journal of the mechanics and physics of solids volume 49 issue 11 2001 viggo tvergaard -- resistance...

7/28/2019 Journal of the Mechanics and Physics of Solids Volume 49 issue 11 2001 Viggo Tvergaard -- Resistance curves for …

http://slidepdf.com/reader/full/journal-of-the-mechanics-and-physics-of-solids-volume-49-issue-11-2001-viggo 1/15

Journal of the Mechanics and Physics of Solids

49 (2001) 2689–2703

www.elsevier.com/locate/jmps

Resistance curves for mixed mode interface crack growth between dissimilar elastic–plastic solids

Viggo Tvergaard ∗

Department of Solid Mechanics, Technical University of Denmark, Building 404, DK-2800 Lyngby,

Denmark

Abstract

For crack growth along an interface joining two elastic–plastic solids resistance curves are

analysed numerically, under conditions of small scale yielding. As mismatch of elastic properties

is accounted for, the corresponding oscillating stress singularity ÿelds are applied as boundary

conditions on the outer edge of the region analysed. The fracture process is represented in terms

of a cohesive zone model, for which the work of separation per unit area and the peak stress

required for separation are basic parameters. Eects of dierent kinds of mismatch between thetwo materials are analysed, including cases where the substrate remains elastic as well as cases

where both materials yield plastically. For all the cases studied it is shown that plastic ow

near the crack tip results in much larger resistance to crack growth when mode II conditions

dominate at the crack tip than when mode I conditions dominate. ? 2001 Elsevier Science Ltd.

All rights reserved.

Keywords: A. Crack tip plasticity; Fracture process; B. elastic–plastic material

1. Introduction

It is well understood that plastic work during crack growth in metals contributes

signiÿcantly to the fracture toughness, so that the macroscopic work of fracture is much

larger than the work absorbed by the the local fracture process required to separate

the crack surfaces. This has been studied in more detail in a number of elastic–plastic

crack growth computations (Tvergaard and Hutchinson, 1992, 1993, 1994a), in which

the fracture process has been modelled by a traction-separation law along the crack

plane with a speciÿed work of separation per unit area, while the material has been

modelled as elastic–plastic. One of these studies (Tvergaard and Hutchinson, 1993)

∗ Tel.: +45-4525-4273; fax: +45-4593-1475.

E-mail addresses: [email protected] (V. Tvergaard).

0022-5096/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 2 2 - 5 0 9 6 ( 0 1 ) 0 0 0 7 4 - 6



2690 V. Tvergaard / J. Mech. Phys. Solids 49 (2001) 2689–2703

has focussed on the special case of crack growth along an interface joining an elastic–

plastic solid to a solid which does not yield plastically.

Experimental studies of interface crack growth, where plastic yielding occurs in at

least one of the solids joined at the interface (Cao and Evans, 1989; Liechti and Chai,1992; O’Dowd et al., 1992), have shown a strong dependence on the mode mixity,

such that the measured fracture toughness is much higher in cases where mode II

loading dominates at the crack tip than in cases where mode I loading dominates. This

was also predicted by Tvergaard and Hutchinson (1993), and it was concluded that

the observed strong dependence on the mode of loading can be explained by plastic

yielding. However, it is noted that also models of asperity interaction, representing

eects of friction due to rough crack surfaces, have shown a somewhat increased

toughness in cases where mode II loading dominates (Evans and Hutchinson, 1989;

Jensen, 1990).

A traction-separation law could be used to represent cleavage fracture by atomic

separation, but the stress peaks during atomic separation are very high, and it has

been found by Tvergaard and Hutchinson (1992, 1993) that crack growth predictions

based on a cohesive zone model of the fracture process are limited by the fact that

according to continuum plasticity crack solutions the maximum stresses near the crack

tip do not exceed about ÿve times the ow stress. However, in these cases the length

scale of the fracture process is typically much smaller than the dislocation spacing or

the elastic cell size within the dislocation structure, and therefore continuum plasticity

would not be realistic near the crack tip, assuming that the crack tip does not emit

dislocations. To model the elastic core region near the crack tip Suo et al. (1993) have proposed the use of a long elastic strip of material, inside which the crack propagates,

