T, T/C, and C/C quadrants. In addition, 12 triaxial tests were performed using this composite material to demonstrate the ability of the triaxial testing facility to perform triaxial tests in triaxial stress space. Tables 3 and 4, respectively, present the average biaxial and triaxial results for all [0/9019~ AS4/3501-6 carbon/epoxy specimens tested in the present study. Each average value shown in Tables 3 and 4 was obtained by testing three and two individual specimens, respectively, unless otherwise noted. All specimen fabrication and testing procedures are presented in Ref. 11.
Discussion ofAS4/3501-6 Carbon~Epoxy Results Although the data presented in Tables 3 and 4 are useful in comparing average results obtained using specific stress ratios, it is difficult to interpret these data in tabular form when evaluating the over-
434
MULTIAXIAL FATIGUE AND DEFORMATION
TABLE 3--Average biaxial ultimate strengths obtained from testing a [0/9019sAS4/3501-6 carbon~epoxy laminate. Average Y-Direction
Average X-Direction
Ultimate Strength
Ultimate Strength Specimen Group I.D.
Stress Ratio
MPa
ksi
CV 2 %
MPa
ksi
JSX1101 JSX3201 JSX310 JSX260 JSX160 JSX170 JSX680 JSX780 JSX660
1/1/0 3/2/0 3/1/0 2 / - 1/0 1/- 1/0 1/-2/0 -1/-3/0 -2/-3/0 -1/-1/0
437 487 465 465 432 252 -210 -428 -519
63.4 70.6 67.5 67.5 62.7 36.6 -30.4 -62.1 -75.3
5.4 12.1 11.5 7.3 4.8 4.2 11.1 3.8 2.6
449 322 157 -232 -433 -507 -633 -654 -519
65.1 46.7 22.7 -33.7 -62.8 -73.5 -91.8 -94.8 -75.3
CV 2
r
3.4 10.1 11.5 7.8 5.7 6.2 10.5 6.7 2.6
1 Specimens failed in loading arm. 2 Coefficient of variation.
all material response in ~rl - tr2 - t~3 stress space. Therefore, Fig. 7 is a graphical representation of these data. Presenting triaxial data on a two-dimensional plot requires the reader to realize that the actual data lies above or below the plane of the plot, as specified by the magnitude of the Z-direction stress. One of the most significant aspects of the biaxial data presented in Fig. 7 is that a reasonable amount of material scatter exists among each of the three specimens tested in a particular stress ratio group. Although each of the specimens failed very near the desired stress ratio, the magnitude of the applied stresses at failure exhibited differences as large as 17% for certain stress ratios. While this level of material scatter is undesirable, similar levels of material scatter for biaxial tests of the same AS4/3501-6 carbon/epoxy material system were obtained in prior studies for a quasi-isotropic laminate configuration [18,19]. The data presented in Fig. 7 also indicates that the present specimen design was successful in generating independent normal stress components in the gage section of the test specimen. Each of the
TABLE 4---Average triaxial ultimate strengths obtained from testing a [0/9019sAS4/3501-6 carbon~epoxy laminate. Average X-Direction
Average Y-Direction
Average Z-Direction
Ult. Strength
Ult. Strength
Ult. Strength
Specimen Group I.D.
Stress Ratio
MPa
JSXI 11 JSX161 JSX661 JSX116 JSX166 JSX666
1/1/11 1/-1/11 -1/-1/11 1/1/- 12 1/- 1/- 12 -1/-1/-12
467 407 -554 468 405 -561
ksi
CV3 %
MPa
67.8 59.1 -80.3 67.9 58.8 -81.4
2.5 2.5 12.4 2.2 11.4 4.4
458 -399 -550 463 -403 -571
ksi
CV3 %
MPa
ksi
CV3 %
66.4 -57.9 -79.7 67.1 -58.4 -82.8
0.5 2.7 16.3 5.2 11.5 4.0
0 0 0 - 146 - 143 -143
0 0 0 -21.2 -20.8 -20.8
0.0 0.0 0.0 3.3 3.5 0.1
1 Z-axis loading attachments debonded prior to ultimate specimen failure. 2 Actual stress ratios at failure were 1/1/-0.3, 1/- 1/-0.4, - 1/- 1/-0.3, respectively. 3 Coefficient of variation.
WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY (ksi)
O-
Y
435
('~a)
800 ~_
Experimental Dam
+
100
Bi~
-
Axis of Syrnm~
Z-Axis Tension O
/
Z-Axis Comp,
400 50
++
/
+
/ -800 _L.
-400 t
I
I
f
f~Lr____
if_
/
-100
+++
/
400
0
IL ]
i
800 (MPa; ~ ~
50
-50
i _
(7
]-
X
100
(ksi)
/
++Jr
/ -50 -400
/
/ /
J +%
,+
§
+
%
-100 -800
FIG. 7--Experimental biaxial and triaxial data obtained by testing a [0/9019s AS4/3501-6 carben~epoxy laminate.
biaxial stress ratios was achieved by directly manipulating the applied force to each of the four cruciform specimen loading arms. Note also that the stress ratios evaluated in the present study reasonably define the entire o-1 - ~2 stress space for a [019019s AS41350l-6 carbon/epoxy laminate configuration demonstrates the potential of the present triaxial test facility to describe the entire biaxial response of a composite material. Unfortunately, an undesirable failure mode was identified for the AS4/3501-6 carbon/epoxy specimens tested in the T/T quadrant of ~1 - ~2 stress space. The majority of these specimens exhibited a transverse tensile failure at the inboard wedge grip alignment hole. This premature failure prevented the test specimens in the T/T quadrant of ~r~ - ~2 stress space from achieving the maximum biaxial stress state in the gage section. As a result, it is recommended that this alignment hole be deleted in future studies to prevent this failure mode. Several additional recommendations, including decreasing the wedge grip taper angle to 10 ~ increasing the width of the cruciform loading arms, and possibly using thicker laminates are believed to be potential future improvements. The data presented in Tables 3 and 4 and Fig. 7 also indicate that the triaxial tension tests were not performed successfully. Because the Z-axis attachments used to generate through-the-thickness tensile forces in the gage section debonded at approximately 10 MPa (1.5 ksi), a true triaxial stress state was not achieved in these specimens at failure. Obviously, this is unacceptable. Damage to the fibers on the surface of the gage section as a result of the thickness-tapering machining procedures are believed to be primarily responsible for this undesirable failure mode [11]. As a result, several modifi-
436
MULTIAXlAL FATIGUE AND DEFORMATION
cations to this specimen configuration, including using end tabs bonded to a thinner composite laminate, are recommended for future studies. Another notable feature of the data presented in Tables 3 and 4 is that the desired stress ratios for the triaxial compression specimens were not achieved. The maximum capacity of the Z-axis force actuators was reached prior to ultimate failure. That is, the stepper motors used to supply power to the Z-axis actuators were only capable of generating approximately - 9 3 kN ( - 2 1 kip) of force before stalling. Those actuators then maintained that level of force for the remainder of the test. Thus, the actual stress ratios at failure for the JSX116, JSX166, and JSX666 specimen groups were approximately 1/1/-0.3, 1/1 - 1/-0.4, and - 1 / - 1/-0.3, respectively. While these tests are believed to be valid and demonstrate the ability of the present test facility to generate triaxial test results, this obviously places limitations on the three-dimensional stress ratios that the present test facility is capable of exploring. However, it is believed that this situation should be