An asymptotic meshless method for sandwich functionally graded circular hollow cylinders with various boundary conditions

Abstract

An asymptotic meshless method using the differential reproducing kernel interpolation and perturbation method is formulated for the three-dimensional bending analysis of sandwich functionally graded circular hollow cylinders and laminated composite ones with clamped and simply-supported edge conditions, in which the effective material properties of the functionally graded layer are evaluated using the Mori–Tanaka scheme. In the formulation, we perform the mathematical processes of nondimensionalization, asymptotic expansion and successive integration to obtain recurrent sets of governing equations for various order problems. Classical shell theory is derived as a first-order approximation of the three-dimensional theory of elasticity, and the governing equations for higher order problems retain the same differential operators as those of classical shell theory, although with different nonhomogeneous terms. Expanding the primary field variables of each order as the Fourier series functions in the circumferential direction, and interpolating these in the axial direction using the differential reproducing kernel interpolation, we can obtain the leading-order solutions of this analysis, and the higher order modifications can then be determined in a systematic manner. Some benchmark solutions for the bending analysis of sandwich functionally graded circular hollow cylinders and laminated composite ones subjected to mechanical loads are given to demonstrate the performance of the asymptotic meshless method.

Keywords

Asymptotic theory meshless method reproducing kernel static functionally graded material cylinders

Introduction

A number of studies have carried out three-dimensional (3D) analyses of the static behaviors and dynamic responses of laminated (or sandwiched) composite plates/shells and functionally graded (FG) ones with various reinforcements, such as the microscaled continuous graphite, grass and boron fibers, and nanoscaled discrete carbon nanotubes and graphenes. Among these, however, relatively few articles consider the 3D exact analyses of the structures with various boundary conditions compared to those that examine structures with fully simply-supported edges. Wu et al. [1] classified the 3D exact approaches for simply-supported structures available in the literature into four different categories as follows: the state space [2 –5], series expansion [6,7], Pagano [8,9], and perturbation methods [10,11]. In addition, the related advanced and refined theories have been surveyed in a number of review articles by Carrera [12 –15] and Carrera and Brischetto [16], that the effect of transverse shear and normal deformations on the thermomechanical bending of FG sandwich plates and fiber-reinforced viscoelastic beams resting on two-parameter elastic foundations have also been conducted by Zenkour [17,18], Zenkour and Alghamdi [19], and that some novelty 2D theories have also been developed to investigate the static, vibration and buckling of sandwich FG structures with and without resting on the Pasternak foundation model, such as the refined zigzag theory [20], four-variable refined theory [21], and refined and advanced higher order shear deformation theories [22 –24].

Due to the mathematical complexity of 3D theories of elasticity/piezoelectricity, and the limitations of various analytical methods with regard to satisfying the boundary conditions, semi-analytical methods have thus been developed for the approximate 3D analyses of the structures with various boundary conditions. In these, the above-mentioned analytical methods are combined with numerical methods, such as the Ritz, differential quadrature (DQ), finite element (FE), and meshless approaches, in order to interpolate or approximate the variations of the field variables in the physical domain.

Based on a discrete layer theory combined with the Ritz method, Ramirez et al. [25,26] developed a semi-analytical method for the static analysis of FG elastic and anisotropic plates and free vibration analysis of FG magneto-electro-elastic ones, and these were shown to be not limited to specific boundary conditions and gradation functions. Zhou et al. [27,28] investigated the 3D vibration of circular, annular and rectangular plates using a Chebyshev–Ritz method, in which the vibrations of annular plates were divided into three distinct categories, axisymmetric, torsional and circumferential vibration ones, and some valuable results were obtained for annular plates with one or both edges clamped. The approach was further extended to the 3D vibration of FG annular plates by Dong [29] and FG rectangular ones by Uymaz and Aydogdu [30].

The differential quadrature (DQ) approach [31,32] has been used in combination with the above-mentioned analytical methods to develop some semi-analytical tools for the analysis of FG circular hollow cylinders with various boundary conditions. Liew and Teo [33] and Liew et al. [34] presented a 3D DQ formulation for the analysis of rectangular isotropic plates with simply-supported and clamped edges. Within the framework of layerwise plate theory, Liew et al. [35] undertook the 3D bending analysis of cross-ply laminated plates with various edge-supports, in which the effects of different boundary conditions, aspect ratios and loading conditions on the static behaviors of the plates were examined, and the applicability, accuracy and stability of the 3D DQ approach were demonstrated. Lü et al. [36 –38] combined the state space and DQ methods to develop a semi-analytical formulation for the 3D analyses of the bending and free vibration of laminated composite plates and multidirectional FG ones with various boundary conditions, in which the DQ method was shown to be applicable to laminates with an arbitrary thickness and for arbitrary edge conditions.

