Complex Material Response Described by Continuum Mechanics with a Deformation Gradient Product Decomposition that has Novel Hyperelastic Implications

Continuum mechanics seeks to describe the mechanical response of materials at size scales in which it is impractical to resolve individual atoms, molecules or their clusters. The theory takes into account the mapping of material points X to new locations x due to the action of physical factors: force, stress, heat, electrical fields, chemical reaction, etc. The gradient of this mapping, denoted by F, is a tensor that naturally enters into the mathematical description by virtue of its mapping of infinitesimal line elements from the reference to the deformed configuration via dx = F dX. For material behavior involving elastic response combined with some other type of response such as plastic flow, viscoelastic relaxation, or biological growth, it is common to posit that a decomposition F = F1 F2, at each material point where the elastic part F1 is, in the equilibrium setting, determined by variational calculus, and the non-elastic part F2 is subject to some kind of time dependent evolution law. The purpose of this research is to critically reexamine these assumptions and, in particular, to consider the case where arguments of variational calculus apply to both products and can be used to determine the decomposition itself. Preliminary indications is that this will offer great benefit in the modeling of complex materials especially as regards the development of singular surfaces that can be interpreted as locations of concentrated microstructural rearrangement.