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Analysis of material instabilities in inelastic solids by incremental energy minimization and relaxation methods: evolving deformation microstructures in finite plasticity. (English) Zbl 1115.74323

Summary: We propose an approach to the definition and analysis of material instabilities in rate-independent standard dissipative solids at finite strains based on finite-step-sized incremental energy minimization principles. The point of departure is a recently developed constitutive minimization principle for standard dissipative materials that optimizes a generalized incremental work function with respect to the internal variables. In an incremental setting at finite time steps this variational problem defines a quasi-hyperelastic stress potential. The existence of this potential allows to be recast a typical incremental boundary-value problem of quasi-static inelasticity into a principle of minimum incremental energy for standard dissipative solids. Mathematical existence theorems for sufficiently regular minimizers then induce a definition of the material stability of the inelastic material response in terms of the sequentially weakly lower semicontinuity of the incremental variational functional. As a consequence, the incremental material stability of standard dissipative solids may be defined in terms of the quasi-convexity or the rank-one convexity of the incremental stress potential. This global definition includes the classical local Hadamard condition but is more general. Furthermore, the variational setting opens up the possibility to analyze the post-critical development of deformation microstructures in non-stable inelastic materials based on energy relaxation methods. We outline minimization principles of quasi- and rank-one convexifications of incremental non-convex stress potentials for standard dissipative solids. The general concepts are applied to the analysis of evolving deformation microstructures in single-slip plasticity. For this canonical model problem, we outline details of the constitutive variational formulation and develop numerical and semi-analytical solution methods for a first-level rank-one convexification. A set of representative numerical investigations analyze the development of deformation microstructures in the form of rank-one laminates in single slip plasticity for homogeneous macro-deformation modes as well as inhomogeneous macroscopic boundary-value problems. The well-posedness of the relaxed variational formulation is indicated by an independence of typical finite element solutions on the mesh-size.

MSC:

74G60 Bifurcation and buckling
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
74G65 Energy minimization in equilibrium problems in solid mechanics
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[1] Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Archive of Rational Mech. Anal, 83, 125-145 (1984) · Zbl 0565.49010
[2] Asaro, R., Micromechanics of crystals and polycrystals, Adv. Appl. Mech, 23, 1-115 (1983)
[3] Ball, J.M., 1977a. Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Knops, R.J. (Ed.), Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. I, Pitman, London.; Ball, J.M., 1977a. Constitutive inequalities and existence theorems in nonlinear elastostatics. In: Knops, R.J. (Ed.), Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, Vol. I, Pitman, London. · Zbl 0377.73043
[4] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal, 63, 337-403 (1977) · Zbl 0368.73040
[5] Biot, M. A., Mechanics of Incremental Deformations (1965), Wiley: Wiley New York
[6] Carstensen, C.; Plecháč, P., Numerical solution of the scalar double-well problem allowing microstructures, Math. Comput, 66, 997-1026 (1997) · Zbl 0870.65055
[7] Carstensen, C.; Roubíček, T., Numerical approximation of Young measures in non-convex variational problems, Numerische Mathematik, 84, 395-415 (2000) · Zbl 0945.65070
[8] Carstensen, C.; Hackl, K.; Mielke, A., Nonconvex potentials and microstructures in finite-strain plasticity, Proc. Roy. Soc. London, Ser. A, 458, 299-317 (2002) · Zbl 1008.74016
[9] Ciarlet, P. G., Mathematical Elasticity (1988), Elsevier Science Publishers B.V: Elsevier Science Publishers B.V Amsterdam · Zbl 0648.73014
[10] Coleman, B.; Noll, W., On the thermostatics of continuous media, Archive of Rational Mech. Anal, 4, 97-128 (1959) · Zbl 0231.73003
[11] Dacorogna, B., Direct Methods in the Calculus of Variations (1989), Springer: Springer Berlin, Heidelberg · Zbl 0703.49001
[12] DeSimone, Dolzmann, 2000. Material instabilities in nematic elastomers. Physica D, 136, 175-191.; DeSimone, Dolzmann, 2000. Material instabilities in nematic elastomers. Physica D, 136, 175-191. · Zbl 0947.76005
[13] Dolzmann, G., Variational Methods for Crystalline Microstructure—Analysis and Computation (2003), Springer: Springer Berlin, Heidelberg · Zbl 1016.74002
[14] Germain, P., Cours de Mécanique des Milieux Continus (1973), Masson et Cie: Masson et Cie Paris · Zbl 0254.73001
