Axisymmetric bending vibration analysis of bidirectional functionally graded rotating micro-disk under thermo-mechanical loading

Abstract

The free vibration behavior of a bidirectional functionally graded rotating micro-disk that is subjected to uniform transverse pressure and high-temperature thermal loading has been studied. The micro-disk is functionally graded along the radial and thickness directions. The problem is mathematically formulated using two distinct but interrelated steps within the framework of Kirchhoff plate theory and modified couple stress theory. The first step determines the time-invariant deformed configuration of the micro-disk under centrifugal, pressure, and thermal loading using minimum potential energy principle. The second step determines the free vibration behavior of the micro-disk in the neighborhood of the deformed configuration using Hamilton’s principle. The solutions of the governing equations for both these steps are obtained using the Ritz method. The mathematical model is successfully validated with various reduced problems. The numerical results for the first four axisymmetric bending vibration modes are presented to investigate the effects of wide range of parameters such as rotational speed, applied pressure, thermal loading, size-dependent thickness, volume fraction indices, and radius ratio. The mode shapes of vibration are illustrated through surface and contour plots.

Keywords

Micro-disk axisymmetric bending modified couple stress theory free vibration bidirectional functionally graded thermo-mechanical loading

1. Introduction

Micro-size rotating disks are found in small-scale steam-turbines, gas-turbines, wind-turbines, pumps, compressors, etc. which are very common in various small-size devices and machines (Kim and Yoon (2016); Reale and Sannino (2021); Tran et al. (2020)) such as micro-CHP (combined heat and power) systems, Power MEMS (micro-electro-mechanical systems), micro-propulsion systems, micro-energy harvesters etc. Theoretical studies on mechanical behavior of micro-size components are mostly based on the modified couple stress theory (MCST) (Yang et al. (2002)). The present work investigates the axisymmetric free vibration behavior of bidirectional functionally graded (BFG) rotating micro-disks under transverse pressure and thermal loading based on MCST. It is known that functionally graded materials (FGMs) (Reddy and Chin (1998)) are suitable for high-temperature (above-ambient) thermal loading, and they are two-phase metal-ceramic composites for which the volumetric proportions of the constituents can be smoothly varied in any desired direction. With widespread applications of FGMs, research studies on functionally graded (FG) rotating disks involving classical theories have gained attention from the researchers. In this regard, some of the studies are found in Refs. Bayat et al. (2009); Tutuncu and Temel (2013); Zheng et al. (2016); Eldeeb et al. (2021) involving static behavior and in Refs. Kermani et al. (2016); Jalali et al. (2018) involving dynamic behavior. It is noted that the circular disk/plate geometries are of two categories: one with a central circular hole (called annular disk/plate) and the other without a central circular hole (called solid circular disk/plate).

The conventional continuum theory (called classical theory) is incapable to predict the small-size effect for micro- and nano-size components, which is experimentally verified through the following works: Lam et al. (2003); Liu et al. (2008); Lei et al. (2016); Li et al. (2019). On the other hand, various continuum-based non-classical theories contain additional length scale parameters to account for the size effect. The notable fundamental theories in this regard are couple stress theory (CST) (Mindlin and Tiersten (1962); Toupin (1962); Koiter (1964)), micro-polar theory (Eringen (1967)), surface elasticity theory (Gurtin and Murdoch (1978)), nonlocal elasticity theory (NET) (Eringen (1983)), and strain gradient theory (SGT) (Fleck et al. (1994); Fleck and Hutchinson (2001)). Based on CST, Yang et al. (2002) proposed MCST that involved only one material length scale parameter and thus simplified the treatment of size effect for micro-size structural elements. Some of the research works on micro- and nano-beams/plates based on various size-dependent theories are found in the following references: Tsiatas (2009); Akgöz and Civalek (2014); Akgöz and Civalek (2015a); Akgöz and Civalek (2015b); Abbaspour and Arvin (2020); Ashrafi et al. (2021); Numanoğlu et al. (2022); Jalaei et al. (2022).

In recent times, many research studies have been undertaken to study the mechanical behavior of circular plate-type structures based on MCST. Research works on static analysis of annular micro-plates were reported in Refs. Zhou and Gao (2014); Karttunen et al. (2017), whereas those involving free vibration analysis of solid circular micro-plates were reported in Refs. Wang et al. (2013); Jomehzadeh et al. (2011). Research works on static behavior of FG solid circular and annular micro-plates were reported in Refs. Nguyen et al. (2017); Ashoori and Vanini (2016); Reddy et al. (2016); Eshraghi et al. (2015); Ji et al. (2017), whereas those involving free vibration analysis of FG solid circular and annular micro-plates were reported in Refs. Eshraghi et al. (2015); Ji et al. (2017); Shojaeefard et al. (2017).

Research works involving rotating micro- and nano-size disks/plates are briefly described here. Based on the classical theory, Pei (2012) studied the damped free vibration of clamped-free annular rotating micro-disks under thermal loading for both axisymmetric and asymmetric modes. Based on the thin axisymmetric plate theory (APT) and SGT, static analysis of annular rotating micro-disks has been conducted by Danesh and Asghari (2014). Employing thin APT and SGT, static stress analysis of FG annular rotating nano-disks of variable thickness under thermal loading has been reported by Hosseini et al. (2016). Static stress and displacement analyses of FG annular rotating nano-disks under thermal loading have been presented by Shishesaz et al. (2017) based on thin APT and SGT. Based on the first-order shear deformation theory (FSDT) and MCST, Mahinzare et al. (2018a) investigated the axisymmetric free vibration of BFG solid rotating micro-plates, which were supported by the Winkler–Pasternak foundation and subjected to thermal loading. They assumed radial and asymmetric through-thickness material gradations and analyzed results for the first two modes each for clamped and hinged boundary conditions. Employing FSDT and MCST, Shojaeefard et al. (2018a) studied the axisymmetric free vibration and critical angular speed of BFG solid rotating micro-plates of non-uniform thickness with radial and asymmetric through-thickness material gradations. They analyzed results for clamped and hinged plates for the first three modes of vibration. Based on FSDT and MCST, Mahinzare et al. (2018b) studied the axisymmetric free vibration response of electro-elastic BFG rotating solid nano-plates considering material gradations in radial and thickness directions. Considering FSDT, and based on both MCST and NET, axisymmetric free vibration of magneto-elastic FG solid rotating nano-plates has been investigated by Shojaeefard et al. (2018b). Based on thin APT and SGT, Hosseini et al. (2019) conducted the static stress analysis of FG annular rotating nano-disks of variable thickness under thermo-mechanical loading. Based on the higher-order shear deformation theory (HSDT) and nonlocal SGT, Al-Furjan et al. (2020) investigated the free vibration and critical angular speed of clamped-free composite annular rotating micro-disks for both axisymmetric and asymmetric modes. Based on the Kirchhoff plate theory and MCST, Pal and Das (2020) investigated the free vibration response of BFG annular rotating micro-disks under thermal loading and considering radial and symmetric through-thickness material gradations. They presented results for both axisymmetric and asymmetric modes as well as for the torsional mode. In the above discussion, except Pal and Das (2020), all the works related to the free vibration of micro- and nano-disks/plates considered the prior effects of loading using the variation of external work through Hamilton’s principle. However, Pal and Das (2020) determined the effect of centrifugal and other loadings separately using minimum potential energy principle, and it has been incorporated into the dynamic problem through tangent stiffness of the deformed micro-disk.

