Continuum Mechanics to Nao Scale

7/26/2019 Continuum Mechanics to Nao Scale

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A New Approach to the Modeling and Analysis of Fracture

through Extension of Continuum Mechanics to the Nanoscale

T. SENDOVAInstitute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN55403, USA

J. R. WALTONDepartment of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA

Abstract: In this paper we focus on the analysis of the partial differential equations arising from a newapproach to modeling brittle fracture based on an extension of continuum mechanics to the nanoscale. It isshown that ascribing constant surface tension to the fracture surfaces and using the appropriate crack surfaceboundary condition given by the jump momentum balance leads to a sharp crack opening profile at the cracktip but predicts logarithmically singular crack tip stress. However, a modified model, where the surfaceexcess property is responsive to the curvature of the fracture surfaces, yields bounded stresses and a cusp-likeopening profile at the crack tip. Further, two possible fracture criteria in the context of the new theory arediscussed. The first is an energy-based crack growth condition, while the second employs the finite cracktip stress the model predicts. The classical notion of energy release rate is based upon the singular solution,whereas for the modeling approach adopted here, a notion analogous to the energy release rate arises througha different mechanism associated with the rate of working of the surface excess properties at the crack tip.

Key Words:fracture, surface excess properties, fracture criteria

1. INTRODUCTION

1.1. Fracture Mechanics: Continuum to Atomistic Approaches

Fracture of brittle materials has been modeled over a broad range of approaches: from clas-

sical continuum theories such as linear elastic fracture mechanics (LEFM) to particulatetheories such as molecular dynamics.Various attempts have been made to supplement the classical continuum approaches

so that the internal inconsistencies in the LEFM theory are circumvented. Cohesive andprocess zone models are among the most widely studied generalizations of the classical

Mathematics and Mechanics of Solids15:368413, 2010 DOI: 10.1177/1081286510362457

11 The Author(s), 2010. Reprints and permissions:

http://www.sagepub.co.uk/journalsPermissions.nav


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FRACTURE THROUGH EXTENSION OF CONTINUUM MECHANICS 369

crack tip model. These types of models require the specification of constitutive properties ofthe cohesive or crack tip process zone, which are very difficult to determine experimentally.Thus, the models used are either based on ad hocchoices for the constitutive behavior of the

cohesive/process zone or on simplified views of the fracture process.1The primary motivation for studying fracture through atomistic scales, in addition to

the fact that they take into account the nanoscale interfacial physics that plays a crucialrole in a neighborhood of the fracture edge, is that the classical continuum models do notcontain the necessary physics to predict fracture. In this sense, molecular dynamics offers anappealing approach to studying the initiation and propagation of fracture, which explains thegrowing literature devoted to this technique [411]. On the other hand, it requires an accuratedescription of the long-range and short-range intermolecular forces in the bulk material,which is a difficult task in the case of liquids and solids [8].

Various multiscale models (so-called atom-to-continuum modeling) have also recentlygained considerable attention. One of the most extensively studied methods of this type,in the context of finite-element method (FEM) approximations to continuum models, is thequasi-continuum method introduced by Tadmor et al. [12]. Based on an atomistic view ofmaterial behavior, its continuum aspect comes from the fact that the FEM is based on en-ergy minimization. A different type of atom-to-continuum modeling is a recently proposedapproach by Xiao and Belytschko [13] which involves the introduction ofbridging domainsbetween regions modeled using bulk (continuum) descriptions of material behavior and re-gions modeled using atomistic descriptions of material behavior. Both of these approachesinvolve adjustable parameters that one needs to fit for every particular application.

Other attempts to incorporate microscale processes into fracture modeling include mod-els based on configurational forces, as considered by Gurtin and Podio-Guidugli [14, 15],Gurtin and Shvartsman [16], Sivaloganathan and Spector [17], Maugin and Trimarco [18],Fomethe and Maugin [19] and others. Also gaining attention is the peridynamic paradigmfor modeling fracture [20].

In contrast to the latter theories, which introduce either an entirely new additional forcesystem (configurational forces) or an entirely new continuum modeling paradigm (peri-dynamics), the theory proposed herein uses conventional ideas of continuum mechanicsthrough the introduction of a dividing surface endowed with excess properties together with amutual force. Our approach builds upon a hybrid theory introduced by Oh et al. [21]. Unlike

a start-from-scratch atom-to-continuum approach, this theory is based on a continuum theoryof material behavior that takes into account effects due to long-range intermolecular forcesfrom adjoining phases in the vicinity of the fracture surfaces. No adjustable parameters areneeded and, as demonstrated in Section 4, the theory using curvature-dependent surface ten-sion predicts a finite crack tip stress amplification of the applied loading, in contrast to thecrack tip stress singularity exhibited by models in classical elastic fracture mechanics. Thisallows us to formulate a fracture criterion based upon the notion of acritical crack tip stress(CCTS) defined to be the minimum crack tip stress level required to propagate the crack inaddition to an energy-based crack growth condition similar to the classical notion of a critical

energy release rate (ERR) defined in the setting of singular crack tip stresses and strains.


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370 T. SENDOVA and J. R. WALTON

1.2. Current versus Reference Configuration

Classical fracture theories are customarily formulated in a reference configuration. However,

aspects of the theory discussed here are more easily described in the current or deformedconfiguration. For example, the correction potential, which determines the mutual bodyforce term and, depending on the model, the crack surface excess properties, is set up onlywhen chemical bonds have been broken and depends on the crack opening profile.

On the other hand, formulating a fracture theory in the current configuration presentscertain complications. First, it is not entirely clear how one should mathematically define afracture in the deformed configuration, where the crack is opened, as traditionally a crack isdefined as a surface in the reference configuration across which the displacement, velocity,stress fields, etc. could sustain a discontinuity. Also, the notions of crack length and crackvelocity in the current configuration are ambiguous, since one needs to separate crack tipmotion due to crack growth from motion due to deformation. (A convenient method fordoing this is presented below in Section 5.) Furthermore, to prove that the proposed modelingapproach leads to bounded stresses and strains, we employ the method of integral transformswhich is most easily applied when using the undeformed configuration as a reference.

2. PRELIMINARY DEFINITIONS

The notation introduced in this section is common to both the quasi-static and the dynamiccases. In Section 4 attention is limited to studying the boundary-value problem arising whenconsidering a classical (quasi-static) Griffith crack. In this case, time is just a parameter anddoes not affect the mechanics, for this reason the time dependence is often suppressed in thenotation. When deriving an energy-based crack growth condition in Section 5, a dynamicproblem needs to be considered.

With subscript1 we denote quantities defined relative to a natural reference (unloaded)configuration. The material body in the reference configuration is denoted by 11 2t3, wherethe time dependence is due to (possible) crack extension (Section 5). Here 11 2t3is viewedas an evolving reference configuration. The boundary of11 2t3, denoted by411 2t3, has the

decomposition

411 2t3 2 213 51 2t3 (1)

where 21 denotes the boundary the body would have in the absence of the crack (whichdoes not evolve with time) and51 2t3denotes the crack. The location of the crack tip in thereference configuration is denoted byc2t3(Figure 1). To that end,4c2t3is thecrack extensionvelocity.

Let Xdenote the position of points in 11 2t3, and62X7 t3, X5 11 2t3a motion of thecracked body which might be accompanied by crack extension. The body, its boundary setsand the crack tip in the current (deformed) configuration are denoted by 1t, 2t,5t, andct,respectively.2 Spatial points are denoted by x5 1t. The material and spatial descriptionsof the displacement are denoted by u1 2X7 t3 2:26 2X7 t3 6 X3andu2x7 t3, respectively, andv2x7 t3is the spatial description of the velocity. LetF2X7 t3denote the deformation gradient


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Figure 1. Edge crack in the reference and current configurations.

Figure 2. Part 1of a body, intersecting the dividing surface5 .

and let J2 det2F3. The gradient and divergence operators are denoted by grad and div inthe spatial frame, and by7and Div in the material frame. Similarly, surface gradient andsurface divergence are denoted by grad28 3and div28 3and by 728 3and Div28 3in the spatial andmaterial frame, respectively.3

If s1 is an oriented curve in the fracture surface 51 2t3, parameterized by arc lengthS5 207L 3, 11 denotes theunit conormal vectorto s1 , i.e. 112 218 N, whereNis the unitnormal to51 2t3and 212 ds1d Sis the unit tangent tos1 .

Let 3be a part of the body intersecting the dividing surface 5 . Let 39 denote thedomain occupied by the first phase (e.g. bulk material) and 36be the domain occupied by

the second (e.g. gas or vacuum). Then32 39 3 2533 336(Figure 2). Let the limit ofa generic bulk field92x3in 3 5be defined by

92x3:2 lims09

92x 6 sn3 for all x 5 5

wherenis the outward (for 3) unit normal vector to the dividing surface5 . Given a field92x3we define the jump of9across5 by

[[9]] :2 99 6 96

Similarly, for a field928 32x3, defined on the fracture surfaces,

222928 31333:2 928 79319 9 928 76316


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Figure 3. The 19 and 16 - unit tangent vectors to59and5 6, respectively, and normal to the crack edgect.

denotes the jump of928 3

1at the crack edge. Here

928 732ct3:2 lims09

928 32ct6 s13

and 1is the unit conormal vector to the crack edgect(tangent to the crack surface5 and,in three dimensions, normal toct), pointing away from the fracture surfaces (Figure 3).

