Safe equilibrium and crack growth in inhomogeneous

materials as a variational problem

K. C. Le1a, M. El Yaagoubib

a Lehrstuhl fur Mechanik-Materialtheorie, Ruhr-Universitat Bochum, Universitatstr. 150,44780 Bochum, Germany

b MS-Schramberg GmbH & Co. KG, Max-Planck-Straße 15, 78713 Schramberg, Germany

Abstract

The variational principle of safe equilibrium for inhomogeneous elastic crackedbodies is formulated. Using the standard calculus of variations, we show thatthe crack remains in safe equilibrium as long as the maximum energy reduc-tion rate of the virtually growing crack is negative. The crack starts to growin the direction of the maximum energy reduction rate when the latter be-comes zero. This energetic criterion implies the criteria proposed by He andHutchinson (Int J Solids Struct 25:1053–1067, 1989). As an application weuse this criterion to predict the growth direction of an interface crack in abimaterial.

Keywords: safe equilibrium, crack growth, energy reduction rate, stressintensity factor, interface crack.

1. Introduction

The main objective of materials engineering is to improve material prop-erties such as strength, high temperature resistance, corrosion resistance,hardness and conductivity. This goal can be achieved by joining differentmaterials in various ways, such as adhesive bonds, protective coatings, thinfilm/substrate systems for electronic packages or composite bodies, to namea few. Based on their individual properties, materials such as ceramics, poly-mers, glasses or metals can be combined. For polycrystalline materials such

1corresponding author: ++49 234 32-26033, email: chau.le@rub.de

Preprint submitted to Elsevier November 9, 2021

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as metals or alloys, which are inherently inhomogeneous, hardening improve-ment can be achieved by metal forming and heat treatment that changeboth the average grain size and the microstructure. All these inhomoge-neous materials consist of at least two different homogeneous components(or grains) with the interface (or grain boundary) between them. Duringtheir manufacturing or use, various defects such as vacancies, dislocations,grain boundaries, microcracks, micropores may occur. With increasing stresslevel, these defects can grow and coalesce, leading to detachment, fracture ordamage of the components or whole bodies.

(i) (ii)

Figure 1: (Color online) Growth of an interface crack in a bimaterial: (i) Along theinterface, (ii) Kinking into the softer component [18].

From the above, we immediately see the important role of fracture me-chanics in predicting safe equilibrium and crack growth in these inhomoge-neous materials. Fig. 1 shows the competing possibilities of an interface crackin a loaded bimaterial: either it grows along the interface, or it kinks out intoone of the components. To predict which of these competing possibilities willoccur, we need a criterion for crack growth. However, because of the differ-ent bulk properties of the components, as well as different molecular bondingwithin the components and at the interface, deriving this criterion from firstprinciples of mechanics is not as straightforward as one might think. Eventhe first step in solving this problem, namely the study of the stress field nearthe interface crack tip, encountered the logical difficulties. As first shown byWilliams [41], the asymptotic displacement and stress fields near the cracktip exhibited oscillatory behavior. Although the zone of oscillation is esti-mated to be of the order of 10−4 of the crack length [12, 11], the oscillation of

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the displacement field leads to unphysical penetration of the materials (seealso the discussion in [32]). Another difficulty associated with this oscillatorybehavior is the decoupling of modes I and II in plane strain problems andthe definition of the corresponding stress intensity factors [37, 13]. Note thatwhen the mismatch parameter β (introduced by Dundurs [9] for bimaterialswhose components are isotropically elastic) vanishes, the oscillatory behaviorof the interface crack disappears and modes I and II can be uniquely sepa-rated. To get rid of the unphysical oscillatory behavior near the tip of aninterface crack in the case β 6= 0, Atkinson [3] and Comninou [7] each pro-posed a modification (see also the review article by Comninou [8]). Atkinsonacknowledged that the interface cracks are not sharp. This leads to a gradualtransition and a non-oscillatory stress field. Comninou stayed with the sharpcrack model, but introduced the impenetration constraint, which leads topartial closure of the crack faces and also to the elimination of stress oscil-lation. It is also worth noting that the oscillatory behavior does not appearin the finite deformation theory, as shown by Knowles and Sternberg for theinterface crack between two neo-Hookean sheets [26].

