The complex Burgers equation as a model for collinear interactions of weakly nonlinear shear plane waves in anisotropic elastic materials

Abstract

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.

Propagation and interaction of weakly nonlinear plane waves in transversely isotropic elastic materials

“…However, anisotropy and in particular the presence of fibres changes this situation. The coupling of shear waves on a quadratically nonlinear level is possible when a special direction of propagation in anisotropic materials is chosen [3][4][5]. Here, we demonstrate that the quadratic nonlinearity may show up even for a single shear wave propagation due to the presence of fibres.…”

“…together with the isotropic part defined by (5). Parameter E z > 0 is interpreted as the initial Young's modulus of the fibre family.…”

“…Parameter E z > 0 is interpreted as the initial Young's modulus of the fibre family. Additionally, the quadratic approximation of the stored energy function (5) shows that the bulk initial modulus and the first Lamé constant of the isotropic matrix are given by…”

“…In Sect. 6, we apply the method of weakly nonlinear asymptotics [5] in order to derive the evolution equation for a single quasi-shear wave for our model of transversely isotropic materials. It turns out that for almost all range of parameters, we obtain the inviscid Burgers' equation.…”

“…Each colour in the table denotes a different nonzero value of the interaction coefficient. (Color figure online) Interaction coefficients Γ5 pq with p, q = 1, 2, . .…”