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Extra resources for Analytical investigation of internally resonant second harmonic lamb waves in nonlinear elastic isotropic plates

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We treat a plane problem in which on all planes parallel to the x − z-plane, the same conditions hold. The most simple way to study elastic waves, under these conditions, is to derive u and w but not v, from potentials. Writing − → − → u = ∇Φ + ∇ × Ψ by components, u = v = w = ∂Φ ∂Ψ2 − ∂x ∂z ∂Ψ1 ∂Ψ3 − ∂z ∂x ∂Φ ∂Ψ2 + , ∂z ∂x 64 CHAPTER 3. BODY WAVES it is obvious that for v two potentials Ψ1 and Ψ3 are required, and these do not occur in u and w. 21) without body forces, which under these conditions becomes a wave equation ∇2 v = 1 ∂2v .

We treat a plane problem in which on all planes parallel to the x − z-plane, the same conditions hold. The most simple way to study elastic waves, under these conditions, is to derive u and w but not v, from potentials. Writing − → − → u = ∇Φ + ∇ × Ψ by components, u = v = w = ∂Φ ∂Ψ2 − ∂x ∂z ∂Ψ1 ∂Ψ3 − ∂z ∂x ∂Φ ∂Ψ2 + , ∂z ∂x 64 CHAPTER 3. BODY WAVES it is obvious that for v two potentials Ψ1 and Ψ3 are required, and these do not occur in u and w. 21) without body forces, which under these conditions becomes a wave equation ∇2 v = 1 ∂2v .

The radiation characteristics in planes other then y = 0 follow from the one shown by multiplication with cos λ. Plane x = 0 is a nodal plane for P - as well as for S-radiation; the plane z = 0 is one only for P . 6) z P y r υ λ x Fig. 14: Double couple in the x-z-plane. 60 CHAPTER 3. BODY WAVES ur = 2ϑ cos λ ′ − sin4πρα t− 3r M r α uϑ = 2ϑ cos λ M′ t − − cos4πρβ 3r r β cos ϑ sin λ ′ 4πρβ 3 r M . 22) is that of one of the two dipoles of the double couple. The radiation characteristics in the x − z-plane are shown in Fig.

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Analytical investigation of internally resonant second harmonic lamb waves in nonlinear elastic isotropic plates by Martin Fritz Mueller


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