Hyperelastic Internal Balance by Multiplicative Decomposition of the Deformation Gradient

Abstract

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.