The conventional finite element methods (FEMs) were also combined with Fourier series expansion to carry out a 3D approximate analysis. Based on the Reissner mixed variational theorem (RMVT) [39,40], Wu and Li [41,42] developed the finite rectangular/cylindrical prism methods (FRPMs and FCPMs) for the static behaviors of multilayered FG plates/cylinders and laminated composite ones with various boundary conditions, in which the single Fourier functions were used to expand the field variables in the circumferential/width coordinate, while the Lagrange polynomials in the thickness-length surface, and the material properties were assumed to obey either the power-law distribution in terms of the volume fraction of the constituent materials, or the exponent-law varying exponentially through the thickness direction.

More recently, in contrast to the above-mentioned numerical methods (i.e. the Ritz, DQ and FE methods), meshless methods were combined with the state space approach to analyze the 3D static and dynamic responses of laminated composite plates/shells and FGM ones, in which the shape functions of various field variables were constructed with the scattered nodes and using the differential reproducing kernel (DRK) [43,44], radial basis (RB) [45,46], and moving least square functions [47]. Sladek et al. [48,49] developed a meshless local Petro–Galerkin method (LPGM) for the static and dynamic analyses of FG circular plates and shallow shells, in which a local weak formulation was derived and the moving least square method was used to construct the approximation functions for the field variables. Based on the first-order shear deformation theory, Zhang et al. [50] and Liew et al. [51] carried out the large deflection geometrically nonlinear analysis and postbuckling analysis of carbon nanotube-reinforced functionally graded cylindrical panels using the kp-Ritz method combined with reproducing kernel particle functions, in which the effective material properties of the panels were evaluated using the Mori–Tanaka approach [52].

To the best of the authors’ knowledge, the perturbation method has never been combined with the meshless methods for use in a semi-analytical approach, nor has such a method been examined to see if it is applicable for the 3D analysis of FG plates/shells with various boundary conditions. An asymptotic DRK-based meshless method is thus developed in this work for the 3D analysis of FG circular hollow cylinders with various boundary conditions. Using direct elimination, we first reduce the fifteen partially differential equations (PDEs) of the 3D theory of elasticity to six PDEs in terms of six primary variables of elastic fields, three displacement components and three transverse shear and normal stress ones. Through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of governing equations for various order problems. Classical shell theory (CST) is derived as a first-order approximation of the 3D theory of elasticity, and the governing equations for higher order problems retain the same differential operators as those of CST, although with different nonhomogeneous terms. Expanding the primary field variables of each order as the Fourier series functions in the circumferential direction, and interpolating these in the axial direction using the DRK interpolation, we can obtain the leading-order solutions of this analysis, and the higher order modifications can then be determined in a systematic and consistent manner. The asymptotic meshless method is apparent to differ from the methods based on the conventional advanced and refined theories. In the latter, a set of kinetic and kinematic models was assumed a priori, and this has to be replaced by using a much higher order model when the accuracy of the original model needs to be improved, resulting the formulation must be totally changed. On the contrary, in the former, the leading-order solutions are identical to the CST ones, and its accuracy can be improved with performing the higher order modifications in a systematic and consistent manner and without changing the whole formulation. Moreover, the asymptotic meshless solutions will approach to the exact 3D solutions level by level.

Some benchmark solutions for the bending analysis of simply-supported, sandwich FG circular hollow cylinders and laminated composite ones subjected to mechanical loads are given to demonstrate the performance of the asymptotic DRK-based meshless method, in which the effective material properties of the FG layer are evaluated using the Mori–Tanaka scheme. The results for the cylinders with various boundary conditions may provide a reference for assessing those obtained using the two-dimensional analytical and numerical methods. Moreover, the effects of the material property gradient index, and different aspect ratios and boundary conditions on the stress and displacement components induced in the pressure-loaded cylinders are also examined.

Basic equations of 3D elasticity

As shown in Figure 1, we consider an FG elastic circular hollow cylinder, of which the thickness is $2 h$ . A set of cylindrical coordinates (x, θ, r) is located at the center of the cylinder, and the thickness coordinate (ζ) is measured from the mid-surface of this. R and L denote the radius and length of the cylinder, the relation between the radial coordinate I and the thickness one (ζ) is $ζ = r - R$ .

Figure 1.

The configuration and coordinates of an FGM sandwich circular hollow cylinder or a laminated composite one.

