Mathematics and Mechanics of Solids

We provide an existence theory for the full thermomechanical quasi-static evolution of a shape memory wire described by the Souza—Auricchio constitutive model. The analysis requires some mild restriction on the choice of the thermomechanical coupling term in the expression on the free energy of the material. This restriction slightly deviates from the original Souza—Auricchio modeling frame, while still being compatible with real situations. We additionally show that, by imposing no such restriction, the original Souza—Auricchio model is ill-posed.

This paper gives new demonstrations of Reynolds’ transport theorems for moving regions in Euclidean space. For moving volume regions the proof is based on differential forms and Stokes’ formula. Moving curves and surface regions are defined and the intrinsic normal time derivative is introduced. The corresponding surface transport theorem is derived using the partition of unity and the surface divergence theorem. A proof of the surface divergence theorem is also given.

The theory of material evolution is used to model anisotropic plasticity. As suggested by the physics of plasticity, whereby the motion of dislocations yields relative slipping along a particular material plane, a shear-like isochoric material deformation is assumed as the admissible kinematic evolution variable, which is constitutively driven by the thermodynamically dual Mandel stress. A simple and intuitive anisotropic evolution law is considered, where slipping is permitted along a fixed material plane which is prescribed as part of the evolution law. An additional yield criterion is proposed to complete the model. It is demonstrated that the anisotropy of the evolution law renders perfect plasticity, hardening or softening and is rate dependent.

We consider an antiplane model which describes the contact between a deformable cylinder and a rigid foundation, under the small deformation hypothesis, for quasistatic processes. The behaviour of the material is modelled using a viscoelastic constitutive law with long memory and the frictional contact is modelled using Tresca’s law. We focus on the weak solvability of the model, based on a weak formulation with dual Lagrange multipliers.

A finite strain model for the mechanical degradation of composite materials with multiple families of fine reinforcing fibers is developed and studied. At any instant of time the matrix material may or may not be degrading with all, some, or none of the interpenetrating fibers also undergoing degradation. This multi-component description of damage is governed by coupled differential equations when more than one damage mechanism is active. These differential equations, and the threshold values of the strain invariants that activate the damage process, emerge naturally from a general framework that describes the response of dissipative systems under a maximum rate of dissipation postulate. In this context we then study uniaxial loadings when either a constant stretch or a constant force is suddenly applied to the composite. It is found that the initial type of degradation (e.g. degrading fibers in a non-degrading matrix) may transition to an alternative type of degradation (e.g. the degradation of all constituents) at some finite time into the process. A rich variety of material and load-dependent transition possibilities are systematically uncovered using a combination of asymptotic and numerical techniques. The resulting macroscopic behavior as the material weakens involves relaxation and creep phenomena that are formally similar to viscoelastic material behavior in solids even though the underlying processes are significantly different. Describing implants and tissue constructs containing biodegradable polymers is one possible area of application.

The aim of the present paper is to establish a convolutional type (Gurtin ME. Variational principles for linear elastodynamics. Arch Ration Mech Anal 1964; 16: 34—50) variational principle and a reciprocal principle in the context of the linear theory of thermoelasticity of type III. To establish the variational and reciprocal principles, we firstly formulate a characterization of the mixed boundary initial value problem in which the initial conditions are incorporated into the field equations.

In this paper we discuss the propagation of torsional surface waves in an anisotropic porous half space in the presence of a gravity field. Although the torsional wave does not propagate in an isotropic elastic half space in the absence of a gravity field, it is found that such a wave propagates in a gravitating isotropic half space. It also propagates in anisotropic half space in the presence or absence of a gravity field if the modulus of elasticity in the horizontal direction is more than that of the vertical direction. It is observed that poro-elasticity has a dominant effect on the velocity of the torsional surface wave. The velocity of the wave increases considerably as the porosity of the medium increases. The presence of a gravity field is also seen to increase the velocity of the torsional surface wave. The velocity depends to a large degree on the anisotropy factor. As a particular case, it is verified that the torsional surface wave does not propagate in perfect fluid under gravity.