Simple models of rapid fracture

PHYSICA[

Physica D 66 (1993) 125-134 North-Holland SDI: 0167-2789(93)E0022-4

Simple models of rapid fracture Michael Marder Center for Nonlinear Dynamics and Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA

The purpose of this article is to introduce the physics of fracture through simple scaling analysis and one-dimensional models. Two one-dimensional models are discussed. The first is a continuum model, for which it is simple to find the steady states and establish linear stability. The second model is a lattice version of the first, and its steady states can be found by the Wiener-Hopf technique. The lattice model steady states involve large quantities of radiation, and may help to explain dynamics observed in some recent unexplained experiments.

I. Introduction T h e study of fracture divides into two parts, quasi-static and dynamic. The first of these studies is vastly m o r e practical. The main question of quasi-static fracture is: given a body with a pre-existing flaw, under what ~tress will the flaw begin to extend? The answer to this question depends upon the precise geometry of the body, the shape and location of the flaw, and the precise amount of energy needed for the flaw to extend in the material at hand. The results can be used to determine what stresses various materials can withstand under practical conditions, so that the discipline of fracture mechanics underlies construction codes [1]. The study of rapid fracture, by contrast, has d e m o n s t r a t e d little practical value. Once an airplane wing is falling off, who has time to argue whether the crack is moving at 200 or 500 meters per second? So the understanding of rapid fracture is mainly to satisfy curiosity, although it may have some use in providing a way to study material response under the extreme conditions of high strain rates. These comments do not diminish the interest in rapid fracture as a basic research problem [2], one which resembles other highly nonlinear pattern-forming systems, but with its own special difficulties and surprises.

The purpose of this article is to set down some simple models relating to rapid fracture, beginning with a simple scaling theory that illustrates the basic principles, and proceeding to two onedimensional models. These models demonstrate how fracture works on a microscopic level, but involve minimal mathematical complication. The simplicity of the arguments may provide a useful introduction to fracture, and in addition provide a promising avenue to answer some puzzling open questions.

2. Scaling analysis, and an open problem The first analysis of rapid fracture was carried out by Mott [3]. It is a scaling analysis which clarifies the basic physical processes, despite being wrong in m a n y details, and consists in writing down an energy balance equation for crack motion. Consider a crack of length l(t) growing at rate v ( t ) in a plate under stress tr~ far from the crack, as shown in fig. 1. When the crack extends, its faces separate, causing the plate to relax within a circular region centered on the middle of the crack and with diameter of order l. The kinetic energy involved in moving a region of this size is guessed to be of the form

0167-2789/93/$06.00 ~) 1993- Elsevier Science Publishers B.V. All rights reserved

126

M. Marder / Simple models o f rapid fracture

Fig. 1. As a crack of length / expands at velocity v in an infinite plate, it disturbs the surrounding medium up to a distance on the order of 1. K = CKI2U 2 ,

(2.1)

a n d the p o t e n t i a l e n e r g y g a i n e d in r e l e a s i n g stress f r o m t h e r e g i o n is g u e s s e d to be o f the form

C o n s i d e r first t h e p r o b l e m o f q u a s i - s t a t i c crack p r o p a g a t i o n . If a c r a c k m o v e s f o r w a r d only slowly, its k i n e t i c e n e r g y will b e negligible, so only t h e q u a s i - s t a t i c p a r t of t h e energy, Eqs, will be i m p o r t a n t . This f u n c t i o n is g r a p h e d in fig. 2. It costs e n e r g y for v e r y s h o r t c r a c k s to e l o n g a t e , a n d in fact such c r a c k s w o u l d heal a n d travel b a c k w a r d s if it w e r e n o t for i r r e v e r s i b l e p r o cesses, such as o x i d a t i o n o f the c r a c k surface, w h i c h t y p i c a l l y p r e v e n t this f r o m h a p p e n i n g . T h a t t h e c r a c k g r o w s at all is d u e to a d d i t i o n a l i r r e v e r s i b l e p r o c e s s e s , s o m e t i m e s c h e m i c a l att a c k on the c r a c k tip, s o m e t i m e s v i b r a t i o n o r o t h e r i r r e g u l a r m e c h a n i c a l stress. It s h o u l d be e m p h a s i z e d t h a t the s y s t e m e n e r g y E i n c r e a s e s as a result of t h e s e p r o c e s s e s . E v e n t u a l l y , at l e n g t h l 0, t h e e n e r g y g a i n e d by r e l i e v i n g elastic s t r e s s e s in t h e b o d y e x c e e d s the cost o f c r e a t i n g n e w s u r f a c e , a n d the c r a c k b e c o m e s a b l e to e x t e n d s p o n t a n e o u s l y . O n e sees t h a t at l 0, the e n e r g y functional Eq~(/) has a q u a d r a t i c maxim u m , so t h a t eq. (2.4) can be r e w r i t t e n E,~(1)