However, the main unresolved issue relates to the criterion of crack growthin inhomogeneous materials and how it should be derived from first principlesof mechanics. So far, a large number of quite different crack growth criteriahave been proposed. They can be roughly divided into two groups: localcriteria involving the stress or crack opening at the crack tip, and globalcriteria related to the energy release rate and the J-integral (see, e.g. [39]and the references therein for a review of widely used criteria). Even the mostcited studies on this subject, by He and Hutchinson [20, 21] and Hutchinsonand Suo [24], contain uncertainties in the choice of criteria. For the crackreaching the interface, the growth direction should be decided in two steps.First, the criterion for the crack growing along the interface is

γ0/γi < G0/Gi for i = 1, 2, (1)

where γ0 and γi are the surface energies (or fracture toughnesses) of theinterface and components, while G0 and Gi are the energy release rates forthe crack growing along the interface and kinking into one of the components,respectively. Although Eq. (1) is energetic in nature, it was not clear whetherit could be derived from the variational principle of fracture mechanics. Oncethe criterion (1) is violated, the next step should be to select the direction ofcrack kinking from either: (i) the crack will grow in the direction of mode Isuch that KII = 0 (local criterion) [1], (ii) the crack will grow in the direction

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of the maximum energy release rate (global criterion) [35, 19]. Note that thelatter criterion is consistent with that derived from the variational principleof fracture mechanics for homogeneous materials [40, 27].

In view of these uncertainties, we aim in this work to derive the safe equi-librium and crack growth criterion from the variational principle of fracturemechanics first formulated in [40, 29] (cf. [16, 15, 6, 5]). We analyze thesituation when the crack tip is located on the interface, which also includesthe interface crack as a special case. The crucial question is: under whatcondition will the crack remain in safe equilibrium and in what directionwill it grow? The variational principle of fracture mechanics states that anelastic body containing a crack will remain in safe equilibrium as long as itsenergy functional in that state has a local minimum. By a local minimum,we mean the minimum among neighboring admissible states, including thosewith a virtually growing crack and with possible crack kinks. As we willsee, this variational principle implies that as long as the maximum energyreduction rate of the virtually growing crack is negative, the crack remainsin safe equilibrium. The crack begins to grow in the direction of the maxi-mum energy reduction rate when the latter becomes zero. We will show thatthis criterion implies Eq. (1) in combination with the global criterion of themaximum energy release rate for the crack kinking. Since the energy releaserate is expressed in terms of the stress intensity factors of the kinked crack,the problem reduces to finding the relationship between the stress intensityfactors before and immediately after crack kinking. This relationship withthe transformation matrix of stress intensity factors was established for ho-mogeneous materials in [2] (see also [42, 43]). For the interface crack, thetransformation matrix of stress intensity factors can be calculated numer-ically by solving the singular integral equation [21] (see also [22, 34]); thismatrix shows a dependence on the kink angle and the Dundurs mismatch pa-rameters mentioned above. Using the results obtained in [21, 22, 34], we canassess the safe equilibrium and predict the growth direction of the interfacecrack.

This article is organized as follows. After this Introduction, we introducein Section 2 the variational principle of safe equilibrium for inhomogeneousbodies containing cracks and derive its consequences. Section 3 deals withthe calculation of the energy release rate of the interface crack as a functionof the kink angle. Section 4 is devoted to evaluating the safe equilibrium andpredicting the growth direction of the interface crack based on the relation-ship between the stress intensity factors before and immediately after crack

4

kinking. We conclude in Section 5 with a brief summary and future researchdirections.

2. Variational principle of safe equilibrium

x1

x2

old crack-tipnewcrack-tip

ε

interface

ωϕ

B

C0

SMaterial 1

Material 2

Figure 2: A virtually growing crack.