LetTdenote the Cauchy stress tensor and T1 the first PiolaKirchhoff stress tensor.Further, let the matrix representation of the first PiolaKirchhoff stress tensor in Cartesiancoordinates be given by

[T1 ] 21 811 812

812 822

2

We assume that there exists a surface Cauchy stress tensor T28 3 which gives contact forceson a curve in the fracture surface 5 . For simplicity and consistency with the literature [22,p. 148] assume that the surface stress can be modeled as Eulerian, i.e.T28 3 23 Pt. HerePtis the perpendicular projection onto the tangent space 4xto 5 tatx. We refer to

3 as the

surface tension. Further, let Hdenote the mean curvature (see (79), Appendix A).

3. FORMULATION OF THE PROBLEM IN THE REFERENCE

CONFIGURATION

We consider a classical Griffith crack, meaning a static, mode I crack of finite length in aninfinite linear elastic body, subjected to far-field tensile loading8.

In view of the fact that the method of integral transforms is most easily applied when theproblem is formulated in a reference configuration where the crack is just a slit, we work in

the unloaded configuration of the body.In order to simplify the presentation, assume thatX2 X17X2 is non-dimensionalized

by crack length, so that the crack in the reference configuration is parameterized by


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51 2 X:61 X1 17X22 0

Consequently, in the current configuration, the upper/lower crack surface can be parameter-

ized by

52 x: x12 X1 9 u12X17 037 x22 u22X17 037 61 X1 1 (2)

wherex 2 x17x2 and u17 u2 are the components ofu1 2X3.Let 31 11 be a part in the reference configuration of the body and let 32 6231 3.

Then, provided that 3does not contain the fracture tip, the force acting on 3is

5233 2 441 Tn da 9 41 b d9 4425 513 T28 3

1 ds

24

41

Tn da 941

b d94

551

div28 3T28 3 da

24

411

JTm F6TN d A 9

411

Jb d V

94

5111J2div28 3T28 3 n63mF6TN6d A (3)

where bis the mutual body force term in the current configuration, nis the outward unitnormal vector to 43 and n6 is the unit normal to the crack profile 5 pointing into thebulk material, N is the outward unit normal vector to 431 andN6is the unit normal to thereference crack profile 51 pointing into the bulk material, 1is the conormal to425 33,whileTm is the material description ofT, i.e. Tm2X32 T262X33. If the fracture tip c is in3, then there is an additional contribution to 5233due to the excess properties at the cracktip, namely

5233

2 441 Tn da 9 41 b d9 4425513 T28 3

1 ds

9b2c3

24

41

Tn da 941

b d94

551

div28 3T28 3 da 6 222T28 313332ct3 9 b2c3

24

411

JTmF6TN d A 9

411

Jb d V

94

51

511

J2div28 3T28 3 n63mF6TN6d A 6 222T28 31 11 3332c3 9 b2c31 (4)

where b2c3 is a mutual force acting at the crack tip, arising due to resistance of chemicalbonds to opening of the fracture surfaces at the fracture tip and b2c31 is the corresponding


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body force in the reference configuration. Here we have used that in the case of plane strainF2c321 2c32 1, which in turn implies b2c3 2 b2c31 . Recall that the first PiolaKirchhoffstress tensor is given by

T12 JTmF6T

Letb1denote the body force in the reference configuration, thenb12 Jb. Using4411

T1 N d A2411

DivT1d V94

51 11[[T1 ]]N6d A7 (5)

we transform (4) to

5233 2 411

2DivT19 b1 3d V

94

51511

6J2div28 3T

28 3 n63mF6TN6 9 [[T1 ]]N67

d A

6 222T28 31 11 3332c3 9 b2c31 (6)

Ifct 5 3, the last two terms in (6) should not be included in 5233. Since equilibriumrequires 5233

2 0, application of the localization theorem to the first term in (6) implies

that the differential momentum balance in the reference configuration can be expressed as

DivT19 b12 07 (7)

the jump momentum balance4 by

J2div28 3T28 3 n63mF6TN6 9 [[T1 ]]N62 07 (8)

and the crack tip momentum balance by

b2c31 2 222T28 31 1 3332c3 (9)

The jump momentum balance in Cartesian component form is

812 2 621 9 u17132 9 u172u2718

21 9 u17132 9 u2271

9 3 2x1321 9 u1713221 9 u17132 9 u2271

9 3 u271u2271u1712 9 u27121 9 u17132u1711 6 u27123 6 21 9 u17132u2711621 9 u17132 9 u2271

72


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822 2 621 9 u17132 9 u172u271

821 9 u17132 9 u2271

9

3 2x1321 9 u1713u271

21 9 u17132 9 u2271

63 21 9 u1713u2271u1712 9 u27121 9 u17132u1711 6 u27123 6 21 9 u17132u27116

21 9 u17132 9 u227172

7 (10)whereui7j2 4ui4xj . A detailed derivation of Equations (10) is provided in Appendix B. The

jump momentum balance provides boundary conditions on the crack surfaces which arehighly non-linear because of the ascribed surface excess properties. Owing to symmetry,it suffices to consider the problem on the upper half plane only. In this case, additionalboundary conditions are needed on

X:

X1

17X2

20. Symmetry implies

u22X17 03 2 07 X1 1

8122X17 03 2 07 X1 1 (11)

We assume that the constitutive behavior of the material can be modeled by Hookes lawin the reference configuration, that is, the PiolaKirchhoff stress tensorT1is given by

T12 2E 9 tr2E3I (12)

where

E 2 12

27u19 7uT1 3

is the infinitesimal strain tensor andandare theLam moduli.Also, a homogeneous tensile far-field loading is assumed, i.e.

limX2

8112X17X23 2 0

limX2

8122X17X23 2 0

limX2

8222X17X23 2 8 (13)

Thus, the problem we are going to consider is formulated in the reference configuration (theupper half plane) and consists of:

1. a differential momentum balance given by Equation (7)12. boundary conditions on

X: X1

17X2

20

given by the jump momentum balance,

Equations (10)13. boundary conditions on X:X1 17X22 0 given by Equations (11)14. a constitutive equation given by Equation (12)15. a far filed loading condition given by Equation (13).


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4. METHOD OF INTEGRAL TRANSFORMS APPLIED TO THE NAVIER

EQUATIONS

4.1. Model with Constant Surface Tension and Zero Mutual Body Force Term

As a first step in our analysis we consider a model with constant surface tension (3 const)and zero mutual body force, i.e. b12 0in (7). The method of integral transforms applied tothe Navier equations is used, a key step in which is the construction of the so-calledDirichlet-to-Neumann and Neumann-to-Dirichlet maps, derived in Appendix A. For the model withno mutual body force correction term, the Dirichlet-to-Neumann and Neumann-to-Dirichletmaps reduce, respectively, to

8122x7 03 2 22

9 3 u2712x7 03 922

923

2 9 33 646 u1712r7 03r6 x dr8222x7 03 2 6

22

9 3 u1712x7 03 922 9 23

2 9 33 64

6

u2712r7 03

r6 x dr (14)

and

u1712x7 03 2 6 9 2

22 9 364

6

8122r7 03

r6 x dr9 1

22 9 3 8222x7 03

u2712x7 03 2 6 9 222 9 364

68222r7 03

r6 x dr6 1

22 9 3 8122x7 03 (15)

where6denotes a Cauchy principal value integral. For simplicity, in this section, the in-dicial notation2X17X23has been replaced by2x7y3. Substituting the first equation of (15)into the second of (14) and using the boundary conditions (11) on x 1, one arrives at

8222x7 03 2 22 9 3

2 9 2364 1

61

u2712r7 03

r6 x dr9

2 9 2364 1

61

8122r7 03

r6x dr

2 E221 6 236

4 161

u2712r7 03

r6 x dr9 1 6 2221 6 3 6

4 161

8122r7 03

r6 x dr (16)

Here Eand denote Youngs modulus and Poissons ratio, respectively:

E2 23 9 23 9 7 2

22 9 3

We now linearize the jump momentum balance boundary conditions (10) under the as-

sumption thatu i7j 2x7 03andu i7j k2x7 03are small. The solution of the linearized problem willthen be checked for consistency with the assumptions made.From (10) it is evident that the asymptotic form of the jump momentum balance equa-

tions is


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8122x7 03 2 0 9 hot7 x 1

8222x7 03 2 63 u27112x7 03 9 hot7 x 1 (17)where hot denotes higher order terms.

