Interfacial and inhomogeneity penalties in phase transitions

Non-convex free energies permit phase transitions to occur. The ensuing state of a body is non-homogeneous and endowed with interphase boundaries. Both the inhomogeneity and the interfaces may contribute to the free energy and thus affect the onset of the phase transition. The paper investigates these effects in a one-dimensional setting and for deformation control. The main conclusion is that the incipient phase mixture is characterized by a stable kernel of small but finite phase fraction. This kernel must not be confused with the unstable nucleus whose energy maximum must be overcome before the kernel can form. We consider also the energy landscape of partial equilibria in which the load is uniform but the phase fraction and the number of interfaces have not yet reached equilibrium.Received: 6 July 2002, Accepted: 18 February 2003, Published online: 9 May 2003PACS: 64.60.-i     

The Influence of Moduli Slope of a Linearly Graded Matrix on the Bulk Moduli of Some Particle- and Fiber-Reinforced Composites

The stored energy functional of a homogeneous isotropic elastic body is invariant with respect to translation and rotation of a reference configuration. One can use Noether's Theorem to derive the conservation laws corresponding to these invariant transformations. These conservation laws provide an alternative way of formulating the system of equations governing equilibrium of a homogeneous isotropic body. The resulting system is mathematically identical to the system of equilibrium equations and constitutive relations, generally, of another material. This implies that each solution of the system of equilibrium equations gives rise to another solution, which describes the reciprocal deformation and solves the system of equilibrium equations of another material. In this paper we derive conservation laws and prove the theorem on conjugate solutions for two models of elastic homogeneous isotropic bodies – the model of a simple material and the model of a material with couple stress (Cosserat continuum). This revised version was published online in August 2006 with corrections to the Cover Date.     

Stability of a thick elastic plate under thrust

When a rectangular plate of incompressible neo-Hookean elastic material is subjected to a thrust, bifurcations of the flexural or barreling types become possible at certain critical values of the compression ratio. The states of pure homogeneous deformation corresponding to these critical compression ratios are states of neutral equilibrium. Their stability is investigated on the basis of an energy criterion, without restriction on the thickness of the plate.The critical state corresponding to the lowest order flexural mode is found to be stable (unstable) if the aspect ratio (thickness/length) is less (greater) than about 0.2. Agreement with the classical Euler theory is established in the limiting case of small aspect ratio.     

Thin-plate theory for large elastic deformations

Non-linear plate theory for thin prismatic elastic bodies is obtained by estimating the total three-dimensional strain energy generated in response to a given deformation in terms of the small plate thickness. The Euler equations for the estimate of the energy are regarded as the equilibrium equations for the thin plate. Included among them are algebraic formulae connecting the gradients of the midsurface deformation to the through-thickness derivatives of the three-dimensional deformation. These are solvable provided that the three-dimensional strain energy is strongly elliptic at equilibrium. This framework yields restrictions of the Kirchhoff-Love type that are usually imposed as constraints in alternative formulations. In the present approach they emerge as consequences of the stationarity of the energy without the need for any a priori restrictions on the three-dimensional deformation apart from a certain degree of differentiability in the direction normal to the plate.     

Equilibrium of Phases with Interfacial Energy: A Variational Approach

The paper proves the existence of equilibrium two phase states with elastic solid bulk phases and deformation dependent interfacial energy. The states are pairs (y,E) consisting of the deformation y on the body and the region E occupied by one of the phases in the reference configuration. The bulk energies of the two phases are polyconvex functions representing two wells of the substance. The interfacial energy is interface polyconvex. The last notion is introduced and discussed below, together with the interface quasiconvexity and interface null Lagrangians. The constitutive theory and equilibrium theory of the interface are discussed in detail under appropriate smoothness hypotheses. Various forms of the interfacial stress relations for the standard and configurational (Eshelby) interfacial stresses are established. The equilibrium equations are derived by a variational argument emphasizing the roles of outer and inner variations.     