For simplicity, we consider a bimaterial consisting of two different isotropiclinearly elastic components that are well bonded along an interface. Let aninitial configuration of this body contain a crack defined as a surface with abroken bond between adjacent material points. We restrict ourselves to the2-D plane strain problem by considering a body of cylindrical shape. Thecross section of the body occupies the region BC0 = B\C0 of the (x1, x2)-plane.The outer boundary of this region, ∂B, is decomposed into two disjoint parts∂s and ∂k, on which the tractions and displacements are given, respectively.The inner boundary is occupied by the initial crack C0 = C0∪∂C0, which, forsimplicity, is assumed to have only one tip lying on the interface S. The caseof a crack with two tips or a multiple crack can be considered in a similarway. Since the bond is broken on the crack, the 2-D displacement field u(x)on C0∪∂C0 is not defined. The limits of u(x) on two sides of the curve C0 aredenoted by u+ and u−. Let C ⊇ C0 be a curve modeling a virtual growingcrack and BC = B\C. We allow for kinking of the crack, so C is generallyassumed to be a piecewise smooth curve. There is also the possibility thatthe initial crack develops into multiple branches, which is very likely in the

5

case of rapid crack propagation [36]; this situation is not considered in thispaper. In the simplest case, the coordinate system is chosen so that locally C0

lies on the negative x1-axis, while the interface S is the straight line inclinedat an angle ω to the x1-axis. Without limiting generality, we can assumethat 0 ≤ ω < π. The virtual growing crack is inclined by an angle ϕ to thex1-axis, as schematically shown in Fig. 2. There are five special cases: (i)ϕ < ω: The crack impinges the interface, (ii) ϕ > ω: The crack is reflectedfrom the interface, (iii) ϕ = ω: The crack deflects into the interface, (iv)ω = ϕ = 0: The interface crack continues to grow along the interface, (v)ω = 0, ϕ 6= 0: The interface crack kinks out of the interface.

The interpenetration of materials on the opposite crack faces is not al-lowed, hence the boundary condition

[[uα]]nα = (u+α − u−α )nα ≥ 0 (2)

must be satisfied everywhere on C, where the Greek indices run from 1 to 2.Here n is the unit normal vector pointing in the + direction. For the interfaceS between two well-bonded materials, the displacements must be continuouseverywhere, unless part of C lies on it, and we choose the unit normal vectorn on S to point to material 1. We define the set of kinematically admissibledisplacements of the body with a virtually growing crack as

D = {u(x) ∈ H2(B\C)| [[uα]]nα ≥ 0 on C,u|∂k = 0}, (3)

with C being an arbitrary piecewise smooth curve containing C0. Functionsu(x) together with their first derivatives with respect to xα are assumed tobe square integrable in B\C and, thus, belong to the Hilbert space H2(B\C).This guarantees the finiteness of the energy per unit length in the x3-directionwhich is defined by

Ψ[u(x)] =

∫BCψ(x, ε(u)) d2x+

∫Cγ(ϕ, [[u]],n) ds−

∫∂s

ταuα ds . (4)

In formula (4) ψ(x, ε) = 12λ(x)(εαα)2 + µ(x)εαβεαβ denotes the free energy

density of the bimaterial, where its elastic moduli are piecewise constantfunctions of x

{λ(x), µ(x)} =

{{λ1, µ1} for material 1,

{λ2, µ2} for material 2.(5)

6

Function γ(ϕ, [[u]],n) in the second (line) integral is the surface (cohesive)energy per unit area which may depend on the direction of crack growth ϕ,on the jump in displacement [[u]], and on the normal vector n. Note that thecrack growth in ductile materials involving the energy dissipation and thecrack resistance (or fracture toughness) can also be reformulated in terms offunction γ as will be shown later. The strain tensor ε is expressed throughthe displacement field by

εαβ(u) =1

2(uα,β + uβ,α). (6)

On the part ∂s of the exterior boundary ∂B the traction vector τ is specifiedso that the last term in (4) corresponds to the work done by τ . We say thatthe body with the pre-existing crack C0 is in safe equilibrium if there exist adisplacement field u(x) whose discontinuity curve is C0 and a neighborhoodDε ⊂ D of it such that the energy functional reaches a local minimum atu(x) [40, 29]

Ψ[u(x)] = minu(x)∈Dε

Ψ[u(x)]. (7)