Letu denote the Fourier transform of u with respect to the x variable. Note that5the Dirichlet-to-Neumann and the Neumann-to-Dirichlet maps (and, consequently, Equa-tion (16)) were derived under the assumption thatu1712p7y3 andu2712p7y3 vanish in thelimit as y , whereas the far-field loading condition for our problem is given by (13). Inorder to reduce the considered problem to a problem which satisfies the above assumptions,we use the linearity of the differential momentum balance and the (linearized) boundary con-ditions and introduceuf andu0 such thatu12 uf 9 u0 withuf being the displacement fieldcorresponding to the homogeneous stress field

Tf121

0 0

0 8

2 (18)

Since the stress and strain tensors corresponding tou f are related constitutively by Hookeslaw, i.e.T f12 2Ef 9 tr2Ef3, whereE f 2 12 27uf 9 27uf3T3, one easily finds

uf1 2x7y3 2 6

8

429

3x

uf2 2x7y3 2

2 9 23842 9 3y (19)

The stress corresponding tou0 vanishes in the limit as y , and consequently

limyu01712p7y3 2 0 and lim

yu02712p7y3 2 0

that is,u0 satisfies the assumptions under which Equation (16) was derived.

From the definition ofu0

and Equations (17) and (18) one concludes that

80122x7 03 2 0 9 hot7 x 1

80222x7 03 2 686 3 u27112x7 03 9 hot7 x 1 (20)Substituting the above equations into (16) one arrives at

686

3 u027112x7 03 2

E

2216

236

4 1

61

u02712r7 03

r

6x

dr (21)

Let us define


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Table 1. Values ofu 271217 03for various values of the (non-dimensional) far-field loading8and (non-dimensional) excess property

3 .

3 8 u271217 03005 0001 6000690002 6001390004 600278

01 0001 6000460002 6000930004 600185

02 0001 6000300002 6000590004 600119

92x3 2 u02712x7 037 2 E221 6 23 (22)

Then Equation (21) takes the form

3 92x3 9

64 1

61

92r3

r6 x dr2 68 7 x5 [617 1] (23)

This is a Cauchy singular, linear integro-differential equation. It arises, for example,when modeling combined infrared gaseous radiations and molecular conduction. Abdou[23] and Badr [24] derive the solution as a series of Legendre polynomials, while Frankelin his 1995 paper [25] derives the solution of an equation of the type (23) as a series ofChebyshev polynomials. It should be noted that while Frankel considers some numericalexperiments, he does not study the convergence of the obtained infinite system of linearalgebraic equations. Various numerical approaches to solving equations of a similar typewere considered in [26, 27].

We take an approach similar to that of Frankel [25] to reduce (23) to an infinite systemof linear equations. For

320 the matrix of the system is eventually diagonally dominant

and its diagonal elements tend to infinity as the row/column index tends to infinity. Using

these key properties and theorems by Farid [28, Theorem 2.1] and Farid and Lancaster [29,Theorem 2.2], one can show that the matrix of the system is invertible and is a compact op-erator froml1intol1, and that the solutions of the truncated systems converge to the solutionof the infinite system.

4.1.1. Numerical Experiments

In Table 1 the values at the crack tip ofu2712x7 03, obtained using the method described above,are compared for various values of the (non-dimensionalized) far-field loading parameter8and surface tension3 . One can observe that the larger the value of the surface excessproperty, the smaller the slope of the crack profile at the crack tip.

From the numerical experiments (Figures 4 and 5) it is clear that the solution u22x7 03for the crack profile is a monotonically increasing function on 2617 03and monotonicallydecreasing on207 13. Furthermore, the slope of the crack profile at the crack tip increases


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Figure 4. Approximation of u 2712x7 03by a finite sum of Chebyshev polynomials (400terms) for 32 005and far-field loading82 0001,0002, and0004.

Figure 5. Approximation of u 2712x7 03 by a finite sum of Chebyshev polynomials (400terms) for far-fieldloading802 0002and 32 0005,001, and002.


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with an increase in the far-field loading. In addition, the surface excess property3 , withthe appropriate boundary condition, given by the jump momentum balance leads to a finiteopening angle at the crack tip. However, (23) implies that if the crack surfaces do not come

together at a cusp, i.e. 92132 u271217 03 is non-zero, then 92x32 u27112x7 03 has alogarithmic singularity at the crack tips. Using (17) one concludes that this leads to a loga-rithmically singular stress at the crack tip. This is an improvement from the classical LEFMmodel which leads to a square-root singularity of the crack tip stress, however, as in LEFM,any singular stress is inconsistent with the assumptions made to linearize the equations andderive (23).

4.2. Model with Curvature Dependence in the Surface Tension and Zero Mutual Body Force

Term

In this section, as a next step, we study a model in which the mutual body force is assumedto be zero but we allow for curvature dependence in the excess property3 of the fracturesurface, i.e.

32 3 2H3 (24)Even though curvature-dependent surface tension models are not common in the fractureliterature, the effect of curvature-dependent surface tension has been widely studied in thecontext of nucleation theory [3032].

Assuming that stresses and strains remain small and combining (85) and (24) one derivesthe following asymptotic expansion for 3 :

3 2x3 2 0 9 1u27112x7 03 9 hot (25)where 0 const and 1 const. After substituting (25) into (10) and linearizing the jumpmomentum balance equations under the assumption that ui7j 2x7 03and u i7j k2x7 03,i7 j7 k217 2 are small, we obtain

8122x7 03 2 1u271112x7 03 9 hot8222x7 03 2 6 0u27112x7 03 9 hot (26)

Note that, unlike in the case of constant surface tension, here the shear stress on the cracksurfaces to first order is not identically zero.

We proceed in a similar way to the approach taken in the case of constant 3(Section 4.1).Using (16) and splitting the displacement vector intou12 u0 9 uf, where the componentsofuf are given by (19), we arrive at the following linear integro-differential equation for92x3 2 u02712x7 03 2 u2712x7 03:

092x3 9 1

64 1

61

192r3 9 2 192r3r6 x dr2 68 7 x5 2617 13 (27)


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where

12 E

2216

23 and 22

1 6 2221

63

First note that unlike in the case with3const (cf. (23)), Equation (27) cannot have asolution 92x3, such that9 2x3is singular at the endpoints. The reason being that this wouldimply92x3 5 L12[617 1]3and consequently the Cauchy principal value integral in (27)would not exist. In turn, this implies that every solution92x3of (27) satisfies

19213 9 2 19213 2 0 (28)

Thus, the solution 92x3adjusts itself so that (28) is satisfied. Note that (28) cannot be

viewed as a boundary condition.Further, taking into account21131we look for a solution of (27) subject to

92613 2 9213 2 0 (29)

so thatu 2712x7 03is continuous at the crack tips. Furthermore, the symmetry of the problemrequires that the crack profileu 22x7 03be an even function ofxand therefore we look for asolution9 of (27) such that

92x3

2 692

6x37 x

52

617 13 (30)

Theorem 1. Problem (27), subject to (29) and (30), has a unique solution for all, apart from

countably many, values of the parameters 0 and 1.

Proof.We use a technique introduced by Mikhlin and Prssdorf [33, Chapter VII] to reduce(27) to canonical form.

Let 2x3:2 92x3. Then

92x32 4

1

6102x7 r32r3 dr

9c1 (31)

and

92x3 24 1

6112x7 r32r3 dr9 c12x9 13 9 c2 (32)

where

02x7 r3 2

17 r5 2617x307 r

52x7 13

7 12x7 r3 2

4 1

61

02x7 t302t7 r3 dt

and c12 92613, c22 92613. After substituting (31) and (32) into (27) we arrive at asingular integral equation for 2x3of the form


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12

64 1

61

2r3

r6 x dr94 1

61k2x7 r32r3 dr2 686 0c1 6

1

64 1

61

c12r9 13 9 c2r6 x dr (33)

where

k2x7 r3 2 002x7 r3 91

64 1

61

12t7 r3

t6x dr

Using the boundary conditions (29), we obtainc220. Also, combining (29), (30), and (31)we find

0 24 1

092x3 d x2

4 10

4 161

02x7 r32r3 dr dx9 c1 (34)

Thus, using (29) and (30), (33) can be written in the (canonical) form

R[]2x3:2 12

64 1

61

2r3

r6 x dr94 1

613k2x7 r32r3 dr2 68 (35)

where

3k2x7 r3 2 002x7 r3 9

1

6

4 1

61

12t7 r3

t

6x

dt6 04

1

002t7 r3 dt

6 1

64 1

61

4 10

02s7 r3t9 1t6x ds dt

Let L22[617 1]3denote the weighted space of functions having a finite norm

24 1

6121 6 x2312 2x32 d x

12

It is well known [27, 33, 34] that the singular integral operatorR[] is a Fredholm operatorfrom L22[617 1]3to L22[617 1]3of index 1. The index depends only on the dominant partof the operator: the singular integral operator

R1[]2x3:2 1

64 1

61

2r3

r6 x dr

and is independent of any compact perturbation, in particular, it is independent of

R2[]2x3:

2 4 1

61 3k2x7 r32r3 drOne can show [35] that the general solution of the singular integral equation


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g2x3 2 1

64 1

61

f2r3

r6 x dr (36)

is given by

f2x3 2 6 11 6 x2

11

64 1

61g2r3

1 6 r2

r6 x dr9 C2

(37)

In essence, formula (37) gives an inverse of the finite Hilbert transform operator (36). Usingthis, we conclude thatR1[]2x3, restricted to the space of functions with

161 2x3 dx2 0,

has a trivial null space. Therefore, (35) is equivalent to

122x3 9 R61

1 R2[]2x3 2 R61

1 [8]2x3

Note thatR611 R2is a compact operator, being the composition of a compact with a continuousoperator, and, consequently, it has a countable spectrum. Furthermore, 12I9 R611 R2,where I is the identity operator, is invertible, unless6 12 is in the spectrum of R611 R2.This concludes the proof. 1

4.2.1. Numerical Experiments

To find a numerical solution to problem (27), subject to (29) and (30), we employ a spline col-location method, similar to the one introduced by Samo23lova in [36], where a first-order sin-gular integro-differential equation (SIDE) is solved. Spline collocation methods for SIDEswere considered by many others, including Schmidt [37]. Using (30), it suffices to solvethe problem on207 13. Further, the boundary conditions (29) combined with (28) imply that92132 92132 0 and (30) yields 92032 92032 0. Consequently we can use a naturalcubic spline S2x3to approximate the solution92x3. Let 02x1 x2 xN9121 bethe evenly spaced spline nodes, i.e.