and Beltz et al. (1996) have proposed a self-consistent way of identifying the width

of the strip. Wei and Hutchinson (1996, 1997) have used this elastic strip model in

steady-state analyses to study interface toughness of metal–ceramic joints under mixed

mode loading, and Tvergaard (1999) has used the model to determine crack growth

resistance curves for the same type of cases. The conclusion of these studies is that also

for cleavage fracture by atomic separation plastic yielding of the material outside the

elastic core region can signiÿcantly increase the fracture toughness. And also in these

situations the predicted fracture toughness is higher in cases where mode II loading

dominates than when mode I loading dominates.In the present paper the resistance curve behaviour is analysed for crack growth along

the interface between two elastic–plastic solids. It is assumed that fracture occurs by a

ductile mechanism, involving voids or small poorly bonded regions on the interface, so

that fracture can be represented by a traction-separation law with a peak stress that does

not exceed either of the two ow stresses by a large factor. The analyses are restricted

to conditions of small scale yielding, with mixed mode loading conditions applied far

away from the crack tip. Thus, the elastic crack tip ÿelds with an oscillating singularity

are applied as boundary conditions, as in the analyses of Tvergaard and Hutchinson

(1993) for a rigid substrate, but in the present studies the stress and strain ÿelds are

determined on both sides of the interface. The analyses are used to study eects of dierences between the elastic properties of the two materials as well as eects of

dierences between the plastic properties.



V. Tvergaard / J. Mech. Phys. Solids 49 (2001) 2689–2703 2691

Fig. 1. Interface crack with dierent elastic–plastic material properties on each side of the crack plane.

2. Problem formulation

In the plane strain interface crack problem analysed by Tvergaard and Hutchinson

(1993) one of the materials is taken to be elastic–plastic while the other material

is rigid. With both materials represented as deformable the present study includes the

previous one as the limit where both Young’s modulus and the yield stress tend towards

inÿnity for the substrate. But the present study also accounts for the symmetric problem

of crack growth through a homogeneous solid (as in Tvergaard and Hutchinson, 1992)and for all mixed mode problems with dierent properties of the two materials.

2.1. Small scale yielding formulation

The equations governing the elastic crack tip singularity ÿeld of a semi-inÿnite in-

terface crack have been given by Rice (1988); see also Tvergaard and Hutchinson

(1993). This elastic crack problem was done long ago by many authors (e.g. England,

1965). The crack has tractions acting on the interface, which are given in terms of the

two stress intensity factor components, K I and K II, by22 + i12 = ( K I + iK II)(2r )−1= 2r i: (2.1)

Here, r is the distance from the tip, i =√ −1; is the oscillation index

=1

2ln

1 − ÿ

1 + ÿ

(2.2)

and ÿ is the second Dundurs’ parameter

ÿ =1

2

1(1 − 22) − 2(1 − 21)

1(1 − 2) + 2(1 − 1)

: (2.3)

The shear moduli are 1 = E 1= ((1 + 1)) and 2 = E 2= (2(1 + 2)), where E 1; 1 are the

elastic constants for material No. 1, i.e. the material located at x2 ¿ 0 (see Fig. 1),




while E 2; 2 are the elastic constants for material No. 2. The relation between the energy

release rate and the magnitude | K | of stress intensity factors is

G = 12

(1 − ÿ2)

1 − 2

1

E 1+ 1 −

2

2

E 2

| K |2; | K | =

K 2I + K 2II: (2.4)

With a reference length L chosen to characterize the remote ÿeld an L-dependent

measure of mode mixity is deÿned by

tan =Im[( K I + i K II) Li]

Re[( K I + i K II) Li](2.5)

which reduces to the more familiar measure, tan = K II=K I, when = 0. By using

Eq. (2.1) in Eq. (2.5) it is seen that tan = 12=22 at r = L on the interface. Thus,

measures the relative proportion of shear to normal stress on the interface a distance L from the tip, as predicted by the elastic solution. The displacement components

associated with the singularity ÿeld, with amplitude | K |, are speciÿed in Tvergaard

and Hutchinson (1993). The mode mixity at the radius r is given in terms of the

L-dependent measure of the mode mixity, , as

= + ln(r=L): (2.6)

