Vibration analysis of quasicrystal sector plates with porosity distribution in a thermal environment

Abstract

An approach to estimating the dynamic characteristic of one-dimensional orthorhombic quasicrystal sector plates with various boundary conditions is presented. These investigated structures have an arbitrary angle span, and the constituent materials of these models possess a certain porosity distribution. The Green–Naghdi generalized thermoelastic equations have been applied to the sector plates in a thermal environment to obtain an explicit expression, from which the heat fluxes equation can be converted into a state equation. The state-space method is conducted in basic equations of phason and phonon fields to capture governing equations of the structures along the r-direction. Fourier series expansions are utilized to control the span of the sector plates, subject to simply supported boundary conditions along the circumferential direction. Meanwhile, the remaining boundary conditions are treated by differential quadrature technique. The global propagator matrix is proposed to overcome the numerical instabilities caused by high-order frequency and large discrete points. Numerical examples show that each order frequency follows an increasing pattern from small to large with the increase of fixed boundary conditions. The changes in angular span can cause variations in stiffness, leading to alterations in structural stability.

Appendix A

Some parameters are written as

$$\begin{gathered} a_{1} = C_{44}^{{}} , \, a_{2} = R_{6} , \, a_{3} = R_{5} , \, a_{4} = C_{33}^{{}} K_{3} - R_{3}^{2} , \, a_{5} = - C_{13}^{{}} K_{3}^{{}} + R_{1} R_{3} , \, a_{6} = - C_{13}^{{}} K_{3}^{{}} + R_{2} R_{3} , \hfill \\ a_{7} = \beta_{1} , \, a_{8} = K_{3} , \, a_{9} = R_{3} , \, a_{10} = - C_{33}^{{}} R_{1} + C_{13}^{{}} R_{3} , \, a_{11} = - C_{33}^{{}} R_{2}^{{}} + C_{13}^{{}} R_{3} , \, a_{12} = \beta_{3} , \, a_{13} = C_{33} , \hfill \\ a_{14} = k_{33}^{{}} , \, a_{15} = \rho , \, a_{16} = C_{13}^{2} K_{3} + C_{33}^{{}} R_{1} R_{2} - C_{13}^{{}} R_{1} R_{3} - C_{13}^{{}} R_{2} R_{3} ,\,a_{17} = C_{12}^{{}} ,\,a_{18} = C_{66}^{{}} , \hfill \\ a_{19} = C_{13}^{2} K_{3} + C_{33}^{{}} R_{2}^{2} - 2C_{13}^{{}} R_{2} R_{3} ,\,a_{20} = C_{11}^{{}} ,\,a_{21} = C_{13}^{{}} ,\,a_{22} = R_{1} , \hfill \\ a_{23} = C_{33}^{{}} R_{1}^{2} - C_{33}^{{}} R_{1} R_{2} - C_{13}^{{}} R_{1} R_{3} + C_{13}^{{}} R_{2} R_{3} ,\,a_{24} = C_{33}^{{}} R_{1} R_{2} - C_{33}^{{}} R_{2}^{2} - C_{13}^{{}} R_{1} R_{3} + C_{13}^{{}} R_{2} R_{3} , \hfill \\ a_{25} = R_{1} R_{3} - R_{2} R_{3} ,\,a_{26} = R_{2} ,\,a_{27} = K_{1}^{{}} ,\,a_{28} = T_{0} ,\,a_{29} = k_{11}^{{}} ,\,a_{30} = k_{22}^{{}} ,\,a_{31} = C_{e} ,\,a_{32} = K_{2} . \hfill \\ \end{gathered}$$

(A1)

The state equation for the QC annular sector plate with CS boundary conditions are

