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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 60 (2012) 1703–1709

0022-50

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/jmps

A nonlinear symmetry breaking effect in shear cracks

Roi Harpaz, Eran Bouchbinder n

Chemical Physics Department, Weizmann Institute of Science, Rehovot 76100, Israel

a r t i c l e i n f o

Article history:

Received 5 March 2012

Received in revised form

13 June 2012

Accepted 26 June 2012Available online 1 July 2012

Keywords:

Dynamic fracture

Friction

Nonlinear crack mechanics

Asymptotic analysis

96/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.jmps.2012.06.010

esponding author. Tel.: þ972 8 9342605; fax

ail address: eran.bouchbinder@weizmann.ac.i

a b s t r a c t

Shear cracks propagation is a basic dynamical process that mediates interfacial failure.

We develop a general weakly nonlinear elastic theory of shear cracks and show that

these experience tensile-mode crack tip deformation, including possibly opening

displacements, in agreement with Stephenson’s prediction. We quantify this nonlinear

symmetry breaking effect, under two-dimensional deformation conditions, by an

explicit inequality in terms of the first and second order elastic constants in the

quasi-static regime and by semi-analytic calculations in the fully dynamic regime. Our

general results are applied to various materials. Finally, we discuss related works in the

literature and note the potential relevance of elastic nonlinearities for various problem,

including frictional sliding.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

A fundamental process that governs the strength and failure of interfaces in a wide range of scientific and technologicalproblems is the propagation of shear cracks. For example, stress and energy release during earthquakes are mediated bythe propagation of shear cracks along fault surfaces (Scholz, 2002). In many situations crack propagation is studied withinthe framework of Linear Elastic Fracture Mechanics (LEFM), where shear driven cracks (i.e. mode II symmetry cracksFreund, 1998; Broberg, 1999) involve interfacial slip, but no crack tip opening displacements, which are characteristic ofmode I symmetry cracks (Freund, 1998; Broberg, 1999). However, the existence of such coupling, which is a symmetrybreaking effect, is of interest and potential importance. This becomes evident when the propagation of shear cracks isaccompanied by frictional processes. Under such circumstances, crack opening, or even a reduction in the confining normalstress at the interface, can significantly reduce the friction force and hence result in weakening the interface.

The most widely invoked mechanism for the coupling between interfacial slip and crack opening (as well as for thecoupling to compressive stresses) is material discontinuity, i.e. the shear crack is assumed to propagate along an interfaceseparating two dissimilar materials (Ben-Zion, 2001; Gerde and Marder, 2001). Alternatively, an intrinsic couplingbetween the different symmetry modes of the linear theory may be obtained by considering material nonlinearity.

Our goal in this paper is to explore the latter physical mechanism. Unlike conventional modeling which almostexclusively considers plastic deformation as the most important material nonlinearity near crack tips (Freund, 1998;Broberg, 1999), we follow recent experimental and theoretical developments (Livne et al., 2008, 2010; Bouchbinder et al.,2008, 2009, 2010; Bouchbinder, 2010; Goldman et al., 2012) and focus on weak, second order, elastic nonlinearities. Thebasic idea here is that weak elastic nonlinearities are expected to be the first physical process that leads to the breakdownof linearity.

ll rights reserved.

: þ972 8 9344123.

l (E. Bouchbinder).

R. Harpaz, E. Bouchbinder / J. Mech. Phys. Solids 60 (2012) 1703–17091704

A related strongly nonlinear elastic mechanism has been discussed in the context of nonlinear elastic fracturemechanics by Stephenson (1982) and also in the context of the sliding of soft, rubber-like, materials by Schallamach(1971). Nevertheless, while the absence of pure shear crack solutions in nonlinear elastic fracture mechanics wasrecognized by Stephenson already in 1982 (Stephenson, 1982), to the best of our knowledge no accurate and self-consistent benchmark weakly nonlinear solutions are presently available.

