Advances in Applied Mechanics Volume 30
Editorial Board T. BROOKEBENJAMIN
Y. C. FUNG
PAULGERMAIN RODNEYHILL PROFESSOR L. HOWARTH C . 4 . YIH (Editor, 1971-1982)
Contributors to Volume 30 JOHN L. BASSANI L. B. FREUND MARKKACHANOV STELIOS
K YRIAKIDES
ADVANCES IN
APPLIED MECHANICS Edited by John W. Hutchinson
Theodore Y. Wu
DIVISION OF APPLIED SCIENCES HARVARD UNIVERSITY CAMBRl DG E. MASSACHUSETTS
DIVISION O F ENGINEERING AND APPLIED SCIENCE CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA. CALIFORNIA
VOLUME 30
ACADEMIC PRESS, INC. Harcourt Brace & Company
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Contents vii ix
CONTRIBUTORS
PREFACE
The Mechanics of Dislocations in Strained-Layer Semiconductor Materials L. B. Freund I. Introduction 11. Selected Properties of Dislocation Fields 111. An Isolated Dislocation itl a Strained Layer IV. Driving Force in More Complex Situations
V. Process Kinetics Acknowledgments References
i 8 18 32 56 64 64
Propagating Instabilities in Structures Srelios Kyriukides 1. Introduction 11. Initiation and Propagation of Bulges in Inflated Elastic Tubes 111. initiation and Propagation of Buckles in Long Tubes and Pipes under
External Pressure IV. Propagating Buckles in Long Confined Cylindrical Shells V. Buckle Propagation in Long, Shallow Panels VI. Summary and Discussion Acknowledgments References
68 70
94 147 169
181 185 185
Plastic Flow of Crystals John L. Bussmi I. Introduction 11. Preliminaries 111. Yield Behavior including Non-Schmid Effects
IV. V. VI. V11.
Flow Behavior including Non-Schmid Effects Hardening Behavior Strain Localization Closure Acknowledgments References V
192 194 198 21 1 217 235 2 54 2 54 2 54
vi
Contents
Elastic Solids with Many Cracks and Related Problems Mark Kachanov 1. Introduction
260
11. Stresses and Crack Opening Displacements Associated with One Crack
in an Infinite Isotropic Linear Elastic Solid I l l . Problems of Many Cracks in a Linear Elastic Solid IV. Various Effects Produced by Crack Interactions V. Interaction of a Crack with a Field of Microcracks VI. Effective Elastic Properties of Cracked Solids VII. On Correlations between Fracturing and Change of Effective Elastic Moduli. Some Comments on Brittle-Elastic Damage Mechanics VIII. Effective Elastic Properties of a Solid with Elliptical Holes Acknowledgments References
26 1 280 301 321 345 412 420 431 438
AUTHORINDEX
441
SUBJECT INDEX
453
List of Contributors Numbers in parentheses indicate the pages on which the authors' contributions begin.
JOHN L. BASSANI (191), University of Pennsylvania, School of Engineering and Applied Science, Department of Mechanical Engineering and Applied Mechanics, 297 Towne Building, 220 South 33rd Street, Philadelphia, Pennsylvania 19104-6315 L. B. FREUND (l), Division of Engineering, Brown University, Providence, Rhode Island 02912 MARKKACHANOV (259), Department of Mechanical Engineering, Tufts University, Medford, Massachusetts 02155 STELIOSKYRIAKIDES (67), Engineering Mechanics Research Laboratory, Department of Aerospace Engineering and Engineering Mechanics, The University of Texas at Austin, Austin, Texas 78712
vii
This Page Intentionally Left Blank
Volume 30 of the Advances in Applied Mechanics is being published 45 years after the first volume in the series appeared in 1948. The series was started by Richard von Mises and Theodore von Karman, who edited the first three volumes. Included in the first volumes were articles by illustrious figures in mechanics such as Biezeno, Burgers, Carrier, Dryden, Geiringer, Lin, Reissner, and von Mises and von Karman themselves. The present volume continues the series’ tradition by containing review articles on problem areas of fundamental importance in mechanics and its applications. The four articles in Volume 30 deal with the mechanics of materials and structures, covering topics as diverse as films, crystals, cracks, and pipes. L. B. Freund writes on dislocations in strained layers. This is a problem of paramount importance in the development and performance of semiconductor materials. Freund treats a number of basic aspects of the problem, summarizing and advancing the current state of knowledge. His article emphasizes the theory of the dislocation mechanics, but contact is made with the experimental state of the subject. The second article, by S. Kyriakides, covers a special type of buckling phenomena wherein an instability mode such as a buckle or bulge propagates along the entire length of a structure. Kyriakides shows that there is a remarkable richness in this class of behaviors. The article is an admirable mix of theory and experiment, accompanied by photographs that clearly elucidate the phenomenon for a number of different structural systems. J. L. Bassani’s article, “Plastic Flow In Crystals,” represents an up-to-date summary of a subject that has received attention for almost a half century. Much of the motivation behind research in this area remains what it always has been, i.e., providing a foundation on which the macroscopic behavior of polycrystalline materials can be understood and predicted. Advances in computing power continue to “up the pressure” for improved constitutive descriptions of single crystals, and Bassani’s article is an important contribution in this direction. The fourth and last article in Volume 30 is an extensive review of elastic crack mechanics by M. Kachanov that focuses on the interaction between
ix
X
Preface
cracks. The article deals with various classes of crack interactions, including multiple microcrack interactions and the effect of microcracks on the stress experienced by a major crack. The author discusses analysis methods, both exact and approximate, and he gives considerable attention to the implication of the theory for understanding and predicting phenomena such as microcrack toughening in brittle materials and the reduction in the effective moduli due to microcracking. John W. Hutchinson and Theodore Y. Wu
ADVANCES IN APPLIED MECHANICS, VOLUME 30
The Mechanics of Dislocations in Strained-Layer Semiconductor Materials L. B. FREUND Division of Engineering Brown University Providence. Rho& Island
. .. . . . .. . . . . .. . . .. . .. . .. . . . . . . . .. . ... . . .. . ... . . . . . .. .. .. ... . .. . . .. . . . . . . . . . .
1
11. Selected Properties of Dislocation Fields . .. . . ... . .. ..... . .. . . ... .. . .. . .. . . .. .. . .. . . . . . A. Stress and Deformation Fields.. . . . . .. . .. . . ... . .... .. . . . . . .... .. . .. . .. . . .... . .. . . . . . B. Force on a Dislocation near a Surface or Interface.. . . . ... . .. . .. . .. . . .... . .. . .. . . C. Net Force on Certain Lines . .... .._.._ . _ _ ..... ... ....... ._.... . . . ......... _..... .
8 9 12 16
111. An Isolated Dislocation in a Strained Layer .......................................... ................................... A. The Threading Dislocation . B. Energy Variations during ..................... C. The Driving Force on the ..... . ................... ,. D. Critical Thickness ........... ,.. ....... ...................................
18 19 20 23 28
IV. Driving Force in More Complex Situations .......................................... A. Nonuniform Elastic Mismatch Strain.. . . .. . ... . ... . .. . . . . .. . . .. . . .. . . . .. . . .. .... . . .. . . ... .. .. . .. B. Unstrained Capp ....................................................... C. Superlattices . .. . . . cations ....... . .. . . ... . . . .. D. Array of Parallel E. Dislocations on Intersecting Glide Planes.. . . .. ... . . . . . . .. ... . .. . ... ..
.
32 32 34 38 39 47
V. Process Kinetics.. .. . ... .. . ... . . . . . . . . . .. . . .. . . . , . . .. . .. .. . . . . ... .. . ... .. . . .... . .. . . . .. . . A. Nonequilibrium Conditions . . . . . . .. . . .. . . .. . . , . . .. .. . .. . .. . ... ... . ... . . . . .... . . .. . . B. Dislocation Nucleation .. . . . . . . . . . .. . . .. . . . .. . . , . . . .. . . . . . .. ... . . .. . ... . .. . . . ... . ... C. Glide through Nonuniform Stress Fields ... . . , .... . .. . . . . . .. . .. . . ... . . . . . . . .. . . . . .
56 56 61 61
Acknowledgments .. . . . . .. . .. . ... . . .. . . .. . . ... . .. . . ... . .. . . , . . ... ... . .. . ... . . . . . . . .. . ... .
64
References. . . . .. . . . .. . .. . .. . ... . . , . . . . . . .. . . . .. . .. . . . . , . . .. . . . . . .... . . . . .. . .. . . . .. . . .. . . . .
64
I. Introduction
,
'
I. Introduction Due to advances in the techniques of crystal growth, it is possible to fabricate bi-material crystalline structures with an extraordinarily high 1 Copyrtght (c)1994 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN 0-12-002030-0
2
L . B. Freund
degree of perfection. For example, a thin layer of one material can be deposited in the form of a single crystal film onto the plane surface of a different material. In the simplest case, the film material has the same crystal structure as the substrate material, and it grows with the same orientation. In more complex situations, the materials may have a different crystallographic orientation, or even a different crystallographic structure, but the lattices are commensurate in some sense across the interface. The growth processes are sophisticated, but they are now routinely performed by any of several techniques in many laboratories. The methods that permit the greatest degree of control over the details of the growth surface are molecular beam epitaxy and organometallic vapor phase epitaxy. In any case, in these methods atoms are deposited one at a time in such a way as to establish the crystalline structure of the growing film, using the crystalline structure of the substrate as a template. Under appropriate conditions, the interface is coherent or epitaxial, at least in the early stages of growth. The term epitaxy derives from the Greek m i for “on” and zol(ia for “arrangement.” A principal motivation for development of this crystal growth technology has been the potential for the use of semiconductor materials in thin-film configurations in electronic and optoelectronic devices. The main function of these devices is to control transport of electrons in a way that permits high spatial density of individual devices and in which the carriers are highly mobile, that is, fast response to electronic inputs with low power consumption. Spatial control of mobile electrons can be facilitated by using a combination of materials, forming a material heterostructure, with one or more of these materials being in a thin-film configuration. Carrier confinement is enforced by a difference in energy band structure across the interface as a barrier and the exploitation of this physical effect in the design of devices has become known as bandgap engineering. A great deal of attention has been focused on film/substrate systems involving the 111-V compounds (InGaAs/GaAs, for example), as well as on 11-VI compounds for optical applications (ZnSe/GaAs, for example). Current efforts are also directed toward SiGe/Si and GaAs/Si systems to exploit the central role of silicon in device technology. Control of the quality of the bi-crystals formed for this purpose has been of central interest. There is evidence that crystal dislocations in an electronically active zone diminish charge carrier density. They also seem to have a strong adverse effect on optical properties of some materials, in some cases being the determining factor in whether a material heterostructure will or will not give off light when stimulated electronically. In the early days of fabrication of
Dislocations in Strained-Layer Semiconductor Materials
3
epitaxial strained layers, complete coherency across the interface was the goal, and it continues to be so in cases involving conduction normal to the interface or other special situations. However, in recent years, the control objective has been shifting for some applications. In many planar devices with lateral transport, dislocations at an interface are not necessarily deleterious, so for such situations the goal has been modified to be elimination of dislocations extending across the thickness of a thin film, the so-called threading dislocations. Such dislocations can act as paths of relatively easy charge transport, as diffusion paths for dopant, or as surface seeds for defects in subsequent overgrowth or other processing steps. Even though the resulting interface may be significantly dislocated, strained-layer epitaxy provides the means to fabricate films of difficult-to-grow materials with low defect density in the electronically active regions. A general introduction to this technology and its promise for electronic applications is given by Bean (1985). Growth with spatially uniform thickness is assumed in these introductory remarks, and attention is focused on points far from the edges of the growing film compared to the thickness. In other words, edge effects are ignored for the time being. By proper selection of materials or alloys, it is sometimes possible to match the lattice parameter of the growing layer to that of the substrate. In such a case, the film grows without intrinsic strain. For example, an aluminum-gallium arsenide compound crystal can be deposited on a gallium arsenide substrate without intrinsic strain in the layer. The crystal structure of gallium arsenide, aluminum-gallium arsenide, and other of the so-called 1x1-IV compounds is the zinc blende structure. The standard notation for this layer-substrate system is Al,Ga, -,As/GaAs(001), where x represents the fraction of group I11 lattice sites occupied by aluminum atoms. The remaining fraction 1 - x of these sites are occupied by gallium atoms. The appendage (001) indicates the direction of the normal to the common interface with respect to the crystal unit cell in the standard index notation of crystallography. In the past few years, the prospect of epitaxial growth of a layer on a substrate in a situation where the lattice parameter of the layer differs from that of the substrate has attracted great interest. For a modest amount of mismatch, the layer will grow with whatever extensional strain is required in the plane of the interface to retain perfect atomic registry of the layer with the substrate. This process has come to be known as strained-layer epitaxy. Not all of the effects of strain on the electronic structure of the material are known. However, it appears that the mechanical strain can be exploited to tailor the
4
L . B. Freund
electronic energy band structure of the material in various ways for specific purposes. For example, strain can be used to separate multiple modes at a certain energy level, to bring maxima and minima in the band structure into alignment (conversion of an indirect bandgap material to a direct bandgap material for optical applications), or to adjust the size of the energy gap between the highest bound states and the lowest conduction states of charge carriers in doped semiconductor materials. A principal difficulty with a strained-layer structure is that the stress associated with the strain gives rise to a driving force for structural defects in the strained-layer lattice. The means of controlling these defects has become a concern of central importance in the field. The main competing effects are illustrated schematically in Fig. 1. The elastic energy stored in a crystal associated with an applied stress field can be reduced by the advance of a dislocation through that crystal, and this effect leads to the concept of an applied force on a dislocation, defined as the reduction of the mechanical energy of the body associated with an advance of the dislocation. On the other hand, the presence of a free surface leads to a countereffect. An elastic material appears to be more compliant near a free surface than far from it. Consequently, in the absence of applied loading, the energy that must be expended to form a dislocation near a free surface is less than that required to form the same dislocation at a greater distance from the surface. This observation leads to the idea of a force tending to pull the dislocation out of the crystal through the free surface, commonly called the imageforce because of a particular analytical method used to represent this force. Just as a dislocation is attracted to a free surface, it is repelled by a rigidly constrained
FIG.1. Schematic diagram illustrating the principal equilibrium stress fields acting on a dislocation. The dislocation tends to move to relieve the applied stress and to reduce its own energy, which are commonly competing effects in strained-layer systems.
Dislocations in Strained-Layer Semiconductor Materials
5
surface, and it may be attracted or repelled for intermediate situations involving bimaterial elastic interfaces. In any case, the competition between applied forces on dislocations resulting from the external elastic fields in which the dislocations exist and the self forces due to configuration or proximity to surfaces is a central issue in understanding the role of dislocations in relaxation of elastic strain in strained-layer systems. A particular defect that has attracted a good deal of attention is the threading dislocation in a strained layer, already mentioned in passing previously. This is a glide dislocation with the dislocation line extending from the free surface of the layer to the interface between the layer and substrate, and then continuing either along the interface or into the substrate. This defect is important because it can move on its glide plane through the layer, leaving behind a segment of misfit dislocation of ever increasing length on the interface between the film and the substrate. The purpose of this chapter is to consider some specific issues concerned with the mechanics of glide and interaction of dislocations in strained-layer systems. Clearly, the preferred situation is to have no dislocations at all in a strained layer being fabricated for electronic purposes. For some purposes, however, it is sufficient to use the intrinsic strain in the layer to push the threading dislocations to the edges of the film, leaving a surface of very high quality in the functional part of the film. A broad view of the means of fabricating and characterizing strained-layer systems involving mainly Si and Ge is given in the proceedings of the Third International Symposium on Silicon Molecular Beam Epitaxy (1989). Suppose a uniform film of some crystal class and lattice parameter af is grown on the surface of a second material of the same class and orientation but lattice parameter a,. In general, a, # a/. If the film grows with the interface perfectly ordered, continuing the structure of the relatively thick substrate, then the elastic mismatch strain between the film and substrate is defined to be E~ = (a, - af)/ar. With reference to the configuration shown in Fig. 2, suppose the film and substrate are both modeled as isotropic elastic materials, each with a shear modulus p and Poisson ratio v, and that the system is constrained against rotation on a remote boundary. In this case, no traction is transmitted across the interface, and both the film and substrate are subject to a state of stress that is essentially an isotropic membrane stress in the x,,x,-plane. If the magnitude of this stress is denoted by CT then
L . B. Freund
6
FIG.2. A film with a free surface bonded to a relatively thick substrate. The configuration extends indefinitely in all directions in the plane of the interface.
where E, and E~ + E, represent the total elastic strain in the substrate and film, respectively. Overall equilibrium of the system, along with the compatibility condition E, = tf imposed by coherence of the interface, imply that E/
= E, =
-Eg
[
1
1
1 - v ; p,h, +7 1 - v, P,h,
-l
For most cases of interest in semiconductor systems, ps x p, and h, >> h,. Consequently, the second term in the denominator of (1.2) is enormous compared to unity and E, = E, FZ 0. If the finite lateral extent of the strained-layer system is taken into account, then an important additional effect arises, even for the case when the layer is very thin compared to the substrate and the elastic properties are similar. Consider the process of removing a circular disk of some radius large compared to h, from the heterostructure, as sketched in Fig. 3. The bending moment per unit length along the boundary of this disk that must be balanced in the process of creating the free edge is essentially ~ o , h , h , . Thus, once the disk is removed from the otherwise unbounded layer-substrate system, it will take on the spherical shape of a thin plate subjected to a uniform radial bending moment along its boundary. The radius of curvature
t
hs>> hf
M O -hf.
Fr
Of
hf h,/2
r3
FIG. 3. Schematic diagram showing the origin of the bending moment causing the film curvature. Measurement of curvature yields an estimate of elastic film strain.
Dislocations in Strained-Layer Semiconductor Materials
7
of this shape can be shown to be
on the basis of small deflection plate theory. This radius of curvature is typically very large. Nonetheless, it can be measured quite accurately by optical methods, and this measurement has been used effectively to monitor the average elastic mismatch strain in strained layers at various stages of relaxation (Nix, 1989). Examples of unit cell dimensions of the crystal lattice at a temperature of 500°C for diamond cubic materials in group IV of the periodic table are aSi = 0.5433 nm and aGe= 0.5660 nm, and for zinc blende cubic compound = 0.5664nm, materials from groups 111 and V are aGaAs = 0.5658nm, aAIAs alnAs= 0.6062 nm, and alnP= 0.5869 nm. Thus, at this temperature, the elastic mismatch strain in a coherent Si, -,Ge, alloy film on a Si(100) substrate is e0 x -O.O42x, where x is the fraction of lattice sites in the film occupied by Ge atoms. The estimate is based on the elementary rule of mixtures, known in z xaGe (1 - x)asi.Similarly, this context as Vegard's law, whereby aGeXS,,_, for a coherent Ga, -,In,As/GaAs(100) heterostructure E~ z -0.071~. By following this same recipe, it can be concluded that Ino,,,Ga,,,,As/InP is a lattice-matched system at 500"C, thus inducing no elastic mismatch strain during coherent growth. Some lattice-matched systems have been important in the development of material heterostructures in microelectronics. However, it is uncommon to find a matched system that also fulfills all of the nonmechanical requirements in selecting materials for a particular application. For example, it may be necessary to add a dopant to one or the other material in a heterostructure to render it semiconducting, but it is found that the chemical properties of the material imposed by the condition of lattice matching are unsuitable for this process. For such reasons, elastically mismatched materials must be admitted in selecting materials that eventually serve a primary function that is nonmechanical. Unless noted to the contrary in the discussion to follow, it will be assumed that the film is much thinner than the substrate, that is, h,/h, * 0, and that the film and substrate have identical elastic properties, that is, ps = p, = p and v, = v, = v. For an assumed isotropic film of thickness h, = h and mismatch strain E ~ the , membrane stress in the plane of the interface is then
+
8
L . B. Freund
FIG.4. Coordinate system used to describe the elastic field of an edge dislocation in a tractionfree half-space.
and all other stress components are zero. The coordinate directions are oriented as indicated in Fig. 4. The extensional strain in the direction of the interface normal is
and the elastic energy per unit area of interface is U o = 2ph$
(-)
l + v 1-v
where p is the elastic shear modulus and v is Poisson’s ratio of both the film and the substrate. For a mismatch strain of one percent in a Si or Ge film of thickness 10nm, this energy density is roughly 1 J/m2. Processes of formation and growth of crystalline defects can draw upon this energy reservoir.
11. Selected Properties of Dislocation Fields
In this section, selected properties of the stress and deformation fields of dislocations in elastic bodies are summarized. Of particular interest are the fields of a long, straight dislocation in a half-space for the case when the dislocation line is parallel to the free surface, a class of problems in which Head (1953a, 1953b) made key contributions. A thorough summary of work on the interaction of dislocations with material inhomogeneities was given by Dundurs (1969). Under the conditions of main interest here, the elastic field is two dimensional. Furthermore, it can be decomposed into a two-dimensional plane strain field associated with the edge or in-plane components of the Burgers vector and a two-dimensional antiplane strain field associated with the screw or out-of-plane component of the Burgers vector. These fields are
Dislocations in Strained-Layer Semiconductor Materials
9
independent, so they can be considered separately. The field of an edge dislocation in a half-space are considered first, and the field of a screw dislocation is considered subsequently. In this section, fields are referred to an underlying rectangular coordinate system with axes labeled x, y , z, which is found to be most convenient for recording general results. In those cases where index notation has some advantage, the same coordinate directions are labeled xl, x2, x3, respectively, but the origin of coordinates may be offset in some cases in a way suggested by the geometry of the configuration being studied. A. STRESSAND DEFORMATION FIELDS
1. Edge Dislocations
a. An lsolated Edge Dislocation Consider the plane strain field of an edge dislocation in an unbounded homogeneous and isotropic elastic solid. A rectangular coordinate system is introduced so that the plane of deformation is the x, y-plane. The origin is taken to coincide with the location of the core of the dislocation, and the components of the Burgers displacement vector in the coordinate directions are bx and b y . The elastic constants of the material are the shear modulus p and the Poisson ratio v. The elastic field is conveniently expressed in terms of the formalism of analytic functions of a complex variable developed by Muskhelishvili (1953), according to which the stress components and displacement components are represented in terms of two functions cp and I,/I of l = x iy, which are analytic over the region of the body, except at isolated points or lines, as
+
3(cry,,+ oxx) = 2 ReCcp’(01 3@,,,,- oxx) + i g x , , = Ccp”(5) + I,/I’(O
(2.1)
The overbar denotes complex conjugate. For plane strain o,, = v(uxx+ cry,,). The displacement components can be found by integration to be 2AUX
+ iu,) = (3 - 4 V ) c p K )
~- Ccp’(l) -
For an edge dislocation at the origin of coordinates,
(2.2)
L . B. Freund
10
+
where b = b, ib,. Integration of (2.3) and substitution into (2.2) reveals the sign convention for Burgers displacement, which is implicit in this representation to be
-b where 8 is the polar angle around the dislocation with respect to the x-axis and do is an arbitrary angle. If the dislocation is located at x = 0, y = - ih in the plane, rather than at the origin of coordinates, then ”(‘)
pib = 474 1 - v)(c
+ ih) ’
*‘a =
according to the coordinate translation rules for this representation. b. Edge Dislocation in a Half-Space Suppose that an edge dislocation exists in the elastic material occupying the half-space y 0. The dislocation line is parallel to the z-axis and at a distance h from the traction-free surface of the half-space, as shown in Fig. 4. The elastic field of this dislocation can be obtained from the field represented in (2.5) by adding the appropriate nonsingular elastic field that negates the traction on the surface y = 0. By substituting (2.5) into (2.1), this traction is readily calculated to be
-=
ib - i 6
T,(x) - iT,(x) =
ib +-+ x + zh
(x - ih)2
Thus, the nonsingular field to be added is the solution of the first fundamental boundary value problem for the half-plane with the surface traction given by (2.6).The solution procedure for this case is outlined by Muskhelishvili (1953, §93), and the result for the present case is P ”(‘)
= 4n(1 - v)
ib
i&[
+ ih)
[t+lh- (5 - if^)^
i(b - 6) - *-ii;]
(2.7)
The corresponding expression for $’([) can also be extracted, but it turns out to be unnecessary. By means of an ingenious argument based on analytic continuation, Muskhelishvili (1953, $112) showed that the functions q([)and @ ,I ) are uniquely related throughout the elastic half-plane, provided only that some finite segment of the boundary is traction free. As a result, all components of stress and displacement can be represented in terms of a single
Dislocations in Strained-Layer Semiconductor Materials analytic function of
11
C throughout the half-space according to ~
fJyy - icxy
= cp‘(C) -
cp’(S) + (C - S)cp”(O ~
Ox,
~
+ i f J x y = cp’(i) + cp’(t-1 + 2cp’(i) - (i- S)cp”(C)
c. A Periodic Array of Edge Dislocations in a Half-Space With the solution for the elastic field of an edge dislocation in hand, in the form of (2.7), the solution for a periodic array of identical edge dislocations can be found by superposition. Denote the spacing between dislocations by p , as shown in Fig. 5. The fields for dislocations located at x = np, y = -ih given by (2.7)are simply summed for n = . . . , -2, - 1,0, 1,2,. . . . The details can be carried out by replacing - ih with - ih + np in (2.7)and summing with respect to n over all integers. The trigonometric sums of the resulting infinite series are 1
c
+a)
.=-,[+ih-np
c
+a n= -
[+ih-np - ih - np)’
(c
p 71
p
11’ ,CSC’
P
(2.9) -(
[p”
- ih)]
and it follows that
+- P (2.10) Again, the stress and displacement components are given by (2.8). The same stress and deformation fields were obtained by Willis, Jain, and Bullough (1 990) using Fourier transform methods.
FIG.5. A periodic array of edge dislocations in a half-space with a free surface.
L . B. Freund
12
2. Screw Dislocations The corresponding results for a screw dislocation are much simpler to obtain. For a single screw dislocation at x = 0, y = -ih in the half-space y < 0 with Burgers displacement b,, Pbz
+
cxy iozx= - -
211
[-
-I
1
[+lh
1 [-lh
(2.11)
For a periodic array of screw dislocations in the half-space,
+
azy in,, =
]
n([ + ih) -& 2P {cot[
B. FORCE ON A DISLOCATION NEAR
A
-cot[
n([
]]
- ih)
(2.12)
SURFACE OR INTERFACE
Elastostatics has a number of conservation laws that rest on the invariance of certain energy measures under specific coordinate transformations and on the principle of minimum potential energy. The mathematical statement of each conservation law is that the integral of a certain functional of the elastic field over the bounding surface of a regular closed subregion of the deformed elastic solid equals zero. Budiansky and Rice (1973) considered these integrals over the boundary surface of a stressed elastic solid containing a cavity. In this case, the integration paths enclose a regon that is irregular, and the surface integrals possess nonzero values that can be interpreted as the potential energy release rates associated with translation, rotation, and expansion of the cavity. These observations opened the way for exploiting the conservation laws to extract energetic driving forces on material defects, in the sense of Eshelby (1951), without the need to find full solutions of the underlying boundary value problems. The force on a dislocation near the plane traction-free bounding surface of an elastic solid or near the plane interface of joined elastic solids can be established by applying the conservation law known commonly as the M-integral, defined for plane elastostatics as M(C)=
I
[UniXi - cijnjUi,kXk]dC
(2.13)
where U is the strain energy density, C is a closed path in the plane with unit normal ni, oijis the equilibrium stress field, uiis the displacement field, and x i
Dislocations in Strained-Layer Semiconductor Materials
13
FIG. 6. Integration paths for application of the M-integral conservation law to determine the force on a dislocation due to its proximity to a free surface.
is a point in the plane. If the path C encloses a regular elastic region, then M(C) = 0. Consider the two-dimensional plane strain situation depicted in Fig. 6. An isotropic elastic material with a traction-free plane surface occupies x1 > 0, and a straight dislocation with line parallel to the x,-axis pierces the plane at the point x1 = tl, x2 = tz.The Burgers vector of the dislocation can have any orientation in the plane. Two closed contours C" and Ce are introduced as shown. The loop around the dislocation line is assumed to have vanishingly small radius, and the larger half-circular portion of the path C" is assumed to have an indefinitely large radius. Both contours surround the dislocation, and the region between them is regular. Consequently, the conservation law requires that M(C")
=
M(C)
(2.14)
At any point on C" along xI = 0, both xini and aijnjare zero, so this part of C" makes no contribution to the value of M(Cm).At points very far from the stress field diminishes faster the origin compared to the distance than r-' as r + 00, and there is no contribution to the value of M(C") from the remote part of C" either. Take n, to the normal to the loop Ce pointing away from the dislocation. Then coordinates of points along the circular loop are given by x i = ti eni,
a,
+
L . B. Freund
14 where E
is the radius of the loop. On the basis of (2.14),
M(Ce) ti
+
[Uni - O k j n j U k , i ] dT
j"[6(6)
- 6,#)nji&.&?)]
do = 0
(2.15)
--I
where the tilde (") is used to denote the angular variations of elastic fields for points arbitrarily close to the dislocation; for example,
u x O(e)E-2
(2.16)
along Ce.The second integral has been evaluated in a somewhat more general setting by Freund (1978)and the result is the prelogarithmic energy factor for an edge dislocation in an unbounded body given by p(b: b$)/4~(1- v). The integrand of the first term in (2.15) is Eshelby's energy momentum tensor (Eshelby, 1951), and the value of the integral is the energetic force on the dislocation. If this force is denoted by J i then (2.15) becomes
+
5iJi
+ 4 ~ ( +1-b:)V ) = o
(2.17)
This expression provides the force on the dislocation in the direction of ti. However, the energy of the system is invariant under translation of the dislocation in the x2 direction, so J , = 0. Consequently, the net force is in the x1 direction and given by
(2.18) The force attracts the dislocation to the interface, as anticipated on the basis of heuristic argument in Section I, and its value is independent of the direction of the Burgers vector of the dislocation. This result is well known (Hirth and Lothe, 1982). It turns out, however, that the same approach can be used to determine the force on a dislocation near a bimaterial interface, which is a much more difficult problem to analyze. Consider the bimaterial system formed by joining elastic half spaces along their plane boundaries as shown in Fig. 7. A dislocation exists in material 1 at a normal distance h from the interface. Closed paths Cp and C; are introduced as shown. Because traction and displacement are continuous across the interface, the contributions to M ( C T ) and M(C2) along x1 = 0 cancel each other. The contribution from the small loop Ce around the
Dislocations in Strained-Layer Semiconductor Materials
15
FIG.7. Integration paths for application of the M-integral conservation law to determine the force on a dislocation due to its proximity to a bimaterial interface.
dislocation provides precisely the same two terms as given in (2.15).The main difference between this case and the former is that the contributions from the semicircular arcs of indefinitely large radius are nonzero. For points far from the origin compared to the elastic fields are essentially those of an edge dislocation located in the interface, a much simpler case to consider. Indeed, as shown by Rice (1985), the value of the M-integral taken along the full remote circle is precisely the prelogarithmic energy factor for the bimaterial interface dislocation; see Barnett and Lothe (1974).If the contributions to the M-integral are combined and the conservation law is enforced, the force on the dislocation in the x,-direction is found to be
a
(2.19)
where a and a=
B are the Dundurs (1969) bimaterial constants defined by r(i - vl) - (1 - v2) r ( i - vl) (1 - v,)’
+
p = - 1 r ( i - 2 4 - (I - 2v,) 2 r(i - V J
+ (1 - v 2 )
(2.20)
and r = p2/p1. The main reason for introducing this development in the context of dislocation structures in strained-layer epitaxy is because the models in that area are typically developed under the assumption that the elastic properties of the film and the substrate are identical. Consequently, the force on the
16
L . B. Freund
dislocation due to the free surface is taken into account, but the force due to the interface is neglected, The expression (2.19) makes it possible to estimate the force due to a difference in elastic constants provided that the difference is not too great. This force is discussed in some detail by Dundurs and Sendeckyj (1965). The particular case of a dislocation in a bonded surface layer was considered by Lee and Dundurs (1973).
C. NETFORCE ON CERTAIN LINES 1. Line from the Dislocation to the Surface
A quantity that plays a central role in the description of the driving force on a dislocation in a strained layer is the force resulting from traction that acts on the slipped portion of the glide plane. This traction may arise from an applied stress field or from the stress field djof the dislocation itself. These forces are denoted by f y and f 9, respectively, in the subsequent discussion of the driving force on a threading dislocation. The discontinuity in material displacement across the slipped portion of the glide plane, that is, the Burgers displacement b,, is spatially uniform. During formation of the dislocation, the forcef 9 is constant, so the work done by this force is simplyf yb,. On the other hand, the forceff is proportional to the magnitude of the Burgers displacement, say f 4 = Pijbj,where P , is independent of bi. Then the work done by this force during formation of the dislocation is +fibi (This distinction accounts for the factor of one-half appearing in one of the expressions in (3.17) but not in the other.) With reference to Fig. 8, the forcef: is the force (per unit length in the z or x j direction) due to traction on the line AB. If the boundary of the material BC is traction free then the force acting on AB is equal to that acting on AC. The latter is often simpler to calculate (the line coincides with a coordinate line), and the resultant forces on surfaces extending from the dislocation line in a direction normal to the plane boundary are calculated routinely in this way here. C
C
FIG.8. Contours used to demonstrate that the energy of a dislocation in a bounded body depends on the specific slip surface along which the relative displacement occurs.
Dislocations in Strained-Layer Semiconductor Materials
17
The observed equivalence of the resultant forces on AB and AC just noted is subject to the proviso that the tractions are integrable along these lines, including at the endpoints. Ths is not the case when a dislocation line coincides with the endpoint at A in the figure. In the presence of a dislocation at A, a small cylinder of material of radius r, and centered at A is removed, and the tractions are applied to the newly created cylindrical surface corresponding to the stress field of an isolated dislocation in an unbounded body, as given in (2.3).In this case, the triangular region ABC in Fig. 8(a) is replaced by the slightly more complicated region A’ABC in Fig. 8(b). Now, with the surface BC still traction free, the resultant force on A’C differs from that on AB by an amount equal to the force on the circular arc A’A. For a Volterra dislocation field, this force has a nonzero value for arbitrarily small r,, in general, because the traction is singular as I - ’ at the dislocation core. The effect of this force is usually small, and it is neglected in obtaining all results reported in this article, unless noted explicitly to the contrary. However, it is worthwhile to recognize the existence of this force. For an elastic dislocation in an unbounded body, the energy of the dislocation is independent of the surface on which the Burgers displacement is imposed, but the same is not strictly true for bounded solids. In addition, this force accounts for some differences in mathematical expressions for dislocation energies or driving forces in strained-layer systems in the literature. The resultant force, per unit length along the dislocation line, on the glide plane of a dislocation in a half-space is now determined as the force on the line x = 0, - h + r, y 0. From Section IIA, the components of stress that define the traction on that surface are
-= -=
OXZ(0,
Pbz 2h Y)=2n (h + Y)(h - Y)
(2.21)
Integration of these expressions over - h + r, < y < 0 yields the resultant force with components
R,
Pbz
2h r,
= - ln-
2n
(2.22)
This vector valued function of depth h is used throughout this chapter.
L . B. F r e d
18
FIG.9. Specific lines for calculation of net force due to the traction induced by a dislocation field, as given in (2.22) and (2.23).
2. Straight Contour Normal t o the Boundary It is also useful to record here, for future reference, the force due to a dislocation on an arbitrary plane surface extending a finite distance from the traction boundary, as shown in Fig. 9. If the appropriate components of stress are integrated along the line x = x,, yo < y < 0 then the resultant force is found to have components
Nx(xo7 Yo,
-
- iNy(x0, Yo, h)
[
+ ix,) + 2bx, + 2ibh + ix,)’ ih + iy, + x, ih - iy, + x, 2(bh + 2Eh + ibx,) h2 - 2hy, + y; + x ; + 61n h - yo + ix, h2 + 2hy, + y; + x;
46h(h 4n(1 - v) ( h - y o p
1
(2.23)
111. An Isolated Dislocation in a Strained Layer
The model that has provided the basis of most discussion of the formation of misfit dislocations is illustrated in Fig. 10, which shows a layer of thickness h coherently bonded to a relatively thick substrate of similar elastic properties. The inclined plane in Fig. 10 represents a glide plane for dislocation motion in the film, and if the film and substrate have shared crystal structure and orientation, this glide plane is continued across the interface into the substrate. For example, a common orientation for materials with a diamond cubic or zinc blende structure is with a (100) direction as
Dislocations in Strained-Layer Semiconductor Materials
19
FIG. 10. Schematic diagram of a dislocation in a strained layer bonded to a substrate. The dislocation has a threading segment extending across the thickness of the film,and an interface misfit segment trailing behind the threading segment as it advances along the glide plane.
interface normal and a { 111)-plane as glide plane, so the line of intersection between the glide plane and the interface is a (1 10) direction. The preferred Burgers vectors of glide dislocations on this glide plane are of length b = a,/$ and they lie either along the line of intersection of the glide plane and the interface plane or in the glide plane at 60" to this line. Glide dislocations with Burgers vector parallel to the interface relieve no misfit strain through formation, so only the so-called 60" dislocations are considered. The direction of the Burgers vector is assumed to be such that elastic strain in the film is relieved by dislocation formation, at least locally.
A. THETHREADING DISLOCATION Consider the behavior of an isolated glide dislocation as represented by the heavy line in Fig. 10. This dislocation begins on the traction-free surface x2 = h of the strained layer, threads across the thickness to the interface, and then extends indefinitely along the interface x2 = 0 as a misfit dislocation. The traction on the glide plane induced by the stress in the layer has a component in the direction of the Burgers vector of this dislocation, and consequently, this traction tends to push the threading segment of the dislocation further along the glide plane (in the negative x j direction in the figure), thereby increasing the length of the segment lying in the interface, the so-called misfit dislocation. As noted in (1.6), the elastic energy supply per unit area of interface increases linearly with h. On the other hand, the formation of an additional length of misfit dislocation carries an overhead in the form of energy required to create local material distortions associated with this length, as well as to overcome the tendency for the dislocation to be
20
L . B. Freund
drawn out of the crystal through the free surface. This energy requirement also depends on the film thickness but only weakly for h greater than several times the Burgers displacement b. Thus, for h greater than some value h,, the effect of the elastic mismatch stress dominates the effect of the self-stress on the dislocation, and the threading segment tends to advance along the glide plane. For h < h,, the reverse is true and the threading segment tends to recede. The discriminating thickness h, is the critical thickness for the strained layer, a concept of central interest from both theoretical and practical perspectives. The formation of misfit dislocations is inherently an inelastic process that occurs as a consequence of the discrete nature of the crystalline material. It is thus ironic that, with only a few exceptions, quantitative estimates of associated effects are based on linear elastic continuum models. While this approach suffers from a number of shortcomings, most notably the inability to deal with core effects and the need to introduce an artificial core cutoff radius, it persists as the principal basis for modeling in the area because the driving force on an isolated dislocation derives mainly from its long-range elastic field. The mechanical environment of the dislocation in Fig. 10 is established by the two independent and uncoupled equilibrium stress fields in the body. One of these fields, called the applied stress oyj, arises from the mismatch strain between the film and substrate, as well as from any applied boundary tractions or other inhomogeneities in the material. This field is the total stress in the absence of the dislocation being considered. The second stress field otj is the equilibrium stress field of the dislocation itself, commonly called the self-stress. It is the total stress in the absence of mismatch strain or other applied loading, but with the free surface taken into account. The applied stress exerts a force on the threading dislocation, tending to advance it along the glide plane, while the self-stress exerts an opposed force on the dislocation, tending to cause it to recede. The original argument by Matthews, Mader, and Light (1970) that led to an estimate of critical thickness was based on properties of these stress fields. The work argument to be outlined next leads to essentially the same result for a uniformly strained film, but it has the advantage of broader applicability than the original force analysis. The constructive character of the argument also makes clear the circumstances under which the “equilibrium approach” and the “energy approach” to critical thickness are the same.
B. ENERGYVARIATIONS
DURING
GLIDE
Consider a material element of volume V with bounding surface S. The outward unit normal vector to S is n,. The material is assumed to be a linear
Dislocations in Strained-Layer Semiconductor Materials
21
elastic continuum, except for one or more surfaces interior to the volume across which the displacement field may be discontinuous. Traction is continuous across each such glide surface, and no external force is required to maintain equilibrium. Suppose that some equilibrium stress distribution aij exists throughout the volume. Then the average stress Oij in the material element would be
It is well known that this integral does not exist if the stress field arises from an elastic dislocation due to the strength of the elastic singularity. In such cases, a small tubular region around the dislocation line is excluded from the volume Y This will be done explicitly when particular calculations are done, but it will be adopted only tacitly in the discussion of this section. An average strain for any internal state of deformation of the element specified by a displacement field ut can be defined by means of a virtual work argument. Suppose that the average strain is denoted by 23. Then this average strain is defined by the condition that the virtual work done by the average of any equilibrium stress distribution aijas this strain develops must be equal to the virtual work done by the boundary tractions aijnj of the equilibrium stress field on S working through the displacement field u: for the internal state of interest, that is, r 011.I J3V = Js aijnju: dS A form of this expression which is very useful for present purposes can be obtained by application of the divergence theorem to the right side of (3.2). Because the displacement is not continuous throughout K however, the glide surface(s) must be excluded from the volume. This is accomplished by a standard procedure whereby the element is divided into two (or more). separate regions by the glide surface(s) and the divergence theorem is then applied within each of these regions. Of course, a slip may have occurred over only a portion of the glide surface within an element, but the entire glide surface is taken into account to include the prospect of dislocation motion on the glide surface over time. A unit normal vector is assigned to each glide surface to give it an orientation; the volume of material into which the normal vector points is denoted by V,, and the volume away from the direction of the normal is denoted by V-. Because the normal vector is the outward normal for the volume V-, it is denoted by n;. The entire glide surface is denoted by SgLide, even though slip may have occurred over only a part of that surface at any given time. A depiction for a single glide surface is shown in Fig. 11. In
L . B. Freund
22
FIG.11. A generic material element containing a single glide plane Sllid..The displacement is discontinuous across the glide plane behind a propagating line L. The portion of the glide plane with such a discontinuity is S,,,,.
view of equilibrium of the stress field throughout V and continuity of traction application of the divergence theorem to the right side of (3.2) across Splide, within the separate regions V+ and V- yields
where a superscript (-) or (+) is used to indicate the limiting value of a field as Sglidc is approached from within V- or V+, respectively. Suppose that the displacement discontinuity Cut- - u:+] is denoted by b: over the surface Sglide,and that it is nonzero over a portion of the glide surface denoted by Sslip.Then it follows immediately that the overall average strain of the element is the sum of the two parts (Rice, 1970) (3.4)
For applications to dislocation theory, the vector b t is taken to be constant over a portion of the glide surface denoted by Sslipand to be zero over the remainder. In this case, the expression for slip strain in (3.4) simplifies even further to p l i p ij
=
+(nib!
+ njV)Ss,ip/V
(3.5)
Two observations should be made concerning this result. For one thing, it applies for a single dislocation within the volume of interest. In general, the slip strain will be an expression having the form of a series of terms of the kind appearing on the right-hand side of (3.5), one term for each dislocation involved. Second, the normal vector and the Burgers vector are fixed by the
Dislocations in Strained-Layer Semiconductor Materials
23
crystal structure of the material. Consequently, the slip strain rate is determined by the rate of change of the net slip surface SSlip.
C. THEDRIVING FORCEON
THE
DISLOCATION
When it is interpreted in the context of the dislocation problems at hand, the relationship (3.3) provides a basis for analyzing the driving force on gliding dislocations. Suppose now that aijis the actual total stress distribution in an element in which a dislocation is propagating and that u: is the actual particle velocity ui corresponding to this stress and the propagating dislocation. Then the kinematic jump across the glide plane [u?- - u?'] = [u; - $3 = [lai]is zero everywhere except very near to the moving portion of the dislocation line, say L. Under these conditions,
aijiijV=
jv
aijiijdV-
aijn,: [UJdS
(3.6)
where the interpretation of the kinematic variables as rates has been made explicit. For a dislocation configuration, the integrand of the last term in (3.6) is zero everywhere over the glide surface except right along the moving portions of the dislocation line L. The essential interpretation of (3.6) is that the rate of internal mechanical energy change of the element associated with mean strain equals the rate of strain energy increase reduced by the rate at which work is done on the glide plane by the tractions applied there. The last term in (3.6) expresses the rate of work dissipated by the motion of the dislocation as the inner product of traction with a particle velocity. Suppose mi is a unit vector normal to the dislocation line in the glide plane (see Fig. 11). This term is readily converted to a form involving the inner product of a force with the dislocation line velocity. Without loss of generality, the coordinate system can be oriented so that the glide plane coincides with the x,,x,-plane. Introduce a function, say, x(xl, x,, t), which is defined so the equation of L at any time t is x = 0, with x > 0 over the slipped portion of the glide plane. The parametric form of the motion of the dislocation is expressed as x1 = Xl(s,t), x, = X , ( s , t ) where s is arclength along L. Then
where 6 ( * )is the Dirac delta function. To follow L as time goes on, the
L . B. Freund
24
observation point must move according to dX = 0. Thus,
where the components of the velocity of a point on L are vk = ax,/&.The sifting property of the delta function in (3.6) makes it possible to reduce the area integral in (3.6) to the line integral n
J
n
(3.9) Sdi,
where it is recognized that -V~/lVxl is the unit vector m.The right side of (3.9) is the form of the driving force obtained by Freund (1990a). The unit vector nJ:, which is normal to the glide plane, can be represented in terms of mi and a second unit vector, say T i , which is tangent to L as nJ: = ejklmkzl, where ejrl is the alternating symbol of tensor analysis. Then r
r aijnJ:bimrvrds =
JL
bicijejLltlu("')mkds
(3.10)
JL
where = urmr is the normal velocity of the dislocation line. The integrand is precisely the inner product of the Peach-Koehler force biaijejklTlwith the normal velocity v("')mr of the dislocation line L. Finally, note that the result (3.10) is a scalar equation, and thus, the form obtained is independent of the orientation of coordinates. This last term in (3.6) is the rate of energy flow into the body due to the work of tractions on the glide plane, which is the negative of the rate of energy dissipation associated with dislocation motion. For the strained-layer relaxation processes of interest here, the average strain is fixed by the initial mismatch strain so the rate of change of average strain is zero. The result (3.6) then states that the rate of elastic energy increase plus the rate of energy dissipation in dislocation motion is zero. This observation makes it clear that the expansion of slip surfaces is done at the expense of stored elastic energy in the material. The system is closed, in the sense that no external forces work during the process of strain relaxation. Thus, the rate at which energy is consumed during formation of dislocations in the strained layer is balanced by the rate of elastic energy decrease in the strained layer. On physical grounds, a dislocation will tend to advance if the energy dissipation in doing so is positive, and it will tend to recede if the energy dissipation in an advance is negative. Thus, the discriminating condition that serves as a criterion for whether a change in configuration of a dislocation
Dislocations in Strained-Layer Semiconductor Materials
25
will occur is that the rate of dissipation for the change is zero, Furthermore, the tendency for that advance to occur increases as the rate of energy dissipation associated with the advance increases. This physical idea leads to the notion of driving force on a dislocation as a means of quantifying the tendency to change its configuration. Of course, to calculate a driving force from a change in energy, a kinematic description of that motion must be provided with the driving force components being work conjugate to the kinematic variables employed in that description. For example, if a finite number of kinematic variables, say t k ( t ) for k = 1,. . . , M , are used to characterize the shape of a dislocation, then
(3.1 1) where g k is the generalized force work conjugate to &. If the motion of a dislocation is a steady translation of a configuration in the negative x3 direction at speed u, then the position is completely defined by x 3 = - <(t)= - v t and the dissipation integral (3.11) has the form F v where 9 is the net energetic driving force for the process, which is work conjugate to If a strained layer is bonded to a relatively large and thick substrate, then the substrate is unstrained before, during, and after any strain relaxation. It follows that, under these conditions, the average strain in the layer is constant during the strain relaxation process. In other words, the elastic strain may be reduced due to dislocation motion but the slip strain increases so that the total strain, averaged over the layer, remains unchanged in the sense of (3.6). This is not true pointwise in the film, but only in the average sense. It follows immediately that if the configuration and dislocation distribution are periodic in some way, then the total strain within each period of the configuration is necessarily constant during relaxation, as well. Consider the particular element shown in Fig. 12 and suppose that the average strain is constant or iij= 0. If the glide is steady state at speed v in the negative x 3 direction, then the strain rate kij = vaeij/ax3. The first term on the right side of (3.6) can then be integrated to yield
r.
=
!A+
IgY&-.E-dA -v 2 1J 1J
%+
lg7&;7dA 2 CJ I J +0
9
(3.12)
where c (+) indicates evaluation of a variable far behind (ahead of) the threading dislocation along the glide plane. If the common factor v is
L . B. Freund
26
FIG.12. A slice of a layer-substrate system with a single threading dislocation. The slice may be indefinitely wide, as in the case of an isolated dislocation,or of width equal to the period in the case of a periodic array of dislocations.
cancelled from (3.12), then it takes the form
I+
4o;E;dA
=
-
I+
'207.E7.dA 1J 1J
(3.13)
Evidently, this is a statement that the driving force for the glide of the dislocation equals the energy density at the end of the element far ahead of the advancing dislocation minus the energy density far behind the dislocation. But these are precisely the quantities that enter the two critical thickness criteria known commonly as the energy criterion and the equilibrium criterion (Matthews et al., 1970). The driving force B that appears on the left side of (3.13) is the net force acting on the threading dislocation in the direction of its propagation, which is calculated in the equilibrium criterion. On the other hand, the energies appearing on the right side of (3.13) are the two energies compared in the energy criterion (van der Merwe and van der Berg, 1972). One is the energy per unit length in the x3 direction in the absence of the dislocation, and the other is the energy per unit length in the x3 direction in the presence of the fully formed dislocation, without regard for the mechanism by which the dislocation was formed. The result (3.13) shows that, if the process by which the dislocation is formed is steady state, then the two criteria are the same. This conclusion was established by Freund (1987) who gave a rudimentary proof on the basis of a cut-and-paste argument for the specific case of an isolated dislocation. More recently, Willis et al. (1990)
Dislocations in Strained-Layer Semiconductor Materials
27
demonstrated the equivalence for a periodic array of dislocations. Some examples where this result can be exploited are provided in the next section. The difference of integrals in (3.13) can be recast in a form that simplifies calculation of 9 in particular cases. The domains of integration of the two integrals are the same area in the x,,x,-plane, that is, A , = A , = A, but the integrands are different limiting values of certain fields, so that
(3.14) The two-dimensional equilibrium field 0; is the initial state prior to appearance of the dislocation whose motion is being considered. This stress field provides the loading on the growing dislocation, and it was defined to be the applied stress flj in the Introduction. The two-dimensional equilibrium field 0; is this same field perturbed by the equilibrium dislocation field, denoted by .fj, so that (3.15) 0 :1J = (J?. V + ad. V over the area A in (3.14). The area is that shown in Fig. 13, and (3.14) can then be rewritten in the form r
(3.16) If it is recalled that oijeij = ( r ~ ~ for ~ u any ~ ) equilibrium ,~ field, then application of the divergence theorem to (3.16) leads immediately to the form
9 = -[f: + +ff]bi
(3.17)
where bi = [u; - u'] is the Burgers displacement of the dislocation, andf, is the force on the slip surface per unit length in the x3 direction given by
(3.18)
FIG.13. Area of integration A in (3.14).
28
L . B. F r e d
for either equilibrium stress field. The angle a is the angle of inclination of the slip surface as defined in Fig. 13. The result (3.17) reduces the calculation of the driving force on the dislocation and the estimate of critical thickness to the calculation of the resultant force on a slip surface due to a twodimensional dislocation field. Although the foregoing discussion was tacitly based on the assumption of like properties for the isotropic elastic film and substrate, it goes through without modification for general anisotropy and for distinct elastic properties. Likewise, it does through with modest modification for nonlinear elastic material behavior. It is common practice to add a dislocation core contribution to the right side of (3.17). This core contribution is usually estimated in the following way. The material around the dislocation line within a tubular surface of radius ro is removed and the effect of the removed material is replaced by traction on the newly created surface, which leaves the stress field outside the core region unaffected. If the core radius is small compared to all other continuum dimensions in the configuration, including the curvature of the dislocation line in the glide plane, then this traction is that corresponding to a long straight dislocation in an unbounded body. With this point of view, the core contribution is estimated to be p(1 - 2v)(b: + bi)/16n(l - v)’ per unit length of dislocation line. The two-dimensional equilibrium stress fields for elastic dislocations, which are required to yield the driving force for any of a host of misfit formation problems, are commonly expressed in terms of the real Airy stress function, the complex Kolosoff- Muskhelishvili stress functions, or exponential Fourier transform integrals. The results in terms of the Airy function and the analytic functions are particularly well suited for the general approach being outlined here because each representation includes expressions for the resultant force of an equilibrium stress field on a contour in terms of a combination of the relevant stress functions evaluated at the end points of the contour, thus precluding the need to actually integrate the stress components in either case to determine this force.
D. CRITICAL THICKNESS The notion of a critical thickness for the strained layer was introduced in Section I. It is essentially the discriminating thickness between the tendency for the threading dislocation to spontaneously advance or retreat along its glide plane. This tendency is reflected in the driving force; that is, the
Dislocations in Strained-Layer Semiconductor Materials
29
dislocation will tend to advance or retreat if the energetic driving force acting on it is positive or negative, thus the discriminating condition is seen to be that 9'= 0, which provides a means of calculating the critical thickness, given the mismatch strain, material properties, and Burgers vector. If the angle o! is taken to be zero, then the components of force on the slip plane x1 = 0,O < x2 c h per unit length in the x,-direction due to the elastic mismatch far ahead of the threading segment are f:
= 2phc0
(-),
fz
l+v 1-v
=
1 ;= 0
(3.19)
The field of a straight dislocation in a half-space with traction-free surface, with the dislocation line parallel to the free surface, is given in Section 11. The traction on the plane x1 = 0, r, c x2 < h is readily calculated from this stress field, as illustrated in Section IIC, from which it follows that
According to (3.17), the driving force on the threading dislocation is then
(i ":>
9= - 2 / . ~ h b l ~ g ~
-
P 4 4 1 - v)
For reasons outlined in Section IIC, this result is accurate only to terms of order pb2. Nonetheless, all terms obtained through evaluating the force defined by the integral are retained, including terms of this order, so that the expressions appearing here for different cases are consistent with each other. The critical thickness condition is established by setting 9 = 0. Otherwise, 9 represents the driving force tending to change the length of the interface misfit dislocation. Following some algebraic manipulation, the critical thickness condition takes the form
8nh,b1(l
+ v)c0 + [ b f + b: + (1
-
2h v)b;] In 2 - +(b: L r7
+ b:) = 0
(3.22)
This is essentially the result obtained by Matthews et al. (1970), and it has become known as the Matthews-Blakeslee critical thickness condition. For the case of cubic materials with a [OlO] interface, the Burgers vector is given
L . B. Freund
30
by
bl = - b sin a sin j3,
b3 = b cos j3
b, = b cos a sin j3,
(3.23)
where B = f4 3 or & 2 4 3 and sin a = 1 1 3 . With the sign convention for Burgers displacement being followed here, strain relief occurs with Eob, < 0. For the case when the cutoff radius is chosen to be ro z b, the dependence of critical thickness on mismatch strain is shown in Fig. 14. The data points in this figure show the results of observations on a SiGe/Si(100)system in which initial relaxation occurred almost exclusively by formation of 60" misfit dislocations (Houghton, 1991). In these observations, a dislocation etching technique developed by Tuppen, Gibbings, and Hockly (1989) provided sensitive detection of threading dislocations at extremely low densities, on the order of one threading dislocation per square centimeter. The Matthews critical thickness theory is a fundamentally sound, selfconsistent mechanistic theory that has played a central role in understanding strain relaxation in heterostructures (Matthews and Blakeslee, 1974, 1975, 1976). Certain assumptions underlie the theory that must be kept in mind in applying the criterion to assess a particular set of observations. Among the limitations of the theory are as follows: 1. The criterion refers to the conditions that prevail when it first becomes
103
Equation (3.22) SiGe/Si(l 00)
0
102
1 01
100'
0.0
'
'
'
'
0.3
'
'
'
0.6 EO
'
' 0.9
'
'
' ' 1.2
'
' 1.5
(%)
FIG. 14. Critical thickness h, versus elastic mismatch strain e, for glide of an isolated threading dislocation according to (3.22). The discrete points indicate observations of onset of glide of threading dislocations in a SiGe/Si(l00) system as reported by Houghton (1991).
Dislocations in Strained-Layer Semiconductor Materials
31
possible for any threading dislocation to advance. The ability to detect the onset of relaxation depends on the resolution of the technique used to observe dislocation motion (Fritz, 1987; Gourley, Fritz, and Dawson, 1988). X-ray diffraction methods detect average strain over an area on the order of a square millimeter, and a detectable change in average strain occurs only after extensive dislocation activity at thicknesses well in excess of the true critical thickness. Consequently, such methods are ineffective in efforts to actually observe the onset of dislocation glide in strained layers. 2. It is assumed a priori that a threading dislocation exists, and the criterion addresses only the tendency for that threading segment to advance or recede. It has nothing to say about the nucleation of such threading segments. From time to time, it has been observed that films could be grown to a thickness in excess of the critical thickness under certain circumstances, and it has been argued that such observations imply that the critical thickness theory is incorrect. This is not the case, however, because the premises of the critical thickness theory are not met in such cases. In particular, if no threading dislocation is present then the question of whether or not such a dislocation will glide cannot be considered. 3. The tendency for an existing threading segment to advance or recede does not imply that it will actually do so in any particular way. A kinetic model of the glide process is required to consider this aspect, including effects that cannot be taken into account in a continuum model. This issue will be discussed briefly in Section IV. 4. The model also presumes that the threading dislocation exists in a layer that is otherwise spatially uniform. However, the behavior of a dislocation can be strongly influenced by other dislocations present, either on parallel or intersecting glide planes. 5. At the level of the preceding discussion, the critical thickness condition % = 0 is based on elastic continuum concepts only. This overlooks a variety of effects that may bear on the process. Among these are the glide resistance due to the Peierls-Nabarro stress of the material (Matthews, 1975), the role of surface or interface energy (Cammarata and Sieradzki, 1989), the resistance due to creation of a surface ledge in the wake of the threading dislocation (Matthews, 1975), and the possibility of dissociation of the dislocations into Shockley partial dislocations (Alexander, 1986). Thus, the critical thickness condition could be modified to read % = 9,where 9 is a resisting force representing these subcontinuum effects.
32
L . B. Freund
IV. Driving Force in More Complex Situations
The study outlined in Section 111 was concerned with the canonical problem in the area of strain relaxation in epitaxial films by means of dislocation glide, namely, the isolated threading dislocation in a single, uniformly strained layer. The ideas introduced in discussing the problem, however, have much broader applicability than just in this particular configuration. Among the configurations that can be studied by generalizing the approach are those involving gradients in strain through the thickness of a single film, the presence of more than one layer on the surface of the substrate, and the interaction of dislocations moving on parallel guide planes through a strained layer. These are discussed separately in the subsections to follow. Of course, these effects can also be combined in various ways, but the generalizations are not pursued beyond the extensions outlined in this section. A. NONUNIFORM ELASTICMISMATCH STRAIN
Some epitaxial film growth processes result in a mismatch strain that is not uniform through the thickness of the film but which varies continuously with profile E&). For example, in the case of growth of a SiGe film on Si by molecular beam epitaxy, the Ge content of the incident beam can be varied during the growth process to produce a film with a very well-controlled strain gradient, say increasing linearly from no mismatch at the film-substrate interface as the film grows (LeGoues, Meyerson, and Morar, 1989). On the other hand, in a growth method known as limited reaction processing, Ge is implanted in a Si crystal with a depth profile determined by the incident beam energy of the implantation process (Paine, Howard, and Stoffel, 1990). High concentrations of Ge and high implantation energy result in the region of implantation being essentially amorphized. The compound material is subsequently recrystallized by selective thermal processing, thereby causing a crystallization front to propagate through the material, leaving a strained layer in its wake. The profile of the strain in the strained layer in this case is determined by the distribution of implanted Ge atoms. Mismatch dislocations can form in these graded strain layers, but it cannot be assumed a priori that the interface is the equilibrium position of these resulting misfit dislocations, as was done tacitly in the case of a uniformly strained layer. Suppose that a misfit dislocation forms at some distance '1
Dislocations in Strained-Layer Semiconductor Materials
33
from the interface, as a result of glide of the threading segment of the dislocation. Then the driving force on the threading segment can be calculated by the same procedure followed in Section IID, except that the forcef; appearing in (3.17) is given by
The net energetic driving force on the threading segment is then
where h - q is now the depth of the misfit dislocation from the free surface at points far behind the advancing threading segment. A term of order ,ub2 that is independent of h has been ignored in writing (4.2). A necessary condition for the formation of a misfit dislocation in a graded strain layer is that F ( q ) 2 0. For a particular depth of formation q to be admissible, the misfit dislocation must be in equilibrium there; that is, at that depth the condition
a9
-=0
(4.3)
all
must hold. Furthermore, the equilibrium position must be stable under variations in position; that is, at the equilibrium depth the condition
must also hold. For a given strain profile eO(x2)the smallest value of h for which all of these conditions can be satisfied provides the analog of the critical thickness introduced previously. Enforcement of the various con-. ditions provides a value of q as well as the critical thickness. For example, suppose that the strain gradient is constant, say, cO(x2)= g0x2. Then, the form of the condition 9 = 0 is readily determined from (4.2), and the equilibrium condition is found from (4.3) to be
a 9-
---[2(1 aq
CLb2
(1 - v)h
+ v ) - -b,- - q h2go
b h
b
[b:
+ bf + ( 1 - v)b:] ] = o 4zb2(1 - q/h)
(4.5)
For a given value of go, there is no equilibrium position for very small values of h, as illustrated in Fig. 15. As h increases, P value is reached at which there
34
L . B. Freund
cu
2
0.1
-
\
c
5
7
0.0
7 v
-0.1
-
-0.2 0.0
1
0.2
0.4
0.6
0.8
1 .o
77/h FIG. 15. Graph of a.F/aq from (4.5) versus distance from the interface
is a root of (4.5) of multiplicity 2, and for greater values of h there are two equilibrium positions. In the case of two equilibrium positions, the stability condition reveals that the position closer to the free surface is unstable whereas the position further from the surface is stable, as indicated in the figure.
B. UNSTRAINED CAPPING LAYER Suppose that a uniformly strained layer is grown to a thickness, say h,,, in excess of the critical thickness h, for the uniform elastic mismatch strain E ~ and that this layer contains an isolated threading dislocation. Film growth is then continued beyond the thickness h,, with a material that is lattice matched to the substrate, possibly the same material as the substrate. In the absence of any dislocation, this capping layer is unstrained or free of residual stress. However, the threading dislocation is extended by the growth process so that it threads across the thickness of the composite film consisting of the strained layer and capping layer as illustrated in Fig. 16. The approach developed above can again be applied to calculate the driving force on the threading dislocation (Houghton et al., 1989; Nix, Noble, and Turlo, 1990). Suppose that the total film thickness, that is, strained layer plus capping layer, is denoted by h, so that the thickness of the capping layer alone is
,
Dislocations in Strained-Layer Semiconductor Materials
35
FIG.16. Schematic diagram of a threading dislocation extending across both a strained layer of thickness h,, on a substrate and an unstrained capping layer.
h - hsl. The force f 4 is given for this case by (3.19) with h replaced by hsl to account for the fact that elastic energy can be recovered from only a part of the total thickness of the film. The other forcef! appearing in (3.17) is still given by (3.21),assuming that the misfit dislocation will form at the interface between the strained layer and the substrate. Thus, the net driving force on the threading dislocation segment in this case is
In view of the critical thickness condition on the strained layer alone, this result can be put into the form -%apped
- - & ~- lI )E -O hsI
[b:
+ 6: + (1 - v)b:] In h 8n(l
e e r
eef
+ v)bh,
-
hC
(4.7)
where the reference force is = 2pbh,(l + v)/(l - v), h, is the critical thickness of the strained layer in the absence of a capping layer, and b= Typically, the product blEOis negative. It should be noted that h, and c0 in (4.7) are not independent; if .z0 is fixed then the value of h, is determined from (3.22). From the fact that the first term on the right side of (4.7)is constant, it is evident that the driving force decreases continuously as the thickness of the unstrained capping layer increases. The reason for this outcome is that energy can be extracted only from the strained layer of thickness h,, to drive the threading dislocation forward, but the full thickness h must be distorted to increase the length of the interface misfit dislocation. The dependence represented by (4.7)is illustrated in Fig. 17 for the particular case when h,/b = 20 and hs,/hc= 2.5. The driving force increases as the
Jb,b,.
36
L . B. F r e d
1
3
2
4
h I hc FIG.17. Normalized driving force versus normalized layer thickness h/h, for a threading dislocation in an uncapped strained layer, a threading dislocation in a strained layer with an unstrained capping layer, and a threading dislocation in a buried layer. Results are for E,, = 0.01 and v = 0.3.
uncapped strained layer grows beyond critical thickness to 2Sh, at point p in the figure. Deposition of an unstrained capping layer results in a diminishing driving force along the path from point p to point q in the figure. The driving force continues to decrease as the capping layer becomes thicker, and eventually the burden of pulling along the threading segment in the unstrained capping layer without recovering any stored elastic energy becomes too great (Nix et al., 1990). This segment falls behind the segment within the strained layer. The resulting configuration is essentially a threading dislocation segment in a buried strained layer, leaving behind two interface misfit dislocations, one on each face of the buried strained layer, as it advances, as illustrated in Fig. 18. The influence of the free surface is greatly diminished in this case, but the role of the image force is played here by the attractive force tending to pull the two misfit dislocations on opposite faces of the buried layer toward each other. The driving force on the threading segment within the buried strained layer can also be determined by means of the argument advanced previously. The components of the forcef 9 on the slip plane far ahead of the threading segment are
Dislocations in Strained-Layer Semiconductor Materials
37
FIG. 18. Schematic diagram of a dislocation threading across the thickness of a buried strained layer, along with two trailing interface misfit dislocations, one on each face of the strained layer.
The net force on the portion of the glide plane between the two dislocations far behind the threading segment is determined simply by superimposing the stress fields of the two dislocations. With reference to the definitions in Section IIC, it follows that
ff= Ri(h)+ Ni(h,, tan a, -h,,, h - h,,) + R,(h - hsl) +Ni(-hh,ltana,h,,, h)
(4.9)
While the expression for f f can be written explicitly, the results are algebraically too complicated to provide any insight into the behavior. However, an approximate expression valid for h/h,, >> 1 is readily obtained, and it provides the essential information. Thus, for values of h large compared to h,, it is found that the driving force on a buried threading segment is
h: 14b: +$
+ 4b: + 8b,b, tanu + (1 - v)b:] 4nbhc(l + v)
(4.10)
accurate to terms of order h?,/h3. It is clear from this result that an unstrained capping layer has a strong stabilizing effect on a threading dislocation. For example, it is evident from (4.10) that, when h >> h,,, the driving force on a threading dislocation in a buried layer will be positive only if h,, > 2h,. In other words, for a given amount of mismatch strain cO, the critical thickness for a buried layer is twice that of a surface layer.
L . B. Freund
38
C. SUPERLATTICES
In its simplest form, a strained-layer superlattice is a thin composite film of many layers in which the material properties vary periodically in the thickness direction of the film, with the period being on the order of the thickness of the individual layers. The general approach outlined in the preceding section can be followed to estimate the driving force on a threading dislocation in a superlattice, either for the case when the threading dislocation extends through some of the layers in the structure or when it threads all of the layers. For example, suppose a SiGe superlattice is formed of a Si substrate by alternately growing n layers of SiGe, each of thickness hsic,, and n layers of Si, each of thickness hSi. If the total thickness of the superlattice is small compared to the thickness of the substrate, then the SiGe alloy layers will each have some mismatch strain c0, depending on the Ge content, and the Si layers will be unstrained prior to formation of any dislocations. Following the arguments of Section 111, the driving force on a threading dislocation extending across the thickness of all 2n layers in the superlattice and then continuing as a misfit dislocation at the interface between the superlattice and the substrate, can be shown to be
(4.1 1)
Not surprisingly, the driving force for the superlattice (and consequently the critical thickness condition) is the same as that for a single layer of thickness h (cf. Eq. (3.22)) if the thickness parameter heff
= n(hSiGe
+hi)
(4.12)
is identified with h and the strain averaged over the thickness &we = EOhSiGe/(hSiGe
-k
hi)
(4.13)
is identified with the mismatch strain c0 in the uniform layer. In the case of a superlattice, misfit dislocations can form at any of the interior interfaces. The case of a buried threading dislocation discussed in the preceding section provides an example of just such a configuration. However, the first misfit dislocations are more readily formed at the superlattice-substrate interface than at any of the interior interfaces. Extensive studies of arrays of dislocations in strained multilayers have been reported by Hirth and Feng (1990).
Dislocations in Strained-Layer Semiconductor Materials
39
D. ARRAYOF PARALLEL MISFITDISLOCATIONS As a misfit dislocation forms at the interface on a particular glide plane at some thickness beyond the critical thickness, it relaxes the elastic strain on adjacent glide planes in the crystal, thereby reducing the likelihood that parallel misfit dislocations will form in close proximity. The state of stress at more remote points is unaffected, so additional misfit dislocations can form readily on more distant glide planes. This interaction process leads to a more or less regularly spaced array of parallel misfit dislocations on a particular glide system. For a cubic crystal with a [00l] interface, such arrays commonly form along both the [llO] and [lfo] directions in the interface, although not symmetrically for InGaAs/GaAs( 100) or other 111-V compounds (Fitzgerald et al., 1989). If the thickness of the film is further increased through epitaxial growth, then the density of the array likewise increases. If the mean spacing of the identical dislocations in an array is denoted by p , as shown in Fig. 5, then the relaxation strain, that is, the reduction of elastic strain from the initial mismatch strain, is roughly b/p. This, in turn, implies that the spacing of dislocations in an array that completely relaxes an elastic mismatch strain e0 is roughly b/Eo. In making this estimate, of course, nondimensional geometrical factors of order unity have been overlooked. More detailed quantitative estimates are considered in the remainder of this section. The driving force on individual threading dislocations as they form increasing lengths of misfit dislocation in an array can be estimated by the procedure introduced in Section IIB. An attempt to define this driving force in terms of a particular physical mechanism highlights an uncertainty in the standard procedure for estimating the spacing of dislocations in a parallel array. In the approach pioneered by Frank and van der Merwe (1949), the total energy of a periodic array of dislocations in a uniformly strained layer is first determined. This energy is then tacitly assumed to be a state function that is to be minimized with respect to the spacing of the periodic array, the state variable. This approach does not necessarily capture the essence of the process by which such an array is formed, mainly because it assumes that all dislocations in the array are formed simultaneously, each subject to conditions identical to all others. In fact, the dislocations in the array form sequentially, so the growth of any particular member is influenced by those formed earlier. If this feature is included in modeling then estimates of array spacing at a given thickness are different from those obtained from the simultaneous formation hypothesis. Some progress on the sequential formation of the dislocations in a parallel array is also reported here.
L . B. Freund
40
A first estimate of the equilibrium spacing of the dislocations in a periodic array can be obtained by basing the calculation on the mean elastic strain in the film, rather than on the detailed elastic field of a dislocation array. If the elastic mismatch strain is initially E,, and then a single periodic array of dislocations lying parallel to the x3 axis is formed, the spatial average of the elastic strain component is simply
&?Yn= E, + b,/p
(4.14)
(Recall that the algebraic signs of E, and b , are typically opposite.) The spatial average of the strain component cj3 is still E,. The driving force on the threading segment of a dislocation gliding through the partially relaxed film, thereby forming an additional interface misfit dislocation, can be estimated from (3.17) simply by replacing the elastic mismatch strain E, with E??, that is,
' [
-
4 ~ ( 1- V )
[b:
+ bt + ( 1 - v)b:)]In
2h --
$b:
I0
+ b:)]
(4.15)
The equilibrium spacing for a given thickness follows from the condition Fmean(h, p ) = 0,and it is illustrated in Fig. 19 for an initial mismatch strain of 2.5
-
,-,-,-
simultaneous
h
c! Q 0 -0
Y
----_ , - t - , - ,
1.6
1 .o
I
1.3
1.6
1.9
2.2
2.5
log(h/b) FIG. 19. The equilibrium spacing p of a periodic array of dislocations in a cubic material versus thickness h of the strained layer, for E~ = 0.01 and v = 0.3, as predicted by three different models.
Dislocations in Strained-Layer Semiconductor Materials
41
1 percent. Because the elastic field of a dislocation is quite localized, the use of simple averaging methods can produce results that are misleading. For example, in the present instance, the use of simple averaging suggests that the mean elastic strain gradually approaches zero as the density of a dislocation array becomes higher. In fact, as will be shown later, the region over which a dislocation actually relieves background elastic strain is highly localized, and it is energetically possible to introduce enough dislocations by this mechanism to cause the average elastic strain to change sign during the process. To pursue a less approximate estimate, consider first the case of formation of a periodic array of identical misfit dislocations. The driving force on each of the threading dislocations that must propagate to form the misfits is again calculated by means of (3.17). Consider once again the general energy rate expression (3.6) and the associated schematic diagram in Fig. 20. For the case of an isolated dislocation, the block of material represented in Fig. 12 is assumed to be of indefinitely large extent in the x1 direction. There is then no energy flow into or out of the body through the remote planes normal to the xI axis, which was an observation central to recognizing that the left side of (3.6) is equal to zero during the formation process.
FIG.20. Schematicdiagram of the formation of a periodic array of interface misfit dislocations with period p by simultaneous glide of identical threading dislocations.
42
L . B. Freund
In the present case, the same surfaces in Fig. 12 are understood to be a distance p apart, say at x1 = _+p/2.The displacement distributions and stress distributions on the two surfaces are identical due to periodicity of the elastic field. The traction distribution on the surfaces are equal in magnitude but opposite in sign. Consequently, the net work done on these faces by the tractions imposed there is zero for any deformation process, which is an essential feature of periodic boundary conditions. It follows immediately that the term on the left side of (3.6) is zero for each slice of the strained layer system of width p in the x,-direction and that the driving force on the threading dislocation traveling along this slice is again given by (3.17). For the case of a film of thickness h with uniform mismatch strain E ~ the , componentsf; of the applied force on the glide plane are given by (3.19). The componentsf: of the force due to the dislocation are readily found from the results in Section IIA to be f! - i f $ =
4n(l - V)
2(b1 - ib,) In
p(1 - e - 4 n h i P ) 27tr,
+ 8bl-7th P
(4.16)
2n
p ( 1 - e -4 n h 9 2nr,
+ 2 2P 1
From (3.17), the driving force on each threading dislocation in the periodic array is
+4b:
nh
--
P
h:
+ b: + 2(1 - v)bi
-
(4.17)
This result was first obtained by Willis et a!. (1990, 1991)in a slightly different
Dislocations in Strained-Layer Semiconductor Materials
43
form. It is easily verified that, in the limit as h / p -+ 0, this expression for driving force approaches that for an isolated threading dislocation given in (3.211, lim E i m ( p ,h) = F ( h ) (4.18) hlp-0
The necessary condition for formation of this array is that Fsim(p,h) = 0, and the relationship between spacing p and thickness h implied by this necessary condition is shown in Fig. 19. Note that, for any film thickness, the spacing of dislocations is less than that predicted by the elementary mean strain model. This implies that it is energetically possible to introduce enough dislocations into the array to produce an average elastic strain at a fully relaxed condition that has a sign opposite to the original elastic mismatch strain. The foregoing analysis is based on the assumption that all the parallel dislocations are formed simultaneously. This is not what actually takes place in a relaxation process, of course (Thouless, 1990). The misfit dislocations are observed to form in an irregular order, with those formed earlier in the process generally diminishing the driving force on those formed later. To quantify the influence of this aspect of the relaxation process, consider the following sequence. A periodic array of misfit dislocations with a period of 2p is formed, partially relieving the initial elastic misfit strain co. The manner in which they form is immaterial for purposes of this discussion, although if they do form simultaneously then the driving force on each threading dislocation in the array as it forms is given by Shim(2p,h).After the first array is fully formed, a second set of misfit dislocations is introduced by glide of threading dislocations along the planes midway between the glide planes of the first set, as illustrated in Fig. 21. Again, the driving force on each dislocation in the second set can be determined by application of the general results recorded in Section IIC. For the moment, it is assumed that the Burgers vectors of the two sets are identical, but the implications of relaxing this restriction will be discussed later. The driving force on each threading dislocation in the second set as it forms is given by 9 & ( 2 p , h) plus a second contribution to account for the fact that the stress field of the first set induces a traction on the glide plane of the second set, which can influence the formation process. Suppose the first set forms at x1 = f 2 n p , n = .. . , - 1,0, 1,2,. . . . The additional contribution to f; due to the effect of interaction between the two sets is established by evaluating the net force due to the traction (2.8), on x1 = p , 0 < x2 < h. The result of this complicated but straightforward calculation provides the
L . B. Freund
44
FIG.21. Schematicdiagram of the formationof a periodic array of interface misfit dislocations with period p by formation of one set with period 2 p followed by glide of a second set of threading dislocations to result in a final configuration of period p .
expression 2 In $( 1
nh + e-4"h/p)+ 8 1+
nh
/lb2
f2
= 4n(l
- v)
[ 2 In &l
1 e-4nh/~
,-4nhlp
+ e-4nhip) + 8 - 1 + e - 4 n h / ~ (4.19)
as additional contributions to f;. The total driving force on each threading dislocation is then
Dislocations in Strained-Layer Semiconductor Materials
{
x 2[b:
45
+ b: + (1 - v)b:]ln3(1 + e-4"h/p)
nh +8b: - + 4bj(l - V) P P
(4.20)
This expression, too, reduces to (3.21) in the limit as h/p + 0. For a given elastic mismatch strain e0, the smallest thickness of the film at which additional dislocations can be formed in the gaps of an array with spacing 2p is given by the roots of Fseq(p,h) = 0. A graph of the relationship between h and p necessary for this to be the case is given in Fig. 19. It is interesting to note that the assumption of sequential formation leads to an estimate of spacing that is closer to the mean strain model than to the simultaneous formation model, although the differences are not particularly large. The expression for Fsim is a quadratic form in the components of the Burgers vector of a particular dislocation. This feature results in a definite algebraic sign for each of the quadratic terms. On the other hand, the expression beyond appearing on the right side of (4.20)is a bilinear form in the components of the Burgers vectors of two different dislocations. In writing (4.20),it was assumed that the Burgers vectors were, in fact, the same. However, they need not be so. The sign of b , is essentially fixed by the sign of the mismatch strain. However, the signs of the x2 components of Burgers vectors of the two interacting sets may be different, in which case the sign of all terms involving b: in (4.20)should be changed. The same is true for the x3 components. Thus, even though the spacing of the final array is fixed and the crystallography is fixed, there is some flexibility in the selection of components of Burgers vector and, consequently, in the level of driving force on dislocations tending to reduce the spacing of an array. Up to this point in the discussion of formation of parallel arrays of interface misfit dislocations, it has been tacitly assumed that all dislocations form on parallel guide planes. In reality, for most crystal orientations, such arrays can form simultaneously on more than one set of crystallographic planes. For example, in cubic materials with a [OOl] interface, arrays form in
eScq
L . B. Freund
46
both the [ l l O ] and [ti01 directions. Consequently, the traction on any potential glide plane is reduced by both the formation of dislocations on parallel glide planes and on intersecting glide planes, the latter reduction being due to the Poisson effect. This effect is often included in modeling in an approximate way, based on simple averages, and no complete modeling of this interaction is yet available. For a film with thickness that exceeds the critical thickness, but only by a factor of 2 or 3, the dislocation spacing is commonly found to be substantially greater than that predicted by the condition 9 = 0 (Nix, 1989; Paine et al., 1990) as illustrated for the case of ZnSe/GaAs(100) in Fig. 22 (Petruzzello er nl., 1988). There are two main reasons for this outcome, both of which were listed as limitations to the elementary critical thickness theory earlier. One reason concerns the kinetics of dislocation motion under positive driving force. In effect, a misfit dislocation does not form spontaneously whenever it becomes energetically favorable for it to exist. Instead, it must evolve according to some kinetic process involving time, temperature, and driving force. A second reason concerns dislocation interactions. The estimate based on (4.15) involves only average elastic strain in the film. The actual elastic
I ZnSelGaAs(lO0) h
mean strain model
-
0
103
102' 102
I
I
I 03
I 04
log(h/b) FIG. 22. Observed average spacing of interface misfit dislocations in the (110) and (170) directions of a ZnSe/GaAs(lOl) system with E,, = 0.003 (Petruuello et al., 1988). The curve is a prediction based on the elementary mean strain model.
Dislocations in Strained-Layer Semiconductor Materials
47
strain field, and therefore the actual applied stress field, an advancing threading dislocation encounters is highly nonuniform, especially for thinner films. The nonuniformity arises from the presence of other dislocations on the same glide plane, on parallel glide planes, or on intersecting glide planes (Willis et al., 1991; Hirth and Feng, 1990; Freund, 1990b). The situation of the interaction of dislocations on intersecting glide planes is considered next.
E. DISLOCATIONS ON INTERSECTINGGLIDEPLANES A possible impediment to the relaxation of elastic mismatch strain by glide of threading dislocations is the interaction with dislocations on intersecting glide planes. For the case of 35nm Ge0.25Sio.75 layers it has been reported that the lengths of individual interface dislocation lines did not increase significantly during relaxation, even though the total number of dislocations did increase significantly (Hull et al., 1989). Furthermore, the hesitation of a threading dislocation as it encounters a misfit dislocation in its path is evident in the real-time video recordings of threading dislocation motion in SiGe/Si films during an anneal obtained by means of transmission electron microscopy (Hull and Bean, 1990). It has been suggested that this behavior may be due to the interaction of dislocations on intersecting glide systems. A compelling case for the existence of this mechanism of glide resistance has also been shown in the work of Paine et al. (1990). The purpose in this section is to describe an estimate of the strength of this interaction. The energetic driving force for glide of a threading dislocation in a uniformly strained layer is adopted as the basis for comparison. The conditions necessary for steady glide of a threading dislocations are clearly not present in this case. Nonetheless, the definition of driving force provides a basis for making a quantitative estimate of the influence of this blocking effect. In this section, a threading dislocation is assumed to glide in the negative x,-direction on the xi, x,-plane (see Fig. 23). Furthermore, a second dislocation lies along the x1 axis, extending indefinitely in both directions. This dislocation could have been formed by glide of a threading dislocation at some earlier time. In any case, the Burgers vector of this dislocation is denoted by a,. The presence of the straight dislocation results in an additional traction on the glide plane of the threading dislocation, and consequently the driving force will be affected. To make the discussion fairly specific, the shaded plane in Fig. 24 is assumed to be a { 111)-plane in a cubic crystal, and the normal to the interface
48
L . B. Freund
FIG. 23. Schematic diagram of a threading dislocation approaching an interface misfit dislocation lying across its path.
is the [OOl] direction. This plane is identified as abq on the pyramids of { 111)planes in a cubic material in Fig. 24. The straight dislocation lies along the line ad in Fig. 24, and its glide plane is either adp and adq. For a threading dislocation gliding along the plane abq and interacting with a straight dislocation along the line ad, eight possible combinations of Burgers vectors will result in strain relief. The threading dislocation on plane abq may have a Burgers vector along either of the two lines aq or bq, and the straight dislocation along the line ad may have a Burgers vector along any of the lines aq, dq, up, and dp. Results are included here for four different combinations of Burgers vectors. In all four cases the Burgers vector of the threading dislocation coincides with bq in Fig. 24, and the Burgers vector of the misfit dislocation coincides with (i) dp, (ii) up, (iii) aq, or (iv) dq. The two Burgers vectors have the same magnitude, so that a = b. For purposes of computing the driving force on the gliding dislocation,
FIG.24. Pyramids of { 1 1 1)-planes in a cubic crystal showing both the crystallographicaxes and the spatial axes. With reference to Fig. 23, the interface normal is the [Ool] direction, and the Burgers vectors of threading dislocations must coincide with the edges of the pyramids.
Dislocations in Strained-Layer Semiconductor Materials
49
denote by AGj the additional contribution to the applied stress field Gjdue to dislocation interaction. With reference to (3.17), the quantity representing the additional contribution to the driving force due to the interaction is P
A 9=
A4jnJ:bidx;
-
(4.21)
j L
The stress field corresponding to the edge component of the fixed straight dislocation in Fig. 4 depends only on the position in the x 2 , x,-plane, and it can be computed from the results in Section IIA. It is evident that the stress components may be written as explicit but lengthy functions of position. The stress A 4 j is a linear function of ai so A4jnJ.nibi is a bilinear function of a, and b,. By symmetry, it can be seen that only four of the eight possible cases are independent. Graphs of the nondimensional function (4.22) as a surface over the plane of x2/h, x,lh for the Poisson ratio v = 0.3 and the four independent combinations of Burgers vectors a, and b, are shown in Fig. 25. In view of its role in the definition of the driving force in (4.21), the distribution represented by w(x2,x 3 ) is termed the force density on the surface. I t is obviously the inner product of the traction on the glide surface with the Burgers vector of the dislocation. It is also the inner product of the Peach-Koehler force with the unit vector normal to L in the glide plane, a quantity independent of the orientation of the dislocation line. The effect of the interaction is greatest near the line of the stationary dislocation, which is at the origin of coordinates. The surfaces are arbitrarily truncated at values of normalized force density of _+ 2, and the outlines of the resulting plateaus on the surfaces reveal level curves of force density. The surfaces are shown in terms of the independent variables (x2,x 3 ) but they could be shown equally well in terms of ( x i , x3), where x2 = x; cos a. In the course of advancing down its glide plane, the threading dislocation must glide through this force density distribution. If the force density is positive (negative) at a point on the glide plane, then the glide of a small segment of the threading dislocation past that point results in an increase (decrease) in the elastic energy of the body, and consequently, the energetic driving force is negative (positive). The interaction driving force A 9 is obtained by integrating the force density along the line of the threading dislocation in the way prescribed by (4.21).The value of A 9 in any particular case depends on the shape of L, and
50
L . B. Freund
a rough estimate can be obtained by assuming a particular shape, say a straight line perpendicular to the x,-axis and extending to the traction free surface at x i = hseca. An alternate approach to a quantitative estimate of the effect is suggested by the surfaces in Fig. 25. It is evident that, in the absence of an interface misfit dislocation in its path, the motion of a threading dislocation is controlled by two competing effects. One effect is the traction on the glide plane due to a uniform background strain E~ in the layer, and the other is the traction on the glide plane due to the stress field of the dislocation itself near the free surface. During interaction of the threading dislocation with a misfit dislocation in its path, the stress field
Dislocations in Strained-Layer Semiconductor Materials
51
FIG.25. Four cases of force density as defined in (4.22)versus position on the glide plane of the threading dislocation (see Fig. 23).
of the misfit dislocation can completely negate the traction due to the uniform background stress over some part of the glide plane. Furthermore the influence of the background stress is diminished, if not completely cancelled, over a larger part. This reduces the area of the glide plane within the layer over which a positive driving force may act, and thus, the threading dislocation must advance through a “channel” of width less than the layer thickness if it is to bypass the stationary misfit dislocation. This interpretation is suggested by the shapes of the surfaces in Fig. 25, and it is illustrated
L . B. Freund
52
Free surface
I
t
t
Substrate FIG.26. Schematic diagram showing the main effect of the misfit dislocation is to negate the traction of the background strain on the glide plane of the threading dislocation, thereby impeding its progress.
schematically in Fig. 26. In this diagram, the curve C, represents the level curve of force density along which the force density has the same value as the spatially uniform background force density. The curve C , represents a level curve of force density with a lower value. If the threading dislocation is to bypass the misfit dislocation, it must be able to glide through the remaining gap or channel of thickness h, between the point of closest approach of C, to the free surface and the surface itself. An estimate of the conditions under which this is possible is next extracted. To examine the quantitative implications of this interpretation, contours of the force density surface for case (d) in Fig. 25 is shown in Fig. 27. The contours are projections of the level curves of [27c(l - ~ ) h / p b ~ ] A 4 ~bin onto ,:
x3
FIG.27. Level curves of force density for case (d) from Fig. 25. Values of w on the level curves vary from 0 to 3 at intervals of 0.5.
Dislocations in Strained-Layer Semiconductor Materials
53
the glide plane; only level curves corresponding to positive values of force density are shown. The particular level curve enclosing the area of the glide plane over which the misfit dislocation stress field AGj completely cancels the uniform force density is denoted by C,. For the case of cubic materials under consideration, the latter force density reduces to wg
=
2741 - v)h
+
o?.n.b. = 4 ~ ( 1 v ) g
Pb2
’’
l’
(4.23)
where 0: is given in (1.4).A contour corresponding to a lower magnitude of background strain, say, E,, results from a level curve with value 6.7hsJb. The distance h, between the point of closest approach of the contour C, and the free surface is determined as follows. This distance is h, = h - x?, where xz is the value of x2 that satisfies the simultaneous equations W(X?,
x3) =
4x(1
+ v ) -h
fi
aw ~
b
&*’
(x?, x?) = 0
(4.24)
ax2
The first condition in (4.24)ensures that the point (xz, x3) is on the level curve of interest, and the second condition requires this point to be the particular point on the level curve closest to the free surface. These equations have been solved numerically for x f / h versus 6,. Note that E, is a measure of the reduction in driving force in the channel due to the misfit dislocation, so that E, - E , is a measure of the residual driving force at a distance h, from the free surface. (A true average of the residual driving force density in the channel can be calculated, but this introduces an unwarranted complication in pursuing the model.) Thus, it is assumed that the distance h, is the critical thickness, according to (3.22), corresponding to the reduced strain level E, - E,; that is, E,
- &* =
b(4 - V ) 8h In 2 16nhJl + v ) b
(4.25)
for the case of cubic materials being considered here. For any pair of values of E, and h,, this equation provides a critical condition on values of E, and h/b for the threading dislocation to bypass the misfit dislocation in its path. Of particular interest is the smallest value of background strain E~ in a layer of normalized thickness h/b for which the threading dislocation can get by the barrier. This value of has been found numerically and the results are shown for the two cases labeled (b) and (d) in Fig. 28. The interpretation of this figure is that the blocking mechanism will not be important in strain
54
L . 5.Freund
ln(h/b) FIG.28. The solid curves represent the minimum values of uniform background elastic strain for which the threading dislocation can successfully bypass the interface misfit dislocation for cases (b) and (d) from Fig. 25. The dashed curve is the critical thickness condition from (3.22).The discrete points are discussed in the text.
relaxation for systems with characteristics above and to the right of the curve, but that it will play a role in the strain relaxation process for characteristics that are below and to the left of the curve. While reports of direct observations of this blocking mechanism are scarce, there is no doubt that it is operative in some cases. Three data points based on transmission electron microscopic observations are included in Fig. 28. The points marked by square symbols are adapted from the work of Hull et al. (1989) on Ge,Si, -JSi( 100). Specimens were grown by molecular beam epitaxy, and dislocation motion during a subsequent high-temperature anneal was observed. For x = 0.15 and h = 300nm, it was reported that a threading dislocation was only rarely blocked by a misfit dislocation, so that this mechanism did not seem to be important in the stress relaxation process. These data correspond to the filled square symbol in Fig. 28, which is well above the curve on the “no blocking” side. For x = 0.25 and h = 35 nm, on the other hand, it was reported that a threading dislocation was 100 times more likely to be blocked by a misfit dislocation than in the former case, and that blocking seemed to be an important mechanism in strain relaxation.
