Second-order constitutive approximation in electroelasticity: Cauchy stress formulation using Biot strain

This article proposes a second-order constitutive approximation for electroelastic materials using the Biot strain measure. The deformation gradient tensor and the electric field vector, both expressed in terms of the undeformed reference configuration, are used to express the strain energy density. From this expansion, the Cauchy stress tensor is obtained by retaining terms up to second order in the gradient of the strain energy function, which introduces the electromechanical coupling contributions. As a result, the constitutive relation contains the usual mechanical response together with additional terms arising from the electric field. In the absence of the electric field, the formulation is consistent with material frame indifference and reduces to the conventional second-order elasticity model.