In this paper, we derive general weakly nonlinear asymptotic solutions for two-dimensional frictionless shear crackspropagating steadily along an interface separating two identical homogeneous isotropic materials. The conditions for cracktip opening, as a function of the first and second order elastic constants and crack propagation velocity, are quantified andthe theory is applied to polymeric, glassy and metallic materials. Finally, we discuss the possible relevance of our results tofrictional shear cracks and to the super-shear transition (Rosakis et al., 1999; Abraham and Gao, 2000; Xia et al., 2004).

The structure of this paper is as follows. In Section 2 we describe our general theoretical procedure and derive generalweakly nonlinear elastic solutions for shear cracks. In Section 3 we derive explicit results for the opening displacement ofshear cracks and apply them to various materials. In Section 4 we discuss some available results in the literature andexplain how these differ from our solution. Section 5 offers a summary and discussion of the potential relevance of ourresults to other problems.

2. Theory

We start by considering the motion u, which is assumed to be a continuous, differentiable and invertible mappingbetween a reference configuration x and a deformed configuration x0 such that x0 ¼uðxÞ ¼ xþuðxÞ, where uðxÞ is thedisplacement field. The Green–Lagrange (metric) strain tensor E is defined in terms of the deformation gradient tensorF ¼ru¼ IþH as E¼ 1

2ðFT F�IÞ (Murnaghan, 1951). Here I is the identity tensor and H ¼ru is the displacement gradient

tensor.The elastic energy per unit reference volume UðEÞ of isotropic and compressible materials can be expressed in terms of

the scalar invariants of E, which are conveniently written as tr E, tr E2 and tr E3. To third order in E, UðEÞ takes the form

UðEÞ ¼l2ðtr EÞ2þm tr E2

þa1

3ðtr EÞ3þa2 tr E tr E2

þ2a3

3tr E3

þOðE4Þ, ð1Þ

where fl,mg are the first order (Lame) constants and faigi ¼ 123 are second order elastic constants. The latter are importantphysical constants that are closely related to the leading anharmonic contributions to interatomic interaction potentials.These anharmonic contributions are known to be the origin of many basic physical phenomena and properties such as theGruneisen parameters, deviations from the Dulong–Petit law at high temperatures, thermal expansion and the existence ofthermal resistance. While the second order elastic constants are not as well-studied and well-documented as their firstorder counterparts, methods for measuring them are well-developed and rather widely used, see for example (Bridgman,1929; Hughes and Kelly, 1953; Crecraft, 1967; Powell and Skove, 1968; Gauster and Breazeale, 1968; Riley and Skove,1973; Yost and Breazeale, 1973; Hiki, 1981; Kruger et al., 1991; Cavaille et al., 2009; Kobelev et al., 2007; Payan et al.,2009). The implications of weakly nonlinear elasticity for materials failure remain largely unexplored (but see Knowles,1981; Chow et al., 1986; Chen et al., 2004a,b; Livne et al., 2008, 2010; Bouchbinder et al., 2008, 2009; Bouchbinder, 2010;Goldman et al., 2012), a situation that the present work aims to at least partially improve.

For simplicity, in what follows we focus on plane strain conditions (plane stress calculations revealed no qualitativechange in the results) for which Eq. (1) reduces to

U2DðEÞ ¼

l2ðtr EÞ2þm tr E2

þl

3ðtr EÞ3þ

2m

3tr E3, ð2Þ

where E is understood hereafter as a 2D tensor and l¼ a1þa2 and m¼ a3þa2 are two of the Murnaghan coefficients(Murnaghan, 1951). It is important to note that Eq. (2) goes beyond linear elasticity in two ways. First, the Green–Lagrange(metric) strain tensor E¼ 1

2ðHþHTþHT HÞ is itself intrinsically nonlinear in the displacement gradient H. This is termed

geometric nonlinearity and it is essential for ensuring proper invariance under finite rotations. Second, there existconstitutive (material) nonlinearities which are quantified by the terms proportional to l and m in Eq. (2). All of the resultsto follow can be separated into the different contributions emerging from geometric and constitutive nonlinearities bysimply setting l¼m¼ 0. This important distinction is further discussed in Section 3.