S2x3 2

S12x37 x5 [x17x2]

S22x37 x5 [x27x3]

SN2x37 x5 [xN7xN91]

with [38]

Si 2x3 2 zi912x6xi 33 9 zi 2xi91 6 x33

6h9

yi91h

6 h6

zi912x6xi 3

9

yi

h6 h

6zi

2xi91 6 x3 (38)


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Here h2 1N, yi approximates92xi 3and the coefficients zi can be found by solving thetridiagonal system of equations

z1 2 0zi61 9 4zi9 zi91 2

6h2

2yi91 6 2yi9 yi613

zN91 2 0

Using (30), we transform (27) into

092x3

92

64 1

0

2192r3 9 2 192r33rr

2

6 x2

dr

2 68 7 x

5207 13 (39)

The Cauchy principal value integral is calculated with the help of a product integrationmethod[39, 40], i.e.

64 1

0

21 S2r3 9 2 1 S2r33rr2 6 x2 dr2

Ni21

64 xi91

xi

21 Si 2r3 9 2 1 Si2r33rr2 6 x2 dr7

and, using (38), each of the integrals on [xi 7xi91] is evaluated exactly.In the end, the following2N

613

82N

613linear system of equations for the unknowns

y27y37 7yNis solved

0 Si 2ti 3 9

Nj21

64 xj91

xj

21 Sj 2r3 9 2 1 Sj 2r33rr2 6 t2i

dr2 68 7 i2 27 7N (40)

wheretiis the midpoint of the interval [xi 7xi91].Figures 69 are graphs of the slope of the crack profile u2712x7 03and of 1u27112x7 03

for2 033 and various values of the parameters 0, 1and the far-field loading8. Notethat the parameters are non-dimensionalized (e.g.1

21E), but for simplicity of notation

the superscriptis dropped.From the numerical experiments (Figure 6) it is clear that822217 03, the stress at the

crack tip (in absolute value), is an increasing function of the far-field loading (cf. (26)). Fur-thermore, the larger the value of 1, the smaller the crack tip stress (Figure 7). Interestingly,unlike in the constant surface tension model (Section 4.1), the crack tip stress is an increas-ing functionof 0(Figure 8). In Table 2 the values at the crack tip of9

2x32 u27112x7 03,in the case of curvature-dependent surface tension, are compared for various values of the(non-dimensionalized) far-field loading parameter 8and the parameters 0 and 1 whichdetermine the surface excess property

3 2x3.

It should be noted here that for certain values of the parameters, namely when 0is notmuch smaller than 1, the model yields unphysical solutions and predicts interpenetration ofthe upper and lower crack surfaces (Figure 9).


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Figure 6. Graph of u2712x7 03 and of 1u27112x7 03 for 0 2 01, 1 2 1 and far-field loading82 00017000270004.


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Figure 7. Graph of u2712x7 03 and of 1u27112x7 03 for 0 2 01, far-field loading 8 2 0002 and 12 0971711.


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Figure 8. Graph of u2712x7 03 and of 1u27112x7 03 for 1 2 1, far-field loading 8 2 0002 and 02 005701702.


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Figure 9. Graph of u2712x7 03 and of 1u27112x7 03 for 0 2 01, 1 2 1 and far-field loading82 00017000270004.


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Table 2. Value of 92132 u2711217 03 for various values of the (non-dimensional) far-fieldloading8and (non-dimensionalized) 0 and 1.

0 1 8 9213 0 1 8 9

213 0 1 8 9213

0.05 0.9 0.001 0.0060 0.1 0.9 0.001 0.0079 0.2 0.9 0.001 0.02020.002 0.0120 0.002 0.0158 0.002 0.04040.004 0.0240 0.004 0.0315 0.004 0.0808

1 0.001 0.0050 1 0.001 0.0063 1 0.001 0.01250.002 0.0101 0.002 0.0126 0.002 0.02500.004 0.0201 0.004 0.0252 0.004 0.0500

1.1 0.001 0.0043 1.1 0.001 0.0053 1.1 0.001 0.00900.002 0.0087 0.002 0.0105 0.002 0.01800.004 0.0173 0.004 0.0210 0.004 0.0361

Further, models for which 0is much larger than 1predict highly oscillatory solutions.Thus, physically realistic solutions require that the dependence of surface tension upon cur-vature not be too weak relative to its baseline value in the limit as crack surface curvaturevanishes.

Most importantly, introducing curvature dependence into the surface tension removesthe crack tip stress singularity and leads to a crack profile such that the two crack surfacesmeet at a cusp at the crack tip. Moreover, models with curvature-dependent surface tensionyield solutions such that u2712x7 03and u27112x7 03remain small (when8is small enough),which is consistent with the assumptions made to derive (27).

4.3. Model Including Mutual Body Force Correction

In order to construct the Dirichlet-to-Neumann (99) and the Neumann-to-Dirichlet (101)maps for the model with non-zero mutual body force correction, one needs to find a particularsolution of (91), which can be done using standard techniques.

We take an approach similar to that presented in Sections 4.1 and 4.2. Specifically, usingthe Dirichlet-to-Neumann map (99), the Neumann-to-Dirichlet map (101) and the boundaryconditions on the crack surfaces, we construct an equation relating the tensile stress8222x7 03

and the slope of the crack profile u2712x7 03 forx5 [617 1], after which we linearize the com-ponents of the body force term b1 2x7 t3under the assumption that ui7j 2x7 03and ui7j k2x7 03are small.

We adopt a model for the mutual body force term analogous to the model proposed byOh et al. [21], where the body force in the current configuration is given by b2 6gradwith 2x17x23, a correction potential of the form

2x17x23 24 1

61

4 h2a36h2a3

46

2

2x1 6 a32 9 2x2 6 b32 9 c23 d c d b d a (41)

Hereis an interatomic potential, for example of Morse or LennardJones type.6


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4.3.1. Model with Constant Surface Tension and a Mutual Body Force Term

First consider the case where the surface excess property 3 is a constant. As in Section 4.1,we splitu12 u0 9 uf whereuf is the displacement field corresponding to the homogeneousstress field determined by the far-field loading8.One can show that introducing a mutual body force term, based on a correction poten-

tial as given in (41), leads to the addition of a compact operator to the original singularintegro-differential operator. More precisely, proceeding in the same way as for the deriva-tion of (21), after linearization of the body force in the reference configuration b1under theassumption of small displacement gradient, one concludes thatu 02712x7 03satisfies

686 3 u027112x7 03 2

E

2216

2364

1

61

u02712r7 03

r6

xdr96[u0271]2x3 (42)

where

6[u0271]2x3 24 1

61

4 a61

u02712r7 03 dr K2x7 a3 da (43)

withK : [617 1]8[617 1] 1 - a continuous function. Therefore,6 is a compact operatoronC[617 1], being the composition of a Fredholm integral operator of the first kind [41, p.55] and a bounded operator onC[617 1].

Let

92x3:2 u02712x7 03

and let the singular integro-differential operator 2be defined by

2[9]2x3:2 3 92x3 9 E221 6 236

4 161

92r3

r6 x dr

Then (42) is equivalent to

2[9]2x3 96[9]2x3 2 68 (44)

4.3.2. Model with Curvature-dependent Surface Tension and a Mutual Body Force Term

In the case when the surface excess property is curvature-dependent and the model includesa body force correction term, after linearizing the differential and jump momentum balances,using arguments similar to those given in the previous section, it is straightforward to showthatu 0

271

2x7 03satisfies the following equation

0u027112x7 03 9

1

64 1

61

1u02712r7 03 9 2 1u0271112r7 03

r6 x dr96[u0271]2x3 2 68 (45)


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Let

32[9]2x3:2 0u027112x7 03 9 164 1

61

1u02712r7 03 9 2 1u0271112r7 03

r6 xdr

Here the notation introduced in Section 4.2 is used. Then (45) can be written in the form

32[9]2x3 96[9]2x3 2 68 (46)4.3.3. Discussion

We conjecture that if3 2 0, then for any physically reasonable correction potential thesolution9of the singular integral equation (44) exhibits a square root singularity at the cracktip. Furthermore, if3 2x3 0 9 1u27112x7 03with 120, i.e. there is non-zero curvaturedependent surface tension introduced as an excess property of the fracture surfaces, then(45) has a unique solution92x3 2 u02712x7 03for all, apart from countably many values of theparameters 0and 1. Moreover, 92x3and 9

2x3are bounded on [617 1], i.e. the operator32[] 96[], where 6 is the compact operator given by (43), behaves in a similar way to thesingular integro-differential operator32[].

