Advances in Applied Mechanics, Volume 29

Advances in Applied Mechanics Volume 29 Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN RODNEY HILL L. HOWA...

Author: John W. Hutchinson

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Advances in Applied Mechanics Volume 29

Editorial Board T. BROOKE BENJAMIN Y. C. FUNG PAULGERMAIN RODNEY HILL L. HOWARTH C . 4 . Ym (Editor, 1971-1982)

Contributors to Volume 29 J. W. HUTCHINSON TUNGHUALIN z. s u o HANXIN ZHANG FENGGAN ZHUANG

ADVANCES IN APPLIED MECHANICS Edited by John W. Hutchinson

Theodore Y. Wu

DIVISION OF APPLIED SCIENCES HARVARD UNIVERSITY CAMBRIDGE, MASSACHUSETTS

DIVISION OF ENGINEERING AND APPLIED SCIENCE CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA

VOLUME 29

W

ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers

Boston San Diego New York London Sydney Tokyo Toronto

THISBOOK

IS PRINTED ON ACID-FREE PAPER.

COPYRIGHT

@

@ 1992 BY ACADEMIC PRESS, INC.

ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY

INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101

United Kingdom Edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NWI 7DX

LIBRARY OF CONGRESS CATALOG CARDNUMBER:48-8503 ISBN 0-12-002029-7 PRINTED IN THE UNITED STATES OF AMERICA

91929394

9 8 7 6 5 4 3 2 1

Contents CONTRIBUTORS

vii

PREFACE

ix

Micromechanics of Crack Initiation in High-Cycle Fatigue T. H. Lin I. Introduction 11. Dislocations and Plastic Strain 111. Slip Bands under Monotonic Loadings

IV. V. VI. VII.

A Micromechanic Theory of Fatigue Crack Initiation A Quantitative Theory of Fatigue Crack Initiation Effects of Mean Stress, Grain Size, Strain Hardening, and Overload Combined Cyclic Axial and Torsional Loadings Acknowledgments References

2 2 7 19 30 45 51 59 59

Mixed Mode Cracking in Layered Materials J. W. Hutchinson and Z . Suo I. Introduction 11. Mixed Mode Fracture: Crack Tip Fields and Propagation Criteria 111. Elasticity Solutions for Cracks in Multilayers IV. Laminate Fracture Test

V. Cracking of Pre-tensioned Films

V1. Buckle-Driven Delamination of Thin Films VII. Blister Tests VIII. Failure Modes of Brittle Adhesive Joints and Sandwich Layers Acknowledgments References

64 65 90 112 126 147 167 172 186 187

NND Schemes and Their Applications to Numerical Simulation of Two- and Three-Dimensional Flows Hanxin Zhang and Fenggan Zhuang I. Introduction 11. The Importance of the Role of the Third-Order Dispersion Term Ill. A Formulation of the Semi-discretized NND Scheme V

193 194

200

Contents

vi IV. V. VI. VII.

Explicit NND Schemes Implicit NND Scheme Applications to Solutions of Euler and Navier-Stokes Equations Concluding Remarks References

AUTHOR

INDEX

SUBJECTI N D E X

206 21 1 213 254 256

257 260

List of Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

J. W. HUTCHINSON (63), Division of Applied Sciences, Harvard University, Cambridge, MA 02138 TUNGHUALIN (l), Department of Civil Engineering, 4531 Boelter Hall, University of California-Los Angeles, Los Angeles, CA 90024 Z. Suo (63), Department of Mechanical Engineering, University of California, Santa Barbara, CA 93 106

HANXINZHANG(193), China Aerodynamics Research and Development Center, PO Box 849, Beijing, 100830, China FENGGAN ZHUANG (193), China Aerodynamics Research and Development Center, PO Box 849, Beijing, 100830, China

vii

This Page Intentionally Left Blank

Preface This volume of Advances in Applied Mechanics contains two chapters dealing with the micromechanics of materials and one chapter on the computational aspects of fluid flow. T. H. Lin brings to his article a lifetime of research experience in solid and structural mechanics. Some of his work on the difficult and important problem of microcrack initiation in fatigue is summarized in the first chapter. This is a basic article that reviews physical mechanisms and provides the micromechanical analysis of these mechanisms. The editors believe that this article will serve as a foundational paper in the field and help stimulate further progress in this aspect of fatigue. One of the editors, J. W. Hutchinson, joins Z. Suo as a co-author of the second chapter on the fracture of layered material systems. This chapter first presents the mechanics of the elastic fracture of interfaces. It then gives a fairly extensive catalogue of solutions to interface and multilayer crack problems. Applications of the mechanics are also included for a wide range of problems involving various modes of cracking such as delamination, spalling, tunneling, and blistering. The paper by H. Zhang and F. Zhuang presents some results of the development of computer methods based on the Euler and the NavierStokes models of evaluating supersonic and hypersonic flows around obstacles that contain shock waves. One may find successive improvements gained in devising the NND (non-oscillatory, no free parameters employed, and dissipative) schemes whose effectiveness is tested and whose results are reported here. The editors wish to express their appreciation to the authors of these papers. Review papers such as these are especially valuable in this age of frenetic paper publishing. Theodore Y. Wu John W. Hutchinson

ix


ADVANCES IN APPLIED MECHANICS. VOLUME 29

Micromechanics of Crack Initiation in High-Cycle Fatigue T. H . LIN Department of Civil Engineering University of California Los Angeles

I . Introduction .........................................................................................

2

I1 . Dislocations and Plastic Strain ................................................................. A . Dislocations in Crystalline Solids ......................................................... B. Dislocation Displacement and Plastic Strain ........................................... C . Force on a Dislocation and Schmid’s Law ............................................. 111. Slip Bands under Monotonic Loadings

A. B. C. D.

......................................................

Polarization of Dislocations ............................................................... Analogy between Inelastic Strain and Applied Force ................................ Stress Field of a Uniformly Slid Slice ................................................... Free Surface Effect ...........................................................................

IV . A Micromechanic Theory of Fatigue Crack Initiation ....................... A . Some Previous Theories of Fatigue Crack Initiation ................................ B. A Polycrystal Model ......................................................................... C . Gating Mechanism Provided by Stress Field ........................................... D. Secondary Slip and Extent of Extrusion and Intrusion ............................. E Supporting Experimental Observations .................................................

.

V . A Quantitative Theory of Fatigue Crack Initiation ....................................... ................. A . Analytical Method ................................................. B. Numerical Calculations of Fatigue Bands .............................................. C . Fatigue Bands with Creep ..................................................................

VI . Effects of Mean Stress, ’ze, Strai A . Mean Stress .......... ........... B. Grain Size and Micr Strain H C . Overload ................................................

VII . Combined Cyclic Axial and Torsional Loadings

7 9 9 11

16 19 20 21 23 25 25 30 30 32 39

ing, and Overload . ..... .........................................

45 46 47 47

..........................................

51

Acknowledgments .................................................................................

59

References ...........................................................................................

59

1 Copyright 0 1992 by Academic Press. Inc . All rights of reproduction in any form reserved . ISBN 0-12-W2029-7

2

T. H. Lin

I. Introduction More than 90% of all catastrophic failures of machines, vehicles, and other structures occurring in practice are cause by fatigue of materials. Fatigue failures occur under loadings substantially lower than the yield strength of the material. Fatigue cracks occur in two stages: crack initiation and crack propagation. Crack propagation predominates fatigue life in low-cycle fatigue, while crack initiation predominates the life in high-cycle fatigue and is also a necessary stage prior to crack propagation. Hence, fatigue crack initiation is of both scientific interest and practical importance. MacCammon and Rosenberg (1957) and MacCone et al. (1959) showed that metals are subject to failure at temperatures as low as 1.7 K. This indicates that although surface corrosion, gas adsorption, gas diffusion into the metal, and vacancy diffusion to form voids can have an important effect on fatigue deformation, they are not necessary for fatigue failure. This seems to leave mechanics, i.e., the local stress and strain, as a basic mechanism of fatigue. Slip bands consisting of highly localized plastic deformation have been observed in single and polycrystals. In the following, the mechanism of the formation of slip bands under monotonic loadings is first given, then the mechanism of the build-up of fatigue bands is presented. Plastic strain is mainly caused by the displacement of dislocations. The relation between dislocation and plastic strain is reviewed. It is then shown how the localized plastic strain can occur to form slip bands. Some previous theories of fatigue crack and their drawbacks are briefly shown. Then, the gating mechanism proposed by the writer and his colleagues is presented. Experimental verifications of this theory are given. The quantitative effects of mean stress, grain size, strain hardening, overload, and creep on fatigue crack initiation are shown. This theory is shown to be applicable to combined axial and torsional loadings.

11. Dislocations and Plastic Strain

A. DISLOCATIONS IN CRYSTALLINE SOLIDS Many crystalline solids are used to withstand high mechanical stresses. These crystals consist of atoms arranged in a pattern repeated periodically in three dimensions. This regular arrangement of atoms is called a space

Micromechanics of Crack Initiation in High-Cycle Fatigue

3

FIG. 1 . A face-centered cubic lattice.

lattice. The smallest lattice representing a lattice type or the pattern of arrangement is called a unit cell. The lattice space is built of unit cells. The geometrical patterns of the atoms create anisotropic elastic moduli and other anisotropic physical properties. The most common lattices in metals used in engineering are face-centered cubic, body-centered cubic, and closely packed hexagonal lattices. The unit cell of a face-centered cubic crystal (f.c.c. crystal) has one atom located at each corner and one atom at the center of each face of the cubic as shown in Fig. 1. A basic principle of plastic deformation in crystals is that the deformation does not destroy the crystallinity of the metal. The external shape of a piece of metal may be greatly changed by deformation, but the crystals in it remain as crystals. Consider the shearing of two rows of atoms in a homogeneously strained crystal as shown in Fig. 2a. Let the spacing between the rows be “a,” that between two adjacent atoms in a row be “b,” and the shear displacement of the upper row over the lower one denoted by x. From the symmetry of the arrangement of atoms (Fig. 2a), the shear stress is zero when the displacement x is zero or b/2 or b. Each atom of the row above A B is attracted towards its nearest lattice site defined by the atoms of the row below AB, so the shearing stress t must be a periodic function of x with period b. Assume this function to be t=

2nx Ksin-, b

T. H . Lin

4

(C)

FIG.2. Balanced forces at a moving dislocation.

where K is the maximum shear stress that the crystal can take. When x / b is small, this can be expressed as 2na x 7=K-b a’

where x / a is the shear strain. When the strain is small, it is elastic and K(2na/b) corresponds to elastic shear modulus G . This gives b K=G-. 2na

For a f.c.c. crystal, b / a = fl/fi.So K = G / 2 n . For aluminum, this is roughly equal to 7300 MPa (lo6psi), whereas the experimentally observed value for single crystals is of the order of 7.3 MPa (1000 psi). A refined calculation of the theoretical shear strength by Frankel (1926) gives an approximate value of G/30, which is still immensely greater than the observed strength of crystals. This large disparity has been explained by Taylor (1934), Orowan (1934), and Polanyi (1934) by their ingenious concept of dislocations.


5

E

3 FIG.3.

Unit slip in area ABCD producing dislocation line ABCD.

A crystal subject to shearing stress deforms elastically until gliding starts in some region of the crystal. Part of the crystal slides with respect to the rest by unit atomic spacing. This slip can occur in a portion of a plane separating two parts of a crystal. The line separating the slipped surface from an unslipped surface is called a dislocation line. The direction and the amount of slippage is represented by a vector b known as Burger’s vector (Burger, 1939). The dislocation line A B (Fig. 3) parallel to the Burger’s vector is called a screw dislocation. The dislocation line BC normal to the Burger’s vector is called an edge dislocation. This slip initiates over a small local area, and spreads over a larger area. Referring to Figs. 2b and 2c, when a dislocation is present, this resistive and attractive forces act at the same time and approximately cancel each other. The resistive forces on the left side of this dislocation are balanced by the attractive forces on the right side. Thus, a dislocation almost always has a system of balance forces acting on it. Only a small biasing stress needs to be applied to the crystal to cause a dislocation to move forward or backward (Gilman, 1960). This explains why the yield strength is much smaller than the theoretical shear strength of the crystal.

B. DISLOCATION DISPLACEMENT AND PLASTIC STRAIN(Mura, 1967)

Consider a number of dislocation lines of various Burger’s vectors b(’), b‘”, b(”) with directions @, t‘”, t(“) and displacements V(l), V(2), VC3’ passing through a small surface area A S in the deformed material. The

T. H. Lin

6

number of dislocations through the surface AS is E: tf”vh@’and the total Burger’s vector of n dislocations threading through surface A S is

where v is the normal vector to AS. (Yhi is called the dislocation density tensor giving the i-component of the Burger’s vector per unit area normal to the xh-axis. The flux of dislocation is defined as n

ulhi vh

As =

1 m=

&(m’t~rn’b~rn’Vh,

(2.4)

1

where ulhj is called the dislocation flux tensor (Mura, 1967). The displacement gradient ui,jof an elastic-plastic body can be considered to consist of the elastic distortion and plastic slip: u1.J . . = u1! J. + u!’ . I,J 9

(2.5)

where the single prime denotes the elastic part, and the double prime, the plastic part. The subscript j after comma denotes differentiation with respect to the xj-axis. The plastic and elastic linear strains are expressed as

e$

=

$(u!,~+ u;,~)

+

and

eb = + ( ~ f , ~u,!,~).

(2.6)

In the unit cube shown in Fig. 3, the Burger’s vector is along x,-axis. Slip on the shaded area causes u’;,* in the body. The average u ’ ; ,increases ~ with the size of the shaded area. This size increases with the displacement of AB along the negative x,-axis and that of BC along x,-axis. AB corresponds to a l l and BC corresponds to ( ~ 3 1 The . plastic distortion is hence caused 1 the x,-direction and/or by the displacement of by the displacement of ( ~ 3 in a l l in the negative direction of x,: ug,l

= u131

-

u311.

With the repetition of subscripts denoting summation from 1 to 3, we can write u!’. = &.i r n n u r n n j , (2.7) i,j where cimnis the skew-symmetric tensor, i.e., &uk

=

0,

= 1, = -1,

when any two of the indices are equal, when i , j , k is a cyclic permutation of 1 , 2 , 3 , when i, j , k is a cyclic permutation of 1 , 3 , 2 .


7

The plastic strain component

e$ = +(u:!,~ + u:,~)= -+(E~,,,,,U~,,~ + E~,,,,,u,,,~~).

C . FORCEON

A

DISLOCATION AND SCHMID’S LAW

The crystal shown in Fig. 3 is taken to be a unit cube. When it is subjected to an external stress t, the dislocation line BC tends to move along x,-axis. Let the shear stress acting in the slip direction on the slip plane be denoted by t. When BC of length s with dislocation strength b in this plane moves forward a small distance dl, this movement causes an average shear displacement of EFGH relative to IJKL of the amount bsdl. The external work done on this crystal is tbsdl. Let the force on the dislocation per unit length be denoted by f . The work done on the dislocation is fsdl. Hence,

rbsdl

= fsdl,

f = tb.

This stress t is the shear stress along the slip direction on the slip plane, and is called the resolved shear stress. Dislocation movement depends on the resolved shear stress, and plastic strain is mainly caused by dislocation movement; hence, plastic slip depends on the resolved shear stress. This explains why, in single-crystal tests, slip occurs along certain directions on crystal planes and depends on the resolved shear stress and not on the normal stress on the sliding plan. This dependency of slip on the resolved shear stress is known as Schmid’s law. This dependency holds for both monotonic and reversed loadings.

111. Slip Bands under Monotonic Loadings

Rosenhain (1905) showed that dark bands seen on the surface of a plastically deformed crystal consisted of slips formed by shearing slabs of atoms over each other along well-defined planes, just like a deck of cards. The thickness of each slab is of the order of a few microns in most metals. There seems to be no relative movement of the atomic planes within a slab. However, Heidenreich and Shockley (1947) found that under a light microscope the slip bands resolved into clusters of fine “lines.” This indicates the steps found by Rosehnain are not formed by slip on one plane, but are

T. H. Lin traces of small steps formed by slip over several parallel slip planes spaced some 100 atoms apart. These slip planes have been considered as the boundaries of elementary slip lamillae. Each lamilla has been found to have microns). slid over its neighbors a distance of about 2,000 angstroms Similar observations were made by Brown (1949), Chen and Pond (1952), and Pond (1971). In general during deformation, the number of slip bands increase. Using transparent crystals, Nye (1949) found coarse slip bands that run through the crystal. Markings, obtained by special etchants on the surface of deformed crystals from which the slip bands had been erased by polishing, have been shown by McLean (1948) and Burke and Barrett (1948) to be traces of slip bands that run through the specimens. From the slip lines shown on both the top and bottom faces of a quenched hardened aluminum crystal, Mori and Meshii (1969) also inferred that slip bands run right through the crystal. To explain the observed heterogeneity of plastic deformation, it has been suggested that some weak slip plane may exist in a crystal, and slip occurs on this plane under an applied stress. The fact that slip does not continue on this plane until fracture occurs but shifts to other planes has been commonly explained by work hardening; i.e., the lattice around the active slip plane is assumed to be severely distorted, so a higher stress is required to cause slip to continue on the original plane. As pointed out by Brown (1949), this hypothesis fails to account for some experimental observations. The first is that slip occurs by visible amounts in each band over distances of 1000 atoms. It seems unreasonable to have so much movement before the slip plane becomes harder than some other plane still inactive. Secondly, slip bands, once formed, can increase in shear while new bands are formed. If work hardening were due to damage to the active slip plane, the planes on which slip has not occurred being more perfect should facilitate easier slipping there than on those which have already slid. Hence, this hypothesis does not explain the formation of slip bands satisfactorily. The slip bands on a crystal under monotonic loading appear to be regular. The observed mean slip band spacing on an aluminum crystal (Brown, 1949) is about 25 p at an extension of 0.7%, and about 1 p at 2.5%. It was suggested by Brown (1949) that large numbers of dislocations accumulate on slip planes, and these dislocations produce stresses in the lattice tending to block slip within a zone on either side of the slip plane, thus causing the observed slip band spacing. However, Mott (1951) pointed out that the range of the stresses due to dislocations in a plane array is much too small to explain even the spacing of closest slip bands. Consider an edge dislocation


9

of the origin in an elastically isotropic infinite body. The stress field due to this dislocation gives (Hirth and Lothe, 1968) 511

=

Gb y(3x: + x i ) 2n(l - v) ( x f + x y ’

522

=

Gb x2(x: + x,”) 2z(1 - v ) (x: + x y ’

512

=

Gb x,(x: + x i ) 2n(1 - v) (x: + x y ’

(3.1)

where G is the shear modulus, v is the Poisson’s ratio and b is the spacing between atoms. From these expressions, it is seen that the stress at a distance of 100 lattice spacings from the dislocation is only 1/100 of the stress intensity at r = b. Thus, for a lattice spacing of 3 A (Angstrom), the stress field of a dislocation becomes rather small at a distance of 300 A. The observed slip band spacing is much larger than this. Hence, this leads Mott (1951)to conclude that the range of stress due to the presence of dislocation in a plane array is much too small to account for the observed slip-band spacings.

A. POLARIZATION OF DISLOCATIONS During plastic deformation, dislocation lines not only increase in number but also move within the crystal. The positive and negative dislocations move in opposite directions on the slip plane. The distance of movement is much larger than the lattice spacing. The concentration of the positive dislocation in one side of the crystal and negative dislocations in the other side, referred to as the polarization of dislocations, facilitates the development of long-range internal stresses, which is the main determinant of the observed slip-band spacing.

B. ANALOGY BETWEEN INELASTIC STRAIN

AND APPLIED

FORCE

When a rod is stretched beyond the elastic limit, a time-independent plastic strain e p occurs. If the temperature of a metal rod is raised, the length of the rod increases. This gives a thermal strain e‘. It the rod is

T. H . Lin

10

stressed at an elevated temperature, a time-dependent creep strain ec occurs. The analysis of a body with only thermal and elastic strain is generally known as thermoelastic analysis. Plastic strain, thermal strain, and creep strain can be treated mathematically in the same manner in examining how they cause stresses in a body; hence, these three strains grouped together are called inelastic strain, and are denoted by double prime. This inelastic strain has been called as the eigenstrain by Mura (1982), in his well-known book “Micromechanics of Defects in Solids.” We have e” = ep

+ e‘ + ec.

Denoting the elastic strain by e’, the total strain is the sum of the elastic and inelastic strains: e = e’

+ e”.

Writing this in terms of strain components in rectangular coordinates, e.. IJ = e!. IJ + e!’. IJ *

(3.2)

Neglecting the anisotropy of the elastic constants, the stress is related to the elastic strain as

r.. fJ = 6..18‘ IJ + 2pe!. JJ

=

d..d(s IJ - 8”) + 2p(e.. rJ - e!’.) IJ ’

(3.3)

where 1 and p are Lame’s constants, 8 is the dilatation, and 8” is the inelastic dilatation. Both p and G denote the shear modulus. The condition of equilibrium within a body of volume V is

r.. 1JJ. + F. I = 0

in V,

(3.4)

where the subscript after the comma denotes differentiation, the repetition of the subscript denotes summation from 1 to 3, and 6 denotes the body force per unit volume along the xj-axis. At any point on the boundary r with normal v, the i-component of the surface traction per uniter area T(”) can be written from the condition of equilibrium as

T(”) = rijvj

on

r,

(3.5)

where vj is the cosine of the angle between the normal v and the xj-axis. Substituting (3.3) into (3.4) and ( 3 . 9 , we obtain

+ 2pejj,j - (aijA82 + 2 ~ e ; , +~ )4 = 0, T(”)= Vj[6,Le + 2peij - (6,ne” + 2peij)].

6, LO,

(3.6) (3.7)


11

Writing the parenthesis term in (3.6) as a -and that in (3.7) as q",we have

+ 2peijSj+ 4 + 4 -= 0, T(")+ q(') = v,(6, LO + 2peij).

(3.8) (3.9)

F;. -and q' are respectively called the equivalent body and surface forces. Hence, the strain distribution in a body with inelastic strain under external load is the same as that in an elastic body (no inelastic strain) with the additional equivalent body and surface forces 4 and q'. The stress field caused by the equivalent forces is denoted by r; ,ihichyquals 6, LO + 2peu. The stress caused by the inelastic strain is referred to as the residual stress T;. From Eq. (3.3), ~ IJf = . r?. IJ - 8.. II LO'' - 2pe'. 1J * (3.10) This reduces the solution of stress field of a body with known inelastic strain distribution to the solution of an identical elastic body with an additional set of equivalent body and the surface forces (Lin, 1968). Due to thermal strain with thermal coefficient of expansion a and temperature T , we have e;

=

SijaT,

0'' = e$ = 3aT.

Then, the equivalent body and surface forces become

+ 2p)aTi, q'") = vi(3A + 2p)aT. -= -(3A F;.

(3.11)

(3.12)

This is the well-known Duhammel's analogy ( 1 938) between temperature gradient and the body force in an elastic medium. Hence, Duhammel's analogy is a special case of the general analogy for inelastic strain. Eqs. (3.6) to (3.9) yield the same results in Eshelby's ingenious method of cutting, restoring, welding, and relaxing in his famous paper on ellipsoidal inclusions (Eshelby, 1957).

C. STRESS FIELD OF A UNIFORMLY SLIDSLICE Consider a thin slice of metal of a uniform rectangular cross-section, shown in Fig. 4, experiencing a uniform plastic shear strain efz.This slice is embedded in an infinite isotropic elastic medium. The length of the slice is much larger than the width 2d and the thickness 2w. To calculate the

T. H . Lin

12

I

xz

FIG.4. Thin slice cross-section.

stress field caused by this uniform slip, the strain distribution in the major central portion of the slice away from the two ends is assumed to be under plane strain. Imagine that this thin slice ABCD is cut out of the medium, leaving in it a long hole of uniform rectangular cross-section. This slice after the cut is stress free and has a uniform plastic strain e&. Imagine that a uniform shear stress -2GeF2 is applied to the slice to restore it back to the original shape and size before the occurrence of the plastic strain, and that this slice is inserted back to the hole and welded back to the medium. Since actually there is no such stress applied, it is relaxed by applying an equal and opposite force 2Gef” per unit area of the boundary of the slice as shown in Fig. 5 . This force is the equivalent surface force T’’ given in the previous section. The stress field caused by this equivalent force is denoted by r$ and the residual stress field rb is obtained by Eq. (3.10) as

rf. JJ = rs. 1J - 2Gef” =

r?. IJ

within the slice, outside the slice.

(3.13)

To calculate the stress field, this equivalent force is considered to be applied in an infinite medium. The shear stress field r12caused by a uniformly distributed force Fl along the x,-direction per unit length along the x,-axis

D

--

x1

C

FIG. 5 . Equivalent surface force caused by the plastic strain.


13

at (Xl ,x2) is given by Sneddon (1951) as ‘512

=

- X2)

-FAX2

4 ~ ( 1-

V)[(XI

+ (xZ

-

- X2)2]

2(x, - XJ2 (XI

+ x,

- RJ2

(3.14)

- R2)2

Since the equivalent force per unit area on the (xl, x,)-plane A B is 2pef;, the shear stress field caused by this 2pef2 on A B is obtained by replacing Fl by 2peE dXl, and integrating this expression over X, from - d to + d :

+

(x2

-

(XI -

- W)(Xl - d ) ( ~ 2 w)’

d)2+

(3.15) Similarly, the shear stress field caused by this fictitious force on DC, CB, and DA are calculated. Adding up the resolved shear stresses caused by these equivalent forces on AB, BC, CD, and DA, we obtain the shear stress ti2 due to these forces. From Eq. (3.10), we obtain the residual shear stress field (x2

-

‘512 =

(xZ

+ At x1 = 0, x2

(x2 (XI -

=

+ wNx1 -_d _ ) -

d)2+

(x2

-

-

w) +

- d) w ) +~ (XI- d ) 2 - W)(XI

(x1

-

d)

+ w ) ~ (XI + d ) 2 + ( ~ +2 w

( ~ 2

1

.

) ~

(3.16)

0, and taking w 4 d , Ti2

=-

2peP2w n(l - v)d *

Since the stress and strain components are for the shear stress rI2 and shear strain e12only, the subscripts are dropped. Along x,-axis (x2 = O ) , y‘ =

4pePwd n(l - V) [w’+

(x: - d 2 ) - w2

+ d ) 2 ] [ ~+ 2

(XI

(XI-

d ) 2 ]*

(3.17)

This shows that the relief of resolved shear stress is directly proportional to the product of the elastic strain e p and the ratio w/d. Since w/d is very small, extremely large plastic strain e p is required to give a finite value of TI.

T. H . Lin

14

Outside of the slice, T' is positive along XI-axis. Plastic strain in this slice causes the positive resolved shear stress to increase and, consequently, the width of this thin slice (2d)to increase. This seems to explain why, in general, a slip band rapidly widens to cover the whole crystal in a polycrystal. Along the x2-axis (xl = 0), T' =

(x: - w2) - d2 4pePwd n(l - V) [(xZ - w ) ~+ d2][(x2 + w ) ~+ d2]*

(3.18)

Near the band x2 < d, w 4 d, t '

-

~ ( 1 V) d

(3.19)

4pePw 2x2 --2: - 4 0. dx2 n(l - v)d d dt'

This shows that, in those regions above A B and below CD, the relief of the resolved shear stress t' is practically constant and is the same as that in the slice. These regions did not slide when the slice first slid. This indicates that the initial resolved shear stress ti in those regions was less than that in the slice. After this slice slides, t = ti + T' + ta would be still less in these regions than that in the slice. Assuming no strain hardening, the resolved shear stress in the slice equals the critical shear stress tC,and that in those neighboring regions would still be less that tC;hence, no slip occurs in these neighboring regions. This explains why large localized plastic strain occurs in shear band. When w is small, Eq. (3.17) reduces to (3.20) Consider a slip band divided into 2N equal thin slices along its width. Each slice has a width 2d. The distance x at the centroid of the mth slice is (2m - 1)d. Let e i be the plastic strain caused by slip in the mth slice. We have N 1 4P w (3.21) t:, = e,". n(l - v)d n = -N 4(m - n)2 - 1

c

The resolved shear stresses in all the slices must be equal to ta

+ th = t c ,

m

=

1 to N .

tc:

(3.22)


15

Distance along band width Total band width FIG.6. Slip distribution in a single slip band.

Solving Eqs. (3.21) and (3.22), we obtain the plastic distribution el’s in the slip band to give a uniform resolved shear stress in the band. This distribution of e p is shown in Fig. 6. Rearranging Eq. (3.16), we have =

(XI + d)[(x; - w2) - (XI + d)2] [(xz - w)2 + (XI + d)2][(x2 + w)2 + (XI + d)2]

1.

d)[(x, - dI2 - (x2” - WZ)1 + [(XI - d(XI) 2 +- (x2 + w)2][(x1- d ) 2 + (x2 - w)21

(3.23)

Let epw be kept constant; let e p increase and let w decrease to zero; then, 51

=

(xl - d)[(x, - d)’ - x:] - (xl + d)[(x, + d)’ - x;] [(XI - d)2 + xi], [(XI + d)2 + xi12 (3.24)

1.

The resolved shear stress due to the presence of an edge dislocation along the x,-axis (0, 0, xj), with a Burger’s vector point in the x,-direction, is given by Hirth and Lothe (1968) as (3.25)

Substituting for 2wep in Eq. (3.24) the magnitude of the Burger’s vector b, it is seen that the resolved shear stress caused by the displacement of an edge dislocation of this Burger’s vector from (4, 0) to (d, 0) is exactly the same as given by Eq. (3.25). Hence, this plastic strain in this thin slice can be caused by the displacement of dislocations. This can also be interpreted as having a positive dislocation at ( d , 0) and a negative one at (-d, 0) in the solid.

16

T. H. Lin D. FREESURFACE EFFECT(Lin and Ito, 1967)

Free surface has a large effect on the plastic deformation in a slip band. Consider a most favorably oriented crystal located at a free surface of an aluminum polycrystal loaded in tension. The anisotropy of elastic constants of an aluminum crystal is small. Neglecting this anisotropy, we consider this polycrystal to be elastically isotropic and homogeneous. When it is uniformly loaded, the stress field is uniform throughout. However, the differently oriented crystals have different resolved shear stresses. All crystals are taken to have the same critical shear stress. The crystal with the highest resolved shear stress, known as the most favorably oriented crystal, will first reach the critical shear stress and slides. Under uniaxial tension, this crystal has a slip direction CY and a normal jl to the slip plane making 45"

flllffltllll-

Xlr

x;

FIG. 7. Division of the thin slice into parallelogram grids. Reproduced from Journal of Applied Physics, Vol. 38, p. 777, 1967, courtesy of the American Institute of Physics.


17

with this loading axis. A small uniform initial resolved shear stress r t p is assumed to exist in a thin slice of this crystal, and to be negligible elsewhere. Let x1, x, , and x3 be one set of rectangular coordinates and let a,p, x3 be another set, as shown in Fig. 7 . Due to this small initial shear stress in this slice, it will slide first. To determine the slip distribution along the width of the slice, this slice is divided along the width into N parallelogram grids S,, , n = 1 , 2 , ..., N , within each of which this plastic strain e& is taken to be constant. The stress fields caused by a uniform plastic strain in each grid were first determined. Then, the stress field caused by the plastic strain in the whole slice was obtained by summing those contributed by all the grids. Within each grid, this equivalent body force is zero. However across the boundary of the nth grid towards the exterior, this plastic strain drops from e:s. to zero. This induces an equivalent surface force T" of 2Ge$vj. These forces are in (xl, x,)-plane. The polycrystal is of fine grain and the equivalent force may be considered as acting in a semi-infinite solid. The thickness of the slice is much smaller than its length (dimension along x3-direction in Fig. 7). The plastic strain due to the slip is assumed to be constant along the length, so the deformation caused by the equivalent forces are taken to be of plane strain. The plane stress solution of the stress field due to a point force applied in a semi-finite plate has been given by Melan (1932). His solution was modified by Tung and Lin (1966) for plane strain. Let t[(x, X) be the stress at x due to a unit force along the xk-axis, applied at X, of this semi-infinite medium, and let &(X, a) be the corresponding Airy stress function, where x denotes (xl ,x,, x3). Lin and Lin (1974) have expressed the stress components in terms of the stress function as

(3.26)

T. H. Lin

18

with

1

0, = a r c t a n t s ) ,

I

X] =

X,

(XI

= (XI

-R 5 2

e2 < -R

(3.28)

2

x,y + (x2 - ls2)2, + x1)2+ (x2 - x,)2 -

Using this plane strain solution, we calculate the average resolved shear stress t,@ in the rnth grid caused by a unit uniform plastic strain e,Ppnin the nth grid, and denote this average stress by G,, . Then, the residual resolved shear stress tip,in the rnth grid caused by plastic strains in all grids is

where n is summed over all grids with plastic strain. Since the resolved shear stress and strain in this section refer to the a/3-slip system only, the subscripts a/3 are dropped, giving (3.29)

After plastic strain occurs, the total resolved shear stress t = ti

+ tr + ta,

where tais the resolved shear stress caused by the applied load. For a grid to start or continue sliding, this total resolved shear stress must be equal to the critical shear stress: t = ti

+ tr+ ra = t".

