Abstract
Collinear interactions of weakly nonlinear quasi-shear plane waves in anisotropic (in particular fiber-reinforced) 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.
Highlights
We are interested in revealing the manifestation of the anisotropic properties of elastic materials by a collinear propagation of weakly nonlinear quasi-shear plane waves
It is known that changes in the amplitudes of weakly nonlinear transverse or quasi-transverse waves in isotropic elastic media are governed by cubically nonlinear evolution equations [1], and quadratic nonlinearity cannot occur in these equations [1]
In the language used in acoustics, this fact is often formulated as follows: second harmonic generation is impossible for shear waves in isotropic materials
Summary
We are interested in revealing the manifestation of the anisotropic properties of elastic materials by a collinear propagation of weakly nonlinear quasi-shear plane waves. It turns out that choosing a special direction of transverse wave propagation in an anisotropic material may cause the generation of a second harmonic This manifests itself with the presence of quadratically nonlinear terms in the evolution equations for shear wave amplitudes. When shear waves propagate along such chosen fiber directions, a quadratically nonlinear coupling occurs in a special way: evolution equations that describe the coupling of a pair of transverse. The modeling equations for a pair of shear waves propagating along a special fiber direction, which is the threefold symmetry acoustic axis, are reduced to a single inviscid complex Burgers equation. Some properties of this equation are discussed.
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