In other words, it is the surface tension

3 of the fracture surfaces, together with the ap-

propriate fracture surface boundary conditions in the form of the jump momentum balance

that is responsible for removing the square root singularities at the fracture tips, character-istic of the classical LEFM model. Moreover, including a mutual body force term in themodel, after linearization of the jump momentum balance conditions, results in a compactperturbation of the SIDE. We conjecture that this compact perturbation does not affect thefundamental result, namely a model with curvature-dependent surface tension ascribed to thecrack surfaces yields bounded stresses and strains for any physically reasonable body forcecorrection. However, these are still open questions, subject to future investigation.

5. ENERGY-BASED CRACK GROWTH CONDITION

Various approaches to the thermodynamic analysis of fracture have been studied in the liter-ature, with or without consideration of temperature effects. These approaches have incorpo-rated classical singular theories with singular stresses and singular power flux into the cracktip (see Gurtin [42, 43] and Gurtin and Yatomi [44]) or with cohesive zones designed to re-move the singularities (see Gurtin [45]). Others have included the notion of aconfigurationalforce system, with or without cohesive zone (see Gurtin and Shvartsman [16], Costanzo [46],Gurtin and Podio-Guidugli [14, 15]) or excess surface properties, with or without cohesive

zone.Separately, Gurtin and Murdoch [47, 48], Murdoch [49] and Fried and Gurtin [50] havedeveloped a theory of elastic material surfaces, incorporating models with excess surfaceproperties, not necessarily directed towards fracture.


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The idea of ascribing excess properties to a dividing surface between two phases datesback to Gibbs. In the development of fracture theory, Griffith was the first to introduce sur-face excess properties in the context of solids, but he did not build it into a model of fracture

in any concrete way. To the best of the authors knowledge, the first comprehensive attemptto develop a fracture theory including excess properties was offered by Eftis and Liebowitz[51] (see also Zhang and Karihaloo [52], Van der Varst and De With [53]). Unfortunately,their development contains serious conceptual and technical flaws.

Our approach bears resemblance to several of the above modeling approaches in that itincludes a detailed description of the surface excess properties. What is new in this approachis that, as shown in Section 4, curvature-dependent excess properties together with the ap-propriate jump momentum balance, which defines the boundary condition on the fracturesurface, lead to a theory with bounded stresses and strains. Further, even though the modelin which the surface tension is taken to be constant exhibits a logarithmic crack tip stress sin-gularity, this singularity does not lead to an influx of energy into the crack tip, and thereforethe theory outlined below is applicable in this case as well.

An energy-based crack growth condition is formulated, including terms similar to theclassical notion of a critical ERR, defined in the setting of singular crack tip stresses andstrains. Classically the ERR arises due to singular fields, whereas in the case of the modelingapproach adopted here, a notion analogous to the ERR arises through a different mechanism,associated with the rate of working of the surface excess properties at the crack tip.

5.1. Problem Statement

Here we consider a plate, notched on one edge (two planes intersecting at a relatively largeangle). The plate is undergoing dynamic, mode I fracture, the fracture having formed at thetip of the notch.

In our analysis of this problem, we make the following assumptions.

1. Temperature is independent of position and time.2. The rates of external and mutual energy transmission, and the rate of contact energy trans-

mission are negligible.3. Mass transfer is negligible at all phase interfaces.

4. Pressure in the gas phase between crack surfaces is the atmospheric pressure, considerednegligible when compared with the stresses generated in the system by deformation.

Owing to crack growth, in the modeling approach taken herein, we consider a continuoussequence of reference configurations (where the slit has different lengths). The mass transferfrom the bulk material to the fracture edge and from the fracture edge to the fracture surfaceis more conveniently accounted for in the reference configuration using evolving naturalconfigurations.7

5.2. Analysis

To simplify the discussion, the derivation of the crack growth condition given in Section 5.2is in the context of a straight edge crack (in the reference configuration) in a bounded two-


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dimensional body. An expression for the surface first PiolaKirchhoff stress tensor T28 31 isderived8 in Appendix A, namely

T28 31 2 F21PtF6T11T28 3PtF

6T (47)

From here on, 1tis assumed to be a bounded two-dimensional body. Let 61tde-note the kinetic energy of the body, expressed in terms of the current and the referenceconfiguration:

61t 2 12

42t

v2 d912

442t

28 3 v2 da 912

2c3 v2ct7 t32

2 12421 2t3

1 4x2 d V9 124

421 2t3

28 31 4x2 d A 9122c31 4ct2 (48)

where is the mass density of1t, 28 3 is the surface mass density and 2c3 is the massdensity associated with the fracture tip ct. In a similar way, 1 ,

28 31 , and

2c31 are the mass

density, surface mass density, and mass density associated with the fracture tip in the refer-ence configuration.9

Let 71tbe the internal energy of1twith 71t 2 81t 9T21twhere 81tdenotes the stored energy, 21tdenotes the entropy, and Tdenotes the constant absolutetemperature. LetA andA28 3 be respectively the Helmholtz free energy density and thesurface Helmholtz free energy density per unit mass, then

81t 242t

A d9 442t

28 3A28 3 da2

421 2t3

1A d V9 4421 2t3

28 31A28 3 d A (49)

In the current model there is no free energy density associated with the crack tip, although itcan easily be added, if needed. Further, let 3

1

tbe the power input to the body

31t 2421 2t3

b1 4x d V9 b2c31 4ct9431

se1 4x d A

9 2T28 31 2c2t3311 2c2t33 F2c2t334c2t3 (50)

Here se1 are the external tractions per unit area in the reference configuration acting on thebody and b2c31 , as in (4), is the mutual body force acting at the crack tip. Note that 31tincludes the power input not only through the external force system, but also from (possible)

mutual body forces and surface tractions arising from the material response of the body(through the jump momentum balance). The last term in (50) represents the rate of workingof the crack surface stresses at the crack tip and it is non-zero only when there is crack


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extension (only the part of the crack tip velocity which is due to crack extension (bondbreaking) is taken into account, Equation (68)).

The fundamental power balance can be written in the form

d

dt61t 9 d

dt71t 2 31t 691t (51)

where 91t is the fracture energy dissipation rate.The entropy inequality in the form of the ClausiusDuhem inequality (see [54, p. 728]

and [55, p. 130]) in the context of assumptions 1 and 2 reduces to

d

dtT21t 0 (52)

Next, following the analysis of Gurtin and Podio-Guidugli [14], we derive the transporttheorems appropriate for the current setting. It is important to keep in mind that in the settingof [14], the mechanical power flux into the crack tip is not zero due to the singular crack tipstress and strain fields, whereas here stresses and strains are bounded at the crack tip. Forthis reason, the transport relations needed here differ from those derived in [14, 15].

Lemma 1 (Transport relation for a bulk function in the reference configuration). Let

2X7 t3be a field defined on 11 2t3 819which is bounded and sufficiently smooth up to thecrack from either side. Then

d

dt

421 2t3

2421 2t3

4 (53)

where4 is the material time derivative of .

Lemma 2 (Transport relation for a function defined on a growing surface in the ref-

erence configuration). Consider11 2t3 12. Let 28 32X7 t3 be a field defined on the(possibly growing) crack surface51 2t3

81

9 which is bounded, and sufficiently smooth on251 2t3 8 193 2c2t3 8 193. Then

d

dt

4421 2t3

28 3 24

421 2t3

428 3 9 228 32c2t33 4c2t3 (54)

Proofs of Lemmas 1 and 2 are provided in Appendix E.

5.2.1. Momentum Balance Relations

Invoking the transport theorem (which has the usual form for a bulk control volume due tothe fact that stresses and strains remain bounded) and standard localization arguments, onederives the local form of the balance of linear momentum in the reference configuration


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1 x 2 DivT19 b1 7 X 5 11 2t3 (55)

and jump momentum balance

28 31 x 2 Div28 3T28 31 6 [[T1 N]]7 X 5 51 2t3

28 31 x 2 Div28 3T28 31 6 T1 N 9 se1 7 X 5 21 (56)

HereN is the outward unit normal. For the problem considered herein, 5 1 2t32 51 2t39351 2t3

6is not just a dividing surface in the body, but rather a part of its boundary and the firstequation in (56) could also be written in the form

28 3

1 x

2Div28 3T28 3

1 6T

1

N7 X5

51

2t3 (57)

In addition to these, there is a momentum balance at the crack tip given by

2c31 ct2 b2c31 9 2222c31 24c 11 2c2t3334c 6 T28 31 2c2t3311 2c2t33333 (58)

where 11 2c2t33is the conormal at c2t3(in two-dimensional space this is the unit tangent tothe fracture curve) pointing away from the fracture surface. Equation (58) is based on [22,(2.1.9-15)], stated with respect to the reference configuration and modified for the case of apropagating crack in a two-dimensional body.