corrected through specimen modifications as well as by increasing the capacity of the stepper motors used to power these actuators [11]. Conclusions A triaxial testing facility that is capable of testing composite materials in both biaxial and triaxial configurations has been developed. This facility was evaluated by performing uniaxial and biaxial tests on 6061-T6 aluminum before investigating the biaxial and triaxial response of an AS4/3501-6 carbon/epoxy cross-ply laminate. Although difficulties associated with the triaxial testing facility and current cruciform specimen design have been identified, the authors believe that the potential of this facility to successfully perform biaxial tests on composite materials in any region of o"1 - o-2 stress space has been demonstrated. The authors believe that a similar statement can be made regarding the potential of the testing facility to successfully perform triaxial tests on composite specimens. Several modifications to the triaxial test facility and present thickness-tapered cruciform test specimen geometry have been identified that should reduce the difficulties encountered in the present study. The authors believe that once these issues have been properly addressed, the resulting test facility will be capable of generating accurate and consistent biaxial and triaxial experimental data for composite materials. Acknowledgments
The authors are grateful for the continuing support of the Federal Aviation Administration, Office of Research and Technology Application, through FAA Grant No. 94-G-009. The technical direction and encouragement of Dr. Donald W. Oplinger, Technical Monitor, FAA Technical Center, and Mr. Joseph R. Soderquist, FAA Headquarters, is sincerely appreciated. In addition, the authors are grateful for the support of the University of Wyoming, Major Equipment Grant Program, for funding a portion of this study. References [1] Welsh, J. S., and Adams, D. F., "Unidirectional Composite Compression Strengths Obtained by Testing Mini-Sandwich, Angle-, and Cross-Ply Laminates," Report No. UW-CMRG-R-95-106, Composite Materials Research Group, University of Wyoming, Laramie, WY, April 1995. [2] Wegner, P. M. and Adams, D. F., "Composite Lamina Compressive Properties Using the Wyoming Combined Loading Compression Test Method," Report No. UW-CMRG-R-98-116, Composite Materials Research Group, University of Wyoming, Laramie, WY, September 1998. [3] Traceski, F. T., Specifications & Standards for Plastics and Composites, ASM International, Materials Park, OH, 1990. [4] Hart-Smith, L. J., "Some Observations About Test Specimens and Structural Analysis for Fibrous Composites," Proceedings of the 9th ASTM Symposium on Composite Materials: Testing and Design, ASTM STP 1059, S. P. Garbo, Ed., American Society for Testing and Materials, West Conshohocken PA, 1990, pp. 86-120.
WELSH AND ADAMS ON TRIAXIAL TESTING FACILITY
437
[5] Camponeschi, E. T., Jr., "Compression Responses of Thick-Section Composite Materials," Report DTRCSME-90/60, David Taylor Research Center, Annapolis, MD, October 1990. [6] Chen, A. S. and Matthews, F. L., "A Review of Multiaxial/Biaxial Loading Tests for Composite Materials," Composites, Vol. 24, No. 5, 1993, pp. 395-405. [7] Jones, R. F., Jr., Ed., "Report of ONR Workshop on Multiaxial Evaluation of Fibrous Composite Materials," Office of Naval Research/DTRC, November 1990, [8] Hart-Smith, L. J., "Use of the