(3.30)

Writing this in incremental form: A t r + A t a = At".

(3.31)

Plastic deformation in metals is highly localized (Brown, 1952). Plastic strain concentrates in thin slices. The microscopic plastic strain in the thin slice is much larger than the macroscopic plastic strain in the metal. Hence,


19

the strain hardening of the thin slice is much less than that of the metal and is neglected. This gives a constant critical resolved shear stress of T', since the initial shear stress does not change with loading. With AT' = 0, we have, in any sliding grid, G,, Ae:

+ AT; = 0.

(3.32)

n=l

This equation is applied to all grids with incremental plastic strain. We have as many equations as the unknowns. Thus, the plastic strains in all grids were solved. With Poisson's ratio v = 0.3 and b / d in Fig. 7 equal to 1/10, the resulting distribution of ei0 is shown in Fig. 8. It shows that slip near the free surface is much more extensive than at the interior end of the slice. This explains why cracks generally initiate at a free surface.

IV. A Micromechanic Theory of Fatigue Crack Initiation When a fatigue loading is applied to a metal, the individual grains begin to show fine slip markings after certain cycles. As loading continues, some of the lines intensify and become dark. These intense slip bands appear to be

Distance from Free Surfacelb FIG.8. Plastic strain distribution in a slice at the free surface: is the applied residual shear stress, 'T is the critical shear stress, and p is the shear modulus of elasticity.

20

T. H. Lin

the source of fatigue cracks. If the surface of the fatigued metal is electropolished, the fainter slip markings can be removed, but not the darker bands. These are known as the persistent slip bands. The life of a specimen can be prolonged greatly by removing the surface layer periodically. A microscope study of the development of cracks in copper by Thomson el al. (1955)verified the initiation of fatigue cracks in these bands. This was also verified by straining in tension specimens prefatigued to the state where persistent bands were formed. Forsyth (1953) and Hull (1958) found thin ribbons of metals were extruded from the slip bands during fatigue loading. Extrusions of about 0.1 p thick sometimes reached a height of lop, and varied in width from about 1 p to a substantial fraction of the grain dimension. This formation of extrusions clearly is associated with shears of opposite signs on the opposite sides of an extrusion. The reverse of extrusion, i.e., intrusion, has also been observed. Both extrusions and intrusions grow monotonically in depth and width with cycles of loading. These observations are pertinent in developing the theory of fatigue crack initiation.

A. SOMEPREVIOUS THEORIES OF FATIGUE CRACKINITIATION

Following the clues that the observations on extrusions and intrusions in slip bands have provided, a number of theories of fatigue crack initiation have been proposed by different distinguished investigators. For example, Cottrell and Hull (1957) assumed Frank Read sources to exist on two intersecting slip planes, and a complete cycle of forward and reversed loading results in an extrusion and intrusion. Such a model would predict the extrusion and intrusion to form in neighboring slip bands and to be inclined to each other, but they have been found to occur in the same slip band and to be parallel to each other. Mott (1958)proposed that a column of metal containing a single screw dislocation intersecting a free surface travels a complete circuit. The volume contained in the circuit is translated parallel to the dislocation; this causes the metal to extrude. This mechanism does not explain why the dislocation under cyclic stressing does not oscillate back and forth along the same path rather than traversing a closed circuit. Clearly, some form of gating mechanism is required to convert the back and forth oscillations of screw dislocation into unidirectional circuits. McEvily and Machlin (1959) proposed a model in which two screw dislocations terminate in a surface and intersect a node where these dislocations meet.


21

Under an alternating shear stress, the two screw dislocations are assumed to shift around a circuit, causing an intrusion and an extrusion formed in the same slip band. However this model, as pointed out by Kennedy (1963), does not explain why these two dislocations travel around a circuit instead of back and forth. Wood (1956) proposed a simple model of a single operative slip system. A unidirectional stressing causes layers of metal to slide in the same direction, but forward and reverse stressing causes different amounts of net slip on different planes, and results in hills and valleys. However, this model does not explain why, under an alternate loading, the slip continue to monotonically deepen the valley and raise the peaks, as observed in experiments. Drawbacks of other theories have also been discussed by Kennedy (1963). For a dislocation to glide, first the glide must be along a certain direction on a certain crystal plane, and secondly the metal must be subjected to a resolved shear stress equal to or greater than the critical shear stress. The previously mentioned theories show the possible paths of dislocation movement that satisfy the first condition, but the resolved shear stress field caused by the dislocation movement, which has important effect on the second condition, was not considered. In the present study, this effect of this stress field, which supplies a natural gating mechanism, is shown.

B. A POLYCRYSTAL MODEL(Lin, 1981; Lin and Ito, 1969b) Fatigue cracks generally initiate at a free surface. As shown in Fig. 8, to relieve the same amount of resolved shear stress in a thin slice, a greater amount of slip is required near the free surface than at the interior of metals. For the present study, we consider a thin slice of metal in a most favorably oriented crystal at a free surface of a polycrystal that is subject to a cyclic tension and compression of low amplitude. The slip plane and slip direction of this crystal form an angle of 45" with the specimen axis. Under this low amplitude of loading, plastic deformation mainly occurs in this most favorably oriented crystal. Imperfections like dislocations exist in all metals and cause initial stress fields. For a slice of metal to extrude out of a surface, positive shear deformation must occur on one side of the extrusion and negative shear on the other. The initial stress field T~ near the free surface favorable for the initiation of extrusion is one with positive resolved shear stress above and a negative one below the slice. Referring to Fig. 9, x , and x, are a set of

T. H. Lin

22 x2

T22

FA

Oriented Most Crystal

PRO

Y

u (Along Slip Direction)

- d

FIG. 9. Most favorably oriented crystal at a free surface. Reproduced from Journal of Applied Mechanics, Vol. 105, p. 368, 1983, courtesy of ASME.

rectangular axes on a longitudinal section of polycrystalline metal subject to alternate tension and compression along the x,-axis, a and B are another set of rectangular axes with a along the slip direction and /3 along the normal to the slip plane of the most favorably oriented crystal at the free surface. From the analogy of applied force and plastic strain shown in Section III.B, an initial stress field caused by a linear variation of the initial strain e;a, from zero at the free surface to maximum at the interior boundary in the thin slice R of this crystal, was calculated by (Lin and Lin, 1983) and was


23

i

I I 1

I

I

I I

I &

B

FIG. 10. A dislocation interstitial dipole.

found to give a positive shear stress in P,and negative in Q. This initial shear stress field clearly is favorable for the initiation of an extrusion. Consider a perfect crystal. If we cut a slit through this crystal and force a sheet of metal of one atom thick into the slit, a pair of parallel edge dislocations A and B of opposite signs, forming an interstitial dipole, is produced as shown in Fig. 10. If we cut a rectangular block along the dotted line, the free length of this block will be one atomic spacing more than the corresponding length of the hole. If there are n such dipoles in a length of N atomic spacings, this will give an initial strain eLa of n / N . Hence, this initial strain can be caused by an array of dislocation dipoles. This array of dipoles was suggested by Lin and Ito (1969a) as a possible way of providing the initial strain to cause the favorable initial stress field. Recently these dislocation dipoles were observed in fatigue specimens as ladder structures (Fig. 11) in persistent slip bands, Mughrabi et al. (1981).

C. GATING MECHANISM PROVIDED BY STRESSFIELD A tensile loading causes a positive resolved shear stress tain the whole crystal. In P, the resolved shear stress will first reach the critical shear stress t Cto cause a slip. This slip causes a residual stress rr.The plastic strain and its equivalent forces caused by this slip are taken to be constant along

24

T. H . Lin

FIG. 11. Dislocation arrangement in fatigued copper single crystals (121)-section showing matrix (M) and persistent slip bands (PSB). Reproduced from ASIM ST9 675, 1975.

x,-axis; hence, ar,,/dx, = 0. From the equilibrium condition, Eq. (3.4) with no body force, we have

(The repetition of Greek subscripts does not denote summation.) Since drm,/aa is finite, arap/d/3must be also finite, and the change in rapacross the small distance between P and Q is very small. Therefore, the slip in P relieves not only the positive shear stress in P , but also in its neighboring region, including Q. This keeps the positive shear stress in the neighboring region from reaching that of P during the forward loading. Hence, only P slides in the forward loading. The relief of positive resolved shear stress has the same effect as increasing negative resolved shear stress. During the reversed loading, Q has the highest negative resolved shear stress, and hence slides. This slip causes the relief of negative shear stress not only in Q but also in its neighboring region, including P . This relief of negative resolved shear stress has the same effect as increasing positive resolved shear stress, thus causing P to be more ready to slide in the next forward loading. During the next forward loading, P has the highest positive shear stress, and hence slides. This slip again relieves the positive shear stress and increases the negative shear stress in Q, thus causing it to slide in the next reversed loading. This process is repeated and gives the natural gating mechanism to cause alternate sliding in P and Q. As a result, positive slip in P and negative


25

slip in Q increase monotonically with cycles of loading, and produce an extrusion. The interchange of the signs of the initial resolved shear stresses in P and Q will yield an intrusion instead of an extrusion. This theory explains the observed monotonic raising of extrusions and deepening of intrusions. Such an initial stress field can be obtained by a given initial strain distribution that can be caused by a distribution of dipoles (Lin and Ito, 1969a) in the slice R. This theory does not depend on cross slip and is applicable to hexagonal crystals as well as f.c.c. metals. The development of such a fatigue band depends on the existence of an initial strain eta in R . The spacing of such slices R is not expected to be regular. This seems to explain why the spacing of fatigue bands is not as regular as that of slip bands under monotonic loadings (Kennedy, 1963). D. SECONDARY SLIPAND EXTENT OF EXTRUSION AND INTRUSION

The build-up of the slip strain eLs in P and Q is caused by efa in R . If R were cut out, the free length of R would be longer than the slot by an amount referred to as the static extrusion by Mughrabi et al. (1983). This eta causes an initial compression r;, in R. Under cyclic loading, the extrusion grows and the thin slice R increases in length. This elongation causes the compression to decrease. A question raised by Mughrabi (1980) and Essmann et al. (1981) is whether the extrusion growth will cease after the extrusion has reached the magnitude of the static extrusion. The change of the direct stress T~~ in R causes changes of resolved shear stresses in all slip systems. When the decrease of compression R becomes large, its resulting residual stresses combined with the applied stress can cause a second slip system to have shear stress reaching the critical and slide. Let this second slip system be denoted by rq. The plastic strain egv caused by this slip has a tensor component eLa, just like e l in causing the positive and negative rfs in P and Q, respectively. Hence, with secondary slip the extrusion can grow considerably beyond the static extrusion as shown in Fig. 23. E . SUPPORTING EXPERIMENTAL OBSERVATIONS The preceding theory has much experimental evidence. Some of this is given in what follows. New slip lines have been observed to form in reversed loading. Tests on single aluminum crystals under cyclic loading in tension and compression by

26

T. H. Lin

Charsley and Thompson (1965) have shown that a reversal of stress after a prior forward loading gives rise to new parallel slip lines. Buckley and Entwistle (1956) also found that, on an aluminum crystal, slip lines formed during compressive loading lie between those formed in prior tensile loading. These and other tests show the occurrence of slip lines in the reversed loading to be very close to, but distinct from, those formed in forward loading like P and Q in the proposed theory. Gough (1933) tested two single crystals in reversed torsion with superimposed static tensile load. This test aims to determine whether the maximum shear stress or the maximum range of shear stress determines slip under cyclic loading. The maximum shear stress in this case acted on a plane different from that with the maximum range of shear stress. It was found that the maximum shear stress determined the slip system only in the very early stages of the test and very soon the slip changed the slip system along with maximum range of stress. The dependence of the build-up of plastic strain on the range of stress agrees with the present theory. X-ray reflection patterns of monotonically and cyclically loaded specimens are very different (Wood, 1956). The latter retain the discrete spots like that of annealed metals while the former do not (Fig. 12). This shows that slip occurrence in alternate loadings does not cause lattice straining in the bulk of the metal. Under cyclic loading, the positive shear slip lines (like P)are located close to the negative one (like Q). At some distance from the slip

FIQ.12. (a) Sharp X-ray reflection from annealed a-brass. (b) From same specimen as (a) after a unidirectional strain 150 x 0.5” twist. (c) From same specimen as (a) after 1500 reversals of plastic strain 0.5” twist and showing same reflections as (a). Reproduced from the book “Fracture,” 1959, courtesy of Technological Press, Massachusetts Institute of Technology.


27

FIG. 13. Initially straight scratches (I, b, c are displaced unidirectionally by static slip band AB. Reproduced from Trans. Metal SOC. AIME, 1962, courtesy of AIME.

lines, the stress field caused by positive slip in P is balanced by that caused by negative slip in Q . Hence, the stress field and the lattice strain is small in the bulk of the metal. Under monotonic loadings, the slip in all slip lines tends to be all of the same sign, and causes a significant average plastic strain, which causes an appreciable stress field and a lattice strain in the bulk of the metal. The preceding theory accounts for the different X-ray reflection patterns of the monotonically deformed and cyclically deformed metals. An informative experiment on slip band formation was made by Wood and Bender (1962). They tested copper circular rod specimens subject to torsion. The specimens were electro-polished and then scratched as markers with a pad carrying 0 . 5 ~ diamond dust. Some specimens were subject to alternate torsion and some subject to single twist through large angles. The deformation in a typical slip band AB of a specimen subject to single twist is shown in Fig. 13; a, b, care typical scratches, which were initially straight and continuous. It is seen that the single twist causes the scratches above A B to displace relative to those below. Figure 14 shows the deformation under cyclic torsion with scratch d , e, f and a typical fatigue band DC. It is seen that the cyclic deformation caused no relative displacement of the scratches left and right of the fatigue bands but, within the band, the scratches have displaced equally up and down, producing a zigzag. A severley slid line with positive shear such as P is sandwiched by two less severely slid lines with negative shear such as Q. This clearly agrees with the theory proposed.

28

T. H. Lin

FIG. 14. Cyclic slip band CD produces no overall displacement of scratches d , e , f within the slip band; the scratches are displaced equally backward and forward. Reproduced from Trans. Metal SOC.AIME, 1962, courtesy of AIME.

Forsyth (1953) has given a picture showing the diplacement of a grain boundary at the end of a slip band. It is seen in Fig. 15 that shears of opposite signs are closely associated. Recent tests by Woods (1973) and Winter (1974) of single copper crystals have shown that from the very early stages of fatigue tests the specimens develop into a state that contains two phases: a soft phase and a hard phase.

FIG. 15. Grain boundary at the ends of slip bands in fatigued aluminum. Reproduced from the Journal of the Institute of Metals, Vol. 2 , 1953, courtesy of the Institute of Metals.


29

FIG. 16. Extruding on two opposite surfaces of a single crystal. PSB, persistent slip band, and SGB, subgrain boundary, by Meke and Blochwitz (1980). Reproduced from Phys. Stat. Solidi,61 K5, 1980, courtesy of Physicia Status Solidi.

The regions in slices P and Q of the present model correspond to the observed soft phase, and those outside P and Q correspond to the hard-phase regions. Meke and Blochwitz (1980) have indicated that persistent slip bands protrude out in two sides of a single crystal under cyclic loading, as shown in Fig. 16. A positive initial shear stress t& in P' and Q and a negative one in P" and Q' in a single crystal, as shown in Fig. 17, would give protrusions on both faces of the single crystal.

I I ldz 4,000~

#1 10, ooop

k L 1 i 1 FIG. 17. A single crystal with slip bands.

T. H. Lin

30

V. A Quantitative Theory of Fatigue Crack Initiation* As shown in the previous section, the physical concept of the micromechanic theory of fatigue band formation has extensive experimental support. Development of these concepts into a quantitative theory of fatigue crack initiation is presently shown.

A. ANALYTICAL METHOD

Consider a fine grained polycrystal loaded along x2-axis, parallel to the plane of the free surface, as shown in Fig. 9. The thickness of the slices P, Q, and R is much smaller than the length (dimension along x3-axis). The plastic strain caused by slip is assumed to be constant along this length. Hence, the equivalent forces caused by this plastic strain are considered to act in a semi-infinite elastic medium under plane deformation. Slip in the Cup-slip system causes the equivalent forces Fl and F2 in the (xl, x,)-plane. The method of calculation of the stress field caused by slip in this slip system in one thin slice has been shown in Section 1II.D. Now, slip occurs in both slices P and Q. The same method can be readily applied. However, when the extrusion grows, causing a second slip system to slide in R , the plastic strain e:, caused by this slip has a tensor component eLu, which induces an equivalent force component F3. The presence of this F3 requires the modification of the solution of theplane-strain problem.% similar problem was shown in the analysis of prismatic bars by Lekhnitskii (1963), and is referred to as the generalized plane-strain problem. This generalized plane strain is defined as ui = ui(xix2),

This -gives tij

where 8 = ul,l

= 2G

= 1,2,3.

1

V 1 6,e + - ( u ~ + , ~u j , ~ ), 2

[1+2v

+ u2,2and

i

v is Poisson's ratio. The substitution of this

'Figures 18 to 23 are reproduced from Phil. Mag. A . , Vol. 59, No. 6, 1989, courtesy of Taylor and Francis Ltd. Figures 24 to 28 are reproduced from Journal of Applied Mechanics, to be published, courtesy of ASME.


31

expression into the condition of equilibrium in Eq. (3.4) yields

v2u,

1 ae F, +-+ - - 0 , 1 - 2~ ax, G

a+l,2,

3 v2u3 + F - = 0, G

(5.3) (5.4)

where

v = -a2 + - a2 ax:

ax;'

(5.5)

The differential equations (5.3) and (5.4) are not coupled, and can be solved separately. The solution of Eq. (5.3) for a semi-infinite solid is shown in Section 1II.D. To solve Eq. (5.4) for F3, we write 43(x, X) as Gu3(x,X). The stress is then

All other stress components are zero. For a unit concentrated force F3 at X, Eq. (5.3) gives v24, + 6(x - X) = 0, (5.7) where 6(x, X) is the Dirac delta function. With the boundary conditions of

and

(5.8)

we have

4&,

t) =

-$n(lnX,

+ lnX2),

(5.9)

where X1,X2 are given in Eq. (3.28). The three slip direction of each of the four slip planes of a f.c.c. crystal are shown in Fig. 18. The most favorably oriented crystal of a polycrystal loaded under alternate tension and compression along the x2-axis (Fig. 9) has a slip plane and a slip direction making an angle of 45" with the direction of loading. Let the a2 (Fig. 18) correspond to this system. During fatigue loading, the build-up of large local plastic shear strain in the primary slip system, positive in P and negative in Q, tends to start an extrusion or an intrusion. Consequently, an appreciably direct stress T,, will be

T. H. Lin

32

x3

E

x1

FIG. 18. Crystallographic directions of an f.c.c. crystal.

built up in R.The Schmid’s factors of all the 12 slip systems under the stress r22and those under r,, are listed in Table 1. It is seen that there are four slip systems c1, c 3 ,d l , and d2 equally favorable under T,, . Of these four, c3 is most favorably oriented under 522rwhere 522 is the alternate loading. Hence, c3 is the active second slip system in R.

B. NUMERICAL CALCULATIONS OF FATIGUE BANDS

The thickness of the slices P , Q, and R are very small as compared to their lengths. The slip in these slices are taken to be constant across the thickness. Each slice is divided into a number of parallogram grids along the length in the a direction. For numrical calculation, the plastic strain in a grid was taken to be uniform. From the plane strain solution of a semi-infinite medium, the stress field caused by a uniform plastic strain ela, in the nth grid was calculated. The average resolved shear stress r&(x) over the mth


33

TABLE 1 RESOLVEDSHEAR STRESSES IN DIFFERENT SLIPSYSTEM CAUSED BY CYCLIC LOADINGT~~ AND (a DENOTES a, DIRECTION)

T,,

Schmid factors Slip direction

Normal to slip plane 3/

Slip system

a

rd722

( -fTi,f1 i, O )

0.500

a2

a3

\iz + -Adz - -A- -4

3 ' 4

( -f2i,-f2 .i0 )

b3

CI

c2

(-A

A A fi 3 + -6 ' -3 - - 6 '

43\15

--

\ i z A f i A

(-d d2

,: +: -

-

9 -9

(d Ti,dT .

!?3 '

6

0

-0.250

0

-0.053

0

-0.164

0

0.121

0.408

0.197

0

-0.318

c3

r*x/raa

0.182

-0.408

-0.408

0.288

0.408

0.469

0

T. H. Lin

34

grid, denoted by r i K mwas , computed and is written as riKm

= -G(m,

4 ~n,;d M s n .

(5.10)

As the cyclic loading proceeds, slip occurs in P and Q and causes a residual stress field. The total resolved shear stress in the afl-slip system of the mth grid is the sum of the initial, residual, and applied stresses:

raOm= raPm i + rza,

-

Cn G(m, d;n, a P E p n ,

(5.11)

where the sum over n means the sum over all the sliding grids with plastic strain e:,". As discussed previously in Section 111, we neglect the microscopic strain hardening. This gives a constant critical resolved shear stress Arc = 0. Sliding occurs in the grids where r(x) = + r C .For an incremental applied shear stress Ar;, Arm = Arc = 0. Equation (5.11) yields G(m, a!P; n, a!@) AeLPn.

=

(5.12)

n

There are as many unknown AeL,n's as the number of equations. The plastic strain increments Ae:,n in the sliding grids for an incremental loading Ara can be readily calculated. Similarly, the stress components ria and rip in R can be calculated (Lin, 1972), and written as raa, =

C C(m, act; n, a!P)e:,,, n

(5.13)

rip, = Cn C(m, PB; n, crB)eLbn. From these calculations, it was found that rkg, in R is quite small. To find G(m, aa!;n , ap), a different approach, which gives a better physical insight, is also shown. The elastic shear strain e;,, which equals rc/2G in the sliding grids, is much less than the cumulative plastic strain e;,. The cumulative plastic shear strains e:, in P and Q are about the same. Denoting the thickness of the slices P and Q by t , we can express the displacement in R along the a! direction as -2te,,,,, or approximately -2te;,,, . The negative sign is due to the fact that the direction of the extrusion is opposite to the direction of the a! direction. This causes a tensile strain in R: (5.14)


35

Since tiSis very small and can be neglected, we have the residual tensile stress ?&

=

Et aeLp --. 1 - v 2 aa

(5.15)

With the residual stress increasing with cycles of loading, the initial compressive stress ria is gradually relieved. As discussed earlier, c3 (Table 1) is the second slip system to become active. The resolved shear stress in c3 is denoted by T~,.The initial resolved shear stress T;, varies with the initial stress components ti3, r i 3 , 5 f 3 (Lin et al., 1989). For our numerical calculation, rls is assumed to be zero. From Table 1, AtE, = -0.318 Ar22 - 0.408 (5.16) When rt., increases to 7‘ or decreases to -rC, this slip system slides and causes e&. Now, we have slip in the Cup-slip system in P and Q and in the &slip system in R . To calculate the relief of resolved shear stress T& in the cupslip system in the ith grid in P and Q, and ?& in the (tl-slip system in the kth grid in R , we divide the two slices P and Q into 2N grids, and R into M grids. The resolved shear stress in the @-slip system of the ith grid is written as ?:Pi = -GG, ski, cYP)eLaj - G(i, CYB; I , Ttl)e;,,,, (5.17) and that in the &slip system in the kth grid as

?iVk = -G(k,

01;j , cYP)eLpj - G(k, Ttl; 1, ttl)eg,,.

(5.18)

The repetition of subscript j denotes summation from 1 to 2N and that of subscript 1 denotes summation from 1 to M. G(i, t q ; j ,C Y ~is) the influence coefficient of the average residual stress T;, in ith grid due to a unit uniform plastic strain e& in thejth grid. With residual stresses given by Eqs. (5.17) and (5.18), the incremental plastic strains in all grids are calculated by Eq. (5.12). This procedure was applied to calculate the growth of an extrusion of an aluminum polycrystal. The dimensions of the most favorably oriented crystal at the free surface are shown in Fig. 9. The thickness b in P and Q is taken to be 0.01 p along the x2 direction and the thickness a in R , 0.1 p along the same direction. The initial strain eia in R is assumed to vary linearly from zero at the free surface to 3.64 x lo-’ at x , = 50p. The static extrusion is then 1/2 x 3.64 x lo-’ x 50 x f i p = 1.28 x 10-3p. Since e:s = au,/ap, the

T. H . Lin

36

static extrusion gives a plastic strain eLa of 0.091 in P and Q at the free surface. The initial strain field causes a uniform initial shear stress of 1.46 x MPa (0.20 psi) in P and - 1.46 x MPa (0.20 psi) in Q. The polycrystal is subject to a cyclic tensile and compressive loading in the x, direction. Let rEbe the excessive shear stress, defined as the initial shear stress ri plus the maximum applied shear stress ra minus the critical shear stress rc: tE

= 5’

+ ra

(5.19)

- Tc.

Here, rc is taken to be 0.370 MPa (53.5 psi). For a maximum applied shear stress 7a of 0.370MPa (53.5 psi), the excessive shear stress rEis 1.46 x MPa (0.20 psi). The plastic strain distributions etvin the second slip system in R at different cycles were calculated and are shown in Fig. 19, and the corresponding resolved shear stress (7t,J distributions are shown in Fig. 20. The plastic strain etv in the second slip system causes some cia, which in turn, causes positive ra8 in P and a negative rap in Q , just like the initial strain cia. This eivcauses additional +rLain P and -ria in Q. The sum of the initial plus residual shear stress rLa + &, on which the extrusion growth depends, in P and Q is shown in Fig. 21.

.-P

Gi

5% Distance from Free Surface

F I ~ 19. . Plastic strain distribution of the secondary slip system.


31

x1 10

40 Distance from Free Surface

20

30

5OP

FIG.20. Residual stress distribution in the secondary slip system.

1 Cycle

0.10

500 Cycles

10

20

30

40

5OP

Distance from Free Surface

Ro. 21. Initial and residual stress distribution in P and Q ( P and Q opposite sign). Key: -with secondary slip; - - - without secondary slip.

T. H. Lin

38 e 00

0.4

r9:

2500 Cycles

0.3

9)

.!! 6i C

0

.$ 0.2 m b

0.1

0

500 Cycles

----------10

20 30 40 Distance from Free Surface

501.1

x1

FIG.22. Plastic strain distribution. Key: -with secondary slip; - - - without secondary slip.

N

1000

2000 Number of Cycles

3000

Fro. 23. Plastic strain build-up at the free surface. Key: - with secondary slip; - - - without secondary slip.


39

It is seen that this sum is considerably more in the cases with the secondary slip than in those without. The plastic shear strain distributions e& in P and Q at different cycles of loading were calculated for the cases with and without the secondary slip, and are shown respectively as the solid and dotted lines in Fig. 22. The eta's at the free surfaces versus cycles of loading are shown in Fig. 23. It is seen that, for the case without considering the secondary slip, extrusion growth ceases as the extrusion approaches the static extrusion. The plastic strains eLa in P and Q at the free surface, which represent the amount of extrusion or intrusion, with the secondary slip are about four times those without the secondary slip. This explains the observed extrusion growth beyond the static extrusion as reported by Mughrabi et al. (1983), and explains that a face-centered polycrystal has greater extrusion height and intrusion depth than hexagonal polycrystals. The stress intensity factor of crack depends on the extent of extrusion and/or intrusion. Hence, this extent is important for the study of crack initiation.

C. FATIGUE BANDSWITH CREEP(Lin et al., 1990a) Now we consider the extrusion growth in an aluminum polycrystal with creep: A pure aluminum polycrystal subject to cyclic tension and compression at 204.4"C (400°F) is considered. The most favorably oriented crystal at the free surface is shown in Fig. 9. The thickness b in P and Q is taken to be 0.01 p (micron), and the thickness a in R is taken to be 0.1 p along the x, direction. The initial shear stress in P , Q is taken to vary linearly from 0 at the interior boundary to 1 . 1 x lo4 Pa (1.6 psi) at the free surface. This is expressed as (5.20)

where s is the distance from the free surface along the equilibrium conditions, 0.1 p

fi

j

7,,(s)

= -2

e;,(s)

=

s:

7;&)ds = -2.2

x 104

-(1 - v2) 7,, = 0.412 x E

(

s-

Q!

s2

direction. From

looxfip

)Pa,

(5.21)

(5.22)

T. H. Lin

40

The static extrusion is then SOJZP

,

ef,(s) ds = 0.687 x 10-2p.

(5.23)

10

The creep strain e& at the free surface corresponding to the above static extrusion is 0.486. The single crystal creep property is taken from tests of pure aluminum at this temperature performed by Johnson et al. (1953, 1955) and is represented approximately by a linearly relation (Lin and Lin, 1978, 1980).

e& = 3.26 x lo-’

m2/N [

T , ~-

1.3167 x lo6 N/m2 (190.1 psi)]/min, (5.24)

where the dot on top denotes the time derivative and ‘suO is the resolved shear stress in N/m2 in the active slip system. Creep strain is highly localized in slip bands (Brown, 1952). The creep rate in this band is much higher than the average rate of the whole crystal, and is assumed to be 1000 times this average rate, giving

e&

= 3.26

x

m2/N(’sa0 - 1.3167 x lo6 N/m2)/min.

(5.25)

Referring to Fig. 9, the thin slices P , Q, and R are divided along their lengths into thin paralellogram grids. For numerical calculation, the creep strain in each grid is assumed to be constant. The average residual resolved shear stress influence coefficients G(i,a&j, $k), denoting the average tab in the ith grid caused by a unit uniform plastic strain e& in the j t h grid, have been calculated previously using the generalized plane strain solution of a semi-infinite solid. Since creep strain induces the same equivalent forces and hence the same stress fields as the plastic strain, the influence coefficients calculated for plastic strain are identical to those for creep strain. With creep strain e& occurring only in P , Q, and e;,, only in R , the residual resolved shear stress in the ith grid due to creep strain e$ and e;,, is then written as LOi -

T:,,~ =

-W, aP;j, 4WOj - G(i, UP; k , -GUS & ; j , cWa)e&, - G(i, 01;k,ttlk;,,, .

(5.26)

The repetition of subscript j denotes summation covering all grids with e& , and k summation covering all grids with e;,, . The resolved shear stress in the ith grid is written as (5.27)

41

Micromechanics of Crack Initiation in High-Cycle Fatigue From Eqs. (5.25) through (5.27), we can write

P& = 3.26 x =

3.26 x -

m2/N(tWs- 1.3167 x lo6 N/m2(190.1 psi)/min mZ/N [&,

G(i, ~$3; k , &)e&,

-

+ ttPi- G ( i , Cyp;j , olp)ez, 1.3167 x lo6 N/m2 (190.1 psi)].

(5.28)

A similar equation can be written for e&. From the preceding, the creep strain rates in grids with resolved shear stress greater than this critical 1.3167 x 106N/m2 (190.1 psi) can be determined. Write Eq. (5.28) in matrix form:

I4

= [AlIeI

+ IB(t)I,

(5.29)

subject to the initial condition (e(t,)) = (eo),where ( ] denotes a column matrix and [ ] a square matrix. m2/N(tLsi + t& - 1.316 x lo6 N/m2).

Bi = 3.26 x

(5.30)

The series solution of Eq. (5.29) is given by Frazer et al. (1963). The complementary solution of Eq. (5.29) is given as let) = & [ A l W O )1%)

where e, is the creep strain at to, E is the base of the natural logarithm and [ I ] is the identity matrix. The particular solution is obtained by multiplying Eq. (5.29) by This gives

d dt

- (~-[~l'(e,)) =

(e,) =

e-[Al'(B(t)),

s,:

(5.32)

E - [ ~ I ( ~ - ' ) ( Bds( S ) )

The cyclic loading is taken to be rectangular; i.e., tension and compression are applied instantaneously and remain constant during each half cycle. (5.33)

T. H. Lin

42

To calculate the creep strain distribution in a given time interval 0 to T, this interval is divided into N steps with At = T / N . The A t is chosen to be small enough so the maximum eigenvalue of [A]At is less than unity. Then, the matrix [A]"(t- to)"approaches a zero matrix as n approaches infinity. The series solution of Eqs. (5.32) and (5.33) will converge. For each time step t, take the first p terms of [eo) and (ep)to obtain the creep strains. The smaller the At and the more terms used, the more accurate will be the results. In previous work (Lin and Lin, 1979, 1982), the creep strain rate was assumed to be constant in At. The solution corresponds t o p = 1. It has been found that using this series solution reduces the numerical calculation by more than six times for the same accuracy in evaluating the creep strain distributions. In the present example, the magnitude of applied load was 2.7326 x 106N/m2 (380psi), giving ta = 1.366 x lo6N/m2 (190 psi). Two loading frequencies were calculated; one is 0.345 cycles/min and the other is 345 cycledmin. In the high-frequency loading, the half period was equally divided into 10 time increments and three terms in the series solution were used. The creep strain in the second slip system ei,, in R versus the distance from the free surface at different cycles of loading under a frequency of 345 cycles per minute was calculated, and is shown in Fig. 24. The creep strain ei,, in the second slip system causes some e:a, which in turn causes additional positive +ria in P and negative in Q. The creep rate in P and Q depends on the sum of the initial and

3.0

u9)s C

'C G

2.0

$

1.0

.-a m

10' Cycles

U

C 0 0

x1

v)

0

25

5 0 ~

Distance from Free Surface FIG.24. Creep strain distribution of secondary slip. Loading: 345 c.p.m.


