Notes on Fracture

Notes on Fracture Michael P. Marder Center for Nonlinear Dynamics and Department of Physics The University of Texas at Austin Austin Texas, 78712, USA February 7, 2003

Contents 1

2

Continuum Theories of Fracture: Scaling 1.1 Introduction . . . . . . . . . . . . . . 1.2 Scaling theory I: Things fall apart . . 1.3 Scaling Theory II: All together . . . . 1.4 Scaling theory III: Cracks at last . . . 1.5 The scaling of dynamic fracture . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

Continuum fracture mechanics 2.1 Structure of fracture mechanics. . . . . . . . . . . . . 2.2 Dissipation and the process zone . . . . . . . . . . . . 2.3 Conventional fracture modes . . . . . . . . . . . . . . 2.4 Stresses Around an Elliptical Hole . . . . . . . . . . . 2.5 Stress Intensity Factor . . . . . . . . . . . . . . . . . . 2.6 Universal sin singul gularities nea near the crack tip . . . . . . . . 2.7 2.7 Th Thee rela relati tion on of the the stre stress ss inte intens nsit ity y fact factor or to ener energy gy flux flux 2.8 Elasticity in Two Dimensions . . . . . . . . . . . . . . 2.9 Steady States in Plane Stress . . . . . . . . . . . . . . 2.10 Energy flux and limiting crack speed . . . . . . . . . . 2.11 Crack paths . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

. . . . . . . . . . .

. . . . .

2 2 3 3 4 6

. . . . . . . . . . .

9 9 9 11 12 14 15 16 16 19 22 23

3

Incorporating Plasticity

25

4

Atomic Theories of Fracture 4.1 Physical argument for velocity gap . . . . . . . . . . . . 4.2 4.2 Dy Dyna nami micc frac fractu ture re of a latt lattic icee in anti anti-p -pla lane ne shear shear,, Mod Modee III III 4.3 Definition and energetics of the model del . . . . . . . . . . 4.4 Symmetries: . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Elimin Eliminati ating ng the spatia spatiall inde index x m: . . . . . . . . . . . . .

26 28 31 31 34 35

1

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

4.6 4.7 4.8 4.9 4.10 4.11 4.12 5

Equati Equa tion onss solve solved d in term termss of a sing single le mass mass poin pointt on crac crack k line line:: Relati Relation on betwee between n and . . . . . . . . . . . . . . . . . . . Phonon emission: . . . . . . . . . . . . . . . . . . . . . . . . Forbidden velocities . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Instabilities . . . . . . . . . . . . . . . . . . . . . . The connection to the Yoff offe insta stability . . . . . . . . . . . . . Generalizing to Mode I: . . . . . . . . . . . . . . . . . . . . .

Scaling ideas and molecular dynamics 5.1 Molecular dynamics . . . . . . . . . . . . . . . . 5.2 Interatomic potentials . . . . . . . . . . . . . . . 5.3 Realistic potentials for silicon? . . . . . . . . . . 5.4 5.4 Resul esults ts of zero zero Kelvi elvin n cal calcula culati tion onss in sil silicon icon . . . 5.5 5.5 Along long 110 . . . . . . . . . . . . . . . . . . . . 5.6 5.6 Along long 111 : Crackons . . . . . . . . . . . . . . 5.7 5.7 Crack rack beha behavi vior or at nonz nonzeero temp temper eraature turess in sili silico con n 5.8 Comparisons with experiment . . . . . . . . . . 5.9 Final thoughts . . . . . . . . . . . . . . . . . . .

1 1.1 1.1

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . .

. . . . . . . . .

. . . . . . .

. . . . . . . . .

. . . . . . .

. . . . . . . . .

. . . . . . .

. . . . . . . . .

. . . . . . .

. . . . . . . . .

. . . . . . .

. . . . . . . . .

. . . . . . .

35 38 40 43 43 45 46

. . . . . . . . .

48 49 49 50 53 55 55 57 58 61

Continuu Continuum m Theories Theories of Fracture Fracture:: Scali Scaling ng Intr Introd oduc ucti tion on

The world is full of phenomena that are completely familiar but become more and more unexpected unexpected the more one thinks thinks about them. Fracture Fracture is one one of them. Who would would guess guess that that throwing a brick at a pane of glass would lead a few atomically thin regions of separation to run at the speed of sound from end to end? It is common to say at the outset of papers by physicists on fracture that little is understood. In fact, few problems problems have been studied more exhaustively exhaustively,, or solved solved for practical practical purposes more completely. Still, it is true that much remains to be solved. The problem has many features that make it particularly challenging. Cracks extract energy from the macroscopic reaches reaches of a solid and funnel funnel it spontane spontaneously ously to regions regions as small small as the atomic atomic scale. scale. Cracks can creep at rates less than an atom per second, or run at the speed of sound. They travel in solids well described by linear elasticity, but near the tip, where the action is, they are far from equilibrium. The crack connects atomic and macroscopic scales, and connot be understood without including both scales at once. The goal of these lectures is to lay out some of the elements of the physics of fracture. For the purposes of engineering, the most important concern is to prevent objects from breaking. breaking. From the perspective perspective of physics, other issues are also important. important. In particular, particular, before a physics audience, I will want to ask a set of questions that is not settled How can one calculate the energy needed for a crack to propagate? When is the motion stable stable or unstable? unstable? How does a crack choose its path? 2

The lectures will begin by establishing the basic stability criteria that determine when fractures fractures to begin moving. moving. They will proceed proceed to simple scaling scaling theories for rapid crack  motion, and then to a brief description of linear elastic fracture mechanics, an ingenious framework that allows a huge number of practical questions about fractures to be answered without without needing to resolve resolve how they behave near the tip. Howeve However, r, linear elastic fracture mechanics is not complete, and the lectures will continue with a discussion of current attempts to provide the missing information, including continuum theories of crack tips, atomic theories of crack tips, and techniques to consider cracks simultaneously at many scales.

1.2

Scali Scaling ng theo theory ry I: I: Thing Thingss fall fall apart apart

Consider a piece of rock, of area A and height h. According to equilibrium principles the rock should not be able to sustain its own weight under the force of gravity if it becomes tall tall enough. enough. I begin with with a simple simple estima estimate te of what what the critical critical height height should should be. The 2 gravitational potential energy of the rock is Ah g 2 where is the density density.. By cutting cutting the rock into two equal blocks of height h 2 and setting them side by side, this energy can be reduced to Ah2 g 4, for an energy gain of  Ah2 g 4. The cost of the cut is the cost of  creating new rock surface. Estimate that there is a atom per square angstrom at the surface, and that that the the cost of cuttin cutting g a bon bond d at the the surface surface is is 1 eV. eV. The The the cost of creati creating ng new new surface surface 2 2 3 ˚ , or G 10 J/m . Taking the density to be is G 1 eV/ A 2000kg/m , the critical height at which it pays to divide the rock in two is 4G g

h

14cm

(1)

So every block of stone more than 20 cm tall is unstable under its own weight. A similar estimate estimate applies applies to steel, concrete, or the bones in our legs. Almost every every feature feature in the solid world around us is out of mechanical equilibrium. Mountains, buildings, and even our own bodies should fall apart in the long run under the influence of gravity. Yet they are apparently very stable, something which can only be explained if strong energy barriers keep them intact. The next task is to estimate the size of  these barriers.

1.3

Scalin Scaling g Theo Theory ry II: All All toge togethe therr

It appears that an easy way to obtain a rough value would be by imagining what happens to the atoms of a solid as one pulls it uniformly at two ends. ends. At first, the forces between between the atoms increase, but eventually they reach a maximum value, and the solid falls into pieces, pieces, as shown shown in Figure 1. Interatomic Interatomic forces vary greatly between between different elements elements and compounds, but the forces typically reach their maximum value when the distance between atoms increases by around 20% of their original separation. The force needed to stretch stretch a solid slightly slightly is (Figure (Figure 1) F 

EA  L

3

(2)

F 

      L        d       L

      L

 A F 

Figure 1: Mechanically stable configurations are often far from their lowest energy state. For example, a solid completely free of flaws would only pull apart when all bonds as long a plane snapped simultaneously, despite the fact that it would generally be energetically advantageous for the solid to be split. Material Material Young’s oung’s Modulus Modulus GPa Iron 195-205 Copper 110-130 Titanium 110 Silicon 110-160 Glass 70 Plexiglas 3.6

Theoretical Theoretical Strength Strength GPa 43-56 24-55 31 45 37 3

Practical Practical Strengt Strength h GPa 0.3 0.2 0.3 0.7 0.4 0.05

Practical/ Practical/Theore Theoretical tical 0.006 0.005 0.009 0.01 0.01 0.01

Table 1: The experimental experimental strength strength of a number of materials materials in polycrystall polycrystalline ine or amorphous form, compared to their theoretical strength, from [1] and [2] where E  is the material’s Young’s modulus, so the force per area needed to reach the breaking point is around F   E  (3)  A 5 As shown in Table Table 1, this estimate is in error by orders of magnitude. magnitude. The problem does not lie in the crude estimates used to obtain the forces at which bonds separate, but in the conception conception of the calculation. calculation. The first scaling argument greatly greatly underestimated underestimated the practical resistance of solid bodies to separation, while this one greatly overestimates it. The only way to uncover correct orders of magnitude is to account for the actual dynamical mode by which brittle solids fail, which is by cracking. cracking.

1.4

Scali Scaling ng theo theory ry III: Cracks Cracks at last last

Imagining that solid objects quickly reach states of lowest energy leads to the conclusion that nothing nothing over 1 m high is stable. Imagining Imagining that solid objects objects fail through the simul4

taneous taneous failure of a macroscopic macroscopic plane’s plane’s worth of bonds leads to a huge overestimation overestimation of  practical practical strength. strength. To obtain even even the correct correct order of magnitude in describing the resistance of solids to failure, failure, one must consider consider the presence presence of cracks. A crack crack is a region region of consecu consecuti tive ve broken broken bonds. bonds. A crack crack in an otherwis otherwisee perfect perfect material material creates creates a stress singularity singularity at its tip. For an atomically atomically sharp crack tip, a single crack a few microns long is sufficient to explain the large gaps between the theoretical and experimental experimental material material strengths as shown shown in Table 1. The theoretical theoretical strength of glass fibers, for example, can be approached by means of acid etching of the fiber surface. The etching process serves to remove any initial microscopic flaws along the glass surface. By removing its initial flaws, a 1mm 2 glass fiber can lift a piano (but during the lifting process, I wo woul uldn’ dn’tt advi advise se stan standi ding ng unde underr the the pian piano). o). Th Thee stre stress ss sing singul ular arit ity y that that deve develo lops ps at the tip tip of  a crack serves to focus the energy that is stored in the surrounding material and efficiently use it to break one bond after another. another. Thus, the continuous continuous advance advance of the crack tip, or crack propagation, provides an efficient means to overcome the energy barrier between two equilibrium states of the system having different amounts of mechanical energy.

 L

dl  L dl

 y  x

Figure Figure 2: The upper upper surface surface of a strip strip of height height L and thickness is rigidly displaced upwards by distance . A crack is cut through through the center center of the strip, and reliev relieves es all stresses stresses in its wake. When the crack moves distance dl from (A) to (B), the net effect is to transfer length dl of strained material into a length dl of unstrained material. Consider Consider the crack moving in a thin strip depicted in Figure 2. Far to the right of the crack tip the material is under tension, and stores an elastic elastic energy right per area 1 2 E  2  L

right

(4)

Far to the left of the crack, the material is completely relaxed, and therefore if the boundaries of the system are rigid as the crack proceeds in a steady fashion, the energy dissipated by crack motion must be 1 2 E  G (5) 2  L per unit area crack advance. advance. The reason for this assertion assertion is that when the crack moves ahead in a steady fashion the stress and strain fields around it do not change; as shown in Figure 2, the fields translate in the direction of crack motion, and the change in energy 5

comes comes from from the transfo transforma rmatio tion n of stresse stressed d materi material al ahead ahead of the crack crack to unstres unstressed sed materi material al behind. This conclusion rests upon symmetry, and does not even demand that strains ahead of the crack crack tip to be so small that linear linear mechanics mechanics be applic applicabl able. e. Steady Steady velocity velocity and a corresponding energy flux G are achieved in the long time limit, the natural scale in experiments as well as in analytical calculations, where the crack tip reaches dynamic equili equilibri brium um with waves waves reflectin reflecting g from from top and bottom bottom boundari boundaries. es. That That is, the return of acoustic waves from the system boundaries actually simplifies the task of understanding energy energy balance in the thin strip geometry geometry.. According According to fracture fracture mechanics, mechanics, the relationship relationship between energy flowing to a crack tip and crack velocity is, for a given lattice direction, universal. universal. Having Having obtained obtained the relation in a strip, one knows it for any of the vast range of geometries to which fracture mechanics is applicable, such as a long crack in a large plate [3].

1.5

The scalin scaling g of of dynam dynamic ic fractu fracture re

l

Figure 3: As a crack of length l expands at velocity in an infinite plate, it disturbs the surrounding medium up to a distance on the order of  l . The first analysis of rapid fracture was carried out by Mott [4], and then slightly improved proved by Dulaney and Brace [5]. It is a dimensional dimensional analysis analysis which clarifies clarifies the basic physical processes, despite being wrong in many details, and consists of writing down an energy balance equation for crack motion. Consider a crack of length l t  growing at rate t  in a very large plate where a stress is applied at the far boundaries of the system, as shown in Figure 3. When the crack  extends, its faces separate, causing the plate to relax within a circular region centered on 6

the middle of the crack and with diameter diameter of order l . The kinetic energy involved involved in moving 2 a region of this size is M  2, where M  is the total mass, and is a characteristic velocity. Since the mass of material that moves is proportional to l 2 , the kinetic energy is guessed to be of the form 2 2 KE C K  (6) K l The region of material that moves is also the region from which elastic potential energy is being released as the crack opens. Therefore, the potential energy gained in releasing stress is guessed to be of the form C P l 2 PE (7) 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 diverge.. This divergenc divergencee will be demonstrated demonstrated later, later, but for the moment, let us proceed proceed fearlessly fearlessly.. The final process contributing contributing to the energy energy balance balance equation is the creation of  new crack surfaces, which takes energy l . The fracture energy  accounts not only for the minimal energy needed to snap bonds, but also for any other dissipative processes that may be needed in order for the fracture to progress; it is often orders of magnitude greater than the thermodynamic surface energy. However, for the moment, the only important fact is that creating new surface scales as the length l of the crack. crack. So the total total energy energy of the system containing a crack is given by 2 C K  K l

2

l

C P l 2

qs

(8)

l

with qs

l

(9)

Consider first the problem of quasi-static crack propagation. If a crack moves forward only slowly, its kinetic energy will be negligible, so only the quasi-static part of the energy, qs , will be important. For cracks where l is sufficiently small, the linear cost of fracture energy is always greater than the quadratic gain of potential energy, and in fact such cracks would heal and travel backwards if it were not for irreversible processes, such as oxidation of the crack surface, surface, which which typicall typically y preven preventt it from happeni happening. ng. That That the crack grows grows at all is due to additional irreversible processes, sometimes chemical attack on the crack  tip, sometimes sometimes vibration or other irregular irregular mechanical mechanical stress. It should be emphasized emphasized that the system energy increases as a result of these processes. Eventually, at length l 0 , the energy gained by relieving elastic stresses in the body exceeds the cost of creating new surface, and the crack becomes able to extend spontaneously. One sees that at l 0 , the energy functional qs l has a quadratic quadratic maximum. The Griffith Griffith criterion for the onset of fracture fracture is that fracture occurs when the potential energy released per unit crack extension equals the fracture energy, . Thus fracture fracture in this system will occur occur at the crack length l 0 where d 

qs

0

dl

(10)

Using Eq. (9) I find that l0

2C P 7

(11)

     s      q

      E

    y     g     r     e     n     e     c      i      t     a      t     s      i     s     a     u      Q

l0

Crack length l Figure 4: 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 l 0 the crack begins to move rapidly, and energy is conserved. Eq. (9) now becomes qs

l

qs

l0

C P l

l0

2

(12)

This function is depicted in Figure 4. Much of engineering fracture mechanics boils down to calculating l0 , given things such as external stresses, which in the present case have all been condensed into the constant Dynamic fracture fracture starts in the next instant, instant, and because it is so rapid, the energy of the C P . Dynamic system is conserved, remaining at qs l0 . Using Using Eq. (8) and and Eq. (12), (12), with qs l0 gives C P l0 l0 1 (13) t  max 1 C K  l l K  This equation predicts predicts that the crack will accelerate accelerate until it approaches approaches the speed max . The maximum speed cannot be deduced from these arguments, but Stroh [6] correctly argued that max should be the Rayleigh wave speed, the speed at which sound travels over a free surface. A crack is a particularl particularly y severe distortion distortion of a free surface, but assuming that it is legitimate to represent a crack in this way, the Rayleigh wave speed is the limiting speed to expect, and this result was implicit in the calculations of Yoffe [7]. In this system one needs only to know the length at which a crack begins to propagate in order to predict all the following dynamics. dynamics. As we will see presently presently,, Eq. (13) comes astonishingly close to anticipating the results of a sophisticated mathematical analysis developed veloped over fifteen years. This result is especially especially puzzling puzzling since the forms (6) and (7) for the kinetic and potential energy are incorrect; these energies should actually diverge as the crack begins to approach the Rayleigh Rayleigh wave speed. The reason that the dimensional dimensional analysis succeeds anyway is that to find the crack velocity one takes the ratio of kinetic and potential energy; their divergences are of exactly the same form, and cancel out.