= E,s(lo)

- c,,(1-

t,,) e ,

lo = -~/c,, .

(2.5) P = -cpl

2.

(2.2)

These guesses are correct for slowly moving cracks, but fail qualitatively as the crack velocity approaches the speed of sound, in which case both kinetic and potential energies diverge. This divergence will be demonstrated in a later section of the article, but for the moment, let us proceed fearlessly. The final process contributing to the energy balance equation is the creation of new crack surfaces, which takes energy 23'l, where 3" is the surface energy. So the total energy of the system containing a crack is given by E = c K l 2 o 2 + Eqs(/) ,

(2.3)

with E,as(l ) = - C e 12 + 2 3 ' l .

1 ,,

(2.4)

Distance t Fig. 2. The energy of a plate with a crack as a function of length. In the first part of its history, the crack grows quasi-statically, and its energy increases. At 10 the crack begins to move rapidly, and energy is conserved.

M. Marder / Simple models of rapid fracture The whole study of engineering fracture mechanics boils down to calculating lo, given things such as external stresses, which in the present case have all been condensed into the constant Cp. Dynamic fracture starts in the next instant, and because it is so rapid, the energy of the system is conserved, remaining at Eqs(10). Using eqs. (2.3) and (2.5), with E = Eqs(lo) gives

127

1000 /

800

600

~

4oo

>

/)(t)=~

(1--~)=/)max(1--~).

(2.6)

This equation predicts that the crack will accelerate until it approaches the speed/)max The maximum speed cannot be deduced from these arguments, but Stroh [4] correctly argued that Umax should be the Rayleigh wave speed, the speed at which sound travels over a free surface. One needs only to know the length at which a crack begins to propagate in order to predict all the following dynamics. Although the forms of kinetic and potential energy, eq. (2.1) and eq. (2.2), are wrong, the errors somehow cancel themselves out, and eq. (2.6) is quite accurate. There is the remarkable result, due to Kostrov [5], Eshelby [6], and Freund [2, 7], that for a semi-infinite crack in an infinite two-dimensional plate, with a constant force pushing the faces of the crack apart, an exact solution of the partial differential equations for elasticity gives a complicated analytical expression, which differs from eq. (2.6) by at most a few percent. There is no exact solution for the motion of a crack starting at the edge of a finite plate (experiments tend to be carried out in finite plates) but everyone's best guess seems to be that eq. (2.6) applies in that case as well, at least until reflected sound waves from the top and bottom of the plate become important. Experiments to test this theory were first carried out in the late 1950's and have continued with increasing accuracy to the present day. The verdict is unanimous: there is no amorphous material in which cracks obey Mott's scaling law. To understand the nature of the divergence, see

200

I

I

I

I

8

12

16

20

Distance (era) Fig. 3. Data of Kobayashi et al,, ref. [8], showing that cracks accelerate much more slowly than the scaling theory predicts.

fig. 3, which shows data of Kobayashi et al. [8] in comparison with the scaling theory, and table 1, compiled by Ravi-Chandar and Knauss [9], which shows observed maximum velocity divided by predicted maximum velocity for a variety of amorphous brittle materials. The reason for this discrepancy is not understood, and this article will not resolve it. However, some recent experiments [10] have indicated that cracks become dynamically unstable past about thirty percent of the Rayleigh wave speed, and that understanding this instability is the key to resolving the problem. Therefore, the balance of this article will be devoted to some simple models of fracture which can be used as a theoretical laboratory in which to test various mechanisms to destabilize a crack. Similar modTable 1 Survey by Ravi-Chandar and Knauss showing that cracks in amorphous brittle materials never reach the predicted limiting velocity. Material