43

+ rQ

r:p

I

0

25

5%

Distance from Free Surface

FIG. 25. Initial and residual stress distribution in P and Q ( P and Q opposite sign). Loading: 345 c.p.m. Key: - with secondary creep; - - - without secondary creep.

1 .c

0.5

0

x1

25 Distance from Free Surface

FIG. 26. Creep strain distribution. Loading: 345 c.p.m. Key: - with secondary creep; - - - without secondary creep.

T. H . Lin

44 e

0.1

Q 0

f

v)

0.5 U

+

.-C E fx n

F u 0

10

N(1 0 5 )

30

20 Number of Cycles

FIG.21. Creep strain at free surface versus number of cycles. Loading: 345 c.p.m. Key: -with secondary creep; - - - without secondary creep; - - - - static extrusion.

0.5

0.4 Q

m 't u, 0

0.3

2 U 40%

0.2

.-C

E

fx

a al

2

0.1

0

0

20

40

60

80

100

120

140

FIG. 28. Creep strain build-up at free surface versus loading time N / J Key: secondary creep; - - - without secondary creep.

- with


45

the residual shear stresses ria + ria in P and Q. The sum at different stages of loadings is shown in Fig. 25. Note that this sum decreases considerably more slowly in the cases with the secondary slip e& than the case without. The creep shear strain distribution e& in P and Q at different cycles of loading were calculated for the cases with and without secondary slip and are shown in Fig. 26. The e&’s at the free surface, versus cycles of loading are shown in Fig. 27. This strain versus loading time N / f is shown in Fig. 28. Note that with no secondary slip, the extrusion growth ceases when the extrusion approaches the static extrusion. The creep strain e& in P and Q at the free surface, which represents the amount of extrusion or intrusion with the secondary slip, is much higher than that without the secondary slip. Calculations also have been made at a frequency of loading of 10 cycledmin, and the creep strain in P and Q versus N / f have been found to be essentially the same as that at 345 cycles/min. In the high-cycle range, the creep strain depends mainly on the time of loading. Similar calculations were made on a low-frequency loading 0.345 cycles/min. The creep strain at this low frequency depends on the number of cycles as well as the time of loading N / f .

VI. Effects of Mean Stress, Grain Size, Strain Hardening, and Overload In order to evaluate the effects of mean stress, grain size, strain hardening, and overload on the fatigue crack initiation of polycrystals, we first calculate the crack initiation of a polycrystal with none of these effects. Then, the fatigue crack initiation of the same polycrystal with these effects, one by one, is calculated. A most favorably oriented crystal at the free surface of an aluminum polycrystal under cyclic tension and compression loading as shown in Fig. 9 is considered, the thickness b of P and Q is 0.02 p, that of R , denoted by a, is 0.1 p, and the size of crystal d is 50p. The critical shear stress rc is 0.370 MPa (53.5 psi), Poisson’s ratio v is 0.3, and the shear modulus G = 26.6 x lo3MPa (3.85 x lo6 psi). The initial resolved shear stress is assumed to be 0.0346 MPa (5 psi) = =

-0 2) -

0

0.0346 MPa

elsewhere.

(5 psi)

in P , in Q,

T. H. Lin

46

The build-up of the plastic shear strain e:@ in the most favorably oriented slip system in P and Q at different cycles of loading were calculated. Without any of these effects, this plastic strain e:@ reaches 38% in 460 cycles.

A. MEANSTRESS

Two mean tensile stresses, rM of 0.0346 MPa (5 psi) and then 0.0692 MPa (10 psi), are considered. The applied shear stress is then rM + TA. Under this loading, the local plastic strain build-ups at the free surface with these two mean stresses were calculated. The variation of the number of loading cycles with mean stress necessary to give 38% local plastic strain at the free surface was computed. From these results, the ratio of alternating stress to the alternating fatigue strength plotted against the ratio of the mean tensile stress to the ultimate tensile strength is shown in Fig. 29. It is seen that the mean stress reduces the number of cycles to yield the same local plastic strain and hence the fatigue life. Two methods have been commonly used to predict fatigue failure with mean stress; one is Goodman's linear law and the other is Gerber's Parabola law (Forest, 1962). Using these two methods to calculate the local plastic strain needed to reach 38070,we found that the present theory gives results lying between Gerber's and Goodman's methods, Gerber's Parabola

I2 I

0

.005

.01

Mean Tensile Stress t~ Tensile Strength TP

FIG.29. Alternating tension-compression stress versus mean tensile stress needed to yield a 38% local plastic strain.


47

d = 50p

107.

d = 50p c =

0.5

106.5.

106,

I

N

100

200

300

400

Number of Cycles to Build Up 62% Local Plastic Strain

FIG.30. Effect of grain size and strain hardening on plastic strain build-up.

but is closer to Gerber's parabola, as shown in Fig. 29. This agrees with the majority of experimental data of fatigue failure (Forest, 1962), which fall between the Goodman line and the Gerber's parabola. B. GRAINSUE AND MICROSCOPIC STRAINHARDENING (Lin and Ito, 1971)

The two slices P and Q are taken to span the whole crystal so their widths are directly proportional to the grain size. The slice thickness and the distance between the two slices are assumed to be independent of the grain size. Thus, the effect of grain size on the local plastic strain growth in a slip band is calculated by varying the width of the slice while keeping the other dimensions the same. Two grain sizes are considered; one is 50p and the other is 1OOp. One case with zero strain hardening and one with a hardening rate C = 0.5 psi are considered. The variation of the number of cycles necessary to produce 62% local plastic strain under various alternating stresses for different rates of strain hardening and grain sizes are shown in Fig. 30. It is seen that the increase of strain hardening rate and/or the decrease of grain size decreases the rate of growth of local plastic strain. C. OVERLOAD (Lin et al., 1990b)

The retardation effect of an overload on fatigue crack growth has been extensively studied, but this effect on fatigue crack initiation is not found in the literature. From the variation of surface plastic strain versus the cycles

48

T. H. Lin

of loading calculated for a single fatigue band (Fig. 23) in 2 polycrystal, the rate of this strain decreases with cycles of loading and with the extent of extrusion. An overload causes a large increment of extrusion. After the application and removal of the overload, the rate of extrusion decreases. Generally, the number of cycles to reach a given surface plastic strain for a single fatigue band is not increased by this overload. But in the most favorably oriented crystal, there can be a number of persistent slip bands. Slip in one band can relieve resolved shear stress in other bands (Lin and Chen, 1989). This interaction effect can give a retardation effect on the fatigue crack initiation. Softening of persistent slip bands have been reported. Copper single crystal fatigue tests by Hunsche and Neumann (1986) showed a drop of critical shear stress from 35 MPa to 32 MPa, and then it remains at 32 MPa in the subsequent loading, indicating a material softening from the peak critical stress TF to a lower critical stress T;. The drop of critical shear has a large retardation effect on fatigue crack initiation. Seven persistent slip bands with different initial resolved shear stresses positive in P and negative in Q are assumed to exist in the most favorably oriented crystal at a free surface of a nickle polycrystal (Fig. 31). The elastic constants of the crystal are assumed to be isotropic. The sum of the initial and applied resolved shear stresses is assumed to exceed the critical shear stress TE only in mid-band IV at the initial loading. After this band A

P

FIG.3 1 . A most favorably oriented crystal with seven slip bands.


49

slides, its critical shear stress drops to ti. Then, the excessive shear stress tE = ti+ ta + tT- ti becomes relatively large, causing a relatively large increase of e& in P and Q. This e& in band IV not only relieves the resolved shear stresses in P and Q in this band but also relieves those in other bands. This keeps the shear stresses of other bands from reaching ti to initiate sliding. Now, if an overload is imposed, other bands will have their resolved shear stresses reaching t; and will slide. Then, their critical shear stress will drop to ti. Since ti is less than ti, even after overload removal all or part of these bands will continue to slide. Due to the interaction of slip bands, slip in one band also relieves shear stresses in other bands. Hence, the rate of slip in the band with the highest rate of slip decreases. This causes the retardation of extrusion growth and the fatigue crack initiation. As shown in Section V.B, slip in P and Q causes extrusion to grow and relieves the initial direct stress t,, in R . This change of the direct stress tua combined with the cyclic tensile and compressive stress t22can cause a second slip system to slide in R , giving a plastic strain e&. This plastic strain has a tensor component e;,, which has the same effect as the initial tensile strain eta causing the positive and negative resolved shear stresses in P and Q, respectively. The method of calculating the distributions of e& in P and Q and e{q in R for a single fatigue band has been shown in Section V.B. The same method is used to calculate these distributions of plastic strains for the multiple fatigue band. Following the same procedure as given in Section V, the growths of e& with cycles of loading with and without overload were calculated. A nickel alloy specimen was assumed to have a peak critical shear stress T; of 6.99 MPa (1013 psi) and a ti of 6.98 MPa (1012 psi). The applied cyclic shear stress used was 6.98 MPa (1012 psi). The initial stresses were assumed to be linearly varying from the maximum at the free surface to zero at the innermost grid. These maximum values were assumed to be k3.45 KPa (0.5 psi) in bands I, 11, 111, V, VI, and VII, and k13.85 KPa (2 psi) in band IV. The in R was assumed to be zero. This loading initial resolved shear stress causes the resolved shear stress in band IV to reach the tT first and slide. Then, the critical shear stress drops to 6.98 MPa (1012 psi). Different values of single-cycle overload are applied at the 1lth cycle. The variation of the numbers of cycles necessary to reach two specific values of surface plastic strain versus the overload intensity is shown in Fig. 32. Calculations were also made for single-cycle overload of 69 KPa (10 psi) applied at 11th cycle, 501st cycle, and lOOlst cycle. The results are shown in Fig. 33. It is seen that the earlier the application of the overload, the greater is its retardation effect.

tiq

T. H . Lin

50

Number of Cycles

Fia. 32. Cycles of loading versus overload necessary to yield given surface strain e&.

19.3 W a Overioad 0.03

-

at rhc 1la Cycle

68.95 KPa Overload 0.02'

68.95 P a 0verio.d at the 501" Cycle 0.01

o

68.95 W a Overload ar the 1001*Cycie

N 0

xw)

1030

wo

Number of Cycles

FIG. 33. Surface plastic shear strain versus cycles of loading.


51

VII. Combined Cyclic Axial and Torsional Loadings A theory considering free surface effect has been proposed by Brown and Miller (1973) for fatigue failure under multiaxial stress conditions. Their theory has been shown to correlate with experimental data considerably better than previous theories. Their theory is mainly for fatigue crack growth. The theory presented here is for fatigue crack initiation, and is based on the micromechanics of this initiation. A circular shaft under a combined axial and torsion has maximum combined axial and shear stresses in the outer layer of the shaft. This combined stress gives two principal stresses o1and o2on two principal planes as shown in Fig. 34. The third principal stress o3along the radial direction is taken to be zero.

(7.1)

6 3

=

0.

The three extreme values of shear stresses are then

r1 has a shear direction parallel to the free surface; T~ and r3 have a shear direction making an angle of 45" with the free surface. Single crystal tests have shown that extrusion occurs on the slip plane along the highest stressed slip direction, but does not occur when this direction is parallel to the free surface (Thompson and Wadeworth, 1958). t2and r3have maximum shear stress direction at 45" to the free surface. This gives the same inclination angle as specimens under cyclic tension and compression. Hence, the analysis for cyclic tension and compression (Section V) is applicable to evaluate the fatigue crack initiation caused by r2 and r 3 .

T. H. Lin

52

FIG.34.

Combined axial and shear loading.

However, r1 has the maximum shear stress direction parallel to the free surface. The extrusion or intrusion process will not occur in this maximum shear stress direction. But, this process may occur in some crystal with a slip direction making a small angle /3 with the free surface (Cooley and Lin, 1986), as shown in Fig. 35. Consider such a crystal with a slip plane with a normal along the y direction: t,,, =

rl cosp.

(7.3)

Another crystal with a slip plane with normal along the z direction and a slip direction making an angle /3 with the surface has

r,,

=

r1cosp.

(7.4)

The treatment of the growth of e,",is the same as that of e;,, and hence is not repeated here.

\''n

de9ree'

'

Free Surface

z

FIG.35. Normal view of the active glide plane. The slip direction /3 forms an angle /3 with the free surface. Reproduced from the Journal of Applied Mechanics, Trans. of ASME, Vol. 53, p. 551, 1986.


53

FIG. 36. Surface crystal with two slip lines P and Q. Reproduced from the Journal of Applied Mechanics, Trans. of ASME, Vol. 53, p. 551, 1986.

Two closely located thin slices P and Q having a slip direction making an angle p with the free surface are assumed to exist in a polycrystal as shown in Fig. 36. As before (Section V), P is assumed to have a positive initial resolved shear stress, and Q a negative one. Under a positive loading, the resolved shear stress in P first reaches the critical shear stress and slides. Slip in P relieves the positive shear stress not only in P , but also in Q, causing Q to more readily slide in the reversed loading. During the reversed loading, Q slides. This slip causes the relief of negative shear stress both in Q and in P , thus causing P to be more ready to slide in the positive loading. This process is repeated. In this way, positive slip in P and negative slip in Q increase monotonically with cycles of loading to initiate an extrusion or intrusion. The displacement induced by plastic strain eLDis along the p direction. The component of the displacement normal to the free surface is taken as a measure of the amount of extrusion or intrusion and the crack initiation. To calculate the residual shear stress tr,we consider the formation of an extrusion or intrusion in the center portion of the region of the crystal near the free surface. In this portion, the strain fields and the plastic strain are assumed to be independent of the z-axis (Fig. 35). As in Section 111, the ~ deformation is assumed to be of generalized plane strain ( u ~=, 0; i = x, y , 2 ) . Consider slip to occur on the xz plane along the p direction. This gives plastic strain eJs. This strain resolves into e& = eJs sin p and eJz = eJs cos p. eyz,z= 0, and the equivalent body forces are

F,

=

-2~4e;[,,~; Fy

=

- 2 ~ 4 e & ~ ; F,

=

-2~4eJ,,~.

(7.5)

T. H. Lin

54

For a line force F, per unit length along z direction, applied at x = R, y = p in a semi-infinite medium, the stress given in polar coordinates was shown by Cooley (1984) to be

where

Let g,,(x, y ; R, p) be the Green's function of the shear stress T, at (x, y) due to a unit line force applied at (R, 7)along the x direction; g,,(x, y, 2,y), the same stress due to a unit line force applied at (X,p) along the y direction; gwz(x,y , R, p), the same stress rw at (x, y ) due to a unit line force applied at (X,p) along the z direction. From Eq. (7.6), we have

The functions g,,(x, y; R, p) and g,,(x, y; X,p) were shown by Cooley (1984) to be

(1 - 2v)

+ 2(yr;

"1

x [(x + q2- 3(y - Y)']

-

fl(x

+ X)

~ (1 v)r:

[

+7 x + nr2 2(LX ~

v,l


55

Using the equivalent forces caused by plastic strain (Eq. 7.5) and these functions, the residual stresses (Cooley, 1984) are then

s

Y ; x, Y ) W ,Y ) dx dY

T&(x,Y ) = g,,(x,

(7.10) (7.11)

(7.12)

For numerical calculation of the plastic strain distributions e&,, the thin slices P , Q are divided along the x direction into the rectangular regions R , (Fig. 37), where n = 1,2, ...,UV,and N is the number of grids in one slice. The influence coefficient of the average resolved shear stress zv in the mth grid caused by a unit uniform plastic strain in the nth grid, denoted by G,, , was calculated for all grids in P and Q. As in Section V.B, sliding will not occur in the grids, where the magnitude of the resolved shear stress is less than the critical magnitude, and can occur only in grids, where the resolved shear stress equals to the critical shear stress. Neglecting strain hardening AT' = 0, we have AT, = Aza

+ 1 G,,

Aei

=

0.

(7.13)

This gives one equation for each sliding grid. Hence, there are as many equations as their are unknowns. This procedure was used in calculating the ' Y

Free Surface -1 0.lOpm

40000pm

I

I

-.----------1

1

I

---------(

I I

2

5

I 2 13

4 14

.....-

S

b

7

&llr;SliP 8

9

1

I 5 I b 17 I 8 I 9

400pm

0 X

Slip Line 0

FIG.37. Slip line grids and model dimensions. Reproduced from the Journal of Applied Mechanics, Trans. of ASME, Vol. 53, p. 551, 1986.

T. H. Lin

56 0.010f

2:

0

.-c a

After 4 cycles

0.001l

o

After 1 cycle

A

.

0

Distance from the Free Surface (pm)

FIG.38. Plastic shear strain development e; in the lower slip line Q; /3 = 12.75 degrees. Reproduced from the Journal of Applied Mechanics, Trans. of ASME, Vol. 53, p. 551, 1986.

build-up of plastic strain in P , Q under cyclic loadings. In the following, the formation of a fatigue band at the free surface of a circular shaft subject to cyclic torsion is considered (Cooley and Lin, 1986). For our calculation, as before, the critical shear stress of an annealed aluminum single crystal, tC, is taken to 369.0 KN/m2 (53.5 psi), Poisson's ratio v = 0.3, and the shear modulus p = 26.5 x 106KN/m2 (3.84 x 106psi). For one case, the initial shear stress ti ws assumed to be 27.6 KN/m2 (4 psi) in P , -27.6 KN/m2 (-4psi) in Q, and very small elsewhere. The applied shear stress is k369.0 KN/m2 cos fi (53.5 psi cos p). For this case, the plastic strain distributions in P , Q after different numbers of cycles of loadings were calculated, and are shown in Fig. 38. The shear stress distributions in P and Q at 16 cycles are shown in Fig. 39. In this figure, it is seen that the shear stress in P and Q remain nearly uniform as the initial shear stress, and the differences in shear stress in P and Q remain essentially the same over the main part of the crystal, and the plastic strain build-up is maximal at the free surface. The build-up of plastic strain at the free surface varies linearly with the excessive shear stress, i.e., tE = ta + ti + T I - tC,and varies with the direction of slip 8. The variation of this plastic strain at the free surface with p is plotted for different excessive shear stresses t E ' s in Fig. 40.For the case t E= 27.6 KN/m2, the value of /3 giving the maximum e& is 12.5 degrees, and the growth rate of this e& is 0.00061/cycle. The surface plastic strain et6 of a fatigue specimen under cyclic tension and compression calculated by Lin and Lin (1983) gives a rate of ets of 0.000016/cycle, at a tE of 0.14KN/m2 (0.02psi). In cyclic tension and


'

51

- - - . Initial Shear Stress Shear Stress After 100 Cycles .

a,

4-

-501 0

10

20

30

40

FIG.39. Internal shear stress T& of the upper and lower slip lines. Key: - - - initial shear stress; -shear stress after 100 cycles. Reproduced from the Journal of Applied Mechanics, Trans. ofASME, Vol. 53, p. 551, 1986.

("Excessive Resolved Shear Stress)

Angle of the Slip Direction, /3 (Degrees)

FIG. 40. Plastic shear strain in the lower slip line Q after 16 loading cycles. Reproduced from the Journal of Applied Mechanics, Trans. of ASME, Vol. 53, p. 551, 1986.

T. H. Lin

58

compression, the slip direction giving maximum extrusion rate does not change with T ~ The . plastic strain varies linearly with T ~ This . gives a rate of ego = 0.0032/cycle at gE = 27.6 KN/m2 (4 psi). The thickness of the slices P and Q were taken to be 0.02pm. The displacement increment normal to the free surface per cycles is

Au; = eaS x 0.02 sin 45" = 0.02 x 0.707 x 0.0032 =

4.525 x lO-'p/cycle.

Under torsional loadings with the same T ~ , Au!

= eJx x

0.02

=

0.00061 x 0.02 = 1.22 x 10-5p/cycle.

u; is considered as a measure of intrusion or extrusion, and hence a measure of crack initiation. We see that for the same excessive resolved shear stress, the rate of crack initiation under cyclic tension and compression is much higher than that for torsion for high-cycle fatigue loadings. This agrees with the experimental data for endurance limits under bending stress and torsional stress as reported by Dolan (1953). Under a combined axial and shear stress, we have two planes of maximum shear stresses. One is a1/2 making an angle of 45" with the xy-plane, causing a fatigue crack initiation rate just like that under cyclic tension and compression. This rate can be obtained from the linear line in Fig. 41. The other is T~ acting along the x direction making an angle of p with the free surface (Fig.35). This gives a fatigue crack initiation rate represented by the

1

f

- 2.00

Cyclic Tanum and Cornprawlon

LL

r.. €

t

nn

!"""I / r--

Cvclic Tinsion

>J

0.015 0.030 0.045 0.060 0.075 0.090 0.115

Ratio 01 Excessive Shear Strarr 10 Critical Shear Stress r'lr"'

FIG.41. Plastic strain rate of growth at free surface e& versus ratio of excessive shear stress to critical shear stress.


59

lower curve in Fig. 41. Among these two curves (one is a straight line), the one giving the higher crack initiation rate represents the rate of the polycrystal. This gives a method to evaluate the crack nucleation under high-cycle fatigue.

Acknowledgments

This research was supported by the Office of Naval Research through contract N00014-86-K-0153, and by the Air Force Office of Scientific Research through Grant AFOSR-89-0096. The interest of the scientific officer, Dr. Yapa Rajapakse of ONR, and the program manager, Dr. G . K. Hairtos, Col. Steve Boyce and Dr. John Botsis of AFOSR, are gratefully acknowledged. The author wishes to thank Dr. S. R. Lin, Dr. Y. M. Ito, Dr. K. S. Chi, Dr. X. Q. Wu, and Mr. Q. Y. Chen for many valuable discussions.

References Brown, A. F. (1949). Fine structure of slip zones. Nature (London) 163, 961. Brown, A. F. (1952). “Surface Effects in Plastic Deformation of Metals.” Advances in Physics, Vol. 1. Taylor and Francis, London. Brown, M. W., and Miller, K. J. (1973). A theory for fatigue failure under multiaxial stress-strain conditions. Proc. Instn. Mech. Engrs. 187, 745-755. Buckley, S. N., and Entwistle, K. M. (1956). The Bauschinger effect in super pure aluminum single crystal and polycrystals. Acta Metalhrgica 4, 352. Burger, J. M. (1939). Some considerations on the fields of stress connected with dislocations in a regular crystal lattice. Proc. Kon. Ned. Akad. Wetensch 42, 296 and 378. Burke, J. E., and Barrett, C. S. (1948). Trans AIME 175, 107. Charsley, P., and Thompson, N. (1965). The behavior of slip lines on aluminum crystals under reversed stresses in tension and compression. Phil. Mag. 8, 77. Chen, N. K., and Pond, R. B. (1952). Dynamic Formation of Slip Bands in Aluminum. J. Metals 4, 1085. Cooley, W. U. (1984). Plastic Strain Development at the Free Surface of an Aluminum Bar in Torsional Fatigue. M.S. Thesis, Department of Civil Engineering, UCLA. Cooley, W. U., and Lin, T. H. (1986). Fatigue crack initiation under cyclic torsion. J. Appl. Mech. 13, 550-554. Cottrell, A. H., and Hull, D. (1957). Extrusions and intrusions by cyclic slip in copper. Proc. Roy. SOC. London A 242, 21 1. Dolan, T. J. (1953). Stress range. In “ASME Handbook, Metals Engineering.Design,” pp. 82-88. Duhammel, M. J. C. (1938). Memore sur le calcul des actions moleclaires developees par less changements de temperature dans le corpe solides. Mem. Inst. France 5 , 440-498. Eshelby, J. D. (1957). The determination of the elastic field of an ellipsoidal inclusion and related problems. Proc. Roy. SOC. A . 241, 396.

T. H. Lin Essmann, V., Gosele, V., and Mughrabi, H. (1981). A model of extrusion and intrusion in fatigued metals. I. Point-defect production and the growth of extrusions. Phil. Mag. A . 44, 405-426. Forest, P. G. (1962). “Fatigue of Metals,” Pergamon Press, London, p. 107. Forsyth, P. J. E. (1953). Some further observations on the fatigue process in pure aluminum. J. Inst. Metals 82, 449. Forsyth, P. J. E., and Stubbington, C. A. (1955). The slip band extrusion effect observed in some aluminum alloys subjected to cyclic stresses. J. Inst. Metals 83, 395. Frankel, J . (1926). Zur Theorie der Elastizitatsgrenze under der Festigkeit Kristallinishcher Korper. Z. Phys. 31, 572. Frazer, R. A., Duncan, W. J., and Coller, A. R. (1963). “Elementary Matrices and Some Applications to Dynamics and Differential Equations,” Cambridge University Press, Cambridge, p. 45. Gilman, J. J. (1960). Physical nature of plastic flow and fracture plasticity. In “Proc. 2nd Symp. Nav. Struct. Mech.,” Pergamon Press, London, pp. 43-99. Gough, H. H. (1933).Crystalline structure in relation in failure of metals especially by fatigue. Proc. Am. SOC.Test. Mater. 33, 3. Heidenreich, R. D., and Shockley, W. (1947). Electron microscopic and electron-deformation study of slip in metal crystals. J. Appl. Phys. 18, 1029. Hirth, J. P., and Lothe, J. (1968). “Theory of Dislocations,” McGraw-Hill Book Company, New York, pp. 71-75. Hull, D. (1958). Surface structure of slip bands in copper fatigued at 293”, 90”, 20”, and 4.2”K. J. Inst. of Metals 89, 425. Hunsche, A., and Neumann, P. (1986). Quantitative measurement of persistent slip band profiles and crack initiation. ACTA Metallurgica 34, 207-217. Johnson, R. D., Shober, F. R., and Schwope, A. D. (1953). The creep of single crystals of aluminum. NACA (Nut. Adv. Comm. for Aeronautics), Tech. Note 2945. Johnson, R. D., Young, A. P., and Schwope, A. D. (1955). Plastic deformation of aluminum single crystals at elevated temperatures. NACA, Tech. Note 3351. Kennedy, A. J . (1963). “Processes of Creep and Fatigue in Metals,” John Wiley and Sons, Inc., New York. Lekhnitskii S. G . (1963). “Theory of Elasticity of an Anisotropic Plastic Body,” Holden-Day Inc., San Francisco, pp. 129-134. Lin, S. R., and Lin, T. H. (1974). Effect of secondary slip systems on early fatigue damage. J. Mech. and Phys. Solids 22, 177-192. Lin, S. R., and Lin, T. H. (1983). Initial strain field and fatigue crack initiation mechanics. J. Appl. Mech. 50, 367-372. Lin, T. H. (1968). “Theory of Inelastic Structures,” John Wiley and Sons, New York, pp. 43-55. Lin, T. H. (1972).Microstress fields of slip bands and the inhomogeneity of plastic deformation of metals. In “Proc. Symp. on Foundation of Plasticity.” Lin, T. H. (1977). Micromechanics of deformation of slip bands under monotonic and cyclic loadings. In “Reviews of the Deformation Behavior of Materials” (P. Felham, ed.), Freund Publishing House, Tel Aviv, Israel, pp. 317-351. Lin, T. H. (1981). Micromechanics of fatigue crack initiation: Theory and experimental observation. In “Proc. Mechanics of Fatigue” (T. Mura, ed.), ASME-AMD-Vol. 4T, pp. 91-100. Lin, T. H., and Chen, Q. Y. (1989). Interaction of fatigue bands. In “Micromechanics and Inhomogeneity, The Toshio Mura Anniversary Volume,” (G. J . Weng, M. Taya, and H. Abe, eds.), Springer-Verlag Publishers, New York, pp. 231-241.


61

Lin, T. H., and Ito, Y. M. (1967). Slip distribution in a thin slice of a crystal at a free surface. J. Appl. PhyS. 3, 775-780. Lin, T. H., and Ito, Y. M. (1969a). Fatigue crack nucleation in metals. Proc. U.S. Nut. Academy of Sciences 62, 631-635. Lin, T. H., and Ito, Y. M. (1969b). Micromechanics of a fatigue crack nucleation mechanism. J. Mech. Phys. Solids 17, 51 1-523. Lin, T. H.,and Ito, Y. M. (1971). The influence of strain-hardening and grain size on early fatigue damage based on a micromechanics theory. J. Mech. Phys. Solids 19, 31-38. Lin, T. H.,and Lin, S. R. (1979). Micromechanics theory of fatigue crack initiation applied to time-dependent fatigue. ASTM STP 675, pp. 707-728. Lin, T. H., and Lin, S. R. (1982). Fatigue crack initiation with creep. In “Proc. Intl. Symp. on Defects and Fracture,” Tucsno, Poland (G. C. Sih, and H. Zorski, eds.), Martinus Nijhoff Publishers, The Hague, pp. 3-13. Lin, T. H., Lin, S. R., and Chen, Q. Y. (1990b). Overload effects on the retardation of fatigue crack initiation. In “Proc. 4th Int. Conf. on Fatigue and Fatigue Thresholds,” Vol. I (H. Kitagawe, and T. Tanaka, eds.), Materials and Component Engineering Publications Ltd., Birmingham, U.K., pp. 487-492. Lin, T. H., Lin, S. R., and Cooley, W. U.(1987). Micromechanics of fatigue crack initiation under axial and torsional loadings. In “Proc. 3rd Int. Conf. Fatigue and Fatigue Thresholds,” Vol. I1 (R. D. Ritchie, and E. A. Starke Jr., eds.), Engineering Materials Advisory Services Ltd., West Midland, U.K., pp. 941-950. Lin, T. H., Lin, S. R., and Wu, X. Q.(1989). Micromechanics of an extrusion in high-cycle fatigue. Phil. Mag. A 59(6), 1263-1276. Lin, T. H., Lin, S. R., and Wu, X. Q. (1990a). Micromechanics of an extrusion in high-cycle fatigue with creep. J. Appl. Mech. 57, 815-820. MacCone, R. K., McCammon, R. D.. and Rosenberg, H. H. (1957). The fatigue of metals at 1.7’K. Phil. Mag. 4, 267. McCammon, R. D., and Rosenberg, H. M. (1957). The fatigue and ultimate tensile strengths of metals between 4.2“ and 293”. Proc. Roy. SOC. London A 242, 203. McEvily, A. J., Jr., and Machlin, E. S. (1959). Critical experiments on the nature of fatigue in crystalline materials. In “Proc. Int. Conf. on the Atomic Mechanisms of Fracture,” Technology Press, MIT and John Wiley and Sons. McLean, D. (1948). Striations: Metallographic evidence of slip. J. Znst. Metals 74, 95. Meke, K., and Blochwitz, C. (1980). Internal displacement of persistent slip bands in cyclically deformed nickel single crystals. Phys. Stat. Sol. (a) 61,5. Melan, E. (1932). “Der Spannugszustand der durch eine Einzelkraft in Innern beansprochten Halbscheibe,” Zeitschrift fur angewandte Mathematik und Mechanik 12, 343-346; correction in 20, 1940, p. 368. Mori, T., and Meshii, M. (1969). Plastic deformation of quench-hardened aluminum single crystals. Acta Met. 17, 167. Mott, N. F. (1951). The mechanical properties of metals. Proc. Phys. SOC.B 64, 729. Mott, N. F. (1958). Origin of fatigue cracks. Acta Merallurgica 6 , 195. Mughrabi, H. (1980). Microscopic mechanisms of metal fatigue. “Int. Conf. on Strength of Metals and Alloys,” Vol. 3 (P. Hassen, V. Gerold, and G . Kostone, eds.), Pergamon Press, Oxford and New York, pp. 1615-1638. Mughrabi, H. (1990). Cyclic plasticity of matrix and persistent slip bands in fatigue metals. I n “Continuum Models for Discrete Systems 4” (0. Berlin and R. K. T. Hsieh, eds.), North-Holland Publishing Company, Amsterdam, New York, Oxford, pp. 241-257. Mughrabi, H., Wang, R., Differet, K.,andEssmann, V. (1983). Fatiguecrack initiation by cyclic slip irreversibilities in high-cycle fatigue. Fatigue Mechanism STM STP 81 1, pp. 5-45.

