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Register a new userDeformation concentration for martensitic microstructures in the limit of low volume fraction
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We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $Gamma$-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $SBV^p$ functions whose jump sets have a prescribed orientation.
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We study microstructure formation in two nonconvex singularly-perturbed variational problems from materials science, one modeling austenite-martensite interfaces in shape-memory alloys, the other one slip structures in the plastic deformation of crystals. For both functionals we determine the scaling of the optimal energy in terms of the parameters of the problem, leading to a characterization of the mesoscopic phase diagram. Our results identify the presence of a new phase, which is intermediate between the classical laminar microstructures and branching patterns. The new phase, characterized by partial branching, appears for both problems in the limit of small volume fraction, that is, if one of the variants (or of the slip systems) dominates the picture and the volume fraction of the other one is small.
Imposing either Dirichlet or Neumann boundary conditions on the boundary of a smooth bounded domain $Omega$, we study the perturbation incurred by the voltage potential when the conductivity is modified in a set of small measure. We consider $left(gamma_{n}right)_{ninmathbb{N}}$, a sequence of perturbed conductivity matrices differing from a smooth $gamma_{0}$ background conductivity matrix on a measurable set well within the domain, and we assume $left(gamma_{n}-gamma_{0}right)gamma_{n}^{-1}left(gamma_{n}-gamma_{0}right)to0$ in $L^{1}(Omega)$. Adapting the limit measure, we show that the general representation formula introduced for bounded contrasts in citep{capdeboscq-vogelius-03a} can be extended to unbounded sequencesof matrix valued conductivities.
In this work we introduce a new system of partial differential equations as a simplified model for the evolution of reversible martensitic transformations under thermal cycling in low hysteresis alloys. The model is developed in the context of nonlinear continuum mechanics, where the developed theory is mostly static, and cannot capture the influence of dynamics on martensitic microstructures. First, we prove existence of weak solutions; secondly, we study the physically relevant limit when the interface energy density vanishes, and the elastic constants tend to infinity. The limit problem provides a framework for the moving mask approximation recently introduced by the author. In the last section we study the limit equations in a one-dimensional setting. After closing the equations with a constitutive relation between the phase interface velocity and the temperature of the one-dimensional sample, the equations become a two-phase Stefan problem with a kinetic condition at the free boundary. Under some further assumptions, we show that the phase interface reaches the domain boundary in finite time.
In this work we introduce a moving mask approximation to describe the dynamics of austenite to martensite phase transitions at a continuum level. In this framework, we prove a new type of Hadamard jump condition, from which we deduce that the deformation gradient must be of the form $mathsf{1} +mathbf{a}otimes mathbf{n}$ a.e. in the martensite phase. This is useful to better understand the complex microstructures and the formation of curved interfaces between phases in new ultra-low hysteresis alloys such as Zn45Au30Cu25, and provides a selection mechanism for physically-relevant energy-minimising microstructures. In particular, we use the new type of Hadamard jump condition to deduce a rigidity theorem for the two well problem. The latter provides more insight on the cofactor conditions, particular conditions of supercompatibility between phases believed to influence reversibility of martensitic transformations.
We analyze generic sequences for which the geometrically linear energy [E_eta(u,chi):= eta^{-frac{2}{3}}int_{B_{0}(1)} left| e(u)- sum_{i=1}^3 chi_ie_iright|^2 d x+eta^frac{1}{3} sum_{i=1}^3 |Dchi_i|(B_{0}(1))] remains bounded in the limit $eta to 0$. Here $ e(u) :=1/2(Du + Du^T)$ is the (linearized) strain of the displacement $u$, the strains $e_i$ correspond to the martensite strains of a shape memory alloy undergoing cubic-to-tetragonal transformations and $chi_i:B_{0}(1) to {0,1}$ is the partition into phases. In this regime it is known that in addition to simple laminates also branched structures are possible, which if austenite was present would enable the alloy to form habit planes. In an ansatz-free manner we prove that the alignment of macroscopic interfaces between martensite twins is as predicted by well-known rank-one conditions. Our proof proceeds via the non-convex, non-discrete-valued differential inclusion [e(u) in bigcup_{1leq i eq jleq 3} operatorname{conv} {e_i,e_j}] satisfied by the weak limits of bounded energy sequences and of which we classify all solutions. In particular, there exist no convex integration solutions of the inclusion with complicated geometric structures.
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