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We propose, analyze, and compare several numerical methods for the
computation of the deformation of a pressurized martensitic thin
film. Numerical results have been obtained for the hysteresis of
the deformation as the film transforms reversibly from austenite to
martensite.
A way of geometrically representing symmetric 2 × 2-gradients is proposed, and a general theorem characterizing sets of gradients is proved. We believe this perspective may help in understanding the structure of gradients and visualizing it. Several non-trivial examples are discussed.
The propagation of fractures in a solid undergoing cyclic loadings
is known as the fatigue phenomenon. In this paper, we present a time continuous
model for fatigue, in the special situation of the debonding of thin layers,
coming from a time discretized version recently proposed by
Jaubert and Marigo [C. R. Mecanique333 (2005) 550–556].
Under very general assumptions on the surface energy density and
on the applied displacement, we discuss the well-posedness of our
problem and we give the main...
Let be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set of all by matrices. Based on conditions for the rank 1 convexity of in terms of signed invariants of (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.
We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to use velocity...
We present a new class of averaging lemmas directly motivated by the question of regularity for different nonlinear equations or variational problems which admit a kinetic formulation. In particular they improve the known regularity for systems like γ = 3 in isentropic gas dynamics or in some variational problems arising in thin micromagnetic films. They also allow to obtain directly the best known regularizing effect in multidimensional scalar conservation laws. The new ingredient here is to...
The aim of this paper is to study the problem of regularity of solutions in Hencky plasticity. We consider a non-homogeneous material whose elastic-plastic properties change discontinuously. We prove that the displacement solutions belong to the space if the stress solution is continuous and belongs to the interior of the set of admissible stresses, at each point. The part of the functional which describes the work of boundary forces is relaxed.
The aim of this paper is to study the problem of regularity of displacement solutions in Hencky plasticity. We consider a plate made of a non-homogeneous material whose elastic-plastic properties change discontinuously. We prove that the displacement solutions belong to the space if the stress solution is continuous and belongs to the interior of the set of admissible stresses, at each point. The part of the functional which describes the work of boundary forces is relaxed.
The aim of this paper is to find the largest lower semicontinuous minorant of the elastic-plastic energy of a body with fissures. The functional of energy considered is not coercive.
In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the norm of the deformation gradient and a nonlinear function of the determinant of the gradient. Non–interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some -norm of the gradient with is controlled...
Some properties of nonlinear partial differential equations are naturally associated with the geometry of sets in the space of matrices. In this paper we consider the model case when the compact set is contained in the hyperboloid , where , the set of symmetric matrices. The hyperboloid is generated by two families of rank-one lines and related to the hyperbolic Monge-Ampère equation . For some compact subsets containing a rank-one connection, we show the rigidity property of by imposing...
Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the proper...
Recall that a smooth Riemannian metric on a simply connected domain can
be realized as the pull-back metric of an orientation preserving deformation if
and only if the associated Riemann curvature tensor vanishes identically.
When this condition fails, one seeks a deformation yielding
the closest metric realization.
We set up a variational formulation of this problem by
introducing the non-Euclidean version of the nonlinear
elasticity functional, and establish its Γ-convergence under the proper
scaling....
In this paper we investigate the equivalence of the sequential
weak lower semicontinuity of the total energy functional and the quasiconvexity of the
stored energy function of the nonlinear micropolar elasticity. Based on techniques of Acerbi and Fusco [Arch. Rational Mech. Anal.86 (1984) 125–145] we extend the result from Tambača and Velčić [ESAIM: COCV (2008) DOI: 10.1051/cocv:2008065] for energies that
satisfy the growth of order p≥ 1. This result is the main
step towards the general existence...
We introduce an intrinsic notion of perimeter for subsets of a general Minkowski space ( a finite dimensional Banach space in which the norm is not required to be even). We prove that this notion of perimeter is equivalent to the usual definition of surface energy for crystals and we study the regularity properties of the minimizers and the quasi-minimizers of perimeter. In the two-dimensional case we obtain optimal regularity results: apart from a singular set (which is -negligible and is empty...
We introduce an intrinsic notion of perimeter for subsets of
a general Minkowski space (i.e. a finite dimensional Banach space in which the
norm is not required to be even).
We prove that this notion of perimeter is equivalent to
the usual definition of surface energy for crystals and
we study the regularity properties of
the minimizers and the quasi-minimizers of perimeter.
In the two-dimensional case we obtain optimal regularity results:
apart from a singular set (which is -negligible and is...
A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem...
A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret (1995). Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitté et al. (2002), the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem...
We prove the periodicity of all H2-local minimizers with low energy
for a one-dimensional higher order variational problem.
The results extend and complement an earlier work of Stefan Müller
which concerns the structure of global minimizer.
The energy functional studied in this work is motivated by the
investigation of coherent solid phase transformations and the
competition between the
effects from regularization and formation of small scale structures.
With a special choice of a bilinear double...
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