- No tags were found...

Grinfeld instability on crack surfaces - JuSER - Forschungszentrum ...

PHYSICAL REVIEW E, VOLUME 64, 046120

**on** **crack** **surfaces**

R. Spatschek and Efim A. Brener

Institut für Festkörperforschung, **Forschungszentrum** Jülich, D-52425 Jülich, Germany

Received 1 February 2001; published 24 September 2001

The surface of a propagating **crack** is shown to be morphologically unstable because of the n**on**hydrostatic

stresses near the surface Asaro-Tiller-

**crack** becomes smaller than the energy of a straight **crack** if the **crack** length exceeds a critical length L c

5.18L G (L G is the Griffith length. We analyze the dynamic evoluti**on** of this

diffusi**on** or c**on**densati**on** and evaporati**on**. It turns out that the curvature of the **crack** surface becomes divergent

near the **crack** tips. This implies that the widely used c**on**diti**on** of the disappearance of K II , the stress

intensity factor of the sliding mode, is replaced by the more general requirement of matching chemical

potentials of the **crack** **surfaces** at the tips. The results are generalized to situati**on**s of different external loading.

DOI: 10.1103/PhysRevE.64.046120

PACS numbers: 62.20.Mk, 46.50.a, 81.40.Np

I. INTRODUCTION

The uniform moti**on** of a straight **crack** is well understood

1: A **crack** exceeding a certain critical length, the Griffith

length, starts to grow, since the energy gain due to elastic

relaxati**on** is bigger than the loss of surface energy that appears

as a c**on**sequence of the el**on**gati**on** of **crack** **surfaces**.

However, in experiments the **surfaces** of a **crack** are often

rough 2. Some of these results are interpreted in the framework

of models of **crack**s propagating in heterogeneous media.

The other possibility for the roughening of the **crack**

**surfaces** is the **on** of the **crack**

tip. Recent experiments revealed that many puzzling phenomena

in brittle fracture are related to an oscillatory

at velocities appreciably below the Rayleigh speed 3.

There were several attempts in literature to investigate the

stability of a propagating **crack**. The linear stability analysis

of the quasistatic **crack** subject to mode I opening mode

loading has been performed by Cotterell and Rice 4 with

subsequent refinement by Adda-Bedia and Ben Amar 5.

They employ the Griffith theory and the so-called principle

of local symmetry, i.e., the c**on**diti**on** that mode II sliding

mode stress intensity factor K II vanishes at the tip of the

**crack**. They found that the straight moti**on** of the **crack** becomes

unstable if the tangential loading exceeds a critical

value.

A full dynamical model, including the microscopic descripti**on**

of the cohesive z**on**e around the **crack** tip, has been

developed by Ching, Langer, and Nakanishi 6. The cohesive

force in the neighborhood of the tip provides a fracture

energy and a mechanism for regularizing the stress singularity;

this model removes the need to speculate about a principle

of local symmetry. In additi**on** to Refs. 4,5, they

found a str**on**g microscopic

**crack** velocities, which depends very sensitively **on** tiny details

of the cohesive-z**on**e model. However, later Langer and

Lobkovsky 7 showed that these cohesive-z**on**e models in

the framework of a sharp-tip representati**on** lead to unphysically

unreas**on**able features of the elastic stresses in spite of a

regularizati**on** by cohesive forces.

We str**on**gly point out that in all these descripti**on**s a **crack**

is recognized as the trace left behind by the propagating

**crack** tip. In this sense its **surfaces** are ‘‘frozen’’ and not

subject to any additi**on**al dynamics.

On the other hand, another elastic

much interest in the recent time:

Tiller 8 discovered that the energy of a n**on**-hydrostatically

stressed solid with a flat surface can be diminished by a

change of its shape and formati**on** of deep grooves 9. This

deformati**on** is not due to elastic strain but to redistributi**on**

of matter al**on**g the surface. Apart from surface diffusi**on**

other transport mechanisms can be taken into account as

well: For example, a solid phase that is in c**on**tact with its

melt grows due to melting and recrystallizati**on**. Similarly an

evaporati**on**-c**on**densati**on** mechanism is also c**on**ceivable.

In all cases the central reas**on** for this

drastic decrease of elastic energy during the deformati**on** process.

This decrease is bigger than the accompanying increase

of surface energy for relatively l**on**g-wave interface perturbati**on**s.

where he described the temporal evoluti**on** of a curved interface

shape y(x,t)y 0 exp(ikxt). The line y0 corresp**on**ds

to the initial unperturbed interface of a twodimensi**on**al

body in the x-y plane.

In the case of surface diffusi**on** the time-evoluti**on** is governed

by the dispersi**on** relati**on**

2 Dv s k 2 0 2 1 2

kk 2,

E

where D is proporti**on**al to the surface diffusivity and v s the

atomic volume. E and are the Young and Poiss**on** coefficients

respectively, is the surface tensi**on**. 0 nn

, with normal and tangent directi**on**s n and , reflects

the n**on**hydrostaticity of the stress tensor . Here **on**e can

easily see that l**on**g-wave perturbati**on**s lead to the

**on**s are hampered

by surface tensi**on**. The most unstable mode k m 3 2 0 (1

2 )/2E evolves with m Dv s k 4 m /3.

Recently, it was emphasized in Ref. 10 that the c**on**diti**on**

of n**on**vanishing 0 is fulfilled **on** the cut interfaces of a

straight **crack** that is loaded perpendicular to the **crack** at

infinity. In this sense a straight **crack** cannot be a stable c**on**figurati**on**

under all circumstances, because slight deforma-

1

1063-651X/2001/644/04612013/$20.00

64 046120-1

©2001 The American Physical Society

R. SPATSCHEK AND EFIM A. BRENER PHYSICAL REVIEW E 64 046120

FIG. 2. Deformati**on** of the **crack** shape as a result of a reshuffling

of matter. This can happen either directly through the **crack** or

al**on**g the **crack** **surfaces** and around the tips.

FIG. 1. Geometry of a wavy **crack** in a two-dimensi**on**al solid.

ti**on**s may reduce the total energy. In Ref. 10 it was predicted

that this can happen provided that a certain critical

**crack** length is exceeded.

The aim of the current paper is a deeper understanding of

this

expressi**on**s to compare the energy of a static straight

and a wavy **crack**. In Sec. III the dynamics of **crack** deformati**on**

bey**on**d the threshold of

more detailed investigati**on** of the behavior near the singular

**crack** tips is performed, and the situati**on** is generalized to

not necessarily parallel **crack** **surfaces**. In Sec. IV we c**on**sider

different loading mode situati**on**s. Appendix A c**on**tains

a soluti**on** of the elastic problem of a **crack** with independent

**surfaces**. Appendix B is a proof for the equivalence of two

representati**on**s of the elastic energy that are derived in

Sec. II.