Dislocations in Strained-Layer Semiconductor Materials
55
These data correspond to the unfilled square symbol in Fig. 28, which is a marginal case according to the theoretical blocking criterion. The x-shaped symbol represents the system considered by Paine et al. (1990). Blocking was evident in the transmission electron microscopy images reported, and the criterion indicates that many interactions should result in blocking. Thus, the available data are consistent with the criterion but the evidence is too meager to permit stronger conclusions. Finally, it is noted that the shape of the threading dislocation will probably change as it tries to overcome the impediment, but this effect is not considered here. Furthermore, the assumption that the straight interface misfit dislocation remains straight requires further consideration. The blocking criterion illustrated in Fig. 28 has been based on the threedimensional stress field associated with the misfit dislocation. With this result in hand, it is possible to offer a simple, more qualitative argument that leads to essentially the same result. Very roughly, the strain in the layer in Fig. 23 at a distance x t from the interface due to a misfit dislocation in the interface with Burgers displacement b is &
*
1 b 211 x:
=--
(4.26)
If the remaining distance to the free surface is denoted by h, previously, then
= h - xf, as
(4.27) The rudimentary critical condition on thickness h for layer strain analogous to (3.22)but stripped of all but the most essential details, is
b 2h In 4nh b
E~ = __
E ~ ,
(4.28)
where the value of core cutoff radius r , = b has again been incorporated. The blocking criterion invoked in the preceding section presumes that blocking occurs only if the thickness h, is less than the critical thickness for the strain difference E~ - E*. If the same criterion is imposed here, then Eo
If
E*
- E*
b 4nh,
= -In
2h, b
-.
(4.29)
is eliminated by means of (4.27), then E0=--
1 b 2h 1 b +--lnL b 2~ h - h, 411 h,
(4.30)
56
L . B. Freund
For a given thickness h, this relationship provides an estimate of the minimum value of E~ for which blocking occurs. The relationship still involves the parameter h,, however, which can take values between 0 and h. The estimate is thus minimized with respect to h,, which requires that h, must satisfy (4.31) If the appropriate root of (4.31) is substituted into (4.30), the resulting estimate of the minimum c0 for blocking is only slightly higher than that indicated in Fig. 28. Several general observations can also be made. The nature of the interaction depends on the directions of the Burgers vectors of the two interacting dislocations. The two cases for which numerical results are presented here are (b), the Burgers vectors of the interacting dislocations are parallel, and (d), the Burgers vectors lie along diagonally opposite edges of the pyramid of {111}-planes of the material. It is also noted that, if the threading dislocation encounters a cluster of n dislocations with identical Burgers vectors, then the interaction force is amplified by a factor of n.
V. Process Kinetics A. NONEQUILIBRIUM CONDITIONS Early observations of elastic strain relaxation during growth of epitaxial layers led to paradoxical results. An attempt to interpret the observations on the basis of the critical thickness theory in its most elementary form suggested that, once the thickness of a film exceeded the critical thickness, the final elastic strain of the film should be determined by the thickness of the film alone, independent of the original, or fully coherent, mismatch strain. This is implied by the result in (4.15), which states that the mean elastic strain predicted by the equilibrium condition RSim = 0 is completely determined by h beyond critical thickness, no matter what the value of c0. However, it was found that the postgrowth elastic strain as measured by x-ray diffraction methods did indeed vary with the initial elastic mismatch strain, and it did so in different ways for different film thicknesses, as shown in Fig. 29. As a consequence, the critical thickness theory came under question, and a variety of alternate models were proposed to replace it. However, further study of the
57
Dislocations in Strained-Layer Semiconductor Materials h=lOnrn
.
h=50nrn h=lOOnm h=250nrn
0.0
0
1
3
2
Eo
4
5
(Yo)
FIG.29. Observed final elastic strain measured by x-ray diffraction versus initial mismatch strain following growth of Ge,Si,-,/Si(100) at a temperature of 550°C (Bean et al., 1984).
problem has revealed the relaxation process to be much richer in physical phenomena than anticipated, with the critical thickness theory revealing only part of the story. In the interpretation of the observations, it had been tacitly assumed that dislocations would appear fully formed whenever it was energetically possible for them to exist, but this is not the case. Instead, the rate of dislocation nucleation in materials with low initial dislocation densities and the rate of dislocation motion in semiconductor materials emerged as important features. Roughly, the time for a threading dislocation to glide a distance equal to the lateral extent of a film at a typical growth temperature under the action of the mismatch strain is often on the same order of magnitude as the growth time itself. Consequently, the variability observed in the relaxation data could be associated with the kinetics of nucleation and glide of dislocations. This realization led to the use of postgrowth annealing experiments to study relaxation (Fiory et al., 1984). In this approach, a strainedlayer material is grown beyond critical thickness with minimal relaxation, due to low growth temperatures or other controls, and then cooled. In the resulting structures, any threading dislocation that exists is still subjected to a positive driving force but it is kinetically restrained from relaxing the elastic strain in the film. The resulting structures have been labeled as metastable, an appropriate term if thermal fluctuation is understood to be the external
58
L. B. Freund
stimulus by which stability is probed. In any case, under well-controlled conditions, the film is heated and observed, sometimes simultaneously and sometimes alternately, to study the mechanisms of strain relaxation (Hull et al., 1988; Nix et al., 1990; Houghton, 1991). The results of these experiments have been of central importance in resolving some of the paradoxes. The annealing or temperature cycling experiments are also relevant to industrial fabrication technology, where a device incorporating a strained-layer heterostructure may be subject to several temperature excursions in a manufacturing sequence. Virtually all observations of dislocation glide in bulk covalent crystals at stress levels above some modest threshold level reveal that the normal glide velocity for a given material varies with resolved shear stress on the glide plane z and absolute temperature T according to
where vo is a material constant with dimensions of speed, Qo is the (assumed constant) activation energy, and k is Boltzmann's constant (Alexander, 1986). For Si and Ge, the exponent m appears to be slightly temperature dependent, but it is usually assumed to be a constant in the range 1 6 m < 2. The velocity factor uo is about 1.36 x lo5m/s for Si and 1.15 x 106m/s for Ge. It follows immediately that the speed of self-similar advance of an isolated threading dislocation is
where,,z, is an effective, or excess, shear stress proportional to 9cos a/bh for the threading dislocation (Tsao and Dodson, 1988; Tsao et al., 1987; Freund and Hull, 1992). The excess stress is defined so that (5.2) reduces to (5.1) in the limit as the film becomes very thick. Most likely, glide is accomplished by the thermally activated motion of kinks along the threading segment (Tuppen and Gibbings, 1990; Hull et al., 1991). A free surface or bimaterial interface may offer a better site for kink formation than interior points along the dislocation line, and this is an influence on relaxation that cannot be taken into account by approaches based on continuum mechanics. The results of in situ high-voltage transmission electron microscopy measurements of threading dislocation velocities in Si,Ge, -JSi( 100) films are shown in Fig. 30, interpreted according to the relationship (5.2). To obtain these data, an isolated threading dislocation was found in the film at low temperature, and the motion of this dislocation was then monitored as the sample was heated to progressively higher temperatures up to 675°C. The
Dislocations in Strained-Layer Semiconductor Materials
59
prospect of following this approach for still higher temperatures was precluded by the fact that dislocations moved too rapidly to be tracked, so the data in this range were obtained by chance observations over a fixed area. When plotted in the form of lnu,, versus l/kT, the slope of the best-fit line provides a direct estimate of the activation energy Qo. The inferred value of Qo is about 2.2eV. Similar values have been obtained by other means. The data in Fig. 30 cover a speed range from about 5 nm/s to 150 pm/s. The connection between the motion of individual dislocations and bulk strain relaxation is tenuous at this time, but some progress has been made (Tsao et al., 1987; Dodson and Tsao, 1987). For each dislocation, suppose that the area of slip on any glide plane is SSlip, the Burgers vector is bi, and the unit normal vector to the glide plane is n,. Then the average plastic strain, or relaxed strain, in a volume V that includes surfaces of slip of N dislocations is given by the tensor (Rice, 1970) N &P. CJ = L 2 v-1
a= 1
S$!p(bp)ny)+ $)by))
(5.3)
If the relaxation is isotropic in the plane of the interface, if V corresponds to
K
.-
: 1
Q7\
v
h
r
5 -1
Y
K -
-3 - 0 x-0.226 h d 1 8 nm x=0.226 h=62nm
A
~ 1 0 . 3 4 4h=45nm
B,1A &
-5 10
11
12
13
14
15
1 /kT (eV-1) FIG. 30. Glide speed of an isolated threading dislocation in a SiGe/Si(100) film versus temperature from in situ transmission electron microscopic observations (Nix et al., 1990 with permission). The slope of the line fitted to the data provides an estimate of the activation energy according to (5.2).
60
L . B. Freund
some fixed area A of the interface times film thickness h, and if S$/p corresponds to length of interface misfit dislocation I, times thickness h, then the essential information from (5.3) is contained in the reduced scalar form N
eP = C(b/A)
C
1,
(5.4)
a= 1
where C is a geometrical factor of order 1. Because A and b are constant, the corresponding rate of strain relaxation is k p = cplhbi
(5.5)
if it is assumed that all Nth <. N threading dislocations moving across the area A have the same speed 1 at the current density and Pth = Nth/A. But i = v,h so the rate of relaxation strain is
where 9 is a driving force appropriate for glide of dislocations through the partially relaxed film. In the early stage of relaxation at a fixed temperature the strain rate is low because the dislocation density Prh is low, while in the late stage it is low because the driving force is low. Further progress along these lines requires a more detailed understanding of the evolution of the density Pth than is currently available, although the implications of several empirical evolution laws have been examined (Nix et al., 1990; Dodson and Tsao, 1987). For example, if the driving force is taken to be (3.21) with the mismatch strain e0 replaced by the residual elastic strain after some isotropic relaxation, say e0 - ep, and if the density of mobile threading dislocations is taken to be proportional to a constant plus the current level of plastic strain, then (5.6) reduces to the differential equation k" = (Uo/b)C1(&0 - e p - C~)'"(E''
+ cJ)
(5.7)
where cl, c2, c3 are dimensionless constants. The differential equation is subject to the condition that eP = 0 initially. The differential equation can be integrated in terms of elementary functions for rn = 1 or 2, but it is evident without doing so that the features of the solution are consistent with the behavior exhibited in Fig. 29. While various mechanisms for dislocation multiplication during glide have been proposed, available evidence indicates that multiplication (as opposed to nucleation) plays a relatively minor role compared to that of nucleation in the relaxation process, especially for SiGe/Si systems.
Dislocations in Strained-Layer Semiconductor Materials
61
B. DISLOCATION NUCLEATION The relaxation mechanisms just discussed rely on a supply of threading dislocations to operate, especially in the very low-density materials now available, but the search for mechanisms by which these dislocations are nucleated has been inconclusive. A theory of long-standing interest is based on homogeneous nucleation of glide dislocation half-loops at the free surface of the film, say loops on { 111)-planes in a cubic system with a (100) interface (Fitzgerald et al., 1989; Kamat and Hirth, 1990). According to this theory, the loop nucleus is formed by thermal activity of the lattice at a size sufficiently large that the elastic energy drawn from the strained layer during subsequent expansion exceeds the distortional energy of the lattice associated with the dislocation itself. An assessment of this mechanism within the framework of elastic dislocation theory suggests that the elastic mismatch strain must be at least several percent for it to be operative, which is a much larger strain than actually exists in virtually all films relaxed through dislocation glide. This severe restriction can be relaxed somewhat by assuming that the loop nucleates from a surface ledge or from a macroscopic stress concentrator such as a topographical feature. Various sources of heterogeneous nucleation have also been identified or proposed by various investigators, including (i) foreign matter on the growth surface of the substrate in the form of chemical contamination or handling debris (Perovic et al., 1989),(ii) crystallographic ledges or steps on the growth surface of the substrate, (iii) stacking fault loops within the film resulting as natural growth defects or from the coalescence of vacancies (Eaglesham et al., 1989), and (iv) compositional gradients in alloying elements (Gravesteijn et a/., 1989).To circumvent some of the uncertainties associated with the issue of nucleation, several investigators have conducted annealing experiments in which large numbers of dislocations were created artificially by scratching or indenting the surface of the film prior to heating (Nix et al., 1990; Tuppen and Gibbings, 1990).
C . GLIDETHROUGH NONUNIFORM STRESS FIELDS In the models developed in this chapter, the detailed shape of a gliding dislocation had little influence on the conclusions drawn. Either glide occurred under steady state conditions for uniform stress field or inferences were drawn for nonuniform fields that were insensitive to the shape. In some cases, however, it may be useful or necessary to actually establish the detailed
62
L . B. Freund
shape of a dislocation line. The purpose of this section is to propose a means of doing so that appears to be promising on the basis of a test calculation. The approach is based on the capability to express the energy of a piecewise linear dislocation line in terms of the positions in space of the vertices of the line. For example, consider a dislocation line in the unbounded plane shown in Fig. 31. The piecewise linear configuration may be viewed as an approximation to a smooth line. The shape is completely specified by the coordinates in the plane of the vertices. In the simplest prescription, the nodes are restricted from displacing in the y direction, and their positions are given simply in terms of their x coordinates; say, tk,k = 1,. . . ,n. The segments at the extremes of the line are assumed to extend to infinity. The interaction energy for two straight dislocation segments is known, so it is possible to express the total energy of the line in terms of these generalized coordinates ti;say, E(t,,. . 5.). The variation of energy of the system with respect to variations in the shape of the line can be viewed as the force tending to drive that change of shape. As is customary, the force is defined as the negative gradient of the total energy with respect to the net interaction energy in the space of . ?
TY
\
FIG.31. An approximation of a dislocation line as a piecewise linear configuration,allowing representation of the configuration in terms of n generalized coordinates &, k = 1,. ..,n. Fkis the generalized force work conjugate to lk.
Dislocations in Strained-Layer Semiconductor Materials
63
generalized coordinates of the system; that is,
This force iS work conjugate to ( k , and it acts in the direction of increase of ( k . A necessary condition for equilibrium of the system is that F k = 0, k = 1,. . . ,n. However, this concept of generalized force is perhaps more useful in considering the kinetics of nonuniform processes. In view of the kinetic equation (5.2) describing glide of a dislocation in Si or Ge, it is reasonable to expect that the rate of change of a coordinate will depend on the corresponding generalized force according to
where the coefficient u,(k) is unknown a priori. These factors, which vary from vertex to vertex, are determined by requiring that the total energies dissipated through dislocation glide according to the discrete and continuous descriptions are identical; that is, n
JL
aijnfbim,v,ds = C
*ik
(5.10)
k
But this condition is simply a restatement of the contents of the general expressions (3.11). To see how the coefficients are evaluated, first note that the complete configuration of the dislocation line is specified by the values of t k ( t ) .Thus, v, appearing on the left side of (5.10) is linear in the quantities t k . From the kinetic equation, the resolved shear stress aijn,: bi can be expressed in terms of u,m, at each point along L. Likewise, the force F k appearing on the right side can be expressed in terms of the rates &. Then, k equations for the parameters vo(k)are obtained by requiring that the condition (5.10) must hold for every choice of coordinate rates 4,. This procedure has not yet been fully developed. It has been applied to the case of glide of a threading dislocation in a buried layer, and the results on critical thickness and steady state speed of glide are virtually identical with the corresponding steady state results obtained directly (Ramirez, Bower, and Freund, 1990). The method seems to hold promise for dealing with multiple dislocation configurations, including cases when the dislocations glide on different glide planes, either parallel or intersecting.
64
L . B. Freund
Acknowledgments Research in the area of mechanics of electronic materials has been supported at Brown University by the Office of Naval Research through contract NOOO14-90-5-04051 and by an IBM Research Award in Materials Science. This support is gratefully acknowledged.
References Alexander, H. (1986). Dislocations in covalent crystals. In: Dislocations in solids, 7 (F. R. N. Nabarro, ed.). North-Holland, New York, pp. 115-234. Barnett, D. M., and Lothe, J. (1974). An image force theorem for dislocations in anisotropic bicrystals. J. Phys. F: Metal Phys. 4, 1618-1635. Bean, J. C. (1985). Strained-layer epitaxy of germanium-silicon alloys. Science 230, 127- 131. Bean, J. C., Feldman, L. C., Fiory, A. T., Nakahara, S. and Robinson, I. K. (1984), Ge,Si,-,/Si strained layer superlattice grown y molecular beam epitaxy. J. Vac. Sci. Tech. A2,436-440. Budiansky, B., and Rice, J. R. (1973). Conservation laws nd energy release rates. J . Appl. Mech. 40,201-203. Cammarata, R. C., and Sieradzki, K. (1989). Surface stress effects on the critical thickness for epitaxy. Appl. Phys. Lett. 55, 1197- 1198. Codson, B. W., and Tsao, J. Y. (1987). Relaxation of strained-layer semiconductor structures via plastic flow. Appl. Phys. Lett. 51, 1325-1327. Dundurs, J. (1969). Elastic interaction of dislocations with inhomogeneities. in: Mathematical theory of dislocutions. A S M E, New York, pp. 70- 1 15. Dundurs, J. and Sendeckyj, G. P. (1965). Behavior of an edge dislocation near a bimetallic interface. J. Appl. Phys. 36, 3353-3354. Eaglesham, D. J., Kvam, E. P., Maher, D. M. Humphreys, C. J., and Bean, J. C. (1989). Dislocation nucleation near the critical hickness in GeSi/Si strained layers. Phil. Mag. A59, 1059-1073. Eshelby, J . D. (1951). The force on an elastic singularity. 7kans. Roy. SOC.(London)A244,87-112. Fiory, A. T. Bean, J. C. Hull, R.,and Nakahara, S. (1984). Thermal relaxation of metastable strained layer Ge,Si, -JSi epitaxy. Phys. Rev. B31,4063-4065. Fitzgerald, E. A., Watson, G. P., Proano, R. E., Ast, D. B. Kirchner, P. D., Pettit, G. D., and Woodall, J. M. (1989). Nucleation mechanisms and the elimination of misfit dislocations at mismatched interfaces by reduction in growth area. J. Appl. Phys. 65, 2220-2237. Frank, F. C., and van der Merwe, J. H. (1949a). One-dimensional dislocations: I. Static theory. Proc. Roy. SOC.(London)A198, 205-216. Frank, F. C., and van der Merwe, J. H. (1949b). One-dimensional dislocations: 11. Misfitting monolayers and oriented growth. Proc. Roy. SOC.(London)A198, 216-225. Freund, L. B. (1978). Stress fntensity factor calculations based on a conservation integral. Int. J. Solids. Struc. 14, 241-250. Freund, L. B. (1987). The stability of a dislocation threading a strained layer on a substrate. J. Appl. Mech. 54, 553-557. Freund, L. B. (1990a). The driving force for glide of a threading dislocation in a strained epitaxial layer on a substrate. J . Mech. Phys. Solids 38, 651-679.
Dislocations in Strained-Layer Semiconductor Materials
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Freund, L. B. (1990b). A criterion for arrest of a threading dislocation in a a strained epitaxial layer due to an interface misfit dislocation in its path. J. Appl. Phys. 68, 2073-2080. Freund, L. B., and Hull, R. (1992). On the Dodson-Tsao excess stress for glide of a threading dislocation in strained epitaxial layer. J . Appl. Phys. 71, 2054-2056. Fritz, I. J. (1987).Role of experimental resolution in measurements of critical layer thickness for strained-layer epitaxy. Appl. Phys. Lett. 51, 1080- 1082. Gravesteijn, D. J., Zalm, P. C., van de Walk, G. F. A., Vriezema, C. J., van Gorkum, A. A., and van Ijzendoorn, L. J. (1989). Ge segregation during molecular beam epitaxial growth of Si, -,Ge,/Si layers. Thin Solid Films 183, 191-196. Gourley, P. L., Fritz, I. J., and Dawson, L. R. (1988).Controversy of critical layer thickness for InGaAslGaAs strained layer epitaxy. Appl. Phys. Lett. 52, 377-379. Head, A. K. (1953a). The interaction of dislocations and boundaries. Phil. Mag. 44, 92-94. Head, A. K. (1953b). Edge dislocations in inhomogeneous media. Proc. Roy. Soc. (London) B66, 793-801. Hirth, J. P., and Feng, X. (1990). Critical layer thickness for misfit dislocation stability in multilayer structures. J. Appl. Phys. 67, 3343-3349. Hirth, J. P., and Lothe, J. (1982). Theory of dislocations, Wiley, New York. Houghton, D. C. (1991).Strain relaxation kinetics in Si, -,Ge,/Si heterostructures. J. Appl. Phys. 70, 2136-2151. Houghton, D. C., Gibbings, C. J., Tuppen, C. G., Lyons, M. H., and Halliwell, M. A. G. (1989). The structural stability of uncapped versus buried Si, -xGe, strained layers through high temperature processing. Thin Solid Film 183, 171- 182. Houghton, D. C., Perovic, D. D. Baribeau, J.-M., and Weatherly, G. C. (1990). Mistit strain relaxation in Ge,Si, -JSi heterostructures: The structural stability of buried strained layers and strained-layer superlattices. J . Appl. Phys. 67, 1850- 1862. Hull, R., and Bean, J. C. (1989).Thermal stability of Si/Ge,Si -JSi heterostructures. Appl. Phys. I Lett. 55, 1900- 1902. Hull, R., Bean, J. C., Werder, D. J., and Leibenguth, R. E. (1988). In situ observations of misfit dislocation propagation in Ge,Si, -,/Si(lOO) heterostructures. Appl. Phys. Lett. 52, 16051607. Hull, R., Bean, J. C. Eaglesham, D. J. Bonar, J. M., and Buescher, C. (1989). Strain relaxation phenomena in Ge,Si, - ,/Si strained structures. Thin Solid Films 183, 117- 132. Hull, R., Bean, J. C., Bahnck, D., Peticolas, L. J. Short, K. T., and Unterwald, F. C. (1991). Interpretation of dislocation propagation velocities in strained Ge,Si, _,/Si(lOO) heterostructures by the diffiisive kink pair model. J. Appl. Phys. 70, 2052-206s. Kamat, S. V., and Hirth, J. P., (1990).Dislocation injection in strained multilayer structures. J . Appl. Phys. 67, 6844-6850. LeGoues, F. K., Meyerson, B. S., and Morar, J. F. (1991).Anomalous strain relaxation in SiGe thin films and superlattices. Phys. Rev. Lett. 66, 2903-2906. Lee, M. S., and Dundurs, J. (1973).Edge dislocation in a surface layer. Int. J. Engrg. Sci. 11,8794. Matthews, J. W., and Blakeslee, A. E. (1975).Defects in epitaxial multilayers 11. Dislocation pileups, threading dislocations, slip lines and cracks. J. Crys. Growth 29, 273-280. Matthews, J. W. (1975).Defects associated with the accommodation of misfit between crystals. J. Vac. Sci. Tech. 12, 126- 133. Matthews, J. W., and Blakeslee, A. E. (1974). Defects in epitaxial multilayers I. Misfit dislocations. J . Crys. Growth 27, 118- 125. Matthews, J. W.. and Blakeslee, A. E. (1975).Defects in epitaxial multilayers 11. Dislocation pileups, threading dislocations, slip lines and cracks. J. Crys. Growth 29, 273-280. Matthews. J. W., and Blakeslee, A. E. (1976). Defects in epitaxial multilayers 111. Preparation of almost perfect multilayers. J. Crys. Growth 32, 265-273.
L. B. Freund
66
Matthews, J. W., Mader, S., and Light, T. B. (1970).Accommodation of misfit across the interface between crystals of semiconducting elements or compounds. J . Appl. Phys. 41,3800-3804. Muskhelishvili, N. I. (1953).Some basic problems in the mathematical theory of elasticity. Noordhoff, Groningen, Germany. Nix, W. D. (1989).Mechanical properties of thin films. Met. 'Runs. 20A, 2217-2245. Nix, W. D. Noble, D. B., and Turlo, J. F. (1990).Mechanisms and kinetics of misfit dislocation formation in heteroepitaxial thin films. Materials Research Society Symposium Proceedings 188, 315-330.
Paine, D. C., Howard, D. J., and Stoffel, N. G. (1990).Formation of strained Si,-,Ge, by solid phase epitaxy. J . Electr. Mat. 20, 735-746. Perovic, D. D., Weatherly, G. C., Baribeau, J. M., and Houghton, D. C. (1989).Heterogeneous nucleation sources in molecular beam epitaxy grown GexSi, -JSi strained layer superlattices. Thin Solid Films 183, 141-156. Petruuello, J., Greenberg, 8. L., Carnrnack, D. A,, and Dalby, R.(1988).Structural propertiesof the ZnSelGaAs system grown by molecular beam epitaxy. J. Appl. Phys. 63, 2299-2303. Ramirez, J. C., Bower, A. F., and Freund, L. B. (1990).Simulation of transient glide of a threading dislocation in an embedded strained epitaxial layer. Brown University Report. Rice, J. R. (1970). On the structure of stress-strain relations for time dependent plastic deformation in metals. J . Appl. Mech. 37, 728-737. Rice, J. R. (1985). Conserved integrals and energetic driving forces. In: Fundamentals of Deformation and fracture (8. A. Bilby, K. J. Miller, and J . R. Willis, eds.). Cambridge University Press, Cambridge, pp. 33-56. Third International Symposium on Silicon Molecular Beam Epitaxy. (1989).Strasbourg, France, Thin Solid Films 183. Thouless, M. D. (1990).Crack spacing in brittle films on elastic substrates. J. Amer. Ceramic Soc. 73, 2144-2146. Tsao, J. Y., and Dodson, B. W. (1988).Excess stress and the stability of strained heterostructures. Appl. Phys. Lett. 53, 848-850. Tsao, J. Y.Dodson, B. W. Picraux, S. T., and Cornelison, D. M. (1987).Critical stresses for Si,Ge, - x strained layer plasticity. Physical Review Letters 59, 2455-2458. Tuppen, C. G., and Gibbings, C. J. (1990).A quantitative analysis of strain relaxation by misfit dislocation glide in Si, -,Ge,/Si heterostructurs. J. Appl. Phys. 68, 1526-1534. Tuppen, C. G., Gibbings, C. J., and Hockly, M. (1989).The effects of misfit dislocation nucleation and propagation on Si/Si, -,Ge, critical thickness values. J. Crys. Growth 94,392-404. van der Merwe, J. H. (1991).Strain relaxation in epitaxial overlayers. J . Electr. Mat. 20,793-803. van der Menve, J. H., and van der Berg, N. G. (1972).Misfit dislocation energy in epitaxial overgrowths of finite thickness. SurJ Sci. 32, 1-15. Willis, J. R., Jain, S. C. and Bullough, R. (1990).The energy of an array of dislocations: Implications for strain relaxation in semiconductor heterostructures. Phil. Mag. A62, 11 5-
129. Willis, J. R., Jain, S. C., and Bullough, R. (1991).Work hardening and strain relaxation in strained-layer buffers. Appl. Phys. Lett. 59,920-922.
ADVANCES IN APPLIED MECHANICS, VOLUME 30
Propagating Instabilities in Structures STELIOS KYRIAKIDES Engineering Mechanics Research Laboratory Department of Aerospace Engineering and Engineering Mechanics The University of Texas ai Austin Austin, Texas
.... . . . ... ..... . .... ... .. . . .. . . . . . . ... . . .. . . . . . . .. . ... . ..... . .... . ... ...... ..
68
11. Initiation and Propagation of Bulges in Inflated Elastic Tubes. .. . .. . ... . .... ..... . . A. The Problem ...... ................................................................... B. Experiments: Initiation and Propagation of a Bulge ...... C. Analyses.. . . . ........................................................ D. Discussion and Conclusions .... . . .. ... . . .. . . .. . . . . . . . .. . .. . .. . ... .. . . .. . ... . . .... . .
70 70 72 78 92
1. Introduction
111. Initiation and Propagation of Buckles in Long Tubes and Pipes under.. . .. . .. . .
External Pressure.. ..... A. The Problem .................. .......... ..... B. Propagating Buckle Experiments.. . C. Analyses.. . . .. . .. . ... .. . . .. ... . . ... . .. . ... . . .. . . . , . . . . . .. . . . .. . ... ...... . .. . . ... . . .. . . D. Effect of Tension on the Buckle Propagation Pressure ..... .................... . . . . . .. . . E. Discussion and Conclusions.. . . . . . . .. . . . . .. . .. . .. . ... .. . .. . .
94 94 99 108 134 145
IV. Propagating Buckles in Long, Confined Cylindrical Shells., . . . . . . . . . . . . . . . . . . .. . . A. The Problem .................. ............ B. Confined Propagating Buckle Experime C. Analyses .. . . . . . .. .... ... . .. . .. . . .. . .. . . .... . .. . ... . . . . . .. . . . .. . . .. . .. . .. . .. ..... . .. . .. D. Discussion and Conclusions . . . . . .. . .. . . ... . . .. . .. . . . . . . . .. . .. ....... ... . . .. . . ... . ..
147 147 150 156 168
V. Buckle Propagation in Long, Shallow Panels.. . . . . . . . . ... . . .. . ... .. . .. . .. . . . ... . .. . . . A. Buckling of Shallow Arches.. . . .. . . .. . . . . . . . . . . .. . . .. . .. . . .. ....... . .. . .. . . .... . ... . B. Propagating Buckles in Long, Shallow Panels ... . . . . . .. . .. . ... . .. . . .. . . .. . . . .. . . . C. Discussion and Conclusions . . . .. . . ... . . .. . .. . . . .. . .. . .... . ... ... .. . .. . .. .. . . .. . ... .
169 169 172 180
VI. Summary and Discussion.. . .. . . . . . . , . . ... . . .. . . . . . ... . .. . ... . ... ... . .. . ... . . . . . . ... . . ..
181
Acknowledgments .... . ..... . .. . .. . . ... . . .. .... . . . . ... . ... . .. . . .. . . . . . .. . .. . . .... ... . ... .
185
References.... . . . . . ... . . ... . .. . .. . . .. . ... . ... . ... . .. . . . . . .. . . . . . ... . . . . . .. . ... . . .. . ... ... .
185
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67 Copyright ig1994 by Academic Prcss, Inc. All rights or rcproduction in any Corm rcscrvcd. ISBN 0-1?-002030-0
68
Stelios Kyriakides
I. Introduction Instability is one of the factors that limit the extent to which structures can be loaded or deformed and plays a pivotal role in the design of most structures loaded primarily in compression, but often also structures loaded in tension. The vast literature on the subject, developed over the period of 250 years since Euler’s original work, attests to the importance of the subject. The major issue of practical concern is the establishment of the critical buckling load and deformation; that is, the lowest load or smallest deformation at which alternate equilibria become possible. This is usually established through linearized bifurcation analyses. In practice, the critical condition can be affected by imperfections, material nonlinearities and other factors. Incorporating these usually requires solution of the more complete nonlinear problems. This classical approach, that is, concern about the onset of instability, is sufficient for the design of most structures. In the past 15 years, a class of structural instabilities, known as propagating instabilities, have been identified in which the classical approach is not sufficient. In general, this instability affects structures of larger size in which, following the onset of instability, collapse is confined to a relatively small part of the structure; that is, it localizes. However, under prevailing conditions, the initially local collapse can propagate or spread, often in a dynamic fashion, over the rest of the structure. The load required to propagate such instabilities is often substantially lower than that required to initiate them in the original structure. In practice, propagating instabilities are initiated from local imperfections that locally weaken the structure and cause local change in geometry (collapse). Very much like a set of dominoes (see Bert, 1986; Stronge and Shu, 1988), once the geometric integrity of such structures is compromised, the instability has the potential of spreading over the whole structure. The aim of this chapter is to demonstrate the breadth of relevance of the phenomenon by pulling together results from studies of four structural examples that exhibit this behavior. Although the practical relevance of the examples varies from insignificant to very significant, each demonstrates different aspects of this behavior. The problem of finite inflation of an elastic tube, discussed in Section 11, demonstrates the complexity of a localization-type instability, which in this case takes the form of a bulge and the subsequent propagation or spreading of the bulge. The problem, once understood, can be solved quite adequately with relative ease using
Propagating Instabilities in Structures
69
finite-deformationmembrane theory. Thus, it provides a model problem for introducing some of the features of this type of behavior. The author’s initial experience with propagating instabilities was developed through work on the problem of propagating buckles in offshore pipelines used for transporting oil and gas. Under some circumstances,a local disturbance can initiate a collapse that can propagate dynamically along the length of the line and totally destroy it. The understanding of many aspects of this problem over the last 15 years has reached a certain degree of maturity. Section I11 summarizes much of the progress achieved in this problem to date. Section IV addresses localized collapse and the propagation of collapse in shells lining stiffer cylinders. In Section V, we revisit the classical problem of buckling of arches and demonstrate how a long panel with an arch cross section, loaded by pressure, develops a localized buckle that then can propagate at a relatively low pressure and collapse the whole panel. All the problems discussed can develop instabilities that propagate dynamically. In this first expose of this class of nonlinear behavior, discussion is limited to the conditions of initiation and quasi-static propagation of the instabilities. The dynamics of this unstable behavior will be left for future discussion. The same can be said for the practical problem of the arrest of these instabilities. We hope that once attention is given to the potential for catastrophic spread of collapse the design of practical means of limiting the extent of collapse, where this is appropriate, will follow. It will be shown that the underlying, common characteristic of structures exhibiting this class of instabilities is a “local,” nonlinear, “load-deformation” response with two branches with positive slopejoined by an intermediate one with negative slope, as shown in Fig. 1. Thus, for a range of loads, such
Deformation
FIG.1. Local load-deformationresponse characteristic of this class of problems.
70
Stelios Kyriakides
structures have three possible equilibria for each value of load. The one on the first ascending branch will represent prebuckling deformations. The one on the second ascending branch will represent buckled configurations. Under some conditions, the two states can coexist in the same structure. This underlying fundamental characteristic of the force-displacement response is similar to some of the features of phase transitions in liquidvapor systems (see $43-46 in Guggenheim, 1959) and to those of phase transitions in solids (see Ericksen, 1975). The connection, although of great interest, will be left for future work. The objectives of this chapter are best achieved through a thorough description of the phenomena with the help of results from experimental and analytical-numerical simulations. Although the novelties of both of these types of investigatory tools will be described, the reader will often be referred to the literature for details. 11. Initiation and Propagation of Bulges in Inflated Elastic Tubes
A. THEPROBLEM The inflation of a thin-walled tube made from a rubberlike material is a problem that vividly illustrates the initiation and subsequent propagation of propagating instabilities. Furthermore, the phenomena can be demonstrated experimentally and analytically with relative ease. As a result, this example will be used to identify the phenomena that characterize this class of problems, describe the experimental and analytical challenges that must be met to study these phenomena, and establish the terminology used throughout this chapter. With the aid of Fig. 2, let us consider the sequence of events that takes place when inflating a section of a tube made from a rubber compound that can undergo finite elastic deformations. During inflation, the tube first expands uniformly, growing both in the radial and axial directions (ungoforminjation). This increase in deformation occurs with increasing pressure. However, the extent to which the tube can grow uniformly is limited. At time tl in the figure, the pressure reaches a local maximum and the cylindrical configuration becomes unstable. With dropping pressure, a bulge a few tube diameters long appears somewhere along the length of the tube. What happens next depends very much on the method used to inflate the tube. Let us first consider a pressurization scheme in which the volume of fluid in the
Propagating Instabilities in Structures
71
-z 2 u)
'u,
n
FIG 2. Schematic representation of volume-controlled inflation of an elastic tube.
tube is prescribed (volume-controlled pressurization). Let us also assume that we choose to increase the internal volume at a slow, steady rate. Under these conditions, following an initial transient, the bulge grows both radially and axially in a controlled fashion while the pressure in the tube continues to decrease. When the diameter of the bulge reaches a critical value ( t z in figure) the radial growth stops and the bulge starts to spread axially. By selecting the rate at which fluid is supplied into the tube, the axial growth can occur under steady-state conditions in a quasi-static fashion. The pressure required to maintain quasi-static propagation of the bulge is known as the propagation pressure (P,). This pressure is usually much lower than that required to initiate the bulge (initiation pressure, P,). A characteristic of the propagation phase of this process is that inflated and uninflated configurations of the structure coexist. The length of the inflated section grows while the pressure is maintained at P,. This continues until the whole length of tube is inflated ( t 3 in figure). Inflation beyond this point results again in uniform, cylindrical expansion of the tube, which occurs in a stable (increasing pressure) fashion. The response of the structure between t , and t 3 is unstable under prescribed pressure. Thus, if the tube is connected to a constant pressure source, the axial spread of the bulge will occur dynamically. Under the conditions described, the rate of expansion of the bulge will be strongly influenced by the internal flow of the fluid supplied.
72
Stelios Kyriakides
In summary, we have seen that the problem has three aspects. First is the issue of initiation of the bulge in a tube that was previously deforming uniformly. Second is the propagation of the bulge and in particular the lowest pressure at which the bulge will grow. Third is the issue of dynamic growth of the bulge if inflated under prescribed pressure conditions. In this chapter we will be concerned primarily with the first two aspects of the problems.
B. EXPERIMENTS: INITIATION
AND PROPAGATION OF A
BULGE
The characteristics of the problem just described were illustrated in a series of experiments on commercially available natural latex rubber tubes with nominal outside diameter of 0.50 in. (12.7 mm) and wall thickness of 0.063 in. (1.60mm). Two sets of experiments were conducted. The first set (see Kyriakides and Chang, 1991) involved tubes inflated under volumecontrolled conditions with the objective of capturing the details of the initiation process. In the second set of experiments (see Kyriakides and Chang, 1990), the main objective was the measurement of the propagation pressure of the bulge under steady-state quasi-static conditions. Therefore, the tubes were inflated with compressed air. 1. Volume-Controlled Inflation Experiments
In this set of experiments, the lengths of the tubes tested ranged between 5 to 36 tube diameters. The experimental set-up used is shown schematically in Fig. 3. One end of the tube was clamped to a rigid manifold, while the other was sealed but left free. Axial load was applied by connecting the free end to a
Containment Vessel Pressure Transducer I
.
-
I
Manifold,
Weight
FIG.3. Experimental apparatus for inflating tubes under volume control.
Propagating Instabilities in Structures
13
weight, as shown in the figure. The tube and all connected to it were filled with water. The effect of the weight of the water inside the tube was negated by immersing the tube in a bath of water. The walls of the bath were transparent to allow observation of the experiment. A positive-displacement precision metering pump was used to inflate the tube. The pump delivered water at the rate of 0.062 uo/sec., where uo is the initial internal volume of each test specimen. This was done to maintain some parity in loading rate (at least up to the onset of instability) between the different experiments. A one-way valve was used at the inlet of the manifold to reduce the possibility of backflow. This water supply arrangement is a good approximation of volume-controlled loading conditions. A high-resolution video system was used to monitor and record the deformed configurations of the tube during inflation. Events were recorded at the rate of 30 frames/sec. The pressure in the tube was monitored with an electrical pressure transducer. The transducer output was recorded on a common time base with the video recording. A video analyzer was used to obtain specific dimensional measurements from the recorded configuration history. These were matched with the corresponding pressure measurements. a. Experimental Results
A typical set of results from this set of experiments is shown in Fig. 4. A tube with length of L/R = 20.6 (length of tube = 2L, diameter = 2R) was inflated in the presence of an axial load of F / p A = 0.383 (A = initial crosssectional area of tube, ,u = initial shear modulus). The initial part of the recorded pressure-time history and a sequence of tube configurations obtained from the video recording are shown in the figure. The pressure corresponding to each tube configuration is marked on the pressure-time recording. The following sequence of events was observed during the experiment. At the early stages of the inflation process, the tube deformed essentially uniformly (except at the ends), growing both diametrically and axially. The initial stiffness of the structure was high and the pressure rose sharply with time. As the tube deformed, its stiffness decreased and eventually the pressure reached a maximum value. Very close to the pressure maximum, a bulge three to four diameters long developed somewhere along the length of the tube. With the appearance of the bulge, the pressure in the tube started to drop. The initial pressure drop occurred in a jump fashion (indicated by a dashed line in Fig. 4), but eventually the deformation process returned to a controlled quasi-static rate of growth.
Stelios Kyriakides
74
1
024P P -
-8
1 0 16-
0
Latex Rubber 0 08<=O 0 0
I
2
383
3
r
4
5
1 , 1
6
7
-Time(min)
FIG.4. Recorded pressure-time history and sequence of corresponding tube configurations.
The pressure jump can be explained as follows. The downturn in pressure caused unloading in most of the tube, except in the bulged section, which continued to experience an increase in deformation. As a result of the unloading, the volume of the cylindrical sections was reduced. By contrast, the volume in the bulging section increased. Since the instantaneous volume of fluid inside the test specimen was prescribed, for a shorter tube the volume increase in the bulge could be larger than the volume decrease in the
Propagating Instabilities in Structures
75
cylindrical ends, and the postlimit load response could progress in a controlled fashion; i.e., without “jumps” in deformation. (For our parameters this was possible only for tubes with an initial length of a few diameters.) If the tube were longer, so that the opposite would be true, the initial unloading would occur dynamically as the structure “snaps” to a configuration that has the same internal volume. The longer is the initial length of the tube, the greater the jump. This issue will be further discussed with the help of the numerical results presented later. Continued pumping of water into the tube caused growth of the bulge, as shown in Fig. 4. At the same time, the pressure dropped gradually and asymptotically approached a steady-state value. The bulge eventually stopped growing diametrically and started growing axially. During the axial spreading of the bulge, the shape of the transition between the inflated and “uninflated sections was maintained and the pressure remained constant. This constant value of pressure is the bulge propagation pressure. Configuration 10 in Fig. 4 is at this pressure. Once this steady-state condition was reached, the experiment was accelerated by increasing the pumping rate by a factor of 3. The bulge gradually spread over the whole length of the tube. A series of similar experiments was conducted in which the length of the test specimen and the applied axial load were varied. A summary of the results can be found in Kyriakides and Chang (1991). Varying the test specimen length was found to affect the transient part of the initiation process but not the rest of the response. Shorter specimens experienced a smaller jump in pressure, and longer ones, a larger jump. In the shortest specimen tested ( L / R = 6.86), the response had no discontinuity, indicating that the volume increased monotonically during the loading process. Increasing the axial load was found to reduce the initiation pressure ( P , ) as well as the propagation pressure recorded. In addition, the shape of the bulges formed was found to be significantly altered by axial load. This difference is illustrated in Fig. 5, where bulge profiles, with the same crown point radial stretch ratios (A2= = 2.36) from tubes inflated at axial loads of F / p A = 0.383,0.766, 1.149 and 1.531, are compared. The four test specimens had initial lengths of L/R = 20.6. 2. Determination of Propagation Pressure of Bulges In this set of experiments, the speciment lengths ranged between 4 and 6 ft (1.2- 1.8m). One end of the tube was fixed to an outlet of a rigid manifold, and the tube was allowed to hang freely. The lower end of the tube was sealed and
76
Stelios Kyriakides
FIG.5. Initial bulges from tubes inflated at diteerent values of axial tension to the same maximum circumferential deformation.
a calibrated weight was suspended from it as shown in Fig. 6. The tube was inflated with compressed air supplied at constant pressure using a pressure regulator. The rate of inflation was controlled manually through a needle valve downstream of the pressure regulator. The pressure in the system was monitored by a pressure transducer and a pressure gage. During the experiment, the output of the pressure transducer was recorded on an analog strip-chart recorder. Changes in length were recorded by measuring the distance between markings placed along the length of the undeformed tube. In this case the initiation of the bulge occurred dynamically. The size of the initial bulge formed depended on the total volume of air available in the system (i.e., on the length of the tube and the internal volume of the manifold and accessories). For some of the longer test specimens used, the dynamic initiation process resulted in a bulge that was at the propagation stage of deformation; i.e., a bulge whose diameter is four to five times its length. Once the bulge was initiated, air was supplied at a rate that resulted in axial expansion of the bulge by approximately one-half a bulge diameter per minute. The inflation was continued until the bulge grew to a length of 12 to 15 diameters. Figure 7 shows a sequence of pictures from the initiation and steady-state propagation of a bulge in one of these experiments.
FIG.6. Experimental set-up used for measuring the propagation pressure of bulges in elastic tubes under prescribed values of axial tension.
FIG.7. Steady-state inflation of an elastic tube.
77
78
Stelios Kyriakides
A series of such experiments was conducted in which the propagation pressure was measured for different values of axial tension. The propagation pressures recorded are plotted as a function of the applied axial force in Fig. 8. The propagation pressure is seen to decrease with tension. Profiles of propagating bulges from experiments at three different values of tension are compared in Fig. 9. The tension is seen to cause elongation of the profiles and reduction of the diameters of both the inflated and uninflated parts.
a. Material Properties Independent experiments were conducted to measure the material properties of the latex rubber tubes used in the inflation experiments. Details of the experimental procedure followed can be found in Kyriakides and Chang (1990) and Chang (1990). The results were used to obtain material constants for the strain energy density function suggested by Ogden (1972) and given by
where li are the principal stretches and a, and pn are material constants. The material was assumed to be incompressible, and the following values of a, and p,,were found to best fit the measured material responses: p 1 = 89.4 psi (617 kPa),
a, = 1.30,
pz = 0.27 psi (1.86 kPa), p3 = - 1.42 psi (- 9.79 kPa),
at = 5.08, a3 = - 2.00.
(2.2)
The initial shear modulus of the material, used for normalizing the variables with dimensions of stress in the figures, is given by 1
p =-
3
2 n=l
p,a, = 60.4psi (416 kPa).
(2.3)
C. ANALYSES 1. Uniform Inflation Significant insight into the mechanics of this problem can be gained, with relatively small effort, from an analysis of the finite inflation of a section of tube restricted to deform cylindrically. Various aspects of this classical problem have been studied by Corneliussen and Shield (1961), Shield (1971, 1972), Alexander (1971), Haughton and Ogden (1979) and others. For our purposes, it is sufficient to idealize the tube as an elastic cylindrical membrane
79
Propagating Instabilities in Structures 0.25
-Predictions: Maxwell Construction
PP
a Experiments
0.05
1 Latex Rubber
0
I
0
0.5
I
10
r
I
1.5
2.0
25
3
F/pA
FIG.8. Propagation pressures of tubes as a function of applied tension.
with internal pressure P and axial tensile force F. The material is assumed to be incompressible and represented by a strain energy density function
W n , , 1,) = W@l, b,(n,n,)- 9.
(2-4)
Furthermore, we adopt Ogden’s representation of W from (2.1) with the
FIG.9. Profiles of propagating bulges at different values of axial tension.
Stelios Kyriakides
80
measured values of the constants given in (2.2). We let (R, r) and ( S , s) be the undeformed and deformed radii and axial measures of length, respectively. The principal stretches can then be written as follows: A1 =
ds
-, dS
A2
=
r
and A3 = (A1A2)-l
We denote the axial and circumferential stress resultants by TI and T2. Equilibrium of forces yields A2RP TI=-+2
F and T, = A,RP. 211RA2
Finally, the stress resultants can be related to the principal stretches as follows (see Green and Adkins, 1960): TI
H -
HA
T2 = - W 2
and
(2.7)
A1
12
where
and H is the undeformed wall thickness of the tube. The preceding equations can be solved numerically. A set of results
P
Latex Rubber Uniform Inflation
O! 0
I
50
I
100
I
I50
I
I
200
250
300
-wuo FIG. 10. Pressure-change in volume responses for uniform inflation of a tube at different values of axial tension.
Propagating Instabilities in Structures
81
calculated for the parameters of the tube used in the experiments is shown in Fig. 10. The applied pressure ( P / p ) is plotted against the change in internal volume of the tube (Av/uo)for four values of axial force F / p A The responses are initially relatively stiff reaching a limit pressure ( P L )with a relatively small change in volume. Following the limit value, the pressure drops to a local minimum (P,,,), which occurs at a change of volume of 50 to 100 times. Further increase in deformation requires an increase in pressure. The limit load instability can be explained as follows. The initial cylindrical deformation causes a reduction in the wall thickness. Concurrently, as the stresses increase, the material modulus decreases. The combined effect of the geometric and material reductions in stiffness limits the maximum pressure that can be initially sustained by the tube (just as the area reduction and the relatively small modulus of many structural materials in the plastic range limit the maximum load that can be applied to a bar under uniaxial tension). In a cylindrically deforming tube, the deformation and stresses continue to grow monotonically after the pressure maximum, but with a decreasing pressure. At higher values of deformation, the material response stiffens again. Further increase in stress and deformation again requires higher pressure, resulting in the upturn observed in the pressure-volume response. The results in Fig. 10 demonstrate that, when axial tension is applied simultaneously, the pressure required for inflation is lower but the other features of the P - Av response remain the same. In view of the characteristics of these responses the following comments can be made: 1. The presence of a load maximum in the response indicates that for long
tubes noncylindrical, localized modes of deformation will be energetically preferable after the limit load. Indeed, a linearized bifurcation check using axisymetric perturbations (Shield, 1972; Kyriakides and Chang, 1991) reveals that for relatively long tubes, the descending parts of the responses in Fig. 10 are unstable. 2. The calculated up-down-up responses show that, for a range of pressure, the structure has three possible equilibria. Thus, in this range of pressure, given a large enough disturbance, it is possible to cause a jump from the small deformation equilibrium (along OL in Fig. 11) to the large deformation equilibrium (along M N in Fig. 11). a. Calculation of the Propagation Pressure The experiments demonstrated that uniform (cylindrical) inflation is possible only for some deformation regimes. Following some initial uniform inflation, deformation localizes at a section with the largest imperfections
Stelios K yriakides
82
FIG. 11. Maxwell construction for calculating the propagation pressure of a bulge.
(lowest PL).However, under volume control, the tube can be inflated in a quasi-static fashion, and a steady-state condition can be developed where “inflated” and “uninflated” sections of the tube coexist as shown in Fig. 11. Thus, on the left hand side, the tube is at equilibrium A, and on the right hand side, at equilibrium C. The transition region joining them can be viewed as the “disturbance” that causes a section of tube originally at A to move to C. Chater and Hutchinson (1984), showed how the propagation pressure can be evaluated using a simple but exact energy balance of the steady-state propagation process. Their argument is as follows. Consider a bulge that has been initiated and is propagating under steady-state, quasi-static conditions at pp. Refering to Fig. 11, let the bulge grow by a unit length. The external work done is given by Pp(Avc - AvA).
Since the material is elastic, the change in internal work is strictly a function of the initial and final configurations of the cross section, that is, states A and C in the figure. The change in internal work will be equal to the external work done; thus, A
Pp(Avc - Av,) =
Expression (2.8) implies that
pp is at a level that makes the shaded areas A,
Propagating Instabilities in Structures
83
and A,, in Fig. 11, equal. (This is known as Maxwell construction due to its similarity to the work of Maxwell in liquid-vapor phase transitions.) The Maxwell construction was used by Kyriakides and Chang (1990) to evaluate the propagation pressures of tubes at different values of axial tension. The predictions for the cases analyzed experimentally are included in Fig. 8. The predictions are seen to be in good agreement with the measured values. 2. Simulation of the Initiation and Propagation of a Bulge
Simulation of the initiation and propagation of a bulge as observed in the experiments requires a formulation that can capture the variation of deformation along the length of the tube. The simplest, and yet sufficient, formulation is one where the tube is treated as a membrane which can undergo finite, axisymmetric deformations as proposed by Kyriakides and Chang (1991). Their treatment of the problem can be summarized as follows. The tube (see Fig. 12) has an initial length 2L, radius R and wall thickness H.
l
Undeformed
FIG.12. Axisymmetric membrane geometric variables.
Stelios K yriakides
84
The ends are closed with rigid discs of radius R. It is inflated with internal pressure P in the presence of an external axial force F. The deformations are axisymmetric and symmetric about the mid-span. The undeformed and deformed coordinates used are (R, Z ) and (r, z), respectively. Following Green and Adkins (1960), the field equations governing the problem can be summarized as follows. The deformed geometry is described by dr _ -- sin o, ds
dz ds
-=
and the principal curvatures K~ and K1=
do ds
--
K~
-cos o,
are given by
cos w and u2 =r
(2.10)
where o is defined in Fig. 12. The principal stretches for incompressible membrane material are ds dS
11 = -,
1, =
r
(2.1 1)
and A3 =
The equilibrium equations are as follows: 211R12T1cosw = rcR21Z:P
+ F,
T,K, + T 2 ~=2P
(2.12)
where TI and T, are the meridional and circumferential stress resultants, respectively. The material is assumed to have the same constitutive behavior as described in the previous section, and TI and T2are related to the principal stretches through (2.7).The field equations can be rearranged into a system of ordinary differential equations as follows:
dl,=-[AlsinoS’-A,R’], 1 dZ R do 1 -= T d Z RW, dz-dZ
[
COS o(
- 11 cos
O
w2- 2 2 w A2
,) + F h] s, A2
(2.13)
nRH
os,
where R’ = dR/dZ and S’ = dS/dZ. In developing theese equations dP/dZ = 0 was used, and the value of F is
Propagating Instabilities in Structures
85
assumed to be prescribed. The boundary conditions appropriate for the problem are as follows:
40)= 0,
z(0) = 0,
&(L) = 1
and
A2(0)= Aly
or z(L) = I*,
where A: and I* are prescribed quantities. Equations (2.13) were solved numerically. Details about the solution procedure used can be found in Kyriakides and Chang (1991). The major characteristics of the solution will be illustrated through an example involving a tube like those used in the experiments, with a length of L/R = 15 and F = 0. Figure 13(a) shows the calculated pressure-change in volume response and the corresponding tube configurations during the early stages of the initiation of the bulge. Subsequent configurations and the corresponding pressure-deformed length (0 response are shown in Fig. 13(b). In Fig. 13(a), we observe that the tube initially undergoes cylindrical deformations very close to those calculated by the uniform inflation analysis. However, the radially fixed end condition at Z = L causes a small deviation from the cylindrical shape in the form of a gradually varying curvature along the length of the tube. The restraint results in a slightly higher response than that from the uniform inflation analysis. The limit pressure on the actual response (identified with ‘‘ v ”) is seen to precede the limit pressure of the trivial solution (identified with “ A ”). Soon after the limit pressure, a sharp cusp develops in the P- Au response. Concurrently, the deformation starts to localize. The central section of the tube, which is already more deformed, experiences an accelerated radial growth. The ends remain cylindrical, and as the pressure starts to drop, they unload cylindrically. In summary, after the limit load, part of the structure experiences an increase in deformation and part of it a decrease otherwise known as localization (see Tvergaard and Needleman, 1980, for a broad discussion of structural problems with this behavior). This turn of events leads to a reduction of the volume in the cylindrical sections and an increase within the bulged section. In the particular case shown in the figure, the structure initially experiences a net decrease in volume that causes the cusp observed in the P-Au response. In a volumecontrolled experiment, a jump in pressure bridging the cusp would be experienced (this explains the behavior observed in the experiments reported earlier). As the bulge grows, the volume required for continuing its gmwth becomes
-P P
Latex Rubber Uniform Inflaiion
F=O
1.0
0
20
15
0.5
-
3.0
2.5
*u/vo
0.241
01
0
I
2
3
4
5
6
IIL
FIG. 13. (a) Calculated pressure-volume response and corresponding sequence of configurations during initial stages of inflation. (b) Pressure-length response and sequence of configurationsduring propagation of bulge.
86
87
Propagating Instabilities in Structures
larger than the decrease in volume in the cylindrical ends. Thus, the net volume of the tube must now be increased. Beyond the cusp, the volume increases monotonically, while the pressure drops gradually. The bulge continues to grow both axially and diametrically, as shown in Fig. 13(b). Eventually it stops growing diametrically and starts spreading axially. This spreading occurs at a well-defined pressure plateau, which corresponds to the propagation pressure of the tube. This continues until the whole length of the tube is inflated to the same diameter (A, x 5.6). At the same time, the tube increases its length by approximately 4.1 times. Inflation beyond this point results again in cylindrical deformations of the tube, as shown in Fig. 13(b). This growth now requires an increase in pressure. Inflation can be continued until the material fails. This analysis was carried out without regard to such material limitations. For this particular case, the propagation pressure calculated from various types of analyses were as follows: Pdic Complete membrane analysis 0.1229 Maxwell construction: membrane 0.1227 Maxwell construction: thick-walled tube 0.1262 Experiment 0.123 Figure 14 shows pressure-circumferential stretch ratio histories for tubes with L / R = 30 for the point corresponding to the crown of the bulge and for a point far away from the bulge. The P-A, response from the uniform inflation solution is included for comparison purposes. Initially, the tube deforms
P -
I024
0 16
Latex Rubber -L --30
0 08
F=O
0 00
10
I
I
I
I
20
30
40
50
I
60
70 -A2
FIG.14. Pressure-& response at two different points along an inflated tube.
Stelios Kyriakides
88
cylindrically, and the responses from the two points coincide and agree with that from the uniform inflation solution. After the limit load, the deformation in the bulged section grows whereas the cylindrical sections experience reduction in deformation. The “bifurcation” into these two types of behavior occurs very close to the pressure maximum as illustrated in the figure. a. Effect of Thbe Length The length of the specimen analyzed was found to have a significant effect on the initiation of the bulge. Calculated P-Au responses for tubes with L/R of 10, 15, 20, 30 and lo4 are shown in Fig. 15. Initially, the pressure and volume increase monotonically. The tube length has a relatively small effect on this part of the response for the range of L / R values considered. However, the volume at which the maximum pressure is achieved (marked with “ v ”) and the size of the cusp that follows are significantly affected by L/R. The shortest of the five tubes analyzed develops a pressure maximum at the smallest value of Au/u,. Following the maximum, the response first experiences a very gradual decrease in pressure followed by a sudden pressure drop. The latter is associated with the initiation of the bulge. In this case, the
P
t
Latex Rubber
Propagating Instabilities in Structures
89
length is such that the increase in volume in the bulged section is larger than the decrease in volume experienced elsewhere at all times. As the length of the tube increases, the limit load occurs at a larger Au/u,. A t the same time, the delay between the pressure maximum and the instability is reduced. For L / R 3 15, a cusp develops in the response following the onset of instability. The size of the cusp grows with L / R for the reasons already given. In spite of differences in the bulge initia'tion mechanism, the pressure asymptotically drops to the tube propagation pressure for all tube lengths considered as shown in Fig. 16. However, for tubes with shorter lengths the minimum pressure in the response can be higher than P,. Figure 17 shows the initial configurations of the bulging tubes with lengths of L / R = 15 and 30. No visible difference in the bulge shapes of the two tubes was detected. This was found to be true for all tube lengths analyzed. This result confirms the experimental observations on the subject. A very long tube (L/R = lo4) was also analyzed as described in Kyriakides and Chang (1991). In this case, the limit load coincides with that calculated from the uniform inflation analysis. The instability is seen to occur at the limit load, much in agreement with the results from bifurcation analyses. Initially, the volume increase experienced in the bulged section is relatively small. Most of the tube unloads, and as a result, the unloading follows nearly the same P-Au path as the loading. More significantly, the pressure stops dropping when the value of the propagation pressure is reached. In view of the preceding, in a volume-controlled experiment on a long tube, the pressure can be expected to suddenly drop from P L to P,. The change will
FIG.16. Pressure-volume responses for tubes of different lengths.
90
Stelios Kyriakides
....
_---
FIG. 17. Initial configurations of bulges in tubes of different lengths and axial tensions.
occur dynamically as the tube snaps from the cylindrical configuration to one that has a bugle of radius r N 5.6R. The length of the bulge will be such that it accommodates the volume reduction caused by unloading the cylindrical part of the structure from P , to P,. This behavior is illustrated with an example of a tube with initial diameter of 2R and length 1 , W R in Fig. 18. When inflated to the limit pressure, the tube elongates to 1,115R and its diameter expands to 3.22R. In a volume-controlled experiment, on reaching the limit pressure, the tube snaps into a configuration that has a bulge with a diameter of 11.15R and length of 51R. The pressure drops to the propagation pressure and as a result the remainder of the tube unloads to a diameter of 2.02R. b. Efect of Tension The uniform inflation solution showed that axial tension has the effect of lowering the pressure required to inflate the tube. The more complete analysis was used to simulate the inflation of a tube of length L/R = 27.5 for various values of axial tension. A set of P-Au responses calculated are shown in Fig. 19. The responses shown have the same characteristics as described earlier. It is interesting to observe that the size of the cusp in these responses is reduced
Propagating Instabilities in Structures
91
FIG.18. Example of initiation of bulge in a long tube.
as the tensile force is increased. The propagation pressure of the bulge is also reduced as discussed earlier. The values of P , predicted by the Maxwell construction are again reproduced very well. A set of calculated bulge configurations for a tube inflated in the presence of axial tension F / p A = 1.149 are shown in Fig. 17. As mentioned earlier, tension causes elongation of the bulge.
008i
#Y
I
Lotex Rubber
k.27.5 0 0
I
2
3
4
5
6
7
-
8
wl,
FIG.19. Pressure-volume responses of tubes at different values of axial tension.
92
Stelios Kyriakides
c. Projle of Bulge Propagating Quasi-Statically at a Steady State The condition of steady-state propagation of the instabilities for all problems in this class is an important one, worth special attention. In this case, one way of reaching this state is to follow the development of the bulge as described previously until the pressure plateau is reached. A much simpler approach, suggested by Kyriakides and Chang (1990), is to consider the equilibrium of the transition region at the steady-state condition directly. Referring to Fig. 11 we know the equilibrium pressure to be P,. In addition, we know the stretches at the two ends of the profile, i.e., A and C. The profile must satisfy the equilibrium equations, which can be expressed as A
W
- l,W1 = const.
KR’A:P,
= C,,,
+ F = 2zRHW1 COSO.
(2.14)
The first part of (2.14) is a special form of one of the equilibrium equations suggested by Pipkin (1968), and the second part comes directly from (2.12). Since equilibria A and C are known Co can be evaluated from either of them. At intermediate points the solution can be obtained as follows (see also Yin, 1977). Select a value of rA < r* c rc (i.e., 1, c 1* c AC). Find the corresponding value of A: using (2.4)to (2.6)and the first part of (2.14).Obtain o from the first part of (2.14) and calculate the axial position of this point from (2.15) Thus, starting from say A, we can evaluate the shape of the profile using (2.15). Figure 20 shows a set of profiles calculated numerically for the parameters of the problem for several values of axial tension. The elongation of the profile with increasing tension is quite clear.
D. DISCUSSION AND CONCLUSIONS It has been demonstrated that, at least for rubberlike material properties, the inflation of a long tube has three regimes of behavior. For relatively small deformations associated with the initial part the inflation process, the tube deforms homogeneously into a cylindrical shape. The cylindrical deformation is terminated by a limit load instability. This instability results in localization of deformation in the form of a bulge. The bulge initially grows radially and axially. When it reaches a critical diameter the radial growth stops and the
93
Propagating Instabilities in Structures Flp A
FIG.20. Profiles of the bulge propagating at a steady state at different values of axial tension.
bulge starts to spread or propagate along the length of the tube. The minimum pressure required for the propagation of the bulge is significantly lower than that required to initiate it. The propagation of the bulge continues until the whole length of the tube is inflated to the same diameter. Inflation beyond this point is cylindrical and requires an increase in the applied pressure. A simple, uniform inflation analysis of a section of the tube reveals that the pressure-change in volume response has an up-down-up behavior. This behavior has been shown to be a requirement for the propagation of the instability. Barring significant local imperfections in the tube, the limit load in this response is quite representative of the initiation pressure of the bulge in a long tube. Indeed, the limit load is responsible for the initial formation of the bulge. The arrest of the local deformation and the propagation of the bulge can be attributed to the upturn in the P-Au response. Further discussion on how the geometric and material parameters of the problem can alter both the initiation and propagation phases of the inflation process is given in Kyriakides and Chang (199 1). Chater and Hutchinson (1984) showed that for elastic materials, the Maxwell construction applied to the P- Au response from the uniform inflation analysis yields an exact value of P,. Other features of the response require a more complete solution of the problem in which axial deformations are accounted for, such as the one outlined in Section II.C.2.
94
Stelios Kyriakides
111. Initiation and Propagation of Buckles in Long Tubes and Pipes under External Pressure
A. THE PROBLEM In a number of practical applications, long tubes and pipes must be designed to resist high external pressure. For example, pipelines and flow lines installed on the sea floor for gathering and transporting oil and gas must withstand the external pressure due to water depths ranging from a few feet to several thousand feet. Risers, that is, long tubular structures that connect sea floor wells to mother vessels, platforms, etc., on the surface of the sea, must be capable of withstanding the ambient external pressure and other (dynamic) loads induced by waves and currents. Casings, that is, long tubes installed in wells to protect production and hydraulic tubing and equipment, are also designed to withstand the external pressure due to water that often fills the well, which can be several thousand feet deep, from surrounding formations. The structures mentioned are typically made from various steels with yield stress ranging from 40 to 65 ksi (276 to 448 MPa) for pipelines and risers and 70 to 100 ksi (483 to 690 MPa) for casing. Naturally, buckling and collapse due to external pressure play important roles in their design. Typically, such structures have diameter-to-thickness ratios ( D / t ) ranging from 70 to 10 with casing usually being in the lower half of this range. For these combinations of geometric and material parameters, buckling can occur either in the elastic or plastic range of the material. In the elastic range, the buckling pressure ( P c ) is given by 2E
3
where E is the Young’s modulus and v the Poisson’s ratio of the material, D the diameter and t the wall thickness of the pipe (for the ring case see Levy, 1884). For buckling in the plastic range, EM1 - v2) in (3.1) is replaced by the corresponding plastic modulus (Dubey, 1969; Ju and Kyriakides, 1991). It is important to point out that, for the range of material and geometric parameters of interest, buckling results in catastrophic collapse of the structures. Consequently, the collapse pressure can be significantly reduced if the geometric integrity of the structure is altered. For example, a dent or geometric imperfection induced in a pipe under external pressure can result in local collapse at a pressure much lower than the ones calculated from the expressions mentioned earlier. Clearly, such a collapse would at first be local
Propagating Instabilities in Structures
95
in nature. In the early 1970s, it was discovered that such a local collapse can initiate a more global instability where, driven by the external pressure, the buckled (collapsed) section spreads (propagates) along the long structure, often at high velocity, and flattens it. The buckle stops only if it encounters a physical obstacle that resists the flattening or when it reaches an area of low pressure where propagation cannot be sustained. The phenomenon is known as a propagating buckle. The lowest pressure that can sustain such a buckle in propagation is known as the propagation pressure ( P p ) .For the type of structures discussed here, 0.15 P,/P, < 0.33 (approximately). The big difference between these two critical pressures, the potential of catastrophic failure to miles of structure in a rather short time and the potential of damage to the environment made the problem a very important one. The first concern for the problem came in the early 1970s from a study conducted at the Battelle laboratories dealing with a variety of mechanics problems affecting offshore pipelines. At the time, this was an emerging technology and the continental shelf was viewed as a new frontier. The study was cosponsored by a group of companies with offshore interests and was conducted at the Battelle laboratories in collaboration with technical personnel from the sponsors. The first public airing of the name propagating buckle came in a brief description of the problem given by Mesloh, Sorenson, and Atterbury in July 1973. Parallel to the work at Battelle, various aspects of the problem were being investigated by some of the sponsors. For example, in April 1972, Lockridge and Gibson (of Brown and Root, Inc.) and Broussard, Ayers and Walker (of Shell Oil Company) submitted a number of buckle arrestor designs for patents. In their applications, they demonstrated significant insight into the problem. (The patents were awarded in 1973.) Concurrently, it seems, British Petroleum (1972) was conducting full-scale tests on 32 in (0.81 m) pipes in the sea to determine the efficiency of buckle arrestors (Palmer, 1980). Although reports from the time are rather sketchy, propagating buckles seem to have been first encountered during experiments to establish the collapse capacity of tubes under combined bending and pressure. It was observed that, below a certain pressure, buckling tended to remain local, whereas above it, it tended to spread and collapse the whole length of the test specimen. Subsequent work on the problem will be discussed in the following sections. The importance of the problem can best be demonstrated by going through some possible scenaria for its occurrence in offshore pipelines. The most severe loading conditions seen by such a pipeline often occur during
-=
96
Stelios Kyriakides
installation. Pipelines are usually installed empty to take advantage of buoyancy. Figure 21 shows schematically a line being installed on the sea floor by the so-called s-lay method using a specially equipped vessel. On the way to the sea floor, the line acquires the configuration shown, which induces bending in the pipeline. The most critical combination of bending and pressure occurs in the sagbend, which is close to the sea floor. To avoid collapse, the induced bending curvature is maintained at a safe level by tension applied at the vessel (pipeline behaves approximately like a catenary). The operation just described is rather delicate. If tension is lost, either due to sudden movement of the vessel, loss of an anchor or malfunction of a thruster or the tensioner, the pipe can buckle in the sagbend due to the combined effect of bending and pressure. Such buckles are initially local in nature as shown in Fig. 22(a). However, if the external pressure is “high enough,” a propagating buckle can be initiated (see Fig. 22(b)).It is important to point out that bending plays a role only in causing the initial incident. Once the pipe buckles, the local bending moment drops by as much as an order of magnitude (see Corona and Kyriakides, 1988). Thus, the primary loading during the propagation process is the external pressure. A propagating buckle can also be initiated during the operation of a pipeline from a local dent or imperfection induced by a foreign object dropped on the line. Many incidents involving dropping equipment or anchors on pipelines from a ship operating in the area have been recorded (see Fig. 23). The impact can cause a dent that reduces the local collapse pressure of the
Tensioner
.:
Propagafion Critbl-Depth
Sagbend
\ 1’
’
I
Lay Vessel
A3 Propagating Buckle
Sea Floor
FIG.21. Initiation of a propagating buckle during the installation of an onshore pipeline
Propagating Instabilities in Structures
FIG.22. (a) Bending buckle (AI-6061-T6, D/t bending buckle.
= 68.7). (b) Propagating buckle
97
initiated from
pipe. If the line is operating at a high enough internal pressure, catastrophic failure may be averted. However, the pipe can subsequently collapse during a shutdown or during one of the periodic piggings of the line. Again, if the pressure is high enough, the local collapse can initiate a propagating buckle with the catastrophic consequences described earlier. Local collapse can occur due to a number of other causes. For example, sections of pipe with
FIG.23. Initiation of a propagating buckle in an offshorepipeline by impact of a foreign object.
98
Stelios Kyriakides
thinner wall thickness as a result of corrosion or wear (internal erosion of the wall) will have a reduced collapse pressure. I t suffices to say that any such local collapse, irrespective of its cause, has the potential of initiating a propagating buckle. Similar scenaria also exist that can lead to the initiation of a propagating buckle in risers and casing. From this brief review of the problem as it occurs in practice, it is clear that two conditions are necessary for a propagating buckle to develop. The first is that the pipe or tube must be under the action of a sufficiently high external pressure. The second is that the buckle must be initiated somewhere along the length of the structure. The highest pressure circular or nearly circular pipes or tubes can usefully sustain is the buckling (P,) or collapse (Pco) pressure. The lowest pressure at which an initiated buckle will propagate is defined as the propagation pressure (P,) of the pipe. Thus, a propagating buckle can occur at pressures P p < P < Pc. Two examples of buckles propagated at the propagation pressure of a test specimen in the laboratory are shown in Fig. 24. The catastrophic nature of the induced collapse is quite clear. Conceivably, local defects can be large enough to reduce the local collapse pressure to P,. As a result, a buckle can be initiated at any pressure P, < P < P,. A buckle initiated in a constant pressure environment of P > P , will propagate
FIG.24. Propagatingbuckles:(a) X-42 line grade steel;D
6061-T6;D = 1.25in. (31.8mm), D / t = 35.7.
= 4.0in. (102 mm),D / t =
27.7;(b) Al-
Propagating Instabilities in Structures
99
in a dynamic fashion. If it is initiated at the propagation pressure, it will propagate in a quasi-static manner. The problem as described has various aspects. First is the issue of initiation of a propagating buckle, which clearly is of great practical importance. Second is the determination of the lowest pressure that can sustain a propagating buckle; i.e., the propagation pressure. Third is the dynamics of buckles initiated at pressures above P,. Fourth is the practical issue of arresting such a buckle. The four problems have been studied to varying degrees by various investigators. The main thrust of what follows will be on the first and second aspects of the problems as they best fit the goals set for this chapter in the introduction. Discussion of other aspects of the problem will be limited.
B. PROPAGATINGBUCKLEEXPERIMENTS
1. Experimental Procedure The propagation pressure of a given pipe can be measured in a number of different ways. One possible way is to initiate a buckle in an environment of varying pressure (initially P > P,) and watch for the pressure at which the buckle will be arrested. This could be achieved for example by initiating a buckle at the lower end of a long pipe placed vertically in the ocean. Another approach is to develop experimental conditions under which the buckle can be initiated at any pressure, but is allowed to propagate only under controlled, quasi-static conditions. This can be achieved by volumecontrolled pressurization of a long pipe specimen using an apparatus like the one shown in Fig. 25, described by Kyriakides (1980) and Kyriakides and Babcock (1979, 1981a).' The procedure can be summarized as follows. A section of pipe, usually more than 50 pipe diameters long, is sealed at both ends and placed in a stiff pressure vessel. To ensure that the internal pressure of the pipe is constant during the collapse process, the pipe is vented to the atmosphere through an umbilical. The vessel is completely filled with water, sealed and pressurized using a positive-displacement hydraulic pump. In the experiments conducted by Kyriakides and coworkers reported here, a
' This method was developed and used concurrently by the early (1972-79) investigators of this problem such as the Battelle group (see Johns, Mesloh, and Sorenson, 1976; Mesloh, Johns, and Sorenson, 1976) and Kyriakides and Babcock at Caltech. It has subsequently been widely used by many others (e.g., Steel and Spence, 1983; Kamalarasa and Calladine, 1988), where today it may be considered as a standard test.
100
Stelios Kyriakides
FIG.25. Experimental apparatus for determinationof the propagation pressure of long tubes.
dent approximately one diameter long was induced in the test specimens three to four diameters away from one of the end plugs. This served as the initiator of the propagating buckle. Differentcapacity (and stiffness)pressure vessels (200, 1,000, 3,000 and 10,000psi-14, 69, 207 and 690 bar) were used, depending on the D/t of the tube tested and other requirements of the experiments. (The smaller capacity vessel has transparent walls for visual observations.) In most experiments, the rate of fluid pumped into the vessel was controlled manually using a needle valve. A positive displacement LDC Analytical pump (capacity 9,000 psi-620 bar) with calibrated, selectable, discharge volume rate was used in some experiments to capture details of the initiation process (see Dyau and Kyriakides, 1993a). During the experiment, the pressure in the vessel was monitored by an electrical transducer. The signal was recorded on a strip-chart recorder or a digital data acquisition system. The results from a typical experiment are shown schematically in Fig. 26 and are described next. The pressurization starts at time to by pumping water into the system at a relatively slow, constant rate. The system initially deforms elastically and is relatively stiff. As the pressure increases, the dented section of the tube becomes plastic and the response becomes “softer.” Eventually, at t , , a pressure maximum develops and the dented section collapses. The collapse leads to a significant drop in pressure and often occurs in an uncontrolled, “jump” fashion. The extent of the pressure jump and the length of tube collapsed dynamically depend on the severity of the induced dent, the postbuckling stiffness of the test specimen and the stiffness of the test apparatus. As the pressure decreases, energy stored in the test facility becomes available and contributes to the extent of the local collapse.
101
Propagating Instabilities in Structures -Initial
Dent
P
I
0 0
1
Cross Sections
1 PP
I I I
*
3 '
Time(t)
FIG. 26. Typical pressure history in experiment for the determination of the propagation pressure.
As more water is pumped into the vessel, the collapsed section and its neighborhood gradually acquire the configuration of the propagating buckle front, shown in Fig. 24, and at time t, the pressure stabilizes at the propagation pressure of the pipe. If the pumping rate is maintained at a relatively low, constant value, the buckle gradually spreads along the length of the tube at a slow, steady rate (quasi-static) while the pressure in the vessel remains at P , . For rate-dependent materials such as SS-304, P p was found to depend to some degree on the rate of the pumping of water. In the experiments described here, buckles were propagated at the rate of one to three tube diameters per minute. The propagation pressure was also found to be sensitive to variations in the wall thickness of the tubes. Wall thickness variations around the circumference can break the symmetry of the collapsed cross section and alter the measured value of P,. Variations of wall thickness along the length make P, position dependent. Careful measurement of the wall thickness along the length and around the circumference of the test specimens partly negated these variations. In general, the pressure in the vessel remained at P , until the whole length of the test specimen was collapsed. Once this occurred (time t , in Fig. 26), the deformation process became stable again, and the pressure in the tank started rising The transient part of the response (tl-t2) represents the initiation of the propagating buckle. The local pressure maximum is known as the initiation
102
Stelios Kyriakides
pressure (P,). The value of P, is strongly influenced by the shape and amplitude of the initial dent, whereas the extent of the transient is also influenced by the stiffness of the test system as a whole. In the experiments described here, the initiation pressure was typically 1.5-3 times the propagation pressure. A sequence of pictures illustrating the steady-state propagation of a buckle as seen by a stationary observer is shown in Fig. 27 (experiment performed in a transparent pressure vessel). The length of the transition region (projle of propagation) from the circular to the collapsed cross section, as well as the exact geometry of the collapsed cross section, depend on the geometric and material parameters of the tube and on the pressure at which the propagation takes place. At the propagation pressure, profiles of propagation ranging from 5 to 10 diameters in length have been measured (Table 4.1 in Kyriakides, 1980). The lengths of the profiles become significantly shorter for buckles propagating dynamically at pressures higher than P p . Figure 28(a) shows a set of rings cut through the profile of a buckle propagated at P, (tube same as in Fig. 24(b)).The rings illustrate the deformation history of a cross section of
FIG.27. Quasi-static propagation of buckle (AI-6061-T6, D = Loin. (25mm)and D/t = 50).
Propagating Instabilities in Structures
103
FIG.28. (a) Sections through profile of buckle propagated quasi-statically.(b) Cross sections of tubes collapsed by buckles propagated at Pp (first on left) and progressively higher pressures.
the tube as it is collapsed by a propagating buckle and the severity of the induced collapse. Figure 28(b) shows cross sections from similar tubes that were collapsed by buckles propagated at pressures higher than P,. The severity of the induced collapse is seen to increase with pressure. The experiments confirmed that, during quasi-static propagation of a buckle, the walls of the collapsed section touch at least along a line (axial). The small gaps seen between the two crown points in Figs. 24 and 28 are due to elastic springback after the pressure was removed. Propagation at pressures higher than P, results in progressively more contact between the opposite walls of the collapsed cross section, as implied by Fig. 28(b).
2. Propagation Pressure Results Two sets of experimental results obtained by Kyriakides and coworkers (e.g., Kyriakides and Babcock, 1981a; Kyriakides and Yeh, 1985; Kyriakides,
104
Stelios Kyriakides
Yeh, and Roach, 1984; Dyau and Kyriakides, 1993b will be used to illustrate the parametric dependence of the propagation pressure. The first set came from 46 experiments on stainless steel (SS-304) tubes with values of D l t ranging from approximately 100 to 12. The second set came from 26 experiments on aluminum (A1-6061-T6) tubes with values of D / t from approximately 100 to 17. All test specimens were of commercial quality seamless tubing with diameters ranging from 1.0in. (25.4mm) to 2.5in. (63.5mm). The geometric tolerances were max IAD/DJx 0.2% and max IAt/t( x 7%. The stress-strain response of each tube tested was obtained from uniaxial tests on axial test coupons. Variations in the material properties can be found in the original references. The measured propagation pressures normalized by the individual material yield stresses (ao= stress at 0.2% strain offset) are plotted against D l t on log-log scales in Fig. 29. The two plots indicate that
The following values for A and fi were obtained from least squares fits of the experimental results: AI-6061-T6 ss-304
A 14.178 35.547
P 2.339 2.47 1
These results indicate that the stress-strain behavior of the material cannot be adequately captured by just one parameter (ao).We note that the two sets of tubes tested had the following additional (typical) material characteristics: A1-6061-T6 ss-304
El00
E'lao
220 670
3-4 7- 10
where E' is an average postyield tangent modulus (up to strain x 10%). Kyriakides and Babcock (1981a) showed that the next level of improvement of (3.2) required that A = A(E'/ao). Larger variations in E'/uoand in the yield anisotropy in the stainless steel test specimens used are mainlv responsible for the greater scatter seen in Fig. 29 for this set of results. 3. Wet Buckles Figure 28 clearly shows that the deformation induced in a tube by a propagating buckle is mainly concentrated at four equally spaced locations on its original cross section. The bending strains developed at these points
Propagating Instabilities in Structures
105
'Q
I
8 "0
10-
5-
I-
0.5-
-
10
D/t
FIG.29. Propagation pressures measured for ss-304and A1-6061.-T6 tubes as a function of tube D / t values.
can be quite large (10-25%), and increase for pipes with lower D/t values. If the ultimate strain of the material is exceeded, the initiation of a propagating buckle results in cracking of the wall, as in the case shown in Fig. 30. As a consequence, the pipe is flooded, and the buckle does not propagate any further. This type of failure is known as a wet buckle and is highly undesirable in offshore pipelines. In the case of the aluminum alloy used in the experiments, wet buckles occurred for tubes with D / t 17. The stainless steel alloy used was much more ductile, and as a result, buckles could be propagated in tubes with much lower values of D/t. It is clear from Fig. 28(b) that, for each material, the cutoff D/t for which wet buckles start to occur must be increased if pipelines are designed to operate at pressures higher
-=
~-
106
Stelios Kyriakides
FIG.30. Cracking of tube wall as a result of deformationinduced by propagating buckle (AI6061-T6, D = 1.5in. (38.1 mm), D / t = 18.1): (a) cross-sectional view; (b) axial view.
4. E$ect of Propagating Buckles on Pipeline Design
Using the experimental results for P p , we can now further illustrate the design dilemma caused by the potential of initiation for a propagating buckle in an offshore pipeline. The P p / a o results for SS-304 are plotted against D/t again in Fig. 31 together with collapse pressures (Pco/ao)from 32 collapse experiments on tubes from the same set (from Kyriakides and Yeh, 1985; Yeh and Kyriakides, 1986). Collapse pressures for tubes with D/t values ranging from approximately 50 to 13 are included. The collapse pressure predicted numerically by using average material properties and geometric imperfection amplitudes is also included (from same references) for the whole range of D / t values shown. The big difference between P , and the corresponding Pco observed in Fig. 31 is quantified for specific values of D / t in Table I. The influence of this difference on design can be illustrated through an example. Let the water depth be such that Pco/ao = 71.5 x From Fig. 31, this requires a pipe with D l t x 2 1 . The propagation pressure for this D/t is P , / a , = 18.6 x Now suppose that the propagation pressure is used as the design criterion; then the pipe must have a D / t x 12.3. This implies an increase of approximately 70% in the material required and a significant increase in the installation cost. Thus, for deeper waters, it can be economically and practically necessary to design pipelines above Pp. In such a case, the use of buckle arrestors would be prudent.2 For buckle arrestors to be effective, the
’
See Broussard et at., 1973; Lochridge and Gibson, 1973; Johns et al., 1976; Kyriakides and Babcock, 1980, 1982; Steel and Spence, 1983.
Propagating Instabilities in Structures
107
-
O/I
FIG.31. Comparison of measured propagation and collapse pressures for SS-304 tubes for a range of D/t values.
TABLE I COMPARISON OF COLLAPSE ANV
PROPAGATION
PRESURES OF
FIVE
S S - 3 0 4 TUBES -
~
Dlt
50.8
37.2
25.5
p , Psi (bar) P p psi (bar)
356 (24.6) 66 (4.55)
1,124 (77.5) 20 1 (13.9)
2,498 (172.3) 502 (34.6)
18.2 4,360 (300.7) 1,202 (82.9)
12.8 7,200 (497.6) 3,080 (212.4)
108
Stelios Kyriakides
pipe material ductility must be high enough to ensure that the propagation of buckles at the design pressure selected will occur free of wet buckles. C. ANALYSES
1 . Uniform Collapse of lhbes under External Pressure: Shell Analysis Significant insight into the mechanics of propagating buckles can be gained by examining large deflection response of a section of a tube under external pressure. This was demonstrated by Kyriakides and Babcock in their analysis of an elastoplastic ring (1981b) and in their first paper on the propagation pressure (1981a). For our purposes here, a section of a tube collapsing uniformly under plane strain conditions is somewhat more appropriate. A formulation and solution procedure suitable for such calculations was presented by Yeh and Kyriakides (1986).The tube is treated as a thin-walled structure with the customary assumptions, but the kinematics adopted (Reissner, 1972) allow for arbitrarily large rotations of the normals to the mid-surface. The elastic-plastic material behavior was modeled through the J 2 flow theory of plasticity with isotropic hardening. The resultant two-point boundary value problem was discretized through a finite difference scheme and solved numerically. Pressure-displacement responses (P-A) calculated for an example involving an aluminum tube with D / t = 35.0 (tube I in Table 11) are shown in Fig. 32. For simplicity, the stress-strain response of the material was idealized as elastic-linear strain hardening with parameters E, o,, and a (see inset in Fig. 32(a)). For the parameters of this example, a geometrically perfect tube will buckle elastically at P,, given in (3.1). The buckling mode, in terms of the TABLE I1
GE~METRIC AND MATERIAL PARAMETERS OF TUBES I, I1 AND 111 Tube
Material
I
A1-6061-T6
I1
ss-304
I11
ss-304
D in. (mm)
1.2522 (31.81) 1.2555 (31.89) 1.3750 (34.93)
D/t
35.0 25.8 18.2
E msi (GPa) 10.1 (69.6) 29.2 (201.3) 29.3 (202.0)
44.79 (308.8) 39.8 (274.4) 41.8 (288.2)
44.5 (306.8) 33.0 (227.5) 37.0 (255.1)
n
a z 0 x lo3
34
0.52
9
0.44
LO
0.50
Propagating Instabilities in Structures
0
0
0.2
0.4 0 4
109
06 (b)
-
1
0.8 0 8 A/R
FIG.32. Pressuredisplacementresponses for uniform collapse of tube under external pressure; comparison of elastic and eiastic-plastic cases: (a) shell analysis; (b) discrete model.
radial (w)and tangential displacements (u) (see Brush and Almroth, 1975), is and v = b sin 20 (3.3) which represent ovalization of the cross section. Buckling is accompanied by loss of “stiffness” in the structure. As long as the material remains elastic (modulus E), the response will maintain a small but positive slope. For very large deflections, some stiffness is recovered (see figure). However, for the parameters of this problem, soon after buckling occurs, the combined membrane and bending stresses, acting at the four points with the largest radial deflections, cause yielding of the material. This additional reduction in local stiffness causes a downturn in the response, as shown in Fig. 32(a). This unstable branch (negative slope) continues until the first two diametrically opposite points on the cross section touch. Collapse configurations associated with this response are shown in Fig. 33(a). Included in Fig. 32 are responses for tubes with initial imperfections of the type w
= a cos 28
WO
= A0 cos 20.
(3.4)
110
Stelios Kyriakides
FIG.33. (a) Sequence of calculated colllapse configurations up to first contact (tube I). (b) Collapse configurations following first contact (tube I).
If the material remains elastic, the main effect of such imperfections is an amplification of nontrivial deformations in the neighborhood of Pc (see Fig. 32(a)). For the actual material parameters, a limit load instability develops in the response that can be viewed as the collapse load of the structure. This limit load is seen to be sensitive to the amplitude of the initial imperfections. For tubes that buckle in the elastic range, the limit load can be predicted, with engineering accuracy, by the “first yield” criterion of Timoshenko (1933; see also Yeh and Kyriakides, 1986). From the preceding we see that plastic effects, first, make the structure imperfection sensitive and, second, make the postbuckling response unstable under prescribed pressure. Unfortunately, the amount of deviation from the linearly elastic material response required for this drastic change in behavior is rather small. This is illustrated in Fig. 34, which shows P-A responses for the same tube geometry, E and o,,, but different postyield slopes E/u. We observe that a limit load instability occurs even for c1 = 1.2. For c1 > 5, the structure experiences what can be viewed as catastrophic collapse after the limit load. It is also of interest to consider the response beyond first contact of the tube walls. Contact, even at just one point (axial line), has a stabilizing effect on the structure. The four collapsed quadrants now behave, in essence, as arches. As
111
Propagating Instabilities in Structures p
18 16
14
12
10 08 06
04
02 O l .1 0 0
02
04
08
06 (4
10
-A!
FIG.34. Pressuredisplacement responses for uniform collapse of tube for various postyield moduli: (a) shell analysis; (b) discrete model.
a result, further deformation requires an increase in pressure. This was demonstrated in the numerical and experimental results of Kyriakides and Arikan (1983) (additional results appeared in Kyriakides, Yeh, and Roach, 1984). The pressure-displacement ( P - 6 ) and pressure-change of volume (P-Au) responses for the example considered earlier are given in Fig. 35 (uo is the initial volume of the tube). The sharp rise in pressure following first contact can be clearly seen in the two figures. A sample of configurations associated with this part of the response is shown in Fig. 33(b). Initially, contact is limited to one point. As the pressure increases, and depending on the parameters of the tube, the contact can spread over a small part of the central section of the tube. For thinner tubes, the four archlike sections can subsequently collapse, as in Fig. 33(b). This causes some loss of stiffness in the response and results in two additional regions of contact. The additional support increases the slope of the response again (see comparison between experimental and numerical results in Fig. 7 of Kyriakides and Arikan, 1983).3 In the collapse process described, parts of the wall undergo significant, and in some sections repeated, unloading and reverse loading. As a result, constitutive models able to treat such loading histories are required for accurate analysis of this part of the response.
Stelios Kyriakides
112 P
12~
E
t
lo-
08-
060 4-
0.2First Contoc
O+ 0
5
01
02
03
04
--%
AI-6061-T6
9 = 0 . 5 2X I O - ~
0.8-
Figure 36 shows P-6 and P-Ao responses calculated for an SS-304tube with D/t = 18.2 (tube I11 in Table 11). In this case, the geometric and material parameters are such that buckling occurs in the plastic range ( P o = 2o0t/DO, yield pressure). Although the onset of instability is different, the main characteristics of the responses and of the corresponding configurations are essentially the same as those of the previous case. Collapse configurations corresponding to the calculated equilibrium paths are shown in Fig. 37. They are also approximately the same as those presented in Fig. 33. In summary, we have seen that, for tubes from two structural metals, the combined effect of reduction in stiffness caused by buckling, coupled with the progressive reduction in the modulus of the material as plastic deformations
0
02
04 (b)
06 --All/
08 uo
FIG.36. Calculated responses for tube 111: (a) P-6 response; (b) P-Ao response.
FIG.37. Sequence of calculated collapse configurations for tube 111.
113
114
Stelios Kyriakides
develop, result in the development of a limit load instability. For values of D/t < 100 of interest here, the limit load instability occurs also for tubes for which first buckling occurs in the elastic range. If uniform collapse is assumed, the limit load is followed by a sharp drop in pressure down to a pressure plateau, at which the tube collapses catastrophically. Following first contact of the walls, the response becomes stable again and a sharp rise in pressure is required for an increase in deformation. In view of these characteristics, two additional observations can be made: 1. The presence of the load maximum in the response indicates that, for
long tubes like the ones of interest here, localized collapse will be energetically preferable to the uniform collapse assumed so far. Since initial imperfections can be expected to vary along the length of a tube, local collapse may indeed be triggered at a location along the length where their amplitude is the greatest. 2. The up-down-up responses calculated indicate that such structures have three possible equilibria for each value of pressure P , < P < P L (see Fig. 38). Thus, if a tube under constant pressure, P, < P < PL, is sufficiently disturbed it can jump from its prebuckling equilibrium on OL to one on MN (i.e., collapse). In a long enough tube, it is possible for part of the structure to be at the collapsed equilibrium and for part of it to be at the prebuckling one corresponding to the same pressure.
c
0'
-
nv
FIG.38. Pressure-volume response of section of tube and the Maxwell construction.
