Shaun Sellers

Configurational stress, yield and flow in rate-independent plasticity

The role of configurational stress in yield and plastic flow is discussed for a macroscopic model of rate–independent, finite–strain plasticity. The model is based on the traditional elastic–plastic decomposition of the deformation gradient, on integral balance laws and on thermodynamically restricted, rate–independent constitutive relations. Its formulation emphasizes the intermediate configuration in both the development of constitutive relations and the expression of balance laws. In addition to the usual balance laws, a couple balance is included to represent the action of plastic couples in the intermediate configuration. In particular, it is shown that the internal couple decomposes into a non–dissipative configurational stress and a dissipative couple that resists plastic flow. The couple balance thus determines a relation between the configurational stress and the plastic–flow resistance, a relation that can be interpreted as a generalized yield condition. A dissipation function is introduced and a maximum–dissipation criterion is used to obtain additional constitutive restrictions, which lead to a counterpart in the intermediate configuration of the classical normality conditions. The versatility of the framework is illustrated by applying it to rigid–plastic flow, in which case a nonlinear generalization of the classical Levy–von Mises theory is obtained.

Orientational order and finite strain in nematic elastomers

Nematic elastomers exhibit large, spontaneous shape changes at the transition from the high-temperature isotropic phase to the low-temperature nematic phase. These finite deformations are studied here in the context of a nonlinear, properly invariant, variational theory that couples the orientational order and elastic deformation. The theory is based on the minimization of a free-energy functional that consists of two contributions: a nematic one due to the interaction of the mesogenic units and an elastic one arising from the stretching of the cross-linked polymer chains. Suitable choices for these two contributions allow for large, reversible, spontaneous shape changes in which the elastic deformation can affect the isotropic-nematic transition temperature. The change in transition temperature as well as the magnitude of the resulting spontaneous deformation is illustrated for various parameter values. The theory includes soft elasticity as a special case but is not restricted to it.

Soft elasticity is not necessary for striping in nematic elastomers

The neoclassical model for nematic elastomers, which displays soft elasticity, predicts striping in stretched sheets. Thus the experimental observation of striped domains has been suggested as evidence for soft elasticity. Here we show that the postulated director rotations and shears in the domain regions are also predicted by more general constitutive models that do not involve any notion of softness. Striping in nematic elastomers may therefore be a more general phenomenon that is not necessarily an indication of soft elasticity. Furthermore, constitutive models more general than the neoclassical model may also explain the behavior of some nematic elastomers that do not appear to exhibit striping.

Force-Free States, Relative Strain, and Soft Elasticity in Nematic Elastomers

The remarkable ability of nematic elastomers to exhibit large deformations under small applied forces is known as soft elasticity. The recently proposed neo-classical free-energy density for nematic elastomers, derived by molecular-statistical arguments, has been used to model soft elasticity. In particular, the neo-classical free-energy density allows for a continuous spectrum of equilibria, which implies that deformations may occur in the complete absence of force and energy cost. Here we study the notion of force-free states in the context of a continuum theory of nematic elastomers that allows for isotropy, uniaxiality, and biaxiality of the polymer microstructure. Within that theory, the neo-classical free-energy density is an example of a free-energy density function that depends on the deformation gradient only through a nonlinear strain measure associated with the deformation of the polymer microstructure relative to the macroscopic continuum. Among the force-free states for a nematic elastomer described by the neo-classical free energy density, there is, in particular, a continuous spectrum of states parameterized by a pair of tensors that allows for soft deformations. In these force-free states the polymer microstructure is material in the sense that it stretches and rotates with the macroscopic continuum. Limitations of and possible improvements upon the neo-classical model are also discussed.

Multi-phase equilibrium of crystalline solids

Abstract A continuum model of crystalline solid equilibrium is presented in which the underlying periodic lattice structure is taken explicitly into account. This model also allows for both point and line defects in the bulk of the lattice and at interfaces, and the kinematics of such defects is discussed in some detail. A Gibbsian variational argument is used to derive the necessary bulk and interfacial conditions for multi-phase equilibrium (crystal–crystal and crystal–melt) where the allowed lattice variations involve the creation and transport of defects in the bulk and at the phase interface. An interfacial energy, assumed to depend on the interfacial dislocation density and the orientation of the interface with respect to the lattices of both phases, is also included in the analysis. Previous equilibrium results based on nonlinear elastic models for incoherent and coherent interfaces are recovered as special cases for when the lattice distortion is constrained to coincide with the macroscopic deformation gradient, thereby excluding bulk dislocations. The formulation is purely spatial and needs no recourse to a fixed reference configuration or an elastic–plastic decomposition of the strain. Such a decomposition can be introduced, however, through an incremental elastic deformation superposed onto an already dislocated state, but leads to additional equilibrium conditions. The presentation emphasizes the role of configurational forces as they provide a natural framework for the description and interpretation of singularities and phase transitions.