The linear constitutive equations valid for the symmetrical class of elastic materials are given by

{σ x σ θ σ r τ θ r τ x r τ x θ} = [c 11 c 12 c 13 000 c 12 c 22 c 23 000 c 13 c 23 c 33 000000 c 44 000000 c 55 000000 c 66] {ɛ x ɛ θ ɛ r γ θ r γ x r γ x θ}

(1)

where

σ x, σ θ, σ r, τ xr, τ θ r, τ x θ, and ɛ x, ɛ θ, ɛ r, γ xr, γ θ r, γ x θ

denote the stress and strain components, respectively.

c ij

(i, j = 1–6) are the elastic coefficients relative to the geometrical axes of the cylinder, and these are considered to be heterogeneous through the thickness of the cylinder (i.e.

c ij = c ij (ζ)

The kinematic equations in terms of the cylindrical coordinates x, θ, and r are

ɛ x = \frac{\partial u x}{\partial x}, ɛ θ = \frac{1}{r} (\frac{\partial u θ}{\partial θ} + u r), ɛ r = \frac{\partial u r}{\partial r} γ xr = \frac{\partial u x}{\partial r} + \frac{\partial u r}{\partial x}, γ θ r = \frac{\partial u θ}{\partial r} - \frac{u θ}{r} + (\frac{1}{r}) \frac{\partial u r}{\partial θ}, γ x θ = (\frac{1}{r}) \frac{\partial u x}{\partial θ} + \frac{\partial u θ}{\partial x}

(2a-2f)

in which u_x, u_θ, and u_r are the displacement components.

The stress equilibrium equations of an elastic body without accounting for body forces are given by

r \frac{\partial σ x}{\partial x} + \frac{\partial τ x θ}{\partial θ} + τ x r + r \frac{\partial τ xr}{\partial r} = 0

(3)

r \frac{\partial τ x θ}{\partial x} + \frac{\partial σ θ}{\partial θ} + r \frac{\partial τ θ r}{\partial r} + 2 τ θ r = 0

(4)

r \frac{\partial τ x r}{\partial x} + \frac{\partial τ θ r}{\partial θ} + r \frac{\partial σ r}{\partial r} + σ r - σ θ = 0

(5)

The boundary conditions of the problem are specified as follows.

On the lateral surface the transverse loads are given by

[τ x r τ θ r σ r] = [0 0 {\bar{q}}_{r}^{\pm}] on ζ = \pm h

(6)

The edge boundary conditions of the cylinder are considered as either simply-supported or clamped ones, and are given as follows.

For the simply-supported edge

σ x = u θ = u r = 0

(7a)

For the clamped edge

u x = u θ = u r = 0

(7b)

Nondimensionalization

A set of dimensionless coordinates and elastic field variables is defined as follows

x 1 = x / (R \in), x 2 = θ / \in, x 3 = ζ / h, u 1 = u x / (R \in), u 2 = u θ / (R \in), u 3 = u r / R, σ 1 = σ x / Q, σ 2 = σ θ / Q, τ 12 = τ x θ / Q, τ 13 = τ x r / (Q \in), τ 23 = τ θ r / (Q \in), σ 3 = σ r / (Q \in 2)

(8a-8l)

where

\in 2 = h / R

, and Q denotes a reference elastic modulus.

As we previously listed in equations (1) to (5), there are 15 basic equations of 3D elasticity theory for the bending analysis of FG elastic circular hollow cylinders. In order to make the previous complicated formulation suitable for mathematical treatment, we eliminate the in-surface stress ( $σ x, σ θ, τ x θ$ ) and strain ( $ɛ x, ɛ θ, ɛ r, γ x r, γ θ r, γ x θ$ ) components from equations (1) to (5), introduce equation (8) in the resulting equations, and then express the 3D basic equations in terms of the dimensionless forms of displacement ( $u 1, u 2, u 3$ ) and transverse stress ( $τ 13, τ 23, σ 3$ ) components, as follows

u 3, 3 = - \in 2 L 1 u - \in 2 \tilde{l} 33 u 3 + \in 4 ({Qc}_{33}^{- 1}) σ 3

(9)

u, 3 = - D u 3 + \in 2 L 2 u + \in 2 S σ s + \in 4 L 3 σ s

(10)

σ s, 3 = - L 4 u - L 5 u 3 - \in 2 L 6 σ s - \in 2 L 7 σ 3

(11)

σ 3, 3 = L 8 u + \tilde{l} 63 u 3 - D T σ s - \in 2 L 9 σ s - \in 2 \tilde{l} 64 σ 3