Propagating Instabilities in Structures
115
a. Estimation of the Propagation Pressure Let us now return to the steady-state propagation of a buckle along a long tube under external pressure. Let the collapse process be initiated locally at a section with the “largest” imperfection. Further collapse of the tube will depend strictly on whether the applied pressure is subsequently lower or higher than the tube propagation pressure. If the pressure is at or above P,, then the locally buckled section acts as the disturbance that progressively forces the tube section adjacent to it to jump from its equilibrium state on OL to one on M N , collapsing that section of tube in the process. The profile of propagation can, in fact, be viewed as a series of rings, each having a progressively higher degree of ovality and each assisting the one downstream of it to cross the energy barrier between A and B. From this discussion, it can be concluded that the propagation pressure of the tube must be between P , and P,. In fact, P , can be viewed as the lower bound for P,, as suggested by Kyriakides and Babcock (1981a). Chater and Hutchinson (1984) proposed that a good estimate of P, can be obtained by using a simple energy balance, which results in the Maxwell construction in Fig. 38. Their proposal was motivated by their work on the prediction of the propagation pressure of bulges in inflated elastic tubes and the lowest load at which necks propagate in a polymeric tensile specimen (Hutchinson and Neale, 1983; see also G’Sell, Aly-Helal, and Jonas, 1983; Neale and Tugcu, 1985). In the case of the elastic tube, the Maxwell construction yields the exact answer, and for the second, it yields a good first estimate of the actual value. An energy balance of this type can be applied provided the material behavior is path independent. If this is true, then the internal work done when the buckle propagates quasi-statically over a unit length of the tube can be calculated strictly from the difference in strain energy between the initial and final configurations. Since one path that can cause this collapse has been calculated (OALEMC in Fig. 38), it can be used to calculate the difference in the internal energy between states A and C. P , can be estimated as follows. The external work done when the buckle propagates along a unit length of the tube is given by Fp(Avc- AvA)
We equate this to the change in internal work between states A and C to obtain
(3.5)
116
Stelios Kyriakides
It can be easily seen that expression (3.5) requires the line ABC (Maxwell line) to be at a level which makes A, = A, in Fig. 38. Chater and Hutchinson (1984) used the J , deformation theory of plasticity (a potential theory) to model the inelastic behavior of the material in calculating the P-Au responses of tubes. This validated the use of (3.5) to estimate P p . Kyriakides et al. (1984) argued that if the deformation theory can be used, the corresponding J , flow theory can also be used equivalently. The stress paths induced to the material during quasi-static propagation of the buckle are not proportional. However, both papers discussed various helpful factors that make this energy balance a good approximation. The strengths and weaknesses of these two essentially equivalent methods for estimating P p will be discussed subsequently. 2. Uniform Collapse: Discrete Models Many of the essential characteristics of buckling and uniform collapse of tubes can be captured, at least qualitatively, with relative ease by using simplified models like the one shown in Fig. 39. With the hindsight of the experimentalresults, the tube is idealized by four rigid members connected by four springs. For simplicity, the springs are assumed to have a bilinear moment-rotation response, as shown in the inset of Fig. 32(b). We consider strictly the doubly symmetric mode of deformation represented by a single degree of freedom (0). The model tube can have an initial imperfection represented by O0. From geometry, the maximum deflection (measured from the imperfect configuration) of the cross section is given by
A = R[(cos 8,
- sin 8,)
- (cos 8 - sin O)].
(3.6)
The change in volume in the deformed cross section is given by AU
= 2~2(cos 2e0 - cos 28).
FIG.39. Discrete model of tube collapse.
(3.7)
Propagating Instabilities in Structures
117
Let us first consider the linearly elastic case (i.e., a = 1). The equations of equilibrium for the perfect case can be shown to be P
- < 1,
8 = 0,
PC
P 28 P , sin28’
P Pc
-21
where Pc = 2k/R2. For the imperfect case
P 2(8 - 8,) _ Pc
(3.9)
sin28
If the M-8 response of the spring is bilinear (a > l), then the equilibrium equations for the perfect case become P
- < 1,
8 = 0,
p,
P - 20 P , sin28’
For the imperfect case they are P P,
- --
2(8 - 8,) sin28 ’
Some parity between the results of the discrete and continuous models can be achieved by selecting the spring constants as follows:
k=- E t 3
40 ’
8=-
COD
Et
and a = -
E
(3.12)
Et
where E‘ = E/(1 - vz). Figure 32(b) shows a set of results obtained for the bilinear spring using (3.6) to (3.12). The qualitative similarity with the corresponding results from the continuous model, which appear in Fig. 32(a), is quite obvious. All
118
Stelios Kyriakides
features of the responses from the continuous model are seen to be captured by the discrete model for the linearly elastic and elastic-plastic cases for initially perfect and imperfect structures. The limit load instability occurs at first yield; i.e., when 2(0 - 0,) = 6,. The two sets of limit loads are seen to be in quite good agreement. The postlimit load response, important for propagation pressure studies, is also seen to be in reasonable agreement with the continuous model. Clearly, the model as presented has a rigid response following first contact (ie., A = R). With all other parameters fixed to the values given in Fig. 32(b), we now vary a in the same range of values as was done for the continuous model. The results of the model in Fig. 34(b) are again seen to have a very satisfactory performance in predicting the P - A responses (compare (a) and (b) in Fig. 34). A direct comparison between the P-6 response from the discrete and continuous models for tube I is made in Fig. 35(a). The results from the discrete model are seen to be in excellent agreement with those of the continuous model. A similar comparison for the P-Av responses, in Fig. 35(b), shows the discrete model to be somewhat deficient. The simple discrete model just described assumes that the main resistance to deformation is the bending stiffness of the structure. In the case of thicker tubes, which buckle beyond the elastic limit, membrane stresses and deformations influence the onset of instability and, to some degree, the postlimit load collapse. In such cases, this model and the particular choice of spring constants given in (3.12) are inappropriate, as they assume that the original tube buckles elastically. In the case of tubes that buckle beyond the elastic limit, a modification of the model along the lines of the Shanley model would alleviate this difficulty. This will not be demonstrated here. a. Estimation of the Propagation Pressure The same reasoning that allowed us to carry out the energy balance to obtain an estimate for P p from the continuous model can also be applied to the discrete model. For most collapse critical tubes and pipes used in practice, initial imperfections are quite small (0, 0.01);thus cos 20, can be approximated by 1 in (3.6) to (3.11).The external work done when a buckle propagates by a unit length is
-=
FPAv = Pp2R2 If we equate this to the internal work, then
(k 2 +
where ub = uo/(l - v2).
-
91
(3.13)
Propagating Instabilities in Structures
119
If the material can be idealized as elastic-perfectly plastic, LY -,00 and (3.13) reduces to
F,
= nub
(;y 2). (1 -
(3.14)
For some cases, a rigid-linear hardening idealization of the material may be appropriate, in which case (3.13) reduces to
F, = zoo
(;y
(1
+ 4n a, E, E) t
(3.15)
If we further idealize the material as rigid-perfectly plastic, then both (3.14) and (3.15) reduce to (3.16) (It can be argued (Kyriakides and Chang, 1992) that, in the last two cases, a plane strain collapse mechanism is more representative, in which case uo is replaced by 200/$.) Equation (3.16) is the well-known estimate of P , first reported by Palmer and Martin in 1975. The effectiveness of these models in predicting P, will be evaluated in Section 111.C.4. A number of other models for estimating P , strictly from the twodimensional response of a ring have been proposed (e.g., Steel and Spence, 1983; Croll, 1985; Whiersbicki and Bhat, 1986). The emphasis has been to use various approximate techniques, such as moving hinges, to improve the estimation of the internal work. Unquestionably, closed form solutions, even when approximate, play an important role in engineering practice. However, as such models are made more realistic, they lose their main attraction, which is simplicity. As a result, beyond a certain degree of complexity, more accurate numerical solutions of the type presented in Sections III.C.l and C.3 may be more effective.
3. Simulation of lnitiation and Propagation Process As in the problem of the inflation of elastic tubes, simulation of the complete process of initiation and propagation of a buckle requires a solution procedure that can capture the variation of deformation along the length of the tube. Indeed, the motivation for such a solution procedure is stronger in this problem since this is the only way to correctly account for the pathdependent nature of the material behavior.
120
Stelios Kyriakides
The problem of initiation of a propagating buckle from a bending buckle was simulated through a finite deformation, finite element formulation by Remseth et al. (1978). The solution procedure was stopped after the bending buckle localized. More recently, Jensen (1988) and Tassoulas, Katsounas, and Song (1990) have developed numerical schemes for simulating the propagation process directly. Jensen used a hybrid finite eIement/Rayleigh-Ritz procedure and shell-type nonlinear kinematics. The contact of the collapsing walls of the tube was assumed to occur along an axial line. Tassoulas et af. used the nine-node, isoparametric, shell finite element of Ahmad to discretize the problem. The contact of the walls was allowed to develop on the whole plane of symmetry using contact elements based on the penalty method. In this chapter, we will present results from a similar formulation developed by Dyau and Kyriakides (1993a, 1993b). The formulation incorporates Sanders’ (1985) nonlinear shell kinematics, which allow for large rotation of the normals to the mid-surface and about the normals (as in Jensen, 1988). The results presented here were obtained by using the smalldeformation J , flow theory of plasticity with isotropic hardening to model the material behavior. Motivated by the symmetries of the deformed shell observed in the experiments (see Fig. 24), only one-eighth of a tube of length 2L, radius R and wall thickness t was analyzed (see Fig. 40). The structure was discretized using the following trigonometric expansions of the displacements u, u and w: N
M
L
n=O m = l
N
v =R
M
1 1 b,,,cos- mLz x sin2n0,
n = l m=O
N
w =R
(3.17)
M
mzx 1 1 amncosIcos2nB.
n=O m=O
L
The problem was solved incrementally using the principle of virtual work. The contact was assumed to occur at discrete points along the line 6 = 0. For efficiency in the computations, a different parameter was selected to prescribe the “loading” of the structure depending on the stage of deformation: the pressure, the displacement at (x, 0) = (0,O)or the volume enclosed by the shell are three loading parameters used. Details about the formulation and numerical procedure can be found in Dyau and Kyriakides (1993a).
Propagating Instabilities in Structures
121
FIG.40. Geometry of shell analyzed. (1,2), (1,3) and (2,3) are planes of symmetry.
a. Initiation of Collapse In offshore pipelines, propagating buckles are initiated at a section along the length that, for some reason, has reduced resistance to collapse. The causes of degradation of the local resistance to collapse are multiple and varied. Thus, the initiation process cannot be addressed in general terms. Particular causes of local collapse, like those discussed in the introduction, have been studied, for example, by Kyriakides, Elyada, and Babcock (1984). It is worthwhile to consider how a long, circular (or nearly circular) tube collapses, how the collapse localizes and, in the process, initiates a propagating buckle. This will be done by considering the collapse of the two tubes, for which some results were presented in the previous sections and a third one. The geometric and material parameters of the tubes, referred to henceforth as tube I, I1 and 11, are given in Table 11. We first consider the early stages of collapse of a section of tube I having a length of 32 diameters and a perfectly circular cross section. Figure 41(a) shows the calculated P - Au response and Fig. 41(b) shows the deflection w of the generator at 0 = 0 at various stages of the loading history. Initially (0 to l), the tube deforms homogeneously ( w = const.). The tube buckles elastically at P = P , given in (3.1). Buckling has the form of uniform ovalization of the cross section and can be represented by (3.3). The initial postbuckling P - Au response has a very small positive slope and the
Stelios Kyriakides
122
-
t
1.2 1
pc
1.0-
0.80.6 -
Al - 606l-T6
0.4-
I
#
0
0.1
0.2
0.3
0.4
0.5
I
0.6
(
0 I8 w&& R
10 12
006
0
-05
0
05
-
“L
I0
(b)
FIG. 41. Full shell simulation of collapse for tube I: (a) pressure-volume response; (b) deformed configurations of generator at fJ = 0.
generator 0 = 0 is seen to remain straight in configurations 1 to 4.Thus, the deformation is uniform along the length. The change in the volume of the tube and of the deflection at 0 = 0 between 1 and 4 are both due to the growth of ovalization (during this phase, the deflection w(0,O) was prescribed). As a result of the ovalization, the bending stresses grow and eventually the material yields. Yielding induces a limit load instability as described earlier. In view of what has been described previously, the limit load instability can be predicted accurately by a uniform collapse analysis.
123
Propagating Instabilities in Structures
The downturn in pressure triggers localization of the ovalization. In Fig. 41(b), localization is first observed in configuration 5. Figure 42(a) shows distinct deviation of the current solution from the corresponding uniform one, after configuration 6. Localization implies growth of deformation and stresses in a section of the tube at a dropping pressure. At the same time, sections of the tube away from the locally collapsing section undergo unloading. This is demonstrated in Fig. 42, in which the pressuredisplacement responses at point C (0,O)and at point A (15D, 0) are compared. The unloading at point A is elastic; i.e., the response follows approximately the same path followed during loading. As a result of this, a significant portion of the change in volume induced at points like A during loading is recovered. We thus again have a situation like the one described in Section 11; that is, we have sections of the structure that experience increase in volume and the section experiencing local collapse in which the volume decreases. Depending on the overall length of the tube, the net change in volume may be initially negative. This indeed was the case for 2L/D = 32, which results in the formation of the cusp seen in Fig. 41(a). The localization grows and eventually, in configurations 11 and 12, reaches a full axial length of approximately 12 diameters. By this time the rest of the tube has unloaded to a nearly circular shape. It is interesting to observe that
e
12
-k_l
11.0
-I
0.8
-------
C
A
AI-6061-T6
0.6
D
7~35.0
+=32
0.4
01 0
001
002
0.03
0.04
-
0.05
I
0.06
W(.,o)
R FIG.42. Comparison of pressure-deflection response at two points along tube I.
124
Stelios Kyriakides
sections of the tube adjacent to the collapsed region undergo reverse ovalization. In this case, the length of these sections is approximately 4 This diameters and the maximum amplitude is approximately 7R x feature of the deformation had been observed earlier in experiments (see Kyriakides, 1980). The size of the cusp depends on the length of the tube and the amplitude and nature of initial imperfections present in the tube. The presence of the cusp shows that the collapse process cannot be completely controlled in an experiment in which the volume is prescribed. This was indeed proved experimentally using a very stiff testing system and an accurate metering pump to pressurize the tubes. On reaching the limit load, the tube collapsed dynamically and the pressure dropped by approximately 20 to 30%. Figure 43(a)shows pictures of the resultant local buckle. Figure 43(b) shows a view of a calculated configuration with the same maximum deflection. (For details on the sensitivity of the initial collapse on the tube length, the initial imperfections and the stiffness of test system, see Dyau and Kyriakides, 1993a.)
FIG.43. Localization of collapse in tube I: (a) experiment; (b) prediction.
125
Propagating Instabilities in Structures
It is of interest to compare the behavior of tube I with that of tube 111, which buckles in the plastic range. The P-Ao response and deflection profiles of the generator at 0 = 0 calculated for this tube are shown in Fig. 44, whereas the radial deflection at two points along the same generator are compared in Fig. 45. In this case, the bifurcation occurs in the plastic range at an increasing load. The buckling mode is again one of uniform ovalization (see Ju and Kyriakides, 1991). The uniform ovalization grows stably until a limit load develops. With the downturn in pressure, the deformation again
P
Cylindrical Deformation
t
>-,-----
I
0 0
0.1
0.2
0.3
0.4
0.5
-Av/
0.7
0.6
(70)
vo
(a)
a
i
SS-304
0 12
0 06
0
FIG. 44. Full shell simulation of collapse for tube 111: (a) pressure-volume response; (b) deformed configurations of generator at 8 = 0.
126
Stelios Kyriakides
localizes as seen in Fig. 44(b). In this case, the sections of the tube away from the localized region unload elastically, as seen in Fig. 45; i.e., not all of the prebuckling deformation is recovered. As a result, the cusplike behavior in the P-Av response does not occur in this case. For a tube of length 2L/D = 29, the volume monotonically decreases during the collapse process. For longer tubes, the initial postbuckling slope is reduced and asymptotically approaches a slope equal to that of the initial loading. An interesting feature of this case is that the length of the localized section is significantly reduced compared to that of tube I. It has been observed that for most collapse critical tubes and pipes used in practice, initial geometric imperfections left by the manufacturing process have relatively small amplitudes and characteristic axial wavelengths that are long (typical azo 1 x typical wavelengths are longer than 1OD (see Yeh and Kyriakides, 1986, 1988). As a result, the collapse pressure can be calculated with good engineering accuracy by using an axially uniform imperfection given by
-
wo(0) =
- azoR cos 20
(3.18)
where the amplitude is obtained from the maximum and minimum diameters
0'41
C
A
SS-304 f)=18.2
0.2
O! 0
9.29 I
0.01
1
0.02
I
I
0.03 0.04 0.05 C -2 W(* 0 )
R FK. 45. Comparison of pressuredeflection response at two points along tube 111.
Propagating Instabilities in Structures
127
measured along the length of the tube (a,, corresponds to Ao/R in (3.6)). The success of this scheme was demonstrated in the two references just cited. Table 111 shows a comparison of the collapse pressure calculated with this assumption (pco)with the experimental values (Pco).The values of a,, used are given in Table 11. The predictions are in good agreement with the experiments. The critical bifurcation buckling pressure (pc)calculated for each tube is also listed in the table for comparison. (Dyau and Kyriakides (1993a) address how the wavelength of imperfections affects the collapse pressure.) The following conclusions can be drawn from these results: 1. The postbuckling response of tubes and pipes from the class of material and geometric parameters considered here has a negative slope. This results in localization of the collapse to a length of a few diameters. The characteristic length of localization is governed by the problem parameters. 2. The cusplike behavior in the postbuckling part of the pressure-change in volume response is a feature of tubes that buckle in the elastic range but collapse due to the additional loss of stiffness caused by plastic deformations.
b. Steady-State Quasi-Static Propagation of Buckles The analysis just described was also used to simulate the spreading of the initially local collapse to the rest of the tube. In fact, the lengths of the tubes used in these analyses were sufficient for the buckle to reach a condition of
TABLE I11 MEASURED COLLAPSEPRESSURESCOMPARED CALCULATED BUCKLING AND COLLAPSE PRESSURESOF THREE TUBES
WITH
I I1 I11
560
(38.6) 2,498 (168.9) 4,360 (300.7)
565 (39.0) 2,600 (179.3) 4,712 (325.0)
557 (38.4) 2,436 (168.0) 4,483 (309.2)
128
Stelios Kyriakides
steady-state propagation. The response of the structure following initial collapse was simulated by prescribing the change in internal volume of the tube. Without loss of the generality of the solution, we continued to assume that the collapse spreads symmetrically from mid-span outward along two fronts. Figure 4qa) shows the P-Ao response calculated for tube I. This tube had an overall length of 32 diameters and an initial axially uniform imperfection Figure 46(b) shows the deflected as defined in (3.18) with uz0 = 0.52 x shape of the generator at 0 = 0 at various points along the calculated response. Collapse was initiated at P/P, = 0.986. Following the limit pressure, the collapse localized as described earlier. The minimum pressure in the response occurs just before the walls of the tube touched for the first time (configuration 3). At first contact, we can observe that a length of approximately 13.5 tube diameters has undergone significant ovalization. Contact of the walls, even at one point, has a stabilizing effect in the response, and the pressure starts to rise. This feature has been observed systematically in
F;
f 10
81-6061-T6
08 o2.=052 ~10.'
06
04 02
0
0
004
008
012
016
020
024
028
-
032
0 3
036 Pv/
V,
(a)
t
',
i
\
\
i
\,
p.350
=32
0
01
02
03
05
04
(b)
06
07
08
-
09
y
I
10
FIG.46, Simulation of initiation and propagation of buckle in tube I: (a) pressure-volume response; (b) deformed configurations of generator 0 = 0.
Propagating Instabilities in Structures
129
FIG.47. Sequence of calculated configurations showing propagation of buckle in tube I.
experiments. Following this initial small upturn in pressure, the buckle starts to spread along the length of the tube as can be observed in Fig. 4qb). In the process, the section over which the walls of the tube come into contact increases. In the cases presented here, it was sufficient to assume that contact occurred along an axial line only. A steady-state propagation condition is developed in a relatively small distance following first contact. Subsequently the pressure stabilizes to the relative pressure plateau identified in the figure as p p (the small periodic variation observed in the pressure “plateau” is due to the combined effect of the periodic nature of the displacement shape functions used (see (3.17)) and of the way the contact was simulated). Configurations 4 to 7 can be considered to be in the steady-state propagation condition, since the pressure remains essentially constant and the displacement profiles are seen to remain unchanged as the buckle propagates to the right. At this stage, the profile of the propagating buckle is approximately 7.5 diameters long. The propagation pressure obtained from this calculation was 151 psi (10.41 bar), which compares with the experimental value of 149 psi (10.28 bar). The calculation was terminated before the ends started having a significant effect on the solution. The maximum strain induced by the
130
propagating buckle to the circumference of the collapsed tube was found to be approximately 14%. Graphical reproductions of a sequence of collapse configurations are shown in Fig. 47 (generated from the calculated configurations using a Sun Vision image processing system). The positions of these configurations on the P-Ao history are identified in Fig. 46. The geometric similarity of the calculated configurations to those seen in the experiments was found to be of exceptional quality. Results from a similar calculation involving tube 111 are shown in Fig, 48. In this case, the length of the tube analyzed was 29 diameters. Collapse was initiated at PIP, = 0.971. The collapse localized as described in Fig. 44. First contact, represented by configuration 3 in Fig. 48, occurred at PIP, = 0.189. In this case, the collapse is more localized than in tube I and, in configuration 3, is seen to affect a length of approximately 8.5 tube diameters. After first contact, the pressure rises and the buckle starts to spread along
0
I
0
01
0'2
'
0'3
'
09
'
05
0'6
0'7
0'8
-
09
I
10
"L
(b)
FIG.48. Simulation of initiation and propagation of buckle in tube 111: (a) pressure-volume response; (b) deformed configurations of generator 0 = 0.
131
Propagating Instabilities in Structures
the length of the tube. Following an initial transient, the pressure stabilizes to a plateau value of 1207 psi (83.3 bar = mean value of “plateau”), which compares with the measured value of 1202 psi (82.9 bar). As the buckle propagates to the right, its profile is seen to retain its shape. However, it is interesting to observe that, in this case, the profile is only 5.5 diameters long, which is significantly shorter than that of tube I. 4. Evaluation of the Propagation Pressure Predictions
We have now seen three levels of analysis of varying complexity for estimating the propagation pressure of tubes. The first two attempt to solve the problem in an approximate fashion by using results from twodimensional uniform collapse analyses. A n energy balance argument, resulting in the Maxwell construction, is employed to obtain approximate predictions of P,. This is done even though it is recognized that the path dependence of elastic-plastic material behavior makes this approach questionable. In the first method, the uniform collapse process is simulated numerically; in the second, a four-hinge, admissible, discrete mechanism is used to approximate the collapse process. As a result, the predictions can be viewed as approximations of the first method. The third method involves a large-scale numerical effort that simulates the process of initiation and steady-state propagation of the buckle. Numerical predictions of P , from the three levels of analysis for the three tubes discussed previously are given in Table IV. The corresponding experimental values are included for comparison. These three examples were selected because they can illustrate the strengths and weaknesses of the three levels of analysis. TABLE IV COMPARISON OF EXPWMENTAL AND PREDICTED VALUESOF PROPAGATION PRBSLJRES FOR THRF.E TUBES
Tube I
I1 111
Exper. P p psi (bar)
2-D/Shell/RO Fp psi (bar)
2-D/Shell/BL Fp psi (bar)
2-DPiscr. Fp psi (bar)
P_almer pP psi (bar)
3-D/Shell Fp psi (bar)
149 (10.3) 503 (34.7) 1202 (82.9)
152 (10.5) 348 (24.0) 83 1 (57.3)
148 (10.2) 340 (23.4) 813 (56.1)
122 (8.41) 261 (1 8.0)
115 (7.93) 21 1 (14.6) 469 (32.3)
151 (10.4) 480 (33.1) 1207 (83.3)
8
132
Stelios Kyriakides
All predictions of P, are sensitive to the way the material properties are represented. It is important to note that the propagation of buckles can induce significant strains, at least in the four regions on the cross section that undergo large rotations. Thus, material tests were conducted up to the required strain levels and care was taken to appropriately fit the measured properties. Good predictions require accurate representation of the initial yielding (rounded knee) of the stress-strain response as well as of the tangent modulus at larger values of strain. The designation RO in the table implies use of the Ramberg-Osgood fit to represent at least the initial yielding (in the case of SS-304, it was found more accurate to truncate the RO fit in the range of 1.53% strain and to use a linear fit for higher strains). The designation BL implies a bilinear approximation of the stress-strain response made for the purposes of the discrete models. In the 2-D/Shell/RO and 3-D/Shell analyses the material representation used was the same. Comparisons of the predictions with the experimental values show that the full simulation (3-D/Shell) of the problem yields results very close to those measured in the experiments for all three cases, in spite of the relative simplicity of the constitutive model used and the way the contact was modeled. The two most important factors influencing the predicted values of P p were found to be the discretization scheme adopted, represented in this case by the displacement series expansion given in (3.17), and the accuracy of the fit of the material stress-strain response. Detailed discussion of these can be found in the original reference. The quality of the predictions from the 2-D/Shell analyses is material and D/t dependent. For example, in the case of tube I, which happens to be aluminum, the prediction differs by 2% from the experimental value. Good predictions were also obtained from such an analysis for several tubes of this material by Chater and Hutchinson (1984). Kyriakides, Yeh, and Roach (1984) confirmed this success, but used a larger sample of data to show that the predictions deteriorate somewhat for lower D/t values. They also pointed out that the same analysis, applied to stainless steel tubes, underestimated P , by 10-30% This is demonstrated again here for tubes 11 and 111, where the predictions are on the order of 30% lower than the measured values. This difference in the performance of this analysis is due to differences in the stress-strain response of the two materials. Stainless steel-304 has a lower a,/E and a healthier postyield hardening. As a result, the initial collapse was seen in the experiments, as well as in the 3-D simulation, to be more localized and the profiles of the propagating buckles to be significantly shorter than in the aluminum tubes. More localized deformation induces
Propagating Instabilities in Structures
133
more complex stress histories, including more severe loading and unloading in the propagating front of the buckle. In addition, axial effects play a stronger role (see Bhat and Wierzbicki, 1987; Kamalarasa and Calladine, 1988). The net effect is that the propagating buckle dissipates more energy, which results in higher values of P , seen in Fig. 29. These complexities obviously cannot be captured by a model based on path independence of the material. The relatively small contribution of axial effects to the stress history of the profile and the limited unloading observed in the original uniform collapse calculations of aluminum tubes by Kyriakides and Babcock (1981a, 1981b) and in subsequent analyses were the main reasons that allowed Chater and Hutchinson (1984) and Kyriakides, Yeh, and Roach (1984) to use the Maxwell construction to calculate P,. Quite clearly, these helpful, special circumstances are material dependent and do not, for instance, develop for SS-304. The stress-strain behavior of actual pipeline steels varies from ones with very low hardening to others with hardening comparable to that of SS-304. In view of the previous discussion, design engineers should use caution when using simplified models to estimate P,. Good model performance demonstrated in any one material does not guarantee good performance in other materials. It is interesting to observe from the results in Table IV that the predictions from the same analysis using a bilinear approximation of the stress-strain response of the material (2-D/Shell/BL) do not differ significantly from those with the more accurate stress-strain fit. The predictions from the discrete models are significantly worse than the ones from the 2D/Shell analyses. The original four-hinge model of Palmer is dependable only for order-ofmagnitude estimates of P , for most cases. As the D/t becomes lower, its predictions progressively diverge from the measured values, as was observed by Palmer and Martin (1975). Since material hardening is neglected, this model yields poorer predictions for stainless steel tubes that have a healthier hardening. This deficiency is alleviated, to some degree, by the inclusion of linear hardening in the spring stiffness (i.e., (3.13)),as was suggested in Section III.C.2. However, the poor performance at lower tube D/t values persists. (Note that (3.13) is not applicable in the case of tube I11 as it buckles in the plastic range.) It is to be understood that the best that can be expected from any model based on a uniform collapse analysis is to approach the predictions of the numerical solution depicted as 2-D/Shell/RO. Approximate models that take into account the full three-dimensionality of the problem, for instance, that of Kamalarasa and Calladine (1988), can have improved performance.
134
Stelios Kyriakides
D. EFFECTOF TENSION ON
THE
BUCKLEPROPAGATION
PRESSURE
In a number of practical applications involving long, suspended tubular structures under external pressure, the tensile loads developed can be significant. For example, Fig. 49 shows a pipeline suspended from a lay vessel in a characteristic J-configuration proposed for use in deep water applications (3,000-8,OOOft [l,OOO-2,5OO m],see Langner and Ayers, 1985). Pipes installed in such waters will be negatively buoyant. As a result, significant tension must be applied to hold the structure in the required configuration. Pipelines suspended from floating production systems, tubular tethers for deepwater tension-leg platforms and deep well casing are other examples that experience combined tension and pressure loads. All of these structures are susceptible to local dents induced either during their installation or their operation. As a result, the possibility of initiation of a propagating buckle must be considered. It is well known that tension can lead to a reduction of the buckling and collapse pressures of long tubes (Babcock and Madhavan, 1987).The primary mechanism of interaction between pressure and tension comes about through a reduction of the yielding of the material as a result of the induced biaxial stress state. The same effect was found by Kyriakides and Chang (1992) to
Suspended Pipe
FIG.49. Schematic of J-pipe installation method proposed for deepwater applications.
135
Propagating Instabilities in Structures
reduce the pressure at which buckles propagate in a pipe under tension. Some of their results are summarized here. Emphasis will again be primarily given to quantifying the effect of tension on the pressure at which buckles that have been initiated propagate quasi-statically. 1. Experimental Procedure
To measure the propagation pressure of tubes loaded concurrently in axial tension the experimental procedure described in Section III.B.l was modified as follows. A combined tension-pressure loading test facility, shown in Fig. 50, was developed. The test facility consists of a long and narrow reaction frame that can slide into a pressure vessel. The load frame consists of two rods 80in. (2m) long, which are fastened 5in. (0.127m) apart onto one of the “blind” flange closures of the pressure vessel. Axial tension is applied through a single-ended hydraulic actuator located outside the pressure vessel. The actuator is connected to the pressure vessel flange through four posts 20in. (0.5m) long. The load is transferred to the test specimen by a rod that penetrates the vessel flange. Relatively low-friction, dynamic seals are used to seal the rod. A photograph and a detailed schematic view of the loading mechanism are shown in Fig. 51. The test specimen is connected to the cross head of the load frame at one end and to the loading rod through a universal joint at the other. A load cell is placed in the load loop as shown in Fig. 51. A linear voltage differential transformer (LVDT) is used to monitor the overall axial displacement. The test facility has a load capacity of 10,000lbs (45 kN) and a maximum displacement of 6 in. (0.15m). A closed-loop servo-controlledsystem was used to prescribe the axial tension. The loading frame was stiff enough so that the energy stored was at least one order of magnitude less than that in the specimen. The complete axial loading system rests on wheels so that the test section can be rolled into the pressure vessel with relative ease. Buckles were initiated Pressure Tmnrdu
FIG.50. Schematic of combined pressure-tension test facility used to determine the effect of tension on the propagation pressure of tubes.
136
Stelios Kyriakides
FIG.51. (a) Photograph of loading device. (b) Major components of loading device.
with a dent induced at one end of the test specimen. The pressurization process used was the same as described in Section III.B.l. During the experiment, the pressure in the vessel, the applied axial load and the elongation of the tube were monitored by electrical transducers. The three signals were recorded on a common time base, either on an analog strip chart recorder or, in digital form, using a computer-based data acquisition system. The results from a typical experiment are shown schematically in Fig. 52 and are described next. Initially, the tube was pressurized at zero prescribed axial load. Thus, the initiation and initial propagation of the buckle are essentially the same as in Fig. 26. To confirm the value of P p recorded, the buckle was propagated by approximately six to eight diameters beyond the collapsed length corresponding to t2. The rate of pumping of water was again such that the buckle propagated at one to two tube diameters per minute. At time t , and withut interrupting the pumping of water into the system axial tension was applied to the test specimen for the first time. The required axial load level (7') was typically reached in 10-20sec. The application of axial load caused an immediate drop in the pressure in the vessel. The lowest pressure recorded was found to correspond approximately with the completion of the load ramp at t4. Again, the pressure response recorded exhibits
Propagating Instabilities in Structures
T
137
I I
I I
1st.
Time
10
1, 1; I* 1, '7
T
FIG. 52. Typical history of experiment in which a buckle is initiated and propagated in the presence of tension.
a transient (t3-t5), beyond which the propagating buckle reaches a new steady state. This is associated with the lower pressure plateau in Fig. 52. This value of pressure is defined as the propagation pressure of the tube at tension T and will be depicted as P,. The application of tension leads to a change in the geometry of the buckle profile. The profile length increases with tension as can be seen by comparing the profiles shown in Fig. 53. The extent of the transient ( t 3 - t 5 )depends on the geometry and material of the tube, as well as on the value of the axial load applied. Clearly, in view of the differences in the shape, as well as in the stress states in the profiles before and after axial load is applied, steady-state propagation can occur only after the buckle has moved "far enough" away from its initiation point. This transient region was found to extend from 5 to 10 tube diameters. Once achieved, the new steady-state propagation was continued typically for 8 to 10 additional tube diameters, and the experiment was terminated. This particular loading history was adopted primarily for experimental convenience. Its use does not affect the accuracy or uniqueness of the measured values of PpT.It is indeed useful to have P , as well as P,, for each
138
Stelios Kyriakides
FIG.53. (a) Profile of propagating buckle at T = 0 (D/t = 39.9). (b) Profile of propagating buckle at T = T..
tube tested. P p is governed by the same geometric and material parameters as those governing PpT.In a tube 24 ft (7 m) long, such as the ones from which the test specimens originated, these parameters can vary to some extent along the tube length and with the angular orientation of the buckle in the tube cross section. As a result, normalizing PpTby P p should reduce the scatter in the experimental results. Loading history effects on the material properties of the collapsed tubes were minimized by using a long enough test specimen. This provided sufficient separation of the two propagating buckle regimes.
2. Propagation Pressure Results
Experiments were conducted on commercially available 1020 carbon steel (CS) seam-welded tubes. The test specimens had diameters ranging from 1.0 to 1.75 in. (25 to 43mm) and lengths of approximately 70in. (1.7m). Tubes with six different Dlt values ranging from 53.6 to 20.7 were tested. The average geometric and material parameters of each Dlt tested can be found in Table 1 of Kyriakides and Chang (1992). The propagation pressure was measured at various values of axial tension for tubes with D/t values of 45.6, 39.8 and 26.2. Tension values ranging from 0 < T < l.lT, were applied, where T, is the yield tension (0.2% strain offset yield stress). A sufficient number of experiments was conducted in each of the three cases to develop a propagation pressure-tension interaction envelope (numerical values of the results are given in Table 2 of the same reference). The values of P , measured in the three groups of experiments, normalized by P p , are plotted against the applied tension (TIT,) in Fig. 54. With this
Propagating Instabilities in Structures
139
FIG.54. Propagation pressure as a function of applied tension. Comparison of experimental and predicted values for three tube D/t values.
normalization, the three sets of results correlate quite well. Clearly, tension reduces the propagation pressure quite significantly. For the cases considered, a tensile load corresponding to T, causes a reduction in the propagation pressure of more than 50%. For tensile loads higher than approximately 1.06T,, the propagation of the collapse was found to be competing with pure elongation and uniform radial contraction of the intact section of the tube. As a result, the experiments were difficult to control and the buckle difficult to propagate. Because of this difficulty, the experimental propagation pressure-tension interaction envelopes were not closed. Additional experiments were conducted to quantify the effect of D/t on the propagation pressure in the presence of a tensile axial force. Experiments were conducted for tubes with six different Dft values. In each case, the propagation pressure was measured at T = 0 and 0.8T,. The two propagation pressures, normalized by the measured yield stress of the material, are plotted against D/t in log-log scales in Fig. 55. The two sets of data are seen to be approximately linear. The least squares method was used to fit the data as follows: (3.19)
140
Stelios Kytiakides
\
\
\A
\
,
\
\ ' \
cs-1010
\
\\ \a Propagation ,
T=0.8Te
\
Pressure
\ ,
\
,
\
o\
\
0 A Experiments
\
\ '\
- - - Least Squares Fits -Predictions
'
A\\@\ \
\
A\ 30
'
',
-
5'0
' 7'0' 'I(
3
vt FIG.55. Measured propagation pressures and calculated collapse pressures plotted against the tube D/t for two values of tension.
The calculated values of the constant A and /Ifollow:
T/T 0 0.8
A 17.32 9.31
B 2.28 2.20
Although the data sample is rather small for the fits to have enough accuracy, it is clear that tension causes a parallel downward shift (approximately)of the straight-line relationship between the two quantities plotted in Fig. 55 (as discussed earlier A, in general, is not a constant). To illustrate the extent to which propagating buckles can influence the design of collapse-critical structures, the collapse pressures of tubes with D / t values between 50 and 20 were evaluated numerically for T = 0 and
Propagating Instabilities in Structures
141
T = 0.8% (using the average material properties and initial ovality of the six tubes tested). The two sets of calculated results are also included in Fig. 55. The large difference between the collapse pressures and the propagation pressures for both cases is quite clear.
3. Predictions The three levels of analysis discussed in Section 1II.C can also be applied to the problem in the presence of tension with suitable modifications. HOWVG~, the case for using uniform collapse analyses is somewhat weaker in the presence of axial tension due to the more complex (nonproportional) stress histories. Details of the analyses will not be presented here except for some results from the simplest of the discrete models. In spite of its deficiencies, the four-hinge model is again a good starting point that can offer qualitative insight into the problem. As in Palmer and Martin (1975), we approximate the material to be rigid-perfectly plastic but assume that yielding at the four hinges obeys the Von Mises yield criterion. We define MOT to be the full yield moment of the hinges acted upon by an axial tension T per unit length of a tube. M o is defined as the value of M O T at T = 0. If we again equate the external work done by the pressure when a buckle propagates a unit length to the energy expended in the four hinges, it can be easily shown that the propagation pressure at tension T (p,) is aiven by (3.20)
where r = TIT, and
p p is given by (3.16) or by P-
--..(;). 2
2
(3.21)
'-3
The function m(t) can be obtained from any statically admissible stress field satisfying the yield condition. The optimum expression for m(z) was derived by Hodge (1961) and is given by m(t) = -
(in this case, the
13 [cot2 q In
(; ::l)
Pp is given by (3.21)).
- 2 cosec q ]
, (3.22)
142
Stelios Kyriakides
A more approximate, but similar, expression for m(z) can be derived by using an admissible stress field in which the axial stress is assumed to be uniformly distributed across the thickness, and the circumferential membrane stress to be zero. In this case,
1 -T2
=
Jm.
(3.23)
The relationships between ppT/pp and T/T, as given by (3.20) to (3.23) are plotted in Fig. 54 with the experimental results. As mentioned in the previous section, the numerical values predicted by the simple four-hinge models for T = 0 are significantly lower than the experimental results. The same deficiency is, of course, repeated for the predictions of PpT.In spite of this problem, the interaction between axial and circumferential stresses seems to be adequately captured by the models. As a result, with the normalization used, the interaction relationships compare quite well with the experimental results. Expression (3.23),in spite of its approximate nature, is seen to better represent the experiment for T/T, < 0.5. Expression (3.22) is closer to the results for higher values of tension. Both models neglect the material strain hardening and, as a result, are overly conservative for TIT, > 0.9. From the practical point of view, it is rather doubtful that real structures will be designed to carry tensions higher than 0.8T,; thus, expressions (3.22) and (3.23) may prove to be useful design tools, provided the propagation pressure at zero tension is known accurately. The latter can be obtained from an experiment or by one of the analytical methods described in Section 1II.C. The three-dimensional collapse analysis described in Section III.C.3 was utilized to simulate one of the buckle propagation experiments in the presence of tension. The tube analyzed was CS-1020 with D/t = 39.78. The geometric and material parameters used are given in Table V. This tube buckles in the elastic range. Due to the presence of a small, uniform, initial imperfection with amplitude azo= 0.082 x lop3,the collapse pressure was
TABLE V G E ~ M E ~AND U C MATERIAL PARAMETERS OF TUBE ANALYZED
Material cs-1020
D in. (mm)
1.2530 (31.8)
D/t
39.78
Emsi (GPa)
u0 ksi (MPa)
uy ksi (MPa)
28.2 (194.4)
39.1 (269.6)
36.5 (251.7)
n
a z 0 x lo3
34
0.082
143
Propagating Instabilities in Structures
found to be PIP, = 0.984. The collapse localized in the same fashion as in tube I in the previous section. Figure 56 shows the calculated P-Ao response and deflected configurations of the generator at 8 = 0. The position of each configuration on the P-Au loading history is identified. As the initial buckle localizes, the pressure drops and reaches its minimum value of 134psi (9.24 bar) just before first contact (1). At this stage, the local collapse has a length of approximately 13 diameters. Following a small pressure transient, the buckle starts to propagate at steady state at the pressure plateau identified in the figure by fi,,, which corresponds to 151 psi (10.4bar). Configurations 2 to 5 are in the steady-state propagation condition. The profile of propagation is seen to be approximately 7.5 diameters long. Immediately after configuration 5, axial tension corresponding to 0.9q was
07
?/
06 cs-1020
05
?=3978 04
%=32 n,;O
03
OBZxIO-'
02 01
0
I
0
.
.
01
5
,
.
02
03
04
05
06
07
08
09
I
I0
XIL
(b) FIG. 56. Simulation of initiation and propagation of buckle in the presence of tension: (a) pressure-volume response;(b) deformed configurations of generator 0 = 0.
144
Stelios Kyriakides
applied to the tube. The pressure is seen to drop down to a minimum value of 95.5 psi (6.59 bar). The pressure subsequently rises somewhat and stabilizes at a new pressure plateau of 100 psi (6.90 bar), at which the buckle propagates again at a steady state (P,,). With the application of the tension, the profiles of propagation are seen to stretch out to a length of approximately 9.5 tube diameters. Configurations 7 to 10 are in the second steadystate propagation condition. The change in the profile of the propagating buckle induced by tension can also be seen in Fig. 57, in which graphical reproductions of two calculated profiles at T = 0 and T = 0.9T,, are compared. The predicted value of ppT is included in Fig. 54, where it can be seen to be in good agreement with the experimental results. Good agreement with the experiments was also found in all other numerical values quoted earlier. Thus, we conclude that the three-dimensional simulation of the problem can capture the effect of tension on the steady-state propagation of the buckle. However, as mentioned in Section III.C.3, this success requires accurate representation of the properties of the material, at least up to the strain levels that the buckle induces in the collapsed tube. In this particular case, the presence of the seam was found to also influence the calculated propagation pressure values. The area around the seam had a yield stress of approximately 1.5 times higher than that of the rest of the tube. In the experiments, the buckle was initiated with an orientation that placed the seam along one of the four plastic hinges formed by the collapsing tube. This feature had to be
FIG.57. Calculated propagating buckle profiles: (a) T = 0; (b) T = 0.9T,.
Propagating Instabilities in Structures
145
included in the analysis for the predictions to be in good agreement with the experimental values. In closing this section, it can be stated that axial tension lowers the stress at which the material yields in the circumferential direction of the tube. This has the effect of reducing the lowest pressure at which buckles propagate in such tubes. This effect can be included in design by using the results of the simple four-hinge model. The option of complete simulation of the propagation process, which can yield the propagation pressure with accuracy, is also available. E. DISCUSSION AND CONCLUSIONS
It has been demonstrated that long cylindrical shells under external pressure can, under some conditions, develop buckles that propagate in an unstable fashion. Such buckles are usually initiated from local imperfections. Once initiated, the buckle propagates as long as the applied pressure is higher than the so-called propagation pressure. For the metal-alloy tubes considered in this study, the propagation pressure was shown to be significantly lower than the critical buckling pressure of the tubes. The phenomenon is influenced by a strong interaction between geometric and material nonlinearities. This interaction can be demonstrated instructively by categorizing the shells into three broad categories as follows. 1. Category I
In this category we place shells with relatively high values of D/t. Under external pressure such shells buckle elastically at P = P, given in (3.1). Buckling has the form of uniform ovalization of the cross section of the shell. If the shell is thin enough, the postbuckling deformations remain elastic and the shell maintains a relatively small but positive stiffness after buckling. Increase in pressure results in significant, but uniform, ovalization of the cross section. Thus the structure requires an increase in effort for the deformations to grow further. It can thus be stated that the postbuckling response is stable. 2. Category I 1 Let us now consider shells with intermediate values of D/t. Buckling is still elastic; however, following some ovalization, the material now yields due to the combined effect of bending and membrane stresses. This additional loss of stiffness causes a limit load instability. For typical structural metals, this is
146
Stelios Kyriakides
followed by a precipitous downturn in pressure. Under these conditions, the uniform ovalization mode of deformation becomes unstable, and a localized mode of collapse becomes energetically preferable. If the shell material does not have enough ductility, local collapse leads to fracturing of the shell walls and flooding of the structure, in which case the propagating instability never develops.
3. Category I11 In the third category we place shells with relatively low D/t values such that buckling due to external pressure occurs in the plastic range. In this case the growth of ovalization under increasing pressure is very limited. A limit load instability develops followed by localized collapse at a decreasing pressure. The stable nature of the postbuckling response of shells in category I clearly precludes the development of propagating buckles. Shells in categories I1 and I11 differ in the way initial buckling develops, but have similar behavior after the onset of the limit load instability. A key characteristic of the postlimit load behavior is that the collapse localizes to a section of the shell a few diameters long. Given enough ductility in the shell material, the collapse grows until the walls of the structure come into contact. This has a stiffening effect that locally stops the collapse. However, neighboring sections that have undergone significant ovalization continue to collapse until their walls come into contact. As these sections collapse further they ovalize the shell adjacent to them. Thus, much like a set of dominoes, once the buckle is initiated, it continues to propagate provided the pressure is higher than P,. It must be pointed out that the manner in which the initial collapse is initiated does not in any way influence the sequence of events that follow. Thus, a local dent caused by impact with a foreign object reduces the limit pressure of the structure. If this limit pressure is exceeded the structure collapses locally and in the process initiates an instability that propagates. As in the case of the problem considered in Section 11, the potential of developing localized collapse and a propagating instability can be identified by analyzing the uniform collapse of a representative strip of the long shell. The up-down-up pressure-change in volume response calculated for shells in categories I1 (e.g., tube I in Section III.C.2) and I11 (e.g., tube I11 in Section III.C.2) is sufficientto indicate that if the shell is long enough a propagating instability is possible. The lowest pressure that will sustain an initiated buckle in propagation is
Propagating Instabilities in Structures
147
known as the propagation pressure. For a given material, the propagation pressure is proportional to (t/D)s,where N 2.4. Three levels of analysis for estimating this critical pressure have been presented. Analyses in which the collapse is assumed to be uniform along the length were shown to provide a good insight into the problem. The Maxwell construction applied to results from such analyses can provide estimates of the propagation pressure with engineering accuracy. However, the applicability of this technique was shown not to be universal. Large-scale analyses that simulate the initiation and propagation process can provide accurate predictions of P,. Propagating buckles initiated in the presence of axial tension have been shown to have a propagation pressure that is as much as 40% lower at a tension corresponding to 80% of the yield tension.
IV. Propagating Buckles in Long, Confined Cylindrical Shells A. THEPROBLEM Long cylindrical structures are often lined with a thin inner shell to seal them or to protect them from caustic contents. For example, tunnels and ducts used for transporting gases and liquids at power plants are lined with steel shells. Steel casing is used to protect the geometric integrity of oil, gas, water and steam wells. A thin layer of noncorrosive metal is used as cladding in steel pipes to internally protect the steel from corrosive fluids. In all of these applications, conditions can develop that can result in separation of the liner from the outer structure and to the collapse of the liner. Initially, the collapse is local; however, if hydrostatic pressure somehow develops between the outer structure and the collapsed liner, the collapse can spread (propagate) at a relatively low pressure and destory large sections of the structure. The phenomenon, first demonstrated by Kyriakides (1986), is given the name confined propagating buckle. The D/t values of such liners vary from application to application. In the case of large diameter tunnels in power stations, D/t values of 300 to 500 are typical. Casing D/t values vary between 10 to 30, and typical D/t values for noncorrosive cladding is in the range of 40 to 100. The cause of the initial collapse and the potential of developing a confined propagating buckle also vary with applications and must be addressed individually. The purpose of steel casing is to maintain the geometric integrity of the well
148
Stelios Kyriakides
and protect tubes inside it from creep of the ground formation and the external pressure developed by water or gas around the well. Casing is typically required to support its weight and resist collapse under hydrostatic pressure equivalent to a water head equal to the well depth. In active formations, casing is grouted with cement either along the whole or part of its length, as shown in Fig. 58. The grout increases the collapse resistance of the pipe but collapse is still possible. It is usually initiated at a section along the length whose resistance to collapse has degraded, either due to corrosion or due to local out-of-roundness produced by ground movement and other causes. Texter (1955) gave a very graphic report of such collapse failures. He identified two modes of collapse: the “ribbon” and “trough” modes. The “ribbon” mode has the familiar “dogbone” collapsed cross section reported in Section 111. An example of the “trough” mode, as it affected a string of casing 60ft (18m) long, is shown in Fig. 59. It has a characteristic U-shaped collapsed cross section common to circular cylinders that buckle in a circular cavity. The length of the collapsed section shown and reports that collapse often affects “several strings” indicates that enough fluid was available from the surrounding formations to propagate the buckle. Shell liners used in large diameter underground shafts at power stations
FIG.58. Schematic of grouted well casing with a defect.
149
Propagating Instabilities in Structures
FIG. 59. Collapsed string of well casing (Texter, 1955; courtesy of American Petroleum Institute).
and hydroelectric plants lie at the other extreme of the D / t range of interest. McCaig and Folberth (1962) and Ullmann (1964) give some of the collapse considerations affecting these applications. Due to the large diameter (several meters) of these structures, it has been found more economical to drill circular tunnels into the ground. The tunnels are lined with a thin-walled ( D / t 300500) steel shell. The gap between the tunnel wall and the liner is filled with concrete grout. Thus, the liner’s main purpose is to contain the flow. The loads caused by the internal pressure are reacted mainly by the concrete and the rock around it. Although the shafts usually operate under internal pressure they must often be emptied for maintenance. The porous nature of the surrounding ground formations allows external water pressure to develop between the liner and the cavity which can lead to buckling. Given a sufficiently high water flow, the buckle can again spread and damage a significant section of the liner. Similar buckling failures have been reported in smaller diameter lined ducts by Monte1 (1960) and Yamamoto and Matsubara (1977, 1981). Bimetallic pipes and tubes are being used increasingly for “sour gas” pipelines (flows contain H,S, CO, and chlorides). The pipes are made from low-alloy steels but are lined with a thin inconel shell. Liner D / t values range between 40 and 100. The bimetallic pipe offers the required corrosion resistance and strength at economically acceptable values. It has been found that hydrogen gas can sometimes be generated at the bimaterial interface at sufficiently high pressures to cause separation of the liner from the steel and cause its collapse. The hydrogen is generated by corrosion either at the interface or at the outer surface of the steel and subsequent permeation to the interface (see Miyasaka, Ogawa, and Mimaki, 1991). Colwell, Martin, and Mack (1989) conducted experiments under accelerated corrosion conditions on clad pipes in which the liner was installed by different manufacturing processes. For some cases the amount of hydrogen generated was sufficient to develop an axial blister several pipe diameters long, again indicating propagation of the instability (see Fig. 9 in reference).
-
150
Stelios Kyriakides
From this brief review of the problem as it occurs in practice, we again see that like the other instabilities discussed in the chapter, this problem also has an initiation phase and a propagation phase. The initiation of such a buckle is of great practical importance. It has received some attention in the literature, which will be reviewed in Section 1V.C. However, the main objective here will be to discuss the phenomenon of confined buckle propagation, present methods for determining the lowest pressure at which it will occur and methods for its prediction. It will be shown that, unlike the propagating buckle discussed in Section 111, this one can also occur for strictly elastic liner deformations. Inelastic deformations aggravate the instability.
B. CONFINED PROPAGATING BUCKLEEXPERIMENTS 1. Experimental Procedure
In the experiments the problem was idealized as a long, circular cylindrical shell lining the wall of a cavity in a stiff medium. These conditions were simulated experimentally (Kyriakides, 1986) as will be described. To illustrate the influence of geometric and material parameters on the phenomenon of confined buckle propagation, two sets of experiments were conducted. In the first set, thin-walled mylar tubes were used as the liners. In these cases, the phenomenon occurred strictly in the elastic range. The second set of experiments involved aluminum and steel alloy tubes for which plastic deformations were unavoidable. Somewhat different experimental procedures were used in the two sets of experiments. a. Linearly Elastic Shells The tubes were made by rolling thin strips of mylar on a circular mandrel and lap-joining the ends using double-sided adhesive tape. The mandrel was removed and the tube was fitted inside a thick, transparent, acrylic tube with essentially the same inside diameter as the outer diameter of the mylar tubes. The ends of the mylar tube were sealed, arranging for access to a vacuum pump and pressure measuring transducers, as shown in Fig. 60. The dimensions of the tubes used were such that the atmospheric pressure was enough to cause collapse and propagation. Thus, external pressure was applied by reducing the internal pressure in the tube using a vacuum pump. The buckle was initiated at one end, using a mechanical plunger to locally separate the liner from the cavity wall. This reduced the local collapse pressure and resulted in local buckling of the liner, as shown in Fig. 61(a).
Propagating Instabilities in Structures
FIG.
151
60. Experimental apparatus used for propagation of confined buckles in mylar shells.
As a result of the “flexible” nature of the whole system the initial collapse
occurred in a dynamic fashion and resulted in a significant pressure transient. The initial buckle was 8 to 10 diameters long. Steady-state quasi-static propagation was achieved within 10 tube diameters from the initiation point. Following this, the rate of propagation was controlled by selecting the rate at which the tube was evacuated using a needle valve. (In all experiments, the tube seam was positioned diametrically opposite the initiation point.) A picture showing a buckle at the propagation stage is shown in Fig. 61(b).
b. Elastic-Plastic Shells In the second set of experiments the shell liners were made from seamless, drawn metal tubes with diameters ranging from 1.25in. (32mm) to 4.0in.
FIG.61. (a) Initiation of buckle in mylar shell. (b) Steady-state propagation of confined buckle.
152
Stelios Kyriakides
(102mm) with D/t values ranging from approximately 20 to 100. The pressures required to initiate and propagate confined propagating buckles for these combinations of geometric and material parameters are relatively high. As a result an alternative experimental scheme had to be devised. The circular confinement was established by molding a thick, concentric, cylinder of plaster of paris around each liner. The tubes were well lubricated prior to pouring the plaster to reduce friction between the two surfaces. A steel outer tube was used as the mold and the assembly was kept concentric by two ring spacers placed at each end. The final assembly is shown schematically in Fig. 62. Plaster of paris was selected because it retains its strength in water and dries much faster than cement (approximately 12hrs). The hardness of this type of plaster is quite high. Results from a comparative experiment using portland cement showed no quantifiable effect on the measured variables from the two grout materials. During the experiment the steel mold was left in place to ensure that no cracking of the plaster occurred. Since the goal of this study was to demonstrate the phenomenon and establish the confined propagation pressure, no attempt was made to simulate the initiation process as it occurs in practice. Instead, it was decided to initiate the buckle on a section of tube left outside the confinement by physically denting it, as shown in Fig. 62. The confined length of the test specimen was always longer than 25 tube diameters. The unconfined section was 5 to 15 diameters long. In this case the experiments were conducted under volume-controlled pressurization using a 10,OOOpsi (690 bar) pressure vessel and the experimental set up shown in Fig. 25. The inside of the test specimen was vented to
Vent
Concentric Spacer A-A
FIG.62. Assembly of metal shell specimen and confinement.
153
Propagating Instabilities in Structures
the atmosphere to ensure that the internal pressure remained constant during the experiment. A typical pressure history of such an experiment is shown in Fig. 63. The collapse process was initiated from a dent on the unconfined section. P,, which occurs at t , , represents the initiation pressure of this dent. The initiation process is completed by t,. Subsequently, the buckle propagates at the propagation pressure ( P p )of the “dogbone” collapse mode. This buckle stops propagating once it reaches the edge of the confined section at t 3 . Continued pumping of water into the vessel leads to a relatively sharp rise in pressure. (The rise in pressure is not instantaneous because a finite volume of water is required to further flatten the already collapsed section of tube.) The confined tube remains virtually undisturbed until time t , in the figure, when the flattened section of tube adjacent to the edge of the confinement snaps into a U shape, enabling the buckle to start penetrating the confinement. The pressure at which this occurs (Pic), is usually the highest pressure experienced during such an experiment. It represents the initiation pressure of the confined propagation process under the particular experimental conditions described here. The profile of confined propagation is fully
,Initial
Dent
rConf ining Cylinder
End Plug
P
FIG.63. Typical pressure-time history of experiment for determination of PK and corresponding shell collapse configurations.
154
Stelios Kyriakides
developed by time t5 (typically five tube diameters from the edge of the confinement). The pressure stabilizes at a new plateau as the buckle propagates at a steady state. In these experiments, the rate of pumping of water was maintained at such a level so that the buckle propagated at two to three diameters per minute. The propagation of the buckle continues at this pressure plateau until the whole length of the shell has been collapsed. The value of the pressure plateau is defined as the confined propagation pressure ( P K ) of the tube. The collapse process was visually recorded in a separate experiment. A 1.5in. (38 mm) cavity was bored into a transparent acrylic rod with diameter of 4 in. (100mm). A thin-walled aluminum tube was fitted into the cavity. The assembly was placed into a transparent pressure vessel and a propagating buckle was initiated and propagated in the way described previously. This arrangement allowed photographic recording of the propagation process. Edited results from this experiment are shown in Fig. 64. The development of the steady-state propagation process is clearly visible. A typical profile of a confined propagating buckle is shown in Fig. 65. The profile transforms the originally circular cross section of the tube into a characteristic U-shaped collapsed cross section. The profile is typically three to five tube diameters long. Another view of the sequence of configurations a
FIG.64. Confined buckle propagation in an acrylic cavity (A1-6061-T6, D / t
=
53.4).
Propagating Instabilities in Structures
155
FIG.65. Profile of confined propagating buckle (Al-6061-T6, D / t = 25.8).
cross section must go through as the buckle goes by it can be seen in the set of thin rings cut through the profile of such a buckle in Fig. 66. 2. Conjned Propagation Pressure Results
a. Linearly Elastic Shells Only a limited number of mylar tube experiments was conducted. The tubes had diameters of 1.82in. (46.3mm) and wall thickness of 0.010in. (0.25mm). The measured confined propagation pressure was 0.55 psi (0.038 bar). The main difference between these experiments and those on metals was the much longer profile of the propagating buckle and the reversibility of the process. b. Elastic-Plastic Shells Two sets of results from aluminum-6061-T6 and stainless steel 304 tubes reported by Kyriakides (1986) will be used to illustrate the parametric
FIG.66. Ring sections through buckle profile (A1-6061-T6, D / t = 53.4).
156
Stelios Kyriakides
dependence of P,. Eleven experiments were reported for each set. The tubes were of commercially available quality and seamless, with D/t valves ranging from 20 to 115 and diameters from 1.0 to 4.0 in. (25 to 100mm). The diameters, wall thicknesses and confined propagation pressures recorded are given in Tables 1 and 2 of the reference. The stress-strain response obtained from a uniaxial test on an axial test coupon cut from each test specimen was also recorded, and the results are summarized in the same reference. The measured propagation pressures normalized by the individaul yield stresses (ao = stress at 0.2% strain offset) are plotted against D / t on log-log scales in Fig. 67. We again fit the data with a power law representation as follows:
The values of A and j obtained are as follows: Material A1-606 1-T6 ss-304
A 26.07 18.29
B 2.15 1.92
The aluminum results are seen to be well represented by this relationship. In the case of the stainless steel, the results deviate from this relationship for D/t values higher than approximately 60. It is instructive to compare values of the confined propagation pressure and those of the unconfined one for the same tubes. Measured values of P , and P p from pairs of aluminum and stainless steel tubes with approximately the same values of D/t are compared in Table VI. The confined propagation pressure is seen to be systematically three to four times higher than the propagation pressure at which buckles propagate if not confined radially. This reflects the much larger energy required to deform the buckled tube in a circular cavity (partly due to the final shape of the buckled cross section and partly due to the significant energy required to reverse the curvature of the profile-see Fig. 65).
C. ANALYSES 1. Uniform Collapse of Conjned Shells The problem can again be seen to have an initiation stage and a propagation stage. Both issues are rather challenging due to the geometric nonlinearities, the material nonlinearities in the plastic range, and the
Propagating Instabilities in Structures
157
50
\
0
\
SS-304
AI-6061-T6 - Least Squares Fits A
\
0.5 f
10
I
I
20
I
I
40
l
I
I
60 80
l
IbOD,+
FIG.67. Confined propagation pressures measured for SS-304 and A1-6061-T6 shells as a function of shell D/r values. TABLE VI ~ M P A R l S O NOF VALUES OF
P,
AND
Pp
Material A1-6061-T6
AI-6061-T6 AI-6061-T6 Al-6061-T6 ss-304 ss-304 ss-304 ss-304
35.0 35.7 25.1 25.8 34.1 34.8 50.8 50.0
3.33 8.33 4.87 1.94 -
12.33 24.0 -
23.2 -
11.9
158
Stelios Kyriakides
nonlinearity of the contact between the liner and outer support structure. As in the previous two problems, it is again instructive to start by considering the axially uniform collapse problem first. The complexity of the mechanism of buckling of confined rings and shells has been demonstrated in the past. For example, Pian and Bucciarelli (1967), Zagustin and Herrrnann (1967), and McGhie and Brush (1971) considered buckling of confined rings due to inertial loads. Bucciarelli and Pian (1967) and El-Bayourny (1972) considered buckling under thermal loading, whereas Bottega (1988) studied the case of a point radial load. The pressure-loaded case was studied by Yamamoto and Matsubara (1977, 1981), whereas Kyriakides and Youn (1984) considered the initial buckling and large deflection collapse of such rings. Here we will consider the more general problem, treated in Li and Kyriakides (1991), of a thin-walled cylinder consisting of a circular outer shell of radius R and wall thickness to, and an inner shell smoothly in contact with the outer one with wall thickness t,. The inner shell has a small geometric imperfection that causes a length of 2s to be detached from the outer one, as shown in Fig. 68. Two loading cases were considered: in type I the complete structure and the cavity formed by the imperfection were subjected to hydrostatic pressure; for load type I1 only the cavity was pressurized. The related problem, in which the composite shell has two initial imperfections symmetrically located, as shown in Fig. 69, will also be considered for completeness. The problem was solved numerically using nonlinear beam kinematics, which allow arbitrarily large rotations (Reissner, 1972). Details of the numerical solution procedure, and of how the varying contact between the walls of the shells (rings) was treated, can be found in Li and Kyriakides (1991) for the elastic case and in Li (1991) for the elastic-plastic case. a. Linearly Elastic Shells We first consider the case in which the two shells are linearly elastic with Young’s modulus and Poisson’s ratio, (Eo7vo) and (E17vI), respectively. Sequences of collapse configurations for deformations that are symmetric about one and two planes for load type I and rigidity ratios of (Eot2/EIt:) = 3 are shown in Fig. 70. The pressure deforms the two shells and causes the inner one to detach from the outer one and collapse. Figure 71 shows pressure-volume responses for the two types of loads considered for the singly symmetric case [Au is the increase in the volume of the cavity formed between the two rings during deformation; uo = aR2;
FIG.68. Initial geometry of shell with one plane of symmetry: (a) load type I; (b) load type 11.
FIG.69. Initial geometry of shell with two planes of symmetry.
159
(b)
FIG.70. Sequence of calculated collapse configurations for case; (b) doubly symmetric case.
= 3: (a) singly
symmetric
18
Sinply Symmetric
S CS’ E, 1% , 6
Load I
14
q,,, E:
12
‘“R If
12
1,
10
08
08 06 04
0 02
4
0
025
W
050
(0)
-~x,
02
075
0
025 (b)
050 -dU&
075
FIG. 71. Pressure-volume responses for various shell bending rigidity ratios for singly symmetric deformations:(a) load type I; (b) load type 11.
160
Propagating Instabilities in Structures
161
E: = EJ(1 - vt)]. A particular imperfection, defined in Li and Kyriakides (1991) with amplitude A. = 0.025R, was used in these calculations. Results for various shell rigidity ratios are presented. The presence of the outer shell clearly has a stiffening effect on the response of the composite structure for both types of loads considered. In the case of load type I, if the bending rigidity of the outer shell is relatively small, the response is similar to that of a single tube under external pressure. Beyond a certain pressure (the critical buckling pressure given in (3.1)),the structure experiences a substantial loss of stiffness but continues to deform under increasing pressure. For higher bending rigidity ratios, the response becomes initially stiffer, but a limit load instability develops. The value of the limit pressure increases as the rigidity of the outer shell increases. For any given imperfection there is an upper bound to the limit pressure. This is achieved when the outer shell is rigid. Thus, in the presence of initial imperfections the liner will buckle even if the cavity is rigid, as reported in Kyriakides and Youn (1984). Following the maximum, the pressure drops sharply to a pressure plateau at which most of the deformation takes place. These general features are also exhibited in the P- Au responses calculated for load type I1 shown in Fig. 71(b). However, for this loading, a limit load instability was found to occur even for relatively small bending rigidity ratios. The P-Au responses for doubly symmetric deformations of the same shell geometries and loads are shown in Fig. 72. They are seen to be qualitatively similar to those in Fig. 71. For this mode of collapse the limit pressure was found to be lower for smaller bending rigidity ratios, whereas the structure was found to collapse in the singly symmetric mode for higher bending rigidity ratios. The limit pressure is strongly dependent on the initial imperfection used. Details of this dependence can be found in the original reference. Figure 73 shows how the limit pressure ( P L ) varies with the thickness ratio ( t o / t , ) of the two shells for imperfections with amplitude A. = 0.075R for load type I. Results from both modes of buckling are included. The limit load of the singly symmetric mode of buckling becomes lower than that of the doubly symmetric mode for t,/t, larger than approximately 2.5. Thus for thickness ratios higher than this transitional value, the tube can be expected to collapse in the mode with one plane of symmetry. Similar results, reported by Li (1991) for load type 11 show the transitional value of t,/t, to be approximately 2.0. These results stop when the opposite walls of the collapsing shells touch for the first time. Kyriakides and Youn (1984)continued the solution beyond this point. It was found that the contact has a stabilizing effect on the response.
0
0.25
0
0.75
0.50
0.25
-Auhe
(a)
0.50 (bl
0.75
--AVUo
FIG. 72. Pressure-volume responses for various shell bending rigidity ratios for doubly symmetric deformations:(a) load type I; (b) load type 11.
1.00.8 0.6 -
ii ,
0 0
;
A,=0.075 R Load Type I
2.0
4.0
6 0
-(t,/t,)
FIG.73. Limit pressure as a function of thickness ratio for the two modes of deformation.
162
Propagating Instabilities in Structures
163
Singly Symmetric Load Type I
A,=0.075R
I
0.4
k
Elastic-Plastic
-
0 0
0.2
0.4
0.6
0.8
3
FIG. 74. Comparison of pressure-crown displacement responses of linearly elastic and elastic-plastic shells.
The pressure required to continue deforming the shells beyond first contact rises sharply (i.e., the response becomes stable again under prescribed pressure), The same is true for all cases presented here. However, due to the steepness of this part of the response, for simplicity we opt to approximate it as rigid (i.e., as a vertical line in P-Au plots). b. Elastic-Plastic Shells In the majority of the shells tested in the experiments, plastic material effects played a significant role in the behavior observed. Material yielding and the additional loss of stiffness as plastic deformations grow cause an earlier onset of the limit load instability. In fact, the limit load occurs soon after first yield of the material. This is demonstrated for a special case in Fig. 74. The postlimit load pressure drops more sharply and to a lower pressure plateau than the corresponding elastic case. In addition, the collapsed cross sections have regions with much higher curvatures than in the elastic case (compare Fig. 66 and Fig. 7qa)). The main qualitative features of the updown-up response do however remain the same and will not be discussed further here (additional elastic-plastic results can be found in Kyriakides and Youn (1984) and Li (1991)). In summary, we observe that if the composite structure is assumed to collapse uniformly along its length, a limit load instability occurs for both the
Stelios Kyriakides
164
elastic and elastic-plastic cases. The limit load is very imperfection-sensitive and in addition depends on the relative thickness of the inner and outer shells and on the yield stress of the material. The limit load is followed by a sharp drop in pressure down to a relative pressure plateau at which most of the deformation takes place. The drop in pressure increases as the relative stiffness of the outer ring is increased. In the elastic case for load type I, the up-down-up response is no longer possible for rigidity ratios lower than approximately 1. Shells that buckle in the plastic range have a limit load instability for all thickness ratios of the two shells. Plastic effects lower the pressure of the whole response, but the up-down-up behavior is maintained. In view of these characteristics the following observations can be made. 1. The presence of the pressure maximum in the response indicates that, for long tubes like the ones of interest here, local collapse will be energetically preferable to the uniform collapse. Such a collapse will be initiated at a location along the length with the largest imperfections. 2. The up-down-up responses calculated indicate that such structures have three possible equilibria for pressure values P , .c P P , (see Fig. 75). Equilibria on O L and M N are stable, whereas those on L M are unstable under prescribed pressure. Thus, if a structure in a stable equilibhum at pressure P is given a large enough disturbance, it can experience snap buckling from a point like A to a point like C. In a long enough tube it is possible for part of the structure to be at the collapsed
-=
P
I
I.
OI”,
”c
Av
FIG.75. Pressure-volume response of elastic system and Maxwell construction.
Propagating Instabilities in Structures
165
equilibrium and for part of it to be at the prebuckling one corresponding to the same pressure. 2. Estimation of the Con$ned Propagation Pressure a. Linearly Elastic Shells For elastic shell materials the confined buckle propagation pressure can be evaluated exactly by the Maxwell construction, using the P - Au response from the uniform collapse analysis. We consider a buckle that has been initiated somewhere along a long tube and is propagating in a steady-state, quasi-static fashion at a pressure of P,. From Fig. 75, the external work done when the buckle propagates along a unit length of the tube is given by
~ , , ( A v C- AvA). Since the material is elastic, the change in internal work is strictly a function of the initial and final configurations of the cross section; i.e., states A and C. The change in internal work will be equal to the external work done; thus,
:*I
P ~ J A v ,- AvA)=
P(Av) dAu.
(4.2)
Expression (4.2) is satisfied when the two shaded areas in Fig. 75 are equal. This method was used (Kyriakides, 1986) to estimate P,, for the mylar shells tested in the experiments. The following expression was found by making the outer shell rigid: PPC
(4.3)
The mylar shell used had R = 0.91 in. (23.1 mm), t = 0.010in. (0.25 mm), E = 735 ksi (5.07Nmm-*) and v z 0.3. With these parameters (4.3) yields P,, z 0.5 psi (0.035 bar), which is 10% lower than the value measured. Expression (4.2) was used to calculate the confined propagation pressure as a function of the bending rigidity ratio of the two shells for the two types of loads considered (Li and Kyriakides, 1990). The calculated values of ppcare plotted against the bending rigidity ratios in Fig. 76. The confined propagation pressure yielded by the singly symmetric and the doubly symmetric analyses are included for each load type. The following observations can be made from these results: 1. The confined propagation pressure for the singly symmetric case is seen to be lower for nearly all values of variables considered. Thus it can be
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07
07 06
f-Doubly Symmetric
t
05
O4
Singly Symmetric
4
€1
I
05
04
Singly Symmetric
03
03
02
06
$(Ef
Load I
Load II
02
01
01
0
0
FIG.76. Confined propagation pressure for two modes of collapse as a function of bending rigidity ratio: (a) load type I; (b) load type 11.
concluded that this is the prevalent mode of deformation for the phenomenon. The propagation pressure for the doubly symmetric mode of deformation is lower only for very small values of bending rigidity ratios for load type I. 2. In contrast to the propagation pressure, the limit load was found to have transition thickness ratios ( t a / t I )of approximately 2.5 and 2.0 for load types I and 11, respectively. In view of the results in Fig. 76, it can be deduced that for low thickness ratios the collapse may initially have two planes of symmetry, but somewhere in the postlimit load regime the singly symmetric mode of deformation will be preferred. Thus, the collapse process can start with one mode of deformation but transfer to another as the deformations grow. A similar behavior will also be demonstrated in Section V in the case of arches with intermediate values of the shallowness parameter 1. 3. The normalized, confined propagation pressure for the singly symmetric mode of collapse is relatively insensitive to the bending rigidity ratio of the two shells. 4. If the rigidity of the outer shell is relatively small, the pressure-volume response of the structure for load type I was found to be monotonically increasing. In such cases, the collapse process can occur in a stable manner and propagating buckles are no longer of concern. For load type I1 this change does not occur.
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5. The values of P , for the two loads differ by only a small amount, with
load type II yielding slightly lower values. 6. The amplitude of the initial geometric imperfection used in the analysis strongly influences the limit pressure exhibited in the response, but was found to have only a small effect on the calculated confined propagation pressure. 7. Once P , is evaluated from (4.2), the profile of the buckle in the steadysteady condition can be evaluated by using a “shooting” approach like the one used to evaluate the profiles of the bulges in Section 11. This was not attempted here, but the expected profiles for the two modes of deformation considered are shown in the assemblages of twodimensional collapse configurations in Fig. 77. b. Elastic- Plastic Shells
A11 metal tubes tested exhibited plastic, irreversible deformations. Plastic material behavior is loading path dependent. Loading path independence is a necessary condition for the application of the energy balance argument developed previously. However, for certain loading paths (proportional or “nearly” proportional stress paths), the assumption of path independence of
FIG.77. Schematics of profiles of confined buckle propagation: (a)singly symmetric case; (b) doubly symmetric case.
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Stelios Kyriakides
material behavior is correct even for elastic-plastic materials. The energy balance was shown in Section 111 to yield estimates of the propagation pressure of some unconfined shells with engineering accuracy. This was due to the relatively limited role of the axial stresses in the buckle propagation process and the limited unloading and reverse loading experienced in those cases. Unfortunately, that success was not repeated in the case of the confined propagating buckle (Kyriakides, 1986). This difference is due to the fact that large sections of the circumference undergo unloading and reverse bending after being bent into the plastic range. Examination of the profile of confined buckle propagation (Figs. 64 to 66) clearly shows that the buckled section also undergoes rather severe bending and stretching in the axial direction. These deformation patterns indicate that the stress paths, in a good part of the shell, are highly nonproportional. As a result, the energy balance argument fails (see Calladine, 1986, for an alternative approach).
D. DISCUSSION AND CONCLUSIONS It has been demonstrated that thin-walled shells used to line long, stiff cylinders or cylindrical cavities and loaded by hydrostatic pressure applied at the interface of the contacting cylinders can develop propagating instabilities. It has been shown that in the presence of small initial imperfections such composite structures have limit load instabilities. The limit loads are very imperfection-sensitive and occur for linearly elastic and elastic-plastic material behavior. If the thickness ratio of the two cylinders is higher than approximately 1, the limit load is followed by a precipitous drop in pressure. As a result of this postbuckling behavior, collapse of a long liner is local to a section a few shell diameters long. The local collapse, which has a U-shaped cross section, grows until the walls of the liner touch. Contact of the collapsed walls has a stiffening effect and arrests the development of localization. However, as a consequence of the local collapse, the geometric integrity of sections of the shell adjacent to it is compromised. The walls of the liner partially detach from the outer shell and provide the weak link for the collapse to propagate. Given a high enough pressure and sufficient fluid flow, the collapse can spread to the rest of the shell. The propagation of the collapse will continue as long as the applied pressure is higher than the confined propagation pressure of the composite structure. As in the other problems in this chapter, the way initial collapse is initiated does not
Propagating Instabilities in Structures
169
influence the propagation of the instability. Thus, local dents and other imperfections that locally reduce the collapse pressure of the structure can be viewed as factors that facilitate the initiation of the propagating instability. As in the case of the other problems in this chapter, the potential of developing localized collapse and a propagating instability can be identified by analyzing the uniform collapse of a representative strip of the long composite shell. This yields an up-down-up pressure change in volume response characteristic of this class of problems, which is sufficient to confirm that, given a long enough structure, both localization and propagation types of instabilities are possible. The lowest pressure that will sustain an initiated buckle in propagation is the confined propagation pressure. In the case of elastic shells, this has been shown to be obtained exactly from the Maxwell construction applied to the pressure-change in volume response from a uniform collapse analysis. Unlike the problem in Section 111, this technique did not yield good predictions of P , for shells that undergo elastic-plastic deformations. It is worth noting that confined propagating buckles can also occur for the same problem geometry with different dead loading. For example, confined rings with a point load (Bottega, 1988), a uniform inertial gravity load (Pian and Bucciarelli, 1967) and a thermal load (El-Bayoumy, 1972) have been shown to have a limit load instability similar to the one established here for hydrostatic pressure. If we replace the rings with long shells with axially uniform dead loading then collapse can be expected to localize and subsequently propagate provided the confined propagation “load” of each case is exceeded. V. Buckle Propagation in Long, Shallow Panels
A. BUCKLING OF SHALLOW ARCHES Laterally loaded shallow arches and shallow conical caps represent classic examples of structures that exhibit “snap-through” buckling or “oil canning.” For some load and deformation regimes these structures, given a finite disturbance, can “snap” to an alternate equilibrium configuration. Their prebuckling response is strongly influenced by membrane deformations that are induced by the lateral loading. The membrane deformations cause a progressive softening of the response of the structure that eventually results in the development of a limit load instability. Under “displacement” controlled
170
Stelios Kyriakides
loading, the deformation continues to grow at a decreasing load, and in the process the curvature of part of the structure reverses sign. Eventually, further increase in deformation requires membrane stretching. This has a stiffening effect and leads to an upturn in the required load. Beyond the load minimum (local), the stiffness of the structure increases as membrane stretching becomes the more dominant part of the induced deformation. Thus, these structures are also characterized by an up-down-up load deformation response that is similar to the response of the other structures discussed in this chapter. A characteristic example of this group of structures is a shallow, clamped arch under uniform external pressure like the circular arch shown in Fig. 78. The pressure-volume response of this case, for deformations symmetric about mid-span, is shown schematically in Fig. 79(a). Early workers in elastic buckling of shallow arches include Timoshenko (1933, Bienzeno (1938), Marguerre (1938), Fung and Kaplan (1952), Hoff and Bruce (1955), Gjelsvik and Bodner (1962) and others. The particular problem of Fig. 78 was very clearly analyzed by Schreyer and Masur (1966) and further explored by Schreyer (1972) and others. The shallowness of the arches they considered allowed them to adopt first-order nonlinear kinematics (moderate rotations) and an approximate expression for the change of volume under the arch in their formulation. This enabled them to obtain a closed form solution of the problem. Cheung and Babcock (1970) confirmed the validity of these assumptions (at least up to the first instability) in experiments involving the related problem of a point-loaded arch. As in other arches (see Fung and
FIG.78. Clamped, circular, shallow arch under uniform pressure.
171
Propagating Instabilities in Structures
P
I
-
Av
(bl
FIG. 79. (a) Pressure-volume response of an arch that buckles symmetrically. (b) Pressurevolume response of arch in which unsymmetric buckling occurs for part of the postbuckling history.
Kaplan, 1952), the behavior is strongly influenced by the shallowness of the arch in this case represented by
R t
;1=U2-
where 2a is the arch angle, R its radius and t its wall thickness. As expected, for very shallow arches (A < 2.85), the arch effect on the response is small. The load increases monotonically with deformation and the structure has no instability. For higher values of A, the pressure-volume response is as shown in Fig. 79(a). Thus, if the pressure is prescribed, the structure, on reaching the limit load (P,.), “snaps” from equilibrium point b to d. If the structure is elastic, unloading from d to pressure P , takes place along dc. On reaching P,,
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Stelios Kyriakides
the structure “snaps” to point a on the prebuckling branch. (This snapping back and forth is known as the oil-can efect.) For even higher values of I, part of the symmetric response is unstable in that an unsymmetric buckling mode is preferred, for example, from point b to point c in Fig. 79(b). Thus, on reaching the bifurcation pressure ( P b ) the structure buckles into an unsymmetric mode. If the pressure is prescribed, then the arch will snap from b to e. Along the way, the unsymmetric deformation progressively changes to the symmetric one, represented by equilibrium point e. Again, if the structure is unloaded from e to pressure P , path ed is followed. At d, the arch snaps back to point a on the prebuckling equilibrium branch. Thus, for prescribed pressure, the main difference between this and the previous case is that the maximum pressure reached is Pb rather than P,. In some cases, an observer of the buckling process may, in fact, miss that the initial cause of buckling is due to an unsymmetric mode because all the stable equilibrium configurations observed, i.e., along ob and df, are symmetric. Similar behavior to what has just been discussed can be obtained for shallow arches of other geometries, boundary conditions and loads (e.g., P = P(0)).(We remind the reader that “high” arches, i.e., arches where the rise is on the order of the span, buckle in an inextensional, unsymmetric bending mode not of interest here; e.g., see Timoshenko and Gere, 1961.) B . PROPAGATING
BUCKLES IN LONG,SHALLOW
PANELS
The preceding discussion clearly demonstrates that shallow arches have one of the characteristics necessary for a propagating instability to develop; that is, they have the required local load-deformation response. A second requirement for such an instability to develop is on the structure size. Clearly, if instead of the narrow, curved, beamlike arch analyzed in the references mentioned, we consider a long, uniform panel with an arch cross section, this requirement will have been met. Any cross section along the length of the panel still has, in essence, the same required load-deformation response (a plane strain condition in the axial direction is more appropriate in this case). However, following initial buckling, either at P , or Pb, one can reasonably expect that the deformation will cease to be uniform along the length of the panel. The induced buckling deformation will tend to remain local. In fact, the exact sequence of events that follow initial buckling will depend on the nature of the applied loading. Looking at the problem from a different perspective, in the case of a long
Propagating Instabilities in Structures
173
panel, buckling will first occur in the section of the structure with the largest imperfections. If the load is allowed to drop after first buckling (as would occur in volume-controlled pressurization) the buckle can be expected to remain local. However, this local buckle can spread (propagate) at a pressure much lower than the one at which the instability was initiated. If the pressure is prescribed, the local buckle will spread dynamically and quickly collapse the whole structure (i.e., to equilibrium point d in Fig. 79(a)), very much like a propagating buckle in a long pipeline discussed in Section 111. In what follows, we will demonstrate the initiation of a buckle in a long panel and its steady-state quasi-static propagation. Results from a Maxwell-type energy balance will then be used to calculate the propagation pressure of such buckles. 1. Experimental Demonstration of Initiation and Propagation of Buckles
A simple test apparatus, shown schematically in Fig. 80(a), was developed to demonstrate the initiation and steady-state propagation of such buckles. A long, shallow panel was formed by bending a strip of thin mylar into the cross section shown in Fig. 8qb). The ends of the mylar were positioned into
FIG. 80. (a) Experimental apparatus used to demonstrate the initiation and propagation of buckles in long panels. (b) Cross section of experimental apparatus.
174
Stelios Kyriakides
oblique slits cut in two long, metal end support beams, as shown in the figure. The mylar was bonded into the slits, thus providing, in essence, clamped end conditions. The two end beams were mounted onto a U-shaped aluminum support structure that was made by removing the top of a 4in. diameter aluminum tube. The arrangement is such that the mylar has a circular arch shape with arch angle a = 0.443 and radius of R = 3.42in. (86.9 mm) (both could be varied by using slits with different orientation). Mylar strips of wall thickness of 0.015 and 0.020in. (0.38 and 0.51 mm) were used. The tubelike structure has a length of loft (3 m). Its ends are closed with flanges, and provisions were made for the inside to be hermetically sealed. To augment visualization of the buckling process, grid lines were drawn on the mylar panel before it was installed into the device. The dimensions of the arches were such that snap-through buckling occurred at external pressures only a small fraction of an atmosphere. Thus, external pressure loading was applied by evacuating the air inside the closed tube with a vacuum pump. The rate of evacuation was controlled manually by using a needle valve, shown in Fig. 8qa). The pressure was monitored with an electrical pressure transducer and recorded on a strip-chart recorder. The parameters of the problem were such (A = 44.7) that initial buckling was initiated by unsymmetric buckling (i.e., behavior as in Fig. 79(b)). In addition, the demonstration structures used had imperfections as well as residual stresses that influenced the buckling pressure. Initial buckling was always local and occurred in a dynamic fashion. The local buckle that developed was symmetric and extended over a length of approximately seven to eight times the arch span as shown in Fig. 81(a). Under the loading conditions used, the system was relatively compliant, and as a result, the snap-through buckling produced a significant pressure transient (it also affected the length of the local buckle produced). However, the local buckle could be “frozen” by interrupting the evacuation of air from the closed tube. In the experimentsconducted, the local buckles were usually close to the midspan of the long panel. Once the local buckle was fully developed, further propagation or spreading of the buckle occurred at a well-defined pressure that was significantly lower than the pressure required to initiate local collapse. Under these circumstances, the buckle propagated in both directions until it reached one of the ends, where it was arrested. The buckle continued propagating on the other side until the whole panel was collapsed. In experiments in which the main objective was the demonstration of the steady-state propagation of buckles in such panels, the structure was usually
Propagating Instabilities in Structures
175
FIG.81. (a) Local symmetric buckle in long panel. (b) Profile of propagating buckle in a long panel (quasi-staticpropagation).
loaded to a pressure just below the critical one. A local buckle was then initiated close to one of the end flanges by applying a point load at the center of the arch. The buckle was then propagated in a quasi-static static fashion by controlling the rate at which the closed system was evacuated using the needle valve. The propagation progressed at a steady state, and a well-defined propagation pressure could be recorded. Figure 8 l(b) shows the transition between the buckled and unbuckled sections during such an experiment. Once the buckle reached the end of the panel, further deformation required a significant increase of the applied pressure.
Stelios Kyriakides
176
2. Analysis
a. lbo-Dimensional Response The pressure at which buckles initiated in such long panels will propagate quasi-staticallycan again be evaluated from the solution of uniform collapse of the panel by using the Maxwell construction. The closed form solution of Schreyer and Masur (1966) can be used for this purpose if the modulus E is replaced by E' = E/(1 - v2) (v = Poisson's ratio). Figure 82 shows a set of pressure-volume change responses calculated using the solution of Schreyer and Masur. The appropriate nondimensional pressure (4and volume change (AO) are given by
-
P
P = 12-(-) E'
a 2 II
R 3
(T)
1 Av
and A6 =--.
a3 R2
Using (5.2) the solution becomes strictly a function of the geometric parameter I given in (5.1). As the original paper states, buckling can occur if I > 2.85. Results representing the symmetrical deformation of panels with I values of 4.4, 5.2, 6.9, 12.0 and 27.4 are shown in Fig. 82. A linearized bifuraction analysis presented in the same reference yields the result that for 1 > 5.02 unsymmetrical buckling becomes possible for part of the response. If 5.02 < I < 5.74, the two bifurcation points occur on the descending part of P
t
-Symmetric
Deformation Stable
_ _ _ _ _ Symmetric Deformation Unstable
30$\ I \
Points
20 4.4 5.2
Propagation Pressure
6.9 12.0 27 4
O! 0
0.2
0.4
0.6
0.8
-At
FIG.82. Calculated pressure-volume responses for various values of 1 (solution by Schreyer and Masur, 1966).
Propagating Instabilities in Structures
177
the symmetric response as seen in the figure for 1= 5.2. For 1> 5.74, the first bifurcation point occurs on the ascending part of the response and the second after the limit load (see A = 6.9, 12.0, 27.4). Dashed lines in the figure represent the parts of the responses in which the symmetric mode of deformation is unstable. In all cases shown, the ascending parts of the calculated symmetric responses following the limit loads were found to be stable. For completeness, the problem was also solved numerically (see Kyriakides and Arseculeratne, 1993) using kinematics that allow arbitrarily large rotations (Reissner, 1972). For smaller values of A (1 5), the solution of Schreyer and Masur was found to be very accurate. As expected, as 1 was increased, the solution was found to progressively deviate from the numerical results. The main difference was found to occur in the part of the response that follows the local minimum (i.e., for larger displacements), where the pressure was underpredicted. The difference in the predicted limit pressures was found to remain small for the range of parameters examined (1< 100). The difference in the two solutions is well illustrated in Fig. 83(a), which includes plots of the P- Av responses predicted by the two methods. These
-
Large Rotation
20
\
10
--Propogation Pressure
1 \
0
I 7
0
02
06
04
08
-
Llu
(bl
FIG. 83. (a) Comparison of P-Au responses from two solution procedures (2 = 44.7). (b) Sequence of calculated collapse configurations(1= 44.7) (vertical coordinate amplified by 1.5).
178
Stelios K yriakides
results represent one of the panels used in the experiments for which R/t = 228, a = 0.443 and R = 44.7 (in the results from the large rotation kinematics, 1 is no longer a unique parameter; R/t and a must be varied independently). It is interesting to observe that the difference in the two solutions up to the local minimum is rather small. Figure 83(b) shows a sequence of collapse configurations calculated by the numerical solution procedure for this case. b. Estimation of the Propagation Pressure The propagation pressure can be evaluated again exactly by the Maxwell construction, using the P-Au responses calculated earlier. We consider a buckle that has been initiated in a long panel. The buckle has developed to such an extent that it can propagate under steady-state conditions in a quasistatic fashion (see Fig. 81(b)) at a pressure of P,. Referring to Fig. 84, the external work done when the buckle propagates along a unit length of the panel is given by
Fp(Auc
- Au,,).
Since the material is elastic, the change in internal work is strictly a function of the initial and final configurations of the cross section, i.e. states A and C. The change in internal work will be equal to the external work done, thus Pp(Auc - A u ~=)
(5.3.a)
PL
PP
0 FIG.84. Maxwell constructionfor a long, shallow panel under unlform pressure.
Propagating Instabilities in Structures
179
Expression (5.3a) is satisfied when the two shaded areas in Fig. 84 are equal. An equivalent expression is Pp(AuC-AvA)= V(Av,)- U(Av,)
(5.3.b)
where V ( is the strain energy of a given equilibrium state.4 For the closedform solution of Schreyer and Masur (1966), it can be shown that (5.3b) is satisfied when Fp = 1 for all 1.(In their analysis this is represented by the limiting case p = n. Equation (5.3b) can be shown to imply p p = 1 by using their equations (24) and (26)). Thus, from (5.2) a )
pP-
(5.4)
As demonstrated in Fig. 82, depending on the value of A, part of the response OABCD in Fig. 84 can be unstable. However, since (5.3) depends strictly on equilibria A and C and not on the path followed to get from A to C, the only requirement for its validity is that A and C be stable. (A more practical requirement is that OA and CD be stable.) This indeed happens to be < 100). satisfied for reasonable values of In the case of the numerical solution to the problem, p p was evaluated numerically by using the calculated P-Av response and (5.3a). The values of p p obtained from the two solutions for the case shown in Fig. 83 were Fp = 1 (0.287psi) and 1.034 (0.297psi), which are in good agreement with the experimental value measured. In general, the values of p pestimated by the two solutions were found to be in close agreement for smaller values of 1, but progressively diverged for larger values of A, as shown in Fig. 85 ( R / t = 100). The limit and bifurcation pressures predicted are also plotted in the same figure for comparison. Some of the transitional values of 1,as calculated by Schreyer and Masur (1966), are also included. For values of 1 larger than approximately 4, p p is seen to be significantly lower than the critical buckling pressure of the structure. This is a characteristic common to structures that exhibit propagating instabilities. For arches with low values of 1 (-3), pL and p p converge and the problem of buckle propagation becomes irrelevant. It is of historical interest to point out that Fp as defined by (5.3b) corresponds to the socalled energy buckling load of Friedricks (1941) and Tsien (1942). Of course the physical implication of the two critical pressures is different.