It is evident that the free vibration behavior of BFG rotating micro-disks under combined effects of transverse pressure and thermal loading is not yet investigated. The present work attempts to bridge this research gap. The mathematical formulation is based on the Kirchhoff plate theory and MCST for axisymmetric bending of the micro-disk. The micro-disk is assumed BFG with gradations along the radial and thickness directions, and the temperature-dependence of the material properties is considered employing the Touloukian model. The analysis is based on two distinct but interrelated steps. In the first step, the deformed configuration of the micro-disk under combined centrifugal, pressure, and thermal loadings is determined employing the minimum potential energy principle. In the second step, the free vibration response of the deformed micro-disk is determined employing Hamilton’s principle and incorporating the tangent stiffness of the deformed micro-disk. The solutions of the governing equations are obtained using the Ritz method.

2. Mathematical formulation

An annular disk, as shown in Figure 1, having outer radius $b$ , inner radius $a$ , and thickness $t$ has been considered. The set of unit vectors for the set of coordinate directions $(r, θ, z)$ is given as $(u_{r}, u_{θ}, u_{z})$ , and the origin $O$ of the coordinate system lies at the geometric center of the disk. The disk along with the coordinate axes is assumed to be rotating with uniform angular speed $Ω$ about the $z -$ axis. The disk is subjected to centrifugal loading due to rotation, uniform transverse pressure $p_{0}$ , and thermal loading $Δ T = T - T_{0}$ , where $T$ and $T_{0}$ (=300 K) are the current disk temperature and stress-free temperature, respectively.

Figure 1.

Schematic diagram of a rotating disk.

The material of the disk is functionally graded along the radial and thickness directions. Following the rule of mixture, any generic effective material property $(R_{f})$ is determined as given below

R_{f} (r, z, T) = R_{m} (T) + {R_{c} (T) - R_{m} (T)} {(\frac{r - a}{b - a})}^{k_{r}} {| \frac{2 z}{t} |}^{k_{t}} .

(1)

Here,

R_{m}

and

R_{c}

are the properties of the metal and ceramic constituent, respectively;

k_{r} (0 \leq k_{r} \leq \propto)

and

k_{t} (0 \leq k_{t} \leq \propto)

symbolize the volume fraction indices along the radial and thickness directions, respectively. The disk material is pure ceramic (i.e.,

R_{f} = R_{c}

) when

k_{r}

k_{t}

= 0 and pure metal (i.e.,

R_{f} = R_{m}

) when

k_{r}

k_{t}

\propto

. Furthermore, the disk material is pure metal (i.e.,

R_{f} = R_{m}

) at the inner edge (

r = a

) and also at the mid-plane (

z = 0

). Hence, at any non-zero finite values of

k_{r}

and

k_{t}

, moving along from the inner edge towards the outer edge, or moving across from the mid-plane towards the top or bottom surface, the disk material becomes less richer in metal constituent and more richer in ceramic constituent. Following the Touloukian model (Reddy and Chin (1998)), the temperature-dependence of any property at high-temperature environment is determined using the following relationship

R_{m} (T) or R_{c} (T) = R_{0} [R_{- 1} T^{- 1} + 1 + R_{1} T + R_{2} T^{2} + R_{3} T^{3}],

(2)

where

R_{0}

R_{- 1}

R_{1}

R_{2}

, and

R_{3}

are property-specific temperature coefficients. The temperature coefficients of stainless steel and silicon nitride for Young’s moduli

(E_{m}, E_{c})

, Poisson’s ratios

(ν_{m}, ν_{c})

, thermal expansion coefficients

(α_{m}, α_{c})

, and densities

(ρ_{m}, ρ_{c})

are given in Table 1. The effective shear modulus

(G_{f})

is obtained using the well-known relation

G_{f} = E_{f} / {2 (1 + υ_{f})}

. Equation (2) has been obtained through cubic fit relation from the experimental data published in Touloukian (1967) and provides higher-order effects of temperature at high-temperature environment. Equation (1) shows that the material gradation is symmetric across the thickness and thus avoids bending due to the rotational effect (Bhattacharya and Das (2019)).

Table 1.

Temperature coefficients of stainless steel and silicon nitride.

Material	Property	$R_{0}$	$R_{1}$	$R_{2}$	$R_{3}$
Stainless steel (SUS304)	$E_{m}$ (Pa)	201.04 × 10⁹	3.079 × 10⁻⁴	−6.534 × 10⁻⁷	0
	$α_{m}$ (1/K)	12.330 × 10⁻⁶	8.086 × 10⁻⁴	0	0
	$υ_{m}$	0.3262	−2.002 × 10⁻⁴	3.797 × 10⁻⁷	0
	$ρ_{m}$ (kg/m³)	8166	0	0	0
Silicon nitride (Si₃N₄)	$E_{c}$ (Pa)	348.43 × 10⁹	−3.070 × 10⁻⁴	2.160 × 10⁻⁷	−8.946 × 10⁻¹¹
	$α_{c}$ (1/K)	5.8723 × 10⁻⁶	9.095 × 10⁻⁴	0	0
	$υ_{c}$	0.2400	0	0	0
	$ρ_{c}$ (kg/m³)	2370	0	0	0

2.1. Derivation of strain, curvature, stress, and couple stress components

According to the Kirchhoff plate theory for axisymmetric bending, the displacement components ${u, v, w}$ along the coordinate directions $(r, θ, z)$ of any point $(r, z)$ are expressed as

u (r, z) = u_{0} (r) - z \frac{\partial w_{0} (r)}{\partial r}, v (r, z) = 0, w (r, z) = w_{0} (r),

(3)

where

u_{0}

and

w_{0}

are the displacement components of

(r, 0)

. Hence, the non-zero components of the classical strain tensor

(ε)

are derived to the following form (Reddy et al. (2016))

ε_{r r} = [\frac{\partial u_{0}}{\partial r} - z \frac{\partial^{2} w_{0}}{\partial r^{2}} + \frac{1}{2} {(\frac{\partial w_{0}}{\partial r})}^{2}], ε_{θ θ} = [\frac{u_{0}}{r} - \frac{z}{r} \frac{\partial w_{0}}{\partial r}] .

