On Higher Order Effective Boundary Conditions for a Coated Elastic Half-Space

Abstract

Higher order effective boundary conditions are derived for a coated half-space. Comparison with the long wavelength expansion of the exact solution of a plane time-harmonic problem for the coating demonstrates the validity of the proposed formulation. At the same time the corrections to the simplest leading order effective conditions, earlier obtained in the widely cited paper (Bövik (1996). J. Appl. Mech. 63(1), 162–167.) [1], are proven to be asymptotically inconsistent.

Appendix

The constants in (49) are

$$\begin{aligned} A_1=h\dfrac{N_1}{D}, \quad A_2={\mathrm e}^{kh\alpha }h\dfrac{N_2}{D}, \quad A_3=-h\dfrac{N_3}{D}, \quad A_4=-{\mathrm e}^{kh\beta }h\dfrac{N_4}{D}, \end{aligned}$$

(59)

where

$$\begin{aligned} \begin{array}{l} N_1=iB_1\left( {\mathrm e}^{kh\alpha }(D_1\alpha \beta +D_2\gamma ^4)-2{\mathrm e}^{kh\beta }\alpha \beta \gamma ^2\right) \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad -B_3\beta \left( {\mathrm e}^{kh\alpha }(D_2\alpha \beta +D_1\gamma ^4)-2{\mathrm e}^{kh\beta }\gamma ^2\right) ,\\ N_2=iB_1\left( D_1\alpha \beta -2{\mathrm e}^{kh(\alpha +\beta )}\alpha \beta \gamma ^2-\gamma ^4D_2\right) \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad +B_3\beta \left( D_1\gamma ^4-D_2\alpha \beta -2{\mathrm e}^{kh(\alpha +\beta )}\gamma ^2\right) ,\\ N_3=iB_3\left( {\mathrm e}^{kh\beta }(D_3\alpha \beta +D_4\gamma ^4)-2{\mathrm e}^{kh\alpha }\alpha \beta \gamma ^2\right) \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad +B_1\alpha \left( {\mathrm e}^{kh\beta }(D_4\alpha \beta +D_3\gamma ^4)-2{\mathrm e}^{kh\alpha }\gamma ^2\right) ,\\ N_4=iB_3\left( D_3\alpha \beta -2{\mathrm e}^{kh(\alpha +\beta )}\alpha \beta \gamma ^2-\gamma ^4D_4\right) \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad -B_1\alpha \left( D_3\gamma ^4-D_4\alpha \beta -2{\mathrm e}^{kh(\alpha +\beta )}\gamma ^2\right) ,\\ \text {and}\\ D=k\left[ 8{\mathrm e}^{kh(\alpha +\beta )}\alpha \beta \gamma ^2+D_2D_4(\alpha ^2\beta ^2+\gamma ^4)-D_1D_3\alpha \beta (1+\gamma ^4)\right] , \end{array} \end{aligned}$$

(60)

with

$$\begin{aligned} D_1=1+{\mathrm e}^{2kh\beta },\quad D_2=1-{\mathrm e}^{2kh\beta },\quad D_3=1+{\mathrm e}^{2kh\alpha },\quad D_4=1-{\mathrm e}^{2kh\alpha }. \end{aligned}$$

(61)