The materials considered here are taken to be elastic–plastic with the true stress-logarithmic

strain curve in uniaxial tension speciÿed by

=

=E for 6Y ;

(Y=E )(=Y)1=N for ¿Y :(2.7)

Here, Y is the initial yield stress and N is the power hardening exponent, while E

and are Young’s modulus and Poisson’s ratio, respectively. The tensile behaviour is

generalized to multiaxial stress states assuming isotropic hardening and using the Mises

yield surface. For material No. 1 the parameters are denoted E 1; 1; Y1 and N 1, while

for material No. 2 they are E 2; 2; Y2 and N 2 (see Fig. 1).

In the numerical analyses ÿnite strains are accounted for, using a convected coor-

dinate, Lagrangian formulation of the ÿeld equations, in which gij and G ij are metrictensors in the reference conÿguration and the current conÿguration, respectively, with

determinants g and G , and Áij = 1= 2(G ij − gij) is the Lagrangian strain tensor. The

contravariant components ij of the Kirchho stress tensor on the current base vectors

are related to the components of the Cauchy stress tensor ij by ij =

G=gij. Then,

in the ÿnite-strain generalization of J 2-ow theory discussed by Hutchinson (1973),

an incremental stress–strain relationship is obtained of the form ij = LijklÁkl (e.g. see

Tvergaard and Hutchinson, 1993).

2.2. Traction-separation law at interface

A traction-separation law proposed by Needleman (1987) for separation due to the

normal stress on an interface was generalized by Tvergaard (1990) to also account for




separation law are relatively unimportant, and that the two most important parameters

characterizing the fracture process in this model are 0 and .

2.3. Reference values

As mentioned above, the remote loading is speciÿed in terms of the three parameters

| K |; , and L. Using Eqs. (2.4) and (2.11) a reference stress intensity factor is deÿned

as

K 0 =

1 − 2

1

E 1+

1 − 22

E 2

−1= 2 20

1 − ÿ2

1= 2

: (2.12)

Here, K 0 represents the value of

| K

|needed to advance the interface crack in the

absence of any plasticity. This value is independent of since a potential is used togenerate the relation of tractions to crack face displacements of the interface. A length

quantity R0, which scales with the size of the plastic zone in material No. 1 (when

| K | ∼= K 0), is deÿned as

R0 =1

3

K 0

Y1

2

=2

3

1 − 2

1

E 1+

1 − 22

E 2

−10

(1 − ÿ2)2Y1

: (2.13)

While the mode mixity measure refers to the distance L from the tip, it is natural to

deÿne a reference measure of mixity, 0, based on the reference length R0. By using

Eq. (2.6), and 0 are related by

0 = + ln( R0=L): (2.14)

In the case to be analysed here the material parameters in materials 1 and 2, and thus

the value of the oscillation index according to Eq. (2.2), vary from case to case, and

this means that the phase shift ln( R0=L) in Eq. (2.14) will dier from case to case.

3. Numerical procedure

The numerical solutions are obtained by a crack growth procedure analogous to that

used by Tvergaard and Hutchinson (1993). Thus, a ÿnite element approximation of the

displacement ÿelds is used in a linear incremental method, with a Cartesian coordinate

system xi as reference, and with the displacement components on the reference base

vectors denoted by ui. The Lagrangian strain increments are given by

Áij =1

2(u i; j + u j; i + uk

; juk; j + uk ; iu k; j); (3.1)

where ( ); j denotes the covariant derivative in the reference frame. Furthermore, the

incremental principle of virtual work takes the form V

{ijÁij + ijuk ; iuk; j} dV =

S

T ui dS −

V

ijÁij dV −

S

T iui dS

; (3.2)




Fig. 3. Mesh used for some of the crack growth analyses.

where V and S are the reference volume and surface of the region analysed, T i are the

nominal traction components, and the bracketed terms are included to prevent drifting

of the solution away from the true equilibrium path.An example of the mesh used for the computations is shown in Fig. 3, where it

is seen that a uniform mesh region is used in the range where crack growth along

the interface is studied. The length of one square element inside the uniform mesh is

denoted 0, and the initial crack tip is located at x1 = 0. Most of the computations

are carried out with 60 × 6 quadrilaterals in the uniform mesh along the interface, and

a few computations have used a 120 × 6 mesh to allow for more crack growth. The

elements used are quadrilaterals each built-up of four triangular, linear-displacement

elements. The outer radius of the region analysed is chosen to be A0=0 =8000, in

order that the plastic zone size should not exceed A0= 10.

