By Martin Fritz Mueller
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Extra resources for Analytical investigation of internally resonant second harmonic lamb waves in nonlinear elastic isotropic plates
If, on the other hand, a P − (SV −)wave interacts with an interface, reﬂected and refracted SV −(P −)waves occur, but no SH -waves occur. These statements hold, in general, only for the case of an interface between two solid half-spaces. In liquids, neither SH- nor SV-waves propagate; in a vacuum a rigid half-space, or no waves propagate at all. Correspondingly, the situation is even more simple if such half-spaces are involved. The decoupling of P-SV- and SH-waves holds for plane problems not only in the simple case of an interface z=const between two homogeneous half-spaces, but also in the more complicated case of an inhomogeneous medium, as long as density, wave velocity, and module are only functions of x and z.
9). 15) can also be computed in non-Cartesian coordinates, something we now use. Application of the wave equation for Φ We introduce the spherical coordinates (r′ , ϑ, λ) relative to point P . λ is deﬁned, see sketch, via an additional Cartesian coordinate system (x, y, z). Fig. 6: Additional Cartesian coordinate system (x, y, z). The x-axis of of this coordinate system is identical to the line OP. The z-axis is in the plane deﬁned by the zandζ-axis and the line OP. Then the z, ζ-axis has then in the x − y − z-system the direction of the unit vector − → n = (cos γ, 0, sin γ) = z z2 , 0, 1 − 2 r r 1 2 .
The integration covers the whole volume. − → For Ψ , there is the additional requirement that − → ∇ · Ψ = 0. − → Ψ has to be determined in Cartesian coordinates. 6. 26) is called compressional part of − u . It is free of rotation and curl. 1). In the main axis system of the deformation tensor, only volume changes occur (compression or − → dilatation). 26) is called shear component. It is free of divergence and source contributions; the volume elements suﬀer no volume change, but shear deformation and rigid rotation.
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