% ****** Start of file template.aps ****** % % % This file is part of the APS files in the REVTeX 3.0 distribution. % Version 3.0 of REVTeX, November 10, 1992. % % Copyright (c) 1992 The American Physical Society. % % See the REVTeX 3.0 README file for restrictions and more information. % % % This is a template for producing files for use with REVTEX 3.0. % Copy this file to another name and then work on that file. % That way, you always have this original template file to use. % \documentstyle[preprint,aps]{revtex} \begin{document} %\draft command makes pacs numbers print \draft \title{Crack Stability and Branching at Interfaces.} % repeat the \author\address pair as needed \author{Robb Thomson, NIST Fellow, Emeritus} \address{Materials Science and Engineering Laboratory\ National Institute of Standards and Technology\ Gaithersburg, MD 20899} \date{\today} \maketitle \begin{abstract} % insert abstract here The various events which occur at a crack on an interface are explored, and described in terms of a simple graphical construction called the crack stability diagram. For simple Griffith cleavage in a homogeneous material, the stability diagram is a sector of a circle in the space of stress intensity factors, $K_I/K_{II}$. The Griffith circle is limited in both positive and negative $K_{II}$ directions by nonblunting dislocation emission on the cleavage plane. For a branching plane inclined at an angle to the original cleavage plane, both cleavage and emission (which blunts the crack) can be described as a balance between an elastic driving force and a lattice resistance for the event. We use an analytic expression obtained by Cotterell and Rice for cleavage, and show that it is an excellent approximation, but show that the lattice resistance includes a cornering resistance, in addition to the standard surface energy in the final cleavage criterion. Our discussion of the lattice resistance is derived from simulations in 2D hexagonal lattices with UBER force laws with a variety of shapes. Both branching cleavage and blunting emission can be described in terms of a stability diagram in the space of the remote stress intensity factors, and the competition between events on the initial cleavage plane and those on the branching plane can be described by overlays of the two appropriate stability diagrams. The popular criterion that $k_{II}=0$ on the branching plane is explored for lattices and found to fail significantly, because the lattice stabilizes cleavage by the anisotropy of the surface energy. Also, in the lattice, dislocation emission must always be considered as an alternative competing event to branching. \end{abstract} % insert suggested PACS numbers in braces on next line \pacs{PACS numbers: 07.05.T, 62.20.M, 61.72.B, 61.72} % body of paper here \section {Introduction} A crack on an interface in a material can take any of four different event paths: It can cleave straight ahead on the interface plane, emit a dislocation on this plane, or it can branch in cleavage on an inclined cleavage plane, or it can emit a dislocation on an inclined slip plane. Depending on the symmetry of the lattice and the geometry of the crack line and cleavage plane, a number of slip or branching planes may be available. The subject of this paper is to give a unified presentation of the criteria for these various events, and introduce a graphical representation from which it is possible to pick off what path a crack under a particular (mixed) load will take, given that the force laws and crystal type are known. Since it is impossible to give criteria for the general crystal including all possible bonding types, we will illustrate the process with a generic 2D hexagonal lattice with a coherent interface, using UBER and associated pair forces. The criterion for branching is presumably associated with a Griffith condition with an appropriate crack driving force (energy release rate) for the particular plane. The emission condition is more complicated, and will be analyzed in terms of Rice's unstable stacking fault energy\cite{R}, $\gamma_{us}$ and its extensions\cite{ZCT,TC,T}. Both emission and cleavage events must be a balance between a driving force for the event, and a lattice resistance to the event. Since branching is a cleavage process, the driving force must be the standard energy release rate on the branching plane, written in terms of the local stress intensity factors, $k_i$, for the branching crack in the limit of zero length for the kink. (We will alternatively refer to the branching event as a kinking of the original crack, and the emerging crack on the branching plane as a kink.) The limiting local stress intensity factors for the kink are not obtainable from the stress intensity factors of the main crack in an analytic form, but must be obtained numerically\cite{BC,BCH,DR,L}. The more complicated interfacial case has been analyzed by He and Hutchinson\cite{HH}. Even though analytic results are not possible, Cotterell and Rice\cite{CR} have shown that the kink stress intensity factors are closely related to the stress fields surrounding the main crack, and have written an analytic approximation for the local $k$'s which is remarkably accurate. Since the continuum approximation introduces inaccuacies of its own, the Cotterell/Rice expressions may be as close as one can come to these quantities in the continuum approximation, and will be used as the analytic basis for our lattice comparisons. The correct driving force for cleavage