To develop a fracture theory based on the general expansion in Eq. (2) we write down the momentum balance equation

r � s¼ r@ttu, ð3Þ

where r is the reference mass density and s¼ @F U2D is the first Piola–Kirchhoff stress tensor, that is work-conjugate to thedeformation gradient F (Murnaghan, 1951). A frictionless crack is introduced by the usual traction-free boundaryconditions

sxyðr,y¼7pÞ ¼ syyðr,y¼ 7pÞ ¼ 0, ð4Þ

where ðr,yÞ is a polar coordinate system that moves with the crack tip and is related to the reference frame by

r¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx�vtÞ2þy2

qand y¼ tan�1½y=ðx�vtÞ�. x is the propagation direction, y is the perpendicular direction and v is the tip

R. Harpaz, E. Bouchbinder / J. Mech. Phys. Solids 60 (2012) 1703–1709 1705

propagation velocity. In what follows we neglect crack acceleration effects, implying that all of the fields depend on x and t

through the combination x�vt and therefore @t ¼�v@x.To proceed, we expand s order by order in powers of the displacement gradient H to obtain

sðHÞCl tr eIþ2meþ12l tr ðHT HÞIþm HT Hþl tr eHþ2mHeþ lðtr eÞ2Iþ2me2þOðH3

Þ, ð5Þ

where e¼ 12ðHþHT

Þ is the infinitesimal strain tensor. We then follow Truesdell and Noll (1965), Knowles (1981), Chowet al. (1986), Chen et al. (2004a,b), Livne et al. (2008, 2010), Bouchbinder et al. (2008, 2009), and Bouchbinder (2010) andwrite the displacement field as

uðr,yÞCuð1Þðr,yÞþuð2Þðr,yÞ, ð6Þ

where the superscripts denote different orders in H. Substituting Eqs. (5) and (6) into Eqs. (3) and (4), we obtain theequations and boundary conditions for uð1Þ and uð2Þ.

To linear order, we obtain the well-studied equations of isotropic Linear Elastic Fracture Mechanics (LEFM) (Freund,1998). To second order, we obtain

mr2uð2Þ þðlþmÞrðr � uð2ÞÞþF ½uð1Þ� ¼ r €uð2Þ, ð7Þ

where the boundary conditions at y¼ 7p read

�r�1@yuð2Þx �@ruð2Þy �Sx½u

ð1Þ� ¼ 0,

�ðlþ2mÞr�1@yuð2Þy �l@ruð2Þx �Sy½uð1Þ� ¼ 0: ð8Þ

The functionals F ½uð1Þ� and S½uð1Þ� are easily obtained from Eq. (5). As the structure of a perturbation theory implies, F ½uð1Þ�and S½uð1Þ� are quadratic in uð1Þ, i.e. they are second order in the smallness, and hence appear together with linear terms inuð2Þ in the perturbative hierarchy. Terms that couple uð1Þ and uð2Þ appear only to higher orders.

Asymptotic near-tip solutions of uð2Þ for mode I (tensile) symmetry were derived in Bouchbinder et al. (2008, 2009),Livne et al., 2010, Bouchbinder, 2010, with a focus on a particular elastic functional.

Here we focus on global mode II (shear) symmetry loading for a general elastic functional. A schematic sketch of thecrack problem under consideration is shown in Fig. 1.

Under these conditions, the linear order solution uð1Þ possesses the following symmetry properties:

uð1Þx ðr,�yÞ ¼�uð1Þx ðr,yÞ and uð1Þy ðr,�yÞ ¼ uð1Þy ðr,yÞ: ð9Þ

The leading order near-tip asymptotic solution reads (Freund, 1998)

uð1Þx ðr,yÞ ¼2KII

mffiffiffiffiffiffi2pp

DðvÞ½2asr

1=2d sinðyd=2Þ�asð1þa2

s Þr1=2s sinðys=2Þ�,

uð1Þy ðr,yÞ ¼2KII

mffiffiffiffiffiffi2pp

DðvÞ½2asadr1=2

d cosðyd=2Þ�ð1þa2s Þr

1=2s cosðys=2Þ�, ð10Þ

here a2d,s ¼ 1�v2=c2

d,s, tan yd,s ¼ ad,s tan y, rd,s ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ðv sin y=cd,sÞ

