Image force in cubic piezoelectric quasicrystal half-space and bi-material composite space

Abstract

This paper is concerned with the image force in cubic piezoelectric quasicrystal semi-infinite space and infinite space containing two dissimilar quasicrystal half-spaces. On the basis of the Stroh formalism, the expressions of the Green’s function for generalized displacement and stress under multi-physical loading conditions are obtained exactly. Also, the image force applied to a generalized line dislocation with different boundary conditions is taken into account. The image force for the traction-free and electrically open surface always attracts dislocation to the boundary, while that for clamped and electrically closed surface shows the opposite tendency. Illustrative examples, such as generalized line force and line charge, are given to present the mechanical behaviors of quasicrystals under different loading conditions and investigate the influences of material parameters and dislocation scheme on image force. The results show that generalized line force has little effect on atomic configurations, and image force F is the strongest when phonon and phason dislocations interact together.

Appendix

By using Eqs. (1), (2), and (3), we have

$${\mathbf{Q}} = \left[ {\begin{array}{*{20}c} {C_{11} } & 0 & 0 & {R_{1} } & 0 & 0 & 0 \\ 0 & {C_{44} } & 0 & 0 & {R_{3} } & 0 & 0 \\ 0 & 0 & {C_{44} } & 0 & 0 & {R_{3} } & 0 \\ {R_{1} } & 0 & 0 & {K_{11} } & 0 & 0 & 0 \\ 0 & {R_{3} } & 0 & 0 & {K_{44} } & 0 & 0 \\ 0 & 0 & {R_{3} } & 0 & 0 & {K_{44} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - \xi_{11} } \\ \end{array} } \right], \, {\mathbf{R}} = \left[ {\begin{array}{*{20}c} 0 & 0 & {C_{12} } & 0 & 0 & {R_{2} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {d_{14} } \\ {C_{44} } & 0 & 0 & {R_{3} } & 0 & 0 & 0 \\ 0 & 0 & {R_{2} } & 0 & 0 & {K_{12} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & {d_{123} } \\ {R_{3} } & 0 & 0 & {K_{44} } & 0 & 0 & 0 \\ 0 & {d_{14} } & 0 & 0 & {d_{123} } & 0 & 0 \\ \end{array} } \right], \, {\mathbf{T}} = \left[ {\begin{array}{*{20}c} {C_{44} } & 0 & 0 & {R_{3} } & 0 & 0 & 0 \\ 0 & {C_{44} } & 0 & 0 & {R_{3} } & 0 & 0 \\ 0 & 0 & {C_{11} } & 0 & 0 & {R_{1} } & 0 \\ {R_{3} } & 0 & 0 & {K_{44} } & 0 & 0 & 0 \\ 0 & {R_{3} } & 0 & 0 & {K_{44} } & 0 & 0 \\ 0 & 0 & {R_{1} } & 0 & 0 & {K_{11} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & { - \xi_{33} } \\ \end{array} } \right].$$

(A1)

By Eq. (29), we introduced a new function

$$F^{*} { = }{\text{Re}} \sum\limits_{\beta = 1}^{7} {\sum\limits_{\alpha = 1}^{7} {\left[ {\frac{{p_{\alpha } }}{{p_{\alpha } - \overline{p}_{\beta } }}\left( {{\mathbf{BI}}_{\alpha } {\mathbf{B}}^{ - 1} } \right){\mathbf{L}}\left( {{\overline{\mathbf{B}}\mathbf{I}}_{\beta } {\overline{\mathbf{B}}}^{ - 1} } \right)^{T} } \right]} }$$

(A2)

For Eq. (A2), first, the right side is replaced by its complex conjugate, then the $\alpha$ and $\beta$ are interchanged, and finally, matrix transpose is applied on the right. Based on these operations, we can obtain that Eq. (A3) is equal to Eq. (A2).

