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- ItemEnergy release rate for cracks in finite-strain elasticity(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Knees, Dorothee; Mielke, AlexanderGriffith's fracture criterion describes in a quasistatic setting whether or not a pre-existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: If the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we derive rigorously the well-known Griffith formula and the $J$-integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors.
- ItemInteraction of modulated pulses in the nonlinear Schrödinger equation with periodic potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Giannoulis, Johannes; Mielke, Alexander; Sparber, ChristofWe consider a cubic nonlinear Schrödinger equation with periodic potential. In a semiclassical scaling the nonlinear interaction of modulated pulses concentrated in one or several Bloch bands is studied. The notion of closed mode systems is introduced which allows for the rigorous derivation of a finite system of amplitude equations describing the macroscopic dynamics of these pulses.
- ItemExistence results for a contact problem with varying friction coefficient and nonlinear forces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Schmid, Florian; Mielke, AlexanderWe consider the rate-independent problem of a particle moving in a three - dimensional half space subject to a time-dependent nonlinear restoring force having a convex potential and to Coulomb friction along the flat boundary of the half space, where the friction coefficient may vary along the boundary. Our existence result allows for solutions that may switch arbitrarily often between unconstrained motion in the interior and contact where the solutions may switch between sticking and frictional sliding. However, our existence result is local and guarantees continuous solutions only as long as the convexity of the potential is strong enough to compensate the variation of the friction coefficient times the contact pressure. By simple examples we show that our sufficient conditions are also necessary. Our method is based on the energetic formulation of rate-independent systems as developed by Mielke and co-workers. We generalize the time-incremental minimization procedure of Mielke and Rossi for the present situation of a non-associative flow rule.
- ItemReverse approximation of energetic solutions to rate-independent processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Mielke, Alexander; Rindler, FilipEnergetic solutions to rate-independent processes are usually constructed via time-incremental minimization problems. In this work we show that all energetic solutions can be approximated by incremental problems if we allow approximate minimizers, where the error in minimization has to be of the order of the time step. Moreover, we study sequences of problems where the energy functionals have a Gamma limit.
- Item[Gamma]-limits and relaxations for rate-independent evolutionary problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, Alexander; Toubíček, Tomáš; Stefanelli, UlisseThis work uses the energetic formulation of rate-independent systems that is based on the stored-energy functionals ε and the dissipation distance D. For sequences (ε k)k ∈ ℕ and (D k)k ∈ ℕ we address the question under which conditions the limits q∞ of solutions qk: [0,T] → Q satisfy a suitable limit problem with limit functionals ε∞ and D∞, which are the corresponding Γ-limits. We derive a sufficient condition, called emphconditional upper semi-continuity of the stable sets, which is essential to guarantee that q∞ solves the limit problem. In particular, this condition holds if certain emphjoint recovery sequences exist. Moreover, we show that time-incremental minimization problems can be used to approximate the solutions. A first example involves the numerical approximation of functionals using finite-element spaces. A second example shows that the stop and the play operator convergece if the yield sets converge in the sense of Mosco. The third example deals with a problem developing microstructure in the limit k → ∞, which in the limit can be described by an effective macroscopic model.
- ItemTwo-scale homogenization for evolutionary variational inequalities via the energetic formulation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Mielke, Alexander; Timofte, AidaThis paper is devoted to the two-scale homogenization for a class of rate-independent systems described by the energetic formulation or equivalently by an evolutionary variational inequality. In particular, we treat the classical model of linearized elastoplasticity with hardening. The associated nonlinear partial differential inclusion has periodically oscillating coefficients, and the aim is to find a limit problem for the case that the period tends to 0. Our approach is based on the notion of energetic solutions which is phrased in terms of a stability condition and an energy balance of an energy-storage functional and a dissipation functional. Using the recently developed method of weak and strong two-scale convergence via periodic unfolding, we show that these two functionals have a suitable two-scale limit, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. Moreover, relying on an abstract theory of Gamma convergence for the energetic formulation using so-called joint recovery sequences it is possible to show that the solutions of the problem with periodicity converge to the energetic solution associated with the limit functionals.
- ItemModeling solutions with jumps for rate-independent systems on metric spaces(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Mielke, Alexander; Rossi, Riccarda; Savaré, GiuseppeRate-independent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rate-independent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of generalized gradient flows in metric spaces, and comparison with other notions of solutions is given.
- 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.
- 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.
- ItemOn the inviscid limit of a model for crack propagation(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Knees, Dorothee; Mielke, Alexander; Zanini, ChiaraWe study the evolution of a single crack in an elastic body and assume that the crack path is known in advance. The motion of the crack tip is modeled as a rate-independent process on the basis of Griffith's local energy release rate criterion. According to this criterion, the system may stay in a local minimum before it performs a jump. The goal of this paper is to prove existence of such an evolution and to shed light on the discrepancy between the local energy release rate criterion and models which are based on a global stability criterion (as for example the Francfort/Marigo model). We construct solutions to the local model via the vanishing viscosity method and compare different notions of weak, local and global solutions.