Cruciform Sandwich Beam Test to Approximate the Bia• Strengths of 0~ ~ Composite Laminates," Proceedings of the 39th International SAMPE Symposium, K. Drake, et al. Eds., SAMPE International, April 1994, pp. 3248-3259. [9] Arnold, W. S., Robb, M. D., and Marshall, I. H., "Failure Envelopes for Notched CSM Laminates under Biaxial Loading," Composites, Vol. 26, No. 11, 1995, pp. 739-747. [10] Mahishi, J. M. and Adams, D. F., "Three-Dimensional Elastoplastic Stress Analysis of Unidirectional Boron/Aluminum Composites Containing Broken Fibers," Report No. UWME-DR-201-107-1, Department of Mechanical Engineering, University of Wyoming, Laramie, WY, October 1982. [11] Welsh, J. S. and Adams, D. F., "Development of a True Triaxial Testing Facility for Composite Materials," Report No. UW-CMRG-R-99-102, Composite Materials Research Group, University of Wyoming, Laramie, WY, May 1999. [12] Welsh, J. S. and Adams, D. F., "The Development of an Electromechanical Triaxial Test Facility for Composite Materials," submitted for publication in Experimental Mechanics, July 1999. [13] Welsh, J. S. and Adams, D. F., "Biaxial and Triaxial Failure Strengths of 6061-T6 Aluminum and AS4/3501-6 Carbon/Epoxy Laminates Obtained by Testing Thickness-Tapered Cruciform Specimens," submitted for publication in the Journal of Composites Technology & Research, July 1999. [14] Youssef, Y., Laborite, S., Roy, C., and Lefebvre, D., "Validation of an Effective Flat Cruciform-Shaped Specimen to Study CFRP Composite Laminates under Biaxial Loading," Canadian Aeronautics and Space Journal, Vol. 40, No. 4, December 1994, pp. 158-162. [15] Demmerle, S. and Boehler, J. P., "Optimal Design of Biaxial Tensile Cruciform Specimens," Journal of Mechanics andPhysics of Solids, Vol. 41, No. 1, 1993, pp. 143-181. [16] Makinde, A., Thibodeau, L., and Neale, K. W., "Development of an Apparatus for Biaxial Testing using Cruciform Specimens," ExperimentalMechanics, Vol. 32, No. 2, 1992, pp. 138-144. [17] ASM Specialty Handbook, J. R. Davis and Associates, Eds., Aluminum and Aluminum Alloys, ASM International, Materials Park, OH, 1993, p. 72. [18] Swanson, S. R. and Christoforou, A. P., "Response of Quasi-Isotropic Carbon/Epoxy Laminates to Biaxial Stress," Journal of Composite Materials, Vol. 20, No. 5, September 1986, pp. 457-471. [19] Swanson, S. R. and Colvin, G. E., Jr., "Compressive Strength of Carbon/Epoxy Laminates under Multiaxial Stress," Report No. UCRL-21235, Mechanics of Composites Laboratory, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, September 1989.
STP1387-EB/Oct. 2000
Author Index L
A Lerch, Bradley A., 99 Lagoda, Tadeusz, 173 Li, B., 139 Lissenden, Cliff J., 99 Lohr, Raymond D., 355 L6wishc, Gtinther, 232
Adams, Donald F., 423 Ahmad, Jalees, 41 Amstutz, Hans, 405 Arzt, Markus, 126 Au, Y. H. J., 54
Bartolotta, Paul A., 26, 369 Bayyari, M., 382 Bomas, Hubert, 232 Bonacuse, Peter J., vii, 281 Bonnen, John J. F., 213 Brocks, Wolfgang, 126 Buczynski, A., 82
M
Macha, Ewald, 173 Mayr, Peter, 232 Mohr, Tainer, 126
N
C Covey, Steven J., 26
Nelson, Drew V., 246 Newaz, Golam M., 41 Nicholas, Theodore, 41
D
de Freitas, M., 139 Donne, Claudio Dalle, 405
P
Park, Jinsoo, 246 P6tter, Kurt, 157, 323
E R
Ellis, J. R., 382 G
Rees, David W. A., 54 Renauld, Mark L., 266
Glinka, G., 82
I
lsobe, Nobuhiro, 340
J
Jenkins, M. G., 13
Sakurai, Shigeo, 340 Salem, J. A., 13 Sandlass, G. S., 382 Santos, J. L. T., 139 Schram, A., 323 Socie, Darell, 3 Suhartono, H. A., 323