Poisson ratio ~,

V,er,,,na,IVmax

Glass Plexiglas Homalite

0.22 0.35 0.31

0.47-0.66 0.58-0.62 0.33-0.41

128

M. Marder / Simple models of rapid .fracture

els have been used before, by many authors [11-19]. T h e first model is a continuum model, whose steady-state solutions are particularly easy to find. Establishing the linear stability of these states gives and example of how to perform perturbation theory about fracture solutions. Finally, the lattice version of the model is introduced, and the W i e n e r - H o p f technique used to find the steady states. The steady states of the lattice model are qualitatively different from those of the continuum model, involving large quantities of radiation.

3. O n e - d i m e n s i o n a l m o d e l

The basic model consists of a series of horizontal and vertical springs, connected to unit masses, and arranged as in fig. 4. The thin springs are weak, and have a spring constant E 2. The thick springs are strong, and have a spring constant of unity. When the vertical thick springs are stretched to a distance of more than 1 they snap, and at the instant they snap they dissipate a small a m o u n t of energy, proportional to the

instantaneous velocity of the spring, and a phenomenological constant b. Passing first to the continuum limit, one has the equation a:u

/ -- - - -

E2(u - J ) - uO(l - u) - bu 6(1 - u ) . (3.1)

3x2

where u(x) represents the height of the mass at distance x along the horizontal axis, O(x) is the step function, and 6(x) is the Dirac delta function. When b = 0, the model is perfectly brittleelastic, and otherwise it includes a bit of viscoelasticity. One may easily find steady states of this model, cracks moving at constant velocity v from left to right. First rewrite eq. (3.1) in a coordinate system that moves with the crack tip. One has 02U

ax 2 ( 1 - v 2 ) - e 2 ( u - j ) - u O ( 1 - u ) OH

+ v ~xx b 6(1 - u) = O.

(3.2)

Taking the fracture to move from left to right, consider the solution to the left of the point where the thick springs snap, u = 1, which may be taken to be x = O. In the left region q~

U = A + A ~ e xq' ,

---

Ey ,

(3.3)

where 1

Y- Vrl-v z

(3.4)

In the right hand region, u = u~ + A r e xqr

,

qr = --X/I + e2y ,

(3.5)

with Fig. 4. S p r i n g a n d m a s s c o n f i g u r a t i o n for a o n e - d i m e n s i o n a l m o d e l of fracture. T h e p o s i t i o n s of the m a s s e s are given by a s o l u t i o n of eq. (5.10) for v = (/.5.

At~ 2

us

l+e 2'

(3.6)

M. Marder I Simple models o f rapid fracture

By assumption one has that u(0) = 1,

(3.7)

so that A e = -(A-

1)

(3.8)

and A r = 1 - u~.

(3.9)

By integrating eq. (3.2) from a little before x = 0 to a little after, one finds that ( A r q ~ -- A e q e ) ( 1

-

v 2) = v b ,

(3.10)

which may be solved to give v=

(A - Ac)(~f~e: + 1 + , ) Vzl2b: + (A - Ac)2(V~-E2+ 1 + ,)2

129

passes, the energy must be completely absorbed by snapping the springs. In the case b = 0, it is easy to work out an energy balance equation. The process of snapping a spring absorbs energy __ 1 2 u=, which is what is required to stretch the spring from its value far ahead of the crack to the breaking point. If the energy stored in the springs far ahead of the crack, I(A - u s ) , 2 2 is less than this value, the crack cannot travel forward. If it is exactly equal, the crack can travel at any velocity. If it is greater, more energy is released per unit crack extension than the springs can absorb, and the system must absorb the extra energy by having the crack continually accelerate towards v = 1. Therefore, steady states are only generic if the energy needed to form new crack surface increases with velocity; a simple way to introduce this behavior is through the phenomenological constant b.