8

2

Conti Continuu nuum m fractu fracture re mechan mechanics ics

In this section I will attempt to review briefly the background, basic formalism and underlying assumptions assumptions that form the body of continuum fracture fracture mechanics. mechanics. There are many textbooks on the subject from many different points of view [8–14] , and I will only cover a small subset of topics topics in them. I will first discuss the reduction reduction of linear linear elasticity to two dimensions. Some basic concepts common to both static and dynamic cracks will then be discuss discussed. ed. These These will include include the creati creation on of a singula singularr stress stress field at the tip of a static static crack together together with criteria criteria for the onset and growth growth of a moving moving crack. I then turn to a description description of moving cracks. cracks. I will first describe the formalism used to describe a crack  moving at a given velocity and, using this formalism, look at the dependence of the near tip stress fields as a function of the crack’s velocity. I then discuss criteria for determination of a crack’s crack’s path and conclude the section by discussing a number of points that the theory can not address.

2.1

Structure Structure of fracture fracture mechanics mechanics..

The structure of fracture mechanics follows the basic ideas used in the scaling arguments described described in the previous section. section. The general strategy strategy is to solve for the displacement displacement fields in the medium subject to both the boundary conditions and the externally applied stresses. stresses. The elastic energy energy transported by these fields is then matched matched to the amount of  energ energy y dissipa dissipated ted througho throughout ut the system, system, and an equati equation on of motion motion is obtain obtained. ed. For a single moving crack, as in the scaling argument above, the only energy sink existing in the system system is at the tip of the crack crack itself. itself. Thu Thus, s, an equation equation of motion can be obtained obtained for a moving crack if one possesses detailed knowledge of the dissipative mechanisms in the vicinity of the tip.

2.2

Dissip Dissipati ation on and the pr proce ocess ss zone zone

Unfortunately, the processes that lead to dissipation in the tip vicinity are far from simple. Depending Depending on the type of material, material, there is a large number of complex dissipati dissipative ve processes processes ranging from dislocation formation and emission in crystalline materials to the complex unravelin unraveling g and fracture fracture of intertangled intertangled polymer polymer strands in amorphous polymers. polymers. At first glance, the many different (and in many cases, poorly understood) dissipative processes that that are observed observed would would appear appear to preclude preclude a univer universal sal descrip descriptio tion n of fractu fracture. re. A way way around this problem was proposed by Irwin and Orowan [15,16]. Fracture, together with the complex dissipative processes occurring in the vicinity of  the tip, occurs due to intense values of the stress field that occur as one approaches the tip. As I will show shortly, if the material surrounding the tip were to remain linearly elastic until fracture, a singularity of the stress field would result at the mathematical point associated sociated with the crack tip. Since a real material material cannot support singular singular stresses, in this vicinity the assumption of linearly elastic behavior must break down and material dependent, dissipativ dissipativee processes processes must come into play. play. Irwin and Orowan Orowan independently independently proposed proposed that the medium around the crack tip be divided into three separate region as follows.

9

Figure 5: Structure of fracture mechanics. The crack tip is surrounded by a region in which the physics is unknown. unknown. Outside Outside this process zone is a region in which elastic elastic solutions solutions adopt a universal form. The process zone: In the region immediately surrounding the crack tip, called the process zone (or cohesive zone), all of the nonlinear dissipative processes that ultimately allow allow a crack to move forward, forward, are assumed to occur. occur. Fracture Fracture mechanics mechanics avoids any sort of detailed description of this zone, and simply posits that it will consume some energy per unit area area of crack extensi extension. on. The size of the process process zone zone is material dependent, ranging from nanometers in glass to microns in brittle polymers, to centimeters or larger in ice near its melting point. The typical size of the process zone can be estimated by using the radius at which an assumed linear elastic stress field surrounding the crack tip would equal the yield stress of the material.

Everywhere outside of the process zone the response response of the The universal elastic region: Everywhere material material can be described described by continuum continuum linear elasticity elasticity.. In the vicinity vicinity of the tip, but outside of the process zone, the stress and strain fields adopt universal singular forms which solely depend on the symmetry of the externally applied loads. In two dimensions the singular fields surrounding the process zone are entirely described by three constants, called called stress intensity factors. The stress intensity factors incorporate all all of the the info inform rmat atio ion n rega regard rdin ing g the the load loadin ing g of the the mate materi rial al and and are are rela relate ted, d, as we shal shalll see, to the energy flux into the process zone. The larger the overall size of the body in which the crack lives, the larger this region becomes. becomes. In rough terms, for given given values of the stress intensity factors, the size of this universal elastic region scales as L, where L is the macroscopic scale on which forces are applied to the body. Thus the assumptions of fracture mechanics become progressively better as samples

10

become larger and larger. Outer elastic region: Far from the crack tip, stresses and strains are described by linear elasti elasticit city y. There There is nothin nothing g more more general general to relate relate;; deta details ils of the soluti solution on in this this region region depend upon the locations and strengths of the loads, and the shape of the body. In special cases, analytical solutions are available, while while in general one can resort to numerical merical solution. solution. At first glance, glance, the precise linear linear problem problem that must be solved might seem inordinately complex. How can one avoid needing explicit knowledge of complicated boundary conditions on some complicated loop running outside the outer rim of the process zone? The answer is that so far as linear elastici elasticity ty is concerned, viewed on macroscopic scales the process zone shrinks to a point at the crack tip, and the crack becomes a branch cut. Replacing the complicated domain in which linear elasticity actually holds with an approximate one that needs no detailed knowledge of the process zone is another approximation that becomes increasingly accurate as the dimensions of the sample, hence the size of the universal elastic region, increase.

The dissipative processes within the process zone determine the fracture energy, , defined as the amount of energy required required to form a unit area of fracture surface. surface. In the simplest case, where no dissipative processes other than the direct breaking of bonds take place, place, is a constant, depending on the bond energy. In the general case may well be a complicated function of both the crack velocity and history and differ by orders of magnitude from the surface energy defined as the amount of energy required to sever a unit area of atomic bonds. bon ds. No general general first princi principle pless descri descripti ption on of the process process zone exists, exists, althou although gh numerou numerouss models have been proposed (see e.g. [13]).

2.3

Conventi Conventional onal fracture fracture modes modes

It is conventional to focus upon three symmetrical ways of loading a solid body with a crack. These are known as modes, and are illustrated in Figure 6. A generic loading situation produced by some combination combination of forces without any particular particular symmetry symmetry is referred referred to as mixed mode  fracture. fracture. Understanding Understanding mixed-mode mixed-mode fracture is obviously obviously of practical practical importance, but since our focus will be upon the physics of crack propagation rather than upon engineering applications, we will restrict attention to the special cases in which the loading loading has a high high degree degree of symmet symmetry ry.. The fracture fracture mode that I will will mainly mainly deal with here is Mode I, where the crack faces, under tension, are displaced in a direction normal to the fracture plane. plane. In Mode II, the motion of the crack faces is that of shear along the fracture plane. Mode III fracture corresponds to an out of plane tearing motion where the direction direction of the stresses at the crack faces is normal to the plane of the sample. One experimental difficulty of Modes II and III is that the crack faces are not pulled away from one another. It is unavoidable that contact along the crack faces will occur. The resultant friction between the crack faces contributes to the forces acting on the crack, but is difficult to measure precisely. For these reasons, of the three fracture modes, Mode I corresponds most closely to the conditions conditions used in most experimental experimental and much theoretical theoretical work. In two–dimensional two–dimensional isotropic materials Mode II fracture cannot easily be observed, since slowly propagating 11

Mode I Mode II

 y

 x  z

Mode III

Figure 6: Illustration of the three conventional fracture modes, which are characterized by the symmetry of the applied applied forces about the crack plane crac cracks ks spont spontan aneo eousl usly y orie orient nt them themse selv lves es so as to make make the the Mod Modee II comp compon onen entt of the the load loadin ing g vanish near the crack tip [17], as I will discuss in Section 2.11. Mode II fracture is however observed observed in cases where material material is strongly anisotropic. anisotropic. Both friction friction and earthquakes earthquakes along a predefined fault are examples of Mode II fracture where the binding across the fracture interface is considerably weaker than the strength of the material that comprises the bulk material. material. Pure Mode III fracture, although experiment experimentally ally difficult difficult to achieve, is sometimes used as a model system for theoretical study since, in this case, the equations of  elasticity simplify considerably. Analytical solutions, obtained in this mode, have provided considerable insight to the fracture process.

2.4

Stresses Stresses Around Around an Elliptic Elliptical al Hole

It is possible to find the stresses surrounding an elliptical hole sitting within a plate that has been placed under remote tension, tension, as illustrated illustrated in Figure 7. To simplify matters matters as much as possible, assume that the stresses applied to the plate are such as only to create a displacement u u z in the z direction, which obeys Laplace’s equation, 2

0

u

(14)

The theory of complex variables can once again be brought to bear in order to find solutions. For the boundary problem problem at hand, conformal mapping  is the appropriate technique. Because u is a solution of Laplace’s equation, it can be represented by u

2 12

(15)

where is analytic, and x iy. The asymptotic behavior of  is easy to determin determine. e. Far Far from from the hole, hole, the stress stress goes to a constant value , where is dimensionless. So u  yz

 y i

i 2 for

x

iy

i

x

yz

(16)

iy

(17)

How does the presence of the hole affect the stress field? Because the edges of the hole are free, all stresses normal to the edge must vanish. Let t  be a variable that parameterizes the edge of the hole, so that  x t  y t  (18) travels travels around the boundary of the hole as t  moves moves along the real axis. Then T 

x

 y

t 

t 

and

N 

y

 x

t 

t 

It is easy to verify that N  T  =0, and because T  is tangent N  must be normal.

(19)

are tangent vector and normal vectors along the edge of the hole, so requiring normal stress to vanish means that  xz

N 

yz

0

(20)

u

u

 y

 x

 x

 y

t 

t 

0

u  x

u  y

 y t 

 x t 

 x ix t 

ix t 

 y

t 

iy

iy

t 

0

(21) (22)

Insert (15) for u.

This equality holds only on the boundary. boundary.

(23)

Because is arbitrary up to a constant anyway, there is no worry about dropping a constant of integration. integration.

(24)

when lies on the boundary. Equation Equation (24) appears appears innocent, innocent, but is in fact a powerful powerful relation that can lead to rapid solution solution of complicate complicated d boundary-va boundary-value lue problems. Suppose, Suppose, to be definite, definite, that the hole is elliptical and thus can be described by  p

with

(25)

 p is a number lying between 0 and 1.

lying on the unit circle, ei

and real. real. When When p 0, the boundary is circular, and when p along the real axis. Considering as a function of  , one has 1

(26) 1, the boundary is a cut

The notation means that if  is expanded as a power series in , one should should take the complex conjugate of all the coefficients in the expansion.

13

(27)

because 1 on the unit circle. circle. Equation Equation (27) is a relation between between two analytic analytic functions that holds over the whole unit circle. The difference between and 1 is an analytic function; the Taylor series of the difference vanishes on the unit circle, and by analytic continuation the difference vanishes everywhere, so the two functions must be equal everywhere in the complex plane. Now can be determined completely by analyzing its asymptotic behavior. behavior. Outside of the hole must be completely regular, except for the fact that it diverges as i for large . The relation between and is 2

If the other sign of the square root were chosen here, the argument below would have to be modified in several places, but the final answer would be the same.

4 p

2 Therefore, when

is large, large,

(28)

, and in accord with Eq. (17) one obtains for

i

(29)

Consulting Eq. (27) one concludes also that 1

for

i

(30)

which means that i i

for

0

(31)

for

0

(32)

However, can have no other singularities within the unit circle, or else would have corresponding singularities outside the unit circle, which is forbidden if  u is to be smooth away from the hole. Having determined all the possible singularities of  within and without the unit circle, it is determined up to a constant. constant. It must be given by i i

i

2

1

(33)

Add Eqs. (32) and (29).

1

Use Eq. (28) and 1

4 p

2 1

i 1

2 p 4 p

1 2

1

4 p

2

(34)

2 p.

Displa Displacem cement entss and stres stresses ses can now now be determ determine ined d from Eqs. Eqs. (15) (15) and (16). (16). Figure 7 shows shows a plot of  yz along the line y 0 for a narrow ellipse with p 0 9.

2.5

Stress Stress Intens Intensity ity Factor actor

The limit p 1 is particularly particularly interesting interesting.. The elliptical elliptical hole closes down and becomes a thin crack reaching from x 2 to x 2. The funct function ion acquires a branch cut over

14

Radius of curvature R l

 yz        0

 y

     y

 x  z

     g      n      o         l      a   ,

20 15 10

     z      y

     s      s      e      r       t

       S

5 0

3 2 Distance x

1

4

Figure 7: The stresses at the tips of an elliptical elliptical hole in a solid are much greater than those applied off at infinity. The plot shows a solution of  yz derived from Eq. (35) for 1 and m 0 9. Note the 20-fold increase increase in stress near the crack tip. exactl exactly y the same same region region.. The displac displaceme ement nt u is finite finite every everywhe where, re, but the stress stress singular, diverging on the x axis as u  yz

 x

 y

 x

2 x

2

x

2

as x

2

yz

becomes

(35)

The stress intensity factor  K  is a coefficient of the diverging stress, defined by K 

lim

r  0

In the case of Eq. (35), K 

2.6

2 r  2

yz

r 

2 is the distance from the crack tip.

x

(36)

.

Universa Universall singularit singularities ies near the crack tip

The calculations outlined in the previous section demonstrate that stresses become singular near near the tip of a long long thin thin hole. The computa computatio tions ns were restric restricted ted to the case case of Mode III, antiplane antiplane shear, shear, for which the algebra algebra is relatively relatively simple. simple. The cases of Mode I and Mode II, which are of greater practical importance, require lengthier expressions, but are not radically different. As one approaches the tip of a crack in a linearly elastic material, the stress field surrounding surrounding the tip develops develops a square root singularity singularity.. As first noted by Irwin [18], the stress field at a point r  near the crack tip, measured in polar coordinates with the crack line corresponding to 0, takes the form i j

K 

1 2 r  15

 f i

j

(37)

where is the instantaneous crack velocity, and is an index running running through Modes I, II, and III. For each of these three symmetrical loading configurations, f i j in Eq. (37) is a known universal function. The coefficient, K  , called the Stress Intensity Factor , contains all of the detailed information regarding sample loading and history. K  will, of course, be determined by the elastic fields that are set up throughout the medium, but the stress that locally locally drives drives the crack is that which which is present at its tip. Thus, this single quantity quantity will entirely determine the behavior of a crack, and much of the study of fracture comes down to either the calculation or measurement of this quantity. One of the main precepts of fracture mechanics in brittle materials is that the stress intensity factor provides a universal versal description description of the fracture process. In other words, no matter matter what the history or the external conditions in a given system, if the stress intensity factor in any two systems has the same value, the crack tip that they describe will behave in the same way. The universal form of the stress intensity factor allows a complete description of the behavior of the tip of  a crack where one need only carry out the analysis of a given problem within the universal elastic region (see 2.2). For arbitrary loading configurations, the stress field around the crack tip is given by three stress intensity factors, K  , which lead to a stress field that is a linear combination of  the pure Modes: 3 1 K   f i j (38) i j r  2 1

2.7

The relati relation on of the stress stress intensity intensity factor to energy energy flux

How are the stress intensity factors related to the flow of energy into the crack tip? Since one may view a crack as a means of dissipating built-up energy in a material, the amount of energy flowing into its tip must influence its behavior. Irwin [19] showed that the stress intensity factor is related to the energy release rate, G, which is defined as the quantity of  energy flowing into the crack tip per unit fracture surface formed. The relation between the two quantities has the form 3 2 1 (39) G  A K 2  E  1 where is the Poisson ratio of the material and the three functions A depend only upon the crack veloci velocity ty . This relati relation on between between the stress intensity intensity factor factor and the fracture fracture energy energy is accurate whenever the stress field near the tip of a crack can be accurately described by Eq. (37). (37). The near field field approxi approximat mation ion of the stress stress fields fields surroun surroundin ding g the crack tip embodied in Eq. (38) becomes increasingly more accurate as the dimensions of the sample increase.

