1 Introduction

The phenomenon of thermoelastic material finds a wide range of applications in all fields of science including atomic physics, industrial engineering, thermal power plants, submarine structures, pressure vessel, aerospace, chemical pipes and metallurgy. The spherically curved plate, like structures are used in pressure vessels, spherical domes of power plants in addition to many other industrial applications. For non-destructive evaluation of such spherical structures, the mechanics of elastic wave propagation in spherical curved plates must be understood. The current literature shows some valuable studies on Rayleigh surface wave propagation in isotropic solids with spherical boundaries.

The phenomenon of elastic wave propagation in cylindrical structures has received a significant attention recently [1–5]. Shah et al. [6] have analyzed three-dimensional hollow spheres using shell theory and found that the characteristic frequency equation is independent of the longitudinal wave number. Buldyev and Lanin [7] studied the surface wave propagation in solids with curved boundary conditions. Wang et al. [8] studied the stress wave- propagation in orthotropic laminated spherical shells subjected to arbitrary radial dynamic load with the help of finite Hankel transforms and Laplace transforms. Towfighi et al. [9, 10] studied the guided wave propagation problem in circumferential direction of cylindrical curved plate. Towfighi and Kundu [11] studied wave propagation of anisotropic spherical curved plates. Sharma and Pathania [12] investigated the generalized wave propagation in circumferential direction of transversely isotropic cylindrical curved plates. Yu et al. [13] used an orthogonal polynomial series expansion approach for determining the guided wave dispersion curves and the distribution of displacements in homogeneous anisotropic spherical curved plates based on three-dimensional elasticity and compared the results with Towfighi and Kundu [11]. Yu et al. [14] used a Legendre orthogonal polynomial series expansion approach for determining the characteristics of guided waves in continuous functionally graded piezoelectric materials as spherical curved plates. They reported the influence of radius to thickness ratio on dispersion curves. Yu et al. [15] presented an elastodynamic solution for the stress wave propagation in spherical curved plates composed of functionally graded materials. Yu and Xue [16] investigated the propagation of thermoelastic waves in orthotropic spherical curved plates subjected to stress-free, isothermal boundary conditions in context of the Green-Naghdi (GN) generalized thermoelastic theory. Yu and Dong [17] solved linearized three-dimensional piezoelasticity equations through orthogonal polynomial approach and determined the elastic wave propagation modes in a piezoelectric spherical curved plate. Sharma et al. [18] focused on the analysis of free vibrations of axisymmetric functionally graded hollow spheres. The material is assumed to be graded in radial direction with a simple power law.

The spherically curved plate like structures are used in pressure vessels, spherical domes of power plants in addition to many other industrial applications. From the perspective of non-destructive evaluation of such spherical structures, it is important to understand the mechanics of elastic wave propagation in spherical curved plates through various mathematical techniques. The current literature shows some valuable studies on Rayleigh surface wave propagation in isotropic solids with spherical boundaries. Considering various applications of spherical curved plates in industrial applications, it is proposed to model and study the propagation of circumferential waves in transradially isotropic and thermally conducting spherically curved elastic plates. The partial differential equations of motion and heat conduction along with boundary conditions on the inner and outer surfaces of a spherical curved plate constitute the mathematical model for this problem. The model has been solved using matrix Fröbenius method. The numerical solutions have also been obtained and presented for zinc, cobalt and silicon nitride material plate. It is expected that the wave characteristics may remain coupled and may be affected by temperature changes except for purely shear-harmonic (SH) wave motion wherein they get decoupled and have no effect of temperature. The obtained results compared with those existing in literature to validate the present approach and brief summary of the present work is given at the end of the paper.

2 Formulation and Solution

We consider a homogeneous, transversely isotropic, thermally conducting, elastic spherical curved plate with inner and outer radii \( a \) and \( b \) respectively. The plate is assumed initially at uniform temperature \( T_{0} \) in the undisturbed state. The geometry of the problem is shown in Fig. 1 and we considered the problem of wave propagation in the direction of the curvature. Here we represent wave carrier by different names like a curved plate/a spherical plate/a pipe segment or simply a pipe (all these nouns represent wave carrier only). Moreover, we do not include the reflected guided waves from the plate boundary. The considered geometry of the problem can be a segment of a sphere or a complete sphere. But we focus on analysis of the dispersive waves in the curved plate for waves propagating from section \( S_{1} \) to \( S_{2} \) (see Fig. 1). Clearly, the wave speed is proportional to radius of curvature. According to Towfighi and Kundu [11] the wavefront on the surface of a spherical shell is assumed to be toroidal. To study wave propagation in a spherical plate segment, the points \( {\text{S}}_{1} \) and \( {\text{S}}_{2} \) is aligned along the equator of a sphere by adjusting the positions of north and south poles. Therefore, in order to study the wave propagation between two points in a spherical plate segment, it is sufficient to solve the governing equations for \( \theta = \frac{\pi }{2} \) only. The linear governing equations of motion and heat conduction in the absence of body forces and heat sources for a thermoelastic spherical structure have been considered here. These govern the displacement \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {u} = \left( {u_{r} ,u_{\theta } ,u_{\phi } } \right) \) and temperature change \( T\left( {r,\phi ,t} \right) \) in the plate and are given by Dhaliwal and Singh [19]:

$$ \sigma_{rr,r} + \frac{ 1}{\text{r}}\sigma_{{{\text{r}}\phi ,\phi }} + \frac{ 1}{\text{r}}\left[ { 2\sigma_{rr} - \sigma_{\theta \theta } - \sigma_{\phi \phi } } \right] \, = \rho {{\ddot{\textit u}}}_{\text{r}} \, $$
(1)
$$ \sigma_{r\phi ,r} + \frac{ 1}{\text{r}}\sigma_{\phi \phi ,\phi } + \frac{ 3}{\text{r}}\sigma_{r\phi } \, = \rho {{\ddot{\textit u}}}_{\phi } \, $$
(2)
$$ \sigma_{r\theta ,r} + \frac{ 1}{\text{r}}\sigma_{\phi \theta ,\phi } + + \frac{ 3}{\text{r}} \, \sigma_{r\theta } = \rho {{\ddot{\textit u}}}_{\theta } \, $$
(3)
$$ K_{3} \left( {T_{rr} + \frac{2}{r}T_{,r} } \right) + K_{1} \frac{1}{{r^{2} }}T_{,\phi \phi } \, - \rho C_{e} \dot{T}\,\, = T_{0} \left[ {\beta_{1} \left( {\dot{e}_{\theta \theta } + \dot{e}_{\phi \phi } } \right) + \beta_{3} \dot{e}_{rr} } \right] $$
(4)

where

$$ \begin{aligned}& \sigma_{\theta \theta } = c_{11} e_{\theta \theta } + c_{12} e_{\phi \phi } + c_{13} e_{rr} - \beta_{1} T,\quad \sigma_{r\theta } = 2c_{44} e_{r\theta } \hfill \\ & \sigma_{\phi \phi } = c_{12} e_{\theta \theta } + c_{11} e_{\phi \phi } + c_{13} e_{rr} - \beta_{1} T,\quad \sigma_{r\phi } = 2c_{44} e_{r\phi } \hfill \\ \end{aligned} $$
(5)
$$ \begin{aligned} \sigma_{rr} = c_{13} e_{\theta \theta } + c_{13} e_{\phi \phi } + c_{33} e_{rr} - \beta_{3} T,\quad \sigma_{\theta \phi } = 2c_{66} e_{\theta \phi } \hfill \\ e_{rr} = \frac{{\partial u_{r} }}{\partial r},\quad e_{\theta \theta } = \frac{{u_{r} }}{r},\quad e_{\phi \phi } = \frac{1}{r}\frac{{\partial u_{\phi } }}{\partial \phi } + \frac{{u_{r} }}{r} \hfill \\ e_{r\phi } = \frac{1}{2}\left[ {\frac{1}{r}\frac{{\partial u_{r} }}{\partial \phi } - \frac{{u_{\phi } }}{r} + \frac{{\partial u_{\phi } }}{\partial r}} \right],\quad e_{r\theta } = \frac{1}{2}\left[ {\frac{{\partial u_{\theta } }}{\partial r} - \frac{{u_{\theta } }}{r}} \right] \hfill \\ e_{\phi \theta } = \frac{1}{2}\left[ {\frac{{\partial u_{\theta } }}{\partial \phi }} \right]. \hfill \\ \end{aligned} $$
(6)

Here \( c_{11} ,\,c_{12} ,\,\,c_{13} ,\,\,c_{33} \,{\text{and}}\,\,c_{44} \) are isothermal elastic parameters; \( \alpha_{1} \), \( \alpha_{3} \) are the coefficient of Linear thermal expansion, \( K_{1} \), \( K_{3} \) are the coefficients of thermal conductivities, along and perpendicular to the axis of symmetry, \( \rho \) and \( C_{e} \), are the density and specific heat at constant strain respectively. The comma notation is used for spatial derivatives and the superposed dot denotes time differentiation. Sharma and Sharma [20] proved thermodynamically that \( K_{1} > 0,\;K_{3} > 0 \) and of course \( \rho > 0 \) and \( T_{0} > 0 \). In addition it is proved that \( C_{e} > 0 \) and the isothermal elasticities are components of a positive definite fourth-order tensor. The necessary and sufficient conditions for the satisfaction of latter requirements are

$$ c_{11} > 0,\,\,\,c_{11} > c_{12} ,\,\,c_{11}^{2} > c_{12}^{2} ,\,\,c_{44} > 0,\,c_{33} \left( {c_{11} + c_{12} } \right) > c_{13}^{2} . $$
(7)
Fig. 1
figure 1

Geometry of the problem

Full size image

On substituting Eqs. (5) and (6) by Eqs. (1)–(4) the governing differential equations in non-dimensional form are obtained and are given as:

$$ u_{\theta ,rr} - \frac{2}{r}u_{\theta ,r} + \frac{1}{{r^{2} }}u_{\theta } + \frac{{\left( {c_{1} - c_{2} } \right)}}{{2r^{2} }}u_{\theta ,\phi \phi } + \frac{3}{r}\left( {u_{\theta ,r} - \frac{{u_{\theta } }}{r}} \right) - {{\ddot{\textit u}}}_{\theta } = 0 $$
(8)
$$ \begin{aligned} \,c_{4} u_{r,rr} + \frac{{2c_{4} }}{r}u_{r,r} + \frac{{2\left( {c_{3} - c_{1} - c_{2} } \right)}}{{r^{2} }}u_{r} + \frac{1}{{r^{2} }}u_{r,\phi \phi } + \frac{{1 + c_{3} }}{r}u_{\phi ,r\phi } \hfill \\ \quad + \frac{{c_{3} - c_{1} - c_{2} - 1}}{{r^{2} }}u_{\phi ,\phi } - {{\ddot{\textit u}}}_{r} - \beta *\left( {\frac{\partial }{\partial r} + \frac{2}{r} - \frac{{2\bar{\beta }}}{r}} \right)T = 0 \hfill \\ \end{aligned} $$
(9.1)
$$ u_{\phi ,rr} + \frac{2}{r}u_{\phi ,r} - \frac{2}{{r^{2} }}u_{\phi } + \frac{{c_{1} }}{{r^{2} }}u_{\phi ,\phi \phi } + \frac{{\left( {1 + c_{3} } \right)}}{r}u_{r,r\phi } + \frac{{2 + c_{1} + c_{2} }}{{r^{2} }}u_{r,\phi } - {{\ddot{\textit u}}}_{\phi } - \frac{{\bar{\beta }\beta *T_{,\phi } }}{r} = 0 $$
(9.2)
$$ T_{,rr} + \frac{2}{r}T_{,r} + \frac{1}{{r^{2} }}T_{,\phi \phi } - \Omega^{*} \dot{T}\, - \varepsilon^{*} \Omega^{*} \left( {\dot{u}_{r,r} + \left( {\frac{2}{r}\dot{u}_{r} + \frac{1}{r}\dot{u}_{\phi ,\phi } } \right)\bar{\beta }} \right) = 0 $$
(9.3)

