Elastodynamic Green’s functions of transversely isotropic n-layer half- and full-spaces subjected to a surface or buried time-harmonic annular loading and associated material degeneracy

Abstract

The elastodynamic behavior of layered media consisting of an arbitrary combination of isotropic and transversely isotropic layers is of great importance for many engineering applications. In this work, some appropriate elastodynamic Green’s functions pertinent to both the displacement and the stress fields are devised so that the problems associated with the n-layer semi-infinite and infinite media with any combinations of transversely isotropic and isotropic layers subjected to surface and buried oblique time-harmonic annular loading can all be treated in a unified manner. The material degeneracy arising due to the scenarios where one or more regions are isotropic is also discussed and treated. The proposed Green’s functions are then utilized to address a number of illustrative examples involving different combinations of transversely isotropic and isotropic layers subjected to such loading conditions as point load, ring as well as full circularly distributed loads, and loading along a circle perimeter including normal, horizontal, and torsional type loads. The robustness of the current formulations is shown through the verification of the available results pertinent to several problems with diverse topologies and loading conditions. Additional examples which have not been addressed in the literature are also treated herein and verified either by using finite element analysis (FEA) or by considering the pertinent analytical limiting cases.

Appendices

Appendix A: Material degeneracy

As it was alluded to, the determination of the unknown constants in Eqs. (26)–(31) require the coefficient matrices ${\mathscr {M}}$($\xi$) and ${\mathscr {N}}$($\xi$) which appear, respectively, in Eqs. (32) and (33) be invertible. The requirement is met so as long as all layers I, II, and III are transversely isotropic. But, if any of the layers is made of an isotropic material then ${\mathscr {M}}$($\xi$) will be non-invertible, as its determinant will be identically equal to zero. It should be noted that in the case of isotropic materials, $\lambda _{1}$ and $\lambda _{2}$ in Eq. (19) will become equal and cause Eq. (16) to have a repeated root. More specifically: (1) if layer I is isotropic then the first two columns of ${\mathscr {M}}$($\xi$) become identically the same; (2) if layer II is isotropic then columns 3 and 5 as well as columns 4 and 6 become identical; and (3) if layer III is isotropic then columns 7 and 9 and also columns 8 and 10 become identical. Thus under any of the above-mentioned conditions ${\mathscr {M}}$($\xi$) becomes singular. Hence, for isotropic layers, Eqs. (26), (28), (30), (34), and (36) should be modified to prevent singularity in Eq. (34). Assume the proposed medium consists of two finite and one semi-finite isotropic layers, then due to the equivalency of $\lambda _{1}^\Sigma$ and $\lambda _{2}^\Sigma \; (\Sigma =I,II,III)$, mentioned equations are changed as follows:

$$\begin{aligned}&{\tilde{F}}_m^m(\xi ,z)=B_m^I(\xi )\mathrm{{e}}^{-\lambda _1^{I}z}+zD_m^I(\xi )\mathrm{{e}}^{-\lambda _2^{I}z}, \end{aligned}$$

(A1)

$$\begin{aligned}&{\tilde{F}}_m^m(\xi ,z)=A_m^{II}(\xi )\mathrm{{e}}^{\lambda _1^{II}z}+B_m^{II}(\xi )\mathrm{{e}}^{-\lambda _1^{II}z}+zC_m^{II}(\xi )\mathrm{{e}}^{\lambda _2^{II}z}+zD_m^{II}(\xi )\mathrm{{e}}^{-\lambda _2^{II}z}, \end{aligned}$$

(A2)

$$\begin{aligned}&{\tilde{F}}_m^m(\xi ,z)=A_m^{III}(\xi )\mathrm{{e}}^{\lambda _1^{III}z}+B_m^{III}(\xi )\mathrm{{e}}^{-\lambda _1^{III}z}+zC_m^{III}(\xi )\mathrm{{e}}^{\lambda _2^{III}z}+zD_m^{III}(\xi )\mathrm{{e}}^{-\lambda _2^{III}z}, \end{aligned}$$

(A3)