K
T
KaUuri, Sreeramesh, vii, 281 Krause, David L., 369
Topper, T. H., 213, 305 Trautmann, Karl-Heinz, 405
Copyright9
by ASTM lntcrnational
439 www.astm.org
440
AUTHORINDEX Y
V
Varvani-Farahani, A., 305
Yousefi, Farhad, 157, 191
W
Walker, Mark A., 99 Wang, Jerry, 3 Welsh, Jeffry S., 423 Witt, Mario, 191
Z
Zamrik, Sam Y., 266 Zenner, Harold, 157, 191,323
STP1387-EB/Oct. 2000
Subject Index A ABAQUS, 126 Aeropropulsion materials, 382 AIS| 1015, 323 2124 aluminum alloy metal matrix composite, 54 ANSYS finite-element code, 382 Attachment methods, 382 Axial fatigue, 281 Axial-torsional load effects, Haynes 188, 99
Critical plane, 305 Critical plane approach, 173, 191 Crossland fatigue criteria, 139 Cruciform, 355,369, 382, 405 thickness-tapered, 423 Crystals, single, 13 Cumulative fatigue, 281 Cyclic hardening, 281 Cyclic loading, 126 Cyclic plasticity, 54 Cyclic testing, 369 D
Biaxial fatigue effect of periodic overloads, 213 in- and out-of-phase combined bending-torsion, 246 isothermal model, 266 microcrack growth, 323 Biaxial isothermal fatigue model, 266 Biaxial loading, in-plane, 369, 405 Biaxially loaded cruciform-shaped specimen, 41 Biaxial strain ratio, 355 Biaxiai strength testing, isotropic and ansiotropic monoliths, 13 Biaxial testing, 423 in-plane, 382
C Ceramics, 13 Cobalt-base superalloy, 99, 188, 281 Cold forming, stainless steel, 26 Combined loading, 191 Complex loading, 126 Composite materials, 423 Composite strength, 3 Compression testing, 405 Constitutive equations, 126 Crack closure, effect of periodic overloads, 213 Crack face interface, effect of periodic overloads, 213 Crack growth in- and out-of-phase combined bending-torsion, 246 rate model, 305 Cracking behavior, effect of periodic overloads, 213 Crack initiation high-cycle fatigue, 139 in- and out-of-phase combined bending-torsion, 246 Crack propagation mulfiaxial high-strain fatigue, 355 multiaxial low-cycle fatigue, 340
Damage curve approach, 281 Deep drawing, 405 Deformation, 369, 405 metal matrix composite, 54 Displacement, isotropic and ansiotropic monoliths, 13 Dissipation potential, 99 E
Effective fatigue, 305 Effective intensity factor range, 305 Elastic-plastic notch tip stresses, 82 Elastic-plastic strain analysis, 82 Elasto-viscoplastic material, 126 Electromechanical testing facility, 423 Energy, 305 Extensometer, in-plane biaxial contact, 369 External pressure, 355
Failure envelope, 423 Failure loads, 26 Failure surface, 423 Failure theories, 3 Fatigue criteria, 232 Fatigue life prediction, 139 Fatigue lifetime prediction combined tension-torsion in- and out-of-phase, 232 cumulative axial and torsional, 281 effect of periodic overloads, 213 in- and out-of-phase combined bending-torsion, 246 under multiaxial random loading, 157 welded joints, 191 Fiber metal laminate, 405 441
442
MULTIAXIALFATIGUE AND DEFORMATION
Finite-element analysis in-plane biaxial loading, 382 non-linear problems, 126 Fracture, metal matrix composite, 54
G G-10 composite laminate tube, 3 Generalized strain energy density crition, 173 Glass fiber-reinforced epoxy laminate, 3 Grain boundaries, effect on microcrack growth, 323
H
Haynes 188, 281 axial-torsional load effects, 99 High-cycle fatigue prediction, 139 Hoop compression, 3 Hypothesis of the integral approach, 157