Stelios Kyriakides
180
P
3.5
2.5 -
------
I
I
I 1
2
, 4
I
I
6
,
l
,
l
810
I
20
I
I
40
I
,
,
,
,
60 80100 -A
FIG.85. Critical pressures of clamped panel as a function of 2..
C. DISCUSSION AND CONCLUSIONS It has been demonstrated that, in addition to classical buckling, long, shallow panels can also develop propagating buckles. Such buckles are usually initiated from local imperfections. Once initiated, these buckles will propagate as long as the pressure is higher than the propagation pressure. This new critical pressure is given approximately by (5.4). A more accurate estimate can be obtained using the more exact numerical solution of the problem presented in Kyriakides and Arseculeratne (1992). For arches with realistic parameters, P p was shown to be significantly lower than the classic critical buckling pressure (i.e., P, or Pb). A buckle initiated under constant pressure loading that is higher than P , will propagate dynamically down the length of the panel. Quite clearly, one way of avoiding the possible propagation of such a buckle in an actual structure, is not to allow the maximum pressure to exceed P,. Another way might be to add periodic reinforcement to the structure (e.g., bulkheads) that could arrest the spread of the collapse. The results presented are based on strictly elastic material behavior at least up to deformations corresponding to equilibrium state C in Fig. 84. As was the case in the problem discussed in Section IV, inelastic material behavior
Propagating Instabilities in Structures
181
FIG.86. Propagation of a buckle in a long panel under line load.
during any part of this response (i.e., OABC) will make the Maxwell construction invalid and lower the actual value of P , (i.e., the problem will persist and become more severe). The propagation pressure can then be evaluated by analyses similar to those presented in Section 111. The objective of this study was to demonstrate the phenomenon using a practical example. Quite clearly, many other panel geometries, boundary conditions and loads are of practical importance. The problem of buckle propagation will be relevant as long as the load-deformation response of a section of the structure is characterized by the up-down-up behavior shown in Fig. 84, the load is prescribed (e.g., dead loading) and the structure is long. For example, Fig. 86 schematically shows a panel loaded by a line load (dead weight) that can also develop a propagating buckle. The propagation load ( Q p )of this case can again be evaluated from the corresponding solution also in Schreyer and Masur (1966) in a similar fashion as described previously. VI. Summary and Discussion
This chapter has been concerned with the subject of propagating instabilities as they affect structures of larger size. Emphasis has been given to the
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182
conditions under which such instabilities are initiated and the conditions required for the instabilities to propagate. The characteristics of the phenomenon were illustrated through four examples that have been shown, in experiment and analysis, to have the potential of developing propagating instabilities. Although the details of the mechanical behavior of each example considered are influenced by significantly different factors, their global mechanical behavior is quite similar. One common feature of the global mechanical behavior of such problems is the global “load-deformation” response in a “deformation”-controlled experiment (deformation here is meant to be the work conjugate of the applied “load”). In all cases, the response has been shown to have the characteristics shown schematically in Fig. 87. Initially (do-dl), the structure deforms uniformly at an increasing load. Uniform deformation is limited by a limit load instability (LI).(The fact that in some cases this occurs naturally and in others it is induced by bifurcation buckling is of secondary importance.) Following the limit load, and because the effort required to deform the structure further is decreasing, localized deformation modes are energetically preferred. Thus, in the case of the inflated elastic tube considered in Section 11, a bulge develops. In the case of the three shell problems considered in the other sections, localized collapse occurs. The details of localization can be quite complex where for larger structures this part of the response cannot be controlled by prescribing global load or deformation variables. In each case, and for different reasons, the growth of the localized instability is eventually arrested ( d J . In the case of the latex rubber tube, this is due to an increase in the stiffness of the material at high stretch ratios (as a result of alignment of the long molecular chains of this polymeric material). Uniform DeTormation Localization
-0 0
3
-... .-. ...
I I
I I
l
l
I I L
do
d, d, d,
d, Deformation
FIG.87. Global load-deformation response characteristic of this class of problems.
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183
In the case of the collapsing cylindrical shells considered in Sections 111 and IV, localized collapse is limited due to contact of the collapsing walls of the shells. In the case of the panel of Section V, this is due to the development of tensile membrane forces in the buckled section of the panel. The arrest of localization usually corresponds to a local minimum (at d 2 ) in the recorded load. Once localization is halted, the instability can spread to neighboring sections of the structure. In a “deformation”-controlled experiment, following a load transient, the spread or propagation of the instability can occur in a steady-state, quasi-static fashion (d3-d4). The load required to spread the instability under these conditions is the propagation load of the structure ( L J . In the problems considered, the propagation load was significantly lower than the load required to initiate the instability in a geometrically intact structure. The reason the instability propagates at a lower load than L, can be described as follows. The limit load instability characteristic of such structures shows that they have the propensity to collapse. The limit load is usually strongly influenced by geometric imperfections. We now remember that the sequence of events leading to a propagating instability is triggered by localization. Localization compromises the geometric integrity of adjoining sections of the structure and lowers their local resistance to deformation (i.e., lowers their local limit load). Thus, once the geometry of the structure is altered locally by an instability, the instability can be expected to propagate at a load that is lower than that required to initiate it in a geometrically intact structure. This just confirms the intuitive notion that such structures carry their highest load when they are in their original, undisturbed, geometric configuration. A characteristic of the propagation stage of this process is that highly deformed (bulged, collapsed) regimes of the structure and relatively undeformed regimes coexist at the same load. In a structure of finite size, the instability can eventually spread over the whole structure (d, in Fig. 87). Once this occurs, further deformation is usually uniform over the whole structure and requires an increase in load. Thus, the response of the structure becomes stable again. The potential of a structure to develop a propagating instability can usually be recognized with a relatively small effort. For the structures considered, this involved analyses in which the deformation was assumed to be uniform over the domain (length). This can be viewed as the “local” response of the structure. In all cases, the “local” response had the characteristic up-down-up shape introduced in Fig. 1.
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The presence of the load maximum in the response indicates that, if the structure were sufficiently large, localized modes of deformation would be preferred after the limit load. The second stable branch (ascending) on the right indicates the existence of alternate equilibrium points to those on the first ascending branch. Thus, for a range of loads given a large enough disturbance, the structures can jump from equilibria on one branch to equilibria on the other. It has been demonstrated experimentally that, for structures undergoing elastic deformations, the propagation pressures can be obtained from the response from such two-dimensional analyses by implementing the equal area rule of Maxwell. In the three shell problems considered, inelastic deformations can play a role. In the case of the liner shell and the long panel loaded by pressure, the up-down-up response is a characteristic of the geometry. Inelastic deformations simply aggravate the situation (i.e., lower the limit and propagation loads). In the case of the long shell under external pressure, elasticplastic material behavior was shown to be an essential component for developing the up-down-up response. In spite of the elastic-plastic deformations, the up-down-up response is still a necessary characteristic, which indicates that the structure has the potential of developing a propagating instability. However, the path dependence of the material invalidates use of the Maxwell construction to predict Lp. In such cases, more complicated analyses that take full account of the path dependent nature of the material behavior can be used to calculate the propagation load. In all problems of this class, an instability initiated in a structure under a prescribed load higher than L, will propagate dynamically. The potential of dynamic propagation can easily be seen by comparing the energy available due to the jump in deformation in the structure to the energy required for this deformation (proportional to (A,-A,) in Fig. 88). Some experimental results on the dynamics of propagating buckles in pipelines can be found in Kyriakides and Babcock (1979) and Kyriakides (1980). Some recent numerical results on the subject have been presented by Song and Tassoulas (1991). The general subject of dynamic behavior of such instabilities remains however relatively unexplored. In concluding this chapter it is worth noting that the global mechanical characteristics of this class of structural instabilities more than resemble characteristics of instabilities seen at the material level. A good example of this relationship is the behavior of shape memory metals, at least in some temperature regimes (see,for example, Abeyaratne and Knowles, 1991).What
Propagating Instabilities in Structures
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Maxwell Load
, I
I
I
d,
FIG.88. Local load-deformation response: (A,-A energy if instability is initiated at L > L,.
dz
Deformation
represents energy available for kinetic
can be experimentally shown to relate this class of structural behavior to such materials is the global response shown in Fig. 87. Whereas in the present context this global behavior can be directly and unequivocally related to a local up-down-up response, in shape memory metals at this time this is a supposition requiring proof.
Acknowledgments The work reported was supported in part by the National Science Foundation through the PYI award MSM-835207 (1984-1990) and by the Office of Naval Research through grant N-00014-91-J-1103. The author wishes to thank K. M. Liechti for a number of comments that improved the manuscript. The assistance of T. Valdez in producing the line drawings and D. P. Francis in proofreading is also acknowledged with thanks.
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Hodge, P. G . (1961). The Mises yield condition for rotationally symmetric shells. Quurt. Applied Math. 18, 305-31 1. Hoff, N. J., and Bruce, V. G. (1954). Dynamic analysis of the buckling of laterally loaded flat arches. J. Math. & Physics 32, 276-288. Hutchinson, J. W., and Neale, K. W. (1983). Neck propagation. J. Mech. Phys. Solids 31, 405426.
Jensen, H. M. (1988). Collapse of hydrostatically loaded cylindrical shells. Intl. J. Solids & Structures 24, 51 -64. Johns, T. G., Mesloh, R. E.,and Sorenson, J. E. (1976). Propagating buckle arrestors for offshore pipelines. In: Proc. Ofshore Technology Confwence, OTC 2680, 3, 721-730. Ju, G. T., and Kyriakides, S. (1991). Bifurcation buckling versus limit load instabilities of elasticplastic tubes under bending and external pressure. ASME J . Ofshore Mechanics and Arctic Engineering 113, 43-52. Kamalarasa, S., and Calladine, C. R. (1988). Buckle propagation in submarine pipelines. Intl J . MechunicUl Sci. 30,2 17-228. Kyriakides, S . (1980). On the propagating buckle and its arrest. Ph.D. Dissertation, California Institute of Technology, Pasadena, CA. Kyriakides, S. (1986). Propagating buckles in long confined cylindrical shells. Intl. J. Solids & Structures 22, 1579- 1597. Kyriakides, S., and Arikan, E. (1983). Postbuckling behavior of inelastic inextensional rings under external pressure. ASME J. of Applied Mechanics 105, 537-543. Kyriakides, S., and Arseculeratne, R. (1993). Propagating instabilities in long shallow panels. ASCE J . Eng. Mech. 119, 570-583. Kyriakides, S., and Babcock, C. D. (1979). On the dynamics and the arrest of the propagating buckle in offshore pipelines. In: Proc. Ofshore Technology Conference, OTC 3479,2, 10351045.
Kyriakides, S. and Babcock, C. D. (1980). On the slip-on buckle arrestor for offshore pipelines. ASME J . Pressure Vessel Technology 102, 188-193. Kyriakides, S., and Babcock, C. D. (1981a). Experimental determination of the propagation pressure of circular pipes. ASME J . Pressure Vessel Technology 103, 328-336. Kyriakides, S., and Babcock, C. D. (1981b). Large deflection collapse analysis of an inelastic inextensional ring under external pressure. Intl. J. Solids & Stntctures 17, 981 -993. Kyriakides, S., and Babcock, C. D. (1982). “The Spiral Arrestor”-a new buckle arrestor design for offshore pipelines. ASME J . Energy Resources Technology 104, 73-77. Kyriakides, S., and Chang, Y.-C. (1990). On the inflation of a long elastic tube in the presence of axial load. f n t l . J . Solids & Strucrures 26, Babcock Memorial Volume, 328-336. Kyriakides, S., and Chang, Y.-C. (1991). The initiation and propagation of a localized instability in an inflated elastic tube. Intl. J . Solids & Structures 27, 1085- 1111. Kyriakides, S., and Chang, Y.-C. (1992). On the effect of axial tension on the propagation pressure of long cylindrical shells. Intl. J . Mechanical Sci. 34, 3-15. Kyriakides, S., Elyada, D., and Babcock, C. D. (1984). Initiation of propagating buckles from local pipeline damages. ASME J . Energy Resources Technology 104,79-87. Kyriakides, S., and Yeh, M.-K. (1985). Factors all‘ecting pipe collapse. Project Report to the University of Texas at Austin American Gas Association (PRC) for project PR-106-404, EMRL Rep. No. 85/1. Kyriakides, S., Yeh, M.-K., and Roach, D. (1984). On the determination of the propagation pressure of long circular tubes. ASME J . Pressure Vessel Technology 106, 150-159. Kyriakides, S., and Youn, S.-K. (1984). On the collapse of circular confined rings under external pressure. Inrl. J. Solids & Structures 20, 699-713. Langner, C. G., and Ayers, R. R. (1985). The feasibility of laying pipelines in deep waters. In: Proc. Ofshore Mechanics and Arctic Engineering Con/: ASME, Dallas, 1, 478-489. Levy, M. (1984). Memoire sur un nouveau cas integrable du probleme de I’elastique et rune de ses applications. Journal de Mathematiques Pures et Appliqukes 10 (3), 5-42.
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Li, F. S. (1991). On the stability of confined shells under external pressure. Ph.D. dissertation, University of Texas at Austin. Li, F. S., and Kyriakides, S. (1990). On the propagation pressure of buckles in cylindrical confined shells. ASME J. Applied Mechanics 112, 1091 - 1094. Li, F. S., and Kyriakides, S. (1991). On the response and stability of two concentric, contacting rings under external pressure. Intl J. Solids & Structures 27, 1 - 14. Lochridge, J. C., and Gibson, T. L. (1973). Method of arresting the propagation of a buckle in a pipeline. United States Patent 3,747,356, July 24, 1973. Marguerre, K. (1938). Uber die Anwendung der energetischen Methode auf stabilitatsprobleme. DVL 252-262; also N A C A T M 1138, 1947. McCaig, 1. W.,and Folberth, P. J. (1962). The buckling resistance of steel liners for circular pressure tunnels. Water Power 14, 272-278. McGhie, R. D., and Brush, D. 0.(1971). Deformation of an inertia-loaded thin ring in a cavity with initial clearance. Intl. J. Solids & Structures 7, 1539- 1553. Mesloh, R. E., Sorenson, J. E., and Atterburry, T. J. (1973). Buckling and offshore pipelines. Gas Magazine 7 (July), 40-43. Mesloh, R., Johns, T. E., and Sorenson, J. E. (1976). The propagating buckle. In: Proc. BOSS '76 1, 787-797. Miyasaka, A., Ogawa, H., and Mimaki, T. (1991). Consideration for liner collapse risk of offshore bimetallic pipelines manufactured by thermohydraulic process. In: Proc. Corrosion 91, NACE Annual Conference, paper 9. Montel, R. (1960). A semi-empirical formula for determining the limiting external pressure for the collapse of smooth metal pipes embedded in concrete. LA Houille Blanche No. 5 (Sept.Oct.), 560-569. Neale, K. W., and Tugcu, P. (1985). Analysis of necking and neck propagation in polymeric materials. J. Mech. Phys. Solids 33, 323-337. Ogden, R. W. (1972). Large deformation isotropic elasticity-on the correlation of theory and experiment for incompressible rubber-like solids. Proc. Roy Society London A326,565-584. Palmer, A. C. (1980). Private communication. Palmer, A. C., and Martin, J. H. (1975). Buckle propagation in submarine pipelines. Nature 254, 46-48. Pian, T. H.H., and Bucciarelli, L. L. (1967). Buckling of radially constrained circular ring under distributed loading. Intl. J . Solids & Structures 3, 715-730. Pipkin, A. C. (1968). Integration of an equation in the membrane theory. J. Appl. Math. & Phys. ( Z A M P ) 19, 818-819. Reissner, E. (1972). On one-dimensional finite strain beam theory: The plane problem. J. Applied Math. Phys. Z A M P 23, 795-804. Remseth, S. N., Holthe, K., Bergan, P. G., and Holand, I. (1978). Tube buckling analysis by the finite element method. In: Finite elements in nonlinear mechanics. TAPIR, Trondheim, Norway. Sanders, J. L. (1963). Nonlinear theories of thin shells. Quart. Appl. Moth. 21, 21-63. Schreyer, H. L. (1972). The effect of initial imperfections on the buckling load of shallow circular arches. ASME J. Applied Mechanics 94,445-450. Schreyer, H. L., and Masur, E. F. (1966). Buckling of shallow arches. ASCE J. Eng. Mech. Diu. 92, 1-19. Shield, R. T. (1971). On the stability of finitely deformed elastic membranes. Part I: Stability of a uniformly deformed plane membrane. J. Appl. Math. Physics ( Z A M P ) 22, 1016-1028. Shield, R. T. (1972). On the stability of finitely deformed elastic membranes. Part 11: Stability of inflated cylindrical and spherical membranes. J. Appl. Math. Phys. ( Z A M P ) 23, 16-34.
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ADVANCES IN APPLIED MECHANICS, VOLUME 30
Plastic Flow of Crystals JOHN L. BASSANI Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania Philadelphia, Pennsylvania
I. Introduction
........,...... ................._................_........._...... ......... ..
11. Preliminaries.. .... . .. ..
........................................
........................... A. Kinematics. ... .......... .. ...._........... B. Elasticity.. .. .............................................. ............... ........... . ............................ C. Schmid Stresses.. ..... ... . .. . ....
192 194 194 196 196
111. Yield Behavior includi A. Yield Criteria ............... ........................... B. Yield Surfaces ... .... .... ... . .. . ...... . . ... . ... . . C. Restricted Slip . . .... .. ... . ... . . .. . .. . . .. . . ... . . .. . . . . . ... . .................
198 199 208 210
IV. Flow Behavior includi ...................... A. Slip-System Flow Rules ... . .... ... . .. . . . .. . ... . B. Uniqueness of Slips ........................................ C. An Example for L 1 Intermetallic Compounds . . ... . .. . . . . ... ... . ... . ... ... .. ....
21 I 21 1 214 215
........................................ V. Hardening Behavior.. .. . . A. Experimental Observa ... ...................... B. Restrictions on h,, ............ ........... ....... C. Analytical Characterization of ha).. . ... . .. . . . .. . .. .. .... ... . ... ... ... . . ... . . ........................... D. Simulations under Uniaxial Stressing . . ... . ...
217 219 225 226 230
........................................ VI. Strain Localization.. . . . ... ........................... A. Shear Bifurcations . . ... ... .. . . .. . .... . ... .. B. Crystals Under ................. C. Planar Double D. A Three-Dimen ..................................... E. Yield Vertices a ............ ...........................
235 231 238 240 241 252
VII. Closure.. .. . .. .. . .... ... ... . ... . . ...... . .. . .... . ... . . ... . ... .... . ... ... . ....... . .... . . ... .
2 54
Acknowledgments . ... . ... ... .... . ... ... . .... ... . ... . . . ... .. . .. . . . ... ... . .. . .. ... . .. . ....
254
References...... .. . ........ ..................... .................... . ............,........
254
191 Copyright el1994 by Academic Prcss, Inc. All rights of reproduction in any iorm rescrved. ISBN 0-11-001030-0
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John L. Bassani
I. Introduction Plastic flow under multiple slip, particularly in metallic crystals, has been widely studied by materials scientists and applied mechanicians since the early 1900s. Many books and reviews, most of which are slanted toward the physical point of view, are available in the literature. Examples include Schmid and Boas (1935), Taylor (1938a), Cottrell(1953), Seeger (1957),Kelly and Nicholson (1971), Basinski and Basinski (1979), Asaro (1983), and Havner (1992). Therefore, no attempt will be made in this chapter to provide a comprehensive survey. Instead, recent studies by the author of multiple-slip interactions and hardening (Wu, Bassani, and Laird 1991; Bassani and Wu, 1991) and of non-Schmid effects (Qin and Bassani 1992a, 1992b) will be brought together within a time-independent theory, and their influence on strain localization explored. A rigorous finite-strain continuum theory of plastically deforming crystals has been developed during the last few decades; the works of Mandel(1965), Hill (1966), Rice (1971), Hill and Rice (1972), Asaro and Rice (1977) and Hill and Havner (1982) are noteworthy. This theory has been applied successfully to a wide range of problems, mostly for face-centered cubic (FCC) crystals displaying Schmid-type behavior, and many examples are referenced in Asaro (1983) and Havner (1992). Havner’s (1992) treatise is rather comprehensive on the basics of finite-strain crystalline mechanics although it includes only a small discussion of multislip hardening and no mention of non-Schmid effects or strain localization. Highly nonuniform flows in crystals at relatively small plastic strains, such as those involving coarse slip bands under monotonic uniaxial stressing preceding the formation of macroscopic shear bands, loop-patch dislocation structures in cyclic deformation preceding the formation of persistent slip bands, as well as microlocalizationsaround particles sometimes giving rise to debonding, commonly are observed. In fact, they may be regarded as the rule rather than the exception. The occurrence of small secondary slips in the early stages of deformation, which are known to be very important in work hardening, also are known to be important in triggering such nonuniform flows (see, for example, Basinski and Basinski, 1979; Kulhmann-Wilsdorf and Laird, 1980; Basinski and Basinski, 1992). These phenomena provide a significant motivation for the theoretical problems addressed in this chapter. With observations of coarse slip bands in the early stages of uniaxial
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tension or compression deformations in mind, Asaro and Rice (1977) analyzed shear band formation (a bifurcation analysis) under single-slip conditions. If the normality structure governs flow, so-called Schmid behavior, they found that bifurcations are precluded at positive values of the (instantaneous) slip-system hardening rates. Under truly single-slip deformations, non-Schmid effects (i.e., nonnormality) are necessary to trigger shear bifurcations at positive hardening rates (Asaro and Rice 1977; Qin and Bassani, 1992b).On the other hand, based on his own experiments and those of Price and Kelly (1964) on aluminum single crystals, which are thought to obey a normality flow rule, Asaro (1983) stated that material in these coarse slip bands generally hardens during flow localization. As noted previously, secondary slips may be the missing link-the analyses presented in Section VI suggest they are. Secondary slips, in fact, are the key to Wu, Bassani, and Laird’s (1992) reinterpretation of a large body of experimental data, as well as their latent hardening experiments on copper, and to Bassani and Wu’s (1992) multipleslip hardening theory, which is expounded in Section V. The implications of this new theory on the development of coarse slip bands and macroscopic shear bands is analyzed in Section VI for both two- and three-dimensional deformations. Within the normality structure, secondary slips provide the additional kinematical degree of freedom required to trigger these localized bands of deformation in accord with observations. Nevertheless, non-Schmid effects are important in their own right. Qin and Bassani (1992a, 1992b) developed a multiple-slip theory that is motivated by well-known behaviors of various intermetallic compounds, particularly those of widely studied NiJAI, which has the so-termed L1, structure based on the FCC lattice. A theory of yielding and flow that can include a dependence of the critical resolved shear stress on the orientation of the axis of uniaxial stressing of FCC single crystals as well as a tension-compression asymmetry is outlined in Sections I11 and IV, respectively. Non-Schmid behaviors are predicted to significantly influence the formation of shear bands under both single- and multiple-slip conditions. A planar, nonsymmetric double-slip model is developed in Section VI that incorporates both the hardening and non-Schmid effects on shear localizations. Standard notation is used throughout: bold-faced letters denote tensors and Latin subscripts denote Cartesian components; Greek superscripts and subscripts denote slip systems and in some instances non-Schmid components.
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II. Preliminaries A. KINEMATICS
Plastic flow in the single crystal is assumed to result from continuous shearing or slip, rather than the motion of dislocations, on well-defined lattice planes in well-defined directions. The underlying lattice is assumed to be unaffected by the plastic slips. Following Rice (1971), Hill and Rice (1972), and Hill and Havner (1982), the multiplicative decomposition of the deformation gradient is assumed
where the plastic part FPcorresponds to the intermediate configuration (or isoclinicconfiguration in the language of Mandel, 1973)in which the lattice is undeformed. The elastic lattice deformation (stretch and rotation), therefore, results from F'. Furthermore, since FParises solely from slips, IFPI = 1 and IF1 = IF'I. From (2.1), the Eulerian velocity gradient in the current state is
L =D
+ W = F.F-1 = P . F e -1 +
Fe. FP.FP-1.Fe-1
(2.2)
where D and W are the symmetric and antisymmetric parts, respectively;i.e., the rate of stretching tensor and the spin tensor. As a further decomposition, L, D and W are written as the sum of elastic and plastic parts corresponding to the first and second terms, respectively, on the righthand side of (2.2). Rates of slip jf' on well-defined crystallographic planes with normals nuand directions ma in the current configuration (ma n" = 0) give rise to the plastic part of the velocity gradient:
where the sum is over all slip systems for a particular crystal class, e.g., 24 for FCC crystals where the positive and negative sense of slip are counted separately so that jf' 2 0. The two crystallographic tensors
+
d" = (ma @ nu + nu 0 ma)
(2.4a)
wa = f (ma@ nu - n" 0 ma)
(2.4b)
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enter prominently throughout, for example, (2.5a) (2.5b)
Let ii" and ma be the slip vectors in the undeformed or reference configuration. Since the lattice is assumed to be unaffected by slip, these are also the slip vectors in the intermediate (or isoclinic) configuration. For simplicity (particularly in leading to Equations 2.20 and 2.21 later), the slip vectors are taken to convect with the elastic (lattice) deformation:
With this choice the slip vectors not only rotate with the deformation, they also stretch. Furthermore, ma = (De
+ We).ma
pa = (-De
+ We)-na
(2.7a) (2.7b)
It is straightforward to adopt unit slip vectors in the current configuration as, for example, in Hill and Havner (1982). Here, since the slip vectors in the current configuration are not unit vectors, the slip rates jf' that enter (2.3) and (2.5) are the actual slip rates in the current (lattice) configuration times (Im"Ilnal)- Furthermore, from (2.2), (2.3) and (2.6), in the intermediate configuration,
where these slip vectors are unit vectors. Therefore, the v adopted here have the meaning of the in Hill and Havner (1982). In cases where the elastic strains are small (see Aravas and Aifantis (1991) for general kinematical consequences), which is typical of metals, these distinctions are inconsequential (the examples of shear localizations in copper single crystals discussed later involve elastic strains of roughly for coarse slip bands and l o p 2for macroscopic shear bands).
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John L. Bassani
B. ELASTICITY Hyperelastic behavior on the intermediate configuration is assumed. This can be expressed, for example, from a scalar potential function of the elastic Green strain, the gradient of which gives the work conjugate (elastic) second Piola-Kirchhoff stress (see Hill and Havner, 1982, Section 5). Alternatively, in terms of the lattice corotational rate of the Kirchhoff stress Ve t =t
- W'.t
+ T'We,
(2.9)
where t = IFelu and u is the Cauchy stress, and the elastic rate of stretching in (2.2) De = D - DP = t ( p . F e - 1 + F e - T . P T ) , (2.10) and using direct relationships with the former work-conjugate measures, hyperelasticity on the intermediate configuration can be expressed as V'
t = 9':D'
(2.11)
where 9' is the corresponding instantaneous elastic moduli with the usual symmetries Yfjkl = 9 T i k l = 9Fjl= kYgrij (see Havner, 1982, Sections 3.4 and 4.1 for a discussion of these moduli). In terms of the Jaumann-Zaremba derivative, which is corotational with the material spin, V
t =t -
w * t + 2 ' w,
(2.12)
and with We = W - Wp, D' = D - Dp, and (2.5a, b), the instantaneous elastic relation (2.1 1) can be expressed as V
t=5?e:D-Ci)"(9e:da+f3@)
(2.13)
a
where
v t-t=Ci)"B"
vt
(2.14a)
a
C . SCHMID STRESSES The Schmid stress f,or generalized Schmid stress in the terminology of Hill and Havner (1982),is defined such that T'Y is the rate of plastic working
Plastic Flow of Crystals
197
per unit reference volume due to slip on system a. Under multiple slip, the rate of working is (2.15) where tr denotes the trace and (2.5a) has been used. Consequently, the Schmid stress for slip system a is defined as t" =
t:d"
(2.16a)
or
ra = n".r.m" = IFelnu.a.m"
(2.16b)
Substitution of (2.6) into (2.16b) leads to the following (see Hill and Havner, 1982, Eq. 4.3): p
= i"
.z .
m u
(2.17a)
z = F'-'.t.F"
(2.17b)
where the nonsymmetric stress
ep
given in (2.8). is the plastic-work-rate conjugate to This plastic-work-rate conjugate Schmid stress is equal to the conventional Schmid stress, which is the shear component of the Cauchy stress on the slip system in the current configuration (deformed lattice), times IF"Ilm"Iln"l. For small elastic strains, these two Schmid stresses differ only by the order of the elastic strain. Finally, an important result for the rate of change of the Schmid stress (2.16), which enters prominently in the time-independent associated flow rule (i.e., the normality structure), is derived based on kinematical considerations (see Hill and Havner, 1982, Section 3; and Asaro, 1983, Section 3B). We begin with V'
2" = t :d"
V'
+ t :d",
(2.18)
and note, from (2.4a), (2.7) and the definition of the lattice corotational rate of a second-order tensor implied by (2.9), that (2.19) from which it follows that V'
t:d* = P:D'
(2.20)
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John L. Bassani
This relation is a variant of the "pivotal connection" given by Hill and Havner (1982) in their equation (3.3). Substitution of (2.11) and (2.20) into (2.18) gives + " = p a :Ve t = X":De,
(2.21)
pa = d" + Y e - ' : P
(2.22a)
La = Y e:d" +
(2.22b)
where
with the connection Xu = Y e : p a .With (2.22b) the constitutive relation (2.13) becomes
-c j " X "
V z = Ye:D
(2.23)
a
For slip-system flow rules governed by the Schmid stress where (2.21) enters the criteria for plastic loading versus elastic unloading, the normality structure of Hill and Rice (1972) is implied with (2.23). Given the symmetries of Zeand P, we see that pa and I"are symmetric.
111. Yield Behavior including Non-Scbmid Effects
In the setting of a time-independent flow theory of plasticity, Schmid's law states that a slip system is potentially actioe when the resolved shear stress on the slip plane in the slip direction reaches or attains the value of the critical resolved shear stress (CRSS). When the rate of shear stressing equals the rate of hardening of the CRSS then the slip system is said to be active. Two important consequences of Schmid's law for well-annealed cubic crystals are that the CRSS is independent of orientation of the uniaxial loading axis with respect to the lattice and there is no tension-compression asymmetry. Exceptional behaviors have been observed, e.g., in body-centered cubic (BCC) single crystals (Christian, 1983). Asaro and Rice (1977) recognized the importance of non-Schmid effects on strain localization and explored these effects based on a model of cross slip. Strong non-Schmid yield behaviors have been observed for intermetallic compounds having the L1, structure, including Ni3Al (the most widely studied), Ni,Ga and Co,Sn (En,Pope,and Vitek, 1987; Takasugi et al., 1987; etc.). For example, in uniaxial stressing of well-annealed L1, crystals the
Plastic Flow of Crystals
199
CRSS depends both upon the orientation of the loading axis as well as the sense of the load; i.e., tension or compression. Paidar, Pope, and Vitek (1984) proposed a microscopic model that successfully explains various aspects of the anomalous behavior in Ni,AI and materials with similar structure. Due to the complex core structure of dislocations in the crystal, they suggest that not only does the resolved shear stress on the primary slip system (the Schmid stress) control the dislocation motion on that system, but other shear stress components (non-Schmid stresses) also affect the mobility of dislocations. Stouffer et al. (1990)and Sheh and Stouffer (1990) have adapted their model to describe viscoplastic behavior of certain superalloys.
A. YIELD CRITERIA To include non-Schmid behaviors, Qin and Bassani (1992a) proposed that slip system a is potentially active when the yield function for that system reaches a critical value: z*'(t;
lattice parameters) = ZS
(3.1)
where z*' is homogeneous function of degree one in the Kirchhoff stress tensor t and zrp is the current measure of hardness, i.e., the critical stress, for that system. If t*l"= zu then (3.1) reduces to Schmid's law. In general, z*l" can be distinct from the (work conjugate) Schmid stress z" and will depend on many aspects of the crystal structure including the arrangement of dislocations as well as elastic distortion of the lattice. A simple generalization of Schmid's law would be to take z*' to depend linearly on stress:
where 7: are the non-Schmid stresses associated with slip system a and a; are the material parameters that could depend on the slips; summation is over the total number of non-Schmid components entering the yield criterion. In this case, the yield criterion for slip system a can be expressed as t:d*a = zcr *a (3.3) 9
where the symmetric second-order lattice tensor that includes non-Schmid effects is expressed as
d*" = d" + 1 a:d; 9
(3.4a)
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John L. Bassani
Motivated by cross-slip phenomena and behaviors of L1 intermetallic compounds, both of which are discussed later, attention will be focused on non-Schmid stresses T:, which are also shear stresses. In this case, d: can be expressed in dyadic form:
+
d: = (m;€3 n:
+ n; €3 m:),
(3.5a)
where the vectors m; and n: can be regarded as lattice vectors on the same footing as the slip-system vectors m uand n"(rn; n: = 0). The skew-symmetric non-Schmid lattice tensors
w:
=
+ (mi€3 n;
- n;
€3 m;)
(3.5b)
will also enter prominently through w*" = w"
+ 1 a:w:
(3.4b)
'1
With T; a shear component of stress, the orthogonal vectors m; and n; can be taken to convect with the lattice deformation like m uand,'n respectively (see (2.6)):
mi = Fe-m:
(3.6a)
=' fia. 1
(3.6b)
9
Fe-1
where m; and 5:: are the corresponding lattice vectors in the undeformed or reference configuration (as well as in the intermediate or isoclinic configuration). As with the generalized Schmid stress, the relative difference between this shear stress and the physical one would be through terms on the order of the elastic strain. Finally we note that since (3.6) leads to relations analogous to (2.7) for the rates of change of mi and n:, the rate of change of the generalized yield stress T*' follows relations analogous to (2.18) to (2.22): p a
= p*u: V = l * a : D e ,
(3.7)
where
and $*' = w*".r - 7. w*'
(3.9)
Plastic Flow of Crystals
20 1
Note the connection I*" = 9' :p*'. Relations (3.7) to (3.9) enter in the flow rules for the slip rates which are developed in Section IV. Each of the secondorder tensors )(*', I*' and fJ*' is symmetric. 1. Cross-Slip Efects
The yield behavior of a FCC crystal composed of a single element or a disordered alloy is well approximated by Schmid's law. To demonstrate deviation from the conventional Schmid's law, an example is considered involving only one non-Schmid stress in (3.3) corresponding to a shear stress on a cross-slip plane and in the direction of the primary Burger's vector. Consider shear stresses on {100)-type planes in the direction of the (011) slips (Burger's vectors), which shall be referred to as cross-slip stresses, since cross slip onto planes {lOO}has been observed in some FCC crystals and L1 intermetallic compounds; see, e.g., Pope and Ezz (1984). In this case, (3.3) becomes T*'
= T"
+ B7Eb =
(3.10)
where ra is the resolved shear stress on a primary slip system ( O i l ) { 11I}, & is the resolved shear stress on a { lOO} cross-slip plane in the direction of the primary slip ( O i l ) and B is a material parameter associated with the strength of the non-Schmid effect. The directions maand plane normals n" for Z~and zzb are listed in Table I. With crystallographic shears in opposite directions TABLE I ORIENTATIONS OF THE SHEAR STRESSES ENTERING (3.10)AND
a
?=
1 2 3 4
5 6 7 8
9 10 11 12
(1 ii)[oii] (1 1i)[ioi] (1 1I)[ iio] (iii)[oii] (iii)[ioi] (iii)[ilo] (iii)[oi 11 (iii[ioi] (iii)[iio]
b:T
Tic
(ioo)[oii]
(1 11)[Z11]
(oio)[Toi]
(1 ll)[121] (111)[11Z] (1ii)[Zif] (iii)[izi] (iii)[ii2] (iii)pii] (iii)[iIi] (iii)[iiz] (Ti 1)[2i 11 (iii)[i2i] (iii)[iiZ]
(ooi)[ iio] (ioo)[oii]
(oio)[Toi]
(mi)[1101 (ioo)[oi 11 (010)~ ioi]
(iii)[oii] (iiI)[ 1011
(ooi)[iio] (ioo)[oii] (oio)[1011
(iii)[iio]
(ooi)[iio]
(3.12).
ee
(iii)[i~i] (iii)[ii~] (ii1)[2i 11 (1 11)[151] (iii)[1i2] (i1i)[2 1f] (iii)[izi] (111)[11Z]
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John L. Bassani
counted as if on distinct slip systems, for the (12 + a)th slip systems of the FCC crystal the slip vectors m uhave the opposite sense of those on the ath system, while plane normals n" remain the same. In uniaxial tensile or compressive stressing of FCC single crystals, any orientation of the loading axis with respect to the crystal lattice can be represented by a point in a standard spherical triangle. Due to the symmetry of an FCC lattice and assuming well-annealed crystals, the initial1 critical value z,*," is taken to be the same for each slip system and denoted by z& In this case, according to the conventional Schmid's law, i.e., B = 0 in (3.10),the critical value of the resolved shear stress on the primary slip system (the Schmid stress) at initial yield is the same for all orientations of the tensile axis in the spherical triangle. With the introduction of the non-Schmid stress zZb in (3.10), z*', and not z' alone, is the driving force for slip. In this case, with T*" independent of orientation as the yield criteria imply, then 'T is not independent of orientation; i.e., the CRSS is not constant over the spherical triangle. For various values of B in (3.10), Fig. 1 is a plot of the CRSS under uniaxial stressing within the spherical triangle with [Ool], [Oll] and [ i l l ] vertices. The most highly stressed slip system for this spherical triangle is (111) [ioi].
FIG.1. Resolved shear stresses on the primary slip system at yield for all orientations of the tensile axis for FCC crystalsobeying (3.10)for cross-slipfactors B = -0.5, -0.25,0,0.25 and 0.5.
Plastic Flow of Crystals
203
Simple expressions for the CRSS under uniaxial stressing can be derived for high-symmetry orientations corresponding to corners of the spherical triangle, e.g., [Ool], [011] and [ i l l ] . Qin and Bassani (1992a) have given tables of the ratios of the Schmid stress and the cross-slip stress to the uniaxial stress (the former is known as the Schmid factor) for each slip system for these special orientations of the uniaxial-stress axis. Corresponding to the most highly stressed system, at initial yield with::7 = 7; for each system, the CRSS is (3.11) where A = 0, $/2 and $ for the [Ool], [Oll] and [ i l l ] orientations, respectively. Note that B c 0 implies that crystals oriented toward the [Ool] orientation are weaker than those toward the [Ol 1]-[i11] boundary line as plotted in Fig. 1. It is also interesting to note that crystals oriented near [ i l l ] are affected more since the magnitude of L is greatest there. As with FCC crystals obeying Schmid's law, based on the yield criterion 7,:" = 7; for each slip system, several systems are simultaneously activated for these special orientations: eight for [Ool], four for [Oll] and six for [ i l l ] . As will be seen later, 7ab also enters the yield criterion for intermetallic crystals with the L1 structure. The latter also display a tension-compression asymmetry that does not arise from (3.10). 2. L1, Zntermetallic Compounds Many intermetallic compounds and particularly the L1 compounds, such as Ni,Al, have anomalous mechanical behaviors in the sense that the CRSS at initial yield depends on the sense of the load as well as the orientation of loading axis (see Paidar et al., 1984; Pope and Ezz, 1984; and Salah, Ezz, and Pope, 1985). Furthermore, the CRSS increases with temperature in a certain regime (this can be associated with temperature dependence in the nonSchmid material parameters a; and the critical hardnesses rzr in (3.3) but will not be addressed here). The CRSS in tension can be larger or smaller than that in compression depending upon the orientation of the loading axis unlike the simple case of cross slip considered earlier where the CRSS in tension is always equal to that in compression while its magnitude depends on orientation (see (3.11) and Fig. 1). For these compounds, according to the so-called PPV theory (Paidar, Pope, and Vitek, 1984), superpartials on the primary slip system dissociate
204
John L. Bassani
into Shockley partials with complex stacking faults between them. The core of the screw dislocation tends to move onto (100) cross-slip planes and becomes immobile. This cross-slip-like core transformation is affected by the shear stress component on the primary slip plane in the edge direction of the superpartial (zpc), the shear stress on the secondary slip plane in the edge direction of the corresponding superpartial (zse) and the shear stress on the cross-slip plane in the direction of Burger’s vector (Tcb). Therefore, in addition to the resolved shear stress on the primary slip system z (the Schmid stress), three other shear stresses also affect dislocation motion in L1, crystals: zpe, z, and Tcb, Because of the configuration of the superpartials, the effects of T~~ and z, on dislocation motion depend on their sense. In one sense they tend to pull the two Shockley partials together, which enhances the cross-slip-like transformation of the dislocation core, whereas in the opposite sense they would push the Shockley partials apart, which hinders the transformation. The slip system on the (111) plane in [Oli] direction (a = 1) is illustrated geometrically in Fig. 2. The primary slip plane is ABC, the secondary slip plane is ABO. The primary slip direction (Burger’s vector) is AB; A P and P B are the Shockley partials; and EP is their edge direction, which defines the
FIG.2. Directions of the non-Schmid shear stresses for the (111)[OlT] slip system in L1, intermetallic compounds.
Plastic Flow of Crystals
205
sense of rpe for the slip system. Similarly, ES defines the direction of rSe. The cross-slip plane is the curve face that contains the slip direction AB, i.e., the [1001plane, which defines the sense for z c b . The direction of z,,, is the Burger's vector AB. The planes and directions that define these shear stresses, all of which enter the yield criterion proposed later for L1, intermetallic compounds, are listed in Table 1 for the 12 FCC slip systems. In defining systems 13-24, the plane normals are kept the same as for slip systems 1-12, while all the directions, i.e., sense of the shear stresses, are reversed. This convention is adopted here. The slip-system yield criteria for L1, intermetallic compounds that is consistent with the linear form of (3.3) and incorporates zpe, z,, and rcb as described previously are (Qin and Bassani, 1992a)
where A, B and k are material parameters that depend on temperature and, in principle, can evolve with the deformation although here they are taken as the same constant for each slip system (e.g., for well-annealed FCC crystals). The magnitudes of these non-Schmid factors A, B and k are typically less than unity. The sign of the term involving rpe and zse reverses for systems a = 13,24 because of their effects on the superpartials associated with the opposite senses of slip relative to systems a = 1,12.(Note that (3.12) are associated with the labelling of systems a = 1,12 given in Table 1. Some other choice for those first 12 systems such as the classical one adopted by Wu, Bassani, and Laird (1991) would not lead to the simple partitioning with respect to & A ) Equations (3.12) can be written in a more compact form for the case where TC, *('+ 1 2 ) = 7:; (which does not preclude a different CRSS with sense of slip on a given system):
+
r' -k A ( z & + krze!,l
B?:b
=
&7:;
(a = 1, 12)
(3.13)
where activation of a slip system in the negative sense is implied when equality is met with the righthand-side negative. This form clearly indicates the dependence on the sense of the stress. Uniaxial stressing for oeientations of the loading axes in the spherical triangle can readily be investigated as in the example of cross-slip effects considered above. Again, for simplicity, the critical values are assumed to be the same for each slip systems, i.e., 7:; = 75, which characterizes a well-
John L. Bassani
206
annealed crystal. Initial yield is reached when one of the 24 slip systems is activated; i.e., at least one equation in (3.13) is satisfied where the various shear stresses arising in uniaxial stressing are readily computed using Table I. The resolved shear stress on the primary slip system at yield (CRSS)is plotted in Fig. 3 for all orientations of the uniaxial stressing axis over the spherical triangle for various values of A , B and k. The degree of the tensioncompression asymmetry is controlled by the factor A, and increasing its magnitude produces a larger separation of the tension and compression surfaces. Changing the sign of A exchanges the two surfaces. The factor k determines the orientations where the CRSS at initial yield in tension is the
TJ
z I Yo
2.0 A=-0.6
2.0
B=-O.l k = -0.4
A~0.3
20
B--0.1
k = -0.4
1.0
I .o
0.0
0.0
I
I
7/T, A = -0.3
B 10.2
A = -0.3
20
B1-0.1 k = 4.8
k=-0.4
1.o
1.0
1111
0.0
1111
0.0
I
I (C)
(4
FIG.3. Effects of non-Schmid factors on initial CRSS with respect to the orientation of the uniaxial stressing axis.
207
Plastic Flow of Crystals
same as in compression. The cross-slip factor E influences those crystals oriented near the [011]-[~11] boundary as also seen in Fig. 1. The prediction for the CRSS in uniaxial stressing plotted in Fig. 4(a) corresponds to values of the non-Schmid parameters (A = -0.3, B = -0.1 and k = -0.4) chosen to qualitatively agree with the experimentally observed tension-compression asymmetry for Ni,Al shown in Fig. 4(b) (Pope and Ezz, 1984). Once again, z,*p = 7;. As before, high-symmetry orientations of the uniaxial loading axis at the corners of the spherical triangle can be examined analytically, but now the CRSS also depends on the sense of the applied uniaxial stress. For the particular choice of the parameters A, E and k just noted describing Ni3Al, the most highly stressed slip systems that correspond to those activated at initial yield for each of these special orientations are the same as for both
W A L A-03
B--0.1 k-44
1.0
0.0
I
(b) FIG.4. Resolved shear stress on the primary slip system at yield for both tension ( T )and compression (C) with respect to the orientation of loading axis for Ni,AI.
John L. Bassani
208
FCC single crystals obeying Schmid's law, i.e., A = B = 0 in (3.13) and for the simple cross-slip model of (3.10). One such system is (1 1 l)[TOl]; i.e., a = 2. For values of the non-Schmid parameters that differ significantly from these, some other system(s) may become the critical one(s). The ratio of CRSS in tension and compression for uniaxial stressing at the corners of the spherical triangle can be easily found as a function of A, B and k from (3.13) with Table I:
+ k)/$ + + k)/$ 1 + A/$ +dB/2 [otti 1 - A/$ + $B/2 = 1 + A(l + k)/$ + $B 1 - A(l 1 A(l
T
1'
T
[itti
1 - A(l
+ k)/$ + $B
(3.14a)
(3.14b)
(3.14~)
From these expressions, the three non-Schmid factors A, B and k can be readily determined by the measurements of the tension-compression asymmetry at the comers of the spherical triangle.
B. YIELDSURFACES The yield criterion (3.1) for each slip system can be represented as a surface in a six-dimensional hyperspace. The inner envelope of these surfaces from all slip systems produces the complete yield surface for the crystal with vertices that correspond to the activation of more than one system. Since (3.2) and (3.3) are linear in stress, the surface corresponding to each slip system is a hyperplane and the inner envelope is polygonal (Bishop and Hill, 1951). To get a geometrical visualization of this surface we can consider a threedimensional subspace whose orthonormal base vectors coincide with a particular principal stress state defined with respect to the crystal lattice. When yielding is independent of hydrostatic stress, which is the case when all the non-Schmid stresses are shear stresses, the subyield surface for any such principal stress space is a cylinder whose generator is parallel to the o1 = a2 = a3 direction. The projection onto a plane perpendicular to a generator, the so-called a-plane, is a two-dimensional locus that completely specifies the subyield surface. We emphasize that the shape of this locus depends on the particular principal stress space under consideration.
Plastic Flow of Crystals
209
Various yield loci are plotted in Qin and Bassani (1992a, 1992b). Figure 5 shows only one such result, which depicts the di5erence between an FCC crystal such as copper or gold and the L1, crystal Ni,Al. The arbitrarily chosen orientation of the principal stress axes with respect to the crystal axes in Fig. 5 is obtained by rotating the crystal axes through an angle of - 18.6" about the vector whose components in the crystal axes are [-0.7, 0.487, -0.5271. Relative to the high-symmetry locus for the FCC case, the
4.0
L
FCC
4.0
I
62
L12 FIG. 5. Comparison between yield loci (normalized by 7 ; ) of FCC single crystals obeying Schmid's law and the L1, intermetallic compounds Ni,AI.
210
John L. Bassani
Ni,Al(Ll 2) is significantly distorted and does not possess 180” rotational symmetry due to the effects of zpe and z, in (3.13), i.e., the non-Schmid effects that produce a tension-compression asymmetry. Again, z,*p = z$ is assumed for each slip system. In general, this locus shrinks with increasing non-Schmid parameter B. C. RESTRICTEDSLIP Geometrically, the slip systems associated with lattices based on the FCC structure possess a high degree of symmetry. As Taylor (1938a, 1938b), Bishop and Hill (1951) and Bishop (1953) observed, due to the symmetry of the ordinary FCC lattice many special stress states can accommodate an arbitrary plastic strain by simultaneously activating five or more linearly independent slip systems. These correspond to vertices on the single-crystal yield surface. In particular, there are 28 such stress states for FCC crystals obeying Schmid’s law with zrp = z$ on each slip system, and each state activates either six or eight slip systems. Each of these stress states are of four types (A, B, C, or D) (see Bishop (1953)for FCC crystals obeying Schmid’s law and Qin and Bassani (1992a) when non-Schmid effects are included). On the other hand, yield criteria containing non-Schmid terms, e.g., (3.3) or (3.12),do not possess the high degree of symmetry of those based on Schmid‘s law. As a consequence, when non-Schmid stresses enter the yield criteria not only are yield surfaces distorted, but it also is more difficult to activate many systems simultaneously as generally required in large plastic deformations of pol ycrystais. Although the most general stress state corresponding to vertices on the yield surface can be readily analyzed for FCC crystals obeying Schmid’s law (see, e.g., Bishop 1953), the presence of non-Schmid stresses significantly complicates matters. Instead, Qin and Bassani (1992a)considered only the 28 Bishop-Hill states and found the following for L1, compounds. For six of the stress states at eightfold vertices of types A and B, the multiplicity of vertices is unchanged even though the shape of the yield surface is distorted. For type C stress states, however, half of the eightfold vertices split into two fourfold vertices, i.e., only four slip systems can be activated simultaneously, while for the rest, spreading of the hyperplanes is observed but the vertices remain eightfold. Similarly, for the type D stress states, the sixfold vertices turn into fourfold ones; i.e., hyperplanes separate. The question naturally arises as to whether there are other stress states with high multiplicities. Numerous numerical investigations indicated
Plastic Flow of Crystals
21 1
otherwise. For example, this has been demonstrated utilizing an extension of the elastic-plastic self-consistent approximation for polycrystals introduced by Hill (1965)and applied by Hutchinson (1970).For a polycrystal composed of randomly oriented single crystals undergoing non-Schmid behaviors the overall response is significantly stiffer than for constituent crystals obeying Schmid's law (Qin, 1990).In general, the compatibility requirement between individual grains in the polycrystal requires extensive multiple slip once the overall plastic strains exceed the elastic ones. This stiffer response, i.e., higher stresses to generate a given level of plastic deformation, may contribute to the brittle behavior of certain intermetallic polycrystals such as Ni,Al as compared with the single-crystal response.
IV. Flow Behavior including NonlSchmid Effects
As noted from the outset, the emphasis in this chapter is on timeindependent behavior. A rate-dependent framework consistent with generalized slip-system stress measures introduced in Section 111 can readily be established along the lines of Asaro (1983) and Stouffer et al. (1990). For example, one could adopt a power-law description: jP a (t*"/.r::d)". A. SLIP-SYSTEM FLOWRULES In the framework of time-independent plasticity, a slip system is said to be potentially active if the yield criterion (3.1) is satisfied. Whether slip system c1 is actually active or not depends on whether the rate of effective stressing for that system, i.e., 2*", keeps up with the rate of hardening on that system, i.e., trp. In that framework, system of equations involving inequalities determines the slip rates jP, a = 1, N, on all systems, although questions of uniqueness must be addressed. Based on the yield criteria (3.1) or (3.3), slip-system c1 is inactive if the effective yield stress t*" is less than a critical value: .r**
<; :7
=0
(inactive)
(4.1)
Otherwise, system a is potentially active. In this case, the slip system is active only if the rate of effective stressing on that system is equal to the rate of hardening: p a = t, *, a and 2*" = :2 => 9" 2 0 (active) (4.2)
John L. Bassani
212 otherise it is inactive: T*'
= T~~ +a
and
+*' < frp
=0
(inactive)
(4.3)
Classical flow rules for single crystals obeying Schmid's law are based on a normality structure where T*" = T' and, from (2.5a) and (2.16a), DP=
1y a T a p r = c 01
j"da
(I
(see Koiter 1960; Mandel 1965; Hill 1966 Rice 1971; Hill and Rice, 1972; Hill and Havner 1982). In contrast, the flow rule (4.1) to (4.3)with (2.5a) and (3.2) clearly does not follow a normality structure (unless, of course, each of the non-Schmid factors a; = 0) since for the particular linear form of (3.2)or (3.3)
In this case, the flow is said to be nonassociative. This is known to have a significant effect on postyield behavior, particularly bifurcations, as will be discussed. To complete the flow rules (4.1) to (4.3) the evolution of the slip-system hardening measures T:; must be specified. A multiple-slip hardening law that can incorporate the influence of slip from all systems on the hardening of each system is (Hill 1966)
where the instantaneous hardening moduli ha, are, in general, functionals of the entire deformation history. The hardening relation adopted by Asaro and Rice (1977),which introduces non-Schmid stresses in single slip, is an example of (4.5) with (3.3). Many simple hardening rules have been proposed beginning with the isotropic rule of Taylor (1938a),where all components of ha, are equal. Koiter (1953) proposed an independent hardening rule where h,, = ha,, is a diagonal matrix. (A diagonal form has been used successfully by Bassani and Wu (1991), where each diagonal element is a function of all the slips; see Section V.) Budiansky and Wu (1962) adopted a "kinematical" hardening rule of the form ha, = hd" :d@,which does incorporate a weak latent hardening. Hutchinson (1970) adopted a latent-hardening rule introduced where ha, = (ha - h,)d,, h,; and more recently, Pierce, Asaro, and Needleman (1982) proposed a variant of the latter to eflect certain features in single-slip hardening as well as a latent-hardening effect.
+
Plastic Flow of Crystals
213
With (3.7) and ( 4 4 , for each potentially-active slip system, (4.2) and (4.3) are rewritten as p*a:
V'
r
= A*':
D'=
C haBjfl* 7'
20
(4.6a)
C haBjp+ja = 0
(4.6b)
B
V'
p*': z = A*":D'<
B
where p*a and A*' are defined in (3.8a, b). In general, with coupling between slip systems through hardening in terms of the slip rates, the set of equations (4.6a, b) for all potentially active slip systems cannot be directly solved for the nonzero slip rates. Havner and Varadarajan (1973), Guldenpfennig and Clifton (1977) and Bassani and Wu (1991) have utilized variants of quadratic programming algorithms to solve these equations. On the other hand, in the special case when the hardening matrix ha, is purely diagonal, i.e., independent hardening, these flow equations are easily solved since for any system with 2*a > 0 (see (3.7) for the connection), = t*'/h,, > 0 (no sum on a). Nevertheless, strong interactive hardening effects can readily be accounted for with a diagonal form of the hardening matrix as demonstrated later in Section V. Equations (4.6) can be transformed by using (2.5a) and D' = D - DP,for potentially active slip systems: A*a :D
=
C gaajB =s p" 2 0
(4.7a)
B
(4.7b) where gap = hap
+ A*'
:ds = hap
+ d*a :2 :dB + p*' :da
(4.8) Alternatively, in terms of the Jaumann rate of Kirchhoff stress, with (2.1 1) and (2.14a), (4.6) for potentially active slip systems can be rewritten as (4.9a) V p*ll:t
<x
=0
(4.9b)
= haB - d*":fl@- $ * a : y - l : p
(4.10)
kapj@*p"
B
where k,, = ha, -
The instantaneous elastic relation given in the form (2.13) or (2.23) can be cast in a more compact form by using (4.7). Let gap be the square matrix
John L. Bassani
214
associated with the active systems only and let gas' denote its left inverse. Then solving (4.7a) for the nonzero pol gives V
(4.11)
z=9 : D
where Y = Y' -
c a
(4.12)
g a j ' l " @ 1*fl B
In general, Yijk, = Yjikl = 6 4 i j l k # Yk1ij. Furthermore, gap symmetric is not sufficient to ensure that the instantaneous elastic-plastic modulus Y is diagonally symmetric when non-Schmid effects are present; i.e., when k*' # 1"(Qin and Bassani, 1992b).
B. UNIQUENESS OF SLIPS
Sufficientconditions that guarantee the solution to the flow equations (4.6) for the slip rates with Or D' prescribed, or to (4.7) with D prescribed, or to (4.9) with prescribed yield a unique set of slips depending on the positive definiteness of the matrices ha, or gapor k,,, respectively, where each square matrix is understood to be associated only with the potentially active slip systems that enter those equations. Hill (1966) and Hill and Rice (1972) provide proofs for Schmid-type flow behavior. A proof follows that includes non-Schmid behavior. Let A denote differences in pairs of quantities associated with two distinct ADe = D; - D;. Beginning with (4.6a, b), sets of slips; e.g., A V = fi consider the quantity
7
x,
1
:ADe)Ai)"
a
If for some (slip-system) a, both sets 1 and 2 are active, then from (4.6a)
AD')Aj" =
c h,,A.jl"Affl
(4.13)
B
Clearly, (4.13) also holds if system a is inactive in both sets (AV = 0). If, on the other hand, system a is active in set 1 but not in set 2(Ajl" 2 0), from (4.6b)
ha8fi - X*":D; A T 2 >
c h,,AvAfS I
(4.14)
215
Plastic Flow of Crystals
Similarly, if system a is active in set 2 but not in set l(A)"
L*":D', -
7
h,,f;)
Therefore,
AV 2
< 0), from (4.6b)
c h,,Aj"AfS
c
(I*' :ADe)AV 2 1 h,,Aj"Aj8 a
(I
(4.15)
B
(4.16)
,
A sufficient restriction on the hardening moduli h,, to ensure uniqueness of slips satisfying (4.6a, b) now follows from (4.16). If De (or = 2 ' : : ' ) is prescribed, that is, rates formed on the (rotating) lattice are prescribed, then ADe = 0 and the lefthand-side of (4.16) identically vanishes. In this case, if the square matrix h,, associated with the potentially active slip systems is positive definite, then the righthand-side must also vanish, in which case A V = 0, a = 1, N. Since the slips are unique determined and D' is prescribed, then D and i are also uniquely determined. With similar arguments applied to (4.7), a sufficient condition that the slips are uniquely determined with D prescribed is that gas is positive definite. From (4.9), the slips are uniquely determined with prescribed if k,, is positive definite. C. AN EXAMPLE FOR L1, INTERMETALLIC COMPOUNDS
,
Many L1 intermetallic compounds display strong non-Schmid yield behavior, in which case, as discussed in Section 111, the slip-system yield criteria (3.13) depend on both non-Schmid stresses and the sense of stress. Plots of variations in the initial CRSS with orientation and sense of the uniaxial stress are given in Figs. 3 and 4, and an initial yield surface is plotted in Fig. 5. Based on a simple hardening rule, Qin and Bassani (1992b) demonstrated the evolution of these effects with continued plastic deformation. An example is reproduced here. A simple form of the hardening matrix is chosen where the instantaneous hardening moduli monotonically decrease with increasing slip:
h, =
h0 1 + Y"/Yo
+ h,
(a = /3, no sum)
(4.17)
216
John L. Bassani
where ho, h,, yo and q are taken to be positive constants and y" 2 0. Specifically, h, and h, characterize the initial and the saturated values of the incremental hardening moduli, respectively, and q is the latent hardening parameter. This is a strain-dependent variant of the Hutchinson (1970) hardening rule, which is similar to the one adopted by Pierce et al. (1982). Obviously, the hardening matrix is positive definite when q = 0. When ho = 0 and q = 1, (4.17)reduces to the isotropic Taylor hardening law. In the simple form given in (4.17), slip on other systems is assumed not to affect selfhardening (see Section V for a more complete discussion of hardening). Uniaxial stressing of a L12 single crystal is considered where, for simplicity, V' the stressing axis is taken to rotate with the crystal axis; i.e., r is prescribed. The crystal is assumed to be elastically isotropic, therefore the elastic response is independent of the orientation of the uniaxial stressing axis. On the other hand, plastic response depends upon the orientation of the loading axis with respect to the lattice, which in this case is represented by its stereographic projection onto the spherical triangle. Uniaxial stress-strain curves are plotted in Fig. 6 in both tension and compression for four orientations of loading axis, three of which are near the high-symmetry orientations. The material parameters in the yield functions (3.13) are chosen as before to describe Ni,Al(A = -0.3, B = -0.1 and k = -0.4), and isotropic and incompressible elasticity is assumed with Young's modulus E/z$ = 1OOO. The initial critical stresses are taken to be the same for all systems, i.e., zr," = 7: (the stress axes in Fig. 6 are normalized by z);, while the and hardening implied by (4.17)with h, = 5000z,*,h, = 50z& yo = 1.6 x q = 0 is highly anisotropic. For this independent hardening law (q = 0) multiple-slip deformation generally develops under uniaxial stress even when initial yield involves only one slip system (Section V). The tension-compression asymmetry in initial yield (also see Fig. 4) persists during postyield deformation as seen in Fig. 6. For stressing axes oriented near the [OllJ-fill] boundary, a Ni,Al crystal is stiffer in compression than in tension, whereas the opposite holds for axes oriented near [Ool], which follows with the initial yield behavior. Even though it is common practice to identify the initial yield point either from an offset plastic strain or from back extrapolation rather than from the precise onset of irreversible straining, in this example, even with strong non-Schmid effects either is found to give a reasonable estimate of the relative magnitude of the tension-compression asymmetry.
Plastic Flow of Crystals
217
a /z; 8.0
-
T
T [-Ill]
[-I111
4 [Olll
I001
0.05
0.0
1~ 1 o
0.0 0.0
l
A
0.3
& (c)
l
&
(4
FIG. 6. Uniaxial stress-strain curves in tension (7') and compression (C) for various orientations loading axis for Ni,AI: A = -0.3, B = -0.1 and k = -0.4.
V. Hardening Behavior
The instantaneous hardening matrix ha, that enters the flow rules-and strongly influences whether the constitutive equations summarized in the previous section guarantee unique solutions for the active set of slips-is dependent on crystal structure, material composition and preconditioning and is the least well-characterized aspect of the theory. Even though uniqueness is not an issue in the time-dependent framework, these moduli
l
218
John L. Bassani
(and specifically their evolution with deformation) strongly correlate with various fundamental behaviors that either a rate-independent or ratedependent theory should be able to capture (Wu et d., 1991). One such example that is particularly central to the characterization of these moduli is the occurrence of secondary slips, which generally are small compared to the primary slip@)but significantly influence overall hardening, under uniaxial stressing in a so-called single-slip orientation. These secondary slips are not predicted by either the simple Taylor (1938a) hardening rule, where all components of ha, are equal, or the latent hardening rule of Hutchinson (1970), where ha, = (h, - h,)d,, + hf with the off-diagonal element hl not taken to be small compared to the diagonal element ha (in fact, hf > h, has often been adopted from observed latent hardening measurements at finite slip in the secondary test). Havner and Shalaby (1977) have proposed a mathematical theory of hardening at finite strain within the normality structure, i.e., for Schmid-type behavior (d*" = d", p*" = and A*" = A"), that leads to a symmetric form for the instantaneous slip-system moduli gas and k,, entering (4.7), (4.9), and (4.12).This is accomplished by choosing ha, = Hap + d": p p with Ha, taken to be symmetric. Note that this introduces an explicit stress dependence in the physical slip-system moduli ha, through flfi and, thereby, introduces terms that can be large compared to the physical moduli, for example, in easy glide. In their "simple" theory, Ha, = constant is adopted. Pierce et al. (1982) suggested a modification that lessens the strength of the stress-induced latent hardening and also used a variant of the two-parameter Hutchinson form Ha, = h(y)[(l - q)6,, + q ] , where q is a latent-hardening parameter and y is taken to be the total accumulated slip on all systems (also see Asaro, 1983). When viewed in terms of observations from uniaxial stressing of single crystals including orientation dependence of hardening, secondary slips before overshoot and coarse slip band formation, the degree of latent hardening introduced in these hardening descriptions appears to be too high. Instead, following Wu et al. (1991) and Bassani and Wu (1991) (also see Bassani, 1990), a description of hardening is developed from physical considerations. The experimental basis for this hardening theory is limited primarily to pure elements and disordered alloys that display Schmid-type behaviors. Observations of orientation dependence of hardening, various stages of hardening, secondary slips, etc., abound for these materials. On the other hand, for materials displaying strong non-Schmid effects such as the L1, intermetallic compound Ni,Al, which is one of the most widely studied
Plastic Flow of Crystals
219
ordered alloy, experiments that provide direct insights into multiple-slip hardening are still needed. For Schmid-type behavior, only the evolution with deformation of the critical value of the Schmid stress, i.e., zZr, the hardness of system a, needs to be characterized for each system. This itself is a formidable task since, for system a, the hardness generally depends on the slip on all systems as given by (4.5)so that each component of ha, depends on the slip history. For materials displaying non-Schmid behavior,: :7 is proposed as the hardness measure relative to the effective stress z*’, which in turn depends on several material parameters, e.g., the non-Schmid factors a: in (3.2),each factor may independently depend on details of the multiple-slip history (recall that in the PPV yield theory for L1, intermetallics that the non-Schmid terms arise due to complex dislocation core effects). The simulations we have carried out (see Fig. 6 and the shear band calculations in Section VI) assume that the non-Schmid factors a: in (3.2)do not vary with deformation. A. EXPERIMENTAL OBSERVATIONS
1. Uniaxial Stressing of FCC Crystals Observations of single crystals undergoing predominantly single slip under uniaxial stressing are based primarily on load-displacement curves that reflect the macroscopic resistance of the specimen to continued deformation. From the measured load the resolved shear stress on the most highly stressed slip system is calculated, while from the measured axial extension the shear strain on this primary system is approximated in the sense that the total plastic strain in the specimen is assumed to be accommodated by this slip system alone. Figure 7 is a typical (schematic) z-y curve for a FCC single crystal obeying Schmid’s law with the tensile axis oriented within the standard stereographic triangle (see, for example, Piercy, Cahn, and Cottrell, 1955;Wu and Drucker 1967;Franciosi and Zaoui 1982b;Honeycombe 1984). For a well-annealed single crystal the critical resolved shear stress zo at the initiation of plastic flow is the same for all orientations, while the extent of easy glide in Stage I, the stress q,at the beginning of Stage 11, and the Stage I1 hardening rate, which is much higher than in Stage I, depends strongly on orientation. Many studies are found in the literature: the reader is referred to Rosi (1954);Garstone, Honeycombe, and Greetham (1956);Diehl (1956); Davis et al. (1957);and Honeycombe (1984)for typical examples. Often T~ is
John L. Bassani
220
Stage1
,
,Stagen Stagem
In In
w
a
Li
rm
a
Q W
I
In
d
3 E
rII TI
To
B SHEAR STRAIN
FIG. 7. Typical 7-y curve of a pure FCC single crystal loaded in uniaxial tensioncompression with an initial orientation for single slip: point B denotes where secondary slip commences.
approximated by backward extrapolation from the linear region in Stage I to zero strain, i.e., by z,. Backward extrapolation has also been used in the secondary test of a latent hardening experiment that has led to a source of confusion in characterizing the instantaenous hardening moduli h,, . Rosi (1954)and Garstone et al. (€956)observed that zl,-zl is nearly a constant for all orientations undergoing easy glide, while the Stage I hardening rate is typically at least an order of magnitude lower than the Stage I1 hardening rate. The high Stage I1 hardening rate is generally associated with strong dislocation interactions arising between the primary dislocations and the Stage I1 secondary dislocations (see, for example, the early references by Seeger, 1957;Basinski, 1959; Hirsch, 1959; Mitchell and Thornton, 1964; Basinski and Basinski, 1970 and 1979; as well as a recent paper by Kuhlmann-Wilsdorf, 1989). Although nearly linear z-y behavior in both Stage I and Stage I1 suggests a constant work hardening rate, the possibility of secondary slips in Stage I1 giving rise to strong dislocation interactions suggests a more complex hardening of individual systems. In Stage I1 deformation, the magnitude of primary slip is still a good measure of the overall strain, but the Schmid stress on that system is not a good measure of the overall hardening rate. Studies of orientation dependence of hardening in FCC crystals, e.g., Diehl (1956) on copper and Davis et al. (1957)on aluminum, have common features. It is seen that, as the orientation approaches the (loo)-( 1 1 1) symmetry boundary, the length of Stage I decreases while the Stage I hardening rate increases. Under high-symmetry orientations of the tensile axis for which multiple slip (Stage 11) occurs from the onset of plastic deformation, Stage I
Plastic Flow of Crystals
22 1
easy glide is suppressed (Kocks, 1964; Kocks, Nakada, and Ramaswami, 1964). This dependence of the hardening rate and extent of Stage I on initial orientation is an important characteristic of the multislip hardening through the matrix hap. A generally accepted notion is that the rate of work hardening is correlated with dislocation interactions. Various models have been proposed to describe work hardening by micro-mechanisms such as dislocation pile-ups (Seeger, 1957), forest dislocations (Basinski, 1959), and dislocation tangles (Hirsch, 1959). Basinski and Basinski (1979) offered the following description of the evolution of dislocation substructures in FCC materials during room temperature uniaxial stress deformation: 1. In Stage I deformation (easy glide) isolated long bundles of primary dislocations of predominantly edge character are formed and these lie in extensive regions of very low dislocation density. Layered structures are formed, and the forest (secondary) dislocation density is about an order of magnitude lower than the primary one throughout Stage I deformation. 2. In Stage I1 deformation (rapid hardening) bundles containing shorter segments and a high portion of secondary dislocations are linked together by secondary tangles. Lomer-Cottrell locks also are observed (Lomer, 1951). A more isotropic structure develops as the forest density approximately equals the primary. 3. In Stage I11 deformation (parabolic hardening) the development of cell structure becomes more marked. Thermally activated cross slip occurs, which allows dislocations to circumvent obstacles and thereby decreases the hardening rate. Dislocation tangles and double cross slip lead to the formation of slip bands. As noted, secondary dislocations or slips are the key to understanding the complex hardening under multiple-slip conditions. With the uniaxial stressing axis within a standard triangle, in which case the primary slip system is most highly stressed, the only way that the (previously latent) secondary slip systems can become active is if those systems actually harden less than the primary systems.
2. Latent Hardening Experiments and Overshoot Direct measurements of the evolution of the hardening matrix ha, during plastic deformation is formidable, if not impossible, due to limitations in simultaneously measuring infinitesimal stress and strain increments on
222
John L. Bassani
different slip systems. Instead, indirect measurements have been used to infer hardening rates on different systems based on implicit assumptions of single slip in certain regimes of deformation and a simple dependence of hardening on strain history. For example, in latent hardening experiments (see, for example, Edwards and Washburn, 1954; Kocks, 1964; Ramaswami, Kocks, and Chalmers, 1965; Jackson and Basinski, 1967; Franciosi, Berveiller, and Zaoui, 1980), first the most highly stressed slip system is activated up to some stress level zp(yp)on that system-the primary test. Then, in a successive secondary test a previously latent system is activated, and the amount of latent hardening during the primary test is estimated from the apparent initial yield stress z, in this secondary test, typically as measured based on an offset secondary strain (ys) or back extrapolation. This procedure, which involves activating the previously latent system, leads to an estimate of its hardness at the end of the primary test that typically depends on the amount of slip accumulated in the secondary test as well as the assumption of single slip in the primary test. For sufficiently large y,, typically with y, < y p , 7JyP, 7,) > z p ( y p ,y.) is generally observed. As another example of an indirect observation, when a single crystal is uniaxially loaded in tension by a stiff testing machine, as slip progresses the lattice tends to rotate with respect to the tensile axis. The tendency for the tensile axis to overshoot the symmetry boundary between the stereographic triangles (Bell and Green, 1967; Joshi and Green, 1980) instead of stopping there as predicted by Taylor and Elam (1925) is regarded as a measure of the hardening anisotropy between different slip systems (Ramaswami et al., 1965). If leading up to the symmetry boundary and thereafter all systems hardened equally, then equal slip rates at the boundary would preclude overshoot. If, on the other hand, the latent systems hardened more than the active ones then secondary slips would be precluded (as noted previously) and overshoot would follow. Once again, this indirect measure of latent hardening assumes single slip, which is not consistent with the observations summarized in the previous subsection. A precise interpretation of these indirect measurements is difficult. Nevertheless, simple hardening descriptions have been proposed that are consistent with some of these observations, including, for example, that the hardness of the previously latent system after some amount of slip on that system in the secondary test is typically 1 to 1.6 times (Kocks, 1970) the hardness of the primary system and that the active hardening rate is on the same order of magnitude regardless of the straining history. However, within the framework of time-independent plasticity, the constitutive equations
Plastic Flow of Crystals
223
based upon the simple hardening descriptions do not lead to a unique prediction of the set of active slip systems as noted by Hill (1966), Hill and Rice (1972), and Asaro (1983). To circumvent this nonuniqueness problem Asaro (1983) has proposed a time-dependent formulation while Fuh and Havner (1989) have proposed an additional minimum plastic spin requirement. Nevertheless, these alternative approaches, which generally but no necessarily adopt the simple hardening description, have not addressed fundamental issues such as the transition from Stage I to Stage I1 deformation or the orientation dependence of the uniaxial stress-strain curve (see, e.g., Honeycombe, 1984). Latent hardening tests on copper single crystals (with a purity of 99.99%) loaded in uniaxial compression were carried out by Wu et al. (1991) to obtain direct estimates of the slip-system hardening moduli during the primary and secondary tests. Using a computer data acquisition system, data was numerically smoothed and differentiated to determine the overall (apparent) tangent modulus dzldy = m2 dalde where m is the Schmid factor. In their study the initial yield point in the secondary test was associated with a precipitous drop in the tangent modulus. The resolution using this technique is sufficient to demonstrate that the initial flow stress on the secondary (previously latent) system tends to be lower than the current flow stress on the primary system. The reader is referred to Wu et al. (1991) for experimental details. Figure 8 is a schematic of the z-y curves from both the primary and secondary tests of a latent hardening experiment. Note the high initial rate of hardening in the secondary test. In keeping with the time-independent
ln ln
g
ss (from bockword extrapolation)
$ I
a
a
W
I
ln
a w
2
ln
Primary
W
a SHEAR STRAIN
FIG.8. Typical T - y curves from the latent hardening experiment: r p denotes the current yield stress of the primary system corresponding to the flow stress before unloading. Backward extrapolation is often adopted as a measure of the initial yield stress of the secondary system 5,.
224
John L. Bassani
(incremental) formulation of the plastic constitutive relations of Section IV, a more precise measure of T, is based on a measurable decrease of the tangent modulus (from the linear elastic modulus),which indicates the onset of plastic deformation in the secondary test. Figure 9 is a typical result from the experiments of Wu et al. (1991),where only the curves from the secondary test are plotted although the magnitude of the CRSS at the end of the primary test, t p= 4.2MPa, is noted on the t s - y s curve. Two secondary tests (S, and S,) from the same primary test (PI),which was carried out to a primary slip of 2%, are shown. With the initial yield stress on the secondary system associated with a precipitous drop in the tangent modulus (Fig. 9), these experiments indicate that plastic flow in the secondary test commences at a flow stress lower than the current hardness of the primary system. Furthermore, the initial high hardening rate on the previously latent system leads to a greater flow stress on that system after only a small amount of secondary slip. This is consistent with traditional views of latent hardening. That is, at initial yield on the secondary system z, < tp,but as that system continues to slip z, is observed to increase rapidly. Typically, for ys > 0.005 then z, > t p .This behavior has been seen before, for example, by Edwards and Washburn (1954), Kocks (1964), Ramaswami et al. (1965), and Jackson
-a m
E
O
Y
-2010-
'
orlo2
o.dc.4 o.& o.dos
0.;10
Y
-2010-
Y
Y
FIG. 9. From two secondary tests of a latent hardening experiment (a) r-7 curves and (b) tangent modulus 0 = ds/dy versus shear strain y. Note that in both tests 0 begins to drop at a flow stress sS, which is lower than r,,.
Plastic Flow of Crystals
225
and Basinski (1967), who noted that back-extrapolated measures of the initial yield stress in the secondary test neglected the initial rapid hardening.
B. RESTRICTIONS ON ha, Based on a careful interpretation of the transition from Stage I to Stage I1 deformation and the orientation dependence of hardening in the 7 - 3curve under uniaxial tension as well as a reinterpretation of the latent hardening experiment that includes the variation with strain (both y p and y s ) of ratio of the flow stress in the secondary test, Wu et al. (1991) observed that: 1. The total hardening on any truly latent, secondary system is lower than the total active hardening on the primary system, at least before the activation of the secondary slips. 2. The interaction of slips on different systems has a stronger effect on work hardening than slip from the same system. 3. The activation of a new slip system causes a greater hardening rate increase on systems with low slip intensity than on ones with high slip intensity. From these observations they propose inequality restrictions on the instantaneous hardening moduli. These inequalities are expressed most simply in the case of two slip systems, although the generalization to multiple slip is straightforward and is accounted for in the proposed forms for the hardening matrix. In terms of the critical stresses evolving, for example, under monotonic uniaxial stressing with the tension or compression axis within a spherical triangle, with 1 denoting the primary slip system and 2 denoting the secondary system,
C(y',
7' = 0)< rdr(y1, y 2 = 0 )
(5.1)
r:r(y', 'Y = YO) > 7dr(y1, 'Y = 0)
(5.2)
where yo need only be a small (secondary) strain, typically on the order of 0.005 in the latent hardening experiments of Wu et al. (1991). The first of these inequalities accounts for the activation of secondary slips at the end of Stage I easy glide and the subsequent high level of hardening leading to the second inequality in Stage 11.
226
John L. Bassani
In terms of the instantaneous hardening moduli, with T&(O, 0) = T:~(O,O)= q, for well-annealed crystals, Wu et al. (1991) have shown that (5.1) and (5.2) are derived from h21(Y1,0)
< h,1(y1,O)
(5.3a) (5.3b)
Clearly, the simple hardening rules noted earlier that others have adopted, including the one proposed by Pierce et al. (1982),which has a dependence on the total slip accumulated on all systems, cannot satisfy all of these inequalities.
c. ANALYTICALCHARACTERIZATION OF ha, To describe the complex multiple-slip phenomena during plastic flow, Bassani and Wu (1991) begin with the notion that the instantaneous hardening moduli depend on all the slips; i.e., h,,({y"; K = l,N}), a, p = 1,N. That is, a particular component of the hardening matrix depends on the total accumulated slip on all systems and, possibly, the history of slip. A particular multiplicative form is proposed where each diagonal component is taken as the product of a self-hardening term F times an interactive hardening term G:
h,,
= F(yu)G({yK; K =
1, N ,
K
# a))
(5.4)
The experimental observations cited previously correspond to monotonic loading conditions and, therefore, typically involve no slip reversals. In this case there is no ambiguity in defining the total nonzero slips even if opposite senses of slip are counted independently. In the case of cyclic deformations, an interpretation of the slips entering (5.4) is given later in a separate subsection. With G({yK= 0; K = 1, N , K # a } ) = 1 under single-slip (easy glide) conditions, F(y") is the instantaneous hardening modulus. In general, F and G may be functionals of their arguments. During Stage I1 deformation when seccndary slips are activated, i.e., y K > 0 for some K # a, the function G characterizes the effects caused by slip interactions between the primary system LY and the secondary systems (e.g., forest hardening). The hardening of system fl due to slip on system a is simply taken to be a
Plastic Flow of Crystals
227
fraction q of the active modulus, and the off-diagonal components are given as
h,, = 4h.m
a
zB
(5.5)
Other choices for the off-diagonal terms, e.g., the symmetric one ha, = q(h., + h,,), could easily be adopted, but recall that symmetry in ha, generally means that g., and k,, given in (4.8) and (4.10) are not symmetric. Nevertheless, excellent agreement with experimental observations is obtained with q = 0, which leaves ha, symmetric. 1. Stages I and I1
A simple form for F (single slip) that gives a monotonically decreasing modulus at small strains and a finite (or possibly zero) rate of hardening at large y, is (Bassani and Wu, 1991)
F(y") = (h, - h,)sech2
[
(ho - h,)y" 7,- T o
-k hs
The constants in (5.6), some of which appear in Fig. 7, have the following interpretations: z, is the initial critical resolved shear stress, T, is the Stage I stress (or the breakthrough stress where large plastic flow initiates), ho is the hardening modulus just after initial yield and h, is the hardening modulus during easy glide. A slightly modified form for F is proposed later that can include a Stage I11 saturation modulus that is distinct from h,. A form for the interactive hardening function G that equals unity when its arguments are all zero and asymptotes to finite values when all slips yK, K # a, are large is G({y" 2 0; K = 1, N ,
K
# a}) = 1
+
f,,tanh(yK/yo)
(5.7)
KfU
where yo represents the amount of slip after which a given interaction between slip system a and K reaches peak strength and each fur represents the magnitude of the strength of the interaction. For example, coplanar interactions tend to be weaker than noncoplanar ones. The amplitude factors fuK depend on the type of dislocation junction formed between slip system a and K, which in turn depends on the geometric relation between the two slip systems (Wu and Drucker, 1967; Zarka, 1975; Franciosi et al., 1980; Franciosi and Zaoui, 1982a; Bassani and Wu,1991). In the present study on FCC crystals, fdKare classified into five groups and represented by five constants, ai, i = 1,5:
228 a, a2
a3 a4 as
John L. Bassani No junction: the resultant Burger’s vector is parallel to the original one. Hirth lock: the resultant Burger’s vector is not energetically admissible. Coplanar junction: the resultant Burger’s vector is on the same slip plane as the original ones. Glissile junction: the resultant Burger’s vector is energetically admissible and on one of the two slip planes. Sessile junction: the resultant Burger’s vector is energetically admissible but not on either of the two slip planes.
In general, a, > a4 > a3 > a, > a1 is expected. For each pair of slip systems in the FCC single crystal, Table I1 lists the type of Junction and relates each value off,, to one of the a, values. Note that the slip system ordering in Table I1 is the same as in Table I, which is motivated by the compelling geometric relationships of non-Schmid stresses that lead to the compact forms of (3.12) and (3.13). This ordering is different than that adopted by Wu et al. (1991) although the classical Schmid and Boas (1935) labels, e.g., B4, are preserved; Table I1 in this chapter is a transposed version of Table 2 in Bassani and Wu (1991).
TABLE I1
JUNCTION Types AND I~-rwucnw HARDENING Pmmm~j& IN (5.7) FOR FCC CRYSTALS WITH SCHMID-TYPE BEHAVIOR. 1 (A6) (A3) (A2) (D6) (D4) (Dl) (C5) (C3) (Cl) (B5) (B4) (B2)
1 2 3 4 5 6 7 8 9 1 0 11 12
0 C C N G G H G S H S G
2
0 C G H S G N G
S H G
3
0 G S H
S G H G G N
4
0 C C H S G H G S
5
0 C
S H G G N G
6
O G G N
S G H
7
8
9 1 0 1 1 1 2
0 C C N G G
O C G H
O G
S
H
S
0 C C
0 C
0
Note: N (no junction), f.8 = a,; H (Hirth lock), fa8 = a,; C (coplanar junction),L8 = a,; G (glisslejunction), fm8 = a,; S (sessilejunction),fat = a,.
Plastic Flow of Crystals
229
2. Stages I , 11 and 111 In general, the hardening modulus during Stage I easy glide is distinct from that in Stage 111, when plastic strains are relatively large and the effects of interactive hardening have saturated. A simple way to take this possibility into account is to assume that h, depends on the total accumulated slip y = Eaya on all systems:
h, = hf + (h:" - hf) tanh(y/yg')
(5.8)
where yg' is approximately the accumulated slip at the onset of Stage I11 deformation. In using (5.8) with (5.4) and (5.6) it is tacitly assumed that the relative strengths of interactions among slips in Stages I1 and I11 are the same, although this has not been investigated experimentally. This form has been used by Mohan and Bassani (1992) in three-dimensional studies of shear bands under uniaxial stressing. 3. Cyclic Deformation
Wu et al. (1992) have modified (or reinterpreted) the hardening rule given in (5.4) to (5.7) to include a cycle-dependent hardening as well as a Bauschinger effect, both of which are associated with slip reversals. This is accomplished through a simple modification in the interactive-hardening function G that enters ha, and in the off-diagonal terms in ha,, otherwise the basic form of those equations is preserved. The self-hardening function F is unchanged with the understanding that it depends on the actual integrated slip where positive and negative senses of slip tend to cancel each other. The interactive term G is now expressed as a function of total accumulated slips without regard to sign; i.e., y" is replaced by
1
Y 2 = IdY"l
(5.9a)
Or, with the positive and negative sense of slip counted separately, for FCC crystals this can be rewritten as
(5.9b) where it is understood the sum in the interactive hardening term G involves the first 12 systems.
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John L. Bassani
A Bauschinger effectis introduced in the off-diagonal terms of (5.5) through a cycle-dependent term B(n): h,, = qh,,
a#
h,, = qB(n)h,,,
B,
la - BI # 12
la - BI = 12
(5.10a)
(5.10b)
A simple form for B(n) is suggested by Wu et al. (1992):
B(n) = B, - (B,
+ 1)tanh(n/n,)
(5.11)
where n is the loading-cycle number and B, and no are material parameters. Even with q = 0, simulations of cyclic deformation are able to reproduce the evolution of hysteresis loops as well as the cyclic stress-strain curve under plastic-strain-amplitude control. These results will be presented elsewhere.
D. SIMULATIONS UNDER UNIAXIAL STRESSING The hardening rules proposed earlier in Eqs. (5.4)to (5.7) are used to predict the plastic response of FCC single crystals under uniaxial stressing. For the general case of q # 0, Bassani and Wu (1991) have used a quadratic programming algorithm to calculate the active set of slips from (4.6). For q = 0, which can adequately reproduce the complex hardening behaviors just described, those equations can be solved directly given an imposed stress rate or deformation rate on the lattice as noted earlier. Selected results will be presented that display: variations in the overall stress-strain curves for crystals with initial orientation of the tensile axis, variations in the flow stress on secondary systems with both primary and secondary slip in the latent hardening experiment and secondary slip activity before tensile axis reaches the symmetry boundary followed by tensile overshoot. The results that follow are for copper single crystals. The elastic constants possess cubic symmetry and are normalized by the magnitude of the initial critical resolved shear stress, 70, which is the same for all slip systems: C1, = 10 x 103.r0,C, = 3.8 x 1037, and C12= 7.35 x lo3?,. The constants in the hardening law (5.4) to (5.7) are 7, = 1.57,, h, = 1.5z0,yo = 1O-j and the slip interactions governed by the factors fmfl are a , = a2 = a , = 8, a4 = 15 and a , = 20 (see Table 11). These values are based on the experimental results of Wu et al. (1991) as well as those of Basinski and Basinski (1970) and Franciosi et al. (1980) on single crystal copper with purity of at least 99.99%. For these simulations q = 0 is assumed, which ensures positive definiteness of h,, and, at each stage in the deformation, that the set of slips is uniquely
Plastic Flow of Crystals
231
determined when the stress rate is imposed relative to the lattice frame as discussed in Section 1V.B. Complete details of the calculation are given in Bassani and Wu (1991). The predicted uniaxial stress-strain curve for a single crystal with the “single-slip” orientation where the tensile axis is initially along the [632] crystal direction is plotted in Fig. 10. The material fiber along that direction is held fixed so that the lattice rotates with respect to the tensile axis. So long as the tensile axis is within the spherical triangle the Schmid factor on the primary system is nearly 0.5 so that the uniaxial stress is approximately twice shear stress on the primary system. For all “single-slip’’ orientations considered, the primary system is B4 (see Tables I and 11). As the tensile axis rotates towards the [lOO]-[ 1113 symmetry boundary the Schmid factor decreases. During easy glide in Stage I only one slip system is active, and it hardens faster than the others. For the present case where 4 = 0 in (5.5) only the primary system hardens during Stage I. Eventually, the critical shear stress is exceeded on one or more secondary systems before the tensile axis reaches the symmetry boundary, and this leads to the high multiple-slip hardening of Stage 11. Thereafter, the deformation is no longer single slip. Nevertheless, the tensile axis eventually overshoots the symmetry boundary as seen in experiments. That is, even though latent hardening is absent in this calculation, overshoot is still predicted. In Stage I deformation the very fine secondary slips that have been reported, for example, see Kulhmann-Wilsdorf (1989), are precluded in the time-independent framework adopted here. Although these fine slips do not
lOOr
I
“0
6.05
0.1
0.15
0.2 0.25
e FIG. 10. Simulation of uniaxial tension for a single crystal with the “single-slip” orientation [632]. Note Stage I and Stage I1 deformations and overshoot of the symmetry boundary.
John L. Bassani
232
contribute significantly to the overall hardening behavior, they could naturally arise in a time-dependent framework. Figure 11 shows the stress-strain curves for various orientations of the tensile axis. The dependence of the overall hardening rate and the extent of Stage I easy glide on the initial orientation is in reasonable agreement with experiments (see Diehl, 1956). Note that as the orientation approaches the [1001- [1113 symmetry boundary, the length of easy glide decreases whereas hardening rate increases. This is due to secondary slip@)and the resulting slip interactions, which are captured by the interactive hardening function G given in (5.7). For the high-symmetry [ill] orientation of the tensile axis where multislip commences immediately after plastic deformation, easy glide and lattice rotation are both suppressed as observed in experiments. The initial high hardening rate is due to slip interactions. An important test of the characterization of slip interactions is the behavior corresponding to the high symmetry orientations. From Fig. 11, note that the [lll] orientation has a higher hardening rate than [lo01 even though [111] has a fewer number of active slip systems; namely, six versus eight. This is due to different types of dislocation locks that are formed between the active slip systems, which are characterized by the f,@ factors (see Table 11) in the interactive hardening function. It is also seen that even though cross slip and other thermally activated mechanisms are not included in this formulation, a trend toward the Stage TI1 deformation (parabolic hardening) is still predicted for some orientations. The hardening rates of all slip systems that are activated and the individual slips are plotted in Fig. 12 as a function of overall strain for a single-slip
0
OD5
0.1 0.15
0.2 0.25
E
FIG. 11. Simulation of uniaxial tension for single crystals with various Orientations.
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233
final
\
50
OO
0.1
0.2
E
A 0.1 0.2
1 ooo
:::K; (b)
0.5
r
0.2
0.1
0
0
0.1
0.2
E
(c)
FIG.12. (a) The (T-E curve for uniaxial tension with a “single-slip”orientation. (b) Variation of the active hardening rate (drldy) on slip systems B4, B5, C1 and A3. (c)Accumulated slip on slip systems B4, B5, C1 and A3 during the deformation.
orientation of the tensile axis (the standard notation of Schmid and Boas (1935) for the slip systems is used). Note that even before the tensile axis reaches the symmetry boundary significant amounts of secondary slips are predicted in accord with the measurements by Basinski and Basinski (1970). It is also seen that the hardness of the slip systems becomes very anisotropic.
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John L. Bassani
For example, among the seven active slip systems, B4 is the primary system and A3 is the least stressed system, and after a uniaxial strain of 25%, z,,(B4)/zC,(A3) x 25. This ratio is consistent with the variation in the ratio of the Schmid factors for these two slip systems throughout the deformation. Finally we note that the latent hardening theory of Havner and Shalaby (1977) (see also Havner and Shalaby 1978; Havner, Baker, and Vause 1978) applied to uniaxial stressing in a single-slip orientation leads to a greater hardening of latent systems compared to that of the active (primary) system so that these secondary slips are precluded even beyond overshoot. A typical simulation for the latent hardening experiment is shown in Fig. 12. In Fig. 13(a)the stress-strain curves corresponding to the primary (z,-y,) and secondary (~,-y,) tests are plotted. Figure 12(b) is a plot of the “flowstress ratio,” which is defined as FSR = zs(yp, yso)/zp(yp,0), versus the amount of slip in the primary test y p at four values of off-set slip ys = yso in the secondary test. This ratio is commonly referred to as the latent-hardening ratio (LHR), but since these results include the behavior when the secondary
Scconda ry
0
0.02
Y
0.04
(a)
0.005.
rso
0.002
’
‘0
0.001 0
0.02
0.04
YP (b)
FIG.13. Simulation of the latent hardening experiment on non-coplanar slip systems that form glissile dislocations;e.g., B4 and A2. (a) The 1-7 curves for the primary and secondary tests. (b) Flow stress ratio ( T . / T ~ )versus primary slip (y,) for different offset secondary strains (y.,).
