SUPPLEMENTARY INFORMATION

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1 Supplementary material How things break under shear and tension? How does a crack propagate in three dimensions when a material is both under tension and sheared parallel to the crack front? Numerous experimental observations for over half a century have shown that mixing those two fracture modes (antiplane shear and tension) produces stepped fracture surfaces with characteristic lance-shaped markings. Those markings are ubiquitous in both engineering and geological materials including glasses, polymers, ceramics, metals, and rocks. Although it is known that stepped surfaces result from a complex segmentation of the crack front into partial fronts, the mechanism of this segmentation has remained elusive. Besides these crack-front evolutions are hard to track during the material failing, especially in three dimensions. Movies of three-dimensional crack-front instability The accompanying movies of phase-field simulations reveal for the first time the complex path followed by a crack-front in mixed mode fracture. Those movies were obtained by numerical simulations of crack propagation using the phase-field model described in this supplementary material. The first two movies show frontal (frontal.mov) and lateral (lateral.mov) views of the propagating crack-front that produced the fracture surfaces that are displayed at different times in Fig. 1(c-d) of the article. The third movie (helical.mov) is an animation of Fig. 2(a) that highlights the helical nature of the crack-front instability. It shows that the front deforms into a helix that subsequently evolves nonlinearly into a more complex sawtooth wave shape as the crack-front segments into partial fronts to produce the observed stepped surfaces. 1

2 Phase-field method Crack propagation was studied using the phase-field method 1-4. This method has the advantage that it tracks automatically the crack-front evolution by the introduction of a scalar phase field, which distinguishes between broken and unbroken states of the material. This field varies smoothly in space on the fracture process zone scale, thereby providing a smooth cut-off for the divergence of the singular stress fields. This method has recently been tested numerically in two dimensions through benchmark comparisons with predictions of continuum fracture mechanics for kink cracks under mixed mode I+II plane strain loading 1 and for wavy cracks under biaxial loading 2. In addition, a crack propagation law consistent with the standard principle of local symmetry 5 was derived analytically from the phase-field model for twodimensional propagation in isotropic media 2. The total energy of the material is represented by the functional E = d 3 x ρ 2 tu φ 2 + g(φ) ( e strain e c ) (s1) ( ) 2 + κ 2 where is the strain energy density and is the usual strain tensor of linear elasticity with corresponding to, respectively. The broken state of the material becomes energetically favoured when the strain energy exceeds the threshold and is a monotonously increasing function that describes the softening of the elastic energy at large strain. The evolution equations for and the three components of the displacement are derived variationally from equation (s1) and are given by (s2) (s3) 2

3 The resulting set of four coupled partial differential equations provides a self-consistent description of both macroscopic linear elasticity and material failure without the need to track explicitly the crack front location, which can be defined conveniently as the leading edge of the fracture surface. Moreover, this gradient dynamics guarantees that the total energy is a monotonously decreasing function of time with all the energy dissipated inside the fracture process zone on a characteristic time scale. After writing the phase-field equations in dimensionless form by measuring length in units of the fracture process zone scale and time in units of, the remaining parameters were chosen as, ( ), and, where is the shear wave speed. Slab geometry and loading conditions. Crack propagation was simulated in a large strip of width, thickness, and length along the propagation axis. Mixed mode I+III loading was imposed by choosing initially (mode I) and (mode III) corresponding to stress intensity factors and displacements fixed at all time on the, respectively, and by keeping the boundaries. Periodic boundary conditions were imposed in the direction. The strip length and width ( and ) were chosen large enough to eliminate boundary effects as much as possible and was varied over a wide range (see Figure captions) to study the development of instability for different extensions of the crack front. Aside from the strip size, the control parameters of the simulations include the ratio and the ratio where is the energy release rate at planar crack propagation and the Griffith threshold for propagation of a semi-infinite crack. The phase-field model describes an ideally brittle limit where this threshold is simply twice the surface energy,, where the surface energy is given by the expression 1,4 (s4) 3

4 For the given choice of, the above expression yields. The ratio was chosen in the range 1.25 to In this range, the speed of the parent and daughter cracks do not exceed about one third of the shear wave speed and inertial effects do not suffice to produce a crack branching instability. Therefore the crack segmentation instability studied here only occurs with the superposition of mode III for this range of. This was checked explicitly by repeating a few test simulations with the only modification that the quasistatic equations without inertia were solved with a successive over-relaxation (SOR) method instead of the wave equations. The main characteristics of the instability were unchanged. Numerical implementation. The equations were discretized with a second-order accurate finite difference scheme on a uniform mesh with grid spacing (which yields a slightly larger surface energy ) and integrated with an explicit scheme that handles accurately the energy equation with a time step. The FORTRAN code was parallelized with MPI to handle the computing and memory requirements of large strip sizes with up to grid points and the simulations were carried out on Linux Clusters at Northeastern University. Relationship of phase-field and materials parameters. The parameters that enter the phase-field model include both known material parameters and parameters specific to the phase-field model that can be related to physical quantities. The known material parameters include the density and the Lamé coefficients in the elastic energy, from which we can define Poisson s ratio and the shear wave speed. The three parameters specific to the phase-field model include the coefficient of the phase-field gradient-square term in the energy functional (equation (s1)), the threshold elastic energy density for bond breaking, and the kinetic coefficient that governs the rate of evolution of the phase-field in equation (s2). As discussed above, those last three parameters can be combined to define (i) the physical 4