We define:

$$ \left. \begin{aligned} &\zeta = r/R,\,\,\,U_{i} = u_{i} /R,\,\,\tau_{ij} = \frac{{\sigma_{ij} }}{{c_{44} }},\,\,\,\varepsilon_{T} = \frac{{\beta_{3}^{2} T_{0} }}{{\rho C_{e} c_{44} }},\,\,\,\varsigma_{1} = a/R, \hfill \\ & \varsigma_{2} = b/R,\,\,c_{1} = \frac{{c_{11} }}{{c_{44} }},c_{2} = \frac{{c_{12} }}{{c_{44} }},\,\,c_{3} = \frac{{c_{13} }}{{c_{44} }},\,\,c_{4} = \frac{{c_{33} }}{{c_{44} }},\,\,\,\varepsilon^{*} = \frac{{\varepsilon_{T} \Omega^{*} }}{{\beta^{*} }}, \hfill \\& \bar{\beta } = \frac{{\beta_{1} }}{{\beta_{3} }},\,\,\bar{K} = \frac{{K_{1} }}{{K_{3} }},\,\,\tau = \frac{{v_{s} }}{R}t,\,\,T^{\prime} = \frac{T}{{T_{0} }},\,\,\,\,\Omega^{*} = \frac{{\omega^{*} R}}{{v_{s} }},\beta^{*} = \frac{{\beta_{1} T_{0} }}{{c_{44} }} \hfill \\ \end{aligned} \right\} $$
(10)

Here dashes are ignored for convenience, \( \omega^{ \cdot } = \frac{{C_{e} c_{44} }}{{K_{3} }} \) is the thermoelastic characteristic frequency of the plate and \( v_{s}^{2} = \frac{{c_{44} }}{\rho } \) shear wave velocity in the medium respectively and the quantity \( \varepsilon \) is called the thermoelastic coupling parameters.

Introducing quantities (10) in Eqs. (8) and (9), we obtain

$$ \left( {\frac{{\partial^{2} }}{{\partial \zeta^{2} }} + \frac{2}{\zeta }\frac{\partial }{\partial \zeta } + \frac{1}{{\zeta^{2} }} - \frac{{\partial^{2} }}{{\partial \tau^{2} }}} \right)U_{\theta } + \frac{{\left( {c_{1} - c_{2} } \right)}}{{2\zeta^{2} }}\frac{{\partial^{2} }}{{\partial \phi^{2} }}U_{\theta } + \frac{3}{\zeta }\left( {\frac{\partial }{\partial \zeta } - \frac{1}{\zeta }} \right)U_{\theta } = 0 $$
(11)
$$ \left( {c_{4} \left( {\frac{{\partial^{2} }}{{\partial \zeta^{2} }} + \frac{2}{\zeta }\frac{\partial }{\partial \zeta }} \right) + \frac{{2\left( {c_{3} - c_{1} - c_{2} } \right)}}{{\zeta^{2} }} - \frac{{\partial^{2} }}{{\partial \tau^{2} }}} \right)\,U_{\zeta } + \frac{1}{{\zeta^{2} }}\frac{{\partial^{2} }}{{\partial \phi^{2} }}U_{\zeta } + \frac{{1 + c_{3} }}{\zeta }\frac{{\partial^{2} }}{\partial \zeta \partial \phi }U_{\phi } + \frac{{c_{3} - c_{1} - c_{2} - 1}}{{\zeta^{2} }}\frac{\partial }{\partial \phi }U_{\phi } - \beta^{*} \left( {\frac{\partial }{\partial \zeta } + \frac{2}{\zeta } - \frac{{2\bar{\beta }}}{\zeta }} \right)\Theta = 0 $$
(12)
$$ \left( {\frac{{\partial^{2} }}{{\partial \zeta^{2} }} + \frac{2}{\zeta }\frac{\partial }{\partial \zeta } + \frac{1}{{\zeta^{2} }} - \frac{{\partial^{2} }}{{\partial \tau^{2} }}} \right)U_{\phi } + \frac{{c_{1} }}{{\zeta^{2} }}\frac{{\partial^{2} }}{{\partial \phi^{2} }}U_{\phi } + \frac{{\left( {1 + c_{3} } \right)}}{\zeta }\frac{\partial }{\partial \zeta \partial \phi }U_{\zeta } + \frac{{2 + c_{1} + c_{2} }}{{\zeta^{2} }}\frac{\partial }{\partial \phi }U_{\zeta } - \frac{{\bar{\beta }\beta *}}{\zeta }\frac{\partial \Theta }{\partial \phi } = 0 $$
(13)
$$ \left( {\frac{{\partial^{2} }}{{\partial \zeta^{2} }} + \frac{2}{\zeta }\frac{\partial }{\partial \zeta } + \frac{1}{{\zeta^{2} }} - \Omega^{*} \frac{\partial }{\partial \tau }} \right)\Theta - \varepsilon^{*} \Omega^{*} \frac{\partial }{\partial \tau }\left( {U_{\zeta ,\zeta } + \left( {\frac{2}{\zeta }\frac{{\partial U_{\zeta } }}{\partial \zeta } + \frac{1}{\zeta }\frac{{\partial U_{\phi } }}{\partial \phi }} \right)\bar{\beta }} \right) = 0 $$
(14)

For further simplification of the Eqs. (11)–(14), we introduce the following transformations

$$ \left. \begin{aligned} &U_{\zeta } = \bar{U}/\sqrt \zeta \hfill \\ &U_{\theta } = \bar{V}/\sqrt \zeta \hfill \\ &U_{\phi } = \bar{W}/\sqrt \zeta \hfill \\ &\Theta = \bar{\Theta }/\sqrt \zeta \hfill \\ \end{aligned} \right\} $$
(15)

In light of the transformation (15), the Eqs. (11)–(14), upon simplification provide us

$$ \left( {\frac{{\partial^{2} }}{{\partial \zeta^{2} }} + \frac{1}{\zeta }\frac{\partial }{\partial \zeta } - \frac{9}{{4\zeta^{2} }} + \frac{{c_{1} - c_{2} }}{{\zeta^{2} }} + \frac{{c_{1} - c_{2} }}{{2\zeta^{2} }}\frac{{\partial^{2} }}{{\partial \phi^{2} }} - \frac{{\partial^{2} }}{{\partial \tau^{2} }}} \right)\bar{V} = 0 $$
(16)
$$ \left[ {c_{4} \left( {\frac{{\partial^{2} }}{{\partial \zeta^{2} }} + \frac{1}{\zeta }\frac{\partial }{\partial \zeta } - - \frac{1}{{4\zeta^{2} }}} \right) - \frac{{2\left( {c_{1} - c_{3} + c_{2} } \right)}}{{\zeta^{2} }} + \frac{1}{{\zeta^{2} }}\frac{{\partial^{2} }}{{\partial \phi^{2} }} - \frac{{\partial^{2} }}{{\partial \tau^{2} }}} \right]\bar{W} - \left[ {\frac{{1 + c_{3} }}{\zeta }\left( {\frac{\partial }{\partial \zeta } - \frac{1}{\zeta } - \frac{1}{2\zeta }} \right) + \frac{1}{{\zeta^{2} }}\left( {2c_{3} - c_{1} - c_{2} } \right)} \right]\frac{{\partial^{2} }}{{\partial \phi^{2} }}\bar{U} - \beta^{*} \left( {\frac{\partial }{\partial \zeta } - \frac{1}{2\zeta } + \frac{2}{\zeta } - \frac{{2\bar{\beta }}}{\zeta }} \right)\bar{\Theta } = 0 $$
(17)
$$ \left( {\frac{{\partial^{2} }}{{\partial \zeta^{2} }} + \frac{1}{\zeta }\frac{\partial }{\partial \zeta } - \frac{1}{{4\zeta^{2} }} + \frac{{c_{1} }}{{\zeta^{2} }}\frac{{\partial^{2} }}{{\partial \phi^{2} }} - \frac{{2 - c_{1} + c_{2} }}{{\zeta^{2} }} + \frac{{\partial^{2} }}{{\partial \tau^{2} }}} \right)\bar{U} + \left( {\frac{{1 + c_{3} }}{\zeta }\left( {\frac{\partial }{\partial \zeta } - \frac{1}{2\zeta }} \right) + \frac{{2 + c_{1} + c_{2} }}{{\zeta^{2} }}} \right)\bar{W} - \frac{1}{\zeta }\bar{\beta }\beta^{*} \bar{\Theta } = 0 $$
(18)
$$ \left( {\frac{{\partial^{2} }}{{\partial \zeta^{2} }} + \frac{1}{\zeta }\frac{\partial }{\partial \zeta } - \frac{1}{{4\zeta^{2} }} + \frac{{\bar{K}}}{{\zeta^{2} }}\frac{{\partial^{2} }}{{\partial \phi^{2} }} - \Omega^{*} } \right)\bar{\Theta } - \Omega^{*} \varepsilon^{*} \left[ {\left( {\frac{\partial }{\partial \zeta } - \frac{1}{2\zeta } + \frac{{2\bar{\beta }}}{\zeta }} \right)\bar{W} + \frac{\beta }{\zeta }\frac{{\partial^{2} }}{{\partial \phi^{2} }}\bar{G}} \right] = 0 $$
(19)

According to Yu et al. [13] in toroidal wave travel path, the wave front position is only a function of \( \phi \) because, all points with the same \( \phi \) and for different values of \( \theta \) are in phase and hence the motion is independent of \( \theta \). In a spherical geometry, waves travel a constant angle \( \left( \phi \right) \) rather than a constant linear distance in a given time interval. Therefore the toroidal wave travel path length is defined as the product \( m\varsigma_{2} \phi \) which is dimensionally identical to \( \Omega \tau \). Thus, the displacement components and temperature of this toroidal has been written as Yu et al. [13]:

$$ \left. \begin{aligned} \bar{U}\left( {\zeta ,\phi ,\tau } \right) = U\left( \zeta \right)\exp \left( {im\varsigma_{2} \phi + i\Omega \tau } \right) \hfill \\ \bar{V}\left( {\zeta ,\phi ,\tau } \right) = V\left( \zeta \right)\exp \left( {im\varsigma_{2} \phi + i\Omega \tau } \right) \hfill \\ \bar{W}\left( {\zeta ,\phi ,\tau } \right) = W\left( \zeta \right)\exp \left( {im\varsigma_{2} \phi + i\Omega \tau } \right) \hfill \\ \bar{\Theta }\left( {\zeta ,\phi ,\tau } \right) = \Theta^{*} \left( \zeta \right)\exp \left( {im\varsigma_{2} \phi + i\Omega \tau } \right) \hfill \\ \end{aligned} \right\} $$
(20)

where \( U\left( \zeta \right), \, V\left( \zeta \right),\,\,\,W\left( \zeta \right) \) and \( \Theta^{*} \left( \zeta \right) \) represent the amplitude of vibration in the radial and two tangential directions, \( m \) is the magnitude of wave vector along wave propagation direction, and \( \varsigma_{2} \) is the non-dimensional outer radius of the plate.

