Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance

@article{Conti2006RigidityAG,
  title={Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance},
  author={Sergio Conti and Ben Schweizer},
  journal={Communications on Pure and Applied Mathematics},
  year={2006},
  volume={59}
}
  • S. Conti, B. Schweizer
  • Published 1 June 2006
  • Mathematics
  • Communications on Pure and Applied Mathematics
The singularly perturbed two‐well problem in the theory of solid‐solid phase transitions takes the form $$I_{\varepsilon}[u] = \int\limits^{}_{\Omega} {1 \over {\varepsilon}} W(\nabla u) + \varepsilon|\nabla^{2}u|^{2},$$ where u : Ω ⊂ ℝn → ℝn is the deformation, and W vanishes for all matrices in K = SO(n)A ∪ SO(n)B. We focus on the case n = 2 and derive, by means of Gamma convergence, a sharp‐interface limit for Iε. The proof is based on a rigidity estimate for low‐energy functions. Our… 
A Γ-Convergence Result for Thin Martensitic Films in Linearized Elasticity
TLDR
This work proves compactness of displacement sequences and derives the $\Gamma$-limit of the functionals $\frac{1}{h^2} E^h$ as $h\to 0$.
Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions
We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel
Two-well linearization for solid-solid phase transitions
In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we
Rigidity of Branching Microstructures in Shape Memory Alloys
  • Thilo M. Simon
  • Art, Chemistry
    Archive for Rational Mechanics and Analysis
  • 2021
TLDR
It is shown how generic sequences for which the geometrically linear energy equations are analyzed can be transformed into discrete discrete-time solutions.
Austenite as a Local Minimizer in a Model of Material Microstructure with a Surface Energy Term
  • J. Bevan
  • Mathematics
    SIAM J. Math. Anal.
  • 2011
A family of integral functionals $\mathcal{F}$ which model in a simplified way material microstructure occupying a two-dimensional domain $\Omega$ and which take account of surface energy and a
Closure and Commutability Results for Γ-Limits and the Geometric Linearization and Homogenization of Multiwell Energy Functionals
TLDR
A $\Gamma-closure theorem is obtained by proving that geometric linearization and homogenization of multiwell energy functionals commute.
Integral Representation for Functionals Defined on SBDp in Dimension Two
We prove an integral representation result for functionals with growth conditions which give coercivity on the space $${SBD^p(\Omega)}$$SBDp(Ω), for $${\Omega\subset\mathbb{R}^{2}}$$Ω⊂R2, which is a
Local minimizers and planar interfaces in a phase-transition model with interfacial energy
Interfacial energy is often incorporated into variational solid-solid phase transition models via a perturbation of the elastic energy functional involving second gradients of the deformation. We
The Asymptotically Sharp Geometric Rigidity Interpolation Estimate in Thin Bi-Lipschitz Domains
This work is part of a program of development of asymptotically sharp geometric rigidity estimates for thin domains. A thin domain in three dimensional Euclidean space is roughly a small neighborhood
On spaces of bounded $q$-variation in dimension $N$
Motivated by the formula, due to Bourgain, Brezis and Mironescu, \begin{equation*} \lim_{\varepsilon\to 0^+} \int_\Omega\int_\Omega
...
...

References

SHOWING 1-10 OF 43 REFERENCES
A Γ‐convergence result for the two‐gradient theory of phase transitions
The generalization to gradient vector fields of the classical double‐well, singularly perturbed functionals, $$ I_{\varepsilon} ( u;\Omega ) :=\int_{\Omega}{{1}\over{\varepsilon}} W(\nabla u)
New integral estimates for deformations in terms of their nonlinear strains
AbstractIf u is a bi-Lipschitzian deformation of a bounded Lipschitz domain Ω in ℓn (n≧2), we show that the LP norm (p≧1, p≠n) of a certain “nonlinear strain function” e(u) associated with u
Singular perturbations of variational problems arising from a two-phase transition model
AbstractGiven thatα, β are two Lipschitz continuous functions of Ω to ℝ+ and thatf(x, u, p) is a continuous function of $$\bar \Omega $$ × ℝ × ℝN to [0, + ∞[ such that, for everyx, f(x,·, 0) reaches
Higher integrability of determinants and weak convergence in L1.
Let Ω be a bounded, open subset of M and assume that u : Ω — > lf$ is in the Sobolev space W ̂ "(Ω ; AP"), i.e. \\u\\w,,n = \ \u\ + \Du\dx<vo, where Du denotes the Ω distributional gradient. Then
A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity
The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a Γ‐limit of
The gradient theory of phase transitions for systems with two potential wells
  • I. Fonseca, L. Tartar
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 1989
Synopsis In this paper we generalise the gradient theory of phase transitions to the vector valued case. We consider the family of perturbations of the nonconvex functional where W:RN→R supports two
An Introduction to-convergence
1. The direct method in the calculus of variations.- 2. Minimum problems for integral functionals.- 3. Relaxation.- 4. ?-convergence and K-convergence.- 5. Comparison with pointwise convergence.- 6.
A TWO WELL LIOUVILLE THEOREM
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
Functions of bounded deformation
We study the space BD(Ω), composed of vector functions u for which all components εij=1/2(ui, j+uj, i) of the deformation tensor are bounded measures. This seems to be the correct space for the
...
...