A non-classical model for first-ordershear deformation circular cylindrical thin shells incorporating microstructure and surface energy effects

Abstract

A new non-classical model for first-order shear deformation circular cylindrical thin shells is developed by using a modiﬁed couple stress theory and a surface elasticity theory. Through a variational formulation based on Hamilton’s principle, the equations of motion and boundary conditions are simultaneously obtained, and the microstructure and surface energy effects are treated in a unified manner. The newly developed non-classical shell model contains one material length-scale parameter to account for the microstructure effect and three surface elastic constants to capture the surface energy effect. The new model includes shell models considering the microstructure effect only or the surface energy effect alone as special cases and recovers the first-order shear deformation circular cylindrical thin shell model based on classical elasticity as a limiting case. In addition, the current shell model reduces to the non-classical model for Mindlin plates incorporating the microstructure and surface energy effects when the thin shell radius tends to infinity. To illustrate the new model, the static bending and free vibration problems of a simply supported circular cylindrical thin shell are analytically solved. The numerical results reveal that the inclusion of the microstructure and surface energy effects leads to reduced shell deflections and rotation angles and increased natural frequencies. The differences are significant when the shell is very thin, but they diminish as the shell thickness increases. These predicted size effects at the micron scale agree with the general trends observed in experiments.

Keywords

Circular cylindrical thin shell first-order shear deformation shell size effect couple stress theory surface elasticity microstructure free vibration natural frequency

1. Introduction

Thin structural components often exhibit microstructure- and surface energy-dependent size effects (e.g., [1 –3]). The classical elasticity theory cannot be used to interpret such size effects due to a lack of any material length-scale parameter.

Higher-order elasticity theories containing microstructure-dependent material constants have been applied to study shell deformations (e.g., [4 –7]). For example, stability and free vibrations of circular cylindrical thin shells were investigated in [8,9] using a simplified strain gradient elasticity theory (SSGET) (e.g., [10 –13]). By applying a non-local constitutive relation proposed in [14], Hu et al. [15] investigated transverse and torsional wave dispersions in carbon nanotubes. Based on the SSGET and the non-local elasticity model of Eringen [14], Ghavanloo and Fazelzadeh [16] analyzed free vibrations of orthotropic doubly-curved shallow shells. By using a modified couple stress theory (MCST) [17,18], free vibrations of thin shells were studied in [19,20] based on the Kirchhoff and first-order shear deformation (FSD) shell models, respectively.

Surface elasticity theories (e.g., [21 –24]) have also been used to develop non-classical thin shell models incorporating surface energy-dependent size effects (e.g., [1,25,26]). For example, Altenbach et al. [27] derived two-dimensional (2D) equilibrium equations for shells by including transverse shear and surface energy effects. Chen et al. [28] developed a model for wave propagation in a transversely isotropic cylinder by considering the surface elasticity effect. Rouhi et al. [29] provided a surface energy-dependent FSD cylindrical shell model for free vibrations.

However, very few models have been developed for thin shells by incorporating both the microstructure and surface energy effects (e.g., [30,31]). Recently, one such model was proposed in [32] for Love–Kirchhoff thin shells based on the MCST and the surface elasticity theory. For cylindrical shells that have small length to diameter ratios and exhibit microstructure- and surface energy-dependent size effects, new non-classical models, which include cross-sectional rotations to account for the shear deformation effect, material length-scale parameters to capture the microstructure effect in the bulk and surface elastic constants to describe the surface energy effect, are very desirable. This motivated the current work.

The aim of the present study is to develop a non-classical FSD model for circular cylindrical thin shells using the MCST [17,18] and the surface elasticity theory [21,22]. The MCST is one of the simplest higher-order elasticity theories and contains only one material length-scale parameter. It was recently shown in [13] that the MCST can be simplified from Form I or III of Mindlin’s strain gradient elasticity theory [33,34]. In addition, the MCST has been given a direct physical interpretation based on material microstructures [35].

The rest of this paper is organized as follows. In Section 2, the new non-classical FSD circular cylindrical thin shell model is developed. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and boundary conditions (BCs) and provides a unified treatment of the microstructure and surface energy effects. The newly obtained model includes the shell models considering the microstructure effect only or the surface energy effect alone as special cases and recovers its classical elasticity-based counterpart as a limiting case. In addition, it reduces to the Mindlin plate model when the shell mean radius tends to infinity. In Section 3, static bending and free vibration problems of a simply supported closed circular cylindrical thin shell are analytically solved by directly applying the current shell model. The numerical results are also presented to illustrate the new model. The paper concludes in Section 4 with a summary.

2. Formulation

According to the MCST [17,18], the constitutive equations for an isotropic linear elastic material are given by

σ = λ tr (ε) I + 2 μ ε,

(1)

m = 2 l^{2} μ χ,

(2)

where σ is the Cauchy stress tensor, m is the deviatoric part of the couple stress tensor, I is the second-order identity tensor, λ and μ are the Lamé constants in classical elasticity, l is a material length-scale parameter measuring the couple stress effect (e.g., [36,37]), and ε and χ are, respectively, the infinitesimal strain tensor and the symmetric curvature tensor expressed as

ε = \frac{1}{2} [\nabla u + {(\nabla u)}^{T}],

(3)

χ = \frac{1}{2} [\nabla φ + {(\nabla φ)}^{T}],

(4)

in which ∇ denotes the gradient, the superscript T represents the transpose, u is the displacement vector, and $φ$ is the micro-rotation vector defined by

φ = \frac{1}{2} curl u .

(5)

The constraint in Equation (5) is imposed in both the classical couple stress theory (e.g., [38,39]) and the MCST. The use of this constraint makes the couple stress theories differ from the Cosserat theory, in which the displacement vector and the micro-rotation vector are regarded as independent (e.g., [4]).

According to the surface elasticity theory (e.g., [21 –24]), the governing equation for the surface layer of zero thickness can be written as

di v_{S} τ = σ n,

(6)

where n is the outward-pointing unit normal vector to the surface, div_S denotes the surface divergence, τ is the in-plane surface elasticity tensor given by (e.g., [21,22,40 –42])

τ = [τ_{0} + (λ_{0} + τ_{0}) (tr ε^{s})] I_{S} + 2 μ_{0} ε^{s} - τ_{0} (\nabla_{S} u)^{T},

(7)

where μ₀ and λ₀ are the surface elastic constants, τ₀ is the residual surface stress, $I_{S}$ is the projection tensor with $I_{S} = I - n \otimes n$ , $ε^{s}$ is the surface strain tensor given by

ε^{s} = \frac{1}{2} [\nabla_{S} u + {(\nabla_{S} u)}^{T}],

(8)

and $\nabla_{S}$ is the surface gradient operator defined as $\nabla_{S} (\cdot) = (\nabla - n \frac{\partial}{\partial z}) (\cdot)$ , with z being the coordinate along the direction of the surface normal n (e.g., [43]). It is clear from Equation (7) that τ is not symmetric.

The out-of-plane components of the surface stress tensor read [22]

τ_{3 β} = τ_{0} u_{3, β} .

(9)

Note that throughout this paper, the summation convention and standard index notation are used, with the Greek indices running from 1 to 2 and the Latin indices from 1 to 3 unless otherwise indicated.

Consider a circular cylindrical thin shell of middle surface radius R and uniform thickness h. Based on the coordinate system (x, θ, z) shown in Figure 1, the displacement field in a FSD thin shell of uniform thickness h can be written in terms of physical components as (e.g., [44,45])

u_{x} (x, θ, z, t) = u (x, θ, t) + z ψ_{x} (x, θ, t),

(10a)

u_{θ} (x, θ, z, t) = v (x, θ, t) + z ψ_{θ} (x, θ, t),

(10b)

u_{z} (x, θ, z, t) = w (x, θ, t),

(10c)

where u_x, u_θ and u_z are, respectively, the x-, θ- and z-components of the displacement vector u of a point (x, θ, z) in the shell at time t, and u, v and w are, respectively, the x-, θ- and z-components of the displacement vector of the corresponding point (x, θ, 0) on the shell middle surface at time t, and ψ_x and ψ_θ are, respectively, the rotation angles of a transverse normal about the θ-axis and x-axis.

Note that the position of a point in the circular cylindrical shell (see Figure 1) can also be represented by using the three orthogonal curvilinear coordinates defined as

x^{1} = x, x^{2} = θ, x^{3} = r,

(11)

which are related to the Cartesian coordinates x_i through

x_{1} = x^{1}, x_{2} = x^{3} \sin x^{2}, x_{3} = x^{3} \cos x^{2} .

(12)

From Equations (11) and (12), the metric and reciprocal metric tensors $g_{ij}$ and $g^{ij}$ can be determined as

[g_{ij}] = [\begin{matrix} 1 0 0 \\ 0 r^{2} 0 \\ 0 0 1 \end{matrix}], [g^{ij}] = [\begin{matrix} 1 0 0 \\ 0 \frac{1}{r^{2}} 0 \\ 0 0 1 \end{matrix}],

(13)

and the Christoffel symbols of the second kind given by (e.g., [46])

Γ_{jk}^{i} = \frac{1}{2} g^{im} (\frac{\partial g_{mk}}{\partial x^{j}} + \frac{\partial g_{jm}}{\partial x^{k}} - \frac{\partial g_{jk}}{\partial x^{m}}) = Γ_{kj}^{i}

(14)

can be obtained as

Γ_{22}^{3} = - r, Γ_{23}^{2} = Γ_{32}^{2} = \frac{1}{r}, all others = zero .

(15)

In terms of the current orthogonal curvilinear coordinates $x^{i}$ , the gradient of a vector field F can be written as

\nabla F = F_{i, j} g^{i} \otimes g^{j} = (\frac{\partial F_{i}}{\partial x^{j}} - Γ_{ij}^{k} F_{k}) g^{i} \otimes g^{j} = \sum_{i, j, k}^{3} (\frac{\partial F_{i}}{\partial x^{j}} - Γ_{ij}^{k} F_{k}) \frac{1}{h_{i} h_{j}} {\overset{⌢}{e}}_{i} \otimes {\overset{⌢}{e}}_{j} \equiv \sum_{i, j}^{3} {\overset{⌢}{F}}_{i, j} {\overset{⌢}{e}}_{i} \otimes {\overset{⌢}{e}}_{j},

(16)

where F_i and F_i,_j are, respectively, the covariant components of F and $\nabla F$ , gⁱ are the reciprocal base vectors, ${\overset{⌢}{e}}_{i}$ are the unit base vectors, ${\overset{⌢}{F}}_{i, j}$ are the physical components of $\nabla F$ , and h_i are the scale factors given by

h_{1} = \sqrt{g_{11}} = 1, h_{2} = \sqrt{g_{22}} = r, h_{3} = \sqrt{g_{33}} = 1,

(17)

where use has been made of Equation (13).

The covariant components of the displacement vector u are

u_{1} = u \cdot g_{1} = u_{x}, u_{2} = u \cdot g_{2} = r u_{θ}, u_{3} = u \cdot g_{3} = u_{z},

(18)

where $g_{i}$ are the natural base vectors, and u_x, u_θ and u_z are the physical components of u introduced in Equations (10a–c).

From Equations (10a–c), (11) and (15)–(18), it follows that

\nabla u = {[\begin{matrix} \frac{\partial u}{\partial x} + z \frac{\partial ψ_{x}}{\partial x} & \frac{1}{r} (\frac{\partial u}{\partial θ} + z \frac{\partial ψ_{x}}{\partial θ}) & ψ_{x} \\ \frac{\partial v}{\partial x} + z \frac{\partial ψ_{θ}}{\partial x} & \frac{1}{r} (\frac{\partial v}{\partial θ} + z \frac{\partial ψ_{θ}}{\partial θ} + w) & ψ_{θ} \\ \frac{\partial w}{\partial x} & \frac{1}{r} (\frac{\partial w}{\partial θ} - v - z ψ_{θ}) & 0 \end{matrix}]}_{{e_{x}, e_{θ}, e_{r}}} .

(19)

Using Equation (19) in Equation (3) then yields

\begin{matrix} ε_{xx} = \frac{\partial u}{\partial x} + z \frac{\partial ψ_{x}}{\partial x}, ε_{θ θ} = \frac{1}{r} (\frac{\partial v}{\partial θ} + w + z \frac{\partial ψ_{θ}}{\partial θ}), ε_{zz} = 0, \\ ε_{x θ} = \frac{1}{2} (\frac{1}{r} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x} + \frac{z}{r} \frac{\partial ψ_{x}}{\partial θ} + z \frac{\partial ψ_{θ}}{\partial x}) = ε_{θ x}, \\ ε_{xz} = \frac{1}{2} (\frac{\partial w}{\partial x} + ψ_{x}) = ε_{zx}, ε_{θ z} = \frac{1}{2} (- \frac{v}{r} + \frac{1}{r} \frac{\partial w}{\partial θ} + ψ_{θ}) = ε_{z θ} \end{matrix}

(20)

as the strain components in the FSD circular cylindrical thin shell.

From Equation (5), the rotation vector can be written as

φ = - \frac{1}{2} \sqrt{g} ε_{ijk} u^{j, k} g^{i} \equiv φ_{i} g^{i},

(21)

where g = det [g_ij] = r² (see Equation (13)), $ε_{ijk}$ is the Levi-Civita symbol, and $u^{i, j}$ are the contravariant components of $\nabla u$ , which are related to the covariant components $u_{i, j}$ through

u^{i, j} = g^{ik} g^{jl} u_{k, l} .

(22)

From Equations (10a–c), (13), (15)–(18), (21) and (22), it follows that

\nabla φ = \frac{1}{2} {[\begin{matrix} - \frac{1}{r} \frac{\partial v}{\partial x} + \frac{1}{r} \frac{\partial^{2} w}{\partial x \partial θ} - \frac{\partial ψ_{θ}}{\partial x} & - \frac{1}{r^{2}} (\frac{\partial v}{\partial θ} - \frac{\partial^{2} w}{\partial θ^{2}} + r \frac{\partial ψ_{θ}}{\partial θ}) & \frac{1}{r^{2}} (v - \frac{\partial w}{\partial θ}) - \frac{ψ_{θ}}{r} \\ - (\frac{\partial^{2} w}{\partial x^{2}} - \frac{\partial ψ_{x}}{\partial x}) & \frac{1}{r} (- \frac{1}{r} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x} - \frac{\partial^{2} w}{\partial x \partial θ} + \frac{\partial ψ_{x}}{\partial θ} + z \frac{\partial ψ_{θ}}{\partial x}) & 0 \\ - \frac{1}{r} \frac{\partial^{2} u}{\partial θ \partial x} + \frac{\partial^{2} v}{\partial x^{2}} - \frac{z}{r} \frac{\partial^{2} ψ_{x}}{\partial θ \partial x} + z \frac{\partial^{2} ψ_{θ}}{\partial x^{2}} & \frac{1}{r} (- \frac{1}{r} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x} - ψ_{x} - \frac{z}{r} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + z \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) & \frac{1}{r^{2}} \frac{\partial u}{\partial θ} - \frac{1}{r} \frac{\partial ψ_{x}}{\partial θ} + \frac{\partial ψ_{θ}}{\partial x} \end{matrix}]}_{{e_{x}, e_{θ}, e_{r}}} .