II. STABILITY ANALYSIS

As menti**on**ed above a **crack** with a length different from

the Griffith length wants either to grow until the whole material

is fractured into pieces or to shrink until it disappears

completely. In order to study the quasistatic kinetics of the

**on** must be suppressed.

Formally we fix the tip positi**on**s of the **crack** and

**on**ly discuss shape deformati**on**s.

The key questi**on** is whether a straight **crack** is stable with

respect to small perturbati**on**s of its shape.

For the moment we restrict our c**on**siderati**on**s to the case

of parallel **crack** **surfaces** mathematical **crack**. Thus we can

describe the **crack** shape by a functi**on** y(x), with Lx

L see Fig. 1. The tips are located at xL, y0 and

the straight **crack** corresp**on**ds to the shape functi**on** y(x)

0. The goal of this secti**on** is to derive expressi**on**s for the

energy change due to shape deformati**on**s UyU0. This

functi**on**al depends in a complicated, n**on**local way **on** the

functi**on** y(x). We normalize U00. Since the homogeneous

external stress yy P acts perpendicular to the

straight **crack**, it is clear from symmetry that two **crack**s y(x)

and y(x) must have the same energy. Thus O(y 2 ) is the

lowest n**on**vanishing c**on**tributi**on** to Uy.

From now **on** we assume the special case of a twodimensi**on**al

plane-strain situati**on**.

The basic idea to derive the change of energy during deformati**on**

of the **crack** shape is founded **on** the expressi**on** for

the chemical potential of the solid phase at an interface 11

s v s f s0 12

2E nn 2 . 2

Here v s is the atomic volume of the solid phase; f s0 is the

free energy density for a hydrostatic situati**on**; is the surface

energy; is the curvature of the interface counted positive

for a c**on**vex solid.

In principle **on**e has to remove matter from **on**e fr**on**t of

the **crack** and deposit it at the opposite **on**e. In this way the

originally straight **crack** is deformed to its final wavy shape.

This procedure is outlined in Fig. 2.

Later different mechanisms for this transport process will

be discussed: The removed matter can either cross directly

through the interior of the **crack** or diffuse al**on**g the **surfaces**.

The first case corresp**on**ds to a evaporati**on**-c**on**densati**on**

process; in the sec**on**d case of surface diffusi**on** the matter

must wander around the **crack** tips. For energetic c**on**siderati**on**s

the precise transport process is of course irrelevant; it

becomes important later for dynamical approaches.

An easy way to calculate the energy of a perturbed **crack**

was proposed in Ref. 10. The total energy change c**on**sists

of a change of both the surface energy and the elastic energy.

First a deformati**on** increases the arc length of the **crack**

and therefore the surface energy. To the lowest n**on**vanishing

order this change is given by

L y

U s 2 2

dx.

L 2

The factor 2 appears because the **crack** c**on**sists of two interfaces.

Additi**on**ally, the change of geometry also alters the

stored elastic energy. We calculate this energy change in two

3

046120-2

GRINFELD INSTABILITY ON CRACK SURFACES PHYSICAL REVIEW E 64 046120

steps: First we keep the stress field of the straight **crack** artificially

fixed in the whole solid while the material is reshuffled

al**on**g the two sides of the **crack**. One uses the expressi**on**

for the chemical potential 2 to calculate the

required energy. When the final shape is reached, the usual

c**on**diti**on** of vanishing normal and shear stress al**on**g the

**crack** **surfaces**, nn n 0, is clearly violated. Therefore

**on**e has to adjust the stress appropriately by adding a compensati**on**

field (1) and book keeping the relaxati**on** of energy.

Let us begin with the first c**on**tributi**on** to the elastic energy.

The analytic c**on**tinuati**on** of the stress field in the vicinity

of a straight **crack** up to first order is given by see for

example 12

xx P 1

2L2 y

L 2 x 2 3/2Oy 2 ,

We imagine to create this **crack** by cutting the solid, starting

at the left tip xL and proceeding to xL. The

associated mode II stress intensity factor K II is calculated in

4. Assume that the current **crack** tip positi**on** is xL with

LLL, it is given by

K II L 2

LL P L

xL

Lx

yxdx. 7

Lx

The energy release rate coming al**on**g with an increase of the

**crack** length by dL is given by Irwin’s formula 13

U elu 12

E

L

2 KII LdL. 8

L

Finally the total energy of the wavy **crack** is given by the

sum

yy 0Oy 2 ,

UyU s U els U elu .

9

xy 0Oy 2 .

The minus branch in xx corresp**on**ds to the upper, the plus

branch to the lower **crack** surface. Therefore the elastic c**on**tributi**on**

to the chemical potential 2 at the upper **crack** surface

is the shape-independent parts are irrelevant and therefore

neglected

4

s u 2v sP 2 1 2 L 2

EL 2 x 2 3/2 y. 5

At the lower interface s l s u . We note that due to

the fixing of the stress field the chemical potential is also

fixed during the redistributi**on** of matter. It results in

L

U els 2

L

y(x)

s u

y0 v s

dy dx

L 4P

2 1 2 L 2 y 2

dx

L EL 2 x 2 3/2 2

2P2 1 2 L

dxy 2 L 2 x 2 .

E L

As **on**e can see from the sec**on**d representati**on** this energy

c**on**tributi**on** is always positive, i.e., stabilizing. The last representati**on**

results from integrati**on** by parts which requires

the boundary c**on**diti**on**s y(L)0 and corresp**on**ds to fixed

**crack** tips.

The sec**on**d c**on**tributi**on** to the change of the elastic energy

comes from the adjustment of the stress field. From Eq.

4 it follows that after performing the first step the shear

stress al**on**g the new **crack** surface y(x) is n Py

O(y 2 ). To this order the normal comp**on**ent is already correct:

nn 0. The compensati**on** field (1) introduced above

must cancel these boundary values and vanish at infinity. It

corresp**on**ds to the stress field of a straight **crack** with surface

tracti**on**s T s Py and T n 0.

6

Alternatively, and more intuitively, the total energy of the

**crack** can be calculated as follows: As before **on**e can imagine

to reshuffle the matter al**on**g the **crack** **surfaces** to obtain

the final wavy shape, but this time the stress field is not fixed

during the redistributi**on**. At each step during this process the

c**on**diti**on** of vanishing normal and shear stress at the **crack**

**surfaces** must be fulfilled. This requires the soluti**on** of the

elastic problem of a wavy **crack**; its soluti**on** is derived in

Appendix A for the generalized situati**on** of not necessarily

parallel **crack** **surfaces**. Inserting these expressi**on**s A14–

A16 into the chemical potential 2 gives to first order in y

s u/l v s f s0 12 4P

2E 2

P 2

L 2 x 2 1/2

L ytL

P 2 t 2

dt

L tx

4P2 L 2

L 2 x 2 3/2 yx yx

10

P denotes the principal value of the integral. In c**on**trast to

the former approach, the stress distributi**on** changes during

the rearrangement, and therefore the chemical potential 10

depends **on** the ‘‘intermediate shape’’ y(x). The total energy

can now be obtained by integrati**on**

L

U

L

y(x)

s x u s x l

dy dx, 11

y0 v s

similar to Eq. 6. This results in a completely different representati**on**

of U elu .