[15] Govindjee, S.; Mielke, A.; Hall, G. J., The free-energy of mixing for n-variant martensitic phase transformations using quasi-convex analysis, J. Mech. Phys. Solids, 50, 1897-1922 (2002) · Zbl 1116.74399
[16] Hackl, K.; Hoppe, U., On the calculation of micro-structures for inelastic materials using relaxed energies, (Miehe, C., Computational Mechanics of Solid Materials at Large Strains (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 77-86 · Zbl 1040.74006
[17] Hadamard, J., Leçons sur la propagation des ondes et les équations de l’hydrodynamique (1903), Hermann: Hermann Paris · JFM 34.0793.06
[18] Halphen, B.; Nguyen, Q. S., Sur les matéraux standards généralisés”, J. Méc, 40, 39-63 (1975) · Zbl 0308.73017
[19] Havner, K. S., Finite Plastic Deformation of Crystalline Solids (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0774.73001
[20] Hill, R., On uniqueness and stability in the theory of finite elastic strains, J. Mech. Phys. Solids, 5, 229-241 (1957) · Zbl 0080.18004
[21] Hill, R., Acceleration waves in solids, J. Mech. Phys. Solids, 6, 1-16 (1962) · Zbl 0111.37701
[22] Kohn, R. V., The relaxation of a double-well problem, Continuum Mech. Therm, 3, 193-236 (1991) · Zbl 0825.73029
[23] Kohn, R. V.; Strang, G., Explicit relaxation of a variational problem in optimal design, Bull. Am. Math. Soc, 9, 211-214 (1983) · Zbl 0527.49002
[24] Kohn, R.V., Strang, G., 1986. Optimal design and relaxation of variational problems I, II, III. Commun. Pure Appl. Math. 39 113-137, 139-182, 353-377.; Kohn, R.V., Strang, G., 1986. Optimal design and relaxation of variational problems I, II, III. Commun. Pure Appl. Math. 39 113-137, 139-182, 353-377. · Zbl 0609.49008
[25] Krawietz, A., Materialtheorie: Mathematische Beschreibung des phänomenologischen thermomechanischen Verhaltens (1986), Springer: Springer Berlin · Zbl 0593.73001
[26] Lambrecht, M.; Miehe, C.; Dettmar, J., Energy relaxation of non-convex incremental stress potentials in a strain-softening elastic-plastic bar, Int. J. Solids Struct, 40, 1369-1391 (2003) · Zbl 1065.74543
[27] Luskin, M., On the computation of crystalline microstructure, Acta Numer, 36, 191-257 (1996) · Zbl 0867.65033
[28] Mandel, J., Plasticité Classique et Viscoplasticité. CISM Courses and Lectures No. 97 (1972), Springer: Springer Berlin
[29] Marsden, J. E.; Hughes, T. J.R., Mathematical Foundations of Elasticity (1994), Dover Publications: Dover Publications New York
[30] Martin, J. B., Plasticity. Fundamentals and General Results (1975), MIT Press: MIT Press Cambridge, Massachusetts
[31] Maugin, G. A., The Thermodynamics of Plasticity and Fracture (1992), Cambridge University Press: Cambridge University Press Cambridge
[32] Miehe, C., Exponential map algorithm for stress updates in anisotropic multiplicative elastoplasticity for single crystals, Int. J. Num. Meth. Eng, 39, 3367-3390 (1996) · Zbl 0899.73136
[33] Miehe, C., Strain-driven homogenization of inelastic microstructures and composites based on an incremental variational formulation, Int. J. Num. Meth. Eng, 55, 1285-1322 (2002) · Zbl 1027.74056
[34] Miehe, C.; Lambrecht, M., A two-scale finite element relaxation analysis of shear bands in non-convex inelastic solidssmall-strain theory for standard dissipative materials, Comput. Meth. Appl. Mech. Eng, 192, 473-508 (2003) · Zbl 1088.74044
[35] Miehe, C.; Lambrecht, M., Analysis of microstructure development in shearbands by energy relaxation of incremental stress potentialslarge-strain theory for standard dissipative solids, Int. J. Num. Meth. Eng, 58, 1-41 (2003) · Zbl 1032.74526
[36] Miehe, C.; Schotte, J.; Lambrecht, M., Homogenization of inelastic solid materials at finite strains based on incremental minimization principlesApplication to the texture analysis of polycrystals, J. Mech. Phys. Solids, 50, 2123-2167 (2002) · Zbl 1151.74403
[37] Mielke, A., Finite elastoplasticity, Lie groups and geodesics on SL(d), (Newton, P.; Weinstein, A.; Holmes, P. J., Geometry, Dynamics, and Mechanics (2002), Springer: Springer Berlin), 61-90 · Zbl 1146.74309
[38] Moreau, J. J., On unilateral constraints, friction and plasticity, (Capriz, G.; Stampacchia, G., New Variational Techniques in Mathematical Physics. CISM Course (1974), Springer: Springer Berlin) · Zbl 0703.73070
[39] Moreau, J. J., Application of convex analysis to the treatment of elastoplastic systems, (Germain, P.; Nayroles, B., Application of Methods of Functional Analysis to Problems in Mechanics (1976), Springer: Springer Berlin) · Zbl 0337.73004
[40] Morrey, C. B., Quasiconvexity and the semicontinuity of multiple integrands, Pacific J. Math, 2, 25-53 (1952) · Zbl 0046.10803
[41] Müller, S., 1998. Variational models for microstructure and phase transisitions. Lecture Notes 2/1998, Max Planck Institute for Mathematics in the Sciences, Leipzig.; Müller, S., 1998. Variational models for microstructure and phase transisitions. Lecture Notes 2/1998, Max Planck Institute for Mathematics in the Sciences, Leipzig.