On the modelling of anisotropic material behaviour in viscoplasticity

A uniaxial viscoplastic deformation is motivated as a discrete sequence of stable and unstable equilibrium states and approximated by a smooth family of stable states of equilibrium depending on the history of the mechanical process. Three-dimensional crystal viscoplasticity starts from the assumption that inelastic shearings take place on slip systems, which are known from the particular geometric structure of the crystal. A constitutive model for the behaviour of a single crystal is developed, based on a free energy, which decomposes into an elastic and an inelastic part. The elastic part, the isothermal strain energy, depends on the elastic Green strain and allows for the initial anisotropy, known from the special type of the crystal lattice. Additionally, the strain energy function contains an orthogonal tensor-valued internal variable representing the orientation of the anisotropy axes. This orientation develops according to an evolution equation, which satisfies the postulate of full invariance in the sense that it is an observer-invariant relation. The inelastic part of the free energy is a quadratic function of the integrated shear rates and corresponding internal variables being equivalent to backstresses in order to consider kinematic hardening phenomena on the slip system level. The evolution equations for the shears, backstresses and crystallographic orientations are thermomechanically consistent in the sense that they are compatible with the entropy inequality. While the general theory applies to all types of lattices, specific test calculations refer to cubic symmetry (fcc) and small elastic strains. The simulations of simple tension and compression processes of a single crystal illustrates the development of the crystallographic axes according to the proposed evolution equation. In order to simulate the behaviour of a polycrystal the initial orientations of the anisotropy axes are assumed to be space-dependent but piecewise constant, where each region of a constant orientation corresponds to a grain. The results of the calculation show that the initially isotropic distribution of the orientation changes in a physically reasonable manner and that the intensity of this process-induced texture depends on the specific choice of the material constants.     

Mechanical experiments to identify homogeneous bodies

All bodies are inhomogeneous at some scale but experience has shown that some of these bodies can be idealized as a homogeneous body. Here we examine which bodies can be idealized as a homogeneous body when they are subjected to a non-dissipative mechanical process. This is done by studying circumstances in which an inhomogeneous body admits pure stretch homogeneous deformations. Then, we devise experiments wherein these circumstances are prevented. If homogeneous deformation is observed in these devised experiments, the body could be modeled as a homogeneous body. We limit our analysis to a class of isotropic elastic bodies deforming from a stress free reference configuration whose Cauchy stress is explicitly related to left Cauchy–Green deformation tensor. It is further assumed that the constitutive relation is differentiable function of the position vector of material particles in the stress free reference configuration. Then, we find that a cuboid made of compressible and isotropic material could be modeled as a homogeneous body if it deforms homogeneously due to the application of the normal stresses on all of its six faces and the magnitude of the normal stresses on three orthogonal faces are different. A cuboid made of incompressible and isotropic material could be modeled as a homogeneous body, if it deforms homogeneously in two different biaxial experiments, such that the plane in which the forces are applied in the two biaxial experiments is mutually orthogonal.     

Interaction between phase transformations and dislocations at the nanoscale. Part 1. General phase field approach

Thermodynamically consistent, three-dimensional (3D) phase field approach (PFA) for coupled multivariant martensitic transformations (PTs), including cyclic PTs, variant–variant transformations (i.e., twinning), and dislocation evolution is developed at large strains. One of our key points is in the justification of the multiplicative decomposition of the deformation gradient into elastic, transformational, and plastic parts. The plastic part includes four mechanisms: dislocation motion in martensite along slip systems of martensite and slip systems of austenite inherited during PT and dislocation motion in austenite along slip systems of austenite and slip systems of martensite inherited during reverse PT. The plastic part of the velocity gradient for all these mechanisms is defined in the crystal lattice of the austenite utilizing just slip systems of austenite and inherited slip systems of martensite, and just two corresponding types of order parameters. The explicit expressions for the Helmholtz free energy and the transformation and plastic deformation gradients are presented to satisfy the formulated conditions related to homogeneous thermodynamic equilibrium states of crystal lattice and their instabilities. In particular, they result in a constant (i.e., stress- and temperature-independent) transformation deformation gradient and Burgers vectors. Thermodynamic treatment resulted in the determination of the driving forces for change of the order parameters for PTs and dislocations. It also determined the boundary conditions for the order parameters that include a variation of the surface energy during PT and exit of dislocations. Ginzburg–Landau equations for dislocations include variation of properties during PTs, which in turn produces additional contributions from dislocations to the Ginzburg–Landau equations for PTs. A complete system of coupled PFA and mechanics equations is presented. A similar theory can be developed for PFA to dislocations and other PTs, like reconstructive PTs and diffusive PTs described by the Cahn–Hilliard equation, as well as twinning and grain boundaries evolution.     