Plastic Flow of Crystals
235
slip is small (or even zero) and the FSR tends to be less than unity, the term FSR is used to avoid confusion with the notion that the LHR is greater than unity for finite secondary slips. Note, when yso = 0, which precisely corresponds to initial yield of secondary system, that FSR < 1 is consistent with our conjecture that the hardening of a truly latent or inactive system is less than that of an active system. For yso on the order of or greater than about FSR > 1. Bassani and Wu (1991) also demonstrate that the extent of hardening of the secondary system (“latent hardening”) depends on the type of secondary system relative to the primary system; that is, the strength of the particular slip interactions. These predictions reasonably reproduce various experimental findings (see, e.g., Edwards and Washburn, 1954; Kocks, 1964; Franciosi et al., 1980; and Wu et al., 1991). In summary, the significant influence of total slips in all systems on the hardening of each system has been experimentally verified and successfully incorporated into a new hardening law. In the next section, the significant influence of secondary slips, which have been precluded in earlier studies, on strain localization in the form of coarse slip bands and macroscopic shear bands is demonstrated. Non-Schmid effects on localization are also addressed.
VI. Strain Localization After some more or less uniform straining, plastic deformation often concentrates within narrow bands. Upon further straining, the deformation may become diffuse again or it may continue to concentrate, leading to fracture in the vicinity of the bands. Localized plastic flow usually initiates at inhomogeneities in the material and can occur at various microstructural scales. Within single crystals, inhomogeneities such as fine precipitates can lead to localization early in the deformation in the form of coarse slip bands ( E w 0.005-0.01 in the logarithmic measure) that are crystallographic in nature and appear to be the precursor to the noncrystallographic macroscopic shear bands (E x 0.1-0.03). These two types of shear bands are seen in Fig. 14, which is taken from the work of Chang and Asaro (1981) on Al2.8%Cu crystals. Other examples are found in Elam (1927), Cahn (1951), Piercy, Cahn, and Cottrell(1955),and Price and Kelly (1964) to cite a few. It is important to note that coarse slip bands tend to harden so that some (and possibly considerable) stable deformation, albeit inhomogeneous, can continue beyond the onset of localization.
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John L. Bassani
FIG.14. Coarse. slip band (CSB) and macroscopic shear band (MSB) formation in Aluminum2.8% wt. copper alloys. The CSBs are oriented on the slip planes while the MSBs are misoriented by a few degrees. (Reprinted with permission from Acta Metullurgicu 29, Chang, Y. W., and Asaro, R. J., An experimental study of shear localization in aluminum-copper single crystals. Copyright 1981, Pergamon Press Ltd.)
Strain localization is analyzed in this section in the form of a shear band bifurcation that results from a constitutive and geometric instability as Rudnicki and Rice (1973),Hill and Hutchinson (1975), Rice (1977),Asaro and Rice (1977), Peirce (1983), and Qin and Bassani (1992b) among others have viewed it. With coarse slip band formation in the early stages of uniaxial tension or compression deformations in mind, Asaro and Rice (1977) analyzed shear befurcations under single-slip conditions. When the normality structure governs flow (Schmid behavior) they found that bifurcations are precluded at positive values of the (instantaneous) slip-system hardening modulus. Under truly single-slip flow, non-Schmid effects are necessary to trigger shear bifurcations at positive hardening (Asaro and Rice, 1977; Qin and Bassani, 1992b).Based on his own experiments and those of Price and Kelly (1964) on aluminum single crystals, Asaro (1983) noted that material in these coarse slip bands generally hardens during flow localization. Subtle details of slip interactions, which have been commonly neglected in mechanics analyses, are essential in understanding overall hardening of single crystals. Secondary slips are not only the key to fundamenal hardening
Plastic Flow of Crystals
237
behaviors under uniaxial stressing in single-slip orientations, they are also the key to coarse slip band formation and, to a lesser extent, the details of macroscopic shear band formation. In this section, three shear band analyses are presented for single slip with non-Schrnid effects, nonsymmetric double slip with the effects of non-Schmid stresses as well as secondary slips and a three-dimensional crystal model with the effects of secondary slips for Schmid-type behavior. As noted earlier, with the effects of non-Schmid stresses in the yield criteria, single-crystal plastic flow is not associated with the yield criterion in the sense of Hill (1950). As previous investigators (e.g., Rudnicki and Rice, 1973; Asaro and Rice, 1977; Needleman, 1979; Raniecki and Burhns, 1981; Qin and Bassani, 1992b) noted, such nonnormality in the constitutive description can significantly ease the condition at which the shear bands develop. A. SHEAR BIFURCATIONS Without loss in generality, in analyzing bifurcations from a homogeneously strained state we can take the reference state to coincide instantaneously with the current state. For strain localization within a thin planar band, the difference in the velocity gradient inside and outside the band with unit normal k is (Hill, 1962; Rice, 1977) AL=g@k
(6.1)
with symmetric and antisymmetric parts, respectively, AD=3(g@k+kkg)
(6.2a)
AW = $ (g@ k - k @ g )
(6.2b)
where g is a nonzero velocity vector and A denotes the difference in quantities inside and outside the localized band. If the rate of dilatation is the same both inside and outside the band at the localization, then g lies in the plane with normal k since A tr(D) = giki = 0. Equilibrium requires that the traction rate is continuous across the band
k.Ab
=0
(6.3)
With the reference and current state instantaneously coinciding and since A tr(D) = 0, we begin by rewriting (4.11) with (4.12) and (2.12) in terms of the velocity gradient L: u=9:L.
(6.4a)
John L. Bassani
238
or in component form with Lij = &+/axjand v is the velocity vector: bij= 9 i j k l aok/ax,
(6.4b)
where g j k l
= %jkl - 3 ( a i k s j l - ailsjk
+ ajk6il
- @jI6ik)
(6.5)
Note that g j k j = 4 i k I # 2 i j l k and that $jkl # 9 k l i j . Assuming that the same instantaneous hardening moduli (6.5) are valid both inside and outside the band, taking differences, and substituting the result into (6.3) with (6.1), we arrive at (k*&-k)*g= 0
(6.6)
Therefore, the condition for localization on a plane with normal k, i.e., a nontrivial solution g to (6.6), is det(k.9.k) = 0
(6.7)
Several examples of shear localizations will be presented that emphasize the effects of small secondary slips, the effects of multiple-slip hardening and non-Schmid effects. The problems essentially reduce to determining the critical states satisfying the determinantal equation (6.7), although other approaches have been taken. In the case of the single-slip model, the calculation follows Asaro and Rice (1977) and Qin and Bassani (1992b). In the case of the double-slip model, the calculation follows Asaro (1979) and Qin and Bassani (1992b). Some recent results based on a nonsymmetric double-slip model (Qin and Bassani, 1992c) are also presented that bring in the effects of small secondary slips. Finally, a three-dimensional crystal undergoing Schmid-type behavior is analyzed under uniaxial stress (Mohan and Bassani, 1992). The latter not only predicts the formation of coarse slip bands, but it also resolves some issues relating to macroscopic shear bands. For brevity and to emphasize the results, the details of the analyses are omitted since the essentials already appear in the literature and two other papers are forthcoming that detail these new results. B. CRYSTALS UNDERGOING SINGLESLIP This case was first analyzed by Asaro and Rice (1977). An analysis that builds on the non-Schmid flow behavior described in Section IV is given by Qin and Bassani (1992b). Let j~ = 2:Jh = 2*/h under single slip, where 2* is given by (3.7) with (3.9, (3.8) and (3.9); in this subsection, which is limited to
Plastic Flow of Crystals
239
single slip, the slip-system index is omitted. Neglecting terms on the order of stress relative to the elastic modulus, the hardening modulus at localization satisfying (6.7) is (see Eq. (76) in Qin and Bassani, 1992b)
h = -d*:3ye:d where
N
= 9'-
(~e.k).(k.~e.k)-'.(k-~e)
(6.9) and d is given in (2.4a) and d* is given in (3.4a), which involves the nonSchmid terms. The fourth-order tensor 3ye has the same symmetries as Y e and is positive semi-definite. If the non-Schmid effects are neglected, in which case a,, = 0 and d* = d, and since M e is positive semi-definite, one can show that the greatest value of h satisfying the localization criterion under Schmid-type behavior is h,, = 0 corresponding to k = n or k = m,where n and m are the slip vectors (Asaro and Rice, 1977). For a small departure from Schmid-type behavior corresponding to la, 1 small compared to unity, let k=n+K
(6.10)
where 1uI also is assumed to be small. Substituting (6.10) into (6.9) and the ) O(a,,PK) (see Eq. result into ( 6 4 , and retaining terms up to O ( 9 e ~ 2and (78) in Qin and Bassani, 1992b)
h=
-K'(m*N"'m).K
+ c (a,,d,,:JY"-m)*w
(6.11)
B
where the sum is over the non-Schmid terms as in (3.4a) and m is the slip direction. The maximum h for nonzero K is readily determined from (6.1 1):
h,,
1
=-
(6.12) c c (a,,d,,:3Ye-m)-(m~3Ye-m)-'~(a,Ke~m:du)
4 v
u
which is positive and achieved at the critical orientation of the shear band normal (6.1 3) Note that m . .".in has no components along the direction n, and therefore, its inverse is defined on the plane whose normal is n.
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John L. Bassani
To summarize, for a single crystal with one active slip system the hardening modulus at the onset of strain localization is nonpositive for the classical normality flow rule involving only the Schmid stress, and it is zero when the localization plane coincides with the slip plane, which is the critical situation. With the effects of non-Schmid stresses on flow included, i.e., nonnormality, the critical hardening modulus can be positive for localization under single slip and the critical plane slightly deviates from the slip plane. Finally, we note that significant shear banding has been observed prior to failure in Ni3Al by Heredia and Pope (1991).
C. PLANAR DOUBLE-SLIP MODELS It is well-known that the kinematical freedom associated with a vertex on the yield surface at the stress state, which is inevitable for single crystals undergoing plastic deformation beyond easy glide with or without nonSchmid effects, enhance the possibility for strain localization (or bifurcations). Asaro (1979) developed a symmetric double-slip model in two dimensions and demonstrated that with Schmid-type behavior, i.e., the normality flow rule, the critical hardening modulus at which localization occurs is positive although rather small. The latter implies a later stage of plastic deformation, and his results have features consistent with macroscopic shear bands. Peirce (1983)considered the actual three-dimensionalconfiguration, from which the two-dimensional (2-D) model is derived, for symmetric double slip (Schmidtype behavior) and found trends similar to those of the 2-D model, except that the critical hardening modulus in his 3-D case tended to be much smaller in magnitude and the band angle in poorer agreement with experiment. This unexpected result is resolved in the next subsection. Qin and Bassani (1992b) extended Asaro’s (1979) symmetric double-slip model to include non-Schmid effects. As expected, and consistent with the single slip results, non-Schmid effects tend to enhance localization as compared with Schmid-type behavior. Another variant of this model has recently been developed by Qin and Bassani (1992~)that allows asymmetries through different active hardening rates on the two systems. The latter model can incorporate the effects of small secondary slips, with and without nonSchmid effects, which begins to explain the formation of coarse slip bands. The geometry of these models is shown in Fig. 15 where the loading stress is uZ2.The two slip systems are taken to be symmetrically disposed about the uniaxial loading axis, which can represent the behavior in uniaxial stressing
Plastic Flow of Crystals
24 1
2
022
FIG.15. The 2-D symmetric double-slip model.
of a single crystal where lattice rotations tend to bring the primary and secondary slip systems into such a configuration. In Fig. 15, the angle 4 gives the orientation of the two slip systems (and, therefore, the Schmid stress) and 19 represents the direction of localization band. A hypothetical non-Schmid shear stress is introduced for each slip system, whose orientation is characterized by the angle $ (dashed lines in Fig. 15). Qin and Bassani (1992~)have also carried out a calculation for the case when the two slip systems are not symmetrically disposed about the loading axis-these results are qualitatively similar to those for the symmetric configuration and will not be presented here. In the double-slip model, the yield criterion for the two slip systems that includes non-Schmid effects when a, # 0 is taken from (3.3): .pa
= Ta + a 1z~1
- T,,*a
(a = 1 9 2 )
(6.14)
where T~ are the Schmid stress components on the two systems at an angle of + 4 to the loading axis and 7; are the non-Schmid stress components oriented at fh ,t. The non-Schmid factor a , characterizes the deviation from normality. For analytical simplicity, the crystal is assumed to be elastically isotropic
242
John L. Bassani
and incompressible, therefore only the elastic shear modulus G enters. The hardening matrix in 2: = Zflh,,ja is chosen to be symmetric: (6.15) where H characterizes the magnitude of hardening, q is the latent hardening ratio and p measures the relative hardening rates on the primary and secondary systems (Qin and Bassani, 1992~).Since the slip systems are symmetrically disposed, if lattice spin is ignored, i.e., .5*' = 2*2, then at bifurcation (6.16) Asaro (1979)and Qin and Bassani (1992b)took p = 1 so that the slip rates on the two systems at bifurcation are equal. The motivation here in taking p 2 1 is to account for the high rate of hardening associated with the onset of small secondary-slip activity in a predominantly single-slip orientation as indicated in inequality (5.3~).Note, with p large and q < 1, that p = 1. Symmetric Double Slip For an incompressible single crystal with two symmetrically oriented systems undergoing equal slip, which corresponds to p = 1, analytical solutions for the bifurcation condition can be obtained. Asaro (1979) solved this problem for a normality flow rule and Qin and Bassani (1992b) included non-Schmid effects. In this case, the conditions for shear bifurcations given in (6.1) to (6.7) are satisfied when R , sin228
+ R 2 cos228 = (az2/2G)cos 28
(6.17)
where the two nondimensional functions R , and R 2 that depend upon 022, H,q, alG, 4 and $ are R,
=
+ a, sin 2JI) S + cos 2JI) - (1 + ~ , ) a i , / 2 G (1 - q)H + O ~ ~ ( C O24 R2 = (1 - q)H + 2G cos ~ ~ ( C 24 O S+ a, cos 2$) - (1 + cos 2$ (1
+
(1 + 4)H q)H + 2G sin 2$(sin 24
(6.18a) (6.18b)
~,)022
Equations (6.17) and (6.18) are similar to Eq. (101)in Qin and Bassani (1992b) except for differences arising from the choice for convection of the lattice
Plastic Flow of Crystals
243
vectors. Due to elastic stretching of lattice vectors, which is assumed in this chapter (see Eqs. (2.6)) but not in the other, the last terms involving 1 + a , in the numerator and denominator of (6.18) do not enter the other formulation where the lattice vectors are assumed to rotate rigidly. Nevertheless, when the magnitude of the elastic modulus is much larger than that of the stress (laz2/GJ<< l), the aforementioned differences in the hardening modulus, stress and shear band angle at bifurcation are very small. In fact, for rigid-plastic crystals, G -P 00, (6.18) reduces to (6.19) where S=
sin228 sin 24(sin 2 4 + a , sin 2$)
(6.20a)
C=
cos22e cos 24(cos 2 4 + a , cos 2lj)
(6.20b)
For metals, since H tends to monotonically decrease as 0 2 2increases, with q, a , , 4, and $ prescribed, the maximum value of H / ~ from Y ~ (6.19) ~ for some band angle 8 would correspond to the critical condition for the first bifurcation. For example, with 4 = 35", $ = 10" and a , = q = 0, (H/O& z 0.065 and the shear band is at an angle of 8 z 49", which is about 5-6" off the slip plane. As suggested by Chang and Asaro (1981) and Asaro (1983), these results are consistent with the occurrence of macroscopic shear bands seen in Fig. 14. There is a small and approximately linear increase in ( H / ~ Ywith ~ ~ ) ~ ~ increasing a , due to non-Schmid effects. Increasing the latent hardening ratio q tends to decrease ( H / C T , ~ ) Examples ~,. of these and other results that emphasize non-Schmid effects and include elasticity are given in Qin and Bassani (1992b). Selected results follow.
2. Nonsymmetric Double Slip With p # 1 in (6.15) there are unequal slip rates at bifurcation (see (6.16) even though the slip systems are symmetrically disposed about the loading axis. Once again (6.1) to (6.7) govern, although now a simple analytical solution for bifurcation cannot be found. Qin and Bassani (1992~)have obtained numerical solutions for the geometry of Fig. 15 with 4 = 35" and
John L. Bassani
244
I) = 10" for various values of p, q and a, and G >> uzz.These results agree with (6.17) to (6.20) for p = 1. As noted previously, p measures the relative hardening rates on the primary and secondary systems. Recall the extensive discussion in Section V about the relatively high rate of hardening associated with the onset of secondary slip activity in a predominantly single-slip orientation. For double slip, this is stated in inequality (5.3c), which is consistent with p >> 1 and
P
=
Pl/?Z.
With p, q, a,, 4 and I) (and G >> oz2)prescribed, for a certain range of band angles positive values of H / u 2 , exist satisfying (6.1) to (6.7); the maximum value corresponds to the critical condition for the first bifurcation. Figure 16(a)is a plot of H / o z 2 versus 8 for q = 0 (i.e., no latent hardening), a , = 0 (Schmid behavior), and p = 1 (symmetric slip), 10, 100 and lo00 (very small secondary slips). For a given p, the maximum on the H / a z z - B curve corresponds to the first or critical bifurcation at a well-defined band angle. In the case of symmetric slip (p = l), which agrees with Asaro (1979) and Qin and Bassani (1992b), (H/uzz)crx 0.065 and the shear band is at an angle of 8 = 49", which is about 5-6" off the slip plane. As p increases and secondaryslip activity decreases relative to the primary activity, (H/U& decreases, as plotted in Fig. lqb), and the critical angle moves toward the primary slip system. The overall (observed) hardening is greater due to the higher hardening rate on the secondary system; this is seen in the three-dimensional results presented later. The influence of non-Schmid effects are seen in Fig. 17. For p = 1, increasing a , increases (H/uzz),, while the critical band angle changes little as seen in Figs. 17(a) and 17(b). These results are for q = 0. Qin and Bassani (1992b) have shown that the critical band angle generally varies little with either a, or q. The combination of non-Schmid effects and hardening associated with small secondary slips, i.e., a, > 0 and p > 1, is interesting. From Fig. 17(c)it can be seen for 1 < p < 10 that the non-Schmid effects tend ~ ) , ~ the corresponding value for Schmid behavior (a, = 0) to raise ( H / U ~ above while for p > 10 the opposite is true. As in the case of a , = 0, for a, > 0 as p increases and secondary-slip activity decreases relative to primary slip, (H/oz2)crdecreases and the critical angle moves toward the primary slip system. Based on the multislip hardening phenomena described in Section V, we can speculate on the relative stability of the shear bands corresponding to p = 1 and p >> 1. In the former case, equal hardening rates and slip activity on the two systems would be typical of large-strain, Stage I11 deformations
245
Plastic Flow of Crystals H/UZ
-
.I
p=looO p = 100
0
p = 10
- 1
t
I
10
100
lea3
P (b) FIG. 16. The effects of a secondary slip activity: (a) localization condition for different bifurcation modes (shear band orientation) and (b) (H/u2Jcrat band orientation, corresponding to maximum H/u,, from (a) for a given ratio p = h,,/h, with q = al = 0.
where the overall hardening rate tends to monotonically decrease and is much lower than in Stage 11. Consequently, the postbifurcation flow is likely to remain concentrated in these bands, eventually leading to fracture, which is consistent with observations of macroscopic shear bands (Asaro, 1979,
John L. Bassani
246
-
0.08
---
ll -0.0
.,-0.1
---- r1-0.2
0.06
__--.-. ilmo.5
0.04
0.02
0
1
10
100
let03
Fic. 17. Non-Schmid effects: (a) localization condition for different bifurcation modes (shear band orientation);(b) (H/u& at band orientation corresponding to maximum H/uz2 from (a) for a given with p = 1, q = 0; and (c) combined effects of non-Schmid factor and secondary slip activity with q = 0.
1983). In the latter case, p >> 1, the very high self-hardening rate on the secondary slip system compared to that on the primary system is observed at relatively small strains at the onset of Stage I1 deformations (see inequality 5.3~). The subsequent high overall hardening rate in Stage I1 would tend to stabilize these localized bands as observed in the case of coarse slip bands. This conclusion is consistent with the three-dimensional results presented later that are based on the hardening laws developed in Section V.
Plastic Flow of Crystals
247
Finally, we note that Qin and Bassani (1992b) have shown that latent hardening tends to lower ( H / ~ Y ~for ~ )both ~ , Schmid and non-Schmid behavior with p = 1. The same is true for a given p # 1. D. A THREE-DIMENSIONAL CALCULATION In this study a three-dimensional FCC crystal is loaded in uniaxial tension from an unstressed state. Only Schmid-type flow behavior is considered. The tensile axis is assumed to remain aligned with a given material fiber so that the lattice rotates with respect to that axis during loading. The complete evolution of slip activity, including secondary slips, of hardening according to the theory developed in Section V and the instantaneous moduli entering the bifurcation condition (6.7),are determined throughout the homogeneous part of deformation. Both coarse slip band formation at the transition from Stage I to Stage I1 for “single-slip” orientations and macroscopic shear band formation in Stage 111 for “double-slip” orientations are predicted. The precise form of hardening law used is given by (5.4) with (5.6) to (5.8). Both isotropic and cubic elasticity are considered. The constitutive equations are explicitly integrated up to the first bifurcation satisfying (6.7) which determines (h/a),, and the shear plane orientation. That state is determined by tracking the plane that minimizes the determinant in (6.7) as the load is increased (see Ortiz et al., 1987). In the single-slip cases considered, the integration led to other components of stress that, at the bifurcation state, are at most three orders of magnitude smaller than the applied uniaxial stress. In the symmetric double-slip orientation, secondary stresses are essentially nonexistent. Details appear in Mohan and Bassani (1992). The parameters in the hardening law are taken to describe copper with q = 0 (i.e., no latent hardening);most have been given in Section V.D, while the additional parameters that enter (5.8), which distinguish the saturation hardening rate in Stages I and I11 are h l = 1 . 3 ~and ~ hi” = 0 . 0 7 5 ~ ~ To set the stage for these three-dimensional bifurcation results, we first recall the macroscopic shear band analysis of Peirce (1983),which accounted for the actual 3-D geometry, with symmetric slip restricted to the primary and conjugate systems. In fact, this is the geometry from which Asaro’s approximate 2-D double-slip model is derived. An essentially identical bifurcation analysis for the 3-D configuration predicted the formation of a (macroscopic)shear band at very small values of ( H / ~ Y much ~ ~ ) smaller ~ ~ , than in the 2-D model. and there was little or no misorientation between the
John L. Bassani
248
primary slip system and the shear band. Based upon the calculations presented later, we conclude that small secondary slips also are very important in the 3-D symmetric slip case. Why they apparently are not as important in the 2-D model is not clear, but it does raise questions about the validity of this model in the case of symmetric double slip. 1. Coarse Slip Bands
Coarse slip bands are observed early in uniaxial deformations under “single-slip” orientations. Three results for two orientations are plotted in Fig. 18, where the terminus of each curve corresponds to the instant of bifurcation. The curve for orientation 1 assumes elastic isotropy (E = lo4 T,, and v = 0.3), while for orientation 2, results for both elastic isotropy and anisotropy are given; for the latter case the constants are given in Section V.D. First, we note that the localization condition for these single-slip orientations is satisfied at small overall logarithmic strains on the order of 1%. Furthermore, the critical plane orientation deviates very slightly, on the order of 0.001” as Chang and Asaro (1981) (see Fig. 14) and others have observed. These results are summarized in Table 111, which also includes the ratio of the instantaneous hardening on the primary system (B4) to the overall applied stress. (The notations for the slip systems are those given in Table 11.) Since the strains are small, the term h: in (5.8) dominates. For each of the three cases in Fig. 18 and Table 111, secondary slips initiate before bifurcation. Recall that in pure single slip, for Schmid-type behavior
0
0
,002
.OM .006 ,008 .010 ,012 ,014 &
FIG.18. True stress-logarithmic strain curves up to coarse slip band initiation (terminus)in “single-slip”orientations. Table 111 defines parameters for each curve.
Plastic Flow of Crystals
249
TABLE 111
THREE-DIMENSIONAL COARSE SLIP BANDCALCULATION. Case 1 2i 2a
Elastic
Misori.
(hBl/4cr
Ecr
Isotropy Isotropy Anisotropy
0.001" 0.001 0.002"
0.95
0.7% 1.1% 1.3%
0.41 0.28
the slip system hardening must go to zero for the first bifurcation (see (6.12) with a,, = 0). The variation of the hardening moduli on the primary as well as the secondary systems are plotted in Fig. 19; the arrows indicate when the secondary system(s) is activated. For Case 1, one secondary system (Cl) is
hlo;:L 10
,
84
0
,004
,008
1
I
I
I
(a)
Case 28-single slip
0
,010
,005
.015
E
(C) FIG. 19. Evolution of the slip-system hardening moduli on the primary (B4) and secondary systems (Cl), (A3) or (BS)corresponding, respectively, to the three cases in Fig. 18 and Table 111: (a) Case 1, (b) Case 2i; (c) Case 2a.
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John L. Bassani
activated when the overall strain reaches 60% of the bifurcation strain. For Cases 2i and 2a, two secondary slip systems (A3) and (B5) are activated and, just after that, the localization criterion (6.7) is met. Since the secondary slips are very small, inequalities (5.3a-c) are satisfied. For example, hg4 decreases to a level three orders of magnitude less than the hardening rates on the secondary systems. The important role of (small) secondary slips is now clearly demonstrated for the formation of coarse slip bands. In the examples given, these form during the beginning of the transition from Stage I to Stage I1 hardening, when slip interactions are complex. Their role has been suggested from experimental observations as well (see, for example, Basinski and Basinski, 1979; Kulhmann-Wilsdorfand Laird, 1980; and Basinski and Basinski, 1992), but further investigation is needed. This small slips and high hardening rates are dimcult to detect, and the latter can be mistaken for elastic as KulhmannWilsdorf (1989) has suggested. 2. Macroscopic Shear Bands Macroscopic shear bands are observed at relatively high levels of overall strain. Four results for a symmetric “double-slip” orientation are plotted in Fig. 20, where the terminus of each curve corresponds to the instant of bifurcation. Each curve corresponds to the same initial orientation [T12] with different elastic assumptions and values of the saturation slip parameter #. Basically, with a given value of h:” in (5.8) the larger is the value of $, the
01 0
I
.04
I
I
.I2
.M,
I
.I6
I
20
E
FIG. 20. True stress-logarithmic strain curves up to macroscopic shear band initiation (terminus) in “double-slip”orientations. Table IV defines parameters for each curve.
25 1
Plastic Flow of Crystals
larger the critical strains that, for the material parameters chosen, are on the order of 0.1; i.e., an order of magnitude larger than that for the coarse slip bands. Furthermore, the critical plane orientation deviates by 2", which is similar to the surface observations Chang and Asaro (1981) (see Fig. 14). The results are summarized in Table IV, which includes the ratio of the instantaneous hardening on the primary systems (B4)and (Cl) to the overall applied stress. Since the strains at bifurcation are relatively large, the term in (5.8) dominates. For each of the four cases in Fig. 20 and Table IV, secondary slips initiate before bifurcation. Recall that in Peirce's (1983) 3-D double-slip calculation for Schmid-type behavior, (h/a,,) is predicted to be very small, much lower than the value of 0.65 predicted in the 2-D double-slip model; moreover, the shear band is essentially oriented with the primary slip systems in disagreement with Chang and Asaro's (1981) observations and the results summarized in Table IV. The variation of the hardening moduli on the primary as well as the most highly strained secondary systems (A3) and (B5)are plotted in Fig. 21. Four other secondary systems-(A2), (C5), (B2) and (C4)-are activated before bifurcation, but their slips are one to two orders of magnitude smaller than those other two secondary systems-(A3) and (B5). As in the case of coarse slip bands, since the secondary slips are very small, inequalities (5.3a-c) are satisfied; e.g., the hardening rates on the secondary systems are three orders of magnitude greater than those on the primary systems. Once again, the important role of (small) secondary slips is demonstrated, this time for the formation of macroscopic slip bands. In the examples given, these form during Stage 111 hardening. The only apparent differencebetween the calculation presented here and that of Peirce (1983) is the role of secondary slips. TABLE IV THREE-DIMENSIONAL MACROSCOPIC SHEAR BANDCALCULATION. Case 1
2 3 4
Y6' 0.1 0.075 0.075 0.15
Elastic
Misori.
Isotropy Isotropy Anisotropy Isostropy
2.05" 2.03" 1.9" 1.9"
(hB4/4,,
0.034 0.035 0.032 0.031
EC,
12.1% 9.3% 9.7% lE%
252
John L. Bassani Case 1
Case 2 A3,85
h
10
I .10
I
1
.05
0
E
E
(a)
Case 3
Case 4
h
0
.05
0 .fO
E (C)
.1
.2
E
(4
FIG.21. Evolution of the slip-system hardening moduli on the primary systems (B4)and (Cl) and two of the secondary systems (A3) and (B5)corresponding,respectively, to the four cases in Fig. 20 and Table I V (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4.
E. YIELDVERTICES AND SECONDARY SLIPS As demonstrated both from the nonsymmetric double-slip model in the case of small secondary slips ( p >> 1) and from the three-dimensional calculation, secondary slips have a significant effect on the shear band predictions. This can be understood in terms of corners on yield surfaces, which are known to be generally important for phenomena associated with bifurcations in the plastic range. For example, bifurcation predictions of
Plastic Flow of Crystals
253
plastic buckling based on the classical J , flow theory, in which case the yield surface is smooth, tend to overestimate buckling loads whereas the incremental version of J 2 deformation theory, which gives rise to a corner on the yield surface at the active stress state, is in better agreement with experiment. In fact, this well-known discrepancy motivated Sewell (1973, 1974) and Christoffersen and Hutchinson (1979) to develop phenomenological corner theories of plasticity. Recall, in single slip governed by Schmid-type behavior (d* = d) that shear bifurcations are precluded from (6.8) for positive values of the slip-system hardening rate h. In this case the active stress state is on the flat of a hyperplane of the single-crystal yield surface and the plastic strain rate is constrained to be normal to that plane. Non-Schmid effects relieve the normality constraint and permit bifurcations at positive values of h. In the planar double-slip model the active stress state is at the vertex of two intersecting hyperplanes, i.e., a twofold vertex, and for Schmid behavior the plastic strain rate lies in the forward cone of normals. Consequently, the change in strain rate at bifurcation can be accommodated at positive levels of hardening as Asaro (1979) demonstrated in the case of symmetric double slip. The active stress state is also at a twofold vertex in the case of nonsymmetric double slip. Even when p >> 1, which corresponds to nearly single-slip conditions, the fact that the plastic strain rate can lie in the forward cone of normals and is not constrained to the direction of the normal to the hyperplane of the primary system is sufficient to give bifurcations a positive slip-system hardening rate. The same argument holds for the threedimensional calculation of coarse slip bands that form under nearly singleslip conditions. Referring to Table I11 and Figs. 18 and 19 we note that Case 1 involves a twofold vertex at bifurcation while Cases 2i and 2a, which correspond to a different initial orientation of the crystal from that in Case 1. involve a threefold vertex. Finally, recall the three-dimensional calculation for the symmetric orientation where the tensile axis in the [i12] crystal direction. In Peirce’s (1983) calculation that accounted for only the two primary slip systems, the active stress state at bifurcation is at a twofold vertex. In this case the predicted hardening rate at bifurcation is very low, and the shear band is nearly oriented on a slip system, both of which are in conflict with experiments. With the new description of hardening, which takes truly latent hardening to be less than the active hardening, six additional secondary systems are activated before bifurcation, two of which are more highly strained than the other four. Therefore, at bifurcation the stress state is at an
254
John L. Bassani
eightfold vertex, and the hardening rate and shear band orientation are significantly different and in much better agreement with experiments. VII. Closure
Finite strain crystal plasticity is a rigorous nonlinear continuum theory. Over the last few decades, when much progress was made, the emphasis has been on the mathematical formulation and all its nuances. With few exceptions, subtleties of this theory such as multislip hardening and nonSchmid effects have not been tested sufficiently in critical experiments. In particular, through the interplay between small secondary slips and hardening, there is now compelling evidence that important effects largely have been ignored. Furthermore, since intermetallic compounds are currently of growing interest in high-temperature applications, non-Schmid effects, which are also important in BCC metals and alloys, should also be considered more seriously. This chapter has focused on single crystal behavior. Until these and other single crystal issues are resolved, the understanding of and ability to predict polycrystalline behaviors is certain to be limited. The connection seems intuitive. For example, in deformation processing of polycrystalline aluminum alloys, such as sheet rolling and cup drawing, the occurrence of microscopic shear bands is a significant and costly problem.
Acknowledgments The author gratefully acknowledges many fruitful discussions with N. Aravas, 2.Basinski, C. Laird, R. Mohan, D. P. Pope, Q. Qin, J. R. Rice, 0.Richmond, and V. Vitek. This research was supported by the United States Department of Energy under Grant DE-FG02-85ER45188 and the NSF/MRL program at the University of Pennsylvania under Grant DMR88-19885.
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Asaro, R. J. (1983). Micromechanics of crystals and polycrystals. Adu. Appl. Mech. 23, 1. Asaro, R.J., and Rice, J. R. (1977). Strain localization in ductile single crystals. J. Mech. Phys. Solids 25, 309-338. Basinski, S. J., and Basinski, Z. S. (1970). Secondary slip in copper single crystals deformed in tension. Second International Conference on the Strength of Metals and Alloys, ASM, Metal Park, Ohio, I, 189. Basinski, S. J., and Basinski, Z. S. (1979). Plastic deformation and work hardening. In: Dislocations in solids (F. R. N. Nabarro ed.). North-Holland, Amsterdam, p. 262. Basinski, Z. S. (1959).Thermally activated glide in face-centered cubic metals and its application to the theory of strain hardening. Phil. Mag. 4,343. Basinski, Z. S., and Basinski, S. J. (1992). Fundamental aspects of low amplitude cyclic deformation in face-centered cubic crystals. Progress in Materials Science 36, 89- 148. Bassani, J. L. (1990). Single crystal hardening. Appl. Mech. Rev. 43,S320-S327. Bassani, J. L., and Wu, T. Y. (1991). Latent hardening in single crystals 11. Analytical characterization and predictions. Proc. Roy. Soc. Lond. A435, 21-41. Bell, J. F., and Green, R. E., Jr. (1967). An experimental study of the double slip deformation hypothesis for face-centered cubic single crystals. Phil-Mag. Ser.8 15,469. Bishop, J. F. W. (1953). A theoretical examination of the plastic deformation of crystals by glide. Phil. Mag. 44, 51. Bishop, J. F. W.. and Hill, R. (1951). A theoretical derivation of the plastic properties of a polycrystalline face-centered metal. Philos. Mag. 42,414-427. Budiansky, B., and Wu, T. Y. (1962). Theoretical prediction of plastic strains of polycrystals. Proc. 4th US.Nar. Congr. Appl. Mech., 1175. Cahn, R. W. (1951). Slip and polygonization in aluminum. J. Inst. Metals 79, 129-158. Chang, Y.W., and Asaro, R. J. (1981).An experimental study of shear localization in aluminumcopper single crystals. Acta Metall. 29, 241-257. Christian, J. W. (1983). Some surprising features of the plastic deformation of body-centered cubic metals and alloys. Metall. Pans. 14A,July, 1237. Christoffersen, J., and Hutchinson, J. W. (1979). A class of phenomenological corner theories of plasticity. J. Mech. Phys. Solids 27,465-487. Cottrell, A. H. (1953). Dislocations and plasticjow ofcrystals. Oxford University Press, London. Davis, R.S., Flesicher, R. L., Livingston, J. D., and Chalmers, B. (1957). Effect of orientation on the plastic deformation of aluminum single crystals and bicrystals. J. Inst. Metals 209, 136140. Diehl, J. (1956). Zugverformung von Kupfer-Einkristallen. Z . Metallk. 47,331. Edwards, E. H., and Washburn, W. (1954). A theory of minimum plastic spin in crystal mechanics. Trans. Metall. Soc. A I M E C200, 1239. Elam, C. F. (1927). Tensile tests on crystals. Part IV. A copper alloy containing five per cent aluminum. Proc. R . SOC.London, Ser. A 115,694-702. Ezz, S. S., Pope, D. P., and Vitek, V., (1987).Asymmetry of plastic flow in Ni,Ga single crystals. Acta metall. 35, 1879. Franciosi, P., Berveiller, M., and Zaoui, A. (1980). Latent hardening in copper and aluminum single crystals. Acta Metall. 28, 273-283. Franciosi, P., and Zaoui, A. (1982a). Multislip in FCC crystals a theoretical approach compared with experimental data. Acta Metall. 30, 1627-1637. Franciosi, P., and Zaoui, A. (1982b). Multislip tests on copper crystals: a junctions hardening effect. Acta Metall. 30, 2141-2151. Fuh, S., and Havner, K. S. (1989). Strain hardening of latent slip systems in zinc crystals. Proc. R. SOC.Lond. A422, 193-239. Garstone, J., Honeycombe, R.W. K., and Greetham, G. (1956).Easy glide of cubic metal crystals. Acta Metall. 4,485.
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Lomer, W. M. (1951). A dislocation reaction in the face-centered cubic lattice. Philos. Mag. 42, 1327- 133 1.
Mandel, J. (1965). Generalisation de la theorie de plasticite De W. T. Koiter. Int. J. Solids Struct. 1, 273.
Mandel, J. (1973). Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques, Int. J. Solids Struct. 9, 725. Mitchell, T. E., and Thornton, P. R. (1964). The detection of secondary slip during the deformation of copper and a-brass single crystals. Philos. Mag. 10, 315-323. Mohan, R., and Bassani, J. L.(1992). Influence of hardening and multiple slip behavior on strain localization in ductile single crystals. To be published. Needleman, A. (1979). Non-normality and bifurcation in plane strain tension and compression. J. Mech. Phys. Solids 21, 231. Ortiz, M., Leroy, Y., and Needleman, A. (1987). A finite element method for localized analysis. Comp. Mrth. Appl. Mrch. Eng. 61, 189-214. Paidar, V., Pope, D. P., and Vitek, V. (1984). A theory of the anomalous yield behavior in Ll, ordered alloys. Acta metall. 32,435-438. Peirce, D. (1983). Shear band bifurcation in ductile single crystals. J. Mech. Phys. Solids 31, 133. Peirce, D., Asaro, R. J., and Needleman, A. (1982). An analysis of nonuniform and localized deformation in ductile single crystals. Acta Metall. 30, 1087. Piercy, G. R., Cahn, R. W., and Cottrell, A. H. (1955). A study of primary and conjugate slip in crystals of alpha-Brass. Acta Metall. 3, 331-338. Pope, D. P., and E n , S. S., (1984). Mechanical properties of Ni,AI and nickel-base alloys with high volume fraction of y'. Int'l Metal Reo. 29, 136. Price., R. J., and Kelly, A. (1964). Deformation of age-hardened aluminum alloy crystals-11. Fracture. Acta Metall. 12, 979-991. Qin, Q. (1990). Crystal plasticity with non-associated flow. PhD thesis, University of Pennsylvania, Philadelphia. Qin, Q., and Bassani, J. L. (1992a). Non-Schmid yield behaviors in single crystals. J. Mech. Phys. Solids 40,813-833. Qin, Q., and Bassani, J. L. (1992b). Non-associated plastic flow in single crystals. J. Mech. Phys. Solids 40,835-862. Qin, Q., and Bassani, J. L. (1992~). Strain localization in single crystals undergoing nonsymmetric double slip. To be published. Raniecki, B., and Burhns, 0. T. (1981). Bounds to bifurcation stresses in solids with nonassociated plastic flow law at finite strain. J. Mech. Phys. Solids 29, 153. Ramaswami, B., Kocks, U. F., and Chalmers, B. (1965). Latent hardening in silver and an Ag-Au alloy. Pans. AIME 233,927-931. Rice., J. R., (1971). Inelastic constitutive relations for solids: An internal-variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433. Rice, J. R. (1977). The localization of plastic deformation. Theoretical and applied mechanics (W. T.Koiter, ed.). North-Holland, Amsterdam, pp. 207-220. Rosi, F. D. (1954). Stress-strain characteristics and slip-band formation in metal crystals: Effects of crystal orientation. J. of Metals 6, 1009. Rudnicki, J. W.,and Rice, J. R., (1973). Conditions for the localization of deformation in pressure-sensitive dilatant materials. J. Mech. Phys. Solids 23, 371. Salah, S., Ezz, S., and Pope, D. P., (1985). The asymmetry of cyclic hardening in Ni,(AI, Nb) single crystals. Scripta Metellurgic 19, 741.
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Schmid, E., and Boas, W. (1935). Plasticity of crystals. Chapman and Hall, London (translation of 1935 Springer edition). Seeger, A. (1957). Dislocations and mechanical properties of crystals. Wiley, New York, p. 243. Sewell, M. J. (1973). A yield-surface corner lowers the buckling stress of an elastic-plastic plate under compression. J . Mech. Phys. Solids 21, 19-45. Sewell, M. J. (1974). A plastic flow rule at a yield vertex. J. Mech. Phys. Solids 22, 469-490. Sheh, M. Y., and Stouffer, D. C. (1990). A crystallographic model for the tensile and fatigue response for RenC N4 at 982°C. J. of Appl. Mech. 57, 25. StouEer, D. C., Ramaswamy, V. G.,Laflen, J. H., Van Stone, R. H., and Williams, R. (1990). A constitutive model for the inelastic multiaxial response of Rent 80 at 871C and 982C. J. Engineering Materials and Technology 112, 241. Takasugi, T., Hirakawa, S., Izimi, O., Ono, S., and Watanabe, S. (1987). Plastic flow of Co,Ti single crystal. Acta metall. 35, 2015. Taylor, G. I. (1938a). Plastic strain in metals. J. Inst. Met. 62, 307-325. Taylor, G. I. (1938b). Analysis of plastic strain in a cubic crystal. In: Stephen Timoshenko sixtieth anniversary volume (J. M. Lessels, 4.) Maanillan, . New York. Taylor, G. I., and Elam, C. F. (1925). The plastic extension and fracture of aluminum crystals. Proc. R. SOC.London, Sec. A 108, 28-51. Wu,T. T., and Drucker, D. C. (1967). Continuum plasticity theory in relation to solid solution, dispersion, and precipitation hardening. J . Appl. Mech. 34, 195- 199. Wu, T.-Y., Bassani, J. L., and Laird, C. (1992). Hardening of single crystals under cyclic deformation. Submitted for publication. Wu, T.-Y. Bassani, J. L., and Laird, C. (1992). Hardening of single crystals under cyclic deformation. Submitted for publication. Zarka, J. (1975).Modelling of changes of dislocation structures in monotonically deformed single phase crystals. In: Constitutive equations in plasticity (A. S. Argon, ed.). MIT Press, Cambridge, Mass., pp. 359-385.
ADVANCES IN APPLIED MECHANICS, VOLUME 30
Elastic Solids with Many Cracks and Related Problems MARK KACHANOV Department of Mechanical Engineering Tufts University Medford, Massachusetts
I. Introduction ... . .... ... ....... . .... .. . . . ... . .. . .... . .. . . ..... .. . .. . ... ... ..... ... ..... . ..
260
11. Stresses and Crack Opening Displacements Associated with One Crack in an Infinite Isotropic Linear Elastic Solid.. .. ..... .... ..,. . ... . ... . ._.. .. ._.... . A. Two-Dimensional Fields ................... .......................... ............... B. Three-Dimensional Fields (Circular Cracks). .. . ... . .. . . .......... , . ......... .... . . C. Far-Field Asymptotics. ......., .. . .............. D. Crack Tip Fields . ....... ....., .. . ... ..... . .. . . ... . . .. . ... . ... . .. . .. . .. . . .. . .... . ....
26 1 262 27 1 275 279
__
..............
280 280 283
..............
296
IV. Various EBects Produced by Crack Interactions.. ... . . .. . .... ...... ... . ... .,.. . .... .. A. Stress Shielding a Amplification. Two- and Three-Dimensional ............................................ Interactions ... . ... . B. Range of Influence in an Array of Interacting Cracks. C. Extremal Properties of Slightly Asymmetric Arrays .............................. D. Intersecting Cracks (Two-Dimensional Configurations)............ E. Intersecting Cracks (Three-DimensionalConfigurations) .. . .. . .. . .. . .... . . .. .. . .. F. Interaction of Cracks Filled with Compressible Fluid .. . ... .. . ... .... . . .. . .... . .. G. Interaction of Cracks in an Anisotropic Matrix . .... ..... . ... ... . .... ........ . . ..
30 1
V. Interaction of a Crack with a Field of Microcracks . .. .... . .. . .. . ... __.. ... . . ... . .. .. A. Introduction.... .... .......... . ........ .. . .... .... .. . . ... . . .. . ... . .. . ... .... . ... . . .. . . B. Basic Equations for Crack-Microcrack Interactions ....... ... . ... ..... . .. .. . ... . C. Interaction of a Crack with Some Simple Microcrack Configurations .. . . ... . .. D. Interaction of a Crack with Large Microcrack Arrays: Two-Dimensional Configurations..... . ....... .... . .. . ....... . ... . .. . .,.. . ... ... .... ..,. ... . ..... . ...... E. Interaction of a Crack with Large Microcrack Arrays: Three-Dimensional Configurations..... .... .. . .... ... . . .. . ... ..... ... . .... . .. . .. . .... .... . .. . .... . . .. .... F. Is Modeling of the Microcracked Zone by Effective Elastic Material Adequate? G. On Toughening by Microcracking. Conclusions.. ................-...............
321 321 324 329
111. Problems of Many Cracks in a Linear Elastic Solid
A. General Considerations . .... .,. .... .... . ... . .. B. A Method of Analysis of Crack Interactions C. Approximation of Small Transmission Facto Widely Spaced Cracks .........................................
259
30 1 306 308 311 314 315 320
333 341 344 344
Copyright 0 1 9 9 4 by Academic Prcss, Inc.
All rights or rcproduction in any form reserved. ISBN 0-12-002030-0
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Mark Kachanov
VI. Effective Elastic Properties of Cracked Solids.. ... . .... . . ..... . . .... . ..... . ...... . .... A. Introduction.. . .......... .... . . . .................. B. Noninteracting Cracks. Crack C. Noninteracting Cracks in an Anisotropic Matrix . . . ..... . .... .. . . ... . . . .. .. . ... . . D. Noninteracting Cracks Constrained against the Opening (but Allowed to Slide) E. Noninteracting Cracks Filled with Compressible Fluid.. . . . . ... . . . .... . . ..... .... F. Summary on Noninteracting Cracks.. .... ..... . ..... . . ..... . . . ... . . .... .. ..... . ... G. Impact of Interactions on the Effective Moduli: General Considerations.. . . . .. H. Approximate Schemes.. .... . ,... . ..... . ... ...... . .... . . .... . . . ... . . . ..... . ..... . .... 1. Computer Experiments on Deterministic Arrays (Solution of the Interaction Problem for Sample Crack J. Computer Experiments on ............. K. On Bounding the Effective ..................................... L. Alternative Crack Density Parameter.. .. ... .... . ..... . ..... . . . . .. . . ..... . ..... . ...
.
VII. On Correlations between Fracturing and Change of ElTective Elastic Moduli: Some Comments on Brittle-Elastic Damage Mechanics.. . .. . . ... . . . ..... . . . .... . .... .. . .. . . A. Microcracking and the Change of Elastic Moduli ... . .... . . . .... . . ... . . . .. .. . . ... B. On the Construction of Elastic Potentials in Terms of Damage Parameters.. .. ml. Effective Elastic Properties of a Solid with Elliptical Holes ................ ........ A. One Elliptical Hole in a Uniform Stress Field ... ....... B. Noninteracting Elliptical Hole ...................................... C. Interacting Elliptical Holes.. . D. Mixture of Circular Holes and Cracks. .......................... E. Nonequivalence of the Approx .................. Density of Holes.. ... .. . . . . ... . .. . .. . . . F. Comparison with the Self-Consistent
345 346 352 372 376 380 385 386 389 402 407 408 410
412 413 417 42 I 422 427 43 1 433 436 436
Acknowledgments ...... .... . .... . . .... . ... . ...... .. . .. ..... . . .... . ..... . ... . . . ..... . ... .
437
References. ..... ... .. .... ... .. . .. . . . .... . .... . ....... . .. . .. . . . . .. . . . . .. . . . ... . . . .. . . . . ... .
438
I. Introduction This review discusses some basic problems in mechanics of elastic solids containing multiple cracks. Being relatively short (compared to the volume of knowledge in the field), it does not cover all the related problems. A number of mathematical aspects that frequently constitute fields of their own (like various numerical techniques) are discussed very briefly or not mentioned at all. Instead, we focus on physically important effects produced by crack interactions and attempt to present results in the simplest form possible. The selection of topics reflects, to some extent, interests of the author. The problems considered in this chapter can be divided into two groups. 1. The impact of interactions on individual cracks, particularly on the stress intensity factors (SIFs). Interactions can produce a variety of phenomena-stress shielding and stress amplification, coupling ot the
Elastic Solids with Many Cracks and Related Problems
26 1
normal and shear modes, etc. The configuration of a macrocrack interacting with a field of microcracks is of particular interest for materials science applications. 2. The eflectiue elastic properties of solids with many cracks. This is a classical problem of continuum mechanics; it also has applications in materials science, structural mechanics, geophysics and serves as a background of some nondestructive evaluation (NDE) techniques. Degradation of stiffness, development of anisotropy, and changes in wavespeeds caused by microcracking are of obvious importance for these fields. This problem and most of the approaches to it (approximation of noninteracting cracks, various self-consistent schemes, etc) have roots in the effective media theories of physics. At the same time, cracks constitute a type of defect distinctly special: they occupy no volume (in contrast with inclusion-type inhomogeneities); the stress field generated by a single crack is quite complex, particularly in three dimensions (in contrast with particle-generated fields that are, typically, spherically symmetric). As a result, the theory of effective properties of cracked solids has many special features: the choice of an appropriate crack density parameter becomes nontrivial; the approximation of noninteracting cracks has a wider, than can be expected, range of applicability; bounds for the effective moduli cannot, generally, be established. Problems of the first group are, generally, relevant for the fracture-related considerations; solutions are sensitive to the positions of individual cracks. Problems of the second group dea1 with the volume average quantities; they are relatively insensitive to the information on individual cracks. We discuss, in this connection, whether correlations exist between these two groups of quantities; in particular, whether microcracking can be reliably monitored by measuring changes in the effective elastic moduli. The reader is expected to be familiar with basics of linear elastic fracture mechanics.
11. Stresses and Crack Opening Displacements Associated with One Crack
in an infinite Isotropic Linear Elastic Solid
This section summarizes results for stresses and crack opening displacements associated with one crack in an infinite isotropic linear elastic solid. The structure of these fields has direct implications for the problems of crack interactions and effective elastic properties of solids with many cracks. The
262
Mark Kachanov
far field and the near tip asymptotics are also given and the regions of their applicability are discussed. In the two-dimensional (2-D) case (rectilinear cracks), solutions constitute a common knowledge. In the 3-D case (circular, or “penny-shaped” crack), solutions in elementary functions (rather than in integrals of Bessel’s functions) were obtained only recently (Fabrikant, 1990). As is well known, the problem of a linear elastic solid loaded on the boundary and containing a traction-free crack is equivalent (to within a homogeneous stress state) to the problem where tractions are applied to crack faces and the boundary of the solid is traction free. This latter formulation will be assumed in the text to follow.
A. TWO-DIMENSIONAL FIELDS
The fields generated by a loaded crack can be represented as a superposition of the fields produced by tensile (I), shear (11) and antiplane (111) modes of loading. In the case of a crack in an inJnite solid, these modes are uncoupled, in the sense that a certain mode of loading produces the crack opening displacements (COD) of the same mode only. Solutions for the stress and displacement fields can be obtained by the complex variable methods; they are given below. The crack occupies the interval ( - 1 , 1) of the x-axis and r 3 (x’ + y2)’I2. 1. Stress Fields Generated by a Uniform Loading of Unit Intensity
a. Normal Loading
b. Shear Loading a,, = 3 3 ~ 1 ,- 3xy1, - 4y31, bXy= I 2 bYY=
+ 4xy31,)
- 8yzZ4 + 8y4Z6
3- yz,
+ xyz, + 4y3z,
- 4xy31,)
Elastic Solids with Many Cracks and Related Problems
263
where it is denoted
+ y3I2)6 + 3
j2
I -6 - 2 a = (x
4 3 6
+4
4 3
(ay)3/285’2
- 1)’
+ y2
+ y2 y = (x + + y2
/?= 2(x’
12)
s=/?+2&. c. Antiplane Loading
+ (2- l2 - y2)]1/2
+ y[M’ 1
O ~ . ( X , ~ )= - { y [ M
- (x’ - l2
- J J ~ ) ] ~’ ~1}
+ (x’ - 1’
- Y ~ ) ] ”-~ x [ M 2 - (x’
- 1’
-J J ~ ) ] ~ ’ ~ }
fiM
(2.3) where M = [(x - 1)’ + Y ~ ] ~ / ~ + [ (1)2 x +~7~1’~~. Figures 1-3 depict these fields. Figure 4 shows the regions of tensile and compressive principal stresses of the modes I and I1 fields. We note that, in the mode 1 field, the region of negative (compressive)stress cyyis substantially larger than the region of tensile nYy:a crack generates an extended “shadow”
Mark Kachanov
264
IY
Ox, \
t
1 -0.60
- 2 -0.40 - 3 -0.20 .......... 4 +o.oo
-
f
5 +0.20
i
6 +0.40
I
7 +0.60
'\4.
OYY
I 4
4
'*,
t
'\
/
4;'
4
FIG. 1. Mode I standard stress field (produced by uniform loading). Note that the shielding zone for uyy(where uyy< 0) is less intense but larger than the amplificationzone (uyy> 0).
and a relatively small "amplification" zone. This observation explains certain features of crack interactions (amplifying interaction effect in collinear arrangements has a relatively short range, whereas the shielding effect in stacked arrangements has a substantially longer range) and has implications for the problem of effective elastic properties of cracked solids (Section VI).
Elastic Solids with Many Cracks and Related Problems
265
IY 1 -0.60
- 2 -3 4
- 5
-0.40 -0.20
+o.oo
+0.20 6 +0.40
4,,
4
4 4
‘.> 4
OYY 4
oxx
4
4 4
I
4
4 4
4
‘ 4..--.4I
4 A
FIG.2. Mode I1 standard stress field.
The mode IZ field has an unusual feature: the shear stress oxyexceeds the one applied on the crack face (by about 9%) in a small region above the crack line, at the distance from the crack of about one-half the crack length (this region is not shown in Fig. 2 because of its smallness). Thus, a shear “source” generates a small region of shear stresses that are more intense than the applied stress. The mode ZZZ field does not contain the ox,, oxy,oyycomponents and, thus,
266
Mark Kachanov
QXZ
?
i
-
1 -0.60
2 -0.40 - 3 -0.20 4
- 5 -6 - 7
I
+o.oo
+0.20
+0.40 +0.60
4
4
FIG.3. Mode I11 standard stress field.
does not interact with modes I and 11. Therefore, in the case of interacting cracks, mode 111 analysis can be done separately. 2. Stress Fields Generated by a Pair of Equal and Opposite Point Forces
These fields, due to forces applied at an arbitrary point x are as follows.
= a of
the crack,
Elastic Solids with Many Cracks and Related Problems
267
FIG.4. Modes I and I1 standard stress fields. Regions where the principal stresses are positive and negative (marked + and -).
a. Normal forces P
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Mark Kachanov
b. Shear Forces T
where the coordinate system of Fig. 5 is used. c. Antiplane Forces S
- x ) ( l - N / M ) ' / ~- y(1 + N / M ) ' / ~ [(a - x ) + ~ y 2 ]M 112
a
- x)(l
= [(x'
+
(2.6)
[(a - x ) ~ y2]M1I2
a
where M
+ N/M)'l2 + y(1 - N/M)'i2
- y 2 - (2)2
+ 4 x 2 y 2 ] * / zand N
(a - x ) + ~ y2.
3. Stress Intensity Factors A pair of equal and opposite point forces P, T and S (normal, shear and antiplane, respectively) applied to opposite crack faces at the point x = a produces the following stress intensity factors (SIFs) at the right and left tips + I : { K , , KII,K , , , } ( f . I )= ( x ~ ) - ' / ~ { T, P ,S}(I k a)'/'(I
SIFs due to a uniform loading are {KI, KII, KIld = {P, 2, S for modes I, 11 and 111, respectively.
} ( W 2
T u)-'/~
(2.7)
269
Elastic Solids with Many Cracks and Related Problems
b
0
FIG.5. Coordinate system for the stress field generated by a pair of point forces applied at point b of a crack ( - 1, I ) .
4. Crack Opening Displacement and Average COD due to a Pair of Point Forces
The COD, or displacement discontinuity, b(x) = u+(x) - u-(x) due to a pair of equal and opposite point forces P, T and S (normal, shear, and antiplane, respectively) applied to opposite crack faces at the point x = a is given by the following expression:
where E’ denotes Young’s modulus E for plane stress and E(l - v 2 ) - l for plane strain (this notation is assumed throughout this chapter), and a = E( 1 + v)- ‘ / E . The average COD due to this pair of point forces (quantity relevant for the effective elastic properties of a cracked solid) is {(bn),
41
{ P , 7: S}[ 1 - (a/021”2
(2.10)
5. Crack Opening Displacement and Average COD due to a Uniform Loading
Under uniform loadings p, t and S (normal, shear and antiplane) applied to opposite crack faces, the COD has an elliptical shape and is given by the expression (2.1 1)
Mark Kachanov
270
The average COD due to a uniform loading is (2.12) 6. Collinearity of CODs and the Applied *actions
As seen from the formulas (2.7) to (2.10), “compliances” (i.e., CODs produced by unit loads) of a two-dimensional crack in normal (I) and shear (11) modes coincide (in contrast with the three-dimensional penny-shaped crack, see later). This means that, under combined modes I and I1 loading, formulas (2.9) to (2.12) can be written in the vectorial form; i.e., b(x) = -Fln 4
12
- ax
4 E
= yFl[1
- a2X12- x21
llx - a1
nE’
(b)
+ J
-( ~ 2 / 1 ) ~ ] ” ~
(2.9a) (2.10a)
where F is a pair of point forces of arbitrary direction, and c
(2.11a) (2.12a)
where t is a uniform traction of arbitrary direction. As shown in Section VI, this collinearity has important implications for the problem of effective elastic properties of cracked solids. 7. Nonunformly Loaded Crack: Relation Between Average COD and Average *action Calculations for the case of variable traction on a crack show that, although the average COD is not exactly proportional to the average traction on the crack (otherwise a traction distribution with a zero average would have produced a zero average COD-an incorrect statement),one can assume that, typically, such proportionality holds with good accuracy and, for modes I and I1 loading, has the form (with (b) changed to a(b) for mode 111)
Elastic Solids with Many Cracks and Related Problems
27 1
Indeed, if one assumes that the traction distribution on a crack (normal load, for example) has the form of cubic polynomial P(x) = P O C 1
+ Wl)+ B ( X / 0 2 + r(x/031
(2.14)
(which, in most cases, provides a sufficient flexibility for approximating the actual tractions), then integration of (2.10) yields the following relation between the average COD and average traction (odd order terms in (2.14), obviously, do not contribute into the average): (b)
+
nl 1 814 E‘ 1 + 8 / 3
(2.15)
In the problems of crack interactions, considered in this work, the nonuniform tractions represent additional tractions on cracks due to interactions. In such problems, the coefficient /?(of the polynomial (2.14)providing the best fit for the actual tractions) may reach values on the order of unity (for extremely strong integrations) but is, usually, substantially smaller than unity. Then the ratio (1 + /?/4)/(1 + 813) in (2.15) is close to unity. (For example, in the problem of two collinear cracks with spacing between cracks of 0.2 of the crack length, B = 0.22. Then this ratio is 0.983.)In conjunction with a similar fact for the mode 11, this means that the proportionality (2.13) can be viewed as a sujiciently accurate approximation. (Note that, if the cubic polynomial (2.14)is upgraded to the seventh degree, with coefficients 6 and o at the fourth and sixth powers, then 618 + 501128 and 615 + 017 should be added to the numerator and denominator of the mentioned ratio; since the coefficients at increasing powers of the (2.14)-type polynomial, typically, rapidly decrease, such an upgrading is not expected to have a strong impact on the proportionality relation (2.19.)
B. THREE-DIMENSIONAL FIELDS(CIRCULAR CRACKS) 1. Stresses Generated by a Crack Loaded by Uniform nactions (Normal
and Shear)
Expressions in terms of integrals of Bessel functions can be found in the book by Sneddon and Lowengrub (1969). More useful expressions in elementary functions (2.16) and (2.17) are due to Fabrikant (1990). Cylindrical coordinates p = (x2 + Y ) ” ~ , 4 and z are used; crack radius is a and
Mark Kachanou
272
r = (x2 + y2 + z ' ) ~ ~Note ~ . that in the axisymmetrical problem of uniform normal loading, the dependence on c$ will disappear if stress components in the cylindrical (rather than Cartesian) coordinate system are used. Formulas (2.7),(2.8),(2.12), and (2.13) assume that z > 0; stresses in the lower half-space z < 0 are found from the symmetry relation oi,(x,y, -z) = aij(-x, -y,z). a. Uniform Normal Loading p
q 1+
61
= It
62
=-
a(13 - a2)1/2
1; - 1:
2 4
2p al:e2'+(1: - a' 71 l;(l; - 1;)
'
lj2
- arc sin
{I - 2 v +
(31
zZ[a2(61: - 21:
(1:
- l:)'(l;
+ p 2 ) - 51:] - a2)
(2.16) where the following notations are used: 61
= ox,
c2 = ox,
+ by, - oYy+ 2io,,
6, = 6 2 2
z, = ox,
I, 12
+
idyz
+ + Z 2 - y - [(a - + z2]1'2} = 1/2{[(a + p)Z + z2-y + [(a + z2]1/2} = 1/2{[(a
p)2
p)2
p)2
b. Uniform Shear Loading z (arbitrarily oriented to the x, y axes: iz,)
t = z,
01
=
+
{
2(iei4 + t e - '4 -2(1 4 2 - v)
+ v) a11(a2 lz
l y 2
Elastic Solids with Many Cracks and Related Problems 02=
2ei+
-~
742 - v)
- v,
al,(a2 - /;)'I2 12(1; - 1;)
z/,(l; - a2)1/2
+
12(1i- 1:)
+
a2(41; - 5p2) + 1; (z (1; - 1:)z = -
+ ?e2'+)
+
2(?ei+ re-'+) zll(l; - ~ ~ ) ' / ~ [ a ~-( 5p2) 4 1 ; + l;] n(2 - v ) i2(i; - 1 3 3
Z(U'
273
(2.17)
+ a2(2a2+ 2z2 - 3p2)]
- l;)'"[l;
(1; - 1 3 3 112
+
z(a2 - l:)'I2[a2(61: - 21:
(1; - 1:)2
+ p 2 ) - 51:]
2. Stress Intensity Factors
SIFs are, generally, variable along the crack edge. Mode I is decoupled from modes I1 and 111, but modes I1 and 111are not decoupled shear loading produces a combination of modes I1 and I11 conditions along the crack edge. A pair of equal and opposite point forces P (normal to the crack) applied to opposite crack faces at the point po, 4oproduces the following mode I SIF at a given point 4 of the crack edge:
Kd4) =
1 ~
n2J2aa2
P
+ P?l - 2apocos(4 - 40)
(2.18)
Modes I1 and I11 SIFs at a given point 4 of the crack edge due to a pair of arbitrarily directed shear forces T = T, + iTy applied to opposite crack faces at the point po, 4o were recently obtained by Fabrikant (1990) in the following convenient complex form (the overbar denotes a complex conjugate):
274
Mark Kachanov
For a uniform normal load p K , = p(2a)’”a-
(2.20)
Uniform shear load 7 produces the following values of K,,and K,,, along the crack edge:
3. Variation of the Energy Release Rate along the Crack Edge under a Uniform Shear Loading
The energy release rate along the crack edge is
(2.22) where the first term in the brackets represents the average along the edge value, and the second term represents the variation; the ratio of G ,, (at the point of pure mode I1 condition) to Gmin(at the point of pure mode I11 condition) is (1 - v)-’. In a very rough first approximation, the second term in the brackets can be neglected (particularly if v is small) and G can be taken as constant around the edge. It should be noted, however, that G, as written, is relevant for the in-plane crack extension only. Actually, as the propagation condition is first met at the point of pure mode 11, the crack will kink at this point, and according to the results for a (locally 2-D plane strain) kinked crack (Cotterell and Rice, 1980), pure mode I conditions exist at the kink tip and K , = 2 / f i K , , . On account of this factor, the G at the point of kinking exceeds the one at the point of pure mode I11 condition by the factor of (4/3)(1 - v)-’, a noticeable difference. 4. COD of a Circular Crack Loaded by a Uniform Paction
COD due to a uniform normal loading p is given by the expression (2.23) In the case of a uniform shear loading, the COD vector is collinear to the direction of loading (note that this would not be true for an elliptical crack, see Section VI) and its magnitude is obtained from (2.23) by dividing it over (1 - v/2).
Elastic Solids with Many Cracks and Related Problems
275
Thus, in contrast with the 2-D case, “compliances” of a crack in the normal and shear directions are different (although not by much). Therefore, in the case of a uniform traction arbitrarily inclined to the crack, the vector of COD is not collinear to the traction vector. Implications of this noncollinearity for the effective elastic properties of a cracked solid are discussed in Section VI. Average normal and shear CODs produced by uniform normal, p , and shear, z, loadings are
C. FAR-FIELD ASYMPTOTICS At distances from the crack that are much larger than the crack size the expressions for stresses simplify. They are given by the following formulas (uniform loading p , 5 and s in modes I, I1 and 111, respectively, is assumed). 1. 2 - 0 Case
Here, l/(x2 + yZ)l/’ is a small parameter. a. Mode I Loading Oxx
+
12x4- 6x2y2 y4 2 (x2 y 2 ) 3
=P-
+
byy=
f2 x4 + 6x2y2 - 3y4 P2 (x2 - y2)3
OXY=
P l2
(2.25)
(x2 - 3y2)xy (x2 - y2)3
b. Mode I1 Loading Oxx =
(3x2 - y2)xy -d2
(x2
dyy=
r12
OXY=
‘y
+
y2)3
(2- 3yZ)xy (x2
+ y2)3
l2 x4 - 6x2y2 + y4 (xZ
+ y2)3
(2.26)
Mark Kachanov
276 c. Mode Ill Loading
3 x2 - y 2 2 (x2 y2)2
XY
oxy= - s12 (x2
oyz= s - l 2
+ y2)2’
+
(2.27)
2. 3 - 0 Case Here, crack radius is a; a/(p2
+ z2)lI2 is a small parameter.
a. Normal Loading p4
+ 8z2p2- 8z4 (p2 + Z2)Z
X
(1
+ 2v + 3(1 - 2v)cos24)p4 - 2(5 + v + 3(2 + v)cos24)z2p2+ 4(1 - v)z4 (p2 + z2)2
=’(
byy
X
)
a
3a
3
Jm
(1 + 2v - 3(1 - 2v)cos24)p4 - 2(5 + v - 3(2 (p2
(1 - 2
+
+ v)cos24)z2p2+ 4(1 - v)z4
z2)2
~ - 2(2 ) +~v)z2p2 ~ (p2
+ z2)2
(2.28)
b. Shear Loading Here, z = ( T ~zy), , and angle 4 is counted counterclockwise from the x-axis. d,,
=-
+
4(Txcos4 rY sin 4) a(2 - v)
a
4pz3 - zp3
Elastic Solids with Many Cracks and Related Problems
-
1 Jm >’ [
5zp3
zxcos 4
( p 2 + z2)2
ffYY
277
=-( 742
2 - v)
+ 4v)zp3 t, cos (b
4(1 - v)pz3 - (1
a
( p 2 + ,2)2
5zp3 - (p2
aXy = -
+ z2)2
r,sinb]
4(1 - v)(t,sin+
+ z,cos(b)
n(2 - v ) 2
a
(2.29)
- (11 - v)p2z2 3(p2
+
vp4
(p2
+
1
22)2
a
(2 - v)p4 - (1 1 - v)p2z2 3(p2
vp4
+ 2(1 +
v)z4
7,
z2)2
+ (v - 5)p2z2 ( t xcos 24 + zy sin 24)
2
+
+
+ z2)2
+ 2( 1 +
v)z4
TY
+ ( v - 5)pZzZ (t,sin 24 - zycos 24)
(p2
+
1
22)2
3. Applicability of the Far-Field Asymptotics
The expressions for the far-field stresses are simpler than the corresponding exact expressions. The question arises, at which distances from a crack do these asymptotic fields provide a sufficiently accurate approximation? Fig. 6, (a) for 2-D and (b) for 3-D, shows the regions where the errors of approximation of the actual fields by their far-field asymptotics are within 20%, 5% and 2% (for all stress components). We note that the far-field asymptotics become applicable at relatively close distances from a crack, particularly in the 3-D case and particularly in the coplanar direction. 4. Comment on Sensitivity of the Far-Field to “Details” of Loading on the
Crack
It is interesting to note that the far-field stresses depend on the exact character of loading on the crack; i.e., substitution of the loading distribution by a statically equivalent one changes the asymptotic far field. The far fields given previously assume uniform loading, and have to be changed if the loading is changed to a statically equivalent one. In the 2-D case, such an
2 0 . MODE I
Z-D, MODE 111
n 1
-
2-
2%
5%
3 - 20%
3 0 . SHEAR MODE (CROSSSECIION NORMAL TU DIRECTION OF S m )
3-D.MODE I
Q:
3-D, %EAR MODE (CROSSSECIION PARALLU TU DIRECnON OF S E A R )
1
-
23-
2% 5% 20%
(b)
FIG. 6. Regions where the far-field asymptotics of the crack-generated stress fields are applicable with the accuracy marked: (a) 2-Dcrack; (b) 3-D (penny-shaped)crack.
278
Elastic Solids with Many Cracks and Related Problems
279
adjustment of the far field is simple; it is based on the fact that for a (statically equivalent) pair of point forces applied at the point x = a of the crack ( - I , I), the far-field stresses given by (2.25) to (2.27) have to be multiplied by (4/lr)[l - (a/1)2]'/2(in all modes). For example, for point forces applied at the median point of the crack the far-field stresses are 4/lr times higher than the ones generated by a (statically equivalent) uniform loading. In this sense, Saint-Venant's principle does not hold for a solid with a loaded crack.
D. CRACK TIPFIELDS Counting the x-coordinate from the crack tip, and denoting 8 = arctan y / x , we have the following asymptotic stress field near the tip. They are shown
in Fig. 7.
a. Mode 1 Field 6
,
$Frcos (:)[I-
ayy
=
5
oxy=
,
=
-sin(')sin(:)]
~
[1 + sin (') sin J2xrcos("> 2
5 J2.r cos
(i)
sin
(31
(3 (y) sin
FIG.7. Angular variation of the asymptotic crack tip fields.
(2.30)
280
Mark Kachanov
b. Mode I1 Field u,, =
cyy=
3 sin @
3
(i)[z + (i) cos
5
uXy= @"OS
cos
I):(
1
sin(;)sin(;)]
(:) (:) ('5)[ -
[sin
cos
(31 (2.31)
sin
c. Mode 111 Field
- KIII
ax== -sin
JG
(i),
uyz=
-
&,OS K1l'
(0
(2.32)
-
d. Zone of Validity of the Asymptotic Crack Tip Fields For a crack offinite size, comparison of the asymptotic crack tip field with the full stress field (generated by the crack loaded by a uniform load) shows that the size of the zone where the asymptotic field approximates the full field (for all stress components and in all modes) is quite small it constitutes 0.01, 0.02 and 0.04 of the crack length, for the accuracies of approximation of 2%, 5% or 20%, respectively. This zone is centered at the crack tip and has an approximately rhombic shape.
111. Problems of Many Cracks in a Linear Elastic Solid
A. GENERAL CONSIDERATIONS We consider an infinite linear elastic solid with N traction-free cracks (having unit normals ni) and stresses ' a prescribed at infinity. This problem can be replaced by an equivalent one: crack faces are loaded by tractions d - a ' and stresses vanish at infinity. (In the following, upper indices indicate crack numbers, and summation over them is not assumed, unless explicitly mentioned.) We further represent this problem as a superposition of N subproblems, each containing one crack but loaded by unknown tractions; the latter consist a and the unknown additional tractions due of the a'-induced tractions ni '
Elastic Solids with Many Cracks and Related Problems
t
28 1
1 k
FIG.8. Stress sul position:reduction of the problem with N cracks to N subproblems with one crack each, but loaded by unknown tractions.
to crack interactions (Fig. 8) so that, being superimposed, stress fields of all subproblems yield the prescribed tractions n i . go on crack faces. Thus, traction on the ith crack, in the ith subproblem, is j+i
<'
where is a current point on the ith crack and Atji is the traction generated (in thejth subproblem, along the site of the ith crack) by thejth crack loaded by traction tj(('). The additional tractions Atji reflect the interaction effect and constitute the unknowns of the interaction problem. Superpositions of the type of (3.1) (usually combined, in the 2-D case, with polynomial expansions of tractions and finding the polynomials' coefficients from a large system of linear algebraic equations) have been used in literature starting since the early 1970s, see Datsyshin and Savruk (1973), Gross (1982), Chudnovsky and Kachanov (1983), Chen (1984). Horii and Nemat-Nasser (1985) called them the method of pseudotractions. In the displacements formulation, similar superpositions were used by Collins (1963). We note that such superpositions also hold for finite solids. In this case, oi*uo should be replaced by the tractions induced along the ith crack site (in a continuous material) by loads at the boundary; they may be variable along the ith crack. Finding these tractions requires solution of the elasticity problem for the finite body without cracks. Several simple configurations involving either a small number of cracks (two or three) or periodic arrangements have been analyzed in literature. In 2D, the analysis is usually done by complex variable methods, and solutions are produced either in closed form (two, or a periodic row, of collinear cracks) or numerically, usually by some sort of polynomial expansions (that,
282
Mark Kachanov
typically, become inapplicable at small spacings between cracks). In 3-D, most of the available solutions were obtained under the assumption (explicit or implicit) that spacings between cracks are not too small and the interactions remain therefore relatively weak (see Kachanov and Laures, 1989 for a review); for closely spaced strongly interacting cracks the available solutions include the work of Fabrikant (1990) on two coplanar circular cracks and solutions that can be produced by the method presented in Section B (Kachanov and Laures, 1989). Surveys of the available solutions can be found in handbooks on stress intensity factors (Tada, Paris, and Irwin, 1973; Rooke and Cartwright, 1976). Several approaches have been developed for arbitrary arrays of cracks. Since computational methods do not constitute the emphasis of our work, we mention them only briefly, referring, for example, to books by Aliabadi and Rooke (1991) and Anderson (1991) for a more detailed presentation. (1) Additional tractions Atj‘ on the ith crack can be represented as a sum Atji(ti)
= AtjU)((i) + AtW(ti)
(3.2)
where Atii(’) is the traction generated by the jth crack loaded by the noinduced traction d.uo.The second (unknown) term Atji’ accounts for the fact that the actual traction on the jth crack includes “feedback” from the other cracks. Neglect of the second term Atji’ constitutes an approximation discussed in Section C (the approximation of small transmission factors). In the next approximation, the feedback traction At”‘ can be taken as generated by thejth crack loaded by tractions that have been found in the previous approximation. Subsequent approximations are constructed as a sequence of feedback corrections. Calculation of the feedback stress fields requires integrations along the crack lines (surfaces,in 3-D) of the point force solutions (stress fields generated by pairs of equal and opposite point forces applied at arbitrary points of the cracks). The technique represents an application of the classic Schwarz-Neumann alternating method; it was applied to crack problems by several authors (Hoagland and Embury, 1980; a series of papers by Atluri and coworkers, see, for example, O’Donoghue, Nisoshika, and Alturi, 1985). It can also be applied to finite solids, by including feedbacks from the boundaries. This method may become quite intensive computationally, particularly in 3-D configurations and particularly when the number of cracks is substantial and spacings between them are small. Taking the far-field asymptotics for Atji(ti) constitutes another approxi-
Elastic Solids with Many Cracks and Related Problems
283
mation discussed in Section C (the approximation of widely spaced cracks). (2) Reduction of the interaction problem to a system of integral equations, either for the CODs (dislocation density distributions) or for the additional tractions Atji(ti)on cracks. One technique for solving such a system is based on expansion of the solution in powers of a small parameter (crack size)/(spacings between cracks); see, for example, Collins (1963) for applications to the 3-D problems. Solutions obtained this way are reliable only when spacings between cracks are not too small. As discussed in Section C, the first term of such expansions (first-order corrections for crack interactions) can actually be obtained by elementary means, with almost no calculations involved, for arbitrary 2-D and 3-D crack arrangements. A technique that has a wider applicability is to expand tractions on cracks in polynomials and reduce the system of integral equations to a system of linear algebraic equations for the polynomial coefficients. This technique is discussed in Section B.6. (3) Other methods include the finite element method with singular elements, the boundary collocation and the boundary integral methods, the weight functions method and various of their modifications and combinations. Our work focusses on physical effects produced by interactions. These effects are analyzed using the method presented in Section B. In the 3-D configurations the method is very accurate; in the 2-D configurations, accuracy is, generally, worse, but remains good at spacings between cracks substantially smaller than the crack sizes. B. A METHOD OF ANALYSIS OF CRACKINTERACTIONS
We present here the method of analysis of problems of many cracks developed by Kachanov (1985,1987).We first illustrate the method on a test problem with a known exact solution and then formulate it for the general case of an arbitrary crack array. 1. Test Problem: 7ivo Collinear Cracks The 2-D problem of two collinear cracks of equal length in an infinite plate loaded by uniform normal traction p o at the crack faces (Fig. 9) has an exact analytical solution (Willmore, 1949) and can therefore, serve as a test for our approximate method. Representing the problem as a superposition of two subproblems, each
284
Mark Kachanov
-lm-KK x m = l
FIG.9. Superposition for two collinear cracks.
containing one crack loaded by unknown tractions, we have, for the traction on crack 1, p'(x) = po
+ Apz'(x)
(3.3)
where ApZ1(x)denotes the stress oYyinduced along the site of crack 1 (in a continuous material) by crack 2, loaded by p o + Ap"(x) in subproblem 2. Tractions ApZ1(x),Ap"(x) due to interactions are the unknowns of the problem (interrelating them through integral equations would constitute a conventional approach to the interaction problem). We represent p'(x) as a sum of its average ( p ' ) = po + (Ap") and the traction nonuniformity Ap" - (ApZ1) having zero average (Fig. 10). The key simplifying assumption of the method is to neglect the impact on crack 2 of the traction nonuniformity on crack 1. Thus, the traction induced on crack 2 by crack 1 is generated by a uniform average traction (p') on crack 1 (unknown as yet). This results in a major simplification of the problem. Indeed, the traction on crack 2 is then (3.4) where the expression in brackets is the oyycomponent of the "standard stress field" (generated by a crack loaded by a uniform normal traction of unit intensity, see Section 11) induced along the site of crack 2 in a continuous material by crack 1. Thus, the average traction (p) constitutes the only
FIG.10. Traction on a crack as a sum of its average and a nonuniformityhaving zero average.
Elastic Solids with Many Cracks and Related Problems
unknown
(P')
=
of
the
problem
(by
symmetry
of
the
285
configuration,
(P2> = (P)).
We introduce a transmission factor A defined as follows. Applying a uniform traction of unit intensity to the faces of crack 1 we take the average traction cYyinduced along the site (k, 1) of crack 2 in a continuous material
The A-factor characterizes attenuation of the average traction in transmission of stress from crack 1 onto crack 2 line; A changes from - 1 to 0 when k changes from 0 to 1. Due to the symmetry of the considered configuration, A is the same in transmissions from crack 1 onto the crack 2 line and vice versa. Finding the transmission factor reduces to calculation of a line integral of an elementary function. Now (p) is readily found in terms of A: taking the average of (3.4)
fi
= P o together with ( p ' )
= (p2)
= (p), (P)
+ A(P')
yields PO
(3.6)
=-
1-A
Equation (3.6) shows an increase of the average traction due to the interaction effect. Using the general formula for SIFs at the tips of a crack loaded by a given traction distribution
and taking this traction distribution from (3.4), one obtains the following expressions for the SIFs at the outer and inner tips, respectively,
I
1 2 8 - k(k ~ ~ (=1K) ; 1 + -~ 1 - A x ( l -k)
1
1-An(1-k)
-28
n + 1 ) X - -(1 2
- k)
n + k(k + 1)X - -(1 2
- k)
where KP = p o J m is the SIF for an isolated crack and X , CW are complete elliptic integrals of the argument k = of the first and second kind, respectively. This completes the solution.
,/-
286
Mark Kachanov TABLE I ITERACTIONOF Two COLLINFAR CRACKS: COMPARISON WITH rn EXACTResu~Ts Kdk)/KP
k
K,(k)/KP
Exact
(inner tip)
Error (%)
Exact
(outer tip)
Error (%)
1.112 1.255 1.473 1.905 2.372
1.112 1.251 1.452 1.809 2.134
0.3 1.4 5.0 10
1.052 1.086 1.120 1.159 1.184
1.052 1.086 1.118 1.154 1.175
0.2 0.4 0.8
~~
0.2 0.1 0.05 0.02 0.01
These results are compared with the exact ones in Table I. The agreement is quite good. The error becomes noticeable for closely spaced cracks but even at k = 0.05 (distance between the cracks is one order of magnitude smaller than the crack length), the error is only 1.4% for K,(k) and 0.2% for Kl(l). A slight underestimation of KI is due to the neglected influence of the traction nonuniformity p f f-
(3.9)
Elastic Solids with Many Cracks and Related Problems
287
where 4, 4 are the “standard” stress fields discussed in Section I1 (fields generated by the jth crack loaded by uniform tractions, normal and shear, of unit intensity) and I is a 2-D unit tensor. The problem is thus reduced to finding the average tractions (t’) on cracks. They are found by averaging (3.9) along the ith crack line: (pi) = pi(”
+ 1 [A”.($) + A ~ ~ ( z j ) ] j#i
(3.10)
where the transmission A-factors characterize attenuation of the average normal and shear tractions in transmission of stresses from one crack onto the other crack sites; for example, A: is the average shear traction induced at the site of crack 1 in a continuous material by the normal uniform unit load o n crack 3. Note that, generally, AiJ # Aji (for example, the impact of a small crack on a larger one is smaller than vice versa). Equations (3.10) constitute a system of 2N linear algebraic equations for the average tractions. They can be written in a compact vectorial form:
-
(2dikI - Aik) (ti)
= t‘(O)
(3.11)
where summation over all crack numbers i = 1,2,..., N is assumed. Tensorial element Aikrepresents the average traction vector generated along the kth crack line by the ith crack loaded by a uniform traction of arbitrary direction and unit intensity (Fig. 11); i.e., it transforms the vector of average traction on the “source” crack into the average traction vector on the “recipient” crack line. Diagonal element Aii (characterizing “interaction of a crack with itself‘) is defined as a unit tensor I (consistentlywith the fact that tractions induced by the ith crack on lines close to I’ are close to the ones applied on the crack). The righthand part of (3.1 1) represents the remote loading conditions. The lefthand part characterizes the intrinsic geometry of the crack array, which is
FIG.11. Tensorial transmission factor.
288
Mark Kachanov
characterized, in the framework of our method, by the crack interaction matrix [A"]. Solution of the system (3.1 1) has the form (ti)
= nji.@ O .
,+ = nii,+:
(3.12)
where summation over all crack numbers j = 1,2,. . . , N is assumed, and where it is denoted (3.13)
After the average tractions (ti) are determined from this system, the actual (variable) tractions pi(t), ~'( on 0cracks are found from (3.9). The SIFs at the crack tips are then readily obtained as generated by these tractions, from (3.7). Mode I11 loading can be analyzed along the same lines. Since it does not interact with modes I and 11, the mode 111 analysis is simpler and can be done separately.In the present work, 2-D mode 111 interactions are not considered (see Huang, 1990).
3. Construction of a Full Stress Field
The obtained values of SIFs and the average tractions ( t i ) can be used to construct, in elementary functions, an approximate but quite accurate expression for afull stressjield in a 2-D solid with an arbitrary arrangement of interacting cracks. The displacement field generated by a single crack can be represented as an integral of the double layer potential type (also known as a representation of a crack by dislocations): (3.14)
where b(t) is the displacement discontinuity (crack opening displacement, COD); it is vectorial since both modes I and I1 may be present; and @ is the second Green's tensor of elasticity for an infinite plane 4nR
(1 - 2vXnR - Rn
RR1
- n-RI) - 2n-R-RZ
(3.15)
where R = 5 - x and plane stress is assumed (in the case of plane strain, the factor 1 + v is to be changed to (1 - v)-'). Differentiation of u(x) (can be applied to @ under the integral) and multiplication by the appropriate elastic
Elastic Solids with Many Cracks and Related Problems
289
constants yield the stress field a(x) if b({) is known. In the case of N cracks, we have a sum of integrals (3.14) taken over all I’ so that the problem would have been solved if the CODs were known. We suggest constructing each of the normal b, (tangential b,) components of the COD of a given crack - 1 < 5 < 1 as an ellipse that would correspond to a certain uniform normal (shear) loading multiplied by a quadratic polynomial (“polynomially distorted” ellipse): b , = -E 41 u,
(
l + a 1 , , ~1’ + / ? ~ ) / ~
(3.16)
(E‘ is Young’s modulus E for plane stress and E/(1 - v)’ for plane strain), where the three coefficients of the quadratic polynomials in each of the modes (u,, a,, /?, and u,, a,, /?,) are chosen in such a way as to match the previously found values of the SIFs and average tractions (K,( k I), (p) and Kll(f I), (T)); i.e.,
(3.17)
(Identical formulas hold for the mode I1 quantities.) Substitution of the thus found CODs b(t) into a sum of integral representations (3.14) results in a stress field expressed in elementary functions. Note that the idea of a “polynomially distorted” ellipse can be related to the polynomial conservation theorem stating that a crack loaded by a traction polynomial of degree N assumes the shape of an ellipse times a certain “distortion” polynomial of the same degree N. The shape (3.16) corresponds, therefore, to an approximation of the tractions induced on a given crack line by other cracks by a certain quadratic polynomial; namely,
(3.18)
290
Mark Kachanov
with a, /3 given by (3.17). These tractions match the previously found values of the stress intensity factors and average traction on the given crack. If the polynomial approximation were based on the exact values of SIFs and the traction averages then the difference between the constructed and the actual stress fields would have amounted to a stress field generated by such a system of tractions on the cracks that (1) all the SIFs were equal to zero, and (2) traction averages on each of the crack faces were equal to zero. Such a stress field can be expected to be just a small perturbation of the actual field except, possibly, in the close vicinity of the crack faces (where the boundary conditions are not exactly satisfied). If the values of SIFs and traction averages used in (3.17) are sufficiently close to the actual ones, then the relative error in the stress field can be expected to be small. As an example, consider, again, the problem of two collinear cracks (Fig. 9) for which the exact shape of the COD is given by Sneddon and Lowengrub (1969) for k = 0.1. Figure 12 compares this shape (solid line) with the “quadratically distorted” ellipse, the distorsion polynomial being given by (3.16) and (3.17). These two shapes are very close. This indicates that the “quadratically distorted” ellipse (chosen to match the SIFs and the average tractions) is a reasonable approximation of the actual COD. Applying such representations to a solid with many interacting cracks, one obtains a reasonable approximation of the fields u(x) and a(x).
FIG.12. COD in the problem of two collinear cracks (solid line, actual shape of the cracks; dashed line, “quadratically distorted“ ellipse).
Elastic Solids with Many Cracks and Related Problems
29 1
4. Arbitrary 3 - 0 Array of Circular Cracks in an Injinite Solid The basic idea of the method remains unchanged. The “standard” stress fields are given by much lengthier expressions (2.16) and (2.17) and the directions of shear tractions on cracks have to be further specified in crack planes (it can be done by decomposing the shear tractions into the components along the 0’- induced shear tractions, r, and the normal to it, s, directions). Thus, tractions on the ith crack are:
(3.19)
where p and 4 are the cylindrical coordinates in the ith crack plane (angle 4 is counted from the direction of ricO)); “standard” stress fields @ and 4 correspond to uniform unit tractions applied in two in-plane directions, r and s (note that G; and 0; differ by 90” rotation only); r‘ and mi are unit vectors of the corresponding directions. Taking the average of (3.19) over the ith crack surface yields 3N scalar linear algebraic equations for the traction averages (ti) = ((p‘), (r‘), (si)):
(si) =
j+i
(At*(@)
+ A{;(T~) + Ait(S‘))
wnere the scalar transmission factor A$ characterizes transformation of a vector of a uniform u-traction applied at the faces of the jth (“source”) crack into the resulting average fl-traction induced at the site of the ith (“recipient”) crack and the values 1, 2, 3 of the indices a, fi indicate the normal (n) and two shear components (T and s). Note that finding the transmission factors reduces to calculation of surface integrals of elementary functions. After (ti) are determined from (3.20), the actual (variable) tractions ti(M) on cracks are found from (3.19). The mode I SIF at the given point 4 of the
292
Mark Kachanov
edge of a crack of radius a due to an arbitrary distribution of the normal traction p(po, $o) is given by integration of the formula (2.18)for a point force:
The modes I1 and I11 SIFs due to an arbitrary distribution of the shear traction T (arbitrarily inclined to the x, y axes: z = zX + iz, at each point po, $o) are obtained by integration of the formula (2.19):
where an overbar denotes a complex conjugate. Thus, finding SIFs along the crack edges is reduced to 0
0
calculation of the "crack interaction matrix" [A"] (i.e., averaging of the standard fields generated by ith crack along the kth crack surface) and finding the traction averages from 3N linear algebraic equations (3.20); finding tractions tj(M) on the jth crack surface from (3.19); finding SIFs along the crack edges from (3.21) and (3.22).
The actual computational simplicity of the method depends on how simple the expressions for the standard fields are. The results of Fabrikant (1990) who derived these fields in elementary functions (see Section 11) are, therefore, essential. The numerical accuracy of the method also crucially depends on accuracy of evaluation of the singular integrals (3.21)and (3.22)for SIFs. We eliminate the singularities using the transformation suggested by Laures (1988, 1990) as follows (without it, the integration may yield errors of up to lo%, substantially exceeding accuracy of the method itself). In the integral (3.21) for the mode I SIF, we first transform the integral by changing variables (po,do)in the crack plane to variables (A, 5 ) that characterize position of the point where the load p is applied with respect to the point $ of the crack edge where K, is calculated (as illustrated in Fig. 13):
Elastic Solids with Many Cracks and Related Problems
293
X
Fic. 13. Coordinate system for calculation of SIFs along the edge of a circular crack.
Changing variables again, u = [l
+ (21-'cos5
- 1)1'2]-1
we transform the integral for K, to the form
which is nonsingular and presents no difficulty in numerical integration. In the integral for the modes I 1 and 111 SIFs, following the same steps, we transform (3.22) to the form
(3.24)
which is nonsingular. Several sample 3-D configurations of interacting cracks were considered by Kachanov and Laures (1989).