$$\begin{aligned} \frac{{{\text{d}}\tilde{u}_{{ri}} }}{{{\text{d}}z}} = & - \sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{u}_{{zk}} } - \frac{{a_{2} }}{{a_{1} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{w}_{{zk}} } - \frac{1}{{a_{1} }}\tilde{\sigma }_{{zri}} {\text{ }}\left( {2 \le i \le N} \right), \\ \frac{{{\text{d}}\tilde{u}_{{\theta i}} }}{{{\text{d}}z}} = & \frac{1}{{r_{i} }}p\tilde{u}_{{zi}} + \frac{{a_{3} }}{{a_{1} }}\frac{1}{{r_{i} }}p\tilde{w}_{{zi}} - \frac{1}{{a_{1} }}\tilde{\sigma }_{{z\theta i}} {\text{ }}\left( {2 \le i \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{u}_{{zi}} }}{{{\text{d}}z}} = & \frac{{a_{5} }}{{a_{4} }}\sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \tilde{u}_{{rk}} } + \frac{{a_{6} }}{{a_{4} }}\frac{1}{{r_{i} }}\tilde{u}_{{ri}} - \frac{{a_{6} }}{{a_{4} }}\frac{1}{{r_{i} }}p\tilde{u}_{{\theta i}} + \frac{{a_{8} a_{{12}} }}{{a_{4} }}\tilde{T}_{i} + \frac{{a_{8} }}{{a_{4} }}\tilde{\sigma }_{{zzi}} - \frac{{a_{9} }}{{a_{4} }}\tilde{H}_{{zzi}} {\text{ }}\left( {2 \le i \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{w}_{{zi}} }}{{{\text{d}}z}} = & \frac{{a_{{10}} }}{{a_{4} }}\sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \tilde{u}_{{rk}} } + \frac{{a_{{11}} }}{{a_{4} }}\frac{1}{{r_{i} }}\tilde{u}_{{ri}} - \frac{{a_{{11}} }}{{a_{4} }}\frac{1}{{r_{i} }}p\tilde{u}_{{\theta i}} - \frac{{a_{9} a_{{12}} }}{{a_{4} }}\tilde{T}_{i} - \frac{{a_{9} }}{{a_{4} }}\tilde{\sigma }_{{zzi}} + \frac{{a_{{13}} }}{{a_{4} }}\tilde{H}_{{zzi}} {\text{ }}\left( {2 \le i \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{T}_{i} }}{{{\text{d}}z}} = & - \frac{1}{{a_{{14}} }}\tilde{q}_{{zi}} {\text{ }}\left( {2 \le i \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{zzi}} }}{{{\text{d}}z}} = & - a_{{15}} \omega ^{2} \tilde{u}_{{zi}} + a_{1} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{1ik}}^{{}} \tilde{u}_{{zk}} } + a_{2} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{1ik}}^{{}} \tilde{w}_{{zk}} } - \sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \tilde{\sigma }_{{zrk}} } - \frac{1}{{r_{i} }}\tilde{\sigma }_{{zri}} - \frac{1}{{r_{i} }}p\tilde{\sigma }_{{z\theta i}} {\text{ }}\left( {2 \le i \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{z\theta i}} }}{{{\text{d}}z}} = & \left( { - \frac{{a_{{16}} }}{{a_{4} }} + a_{{17}} } \right)\frac{1}{{r_{i} }}p\sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \tilde{u}_{{rk}} } + a_{{18}} p\sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \frac{1}{{r_{k} }}\tilde{u}_{{rk}} } - \left( {\frac{{a_{{19}} }}{{a_{4} }} - a_{{20}} - 2a_{{18}} } \right)p\frac{1}{{r_{i}^{2} }}\tilde{u}_{{ri}} \\ & - a_{{18}} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 2 \right)}} \tilde{u}_{{\theta k}} } - 2a_{{18}} \frac{1}{{r_{i} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{u}_{{\theta k}} } + a_{{18}} p\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \frac{1}{{r_{k} }}\tilde{u}_{{\theta k}} } + 2a_{{18}} \frac{1}{{r_{i}^{2} }}\tilde{u}_{{\theta i}} - \left( {\frac{{a_{{19}} }}{{a_{4} }} - a_{{20}} } \right)p^{2} \frac{1}{{r_{i}^{2} }}\tilde{u}_{{\theta i}} \\ & - a_{{15}} \omega ^{2} \tilde{u}_{{\theta i}} - \left( {\frac{{a_{6} a_{{12}} }}{{a_{4} }} + a_{7} } \right)p\frac{1}{{r_{i} }}\tilde{T}_{i} - \frac{{a_{6} }}{{a_{4} }}p\frac{1}{{r_{i} }}\tilde{\sigma }_{{zzi}} - \frac{{a_{{11}} }}{{a_{4} }}p\frac{1}{{r_{i} }}\tilde{H}_{{zzi}} {\text{ }}\left( {2 \le i \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{\sigma }_{{zri}} }}{{{\text{d}}z}} = & \left( {\frac{{a_{{19}} }}{{a_{4} }} - a_{{20}} } \right)\sum\limits_{{k = 2}}^{N} {\left( {X_{{ik}}^{{\left( 2 \right)}} - f_{{Nik}} } \right)\tilde{u}_{{rk}} } - \frac{{a_{5} a_{{21}} + a_{{10}} a_{{22}} }}{{a_{4} }}\sum\limits_{{k = 