Connecting weakly nonlinear elasticity theories of isotropic hyperelastic materials

Soft materials exhibit significant nonlinear geometric deformations and stress-strain relationships under external forces. This paper explores weakly nonlinear elasticity theories, including Landau's and Murnaghan's formulations, advancing understanding beyond linear elasticity. We establish connections between these methods and extend strain-energy functions to the third and fourth orders in powers of epsilon, where epsilon is defined as the square root of the inner product of H with itself, epsilon = sqrt(H * H), and epsilon is between 0 and 1. Here, H represents the perturbation to the deformation gradient tensor, where the deformation gradient F is given by F = I + H. Furthermore, we address simplified strain-energy functions applicable to incompressible materials. Through this work, we contribute to a comprehensive understanding of nonlinear elasticity and its relationship to weakly nonlinear elasticity, facilitating the study of moderate deformations in soft material behavior and its practical applications.