Conjugate stress/strain base pairs for planar analysis of biological tissues

Abstract

A theoretical framework is presented for the analysis of planar membranes based upon a triangular (as opposed to a polar) decomposition of the deformation gradient. This leads to a distillation of the deformation gradient into three distinct modes. Each mode can, in principle, be individually activated in an experiment. Measures of stress are shown to exist for each mode of strain so that the stress power can be decomposed into independent additive parts. The outcome is a set of three conjugate stress/strain base pairs (each being a pair of scalars) from which constitutive equations can be constructed for planar solids without relying on tensor invariants to cast the theory. Explicit and implicit elastic models are derived that, when convolved, produce a material model whose stress/strain response is indicative of soft biological tissues. Stress/strain curves for each conjugate pairing are constructed from published experimental data. The model describes these data.