Though it is clear from physical reas**on**s that both approaches

should give the same result, this is not directly

visible from the expressi**on**s. In particular, two **crack**s y(x)

and y(x) should have exactly the same energy. Integrating

the chemical potential 10 clearly reflects this situati**on**.

Also U s and U els obey this symmetry but this property is

046120-3

R. SPATSCHEK AND EFIM A. BRENER PHYSICAL REVIEW E 64 046120

less clear for U elu . Since we started to cut the solid at its left

tip for derivati**on** of U elu , the resulting formulas 7 and 8

seem to violate this parity invariance at a first glance. Nevertheless

it is possible to give a proof of the equivalence of

these two different approaches to calculate the energy of the

system. It is sketched in Appendix B. The idea is to rederive

the expressi**on** for the chemical potential 2 from the total

energy 9, which proves the equivalence of both ways. C**on**sequently

the symmetry c**on**diti**on** is also fulfilled, or, in

other words, the energy operator U commutes with the parity

operator Pˆ , which is defined by (Pˆ y)(x)y(x).

A. Numerical results

Based **on** the energy expressi**on**s derived in the previous

secti**on** we are now able to perform a full stability analysis of

the problem. We remind that we assigned U0 to the

straight **crack** and hence are interested in the occurrence of

Uy0 for a wavy **crack** shape. One can easily check that

all parameters of the problem appear **on**ly in the combinati**on**

L G 2E/(1 2 )P 2 Griffith length in the total energy

9, apart from comm**on** prefactors. Thus the minimum energy

with respect to all possible **crack** shapes with a certain

length L is simply a functi**on** of **on**e single parameter L/L G ,

which easily allows to trace the threshold of

In Ref. 10 minimizati**on** has been performed by a variati**on**al

procedure using **on**ly a limited set of analytical representati**on**s

of shape functi**on**s. Here we solve the problem

numerically and find the real shape without such restricti**on**s

by a full minimizati**on** procedure. For that we have chosen

a Fourier representati**on** of the **crack** functi**on** y(x),

xL,L

yx

k1

b k sin kxL

, 12

2L

where the upper summati**on** limit is replaced by a sufficiently

large cutoff K. Since this is linear in the unknown coefficients

b k and the total energy is quadratic in the amplitude

y(x), we can write the total energy as quadratic form U

D ik b i b k with a real, symmetric KK matrix D that depends

**on**ly **on** L/L G . This matrix can be computed almost

completely analytically.

In order to find the minimum of the free energy U, a

normalizati**on** c**on**diti**on** is needed, since the amplitude is not

restricted in our lowest order calculati**on**. The choice b k b k

L G 2 is c**on**venient but arbitrary. In this case the minimum of

energy is exactly s L G 2 , where s is the smallest eigenvalue

of the matrix D.

It must be remarked that the threshold of

U(L)0 is not affected by the specific choice of normalizati**on**:

Minimizati**on** requires

yx Uy˜ f y0,

13

where f y0 is the arbitrary normalizati**on** c**on**diti**on**,

coupled to the energy by the Lagrange multiplier ˜ . By c**on**structi**on**

this is equivalent to

FIG. 3. Minimum normalized energy of a curved **crack**.

2 v s

s u ˜ f

y 0,

or, by multiplicati**on** with y(x) and integrati**on**

14

U

˜

. 15

L f yx

L y 2 dx

Thus for the threshold of

we have ˜ 0 and the normalizati**on** c**on**diti**on** becomes irrelevant

in Eq. 13. C**on**sequently the critical length of the

**crack** and its corresp**on**ding shape are universal. For any

other length the results depend **on** the normalizati**on** c**on**diti**on**.

Later the physical meaning of the c**on**diti**on** chosen here

will become more obvious.

The c**on**crete normalizati**on** c**on**diti**on** given above is

equivalent to

L

y 2 xdxLL 2 G .

L

Therefore Eq. 15 reads

˜ U

LL G

2

and the minimizati**on** c**on**diti**on** is equivalent to

2 v s

s u/l 2˜ yx0.

16

17

Figure 3 shows the minimum energy versus **crack** length.

For L5.188L G the straight **crack** becomes unstable and favors

a wavy shape. The critical shape is plotted in Fig. 4.

All results turn out to be very robust, and already K

20 harm**on**ics are sufficient to describe the shape quite

accurately. The code has been checked very carefully against

analytically known energy values for special shapes. The result

is c**on**sistent with the predicti**on** L c 6L G in Ref. 10.

As we have already seen, the energy operator commutes

with the parity operator Pˆ and therefore all eigenfuncti**on**s

046120-4

GRINFELD INSTABILITY ON CRACK SURFACES PHYSICAL REVIEW E 64 046120

From the representati**on** of the chemical potential, explicitly

given in Eq. 10, it is clear, that Eq. 18 is of fourth order

with respect to the spatial derivatives. Basically we are interested

in a stability analysis, and therefore we look for

eigenfuncti**on**s of this equati**on**, y(x,t)y(x)exp(t). This

leads to

yxD 2 ˆ sy,

19

where we write ˆ s for the linear operator of chemical potential

that depends n**on**locally **on** the shape y(x).

A simpler dynamics is described by the equati**on**

yx,t

D

t ec s x,t,

20

FIG. 4. Universal shape of the critical **crack**, L5.1882L G .

are either even or odd. It turns out that the most unstable

mode, which bel**on**gs to the smallest eigenvalue, is an even

functi**on**. Thus **on**ly terms with odd k appear in the representati**on**

12. Yet there are still certain length intervals where

the optimum soluti**on** is an odd functi**on**.

III. DYNAMICS OF GRINFELD INSTABILITY

In this secti**on** we go bey**on**d the previous static descripti**on**

where we used the energy to judge whether a certain

c**on**figurati**on** is stable or not. Here we analyze how a given

shape develops in time. If it decays to the straight **crack**, the

**crack** is stable; otherwise a perturbati**on** develops further and

further. This allows to calculate the threshold of

a different way and to compare the results with the predicti**on**s

of the previous secti**on**. Again we calculate the chemical

potential **on**ly up to first order which corresp**on**ds to a

quadratic energy. In this sense we cannot expect to obtain

new results about the l**on**g-time behavior of unstable soluti**on**s.

Especially we cannot describe the known groovelike

structures that are governed by n**on**linear effects 9. Nevertheless

this approach is useful because it allows to study

more carefully the behavior near the **crack** tips. This regi**on**

cannot be described by the previous static approach since the

Fourier representati**on** produces str**on**g oscillati**on**s there. Instead,

we use a real space representati**on** here. By c**on**centrating

more grid points in the sensitive tip regi**on** we are

able to study the peculiarities occurring there quite accurately.

Again we start with the special case of coherent **surfaces**

y u (x)y l (x). Later we will generalize this situati**on**.