2.8

Elasticity Elasticity in Two Dimensio Dimensions ns

The complete complete formal formal devel developm opment ent of fractu fracture re mechani mechanics cs is very very elabor elaborate ate.. For static static cracks, there is a lovely formalism developed by Muskhelishvili [20] that allows one to use conformal mapping to obtain the stress fields around two–dimensional voids in a solid. I will present here a variant of this technique that allows one to find fields surrounding a 16

moving moving crack. I will not take it very far; just far enough to find the universal universal parts of the fields in the vicinity of the crack tip. The starting point is the equation of motion for an isotropic elastic body in the continuum limit, 2 u 2 (40) u u 2 t  originally found by Navier. u is a field describing the displacement of each mass point from its original location in an unstrained body, is the density, and the constants and are called the Lam´e constants, have dimensions of energy per volume, and are typically of  order 1010 ergs/cm3 . The tensor structure of elasticity makes it particularly difficult to solve fully threedimensional problems, and it is difficult to carry out controlled three-dimensional experiments iments as well. Fortunatel Fortunately y, there are cases in which the theory naturally reduces reduces to two dimensions, where most of the analytical results have been obtained. A first case is called called anti-plane anti-plane shear. shear. Imagine Imagine one is tearing tearing a telephone telephone book, with one hand gripping on the left, the other gripping on the right, one pushing up and the other pulling pulling down. The only non-zero non-zero displacement displacement is u z , and it is a function of  x and y alone. The only non-vanishing stresses in this case are u z

 xz

(41)

 x

and

u z

 yz

(42)

 y

The equation of motion for u z is 2

1 c2

u z 2

t 

2

(43)

u z

where (44)

c

Therefore, the vertical displacement obeys the ordinary wave equation. A second second case correspon corresponds ds to pullin pulling g on a thin thin plate, plate, and is called called plane plane stress. stress. Let the z direction direction be the direction direction that goes through the plate. If the scale over which stresses are varying in x and y is large compared with the thickness of the plate, then one might expect that the displacements in the z direction will come quickly into equilibrium with the local x and y stresses. When the material is being stretched, (think of pulling on a balloon), the plate will contract in the z direction, and when it is being compressed, the plate will thicken. Therefore, one guesses that u z

(45)

z f  u x u y

and that u x and u y are independent of  z. One can deduce the function function f  by noticing that must vanish on the face of the plate. plate. This means that u x

u y

 x

 y

2 17

u z  z

0

zz

(46)

at the surface of the plate, which implies that u z

 f  u x u y

2

 z

So u x

u y

u z

 x

 y

 z

2 2

u x

u y

 x

 y

u x

u y

 x

 y

and one can write u

u

u

 x

 x

 x

where

(47)

(48)

(49)

2

(50)

2

and and now range only over x and y. Therefore, Therefore, a thin plate obeys the equations equations of  two-dimensional elasticity, with an effective constant , so long as u z is dependent upon u x and u y according according to Eq. (47). In the following following discussion, discussion, the tilde over will usually be dropped, with the understanding that the relation to three-dimensional materials properties is given by Eq. (49). The equation of motion is still Navier’s equation, Eq. (40), but restricted to two dimensions. A few random useful facts: materials are frequently described by the Young’s modulus  E  and Poisson ratio . In terms of these constants,  E 

1

1

 E 

2

 E  2

1

2 1

(51)

The following following relation relation will be useful in discussing discussing two-dimensional two-dimensional static problems. problems. First note that u 2 (52) Second, taking the divergence of Eq. (40), one finds that 2

2 Therfore,

2 2

t 

u obeys the wave equation, with the longitudinal wave speed

2

cl

Similarly,

(53)

(54)

u also obeys the wave equation, but with the shear wave speed

(55)

ct 

The fact that the local thickness of the plate is tied to stresses in the x and y directions leads to two optical methods to determine stress fields experimentally. The first method relies on the fact that when light reflects off a curved surface, the reflected intensity becomes 18

Figure 8: Interference fringes captured during rapid motion of crack though brittle plastic. K Ravi–Chandar. Ravi–Chandar. singular at certain points that depend on the details of the geometry. In practice, when light is shined on a crack tip, a sharp dark spot surrounds the tip, and its shape and size can be used to deduce the stresses. This technique is known as the method of caustics. A second method, older and more reliable, relies upon the fact that materials under stress will typically ically rotate the plane of polarizatio polarization n of transmitted transmitted light. light. It is possible possible to determine the basic structure of this rotation without detailed calculation. The rotation must depend upon features of the stress tensor which are rotationally invariant, and therefore can depend only upon the two principal stresses, which are the the diagonal elements in a reference frame where the stress tensor is diagonal. diagonal. In addition, there should be no rotation rotation of polarizatio polarization n when the material is stretched uniformly in all directions, in which case the two principal stresses stresses are equal. So the angular rotation rotation of the plane of polarization polarization must be of the form K 

1

2

(56)

where 1 and 2 are the principal stresses (eigenvalues of the stress tensor), and K  is a consta constant nt that that will will have have to be determ determine ined d experi experimen mental tally ly.. Whene Whenever ver stresses stresses of a twotwodimensional problem are calculated analytically, the results can be placed into Eq. (56), and compared with experimental fringe patterns. Fast optical systems have been developed to carry out this procedure procedure for rapidly rapidly moving cracks. Although Although one might wonder about the extent to which Eq. (56) is obeyed when cracks move at speeds on the order of the speed of sound, experimental results match rather well with predictions for crack tip fields.

2.9

Steady Steady States States in Plane Plane Stress Stress

We now proceed to find the form of the singularity around the tip of a Mode I crack. Begin with the dynamical equation for the strain field u of a steady state in the moving frame, 2 u 2 2 u u (57)  x2 Divide u into transverse and longitudinal parts so that u

ut 

19

ul

(58)

with ul

t 

and ut 

l

t 

 y

(59)

 x

It follows immediately that 2

2

2

2

ul

 x2

2

f 

 x2

for some function f  which must be harmonic ( f  x that 2

2

 x2

 y2

2

2

 x2

 y2

2

2

2

2

i f  y is a function of  x 2

2

ut 

(60)

iy). We have then

l

0

(61)

t 

0

(62)

where 2

2

1

(63a)

2 2

2

1

(63b)

Therefore the general form of the potentials is l

0 l

z

0 l

z

1 l

x

i y

1 l

x

i y

(64)

t 

0 t 

z

0 t 

z

1 t 

x

i y

1 t 

x

i y

(65)

subject to the constraint of Eq. (60), which gives a relation between l0 and t 0 . In fact, the purely harmonic pieces l0 and t 0 disappear entirely from the expressions for u They result from the freedom one has to add a harmonic function to l and t  simul1 1 z and z z I have taneously, and can be neglected. Defining z t  z l z for u u x u y

z

z

i

i

z

z

(66a)

z

z

z

(66b)

i y

(67)

z

where  z

x

i y

z

x

Equation Eq. (66) gives a general form for elastic problems which are in steady state moving at velocity . Define also z z and z z Then the stresses are given by  xx

yy

2

 xx

yy

2

1

xy

2

2i

2

z 2

 z z

2

1

z z

2

z

20

(68a)

2i

z

z

(68b)

1

z

z

(68c)

we will also need the rotation, which means u y

u x

 x

 y

2

1

 z

(69)

z

It is worth writing down the stresses directly as well:

 xx

2

1

 yy

1

2

2

 z 2

2i

z  z

z

2i

z

(70)

z z

z

(71)

The definitions of  and in Eq. (63) have been used to simplify the expressions. expressions. To solve solve a general general proble problem, m, one has to find the functi functions ons and which which match match bou bounda ndary ry conditions. conditions. It is interesting interesting to notice that when 0, the right hand side of Eq. (68a) goes goes to zero as well. well. Since Since one will will be finding finding the potenti potentials als from given given stresses stresses at the boundaries, must diverge as 1 , and the right hand side of  E q 68 will turn into a derivati derivative ve of  with respect to . That is why the static theory has a different structure than the dynamic theory. In fact, the dynamic theory is more straightforward. As a first first appl applic icat atio ion, n, I will will show show that that a movi moving ng crac crack k unde underr symme symmetr tric ic load loadin ing g beco become mess unstable at a certain speed. Assume the crack to lie along the negative x axis, terminating at x 0 and moving forward. The problem is assumed symmetric under reflection about the x axis, but no other assumption is needed. This instability was first found in a particular case by Yoffe (1951). We know that in the static case, the stress fields have a square root singularity at the crack tip. I will assume the same to be true in this case (the assumption assumption is verified verified in all cases that can be worked explicitly.) Near the crack tip, I assume that  z

Br 

iBi z1

2

(72)

 z

Dr 

iDi z1

2

(73)

We first appeal to symmetry. Observe that u x

y

u x y

u y

y

(74)

u y

Placing Eq. (73) into Eq. (66) and using Eq. (74), we find immediately that B i Dr  0. Thus Br  iDi  z z (75)  z1 2  z1 2 We also observe that the square roots in Eq. (73) must be interpreted as having their cuts along the negative x axis, corresponding corresponding to the crack. On the crack surface, we have two boundary conditions, which require that xy and yy vanish. vanish. Upon substituting substituting Eq. Eq. (75) into Eq. (71) we find that the condition upon yy is satisfied identically for x 0 y 0. However, substituting into Eq. (69) with y 0 we find that  xy

i 2 Br 

2

21

1 Di

1  x

1  x

(76)

Thus

 Di

2

 Br 

2

(77)

1

This relation is enough to find the maximum velocity at which a crack can proceed stably along the x axis. Using Eq. (77) we find that K   xx

2

1

2 D K 

 yy

K   xy

2 D

2

2i

2

2

2 2

1

2 D

1

1

1  z

2

1  z

1  z

4

1  z

1  z

4

1  z

1  z

1  z

1  z

1  z

1  z

1  z

(78a) (78b) (78c)

with  D

4

2 2

1

(78d)

The constant K  is again called the stress intensity factor, and is given by lim x

K 

0

2 x

(79)

yy

In order to find the direction of maximum stress, one must approach the tip of the crack  along a line at angle to the x axis, and compute the stress perpendicular to that line. So one wants to choose r cos

 z

ri sin

r cos

z

ri sin

(80)

and to evaluate the stress cos2

 yy

sin2

 xx

sin 2

xy

(81)

When this is carried out, one finds that above a certain velocity (for Poisson ratio 1 3, the critical velocity is about .61) the direction of maximum tearing stress points away from the axis, axis, and the crack crack would would presum presumabl ably y become become unstab unstable. le. The point point at which which this this happens happens is referred to as the Yoffe instability.

2.10 2.10

Energy Energy flu flux x and limit limiting ing crack crack spee speed d

Energy flux may be found from the time derivative of the total energy. We have d  dt 

K 

d 

P

dt 

dxdy

2

u u

1 u 2  x

(82)

The spatial integral is taken over an region which is static in the laboratory frame. So d  dt 

K 

P

dxdy

22

u u

u  x

(83)

where the symmetry of the stress tensor under interchange of indices is used for the last term. Using the equation equation of motion motion u

x

we have dxdy

u

u

 x dxdy

S

(85)

 x

(86)

u

 x

u

(84)

x

 x

(87)

n

where the integral is now over the boundary of the system, and n is an outward unit normal. By using the asymptotic forms Eq. (78) for yy and the corresponding corresponding expressions expressions for u y from Eq. (66a), one finds that the total energy flowing into the crack tip per unit time is  J tot

1

2

2 4

1 1

2 2

K 2

(88)

2 2 The factor 4 1 is a function of crack velocity whose roots happen to define the Rayleigh wave speed, which is the speed at which waves travel over a free surface. Because this factor vanishes at the Rayleigh wave speed and becomes negative for larger values of crack velocity, the Rayleigh wave speed is conventionally thought to provide an upper limiting limiting speed for cracks. cracks. The argument seems fairly secure for this loading loading geometry. However, cracks loaded in Mode II, or moving along interfaces can go faster [21,22]. I think Petersan, Deegan, and I have some experimental evidence for cracks traveling faster than the shear wave speed under Mode I loading in rubber, but we are not yet 100% certain.

2.11 2.11

Crac Crack k path pathss

I now now briefly briefly discuss discuss the path path chosen by a moving moving crack. crack. Energ Energy y balanc balancee provid provides es an equation equation of motion for the tip of a crack only when the crack path or propagation propagation direction direction is assumed. Although criteria for a crack’s path have been established for a slowly moving cracks, no such criterion has been proven to exist for a crack moving at high speeds. A slow slow crack crack is one whose whose veloci velocity ty is much much less less than than the Raylei Rayleigh gh wave wave speed speed c R . The path followed by such cracks obeys the “ the principle of local symmetry,” first proposed by Goldstein Goldstein and Salganik [23]. This criterion criterion states that a crack extends so as to set the compon component ent of Mode II loadin loading g to zero. One conseque consequence nce is that that if a statio stationar nary y crack crack is loaded in such a way as to experience Mode II loading, upon extension it forms a sharp kink kink and moves moves at a new new angle. angle. A simple simple explanat explanation ion for this rule is that it means means the crack is moving perpendicular to the direction in which tensile stresses are the greatest. Cotterell and Rice [17] have shown that a crack obeying this principle of local symmetry is also choosing a direction so as to maximize maximize the energy release rate. The distance over over which a crack crack needs to move move so as to set K  II  to zero is on the order of the size of the process 23

zone; Hodgdon and Sethna [24] show how to arrive arrive at this conclusion using little more than symmetry principles. Cotterell and Rice also demonstrated that the condition K  II  0 has the following consequences sequences for crack motion. Consider Consider an initially initially straight straight crack, propagating propagating along the x axis. The stress field components components xx and yy have the following form  xx

 yy

K  I 

2 r  K  I 

1 2

2 r 

1 2

r 1

T  r 1

2

(89a)

2

(89b)

The constant stress T  is parallel parallel to the crack at its tip. Cotterell Cotterell and Rice showed showed that if  T  0, any small fluctuations from straightness cause the the crack to diverge from the x direction, while if  T  0 the crack is stable and continues to propagate along the x axis. They also discuss experimental verification of this prediction. Yuse and Sano [25] and Ronsin, Heslot, and Perrin [26] have conducted experiments by slowly pulling a glass plate from a hot region to a cold one across a constant thermal gradie gradient. nt. The velocity velocity of the crack, crack, driven driven by the stresses stresses induced induced by the nonunifo nonuniform rm therma thermall expansi expansion on of the materia material, l, follo follows ws that of the glass plate. plate. At a critic critical al pulling pulling velocity, velocity, the crack’s crack’s path deviates from straight-line propagation and transverse oscillations develop. develop. This instability instability is completely completely consistent consistent with the principle of local symmetry [27,28]; the crack deviates from straightness when T  in Eq. (89) rises rises above 0. Adda-Bedia Adda-Bedia and Pomeau and Ben-Amar [29] have extended the analysis to calculate the wavelength of  the ensuing oscillations. Hodgdon and Sethna [24] have generalized the principle of local symmetry to three dimensions. dimensions. They show that an equation of motion for a crack line involves involves in principle principle nine different constants. It would be interesting for an experimental study to follow upon their work and try to measure the many constants they have described, but we are not aware of such experiments. Larralde and Ball [30,31] analyzed what such equations would imply for a corrugated crack, and found that the corrugations should decay exponentially. They also performed some simple experiments and verified the predictions. Thus the principle of local symmetry is consistent with all experimental tests that have been been perform performed ed so far on slowly slowly moving moving cracks. cracks. Never Neverthe theles less, s, it does not rest rest upon a particularly solid foundation. There is no basic principle from which it follows that a crack  must extend perpendicular to the maximum tensile stress, or that it must maximize energy release. The logic of the principle of local symmetry says that bonds under the greatest tension break first, and therefore cracks loaded in Mode I move straight ahead, at least until the velocity identified by Yoffe when a crack is predicted to spontaneously break the symmetry inherent inherent in straight-li straight-line ne propagation. propagation. This logic is called into question by a very simple calculation, first described by Rice [32]. Let us look at the ratio xx yy right on the crack line. From Eq. (78) it is  xx  yy

2

1 1 4 24

2

2 1

2 2 2

4

(90)

2

2 4

2

1

2

1

2 2

1

The Taylor expansion of Eq. (91) for low velocities 2

1

ct 4

2c2l ct 2 cl

(91) is

c4l ct  cl

ct 

(92)

and in fact Eq. (91) greater than unity for all . This result is surprising because it states that, in fact, the greatest tensile forces are perpendicular to a crack tip and not parallel to it, as soon as the crack begins to move. Therefore it is hard to understand why a crack is ever supposed supposed to move in a straight straight line. These remarks do not do justice to the calculations performed with the cohesive zone models. In their most elaborate versions, the crack is allowed to pursue an oscillating path, and the cohesive zone contains both tensile and shear components. In most, although not all of these models, crack propagation is violently unstable to very short-length oscillations of the tip. A summar summary y of this work has been been provid provided ed by Langer Langer and Lob Lobko kovsk vsky y [33]. They They consider consider a large large class of models, models, and correct correct subtle errors in previous analyses. analyses. They do find some models in which a crack undergoes a Hopf bifurcation to an oscillation at a critical critical velocity. velocity. However However,, their “general conclusion conclusion is that these cohesive-zone cohesive-zone models are inherently inherently unsatisfactory unsatisfactory for use in dynamical studies. studies. They are extremely extremely difficult mathematically and they seem to be highly sensitive to details that ought to be physically unimportant.” Attempts to find models in this general class continue, with recent attempts by Karma, Kessler and Levine Levine [34] and Aranson, Kalatsky Vinokur Vinokur [35]. Both of these papers model Mode III cracks, and it is not clear how to generalize them to Mode I. The more recent paper cures some problematic features of the first one; I am not convinced that either provides a really useful starting point to address outstanding problems in fracture. One possibility is that cohesive zone models must be replaced by models in which plastic plastic yielding is distribute distributed d across an area, and not restricted restricted to a line. Dynamic Dynamic elasticplastic fracture has not, to our knowledge, considered cracks moving away from straight paths. paths. Anothe Anotherr possibi possibilit lity y is that that these these calcul calculati ations ons signal signal a failur failuree of contin continuum uum theory theory,, and that the resolution resolution must be sought at the atomic or molecular molecular scale. It is not possible right now to decide decide conclusive conclusively ly between these two possibilities. possibilities. However However,, the experimental experimental observation that the dynamic instabilities consist of repeated frustrated branching seems difficult to capture in a continuum description of the process zone.