Upon using solution (20) in Eqs. (16)–(19) and further simplifying, we obtain

$$ \nabla_{2}^{2} + \left( {1 - \eta_{0}^{2} /\xi^{2} } \right)V = 0 $$
(21)
$$ \left[ {\begin{array}{*{20}c} {c_{4} \nabla_{2}^{2} } & {im\varsigma_{2} \left( {\frac{{1 + c_{3} }}{\xi }\frac{\partial }{\partial \zeta } - \mu \,_{1}^{2} /\xi^{2} } \right)} & { - \Omega^{ - 1} \beta^{*} \left( {\frac{\partial }{\partial \xi } + \frac{{3 - 4\bar{\beta }}}{2\xi }} \right)} \\ {im\varsigma_{2} \left( {\frac{{1 + c_{3} }}{\xi }\frac{\partial }{\partial \zeta } + \mu \,_{1}^{2} /\xi^{2} } \right)} & {\nabla_{2}^{2} + \left( {1 - \mu \,_{2}^{2} /\xi \,^{2} } \right)} & { - \frac{{im\,\varsigma_{2} \,\Omega^{ - 1} \bar{\beta }\beta^{*} }}{\xi }} \\ {i\varepsilon^{*} \,\Omega^{*} \left( {\frac{\partial }{\partial \xi } + \frac{{4\bar{\beta } - 1}}{2\xi }} \right)} & {\frac{{i\varepsilon^{*} \,\Omega^{*} m\varsigma_{2} \bar{\beta }}}{\xi }} & {\nabla_{2}^{2} + \left( { - i\Omega^{ - 1} - \mu \,_{4}^{2} /\xi^{2} } \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} U \\ W \\ {\Theta^{*} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ \end{array} } \right] $$
(22)

where

$$ \left. \begin{aligned} \nabla_{2}^{2} = \frac{{\partial^{2} }}{{\partial \xi^{2} }} + \frac{1}{\xi }\frac{\partial }{\partial \xi },\,\xi = \zeta \,\Omega ,\,\,\,\mu_{1}^{2} = \frac{{3 + 2\left( {c_{1} + c_{2} } \right) - c_{3} }}{2} \hfill \\ \mu_{2}^{2} = \frac{{9 + 4c_{1} m^{2} \varsigma_{2}^{2} }}{4},\,\,\,\mu_{3}^{2} = \frac{{c_{4} + 8\left( {c_{1} + c_{2} - c_{3} } \right) + 4m^{2} \varsigma_{2}^{2} }}{4} \hfill \\ \mu_{4}^{2} = \frac{{1 + 4\bar{K}m^{2} \varsigma_{2}^{2} }}{4},\,\,\eta_{0}^{2} = \frac{{9 + 2\left( {c_{1} - c_{2} } \right)m^{2} \varsigma_{2}^{2} }}{4} \hfill \\ \end{aligned} \right\} $$
(23)

Equation (21) gives purely shear wave, which is not affected by temperature change. The solution of Eq. (21) is given as

$$ V = \zeta^{{ - \frac{1}{2}}} \left( {M_{11} J_{{\eta_{0} }} \left( {\zeta \Omega } \right) + M_{12} Y_{{\eta_{0} }} \left( {\zeta \Omega } \right)} \right)\exp \left( {im\varsigma_{2} \phi + i\Omega \tau } \right) $$
(24)

where \( J_{\eta } \) and \( Y_{\eta } \) are the Bessel function of first and second kind and \( \eta_{0}^{2} = \left( {9 - 2\left( {c_{1} - c_{2} } \right)m^{2} \varsigma_{2}^{2} } \right)/4 > 0 \) and \( M_{11} \) and \( M_{12} \) are arbitrary constants to be determined by boundary conditions.

2.1 Matrix Fröbenius Solution

Equation (22) has been solved with the help of matrix Fröbenius method. Clearly the point \( r = 0 \) (i.e.\( \xi \, = \,0 \)) is a regular singular point of Eq. (22) and all the coefficients of this differential equations are finite, single valued and continuous in the interval \( \eta_{1} \le \xi \le \eta_{2} \), where \( \eta_{1} = \varsigma_{1} \Omega \) and \( \eta_{2} = \varsigma_{2} \Omega \). The field quantities satisfy all the necessary conditions to have series expansions and hence the Fröbenius power series method is applicable to solve the coupled system of differential equations. Thus, we have taken the solution vector of the type

$$ Y_{n} = \sum\limits_{k = 0}^{\infty } {Z_{k} } \xi^{s + k} $$
(25)

where

$$ Y_{n} = \left[ {\begin{array}{*{20}c} U & W & {\Theta^{*} } \\ \end{array} } \right]^{\prime } \,\,\, , $$
(26)
$$ Z_{k} = \left[ {\begin{array}{*{20}c} {A_{k} } & {B_{k} } & {D_{k} } \\ \end{array} } \right]^{\prime } . $$
(27)

Here \( s \) is a constant (real or complex) to be determined and \( A_{k} \,,\,\,B_{k} \), \( D_{k} \) are unknown coefficients to be determined. We need solution in the domain \( \eta_{1} \le \zeta \le \eta_{2} \), \( \zeta_{1} > 0 \). The solution (25) is valid in some deleted interval \( 0 < \zeta < R^{\prime} \), \( R^{\prime} > \eta_{2} \) (about the origin) where \( R^{\prime} \) is the radius of convergence.

Upon substituting solution (25) along with its derivatives in Eq. (22) and simplifying, we get

$$ \sum\limits_{k = 0}^{\infty } {\left[ {H_{{\mathbf{1}}} \left( {\upsilon + k} \right)\xi^{ - 2} + H_{2} \left( {\upsilon + k} \right)\xi^{ - 1} + H} \right]\,} \xi^{\upsilon + k} Z_{{\mathbf{k}}} = 0\, $$
(28)

where

$$ \left. \begin{aligned} &H = diag\left( {1,\,\,1,\,\, - im\varsigma_{2} \Omega^{ - 1} \bar{\beta }\,} \right) \hfill \\ &H_{{\mathbf{1}}} {\mathbf{(}}\upsilon + k{\mathbf{)}} = \left( {H_{iq} \left( {\upsilon + k} \right)} \right),\,\,\,i,\,q = \,1,\,\,2,\,\,3 \hfill \\ &H_{2} {\mathbf{(}}\upsilon + k{\mathbf{)}} = \left( {H_{iq} \left( {\upsilon + k} \right)\,} \right),\,\,\,i,\,q = 1,\,\,2,\,\,3 \hfill \\ \end{aligned} \right\} $$
(29)

The non-zero elements, \( H_{iq} \) and \( H^{\prime}_{iq} \), of matrices \( H_{1} \) and \( H_{2} \) are given as under:

$$ \left. \begin{aligned} &H_{11} \left( {\upsilon + k} \right) = \left( {c_{4} \left( {\upsilon + k} \right)^{2} - \mu_{3}^{2} } \right),\,H_{12} \left( {\upsilon + k} \right) = m\varsigma_{2} \left[ {\left( {1 + c_{3} } \right)\left( {\upsilon + k} \right) - \mu_{1}^{2} } \right], \hfill \\ &H_{21} \left( {\upsilon + k} \right) = \left[ {\left( {1 + c_{3} } \right)\left( {\upsilon + k} \right) + \mu_{1}^{2} } \right],\,\,H_{22} \left( {\upsilon + k} \right) = \left( {\left( {\upsilon + k} \right)^{2} - \mu_{2}^{2} } \right), \hfill \\ &H_{33} \left( {\upsilon + k} \right) = \left( {\left( {\upsilon + k} \right)^{2} - \mu_{4}^{2} } \right) \hfill \\ \end{aligned} \right\} $$
(30)
$$ \left. \begin{aligned} &H^{\prime}_{13} \left( {\upsilon + k} \right) = - \varOmega^{ - 1} \beta^{*} \left( {\upsilon + k + \frac{{3 - 4\bar{\beta }}}{2}} \right),\,\,H^{\prime}_{23} \left( {\upsilon + k} \right) = - im\varsigma_{2} \varOmega^{ - 1} \bar{\beta }\beta^{*} , \hfill \\ &H^{\prime}_{31} \left( {\upsilon + k} \right) = i\varOmega^{ - 1} \varepsilon^{*} \varOmega^{*} \left( {\upsilon + k + \frac{{\left( {4\bar{\beta } - 1} \right)}}{2}} \right),\,\,H^{\prime}_{32} = - m\varsigma_{2} \varepsilon^{*} \varOmega^{*} \varOmega^{ - 1} \bar{\beta } \hfill \\ \end{aligned} \right\} $$
(31)

Equating the coefficients of lowest powers of \( \xi \, \) (i.e. coefficient of \( \xi^{\upsilon - 2} = 0 \)) to zero in Eq. (28), we obtain:

$$ H_{1} \left( {\varvec{\upupsilon}} \right)\hat{Z}_{{\mathbf{0}}} = 0\,\, $$
(32)

where

$$ \left. \begin{aligned} &\hat{Z}_{{\mathbf{0}}} = \left[ {\begin{array}{*{20}c} {\,A_{0} } & {\,B_{0} } & {C_{0} } \\ \end{array} \,} \right]^{\prime } \hfill \\ &H_{1} \left( \upsilon \right) = H_{iq} \left( \upsilon \right)\,\,\,\,i,\,q = 1,\,\,2,\,\,3 \hfill \\ \end{aligned} \right\} $$
(33)