Redistributi**on** of matter is driven by spatial variati**on**s of

the chemical potential of the solid phase al**on**g the **crack**

**surfaces**. We analyze two different transport mechanisms in

this secti**on**: Surface diffusi**on** is described by the equati**on**

yx,t

D 2

t

s x,t.

18

with the kinetic coefficient D ec . It corresp**on**ds to a direct

transport of matter through the void of the **crack**. It is similar

to evaporati**on**-c**on**densati**on** EC processes known, e.g.,

from the theory of phase separati**on**. We mainly introduce

this mechanism because of its simplicity that is useful for

testing purposes of the numerical code.

In both languages the threshold of **on**ds

to 0 and should be the same. As before, the equati**on**s of

moti**on** depend **on**ly **on** the single adjustable parameter

L/L G . Thus the different eigenvalues are also **on**ly a functi**on**

of this parameter, and by simple plotting **on**e can easily

detect the crossing point (L c /L G )0.

Both mechanisms are purely dissipative and we expect

that all eigenvalues are real. One can readily check that the

operator ˆ s fulfills the self-adjointness c**on**diti**on** (ˆ sy 1 ,y 2 )

(y 1 ,ˆ sy 2 ) with respect to the standard scalar product

(y 1 ,y 2 ) L

L y 1 (x)y 2 (x)dx. Therefore at least the sec**on**d

mechanism allows **on**ly real lambdas.

For the diffusi**on** process we note that the operator 2 is

positive definite; **on**e can prove that under these circumstances

the compound operator 2 ˆ s indeed has **on**ly real

eigenvalues 14.

Since the eigenvalue equati**on** for the meltingrecrystallizati**on**

mechanism is a sec**on**d order ordinary

integro-differential equati**on**, we require two boundary c**on**diti**on**s.

As before we demand y(L)0. Surface diffusi**on**

requires two additi**on**al boundary c**on**diti**on**s. Since the

chemical potential **on** the upper and lower surface of the

**crack** apart from the trivial c**on**stant c**on**tributi**on**s in Eq.

10 that we ignore from now **on** is the same but with opposite

sign, the c**on**diti**on** of a unique value of s requires

s (L)0. Otherwise a fast redistributi**on** of matter, driven

by the force difference s u (L) s l (L) would take

place in the microscopic regi**on** around the **crack** tips, until

the above c**on**diti**on** is satisfied. In both cases a critical **crack**

is characterized by the c**on**diti**on** s (x)0.

We use the notati**on** Y (y,y, s , s ) and discretize

these functi**on**s **on** the interval L,L. Then the linear

equati**on** 19, together with the boundary c**on**diti**on**s, can be

expressed as A•Y 0 with a real, quadratic matrix A that

depends **on** the c**on**trol parameter L/L G and **on** the eigenvalue

. Since this equati**on** is n**on**local the matrix is not

sparse; n**on**trivial soluti**on**s corresp**on**d to the c**on**diti**on**

det A0, and we use a standard matrix decompositi**on** to

046120-5

R. SPATSCHEK AND EFIM A. BRENER PHYSICAL REVIEW E 64 046120

FIG. 5. Eigenvalues of the surface diffusi**on** vs **crack** length. We

detect n**on**trivial soluti**on**s through a change of sign of the determinant.

Whenever two curves intersect or come close to each other,

this sign does not change. This leads to missing points in the

diagram.

detect it 15: AQ•R, where R is an upper triangular matrix,

Q is orthog**on**al and positive definite, and thus det A

i R ii .

The allowed eigenvalues as a functi**on** of the **crack** length

are shown in Figs. 5 and 6.

To each length bel**on**gs an infinite number of eigenfuncti**on**s

with different eigenvalues; here **on**ly the biggest eigenvalues

near the threshold of

inspecti**on** shows that the first crossing of the 0 axis happens

at L c 5.187L G , which is in excellent agreement with

the previous, static predicti**on**.

Some of the shape functi**on**s are illustrated in Fig. 7.

All eigenfuncti**on**s are even or odd, but it turns out that

not always the even functi**on** is the most unstable **on**e. One

can clearly see that the two biggest eigenvalue functi**on**s of

surface diffusi**on** intersect at around L5.3L G and again at

L6.8L G ; in between the odd branch is the most unstable

**on**e. This phenomen**on** occurs again at bigger **crack** lengths

and also for the EC mechanism.

In both cases the most unstable modes are functi**on**s that

c**on**sist of **on**ly **on**e-half or **on**e full period. With descending

eigenvalue the number of nodes increases. In this sense the

FIG. 7. Some eigenfuncti**on**s of surface diffusi**on**. a L

4.26L G ,0.047, b L5.18L G ,0.096, c L

6.00L G ,0.167, d L7.25L G ,0.167. is given in

units of Dv s /L 4 G .

FIG. 6. The same relati**on** for the EC mechanism.

eigenvalue problem is comparable to simple quantum mechanical

eigenstates, e.g., of a single particle in a box.

It is also instructive to use a simplified scaling analysis of

the situati**on**: The dispersi**on** relati**on** 1 of the free interface

defines a characteristic wavelength L c k c 1 E/(1

046120-6

GRINFELD INSTABILITY ON CRACK SURFACES PHYSICAL REVIEW E 64 046120

2 ) 0 2 . The tangential stress at the **crack** **surfaces** is

0 P. Thus L c L G which underlines the fact that

**on**diti**on** for **crack** growth

are basically of the same origin, namely, the competiti**on**

between surface and elastic energy. Since the tip positi**on**s

are fixed, **on**ly certain perturbati**on** waves fit into this interval.

In particular, a minimum **crack** length LL G must be

exceeded to allow at least for **on**e unstable mode.

For the critical **crack** with LL c , the shape is indistinguishable

from the static picture 4 for both mechanisms. We

note that also for other lengths the shape functi**on**s of the EC

mechanism are very similar to those of the diffusi**on** mechanism.

If **on**e plugs in the ansatz y(x,t)y(x)exp(t) into the

equati**on** of moti**on** 20, **on**e immediately arrives at the c**on**diti**on**

17 for the minimum of the static energy of the last

secti**on** with the amplitude c**on**straint applied there. Now we

see that this arbitrarily chosen c**on**straint is related to the EC

mechanism if we identify

Uv sD ec

LL G

2

21

with the energy U of the normalized **crack**. First, it is clear

from this equati**on** that a stable **crack** in the static sense with

U0 is also stable in the dynamic sense, 0, and vice

versa. Furthermore, the lower the **crack** energy for a given

length, the faster the **on** 21

also holds numerically: Mapping the two graphs Figs. 3 and

6 using Eq. 21, lets the energy curve exactly c**on**ceal the

curve of the most unstable eigenvalue.

In the case of very l**on**g **crack**s the spectrum becomes

more and more c**on**tinuous and finally coincides with the

spectrum 1, since the boundary c**on**diti**on**s become less important.

All eigenvalues that are smaller than the maximum

value of the **on**s.