If this is not the case, we say that the crack begins to grow.We are going now to establish the necessary conditions for the displace-

ment field u(x) of an inhomogeneous elastic body with a pre-existing crackto be in safe equilibrium in accordance with the principle of minimum en-ergy given above. Following Griffith [17], we assume that the surface energydoes not depend on [[u]] and n, so γ(ϕ, [[u]],n) = γ(ϕ), where γ(ϕ) = γ1 ifthe crack is reflected into material 1, γ(ϕ) = γ2 if the crack impinges theinterface and grows in material 2, and γ(ϕ) = γ0 if the crack deflects intothe interface. Let us introduce a one parameter family of admissible dis-placements ε 7→ u(., ε) ∈ D, whose discontinuity curves describe a virtuallygrowing crack Cε (see Fig. 2). Since the crack can only grow, we require that

Cε′ ⊇ Cε ⊇ C0 for ε′ > ε > 0, Cε = C0 when ε = 0, (8)

u(x, 0) = u(x).

After substituting this one parameter family of admissible displacements intothe energy functional (4) it becomes a function of ε, ε ≥ 0. If the bodyis in safe equilibrium, this function has an end-point minimum at ε = 0for arbitrary families of admissible displacements in accordance with ourvariational principle. Therefore the following necessary condition for safe

7

equilibrium must be fulfilled:

δΨ = limε→0

d

dεΨ[u(., ε)] ≥ 0 for arbitrary families of u(., ε) ∈ D. (9)

c

z(x)

C0C0 C0

C

SS

Figure 3: Parametrization and integration domains.

In order to derive consequences from (9), we must be able to calculatethe derivative of Ψ[u(., ε)] with respect to ε and then take the limit as ε→ 0.The difficulty of this calculation is due to the changeable region B\Cε andcurve Cε. In order to overcome it we introduce a one-parameter family ofone-to-one mappings of B onto itself

ε 7→ z(x, ε) (10)

so that

B\C0z7→ B\Cε, C0

z7→ Cε, (11)

z(x, ε) = x, when ε = 0 or x ∈ ∂B.

Since the crack kinking is admitted, functions z(x, ε) are assumed to besmooth everywhere except at the points cε lying on the x1-axis which aremapped to the old crack tip ∂C0. Noticeable are also the curves Sε goingthrough cε that are mapped to the real interface S (see Fig. 3). We choosez(x, ε) such that Sε coincides with S except a small neighborhood of ∂C0.We call z(x, ε) parametrizations of medium. These mappings will be used aschanges of variables for the 2-D and 1-D integrals in (4) which become thenintegrals over the fixed region B\C0 and curve C0 (see Eq. (11)). Accordingto the transformation rule we have∫

B\Cεψ(x, ε(x)) d2x =

∫B\C0

ψ(z(x, ε), ε(z(x, ε)))J d2x , (12)

8

where J = det zα,β is the Jacobian of transformation. Since the region ofintegration, after this change of variables, does not depend on ε, the order ofdifferentiation and integration can be interchanged so that

limε→0

d

dε

∫B\Cε

ψ(x, ε) d2x = limε→0

∫B\C0

(Jδψ + ψδJ) d2x . (13)

Symbol δ under integral signs, called for short variation, is used to denotethe partial derivative with respect to ε at fixed x. The variation of J reads

δJ =∂J

∂zα,βδzα,β = (Cof)αβδzα,β = J

∂xβ∂zα

δzα,β = J∂δzα∂zα

, (14)

where (Cof)αβ is the cofactor of zα,β. Consider now the variation of ψ

δψ =∂ψ

∂zαδzα+

∂ψ

∂εαβδεαβ =

∂ψ

∂zαδzα+

1

2σαβ δ(

∂uα∂zβ

+∂uβ∂zα

) =∂ψ

∂zαδzα+σαβ δ

∂uα∂zβ

.

(15)Here σ is the symmetric stress tensor field with components σαβ = ∂ψ

∂εαβ. In

order to calculate the variation δ ∂uα∂zβ

we recall the following identity

∂uα∂xγ

=∂uα∂zβ

∂zβ∂xγ

. (16)

Applying the product rule of differentiation to this identity and rememberingthat the variation and partial derivatives with respect to xα are commutativewe obtain

∂δuα∂xγ

= δ∂uα∂zβ

∂zβ∂xγ

+∂uα∂zβ

∂δzβ∂xγ

. (17)

Multiplying this equation with ∂xγ/∂zκ and rearranging indices and termswe get

δ∂uα∂zβ

=∂δuα∂zβ

− ∂uα∂zγ

∂δzγ∂zβ

. (18)