(4)

The last term in the first relation of equation (4) appears due to von Kármán type non-linearity.

For any point $(r, z)$ , the relationship between the rotation vector $θ$ and displacement vector $u (= u u_{r} + w u_{z})$ is given by $θ = \frac{1}{2} (\nabla \times u),$ where $\nabla = u_{r} \frac{\partial}{\partial r} + u_{z} \frac{\partial}{\partial z}$ . Furthermore, using $θ$ , the symmetric curvature tensor $(η)$ is derived using the following relation (Yang et al. (2002); Reddy et al. (2016)): $η = \frac{1}{2} [\nabla θ + {(\nabla θ)}^{T}] .$ Hence, the symmetric curvature tensor is related to the displacement vector as follows

η = \frac{1}{4} [\nabla (\nabla \times u) + {(\nabla (\nabla \times u))}^{T}] .

(5)

Substituting equation (3) into equation (5), the non-zero components of the symmetric curvature tensor is obtained in terms of the mid-plane displacement as

η_{r θ} = η_{θ r} = \frac{1}{2} [\frac{1}{r} \frac{\partial w_{0}}{\partial r} - \frac{\partial^{2} w_{0}}{\partial r^{2}}] .

(6)

Following MCST and incorporating the effect of thermal loading $Δ T$ , the non-zero components of the classical stress tensor $(σ)$ and the deviatoric part of the symmetric couple stress tensor $(m)$ are obtained as follows

σ_{r r} = \frac{E_{f}}{1 - {(ν_{f})}^{2}} (ε_{r r} + ν_{f} ε_{θ θ}) - \frac{E_{f} α_{f}}{1 - ν_{f}} Δ T, σ_{θ θ} = \frac{E_{f}}{1 - {(ν_{f})}^{2}} (ε_{θ θ} + ν_{f} ε_{r r}) - \frac{E_{f} α_{f}}{1 - ν_{f}} Δ T .

(7)

m_{r θ} = m_{θ r} = 2 G_{f} l^{2} η_{r θ} .

(8)

In equation (8), $l$ is the material length scale parameter.

2.2. Deformed configuration under centrifugal, pressure, and thermal loading

In the first step of the problem, the deformed configuration of the micro-disk under combined effects of time-invariant centrifugal loading due to constant angular speed, transverse pressure, and thermal loading is determined. The governing equations in this case are derived using minimum potential energy principle given as

δ (U_{c l} + U_{n c l} + V) = 0,

(9)

where

U_{c l}

and

U_{n c l}

are the classical and non-classical strain energies,

V

is the potential energy of the applied loadings, and

δ

is the variational operator. According to MCST, the classical strain tensor

(ε)

conjugates to the classical stress

(σ)

tensor to generate

U_{c l}

, whereas the curvature tensor

(η)

conjugates to the couple stress tensor

(m)

to generate

U_{n c l}

. Using equations (4) and (6)–(8), the expressions of

U_{c l}

U_{n c l}

, and

V

are derived to the following form

\begin{array}{c} U_{c l} = \frac{1}{2} \int_{- t / 2}^{+ t / 2} \int_{0}^{2 π} \int_{a}^{b} (σ : ε) r d r d θ d z \\ = \frac{1}{2} \int_{- t / 2}^{+ t / 2} \int_{0}^{2 π} \int_{a}^{b} [σ_{r r} {ε_{r r} - α_{f} Δ T} + σ_{θ θ} {ε_{θ θ} - α_{f} Δ T}] r d r d θ d z \\ = 2 π \int_{a}^{b} [E_{A} {\frac{r}{2} {(\frac{\partial u_{0}}{\partial r})}^{2} + \frac{{u_{0}}^{2}}{2 r} + \frac{r}{8} {(\frac{\partial w_{0}}{\partial r})}^{4} + \frac{r}{2} \frac{\partial u_{0}}{\partial r} {(\frac{\partial w_{0}}{\partial r})}^{2}} + E_{I} {\frac{r}{2} {(\frac{\partial^{2} w_{0}}{\partial r^{2}})}^{2} + \frac{1}{2 r} {(\frac{\partial w_{0}}{\partial r})}^{2}} \\ + E_{P A} {u_{0} \frac{\partial u_{0}}{\partial r} + \frac{u_{0}}{2} {(\frac{\partial w_{0}}{\partial r})}^{2}} + E_{P I} \frac{\partial w_{0}}{\partial r} \frac{\partial^{2} w_{0}}{\partial r^{2}} - E_{T A} {u_{0} + r \frac{\partial u_{0}}{\partial r} + \frac{r}{2} {(\frac{\partial w_{0}}{\partial r})}^{2}} + r E_{C}] d r, \end{array}

(10)

\begin{array}{c} U_{n c l} = \frac{1}{2} \int_{- t / 2}^{+ t / 2} \int_{0}^{2 π} \int_{a}^{b} (m : η) r d r d θ d z \\ = 2 π l^{2} \int_{a}^{b} G_{A} [\frac{r}{2} {(\frac{\partial^{2} w_{0}}{\partial r^{2}})}^{2} + \frac{1}{2 r} {(\frac{\partial w_{0}}{\partial r})}^{2} - \frac{\partial w_{0}}{\partial r} \frac{\partial^{2} w_{0}}{\partial r^{2}}] d r, \end{array}

(11)

V = - 2 π Ω^{2} \int_{a}^{b} ρ_{A} u_{0} r^{2} d r - 2 π p_{0} \int_{a}^{b} w_{0} r d r .