We indeed observe a very good agreement with this

expectati**on** for L100L G in our numerical calculati**on**s; in

particular, the discrete eigenvalues of the finite geometry become

very dense and hard to separate. Thus we are able to

reproduce the dispersi**on** relati**on** 1 for the case of independent

interfaces.

A. Near-tip behavior

The tip of a **crack** is typically subject to divergencies. For

example, the stress field behaves like r 1/2 , where r is

the distance from the tip. From Eq. 10 it follows readily

that the different c**on**tributi**on**s to chemical potential behave

like

sur yx,

els L 2 x 2 3/2 yx,

elu L 2 x 2 1/2 P

L

L L 2 t 2

yt dt.

tx

22

23

24

FIG. 8. C**on**tributi**on**s to the chemical potential of a critical

**crack** with L5.187L G ,0,M400.

For LL c , they must cancel each other everywhere **on** the

**crack**: s els elu 0. It is difficult to justify this cancellati**on**

for the Fourier representati**on** 12 of the previous

secti**on** because of str**on**g oscillati**on**s near the **crack** tip,

though it holds nicely in the center regi**on** x0. The reas**on**

is that though the full Fourier series with summati**on** cutoff

K→ is complete, K always implies that n**on**vanishing

values of y(L) can **on**ly be achieved asymptotically. With

the real space representati**on** we do not suffer from this problem,

and indeed this cancellati**on** seems to be verified, see

Fig. 8.

In Ref. 10 the guess was made that the divergent c**on**tributi**on**s

of elu and els cancel each other at the **crack** tip.

This is equivalent to a vanishing of the total mode II stress

intensity factor K (tot) II . The expressi**on** for this value has been

derived in Ref. 4 for the case of a slightly wavy **crack**. To

first order in y it is given by 5

K (tot) II K II LLPyL/2.

25

It reflects exactly the decompositi**on** of the elastic field into a

stabilizing and unstabilizing part as used in the first secti**on**

of this paper. The divergent comp**on**ent of the tangential

stress al**on**g the **surfaces** of a wavy **crack** is given by

(sing) 2K (tot)

II

2r 1/2

26

in the close vicinity of the tip. One can easily check that the

sec**on**d c**on**tributi**on** to K (tot) II produces exactly the divergent

part of the stress field 4 and therefore of the chemical potential

els in Eq. 23.

It is clear from the representati**on** 22–24 above that a

cancellati**on** of the divergencies of elu and els is equivalent

to a finite **crack** curvature at the tips and, from Eq. 26,

also to the vanishing of the stress intensity factor K (tot) II . The

latter criteri**on** is widely discussed in literature as a criteri**on**

for the directi**on** of **crack** propagati**on**, referred to as the ‘‘criteri**on**

of local symmetry.’’ It states that a **crack** propagates in

046120-7

R. SPATSCHEK AND EFIM A. BRENER PHYSICAL REVIEW E 64 046120

the directi**on** of maximum energy release, maximum hoop

stress or stati**on**ary Sih energy density factor see references

in 4 for a survey.

Surprisingly the calculati**on**s show that this criteri**on** is not

fulfilled here. It turns out that a divergent curvature c**on**tributi**on**

remains. Calculati**on**s with extreme accuracy raise the

claim that y(r)rr at the tips of a critical **crack** with L

5.187L G , with 1.5 see Fig. 9. For these calculati**on**s

it is necessary to c**on**centrate as many grid points as possible

in the vicinity of the tips where the shape functi**on**s vary

crucially, whereas a moderate accuracy suffices in the middle

part.

The reas**on** for this unexpected behavior is that the

K (tot) II 0 criteri**on** maximizes **on**ly the release of elastic energy,

and does not take surface energy into account; it simply

compares different directi**on**s of el**on**gati**on** and leaves behind

the **crack** as the track of the tip. In our case, the el**on**gati**on**

is completely forbidden, but we allow for a deformati**on**

of the already existing **crack**. Therefore completely

different **crack** shapes are compared to each other to minimize

the total energy. For fast **crack** growth the deformati**on**

can be neglected since it is driven by the slow surface diffusi**on**,

but it is still an unanswered questi**on** how K (tot) II 0 and

c**on**st can be rec**on**ciled for a slow moti**on** of the tip,

when both processes, **crack** propagati**on** and

are present.

B. **on** incoherent **crack** **surfaces**

In this subsecti**on** we give up the restricti**on** of parallel

**crack** **surfaces**. Both **surfaces** are described by independent

shape functi**on**s y u (x) and y l (x), provided that y u/l (L)

0. Of course it must be assured that the two branches of

the **crack** do not overlap, i.e., y u (x)y l (x). However, a

small opening of the **crack** is present due to the applied loading

even in the case of a straight **crack**. C**on**sequently a small

perturbati**on** with a sufficient small amplitude does not lead

to an intersecti**on**.

The dynamics of both interfaces is again described by

surface diffusi**on** ẏ u/l D 2 u/l and the boundary c**on**diti**on**s

u L l L, y u/l L0 27

and also flux c**on**servati**on**

u L l L.

This implies the mass c**on**servati**on** law

L

y u xy l xdxc**on**st,

L

28

29

which is trivially fulfilled for parallel **crack** **surfaces**.

For the moment we restrict our c**on**siderati**on**s to antiparallel

**crack** **surfaces** y u(x) y l(x) y(x). In this case u (x)

l (x)(x) holds everywhere and thus the c**on**diti**on**

27 is satisfied automatically. Equati**on** 28 requires the disappearance

of the tip flux (L)0.

We analyze this problem in the same way as before. It

turns out that the spectrum is very similar to that of coherent

**crack** **surfaces**. The main result is that the critical length is

slightly bigger in this arrangement: L c (anti) 5.212L G . It

means that during a slow el**on**gati**on** of the **crack** the parallel

modes will become visible first.

It is now easy to generalize this statement to arbitrary

**crack** **surfaces** y u/l . Since we are basically interested in the

threshold of

of the last secti**on**. To understand this behavior, we decompose

the shape functi**on**s into a parallel and an antiparallel

c**on**tributi**on**:

y p ªy u y l /2,

y a ªy u y l /2.

From Eqs. 2 and A16 it follows readily that the chemical

potential decomposes similarly,

with

u x p x a x,

l x p x a x,

p xv s 12

2E 4P2 L 2

L 2 x 2 3/2 y px

30

31

4P 2

L 2 x 2 P L yt

p 1/2 L tx L2 t dt

2

y p x,

a xv s f s0 12

2E P 2 4P2 L 2

L 2 x 2 3/2 y ax

4P2

P L y a t

L tx dt y a x.

32

33

The crucial observati**on** is that the total energy

U 1

2v s

L L

u xy u

L

xdx l xy l

L

xdx

34

becomes diag**on**al in this representati**on**:

L

U 1 v s

p y p a y a dx.

L

35

Thus the parallel and antiparallel c**on**figurati**on**s are the

‘‘principal axes’’ of the energy ellipsoid. We can therefore

c**on**clude that the parallel arrangement of **surfaces** indeed

gives the most unstable c**on**figurati**on**.