On the free crack surfaces, for x2 =0 and x1 ¡ 0, zero tractions, T 1 = T 2 = 0, are pre-scribed, both for materials 1 and 2 (see Fig. 1). On the remaining part of the interface,

x2 = 0 and x1 ¿ 0, the displacements and tractions are related by the traction-separation

law (2.10). During the initial part of the crack growth resistance curve an increment of

| K | is prescribed, but this procedure becomes unstable when | K | approaches its asymp-

tote. Then, a Rayleigh–Ritz ÿnite-element method (Tvergaard, 1976) is used to control

a monotonic increase in displacement dierences across the crack tip. Thus, nodal dis-

placements at the top and bottom of the interface are used as mode amplitudes in

the Rayleigh–Ritz ÿnite-element procedure, and for each increment of the numerical

solution an appropriate dierence between such displacements in the fracture process

region is chosen as the parameter to be prescribed.In all computations the properties of material No. 1 are speciÿed as Y1=E 1 = 0:003,

1 = 1= 3 and N 1 = 0:1, while dierent values are considered for the properties of




material No. 2. In the traction-separation law the values cn=c

t = 1; cn = 0:10; 1 = 0:15

and 2 = 0:50 are used, while =Y1 is varied.

At some stages of the deformation the value of the J -integral is calculated on a

number of contours around the crack tip to check agreement with the prescribed amplitude | K | of the edge displacements, representing the | K | ÿeld in the elastic material

far away from the tip. The J -integral (Rice, 1968) was formulated in the ÿnite-strain

context by Eshelby (1970)

J =

{W d x2 − T iui; 1 d s}; W =

Áij

0

ijÁij; (3.3)

where is some path in the reference conÿguration from the lower crack surface to

the upper crack surface, d s is an arc length element along the path, and T i are the

nominal tractions on the boundary of the region enclosed by the contour . The path

independence of J is conÿrmed by these calculations, and good agreement is foundwith the value of the energy release rate G given by Eq. (2.4) in terms of | K |.

It is noted that the present problem formulation allows for switching the material

properties between regions 1 and 2, since full elastic–plastic material behaviour is ac-

counted for in both regions. It has been checked for a few cases that the identical crack

growth resistance curve is predicted when dissimilar material properties are switched

between the two regions, while the sign of the applied mode mixity is changed at the

same time.

4. Results

The resistance curve analyses to be presented ÿrst focus on the eect of elastic mod-

ulus mismatch in cases where material No. 2 does not yield plastically, while material

No. 1 is elastic–plastic. In all these analyses the elastic–plastic material parameters

are taken to be Y1=E 1 = 0:003; 1 = 1= 3 and N 1 = 0:1, while the elastic substrate has

Y2 ∞ and 2 = 1= 3. The values of E 2=E 1 to be considered are 1, 2, 6 and 1000.

For each elastic modulus mismatch a number of dierent values of the peak stress to

ow stress ratio, =Y1, are considered, and the eect of the mode mixity is studied

by considering dierent values of 0 in Eq. (2.14). This mixity measure should give

a good indication of the mode mixity in the process region at the crack tip, since it is based on the reference length R0, which scales with the plastic zone size at the onset

of crack growth.