$$\begin{aligned} \begin{aligned} {\mathscr {M}}(\xi )=\left[ \begin{array}{ccccc} -C_{44}^I\eta _1^I &{}\quad -C_{44}^I\eta _2^{*I} &{}\quad C_{44}^{II}\eta _1^{II} &{}\quad C_{44}^{II}\eta _1^{II} &{}\quad C_{44}^{II}\eta _2^{*II} \\ 0 &{}\quad 0 &{}\quad -C_{44}^{II}\eta _1^{II}e^{\lambda _1^{II}(-h_1)} &{}\quad -C_{44}^{II}\eta _1^{II}e^{-\lambda _1^{II}(-h_1)} &{}\quad -C_{44}^{II}\eta _2^{*II}e^{\lambda _2^{II}(-h_1)} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -C_{33}^{I} \nu _1^{I} &{}\quad C_{33}^{I} \nu _2^{*I} &{}\quad -C_{33}^{II} \nu _1^{II} &{}\quad C_{33}^{II} \nu _1^{II} &{}\quad -C_{33}^{II} \nu _2^{*II} \\ 0 &{}\quad 0 &{}\quad C_{33}^{II} \nu _1^{II}e^{\lambda _1^{II}(-h_1)} &{}\quad -C_{33}^{II} \nu _1^{II}e^{-\lambda _1^{II}(-h_1)} &{}\quad C_{33}^{II} \nu _2^{*II}e^{\lambda _2^{II}(-h_1)} \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -\alpha _3^{I} \lambda _1^{I} &{}\quad \alpha _3^{I} (1-z\lambda _2^{I}) &{}\quad -\alpha _3^{II} \lambda _1^{II} &{}\quad \alpha _3^{II} \lambda _1^{II} &{}\quad -\alpha _3^{II} (1+z\lambda _2^{II}) \\ 0 &{}\quad 0 &{}\quad \alpha _3^{II} \lambda _1^{II}e^{\lambda _1^{II}(-h_1)} &{}\quad -\alpha _3^{II}\lambda _1^{II}e^{-\lambda _1^{II}(-h_1)} &{}\quad \alpha _3^{II}(1+z\lambda _2^{II})e^{\lambda _2^{II}(-h_1)} \\ \vartheta _1^{I} &{}\quad \vartheta _2^{*I} &{}\quad -\vartheta _1^{II} &{}\quad -\vartheta _1^{II} &{}\quad -\vartheta _2^{*II} \\ 0 &{}\quad 0 &{}\quad \vartheta _1^{II}e^{\lambda _1^{II}(-h_1)} &{}\quad \vartheta _1^{II}e^{-\lambda _1^{II}(-h_1)} &{}\quad \vartheta _2^{*II}e^{\lambda _2^{II}(-h_1)} \end{array} \right. \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \\ \left. \begin{array}{ccccc} C_{44}^{II}\eta _2^{*II} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ -C_{44}^{II}\eta _2^{*II}e^{-\lambda _2^{II}(-h_1)} &{}\quad C_{44}^{III}\eta _1^{III}e^{\lambda _1^{III}(-h_1)} &{}\quad C_{44}^{III}\eta _1^{III}e^{-\lambda _1^{III}(-h_1)} &{}\quad C_{44}^{III}\eta _2^{*III}e^{\lambda _2^{III}(-h_1)} &{}\quad C_{44}^{III}\eta _2^{*III}e^{-\lambda _2^{III}(-h_1)} \\ 0 &{}\quad -C_{44}^{III}\eta _1^{III}e^{\lambda _1^{III}(-h_2)} &{}\quad -C_{44}^{III}\eta _1^{III}e^{-\lambda _1^{III}(-h_2)} &{}\quad -C_{44}^{III}\eta _2^{*III}e^{\lambda _2^{III}(-h_2)} &{}\quad -C_{44}^{III}\eta _2^{*III}e^{-\lambda _2^{III}(-h_2)} \\ -C_{33}^{II} \nu _2^{*II} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ C_{33}^{II} \nu _2^{*II}e^{-\lambda _2^{II}(-h_1)} &{}\quad -C_{33}^{III} \nu _1^{III}e^{\lambda _1^{III}(-h_1)} &{}\quad C_{33}^{III} \nu _1^{III}e^{-\lambda _1^{III}(-h_1)} &{}\quad -C_{33}^{III} \nu _2^{*III}e^{\lambda _2^{III}(-h_1)} &{}\quad -C_{33}^{III} \nu _2^{*III}e^{-\lambda _2^{III}(-h_1)} \\ 0 &{}\quad C_{33}^{III} \nu _1^{III}e^{\lambda _1^{III}(-h_2)} &{}\quad -C_{33}^{III} \nu _1^{III}e^{-\lambda _1^{III}(-h_2)} &{}\quad C_{33}^{III} \nu _2^{*III}e^{\lambda _2^{III}(-h_2)} &{}\quad C_{33}^{III} \nu _2^{*III}e^{-\lambda _2^{III}(-h_2)} \\ -\alpha _3^{II} (1-z\lambda _2^{II}) &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ \alpha _3^{II}(1-z\lambda _2^{II})e^{-\lambda _2^{II}(-h_1)} &{}\quad -\alpha _3^{III}\lambda _1^{III}e^{\lambda _1^{III}(-h_1)} &{}\quad \alpha _3^{III}\lambda _1^{III}e^{-\lambda _1^{III}(-h_1)} &{}\quad -\alpha _3^{III}(1+z\lambda _2^{III})e^{\lambda _2^{III}(-h_1)} &{}\quad -\alpha _3^{III}(1-z\lambda _2^{III})e^{-\lambda _2^{III}(-h_1)} \\ -\vartheta _2^{*II} &{}\quad 0 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ \vartheta _2^{*II}e^{-\lambda _2^{II}(-h_1)} &{}\quad -\vartheta _1^{III}e^{\lambda _1^{III}(-h_1)} &{}\quad -\vartheta _1^{III}e^{-\lambda _1^{III}(-h_1)} &{}\quad -\vartheta _2^{*III}e^{\lambda _2^{III}(-h_1)} &{}\quad -\vartheta _2^{*III}e^{-\lambda _2^{III}(-h_1)} \end{array}\right] , \end{aligned} \end{aligned}$$