046120-8

GRINFELD INSTABILITY ON CRACK SURFACES PHYSICAL REVIEW E 64 046120

FIG. 9. Logarithmic plot of the curvature near the tips of a

critical **crack**, L5.187L G ,0. All curves corresp**on**d to the

same number of grid points but different spatial distributi**on**s. The

solid line possesses the least grid point density near the tips, the

dotted **on**e the highest. With increasing accuracy the coefficient

comes closer and closer towards 1.5.

IV. CRACKS UNDER GENERALIZED MODE I LOADING

So far in all situati**on**s a pure uniaxial stress P0, perpendicular

to the **crack**, has been exerted **on** the solid. Now

we add an additi**on**al stress comp**on**ent P x at infinity parallel

to the **crack** (x directi**on**. In the framework of a linear theory

of elasticity this homogeneous field is simply added to the

former field. Since it gives a new c**on**tributi**on** to the shear

tracti**on** of a wavy **crack**, the elastic energy is modified. We

declare P x to be positive for a tensile stress and introduce the

dimensi**on**less parameter P x /P. One easily derives

U els 1U els 0,

U elu 1 2 U elu 0.

36

Hence the critical length becomes a functi**on** of as well;

this dependence is plotted in Fig. 10. For 0 we retain the

former result L c 5.18L G .

The most interesting range is 01: The pure tensile

loading perpendicular to the **crack** causes a tangential stress

xx P al**on**g a straight **crack**. The additi**on**al loading P x

0 reduces this value to 0 xx PP x 0 and thus

hampers the evoluti**on** of the

the critical length increases in comparis**on** to 0. For

→1 the n**on**hydrostaticity vanishes completely and the

**on**sequently the critical length

diverges. In other words, the change of the elastic energy

during redistributi**on** of matter is at least of order O(y 4 )in

the deviati**on** from the straight line y0. This result has

already been derived in Ref. 16. The operator of the chemical

potential therefore c**on**sists to the lowest n**on**vanishing

order **on**ly of the local curvature operator s v s that

has eigenfuncti**on**s y k sin(kx/L). Thus energy becomes diag**on**al

in this Fourier representati**on**. We just remark that

these eigenfuncti**on**s are completely different from the **on**es

obtained above. The most important fact is that they do not

exhibit a divergency of curvature near the tips.

FIG. 10. Dependence of the critical length **on** the loading ratio

P x /P.

For 0 the compressive surface stress is even increased

and the critical length becomes smaller.

The case 1 means that the horiz**on**tal stress becomes

bigger than the vertical loading. Generally the elastic energy

release dU el /dLK 2 I K 2 II during **crack** relaxati**on** is maximized

if the **crack** orients perpendicular to the directi**on** of

the highest stress. A vertical orientati**on** of the **crack** is forbidden

by the boundary c**on**diti**on**s y(L)0; but by taking

**on** a wavy shape the **crack** tries to mimic this optimal shape.

This tendency becomes str**on**ger with higher values of P x .

Therefore L c →0 for →.

V. SUMMARY AND DISCUSSION

We calculated expressi**on**s for the total energy of a wavy

**crack** that is subjected to a mode I loading. It turned out that,

for a loading perpendicular to the **crack**, this energy becomes

smaller, when a critical **crack** length, L c 5.18L G , is exceeded.

Then the **on**

of deep grooves al**on**g the **crack** **surfaces** becomes

possible. This deformati**on** is due to a redistributi**on** of matter

and not due to instabilities of the moving **crack** tip. We determined

the threshold of

descripti**on**, either through the early dynamical evoluti**on**

via surface diffusi**on** or melting and recrystallizati**on**.

We generalized the loading c**on**diti**on** by adding a stress

comp**on**ent parallel to the **crack**; this modifies the threshold

of **on**trivial way. Furthermore the situati**on**

of independent **crack** **surfaces** has been studied, with the result

that the parallel c**on**figurati**on** is the most unstable **on**e.

It turned out that the eigenfuncti**on**s of the equati**on**s of

moti**on** exhibit a singular tip curvature. This corresp**on**ds to a

n**on**vanishing stress intensity factor K (tot) II . The principle of

local symmetry states that during **crack** propagati**on** processes

the directi**on** of extensi**on** is oriented such that K II

(tot)

0 is satisfied. In our model the tip positi**on**s are fixed and

we studied **on**ly the slower surface kinetics.

The most important outstanding problem is the combinati**on**

of **crack** propagati**on** and **on**

that the start of the **on**d the

Griffith threshold, and therefore the deformati**on**s naturally

happen **on**ly in the regime of fast propagating **crack**s. There-

046120-9

R. SPATSCHEK AND EFIM A. BRENER PHYSICAL REVIEW E 64 046120

fore we briefly discuss dynamical aspects of the

However, the stress distributi**on** **on** the **crack** surface remains

qualitatively the same as in the static case, apart from some

factors that depend **on** the tip velocity V 1. Therefore our

predicti**on**s that were performed for a static **crack** remain

qualitatively correct even in the limit of fast **crack** extensi**on**.

Far from the tips of a l**on**g, static **crack** with LL G , the

usual

fast propagating **crack** it is slightly modified due to inertial

effects. In the laboratory frame of reference it can be written

as

4q

Dv s k2 2 kk ,

L G

37

where q is a dimensi**on**less functi**on** that depends weakly **on**

V/V R with the Rayleigh speed V R and the Poiss**on** ratio .

One can expect that the linear

local dispersi**on** relati**on** 37 should be **on**ly c**on**vective in

the frame of reference of the moving tip due to its slow

development compared to the fast tip moti**on**. Indeed the

most unstable mode corresp**on**ds to the values kq/L G and

u Dq 4 /L G 4 . In the moving frame of reference,

should be replaced by (iVk) that c**on**tains a c**on**vective

c**on**tributi**on** of the order c qV/L G . The ratio u / c

q 3 D/(L G 3 V) is expected to be small if the velocity V is of

order of the Rayleigh speed. This corresp**on**ds **on**ly to the

c**on**vective **on** itself

is insensitive to the development of the

tip. Nevertheless, the drastic accelerati**on** of the

and the refining of the length scale in the n**on**linear regime

9 make it still c**on**ceivable that also the tip moti**on** could be

affected by the

We also remark that **crack** growth by tip propagati**on** is

more generally hampered not by surface energy but by the

so-called fracture energy that represents the resistance of

the material to **crack** advance 1. This material parameter is

usually bigger than the surface energy, and therefore for

some materials it is c**on**ceivable that the threshold of

**on** the smaller surface

energy ) and **crack** propagati**on** are quite close to each

other. This increases the chance that the relatively slow tip

moti**on** is yet influenced by the

The main problem in observing the phenomen**on** of the

**crack** propagati**on** for **crack** lengths LL G . It arises from

the fact that the **crack** length or the applied tensi**on** is the

**on**ly tunable parameter, and the two effects cannot be separated.

However, this dilemma can be solved by another experiment

that has already been successfully used in the past

18 and is sketched in Fig. 11. The observed instabilities of

the **crack** shape were interpreted in the framework of the

principle of local symmetry and not as a result of the

the energy of a wavy **crack** is already lower than of a

straight **crack**.