Inelastic deformation, under multiaxial stress, 41 Influencing parameters, 157 In-plane biaxial contact extensometer, 369 In-plane biaxial failure surfaces, 26 In-plane biaxial loading, 405 In-plane biaxial testing, 382 Integral approach, 173, 191 Internal pressure, 355
K
Modeling inelastic deformation under multiaxial stresses, 41 microcrack growth, 323 Monoliths, isotropic and ansiotropic, 13 Multiaxial fatigue, 213 in- and out-of-phase combined bending-torsion, 246 in-plane biaxial contact extensometer, 369 Multiaxial fatigue criteria, 157 energy-based, 173 Multiaxial fatigue life model, 305 Multiaxial high-strain fatigue, 355 Multiaxial loading, 99 G-10 composite laminate tube, 3 high-cycle fatigue prediction, 139 weld joints, 191 Multiaxial low-cycle fatigue, microcrack growth modes and propagation rate, 340 Multiaxial strength, isotropic and ansiotropic monoliths, 13 Multiaxial stress, inelastic deformation, 41 Multiaxial stress-strain notch analysis, 82
Newton algorithm, 126 Nickel aluminide, 13 Non-linear problems, 126 Nonproportional loading, high-cycle fatigue prediction, 139 Nonradial loading metal matrix composite, 54 silicon carbide, 54 Notched specimen, in- and out-of-phase combined bending-torsion, 246 Numerical algorithm, 126 Numerical method, 139
Kinematic hardening, 54 O
Laminate tube, strength, 3 Linear damage rule, 281 Load-type sequencing, 281 Low-cycle fatigue, 355
M
Mean stress effect, 305 Metal matrix composite characterization methods, 41 deformation and fracture, 54 Microcrack growth modes, 340 modelling, 323 propagation rate, 340 Modal control, 355
Off-axis tension tests, 41 Optimization techniques, 382 Out-of-phase loading effects, 139 Overloads, periodic, 213
Phase difference, 157 Phase factors, 266 Plastic limit load, 405 Prediction software, 191 Proportional loading, 213 Prototype fixturing, 382 R
Random load, 157 nonproportional, 173
SUBJECT INDEX Ratcheting, 54 Residual stress, 232 Reusable fixturing, 382 Riveted joint, 405
Sequence effects, 213 Shear energy, 305 Shear plane, multiaxial low-cycle fatigue, 340 Shear stress amplitude, effective, 139 Silicon carbide particulate, 54 Sines fatigue criteria, 139 Stainless steel, 26 austenitic, biaxial isothermal fatigue model, 266 Steel biaxial fatigue, 213 combined tension-torsion in- and out-of-phase, 232 effect of periodic overloads, 213 microcrack growth modes and propagation rate, 340 Strain isotropic and ansiotropic monoliths, 13 principal, 340 Strain hardening, 305 Strain measurement, 369 Strain paths, in- and out-of-phase, 305 Strain rates, equivalent, 99 Strain rate vectors, 99 Stress effect of periodic overloads, 213 equivalent, 26, 99 in- and out-of-phase combined bending-torsion, 246 isotropic and ansiotropic monoliths, 13 superimposed mean, 157 Stress intensity factor, 405 Stress relaxation, 99 Superalloy, 99 System design, multiaxial high-strain fatigue, 355
Temperature, elevated, 369 Tension, combined in- and out-of-phase, 232 Tension-torsion loading, 213 Thermomechanical fatigue, 355 Thermomechanical loading, 266 Thin-walled tube, 355 Torsion, 355 combined in- and out-of-phase, 232 in- and out-of-phase, 246 Torsional fatigue, 281 Torsion stress, 3 Triaxiality factor, 266 Triaxial testing facility, 423 Tungsten carbide, 13 U Unidirectional fiber-reinforced metal matrix composites, 41 V Variable-amplitude tests, 157 Viscoplasticity, 99 models, potential-based, 99 W
Weakest-link model, 232 Weld joints, under multiaxial loading, 191 Y Yield surface, 405
Z-parameter, 266
443
ISBN
0-8031-2865-7
