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

Connecting weakly nonlinear elasticity theories of isotropic hyperelastic materials