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Phase field modeling of fracture in multi-physics problems. II: Coupled brittle-to-ductile failure criteria and crack propagation in thermo-elastic-plastic solids. (English) Zbl 1423.74837
Summary: This work presents a generalization of recently developed continuum phase field models from brittle to ductile fracture coupled with thermo-plasticity at finite strains. It uses a geometric approach to the diffusive crack modeling based on the introduction of a balance equation for a regularized crack surface and its modular linkage to a multi-physics bulk response developed in the first part of this work [Zbl 1423.74838]. This evolution equation is governed by a constitutive crack driving force. In this work, we supplement the energetic and stress-based forces for brittle fracture by additional forces for ductile fracture. These are related to state variables associated with the inelastic response, such as the amount of plastic strain and the void volume fraction in metals, or the amount of craze strains in glassy polymers. To this end, we define driving forces based on elastic and plastic work densities, and barrier functions related to critical values of these inelastic state variables. The proposed thermodynamically consistent framework of ductile phase field fracture is embedded into a formulation of gradient thermo-plasticity, that is able to account for material length scales such as the width of shear bands. It is applied to two constitutive model problems. The first is designed for the analysis of brittle-to-ductile failure mode transition in the dynamic failure analysis of metals. The second is constructed for a quasi-static analysis of crazing-induced fracture in glassy polymers. A spectrum of simulations demonstrates that the use of barrier-type crack driving forces in the phase field modeling of fracture, governed by accumulated plastic strains in metals or crazing strains in polymers, provide results in very good agreement with experiments.

74R10 Brittle fracture
74C05 Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials)
74F05 Thermal effects in solid mechanics
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[1] Francfort, G. A.; Marigo, J. J., Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46, 1319-1342, (1998) · Zbl 0966.74060
[2] Bourdin, B.; Francfort, G.; Marigo, J.-J., The variational approach to fracture, (2008), Springer
[3] Hakim, V.; Karma, A., Laws of crack motion and phase-field models of fracture, J. Mech. Phys. Solids, 57, 342-368, (2009) · Zbl 1421.74089
[4] Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: variational principles and multi-field fe implementations, Internat. J. Numer. Methods Engrg., 83, 1273-1311, (2010) · Zbl 1202.74014
[5] Borden, M. J.; Verhoosel, C. V.; Scott, M. A.; J. R. Hughes, T.; Landis, C. M., A phase-field description of dynamic brittle fracture, Comput. Methods Appl. Mech. Engrg., 217-220, 77-95, (2012) · Zbl 1253.74089
[6] Verhoosel, C. V.; de Borst, R., A phase-field model for cohesive fracture, Internat. J. Numer Methods Engrg., 96, 43-62, (2013) · Zbl 1352.74029
[7] Miehe, C.; Schänzel, L.; Ulmer, H., Phase field modeling of fracture in multi-physics problems. part I. balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids, Comput. Methods Appl. Mech. Engrg., (2014), in press
[8] Miehe, C., Variational gradient plasticity at finite strains. part I: mixed potentials for the evolution and update problems of gradient-extended dissipative solids, Comput. Methods Appl. Mech. Engrg., 268, 677-703, (2014) · Zbl 1295.74013
[9] Miehe, C.; Welschinger, F.; Aldakheel, F., Variational gradient plasticity at finite strains. part II: local-global updates and mixed finite elements for additive plasticity in the logarithmic strain space, Comput. Methods Appl. Mech. Engrg., 268, 704-734, (2014) · Zbl 1295.74014
[10] Kalthoff, J.; Winkler, S., Impact loading and dynamic behavior of materials, 185-195, (1987), DGM Informationsgesellschaft
[11] Song, J.-H.; Areias, P.; Belytschko, T., A method for dynamic crack and shear band propagation with phantom nodes, Internat. J. Numer Methods Engrg., 67, 868-893, (2006) · Zbl 1113.74078
[12] Zhou, M.; Rosakis, A. J.; Ravichandran, G., On the growth of shear bands and failure-mode transition in prenotche plates: A comparison of singly and doubly notched specimens, Int. J. Plast., 14, 435-451, (1998) · Zbl 0945.74671
[13] Gurson, A. L., Continuum theory of ductile rupture by void nucleation and growth, part i—yield criteria and flow rules for porous ductile media, Trans. ASME H, 99, 2-15, (1977)
[14] Needleman, A.; Tvergaard, V., An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids, 32, 461-490, (1984)
[15] Huespe, A.; Needleman, A.; Oliver, J.; Sánchez, A finite thickness band method for ductile fracture analysis, Int. J. Plast., 25, 2349-2365, (2009)
[16] Huespe, A.; Needleman, A.; Oliver, J.; Sánchez, A finite strain, finite band method for modeling ductile fracture, Int. J. Plast., 28, 53-69, (2012)