3

Incorp Incorpora oratin ting g Plasti Plasticit city y

The lectures included an improvised segment on elastic–plastic fracture mechanics. There was no written material prepared, but here is a brief outline: Elastic–plastic fracture mechanics is a large and well–developed subject within engineering, neering, resting on a large large scientific scientific base that considers interactions interactions of cracks and dislocadislocations. Thomson Physics of Fracture Fracture (1986) ; Weertman Dislocation Dislocation–Based –Based Fracture racture MechanMechanics (1996) ; Janssen, Zuidema, and Wanhill Fracture Fracture Mechanics (2002) 25

Basic results: When When plasti plasticc deform deformati ation on at tip is signifi significan cant, t, cracks cracks move move slowly slowly,, and bond bond–br –break eaking ing is irrelevant. Total plastic dissipation can be estimated from irreversible separation of crack faces (Crack Tip Opening Displacement): (93)

c

where

c

is the shear stress at which plastic plastic flow begins.

Distinction between ductile and brittle materials captured by resistance to shear. Roughly, brittle materials are those where 10a

c

(94)

(Rice and Thomson, 1974; Armstrong; Orowan)

4

Atomic Atomic Theori Theories es of Fractu Fracture re

In contrast to cohesive cohesive zone and other continuum continuum models, where the correct correct starting starting equations still are not yet known with certainty, and instabilities in qualitative accord with experiment are difficult to find, calculations in crystals provide a context where the starting point is unambiguous, and instabilities resembling those seen in experiment arise quite naturally. In this section I will record some theoretical results relating to the instability. I will first focus on a description of brittle fracture introduced by Slepyan [9,36], and Marder and Liu [37, 38]. 38]. The aim of this approac approach h to fractu fracture re is to find a case case where where it is possibl possiblee to study the motion of a crack in a macroscopic sample, but describing the motion of every atom in detail. In this way, questions about the behavior of the process zone and the precise nature of crack motion can be resolved without any additional assumptions. This task can be accomplished by arranging atoms in a crystal, and adopting a simple force law between them, one in which forces rise linearly up to a critical separation, and then then abrupt abruptly ly drop to zero. I propose propose to call a solid solid built built of atoms atoms of this this type an ideal  brittle crystal. A force law of this type is not, of course, completely realistic, but has long been thought a sensible approximation in brittle ceramics [13]. It is more realistic in brittle brittle materials materials than, for example, Lennard-Jones Lennard-Jones or Morse potentials. potentials. A surprising fact is that this force law makes it possible to obtain a large variety of analytical results for fracture fracture in arbitrarily arbitrarily large large systems. systems. Furthermore, Furthermore, the qualitativ qualitativee lessons following following from these calculations seem also to be quite general. A summary of results results from the ideal brittle crystal is Lattice trapping: For a range of loads above the Griffith point, a crack can be trapped by the crystal, and does not move, move, although it is energetic energetically ally possible [39, 40]. Steady states: Steady state crack motion exists, and is a stable attractor for a range of  energy flux.

26

Phonons: Steadily moving cracks emit phonons whose frequencies can be computed from a simple conservation law. Fracture energy The relation between the fracture energy and velocity can be computed. Velocity gap: The slowest steady state runs at around 20% of the Rayleigh wave speed; no slower-moving steady state crack exists. Instability: At an upper critical energy flux, steady state cracks become unstable, and generate frustrated branching events events in a fashion reminiscent of experiments in amorphous materials.

One may wonder about the motivation for formulating a theory for the dynamic behavior of a crack in a crystal. Is the lattice essential or can one make the underlying lattice go away away by taking taking a continuum continuum limit? So far as I know, know, all attempts to describe the process process zone of brittle materials in a continuum framework have run into severe difficulties [33]. These problems do not arise in an atomic-scale description. The simplicity of the ideal brittle crystal is somewhat misleading in a number of respects. spects. Therefore, Therefore, before introducing introducing the mathematics, mathematics, I will comment on a number of  natural questions regarding the generality of the predictions it makes.  Does the simple force law employed between atoms neglect essential aspects of the dynamics? I will demonstrate that the same qualitative results observed in the ideal brittle crystal occur in extensive molecular dynamic simulations using realistic potentials. The calculations calculations are are in a strip. Is this geometry too restricti restrictive? ve? To this question the general formalism of fracture mechanics provides an answer. Fracture mechanics tells tells us that that as long long as the condit condition ionss of smallsmall-scal scalee yieldi yielding ng are satisfi satisfied, ed, the behavio behaviorr of a crack is entirely governed governed by the structure structure of the stress fields in the near vicinity of the crack tip. These fields are solely determined determined by the flux of energy energy to the crack  tip. A given energy flux can be provided by an infinite number of different loading configuratio configurations, ns, but the resultant dynamics dynamics of the crack will be the same. As a result, result, the specific geometry used to load the system is irrelevant and no generality is lost by the use of a specific loading configuration. This fact is borne out by the experimental work described in the previous section.  Have lattice trapping and the velocity gap ever been seen experimentally? Maybe. Robert Deegan and I have obtained some results, but the data are not yet conclusive. sive. Molecular Molecular dynamics simulations simulations indicate indicate that lattice lattice trapping disappears disappears at room temperature. temperature. New experiments experiments are needed to obtain obtain a detailed detailed description description of  dynamic fracture in crystals at low temperatures.  Are results in a crystal relevant for amorphous materials? This is still an open question. However However,, the results of the lattice calculations calculations seem to be remarkably robust. robust. Adding quenched noise to the crystal has little qualitative effect. Effects of topological disorder are not known. known. However However,, a certain amount is known about the effects

27

of increasing temperature. temperature. When the temperature temperature of a brittle brittle crystal increases above zero, the velocity gap closes [41], and its behavior is reminiscent of that seen in experiments performed on amorphous materials. Kulakhmetova, Saraikin and Slepyan [9,42] first showed that it is possible to find exact analytical expressions for Mode I cracks moving in a square lattice. They found exact relationships tionships between the energy flux to a crack tip and crack tip velocity velocity.. They also observed that phonons must be emitted by moving cracks, and calculated their frequencies and amplitudes. Later calculations extended these results to other crystal geometries, allowed for a general Poisson ratio, showed that there is a minimum allowed crack velocity, found when steady crack motion is linearly stable, calculated the point at which steady motion becomes unstable to a branching instability, and estimated the spacing between branches [38].

Figure 9: A sketch showing steady state motion of crack moving in an ideal brittle crystalline talline strip loaded in Mode I. These calculations are extremely elaborate. The analytical expressions are too lengthy to fit easily on printed pages, and most of the results were obtained with the aid of symbolic algebra. algebra. For this reason, I will first give a qualitative qualitative argument argument for the most surprising of  the dynamical results, the velocity gap. Then, after summarizing the results of the technical calculations, I will proceed to describe in detail how the calculations work in the case of a simple one–dimensional model where the algebra is much less demanding but most of the ideas are the same.

4.1

Physical Physical argument argument for velocity velocity gap

The velocit velocity y gap is a natura naturall conseq consequen uence ce of rapidl rapidly y snappin snapping g bon bonds. ds. It emerges emerges both both from the simple model depicted in Figure 12, and also in more elaborate models, whose solution requires greater effort, that describe cracks moving through fully two-dimensional 28

crystals. crystals. In the analytically analytically solvable solvable models, models, the bond breaking breaking is instantaneous. instantaneous. In any more more realis realistic tic situat situation ion there there will will instea instead d be be a brief brief period period in which which the force force betw between een atoms atoms falls to zero, but all that is required for the velocity gap is that this time be short compared to the vibrational vibrational period period of an atom. Dynamic fracture is a cascade of bonds breaking, one giving way after another like a toppling toppling line of dominos. Examining Examining Figure Figure 10, watch watch what happens as a crack moves moves ahead. ahead. In frame 2, the bond between between two atoms atoms has just broken. broken. There There is no guaran guarantee tee that the next bond to the right will break. The crack could fall into a static lattice trapped state. The best chance to avoid this fate is for the atom marked in green to deliver enough of a blow to its right-hand neighbor that the bond on that neighbor is broken in turn. This process process had better take place within the first half of the first vibrational vibrational period of the green atom. For the longer it vibrates, the more energy will be distributed to its neighbors in all directions in the form of traveling waves, and the smaller become the chances that there will be enough concentrated energy available to snap the next bond down the line. To make this idea slightly more quantitative, let b be the effective spring constant acting acting on an atom after a bond connected connected to it has snapped. Its oscillatory oscillatory period period is M 

2

(95)

b

where M  is its mass. Assumin Assuming g that the next next bond, at distance distance a, breaks during some fraction 1 of half this oscillatory oscillatory period, the speed of the crack will be a

b

 M 

(96)

By this same rough logic, the speed of sound is given by solving the wave equation a2

 M u

2

u

 x2

(97)

where now is an effective spring constant appropriate for atoms surrounded by unbroken bonds. From Eq. (97) follows follows a wave wave speed c of  c

a

 M 

(98)

One should expect c to be larger than the crack speed . The main main reason reason is that when when one of the springs connected to an atom is snapped, the effective spring constant describing its vibrations should decrease, so There is also an extra extra factor factor of  in Eq. (96). b . There The crack speed estimated here is a lower bound because there is no way that pulling more gently on remote system boundaries can reduce the vibrational period of two atoms once the bond connecting them breaks. Pulling harder on external boundaries can however increa increase se the speed speed of a crack crack becaus becausee it it can supply supply enough enough energy energy that that the atoms atoms are alread already y moving quickly when the bond between them snaps. These arguments presume that the only source of energy available for snapping bonds is contained contained in strained strained material ahead ahead of the tip. This assumption assumption is never completely completely 29

1

5

2

6

Snaps

Snaps

3

7

4

8

Figure 10: Sequence Sequence of eight snapshots snapshots from motion of a crack tip. Bonds break in frames two two and six. The green atom snaps snaps backwa backwards rds each time a bond connect connected ed to it breaks, breaks, provid providing ing a crucia cruciall portion portion of the energ energy y for the next bond to snap. snap. This This process process must occur within the first half of the first vibrational period of the green atom, or it is unlikely ever to occur.

30

correct. correct. At any nonzero temperature temperature,, thermal fluctuations fluctuations can bring extra extra energy in. For this reason, reason, cracks cracks strained strained above above c always always move move ahead at any nonzero nonzero temperat temperature, ure, but the rate is very small when the thermal thermal fluctuations fluctuations are rare. Similarly Similarly,, chemical agents in the environm environment ent can help catalyze bond breaking, breaking, and crack speeds are sometimes sometimes controlled by the rate at which they arrive. Dynamic fracture refers to the motion of cracks when these sorts of effects can be neglected. d 2

d 1 d 1 d 5

d 5 d 3

d 2

d 6

d 6 d 4 d 4

d 3 u2

u1 u0

u5 u3

u6

u4

Figure Figure 11: The geometry of the lattice lattice used for fracture calculations. calculations.

4.2

Dynamic Dynamic fractur fracturee of a lattice lattice in in anti-plan anti-planee shear, shear, Mode Mode III

4.3

Defini Definitio tion n and energe energetic ticss of the the mode modell

I now work in detail through some of the analytical results available for a crack moving through through an ideal ideal brittle brittle crystal crystal loaded loaded in Mode Mode III. The The main results results are the the relation relation between between loading and crack velocity, the prediction of a velocity gap, and the calculation of the point at which steady crack motion becomes unstable. unstable. Linear Linear stability of the steady states states will not be considered here; the result obtained in [38] is that the steady states, when they exist, are always linearly stable. The techniques used to solve problems of this type were first found by Slepyan [9, 36, 42]. There are differences differences between between details details of his solution solution and mine because Eq. (99) describes motion in a strip rather than an infinite plate, and in a triangular rather than a square lattice. lattice. The strip is preferable preferable to the infinite infinite plate when it comes time to compare with numerical simulations, while reducing to the simpler infinite plate results in certain natural natural limits. The velocity velocity gap and nonlinear nonlinear instabilities instabilities were first found in [37, 38]. Slepyan’s plots of crack speed versus driving contain ranges of solutions that turn out to be inconsistent. Consider a crack moving in a strip composed of 2 N  1 rows of mass points, shown in Figure 13. All of the bonds between lattice points are brittle-elastic, behaving as perfect 31

1    

 xm

1

 N

1    

1    

1    

 N  xm

 N 1    

 xm

1

1

1

 xm

 xm

1

1

1

1    

1    

 xm

1

 N

1    

1     1    

 N

 N

2 N 

1

Figure Figure 12: In this this simple simplest st of all solvabl solvablee atomic atomic crack models, models, two two lines lines of atoms atoms are connected by weak springs to a ceiling and a floor, and to horizontal and vertical neighbors by strong springs. The vertical strong strong springs break if they exceed exceed a certain certain extension.

32

linear springs until the instant they snap at a separation of 2 u f , from which point on they exert no force. The location of each mass point is described by a single spatial coordinate u m n , which can be interpreted as the height of mass point m n into or out of the page. The force between adjacent masses is determined by the difference in height between them. N  1 2. The index m takes integer values, while n takes values of the form 1 2, 3 2, The model is described by the equation

Figure 13: Dynamic Dynamic fracture of a triangular triangular crystal in anti-plane anti-plane shear. shear. The crystal has has 2 N  1 rows of atoms, with N  4, and 2 N  1 rows of bonds. Heights of spheres indicate displacements u m n of mass points out of the page once displacements displacements are imposed imposed at the boundaries. The top line of spheres is displaced out of the page by amount u m N  1 2 2 N  1, and the bottom line into the page by amount u m N  1 2 U  N  U  N  2 N  1. Lines connecting connecting mass points indicate whether the displacement displacement between between them has exceeded the critical value of 2u f ; see Eq. (99b)).

u m n

bu

1 2

 f  u m n nearest neighbors m

u m n

(99a)

n

with  f  u

u 2u f 

(99b)

u

describing ideal brittle springs, the step function, and b the coefficient of a small dissipative term. There is no difficulty involved in choosing alternative forms of the dissipation, if  desired. The boundary condition which drives the motion of the crack is (99c) u m N  1 2 U  N  It is important to find the value of  U  N  for which there is just enough energy stored per length to the right of the crack to snap the pair of bonds connected to each lattice site on

33

the crack line. For m

0 one has u m n

nU  N   N 

(100)

1 2

and the height difference between mass points with adjacent values of  n is U right right

U  N   N 

(101)

1 2

Therefore, Therefore, the energy stored per unit length in the 2 N  1 rows of bonds is 1 2 Upper Bonds Site 2 U  N  1 1 2 2 2 N  1 2 2  N  1 2 2Q0 U  N 

Rows

Spring Constant

2 U right

(102) (103)

2

(104)

with Q0

1 2 N  1

(105)

The energy required to snap two bonds each time the crack advances by a unit length is 1 2 1 2

2 Bonds Site 2

1 2

2u f 

2

Spring Constant 2u f 2

Separation at fracture

2

(106) (107)

Therefore, equating Eqs. (104) and (107) the proper dimensionless measure of the external driving is U  N  Q0 (108) u f  a quanti quantity ty which which reache reachess 1 as soon soon as there there is enough enough energy energy to the the right right of of the crack crack to snap snap the bonds along the crack line, and which is linearly related related to the displacement displacementss imposed at the edges of the strip.

4.4 4.4

Symm Symmetr etrie ies: s:

Assume that a crack moves in steady state, so that one by one, the bonds connecting u m 1 2 with u m 1 1 2 or u m 1 2 break. break. They They break because because the distance distance between these points exceeds the limit set in Eq. (99b) and as a consequence of the driving force described described by Eq. (99c). Assuming Assuming that the times at which bonds break are known, known, the original nonlinear problem is immediately transformed into a linear problem. However, one has to come back at the end of the calculation to verify that 1. Bonds break at the time they are supposed to. Imposing this condition determines a relation between crack velocity and loading . 34

2. No bonds break when they are not supposed to. Imposing this condition leads to the velocity gap on the low-velocity end, and to crack tip instabilities above a critical energy flux. Steady states in a crystal are more complicated than steady states in a continuum. In a continuum, continuum, steady state acts as u x t  . The closest one can come in a triangular crystal is by having the symmetry u m n t 

1 n t  1

u m

(109a)

and also u m n t 

which implies in particular that u m 1 2 t  We have defined

4.5

n t 

u m

1 2 t  1 2

0 1 mod n

gn

1 2

u m

1 2 2

(109b)

gn

(109c)

if  n 1 2 5 2 if  n 3 2 7 2 in general

(110)

Elimin Eliminati ating ng the spatia spatiall index index m:

Assuming that a crack is in steady state, we can therefore eliminate the variable m entirely from the equation of motion, by defining un t 

u 0 n t 

(111)

and write the equations of motion in steady state as

un t 

if  n u1

1 2

un 1 t  gn un t  1 un 1 t  gn

1

1

un 1 t  gn un t  1 un 1 t  gn

6un t  1

1

1

bun

(112a)

1

1 2, and

2

t 

1 2

u3 u1 u

2 2

t  t 

1 2

1 t 

4u1 u1 2 t 

2

t  t 

u3 u1 u

2 2

1 1

t  t 

1 2

t 

1

u1

2

t 

1 2

(112b) t  bu1

if  n

1 2 The time at which the bond between u 0 1 2 t  and u 0 1 2 t  breaks has been chosen to be t  0, so that that by symmet symmetry ry the time time the bon bond d betw between een u 0 1 2 t  and u 1 1 2 t  breaks is 1 2 .