$$F^{*} { = }{\text{Re}} \sum\limits_{\beta = 1}^{7} {\sum\limits_{\alpha = 1}^{7} {\left[ {\frac{{ - \overline{p}_{\beta } }}{{p_{\alpha } - \overline{p}_{\beta } }}\left( {{\mathbf{BI}}_{\alpha } {\mathbf{B}}^{ - 1} } \right){\mathbf{L}}\left( {{\overline{\mathbf{B}}\mathbf{I}}_{\beta } {\overline{\mathbf{B}}}^{ - 1} } \right)^{T} } \right]} } .$$

(A3)

Using Eq. (15), we get

$${\mathbf{L}} - {\mathbf{H}}^{ - 1} {\mathbf{SS}} = {\mathbf{H}}^{ - 1} .$$

(A4)

Since H⁻¹S is antisymmetric, Eq. (A4) can be re-expressed as follows:

$${\mathbf{H}}^{ - 1} - {\mathbf{L}} = {\mathbf{S}}^{T} {\mathbf{H}}^{ - 1} {\mathbf{S}}.$$

(A5)

In the above, H⁻¹ and L are positive definite, while S is singular [40]. We can conclude that H⁻¹- L is positive semi-definite.

Substituting Eqs. (37) and (38) into Eq. (36) to yield

$$\begin{gathered} \frac{1}{\pi }{\text{Im}} \left[ {{\mathbf{A}}_{1} < \ln \left( {x_{1} - p_{\alpha }^{(1)} d} \right) > {\mathbf{q}}} \right] + \frac{1}{\pi }{\text{Im}} \sum\limits_{\beta = 1}^{7} {\left[ {{\mathbf{A}}_{1} \left( {\ln \left( {x_{1} - \overline{p}_{\beta }^{(1)} d} \right)} \right){\mathbf{q}}_{\beta }^{(1)} } \right] = \frac{1}{\pi }{\text{Im}} \sum\limits_{\beta = 1}^{7} {\left[ {{\mathbf{A}}_{2} \left( {\ln \left( {x_{1} - p_{\beta }^{(1)} d} \right)} \right){\mathbf{q}}_{\beta }^{(2)} } \right],} } \hfill \\ \frac{1}{\pi }{\text{Im}} \left[ {{\mathbf{B}}_{1} < \ln \left( {x_{1} - p_{\alpha }^{(1)} d} \right) > {\mathbf{q}}} \right] + \frac{1}{\pi }{\text{Im}} \sum\limits_{\beta = 1}^{7} {\left[ {{\mathbf{B}}_{1} \left( {\ln \left( {x_{1} - \overline{p}_{\beta }^{(1)} d} \right)} \right){\mathbf{q}}_{\beta }^{(1)} } \right]} = \frac{1}{\pi }{\text{Im}} \sum\limits_{\beta = 1}^{7} {\left[ {{\mathbf{B}}_{2} \left( {\ln \left( {x_{1} - p_{\beta }^{(1)} d} \right)} \right){\mathbf{q}}_{\beta }^{(2)} } \right].} \hfill \\ \end{gathered}$$

(A6)

Similar manipulation of Eq. (21), we have

$$\frac{1}{\pi }{\text{Im}} \left[ {{\mathbf{A}}_{1} < \ln \left( {x_{1} - p_{\alpha }^{(1)} d} \right) > {\mathbf{q}}} \right] = - \frac{1}{\pi }{\text{Im}} \left[ {{\overline{\mathbf{A}}}_{1} < \ln \left( {x_{1} - \overline{p}_{\alpha }^{(1)} d} \right) > {\overline{\mathbf{q}}}} \right],$$

(A7)

