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Papers by Alexander Mielke

Ann. Sc. Norm. Sup. Pisa Cl. Sci.(5), 2010

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Journal of Dynamics and Differential Equations, 2007

Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 2002

It is well-known in surface wave theory that the secular equation for the surface- wave speed v c... more It is well-known in surface wave theory that the secular equation for the surface- wave speed v can be written as detM = 0 in terms of the surface impedance matrix M. It is shown in this paper that M satisfies the simple identity (M ¡ iR)T¡1(M + iRT) ¡ Q + ‰v2I = 0 in the usual notation in

Multiscale Modeling & Simulation, 2003

We formulate a model describing rate-independent hysteretic re- sponse of shape-memory alloys und... more We formulate a model describing rate-independent hysteretic re- sponse of shape-memory alloys under slow external forcing. Under natural as- sumptions we prove that this model has solution. The microstructure is treated on a \mesoscopic" level, described by volume fractions of particular phases in terms of Young measures. The whole formulation is based on energetic function- als for energy storage and

Archive for Rational Mechanics and Analysis, 1988

Saint-Venant's problem consists in finding elastic deformations of an infinite prismatic body... more Saint-Venant's problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross-sectional resultants of force and moment. Using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite-dimensional manifold. In particular, the flow on this manifold is described by a set of equations having exactly the

Nonlinearity, 1995

We are interested in the long--time behavior of nonlinear parabolic PDEs definedon unbounded cyli... more We are interested in the long--time behavior of nonlinear parabolic PDEs definedon unbounded cylindrical domains. For dissipative systems defined on boundeddomains, the long--time behavior can often be described by the dynamics in theirfinite--dimensional attractors. For systems defined on the infinite line, very little isknown at present, since the lack of compactness prevents application of the standardexistence theory for attractors. We

Meccanica, 2005

A continuum-mechanical description of the stored energy in shape-memory alloys is presented, with... more A continuum-mechanical description of the stored energy in shape-memory alloys is presented, with its multi-well character giving rise to a microstructure described, with a certain approximation, by special gradient Young measures. A rate-independent phenomenological dissipation is then considered to model a hysteretic response. Isothermal simulations with CuAlNi single crystal are presented.

Rate-independent systems allow for solutions with jumps that need additional modeling. Here we su... more Rate-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

In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach... more In this paper we analyze a broad class of abstract doubly nonlinear evolution equations in Banach spaces, driven by nonsmooth and nonconvex energies. We provide some general sufficient conditions, on the dissipation potential and the energy functional,for existence of solutions to the related Cauchy problem. We prove our main existence result by passing to the limit in a time-discretization scheme

In the nonconvex case solutions of rate-independent systems may develop jumps as a function of ti... more In the nonconvex case solutions of rate-independent systems may develop jumps as a function of time. To model such jumps, we adopt the philosophy that rate independence should be considered as limit of systems with smaller and smaller viscosity. For the finite-dimensional case we study the vanishing-viscosity limit of doubly nonlinear equations given in terms of a differentiable energy functional

Analysis, Modeling and Simulation of Multiscale Problems, 2006

We present an overview of recent results concerning wave trains, solitons and their modulation in... more We present an overview of recent results concerning wave trains, solitons and their modulation in FPU chains. We take a thermodynamic perspective and use hyperbolic scaling of particle index and time in order to pass to a macroscopic continuum limit. While strong convergence yields the well-known p-system of mass and momentum conservation, we generally obtain a weak form of it in terms of Young measures. The modulation approach accounts for microscopic oscillations, which we interpret as temperature, causing convergence only in a weak, average sense. We present the arising Whitham modulation equations in a thermodynamic form, as well as analytic and numerical tools for the resolution of the modulated wave trains. As a prototype for the occurrence of temperature from oscillation-free initial data, we discuss various Riemann problems, and the arising dispersive shock fans, which replace Lax-shocks. We predict scaling and jump conditions assuming a generic soliton at the shock front.

We propose a rate-independent, mesoscopic model for the hysteretic evolution of phase transformat... more We propose a rate-independent, mesoscopic model for the hysteretic evolution of phase transformations in shape-memory alloys. The model uses the deformation and phase-indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws. Using the associated functionals, admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality.

We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical lim... more We show that continuum models for ideal plasticity can be obtained as a rigorous mathematical limit starting from a discrete microscopic model describing a viscoelastic crystal lattice with quenched disorder. The constitutive structure changes as a result of two concurrent limiting procedures: the vanishing-viscosity limit and the discrete-to-continuum limit. In the course of these limits a non-convex elastic problem transforms into a convex elastic problem while the quadratic rate-dependent dissipation of visco-elastic lattice transforms into a singular rate-independent dissipation of an ideally plastic solid. In order to emphasize our ideas we employ in our proofs the simplest prototypical system mimicking the phenomenology of transformational plasticity in shape-memory alloys. The approach, however, is sufficiently general that it can be used for similar reductions in the cases of more general plasticity and damage models.

Here rd(z) : [0, T ] → X is defined only µ z -almost everywhere (a.e.). Note that our definition ... more Here rd(z) : [0, T ] → X is defined only µ z -almost everywhere (a.e.). Note that our definition of the differential measure differs from that in Sect.0.1] where ν z ([s, t)) = [s,t) dz and nd(z)(t) = 1 ν z -a.e. in [0, T ] such that dz = nd(z)ν z = rd(z)µ z . The derivative doesn't distinguish between right and left continuous versions, i.e., rd(z + ) = rd(z − ) and

This note deals with a three-dimensional model for thermal stress-induced transformations in shap... more This note 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. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and we study space-time discretizations and establish convergence of these approximations.

PAMM, 2008

We model the evolution of a single crack as a rate-independent process based on the Griffith crit... more We model the evolution of a single crack as a rate-independent process based on the Griffith criterion. Three approaches are presented, namely a model based on global energy minimization, a model based on a local description involving the energy release rate and a refined local model which is the limit problem of regularized, viscous models. Finally we present an example which sheds light on the different predictions of the models.