Thomas J. Pence

A Thermomechanical Model for a One Variant Shape Memory Material

A model is examined for thermoelastic materials, such as those that display the shape memory and pseudoelastic effect. As is common with models for these materials, an internal variable is utilized which gives the phase fraction of austenite at the microstructural level within the continua. Evolution equations are developed that govern the time history of the system based on changes in stress, strain and temperature. Hysteresis is inherent in the system due to the Duhem- Madelung form of one of these equations. Attention is focused on the connection between complete transformation phenomena and arrested loading/unloading (outer loops vs. subloops); the transition from isothermal to adiabatic loading via loading in a heat convective environment; the identification of attracting states associated with both temperature cycling and periodic stressing; and the deter mination of mathematical restrictions on otherwise rather general constitutive entities entering the model so as to ensure well-posedness, proper qualitative behavior and admissible thermodynamic behavior.

Remarks on the behavior of simple directionally reinforced incompressible nonlinearly elastic solids

The effect of directional reinforcing in generating qualitative changes in the mechanical response of a base neo-Hookean material is examined in the context of homogenous deformation. Single axis reinforcing giving transverse isotropy is the major focus, in which case a standard reinforcing model is characterized by a single constitutive reinforcing parameter. Various qualitative changes in the mechanical response ensue as the reinforcing parameter increases from the zero-value associated with neo-Hookean response. These include (in order): the existence of a limiting contractive stretch for transverse-axis tensile load; loss of monotonicity in off-axis simple shear; loss of monotonicity in on-axis compression; loss of positivity in the stress-shear product in off-axis simple shear; and loss of monotonicity for plane strain in on-axis compression. The qualitative changes in the simple shear response are associated with stretch relaxation in the reinforcing direction due to finite rotation.

A constitutive model for hysteretic phase transition behavior

Abstract A model for rate independent hysteresis, based on the Duhem-Madelung framework, is examined in the context of temperature induced phase transitions such as those which occur in shape memory metals. Connection to experiment follows from its dependence upon the hysteresis envelope for complete transitions. Various requirements upon partial transitions, notably that such transitions proceed at a pace that is proportional to the phase fraction of the parent phase, complete the model. Restrictions on the hysteresis envelope are obtained that ensure monotonicity, containment and orientation requirements upon partial transition paths.

Dynamics of Crystal Surfaces and Interfaces

From Atomic Diffusion to Step Dynamics B. Blagojevic, P.M. Duxbury. Atomic Steps in the Decay of 1- and 2-Dimensional Gratings J. Blakely, et al. Morphologies of Periodic Surface Profiles and Small Particles: A Source of Step and Step Interaction Energies H.P. Bonzel, S. Surnev. Anisotropy of Wetting of Pb Crystals by their Own Melt and by Liquid Ga-Pb Alloys D. Chatain, P. Wynblatt. Relaxation of Nanometer-Scale Surface Morphology S.J. Chey, D.G. Cahill. Smoothing of a Grooved Singular Surface Whose Neighboring Orientations are Unstable C. Duport, et al. Step Fluctuations: From Equilibrium Analysis to Step Unbunching and Cluster Diffusion in a Unified Picture T.l. Einstein, S.V. Khare. Kinetic Rate Law Issues in the Morphological Relaxation of Rippled Crystal Surfaces J.D. Erlebacher. Grain Boundary Motion in Aluminum Bicrystals G. Gottstein, et al. An Instability of Heteroepitaxial Interfaces via a Discrete Atom Methods J.K. Lee. Ab-Initio Simulations of the Si (100) Surface: Steps and Melting C.M. Roland, et al. Relaxation of Surface Steps Toward Equilibrium W. Selke. Coarsening of MBE Structures in 2+1 Dimensions E. Somfai, L.M. Sander. 5 Additional Articles. Index.

Models for one-variant shape memory materials based on dissipation functions

Some simple models for the macroscopic behavior of shape memory materials whose microstructure can be described as a mixture of two phases are derived on the basis of a free energy and a dissipation function. Keeping a common expression for the free energy, each model is based on a different expression for the dissipation function. Temperature-induced as well as isothermal, adiabatic and convective stress-induced transformations are studied. Attention is paid to closed form solutions, comparison among the models and parameter identification.

