Existence results for a class of rate-independent material models with nonconvex elastic energies

By Gilles A. Francfort, Alexander Mielke. 2006

227 citations recorded by the source corpus.

Subfields: Biomedical Engineering, Civil and Structural Engineering, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/17e1f4f7c10abfedce7391a4b11eff204a394cca

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In Biomedical Engineering

In Civil and Structural Engineering

In Computational Theory and Mathematics

Citations in the corpus

On the Rate-Independent Limit of Systems with Dry Friction and Small Viscosity (2006, background): “Of course, the energetic formulation (1.2) can also be used in cases where I(t; ) is nonconvex (see for example [Efe03, MiT04, Mie05, FrM05 ]), however this will lead to solutions having jumps which may not correspond to the physically desired jumps.”
A Vanishing Viscosity Approach to Quasistatic Evolution in Plasticity with Softening (2006, unknown): “(a) If µ ≥ µ0 or if µ µ0 and θ0 ≥ β , then in the interval (t0,+∞) the function θ(t) coincides with the unique absolutely continuous solution of the Cauchy problem B0(θ(t)) θ̇(t) = A0(θ(t)) , θ(t0) = θ0 .”
WELL-POSEDNESS AND LONG-TIME BEHAVIOR FOR A CLASS OF DOUBLY NONLINEAR EQUATIONS (2006, background): “1) stem in connection with phase change phenomena [14, 17, 23, 32, 46, 47], gas flow through porous media [68], damage [15, 16, 30, 31, 54], and, in the specific case α(λr) = α(r) for all λ > 0 and r ∈ R (which is however not included in the present analysis), elastoplasticity [25, 50, 51, 52], brit”
Modeling solutions with jumps for rate-independent systems on metric spaces (2008, background): “1, is derivative-free, and thus applies to solutions with jumps and can be used in very general frameworks, like, for example, in a topological space Q, with E and D lower semicontinuous only, see [17, 20, 12].”
A metric approach to a class of doubly nonlinear evolution equations and applications (2008, background): “Rate-independent models indeed occur, for instance, in elastoplasticity [29, 30, 31, 32, 19, 20, 24], in damage [39], in the quasistatic evolution of fractures [21, 28], in shape memory alloys [40, 41, 38], and in several other contexts, see the survey [33] and the references therein.”
Nonsmooth analysis of doubly nonlinear evolution equations (2011, background): “Its analysis has already been developed in [24], in the case of a rate-independent evolution for z.”
The Variational Approach to Fracture (2008, background): “Throughout this subsection, we assume that the Griffith-like potential is positively 1-homogeneous, which amounts to a statement of rateindependence, as explained at length in various works; see (Mielke, 2005) for a general treatment of rate independent processes, and also (Francfort and Mielke, 200”
Γ-limits and relaxations for rate-independent evolutionary problems (2008, methodology): “[14,25,27, 32,34]), there is still a need to develop a theory for parameter dependence and for numerical approximation properties.”