1,721,013 research outputs found
Asymptotic expansions by Gamma-convergence
No full textOur starting point is a parameterized family of functionals (a 'theory') for which we are interested in approximating the global minima of the energy when one of these parameters goes to zero. The goal is to develop a set of increasingly accurate asymptotic variational models allowing one to deal with the cases when this parameter is 'small' but finite. Since Gamma-convergence may be non-uniform within the 'theory', we pose a problem of finding a uniform approximation. To achieve this goal we propose a method based on rectifying the singular points in the parameter space by using a blow-up argument and then asymptotically matching the approximations around such points with the regular approximation away from them. We illustrate the main ideas with physically meaningful examples covering a broad set of subjects from homogenization and dimension reduction to fracture and phase transitions. In particular, we give considerable attention to the problem of transition from discrete to continuum when the internal and external scales are not well separated, and one has to deal with the so-called 'size' or 'scale' effects
Cohesion-decohesion asymmetry in geckos
No full textLizards and insects can strongly attach to walls and then detach applying negligible additional forces. We propose a simple mechanical model of this phenomenon which implies active muscle control. We show that the detachment force may depend not only on the properties of the adhesive units, but also on the elastic interaction among these units. By regulating the scale of such cooperative interaction, the organism can actively switch between two modes of adhesion: delocalized (pull off) and localized (peeling)
A mechanism of transformational plasticity
No full textIn order to understand the phenomenon of reversible plasticity exhibited by shape memoryalloys and other smart materials, we study an elementary prototypical model. Building on anoriginal idea of M ̈uller and Villaggio [17], we consider an inhomogeneous ensemble of bi-stableelements connected in series and loaded in a soft device. To interpret the fine structure of thehysteresis loops observed experimentally, we assume that the dynamics is maximally dissipativeand investigate different evolutionary strategies for a “driven” system with external forcechanging quasi-statically. Our main result is that the inhomogeneity of the elastic propertiesleads to a distinctive hardening with serrations of a Portevin-Le Chatelier type and producesa realistic memory structure characterized by the “congruency” and “return point memory”propertie
Elastic crystals with a triple point
No full textThe 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
Non-hydrostatic stabilization of an orthorhombic phase of zirconia
No full textAn explicit polynomial strain-energy function for tetragonal-orthorhombic-monoclinic zirconia (ZrO2), calibrated from the conventional hydrostatic p-T phase diagram, is used to study the effects of nonhydrostatic loading on the phase equilibria in this material. Several representative sections of the phase diagram of ZrO2 in temperature and stress space, containing both triple and critical points, are computed. A new orthorhombic structure of ZrO2 is predicted to be the most stable phase for a variety of experimentally accessible shear loads, in a wide range of temperatures and pressures
Unified Landau description of tetragonal-orthorhombic-monoclinic zirconia
No full textWe compute an explicit lowest-order polynomial form of the strain-dependent Gibbs potential which provides a unified description of the tetragonal, orthorhombic, and monoclinic phases of zirconia (ZrO2). The resulting energy function interpolates well the available experimental data for this material, reproducing its known elastic moduli, equilibrium strains, and phase diagram to about 1700 K and 8 GPa
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