Fig. 4 shows four calculated crack growth resistance curves, which all correspond

to the same values of the material parameters in the elastic–plastic material, while

material No. 2 does not yield (Y2 ∞). The same traction separation law is used

with =Y1 = 3:0 in all four computations, and the value of the angular measure 0 is

near 0◦

in all cases, so that the crack tip conditions are close to mode I . Thus, the only

dierence between these four cases is the value of Young’s modulus, E 2, in material

No. 2. It is seen that for E 2=E 1 = 1 plasticity has only a small eect on the interface

toughness, increasing the value of | K |=K 0 to 1.12 after a small amount of crack growth,while plasticity has a larger eect for E 2=E 1 =2 and E 2=E 1 = 6. In the case of a nearly

rigid material No. 2, E 2=E 1 = 1000, the resistance curve shows that | K |=K 0 increases to




Fig. 4. Interface crack growth resistance curves for =Y1 = 3:0, with Y1=E 1 = 0:003 and Y2 ∞.

about 2.38, and here this maximum toughness is reached after more crack growth, at

a=R0 = 2:3. For the four dierent values of E 2=E 1 considered in Fig. 4 the inuence

of the bond strength, =Y1, and of the mode mixity, 0, is illustrated in Fig. 5, by

showing the steady state toughness | K |SS=K 0.

In Fig. 5a there is no elastic mismatch, so that ÿ = 0 results in = 0. It is seen that

plasticity gives a strong sensitivity to the mode mixity so that the minimum fracture

toughness for a given value of =Y1 occurs somewhere near 0 = 0◦

, while a notice-

able mode II contribution results in much increased fracture toughness. For the smallestinterface strength analysed, =Y1 = 1:5, there is an interval between 0 −15

◦

and

0 15◦

where the toughness is constant, | K |SS=K 0 = 1, thus illustrating that in the

elastic range a strong mode II contribution does not increase the toughness. The in-

creased toughness found for values of 0 outside this interval is a result of plasticity.

The curves in Fig. 5a also show that for values of =Y1 greater than 4.0 the toughness

increases rapidly with small increases of the peak stress in the traction-separation law.

Figs. 5b–d show a picture very similar to that in Fig. 5a. In these three ÿgures the

values of the oscillation index are 0.0266, 0.0575 and 0.0811, respectively, so in all

these cases the oscillating singularity plays a role, and there is a dierence between

the mode mixity applied at the outer boundary of the region analysed, and that actingin the process region near the crack tip. Fig. 4 has illustrated, for a given value of

=Y1, that the toughness increases when the value of E 2=E 1 is increased, and it is

seen in Fig. 5 that this is a general trend for all values of the interface strength. The

case of E 2=E 1 = 1000 in Fig. 5d approaches that of an elastic–plastic solid bonded to

a rigid substrate, as studied by Tvergaard and Hutchinson (1993), and it is seen that

the results in Fig. 5d are close to those found for the rigid substrate.

The computational model has also been used to study cases where both materials

undergo plastic yielding. For cases where E 2=E 1 = 2, as in Fig. 5b, examples of crack

growth resistance curves are shown in Fig. 6. In these cases the values of 0 are near 0◦

(as in Fig. 4), so that the resistance curves shown correspond to situations where modeI conditions are dominant near the crack tip. The dashed curve for =Y1 = 3:0 and

Y2 ∞ is very similar to one of the resistance curves shown in Fig. 4, while the other




Fig. 5. Steady-state interface toughness as a function of the local mixity measure 0, for Y1=E 1 = 0:003

and Y2 ∞

, considering dierent values of =Y1. (a) E 2=E 1 =1, (b) E 2=E 1 =2, (c) E 2=E 1 =6, (d) E 2=E 1 = 1000.

dashed curve, for Y2=Y1 = 1, illustrates that additional plastic yielding in the substrate

adds noticeably to the fracture toughness. The solid curves in Fig. 6 are obtained for

=Y1 = 4:0, with values 2.0, 1.5 and 1.25 of the ow stress ratio, Y2=Y1, and with

the curve for Y2 ∞ included for reference. The resistance curve for Y2=Y1 = 1:25

shows a large increase in the maximum fracture toughness, compared to the result for

an elastic substrate (Y2=Y1

∞).

The inuence of plasticity in the substrate is further illustrated in Fig. 7 by curvesof steady state toughness vs. mode mixity. As in Fig. 6 the ratio of Young’s moduli

is E 2=E 1 = 2, and a number of dierent values of the ow stress ratio Y2=Y1 are




Fig. 6. Interface crack growth resistance curves for various values of =Y1 and Y2=Y1, with E 2=E 1 = 2

and Y1=E 1 = 0:003.

