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Laws of crack motion and phase-field models of fracture. (English) Zbl 1421.74089
Summary: Recently proposed phase-field models offer self-consistent descriptions of brittle fracture. Here, we analyze these theories in the quasistatic regime of crack propagation. We show how to derive the laws of crack motion either by using solvability conditions in a perturbative treatment for slight departure from the Griffith threshold or by generalizing the Eshelby tensor to phase-field models. The analysis provides a simple physical interpretation of the second component of the classic Eshelby integral in the limit of vanishing crack propagation velocity: it gives the elastic torque on the crack tip that is needed to balance the Herring torque arising from the anisotropic surface energy. This force-balance condition can be interpreted physically based on energetic considerations in the traditional framework of continuum fracture mechanics, in support of its general validity for real systems beyond the scope of phase-field models. The obtained law of crack motion reduces in the quasistatic limit to the principle of local symmetry in isotropic media and to the principle of maximum energy-release-rate for smooth curvilinear cracks in anisotropic media. Analytical predictions of crack paths in anisotropic media are validated by numerical simulations. Interestingly, for kinked cracks in anisotropic media, force-balance gives significantly different predictions from the principle of maximum energy-release-rate and the difference between the two criteria can be numerically tested. Simulations also show that predictions obtained from force-balance hold even if the phase-field dynamics is modified to make the failure process irreversible. Finally, the role of dissipative forces on the process zone scale as well as the extension of the results to motion of planar cracks under pure antiplane shear are discussed.

##### MSC:
 74R10 Brittle fracture 74E10 Anisotropy in solid mechanics
##### Keywords:
fracture; phase field; anisotropy; Eshelby tensor; herring torque
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##### References:
 [1] Adda-Bedia, M.; Arias, R.; Ben Amar, M.; Lund, F., Generalized griffith criterion for dynamic fracture and the stability of crack motion at high velocities, Phys. rev. E, 60, 2366-2376, (1999) [2] Ambrosio, L.; Tortorelli, V., Approximation of functionals depending on jumps by elliptic functionals via gamma-convergence, Commun. pure appl. math., 43, 8, 999-1036, (1990) · Zbl 0722.49020 [3] Amestoy, M.; Leblond, J., Crack path in plane situations. 2. detailed form of the expansion of the stress intensity factors, Int. J. solids struct., 29, 4, 465-501, (1992) · Zbl 0755.73072 [4] Aranson, I.; Kalatsky, V.; Vinokur, V., Continuum field description of crack propagation, Phys. rev. lett., 85, 118-121, (2000) [5] Barenblatt, G.; Cherepanov, G., On brittle cracks under longitudinal shear, Pmm, 25, 1110-1119, (1961) · Zbl 0107.41102 [6] Bourdin, B.; Francfort, G.; Marigo, J., Numerical experiments in revisited brittle fracture, J. mech. phys. solids, 48, 4, 797-826, (2000) · Zbl 0995.74057 [7] Brener, E.A.; Marchenko, V.I., Surface instabilities in cracks, Phys. rev. lett., 81, 5141-5144, (1998) [8] Broberg, K.B., Cracks and fracture, (1999), Academic Press San Diego · Zbl 0423.73064 [9] Corson, F., Adda-Bedia, M., Henry, H., Katzav, E., 2008. Thermal fracture as a framework for crack propagation law. cond-mat.mtrl-sci 0801.2101. · Zbl 1293.74372 [10] Cotterell, B.; Rice, J., Slightly curved or kinked cracks, Int. J. fract., 16, 2, 155-169, (1980) [11] Deegan, R.; Chheda, S.; Patel, L.; Marder, M.; Swinney, H.; Kim, J.; de Lozanne, A., Wavy and rough cracks in silicon, Phys. rev. E, 67, 066209, (2003) [12] Eastgate, L.; Sethna, J.; Rauscher, M.; Cretegny, T.; Chen, C.; Myers, C., Fracture in mode I using a conserved phase-field model, Phys. rev. E, 65, 036117, (2002) [13] Eshelby, J., The force on an elastic singularity, Philos. trans. roy. soc. (London) A, 