(12)

The set of stiffness coefficients ${E_{A}, E_{I}, E_{P A}, E_{P I}, E_{T A}, E_{C}, G_{A}}$ used in equations (10) and (11) and the inertia coefficient $ρ_{A}$ used in equation (12) are defined as

\begin{array}{l} E_{A} = \int_{- t / 2}^{+ t / 2} \frac{E_{f}}{1 - {(ν_{f})}^{2}} d z, E_{I} = \int_{- t / 2}^{+ t / 2} \frac{E_{f} z^{2}}{1 - {(ν_{f})}^{2}} d z, E_{P A} = \int_{- t / 2}^{+ t / 2} \frac{E_{f} ν_{f}}{1 - {(ν_{f})}^{2}} d z, E_{P I} = \int_{- t / 2}^{+ t / 2} \frac{E_{f} ν_{f} z^{2}}{1 - {(ν_{f})}^{2}} d z, \\ E_{T A} = Δ T \int_{- t / 2}^{+ t / 2} \frac{E_{f} α_{f}}{1 - ν_{f}} d z, E_{C} = {(Δ T)}^{2} \int_{- t / 2}^{+ t / 2} \frac{E_{f} {(α_{f})}^{2}}{1 - ν_{f}} d z, G_{A} = \int_{- t / 2}^{+ t / 2} G_{f} d z, ρ_{A} = \int_{- t / 2}^{+ t / 2} ρ_{f} d z . \end{array}

(13)

To derive the governing equations in algebraic form, following the Ritz method, each of the mid-plane displacements ${u_{0}, w_{0}}$ are discretized as finite linear combinations of one-dimensional (1D) orthogonal admissible functions ${ϕ_{j}^{u}, ϕ_{j}^{w}}$ and unknown coefficients $c_{j}$ as

u_{0} (r) = \sum_{j = 1}^{n} c_{j} ϕ_{j}^{u} (r), w_{0} (r) = \sum_{j = 1}^{n} c_{n + j} ϕ_{j}^{w} (r) .

(14)

Here, $n$ is the total number of terms for each function. Considering the inner edge of the disk as clamped and the outer edge as free, the lowest-order admissible functions for ${ϕ_{j}^{u}, ϕ_{j}^{w}}$ are selected as follows

ϕ_{1}^{u} (r) = r; ϕ_{1}^{w} (r) = r^{2} .

(15)

Subsequently, the higher-order functions are numerically generated employing the Gram–Schmidt orthogonalization scheme.

Substituting equations (10)–(12) into equation (9), and employing the approximate displacement fields given by equation (14), the governing equations are obtained in algebraic form, which in matrix form is written as

[S] {c} = {P} .

(16)

Here, $[S]$ is the total stiffness matrix of dimension $2 n$ given by $[S] = [S^{cl}] + [S^{ncl}]$ , where $[S^{cl}]$ and $[S^{ncl}]$ are the classical and non-classical stiffness matrix, respectively; ${c}$ is the vector of unknown coefficients $c_{j}$ ; and ${P}$ is the load vector. The details of $[S^{cl}]$ , $[S^{ncl}]$ , and ${P}$ are provided in APPENDIX A. Due to the presence of von Kármán type non-linearity in the classical strain tensor, $[S^{cl}]$ is non-linear in nature, whereas in the absence of any type of geometric non-linear terms in the curvature tensor, $[S^{ncl}]$ is essentially linear. To solve the system of non-linear algebraic equations given by equation (16), a multi-dimensional secant method known as Broyden’s method (Press et al. (1992)) is employed. The uniqueness of the solution within the realm of the present problem is ensured by proper selection of an initial guess at the beginning of each load step. The present algorithm (Broyden’s method) gives unique solution for almost any initial guess for the initial few load steps where the transverse displacement $w$ is very small, making the problem essentially linear. However, as the transverse displacement grows for the subsequent load steps, the influence of non-linearity becomes significant. In such load steps, the solution for the (i-1)^th load step is considered as the initial guess for the i^th load step, and this has been found to provide unique solution for the present problem. However, occasionally this has failed to converge to a unique solution. In such cases, the load increment for the i^th load step is reduced substantially so that the solution of the (i-1)^th load step can act as a good guess for it to yield a unique solution. The solution of equation (16) when substituted into equation (14) determines the displacement fields and in turn provides the deformed configuration of the micro-disk under combined effects of centrifugal loading, transverse pressure, and thermal loading.

2.3. Free vibration response of deformed micro-disk

In the second step of the problem, the small amplitude linear free vibration behavior in the neighborhood of the deformed configuration of the micro-disk is determined. In this case, the governing equations are derived employing Hamilton’s principle, which is given as follows

δ {\int_{τ_{1}}^{τ_{2}} (T_{k} - U_{c l} - U_{n c l}) d τ} = 0 .

(17)

Here, $T_{k}$ is the kinetic energy of the micro-disk and $τ$ is the time. The expression of $T_{k}$ is derived as follows

\begin{array}{c} T_{k} = \int_{0}^{2 π} \int_{a}^{b} \int_{- t / 2}^{+ t / 2} [\frac{ρ_{f}}{2} {\frac{\partial u (r, z)}{\partial τ}}^{2} + \frac{ρ_{f}}{2} {\frac{\partial w (r, z)}{\partial τ}}^{2}] r d z d r d θ \\ = 2 π \int_{a}^{b} \int_{- t / 2}^{+ t / 2} [\frac{ρ_{f}}{2} {\frac{\partial u (r, z)}{\partial τ}}^{2} + \frac{ρ_{f}}{2} {\frac{\partial w (r, z)}{\partial τ}}^{2}] r d z d r \\ = 2 π \int_{a}^{b} \int_{- t / 2}^{+ t / 2} [\frac{ρ_{f}}{2} {\frac{\partial}{\partial τ} (u_{0} (r) - z \frac{\partial w_{0} (r)}{\partial r})}^{2} + \frac{ρ_{f}}{2} {\frac{\partial (w_{0} (r))}{\partial τ}}^{2}] r d z d r \\ = 2 π \int_{a}^{b} \int_{- t / 2}^{+ t / 2} [\frac{ρ_{f}}{2} {{(\frac{\partial u_{0}}{\partial τ})}^{2} + z^{2} {(\frac{\partial^{2} w_{0}}{\partial τ \partial r})}^{2} - 2 z \frac{\partial u_{0}}{\partial τ} \frac{\partial^{2} w_{0}}{\partial τ \partial r}} + \frac{ρ_{f}}{2} {(\frac{\partial w_{0}}{\partial τ})}^{2}] r d z d r \\ = 2 π \int_{a}^{b} [\frac{ρ_{A}}{2} {{(\frac{\partial u_{0}}{\partial τ})}^{2} + {(\frac{\partial w_{0}}{\partial τ})}^{2}} + \frac{ρ_{I}}{2} {(\frac{\partial^{2} w_{0}}{\partial τ \partial r})}^{2}] r d r . \end{array}

(18)

Here, ${ρ_{A}, ρ_{I}}$ is the set of inertia coefficients in which $ρ_{A}$ is as defined in equation (13), and $ρ_{I}$ is defined as $ρ_{I} = \int_{- t / 2}^{+ t / 2} ρ_{f} z^{2} d z .$ In deriving equation (18), the following simplification due to symmetric through-thickness gradation is considered: $\int_{- t / 2}^{+ t / 2} ρ_{f} z d z = 0 .$ The time-dependent mid-plane displacements ${u_{0}, w_{0}}$ are considered to be originated from the deformed configuration that is already determined in the previous step.