5. Accuracy of the Method As spacings between cracks become smaller, SIFs on a given crack are increasingly influenced by the traction nonuniformities on other cracks, and the presented method becomes less accurate at smaller spacings. We discuss
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Mark Kachanov
here the accuracy of the presented method by comparing solutions for SIFs with ones obtained elsewhere, either analytically or numerically. a. no-Dimensional Problems In the test problem of two collinear cracks considered previously the errors in SIFs are on the order of 1% when spacings between cracks are on the order of lo-’ of the crack length. Note that such accuracy is achieved in spite of the fact that the interaction effect raises SIFs at the inner tips by 47%; i.e., it is quite significant. Even at spacings as small as 0.02 of the crack length (SIFs at the inner tips are increased more than by a factor of 2), the accuracy is still satisfactory for many purposes (on the order of 10%). Similar results hold for a periodic row of collinear cracks (see Kachanov, 1987); the error is 2.9%at the spacings of lo-’ of the crack length (when SIFs are increased by a factor of 2.2 by interactions) and 13% at the spacings of 0.025 of the crack length (when SIFs are increased by a factor of 4.1). These examples are not representative, though, since accuracy in the collinear arrangements is unusually good. In other geometries, accuracy is substantially worse; however, it still remains relatively good. It remains within several percent at spacings between cracks several times smaller than the crack lengths. The worst configuration is the one of partially shifted stacked cracks; in this case, the method remains accurate at spacings of about one-third of the crack lengths. b. Three-Dimensional Problems In 3-D problems, the accuracy is generally much better than in the similar 2-D ones. Referring to Kachanov and Laures (1989)for a detailed discussion, we review here the basic findings. Most of the 3-D solutions available in literature are limited to the case when the spacings between cracks are not too small so that the interactions remain relatively weak. Since our method is asymptotically exact in these cases (see discussion below), these problems do not constitute challenging tests. Comparison with the available results for moderate and strong interactions yields the following results. 1. The configuration analyzed (numerically)by Isida et al. (1985)consisted of parallel “offset” stacked cracks of elliptical shape under mode I loading. In those with a circular shape, the smallest spacing between
Elastic Solids with Many Cracks and Related Problems
295
cracks considered was 0.5 of the crack diameter; the ratios K,/KP were calculated for the “offsets” of 0 (axisymmetrical configuration), 0.5, 1.0 and 1.5 of the crack diameter. The calculated ratios K,/KP were, as read approximately from the graphs, 0.66, 0.83, 0.97 and 1.02. They fully coincide, within these digits, with the ones obtained by our method. 2. Uflyand (1967) obtained results for the axisymmetrical configuration of parallel stacked cracks under mode I loading, at the spacings between cracks of 0.25, 0.5 and 1.0 of the crack diameter. His results are KJKP = 0.7703, 0.8324 and 0.9191 whereas our results are 0.7678, 0.8249 and 0.9176. The disagreement is very small (and may be attributed partly to the procedure of numerical integration used by Uflyand: trapezoidal rule with 10 integration points). 3. Fabrikant (1987a, 1989) considered the problem of very closely spaced, strongly interacting coplanar cracks by reducing it to a new form of integral equations that are nonsingular and can be solved by rapidly converging iterations. For two coplanar cracks of equal size, the results converged up to spacings as small as 2.5 x of the crack diameter (mode I loading) and 0.005 (shear loading). Under mode I loading, the maximal error in SIF along the crack edge (at the point of closest spacing) is only 0.7% and 3.8% at the spacings of 0.05 and 2.5 x of the crack diameter, becoming indistinguishable at the points along the edge that are farther away from the point of the closest spacing. Under mode I1 loading, the results are similar (see Kachanov and Laures, 1989, for details). 6. Comparison with Polynomial Approximation Techniques
The advantages of the method are that (1) it applies to both 2-D and 3-D configurations with equal ease, being particularly accurate in 3-D, and (2)it is sufficiently simple to allow computer experiments on large arrays of interacting cracks, in both 2-D and 3-D (see Section V and VI). It is interesting to compare it with the frequently used 2-D technique of polynomial expansions of tractions on cracks and finding the polynomial coefficients from a system of linear algebraic equations. There are two basic versions of the technique of polynomial expansions: 1. Tractions on each considered crack are approximated by polynomials. 2. When considering tractions on a given crack, polynomial approximation is applied to tractions on the other cracks.
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Mark Kachanov
The best realization of technique 1 is achieved by using Chebyshev’s polynomials. (Gross, 1982). A less efficient realization-by using Taylor’s polynomials-was developed by Chudnovsky et al. (1983, 1984, 1987), Horii and Nemat-Nasser (1985,1987) who called this technique “the method of pseudotractions,” and other authors. Test problems involving collinear crack arrangements (considered in detail by Horii and Nemat-Nasser) show that the degree of approximating polynomials increases rapidly as spacings between cracks become smaller. The closest distance considered by them is one-quarter of the crack length; at this distance the polynomials’ degrees are 28 and 17 for the problems of two collinear cracks and a periodic row of cracks, respectively. The drawback of this technique is that the approximation of the crack-generated fields (having singularities at crack tips) by polynomials becomes inefficient at small spacings between cracks, close to the singularity points. Our method, in contrast, retains the singularities (in the “standard” stress fields). In technique 2, developed by Benveniste et al. (1989), polynomial approximations are applied not to the tractions on a considered crack, but to the tractions on the other cracks that generate additional tractions Atj’ on the considered crack. This technique has the advantage of retaining the singularities in the fields generated by the jth cracks. It appears to be more efficient, as demonstrated by applying it to the test problem of two collinear cracks: SIFs at the inner tips remained accurate at spacings as small as of the crack length. The main drawback of the polynomial techniques is that they are not easily extended to 3-D configurations. In principle, it can be done (polynomials of two variables will have to be used for approximation of tractions on cracks), but the method is expected to become cumbersome, particularly if interactions are strong or the number of cracks is substantial. To our knowledge, this technique has never been actually implemented in 3-D.
c. APPROXIMATIONOF SMALL TRANSMISSION FACTORS A N D APPROXIMATION OF
WIDELYSPACED CRACKS
We outline two approximations that can be formulated in the framework of our method. The first one (approximation of small transmission factors) applies to the situations when all the transmission factors A << 1. This may
Elastic Solids with Many Cracks and Related Problems
297
hold when the spacings between cracks are not too small, but may, nevertheless, depending on the geometry of the configuration, be smaller than the crack sizes. In this approximation, interactions produce only a weak impact on the average CODs (and, therefore, on the effective elastic properties), but may produce a strong impact on SIFs. The second one (approximation of widely spaced cracks) applies under a more restrictive geometrical assumption that spacings between cracks are much larger than crack sizes. Then, the impact of interactions on both effective elastic properties and SIFs is weak. In the first approximation, the method is considerably simplified; in the second one, the simplifications are so major that the results (first-order corrections for crack interactions) can be written down, with almost no calculations involved, for any arrangement of cracks both in 2-D and 3-D. 1. The Approximation of Small Transmission Factors (Interactions Weak in the Average Sense) We consider such crack configurations that the average traction vector (AtJi) induced on each ith crack site by each of the jth crack is much smaller than the traction induced on the ith crack by the remote loading 0': l(Atji) << lti(')l
for all i, j
(3.25)
If (3.25) holds, the interactions can be called weak in the average sense. Note that this definition deals with tractions. It is much less restrictive than the geometric requirement E = (crack
sizes)/(spacingsbetween cracks) << 1
(3.26)
and is, typically, satisfied at spacings between cracks that are comparable to crack sizes or even smaller (particularly, in the collinear and coplanar configurations). It follows from (3.25) that all 1) Aji 1) << 1
(3.27)
(absolute values of all the components of tensors Aji are << 1). Smallness of A-factors implies that, to within small values on the order of A, traction on the ith crack is a sum of the a'-induced traction n i * u Oand the tractions induced at the ith crack site by the other cracks, the latter being loaded by the uO-induced tractions nj- uo (rather than by the traction averages (t')).
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Mark Kachanov
Therefore, instead of (3.9) in 2-D and (3.19) in 3-D we have
(3.28)
(3.29)
in the 2-D and 3-D cases, respectively. Note that this approximation can be interpreted as the first iteration of (3.9) and (3.19), i.e., as truncation of "feedbacks": transmission of the a'-induced tractions from the j t h crack on the ith crack is not followed by feedback. Thus, the major computational simplification in this approximation is that finding the transmission A-factors and the average tractions becomes unnecessary. This approximation is not as restrictive as it may seem: the inequality (3.25) applies to the average tractions and does not exclude the possibility of high local maxima of the tractions At(t) and the resulting sharp local peaks of the SIF-values along the crack edges. Moreover, up to relatively small spacings between cracks, the approximation of weak interactions may actually provide a reasonably accurate description of such peaks. The problem of two coplanar cracks serves as an example (see Kachanov and Laures, 1989). At the distance between cracks as small as 0.005 of the crack diameter the increase of the average traction due to interaction is only 8% (inequality (3.25)holds with a satisfactory accuracy) whereas the maximal value of K,/KP along the crack edge is 1.908. If the approximation of small transmission factors is used, this value changes to 1.842-a satisfactory estimate for many purposes. At the distance 0.05 of the crack diameter the increase of the average traction is only 5% whereas max K J K P = 1.296; this value changes insignificantly-to 1.281 if the approximation of small transmission factors is used. Another example is provided by the problem of two stacked cracks, where the applicability of the approximation is
Elastic Solids with Many Cracks and Related Problems
299
restricted by substantially larger spacings; this approximation results in a 5% error in SIFs only if the spacings between cracks are at least 1.25 and 0.35 of the crack diameter for the normal and shear loadings, respectively. Thus, the interactions may or may not be weak depending on the physical property considered: interactions that are weak in the sense of traction averages are not necessarily weak in the sense of SIFs. This fact has an important implication: the efectioe elastic moduli for a solid with a given arrangement of cracks may differ only slightly from the ones obtained in the approximation of noninteracting cracks, whereas SIFs-fracture-related properties-may be significantly affected by interactions. This serves as yet another illustration of the fact (discussed in Section VII) that there seems to be no stable quantitative correlation between deterioration of the efective elastic stirness of the “damaged” material and its progression toward failure. 2. The Approximation of Widely Spaced Cracks This approximation is defined by quite a restrictive inequality (3.26)-a condition of a geometrical nature. Note that many of the results available in literature have been obtained under this assumption. In this approximation, our method yields solutions (first-order corrections for crack interactions) for arbitrary arrangements of cracks, in both 2-D and 3-D, in an elementary way, with almost no calculations involved. We note, first, that (3.26), obviously, implies that (3.25) is satisfied, so that all the simplifications of the approximation of small transmission factors apply. Moreover, further major simplifications follow. Since the spatial gradient of a crack-generated field attenuates with distance faster than the the impact of the ith crack on the kth one can be characterized, to field within small values of higher order, by the stress that is constant along the kth crack site and can be taken as the value of the ith crack-generated field at the kth crack center. Thus, the only calculation involved is finding the stresses generated by ith cracks (uniformly loaded by the a’-induced tractions n’. 0’). An additional simplification is that, instead of the standard fields, their farfield asymptotics given by much simpler expressions (Section 1I.C) can be used. Note that the interaction correction for a SIF on a given crack depends on the size of the other cracks but not on its own-there is no “feedback” in the framework of the asymptotic solutions (see examples b and d to follow). Several configurations are considered below, as an illustration.
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Mark Kachanov
a. l b o Coplanar Cracks of Equal Radius a Under Tensile Remote Loading Here, the distance between the crack centers 2b >> a. Taking the far-field asymptotics of the a,,-component of the standard mode I field and simply adding it to the aO-inducedtraction, one obtains K, =~ ; [ + i (a/b)3/12.3
(3.30)
The second term in the brackets is the first-order correction for crack interaction. This result recovers the one obtained by much lengthier means by Collins (1962). b. %o Coplanar Cracks of Diferent Radii, a, and a,, Tensile Loading Here, the distance between crack centers 2b >> a,, a,. Following the same steps, i.e., evaluating the asymptotic far field generated by crack l(2) at the center of crack 2(1), one obtains = K(1)o I [l
+ (~,/b)~/l2n],
K12’ = K{”O[l
+ (al/b)3/12~]
(3.31)
c. Wo Stacked Cracks of Equal Radius a, Tensile Loading Here, the distance between crack centers 2b >> a. The asymptotic expression for the stress u,, generated by a crack loaded by a uniform pressure p at the far away point b above its center is oZz= -p(2/3n)(b/a) so that KI = KP[l - (2/3~)(a/b)~]
(3.32)
where the minus sign indicates that the effect of interaction is shielding. d. Wo Stacked Cracks of Diflerent Radii under Mode I Loading, Tensile Loading Here, the distance between crack centers 2b >> a,, a, KI1) =
K(1)o I [l - (2/3~)(a,/b)~],
Ki2’ = K”” I c1-
(3.33)
e. lnjinite Stack of Cracks of Equal Radius a, Tensile Loading Here, the distance between crack centers 2b >> a:
[ + (y+ (y ***I
K, = KF[l -(2/3~)(a/b)~] 1 KI = KP[l - (2/3~)(a/b)~]
-
-
+
(3.34)
Elastic Solids with Many Cracks and Related Problems
30 1
where c(3) = 1.202 is the zeta-function of argument 3. Some of these results recover solutions obtained earlier in the literature by much lengthier means. f. Limits of Applicability of the Asymptotic Solutions
Because of the elementary character of these solutions, it is of interest to estimate the limits of their applicability. We note that a characteristic feature of the solutions obtained in the approximation of widely spaced cracks is that SIFs are constant along the edge of each crack (in the 2-D case, they are the same for both crack tips). This indicates a very limited usefulness of such solutions-the results are valid only in the limit of vanishingly weak interactions; see Kachanov and Laures (1989) for discussion and estimates of applicability.
IV. Various Effects Produced by Crack Interactions Using the method of analysis outlined in Section 111, we discuss various effects produced by interactions of cracks. In this section, we focus our attention on SIFs (effective elastic properties are discussed in Section VI). Solutions of various interaction problems were obtained by the method described in Section 111.
A. STRESS SHIELDING AND STRESSAMPLIFICATION. Two- AND THREE-DIMENSIONAL INTERACTIONS Crack interactions may produce either stress amplification (increase of SIFs) or stress shielding (decrease of SIFs),depending on the geometry of the configuration and, for the same geometry, on the mode of loading. Here, we examine these effects by analyzing several representative geometries. 1. 2-0 Geometries
a. Mode I Loading Collinear configurations are characterized by the amplifying effect of interactions (although the amplifying effect is the strongest when the symmetry is slightly “disturbed,” see the discussion in Section C). This is
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Mark Kachanov
1.6 0
a
t
t
t p t
1
1
1
1
1
t \ 0.1
0.5
0.9
1.3
2.1
1.7
FIG.14. Two collinear cracks under mode I loading. SIF at the inner tip.
illustrated in Fig. 14 for the collinear configuration of two cracks of equal length. Stacked conjiyurations, on the other hand, constitute a shielding configuration, as illustrated in Fig. 15. The amplifying and shielding effects of several neighbouring cracks on a 1.o
0.8
0.4
-
i
-1.1 1
1
1
1
1
0.0 0.2
0.5
0.8
1.1
1.4
1.7
2.0
A/] FIG.15. SIFs for two stacked cracks under mode I loading.
Elastic Solids with Many Cracks and Related Problems
303
given crack may compete. The overall effect in such cases depends on the relative distances from the considered crack tip to the amplifying and shielding neighbors. Figure 16 serves as an illustration. Figures 14-16 show that the effect of shielding has a longer range than that of amplification: in the configurations where they compete, the effect of shielding is dominant; i.e., to cancel the impact of shielding neighbors on a given crack tip, the amplifying neighbor should be much closer to the tip. This is a consequence of the structure of the a,,-field associated with one crack under mode I loading: there is a compact amplification zone (a,, > 0) and a substantially larger shielding zone (ayy< 0), see Fig. 1. b. Mode I1 Loading Many configurations that produce the shielding effect under mode I loading become amplifying under mode I1 loading. Figure 17 provides an illustration. 2. Relative Strength of Interaction Efects in 2 - 0 and 3 - 0 Geometries
The 3-D interactions produce effects that are qualitatively similar but substantially weaker than in the similar 2-D configurations: coplanar 2.0
1.5
I
' .
'
.
' I '
'1i @//
0.0 0.0
0.25
'
1
"
' .
'
7
,
0.5
a
'
t
r
'
'
'
tPt
h T I rC.
0.75
.
I
:
f
1.0
Fic. 16. Competition of the amplification and shielding effects (mode I loading) for the inner tips: + and - signs indicate the regions where the amplification and shielding are dominant, respectively; the neutral line (no interaction effect)is shown. Note that the amplifying effect of the collinear interaction has a much shorter range than the shielding effect of the stacked configuration: A must be much smaller than h to neutralize the shielding effect.
Mark Kachanov
304
1.0 0.0 1
"
: 0.5
"
"
"
"
"
1
"
1.5
"
2.0
h?: FIG.17. The same configuration under mode I1 loading (h = I ) .
interactions produce stress amplification whereas stacked ones produce shielding. Similarly to 2-D cases, the shielding interactions under mode I loading have a longer range than the amplifying ones. Figures 21-24, which will appear in Section B, provide an illustration. The interaction efects in 3 - 0 are weaker than in similar 2-D geometries, as illustrated in Fig. 18, which directly compares the strengths of interaction effects in 2-D and 3-D. The difference in strength of the interaction effect depends on the geometry: in configurations of the shielding type (stacked cracks) the difference is moderate and decreases at smaller spacings between cracks; in the amplifying configurations (collinear and coplanar cracks) the difference is very substantial, particularly for closely spaced cracks. For example, in the 3-D configuration of two coplanar cracks, the maximal (along the crack edge) increase of K , is about 30% at the spacing between cracks of 0.05 of the crack size, vs. about 80% for the similar 2-D configuration of two collinear cracks with the same spacing. These facts are directly related to the structure of single crack-generated fields (discussed in Section 11): the field of oyyunder mode I loading extends over comparable distances in the y-direction, in 2-D and 3-D geometries, but attenuates substantially faster with distance in the x-direction in the 3-D geometry.
Elastic Solids with Many Cracks and Related Problems
305
1.8
i l
t
t
t
t p t
t
t p t
1.4 0
LT
\ 1.0
b7 0.6 1
0.2 0.0
1
1
1
0.5
a/2e
1
1.o
1
1
1
1.5
FIG.18. Comparison of interaction effects in similar 2-D and 3-D configurations.
3. Localized Character of 3 - 0 Interactions. Mixed Mode Efects
The 3-D interaction effects vary along the crack edges, and changes in SIFs produced by interactions are usually localized within a relatively small zone. This is illustrated in Fig. 19, which shows variation of the mode I SIF along the edges of two coplanar cracks. The results of Section V on the interaction of a crack with microcracks provide further examples: SIFs along the main crack front are affected by microcracks only in the immediate vicinity of the microcracks. An additional complicating factor is that both modes I1 and I11 are present,
FIG.19. Interaction of two coplanar circular cracks (spacing between cracks is 0.05 of the crack diameter): the solid line shows variation of the SIF along the crack edge.
Mark Kachanov
306
t t t t
1 1 1 1
FIG.20. V-configuration:modes I, I1 and I11 SIFs along the crack edge (SIFs are normalized to their values in absense of interactions, dashed line).
if shear tractions are applied. Figure 20 shows interaction effects in the configuration of two closely spaced inclined cracks (V-configuration),where all the modes are present along the crack edges. An interesting feature of this configuration is a strong mixed mode interaction. Indeed, comparison with two coplanar cracks (having the same spacing between cracks) shows that, although the absolute value of the maximal K, is 55% higher in the coplanar configuration, the increase of max K , due to interactions in the V-arrangement is 1.68 times, vs. 1.30 times in the coplanar configuration. Thus, the interaction effect is much stronger in the V-arrangement. This is explained by the mixed mode interaction: stress field generated by the shear mode on crack 1 enhances the model SIF on crack 2.
B. RANGEOF INFLUENCE
OF A CRACK IN AN
ARRAYOF INTERACTING
CRACKS
We discuss here the range of influence of an individual crack in a crack array. We consider two periodic arrangements of cracks: a collinear (coplanar) one and a stack of parallel cracks. Then, we create a “disturbance”
Elastic Solids with Many Cracks and Related Problems
307
I
0.9
1
2
3
4
Crack Number FIG.21. Changes in SIFs (marked by stars) in a periodic collinear row of cracks generated by a “disturbance.” SIFs are normalized with respect to the one of the undisturbed configuration. (Spacing between cracks in the undisturbed configuration is one-third the crack length.)
(by either removing one crack or rotating it) and examine the zone of influence of this disturbance; i.e., the number of closest neighbors affected. 1. Zone of Influence in 2 - 0
a. Zone of Injluence in the Collinear Direction We consider a collinear crack array with a “disturbance” created by rotating one crack by 45” and calculate the change in SIFs on neighboring cracks caused by the disturbance. Figure 21 shows that the disturbance has a short range of influence: its impact is limited to the closest one or two cracks. b. Zone of InfIuence in the Direction Normal to a Crack A column of parallel cracks is considered; one crack is then removed from the stack. The results (Fig. 22) show that, similarly to the case of collinear array, the disturbance has a short range of influence, confined to the closest one or two neighbors.
2. Zone of Influence in 3 - 0 a. Zone of Influence in the Coplanar Arrangement A row of 11 coplanar cracks is considered, and a “disturbance” is created by rotating a median crack by 45”. The maximal (along the crack edges)
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Mark Kachanov
FIG. 22. Changes in SIFs in a column of cracks generated by a “disturbance.” (Spacing between cracks in the undisturbed configuration is one-half the crack length.)
changes in SIFs at the neighboring cracks are shown in Fig. 23. The range of influence of the disturbance is even shorter than in the 2-D case-its impact is practically confined to the closest neighbor. b. Zone of Influence in the Direction Normal to a Crack A stack of 11 parallel cracks is considered, and a “disturbance” is created by removing a median crack. The results are shown in Fig. 24. Similarly to the previous case, the impact of disturbance is practically confined to one or two closest neighbors. C. EXTREMAL PROPERTIES OF SLIGHTLY ASYMMETRIC ARRAYS
An interesting (and somewhat unexpected) observation is that the interaction effect may be maximal not in the ideally symmetric arrangements but in configurations where the symmetry is slightly disturbed. The simplest example that explains the underlying physical mechanism is the configuration of two collinear cracks loaded under mode I loading: if the right crack is rotated about its center on an angle f#J (Fig. 25), the mode I SIF at the inner tip of the left crack is maximal not at f#J = 0 (as may be intuitively expected) but at x 15”. The reason is that K,at left crack tip is affected by both normal and shear tractions on the right crack (shear traction appears as a result of asymmetry). The impact on K , of the normal traction on the right crack is, obviously, f#J
Elastic Solids with Many Cracks and Related Problems
0.8
309
1
2
3
4
5
Crack Number
FIG.23. Changes in (maximal, along the edge) SIFs in a 3-D periodic coplanar row of cracks generated by a “disturbance.” [Spacing between cracks in the undisturbed configuration is onethird the crack diameter.)
maximal at Cp = 0, but this implies that its rate of change with Q, is zero at Q, = 0 so that the decrease of this impact due to rotation is small at moderately small 4. On the other hand, the impact on K, of the shear traction on the right crack has a nonzero rate of change at Q, = 0. Therefore, the combined impact of both modes is maximal at nonzero Q,. Thus, the underlying mechanism of this phenomenon is the appearance of mixed mode interactions caused by a disturbed symmetry. It is clear,
1.05
\ 1.025
x-
m 10
1
2
3
y 4
5
Crack Number FIG.24. Changes in SIFs in 3-D periodic stack of parallel cracks generated by a “disturbance.” [Spacing between cracks in the undisturbed configuration is one-half the crack diameter.)
Mark Kachanou
3 10
0.0 0.0
30.0
60.0
90.0
d FIG.25. SIFs at the inner tip of the left crack in the collinear configuration with symmetry disturbed by rotation.(Spacing between cracks in the undisturbed configurationis onetenth the crack length.)
therefore, that the phenomenon is general: the “disturbance” of symmetry that gives rise to mixed mode interactions intensifies the interaction effect (if the disturbance remains moderately small). In the case of collinear cracks, for example, disturbances of symmetry ocher than rotations (Fig. 26) lead to a similar intensification of interactions. One can easily construct a number of similar configurations (for example, a rectangular periodic array with slightly disturbed crack orientations or positions). The excess of SIFs in configurations with disturbed symmetry over their values in ideally symmetric configurations is quite noticeable, as seen from the preceding examples. Note that the corresponding reduction of the critical load at which the affected crack tip will propagate becomes even more noticeable if the appearance of mode I1 at the tip (in the disturbed configuration) is also taken into account. A similar phenomenon holds in 3-D configurations. It is less pronounced (due to a general weakness of all interaction effects in 3-D), but has many more manifestations (due to a larger variety of possible geometrical asymmetries). The crack-microcrack configuration of Fig. 36 (in Section V) provides an illustration: the peak of SIF is found not at the symmetry point of the crack front where it may be expected, but at a somewhat shifted position (particularly at 6/a = 0.8); this effect is, again, due to the mixed mode interactions appearing at asymmetric points. These observations on extremal properties of slightly asymmetric
Elastic Solids with Many Cracks and Related Problems
31 1
1.6 1.4
1.2
1.o
0.8 0.6
1
0.0 -0.2
0.0
0.5
1.0
1.5
2.0
A/1 FIG. 26. SIFs at the inner tip in a collinear configuration with symmetry disturbed by translation.
configurations may have some physical implications (although such considerations remain speculative at this point). If one assumes that the system tends to an extremum of the energy release rate (and, thus, of the SIFs), then the initially symmetric configurations will tend to deviate from symmetry in the course of crack propagation, to pick up the mixed mode interactions that increase SIFs. For example, in the configuration of two collinear cracks loaded in mode I, the cracks d o not grow toward each other along the straight line (under mode I loading) but deviate from it, giving the appearance of “avoiding each other.” This problem was discussed by Melin (1983), with a rather complex mathematical discussion involved. It seems that a simpler explanation may be suggested: deviation from symmetry leads to mixed mode interactions that maximizes the energy release rate. Asymmetry of some naturally occurring crack patterns may, possibly, be explained in the same way.
D. INTERSECTING CRACKS(TWO-DIMENSIONAL CONFIGURATIONS) 1. intersecting Cracks
The configurations of intersecting cracks may be of interest in a variety of applications, both micromechanical and structural. Kachanov and Montagut (1989) demonstrated how the methnd of analysis of crack interactions
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Mark Kachanov
outlined in Section I11 can be used for an approximate analysis of such configurations. We apply this method to intersecting cracks by simply ignoring the intersection and treating the “overlapping” cracks according to the general scheme for interacting cracks. Formally, this creates no difficulties in the computational scheme, and SIFs at the outer tips can be found. (Obviously, stress concentration at the point of intersection cannot be analyzed by this technique). The author is unable, at this point, to rigorously justify this procedure; the only justification is that it works surprisingly well for the available test problems. Results for two such test problems are presented in Fig. 27. a. Starlike Configuration The solution obtained by formally applying the method of Section I11 and ignoring the intersection is compared with the exact solution in Fig. 27. The comparison shows that the agreement is very good, up to the number N of cracks in the star of about 20. Note that the actual reason for increasing
1.0
0.8
-
r-
0.6
c
v r
0.4 0.2
0.2 0.0
0.0
2
4
6
8 10 12 16 20
n
0.0
,
I 0.2
,
I 0.4
,
I 0.6
,
I ( 0.8
1.0
Wa
FIG.27. Test problems for intersecting cracks: comparison of our solution (solid line) with the exact results (dashed line).
Elastic Solids with Many Cracks and Related Problems
313
errors at N > 20 is that the crack tips become too close to one another for the method to remain accurate; it is unrelated to the fact of intersection. b. Rectangular Cross The agreement is, again, quite good. This simple approach is no't restricted to any particular symmetric arrangement and can be applied to any nonsymmetric configuration formed by intersecting cracks (and holes; see Kachanov and Montagut, 1988). 2. Crack Touching the Line of Another Crack A somewhat different configuration-with the tip of one crack touching the line of another crack-was considered by Benveniste et al. (1989).Their analysis is based on (1) finding SI[Fs at the tips of very closely spaced cracks by expansions of tractions on cracks into Legendre's polynomials, combined, possibly, with step functions and (2) using these obtained solutions for the outer tips as an approximation of the SIFs at the outer tips of the original configuration. Of the three test problems (a T-crack and two loadings of an H-crack), the accuracy was satisfactory in one of the problems; in the other two problems, the accuracy was satisfactory provided the sizes of cracks were not too different (their ratio not exceeding 5-6 in one configuration and 2.5 in the other). We note that, in the limit when spacings between cracks approach zero, such solutions should indeed converge to the ones with touching cracks. However, this convergence is extremely slow. Consider, as an example, a configuration of two collinear cracks of equal length. In the aforementioned limit, the SIF at the outer tip should increase $ = 1.41 times, as compared to the case of one isolated crack (since crack size increases two times when cracks coalesce). The exact solution of this problem shows that, when the spacing A between cracks is as small as of the crack length, the increase is only 1.28 times, increasing to only 1.32 even when A = spacings on the order of lo-" of the crack length (at which polynomial techniques may become inefficient) are required for a close approach to the $ increase. We note, also, that the configuration of two collinear cracks is the simplest one, and approximation methods, generally, show the best accuracy when applied to this problem. Therefore, for the results for touching cracks obtained by this technique to be reliable, consideration of extremely closely spaced cracks may be necessary.
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Mark Kachanov E. INTERSECTING CRACKS (THREE-DIMENSIONAL CONFIGURATIONS)
The same technique can be applied to three-dimensional configurations of intersecting (penny-shaped) cracks. Unfortunately, no test solutions seem to exist, so that the accuracy of the technique cannot be verified. However, the 3D problems are, generally, less sensitive to simplifying assumptions; the technique, therefore, can be expected to work, generally, better than in 2-D. The restriction remains, of course, that stress concentrations near the line of intersection of crack planes cannot be analyzed by this technique (similarly to the 2-D case, where the stress concentration near the point of intersection could not be analyzed). Another restriction is that the edges of two cracks cannot have common points (this would have created a nonintegrable singularity in expressions for SIFs). Here, results are given for two 3-D configurations: a crack intersected, near its center, at a 45” angle, by a crack three times smaller than it (Fig. 28), and two cracks of equal size intersecting at a 45” angle along a half-diameter (Fig. 29). In the configuration of Fig. 28, SIFs along the edge of the larger crack are almost unaffected by the presence of a “ripple” near its center; the SIFs along
__.. -- .___ ,_....__
FIG.28. SIFs along the edge of the (smaller) crack in the 3-D configuration of intersecting cracks. Remote tension normal to the larger crack is applied. The ratio of the crack sizes is 1/3. SIFs along the edge of the larger crack are almost unaffected by the presence of a “ripple” near its center. SIFs are normalized to their values for an isolated crack of the same orientation.
Elastic Solids with Many Cracks and Related Problems
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B
FIG.29. SIFs along the edge of crack 2 in the 3-D configurationof intersecting cracks (of equal size). Remote tension normal to crack 1 is applied. SlFs are normalized to their values for an isolated crack of the same orientation.
the edge of the smaller crack experience a strong shielding (shown in Fig. 28) at all points of the crack edge, due to the presence of the larger crack. In the configuration of Fig. 29, SIFs along the edge of crack 2 are shown; they experience shielding in the zone of overlapping and almost no effect of interactions outside of this zone. In both configurations, the SIFs are normalized to their values for an isolated crack (dashed line). The results may not be reliable in the vicinity of the line of intersection of the crack planes.
F.
INTERACTION OF CRACKS
FILLED WITH COMPRESSIBLE FLUID
We analyze the situation when crack cavities are filled with a compressible fluid. It is assumed that the fluid is nonviscous; i.e., it resists compression but does not resist shear motions of the crack faces. This situation may be of interest for the mechanics of fluid-saturated geomaterials. We assume the “undrained” conditions, i.e., that the total mass of fluid in each cavity remains constant; if fluid diffusion across crack faces does take place, this approximation corresponds to a “short time” response. (In the opposite limit of “drained” approximation, when crack cavities freely
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Mark Kachanou
exchange fluid with the surroundings, the mechanics of crack interactions is the same as in the case of dry cracks.) The analysis of this section focusses on the fluid’s impact on stress interactions, in particular, on SIFs. (Efectiue elastic properties of a solid with fluid-filled cracks and pressure polarization are discussed in Section V1.E.) First, the behavior of one fluid-filled crack is analyzed; the consideration is similar to the analysis of Budiansky and OConnell (1976). Then, crack interactions and their coupling with fluid pressures are analyzed, using the method of Section 111. 1. One lsolated Crack
If fluid is present in a crack cavity, and to = ( p o , r o ) is the traction corresponding to the remote loading, then the overall normal traction on crack faces is
+ Aq,
p = po
T = TO
(4.1)
where Aq is the fluid pressure (that develops as a reaction to the applied normal traction). Its sign is opposite to the sign of po and thus Aq has a “dampening” (counteracting) effect. The value of Aq is determined by relating it, through a constitutive equation of the fluid, to the change of density of the fluid, which, in turn, is related to the change in the cavity’s volume. For simplicity of calculation, we assume that the cavity has the shape of a relatively thin platelet, so that its normal compliance can be approximated by that of a crack. We first consider the case of a cavity circular in plane of radius 1 that, in the absence of applied stresses, has average opening ( b , ) and volume n13c,, where lo= (b,)/l can be called the auerage aspect ratio. Since the cavity is thin, its volume change is due mainly to changes in ( b , ) ( I remains approximately constant). Thus, the relative volume change of the cavity is AVIV, = ( A b J / ( b , ) . Since, under undrained conditions, the fluid mass within a cavity is constant: qoV, = ql! the change of fluid density is A‘tl‘lo = -AV/Vo = -
(4.2)
The corresponding change of fluid pressure is given by the constitutive equation for a compressible fluid, which, in the linearized formulation (small density changes), has the form W
l
O =
-uAq
(4.3)
where u is the compressibility of a fluid (generally, a function of q and temperature; treated as a constant in the considered linearized approxima-
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317
tion); in conjunction with (4.2), this equation relates Aq to the change of opening (Ab,,). On the other hand, (Ab,) is related to the overall normal traction on a cavity p o + Aq by a “cavity compliance” relation: (AbJ = (PVEoXpO -t A d
(4.4)
where the compliance of a (thin) cavity is taken to be approximately the same as the one of a circular crack of the same diameter: p = 1q1 - v , 3 / 3 ~ In . the 2-D case (narrow elliptical holes of length 2/), B = n/. These equations also yield the change of fluid pressure induced by the applied traction PO:
where
s = p-’KEo[o
(4.6)
is the key constant of the problem that determines coupling between fluid pressure and applied stress. It is similar to the one used by Budiansky and O’Connell. Physically, (4.4) means that the applied compressive stress is counteracted by two factors: the “cavity’s stiffness” and the fluid pressure. Parameter 6 determines relative shares of these two factors. In the limit of highly compressible fluid (K -+ 00, “air”), the change in the fluid pressure in cavities caused by applied stresses is 0, and the case of a dry crack is recovered. In the opposite limit K + 0 (incompressible fluid), Aq = --po and the crack is fully constrained against normal opening; the same is true if K is finite but cavities are infinitesimally thin, lo -+0, indicating that coupling between stresses and fluid pressure is stronger for thinner cracks. As an example that may be relevant for fluid-saturated rocks, we consider MPa-’ (water) and E , x 6 x 10” MPa (granite). the case of K = 0.5 x Then, for the cavity aspect ratios lo < 0.006 the impact of fluid is so strong that the normal COD constitutes less than 10% of that of a dry crack. The impact of fluid becomes small (COD differs by less than 10% from that of a > 0.5-0.6 (note that medium with dry cracks) only for very wide cavities, lo the calculation of the cavity’s volume change used earlier becomes then inaccurate). Thus, in the entire range of interest [,, < 0.5, the effect of stress-fluid pressure coupling is strong.
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Mark Kachanov
2. Interacting Cracks Here, we analyze some physical effects produced by coupling between stress interactions and Jiuid pressures. The basic observation is that normal and shear loading modes on cracks play different roles in the coupling. Since the fluid does not resist shear CODs, mixed mode interactions of the type shear-shear and shear-normal (abbreviated as s - s and S - n ) are unaffected by the fluid, whereas interactions of the type n -+ s and n -+ n are “dampened.” For brevity, we combine, in the 3-D case, both shear modes T and s, into one (in order to use, instead of (3.20),the shorter formulas (3.10)). The formulas (3.10) that interrelate averages tractions in dry cracks, are to be modified as follows:
where, in the spirit of (4.9, we have
so that equations for the average tractions take the form
We observe that fluid produces “dampening” effects of two kinds: (1) it reduces the overall traction on a given cavity, as described by (4.1), independently of interactions, and (2) it dampens the interactions, in the modes n - s and n + n . Another observation is that both (1) and (2) are dampened to the same extent, as shown by the common dampening multiplier (1 ai)The first equation of (4.9)can be rewritten in a form that resembles the one for dry cracks; the difference is indicated by overbars:
+
’.
(4.10)
Elastic Solids with Many Cracks and Related Problems
3 19
(4.11)
Note that the transmission &-factors have the structure of products (geometrical factor A) x (damping factor). In the limit of S -,0 (incompressible fluid or very thin cavities), (pi) = 0 and
(fi) = fi(0)
+ 1 Ajf(~j)
(4.12)
j+i
indicating that the normal mode on cracks is fully suppressed,and only shear interactions (unaffected by fluid) take place. In the opposite limit of 6 + CO (“air” or wide cavities), the case of dry cracks is recovered. a. Impact of Fluid on SIFs of Interacting Cracks According to the method of Section 111, the actual (variable)tractions on a given crack are found from (3.9), as a sum of the no-induced tractions and the impact of the other cracks loaded by the average tractions on them. Applying this superposition to our case, we represent the mode I SIFs on a given crack in the form K , = KP,dlY
+ KP.ICaC1 +
KP
(4.13)
where the first term is the SIF of an isolated dry crack, the second term characterizes the reduction of SIF due to the fluid pressure arising in an isolated crack as a reaction to the applied stress and the third term represents the interaction effect(partly dampened by fluid, as discussed above). In shear modes, the second term is absent: (4.14)
An important observation is that fluid dampens both amplifying and shielding interactions. b. Periodic Row of Collinear Cracks This simple 2-D configuration illustrates the mechanics of stress interaction-fluid pressure coupling. Under mode I1 loading, fluid has no impact on interactions, and the problem has the same solution as for dry cracks. Under
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Mark Kachanov
mode I loading, the equation for the average normal traction (the same on all cracks) has the form ( p ) = 6(1 + 6 ) - ' p o
+ 6(1 + d)-' C A(k)(p)
(4.15)
where A(k)is the transmission factor characterizing interaction of a given crack with the kth neighbor; thus, (4.16) Assume, for illustration, that parameter 6 = 1, and that the spacing between cracks is one-half of the crack length. Then, denoting by KF the SIF for an isolated dry crack (reference value), we obtain that, for an isolated Juid-jlled crack, K , = 0.50, fluid strongly dampens the SIF. This represents the first mechanism through which fluid affects SIFs. The second mechanism is the impact of fluid on interactions: for interacting fluid-filled cracks, K , = 0.56, interactions increase the SIF by 12% It is interesting to compare this increase due to interactions to the one for dry cracks: for the latter, interactions increase the SIF by 29%. Thus, fluid significantly dampens the interaction effect. In this example, interactions have an amplifying effect, and their dampening means reduction of the SIF. In the configurations when interactions have a shielding effect (for example, stacked cracks), their dampening by fluid means an increase in the SIF.
G. INTERACTION OF CRACKS IN AN ANISOTROPIC MATRIX This section is a brief summary of the work of Mauge and Kachanov (1990,1992) and Mauge (1993), where the stated problem was investigated in detail (in the 2-D case). The method of Section I11 was used; in the available test problem (two inclined cracks) the method's accuracy remained high at the spacings between cracks as small as 0.05 of the crack length. The "standard" stress fields generated by uniformly loaded cracks are significantly affected by the matrix anisotropy. Therefore, the transmission Afactors are functions not only of the geometry of the crack array (as in the isotropic case), but on the matrix moduli as well. Our analysis is restricted to the case of an orthotropic matrix when solutions for interacting cracks of arbitrary orientations can be obtained in closed form (for a nonorthotropic matrix, the basic procedure remains
Elastic Solids with Many Cracks and Related Problems
32 1
unchanged, but solutions involve a numerical procedure of solving a fourthorder algebraic equation). With the exception of the problem of collinear cracks (when the interaction problem is identical to that for the isotropic matrix), the interactions are affected by the matrix anisotropy. Three parameters that affect the interactions are the ratio of Young’s moduli E J E , in the principal directions of orthotropy, the shear modulus G I , and Poisson’s ratio v I 2 . Of these parameters, the ratio E J E , plays the dominant role. If E,/E, > 1 (or < l), then interactions under a uniaxial loading in the x2 direction are enhanced (or weakened) by the matrix anisotropy, i.e., are more (less) intense, compared to the case of isotropy. For example, the shielding effect for stacked cracks is enhanced (or weakened) if matrix stiffness in the direction normal to cracks is higher (lower) than in the perpendicular direction. This has a clear physical explanation: transmission of stresses in a direction of higher (lower) stiffness is more (less) “efficient.” Note that the impact of the ratio E J E , is strongly “asymmetric”; enhancement of the interaction effects is much stronger than their weakening. The impact of the shear modulus G,, becomes noticeable only when this modulus is small (compared to the smallest of Young’s moduli). Impact of Poisson’s ratio v l t on crack interactions is small. The problem of intersecting (at a 90” angle) cracks in an orthotropic matrix was also analyzed, using the approach of Section 1V.D (such crack configurations in an orthotropic matrix may be of interest for the mechanics of composites). It was found that, in the case of sufficiently high contrast between matrix stiffnesses in two perpendicular directions, SIFs at the tips of one of the cracks may be negative under the applied hydrostatic tension.
V. Interaction of a Crack with a Field of Microcracks A. INTRODUCTION
Microcracking accompanies crack propagation in many brittle materials (see, for example, observations of Han and Suresh (1989) on high-temperature fracture of ceramics). This may substantially affect the overall mechanics of fracture propagation.
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Mark Kachanov
There are several physically different mechanisms by which microcracking may affect the overall resistance to fracture propagation. Among them 1. Expenditure of energy on nucleation of microcracks near the main crack.
This may (or may not, depending on strength of the intergranular cohesion, the level of residual stresses and other factors) constitute a significant energy “sink” and thus noticeably increase the apparent fracture toughness. 2. Performation of the material. The crack path passes through material only partly; another part of it passes through empty space. This factor reduces the apparent fracture toughness. 3. Release of residual stresses. 4. Elastic interaction of the main crack with microcracks that changes the stress intensity factors at the main crack. A number of papers have suggested that such an interaction produces stress shielding; i.e., reduces SIFs at the main crack tip. This result was obtained, in most cases, by modelling the microcracked zone by an effective elastic material of reduced stiffness. This work analyzes mechanism 4 only. In particular, we discuss whether mechanism 4 leads to an increased resistance to fracture propagation (“toughening by microcracking”)-a phenomenon suggested in a number of papers. We also discuss whether modeling the microcracked zone by an effectivehomogeneous material of reduced stiffness is adequate. We first consider several simple geometries involving one or several microcracks and then examine the effects produced by large arrays of randomly located microcracks: these arrays represent samples of certain microcrack statistics. In the literature, a number of solutions (analytical and numerical) have been obtained for relatively simple configurations involving one or several (symmetrically located) microcracks, in 2-D (Turska-Klebek and Sokolowski, 1984; Chudnovsky et al., 1984; Rubinstein, 1985, 1986; Rose, 1986; Hori and Nemat-Nasser, 1987; Kachanov and Montagut, 1986; Fischer, Maurer, and Daves, 1988; Dolgopolsky, Karbhari, and Kwak, 1988; Gong and Horii, 1989) and in 3-D (Laures and Kachanov, 1991). “Optimal” 2-D configurations involving one or two microcracks that provide maximal shielding were considered by Shum and Hutchinson (1990). The asymptotic 2-D case of very small spacing between crack tips was analyzed by Horii and Nemat-Nasser (1987) by power series expansions in terms of ratios of the spacing between crack tips and the microcrack size to
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323
the size of the macrocrack. Their approach seems to be limited to the case of a finite main crack. A n alternative approach to the same asymptotic situation (which can be considered as complementary to the aforementioned one) was developed by Gorelik and Chudnovsky (1992); it was applied to the case of a semi-injinite main crack (microcracks are embedded into the crack tip field of the macrocrack). From the physical point of view, interactions with large microcrack arrays that represent certain microcrack statistics are of primary interest. In this respect, several relevant works should be mentioned. Hoagland and Embury (1980) were, probably, the first to consider interaction of a crack with a large microcrack array; they performed several computer simulations. The limitations of this work are that the analysis was two-dimensional, the microcrack array was assumed ideally symmetric with respect to the main crack (so that some important effects produced by stochastic asymmetries in the microcrack field were missed, see discussion later) and the procedure of evaluating tractions on microcracks seems to implicitly assume that the interactions are weak. Tamuzh and Romalis (1984) considered a 2-D problem of a crack interacting with an infinite doubly periodic system of two families of parallel microcracks; the results were obtained for sufficiently wide spacings between cracks when the interactions are weak. Romalis and Tamuzh (1988) analyzed a 3-D problem of a penny-shaped macrocrack interacting with penny-shaped microcracks; their iterative procedure assumes that the distance between a current point M on the macrocrack and a current point on a microcrack can be identified with the distance between M and the microcrack center, which may yield substantial errors at close spacings between cracks; actual results (obtained with interactions between microcracks being neglected) were reported for several simple geometries that do not represent physically realistic microcrack statistics. Chudnovsky and Wu (1990) considered a special case of an extremely dense 2-D array of parallel microcracks and gave further analysis in terms of continuous distributions of the microcracks openings (1991). Substantial literature exists on modelling the microcracked zone b y an “equivalent” homogeneous elastic material of reduced stiffness; see, for example, Buresch (1984), Clarke (1984), Evans and Fu (1985), Charalambides and McMeeking (1987), Laws and Brockenbrough (1987), Ortiz (1987), Hutchinson (1987), Ruhle et al. (1987); such modelling was used by Gong and Meguid (1991) in the analysis of the effect of release of residual stresses due to microcracking. Adequacy of this modelling is discussed in Section F. We reformulate the method of analysis of crack interactions presented in
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Mark Kachanov
Section 111for the crack-microcracks configurations. The method allows one to consider arbitrary arrangements of large numbers of cracks by relatively simple means, both in 2-D and 3-D. We concentrate, mainly, on SIFs at the main crack, since they are, typically, higher than SIFs at the microcrack tips (the latter can also be found, if they are of interest).
B. BASICEQUATIONS FOR CRACK-MICROCRACK INTERACTIONS The method of analysis of crack interactions presented in Section 111 is reformulated here for crack-microcrack configurations, by assuming that the microcracks are embedded into the asymptotic crack tip field of the main crack. (Although the method may, in principle, be applied directly to the crack-microcrack configurations by simply taking the main crack to be much larger than the others, such a formulation has certain limitations, see the discussion later). 1. Two-Dimensional Configuration (Semi-lnjiniteCrack and N Microcracks) The stress field in the microcracked zone can be represented as a superposition a(x) = K pJl(X) + K,,a,,(x) +
c a’(x)
(5.1)
where K,, K,, are SIFs at the main crack tip (to be determined); aI= f,(@O(~r)-’~* and uI,= f,I(8)(2ar)-”2 denote modes I and I1 asymptotic crack tip fields and a’ is the ith microcrack-generated stress field (which would have been generated by an isolated ith crack loaded by the tractions t’(t) induced at its site -1’ < t < I’ in a continuous material by the other microcracks and the main crack tip). According to the key simplification of the method, we approximately represent the traction t‘(t) on the ith microcrack as a sum of tractions generated along its line in a continuous material by the main crack tip and the other microcracks, the latter being loaded by uniform average tractions on them:
+ K,IUll(t) + Cai(t)> {Krai(t) + KIIaii(t) + C { < p’>dXt) +
t’(0 = n i * { & ~ l ( t )
n‘ *
(5.2)
where (( p’), (z’)) = (tk) are the average normal and shear tractions on the kth microcrack and d,t$are the “standard” stress fields, normal and shear.
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325
The first two terms characterize the impact of the main crack on the ith microcrack and the second is the impact of the other microcracks. Taking the average of (5.2) over the ith microcrack yields N vectorial linear algebraic equations interrelating N average tractions on microcracks and K , , K,, at the main crack tip:
where the transmission A-factors interrelate average tractions on different microcracks and vectors gi = n'-(u,)' and JIi = n ' - ( ~ , , ) characterise ~ the average tractions produced by the modes I and I1 components of the main crack tip field at the site of the ith microcrack. N vectorial equations (5.3)can be further expanded into 2N scalar equations. Two additional scalar equations express the microcracks' impact on the main crack: (5.4a) (5.4b) where KP, K: denote SIFs at the main crack tip in the absence of microcracks, and ui are the stresses generated by microcracks loaded by the uniform average tractions on them. Solving the system of 2N + 2 scalar equations (5.3) and (5.4) yields K , , K,, at the main crack tip and traction averages (tk) on microcracks. If the SIFs at the microcrack tips are of interest, they are found as generated by the (variable) tractions ti(<)obtained by substituting ( t k ) into (5.2). 2. Three-Dimensional Configuration (Rectilinear Crack Front aiid N Penny-Shaped Microcracks) We assume that the microcracks are embedded into the asymptotic crack front field. In 2-D, such a formulation reduces the problem to a system of 2N 2 linear algebraic equations (5.3) and (5.4) interrelating the SIFs K,, K,, at the main crack tip and the traction averages (pi), (zi) on the microcracks. In 3-D, however, such a reduction is not straightforward, since the SIFs are variable along the macrocrack front (due to the interaction with microcracks) and cannot therefore immediately enter a system of linear algebraic equations. One alternative is to revert to the general method of analysis of interacting cracks outlined in Section I11 and simply assume that one of the cracks
+
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326
(macrocrack) is much larger than the others. Such a formulation has two limitations. The first one is technical: at the spacings between cracks of the order of lo-’ of the main crack size or smaller, numerical difficulties (related to handling expressions for the crack-generated stresses) arise; the limiting case of a rectilinear main crack front cannot be handled. The second difficulty is more substantial: if the difference in crack sizes is very large, the basic simplification of our method-taking the impact of the macrocrack on the microcrack by assuming a uniform average traction on the macrocrackmay become inadequate. An alternative is to retain modelling of the macrocrack by a rectilinear crack front, but to take its impact on microcracks in a simplified way: to assume that the microcracks are affected by the average (along the front) main crack SIFs (KI, KII,KIII).The errors due to this simplification will be small if (1) the nonuniformities K - (K) of the “primary” SIFs (modes I, 11, I11 SIFs due to the modes I, 11, I11 of applied remote loading, respectively) along the front are not overly large as compared to (K) itself, so that the “double feedback” correction K --+ (K) --+ average tractions on microcracks + K is small and (2) the microcrack field remains statistically homogeneous along the front so that (K) is stable along different intervals of the front (physically, such situation is of primary interest). Results of Laures and Kachanov (1991) show that when the microcrack size constitutes 1/100 of the circular macrocrack, the difference in results produced by both formulations is very small; this justifies, to some extent, the formulation for the rectilinear crack front. Then the average SIFs and the average tractions on microcracks can be interrelated through a system of 3N + 3 linear algebraic equations:
N
(s’)
=
j+i
[AC ( p’)
+ A!:
(z’)
+ A;:
(d)]
Elastic Solids with Many Cracks and Related Problems
+c N
( K J = K?
[s’..
+ W.<7’> + @,,(&]
CO’,,(p’>
+ %(7’>
j= 1
N
+ @L(S’>l
(K,J = K,q
+
(Kill) =
+ j = 1 Cs’.,,(Pi>+ % ( 7 J > + @k(S’>l
j= 1
327
c N
where the @- (and @-)transmissionfactors characterize the average tractions induced by the main crack front at the microcrack sites (and vice versa) so that, for example, @i,, characterizes the impact of the mode I1 main crack tip field on the normal traction on the ith microcrack and Wi, the impact of shear loading on the j t h microcrack on the mode I SIF on the main crack. Note that shear tractions on microcracks are decomposed into the 7 - and scomponents, along the direction of the shear traction generated by the main crack front and the direction normal to it (thus, the s-components reflect the interaction between microcracks only). To find the (variable along the crack front) K , , K,,, K,,, one needs the expression for SIFs along the rectilinear crack front due to a pair of opposite point forces applied at the crack surface. They are given by these formulas (Fabrikant, 1990):
‘ K, = ( z - ~ ) ~ ( c ot)1/zA-3/2 s K,, + XI,,= (K2)p(cos 4;)1’212-3/2{7
+ v(2 - V ) - ’ Z ~ - ~ 1’ ~
(5.6)
where p = p(A, 4;) is the normal point load at the point (A, 5) and 7(A, 4;) denotes a shear point load written in a complex form z = 7, + i ~Integration ~ . of (5.6) along the main crack surface, with distributed tractions p(A, 4;), 7(A, t) produced by the microcrack-generated stress fields
5) = &.(A, 4;)($>
+ 4 4 O ( 7 9 + M,4;)(s’>
(5.7)
will yield the SIFs along the main crack front. To find SIFs along the microcrack edges (if they are of interest), the actual (variable) tractions t i ( M ) on microcracks are found from (5.7) and the SIFs from (3.21) and (3.22) (see Section I11 for proper handling of the integrals for SIFs and other mathematical details). Thus, finding the SIFs is reduced to 0
calculation of the interaction matrix [ A i k ] i.e., , averaging the “standard” field generated by a uniformly loaded ith crack along the kth crack line (surface, in the 3-D configurations);
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finding tractions averages (t‘) on microcracks from a system of linear algebraic equations-note that the size of this system is moderate (2N + 2 equations in 2-D configurations and 3N + 3 equations in 3-D configurations); finding the (variable) tractions and the SIFs on the main crack; finding the (variable) tractions and the SIFs on microcracks, if they are of interest. 3. Accuracy of the Method
We check the accuracy of the method by comparing the results for SIFs with the solutions available in literature. Such test problems exist in 2-D only. The simplest configuration for which the test solution exists is a collinear arrangement of one microcrack and a semi-infinite main crack (Fig. 30). In this case, the SIF at the main crack tip produced by our method differs from the exact one by 1%, 2% and 7% at A121 = 0.25, 0.20 and 0.10, respectively; A121 denotes the ratio of the spacing between the crack tips to the microcrack length. (We used, as a test, results for two collinear cracks of unequal lengths, with the ratio of the lengths equal to lo3,further increase of the ratio leaves the results almost unchanged, with the appropriate asymptotic expressions for the elliptic integrals in the solution. The results of Rubinstein (1985),who considers the case of a semi-infinite main crack, were not used, since his formulas and graphs were found at some variance with each other.) In the case of two inclined microcracks (“focused” at the main crack tip and symmetric with respect to the main crack line), comparison with the results produced numerically by Hutchinson and Shum shows that the maximal (over all inclination angles) error of our method is 1% and 4% at A121 = 0.5 and 0.25, respectively, increasing to 20% at 6/21 = 0.10 (Kachanov and Montagut, 1989). Comparison with the results of Rubinstein (1986) for a microcrack rotated around the macrocrack tip at the distance A121 = 0.75
OUR METHOD
-
* 60.1
0.3
21
FIG.30. Macrocrack interacting with a collinear microcrack (mode I conditions).
Elastic Solids with Many Cracks and Related Problems
329
shows that, at this spacing, the errors of our method are almost indistinguishable. Dutta, Maiti, and Kakodkar (1991) calculated, using singular finite elements, the SIF in the shielding configuration involving two parallel stacked microcracks; the smallest spacing A121 at which they produced results was 0.25. Their results are in perfect agreement with the ones produced by our method. These examples indicate that, in the 2-D configurations, our method remains accurate (within several percent) at the spacings between crack tips as small as one-quarter of the microcrack length and, in the configurations close to the collinear one, at even smaller spacings; at the spacings on the order of the microcrack length, the errors are indistinguishable in all geometries. In the 3-D crack-microcrack configurations, no test solutions are available, to our knowledge. Test solutions exist, however, for several configurations involving two circular cracks of equal size; comparison with these solutions shows (Kachanov and Laures, 1989) that accuracy of our method in 3-D is much higher than in 2-D. Therefore, the results for crackmicrocrack configurations will be accurate at spacings substantially smaller than in corresponding 2-D configurations; they are expected to retain accuracy at spacings on the order of lo-' of the microcrack size and, in the case of configurations that are close to coplanar arrangements, at even closer spacings. OF A CRACK WITH SOMESIMPLE MICROCRACK C. INTERACTION CONFIGURATIONS
We present here the results for several relatively simple arrangements involving one or several microcracks. Although such configurations do not represent any realistic situations, solutions for them illustrate the influence of various geometrical factors. 1. Macrocrack Interacting with a Collinear Microcrack ( 2 - 0 ) This is an example of an amplifying configuration, in both modes I and I1 loading conditions. Figure 30 shows the results for the main crack SIF and mode I loading conditions. 2. Macrocrack Interacting with Two Parallel Microcracks (2-D) This presents an example of a configuration where the interaction effect depends, even qualitatively, on the geometrical parameters. If the microcracks
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Mark Kachanov
I
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are located sufficiently far from the main crack tip (ahead of it), the effect of such interaction is stress amplification (since the main crack "sees" them as one collinear microcrack). If, on the other hand, the microcrack centers are located above and underneath the main crack tip, the impact of interactions is the one of shielding. Figures 31 and 32 show the results for the main crack SIF and mode I loading conditions.
FIG.32. Macrocrack interacting with three parallel microcracks:competition of the shielding and amplificationeffects. Note the short range of the amplifying collinear interaction(similar to the configuration of Fig. 16).
Elastic Solids with Many Cracks and Related Problems
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9
1.0 0.0
60.0
30.0
90.0
$ FIG.33. Macrocrack interacting with a totated microcrack, mode I loading (spacing between cracks in the undisturbed configuration is one-tenth the microcrack length).
3. Main Crack Interacting with a Rotated Microcrack ( 2 - 0 ) The microcrack center is on a continuation of the main crack line. Results for both modes I and 11 loading conditions are shown in Figures 33 and 34. Note that, as Fig. 33 shows, the impact of the microcrack on the macrocrack is maximal not when the cracks are aligned, but in the configuration with a slightly “disturbed” symmetry; this is a manifestation of the general
0.0
30.0
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$ FIG.34. Same as Fig. 33, mode I1 loading.
90.0
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Mark Kachanov
1 0.85 0.0
30.0
90.0
60.0
$ FIG.35. Macrocrack interacting with a microcrack located in the “wake” zone, mode I loading (microcrack center is shifted behind and above the macrocrack tip on the distance of 0.8 of the microcrack length). Note the weakness of the interaction effect.
phenomenon (extremal properties of slightly asymmetric arrangements) discussed in Section 1V.C.
4. Microcrack in the “Wake’’ Zone Produces a Very Small Impact
on the
Main Crack (Figs. 35 and 36)
This can be explained by the low level of stresses in the wake zoneintroduction of a microcrack into such a zone produces only a small effect.
L .f.*
0.950.0
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FIG.36. Same as Fig. 35, mode I1 loading.
9
.o
Elastic Solids with Many Cracks and Related Problems
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5. Three-Dimensional Coplanar Crack- Microcrack Configuration
The amplifying impact of the microcrack shown in Fig. 37 is much weaker than in the similar 2-D configuration (Fig. 30), consistent with the fact that 3D interactions are generally weaker than the 2 - 0 ones (Section IV). Note that in this example the main crack is modelled as a circular one (rather than a rectilinear crack front); the results for both models are very close in this case, consistent with the discussion of Section B. 6. Asymmetric Three-Dimensional Crack- Microcrack Conjiguration
The configuration of Fig. 38 is subjected to mode I loading. Note appearance of the “secondary” modes (K,,, K,,,), as a consequence of asymmetry of the configuration.
D. INTERACTION OF A CRACK WITH LARGEMICROCRACK ARRAYS: TWO-DIMENSIONAL CONFIGURATIONS We solved the interaction problem for a number of sample microcrack arrays representing different realizations of a certain microcrack statistics. SIFs at the main crack tip K,,K,,and at the microcrack tips k,, kll were found.
FIG.37. Mode I stress intensity factors along the edges of a circular macrocrack and a circular microcrack in the coplanar configuration,mode I loading. The ratio of the crack sizes is 20 and closest spacing between the cracks is one-tenth the microcrack diameter.
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00
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-02 -20
00
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-2 0
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FIG.38. Mode I loading of the parallel noncoplanar crack-microcrack configuration:stress intensity factors (primary, K,,and secondary, K,,, K,,,)along the main crack edge.
1. Microcrack Arrays
Shape of the microcracking zone was determined from the condition that the maximal tensile stress, as induced by the main crack tip field, reaches certain critical value: a,,, = a*. This criterion was used by several authors (Evans and Fu, 1985; Hutchinson, 1987; Montagut and Kachanov, 1988).The shapes of microcracking zones for modes I, 11, and I11 are shown in Fig. 39. Crack arrays were generated with the help of the random number generator. Microcrack centers were assumed to be randomly located, with an additional restriction that the spacings between cracks were not allowed to be overly small (so that calculations based on our method remained reliable); this was achieved by surrounding each crack by an ellipse having semiaxes 1.11 and 0.371 in the directions collinear and normal to the crack and not allowing the ellipses of different cracks to intersect (the intersections were
Elastic Solids with Many Cracks and Related Problems
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K*=K n =Krn
FIG. 39. The shapes of the microcracked region for the modes I, 11, 111 and mixed mode ( K I= K,,= KIII)loadings, as determined by the condition u,,, = u-.
avoided by introducing the new cracks successively and discarding any crack whose ellipse intersected the already existing ones). At such spacings between cracks, the maximal error in SIFs is within 5% for all geometries (and is, typically, smaller). Similarly, the distance between the main crack and the ellipses surrounding the microcracks was kept larger than one-quarter the microcrack length. Such avoidance of overly small spacings eliminates the extreme situations of very strong interactions; without this artificial restriction, fluctuation of the interaction effects from one microcrack sample to another (discussed in the next subsection) would have been even greater. Two microcrack orientation statistics were examined: random orientations and the a,,,-statistics (microcracks are normal to the direction of the maximal tensile stress induced at the microcrack center by the main crack). The actual orientational distribution is, probably, in between these two extremes. All the microcracks were assumed to be of the same length, 21 (this assumption was used to limit the number of parameters involved and is not essential for the method of analysis used). The microcrack density p = N12/A ( N is the number of cracks in a representative area A ) was first assumed to be uniform in the microcracked zone and then the effect of variable p , decreasing with the increasing distance from the main crack tip, was examined. Six sample microcrack arrays (consisting of up to 48 microcracks each) were generated for each combination of 0
two orientation statistics, a random one and the (r,,,-statistics; modes I and I1 remote loading; Uniform) microcrack densities p = 0.06,O.12 and 0.23 (in 2-D configurations they can be considered as low, moderate and relatively high).
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FIG.40. Some of the sample microcrack arrays for which the interaction problem was solved. Microcrack densities are 0.06,0.12 and 0.23. The shape of the microcracking zone correspondsto the mode I loading. These are random orientation statistics.
Figures 40-43 show some of the sample microcrack arrays for which the interaction problem was solved.
2. Results a. “Short-Range” and “Long-Range” Znteractions The zone of “short-range” interactions, which consists of several microcracks closest to the main crack tip, produces a dominant effect (as compared to the overall combined effect of the “long-range” ones). The interaction effects are highly sensitiue to exact positions of microcracks in this zone. This is further illustrated by the 3-D examples of the next section, where Fig. 45
\-. -- . .FIG.41. Same as Fig. 40: cT,,,-orientation statistics.
-
\ \ '
FIG.42. Same as Fig 40, mode I 1 loading, random orientation statistics.
337
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Mark Kachanov
FIG.43. Same as Fig. 40, mode I 1 loading, cT,,,-orientation statistics.
will show that removal of all microcracks that are farther away from the main crack front than 1.5 of the microcrack radius produces only a small effect. b. Fluctuation of the Interaction Effectfrom One Microcrack Sample to Another Due to the high sensitivity of the interaction effect to the exact positions of several microcracks closest to the main crack tip, the SIFs at the main crack tip fluctuate significantly from sample to sample, changing from K / K o < 1 (shielding) to K / K o > 1 (amplification). Thus, interaction with microcracks appears to produce no statistically stable effect of either shielding or amplification, at least when the microcrack centers are randomly located. c. “Primary” and “Secondary” Modes Stochastic asymmetries in the microcrack arrays produced noticeable “secondary” modes; i.e., KII(K,)under mode I (11) loading: the ratio K,,/KP of the “secondary” SIF to the “primary” one reached 0.16 for some arrays. Since mode I1 SIF promotes kinking, this effect may be partially responsible for the irregularity of a crack path in brittle microcracking materials. Note that this phenomenon cannot be reproduced by modelling the microcracked zone by a region of “effective” material of reduced stiffness.
Elastic Solids with Many Cracks and Related Problems
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d. S l F s at the Microcrack Tips These were significantly lower than the ones at the main crack tip for all the microcrack arrays generated. (This conclusion should be viewed with caution, however, since spacings between all cracks were not allowed to be overly small.) e. Mode I1 Remote Loading Here, the interactions tended to be, on average, more amplifying, compared to the mode I conditions. This is in agreement with several simple crack-microcrack problems considered by Kachanov and Montagut (1986, 1989).
f. Nonuniform Microcrack Density Consistent with the dominant role of the short-range interactions, the impact of a nonuniform microcrack density (decreasing with the distance from the main crack) on SIFs at the main crack was found to be small: reduction of the density in the part of the microcrack array farther away from the tip produced relatively small changes (smaller than fluctuations of the interaction effect from one sample to another) and, therefore, can be considered statistically insignificant. g. Impact of the “Wake” Region of Microcracking We examined the hypothesis that microcracking in the wake region (“behind” the main crack tip) produces a substantial shielding effect. In several crack configurations with a frontal microcracked zone, a large number of additional microcracks (up to 75) was gradually added into the wake region. The wake region had length of up to nine microcrack lengths. Introduction of additional microcracks into the wake region produced some shielding in our computer experiments, but it was, typically, smaller than the fluctuations of the interaction effect from one sample microcrack array to another. For example, in one of the configurations, introduction of a wake produced a change in K,/KP from 1.05 to 1.00 (this case is shown in Fig. 44)whereas in another K,/KP changed from 0.77 to 0.75 (the latter example represents a relatively rare occurrence of a significant shielding); as seen from these examples, the additional shielding produced by the wake is relatively weak and dominated by the juctuation of the interaction efect from one realization of the microcrack statistics to another. Similar results were
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.
-
/I
-
FIG. 44. Gradual introduction of microcracks into the “wake” region. The interaction problem was solved for each arrangement shown.
obtained for the microcrack fields with the a,,,-orientation statistics. It appears, therefore, that the effect of the wake is not statistically significant, at least for the wake lengths considered. The question that remains open is whether a wake of a much larger length produces a substantially stronger effect. We note that stresses in the region adjacent to the (traction-free) macrocrack are quite low, so that introduction of a microcrack into the wake region, particularly at a distance from the main crack tip that is an order of magnitude (or more) larger than the microcrack length, produces a virtually indistinguishable effect. One can expect, therefore, that a further increase of the wake Iength is unlikely to produce a strong effect. h. “Clusters” of Microcracks The effect of a microcrack “cluster” on the main crack tip has been analyzed by placing the microcracks in a circular zone having diameter D of 5.2 microcrack lengths; the cluster was located on the continuation of the main crack line (see Montagut, 1989, for details). The spacing between the main crack tip and the cluster was equal to D;at such spacings, sensitivity to
Elastic Solids with Many Cracks and Related Problems
34 1
the positions of the individual microcracks within the cluster is low. Microcrack orientation statistics was assumed to be random. It was found that the effect of the cluster on the main crack tip can be simulated by one “effective” collinear microcrack; the length of this “effective” microcrack increases with the microcrack density p inside the cluster. At p on the order of 0.32-0.35 (about 30 microcracks in the cluster), the length 2L of the “effective” microcrack approaches D. Computer simultations seem to indicate that, at lower p, L is approximately proportional to D’12.
3. Crack- Microcrack Interactions in an Anisotropic Matrix Interaction of a crack with a field of arbitrarily oriented and located microcracks in an orthotropic matrix was analyzed by Mauge and Kachanov (1990,1993) and Mauge (1993). The anisotropy of the matrix significantly affected the shape of the microcracking zone but otherwise left the basic conclusions (fluctuation of the results, appearance of the secondary modes, etc.) unchanged. E.
LARGEMICROCRACK ARRAYS: THREE-DIMENSIONAL CONFIGURATIONS
INTERACTION OF A CRACK WITH
1. Microcrack Arrays We solved the problem of a rectilinear main crack front interacting with an array of penny-shaped microcracks, for a number of sample microcrack arrays. SIFs along the main crack front were found (SIFs along the microcrack edges were, typically, substantially smaller). Similar to the 2-D case, two microcrack orientation statistics (a random one and the D,,, one) were considered; locations of the microcrack centers were assumed to be random. Crack intersections and overly small spacings between cracks were avoided by surrounding each crack by a cylinder of a height 0.05 of the microcrack diameter and a diameter of 1.01 of the microcrack diameter and not allowing the cylinders to intersect. (This actually allowed spacing between cracks to be as small as lo-’ of the microcrack size or smaller.) Three microcrack densities were assumed: p = 0.15; 0.26 and 0.35, which can be considered, in the 3-D case, as low, moderate and relatively high. For each combination of the orientation statistics, crack density and mode of loading, three different microcrack samples (containing 50 microcracks) were generated and the interaction problem was solved for each. The microcrack density in the zone of microcracking was assumed constant (since, as discussed earlier, the short-range interaction zone plays a dominant role
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and the decrease of p with increasing distance from the main crack produces only a small effect). 2. Results
The results can be summarized as follows. a. Fluctuation of the Interaction E$ect Along the Crack Front Interaction with a microcrack field produces variable along the main crack front SIFs and intervals of shielding alternate with local peaks of amplification. Since crack propagation initiates at local peaks, their presence enhances crack propagation. This may indicate that interactions with microcracks promote the tendency of a crack front to assume an irregular shape as it propagates. Fig. 45 shows a typical profile of K , along the front under mode I loading. b. “Secondary” Modes Along the Crack Front Similar to the 2-D configurations, stochastic asymmetries in microcrack arrays produce secondary modes and thus seem to promote the out-of-plane crack propagation under mode I loading. Figure 45 shows typical profiles of the secondary modes I1 and I11 along the main crack front under mode I loading. c. Numerical Results For the random orientation statistics and mode I loading, the interaction problem was solved for three microcrack samples for each of the two crack densities, p = 0.15 and 0.35. The maximal (along the front) local peaks of K,/KP were > 1 in all samples; max(K,/KP) over all samples was 1.12. The “secondary” SIFs reached up to 15% of the “primary” ones. For the shear loading conditions, the interaction problem was solved for three microcrack samples of randomly oriented microcracks ( p = 0.26). The local amplifications of KIIand K,,, reached 1.10 and 1.14, respectively. The “secondary” SIFs reached local peaks of 16% of the primary SIFs. The results for p = 0.15 and 0.35 (one microcrack array for each of these densities was examined) were similar. For the a,,,-orientation statistics and mode I loading, the interaction problem was solved for three microcrack samples for each of the densities p = 0.15 and 0.35. The value of K,/KP remained < 1 along the most part of
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\ -
MODE I MADING
I a0
5
075
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0 50 -20
00
20
YJa FIG.45. Three-dimensional interaction of a crack front with an array of 50 penny-shaped microcracks (mode I loading); urn,,-orientation statistics. Typical profile (its “slice”) of a primary SIF K, and “secondary” SlFs K,, and K,,,. The dotted line corresponds to the “truncated” microcrack array (microcracks farther away from the crack front than 1.5 of the microcrack radius are removed). Note that the two lines are quire close; i.e., the short-range interactions are dominant.
the front, barely reaching 1.00-1.02 at the local peaks. Although higher peaks of K,/Kt (corresponding to coplanar configurations and close spacings) can, of course, occur, this limited data seems to indicate that the om,,-microcrack arrays tend to be more shielding than the random arrays. This may be related to the tendency of the om,, microcrack arrays to form relatively small angles with the main crack. For the om,,-orientation statistics and shear loading, the problem was solved for three samples (p = 0.26). The maximal local amplifications of K,, and K,,, were 1.08 and 1.03, respectively.The “secondary” modes (KIII,K, and K,,,K , under modes XI and 111 loading) appeared; their peaks were (0.05,0.15) and (0.06, 0.09) of the primary SIFs, respectively.
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F. IS MODELLING OF THE MICROCRACKED ZONE BY EFFECTIVE ELASTIC MATERIAL ADEQUATE? As discussed in Sections D and E, the microcracks closest to the main crack front (“short-range” interactions) produce a dominant impact on the main crack SIFs. The short-range interactions are quite sensitive to the individual microcrack positions. Their modelling by a homogeneous elastic material of reduced stiffness would not reflect this sensitivity and, therefore, the inherent statistical instability of the interaction effect. Such a modelling would also miss other important effects, like presence of local peaks of SIFs along the crack front (that makes, in 3-D cases, the overall effect of interactions one of enhancement, rather than shielding) and appearance of “secondary” SIFs that enhance crack kinking. This can be further illustrated by the following example. Laws and Brockenbrough (1987) considered a 2-D interaction of a crack with a hexagonal microcrack array. By modelling the microcracked zone by a weakened elastic material they obtained a noticeable shielding: K,/KP = 0.86. However, a direct solution of the interaction problem for the configuration shown in their article yields the opposite result-a mild amplification: K,/KP = 1.05. (Note that this result would change considerably with the parallel “shift” of the microcrack array with respect to the main crack, whereas the predictions obtained by modelling of the microcracked zone by the effective material would remain the same; see Montagut, 1989,for details). Similar conclusions on inadequacy of modelling of a microcracked zone by an effective elastic material were reached by Curtin and Futamura (1990), who used an entirely different modelling (2-D spring network). Physically, the lack of correlation between the impact of microcracks on SIFs and the fact that microcracking reduces the effective stiffness has a clear explanation: the SIFs are governed by local Jluctuations of the microcrack field geometry whereas the effective moduli are volume auerage quantities that are relatively insensitive to such fluctuations. (See Section VII for a general discussion of this issue).
G.
ON
TOUGHENING BY MICROCRACKING. CONCLUSIONS
As discussed earlier, ifthe microcrack locations are more or less random, the interactions do not produce any statistically stable effect of shielding, due to the fact that the interaction effect is dominated by the short-range interactions. The interaction effect fluctuates, even qualitatively (from shielding to
Elastic Solids with Many Cracks and Related Problems
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amplification), from one sample of the microcrack statistics to another. Moreover, in the 3-D configurations, the local peaks of SIFs along the main crack front enhance the front advances, making the overall effect of interactions the one of amplijcation. Therefore, elastic interactions with microcracks do not appear to produce any toughening. The analysis reveals some other interesting aspects of the problem. Stochastic asymmetries in the microcrack field produce noticeable “secondary” mode SIFs on the main crack; appearance of the mode I1 SIFs under mode I loading may be partially responsible for crack kinking and irregular crack path. Obviously, these conclusions do not rule out other mechanisms of toughening by microcracking. First, shielding may be produced by elastic interactions with some special microcrack conjigurations (for example, extremely dense microcrack arrays parallel to the main crack, see Chudnovsky and Wu, 1990). We note, also, that the locations of naturally occurring microcracks may not necessarily be random: one may expect that the main crack tip field enhances nucleation of microcracks in its amplification zone and thus creates an “amplification bias” in the microcrack patterns. On the other hand, an opposite “bias” appears to have been observed as well: experimental observations of Han and Suresh (1989) seem to indicate that some (but not all) of the observed microcrack patterns are “biased” toward shielding. Second, toughening by microcracking may be due to mechanisms other than elastic interactions (some of them are mentioned in Section A). Our analysis is an elastostatic one and does not predict the evolution of SIFs as the main crack front propagates. For example, one may argue that local advances of the crack front, enhanced by local peaks of SIFs, will eventually be “trapped” by a shielding local arrangement of microdefects; examination of this hypothesis would require analyses of crack fronts of complex shapes. These problems are beyond the scope of the present work and need further studies.
VI. Effective Elastic Properties of Cracked Solids The theory of effective elastic properties of cracked solids predicts degradation of stiffness, development of anisotropy and changes in wavespeeds caused by microcracking. Therefore, it is of obvious interest for materials science, structural mechanics and geophysics. Most of the approaches to this problem have roots in the effective media
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theories of physics. For example, the approximation of an efectiue matrix (a self-consistent scheme, in the terminology of the mechanics of solids with inhomogeneities),in which a representative inhomogeneity is placed into the effective matrix and its modification-the differential scheme-were first used in the problems of electrostatics, by Maxwell and by Bruggeman in 1935. The method of an efectiue field (Mori-Tanaka’s method being its simplest version), in which a representative inhomogeneity is embedded into the effective field, was first developed in 1940s and 1950s in connection with wave propagation problems (Lax, 1951). At the same time, cracks constitute a distinctly special kind of inhomogeneities: they occupy no volume; stress fields generated by them are quite complex and have strong orientational dependence. As a result, cracked solids have many special features: the choice of crack density parameter is nontrivial; the effective properties are, generally, anisotropic; bounds for the effective moduli generally cannot be established; the approximation of noninteracting cracks has a wider than expected range of applicability. Several approximate schemes have been suggested in literature (and a number of their modifications and combinations can, probably, be added). Their predictions are substantially different. A researcher, trying to use the theory, may well be confused by the choice of several models, yielding different results. Here, we attempt to introduce some clarity into the problem. We start with a detailed review of the approximation of noninteracting cracks, not only because it is the only noncontroversial approximation, but also because our direct computer experiments (as well as the method of effective field and Mori-Tanaka’s method) show that it remains accurate at high crack densities, provided the locations of crack centers are random. We pay particular attention to anisotropy caused by non-randomly oriented cracks. We then discuss various approximate schemes and the possibility of bounding the effective moduli. When locations of crack centers are nonrandom, we suggest an alternative crack density parameter, which is sensitive to mutual positions of cracks.
A. INTRODUCTION 1. Solid with Cracks us. Solid with Inclusions
The problem of effective moduli of cracked solids can formally be regarded as a limiting case of the more general problem for a matrix containing
Elastic Solids with Many Cracks and Related Problems
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inclusions: in this limiting case, (a) inclusions shrink to surfaces and (b) their elastic moduli tend to zero. Such an approach leads to some difficulties, however: (1) The results may depend on the order of the mentioned transitions to limit. The proper one first sets the inclusion’s moduli to zero (thus creating a cavity) and then shrinks the cavity to a crack. An attempt to reverse the order leads to difficulty: in the framework of linear elasticity, infinitesimally thin platelets of finite stiffness do not affect the effective moduli, thus becoming “invisible,” and the second transition to limit cannot be made (unless it is combined with the first one and a special constraint is imposed). (2) The parameter of concentration used in mechanics of composite materials-relative volume of inclusions-should be changed, since it is zero for cracks. The commonly used crack density parameter (introduced by Bristow, 1960, and used by subsequent authors, starting with Walsh, 1965) is: 1 p =-
A 1
p =V
1 1(i)2 in 2-D (rectilinear cracks of lengths 21’, A is the representative area) (6.1) P 3in 3-D (circular cracks of radii li, V is the representative volume)
1
Some definitions include a multiplier A in the 2-D case (or 4n/3, in the 3-D case), so that p is the relative area (volume) of circles (spheres) with cracks as diameters. For cracks of iioncircular shapes in 3-D, p was generalized by Budiansky and O’Connell(l976) to 2 1 p =-nV
1(S/P)‘”
(6.1a)
where S and P are the crack area and perimeter. (3) Some approaches of mechanics of composite materials degenerate for cracked solids. Bounding the effective moduli constitutes an example: neither upper nor lower nontrivial bounds (which would apply to any sample of crack statistics of a given density and given orientational distribution) can be established for cracked solids (see the discussion of Section I). Another example is Mori-Tanaka’s method for materials with interacting inclusions: being applied to cracked solids, it yields results corresponding to noninteracting cracks. A direct approach to the problem of effective moduli of cracked solids (rather than the one based on mechanics of two components materials) appears, therefore, more efficient.
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2. Average Strains and Stresses in a Cracked Solid Since displacements are, generally, discontinuous across crack surfaces S', the strains are singular at S'. Thus, the strain field in a solid with cracks is a sum of regular and singular parts: E(X)
= M:u(x)
+1
(bn i
+ nb)'d(S')
where, in representing the regular part as M :u, the matrix material is assumed to be linear elastic (characterized by a compliance tensor M), and 6(Si)is a delta function concentrated on S' (as a derivative of a unit step, it has the dimension of length - '). A colon denotes contraction over two indices; bn, nb denote dyadic (tensor) products of the displacement discontinuity vector b = u+ - u- (crack opening displacement) and unit normal n to a crack. Vectors b and n are, generally, variable along cracks; their directions are coordinated by defining the + side of S as corresponding to the positive sense of n (then change of sign of n changes the sign of b and the products bn, nb remain invariant). In the following, we assume that the matrix material is elastically homogeneous: M(x) = constant = Mo. Averaging (6.2) over V and using the property of 6(S) that Jvf(x)6(S)dV reduces to the surface integral Jsf(x) dS yields the volume average of strain: E = Mo:a + -
iv
(bn i
+ nb)dS = (Mo + AM):u 5 M : a
(6.3)
[st
where AM is the change in compliance due to cracks and M is the effective compliance. Linear dependence of b' on ij (used in (6.3)) assumes linear elasticity of the matrix and absence of friction along crack faces (if friction is present, then (6.3)is to be rewritten in the incremental form and AM depends not only on 6 but, also, on the direction of d6, see Section D). Volume averages 5 and a can be expressed in terms of quantities defined on the outer boundary r of V; i.e., in terms of experimentally accessible (measurable)quantities. Since, as follows from the divergence theorem, ii for a simply connected region is given by the integral (1/2V) Jr (un + nu) d r , this integral defines E for a volume element of the efective (homogenized)material. Thus, if displacements are prescribed at the boundary of r! then, by definition, introduction of cracks into V does not change ii. At the microstructural level, introduction of cracks produces redistribution of strains: due to the contribution from displacement discontinuities into the unchanged total E, strains in the matrix decrease.
Elastic Solids with Many Cracks and Related Problems
349
Average stresses are defined in terms of quantities on r as (l/V)JrXTdT, where x is a position vector (counted from the centroidal point) and T is a traction vector. This definition is justified by the fact that, as follows from the divergence theorem, u = (l/V) Jr xT d r , for both a simply connected region and a region with traction-free cracks, provided the body force is constant in V (as noted by Lehner, 1992, the latter requirement can be relaxed). Thus, by definition, introduction of cracks does not change the average stress, if tractions are prescribed on the boundary. This means that, if a certain stress component is locally amplified by a crack (say, has a singularity near the tip), this is compensated for by “unloading” at other points of a solid. If stress is constant (=ao)along r (“homogeneous boundary conditions,” Hashin, 1983), then another application of the divergence theorem yields ao = a. If the RVE constitutes a part of a statistically homogeneous field of cracks and is sufficiently large to be representative, then it can be reasonably hypothesized that substitution of the actual stress on r (generally, fluctuating along r) by its constant average will affect only a “boundary layer” of V (having thickness on the order of the fluctuation wavelength) and will not significantly change u, so that no z 0. If, in addition, the remotely applied om is such that, in the absence of cracks, the stress field would have been homogeneous, then uo can be approximately identified with am(for a more detailed discussion of statistical homogeneity and related issues, see Hashin, 1983, and Beran, 1968). We add only that the identification uo = u = amis exact for noninteracting cracks, when cracks inside the RVE experience no influence of those outside. In the following, we omit the overbar for stresses and strains, assuming E = Eand n = 8 = om. Forpat cracks, n = constant within each crack, and (6.3) takes the form E
= Mo:a
-1 +1 ((b)n + n(b))’S’ 2v ’
(volume V and areas s’are to be changed to area A and lengths 21 in the 2-0 case) where (b’) is b‘ averaged over S’. Thus, contribution of a given crack into the overall strain is proportional to the product (crack area)(b). The unknowns are vectors (b’) as functions of u. Representations of the type of (6.3) or (6.4) have been routinely used since at least the 1970s as a starting point for analysis of cracked solids (see, for example, Varakin and Salganik, 1975). They directly follow from a more general case of arbitrary cavities; as follows from the divergence theorem,
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Mark Kachanov
the cavity’s contribution into the overall strain is an integral over the cavity surface (- 1/2V)
b
(un
+ nu) dS
It reduces to the second term of (6.3) when cavities shrink to cracks.
3. Elastic Potential (Complementary Energy Density) of a Cracked Solid Elastic potential f of the effective material is obtained by contracting aij/2 with taking the latter from (6.4) yields f in terms of “microstructural” quantities defined on S‘: 1
1
1
~(u)=-u:M~~~:u=-u:M +-2V~ : u~i ( n . e . ( b ) ) ’ S i - f 0 ( a ) + A f 2 2
(6.5)
(to be appropriately modified in the 2-D case) where fo(e)is the potential of a matrix without cracks. For an isotropic matrix with Young’s modulus Eo and Poisson’s ratio vo
where, in the 2-D case, E,, changes to E, (= E, for plane stress and Eo/(1 - vg) for plane strain). Expression (6.5) plays a central role in further analysis.
4. COD Tensor
We introduce tensor B that interrelates a vector of uniform traction t applied at the crnck faces and the vector of resulting average COD:
(b) = t . B
(6.7)
The second-rank tensor B can be called a COD tensor. It depends on the crack size and shape, on the elastic properties of the matrix and, in the case of anisotropic matrix, on the orientation of the crack with respect to the anisotropy axes of the matrix. In the case of a body of finite size, it also depends on the body’s geometry. In the case of an isolated crack in an infinite body with stress e at infinity
Elastic Solids with Many Cracks and Related Problems
351
(such that, in the absence of a crack, the stress state is uniform), (6.7) can be rewritten as
(a)
= n.a.B
(6.7a)
In the coordinate system (n, t) where t is tangential to the crack, the diagonal components B,,, B,, characterize the normal (shear) CODs produced by the normal (shear) tractions. The off-diagonal components characterize coupling of the modes (generally present for cracks in finite bodies; for cracks in an injnite body, it is generally present in the case of anisotropic matrix; and in the case of noncircular cracks in an isotropic matrix, between any two inplane directions, unless these two directions are the principal directions of the crack shape). Betti's reciprocity theorem implies that Btn = B,; ie., B is symmetric.
5. COD Tensor of a Crack in an Injnite Isotropic Solid
In the case of an infinite isotropic matrix, normal and shear modes are uncoupled. In 2-D, B is proportional to a unit tensor: B = (nl/Eb)I
(6.8)
reflecting equal compliances of a crack in normal and shear modes. In 3-D, for a circular crack (radius I ) , B = pnn + y(tt + ss), where s and t are any two orthogonal vectors in the crack plane, and B = 161(1 - vg)/3nE0; y = B/(1 - v0/2).Note that B is not proportional to I, since normal and shear compliances of a crack are different. Since tt + ss + nn = I, we can rewrite B in the form
(:)
B = y I--nn
The I-term corresponds to the part of COD that is collinear to the traction n * a;the on-term characterizes the deviation from collinearity. An important observation is that the coefficient of the nn-term is substantially smaller than the one of the I-term, reflecting a relatively small difference in normal and shear compliances of a circular crack (vanishing at v,, = 0). For an elliptical crack, tensor B can be constructed by utilizing Eshelby's results on ellipsoidal inclusions (that were specialized for elliptical cracks by Budiansky and OConnell, 1976). Tensor B has the form B = (nn
+ qss + ctt
(6.10)
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Mark Kachanov
where s and t are unit vectors of the major, 2a, and minor, 2b, ellipse’s u, ( are expressed in terms of elliptic integrals: axes. Coefficients =4(1- v$)b/(3E0E(k)),where E(k) is the complete elliptic integral of the first kind of the argument k = [l - (b/a)2]”2 and 9, [ are obtained from 5 by changing E(k) to expressions Q-’and R - ’ , respectively, where Q and R are given by formula (31) of Budiansky and O’Connell. Using the identity on + ss + tt = I, we transform (6.10) to the form
<
r,
where the first two terms are similar to B of a circular crack; the third term reflects deviation of the crack shape from circular and is responsible for the noncollinearity of the shear traction and the resulting shear COD. B-tensors are particularly useful in the cases of noncircular cracks and cracks in anisotropic matrix.
B. NONINTERACTING CRACKS. CRACK DENSITY TENSOR. ORTHOTROPY 1. General Remarks. Nonequivalence of the Assumptions of Noninteracting Cracks and Small Crack Density In the assumption of noninteracting cracks (in the isotropic matrix material) the effective moduli can be found exactly, for an arbitrary crack orientation statistics, in both 2-D and 3-D. It is the simplest (and the only noncontroversial) approximation to the problem; at the same time, it has a wider than expected range of applicability (Section I). It is often called the approximation of small crack density (in fact, the two names are usually used as synonyms). We refrain, however, from using this term, since these two assumptions are generally not equivalent. Indeed, if locations of crack centers are nonrandom, interactions can be strong (and the discussed approximation inapplicable) at vanishingly small density (see the examples of Section J). On the other hand, in the case of random locations, predictions of the approximation of noninteracting cracks remain accurate at high crack densities (due to cancellation of the competing effects of shielding and amplification, see Section I). In the approximation of noninteracting cracks, each crack is regarded as isolated it is embedded into the a-field and does not experience any influence of other cracks. Then (b) for each crack is expressed, in a simple way, in terms of o*u.Mutual positions of cracks do not matter in this approxi-
Elastic Solids with Many Cracks and Related Problems
353
mation, hence averaging over a crack array is reduced to summation over orientations. Since the change in compliance AM due to cracks is a sum of the isolated cracks' contributions, the compliance is linear in the crack density parameter. Elastic strfnesses, obtained by inversion of compliances, will be functions of the crack density of the form (1 + C p ) - ' . Linearization of this form corresponds to the assumption of small crack density. If, however, this form is left as it is, then, as seen later, the results remain accurate at high crack densities. The problem of effective elastic moduli in the approximation of noninteracting cracks was first solved, for randomly oriented cracks, in both 2-D and 3-D, by Bristow (1960). His calculations were repeated and extended to the case of frictional contact between crack faces by Walsh (1965). Since averaging is reduced simply to integration over orientations, such calculations are easily repeated for an arbitrary (nonrandom) orientational distribution. Such calculations were done by a number of authors. For one or several families of parallel cracks in 3-D, the moduli were explicitly derived by Piau (1980), where the derivation was done in connection with the wave propagation problems (and, from the point of view of finding the effective properties, was not the simplest); for the sine and cosine orientational distributions of penny-shaped cracks, by Zhao et al. (1989). They are implicitly (in the form of elastic potential) given, for arbitrary orientation distributions, by Kachanov (1980). They are also contained, explicitly or implicitly, in papers on self-consistent and differential schemes (see, for example, Hoenig, 1979). The results can be written in the most general form (arbitrary orientational distribution and crack shapes) by using the B-tensor and introducing (6.7a) into (6.5). This yields the elastic potential
Af = (1/2)a:AM:~
(6.12)
where AM = (l/A) = (1/2V)
li(nBn)'
in 2-D
C S'(nBn)'
in 3-D
(6.13)
is an additional compliance due to cracks (AMijklis to be appropriately symmetrized, with respect to i w j , k CI I , ij c,kl). The presentation that follows is based on the concept of the crack density tensor. It leads to results for arbitrary crack orientational distribution in a simple unified way, avoiding averaging over orientations. This tensor
3 54
Mark Kachanou
(introduced by Vakulenko and Kachanov, 1971; Kachanov, 1972 and, in a more complete and corrected form, by Kachanov, 1980a) emerges naturally as a term in the elastic potential. We emphasize that the choice of this tensor as a measure of crack density is dictated by the structure of the elastic potential. 2. Invo-Dimensional Case Vectors (b') and n'*a are collinear:
(b') = (nl'/Eb)ni-a
(6.14)
since "compliances" of a crack are the same in shear and normal modes. This collinearity has important implications for symmetry of the effective properties, as will be seen. Substituting (6.14) into potential (6.5) (or, equivalently, B-tensor (6.7a) into (6.12) and (6.13)), one obtains Af = (n/Eb)((~.(~):a = (n/Eb)oijojkaik
(6.15)
1 a = - C (i2nn)' A i
(6.16)
where
is a second-rank crack density tensor (summation may be substituted by integration over crack orientations? if computationally convenient). Since n - n = 1, the linear invariant t r a = akk is the conventional scalar crack density: (6.17) so that a is a natural censorial generalization of p that accounts for crack orientations. Thus, elastic potential is a function of (I and a:
where Af is a simultaneous invariant of (I and a, quadratic in (I (linear elasticity at fixed a) and linear in a (noninteracting cracks). This invariant structure reflects isotropy of the matrix: if both (I and a undergo the same orthogonal transformation (say, rotation) then the potential is insensitive to any such transformation.