4.6

Equations Equations solve solved d in terms of a single single mass mass point point on on crack crack line: line:

Above the crack line, the equations of motion (112a) are completely linear, so it is simple to find the motion of every atom with n 1 2 in terms of the behavior of an atom with 35

2

n

1 2. Fourier transforming Eq. (112a) in time gives 2

un

1 u 2 n 1 1 u 2 n 1 u 2 n 1

ib

ei ei ei

gn

1

1

6 gn

gn

i

gn

1

i

e 1

1

ei

(113) e

1

Let un

u1

1 2

ek  n

2

2

i gn

Substituting Eq. (114) into Eq. (113), and noticing that g n 2

u1

ib u1

2

1 u 2 1 2 1 u 2 1 2 1 2 u1 2

2

2

ek  k 

e

2cosh k  cos

ib

ei ei ei

gn

6

2

1

2

2

ei

gn

1

gn

2

2

ei

gn

1

gn

2

i

e gn

1

1 gives

gn gn

1

gn

(114)

2

cos

3

(115) (116)

0

Defining 3

 z

2

cos 2cos

ib

(117)

2

one has equivalently that  y

z2

z

1

(118)

with ek 

 y

(119)

One can construct a solution which meets all the boundary conditions by writing un

u1

2

y N 

i gn 2

e

1 2 n

N  1 2 n

 y

 y N   y

1 2

U  N  n

N 

 N 

2 2

2

(120)

2 2 This solution solution equals u1 2 for n 1 2, and equals U  N 2 for n N  1 2. The U  N . The Fourie reason to introduce is that for n N  1 2, u m n t  Fourierr transform transform of  this boundary condition condition is a delta function, function, and hard to work with formally formally.. To resolve uncertainties, it is better to use instead the boundary condition

u N 

1 2

t 

t 

U  N e

(121)

and send to zero the end of the calculation. In what follows, frequent use will be made of the fact that is small. The most interesting variable is not u1 2 , but the distance between the bonds which will actually snap. For this reason define U  t 

u1

2

t 

u

1 2

t 

u1

2

t 

2

u1

2

t 

1 2

(122)

2

Rewrite Eq. (112b) as u1

2

t 

1 2

u3 2 t  u1 2 t  2U  t 

1

4 t 

u1 2 t  2U  t  1 2

36

u3 2 t  1 u1 2 t  1 t  1 2

bu1

2

(123)

Fourier transforming this expression using Eq. (120) and defining U 

d  ei t U  t 

t 

U  N 

(124)

now gives u1

2

F 

with

y N 

F 

1

ei

2

1

 y

N  1

 y N   y

U 

2

 N 

2 z cos

N 

2 2

2

1

(125)

(126)

Next, use Eq. (122) in the form 1

U 

i

e

2

u1

2

to obtain U 

2 cos2

F 

(127)

2

U  N 

4 U 

 N 

2 2

2

(128)

Writing U 

U 

(129)

U 

finally gives U 

Q

U 

1

U  N Q0

with

1 i

i

F 

Q

2cos2

F 

(131)

4

To obtain the right hand side of Eq. (130) one uses the facts that F  0 is very small, so that the right hand side of Eq. (130) is a delta function. The Wiener-Hopf technique [46] directs one to write Q

Q

(130)

1 N , and that

(132)

Q

where Q is free of poles and zeroes in the lower complex plane and Q is free of poles and zeroes zeroes in the upper upper comple complex x plane. plane. One can carry carry out this decomposi decompositio tion n with with the explicit formula d 

exp lim

Q

2

0

ln Q i

i

(133)

Now separate Eq. (130) into two pieces, one of which has poles only in the lower half  plane, and one of which has poles only in the upper half plane: U 

Q0U  N 

Q

Q

Q0U  N 

1

Q

0

i

0

1 i U  Q

37

(134)

Because the right and left hand sides of this equation have poles in opposite sections of the complex plane, they must separately equal a constant, . The constant must vanish, or U  and U  will behave as a delta function near t  0. So Q0 Q

U 

U  N 

U 

U  N 

0

Q

(135a)

i

and Q0 Q

0

Q

(135b)

i

One now has an explicit solution for U  . Numerical Numerical evaluati evaluation on of Eq. (133), and U  t  from from Eq. (135) (135) is fairly fairly straig straightf htforw orward ard,, using fast fast Fourier Fourier transfo transforms. rms. However However,, in carryi carrying ng out the numerical transforms, it is important to analyze the behavior of the functions for large values of  . In cases where functions to be transformed decay as 1 i , this behavior is best subtracted off before the numerical transform is performed, with the appropriate step function added back analytically afterwards. Conversely, in cases where functions to be transformed have a step function discontinuity, it is best to subtract off the appropriate multiple of  e t  t  before the transform, adding on the appropriate multiple of 1 1 i afterwards. afterwards. A solution of Eq. (135) constructed constructed in this manner appears appears in Figure 15.

4.7

Relati Relation on betwee between n

and

Recall that making the transition from the nonlinear problem originally posed in Eq. (99a) to the the linear linear proble problem m in Eq. Eq. (112) (112) relies relies on suppo supposing sing that that bonds bonds along along the the crack crack line line snap at time intervals of 1 2 . Because Because of the symmetries symmetries in Eq. (109), it is sufficient sufficient to guarantee that (136) u t  u f  at t  0 All displacements are simply proportional to the boundary displacement U  N , so Eq. (136) fixes a unique value of U  N , and its dimensionless counterpart, . Once one assumes that the crack moves in steady state at a velocity , there is a unique to make it possible. To obtain Eq. (136), one needs to require that d 

lim

t 

e

2

0

i t 

U 

This integral can be evaluated by inspection. One knows that for positive t  d  exp

(137)

u f 

0

i t  U 

0, (138)

and that any function whose behavior for large is 1 i has a step function discontinuity discontinuity at the origin. Therefore, Therefore, Eq. (137) and Eq. (135a) become u f 

Since from Eq. (131) it follows that Q

U  N Q0

Q Q

0

1, one sees from Eq. (133) that 38

(139)

Q

1

Q

(140)

As a result, one has from Eq. (139) and the definition of  Q

given in Eq. (108) that

0 Q0

(141)

To make this result more explicit, use Eq. (133) and the fact that Q write Q

0

2

0

1 2i

d 

exp Q

1 ln Q 2 i

d 

exp

2

2

to

(142)

i

ln

d 

Q0 exp

ln Q

Q

Q Q

2

2

ln Q 0

Q 1 ln 2i Q

(143)

Q 1 ln 2i Q

(144)

Placing Eq. (143) into Eq. (141) gives d 

exp

2

In order to record a final expression that is correct not only for the Mode III model considered here, but for more general cases, rewrite Eq. (144) as d 

C exp

2

1 2i

ln Q

ln Q

(145)

where C  is a constant of order unity that is determined by the geometry of the lattice, equaling 1 for the triangular lattice loaded in Mode III, but 2 3 for a triangular lattice loaded loaded in Mode I [38]. [38]. When When writte written n in this form, Eq. (145) (145) is suitab suitable le for numeric numerical al evaluation, since there is no uncertainty relating to the phase of the logarithm. When b becomes becomes sufficientl sufficiently y small, small, Q is real real for for real real except except in the small small neighbo neighborho rhood od of isol isolat ated ed root rootss and and pole poless that that sit sit near near the the real real axis. Let r i be the the root rootss of Q with negative negative imaginary part (since they belong with Q ), r i the roots of  Q with positive positive imaginary imaginary part, and similarly pi the poles of  Q. Then one can rewrite Eq. (145) as r i pi

C 

for b

r i pi

0

(146)

One may derive Eq. (146) as follows: away from a root or pole of  Q, the integrand of  Eq. (145) vanishes. vanishes. Consider Consider the neighborhood neighborhood of a root r  of  Q which falls to the real axis from the negative side as b 0. For the sake of argument, take the imaginary part of  this root to be ib. In the neighborhood of this root, say within a distance b, the integral to compute for Eq. (145) is 1 2

r  r 

b b

d 

ln

2i

r 

ln 39

ib r 

ib

(147)

Defining

r  , and integrating by parts gives

1 2 1 ln r  2

b b

d 

2i

ln

r 

2ib 2

(148)

b2

(149)

b

Similar integrals over other roots and poles of  Q finally produce Eq. (146). Together with Eq. (135), Eq. (145) and Eq. (146) constitute the formal solution of the model. Since Q is a function of the steady state velocity , Eq. (144) relates the external driving force on the system, , to the velocity of the crack  . The results of a calculation appear in Figure 14. 0.8 Steady states go unstable 0.6     c

0.4

Slowest steady state

0.2

Velocity gap

0.0

Linearly stable lattice-trapped states

1.0

1.1

1.2

1.3

1.4

Figure 14: The velocity of a crack  c, scaled by the sound speed c 3 2, is plotted as a function of the driving force . The calculation calculation is carried out using Eq. (146) for N  9.

4.8

Phonon Phonon emissi emission: on:

Right at 1 just enough energy is stored to the right of the crack tip to break all bonds along the crack line. However, all steady states occur for 1, so not all energy stored to the right of the crack tip ends up devoted to snapping bonds. The fate of the remaining energy depends upon the amount of dissipation b, and the distance from the crack tip one inspects. inspects. In the limit of vanishing dissipatio dissipation n b, traveling waves leave the crack tip and carry energy off in its wake; the amount of energy they contain becomes independent of  b. Such a state is depicted in Figure 15, which shows a solution of Eq. (135) for 0 5,  N  9, and b 0 01. For all nonzero b, these traveling waves waves will eventually eventually decay, and the extra energy will have been absorbed by dissipation, but the value of  b determines whether one views the process as microscopic or macroscopic. The frequencies of the radiation emitted by the crack have a simple physical interpretation as Cherenkov radiation. Consider the motion of a particle through a lattice, in which 40

5.0

4.0

3.0    )    t    (   u

2.0

1.0

0.0

-50

-25

0

25

50

t

Figure 15: A plot of U  t  for 5, N  9, and b 0 01, produced by direct evaluation of  Eq. (135). Note that mass points are nearly motionless until just before the crack arrives, and that they oscillate afterwards for a time on the order of 1 b.

3.0 5 2.5

        k

2

2.0 1.5

        k

1.0 0.5 0.0 0

2

6

4

8

10

k 

Figure 16: Graphical solution of Eq. (153), showing that for low velocities, a large number of resonances may be excited excited by a moving crack.

41

the phonons are described by the dispersion (150)

k

If the particle moves with constant velocity , and interacts with the various ions according to some function , then to linear order the motions of the ions can be described by a matrix  D which describes their interactions with each other as mul

Rl

 D

 Rl ul

 Rl

l

t 

(151)

l

l

Multiplying everywhere by eik R , summing over l , letting K  be reciprocal reciprocal lattice vectors, vectors, and letting be the volume of a unit cell gives mu k

k u k

D

1

ei

k

K  t 

k

K 

(152)

K 

Inspection shows that the lattice frequencies excited in this way are those which in the extended zone scheme[ [47]] obey k

(153)

k

Pretending that the crack is a particle, one can use Eq. (153) to predict the phonons that the crack emits. There is another version of this argument argument that is both simpler and more general. The only only way way to transp transport ort radiati radiation on far from from a crack crack tip is in traveli traveling ng waves. waves. Ho Howe wever ver,, in steady state, state, the traveling traveling waves waves must obey symmetry (109a). (109a). In a general general crystal with lattice vectors R and primitive vectors a, applying this requirement to a traveling wave exp ik  R i t  gives e ik 

R na

i

k  t  na

e ik  R

i

k  t 

(154)

Assuming a and are parallel, Eq. (153) results again. There are two phonon dispersion relations relations to consider. consider. One gives the conditions conditions for propagating radiation far behind the crack tip, and the other gives the conditions for propagating gating radiat radiation ion far ahea ahead d of the crack crack tip. tip. Far behind behind the the crack crack tip, tip, all all the bond bondss are broken broken.. Finding waves that can travel in this case is the same as repeating the calculation that led to Eq. (128), but with U  set to zero, since all bonds are broken, and with U  N  0, since phonons can propagate without any driving term. Examining Eq. (128), one sees that the condition for surface phonons to propagate far behind the crack tip is F  0. Similarly, far ahead of the crack tip no bonds are broken U  should be set equal to U , and the condition for phonons is F  2cos 2 4 0. According According to Eq. (131), the roots and poles of  Q are therefore the phonon frequencies behind and ahead of the crack, and these are the quantities appearing in Eq. (146). I do not know if Eq. (146) is more than approximately correct for particle interactions more general than ideally brittle bonds. 42

5.0 4.0       U

    t    n    e    m    e    c    a      l    p    s      i      D

3.0 2.0 1.0 0.0 -50

-25

0

25

50

Time t 

Figure 17: Behavior of U  t  for 2, b 0 05, and N  9. Notice Notice that at t  0, indicated by the dashed line, u is decreasing, and that it had already already reached height 1 earlier. earlier. This state is not physical.

4.9

Forbi Forbidd dden en veloci velocitie tiess

After making sure that bonds along the crack line break when they are supposed to, it is necessary to verify that they have not been stretched enough to break earlier. That is, not only must the bond between u0 and u0 reach length 2u f  at t  0, but this must be the first time at which that bond stretches to a length greater than 2 u f . For 0 03 (the precise value of the upper limit varies with b and N ) that condition is violated. The states have have the unphysical unphysical character character shown in Figure 17. Masses rise above height u f  for t  less than 0, the bond connecting them to the lower line of masses remaining however intact, and then then they descend, descend, where whereupo upon n the bond snaps. snaps. Since Since the soluti solution on of Eq. (112) (112) is unique unique,, but does not in this case solve solve Eq. (99a), no solutions solutions of Eq. (99a) exist at all at these these velocities. velocities. Once the crack velocity has dropped below a lower critical value, all steady states one tries to compute have this character. This argument shows that no steady state in the sense of Eq. (109a) can exist. It is also possible possible to look analytically analytically for solutions solutions that are periodic, periodic, but travel travel two lattice lattice spacings spacings before before repeatin repeating. g. No solutions solutions of this this type have yet been found. Numeri Numerical cally ly,, one can verify that if a crack is allowed to propagate with right above the critical threshold, and is then very slowly lowered through the threshold, the crack stops propagating. It does not slow down noticeably; suddenly the moving crack emits a burst of radiation that carries off its kinetic kinetic energy energy, and stops in the space of an atom. That is why Figure 14 shows a velocity gap.

4.10 4.10

Nonli Nonlinea nearr Instabi Instabili litie tiess

It was assumed in the calculations predicting steady states that the only bonds which break  are those which which lie on the crack crack path. path. From From the numeri numerical cal solutio solutions ns of Eq. (135), (135), one can test this assumption; it fails above a critical value of  . Th Thee sound sound spee speed d c equals 43

3 2, and veloci velocitie tiess will be scaled scaled by this this value. value. For N  9, at a velocity of  c c 666 , c 1 158 , the bond between u 0 1 2 and u 1 1 2 reaches a distance of  2u f  some short time after the bond between u 0 1 2 and u 0 1 2 snaps. snaps. The steady steady state solutions solutions strained strained with larger values of  are inconsistent; only dynamical solutions more complicated than steady states, involving the breaking of bonds off the crack path, are possible. To investigate these states, one must return to Eq. (99) and numerically solve the model directly directly. These simulations simulations have have been carried carried out [37, 48] and some results results are contained in Figure 18. These theoretical theoretical calculations calculations were done in response to experiments experiments [49–51, 51] that showed cracks The geometry of the instability is the same as the geometry of instabilities found experimentall experimentally y in Plexiglas, Plexiglas, shown in Figure 19. However However,, a detailed connection connection between the theory, theory, which describes the onset of branching in a crystal, and the experiment, which observes observes mesoscopic mesoscopic branching branching in an amorphous amorphous polymer, polymer, has not not fully been been worked worked out. 1 147 c

0 645

1 165 c

0 630

1 376 c

0 624

1 835 c

0 775

Figure 18: Pictures of broken bonds left behind the crack tip at four different values of  . The top figure shows the simple pattern of bonds broken by a steady-state crack. At a value of  slightly above the critical critical one where horizontal bonds occasionally snap, the pattern is periodic. All velocities are measured relative to the sound speed c 3 2. Notice Notice that the average velocity can decrease relative to the steady state, although the external strain has increased. As the strain increases further, other periodic states can be found, and finally states states with complicated complicated spatial structure. structure. The simulations simulations are carried out in a strip with half-width N  9, of length 200 and b 0 01. The front front and back back ends of the strip strip have short energy-absorbing regions to damp traveling waves. waves. The simulation was performed by adding unbroken unbroken material to the front and lopping it off the back as the crack advanced. advanced. The diagram shows shows patterns of broken bonds left behind the crack tip. Just above the 44

Figure 19: 19: Images of local crack crack branches branches in the x y plane plane in PMMA. PMMA. The arrow arrow, of length length 250 m, indicates the direction of propagation. All figures are to scale with the path of the main crack in white. (top) 1 18 c (bottom) 1 45 c . (from [52]) c (center) threshold at which horizontal bonds begin to break, one expects the distance between these extra broken bonds to diverge. The reason is that breaking a horizontal bond takes energy from the crack and slows it below the critical critical value. The crack then tries once more to reach the steady state, and only in the last stages of the approach does another horizontal bond snap, beginning beginning the process again. This scenario scenario for instability instability is similar similar to that known known as intermittency intermittency in the general framework framework of nonlinear nonlinear dynamics [53] ; the system spends most of its time trying to reach a fixed point which the motion of a control parameter has caused to disappear. Heizler, Kessler, and Levine [43] have shown that when the snapping bond is replaced by a rounded potential, the steady crack states are linearly unstable in a more conventional way, the the instability is a Hopf bifurcation.

