Ultrasonic-based characterization of soft tissue is an emerging technology with great potential as a clinical diagnostic tool. There is evidence that abnormalities in the structural architecture of soft tissues are linked to a broad range of pathologies including tumors, liver fibrosis, preterm birth...The present work derives different theories of non-linear formulation of wave propagation using generalized and particular coordinates on a material characterized by different strain energy

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... ons. Afterwards, these formulations are adapted to simulate the propagation of shear waves on nonlinear elastic materials to explain the properties of soft tissue when is measured under these conditions. The resultant system of equations could be solved under a finite differences framework and probabilistic inverse problem to explore how improves the understanding of soft tissues. The response of the system varying the linear/non-linear parameters of the strain energy is obtained via Finite Differences Time Domain simulation showing significant variations among the different models. Nonlinear wave propagation approach implies a new paradigm to model the nonlinear biomechanical behavior of soft tissue improving the plausibility versus previous elastic or viscoelastic predictions.