5 length that measures the size of the process zone, which is the region around the crack edge where the phase-field increases from a value close to zero in the broken material to one in the unbroken material, (ii) the characteristic time scale of energy dissipation in this zone, and (iii) the surface energy via equation (s4). With length and time measured in units of and, respectively, the crack dynamics in the phase-field model is controlled solely by Poisson s ratio, the dimensionless combination, and the external loading conditions through the ratios and, where is the energy release rate. In the quasi-static limit where inertial effects are small, the geometrical evolution of the crack surface becomes insensitive to the wave speed and thus to the ratio. Consequently, this evolution is controlled predominantly by Poisson s ratio and external loading conditions with the process zone scale acting only as a scaling length. Namely, if a material A has a process zone scale a hundred times larger than some other material B, the initial scale of instability would be predicted by equation (2) to be a hundred times larger in A than B, but the crack evolutions would be identical in the two materials up to a change of scale. Thus estimates of, which can vary by several orders of magnitude for different materials, can be used to compare phase-field simulations to experiments. This was done in the main text when comparing phase-field predictions for the initial instability wavelength (equation 2) to experimental observations in glass and PMMA. Instability wavelength An analytical expression for the instability wavelength can be derived heuristically by considering the effects of both configurational and cohesive forces acting on a helical crack front. Configurational forces acting perpendicular to the local crack plane have been treated by Eshelby 6 in the traditional theoretical framework of continuum fracture mechanics. The configuration force originates from the directional dependence of the 5

6 energy release rate that is present when is non-vanishing at some point along the Figure s1. Schematic representation of the in-plane (green line) and the outof-plane (red line) projections of a helical crack-front. The arrows indicate the direction of forces acting perpendicularly to the plane of the parent crack at points (black filled circles) separating leading A and lagging B zones of the front. The forces include the destabilizing configurational forces originating from the directional dependence of the energy release rate (orange arrows) and the stabilizing cohesive forces (gray arrows). front. In this case, the stress distribution is non-symmetrical about the instantaneous crack propagation axis and the crack can release energy at a faster rate by changing its growth direction. This force has magnitude 1,6, where the prime symbols indicate that the stress intensity factors are evaluated locally along the front and are distinct from the unprimed ones that are fixed by external loading conditions. More recently, Hakim and Karma derived laws of crack motion from the phase-field model of fracture in two dimensions. They obtained a condition for at the crack tip that reflects the balance between Eshelby s configurational force 6 and cohesive forces acting perpendicularly to the crack front. The latter are non-vanishing when the surface energy is anisotropic. When the surface energy is isotropic, which is the case considered here, 6

7 this force balance reduces to the standard principal of local symmetry 5 ( ). For propagation in three dimensions, however, the crack front is a curved line. In this case, Eshelby s configurational force 6 perpendicular to the local crack plane can be balanced by a local cohesive force generated by the out-of-plane curvature of the front. This cohesive force has a magnitude where is the surface energy and is the local out-of-plane radius of curvature of the front as illustrated schematically in Figure s1 of this supplemental material. The balance of these forces yields the condition of equation (1) in the main text, which extends the principle of local symmetry ( three dimensions. A scaling relation for the marginal instability wavelength ) to, i.e. the wavelength for which a helical perturbation neither grows nor decays, can now be derived by requiring that those forces balance on the sides of protruding A facets, which correspond to the points separating A and B zones marked by filled circles in Figure s1. At those points, we have and, where is the amplitude of a helical perturbation of wavenumber. Furthermore, we use the fact that for a small amplitude perturbation close to the onset of propagation and that in the limit cohesive forces yields the equality. Balancing the configurational and. Substituting in this equality the above relations,,, and, we obtain the prediction for the marginally unstable wavelength. Finally, we use the fact that the numerically computed linear stability spectrum (Fig. 2(c) of the main text) is well fitted by a quadratic polynomial of the form. This form implies that the fastest growing wavelength is approximately twice larger than the marginally stable wavelength, yielding. 7