For the existence of non-trivial solution of Eq. (32) one must have \( \left| {H_{1} \left( {\varvec{\upupsilon}} \right)} \right| = 0 \), This results in the following system of indicial equations

$$ \left. \begin{aligned} &\upsilon^{4} - A^{*} \upsilon^{2} + D^{*} = 0 \hfill \\ &\upsilon^{2} - \mu_{4}^{2} = 0 \hfill \\ \end{aligned} \right\} $$
(34)

where

$$ \begin{aligned} &A^{*} = \frac{{\mu_{3}^{2} + c_{4} \mu_{2}^{2} - m\varsigma_{2} \left( {1 + c_{3} } \right)^{2} }}{{c_{4} }} \hfill \\ &D^{*} = \frac{{\mu_{2}^{2} \mu_{3}^{2} - m^{2} \varsigma_{2}^{2} \mu_{1}^{4} }}{{c_{4} }} \hfill \\ \end{aligned} $$
(35)

The roots of indicial Eqs. (34) are given as

$$ \left. \begin{aligned} &\upsilon_{1}^{2} = \frac{{A^{*} + \sqrt {A^{*2} - 4D^{*} } }}{2}\,\,\, \hfill \\ &\upsilon_{2}^{2} = \frac{{A^{*} - \sqrt {A^{*2} - 4D^{*} } }}{2}\, \hfill \\ &\upsilon_{3}^{2} = \mu_{4}^{2} \hfill \\ \end{aligned} \right\} $$
(36)

The roots \( \upsilon_{j} \,\,(j = 1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6) \) of Eq. (36) are related through the relation \( \upsilon_{4} = - \upsilon_{1} ,\,\,\upsilon_{5} = - \upsilon_{2} ,\,\,\upsilon_{6} = - \upsilon_{3} \). Out of these \( \upsilon_{3} \) is real and the roots \( \upsilon_{1} \) and \( \upsilon_{2} \) may be, in general, complex. If s is complex, then leading terms in the series solution (28) are of the type:

$$ \left[ {\begin{array}{*{20}c} {A_{0} } \\ {B_{0} } \\ {D_{0} } \\ \end{array} } \right]\xi^{s} = \,\,Z_{0} \xi^{{s_{R} + s_{I} }} = \,Z_{0} \xi^{{s_{R} }} \left[ {\cos \left( {s_{I} \log \xi } \right) + i\sin \left( {s_{I} \log \xi } \right)} \right] $$
(37)

According to Neuringer [21], in order to obtain two independent real solutions, it is sufficient to use any one of the complex roots in a part and then taking its real and imaginary parts. The treatment of complex case is unlike that of the real root case with the advantage that the differential equation is required to be solved only once in the former case rather than twice in latter one. For the choice of roots of the indicial equations, Eq. (28) leads to following eigen-vectors:

$$ \left. \begin{aligned} &Z_{0} \left( {\upsilon_{1} } \right) = Z_{0} \left( {\upsilon_{4} } \right) = \left[ {\begin{array}{*{20}c} 1 \\ {Q_{B} \left( {\upsilon_{1} } \right)} \\ 0 \\ \end{array} } \right]V_{0} ,\,\,\,\,\,Z_{0} \left( {\upsilon_{2} } \right) = Z_{0} \left( {\upsilon_{5} } \right) = \left[ {\begin{array}{*{20}c} 1 \\ {Q_{B} \left( {\upsilon_{2} } \right)} \\ 0 \\ \end{array} } \right]V_{0} ,\, \hfill \\ &Z_{0} \left( {\upsilon_{3} } \right) = Z_{0} \left( {\upsilon_{6} } \right) = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 1 \\ \end{array} } \right]V_{0} \,\,\,\, \hfill \\ \end{aligned} \right\} $$
(38)

where

$$ Q_{B} \left( {\upsilon_{j} } \right) = - \frac{{\left( {1 + c_{3} } \right)\upsilon_{j} + a_{1}^{2} }}{{\upsilon_{j}^{2} - a_{2}^{2} }} = - \frac{{c_{4} \upsilon_{j}^{2} - a_{3}^{2} }}{{im\varsigma_{2} \left[ {\left( {1 + c_{3} } \right)\upsilon_{j} - a_{1}^{2} } \right]}},\quad \left( {j = 1\,\,,\,\,2,\,4,\,\,5} \right) , $$

\( V_{0} \) is a constant.

Thus we have

$$ A_{0} = [ \begin{array}{ccc} 1 & 1 & 0 \end{array}]V_{0} ,\quad\quad B_{0} = [\begin{array}{ccc} Q_{B} (1) & Q_{B} (2) & 0 \end{array} ]V_{0} ,\quad\quad D_{0} = [ \begin{array}{ccc} 0 & 0 & 1 \end{array}]V_{0}$$
(39)

Again equating to zero the coefficients of next lowest degree term \( \xi^{s - 1} \), which corresponds to \( k = 1 \), in Eq. (28), we get:

$$ H_{1} \left( {\upsilon_{j} + 1} \right)Z_{1} + H_{2} \left( {\upsilon_{j} } \right)Z_{0} = 0,\quad j = \,1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6 $$
(40)

Clearly \( H_{1} \left( {\upsilon_{j} + 1} \right) \) is non singular for each \( \upsilon_{j} \), therefore we have:

$$ Z_{1} = \,\, - H_{1} \left( {\upsilon_{j} + 1} \right)^{ - 1} H_{2} \left( {\upsilon_{j} } \right)Z_{0} = D_{1}^{*} Z_{0} $$
(41)

where

$$ D_{1}^{*} = - H_{1} \left( {\upsilon_{j} + 1} \right)^{ - 1} H_{2} \left( {\upsilon_{j} } \right) = \left[ {\begin{array}{*{20}c} 0 & 0 & {A_{13} } \\ 0 & 0 & {A_{23} } \\ {A_{31} } & {A_{32} } & 0 \\ \end{array} } \right] $$
(42)

where \( A_{32} \) is defined in Appendix A1.

Now equating the coefficients of powers of \( \xi^{s + k} \) equal to zero, we have

$$ H_{1} (s + k + 2)Z_{k + 2} = - H_{2} ( s + k + 1)Z_{k + 1} - HZ_{k} \quad k = 0,1,2 $$
(43)

where the matrices \( H_{1} ,\,\,H_{2} \) and \( H \) are defined in Eq. (29).

The Eq. (43) implies that:

$$ Z_{k + 2} = - \left( {H_{1} \left( {\upsilon_{j} + k + 2} \right)} \right)^{ - 1} \left[ {H_{2} \left( {\upsilon_{j} + k + 1} \right)Z_{k + 1} + HZ_{k} } \right] $$
(44)

Now putting \( k = 0,\;1,\;2,\;3 \ldots \) successively we get

$$ \begin{aligned} Z_{2}&= - \left( {H_{1} \left( {\upsilon_{j} + 2} \right)} \right)^{ - 1} \left[ {H_{2} \left( {\upsilon_{j} + 1} \right)D_{1}^{*} + H} \right]Z_{0} = D_{2}^{*} Z_{0} \hfill \\ Z_{3} &= - \left( {H_{1} \left( {\upsilon_{j} + 3} \right)} \right)^{ - 1} \left[ {H_{2} \left( {\upsilon_{j} + 2} \right)D_{2}^{*} + HD_{1}^{*} } \right]Z_{0} = D_{3}^{*} Z_{0} \hfill \\ Z_{4} &= - \left( {H_{1} \left( {\upsilon_{j} + 4} \right)} \right)^{ - 1} \left[ {H_{2} \left( {\upsilon_{j} + 3} \right)D_{3}^{*} + HD_{2}^{*} } \right]Z_{0} = D_{4}^{*} Z_{0} \hfill \\ &\vdots \hfill \\ Z_{k + 2} &= - \left( {H_{1} \left( {\upsilon_{j} + k + 2} \right)} \right)^{ - 1} \left[ {H_{2} \left( {\upsilon_{j} + k + 1} \right)D_{k + 1}^{*} + HD_{k}^{*} } \right]Z_{0} = D_{k + 2}^{*} Z_{0} \hfill \\ \end{aligned} $$

where \( D_{{_{0} }}^{*} = I \), \( D_{k + 2}^{*} = - \left( {H_{1} \left( {\upsilon_{j} + k + 2} \right)} \right)^{ - 1} \left[ {H_{2} \left( {\upsilon_{j} + k + 1} \right)D_{k + 1}^{*} + HD_{k}^{*} } \right],\;k = 1,\,\,2,\,\,3 \ldots \)

It can be shown that the matrix \( D_{k + 2} \) has similar form as that of \( H_{1} \left( {\upsilon_{j} + k + 2} \right) \) for even values of \( k \) and it is alike \( H_{2} \left( {\upsilon_{j} + k + 2} \right) \) for odd values of \( k \). Thus, we have:

$$ \left. \begin{aligned} Z_{2k + 2} = D_{2k + 2}^{*} Z_{0} \hfill \\ Z_{2k + 1} = D_{2k + 1}^{*} Z_{0} \hfill \\ \end{aligned} \right\} $$
(45)

where

$$ \left. \begin{aligned} &D_{2k + 2}^{*} = - \left( {H_{1} \left( {\upsilon_{j} + 2k + 2} \right)} \right)^{ - 1} \left[ {H_{2} \left( {\upsilon_{j} + 2k + 1} \right)D_{2k + 1}^{*} + HD_{2k}^{*} } \right], \hfill \\ &D_{2k + 1}^{*} = - \left( {H_{1} \left( {\upsilon_{j} + 2k + 1} \right)} \right)^{ - 1} \left[ {H_{2} \left( {\upsilon_{j} + 2k} \right)D_{2k}^{*} + HD_{2k - 1}^{*} } \right] \hfill \\ \end{aligned} \right\} $$
(46)

Upon simplification, we get

$$ D_{2k + 2}^{*} = \left[ {\begin{array}{*{20}c} {K_{11} } & {K_{12} } & 0 \\ {K_{21} } & {K_{22} } & 0 \\ 0 & 0 & {K_{33} } \\ \end{array} } \right] $$
(47)
$$ D_{2k + 1}^{*} = \left[ {\begin{array}{*{20}c} 0 & 0 & {K^{\prime}_{13} } \\ 0 & 0 & {K^{\prime}_{23} } \\ {K^{\prime}_{31} } & {K^{\prime}_{32} } & 0 \\ \end{array} } \right] $$
(48)

where \( K_{ij} ,\,\,K^{\prime}_{ij} \,\,\,\left( {i,\,\,j\, = 1,\,\,2,\,\,3} \right) \) given by Eqs. (A.2)–(A.3) as are defined in Appendix.

2.2 Convergence of the Series

According to Cullen [22], in case of a matrix sequence \( \left\{ {P_{k} } \right\} \) in \( C_{k \times k} \), we have \( Lim_{k \to \infty } \,P_{k} = \,P\left( {\left\{ {P_{k} } \right\} \to P} \right) \) if each of \( k^{2} \) component sequence converges. That is \( Lim_{k \to \infty } \left( {ent_{ij} \left( {P_{k} } \right)} \right) = p_{ij} ,\,\,\,i,\,\,j = \,1,\,\,2,\,\,3, \ldots ,k \). Moreover, if the matrix sequence \( \left\{ {P_{k} } \right\} \) and \( \left\{ {Q_{k} } \right\} \) converge to matrices \( P \) and \( Q \), respectively, then the sequence \( \left\{ {P_{k} ,\,\,Q_{k} } \right\} \to PQ \) and \( \left( {\alpha P_{k} + \beta Q_{k} } \right) \to \alpha \,P + \beta \,Q \) for any \( \alpha ,\,\,\beta \in C \).