By Eqs. (A7), (A6)₁, and Eq. (22)₁, we derive

$$- {\text{Im}} \sum\limits_{\beta = 1}^{7} {\left[ {{\overline{\mathbf{A}}}_{1} \left( {\ln \left( {x_{1} - \overline{p}_{\beta }^{(1)} d} \right)} \right){\mathbf{I}}_{\beta } {\overline{\mathbf{q}}}} \right]} + {\text{Im}} \sum\limits_{\beta = 1}^{7} {\left[ {{\mathbf{A}}_{1} \left( {\ln \left( {x_{1} - \overline{p}_{\beta }^{(1)} d} \right)} \right){\mathbf{q}}_{\beta }^{(1)} } \right]} = - {\text{Im}} \sum\limits_{\beta = 1}^{7} {\left[ {{\overline{\mathbf{A}}}_{2} \left( {\ln \left( {x_{1} - \overline{p}_{\beta }^{(1)} d} \right)} \right){\overline{\mathbf{q}}}_{\beta }^{(2)} } \right],}$$

(A8)

hence, Eq. (A8) has the expression

$${\mathbf{A}}_{1} {\mathbf{q}}_{\beta }^{(1)} + {\overline{\mathbf{A}}}_{2} {\overline{\mathbf{q}}}_{\beta }^{(2)} = {\overline{\mathbf{A}}}_{1} {\mathbf{I}}_{\beta } {\overline{\mathbf{q}}}.$$

(A9)

With the aid of Eq. (A6)₂, the following equation is obtained due to similarity

$${\mathbf{B}}_{1} {\mathbf{q}}_{\beta }^{(1)} + {\overline{\mathbf{B}}}_{2} {\overline{\mathbf{q}}}_{\beta }^{(2)} = {\overline{\mathbf{B}}}_{1} {\mathbf{I}}_{\beta } {\overline{\mathbf{q}}}.$$

(A10)

Equation (A9) has an alternative expression

$$\left( {{\mathbf{A}}_{1} {\mathbf{B}}_{1}^{ - 1} } \right){\mathbf{B}}_{1} {\mathbf{q}}_{\beta }^{(1)} + \left( {{\overline{\mathbf{A}}}_{2} {\overline{\mathbf{B}}}_{2}^{ - 1} } \right){\overline{\mathbf{B}}}_{2} {\overline{\mathbf{q}}}_{\beta }^{(2)} = \left( {{\overline{\mathbf{A}}}_{1} {\overline{\mathbf{B}}}_{1}^{ - 1} } \right){\overline{\mathbf{B}}}_{1} {\mathbf{I}}_{\beta } {\overline{\mathbf{q}}}.$$

(A11)

Making use of Eqs. (A10) and (A11) give

$$\left( {{\mathbf{A}}_{1} {\mathbf{B}}_{1}^{ - 1} - {\overline{\mathbf{A}}}_{2} {\overline{\mathbf{B}}}_{2}^{ - 1} } \right){\overline{\mathbf{B}}}_{2} {\overline{\mathbf{q}}}_{\beta }^{(2)} = \left( {{\mathbf{A}}_{1} {\mathbf{B}}_{1}^{ - 1} - {\overline{\mathbf{A}}}_{1} {\overline{\mathbf{B}}}_{1}^{ - 1} } \right){\overline{\mathbf{B}}}_{1} {\mathbf{I}}_{\beta } {\overline{\mathbf{q}}},$$

(A12)

or

$$\left( {{\mathbf{M}}_{1}^{ - 1} + {\overline{\mathbf{M}}}_{2}^{ - 1} } \right){\overline{\mathbf{B}}}_{2} {\overline{\mathbf{q}}}_{\beta }^{(2)} = 2{\mathbf{L}}_{1}^{ - 1} \left( {{\overline{\mathbf{B}}}_{1} {\mathbf{I}}_{\beta } {\overline{\mathbf{q}}}} \right),$$

(A13)

where

$${\mathbf{M}} = - i{\mathbf{BA}}^{ - 1} = {\mathbf{H}}^{ - 1} + i{\mathbf{H}}^{ - 1} {\mathbf{S}}, \, {\mathbf{M}}^{ - 1} = i{\mathbf{AB}}^{ - 1} = {\mathbf{L}}^{ - 1} - i{\mathbf{SL}}^{ - 1} .$$

(A14)