8 Facet coarsening The coarsening of the wavelength by the elimination of daughter cracks is directly analogous to the coarsening of the wavelength of finger fronts in other well-known interfacial pattern forming instabilities, including Saffman-Taylor viscous fingers 7, directional solidification fingers 8 and Laplacian needle growth 9. In those examples, the fastest unstable mode of an array of fingers typically has a wavelength equal to twice the finger spacing 7,8. The symmetry is spontaneously broken by small perturbations and the spatial period doubling instability has generically no threshold in situations where the system is invariant under translation along the growth direction. The absence of threshold stems from this translational invariance (i.e. the standard goldstone mode argument). With an imposed gradient as in directional solidification 8, translation symmetry is broken and coarsening stops when the finger spacing is large enough. The rule is that the number of fingers decreases roughly inversely proportionally to the total growth length of the fingers up to small logarithmic corrections 9. This inverse relation follows from the simple picture that fingers interact through a long range field (here the stress field) without any intrinsic length scale. Therefore, there are no other scales for the coarsening than the spacing between actively growing fingers. This rule appears to describe well the coarsening of daughter cracks in Figure 3. Facet rotation angle Analytical expressions for the facet rotation angle can be obtained using the expressions for the stress intensity factors at the edge of daughter cracks

9 The assumptions that facets orient to be free of shear stress (shear-free), is mathematically equivalent to the assumption that they orient to maximize the mode I stress intensity factor,. Both principles yield at once the prediction of equation (3) in the main text. The assumption that the facets orient to maximize the energy release rate yields the condition where. This condition gives the same prediction as the first two principles up to a threshold value of beyond which it bifurcates into low and high angle branches with equal maximum energy release rate as shown in Fig. 4(a) in the main text. Above this threshold, which depends on Poisson s ratio, the shear- free branch has minimum energy release rate. As shown in Figure 4(a) of the manuscript, the shear-free (SF) and maximum energyrelease-rate (maximum G') conditions give identical predictions up to a critical value of /, beyond which the latter predicts a bifurcation into separate low and high angle branches (dashed lines). Experiments to date have explored ratios of / that fall below the threshold value of this bifurcation. Hence, they have been so far unable to test whether such a bifurcation occurs. The present phase-field simulations explore a much larger range of / and show that this bifurcation is absent. The phase field facet angle increases smoothly with /, in qualitative agreement with the shearfree prediction. After reaching a maximum, it then decreases at large. This decrease, which is not predicted by previous theories, can be understood by noting that 9

10 the magnitude of the destabilizing configurational force is proportional to K I K III because along a helical crack front is induced by mode III. Since vanishes at large for fixed energy release rate, this destabilizing force also vanishes for large. Thus cohesive forces that stabilize planar crack growth ultimately dominate for large, thereby causing the facet angle to decrease. We note that Sommer s experiments in glass 11 only explored a range of / much smaller than unity. Consequently, those experiments produced very small rotation angles (less than 3 degrees) that are not shown in Fig. 4(b) of the manuscript since they do not provide an adequate basis of comparison with our phase-field results. Only experiments in steel and PMMA that explore a larger range of /, and hence a larger range of angles, are shown. Also, to make the comparison of phase-field results and experiments quantitative, we have plotted the ratio versus / to scale out the dependence on Poisson s ratio that is material-dependent. REFERENCES 1. Hakim, V. & Karma, A., Laws of crack motion and phase-field models of fracture. J. Mech. Phys. Solids 57, (2009). 2. Henry, H. & Levine, H., Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys. Rev. Lett. 93, (2004). 3. Karma, A., in Handbook of Materials Modeling (ed. Yip, S.) (Springer, Netherlands, 2005). 4. Karma, A., Kessler, D. A., & Levine, H., Phase-field model of mode III dynamic fracture. Phys. Rev. Lett. 87, (2001). 5. Goldstein, R. V. & Salganik, R. L., Brittle fracture of solids with arbitrary cracks. Int. J. Fract. 10, (1974). 6. Eshelby, J.D., The elastic energy-momentum tensor. J. Elast. 5, (1975). 7. Kessler, D. A. & Levine, H.,Coalescence of Saffman-Taylor fingers: A new global instability. Phys. Rev. A 33, (1986). 10

11 8. Losert, W., Shi, B. Q. & Cummins, H. Z., Spatial period-doubling instability of dendritic arrays in directional solidification. Phys. Rev. Lett. 77, (1996). 9. Krug, J., Kassner, K., Meakin, P. & Family, F., Laplacian needle growth. Europhys. Lett. 24, (1993). 10. Cooke, M. L. & Pollard, D. D., Fracture propagation paths under mixed mode loading within rectangular blocks of polymethyl methacrylate. J. Geophys. Res. 101, (1996). 11. Sommer, E. Formation of fracture `lances' in glass. Eng. Fract. Mech. 1, (1969). 11