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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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