(23)

Substituting Equation (23) into Equation (4) gives

\begin{matrix} χ_{xx} = \frac{1}{2} (- \frac{1}{r} \frac{\partial v}{\partial x} + \frac{1}{r} \frac{\partial^{2} w}{\partial x \partial θ} - \frac{\partial ψ_{θ}}{\partial x}), χ_{θ θ} = \frac{1}{2 r} (- \frac{1}{r} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x} - \frac{\partial^{2} w}{\partial x \partial θ} + \frac{\partial ψ_{x}}{\partial θ} + z \frac{\partial ψ_{θ}}{\partial x}), \\ χ_{zz} = \frac{1}{2} (\frac{1}{r^{2}} \frac{\partial u}{\partial θ} - \frac{1}{r} \frac{\partial ψ_{x}}{\partial θ} + \frac{\partial ψ_{θ}}{\partial x}), χ_{x θ} = \frac{1}{4} [\frac{1}{r^{2}} (- \frac{\partial v}{\partial θ} + \frac{\partial^{2} w}{\partial θ^{2}}) - \frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ψ_{x}}{\partial x} - \frac{1}{r} \frac{\partial ψ_{θ}}{\partial θ}] = χ_{x θ}, \\ χ_{xz} = \frac{1}{4} (- \frac{1}{r} \frac{\partial^{2} u}{\partial θ \partial x} + \frac{v}{r^{2}} + \frac{\partial^{2} v}{\partial x^{2}} - \frac{1}{r^{2}} \frac{\partial w}{\partial θ} - \frac{z}{r} \frac{\partial^{2} ψ_{x}}{\partial θ \partial x} - \frac{ψ_{θ}}{r} + z \frac{\partial^{2} ψ_{θ}}{\partial x^{2}}) = χ_{zx}, \\ χ_{θ z} = \frac{1}{4 r} (- \frac{1}{r} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x} - ψ_{x} - \frac{z}{r} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + z \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) = χ_{z θ} . \end{matrix}

(24)

The total strain energy in the isotropic elastic thin shell satisfying the MCST in the bulk and the surface elasticity theory in the surface layers has the form

U_{T} = U_{C} + U_{M} + U_{S} = \frac{1}{2} \int_{Ω} σ_{ij} ε_{ij} d V + \frac{1}{2} \int_{Ω} m_{ij} χ_{ij} d V + (\frac{1}{2} \int_{S^{+}} τ_{α β} ε_{α β} d A + \frac{1}{2} \int_{S^{-}} τ_{α β} ε_{α β} d A),

(25)

where Ω is the region occupied by the shell, S⁺ and S⁻ represent, respectively, the outer and inner surface layers of the shell (see Figure 1), $τ_{α β}$ and $ε_{α β}$ are, respectively, the components of the surface stress and surface strain tensors defined in Equations (7) and (8), dV is the volume element, and dA is the area element. In addition, U_C (the first volume integral) is the classical part of the strain energy, U_M (the second volume integral) is the microstructure-dependent part of the strain energy, and U_S (the two surface integrals) is the strain energy in the surface layers.

From Equation (25), it follows that the first variation of the total strain energy in the deformed circular cylindrical thin shell over the time interval [0, T] is given by

\begin{matrix} δ \int_{0}^{T} U_{T} d t = δ \int_{0}^{T} (U_{C} + U_{M} + U_{S}) d t = \int_{0}^{T} \int_{Ω} σ_{ij} δ ε_{ij} d V d t + \int_{0}^{T} \int_{Ω} m_{ij} δ χ_{ij} d V d t \\ + \int_{0}^{T} \int_{S^{+}} (τ_{α β}^{+} - \frac{1}{2} τ_{0} δ_{α β}) δ ε_{α β}^{+} d A d t + \int_{0}^{T} \int_{S^{-}} (τ_{α β}^{-} - \frac{1}{2} τ_{0} δ_{α β}) δ ε_{α β}^{-} d A d t, \end{matrix}

(26)

where $τ_{α β}^{+}$ , $ε_{α β}^{+}$ and $τ_{α β}^{-}$ , $ε_{α β}^{-}$ represent, respectively, the components of surface stress and strain on the outer (S⁺) and inner (S⁻) surfaces.

Note that the volume integral of a sufficiently smooth function D(x, θ, z, t) over the region Ω occupied by a uniform-thickness shell can be written as

\int_{Ω} D (x, θ, z, t) d V = \int_{S} \int_{- h / 2}^{h / 2} D (x, θ, z, t) d z d A,

(27)

where h is the shell thickness, and S is the middle surface of the shell.

From Equations (20), (24) and (27), the three parts of the first variation of the strain energy can be determined as, with r = R on S,

\begin{matrix} δ \int_{0}^{T} U_{C} d t = \int_{0}^{T} \int_{Ω} σ_{ij} δ ε_{ij} d V d t = \int_{0}^{T} \int_{S} [- (N_{xx, x} + \frac{N_{x θ, θ}}{R}) δ u - (N_{x θ, x} + \frac{N_{θ θ, θ}}{R} + \frac{N_{θ z}}{R}) δ v \\ - (N_{xz, x} - \frac{N_{θ θ}}{R} + \frac{N_{θ z, θ}}{R}) δ w - (- N_{xz} + M_{xx, x} + \frac{M_{x θ, θ}}{R}) δ ψ_{x} - (- N_{θ z} + M_{x θ, x} + \frac{M_{θ θ, θ}}{R}) δ ψ_{θ}] d A d t \\ + \int_{0}^{T} \oint_{\partial S} [(N_{xx} n_{x} + \frac{N_{x θ}}{R} n_{θ}) δ u + (N_{x θ} n_{x} + \frac{N_{θ θ}}{R} n_{θ}) δ v + (N_{xz} n_{x} + \frac{N_{θ z}}{R} n_{θ}) δ w \\ + (M_{xx} n_{x} + \frac{M_{x θ}}{R} n_{θ}) δ ψ_{x} + (M_{x θ} n_{x} + \frac{M_{θ θ}}{R} n_{θ}) δ ψ_{θ}] d s d t, \end{matrix}

(28)

where

\begin{matrix} N_{xx} \equiv \int_{- h / 2}^{h / 2} σ_{xx} d z, N_{x θ} \equiv \int_{- h / 2}^{h / 2} σ_{x θ} d z, N_{xz} \equiv \int_{- h / 2}^{h / 2} σ_{xz} d z, N_{θ θ} \equiv \int_{- h / 2}^{h / 2} σ_{θ θ} d z, \\ N_{θ z} \equiv \int_{- h / 2}^{h / 2} σ_{θ z} d z, M_{xx} \equiv \int_{- h / 2}^{h / 2} σ_{xx} z d z, M_{x θ} \equiv \int_{- h / 2}^{h / 2} σ_{x θ} z d z, M_{θ θ} \equiv \int_{- h / 2}^{h / 2} σ_{θ θ} z d z \end{matrix}

(29)

are the Cauchy stress resultants through the shell thickness,

\begin{matrix} δ \int_{0}^{T} U_{M} d t = \int_{0}^{T} \int_{Ω} m_{ij} δ χ_{ij} d V d t \\ = \frac{1}{2} \int_{0}^{T} \int_{S} [(- \frac{Y_{xz, x θ}}{R} + \frac{Y_{θ θ, θ} - Y_{θ z, θ θ} - Y_{zz, θ}}{R^{2}}) δ u + (\frac{Y_{xx, x} - Y_{θ θ, x} + Y_{θ z, x θ}}{R} + \frac{Y_{x θ, θ} + Y_{xz}}{R^{2}} + Y_{xz, xx}) δ v \\ + (\frac{Y_{xx, x θ} - Y_{θ θ, x θ} - Y_{θ z, x}}{R} - Y_{x θ, xx} + \frac{Y_{x θ, θ θ} + Y_{xz, θ}}{R^{2}}) δ w + (- Y_{x θ, x} + \frac{- Y_{θ θ, θ} - Y_{θ z} + Y_{zz, θ} - H_{xz, x θ}}{R} - \frac{H_{θ z, θ θ}}{R^{2}}) δ ψ_{x} \\ + (Y_{xx, x} + \frac{Y_{x θ, θ} - Y_{xz}}{R} - Y_{zz, x} + H_{xz, xx} - \frac{H_{θ θ, x} - H_{θ z, x θ}}{R}) δ ψ_{θ}] d A d t + \frac{1}{2} \int_{0}^{T} \oint_{\partial S} [(\frac{Y_{xz, x}}{2 R} n_{θ} + \frac{Y_{xz, θ}}{2 R} n_{x} \\ + \frac{- Y_{θ θ} + Y_{θ z, θ} + Y_{zz}}{R^{2}} n_{θ}) δ u + (- \frac{Y_{xx} - Y_{θ θ}}{R} n_{x} - \frac{Y_{x θ}}{R^{2}} n_{θ} - Y_{xz, x} n_{x} - \frac{Y_{θ z, x}}{2 R} n_{θ} - \frac{Y_{θ z, θ}}{2 R} n_{x}) δ v \\ + (- \frac{Y_{xx, x} - Y_{θ θ, x}}{2 R} n_{θ} - \frac{Y_{xx, θ} - Y_{θ θ, θ}}{2 R} n_{x} + Y_{x θ, x} n_{x} - \frac{Y_{x θ, θ} + Y_{xz}}{R^{2}} n_{θ} + \frac{Y_{θ z}}{R} n_{x}) δ w + (Y_{x θ} n_{x} \\ + \frac{Y_{θ θ} - Y_{zz}}{R} n_{θ} + \frac{H_{xz, x}}{2 R} n_{θ} + \frac{H_{xz, θ}}{2 R} n_{x} + \frac{H_{θ z, θ}}{R^{2}} n_{θ}) δ ψ_{x} + (- Y_{xx} n_{x} - \frac{Y_{x θ}}{R} n_{θ} + Y_{zz} n_{x} - H_{xz, x} n_{x} \\ + \frac{H_{θ θ}}{R} n_{x} - \frac{H_{θ z, x}}{2 R} n_{θ} - \frac{H_{θ z, θ}}{2 R} n_{x}) δ ψ_{θ} - \frac{Y_{xz}}{2 R} n_{θ} δ u_{, x} - (\frac{Y_{xz}}{2 R} n_{x} + \frac{Y_{θ z}}{R^{2}} n_{θ}) δ u_{, θ} + (Y_{xz} n_{x} + \frac{Y_{θ z}}{2 R} n_{θ}) δ v_{, x} \\ + \frac{Y_{θ z}}{2 R} n_{x} δ v_{, θ} + (\frac{Y_{xx} - Y_{θ θ}}{2 R} n_{θ} - Y_{x θ} n_{x}) δ w_{, x} + (\frac{Y_{xx} - Y_{θ θ}}{2 R} n_{x} + \frac{Y_{x θ}}{R^{2}} n_{θ}) δ w_{, θ} - \frac{H_{xz}}{2 R} n_{θ} δ ψ_{x, x} \\ - (\frac{H_{xz}}{2 R} n_{x} + \frac{H_{θ z}}{R^{2}} n_{θ}) δ ψ_{x, θ} + (H_{xz} n_{x} + \frac{H_{θ z}}{2 R} n_{θ}) δ ψ_{θ, x} + \frac{H_{θ z}}{2 R} n_{x} δ ψ_{θ, θ}] d s d t, \end{matrix}

(30)

where

\begin{matrix} Y_{xx} \equiv \int_{- h / 2}^{h / 2} m_{xx} d z, Y_{x θ} \equiv \int_{- h / 2}^{h / 2} m_{x θ} d z, Y_{xz} \equiv \int_{- h / 2}^{h / 2} m_{xz} d z, Y_{θ θ} \equiv \int_{- h / 2}^{h / 2} m_{θ θ} d z, Y_{θ z} \equiv \int_{- h / 2}^{h / 2} m_{θ z} d z, \\ Y_{zz} \equiv \int_{- h / 2}^{h / 2} m_{zz} d z, H_{xz} \equiv \int_{- h / 2}^{h / 2} m_{xz} z d z, H_{θ θ} \equiv \int_{- h / 2}^{h / 2} m_{θ θ} z d z, H_{θ z} \equiv \int_{- h / 2}^{h / 2} m_{θ z} z d z \end{matrix}

