Hyperelastic constitutive relations for soft elastomers with thermally-induced residual stress

Abstract

Residual stress widely exists in soft materials. Besides growth, inhomogeneous thermal expansion is also a primary cause of residual stress. However, establishing a proper hyperelastic constitutive relation is a great challenge since the existing theories cannot capture the change of underlying mechanical responses triggered by temperature variations. In this paper, a general hyperelastic constitutive relation for soft elastomers with thermally-induced residual stress is developed. We first reveal the initial temperature dependence of conventional thermoelastic models. This property attributes the alteration of the underlying thermoelastic response to free thermal expansions. Then, a compatibility-broken curvature compensation (CBCC) framework is established based on finite thermoelasticity. It generates a free thermal expansion to eliminate the Riemannian curvatures of the virtual stress-free configuration derived from the isothermal stress release. Such a mechanism indicates the non-local effect of the residual stress, which fundamentally modifies the traditional view that invariant formulations cover all the possible functional dependence of residual stress. Also, the obtained governing equations are similar to Einstein field equations of the general theory of relativity. This similarity may deeply imply a standard mechanism concerning the curvature compensation leading to residual stress genesis. We finally conduct comparative analyses of the spherically symmetric and axisymmetric problems between the current constitutive relation and the existing models. The influences of adopting distinct residual stresses, the performance of the non-local effect, and the availability of the new constitutive relation are investigated in detail. This framework can shed some light on the constitutive modeling of soft materials.