5.2.2. Crack Growth Condition

We now proceed with a derivation of a necessary condition for crack propagation, workingin the reference configuration. Substitution of (48), (49), and (50) into (51) yields

d

dt

12

421 2t3

1 4x2 d V912

4421 2t3

28 31 4x2 d A 912

2c31 4ct2

9 421 2t3

1A d V9 4421 2t3

28 31A28 3 d A

2421 2t3

b14x d V9 b2c31 4ct9431

se14x d A

9 2T28 31 2c2t3311 2c2t33 F2c2t334c2t3 6 d

dtT21t 691t (59)

Lemmas 1 and 2 together with (55), (56), and (59) lead to


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2

12

28 31 2c2t33 4ct2 9 28 31 2c2t33A28 32c2t33 4c2t3 11 2c2t339 12 d

2c3

1

dt4ct2 9 2c31 ct 4ct9 4

21 2t3DivT14x 9 1 4A d V

94

421 2t3

Div28 3T28 31 4x 6 T1 N 4x 9 28 31 4A28 3 d A

2 b2c31 4ct9 2T28 31 2c2t3311 2c2t33 F2c2t334c2t3

6 ddt

T21t 691t (60)

Use of the divergence theorem yields421 2t3

DivT14x d V2 6421 2t3

T14F d V94

421 2t3

T1 N 4x d S (61)

and similarly from the surface divergence theorem (see [22, (A.6.3-7) and (2.1.9-3)])4421 2t3

Div28 3T28 31 4x d S

2 64421 2t3

T28 31 728 3 4x d S9 222T28 31 2c2t331 2c2t33 4ct333

2 64

421 2t3

T28 31 4FP1d S9 2T28 31 2c2t331 2c2t33 4ct (62)

withP1the perpendicular projection operator onto411 2t3, where Equation (71) is used.In virtue of (58), (61), and (62), Equation (60) reduces to

421 2t3

1 4A 6 T14F d V9 4421 2t3

28 31 4A28 3 6 T28 31 4FP1 d A9

28 31

2c2t33 4ct2 9 22c2t33A28 32c2t339 22c31 4c 4ct 4c2t3 11 2c2t33

9 12

d2c31dt

4ct2

2 2T28 31 2c2t3311 2c2t33 F2c2t334c2t3 6 d

dtT21t 691t (63)

Since in any isothermal process a thermoelastic material is hyperelastic (T12 1 4FA)with free energy function equal to the stored energy [55, p. 134], we have 1

4A2 T14F.


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By assumption, the surface Cauchy stress is of the form T28 3 23 Pt. LetA28 32X32A

28 32F2X332

3A28 32j23where j22 J

F6TN

is the RadonNikodym derivative of the area

measure on5 twith respect to that on51 2t3(see Proposition 1, Appendix D). Then

28 314A28 3 2 28 31 dd j2 3A28 32j234Fj2 4F

Now,

4Fj2 2F6TN 4FJ9 J4F F6TN

2 J

F6TN

F6T 6 J

F6TN

2F6TN F6TN3F6T

2 JF6TNPtF6T 2 j2PtF6TOn the other hand, by Proposition 1 in Appendix D

T28 31 4FP12 j23 PtF6T 4FP12 j23 PtF6T 4FSince (see [22, p. 325])

32 28 31 dd j2 3A28 32j237Equation (63) reduces to

28 31

2c2t33 4ct2 9 22c2t33A28 32c2t339 22c31 4c 4ct 4c2t3 11 2c2t33

9 12

d2c31dt

4ct2

2 2T28 31 2c2t3311 2c2t33 F2c2t334c2t3 6 d

dtT21t 691t (64)Note that the terms remaining in (64) are non-zero only when the crack starts to propagate.UsingT28 3 23 Ptand the results in Appendix D, one can show that T28 31 112 j3 1, wherej2F21. Assuming that the kinetic energy of the crack tip is negligible and appealing tothe entropy inequality, (64) reduces to

28 31 2c2t33

A28 32c2t33 j

34c2t3 FT2c2t3312ct3

4c2t3 11 2c2t33 7 (65)

where we have used4c2t3 11 2c2t33 0, which holds true since 11 2c2t33is the unit tangent tothe fracture surface pointing away from it and provided the crack can only change directionat an angle smaller than 2. Note that in the case of plane strain j2 F2c2t3321 2c2t33 1.


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Now letG28 3c be the critical value of the surface Gibbs free energy per unit mass, whichdepends only on the atomic bond strength, and let be the surface energy. Here

G28 3c can be

interpreted as the energy required to break the chemical bonds [56], while as the energy

required to overcome the long-range intermolecular forces [57]. Then

28 31A28 3 2 28 31G28 3c 9 (66)

If the crack does not change direction, i.e. ifP1 2c2t334c2t324c2t3, using 112 P1 FT1P1 FT1, Equation (65) can be written in the form28 31 2c2t33

G28 3c 2c2t33 9 2c2t33

3

P1 2c2t33FT2c2t3312ct3

(67)

Equation (67) gives a necessary condition for crack propagation. The left-hand side of(67) depends only on the material properties at the crack tip, while the ratio on the right-handside is related to the deformation gradient and depends on the far-field loading. The reasonwe refer to (67) as a crack growth condition rather than a fracture criterion has to do withthe energy dissipation921t3and the entropy production which are not being modeled in thepresent discussion.

Note:The analysis in Section 5.2 can be performed in the current configuration, if needed,in a similar way, using the analogs of Lemmas 1 and 2 for the current configuration, as wellas

112 P1 FT1

P1 FT1

and

T28 3 2 1j2

T28 31 P1 FT

In this case one has to be careful to distinguish the motion of the crack tip due to crack exten-

sion (bond breaking) from the motion of the crack tip due to deformation. A straightforwardcalculation gives

4ct 2 v2ct7 t3 9 F2c2t37 t34c2t3

2 v2ct7 t3 9 2I 6 grad u2ct7 t3361 4c2t3 (68)

in whichIdenotes the identity tensor. It follows that

4c2t3

22I

6grad u2ct7 t332

4ct6

v2ct7 t33 (69)

which gives the spatial description of the crack extension velocity. In the absence of crackextension (bond breaking),4c2t3 2 0while, in general,4ct2 0.


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6. CRACK TIP STRESS CRITERION

Since the theory proposed herein predicts a finite crack tip stress (Section 4.2), there is an

alternative fracture criterion, based on crack tip stress rather than energy. This criterionappeals to the assumption that the crack will start to propagate once the cleavage stressexceeds the stress required to overcome the short-range (chemical bonds) and long-rangeintermolecular forces. The critical stress for a given material can (in principle) be estimatedthrough ab initiomolecular dynamics calculations.

For example, for the model with curvature-dependent surface tension and no body forcecorrection term, we estimate the tensile stress at the crack tip from (26):

822217 03 2 6 0u2711217 03 9 hot

Since the considered model leads to finite stresses and strains, we conclude that the cleavagestress is well defined and can be calculated using the results from Section 4.2. Thus, the crackstarts to propagate once the stress at the crack tip reaches the value of the critical stress10, i.e.

822217 03 8crit

E

This new approach to formulating a fracture criterion is very appealing with its straight-forward physical interpretation and simple implementation.

7. CONCLUSIONS

In this paper we study several types of models arising from a new approach to modeling brit-tle fracture based on an extension of continuum mechanics to the nanoscale, first proposedby Slattery et al. [22] and then applied in the context of fracture by Oh et al. [21]. The mainidea of the theory is to correct bulk material behavior in a neighborhood of the fracture sur-faces for effects of long-range intermolecular forces from adjoining phases. This, however,

leads to a non-linear, non-local boundary-value problem.The method of integral transforms applied to the Navier equations is employed to resolve

the fracture profile and tensile stress at the crack surfaces. First we consider a model offracture which incorporates constant surface tension ascribed to the fracture surfaces and usethe jump momentum balance as a boundary condition at the crack surfaces, and then showthat it leads to a sharp crack profile at the crack tip (as opposed to the blunt one predicted bythe classical LEFM model) and stresses which exhibit a logarithmic singularity at the cracktip.

Further, a modified model is studied in which the surface excess property includes cur-vature dependence. We show that this model yields bounded stresses and strains. This lendssupport to having a stress-based fracture criterion which amounts to a simple strength of ma-terials argument. Moreover, in the case of the curvature-dependent surface tension model,the two fracture surfaces form a cusp at the crack tip. The resulting second-order linear SIDE


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is solved numerically using spline collocation methods combined with product integrationtechniques.

The second part of the paper is devoted to the derivation of a crack growth condition

based on the global energy balance and the second law of thermodynamics. The analysisis in the spirit of Gurtin [42, 43], Gurtin and Yatomi [44] and Gurtin and Podio-Guidugli[14, 15] with the major exception that in the theory presented herein there are no stress andstrain crack tip singularities in contrast to those previous studies. For this reason, the energy-based crack growth condition developed in this paper arises in a very different way from theclassical notion of ERR used in the papers cited above.

The theoretical results derived for the fracture modeling paradigm studied in this paperoffer a number of potentially important benefits to practical fracture mechanics analyses.In addition, the intriguing prospect of using a CCTS fracture criterion enabled by the finitecrack tip stress field predicted by the model, as shown above when one uses the jump mo-mentum balance boundary condition on fracture surfaces along with a curvature-dependentsurface tension, there is no need in finite-element calculations to employ singular elementsat crack tips, cohesive zones, or process zones, all of which entail difficulties to implementefficiently and accurately. With bounded crack tip stresses and strains resulting from useof the appropriate conditions, finite-element implementation is made a relatively straightfor-ward and simple affair.