[17] Argon, A. S., A theory for the low-temperature plastic deformation of glassy polymers, Phil. Mag., 28, 839-865, (1973)
[18] Boyce, M. C.; Parks, D. M.; Argon, A. S., Large inelastic deformation of glassy polymers. part I: rate dependent constitutive model, Mech. Mater., 7, 15-33, (1988)
[19] Wu, P. D.; van der Giessen, E., Analysis of shear band propagation in amorphous glassy polymers, Int. J. Solids Struct., 31, 1493-1517, (1994) · Zbl 0946.74511
[20] Wu, P. D.; van der Giessen, E., On neck propagation in amorphous glassy polymers under plane strain tension, Int. J. Plast., 11, 211-235, (1995) · Zbl 0823.73023
[21] Miehe, C.; Göktepe, S.; Mendez, J., Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space, Int. J. Solids Struct., 46, 181-202, (2009) · Zbl 1168.74324
[22] Miehe, C.; Méndez Diez, J.; Göktepe, S.; Schänzel, L., Coupled thermoviscoplasticity of glassy polymers in the logarithmic strain space based on the free volume theory, Int. J. Solids Struct., 48, 1799-1817, (2011)
[23] Estevez, R.; Tijssens, M. G. A.; van der Giessen, E., Modeling of the competition between shear yielding and crazing in glassy polymers, J. Mech. Phys. Solids, 48, 2585-2617, (2000) · Zbl 1012.74064
[24] Tijssens, M.; van der Giessen, E.; Sluys, L., Modeling of crazing using a cohesive surface methodology, Mech. Mater, 32, 19-35, (2000)
[25] Tijssens, M.; van der Giessen, E.; Sluys, L., Simulation of mode I crack growth in polymers by crazing, Int. J. Solids Struct., 37, 7307-7327, (2000) · Zbl 0992.74074
[26] Gearing, B. P.; Anand, L., On modeling the deformation and fracture response of glassy polymers due to shear-yielding and crazing, Int. J. Solids Struct., 41, 3125-3150, (2004) · Zbl 1119.74574
[27] Borden, M. J.; Hughes, T. J.R.; Landis, C. M.; Verhoosel, C. V., A higher-order phase-field model for brittle fracture: formulation and analysis within the isogeometric analysis framework, Comput. Methods Appl. Mech. Engrg., 273, 100-118, (2014) · Zbl 1296.74098
[28] Miehe, C., A multi-field incremental variational framework for gradient-extended standard dissipative solids, J. Mech. Phys. Solids, 59, 898-923, (2011) · Zbl 1270.74022
[29] Frémond, M.; Nedjar, B., Damage, gradient of damage, and principle of virtual power, Int. J. Solids Struct., 33, 1083-1103, (1996) · Zbl 0910.73051
[30] Frémond, M., Non-smooth thermomechanics, (2002), Springer · Zbl 0990.80001
[31] Pham, K.; Amor, H.; Marigo, J.; Maurini, C., Gradient damage models and their use to approximate brittle fracture, Int. J. Damage Mech., 20, 618-652, (2011)
[32] Tvergaard, V.; Needleman, A., Analysis of the cup-cone fracture in a round tensile bar, Acta Metall., 32, 157-169, (1984)
[33] Capriz, G., Continua with microstructure, (1989), Springer · Zbl 0676.73001
[34] Mariano, P. M., Multifield theories in mechanics of solids, Adv. Appl. Mech., 38, 1-93, (2001)
[35] Coleman, B.; Gurtin, M., Thermodynamics with internal state variables, J. Chem. Phys., 47, 597-613, (1967)
[36] Miehe, C., A formulation of finite elastoplasticity based on dual co- and contra-variant eigenvector triads normalized with respect to a plastic metric, Comput. Methods Appl. Mech. Engrg., 159, 223-260, (1998) · Zbl 0948.74009
[37] Miehe, C.; Apel, N.; Lambrecht, M., Anisotropic additive plasticity in the logarithmic strain space. modularkinematic formulation and implementation based on incremental minimization principles for standard materials, Comput. Methods Appl. Mech. Engrg., 191, 5383-5425, (2002) · Zbl 1083.74518
[38] Miehe, C.; Lambrecht, M., Algorithms for computation of stresses and elasticity moduli in terms of seth-hill’s family of generalized strain tensors, Commun. Numer. Methods Engrg., 17, 337-353, (2001) · Zbl 1049.74011
[39] Miehe, C.; Aldakheel, F.; Mauthe, S., Mixed variational principles and robust finite element implementations of gradient plasticity at small strains, Internat. J. Numer Methods Engrg., 94, 1037-1074, (2013) · Zbl 1352.74408
[40] Boyce, M. C.; Montagut, E. L.; Argon, A. S., The effects of thermomechanical coupling on the cold drawing process of glassy polymers, Polym. Eng. Sci., 32, 1073-1085, (1992)
[41] Arruda, E. M.; Boyce, M. C., A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, J. Mech. Phys. Solids, 41, 389-412, (1993) · Zbl 1355.74020
[42] Treloar, L. R.G., The physics of rubber elasticity, (1975), Clarendon Press · Zbl 0347.73042
[43] Raha, S.; Bowden, P. B., Birefringence of plastically deformed poly(methyl methacrylate), Polymer, 13, 174-183, (1972)
[44] Arruda, E. M.; Boyce, M. C.; Jayachandran, R., Effects of strain rate, temperature and thermomechanical coupling on the finite strain deformation of glassy polymers, Mech. Mater., 19, 193-212, (1995)
[45] Basu, S.; van der Giessen, E., A thermo-mechanical study of mode i, small-scale yielding crack-tip fields in glassy polymers, Int. J. Plast., 18, 1395-1423, (2002) · Zbl 1032.74522
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