(31)

are the couple stress resultants through the shell thickness, and

\begin{matrix} δ \int_{0}^{T} U_{S} d t = \int_{0}^{T} \int_{S^{+}} (τ_{α β}^{+} - \frac{1}{2} τ_{0} δ_{α β}) δ ε_{α β}^{+} d A d t + \int_{0}^{T} \int_{S^{-}} (τ_{α β}^{-} - \frac{1}{2} τ_{0} δ_{α β}) δ ε_{α β}^{-} d A d t \\ = - \int_{0}^{T} \int_{S} [(τ_{xx, x}^{+} + τ_{xx, x}^{-} + \frac{τ_{x θ, θ}^{+} + τ_{θ x, θ}^{+}}{2 r^{+}} + \frac{τ_{x θ, θ}^{-} + τ_{θ x, θ}^{-}}{2 r^{-}}) δ u + (\frac{1}{2} τ_{x θ, x}^{+} + \frac{1}{2} τ_{x θ, x}^{-} \\ + \frac{1}{2} τ_{θ x, x}^{+} + \frac{1}{2} τ_{θ x, x}^{-} + \frac{τ_{θ θ, θ}^{+}}{r^{+}} + \frac{τ_{θ θ, θ}^{-}}{r^{-}}) δ v - (\frac{τ_{θ θ}^{+} - \frac{τ_{0}}{2}}{r^{+}} + \frac{τ_{θ θ}^{-} - \frac{τ_{0}}{2}}{r^{-}}) δ w + \frac{h}{2} (τ_{xx, x}^{+} - τ_{xx, x}^{-} \\ + \frac{τ_{x θ, θ}^{+} + τ_{θ x, θ}^{+}}{2 r^{+}} - \frac{τ_{x θ, θ}^{-} + τ_{θ x, θ}^{-}}{2 r^{-}}) δ ψ_{x} + \frac{h}{2} (\frac{1}{2} τ_{x θ, x}^{+} - \frac{1}{2} τ_{x θ, x}^{-} + \frac{1}{2} τ_{θ x, x}^{+} - \frac{1}{2} τ_{θ x, x}^{-} \\ + \frac{τ_{θ θ, θ}^{+}}{r^{+}} - \frac{τ_{θ θ, θ}^{-}}{r^{-}}) δ ψ_{θ}] d A d t + \int_{0}^{T} \oint_{\partial S} [(- τ_{0} n_{x} + τ_{xx}^{+} n_{x} + τ_{xx}^{-} n_{x} + \frac{1}{2} \frac{τ_{x θ}^{+}}{r^{+}} n_{θ} + \frac{1}{2} \frac{τ_{x θ}^{-}}{r^{-}} n_{θ} + \frac{1}{2} \frac{τ_{θ x}^{+}}{r^{+}} n_{θ} \\ + \frac{1}{2} \frac{τ_{θ x}^{-}}{r^{-}} n_{θ}) δ u + (- \frac{τ_{0}}{2 r^{+}} n_{θ} - \frac{τ_{0}}{2 r^{-}} n_{θ} + \frac{1}{2} τ_{x θ}^{+} n_{x} + \frac{1}{2} τ_{x θ}^{-} n_{x} + \frac{1}{2} τ_{θ x}^{+} n_{x} + \frac{1}{2} τ_{θ x}^{-} n_{x} + \frac{τ_{θ θ}^{+}}{r^{+}} n_{θ} \\ + \frac{τ_{θ θ}^{-}}{r^{-}} n_{θ}) δ v + \frac{h}{2} (τ_{xx}^{+} n_{x} - τ_{xx}^{-} n_{x} + \frac{1}{2} \frac{τ_{x θ}^{+}}{r^{+}} n_{θ} - \frac{1}{2} \frac{τ_{x θ}^{-}}{r^{-}} n_{θ} + \frac{1}{2} \frac{τ_{θ x}^{+}}{r^{+}} n_{θ} - \frac{1}{2} \frac{τ_{θ x}^{-}}{r^{-}} n_{θ}) δ ψ_{x} \\ + \frac{h}{2} (\frac{1}{2} τ_{x θ}^{+} n_{x} - \frac{1}{2} τ_{x θ}^{-} n_{x} + \frac{1}{2} τ_{θ x}^{+} n_{x} - \frac{1}{2} τ_{θ x}^{-} n_{x} + \frac{τ_{θ θ}^{+} - \frac{τ_{0}}{2}}{r^{+}} n_{θ} - \frac{τ_{θ θ}^{-} - \frac{τ_{0}}{2}}{r^{-}} n_{θ}) δ ψ_{θ}] d s d t, \end{matrix}

(32)

where $r^{+} = R + h / 2$ and $r^{-} = R - h / 2$ . Note that in reaching Equation (32), use has been made of the approximate relations S⁺≈S≈S⁻ and ∂S⁺≈∂S≈∂S⁻ for the uniform-thickness cylindrical thin shell.

The kinetic energy of the shell has the form (e.g., [45,47,48])

K = \frac{1}{2} \int_{Ω} ρ [{({\overset{\cdot}{u}}_{x})}^{2} + {({\overset{\cdot}{u}}_{θ})}^{2} + {({\overset{\cdot}{u}}_{z})}^{2}] d V,

(33)

where ρ is the mass density of the shell material. Note that here and in the sequel, the overhead “·” and “··” denote, respectively, the first and second time derivatives (e.g., ${\overset{\cdot}{u}}_{x} = \partial u_{x} / \partial t,$ ${\overset{\cdot\cdot}{u}}_{x} = \partial^{2} u_{x} / \partial t^{2}$ ). Note that the kinetic energy expression in Equation (33) is in a conventional form without contributions from the surface layers, which are regarded to have zero thickness in the surface elasticity theory employed in the current study. This has also been done in other studies on thin structures and membranes (e.g., [22,49]).

From Equations (10a–c), (27) and (33), the first variation of the kinetic energy, over the time interval [0, T], can be obtained as

δ \int_{0}^{T} K d t = - \int_{0}^{T} \int_{S} (m_{0} \overset{\cdot\cdot}{u} δ u + m_{0} \overset{\cdot\cdot}{v} δ v + m_{0} \overset{\cdot\cdot}{w} δ w + m_{2} {\overset{\cdot\cdot}{ψ}}_{x} δ ψ_{x} + m_{2} {\overset{\cdot\cdot}{ψ}}_{θ} δ ψ_{θ}) d A d t,

(34)

where

m_{0} \equiv \int_{- h / 2}^{h / 2} ρ d z = ρ h, m_{2} \equiv \int_{- h / 2}^{h / 2} ρ z^{2} d z = \frac{ρ h^{3}}{12} .

(35)

In reaching Equation (34), it has been assumed that the initial (t = 0) and final (t = T) configurations of the shell are prescribed so that the virtual displacements vanish at t = 0 and t = T. In addition, ρ is taken to be constant along the shell thickness and over the time interval [0, T] such that ${\overset{\cdot}{m}}_{0} = 0$ and ${\overset{\cdot}{m}}_{2} = 0$ .

Based on the general expression of the work done by external forces in the MCST [18] and in the surface elasticity theory [21,22], the virtual work done by the forces applied on the current shell over the time interval [0, T] can be written as (e.g., [50,51])

\begin{matrix} δ \int_{0}^{T} Wdt = \int_{0}^{T} \int_{S} [f_{x} δ u + f_{θ} δ v + f_{z} δ w + \frac{1}{2} c_{x} δ (- \frac{v}{R} + \frac{1}{R} w_{, θ} - ψ_{θ}) - \frac{1}{2} c_{θ} δ (w_{, x} - ψ_{x}) \\ + \frac{1}{2} c_{z} δ (- \frac{1}{R} u_{, θ} + v_{, x})] d A d t + \int_{0}^{T} \oint_{\partial S} [\bar{t_{x}} δ u + \bar{t_{θ}} δ v + \bar{t_{z}} δ w + \bar{M_{x}} δ ψ_{x} + \bar{M_{θ}} δ ψ_{θ} \\ + \frac{1}{2} \bar{s_{x}} δ (- \frac{v}{R} + \frac{1}{R} w_{, θ} - ψ_{θ}) - \frac{1}{2} \bar{s_{θ}} δ (w_{, x} - ψ_{x}) + \frac{1}{2} \bar{s_{z}} δ (- \frac{1}{R} u_{, θ} + v_{, x})] d s d t \\ + \int_{0}^{T} \int_{S^{+}} (τ_{3 x, x}^{+} + \frac{1}{r^{+}} τ_{3 θ, θ}^{+}) δ w d A d t + \int_{0}^{T} \int_{S^{-}} (τ_{3 x, x}^{-} + \frac{1}{r^{-}} τ_{3 θ, θ}^{-}) δ w d A d t, \end{matrix}

(36)

where f_i and c_i (i = x, θ, z) are, respectively, the components of the body force resultant (force per unit area) and body couple resultant (moment per unit area) through the shell thickness acting in the area S (i.e., the middle surface of shell), $\bar{t_{i}}$ and $\bar{s_{i}}$ (i = x, θ, z) are, respectively, the Cauchy traction resultant (force per unit length) and the surface couple resultant (moment per unit length) through the shell thickness acting along ∂S (i.e., the boundary of S), and $\bar{M_{x}}$ and $\bar{M_{θ}}$ are, respectively, the applied moments per unit length about the θ-axis and x-axis acting along ∂S.

According to Hamilton’s principle (e.g., [45,52])

δ \int_{0}^{T} [K - (U_{T} - W)] d t = 0 .

(37)

Substituting Equations (28), (30), (32), (34) and (36) into Equation (37) and applying the fundamental lemma of the calculus of variations (e.g., [53,54]) will yield, with the arbitrariness of δu, δv, δw, δψ_x and δψ_θ and the relations S⁺≈S≈S⁻ and ∂S⁺≈∂S≈∂S⁻

\begin{matrix} N_{xx, x} + \frac{N_{x θ, θ}}{R} + \frac{Y_{xz, x θ}}{2 R} + \frac{- Y_{θ θ, θ} + Y_{θ z, θ θ} + Y_{zz, θ}}{2 R^{2}} + τ_{xx, x}^{+} + τ_{xx, x}^{-} + \frac{τ_{x θ, θ}^{+} + τ_{θ x, θ}^{+}}{2 r^{+}} + \frac{τ_{x θ, θ}^{-} + τ_{θ x, θ}^{-}}{2 r^{-}} \\ + f_{x} + \frac{1}{2 R} c_{z, θ} = m_{0} \overset{\cdot\cdot}{u}, \end{matrix}

(38a)

\begin{matrix} N_{x θ, x} + \frac{N_{θ θ, θ} + N_{θ z}}{R} + \frac{- Y_{xx, x} + Y_{θ θ, x} - Y_{θ z, x θ}}{2 R} - \frac{Y_{x θ, θ} + Y_{xz}}{2 R^{2}} - \frac{1}{2} Y_{xz, xx} + \frac{1}{2} (τ_{x θ, x}^{+} + τ_{x θ, x}^{-} + τ_{θ x, x}^{+} + τ_{θ x, x}^{-}) \\ + \frac{τ_{θ θ, θ}^{+}}{r^{+}} + \frac{τ_{θ θ, θ}^{-}}{r^{-}} + f_{θ} - \frac{1}{2 R} c_{x} - \frac{1}{2} c_{z, x} = m_{0} \overset{\cdot\cdot}{v}, \end{matrix}

(38b)

\begin{matrix} N_{xz, x} - \frac{N_{θ θ} - N_{θ z, θ}}{R} + \frac{- Y_{xx, x θ} + Y_{θ θ, x θ} + Y_{θ z, x}}{2 R} + \frac{1}{2} Y_{x θ, xx} - \frac{Y_{x θ, θ θ} + Y_{xz, θ}}{2 R^{2}} - \frac{τ_{θ θ}^{+} - \frac{τ_{0}}{2}}{r^{+}} \\ - \frac{τ_{θ θ}^{-} - \frac{τ_{0}}{2}}{r^{-}} + τ_{3 x, x}^{+} + \frac{1}{r^{+}} τ_{3 θ, θ}^{+} + τ_{3 x, x}^{-} + \frac{1}{r^{-}} τ_{3 θ, θ}^{-} + f_{z} - \frac{1}{2 R} c_{x, θ} + \frac{1}{2} c_{θ, x} = m_{0} \overset{\cdot\cdot}{w}, \end{matrix}

(38c)

\begin{matrix} - N_{xz} + M_{xx, x} + \frac{M_{x θ, θ}}{R} + \frac{1}{2} Y_{x θ, x} + \frac{Y_{θ θ, θ} + Y_{θ z} - Y_{zz, θ}}{2 R} + \frac{H_{xz, x θ}}{2 R} + \frac{H_{θ z, θ θ}}{2 R^{2}} + \frac{h}{2} (τ_{xx, x}^{+} - τ_{xx, x}^{-}) \\ + \frac{h}{4} (\frac{τ_{x θ, θ}^{+} + τ_{θ x, θ}^{+}}{r^{+}} - \frac{τ_{x θ, θ}^{-} + τ_{θ x, θ}^{-}}{r^{-}}) + \frac{1}{2} c_{θ} = m_{2} {\overset{\cdot\cdot}{ψ}}_{x}, \end{matrix}

(38d)

\begin{matrix} - N_{θ z} + M_{x θ, x} + \frac{M_{θ θ, θ}}{R} - \frac{1}{2} Y_{xx, x} - \frac{Y_{x θ, θ} - Y_{xz}}{2 R} + \frac{1}{2} Y_{zz, x} - \frac{1}{2} H_{xz, xx} + \frac{H_{θ θ, x}}{2 R} - \frac{H_{θ z, x θ}}{2 R} \\ + \frac{h}{2} (\frac{τ_{θ θ, θ}^{+}}{r^{+}} - \frac{τ_{θ θ, θ}^{-}}{r^{-}}) + \frac{h}{4} (τ_{x θ, x}^{+} - τ_{x θ, x}^{-} + τ_{θ x, x}^{+} - τ_{θ x, x}^{-}) - \frac{1}{2} c_{x} = m_{2} {\overset{\cdot\cdot}{ψ}}_{θ} \end{matrix}

(38e)

as the equations of motion of the circular cylindrical thin shell for any (x, θ) ∈S and t∈ (0, T), and

\begin{matrix} 2 N_{xx} n_{x} + 2 \frac{N_{x θ}}{R} n_{θ} + \frac{Y_{xz, θ}}{2 R} n_{x} + (\frac{Y_{xz, x}}{2 R} - \frac{Y_{θ θ} - Y_{θ z, θ} - Y_{zz}}{R^{2}}) n_{θ} + 2 (τ_{xx}^{+} + τ_{xx}^{-} - τ_{0}) n_{x} \\ + (\frac{τ_{x θ}^{+} + τ_{θ x}^{+}}{r^{+}} + \frac{τ_{x θ}^{-} + τ_{θ x}^{-}}{r^{-}}) n_{θ} + \frac{c_{z}}{r} n_{θ} - 2 {\bar{t}}_{x} = 0 or u = \bar{u}, \end{matrix}

(39a)

\begin{matrix} 2 N_{x θ} n_{x} + \frac{2 N_{θ θ}}{R} n_{θ} - \frac{Y_{xx} - Y_{θ θ}}{R} n_{x} - \frac{Y_{x θ}}{R^{2}} n_{θ} - Y_{xz, x} n_{x} - \frac{Y_{θ z, x}}{2 R} n_{θ} - \frac{Y_{θ z, θ}}{2 R} n_{x} \\ + 2 (\frac{τ_{θ θ}^{+} - \frac{τ_{0}}{2}}{r^{+}} + \frac{τ_{θ θ}^{-} - \frac{τ_{0}}{2}}{r^{-}}) n_{θ} + (τ_{x θ}^{+} + τ_{x θ}^{-} + τ_{θ x}^{+} + τ_{θ x}^{-}) n_{x} - c_{z} n_{x} - 2 {\bar{t}}_{θ} + \frac{{\bar{s}}_{x}}{R} = 0 or v = \bar{v}, \end{matrix}

(39b)

\begin{matrix} 2 N_{xz} n_{x} + \frac{2 N_{θ z}}{R} n_{θ} - \frac{Y_{xx, x} - Y_{θ θ, x}}{2 R} n_{θ} - \frac{Y_{xx, θ} - Y_{θ θ, θ}}{2 R} n_{x} + Y_{x θ, x} n_{x} - \frac{Y_{x θ, θ} + Y_{xz}}{R^{2}} n_{θ} + \frac{Y_{θ z}}{R} n_{x} \\ - \frac{c_{x}}{R} n_{θ} + c_{θ} n_{x} - 2 {\bar{t}}_{z} = 0 or w = \bar{w}, \end{matrix}