has not been clearly identified until lattice studies\cite{AR} showed it to be given by the standard mechanics expression \begin{equation} {\cal G}_c={1\over 2\mu'}(k_I^2+k_{II}^2), \label{eqn1} \end{equation} where $\mu'$ is an appropriate elastic shear modulus. It has sometimes been claimed\cite{T1} that the cleavage condition should be predominantly a matter of the Mode I loading, because the opening mode is the only one to actually force the cleavage planes apart; otherwise the crack should simply reclose itself. But the lower limit for $k_I$ is actually determined by the fact that the crack will break down in shear and emit a dislocation when the shear load is sufficiently great on the Griffith circle, and for shear loads below this limit, the criterion is given by the $\cal G$ formula above. Thus, contrary to He and Hutchinson\cite{HH}, we will always take the local $\cal G$ on the kinking crack as the appropriate driving force. When this quadratic expression is transformed back to the remote loading stress intensity factors, of course, a more complex quadratic form in $k_I$ and $k_{II}$ results. The appropriate driving force for emission has been proposed by Rice\cite{R} to be the Mode II component of $\cal G$, or \begin{equation} {\cal G}_e={1\over 2\mu'}k_{II}^2. \label{eqn2} \end{equation} Again, when this equation is transformed back to the remote or lab system of stress intensity factors, a more complex form results, to be discussed in \S III. The lattice resistance for an event will be computed in the simulations to be reported. One would expect that the lattice resistance for cleavage should simply be the intrinsic surface energy of the lattice, but we will show that a special resistance due to turning the corner of the kink apparently comes into play, so that the standard Griffith relation is not satisfied exactly in this case. For emission, the lattice resistance has been the subject of intense study, and it has been shown\cite{ZCT2} that when the crack is not blunted by the emission, the lattice resistance is simply given by the unstable stacking fault, $\gamma_{us}$, as proposed by Rice\cite{R}. When the crack is blunted by the emission, however, the lattice resistance is more complex, and involves a product of $\gamma_s\gamma_{us}$, as shown by Zhou, Carlsson and Thomson\cite{ZCT} and by Thomson and Carlsson\cite{TC}. Blunting emission for the interface was studied by Thomson\cite{T}. The lattice modeling will be done with the methodology described elsewhere\cite{T,TZCT} for the 2D hexagonal lattice with nearest neighbor central force laws, and the reader is referred to the references for the details of the simulations. The reason for choosing the hexagonal lattice is that it is elastically isotropic in the continuum limit, so anisotropic corrections need not be considered for the analysis to be carried out. Also, our purpose is to illucidate the generic physics of cracks in lattices, and a method which gives quick results for a variety of different force laws in a simulation where the load mix and lattice mismatch can be varied easily is desired. \section{Analysis.} The Cotterell/Rice\cite{CR} prescription for the local stress intensity factor of a kinking crack when applied to an interface is given by\cite{RSW} \begin{eqnarray} k_I&&=\sigma_{\theta\theta}\sqrt{2\pi r}={\cal K}_I\Sigma^I_{\theta\theta} +{\cal K}_{II}\Sigma^{II}_{\theta\theta}\nonumber\\ k_{II}&&=\sigma_{r\theta}\sqrt{2\pi r}={\cal K}_I\Sigma^I_{r\theta} +{\cal K}_{II}\Sigma^{II}_{r\theta}\nonumber\\ {\cal K}&&={\cal K}_I+i{\cal K}_{II}=|K|e^{i(\psi-\eta)}\nonumber\\ e^{i\eta}&&=\left({2a\over r}\right)^{i\epsilon}\nonumber\\ \epsilon&&={1\over 2\pi}\ln{\kappa_1\mu_2+\mu_1\over\kappa_2\mu_1+\mu_2} ={1\over 2\pi}\ln{11(c_1/c_2)+5\over 11+5(c_1/c_2)}. \label{eqn3} \end{eqnarray} $K$ is the (complex) remote load stress intensity factor of the bulk material (i.e. without the interface) written as $K=K_I+iK_{II}=|K|\exp{i\psi}$. $\psi$ is the phase angle of the remote load. The stress intensity factor of the unkinked interfacial crack is written as $\cal K$ with the connection given above to the local stress intensity, $k_{II}$, and to the shear stress, $\sigma_{r\theta}$. This definition for the stress intensity factor differs slightly from that in common use\cite{RSW}, but is appropriate for the crack and load geometry in use in this work. In these equations, the additional phase angle at the crack tip generated by the elastic mismatch at the interface is given by $\eta$. $\eta$ is a singular logarithmic function of the distance, $r$, from the crack tip---the mode mixing anomaly characteristic of interfacial cracks in the continuum limit. $\epsilon$ is a constant which depends on the elastic mismatch, where $\kappa$ and $\mu$ are the standard isotropic elastic parameters for the two materials. The second form for $\epsilon$ in the last equation in (\ref{eqn3}) refers to the 2D hexagonal lattice, with the two spring constants, $c_1$ and $c_2$. The approximation by Cotterell and Rice\cite{CR} is to recognize that the shear stress when normalized by the square root of the radius has the dimensions of the stress intensity factor, and when written in terms of the angular variables, $r$ and $\theta$, has the actual character of the appropriate tensile or shear stress intensity factor for the branching crack. This approximation is rigorous, of course, in the limit of small $\theta$. Explicit expressions for the $\Sigma$'s are given