Fig. 7. Steady-state interface toughness as a function of the local mixity measure 0, for E 2=E 1 =2 and

Y1=E 1 = 0:003. Dierent values of Y2=Y1 and =Y1 are considered.

considered. The three curves for Y2 ∞ in Fig. 7 are identical to three of the curves

shown in Fig. 5b, while the remaining curves in Fig. 7 illustrate the increase in fracture

toughness resulting from plastic yielding in the substrate. If we focus on Y2=Y1 = 2:0,

no dierence from Y2 ∞ is found for =Y1 = 3:0, whereas a small dierence is

found for =Y1 = 3:5, and a clear increase of the toughness is found for =Y1 = 4:0

in the whole range of 0 values analysed. For Y2=Y1 = 1:5 the same tendency is

observed, with a much stronger toughness increase at the higher interface strength.For Y2=Y1 = 1:25 the curve corresponding to the highest interface strength is just

visible at the top of the diagram in Fig. 7. Thus, with the same elastic–plastic material




Fig. 8. Steady-state interface toughness as a function of the local mixity measure 0, for E 2=E 1 =1 and

Y1=E 1 = 0:003. Dierent values of Y2=Y1 and =Y1 are considered.

behaviour for material No. 1 the curves in Figs. 6 and 7 show a strong sensitivity to

whether or not plastic yielding takes place in the substrate. The toughness is increased

by plastic ow in the substrate, and this increase is larger the lower the initial yieldstress in the substrate. Also, the sensitivity to yielding in the substrate is much larger

in cases where the bond strength is higher, i.e. for a higher value of =Y1.

Curves of steady state toughness vs. mode mixity, analogous to those in Fig. 7, are

shown in Fig. 8 for a case where there is no elastic mismatch, E 2=E 1 = 1, so that the

only mismatch is that due to a dierence in ow stress. Again, the two curves for

Y2=Y1 ∞ are identical to two of the curves shown in Fig. 5a, and plastic ow in

the substrate results in an increased fracture toughness. The curves for Y2=Y1 = 1:5

illustrate that also here the eect of plastic yielding in the substrate is much larger if

the interface strength is higher.

The dependence of the steady-state fracture toughness on the value of =Y1 isshown in Fig. 9 for a number of dierent cases of material mismatch. The values

used to plot these curves correspond to the minima of the toughness vs. mode mixity

curves as those shown in Figs. 5, 7 and 8. It is noted that analogous curves have been

calculated by Tvergaard and Hutchinson (1994b, 1996) in studies of the toughness of

an interface along a thin ductile layer joining elastic solids, but these results represent

remote mode I loading rather than the minimum of the toughness vs. mixity curve.

The solid curves in Fig. 9, corresponding to Y2=Y1 ∞, illustrate how increasing

elastic stiness of the substrate gives increasing fracture toughness, for a given value of

=Y1. Curves representing plastic yielding of the substrate are shown for Y2=Y1 = 2

and Y2=Y1 = 1. Both sets of curves illustrate the trends found in Figs. 7 and 8 that plastic yielding of the substrate increases the toughness, and the curves quantify how

this behaviour varies with the value of =Y1.




Fig. 9. Steady-state interface toughness as a function of =Y1 , for various values of E 2=E 1 and Y2=Y1 .The values shown correspond to minima of curves like those in Figs. 7 and 8.

5. Discussion

The mixed mode toughness of an interface joining an elastic–plastic solid to a

solid which does not yield plastically has been analysed by Tvergaard and Hutchinson

(1993), based on the simplifying assumption that the non-yielding substrate is rigid.

The present study extends this work to account for deformations of the substrate, still

with full consideration of mode mixity eects and of the oscillating singularity dueto elastic mismatch, and still using a traction separation law to represent the fracture

process along the interface. One of the results obtained is the quantiÿed understanding

of the eect of the elastic stiness of a non-yielding substrate. In addition, results are

obtained for the eect of plastic yielding of the substrate.

