In this paper we analyze mathematical properties of a soft solids model which was proposed to interpret measurements of nonlinear shear waves in tissue-like media. In our analysis we pay special attention to the third order nonlinear terms. We present for the first time the full equations of elastodynamics for this model with third order nonlinear terms. These third order terms are also included in the equations for plane waves. Next the interaction coeffcients for cubically nonlinear interactions of plane waves in this model are presented. These coefficients show which wave interact with each other and determine how strong cubically nonlinear interactions are.

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