(39c)

\begin{matrix} 2 M_{xx} n_{x} + \frac{2 M_{x θ}}{R} n_{θ} + Y_{x θ} n_{x} + \frac{Y_{θ θ} - Y_{zz}}{R} n_{θ} + \frac{H_{xz, x}}{2 R} n_{θ} + \frac{H_{xz, θ}}{2 R} n_{x} + \frac{H_{θ z, θ}}{R^{2}} n_{θ} + h (τ_{xx}^{+} - τ_{xx}^{-}) n_{x} \\ + \frac{h}{2} (\frac{τ_{x θ}^{+} + τ_{θ x}^{+}}{r^{+}} - \frac{τ_{x θ}^{-} + τ_{θ x}^{-}}{r^{-}}) n_{θ} - 2 \bar{M_{x}} - \bar{s_{θ}} = 0 or ψ_{x} = \bar{ψ_{x}}, \end{matrix}

(39d)

\begin{matrix} 2 M_{x θ} n_{x} + \frac{2 M_{θ θ}}{R} n_{θ} - (Y_{xx} - Y_{zz}) n_{x} - \frac{Y_{x θ}}{R} n_{θ} - H_{xz, x} n_{x} + \frac{H_{θ θ}}{R} n_{x} - \frac{H_{θ z, x}}{2 R} n_{θ} - \frac{H_{θ z, θ}}{2 R} n_{x} \\ + h (\frac{τ_{θ θ}^{+} - \frac{τ_{0}}{2}}{r^{+}} - \frac{τ_{θ θ}^{-} - \frac{τ_{0}}{2}}{r^{-}}) n_{θ} + \frac{h}{2} (τ_{x θ}^{+} - τ_{x θ}^{-} + τ_{θ x}^{+} - τ_{θ x}^{-}) n_{x} - 2 \bar{M_{θ}} + {\bar{s}}_{x} = 0 or ψ_{θ} = \bar{ψ_{θ}}, \end{matrix}

(39e)

\frac{Y_{xz}}{2 R} n_{θ} = 0 or u_{, x} = \bar{u_{, x}},

(39f)

\frac{Y_{xz}}{2} n_{x} + \frac{Y_{θ z}}{R} n_{θ} - \bar{s_{z}} = 0 or u_{, θ} = \bar{u_{, θ}},

(39g)

Y_{xz} n_{x} + \frac{Y_{θ z}}{2 R} n_{θ} - \bar{s_{z}} = 0 or v_{, x} = \bar{v_{, x}},

(39h)

\frac{Y_{θ z}}{2 R} n_{x} = 0 or v_{, θ} = \bar{v_{, θ}},

(39i)

\frac{Y_{xx} - Y_{θ θ}}{2 R} n_{θ} - Y_{x θ} n_{x} + \bar{s_{θ}} = 0 or w_{, x} = \bar{w_{, x}},

(39j)

\frac{Y_{xx} - Y_{θ θ}}{2} n_{x} + \frac{Y_{x θ}}{R} n_{θ} - \bar{s_{x}} = 0 or w_{, θ} = \bar{w_{, θ}},

(39k)

\frac{H_{xz}}{2 R} n_{θ} = 0 or ψ_{x, x} = \bar{ψ_{x, x}},

(39l)

\frac{H_{xz}}{2 R} n_{x} + \frac{H_{θ z}}{R^{2}} n_{θ} = 0 or ψ_{x, θ} = \bar{ψ_{x, θ}},

(39m)

H_{xz} n_{x} + \frac{H_{θ z}}{2 R} n_{θ} = 0 or ψ_{θ, x} = \bar{ψ_{θ, x}},

(39n)

\frac{H_{θ z}}{2 R} n_{x} = 0 or ψ_{θ, θ} = \bar{ψ_{θ, θ}}

(39o)

as the BCs for any (x, θ) ∈∂S and t∈ (0, T), where the overhead bar denotes the prescribed value.

Based on Equations (1), (20) and (29), the Cauchy stress resultants can be written in terms of u, v, w, ψ_x and ψ_θ, which are given in Equations (A.1a–h) in Appendix A.

Similarly, from Equations (2), (24) and (31), the couple stress resultants can be obtained in terms of u, v, w, ψ_x and ψ_θ, which are listed in Equations (A.2a–i) in Appendix A.

In addition, the surface stress components can be expressed in terms of u, v, w, ψ_x and ψ_θ by using Equations (7)–(9), (19) and (20), which are provided in Equations (A.3a–f) in Appendix A.

Using Equations (A.1a–h), (A.2a–i) and (A.3a–f) in Equations (38a–e) then yields the equations of motion for the FSD circular cylindrical thin shell in terms of u, v, w, ψ_x and ψ_θ as

\begin{matrix} h [(λ + 2 μ) \frac{\partial^{2} u}{\partial x^{2}} + \frac{λ}{R} (\frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x})] + \frac{μ h}{R} (\frac{1}{R} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ}) + \frac{μ l^{2} h}{4 R} \frac{\partial^{4} v}{\partial x^{3} \partial θ} \\ + \frac{μ l^{2} h}{4 R^{2}} (- \frac{\partial^{4} u}{\partial x^{2} \partial θ^{2}} + \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) + \frac{μ l^{2} h}{4 R^{3}} (\frac{\partial^{4} v}{\partial x \partial θ^{3}} - \frac{\partial^{2} v}{\partial x \partial θ} + 2 \frac{\partial^{3} w}{\partial x \partial θ^{2}} - 5 \frac{\partial^{2} ψ_{x}}{\partial θ^{2}}) + \frac{μ l^{2} h}{4 R^{4}} (4 \frac{\partial^{2} u}{\partial θ^{2}} - \frac{\partial^{4} u}{\partial θ^{4}}) \\ + \frac{λ_{0} + τ_{0}}{r^{+}} (\frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x} + \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) + \frac{λ_{0} + τ_{0}}{r^{-}} (\frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x} - \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) \\ + 2 (λ_{0} + 2 μ_{0}) \frac{\partial^{2} u}{\partial x^{2}} + \frac{2 μ_{0} - τ_{0}}{2 r^{+}} (\frac{1}{r^{+}} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ} + \frac{h}{2 r^{+}} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) \\ + \frac{2 μ_{0} - τ_{0}}{2 r^{-}} (\frac{1}{r^{-}} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ} - \frac{h}{2 r^{-}} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} - \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) + f_{x} + \frac{1}{2 R} \frac{\partial c_{z}}{\partial θ} = ρ h \overset{\cdot\cdot}{u}, \end{matrix}

(40a)

\begin{matrix} \frac{h}{R} [λ \frac{\partial^{2} u}{\partial x \partial θ} + \frac{λ + 2 μ}{R} (\frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial w}{\partial θ})] + μ h (\frac{1}{R} \frac{\partial^{2} u}{\partial x \partial θ} + \frac{\partial^{2} v}{\partial x^{2}}) + \frac{k_{s} μ h}{R} (- \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + ψ_{θ}) \\ - \frac{1}{4} μ l^{2} h \frac{\partial^{4} v}{\partial x^{4}} + \frac{1}{4 R} μ l^{2} h (\frac{\partial^{4} u}{\partial x^{3} \partial θ} + 3 \frac{\partial^{2} ψ_{θ}}{\partial x^{2}}) - \frac{μ l^{2} h}{4 R^{2}} (- 2 \frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{4} v}{\partial x^{2} \partial θ^{2}} + 3 \frac{\partial^{3} w}{\partial x^{2} \partial θ} - 2 \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}) \\ + \frac{μ l^{2} h}{4 R^{3}} (- \frac{\partial^{2} u}{\partial x \partial θ} + \frac{\partial^{4} u}{\partial x \partial θ^{3}} + ψ_{θ} + \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}}) - \frac{μ l^{2} h}{4 R^{4}} (v - \frac{\partial^{2} v}{\partial θ^{2}} - \frac{\partial w}{\partial θ} + \frac{\partial^{3} w}{\partial θ^{3}}) \\ + \frac{1}{r^{+}} [(λ_{0} + τ_{0}) (\frac{\partial^{2} u}{\partial x \partial θ} + \frac{h}{2} \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}) + \frac{λ_{0} + 2 μ_{0}}{r^{+}} (\frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial w}{\partial θ} + \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}})] \\ + \frac{1}{r^{-}} [(λ_{0} + τ_{0}) (\frac{\partial^{2} u}{\partial x \partial θ} - \frac{h}{2} \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}) + \frac{λ_{0} + 2 μ_{0}}{r^{-}} (\frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial w}{\partial θ} - \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}})] \\ + \frac{2 μ_{0} - τ_{0}}{2} [(\frac{1}{r^{+}} + \frac{1}{r^{-}}) \frac{\partial^{2} u}{\partial x \partial θ} + 2 \frac{\partial^{2} v}{\partial x^{2}} + \frac{h}{2} (\frac{1}{r^{+}} - \frac{1}{r^{-}}) \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}] + f_{θ} - \frac{1}{2} \frac{c_{x}}{R} - \frac{1}{2} \frac{\partial c_{z}}{\partial x} = ρ h \overset{\cdot\cdot}{v}, \end{matrix}

(40b)

\begin{matrix} k_{s} μ h (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ψ_{x}}{\partial x}) + \frac{k_{s} μ h}{R} (- \frac{1}{R} \frac{\partial v}{\partial θ} + \frac{1}{R} \frac{\partial^{2} w}{\partial θ^{2}} + \frac{\partial ψ_{θ}}{\partial θ}) - \frac{h}{R} [λ \frac{\partial u}{\partial x} + \frac{λ + 2 μ}{R} (\frac{\partial v}{\partial θ} + w)] \\ + \frac{μ l^{2} h}{4} (- \frac{\partial^{4} w}{\partial x^{4}} + \frac{\partial^{3} ψ_{x}}{\partial x^{3}}) + \frac{μ l^{2} h}{4 R} \frac{\partial^{3} ψ_{θ}}{\partial x^{2} \partial θ} + \frac{μ l^{2} h}{4 R^{2}} (3 \frac{\partial^{3} v}{\partial x^{2} \partial θ} + \frac{\partial^{2} w}{\partial x^{2}} - 2 \frac{\partial^{4} w}{\partial x^{2} \partial θ^{2}} - \frac{\partial ψ_{x}}{\partial x} + \frac{\partial^{3} ψ_{x}}{\partial x \partial θ^{2}}) \\ + \frac{μ l^{2} h}{4 R^{3}} (- 2 \frac{\partial^{3} u}{\partial x \partial θ^{2}} + \frac{\partial ψ_{θ}}{\partial θ} + \frac{\partial^{3} ψ_{θ}}{\partial θ^{3}}) - \frac{μ l^{2} h}{4 R^{4}} (\frac{\partial v}{\partial θ} - \frac{\partial^{3} v}{\partial θ^{3}} - \frac{\partial^{2} w}{\partial θ^{2}} + \frac{\partial^{4} w}{\partial θ^{4}}) \\ - \frac{1}{r^{+}} [\frac{τ_{0}}{2} + (λ_{0} + τ_{0}) (\frac{\partial u}{\partial x} + \frac{h}{2} \frac{\partial ψ_{x}}{\partial x}) + \frac{λ_{0} + 2 μ_{0}}{r^{+}} (\frac{\partial v}{\partial θ} + w + \frac{h}{2} \frac{\partial ψ_{θ}}{\partial θ})] \\ - \frac{1}{r^{-}} [\frac{τ_{0}}{2} + (λ_{0} + τ_{0}) (\frac{\partial u}{\partial x} - \frac{h}{2} \frac{\partial ψ_{x}}{\partial x}) + \frac{λ_{0} + 2 μ_{0}}{r^{-}} (\frac{\partial v}{\partial θ} + w - \frac{h}{2} \frac{\partial ψ_{θ}}{\partial θ})] + 2 τ_{0} \frac{\partial^{2} w}{\partial x^{2}} \\ + \frac{τ_{0}}{{(r^{+})}^{2}} (- \frac{\partial v}{\partial θ} + \frac{\partial^{2} w}{\partial θ^{2}} - \frac{h}{2} \frac{\partial ψ_{θ}}{\partial θ}) + \frac{τ_{0}}{{(r^{-})}^{2}} (- \frac{\partial v}{\partial θ} + \frac{\partial^{2} w}{\partial θ^{2}} + \frac{h}{2} \frac{\partial ψ_{θ}}{\partial θ}) + f_{z} - \frac{1}{2 R} \frac{\partial c_{x}}{\partial θ} + \frac{1}{2} \frac{\partial c_{θ}}{\partial x} = ρ h \overset{\cdot\cdot}{w}, \end{matrix}

(40c)

\begin{matrix} \frac{1}{12} h^{3} [(λ + 2 μ) \frac{\partial^{2} ψ_{x}}{\partial x^{2}} + \frac{λ}{R} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}] + \frac{μ h^{3}}{12 R} (\frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) - k_{s} μ h (\frac{\partial w}{\partial x} + ψ_{x}) \\ + \frac{μ l^{2} h^{3}}{48 R} (\frac{\partial^{4} ψ_{θ}}{\partial x^{3} \partial θ} + \frac{1}{R^{2}} \frac{\partial^{4} ψ_{θ}}{\partial x \partial θ^{3}} - \frac{1}{R} \frac{\partial^{4} ψ_{x}}{\partial x^{2} \partial θ^{2}} - \frac{1}{R^{3}} \frac{\partial^{4} ψ_{x}}{\partial θ^{4}}) + \frac{1}{4} μ l^{2} h (- \frac{\partial^{3} w}{\partial x^{3}} + \frac{\partial^{2} ψ_{x}}{\partial x^{2}}) \\ - \frac{3 μ l^{2} h}{4 R} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ} + \frac{μ l^{2} h}{4 R^{2}} (2 \frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x} - \frac{\partial^{3} w}{\partial x \partial θ^{2}} - ψ_{x} + 4 \frac{\partial^{2} ψ_{x}}{\partial θ^{2}}) - \frac{5 μ l^{2} h}{4 R^{3}} \frac{\partial^{2} u}{\partial θ^{2}} \\ + \frac{h (λ_{0} + τ_{0})}{2} [(\frac{1}{r^{+}} - \frac{1}{r^{-}}) (\frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x}) + \frac{h}{2} (\frac{1}{r^{+}} + \frac{1}{r^{-}}) \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}] + \frac{h^{2} (λ_{0} + 2 μ_{0})}{2} \frac{\partial^{2} ψ_{x}}{\partial x^{2}} \\ + \frac{h (2 μ_{0} - τ_{0})}{4 r^{+}} (\frac{1}{r^{+}} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ} + \frac{h}{2 r^{+}} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) \\ - \frac{h (2 μ_{0} - τ_{0})}{4 r^{-}} (\frac{1}{r^{-}} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ} - \frac{h}{2 r^{-}} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} - \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) + \frac{1}{2} c_{θ} = \frac{ρ h^{3}}{12} {\overset{\cdot\cdot}{ψ}}_{x}, \end{matrix}

