Rohan Abeyaratne

On the Driving Traction Acting on a Surface of Strain Discontinuity in a Continuum

The notion of the driving traction on a surface of strain discontinuity in a continuum undergoing a general thermomechanical process is defined and discussed. In addition, the associated constitutive notion of a kinetic relation, in which the normal velocity of propagation of the surface of discontinuity may be a given function of the driving traction and temperature, is introduced for the special case of a thermoelastic material.

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.

A Continuum Model of a Thermoelastic Solid Capable of Undergoing Phase Transitions

We construct explicitly a Helmholtz free energy, a kinetic relation and a nucleation criterion for a one-dimensional thermoelastic solid, capable of undergoing either mechanically- or thermally-induced phase transitions. We study the hysteretic macroscopic response predicted by this model in the case of quasistatic processes involving stress cycling at constant temperature, thermal cycling at constant stress, or a combination of mechanical and thermal loading that gives rise to the shape-memory effect. These predictions are compared qualitatively with experimental results.

Deformation of a Peridynamic Bar

The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venants principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Greens function for general loading problems.

Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids

This paper is concerned with the propagation of phase boundaries in elastic bars. It is known that the Riemann problem for an elastic bar capable of undergoing isothermal phase transitions need not have a unique solution, even in the presence of the requirement that the entropy of any particle cannot decrease upon crossing a phase boundary. For a special class of elastic materials, the authors have shown elsewhere that if all phase boundaries move subsonically with respect to both phases, this lack of uniqueness can be resolved by imposing a nucleation criterion and a kinetic relation for the relevant phase transition. Others have singled out acceptable solutions on the basis of a theory that adds effects due to viscosity and second strain gradient to the elastic part of the stress. It is shown that, for phase boundaries that propagate subsonically, this approach is equivalent to the imposition of a particular kinetic relation at the interface between the phases.

Cavitation in elastic and elastic-plastic solids

Abstract I n this study, we examine the phenomenon of cavitation under non-symmetric loading. We seek all points in (τ 1 , τ 2 , τ 3 )-stress space, such that, when the local principal true stress components (τ 1 , τ 2 , τ 3 ) at a particle reach a point on that set, cavitation ensues. This set can be described by a surface (τ 1 , τ 2 , τ 3 ) = 0 in stress space, which we refer to as a cavitation surface , and corresponds to a cavitation criterion that arises naturally from the analysis. By considering a particular subclass of the set of all kinematically admissible deformation fields, we determine an approximate analytical expression for by using the principle of virtual work. We explicitly determine and discuss the cavitation surface for a neo-Hookean material. We then consider the special case of axisymmetric cavitation corresponding to a stress state (τ 1 , τ 2 , τ 3 ), and illustrate our results for a neo-Hookean material and for a piccewisc power-law clastic-plastic material of the deformation theory . If cavitation occurs before yielding, we find that a good approximate criterion for cavitation is that it occurs when the mean stress τ m = (τ 1 + 2τ 2 )/3 reaches a critical value, even if τ 1 ≠ τ 2 ; however, if cavitation is preceded by yielding, we find that this is not a good approximation. The accuracy of our approximate analytical results is assessed by comparing them with finite element results and the results of other researchers. The utility of the cavitation surface is illustrated by applying the cavitation criterion = 0 to two experimental settings.

A One-Dimensional Continuum Model for Shape-Memory Alloys

In this paper we construct an explicit one-dimensional constitutive model that is capable of describing some aspects of the thermomechanical response of a shape-memory alloy. The model consists of a Helmholtz free-energy function, a kinetic relation and a nucleation criterion. The free-energy is associated with a three-well potential energy function; the kinetic relation is based on thermal activation theory; nucleation is assumed to occur at a critical value of the appropriate energy barrier. The predictions of the model in various quasi-static thermomechanical loadings are examined and compared with experimental observations.

On the dissipative response due to discontinuous strains in bars of unstable elastic material

Some elastic materials are capable of sustaining finite equilibrium deformations with discontinuous strains. Boundary-value problems for such “unstable” elastic materials often possess an infinite number of solutions, suggesting that the theory suffers from a constitutive deficiency. In the setting of the one-dimensional theory of bars in tension, the present paper explores the consequences of supplementing the theory with further constitutive information. This additional information pertains to the surface of strain discontinuity and consists of a ‘kinetic relation’ and a criterion for the “initiation” of such a surface. We show that the quasi-static response of the bar to a prescribed force history is then fully determined. In particular, we observe how unstable clastic materials can he used to model macroscopic behavior similar to that associated with viscoplasticity.

Cyclic effects in shape-memory alloys: a one-dimensional continuum model

Abstract We generalize the thermoelastic constitutive model of Abeyaratne and Knowles (1993, J. Mech. Phys. Solids41, 541–571), and Abeyaratne et al. (1994, Int. J. Solids Structures31, 2229–2249) so as to qualitatively model a variety of phenomena exhibited by shape-memory alloys in cyclic loading. The internal variable that is added to the model is meant to capture the idea that defects are precipitated during transformation and that these defects tend to make the nucleation of martensite easier.

Dynamics of propagating phase boundaries: Thermoelastic solids with heat conduction

This paper is concerned with the incorporation of thermal effects into the continuum modeling of dynamic solid-solid phase transitions. The medium is modeled as a one-dimensional thermoelastic solid characterized by a specific Helmholtz free-energy potential and a specific kinetic relation. Heat conduction and inertia are taken into account. An initial-value problem that gives rise to both shock waves and a propagating phase boundary is analyzed on the basis of this model.