Geometrically consistent nonlinear plane strain and stress constitutive models: Application to soft-material oscillations

Abstract

Position-gradient constraints are used in this study to develop new geometrically consistent plane strain and stress constitutive models for the nonlinear large-displacement analysis of soft materials. The position gradients are used to define strain and stress constraints that enter into the formulation of the deformation tensor, strain-energy density function, and new constitutive models. The new approach is applied to three nonlinear hyperelastic constitutive models for incompressible materials: Neo-Hookean, Mooney Rivlin, and Yeoh Models. The underlying geometric plane-stress and plane-strain assumptions are preserved by using the finite elements (FE) of the absolute nodal coordinate formulation (ANCF). Limitations of widely used incompressibility penalty functions that produce stresses at initial stress-free configurations are discussed. Several numerical examples developed using planar ANCF shear-deformable beam and rectangular elements are used to assess the proposed formulation and verify its results using spatial solid elements implemented in commercial FE software. It is demonstrated that the ANCF plane-stress and plane-strain formulations for the incompressible materials predict accurately soft-material oscillations, demonstrating that stiffer behavior can be attributed to material-model inconsistencies and cannot always be attributed to locking.