(A4)

$$\begin{aligned} \begin{aligned}&\eta _i^*=\left( \alpha _3-\alpha _2\right) \left( 2\lambda _i+z\lambda _i^2\right) +\xi ^2\left( 1+\alpha _1\right) z-\frac{\rho \omega ^2}{C_{66}}z, \quad \quad \quad \quad \quad \quad \quad \\&\vartheta _i^*=\alpha _3 \left( 2\lambda _i+z\lambda _i^2\right) -\eta _i^*, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \\&\nu _i^*=-\alpha _3\frac{C_{13}}{C_{33}}\xi ^2(1+z\lambda _i)- \left( \frac{\rho \omega ^2}{C_{66}}-\xi ^2(1+\alpha _1)\right) (1+z\lambda _i) -\alpha _2\left( 3\lambda _i^2+z\lambda _i^3\right) , \; \; \; (i=2). \end{aligned} \end{aligned}$$

(A5)

One of the applicabilities of the proposed formulation is that a combination of layers with isotropic and transversely isotropic materials can be modeled. For example, if only layer II (the middle layer in the medium made of two finite and one semi-finite layers) is isotropic, then Eq. (A2) instead of Eq. (28), and all of the components of columns 5 and 6 in the medium of Eq. (A4) with the parameters of Eq. (A5) instead of components of columns 5 and 6 in the medium of Eq. (34) with the parameters of Eq. (36) should be utilized.

Appendix B: Green’s function associated with axisymmetric loadings

In general, axial loading in z-direction is an axisymmetric loading, while the plane loading is not axisymmetric (Fig. 2a and b). As a special case, one can define axisymmetric loading cases in the radial and rotational directions in the $x-y$ plane. In this case, the value of $\theta -\theta _0$ for P and Q in Eq. (41) should be zero and $\pi /2$ for axisymmetric radial and rotational plane loadings, respectively (Fig. 2d and e). The displacement Green’s function of axisymmetric loading conditions (Fig. 2b, d, and e) are as follows:

$$\begin{aligned} \begin{aligned}&G_{rj}(r,\theta ,z,t)=\frac{1}{2} \left\{ \int _{0}^{\infty }\left( \alpha _3\xi \frac{\mathrm{{d}}{\hat{F}}_{0}^{0}}{\mathrm{{d}}z}-\text {i}\xi {\hat{\chi }}_{0}^{0}\right) \xi J_{1}(r\xi )\mathrm{{d}}\xi \right. \quad \quad \quad \quad \\&\quad \left. +\int _{0}^{\infty }\left( -\alpha _3\xi \frac{\mathrm{{d}}{\hat{F}}_{0}^{0}}{\mathrm{{d}}z}-\text {i}\xi {\hat{\chi }}_{0}^{0}\right) \xi J_{-1}(r\xi )\mathrm{{d}}\xi \right\} \mathrm{{e}}^{\text {i}\omega (t-t_0)}, \quad \quad \quad \quad \\&G_{\theta j}(r,\theta ,z,t)=\frac{-\text {i}}{2} \left\{ \int _{0}^{\infty }\left( \alpha _3\xi \frac{\mathrm{{d}}{\hat{F}}_{0}^{0}}{\mathrm{{d}}z}-\text {i}\xi {\hat{\chi }}_{0}^{0}\right) \xi J_{1}(r\xi )\mathrm{{d}}\xi \right. \quad \quad \quad \quad \\&\quad \left. -\int _{0}^{\infty }\left( -\alpha _3\xi \frac{\mathrm{{d}}{\hat{F}}_{0}^{0}}{\mathrm{{d}}z}-\text {i}\xi {\hat{\chi }}_{0}^{0}\right) \xi J_{-1}(r\xi )\mathrm{{d}}\xi \right\} \mathrm{{e}}^{\text {i}\omega (t-t_0)}, \quad \quad \quad \quad \\&G_{zj}(r,\theta ,z,t)=\left\{ \int _{0}^{\infty }\left( \alpha _2 \frac{\mathrm{{d}}^2}{\mathrm{{d}}z^2}+\frac{\rho \omega ^2}{C_{66}}-\xi ^2(1+\alpha _1)\right) {\hat{F}}_{0}^{0}\xi J_{0}(r\xi )\mathrm{{d}}\xi \right\} \mathrm{{e}}^{\text {i}\omega (t-t_0)}, \end{aligned} \end{aligned}$$