(40d)

\begin{matrix} \frac{h^{3}}{12 R} [λ \frac{\partial^{2} ψ_{x}}{\partial x \partial θ} + \frac{λ + 2 μ}{R} \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}}] + \frac{1}{12} μ h^{3} (\frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial x \partial θ} + \frac{\partial^{2} ψ_{θ}}{\partial x^{2}}) - k_{s} μ h (- \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + ψ_{θ}) \\ - \frac{μ l^{2} h^{3}}{48} (- \frac{1}{R^{3}} \frac{\partial^{4} ψ_{x}}{\partial x \partial θ^{3}} - \frac{1}{R} \frac{\partial^{4} ψ_{x}}{\partial x^{3} \partial θ} - \frac{2}{R} \frac{\partial^{2} ψ_{θ}}{\partial x^{2}} + \frac{\partial^{4} ψ_{θ}}{\partial x^{4}} + \frac{1}{R^{2}} \frac{\partial^{4} ψ_{θ}}{\partial x^{2} \partial θ^{2}}) + μ l^{2} h \frac{\partial^{2} ψ_{θ}}{\partial x^{2}} \\ + \frac{μ l^{2} h}{4 R} (3 \frac{\partial^{2} v}{\partial x^{2}} - \frac{\partial^{3} w}{\partial x^{2} \partial θ} - 3 \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}) + \frac{μ l^{2} h}{4 R^{2}} (\frac{\partial^{2} u}{\partial x \partial θ} - ψ_{θ} + \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}}) + \frac{μ l^{2} h}{4 R^{3}} (v + \frac{\partial^{2} v}{\partial θ^{2}} - \frac{\partial w}{\partial θ} \\ - \frac{\partial^{3} w}{\partial θ^{3}}) + \frac{h}{2 r^{+}} [(λ_{0} + τ_{0}) (\frac{\partial^{2} u}{\partial x \partial θ} + \frac{h}{2} \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}) + \frac{λ_{0} + 2 μ_{0}}{r^{+}} (\frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial w}{\partial θ} + \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}})] \\ - \frac{h}{2 r^{-}} [(λ_{0} + τ_{0}) (\frac{\partial^{2} u}{\partial x \partial θ} - \frac{h}{2} \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}) + \frac{λ_{0} + 2 μ_{0}}{r^{-}} (\frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial w}{\partial θ} - \frac{h}{2} \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}})] \\ + \frac{h (2 μ_{0} - τ_{0})}{4} [(\frac{1}{r^{+}} - \frac{1}{r^{-}}) \frac{\partial^{2} u}{\partial x \partial θ} + (\frac{h}{2 r^{+}} + \frac{h}{2 r^{-}}) \frac{\partial^{2} ψ_{x}}{\partial x \partial θ} + h \frac{\partial^{2} ψ_{θ}}{\partial x^{2}}] - \frac{1}{2} c_{x} = \frac{ρ h^{3}}{12} {\overset{\cdot\cdot}{ψ}}_{θ} . \end{matrix}

(40e)

The initial-boundary value problem for determining u, v, w, ψ_x and ψ_θ is defined by the differential equations in Equations (40a–e), the BCs in Equations (39a–o), and given initial conditions at t = 0.

When l = 0 and c_i = 0, Equations (40a–e) will reduce to the governing equations for the FSD circular cylindrical thin shell in the absence of the microstructure (or couple stress) effect.

When λ₀ = µ₀ = τ₀ = 0, Equations (40a–e) will become the governing equations for the FSD circular cylindrical thin shell without the surface energy effect.

When l = 0, c_i = 0 and λ₀ = µ₀ = τ₀ = 0, Equations (40a–e) reduce to the governing equations for the FSD circular cylindrical thin shell based on classical elasticity, which are given in Equations (B.1a–e) in Appendix B.

When $\partial y = R \partial θ$ and R→∞, with $r^{+} \approx r^{-} \approx R$ , Equations (40a–e) recover the governing equations for non-classical Mindlin plates incorporating the microstructure and surface energy effects, which are listed in Equations (B.2a–e) in Appendix B.

Finally, for thin shells with a large mean radius R and/or a small thickness h, the following approximations hold

\frac{1}{r^{+}} \approx \frac{1}{R} \approx \frac{1}{r^{-}} .

(41)

The use of Equation (41) in Equations (40a–e) then gives the equations of motion for such FSD circular cylindrical thin shells as

\begin{matrix} h [(λ + 2 μ) \frac{\partial^{2} u}{\partial x^{2}} + \frac{λ}{R} (\frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x})] + \frac{μ h}{R} (\frac{1}{R} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ}) + \frac{μ l^{2} h}{4 R} \frac{\partial^{4} v}{\partial x^{3} \partial θ} + \frac{μ l^{2} h}{4 R^{2}} (- \frac{\partial^{4} u}{\partial x^{2} \partial θ^{2}} \\ + \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) + \frac{μ l^{2} h}{4 R^{3}} (- \frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial^{4} v}{\partial x \partial θ^{3}} + 2 \frac{\partial^{3} w}{\partial x \partial θ^{2}} - 5 \frac{\partial^{2} ψ_{x}}{\partial θ^{2}}) + \frac{μ l^{2} h}{4 R^{4}} (4 \frac{\partial^{2} u}{\partial θ^{2}} - \frac{\partial^{4} u}{\partial θ^{4}}) + 2 (λ_{0} \\ + 2 μ_{0}) \frac{\partial^{2} u}{\partial x^{2}} + \frac{2 (λ_{0} + τ_{0})}{R} (\frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x}) + \frac{2 μ_{0} - τ_{0}}{R} (\frac{1}{R} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ}) + f_{x} + \frac{1}{2 R} \frac{\partial c_{z}}{\partial θ} = ρ h \overset{\cdot\cdot}{u}, \end{matrix}

(42a)

\begin{matrix} \frac{h}{R} [λ \frac{\partial^{2} u}{\partial x \partial θ} + \frac{λ + 2 μ}{R} (\frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial w}{\partial θ})] + μ h (\frac{1}{R} \frac{\partial^{2} u}{\partial x \partial θ} + \frac{\partial^{2} v}{\partial x^{2}}) + \frac{k_{s} μ h}{R} (- \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + ψ_{θ}) \\ - \frac{1}{4} μ l^{2} h \frac{\partial^{4} v}{\partial x^{4}} + \frac{1}{4 R} μ l^{2} h (\frac{\partial^{4} u}{\partial x^{3} \partial θ} + 3 \frac{\partial^{2} ψ_{θ}}{\partial x^{2}}) - \frac{μ l^{2} h}{4 R^{2}} (- 2 \frac{\partial^{2} v}{\partial x^{2}} + \frac{\partial^{4} v}{\partial x^{2} \partial θ^{2}} + 3 \frac{\partial^{3} w}{\partial x^{2} \partial θ} - 2 \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}) \\ + \frac{μ l^{2} h}{4 R^{3}} (- \frac{\partial^{2} u}{\partial x \partial θ} + \frac{\partial^{4} u}{\partial x \partial θ^{3}} + ψ_{θ} + \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}}) - \frac{μ l^{2} h}{4 R^{4}} (- \frac{\partial w}{\partial θ} + \frac{\partial^{3} w}{\partial θ^{3}} + v - \frac{\partial^{2} v}{\partial θ^{2}}) + \frac{2 μ_{0} + 2 λ_{0} + τ_{0}}{R} \\ \times \frac{\partial^{2} u}{\partial x \partial θ} + \frac{2 (λ_{0} + 2 μ_{0})}{R^{2}} (\frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial w}{\partial θ}) + (2 μ_{0} - τ_{0}) \frac{\partial^{2} v}{\partial x^{2}} + f_{θ} - \frac{1}{2 R} c_{x} - \frac{1}{2} \frac{\partial c_{z}}{\partial x} = ρ h \overset{\cdot\cdot}{v}, \end{matrix}

(42b)

\begin{matrix} k_{s} μ h (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ψ_{x}}{\partial x}) + \frac{k_{s} μ h}{R} (- \frac{1}{R} \frac{\partial v}{\partial θ} + \frac{1}{R} \frac{\partial^{2} w}{\partial θ^{2}} + \frac{\partial ψ_{θ}}{\partial θ}) - \frac{h}{R} [λ \frac{\partial u}{\partial x} + \frac{λ + 2 μ}{R} (\frac{\partial v}{\partial θ} + w)] \\ + \frac{μ l^{2} h}{4} (- \frac{\partial^{4} w}{\partial x^{4}} + \frac{\partial^{3} ψ_{x}}{\partial x^{3}}) + \frac{μ l^{2} h}{4 R} \frac{\partial^{3} ψ_{θ}}{\partial x^{2} \partial θ} + \frac{μ l^{2} h}{4 R^{2}} (\frac{\partial^{2} w}{\partial x^{2}} - 2 \frac{\partial^{4} w}{\partial x^{2} \partial θ^{2}} + 3 \frac{\partial^{3} v}{\partial x^{2} \partial θ} - \frac{\partial ψ_{x}}{\partial x} + \frac{\partial^{3} ψ_{x}}{\partial x \partial θ^{2}}) \\ + \frac{μ l^{2} h}{4 R^{3}} (- 2 \frac{\partial^{3} u}{\partial x \partial θ^{2}} + \frac{\partial ψ_{θ}}{\partial θ} + \frac{\partial^{3} ψ_{θ}}{\partial θ^{3}}) - \frac{μ l^{2} h}{4 R^{4}} (\frac{\partial v}{\partial θ} - \frac{\partial^{3} v}{\partial θ^{3}} - \frac{\partial^{2} w}{\partial θ^{2}} + \frac{\partial^{4} w}{\partial θ^{4}}) - \frac{τ_{0}}{R} - \frac{2}{R} [(λ_{0} + τ_{0}) \frac{\partial u}{\partial x} \\ + \frac{λ_{0} + 2 μ_{0}}{R} (\frac{\partial v}{\partial θ} + w)] + 2 τ_{0} (- \frac{1}{R^{2}} \frac{\partial v}{\partial θ} + \frac{\partial^{2} w}{\partial x^{2}} + \frac{1}{R^{2}} \frac{\partial^{2} w}{\partial θ^{2}}) + f_{z} - \frac{1}{2 R} \frac{\partial c_{x}}{\partial θ} + \frac{1}{2} \frac{\partial c_{θ}}{\partial x} = ρ h \overset{\cdot\cdot}{w}, \end{matrix}

(42c)

\begin{matrix} \frac{1}{12} h^{3} [(λ + 2 μ) \frac{\partial^{2} ψ_{x}}{\partial x^{2}} + \frac{λ}{R} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}] + \frac{μ h^{3}}{12 R} (\frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) - k_{s} μ h (\frac{\partial w}{\partial x} + ψ_{x}) \\ + \frac{μ l^{2} h^{3}}{48 R} (- \frac{1}{R} \frac{\partial^{4} ψ_{x}}{\partial x^{2} \partial θ^{2}} - \frac{1}{R^{3}} \frac{\partial^{4} ψ_{x}}{\partial θ^{4}} + \frac{1}{R^{2}} \frac{\partial^{4} ψ_{θ}}{\partial x \partial θ^{3}} + \frac{\partial^{4} ψ_{θ}}{\partial x^{3} \partial θ}) + \frac{1}{4} μ l^{2} h (- \frac{\partial^{3} w}{\partial x^{3}} + \frac{\partial^{2} ψ_{x}}{\partial x^{2}}) \\ - \frac{3 μ l^{2} h}{4 R} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ} + \frac{μ l^{2} h}{4 R^{2}} (2 \frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x} - \frac{\partial^{3} w}{\partial x \partial θ^{2}} - ψ_{x} + 4 \frac{\partial^{2} ψ_{x}}{\partial θ^{2}}) - \frac{5 μ l^{2} h}{4 R^{3}} \frac{\partial^{2} u}{\partial θ^{2}} \\ + \frac{h^{2} (λ_{0} + 2 μ_{0})}{2} \frac{\partial^{2} ψ_{x}}{\partial x^{2}} + \frac{h^{2} (2 μ_{0} - τ_{0})}{4 R^{2}} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + \frac{h^{2} (2 λ_{0} + 2 μ_{0} + τ_{0})}{4 R} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ} + \frac{1}{2} c_{θ} = \frac{ρ h^{3}}{12} {\overset{\cdot\cdot}{ψ}}_{x}, \end{matrix}

(42d)

\begin{matrix} \frac{h^{3}}{12 R} [λ \frac{\partial^{2} ψ_{x}}{\partial x \partial θ} + \frac{λ + 2 μ}{R} \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}}] + \frac{1}{12} μ h^{3} (\frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial x \partial θ} + \frac{\partial^{2} ψ_{θ}}{\partial x^{2}}) - k_{s} μ h (- \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + ψ_{θ}) \\ - \frac{μ l^{2} h^{3}}{48} (- \frac{1}{R^{3}} \frac{\partial^{4} ψ_{x}}{\partial x \partial θ^{3}} - \frac{1}{R} \frac{\partial^{4} ψ_{x}}{\partial x^{3} \partial θ} - \frac{2}{R} \frac{\partial^{2} ψ_{θ}}{\partial x^{2}} + \frac{\partial^{4} ψ_{θ}}{\partial x^{4}} + \frac{1}{R^{2}} \frac{\partial^{4} ψ_{θ}}{\partial x^{2} \partial θ^{2}}) + μ l^{2} h \frac{\partial^{2} ψ_{θ}}{\partial x^{2}} + \frac{μ l^{2} h}{4 R} \\ \times (3 \frac{\partial^{2} v}{\partial x^{2}} - \frac{\partial^{3} w}{\partial x^{2} \partial θ} - 3 \frac{\partial^{2} ψ_{x}}{\partial x \partial θ}) + \frac{μ l^{2} h}{4 R^{2}} (\frac{\partial^{2} u}{\partial x \partial θ} - ψ_{θ} + \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}}) + \frac{μ l^{2} h}{4 R^{3}} (v + \frac{\partial^{2} v}{\partial θ^{2}} - \frac{\partial w}{\partial θ} - \frac{\partial^{3} w}{\partial θ^{3}}) \\ + \frac{h^{2} (2 μ_{0} + 2 λ_{0} + τ_{0})}{4 R} \frac{\partial^{2} ψ_{x}}{\partial x \partial θ} + \frac{h^{2} (2 μ_{0} - τ_{0})}{4} \frac{\partial^{2} ψ_{θ}}{\partial x^{2}} + \frac{h^{2} (λ_{0} + 2 μ_{0})}{2 R^{2}} \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}} - \frac{1}{2} c_{x} = \frac{ρ h^{3}}{12} {\overset{\cdot\cdot}{ψ}}_{θ} . \end{matrix}

(42e)

The simplified equations of motion in Equations (42a–e) can be applied to thin shells with a large R/h ratio.