2q

, DðvÞ ¼ 4asad�ð1þa2s Þ

2; KII is the mode II ‘‘stressintensity factor’’ and cd ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðlþ2mÞ=r

p, cs ¼

ffiffiffiffiffiffiffiffiffim=r

pare the dilatational and shear wave speeds, respectively (Freund, 1998).

Using Eq. (10) to calculate F ½uð1Þ� and S½uð1Þ� in Eqs. (7) and (8), we obtain

F IIðr,yÞ ¼K2

IIgIIðy;vÞm2r2

, SIIx ðr,7pÞ ¼ 0, SII

y ðr,7pÞ ¼ K2IIkIIðvÞ

m2r, ð11Þ

where gIIðy;vÞ and kIIðvÞ are functions of stress dimension that are too lengthy to report here (the superscript denotes thatthese quantities correspond to mode II loading conditions). F IIðr,yÞ and SII

y ðr,7pÞ, when substituted into Eqs. (7) and (8),

Fig. 1. A schematic sketch of the crack problem under consideration. ðr,yÞ is a polar coordinate system moving with the crack tip. The crack propagates at

a velocity v in the y¼ 0 direction. The global shear loading (mode II) is represented by the upper and lower oppositely pointing arrows. D, defined in Eq.

(17), is the possible crack tip opening displacement, which is the symmetry breaking effect studied in this paper.

R. Harpaz, E. Bouchbinder / J. Mech. Phys. Solids 60 (2012) 1703–17091706

can be respectively interpreted as an effective body force and a surface force in an effective linear elastic crack problem.Their symmetry properties dictate the symmetry properties of the weakly nonlinear solution uð2Þ. Eq. (11) indicates thatSIIðr,pÞ ¼SII

ðr,�pÞ and an explicit calculation shows that

gIIx ðy;vÞ ¼ gII

x ð�y;vÞ and gIIy ðy;vÞ ¼ �gII

y ð�y;vÞ: ð12Þ

These properties imply that the weakly nonlinear problem is of pure mode I nature, even though the loading is of puremode II nature (Knowles, 1981; Chow et al., 1986; Chen et al., 2004a,b). Therefore, while uð1Þ possesses mode II symmetry,cf. Eq. (9), uð2Þ satisfies

uð2Þx ðr,�yÞ ¼ uð2Þx ðr,yÞ and uð2Þy ðr,�yÞ ¼ �uð2Þy ðr,yÞ, ð13Þ

which is the nonlinear symmetry breaking effect we aim at studying here.To quantify the effect we need to solve the problem posed in Eqs. (7), (8) and (11). Since this is a pure mode I problem,

we can follow Bouchbinder et al. (2008, 2009), Livne et al. (2010), and Bouchbinder (2010) to obtain

uð2Þx ðr,yÞ ¼K2

II

m2A log rþ

A

2log 1�

v2 sin2yc2

d

!þBas log rþ

Bas

2log 1�

v2 sin2yc2

s

!þUII

x ðy;vÞ

" #,

uð2Þy ðr,yÞ ¼K2

II

m2½�Aadyd�BysþUII

y ðy;vÞ�: ð14Þ

!IIðy;vÞ is an r-independent solution of Eq. (7) that does not satisfy the boundary conditions in Eq. (8). It can readily

obtained semi-analytically using a Fourier series method (Bouchbinder et al., 2008, 2009). The rest of the solution satisfiesEq. (7) when the effective body force is omitted. It contains two parameters, A and B, that should be determined by theboundary conditions. However, the first equation in (8) is satisfied automatically and only the second one imposes aconstraint of the form

