Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-04T21:02:21.128Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical approaches to rate-independent processes and applications in inelasticity

Published online by Cambridge University Press:  08 April 2009

Alexander Mielke  and
Tomáš Roubíček
Alexander Mielke
Affiliation:
Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstr. 39, 10117 Berlin, Germany. Institut für Mathematik, Humboldt Universität zu Berlin, Rudower Chaussee 25, 12489 Berlin, Germany.
Tomáš Roubíček
Affiliation:
Mathematical Institute, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic. roubicek@karlin.mff.cuni.cz Institute of Thermomechanics of the ASCR, Dolejškova 5, 182 00 Praha 8, Czech Republic.
Get access

Abstract

A conceptual numerical strategy for rate-independent processes in the energetic formulation is proposed and its convergence is proved under various rather mild data qualifications. The novelty is that we obtain convergence of subsequences of space-time discretizations even in case where the limit problem does not have a unique solution and we need no additional assumptions on higher regularity of the limit solution. The variety of general perspectives thus obtained is illustrated on several specific examples: plasticity with isotropic hardening, damage, debonding, magnetostriction, and two models of martensitic transformation in shape-memory alloys.

Keywords

Rate-independent evolutionenergetic solutionapproximationplasticitydamagedebondingmagnetostrictionmartensitic transformation.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 3 , May 2009 , pp. 399 - 428
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acerbi, E. and and Fusco, N., Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal. 86 (1984) 125–145. CrossRef
Alberty, J. and Carstensen, C., Numerical analysis of time-dependent primal elastoplasticity with hardening. SIAM J. Numer. Anal. 37 (2000) 1271–1294. CrossRef
Arndt, M., Griebel, M. and Roubíček, T., Modelling and numerical simulation of martensitic transformation in shape memory alloys. Continuum Mech. Thermodyn. 15 (2003) 463–485. CrossRef
Arndt, M., Griebel, M., Novák, V., Roubíček, T. and Šittner, P., Martensitic transformation in NiMnGa single crystals: numerical simulations and experiments. Int. J. Plasticity 22 (2006) 1943–1961. CrossRef
Auricchio, F. and Petrini, L., Improvements and algorithmical considerations on a recent three-dimensional model describing stress-induced solid phase transformations. Int. J. Numer. Meth. Engng. 55 (2002) 1255–1284. CrossRef
Auricchio, F., Mielke, A. and Stefanelli, U., A rate-independent model for the isothermal quasi-static evolution of shape-memory materials. Math. Models Methods Appl. Sci. 18 (2008) 125–164. CrossRef
Bourdin, B., Francfort, G. and Marigo, J.-J., Numerical experiments in revisited brittle fracture. J. Mech. Phys. Solids 48 (2000) 797–826. CrossRef
Boyd, J.G. and Lagoudas, D.C., A thermodynamical constitutive model for shape memory materials, Part I. The monolithic shape memory alloys. Int. J. Plasticity 12 (1996) 805–842. CrossRef
W.F. Brown, Magnetoelastic interactions, in Springer Tracts in Natural Philosophy 9, C. Truesdel Ed., Springer (1966).
Colli, P. and Sprekels, J., Global existence for a three-dimensional model for the thermo-mechanical evolution of shape memory alloys. Nonlinear Anal. 18 (1992) 873–888. CrossRef
Colli, P., Frémond, M. and Visintin, A., Thermo-mechanical evolution of shape memory alloys. Quarterly Appl. Math. 48 (1990) 31–47. CrossRef
Conti, S. and Ortiz, M., Dislocation microstructures and effective behaviour of single crystals. Arch. Ration. Mech. Anal. 176 (2005) 103–147. CrossRef
Falk, F., Model free energy, mechanics and thermodynamics of shape memory alloys. Acta Metall. 28 (1980) 1773–1780. CrossRef
Francfort, G. and Mielke, A., An existence result for a rate-independent material model in the case of nonconvex energies. J. Reine Angew. Math. 595 (2006) 55–91.
Frémond, M., Matériaux à mémoire deforme. C.R. Acad. Sci. Paris Sér. II 304 (1987) 239–244.
M. Frémond, Non-Smooth Thermomechanics. Springer, Berlin (2002).