The results for the inuence of elasticity of the substrate show that the resistance

curves are much like those found for a rigid substrate. Thus, for material parameters

where the interface toughness is only slightly higher than the elastic toughness the

peak toughness is reached after a very small amount of crack growth, but when the

normalised toughness

| K

| R=K 0 attains values higher than 3.0 or 4.0 the peak toughness is

reached after a great deal of crack growth, corresponding to several times the referencesize R0 for the plastic region. However, as shown by Fig. 4, and also by the solid

curves in Fig. 9, the interface toughness is strongly dependent on the elastic modulus

ratio E 2=E 1, for ÿxed values of all other material parameters. A similar dependence

has been noted in studies of interface crack growth along a thin ductile layer joining

elastic solids (Tvergaard and Hutchinson, 1994b, 1996), but those studies are limited

to remote mode I loading.

Crack growth along an interface joining an elastic–plastic solid to an elastic solid

is relevant to metal–ceramic systems, and the elastic modulus ratio E 2=E 1 =2 and

E 2=E 1 = 6 considered in Figs. 5b and c, respectively, are of an order of magnitude

representative of such material systems. The case for E 2=E 1 = 1 in Fig. 5a could repre-sent two blocks of the same metal bonded together, where one of the blocks has been

hardened to have a much higher yield stress. The last case, E 2=E 1 = 1000 (Fig. 5d),




has been included here to closely approach the results obtained previously for a rigid

substrate. A common feature of these studies of the eect of mode mixity is that a

signiÿcant mode II contribution near the crack tip gives a signiÿcant increase of the

toughness, as was also found in the case of a rigid substrate. In all four ÿgures thecurve for =Y1 = 1:5 has a central interval around 0 = 0

◦

, in which | K |ss=K 0 = 1:0.

These are intervals in which the behaviour is entirely elastic, but as soon as plasticity

appears at the ends of these intervals, it is again seen that the toughness increases with

increasing mode II contribution in the near tip region.

When also the substrate undergoes plastic yielding, more energy is dissipated during

crack growth as the near-tip plastic regions sweep over the material, and this results in

a higher interface toughness. This is illustrated by the resistance curves in Fig. 6 and

is seen in the curves of steady-state toughness vs. local mode mixity (Figs. 7 and 8),

which also show that the minima of the curves remain near 0 = 0◦

, i.e. in the range

where the near-tip behaviour is completely dominated by mode I conditions. These

curves also show that the sensitivity to reductions of the yield stress ratio, Y2=Y1, is

much stronger for higher values of the interface strength, =Y1, which is associated

with the increasing slope of the curves in Fig. 9 for increasing value of =Y1.

The conÿgurations with plastic yielding on both sides of the interface include as a

special case the symmetric solution for mode I crack growth along an interface be-

tween blocks of identical elastic–plastic solids, as studied by Tvergaard and Hutchinson

(1992). Indeed, the curves in Fig. 9 for E 2=E 1 = 1 tend clearly towards this previous

result for the symmetric case, when considering the trend from Y2=Y1 = 2:0 through

Y2=Y1 = 1:5 and Y2=Y1 = 1:25. But the non-symmetric cases in Figs. 6–9 are alsorelevant to a number of applications, where crack growth can occur along an interface

between blocks of dissimilar metals. Such dissimilar metal blocks can be bonded by

a number of dierent methods, such as cold welding, thermal diusive bonding, or a

thin adhesive layer. The non-symmetric analyses can also be relevant to some cases

of crack growth along a weld, where signiÿcant overmatching of the weld material

gives highly dissimilar yield stress but no elastic dierences on either side of the crack

(Niordson, 1999).

It is ÿnally noted that crack growth predictions by the present model are limited by

the fact that according to continuum plasticity crack solutions the maximum stresses

near the tip do not exceed about ÿve times the ow stress. Therefore, dierent models(Suo et al., 1993) must be used when fracture occurs by atomic separation. Such

procedures using a thin elastic strip of material around the crack tip have been employed

by Tvergaard (1999) to determine the resistance curve behaviour under mixed mode

loading, at an interface between an elastic–plastic solid and a rigid solid.