[42] Nguyen, A., Stability and Nonlinear Solid Mechanics (2000), Wiley: Wiley New York
[43] Ortiz, M.; Repetto, E. A., Nonconvex energy minimization and dislocation structure in ductile single crystals, J. Mech. Phys. Solids, 47, 397-462 (1999) · Zbl 0964.74012
[44] Ortiz, M.; Stainier, L., The variational formulation of viscoplastic constitutive updates, Comput. Meth. Appl. Mech. Eng, 171, 419-444 (1999) · Zbl 0938.74016
[45] Ortiz, M.; Repetto, E. A.; Stainier, L., A theory of subgrain dislocation structures, J. Mech. Phys. Solids, 48, 10, 2077-2114 (2000) · Zbl 1001.74007
[46] Petryk, H., The energy criteria of instability in time-independent inelastic solids, Arch. Mech, 43, 519-545 (1991) · Zbl 0757.73029
[47] Petryk, H., Material instability and strain-rate discontinuities in incrementally nonlinear continua, J. Mech. Phys. Solids, 40, 1227-1250 (1992) · Zbl 0763.73028
[48] Petryk, H., 2000. Theory of material instability in incrementally nonlinear plasticity. In: Petryk, H. (Ed.), Material Instabilities in Elastic and Plastic Solids. CISM Courses and Lectures No 414, Springer.; Petryk, H., 2000. Theory of material instability in incrementally nonlinear plasticity. In: Petryk, H. (Ed.), Material Instabilities in Elastic and Plastic Solids. CISM Courses and Lectures No 414, Springer. · Zbl 1031.74015
[49] Petryk, H.; Thermann, K., Post-critical plastic deformation in incrementally nonlinear materials, J. Mech. Phys. Solids, 50, 925-954 (2002) · Zbl 1072.74511
[50] Rice, J. R., Inelastic constitutive relations for solidsan internal-variable theory and its application to metal plasticity, J. Mech. Phys. Solids, 19, 433-455 (1971) · Zbl 0235.73002
[51] Rice, J. R., The localization of plastic deformation, (Koiter, W. T., Theoretical and Applied Mechanics (1976), North-Holland: North-Holland Amsterdam), 207-220
[52] Rockafellar, R. T., Convex Analysis (1970), Princeton University Press: Princeton University Press Princeton · Zbl 0202.14303
[53] vSilhavý, M., The Mechanics and Thermodynamics of Continuous Media (1997), Springer: Springer Berlin, Heidelberg, New York
[54] Simó, J. C.; Armero, F., Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, Int. J. Num. Meth. Eng, 33, 1413-1449 (1992) · Zbl 0768.73082
[55] Smyshlyaev, V. P.; Willis, J. R., On the relaxation of a three-well energy, Proc. Roy. Soc. London, Ser. A, 455, 779-814 (1999) · Zbl 0960.74027
[56] Thomas, T. Y., Plastic Flow and Fracture in Solids (1961), Academic Press: Academic Press London · Zbl 0081.39803
[57] Truesdell, C.; Noll, W., The nonlinear field theories of mechanics, (Flügge, S., Handbuch der Physik Bd. III/3 (1965), Springer: Springer Berlin) · Zbl 0779.73004
[58] Weber, G.; Anand, L., Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic-viscoplastic solids, Comput. Meth. Appl. Mech. Eng, 79, 173-202 (1990) · Zbl 0731.73031
[59] Young, L. C., Lectures on the Calculus of Variations and Optimal Control Theory (1969), Saunders: Saunders London · Zbl 0177.37801
[60] Ziegler, H., 1963. Some extremum principles in irreversible thermodynamics with application to continuum mechanics. In: Sneddon, I.N., Hill. R. (Eds.), Progress in Solid Mechanics, Vol. IV. North-Holland, Amsterdam.; Ziegler, H., 1963. Some extremum principles in irreversible thermodynamics with application to continuum mechanics. In: Sneddon, I.N., Hill. R. (Eds.), Progress in Solid Mechanics, Vol. IV. North-Holland, Amsterdam.
[61] Ziegler, H., Wehrli, C., 1987. The derivation of constitutive relations from the free energy and the dissipation function. In: Wu, Th.Y., Hutchinson, J.W. (Eds.), Advances in Applied Mechanics. 25. Academic Press, New York.; Ziegler, H., Wehrli, C., 1987. The derivation of constitutive relations from the free energy and the dissipation function. In: Wu, Th.Y., Hutchinson, J.W. (Eds.), Advances in Applied Mechanics. 25. Academic Press, New York. · Zbl 0719.73001
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