A multi-field incremental variational framework for gradient-extended standard dissipative solids.

*(English)*Zbl 1270.74022Summary: The paper presents a constitutive framework for solids with dissipative micro-structures based on compact variational statements. It develops incremental minimization and saddle point principles for a class of gradient-type dissipative materials which incorporate micro-structural fields (micro-displacements, order parameters, or generalized internal variables), whose gradients enter the energy storage and dissipation functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro-structural fields are governed by additional balance equations including micro-structural boundary conditions. They describe changes of the substructure of the material which evolve relatively to the material as a whole. Typical examples are theories of phase field evolution, gradient damage, or strain gradient plasticity. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure. We outline a unified framework for the broad class of first-order gradient-type standard dissipative solids. Particular emphasis is put on alternative multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These three-field settings are suitable for models with threshold- or yield-functions formulated in the space of the driving forces. It is shown that the coupled macro- and micro-balances follow in a natural way as the Euler equations of minimization and saddle point principles, which are based on properly defined incremental potentials. These multi-field potential functionals are outlined in both a continuous rate formulation and a time-space-discrete incremental setting. The inherent symmetry of the proposed multi-field formulations is an attractive feature with regard to their numerical implementation. The unified character of the framework is demonstrated by a spectrum of model problems, which covers phase field models and formulations of gradient damage and plasticity.

##### MSC:

74A99 | Generalities, axiomatics, foundations of continuum mechanics of solids |

74G65 | Energy minimization in equilibrium problems in solid mechanics |

##### Keywords:

variational principles; size effects; phase field models; gradient damage; strain gradient plasticity##### References:

[1] | Aifantis, E.C., The physics of plastic deformation, International journal of plasticity, 3, 211-247, (1987) · Zbl 0616.73106 |

[2] | Armero, F.; Pérez-Foguet, A., On the formulation of closest-point projection algorithms in elastoplasticity – part I: the variational structure, International journal for numerical methods in engineering, 53, 297-329, (2002) · Zbl 1051.74056 |

[3] | Arzt, E., Size effects in materials due to microstructural and dimensional constraints: a comparative review, Acta materialia, 46, 5611-5626, (1998) |

[4] | Ashby, M.F., The deformation of plastically non-homogeneous materials, The philosophical magazine A, 21, 399-424, (1970) |

[5] | Biot, M.A., Mechanics of incremental deformations, (1965), John Wiley & Sons Inc New York |

[6] | Capriz, G., Continua with microstructure, (1989), Springer Verlag · Zbl 0676.73001 |

[7] | Capriz, G.; Podio-Guidugli, P.; Williams, W., On balance equations of materials with affine structure, Meccanica, 17, 80-84, (1982) · Zbl 0505.73002 |

[8] | Carstensen, C.; Hackl, K.; Mielke, A., Non-convex potentials and microstructures in finite-strain plasticity, Proceedings of the royal society London A, 458, 299-317, (2002) · Zbl 1008.74016 |

[9] | Coleman, B.; Gurtin, M., Thermodynamics with internal state variables, The journal of chemical physics, 47, 597-613, (1967) |

[10] | Cosserat, E.; Cosserat, F., Sur la théorie des corps deformables, (1909), Dunod Paris · JFM 40.0862.02 |

[11] | Eringen, A.C.; Kafadar, C.B., Polar field theories, () |

[12] | Evers, L.P.; Brekelmans, W.A.M.; Geers, M.G.D., Non-local crystal plasticity model with intrinsic SSD and GND effects, Journal of the mechanics and physics of solids, 52, 2379-2401, (2004) · Zbl 1115.74313 |

[13] | Fleck, N.A.; Hutchinson, J.W., Strain gradient plasticity, Advances in applied mechanics, 33, 295-362, (1997) · Zbl 0894.73031 |

[14] | Fleck, N.A.; Muller, G.M.; Ashby, M.F.; Hutchinson, J.W., Strain gradient plasticity: theory and experiment, Acta materialia, 42, 475-487, (1994) |

[15] | Fleck, N.A.; Willis, J.R., A mathematical basis for strain-gradient plasticity theory. part I: scalar plastic multiplier, Journal of the mechanics and physics of solids, 57, 161-177, (2009) · Zbl 1195.74020 |

[16] | Fleck, N.A.; Willis, J.R., A mathematical basis for strain-gradient plasticity theory. part II: tensorial plastic multiplier, Journal of the mechanics and physics of solids, 57, 1045-1057, (2009) · Zbl 1173.74316 |