(3.11a)

4. Stability and A c = v-i-+ •-2

(3.11b)

What are the lessons to be learned from this simple calculation? First, notice that the kinetic energy diverges as v approaches the wave speed, v = 1, since K=/j

f

-1 / j 2 d x

= 1 v 2( A e2q e + A r2l q r l ) ,

(3.12)

and both qe and qr diverge as v ~ 1. The source of the divergence is in the large gradients which develop in the vicinity of the tip. This calculation demonstrates why Mott's guess, eq. (2.1), does not provide a realistic account of the kinetic energy of a crack. Second, notice that when b = 0 it appears from eq. (3.11) that v = 1, but appearances are deceptive. The reason is that when b - - 0 there is no dissipation. Since potential energy stored in the springs ahead of the crack tip is completely released after the crack

As a next task, consider the linear stability of these steady states. One cannot do perfectly straightforward perturbation theory because the tip of the crack is too singular to allow it. That is, if one simply writes u(x, t) = u~(x - vt) + a a(x)

e i'°' ,

and linearizes in the small parameter a, there are divergent operators which develop at the crack tip and make nonsense of the perturbation theory. The way around this problem is to define a reference frame in which the crack tip is always at the origin, so that by definition the perturbations vanish just where the problem becomes most singular [20]. The formal way to achieve this is adopt a coordinate X ' = X - - V t - - a e °'t ,

(4.1)

which is defined to be the location of the crack tip, u = 1. In this coordinate system, dropping

M. Marder / Simple models of rapid fracture

130

the p r i m e s , one has O ---2(v+ae

~'w

Ot2

)

02 ---aw'e Ox Ot

0 °~'-

Ox

[A~q,(q~-

-

e 2 ( u - A) - u 0(1 - u)

O

~t-(v+°Ja

e')

O)

~x

(4.2)

uba(1-u).

Let u = u~ + a ~ e ~' ,

(4.3)

w h e r e u~ is the steady state solution at s o m e velocity v. T h e n using the fact that the coordinate s y s t e m has b e e n defined to k e e p the crack tip at x = 0, 07(0) = 0), and linearizing in a, one has

Ox

44)]( 1 - v2)

= wb + 2vo~(qrA ~- qtA~).

e ~') -~x2

02U Ox 2

4~) - A , q , ( q t -

\

O2 ~- (U 2 -1- 2 y a w

,

T h e discontinuity of the derivative of ~ at x = 0 is d e t e r m i n e d by

(4.7)

A n y solutions of eq. (4.7) where ~o has positive real part will c o r r e s p o n d to unstable modes. H o w e v e r , the solutions always occur for o) negative and real, so the steady state cracks are quite stable. T h e r e is at most one such solution, which m a y s e e m a difficulty, since solutions of the linearized p r o b l e m a b o u t the steady states should constitute a c o m p l e t e set of functions. T h e resolution is that w h e n w is purely imaginary and sufficiently large, one or both of c~er will be purely imaginary, in which case the solution eq. (4.5) is not sufficiently general, since both signs of c~r b e c o m e acceptable. Thus an arbitrary pert u r b a t i o n a b o u t the steady state is resolved mainly in t e r m s of u n d a m p e d traveling waves.

Ox

5. Discrete one-dimensional model

02a Ox 2 (1 - v 2 ) -

e2a - a O(-x)-

o)b 6 ( x ) .

(4.4)

u t = A e~/e ~ (e xq~ - e ~4t ) ,

(4.5a)

T h e final calculation is for the steady states of the lattice version of eq. (3.1). T h e possibility of solving such models was discovered by Slepyan [21]; similar m o d e l s had also been solved previously by A t k i n s o n and C a b r e r a [11], and Celli and Flytzanis [12]. T h e starting point is o b t a i n e d by interpreting the a r r a n g e m e n t in fig. 4 literally, and giving all thick springs the s a m e strength, to obtain the e q u a t i o n

ar = A rqr(e xqr - eXqr) ,

(4.5b)

iim = u,n+l -- 2Um + Um ~ -- e (Um -- A )

It is clear by inspection that OUs/OX is a solution of eq. (4.4), except for the fact that it does not vanish at the crack tip as ~7. T h e correct solution is o b t a i n e d by subtracting off the solution of the h o m o g e n e o u s part of eq. (4.4):

-- Um 0(1 -- Urn) -- b~im.