Constitutive branching analysis of cylindrical bodies under in-plane equibiaxial dead-load tractions

Finite homogeneous deformations of hyperelastic cylindrical bodies subjected to in-plane equibiaxial dead-load tractions are analyzed. Four basic equilibrium problems are formulated considering incompressible and compressible isotropic bodies under plane stress and plane deformation condition. Depending on the form of the stored energy function, these plane problems, in addition to the obvious symmetric solutions, may admit asymmetric solutions. In other words, the body may assume an equilibrium configuration characterized by two unequal in-plane principal stretches corresponding to equal external forces. In this paper, a mathematical condition, in terms of the principal invariants, governing the global development of the asymmetric deformation branches is obtained and examined in detail with regard to different choices of the stored energy function. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. With reference to neo-Hookean, Mooney-Rivlin and Ogden-Ball materials, a broad numerical analysis is performed and the qualitatively more interesting asymmetric equilibrium branches are shown. Finally, using the energy criterion, a number of considerations are put forward about the stability of the computed solutions.     

Thermomechanical modeling of the thermo-order-mechanical coupling behaviors in liquid crystal elastomers

Liquid crystal elastomer is a kind of anisotropic polymeric material, with complicated micro-structures and thermo-order-mechanical coupling behaviors. In this paper, we propose a method to systematically model these coupling behaviors. We derive the constitutive model in full tensor structure according to the Clausius-Duhem inequality. Two of the constitutive equations represent the mechanical equilibrium and the other two represent the phase equilibrium. Choosing the total free energy as the combination of the neo-classical free energy and the Landau-de Gennes nematic free energy, we obtain the Cauchy stress-deformation gradient relation and the order-mechanical coupling equations. We find the analytical homogeneous solutions of the deformation for the typical mechanical loadings, such as uniaxial stretch, and simple shear in any directions. We also compare the compression behavior of prolate liquid crystal elastomers with the stretch behavior of oblate liquid crystal elastomers. As a result, the stress, strain, temperature, order parameter, biaxiality and the direction of the director of liquid crystal elastomers couple with each other. When the prolate liquid crystal elastomer sample is stretched in the direction parallel to its director, the deviatoric stress makes the mesogens more order and increase the transition temperature. When the sample is sheared or stretched in the direction non-parallel to the director, the director of the liquid crystal elastomer will rotate, and the biaxiality will be induced. Because of the order-mechanical coupling, under infinitesimal deformation, liquid crystal elastomer has anisotropic Young’s modulus and zero shear modulus in the direction parallel or perpendicular to the director. While for the oblate liquid crystal elastomers, the stretch parallel to the director will cause the rotation of the director and induce the biaxiality.     

Constraints, Constitutive Limits, and Instability in Finite Thermoelasticity

This paper examines all the possible types of thermomechanical constraints in finite-deformational elasticity. By a thermomechanical constraint we mean a functional relationship between a mechanical variable, either the deformation gradient or the stress, and a thermal variable, temperature, entropy or one of the energy potentials; internal energy, Helmholtz free energy, Gibbs free energy or enthalpy. It is shown that for the temperature-deformation, entropy-stress, enthalpy-deformation, and Helmholtz free energy-stress constraints equilibrium states are unstable, in the sense that certain perturbations of the equilibrium state grow exponentially. By considering the constrained materials as constitutive limits of unconstrained materials, it is shown that the instability is associated with the violation of the Legendre–Hadamard condition on the internal energy. The entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints do not exhibit this instability. It is proposed that stability of the rest state (or equivalently convexity of internal energy) is a necessary requirement for a model to be physically valid, and hence entropy-deformation, temperature-stress, internal energy-stress, and Gibbs free energy-deformation constraints are physical, whereas temperature-deformation constraints (including the customary notion of thermal expansion that density is a function of temperature only), entropy-stress constraints, enthalpy-deformation constraints, and Helmholtz free energy-stress constraints are not.     

