Abstract
The paper presents incremental minimization principles for the broad class of standard dissipative materials at finite strains. Starting with a minimization principle for the local constitutive material response we define a minimization problem for the incremental boundaryvalue problem of standard dissipative solids. The existence of this principle allows the determination of micro-structure developments in non-stable dissipative materials based on a convexification analysis. Finally, we propose a minimization principle for the boundary-value problem of homogenization in heterogeneous microstructures.
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References
Ball J.M. (1977) Convexity Conditions and Existence Theorems in Nonlinear Elasticity. Archive of Rational Mechanics and Analysis 63: 337 - 403
Biot M.A. (1965) Mechanics of Incremental Deformations. John Wiley and Sons, New York
Bronkhorst C.A., Kalidindi S.R., Anand L. (1992) Polycrystalline Plasticity and the Evolution of Crystallographic Texture in f.c.c. Metals. Philosophical Transactions Royal Society London A 341: 443-477
Carstensen C., Hackl K., Mielke A. (2002) Nonconvex potentials and microstructures in finite-strain plasticity. Royal Society London, Proc. Ser. A 458: 299-317
Dacorogna B. (1989) Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin
Halphen B., Nguyen Q.S. (1975) Sur les Matériaux Standards Généralisés. Journal de Mécanique 40: 39 - 63
Hill R. (1972) On Constitutive Macro-Variables for Heterogeneous Solids at Finite Strain. Proceedings of the Royal Society London A 326: 131 - 147
Kohn R.V., Strang G. (1983) Explicit Relaxation of a Variational Problem in Optimal Design. Bulletin of the American Mathematical Society 9: 211 - 214
Martin J.B. (1975) Plasticity. Fundamentals and General Results. MIT press, Cambridge, Massachusetts
Miehe C. (2002) Strain-Driven Homogenization of Inelastic Microstructures and Composites Based on an Incremental Variational Formulation. International Journal for Numerical Methods in Engineering 55: 1285 - 1322
Miehe C., Schröder J., Schotte J. (1999) Computational Homogenization Analysis in Finite Plasticity. Simulation of Texture Development in Polycrystalline Materials. Computer Methods in Applied Mechanics and Engineering 171: 387-418
Miehe C., Schröder J., Becker M. (2002) Computational Homogenization Analysis in Finite Elasticity. Material and Structural Instabilities on the Micro-and Macro-Scales of Periodic Composites and their Interaction. Computer Methods in Applied Mechanics and Engineering 191: 4971-5005
Miehe C., Schotte J., Lambrecht M. (2002) Homogenization of Inelastic Solid Materials at Finite Strains Based on Incremental Minimization Principles. Journal of the Mechanics and Physics of Solids 50: 2123 - 2167
Miehe C, Lambrecht M. (2000) Analysis of Microstructure Development in Shearbands by Energy Relaxation of Incremental Stress Potentials: Large-Strain Theory for Generalized Standard Solids. International Journal for Numerical Methods in Engineering, in press
Miehe C., Lambrecht M., Gürses E. (2002) Analysis of Material Instabilities of Inelastic Solids based on Incremental Variational and Relaxation Methods: Application to Single-Slip Crystal Plasticity. Journal of the Mechanics and Physics of Solids, submitted
Mielke A. (2002) Finite elastoplasticity, Lie Groups and geodesics on SL(d). In: Newton P., Weinstein A., Holmes P. (Eds.), Mielke A, 61 - 90, Springer-Verlag
Moreau J.J. (1976) Application of Convex Analysis to the Treatment of Elastoplastic Systems. In: Germain P., Nayroles B. (Eds.), Application of Methods of Functional Analysis to Problems in Mechanics. Springer-Verlag, Berlin
Morrey C.B. (1952) Quasiconvexity and the Semicontinuity of Multiple Integrands. Pacific Journal of Mathematics 2: 25 - 53
Müller S. (1987) Homogenization of Nonconvex Integral Functionals and Cellular Elastic Materials. Archive of Rational Mechanics and Analysis 99: 189 - 212
Müller S. (1999) Variational models for microstructure and phase transitions. In: Bethuel F., Huisken G., Müller S., Steffen K., Hildebrandt S., Struwe M. (Eds.), Lectures given at the 2nd Session of the Centro Internazionale Matematico Estivo held in Cetaro, Italy, June 15-22, 1996, Springer, Berlin
Nguyen Q.S. (2000) Stability and Nonlinear Solid Mechanics. Wiley and Sons, Chichester
Ortiz M., Repetto E.A. (1999) Nonconvex Energy Minimization and Dislocation Structures in Ductile Single Crystals. Journal of the Mechanics and Physics of Solids 47: 397 - 462
Ortiz M., Repetto E.A., Stainier L. (2000) A Theory of Subgrain Dislocation Structures. Journal of the Mechanics and Physics of Solids 48: 2077 - 2114
Ponte Castaneda P., Suquet P. (1998) Nonlinear Composites. Advances in Applied Mechanics 34: 171 - 303
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 I V. North-Holland Publishing, Amsterdam
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Miehe, C., Lambrecht, M., Schotte, J. (2003). Computational Homogenization of Materials with Microstructures Based on Incremental Variational Formulations. In: Wendland, W., Efendiev, M. (eds) Analysis and Simulation of Multifield Problems. Lecture Notes in Applied and Computational Mechanics, vol 12. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-36527-3_10
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DOI: https://doi.org/10.1007/978-3-540-36527-3_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-05633-8
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