where ~/co 2 + e 2 ( 1 - v 2 ) - v w qe = 1 - v2 '

(4.6a)

go e+(2+l)(a_v 2)+v~o qr = -

2

1-v

2

(4.6b)

(5.1)

T h e dissipation b is introduced in a different f o r m than b e f o r e , and in addition it will be t a k e n to be infinitesimal in all the a r g u m e n t s which follow. T h e r e is no unique way to define a steady state for a lattice m o d e l , but the simplest possible

131

M. Marder / Simple models o f rapid fracture

assumption is that a state traveling at velocity v completely reproduces itself, apart from translation of one lattice spacing, at time intervals of 1/u. Stated mathematically, one has that Um+l(t + 1 / V ) = u m ( t )

(5.2)

Urn(t) = Uo(t -- m / v ) =-- u ( r ) .

Notice that u-(to) has no singularities when to is in the lower half of the complex plane, and that u+(to) has no singularities when to is the upper half of the complex plane, since the integrals in eq. (5.6) are guaranteed to converge in these cases. The problem may be solved by the Wiener-Hopf technique [22]. Decompose F and G as

All information about this state is contained in the behavior over time of any single lattice site. More complicated candidates for steady states could be constructed, in which for example the state reconstructs itself only after passing two lattice sites, but I have been unable to solve them. The equation for the steady state takes the form

where F - ( t o ) and G (to) are free of singularities in the lower half of the complex plane, while G+(to) and F+(to) are free of singularities in the upper half of the complex plane. Then one can rewrite eq. (5.6) as

ii = u ( z + 1 / v ) - 2 u + u ( z - 1 / v ) - e2u

G-(to)

F-

G-

F = F--7 and

G-

G+ ,

G +(0)

(5.8)

1

F - ( to------ff u - - A F + ( O~ a + i t o + e2A e -~ITI - u O ( - z ) -

(5.3)

bu,

with a to be taken to zero at the end of the calculation. Taking the Fourier transform of eq. (5.3) by u(to) =

d r u ( z ) e ''°" ,

f

•

t"

(5.4)

(5.5)

F-(to) G+(0) A u - ( t o ) - G - ( t o ) F+(0) a + i t o '

one obtains

G+(0) A G+(to) F+(0) a - i t o "

2(

1

q-

--

Of

with F ( t o ) = to2 _ 4 s i n 2 ( t o / 2 v ) -

2 .q._itob ,

(5.7a)

and to2

_

4 sinZ(to/2v) -

(5.10b)

,

(5.6)

=

(5.10a)

u+(to) = F+(to)

F(to) u + + a(to) u- = -a

G(to)

(5.9)

Since the left side of this expressions is free of singularities in the lower half plane, and the right side is free of singularities in the upper half plane, both sides must equal a constant. One finds that u(r) is not continuous at r = 0 unless the constant is zero. Therefore

and defining u÷-(to) = J d r O( +-r) u ( r ) e i°'" ,

G +(to) u + G+(0) 1 F+(to) + a ~F+(0 a - i k "

2 _ 1 + itob .

(5.7b)

The feature of the discrete model which is quite different from the continuous model lies in the fact that F and G have roots lying almost exactly on the real axis, pushed off it only by the infinitesimal damping b. These roots turn into real poles of u, correspond to traveling waves, and because of this special significance, one should separate them out. Let w~ be the real roots of F, with the -+ sign indicating whether the root will belong to F - or F ÷ (w~- will have an

M. Marder / Simple models of rapid fracture

132

infinitesimal imaginary part above the real o2 axis), and let v,-+ be the corresponding real roots of G. One can tell which camp a root belongs to by computing, for example,

wi

dF(w~) dw

(5.l 1)

If this quantity is positive, the root in question belongs with F +, and otherwise it belongs with F -. Next, define

[~(o,) -

F(w) u~ (~o - 0,) ) "

G(,o) (~(oJ) - II, (o) - v ~ )

"

(5.12)

T h e formal reason to divide out the roots in this m a n n e r has to do with the identities

F(~o; b) = F ( - w ;

b) ,

G(~o; b) = G ( - ~ o ; - b ) .

(5.13b)

/~(w) = / 3 ( - w ) ,

(5.14a)

G(w) = G ( - w ) .