The dynamic displacement components are discretized using the Ritz method using the same set of functions ${ϕ_{j}^{u}, ϕ_{j}^{w}}$ as used in the previous step but with a new set of unknown coefficients $d_{j}$ as follows

\begin{array}{l} u_{0} (r, τ) = \sum_{j = 1}^{n} (d_{j} e^{i ω τ}) ϕ_{j}^{u} (r), \\ w_{0} (r, τ) = \sum_{j = 1}^{n} (d_{n + j} e^{i ω τ}) ϕ_{j}^{w} (r) . \end{array}

(19)

Here, it is considered that the vibratory motion is harmonic with $ω$ as the frequency of vibration and $i = \sqrt{- 1}$ .

The tangent stiffness $([K])$ of the deformed micro-disk is responsible for small amplitude linear free vibration that occurs in the neighborhood of the deformed configuration. At any deformed state given by ${c}$ , the components of $[K]$ are determined by taking partial derivative of each of the elements of the restoring force vector with respect to each component of ${c}$ (Das (2018)). Hence, the components of the tangent stiffness matrix are derived using the following relation

K_{i k} = \frac{\partial {\sum_{j = 1}^{2 n} S_{i j} c_{j}}}{\partial c_{k}} for i = 1,2, . . . . ., 2 n; k = 1,2, . . . . ., 2 n .

(20)

The tangent stiffness matrix consists of classical $([K^{cl}])$ and non-classical $([K^{ncl}])$ parts. The details of $[K^{cl}]$ are provided in APPENDIX B. The non-classical part of the total stiffness matrix ( $[S^{ncl}]$ ) is essentially linear due to the absence of geometric non-linear terms in the curvature tensor. Hence, by virtue of equation (20), the non-classical part of the tangent stiffness matrix ( $[K^{ncl}]$ ) becomes linear and identical to $[S^{ncl}]$ , that is, $[K^{ncl}] = [S^{ncl}]$ . However, the tangent stiffness matrix is non-linear as a whole because the classical part of the tangent stiffness matrix $([K^{cl}])$ is non-linear.

Substituting equation (19) into equation (18) and using the tangent stiffness, the governing equations are derived, employing equation (17), as given below

[[K] - ω^{2} [M]] {d} = 0,

(21)

where

[M]

is the mass matrix. Equation (21) represents an eigen-value problem where

{d}

is the eigen-vector. The tangent stiffness matrix in equation (21) is linearized using the already known coefficients

{c}

of the deformed state. The square root of the eigen-values provides the frequencies of free vibration of the deformed micro-disk for different modes and the corresponding set of eigen-vectors determine the relevant mode shapes. The components of

[M]

are provided in APPENDIX C.

3. Results and discussion

The numerical results are generated for stainless steel (metal) and silicon nitride (ceramic) FG composition. If not mentioned otherwise, the following values of the parameters are used: $l$ = 17.6 × 10⁻⁶ m (Park and Gao (2006); Lam et al. (2003)), $t / l$ = 1, $b / t$ = 50, $b / a$ = 10.0, $k_{r} = k_{t} = 1$ , $p^{*}$ = 5, and $Δ T$ = 200. The non-dimensional speed $(Ω^{*})$ , pressure $(p^{*})$ , and vibration frequency $(ω^{*})$ are defined as follows: $Ω^{*} = Ω b^{2} \sqrt{ρ_{0} t / D}$ , $p^{*} = p_{0} b^{4} / (E_{0} t^{4})$ , and $ω^{*} = ω b^{2} \sqrt{ρ_{0} t / D}$ , where $D = E_{0} t^{3} / {12 (1 - {υ_{0}}^{2})}$ . The material properties with subscript “0” refer to the ceramic phase, determined at $T_{0}$ = 300 K.

The variation of the vibration frequency with rotational speed for the fundamental mode of a homogeneous $(k_{r} = 0, k_{t} = 0)$ annular clamped-free rotating disk, subjected to uniform transverse pressure, has been compared with Das et al. (2010), and it is shown in Figure 2. The non-dimensional speed $\bar{Ω}$ in Figure 2 is defined as $\bar{Ω} = Ω b \sqrt{ρ / σ_{y}}$ , and the results are generated with the following values: $l$ = 0, $E$ = 210 × 10⁹ N/m², $ν$ = 0.3, $ρ$ = 7850 kg/m³, $σ_{y}$ = 300 $\times$ 10⁶ N/m², $t$ = =0.006 m, $b / t$ = 50, $b / a$ = 12, $p^{*}$ = 0.3571, and $Δ T$ = 0.

Figure 2.

Comparison of non-dimensional speed versus frequency behavior for the first mode of a homogeneous annular rotating disk under uniform transverse pressure.

The non-dimensional load-deflection behavior of a thickness-FG $(k_{r} = 0)$ annular clamped-free micro-plate under uniform transverse pressure has been compared with Reddy et al. (2016) in Figure 3 for different values of $k_{t}$ . With respect to Figure 3, the normalized deflection $(w^{*})$ is defined as $w^{*} = w_{\max} / t$ . The results of Figure 3 are produced using the following property relation: $E_{f} (z) = E_{2} + (E_{1} - E_{2}) (\frac{z}{t} + \frac{1}{2})$ , and the following values: $E_{1}$ = 10⁶ N/m², $E_{2}$ = 10⁵ N/m², $ν_{f}$ = =0.25, $t$ = 0.1 $\times$ 10⁻⁶ m, $l / t$ = =0.6, $b / t$ = 10, $b / a$ = 4, $Ω^{*}$ = 0, and $Δ T$ = 0.

Figure 3.

Comparison of non-dimensional pressure versus deflection behavior of a thickness-FG annular clamped-free micro-plate.