Fried, E. and Gurtin, M.E., Dynamic solid-solid transitions with phase characterized by an order perameter. Physica D 72 (1994) 287–308. CrossRef
Giacomini, A. and Ponsiglione, M., Discontinuous finite element approximation of quasistatic crack growth in nonlinear elasticity. Math. Models Methods Appl. Sci. 16 (2006) 77–118. CrossRef
Govindjee, S. and Miehe, Ch., A multi-variant martensitic phase transformation model: formulation and numerical implementation. Comp. Meth. Appl. Mech. Engr. 191 (2001) 215–238. CrossRef
Govindjee, S., Mielke, A. and Hall, G.J., Free-energy of miixng for n-variant martensitic phase transformations using quasi-convex analysis. J. Mech. Phys. Solids 50 (2002) 1897–1922. CrossRef
Halphen, B. and Nguyen, Q.S., Sur les matériaux standards généralisés. J. Mécanique 14 (1975) 39–63.
W. Han and B.D. Reddy, Plasticity. Mathematical theory and numerical analysis. Springer, New York (1999).
Han, W. and Reddy, B.D., Convergence of approximations to the primal problem in plasticity under conditions of minimal regularity. Numer. Math. 87 (2000) 283–315. CrossRef
Hoffmann, K.-H., Niezgódka, M. and Songmu, Z., Existence and uniqueness of global solutions to an extended model of the dynamical development in shape memory alloys. Nonlinear Anal. Theory Methods Appl. 15 (1990) 977–990. CrossRef
Huber, J.E., Fleck, N.A., Landis, C.M. and McMeeking, R.M., A constitutive model for ferroelectric polycrystals. J. Mech. Phys. Solids 47 (1999) 1663–1697. CrossRef
James, R.D. and Kinderlehrer, D., Theory of magnetostriction with applications to Tb x Dy1-xFe2. Phil. Mag. 68 (1993) 237–274. CrossRef
James, R.D. and Wuttig, M., Magnetostriction of martensite. Phil. Mag. A 77 (1998) 1273. CrossRef
Jung, Y., Papadopoulos, P. and Ritchie, R.O., Constitutive modeling and numerical simulation of multivariant phase transformation in superelastic shape-memory alloys. Int. J. Numer. Meth. Engng. 60 (2004) 429–460. CrossRef
Kinderlehrer, D. and Pedregal, P., Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994) 59–90. CrossRef
Kočvara, M., Mielke, A. and Roubíček, T., A rate-independent approach to the delamination problem. Math. Mech. Solids 11 (2006) 423–447. CrossRef
Kružík, M., Mielke, A. and Roubíček, T., Modelling of microstructure and its evolution in shape-memory-alloy single-crystals, in particular in CuAlNi. Meccanica 40 (2005) 389–418. CrossRef
Levitas, V.I., The postulate of realizibility: formulation and applications to postbifurcational behavior and phase transitions in elastoplastic materials. Int. J. Eng. Sci. 33 (1995) 921–971. CrossRef
A. Mainik, A rate-independent model for phase transformations in shape-memory alloys. Ph.D. Thesis, Fachbereich Mathematik, Universität Stuttgart, Germany (2004).
Mainik, A. and Mielke, A., Existence results for energetic models for rate-independent systems. Calc. Var. 22 (2005) 73–99. CrossRef
A. Mainik and A. Mielke, Global existence for rate-independent gradient plasticity at finite strains. J. Nonlinear Science (2008) DOI: 10.1007/s00332-008-9033-y (published online).
Mielke, A., Energetic formulation of multiplicative elasto-plasticity using dissipation distances. Cont. Mech. Thermodynamics 15 (2003) 351–382. CrossRef
A. Mielke, Evolution of rate-independent systems, in Handbook of Differential Equations: Evolutionary Equations, C. Dafermos and E. Feireisl Eds., Elsevier, Amsterdam (2005) 461–559.
Mielke, A. and Müller, S., Lower semicontinuity and existence of minimizers for a functional in elastoplasticity. Z. Angew. Math. Mech. 86 (2006) 233–250. CrossRef
Mielke, A. and Roubíček, T., Rate-independent model of inelastic behaviour of shape-memory alloys. Multiscale Model. Simul. 1 (2003) 571–597. CrossRef
Mielke, A. and Roubíček, T., Rate-independent damage processes in nonlinear inelasticity. Math. Models Methods Appl. Sci. 16 (2006) 177–209. CrossRef
A. Mielke and F. Theil, A mathematical model for rate-independent phase transformations with hysteresis, in Models of Continuum Mech. in Anal. and Engineering, H.-D. Alber, R. Balean and R. Farwig Eds., Shaker Ver., Aachen (1999) 117–129.
Mielke, A. and Theil, F., On rate-independent hysteresis models. Nonlin. Diff. Eq. Appl. 11 (2004) 151–189.