A¼2mBas�ðlþ2mÞ@yUII

y ðp;vÞ�kIIðvÞ

l�ðlþ2mÞa2d

: ð15Þ

To complete the solution, we need an additional condition that allows the calculation of both A and B. To see where thisadditional condition emerges from, substitute the solution in Eq. (14) in s of Eq. (5) and write s¼ sð1Þ þsð2Þ, where thesuperscripts denote different orders of KII. sð2Þ is characterized by a 1=r spatial singularity that was shown to give rise to a(spurious) force in the crack parallel direction, f ð2Þx , a force that cannot be balanced by material inertia which vanishes tothis order (Rice, 1974; Bouchbinder et al., 2009; Bouchbinder, 2010). Therefore, we need to impose the condition

f ð2Þx ¼

Z p

�psð2Þxj njr dy¼ 0, ð16Þ

ensuring that our solution is consistent with the conservation of momentum, which is otherwise violated. Eq. (16), whichis a linear relation between A and B, together with Eq. (15), allows us to fully determine the solution in Eq. (14).

3. Results

A natural and physically transparent way to quantify the symmetry breaking effect is through the dimensionless cracktip displacement defined as

D�m2½uð2Þy ðr,pÞ�uð2Þy ðr,�pÞ�

K2II

: ð17Þ

Crack tip opening emerges when D40, cf. Fig. 1. The complementary situation, Do0, obviously does not imply that thecrack’s faces penetrate each other, but rather that the traction-free boundary conditions are no longer valid and otherphysical effects should be taken into account. Eq. (14) immediately tells us that D is r-independent. Hence, we are mainlyinterested in the dependence of D on the first and second order elastic constants and on the crack velocity, i.e. Dðl,m,l,m,vÞ.To study this function we first focus on the quasi-static limit v-0, and later examine the behavior in the fully dynamicregime.

The problem for v¼ 0 can be solved analytically and leads to

Dðv¼ 0Þ ¼ð1�2nÞ2

4þð1�2nÞ3

4

l

m�nð1�nÞð1�2nÞ

2

m

m , ð18Þ

where n¼ l=½2ðlþmÞ� is Poisson’s ratio. We first note that D depends only on three dimensionless material parameters, n,l=m and m=m. In addition, D contains a common factor inversely proportional to the bulk modulus K � ð1�2nÞ�1 and maypossibly feature an interesting behavior as the incompressibility limit is approached, n-1=2. We are mainly interested inthe sign of D, and if positive, in its magnitude. Therefore, Eq. (18) can in fact serve as an explicit inequality that n, l=m andm=m should satisfy in order to observe crack tip opening, D40.

R. Harpaz, E. Bouchbinder / J. Mech. Phys. Solids 60 (2012) 1703–1709 1707

In order to better understand this inequality, let us consider the different terms in Eq. (18). The first term isindependent of the second order elastic constants l and m, and hence is associated with a geometric nonlinearity. This termexists even in the absence of constitutive nonlinearities, i.e. when l¼m¼ 0, and emerges because E is nonlinear in H (seeabove). Most importantly, it is positive independently of n. Therefore, we conclude that geometric nonlinearitiesuniversally tend to induce opening of shear cracks in the quasi-static regime, v¼ 0. The last two terms on the right-hand-side of Eq. (18) depend on constitutive nonlinearities quantified by l and m. Since l and m can be both positive andnegative (typically l,mbm), we cannot say something definite about these contributions and whether the inequality issatisfied or not depends on the specific values for each material.

It would be interesting to analyze the dependence of D on the crack propagation velocity v. To extend Eq. (18) to thedynamic regime we should introduce another dimensionless parameter, v=cs, and write

DðvÞ ¼ C0ðn,v=csÞþClðn,v=csÞl

m þCmðn,v=csÞm

m , ð19Þ

where C0, Cl and Cm are dimensionless functions that reduce to their counterparts in Eq. (18) when v-0. DðvÞ wascalculated semi-analytically following the procedure described in Bouchbinder et al. (2008, 2009), Bouchbinder (2009,2010), and Livne et al. (2010). In Fig. 2 we show DðvÞ for a polymer (Polystyrene), a metal (Copper) and a glass (Pyrex). Thefirst and second elastic constants of these materials, used in this calculation, can be found in Table 1.