by\cite{RSW} \begin{eqnarray} \Sigma_{\theta\theta}^I&&={1\over\cosh\pi\epsilon} \left[\sinh((\pi-\theta)\epsilon)\cos{3\theta\over 2}+ e^{-(\pi-\theta)\epsilon}\cos{\theta\over2}(\cos^2{\theta\over 2} -\epsilon\sin\theta)\right]\nonumber\\ \Sigma_{\theta\theta}^{II}&&=-{1\over\cosh\pi\epsilon} \left[\cosh((\pi-\theta)\epsilon)\sin{3\theta\over 2}+ e^{-(\pi-\theta)\epsilon}\sin{\theta\over 2} (\sin^2{\theta\over 2}+\epsilon\sin\theta)\right]\nonumber\\ \Sigma^I_{r\theta}&&={1\over\cosh\pi\epsilon}\left[\sinh((\pi-\theta)\epsilon) \sin{3\theta\over 2}+ e^{-(\pi-\theta)\epsilon}\sin{\theta\over 2}(\cos^2{\theta\over 2} -\epsilon\sin\theta)\right]\nonumber\\ \Sigma^{II}_{r\theta}&&={1\over\cosh\pi\epsilon} \left[\cosh((\pi-\theta)\epsilon) \cos{3\theta\over 2}+ e^{-(\pi-\theta)\epsilon}\cos{\theta\over 2}(\sin^2{\theta\over 2} +\epsilon\sin\theta)\right]. \label{eqn4} \end{eqnarray} It is more useful to write these complicated expressions in terms of a set of angular phase shifts in the form \begin{eqnarray} k_I'&&= \Sigma_{\theta\theta}K\,\cos(\psi-\eta_e+\lambda_{\theta\theta})\nonumber\\ (\Sigma_{\theta\theta})^2&&=(\Sigma_{\theta\theta}^I)^2 +(\Sigma_{\theta\theta}^{II})^2\nonumber\\ \lambda_{\theta\theta}&&=-\tan^{-1}(\Sigma_{\theta\theta}^{II}/ \Sigma_{\theta\theta}^I)\nonumber\\ k_{II}'&&= \Sigma_{r\theta}K\sin(\psi-\eta_e+\lambda_{r\theta})\nonumber\\ (\Sigma_{r\theta})^2&&=(\Sigma^{II}_{r\theta})^2+(\Sigma^I_{r\theta})^2 \nonumber\\ \lambda_{r\theta}&&=\tan^{-1}(\Sigma^I_{r\theta}/\Sigma^{II}_{r\theta}) \nonumber\\ e^{i\eta_e}&&=\left({2a\over r_0}\right)^{i\epsilon}. \label{eqn5} \end{eqnarray} These last equations are written for a critical value of the local stress intensity factor (with a prime) corresponding to emission, cleavage, etc. The total local phase shift for the crack is composed of the separate contributions from the phase shift for the remote load, $\psi$; the geometrical contribution due to the kink angle, $\lambda$; and the interfacial phase shift, $\eta$, written in terms of the core size, $r_0$. $\lambda$ is a function of the interface mismatch, as well as the kink angle, $\theta$. Earlier\cite{T}, we have found that the core size, $r_0$, can simply be set equal to the range parameter in the force law. The reader should note that the stress field of the kinked crack is not phase shifted by the length of the kink, because the crack is now off the interface in homogeneous material. However, the stress intensity factor in (\ref{eqn3}) does contain the phase shift\cite{HH}. The physical reason why the local stress intensity of the kinked crack is phase shifted by the interfacial mismatch is that the kinked crack is ``loaded'' by the phase shifted stress field of the parent interface crack. The criteria for critical events in terms of the local stress intensity factors are written with the subscripts corresponding to cleavage or emission. Thus \begin{equation} {\cal G}_c={(k'_{Ic})^2+(k'_{IIc})^2\over 2\mu'_{eff}} \label{eqn6} \end{equation} for cleavage, and \begin{equation} {\cal G}_{IIe}={(k'_{IIe})^2\over 2\mu'_{eff}}, \label{eqn7} \end{equation} for emission. In these equations, an appropriate shear modulus must be taken. If the crack is kinked off the interface into material 2, then $\mu'_{eff}=\mu'_2$, while if the crack is on the interface, then $\mu'_{eff}=2\mu'_1\mu'_2/(\mu'_1+\mu'_2)$, and the prime on the $\mu$ represents taking the appropriate plane strain or plane stress elastic modulus. In plane stress, which must be used in the 2D simulations, $\mu'=\mu(1+\nu)$, where $\mu$ and $\nu$ have their standard elasticity definitions. The appropriate ``right hand sides'' for the lattice resistance in these equations are to be determined by computer simulations. The accuracy of these Cotterell/Rice expressions are easily checked in the zero mismatch case from their paper, where comparisons are graphed as a function of the branching angle\cite{CR}. The main deviation appears in the value for $\Sigma_{r\theta}^{II}$, where the analytic expression is about 10\% higher than the correct numerical value at $\theta=60^\circ$. All other $\Sigma$'s are accurate to within a percent or so. \section{Crack Stability Diagram.} The crack stability diagram is a generalization of a graphical description of the Griffith criterion for simple homogeneous cracks. That is, for a straight crack in homogeneous material, the Griffith condition is \begin{equation} {\cal G}_c={k_{Ic}^2+k_{IIc}^2\over 2\mu'}=2\gamma_s, \label{eqn8} \end{equation} and is graphed in $k_I/k_{II}$ space as a simple circle. In continuum elasticity, the crack is stable so long as the loads correspond to stress intensity factors lying on the Griffith circle. In the lattice, a crack will be stable over a region in $k_I/k_{II}$--space containing the Griffith circle, because of lattice trapping. More important, the stability of the crack will be limited by the ability of the core of the crack to sustain shear stresses, and beyond some critical shear, the lattice will break down with emission of a dislocation. A diagram showing such a stable region of loading is shown in Fig. \ref{f1} (Use axes $k_I/k_{II}$). As explained in the introduction, when the crack is on an interface, the critical parameter for cleavage or emission is not the bare remote or lab stress intensity factor, but the local quantity for the core, Eqn. (\ref{eqn5}). A generalization of the stability diagram valid for the interface, for simple straight cleavage on the initial cleavage plane, or for