Acknowledgments. The authors would like to thank Dr John Slattery and Dr Kaibin Fu for the numerous fruitfuldiscussions. This work was supported in part by the Air Force Office of Scientific Research through Grant FA9550-06-0242 and in part by award number KUS-C1-016-04 made by King Abdullah University of Science and Technology(KAUST).

APPENDICES

A. SURFACE GRADIENT AND SURFACE DIVERGENCE

Let 92x3be a scalar field defined in a neighborhood of a dividing surface 5 , nbe a unitvector normal to5 and letPdenote the projection tensor onto the tangent space to 5 . Thenthe surface gradient of92x3is given by (see [22, p. 632])

grad28 392 Pgrad9 (70)

In a similar way, the surface gradient of a vector fieldv2x3may be expressed in the followingform (see [22, p. 648])

grad28 3v 2 2gradv3 P (71)

and consequently for the surface divergence ofvone obtains

div28 3v 2 tr22gradv3 P3 (72)


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As for the surface divergence of a second-order tensor fieldA2x3, it can be easily shown (see[22, p. 661]) that it satisfies

c div28 3A 2 div28 32ATc3 (73)for any constant vectorc. Equation (73) can be used as a definition of div28 3A. The surfacedivergence and surface gradient satisfy product rules analogous to the standard rules (for divand grad).

Lemma 3. Let9 ,v,w,andA be smooth fields with9 scalar valued,vandw vector valued,andA tensor valued. Then

div28 329A3 2 Agrad28 39 9 9div28 3A (74)div28 32v w3 2 vdiv28 3w 9 2grad28 3v3w (75)

For a detailed discussion of the theory of elastic material surfaces, see [22, 4749]. Wecan now find an explicit expression for div28 3T28 3. If we assume that the surface stressT28 3

is given byT28 3 2 3 P, wherePis the projection tensor defined byP 2 I 6 n n (76)

then (74) yields

div28 3T28 3 2 grad28 339 3 div28 3P (77)Now, combining (76) and (75) one has

div28 3P 2 6div28 3n n 2 6ndiv28 3n (78)

Here we have used the fact that

2grad28 3n3n 2 2gradn3 Pn 2 0

Let us now define thecurvatureof5 by

H2 612

div28 3n6 (79)

Here, in order to avoid ambiguity of the definition, we have chosen the unit normal vectorn6to the fracture surface5 , pointing into the bulk material. Note that div28 3Pis independentof how we choose the direction ofn. Then we can express div

28 3Pin the following way

div28 3P 2 2Hn6 (80)


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Using the equivalent of (5) in the current configuration and combining (3), respectively (4),(77), and (80) one concludes that the jump momentum balance in the deformed configurationtakes the form

grad28 339 23Hn6 9 [[T]]n62 0 (81)

B. COMPONENT FORM OF THE JUMP MOMENTUM BALANCE EQUATIONS

IN THE REFERENCE CONFIGURATION

From Equation (2) one concludes that the unit normal vector to 5 pointing into the bulk

material has the following component form

n62 1821 9 u17132 9 u2271

6u2717 1 9 u171T7 (82)

consequently

P 2 121 9 u17132 9 u2271

1 21 9 u17132 21 9 u1713u27121

9u1713u271 u

2271

2 (83)

Hereu i7jare evaluated at pointsXon51 , i.e. 61 X1 1,X22 0. Whenever this is clear,for simplicity of notation, this dependence is suppressed.

Combining Equations (70) and (83), one arrives at the following expression forgrad28 33 :

grad28 332 3 2x1321 9 u17132 9 u2271 21 9 u171327 21 9 u1713u271T (84)Further, the definition of a surface divergence of a smooth vector field (72) and Equation(83) yield

div28 3n62u2271u1712 9 u27121 9 u17132u1711 6 u27123 6 21 9 u17132u27116

21 9 u17132 9 u2271732 (85)

It only remains to evaluate the term JF6TN6 n6. The matrix of the deformation gradient inCartesian coordinates can be written in the following form

[F] 2 1 1 9 u171 u172u271 1 9 u272 2


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Using this, the fact that N62 07 1T and Equation (82) forn6, one arrives at

JF6T

N6 n62 21

9u1713

2

9u172u271821 9 u17132 9 u2271 (86)

Finally, Equations (77), (78), (82), (84), and (86) together with (8) lead to the following ex-pression for the jump momentum balance equations formulated in the reference configuration:

812 2 621 9 u17132 9 u172u2718

21 9 u17132 9 u2271

9 3 2x1321 9 u1713221 9 u17132 9 u2271

93 u271u2271u1712 9 u27121 9 u17132u1711 6 u27123 6 21 9 u17132u27116

21 9 u17132 9 u227172 7

822 2 621 9 u17132 9 u172u2718

21 9 u17132 9 u2271

93 2x1321 9 u1713u27121 9 u17132 9 u2271

6 3 21 9 u1713

u2271u1712 9 u27121 9 u17132u1711 6 u27123 6 21 9 u17132u2711

621 9 u17132 9 u227172 (87)

C. CONSTRUCTION OF THE DIRICHLET-TO-NEUMANN AND THE

NEUMANN-TO-DIRICHLET MAPS

Following [35], we proceed as follows. The component form of the differential momentumbalance (7) is given by

81171 9 81272 9 b11 2 0

82171 9 82272 9 b12 2 0 (88)

where, from Hookes law (12),

811 2 2 9 23u171 9 u272812 2 2u172 9 u2713

822 2 u171 9 2 9 23u272 (89)

After substituting (89) into (88) and differentiating with respect to x, one obtains


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2 9 23u17111 9 u17122 9 2 9 3u27112 9 b1171 2 0

2 9 3u17112 9 u27111 9 2 9 23u27122 9 b1271 2 0 (90)

Taking the Fourier transform of (90) with respect to xresults in the following system ofordinary differential equations

d2

d y2u171 9 i p2 9 3 d

dyu271 6 p22 9 23u171 9 i pb11 2 0

2 9 23 d2

d y2

u271 9 i p2 9 3 d

dy

u171 6 p2

u271 9 i p

b12 2 0 (91)

where the Fourier transform of an integrable function f on 1 is defined by

5[f]2p3 2f2p3 2 46

f2x3e6i px d x (92)

and use is made of the property

5[f] 2 i p5[f] (93)

for f a continuous and piecewise smooth function such that f

5 L1213(see [58]). System

(91) is equivalent to a first-order system of ordinary differential equations

Y2 AY 9 B (94)

where

Y 2u1717u2717 d

d yu1717 d

dyu271T

B 2 07 07 6i pb117 6i pb12T

and

A 2

9

0 0 1 0

0 0 0 1

p22 9 23

0 0 6 i p2 9 3

0 p2

9 2 6 i p2 9 3 9 2 0

The general solution of the homogeneous system is


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Yh1 2 i

6A1 9 9 3p2 9 3A2

e6py 6 i A2ye6py

9 i A3 9 9 3p2 9 3A4 epy 9 i A4yepyYh2 2 A1e6py 9 A2ye6py 9 A3epy 9 A4yepy

Yh3 2 i

p A1 622 9 23

9 A2

e6py 9 i p A2ye6py

9 i

p A3 922 9 23 9 A4

epy 9 i p A4yepy

Yh4 2

26

p A19

A23 e6py

6p A2ye

6py

92p A3

9A43 e

py

9p A4ye

py

with Ai2 Ai 2p3,i217 7 4. Then, the general solution of (94) is given by Y2Yh9 P,whereP2p7y32 12p7y37 22p7y37 32p7y37 42p7y3T is a particular solution such thatlimyP2p7y32 0. Thus, the general solution of (91), defined on the upper half plane,which vanishes as y is

u1712p7y3 2 i 6sgn 2p3A1 9 9 3p2 9 3A2

e6py 6 isgn 2p3A2ye6py 9 12p7y3

u2712p7y3 2 A1e6py 9 A2ye6py 9 22p7y3 (95)From (93) we have

u1712p7y3 2 i pu12p7y3 u1722p7y3 2 6 ip

d

d yu1712p7y3

u2712p7y3 2 i pu22p7y3 u2722p7y3 2 6 ip

d

d yu2712p7y3

The above equations together with (89) and (95) lead to

8122p7y3 2 6 ip

d

d yu171 9u271

2 2

A1 6 9 2p 2 9 3A2 9 A2y

e6py 6 i

p

d

dy12p7y3 9 22p7y3

8222p7y3 2 u171 6 ip

2 9 23 dd yu271

2 2i sgn 2p3A1 6 p2 9 3A2 9 sgn 2p3A2y e6py9 12p7y3 6

i

p2 9 23 d

d y22p7y3 (96)


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We can solve (95) for A12p3and A22p3

A12p3

2u2712p7 03 6 22p7 03A22p3 2 6 i p

2 9 3 9 3

6u1712p7 03 9 isgn 2p3u2712p7 03 6 i sgn 2p322p7 03 6 12p7 037and substitute these into (96) to arrive at