244, 877, 87-112, (1951) · Zbl 0043.44102 [14] Eshelby, J., Elastic energy-momentum tensor, J. elasticity, 5, 3-4, 321-335, (1975) · Zbl 0323.73011 [15] Francfort, G.; Marigo, J., Revisiting brittle fracture as an energy minimization problem, J. mech. phys. solids, 46, 8, 1319-1342, (1998) · Zbl 0966.74060 [16] Goldstein, R.; Salganik, R., Brittle-fracture of solids with arbitrary cracks, Int. J. fract., 10, 4, 507-523, (1974) [17] Griffith, A., The phenomena of rupture and flows in solids, Philos. trans. roy. soc. (London) A, 221, 163-198, (1920) [18] Gurtin, M.; Podio-Guidugli, P., Configurational forces and a constitutive theory for crack propagation that allows for kinking and curving, J. mech. phys. solids, 46, 8, 1343-1378, (1998) · Zbl 0955.74004 [19] Hakim, V.; Karma, A., Crack path prediction in anisotropic brittle materials, Phys. rev. lett., 95, 235501, (2005) [20] Hauch, J.; Holland, D.; Marder, M.; Swinney, H., Dynamic fracture in single crystal silicon, Phys. rev. lett., 82, 3823-3826, (1999) [21] Henry, H., Study of the branching instability using a phase field model of inplane crack propagation, Europhys. lett., 83, 16004, (2008) [22] Henry, H.; Levine, H., Dynamic instabilities of fracture under biaxial strain using a phase field model, Phys. rev. lett., 93, 105504, (2004) [23] Herring, C., 1951. in: Kingston, W.E. (Ed.), The Physics of Powder Metallurgy. McGraw-Hill, New York. [24] Hodgdon, J.A., Sethna, J.P., 1993. Derivation of a general 3-dimensional crack propagation law – a generalization of the principle of local symmetry. Phys. Rev. B 47, 4831-4840. [25] Hutchinson, J.W.; Suo, Z., Mixed mode cracking in layered materials, Adv. appl. mech., 29, 63-191, (1992) · Zbl 0790.73056 [26] Irwin, G., 1957. J. Appl. Mech. 24, 361. [27] Karma, A.; Lobkovsky, A., Unsteady crack motion and branching in a phase-field model of brittle fracture, Phys. rev. lett., 92, 245510, (2004) [28] Karma, A.; Kessler, D.; Levine, H., Phase-field model of mode III dynamic fracture, Phys. rev. lett., 8704, 045501, (2001) [29] Katzav, E.; Adda-Bedia, M.; Arias, R., Theory of dynamic crack branching in brittle materials, Int. J. fract., 143, 245-271, (2007) · Zbl 1197.74111 [30] Landau, L.D.; Lifshitz, E.M., The classical theory of fields, (1975), Pergamon Press Oxford · Zbl 0178.28704 [31] Leblond, J., 2005. Private communication. [32] Marconi, V.; Jagla, E., Diffuse interface approach to brittle fracture, Phys. rev. E, 71, 036110, (2005) [33] Marder, M., Cracks cleave crystals, Europhys. lett., 66, 3, 364-370, (2004) [34] Marder, M., Effect of atoms on brittle fracture, Int. J. fract., 130, 517-555, (2004) [35] Mumford, D.; Shah, J., Optimal approximations by piecewise smooth functions and associated variational-problems, Commun. pure appl. math., 42, 5, 577-685, (1989) · Zbl 0691.49036 [36] Noether, E., 1918. Invariante Variationsprobleme. Nachr. v. d. Ges. d. Wiss. zu Göttingen, pp. 235-257. · JFM 46.0770.01 [37] Oleaga, G., Remarks on a basic law for dynamic crack propagation, J. mech. phys. solids, 49, 10, 2273-2306, (2001) · Zbl 1017.74061 [38] Pons, A., Karma, A., 2008, in preparation. [39] Rice, J., A path independent integral and approximate analysis of strain concentration by notches and cracks, J. appl. mech., 35, 2, 379, (1968) [40] Sih, G., Stress distribution near internal crack tips for longitudinal shear problems, J. appl. mech., 32, 1, 51, (1965) · Zbl 0127.14805 [41] Sommer, E., Formation of fracture “lances” in Glass, Eng. fract. mech., 1, 539-546, (1969) [42] Spatschek, R.; Hartmann, M.; Brener, E.; Müller-Krumbhaar, H.; Kassner, K., Phase field modeling of fast crack propagation, Phys. rev. lett., 96, 015502, (2006) [43] Wang, Y.; Jin, Y.; Khachaturyan, A., Phase field microelasticity theory and modeling of elastically and structurally inhomogeneous solid, J. appl. phys., 92, 3, 1351-1360, (2002) [44] William, M., On the stress distribution at the base of a stationary crack, J. appl. mech., 24, 109-114, (1957) · Zbl 0077.37902
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