[17] | Forest, S., Micromorphic approach for gradient elasticity, viscoplasticity, and damage, Journal of engineering mechanics, 135, 117-131, (2009) |

[18] | Forest, S.; Sievert, R., Elastoviscoplastic constitutive frameworks for generalized continua, Acta mechanica, 160, 71-111, (2003) · Zbl 1064.74009 |

[19] | Francfort, G.; Mielke, A., Existence results for a class of rate-independent material models with nonconvex elastic energies, Journal für die reine und angewandte Mathematik, 595, 55-91, (2006) · Zbl 1101.74015 |

[20] | Frémond, M., Non-smooth thermomechanics, (2002), Springer · Zbl 0990.80001 |

[21] | Frémond, M.; Nedjar, B., Damage, gradient of damage and principle of virtual power, International journal of solids and structures, 33, 1083-1103, (1996) · Zbl 0910.73051 |

[22] | Fried, E.; Gurtin, M.E., Continuum theory of thermally induced phase transitions based on an order parameter, Physica D, 68, 326-343, (1993) · Zbl 0793.35049 |

[23] | Gao, H.; Huang, Y.; Nix, W.D.; Hutchinson, J.W., Mechanism-based strain gradient plasticity—I. theory, Journal of the mechanics and physics of solids, 47, 1239-1263, (1999) · Zbl 0982.74013 |

[24] | Germain, P., La méthode des puissances virtuelles en mécanique des milieux continus, Journal de Mécanique, 12, 235-274, (1973) · Zbl 0261.73003 |

[25] | Gudmundson, P., A unified treatment of strain gradient plasticity, Journal of the mechanics and physics of solids, 52, 1379-1406, (2004) · Zbl 1114.74366 |

[26] | Gurtin, M.E., Generalized ginzburg – landau and cahn – hilliard equations based on a microforce balance, Physica D: nonlinear phenomena, 92, 178-192, (1996) · Zbl 0885.35121 |

[27] | Gurtin, M.E., On the plasticity of single crystals: free energy, microforces, plastic-strain gradients, Journal of the mechanics and physics of solids, 48, 989-1036, (2000) · Zbl 0988.74021 |

[28] | Gurtin, M.E., A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations, Journal of the mechanics and physics of solids, 50, 5-32, (2002) · Zbl 1043.74007 |

[29] | Gurtin, M.E., On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients, International journal of plasticity, 19, 47-90, (2003) · Zbl 1032.74521 |

[30] | Gurtin, M.E.; Anand, L., A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. part i: small deformations, Journal of the mechanics and physics of solids, 53, 1624-1649, (2005) · Zbl 1120.74353 |

[31] | Halphen, B.; Nguyen, Q.S., Sur LES matéraux standards généralisés, Journal de Mécanique, 14, 39-63, (1975) · Zbl 0308.73017 |

[32] | Han, W.; Reddy, B.D., Plasticity. mathematical theory and numerical analysis, (1999), Springer New York · Zbl 0926.74001 |

[33] | Hill, R., The mathematical theory of plasticity, (1950), Clarendon Oxford · Zbl 0041.10802 |

[34] | Kröner, E., Allgemeine kontinuumstheorie der versetzungen und eigenspannungen, Archive for rational mechanics and analysis, 4, 273-334, (1960) · Zbl 0090.17601 |

[35] | Liebe, T.; Steinmann, P., Theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity, International journal for numerical methods in engineering, 51, 1437-1467, (2001) · Zbl 1065.74516 |

[36] | Mariano, P.M., Multifield theories in mechanics of solids, Advances in applied mechanics, 38, 1-93, (2001) |

[37] | Martin, J.B., Plasticity. fundamentals and general results, (1975), MIT Press Cambridge |

[38] | Maugin, G.A., The method of virtual power in continuum mechanics: application to coupled fields, Acta mechanica, 35, 1-70, (1980) · Zbl 0428.73095 |

[39] | Maugin, G.A., Internal variables and dissipative structures, Journal of non-equilibrium thermodynamics, 15, 173-192, (1990) |

[40] | Maugin, G.A.; Muschik, W., Thermodynamics with internal variables part I. general concepts, Journal of non-equilibrium thermodynamics, 19, 217-249, (1994) · Zbl 0808.73006 |