Combining all these formulas we obtain finally

limε→0

d

dε

∫B\Cε

ψ(x, ε(u(x, ε)) d2x = limε→0

∫B\C0

(σαβ

∂δuα∂zβ

+∂ψ

∂zαδzα+µαβ

∂δzα∂zβ

)J d2x

= limε→0

∫B\Cε

(σαβδuα,β +∂ψ

∂xαδzα + µαβδzα,β) d2x , (19)

9

C0 C

S

Figure 4: Regularized integration domain Bε.

whereµαβ = −σγβuγ,α + ψδαβ (20)

is the Eshelby tensor [14], while ∂ψ/∂xα denotes the partial derivatives of ψat fixed ε. For the piecewise constant functions λ(x) and µ(x) the derivative∂ψ/∂xα must be understood as the generalized function. Since the stressfield σ and the displacement gradients uα,β are singular at ∂C0 and ∂Cε andmay suffer jumps on S, Gauss’ theorem cannot be applied to the right-handside of (19) directly. To do this properly we replace the region B\Sε by Bε,whose interior boundary is shown in Fig. 4. It turns out that formula (19)remains valid for the integral taken over Bε. Applying Gauss’ theorem andletting ε approach zero, we obtain

limε→0

d

dε

∫B\Cε

ψ(x, ε(u)) d2x =

∫B\C0

[−σαβ,βδuα + (∂ψ

∂xα− µαβ,β)δzα] d2x

+

∫C0

[(−σ+αβδu

+α + σ−αβδu

−α )nβ + (−µ+

αβ + µ−αβ)nβδzα] ds

+

∫S(−σ+

αβ + σ−αβ)δuαnβ ds+

∫∂s

σαβδuαnβ ds−G(ϕ)δl. (21)

In the last term of (21) G(ϕ) is given by the J-integral [38]

G(ϕ) = να limε→0

∫Γε

µαβκβ ds = να limε→0

∫Γε

(−σγβuγ,ακβ + ψκα) ds , (22)

with ν being the unit vector pointing to the direction of crack extension, Γεthe contour of radius ε1 � ε surrounding the crack tip ∂Cε, κ the outward

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unit normal vector on Γε, and δl = να limε→0 δzα|∂Cε the virtual crack ex-tension length. Note that, away from the crack tip, the stress field σ, thedisplacement gradients uα,β and the energy density ψ approach the corre-sponding quantities calculated for u = u in the limit ε → 0. We drop forshort the check over these quantities. Note also that the integral along thecontour surrounding the point ∂C0 tends to zero as ε→ 0, because the stressfield and the displacement gradients have in its neighborhood the corner sin-gularity, which turns out to be weaker than the square root singularity atthe crack tip [23]. When deriving (21) the following asymptotic property istacitly used

limε→0

∫Γε

σαβκβ ds = 0. (23)

This is due to the square root singularity of the stress field [41]. The integral(22) describes the elastic energy release per unit crack extension length whichis expected to be a function of the kink angle ϕ.

The variation of the surface energy is given by

limε→0

d

dε

∫Cεγ(ϕ) ds = γ(ϕ)δl. (24)

Note that, if there is resistance to crack growth and, consequently, nonzeroenergy dissipation, the variational inequality (9) must be extended as follows

δΨ + r(ϕ)δl ≥ 0 for arbitrary families of u(., ε) ∈ D, (25)

where r(ϕ) is the fracture toughness. However, if the latter does not dependon the crack tip velocity (rate-independent theory), the second term on theleft-hand side of (25) can be written as

r(ϕ)δl = limε→0

d

dε

∫Cεr(ϕ) ds , (26)

so it can be combined with the variation of the surface energy term, withγeff = γ + r being interpreted as the effective surface energy density. Inthis sense, crack growth in ductile materials involving energy dissipation andrate-independent fracture toughness can also be reformulated in terms of γeff.The variation of the external work is equal to

limε→0

d

dε

∫∂s

ταuα ds =

∫∂s

ταδuα ds . (27)

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Combining formulas (21)-(27), one transforms the inequality (9) to∫B\C0

