# Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions

@article{Davoli2020TwowellRA,
title={Two-well rigidity and multidimensional sharp-interface limits for solid–solid phase transitions},
author={Elisa Davoli and Manuel Friedrich},
journal={Calculus of Variations and Partial Differential Equations},
year={2020}
}
• Published 15 October 2018
• Mathematics
• Calculus of Variations and Partial Differential Equations
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 rigidity result, we then analyze solid-solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of $\Gamma$-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an…
14 Citations
Two-well linearization for solid-solid phase transitions
• Mathematics
• 2020
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
Homogenization in BV of a model for layered composites in finite crystal plasticity
• Mathematics
• 2019
Abstract In this work, we study the effective behavior of a two-dimensional variational model within finite crystal plasticity for high-contrast bilayered composites. Precisely, we consider materials
Minimal Energy for Geometrically Nonlinear Elastic Inclusions in Two Dimensions
• Mathematics
• 2022
. We are concerned with a variant of the isoperimetric problem, which in our setting arises in a geometrically nonlinear two-well problem in elasticity. More precisely, we investigate the optimal
Asymptotic Analysis of Deformation Behavior in High-Contrast Fiber-Reinforced Materials: Rigidity and Anisotropy
• Mathematics
Mathematical Models and Methods in Applied Sciences
• 2022
We identify the restricted class of attainable effective deformations in a model of reinforced composites with parallel, long, and fully rigid fibers embedded in an elastic body. In mathematical
Emergence of Rigid Polycrystals from Atomistic Systems with Heitmann–Radin Sticky Disk Energy
• Physics
Archive for Rational Mechanics and Analysis
• 2020
We investigate the emergence of rigid polycrystalline structures from atomistic particle systems. The atomic interaction is governed by a suitably normalized pair interaction energy, where the
Gradient Polyconvexity and Modeling of Shape Memory Alloys
• Mathematics
• 2021
We show existence of an energetic solution to a model of shape memory alloys in which the elastic energy is described by means of a gradient-polyconvex functional. This allows us to show existence of
Geometric rigidity in variable domains and derivation of linearized models for elastic materials with free surfaces
• Mathematics
• 2021
We present a quantitative geometric rigidity estimate in dimensions d = 2, 3 generalizing the celebrated result by Friesecke, James, and Müller [49] to the setting of variable domains. Loosely
Convex integration solutions for the geometrically nonlinear two-well problem with higher Sobolev regularity
• Mathematics
Mathematical Models and Methods in Applied Sciences
• 2020
In this paper, we discuss higher Sobolev regularity of convex integration solutions for the geometrically nonlinear two-well problem. More precisely, we construct solutions to the differential
Rigidity of Branching Microstructures in Shape Memory Alloys
• Thilo M. Simon
• Art, Chemistry
Archive for Rational Mechanics and Analysis
• 2021
It is shown how generic sequences for which the geometrically linear energy equations are analyzed can be transformed into discrete discrete-time solutions.

## References

SHOWING 1-10 OF 69 REFERENCES
A Sharp-Interface Limit for a Two-Well Problem in Geometrically Linear Elasticity
• Mathematics
• 2006
AbstractIn the theory of solid-solid phase transitions the deformation of an elastic body is determined via a functional containing a nonconvex energy density and a singular perturbation. We study
Multiwell Rigidity in Nonlinear Elasticity
• Mathematics
SIAM J. Math. Anal.
• 2010
A quantitative rigidity estimate is derived for a multiwell problem in nonlinear elasticity that holds for any connected subdomain and has the optimal scaling.
A quantitative rigidity result for the cubic-to-tetragonal phase transition in the geometrically linear theory with interfacial energy
• Materials Science
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2012
We are interested in the cubic-to-tetragonal phase transition in a shape memory alloy. We consider geometrically linear elasticity. In this framework, Dolzmann and Müller have shown that the only
A Compactness and Structure Result for a Discrete Multi-well Problem with SO(n) Symmetry in Arbitrary Dimension
• Mathematics
Archive for Rational Mechanics and Analysis
• 2018
In this note we combine the “spin-argument” from Kitavtsev et al. (Proc R Soc Edinb Sect A Mater 147(5):1041–1089, 2017) and the n-dimensional incompatible, one-well rigidity result from Lauteri and
Surface Energies Arising in Microscopic Modeling of Martensitic Transformations
• Mathematics
• 2014
In this paper we construct and analyze a two-well Hamiltonian on a 2D atomic lattice. The two wells of the Hamiltonian are prescribed by two rank-one connected martensitic twins, respectively. By
The Cubic-to-Orthorhombic Phase Transition: Rigidity and Non-Rigidity Properties in the Linear Theory of Elasticity
In this paper we investigate the cubic-to-orthorhombic phase transition in the framework of linear elasticity. Using convex integration techniques, we prove that this phase transition represents one
Surface energies emerging in a microscopic, two-dimensional two-well problem
• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2017
In this paper we are interested in the microscopic modelling of a two-dimensional two-well problem that arises from the square-to-rectangular transformation in (two-dimensional) shape-memory
The gradient theory of phase transitions and the minimal interface criterion
In this paper I prove some conjectures of GURTIN [15] concerning the Van der Waals-Cahn-Hilliard theory of phase transitions. Consider a fluid, under isothermal conditions and confined to a bounded
Rigidity and gamma convergence for solid‐solid phase transitions with SO(2) invariance
• Mathematics
• 2006
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) +
Geometric rigidity for incompatible fields, and an application to strain-gradient plasticity
• Mathematics
• 2014
In this paper, we show that a strain-gradient plasticity model arises as the Γ -limit of a nonlinear semi-discrete dislocation energy. We restrict our analysis to the case of plane elasticity, so