A l**on**g thin glass strip is pulled from a hot regi**on** heater

to a cold **on**e water bath at a slow and c**on**stant velocity V.

FIG. 11. Suggesti**on** of an experiment to observe the

**on** the **surfaces** of a slowly propagating **crack**.

With a suitable choice of the c**on**trol parameters, the temperature

decrease T between the hot and cold regi**on** and

the distance b between them, a stati**on**ary semi-infinite **crack**

growth with velocity V becomes possible. All elastic strains

arise here from a thermal gradient TT/b; no external

tensi**on** is applied here. For simplicity we choose the strip

width also to be of order b; in other words, b is the **on**ly

relevant length scale in the problem. Far from the tip in the

cold and hot regi**on**s the material is completely relaxed. In

the transiti**on** regi**on** the characteristic stresses are

E T T ( T is the coefficient of thermal expansi**on**. They

result in a stress intensity factor Kb 1/2 . By Irwin’s theorem

an advance of the **crack** tip by the length ds reduces the

elastic energy by W el K 2 /Eds. It is accompanied by an

increase of the surface energy by W s ds. Near the Griffith

threshold W el W s the propagati**on** velocity is arbitrarily

small; it corresp**on**ds to the temperature difference (T) 2

/(bE T 2 ). On the other hand, the characteristic wavelength

of the

E/ 2 b, in agreement with the above statement that all

length scales are of order b. By this means it should be

possible to observe the

slowly propagating or even stati**on**ary **crack**.

Alternatively, it could also be possible to observe the

Sec. IV, it is possible to alter the critical length by applying

an additi**on**al stress in l**on**gitudinal directi**on** together with

the perpendicular loading. This allows us, for example, to

keep the **crack** exactly at the Griffith threshold, but still exceeding

the critical length for the

completely unanswered questi**on** how the criteri**on** of local

symmetry and the c**on**tradicting result of divergent tip curvature

come together in this regime. Further analysis of this

problem is required in the future.

Another unclear point is the exact behavior near the **crack**

tips. Basically we analyzed l**on**g wave perturbati**on**s of the

**crack** shape, where the wavelength is of the order of the

length of the **crack** itself. However, in our model of a mathematical

**crack** we observed a divergence of the curvature

near the **crack** tips. This stimulates the suspici**on** of the importance

of this regi**on**. In particular, **on**e can speculate that

perturbati**on**s of a rounded tip, with a wavelength of the order

of the tip radius, may lead to important new features,

including a tip splitting as a small scale

the introducti**on** of a new degree of freedom, the radius

of curvature of the blunt tip. We stress that even if

surface diffusi**on** is slow **on** the macroscopic scale, it can still

be very efficient **on** the microscopic scale in the vicinity of

046120-10

GRINFELD INSTABILITY ON CRACK SURFACES PHYSICAL REVIEW E 64 046120

FIG. 12. Geometry of the curved **crack**. The

branch cut of the analytic functi**on**s is indicated

by the thick line.

the tip. A similar idea has been introduced by Langer 19 to

describe the plastic flow in the vicinity of a **crack** tip, where

he used an elliptic **crack** to incorporate the tip curvature. A

future goal is to include such a descripti**on** into our model.

However, in the brittle theory of fracture a mechanism to

regularize the tip curvature is not present; c**on**sequently further

research is necessary in this directi**on** to clear up this

point.

APPENDIX A: SLIGHTLY CURVED CRACKS

The aim of this secti**on** is to derive expressi**on**s for the

stress field around a curved **crack** that is subjected to a mode

I loading. We generalize the results from 4 to describe the

case of n**on**parallel **crack** **surfaces** as depicted in Fig. 12.

Since we deal with a two-dimensi**on**al plane strain situati**on**,

Muskhelishvili’s analytic functi**on** method can be used

20. The whole informati**on** is c**on**tained in two analytic

functi**on**s (z) and (z). Stresses can be expressed as follows:

analysis. Therefore we expand and with respect to the

‘‘smallness parameter’’ y u/l and retain **on**ly first order c**on**tributi**on**s,

zF 0 zF 1 z, zW 0 zW 1 z, A4

where the functi**on**s F i (z) and W i (z) are everywhere analytic

apart from the straight **crack** line L,L and are of order i

in the expansi**on** parameter y u/l . We introduce the comm**on**

abbreviati**on**

To first order **on**e can write

x lim xi.

→0

„xiy u x…F 0 xiy u x F 0 xF 1 x,

„xiy l x…F 0 xiy l x F 0 xF 1 x,

A5

xx yy 4Rez,

yy xx 2i xy 2z¯zz.

A1

A2

and similarly for . Noting that exp(2i)12iy u/l (x) to

first order, the boundary values **on** the straight cut, Eq. A3,

become

As usual we reduce the elastic problem to the case of vanishing

stresses at infinity and given loadings nn u/l and

n u/l al**on**g upper and lower **crack** **surfaces** y u/l (x). For

c**on**venience we introduce the functi**on**

zª¯zzz¯z,

where we used the notati**on** ¯(z)ª(z¯). The boundary values

can be expressed by

nn i n zz

e 2i zz¯zz¯z.

A3

As in Ref. 4 we allow **on**ly small deviati**on**s from the

straight **crack**, y u/l (x)L, and perform a perturbati**on**

nn i n u/l F 0 xiy u/l x F 0 xF 1 x

2iy u/l x F 0 xW 0 x

iy u/l x W 0 xW 1 x

2iy u/l x„W 0 xF 0 x….

Thus separating zero and first order give

(0) nn i (0) n u/l F 0 xW 0 x,

F 1 xW 1 x (1) nn i (1) n u/l „iy u/l xF 0 xW 0

2iy u/l xF 0 xW 0 x….

Here (i) u/l are the c**on**tributi**on**s of order i with respect to

y u/l to the prescribed stresses u/l . The boundary values of

F 0 (x)W 0 (x) and F 0 (x)W 0 (x) are given by

046120-11

R. SPATSCHEK AND EFIM A. BRENER PHYSICAL REVIEW E 64 046120

F 0 xW 0 x F 0 xW 0 x

(0) nn i (0) n u (0) nn i (0) n l ,

F 0 xW 0 x F 0 xW 0 x

(0) nn i (0) n u (0) nn i (0) n l .

s (i) u/l (i) 1

nn u/l

L 2 x 2 1/2

L

P (i) n u n

L tx

(i) l

L 2 t 2 dt

1

P L (i) n u (i) n l

dt.