References

Beltz, G.E., Rice, J.R., Shih, C.F., Xia, L., 1996. A self-consistent model for cleavage in the presence of

plastic ow. Acta Metall. Mater. 44, 3943–3954.Cao, H.C., Evans, A.G., 1989. An experimental study of the fracture resistance of bimaterial interfaces.

Mech. Mater. 7, 295–304.

England, A.H., 1965. A crack between dissimilar media. J. Appl. Mech. 32, 400–402.




Eshelby, J.D., 1970. In: Kanninen, M.F. et al. (Eds.), Inelastic Behavior of Solids. McGraw-Hill, New York,

pp. 77–115.

Evans, A.G., Hutchinson, J.W., 1989. Eect of non-planarity on the mixed mode fracture resistance of

bimaterial interfaces. Acta Metall. 37, 909–916.Hutchinson, J.W., 1973. Finite strain analysis of elastic–plastic solids and structures. In: Hartung, R.F. (Ed.),

Numerical Solution of Nonlinear Structural Problems. ASME, New York, pp. 17.

Jensen, H.M., 1990. Mixed mode interface fracture criteria. Acta Metall. Mater. 38, 2637–2644.

Liechti, K.M., Chai, Y.S., 1992. Asymmetric shielding in interfacial fracture under in-plane shear. J. Appl.

Mech. 59, 295.

Needleman, A., 1987. A continuum model for void nucleation by inclusion debonding. J. Appl. Mech. 54,

525–531.

Niordson, C.F., 1999. Analysis of steady-state ductile crack growth along a laser weld. DCAMM Rep. 626.

O’Dowd, N.P., Stout, M.G., Shih, C.F., 1992. Fracture toughness of alumina= niobium interfaces: experiments

and analyses. Philos. Mag. A66, 1037.

Rice, J.R., 1968. A path independent integral and the approximate analysis of strain concentration by notches

and cracks. J. Appl. Mech. 35, 379–386.Rice, J.R., 1988. Elastic fracture mechanics concepts for interfacial cracks. J. Appl. Mech. 55, 98–103.

Suo, Z., Shih, C.F., Varias, A.G., 1993. A theory for cleavage cracking in the presence of plastic ow. Acta

Met. Mater. 41, 1551–1557.

Tvergaard, V., 1976. Eect of thickness inhomogeneities in internally pressurized elastic–plastic spherical

shells. J. Mech. Phys. Solids 24, 291–304.

Tvergaard, V., 1990. Eect of ÿbre debonding in a whisker-reinforced metal. Mater. Sci. Eng. A125,

203–213.

Tvergaard, V., 1999. Eect of plasticity on cleavage crack growth resistance at an interface. J. Mech. Phys.

Solids 47, 1095–1112.

Tvergaard, V., Hutchinson, J.W., 1992. The relation between crack growth resistance and fracture process

parameters in elastic–plastic solids. J. Mech. Phys. Solids 40, 1377–1397.

Tvergaard, V., Hutchinson, J.W., 1993. The inuence of plasticity on mixed mode interface toughness. J.

Mech. Phys. Solids 41, 1119–1135.

Tvergaard, V., Hutchinson, J.W., 1994a. Eect of T -stress on mode I crack growth resistance in a ductile

solid. Int. J. Solids Struct. 31, 823–833.

Tvergaard, V., Hutchinson, J.W., 1994b. Toughness of an interface along a thin ductile layer joining elastic

solids. Philos. Mag. A 70, 641–656.

Tvergaard, V., Hutchinson, J.W., 1996. On the toughness of ductile adhesive joints. J. Mech. Phys. Solids

44, 789–800.

Wei, Y., Hutchinson, J.W., 1996. Mixed mode interface toughness of metal = ceramic joints. Material Research

Society Symposium, Vol. 409, Materials Research Society, Pittsburgh, pp. 163–170

Wei, Y., Hutchinson, J.W., 1997. Nonlinear delamination mechanics for thin ÿlms. J. Mech. Phys. Solids

45, 1137–1159.

journal of the mechanics and physics of solids volume 49 issue 11 2001 viggo tvergaard -- resistance...

Documents