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- ItemConvergence of solutions of kinetic variational inequalities in the rate-independent quasi-static limit(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Mielke, Alexander; Petrov, Adrien; Martins, João A.C.This paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to three-dimensional elastic-plastic systems with hardening is given.
- ItemOn existence and approximation for a 3D model of thermally-induced phase transformations in shape-memory alloys(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Mielke, Alexander; Paoli, Laetitia; Petrov, AdrienThis paper deals with a three-dimensional model for thermal stress-induced transformations in shape-memory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and temporal regularity is obtained in case of uniform convexity. We study also space-time discretizations and establish convergence of these approximations.
- ItemExistence result for a class of generalized standard materials with thermomechanical coupling(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Paoli, Laetitia; Petrov, AdrienThis paper deals with the study of a three-dimensional model of thermomechanical coupling for viscous solids exhibiting hysteresis effects. This model is written in accordance with the formalism of generalized standard materials. It is composed by the momentum equilibrium equation combined with the flow rule, which describes some stress-strain dependance, and the heat-transfer equation. An existence result for this thermodynamically consistent problem is obtained by using a fixed-point argument and some qualitative properties of the solutions are established.
- ItemOn the numerical approximation of a viscoelastodynamic problem with unilateral constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Petrov, Adrien; Martins, J.A.C.The present work is dedicated to the study of numerical schemes for a viscoelastic bar vibrating longitudinally and having its motion limited by rigid obstacles at the both ends. Finite elements and finite difference schemes are presented and their convergence is proved. Finally, some numerical examples are reported and analyzed.
- ItemError estimates for space-time discretizations of a rate-independent variational inequality(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Mielke, Alexander; Paoli, Laetitia; Petrov, Adrien; Stefanelli, UlisseThis paper deals with error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, we address a fully-discrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit space-time convergence rates for the isothermal Souza-Auricchio model for shape-memory alloys.
- ItemThermodynamics of multiphase problems in viscoelasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Paoli, Laetitia; Petrov, AdrienThis paper deals with a three-dimensional mixture model describing materials undergoing phase transition with thermal expansion. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. A global solution for this thermodynamically consistent problem is obtained by using a fixed-point argument combined with global energy estimates.
- ItemGlobal existence result for phase transformations with heat transfer in shape memory alloys : dedicated to 75th birthday of K. Gröger(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Paoli, Laetitia; Petrov, Adrien; Gröger, K.We consider three-dimensional models for rate-independent processes describing materials undergoing phase transformations with heat transfer. The problem is formulated within the framework of generalized standard solids by the coupling of the momentum equilibrium equation and the flow rule with the heat transfer equation. Under appropriate regularity assumptions on the initial data, we prove the existence a global solution for this thermodynamically consistent system, by using a fixed-point argument combined with global energy estimates.
- ItemGlobal existence result for thermoviscoelastic problems with hysteresis : dedicated to the memory of M. Schatzman(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Paoli, Laetitia; Petrov, Adrien; Schatzman, M.We consider viscoelastic solids undergoing thermal expansion and exhibiting hysteresis effects due to plasticity or phase transformations. Within the framework of generalized standard solids, the problem is described in a 3D setting by the momentum equilibrium equation, the flow rule describing the dependence of the stress on the strain history, and the heat transfer equation. Under appropriate regularity assumptions on the data, a local existence result for this thermodynamically consistent system is established, by combining existence results for ordinary differential equations in Banach spaces with a fixed-point argument. Then global estimates are obtained by using both the classical energy estimate and more specific techniques for the heat equation introduced by Boccardo and Gallouet. Finally a global existence result is derived.
- ItemThermally driven phase transformation in shape-memory alloys(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Mielke, Alexander; Petrov, AdrienThis paper analyzes a model for phase transformation in shape-memory alloys induced by temperature changes and by mechanical loading. We assume that the temperature is prescribed and formulate the problem within the framework of the energetic theory of rate-independent processes. Existence and uniqueness results are proved.
- ItemMathematical results on existence for viscoelastodynamic problems with unilateral constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Petrov, Adrien; Schatzman, M.We study a damped wave equation and the evolution of a Kelvin-Voigt material, both problems have unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution.