Elastic field of a rotating cubic quasicrystal disk

Abstract

Owing to anisotropy in the phonon and phason fields, the analysis of the elastic problems in quasicrystals is more difficult than conventional crystals. A rotating cubic quasicrystal disk is considered. Due to the nature of cubic quasicrystals, the associated problem is not axisymmetric. The semi-inverse solution method is applied to derive the closed-form solution. Explicit expressions for the phonon and phason displacements and stresses are obtained. With reference to polar coordinates, the shear stress components of the phonon and phason fields vanish everywhere in the disk, and the radial and hoop stress components of the phonon and phason fields are independent of the angular coordinate $\theta$. However, the radial and hoop displacements of the phonon and phason fields are related to the angular coordinate. The influence of the material properties on the distribution of the elastic fields is discussed.

Appendices

Appendix A

Here, the relationships between the stiffness constants in plane stress state and in a three-dimensional space are given below. For the former case, $\sigma _{zz}$ and $\tau _{zz}$ are equal to 0. By virtue of Eqs. (1) and (2), we can obtain

$$\begin{aligned} \sigma _{zz}&=C_{12}\left( \varepsilon _{xx}+\varepsilon _{yy}\right) +C_{11}\varepsilon _{zz}+R_{12}\left( w_{xx}+w_{yy}\right) +R_{11} w_{zz}=0 \end{aligned}$$

(A.1)

$$\begin{aligned} \tau _{zz}&=R_{12}\left( \varepsilon _{xx}+\varepsilon _{yy}\right) +R_{11}\varepsilon _{zz}+K_{12}\left( w_{xx}+w_{yy}\right) +K_{11}w_{zz}=0 \end{aligned}$$

(A.2)

From the above, we can get the expressions for $\varepsilon _{zz}$ and $w_{zz}$ as follows by solving Eqs. (A.1) and (A.2).

$$\begin{aligned} \varepsilon _{zz}&=-\frac{\left( K_{11}R_{12}-K_{12}R_{11}\right) \left( w_{xx}+w_{yy}\right) +\left( C_{12}K_{11}-R_{11}R_{12}\right) \left( \varepsilon _{xx}+\varepsilon _{yy}\right) }{\left( C_{11}K_{11}-R_{11} R_{11}\right) } \end{aligned}$$

(A.3)

$$\begin{aligned} w_{zz}&=\frac{\left( R_{11}R_{12}-C_{11}K_{12}\right) \left( w_{xx}+w_{yy}\right) +\left( R_{11}C_{12}-C_{11}R_{12}\right) \left( \varepsilon _{xx}+\varepsilon _{yy}\right) }{\left( C_{11}K_{11}-R_{11} R_{11}\right) } \end{aligned}$$

(A.4)

Then one substitutes the expressions for $\varepsilon _{zz}$ and $w_{zz}$ above into Eqs. (1) and (2) to derive Eqs. (4) and (5). In addition, $\sigma _{yz},\sigma _{xz},\tau _{yz}$ and $\tau _{xz}$ are equal to 0 in plane stress state. So the relationships between the stiffness constants in plane stress state and those in a three-dimensional space can be given as follows:

$$\begin{aligned} C_{11}^{*}&=C_{11}+\frac{2R_{11}R_{12}C_{12}-R_{12}^{2}C_{11} -C_{12}^{2}K_{11}}{C_{11}K_{11}-R_{11}R_{11}} \end{aligned}$$

(A.5)

$$\begin{aligned} C_{12}^{*}&=C_{12}+\frac{2R_{11}R_{12}C_{12}-R_{12}^{2}C_{11} -C_{12}^{2}K_{11}}{C_{11}K_{11}-R_{11}R_{11}} \end{aligned}$$

(A.6)

$$\begin{aligned} C_{44}^{*}&=C_{44} \end{aligned}$$

(A.7)

$$\begin{aligned} K_{11}^{*}&=K_{11}+\frac{2R_{11}R_{12}K_{12}-C_{11}K_{12}^{2} -R_{12}^{2}K_{11}}{C_{11}K_{11}-R_{11}R_{11}} \end{aligned}$$

(A.8)

$$\begin{aligned} K_{12}^{*}&=K_{12}+\frac{2R_{11}R_{12}K_{12}-C_{11}K_{12}^{2} -R_{12}^{2}K_{11}}{C_{11}K_{11}-R_{11}R_{11}} \end{aligned}$$

(A.9)

$$\begin{aligned} K_{44}^{*}&=K_{44} \end{aligned}$$

(A.10)

$$\begin{aligned} R_{11}^{*}&=R_{11}+\frac{R_{12}\left( R_{11}R_{12}-C_{11} K_{12}\right) -C_{12}\left( R_{12}K_{11}-R_{11}K_{12}\right) }{C_{11} K_{11}-R_{11}R_{11}} \end{aligned}$$

(A.11)

$$\begin{aligned} R_{12}^{*}&=R_{12}+\frac{R_{12}\left( R_{11}R_{12}-C_{11} K_{12}\right) -C_{12}\left( R_{12}K_{11}-R_{11}K_{12}\right) }{C_{11} K_{11}-R_{11}R_{11}} \end{aligned}$$

(A.12)

$$\begin{aligned} R_{44}^{*}&=R_{44} \end{aligned}$$

(A.13)

In other words, for plane stress state, the material constants with superscript asterisk can be determined by those for plane strain state. The results for plane strain state can be directly transformed to those for plane stress state through a direct substitution of their relationships. Due to this reason, we omit the asterisk.