3. Examples: static bending and free vibration of a closed circular cylindrical shell

To demonstrate the new FSD circular cylindrical thin shell model developed in Section 2, static bending and free vibration problems of a closed circular cylindrical shell of length a, thickness h and middle surface radius R with simply supported BCs (see Figure 2) are analytically solved in this section by directly applying the new model.

In view of the general form of the BCs in Equations (39a–o), the BCs for the current problem with a large R/h ratio can be identified as, with $c_{x} = c_{θ} = c_{z} = 0,$ ${\bar{t}}_{x} = {\bar{t}}_{θ} = {\bar{t}}_{z} = 0$ , ${\bar{s}}_{x} = {\bar{s}}_{θ} = {\bar{s}}_{z} = 0$ , ${\bar{M}}_{x} = {\bar{M}}_{θ} = 0$ and the relations in Equation (41),

2 N_{xx} + \frac{Y_{xz, θ}}{2 R} + 2 (τ_{xx}^{+} + τ_{xx}^{-} - τ_{0}) = 0 at x = 0, a,

(43a)

v (0, θ) = v (a, θ) = 0,

(43b)

w (0, θ) = w (a, θ) = 0,

(43c)

2 M_{xx} + Y_{x θ} + \frac{H_{xz, θ}}{2 R} + h (τ_{xx}^{+} - τ_{xx}^{-}) = 0 at x = 0, a,

(43d)

ψ_{θ} (0, θ) = ψ_{θ} (a, θ) = 0,

(43e)

u_{, x} (0, θ) = u_{, x} (a, θ) = 0,

(43f)

Y_{xz} = 0 at x = 0, a,

(43g)

v_{, θ} (0, θ) = v_{, θ} (a, θ) = 0,

(43h)

Y_{x θ} = 0 at x = 0, a,

(43i)

w_{, θ} (0, θ) = w_{, θ} (a, θ) = 0,

(43j)

ψ_{x, x} (0, θ) = ψ_{x, x} (a, θ) = 0,

(43k)

H_{xz} = 0 at x = 0, a,

(43l)

ψ_{θ, θ} (0, θ) = ψ_{θ, θ} (a, θ) = 0

(43m)

for the boundaries x = 0, 0 ≤θ≤ 2π and x = a, 0 ≤θ≤ 2π, where n_θ = 0 and n_x = −1 (on x = 0) or n_x = 1 (on x = a).

Using Equations (A.1a, f), (A.2b, c, g) and (A.3a) in Equations (43a, d, g, i, l) yields

\begin{matrix} 2 h [(λ + 2 μ) \frac{\partial u}{\partial x} + \frac{λ}{R} (\frac{\partial v}{\partial θ} + w)] + \frac{μ l^{2} h}{4 R} (- \frac{1}{R} \frac{\partial^{3} u}{\partial x \partial θ^{2}} + \frac{1}{R^{2}} \frac{\partial v}{\partial θ} + \frac{\partial^{3} v}{\partial x^{2} \partial θ} - \frac{1}{R^{2}} \frac{\partial^{2} w}{\partial θ^{2}} - \frac{1}{R} \frac{\partial ψ_{θ}}{\partial θ}) \\ + \frac{4 (λ_{0} + τ_{0})}{R} (\frac{\partial v}{\partial θ} + w) + 4 (λ_{0} + 2 μ_{0}) \frac{\partial u}{\partial x} + 2 τ_{0} = 0, \end{matrix}

(44a)

\begin{matrix} \frac{1}{6} h^{3} [(λ + 2 μ) \frac{\partial ψ_{x}}{\partial x} + \frac{λ}{R} \frac{\partial ψ_{θ}}{\partial θ}] + \frac{1}{2} μ l^{2} h (- \frac{1}{R^{2}} \frac{\partial v}{\partial θ} - \frac{\partial^{2} w}{\partial x^{2}} + \frac{1}{R^{2}} \frac{\partial^{2} w}{\partial θ^{2}} + \frac{\partial ψ_{x}}{\partial x} - \frac{1}{R} \frac{\partial ψ_{θ}}{\partial θ}) \\ + \frac{μ l^{2} h^{3}}{48 R} (- \frac{1}{R} \frac{\partial^{3} ψ_{x}}{\partial x \partial θ^{2}} + \frac{\partial^{3} ψ_{θ}}{\partial x^{2} \partial θ}) + h^{2} (λ_{0} + 2 μ_{0}) \frac{\partial ψ_{x}}{\partial x} + \frac{h^{2} (λ_{0} + τ_{0})}{R} \frac{\partial ψ_{θ}}{\partial θ} = 0, \end{matrix}

(44b)

- \frac{1}{R} \frac{\partial^{2} u}{\partial θ \partial x} + \frac{v}{R^{2}} + \frac{\partial^{2} v}{\partial x^{2}} - \frac{1}{R^{2}} \frac{\partial w}{\partial θ} - \frac{ψ_{θ}}{R} = 0,

(44c)

- \frac{1}{R^{2}} \frac{\partial v}{\partial θ} - \frac{\partial^{2} w}{\partial x^{2}} + \frac{1}{R^{2}} \frac{\partial^{2} w}{\partial θ^{2}} + \frac{\partial ψ_{x}}{\partial x} - \frac{1}{R} \frac{\partial ψ_{θ}}{\partial θ} = 0,

(44d)

- \frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial θ \partial x} + \frac{\partial^{2} ψ_{θ}}{\partial x^{2}} = 0

(44e)

at x = 0 and x = a (with 0 ≤θ≤ 2π). Note that free vibrations of circular cylindrical shells with BCs of this and other types have been extensively studied using shell theories based on classical elasticity (e.g., [55,56]).

3.1. Static bending

For static bending problems, u, v, w, ψ_x and ψ_θ are independent of time t so that all of the time derivatives involved in Equations (42a–e) vanish.

The boundary value problem (BVP) for the static bending of the simply supported closed circular cylindrical shell shown in Figure 2 is defined by the governing equations in Equations (42a–e) and the BCs in Equations (43b, c, e, f, h, j, k, m) and (44a–e), with u = u(x, θ), v = v(x, θ), w = w(x, θ), ψ_x = ψ_x(x, θ) and ψ_θ = ψ_θ(x, θ). For the current case with f_x = f_θ = 0, c_x = c_θ = c_z = 0 and $τ_{0} = 0$ , the displacements u, v and w and the rotation angles ψ_x and ψ_θ can be obtained from solving this BVP analytically.

Consider the following Fourier series solutions for u, v, w, ψ_x and ψ_θ:

u = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U_{mn} \cos (\frac{m π x}{a}) \sin (\frac{n θ}{2}),

(45a)

v = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V_{mn} \sin (\frac{m π x}{a}) \cos (\frac{n θ}{2}),

(45b)

w = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W_{mn} \sin (\frac{m π x}{a}) \sin (\frac{n θ}{2}),

(45c)

ψ_{x} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Ψ_{mn}^{x} \cos (\frac{m π x}{a}) \sin (\frac{n θ}{2}),

(45d)

ψ_{θ} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Ψ_{mn}^{θ} \sin (\frac{m π x}{a}) \cos (\frac{n θ}{2}),

(45e)

where $U_{mn}$ , $V_{mn}$ , $W_{mn}$ , $Ψ_{mn}^{x}$ and $Ψ_{mn}^{θ}$ are the Fourier coefficients to be determined for each pair of m and n. It can be readily shown that u, v, w, ψ_x and ψ_θ in Equations (45a–e) satisfy the BCs in Equations (43b, c, e, f, h, j, k, m) and (44a–e) at x = 0, a for any $U_{mn}$ , $V_{mn}$ , $W_{mn}$ , $Ψ_{mn}^{x}$ and $Ψ_{mn}^{θ}$ .

The load f_z (x, θ) involved in Equation (42c) can also be expanded in a Fourier series as

f_{z} (x, θ) = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Q_{mn} \sin (\frac{m π x}{a}) \sin (\frac{n θ}{2}),

(46)

where Q_mn are the Fourier coefficients given by

Q_{mn} = \frac{4 p_{0}}{mn π^{2}} [1 - \cos (m π)] [1 - \cos (n π)]

(47)

for the current shell loaded by the uniformly distributed internal pressure p₀ (i.e., $f_{z} (x, θ) = p_{0}$ ).

Using Equations (45a–e) and (46) in Equations (42a–e) results in

[C] {[U_{mn}, V_{mn}, W_{mn}, Ψ_{mn}^{x}, Ψ_{mn}^{θ}]}^{T} = {[0, 0, - Q_{mn}, 0, 0]}^{T},

(48)

where $[C]$ is a 5-by-5 matrix whose components are

C_{11} = - \frac{h (λ + 2 μ) π^{2} m^{2}}{a^{2}} - \frac{μ h n^{2}}{4 R^{2}} - \frac{μ l^{2} h}{16 R^{2}} \frac{π^{2} m^{2} n^{2}}{a^{2}} - \frac{μ l^{2} h n^{2}}{4 R^{4}} - \frac{μ l^{2} h n^{4}}{64 R^{4}} - \frac{2 (λ_{0} + 2 μ_{0}) π^{2} m^{2}}{a^{2}} - \frac{μ_{0} n^{2}}{2 R^{2}},

C_{12} = - \frac{[(λ + μ) h + 2 (λ_{0} + μ_{0})]}{2 R} \frac{π mn}{a} + \frac{μ l^{2} h}{8 R} \frac{π^{3} m^{3} n}{a^{3}} + \frac{μ l^{2} h}{32 R^{3}} \frac{π m n^{3}}{a} + \frac{μ l^{2} h}{8 R^{3}} \frac{π mn}{a},

C_{13} = \frac{λ h + 2 λ_{0}}{R} \frac{π m}{a} - \frac{μ l^{2} h}{8 R^{3}} \frac{π m n^{2}}{a}, C_{14} = \frac{5 μ l^{2} h n^{2}}{16 R^{3}}, C_{15} = - \frac{μ l^{2} h}{8 R^{2}} \frac{π mn}{a},

C_{21} = C_{12},

\begin{matrix} C_{22} = - \frac{h (λ + 2 μ) n^{2}}{4 R^{2}} - \frac{μ h π^{2} m^{2}}{a^{2}} - \frac{k_{s} μ h}{R^{2}} - \frac{μ l^{2} h}{4} \frac{π^{4} m^{4}}{a^{4}} - \frac{μ l^{2} h}{16 R^{2}} \frac{π^{2} m^{2} n^{2}}{a^{2}} - \frac{μ l^{2} h}{2 R^{2}} \frac{π^{2} m^{2}}{a^{2}} \\ - \frac{μ l^{2} h n^{2}}{16 R^{4}} - \frac{μ l^{2} h}{4 R^{4}} - \frac{(λ_{0} + 2 μ_{0}) n^{2}}{2 R^{2}} - \frac{2 μ_{0} π^{2} m^{2}}{a^{2}}, \end{matrix}

C_{23} = [(λ + 2 μ) h + k_{s} μ h + 2 (λ_{0} + 2 μ_{0}) + \frac{μ l^{2} h}{4 R^{2}}] \frac{n}{2 R^{2}} + \frac{3 μ l^{2} h}{8 R^{2}} \frac{π^{2} m^{2} n}{a^{2}} + \frac{μ l^{2} h n^{3}}{32 R^{4}},

C_{24} = - \frac{μ l^{2} h}{4 R^{2}} \frac{π mn}{a}, C_{25} = \frac{k_{s} μ h}{R} + \frac{μ l^{2} h}{4 R^{3}} - \frac{3 μ l^{2} h}{4 R} \frac{π^{2} m^{2}}{a^{2}} - \frac{μ l^{2} h n^{2}}{16 R^{3}},

(49)

C_{31} = C_{13}, C_{32} = C_{23},

\begin{matrix} C_{33} = - \frac{h (λ + 2 μ)}{R^{2}} - \frac{2 (λ_{0} + 2 μ_{0})}{R^{2}} - (k_{s} μ h + \frac{μ l^{2} h}{4 R^{2}}) \frac{π^{2} m^{2}}{a^{2}} - (k_{s} + \frac{l^{2}}{4 R^{2}}) \frac{μ h n^{2}}{4 R^{2}} \\ - \frac{μ l^{2} h}{4} \frac{π^{4} m^{4}}{a^{4}} - \frac{μ l^{2} h}{8 R^{2}} \frac{π^{2} m^{2} n^{2}}{a^{2}} - \frac{μ l^{2} h n^{4}}{64 R^{4}}, \end{matrix}

C_{34} = - \frac{k_{s} μ h π m}{a} + \frac{μ l^{2} h}{4} \frac{π^{3} m^{3}}{a^{3}} + \frac{μ l^{2} h}{16 R^{2}} \frac{π m n^{2}}{a} + \frac{μ l^{2} h}{4 R^{2}} \frac{π m}{a},

C_{35} = - \frac{k_{s} μ hn}{2 R} + \frac{μ l^{2} h}{8 R} \frac{π^{2} m^{2} n}{a^{2}} + \frac{μ l^{2} h n^{3}}{32 R^{3}} - \frac{μ l^{2} hn}{8 R^{3}},

C_{41} = C_{14}, C_{42} = C_{24}, C_{43} = C_{34},

\begin{matrix} C_{44} = - [\frac{(λ + 2 μ) h^{3}}{12} + \frac{μ l^{2} h}{4} + \frac{h^{2} (λ_{0} + 2 μ_{0})}{2}] \frac{π^{2} m^{2}}{a^{2}} - (\frac{μ h^{2}}{12} + μ l^{2} + \frac{μ_{0} h}{2}) \frac{h n^{2}}{4 R^{2}} \\ - \frac{μ l^{2} h^{3}}{192 R^{2}} \frac{π^{2} m^{2} n^{2}}{a^{2}} - \frac{μ l^{2} h^{3} n^{4}}{768 R^{4}} - k_{s} μ h - \frac{μ l^{2} h}{4 R^{2}}, \end{matrix}

C_{45} = - [\frac{(λ + μ) h^{2}}{6} - \frac{3 μ l^{2}}{2} + (λ_{0} + μ_{0}) h] \frac{h}{4 R} \frac{π mn}{a} + \frac{μ l^{2} h^{3}}{96 R} \frac{π^{3} m^{3} n}{a^{3}} + \frac{μ l^{2} h^{3}}{384 R^{3}} \frac{π m n^{3}}{a},

C_{51} = C_{15}, C_{52} = C_{25}, C_{53} = C_{35}, C_{54} = C_{45},

\begin{matrix} C_{55} = - (\frac{μ h^{2}}{12} + μ l^{2} + \frac{μ_{0} h}{2} + \frac{μ l^{2} h^{2}}{24 R}) \frac{h m^{2} π^{2}}{a^{2}} - [\frac{(λ + 2 μ) h^{2}}{6} + \frac{μ l^{2}}{2} + (λ_{0} + 2 μ_{0}) h] \frac{h n^{2}}{8 R^{2}} \\ - k_{s} μ h - \frac{μ l^{2} h}{4 R^{2}} - \frac{μ l^{2} h^{3}}{48} \frac{π^{4} m^{4}}{a^{4}} - \frac{μ l^{2} h^{3}}{192 R^{2}} \frac{π^{2} m^{2} n^{2}}{a^{2}} . \end{matrix}

Solving the linear algebraic equation system in Equation (48) will yield $U_{mn}$ , $V_{mn}$ , $W_{mn}$ , $Ψ_{mn}^{x}$ and $Ψ_{mn}^{θ}$ . Then, substituting them into Equations (45a–e) will give the exact solutions for u, v, w, ψ_x and ψ_θ based on the current non-classical model for the simply supported closed circular cylindrical thin shell subjected to uniform load.