(B1)

in which, ${\hat{F}}_m^m={\tilde{F}}_m^m \mathrm{{e}}^{\text {i}m\theta _0}$ and ${\hat{\chi }}_m^m={\tilde{\chi }}_m^m \mathrm{{e}}^{\text {i}m\theta _0}$. In fact, $G_{ij}(r,\theta ,z,t)$ represents the displacement in the i-direction at point ($r, \theta ,z$) and time t due to a uniform annular loading with inner radius “a” and outer radius “A” in the j-direction at $z=z_0$ plane and time $t_0$. From Eqs. (41)–(43), it can be shown that $G_{r \theta }=G_{\theta r}$. It should be noted that ${\hat{F}}_{0}^{0}$ and ${\hat{\chi }}_{0}^{0}$ in Eq. (B1) are obtained from Eqs. (26)–(38) for each loading direction. Moreover, for different radial, rotational, and axial loading directions, the parameters $X_m, Y_m,$ and $Z_m$ in Eq. (38) are as follows:

Radial Loading:

$$\begin{aligned} \begin{aligned}&X_0=-\frac{\xi A}{6} \sum _{k=0}^{\infty }\frac{(\frac{3}{2})_k}{(2)_k(\frac{5}{2})_k}\frac{(-A^2 \xi ^2)^k}{4^k k!}\,+\frac{\xi a}{6} \sum _{k=0}^{\infty }\frac{(\frac{3}{2})_k}{(2)_k(\frac{5}{2})_k}\frac{(-a^2 \xi ^2)^k}{4^k k!},\, X_m=0\quad (m\ne 0), \\&Y_0=\frac{\xi A}{6} \sum _{k=0}^{\infty }\frac{(\frac{3}{2})_k}{(2)_k(\frac{5}{2})_k}\frac{(-A^2 \xi ^2)^k}{4^k k!}\, -\frac{\xi a}{6} \sum _{k=0}^{\infty }\frac{(\frac{3}{2})_k}{(2)_k(\frac{5}{2})_k}\frac{(-a^2 \xi ^2)^k}{4^k k!},\, Y_m=0\quad (m\ne 0), \\&Z_m=0. \quad \quad \quad \quad \quad \quad \quad \end{aligned} \end{aligned}$$

(B2)

Rotational Loading:

$$\begin{aligned} \begin{aligned}&X_0=\text {i} \left\{ \frac{\xi A}{6} \sum _{k=0}^{\infty }\frac{(\frac{3}{2})_k}{(2)_k(\frac{5}{2})_k}\frac{(-A^2 \xi ^2)^k}{4^k k!}\,-\frac{\xi a}{6} \sum _{k=0}^{\infty }\frac{(\frac{3}{2})_k}{(2)_k(\frac{5}{2})_k}\frac{(-a^2 \xi ^2)^k}{4^k k!}\right\} ,\, X_m=0\quad (m\ne 0), \\&Y_0=\text {i} \left\{ \frac{\xi A}{6} \sum _{k=0}^{\infty }\frac{(\frac{3}{2})_k}{(2)_k(\frac{5}{2})_k}\frac{(-A^2 \xi ^2)^k}{4^k k!}\,-\frac{\xi a}{6} \sum _{k=0}^{\infty }\frac{(\frac{3}{2})_k}{(2)_k(\frac{5}{2})_k}\frac{(-a^2 \xi ^2)^k}{4^k k!}\right\} ,\, Y_m=0\quad (m\ne 0), \\&Z_m=0. \quad \quad \quad \quad \quad \quad \quad \end{aligned} \end{aligned}$$

(B3)

Axial Loading:

$$\begin{aligned} \begin{aligned}&X_m=0, \\&Y_m=0, \\&Z_0=\frac{J_1(A\xi )A}{\xi }-\frac{J_1(a\xi )a}{\xi },\quad Z_m=0\quad (m\ne 0). \end{aligned} \end{aligned}$$

(B4)

In the Eqs. (B2) and (B3), $(x)_k$ represents the Pochhammer symbol which can be expressed as:

$$\begin{aligned} \begin{aligned} (x)_k \equiv \frac{\Gamma (x+k)}{\Gamma (x)}=x (x+1) \ldots (x+k-1). \end{aligned} \end{aligned}$$

(B5)