Further from Eqs. (47)–(48), it can be shown that

$$ D_{2k + 2}^{*} \approx o\left( {k^{ - 2} } \right)D^{*} \quad {\text{and}}\quad D_{2k + 1}^{*} \approx o\left( {k^{ - 1} } \right)D^{**} $$
(49)

where \( D^{*} = \frac{{i\varepsilon^{*} \Omega^{*} \Omega^{ - 1} }}{{c_{4} }}{\text{diag}}\left[ {\begin{array}{*{20}c} 1 & 0 & 1 \\ \end{array} } \right] \) and \( D^{**} \) is a \( 3 \times 3 \) null matrix.

By using above facts, both the matrices \( D_{2k + 2}^{*} \to 0 \) and \( D_{2k + 1}^{*} \to 0 \) as \( k \to \infty \). This implies that the series (25) are absolutely and uniformly convergent having infinite radius of convergence. Therefore, the considered series in Eq. (25) is analytic and hence can be differentiated term by term.

Thus the general solution of Eq. (18) has the form

$$ \left. \begin{array}{l} (U_{\zeta } ,U_{\phi } ,\Theta) ( \zeta,\phi,\tau ) = \zeta^{{\frac{- 1}{2}}} \sum\limits_{j = 1}^{6} \sum\limits_{k = 0}^{\infty } E_{jk} (A_{k} \left( {\upsilon_{j} } \right), B_{k} ( \upsilon_{j}),C_{k} (\upsilon_{j}))( \Omega \zeta)^\upsilon_{j} + k \exp ( i ( m \varsigma_{2} \phi + \Omega \tau ) ) \\U_{\theta } (\zeta,\phi,\tau) = \zeta^{{\frac{ - 1}{2}}} (M_{11} J_{\eta } \left( {\Omega \zeta } \right) + M_{22} Y_{\eta } \left( {\Omega \zeta } \right))\exp \left( {i\left( {m\varsigma_{2} \phi + \Omega \tau } \right)} \right) \end{array} \right\} $$
(50)

The unknowns \( M_{11} \), \( M_{22} \) and \( E_{jk} \,\left( {j = 1,\,2,\,3,\,4,\,5,\,6} \right) \) can be evaluated by using boundary conditions.

The formal solution for displacements and stresses is given by

$$ U_{\zeta } = \zeta^{{\frac{ - 1}{2}}} \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} A_{k} \left( {\upsilon_{j} } \right)} } \left( \xi \right)^{{\upsilon_{j} + k}} \exp \left( {i\left( {m\varsigma_{2} \phi + \Omega \tau } \right)} \right) $$
(51)
$$ U_{\theta } = \zeta^{{ - \frac{1}{2}}} \left( {M_{11} J_{\eta } \left( \xi \right) + M_{12} Y_{\eta } \left( \xi \right)} \right)\exp \left( {i\left( {m\varsigma_{2} \phi + \Omega \tau } \right)} \right) $$
(52)
$$ U_{\phi } = \zeta^{{\frac{ - 1}{2}}} \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} B_{k} \left( {s_{j} } \right)} } \left( \xi \right)^{{s_{j} + k}} \exp \left( {i\left( {m\varsigma_{2} \phi + \Omega \tau } \right)} \right) $$
(53)
$$ \begin{aligned} \tau_{\zeta \zeta } &= \zeta^{{\frac{ - 1}{2}}} \{ \sum\limits_{j = 1}^{6} {E_{j0} } \left\{ {\left( {c_{34} + c_{4} \left( {\upsilon_{j} + 1} \right)} \right)A_{0} \left( {\upsilon_{j} } \right) + im\varsigma_{2} c_{3} B_{0} \left( {\upsilon_{j} } \right)} \right\}\xi^{{\upsilon_{j} - 1}} \hfill \\ &\quad + \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} \left\{ \begin{aligned} \left( {c_{34} + c_{4} \left( {\upsilon_{j} + k + 1} \right)} \right)A_{k + 1} \left( {\upsilon_{j} } \right) \hfill \\ + im\varsigma_{2} c_{3} B_{k + 1} \left( {\upsilon_{j} } \right) - \Omega^{ - 1} \beta^{*} C_{k} \left( {\upsilon_{j} } \right) \hfill \\ \end{aligned} \right\}} } \left( \xi \right)^{{\upsilon_{j} + k}} \} \exp \left( {i\left( {m\varsigma_{2} \phi + \Omega \tau } \right)} \right) \hfill \\ \end{aligned} $$
(54)
$$ \tau_{\zeta \theta } = \zeta^{{ - \frac{1}{2}}} \left( {M_{11} \left( {J^{\prime}_{\eta } \left( \xi \right) - \frac{3}{2\xi }J_{\eta } \left( \xi \right)} \right) + M_{12} \left( {Y^{\prime}_{\eta } \left( \xi \right) - \frac{3}{2\xi }Y_{\eta } \left( \xi \right)} \right)} \right)\,\exp \left( {i\left( {m\varsigma_{2} \phi + \Omega \tau } \right)} \right) $$
(55)
$$ \tau_{\zeta \phi } = \zeta^{{\frac{ - 1}{2}}} [\sum\limits_{j = 1}^{6} {E_{j0} \left\{ {im\varsigma_{2} A_{0} + \left( {\upsilon_{j} - \frac{3}{2}} \right)B_{0} } \right\}\left( \xi \right)^{s - 1} + } \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} } } \left\{ im\varsigma_{2} A_{k + 1} + \left( {\upsilon_{j} + k - \frac{1}{2}} \right)B_{k + 1} \right\}\left( \xi \right)^{{\upsilon_{j} + k}} ]\exp \left( {i\left( {m\varsigma_{2} \phi + \Omega \tau } \right)} \right) $$
(56)
$$ \frac{\partial \Theta }{\partial \zeta } = \zeta^{{\frac{ - 1}{2}}} \left[ \begin{aligned}&\sum\limits_{j = 1}^{6} {E_{j0} \left( {\upsilon_{j} - \frac{1}{2}} \right)C_{0} \left( {\upsilon_{j} } \right)\left( \xi \right)^{{\upsilon_{j} - 1}} } \hfill \\ &\quad + \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} \left( {\left( {\upsilon_{j} + k + 1} \right)C_{k + 1} \left( {\upsilon_{j} } \right)} \right)} } \left( \xi \right)^{{\upsilon_{j} + k}} \hfill \\ \end{aligned} \right]\exp i\left( {m\varsigma_{2} \phi + \Omega \tau } \right) $$
(57)

Equations (51)–(57) constitute the formal solution of the system of coupled partial differential equations.

3 Boundary Conditions

The following types of boundary conditions are taken on the surfaces \( \zeta = \zeta_{1} \) (inner surface) and \( \zeta = \varsigma_{2} \) (outer surface) of spherical curved plate. The boundary conditions are:

  • Set I: Stress free and thermally insulated condition

$$ \tau_{\zeta \zeta } = 0,\,\,\,\tau_{\zeta \theta } = 0,\,\,\,\tau_{\zeta \phi } = 0,\quad \frac{\partial \Theta }{\partial \zeta } = 0 $$
(58)

  • Set II: Stress free and isothermal condition

$$ \tau_{\zeta \zeta } = 0,\,\,\,\tau_{\zeta \theta } = 0,\,\,\,\tau_{\zeta \phi } = 0,\quad \Theta = 0 $$
(59)

  • Set III: Rigidly fixed and thermally insulated condition

$$ U_{\zeta } = 0,\quad U_{\theta } = 0,\quad U_{\phi } = 0,\quad \frac{\partial \Theta }{\partial \zeta } = 0 $$
(60)

  • Set IV: Rigidly fixed and isothermal condition

$$ U_{\zeta } = 0,\quad U_{\theta } = 0,\quad U_{\phi } = 0,\quad \Theta = 0 $$
(61)

4 Dispersion Relations

In this section we derive the secular equations for a spherical curved plate subjected to traction-free thermally insulated/isothermal or rigidly fixed, thermally insulated/isothermal boundary conditions at the surfaces.

4.1 Stress Free Curved Plate

In this subsection we derive the characteristic equations for stress free, thermally insulated and stress free isothermal spherical curved plate.

  • Set I:

Upon using the boundary condition (58) in the expressions (54)–(57) at the surface \( \xi = \eta_{1} \) and \( \xi = \eta_{2} \) we get the following equations

$$ \sum\limits_{j = 1}^{6} {E_{j0} } \left\{ {\left( {c_{4} \left( {\upsilon_{j} + 1} \right) + c_{34} } \right)A_{0} \left( {\upsilon_{j} } \right) + im\varsigma_{2} c_{3} B_{0} \left( {\upsilon_{j} } \right)} \right\}\left( {\eta_{i} } \right)^{{\upsilon_{j} - 1}} + \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} \left\{ \begin{aligned} \left( {c_{4} \left( {\upsilon_{j} + k + 1} \right) + c_{34} } \right)A_{k + 1} \left( {\upsilon_{j} } \right) \hfill \\ + im\varsigma_{2} c_{3} B_{k + 1} \left( {\upsilon_{j} } \right) - \Omega^{ - 1} C_{k} \left( {\upsilon_{j} } \right) \hfill \\ \end{aligned} \right\}} } \left( {\eta_{i} } \right)^{{\upsilon_{j} + k}} = 0 $$
(62)
$$ M_{11} \left( {J^{\prime}_{\eta } \left( {\eta_{i} } \right) - \frac{3}{{2\eta_{i} }}J_{\eta } \left( {\eta_{i} } \right)} \right) + M_{12} \left( {Y^{\prime}_{\eta } \left( {\eta_{i} } \right) - \frac{3}{{2\eta_{i} }}Y_{\eta } \left( {\eta_{i} } \right)} \right)\, = 0 $$
(63)
$$ \sum\limits_{j = 1}^{6} {E_{j0} i\,m\,\varsigma_{2} A_{0} \left( {\upsilon_{j} } \right) + \left( {\upsilon_{j} - \frac{3}{2}} \right)B_{0} \left( {\upsilon_{j} } \right)\left( {\eta_{i} } \right)^{{\upsilon_{j} - 1}} + } \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} } } i\,m\,\varsigma_{2} A_{k + 1} \left( {\upsilon_{j} } \right) + \left( {\upsilon_{j} + k - \frac{1}{2}} \right)B_{k + 1} \left( {\upsilon_{j} } \right)\left( {\eta_{i} } \right)^{{\upsilon_{j} + k}} = 0 $$
(64)
$$ \sum\limits_{j = 1}^{6} {E_{j0} \left( {\upsilon_{j} - \frac{1}{2}} \right)C_{0} \left( {\upsilon_{j} } \right)\left( {\eta_{i} } \right)^{{s_{j} - 1}} + } \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} \left( {\left( {\upsilon_{j} + k + 1} \right)C_{k + 1} \left( {\upsilon_{j} } \right)} \right)} } \left( {\eta_{i} } \right)^{{\upsilon_{j} + k}} = 0 $$
(65)

where \( i = 1 \) for inner surface and \( i = 2 \) for outer surface of the spherical curved plate.

Equations (62)–(65) is a system of simultaneous linear algebraic equations in eight unknowns \( M_{11} ,\,\,\,M_{12} \) and \( E_{jk\,} ,\,\,\left( {j = 1,\,2,\,3,\,4,\,5,\,6} \right) \). These are uniformly and absolutely convergent series. Thus the above system of equations can be expressed in compact form as given below:

$$ {\mathbf{G}}^{{\mathbf{0}}} {\mathbf{X}}_{{\mathbf{0}}} + {\mathbf{G}}^{{\mathbf{k}}} {\mathbf{X}}_{{\mathbf{k}}} = 0 $$
(66)

where

$$ X_{0} = \left[ {\begin{array}{*{20}c} {E_{10} } & {E_{20} } & {E_{30} } & {E_{40} } & {E_{50} } & {E_{60} } & {M_{11} } & {M_{12} } \\ \end{array} } \right],\quad {\mathbf{G}}^{{\mathbf{0}}} = \left( {G_{ij}^{0} } \right)_{8 \times 8} $$
(67)
$$ X_{k} = \left[ {\begin{array}{*{20}c} {E_{1k} } & {E_{2k} } & {E_{3k} } & {E_{4k} } & {E_{5k} } & {E_{6k} } & {M_{11} } & {M_{12} } \\ \end{array} } \right],\quad {\mathbf{G}}^{{\mathbf{k}}} = \left( {G_{ij}^{k} } \right)_{8 \times 8} $$
(68)
$$ \left. \begin{aligned} &G_{11}^{0} = \left( {\left( {c_{4} \left( {\upsilon_{1} + 1} \right) + c_{34} } \right)A_{0} \left( {\upsilon_{1} } \right) + im\varsigma_{2} c_{3} B_{0} \left( {\upsilon_{1} } \right)} \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} - 1}} \hfill \\ &G_{51}^{0} = \left( {im\varsigma_{2} A_{0} \left( {\upsilon_{1} } \right) + \left( {\upsilon_{1} - \frac{1}{2}} \right)B_{0} \left( {\upsilon_{1} } \right)} \right)\left( {\eta_{1} \,} \right)^{{\upsilon_{1} - 1}} \hfill \\ &G_{71}^{0} = \left( {\upsilon_{1} - \frac{1}{2}} \right)C_{0} \left( {\eta_{1} } \right)^{{\upsilon_{1} - 1}} \, \hfill \\ \end{aligned} \right\} $$
(69)
$$ \left. \begin{aligned} G_{37}^{0} = \left( {\frac{{J_{\eta - 1} \left( {\eta_{1} } \right) - J_{\eta + 1} \left( {\eta_{1} } \right)}}{{\eta_{1} }} - \frac{3}{{2\eta_{1}^{2} }}J_{\eta } \left( {\eta_{1} } \right)} \right) \hfill \\ G_{47}^{0} = \left( {\frac{{J_{\eta - 1} \left( {\eta_{2} } \right) - J_{\eta + 1} \left( {\eta_{2} } \right)}}{{\eta_{2} }} - \frac{3}{{2\eta_{2}^{2} }}J_{\eta } \left( {\eta_{2} } \right)} \right) \hfill \\ \end{aligned} \right\} $$
(70)
$$ \left. \begin{aligned} &G_{11}^{k} = \left( {\left( {c_{4} \left( {\upsilon_{1} + k + 1} \right) + c_{34} } \right)A_{k + 1} \left( {\upsilon_{1} } \right) + im\varsigma_{2} c_{3} B_{k + 1} \left( {\upsilon_{1} } \right) - \Omega^{ - 1} \beta^{*} C_{k} \left( {\upsilon_{1} } \right)} \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &G_{51}^{k} = \left( {im\varsigma_{2} A_{k + 1} \left( {\upsilon_{1} } \right) + \left( {\upsilon_{1} + k - \frac{1}{2}} \right)B_{k + 1} \left( {\upsilon_{1} } \right)} \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &G_{71}^{k} = \left( {\upsilon_{1} + k + 1/2} \right)C_{k + 1} \left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &G_{ij}^{k} = G_{ij}^{0} ,\,\,i = 3,\,4\,\,\,\,{\text{and}}\,\,\,\,j = 7 ,\,\, 8\hfill \\ &G_{ij}^{k} = G_{ij}^{0} = 0,\,\,i = 3,\,4;\,\,\,j = 1 ,\, 2 ,\, 3 ,\, 4 ,\, 5 ,\, 6 {\text{ or }}i = 1,\,2,\,5,\,6,\,7,\,8\,;\,\,j = 7,\,\,8 \hfill \\ \end{aligned} \right\} $$
(71)

where \( \eta_{1} = \Omega \varsigma_{1} \) and \( \eta_{2} = \Omega \varsigma_{2} \).

Here the elements \( G_{ij}^{0} \) and \( G_{ij}^{k} \) \( \left( {i = 1,\,\,5,\,\,7\,;\,\,j = 2,\,3,\,4,\,5,\,6} \right) \) in Eq. (66) are obtained from \( G_{ij}^{0} \) and \( G_{ij}^{k} \) \( \left( {i = 1,\,\,5,\,\,7} \right) \) in Eqs. (70) and (71) by replacing \( \upsilon_{1} \) with \( \upsilon_{j} ,\,\,\,\,(j = 2,\,\,3,\,\,4,\,\,5,\,\,6) \), respectively. The elements \( G_{ij}^{k} \),\( G_{ij}^{k} \), \( 3,\,\,4,\,\,\,5,\,\,6) \) are written from \( G_{ij}^{0} \) and \( G_{ij}^{k} \,\,\left( {i = 1,\,\,\,5,\,\,7;\,\,j = 1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6} \right) \) by replacing \( \eta_{1} \) with \( \eta_{2} \) therein. The elements \( G_{38}^{0} ,\,\,G_{48}^{0} \) are obtained from \( G_{37}^{0} ,\,\,G_{47}^{0} \) respectively by replacing Bessel functions of first kind with Bessel function of second kind.

Equation (66) holds if and only if each term vanishes separately. This implies that

$$ {\mathbf{G}}^{{\mathbf{0}}} {\mathbf{X}}_{{\mathbf{0}}} = 0 = 0 \quad {\text{for }}\,k = 0 $$
(72)
$$ {\mathbf{G}}^{{\mathbf{k}}} {\mathbf{X}}_{{\mathbf{k}}} = 0 \quad {\text{ for}}\,k > 0 $$
(73)

Equations (72) and (73) has a non-trivial solution if and only if

$$ \left| {{\mathbf{G}}^{{\mathbf{0}}} } \right| = 0 \quad {\text{for}}\,k = 0 $$
(74)
$$ \left| {{\mathbf{G}}^{{\mathbf{k}}} } \right| = 0 \quad {\text{for}}\,k > 0 $$
(75)

After lengthy but straightforward simplifications and reductions, the determinantal Eqs. (74) and (75) lead to the following secular equations

$$ \left| {G^{0} } \right| = 0,\,\,i,\,\,j = 1,\,\,2,\,\,3,\,4,\,\,5,\,\,6\quad {\text{for}}\,\,k = 0 $$
(76)
$$ \left| {G^{k} } \right| = 0,\,\,i,\,\,j = 1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6\quad {\text{for}}\,\,k > 0 $$
(77)
$$ \left( {J_{\eta - 1} \left( {\eta_{1} } \right) - J_{\eta + 1} \left( {\eta_{1} } \right) - \frac{3}{{2\eta_{1} }}J_{\eta } \left( {\eta_{1} } \right)} \right)\left( {J_{\eta - 1} \left( {\eta_{2} } \right) - J_{\eta + 1} \left( {\eta_{2} } \right) - \frac{3}{{2\eta_{2} }}J_{\eta } \left( {\eta_{2} } \right)} \right) - \left( {Y_{\eta - 1} \left( {\eta_{1} } \right) - Y_{\eta + 1} \left( {\eta_{1} } \right) - \frac{3}{{2\eta_{1} }}Y_{\eta } \left( {\eta_{1} } \right)} \right)\left( {Y_{\eta - 1} \left( {\eta_{2} } \right) - Y_{\eta + 1} \left( {\eta_{2} } \right) - \frac{3}{{2\eta_{2} }}Y_{\eta } \left( {\eta_{2} } \right)} \right) $$
(78)

where

$$ \left. \begin{aligned} &G_{11}^{0} = \left( {\left( {c_{4} \left( {\upsilon_{1} + 1} \right) + c_{34} } \right)A_{0} \left( {\upsilon_{1} } \right) + im\varsigma_{2} c_{3} B_{0} \left( {\upsilon_{1} } \right)} \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} - 1}} \hfill \\ &G_{21}^{0} = \left( {im\varsigma_{2} A_{0} \left( {\upsilon_{1} } \right) + \left( {\upsilon_{1} - \frac{1}{2}} \right)B_{0} \left( {\upsilon_{1} } \right)} \right)\left( {\eta_{1} \,} \right)^{{\upsilon_{1} - 1}} \hfill \\ &G_{31}^{0} = \left( {\upsilon_{1} - \frac{1}{2}} \right)C_{0} \left( {\eta_{1} } \right)^{{\upsilon_{1} - 1}} \, \hfill \\ \end{aligned} \right\} $$
(79)
$$ \left. \begin{aligned} &G_{11}^{k} = \left( {\left( {c_{4} \left( {\upsilon_{1} + k + 1} \right) + c_{34} } \right)A_{k + 1} \left( {\upsilon_{1} } \right) + im\varsigma_{2} c_{3} B_{k + 1} \left( {\upsilon_{1} } \right) - \Omega^{ - 1} \beta^{*} C_{k} \left( {\upsilon_{1} } \right)} \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &G_{31}^{k} = \left( {im\varsigma_{2} A_{k + 1} \left( {\upsilon_{1} } \right) + \left( {\upsilon_{1} + k - \frac{1}{2}} \right)B_{k + 1} \left( {\upsilon_{1} } \right)} \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &G_{51}^{k} = \left( {\upsilon_{1} + k + 1/2} \right)C_{k + 1} \left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ \end{aligned} \right\} $$
(80)

The elements \( G_{ij}^{0} \left( {j = 2,\,3,\,4,\,5,\,6} \right) \) of determinant Eqs. (76) and (77) are obtained by just replacing \( \upsilon_{j} ,\,\,j = 1 \) in \( G_{ij}^{0} \left( {j = 1,\,3,\,5} \right) \) with \( \upsilon_{j} ,\,\,\,\,j = 2,\,3,\,4,\,5,\,6 \) while \( G_{ij}^{0} \left( {i = 2,\,4,\,6} \right) \) are obtained by replacing \( \eta_{1} \) in \( G_{ij}^{0} \left( {i = 1,\,\,3,\,\,5} \right) \) with \( \eta_{2} \).