T. H.Lin Mura, T. (1967). “Continuum Theory of Dislocations and Plasticity,” Springer, Berlin. Mura, T. (1982). “Micromechanics of Defects in Solids,” Chap. 1, M. Nijhoff, Boston. Nye, J. F. (1949). Plastic deformation of silver chloride-internal stress and slide mechanism. Proc. Roy. Soc. A 198, 190. Orowan, E. (1934). Zur Kristallplastizitat I, Tieftemperaturplastizitat und Beckersche Former]. Z. Phys. 98, 605. Polyanyi, M. (1934). Uber eine Art Gitterstorung, kie einen Kristall plastisch mechen konnte. Z . Plys. 89, 660. Pond, R. B. (1971). The non-homogeneous development and growth of slip bands. American Society of Metals, pp. 1-18. Rosenhain, W. (1905). Further observations on slip bands in metallic fractures-Preliminary note. Proc. Roy. Soc. 74, 557. Sneddon, I. N. (1951). “Fourier Transforms,” McGraw-Hill, New York, p. 402. Taylor, G. I. (1934). The mechanism of plastic deformation of crystals. Proc. Roy. Soc. A 165, 362-404.

Thompson, N., and Wadsworth, N. J. (1958). Metal Fatigue. Advances in Physics, London 7, 72-170.

Thompson, N., Wadsworth, N. J., and Louat, N. (1955). The origin of fatigue fracture in cooper. Phil. Mag. 1, 113. Thompson, N. (1959). Some observations in the early stages of fatigue fracture. Zn “Proc. Int. Conf. on the Atomic Mechanisms of Fracture,” Technology Press MIT and John Wiley and Sons, New York. Tung, T. K.,and Lin, T. H. (1966). Slip strains and stresses in polycrystalline aggregate under cyclic load. J. Appl. Mechanics 33, 363-370. Winter, A. T. (1974). A model for the fatigue of copper at low plastic strain amplitudes. Phil. Mag. 30(4), 719-738. Wood, W. A. (1956). Mechanisms of fatigue. Zn “Fatigue in Aircraft Structures” (A. M. Freundental, ed.), Academic Press, New York, pp. 1-19. Wood, W. A., and Bender, A. M. (1962). The fatigue process in copper as studies by electron metallography. Trans. Metallurgical Society AZME 244, 180-1 86. Woods, P. T. (1973). Low-amplitude fatigue of copper and copper at 5% aluminium single crystals. Phil. Mug. 28(1), 155-191. Yakutovitch, M. V., Yakovleva, E. S., Lerinman, R. M., and Buisov, N. N. (1951). Zzv. Akad. SSSR (Physical Seminars) 15, 383.


Mixed Mode Cracking in Layered Materials J . W . HUTCHINSON Division of Applied Sciences Harvard University Cambridge. Massachusetts

and

z . suo Mechanical Engineering Department University of California Santa Barbara. California

I . Introduction

.......................................................................................

64

I1. Mixed Mode Fracture: Crack Tip Fields and Propagation Criteria ................. A . Isotropic Elastic Solids ..................................................................... B. Homogeneous, Orthotropic Elastic Solids ............................................ C . Interface Cracks .............................................................................

65 66 69 72

I11. Elasticity Solutions for Cracks in Multilayers ............................................. A . Concept of Steady-State Cracking ...................................................... B. Cracks in Layers Loaded along Edges ................................................. C . A Bilayer Held between Rigid Grips .................................................... D . Small-Scale Features ........................................................................

95 105 107

IV . Laminate Fracture Test ......................................................................... A Delamination Beams ........................................................................ B. Interfacial Fracture Specimens .................. .................................... C Brazil-Nut Sandwiches ..................................................................... D . Delamination R-Curves ....................................................................

112 112 116 120 122

V . Cracking of Pre-tensioned Films ............................................................. A . Controlling Quantities and Failure Modes ............................................ B. Cracking in Films ............................................................................ C . Substrate Cracking .......................................................................... D . Interface Debond ............................................................................ E . Thermal Shock Spalling ....................................................................

126 127 131 137 143 146

90

90

. .

VI . Buckle-Driven Delamination of Thin Films ............................................... A . The One-Dimensional Blister ............................................................. B. The Circular Blister ......................................................................... C . Conditions for Steady-State Propagation of a Straight-Sided Blister .......... 63

147 149 158 163

.

Copyright 0 1992 by Academic Press Inc . All rights of reproduction in any form reserved. ISBN 0-12-002029-7

J. W. Hutchinson and Z . Suo

64

VII. Blister Tests ........................................................................................ A. Pressure Loading ............................................................................ B. Point Loading ................................................................................

167 168 171

VIII. Failure Modes of Brittle Adhesive Joints and Sandwich Layers ..................... A. Basic Results for Straight Cracks........................................................ B. Crack Trapping in a Compliant Layer under Non-zero KG ...................... C. Configurational Stability of a Straight Crack within the Layer ................. D. Interface or In-layer Cracking? .......................................................... E. Alternating Morphology ................................................................... F. Tunneling Cracks ............................................................................ Acknowledgments................................................................................

172 174 175 176 178 180 182

References ..........................................................................................

187

186

I. Introduction

The aim of this chapter is to pull together recent work on the fracture of layered materials. Many modern materials and material systems are layered. Interfaces are intrinsic to these materials, as are heterogeneities such as residual stresses and discontinuities in thermal and elastic properties. The structural performance of such materials and systems generally depends on just these features. The potential applications of fracture mechanics of layered materials ranges over a broad spectrum of problem areas. Included are: protective coatings, multilayer capacitors, thin filmlsubstrate systems for electronic packages, layered structural composites of many varieties, reaction product layers, and adhesive joints. Attention is confined in this chapter to elastic fracture phenomena in which the extent of the inelastic processes is small compared with the relevant geometric length scales, such as layer thickness. For the most part, the separate sections are designed so that they can be read independently. The main exceptions are Section 11, which presents the theory of mixed mode interfacial fracture underlying many of the applications, and Section 111, which catalogues a number of basic elasticity solutions for layered systems referred to throughout this chapter. Then follow sections on test specimens for determining interfacial toughness, fracture modes in thin films under either tension or compression, blister tests, and, lastly, failure modes of adhesive joints. We believe that most of the important fracture concepts for layered systems emerge in the analysis of these examples. One concept, in particular, that plays a central role is the idea of steady-state cracking. In almost every application considered here, a steady-state analysis provides a simplified solution that is directly relevant to design against fracture.

Mixed Mode Cracking in Layered Materials

65

This chapter builds on earlier work by many researchers, but specifically the contributions to the elasticity theory of cracks in layered materials of Erdogan and coworkers in the 1970s, which comprised most of the available solutions until recently. Special mention must also be made of the article on thin films and coatings by Gille (1985), which gives a comprehensive treatment of fracture modes without the insights from the recent developments in interfacial fracture. It is especially these recent developments that have transformed the subject. We have been fortunate to have been involved with one of the groups (that centered at the University of California, Santa Barbara) that have been concerned with the extension of both experimental and theoretical aspects of fracture mechanics to interfaces. This involvement is reflected in our approach as well as the topics that have been chosen for presentation. Structural reliability of multilayers is a fast growing field. An article written at this point is most likely transitory work, although we have tried to put various aspects into perspective, and we believe some of them are of permanent nature. Like most review articles of this kind, subject matter with various degrees of novelty that has not been published previously is incorporated. Some fill gaps, others are ready extensions, and still others are simply speculations. The writers sincerely urge the practitioners in the related disciplines to use the article critically, so that the results can be validated, expanded, or modified. A more consolidated version of the article could then emerge on a later occasion.

11. Mixed Mode Fracture: Crack Tip Fields and Propagation Criteria

There is ample experimental evidence that cracks in brittle, isotropic, homogeneous materials propagate such that pure mode I conditions are maintained at the crack tip. This appears to be true for fatigue crack growth and stress corrosion cracking as well as crack advance under monotonic loading. An unloaded crack subsequently subject to a combination of modes I and I1 will initiate growth by kinking in a direction such that the advancing tip is in mode I. A crack in a material with strongly orthotropic fracture properties, or a crack in an interface with a fracture toughness that is distinct from the materials joined across it, can experience either kinking or straight-ahead propagation under mixed mode loading depending on a number of factors, including the relative toughnesses associated with the competing directions of advance. This section gives results from studies of


66

crack tip fields for specifying criteria for straight-ahead propagation or kinking under mixed mode loading. An assessment of the competition between different directions of advance can also be made. Homogeneous materials are considered first, starting with the isotropic case and going on to orthotropic symmetry. Cracks on interfaces between dissimilar isotropic elastic solids are dealt with last. A. ISOTROPICELASTIC SOLIDS

The stress fields at the tip of a crack in plane stress or plane strain for a homogeneous, isotropic elastic solid have the well-known general form

uij= ~,(2nr)-~/~a;(e) + K , ~ ( ~ ~ C ~ ) - ~ /+~mi, C T ;sjl, (O)

(2.1)

where 6, is the Kronecker delta and r and 0 are polar coordinates centered at the tip as shown in Fig. 1. The &variations are given in many texts on fracture. They are the same for plane stress and plane strain, except 0 3 3 , which vanishes in plane stress and is given by v(oI1+ 022)in plane strain, where v is Poisson's ratio. Mode I fields are symmetric with respect to the crack line with 0i2 = 1 and 0:2= 0 on 8 = 0, while the mode I1 fields are antisymmetric with 0:; = 1 and 0;:= 0 on 8 = 0. The higher order contributions not included in (2.1) all vanish as r -,0. The T-stress, oI1= T, arises in discussions of crack stability and kinking. Thus, the singular tractions on the line ahead of the crack tip (d = 0) have the mode I and I1 stress intensity factors as amplitudes according to oZ2= ~ , ( 2 n r ) - ' / ~ , oI2= ~ ~ , ( 2 n r ) - ' / ~ .

The relative displacements of the crack faces behind the tip,

si = ui(r, e = n) - u i ( r , e = -n), X 2i I I

(G' FIG. 1 . Conventions at a crack tip and the geometry of a kinked crack.

(2.2)


67

in the region dominated by the singular fields are given by (62

9

81)= (KI,K,,)(8/E)[r/(2n)]'12,

(2.3)

where

I?

= E/(1 =

E

v2)

(plane strain)

(plane stress)

(2.4)

and E is Young's modulus. Irwin's relation between the energy release rate G for straight-ahead quasi-static crack advance and the stress intensity factors is G = (KI" K:,)/E. (2.5)

+

Next, consider a putative crack segment of length a kinking out the plane of the crack at an angle SZ with the sense shown in Fig. 1. When a is sufficiently small compared with all in-plane geometric lengths, including the crack length itself, there exists a relation between the stress intensity factors K: and Ki, at the tip of the putative crack and the stress intensity factors K , and K,, and the T-stress acting on the parent crack tip when a = 0. The relation has the form

+ c12KII+ blTa112, K:, = cZlKI+ C ~ ~ K+, ,b2Ta'12. K: = cllKI

The SZ-dependences of the c's are given by Hayashi and Nemat-Nasser (1981) and by He and Hutchinson (1989b), while the a-dependence of the b's is given by He et al. (1991). The ratio of the energy release rate of the parent crack when it advances straight-ahead to that of the kinked crack, G' = (Kj2 + K::)/E, is of the form GIG' = F W , W , v), (2.7) where F depends on the coefficients in (2.6). In addition, v/ is the measure of mode I1 to mode I loading acting on the parent crack defined by and The ratio (2.7) applies to both plane strain and planes stress. With GLaxdenoting the value of G' maximized with respect to SZ for a given W , the ratio G/G& is plotted as a function of IU, for various values of q in Fig. 2, which was taken from He et al. (1991). The kinking angle fi at


68 1

.6

4

.2

0

0'

10'

20'

30'

40'

50'

60'

70'

80'

90'

w FIG.2. Ratio of energy release rate for straight-aheadadvance to maximum energy release rate for a kinked crack as a function of v/ = tan-'(K,,/K,). Reproduced from He el a/. (1991).

which G' is maximized is plotted as a function of ty in Fig. 3 for the limit q = 0. The ratio in Fig. 2 corresponds to P(ty, q ) = F(h, ty, q). The kinking angle that maximizes G' is nearly coincident with the kinking angle for which KiI = 0, as can be seen in Fig. 3. Only for ty greater than about 50" is the difference more than one degree, and the difference between the energy release rates for the two directions is numerically insignificant. Thus,

Fro. 3. Kink angle as predicted by two criteria.


69

for all practical purposes, there is no distinction between a criterion for kinking based on maximizing G' or one based on propagation in the direction in which K,, = 0. With r = K k / E denoting the mode I toughness, kinking will initiate at a crack tip in a brittle material subject to monotonic mixed mode loading when

G

=

&I,

=

o)r,

(2.10)

where k is the ratio in Fig. 2. Once initiated, the advancing tip will be influenced by the T-stress through the q-dependence of E.

B. HOMOGENEOUS, ORTHOTROPIC ELASTIC SOLIDS Consideration will be restricted to plane cracks aligned with the principal axes of orthotropy and crack advance that is either straight-ahead or kinked at 90" parallel to the second in-plane orthotropy axis. With reference to Fig. 1 , let the orthotropy axes coincide with the xi-axes and take the plane of the crack to be x2 = 0 with its edge along the x,-axis. Introduce elastic compliances of the solid in a standard way according to 6

si,a,,

E; =

i = 1 to 6,

(2.11)

j = 1

where [Eil =

[ail=

>

(011

9

E Z Z 9 E33 9

2%3 2E13 3

5

2&121r

022 9 0 3 3 9 023 9 0 1 3 3 0121.

For the orthotropy assumed here, deformations in the (1,2) plane satisfy (Lekhnitskii, 1981)

,

bjjaj,

&. =

i

=

1,2,6,

(2.12)

i = 1,2,6

where, for i, j

=

1,2, 6, (plane stress) b.. = Si; 'J (si; - si3s,3/s33, (plane strain) 7

(2.13)

with only four independent elastic constants: b,, , b,, = bzl , bZ2,and b66 (bl6 = b26 = 0). For simply connected domains with traction boundary conditions, Suo (1990~)has shown that the stresses depend on only the following two


70

(rather than three) nondimensional elastic parameters:

1=

bldb22,

p =

(bl2

+ +b66)/(bllb22)1/2.

(2.14)

This particular choice of parameters is particularly useful for reasons that will emerge shortly. When 1 = p = 1, the in-plane behavior is isotropic (Lee,the material is transversely isotropic with respect to the x,-axis), and when just I = 1, the material has cubic in-plane symmetry. Positive definiteness of the strain energy density requires I > 0 and -1 < p < 00. The singular crack tip fields are contained in the work of Sih et al. (1965). Here, mode I and I1 stress intensity factors are defined such that (2.1) and (2.2) remain in effect, where the functions 6;and 6: now depend on 1 and p as well as 8. The displacements of the crack faces behind the tip are

(6, ,6,)= (1-3/4K1,1-'~4KII)8nbl,[r/(2~)]'~2,(2.15) where n = [(l advance is

+ p)/2]'/2.

The energy release rate for straight-ahead crack

+

G = b11n(l-3/4Kf 1-''4K6)

(2.16a)

or, equivalently, in a notation used in the composites literature, as G GI + GII, where

GI = bl,n1-3'4K:,

GI, = b l l n l - 1 / 4 ~ 2

=

(2.16b)

A crack kinking analysis as extensive as that described for the isotropic material has not been performed for orthotropic materials. Many such materials have strongly orthotropic fracture properties, wood and laminated composites being well-known examples. When kinking occurs, it often does so at a right angle to the plane of the crack (ie., Q = 90" in Fig. 1) along the plane of the grain or a laminate. Suo et af. (1990b) have shown that for a = 90" the generalization of (2.6) is (neglecting T)

KiI =

+ c~ZA'/~K,,.

(2.17)

The c's depend on p, but this dependence is rather weak. The energy release rate of the kinked crack tip, G', is related to Ki and K;, by an expression similar to (2.16), i.e.,

G' = b22n(A3/4K~2 + 11/4K:,2),

(2.18)


71

7

6

5

4

3

2

1

00

200

40'

60'

80'

tan-' (h''4 K ,,/K ,) FIG.4.

Normalized ratio of energy release rates for orthotropic material.

where I , n, and p remain defined as before. Thus, the ratio of the energy release rates for the competing trajectories can be obtained from (2.16)(2.18) as

where ( = I'/4K,,/K,. This ratio is plotted in Fig. 4.Note that it depends on the relative proportion of K,, to KI but not on their magnitudes. Suppose the main crack tip is subject to a mode I loading (K, > 0, K,, = 0). Let robe the material toughness associated with straight-ahead crack advance, and r9, be that associated with crack advance by kinking with = 90". (Note from (2.17) that the tip of the kinked crack is subject to mixed mode with Ki,/K: = 11'4c21/c11. Thus r90 must represent the mixed mode toughness for cracking parallel to the xz-plane.) If

-G> - , ro G'

r90

the crack will advance straight ahead since the condition G = ro will be reached before G' = r90. The crack will advance by kinking at 90" if the inequality is reversed. From Fig. 4, it can be seen that the condition on the

12

J. W . Hutchinson and Z . Suo

toughness ratio for kinking is (2.20) wheref(1) = 0.26,f(1/10) = 0.29, andf(10)

=

0.16.

C. INTERFACE CRACKS The emphasis of much of this chapter is on the mechanics of interfacial fracture and applications. This section introduces some of the basic results on the characterization of crack tip fields and on specification of interface toughness. If an interface is a low-toughness fracture path through joined solids, then one must be concerned with mixed mode crack propagation since the crack is not free to evolve with pure mode 1 stressing at its tip, as it would in an isotropic brittle solid. The asymmetry in the moduli with respect to the interface, as well as possible nonsymmetric loading and geometry, induces a mode 2 component. The competition between crack advance within the interface and kinking out of the interface depends on the relative toughness of the interface to that of the adjoining material. This competition will be addressed at the end of this section, but first it is necessary to consider how mixed mode conditions affect crack propagation in the interface. The article will focus on isotropic materials. Extensions to anisotropic materials are reviewed in Suo (1990a) and Wang et al. (1990). 1. Crack Tip Fields Consider two isotropic elastic solids joined along the xl-axis as indicated in Fig. 5 with material 1 above the interface and material 2 below. Let p i , E i , and vi (i = 1,2) be the shear modulus, Young’s modulus, and Poisson’s ratio of the respective materials, and let K~ = 3 - 4vi for plane strain and K~ = (3 - vi)/(l + vi) for plane stress. Dundurs (1969) has observed that wide class of plane problems of elasticity for bimaterials depend on only two (rather than three) nondimensional combinations of the elastic moduli. With the convention set in Fig. 5 , the Dundurs’ elastic mismatch parameters are

(2.21)


73

#2 FIG.5. Geometry and conventions for an interface crack.

A more revealing expression for a is a =

(El - E2)/(E, + &),

(2.22)

where Ej = EJ(1 - v;) in plane strain and Ei = Eiin plane stress. Thus, a measures the mismatch in the plane tensile modulus across the interface. It approaches + 1 when material 1 is extremely stiff compared to material 2, and approaches - 1 when material 1 is extremely compliant. Both a and vanish when there is no mismatch, and both change signs when the materials are switched. The parameter p is a measure of the mismatch in the in-plane bulk modulus. In plane strain, (2.23)

Thus, in plane strain, p vanishes when both materials are incompressible (v, = v2 = 1/2), and p = a/4 when v1 = v2 = 1/3. In plane stress, p = a/3 when v1 = v2 = 1/3. When v1 = v2, a is the same in plane strain and plane stress. In plane strain, the physical admissible values of a and p are restricted to lie within a parallelogram enclosed by a = + 1 and a - 4/3 = + 1 in the (a,p) plane, assuming nonnegative Poisson's ratios. The range of a and p in plane stress is somewhat more restricted. Representative material combinations are plotted for plane strain in Fig. 6 , in every case with the stiffer material as material 1 so that a is positive. This plot is similar to one given by Suga et al. (1988). Note that most of the (a,p) combinations in Fig. 6 fall between /3 = 0 and fi = a / 4 . Combinations that satisfy fi = 0 give rise to simpler crack tip fields than combinations with /3 # 0, and special attention will be paid to this restricted family of bimaterials in a separate section following this one.

74


0

FIG.6. Values of Dundurs' parameters in plane strain for selected combinations of materials.

Solutions to bimaterial interface crack problems were presented in the earliest papers on the subject by Cherepanov (1962), England (1965), Erdogan (1965), and Rice and Sih (1965). Williams (1959) investigated the singular crack tip fields. Here, the notations and definitions of Rice (1988) for the crack tip fields will be adopted since these reduce to the conventional notation when the mismatch vanishes. Take the origin at the crack tip, as in Fig 5 , with the crack flanks lying along the negative x,-axis. The dominant stress singularity for any plane problem in which zero tractions are prescribed on a portion of the negative x,-axis ending at the origin is of the form ow9 -

where i =

Re [Krie](2nr)- '/2at,(8, E)

+ Im [Krie](274-

(8, E ) ,

(2.24)

a, r and 8 are defined in Fig. 5 , and E

= -In(%) 1

2n

(2.25)

The complex interface stress intensity factor K = K , + iK2 has real and imaginary parts K , and K , ,respectively, which play similar roles to the conventional mode I and Mode I1 intensity factors. The quantities & and 0:; are given by Rice et al. (1990); they reduce to the corresponding quantities in (2.1) when E = 0.


75

The singular fields are normalized so that the tractions on the interface directly ahead of the tip are given by r~~~

+ ia12= ( K , + i ~ ~ ) ( 2 n r ) - ' / ~ r ' &

(2.26a)

or t722=

Re[Kr"](2nr)-'/2,

o12= Im[Krie](2nr)-"2

(2.26b)

where rre= COS(E In r) + i sin(&In r). This is a so-called oscillatory singularity, which brings in some complications that are not present in the elastic fracture mechanics of homogeneous solids, as will be discussed in detail later. The associated crack flank displacements a distance r behind the tip, di = ui(r,8 = n) - ui(r, 8 = -n), are given by

where (2.28) The energy release rate for crack advance in the interface is (Malyshev and Salganik, 1965) (2.29) which reduces to (2.5) in the absence of mismatch. Equations (2.27) and (2.29) can be re-expressed using the connection 1 - b2 = l/cosh2(ne). To help motivate the application of the crack tip fields to characterize interface toughness, it is useful to give two examples of stress intensity factors for solved problems. The problem of the isolated crack of length 2a lying on the interface between two remotely stressed semi-infinite blocks (see Fig. 7a) was solved in the early papers cited previously. For the right hand tip of the crack, K,

+ iK2 = (0; + iaz)(l + 2ie)(na)"2(2a)-i&.

(2.30)

This particular set of intensity factors depends on the elastic mismatch only through E and, by (2.25), is independent of a. The problem of the infinite double cantilever beam (see Fig 7b) loaded with equal and opposite moments (per unit thickness perpendicular to the (1,2) plane) was solved by Suo and Hutchinson (1990) as the special case of a more general solution


76

M M

Fro. 7. Two basic interface crack problems.

presented in Section 111. The solution is K , + i ~ =, 2 f i ~ h - 3 / 2 - i e ( l- p 2 ) - 1 / 2 e i w * ( u , P )

Y

(2.31)

where the function &(a, j3) is displayed in Fig. 8. 2. Crack Tip Fields and Interface Toughness with j3 = 0

When j3 = 0 (and thus

E =

0 by (2.25)),(2.26) becomes 9

(2.32)

(82,6,) = (8/E*)(KlY K2)[r/(27c)11/2.

(2.33)

((722,(712)

=

(K, K2)(27rr)-1/2,

and (2.27) reduces to

The interface stress intensity factors K , and K2 play precisely the same role as their counterparts in elastic fracture mechanics for homogeneous, isotropic solids. The mode 1 component K l is the amplitude of the singularity of the normal stresses ahead of the tip and the associated normal separation of the crack flanks, while the mode 2 component K2 governs the shear stress on the interface and the relative shearing displacement of the flanks. When j3 # 0, the decoupling of the normal and shear components of stress on the interface and associated displacements behind the tip within the zone dominated by the singularity does not occur. When j 3 # 0, the notions of mode 1 and mode 2 require some modification. In addition, the traction-free line crack solution for the displacements (2.27) implies that the crack faces interpenetrate at some point behind the tip. Both of these features have caused conceptual difficulties in the development of a mechanics of interfaces. For this reason, we have chosen to introduce the elastic fracture mechanics for bimaterial systems with j3 = 0, either exactly


77

FIG. 8. Phase factor w* for the problem of Fig. 7b.

or as an approximation. The extension for systems with /3 # 0 will be given in the following section, where it will also be argued that the effect of nonzero p is often of secondary consequence. When p = 0, take the measure of the relative amount of mode 2 to mode 1 at the crack tip to be w = tan-'(K,/K,). (2.34) The finite crack in the infinite plane, (2.30), gives

w

= tan-'(a;/a$),

(2.35)

while the double cantilever beam loaded by equal and opposite moments, (2.31), has w = w*((Y,O). (2.36) The double cantilever has symmetric geometry and loading; the asymmetry is due entirely to the elastic mismatch. Note from Fig. 8 that the specimen is in mode 1 when Q! = 0, as it must by symmetry, but develops a substantial mode 2 component when the elastic mismatch becomes significant. Efforts to measure interfacial toughness under mixed mode conditions go back some years (e.g., Trantina, 1972, and Anderson et al., 1974), as reviewed by Liechti and Hanson (1988). Parallel efforts have also been

78


underway to develop mixed mode fracture specimens designed to measure the delamination toughness associated with ply separation in polymermatrix composites (e.g., Kinloch, 1987). A series of recent experiments (Cao and Evans, 1989; Wang and Suo, 1990; and Liechti and Chai, 1990a) have focussed on the interface between epoxy and glasses, metals and plastics. Thouless (1990b) has carried out mixed mode toughness experiments for crack propagation in the interface between a brittle wax and glass. In all these systems, the interface toughness is not a single material parameter, rather it is a function of the relative amount of mode 2 to mode 1 acting on the interface. The criterion for initiation of crack advance in the interface when the crack tip is loaded in mixed mode characterized by iy is G = qv).

(2.37)

The toughness of the interface, T ( i y ) , can be thought of as an effective surface energy that depends on the mode of loading. Condition (2.37) is also assumed to hold for quasi-static crack advance when crack growth resistance effects can be disregarded. Data from Wang and Suo (1990) for a crack in a plexiglass/epoxy interface is shown in Fig. 9. This data was obtained using a layer of epoxy sandwiched between two halves of a Brazil nut specimen. The specimen, which will be considered later in Section IV.C.2, enables the experimentalist to vary the mix of loading from pure mode 1 to pure mode 2 by varying the

O0

I

I

I

I

20°

40'

60'

80'

\I' FIG.9. Interface toughness function for a plexiglass (#l)/epoxy(#2) interface. Obtained using a Brazil nut specimen by Wang and Suo (1990).


79

angle 6 of the compression axis (see the insert in Fig. 9). For the plexiglass (#l)/epoxy(#2) interface in plane strain, (Y

= -0.15,

p

= -0.029,

E

= 0.009.

(2.38)

The error in taking p = 0 is negligible for this system as will be clear in the next section. Note, for example, that the error in G in (2.29) from this approximation is less than 0.1 To. 3 . Phenomenological Characterization of Interface Toughness

A micromechanics of interface toughness is not far advanced. An overview of various mechanisms responsible for the strong dependence of interfaced toughness on mode mixity is given by Evans et al. (1990). Two primary mechanisms are asperity contact and plasticity. Asperities on the fracture surfaces will tend to make contact for some distance behind the tip when mode 2 is present along with mode 1. A micromechanics model of shielding of the tip due to asperity interaction was presented by Evans and Hutchinson (1989). That model led to a prediction of T(w) in terms of a nondimensional measure of fracture surface roughness. Crack tip plasticity also depends on ty, with the plastic zone in plane strain increasing in size as ltyl increases, with G held fixed (Shih and Asaro, 1988). When an interface between a bimaterial system is actually a very thin layer of a third phase, the details of the cracking morphology in the thin interface layer can also play a role in determining the mixed mode toughness. Some aspects of cracking at the scale of the interface layer itself will be discussed in the final section of this chapter. The approach for the time being is that the interface has zero thickness and is modeled by the toughness function T(w) which, in general, must be determined by experiment. A simple, one parameter family of mixed mode fraction criteria that captures the trend illustrated by the data in Fig. 9 is

Ei‘(K:

+ 1K;) = GF.

(2.39)

The parameter A adjusts the influence of the mode 2 contribution in the criterion. The limit A = 1 is the “ideally brittle” interface with initiation occurring when G = G,C for all mode combinations. This limit coincides with the classical surface energy criterion. When 1 = 0, crack advance only depends on the mode 1 component. For any value of A , G,C is the pure mode 1 toughness. The criterion can be cast in the form (2.37) where the mixed mode toughness function is

T(v)

=

G,C[1 + ( A - 1) sin2ty]-’.

(2.40)


80

-

r(y)=G:[ l + ( ~ - l ) s i n * ~ ] ~ ’

6-

-

_________trend line of plexiglasslepoxy data

2-

7

1

1

I

I

I

I

I

I

I

I

FIG. 10. A family of interface toughness functions and comparison with data for a plexiglass/epoxy interface (represented by the broken line).

The toughness is plotted as a function of tp in Fig. 10 for various values of 1. Included in this figure is the data for the plexiglass/epoxy interface, which is approximately represented by the choice 1 = 0.3. This particular interface displays a toughness that is far removed from ideally brittle behavior. The family of criteria (2.39) was extended to include a mode 3 contribution by Jensen et al. (1990). In a slightly different form, this family of criteria has been used for some time to characterize interlaminar failure in fiber reinforced composites (cf. Kinloch, 1987). When /?= 0, one can introduce “components” of G according to

a,

(GI G2) = J%’(G 9

(2.41)

such that G = G1 + G,.+ Alternatively, for a crack in a homogeneous orthotropic material, GI and G2 can be defined using (2.16). The criterion (2.39) can be rewritten as (Gl/Gf) + ( G 2 / G 3= 1 ,

(2.42)

where G,C = G f / 1 has the interpretation as the pure mode 2 toughness. Other phenomenological criteria have been proposed to characterize mixed mode toughness data for interlaminate fracture (e.g., Kinloch, 1987). Two alternatives to (2.40) are now given which have qualitative features ‘The components can be regarded as the work of the normal and shear tractions on the interface through their respective crack face displacements as the crack advances. This decomposition does not exist when # 0.


4

81

b)

FIG.11. Alternative families of interface toughness functions.

that may more realistically reproduce data trends for interfacial fracture: T(ty) = Gf(1 + tan2[(1 - A)ty])

(2.43)

T(ty) = Gf[1 + (1 - A)tanZty].

(2.44)

and These are plotted in Fig. 1 1 . Both coincide with (2.40) in the limit A = 0, i.e., they reduce to a criterion based on a critical value of K , , independent of K 2 . Both are ideally brittle with A = 1 . According to (2.43), the toughness increases sharply as ty 90" (mode 2), as opposed to (2.40), which has the toughness leveling off as ty 90". Equation (2.44) models the toughness as unbounded as ty -,90" for all A < 1. While this feature should not be taken literally, it did emerge in the simple model of mixed mode interface toughness due to asperity contact of Evans and Hutchinson (1989). Of the three formulas for r(ty),(2.44) most accurately reflects the trends of that model. All three of the interface toughness functions T(ty) are symmetric in ty. In general, symmetry of interface toughness with respect to ty should not be expected. Some evidence that r(ty)is asymmetric for an epoxy/glass interface will be presented in the next section. -+

-+

4. Interface Toughness with /3 # 0

When /3 # 0, the notion of a mode 1 or a mode 2 crack tip field must be defined precisely, and the possibility of contact of the crack faces within the region dominated by the near tip K-fields must be considered. As noted by Rice (1988), a generalized interpretation of the mode measure is the most important complication raised by the oscillatory singularity, and the

82

J. W. Hutchinson and 2. Suo

approach recommended here follows largely along the lines of one of his proposals. First, a definition of a measure of the combination of modes is made that generalizes (2.34). Let I be a reference length whose choice will be discussed later. Noting the stress distribution (2.26b) on the interface from the K-field, define v/ as (2.45) where K = K 1 + iK2 is the complex stress intensity factor. For a choice of I within the zone of dominance of the K-field, (2.45) is equivalent to (cf. (2.26b))

w

= tan-’[

(“>

022 r = i

1.