The variation of the vibration frequency with volume fraction index ( $k_{t}$ ) for the first two modes of a thickness-FG $(k_{r} = 0)$ solid circular $(a = 0)$ clamped micro-plate has been compared with Shojaeefard et al. (2017) in Figure 4. The results in Figure 4 correspond to stainless steel–silicon nitride plate under non-linear through-thickness temperature variation given by $T (z) = 300 + 90 {(\frac{z}{t} + \frac{1}{2})}^{0.9}$ K. The results of Figure 4 are generated using the following relation: $R_{f} (z) = R_{m} + (R_{c} - R_{m}) {(\frac{z}{t} + \frac{1}{2})}^{k_{t}}$ and with the following values: $l$ =17.6 $\times$ 10⁻⁶ m, $t / l$ = 1.0, $b / t$ = 20, $Ω^{*}$ = 0, and $p^{*}$ = 0. The comparisons illustrated through Figures 2–4 show excellent matching of the results and thus validate the present model.

Figure 4.

Comparison showing variation of non-dimensional vibration frequency with volume fraction index for the first two modes of a thickness-FG solid circular clamped micro-plate under thermal loading.

The centrifugal loading due to rotation and the thermal loading are in-plane in nature and do not induce any bending of the micro-disk. However, it is subjected to static bending due to applied transverse pressure. To illustrate this, the deflection-pressure curves in high-temperature environment are shown in non-dimensional plane in Figure 5 for different rotational speed where the normalized deflection is defined as $w^{*} = w_{\max} / t$ . Figure 5 clearly shows the hardening type non-linear static behavior at low to moderate rotational speeds and linear static behavior at higher values of rotational speed. This non-linearity arises out of the geometric stiffening due to von Kármán type non-linearity. Furthermore, the deflection is found to decrease with increasing rotational speed as a result of enhanced centrifugal stiffening.

Figure 5.

Variation of transverse deflection with applied transverse pressure for different rotational speed under constant thermal loading.

Figures 6(a)–(d) show the non-dimensional speed–frequency behavior in high-temperature environment for the first four modes, each for different transverse pressure. The frequencies are found to increase with speed because of centrifugal stiffening. However, exception to this occurs for the first and second modes (Figures 6(a) and (b)) at lower speed ranges for moderate and higher pressure values where the frequency remains constant or slightly decreases with speed. This clearly indicates softening type behavior for these modes at these loading conditions. Further, the increase in transverse pressure exhibits increasing frequency values for all the four modes, indicating enhanced geometric stiffening at higher pressure values. However, the effect of transverse pressure diminishes with increasing speed, and it almost ceases to exist at higher rotational speeds for higher modes of vibration. For visualization of the modes, Figures 7(a)–(d) illustrate the mode shapes and the corresponding contour plots of the first four axisymmetric bending vibration modes for $Ω^{*}$ = 10, $p^{*}$ = 5, and $Δ T$ = 200. As the disk is clamped at the inner edge and free at the outer edge, the first mode of vibration does not contain any nodal circle. However, as the mode number increases, the number of nodal circles increases by one for each subsequent mode. It is worthwhile to mention that none of the axisymmetric vibration modes contains nodal diameters. To investigate the effect of size, Figure 8 has been presented where the normalized modal deflection $(\bar{w})$ has been plotted against the non-dimensional radial coordinate $(\bar{r})$ for each of the first four modes. For each mode, plots with ( $t / l$ = 1) and without ( $l$ = 0) size effects are shown. In Figure 8, the modal deflections have been normalized by the maximum modal deflections for the corresponding modes, and the non-dimensional radial coordinate has been defined as $\bar{r} = (r - a) / (b - a)$ . Figure 8 clearly shows the significant effect of size on the mode shapes of vibration, and this effect has been found to diminish as the mode number increases. Figures 9(a)–(d) show the non-dimensional speed–frequency behavior at a constant transverse pressure for the first four modes, each for different thermal loading. The frequencies are found to decrease with increasing thermal loading due to the thermo-elastic degradation of the material properties, and this effect becomes enhanced as the mode number increases. A reverse trend is observed for the first mode at lower speeds where the frequency increases with increasing thermal loading.

Figure 6.

Variation of vibration frequency with rotational speed for different transverse pressure under constant thermal loading: (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode.

Figure 7.

Mode shapes and contour plots of the axisymmetric bending vibration modes: (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode.

Figure 8.

Effect of size on the normalized modal deflection for the first four axisymmetric bending vibration modes.

Figure 9.

Variation of vibration frequency with rotational speed for different thermal loading under constant transverse pressure: (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode.

Table 2 lists the vibration frequencies in high-temperature environment for different size-dependent thickness (

t / l

) values, each for the first four modes. Table 2 shows the frequency values for different combinations of rotational speed and transverse pressure. It is to be mentioned that lower the size-dependent thickness, higher is the size effect and vice-versa, and the classical theory (i.e., where no size effect is considered) is restored by putting

l

= 0. As such, the increase in the size-dependent thickness leads to decrease in the frequency values for all the modes, indicating diminishing stiffness when the size effect is reduced. Exception to this occurs for the first mode for

p^{*}

= 5 and 10 at low speed (

Ω^{*}

= 1), where the frequency values slightly increase when

t / l

is increased as 2, 5 up to 10. However, the classical theory provides the least values of frequencies for all the modes, and for any combination of speed and pressure. It is found that the difference in the frequency values between the classical theory and that corresponding to

t / l

= 10 varies widely depending on the loading conditions and the mode numbers. Further, the extent of size effect varies with the mode number because it is found that the higher the mode number, the higher is the reduction of the frequency values found with increase in the

t / l

values. Table 3 lists the natural frequency of vibration (

Ω^{*}

= 0,

p^{*}

= 0,

Δ T

= 0) for the first four modes, each for different size-dependent thickness values. The values of Table 3 can be compared to the corresponding values in Table 2 for

Ω^{*}

= 1,

p^{*}

= 0, and

Δ T

= 200. The comparison shows that the frequency values of Table 3 are greater than those of Table 2 corresponding to

Ω^{*}

= 1,

p^{*}

= 0, and

Δ T

= 200. This is as per the observations discussed for Figures 6 and 9 with regard to the effect of speed, pressure, and thermal loading.

Table 2.

Effect of size-dependent thickness ( $t / l$ ) on vibration frequency ( $ω^{*}$ ) for different transverse pressure and rotational speed under constant thermal loading ( $Δ T$ = 200).