Set II:

Employing the boundary conditions (59) at the surface \( \xi = \eta_{1} \) and \( \xi = \eta_{2} \) via expressions (54)–(56), we obtain a system of simultaneous linear algebraic equations in eight unknowns \( M_{11} \,,\,\,\,M_{11} \) and \( E_{jk\,} ,\,\,\left( {j = 1,\,\,2,\,\,3,\,\,4,\,\,5,\,\,6} \right) \) as Eqs. (62)–(64), along with the equation as given below:

$$ \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} C_{k} \left( {\upsilon_{j} } \right)} } \left( {\eta_{i} } \right)^{{\upsilon_{j} + k}} = 0 $$
(81)

The system of linear algebraic homogeneous Eqs. (62)–(64) and (81) are written in compact form as:

$$ {\mathbf{H}}^{{\mathbf{k}}} {\mathbf{X}}_{{\mathbf{k}}} = 0\, $$
(82)

where \( {\mathbf{H}}^{{\mathbf{k}}} = \left( {H_{ij}^{k} } \right)_{8 \times 8} \).

The elements \( H_{ij}^{k} = G_{ij}^{k} \) are defined in Eqs. (79) and (80) in the form of \( G_{ij}^{k} \) except the element \( H_{51}^{k} \) which is given by

$$ H_{51}^{k} = C_{k} \left( {\upsilon_{1} } \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} $$
(83)

Equation (82) will have a non-trivial solution if and only if the determinant of the coefficients \( {\mathbf{X}}_{{\mathbf{k}}} \) vanishes. This requirement of nontrivial solution leads to a determinantal Eq. (82) along with the following equation:

$$ \left| {{\mathbf{H}}^{{\mathbf{k}}} } \right| = 0\,\,\,\, $$
(84)

The elements \( H_{ij}^{k} = G_{ij}^{k} \) are defined in Eq. (80) except the change in the value of \( H_{51}^{k} \) i.e. given by

$$ H_{51}^{k} = C_{k} \left( {\upsilon_{1} } \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} $$

Equation (84) represents the frequency for stress free spherical curved plate.

4.2 Rigidly Fixed Curved Plate

In this subsection we derive the characteristic equations for rigidly fixed, thermally insulated and rigidly fixed, isothermal spherical curved plate.

Set III:

Upon imposing the boundary conditions (60), we obtain a system of algebraic homogeneous equations as under:

$$ \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} A_{k} \left( {\upsilon_{j} } \right)} } \left( {\eta_{i} } \right)^{{\upsilon_{j} + k}} = 0 $$
(85)
$$ \sum\limits_{k = 0}^{\infty } {\sum\limits_{j = 1}^{6} {E_{jk} B_{k} \left( {\upsilon_{j} } \right)} } \left( {\eta_{i} } \right)^{{\upsilon_{j} + k}} = 0 $$
(86)
$$ M_{11} J_{\eta } \left( {\eta_{i} } \right) + M_{12} Y_{\eta } \left( {\eta_{i} } \right) = 0 $$
(87)

where \( i = 1 \) for inner surface and \( i = 2 \) for outer surface of spherical curved plate.

Equations (85)–(87) and (65) in eight unknowns \( M_{11} \,,\,\,\,M_{12} \) and \( E_{jk\,} ,\,\,\left( {j = 1,\,2,\,3,\,4,\,5,\,6} \right) \) after simplification can be expressed as:

$$ {\mathbf{F}}^{{\mathbf{k}}} {\mathbf{X}}_{{\mathbf{k}}} = 0 $$
(88)

where \( X_{k} \) is defined in (68) and \( {\mathbf{F}}^{{\mathbf{k}}} = \left( {F_{ij}^{k} } \right)_{8 \times 8} \)

$$ \left. \begin{aligned} &F_{11} = A_{k} \left( {\upsilon_{1} } \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &F_{31} = B_{k} \left( {\upsilon_{1} } \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &F_{51} = \left( {\upsilon_{1} + k + 1/2} \right)C_{k + 1} \left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \,\, \hfill \\ \end{aligned} \right\} $$
(89)
$$ F_{57} = J_{\eta } \left( {\eta_{1} } \right) $$
(90)

The elements \( F_{lj} \left( {j = 2,\,3,\,4,\,5,\,6} \right) \) of Eq. (89) are obtained by replacing \( \upsilon_{j} ,\,j = 1 \) in \( F_{lj} \left( {l = 1,\,3,\,5} \right) \) with \( \upsilon_{j} ,\,j = 2,\,3,\,4,\,5,\,6 \) while \( F_{lj's} \left( {l = 2,\,4,\,6} \right) \) are obtained by replacing \( \eta_{1} \) in \( F_{lj} \left( {l = 1,\,3,\,5} \right) \) with \( \eta_{2} \). The element \( F_{il} ,\,\,i = \,7,\,8 \) and \( l = 8 \) in Eq. (90) are obtained by replacing Bessel’s function of first kind \( J_{\eta } \) with that of second kind \( Y_{\eta } \) and the elements \( F_{il} \,\,i = \,8 \) and \( l = 7,\,\,8 \) are obtained by replacing \( \eta_{1} \) in \( F_{il} \,\,\,i = 7 \) and \( l = 7,\,8 \) with \( \eta_{2} \) respectively

Equation (88) has non-trivial solution if and only if we have

$$ \left| {{\mathbf{F}}^{{\mathbf{k}}} } \right| = 0 $$
(91)

Equation (91) can be split into the following equations:

$$ \left| {B_{ij} } \right| = 0,\,\,\,i,\,\,j = 1,\,2,\,3,\,4,\,5,\,6 $$
(92)
$$ \left( {J_{\eta } \left( {\eta_{1} } \right)Y_{\eta } \left( {\eta_{2} } \right) - J_{\eta } \left( {\eta_{2} } \right)Y_{\eta } \left( {\eta_{1} } \right)} \right) = 0 $$
(93)
$$ \left. \begin{aligned} &B_{11} = A_{k} \left( {\upsilon_{1} } \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &B_{31} = B_{k} \left( {\upsilon_{1} } \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \hfill \\ &B_{51} = \left( {\upsilon_{1} + k + 1/2} \right)C_{k + 1} \left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} \,\, \hfill \\ \end{aligned} \right\} $$
(94)

The elements \( B_{lj} \left( {j = 2,\,3,\,4,\,5,\,6} \right) \) of determinantal Eq. (92) are obtained by replacing \( \upsilon_{j} ,\,j = 1 \) in \( B_{lj} \left( {l = 1,\,3\,,\,5} \right) \) with \( \upsilon_{j} ,\,j = 2,3,\,4,\,5,\,6 \) while \( B_{lj's} \left( {l = 2,\,4,\,6} \right) \) are obtained by replacing \( \eta_{1} \) in \( B_{lj} \left( {l = 1,\,3,\,5} \right) \) with \( \eta_{2} \).

Set IV:

Invoking the boundary conditions (61) we obtain a homogeneous system of linear algebraic equations in the eight unknowns \( M_{11} ,\,\,\,M_{12} \) and \( E_{jk\,} ,\,\,\left( {j = 1,\,2,\,3,\,4,\,5,\,6} \right) \) given by Eqs. (81) and (85)–(87) above. Upon simplifying this system of equations are expressed in compact form as:

$$ {\mathbf{F}}^{{ * {\mathbf{k}}}} {\mathbf{X}}_{{\mathbf{k}}} = 0 $$
(95)

where \( {\mathbf{F}}^{{ * {\mathbf{k}}}} = \left( {F_{ij}^{k} } \right)_{8 \times 8} \).

The elements \( F_{ij}^{k} \) are defined in Eqs. (89) and (90) except the change in value of \( F_{ij}^{k} \) given by

$$ F_{51}^{k} = C_{k} \left( {\upsilon_{1} } \right)\left( {\eta_{1} } \right)^{{\upsilon_{1} + k}} $$
(96)

The Eq. (95) has non-trivial solution if and only if we have

$$ \left| {{\mathbf{F}}^{{ * {\mathbf{k}}}} } \right| = 0 $$
(97)

Equation (97) further split into Eq. (93) along with the following equation

$$ \left| {F_{ij}^{k} } \right| = 0,\,\,i,\,j = 1,\,2,\,3,\,4,\,6 $$
(98)

Equations (92) and (98) represent the frequency equation for spherical curved plate in case of rigidly fixed boundary.

5 Homogeneous Isotropic Curved Plate

The analysis reduces to that of an isotropic spherical plate; we make the choice of material parameters as

$$ c_{11} = c_{33} = \lambda + 2\mu \,,\,c_{12} = c_{13} = \lambda ,c_{44} = \mu \,,\,\beta_{1} = \beta = \beta_{3} ,\,K_{1} = K = K_{3} . $$
(99)

where \( \lambda \) and \( \mu \) are the Lame constants. As there is no effect of temperature on SH wave motion. For anisotropic material the Eq. (78) agrees with (14) of Yu et al. [13].

For isotropic materials, Eq. (78) agrees with Eq. (29) of Shah et al. [6]. Equation (78) governs the motion corresponding to the case of shear where only the \( U_{\theta } \) displacement occurs. These modes of vibrations are not affected by temperature change. Equation (93) again governs the motion corresponding to the case of toroidal shear in the rigidly fixed plate where only \( U_{\theta } \) displacement in circumferential direction occurs. These modes of vibrations are not affected by temperature change.

6 Numerical Results and Discussion

In order to illustrate the analytical development we have proposed to carry out some numerical calculations to compute lowest frequency of zinc, cobalt and silicon nitride materials whose physical data are given in Table 1. Due to the presence of dissipation term in heat conduction Eq. (4), the secular equations are, in general, complex transcendental equations which provide us complex values of the frequency \( \left( \Omega \right) \).