(2.46)

Moreover, the definition reduces to (2.34) when /3 = 0, since I’e = 1 when E = 0. When E # 0, a mode 1 crack is one with zero shear traction on the interface a distance I ahead of the tip, and a mode 2 crack has zero normal traction at that point. The measure of the proportion of “mode 2” to “mode 1” in the vicinity of the crack tip requires the specification of some length quantity since the ratio of the shear traction to normal traction varies (very slowly) with distance to the tip when /3 # 0. The choice of reference length I is somewhat arbitrary, as will be made clear in the following. It is useful to distinguish between a choice based on an in-plane length L of the specimen geometry, such as crack length, and a choice based on a material length scale, such as the size of the fracture process zone or a plastic zone at fracture. The former is useful for discussing the mixed mode character of a bimaterial crack solution, independent of material fracture behavior, while the latter is advantageous in interpreting mixed mode fracture data, as will be discussed. When there is the need to keep the two types of choices clearly distinct, the notation (w, I) will be used for a choice based on the specimen geometry and (@,1)will be reserved for a material-based choice. The solution for the complex stress intensity factor to any plane elasticity problem for an interface crack will necessarily have the form

K = (applied stress) x FL1’2-i6,

(2.47)

where L is some in-plane length, such as crack length or uncracked ligament length, and F is a complex-valued, dimensionless function of dimensionless


83

groups of moduli, and in-plane length quantities. Equations (2.30) and (2.31) are two examples. The term KI" in the definition of w will therefore always involve a dimensionless combination such as (I/L)ie = exp[iE ln(l/L)]. For example, the bimaterial double cantilever beam specimen (2.3 1) has v/ = w* ( a,P )

+ ~ln(l/h),

(2.48)

which generalizes (2.36). The freedom in the choice of I in the definition of w is a consequence of the simple transformation rule from one choice to another. Let w1 be associated with I , , and v2 with l z . From the definition in (2.45) one can readily show (2.49) wz = ly, + E 1n(l2/I1). Thus, as noted by Rice (1988), it is a simple matter to transform from one choice to another. In particular, toughness data can readily be transformed, as will be discussed in the following. Let 1 denote a length characterizing the size of the fracture process zone or, perhaps, the typical size of the plastic zone at fracture, and let @beassociated through (2.45). Since small-scale yielding or a small-scale fracture process zone is assumed, 1 necessarily lies within the zone of dominance of the K-field. Given the choice 1, the criterion for interface cracking can again be stated as (2.37), i.e., G = r(@, I), (2.50) where the implicit dependence of the toughness function on has been f ) is the critical value of the energy release rate needed noted. In words, r(@, to advance the crack in the interface in the presence of a combination of tractions whose relative proportion is measured by @. By (2.49), change in one choice of length in the definition of w to another only involves a shift of the y/-origin of I' according to (2.51) as depicted in Fig. 12. When E is small, the shift will generally be negligible even for changes of I of several orders of magnitude. This is the case for the plexiglass/epoxy interface (2.38). An illustration for which the &-effectis not negligible in reporting interface toughness is discussed shortly. In discussing the mixed mode character of a given elasticity solution, it is generally convenient to identify I with an in-plane length of the geometry, such as L in (2.47). For example, if for the double cantilever beam specimen

84


I

I

Lw2

h

l

FIG. 12. Procedure for shifting toughness function from one choice of reference length to another.

one picks 1 = h, then by (2.48),

w

=

0*(%8),

(2.52)

which is independent of the size of the specimen.+ This is necessarily a feature of any choice of 1 that scales with an in-plane length. By contrast, for a choice i that is fixed at some microstructural length, $ varies with specimen size, e.g., for the double cantilever specimen, $ = o*(cy,p)

+ ~ln(l/h).

(2.53)

This reflects the fact that the ratio of oIzto oZ2at a fixed distance r = r^ ahead of the tip varies as the specimen size changes. Standard arguments underlying the mechanics of fracture, based on Irwin’s notion of autonomous crack tip behavior, require that r($,1) be independent of specimen size (assuming, of course, that small scale processes are in effect), while T(w, I ) will depend on specimen size if E # 0 when 1 scales with specimen size. This property, together with the interpretation of mixity in (2.45) in the vicinity of the fracture process zone, favors the choice of a material-based 1for presenting toughness data. Liechti and Chai (1990a, b) have developed a bimaterial interfacial fracture specimen that is capable of generating the interface toughness function r over essentially the full range of w. A schematic of their plane strain specimen is shown in the insert in Fig. 13. The in-plane length of the specimen is long compared to the thickness h of each layer. The bottom ‘The fact that I = h obviously lies outside the zone of dominance of the K-field is of no consequence. The essential point is that any choice of I is acceptable as long as it is recorded along with the result for y , and as long as one is cognizant of the transformation rule.


85

FIG.13. Data of Liechti and Chai (1990a) for an epoxy (#l)/glass(#2) interface: 9, is based on 1, = 12.7 m m and t j i 2 on i, = 127pm. The solid curves are r‘(&), where r is given by (2.44).

surface is rigidly held and the upper surface is attached to a rigid grip that can impose a horizontal, U , and vertical, V , in-plane displacement. The solution to the problem when the layers are infinitely long and the interface crack is semi-infinite was used by Liechti and Chai to obtain the values of K , and K2 (and G and ty) associated with the measured combinations of U and V at which the crack propagated in the interface. For plane strain, the solution is (see Section 1II.C)

where w is a real quantity that depends on p , / p 2 , v, , and v2 and

Let y measure the relative proportion of U to CV applied to the specimen, and define it by y = tan-’[~/(c~)].

(2.56)

Then, with 1 as the reference length, (2.45) gives ty = y

+ o + eln(l/h).

(2.57)

86


The data for r(P, 1) in Fig. 13 was measured by Liechti and Chai for an epoxy(# l)/glass( #2) interface with the following properties for the system: E, = 2.07GPa, E2 = 68.9GPa, v 1 = 0.37, v2 = 0.20, and h = 12.7mm. The plane strain Dundurs' parameters and the oscillation index are a = -0.935,

p

= -0.188,

and

E

= 0.060.

(2.58)

For this system o = 16" (see Section 1II.C). Liechti and Chai took r^ = 12.7 mm in their definition of w, coinciding with the thickness h of the layers. Liechti and Chai recorded plastic zones in the epoxy to be approximately on the order of 1pm when @ = 0" and 140,umwhen @ 90". If instead of I= 12.7mm, r^ is chosen to be two orders of magnitude smaller (i.e., r^= 127pm), the shift in the @-originfrom (2.49) or (2.51) is -15.8". This choice seems somewhat more natural in terms of the interpretation given earlier since now r^ lies well within the zone of dominance of the K-field and has a microstructural identity. This choice also places the origin of the @axis (i.e., "mode 1" for this choice of I) at the approximate minimum of r and roughly centers the data, as can be seen in Fig. 13. Nevertheless, some asymmetry in r with respect to i,D still persists. Included in this figure is the toughness function T(w) from (2.44) for two choices of A, with G,C chosen to coincide with the measured value at = 0. Apart from the asymmetry in the data, a A-value between 0 and 0.5 would seem to give an approximate characterization of the data over the range of I) shown. Other important aspects of the mixed mode fracture behavior of this system have been discussed by Liechti and Chai (1990a). These include possible correlation of the strong increase in toughness with mode 2 with either fracture surface roughness or plasticity, and the role of contact between crack faces when the loading becomes dominantly mode 2. When interpenetration of the crack faces is predicted on the basis of the formulation for a traction-free line crack, the consequences of contact must be taken into account in any application of the solution to fracture. The bimaterial problem with /3 # 0 is unusual in that interpenetration of the faces always occurs according to (2.27). This feature of the interface crack problem was noted in the earliest papers on the subject, and solutions to specific problems posed with allowance for contact have been produced (Comninou, 1977,and Comninou and Schmueser, 1979). Fortunately, under most loadings likely to be of concern, the contact zone predicted by the elasticity solution is tiny compared with relevant near tip physical features such as the fracture process zone or the plastic zone. The larger the

=

e2


87

proportion of mode 2, the more likely is contact of the crack faces to be an issue.+ To see this, a rough estimate of the size of the contact zone is obtained. The estimate is that of Rice (1988), as elaborated on by Wang and Suo (1990). Here, however, emphasis is placed on a definition of I,? in (2.45) based on a microstructural scale length l. For r < 1, it will be assumed that the fracture process or other inelastic effects supercede linear elasticity. Using the definition of ty in (2.45), one can readily show that the normal crack face displacement in the near tip region from (2.27) is

d2 = Id2 + id,[ cos[@ + eln(r/l)

-

tan-'^^)].

(2.59)

Consider the condition for the crack to be open (6, > 0) for 1< r < L/lO. The factor 1/10 is arbitrary, but the near tip fields should not be expected to retain accuracy for r larger than some fraction of L . If contact occurs outside the preceding range, it must be assessed using the full solution. If E > 0, the stated condition is met if R n --+2~ 1 . 5 .

depends on the notch geometry. Secondly, composites usually exhibit Rcurve behavior: Fracture resistance increases as the crack extends. This can be caused by bridging fibers or matrix ligaments in the wake. However, once the delamination is sufficiently long, a steady state should be reached: Both driving force and toughness become independent of the delamination length and initial flaw geometry. The following example establishes the transient zone size for the driving force. The R-curve behavior will be discussed in Section 1V.D. Figure 17 shows a delamination crack nucleated from a sharp notch and driven by an axial tension. Similar problems have been studied by several authors (e.g., O'Brien, 1984, Thouless et al., 1989). The solution that follows is taken from Suo et al. (1990b). The delamination is mixed mode. The energy release rate takes the dimensionless form

94


where EL= l/bll is the effective Young's modulus in the longitudinal direction, L and p are orthotropy parameters, all defined in Section 1I.B. The dimensionless function g depends on the indicated variables. Notice that A and a/h affect the final results only through the product L'14a/h, as identified in the original paper using orthotropy rescaling. This detail turns out to be important in understanding the orthotropy effects, as will be seen shortly. Figure 17 plots the solution obtained by finite elements, with p = 1 . Observe that the energy release rate becomes independent of L'14a/h when the delamination is sufficiently long. An inspection of Fig. 17 suggests that the transient-zone size is given by L'14a/h = 1.5, or

a / h = 1.5(EL/E,)'I4,

(3-4)

where ET = l/bzz is the effective Young's modulus transverse to the fiber = 2. Condirection. For most polymer composites and woods, sequently, a split longer than about three times the notch depth is subject to a constant driving force. Equation (3.4) also reveals that elastic orthotropy tends to prolong the transient zone by a factor of (&/.&)'I4, as compared with the isotropic counterpart. Finite element calculations (not shown here) also indicate that the size of the transient zone is not significantly affected by p within the practical range, so that (3.4) remains valid for general orthotropic materials. An accurate approximation for the steady-state mixed mode energy release rates at the delamination tip in Fig. 17 is (Suo 199Oc)

This steady-state solution G = GI + GII, is indicated in Fig. 17. In conclusion, the steady-state condition can usually be easily attained in practice. These steady-state solutions are of unique significance considering the variety of uncertainties associated with the transient state. Mathematically, the steady-state concept allows one to bypass some messy intermediate calculations. Although an accurate estimate of the transient-zone size may not be available for each steady-state solution described in the rest of the section, we feel that, in conjunction with some heuristic judgment, these solutions can be used to assess technical structures.


95

A

p2

-t

H

M2

B.

v

CRACKS IN

LAYERS LOADEDALONG EDGES

The problems to be disussed in this section are sketched in Fig. 18. The layer can be of one material or bimaterial, the material isotropic or orthotropic, the crack along the interface or in the substrate. The relation is sought between the applied loads and the mixed mode stress intensity factors. 1. A Homogeneous, Isotropic Layer Depicted in Fig. 18 is the cross-section of an infinite layer containing a half-plane crack. The geometry is fully specified by h and H , the thicknesses of the two separated arms. The layer is isotropic, homogeneous and linearly elastic, and is subject, uniformly along the three edges, to axial forces and moments per unit width Piand M i . The problem at various levels of generality has been considered by several authors (Tada et al., 1985; Williams, 1988; Suo and Hutchinson, 1989b; Schapery and Davidson, 1990). The results in the first two of these references contain conceptual errors. The complete solution presented below is taken from Suo (1990~). a. General Solution The near-tip stresses are consistent with the mixed mode crack tip field, with stress intensity factors KI and K,, to be determined. Far from the tip, the three edges are characterized by the linear strain distributions for elementary beams. The energy release rate equals the difference of the strain energy per unit length per unit width stored in the edges far behind and far ahead of the crack tip. Thus,

M: 2‘’ G = -+12-+-+12---2E h h3 H

[“

p3’ H3

h+H

12

M3’

]

(h + H ) 3 ’

(3.6)


96

where E is the effective Young's modulus defined in (2.4). This result may be derived alternatively by using the J-integral (Rice, 1968; Cherepanov, 1979). The preceding energy accounting does not separate the opening and shearing components. The partition is simplified by linearity and dimensionality, coupled with the Irwin relation (2.5). Consequently, the stress intensity factors take the form

All the preceding quantities except o are determined by elementary considerations. Specifically, P and M are linear combinations of the applied loads:

and the geometric factors are functions of 11: 1

-=

U

1

+ 411 + 6q2 + 3q3,

1 V

-=

12(1

sin y + 43, Jvv- 6q2(1 + 11).

(3.9)

Accurate determination of o,which depends only on 11, is nontrivial. The elasticity problem was solved rigorously (with the help of numerical solutions of an integral equation). The extracted w varies slowly with in the entire range 0 Itf I1 , in accordance with an approximate formula =

52.1" - 3'11.

(3.10)

This is a linear fit of the numerical solution, and the error is believed to be within one percent. b. A Mixed Mode Double Cantilever Beam Several special cases are discussed here to illustrate the richness of the solution. First consider a double cantilever beam as in Fig. 19. The specimen is mode I if the crack lies on the mid-plane, but mixed mode if the crack is off the mid-plane. This has been used recently to study mixed mode


97

40'-

30'-

20°-

I

I

I

I

2

I

I

I

I

0.4

0.6

0.8

1.0

FIG. 19. The insert shows a double cantiliver beam with a crack off the mid-plane. The mode mixity v/ = tan-'(K,,/K,) is plotted against the offset y / b .

fracture of an adhesive layer by Thouless (1990b). On Fig. 19, the mode mixity, I,V = tan-'(KII/KI) = o + y - n/2, is plotted against the offset y / b . Focus here is on the configurational stability of an homogeneous specimen when the crack is slightly off the mid-plane as positioned, for example, in the fabrication of the specimen. As indicated by the sign of K,, near y / b = 0, a crack off the mid-plane will be driven further away from the mid-plane. The mid-plane crack is thus configurationally unstable. Crack path stability will be further discussed in Section VIII. c. Exact Solutions for the Case H

=

h

Next consider the crack on the mid-plane and subjected to the general edge loads. The crack path selection is seldom an issue in the composite testing since cracks are usually confined to run along the fiber direction. The exact solution for o can be obtained for this case by considering a special loading M I = M2 = M and all others being zero. By symmetry,


98

6,

Specimen 4

c&rl c

12M

E,h3

G,, 0

M

FIG. 20. Several exact solutions: (a) a pure mode I specimen (double cantilever beam); (b) a pure mode I1 specimen (end-loaded split); (c) a mixed mode specimen (four-point bend); (d) a mixed mode specimen (crack-lap shear).

K,, = 0, which, substituted into (3.7), gives o = c o s - ' w ) = 49.1". The full solution (3-7) can therefore be specialized to K I =GPh-'/'

+ 2fiMh-3/2,

P = PI - 3P3 - $M3/h,

M

K,, = 2Ph-'/', =

MI - iM3.

(3.11)

Several useful edge loads are illustrated in Fig. 20. The mixed mode energy release rates listed are valid for an orthotropic material layer with a principal material axis coincident with the longitudinal direction. Geometries a and b are pure mode I and pure mode 11, respectively. Geometries c and d are mixed mode. d. Surface Layer Spalling As the last example, consider a sub-surface crack in a semi-infinite plate (11 = 0) as illustrated in Fig. 21. The problem was solved by Thouless et al.


99

P

FIG.21. Spalling of a surface layer due to edge loads.

(1987) in a study of impact spalling of ice sheets. The complete solution is KI

=

1 ~ [ p h - cos o

+ 2 f i M h - 3 / 2sin W I , (3.12)

KII

=

1

[Ph-

'/' sin o - 2 f i ~ h - cos ~ /0~1,

where o = 52.07'. Contrary to one's intuition, a significant amount of mode I1 component is caused by pure bending. The solution will be used in Section V to study decohesion of pre-tensioned films and thermal shock spalling. 2. A Homogeneous, Orthotropic Layer The same geometry in Fig. 18 was also analyzed for orthotropic solids (Suo, 1990~).The layer lies in a principal material plane and the crack runs in principal axis-1 of the solid. The energy release rate expression (3.6) remains valid but the longitudinal tensile modulus EL should be used, namely, GI

'[.\lhv

=-

2EL

1' + 1'

M + msin(o

+ y)

,

(3.13a)

M s i n o - mcos(w

y)

,

(3.13b)

coso

where P , M , U, V , and y are given by (3.8) and (3.9). The quantity o depends on q and p, but not A. An integral equation method was used to determine o,and the results indicate that the influence of p within its entire practical range is below one percent, so that (3.10) is an excellent approximation for orthotropic materials. When the stress intensity factors are

100


needed, the Irwin-type relation appropriate for orthotropic materials (2.16) must be used. Notice that all the quantities in the brackets of (3.13) except for w do not depend on material parameters. Further, w may be approximated by (3.10), which is also independent of any material parameters. Consequently, the energy release rates of the two modes are essentially the same as their isotropic counterparts, except that the longitudinal tensile modulus should be used. 3 . A Symmetric Tilt Strip

Imagine two identical layers cut from an orthotropic solid at an angle 4 to principal material axis-1 (Fig. 22). The thickness of the two layers are equal, designated as h. The compliances sll,s22, s12,and s66 are referred to the principal material axes. The two layers are bonded to form a symmetric tilt boundary, with a semi-infinite crack lying along the interface. The tilt angle is 4 and the tilt axis, or the crack front, is one of the principal axes of the orthotropic solid. The general edge loads are applied. The stress field around the crack tip is square root singular. The stress intensity factors are defined such that, asymptotically, traction on the grain boundary varies with the distance r from the crack tip according to a,, = ( 2 n r ) - 1 ’ 2 ~ 1 , T~ =

(3.14)

The Irwin-type relation for a crack on the symmetric tilt grain boundary is (Wang et al., 1990)

GI = [bllb22(1+ p)/2]’/2(A-’/4 cos24 + GII = [b11b2,(l + p)/2]”2(L-’/4sin2$

sin2+)K:,

+ A l l 4 cos24]K&,

(3.15)

where the compliances are referred to the principal axes; see Section 1I.B.

A’

FIG. 22. Two identical grains of an orthotropic crystal form a symmetric tilt boundary, with the principal crystal axis at an angle @ from the interface. The sample is under general edge loads.


101

Solid 1

Solid 2

Ah

M2

Neutral Axis

FIG.23. A bilayer with a half-plane interface crack. The neutral axis of the composite layer is indicated.

The analytical solution is found for the problem. Expressed in energy release rates, it is GI

P

= 3b:,(P =

PI -

+ 2M/h)’/h, iP3 -

$M3/h,

GII = 4btlP2/h, M

=

(3.16)

M 1 - -iM3.

Here, b:, is the compliance in the x direction, which is related to the principal compliances by bf,

=

(b,, b22)1’2(A1’2c0s4+

+ 2p cos2+ sin2+ + A-’”

sin4+). (3.17)

Notice that the energy release rates are identical to the corresponding homogeneous, isotropic results, except that the compliance must be reinterpreted.

4. An Interfacial Crack in a Bilayer Figure 23 is a cross-section of an infinite bilayer with a half-plane crack on the interface. Each layer is taken to be homogeneous, isotropic, and linearly elastic. The uncracked interface is perfectly bonded with continuous displacements and tractions. The bilayer is loaded uniformly along the three edges with forces and moments per unit width. The problem has been studied by Suo and Hutchinson (1990), and the numerical solution has been presented in the entire parameter range. The generality of the edge loads allows the solution to be used to model a variety of delamination processes. A special loading case (a splitting cantilever bilayer) is discussed in Section II.C.4. The solution will be used to calibrate interfacial fracture specimens in Section IV.B.1, and to assess decohesion of pre-tensioned thin films in Section V.D. Focus here is on the presentation of the elasticity solution.


102

Far ahead of the crack tip the bilayer may be regarded as a composite beam. The neutral axis lies a distance hA above the bottom of the beam, with A being (3.18) where (3.19) The composite layer is in a state of pure stretch combined with pure bending. The only nonzero in-plane stress component is a,. The corresponding strain is linear with the distance from the neutral axis, y , according to Ex

=

-k (5+

(3.20)

Z Y )

The dimensionless cross-section A and moment of inertia I are 1 A=-+C, rl

Z=C

[(

A--

;>’

- (A -

); + ;]

+

;

(A -

);

+ $.1 (3.21)

The energy release rate can be calculated in close form:

The energy release rate specifies the magnitude of the near-tip singularity but does not specify the mode mixity. The information is completed by the complex stress intensity factor K , which, to be consistent with linearity, dimensionality and the Irwin-type relation (2.29), takes the form p

1/2

K = h-(-) 1 - a

where i =

(a-

(3.23)

a, and P and M are linear combinations of the edge loads: P = Pi - C1 P3 - C,M3/h,

M = Mi - C3M3.

(3.24)

The geometric factors are given by

c

c

(3.25)


103

a

and by 1

-=

U

1

+ Cq(4 + 6q + 3q2),

1 V

- = 12(1

+ Zq3),

sin y

+ v). Juv= 6 w 7 1 (3.26)

All these formulae are derived from the classical beam theory. The angle w is a function of the Dundurs’ parameters a,p and relative height q . This function was determined by solving the elasticity problem numerically; the computed values are plotted in Fig. 24, and an extensive tabulation can be found in Suo and Hutchinson (1990).

5 . A Substrate Crack in a Bilayer Depicted in Fig. 25 is an infinite bilayer with a semi-infinite crack parallel to the interface. Each layer is isotropic and homogeneous. There are two length ratios: tl = H l / h and ( = H / h . The problem was solved by Suo and Hutchinson (1989b) in the context of substrate spalling of a residual stressed thin film. The details of the application can be found in Section V.C.2, and here we will focus on the solution of the elasticity problem. Of the three edges, one is a homogeneous beam and the other two are composite beams. The positions of neutral axes for the two composite

J. W.Hutchinson and Z . Suo

104

p2

% 2 p

4

3

-_

p3

Material 2

A; h A

MZ MZ

FIG.25. A bilayer with a crack off the interface. The neutral axes for the two composite layers are indicated.

beams are given by (3.27) The three beams may be described by a linear variation like (3.20). The effective cross-section and moment of inertia of the two composite beams are given by

Z = E[(A

A=(+Z.,

-

0'

-

(A - 4 )

+ 1/31

+ A 0. With w defined by (2.45) with I = a, the stress intensity factor is calibrated by K = YTfia-"ee".

(4.7)

Here, T is the nominal bending stress, related to the moment per unit width by T = 6M/ W 2 , (4.8) and Y is the real, positive calibration factor. Both Y and w are dimensionless functions of a!, p, and a/ W. The finite element results are plotted in Fig. 35 (O'Dowd et al., 1990). Observe that the magnitude factor Y is nearly independent of elastic mismatch. The loading phase w varies between 0" to lo", depending on the elastic mismatch.


119

3 . Edge-Notched Shear

Calibration is also available for the specimen shown in Fig. 36 (O'Dowd et al., 1990). The loading phase is controlled by the offset s/ W. The stress intensity factor is K = YT\j;;a-"eiW, (4.9) where

ty

is again defined by (2.45) with I = a, and T =

(A - B)P (A B)W

(4.10)

+

The dimensionless functions Y and

ty

are plotted in Fig. 36. 16

0.2

0.2 c

E

(3

0 0.0

0.2

0.4

0.6

0.8

8.0

1.0

0.2

0.4

0.6

0.8

1.0

Relative Offset sMI

Relative Offset sMI

h

105 I

3 7=

aMI

-0.3 -0.2

a=O

'"1-0.1

-

0 0.0

0.2

0.4

0.6

0.8

1.0

0

Relative Offset sMI

Relative Offset s M I

A

_,_

B

J

FIG. 36. Calibration of the edge-notched shear of a bimaterial bar.


120

C. BRAZENUT SANDWICHES

Any homogeneous specimen can be converted to an interfacial specimen by sandwiching a thin layer of a second material between split halves of the specimen. The general setup is analyzed in Section III.D.2. As an example, here we sandwich the Brazil nut with a layer of second material, and a crack is left on one of the interfaces (Fig. 37). The specimen has been developed to determine interfacial toughness by Wang and Suo (1990). A remarkable feature common to all thin-layer sandwiches is that the residual stress in the layer does not drive the crack, because the strain energy stored in the layer due to residual stress is not released in the process of cracking. Thus, one does not have to measure the residual stress to determine toughness. On the other hand, as discussed in Section VIII, excessive

1.2 r L 1.1

-

I/-0 6

05

m

4 IW 04

tl

03 02 01

5

0

10

15

20

1-06

90

25

30

03 0 1 0 9 0 2 0 1

80

-

5 Y

=

-y1m

-

70

60 50 40

30 20 10

0 0

5

10

15

20

25

30

FIG.37. Driving force and mode mixity of the Brazil-nut sandwich.


121

residual stress may cause complications such as crack tunneling and kinking, so it should be avoided. 1 . Homogeneous Brazil Nuts

A homogeneous Brazil nut is a disk of radius a, with a center crack of length 21 (as illustrated in Fig. 37 but without the interlayer), which has been used for mixed mode testing of brittle solids for years. The loading phase is controlled by the compression angle 8 : It is mode I when 8 = 0", and mode I1 when 8 = 25". The stress intensity factors are K, = fIPa-1'2,

K,,

=

*fIIPa-"2,

(4.11)

where the plus sign is for tip A , and minus for B. The nondimensional calibration factors f, and f,, are functions of 0 and [ / a , and available in fitting polynomial forms in Atkinson et al. (1982). Using the Irwin relation (2.5), the energy release rate is G = (f,"

+ fA)P2/aEs,

(4.12)

where Es is plane strain tensile modulus for the substrate. The loading phases at tips A and B are tan-'(K,,/K,)

= *tan-'( fII/fI),

(4.13)

respectively. Equations (4.12) and (4.13) are plotted in Fig. 37. 2. Sandwiched Brazil Nuts

A sandwich is made by bonding two halves with a thin layer of a second material. Nonsticking mask is supplied on the prospective crack surface prior to bonding. When the layer thickness h is much smaller than other in-plane macroscopic lengths, the energy release rate can still be calculated from (4.12). This is true because of the conservation of the J-integral, and because the perturbation due to the thin layer is vanishingly small in the far field. The mode mixity @, defined by (2.45) with j as the reference length, is shifted from that for the homogeneous specimen, in accordance with @=

+ tan-'(fII/fI) + o + E ln(l/h).

(4.14)

Here, o,plotted in Fig. 28, is the shift due to elastic mismatch (3.38), and the last term is the shift due to the oscillation index E , (2.49).

122


D . DELAMINATION R-CURVES 1. Large-Scale Bridging

Over the last decade, it has become increasingly evident that the toughness of brittle materials can be enhanced by a variety of bridging mechanisms. The mechanics language that describes this is resistance curve behavior (R-curves): Toughness increases as crack grows. Attention here is focussed on the delamination of unidirectional or laminated composites, where cracks nominally propagate in planes parallel to fibers. A comparative literature study shows that for both polymer and ceramic matrix composites, bridging is usually due to intact fibers left behind the crack front, while the crack switches from one fiber-matrix interface to another as it propagates. Additional resistance for polymer matrix composites comes from damage in the form of voids, craze, or micro-cracks. Threedimensional architecture of threading fibers may also give rise to substantial fracture resistance. As the prerequisite for these bridging mechanisms, significant damage must accumulate ahead of the pre-cut tip as an additional energy dissipater. In laminates, for example, the length over which fibers bridge the crack is typically several times lamina thickness. The significance of an R-curve as a material property becomes ambiguous, since the R-curve now depends on specimen size and geometry. The intent of this section is to describe several generic features unique to delamination R-curves, as identified in Suo et al. (1990a); references on the subject can be found in the original paper.

2 . Essential Features of Delamination R-curves Consider a beam with a pre-cut, loaded at the edges by moments (Fig. 38). The damage zone size L can be comparable to or larger than beam thickness h, but the beam and pre-cut are much longer, so that the geometry is fully M

FIG.38. A mode I delamination beam, with a damage zone as an additional energy dissipater. The geometry is specified by L / h .


123

I

V

w

Lss

L

FIG.39. Two generic features of delamination R-curves. The plateau G,, is independent of the beam thickness h. The steady-state damage zone L,,size increases with h .

characterized by the ratio L / h . The material is assumed to be elastically isotropic and homogeneous, and plane strain conditions prevail. The nominal, or global, energy release rate is defined as the J-integral over the external boundary as given by Rice (1968): G

=

CM2,

C

=

12(1 - v2)/Eh3.

(4.15)

Here, M is the applied moment per unit width, C the beam compliance, 2h the thickness, E the Young’s modulus, and v the Poisson’s ratio. The specimen has a steady-state calibration: The global energy release rate does not depend on crack size, nor does it depend on any information of the damage zone (size, constitutive law, etc.). Phenomenological delamination R-curve behaviors are shown schematically in Fig. 39. First, focus on a R-curve measured using a beam of a given thickness, say, h , in the figure. The specimen can sustain the increasing moment, without appreciable damage at the pre-cut front, up to a critical point corresponding to G o .Subsequently, the damage zone size L increases with the applied moment, leading to an increasing curve of resistance GR. The damage zone may attain a steady-state: It maintains a self-similar opening profile and a constant length L,, , translating in the beam, leaving behind the crack faces free of traction. Correspondingly, a plateau G,, would appear on the R-curve. To proceed further, the Dugdale (1960) model is invoked, which, in its generalized form, simulates the homogenized damage reponse with an array of continuously distributed, nonlinear springs. Specifically, at each point in the damage strip, the closure traction 0 depends locally on the separation 6

124


according to a = a(6).

(4.16)

The functional form is related to the nature of damage, but is assumed to be identical for every point in the damage strip, and independent of the specimen geometry. A maximum separation dois specified, beyond which the closure traction vanishes. The spring laws may be measured or modeled using simplified systems. They may also be inferred from experimental R-curves, as will be discussed. For an arbitrary spring law, the following energy balance is due to the J-integral conservation (Rice, 1968):

G = Go

+

s:

~ ( 6d6, )

(4.17)

where 8, referred to as the end-opening of the damage zone, is the separation at the pre-cut tip. Here and later, we will not distinguish the driving force G and the resistance GR, as they can be judged from the context. The two energy release rates G and Go will be referred to as global and local, respectively. The global energy release rate represents the supplied energy, which is related to the applied moment via (4.15); the local one is the energy dissipated at the damage front. The difference given by (4.17) is the energy to create the damage. The steady-state resistance G,, is attained when the end-opening reaches the critical separation, 8 = do. Thus, from (4.17), G,, equals the sum of Go and the area under the spring law. The physical significance is that the plateau G,, does not depend on the beam thickness, and is therefore a property for a given composite laminate. However, it is not yet clear how long the damage strip will be before the steady state is attained. The steadystate damage-zone size L,, indicates the “quality” of a bridging mechanism: Toughness gained from too long a damage zone may not be useful in practice. Qualitatively, a thicker, stiffer, beam is more constrained for deflection, and thus exhibits larger Ls,. These essential features of the delamination R-curves are indicated in Fig. 39. Equation (4.17) suggests a way to determine the damage response. By continuously measuring the end-opening 8, and by using the experimentally determined R-curve, the spring law can be inferred by differentiating (4.17):

~ ( 8 =) aGR/d8.

(4.18)

The intrinsic resistance Go is assumed to be independent of the damage


125

accumulation. This simple method, which bypasses the complexities of large-scale bridging, is one of the advantages of specimens with steady-state calibrations. Large-scale bridging may be used as an experimental probe to study localized (planar) damage response such as polymer craze and interface separation, as uniform separation over a sample may be difficult to accomplish in reality because of the instabilities triggered by inhomogeneities or edge effects. 3. Rigid Plastic Damage Response

To gain some quantitative feel, consider a two-parameter damage response: The closure traction is a, when 6 < do, and vanishes when 6 > 6,. The steady-state toughness is G,, = Go

+ 006,.