$p^{*}$	$t / l$	$ω^{*}$
		First mode			Second mode			Third mode			Fourth mode
		$Ω^{*}$			$Ω^{*}$			$Ω^{*}$			$Ω^{*}$
		1	5	10	1	5	10	1	5	10	1	5	10
0.0	1	2.921	5.170	8.768	25.408	27.525	33.358	82.909	84.926	90.915	167.383	169.510	175.963
	2	2.511	4.750	8.152	17.165	20.224	27.732	53.249	56.362	65.041	106.624	109.940	119.622
	5	2.190	4.471	7.831	13.844	17.539	25.875	41.209	45.188	55.639	81.842	86.131	98.152
	10	2.119	4.416	7.774	13.277	17.105	25.588	39.174	43.349	54.154	77.650	82.161	94.677
	$l$ = 0	2.093	4.396	7.754	13.079	16.956	25.490	38.471	42.717	53.644	76.200	80.794	93.489
5.0	1	5.541	5.754	8.809	27.063	27.986	33.404	83.950	85.239	90.953	168.163	169.745	175.991
	2	5.444	5.452	8.209	19.786	20.949	27.801	55.348	57.013	65.118	108.527	110.560	119.703
	5	5.454	5.314	7.903	17.200	18.487	25.961	44.312	46.178	55.746	84.815	87.139	98.274
	10	5.463	5.293	7.849	16.796	18.104	25.678	42.518	44.420	54.268	80.883	83.265	94.808
	$l$ = 0	3.833	3.922	7.830	14.560	15.388	25.582	39.746	40.498	53.766	77.351	78.088	93.624
10.0	1	7.048	6.746	8.930	28.480	28.875	33.538	84.919	85.863	91.068	168.897	170.216	176.073
	2	6.931	6.542	8.372	21.692	22.225	27.998	57.071	58.205	65.340	110.167	111.714	119.937
	5	6.952	6.501	8.104	19.366	20.009	26.206	46.573	47.822	56.052	87.114	88.844	98.622
	10	6.973	6.499	8.059	19.011	19.674	25.932	44.878	46.160	54.592	83.302	85.091	95.100
	$l$ = 0	3.833	6.498	8.043	14.946	19.561	25.840	39.746	45.594	54.096	77.351	83.803	94.005

Table 3.

Effect of size-dependent thickness ( $t / l$ ) on the natural frequency ( $ω^{*}$ ) of the micro-disk ( $Ω^{*}$ = 0, $p^{*}$ = 0, $Δ T$ = 0).

$t / l$	$ω^{*}$
$t / l$	First mode	Second mode	Third mode	Fourth mode
1	3.230	26.053	84.736	171.034
2	2.796	17.648	54.374	108.817
5	2.458	15.297	42.035	83.404
10	2.383	13.687	39.948	79.100
$l$ = 0	2.355	13.486	39.227	77.612

Figures 10(a)–(d) show the non-dimensional speed–frequency behavior at constant transverse pressure and thermal loading for the first four modes, each for different set of volume fraction indices. The frequencies are observed to increase with speed except for the first mode where they slightly decrease or remain constant with speed at lower speeds, and this is true for all the values of volume fraction indices considered. Further, the frequencies are found to decrease with increasing values of volume fraction indices. This is because increasing volume fraction indices leads to relative increase in volume proportion of the metal content, which is less elastic and lighter, compared to the ceramic phase. This effect of volume fraction indices becomes enhanced with increasing mode number. Figures 11(a)–(d) show the non-dimensional speed–frequency behavior at constant transverse pressure and thermal loading for the first four modes, each for different radius ratio. The frequencies are observed to increase with speed except for the first mode where they slightly decrease or remain constant with speed at lower speeds, and this true for all the values of radius ratio considered. As far as the effect of radius ratio is considered, it shows that the frequencies increase with decreasing radius ratio. This is because the decreasing radius ratio decreases the un-supported span of the disk which is clamped at the inner radius and thus making the disk stiffer for bending. The effect of radius ratio becomes enhanced with higher modes.

Figure 10.

Variation of vibration frequency with rotational speed for different volume fraction indices under constant transverse pressure and thermal loading: (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode.

Figure 11.

Variation of vibration frequency with rotational speed for different radius ratio under constant transverse pressure and thermal loading: (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode.

4. Conclusions

The free axisymmetric bending vibration behavior of a BFG rotating annular micro-disk has been studied based on the Kirchhoff plate theory and MCST. The micro-disk apart from being rotating with constant angular speed is subjected to uniform transverse pressure and thermal loading. The following conclusions are noted based on the presented results:

(i) The transverse pressure leads to non-linear static bending of the micro-disk.

(ii) The vibration frequency increases with the rotational speed, initially gradually and then rapidly indicating dominance of centrifugal stiffening. However, for the first and second modes, the frequency remains constant or slightly decreases with speed at lower speed ranges for moderate and higher pressure values indicating dominance of softening.

(iii) The increase in pressure shows increasing vibration frequency for all the four modes because of enhanced geometric stiffening, though the effect of pressure is found to diminish with increasing speed.

(iv) The thermal loading, in general, provides thermo-elastic degradation of the micro-disk, leading to decreasing frequency values at higher thermal loading.

(v) The size effect is found to have significant influence where the reduced size-dependent thickness provides higher frequency values indicating strong size-dependent stiffening of the micro-disk.

(vi) Increasing the volume fraction indices leads to decreased frequency values, and these parameters effect the frequency values to a large extent.

(vii) Increasing radius ratio leads to significant decrease in the frequency values.

Footnotes

Acknowledgment

The authors gratefully acknowledge that the work reported in this paper is supported by National Doctoral Fellowship (NDF) awarded by All India Council for Technical Education (AICTE), India, under AICTE-NDF scheme-2018.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Debabrata Das

Appendix A

Total stiffness matrix

The sub-matrices of $[S^{cl}]$ , each of dimension $n \times n$ , are as follows: $[S^{cl}] = [\begin{array}{l} [S_{11}^{cl}] [S_{12}^{cl}] \\ [S_{21}^{cl}] [S_{22}^{cl}] \end{array}] .$

The components of these sub-matrices are given below

[S_{11}^{cl}] = 2 π \int_{a}^{b} [E_{A} {r \frac{\partial ϕ_{i}^{u}}{\partial r} \frac{\partial ϕ_{j}^{u}}{\partial r} + \frac{1}{r} ϕ_{i}^{u} ϕ_{j}^{u}} + E_{P A} {ϕ_{i}^{u} \frac{\partial ϕ_{j}^{u}}{\partial r} + \frac{\partial ϕ_{i}^{u}}{\partial r} ϕ_{j}^{u}}] d r,