Table 1 Physical data for zinc, cobalt and silicon nitride crystals
Full size table

We assume that \( \Omega = \Omega_{R} + iD \), where \( \Omega_{R} = \frac{{\Omega_{R} R}}{{v_{s} }} \) and \( D = \frac{{\Omega_{I} R}}{{v_{s} }} \) denote the lowest frequency and dissipation factor of the vibrations, respectively. Then each of the secular Eq. (77) or (92) is rewritten as \( \Omega = f\left( \Omega \right) \), which on separating real and imaginary parts provides us the following system of two real equations:

$$ \Omega_{R} = F\left( {\Omega_{R} ,D} \right),\,\,\,\,\,\,\,\,\,D = G\left( {\Omega_{R} ,D} \right) $$
(100)

The functions \( F \) and \( G \) in Eq. (100) are selected in such a way that they satisfy the conditions

$$ \left| {\frac{\partial F}{{\partial \Omega_{R} }}} \right| + \left| {\frac{\partial F}{\partial D}} \right| < 1,\,\,\,\,\,\left| {\frac{\partial G}{{\partial \Omega_{R} }}} \right| + \left| {\frac{\partial G}{\partial D}} \right| < 1 $$
(101)

for all \( \Omega_{R} ,\,\,\,\,D \) is the neighborhood of the actual root. If \( \Omega_{0} = \left( {\Omega_{{R_{0} }} ,D_{0} } \right) \) be the initial approximation, then we can construct the successive approximations as:

$$ \left. \begin{array}{ccc} \Omega_{R_{1}} = F \left(\Omega_{R_{0}} ,D_{0}\right), & D_{1} = G\left( \Omega_{R_{1}},D_{0} \right)& \\\Omega_{R_{2}} = F \left(\Omega_{R_{1}} ,D_{1} \right), & D_{2} = G\left(\Omega_{R_{2}} ,D_{1} \right) \\ \vdots & \vdots & \\ \Omega_{R_{j}} = F\left( \Omega_{R_{i - 1} } ,D_{i - 1} \right), & D_{j} = G\left( \Omega_{R_{i} } ,D_{i - 1} \right), & i = 0,1,2,3, \ldots \end{array} \right\} $$
(102)

The sequence \( \left( {\Omega_{{R_{n} }} ,D_{n} } \right) \) of approximations to the root will converge to the actual root provided the initial guess \( \left( {\Omega_{{R_{0} }} ,D_{0} } \right) \) lies in its neighborhood. For initial value \( \Omega_{0} = \left( {\Omega_{{R_{0} }} ,D_{0} } \right) \), the indicial roots \( s_{j} \,\,\left( {j = 1,\,\,2,\,\,3} \right) \) given by Eq. (36) are computed and used along with Eq. (38) in the secular Eqs. (77) and (92) to obtain the current values of \( \Omega_{R} \) and \( D \) each time which are further used to generate the sequence (44). The process is terminated as and when the condition \( \left| {\Omega_{i + 1} - \Omega_{i} } \right| < \varepsilon^{\prime} \); \( \varepsilon^{\prime} \) being arbitrarily small number to be selected at random to achieve the accuracy level, is satisfied. The procedure is continuously repeated for different values of inner radius to thickness ratio of the plate. The numerical computations for lowest frequencies and dissipation factor in plate of zinc, cobalt and silicon nitride materials have been presented in Figs. 2, 3, 4, 5, 6, 7, 8 and 9 for stress free or rigidly fixed boundary conditions. In order to illustrate the analytical developments in the previous sections, we now perform some numerical computations and simulations. The secular Eqs. (77) and (92) contain complete information about the effect of different fields’ lowest frequency, ratio of inner radius to thickness \( \eta^{*} = \frac{{\varsigma_{1} }}{{\varsigma_{2} - \varsigma_{1} }} \) and damping factor (\( D \)). The results are presented graphically.

Fig. 2
figure 2

Variation of lowest frequency with wave number for different values of ratio of inner radius to thickness of the plate for silicon nitride material in stress free case

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Fig. 3
figure 3

Variation of lowest frequency with wave number for different values of ratio of inner radius to thickness of the plate for silicon nitride material in rigidly fixed case

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Fig. 4
figure 4

Variation of damping factor with wave number for different values of ratio of inner radius to thickness of the plate for silicon nitride material

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Fig. 5
figure 5

Variation of lowest frequency with wave number for different values of ratio of inner radius to thickness of the plate for cobalt material

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Fig. 6
figure 6

Variation of damping factor with wave number for different values of ratio of inner radius to thickness of the plate for cobalt material

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Fig. 7
figure 7

Variation of lowest frequency with wave number for different values of ratio of inner radius to thickness of the plate for zinc material

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Fig. 8
figure 8

Variation of damping factor with wave number for different values of ratio of inner radius to thickness of the plate for zinc material

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Fig. 9
figure 9

Variation of lowest frequency with wave number for different values of ratio of inner radius to thickness of the plate for cobalt material rigidly fixed

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Fig. 10
figure 10

Variation of damping factor with wave number for different values of ratio of inner radius to thickness of the plate for cobalt material rigidly fixed

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The numerical computations have been performed by employing the iteration technique to the dispersion relation (77) and (92) with the help of MATLAB programming. The computations have been done for different values of inner radius to thickness ratio (\( \eta^{*} \)) for fixed outer radius \( \zeta_{ 2} = 1.0 \).

Figures 2 and 3 show the variations of lowest frequency with respect to the wave number (m) for \( \eta^{*} = 1 \) in case of stress free and rigidly fixed boundary conditions for silicon nitride (\( {\text{Si}}_{ 3} {\text{N}}_{ 4} \)) coupled thermoelastic plate and uncoupled (elastic) thermoelastic plates. From the profiles in these figures, it is noticed that the magnitude of lowest frequency increase with wave number (\( m \)). The magnitude of vibrations is quite high in thermoelastic (TE) plate as compared to that in elastic (E) plate under considered mechanical conditions, which exhibits the effect of thermal variations.

It is also revealed that the magnitude of lowest frequency increases monotonically for \( \eta^{*} \) in the absence and presence of thermal field. Moreover, the magnitude of vibrations is large for stress free thermally insulated plate in comparison to rigidly fixed one, depicting the effect of mechanical constraints. These results are similar to the results of Yu and Xue [16] in case of elastic material (in the absence of thermal variations). Figure 4 shows the variations of damping factor with respect to the wave number (m) for \( \eta^{*} = 1 \) in case of stress free and rigidly fixed boundary conditions for silicon nitride (\( {\text{Si}}_{ 3} {\text{N}}_{ 4} \)) coupled thermoelastic plate and uncoupled (elastic) thermoelastic plates. It is noticed that damping increases with wave number (\( m \)). Also the magnitude of damping is greater for stress free plate in comparison to the rigidly fixed plate.

Figures 5 and 7 show the variation of frequency with wave number for different values of inner radius to thickness ratio (\( \eta^{*} \)) for zinc and cobalt materials respectively. The dispersion behaviour of Lamb- like waves for circumferential spherical curved plates with different values of inner radius to thickness ratio of the spherical curved plate is calculated to analyze its influence on dispersion curves. The thickness to mean radius ratio is defined as \( \eta^{*} \). From both the figures it is observed that with the increase of wave number the lowest frequency increases in linear fashion. One can see that the influence of \( \eta^{*} \) on group dispersion curves is obvious. Here frequency increases with the increase of wave number as the \( \eta^{*} \) decreases the changes are considerably greater for higher value of \( \eta^{*} \) there is curvilinear increase in frequency while as for lower value of \( \eta^{*} \) there is almost linear increase in frequency.

Figures 6 and 8 show the variations of dissipation with wave number for different values of \( \eta^{*} \) for zinc and cobalt materials respectively. The influence of \( \eta^{*} \) on dissipation is more affected for both materials. For zinc in Fig. 6 for higher value of \( \eta^{*} \) dissipation increases with the increase of wave number but for lower value of \( \eta^{*} \) dissipation first increases and then decreases curvilinear for n > 2. In case of cobalt material dissipation increases for higher values of \( \eta^{*} \) and for intermediate value i.e. for \( \eta^{*} = .01 \) it shows fluctuating behaviour. It is noticed that there is constant decrease in dissipation for low values of \( \eta^{*} \). In microscale SAW devices, the operating frequency is usually very high and the wavelength is very small (i.e. large wave number). It can be seen that the thermal effect is so strong that it influences the dissipation. The effect is significant for higher value of \( \eta^{*} \).

Figures 9 and 11 show the variation of frequency with wave number for different values of inner radius to thickness ratio (\( \eta^{*} \)) for zinc and cobalt materials respectively in case of rigidly fixed boundary conditions. From both the figures it is observed that with the increase of wave number the lowest frequency increases. Figures 10 and 12 show the variations of dissipation with wave number for different values of \( \eta^{*} \) for zinc and cobalt materials respectively. The influence of \( \eta^{*} \) on dissipation is more affected for both materials in case of rigidly fixed boundary conditions. It is observed that damping factor increases with increasing value of wave number (m).

Fig. 11
figure 11

Variation of lowest frequency with wave number for different values of ratio of inner radius to thickness of the plate for zinc material rigidly fixed

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Fig. 12
figure 12

Variation of damping factor with wave number for different values of ratio of inner radius to thickness of the plate for zinc material rigidly fixed

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Figures 13 and 14 show the variation of non-dimensional phase velocity \( \left( {v_{ph} = \Omega_{R} /m} \right) \) with circumferential wave number for stress-free and rigidly fixed, thermally insulated boundary for first mode of vibrations for cobalt and zinc materials respectively. From both the figures, it is observed that phase velocities start from large values at vanishing wave number and then exhibit strong dispersion until the velocity flattens out to the value of the thermoelastic Rayleigh wave velocity of the material at higher wave numbers. It is noticed that there is no difference between the plates i.e. spherical curved plate in Fig. 14 with the comparison of cylindrical curved plate given by Sharma and Pathania [6]. Both are made up of zinc material having same outer radius. The dispersion curves are almost similar. The magnitude of phase velocity is noticed to be large for stress free boundary as compared to that of rigidly fixed boundary conditions depicting the effect of mechanical constraints in the discussed problem.

Fig. 13
figure 13

Variation of non-dimensional phase velocity with wave number for different values k Fröbenius parameter for cobalt material stress free and rigidly fixed boundary conditions

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Fig. 14
figure 14

Variation of non-dimensional phase velocity with wave number for different values k Fröbenius parameter for zinc material stress free and rigidly fixed boundary conditions

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7 Concluding remarks

After simplifying the system of governing equations of motion and heat conduction equation for a circumferential waves in transradially isotropic spherical curved plate with the help of extended power series method (matrix Frobenius method) is successfully used to obtain exact solution of the resulting system of equations. The concluding remarks are

  1. a.

    The matrix Fröbenius method has been successfully employed to investigate the vibration characteristics of spherical curved plate structures.

  2. b.

    It is observed that the shear wave motion of spherical curved plate gets decoupled from the rest of the motion and is not influenced by the thermal field and the corresponding results of pure elastic spherical spherical curved plate are in agreement with those of Shah et al. [6] and Yu et al. [13].

  3. c.

    It is noticed that the spherical curved plate structures are highly influenced with inner radius to thickness ratio.

  4. d.

    The lowest frequency has been noticed to increase with the increase of radius to thickness ratio for fixed outer radius.

  5. e.

    There is small change in damping with the increase of radius to thickness ratio.

  6. f.

    Almost similar trends of variations of vibration characteristics have been observed for zinc and cobalt material plates under stress free and rigidly fixed boundary conditions.

  7. g.

    The numerical results for silicon nitride (\( {\text{Si}}_{ 3} {\text{N}}_{ 4} \)) plate compares well with those of Yu and Xue [16] in the absence of thermal variations.

  8. h.

    It is concluded that the behaviour of phase velocity is similar for cylindrical and spherical curved plates except the change for lower values of wave number. Results for zinc material are in good agreement with earlier results of Sharma and Pathania [12].