51 research outputs found
Numerical Computation of Rank-One Convex Envelopes
We describe an algorithm for the numerical computation of the rank-one convex envelope of a function f:\MM^{m\times n}\rightarrow\RR. We prove its convergence and an error estimate in L∞
Stress-modulated growth in the presence of nutrients -- existence and uniqueness in one spatial dimension
Existence and uniqueness of solutions for a class of models for
stress-modulated growth is proven in one spatial dimension. The model features
the multiplicative decomposition of the deformation gradient into an
elastic part and a growth-related part . After the transformation due
to the growth process, governed by , an elastic deformation described by
is applied in order to restore the Dirichlet boundary conditions and
therefore the current configuration might be stressed with a stress tensor .
The growth of the material at each point in the reference configuration is
given by an ordinary differential equation for which the right-hand side may
depend on the stress and the pull-back of a nutrient concentration in the
current configuration, leading to a coupled system of ordinary differential
equations
Optimal rigidity estimates for maps of a compact Riemannian manifold to itself
Let be a smooth, compact, connected, oriented Riemannian manifold, and
let be an isometric embedding. We show that a
Sobolev map which has the property that the differential
is close to the set of orientation preserving
isometries (in an sense) is already close to a global isometry
of . More precisely we prove for the optimal linear
estimate where and where denotes
the group of orientation preserving isometries of .
This extends the Euclidean rigidity estimate of Friesecke-James-M\"uller
[Comm. Pure Appl. Math. {\bf 55} (2002), 1461--1506] to Riemannian manifolds.
It also extends the Riemannian stability result of Kupferman-Maor-Shachar
[Arch. Ration. Mech. Anal. {\bf 231} (2019), 367--408] for sequences of maps
with to an optimal quantitative estimate.
The proof relies on the weak
Riemannian Piola identity of Kupferman-Maor-Shachar, a uniform
approximation through the harmonic map heat flow, and a linearization argument
which reduces the estimate to the well-known Riemannian version of Korn's
inequality
Relaxation of some multi-well problems
Mathematical models of phase transitions in solids lead to the variational problem, minimize ∫_ΩW (Du) dx, where W has a multi-well structure, i.e. W = 0 on a multi-well set K and W > 0 otherwise. We study this problem in two dimensions in the case of equal determinant, i.e. for K = SO(2)U_1 ∪...∪ SO(2)U_k or K = O(2)U_1 ∪...∪ O(2)Uk for U_1,...,U_k ∈ M^(2×2) with det U_i = δ in three dimensions when the matrices U_i are essentially two-dimensional and also for K = SO(3)Û_1 ∪...∪ SO(3)Û_k for U_1,...,U_k ∈ M^(3×3) with (adj U_i^TU_i)33 = δ^2, which arises in the study of thin films. Here, Û_i denotes the (3×2) matrix formed with the first two columns of U_i. We characterize generalized convex hulls, including the quasiconvex hull, of these sets, prove existence of minimizers and identify conditions for the uniqueness of the minimizing Young measure. Finally, we use the characterization of the quasiconvex hull to propose 'approximate relaxed energies', quasiconvex functions which vanish on the quasiconvex hull of K and grow quadratically away from it
Analytical and numerical relaxation results for models in soil mechanics
A variational model of pressure-dependent plasticity employing a
time-incremental setting is introduced. A novel formulation of the dissipation
potential allows one to construct the condensed energy in a variationally
consistent manner. For a one-dimensional model problem, an explicit expression
for the quasiconvex envelope can be found which turns out to be essentially
independent of the original pressure-dependent yield surface. The model problem
can be extended to higher dimensions in an empirical manner. Numerical
simulation exhibit well-posed behavior showing mesh-independent results.Comment: Submitted to Cont. Mech. Thermody
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