Figures 3 and 4 display, respectively, the variations of the shell deflection w and the rotation angle ψ_x predicted by the newly developed model and by its classical elasticity-based counterpart. The numerical values of w and ψ_x are computed using Equations (45c) and (45d), with the coefficients $W_{mn}$ and $Ψ_{mn}^{x}$ determined from Equations (47)–(49). In obtaining the numerical results shown in Figures 3 and 4, the shape of the shell is fixed by letting a = 40R and R = 50h, while the shell thickness h is varying. The uniformly distributed pressure is taken to be p₀ = 1 kPa. The shell material is aluminum with the following properties [42,57,58]: E = 90 GPa, v = 0.23, l = 6.58 μm for the bulk properties, and $μ_{0} = - 5.4251$ N/m, $λ_{0}$ = 3.4939 N/m for the surface layers. Note that Young’s modulus E and Poisson’s ratio ν are related to the Lamé constants λ and μ through (e.g., [59])

λ = \frac{Ev}{(1 + v) (1 - 2 v)}, μ = \frac{E}{2 (1 + v)} .

(50)

The shear correction factor k_s used here is taken to be 0.8, which was shown to be the most accurate value for a FSD thin shell [60].

The numerical values of w and ψ_x are found to be independent of the value of θ, as expected from the axisymmetry of the current problem. The number of terms included in Equations (45c) and (45d) is controlled by adjusting m and n. The numerical results for w and ψ_x obtained with m = 40 and n = 40 are found to be the same as those computed with larger m and n values (up to m = 100, n = 100) to the third decimal place. This indicates that using m = 40, n = 40 in the expansion is sufficient for the convergent numerical solutions of w and ψ_x displayed in Figures 3 and 4. The numerical values of w and ψ_x for the classical elasticity-based model shown in these two figures are also obtained from Equations (45c), (45d) and (47)–(49) but with l = 0, c_i = 0 and λ₀ = µ₀ = τ₀ = 0.

From Figures 3 and 4, it is clearly seen that both the deﬂection w and the rotation angle ψ_x predicted by the current model are always smaller than those predicted by the classical model in all cases considered. Figures 3 and 4 also show that the differences between the two sets of values predicted by the new model and the classical model are significant when the shell thickness h is very small, but they diminish when h becomes large.

Figures 5(a)–5(c) display the deflection curves of the simply supported circular cylindrical shell obtained using the FSD and Love–Kirchhoff thin shell models based on the MCST and the surface elasticity theory. In Figure 5, the deflection values predicted by the current FSD thin shell model are determined from Equations (45c) and (48). The material properties used here are the same as those employed to obtain the values shown in Figures 3 and 4. The deflection values given by the Love–Kirchhoff thin shell model and shown in Figure 5 are obtained using the formulas provided in [32].

It is observed from Figure 5 that the deflection predicted by the current FSD thin shell model is always higher than that by the Love–Kirchhoff thin shell model based on the same couple stress and surface elasticity theories. Also, Figure 5 shows that in all cases considered, the predicted values of the deflection by both the FSD and Love–Kirchhoff shell models are smaller than those predicted by their classical elasticity-based counterparts, with the difference for the FSD models being more significant than that for the Love–Kirchhoff models. In addition, Figure 5 shows that the largest deflection is at the middle span of the shell in all cases considered. Moreover, it is seen that for the cylindrical shell with the same radius and thickness and under the same pressure p₀, the deflection w increases with the increase of the length a, as expected.

3.2. Free vibration

For free vibration problems, the BVP for the simply supported closed circular cylindrical shell shown in Figure 2 is defined by Equations (42a–e) and the BCs in Equations (43b, c, e, f, h, j, k, m) and (44a–e), with all external forces vanished (i.e., $f_{x} = f_{θ} = f_{z} = 0$ , $c_{x} = c_{θ} = c_{z} = 0$ and $τ_{0} = 0$ ).

Consider the solutions of the following Fourier series form:

u = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} U_{mn}^{V} \cos (\frac{m π x}{a}) \sin (\frac{n θ}{2}) e^{i ω_{n} t},

(51a)

v = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} V_{mn}^{V} \sin (\frac{m π x}{a}) \cos (\frac{n θ}{2}) e^{i ω_{n} t},

(51b)

w = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} W_{mn}^{V} \sin (\frac{m π x}{a}) \sin (\frac{n θ}{2}) e^{i ω_{n} t},

(51c)

ψ_{x} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Ψ_{mn}^{x, V} \cos (\frac{m π x}{a}) \sin (\frac{n θ}{2}) e^{i ω_{n} t},

(51d)

ψ_{θ} = \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} Ψ_{mn}^{θ, V} \sin (\frac{m π x}{a}) \cos (\frac{n θ}{2}) e^{i ω_{n} t},

(51e)

where ω_n is the nth natural frequency of vibration of the shell, $U_{mn}^{V}$ , $V_{mn}^{V}$ , $W_{mn}^{V}$ , $Ψ_{mn}^{x, V}$ and $Ψ_{mn}^{θ, V}$ are the Fourier coefficients, and i is the imaginary unit satisfying i²=−1. It can be readily shown that the expansions in Equations (51a–e) satisfy the BCs in Equations (43b, c, e, f, h, j, k, m) and (44a–e) at x = 0 and x = a for any t∈[0, T].

Substituting Equations (51a–e) into Equations (42a–e) yields

[C] {[U_{mn}^{V}, V_{mn}^{V}, W_{mn}^{V}, Ψ_{mn}^{x, V}, Ψ_{mn}^{θ, V}]}^{T} = - [\begin{matrix} ρ h ω_{n}^{2} & 0 & 0 & 0 & 0 \\ 0 & ρ h ω_{n}^{2} & 0 & 0 & 0 \\ 0 & 0 & ρ h ω_{n}^{2} & 0 & 0 \\ 0 & 0 & 0 & \frac{ρ h^{3}}{12} ω_{n}^{2} & 0 \\ 0 & 0 & 0 & 0 & \frac{ρ h^{3}}{12} ω_{n}^{2} \end{matrix}] [\begin{matrix} U_{mn}^{V} \\ V_{mn}^{V} \\ W_{mn}^{V} \\ Ψ_{mn}^{x, V} \\ Ψ_{mn}^{θ, V} \end{matrix}],

(52)

where [C] is the 5-by-5 matrix with the components defined in Equation (49).

For a non-trivial solution of $U_{mn}^{V} \neq 0$ , $V_{mn}^{V} \neq 0$ , $W_{mn}^{V} \neq 0$ , $Ψ_{mn}^{x, V} \neq 0$ and $Ψ_{mn}^{θ, V} \neq 0$ , it is required that the determinant of the coefficient matrix of Equation (52) vanish, which gives

| \begin{matrix} C_{11} + ρ h ω_{n}^{2} & C_{12} & C_{13} & C_{14} & C_{15} \\ C_{12} & C_{22} + ρ h ω_{n}^{2} & C_{23} & C_{24} & C_{25} \\ C_{13} & C_{23} & C_{33} + ρ h ω_{n}^{2} & C_{34} & C_{35} \\ C_{14} & C_{24} & C_{34} & C_{44} + \frac{ρ h^{3}}{12} ω_{n}^{2} & C_{45} \\ C_{15} & C_{25} & C_{35} & C_{45} & C_{55} + \frac{ρ h^{3}}{12} ω_{n}^{2} \end{matrix} | = 0 .

(53)

The characteristic equation in Equation (53) is a fifth-order polynomial equation in $ω_{n}^{2}$ . The smallest (positive) root of Equation (53) gives the nth natural frequency, ω_n, for the free vibration of the current shell.

Figure 6 shows the variation of the first natural frequency ω₁ obtained from Equation (53) (with m = 1, n = 1) with the shell thickness predicted by the current non-classical FSD circular cylindrical thin shell model and its classical counterpart. The numerical values for the current model displayed in Figure 6 are obtained from Equation (53), while those for the classical model are computed from the same equation but with l = 0 and λ₀ = µ₀ = τ₀ = 0. The material properties and geometry of the aluminum shell used here are the same as those employed earlier to obtain the numerical results displayed in Figures 3 –5 (i.e., E = 90 GPa, v = 0.23, l = 6.58 μm, $μ_{0} = - 5.4251$ N/m, $λ_{0}$ = 3.4939 N/m, and τ₀ = 0). In addition, the density for the aluminum shell is taken to be $ρ = 2.7 \times 10^{3} kg / m^{3}$ .

From Figure 6, it is clearly seen that the first natural frequency predicted by the current model is always higher than that predicted by the classical elasticity-based model. The difference between the predictions by the two models is significant only when the shell thickness h is very small (with h < 2l = 13.16 μm here). However, the difference is diminishing as shell thickness h increases (with h > 3l = 19.74 μm here). The size effects at the micron scale shown in Figure 6 as well as in Figures 3 –5 that are predicted by the current non-classical shell model agree with the general trends observed in experiments (e.g., [2,61]).

4. Summary

A new non-classical FSD circular cylindrical thin shell model is proposed using a MCST and a surface elasticity theory. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and BCs and provides a unified treatment of the microstructure and surface energy effects. The new thin shell model includes a material length-scale parameter to represent the microstructure effect and three surface elastic constants to describe the surface energy effect.

The newly developed thin shell model reduces to its classical elasticity-based counterpart as a limiting case when the microstructure and surface energy effects are both ignored. In addition, it is shown that the new model includes models considering the microstructure dependence or the surface energy effect alone as special cases and recovers the non-classical Mindlin plate model incorporating the same microstructure and surface energy effects as a limiting case.

The static bending and free vibration problems of a simply supported circular cylindrical thin shell are analytically solved by directly applying the new model. The numerical results show that the deﬂections and rotations of the simply supported thin shell predicted by the current model are always smaller than those predicted by the classical model. Also, it is observed that the differences in both the deﬂection and rotation predicted by the two shell models are significant when the shell thickness is sufficiently small, but they diminish with the increase of shell thickness. For the free vibration problem, it is found that the first natural frequency predicted by the new shell model is always higher than that predicted by the classical model, and the difference is large for very thin shells.

Footnotes

Appendix A: the Cauchy stress and couple stress resultants and the surface stress components in terms of the kinematic variables

From Equations (1), (20) and (29), the Cauchy stress resultants can be expressed in terms of u, v, w, ψ_x and ψ_θ as

(A.1a)

N_{xx} = h [(λ + 2 μ) \frac{\partial u}{\partial x} + \frac{λ}{R} (\frac{\partial v}{\partial θ} + w)],

(A.1b)

N_{x θ} = μ h (\frac{1}{R} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x}),

(A.1c)

N_{xz} = k_{s} μ h (\frac{\partial w}{\partial x} + ψ_{x}),

(A.1d)

N_{θ θ} = h [λ \frac{\partial u}{\partial x} + \frac{λ + 2 μ}{R} (\frac{\partial v}{\partial θ} + w)],

(A.1e)

N_{θ z} = k_{s} μ h (- \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + ψ_{θ}),

(A.1f)

M_{xx} = \frac{1}{12} h^{3} [(λ + 2 μ) \frac{\partial ψ_{x}}{\partial x} + \frac{λ}{R} \frac{\partial ψ_{θ}}{\partial θ}],

(A.1g)

M_{x θ} = \frac{1}{12} μ h^{3} (\frac{1}{R} \frac{\partial ψ_{x}}{\partial θ} + \frac{\partial ψ_{θ}}{\partial x}),

(A.1h)

M_{θ θ} = \frac{1}{12} h^{3} (λ \frac{\partial ψ_{x}}{\partial x} + \frac{λ + 2 μ}{R} \frac{\partial ψ_{θ}}{\partial θ}),

where k_s is a shear correction factor introduced to account for the non-uniformity of the shear strain components $ε_{xz}$ and $ε_{θ z}$ over the shell thickness (e.g., [60,62]).

From Equations (2), (24) and (31), the couple stress resultants can be written in terms of u, v, w, ψ_x and ψ_θ as

(A.2a)

Y_{xx} = μ l^{2} h (- \frac{1}{R} \frac{\partial v}{\partial x} + \frac{1}{R} \frac{\partial^{2} w}{\partial x \partial θ} - \frac{\partial ψ_{θ}}{\partial x}),

(A.2b)

Y_{x θ} = \frac{1}{2} μ l^{2} h (- \frac{1}{R^{2}} \frac{\partial v}{\partial θ} - \frac{\partial^{2} w}{\partial x^{2}} + \frac{1}{R^{2}} \frac{\partial^{2} w}{\partial θ^{2}} + \frac{\partial ψ_{x}}{\partial x} - \frac{1}{R} \frac{\partial ψ_{θ}}{\partial θ}),

(A.2c)

Y_{xz} = \frac{1}{2} μ l^{2} h (- \frac{1}{R} \frac{\partial^{2} u}{\partial θ \partial x} + \frac{v}{R^{2}} + \frac{\partial^{2} v}{\partial x^{2}} - \frac{1}{R^{2}} \frac{\partial w}{\partial θ} - \frac{ψ_{θ}}{R}),

(A.2d)

Y_{θ θ} = \frac{μ l^{2} h}{R} (- \frac{1}{R} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x} - \frac{\partial^{2} w}{\partial x \partial θ} + \frac{\partial ψ_{x}}{\partial θ}),

(A.2e)

Y_{θ z} = \frac{μ l^{2} h}{2 R} (- \frac{1}{R} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x} - ψ_{x}),

(A.2f)

Y_{zz} = μ l^{2} h (\frac{1}{R^{2}} \frac{\partial u}{\partial θ} - \frac{1}{R} \frac{\partial ψ_{x}}{\partial θ} + \frac{\partial ψ_{θ}}{\partial x}),

(A.2g)

H_{xz} = \frac{1}{24} μ l^{2} h^{3} (- \frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial θ \partial x} + \frac{\partial^{2} ψ_{θ}}{\partial x^{2}}),

(A.2h)

H_{θ θ} = \frac{1}{12} \frac{μ l^{2} h^{3}}{R} \frac{\partial ψ_{θ}}{\partial x},

(A.2i)

H_{θ z} = \frac{1}{24} \frac{μ l^{2} h^{3}}{R} (- \frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) .

