Hyperelastic Internal Balance by Multiplicative Decomposition of the Deformation Gradient

The multiplicative decomposition of the deformation gradient \({{\bf F} = {{\hat{\bf F}}}{\bf F}^*}\) is often used in finite deformation continuum mechanics as a basis for treating mechanical effects including plasticity, biological growth, material swelling, and notions of material morphogenesis. Evolution rules for the particular effect from this list are then posed for F*. The tensor \({{{\hat{\bf F}}}}\) is then invoked to describe a subsequent elastic accommodation, and a hyperelastic framework is put in place for its determination using an elastic energy density function, say \({W({\hat{\bf F}})}\) , as a constitutive specification. Here we explore the theory that emerges if both F* and \({{\hat{\bf F}}}\) are governed by hyperelastic criteria; thus we consider energy densities \({W({{\hat{\bf F}}}, {\bf F}^*)}\) . The decomposition of F is itself determined by energy minimization, and the variation associated with the multiplicative decomposition gives a tensor relation that is interpreted as an internal balance requirement. Our initial development purposefully proceeds with minimal presumptions on the kinematic interpretation of the factors in the deformation gradient decomposition. Connections are then made to treatments that ascribe particular kinematic properties to the decomposition factors—the theory of structured deformations is especially significant in this regard. Such theories have broad utility in describing certain substructural reconfigurations in solids. To demonstrate in the context of the present variational treatment we consider a boundary value problem that involves an imposed twist. If the twist is small then the minimizer is classically smooth. At larger values of twist the energy minimizer exhibits a non-smooth deformation that localizes slip at a singular surface.     

Anti-plane Shear Deformations in Compressible Transversely Isotropic Materials

Anti-plane shear deformations in a compressible, transversely isotropic hyperelastic material are under investigation. The displacement is assumed to be along the direction of the symmetry axis and is independent of the axial position. The resulting equations of equilibrium form an overdetermined system of partial differential equations for which solutions do not exist in general. Necessary and sufficient conditions are derived for such materials to sustain anti-plane shear deformations in the sense that every solution to the axial equation automatically satisfies the other two in-plane equations. Comparison is made with results for isotropic materials. A weaker version of the conditions specialized to axisymmetric anti-plane shear deformations is also obtained.     

On anisotropic compressible materials that can sustain elastodynamic anti-plane shear

We characterize the class of compressible, homogeneous, possibly anisotropic elastic materials that can undergo nontrivial finite elastodynamic anti-plane shear motions.     

Bulk Cavitation and the Possibility of Localized Interface Deformation due to Surface Layer Swelling

We consider the uniform swelling of a compressible hyperelastic surface layer with finite thickness that is attached to an underlying bulk material composed of a non-swelling incompressible hyperelastic material. In addition to classically smooth solutions, two additional phenomena may occur for sufficiently large swelling. One is the formation of cavities in the interior of the underlying bulk material. The other is the disappearance of smooth solutions in the surface layer while the underlying bulk material remains intact. It is conjectured that the latter may be associated with the concentration of deformation at the swelling interface. Both phenomena are investigated by the consideration of solutions to a boundary value problem for a sphere involving radial deformation with a prescribed swelling field that acts as an effective loading device. Specific material models for both the compressible swollen surface layer and the non-swollen incompressible bulk are invoked so as to permit an analytical treatment. Swelling thresholds are obtained that depend on the thickness of the surface layer for the onset of these separate phenomena.     

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.     

Reflection and Refraction of Anti-Plane Shear Waves from a Moving Phase Boundary

The reflection and refraction of anti-plane shear waves from an interface separating half-spaces with different moduli is well understood in the linear theory of elasticity. Namely, an oblique incident wave gives rise to a reflected wave that departs at the same angle and to a refracted wave that, after transmission through the interface, departs at a possibly different angle. Here we study similar issues for a material that admits mobile elastic phase boundaries in anti-plane shear. We consider an energy minimal equilibrium state in anti-plane shear involving a planar phase boundary that is perturbed due to an incident wave of small magnitude. The phase boundary is allowed to move under this perturbation. As in the linear theory, the perturbation gives rise to a reflected and a refracted wave. The orientation of these waves is independent of the phase boundary motion and determined as in the linear theory. However, the phase boundary motion affects the amplitudes of the departing waves. Perturbation analysis gives these amplitudes for general small phase boundary motion, and also permits the specification of the phase boundary motion on the basis of additional criteria such as a kinetic relation. A standard kinetic relation is studied to quantify the subsequent energy partitioning and dissipation on the basis of the properties of the incident wave.