4.11

The connectio connection n to the Yoffe instabili instability ty

The basic reason for the branching instability instability seen above is the crystal analog of the Yoffe Yoffe instab instabili ility ty,, workin working g itself itself out on small small scales scales.. The Mod Modee III calcul calculati ation on finds finds that that the critic critical al velocity for the instability to frustrated branching events is indeed close to the value of  0 6c R predicted predicted by Yoffe in the continuum. continuum. The critical velocity velocity seen experimentally experimentally in amorphous amorphous materials materials is 1/3 of the wave wave speed, not 2/3. This discrepancy discrepancy could could be due to some combination of three factors. 1. The force law between atoms is actually much more complicated than ideal snapping bonds. bon ds. Gao [54], has pointe pointed d out that the Rayleigh Rayleigh wave wave speed in the vicinit vicinity y of  45

a crack tip may be significantly lower than the value of  c R far away from the tip because material is being stretched beyond the range of validity of linear elasticity. 2. The experiments experiments are at room temperature temperature,, while the calculations calculations are at zero. 3. The experiments experiments are in amorphous materials, materials, while the calculations calculations are in crystals.

4.12 4.12

Genera Generaliz lizing ing to Mode Mode I:

Let ui describe describe the displacement displacement of a mass point from its equilibrium equilibrium location. location. Assume that the energy of the system is a sum of terms depending upon two particles at a time, and linearize the energy to lowest lowest order in particle displacements. displacements. Translati Translational onal invariinvariance demands that the forces between particles 0 and 1 depend only upon u 1 u0 , which will be defined to be 1 . Ho Howe wever ver,, the force force can be a genera generall linear linear function functional al of  1 . A way to write such a general linear functional is to decompose the force between particles 0 and 1 into a component along d 1 , and a component that is along d 1 . Th Thee first first component component corresponds corresponds to central forces between between atoms, while the second component component is a non-central non-central force. Non-central Non-central forces between atoms are the rule rather than the exception exception in real materials, materials, as first appreciated appreciated by Born. Their quantum-mecha quantum-mechanical nical origin origin is of no concern here, only the fact that they are not zero. Suppose that the restoring force parallel to the direction of equilibrium bonds is proportional to , while that perpendicular to this direction is proportional to . The force force due to the displac displaceme ement nt of the particl particlee along along ui 1 j 1 ui j is 1 d  1

1

d  1

d 

1

1

d 

1

(155)

1 3 1  x 3  y 3 1 3  x 1  y (156) 1 2 2 2 2 1 2 2 2 1 2 1 Adding up contribution contributionss from other particles particles in this way we get for the force due to neighbors 6

F  m n

q d q j

j

m n

d q j

(157)

 j 1 q

By varying the constants and , one can obtain any desired values of shear and longitudinal wave speeds, which are given by c2l ct 2

3a2 8m 3a2 3 8m

3

(158a) (158b)

where m is the mass of each particle. In addition to forces between neighbors, it is possible to add complicated dissipative functions depending upon particle velocities. In Ref. [38], a term was added to the equations so as to reproduce the experimentally measured frequency depend dependenc encee of sound sound attenu attenuati ation on in Plexig Plexiglas las.. There There is a slight slight technica technicall restric restrictio tion n in the calculations of Ref. [38]; right on the crack line, forces are required to be central. 46

0.8

Microcracking instability

0.6

Mini Minimu mum m velo veloci city ty steady state

Height=80 Height=160

           t

    c

0.4 Velocity gap 0.2 Lattice trapping 0.0 1.0

1.2

1 .4

1.6

1.8

2.0

K  I  K  Ic for strips 80 and 160 Figure 20: Crack velocity velocity versus loading parameter parameter atoms high, in limit of vanishing dissipation, computed for Mode I crack by methods of  following section. will will be defined precisel precisely y in Eq. (108). (108). The spring spring consta constants nts have have values 2 6 and 0 26, so that the non-central non-central forces forces are relatively relatively small. The fact that the results have become independent of the height of the strip for such small number numberss of atoms atoms in the vertic vertical al direct direction ion suggest suggestss that that relati relativel vely y small small molecu molecular lar dynamic dynamicss simulations can be used to obtain results appropriate for the macroscopic limit.

E. Gerde has found a way to overcome this technical limitation, and the results do not change noticeably. There is no universal universal curve curve describing Mode I fracture. Many details details in the relation relation between loading and crack velocity depend upon ratios of the sound speeds, and upon the frequency frequency dependence of dissipation. dissipation. In the limit of central central forces, 0, it turns out to be difficult difficult but not impossible impossible to have cracks in steady state. This means that the range of loads for which cracks can run in a stable fashion is small, and depends in detail upon upo n the amount amount of dissipa dissipatio tion. n. Crack Crack motion motion is greatl greatly y stabil stabilize ized d by havin having g nonzero nonzero . Figure 20 displays a representati representative ve result with some implication implicationss for the design of  molecular dynamics simulations. In the limit of vanishing dissipation, the relation between K  I  K  Ic becomes nearly independent of the crack speed and dimensionless loading number of vertical vertical rows of atoms for strips around 40 rows high. The definition definition of  is that it describes the vertical displacement imposed upon the top and bottom of the strip, but scaled so that 1 when these boundaries have been stretched just far enough apart so that the energy stored per length to the right of the crack tip just equals the energy cost per length length of extending extending the crack. It is defined precisely by Eq. (108) Analytical work has continued on these classes of models, and there are many results I will not refer to much. Kessler Kessler and Levine have conducted conducted the most careful studies of  lattices lattices with nonlinear force laws and dissipation dissipation acting in concert concert [43, 44]. Their findings emphasize emphasize the point that crack dynamics dynamics are terribly terribly sensitive sensitive to features of the tip that do appear in any continuum continuum form of the problem. With With Eric Gerde, I applied applied some of these ideas to interface fracture and to friction [45]. 47

5

Scali Scaling ng ideas ideas and molecu molecular lar dynami dynamics cs

Because fracture involves both large and small scales, because it is of practical importance, and because it became fashionable for reasons that are hard to explain precisely, there is an ongoing effort of many groups to study fracture through numerical methods that reach down to the atoms. These studies fall into roughly two groups: 1. Calculation Calculationss employing classical classical molecular molecular dynamics. dynamics. The largest studies studies employ a billion atoms or more [55–59]. 2. Calculation Calculationss employing employing continuum continuum mechanics, mechanics, but with with special spatial spatial regions regions where the analysis descends to molecular dynamics, and perhaps even, in small regions, quantum mechanics [60–65] My own prejudice is toward carrying out molecular dynamics in the spirit of fracture mechan mechanics. ics. That That is, instead instead of trying trying to simula simulate te a macrosc macroscopi opicc sample sample in its entiret entirety y in the midst of one computer program, I prefer to use computer calculations to tabulate quanti quantitie tiess that that linear linear elastic elastic fractu fracture re mechani mechanics cs needs needs in order order to make make predic predictio tions. ns. If  one assumes that cracks move along a straight line, then what one then has to compute is fracture energy as function of crack velocity. In what follows, I will show how this task can be accomplished with numerical systems of relatively small size. The essential idea is to abstract from the analytical solutions in lattices two scaling laws, one for space and one for time. The time scaling law is easy. Simply look for steady states. Once one finds numerically a steady state obeying the symmetry (109a), one knows what it does forever, and extrapolating up to time scales of milliseconds is no problem. The spatial scaling is slightly trickier. In order to relate samples of different size to one another, let Gc be the Griffith energy density [12,66]; that is, twice the crack surface energy density, a lower bound on the energy per unit area required for a perfectly efficient crack to propagate along a certain plane. One can then define a dimensionless measure of loading to be (159) G Gc where G is the fracture energy energy density already already introduced, introduced, i.e. the elastic strain strain energy stored per unit area (in the fracture plane) ahead of the crack. Analytical solutions for the ideal brittle solid show that the relationship between and crack velocity becomes independent of the height of the strip (number of planes stacked vertically) for surprisingly small strips [67]; a strip 80 atoms high has for all practical purposes reached the infinite limit, limit, Fig 20. Guidance Guidance for conducting conducting computationally computationally expensive expensive,, moderately moderately realistic, realistic, molecular dynamics simulations can be had from this simple result: The very rapid convergence of the main quantity of physical interest allows one to obtain physically meaningful results from simulations that are considerably smaller than many being carried out these days . This approach approach can also be viewed as an alternati alternative ve to methods that join together together atoms and continua [60–65]. If computational resources were infinite, one would still need to choose properly the sample geometry in order to compare with and inform experiment and theory. With limited computational resources, choice of sample geometry and scale are of paramount importance: tance: when performing performing molecular dynamics dynamics simulations simulations of fracture, fracture, one should not ask  48

how large a system one can work with, but rather what is the smallest system that will give give results scalable scalable to the macroscopic macroscopic level. level. A good scaling argument argument can transform transform a ‘grand challenge’ problem into something that’s better focused and less of a challenge. Size does matter: a smaller spatial scale allows one to follow the evolution of systems for longer times; and with a strip geometry one can then analyze fully steady state behavior and connect with theory and experiment.

5.1

Molecu Molecular lar dynami dynamics cs

The molecular dynamics method applies to any system of particles with some prescribed inter-particle potential. It is based upon the following equation: F 

ma

(160)

that is it consists of integratin integrating g Newton’s Newton’s equations equations of motion for all particles particles in lock step over a series of time steps, the size of the step being chosen small enough to give give converged converged dynamics. Time step integration is often done using the Verlet algorithm [68–70]. Lattice theory makes predictions that are hard to observe in experiment, and one of  the reasons reasons for doing doing computer computer simulati simulations ons is to relate relate the two. The simulati simulations ons I will will describe describe here in detail detail were carried out in silicon. silicon. Silicon Silicon is extremely brittle, brittle, and highquality quality macroscopic macroscopic single crystal crystal wafers wafers are cheap. It is therefore therefore an excellent excellent candidate candidate for laboratory fracture fracture experiments experiments.. Silicon Silicon is also technologicall technologically y of great importance, importance, and as a result is one of the most studied studied materials. materials. One measure of this is that there there are over thirty effective (although rather ineffective) interatomic potentials for silicon in the literature literature [71, 72]. Furthermore, Furthermore, transmission transmission electron micrographs micrographs of cracks in silicon silicon wafers reveal atomically sharp crack tips [13,73]. Silicon therefore is an obvious candidate for molecular molecular dynamics investigation investigationss of dynamic dynamic fracture, fracture, and an appropriate appropriate setting for testing atomic–level understanding. If there is any brittle element for which one ought to have the knowledge base making it possible to calculate fracture properties in detail, and the motivation to carry the work through, it would be silicon.

5.2

Interato Interatomic mic potent potential ialss

The equation equation underlyi underlying ng material materialss phy physic sicss is not in doubt. It is Schr¨ Schr¨odinger’s odinger’s equation. This equation can be solved analytically for the hydrogen atom, numerically for the helium atom, and with reliable approximate methods of quantum chemistry for small molecules. For any solid of interest in the study of materials, all hope of controlled approximations must be abandoned, and the Schr¨odinger equation is brutally reduced to tractable form in a way that is refined by comparison with with experiment. Quantitative methods that employ such approximations from the beginning are called ab initio. Despite their origin, the ab inito methods methods are the most reliable technique technique available available for numerical numerical treatment treatment of materials. However However,, they are restricted restricted to perfect perfect crystals, crystals, and become unmanageable when the unit cell contains more than around 1000 atoms. The main motivation, then, for constructing empirical or effective ‘classical’ interatomic potentials is speed of computation and the ability to work with relatively large numbers of particles. 49

With effective potentials, 107 atoms followed for a few tens of nanoseconds is achievable. This difference in computational scales becomes important when modeling processes which require require a minimum minimum of  105 atoms to capture just some of the complex underlying physics physics — processes processes involving fracture, fracture, dislocation dislocation loops, grain grain boundaries, boundaries, or amorphousamorphousto-crystal transitions, for example. Another motivation for constructing effective interatomic potentials is that they make the complex physics of what are fundamentally quantum mechanical phenomena more physically physically intuitive, intuitive, so that one may interpret the results results of atomistic atomistic simulations simulations in terms of simple principles of chemical bonding [72].

5.3

Realistic Realistic potential potentialss for silicon? silicon?

Solid Solid silico silicon n is covalent covalent and has the open open diamon diamond d crysta crystall struct structure ure.. If only only two-bo two-body dy potentials operated among the atoms, one would expect the crystal to collapse in on itself  to form a close-packed structure, thereby reducing its energy. Covalent systems, however, are characterized characterized by restoring restoring forces between between pairs of contiguous interatomic interatomic bonds. That is, pairs of bonds with an atom in common want to maintain a preferred angle between them. These extra forces are what stabilize stabilize the open diamond structure structure in silicon, silicon, carbon, grey-tin, grey-tin, and germanium, for example. example. The lowest-order lowest-order way of capturing this property with effective interatomic potentials is to go beyond binary bonding and include a threebody term in the system’s Hamiltonian. The Stillin Stillinger ger-W -Webe eberr (two(two- and three-bo three-body) dy) potent potential ial [74] [74] has proved proved to be very very popu popular lar and durable in the literature. literature. It gives excellen excellentt elastic properties, properties, and captures well the nonlinear nonlinear physics involv involved ed in heating and melting. It therefore is a reasonable reasonable starting starting point for conducting conducting molecular dynamics dynamics fracture fracture simulations simulations in silicon. silicon. Unfortunatel Unfortunately y,

Figure 21: Crack tip blunting in Stillinger-Weber silicon: two dislocations open up at the tip, preventing preventing it from advancing. The crack is pointing pointing in the direction direction 1 10 in the plane (110), and the system is loaded with a strain parameter parameter 1 6. 50

the potential potential will not yield fracture fracture along along the experimenta experimentally lly preferred preferred fracture fracture planes planes 111 and 110 : At low or moderat moderatee strain strainss a crack crack will will not move move at all. all. What happens happens is is that that two two dislocation dislocationss open up at the crack tip, blunting blunting it and preventing preventing it from advancing, advancing, Fig 21. One can play around with giving a transverse opening velocity to a select select few atoms around the tip. But to no avail. The crack simply will not crack. At very high strains, the crack tip region melts. Stillinger-Weber Stillinger-Weber does give a type of fracture along the 100 plane which is quite rough on the atomic atomic scale, Fig 22. Abraham Abraham et al call this brittle brittle fracture fracture [58]. There is as yet little consensus on a precise definition of brittleness. The experimental results of Lawn and Hockey Hockey [13, 73] for fracture along 111 in silicon show, however, that it is possible to have atomically atomically sharp fracture, fracture, i.e. where the newly created created fracture surfaces are atomically atomically flat. The experimental experimental evidence evidence for fracture along 100 , on the other hand, is scant and inconclusive [75], and that Stillinger-Weber yields fracture along this plane, albeit in a rough manner, might even be yet another indication of the potential’s shortcomings. It is possible to get cracks going, in an ideal brittle manner, along 111 and 110 with the Stillinger-Weber potential by increasing the restoring forces between pairs of bonds, i.e. by increasing increasing the stability of the tetrahedra tetrahedra in the diamond diamond lattice, lattice, and thus making the crystal more brittle. This can be done by scaling a parameter, , the coupling constant constant in the threethree-bod body y term. Once Once has been changed in this way, the potential acquires a new name, and is known as the Inadvertently Modified Stillinger-Weber potential (IMSW). Originally, Stillinger and Weber set 21 (dimensionless). However, by doubling this, as in the pioneering pioneering inadvertent inadvertent modification, modification, one can obtain fast brittle brittle fracture. With With a fast crack running, if one quasistatically decreases , the crack arrests well before one reaches 21, Fig Fig 23. Alth Althou ough gh 42 gives fracture phenomenology in reasonable accord

Figure Figure 22: A two-di two-dimen mension sional al projecti projection on of rough rough cracki cracking ng along along the (100) plane in Stillinger Stillinger-W -Weber eber silicon. silicon. The system is loaded with a strain parameter parameter 1 7, corresponding to a fracture energy density of almost three times the Griffith energy density. The average average crack speed is 1.9 km/s. 51

Figure 23: Crack velocity profile (a) along (110) 110 , and (b) along (211) 111 in modified Stillinger-Weber Stillinger-Weber silicon: s ilicon: scales the strength of angular forces between pairs of bonds that stabilize the diamond lattice structure. Stillinger and Weber’s value for is 21. However, leaving all other parameters unchanged, cracks will not propagate unless one uses a larger value of  . specifies the loading. with experiment, it has the adverse effect of raising the melt temperature above 3500 K, whereas experimentally the melt happens at 1685 K. The Young moduli also get shifted. Moduli results from tensile tests on small samples, for three lattice directions, are given in Table 2 for both the original and modified Stillinger-Weber (SW) potentials, along with the corresponding experimental values. The story of searches for potentials after learning about the Stillinger-Weber potential and its inadvertent modifications is long and alternately depressing and amusing. Dominic Holland and I devoted a fair amount of attention to a potential of Kaxiras and Bazant [72,76–7 [72,76–78], 8], and then then one of Chelik Chelikow owsky sky and Philli Phillips ps [79]. [79]. We We learne learned d that that these these potent potential ialss and many others have a defect that can easily be understood in a qualitative way. Imagine Imagine taking a uniform uniform crystal and simply expanding expanding the lattice lattice constant. This is a

52

computation that can be done easily by density functional theory, and one can compare the results with the same calculation calculation repeated repeated with a classical classical potential. potential. Then repeat the computation computation putting putting the crystal crystal in uniform shear. shear. Most classical classical potentials potentials have too large a peak when the crystal is expanded, and resist shearing too little. The height of the peak  resisting expansion can be as much as 5 times larger than the corresponding peak predicted by density functional theory. Getting these features of atomic force laws wrong is terrible if one wants to predict fracture properties at all accurately. If shearing is easier than pulling apart in tension, a crystal will emit dislocations by shearing at the tip rather than breaking in tension. In searching for a potential that would correctly reproduce the qualitative features obtained from density functional theory, I eventually settled upon the Modified Embedded Atom Method of Baskes Baskes [80, 81]. This potential potential is based upon a number of physical ideas ideas that I find quite persuasve, particularly the idea of screening. The density functional results for uniform expansion are built into the potential from the beginning, Without any tuning or adjustment, it does an acceptable job of describing the resistance to shear, and in computer simulations simulations it does permit permit the motion of brittle cracks. Still, Still, there are some arbitrary arbitrary features of the potential that are a bit unnerving, and from a quantitative point of view it does not turn out in the end to be any better than the IMSW with which our studies began.