nonblunting emission on that plane, will now look like the full Fig. \ref{f1}. Here, the local stress intensity factors are rotated relative to the remote axes by the core phase angle, $\eta_e$, where the core phase angles for emission and cleavage are assumed to be the same. It is important to note that, except for a $\cosh\pi\epsilon$ term, the $\cal G$--force for a straight interface crack is an invariant in the crack length, and does not contain the phase shift\cite{MS}. When the crack branches or kinks off the interface ($\theta=60^\circ$ is assumed), the more general expressions for the local stress intensity factors must be used, and now a contour plot for the values of constant $\cal G$ plotted in the remote or lab system of stress intensity factors, is shown in Fig. \ref{f2}. In the figure, the simple case of zero lattice mismatch (no interface) is shown contrasted with that where the spring constants of the two lattices differ by a factor of 10 ($c_1/c_2=10$). Note that even when the interface mismatch disappears, the maximum gradient of $\cal G$ is rotated off the axes, and the presence of a remote Mode II enhances the tendancy to branch. The crack stability diagram is completed when the limiting values of cleavage corresponding to dislocation emission and shear breakdown are provided by means of emission criteria for the particular force law. These criteria and actual points for specific cases are plotted on these diagrams in the next section. Finally, events on the inclined (kinking) plane will compete with events on the initial interface, and by plotting event loci on the stability diagram, it is possible to determine when, for example, emission of a dislocation on the inclined plane will occur before cleavage on the interface, etc. \section{The Lattice Resistance.} The lattice modeling uses the same lattice Green's function techniques used in previous work\cite{T}, and the general methodology is given in Thomson, etal.\cite{TZCT} Also we use the same 2D hexagonal lattice with the same set of nearest neighbor forces. Figure \ref{f3} shows the lattice with a crack on an interface between atoms of one kind below and a second kind above. In the lattice case, we are at liberty to make the bonding between the layers different from that of either bulk, as in the real physical situation. The elastic constant of material 2 is arbitrarily normalized to unity, since all the physical results scale with the elastic constant. Also, we allow the crack to emit a dislocation into material 2, only, and assume that material 1 is strong, incapable of deforming or cleavage. We also make the assumption that dislocation emission takes place on the ``forward'' slip plane, $\theta=60^\circ$, because the shear stress is largest on this plane. Subsequent emission could conceivably take place on the second slip plane, $\theta=120^\circ$, but multiple emission of dislocations is not explored in this work. The crack is loaded at the center of the crack with a concentrated load, see Fig. \ref{f3}. This method of loading means that the crack system is stable, because the $K$ at the crack tip decreases as the crack length increases by the elastic equation \begin{equation} K={F_0\over\sqrt{\pi a}}, \label{eqn9} \end{equation} where $F_0$ is the point load on upper and lower crack planes, and $a$ is the half length of the parent crack. As in previous work, our simulations are for a bimaterial slab. The slab is $2\times 10^3$ atoms thick, with the interface running down the center. The slab has periodic boundary conditions in the lateral direction, again with repeat distance of $2\times 10^3$ atom spacings. The crack itself is 201 atom spacings in total length, the cohesive zone is 12 atom spacings long on the cleavage plane to the right, and the inclined slip plane is 16 atom spacings long. Thus we have no worries about short crack effects, or interactions with neighboring cracks in the repeating cells, or with the free surfaces. In contrast with our previous work, we have restricted the force law in this work to be the UBER of Rose, etal.\cite{RSF} derived from the energy expression, \begin{eqnarray} E(u)&&=-{c\over 1-e^{-u_0/\alpha}}\left[\alpha^2(1+{u\over \alpha})e^{-u/\alpha} + {u^2-e_0\over2}e^{-u_0/\alpha}\right]\nonumber\\ e_0&&=u_0^2+2u_0\alpha+2\alpha^2. \label{eqn10} \end{eqnarray} $c$ is the lattice spring constant, $u$ is the displacement from the equilibrium distance between two atoms, and $\alpha$ is the range parameter. The lattice parameter will be normalized to unity. This equation has been modified from the standard UBER expression so it has zero energy and force at the second neighbor position, $r=u_0+1=\sqrt{3}$. It will be convenient to normalize the lattice spring constant in sublattice 2 to unity, $c_2=1$. We have performed a set of simulations for a variety of elastic mismatches, Figs. \ref{f4},\ref{f5}. In all these simulations, the range parmeters are held fixed ($\alpha_{12}=0.2,\ \alpha_2=0.13$), and $c_{12}=c_2=1.0$. Thus, for all cases, the interface energy and unstable stacking fault in material 2 are held constant, $\gamma_{us}=0.0116$ and $\gamma_s=0.0284$, in the natural units of the simulation (lattice parameter and spring constant, $c_2$, normalized to unity). Note that with these choices, the bonds in the interface are stronger than in the matrix. This choice amounts to an assumption about the kind of chemical interaction taking place between the two interfaces to the right of the crack tip, and the dislocation