[S_{12}^{cl}] = 2 π \int_{a}^{b} [\frac{E_{A}}{2} r (\frac{\partial w_{0}}{\partial r}) \frac{\partial ϕ_{i}^{u}}{\partial r} \frac{\partial ϕ_{j}^{w}}{\partial r} + \frac{E_{P A}}{2} (\frac{\partial w_{0}}{\partial r}) ϕ_{i}^{u} \frac{\partial ϕ_{j}^{w}}{\partial r}] d r,

[S_{21}^{cl}] = 0,

[S_{22}^{cl}] = 2 π \int_{a}^{b} [E_{A} {r (\frac{\partial u_{0}}{\partial r}) \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{j}^{w}}{\partial r} + \frac{r}{2} {(\frac{\partial w_{0}}{\partial r})}^{2} \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{j}^{w}}{\partial r}} + E_{I} {r \frac{\partial^{2} ϕ_{i}^{w}}{\partial r^{2}} \frac{\partial^{2} ϕ_{j}^{w}}{\partial r^{2}} + \frac{1}{r} \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{j}^{w}}{\partial r}}

+ E_{P A} (u_{0}) \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{j}^{w}}{\partial r} + E_{P I} {\frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial^{2} ϕ_{j}^{w}}{\partial r^{2}} + \frac{\partial^{2} ϕ_{i}^{w}}{\partial r^{2}} \frac{\partial ϕ_{j}^{w}}{\partial r}} - E_{T A} r \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{j}^{w}}{\partial r}] d r .

The sub-matrices of $[S^{n c l}]$ , each of dimension $n \times n$ , are as follows: $[S^{n c l}] = [\begin{array}{l} [S_{11}^{n c l}] [S_{12}^{n c l}] \\ [S_{21}^{n c l}] [S_{22}^{n c l}] \end{array}] .$

The components of these sub-matrices are given below

[S_{11}^{ncl}] = 0, [S_{12}^{ncl}] = 0, [S_{21}^{ncl}] = 0,

[S_{22}^{ncl}] = 2 π l^{2} \int_{a}^{b} G_{A} [- \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial^{2} ϕ_{j}^{w}}{\partial r^{2}} - \frac{\partial^{2} ϕ_{i}^{w}}{\partial r^{2}} \frac{\partial ϕ_{j}^{w}}{\partial r} + r \frac{\partial^{2} ϕ_{i}^{w}}{\partial r^{2}} \frac{\partial^{2} ϕ_{j}^{w}}{\partial r^{2}} + \frac{1}{r} \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{j}^{w}}{\partial r}] d r .

Load vector

The sub-vectors of $[P]$ , each of dimension $n \times 1$ , are as follows: ${P} = {\begin{cases} {P_{1}} \\ {P_{2}} \end{cases}}$ .

The components of these sub-vectors are given below

{P_{1}} = 2 π Ω^{2} \int_{a}^{b} ρ_{A} r^{2} ϕ_{i}^{u} d r + 2 π \int_{a}^{b} E_{T A} [ϕ_{i}^{u} + r \frac{\partial ϕ_{i}^{u}}{\partial r}] d r, {P_{2}} = 2 π p_{0} \int_{a}^{b} r ϕ_{i}^{w} d r .

Appendix B

Tangent stiffness matrix

The sub-matrices of $[K^{cl}]$ , each of dimension $n \times n$ , are as follows: $[K^{cl}] = [\begin{array}{l} [K_{11}^{cl}] [K_{12}^{cl}] \\ [K_{21}^{cl}] [K_{22}^{cl}] \end{array}] .$

The components of these sub-matrices are given below

[K_{11}^{cl}] = 2 π \int_{a}^{b} [E_{A} {r \frac{\partial ϕ_{i}^{u}}{\partial r} \frac{\partial ϕ_{k}^{u}}{\partial r} + \frac{1}{r} ϕ_{i}^{u} ϕ_{k}^{u}} + E_{P A} {ϕ_{i}^{u} \frac{\partial ϕ_{k}^{u}}{\partial r} + \frac{\partial ϕ_{i}^{u}}{\partial r} ϕ_{k}^{u}}] d r,

[K_{12}^{cl}] = 2 π \int_{a}^{b} [E_{A} r (\frac{\partial w_{0}}{\partial r}) \frac{\partial ϕ_{i}^{u}}{\partial r} \frac{\partial ϕ_{k}^{w}}{\partial r} + E_{P A} (\frac{\partial w_{0}}{\partial r}) ϕ_{i}^{u} \frac{\partial ϕ_{k}^{w}}{\partial r}] d r,

[K_{21}^{cl}] = 2 π \int_{a}^{b} [E_{A} r (\frac{\partial w_{0}}{\partial r}) \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{k}^{u}}{\partial r} + E_{P A} (\frac{\partial w_{0}}{\partial r}) \frac{\partial ϕ_{i}^{w}}{\partial r} ϕ_{k}^{u}] d r,

[K_{22}^{cl}] = 2 π \int_{a}^{b} [E_{A} {r (\frac{\partial u_{0}}{\partial r}) \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{k}^{w}}{\partial r} + \frac{3 r}{2} {(\frac{\partial w_{0}}{\partial r})}^{2} \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{k}^{w}}{\partial r}} + E_{I} {r \frac{\partial^{2} ϕ_{i}^{w}}{\partial r^{2}} \frac{\partial^{2} ϕ_{k}^{w}}{\partial r^{2}} + \frac{1}{r} \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{k}^{w}}{\partial r}}

+ E_{P A} (u_{0}) \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{k}^{w}}{\partial r} + E_{P I} {\frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial^{2} ϕ_{k}^{w}}{\partial r^{2}} + \frac{\partial^{2} ϕ_{i}^{w}}{\partial r^{2}} \frac{\partial ϕ_{k}^{w}}{\partial r}} - E_{T A} r \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{k}^{w}}{\partial r}] d r .

Appendix C

Mass matrix

The sub-matrices of $[M]$ , each of dimension $n \times n$ , are as follows: $[M] = [\begin{array}{l} [M_{11}] 0 \\ 0 [M_{22}] \end{array}]$ .

The components of these sub-matrices are given below

[M_{11}] = 2 π \int_{a}^{b} ρ_{A} r ϕ_{i}^{u} ϕ_{k}^{u} d r,

[M_{22}] = 2 π \int_{a}^{b} [ρ_{A} r ϕ_{i}^{w} ϕ_{k}^{w} + ρ_{I} r \frac{\partial ϕ_{i}^{w}}{\partial r} \frac{\partial ϕ_{k}^{w}}{\partial r}] d r .

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