5.4

Results Results of zer zero o Kelvi Kelvin n calculat calculations ions in silico silicon n

I will now describe the results of performing simulations on 100-200,000 atoms in silicon. Most of the calculations are with the modified Stillinger-Weber potential, but toward the end I also quote some results from MEAM. Questions to be answered: (1) Are there loads where cracks are attracted to steady states? states? (2) Do cracks emit phonons at the predicted predicted frequencies? frequencies? (3) Do cracks refuse to travel below a minimum velocity 1 0?, and (4) Do they go unstable above an upper load c ? The answer to all questions is yes. Fig 25, however, which shows atomic motions after the crack has been traveling for over 0.24 ns, as anticipated by the theory of ideal brittle fracture the crack has reached steady state with velocity =3460 m/s, which means that the vertical displacement z R of an atom originally at crystal location R is related to the vertical displacement z R na of an atom n lattice spacings a a x to the right by  z R

 E 100 100  E 110 110  E 111 111

na

t 

na

z R t 

(161)

Silicon Silicon Original Original SW IMSW GPa 130 114 172 GPa 169 139 189 GPa 188 151 201

Table 2: Elastic Elastic constants constants of silicon, silicon, comparing comparing Stillinger-W Stillinger-Weber eber,, modified modified Stillinger Stillinger-Weber potentials, and experiment.

53

6 5      )      ˚      A      /      V     e      (     e     c     r     o      F

Bazant-Kaxiras

4 Stillinger-Weber

3

Density functional

2 1 0 1.0

a

(A) 10 8

     )      V    e      (    m 6    o     t    a    r    e    p 4    y     g    r    e    n      E 2

0 1

(B)

1.4

1.2

1.6

a0

Density Functional Theory Modified Stillinger Weber Original Stillinger Weber Modified Embedded Atom Method

1.5

2 Extension Ratio

2.5

3

Figure Figure 24: (A) Compar Compariso ison n of density density functi functiona onall theory theory,, EDIP, EDIP, and the Stilli Stillinge ngerr-W Weber eber potential for the restoring force during uniform expansion. MEAM is identical to the density functional result by construction. (B) Same as (A), but now repeated for uniform shear. For a range of loads , Eq. 161 applies for any pair of atoms, whatever their separation along along the crack crack surface surface.. In order order to obtain obtain the perfect perfect periodici periodicity ty shown shown in Fig. 25, the crack was allowed to run first for 60,000 time steps so as to come into equilibrium with the waves it sends towards top and bottom boundaries. The longer and/or higher the system the longer it will take to reach a steady state. The 240 ps required to reach the steady state depicted in Fig 25 is about an order of magnitude longer than the duration of most large-scale molecular dynamics simulations of fracture. Using a reduced spatial system size, as validated by the scaling argument, is what makes this possible. possible. This becomes crucially crucially important important when proceeding through through a sequence sequence of  steady states, which requires quasistatic changes in the loading, or G. To obtain obtain a full set of results, like Fig 26, a crack actually will travel tens of microns, or for times on the order of a tenth of a microsecond. microsecond. To achieve this, this, one needs not only efficient efficient code and a high performance computer, but also a physically motivated smallest computational cell — a minimum thin strip on a conveyor belt.

54

Steady State after 0.24 ns at ∆=1.6 vertical profile of two atoms 160 Angstroms apart 5

4

   )   s   m   o    t   r   s   g3   n    A    (    t   n   e   m   e   c   a    l   p   s    i    d2    l   a   c    i    t   r   e   v

1

0

1

2

3 time (ps)

4

5

Figure Figure 25: Crack Crack that that has been traveli traveling ng for over over 0.24 0.24 ns. That That the two overl overlapp apped ed curves are almost completely indistinguishable shows the crack has reached steady state, according according to Eq. 161, and is emitting phonons phonons in accord accord with Eq. 153.

5.5

Along 110

The relation between velocity and load for cracks exposing exposing 110 and traveling along 110 is shown in Fig. 26. The crack velocity smoothly decreases as decreases, decreases, until at 2256 m/s and 1 258, the crack abruptly abruptly comes comes to a halt. Raising Raising again, the crack  does not begin to move until 1 366, a value that is sensitive to residual vibrations in the crystal, but the rising curve then perfectly overlaps overlaps the descending one. Crack speed continues to rise smoothly until 3586 m/s, 2 2, at which point steady state motion becomes unstable. When a crack goes unstable, complicated phenomena such as formation of small branches, emission of dislocations, changes in the plane of propagation can occur, and intermittency where the crack makes repeated attempts at branching.

5.6

Along 111 : Crackons

The relation between velocity and load 011 is shown shown in Fig. Fig. 27. For For 1 44

for cracks along 111 and traveling along 2 2, the crack has stable steady states, and 55

Figure 26: Relation Relation between between crack speed and loading for crack along 110 at 0 K. As descends, velocity drops abruptly to zero at a lower critical value, and as ascends resumption of crack motion is hysteretic. For convergence check, system size was doubled along x and z, and measured.

Figure 27: Relation between crack speed and loading for cracking along 111 01 1 at 0 K. Dotted lines indicate forbidden velocities. The lower figure shows crackons for 1 18 1 44: the crack is able to expose many many different states lying along many hysteresis hysteresis loops. Ideal steady states are unstable above 2 2.

56

for 2 2 it goes unstable in a similar manner to cracks along 110 . Ho Howe wever ver,, for 1 175 1 44 the dynamics of the crack exhibits a number of interesting features that have not been seen previously, and for which there is not yet a complete theoretical description. description. There is a variety variety of different different dynamical dynamical states available available for each value of  , where the crack travels travels at different different speeds. speeds. Each of these states corresponds to a plateau plateau in ; can change by as much as one fifth of the amount needed to go from arrest to instability and the crack velocity does not alter within numerical resolution. When the crack finally decides to accelerate out of the plateau, it may jump by over 1 km/s and reach an upper plateau to within within a few m/s. On cyclical cyclical loading the same plateaus plateaus are always reached. All of these transitions are hysteretic, as depicted in Fig. 27. The different states emit noticeably different phonons; on a given plateau, the phonon frequencies appear fixed and their amplitude changes, while between plateaus the frequencies change in accord accord with Eq. 153. This is crackon behavior behavior. All these phenomena phenomena are easily disguised disguised if strain rates are too high. Resolv Resolving ing all the fine structur structuree visibl visiblee in Fig. Fig. 27 required required 8 s 1 , or h z c 10 5 . 4

unstable

     )    s 3      /    m      k      (      d    e    e    p    s 2      k    c    a    r    c

1

0

1.2

1.4

1.6

1.8

2.0

2.2

Figure 28: Crack velocity profile for Si (111) 211 at 0 K. Dashed lines indicate forbidden velocities. The crack is unstable above 1 9.

5.7

Crack behavior behavior at nonzer nonzero o temperatu temperatures res in silico silicon n

Temperature implies energy fluctuations in time, so that if the crack gets trapped due to an energy fluctuation that reduces the fracture energy, further kinetic fluctuations may subsequently enable the crack to move on, giving rise to the possibility of creep. This suggests that the strength strength of lattice lattice trapping should should be a function of temperature. temperature. A series of simulations at 50 Kelvin intervals were carried out to investigate the effect of temperature on lattice trapping along 111 011 . The results are shown in Fig 29. Note also in Fig 30 that as is decreased below 1 the crack actually actually heals and travels travels backwards. backwards. This This is perfectly perfectly reasonable: reasonable: there there hasn’t hasn’t been any nonuniform nonuniform surface surface damage, damage, and no oxide layer has formed.

57

Figure 29: Velocity elocity gap vanishes vanishes near 200 K. Dotted Dotted lines indicate indicate forbidden velocities. velocities. All low-lying velocity states exist at room temperature. Cracking is along (111) 01 1 .

5.8

Compar Compariso isons ns with with experi experimen mentt

I turn to a number of questions questions that allow allow a comparison comparison of theory and experiment. experiment. First, is there a velocity velocity gap in brittle amorphous amorphous materials materials at room temperature? temperature? The answer seems seems pretty pretty clearly clearly to be “no;” “no;” data data to this effect effect appear appear in Figure Figure 31. There There is no detailed dynamical theory for the motion of cracks in very brittle amorphous materials, so the experiments are the best guide to what happens. However However,, the point of performing performing numerical work in crystalline silicon silicon was the possibility bility of carrying carrying out experiments experiments to compare compare directly. directly. The first were done by Hauch et al. [82], with additional experiments by Cramer, Wanner, and Gumbsch [83], and more recently by Deegan, who has been working with me to obtain data at cryogenic temperatures. Room-temperature runs were performed along 111 112 in order to allow direct comparison with experiment [82] (the zero Kelvin runs for this lattice direction are in Fig 28). ˚ 3, The results are in Fig. 32, and were obtained for a thin strip of size 532 15 154 A periodic periodic along the thin axis. As before, new material was added ahead of the crack tip and ˚ of the old material lopped off at the tail every time the crack advanced to within 200 A forward end of the strip. In this fashion, the crack traveled 7 m during the course of the simulations as G was varied between 5 and 14 J/m 2 . The room-temperatu room-temperature re experiments experiments for fracture in silicon wafers along (111) 112 are described described in [82]. These experiments experiments are difficult to perform because of the high Young’s modulus and brittleness of silicon. 58

Figure 30: Crack velocity profile with respect to quasistatic loading silicon at 300 K. All low-lying velocity states exist.

along 111 01 1 in

300 PMMA 200 100 0 0      )    c 300    e    s      /    m 200      (    y     t      i    c    o      l    e      V

100

200

300

Homalite-100

100 0 0

100

200

300

400

500

75 Glass

50 25 0 0

400

800 Time t  (

1200

1600

sec)

Figure 31: Velocity Velocity records for experiments in the double cantilever beam configuration. In PMMA and Homalite-100 crack arrest was achieved. In both materials the crack appears to decelerate smoothly but with increasing deceleration until it arrests. In glass crack arrest was not achieved in this configuration. However there is no sign of an initial velocity velocity jump. This can be attributed to the sharp seed cracks that can be generated in glass.

59

4

6

   ]   m   c 4    [    h    t   g   n 2   e    L

3      )    s      /    m 2      k      (

0 0

2

4

6

8

10

12

14

16

18

4

   ]   s    / 3   m    k    [   y 2    t    i   c   o 1    l   e    V

0 0

0 0

(A)

Experiment Molecular Dynamics

1

2

4

6

8

10

12

14

16

Time[µs]

5

10

15

18

Energy flow to crack tip

(B)

G

20 2

[J/m ]

5

4

   ]   s    /   m    k    [   y    t    i   c   o    l   e    V

3

2

Experiment (111, Hauch) Experiment (110, Cramer) Experiment (111, Deegan, ramp)

1

0 0

20

40

60

80

2

G [J/m ]

(C) Figure 32: Experimental and numerical determination of the crack velocity as a function of fracture energy G along 111 112 at room temperature. temperature. The fuzz indicates the spread in simulation velocity data. Experimental data from Hauch et al. [82], Cramer Cramer et al. [83] and R. Deegan.

60

The highest experimental and numerical crack velocities shown in Fig. 32 are reasonably close, but the minimum fracture energies at which a crack propagates differ: 2.3 J m 2 in the experiments and 5.2 J m2 in the simulations. Since the scale of crack velocities in a material is bounded by sound speeds [12], which are given correctly by the StillingerWeber potential, it is not surprising that the experimental and computational crack velocity scales scales agree. Furthermore, Furthermore, the potential gives gives the correct correct cohesive energy energy of silicon silicon (but an inaccurate cohesive energy curve — see Sec 5.3), leading to the agreement in numerical and experimental energy scales. However, the nonlinear parts of the potential involved in stretching and rupturing bonds play an important role in determining the actual fracture energies and crack velocities, in particular where the crack arrests and what its highest velocity locity will be. The quantitative quantitative disagreeme disagreements nts shown here point to a shortcoming of the nonlinear nonlinear parts of the potential, potential, which have not received received much attention. The modified Stillinger-Weber potential cannot be expected to reproduce correctly experimental data. But it highlights highlights the control over brittleness brittleness in the three-body term, and the inadequacy inadequacy of  the potential potential tails in the two-body term. These are complicated complicated matters, matters, and yet perhaps only hinting at greater difficulties on the road to a better potential. The lowest fracture energy density at which a crack propagated in the experiments was close to the Griffith energy density for a (111) plane, 2.2 J/m 2 [84]. Since a crack cannot trav travel el with with less less ener energy gy,, ther theree must must be a narr narrow ow rang rangee of frac fractu ture re energ energy y over wh whic ich h the the crac crack  k  velocity rises rapidly from zero to the lowest value measured, 2 km/s. This phenomenon phenomenon is also seen in glass and polymers polymers [38]. Because Because of the extreme extreme precision precision required at the boundaries, experiments with silicon are not yet capable of settling the matter of whether this sharp velocity rise signals a velocity gap. However, as shown in Fig 29, in numerical silicon, the velocity gap is temperature dependent, vanishing above 200 K. At 300 K there do exist ‘steady states’ at all velocities between 0 and 3 km/s. The silicon experiments covered a range of fracture energy densities, 2 16 J m 2 , in which the cracks cracks produced produced very smooth smooth surfaces. surfaces. Thus, cracks cracks in silicon can dissipate dissipate large large amounts of energy, more than seven times the amount needed to create a clean cleavage through the whole crystal, without leaving behind any large scale damage on the fracture surfaces. surfaces. Investiga Investigation tion by atomic force microscopy microscopy shows that for low fracture fracture energies energies the fracture surfaces are flat on the nanometer scale, while at higher energies the surfaces have have pronounced features. features. These features, features, however however,, are smooth on the micron scale, and account for height variations on the order of 30nm over an area of 16 m 2 . The roughness gives an area increase of only 0.1% above above that of a flat cleaved surface. surface. This extra extra surface cannot account for the sevenfold increase in dissipated energy. The simulations, however, indicate that most of the energy can be carried off in lattice vibrations.

5.9 5.9

Fina Finall thou though ghts ts

I begin by posing three questions: questions: How can one calculate the energy needed for a crack to propagate? When is the motion stable stable or unstable? unstable? How does a crack choose its path? 61

Theory and experiment in Silicon

   ]   s 3    /   m    k    [   y    t    i   c 2   o    l   e   v    k   c   a   r 1    C

0

0

2

4

MEAM, 0 kelvin MEAM, 77 kelvin Hauch et al, (111) IMSW, 300 kelvin Cramer et al, (110) Deegan, (111) 10 12

8

6

2

Fracture energy [J/m ]

Figure 33: Molecular dynamics runs at 300 K with Stillinger-Weber potentials and 0K and 77 K runs compared with experiments of Hauch et al, Cramer, Wanner, and Gumbsch [83], and Deegan.

Figure 34: Atomic force microscopy pictures of (111) fracture surfaces in silicon at three different energies: (a) G 4 3 J/m2 , (b) G 7 2 J/m2 , and (c) G 14 4 J/m2 . The crack  has traveled from left to right in a 112 direction. direction. The diagonal surface markings markings in (b) and (c) are approximately in a 110 direction, direction, but it is unclear what causes them. Pictures Pictures by Rachel Mahaffy; experiments and figure composition by Jens Hauch. I will close close by commen commentin ting g on how how well well they they are yet yet answere answered. d. For ideal ideal brittl brittlee crysta crystals, ls, such as silicon, the energy needed to propagate can now be calculated. The scaling problems needed to find the effect of microscopic features on macroscopic behavior have been solved by learning from analytically solvable solvable models. However, quantitative comparison of  theory and experiment is not better than around a factor of two. Furthermore, in amorphous materials such as glass there is no theory, and in more complex materials with intermediate microstructures it is not clear what predictive value atomic–scale considerations bring. The stability of the ideal brittle crystals against microscopic branching can be determined. However, the instabilities instabilities observed in real materials are on much larger larger than atomic scale, and they have not been followed far or explained. 62

Finally concerning crack paths, there are phenomenological rules that appear to have much experimental experimental support for cracks that travel slowly in isotropic isotropic materials. materials. For cracks that move quickly, it is quite uncertain what the rules ought to be, and neither theory nor experiment have gone very far. The interest and difficulty of these problems should be sufficient to ensure that fracture continues to interest physicists for some time to come.