Kink surfaces in a directionally reinforced neo-Hookean material under plane deformation: I. Mechanical equilibrium

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can lose ellipticity in plane deformation if the reinforcing is sufficiently large and the fiber direction is sufficiently compressed. Here we show that the same reinforcing levels can give rise to piecewise smooth plane deformations separated by a plane stationary kink. Attention is restricted to deformations in which, on one side of the kink, the load axis is aligned with the fiber axis. Then the fiber stretch on this side of the kink is a natural load parameter. It is found that such a deformation can support a planar kink for a certain range of this load parameter. This range is dependent on the reinforcing parameter, and can even involve fiber extension if the reinforcing is sufficiently large. The set of all deformation states on the other side of the kink is precisely characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. The results are interpreted in terms of the associated fiber alignment discontinuity and fiber stretch discontinuity.

Loss of Ellipticity in Plane Deformation of a Simple Directionally Reinforced Incompressible Nonlinearly Elastic Solid

Change of type in the governing equations of equilibrium is examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for strength of reinforcement. Plane deformations interpreted in terms of both local and global plane strain are considered. Loss of ordinary ellipticity is found to occur for sufficiently large strength of reinforcement under sufficiently severe deformation which necessarily involves contraction in the reinforcing direction. Loss of ellipticity in local plane strain is easily characterized, and its incipient breakdown is associated with the possible emergence of surfaces of weak discontinuity with orientation normals in the reinforcing direction. Loss of ellipticity in global plane strain is given a two-dimensional manifold characterization in a space involving 2 deformation parameters and the strength of reinforcing parameter. Orientation normals for the associated surfaces of weak discontinuity at incipient breakdown do not in general conform to the reinforcing direction.

Swelling Induced Finite Strain Flexure in a Rectangular Block of an Isotropic Elastic Material

The deformation of a rectangular block into an annular wedge is studied with respect to the state of swelling interior to the block. Nonuniform swelling fields are shown to generate these flexure deformations in the absence of resultant forces and bending moments. Analytical expressions for the deformation fields demonstrate these effects for both incompressible and compressible generalizations of conventional hyperelastic materials. Existing results in the absence of a swelling agent are recovered as special cases.

Kink Surfaces in a Directionally Reinforced neo-Hookean Material under Plane Deformation: II. Kink Band Stability and Maximally Dissipative Band Broadening

Stationary kinks (elastostatic shocks) are examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for the additional fiber reinforcing stiffness. Previous work has shown that such a transversely isotropic material can support stationary kinks in plane deformation if the reinforcing is sufficiently great. If the deformation on one side of the kink involves a load axis aligned with the fiber axis, then the more general plane deformation on the other side of the kink is characterized in terms of a one-parameter family of (kink orientation, kink strength)-pairs. Here, the ellipticity status of the two correlated deformation states is shown to span all four possible ellipticity/nonellipticity permutations. If both deformation states are elliptic, then a suitable intermediate deformation is shown to be nonelliptic. Maximally dissipative quasi-static kink motion is examined and interpreted in terms of kink band broadening in on-axis loading. Such maximally dissipative kinks nucleate only in compression as weak kinks, with subsequent motion converting nonelliptic deformation to elliptic deformation. The associated fiber rotation involves three periods: an initial period of slow rotation, a secondary period of rapid rotation, and a final period of essentially constant orienation.

Void nucleation in tensile dead-loading of a composite incompressible nonlinearly elastic sphere

In this paper, the effect of material inhomogeneity on void formation and growth in incompressible nonlinearly elastic solids is examined. A bifurcation problem is considered for a solid composite sphere composed of two neo-Hookean materials perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the underformed configuration. Such a configuration is the only stable solution for sufficiently large loads. In contrast to the situation for a homogeneous neo-Hookean sphere, bifurcation here may occur either locally to the right orto the left. In the latter case, the cavity has finite radius on first appearance. This discontinuous change in stable equilibrium configurations is reminiscent of the snap-through buckling phenomenon observed in certain structural mechanics problems.Since this paper was written, the authors have carried out further analysis of the class of problems of concern here [11]. In particular the stress distribution in the composite neo-Hookean sphere has been described in [11].