Multiaxial Fatigue and Deformation: Testing and Prediction (ASTM Special Technical Publication, 1387)

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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Printed in Philadelphia,PA October2000

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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FIG. 8--Flow curves and strain path for stress path lb.

(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

-1.2

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-0.8

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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

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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~