emission criterion depends on this chemistry. Nevertheless, the choice we make allows us to explore the form of the stability diagram, which is our purpose. We will return at a later point to discuss the implications of the interfacial chemistry problem. In these simulations, we have arbitrarily cut the bonds to the left of the base of the spur, so that the crack does not have to grow through ``good material'' to reach the branching plane. This corresponds to a chemical knife which has acted over the cleavage plane up to the crack tip, in the time honored manner of the continuum mechanics community. That is, a chemical adsorbate is assumed to have formed over the cleavage plane which has zero interaction with the atoms on the opposite cleavage plane--highly unlikely physically, but perfectly possible, mathematically. It has the advantage of allowing the branching crack to grow out of the tip, without regard for the problem of how it got there. Specifically, the first active bond at the tip is the one at 120$^\circ$ from the base of the right hand spur line where it meets the lower cleavage plane, and is depicted as the first connected bond shown in Fig. \ref{f3}. In Figs. \ref{f4},\ref{f5}, as the solid line, we plot the $\cal G$ appropriate to a simple Griffith law, where $\gamma=\gamma_s$ for material 2. The individual points are the simulation results, and they span the complete range of possible mixed loads, $K$, for which branching cleavage is possible. The upper limits correspond to lattice shear breakdown and dislocation emission on the inclined plane, while the lower values correspond to the lowest loads where the crack is stable on the inclined plane. The results show that the expected Griffith condition is not satisfied for the branching crack, but that an extra ``cornering'' energy is required to turn the crack out of its initial cleavage plane. This cornering energy is increased by increasing the lattice mismatch. The line of points, however, is parallel to the expected $\cal G$--curve, showing that the driving force for cleavage is correctly given by the sum of the squares of the local $k$'s in (\ref{eqn6}). The emission point obtained is consistent with the analytic emission criterion given in the previous paper\cite{T}. We note again, that the lower limit in the line of simulation points simply corresponds to the load where the first bond of the branching crack is not broken. For lower loads, the crack opens up to the artificially created crack tip, but cannot go further. In the case of the zero misfit, the numerical results for ${\cal G}_c$ are known, and can be compared with the actual stability diagram in Fig. \ref{f4}. As noted in \S II, the analytic expression is larger than the numerical value, so the cornering energy is slightly larger than the 30\% shown in Fig. \ref{f4}. Unfortunately, the numerical results of He and Hutchinson\cite{HH} are given for misfit parameters which differ completely from those of the present work, so we cannot comment on what portion of the cornering energy in Fig. \ref{f5} might be due to an error in the analytic Cotterell/Rice approximation for the interface. However, we see no physical reason why the error should be significantly larger than in the zero misfit case, and believe that increasing the misfit increases the cornering energy, as shown in Fig. \ref{f5}. There is another possibly important effect for the interface, however, associated with the dependence of ${\cal G}_e$ on the length of the kink when the elastic mismatch is nonzero. (See the discussion following Eqn. (\ref{eqn5}).) Because of the length dependence of the local $k$ for the kinking crack, the crack may require a larger ${\cal G}_c$ to nucleate than to propagate. This effect could masquerade for the increased cornering energy we observe for the high elastic mismatch case. We would not be able to tell the difference between the oscillation in ${\cal G}_c$ at the growing kink and an increase in cornering resisitance. In He and Hutchinson\cite{HH}, the oscillations in ${\cal G}_c$ are noted, but not quantified, because they have no way to infer the actual phase at the nucleating kink. (They simply suggest that the phase be ignored, which is not a good approximation in our results. In our work, we calculate it in terms of the force law range parameter.) In our simulations, the kinking crack always jumps forward, once it has been nucleated, which would be consistent with an oscillatory ${\cal G}_c$ similar to that shown in Fig. 2 of their paper, but it is also consistent with the existence of a local cornering energy for nucleating the kink. So, it is possible that the additional cornering energy we observe for large elastic mismatch case could be an effect due to oscillations in ${\cal G}_c$. That there is a physical cornering resistance, however, is clear, because it occurs for zero elastic mismatch, where the phase oscillation in ${\cal G}_c$ would disappear. Figs. \ref{f4},\ref{f5} represent the competition between cleavage on the spur plane and emission on that plane. But we noted in the Introduction that there is also competition with events on the cleavage plane to the right of the spur. Depending on circumstances, it is possible for the crack to either cleave or emit on that plane, as well, as shown in Fig. \ref{f1}. The total picture is obtained by overlaying events on the interface plane with events on the spur, as shown scematically in Fig. \ref{f6}. The radius of the Griffith circle for cleavage