Moving interfaces that separate loose and compact phases of elastic aggregates: a mechanism for drastic reduction or increase in macroscopic deformation

Granular materials such as sand may be viewed as continuous bodies composed of much smaller elastic bodies. The multiscale geometry of structured deformations captures the contribution at the macrolevel of the smooth deformation of each small body in the aggregate (deformation without disarrangements) as well as the contribution at the macrolevel of the non-smooth deformations such as slips and separations between the small bodies in the aggregate (deformation due to disarrangements). When the free energy response of the aggregate depends only upon the deformation without disarrangements, is isotropic, and possesses standard growth and semi-convexity properties, we establish (i) the existence of a compact phase in which every small elastic body deforms in the same way as the aggregate and, when the volume change of macroscopic deformation is sufficiently large, (ii) the existence of a loose phase in which every small elastic body expands and rotates to achieve a stress-free state with accompanying disarrangements in the aggregate. We show that a broad class of elastic aggregates can admit moving surfaces that transform material in the compact phase into the loose phase and vice versa and that such transformations entail drastic changes in the level of deformation of transforming material points.     

Effect of Coulomb Damping on Buckling of a Two-Rod System

This paper describes a significant influence of a slight Coulomb damping on buckling, using a simple two rods system. Coulomb damping produces equilibrium regions around the well-known stable and unstable steady states under the pitchfork bifurcation which occurs in the case without Coulomb damping. Also, the stability of the states in the equilibrium regions is examined by using the phase portrait. As a consequence, due to the slight Coulomb damping, it is theoretically clarified that the states in the equilibrium regions are locally stable, even in the neighborhood of the unstable steady states under the pitchfork bifurcation in the case without Coulomb damping, i.e., even in the neighborhood of the unstable trivial steady states in the postbuckling and the unstable nontrivial steady states under the subcritical pitchfork bifurcation. Furthermore, the experimental results are in qualitative agreement with the theoretically predicted phenomena.     

Determination of the ultimate states of elastoplastic bodies

The equations of quasistatic deformation of elastoplastic bodies are considered in a geometrical linear formulation. After discretization of the equations with respect to spatial variables by the finite-element method, the problem of determining equilibrium onfigurations reduces to integration of a system of nonlinear ordinary differential equations. In the ultimate state of a body of an ideal elastoplastic material, the matrix of the system degenerates and the problem becomes singular. A regularization algorithm for determining solutions of the problems for the ultimate states of bodies is proposed. Numerical solutions of test problems of determining the ultimate loads and equilibrium configurations for ideal elastoplastic bodies confirm the reliability of the regularization algorithm proposed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 5, pp. 196–204, September–October, 2000.     

Toward a Field Theory for Elastic Bodies Undergoing Disarrangements

Structured deformations are used to refine the basic ingredients of continuum field theories and to derive a system of field equations for elastic bodies undergoing submacroscopically smooth geometrical changes as well as submacroscopically non-smooth geometrical changes (disarrangements). The constitutive assumptions employed in this derivation permit the body to store energy as well as to dissipate energy in smooth dynamical processes. Only one non-classical field G, the deformation without disarrangements, appears in the field equations, and a consistency relation based on a decomposition of the Piola-Kirchhoff stress circumvents the use of additional balance laws or phenomenological evolution laws to restrict G. The field equations are applied to an elastic body whose free energy depends only upon the volume fraction for the structured deformation. Existence is established of two universal phases, a spherical phase and an elongated phase, whose volume fractions are (1?γ₀)³ and (1?γ₀) respectively, with γ₀:=( $ \sqrt 5 $ ?1)/2 the “golden mean”.     

Complete equilibrium paths for Mises trusses

This paper presents determination of equilibrium paths for Mises trusses with different ratio of height to span. Unsymmetrical deformation modes are considered and the structure is treated as a two DOF system. First, a few special equilibrium configurations are resolved from considerations of free body diagrams. Complete equilibrium paths are determined by solving numerically the governing non-linear equilibrium equations. The stability of possible equilibrium configurations is checked using the second partial derivative test for the total potential energy. The positive definiteness of the appropriate Hessian matrices is checked numerically using the Sylvester criterion.     

