Abstract
We study finite-strain elastoplasticity in a new formulation proposed in [8,1,7]. This theory does not need smoothness and is based on energy minimization techniques. In particular, it gives rise to robust algorithms. It is based on two scalar constitutive functions: an elastic potential and a dissipation potential which give rise to an energy functional and a dissipation distance.
Here we study these dissipation distances in some detail and present situations where they are quite explicitly available. These include isotropic plasticity of Prandtl-Reuß type and examples from two-dimensional single-crystal plasticity. We put special emphasis on the geometric nonlinearities arising from the underlying matrix groups which lead to optimization problems on Lie groups.
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References
C. Carstensen, K. Hackl, and A. Mielke. Nonconvex potentials and microstructure in finite—strain plasticity. Proc. Royal Soc. London Ser. A. 458 (2018), 299–317, 2002.
M. E. Gurtin. On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids, 48 (5), 989–1036, 2000.
K. Hackl and U. Hoppe. On the calculation of microstructures for inelastic materials using relaxed energies. Proc. of IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains (Aug. 2001),to appear.
V. Jurdjevic. Non-euclidean elastica. Am. J. Math, 117, 93–124, 1995.
C. MIEHE AND M. LAMBRECHT. Analysis of Microstructure development in shearbands by energy relaxation of incremental stress potentials: large-strain theory for generalized standard solids. Report No. 01-I-10, Inst. Mech., Univ. Stuttgart., 2001.
C. Miehe, J. Schotte, and M. Lambrecht. Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. Application to the texture analysis of polycrystals. Report No. 01-I-07, Inst. Mech., Univ. Stuttgart., 2001.
A. Mielke. Energetic formulation of multiplicative elastoplasticity using dissipation distances. J. Mech. Physics Solids,2002. Submitted.
A. Mielke. Finite elastoplasticity, Lie groups and geodesics on SL(d). In P. Newton, A. Weinstein, and P. Holmes, editors, Geometry, Dynamics, and Mechanics. Springer—Verlag, 2002. In press.
D. Mittenhuber. Applications of the Maximum Principle to problems in Lie semigroups. In K.H. Hofmann, J.D. Lawson, and E.B. Vinberg, editors, Semi-groups in Algebra, Geometry and Analysis. de Gruyter, 1995.
D. Mittenhuber. The dissipation distance for a 2D single crystal with two symmetric slip systems. In preparation, 2002.
D. Mittenhuber. Dissipation distances for 2D single crystals: The hexagonal lattice. In preparation, 2002.
M. Ortiz and E. Repetto. Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids, 47 (2), 397–462, 1999.
M. Ortiz, E. Repetto, and L. Stainier. A theory of subgrain dislocation structures. J. Mech. Physics Solids, 48, 2077–2114, 2000.
L.S. Pontrjagin, V.G. Boltyanski, R. V. Gamkrelidze, and E.F. Mishchenko. The Mathematical Theory of Optimal Processes. Wiley Interscience, 1962.
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Hackl, K., Mielke, A., Mittenhuber, D. (2003). Dissipation distances in multiplicative elastoplasticity. 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_8
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DOI: https://doi.org/10.1007/978-3-540-36527-3_8
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