on the interface is proportional to $\sqrt{\gamma_s}$, and by changing the chemical bonding between the two interfaces, this circle can be expanded or contracted, relative to branching, as shown. If the events on the spur lie below the Griffith circle for cleavage (or for nonblunting emission) on the interface plane, then events on the spur plane will dominate over those on the interface plane. Alternatively, if the interface bonding is weak, then events on the spur plane can dominate. (The reader will now appreciate why it was necessary to choose a strong force law on the interface to explore events on the spur.) In the Fig. \ref{f6}, the branching plane stability line is shown crossing the smaller Griffith circle. In this case, events on the upper portion of the Griffith circle for the interface will be stable relative to branching, while for loads on the lower portion, branching will be stable relative to events on the interface plane. Such a degenerate situation can easily be set up in the computer with precisely the results predicted by the diagram, but it requires a careful choice of relative bonding between matrix and interface. Note that ``crossover'' from the interface plane to the spur plane takes place over a range of loads, but in the particular situation depicted in Fig. \ref{f6}, there is no combination of loads where the crack will emit a dislocation on the spur plane for the small Griffith circle case. For this particular choice of force laws, then, the crack may cleave or emit on the interface plane, or it may branch onto the spur plane as a cleavage crack. But it will not ever be possible to emit a blunting dislocation, and in this sense, the material is intrinsically brittle. However, there exists a combination of force laws such that the situation depicted in Fig. \ref{f7} is realized, and for a particular choice of loads, the crack stability is degenerate---it can either emit or cleave, corresponding to a crossover from brittle to ductile behavior. In previous papers\cite{RT,ZCT}, we have worked with pure Mode I loads, and this is seen as a special case of the more general criterion for crossover, where the loading mode is not specified. In Fig. \ref{f7}, zero elastic mismatch is assumed for clarity, and the upper limit of the branching curve where emission occurs lies precisely on the $K_I$ axis. In general, of course, the emission point does not lie on the $K_I$ axis. But it is true that negative Mode II enhances cleavage on the branching plane, so the upper limit of the branching stability line will tend to be in the vicinity of the $K_I$ axis, if not precisely on it. (This statement, of course, does not apply to the interface case.) Crossover from cleavage to emission can also be described in terms of the ductility parameter, $\cal D$, defined as \begin{equation} {\cal D}={{\cal G}_c\over{\cal G}_e} \label{eqn11} \end{equation} When ${\cal D}>1$ the material is ductile, brittle otherwise. Because both ${\cal G}_c$ and ${\cal G}_e$ are quantities depending on the mixing of the modes, the crossover condition defined in this way is not unique, but depends somewhat on the mode mixity, $\psi$, as described graphically, above. \section{Crack Path in a Homogeneous Crystalline Material.} In a separate paper\cite{T2}, we have shown that in an amorphous material, where the surface energy is completely isotropic in direction, a crack under mixed modes will deviate out of the initial crack path into the path where $k_{II}=0$ on the branching crack. In such cases, crack deviation is, of course, easy, and widely observed. But in a lattice, where $\gamma_s$ has cusps on the high density planes, only certain special planes are allowed for cleavage. In this paper, we have shown that significant Mode II can be sustained by a crack, up to the limits governed by the stability diagram. Indeed, for any given lattice, branching may be quite difficult, because not only must the Griffith relation on the branching plane be satisfied (with any additional cornering resistance which may be necessary), but dislocation emission on one of the allowed planes might intervene before the relevant Griffith relation is satisfied. We performed a simulation to explore the case of branching in a completely homogeneous lattice where all interfacial and chemical effects were taken out. For the UBER force law, we found that branching was possible only for a range parameter corresponding to the largest possible unstable stacking fault, for the reason quoted above---sufficiently large (negative) $K_{II}$ values were necessary, such that dislocation emission tended to occur on the cleavage plane. When branching was finally induced, however, it became possible only for a very narrow, essentially unique, load range, for which it was found that $k_{II}$ was not zero, but rather $k_{II}/k_I=.25$ when branching occurred. We consider this finding to be a significant violation of the expectation of He and Hutchinson\cite{HH} that $k_{II}=0$ at branching. Presumably the reason for the violation is that the stability diagram on the branching plane covers a range of $k_{II}$ values, and in the UBER law, only the extreme lower limit of the branching stability curve is reachable, and then only for a specific value of the range parameter. Thus, in the lattice, in spite of the continuum prediction that ${\cal G}_c$ is maximized when $k_{II}=0$, the stability line on the branching plane encompasses a range of $k_{II}$ values. Therefore, a complete analysis of