Parametric resonances in a base-excited double pendulum

Two parametrically-induced phenomena are addressed in the context of a double pendulum subject to a vertical base excitation. First, the parametric resonances that cause the stable downward vertical equilibrium to bifurcate into large-amplitude periodic solutions are investigated extensively. Then the stabilization of the unstable upward equilibrium states through the parametric action of the high-frequency base motion is documented in the experiments and in the simulations. It is shown that there is a region in the plane of the excitation frequency and amplitude where all four unstable equilibrium states can be stabilized simultaneously in the double pendulum. The parametric resonances of the two modes of the base-excited double pendulum are studied both theoretically and experimentally. The transition curves (i.e., boundaries of the dynamic instability regions) are constructed asymptotically via the method of multiple scales including higher-order effects. The bifurcations characterizing the transitions from the trivial equilibrium to the periodic solutions are computed by either continuation methods and or by time integration and compared with the theoretical and experimental results.     

Criteria for Floating I

Conditions are determined under which solid bodies will float on a liquid surface in stable equilibrium, under the influence of gravity and of surface tension. These include configurations in which the density of the body exceeds the density of the ambient liquid, so that for an infinitely deep liquid in a downward gravity field there is no absolute energy minimum. Of notable interest are the results (a) that if a smooth body is held rigidly and translated downward into an infinite fluid bath through a family of fluid equilibrium configurations in a downward gravity field, the transition is necessarily discontinuous, and (b) a formal proof that there can be a free-floating locally energy minimizing configuration that does not globally minimize, even if the density of the body exceeds that of the liquid. The present work is limited to the two dimensional case corresponding to a long cylinder that is floating horizontally. The more physical three-dimensional case can be studied in a similar way, although details of behavior can change significantly. That work will appear in an independent study written jointly with T. I. Vogel.     

Spherically symmetric two-phase deformations and phase transition zones

This paper is concerned with characterization and stability assessment of two-phase spherically symmetric deformations that can be supported by a nonlinear elastic isotropic material. We study general properties of equilibrium two-phase spherically symmetric deformations. Then we specialize to phase transformations of a solid sphere that is subjected to an all-round tension/pressure. Two material models are used to demonstrate a variety of transformation behaviours and some common features. For both materials we construct phase transition zones (PTZs) formed in the space of principal stretches by those which can exist adjacently to an equilibrium interface. Then we demonstrate how the PTZ can be used for the prediction of the number of two-phase spherically symmetric solutions and study how the deformation field associated with each solution is related to the PTZ. We show that even in the simplest case of one interface the solution is not unique: two equilibrium two-phase solutions as well as one uniform one-phase solution are found under the same boundary conditions. For the three solutions we construct their load-deformation diagrams and compare the associated total energies. The stability of the two-phase states with respect to radial and small-wavelength perturbations is also examined. We observe how unstable solutions are related with the PTZ.     

On the Lagrangian and instability of medium with defects

The article presents the Lagrangian of defects in the solids, equipped with bending and warp. The deformation of such elastic medium with defects is based on Riemann-Cartan geometry in three dimensional space. In the static theory for the media with dislocations and disclinations the possible choice of the geometric Lagrangian yield the equations of equilibrium. In this article, the assumed expression for the free energy leading is equal to a volume integral of the scalar function (the Lagrangian) that depends on metric and Ricci tensors only. In the linear elastic isotropic case the elastic potential is a quadratic function of the first and second invariants of strain and warp tensors with two Lame, two mixed and two bending constants. For the linear theory of homogeneous anisotropic elastic medium the elastic potential must be quadratic in warp and strain. The conditions of stability of media with defects are derived, such that the medium in its free state is stable. With the increasing strain the stability conditions could be violated. If the strain in material attains the critical value, the instability in form of emergence of new topological defects occurs. The medium undergoes the spontaneous symmetry breaking in form of emerging topological defects.