L tx

A10

Using the formula of Muskhelishvili 20,

1

F 0 z

4zL 1/2 zL 1/2

L

(0) nn (0) n u nn

L

xz

4i 1 L (0) nn i n

L

(0) (0) n l

(0) u (0) nn i n

xz

L 2 x 2 1/2 dx

(0) l

dx,

A6

In the same way **on**e can calculate the first order c**on**tributi**on**,

but the result is quite lengthy. In our case we have a

uniform mode I loading at infinity, i.e., up to the first order

nn u/l (0) nn u/l (1) nn u/l P, A11

n u/l (0) n u/l (1) n u/l Py u/l x.

A12

Thus the normal tracti**on** is c**on**stant and the shear stress has

**on**ly a first order c**on**tributi**on**. This simplifies the expressi**on**s

and **on**e finally obtains, after some simple transformati**on**s,

(0) u/l P, as expected for the straight **crack**, and

1

W 0 z

4zL 1/2 zL 1/2

L

(0) nn (0) n u nn

L

xz

4i 1 L (0) nn i n

L

(0) (0) n l

(0) u (0) nn i n

xz

L 2 x 2 1/2 dx

(0) l

dx.

A7

The sec**on**d integral appearing in both of these two formulas

does not exhibit the usual square root singularity and becomes

**on**ly relevant in the case of n**on**parallel **crack** **surfaces**.

Basically we are interested in the tangential comp**on**ent of

the stress tensor,

Re„2ze 2i zz¯zz¯z….

A8

Evaluating the limiting values of Eqs. A6 and A7, we

obtain to zeroth order

(1) u/l s (1) u/l 2PL2

L 2 x 2 y u/lx.

3/2

A13

The final result is therefore as follows: A curved **crack**

loaded by yy P and ij 0 for all other comp**on**ents at

infinity exhibits total stresses at the interfaces of the **crack**

given by

(tot) nn u/l 0,

(tot) n u/l 0,

A14

A15

(tot) u/l (0) u/l (1) u/l A16

to first order, using the expressi**on**s A9–A13. Here we

have already added the homogeneous stress caused by the

mode I loading at infinity.

APPENDIX B: EQUIVALENCE

OF ENERGY REPRESENTATIONS

To show the equivalence of the different approaches to

calculate the total energy of a wavy **crack** 8 and the direct

integrati**on** of the chemical potential 10, it is sufficient to

derive the chemical potential from the first approach

(0) u/l s (0) u/l , A9

s x u/l v s

2

U

yx

B1

where we have defined the functi**on**

and to compare it with the original expressi**on** 10. Since

this is straightforward for U s and U els , we restrict our c**on**-

046120-12

GRINFELD INSTABILITY ON CRACK SURFACES PHYSICAL REVIEW E 64 046120

siderati**on**s to U elu and give **on**ly some brief hints for deducti**on**.

Using Eqs. 7 and 8, we obtain

U elu 12

E

2P 2

L

L

xL

tL

LL

Lt

yxyt dt dx dL.

Lt B2

L 1 Lx

LL Lx

The main idea is to interpret the triple integral as volume

integral. Rewriting it as

L

LL

L

xL

L

tL

L

•••dt dx dL

tL

L

xL

L

•••dL dx dt,

Lmax(t,x)

B3

allows us to perform the explicit calculati**on** of the innermost

integral. For simplicity we ignore the difficulties arising from

the exchange of the integrati**on** order. In fact, they are resp**on**sible

for the appearance of the principal value integrals,

but we treat all integrals as divergent ordinary integrals,

because we are basically interested in a structural agreement

of the expressi**on**s. However, all calculati**on**s can easily be

extended to overcome this limitati**on**.

We c**on**clude

U elu yyU elu y

2P2 1 2

E

L

tL

L

yt 2yx

xL

ln L2 xtL 2 t 2 L 2 x 2

L

lnxt dx dt,

B4

where **on**ly first order c**on**tributi**on**s have been taken into

account. Another integrati**on** by parts notice that y(L)

0 and further algebraic manipulati**on**s lead to

U elu yyU elu y 4P2 1 2

E

L

yt

tL

L yx

xL tx L2 x 2

dx dt.

L 2 2

t

Now the functi**on**al derivative can be immediately read:

elu u/l 2v sP 2 1 2

EL 2 t 2

L yx

xL tx L2 x 2 dx.

It matches the third term in Eq. 10, of course apart from the

principal value. This completes the proof.

1 L.B. Freund, Dynamic Fracture Mechanics Cambridge University

Press, New York, 1990.

2 E. Bouchau, J. Phys.: C**on**dens. Matter 9, 4319 1997, and

references therein.

3 J. Fineberg, S.P. Gross, M. Marder, and H.L. Swinney, Phys.

Rev. Lett. 67, 457 1991; Phys. Rev. B 45, 5146 1992; J.F.

Boudet, S. Ciliberto, and V. Steinberg, Europhys. Lett. 30, 337

1995; E. Shar**on**, S.P. Gross, and J. Fineberg, Phys. Rev.

Lett. 74, 5096 1995.

4 B. Cotterell and J.R. Rice, Int. J. Fract. 16, 155 1980.

5 M. Adda-Bedia and M. Ben Amar, Phys. Rev. Lett. 76, 1497

1996.

6 E.S.C. Ching, J.S. Langer, and H. Nakanishi, Phys. Rev. E 53,

2864 1996.

7 J.S. Langer and A.E. Lobkovsky, J. Mech. Phys. Solids 46,

1521 1998.

8 R.J. Asaro and W.A. Tiller, Metall. Trans. 3, 1789 1972;

M.A.

Sov. Phys. Dokl. 31, 831 1986.

9 W.H. Yang and D.J. Srolovitz, Phys. Rev. Lett. 71, 1593

1993; K. Kassner and C. Misbah, Europhys. Lett. 46, 217

1999.

10 E.A. Brener and V.I. Marchenko, Phys. Rev. Lett. 81, 5141

1998.

11 C. Herring, J. Appl. Phys. 21, 437 1950; P. Nozières, J. Phys.

I 3, 681 1993.

12 J.R. Rice, in Fracture: An Advanced Treatise, edited by H.

Liebowitz Academic, New York, 1968, Vol. 2, Chap. 3, pp.

191–311.

13 G.R. Irwin, J. Appl. Mech. 24, 361 1957.

14 J.H. Wilkins**on**, The Algebraic Eigenvalue Problem Oxford

University Press, L**on**d**on**, 1988.

15 W.H. Press et al., Numerical Recipes in C Cambridge University

Press, Cambridge, England, 1996.

16 A. Buchel and J. Sethna, Phys. Rev. E 55, 7669 1997.

17 E.M. Lifshitz and L.P. Pitaevskii, Physical Kinetics Pergam**on**

Press, Oxford, 1981.

18 A. Yuse and M. Sano, Nature L**on**d**on** 362, 329 1993; M.

Adda-Bedia and Y. Pomeau, Phys. Rev. E 52, 4105 1995.

19 J.S. Langer, Phys. Rev. E 62, 1351 2000.

20 N.I. Muskhelishvili, Some Basic Problems **on** the Mathematical

Theory of Elasticity Noordhoff, Gr**on**ingen, 1952.

046120-13