# 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} }

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

- MathematicsAdvances in Calculus of Variations
- 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…

Separately global solutions to rate-independent processes in large-strain inelasticity

- MathematicsNonlinear Analysis
- 2022

Asymptotic Analysis of Deformation Behavior in High-Contrast Fiber-Reinforced Materials: Rigidity and Anisotropy

- MathematicsMathematical 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

- PhysicsArchive 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

- MathematicsAdvances in Mechanics and 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

- MathematicsMathematical 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

- Art, ChemistryArchive 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

- MathematicsSIAM 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 ScienceProceedings 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

- MathematicsArchive 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

- Mathematics
- 2016

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

- MathematicsProceedings 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

- Mathematics
- 1987

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…