the stability diagram for the given force law, including competing dislocation emission on all possible planes is necessary, in order to predict when branching will occur. Hutchinson Mear and Rice\cite{HMR} also apply the $k_{II}=0$ criterion for predicting the plane on which cracking will occur when the crack runs parallel to an interface. Presumably similar difficulties will arise in that case when the material is a crystal and not amorphous. Any plane parallel to the interface where the crack satisfies the stability criterion will be a possible plane of cracking. Since $k_{II}/k_I$ can reach values in the approximate range of 0.25 to 0.5 for reasonable force laws for stable cracks, and a crack, once established on a stable plane will not deviate from it, there should be a range of possible planes where a crack will be stable on a plane parallel to an interface. \section{Discussion.} In this paper, we have presented a simple graphical construction in the remote K--space of the crack, with which one can describe the competition between emission and cleavage on two nonequivalent slip/cleavage planes. In general, there may be more planes where the lattice can either cleave or emit dislocations. These more complex situations can be described by simply overlaying the additional diagrams on the ones described here. For events restricted to the initial cleavage plane, the stability diagram is a sector of a Griffith circle, where the circle limits represent lattice breakdown in shear and dislocation emission. For those force laws where lattice trapping is significant, then the Griffith circle fuzzes and develops a finite width, within which the lattice is stable under the relevant loads. When events are possible on an inclined slip plane, then a similar (noncircular) contour line for the critical ${\cal G}_c$ can be constructed in the same space. Dislocation emission corresponds to one limit on this curve, and the other limit corresponds to a critical local Mode I value below which the crack is no longer able to break the relevant bonds in the branching crack core. By overlaying these two plots, one can determine which events are favored, and which are not in a given load range. Moreover, the crossover between simple cleavage on the initial plane and blunting emission on an inclined plane can be determined by varying the interfacial bond strength. This crossover corresponds to the crossover between materials which are intrinsically brittle, and those which are intrinsically ductile. In this paper, we have used a simple lattice and simple force laws to illustrate the physical principles involved. In actual cases, one must carry out calculations for realistic force laws and lattices. For the case of branching, the criterion is quite simple. Branching is expected to occur whenever the extended Griffith relation (including the cornering resistance) with the bulk $\gamma_s$ in the stability diagram drops below the Griffith relation for simple straight crack extension with the interfacial $\gamma_s$. Since negative Mode II translates into a strong ${\cal G}_c$ in the branching plane, mixed modes on the interface strongly influence branching. The magnitude of the cornering resistance can range from about 30\% of the normal Griffith $\gamma_s$ to a factor of about 2 for the highest elastic mismatch interface (where branching would be quite difficult). However, in a crystal lattice, where large Mode II may be necessary in order to induce branching, the competing dislocation emission may intervene. Thus, depending on the details of the force law, increasing Mode II in a lattice may branch the crack, or it may break it down in shear with dislocation emission. It also follows that a crack which emits a dislocation on an inclined glide plane will develop a large mode mixity from the shielding by the emitted dislocation, and this could cause the crack to branch, as is sometimes seen in the computer simulations of Holian, etal.\cite{HZ}. We explored the prospects that branching will take place on the plane where the local $k_{II}=0$. While this criterion does hold for an amorphous material\cite{T2}, we found that significant deviations from this popular criterion take place for crack branching in a crystal lattice. % now the references. delete or change fake bibitem. delete next three % lines and directly read in your .bbl file if you use bibtex. \begin{references} \bibitem{R} J. R. Rice, J. Mech. Phys. Solids, {\bf 40}, 239 (1992). \bibitem{ZCT} S. J. Zhou, A. E. Carlsson, and R. Thomson, Phys. Rev. Lett., {\bf 72}, 852 (1994). \bibitem{TC} R. Thomson and A. E. Carlsson, Phil. Mag. {\bf 70}, 893 (1994). \bibitem{T} R. Thomson, Submitted Phys. Rev. \bibitem{BC} B. A. Bilby and G. E. Cardew, Int. J. Fract., {\bf 11}, 708 (1975). \bibitem{BCH} B. A. Bilby, G. E. Cardew and I. C. Howard, Fracture 1977, vol. 3, Univ. Waterloo Press, (1977) p 197. \bibitem{DR} V. V. Dudukelanko and N. B. Romalis, Izv. An. SSR, {\bf 8}, 129 (1973). \bibitem{L} K. K. Lo, J. App. Mech., {\bf 45}, 797 (1978). \bibitem{HH} M.-Y. He and J. W. Hutchinson, J. Appl. Mech., {\bf 56}, 270 (1989). \bibitem{CR} B. Cotterell and J. R. Rice, Int. J. Fract., {\bf 16}, 155 (1980). \bibitem{AR} P. Anderson and R. Thomson, J. Appl. Phys., {\bf 76}, 843 (1994). \bibitem{T1} R. Thomson, Solid State Physics, {\bf 39}, 1 (1986), ed. D. Turnbull and H. Ehrenreich, Academic Press, New York. \bibitem{ZCT2} S. J. Zhou, A. E. Carlsson and R. Thomson, Phys. Rev. B, {\bf 47B}, 7710 (1993). \bibitem{TZCT} R. Thomson, S. J. Zhou, A. E. Carlsson, and V. K. Tewary, Phys. Rev. B, {\bf 46B}, 10613 (1992). \bibitem{RSW} J. R. Rice, Z. Suo and J.