The origin of nucleation peak in transformational plasticity

A typical stress-strain relation for martensitic materials exhibits a mismatch between the nucleation and propagation thresholds leading to the formation of the nucleation peak. We develop an analytical model of this phenomenon and obtain specific relations between the macroscopic parameters of the peak and the microscopic characteristics of the material. Although the nucleation peak appears in the model as an interplay between discreteness and nonlocality, it does not disappear in the continuum limit. We verify the quantitative predictions of the model by comparison with experimental data for cubic to monoclinic phase transformation in NiTi.     

Martensitic transformations: from continuum mechanics to spin models and automata

We present a new procedure for the systematic reduction of a continuum theory of martensitic transformations to a spin system whose dynamics can be described by an automaton. Our prototypical model reproduces most of the experimental observations in martensites associated with criticality and power-law acoustic emission. In particular, it explains in a natural way why cyclic training is necessary to reach scale-free behavior.      

Finite Scale Microstructures in Nonlocal Elasticity

In this paper we develop a simple one-dimensional model accounting for the formation and growth of globally stable finite scale microstructures. We extend Ericksen's model [9] of an elastic “bar” with nonconvex energy by including both oscillation-inhibiting and oscillation-forcing terms in the energy functional. The surface energy is modeled by a conventional strain gradient term. The main new ingredient in the model is a nonlocal term which is quadratic in strains and has a negative definite kernel. This term can be interpreted as an energy associated with the long-range elastic interaction of the system with the constraining loading device. We propose a scaling of the problem allowing one to represent the global minimizer as a collection of localized interfaces with explicitly known long-range interaction. In this limit the augmented Ericksen's problem can be analyzed completely and the equilibrium spacing of the periodic microstructure can be expressed as a function of the prescribed average displacement. We then study the inertial dynamics of the system and demonstrate how the nucleation and growth of the microstructures result in the predicted stable pattern. Our results are particularly relevant for the modeling of twined martensite inside the austenitic matrix.     

Quasicontinuum Models of Dynamic Phase Transitions

We propose a series of quasicontinuum approximations for the simplest lattice model of a fully dynamic martensitic phase transition in one dimension. The approximations are dispersive and include various non-classical corrections to both kinetic and potential energies. We show that the well-posed quasicontinuum theory can be constructed in such a way that the associated closed-form kinetic relation is in an excellent agreement with the predictions of the discrete theory.     

Roughening Instability of Broken Extremals

We derive a new general jump condition on a broken Weierstrass–Erdmann extremal of a vectorial variational problem. Such extremals, containing surfaces of gradient discontinuity, are ubiquitous in shape optimization and in the theory of elastic phase transformations. The new condition, which does not have a one dimensional analog, reflects the stationarity of the singular surface with respect to two-scale variations that are nontrivial generalizations of Weierstrass needles. The over-determinacy of the ensuing free boundary problem suggests that typical stable solutions must involve microstructures or chattering controls.     

Elastic crystals with a triple point

The peculiar behavior of active crystals is due to the presence of evolving phase mixtures the variety of which depends on the number of coexisting phases and the multiplicity of symmetry-related variants. According to Gibbs’ phase rule, the number of phases in a single-component crystal is maximal at a triple point in the p-T phase diagram. In the vicinity of this special point the number of metastable twinned microstructures will also be the highest—a desired effect for improving performance of smart materials. To illustrate the complexity of the energy landscape in the neighborhood of a triple point, and to produce a workable example for numerical simulations, in this paper we construct a generic Landau strain-energy function for a crystal with the coexisting tetragonal (t), orthorhombic (o), and monoclinic (m) phases. As a guideline, we utilize the experimental observations and crystallographic data on the t-o-m transformations of zirconia (ZrO₂), a major toughening agent for ceramics. After studying the kinematics of the t-o-m phase transformations, we re-evaluate the available experimental data on zirconia polymorphs, and propose a new mechanism for the technologically important t-m transition. In particular, our proposal entails the softening of a different tetragonal modulus from the one previously considered in the literature. We derive the simplest expression for the energy function for a t-o-m crystal with a triple point as the lowest-order polynomial in the relevant strain components, exhibiting the complete set of wells associated with the t-o-m phases and their symmetry-related variants. By adding the potential of a hydrostatic loading, we study the p-T phase diagram and the energy landscape of our crystal in the vicinity of the t-o-m triple point. We show that the simplest assumptions concerning the order-parameter coupling and the temperature dependence of the Landau coefficients produce a phase diagram that is in good qualitative agreement with the experimental diagram of ZrO₂.     

Dynamics of martensitic phase boundaries: discreteness, dissipation and inertia

We study a fully inertial model of a martensitic phase transition in a one-dimensional crystal lattice with long-range interactions. The model allows one to represent a broad range of dynamic regimes, from underdamped to overdamped. We systematically compare the discrete model with its various continuum counterparts including elastic, viscoelastic and viscosity-capillarity models. Each of these models generates a particular kinetic relation which links the driving force with the phase boundary velocity. We find that the viscoelastic model provides an upper bound for the critical driving force predicted by the discrete model, while the viscosity-capillarity model delivers a lower bound. We show that at near-sonic velocities, where inertia dominates dispersion, both discrete and continuum models behave qualitatively similarly. At small velocities, and in particular near the depinning threshold, the discreteness prevails and predictions of the continuum models cannot be trusted.      

Normality Condition in Elasticity

Strong local minimizers with surfaces of gradient discontinuity appear in variational problems when the energy density function is not rank-one convex. In this paper we show that the stability of such surfaces is related to the stability outside the surface via a single jump relation that can be regarded as an interchange stability condition. Although this relation appears in the setting of equilibrium elasticity theory, it is remarkably similar to the well-known normality condition that plays a central role in classical plasticity theory.