(4.19)

The end-opening and crack tip stress intensity factor are given by a2

8 = a L 2 J c c - -L ~ c ~ , ,

(4.20a)

4

(4.20b) where the dimensionless number a depends on L/h only, and the finite element results are listed in Table 2. The R-curve defined by Eq. (4.20b) is plotted on Fig. 40 in a dimensionless form. The plateau G,, in (4.19) should be a horizontal line independent of h and L (not shown in the figure). From material characterization point of view, an inverse problem is of much more interest: how to infer the model parameters from a given experimental R-curve. The quantities Go, G,,, and L,, can usually be read off from the R-curve. Using these, the model parameters a, and a, can be inferred from (4.20). A family of damage responses including softening and hardening have been analyzed in Suo et al. (1990a). The effect of mode mixity has also been discussed. A parallel experimental investigation has been carried out by TABLE 2

u(L/h) L/h U

0.5 4.89

1.o 2.60

1.5 2.01

2.0 1.74

3.0

3.5

1.58

1.48

4.0 1.35

10.0 1.14

00

1.00

J . W.Hutchinson and

126

Z.Suo

4-

0

.5

1

1.5

2

2.5

3

3.5

4

Llh

FIG. 40. Dimensionless R-curves predicted using the rigid plastic damage response (mode I).

Spearing and Evans (1990) with a unidirectional ceramic composite. The experimental data and the model show very similar behavior, suggesting that the model incorporates the controlling features of the toughening mechanism.

V. Cracking of Pre-tensioned Films Thin films of metals, ceramics and polymers, are typically subject to appreciable residual stress, which for ceramic systems can be on the order of a giga-pascal. Such stress can cause cracking of the films. Films under residual tension and compression will be considered in this and the next sections, respectively. In this section, commonly observed fracture patterns in pre-tensioned films are first reviewed, together with a discussion of the governing parameters. These crack patterns are then analyzed in subsequent subsections, with cracking in films, substrates, and along interfaces treated independently. The values of a nondimensional driving force 2 will be documented to assist the practitioners of the field. The last subsection presents a speculative analysis of thermal shock spalling.


127

Interface

FIG.41.

A pre-tensioned film is deposited on a substrate.

A. CONTROLLING QUANTITIES AND FAILUREMODES

Illustrated in Fig. 41 is a film of thickness h on a substrate. Both materials are taken to be isotropic and linearly elastic, with elastic moduli and thermal expansion coefficients (Ef, u f , af)and (E,, v, ,as),respectively. Elasticity mismatch may be characterized by the two Dundurs parameters a and P defined in (2.21); a > 0 when film is stiffer than substrate. A crack will grow as the driving force G attains the fracture resistances rf,r,, ri, depending on whether the crack is propagating in the film, substrate, or along the interface. The mode I fracture resistance is usually appropriate for films and substrates, but mixed mode resistance must be used for interfaces. 1. Driving Force Number and Critical Film Thickness To help visualize the cracking progression, the residual stress is assumed to be due entirely to thermal mismatch. However, with proper interpretation, most of our results would be valid for stress due to other sources. The film-substrate is stress-free at a high temperature & . Upon cooling to the room temperature T,, the contraction strains in the film and substrate, were they unbonded, would differ by (af- a,)(& - T,). A biaxial misfit stress is defined accordingly: B =

(af - a,)(& - T,)Ef/(l- V f ) .

(5.1)

Notice B > 0 when af> a,.This stress is large: Typically, Ea z 1 MPa/K for most materials. For example, the stress would be of order 1 GPa if the temperature drops lo00 K (this is common in processing ceramic systems).

128


FIG. 42. An Eshelby-type superposition to treat a residually stressed film decohering from a substrate.

The thermal stress field can be evaluated by an Eshelby-type superposition. As an example, consider a thin film decohering from a substrate, driven by a biaxial misfit strain (Fig. 42). Problem (a) is trivial: the misfit strain is negated by a mechanical strain corresponding to the tensile stress a; the film is under a uniform biaxial stress, and the substrate is stress-free. In problem (b), a pressure of magnitude 0 is applied on the edge of the film, but no misfit strain is present. The superposition recovers the original problem, with misfit strain but without edge load. Since no stress singularity is present in (a), the crack driving force is entirely due to (b). The latter is a standard elasticity problem, which requires numerical analysis. A unifying dimensionless number 2 is defined such that the energy release rate for a crack is G = Zu2h/&. (5.2) Note that the elastic strain energy stored in a unit area of the film is (1 - v f ) a 2 h / E fThe . number Z is a dimensionless driving force, or order unity, depending on the cracking pattern and elastic mismatch. The practical significance of this dimensionless number was first documented by Evans et al. (1988). Common cracking patterns are sketched in Fig. 43, together with their Z-values, where the film-substrate system is taken to be elastically homogeneous, and the substrate semi-infinite. Equation (5.2) provides a design limit. Given the mechanical properties and misfit stress, a specific cracking pattern is inhibited if the film is thinner than a critical thickness, given by

h, = T E f / Z a 2 ,

(5.3)

where r is the relevant fracture resistance. The following example illustrates a routine application using the information gathered in this section.


Cracking Patterns

129

= ZG2h/E,

Surface Crack

Z = 3.951

Channeling

Z = 1.976

Substrate Damage

Z = 3.951

Spalling

Z = 0.343

1.028 (initiation) 0.5 (steady - state) FIG.43. Commonly observed cracking patterns. The dimensionless driving force for each pattern is listed, assuming that the film-substrate is elastically homogeneous, and the substrate is infinitely thick.

130

J. W. Hutchinson and Z. Suo

Consider channeling cracks in a glass film coated on a thick SiOz substrate. Suppose r, = 7 J/m2, Ef = 70 GPa, 0 = 50 MPa. One reads from Fig. 43 that 2 = 1.976, which is appropriate since glass and SiOz have similar elastic moduli. The critical film thickness computed from (5.3) is h, = 100pm. The channel network is not anticipated if the film is thinner than 100pm. 2. Cracking Patterns Let us go through Fig. 43 to define the various cracking modes. A surface crack is nucleated from a flaw, and arrested by the interface. Yet the stress is not high enough for the crack to channel through the film. Since flaws are necessarily isolated, one would see stabilized, unconnected slits. The driving force available for surface cracks is high, as indicated by the large value of Z. Isolated cracks are detrimental for some applications, such as corrosion protection coatings, but tolerable for others. The channeling process is unstable: Once activated, it would never arrest until it encounters another channel or an edge. Consequently, a connected channel network would emerge, surrounding islands of the intact film. Such cracking may not be acceptable for most applications, but, for example, is common in glaze on fine pottery, and in pavement of roads. Cracks in a film can propagate further to cause substrate damage. This Z-value is the largest on the list. Such a crack may be stabilized at a certain depth, since the misfit stress is localized in the film. However, the crack may divert to run parallel to the interface, leading to the next cracking pattern. Substrate spalling is an intriguing phenomenon: The crack selects a path at a certain depth parallel to the interface, governed by K,,= 0. This is not a localized failure pattern in that extensive flakes can be spalled off. Fortunately, the Z-value for spalling is quite low. If a small amount of substrate damage is acceptable, one gains substantial flexibility in design. Debonding may initiate from edge defects or channel bottoms. The latter can be stable: The driving force for initiation is higher than that for the long debond. This fact is exploited to introduce pre-cracks for certain types of fracture specimens, such as the UCSB four-point flexure specimen. In the following sections, these failure modes will be examined in some detail. Emphasis is placed on the relevant elasticity problems that lead to estimates of the driving force number Z. Experimental efforts will be cited in passing. The writers hope this catalog will be used critically by experimentalists in various disciplines, thereby allowing the catalog to be validated or modified.


131

B. CRACKING IN FILMS Imagine a process with increasing tensile stress in the film, for example, the cooling process. As illustrated in Fig. 43, a surface flaw is activated by the tensile stress, grows towards the interface, and then arrests if the substrate and interface are tough. With further stress increase, the crack may channel through the film. The two stages will be treated separately in the following. 1. Surface Cracks

We model the situation by a plane strain crack; see the insert in Fig. 45. This is appropriate for an initial surface flaw of length several times h, but may not be valid for an equiaxial flaw. The latter is studied by He and Evans (1990a) but is omitted here. The plane strain problem has been solved by Gecit (1979) and Beuth (1990). The following paragraph is a digression to a few mathematical considerations that capture the main features of the solution, and which may be skipped without discontinuity in the content. The dimensionless stress intensity factor K / a G depends only on the relative crack depth a / h and Dundurs’ parameters a and 8. For small a/h, regardless of the elastic mismatch, the stress intensity factor merges to that of an edge crack in a semi-infinite space, i.e., K -, 1 . 1 2 1 5 6 a a as a/h + 0. Asymptotic behavior for another limiting case, a / h --* 1, can be obtained by invoking the Zak-Williams singularity: the stress singularity for a crack perpendicular to, and with the tip on, the interface. Instead of the square root singularity, the stresses near such a crack tip behave like aij K r - x j ( 0 ) , where (r, 0) is the polar coordinate centered at the tip, and theJj are dimensionless angular distributions. The scaling factor plays a part analogous to the regular stress intensity factor, but having different dimensions: [stress][lengthIs. The singularity exponent s (0 c s < 1) depends on elastic mismatch, and is the root to

-

The numerical solution of s is plotted in Fig. 44.For a crack that penetrates the film, l? ohS, with the pre-factor dependent on a and 8 only. As a/h -, 1 but with the crack tip still within the film, the stress field away from the small ligament (h - a) would not feel such a detail, and behaves as if the crack tip were just on the interface, governed by R. Dimensional considerations require the stress intensity factor K to be related to the far

-

132


FIG.44. (a) Zak-Williams singularity. (b) A curve fitting parameter.


133

N

1 3 -

5 -

-

.2

0

.4

.6

.0

1

ah FIG.45. Driving force available for an edge crack at various depths d h .

-

field K according to K K ( h - a)1’2-s.Combination of the preceding gives K / o G (1 - a/h)1/2-sas a h. Motivated by these considerations, Beuth fitted his numerical solution with K / o f i = 1.1215fi(a/h)1~2(1 - ~ / h ) ’ / ~(1- + ’ Ia/h), (5.5)

-

+

where I is taken to be independent of a / h , and is chosen such that the formula agrees with the full numerical solution at a / h = 0.98; the results are plotted in Fig. 44b. The error of ( 5 . 5 ) is within a few percent for intermediate a/h. Equation (5.5) is plotted in Fig. 45 in terms of the dimensionless energy release rate. The energy release rate starts from zero for shallow flaws. As the crack approaches the interface, it drops to zero for relatively compliant films (ac 0), but diverges to infinity for stiffer films (a > 0). One needs apriori knowledge of flaw size to predict a failure stress or the maximum tolerable film thickness. In practice, a plausible flaw depth may be assumed according to the “quality” of the film. Taking, say, a / h = 0.8, one can obtain the nondimensional driving force 2 from Fig. 45 for a known elastic mismatch. The flaws will not be activated if the dimensionless fracture resistance satisfies rfEf/02h> 2.

134


Observe that for relatively compliant films, the driving force attains a maximum at an intermediate depth. The practical significance is that no flaws, regardless of initial depth, can be activated, provided the dimensionless resistance TfEf/a2his greater than the maximum. Such a maximum, depending on the elastic mismatch a and p, provides a fail-safe bound for relatively compliant films. 2. Cracks Channeling through a Film Figure 46 shows a crack channeling through the film. Complications such as substrate penetration, interface debond, and channel interaction are assumed not to occur for the time being. At each instant of the growth, the channel front self-adjusts to a curved shape, such that energy release rate at every point on the front is the same. The elasticity problem is threedimensional in nature, and an accurate solution would require iteration of the front shape. After the length exceeds a few times the film thickness, the channel asymptotically approaches a steady-state: the entire front maintains its shape as it advances; so does &z), the cross-section profile in the wake, which attains the shape of a plane strain through-crack. The steady-state cracking is analogous to that discussed in Section III.A.1,

a. FIG.46. The insert shows a crack channeling across the film, driven by the tensile stress in the film. The available energy at the channel front is plotted for various elastic mismatch.


135

In the steady state, the energy release rate at the channel front can be evaluated using two plane problems-that is, by subtracting the strain energy stored in a unit slice far behind of the front, from that far ahead. The calculation does not require the knowledge of the front shape. Alternative formulae have been developed with this idea. One is

Two plane problems are involved: the stress distribution on the prospective crack plane before cracking, G(z), which, for the present situation, equals the uniform misfit stress a,and the displacement profile for a plane strain crack, G(z).This is particularly convenient for numerical computation. A second formula is

G(a)da, (5.7) hl S oh where G(a) is the energy release rate of a plane strain crack of depth a in Fig. 45. A mathematical interpretation is that G,, is the average of energy release rates for through-cracks at various depths. Both formulae are valid for films and substrates with dissimilar elastic moduli. As an example, suppose the film-substrate is elastically homogeneous, and the substrate occupies a semi-infinite space. The corresponding plane strain problem is an edge crack in a half plane, with energy release rate G(a) = 3.952aza/l? (Tada et al., 1985). The integral (5.7) gives G,, = 1.976azh/E. This pre-factor is listed in Fig. 43. Beuth (1990) carried out an analysis of a thin film on a semi-infinite substrate with dissimilar elastic moduli. The result is reproduced in Fig. 46. If the dimensionless toughness rfEf/a2h is below the curve, a channel network is expected. Observe that a compliant substrate (a > 0) provides less constraint, inducing higher driving force for channeling. The channeling cracks were studied analytically by Gille (1985) using the numerical solutions available at that time, and subsequently by Hu and Evans (1988) with a combination of calculations and experiments. The concept has been extended as a fail-safe bound for cracking in multilayers (Suo, 1990b; Ho and Suo, 1990; Ye and Suo, 1990; Beuth, 1990). Applications include thin films, reaction product layers, adhesive joints, and hybrid laminates. G,,

=-

3. Multiple Channeling The preceding technique can be extended to study interaction among channels. Suppose the biaxial stress is biased so that parallel channels

136

J. W. Hutchinson and 2. Suo

FIG.47. Interaction of multiple channels.

develop in one direction; see the inserts in Fig. 47. For simplicity, attention is restricted to an elastically homogeneous system with a semi-infinite substrate. Consider a periodic set of edge cracks of depth a, spacing L , and subject to an opening stress (T. The energy release rate at each crack tip, G(a), is found in Tada et al. (1985) in a graphic form, which is then fitted by a polynomial. Based on this information, energetic accounting gives the driving force for cracks channeling in the film. The strain energy, per crack, gained in creating a set of cracks of depth h is

1

h

U

=

G(a)da = fa2h2/E,

(5.8)

0

where the dimensionless factor f depends on the crack density h / L . The results obtained by a numerical integration are plotted in Fig. 47. If these cracks are equally extended in the channeling direction, the energy release rate at each front is G,, = U / h = fa2h/E.Thus, f is the dimensionless driving force for this situation. Thouless (1990a) has employed this solution in his discussion of crack spacing in thin films. Next, consider the situation in Fig. 47 where, at a certain stage of loading, the cracks of spacing 2L have already channeled across the film, and the


137

tensile stress in the film has therefore been partially relieved. With further increase of the loading, a new set of cracks are nucleated and grow half-way between the existing channels. The energy release rate at the front of the growing cracks should be computed from G,, = (2UL - U,L)/h = [2f(h/L)- f(h/2L)]a2h/E.

( 5 -9)

This is derived from the strain energy difference far behind and far ahead of the channeling fronts. The result is also plotted in Fig. 47. Given the mechanical properties and with the identification G,, = r,, the plot may be viewed as a relation between the stress level and the channel density. Notice we have assumed that new cracks can always be readily nucleated half-way between existing channels. This might overestimate the crack density for a given stress level. An analysis with aspects similar to the preceding has also been carried out independently by Delannay and Warren (1991).

C. SUBSTRATE CRACKING

Substrate damage may originate from edges or existing channel cracks in the film. The two substrate cracking patterns in Fig. 43 are studied in this section. Observe that the Z-values for the two patterns differ by an order of magnitude. 1. Substrate Damage Caused by Cracks in Films

Suppose the channel cracks in the film have developed at some stage during the cooling but have not yet grown into the substrate, either because the substrate is much tougher or because sufficiently large substrate surface defects are not readily available. The issue is whether these cracks would propagate into the substrate upon further cooling. The problem has been studied by Ye and Suo (1990), and the main results are summarized here. The driving force for a plane strain crack into a substrate was analyzed using finite elements, and the results are plotted in Fig. 48. Observe that the driving force decays for deep cracks, implying stable propagation. For relative compliant films (a c 0), the driving force starts from zero at the interface, and attains a maximum at very small depths. It is difficult for finite elements to resolve these details, so the trend is sketched by dashed lines.


138

r -.9 0 ' " ~ " " ' ~ " ' " ' ~ ' " " " " ~ ~ ' 2

15

1

25

3

3.5

4

a/h

FIG.48. Energy release rate for a plane strain crack with the tip in the substrate.

The plane strain model is not quite correct, since cracks must be renucleated, in a three-dimensional fashion, from a surface flaw on substrate. The insert of Fig. 49 shows a crack growing laterally under an existing channel in the film. The crack arrests at a certain depth because of the decay of the available driving force. The energy release rate at the growing front

"

-1

4.5

0

0.5

1

a FIG.49. Energy release rate for a crack propagating under a channel in the film.


139

may be computed from (5.10)

Again, this is derived from energy accounting. The integral is evaluated using the preceding plane strain results, and the results are summarized in Fig. 49. The plot may be used as a damage tolerance map: Given a damage tolerance a / h , one can read the design number Z . Take the curve for a / h = 1.2 as an example. Provided the dimensionless substrate toughness r,Ef/ha2is above the curve, no channel with depth a / h > 1.2 is anticipated. This holds true even if the initial flaws are deeper than 1.2, as long as they are not channels themselves. Observe that the elastic mismatch plays a significant role. A relatively compliant substrate would provide less constraint, leading to larger driving force. The so-called T-stress in (2.1) has also been computed by Ye and Suo and is found to be positive, unless the film is much stiffer than the substrate and the crack depth is small. As shown by Cotterell and Rice (1980), a positive T-stress results in a tendency for a straight mode I crack to veer off to one side or the other. Further discussion of crack path stability in a related context is given in Section VII1.C. Here, we simply note that the substrate crack will have a strong tendency to branch into a path parallel to the interface, a cracking pattern to be discussed next. 2. Spalling of Substrates

Cracks, originating from either defects in the film or at the edge, have a strong tendency to divert into the substrate, should the latter be brittle, and follow a trajectory parallel to the interface; see Fig. 43. The key insight was provided by Thouless et al. (1987) in a coordinated experimental and theoretical investigation. Their initial intent was to model the impact of ice sheets on offshore structures. The experiments were conducted with PMMA and glass plates, loaded on the edges. Spalling cracks were found to follow a trajectory parallel to the surface, with depth governed by the criterion K,, = 0. (See also Thouless and Evans, 1990.) These authors remarked to the effect that this mechanism would operate in the edge spalling of pre-tensioned films, previously observed by Cannon et al. (1986). As schematically shown in Fig. 43, the crack initiates at the edge, extends along the interface for typically about two times film thickness, then kinks into the substrate, and finally runs parallel to the


140

interface at a depth of a few times the film thickness. In this case, the underlying mechanics is the same for giant ice sheets as for micro-electronic films. Thorough investigations on pre-tensioned films have been conducted, experimentally and analytically, by Hu et al. (1988), Hu and Evans (1988), Drory et al. (1989), Suo and Hutchinson (1989b), and Chiao and Clarke (1990). Focus here is on the steady-state spalling, with the transient stage ignored, since the former provides a well-defined design limit. In the following, the essential mechanics will be elucidated using a simple system, and results will be cited for more general cases. The analysis is arranged separately for spalling originating from edges or channel cracks (planar geometry), and from holes. a. Planar Geometry Inserted in Fig. 50 is a long crack at a depth d in the substrate, driven by the residual tension in the film. Plane strain conditions are assumed. The film is attached to a semi-infinite substrate with the same elastic moduli. Results without these restrictions will be cited later. The equivalent edge force and moment due to rs are P

10

= Oh,

M

=

(5.11)

Oh(d - h)/2.

-

0

8

6

\ A

t

5

6

d/h

dss/h=3.87

Fro. 50. The insert shows a spalling crack. The plot is the mode mixity for crack at various depths.


141

Specialized from (3.12), the stress intensity factors are K,/afi

=

(h/2d)'/2[cosw

+ f i ( 1 - h / d ) sin a],

K I I / a f i = (h/2d)'l2[sino - f i ( 1 - h / d ) cos 01.

(5.12) (5.13)

where o = 52.07'. The mode mixity v = tan-'(K,,/K,) is plotted as a function of crack depth in Fig. 50. Notice K,, > 0 for small depth, but KI, < 0 for large depth and, consequently, a pure mode I trajectory exists at an intermediate depth. This steady-state spalling depth d,, is determined from (5.13) with K,, = 0. Thus, d,, = 3.86h. (5.14) The steady-state, mode I energy release rate can now be readily evaluated from (5.12),which gives G,,

=

0.343a2h/E.

(5.15)

This pre-factor was cited in Fig. 43. Suo and Hutchinson (1989b) carried out an extensive analysis to include elastic mismatch and finite thickness of the substrate. The general solution for arbitrary edge loads is summarized in Section III.B.5. The results for spalling cracks caused by the residual stress in the film are reproduced in Fig. 5 1 . Observe that the spalling depth depends strongly on both elastic mismatch and substrate thickness. However, the dimensionless stress intensity factor is insensitive to the substrate thickness as long as H / h > 10. There has been no formal proof that the spalling trajectory is configurationally stable. One heuristic explanation, as shown in Fig. 5 1 , is that K,, > 0 when d < d,,, implying that a crack above d,, would be driven down. Analogously, a crack below d,, would be driven up. b. Spalling from Circular-Cut Figure 52 shows an axial-section of a spalling crack emanating from the edge of a circular-cut in the film, driven by residual tensile stress in the film. In general, a hole in a pre-tensioned film acts like a stress raiser. However, it differs from an open hole in a plate in that, for the former, cracking is usually confined within a few times hole radius. Other cracking modes around holes include channel cracks in films and decohesion of interfaces. The latter will be treated in the next section. As indicated in Fig. 52, the hole radius is b,, and the crack extends to a radius b. For simplicity, the elastic moduli for the film and substrate are


142

0.0

-

4.8

0.0 Q

4.4

0.4

0.8

1/XO

FIG. 51. Spalling results for film-substrate of dissimilar elastic constants and finite substrate. Both mode I stress intensity factor for the steady-state propagation and the depth of the crack path are given.

I

b

U

FIG.52. An axial-section of spalling from the edge of a circular-cut.


143

taken to be the same, and the substrate semi-infinite. The equivalent edge force P and moment M , per unit length, are still given by (5.11). The stress state in the annulus between b, and b can be determined by the classical plate theory, with the outer boundary clamped. The analysis shows that the moment and force at the crack front are modified by a factor: M(b) = M / k , where

k = i[(l

P(b) = P / k ,

(5.16)

+ V) + (1 - ~ ) ( b / b ~ ) ~ ] .

(5.17)

These loads are indicated in Fig. 52. The results of Section 1II.B.l.d are applicable with moment and force M(b) and P(b) used. In particular, the energy release rate is modified by factor k 2 , i.e.,

z)

1 M2 G =-+ - , 2k2( I

1 I = -Ed3, 12

A =Ed.

(5.18)

This result can also be derived by an energetic accounting, i.e., G = (2d9-l a w a b ,

where U is the strain energy stored in the clamped circular plate. From this latter approach, it would be clear that the solution is the exact asymptote as (b - bo)/h 4 00. The stress intensity factors (5.12) and (5.13) are modified accordingly by a factor of k . Thus, the steady-state depth d,, is independent of b/bo, and is identical to the plane strain result (5.14). The mode I driving force for spalling now becomes 0.343a2h (5.19) G= k2E * Since k increases with b/bo, the spalling crack from a circular-cut would usually arrest. D. INTERFACE DEBOND Pre-tensioned films are susceptible to decohesion or, more precisely, de-adhesion, from substrates. Flaw geometry plays an important role: Debonding emanating from an edge defect, a hole, or a through-cut would behave differently. Analytical results for the first two geometries will be summarized, and the third can be found in Jensen et al. (1990).

144


a =0.6 06

0.4

02 0.0

-.2 -.4

-.6 -.6 40 3 0-0

01

02

h/H

h/H

FIG. 53. Mode mixity and energy release rate for a debonding crack.

1. Decohesion from Edges or Channels Figure 53 illustrates a pre-tensioned film debonding from a substrate. The edge load is a special combination of that studied in Section III.B.4, and the notation there is followed. The misfit stress is equivalent to the mechanical loads (see Fig. 26):

PI = P3 = ah,

A 4 3 = (1/2

+ 1/11 - A)oh2,

A41

=

0. (5.20)

Specialized from Eq. (3.22), the energy release rate is 2: G=I--a2h 2 4 A

[

E(1/2

I.

+ 1/11 - A)2 I

(5.21)

The loading phase I,U is defined by K = IKlh-"exp(iyl), as is consistent with the convention in (2.45) with I = h . Both the driving force and mode mixity are plotted in Fig. 53. Observe that the decohesion process is inherently mixed mode, consisting of somewhat more sliding than opening.


145

The effect of the substrate thickness on the driving force is significant when the film is stiff. Argon et a/. (1989) have used the residual stress as a driving force to measure interface toughness. The result in this section can also be used to calibrate the residual stress effect on some interface fracture specimens, e.g., the UCSB four-point flexure specimen. The complex interfacial stress intensity factor is a superposition of the contribution from residual stress and that from mechanical load. 2 . Decohesion from a Hole Figure 54 illustrates a decohesion crack emanating from the edge of a hole in a pre-tensioned film. Results developed in Section C.2.b for substrate spalling are still valid here. In particular, the energy release rate is given by ha2 G = (5.22) 2Efk2’ where k is given by Eq. (5.17). The result is now valid for films and substrates with dissimilar elastic constants, but the substrate is still assumed to be much thicker than the film. The mode mixity is independent of b/bo when ( b - bo)/h is sufficiently large, and is identical to the plane strain results (Fig. 5 3 , h / H = 0). Decohesion from a circular-cut is stable and has been used to determine interface toughness by Farris and Bauer (1988) and Jensen et al. (1990). This is particularly feasible when the film is transparent, so that the decohesion radius b can be readily measured.

I

FIG.54. An axial-section of a decohesion annulus originated from an edge of a circular-cut.


146

p+=-

IT0 FIG.5 5 . A model for thermal shock spalling.

E. THERMAL SHOCKSPALLING 1. An Idealized Model

Consider a block of brittle material with a thin, pre-tensioned, surface layer. Spalling is possible if the residual stress has a negative gradient with depth. An example is depicted in Fig. 5 5 . A semi-infinite body is initially immersed in a heat bath of temperature To, so that a uniform temperature is established in the body. Upon the removal of the block from the bath, the surface temperature is assumed to drop instantaneously to the room temperature T,. A biaxial tensile residual stress thus develops in a surface layer, as shown schematically in Fig. 55. The equivalent edge force and moment are also indicated. The stress profile changes with the time, and so does the ratio M / P . The problem features a dimensionless number

< = h/&,

(5.23)

where h is the depth of the crack parallel to the surface, ctD the thermal diffusivity, and t the time elapsed after the removal of the heat source. At any given time, a mode I crack path parallel to the surface is available-that is, a number exists where K,, = 0. However, for small t , the depth h is correspondingly small, and therefore the strain energy stored in such a thin layer is insufficient to drive the spalling. Consequently, a certain time elapse is needed before spalling. The following is an attempt to quantify these considerations.

r,,

2 . Spalling Depth and Time Elapse Consider first the temperature and stress field prior to cracking. At a given time t after the removal from the heat bath, the temperature at a


147

depth z is

~ ( zt ), = T, + (T, - T,) erfc(z/2.\ltol,).

(5.24)

The biaxial, tensile, thermal stress field varies with the depth and time, in accordance with ax = a,, = a(z, t ) = a0erfc(z/2JtolD),

(5.25)

where (5.26) T)E/(l - v), and aE is the thermal expansion coefficient. These are classical solutions, which may be extracted from standard textbooks. Next consider the half space with a spalling crack (Fig. 55). The resultant force and moment can be expressed as 00

P

=

= (YE(&

looh,

M

-

=

(*I - J)aoh2,

where erfc(u/2) du,

J(0

=

r

(5.27)

s:

u erfc(u/2) du.

(5.28)

The stress intensity factors can be calculated from Section III.B.l .d. The number corresponding to a mode I trajectory is determined by enforcing K,, = 0. The problem involves a nonlinear algebraic equation, and the numerical solution gives

r

h,, = 6 . 8 2 6 .

(5.29)

The corresponding mode I stress intensity factor is KI = 0 . 1 9 0 a o a .

(5.30)

Given the toughness and stress level, the depth h,, may be predicted from (5.30), and the time elapse for spalling can then be estimated from (5.29). Observe that the spalling depth is independent of the thermal diffusivity, as a feature of this idealized model. As an example, consider a glass with K,, = 0.7 MPa m1’2, a. = 100 MPa, aD = 0.7 x m2/s. The predicted s. crack depth is h,, = 1.4 mm, and the time to spalling is t = 6 x This tiny time elapse is possibly an outcome of the idealized temperature boundary condition that has been adopted. VI. Buckle-Driven Delamination of Thin Films

In many film/substrate systems, the film is in a state of biaxial compression. Residual compression has been observed in thin films that have been

148


FIG. 56. The photograph on the left from Argon et al. (1989) shows a S i c film on a Si substrate delaminating as a wavy circular blister. On the right is a photograph supplied by M. D. Thouless, which shows examples of the straight-sided blister and the telephone cord blister occurring in a multilayered film delaminating from a glass substrate.

sputtered or vapor deposited and it can arise from thermal expansion mismatch. Some remarkable failure modes of such systems have been observed, examples of which are shown in Fig. 56. These pictures reveal regions where the film has been buckled away from the substrate. Various shapes of the buckled regions evolve, including long straight-sided blisters, circular blisters with and without wavy edges, and the so-called telephone cord blister, which is perhaps the most common morphology. The failure entails the film first buckling away from the substrate in some small region where adhesion was poor or nonexistent. Buckling then loads the edge of the interface crack between the film and the substrate, causing it to spread. The failure phenomenon couples buckling and interfacial crack propagation. The straight-sided blister grows at one of its ends. The telephone cord blister grows at its end as if a worm were tunneling beneath the film. This section presents an analysis of the straight-sided and circular blisters and concludes with some speculation about the origin of the telephone cord morphology. It will be seen that a key aspect of the phenomena is the mixed mode fracture behavior of the interface, wherein T(w) increases sharply with increasing mode 2. Formulas relating the energy release rate of the interface crack to the buckling parameters were derived for one-dimensional ply buckles on the surface of laminated composites by Chai et al. (1981). Essentially identical results were obtained by Evans and Hutchinson (1984) and Gille (1985) for the thin-film problem. The energy release rate for the circular blister in biaxially compressed films was given by Evans and Hutchinson (1984) and


149

Yin (1985). The significance of the mixed mode character of the interface crack tip was apparently first appreciated by Whitcomb (1986), who showed that the crack tip becomes predominately mode 2 as a one-dimensional ply buckle spreads. His observation was essential to explain why the buckles do not keep spreading along their edges under constant overall load-that is, why the buckles have a characteristic width. Whitcomb was concerned with compressive failure modes in layered composites. Here, the concern will be with thin films under equi-biaxial compression, but a number of the results and conclusions carry over directly to ply delamination. Storakers (1988) and Rothschilds et al. (1988) deal with various aspects of buckling and delamination in composites, and these authors cite relevant literature in the composites arena. This section starts with a one-dimensional analysis of the infintely long straight-sided blister, closely paralleling the analysis of Whitcomb (1986). Given the availability in Section III.B.4 of the relationships between the interface stress intensity factors and the moment and resultant force change at the edge of the buckle, the one-dimensional analysis can be carried out in closed form. The analysis of the circular blister, which requires some numerical work, is presented next. The two sets of results are then combined in an analysis of steady-state propagation of a straight-sided blister. The steady-state problem gives perhaps the sharpest insights into design constraints on compressed films.