-S. Wang, Metal--ceramics Interfaces, ed. M. Rulhe, A. G. Evans, M. F. Ashby and J. Hirth, Acta Scripta Met. Proc. Series {\bf 4}, 269 (1990) \bibitem{MS} B. M. Malyshev and R. L. Salganik, Int. J. Fract. Mech., {\bf 5}, 261 (1965). \bibitem{RSF} J. Rose, J. Smith and J Ferrante, Phys. Rev., {\bf 28}, 1835 (1983). \bibitem{ZT} S. J. Zhou and R. Thomson, Phys. Rev. B, {\bf 49B}, 44 (1994). \bibitem{RT} J. R. Rice and R. Thomson, Philos. Mag., {\bf 29}, 73 (1974). \bibitem{T2} R. Thomson, to be published. \bibitem{HMR} J. W. Hutchinson, M. E. Mear, and J. R. Rice, J. Appl. Mech., {\bf 54}, 828, (1987). \bibitem{HZ} B. Holian and S. J. Zhou, Unpublished report. \end{references} % figures follow here % % Here is an example of the general form of a figure: % Fill in the caption in the braces of the \caption{} command. Put the label % that you will use with \ref{} command in the braces of the \label{} command. % \begin{figure} \caption{Stability diagram. (For interpretation of the simple homogeneous lattice, consider only the axes $k_I/k_{II}$.) Any load satisfying the Griffith condition will lie on the circle. Dislocation emsission corresponds to the limits of the curve in the $k_{II}$ direction, where the lattice breaks down in shear. The locus of points on the Griffith line between these two limiting points corresponds to possible stable loadings of the lattice. When an interface is present, the criteria for cleavage or emission are given in the local core stress intensity factors ($k_I/k_{II}$ axes), which are rotated from the lab axes ($K_I/K_{II}$ axes) by the core phase angle.} \label{f1} \end{figure} \begin{figure} \caption{Contour plot of $\cal G$ for a branching crack. When the crack branches, then the simple circular Griffith line becomes a more general form, as shown for two cases. In (a), the interface mismatch is zero, while in (b), $c_1/c_2=10$. A stability diagram would correspond to a segment of a critical $\cal G$ contour between allowed values of $k_{II}$.} \label{f2} \end{figure} \begin{figure} \caption{The interfacial crack in a hexagonal lattice. The atoms below the centerline lie in sublattice 1 and those above the line in sublattice 2. The crack line lies on the interface between the two. Sublattice 1 has spring constant $c_1$, sublattice 2 has spring constant $c_2=1$, and the interfacial bonds have a third spring constant, $c_{12}$. Nonlinear bonds are formed at all atoms on the slip plane or the cleavage plane in the cohesive zone of the crack. Different nonlinear bonds can form on lower atoms, upper atoms (including the slip plane), and between atoms facing one another across the interface. The nonlinear bonds are depicted by the zig-zag lines.} \label{f3} \end{figure} \begin{figure} \caption{Simulations plotted as points on the stability diagram for zero mismatch between sublattices ($c_1=c_2$). The solid curve corresponds to the ${\cal G}_c=\gamma_s$ in material 2. If the Griffith relation were satisfied, then the points would lie on the curve. That they lie parallel to it corresponds to the fact that branching the crack off its initial cleavage plane requires an additional energy from that for the straight crack, which we term a cornering energy. The upper limit for the simulation points corresponds to dislocation emission, and the lower limit corresponds to the critical driving force required to break the first bond on the spur plane.} \label{f4} \end{figure} \begin{figure} \caption{Same as previous figure, except $c_1=10c_2$.} \label{f5} \end{figure} \begin{figure} \caption{Superposition of stability diagrams for initial plane and inclined plane for remote stress intensity factors, $K_I$ and $K_{II}$. Two Griffith circles are drawn with different radii, corresponding to different $\gamma_s$ values on the interface plane. The third curve corresponds to the stability line on the branch plane. In this case, only the upper limit of the curve corresponds to dislocation emission. In the case shown, the branching events are all more stable than any events on the large Griffith circle. However, the branch line crosses the small Griffith circle, and the upper part of the small Griffith circle is more stable than the upper part of the branch line, and vice versa for the lower parts of those curves. Note that for the case shown, there is no dislocation emission on the branch plane.} \label{f6} \end{figure} \begin{figure} \caption{Same as above, for ductility crossover. Here, blunting emission is degenerate with the cleavage Griffith circle. This choice of force laws thus corresponds to the crossover between ductile and brittle behavior for the material. This figure would correspond to a homogeneous lattice without interface. It is not expected that blunting emission should always lie exactly on the $K_I$ axis, but it probably does lie close to that axis in most cases.} \label{f7} \end{figure} % tables follow here % % Here is an example of the general form of a table: % Fill in the caption in the braces of the \caption{} command. Put the label % that you will use with \ref{} command in the braces of the \label{} command. % Insert the column specifiers (l, r, c, d, etc.) in the empty braces of the % \begin{tabular}{} command. % % \begin{table} % \caption{} % \label{} % \begin{tabular}{} % \end{tabular} % \end{table} \end{document} % % ****** End of file template.aps ******