Elastic Cross-Anisotropic Constitutive Law

The linear, small-strain, elastic behavior of cross-anisotropic material is governed by the following generalized Hooke's law [4]:

$$\epsilon_{11}=\frac{1}{E}\sigma_{11} - \frac{E-2G}{2GE} \sigma_{22} - \frac{E^{\prime}-2G'}{2G'E'}\sigma_{33}$$ (1)

$$\epsilon_{22}=-\frac{E-2G}{2GE}\sigma_{11}+\frac{1}{E}\sigma_{22}-\frac{E'-2G'}{2G'E'}\sigma_{33}$$ (2)

$$\epsilon_{33}=-\frac{E-2G''}{2G''E}\sigma_{11}-\frac{E-2G''}{2G''E}\sigma_{22}+\frac{1}{E'}\sigma_{33}$$ (3)

$$\epsilon_{23}=\frac{1}{2G''}\sigma_{23} \;\;\; \;\;\; \epsilon_{13}=\frac{1}{2G''}\sigma_{13} \;\;\; \;\;\; \epsilon_{12}=\frac{1}{2G}\sigma_{12}$$ (4)

where, in equation (1) $E = E_{11}$ = in-plane Young's modulus; $E' = E_{33}$ = out-of-plane Young's modulus; $G = G_{12}$ = in-plane shear modulus; $G' = G_{31}$ = out-of-plane shear modulus; $G'' = G_{13}$ = out-of-plane shear modulus. Note that ''in-plane'' refers to the 1-2 isotropy plane (horizontal) and ''out-of-plane'' refers to the 3-direction (assumed vertical).

The above elastic material constants, $E$, $E'$, $G$, $G'$, and $G''$ can be determined from three-dimensional P-wave and shear wave tests. The equation relating in-plane Young's modulus ($E$) and in-plane P-wave velocity ($V_{c_{i}}$; $i=1, 2$) is,

$$E=\rho V^2_{c_{i}}$$ (5)

Similarly, the equation relating out-of-plane Young's modulus ($E'$) and out-of-plane P-wave velocity ($V_{c_{3}}$) is,

$$E'=\rho V^2_{c_{3}}$$ (6)

The equation relating in-plane shear modulus ($G$) and in-plane shear wave velocity with horizontal polarization ( $V^H_{s_{i}}$; $i=1, 2$) is,

$$G=\rho (V^H_{s_{i}})^2$$ (7)

where wave propagation is in the i-direction (i=1,2) with horizontal polarization. The equation relating out-of-plane shear modulus ($G'$) and out-of-plane shear wave velocity ( $V_{s_{3}})$is,

$$G'=\rho (V_{s_{3}})^2$$ (8)

where wave propagation is in the 3-direction with horizontal polarization. Finally, the equation relating out-of-plane shear modulus ($G''$) and in-plane shear wave velocity with vertical polarization ( $V^V_{s_{i}}$; $i=1, 2$) is,

$$G''=\rho (V^V_{s_{i}})^2$$ (9)

where wave propagation is in the i-direction ($i=1, 2$) with vertical polarization.