A. THE ONE-DIMENSIONAL BLISTER

Consider an x-independent segment of the straight-sided blister shown in Fig. 57. The film is taken to be elastic and isotropic with Young’s modulus E l , Poisson’s ratio v l , and thickness h. The substrate is also assumed to be isotropic but with modulus E, and Poisson’s ratio v2. The substrate is modeled as being infinitely deep. The film is assumed to be unattached to the substrate in the strip region -b Iy Ib. A plane strain interface crack of width 2b exists between the film and the substrate. The unbuckled film is assumed to be subject to a uniform, equi-biaxial compressive in-plane stress, ox, = aYY= -0. In the unbuckled state, the stress intensity factors at the crack tips vanish. Only when the film buckles away from the substrate are nonzero stress intensity factors induced. Under the assumption that h 4 b, the film is represented by a wide, clamped Euler column of width 2b. The complex stress intensity factor K at the right-hand


150

tip is related to the moment M and to the change in resultat stress AN at the right-hand end of the column by the relationships given for the 2-layer problem in Section III.B.4. Use of the 2-layer solution to characterize the crack tip field is justified if h / b 4 1, which is, in any case, the condition for the validity of the Euler theory. In what follows, the 2-layer solution is first specialized for the present applications. This solution is also used in the analysis of the blister test discussed in Section VII, and should have fairly wide applicability. Then, the Euler solution is presented and is coupled to the 2-layer solution. 1. General Loading of an Edge Crack on the Interface

between a Thin Film and Substrate Let M and A N be defined with the sign convention in Fig. 57. These quantities will be identified with the momenthnit length and the change in resultant stress at the right end of the wide Euler column. Specializing the solution of Section III.B.4 to the limit of the infinitely deep substrate, one finds for the interface crack:

+

G = 6(1 - v : ) E ; ' ~ - ~ ( M ' hZAN2/12),

tan tp = h

A

Im(KhiE)- G M c o s w + h ANsin w Re(Kh'") - 4 2 M s i n w + h ANcos w

t'

.///A -

/ / / 2b

(6.1)

(6.3) '

-0

-+Y

UNBUCKLED

w

BUCKLED

LOCAL LOADING OF INTERFACE CRACK

Fro. 57. Geometry of the one-dimensional blister, and conventions for the elasticity solution characterizing conditions near the tip of an interface crack between a thin fiim and an infinitely thick substrate. Top left: unbuckled; bottom left: buckled; right: local loading of interface crack.


400

151

t

, , ; . , , -1

-.5

0

i

.5

a

FIG.58. Phase factor w ( a , 8) in (6.2) and (6.3).

Here, w = o ( a ,/3, h / N = 0), which is plotted in Fig. 58 for /3 = 0 and /3 = a/4. The mode mixity parameter is defined using the film thickness h as the reference length 1. 2. Euler Column Solution and Coupling

to Interface Edge Crack Solution The one-dimensional deformation of the wide column in Fig. 57 is characterized by the y- and z-displacements, V(y) and W(y). These are defined to be zero in the unbuckled state with pre-stress a,, = a,, = -a. The wide column is taken to be characterized by von Karman nonlinear plate theory with fully clamped conditions at its edges, i.e., V = W = W,y = 0

at y

=

kb.

(6.4)

The change in the y-component of the stretching strain measured from the unbuckled state is Ey = V,Y + W , Y 2 , (6.5) while the bending strain is W, yy. With N, and N, as the resultant stresses and with AN, = N, + ah and AN, = N, + ah as the changes in the resultant stresses from the unbuckled state, the strain component E, is related to ANy by E, = (1 - v:) ANy/(E1h). (6.6) Since E , = 0, AN, = v1 AN,. The bending moment is related to the bending strain by My = D W, yy, where D = El h 3 / [12( 1 - vf)] is the bending stiffness. In-plane equilibrium requires AN,, y = 0. Therefore, AN, can be

152


taken to be the value at the end of the beam, AN. Moment equilibrium requires D W ,YYJJY - (AN - ah) W ,y y = 0. (6.7) The solution to the preceding system of equations is given in (6.8)-(6.12): W = * 0) is shielded. With K,, as the intensity toughness of the layer material, the apparent toughness measured using the sandwich specimen is

An equivalent statement concerning the elastic shielding is that the load needed to fracture the sandwich specimen differs from that needed to fracture a geometrically similar specimen made entirely from the layer material by the factor [(l + a)/(1 This factor can be quite large when a stiff material is joined by a compliant adhesive, as is the case when, for example, metal of ceramic parts are joined by a polymer adhesive. If the strength of a joint is controlled by crack-like flaws that are on the order of the adhesive thickness or somewhat larger, then this same magnification factor will apply to the strength of the joint compared to the strength of the bulk adhesive with flaws of similar size. If the controlling flaws are much smaller than the layer thickness, the magnification effect is lost. 3 . Crack along the Interface (Fig. 74b)

The relation G" = G also holds for this case, where G is related to the interfacial stress intensity factors K, and K2 by (2.29). The equation relating K1 and K2 to K,"and KZ is (3.38) with K I = K," and K,, = Kg. With y for the tip on the interface defined by (2.45) with I = h, the o-quantity in (3.38) is y - ym.This shift in phase is generally small and even for the largest elastic mismatches never more than about 15". As discussed in Section IV.C, sandwich specimens are attractive for measuring interface toughness. Assuming the crack does advance in the interface, the equation for the interfaces toughness function is simply T(y, 1 = h) = T(w"),

(8.7)

where y = ym+ o.Conversion of T ( y , h) to r($,i) where $ is defined using a material-based length is readily carried out as specified by (2.51). B. CRACKTRAPPINGIN

A

COMPLIANT LAYERUNDER NONZERO K;

Equation (8.3) predicts a strong trapping effect due to elastic mismatch between the layer and the adjoining blocks. When K g is not too large


176

FIG.76. Location of trapped crack in a compliant sandwich layer.

compared to K,”, (8.3) reveals the existence of a straight, mode I crack path within the layer. The condition for KII = 0 from (8.3) gives the relation between the location of the crack, c / h , and tym as

+(c/h,CY, p) = -tym.

(8.8)

+

Since vanishes when the elastic mismatch vanishes, there can be no straight crack paths within the layer unless tym = 0 under this circumstance. A plot of the solution to (8.8) is shown in Fig. 76 for several levels of mismatch, all with p = a / 4 . When there is significant mismatch, a mode I path well within the layer can exist for IKGI as large as 10% of K,“. Such paths exist whether the layer is compliant or stiff, but generally a straight crack in a stiff layer will be configurationally unstable, as will now be discussed. C. CONFIGURATIONAL STABILITY OF A STRAIGHT CRACK WITHIN THE

LAYER

Given the existence of a straight mode I path within the layer, the issue now addressed is whether the path will be insensitive to small perturbations, returning to the straight trajectory, or will be deflected into the interface or possibly into a wavy morphology. Consider a loading with KG = 0 such that the center line through the layer is a mode I crack trajectory. Two results are presented that indicate whether or not the centerline path is configurationally stable. First, suppose at the start of propagation the crack lies off the centerline (i.e., c / h # 1/2 in Fig. 74a). From (8.3), with KG = 0,

K,, = [(I

-

+

a)/(l - a)]”’ sin K;.

(8.9)


177

The offset crack will kink toward the centerline if K,, > 0 when c / h > 1/2, and if K,, < 0 when c / h c 112. The function 4 given by (8.4) is odd with respect to the centerline at c / h = 112. For a compliant layer (a > 0) with /3 = a / 4 , 4 is positive when c / h > 1/2, implying by (8.9) that the crack will kink toward the centerline. By contrast, when the layer is stiff (a < 0), 4 is negative when c / h > 1/2 and the crack will kink away from the centerline. A compliant layer with /3 = 0 has a small negative 4 when c / h > 1/2 and would also cause the offset crack to kink away from the centerline. This particular test of configurational stability requires both a and /3 be positive, This same test has been used in Section 1II.B.l.b for the double cantilever beam, and in Section V.C.2.a for substrate spalling driven by residual tension in the film. Another insight is provided by the condition for stability of a straight, mode I crack path to small perturbations derived by Cotterell and Rice (1980). Their necessary condition for straight cracking is T c 0, where T is the second-order term in the crack tip expansion (2.1). Fleck et al. (1991) have solved for T for the crack problem of Fig. 74a. For the centerline crack ( c / h = 1/2) under K z = 0, they give T = [(l - a)/(l

+ a)]T" + OR + ~ I ( a , / 3 ) K y h - ' / ~ .

(8.10)

Here, T" is the T-stress for the homogeneous specimen in the absence of the layer, cR is the residual stress in the layer acting parallel to the centerline, and c, is tabulated by Fleck ei al. and presented here in Fig. 77. Residual compression parallel to the crack plane contributes to stability, as does a compliant mismatch of the layer relative to the rest of the specimen through the last term in (8.10). The last term in (8.10) is destablizing when the film is stiff. Note that the residual stress aR has no effect on the existence of a mode I path in the layer, just on its stability. When there is significant elastic mismatch, the first term in (8.10) will usually be insignificant compared to the third term, since T" is typically on the order of KTL-1'2, where L is a length characterizing an overall dimension of the specimen, which is assumed to be large compared to h. When this is the case, the T-stress at fracture is Tc

OR

+ [(I + Cr)/(l

- cY)]1/2C~K~ch-1/2,

(8.1 1)

where, by (8.6), K,, is the intensity toughness of the layer material. The requirement T c 0 will always be met for a compliant layer supporting a compressive (or zero) residual stress. When the residual stress is tensile, the sign of T depends on which of the preceding terms is larger. Note that the

178


FIG.77. Coefficient c, in (8.10).

second term in (8.11), which is always negative for a compliant layer, increases in magnitude as the layer thickness diminishes. A number of examples of sandwich systems that have been reported to exhibit straight in-layer cracking are discussed by Fleck et al. (1991), and two of these will be remarked on in what follows. To emphasize the significance of this effect, one can point to the symmetrically loaded, double cantilever beam specimen, which is notoriously unstable in the absence of a layer due to the fact that T" > 0. Because of the stabilizing influence of a thin compliant layer, the specimen can be used successfully to measure the toughness of a material in a sandwich layer.

D. INTERFACE

OR

IN-LAYERCRACKING?

Two sets of toughness data taken using sandwich specimens are shown in Fig. 78. Indicated for each data point is whether the crack propagated along the interface or within the interior of the layer. Thouless's (1990b) data, for a brittle wax layer joining silica glass, is presented as a function of the applied phase angle of loading @'. Only for wm = 0 did the crack propagate within the brittle wax layer. The toughness of the wax was about one half


N

-i

SteeVEpoxy =o.l

179

as-milled o polished 0 in-layer fracture

50

10

I

I

FIG. 78. a) Data of Thouless (1990b) for brittle wax layer sandwiched between glass substrates. b) Data of Wang and Suo (1990) for an epoxy layer between steel substrates.

that of the interface for near-mode-1 fracture. The data of Wang and Suo (1990) for an epoxy adhesive layer joining two halves of a steel Brazil nut shows instances of in-layer propagation for I @[-valuesas large as about 10". Moreover, the epoxy is significantly tougher than the interface at low values of @, and the in-layer path involves substantially higher energy dissipation and applied load than the interface path. Nevertheless, a number of specimens did exhibit in-layer propagation. This preference for a high energy path over a low energy path in close proximity highlights the importance of understanding the mechanics of crack path selection. It remains an open question as to why a path down the interface was not selected, especially

180


since Wang and Suo started their cracks on the interface and observed a small amount of interface crack growth prior to the crack kinking into the interior of the layer. Condition (2.65) for kinking out of the interface, including the influence of the u-contribution, does not appear to be satisfied. This is the feature of the behavior that remains to be explained. Both sandwich systems in Fig. 78 have a highly compliant layer, and both systems have a tensile residual stress oRin the layer. But in each case the second term in (8.11) is at least twice as large in magnitude as oR (cf. Fleck et al., 1991, for complete details). Thus, the mode I specimens have a distinctly negative T-stress, and straight cracking within the layer is consistent with the stability theory. By contrast, the plexiglass/epoxy sandwich system of Fig. 9 has relatively small elastic mismatch and a positive T-stress under mode I loading. For this system, the crack always followed one of the two interfaces. For values of I,v"~ outside the range of possible trapping of the crack within the layer (e.g., l y ~ " l greater than 0" to lo", depending on the mismatch), the crack will be driven toward one interface or the othertoward the lower interface if v/" > 0 and toward the upper if I,P< 0. If material # 1 is sufficiently tough to resist any attempts for the interface crack to kink into it, the crack will follow the interface and the test will generate the interface toughness according to (8.7). This is the case for both sets of test data presented in Fig. 78, other than those data points mentioned in the preceding. Various micro-morphologies of interface fracture have been observed, some of which have been discussed by Evans et al. (1989). If the interface toughness is low compared with that of both materials # 1 and #2, then the crack will tend to follow the interface fairly cleanly. If, however, the interface toughness is comparable to that of the layer material, then the interface crack will interact with flaws in the layer adjacent to the interface, and nucleate microcracks. The effect of the mixed mode loading is to grow these microcracks back towards the interface. The resulting fracture surface will be covered with tiny chunks of the layer material. Additional discussion of the micro-morphology of interface fracture is given by Chai (1988), Wang and Suo (1990), and by several authors in the volume on metal-ceramic interfaces (Ruhle et al., 1990).

E. ALTERNATING MORPHOLOGY (Fig. 74c)

In the alternating mode of cracking of Fig. 74c, the crack switches back and forth between interfaces with a fairly regular interval, which is typically


181

several times the layer thickness. The mode has been reported in matrix layers between plies of a composite when the loading is nominally mode I, and it has been fully documented for mode I loading of an aluminum/ epoxy/aluminum sandwich specimen by Chai (1987). Chai used a heat setting epoxy, which gives rise to a relatively high residual tensile stress in the layer (oR= 60MPa), which is more than twice the magnitude of the second term in the T-stress in (8.11) (cf. Fleck et al., 1991). Thus, Chai’s system has a strongly positive T-stress and is not expected to display straight cracking within the layer. Akisanya and Fleck (1990) have carried out a quantitative analysis of the alternating mode of cracking with specific attention to the aluminum/ epoxy system. Central t o the phenomenon is the variation in the proportion of mode 2 to mode 1 of the interface crack as it propagates from the point where it first joins the interface in any given cycle, i.e., at c = 0 in Fig. 79. The trends of the variation in ty found by Akisanya and Fleck are sketched in Fig. 79, where ty is defined by (2.45) with I = h. When oR= 0, ty rapidly approaches the limiting value given by (3.38) with K,, = K z = 0, i.e., = 0.22, E = ty = o. (For the aluminum/epoxy sandwich, a = 0.93, -0.07 and o = -15O.) However, when o R f i/K ? is on the order of unity,

FIG.79. Sketch of trends of phase of loading at interface crack tip for various levels of residual tension uR in the layer. The remote loading is mode I .

182


which is the case for the Chai specimen, the interface crack starts with a large component of mode 2, which then diminishes to the value I,U = w as c/h increases. The large negative mode 2 component forces the crack to remain in the interface, since it cannot penetrate the aluminum. Only when c/h has increased to the point where the magnitude of I,U is sufficiently low does kinking down into the layer become possible. With the aid of the kinking analysis of Section II.C.5, the value of c / h was determined at which a mode I kink crack is possible. For the aluminum/epoxy system of Chai, Akisanya and Fleck found that the kinking condition is met when c / h reaches a value of about 2, in agreement with the intervals observed by Chai.

F. TUNNELING CRACKS An example illustrating the ability of a crack in a brittle adhesive layer to tunnel through the layer was given in Section 1II.A. 1. If the layer material is sufficiently less tough that the interface and the adjoining material, cracking will be confined to the layer as depicted in Fig. 16. Steady-state tunneling results are useful because they provide fail-safe limits on stress levels (or on layer thicknesses) such that extensive cracking can be avoided. The particular example of Section 1II.A. 1 reveals that an initial crack-like flaw whose greatest dimension is equal to the layer thickness (e.g., a pennyshaped crack) will initiate growth at a stress that is only about 10% higher than the steady-state tunneling stress. For many systems where the flaw size is on the order of the layer thickness, the tunneling results should provide realistic upper limits. When the flaw size is much smaller, the stress to initiate crack growth is much higher than that predicted by the steady-state tunneling limit, and the transient tunneling process is then highly unstable. Several steady-state tunneling results for layers are presented in this section. The results and their potential applications have a close resemblance to the results for thin-film cracking in Section V.B. 1. Isolated Tunneling Crack

As previously emphasized, the energy released, hG,, , per unit length of steady-state propagation of a tunneling crack is precisely the energy released by a plane strain crack extended across the layer. Calculations have been performed for G,, by Ho and Suo (1990) for finite thickness sandwiches, as specified in the insert in Fig. 80. In Fig. 80, 0 denotes the uniform tensile

Mixed Mode Cracking in Layered Materials -

183

5

O’h 4

3

2

1

0 1

0

c1

FIG. 80. Steady-state energy release rate for isolated tunneling crack. The crack extends from interface to interface, and is propagating in the direction perpendicular to the crosssection shown.

stress within the central layer prior to introduction of the crack. That stress may be due to a load applied to the sandwich or it may be a self-equilibrated residual stress. Curves of the nondimensional G,, are shown as a function of the elastic mismatch parameter a with /3 = a / 4 for various values of layer thickness to total thickness, h / w . As long as the central layer is not too stiff compared to the adjoining layers, the results for h / w = 0.1 are close to the limiting case h / w = 0. For example, for a = 0, the normalized G,, is 0.788 for h / w = 0.1 and 0.785 for h / w = 0. Observe that a relatively compliant substrate (i.e., small E , and/or w / h ) provides less constraint, inducing higher driving force. It is likely, for the same reason, that higher driving force will be induced by crack-induced plasticity in the substrates, by interface debonding, or by any other source of constraint loss. These effects have been noted in thin film channeling by Hu and Evans (1989). 2 . Multiple Tunneling Cracks The approach to multiple cracking pursued here is identical to that presented in Section V.B.3 for thin films under residual tension. The reader is referred to that section for a more complete discussion of the derivations underlying the results. Here, consideration will be limited to a layer of thickness h sandwiched between two infinitely thick blocks. Elastic mismatch


184 -

1.0

%[

. .

,

,

,

,

,

04

4

,

,

,

,

,

,

,

0.8

0.6

-

0 20

02

06

0.8

h/L 3

FIG.81. Steady-state tunneling cracks with uniform spacing, in the absence of elastic mismatch. The cracks extend from interface to interface.

A 1.0

2 b

‘j,

c

-

0.8 -

0.6

-

0)

n X 0

0.4 -

2

threshold

0

0.2

-

01

0.5

1.0

I

.

1.5

2.0

2.5

’

6JTIrn FIG.82. Relation between tunnel crack density and residual stress in the layer in the absence of elastic mismatch. The curve is obtained from (8.14) with G,, = r,.


185

between the layer and the adjoining blocks is neglected (a = /? = 0). The stress in the layer in the absence of the cracks is a, which may be due to applied load or a residual stress. First, consider an infinite set of cracks periodically spaced a distance L apart as in Fig. 81. If these cracks are equally extended in the tunneling direction, *E a h

=f(h/L).

(8.12)

The function f ( x ) , which can be evaluated using results from Tada et al. (1985), is plotted in Fig. 8 1 . For h / L 0, f = 0.785 and (8.12) reduces to ( 3 . 1 ) . Next, consider the situation in Fig. 82 where one set of cracks spaced a distance 2L apart has already tunneled across the layer, and where a second set bisecting the first set is in the process of tunneling across the layer. The steady-state energy release rate for the cracks in the process of tunneling is +

yl --

2f(h/L) - f(+h/L).

(8.13)

Imagine a process in which (T is monotonically increased, as in application of an overall load or stressing due to temperature change with thermal expansion mismatch. Under the assumption that new cracks will be nucleated half-way between cracks that have already formed and tunneled, the preceding equation gives the relation between 0 and the crack spacing h / L . Wich G,, identified with the mode I toughness of the layer material r,, (8.13)provides the desired relationship, which is plotted in Fig. 82. The ’ threshold corresponds to the lowest stress at which steady-state tunneling can occur, i.e., for h / L 0, a [ h / ( E r ~ ]=~ 1.128. ’~

(8.14)

The effect of elastic mismatch on this threshold level can be determined using the results for the isolated tunneling crack. 3 . Lateral Tunneling of a Kinked Crack

Tunneling appears to be a prevalent mode of cracking in layered materials. When the brittle layer is thin and the flaw size is comparable to the layer thickness, the cracks can be readily nucleated. A variety of applications of these ideas can be found in Ho and Suo (1990) and Ye and Suo (1990). Here, we give one more example to show the versatility.


186

3D local kink crack

lateral spreading of kink crack by tunneling

Y plane strain problem for kink crack

FIO. 83. Spread of a local kink by tunneling. Top left: 3-D local kink crack; top right: lateral spreading of kink crack by tunneling; bottom: plane strain problem for kink crack.

Because it has relevance to the ability of an interface crack to nucleate a kink into the adjoining layer, we mention in passing application of the tunneling concept to crack kinking. Suppose the parent interface crack depicted in Fig. 83 encounters a local, three-dimensional flaw that is capable of nucleating kinking. Consider the process in which the kinked segment of crack “tunnels” laterally along the interface crack front. Formally, this tunneling process can be treated as a steady-state process. The average energy release rate at the laterally spreading crack front can be evaluated using the energy released in the plane strain problem, just as in the previous examples. To simplify the discussion, assume /? = 0 and q = 0, where q is given by (2.63). Since the energy release rate for the plane strain problem for a small kink is independent of kink crack length (cf. Section II.C.5), it follows that the average energy release rate for lateral tunneling along the interface crack front is equal to the plane strain energy release rate. Consequently, there is no barrier to the lateral spread of a locally nucleated kink. This observation may help to explain why crack kinking often appears to occur simultaneously along a more-or-less straight segment of crack front.

Acknowledgments The work of JWH was supported in part by the DARPA University Research Initiative (Subagreement P.O. # VB38639-0 with the University of California, Santa Barbara, ONR Prime Contract N00014-86-K0753),the


187

Materials Research Laboratory (Grant NSF-DMR-89-20490), and the Division of Applied Sciences, Harvard University. The work of ZS was supported in part by the DARPA University Research Initiative (ONR Prime Contract N00014-86-K0753), the National Science Foundation (Grant MSS-9011571), and the University of California, Santa Barbara.

References Akisanya, A. R., and Fleck, N. A. (1990). Analysis of a wavy crack in sandwich specimens. To be published in Int. J. Fracture. Anderson, G . P., DeVries, K. L., and Williams, M. L. (1974). Mixed mode stress field effect in adhesive fracture. Int. J. Fracture 10, 565-583. Argon, A. S., Gupta, V., Landis, H. S., and Cornie, J. A. (1989). Intrinsic toughness of interfaces between Sic coatings and substrates of Si or C fibers. J. Muter. Sci. 24, 1406- 14 12.

Atkinson, C., Smelser, R. E., and Sanchez J. (1982). Combined mode fracture via the cracked Brazilian disk test. Int. J. Fracture. 18, 279-291. Bao, G., Ho, S., Fan, B., and Suo, Z. (1990). The role of material orthotropy in fracture specimens for composites. Int. J. Solids Structures (in press). Beuth, J. (1990). Cracking of thin bonded films in residual tension. To be published. Beuth, J. (1991). Ph.D. thesis research in progress, Harvard University. Cannon, R. M., Fisher, R., and Evans, A. G. (1986). Decohesion of thin films from ceramics. In Proc. Symp. on Thin Films-Interfaces and Phenomena, Boston, Mass. Mater. Res. SOC.1986, pp. 799-804. Cao, H. C., and Evans, A. G. (1989). An experimental study of the fracture resistance of bimaterial interface, Mechanics of Materials 7 , 295-305. Chai, H. (1987). A note on crack trajectory in an elastic strip bounded by rigid substrates. Int. J. Fracture 32, 211-213. Chai, H. (1988). Shear fracture. Int. J. Fracture 37, 137-159. Chai, H. (1990). Three-dimensional fracture analysis of thin film debonding. Znt. J. Fracture.

46, 237-256. Chai, H., Babcock, C. D., and Knauss, W. G. (1981). One dimensional modelling of failure in laminated plates by delamination buckling. Int. J. Solids Structures 17, 1069-1083. Charalambides, P. G., Cao, H. C., Lund, J., and Evans, A. G. (1990). Development of test method for measuring the mixed mode fracture resistance of bimaterial interfaces. Mechanics of Materials 8, 269-283. Charalambides, P. G., Lund, J., Evans, A. G., and McMeeking, R. M. (1989). A test specimen for determining the fracture resistance of bimaterial interfaces. J. Appl. Mech. 56, 77-82.

Cherepanov, G. P. (1962). The stress state in a heterogeneous plate with slits (in Russian). Izvestia A N SSSR, OTN, Mekhan. i Mashin 1, 131-137. Cherepanov, G. P. (1979). “Mechanics of Brittle Materials.” McGraw-Hill, New York. Chiao, Y.-H., and Clarke, D. R. (1990). Residual stress induced fracture in glass-sapphire composites: Planar geometry. A c f a Met. 38, 25 1-258. Comninou, M. (1977). The interface crack. J. Appl. Mech. 44, 631-636. Comninou, M., and Schmueser, D. (1979). The interface crack in a combined tensioncompression and shear field. J. Appl. Mech. 46, 345-348.


188

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England, A. H. (1965). A crack between dissimilar media. J. Appl. Mech. 32, 400-402. Erdogan, F. (1965). Stress distribution in bonded dissimilar materials with cracks. J. Appl. Mech. 32, 403-410. Erdogan, F., and Arin, K. (1972). Penny-shaped interface crack between an elastic layer and a half space. Int. J. Engng. Sci. 10, 115-125. Evans, A. G., Dalgleish, B. J., He, M., and Hutchinson, J. W. (1989). On crack path selection and the interface fracture energy in bimaterial systems. Acta Metal/. 37, 3249-3254. Evans, A. G., Drory, M. D., and Hu, M. S. (1988). The cracking and decohesion of thin films. J. Muter. Res. 3, 1043-1049. Evans A. G., and Hutchinson, J. W. (1984). On the mechanics of delamination and spalling in compressed films. Int. J. Solids Structures 20, 455-466. Evans, A. G., and Hutchinson, J. W. (1989). Effects of non-planarity on the mixed mode fracture resistance of bimaterial interfaces. Acta Metall. Muter. 37, 909-916. Evans, A. G., Riihle, M., Dalgleish, B. J., and Charalambides, P. G. (1990). The fracture energy of bimaterial interfaces. Muter. Sci. Engng. A 126, 53-64. Farris, R. J., and Bauer, C. L. (1988). A self-determination method of measuring the surface energy of adhesion of coatings. J. Adhesion 26, 293-300. Fleck, N. A., Hutchinson, J. W., and Suo, 2. (1991). Crack path selection in a brittle adhesive layer. Int. J. Solids and Structures. 27, 1683-1703. Friedrich, K., ed. (1989). “Application of Fracture Mechanics to Composite Materials.” Elsevier, New York. Gecit, M. R. (1979). Fracture of a surface layer bonded to a half space. Int. J. Engng. Sci. 17, 287-295.

Gille, G. (1985). Strength of thin films and coatings. In “Current Topics in Materials Science.” Vol. 12 (E. Kaldis, ed.), North-Holland, Chap. 7. Hayashi, K. and Nemat-Nasser, S. (1981). Energy-release rate and crack kinking under combined loading. J. Appl. Mech. 48, 520-524. He, M.-Y., Bartlett, A., Evans, A. G., and Hutchinson, J. W. (1991). Kinking of a crack out of an interface: role of in-plane stress. J. A m . Ceram. SOC. 74, 767-771. He, M.-Y., and Evans, A. G. (1990a). Initial cracking of thin films. Manuscript in preparation. He, M.-Y., and Evans, A. G. (1990b). Finite element analysis of beam specimens used to measure delamination resistance of composites. J . Composites Tech. Res. (in press). He, M.-Y., and Hutchinson, J. W. (1989a). Kinking of crack out of an interface. J. Appl. Mech. 56, 270-278. He, M.-Y., and Hutchinson, J. W. (1989b). Kinking of a crack out of an interface: Tabulated solution coefficients. Available for a limited period from Marion Remillard, Pierce Hall 314, Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138. Ho, S., and Suo, Z. (1990). Tunneling cracks in constrained layers. Manuscript in preparation. Hu, M. S., and Evans, A. G. (1989). The cracking and decohesion of thin films on ductile substrate. Acta Met. 37, 917-925.


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Hu, M. S., Thouless, M. D., and Evans, A. G. (1988). The decohesion of thin films from brittle substrates. Acta Met. 36, 1301-1307. Hutchinson, J. W. (1990). Mixed mode fracture mechanics of interfaces. In “Metal-Ceramic Interfaces,” (M. Ruhle, A. G. Evans, M. F. Ashby, and J. P. Hirth, eds.), Pergamon Press, New York, pp. 295-306. Hutchinson, J . W., Mear, M. E., and Rice, J. R. (1987). Crack paralleling an interface between dissimilar materials. J. Appl. Mech. 54, 828-832. Jensen, H. M. (1990). Mixed mode fracture analysis of the blister test. Engin. . I Fracture . Mech. (in press). Jensen, H. M., Hutchinson, J. W., and Kim, K.-S. (1990). Decohesion of a cut prestressed film on a substrate. Int. J. Solids Structures 26, 1099-1 114. Kamada, K., and Higashida, Y. (1979). A fracture model of radiation blistering. J. Appl. Physics 50, 4131-4138. Kinloch, A. J. (1987). “Adhesion and Adhesives,” Chapman and Hall, London. Lekhnitskii, S. G. (1981). “Theory of Elasticity of an Anisotropic Body,” Mir Publishers, Moscow. Liechti, K. M. (1985). Moire of crack-opening interferometry in adhesive fracture mechanics. Experimental Mechanics 25, 255-261. Liechti, K. M., and Chai, Y.-S. (1990a). Asymmetric shielding in interfacial fracture under inplane shear. J. Appl. Mech, in press. Liechti, K. M., and Chai, Y.-S. (1990b). Biaxial loading experiments for determining interfacial toughness. J. Appl. Mech, in press. Liechti, K. M. and Hanson, E. C. (1988). Nonlinear effects in mixed-mode interfacial delaminations. Int. J. Fracture 36, 199-217. Malyshev, B. M., and Salganik, R. L. (1965). The strength of adhesive joints using the theory of cracks. Int. J. fracture. Mech. 5 , 114-128. O’Brien, T. K. (1984). Mixed-mode strain-energy-release rate effects on edge delamination of composites. In “Effects of Defects in Composite Materials,” ASTM STP 836, American Society for Testing and Materials, pp. 125-142. O’Dowd, N. P., Shih, C. F., and Stout, M. G. (1990). Test geometries for measuring interfacial toughness. Submitted for publication. Reimanis, I. E., and Evans, A. G. (1990). The fracture resistance of a model metal/ceramic interface. Acta Metall. Mater. (in press). Rice, J. R. (1968). A path independent integral and approximation analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379-386. Rice, J . R. (1988). Elastic fracture concepts for interfacial cracks. J. Appl. Mech. 55, 98-103. Rice, J. R., and Sih, G. C. (1965). Plane problems of cracks in dissimilar media. J. Appl. Mech. 32, 418-423. Rice, J. R., Suo Z., and Wang, J.-S. (1990). Mechanics and thermodynamics of brittle interfacial failure in bimaterial systems. In “Metal-Ceramic Interfaces”. (M. Ruhle, A. G. Evans, M. F. Ashby, and J. P. Hirth, eds.), Pergamon Press, New York, pp. 269-294. Rothschilds, R. J . , Gillespie, J. W., and Carlsson, L. A. (1988). Instability-related delamination growth in thermoset and thermo-plastic composites. In “Composite Materials: Testing and Design,” ASTM STP 972 (J. D. Whitcomb, ed.), pp. 161-179. Ruhle, M., Evans, A. G. Ashby, M. F., and Hirth, J. P., eds. (1990). “Metal-Ceramic Interfaces,” Acta-Scripta Metallurgica Proc. Series 4, Pergamon Press, New York. Sbaizero, 0.. Charalambides, P. G., and Evans, A. G. (1990). Delamination cracking in a laminated ceramic matrix composite. . I A . m . Ceram. SOC.73, 1936-1940. Shapery, R. A., and Davidson, B. D. (1990). Prediction of energy release rate for mixed-mode delamination using classical plate theory. Appl. Mech. Rev. 43, S2814287.

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NND Schemes and Their Applications to Numerical Simulation of Two- and Three-Dimensional Flows* HANXIN ZHANG and FENGGAN ZHUANG China Aerodynamics Research and Developmen1 Center Beijing. China I . Introduction ......................................................................................... I1 . The Importance of the Role of the Third-Order Dispersion Term ............... A . Analytic Analysis ............................................................................. B . Physical Discussion and Numerical Simulation ....................................... I11. A Formulation of the Semi-discretized NND Scheme ......... A . The Case for a Scalar Equation ................................ B. Extension to Euler and Navier-Stokes Equations . .............................. IV . Explicit NND Schemes ........................................................................... A . NND-I Scheme ................................................................................ B. NND-2 Scheme ................... ......................................... C . NND-3 Scheme ................................................................................ D . NND-4 Scheme ................................................................................ E . NND-5 Scheme ................................................................................ V . Implicit NND Scheme ............................................................................ ...... VI . Applications to Solutions of Euler and Navier-Stokes Equations ...... A . Regular Reflection of Shock Wave ....................................................... B. Supersonic Viscous Flow around a Circular Cylinder ............................... C . Axial Symmetric Free Jet Flows .......................................................... D . Hypersonic Flow around a Blunt Body ................................................. E . Hypersonic Flow around Space-Shuttle-like Geometry ............................. VII . Concluding Remarks .............................................................................. References ............................................... ........................................