2}}^{N} {f_{{1ik}}^{{}} \tilde{u}_{{rk}} } - \left( {a_{{20}} - a_{{17}} - \frac{{a_{{23}} }}{{a_{4} }}} \right)\frac{1}{{r_{i} }}\sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \tilde{u}_{{rk}} } \\ & + \left( {\frac{{a_{{16}} }}{{a_{4} }} - a_{{17}} } \right)\left( {\sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \frac{1}{{r_{k} }}\tilde{u}_{{rk}} - E\tilde{u}_{{ri}} } } \right) + a_{{18}} p^{2} \frac{1}{{r_{i}^{2} }}\tilde{u}_{{ri}} + \left( {a_{{20}} - a_{{17}} + \frac{{a_{{24}} }}{{a_{4} }}} \right)\frac{1}{{r_{i}^{2} }}\tilde{u}_{{ri}} \\ & - a_{{15}} \omega ^{2} \tilde{u}_{{ri}} + \left( {\frac{{a_{{16}} }}{{a_{4} }} - a_{{17}} } \right)p\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \frac{1}{{r_{k} }}\tilde{u}_{{\theta k}} } - a_{{18}} p\frac{1}{{r_{i} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{u}_{{\theta k}} } + \left( {3a_{{18}} + \frac{{a_{{24}} }}{{a_{4} }}} \right)p\frac{1}{{r_{i}^{2} }}\tilde{u}_{{\theta i}} \\ & + \left( {a_{7} + \frac{{a_{6} a_{{12}} }}{{a_{4} }}} \right)\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{T}_{k} } + \frac{{a_{{12}} a_{{25}} }}{{a_{4} }}\frac{1}{{r_{i} }}\tilde{T}_{i} + \frac{{a_{6} }}{{a_{4} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{\sigma }_{{zzk}} } + \frac{{a_{{25}} }}{{a_{4} }}\frac{1}{{r_{i} }}\tilde{\sigma }_{{zzi}} \\ & + \frac{{a_{{10}} }}{{a_{4} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{H}_{{zzk}} } - \frac{{a_{{13}} a_{{26}} - a_{{13}} a_{{22}} }}{{a_{4} }}\frac{1}{{r_{i} }}\tilde{H}_{{zzi}} {\text{ }}\left( {2 \le i \le N} \right), \\ \frac{{{\text{d}}\tilde{H}_{{zzi}} }}{{{\text{d}}z}} = & a_{2} \sum\limits_{{k = 2}}^{{N - 1}} {f_{{1ik}}^{{}} \tilde{u}_{{zk}} } + \frac{{a_{2}^{2} }}{{a_{1} }}\sum\limits_{{k = 2}}^{{N - 1}} {f_{{1ik}}^{{}} \tilde{w}_{{zk}} } + \left( {\frac{{a_{2}^{2} }}{{a_{1} }} - a_{{27}} } \right)\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 2 \right)}} \tilde{w}_{{zk}} } + \left( {\frac{{a_{2}^{2} }}{{a_{1} }} - a_{{27}} } \right)\frac{1}{{r_{i} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{w}_{{zk}} } \\ & - \left( {\frac{{a_{3}^{2} }}{{a_{1} }} - a_{{28}} } \right)p^{2} \frac{1}{{r_{i}^{2} }}\tilde{w}_{{zi}} - a_{{15}} \omega ^{2} \tilde{w}_{{zi}} - \frac{{a_{3} }}{{a_{1} }}p\frac{1}{{r_{i} }}\tilde{\sigma }_{{z\theta i}} - \frac{{a_{2} }}{{a_{1} }}\sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \tilde{\sigma }_{{zrk}} } + \frac{{a_{2} }}{{a_{1} }}\frac{1}{{r_{i} }}\tilde{\sigma }_{{zri}} {\text{ }}\left( {2 \le i \le N - 1} \right), \\ \frac{{{\text{d}}\tilde{q}_{{zi}} }}{{{\text{d}}z}} = & \left( { - \frac{{a_{6} a_{{12}} a_{{28}} }}{{a_{4} }} - a_{7} a_{{28}} } \right)\omega ^{2} \sum\limits_{{k = 2}}^{N} {X_{{ik}}^{{\left( 1 \right)}} \tilde{u}_{{rk}} } - \left( {\frac{{a_{6} a_{{12}} a_{{28}} }}{{a_{4} }} + a_{7} a_{{28}} } \right)\frac{1}{{r_{i} }}\tilde{u}_{{ri}} - \left( {\frac{{a_{6} a_{{12}} a_{{28}} }}{{a_{4} }} + a_{7} a_{{28}} } \right)\frac{1}{{r_{i} }}\tilde{u}_{{\theta i}} \\ & - a_{{29}} \sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 2 \right)}} \tilde{T}_{k} } + a_{{29}} \frac{1}{{r_{i} }}\sum\limits_{{k = 2}}^{{N - 1}} {X_{{ik}}^{{\left( 1 \right)}} \tilde{T}_{k} } - a_{{30}} p^{2} \frac{1}{{r_{i}^{2} }}\tilde{T}_{i} - \frac{{a_{8} a_{{12}}^{2} a_{{28}} }}{{a_{4} }}\omega ^{2} \tilde{T}_{i} - a_{{15}} a_{{31}} \omega ^{2} \tilde{T}_{i} \\ & - \frac{{a_{8} a_{{12}} a_{{28}} }}{{a_{4} }}\omega ^{2} \tilde{\sigma }_{{zzi}} + \frac{{a_{9} a_{{12}} a_{{28}} }}{{a_{4} }}\omega ^{2} \tilde{H}_{{zzi}} {\text{ }}\left( {2 \le i \le N - 1} \right), \\ \end{aligned}$$

(A2)

with $f_{1ik} = g_{i1}^{\left( 1 \right)} g_{1k}^{\left( 1 \right)} ,f_{Nik} = g_{iN}^{\left( 1 \right)} g_{Nk}^{\left( 1 \right)} ,E_{1} = \left[ {\begin{array}{*{20}c} {{\mathbf{0}}_{{\left( {N - 1} \right)\left( {N - 2} \right)}} } & {g_{iN}^{\left( 1 \right)} } \\ \end{array} } \right].$