Mielke, A. and Timofte, A., An energetic material model for time-dependent ferroelectric behavior: existence and uniqueness. Math. Meth. Appl. Sci. 29 (2006) 1393–1410. CrossRef
Mielke, A., Theil, F. and Levitas, V.I., A variational formulation of rate-independent phase transformations using an extremum principle. Arch. Ration. Mech. Anal. 162 (2002) 137–177. CrossRef
Mielke, A., Roubíček, T. and Stefanelli, U., Relaxation and Γ-limits for rate-independent evolution equations. Calc. Var. P.D.E. 31 (2008) 387–416. CrossRef
A. Mielke, L. Paoli and A. Petrov, On the existence and approximation for a 3D model of thermally induced phase transformations in shape-memory alloys. SIAM J. Math. Anal. (submitted) (WIAS Preprint 1330).
A. Mielke, T. Roubíček and J. Zeman, Complete damage in elastic and viscoelastic media and its energetics. Comput. Methods Appl. Mech. Engrg. (submitted) (WIAS preprint 1285).
S. Müller, Variational models for microstructure and phase transitions, in Calculus of variations and geometric evolution problems, S. Hildebrandt et al. Eds., Lect. Notes in Math. 1713, Springer, Berlin (1999) 85–210.
P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997).
Plecháč, P. and Roubíček, T., Visco-elasto-plastic model for martensitic phase transformation in shape-memory alloys. Math. Meth. Appl. Sci. 25 (2002) 1281–1298. CrossRef
Rajagopal, K.R. and Roubíček, T., On the effect of dissipation in shape-memory alloys. Nonlinear Anal. Real World Appl. 4 (2003) 581–597. CrossRef
H. Romanowski and J. Schröder, Modelling of the nonlinear ferroelectric hysteresis within a thermodynamically consistent framework, in Trends in Applications of Math. to Mech., Y. Wang and K. Hutter Eds., Shaker Ver., Aachen (2005) 419–428.
T. Roubíček, A note on an interaction between penalization and discretization, in Proc. IFIP-IIASA Conf., Modelling and Inverse Problems of Control for Distributed Parameter Systems, A. Kurzhanski and I. Lasiecka Eds., Lect. Notes in Control and Inf. Sci. 154, Springer (1991) 145–150.
T. Roubíček, Dissipative evolution of microstructure in shape memory alloys, in Lectures on Applied Mathematics, H.-J. Bungartz, R.H.W. Hoppe and C. Zenger Eds., Springer, Berlin (2000) 45–63.
Roubíček, T. and Kružík, M., Microstructure evolution model in micromagnetics. Z. Angew. Math. Phys. 55 (2004) 159–182. CrossRef
Roubíček, T. and Kružík, M., Mesoscopic model for ferromagnets with isotropic hardening. Z. Angew. Math. Phys. 56 (2005) 107–135. CrossRef
T. Roubíček and M. Kružík, Mesoscopic model of microstructure evolution in shape memory alloys, its numerical analysis and computer implementation, in 3rd GAMM Seminar on microstructures, C. Miehe Ed., GAMM Mitteilungen 29 (2006) 192–214.
Roubíček, T., Kružík, M. and Koutný, J., A mesoscopical model of shape-memory alloys. Proc. Estonian Acad. Sci. Phys. Math. 56 (2007) 146–154.
Rybka, P. and Luskin, M., Existence of energy minimizers for magnetostrictive materials. SIAM J. Math. Anal. 36 (2005) 2004–2019. CrossRef
Shu, Y., Bhattacharya, K., Domain patterns and macroscopic behaviour of ferroelectric materials. Phil. Mag. B 81 (2001) 2021–2054. CrossRef
J.C. Simo, Numerical analysis and simulation of plasticity, in Handbook of Numer. Anal., P.G. Ciarlet and J.L. Lions Eds., Vol. VI, Elsevier, Amsterdam (1998) 183–499.
J.C. Simo and J.R. Hughes, Computational Inelasticity. Springer, Berlin (1998).
R. Temam, Mathematical problems in plasticity. Gauthier-Villars, Paris (1985).
R. Tickle, Ferromagnetic shape memory materials. Ph.D. Thesis, University of Minnesota, Minneapolis, USA (2000).
Visintin, A., Strong convergence results related to strict convexity. Comm. Partial Diff. Eq. 9 (1984) 439–466. CrossRef
Visintin, A., Modified Landau-Lifshitz equation for ferromagnetism. Physica B 233 (1997) 365–369. CrossRef
Visintin, A., Maxwell's equations with vector hysteresis. Arch. Ration. Mech. Anal. 175 (2005) 1–37. CrossRef
Vivet, A. and Lexcellent, C., Micromechanical modelling for tension-compression pseudoelastic behaviour of AuCd single crystals. Eur. Phys. J. Appl. Phys. 4 (1998) 125–132. CrossRef