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In this paper we analyse the structure of approximate solutions to the compatible two well problem with the constraint that the surface energy of the solution is less than some fixed constant. We prove a quantitative estimate that can be seen as a two well analogue of the Liouville theorem of Friesecke James Müller.
Let for . Let . Let . Let be a invertible bilipschitz function with , .
There exists positive constants and depending only on , , such that if and satisfies the...
In this paper we analyse the structure of approximate solutions to the compatible
two well problem with the constraint that the surface energy of the solution
is less than some fixed constant. We prove a quantitative estimate that can be seen as
a two well analogue of the Liouville theorem of Friesecke James Müller.
Let for .
Let . Let .
Let be a invertible bilipschitz
function with , .
There exists positive constants and depending only on σ, ,
such that if
and u satisfies...
In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case....
In order to describe a solid which deforms smoothly in some region, but
non smoothly in some other region, many multiscale methods have recently
been proposed. They aim at coupling an atomistic model (discrete
mechanics) with a macroscopic model
(continuum mechanics).
We provide here a theoretical ground for such a coupling in a
one-dimensional setting. We briefly study the general case of a convex
energy, and next concentrate on
a specific example of a nonconvex energy, the Lennard-Jones case....
We give results for the approximation of a laminate with
varying volume fractions for multi-well energy minimization
problems modeling martensitic crystals that
can undergo either an orthorhombic
to monoclinic or a cubic to tetragonal transformation.
We construct energy minimizing sequences of deformations which satisfy
the corresponding boundary condition, and we
establish a series of error bounds in terms of the elastic energy
for the approximation of the limiting macroscopic
deformation and...
We compare dewetting characteristics of a thin nonwetting solid film in the
absence of stress, for
two models of a wetting potential: the exponential and the algebraic.
The exponential model is a one-parameter (r) model, and the algebraic model is
a two-parameter (r, m)
model, where r is the ratio of the characteristic wetting length to the height
of the unperturbed film,
and m is the exponent of h (film height) in a smooth function that
interpolates the system's surface
energy above and below...
Let be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form applied to rotations controls the gradient in the sense that pointwise . This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math. 55 (2002) 1461–1506; John, Comme Pure Appl. Math. 14 (1961) 391–413; Reshetnyak, Siberian Math. J. 8 (1967) 631–653)] as well as an associated linearized theorem...
Let be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form applied to rotations controls the gradient in the sense that pointwise
.
This result complements rigidity results
[Friesecke, James and Müller, Comme Pure Appl. Math.55 (2002) 1461–1506; John, Comme Pure Appl. Math.14 (1961) 391–413; Reshetnyak, Siberian Math. J.8 (1967) 631–653)] as well as an associated linearized theorem saying...
Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the -limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.
Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the Γ-limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.
Modern physics theories claim that the dynamics of interfaces between
the two-phase is described by the evolution equations involving the
curvature and various kinematic energies. We consider the motion of
spiral-shaped polygonal curves by its crystalline curvature, which
deserves a mathematical model of real crystals. Exploiting the
comparison principle, we show the local existence and uniqueness of the
solution.
Let where are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the -well problem with surface energy. Let , be a convex polytopal region. Defineand let denote the subspace of functions in that satisfy the affine boundary condition on (in the sense of trace), where . We consider the scaling (with respect to ) ofSecondly the finite element approximation to the -well problem without surface...
Let
where are matrices of non-zero determinant. We
establish a sharp relation between the following two minimisation
problems in two dimensions. Firstly the N-well problem with surface energy. Let
, be a convex polytopal region. Define
and let AF denote the subspace of functions in
that satisfy the affine boundary condition
Du=F on (in the sense of trace), where . We consider the scaling (with respect to ϵ) of
Secondly the finite element approximation to the N-well problem
without...
Microstructures in phase-transitions of alloys are modeled by the energy minimization of a nonconvex energy density . Their time-evolution leads to a nonlinear wave equation with the non-monotone stress-strain relation plus proper boundary and initial conditions. This hyperbolic-elliptic initial-boundary value problem of changing types allows, in general, solely Young-measure solutions. This paper introduces a fully-numerical time-space discretization of this equation in a corresponding very...
Microstructures in phase-transitions of alloys are modeled by the
energy minimization of a nonconvex energy density ϕ. Their
time-evolution leads to a nonlinear wave equation
with the non-monotone stress-strain relation
plus proper boundary and initial conditions. This hyperbolic-elliptic
initial-boundary value problem of changing types allows, in general,
solely Young-measure solutions. This paper introduces a
fully-numerical time-space discretization of this equation in a
corresponding...
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