The initial-boundary value problem for the time-dependent Ginzburg-Landau equations that model the macroscopic behavior of superconductors is considered. The convergence of finite-dimensional, semidiscrete Galerkin approximations is studied as is a fully-discrete scheme. The results of some computational experiments are presented.

http://www.personal.psu.edu/qud2/Res/Pre/du94cma.pdf

Finite element methods for the time-dependent Ginzburg-Landau model of superconductivity

On fractional Langevin equation involving two fractional orders

A local inverse function theorem is established for mappings $v \in W^{1,N} (\Omega ,\mathbb{R}^N )$, $\Omega \subset \mathbb{R}^N $ open set, such that $\det \nabla v(x) > 0$ almost everywhere in $x \in \Omega $. Regularity of the local inverse $v^{ - 1} $ is obtained provided that $| {\frac{{{\text{adj}}(\nabla v)}}{{\det \nabla v}}} |^s \det \nabla v \in L^1 (\Omega )$ for some $1 \leq s < + \infty $. The local invertibility property is used to study the weak lower semicontinuity of a functional involving variation of the domain.

Local invertibility of Sobolev functions

On the weak lower semicontinuity of energies with polyconvex integrands

In this paper a new model of pseudo-elastic materials based on experimental obser-vations of one-dimensional motions of SM A materials is introduced. The model incor-porates phase fractions as an internal variable and allows for both reversible (elastic)and irreversible (plastic) modes inside the hysteresis loop. As such, the model accountsfor a wider range of physical phenomena than the traditional plasticity theories. Theconstitutive equations predict transitions between elastic and plastic modes through adependence on the time history of strain and phase fractions. These history effects areincluded in a way that allows for easy computation. 1 Introduction At a certain range of temperatures, one-dimensional motions of shape memory alloys displaypseudo-elastic behavior; i.e., under large strain cycles, specimens go through a hysteresis loopof the form exhibited schematically in Figure 1. Recently Muller and Xu [7] have conductedexperiments on single crystals of CuZnAl which display interesting behavior under partialloading and unloading cycles (i.e., inside the hysteresis loop). Briefly, their experimentsindicate the following sequence of events depicted in Figure 2.1. Initially, a material body is in its undeformed state, point (a). As a tensile load isapplied, the body deforms elastically up to the critical point (b). (In this elastic phasethe deformation is reversible.)2. As the tensile load increases beyond point (b), the body deforms plastically. Thisdeformation is irreversible.3. If the load is increased to the point (c) and the load is then reduced to zero, the usualfull hysteresis loop (cf. Figure 1) is traced.

Keming Wang

Papers

A Gronwall inequality for weakly Lipschitzian mappings

Weakly lipschitzian mappings and restricted uniqueness of solutions of ordinary differential equations