Related papers: Optimization of the branching pattern in coherent …
We study the branching of twins appearing in shape memory alloys at the interface between austenite and martensite. In the framework of three-dimensional non-linear elasticity theory, we propose an explicit, low-energy construction of the…
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…
In materials that undergo martensitic phase transformation, macroscopic loading often leads to the creation and/or rearrangement of elastic domains. This paper considers an example {involving} a single-crystal slab made from two martensite…
We analyse features of the patterns formed from a simple model for a martensitic phase transition. This is a fragmentation model that can be encoded by a general branching random walk. An important quantity is the distribution of the…
We are interested in the energetic cost of a martensitic inclusion of volume $V$ in austenite for the cubic-to-tetragonal phase transformation. In contrast with the work of [Kn\"upfer, Kohn, Otto: Comm. Pure Appl. Math. 66 (2013), no. 6,…
The paper investigates two-phase microstructures of optimal 3D composites that store minimal elastic energy in a given strain field. The composite is made of two linear isotropic materials which differ in elastic moduli and self-strains. We…
We extend a previously studied lattice model of particles with infinite repulsions to the case of finite energy interactions. The phase diagram is studied using grand canonical Monte Carlo simulation. Simulations of dynamical phenomena are…
We discuss how searching for finite amplitude disturbances of a given energy which maximise their subsequent energy growth after a certain later time $T$ can be used to probe phase space around a reference state and ultimately to find other…
We study a simplified two-dimensional model for a cubic-to-orthorhombic phase transition occuring in certain shape-memory-alloys. In the low temperature regime the linear theory of elasticity predicts various possible patterns of martensite…
In the setting of continuum elasticity, phase transformations involving martensitic variants are modeled by a free energy density function that is non-convex in strain space. Here, we adopt an existing mathematical model in which we…
We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling regimes for the minimal energy in terms…
We study the occurrence of domain branching in a class of $(d+1)$-dimensional sharp interface models featuring the competition between an interfacial energy and a non-local field energy. Our motivation comes from branching in uniaxial…
Fine scale elastic structures are widespread in nature, for instances in plants or bones, whenever stiffness and low weight are required. These patterns frequently refine towards a Dirichlet boundary to ensure an effective load transfer.…
We study energy scaling laws for a simplified, singularly perturbed, double-well nucleation problem confined in a half-space, in the absence of gauge invariance and for an inclusion of fixed volume. Motivated by models for boundary…
Needle-like twins are observed experimentally within the transition layer at the martensite-twinned martensite interface. We utilize a phase-field approach to investigate this microstructure. Our goal is to simulate the morphology of the…
We consider two variational models for transport networks, an urban planning and a branched transport model, in which the degree of network complexity and ramification is governed by a small parameter $\varepsilon>0$. Smaller $\varepsilon$…
We analyze generic sequences for which the geometrically linear energy \[E_\eta(u,\chi):= \eta^{-\frac{2}{3}}\int_{B_{0}(1)} \left| e(u)- \sum_{i=1}^3 \chi_ie_i\right|^2 d x+\eta^\frac{1}{3} \sum_{i=1}^3 |D\chi_i|(B_{0}(1))\] remains…
Using a variational model based on non-linear elasticity we investigate whether in a cubic-to-tetragonal phase transformation it is energetically preferable to nucleate martensite within austenite. Under minimal growth assumptions on the…
We present a multi-phase design parameterization to obtain optimized heterogeneous lattice structures. The 3D domain is discretized into a cubical grid wherein each cube has eight distinct unit cell types or phases. When all phases are…
Large-scale 3D martensitic microstructure evolution problems are studied using a finite-element discretization of a finite-strain phase-field model. The model admits an arbitrary crystallography of transformation and arbitrary elastic…