(5.14b)

(,o)¢+(-o,)= ¢+(,,)~ (-,,,).

(5.17a)

d (o,) d ~(-0,)= d (0) d +(0).

(5.17b)

Similar identities do not hold for F and G because when b changes sign, the real roots flip between belonging to F and F +, and so have to be treated separately. The most interesting calculation to perform is of the amount of energy emanating from the crack tip in the form of traveling waves. Far ahead of the crack, there is an energy per site

Eahcad

=

'u: + 2~ e : ( k - u,):

~

1

A

(5.18a)

using eq. (3.6), and eq. (3.11b), while far behind the crack, there is an energy per site of Ebchind

=

2' ,

(5.18b)

which is the total energy needed to bring the lower spring from zero to failure. Any diffcrence between these two quantities E,,,,,~,t,,,,, = ½ ( m e

- 1)

(5.18c)

(5.15)

must be energy carried by traveling waves. The way to proceed is to calculate u ( r - 0 ) , since for the steady state ~, ~.,fion eq. (5.3) to follow from the original m,,dcl eq. (5.1i, one must have

(5.16)

u(~-

These last identities are valuable because employing eq. (5.8) one can see immediately that

~

,~ (,,,),7~(_o,)=,~ (0)~(0).

(5.13a)

The roots of F and G will only move infinitesimally if b changes sign, and so long as the roots are away from the real axis, this will not matter. So one can write

(o~) ,a (-,o) ¢+(o,) />(- o,)

constant. The most convenient form in which to express this relation is

Since F+(-oa) is regular in the lower half plane, on the left side of eq. (5.16) is a function which is regular in the lower half plane, on the right hand side a function which is regular in the u p p e r half plane, and both sides must equal a

0)=1.

(5.19)

Imposing eq. (5.19) will force a particular choice of A, which when used with eq. (5.18) will give the desired result. One need only find the behavior of u for large w, since at r = 0 , u (r) jumps from 0 up to u(0). Such a discontinuity is

133

M. Marder / Simple models of rapid fracture

produced by Fourier transforms that fall off as 1/ioo for large w, so that comparing with eq. (5.10) one sees immediately F-(oo) G+(0) A. (oe) F+(0)

u(r=O)=

1.00

0.80

(5.20)

0.60

G

g >

Since from eq. (5.7) one has _

(5.21)

],

F(~)

0.40 0.20 0.00

using the identities eq. (5.17) it is not hard to show that F (o0) _ X / F - ( 0 ) F+(0) c

(5.22)

Using this expression together with the definitions of F and G in eq. (5.12) gives

J

J

i

i

i

i

0.00

0.20

0.40

0.60

0.80

1.00

ad6 Fig. 5. Velocityof steady state solutions as a function of the inverse of the imposed strain At/A, in the limit of small E. Note that for certain strains, many different steady states are possible, emitting different quantities of radiation. steady states are possible for driving force

-

v,

(5.23)

so that finally the energy involved in snapping the springs is ½uZ(r = 0 ) = 1

(k)

]-I w,_+ v~+ _ 1 , wi vi

(5.24)

- \ 2

~- = wi vi

1

(5.26)

and at this minimum driving force, the steady state has a velocity of 0.38 . . . . In addition, as v -+ 0, one finds

2

A exp[lf ----~ Ac

and the energy carried off in radiation is

1 {[I w, v,+)

A < 1.071Ac,

0

1.6180 . . . . = Eradiatio n .

/ /~e2+]+sin20 lnk~ ~ )dO] + sin20 (5.27)

(5.25)

The analogous expression for a simple twodimensional lattice was obtained by Slepyan [21]. By numerical techniques [23] it is possible to calculate u(r) explicitly, and a snapshot of the lattice at t = 0 for v = 0.5, e = 0.1 is given in fig. 4. It is extremely simple to locate the real roots of the functions F and G, and compute the products appearing in eq. (5.24). The result is given in fig. 5, showing steady state velocity v as a function of the inverse of the imposed strain ac/A, (Zlc being defined by eq. (3.11b)) in the limit of small e. According to this calculation, no