193 194 194 198 200 200 204 206 206 201 208 209 210 211 213 213 224 231 238 246 254 256

.

I Introduction

In the calculation of complex flow fields containing shock waves. the most attention has been paid to the shock-capturing method . In order to capture shock waves smoothly without spurious oscillations near or in the shock regions. mixed dissipative schemes containing free parameters with first-order accuracy near shocks and second-order schemes elsewhere have *This work was supported in part by the China National Natural Science Foundation under Grant 9188010. 193 Copyright 0 1992 by Academic Press. Inc . All rights of reproduction in any form reserved. ISBN 0-12-002029-7

194

H. Zhang and Fenggan Zhuang

been widely used (see Pulliam, 1985; Jameson and Yoon, 1985; and Zhang et al., 1983, and Zheng, 1986). There are inherent disadvantages in employing these schemes. First, the free parameters are basically determined through numerical experiments; second, the resolution of shock is not very satisfactory. Naturally, the development of nonoscillatory dissipative schemes containing no free parameters with high resolution has been much emphasized recently, such as TVD schemes (see Harten, 1984; van Leer, 1979; Chakravarthy and Osher, 1985; Davis, 1984; and Yee, 1987) and E N 0 schemes (see Harten, 1986). In Zhang (1984), Zhang found with a linearized analysis of onedimensional Navier-Stokes equations that when a proper manipulation of the coefficients of third-order derivatives v 3 in the corresponding modified differential equations is used, the spurious oscillation at both upstream and downstream shocks can be totally suppressed. The choice of v3 must be in accordance with the second law of thermodynamics. The numerical experiments with one-dimensional nonlinear Navier-Stokes equations confirm that the conclusion reached by linearized analysis is valid in general. The importance of the role of v3 was not realized before. While it was usually thought that the third-order dispersion term in the modified differential equations would not play an essential role in a linear problem, e.g., in the stability analysis, etc. , they do contribute through nonlinear interactions to the unwanted wave structure, and even lead to chaos if they are not properly treated. Based on these physical concepts, we suggest a new NND scheme (see Zhang, 1988) that is nonoscillatory, contains no free parameters, and is dissipative. The details will be described in this paper. In order to test the effectiveness of the scheme, we have carried out numerical simulations for two- and three-dimensional flows with the Euler equations and NavierStokes equations. The results of calculations are given. Finally, some concluding remarks are given.

11. The Importance of the Role of the Third-Order Dispersion Term A. ANALYTIC ANALYSIS (see Zhang, 1984)

Now we study the problem of one-dimensional shock wave using a timedependent method. The model equation and boundary conditions are

NND Schemes and Their Applications

au at

-

195

a2u + a-au - v- 0, ax ax2 -

where

u2 =

2 y-l(l+ +.1 Y+l

Y-lMm

u, and p are the fluid density, velocity, and the coefficient of viscosity, respectively. The variable t represents the time and x represents a coordinate (Fig. 1). Free-stream conditions are denoted with the sign “00.” M , is the free-stream Mach number and y is the ratio of the specific heat. If the flow is steady and the coefficient of viscosity is constant, Eq. (2.1) represents the exact Navier-Stokes equations governing a normal shock wave. If a finite-difference method is applied to solving Eq. (2.1), the following modified differential equation, which is actually a finite-difference equation, can be written as

p,

au at

au

a2u a2u a3u a4u = v2@ + vj3ax - v4+ ax ax4

-+ a- - v2 ax

...,

(2.3)

where the right-hand side represents the truncation error for the difference method. For a second-order accurate difference method, v2 = 0.

0

FIG.1. One dimensional shock wave.

196


Now, we study the effects of coefficients v 2 , v 3 , ... of the preceding steady solution. In fact, if the flow is steady, Eq. (2.3) can be integrated once to obtain

where

a=-U &

O 2 = Y- (-l 1+ - 4 . Y + l

2 1 Y-lMc

Obviously, 0 = 1 is the nondimensional velocity in the upstream region of the inviscid shock, and a = a2 is the nondimensional velocity in the downstream region. We assume that the numerical oscillations induced by the truncation error of the difference method are small; then, in the upstream region of the shock, a = 1 + u', (2.5a) and in the downstream region of the shock,

a = a2

+ u',

(2.5b)

where u ' < 1 upstream and u' -e a2 downstream. Substituting (2.5a) and (2.5b) into Eq. (2.4) and neglecting the high-order small quantity, we obtain aui ax aul ax

a3ui + V ~a2ui T - ~4= k U' ax ax3 ' a2ui ax2

a3ui ax

v 1 - + v 3 - - v4 7= -k2 u' where v1 =

v

(upstream),

(downstream),

+ v2,

k, = -u,(l - a2) > 0, +

2Y

kz

y + l 2Y

=-

1-02

-> 0. ti2


197

Equation (2.6) is linear and its solution can be determined by following characteristic equations: V , A ~- V , A ~ - V , A

+ kl = o

v4A3 - v3A2 - vlA

-

k2 = 0

(upstream),

(2.7)

(downstream).

(2.8)

To discuss following cases is of some help: Case 1 . v1 > 0, v3 and v4 are very small. This case corresponds to using the first-order difference scheme. The solution of Eq. (2.6) is 1 + Ae(kl/YI)X

u=l+u'=

a,

+

Be-(kdYOX

(upstream x > 0), (downstream x > 0);

(2.9)

A , B are constants. This result shows that there is no numerical oscillation in both sides of the shock. Case 2. v2 = 0, v3 < 0, v4 is very small. For inviscid flow (v = 0) or the flow at high Reynolds number (v is very small), the solution of Eq. (2.6) is

[

1 + A, exp(&x)

( a 2 + Aexp[[

cos(;r\i41v31k,

2- -(v:1 21 V3 I

2IV3 I

-

v:x

+ 4k21v31)1/2

1 (downstream). (2.10)

It is very clear that the spurious oscillations occur in the upstream region of the shock, but not in the downstream region. Case 3. v2 = 0, v3 > 0, v4 is very small. In a similar way, we can prove that the spurious oscillations occur in the downstream region of the shock, but not in the upstream region. Cases 2 and 3 correspond to using the second-order difference scheme. Through the preceding study for one-dimensional Navier-Stokes equations, it is found that the spurious oscillations occurring near the shock with the second-order finite difference equations are related to the dispersion term in the corresponding modified differential equations. If we can keep v3 > 0 in


198

the upstream region of the shock and v 3 < 0 downstream, we may have a smooth shock transition, i.e., the undesirable oscillations can be totally suppressed.

.

B PHYSICAL DISCUSSION AND NUMERICAL SIMULATION (see Zhang and Mao, 1987, and Zhang, 1988)

The preceding conclusion can be also verified by following physical discussion from the second law of thermodynamics. In fact, the one-dimensional Navier-Stokes equations modified by the addition of dispersion terms with coefficient v, are as follows:

-aP+ - =aPu o at au at

ax

p - + p u - +au - = - ap

aH

ax

ap

P - at - - + P Uat- = -

ax

aH ax

adx( );;

(4- p - a,)

+-

4 aH ax 3 P - )ax

+

a

ax 3

a

ax

(

v 3 7

,

(2.11)

$(v35),

where p represents the pressure, H represents the total enthalpy, and the Prandtl number is assumed to be 3/4. From Eq. (2.11), we may obtain an equation of entropy s for the heat-isolated system (2.12) Here, D d D t is the substantial derivative of entropy. For a true physical shock, we have, in the upstream of the shock,

au

- 0 upstream and v 3 < 0 downstream, we may have a smooth shock transition. U

U

(a)

9,8=0.9

a=O.

U 1.2 1.0

0 8

a=O. 9 upstream and a=

(c) 0.6

- 0 . 9 &mm.drm,p=o. 9 0.4 0. 2

G -3.2

-1.G

0

1.G

3.2X

FIG.2 Numerical simulation for normal shock wave with (2.11). a = 2yv,/[(y

+

I)p,u,Ax2]:,

3/ = 3yp/[8(y

+

l)p,u,Ax]:,

M,

= 4.

200

H . Zhang and Fenggan Zhuang

To test the effectiveness of previous conclusion, we calculated onedimensional flow with Eqs. (2.11). Numerical results (see Zhang and Mao, 1987) verify the preceding conclusion. Figure 2a shows the case for v3 > 0 in the whole shock region. Figure 2b is for v3 < 0 in the whole shock region. Figure 2c corresponds with v 3 > 0 upstream and v3 < 0 downstream. In passing, we may note that there might be individual points in the shock region where (au/ax)(a2u/ax2)= 0, and hence when p = 0, the increasingentropy condition is not satisfied at these points. It might be argued that in the limit of mesh size approaching zero, the coefficients v 2 , v 3 , and v, , etc., will become zero anyway, and the solution obtained will seem to converge to the required solution of Navier-Stokes equations. But care must be taken that the correct physical solution can only be arrived from a sequence of solutions that satisfies the second law of thermodynamics. Of course, entropy at certain places may decrease if heat conduction is included; nevertheless, the condition imposed here is necessary. 111. A Formulation of the Semi-discretized NND Scheme (see Zhang, 1988) A. THECASEFOR

SCALAREQUATION

A

To construct a semi-discretized NND scheme, we start with a onedimensional scalar equation (3.1)

Here, f = au and a = af/au, a is the characteristic speed, and we may write where

a = a+ + a - , a+ = (a + la1)/2,

a-

Define we have

'f

= a+u

and

f =' f

=

af+ -au + + -ax= o . afat ax

(a - la1)/2.

f- = a - u ;

+ f-.

Equation (3.1) may be written as

(3.2) (3.3) (3 -4) (3.5)


20 1

It is easy to verify that if the second-order upwind difference scheme is used to evaluate the term af'/ax, the coefficient of the third-order derivative in the right-hand side of the modified equation is positive and the coefficient of the fourth-order dissipative term is negative; if the secondorder central difference is used, the coefficient of the third-order derivative is negative and the coefficient of the fourth-order dissipative term is zero. Similarly, if the second-order upwind scheme is used to evaluate af -/ax, the corresponding coefficient of third-order derivative is negative, while we still have the coefficient of fourth-order dissipative term negative; if the second-order central difference is used, the coefficient of third-order derivative is positive and the coefficient of fourth-order dissipative term remains zero. Based on those findings, we may employ the following difference schemes to evaluate space derivatives. In the upstream region of a shock: second-order upwind difference can be used to replace af + / a x ; second-order central difference can be used to replace af-/ax. In the downstream region of a shock: second-order central difference can be used to replace af + / a x ; second-order upwind difference can be used to replace af -/ax. In so doing, we have made the proper choice of the sign of the coefficient of the third derivative in the modified differential equation, i.e., v 3 is positive upstream of a shock and v3 is negative downstream of a shock, and this will provide us with a sharp transition without spurious oscillations, both upstream and downstream of shocks. At the same time, we notice that the coefficient of fourth-order dissipative derivative is negative in the entire region, and that this helps to suppress odd-even discoupling in smooth regions of physical flows. Now we write down the semi-discretized difference form of Eq. (3.6) as follows: in the upstream region of a shock,

in the downstream region of a shock,


202

Equations (3.7) and (3.8) can be combined together using the expressions for the numerical flux function to form

where hj+1/2 = h: 1/2L + fs1/2R

and &2L

fGl/ZR

=

=

I I

(3.10)

9

h++ iAfi+_,,,

(upstream), (downstream);

(3.11)

&Tl - i A.&1/2

(upstream), (downstream);

(3.12)

fj’ + 3 J+l

f,-

- i AAj;3/2

A h L 2 = h5l - h*.

(3.13)

It is clear from Eq. (3.1 1) that we have to make a choice between and for the upstream and downstream regions. For the shock wave with monotonic u profile, it is easy to convince ourselves that, in general up to second-order accuracy, Afj+_,,, and Afi+cl/2 have the same algebraic sign, and that in the upstream region, IAf g 1/2 I is smaller than IAA: I , and the opposite is true in the downstream region. Similarly, A&, G2 = J(CxE2 + r y p 2 + T Z G 2 ) , E2 = J ( T J 2

5; = (Y,Zf - YfZ,)/J, ry

= (Z,Xf - X,ZfYJ,

T z = (x,Yr - Y,Xf)/J,

t l x = (Z&

- Y€Zf)/J,

'lu = (X$f

- XfZ@J,

tlz

= ( Y g q - XEYfYJ,

(6.5)


=

1 Re TxxU -k TVU 4- t x z W

aT ax

k-

1

le; = Re

,

0

e2 --R 1e

Tzz UTxz 4- UTyz -k WTzz -k

aT az

k-

243


244

and au 2 au - -p(ax 3 ax

rx* = 2p-

av ryy = 2 p -

au

-

2 3

au ax

-), + - + e), av aw ++ az ay av ay

az

where u, v, and w are the velocity components in the Cartesian system (x, Y , 2). The coordinate system ( 0 upstream, and u3 < 0 downstream, the undesirable oscillations can be totally suppressed. Based on this finding, the semidiscretized nonoscillatory, containing no free parameters and dissipative scheme is developed, and five explicit NND schemes (NND 1-5) and an implicit NND scheme are given. We can prove that all NND schemes possess the TVD property. It is very interesting to note the following facts: (1) The expression of NND-1 scheme is formally just the same as the Osher-Chakravarthy scheme based on the second-order upwind method except for the delimiter parameters 8, and NND-1 is of second-order accuracy in regions of steep gradients as well as smooth regions. (2) NND-2 is just the Harten TVD scheme. (3) NND-4 is the TVD scheme that can be looked upon as an extension of MacCormack’s predictor-corrector scheme. (4) From the consideration of the amount of numerical work required, using NND-1, NND-4, NND-5, or implicit NND schemes is suitable. In general, we have seen that the NND schemes developed so far are very flexible, and are easily adapted to solve different kinds of flow problems, either in the time-dependent manner or in the direct treatment of the steady case, e.g., marching in the region of supersonic flow. Besides from the derivation of the scheme at its inception, this kind of consideration may be applied to the calculations of flow region where we have contact discontinuities or shear flows with steep gradients of the flow parameters, as already exemplified here to a certain extent by our numerical simulations. From the preceding calculated results, we have gained confidence in the use of the NND schemes suggested here. The distinguished feature of these schemes is the capability of capturing shock and other contact discontinuities, as examplified by the preceding graphical representations. The schemes possess good stability characteristics and converged accuracy; this is essential to any high-shock resolution scheme. The present form seems to be the simplest; meanwhile, the amount of numerical work is much reduced in comparison with some other high-resolution TVD schemes. The authors are grateful to Mr. Gao Shuchun for many helpful discussions during the course of this work, and also to Mr. Shen Qing and Mr. Ye Youde, who provided many of the numerical results used here. This work was supported in part by the China National Natural Science Foundation under Grant 91888010.


References Chakravarthy, S. R., and Osher, S. (1985). A new class of high accuracy TVD schemes for hyperbolic conservation laws. AZAA paper 85-0363. Davis, S. F. (1984). TVD finite difference schemes and artificial viscosity. ICASE Report 84-20. Harten, A. (1983). A high resolution scheme for the computation of weak solution of hyperbolic conservation laws. J. Cornp. Phys. 49, 351-393. Harten, A. (1984). On a class of high resolution total variation-stable finite difference schemes. SZAM J. Anal. 21, 1-23. Harten, A. (1986). Some results on uniformly high order accurate assentially nonoscillatory schemes. ICASE Report 86-18. Jameson, A., and Yoon, S. (1985). Multigrid solution of the Euler equations using implicit schemes. AZAA Paper 85-0293. Oscher, S., and Chakravarthy, S. R. (1984). Very high order accurate TVD schemes. UCLA Math. Report. Pulliam, T. H. (1985). Artificial dissipation models for the Euler equations. AZAA Paper 85 -043 8. Shen, T. (1989). Applications of NND scheme to the solving Navier-Stokes equations on the fore-head of space-shuttle-like body. Acta Aerodynarnica Sinica 7, (2), 146-155. Teshima, K. (1984). Visualization of freejet by laser induced fluoresence method. J. Japan Soc. Aero. Space Sci. 32, 61-64. Thompson, D. S., and Matus, R. J. (1989). Conservation errors and convergence characteristics of iterative space-marching algorithms. AZAA Paper 89-1935 CP. van Leer, B. (1979). Towards the ultimate conservation difference scheme V , a second order sequel to Godunov’s method. J. Cornp. Phys. 32, 101-136. Weilmuenster, K. J., and Hamilton 11, H. H. (1983). Calculations of inviscid flow over shuttle-like vehicles at high angles of attack and comparisons with experimental data. NASA TP2103. Ye, Y.D. (1989). Numerical computation of inviscid flow around nosetip of space shuttle with NND scheme. Acta Aerodynarnica Sinica 7 , (3), 282-290. Yee, H. C. (1985). On the implementation of a class of upwind schemes for system of hyperbolic conservation laws. NASA TM86839. Yee, H. C. (1987). Upwind and symmetric shock-capturing schemes. NASA TM89464. Zhang, H. X. (1983). A mixed explicit-implicit antidiffusive method of Navier-Stokes equations for supersonic and hypersonic separated flows. Applied Math. and Mech. (China), 4, (l), 54-68. Zhang, H. X. (1984). The exploration of the spatial oscillations in finite difference solutions for Navier-Stokes shocks (in Chinese). Acta Aerodynarnica Sinica 1, 12-19. Zhang, H. X. (1988). NND scheme. Acta Aerodynarnica Sinica 6, (2), 143-165. Zhang, H. X. (1989a). Implicit NND schemes. CARDC Report. Zhang, H. X. (1989b). Explicit NND schemes. CARDC Report. Zhang, H. X., and Mao, M. L. (1987). Numerical simulation for one dimensional Navier-Stokes equations. CARDC Report. Zhang, H. X., and Zheng, M. (1986). A mixed antidissipative method solving three dimensional separated flow. Lecture Notes in Physics, 264, 689-692. Zheng, M., and Zhang, H. X. (1989). Applications of NND scheme to the calculation of freejet flows. Acta Aerodynarnica Sinica 7, (3), 273-281.

Author Index

A

Chen, Q. Y.,47-48, 60-61 Cherepanov, G . P., 74, 96, 187 Chiao, Y.-H., 140, 187 Clarke, D. R., 140, 187 Coller, A. R., 41, 60 Comninou, M., 86, 187 Conway, J. C., 92, 191 Cooley, W. U., 52, 54-56, 59, 61 Cornie, J. A., 145, 148, 187 Cotterell, B., 139, 177, 188 Cottrell, A. H., 20, 59

Akisanya, A. R., 181, 187 Anderson, G. P., 77, 187 Argon, A. S., 145, 148, 187 Arin, K., 169, 188 Asaro, R. J., 79, 190 Ashby, M. F., 98-99, 139, 180, 189-190 Atkinson, C., 121, 187

B Babcock, C. D., 148, 152, 187 Bao, G., 70, 93, 112-114, 122, 125, 187,

D

190

Barrett, C. S., 8, 59 Bartlett, A., 67-68, 88-90, 188 Bauer, C. L., 145, 188 Bender, A. M., 27, 62 Beuth, J., 106, 131, 135, 187 Blochwitz, C., 29, 61 Brown, A. F., 8, 18, 40, 59 Brown, M. W., 51, 59 Buckley, S. N., 26, 59 Buisov, N. N., 62 Burger, J. M., 5 , 59 Burke, J. E., 8, 59

Dalgleish, B. J., 79, 180, 188 Davidson, B. D., 95, 189 Davis, S. F., 4, 256 Delannay, F., 137, 188 DeVries, K. L., 77, 187 Differet, K., 23, 25, 39, 61 Dolan, T. J., 58, 59 Drory, M. D., 128, 140, 188 Dugdale, D., 123, 188 Duhammel, M. J. C., 11, 59 Duncan, W. J., 41, 60 Dundurs. J., 72, 188

C

Cannon, R. M., 139, 187 Cao, H. C., 78, 93, 116-117, 187, 190 Carlsson, L. A., 149, 189 Chai, H., 148, 152, 158, 162, 167, 169, 180-181, 187

Chai, Y.-S.,78, 84-86, 105, 189 Chakravarthy, S. R., 194, 256 Charalambides, P. G., 79, 116-117,

E Elssner, E., 73, 190 England, A. H., 74, 188 Entwistle, K. M., 26, 59 Erdogan, F., 74, 169, 188 Eshelby, J. D., 11, 59 Essmann, V., 23, 25, 39, 59, 61 Evans, A. G., 67-68, 78-79, 81, 88-90,

187-189

98-99, 114, 116-117, 126, 128, 131, 135, 137, 139-140, 148, 152, 159-160, 180, 183, 185, 187-191

Charsley, P., 26, 59 Chen. N. K., 8. 59

257

Author Index

258 F

J

Fan, B., 70, 93, 112-114, 122, 125, 187 190 Farris, R. J., 145, 188 Fisher, R., 139, 187 Fleck, N. A., 174, 177-178, 180-181, 187-188 Forest, P. G., 46-47, 60 Forsyth, P. J. E., 20, 28, 60 Frankel, J., 4, 60 Frazer, R. A., 41, 60 Friedrich, K., 112, 188

Jameson, A., 194, 256 Jensen, H. M., 80, 140, 145, 167, 169, 171, 189 Johnson, R. D., 40, 60

G Gecit, M. R., 131, 188 Gille, G., 65, 135, 148, 152, 155. 188 Gillespie, J. W.,149, 189 Gilman, J. J., 5, 60 Gosele, V., 25, 59 Cough, H. H., 26, 60 Gupta, V., 145, 148, 187

K Kamada, K., 169, 189 Kennedy, A. J., 21, 25, 60 Kim, K.-S., 80, 140, 145, 189 Kinloch, A. J., 78, 80, 189 Kirchner, H. P., 92, 191 Knauss, W. G., 148, 152, 187 L

Landis, H. S., 145, 148. 187 Lekhnitskii, S. G., 30, 60, 69, 189 Lerinman, R. M., 62 Liechti, K. M., 77-78, 84-86, 105, 167, 189 Lin, S. R., 17, 22, 35, 39-40, 42, 47, 56, 60-61

H Hamilton, 11, H. H., 246, 256 Hanson, E. C., 77, 189 Harten, A., 194, 207, 256 Hayashi, k., 67, 188 He, M.-Y., 67-68, 88-90, 114, 131, 180, 188 Heidenreich, R. D., 7, 60 Higashida, y., 169, 189 Hirth, J. P., 9, 15, 60, 180, 189 HO, S., 112-113, 135, 182, 185, 187-188 Hu, M. S., 116-117, 128, 135, 140, 183, 188-189 Hull, D., 20, 59-60 Hunsche, A., 48, 60 Hunt, G. W., 160, I90 Hutchinson, J. W., 67-68, 75, 79-81, 88-90, 95, 98-99, 101, 103, 105, 107, 109-110, 139-141. 145, 148, 152, 159-160, 174, 177-178, 180-181,

188-190 I

Irwin, G. R.,70, 92, 95, 135-136, 185, 190 Ito, Y. M., 16, 21, 23, 25, 47, 60

Lin, T. H., 16-17, 21-23, 25, 34-35, 39-40, 42, 47-48, 56, 59-62

Lothe, J., 9, 15, 60 Louat, N., 20, 62 Lund, J., 116-117, 187

M MacCone, R. K., 2, 61 Machlin, E. S., 20, 61 Malyshev, B. M., 75, I89 Mandell, J. F., 175, I90 Mao, M. L., 198, 200, 256 Mataga, P. A., 93, I90 Matus, R. J., 206, 256 McCammon, R. D., 2, 61 McEvily, Jr., A. J., 20. 61 McGarry, F. J., 175, 190 McLean, D.. 8, 61 Mear, M. E., 107. 189 Meke, K., 29, 61 Melan. E., 17. 61 Meshii, M., 8 , 61 Miller, K. J., 51, 59 Mori, T., 8, 61 Mott, N. F., 8-9, 20, 61 Mughrabi, H., 23, 25, 39.59, 61 Mura, T., 5-6, 10, 61

Author Index N Nemat-Nasser, S., 67, 188 Neumann, p., 48, 60 Nye, J. F., 8, 61 0

O’Brien, T. K., 93, 189 O’DOwd, N. P., 118-119, 189 Orowan, E., 4, 61 Oscher, S . , 256

P Paris, P. C., 70, 92, 95, 135-136, 185, 190 Polyanyi, M., 4, 61 Pond, R. B., 8, 59, 61 Pulliam, T. H., 194, 256

259 T

Tada, H., 92, 95, 135-136, 185, 190 Taylor, G. I., 4, 61 Teshima, K.,238, 256 Thompson, D. S . , 206, 256 Thompson, J. M. T., 160, 190 Thompson, N., 20. 26, 51, 59, 62 Thouless, M. D., 78, 93, 97-99, 136, 139-140, 178-179, 188-189

Trantina, G. C., 77, 190 Tung, T. K., 17, 62

V Van Leer, B., 194, 256

W R Reimanis, I. E., 116, 189 Rice, J. R., 74, 81, 83, 87, 96, 107, 123-124, 139, 177, 188-189 Rosenberg, H. H., 2, 61 Rosenhain, W., 7, 61 Rothschilds, R. J., 149, 189 Riihle, M., 79, 180, 188-189

S Salganik, R. L., 7 5 , 189 Sanchez, J., 121, 187 Sbaizero, O., 116, 189 Schmander, S., 73, 190 Schmueser, D., 86, 187 Schwope, A. D., 40, 60 Shapery, R. A., 95, 189 Shen, T., 238, 256 Shih, C . F., 72, 79, 100, 118-119, 189-190 Shober, F. R., 40,60 Shockley, W., 7, 60

Sih, G. C., 70, 74, 189-190 Smelser, R. E., 121, 187 Sneddon, I. N., 13, 61 Spearing, S . M., 126, 190 Storakers, B., 149, 167, 190 Stout, M.G., 118-119, 189 Stubbington, C. A., 60 Suga, T., 73, 190 SUO, Z., 69-70, 72, 74-75, 78, 87, 90, 93-95, 99-101, 103, 105, 109-110, 112-114, 120, 122, 125, 135, 137, 140-141, 174, 177-182, 185, 187-191

Wadsworth, N. J., 20, 51, 62 Wang, J. S., 74, 78, 87, 120, 179-180, 189-190

Wang, R., 23, 25, 39, 61 Wang, S . S . , 175, 190 Wang, T. C., 70, 72, 93, 100, 114, 190 Warren, P., 137, 188 Weilmuenster, K. J., 246, 256 Whitcomb, J. D., 149, 191 Williams, J. G., 95, 191 Williams, M. L., 74, 77, 187, 191 Winter, A. T., 28, 62 Wood, W. A., 21, 26-27, 62 Woods, P. T., 28, 62 Wu, X. Q . , 35, 39, 61

Y Yakovleva, E. S., 62 Yakutovitch, M. V., 62 Ye, T., 135, 137, 185, 191 Ye, Y. D., 238,256 Yee, H. C., 194, 206,256 Yin, W.-L., 149, 162, 191 Yoon, S . , 194, 256 Young, A. P., 40, 60 Z

Zak, A. R., 191 Zdaniewski, W. A., 92, 191 Zhmg, H. X., 194, 198, 200, 206-211, 214, 224, 231, 256 Zheng, M., 231, 256

Subject Index

A

Crank-Nicolson method, 211 Creep, interaction with fatique, 39

Adhesive joints, 172 Airy’s stress function, 17 Allowable maximum marching step, 222 Applied Force, and inelastic strain, 9, 22

D

Delamination, 92 Delamination specimens, 112 Density contour, 237 distribution, 239 Difference method, 195 Dislocation, 2 Burger’s vector, 5 dipoles, 23 dislocation density tensor, 6 dislocation displacement and plasticity strains, 15 dislocation flux tensor, 6 edge, 5 interstitial dipoles, 23 polarization, 9 screw, 5 vacancy dipoles, 23 Dissipative schemes mixed, 193 nonoscillatory, 194 Double cantilever beam, 96, 113 Drum shock, 237 Duhammel’s Analogy, 11 Dundurs’ parameters, 72

B Blisters circular, 158 one-dimensional, 149 spalling, 163 straight-sided, 163 tests, 167 Blunt body, 238 Body fitted coordinate system, 246 Brazil nut specimen, 120 Buckle-driven delamination, 147 C

Canopy shock, 249 Characteristic equation, 197 Characteristic relation, 245 Configurational stability of inlayer crack, 176

Contact discontinuity, 255 Contact of crack faces, 86 Converged accuracy, 255 Courant number, 206 Crack patterns in films, 130 Cracks bilayer, 101 channeling, 134. 135 edge crack, 150 interface, 72 kinked, 67, 70, 88 sub-interface, 107 substrate, 137 tunneling, 90, 182

E Eigenvalue of matrix, 205 Energy release rates, 67, 75 EN0 scheme, 194 Equation of entropy, 198 Euler equation, 194, 204 Explicit NND scheme, 206 Extrusion, 20 260

Subject Index F Fatigue crack initiation combined cyclic axial and torsional loading, 51 cyclic torsional loading, 5 1 gating mechanism, 2 grain sue, 45, 47 initial stress field, 31 mean stress, 45, 46 overload effect, 45, 47 quantitative theory, 30 strain hardening, 45, 47 Fatigue, interaction with creep, 39 Film debonding, 143 Films in compression, 147 in tension, 126, 130 Fourth-order dissipative term, 201 Frank Read source, 20 Free jet flow, 231

G Gating mechanism, 23 Generalized plane strain, 30 Gerber’s Parabola law, 46 Grid, 245

H High-shock resolution scheme, 255 Hypersonic flow, 238

26 1 M

MacCormack’s explicit scheme, 214 Mach disc, 237 number contour, 247 number distribution, 241 Mixed dissipative scheme, 193 Mixed mode fracture, 65 stress intensity factors, 74 Modified differential equation, 195 Modified Goodman line, 46

N Navier-Stokes equation, 194 NND schemes, 193 explicit, 206 implicit, 21 1 MacCormack’s explicit, 214 NND-1 scheme, 206 NND-2 scheme, 207 NND-3 scheme, 208 NND-4 scheme, 209 NND-5 scheme, 210 semi-discretized, 200 Nonoscillatory dissipative scheme, 194 No-slip condition, 245 Normal shock wave, 199 Numerical simulation, 193

I Implicit NND scheme, 211 Increasing entropy condition, 199 Inelastic strain, 9 and applied force, 9, 22 creep strain, 9 plastic strain, 9 thermal strain, 9 Interface fracture specimens, 116 kinking out of, 88 toughness, 78, 81 Intrusion, 20 Inviscid flow, 242

K Kinking out of interface, 88

0

Odd-even discoupling, 201 Orthotropic elastic solids, 69 layer, 99

P Plastic strain, 15 dislocation displacement, 15 free surface effect, 16 Predictor-corrector scheme, 208 Pressure contours, 237 distribution, 227, 253 Protrusions, 29

262

Subject Index R

R-curves, 122 Rankin-Hugoniot relation, 245 Reflection of shock wave, 213 Runge-Kutta method, 210 S

Sandwich layers, 172 Scalar equation, 200 Schmid’s Law, 7 Second law of thermodynamics, 200 Semi-discretized NND scheme, 200 Shear stress excessive, 36 resolved, 7, 15 Shock capturing method, 193 Shock wave, 193 Slip bands, 2, 18 shear bands, 2 fatigue band, 32 localized plastic strain, 2 persistent, 20 Space lattice, 3 Space-marchingmethod. 246 Space-shuttle-likegeometry, 246 Spalling blisters, 163 substrate, 139 thermal shock, 146

Spurious oscillation near shock, 193 Stability characteristics, 255 Steady-state cracking, 90 Stress fields, 11, 15 initial, 18, 36 residual, 35 Stress influence coefficient, 40 Stress intensity factors, 66, 74 Substrate spalling, 139

T T-stress, 67 Temperature contour, 248 distribution, 239 Thermal shock spalling, 146 Third-order dispersion term, 194 Three-dimensional flow, 194 Time dependent method, 194 Time splitting method, 205 Two-dimensional flow, 194

U Unconditional stability, 211 Upwind scheme, 214

V Viscous flow, 235

Advances in Applied Mechanics, Volume 29

Advances in applied mechanics

Advances in Applied Mechanics