References [1] B. L. Averbac Averbach, h, Fracture, an Advanced Treatise, vol. I, ch. Some physical aspects of  fracture, pp. 441–471. New York: Academic Press, 1968. [2] I. S. Grigor Grigorie iev v and E. Z. Meilkhov Meilkhov,, eds., eds., Handbook of Physical Quantities. Boca Raton: Raton: CRC Press, 1997. [3] J. W. Dally, Dally, “Dynamic “Dynamic photoelastic photoelastic studies of fracture, fracture,”” Experimental Mechanics, vol. 19, pp. 349–361, 1979. [4] [4] N. Mott, Mott, “Bri “Britt ttle le fract fractur uree in mild mild stee steell plat plates es,,” Engineering, vol. vol. 165 165,, pp. pp. 16– 16–18, 18, 1947 1947.. [5] E. Dulaney and W. W. Brace, “Velocity “Velocity behavior of a growing crack,” Journal of Applied  Physics, vol. 31, pp. 2233–2266, 1960. [6] A. Stroh, Stroh, “A theory theory of the fracture fracture of metals metals,,” Philosophica Philosophicall Magazine Magazine, vol. 6, pp. 418–465, 1957. Supplement: Supplement: Advances in Physics. [7] E. H. Yoffe, Yoffe, “The moving Griffith crack,” crack,” Philosophical Magazine, vol. 42, pp. 739– 750, 1951. [8] T. L. Anderson, Anderson, Fracture Mechanics: Fundamentals and Applications. Boca Raton: CRC Press, 1991. Fracture Mechanics. Berlin: Springer, 2002. [9] L. I. Slepyan Slepyan,, Models and Phenomena in Fracture

[10] K. B. B. Broberg, Broberg, Cracks and Fracture. San Diego: Academic Press, 1999. [11] M. F. F. Kanninen and C. Popelar, Popelar, Advanced Fracture Mechanics. New York: York: Oxford, Oxford, 1985. [12] [12] L. B. Freund Freund,, Dynamic Fracture Mechanics. Cambri Cambridge: dge: Cambri Cambridge dge Univer Universit sity y Press, 1990. [13] [13] B. Lawn, Lawn, Fracture in Brittle Solids. Cambridge: Cambridge: Cambridge Cambridge Universit University y Press, second ed., 1993. [14] [14] D. Broek Broek,, The Practical Use of Fracture Mechanics. Dordrecht: Dordrecht: Kluwer Academic Academic Press, 3rd ed., 1994.

63

[15] G. R. Irwin, “Fracture “Fracture mechanics,” mechanics,” in Structural Mechanics (J. N. Goodier and N. J. Hoff, eds.), pp. 557–591, Elmsford, NY: Pergamon Press, 1960. [16] E. Orowan, “Energy criteria of fracture,” fracture,” Weld. Res. Supp., vol. 34, p. 157, 1955. [17] B. Cotterell Cotterell and J. R. Rice, Rice, “Slightly curved curved or kinked cracks,” cracks,” International Journal of Fracture, vol. 14, p. 155, 1980. [18] G. R. Irwin, “Fracture, “Fracture,”” in Handbuch der Physik , vol. vol. 6, p. 551, Berlin: Berlin: Spring Springer er-Verlag, 1958. [19] [19] G. R. Irwin, Irwin, “Anal “Analysis ysis of stresse stressess and strains strains near the end of a crack crack traversin traversing ga plate,” Journal of Applied Mechanics, vol. 24, pp. 361–364, 1957. [20] N. I. Muskhelis Muskhelishvili, hvili, Some Basic Problems of the Mathematical Theory of Elasticity. New York: P. Noordoff and Company, 1933. (English Translation). [21] H. Gao, Y. Y. Huang, and F. F. F. F. Abraham, “Continuum and atomistic studies of intersonic crack propagation,” JMPS, vol. 49, pp. 2113–32, 2001. [22] [22] A. J. Rosaki Rosakis, s, “Inter “Intersoni sonicc shear shear cracks cracks and fault fault rupture ruptures, s,”” Advances in Physics, vol. 51, pp. 1189–257, 2002. [23] R. V. V. Gol’dstein Gol’dstein and R. Salganik, Salganik, “Brittle fracture of solids with arbitrary arbitrary cracks, cracks,”  International Journal of Fracture, vol. 10, pp. 507–523, 1974. [24] J. A. Hodgdon Hodgdon and J. P. P. Sethna, “Derivati “Derivation on of a general general three-dimensional three-dimensional crackcrackpropagation propagation law: a generalizat generalization ion of the principle of local symmetry., symmetry.,”” Physical Physical Review B, vol. 47, pp. 4831–40, 1993. [25] A. Yuse Yuse and M. Sano, “Transiti “Transition on between crack patterns in quenched glass plates,” plates,”  Nature, vol. 362, pp. 329–31, 1993. [26] O. Ronsin, F. F. Heslot, and B. Perrin, “Experimental “Experimental study of quasistatic brittle brittle crack  propagation,” Physical Review Letters, vol. 75, pp. 2252–5, 1995. [27] M. Marder, Marder, “Instability “Instability of a crack in a heated heated strip,” strip,” Physical Review E , vol. R51, pp. 49–52, 1994. [28] H. A. Bahr, Bahr, A. Gerbatsch, U. Bahr, Bahr, and H. J. Weiss, “Oscillatory instability in thermal cracking: cracking: a first-order first-order phase-transit phase-transition ion phenomenon,” phenomenon,” Physical Review E , vol. 52, p. 240, 1995. [29] M. Adda-Bedia and Y. Y. Pomeau, “Crack instabilities in a heated glass strip,” Physical  Review E , vol. 52, pp. 4105–4113, 1995. [30] R. C. Ball and H. Larralde, Larralde, “Three-dimensiona “Three-dimensionall stability analysis analysis of planar planar straight straight cracks propagating quasistatically under type i loading,” International Journal of  Fracture, vol. 71, pp. 365–77, 1995.

64

[31] H. Larralde and R. C. Ball, Ball, “The shape of slowly growing growing cracks, cracks,” Europhysics Letters, vol. 30, pp. 87–92, 1995. [32] [32] J. R. Rice Rice,, Fracture, vol. 2, ch. Chapter Chapter 3, pp. 191–311. Academic Academic Press, 1968. [33] J. S. Langer and A. E. Lobkovsky Lobkovsky, “Critical “Critical examination examination of cohesive-zo cohesive-zone ne models in the theory of dynamic fracture,” Journal of the Mechanics and Physics of Solids , vol. 46, no. 9, pp. 1521–56, 1998. [34] A. Karma, D. Kessler, Kessler, and H. Levine, Levine, “Phase-field “Phase-field model of Mode III dynamic fracture,” Physical Review Letters, vol. 87, p. 045501, 2001. [35] I. S. Aranson, Aranson, V. V. A. Kalatsky Kalatsky,, and V. M. Vinokur, inokur, “Continuum “Continuum field theory of crack  propagation,” Physical Review Letters, no. 85, pp. 118–121, 2000. [36] [36] L. Slepyan, Slepyan, “Dynam “Dynamics ics of a crack crack in a lattic lattice, e,”” Soviet Physics Doklady, vol. 26, pp. 538–540, 1981. Physical Review Review Letters Letters, [37] [37] M. Marder Marder and X. Liu, Liu, “Instab “Instabili ility ty in lattic latticee fractu fracture, re,”” Physical vol. 71, pp. 2417–2420, 1993.

[38] M. Marder and S. Gross, “Origin of crack tip instabilities, instabilities,”” Journal of the Mechanics and Physics of Solids, vol. 43, pp. 1–48, 1995. [39] R. Thomson, C. Hsieh, and V. V. Rana, “Lattice trapping of fracture cracks,” cracks,” Journal of   Applied Physics, vol. 42, no. 8, pp. 3154–3160, 1971. [40] R. Thomson, “The “The physics of fracture, fracture,”” Solid State Physics, vol. 39, pp. 1–129, 1986. [41] D. Holland and M. Marder, Marder, “Ideal brittle brittle fracture of silicon studied with molecular molecular dynamics,” Physical Review Letters, vol. 80, pp. 746–9, 1998. [42] S. A. Kulakhmeto Kulakhmetova, va, V. V. A. Saraikin, Saraikin, and L. I. Slepyan, “Plane “Plane problem of a crack in a lattice,” Mechanics of Solids, vol. 19, pp. 102–108, 1984. [43] S. I. Heizler Heizler,, D. A. Kessler Kessler,, and H. Levine, Levine, “Mode–I fracture in a nonlinear nonlinear lattice with viscoelastic forces,” PRE , vol. 66, pp. 016126/1–10, 2002. [44] D. A. Kessler and H. Levine, “Nonlinear “Nonlinear lattice lattice model of viscoelastic viscoelastic mode III fracture,” PRE , vol. 63, pp. 016118/1–9, 016118/1–9, 2001. [45] E. Gerde and and M. Marder, Marder, “Friction “Friction and fractur fracture, e,”” Nature, vol. 413, pp. pp. 285–288, 285–288, 2001. [46] [46] B. Noble, Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial  Differential  Differential Equations Eq uations . New York: Pergamon, 1958. [47] N. Ashcroft Ashcroft and N. D. Mermin, Mermin, Solid State Physics. Saunders, 1976. [48] [48] X. Liu, Liu, Dynamics of fracture propagation. PhD thesis, University of Texas at Austin, 1993. 65

[49] K. Ravi-Chandar Ravi-Chandar and W. G. Knauss, “Dynamic “Dynamic crack-tip stress under stress wave wave loading-a comparison of theory and experiment,” International Journal of Fracture, vol. 20, pp. 209–222, 1982. [50] K. Ravi-Chandar and W. W. Knauss, “An experimental investigation investigation into dynamic fracture: ture: I. Crack Crack initia initiatio tion n and arrest, arrest,” Internationa ol. 25, 25, Internationall Journal Journal of Fracture racture, vol. pp. 247–262, 1984. [51] K. Ravi-Chandar Ravi-Chandar and W. G. Knauss, Knauss, “An experiment experimental al investigatio investigation n into dynamic dynamic Internationall Journal Journal of Fracture racture, vol. 26, fracture: fracture: II. Microstructural Microstructural aspects,” aspects,” Internationa pp. 65–80, 1984. [52] E. Sharon, S. P. P. Gross, and J. Fineberg, Fineberg, “Local “Local crack branching as a mechanism for instability in dynamic fracture,” Physical Review Letters, vol. 74, pp. 5146–5154, 1995. Weak Turbulence. Boston: Academic Press, [53] P. Manneville, Dissipative Structures and Weak 1990.

[54] [54] H. Gao, Gao, “Surfac “Surfacee rougheni roughening ng and branch branching ing instabi instabilit lities ies in dynamic dynamic fractu fracture, re,”” Journal of the Mechanics and Physics of Solids, vol. 41, pp. 457–486, 1993. [55] [55] P. Gumbsc Gumbsch, h, S. J. Zhou Zhou,, and B. L. Holian Holian,, “Molecu “Molecular lar dynamics dynamics inves investig tigati ation on of  dynamic crack stability, stability,” Physical Review B, vol. 55, pp. 3445–3455, 1997. [56] S. J. Zhou, D. M. Beazley Beazley,, P. P. S. Lomdahl, Lomdahl, and B. L. Holian, “Large-scal “Large-scalee molecular molecular dynamics simulations of three- dimensional ductile failure,” Physical Review Letters, vol. 78, pp. 479–482, 1997. [57] F. F. F. Abraham, “Portrait of a crack: rapid fracture mechanics using parallel molecular dynamics,” IEEE Computational Science and Engineering, vol. 4, pp. 66–77, 1997. [58] F. F. Abraham, Abraham, “Simulatin “Simulating g materials materials failure using parallel molecular molecular dynamics, dynamics,”” in Some New Directions In Science On Computers (G. Bhanot, S. Chen, and P. Seiden, eds.), pp. 91–113, World Scientific, 1997. [59] [59] R. K. Kalia, Kalia, A. Nakano Nakano,, K. Tsuruta, Tsuruta, and P. P. Vashisht ashishta, a, “Morphol “Morphology ogy of pores pores and interfaces interfaces and mechani mechanical cal behavio behaviorr of nanocluster nanocluster-assemb -assembled led silicon silicon nitride nitride ceramic, ceramic,”” Physical Review Letters, vol. 78, pp. 689–692, 1997. [60] J. Q. B. F. F. F. Abraham, N. Bernstein Bernstein and D. Hess, “Dynamic fracture fracture of silicon: concurrent simulation of quantum electrons, classical atoms, and the continuum solid.,”  Mater. REs. Soc Bulletin, vol. 25, no. 5, pp. 27–32, 2000. [61] V. B. Shenoy Shenoy, R. Miller, E. B. Tadmor, Tadmor, and R. Phillips, Phillips, “Quasicontinuum “Quasicontinuum models models of  interfacial structure and deformation,” Physical Review Letters, vol. 80, pp. 742–5, 1998.

66

[62] H. Rafii-Tabar, Rafii-Tabar, L. Hua, and M. Cross, “A multi-scale atomistic-continuum modelling modelling of crack propagation in a two-dimensional macroscopic plate,” vol. 10, pp. 2375–, 1998. [63] [63] E. B. Tadmor admor,, M. Ortiz, Ortiz, and R. Philli Phillips, ps, “Quasi “Quasicon contin tinuum uum analysis analysis of defect defectss in solids,” Philosophical Magazine A, vol. 73, pp. 1529–63, 1996. [64] T. Honglai Honglai and Y. Wei, “Atomistic/c Atomistic/continu ontinuum um simulation of interfacia interfaciall fracture. fracture. ii. atomistic/dislocation/continuum simulation,” simulation,” vol. 10, pp. 237–49, 1994. [65] T. Honglai Honglai and Y. Wei, “Atomistic/c Atomistic/continu ontinuum um simulation of interfacial interfacial fracture. i. atomistic simulation,” vol. 10, pp. 150–61, 1994. [66] A. Griffith, “The phenomena of rupture and flow flow in solids,” solids,” Mechanical Engineering, vol. A221, pp. 163–198, 1920. [67] M. Marder, “Small-scale “Small-scale simulations of fracture,” fracture,” in Fracture – Instability Dynamics, Scaling, and Ductile/Brittle Behavior  (R. L. B. Selinger, J. J. Mecholsky, A. E. Carlsson, and E. R. Fuller, eds.), (Pittsburgh), pp. 289–296, Materials Research Society, 1996. [68] L. Verlet, Verlet, “Compute “Computerr ”experiment ”experiments” s” on classical classical fluids. i. i. thermodynamic thermodynamical al properties properties of lennard-jones molecules,” Physical Review, vol. 159, p. 98, 1967. [69] D. Frenkel Frenkel and B. B. Smit, Understanding Molecular Simulation. New York: Academic Press, 1996. [70] [70] M. Allen Allen and D. Tilde Tildesle sley y, Computer Simulation of Liquids . Clarend Clarendon, on, Oxford, Oxford, 1987. [71] H. Balamane, T. T. Halicioglu, and W. W. A. Tiller, Tiller, “Comparative study of silicon empirical interatomic potentials,” potentials,” Physical Review B, vol. 46, no. 4, pp. 2250–2279, 1992. [72] M. Z. Bazant, E. Kaxiras, Kaxiras, and J. F. Justo, “Environment-dependent “Environment-dependent interatomic interatomic potential for bulk silicon,” Physical Review B, vol. 56, pp. 8542–??, 1997. [73] B. Lawn, B. Hockey Hockey,, and S. Wiederhorn Wiederhorn J. Mater. Sci., vol. 15, p. 1207, 1980. [74] F. H. Stillinger Stillinger and T. A. Weber, Weber, “Computer “Computer simulation simulation of local order in condensed condensed phases of silicon,” Physical Review B, vol. 31, p. 5262, 1985. [75] C. P. P. Chen and M. H. Leipold, Leipold, “Fracture “Fracture toughness of silicon,” silicon,” American Ceramics Society Bulletin, vol. 59, pp. 469–472, 1980. [76] M. Z. Bazant and E. Kaxiras, Kaxiras, “Modeling of covalent covalent bonding bonding in solids by inversio inversion n of cohesive energy curves,” Physical Review Letters, vol. 77, no. 21, p. 4370, 1996. [77] J. Justo, M. Z. Bazant, E. Kaxiras, V. V. Bulatov, Bulatov, and S. Yip Physical Review B, vol. 58, p. 2539, 1998.

67

[78] E. K. M. Z. Bazant and J. F. F. Justo, “The environment-dependent interatomic potential applied applied to silicon silicon disordered disordered structures structures and phase transitions, transitions,”” Mat. Res. Soc. Proc., vol. 491, p. 339, 1997. [79] J. Chelikow Chelikowsky sky and J. Phillips Phillips Physical Review B, vol. 41, p. 5735, 1990. [80] R. Ravelo Ravelo and M. I. Baskes, Baskes, “Free energy calculations calculations of Cu-Sn interfaces, interfaces,”” in Thermodynamics modynamics and Kinetics Kinetics of Phase Transformati Transformations. ons. Symposium (J. S. Im, B. Park, A. L. Greer Greer,, and G. B. Stephen Stephenson son,, eds.), eds.), (Pittsb (Pittsburg urgh), h), pp. 287– 287–93, 93, Materi Materials als Resear Research ch Society, Society, 1996. [81] M. I. Baskes, Baskes, “Modified embedded-atom embedded-atom potentials potentials for cubic materials materials and impurities,” PRB, vol. 46, pp. 2727–2742, 1992. [82] J. Hauch, D. Holland, Holland, M. Marder, Marder, and H. L. Swinney, Swinney, “Dynamic fracture fracture in singlesinglecrystal silicon,” silicon,” Physical Review Letters, vol. 82, pp. 3823–3826, 1999. [83] T. Cramer, A. Wanner, Wanner, and P. Gumbsch, “Energy dissipation and path instabilities in dynamic fracture of silicon single crystals,” PRL, vol. 85, pp. 788–91, 2000. [84] J. C. H. Spence, Y. Y. M. Huang, and O. Sankey Sankey, “Lattice trapping and surface reconstruction for silicon cleavage on (111). ab initio quantum molecular dynamics calculations,” Acta Metallurgica, vol. 41, pp. 2815–2824, 1993.

68