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Wave impedance matrices for cylindrically anisotropic radially inhomogeneous elastic solids. (English) Zbl 1242.74038
The authors consider anisotropic elastic materials whose density $$\rho$$ and elasticities $$c_{ijkl}$$ depend, with respect to cylindrical coordinates $$(r,\theta,z)$$, only on the radial variable $$r$$. They seek displacement fields $$u$$ in the form of time-harmonic cylindrical waves, $$u= U(r) e^{i(n\theta+k_zz-\omega t)}$$. Suppressing obvious dependence on parameters from the notation, the traction in radial direction is of the form $$t_r= Y(r) e^{i(n\theta+k_zz-\omega t)}$$. For this ansatz, the elastodynamic equations reduce to a first-order $$6\times 6$$-system of linear ordinary differential equations, $$r\frac{d}{dr}\eta=iG(r)\eta$$, for the displacement-traction polarization vector $$\eta(r)=(U(r),irY(r))$$. Impedance is a $$3\times 3$$-matrix $$Z(r)$$ which relates displacements to tractions $$rY(r)=-Z(r)U(r)$$.
To correctly handle solid cylinders which have material at the axis $$r=0$$, the authors introduce the solid cylinder impedance $$Z(r)$$, which has the property that the limit $$Z_0=Z(0)$$ exists. The limit is called central impedance. Knowing $$Z(r)$$ is important for finding dispersion equations for guided waves.
The linear system has a regular singular point at $$r=0$$. To apply the Frobenius series method, with indicial roots having positive real parts, is one way the authors propose to calculate $$Z(r)$$. Another is to integrate a differential Riccati equation with initial value $$Z_0$$. Formulas for $$Z_0$$ are obtained by adapting the integral method of D. M. Barnett and J. Lothe [Proc. R. Soc. Lond., Ser. A 402, 135–152 (1985; Zbl 0587.73030)]. For special material symmetries, $$Z_0$$ and $$Z(r)$$ are presented, and a numerical scheme for computing $$Z(r)$$ is proposed.
The solid cylinder impedance $$Z(r)$$ is Hermitian. For cylinders of infinite radius, a radiation impedance is also defined and shown to be non-Hermitian.

##### MSC:
 74J10 Bulk waves in solid mechanics 74E10 Anisotropy in solid mechanics
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