This means that a stationary crack in a noiseless environment will not begin to move forward until A reaches this value; however, since moving cracks are possible at lower driving forces the bifurcation to crack motion is subcritical. Right at zl/A c = 1.6180... a countably infinite number of steady states is possible, each radiating a different quantity of energy from all of the others. The dynamical picture which emerges from direct numerical simulations of the lattice model, eq. (5.1), is that the crack is trapped by the lattice until Zl is raised to a value of around

134

M. Marder / Simple models of rapid fracture

1.5A c, at w h i c h p o i n t it j u m p s to a velocity of a r o u n d 0.9, a c c o m p a n i e d b y r a d i a t i o n . U p o n l o w e r i n g t h e d r i v i n g f o r c e A it is p o s s i b l e to o b t a i n s t a b l e low v e l o c i t y s t e a d y states t h a t involve greater amounts of radiation, The system is e x t r e m e l y h y s t e r e t i c . W h i l e this d y n a m i c a l b e h a v i o r is e x t r e m e l y i n t e r e s t i n g in its o w n right, it d o e s not i m m e d i a t e l y e x p l a i n t h e e x p e r i m e n t a l facts of c r a c k p r o p a g a t i o n in b r i t t l e m a t e r i a l s . Still t h e r e is r e a s o n to h o p e t h a t e x t e n s i o n o f t h e s e i d e a s will e v e n t u a l l y b r i n g t h e o r y a n d e x p e r i m e n t for c r a c k m o t i o n into c l o s e r a c c o r d . W o r k a l o n g t h e s e lines is p r o g r e s s i n g , t o g e t h e r with e x p e r i m e n t s t h a t will p r o b e t h e a c o u s t i c e m i s s i o n s of m o v i n g cracks directly.

Acknowledgement T h i s w o r k was s u p p o r t e d in p a r t by the T e x a s Advanced Research Program, grant number 3658-002, a n d by t h e S l o a n F o u n d a t i o n .

References [1] M.F. Kanninen and C. Popelar, Advanced Fracture Mechanics (Oxford, New York, 1985).

[2] L.B. Freund, Dynamic Fracture Mechanics (Cambridge U.P., New York, 1990). [3] N.F. Mott, Engineering 148 (1948) 16. [4] A.N. Stroh. Adv. Phys. 6 (19571 418. [5] B.V. Kostrov, Appl. Math. Mech. 30 (19661 1241; 38 (19741 511 [Prikl. Mat. Melch]. [6] J.D. Eshelby, J. Mech. Phys. Sol. 17 (1969) 177. [7] L.B. Freund, J. Mech. Phys. Sol. 2(1 (1972) 129, 141; 21 (1973) 47; 22 (1974) 137. [8] A. Kobayashi, N. Ohtani and T. Sato, J. Appl. Polymer Sci. 18 (1974) 1625. [9] K. Ravi-Chandar and W.G. Knauss, Int. J. Fracture 26 (1984) 141. [10] J. Finberg et al. Phys. Rev. Lett. 67 (19911 457; Phys. Rev. B 45 (1992) 5146. [111 W. Atkinson and N. Cabrera Phys. Rev. 138 (1965) A764. [12] V. Celli and N. Flytzanis, J. Appl. Phys. 41 (197(/) 4443. [13] R. Thompson, C. Hsieh and V. Rana, J. Appl. Phys. 42 (19711 3154. [14] L. Knopoff, J.O. Mouton and R. Burridge, Geophys. J. R. Astro. Soc. 35 (19731 169. [15] M.F. Kanninen, Int. J. Fracture 9 (1973) 83. [16] E. Smith, Materials Sci. Eng. 17 (1975) 125. [17] L.B. Freund, in: Mathematical Problems in Fracture, ed. R. Burridge (American Mathematical Society, Providence, RI, 1979) p. 21. [18] K. Hellan, Int. J. Fracture 17 (1981) 311. [19] J.S. Langer, to bc published. [20] M. Marder, Phys. Rev. Lctt. 66 (1991) 2484. [21] L.I. Slepyan, Soy. Phys. Dokl. 26 (1981) 538. [22] B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations (Pergamon, New York, 1958). [23] X. Liu and M. Marder, J. Mech. Phys. Sol. 39 (1991) 947.