For Polystyrene, DðvÞ remains nearly v-independent until it changes sign at high v. For Pyrex glass, DðvÞ increasesmonotonically and quite significantly at high v. For Copper, DðvÞ increases up to a high v where it abruptly decreases andeventually becomes negative. The interplay between material parameters and elastodynamic effects gives rise to this richbehavior.

The extreme elastodynamic limit, v� cs, is of particular interest. This regime might be relevant for phenomena such asordinary earthquakes and the super-shear transition (Rosakis et al., 1999; Abraham and Gao, 2000; Xia et al., 2004). Tobetter understand what determines the sign of D at high v, we present in Fig. 3 C0, Cl, Cm of Eq. (19) as a function of n forv¼ 0:8cs. Several qualitative conclusions can be drawn. All C’s are positive and satisfy C0bCm4Cl, where CmbCl for largevalues of n (n40:3). Therefore, l typically plays a minor role in determining the sign of Dðv¼ 0:8csÞ. In addition, positivevalues of m and l immediately ensure Dðv¼ 0:8csÞ40, which is the case of Pyrex glass, cf. Fig. 2. Finally, since C0 is an orderof magnitude (or more) larger than Cm, a large and negative m is needed in order to have Dðv¼ 0:8csÞo0. This is the case ofPolystyrene, cf. Fig. 2. This analysis allows us to understand the factors controlling shear crack opening at high propagationvelocities.

The scale of the opening displacement D and the distance from the crack tip in which it takes place are inherited fromthe lengthscale K2

II=m2. For example, for a material with m¼ 1 GPa and KII ¼ 5 MPaffiffiffiffiffimp

(which may be relevant for

0 0.2 0.4 0.6 0.8−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Δ

v/cs

PolystyreneCopperPyrex

Fig. 2. The dimensionless crack tip opening D as a function of v=cs for three materials.

Table 1The first and second order elastic constants for Polystyrene (Hughes and

Kelly, 1953), Copper (Crecraft, 1967) and Pyrex glass (Hughes and Kelly,

1953). Measurement accuracy is not reported here.

Material n l=m m=m

Polystyrene 0.34 �13.7 �9.6

Copper 0.35 11.8 �8.1

Pyrex glass 0.17 0.5 3.4

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510−4

10−3

10−2

10−1

100

101

102

v

C0(v=0.8cs)Cl (v=0.8cs)Cm(v=0.8cs)

Fig. 3. A linear-log plot of the coefficient functions C0, Cl , Cm , defined in Eq. (19), vs. n for v¼ 0:8cs .

R. Harpaz, E. Bouchbinder / J. Mech. Phys. Solids 60 (2012) 1703–17091708

polymers) we obtain a lengthscale of � 25 mm. The value of KII (the dynamic mode II fracture toughness) is a materialparameter that is, unfortunately, not well-documented for many materials under dynamic conditions. All of our results canbe recast in a dimensional form once KII is known.

4. Relation to previous works

There have been very few previous attempts to derive specific weakly nonlinear shear crack tip solutions (Knowles,1981; Chow et al., 1986; Chen et al., 2004a,b). Invariably, all of these attempts assumed that uð2Þðr,yÞ is r-independent. Eq.(14) clearly demonstrates that this is not the case as uð2Þx ðr,yÞ contains a contribution proportional to log r, which wasexperimentally verified for mode I fracture (Bouchbinder et al., 2008, 2009; Bouchbinder, 2009; Livne et al., 2010). Moregenerally, Eq. (14) shows that uð2Þðr,yÞ cannot be separated into a product of an r-dependent function and a y-dependentfunction. In essence, the assumption of an r-independent uð2Þðr,yÞ amounts to implicitly adopting the condition AþBas ¼ 0,which annihilates the log r term in Eq. (14). With this condition at hand, indeed a solution of Eqs. (7), (8) and (11) can befound. Unfortunately, such a solution does not satisfy Eq. (16). Therefore, the solutions discussed in Knowles (1981), Chowet al. (1986), and Chen et al. (2004a,b) produce an unphysical force in the crack propagation direction and hence violate theconservation of momentum.