[−σαβ,βδuα + (∂ψ

∂xα− µαβ,β)δzα] d2x+

∫C0

[(−σ+αβδu

+α + σ−αβδu

−α )nβ

+ (−µ+αβ + µ−αβ)nβδzα] ds+

∫S(−σ+

αβ + σ−αβ)δuαnβ ds

+

∫∂s

(σαβnβ − τα)δuα ds− [G(ϕ)− γ(ϕ)]δl ≥ 0. (28)

We shall now analyze the inequality (28). It is obvious that the variationsδu and δz in the region B\C0 as well as δu on ∂τ can be chosen arbitrarily.On the contrary, δu± and δz on C0 should satisfy some constraints. Whenthe crack faces are not in contact with each other, then the variations δu± onC0 can obviously have arbitrary values. If this is not the case, the constraint[[δuα]]nα ≥ 0 must be obeyed. Since Cε ⊇ C0, δzα should satisfy the constraints

δzαnα = 0 on C0, δl ≥ 0 at ∂C0. (29)

Taking all these constraints into account, one can show that the varia-tional inequality (28) leads to

σαβ,β = 0, σαβ =∂ψ

∂εαβin B\C0,

uα = 0 on ∂k, σαβnβ = τα on ∂s,

σ+αβnβ = σ−αβnβ on S,

(u+α − u−α )nα ≥ 0 on C0, (30)

σ+αβnβ = σ−αβnβ = −pnα, p ≥ 0 on C0,

(u+α − u−α )nα > 0⇒ p = 0,

maxϕ

[G(ϕ)− γ(ϕ)] ≤ 0 at ∂C0.

Additionally, we also get the relations

∂ψ

∂xα− µαβ,β = 0, µαβ = −σγβuγ,α + ψδαβ in B\C0, (31)

(−µ+αβ + µ−αβ)nβνα = 0 on C0,

with ν being the tangent vector to the curve C0. However, it is easy to seethat these equations are satisfied identically by virtue of other equations in

12

(30). This is due to the invariant properties of the energy functional withrespect to the group of parametrizations leaving the curve of discontinuityunchanged.

Thus, equations (30) are the necessary conditions for the displacementfield of the cracked body u to be in safe equilibrium. The difference betweenequilibrium and safe equilibrium reduces to the last condition, maxϕ[G(ϕ)−γ(ϕ)] ≤ 0, called the maximum energy reduction rate criterion. Thus, if thismaximum is negative, the crack stays in stable safe equilibrium. The crackstarts to grow in the direction ϕ that maximizes G(ϕ)−γ(ϕ) if the maximumis equal to zero. We want now to show that this criterion implies condition (1)for the crack growing along the interface combined with the maximum energyrelease rate criterion for the crack kinking out into one of the components.Indeed, according to the maximum energy reduction rate criterion, the crackwill grow along the interface if the maximum of G(ϕ) − γ(ϕ), achieved atϕ = ω, is equal to zero. This means

0 = G(ω)− γ0 > G(ϕ)− γ(ϕ) for all ϕ 6= ω. (32)

Bringing the term −γ(ϕ) to the left-hand side of this inequality and dividingby the positive constant γ0 = G(ω), we obtain

γ(ϕ)/γ0 > G(ϕ)/G(ω) for all ϕ 6= ω, (33)

which is equivalent to (1). In case the condition (33) is not fulfilled, themaximum must be sought in one of the components (ϕ > ω or ϕ < ω). Sinceγ(ϕ) is constant there, the maxima of the energy reduction rate G(ϕ)−γ(ϕ)and of the energy release rate G(ϕ) are achieved at the same angle ϕ. Thus,the maximum energy reduction rate criterion implies the maximum energyrelease rate criterion in this case. Thus, to assess the safe equilibrium andpredict the direction of crack growth we must find the relationship betweenG(ϕ) and G(ω). This task will be done for the special case of the interfacecrack with ω = 0 in the next two Sections.