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355
Equivalently to (6.15), one can specify the compliance increment due to cracks AM,, = a2Af/aaijaot, as AMij/cI
= (n/Eb)d{ikaj/)
(6.15a)
where the subscript braces denote all the appropriate symmetrizations (with respect to i w j , k t)I, i j c,kl). We emphasize that, for noninteracting cracks in 2-D, these representations in terms of a are rigorous, and they follow from collinearity of vectors (b') and ni * a. The crack density tensor is symmetric (as a sum of symmetric dyads). Therefore, it has a principal representation: a = Plelel
+ p2e2e2
(6.19)
( p , , p2 and e l , e2 are principal values and principal unit vectors of a).
Each term in (6.19) represents a family of parallel cracks. Therefore, any system of cracks iffully equivalent, in its impact on the effective moduli, t o two mutually perpendicular families of parallel cracks. As a consequence, the effective properties are orthotropic (rectangular symmetry) for any orientational distribution of cracks; the orthotropy axes coincide with the principal axes of a. This result, rigorous for noninteracting cracks in 2-D, may not be intuitively obvious (for example, it covers situations of the type of (6.28), Fig. 46). Moreover, the orthotropy is of a simplijied type: 1. Since a is a second-rank tensor with nonnegative eigenvalues, the variation of elastic compliances with direction is described by ellipses (rather than fourth-order curves that may have complex shapes, as in the case of general orthotropy); 2. The number of independent elastic constants is reduced from four (general orthotropy in 2-D) to three, due to the relation between shear and Young's moduli and Poisson's ratios (it follows from (6.15) and (6.19)): w 1 2
= (1
+ Vl2)lEl + (1 + v21)/E2
(6.20)
(Curiously, this relation was suggested in the general theory of elasticity of anisotropic solids by Saint-Venant, simply as an approximation for reducing the number of constants, see Lekhnitsky, 1963); 3. Under uniaxial loading i n any direction, the presence of cracks does not cause any additional lateral strain: v12lE1 (=v*,lE2) = VOlEb
(6.21)
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Mark Kachanov
It is a consequence of a entering Af through only one invariant (a.a):a (in principle, one may construct one more invariant(a :a)tr a - that is quadratic in a and linear in a,however, it does not
enter A& Therefore, Af does not contain the product ~ 1 1 a 2 2This ). implies that the change of Poisson’s ratio due to cracks is the same as the change of Young’s module: v12/vo = E,/E,. This fact implies (in conjunction with (6.20))that the shear modulus can be expressed as l/GlZ
=
1/E1
+ 1/E2 + 2vo/Eb
(6.20a)
Thus, all elastic constants can be expressed in terms of two Young’s moduli El, E, and one matrix constant vo/Eb. The advantage of the representation (6.15)in terms of a is that the effective moduli are readily obtained for arbitrary orientational distribution of cracks without aoeraging over orientations as soon as this distribution is characterized by a. This is illustrated by the examples that follow. a. Isotropic (Random) Orientation Distribution The crack density tensor is isotropic, and, since tr a = p , a = (PI211
(6.22)
where I is a 2-D unit tensor. Inserting (6.22) into (6.15) yields the elastic potential:
from which the effective moduli follow: E’ = Eo(1
+ np)-l;
v = vo(l + q
I - 1
(6.24)
b. Parallel Cracks (Normal to the x1 Axis) In this case, a = pelel and the potential takes the form
(6.25)
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357
from which one obtains Young's modulus in the x1 direction and the shear modulus E ; = Eo(1
+ 2 7 ~ ~ ) ~GI, ' ; = Go[l + (27~GO/Eb)p]-'
(6.27)
Young's modulus in any other direction (inclined to cracks) is obtained by expressing a in the inclined coordinate system; variation of compliance with the angle of inclination is shown in Fig. 48 (solid line). c. 'Tivo Families of Parallel Cracks
To illustrate convenience of the crack density tensor, we consider an example of two families of parallel cracks inclined at an angle 30" to each other; one has the density twice as high as the other (Fig.46). The crack density tensor is a = (2p/3)n1n1
+ (p/3)n2n2
(6.28)
where nl, n2 are unit normals to cracks of the first and second families. Finding the eigenvalues of a and substituting the principal representation of a into the elastic potential, we obtain that the principal directions of orthotropy are rotated 9.5" to n1 and that Young's moduli in these directions are Eb/(l + 5.86~)and Eb/(l + 0.42~). In the case of several systems of parallel cracks, a
=
C pknknk
(6.28a)
where Pk are partial crack densities, and introduction of (6.28a) into the elastic potential yields results in an elementary way. 3. Three-Dimensional Case (Circular Cracks) Unlike the 2-D case, (b') is not collinear to n'sa, and the proportionality
FIG.46. Orthotropy axes in Example c.
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Mark Kachanov
relation between (b') and n'.u should be written for the normal and shear components separately (see Section 11):
where /3 = 16(1 - vi)/3n and y = 16(1 - vtH1 - v0/2)-'/37r. The compliances of a crack in normal and shear modes are di$erent, p # y (unless vo = 0), although not by much: y = p/(1 - v0/2). Substituting (6.29) into (6.5) (or, alternatively, substituting B-tensor (6.9) of a circular crack into (6.12), (6.13) yields Af =
8(1 - v;) 3(1 - Vo/2)Eo
vo
1
where (6.16a) is the crack density tensor (3-D equivalent of (6.16)) Unlike the 2-D case, Af is not a function of u and a only, but contains an extra term involving the fourth-rank tensor (1/V) (Pnnnn)'. This term causes deviations from orthotropy. Emergence of this term (and, thus, insufficiency of a)is a consequence of the noncollinearity of vectors (b') and n'.u in 3-D. This term's contribution A M into the effective compliance makes the decrease of Poisson's ratio with increasing crack density slower than the decrease of Young's and shear moduli (moreover, Poisson's ratio does not even tend to zero in the limit); see discussion following formula (6.37). Another feature of A M is that it has an unusual symmetry: since (1/V) (Pnnnn)' is symmetric with respect to all rearrangements of indices, AMYlzz= A M Y z l 2and two similar relations, indices 23 and 31, so that AM" contributes equally into the Poisson's ratio effect and the shear moduli. However, the overall impact of this term is small. In particular, deviations from orthotropy are small, due to the following two factors. First, the extra term has a relatively small multiplier v0/2 (this is a consequence of a relatively small difference between normal and shear compliances of a circular crack). If its relative weight is evaluated by the ratio
c
K = (AMYjkiAMGkl)'i2/(AMijkl AMijkr)'/2
(6.31)
of the Euclidean norms of the compliances AM" and AM' corresponding to this term and to the a-term, then K < vo/,,h (Kachanov, 1980a).
Elastic Solids with Many Cracks and Related Problems
359
Second, only a part of this term causes deviations from orthotropy; a substantial part of AM” has the structure of orthotropy coaxial to a and simply adds to the orthotropic moduli A M . The following examples demonstrate the smallness of errors caused by omitting the extra term. These errors can be further reduced if, instead of simply omitting the extra term, we eliminate it in a different way; namely, if we enforce the normal and shear compliances of a crack to be the same and equal to their average, i.e., by replacing @ and y in (6.29) by (B + y)/2, then Af z Af(a, a) =
8(1 - vi) 1 - v0/4 cr*u:a 3Eo 1 - Vo/2
(6.32)
and the fourth-rank tensor is eliminated from the potential. Thus, characterization of a crack array by the crack density tensor a only (that leads to orthotropy of the effective properties) constitutes a good approximation. The examples that follow (in which the effectivemoduli are found both ways-with the extra term retained and omitted-are compared) illustrate this statement. a. Isotropic (Random) Orientation Distribution Both the crack density tensor a and the fourth-rank tensor (1/V)C(13nnnn)’ are isotropic. Since t r a = p, (6.33)
a = (P/3)I
(I is a 3-D unit tensor). Since tensor (l/V)C(l3nnnn)’ is isotropic, its ijklcomponents have the form C1dijdkl C2(6ik6j, + 6 i l 6 j k ) ; since, in addition to being isotropic, it is symmetric with respect to all rearrangements of indices, C , = C, and the ijkl-components are C(6,6,, + dikdj1 6i,6jk),where C is some constant. Since (l/~‘)C(l’n-nn-n)’ = p so that 6 f i 6 k k 2dikdik ( = 15) = p, the constant C = p/15. Inserting this tensor and (6.33) into (6.30) yields
+
+
+
and the effective moduli readily follow:
G -GO =[l+
16(1 - VoX1 - V0/5)
9(1 - vo/2)
V +
8(1 - vi) 45(1 - v0/2) p ] (6.35)
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Mark Kachanou
Note that ifonly the a-term is retained in (6.30) and the term containing the fourth-rank tensor is omitted, the effective moduli (6.35) change to v
G
E
9(1 - vo/2)
9(1 - v,/2)
(6.36) which are close to the rigorous result (6.35). If the potential (6.32), obtained by replacing the normal and shear compliances /3 and y by their average (fl + y)/2 is used, then the effective moduli -E= [ l + EO
G
-GO =[l+
16(1 - vgX1 - v0/4) 9(1 + vo/2) 16(1 - vo)(l - Vo/4) 9(1 - V O P ) PI
v
;
E
vo = -Eo
(6.37)
are even closer to the rigorous result (6.35). We now examine the impact of the fourth-rank tensor on Poisson’s ratio. In contrast with the 2-D case, Poisson’s ratio decreases slower than Young’s or shear moduli, with the increasing crack density (formulas (6.35)). Moreover, Poisson’s ratio does not even tend to zero in the limit (instead, v/vo approaches (10 - 3v0)-’). These features illustrate the impact of the fourthrank tensor on Poisson’s ratio. However, the difference between E/Eo and v/vo is relatively small; this illustrates smallness of the mentioned effects. These features of the Poisson’s ratio behavior disappear if the fourth-rank tensor is removed from the elastic potential, either by simply omitting the corresponding term or by enforcing equal compliances of cracks in normal and shear modes (formulas (6.36) and (6.37)). b. Parallel Cracks (Normal to the x1 Axis) The crack density tensor u = pelel
(6.38)
and the fourth-rank tensor (1/V)~(13nnnn)’= pelelelel. Thus, =O’
+
8(1 - v;) 3(1 - vo/2)Eo
(6.39)
361
Elastic Solids with Many Cracks and Related Problems
so that
(6.40) If only the a-term is retained in (6.30), then the moduli (6.40) change to lql 3(1 - vo/2)
PI-';
G I , remains unchanged
(6.41)
or, if the potential (6.32) is used,
(6.42) Both (6.41) and (6.42) are close to the exact moduli (6.40). c. More Complex Crack Orientation Statistics To illustrate the orthotropic character of anisotropy in more complex crack orientation statistics, we consider an example (similar to (6.28) in 2-D) of two families of parallel cracks inclined at 30" to each other, one having a density twice as high as the other. The crack density tensor is given by (6.28) and the fourth-rank tensor (1/V)C ( I 3nnnn)' = (2p/3)n'n'n1n1 + (p/3)n2n2n2n2. The principal axes of a are the same as in the 2-D case. Unlike the 2-D case, orthotropy, strictly speaking, does not hold, due to the fourth-rank tensor's contribution. However, the deviationfrom orthotropy is small. Indeed, calculated in the principal axes of a, the effective compliances AM,, (obtained by expressing a and the fourth-rank tensor in the principal axes of u and inserting them into (6.30)) are as follows (normalized to p / E o ; vo = 0.25 is assumed):
AMl,,, = 0.167,
AM22,2 = 2.370, AM3333= 0
AM1122 = -0.019,
AM2233 = 0,
AM1212
= 1.400,
AM2323 = 0,
AM3311 = 0 (orthotropic moduli) =
(6.43)
The rest of the moduli would have been exactly zero in the case of rigorous
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Mark Kachanov
orthotropy (the fourth-rank tensor’s contribution neglected). They are quite small, compared to (6.43):the only nonzero ones are A M , , , , = 0.004,
A M z z l z = -0.004
(nonorthotropic moduli)
(6.44)
Their relative weights, evaluated by the ratio (6.3l ) , is negligible: = i.4x 10-3. Note that, as seen from (6.43),A M , , , , is on the order of lo-’ of A M , , , , and AMZ2,,; i.e., the impact of cracks on the lateral strain is small, compared to the impact on the shear and Young’s moduli. This illustrates the accuracy of representation of an arbitrary crack array by the crack density tensor (and of relations (6.45) following from such representation). 4. Simplijied Character of Orthotropy. Number of Independent Constants As shown earlier, the effectiveelastic properties of 3-D cracked solids are approximately orthotropic. Similar to the 2-D case, the orthotropy is of a simplijied type: 1. Directional variation of the compliances is described by ellipsoids
(rather than surfaces of the fourth order, as in the case of general orthotropy). 2. The number of independent constants is drastically reduced, from nine (general orthotropy) to only four, due to the following simplifications (similar to the ones in 2-D): a. Shear moduli can be expressed in terms of the Young’s moduli and Poisson’s ratios through three equations of the type of (6.20); b. Under uniaxial loading in any direction, the presence of cracks does not cause any additional lateral strain: V12/E1 = v n / E z = V3i/E3 = Vo/Eo
(6.45)
This fact, together with the general symmetry relations for the compliance tensor v i j / E i = v j , / E j , implies that all six Poisson’s ratios can be expressed in terms of the Young’s moduli Ei and the matrix constant vo/Eo. It further implies (in conjunction with (6.20)) that the shear moduli, too, can be expressed in terms of Ei and vo/Eo: 1/Gij = 1/E,
+ 1/Ej + 2vo/Eo
(6.46)
Thus, the entire set of elastic constants can be expressed in terms of three
Elastic Solids with Many Cracks and Related Problems
363
Young’s moduli Ei and one matrix constant v o / E 0 . These simplifications may have implications for wave propagation in cracked media, since they may allow one to interrelate various wavespeeds in a medium with crack-induced anisotropy.
5. Loss of “Sharpness” of Anisotropy when Parallel Orientations Are Disturbed. Loss of Isotropy due to Disturbances of Orientational Randomness
Some naturally occurring crack systems have preferential orientations, but they are not ideally parallel. We analyze the impact of disturbance of parallel orientations on the effective moduli. In particular, we examine the loss of “sharpness” of anisotropy and a gradual transition to isotropy as the disturbance increases. We consider one system of parallel cracks. The case of several such systems can be analyzed by using the crack density tensor and summing up contributions of individual systems. We restrict our attention to the 2-D case; analysis in 3-D is similar. The most natural modelling of this situation is to assume that the orientational distribution density p ( 4 ) follows the normal (Gaussian) distribution. However, to present the results in the form of elementary functions and in view of the low sensitivity of the results to the exact form of p(+), we assume a more convenient form of ~ ( 4 )We . consider two cases (Fig. 47): a. Case A Distribution p ( 4 ) has a sharp peak at = 0, and there are no cracks with orientations outside a certain interval 141 < 4o < 742. We model this situation by assuming
P(4) = b o y COSY47 if 141 < 40
and P ( 4 ) = 0, if 141 > 40
(6.46)
where the multiplier (1/2)p0y is due to the normalization condition
1:
p(c$)d4 = po (total density)
(6.47)
and parameter y characterizes the “sharpness” of the peak, i.e., the ratio (equal to ny/2) of the maximal density pmax= p(0) to its average ( p ) = po/n.
Mark Kachanou
364
(*Ar‘:2
-7r/2
n/2
-T/2
n/z
FIG.47. Disturbance of parallel orientations of cracks: (A) Distribution p ( 4 ) with a sharp peak; (B) distribution p(4) with a mild maximum.
b. Case B Distribution p ( 4 ) has a mild maximum at 4 = 0 and cracks of all orientations (- n/2,n/2)are present. We model this situation by assuming 1
P(4) = 5 Po ___ sin yn/2 cosy4
(6.48)
where parameter y characterizes,again, the “sharpness” of the peak: the ratio p,,,/(p) = ny/(2sin 742) and y --+ 0 corresponds to a uniform (isotropic) distribution. The crack density tensor is an integral over orientations (6.49)
In Case A, the components of a and Young’s moduli are all= 3 p o [ 1
{El, E d
-=
E,
+ (1 - 4 ~ - ~ ) -cos’ n/b]
1 1 + 2n{a11, az2} 1
- + npo[l *
and aZ2= p o - a,, 1 (cosn/b)/(l - 4y-2)1
(6.50)
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365
In case B, the components of a and Young’s moduli are
For a numerical illustration, consider the case when the total crack density po = 0.2 (as discussed later, the approximation of noninteracting cracks
remains accurate at such densities, provided locations of crack centers are random). For ideally parallel cracks, the ratio E1/E2= 0.44. It changes to 0.48in the case of a sharp peak in the orientational distribution, characterized by the ratio p,,,/(p) = 9.4. In the case of a somewhat less sharp peak, p,,,/(p) = 4.4,the ratio changes rather noticeably to 0.54; in this case, E2/E0 changes from unity for ideally parallel cracks to 0.88. In the case of a mild maximum, p,,,/(p) = 1.16, the ratio E , / E , = 0.92, dropping to 0.86 for a somewhat sharper maximum p,,,/(p) = 1.32; in this case, EJEo changes from 0.61 for an ideally isotropic distribution to 0.66. The anisotropy becomes practically indistinguishable(E,/E2 = 0.97)when the ratio pmax/pmin = 1.24; i.e., it is noticeably different from unity. This shows a low sensitivity of the effective isotropy to mild disturbances of orientational randomness. 6. “Natural” Coordinate System for Eflective Elastic Moduli. Formulation of the Plane Problem for a Cracked Solid
As discussed previously, the effective elastic properties of solids with noninteracting cracks possess an orthotropy coaxial with a (exactly in 2-D and with good accuracy in 3-D). The principal axes of a constitute, therefore, a “natural” coordinate system for calculation of the effective moduli. In this system, the matrix Mijklhas its simplest form (orthotropic or close to it). The results of computer experiments on sample arrays (Section I) indicate that orthotropic symmetry still holds with satisfactory accuracy for interacting cracks, so that this “natural” coordinate frame remains the system of choice for computing M i j k l . This discussion has implications for the plane problem of elasticity for a cracked solid. In the case of an anisotropic solid, the plane problem can be formulated only in the plane of elastic symmetry. This implies that plane problem for a cracked solid can be formulated only in one of the principal planes of the “natural” coordinate system formed by the principal axes of a.
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Mark Kachanov
7 . On the Effect of Dimensionality Comparison of the stiffness reduction due to cracks in 2-D and 3-D (coefficients at p in formulas (6.24)and (6.35),or in formulas (6.27)and (6.40)) shows that, at the same crack density, the reduction is substantially weaker in 3-0. This conclusion is somewhat relative, though, since each of the definitions (6.1) of p can always be made to incorporate an arbitrary multiplier. A somewhat more solid comparison can be based on adding multipliers a and 4 4 3 into the definitions of p in 2-D and 3-D, respectively, making p a relative area (volume) swept by cracks when such cracks are rotated about their centers (diameters). Comparison based on this modified p still shows a substantially weaker sensitivity to cracks in 3-D: for randomly oriented cracks, for example, the coefficient at p in the formula for E/Eo in 3D constitutes only 0.42 of that in 2-D (assuming vo = 0.25). We note, in this connection, that not only the impact of noninteracting cracks is weaker in 3D, but interactions between cracks are also weaker in 3-D (see Section 1V.A).
8. Noncircular Cracks
We consider now elliptical crack shapes and analyze the impact of the shape factor on the overall effective properties. As discussed in the Introduction, the COD tensor for an elliptical crack has the form
B = -q + ' [ I + ( x ?+' 2
-'
l)nn+-(ss-tt) ? ?+'
]
(6.52)
where s and t are unit vectors of ellipse's axes (2a and 2b) and 5, q, C are given by elliptic integrals. An important factor for anisotropy of the overall properties is whether the vector (b) is collinear to n-a. From this point of view, the three terms of (6.52) play different roles. The first term characterizes the part of (b) that is collinear to the traction vector n u. The second term characterizes deviation from collinearity due to the difference between the shear compliance (averaged between the two principal directions of the ellipse) and the normal compliance of a crack. Therefore, the coefficient at the second term is small if (q + [)/2 is close to the normal compliance 5 (as in the case of a circular crack). The third term describes the situation in the crack plane. It characterizes deviation from collinearity of the in-plane component of n * a from the inplane component of (b) and vanishes if the crack is circular.
-
Elastic Solids with Many Cracks and Related Problems
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Substitution of (6.52) into the general expressions (6.5) and (6.7a) yields the elastic potential as a sum of three terms corresponding to the three terms in (6.52):
+ -n2 0:-1V
xi
(6.53) {ab
n(ss - tt)n}:ts
The third term consists of a sum of the differences nis-u:, where gas,nntare the shear stress components along the s and t directions (of the major and minor ellipses axes). These differences enter the sum with multipliers that depend on the crack sizes and aspect ratios. If the aspect-ratio distribution is uncorrelated with the size and orientation distributions, then the third term vanishes. The first two terms of (6.53) have the same general structure as the potential (6.30) of a solid with circular cracks. They are indeed very close to (6.30); this is seen by introducing equivalent circular cracks, as follows. Guided by the crack density parameter (6.la), we replace the ith crack contribution into (6.53) by the contribution of a circular crack having the same ratio S2/P.The radius 1 of the thus defined equivalent circular crack is expressed in terms of the semiaxes a, b as l3 = nab2/(2E(k)). The elliptic crack and the equivalent circular crack produce exactly the same normal strain; i.e., they have the same product (crack area)(b,). The shear compliance of the equivalent circular crack diyers, however, from the averaged (between two principal directions of the ellipse) shear compliance of the elliptical crack. This difference depends on the aspect ratio a/b of the ellipse, but remains small even at large ratios a/b:it is 0.7%, 2.5% and 3.2% for a / b = 2, 6 and 58, respectively. Therefore, the vector (crack area)(b) is approximately the same for these cracks, in terms of both the direction and the magnitude of this vector. Thus, replacement of an elliptical crack by the equivalent circular crack is adequate with very good accuracy and the potential (6.53) can be replaced by the potential (6.30)of a solid with equivalent circular cracks. Therefore, all the conclusions on orthotropy, descriptions in terms of crack density tensor, etc., hold for a solid with elliptical cracks. This conclusion holds provided the aspect-ratio, size and orientation distributions of elliptical cracks are uncorrelated (so that the third term in (6.53) can be neglected).
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9. Historical Remarks on Crack Density Tensor, Orthotropy and Potentials in Terms of Invariants Crack density tensor and construction of the elastic potential f in terms of simultaneous invariants of u and u were introduced by Vakulenko and Kachanov (1971). The crack density tensor was introduced there in a more general form that covers dislocations as well as cracks and, in the case of cracks, allows one to incorporate information on crack faces’ displacements. For the study of effective elastic properties of a solid with narrow cracks, (6.16), (6.16a) is the appropriate definition; it is dictated by the structure of the elastic potential. The corresponding discussion (with an overview of modifications and applications of the crack density tensor) was given by Kachanov (1980a). Orthotropy of the effective properties was pointed out by Kachanov (1972). Tensors similar to u and construction of the elastic potential (or, equivalently, of the elastic compliance) in terms of invariants were subsequently used by several authors. Oda et a!. (see, for example, 1984) used a “crack tensor” F coinciding with a to within a multiplier 271 (the fourth-rank fabric tensor was also used; however, it was additionally assumed that vectors n - a and (b) are collinear in the 3-D case, although, as discussed earlier, this assumption actually eliminates the fourth-rank tensor’s contribution into the effective moduli. Litewka (see, for example, 1986) considered, primarily, periodic crack arrangements and used a symmetric second rank “damage tensor” D that differsfrom a by carrying information on crack widths (as a consequence, the effectivecompliances may not be fully consistent with the rigorous results for narrow noninteracting cracks, see his formulas (5-6, at z + 0). Krajcinovic (1985) used a similar construction of the potential in terms of invariants, with the difference that uectors, rather than tensors, were used for characterization of cracks (inconsistencies associated with vectorial characterization are discussed in Section VI.B.12). Talreja (1985) applied this construction (with vectors, 1985, changed to tensors in 1987) to the case of cracks in anisotropic matrix (see discussion of Section C). Tensors of the a type were applied by a number of authors to studies of various jiacture processes. The more general version of the crack density tensor that carries information on crack openings (Vakulenko and Kachanov, 1971) was applied by Allen, Harris, and Groves (1987) and Lee, Allen, and Harris (1989) to the description of damage in composites; Dragon and Mroz (1979), Dragon and Chihab (1985), and Chihab and Dragon
Elastic Solids with Many Cracks and Related Problems
369
(1990) used it in a study of damage evolution in rocklike materials and in ductile materials; Talreja (1991) applied it to the study of damage in ceramic composites. Murakami and Ohno (1981) and Murakami (1983) used a “damage tensor” S2, which differs only by a normalizing multiplier from the modification of a used by Kachanov (1972). It is intended to characterize the grain boundary cavitation in creep and is defined in terms of a sum of grain boundary surface elements. (Note that, being normalized to the total area of cavities in a representative volume, S2 does not seem to vanish when damage vanishes.) General discussion of area loss-related tensors was given by Betten (1982). Both Betten and Murakami and Ohno proposed to construct constitutive laws in terms of simultaneous invariants of c and S2, fully similar to the constructions described in this section. We note, as a general comment, that whereas the area loss-related tensors may be appropriate for characterization of creep processes, their use in the problem of elastic properties may lead to inconsistencies (see the discussion of Section VII). For example, it implies that contribution of a given crack into the change of the overall property in 3-D is proportional to its area, rather than to (area)”’. For the elastic properties, tensor (6.16), (6.16a), naturally emerging in the elastic potential, should be used. 10. Applications of Crack Density Tensor t o Other Physical Properties
Tensors of the a type can also be applied to characterization of physical properties other than elastic of cracked solids. These tensors should then be appropriately modified-a multiplier at an individual crack’s dyad nn should be changed from I’ or I’ (as in the case of elastic properties in 2-D or 3-D) to the one that properly reflects the relative contribution of a given crack into the overall property. Analyses of this kind can be done, tor example, forjuidfiltrationthrough a fissured medium and for the electro- and heat conductivity of a medium with cracklike isolators or conductors. Since these physical laws interrelate vectorial quantities (rather than tensors u and E), the tensors that characterize the effective physical property are of the second (rather than fourth) rank. Their specification as functions of a is therefore simpler than specification of the effective elastic moduli. To illustrate the logic of such analyses, we considerfluid filtration through a fissured material following the work of Kachanov, 1975, 1980a (see also a similar work by Oda, 1985;for an earlier analysis of filtration anisotropy due
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Mark Kachanov
to several planar systems of fissures, which is lengthier and does not use tensorial representations, see Romm and Pozinenko, 1963). Darcy’s law for the effective anisotropic permeability, as formulated by Ferrandon (1948), is V =
-p-’K.Vp
(6.54)
where v is the filtration rate, p is the viscosity coefficient and Vp is the externally applied pressure gradient. Symmetric second-rank tensor K characterizes the anisotropy of filtration. Assuming that the “background” permeability in the absence of fissures is isotropic (K = kl), we aim at specifying the change in permeability K - kI due to fissures as a function of the a-type tensor. We note that in rock mechanics it is usually assumed that permeability due to several families of discontinuities is a simple superposition of the individual permeabilities (the underlying assumption-that energy losses at intersections can be neglected-is, typically, adequate, unless discontinuities of one family terminate at the bedding planes of the other family, resulting, possibly, in noticeable energy losses due to sharp changes in flow directions; see Wittke, 1990, for a detailed discussion). It implies that characterization of discontinuities by the a-type tensor (insensitive to mutual positions of fissures) is adequate. Tensor u should, however, be modified, to account for the fact that, according to the hydrodynamic laws, the contribution of a given fissure into v is proportional to its area and to its opening cubed. Thus, a should be changed to a = (1/V) C (Wnn)’ (6.55) i
The dependence of K - kI on a is an isotropicfunction (if both Vp and the array of fissures undergo any orthogonal transformation, say, rotation, then v undergoes the same transformation). Cayley-Hamilton’s theorem then implies that K - kI is a polynomial quadratic in a, with coefficientsfunctions of invariants of a. Since, as remarked previously, v is a sum of the individual fissures’ contributions, the polynomial should be linearized in a (similarly to linearity of the elastic compliance in a for noninteracting cracks). We further note that a family of parallel fissures does not affect permeability in the direction normal to them. These considerations allow one to fully determine K-kI as a function of a:
K - kI = C[(tra)I - a] where C is a constant of a hydrodynamic nature.
(6.56)
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371
Similar analysis can be applied to electric (or heat) conductivity of a medium with cracklike conducting or isolating surfaces, by specifying the corresponding tensor in Ohm’s (or Fourier’s) law as a function of the (appropriately modified) crack density tensor (Kachanov, 1976, 1980a). Note that dyads of the nn-type, formally similar to those comprising a, have been used in the (unrelated to cracks) problem of contacr stresses in a granular material by Satake, 1978 (see also review of Mehrabadi, Nemat-Nasser, and Oda, 1982). 11. Noncoaxiality of Direrent Physical Properties of Cracked Solids Tensors characterizing anisotropy of various effective physical properties-elastic compliance, fluid filtration, electric conductivity, heat conduction-can all, in the approximation of noninteracting cracks, be specified as functions of symmetric second-rank crack density tensors. One may be tempted, therefore, to use the measured anisotropy of one property (say, electric conductivity) to judge anisotropy of another property (say, elastic moduli or fluid permeability); such correlations are sometimes used in geophysics. However, such correlations cannot generally be established, since diflerent modijications of the crack density tensor should be used for characterization of different physical properties. For example, the impact of fissures on fluid permeability is highly sensitive to the fissures’ openings (as reflected in the definition (6.55)of the crack density tensor) whereas the linear elastic moduli are unaffected by them (as reflected in the definition (6.16)).Principal axes of these modifications of a and, therefore, the principal axes of diflerent physical properties have diyerent orientations. 12. Remarks on Characterization of Cracks by Vectors Several authors (Talreja, 1985; Krajcinovic, 1985; Costin, 1985) used vectors (rather than second-rank tensors) to characterize crack arrays; for a given crack, the vector is taken in the direction of a unit normal n, and its length is determined by the crack size. Generally, only even rank tensors may be used as such parameters (see, for example, discussion of Onat and Leckie, 1988). More specifically, the following inconsistencies resulting from the vectorial characterization can be pointed out. 1. It is not clear how to introduce the overall (average)vectorial parameter characterizing the entire crack array. In particular, it is not clear what
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Mark Kachanov
such a parameter would be in the simplest case of randomly oriented cracks (for a second-rank tensor, no such difficulty arises-it is proportional to a unit tensor I). 2. This difficulty is usually avoided by “discretization”,i.e., by retaining the entire set of vectorial parameters ni for each crack (or each family of parallel cracks) and using them as parameters in the elastic potential alongside the stress tensor u (or strain tensor 8). Then, f is constructed in terms of simultaneous invariants of vectors n’ and u (similarly to constructing f in terms of invariants of u and a).However, one still runs into difficulties: a. The number of invariants that should be allowed into f is very large, among them are all the combinations (n’ * u nj)(nk u * 0’) and each entering f with a coefficient to be determined (for (tr uxo’* u * d), example, for three systems of parallel cracks there would be 42 coefficients). b. Formation of simultaneous invariants requires that u, as a secondrank tensor, contracts twice with vectors n’. Thus, there are no terms linear in n’. Therefore, the simplest case-the approximation of noninteractiny cracks (when compliance is linear in crack density)cannot be recovered.
-
-
13. Concluding Remarks The underlying reason for adequacy of the crack density tensor as a parameter of a crack field is collinearity of vectors n*uand (b) (exact in 2-D and approximate in 3D). This implies orthotropy. Therefore, the crack density tensor becomes inadequate to the extent this collinearity is violated. Several physically important situations of this kind are discussed in the next three sections.
c. NONINTERACTING CRACKS IN AN ANISOTROPICMATRIX 1. On the COD Tensor of a Crack in an Anisotropic Solid
Modes I and I1 are coupled: normal or shear traction on a crack produces both normal and shear CODs. This is reflected in the nondiagonality of the COD tensor B B,, = B,, # 0. Also, B = B(n): compliance of a crack depends on its orientation with respect to the anisotropy axes of the matrix. In the case of arbitrary anisotropy of the matrix, B-tensors can be found
Elastic Solids with Many Cracks and Related Problems
313
only numerically. In the case of a 2 - 0 orthotropic matrix, the problem can be solved in a closed form. Referring to Mauge (1993) and Mauge and Kachanov (1992) for analysis of the general case, we consider here one important special case when the effective moduli can be found in a relatively simple form.
2. 2 - 0 Orthotropic Matrix with Cracks Parallel to the Axes of Orthotrory In this case, the normal and shear modes are uncoupled. Using the solution of Ang and Williams (1961), we find, after some algebra, the average normal and shear CODs for a crack normal to the x2 axis:
- 2vy2 + 2(Ey/E!)'/' (b,) =
[{$-
(EyE($'/2a22
= B!,;)a,,
(6.57)
2 4 , -t 2(Ey/E!)"2 }"'/Ey]a12
= B$!)a,,
G12
where B!,:), BI:) are components of the COD tensor (superscript 2 refers to the crack normal to the x2 axis); tensor B(2)is diagonal (B!,? = 0) and is given by the formula
(6.58)
For a crack normal to the x1 axis, tensor B") is obtained from B") by interchanging indices 1 and 2. Introducing tensors B(') and B(2)into (6.13) we obtain the compliance AM due to two families of cracks normal to x1 and x2 (crack densities are p1 and p2, respectively): AM = p,e,B")e,
+ p2e2B'2)e2
(6.59)
(to be appropriately symmetrized) from which all the moduli follow; for example,
E? is obtained from E! by interchanging indices 1 and 2 El
E2
(6.60)
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Mark Kachanov
In the case of one system of parallel cracks, these formuli recover the ones of Gottesman et al. (1980) (which contain a misprint in the expression for the shear modulus). These formulas may serve as a convenient starting point for calculations of the self-consistent and diferential schemes, by changing the matrix’ moduli to the effective ones in the expression for the B-tensor. Such calculations were done, in a somewhat different way, by Gottesman et al. (1980) (self-consistent scheme) and Hashin (1988) (differential scheme). The influence of various elastic constants of the orthotropic matrix on the compliance change due to cracks (aligned with the orthotropy axes of the matrix) was analyzed in detail (with crack interactions taken into account) by Mauge (1990) and Mauge and Kachanov (1990). We now consider the “asymmetry” of the stirness reduction eflect. The impact of cracks on stiffness reduction in the direction normal to them i, stronger (or weaker), compared to the case of the isotropic matrix, if the matrix material is stifer (or sofer) in this direction. This effect is “asymmetric’’: the difference with the case of the isotropic matrix is larger in the case of cracks normal to the stifer direction. As an example, we consider values relevant for glass fiber reinforced plastics: EY/Eg = 4.1; EY/2GY2 = 5; v21 = 0.068; vY2 = 0.277. For parallel cracks normal to the ”softer” direction, we obtain EJE: = (1 + 5.77p2)-’, a small change compared to E/Eo = (1 + 2 4 - l for the isotropic matrix with parallel cracks. For cracks normal to the “stiffer” direction, the difference in the case of the isotropic matrix is much more pronounced: Ei/EY = (1 11.5p1)-’. Thus, the anisotropic matrix is more “sensitiue” (in the sense of stiffnessloss) to cracks normal to the stiffer direction. A somewhat related conclusion has been reached by Mauge (1990) and Mauge and Kachanov (1990, 1992) for interacting cracks in an orthotropic matrix: if cracks are normal to the stiffer (softer) direction, then interactions are enhanced (weakened) compared to the case of an isotropic matrix. Note that, although the overall anisotropy in the considered case is orthotropic, the increment of elastic compliance due to cracks AM cannot be expressed in terms of the crack density tensor a-it depends on tensors B(’) and B”) containing information on matrix moduli.
+
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375
3. On Formulations in Terms of Invariants
For cracks arbitrarily oriented in an anisotropic matrix, B-tensors and the effective moduli can be found only numerically. As an alternative, one can take a “phenomenological”approach and assume that, similar to the case of the isotropic matrix, the potential f is a function of a and a, but they enter f through terms that are invariant only with respect to those orthogonal transjormations that belong to the group of elastic symmetry of the matrix. This approach is by no means rigorous, since there is no justification for excluding from the analysis the following factors (the only reason for disregarding them is that otherwise the analysis becomes intractable). 1. The fourth-rank tensor (discussed in Section B), whose exclusion in the case of anisotropic matrix is difficult to justify because the difference in normal and shear compliances of a crack-the main reason for emergence of such a tensor in the potential-may be significant in the case of anisotropic matrix; 2. B-tensors, containing information on the matrix moduli.
If this approach is taken, then the group of the effective elastic symmetry is an intersection of the groups of symmetry of the matrix and of a (the latter being the group of symmetry of an ellipsoid); for example, a 3-D orthotropic matrix with one family of parallel cracks arbitrarily inclined to the orthotropy axes will have no elements of elastic symmetry (and will require a full set of 21 elastic constants). The expressions for f will then be a sum of a large number of simultaneous invariants of a and (x (with respect to a subgroup of the orthogonal group), each invariant entering with a coefficient to be determined (with the hope that information on the matrix moduli that should, in the proper way, enter f through the B-tensors, can instead be entered through these coefficients-an additional assumption of unclear validity). Such an approach was briefly outlined, as a possibility, by Kachanov (1980a); it was fully developed by Talreja (1987) who wrote expressions for f in several cases of the matrix’ symmetry (in the earlier paper, Talreja, 1985, vectors were used for characterization of crack arrays; inconsistencies associated with vectorial characterization were discussed in Section B). This approach leads to emergence of a large number of coefficients at invariants that have to be found experimentally. Since writing f in the form f (a,a) cannot be rigorously justified in the case of anisotropic matrix, these coefficients are not material constants and may simply play the role of
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Mark Kachanov
adjustable parameters (although Talreja’s data indicate their constancy in some experiments). D. NONINTERACTING CRACKS CONSTRAINED AGAINSTTHE OPENING (BUT ALLOWEDTO SLIDE)
This case is relevant for compressive conditions. The difference in the effective properties between this case and the case of freely opening cracks accounts for the differences in compliance under compressive and tensile conditions. Since the deviation from collinearity of the vectors n - a and (b) is substantial in this case ((b) has only tangential components), the crack density tensor u becomes inadequate and orthotropy does not hold. The situation may be further complicated by friction. We assume here that no secondary (“wing”) cracks form. The first quantitative study of a solid with frictionally sliding cracks under compression was done, probably, by Walsh (1965) who found the effective Young’s and bulk moduli. Stress-induced anisotropy, path dependence of deformation and formation of a vertex on a loading surface were analyzed in detail by Kachanov (1980b, 1982). Self-consistentapproximation was applied to the same problem by Horii and Nemat-Nasser (1983); however, as follows from the results of Section I, the self-consistent scheme may actually be less accurate than the approximation of noninteracting cracks. 1. Sliding Without Friction (“Lubricated” Cracks) In this case the behavior is linear elastic. The potential can be found by introducing terms (b) (tangential to cracks) into (6.5). In both 2-D and 3-D (circular cracks) cases, (b) is proportional to the shear traction n u - n * u nn with the proportionality coefficient d / E 0 in 2D and yl/E, in 3-D (see (6.14) and (6.29)).The potential takes the forms
-
Af = (n/E,){(u.a):u - a:(1/A)~(12nnnn)’:a}
in 2-D
Af = (yn/E,){(u~a):u- a:(1/V)~(13nnnn)’:a} in 3-D
(6.61) (6.62)
This form is similar to (6.30):crack density tensor u becomes insufficient and the fourth-rank tensor (l/A) Z ( P nnnn)’ emerges as a term in the potential. However, in contrast with (6.30), a small multiplier v,/2 at this term is absent; therefore, the contribution of this term into the overall compliance can no longer be neglected. Another difference with the case of freely opening cracks is that
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377
the fourth-rank tensor appears in both 2-D and 3-D. Therefore, orthotropy is significantly violated, in both 2-D and 3-D. The case of elliptical cracks can be analyzed the same way as for freely opening cracks. The two main effects of the constraint are
1. It drastically reduces strains produced by cracks. In the 3-D case of randomly oriented cracks, the constraint reduces the uniaxial compliance due to cracks by the factor of 0.4/(1- 3v0/10). 2. In this case of nonrandom crack orientations, the reduction of strain is even more pronounced. Also, the constraint changes the direction in which the compliance is maximal; i.e., it rotates the anisotropy axes. This is a consequence of the tensor (1/A)Z(I3nnnn)i becoming important. The 2-D case of parallel cracks provides an example (Fig. 48).In the case of freely opening cracks the compliance is maximal in the direction normal to cracks, whereas for constrained cracks the strain response due to cracks is zero in this case, since shear displacements of crack faces (the only source of inelastic strains) are absent. Note that the maximal (over
e FIG.48. Uniaxial compliance (additional, due to cracks) of a 2-D solid with parallel cracks, as a function of direction of loading (forming angle 4 with a normal to cracks), normalized to npa/E,. Dashed and solid lines indicate constrained and unconstrained cracks. Imposition of the constraint shifts the direction of maximal compliance by 45" and sharply reduces strains. This illustrates the difference between compliances of a cracked body in compression and tension.
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Mark Kachanov
all directions) compliance is reduced almost four times when the constraint is imposed. We note that these results can be obtained by using the COD tensor, changed from (6.8) to
B = (d/Eo)(I- nn)
(6.63)
in the 2-D case; from (6.9) to
B = ?(I
- nn)
(6.64)
in the 3-D case of circular cracks, and from (6.10) to B =~
+ [tt
S S
(6.65)
for elliptical cracks.
2. Sliding with Friction If friction is present, vectors (b’) are not unique functions of stress, so that potential does not exist (the tensor of incremental compliance loses symmetry: AMijkl# AMklij).Deformation is loading path dependent, and stressinduced anisotropy emerges; a vertex develops on the loading surface; unloading starts with a “deadband” (stress interval in which no backslidings occur). These issues were investigated by Kachanov (1980b, 1982); here, we briefly review the findings. The “driving force” that produces a shear COD is the shear traction reduced by the frictional resistance. The simplest model for the latter is the Coulomb’s law. Thus, the driving force on a crack is T + po, where T and on are the shear and normal tractions induced on a crack (a, is negative in compression) and p is the friction coefficient;the driving force depends on the crack orientation with respect to loading. The loading path dependence is obvious from the observation that application of shear traction T (while holding on = 0) followed by application of compressive 0, produces a nonzero shear COD, whereas reversing the sequence produces no displacement, if a,, is sufficiently high. Because of the loading path dependence, stress-strain relations have to be formulated in the incremental form (similar to the equations of metal plasticity). However, in contrast with plasticity, the range of loading paths for which deformation is path independent (“fully active” loadings-all cracks that
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379
started to slide, continue to do so, as loading proceeds) is quite wide. For example, in the case of axisymmetric compression u = -q(e,e,
+ e2e2)- pe3e3
where p > 0 and q > 0 are the axial and lateral pressures, with a superimposed stress increment du
=
-dq(e,e,
+ e2e2)- dpe3e3
the range of dp, dq that provides fully active loading is shown in Fig. 49. This range is quite wide: at p = 0.6, for example, the angle ONA differs from n by less than 18". Note the formation of a vertex on the loading surface (in qualitative terms, the development of a vertex in a solid with frictionally sliding cracks was discussed by Rudnicky and Rice, 1975, where a phenomenological model was suggested). The vertex becomes more pronounced as loading proceeds. The strength of this effect depends on friction coefficient, increasing with p and disappearing at p = 0.
q FIG.49. Range of path independence (cone CE) and formation of a vertex on the loading surface in axisymmetrical compression; NA' is parallel to the initial loading surface O A of the most favorably oriented crack. M N shows the loading path (axial compression at constant lateral stress).
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Mark Kachanov
Stress-induced anisotropy develops as loading proceeds, since the strain increment produced by an incremental loading d a superimposed on a current state u depends on orientation of da with respect to a: it is maximal if du is in the direction of u (maximal increase of driving forces z + pnn on the cracks activated at a). It is smaller for directions moderately deviating from a (but still falling in the range of fully active loadings) and is even smaller (or zero) for da forming large angles with u, since such da may “lock” the previously activated cracks. Table I1 shows gradual development of stress-induced anisotropy in the axisymmetric case, as the ratio p / q increases. Note a gradually increasing difference between A M , , , , and AM333, and that AM,,,, #Ah!t331, (indicating that, as expected for path-dependent deformation, the potential does not exist). Although the inelastic strains produced by frictional slidings are, typically, relatively small (compared to the strains of the matrix), they may produce a distinguishable increase of the overall Poisson’s ratio.
E. NONINTERACTING CRACKS FILLED WITH
COMPRESSIBLE
FLUID
1. Basic Equations. Impact of Fluid on Efective Compliance
The impact of fluid on elastic interactions between cracks was analyzed in Section 1V.F. Here, we consider a linear elastic solid with fluid-filled, cracklike cavities in the context of effective elastic properties.
TABLE I1 DEVELOPMENT OF STRESS-INDUCED
AMSOTROPY IN THE AXlSYMMETRICAL LOADING, AS THE RATIO p / 4 OF
~ E INCREASES S (It is assumed that p = 0.6, so that frictional slidings start at p / q = 3.12; compliances are normalized to 8p(l - v;)/[3(2 - vo)Eo] and vo = 0.25 is assumed.)
THE AX~ALAND LATERALS
PI4
M,,,, M3333 M33,1 M1133
3.5
5.0
10
0.190 0.150 -0.241 -0.075
0.340 0.220 -0.382 -0.110
0.458 0.238 -0.439 -0.119
Elastic Solids with Many Cracks and Related Problems
38 1
The collinearity of vectors n - aand (b) is significantly violated, since the presence of fluid reduces the normal compliance of a cracklike cavity, whereas the shear compliance is unaffected. Therefore, orthotropy of the efective properties is noticeably violated. Similar conclusions were reached, on somewhat ditferent grounds, by Sayers (1988), who observed that the elastic anisotropy (as inferred from wave velocities) is reasonably well described by ellipsoids in dry rocks, but deviates substantially from such shapes in puid-saturated rocks. We consider here cracklike cavities filled with compressible fluid, in the approximation when stress interactions between cavities are neglected. Our analysis is somewhat similar to the one of Budiansky and OConnell (1976) but appears to be simpler; it is more general in that it is not restricted to randomly oriented cracks but applies to arbitrary orientational distributions, also it describes dependence of fluid pressures on cavities’ orientation (pressure “polarization”). The analysis of Budiansky and O’Connell differs from ours in that it is done in the framework of the self-consistent approximation, which appears to substantially overestimate the effective compliance (see the discussions of Sections G and H). We assume the “undrained approximation; i.e., that the total mass of fluid in each cavity remains constant (if fluid diffusion across crack faces takes place, this approximation corresponds to a “short-time’’ response). For simplicity of calculation, we also assume that fluid-filled cavities have shapes of relatively thin “platelets.” We first consider the case of cavities circular in plane. The representative cavity has radius 1 and, in the absence of applied stresses, has average opening (b,), average aspect ratio lo = ( b o ) / l and volume d3lO. When stresses are applied, the cavity changes its volume, producing the change of fluid pressure Aq; the latter counteracts the applied traction, reducing the normal COD. Thus, fluid reduces the impact of cracks on the effective compliance, compared to dry cracks. This effect is somewhat similar to imposition of the constraint against opening, but is milder (since the fluid is compressible). Since the cavity is thin, its volume change is mainly due to changes in (b,) ( I remains approximately constant). Thus, the relative change of cavity’s volume AV/V, = ( A b n ) / ( b o ) .Since the fluid mass within a cavity is constant (“undrained” conditions), the change of fluid density q is related to the volume change: qoVo = qY or ArthO
=
-AV/V, = - ( A b ” X b o >
(6.66)
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Mark Kachanoo
The corresponding change of fluid pressure is given by the constitutive equation for a compressible fluid, which, in the linearized formulation (small density changes), has the form
Adlo =
-K&
(6.67)
where K is the compressibility of a fluid (generally, a function of q and temperature, treated as a constant in the considered linearized approximation); in conjunction with (6.66), this equation relates Aq to the change of opening (Ab,). On the other hand, (Ab,) is related to the overall normal traction on a cavity ( n * a * n+ Aq) by a “cavity compliance” relation: (Ab,) = /31/Eo(n.a-n + Aq)
(6.68)
where the compliance of a (thin) cavity is taken to be approximately the same as the one of a circular crack of the same diameter: /3 = 1q1 - v@/3a. These equations yield normal CODs and changes of fluid pressures induced on cracks of different orientations by the applied stress 0: (6.69)
Aq= -
1
1
+ / 3 - l ~ E , l , n-cr’n
(6.70)
Thus, the sensitivity of fluid pressure in a given cavity t o the applied stress is determined by a dimensionless constant
6 = p-lKEOro
(6.71)
This parameter, the key constant of the problem, is similar to the one used by Budiansky and OConnell. It characterizes coupling between stresses andfluid pressures and determines the impact of fluid on the effective compliance. Physically, (6.68) means that the applied compressive stress is counteracted by two factors: “cavity’s stiffness” and the fluid pressure. Parameter 6 determines relative shares of these two factors. In the limit of a highly compressible fluid (K + 00, “air”), change of the fluid pressure in cavities caused by applied stresses is zero and the eflective compliance is the same as in the case of “dry” cracks. In the opposite limit K + 0 (incompressible fluid), Aq = - n u * n and compliance is the same as in the case of cracks constrained against opening (Section D). The same is true if K
-
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383
co
is finite but the cavities are infinitesimally thin, + 0, indicating that coupling between stresses and fZuid pressures is stronger for thinner cracks. In general, the compliance is intermediate between that of a medium with dry cracks and that with cracks constrained against opening. As an example that may be relevant for fluid-saturated rocks, we consider the case of IC x 0.5 x MPa-' (water) and Eo x 6 x lo4 MPa (granite). Then, for the cavity aspect ratios co < 0.006 the impact of fluid on the effective compliance is so strong that the compliance differs by less than 10% from that of a medium with cracks constrained against opening. The impact of fluid becomes small (compliance differs from that of a dry medium by less than 10%)only for very wide cavities, at lo > 0.5-0.6 (note that the calculation of the cavity's volume change used above then becomes inaccurate). Thus, in the entire practically interesting range of lo, the effect of stress-fluid pressure coupling is strong and significantly affects the effective compliance. The elastic potential is obtained by introducing the normal CODs from (6.69) and the shear CODs from the expression for the dry cracks into the general expression (6.5). Thus, for the arbitrary orientational distribution of cracks,
(6.72) At 6 + 00, the expression (6.30) for dry cracks is recovered; at 6 -+ 0, the expression (6.62) for cracks constrained against opening is recovered. Since the coefficient of the term containing the fourth-rank tensor is, generally, not small (compared to that at the a-term), the deviationsfrom orthotropy can be signijcant. Effective compliances follow from (6.72). For example, in the case of randomly oriented cracks,
where p is the scalar crack density. Figure 50 shows Young's modulus as a function of crack density for three values of 6. Results for any other orientational distribution of cracks readily follow
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Mark Kachanov
0.0 0.0
2.0
4.0
6.0
P FIG.50. Young’s modulus of a solid with randomly oriented fluid-filled cracks as a function of crack density, at various values of KEOCO.Note a gradual transition from the “dry” cracks curve ( K E ~ C= ~co) to the curve for cracks constrained against opening ( K E J ~= 0).
from (6.72), by inserting the corresponding crack density tensor a and the fourth-rank tensor. For example, in the case of parallel cracks (normal to the x 1 axis) one obtains
(6.74)
The shear modulus is the same as in the case of dry cracks (see (6.40)), in accordance with the fact that fluid does not resist shear displacements of crack faces. Note that the preceding results can be obtained by using the tensor of a circular crack, changed from (6.9) for a dry crack to B = y I---yo 2
(
,.,..) 6
(6.75)
and that these results can be generalized to elliptical cracks by changing the B-tensor given by (6.1 1) to B=
2
[I+ (-
25
6 -1
tl+r 1 + 6
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385
In the case of parallel cracks with disturbed orientations (Section B.5), the sensitivity of the effective moduli to the disturbance will be higher than for dry cracks, since the shear mode contributions that appear in the disturbed configuration will not be inhibited by the fluid.
2. Polarization of Fluid Pressure In addition to the impact of fluid on the effective compliance, result (6.70) describes ”pressure polarization”: fluid pressure in a given cavity depends on the cavity of orientation n is expressed as Aq = n - Q * n .As seen from (6.70), cavities of different orientations, fluid pressures will be different in different cavities. It has been suggested (Rice, 1978) that a second-rank “pressure polarization tensor” Q can possibly be introduced, so that a fluid pressure change in the cavity of orientation n is expressed as Aq = n Q n. As seen from (6.70), such a tensor can indeed be introduced, provided all the cavities have the same aspect ratio c,-condition that may be too restrictive; if it is satisfied, then Q = -(1 + 6)-’a. Note that Q = -u in the limit of K E , ~ + , 0. 0
-
3. On Drained and Undrained Conditions The preceding analysis is relevant for the undrained conditions. If fluid diffuses across the cavities’ faces, fluid pressures in different cavities gradually equalize and pressure polarization disappears. In the “long-time’’ limit (fully drained response), the effective compliance becomes the same as that of the dry medium. Thus, in the process of Jluid diffusion, the effective compliance changesfrom that found above to that of a dry medium.
F. SUMMARY ON NONINTERACTING CRACKS
For noninteracting cracks in the isotropic matrix, the results are exact for any orientational distribution of cracks, both in 2-D and 3-D, and are free of controversy. They depend on the density and orientations of cracks but are independent of the mutual positions of cracks. As discussed in Sections H and I, this approximation has a wider than expected range of applicability, due to cancellation of the competing effects of stress shielding and amplification in arrays of interacting cracks. In the 2-D case, the second-rank crack density tensor a provides a fully
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adequate characterization of a crack array. We emphasize that a, as a parameter of crack density, is not introduced arbitrarily, but is dictated by the structure of the elastic potential. In the 3-D case, a fourth-rank tensor should be used, in addition to a. However, its impact on the effective moduli is small, so that the characterization by a only is a good approximation. The linear invariant tr a coincides with the conventional scalar crack density p, so that a is a tensorial generalization of p accounting for crack orientation. We note that the assumptions of noninteracting cracks and small crack density are not equivalent: for non-randomly located cracks, interactions may be strong and the approximation of noninteracting cracks inapplicable at vanishingly small density p (see, for example, Fig. 54 in Section K); on the other hand, for randomly located cracks, the approximation of noninteracting cracks remains accurate at high densities (Section I). Since a is symmetric and enters the potential through a simultaneous invariant with u, the effective moduli always possess orthotropy coaxial with the principal axes of a (this result is exact in 2-D and approximate in 3-D). Moreover, the orthotropy is of a simplified type-it is characterized by only four (rather than nine) independent constants. This gives rise to the concept of a “natural” coordinate system-principal axes of a-in which the matrix of effective moduli has an orthotropic form in 2-D and close to such a form in 3-D. (For interacting cracks, as discussed in Section I, orthotropic symmetry still holds with good accuracy, so that the “natural” coordinate frame remains the system of choice). The adequacy of a as a sole parameter of the crack field and the orthotropy of the effective properties are consequences of the collinearity of vectors (b) and n. a; i.e., on equal (or approximately equal) compliances of cracks in the normal and shear modes. If this collinearity is violated (cracks in anisotropic matrix, cracks constrained against the opening and fluid-filled cracks), the crack density tensor becomes inadequate and anisotropy of the effective elastic properties is more complex.
G. IMPACT OF INTERACTIONS ON THE EFFECTIVE MODULI: GENERAL CONSIDERATIONS For interacting cracks, the problem becomes much more complex. In principle, it requires solving the interaction problem for each realization of the crack statistics and subsequent ensemble averaging, which constitutes
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387
averaging over the (a) orientations, (b) positions and (c) sizes of cracks, Since these are interdependent, this presents a nontrivial (particularly in 3-D) problem related to geometrical probabilities. The difficulty of these steps gave rise to a number of approximate schemes (see the discussion of Section H). The predictions of these schemes are substantially different. A researcher trying to use the theory may be confused by the choice of several models, each yielding different results. The question arises: Which of them is most accurate? There is no definite answer to this question, unless the statistics of crack centers (mutual positions of cracks) is specified. Indeed, if the nonrandom location of crack centers is allowed, then crack arrangements of arbitrarily large density p that produce an infinitesimal impact on the moduli can be constructed, as well as configurations of infinitesimal density p that reduce the moduli to zero (see the examples of Section J). Therefore, if the statistics of crack centers is not specified, any approximate scheme is realizable: a crack arrangement that fits any prediction can be found. (A doubly periodic parallel crack array is an example: at any given crack density, the effective stiffness can be varied from E , to zero by changing the ratio of the spacing between cracks in the collinear and stacked directions. Similar examples can be constructed for randomly oriented but nonrandomly located cracks, see Section J). A more meaningful formulation is this: Which model is most accurate, provided mutual positions of cracks (statistics of crack centers) are random? The first basic question that arises in this connection is: Do the interactions increase or reduce the overall stiffness (compared to the stiffness calculated with interactions being neglected)? The question can be reformulated as follows: Which of the two competing modes of interaction-stress shielding or stress amplijication (resulting in stiffening or softening effects, correspondingly)-dominates in the crack array? The answer depends on the geometries of the shielding and amplification zones near cracks. As an example, consider parallel cracks under tension. Some insight into the mechanics of shielding and amplification can be gained by examining the structure of the stress field oyyassociated with one isolated crack, Fig. 1 (although such considerations are by no means rigorous since the field is distorted by the presence of other cracks). This field consists of small and intense amplification zones (near crack tips) and more “diffused shielding zones. (Note that such a structure is a consequence of the fact that the volume average of oYyis not affected by the presence of the crack if tractions are prescribed, see Section A. Since singularities of tensile stress at the tips
Mark Kachanov
provide a large contribution into the average, the zone where ayy is compressive and has no singularity has to be larger.) As a consequence, the (amplifying)collinear interactions become noticeable only at small spacings between cracks, whereas the (shielding) stacked interactions of parallel cracks have a longer range. (This explains the mechanics of interactions in a doubly periodic array of parallel cracks: the amplification effect is dominant only if ligaments between collinear cracks are substantially smaller than the spacings between stacked cracks; otherwise, the shielding effect dominates and interactions increase the effective stiffness; see the solutions for doubly periodic arrays, Filshtinsky, 1974.) Consider now the case of main interest, when locations of crack centers are random. We recall the general observation that, if tractions ae prescribed at the boundary, then introduction of cracks does not change the average stress in the solid (see Section A). Therefore, one may expect that the competing effects of shielding and amplification are balanced and cancel each other; i.e., that the randomness of crack centers ensures the absence of “bias” toward either amplifying or shielding configurations. If this is indeed so, then the approximation of noninteracting cracks remains accurate at high crack densities, in spite of strong interactions. As discussed later, for at least two orientational statistics-parallel and randomly oriented cracks-these expectations are confirmed, by 1. Computer experiments (Section 1)-solution of the interaction problem for a number of sample arrays of interacting cracks; 2. Calculations of the method of effective field, which takes the mutual positions of cracks into account, and by its simplest version-MoriTanaka’s method.
We note that the outcome of the competition between shielding and amplifying effects is sensitive to imposition of various geometrical constraints that violate randomness of crack locations. One such constraint is the prohibition for the cracks to be overly close to each other. Such a constraint (although in a mild form, minimal allowed spacings were quite small) was imposed, for computational accuracy, on our computer experiments; in the case of parallel cracks, it produced a slight bias toward shielding, resulting in the stiffness being slightly higher than in the approximation of noninteracting cracks. Another example of sensitivity to constraint is the experimental data of Vavakin and Salganik (1975) that showed the effective stiffness to be noticeably lower than in the approximation of noninteracting cracks (the authors interpreted it as a confirmation of the differential scheme). A more
Elastic Solids with Many Cracks and Related Problems
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careful analysis of their arrays indicates, however, that crack locations were not random-rather, their statistics were close to those in which crack centers are prohibited from entering the circles described around other cracks. Calculations for these particular statistics, done by the method of effective field (Kanaun, 1980, 1983), show that the observed data are fully explained by the constraint; i.e., this constraint produces some bias against shielding arrangements (probably, by eliminating closely spaced stacked cracks-configurations of strong shielding). We emphasize that the accuracy of the approximation of noninteracting cracks at high crack densities reflects the fact that cracks constitute a distinctly special kind of inhomogeneities. As a consequence, some of the approximate schemes used for inclusions and cavities may yield the wrong results when applied to cracks; their attempts to improve the approximation of noninteracting cracks do not necessarily succeed (see Section H). We note, in this connection, that placing a representative crack into an “effective matrix” of reduced stiffness-the basic idea behind the methods of effective matrix (self-consistent, differential)-ignores the stiffening impact of the shielding mode of interactions; thus, it distorts the actual mechanics of interactions. Such schemes predict that interactions always produce a softening effect; i.e., that stiffness, at a given crack density, is always lower than in the approximation of noninteracting cracks (see the discussion of Section H).
H. APPROXIMATE SCHEMES 1. 7jpes of Schemes
Following the common approaches of the continuum physics, most of the approximate schemes reduce the analysis to one isolated crack, placed into some sort of “effective” environment. The effective environment can be defined in two different ways: as an effective matrix or an effective jield. Correspondingly, the schemes can be divided into two groups. The methods of an effectioe matrix place a representative crack into the matrix with effective moduli. Since the crack is treated as isolated, its contribution into the overall compliance is readily found. In the most frequently used scheme, known as a self-consistent one, the entire crack array is placed into the effective matrix in one step, and its impact on the overall moduli is found by summation (done by averaging over orientations) of the individual crack contributions; it is then required that the resulting moduli
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Mark Kachanov
coincide with the effective ones (the self-consistency condition). In the modification known as a differential scheme, the process of placing the crack array into the effective matrix is done in infinitesimal increments, and the reference matrix is recalculated at each step. In another modification, known as a generalized self-consistent scheme, a “cushion” of undamaged material is placed in between the crack and the effective matrix. One can, probably, add various modifications and combinations of these schemes. For example, a differential version of the generalized self-consistent scheme can be constructed in a straightforward way. A “hybrid” of the selfconsistent and the differential schemes, which introduces a crack array into the effective matrix in two steps rather than one (as in the self-consistent scheme) or infinitely many steps (as in the differential scheme), was proposed by Cai and Horii (1992). The method of an effective j e l d places a representative crack into the undamaged matrix, but subjects it to the effective stressjeld (which does not have to be the same for all cracks). The method appears to be more sound from the physical point of view, since it reflects, in an approximate way, the actual mechanics of interactions and has been demonstrated to be more powerful than the methods of effective matrix (for example, it can take statistics of crack centers into account, and the interactions are not predicted to always produce a softening impact). The simplest version, in which the effective field is homogeneous and equal to its volume average, is known as Mori-Tanaka’s method. Other results and approaches exist, which do not reduce the analysis to one isolated crack. They include constructions of a term second order in crack density, a model for an array of parallel cracks of extremely high density, bounding the effective moduli. We also mention that an exact solution is available for a doubly periodic array of parallel cracks with an arbitrary shift ofrows with respect to each other, see Filshtinsky (1974); this solution covers rectangular and “diamond” arrangements as special cases (the special case of a rectangular array was also considered by Delameter et of., 1975, 1977). Zimmerman (1985) made an interesting observation that, as p increases, the relation between different moduli (say, between the shear and Young’s moduli E and G) is almost independent of the scheme used, so that it is difficult to verify a certain scheme by experimentally measuring the E-G curve. A brief review of these schemes and approaches follows.
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2. Methods of an Effective Matrix These methods model the impact of crack interactions on a given crack by placing the crack into the matrix of reduced stiffness. This hypothesis leads to certain drawbacks, as follows. 1. Crack interactions are predicted to always reduce the effective stiffness, compared to that calculated with the interactions neglected. Thus, the stiffening impact of the shielding interactions (which may fully cancel the softening impact of the amplifying ones) is ignored. This distorts the actual mechanics of crack interactions (see Section G). 2. Since a representative crack is inserted into the effective matrix, the type (and orientation) of the effective anisotropy should be specijied a priori. This leads to difficulties in the case of an arbitrary orientational distribution of cracks. First, the type of the overall anisotropy may not be obvious in cases more complex than randomly oriented or parallel cracks. Second, even if such an a priori judgment on anisotropy can be made, the actual realization of the scheme would require the solution for a single crack arbitrarily oriented in an anisotropic medium-such solutions can, generally, be produced only in numerical form. 3. The schemes are insensitive to mutual positions of cracks, a factor that becomes increasingly important as crack density increases. This seems to limit the schemes to the case of random locations of crack centers. Whether their results are accurate in this case is discussed in Section I. Curves for the effective moduli corresponding to various modifications of the method of an effective matrix and to various orientational distributions of cracks were plotted and compared by many authors; see, for example Kemeny and Cook (1986), Hashin (1988), Sayers and Kachanov (1991). a. Selfconsistent Scheme (SCS) This approach was first used, probably, by Maxwell, in connection with the electrostatics of an inhomogeneous medium. In the mechanics of inhomogeneous materials, this method was used by Hershey and Dahlgren (1954), in connection with effective properties of polycrystals and then in the works of Kroner (1958), Hill (1965b) and Budiansky (1965). For a material with cracks, this scheme was developed by Budiansky and
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O’Connell (1976) for random crack orientations and applied by Hoenig (1979) to crack orientations that are either parallel or have cylindrical symmetry and by Horii and Nemat-Nasser (1983) to frictionally sliding cracks. It was extended to 2 - 0 orthotropic matrices with cracks parallel to the orthotropy axes by Gottesman et al. (1980) and reformulated in a somewhat different form, with particular attention paid to crack shape, by Laws and Brockenbrough (1987). Attiogbe and Darwin (1987) considered, using a numerical procedure (not entirely clear to the present author) for a crack arbitrarily oriented in a transversely isotropic material, the case of the overall transversal isotropy, in which cracks have preferential orientations and all orientations are present. For the purpose of illustration, we review here only the case of random crack orientations. Probably, the simplest way to derive the SCS results is to recall the general expression (6.5) for the elastic potential of a medium with cracks. Vectors (b) depend on the elastic constants. According to the basic idea of SCS, these constants are to be taken as the effective ones. In the 2-0 case, changing E , in (6.15) to the effective modulus E’, one obtains
(instead of (6.23) for noninteracting cracks). Identifying this expression with the potential (6.6), where E,, v, are changed to the egectiue E‘, v, one obtains two algebraic equations for two effective constants. They yield
E = EX1 - np);
v = v,(l
-
np)
(6.78)
At small values of p, these formulas asymptotically coincide with the ones for noninteracting cracks, linearized with respect to p . At larger crack densities, they predict a much “softer” response. At crack density p = l/a the moduli become zeros (“cut-off point). In the 3 - 0 case, changing E,, v, to the effective E, v in (6.34)and identifying thus modified potential with (6.6), where E,, v, are changed to the effective E, v, yields two simultaneous algebraic equations for E(p) and v(p), to be solved numerically. Effective Young’s modulus drops almost linearly with increasing p, to a “cutoff point at p = 9/16.
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b. Comments on SCS 1. The predicted softening effect of interactions is quite strong (substantially stronger than in other approximate schemes), resulting in a “cut-off point for the moduli in the case of random orientations. Even for parallel cracks (for which there is no “cutoff) the moduli rapidly drop to negligible values as p increases: in 3-D, the Young’s modulus drops to about 0.1E, at p = 0.3 (Hoenig, 1979). As discussed in Sections G and I, the actual impact of interactions on the overall stiffness may be quite different. 2. Whether prediction of a “cut-off (at p = l/n in the 2-D case and at p = 9/16 in 3-D) is an advantage of the scheme (since it may be interpreted as the overall loss of integrity at high crack densities) or a disadvantage has been a subject of some discussion (Bruner, 1976; O’Connell and Budiansky, 1976). In our opinion, the scheme that reduces analysis to a single crack (albeit inserted into the effective medium) cannot be expected to predict the loss of integrity, a phenomenon caused by crack intersections. The cut-off point is simply a consequence of the structure of potential (6.77) and seems to have no interpretation compatible with the assumptions of the scheme. Moreover, the cut-off predictions do not seem consistent: if the “loss of integrity” interpretation is accepted, the cut-off cannot be expected in the case of parallel cracks, as indeed shown by Hoenig; on the other hand, for the cylindrical orientation distribution (crack normals lie in parallel planes) the cut-off should be expected, but Hoenig’s calculations do not predict it. 3. Budiansky and OConnell (1976) suggested, as a possibility, that the actual effective stiffness may become very small near the predicted cut-of point. Computer experiments on 2-D arrays (Section I) show, however, that, at p = l/n, Young’s modulus still retains about 40% of E , (see Fig. 53).
c. Formulation of Simplified SCS in Terms of Crack Density Tensor As discussed in Section VI.B.3, the fourth-rank tensor (l/V) X (13nnnn)’in
the potential (6.30) plays a relatively minor role, compared to the a-term, and its neglect leads to relatively small errors (which can be further reduced in formulation (6.32)). Therefore, if one wishes to use the SCS, it can be
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Mark Kachanov
formulated in a simplified way, based on the potential (6.30) with the aforementioned term omitted (or, for a somewhat improved accuracy, based on the potential (6.3.2)).In the case of randomly oriented cracks, this leads to explicit (rather than numerically solved) expressions for the effective moduli E(p), v(p). The results are qualitatively similar to those of the main version (the moduli have the same cut-off point at p = 9/16); quantitatively, the results are very close for E ( p ) and exhibit some variation for v(p). Application of the crack density tensor to the case of non-randomly oriented cracks (anisotropic effective properties) streamlines the calculations of SCS and bypasses the essential difficulty of making an a priori judgment on the type of the effective anisotropy, as well as the necessity to have a solution for a crack arbitrarily oriented in an anisotropic body. Indeed, using the abased potential implies that the effective properties are orthotropic, coaxial with a (as indicated by computer experiments of Section I, this assumption remains valid for interacting cracks). Then, since characterization of a crack array by a means a substitution of the actual crack array by three (two, in the 2-D case) mutually orthogonal families of cracks, the change in the elastic potential due to cracks Af is written by placing a representativecrack into the effective orthotropic matrix, a major simplification being that the crack is normal to one of the orthotropy axes (such solutions exist in closed form in the 2-D case, Ang and Williams, 1961, and in 3-D, Shield, 1951, and Chen, 1966). Finally, the sum f Af is equated to the potential of the effective orthotropic matrix. An even simpler (and less rigorous) technique of formulating the SCS (and other schemes) in the case of nonrandom orientations is suggested in subsection f.
+
d. Dierential Scheme (DS) In this scheme (see Vavakin and Salganik, 1975; McLaughlin, 1977; Hashin, 1988) the analysis is done incrementally: crack density is increased in small steps dp, and the effective matrix moduli are recalculated at each step. The procedure results in differential equations for the effectiveconstants as functions of p (with initial conditions stating that M = Mo at p = 0).In the case of random orientations, there are two equations, for E(p) and v(p), which are coupled in the 3-D case and uncoupled in 2-D. We consider the 2-D case as a simple illustration. For random crack orientations, changing Eb in (6.24) to the “current” E
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and p, to dp and linearizing with respect to (small) dp, results in the differential equation for E' = E'(p): dE'/dp = - n p ,
with initial condition
E'(0)= Eb
(6.79)
which yields: E' = Eoe-"P
(and, similarly, v = voe-"P)
(6.80)
Unlike SCS, there is no "cut-off point at which the stiffness vanishes. For parallel cracks, proceeding similarly to the previous case (using (6.27) as a starting point), we obtain the Young's modulus in the direction normal to cracks: (6.81)
In the case of parallel cracks the predicted stiffness drops much faster with increasing p than in the case of random orientations. At small p, this prediction is correct, since parallel cracks reduce the stiffness more efficiently, but at larger p it seems physically unreasonable, since parallel cracks experience increasing mutual shielding as p increases. e. Comments on DS Unlike the SCS, the DS does not predict a "cut-off point, and thus avoids the problem of its interpretation. On the other hand, the results of DS are not path independent: they may depend on the order of adding cracks of different orientations. This drawback was noted by several authors; we add that, probably, the simplest example of path dependence is provided by two families of parallel cracks forming an acute angle with each other; cracks are widely spaced and the approximation of noninteracting cracks applies. Then the effective properties are rigorously orthotropic (see Section B). If, however, DS is applied and cracks of one family are introduced first, the new reference matrix becomes transversely isotropic; when cracks of the second family are added (at an acute angle to the plane of isotropy), the material becomes nonorthotropic-a wrong prediction. Note that the problem of path dependence surfaces in a simple situation of a 2-D array of low density. A possible remedy for the problem of path dependence is the imposition of the constraint that cracks of different orientations are represented in each crack density increment dp in the same proportion as in the final configuration. Since the DS assumes an imaginary process of gradual addition of cracks, the necessity of placing a restriction on this process seems, however, to
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be a certain disadvantage. Note, also, the conceptual problem that cracks added earlier and cracks added later are placed into different environments (see OConnell and Budiansky, 1976). Similar to SCS, the DS can be reformulated in terms of the crack density tensor. If one wishes to use this scheme, this may simplify calculations in the case of arbitrary orientational distribution of cracks.
f. An Elementary Techniquefor Extension of SCS and DS (and, Possibly, Other Schemes) to Arbitrary Orientations Statistics The aforementioned schemes are difficult to apply to an arbitrary orientational distribution of cracks (particularly in 3-D), since this would require knowledge of a solution for a single crack arbitrarily oriented in an anisotropic solid (available only in numerical form). Sayers and Kachanov (1991) suggested a simple technique for bypassing this difficulty (and eliminating path dependence in DS) by using, as an input, the available results for randomly oriented and parallel cracks and tensorially transforming them to the arbitrary orientational distribution, through the crack density tensor a. In addition to avoiding the mentioned difficulty, this technique is in agreement with orthotropy of cracked solids. The technique automatically transfers some of the functional dependencies of the effective moduli on p from one orientation distribution into another. Such a transfer feature may be seen as a drawback of the scheme; on the other hand, it may be used to correct the undesirable features of the approximate models. Consider, for an illustration, the 2-D case. Recalling the expression (6.15) for the elastic potential Af of a solid with noninteracting cracks, we modify it by assuming that the coefficient at the simultaneous invariant ( a * a ) : ais not n/Eo, but some function of crack density: Af
= g(p)(a.a):a
(6.82)
where g(p) is to be determined from the known results for either randomly oriented or parallel cracks. We then apply this expression to the case of arbitrary orientation statistics. As an example, we consider DS,for which the Young's modulus for randomly oriented cracks E = EOe-'P is used as an input. This implies that Af = (nenP/E0)(aa) :a
(6.83)
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Applying, as a test, potential (6.83) to the case of parallel cracks (a = pelel) we obtain that Young’s modulus in the direction normal to cracks is E/Eo = (1
+ 2zp enp)-
(6.84)
as compared to the actual prediction of DS for parallel cracks (6.85)
The difference between predictions (6.84) and (6.85) (disappearing in the limit p + 0) is moderate: at p = 0.3, it is 0.17 versus 0.15; at p = 0.4, it is 0.10 versus 0.08. At larger densities, the ratio of these two values increases, but this is unimportant, since E becomes very small. Moreover, it is not clear which prediction is actually better, since at large p cracks experience increased mutual shielding and (6.85) may underestimate stiffness (as indeed indicated by the computer experiments of Section I); in this sense, the “transfer” feature turns out to be an advantage-it seems to improve the scheme. Results of DS for any other orientational distribution are obtained by simply introducing the corresponding a into the potential (6.83). In 3-D, the technique is similar and becomes fully identical to that for the 2-D case in a simplified version where the fourth-rank tensor’s contribution is neglected ((6.32), or (6.30) with the second term omitted). The version just presented is somewhat simpler than that of Sayers and Kachanov (1991), due to the fact that the term (a:a)t r a does not enter the potential (it would produce an undesirable effect on Poisson’s ratios, see the comment after formula (6.21); in the work of Sayers and Kachanov the coefficient at this term is actually very small and the results are close to the ones presented here). Being applied to SCS, this technique has the following feature: if the results for randomly oriented cracks are used as an input, then the “cut-off point is transferred to any other orientational distribution; if, on the other hand, the results for parallel cracks are used as an input, then the “cutoff point is eliminated for any orientational distribution. This “transfer” feature may be interpreted as a disadvantage; on the other hand, since the “cut-off’ point itself may be viewed as a disadvantage, the feature may be used for its elimination, by utilizing the results for parallel cracks as an input. g. A Generalized (“Three Phase”) Self-consistent Scheme This scheme was developed by Christensen and Lo (1979) for composite materials. Being applied to a 2-D cracked solid (Aboudi and Benveniste,
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1987), it assumes that the crack is, first, inserted into a circular inclusion of compliance Mo of the undamaged matrix and then this cracked inclusion is placed into the material with eflective compliance M. Thus, a “cushion” (stiffer than the effective matrix) is placed between a crack and the effective matrix. Such modelling is motivated by the fact that the immediate vicinity of a crack is stiffer than the homogenized effective material. Predictably, this scheme produces effective moduli that are stiffer than the ones of SCS; they are quite close to the ones of DS. One of the difficulties encountered in the framework of this model is that the avoidance of overlapping of the circular inclusions containing cracks limits the highest crack density that can be considered (about p = 0.3); it also interferes with the assumption of randomly located cracks. We note that calculations of this model would be difficult in the 3-D case. h. Other Modifcations of the Method of an Effective Matrix One can, probably, add various modifications or combinations of these schemes. For example, a differentialversion of the generalized self-consistent scheme can be constructed; the effective stiffness predicted by such a scheme will be higher than in the generalized self-consisent scheme. A “hybrid of the self-consistent and the differential schemes, which introduces a crack array into the effective matrix in two steps rather than one (as in SCS) or infinitely many (as in DS), was proposed by Cai and Horii (1992); the results are, predictably, in between those of the self-consistent and the differential schemes. 3. Method of an Eflective Field and the Method of Mori-Tanaka
The method of an eflectiuejeld (MEF) was first used, probably, in the wave propagation problems (Lax, 1951). As far as the effective properties of media with various inhomogeneities are concerned, the method was developed and applied to a number of diverse microstructures by Kanaun and Levin. (For references, see Levin, 1976, and Kanaun, 1980, 1983, where a detailed presentation and further references can be found, and their upcoming book on MEF.) The method places a representative inhomogeneity into an effective stress Jield, which, generally, does not coincide with the remotely applied one (this difference accounts for the interaction effect). The condition of selfconsistency is written for this effective field.
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The method possesses substantial advantages, compared to the methods of effective matrix. It takes into account the structure of the stress fields associated with inhomogeneitiesand, thus, incorporates the actual mechanics of interactions. For example, the shielding (stiffening)mode of interactions is not ignored: in contrast with the methods of an effective matrix, MEF does not predict that interactions always reduce effective stiffness. An important feature of MEF is that it takes the mutual positions of defects into account. Being applied to two test configurations-doubly periodic arrangements of parallel cracks-it distinguishes between the effective moduli of the rectangular and diamond arrangements of the same density and reproduces the exact solutions for these two cases very closely. An important result obtained by this method for the cracked medium is that, in the case of random locations of crack centers, the effective moduli coincide with the moduli given by the approximation of noninteracting cracks, at least, for two orientational distributions-randomly oriented cracks and parallel cracks (Kanaun, 1980, 1983, 1992). This result is in full agreement with the results of our computer experiments. It shows that randomness of mutual positions of cracks ensures the absence of “bias” toward either shielding or amplifying configurations. Another interesting result produced by MEF is that imposition of the constraint prohibiting centers of cracks to enter the circles drawn around neighboring cracks lowers noticeably the effective stiffness (i.e., creates a “bias” against shielding arrangements). This result fully explains the experimental data of Vavakin and Salganik (who interpreted their data as a confirmation of the differential scheme). The method allows various refinements, by letting the effective fields to be different for different inhomogeneities, and even allows each defect to be embedded into an inhomogeneous field. The simplest version of MEF, in which the effective stress field is assumed to be the same for all inhomogeneities and equal to the volume average over the matrix is known as MoriTanaka’s method (1973). Mori-Tanaka’s method (MTM) is often used in the mechanics of composite materials (see review of Christensen, 1990) and received a theoretical support in the recent work of Benveniste (1990). It was applied to a 2-D cracked solid by Benveniste (1986); note that a similar analysis for the 3-D cracked solid was done earlier by Kanaun (1980, 1983). In the case of randomly located cracks, predictions of MTM coincide with those of the more general method of effective field and, again, coincide with the predictions of the approximation of noninteracting cracks. The underlying
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physical reason for this result is actually clear: it is a direct consequence of the fact that introduction of cracks does not change the average stress in a solid, provided the boundary conditions are in tractions. Therefore, the homogeneous effective field of MTM (into which a representative crack is inserted) is the same as the volume average one and coincides with the remotely applied one. 4. Other Models
a. Models of the Second Order in Crack Density Such models were developed by Hudson (1980,1981,1986) and by Ju and Chen (1992) and Ju and Tseng (1992). They calculate effective stiffnesses and effectivecompliances, respectively, to the second order in crack density, in the cases of randomly oriented and parallel cracks. The results of Hudson were plotted by Sayers and Kachanov (1991). As seen from these plots, at low crack densities the results follow predictions of DS; at moderate-to-high densities (on the order of p = 0.2 for parallel cracks and p = 0.6 for randomly oriented cracks, in 3-D), they start to exhibit a physically meaningless behavior: the stiffness increases with increasing crack density. The results of Ju et al. are free of this inconsistency. However, it seems that the authors do not properly address the (difficult) problem, related to geometrical probabilities, of avoiding crack intersections; it is unclear whether this inaccuracy introduces errors of the same order of p z or not. Note that, in the case of parallel cracks, the coefficient at the p2-term differs from that calculated by Horii and Sahasakmontri (1990); the reason for the difference is not fully clear to the present author. In view of the results of Section I, as well as results of MEF and MTM, it appears that if locations of the crack centers are random, then there is no need in the second-order theories, since the approximation of noninteracting cracks remains accurate at high crack densities. Such theories can be useful (assuming the second-order term is properly constructed) in the case of nonrandom crack locations, since they are capable of incorporating, in an approximate way, the information on mutual crack positions. If models of this kind are to be used, it is desirable to estimate the interval of p where the second-order term actually provides an improvement over the linear term (i.e., represents a dominant correction, as compared to the impact of higher order terms).
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We suggest that some insight can be gained by examining the cases when the effective properties are known. One of them is the case of cracks with randomly located centers, for which the approximation of noninteracting cracks yields accurate results. For illustration, consider the 2-D case. If the crack orientations are random, then Young's modulus is E/Eo = (1 + np)-'. Expanding into powers of p and retaining terms to the second order yields
E/Eo = 1 - np
+ n2p2
Two comments on this expression should be made. First, at the crack densities higher than 0.12-0.15this formula starts to yield substantial errors. Second, starting from the crack density p = 0.16, this expression shows a physically meaningless behavior: the stiffness increases, as p increases. In the case of parallel cracks, expanding, in a similar way, the expression E/Eo = (1 + 2np)-' for the modulus in the direction normal to cracks yields: E/Eo = 1 - 2np + 47c2p2.This formula starts to yield unacceptable errors at p = 0.06-0.07 and shows an anomalous behaviour (the stiffness increases, as p increases) starting at p = 0.08. One may conclude, on the basis of this data, that the interval of p where the second-order term provides an improvement over the linear one can be expected to be quite narrow; in the 2-D case, the second-order theories may become unreliable at the point where reduction of the effective moduli is of the order of 25%. In the 3-D case, this interval may be wider, since the 3-D configurations are, generally, less sensitive to various simplifications. b. Model for an Array of Parallel Cracks of Very High Density
Wu and Chudnovsky (1990)considered an array of parallel cracks in the limit of very high density. They suggested a beam model for calculation of the Young's modulus in the direction normal to cracks and found upper and lower bounds based on the assumptions of clamped beams and simply supported beams, respectively.
c. On Bounding the Efective Moduli Bounding the effective moduli, used in the mechanics of composites, is usually understood as applicable to any possible arrangement of inclusions of the given density (the latter being the inclusions' volume fraction). In the case
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of cracks, bounds other than the trivial (stating that cracks do not increase stiffness)in this strong sense-as applicable to any possible arrangement of cracks of the given density p-cannot be established. As shown in Section K, crack arrays of infinitely high p that produce an infinitesimal impact on the effective moduli can be constructed, as well as arrays of infinitesimal p that reduce the stiffnessto zero. This may be related to inadequacy ofthe commonly used crack density purumeter p (or its tensorial generalization a) in the case of nonrandom locations of crack centers; see Sections K and L for discussion. Willis (1980)found a bound in a much milder sense: for parallel cracks with uncorrelated positions of crack centers (“well-stirred approximation,” in his terminology), the approximation of noninteracting cracks constitutes the upper bound for the effective Young’s modulus in the sense of bounding the ensemble average. I.
EXPERIMENTS ON DETERMINISTIC ARRAYS(SOLUTION INTERACTION PROBLEM FOR SAMPLE CRACKARRAYS)
COMPUTER
OF THE
Solutions for deterministic arrays of arbitrary geometry can be produced, by relatively simple means and with good accuracy, using the method of analysis of crack interactions outlined in Section 111. The method yields average tractions on cracks (ti) (as a solution of the system (3.11)). Introducing the proportionality relation ((2.13), in the 2-D case) between (ti) and the average CODs (b’) into the representation (6.5) for the elastic potential, we obtain the effective properties for a solid with the given arrangement of cracks. Such computer experiments on sample arrays can be used to verify various approximate schemes. To check the accuracy of the computational scheme, the procedure was supplemented, in several trial runs, by an alternating technique that enhances accuracy; it was found, however, to be unnecessary-the basic method was quite accurate. 1. Computer Experiments
Sample 2-D crack arrays consisting of 25 cracks were generated with the help of random number generator, as realizations of certain crack statistics. Two orientational statistics were examined: randomly oriented cracks and parallel cracks. For each, six crack densities were assumed p = 0.10; 0.15; 0.20; 0.25; 0.30 and 0.35(in 2-D, the densities of 0.30-0.35 can be considered high). Fifteen sample arrays were considered for each density.
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Locations of cracks within the representative area were random; i.e., positions of crack centers were uncorrelated. For parallel cracks, generation of such arrays was straightforward. For randomly oriented cracks, crack intersections had to be avoided; this was achieved by generating cracks successively and discarding the newly generated one if it intersected already existing cracks and generating it again. Although such a procedure, strictly speaking, violates the condition that crack centers are uncorrelated, we assume that it does not create errors of a systematic sign. Accuracy of the method was checked by supplementing it, in several trial runs, by the alternating technique (stress “feedback”) for improved accuracy, until three-digit accuracy for the effective moduli was reached, and examining the correction obtained. The correction was always insignificant (within 2373, confirming accuracy of the basic method. To be confident in accuracy of the basic method in all cases, the spacing between cracks was not allowed to be overly small: space was kept no smaller than 0.02 of the crack length in the case of randomly oriented cracks and, for parallel cracks with significant “overlap,” no smaller than 0.15 of the crack length. (Retaining configurations with arbitrarily small spacing between cracks would have required stress feedbacks, or some other accuracyenhancing technique, applied to a large number of crack arrayscomputationally, an overly intensive procedure). As discussed in Section G, in the case of parallel cracks this constraint may have introduced a slight “bias” against the amplification mode of interactions. In the case of randomly oriented cracks, each array was examined for isotropy: the effective Young’s modulus was calculated in two perpendicular directions and only those arrays were kept for which the difference between these two moduli was smaller than 2%; the Young’s modulus was then taken as the average over these two values. (We note that preferential orientations causing variation of E with direction up to 6-7% are usually not discernible by a “naked eye”). Figure 51 shows two of the generated arrays. As discussed in Section A, the representative area is assumed to constitute a part of a statistically representative, the average tractions along l- will differ insignifiA are assumed constant and equal to the remotely applied ones. This assumption is rigorously correct for noninteracting cracks, when cracks inside A do not experience any influence of those outside A. For interacting cracks, tractions fluctuate along r, but, if A is sufficiently large to be statistically representative, the average tractions along will differ insignificantly from the remotely applied ones (see the discussion of Section A). Substitution of the fluctuating along r tractions by their averages causes a
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FIG.51. Two of the sample crack arrays examined (randomly oriented, p = 0.25, and parallel, p = 0.30).
scatter of the results from one realization of the crack field statistics to another, but is not expected to produce errors of a systematicsign. The results show (Figs. 52 and 53) that, for samples containing 25 cracks, the scatter is relatively small.
2. Results
Figures 52 and 53 present results for the Young’s modulus for randomly oriented and parallel arrays. Vertical bars show scatter of the results from one sample to another. For randomly oriented cracks, the approximation of noninteracting cracks provides surprisingly accurate results, well into the domain of strong
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P FIG.52. Effective Young’s modulus, randomly oriented cracks: results for sample arrays vs. predictionsof the approximation of noninteractingcracks and the self-consistent and differential schemes.
interactions where this approximation is usually considered inapplicable. For parallel cracks, the approximation of noninteracting cracks also provides good results. However, there is a slight but distinguishable tendency for the stiffening overall effect of interactions,indicating a slight dominance of
P FIG. 53. Effective Young’s modulus, parallel cracks: results for sample arrays vs. the approximation of noninteracting cracks and the differential scheme.
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the shielding mode of interactions. We hypothesize that this slight “bias” in favor of shielding was caused by the prohibition for the crack tips to be overly close to each other. We also calculated the shear modulus for parallel crack arrays (for two arrays, with p = 0.1 5 and 0.20). It differed from the predictions for noninteracting cracks by less than 1.5%.
3. Preservation of Orthotropy for Interacting Cracks We examined the deviation from orthotropy caused by interactions (as discussed in Section B, for noninteracting cracks orthotropy coaxial to the crack density tensor a is a rigorous result in 2-D). We solved the interaction problem for the array consisting of two families of parallel cracks of equal density inclined at 30” to each other. At the overall density p =0.24 (significant interaction) the deviation from orthotropy produced by interactions was found to be very small: in the principal coordinate system of a the nonorthotropic compliances A M 1 1 2 and A M z z l 2 were on the order of 10-3-10-4 of the orthotropic compliances; i.e., indistinguishable. This means that not only the values of the moduli predicted by the approximation of noninteracting cracks remain accurate at high densities, but that the symmetry of the effective properties (orthotropy) predicted by this approximation is retained as well. This implies that characterization of crack arrays by the crack density sensor and the approximation of noninteracting cracks remain fully adequate at high crack densities.