Theory for Atomic Diffusion on Fixed and Deformable Crystal Lattices

We develop a theoretical framework, for the diffusion of a single unconstrained species of atoms on a crystal lattice, that provides a generalization of the classical theories of atomic diffusion and diffusion-induced phase separation to account for constitutive nonlinearities, external forces, and the deformation of the lattice. In this framework, we regard atomic diffusion as a microscopic process described by two independent kinematic variables: (i) the atomic flux, which reckons the local motion of atoms relative to the motion of the underlying lattice, and (ii) the time-rate of the atomic density, which encompasses nonlocal interactions between migrating atoms and characterizes the kinematics of phase separation. We introduce generalized forces power-conjugate to each of these rates and require that these forces satisfy ancillary microbalances distinct from the conventional balance involving the forces that expend power over the rate at which the lattice deforms. A mechanical version of the second law, which takes the form of an energy imbalance accounting for all power expenditures (including those due to atomic diffusion and phase separation), is used to derive restrictions on constitutive equations. With these restrictions, the microbalance involving the forces conjugate to the atomic flux provides a generalization of the usual constitutive relation between the atomic flux and the gradient of the diffusion potential, a relation that in conjunction with the atomic balance yields a generalized Cahn–Hilliard equation.

Theory of water transport in melting snow with a moving surface

A model is presented for the transport of water in melting snow where the snow surface and percolation front are treated as propagating singular surfaces. It is based on Colbecks theory of water transport in bulk snow supplemented with boundary conditions that explicitly include the production of water by snow melting at the surface due to a surface heat supply. The consequent motion of the snow surface leads to a free boundary problem, where the snow surface must be determined as part of the solution, which itself depends on the motion of the snow surface. Explicit relations are obtained for the propagation of the melt surface and percolation front. Numerical examples are given of the propagation of one dimensional meltwater waves in deep snowpacks due to periodic heating of the snow surface. It is shown that, for commonly reported parameter values of deep, homogeneous snow packs, small motions of the snow surface generally lead to small corrections in the water saturation and flux.

Incompatible strains associated with defects in nematic elastomers.

In a nematic elastomer the deformation of the polymer network chains is coupled to the orientational order of the mesogenic groups. Statistical arguments have derived the so-called neoclassical free energy that models this coupling. Here we show that the neoclassical model supplemented by the usual Frank energy predicts incompatible network strains associated with the formation of standard nematic textures. The incompatibility is measured by the Riemann curvature tensor, which we find to be nonzero for both radial hedgehog defects and escaped disclinations of strength +1 in circular cylinders. Analogous problems for conventional nonlinearly elastic solids do not possess solutions with such incompatibilities. Compatibility in nematic elastomers would require either more complicated nematic textures in elastomers than in conventional (polymeric and low molecular weight) liquid crystals or a free-energy density more complicated than the neoclassical expression.

Microforces and the theory of solute transport

Abstract. A generalized continuum framework for the theory of solute transport in fluids is proposed and systematically developed. This framework rests on the introduction of a generic force balance for the solute, a balance distinct from the macroscopic momentum balance associated with the mixture. Special forms of such a force balance have been proposed and used going back at least as far as Nernsts 1888 theory of diffusion. Under certain circumstances, this force balance yields a Fickian constitutive relation for the diffusive solute flux, and, in conjunction with the solute mass balance, provides a generalized Smoluchowski equation for the mass fraction. Our format furnishes a systematic procedure for generalizing convection-diffusion models of solute transport, allowing for constitutive nonlinearities, external forces acting on the diffusing constituents, and coupling between convection and diffusion.

On the Nonlinear Mechanics of Bravais Crystals with Continuous Distributions of Defects

Basic ideas in the mechanics of Bravais crystals with continuous distributions of point and line defects are revisited from a perspective that emphasizes the role of configurational forces. In addition, the creation, destruction, and transport of defects are presented in detail. The formulation of the balance laws is given over time-dependent spatial control volumes; furthermore, independent configurational and deformational force balances are postulated. Such a formulation is more versatile than one with standard fixed control volumes and provides the correct Eshelby relation from simple invariance arguments independent of constitutive relations. Two alternative but equivalent constitutive models are proposed, and the corresponding restrictions following from a dissipation inequality are determined. Relations to some previous formulations of continuous distributions of dislocations and vacancies are discussed.