To explicitly demonstrate this point, we consider the solution presented in Chow et al. (1986). These authorsconsidered a nonlinear Hookean material, i.e. a material characterized by the energy functional in Eq. (2) with l¼m¼ 0, inthe quasi-static limit, v-0. As explained above, they assumed that the second order solution takes the form

uð2Þðr,yÞ ¼ uð2ÞðyÞ, ð20Þ

i.e. that uð2Þ is r-independent. Under this assumption, and being unaware of the additional physical condition in Eq. (16),they obtained the following solution:

uð2Þx ðyÞ ¼119�292nþ192n2

32ðn�1Þcos yþ

9�18nþ8n2

16ðn�1Þcos 2yþ

9�12n32ðn�1Þ

cos 3y,

uð2Þy ðyÞ ¼ð1�2nÞð5�4nÞ

8ðn�1Þyþ

1

32ðn�1Þ�

7

8

� �sin yþ

9�18nþ8n2

16ðn�1Þsin 2yþ

9�12n32ðn�1Þ

sin 3y: ð21Þ

By using the boundary condition in Eq. (15), together with the arbitrary assumption that AþB¼ 0 (recall that as ¼ 1 forv¼ 0), one can show that Eq. (14) coincides with Eq. (21). Using Eq. (21) to evaluate sð2Þ and then the integral in Eq. (16),one discovers that f ð2Þx a0, i.e. the above solution produces an unphysical force that violates momentum conservation.Essentially, in Chow et al. (1986) the physical condition in Eq. (16) was replaced by the arbitrary condition AþB¼ 0.

The mistake described above may lead to qualitatively wrong conclusions. For example, the solution in Eq. (21) predictscrack closure, Do0, which is inconsistent with Eq. (18) that shows that D40 for l¼m¼ 0, independently of n.

5. Summary and discussion

The symmetry breaking effect we have quantitatively addressed in this work may have physical consequences forseveral problems. As was already mentioned in the introduction, this effect may influence the strength of frictionalinterfaces. Since the local frictional resistance along an interface depends linearly on the compressive normal stress, cracktip opening or even a reduction in this stress, may locally weaken the interface. Elastic nonlinearities, which are rathergenerally overlooked in frictional analysis, may provide a mechanism for such a local weakening near the crack tip.

R. Harpaz, E. Bouchbinder / J. Mech. Phys. Solids 60 (2012) 1703–1709 1709

A quantitative analysis of this possible effect goes beyond the present work, entailing the introduction of a friction lawacting on the crack faces and the inclusion of a compressive normal stress. We hope to address this problem in the future.

Another potentially related problem is the super-shear transition beyond which a shear crack propagates at a velocitylarger than the shear wave speed (Rosakis et al., 1999; Xia et al., 2004). The transition from a sub-shear state to a super-shear one is believed to be mediated by the nucleation of a ‘‘daughter’’ crack in front of the ‘‘mother’’ crack (Abraham andGao, 2000). The symmetry breaking effect that we have quantitatively studied, manifested by mode I (tensile) crack tipfields, may potentially facilitate the super-shear transition. Indeed, the numerical simulations of Abraham and Gao (2000),which demonstrated the super-shear transition, also feature opening displacement that might be interpreted as thesymmetry breaking effect we addressed (Chen et al., 2004a,b).

We hope that the present work, beyond providing general and self-consistent weakly nonlinear shear crack solutions,will serve as an impetus for more systematic studies of the roles played by elastic nonlinearities in the dynamics of shearcracks in various problems.

Acknowledgments

This work was supported by the James S. McDonnell Foundation, the Minerva Foundation with funding from theFederal German Ministry for Education and Research, the Harold Perlman Family Foundation and the William Z. and EdaBess Novick Young Scientist Fund.

References

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