From Equations (7)–(9), (19) and (20), it follows that the surface stress components can be expressed in terms of u, v, w, ψ_x and ψ_θ as

(A.3a)

τ_{xx}^{\pm} = τ_{0} + \frac{λ_{0} + τ_{0}}{r^{\pm}} (\frac{\partial v}{\partial θ} + w \pm \frac{h}{2} \frac{\partial ψ_{θ}}{\partial θ}) + (λ_{0} + 2 μ_{0}) (\frac{\partial u}{\partial x} \pm \frac{h}{2} \frac{\partial ψ_{x}}{\partial x}),

(A.3b)

τ_{x θ}^{\pm} = μ_{0} (\frac{1}{r^{\pm}} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x}) - τ_{0} \frac{\partial v}{\partial x} \pm \frac{μ_{0} h}{2 r^{\pm}} \frac{\partial ψ_{x}}{\partial θ} \pm \frac{(μ_{0} - τ_{0}) h}{2} \frac{\partial ψ_{θ}}{\partial x},

(A.3c)

τ_{θ x}^{\pm} = μ_{0} (\frac{1}{r^{\pm}} \frac{\partial u}{\partial θ} + \frac{\partial v}{\partial x}) - \frac{τ_{0}}{r^{\pm}} \frac{\partial u}{\partial θ} \pm \frac{(μ_{0} - τ_{0}) h}{2 r^{\pm}} \frac{\partial ψ_{x}}{\partial θ} \pm \frac{μ_{0} h}{2} \frac{\partial ψ_{θ}}{\partial x},

(A.3d)

τ_{θ θ}^{\pm} = τ_{0} + (λ_{0} + τ_{0}) (\frac{\partial u}{\partial x} \pm \frac{h}{2} \frac{\partial ψ_{x}}{\partial x}) + \frac{λ_{0} + 2 μ_{0}}{r^{\pm}} (\frac{\partial v}{\partial θ} + w \pm \frac{h}{2} \frac{\partial ψ_{θ}}{\partial θ}),

(A.3e)

τ_{3 x}^{+} = τ_{3 x}^{-} = τ_{0} \frac{\partial w}{\partial x},

(A.3f)

τ_{3 θ}^{\pm} = \frac{τ_{0}}{r^{\pm}} (- v + \frac{\partial w}{\partial θ} \mp \frac{h}{2} ψ_{θ}) .

Appendix B: two special cases – classical elasticity-based model for first-order shear deformation circular cylindrical thin shells and non-classical model for Mindlin plates

When l = 0, c_i = 0 and λ₀ = µ₀ = τ₀ = 0, Equations (40a–e) reduce to

(B.1a)

h [(λ + 2 μ) \frac{\partial^{2} u}{\partial x^{2}} + \frac{λ}{R} (\frac{\partial^{2} v}{\partial x \partial θ} + \frac{\partial w}{\partial x})] + \frac{μ h}{R} (\frac{1}{R} \frac{\partial^{2} u}{\partial θ^{2}} + \frac{\partial^{2} v}{\partial x \partial θ}) + f_{x} = ρ h \overset{\cdot\cdot}{u},

(B.1b)

\begin{matrix} \frac{h}{R} [λ \frac{\partial^{2} u}{\partial x \partial θ} + \frac{λ + 2 μ}{R} (\frac{\partial^{2} v}{\partial θ^{2}} + \frac{\partial w}{\partial θ})] + μ h (\frac{1}{R} \frac{\partial^{2} u}{\partial x \partial θ} + \frac{\partial^{2} v}{\partial x^{2}}) \\ + \frac{k_{s} μ h}{R} (- \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + ψ_{θ}) + f_{θ} = ρ h \overset{\cdot\cdot}{v}, \end{matrix}

(B.1c)

\begin{matrix} k_{s} μ h (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial ψ_{x}}{\partial x}) + \frac{k_{s} μ h}{R} (- \frac{1}{R} \frac{\partial v}{\partial θ} + \frac{1}{R} \frac{\partial^{2} w}{\partial θ^{2}} + \frac{\partial ψ_{θ}}{\partial θ}) \\ - \frac{h}{R} [λ \frac{\partial u}{\partial x} + \frac{λ + 2 μ}{R} (\frac{\partial v}{\partial θ} + w)] + f_{z} = ρ h \overset{\cdot\cdot}{w}, \end{matrix}

(B.1d)

\begin{matrix} \frac{1}{12} h^{3} [(λ + 2 μ) \frac{\partial^{2} ψ_{x}}{\partial x^{2}} + \frac{λ}{R} \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}] + \frac{μ h^{3}}{12 R} (\frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial θ^{2}} + \frac{\partial^{2} ψ_{θ}}{\partial x \partial θ}) \\ - k_{s} μ h (\frac{\partial w}{\partial x} + ψ_{x}) = \frac{ρ h^{3}}{12} {\overset{\cdot\cdot}{ψ}}_{x}, \end{matrix}

(B.1e)

\begin{matrix} \frac{h^{3}}{12 R} [λ \frac{\partial^{2} ψ_{x}}{\partial x \partial θ} + \frac{λ + 2 μ}{R} \frac{\partial^{2} ψ_{θ}}{\partial θ^{2}}] + \frac{μ h^{3}}{12} (\frac{1}{R} \frac{\partial^{2} ψ_{x}}{\partial x \partial θ} + \frac{\partial^{2} ψ_{θ}}{\partial x^{2}}) \\ - k_{s} μ h (- \frac{v}{R} + \frac{1}{R} \frac{\partial w}{\partial θ} + ψ_{θ}) = \frac{ρ h^{3}}{12} {\overset{\cdot\cdot}{ψ}}_{θ}, \end{matrix}

which are the governing equations for the FSD circular cylindrical thin shell based on classical elasticity.

When $\partial y = R \partial θ$ and R→∞, with $r^{+} \approx r^{-} \approx R$ , Equations (40a–e) become

(B.2a)

\begin{matrix} h [(λ + 2 μ) \frac{\partial^{2} u}{\partial x^{2}} + λ \frac{\partial^{2} v}{\partial x \partial y}] + μ h [\frac{\partial^{2} u}{\partial y^{2}} + \frac{\partial^{2} v}{\partial x \partial y}] + \frac{1}{4} μ l^{2} h (- \frac{\partial^{4} u}{\partial x^{2} \partial y^{2}} - \frac{\partial^{4} u}{\partial y^{4}} + \frac{\partial^{4} v}{\partial x \partial y^{3}} \\ + \frac{\partial^{4} v}{\partial x^{3} \partial y}) + 2 (2 μ_{0} + λ_{0}) \frac{\partial^{2} u}{\partial x^{2}} + (2 μ_{0} - τ_{0}) \frac{\partial^{2} u}{\partial y^{2}} + (2 μ_{0} + 2 λ_{0} + τ_{0}) \frac{\partial^{2} v}{\partial x \partial y} + f_{x} + \frac{1}{2} \frac{\partial c_{z}}{\partial y} = ρ h \overset{\cdot\cdot}{u}, \end{matrix}

(B.2b)

\begin{matrix} h [λ \frac{\partial^{2} u}{\partial x \partial y} + (λ + 2 μ) \frac{\partial^{2} v}{\partial y^{2}}] + μ h [\frac{\partial^{2} u}{\partial x \partial y} + \frac{\partial^{2} v}{\partial x^{2}}] - \frac{1}{4} μ l^{2} h (- \frac{\partial^{4} u}{\partial x \partial y^{3}} - \frac{\partial^{4} u}{\partial x^{3} \partial y} + \frac{\partial^{4} v}{\partial x^{4}} + \frac{\partial^{4} v}{\partial x^{2} \partial y^{2}}) \\ + (2 μ_{0} - τ_{0}) \frac{\partial^{2} v}{\partial x^{2}} + 2 (2 μ_{0} + λ_{0}) \frac{\partial^{2} v}{\partial y^{2}} + (2 μ_{0} + 2 λ_{0} + τ_{0}) \frac{\partial^{2} u}{\partial x \partial y} + f_{y} - \frac{1}{2} \frac{\partial c_{z}}{\partial x} = ρ h \overset{\cdot\cdot}{v}, \end{matrix}

(B.2c)

\begin{matrix} k_{s} μ h (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}} + \frac{\partial ψ_{x}}{\partial x} + \frac{\partial ψ_{y}}{\partial y}) - \frac{μ l^{2} h}{4} (\frac{\partial^{4} w}{\partial x^{4}} + 2 \frac{\partial^{4} w}{\partial x^{2} \partial y^{2}} + \frac{\partial^{4} w}{\partial y^{4}} - \frac{\partial^{3} ψ_{x}}{\partial x^{3}} - \frac{\partial^{3} ψ_{x}}{\partial x \partial y^{2}} \\ - \frac{\partial^{3} ψ_{y}}{\partial x^{2} \partial y} - \frac{\partial^{3} ψ_{y}}{\partial y^{3}}) + 2 τ_{0} (\frac{\partial^{2} w}{\partial x^{2}} + \frac{\partial^{2} w}{\partial y^{2}}) + f_{z} - \frac{1}{2} \frac{\partial c_{x}}{\partial y} + \frac{1}{2} \frac{\partial c_{y}}{\partial x} = ρ h \overset{\cdot\cdot}{w}, \end{matrix}

(B.2d)

\begin{matrix} \frac{1}{12} h^{3} [(λ + 2 μ) \frac{\partial^{2} ψ_{x}}{\partial x^{2}} + λ \frac{\partial^{2} ψ_{y}}{\partial x \partial y}] + \frac{μ h^{3}}{12} (\frac{\partial^{2} ψ_{x}}{\partial y^{2}} + \frac{\partial^{2} ψ_{y}}{\partial x \partial y}) - k_{s} μ h (\frac{\partial w}{\partial x} + ψ_{x}) + \frac{μ l^{2} h^{3}}{48} (- \frac{\partial^{4} ψ_{x}}{\partial x^{2} \partial y^{2}} \\ - \frac{\partial^{4} ψ_{x}}{\partial y^{4}} + \frac{\partial^{4} ψ_{y}}{\partial x \partial y^{3}} + \frac{\partial^{4} ψ_{y}}{\partial x^{3} \partial y}) + \frac{1}{4} μ l^{2} h (- \frac{\partial^{3} w}{\partial x^{3}} - \frac{\partial^{3} w}{\partial x \partial y^{2}} + \frac{\partial^{2} ψ_{x}}{\partial x^{2}} + 4 \frac{\partial^{2} ψ_{x}}{\partial y^{2}} - 3 \frac{\partial^{2} ψ_{y}}{\partial x \partial y}) \\ + \frac{h^{2}}{4} [2 (λ_{0} + 2 μ_{0}) \frac{\partial^{2} ψ_{x}}{\partial x^{2}} + (2 μ_{0} - τ_{0}) \frac{\partial^{2} ψ_{x}}{\partial y^{2}} + (2 λ_{0} + 2 μ_{0} + τ_{0}) \frac{\partial^{2} ψ_{y}}{\partial x \partial y}] + \frac{1}{2} c_{y} = \frac{ρ h^{3}}{12} {\overset{\cdot\cdot}{ψ}}_{x}, \end{matrix}

(B.2e)

\begin{matrix} \frac{h^{3}}{12} [λ \frac{\partial^{2} ψ_{x}}{\partial x \partial y} + (λ + 2 μ) \frac{\partial^{2} ψ_{y}}{\partial y^{2}}] + \frac{1}{12} μ h^{3} (\frac{\partial^{2} ψ_{x}}{\partial x \partial y} + \frac{\partial^{2} ψ_{y}}{\partial x^{2}}) - k_{s} μ h (\frac{\partial w}{\partial y} + ψ_{y}) - \frac{μ l^{2} h^{3}}{48} (- \frac{\partial^{4} ψ_{x}}{\partial x \partial y^{3}} \\ - \frac{\partial^{4} ψ_{x}}{\partial x^{3} \partial y} + \frac{\partial^{4} ψ_{y}}{\partial x^{4}} + \frac{\partial^{4} ψ_{y}}{\partial x^{2} \partial y^{2}}) + \frac{μ l^{2} h}{4} (- \frac{\partial^{3} w}{\partial x^{2} \partial y} - \frac{\partial^{3} w}{\partial y^{3}} - 3 \frac{\partial^{2} ψ_{x}}{\partial x \partial y} + 4 \frac{\partial^{2} ψ_{y}}{\partial x^{2}} + \frac{\partial^{2} ψ_{y}}{\partial y^{2}}) \\ + \frac{h^{2}}{4} [(2 λ_{0} + 2 μ_{0} + τ_{0}) \frac{\partial^{2} ψ_{x}}{\partial x \partial y} + (2 μ_{0} - τ_{0}) \frac{\partial^{2} ψ_{y}}{\partial x^{2}} + 2 (λ_{0} + 2 μ_{0}) \frac{\partial^{2} ψ_{y}}{\partial y^{2}}] - \frac{1}{2} c_{x} = \frac{ρ h^{3}}{12} {\overset{\cdot\cdot}{ψ}}_{y}, \end{matrix}

which are the governing equations for non-classical Mindlin plates incorporating the microstructure and surface energy effects [50].

Acknowledgements

The authors would like to thank Professor David Steigmann and one anonymous reviewer for their encouragement and helpful comments on an earlier version of the paper.

Funding

The authors disclosed receipt of the following financial support for the research, authorship and/or publication of this article: GYZ gratefully acknowledges the support of the National Natural Science Foundation of China [grant # 12002086] and the Fundamental Research Funds for the Central Universities [grant # 2242020R10027].

ORCID iD

X-L Gao

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