3. Energy release rate of the interface crack

We now apply the developed theory to an interface crack (ω = 0). Asshown in Fig. 5, the virtual crack growth can be along the interface (ϕ = 0),kinking out into material 1 (ϕ > 0) or into material 2 (ϕ < 0). We willcalculate the energy release rate G(ϕ), given by Eq. (22), in these cases. To

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(i) (ii)

Figure 5: (Color online) Growth of an interface crack in a bimaterial: (i) Along theinterface, (ii) Kinking-out into one of the components.

focus on the crack growth criterion, we assume for simplicity that one of theDundurs parameters, β, vanishes

β =(1− 2ν2)/µ2 − (1− 2ν1)/µ1

2[(1− ν2)/µ2 + (1− ν1)/µ1]= 0, (34)

where ν1 and ν2 are Poisson’s ratios of the corresponding components. Incontrary, the other Dundurs parameter

α =(1− ν2)/µ2 − (1− ν1)/µ1

(1− ν2)/µ2 + (1− ν1)/µ1

(35)

is not equal to zero. The general case with α 6= 0 and β 6= 0 is considered in[21].

For the interface crack growing along the interface with ϕ = 0 and ν =(1, 0), we have

G(0) = limε→0

∫Γε

(−σγβuγ,1κβ + ψκ1) ds . (36)

One can substitute Williams’ stress and displacement fields around the cracktip into (36) to compute G(0). However, the shorter and more convenientway is that originally proposed by Irwin [25]. Namely,

G(0) =1

2ε

∫ ε

0

σ(1)αβ (r)nβ[[u(2)

α (ε− r)]] dr , (37)

14

where σ(1)αβ (r) are the stresses ahead the crack tip (at θ = 0) prior to the

growth, while [[u(2)α (r′)]] are the jump in displacement behind the crack tip

(at θ′ = ±π, r′ = ε − r) just after the crack advances. Using the stressdistribution ahead the initial crack

σ(1)22 (r) = K1

1√2πr

, σ(1)12 (r) = K2

1√2πr

, (38)

and the jump in displacements behind the advanced crack

[[u(2)2 (ε− r)]] =

8

E∗K1

√ε− r2π

, [[u(2)1 (ε− r)]] =

8

E∗K2

√ε− r2π

, (39)

with E∗ being defined as

1

E∗=

1

4

(1− ν1

µ1

+1− ν2

µ2

), (40)

we find that [32]

G(0) =1

E∗(K2

1 +K22). (41)

For the crack kinking out into material 1 (ϕ > 0), we have

G(ϕ) = limε→0

∫Γε

(−σ′α′β′u′α′,1′κβ′ + ψ′κ1′) ds , (42)

where x′α are the shifted and rotated coordinate system shown in Fig. 5(ii).We substitute the asymptotic formulas of the stress and displacement fieldsnear the extended crack tip ∂Cε given by

σ′α′β′(ε) = K ′I(ε)fIα′β′(ϑ′)√

2πr′+K ′II(ε)

fIIα′β′(ϑ′)√2πr′

+O(1),

u′α′(ε) = K ′I(ε)

√r′

2πvIα′(ϑ′) +K ′II(ε)

√r′

2πvIIα′(ϑ′) +O(r′),

(43)

into (42), with r′ and θ′ the corresponding polar coordinates and fIα′β′(ϑ′),fIIα′β′(ϑ′), vIα′(ϑ′), vIIα′(ϑ′) the well-known angular distributions of thestress and displacement fields (see, e.g., [30]). Computing the integral andletting ε go to zero, we get

G(ϕ) =1− ν1

2µ1

(K2I +K2

II), (44)

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where KI and KII are the limiting values of the stress intensity factors K ′I(ε)and K ′II(ε) of the kinked crack when ε goes to zero. Similarly, for the crackkinking out into material 2 (ϕ < 0),

G(ϕ) =1− ν2

2µ2

(K2I +K2

II). (45)

Thus, the computation of G(ϕ)/G(0) reduces to finding the relationship be-tween (KI , KII) and (K1, K2). This will be done in the next Section.