[41] | Maugin, G.A.; Muschik, W., Thermodynamics with internal variables part II. applications, Journal of non-equilibrium thermodynamics, 19, 250-289, (1994) · Zbl 0808.73006 |

[42] | Menzel, A.; Steinmann, P., On the continuum formulation of higher gradient plasticity for single and polycrystals, Journal of the mechanics and physics of solids, 48, 1777-1796, (2000) · Zbl 0999.74029 |

[43] | Miehe, C., Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation, International journal for numerical methods in engineering, 55, 1285-1322, (2002) · Zbl 1027.74056 |

[44] | Miehe, C.; Lambrecht, M.; Gürses, E., Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity, Journal of the mechanics and physics of solids, 52, 2725-2769, (2004) · Zbl 1115.74323 |

[45] | Miehe, C.; Schotte, J.; Lambrecht, M., Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. application to the texture analysis of polycrystals, Journal of the mechanics and physics of solids, 50, 2123-2167, (2002) · Zbl 1151.74403 |

[46] | Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically-consistent phase field models of fracture: variational principles and multi-field fe implementations, International journal for numerical methods in engineering, 83, 1273-1311, (2010) · Zbl 1202.74014 |

[47] | Mielke, A.; Roubicek, T., Rate-independent damage processes in nonlinear elasticity, Mathematical models and methods in applied sciences, 16, 177-209, (2006) · Zbl 1094.35068 |

[48] | Mindlin, R.A., Micro-structure in linear elasticity, Archive for rational mechanics and analysis, 16, 51-78, (1964) · Zbl 0119.40302 |

[49] | Mindlin, R.A., On the equations of elastic materials with microstructure, International journal of solids and structures, 1, 73-78, (1965) |

[50] | Mühlhaus, H.-B.; Aifantis, E.C., A variational principle for gradient plasticity, International journal of solids and structures, 28, 845-857, (1991) · Zbl 0749.73029 |

[51] | Nix, W.D.; Gao, H., Indentation size effects in crystalline materials: a law for strain gradient plasticity, Journal of the mechanics and physics of solids, 46, 411-425, (1998) · Zbl 0977.74557 |

[52] | Nye, J.F., Some geometrical relations in dislocated crystals, Acta metallurgica, 1, 153-162, (1953) |

[53] | Ortiz, M.; Stainier, L., The variational formulation of viscoplastic constitutive updates, Computer methods in applied mechanics and engineering, 171, 419-444, (1999) · Zbl 0938.74016 |

[54] | Peerlings, R.H.J.; de Borst, R.; Brekelmans, W.A.M.; de Vree, I.H.P., Gradient enhanced damage for quasi-brittle materials, International journal for numerical methods in engineering, 39, 3391-3403, (1996) · Zbl 0882.73057 |

[55] | Peerlings, R.H.J.; Massart, T.J.; Geers, M.G.D., A thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking, Computer methods in applied mechanics and engineering, 193, 3403-3417, (2004) · Zbl 1060.74508 |

[56] | Perzyna, P., Fundamental problems in viscoplasticity, Advances in applied mechanics, 9, 243-377, (1966) |

[57] | Reddy, B.; Ebobisse, F.; McBride, A., Well-posedness of a model of strain gradient plasticity for plastically irrotational materials, International journal of plasticity, 24, 55-73, (2008) · Zbl 1139.74009 |

[58] | Simó, J.C., Numerical analysis and simulation of plasticity, (), 183-499 · Zbl 0930.74001 |

[59] | Simó, J.C.; Honein, T., Variational formulation, discrete conservation laws, and path-domain independent integrals for elasto-viscoplasticity, Journal of applied mechanics, 57, 488-497, (1990) · Zbl 0739.73046 |

[60] | Svendsen, B., On the thermodynamics of thermoelastic materials with additional scalar degrees of freedom, Continuum mechanics and thermodynamics, 4, 247-262, (1999) · Zbl 0944.74005 |

[61] | Svendsen, B., Continuum thermodynamic models for crystal plasticity including the effects of geometrically-necessary dislocations, Journal of the mechanics and physics of solids, 50, 1297-1329, (2002) · Zbl 1071.74554 |

[62] | Toupin, R.A., Theories of elasticity with couple stress, Archive for rational mechanics and analysis, 17, 85-112, (1964) · Zbl 0131.22001 |

[63] | Ziegler, H., Some extremum principles in irreversible thermodynamics with application to continuum mechanics, () |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.