4. Discussion
The underlying reason for accuracy of the approximation of noninteracting cracks is not that the interactions can be neglected, but that the competing effects of stress shielding and stress amplification balance each other (provided locations of crack centers are random). In the language of stress superpositions(Fig. 8), the additional tractions Atji will be of both amplifying and shielding signs on different cracks; on average, their impacts will cancel each other. We emphasize that this conclusion assumes the absence of “bias” in crack statistics toward either amplifying or shielding arrangements. Otherwise(in “ordered” arrays, for example) the impact of interactions can be very large, in the directions of both “softening” and “stiffening.” The reported results are for the 2-D configurations. The approximation of noninteracting cracks is expected to remain accurate in 3-D as well, due to
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the same mechanism of cancellation of shielding and amplification effects. Moreover, we expect the scatter of the results to be smaller, since interactions are, generally, weaker in 3-D (Section 1V.A.).
5. Remarks on Naturally Occurring Crack Arrays The conclusion on accuracy of the approximation of noninteracting cracks is based on the assumption of randomness of crack locations. Such randomness may not always be the case in naturally occurring crack systems. A crack may enhance nucleation of new cracks in its amplification zone and suppress it in the shielding zone, thus creating an amplification “bias.” This may be relevant for the stages of microcracking when localization is about to occur. As an opposite example, the observations of Han and Suresh (1989) on microcracking in ceramics seem to indicate (although not fully conclusively) some shielding “bias” in most (but not all) microcracking geometries (Suresh and Kachanov, to be published). In such cases, the results of this section should be used with caution.
J. COMPUTER EXPERIMENTS ON CRACK ARRAYSIN
AN
ANISOTROPICMATRIX
This section summarizes the results of Mauge (1993) and Mauge and Kachanov (1992) on the effective elastic properties of a 2-D orthotropic matrix with arbitrarily oriented interacting cracks. This problem was solved for a number of sample crack arrays (similarly to the case of isotropic matrix). A strong anisotropy of the matrix was assumed: the ratio of Young’s moduli E,/E, in two principal directions of orthotropy was 10. Two statistics of crack orientations were considered: random orientations and parallel cracks. In the latter case, the cracks were parallel to one of the principal directions of orthotropy. Crack density was increased in steps, from 0.10 to 0.25. The locations of crack centers were random. Fluctuation of the results from one sample of a given crack statistics to another was somewhat higher than in the case of isotropic matrix (this is explained by the fact that the anisotropy of the matrix, which was quite strong in our experiments, enhances the interaction effects and thus increases the scatter), but remained, in r..ost cases, within 5%. For each array, the effective properties were found (1) in the approximation of noninteracting cracks and (2) solving the interaction problem for this
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particular array. The conclusions were similar to those for the isotropic matrix. Namely, in the case of randomly oriented cracks, the interactions did not change the effective stiffness (as compared to predictions of the approximation of noninteracting cracks). In the case of parallel cracks, interactions produced a slight stiffening effect. The methods of an effective matrix, in particular, the self-consistent scheme, substantially underestimate the effective stiffness. The underlying reason is that the impact of interactions on a given crack is simulated by a reduced stiffness of the surrounding material; therefore, interactions are predicted to always reduce stiffness. This ignores the shielding mode of interactions (that balances the amplifying one) and thus distorts the actual mechanics of interactions. On the other hand, our results are in full agreement with the predictions of the method of an effectivefield (which takes the statistics of crack centers into account) for the case of random locations of crack centers, and with the simplest version of this method-Mori-Tanaka’s scheme.
K. ON BOUNDINGTHE EFFECTIVE MODULI Bounding the moduli, widely used in the mechanics of two components materials, degenerates for cracked solids: neither lower nor upper bounds (other than trivial ones stating that cracks do not increase the stiffness) can be constructed, at least, in the sense of bounding that is valid for any particular sample of a given crack statistics. Bounding in a much milder sense-applicable to the ensemble average-was constructed by Willis (1980) for parallel cracks with uncorrelated locations of crack centers (upper bound for the effective stiffness). We note that the opposite statement-that the approximation of noninteracting cracks provides the upper bound for any crack statistics-was made by Gottesman et al. (1980). Such a general conclusion (implying that interactions always reduce stiffness, compared to the approximation of noninteracting cracks) seems to be an overstatement: in statistics biased toward shielding arrangements (stacked cracks, for example), interactions produce the srifening effect. To demonstrate the impossibility of bounds that are valid for any realization of a given orientational distribution, we construct two types of configurations: A. One with crack density p
+ 00 that produces a vanishingly small effect; B. One with p -+ 0 that, nevertheless, makes the stiffness vanishingly small.
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We restrict attention to 2-D geometries and the cases of isotropic and transversely isotropic effective properties. In the case of isotropic effective properties, we first establish a fully symmetric hexagonal mesh (H is a side of a cell) and then consider the following two configurations: A. Radial crack patterns contained within circles of radius r are placed at vertices of the mesh. Making radial patterns arbitrarily dense, while keeping the circle radii sufficiently small (r << H) produces a configuration where an arbitrarily large crack density p produces a vanishingly small impact on the effective moduli (Fig. 54). B. Three families of periodically spaced collinear cracks (crack length is 21 and ligaments between cracks are A) are placed along the mesh (Fig. 55). Both 21 and A are then made progressively smaller, in such a way that A decreases faster than 21. Namely, assuming that H contains N periods: N ( 2 l + A) = H , we set 21 = H / ( N + 1) and A = H / ( N 2 + N). Then considering an elementary cell and calculating the crack density, we obtain p = (4&’N/(N 1)2. We let N + 00, then the total relative length of ligaments N A / H = 1/(N + 1) -+ 0 and, therefore, the efectiue stifness -+ 0. At the same time, p -+0. We further note that, since p and NA/H are independent of H , the parameter H can also be set to -0 as N increases (say, H = H , / N ) . This creates a fine “mist” of infinitesimal cracks that is dense everywhere while possessing the property that the effective Young’s modulus E -+ 0 and, at the same time, p 0. In the case of transversely isotropic effective properties, configuration of type A is constructed as a doubly periodic rectangular pattern of parallel
+
--f
FIG.54. Crack array of infinite crack density, p effective moduli.
+ 00,
producing an infinitesimalimpact on the
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FIG.55. Crack array of infinitesimal crack density, p + 0 that reduces the effective stiffness to zero. Note that this effect is achieved withour violating a coherence of the representative area. The array is dense everywhere (since H + 0).
cracks; spacing between different columns is kept sufficiently wide, while the density of packing within each column + 00. Configuration of type B is constructed as the same pattern, with vanishingly small ligaments between collinear cracks and widely separated rows. We emphasize that the statement on the impossibility of bounds assumes that any possible arrangement of crack centers is allowed. If restrictions are imposed on the statistics of crack centers then nontrivial bounding may be possible. The impossibility of bounding seems to be related to inadequacy of the commonly used crack density parameter p (or its tensorial generalization a) in the case of nonrandom locations of crack centers, due to its insensitivity to mutual positions of cracks. This issue is discussed in the next section.
L. ALTERNATIVE CRACK DENSITY PARAMETER 1. Remarks on the Conventional (Crack Location Insensitive)
Crack Density Parameters
Both the scalar crack density p and its tensorial generalization a are insensitive to mutual positions of cracks. This is fully adequate for either noninteracting cracks (in which case the mutual positions of cracks do not matter) or for interacting cracks provided locations of the crack centers are random. This, however, becomes inadequate for interacting cracks with nonrandom mutual positions, when crack statistics is “biased” toward either shielding or amplifying arrangements. For example, the conventional crack density parameter does not reflect the fact that introduction of a large number of cracks into a shielded zone produces only a small effect.
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This inadequacy manifests itself in the impossibiiity of bounding in terms of p. Another manifestation of inadequacy of p is nonequivalence of the assumption of noninteracting cracks to the assumption of small crack density (as demonstrated by the example of an array of infinitesimal p that reduces the stiffness to zero)-an undesirable feature for a physical theory (and absent in the theory of composites, where volume fractions are used as density parameters). 2. Alternative Crack Density Parameter An alternative parameter that is sensitive to crack positions and, in the case of noninteracting cracks, reduces to the crack density tensor a (or the scalar density p for randomly oriented cracks) can be constructed as follows. We restrict our consideration to the 2-D case; extension to 3-D is straightforward. We recall that the change Af in the elastic potential due to cracks has the form Af = (1/A)
c
(n.a.(b))’l’
(6.86)
i
Making use of the fact that the average COD is approximately proportional of the average traction: ( b ’ ) = ( n l i / E o ) ( t i ) and finding the latter in terms of the transmission factors, we obtain Af
= (n/E0)a : w :u
(6.87)
where the fourth-rank tensor (6.88) which emerges in the elastic potential, accounts not only for the crack sizes and orientations but for the mutual positions as well and can be used as a crack density parameter. In the case of noninteracting cracks, we recover characterization of a crack array in terms of (crack location insensitive)crack density tensor a.Indeed, in this case Roo = 6”I so that U : W : U = @-@:a and Af reduces to the expression (6.15)in terms of a. Thus, w is a generalization of the crack density tensor that accounts for crack interactions and is sensitive to the mutual crack positions. We note that smallness of w means smallness of a sum of products (a-type terms) x (R-terms); the a-terms, as seen from their definition (Section 111),
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characterize average tractions on cracks as fractions of the tractions in the absence of interactions. Thus, the R-terms are small in the case of shielding and large in the case of amplification. Therefore, if o (rather than p or a) is used as a parameter, “paradoxical” situations when infinitesimal crack density reduces the effective stiffness to zero, or an infinite density producing only an infinitesimal effect are eliminated. Indeed, for a dense radial pattern of Fig. 54 (or densely packed stacks of parallel cracks) the conventional density p may be large, but o is small, since the R-terms are small (due to a strong shielding). On the other hand, for the hexagonal pattern of Fig. 55 the parameter o is large (although the conventional density p --* 0) due to a strong amplifying effect of interactions. Another undesirable feature-nonequivalence of the assumptions of noninteracting cracks and of small crack density-is also eliminated, since o is small ifand only ifthe impact of cracks on the effective properties is small. 3. Relevance of o for Randomly Located Cracks and “Ordered” Arrays
Generally, the evaluation of o requires solving the interaction problem (unavoidable, though, if one attempts to incorporate the information on mutual positions). For randomly located cracks, however, the approximation of noninteracting cracks yields good results. Therefore, the description in terms of p or a remains adequate, and switching to the description in terms of o is unnecessary. For “ordered” arrays that may have “bias” toward shielding or amplification, deviations from the predictions of the approximation of noninteracting cracks may be very large. Then, the use of o appears to be important. At the same time, evaluation of o for such arrays seems a more realistic task, since a number of available solutions (as well as approximate results produced by the method of effective field) can be utilized. VII. On Correlations between Fracturing and Change of Effective Elastic Moduli. Some Comments on Brittle-Elastic Damage Mechanics It is argued that, contrary to the spirit of many damage models, there is no stable quantitative correlation between fracturing of a brittle microcracking solid and the change of its effective elastic moduli (with the obvious exception of the immediate vicinity of the fracture point, when the effective stiffness
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abruptly drops to zero). Therefore, the change of effective moduli may not always serve as a reliable indicator of the progression of a microcracking solid toward fracture. Physically, the absence of such a correlation is explained by the fact that the fracture-related quantities (like SIFs) are determined by local details of the microcrack field geometry whereas the effective moduli are volume average quantities, relatively insensitive to such fluctuations. This discussion is relevant for a number of damage models that are aimed at a description of thefracturing of a brittle microcracking solid, but in fact deal with its effective elastic moduli; sometimes damage is even deJined as the change in elastic moduli (see, for example, Wang, Chim, and Suemasa, 1986; for reviews, see Krajcinovic, 1989, and Maugin, 1992, Chapter 10). The underlying idea-that progression toward failure is uniquely correlated with the change of effective moduli-may seem intuitively reasonable; it appears particularly attractive because the moduli can be easily measured. It seems that, for such correlations to be useful, they should be formulated in quantitative terms. An objection may be raised that, as is well known, a small crack in a brittle material has a very small impact on the effective stiffness but may drastically reduce the tensile strength. One may argue, however, that, after a certain initial set of defects has been nucleated and started to grow, the quantitative correlation can be established. This idea is examined here from several points of view. A. MICROCRACKING AND
THE
CHANGE OF ELASTIC MODULI
1 . “Paradoxical” Example
To demonstrate that this idea is far from obvious, we start with a simple example when the correlation between SIFs at the crack tips (fracture-related quantities) and the effective moduli is “paradoxical.” Consider a plate containing stacks of parallel cracks (Fig. 56, solid lines) and assume that additional cracks are introduced in between the neighboring cracks (dashed lines). The introduction of new cracks reduces elastic stiffness in the direction normal to cracks-the material elastically softens. However, the SIFs decrease (due to increased shielding)-the new cracks have a strengthening effect.
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FIG.56. Introduction of new cracks (dashed lines) reduces the SIFs and, at the same time, elastically “softens” the material.
2. Random Crack Arrays It may be argued that the preceding example does not represent any realistic crack statistics (although such parallel crack patterns do occur in rocks and certain composites) and that, for more random arrays, progression toward failure can be quantitatively correlated with the reduction of elastic stiffness. To demonstrate that such a correlation is not obvious, we did the following computer experiments. A number of two-dimensional crack arrays containing randomly oriented cracks of the same length were generated and subjected to a uniaxial loading; statistics of crack centers was also random (subject to the constraint that cracks do not intersect).For each sample array, using the method described in Section 111, we calculated 1. the effective Young’s modulus Ecff, 2. the maximal, among all crack tips, value of Kf + K i . (Strictly speaking, this parameter determines initiation of crack propagation only for rectilinear crack extension; if both K Iand KIIare present, a certain other combination of K , , K,, is a relevant parameter. Typically, however, this combination does not differ much from K f + K i . ) We found that, whereas Eeff was quite stable from one statistical sample Ki)Juctuated signiftcantly, reaching higher values in to another, max(K: larger samples where the probability of occurence of closely spaced cracks is higher. Thus, Eeff stabilizes as the size of the crack sample increases (usually, at the number of cracks on the order of 10) whereas the fracture-related parameter max(K: + K i ) is statistically unstable and increases as the size of the crack sample increases.
+
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415
3. Strong and Weak Interactions. Clustering of Cracks Interactions that produce a strong impact on the SIFs may be weak in their impact on the effective moduli (see the discussion of the case of small transmission factors in Section 111). A closely related issue is that the clustering of cracks strongly affects the SIFs. Figure 57 provides an illustration: max(K: + K i ) is, typically, substantially higher in configurations of the type 57b than in ones of the type 57a. On the other hand, sensitivity of the effective elastic moduli to clustering is low, i.e., the difference in moduli of the configurations 57a and 57b is, generally, small (provided the overall crack density is the same). Thus, monitoring the change in effective elastic constants may not necessarily detect the onset of strong crack interactions and clustering of defects, events that are crucially important from the point of view of progression toward failure.
Fro. 57. Clustering of cracks strongly affects SIFs (fracture-related quantities) and leaves the effective moduli almost unchanged.
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Mark Kachanou
4. Efect of the Crack Shape Consider a 3-D elastic solid containing elliptical cracks. As is well known, the profiles of SIFs along the crack edges and their maximal values depend on the aspect ratios of the ellipses. At the same time, the shapes of cracks have very little impact on the overall effective elastic stiffness (see the discussion of noncircular cracks in Section VI).
5. Crack-Microcrack Interaction Problem In the crack-microcracks problem (discussed in Section V), the absence of correlation between the effective moduli and the fracture-related properties is apparent: microcracks located in the short-range interaction zone (several microcracks closest to the main crack tip) produce a dominant impact on the main crack tip. Due to a high sensitivity of the interaction effect to the exact positions of closest microcracks, the impact of microcracks on the main crack SIFs fluctuates significantly and even qualitatively (from shielding to amplification) from one sample of the microcrack statistics to another. This means that there is no statistically stable effect ofstress shielding, at least, if the microcrack locations are random. On the contrary, being conservative, one may conclude that the overall effect of interactions is the one of enhancement, particularly in 3-D where local peaks of SIFs along the crack front are almost always present. On the other hand, replacing the microcracked region by an “effective” material of reduced stiffness would produce a different result-that interactions result in shielding.
6. Cases when Progression Toward Fracture Can be Monitored by the Change of Effective Moduli The basic underlying reason for the absence of the one-to-one correspondence between fracturing and change of the effective elastic moduli is that the fracture-related quantities (like SIFs) are determined by local fluctuations of the microdefect field geometry whereas the effective elastic properties are uolume auerage quantities, relatively insensitive to such details. Therefore, when the evolution of the defect’s population follows a more or less deterministic reproducible pattern, the progression toward fracture may indeed be monitored by reduction of the effective stiffness. Such situations occur in certain laminated composites where such patterns were called characteristic damage states by Reifsnider and Stinchcomb, 1983. Note, however, that correlations were not straightforward even in these cases. As their data shows, (1) correlation between the fraction T of the remaining lifetime of a specimen and its effective stiffness E held only for a given
Elastic Solids with Many Cracks and Related Problems
417
arrangement of layers: for different ply orientations (and, therefore, different crack patterns), the E-T curve was different; (2) of the three E - T curves, given by the authors, one remained almost p a t (stiffness was almost insensitive to damage accumulation) and another one was changing rather weakly during most of the lifetime. A possible explanation is that, during most of the lifetime, fracturing processes did not raise the overall microcrack density to the levels at which stiffness reduction becomes significant. B.
ELASTICPOTENTIALS IN TERMS OF DAMAGE PARAMETERS
O N THE CONSTRUCTION OF
1. Elastic Potentials and Efective Properties One of the approaches of brittle-elastic damage mechanics is to postulate a certain form of elastic potential in strains-strain energy density-that contains an additional damage parameter D (scalar or tensorial): W = W(E,D);see, for example, Lemaitre and Chaboche (1978); for reviews with further references, see Maugin (1992) and Krajcinovic (1989). Such formulations are, apparently, motivated by the successes of this approach in the theory of creep, where the notion of damage first appeared, in the pioneering work of L. Kachanov (senior) (1958). However, a straightforward application of this idea to the brittle-elastic case may lead to problems. We first note that, if a model for the efective elastic properties is available, such a model fully specijes W(E,D). If the damage is formed by microcracks with randomly located centers, D is the crack density tensor a,reducing to the scalar crack density p for randomly oriented cracks (if the locations of crack centers are not random, their statistics can be incorporated into description through the damage parameter o,see (6.88)). In this case the complementary energy density f = f ( q a ) was constructed in Section VI; it may be Legendre transformed to strain energy density W When damage is formed by cavities, or by a mixture of cracks and cavities, several models for the effective elastic properties are also available (see, for example, Kachanov, 1993), so that W can be fully specified in this case as well. It appears, therefore, that the main motivation for postulating (rather than deriving) W is that the geometry of damage elements is not known, with the hope that the uncertainties can be taken care of by an appropriately chosen form of W We note that any such formulation, regardless of other applications, necessarily implies a certain model for the efective elastic moduli (taking derivative a W/ac at fixed D yields elastic stress-strain relations, with
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Mark Kachanov
moduli-functions of D). Therefore, such constructions should be consistent with the known results on the effective moduli, when damage is formed either by microcracks or cavities. A particular attention should be paid to the case of small D,when the results are rigorous and well known. Here, we comment on several related developments in damage mechanics.
2. Choice of Damage Parameters In some models, the damage parameter is taken as characterizing the “area loss” due to defects (see Murakami and Ohno, 1981; Betten, 1982 gave a general discussion of tensorial parameters of this kind). Such definition of damage is motivated by the aforementioned work of L. Kachanov, where damage accumulation under creep conditions was considered. However, damage parameters that are appropriate in the theory of creep may not be adequate in the theory of effective elastic properties, since microcracking affects creep and the effective moduli in entirely different ways. The process of creep is enhanced by damage due to tensile stress concentrations in zones near defects (these zones can be substantially larger in a creeping solid than in a linear elastic solid). In agreement with this mechanism, the damage parameter in L. Kachanov’s model is introduced in such a way that it couples the processes of damage development and creep and describes a self-accelerating nature of creep accompanied by damage. In the problem of effective elastic properties, on the other hand, stress concentrations are irrelevant-it is the average stress that enters formulation of the problem (and it is not affected by stress concentrations; zones of elevated stress are balanced by shielded zones of reduced stress, see Section V1.A). Therefore, damage parameters in the theory of creep and in the theory of effective elastic properties may not coincide. When damage is formed by microcracks, the proper damage parameter for the problem of effective elastic properties is dictated by the elastic potential: it is the crack density tensor a. If, instead, a diferent damage parameter is used, this may lead to inconsistencies. (For example, the above-mentioned tensor describing cross-sectional area losses seems to possess the property that contribution of a given crack into such a tensor is proportional to the crack area; i.e., to its linear size squared. On the other hand, an individual crack’s contribution to p or a is proportional, in the 3-D case, to its size cubed. This means that use of the area loss-related tensors in the elastic potential would distort relative contributions of individual cracks into the overall elastic properties. In the case of non-randomly oriented
Elastic Solids with Many Cracks and Related Problems
419
cracks, it would lead to incorrect predictions of the overall anisotropy.) Similarly, if damage is formed by spherical cavities, their relative volume is the parameter relevant for the effective elastic properties; altering it may lead to inconsistencies.
3. On Models of the W = (1 - D)Wo n p e It has been suggested in a number of papers (see Lemaitre and Chaboche, 1978, for example) that the elastic potential in strains-the strain energy density of a damaged solid-can be written in the form W(E,D ) = (1 - D)WO(E)
(7.1)
where D is some scalar measure of damage and Wo is the potential without damage. (This form is, obviously, limited to isotropic damage; an application of it to parallel cracks, for instance, yields equal stiffnesses in the directions normal and parallel to cracks). The model (7.1) has some inconsistencies.One of them was pointed out by Ju (1990):the Poisson’s ratio is unaffected by damage-a wrong prediction. (We note two special cases when Poisson’s ratio is, indeed, unaffected by damage. In the first case, damage is formed by spherical cavities and Poisson’s ratio vo of the undamaged matrix is exactly 0.2; then the effective v is unaffected by damage, at least, in the approximation of noninteracting cavities. In the second case, a mixture of cracks and spherical cavities is present and vo < 0.2; since cavities increase v whereas cracks reduce it, one can design a special ratio (dependent on yo) of the crack density to the density of cavities such that it enforces constancy of v.) We add that, in the simplest case of a small D,the model’s predictions of the effective moduli are not consistent with the rigorous results for noninteracting defects of small density, whether these defects are cracks or cavities and D is understood as a usual measure of their density (and the only one acceptable in the elastic potential). Indeed, in the 3-D case of randomly oriented circular microcracks of small density (the case of noncircular cracks can be reduced, with high accuracy, to the case of equivalent circular cracks, see Section VI.B.8) W can be written exactly, as a sum W = Wo+ A W of the potential Wo without cracks and the change A W due to cracks, where WOW =
Mark Kachanov
420
and AW(E)=-
8E0( 1 - v0M1- v0/5)
+
9(1 - ~0/2)(1 vO)
vo(vg - 16v0 EiiE’i
+ 19)
+ 1q1 - v0/5)( 1 - 2v0)’
(Ekk)2]
(7.3)
For the representation Wo + AW
= (1
- D)Wo
(7.4)
to be possible, the coefficients at (ckk)2 in (7.2) and (7.3) should be identical or, at least, close (in which case, D would, of course, be nothing but p , to within a multiplier). This is not the case (unless vo = 0): for vo = 1/3 and vo = 1/4, for example, these coefficients differ 4.4 times and 2.5 times, respectively. A similar problem arises in the case of a small concentration of a mixture of cracks and cavities (although, formally speaking, a special ratio of the crack density to the density of cavities can be designed, such that (7.4) is enforced to be correct, provided vo < 0.2).
4. Damage as an Adjustable Parameter
Models of the type of (7.1) can, of course, be interpreted in a different way: D can be understood as simply an adjustable parameter, defined in such a way as to enforce (7.1) to be correct; i.e., (7.1) serves as a dejnition of damage. In this case, D may not be interpreted in terms of microstructural quantities. Since the geometry of the damage elements may not be known, there is a certain motivation for such an approach. We note that, if such an interpretation is assumed, then Ju’s argument on constancy of Poisson’s ratio still remains valid, indicating that W cannot be written in the form (7.1). Thus, even if D is understood as simply an adjustable parameter, it still cannot be defined in such a way as to make (7.1) correct. It has been suggested to modify (7.1) by “upgrading” scalars 1 and D to fourth-rank isotropic tensors (Ju, 1990). Then the number of adjustable parameters increases from one to two and Poisson’s ratio does not have to be constant. I t seems, however, that a more straightforward way to introduce two adjustable parameters in the elastic potential is to directly treat two efective moduli as such parameters. (General problems that arise from association of fracturing with changes in moduli, will, of course, remain, see Section VILA).
Elastic Solids with Many Cracks and Related Problems
42 1
VIII. Effective Elastic Properties of Solids with Elliptical Holes This Section analyzes the effective moduli of a 2-D elastic solid with elliptical holes. The 2-D modeling is relevant for materials with porous structures that are approximately “tubular.” Besides, results in 2-D allow direct verification by computer experiments on sample arrays of holes, by finite elements, spring force modeling or other techniques (see Section VI and Mauge and Kachanov (1992) for numerical experiments on cracks and Day et al. (1992) for numerical experiments on circular holes). An extensive study of 2-D pores of various shapes was done by Zimmerman (1986, 1991). His work, however, focuses on the compressibility of holes (change of the hole area under the hydrostatic stress). As far as derivation of the full set of efective moduli is concerned, the works of Thorpe and Sen (1985) and Zhao and Weng (1990) should be mentioned. Thorpe and Sen gave results for randomly oriented holes in the approximation of the self-consistent model (which may substantially overestimate the eflective compliance, particularly for narrow, crack-like holes; see Section VI). Zhao and Weng considered composites with elliptical inclusions of “tubular” structure (plane strain) for two cases of orientational distribution (randomly oriented and parallel inclusions); results for holes can be obtained from their results by letting the inclusions’ stiffness go to zero. Our analysis represents a further progress in these studies. A unijied description covering both holes and cracks (or their mixture) is developed. It yields a full set of (generally, anisotropic) moduli for an arbitrary orientational distribution of holes of various aspect ratios. The approach eliminates degeneracies in the models describing the moduli in terms of the area fraction of holes, which arise in the limiting case of cracks (unless special care is taken in the transition to the limit). The key point in our analysis is the identification of proper parameters of density of holes. The choice of such parameters is not arbitrary, but is dictated by the structure of the elastic potential. It is shown that two density parameters are needed: one scalar, the porosity p (which vanishes in the case of cracks), and one second-rank tensor, the “holes density tensor” 0 (which reduces to the crack density tensor in the case of cracks and is proportional to the unit tensor in the case of circular holes). Even in the case of random orientational distribution of holes (isotropy), the effective moduli cannot be characterized in terms of porosity alone-a second parameter, “eccentricity,” is needed.
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The effective moduli are derived in two approximations: (1) the approximation of noninteracting holes, and (2) Mori-Tanaka’s scheme, i.e., by assuming that a representative hole is placed into the aueraye stress in the matrix (which, in the case of holes, can be found exactly). Note that this scheme received theoretical support in the work of Tandon and Weng (1986), Weng (1992), and Benveniste (1990). Two extreme cases-cracks and circular holes-are covered as special cases. For cracks, the hole density tensor fl reduces to the crack density tensor Q (see Section VI) and the known results are recovered. For circular holes, the moduli are expressed in terms of porosity, which remains the only density parameter. A mixture of cracks and holes is analyzed. It is shown that the interaction effect between cracks and holes is “asymmetric”: holes afecr cracks (by changing the average stress environment for cracks) but cracks do not afect holes (since the presence of cracks does not change the average stress, provided the boundary conditions are in tractions). In the case of nonrandom orientations of holes, the effective moduli are anisotropic. The anisotropy is explicitly expressed in terms of the hole density tensor fl and p . The analysis is done in the framework of linear elasticity; nonlinear effects caused by closing of holes by compressive stresses (stiffness increasing with compression, different moduli in compression and tension, etc.) are not covered. This restriction translates, in the case of compressive loading, into a restriction on the magnitude of stresses at which the derived results are valid (see Walsh (1965) and Zimmerman (1991) for estimates of stresses needed to close a hole and produce nonlinear effects). A. ONE ELLIPTICAL HOLEIN
A UNIFORM
STRESS FIELD
We consider a 2-D linear elastic solid containing an elliptical hole with axes 2a and 2b. The macroscopic strain E is given by the usual relation E = M’:u
+ A&
(8.1)
where Mo is the compliance tensor of the matrix. As follows from the divergence theorem (see a footnote in Hill (1963)), the additional strain A& due to the hole is expressed in terms of displacements u of the hole boundary R
Elastic Solids with Many Cracks and Related Problems
423
(n is a unit outward normal to r and A is the total, including the hole, reference area). Consider a uniaxial loading P inclined at an angle a to the axis 2a of the ellipse (Fig. 58). The intergral (2.2) can be calculated from the known solution of elasticity (see Savin (1970) or Muskhelishvili (1963)) or, alternatively, from Eshelby's results for an ellipsoidal cavity in the limiting case of an ellipsoidal cyclinder. Either calculation yields: P 1
=--
Eo A A ~ 2 2=
nb[b
+ (a + b)cos 2al;
P 1
- - na[a - (a + b)cos 2a];
(8.3)
Eo A
P 17t Ac12= - - (a + b)'sin2a E, A 2
The plane strain conditions are assumed hereafter; therefore, Young's modulus E , entering (8.3)should be understood as Eb(1 - v o )- 1 where Eb, 12
vb are the usual 3-D matrix constants. Other plane strain matrix constants used in the text to follow are expressed in terms of the 3-D constants as vo = vb(1 - vb)-', bulk modulus K O = (1/2)Eb(1 - 2vb)-'(1 + vb)-' (for plane stress, E , = Ebv, = vb, K O = (1/2)Eb(l - vb)-'). Note that K O = (1/2)E0(1- vo)- for both plane strain and plane stress. The following observations on strains Acij due to a hole can be made. produced by a hydrostatic loading The volumetric strain Acl + o I I= 0 2 2= P, is
,+
AE,
P I
= --
Eo A
2n(a2 + b2)
J J FIG.58. Elliptical hole subjected to an arbitrarily oriented uniaxial tension.
Mark Kachanov
424
This formula recovers the result of Walsh et al. (1965). We further note that the geometrical parameter a’ + b’ entering (8.4) can be represented as
a’
+ 6’
= 2ab
+ (a - b)’
(8.5)
This simple relation indicates that, among holes of the same area nab, the more “elongated ones have higher compressibility. (Zimmerman (1986) proved this statement for a wider class of hole shapes, showing that the circular hole is the least compliant one). We also remark that the ellipse’s perimeter (given by an elliptic integral) can be approximated by a simple expression 4.22 (a’ + b’)’’’ with the accuracy better than 5.5% for all aspect ratios; therefore, the ellipse’s compressibility can be regarded as approximately determined by its perimeter. Figure 59 shows the compressibility of a hole as a function of the aspect ratio b/a. It is interesting to examine compressibility near the limiting cases of a crack and of a circular hole. At b = 0 (crack), the slope is horizontal, so that a slight “inflation” of a crack leaves the compressibility almost unaffected; the difference is only 1% at bja = 0.1 and 5% at bfa = 0.22. A similar observation in the 3-D case (spheroidal pores) was made by Zimmerman (1986, 1991). (Note that this insensitivity to a small-
2.01
’
’
’
’
’
’
’
’
’
’
circular hole of the same area
I n (v
(d
1.o
0.0
b/a FIG.59. Compressibilityof an elliptical hole as a function of the aspect ratio. Compressibility of the circular hole of equal area is shown for comparison.
Elastic Solids with Many Cracks and Related Problems
425
to-moderate “fin flation” aggravates the problem of degeneracy arising, in the limiting case of cracks, in the models describing the moduli in terms of porosity.) In the opposite extreme of shapes close to circular, the slope of the compressibility curve coincides with the one for a circular hole of equal area. The compressibility can be approximated by that of a circular hole of the equal area with the accuracy better than 5% for b/a > 0.72 (moderately noncircular hole). due to a hole subjected to a uniaxial The volumetric strain A EI ~+ loading P inclined at an angle c1 to the axis 2a is Ac,
P 1 P I n + Ac22 = - x(a2sin2cc + b2 cos*a) = - - - w2 Eo A Eo A 2
(8.6)
where w is the width of the hole’s profile, as seen in the direction of loading. This implies that various ellipses shown in Fig. 60a produce the same volumetric strain, when loaded in the direction shown. The explanation is that the differences in the normal strain in the direction of loading are exactly balanced by the differences in (negative) lateral strain. Therefore, uniaxial loading of a solid with many noninteracting elliptical holes produces a volumetric strain that depends only on the sum X w;, where w,, is the width of the k t h hole profile in the direction of loading (Fig. 60b). produced by The ratio of the normal “compliance” of a hole (strain
due to a hole is determinedby w z ; (b) Volumetric FIG.60. (a) Volumetric strain A c , , + strain produced by noninteracting holes is determined by the sum Z w:.
Mark Kachanov
426
the uniaxial loading produced by o ,),
,
02,of
unit intensity) to the shear “compliance” (BeI2
be,, - (a -Ae2,
+ b)’ + b)
(8.7)
a(2a
is quite sensitive to the eccentricity of the hole: it changes from 1/2 for a crack to 4/3 for a circular hole. This is because the normal compliance of a hole is only moderately larger than the one of a crack coinciding with the ellipse’s major axis (1.5 times, for a circular hole), whereas the shear compliance is substantially larger (4 times, for a circular hole). For a general loading 0, the strain A& can, due to linear elasticity, be expressed as A& = H : a
(8.8)
where the fourth-rank tensor H can be called the hole compliance tensor. Using the result (8.3), H is found as follows: H
= -- n
Eo l A1
a(2a + b)nnnn + b(2b + a)mmmm
[
]
+ -21 (a + b),(mn + nmXmn + nm) - ab(mmnn + nnmm)
(8.9)
where n and m are unit normals to the axes 2a and 2b of the ellipse. The first two terms characterize the normal compliances of the hole in the n and m directions; the third term characterizes the shear compliance, and the fourth term characterizes the Poisson’s ratio effect. (In the case of a crack, H = nBn, where B is the COD tensor of a crack; see (6.7).) The elastic potential in terms of stresses (complementary energy density) of a solid with a hole can be represented as a sum of two terms: f(0) = &:E(o) = $o:Mo:a + $ t ~ : H : a
fo
+ Af
(8.10)
where fo is the potential in the absence of a hole and Af is the change in f due to the hole. Utilizing the expression (8.9) for H and using the identities 0:
IS:
(nnnn - mmmm):
=
20 * CJ:nn - tr ( r *0,
[2(nnnn + mmmm) + (mn + nm)(mn + nm)]: cr = 2 tr O - G
(8.1 1)
that follow from the identities nnnn - mmmm = nnI + Inn - I1 and mmnn + nnmm = I1 - (nnnn + mmmm), we transform Af to a form that
Elastic Solids with Many Cracks and Related Problems
427
explicitly reflects the hole geometry:
Af=-
1
1 (nab[4 t r a - o - (tru)’] 2EO A -
+ 2 x 1 ~a:(a’nn . + b’mm - abI)}
(8.12)
The first term in the braces is expressed in terms of stress invariants, and thus describes the isotropic response; it enters with the ellipse’s area as a multiplier. It vanishes in the case of a crack. The second term is orientationdependent; it vanishes for a circular hole.
B. NONINTERACTING ELLIPTICAL HOLES.PROPER PARAMETERS OF DENSITY OF HOLES In the approximation of noninreractiny holes, each hole is subjected to the same externally applied stress IS. Then the additional strain due to holes is CH‘k):o,so that the overall compliance M = Mo + XH‘k). Elastic potential Af is a sum of terms of the (8.12) type: 1 A ~ = u : C H ‘ ~ ) : U = - (p[4tro~a-(tra)’] + 2 0 . a : ( f i - p 1 ) } 2EO
(8.13)
Thus, the potential is expressed in terms of two dimensionless parameters of density of holes, one scalar and one tensorial: p
1 A
= - nC
(ab)‘k) (porosity)
(8.14)
p = - 1 nC (a’nn + b2mm)‘k’ A
Tensor fl can be called a hole density tensor. For cracks, (1/n)fi reduces to the (2-D) crack density tensor a = (1/A)C(a2nn)‘k), see (6.16). Then (8.13) reduces to the potential of a solid with noninteracting cracks: Af = (n/n/Eo)a*a:a, see (6.15). In the opposite extreme of circular holes, fi = PI. The linear invariant t r p = ( l / A ) Z ( a Z+ b2)(k)characterizes the change in compressibility of the solid due to holes (see (8.4)). Thus, Af is a sum of two terms. The first one (proportional to p) does not depend on the orientational distribution of holes. It vanishes in the case of cracks. The second term vanishes in the case of circular holes; it indicates that, at the same level of porosity, the more “elongated” holes produce a higher impact on the effective moduli. We emphasize that the parameters of density of holes-porosity p and
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Mark Kachanov
the holes density tensor p-are not introduced arbitrarily, but are dictated b y the structure of the elastic potential. Since p is a symmetric second-rank tensor, the effective properties of a solid with elliptical holes are orthotropic, for any orientational distribution of holes. The axes of orthotropy are coaxial with the principal axes of p, which constitute, therefore, a natural coordinate frame for the tensor of elastic constants. This generalizes the fact of orthotropy of a medium with cracks (see Section V1.B). Similar to the case of cracks, the orthotropy is of a simplified type: 1 . The so-called Saint-Venant’s relations (6.20) hold. They reduce the number of independent moduli from four (in the general 2-D case of orthotropy) to three. 2. The orientational dependence of the effective compliances is described by an ellipse (rather than a fourth-order curve, as in the general case of orthotropy).
The difference p - P I entering the potential (8.13) can be called the eccentricity tensor; it can also be written in the form (n/A)X [(a - b)(ann - bmm)](k).Its linear invariant
H
(8.15)
can be called the eccentricity parameter. In the case of circular holes, 4 = 0; in the case of cracks, ( l / n ) q coincides with the conventional scalar crack density p = (1/A)XdkjZ(linear invariant of the crack density tensor a). The effective moduli given by the potential (8.13) are as follows
(8.16)
(the plane strain bulk modulus K is understood as defined by the relation c I + c Z 2 = a/K, where 0 = a1 = oZ2 denotes the applied hydrostatic stress). At p = P I , the results for circular holes are obtained: (8.17)
Elastic Solids with Many Cracks and Related Problems
429
For cracks, p = 0, fi = 7ca and the formulas of Section VI are recovered:
_v -_ - E-
--
E,
1
vo
1
(random orientations);
+ 27cp
(8.18)
_ VI2 El--- 1 -(parallel cracks) VO Eo 1 ZP
+
1. Randomly Oriented Holes (Isotropic Efective Properties)
In this case, fl is proportional to the unit tensor I and, since tr p = 2p + q, 1
fi = ( p + q/2)I so that Af
= - [(4p
2EO
+ 4 ) t r a . o - p(tra)2]
(8.19)
The effective Young's modulus and the bulk modulus K are
E=
Eo . 1+3p+q'
K =
KO 1 + (2P + q ) / ( l - yo)
(8.20)
We emphasize that the effective properties, although isotropic, cannot be expressed in terms of porosity alone: the eccentricity parameter q is also needed. If this factor is ignored (holes are treated as circular), then porosity, as inferred from measurements of the effective moduli, may be substantially overestimated. For example, porosity estimated from the reduction of the bulk modulus K (as compared to its value for the undamaged material) would be exaggerated by a factor of 1 q/2p; if all the holes are identical, this factor is equal to (a/b b/a)/2 - 1 = 1.25 and 1.82 for b/a = 0.5 and 0.3, respectively. On the other hand, if the hole shapes deviate from circles only moderately, the moduli can be approximately expressed in terms of p only. For example, taking K from (8.20) with q = 0 yields errors smaller than 556% if the aspect ratios of all ellipses exceed 0.6-0.7.
+
+
2. Parallel Elliptical Holes (Anisotropic Efective Properties) The hole axes 2 ~ ' and ~ ) 2b(k)are parallel to x2 and x , , respectively (Fig. 61). Then 1
fl = - 7 ~ { [ C a ( ~ ) + ~ ][Ce ~b(k)2]e2e2]} e~
(8.21)
A
We introduce the mean semi-axes and the mean aspect ratio (a2)
1 = - a(k)2,
N
1
( b 2 ) = - C b(k)2; N
-
1:
,I2 = ( b 2 ) / ( a z )
(8.22)
Mark Kachanou
430
0.4 -0.2 -0.0
” ’ : “ “ ’ ” ‘ ” “
where N is the number of holes in the representative area A. The results below can be made more transparent by considering fictitious cracks associated with the ellipses’ axes 2a(k) and using the conventional scalar crack density parameter p = (1/A)Ca(k)2. Then
fi = np(e,e, + 12e,e,) The potential Af is thus expressed in terms of p, p and
Af=-
1
2Ecl
(pC2t1-a-a-(tra)’]
+ 2 x p [ a f l +(1
(8.23)
I: +12a12]f
(8.24)
The ratio of Young’s moduli in the x, and x2 directions is
E, 1 + p + 2xpI2 _ E,
1+p+2xp
(8.25)
In the case when all the holes have the same aspect ratio R (so that p = ZAP), E, _ - 1 + xAp(1
+ 22)
(8.26) 1 xp(2 + A) E, Figure 61 shows this ratio as a function of R; it illustrates a gradual disappearance of anisotropy as the holes are “inflated from the crack-like
+
Elastic Solids with Many Cracks and Related Problems
43 1
shapes to circles. When R is small, this ratio changes rather slowly; for example, it increases from 0.44 at R = 0 (crack) to only 0.49 at R = 0.2, so that the anisotropy due to narrow holes is close to that due to cracks (associated with the ellipses’ major axes). In the opposite extreme, even small deviations of the hole shapes from circles result in a noticeable anisotropy; for example, at A = 0.9 and 0.8, the ratio E J E , is equal to 0.91 and 0.82, respectively. This is in contrast with the case of random orientations, when moderately noncircular holes can be replaced by circles of the same area with good accuracy.
C. INTERACTING ELLIPTICAL HOLES
In the case of cracks, the results of Section VI show that the approximation of noninteracting cracks remains accurate a t high crack densities, provided the mutual positions of cracks are random. The underlying reason-that the competing interaction effects of stress shielding and stress amplification, on average, balance each other-appears related to the fact that introduction of cracks does not change the average stress in the matrix (provided the boundary conditions are in tractions). Therefore, the average stress coincides with the remotely applied one; placing cracks into such a field corresponds to the approximation of noninteracting cracks. Guided by these results, we apply the same approach to elliptical holes with random mutual positions, placing each hole into the average over the solid phase stress cs. In terms of stress superpositions (reduction of a problem with N holes to N problems with one hole each), this means that the interaction tractions induced at the site of a hole (in a continuous material) by the other holes reflect, on average (over all holes), the change of the average stress environment in the solid phase. This approach corresponds to Mori-Tanaka’s model (1973). We note that Mori-Tanaka’s scheme was shown to produce results that lie within Hashin-Strikman’s bounds for randomly oriented ellipsoidal inclusions (Tandon and Weng, 1986) and within Willis’ bounds for parallel inclusions (Weng, 1992). This scheme received further support in the work of Benveniste (1990). For a solid with holes, the average stress in the solid phase Q’ can be found exactly, in terms of the applied (macroscopic) stress c and porosity p . Indeed, application of the divergence theorem to a solid with traction-free holes yields as. 11 /. tJ.. = (1 - p ) -
’
(8.27)
432
Mark Kachanov
Note that all the stress components are raised by the same ratio (1 - p ) - ’ . Thus, placing a representative hole into the average stress, i.e., replacing u by (1 - p)-’u in the hole compliance relation A& = H:u, we obtain the potential Af = (1/2)a:A~ from Af for noninteracting holes by a simple adjustment 1 1 1 Af = - Afnnonin, = - - { p[4 tr u u - (tr a)’] 1 -P l - p 2Eo 0
+ 213 u :(fl - PI)} *
(8.28) where the factor (1 - p ) - ’ reflects the change in the stress environment for holes. The structure of (8.28) shows that the impact of interactions on the efective moduli decreases (defects become, on average, “invisible” for each other) as porosity decreases. In the limiting case of cracks (p = 0), the approximation of noninteracting cracks is recovered, and Af is expressed in terms of the crack density tensor: A! = (n/Eo)u u :a. The effective moduli for interacting randomly oriented elliptical holes are
-
E=
1
+ (3p +EO4x1 - p ) - ’ ’
K =
KO 1 + (1 - p)-’(2p + 4 ) N - V g )
(8.29)
where the factor (1 - p ) - ’ accounts for interactions. Formula (8.29), as well as the special case of the formula (8.30) for parallel holes of identical aspect ratios, can be recovered from the results of Zhao and Weng (1990) on elliptical inclusions, if the inclusions’ stiffness approaches zero. In the case of parallel holes, the anisotropy, as measured by the deviation of the ratio
(8.30) from unity, is enhanced by interactions. In the case of small porosity p (but not necessarily small fl), (8.31) Smallness of p does not necessarily mean that the holes are narrow (p can also be small due to a low density of wide holes). If, however, the holes are
433
Elastic Solids with Many Cracks and Related Problems
narrow, then, to within small values of higher order, ( l / r ~ ) Bcoincides with the crack density tensor a, and 1 rK ---p[2tra*a-(tro)Z] +-(1 2EO EO
71
Af=AfnOnin,+p-a*c:a= EO
+p)o.a:a (8.32)
where the multiplier p at a * a : a reflects the enhancing efect of porosity on cracks.
D. MIXTUREOF
INTERACTING C I R C U L A R
HOLESA N D
CRACKS
This case models microcracking in a porous material and may be of interest for rock mechanics applications. In this case, B is a sum
B = Bholea + Pcracks = P I + za
(8.33)
For a mixture of noninteracting holes and cracks, the potential (8.28) is a sum: Af
=
+
A f ~ ~ AJE:;, ~ ~ ~ = ; &c
+
1
- 0 : ~- - p [ 4 t r a . o
EO
- (tro)’]
2EO
(8.34)
For a mixture of interacting holes and cracks, Af
1
= -Af
1-P
‘I
~[4tro.a--(tra)’]+2n-o.a:a 2E0 1 - p 1- P
. =-
nonlnl
(8.35)
The first term depends on p only (does not depend on crack density a);the second term contains both a and p . This shows that the interaction effect between cracks and holes is “asymmetric”: cracks do not raise the average stress in the solid and, thus, produce, on average, no impact on holes, whereas holes enhance the impact of cracks on the effective moduli (by changing the average stress environment for cracks). Thus, as far as the effective properties are concerned, cracks do not afect holes, but holes afect cracks. If cracks are randomly oriented, a = (p/2)I so that = ( p + 71p/2)I. Then E=
EO
1
+ ( 3 p + ~ p ) ( 1- p ) -
’
K =
K O
1 + (1 - PI-
V P + 71P)/(l - yo)
(8.36)
Mark Kachanov
434
Young’s modulus, as a function of crack density, is plotted for various “background” levels of porosity in Fig. 62. In the case of parallel cracks (normal to the x, axis), u = pelel. Young’s modulus in the direction normal to cracks is (Fig. 63):
E=
Eo
1
+ (3p + 2npX1-
p)-
(8.37)
and the ratio E J E , characterizing the degree of anisotropy is El _ E2
1
1 + 3p(l - p)-’ + (3p + ‘2npX1- p)-’
(8.38)
The results of this section remain valid when holes are moderately
FIG.62. Mixture of circular holes and randomly oriented cracks. Effective Young’s modulus as a function of crack density at different levels of porosity.
Elastic Solids with Many Cracks and Related Problems
43 5
FIG. 63. Mixture of circular holes and parallel cracks. Effective Young’s modulus as a function of crack density at different levels of porosity.
noncircular and randomly oriented (then they can be approximately replaced by circles of equal areas). Results for a mixture of noninteracting holes and cracks are obtained by omitting the factor of (1 - p)- in the formulas (8.35) to (8.38). A n interesting issue is whether the anisotropy due to preferential crack orientations is enhanced or weakened by the background porosity p . The anisotropy is affected by p through two competing mechanisms: (1) in the absence of interactions, porosity reduces the anisotropy; (2) interactions, generally, enhance the impact of cracks; in particular, they enhance the anisotropy. The result (8.37) shows that d(E,/E,)/dp 0, i.e., the first mechanism is dominant: porosity weakens the crack-induced anisotropy.
=-
436
Mark Kachanov
E. NONEQUIVALENCE OF THE APPROXIMATIONS OF NONINTERACTING HOLES AND OF SMALL DENSITY OF HOLES These two approximations are often considered as equivalent, and their names are used as synonyms. In fact, they are not, generally, equivalent. In the approximation of noninteracting holes, equating fo + Af to the potential f of the effective medium yields the following structure for the effective modulus S (denoting either the shear, or Young’s, or bulk modulus; So refers to the matrix without holes):
s/so= ( 1 + c)-’
(8.39)
where p is the appropriate measure of density of holes and C is some constant (formulas (8.16) to (8.18) are examples of the structure (8.39)). Linearization of (8.39) yields the approximation of small density (“dilute 1im it” )
s/so= 1 - cp
(8.40)
(We note that (8.40) coincides, formally, with the approximation of the self-consistent scheme). For circular holes, results in the form (8.40) were given by Vavakin and Salganik (1975) and Day et a/. (1992). There seems to be no need to linearize (8.39). Such a linearization narrows the range of applicability of the approximation of noninteracting holes. (In the case of cracks, the approximation (8.39) remains accurate at high crack densities, see Section VI; the linearization renders this approximation inapplicable at relatively small crack densities). Thus, the linearized “dilute limit” appears to be an unnecessary construction. In the case with nonrandom mutual positions, the approximation of noninteracting defects and the linearized (“dilute limit”) approximation become entirely unrelated. For example, a crack array of an infinitesimally small density can be constructed, such that the interactions are very strong (so that the approximation of noninteracting cracks is therefore inapplicable) and the effective stiffness is reduced to zero (Section V1.K).
F. COMPARISON WITH THE SELF-CONSISTENT AND THE DIFFERENTIAL SCHEMES The self-consistent scheme (SCS) and the differential scheme (DS) place a representative hole into the effective matrix (rather than the effective stress, as in Mori-Tanaka’s scheme); see Section V1.H for discussion. The results are obtained from the approximation of noninteracting holes in a
Elastic Solids with Many Cracks and Related Problems
437
straightforward way, similar to the derivations for cracks. (We ignore the adjustment of the hole density parameter due to the finite probability of “overlapping” of holes that was used by McLaughlin (1977) and Zimmerman (1991b), since consideration of “overlapping” holes is not, in our opinion, fully compatible with the basic assumption that the representative defect is an elliptical hole.) For illustration, we analyze the case of randomly oriented holes (isotropic effective properties) and consider the effective Young’s modulus. Then, instead of (8.20a) for the approximation of noninteracting holes and (8.29a) for Mori-Tanaka’s scheme, we have
E/E,
=
1 - 3 p - q (SCS)
and
E/E,
=
e-(3p+q)(DS) (8.41)
For circular holes, the predictions of DS and of Mori-Tanaka’s scheme are close, but, as the eccentricity g increases, they become increasingly divergent. In the case of cracks, they are quite different (see Figs. 5 2 and 53). As discussed in Section VI.H, the method of Mori-Tanaka (MT) appears to be more physically consistent than the other approximate schemes for interacting defects. In the case of cracks, the MT results (coinciding with the approximation of noninteracting cracks) are confirmed by computer experiments (Figs. 5 2 and 53). Generally, the MT approximation can be shown to follow from a more sophisticated method of effective field that explicitly takes into account the statistics of mutual positions of defects, in the case when these mutual positions are random (Kanaun, 1992). It appears, therefore, that the DS, and, particularly, the SCS, may overestimate the effective com pliance.
Acknowledgments The author is grateful to A. Chudnovsky for the original motivation of the work and discussions and to S. Kanaun, F. Lehner, I. Tsukrov, and C. Mauge for a number of useful remarks. The author is also grateful to C. Mauge for substantial computational help and to I. Tsukrov and B. Shafiro for help in preparation of the manuscript. This work summarizes the research supported, at various times, by the U.S. Department of Energy, the U.S. Army Research Office and the Air Force Office for Scientific Research. Access to the Cornell Supercomputer Facility is acknowledged. Finally, the author is grateful to Victoria Jennings, Academic Press, for her patience and cooperation.
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Author Index
A
Benveniste, Y., 296, 313. 349. 397-399.422. 43 1,438 Beran, M. J., 296,313,349,438 Bergan, P.G., 120,188 Bert, C. W., 68,185 Berthaud, Y.,443 Berveiller, M., 219,222,227,230, 235,255 Betten, J., 369,418,438 Bhat, S. U., 115, 133,186 Biezeno, C. B., 170,186 Bishop: J. F., 210,255 Blakeslee, A. E., 30.65 Boas, W., 192,258 Bodner. S. R., 170,186 Bonar, J. M., 47,54,65 Boltega, W. J., 169, 186 Bower, A. F,, 63, 66 Brace, W. F., 424,445 Bristow, J. R., 347,353,438 Brockenbrough, J. R., 323,344,392,442 Broussard, D. E., 95, 106,186 Bruce, V.G., 170,187 B ~ l lM. , A,, 374, 392,408,440 Bruner, W. M., 393,438 Brush, D. 0..109, 158,186,188 Bucciarelli, L. L.. 158, 169, 186, 188 Budiansky, B., 12,64,212,55,347,35 1,381, 391-393,396,438,443 Buescher, C., 47.54.65 Bullough, R., 1I. 26,42,47, 66 Buresch, F. E., 323,438 Burhns, 0.T., 237,257
Abeyaratne, R., 184,185 Aboudi, J., 397-398,438 Adkins, J. E., 80,84,186 Aifantis, E. C., 195,255 Alexander, H., 31,58,64,78,185 Aliabadi, M. H., 282,438 Allen, D. H., 368,438,442 Almroth, B. 0.. 109,186 Alturi, S. N., 282,443 Aly-Helal, N. A,, 115,186 Anderson, T. L., 282,438 Andreikiv, A. E., 438 Ang, D. D., 394,438 Aravas, N., 195,255 Arikan, E., 1 1 I , 187 Arseculeratne, R., 177, 187 Asaro, R. J., 192-193, 198,211-212,218,223, 235-236,238,240,242-246.248.255 Ast, D. B., 39, 6 I , 64 Atterbuny, T. I., 95,99,188 Attiogbe, E. K., 392,438 Ayers, R. R., 95, 134,186-187
B Babcock,C. D.,99, 103-104, 106, 108, 115, 121, 133-134, 170, 184-185, 186-187 Bahnck, D., 58.65 Baker, G. S., 234,255 Baribeau, J.-M., 6 1 , 6 5 4 6 Bamett, D. M., 15.64.390.439 Basinski, S. J., 192-193,221-222,224,230, 233,250.255-256 Basinski, Z. J., 193,221-222.233, 250,255 Bassani, J. L., 192-194, 199,203,205,209-210, 212-215,218,223,22%231,235-238,240-245, 247,255,257 Bazant, Z. P., 443 Bean, J. C., 3,47,54,57-58,61,6445 Bell, J. F., 222,255
C Cahn, R. W., 219,235,255,257 Cai, M., 390, 398, 438 Calladine, C. R., 99, 133, 168, 186-187 Cammack, D. A., 46.66 Cammarata, R. C., 3 1.64 Cartwright, D. J., 282,444 Chaboche, J. L., 417,442 447
Author Index
448
Chalmers, B., 218, 220, 222, 224,257 Chang, Y.-C., 72,75,78,81,83,85,89,92-93, 119, 134, 138,186-187 Chang, Y. W., 235,247,255 Charalambides. P. G., 323,438,444 Chater, E., 82.93, 115-1 16, 132,186 Chen, T. M., 400,441 Chen, W. T., 394,438 Chen, Y. Z., 28 1,438 Cheung, M. C., 170,186 Chihab, A., 368-369,438,439 Chim, E. S . M., 413,445 Christensen, R. M., 397, 399,438439 Christian, J. W., 198,255 Christoffersen,J., 253,255 Chudnovsky, A,, 281,296,322-323,345,401, 439440,445 Clarke, D. R., 323,439 Clinton, R. I., 213,256 Collins, W. D., 281,283, 300,439 Colwell, J. A,, 149, 186 Cook, N. G. W., 391,442 Cornelison, D. M.. 58-59.66 Comeliussen, A. H., 78,186 Corona, E., 96,186 Costin, L. S., 371,439 Cotterell, B., 274,439 Cottrell, A. H., 219, 235,255 Croll, J. G. A., 115,186 Curtin, W. A,, 344,439
D Dahlgren, V. A,, 391,440 Dalby, R., 46,66 Darwin, D., 392,438 Datsyshin, A., 281,439 Daves, W., 322,440 Davis, R. S . , 218,220,255 Dawson, L. R., 31,65 Day, A. R., 421,436,439 Delameter, W. R., 390,439 Diehl, J., 220, 232,255 Dodson, B. W., 5 8 4 0 , 6 4 , 6 6 Dolgopolsky, A,, 296, 322,439 Dragon, A,, 368-369.438-439 Drucker, D. C., 219,227,258 Dubey, R. N., 94,186 Dundurs, J., 8, 15-16.64-65 Dutta, B. K., 329,439 Dvorak, G. J., 296,313,349,438
Dyau,J. Y., 104, 120, 124, 127, I86
E Eaglesham, D. J., 47,54,61,64-65 Edwards, E. H., 222,234-235.255 El-Bayoumy, L., 158, 169,186 Elam, C. F., 222,235,255 Elyada, D., 121,187 Embury, J. D., 282,323,440 England, A. W., 424,445 Ericksen, J. L., 70,186 Eshelby, J. D., 12, 14.64 Evans, A. G., 323,334,439,444 Ezz, S. S . , 198,203,297,255
F Fabrikant, V. I., 262,271,273,282,292,295, 327,437,439440 Feldman, L. C., 57, 64 Feng, X., 38,47,65 Ferrandon, J., 370,440 Filshtinsky, L. A,, 388, 390,440 Fiory, A. T., 57,64 Fischer, F. D., 322,440 Fitzgerald, E. A., 39.64 Fleischer, R. L., 218,220,255 Folberth, P. J., 149,188 Franciosi, P., 219,222,227,230,235,255 Frank, F. C., 61,64 Freund, L. B., 14,24,26,47,58,63,64-66 Friedrichs, K. O., 179,186 Fritz, I. J., 31, 65 Fu, W. S . , 440 Fu, Y., 323,334,439 Fuh, S., 223,255 Fung, Y. C., 170, I86 Futamura, K., 344,439 G Garboczi, E. J., 421,436,439 Garstone, J., 219-220.255 Gere, J. M., 172,189 Gibbings, C. J., 30.34.58.61.65-66 Gibson, T. L., 95, 106,188 Gjelsvik, A., 170, 186 Gong, S. X.,322-323.440 Gorelik, M., 323,440 Gottesman, T., 374,392,408,440 Gourley, P. L., 31.65
Author Index Gravesteijn, D. J., 61.65 Green, A. E., 80,84,186 Green, Jr., R. E., 222, 255 Greenberg, B. L., 46,66 Greetham, G., 219-220,255 Gross, D., 28 I , 296,440 Groves, S. E., 368,438 G’Sell, C., 115,186 Guggenheim, E. A,, 70,186 Guldenpfennig, J., 213,256
H Hahn, G., 440 Halliwell, M. A. G., 34, 65 Han, L. X., 321, 345,407,440 Hams, C. E., 368,438,442 Hashin, Z., 349, 374, 391-392, 394, 408,440 Haughton, D. M., 78,186 Havner, K. S., 191,212-213,218.223-234.256 Head, A. K., 8.65 Heredia, F., 239, 256 Henmann, G.. 158,189,390,439 Hershey, A. V., 391,440 Hill, R., 191, 194-198,211-212.214, 223, 236237,256,391,422,440 Hirakawa, S., 198,258 Hirota, K., 294,441 Hirsh, P. B., 220-221,256 Hirth, J. P., 14,38,47, 61.65 Hoagland, R. G., 282,323,440 Hodge, P. G., 141,187 Hoenig, A,, 353,440 Hoff. N. J., 170, 187 Holand, I., 120,188 Holthe, K., 120, 188 Honeycombe, R., 219-220,223,255-256 Horii, H., 281,296,322, 376,390,392,398, 400,438,440 Houghton, D. C., 30,34,58,61,6546 Howard, D. J., 3 2 , 4 6 4 7 , 5 5 , 6 6 Huang, Z., 288,441 Hudson, J. A,, 400.441 Hull, R., 47.54.57-58,6445 Humphreys, C. J., 61.64 Hutchinson, J. W., 82,93, 115-1 16, 132, 186-187,211-212,216,218,222,236,253, 255-256,322.323.334,441,444
I Irwin, G. P., 282,444
449
Isida, M., 294,441 Izimi, O., 198, 258
J Jackson, P. J., 222,224,256 Jain, S. C., 11,26,42,47.66 Jensen, H. M., 120,187 Johns, T. G., 99.187-188 Jonas, J. J., 115, 186 Joshi, N. R., 222,256 Ju, G. T., 94, 125, 187 Ju, J. W., 400,419-420,441
K Kachanov, L., 417,441 Kachanov, M., 28 1-283,293-296,298,301, 311, 313, 320, 322,326,328,334, 339,341, 349,354,358,368,369,371,373-376,378, 391,396-397,400,407,417,421,439. 441445 Kakodkar, A,, 329,439 Kamalarasa, S., 99, 133,187 Kamat, S. V., 61.65 Kamel, M., 441 Kamemura, K., 443 Kanatani, K., 441 Kanaun, S., 389,398-399.437.442 Kaplan, A,, 170,186 Kapshyvij, A. A,, 442 Karbhari, V., 322,439 Katsounas, A. T., 120,189 Keer, L. N., 440 Kelly, A,, 192,256 Kemeny, J., 391,442 Kirchner, P. D., 39,61,64 Knowles, J. K., 184,185 Kocks, U. F., 221-222,224,235,256 Koiter, W., 212,256 Kopystyra, N., P., 442 Krancinovic, D., 368,371,413,417,442 Kroner, E., 442 Kuhlmann-Wilsdorf, D., 220,256-257 Kuzmin, Y. N., 442 Kvam, E. P., 61.64 Kwak, S. S., 322,439 Kyriakides, S., 72,75,78,8 1,83,85,89,92-94, 96,99, 102-104, 106, 108, 110-111, 115-116, 119-121, 124-127, 132-134, 138, 147, 150,155, 158, 161,163,165,168,177,180,184,186-189
450
Author Index L
Laflen, J. H., 199,211,258 Laird, C., 205.218-220,223,225-230,235. 256-258 Langner, C. G., 134,187 Laures, J.-P., 282,292-295,298,301, 322,326, 441 -442 Laws, N., 323,344,392,442 Lax, M., 346,398,442 Lebedev, N. N., 442 Leckie, F. A., 371,443 Lee, J. W., 368,442 Lee, M. S., 16,65 LeGoues, F. K., 32,65 Lehner, F., 349,442 Leibenguth, R. E., 58.65 Lekhnitsky, S. G., 355,442 Lemaitre, J., 417,442 Leroy, Y., 247,257 Levin, V., 398,442 Levy, M., 94,188 Li, F. S., 158, 161, 163, 165,188 Liaw, B. M., 441 Lietzau, H. P., 442 Light, T. B., 20,26,29,66 Litewka, A., 368,443 Livingston, J. D., 222,255 Lo, K. H., 397,439 Lochridge, J. C., 95,106,188 Loethe, J., 1 4 1 5 , 6 4 6 5 Lomer, W. M., 221,257 Lowengrub, M., 271,444 Lyons, M. H., 34.65
M Mack, R. D., 149,186 Mader, S., 20,26,29,66 Madhavan, R., 134,185 Maeshibu, T., 368,443 Maher, D. M., 61,64 Maiti, S. K., 329,439 Mandel, J., 192-193, 212,257 Marguerre, K., 170,188 Martin, C. J., 149,186 Martin, J. H., 119, 133. 141, 188 Masur, E. F., 170, 176177, 179, 181,188 Matsubara, N., 149, 158,189 Matthews, J. W., 20,26, 29-31,6566 Mauge, C., 320,341,373-374,407,421,443
Maugin, G. A., 413,417,443 Maurer, K. L., 322,440 McCaig, I. W., 149, 188 McGhie, R. D., 158,188 McLaughlin, R. A., 394,437,443 McMeeking, R. M., 323,438,444 Meguid, S. A,, 323,440 Mehrabadi, M. M., 371,443 Melin, S., 3 1 1,443 Mesloh, R. E., 95.99.187-188 Meyerson, B. S., 32.65 Mimaki, T., 149,188 Mitchell, T. E., 220,257 Miyasaka, A., 149,188 Mohan, R., 229,238,247,257 Montagut, E., 31 1, 313,322,328, 334, 339-340, 344,441,443 Montel, R., 149, 188 Morar, J. F., 32.65 Mori, T, 399,43 1,443 Mroz, Z., 368,439 Murakami, S., 369,418,443 Muskhelishvili, N. I., 9-10.66,423,443
N Nakada, Y., 221,224,235,256 Nakahara, S., 57.64 Neale, K. W., 115,187-188 Needleman, A,, 85,189,237,247,257 Nemet-Nasser, S., 281,296,322,371, 376, 392, 440,443 Nicholson, R. B., 192,256 Nisoshika, T., 282,443 Nix, W. D., 7.34,36,46,58-61,66 Noble, D. B., 34,36,58-61,66 Noguchi, H., 294,441 0 O’Connell, R. J., 347, 351,381,391-393, 396, 438,443 Oda. M., 368-369,371,443 O’Donoghue, P. E., 282,443 Ogawa, H., 149,188 Ogden, R. W., 78,186,188 Ohno, N., 369,418,443 Onat, E. T., 371,443 Ono, S., 198,258 Ortiz, M., 247,257, 323,443
Author Index P Paine, D. C., 32,4647,55,66 Palmer, A. C, 95, 119, 133, 141,188 Panasyuk, V. V., 438 Paris, P. C., 282,444 Pasha, M. L., 443 Peirce, D., 212,257 Perovic, D. D., 6 1 , 6 5 4 6 Peticolas, L. J., 58,65 Petruzzello, J., 46,66 Pettit, G. D., 39, 61, 64 Pian,T.H.H., 158, 169,186,188 Piau, M., 353,443 Picraux, S. T., 58-59,66 Piercy, G. R., 219,235,257 Pijaudier-Cabot, G., 443 Pipkin, A. C., 92,188 Pope, D. P., 198-199,201,203,207,239,297, 255,257 Pozinenko, 9. V., 370,444 Price, R. J., 193,235,257 Proano, R. E., 39.6 I , 64
Q Qin, Q., 192-193, 199,203,205,209-210, 214-215,236-238,240-244,247.257
R Ramaswarni, B., 222,224,257 Ramirez, J. C., 63.66 Raniecki, B., 237,257 Reifsnider, K., 4 16,444 Reissner, E., 108, 158,188 Rernseth, S. N.. 120, I88 Rice, J. R., 12. 15, 22, 59,64,66, 192-194, 198 212,23&238,240,257,274,379,385,439, 444 Roach, D., 104, 116, 132-133,187 Robinson, I. K., 57,64 Romalis, N. B., 323,444 Rornrn, E. S., 370,444 Rooke, D. P., 282,438,444 Rose, L. R. F.. 322,444 Rosenfield, A., 440 Rosi, F. D., 220,257 Rubenstein, A,, 322, 328,444 Rudnicki, J. W., 236237,257 Rudnicky, J. W., 379,444
45 I
Ruhle, M.,323,444
S Sahasakmontri, K., 400,440 Salah, S., 203,257 Salganik, R. L., 349, 388,394,436,445 Sanders, J. L., 120,188 Satake, M., 371,444 Savin, G. N., 423,444 Savruk, M., 281,439 Sayers, C., 381,391,396-397,400,444 Schmid, E., 192,258 Schreyer,H. L., 170, 176-177, 179, 181,188 Seeger, A,, 192,22&22 I , 258 Sen, P. N., 421,444 Sendeckyj, G. P., 16, 64 Sewell, M. J., 253,258 Shalaby, A. H., 218,235,255 Sheh, M. Y., 199,258 Shield, R. T., 78,81,186,188,444 Short, K. T., 58.65 Shu, D., 68,189 Shurn, D., 322,444 Sieradzki, K., 31, 64 Sneddon, I. N., 271,444 Snyder, K.A,, 421,436,439 Sokolowski, M., 322,445 Song, H.-W., 120, 184.189 Sorenson, J. E., 95,99,187-188 Spence, J., 99, 106, 115,189 Steel, W. J. M., 99, 106, 115,189 Stinchcomb, W., 416,444 Stoffel, N. G., 32,4647.55.66 Stonge, W. J., 68, 189 Stouffer, D. E., 199,21I , 258 Suemasu, H., 413,445 Suresh, S., 321, 345,401,440 Suzuki, K., 368,443
T Tada, H., 282,444 Takasugi, T., 198,258 Talreja, R., 371, 315,444 Tamuzh, V. P., 323,444 Tanaka, K., 399.43 I , 443 Tandon, G. P., 422,431,444 Tassoulas, J. L., 120, 184.189 Taylor, G. I., 192,210,212,218,222,258 Texter, H. G., 148-149.189
452
Author Index
Thornton, P. R., 220,257 Thorpe, M. F., 421,436,439,444 Thouless, M. D., 43,66 Timoshenko, S., 110, 170, 172,189 Tsao, J. Y., 58-60.64,66 Tseng, K. H.,400,441 Tsien, H . S . , 179,189 Tugcu, P., 115,188 Tuppen, C. G.. 30,34,58,61,6546 Turlo, J. F., 34.36, 58-61,66 Turska-Klebek, E., 322,445 Tvergaard, V., 85,189
U Uflyand, Y. S., 295,442,445 Ullman, F., 149,189 Untenvald, F. C., 58.65
V Vakulenko, A,, 354, 368.445 van de Walle, G. F. A., 61, 65 van der Berg, N. G., 26.66 van der Merwe, J. H.. 26, 39,64,66 van Gorkum, A. A,, 61,65 van Ijzendoom, L. J., 61.65 Van Stone, R. H., 199,211,258 Varadarajan, R., 2 13,255 Vause, R. F., 234,256 Vavakin, A. S., 349,388,394,436,445 Vitek, V., 198,203,297,255 Vriezema, C. J., 61.65 W
Walker, G. E., 95,186 Walpole, L. J., 445 Walsh, J. B., 347, 353, 376,422,424,445 Wang, S. S., 413,445
Washbum, W., 222,236235,255 Watanabe, S., 198,258 Watson, G. P.,39,61,64 Weatherly, G. C., 6 1 , 6 5 4 6 Weng, G. J., 42 1422.43 1,432,444445 Werder, D. J., 58,65 Wierzbicki, T., 115, 133,186, 189 Williams, M. L., 394,438 Williams, R., 199.21 1,258 Willis, J. R., 11, 26, 42, 47, 66, 402,408,445 Willmore, T. J., 283,445 Wittke, W., 370,445 Woodall, J. M., 39, 61. 64 Wu, S., 323, 345,401,439,445 Wu, T., 193,229,258 Wu, T. T., 219,227,258 Wu, Tien-Yue, 192,205,218,223,225-226, 228,230,235,258 Wung, E. C. J., 296,313,349,438
Y Yarnabe, T., 443 Yamamoto, Y., 149, 158,189 Yeh, M.-K., 103, 106, 108. 110, 116, 126, 132-133,187,189 Yin, W.-L., 92,189 Yoshida, T.. 294,441 Youn, S. K., 158, 161, 163,187 Z
Zagustin, E. A., 158, I89 Zalm. P. C., 6 1.65 Zaoui, A,, 219,222,221,230,235,255 Zarka, J., 227,258 Zarzour, J., 296,313,349,438 Zhao, Y. H., 421,432,445 Zimmerman, R. W., 390,422,424,445
Subject Index
A
pressure of tubes, 94, 121, 125 Bimetallic pipe, see Cladded pipe Blocking criterion, 52 Blocking dislocation, 47 Bounding effective moduli, 401-402.408410 Buckle arrestor, 95, 106 Buckling pressure of arches, 172 of elastic-plastic tubes, 125 of elastic tubes, 94 of panels, 180 Burgers displacement, 9 Buried strained layer, 36
Activation energy, dislocation glide, 58 a-type tensor, 370 Alternative crack density parameter, 41 1 4 1 2 Anisotropic matrix crack arrays, computer experiments, 4 0 7 4 0 8 crack interaction, 320-321 crack-microcrack, 338 noninteracting cracks, 372-376 Anisotropy, stress-induced, 380 Antiplane forces, 268 Antiplane loading, 263 Approximation of noninteracting cracks, 352-353, 389,399 randomly oriented cracks, 404,406 Approximation of noninteracting holes comparison with self-consistent and differential schemes, 43&437 nonequivalence with approximation of small density of holes, 436 Approximation of small crack density, 352 Approximation of small transmission factors, 297-299 Approximation of widely spaced cracks, 299-301 Arch high, 172 shallow, 171 symmetric buckling of, 172 unsymmetric buckling of, 172, 174, 176 Aspect ratio, 3 16
C Capping layer, 34 Casing, 94, 147-148 Circular cracks arbitrary 3-D array infinite solid, 291-293 cracked solids, 357-362 Cladded pipe, 147,149 Coarse slip band, 192,235,246,248-250 COD tensor cracked solids, 350-351 crack in anisotropic solid, 372-373 infinite isotropic solid, 35 1-352 Collapse configurations of confined shells, 154,160, 163 of tube, 129 Collapse of confined shells under pressure, uniform (axially) analysis of, 156 Collapse of tubes under external pressure analysis, 108 first yield criterion, 110 uniform, 108 uniform discrete models, 117 Collapse pressure of tubes, 94, 10&107 Collinear cracks, 301-302 interaction analysis, 283-286
B Battelle Laboratories, 95 Bauschinger effect, 229 Bicrystal, 1, 14 elastic constants, 15 mismatch strain, 5 Bifurcation buckling check for uniformly inflated tubes, 81 pressure of arches, 172 453
454
Subject Index
periodic row, interacting, 3 19-320 zone of influence, 307 Complimentary energy density, 426 Confined cylindrical shells buckling of, 149 localization in, 15 1 propagating buckles in, 149, 151-154 Coplanar crack, zone of influence, 307-308 Core, dislocation, 28 Core cutoff radius, 1 1 Crack clustering of, 415 elliptical, 366-367 interacting with circular holes, 433-435 lubricated, 376-378 noninteracting, 352-372 alternative crack density parameter, 41 1 in anisotropic matrix, 372-376 characterization by vectors, 37 1-372 character of orthotropy, 362-363 constrained against opening, 376-380 crack density tensor, 368-371 disturbances of orientational randomness, 363-365 drained and undrained conditions, 385 elastic potential, 383-386 filled with compressible fluid, 380-385 fluid pressure polarization, 385 noncircular cracks, 366-367 plane problem, 365 sliding with friction, 378-380 sliding without friction, 376-378 3-D case, 357-362 2-D case, 354-357 parallel, high density, model, 401 random arrays, microcracking and elastic moduli change, 414 rectilinear front, 325-328 Crack array, arbitrary 2-D, infinite plate, 286-288 Crack density, second order, models, 400401 Crack density parameter, 347,41041 1 alternative. 41 1 4 1 2 Crack density tensor, 353-355 applications to physical properties, 369-37 1 families of parallel cracks, 357 parallel cracks, 360-361 self-consistent scheme in tenns of, 393-394 in terms of invariants, 368-369 Cracked solids, effectiveelastic properties, 345-412 alternative crack density parameter. 41 1-412 approximate schemes, 389402
average strains and stresses, 348-350 bounding effective moduli, 401402,408-410 characterization by vectors, 371-372 character of orthotropy, 362-363 circular cracks, 357-362 COD tensor, 350-351 computer experiments on crack arrays in anisotropic matrix, 407-408 deterministic arrays, 402-407 crack density tensor, 353-355,368-369 crack orientation statistics, 361-362 cracks parallel to axes of orthotropy, 373-374 differential scheme, 394-396 dimensionality effect, 366 effective matrix methods, 391 modifications, 398 effective moduli, interaction impact, 386-389 elastic potential, 350, 368-369 generalized self-consistent scheme, 397-398 infinite isotropic solid, COD tensor, 35 1-352 isotropic orientation distribution, 356-357, 359-360 loading path dependence, 378 method of an effective field, 398-399 Mori-Tanaka’s method, 399-400 naturally occumng crack arrays, 407 noncircular cracks, 366-367 noncoaxiality of physical properties, 37 1 noninteracting cracks, 352-372 constrained against opening, 376-380 filled with compressible fluid, 380-385 in anisotropic matrix, 372-376 parallel cracks, 360-361 path independence, 378-379 plane problem, 365 preservation of orthotropy for interacting cracks, 406 self-consistent scheme, 391-394 sliding with friction, 378-380 without friction, 376-378 solids with cracks versus inclusions, 346-347 terms of invariants, 375-376 two families of parallel cracks, 357 uniaxial compliance, 377 Crack interaction matrix, 288 Crack interactions, 271,301-321 analysis, 283-296 accuracy, 293-295 arbitrary 3-D array of circular cracks, infinite solid, 291-293
Subject Index arbitrary 2-D crack array, infinite plate, 286-288 collinear cracks, 283-286 comparison with polynomial approximation techniques, 295-296 full stress field construction, 288-290 in anisotropic matrix, 320-321 cracks filled with compressible fluid, 315-320 interacting cracks, 3 18-320 one isolated crack, 316-317 crack touching line of another crack, 3 13 with field of microcracks, 32 1-345 asymptotic 2-D case, 322-323 crack-microcrack configurations, 333-334 equations, 324-329 interaction effect fluctuation, 342 large microcrack arrays, 323,333-343 macrocrack interactions, 329-33 1 microcrack in wake zone, 332 modelling by effective elastic, 344 primary and secondary modes, 338 rotated microcrack interactions, 33 1-332 short- and long-range interactions, 336,338 toughening by microcracking, 344-345 fluid impact, 3 19 isolated fluid-filled crack, 320 range of influence of crack in an array, 3 6 3 0 8 relative strength of interaction effects, 303-305 slightly asymmetric arrays, extremal properties, 308-3 1 1 stress shielding and amplification, 301-306 3-D configurations, 305-306, 314-3 15 2-D configurations, 31 1-313 Crack-microcrack configuration asymmetric, 333-334 coplanar, 333 Crack-microcrack interaction, 4 16 in anisotropic matrix, 341 equations, 324-329 accuracy, 328-329 3-D configuration, 325-328 2-D configuration, 324-325 Crack opening displacements circular crack loaded by uniform traction, 274-275 collinearity, 270 from pair of point forces, 269 relation with traction, 270-271 shear, 378 uniform loading, 269-270 Crack tip fields, elastic solids, 279-280
455
Creep, 418 Critical thickness, 20, 28 buried layer, 36 with capping layer, 32 energy criterion, 20,26 equilibrium criterion, 20,26 superlattice, 38 Cross slip effect, 201-205 Crystal critical resolved shear stress, 198 cross slip effect, 201 elasticity, 196-197 flow, 211-217 nonassociative. 212 normality structure, 2 12 slip rate, 21 1 uniqueness, 214-2 15 kinematics, 194 non-Schmid effect, 198-199.201 shear stress, 2OC-201 yield behavior, 198-21 1 plasticity, 194-216 Schmid law, 299,203 Schmid stress, generalized, 196-200 plastic-work-rate conjugate, 197 slip vector, 194,20&201 spherical triangle, 202.206-207,216,231 yield criteria, 199-208 potentially active, inactive, 198-199 yield surface, 208-21 1 vertex effect, 210,240 Cubic materials, 29 Curvature, film, 6 Cyclic deformation, 229-230
D Damage parameters adjustable, 420 choice, 418-419 Darcy’s law, 370 Differential scheme, 394-396 comparison with approximation,4 3 W 3 7 extension, 396-397 Dislocation forest, 221 Hirth lock, 228 interaction, 220 junction, 227-228 coplanar, 228 glissile, 228
Subject Index
456
sessile, 228 Lomer-Cottrel lock, 221 pile-up, 221 Dissociation, dislocation, 3 1 Driving force, threading dislocation, 23 Dynamic propagation of buckles in tubes, 184 of instabilities, 184
E Eccentricity parameter, 428 Eccentricity tensor, 428 Edge dislocation isolated, 9 periodic array, 11 Effective elastic material, microcracked zone modelling, 344 Elasticity, 196 Cauchy stress, 196197 Kirchhoff stress, 196 second Piola Kirchhoff stress, 196 Elastic mismatch strain, 5 nonuniform, 32 Elastic moduli, change and micmracking, 413417 Elastic potential, 354 cracked solids, 350 damage parameter choice, 418-419 effective properties and, 417418 noninteracting cracks, 383-384 noninteracting elliptical holes, 4 2 7 4 3 1 in strains, 419420 structure, 421 in terms of invariants, 368-369 Elastic properties, see atso Solids, with elliptical holes effective, elastic potentials and, 417418 Elastic solids, linear, 259-437 crack opening displacements,269-270 many cracks, 280-301 arbitrary arrays, 282 crack interaction analysis, see Crack interactions Schwarz-Neumann alternating method, 282 small transmission factor approximation, 297-299 widely spaced crack approximation, 29S301 nonuniformly loaded crack, 270-27 1 one crack, 26 1-280 crack tip fields, 279-280 energy release rate along crack edge, 273
far-field asymptotics, 275-279 stresses from uniform tractions, 27 1-273 three-dimensionalfields, 27 1-275 stress intensity factors, 268-269,273-274 stress fields, 262-268 two-dimensional fields, 262-27 1 Electric conductivity, crack density tensor applications, 37 1 Elliptical holes, see Solids, with elliptical holes Energy, release rate along crack edge, 274 Energy buckling load. 179 Epitaxy, 2 Excess shear stress. 58
F Far-field asymptotics, 282-283 elastic solids, 275-279 Flow, see Plastic flow Flow stress ratio, 234 Fluid compressible, in cracks, 315-320 pressure polarization, 385 Fluid filtration, crack density tensor application, 369-370 Fluid pressures, coupling with stress interactions, 3 18-320 Force on dislocation, 4 driving force, 4 image force, 4
G GaAs/Si film, 2 Generalized force, 25,63 Glide surface, 21 Green’s tensor of elasticity, second, 288
H Hardening, 212,217-235,249,252 analytical characterization,226-230 backward extrapolation, 223 Bauschinger effect, 229 cyclic deformation, 229-230 easy glide, 22 1 experiments, 221-225 inequality restriction, 225 interaction factor, 227-228 isotropic, 212 kinematic, 212 latent, 212
Subejct Index multiple slip, 226-230, 244 observations, 21 9-225 orientation dependence, 218-219.232 stage I, 11,111. 22&221, 227-229 Taylor's law, 216 tension/compression asymmetry, 21 5-217 Hole compliance tensor, 426
457
superpartial, 204 tension-compression asymmetry, 2 16 Localization in a panel, 174-175 in collapsing tube, 123, 127 in inflated tube, 85 of instability, 182-183
I Inflation of elastic tubes, 70. 77 localized, 74 uniform, 70,78,80 InGaAs/GaAs film, 2.39 Initial imperfections, 109, 126, 158, 161 Initiation of buckles in confined shells, 151, 153 of buckles in panels, 174-175 of buckles in tubes, 96-97 of bulges in tubes, 70 of collapse in tubes, 119, 121 of propagating buckles in pipeline, 96-97 of propagating instabilities, 182 Initiation pressure of propagating buckles in confined shells, 153 of propagating buckles in panels, 173-174 of propagating buckles in tubes, 94, 102 of propagating bulges in tubes, 7 1.75 of propagating instabilities, 182 Installation, offshore pipeline, 96 Isotropic solid, infinite, COD tensor, 35 1-352
K Kinematics, 194 elastic, plastic, 194-196 intermediated configuration, 195 Kinetics, dislocation glide, 56
L Latent hardening ratio, 234 Ledge, surface, 3 1 Limit load instability, 81,85, 114, 122, 161, 164, 169. 182 Limit pressure of confined shells, 162 L12 intermetallic compound, 203-208, 215-217 cross slip, 204 multiple slip, 216,219 non-Schmid stress, 203-2 I 1 PPV theory, 203 Shockley partial, 204
M Macrocrack interactions with collinear microcrack, 329 with parallel microcracks, 329-33 1 modelling, 326 Macroscopic shear bands, 192,235,243,25&252 Maxwell construction for propagation of buckles in confined shells, 165 of buckles in panels, 178 of buckles in tubes, 116, 1 18-1 19 of bulges in tubes, 83 of instabilities, 184 Membrane axisymmetric, 83-84 circular cylindrical, 78 Metastable state, 57 Method of an effective field, 398-399 Microcrack, see also Crack interactions clusters, 340-341 collinear, interaction with macrocrack, 329 interaction effect fluctuation, 338 large array, interaction with crack, 323,333-341 nonuniform density, 339 parallel, interaction with macrocrack, 329-331 resistance to fracture propagation, 322 rotated, interaction with main crack, 331-332 damage parameter, 418 wake region impact, 339-340 in wake zone, 332 zone shape, 334-335 Microcracking change of elastic moduli and, 4 1 3 4 1 7 toughening by, 344-345 M-integral, 12 Misfit dislocation, 5 parallel array, 39 sequential formation, 43 simultaneous formation, 41 Mori-Tanaka model, 43 1 Mori-Tanaka's method, 3 9 9 4 0 0
45 8
Subject Index N
experimental procedure, 72 numerical simulation, 85-87 Ni3AI under tension, 77, 79,91 tension, compression, 207 Propagating buckles in panels, 169-181 uniaxial stress-strain curve, 217 expenmental demonstration, 173-1 75 yield loci, 209 Propagating buckles in tubes and pipes, 94-147 Non-Schmid stress, 198-21 1,238-247 effect of, in pipeline design, 106 Nucleation, dislocation, 31, 61 experimental procedure, 10&101 heterogeneous, 61 numerical simulation, 128, I30 homogeneous, 61 Propagating buckles in tubes under tension, 134-144 0 experimental procedure, 135- 137 numerical stimulation, 143 Offshore pipelines, 94 predictions of propagation pressure, 141-144 design of, 106 propagation pressures, 138-14 1 propagating buckles in, 69 Propagating instability, 68, 183 Oil can effect, defined, 172 Propagation pressure of buckles in confiied shells definition, 154 P experimental determination, 153 Panels, shallow, 172-181 experimental values (elastic-plastic shells), buckling, 174 155-1 57 localization in, 175 experimental values (elastic shells), 155 propagating buckles in, 174175 predicted values (elastic shells), 165-166 Peach-Koehler force, 24,49 Propagation pressure of buckles in panels Peierls-Nabarro stress, 3 1 definition, 174 Phase transitions experimental determination, 173 in liquid-vapor fluids, 83 predicted values, 179 in solids, 70. 184 Propagation pressure of buckles in tubes Plastic flow, 211-217, 230-235 definition, 95, 101 L12 intermetallic compound, 215 experimental determination, 99 localized, 235-237 experimental values, 104105, 107, 131 slip system, 21 1 Palmer-Martin estimate, 119 uniqueness. 214 predicted values (numerical simulation), 128, Poisson’s ratio, effect of crack density tensor, 360 130-131 Pressure-time history predicted values (uniform collapse), 115-1 19 of confined buckled propagation experiment, 153 Propagation pressure of buckles in tubes under of tube collapse and buckle propagation tension experiment, 101 definition, 137 of tube inflation experiment, 74 experimental determination, 137 Profile of experimental values, 138-140 confined propagating buckle, 155 predicted values (numerical simulation), 143-144 propagating buckle in panel, 175 predicted values (uniform collapse), 141 propagating buckle in tube, 102, 144 Propagation pressure of bulges in tubes, 75, 87, propagating bulge in tube, 79,92-93 definition, 7 I Propagating buckles in confined shells, 147-169 experimental determination, 72, 75-78 definition, 147 experimental values, 79 elastic, 150 predicted values (numerical simulation), 87 elastic-plastic, 151 predicted values (uniform inflation), 79.82, 87 experimental procedure, 150-154 R uniform collapse analysis, 156 Propagating buckles in inflated tubes, 70-93 Riser, 94
Subject Index S
Saint-Venant’s relations, 355,428 Schwarz-Neumann alternating method, 282 Screw dislocation, 12 periodic array, 12 Self-consistent scheme, 39 1-394 extension, 396397 formulation in terms of crack density tensor, 393-394 generalized, 397-398 2-D, 392 Shear band, 235-254 coarse, 248-250 conjugate, system, 247, 25 1 crystallographic, 235 first bifurcation, 249 primary, secondary, system, 247-248,251 secondary slip, 236,248 single slip, 236, 248 macroscopic, 25@2-54 double-slip, 250 noncrystallographic, 235 Shear forces, 268 Shear loading, 262-263 Shell liner, 148 SiGe/Si film, 2, 30,47,54,58 Slip double cross, 221 planar model, 24G242 multiple, 226 orientation, 250 overshoot, 222 primary, conjugate system, 247 single, 23 1-234,238,249 Solids, see also Cracked solids with cracks versus with inclusions, 346-347 with elliptical holes, effective elastic properties, 421-437 interacting elliptical holes, 43 1 4 3 3 interacting circular holes and cracks, 433435 noninteracting elliptical holes, 4 2 7 4 3 1 one elliptical hole in uniform stress field, 422-427 parameters of density of holes, 42 1 Spacing, dislocation, 40.46 Stacked configurations, 302 Starlike configuration, 312-3 13 Stiffness reduction effect, asymmetry, 374
459
Strain in cracked solids, 348-350 elastic potential in, 4 1 W 2 0 volumetric, due to hole, 425 Strained-layer epitaxy, 3 Strain energy density function incompressible material, 79 of Ogden, 78 Strain localization, 235-254 constitutive, geometric instability, 236 multislip hardening, 244 non-Schmid effect, 238-247 planar double slip model, 240 nonsymmetric, 243 symmetric, 242 post-bifurcation, 245 secondary slip, 238,247 shear bifurcations, 237 single slip, 238 3-D calculation, 247 coarse slip bands, 248-250 double slip orientation, 250 elasticity, effect, 247 macroscopic shear band, 25C-254 single slip orientation, 248 vertex effect, 240 Stresses, in cracked solids, 348-350 Stress fields, 262-266 in microcracked zone, 324 one elliptical hole in, 422-427 Stress intensity factors, 268-269.273-274 Stress interactions, coupling with fluid pressures, 3 18-320 Surface energy, 31
T Tensionlcompression asymmetry, 198,206-208, 2 15-2 17 Threading dislocation, 3, 5, 19 applied stress, 20 driving force, 23 self stress, 20 sequential formation, 43 simultaneous formation, 41 Toughening, by microcracking, 344-345 Transmission factor, 285
U Uniqueness. 214-2 15
460
Subject Index
Up-down-up local response, 81, 114, 164, 170, 183-184
V Vegard’s law, 7 Volume controlled collapse of shells, 128 experiments, 72, 101 inflation, 72 pressurization, 70.99, 153
cross-slip effect, 201-203 L12 intermetallic compound, 201-21 1 non-Schmid stress, 199 Yield locus, surface, 209 Yield moment, 141 Yield pressure, 112 Yield tension, 138 Yield vertex, 252 Young’s modulus, 3 9 6 3 9 7 as function of crack density, 4 3 3 4 3 5 parallel cracks, 4 0 3 4 0 5 randomly oriented cracks, 4 0 3 4 0 5
W Z
Wet buckle, 104
Y Yield criterion, 199-21 1
ZnSelGaAs film, 2 . 4 6
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