8122p7 03 2 22 9 3u2712p7 03 9 isgn 2p3 22 9 23 9 3 6u1712p7 03 6 12p7 037

9 2

93

9 3 22p7 03 6 i

p

d

d y 12p7 03

8222p7 03 2 6 22 9 3u1712p7 03 9 i sgn 2p3 22 9 23 9 3 6u2712p7 03 6 22p7 037

9 2 9 232 9 3 9 3 12p7 03 6

i

p2 9 23 d

d y22p7 03 (97)

Next, we apply the inverse Fourier transform to Equations (97), using

561[isgn 2p3f2p3]2x3 2 1

64

6

f2r3

r6 x dr2 [f]2x3 (98)

The operator [f] defined in (98) is known as the Hilbert transform. This leads us to theso-called Dirichlet-to-Neumannmap:

8122x7 03 2 22

9 3 u2712x7 03 922 9 23

2 9 33 64

6

u1712r7 03 6 12r7 03r6 x dr

9 2

93

9 3 22x7 03 9 d

d y 4 x0 12s7 03 ds8222x7 03 2 6

22

9 3 u1712x7 03 922 9 23

2 9 33 64

6

u2712r7 03 6 22r7 03r6 x dr

9 2 9 232 9 3 9 3 12x7 03 9 2 9 23

d

d y

4 x0

22s7 03 ds (99)

where f2

561[f] denotes the inverse Fourier transform of f. In order to construct the

inverse map of (99), one solves Equations (97) foru171andu172:


40/46


u1712p7 03 2 6isgn 2p3 9 222 9 3 28122p7 03 6 22p7 033 9 122 9 38222p7 03

9

92

22 9 3 12p7 03 9sgn 2p3p dd y 12p7 03 9 ip ddy 22p7 03u2712p7 03 2 6isgn 2p3 9 2

22 9 3 28222p7 03 6 12p7 033 6 122 9 38122p7 039 2 9 3

22 9 3 22p7 03 9sgn 2p3

p

2 9 23222 9 3

d

d y22p7 03

6 ip

229

3

d

d y12p7 03 (100)

After applying the inverse Fourier transform to (100) one obtains the Neumann-to-Dirichletmap:

u1712x7 03 2 6 9 2

22 9 364

6

8122r7 03 6 22r7 03r6 x dr9

122 9 3 8222x7 03

9 9 222 9 3

12x7 03 9 1

d

d y64

6

4 r0

12s7 03 ds drr6x

6 dd y

4 x0

22s7 03 ds

u2712x7 03 2 6 9 2

22 9 364

6

8222r7 03 6 12r7 03r6 x dr6

122 9 3 8122x7 03

9 2 9 322 9 3 22x7 03 9

2 9 23222 9 3

d

d y64

6

4 r0

22s7 03r6 x ds dr

9

22 9 3d

dy 4 x0 12s7 03 ds (101)D. SURFACE FIRST PIOLAKIRCHHOFF STRESS

For purposes of deriving an expression for the surface first PiolaKirchhoff stress tensor, itis most natural to work in a three-dimensional setting. Thus, within this section, we assumethat 1t 13 and that5 tis a two-dimensional surface. The results derived herein are validin the two-dimensional context as well, since the problem we consider can be viewed as athree-dimensional problem, reduced to a two-dimensional problem.


41/46


The notation introduced in Section 2 is used, with s1 an oriented curve in the fracturesurface 51 2t3, parameterized by arc lengthS5 207L 3. Further, letdsandd Sbe small lengthelements in the current and reference configuration, respectively. Then

ds2 dd S6 s1 2S3

d S2 F2s1 2S3321 2s1 2S33 d S2 j d S (102)Here j2F21 is sometimes called the RadonNikodym derivative of the arc length mea-sure ons 2 6 s1with respect to that ons1(see [47]).

One can show that the unit conormal to the image s of the curve s1 in the currentconfiguration is given by the expression

1 2 PtF

6T11

PtF6T11 (103)Indeed, letndenote the unit normal to the surface5 t7and 2tbe the unit tangent to the curves. To show that 1, as given by (103), is conormal to s7it suffices to prove that 1 n20 and1 2t2 0. The first is clearly satisfied since 15 4x, whilen5 4x . For the latter note that,Pt2 PTt andPtF212 F21 , consequently

1 2t2 PtF6T

11

PtF6T11 F21F21

2 11PtF6T11 21F21

2 0

The total force exerted by the material ins9(the part of5 tinto which 1points) on thematerial ins6is4

s

T28 31 24 L

0T28 326 s1 2S33 PtF

6T11

PtF6T11j d S2

4s1

T28 31 11

where

T28 31

2

F21

PtF6T1

1T28 3PtF

6T (104)

is the surface first PiolaKirchhoff stress tensor.11

Proposition 1.Let J2 det2F3, then

F21PtF6T11

2 JF6TN 7 (105)

in particular, (47) is independent of the conormal 11 . Furthermore, ifn2

F6TNF6TN(i.e.nis the unit normal to5t) andd anandd ANare area elements for surfaces in the current(respectively, reference) configuration, normal to nandN, respectively, then


42/46


dan2 JF6TN d AN2 j2d AN7

i.e. j22

JF6TN is the RadonNikodym derivative of the area measure on 5twith respectto that on51 2t3.

Proof.By the spectral theorem [59], there exists a representation ofUin the form

U 2

i ei ei

where B2 ei 7 i2 17 27 3is an orthonormal basis for the vector space , consisting ofeigenvectors ofU, andi are the corresponding eigenvalues. Let [11 ]2 a7 b 7 cT and[N]

2 aN7 bN7 cN

T be the representations of11andN, respectively, relative to the basis B.

Recall that 12 N 8 11 , thereforej 2 F 1

28

212bNc6 b cN32 9 222cNa6 c aN32 9 232cNa6 c aN32 (106)

Furthermore,

PtF6T

11 2 I 6F6TN F6TN

F6T

N2 F

6T11

2U61116 U62N 11U61N2U61N

After some straightforward manipulations one concludes

F 1PtF6T11

28

2223a

2N9 2123b2N9 2122c2N

2 123a2N21

9 b2N22

9 c2N23

2 JF6TN Now, by Nansons formula,

ndan2 JF6TN d AN

Taking the inner product of the above equation with n 2 F6TN

F6TN

yields

dan2 JF6TN d ANwhich concludes the proof. 1


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E. PROOFS OF LEMMAS 1 AND 2

Proof (Lemma 1). Let 91 2t3denote a disk of radius centered at the crack tip c2t3and

moving with it. Let 11 2t3 2 11 2t3 91 2t3. Thend

dt

421 2t3

2421 2t3

464

441 2t3

4c2t3 N

HereNis the unit normal vector to 491 2t3pointing into the bulk material. Using the regu-larity and boundedness of one concludes that

d

dt 421 2t3 2lim0d

dt 421 2t3 2 421 2t3 41

Proof (Lemma 2). Let l2t3 be a curve in 12 parameterized by l2s7 t3, 0 s 1 witha2t3 2 l207 t3andb2t3 2 l217 t3. Then the standard transport theorem yields

d

dt

4l2t3

24

l2t3

49 2a2t33 4a Na9 2b2t33 4b Nb

whereNa andNbare the unit tangent vectors to l 2t3at a2t3and b2t3respectively. Now letl2t3 2 21

51 2t3. Thena2t3 2 b2t3 2 c2t3and4a Na24b Nb2 4c2t3, which concludes

the proof. 1

NOTES

1. An interesting exception is the recent work by Ortiz and colleagues [13] to derive universal forms forcohesive laws in crystalline materials. However, their studies consider the work of decohesion requiredto separate infinite, parallel planes of atoms in the renormalized limit of infinitely many layers of atoms.It is not clear how appropriate that is for modeling the work required to open the short (atomic-scale)cohesive zone in a brittle crystal for which the assumption of infinite parallel planes of atoms seems ofdubious validity.

2. The subscripttis suppressed in the static case.3. See Appendix A for the definitions of surface gradient and surface divergence.4. For a derivation of the jump momentum balance in the current configuration, see Appendix A.5. See Appendix A.6. If the potential becomes singular as the distance between the points tends to zero, hard-sphere approxi-

mation is used.7. An alternative approach to modeling crack propagation takes the body to be the bulk material together

with the gas phase. In this case no points are added to the boundary and thus one does not need

to consider an evolving reference configuration. However, in this approach one must account for themass transfer between the bulk material, the dividing surface, and the fracture tip. In essence, fracturepropagation is viewed as a chemical reaction occurring at the fracture edge.


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8. The derivation forT28 31 presented herein corrects a small mistake in [47] in the formula, correspondingto (103), relating the conormal in the current to the conormal in the reference configuration. For thisreason, the expression for T28 31 derived here differs slightly from the formula given in [47].

9. Owing to the fact that the upper and lower fracture surfaces meet at an angle (strictly less than ), thefracture tip (in the case of three dimensions, the fracture edge) is viewed as a common point (in thecase of three dimensions, a common line), endowed with mass density.

10. Here822denotes the non-dimensionalized tensile stress.11. The formulas (103) and (104) should be compared with (5.23) and (5.24) in [47] which we believe to be

in error.

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