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Collinear interactions of weakly nonlinear quasi-shear plane waves in anisotropic (in particular fiberreinforced) compressible elastic materials are analyzed. Evolution equations for quasi-shear wave amplitudes are derived with the help of the asymptotic method of a double-scale expansion. It is shown that quadratically nonlinear coupling is possible when shear waves propagate along a special fiber direction in anisotropic materials. The evolution equations are reduced to a single inviscid complex Burgers equation when the fiber direction is a threefold symmetry acoustic axis. Some properties of this equation are analyzed. General considerations are illustrated on examples of shear waves propagating along a threefold symmetry acoustic axis in a cubic crystal and in an icosahedral quasicrystal.

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In this work, we use the MCMV slightly compressible hyperelastic constitutive model of rubber-like materials that was proposed in the authors’ work and implemented in the MES ABAQUS / Standard program. The stored energy function of the MCMV model is approximated by polynomials relative to the Lagrange strain tensor, attempting to evaluate the scope of applicability of Murnaghan’s models in the boundary-value problem of statics. We analyze in detail the homogeneous deformation problems. The point is to determine the approximate range of deformation, in which we get similar results of the problems mentioned, using the MCMV material model and its approximation.