Appendix B

The material constants in polar coordinates can be expressed in terms of those in Cartesian coordinates below

$$\begin{aligned} c_{rr}&=c_{\theta \theta }=\frac{1}{2}\kappa _{1}+\eta _{1}+h_{1}\cos 4\theta , \end{aligned}$$

(B.1)

$$\begin{aligned} c_{r\theta }&=\frac{1}{2}\kappa _{1}-\eta _{1}-h_{1}\cos 4\theta , \end{aligned}$$

(B.2)

$$\begin{aligned} c_{\theta s}&=-c_{rs}=h_{1}\sin 4\theta , \end{aligned}$$

(B.3)

$$\begin{aligned} c_{ss}&=\eta _{1}-h_{1}\cos 4\theta , \end{aligned}$$

(B.4)

$$\begin{aligned} r_{rr}&=r_{\theta \theta }=\frac{1}{2}\kappa _{2}+\eta _{2}+h_{2}\cos 4\theta , \end{aligned}$$

(B.5)

$$\begin{aligned} r_{r\theta }&=\frac{1}{2}\kappa _{2}-\eta _{2}-h_{2}\cos 4\theta , \end{aligned}$$

(B.6)

$$\begin{aligned} r_{\theta s}&=-r_{rs}=h_{2}\sin 4\theta , \end{aligned}$$

(B.7)

$$\begin{aligned} r_{ss}&=\eta _{2}-h_{2}\cos 4\theta , \end{aligned}$$

(B.8)

$$\begin{aligned} k_{rr}&=k_{\theta \theta }=\frac{1}{2}\kappa _{3}+\eta _{3}+h_{3}\cos 4\theta , \end{aligned}$$

(B.9)

$$\begin{aligned} k_{r\theta }&=\frac{1}{2}\kappa _{3}-\eta _{3}-h_{3}\cos 4\theta , \end{aligned}$$

(B.10)

$$\begin{aligned} k_{\theta s}&=-k_{rs}=h_{3}\sin 4\theta , \end{aligned}$$

(B.11)

$$\begin{aligned} k_{ss}&=\eta _{3}-h_{3}\cos 4\theta , \end{aligned}$$

(B.12)

where

$$\begin{aligned} h_{1}&=\frac{1}{4}\left( C_{11}^{*}-C_{12}^{*}-2C_{44}^{*}\right) ,\ \ \kappa _{1}=C_{11}^{*}+C_{12}^{*}, \end{aligned}$$

(B.13)

$$\begin{aligned} \eta _{1}&=\frac{1}{4}\left( C_{11}^{*}-C_{12}^{*}+2C_{44}^{*}\right) . \end{aligned}$$

(B.14)

$$\begin{aligned} h_{2}&=\frac{1}{4}\left( R_{11}^{*}-R_{12}^{*}-2R_{44}^{*}\right) ,\ \ \kappa _{2}=R_{11}^{*}+R_{12}^{*}, \end{aligned}$$

(B.15)

$$\begin{aligned} \eta _{2}&=\frac{1}{4}\left( R_{11}^{*}-R_{12}^{*}+2R_{44}^{*}\right) . \end{aligned}$$

(B.16)

$$\begin{aligned} h_{3}&=\frac{1}{4}\left( K_{11}^{*}-K_{12}^{*}-2K_{44}^{*}\right) ,\ \ \kappa _{3}=K_{11}^{*}+K_{12}^{*}, \end{aligned}$$

(B.17)

$$\begin{aligned} \eta _{3}&=\frac{1}{4}\left( K_{11}^{*}-K_{12}^{*}+2K_{44}^{*}\right) , \end{aligned}$$

(B.18)

The specific expressions for$\ \bar{h}_{1},$ $\bar{\kappa }_{1},$ $\bar{\eta }_{1}$ are as follows

$$\begin{aligned} \bar{h}_{1}&=\frac{1}{4}\left( C_{11}-C_{12}-2C_{44}\right) ,\ \ \bar{\kappa }_{1}=C_{11}+C_{12}, \end{aligned}$$

(B.19)

$$\begin{aligned} \bar{\eta }_{1}&=\frac{1}{4}\left( C_{11}-C_{12}+2C_{44}\right) , \end{aligned}$$

(B.20)

where $C_{11},$ $C_{12},$ $C_{44}$ represent three basic elastic parameters of cubic anisotropic materials.

The constants $\bar{a}_{1},$ $\bar{b}_{1},$ $\bar{c}_{1}$ can be obtained by the following matrix equation

$$\begin{aligned} \left[ \begin{array}{ccc} 0 &{} \bar{h}_{1} &{} \bar{\eta }_{1}\\ 0 &{} \frac{1}{2}\bar{\kappa }_{1}+\bar{\eta }_{1} &{} \bar{h}_{1}\\ \bar{\kappa }_{1} &{} \bar{\kappa }_{1} &{} 0 \end{array} \right] \left\{ \begin{array}{c} \bar{a}_{1}\\ \bar{b}_{1}\\ \bar{c}_{1} \end{array} \right\} =\left\{ \begin{array}{c} 0\\ -\frac{1}{8}\rho \omega ^{2}a^{2}\\ \sigma _{0}+\frac{1}{4}a^{2}\rho \omega ^{2} \end{array} \right\} \end{aligned}$$

(B.21)