4. Growth of the interface crack

Let us assume for definiteness that

maxϕ>0

[G(ϕ)− γ1] < maxϕ<0

[G(ϕ)− γ2]. (46)

In this case, the kinking of the crack in material 1 is excluded, and we needjust to find G(ϕ) for negative ϕ. Let us redefine ϕ as the positive kink anglewhen the crack kinks out into material 2. The relationship between (KI , KII)and (K1, K2) in the complex form reads [21]

KI + iKII = c(ϕ, α)K + d(ϕ, α)K, (47)

where the bar indicates complex conjugation, K = K1 + iK2, and c and dare complex-valued functions of ϕ. Substitution of (47) into (45) yields

G(ϕ) =1− ν2

2µ2

[(|c|2 + |d|2)KK + 2 Re(cdK2)]. (48)

Using the representation K = |K|eiγ, with γ = arctan(K2/K1) being theparameter characterizing the load combination, we bring the energy releaserate ratio to the following form

G(ϕ)

G(0)= (1 + α)[(|c|2 + |d|2) + 2 Re(cdeiγ)] (49)

Note that, except ϕ, this ratio depends also on α and γ through the coeffi-cients c and d.

The finding of coefficients c and d in (47) is based on the numericalsolution of the singular integral equation (see [21, 22, 34]). Using their results

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(i) (ii)

Figure 6: Dependence of the energy release rate ratio G(ϕ)/G(0) as a function of the kinkangle ϕ and load combination γ for different Dundurs parameter α: (i) α = 0.25, (ii)α = −0.25.

(i) (ii)

Figure 7: Dependence of the energy release rate ratio G(ϕ)/G(0) as a function of the kinkangle ϕ and load combination γ for different Dundurs parameter α: (i) α = 0.75, (ii)α = −0.75.

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(i) (ii)

Figure 8: (Color online) 3D-plots of the energy release rate ratioG(ϕ)/G(0) as a function ofthe kink angle ϕ and load combination γ for different Dundurs parameter α: (i) α = 0.25,(ii) α = −0.25.

(i) (ii)

Figure 9: (Color online) 3D-plots of the energy release rate ratioG(ϕ)/G(0) as a function ofthe kink angle ϕ and load combination γ for different Dundurs parameter α: (i) α = 0.75,(ii) α = −0.75.

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we can find the energy release rate ratio G(ϕ)/G(0), whose plots are shown inFigs. 6(i)-(ii) for α = ±0.25 and 7(i)-(ii) for α = ±0.75. The corresponding3D plots of G(ϕ)/G(0) as function of two variables ϕ and γ are shown inFigs. 8(i)-(ii) for α = ±0.25 and 9(i)-(ii) for α = ±0.75. From these plots,one can see how the variation of the Dundurs mismatch parameter α affectsthe change in the maximum of G(ϕ)/G(0) with respect to ϕ. In particular,the slopes of the curves G(ϕ)/G(0) as a function of ϕ for positive and negativeα are different. For large negative α, the maximum is less than 1, as can beseen in Fig. 7(ii). Using Eq. (33) in combination with the maximum energyrelease rate criterion, we can easily predict the direction of crack growth interms of the Dundurs parameter α and the parameter γ characterizing theload combination.

Figure 10: Kink angle ϕ as a function of the loading combination γ at different α.

The plot of the kink angle ϕ that maximizes the energy release rate asfunction of the load combination γ at different α is shown in Fig. 10. Notethat the deviation from that angle determined by the criterion KII = 0 issmall except for large γ.

5. Conclusion

The main result of this work is the derivation of the maximum energyreduction rate criterion from the variational principle of fracture mechanics.The derived criterion can be applied both to the interface crack and to thecrack whose tip reaches the interface. Although this criterion reduces to thewell-known and widely accepted criteria of crack growth formulated by He

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and Hutchinson [20], the derivation from first principles brings two advan-tages. First, it removes the uncertainties in the choice of criteria mentionedin the Introduction. Second, the variational formulation opens the way fordirect numerical methods, e.g., the computationally effective non-remeshingfinite elements proposed, for example, in [33], which can be applied to inho-mogeneous cracked bodies of arbitrary geometry.

There are several possible extensions to the variational formulation givenin this paper. First, the extended variational problem can be studied forlaminated composites with imperfect bonding or thin film adhesives, wherethe surface energy could depend on the displacement jump. Second, it canbe extended to inhomogeneous materials that deform at finite strain, withthe goal of applying it to filled elastomers (cf. [40, 28, 31, 10] for nonlinearfracture mechanics of homogeneous rubbery materials). Third, an interestingtopic closely related to the one considered here is crack nucleation in inho-mogeneous solids. To address this issue, thermal fluctuation in the spirit of[4] must be considered in addition to the energy of microcracks.

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