(12)

where

u = {u 1 u 2}, D = {\partial 1 \partial 2}, S = [(Q / c 55) 00 (Q / c 44)], σ s = {τ 13 τ 23} L 1 = [\tilde{l} 31 \tilde{l} 32], L 2 = [000 \tilde{l} 22], L 3 = [000 \tilde{l} 23] L 4 = [\tilde{l} 41 \tilde{l} 42 \tilde{l} 42 \tilde{l} 52], L 5 = [\tilde{l} 43 \tilde{l} 53], L 6 = [\tilde{l} 44 00 \tilde{l} 55], L 7 = [\tilde{l} 46 \tilde{l} 56] L 8 = [\tilde{l} 61 \tilde{l} 62], L 9 = [x 3 \partial 1 0]

and the relevant functions

\tilde{l} ij

L i

(i = 1–8) are given in Appendix 2.

Following a similar derivation process, we rewrite the in-surface stresses in the dimensionless form as

σ p = B 1 u + B 2 u 3 + \in 2 B 3 σ 3

(13)

where

σ p = {σ 1 σ 2 τ 12}, B 1 = [\tilde{b} 11 \tilde{b} 12 \tilde{b} 21 \tilde{b} 22 \tilde{b} 31 \tilde{b} 32], B 2 = [\tilde{b} 13 \tilde{b} 23 0], B 3 = [\tilde{b} 14 \tilde{b} 24 0], and \tilde{b} ij

are the relevant functions, and are also given in Appendix 3.

The dimensionless forms of the boundary conditions of the problem are specified as follows.

On the lateral surface the transverse loads are given by

[τ 13 τ 23 σ 3] = [0 0 {\bar{q}}_{3}^{\pm}] on x 3 = \pm 1

(14)

in which

{\bar{q}}_{3}^{\pm} = {\bar{q}}_{r}^{\pm} / (Qh / R)

At the edges ( $x 1 = 0$ and $x 1 = L / \sqrt{Rh}$ ), the boundary conditions are given as follows.

For the simply-supported edge

σ 1 = u 2 = u 3 = 0

(15a)

For the clamped edge

u 1 = u 2 = u 3 = 0

(15b)

Asymptotic expansion

By observation of equations (9) to (13), we find that these contain terms involving only even powers of ∈. We thus asymptotically expand the field variables in the powers $\in 2$ , as given by

f = f (0) (x 1, x 2, x 3) + \in 2 f (1) (x 1, x 2, x 3) + \in 4 f (2) (x 1, x 2, x 3) + \dots

(16)

Substituting equation (16) into equations (9) to (13) and collecting coefficients of equal powers of ∈, we obtain the following sets of recurrence equations.

Order $\in 0$

u_{3}^{(0)}, 3 = 0

(17)

u (0), 3 = - D u_{3}^{(0)}

(18)

σ_{s}^{(0)}, 3 = - L 4 u (0) - L 5 u_{3}^{(0)}

(19)

σ_{3}^{(0)}, 3 = L 8 u (0) + \tilde{l} 63 u_{3}^{(0)} - D T σ_{s}^{(0)}

(20)

σ_{p}^{(0)} = B 1 u (0) + B 2 u_{3}^{(0)}

(21)

Order $\in 2 k$ (k = $1, 2, 3, \dots$ )

u_{3}^{(k)}, 3 = - L 1 u (k - 1) - \tilde{l} 33 u_{3}^{(k - 1)} + ({Qc}_{33}^{- 1}) σ_{3}^{(k - 2)}

(22)

u (k), 3 = - D u_{3}^{(k)} + L 2 u (k - 1) + S σ_{s}^{(k - 1)} + L 3 σ_{s}^{(k - 2)}

(23)

σ_{s}^{(k)}, 3 = - L 4 u (k) - L 5 u_{3}^{(k)} - L 6 σ_{s}^{(k - 1)} - L 7 σ_{3}^{(k - 1)}

(24)

σ_{3}^{(k)}, 3 = L 8 u (k) + \tilde{l} 63 u_{3}^{(k)} - D T σ_{s}^{(k)} - L 9 σ_{s}^{(k - 1)} - \tilde{l} 64 σ_{3}^{(k - 1)}

(25)

σ_{p}^{(k)} = B 1 u (k) + B 2 u_{3}^{(k)} + B 3 σ_{3}^{(k - 1)}

(26)

The boundary conditions for various order problems are specified as follows.

On the lateral surface, the transverse loads are given by the following.

Order $\in 0$

[τ_{13}^{(0)} τ_{23}^{(0)} σ_{3}^{(0)}] = [0 0 {\bar{q}}_{3}^{\pm}] on x 3 = \pm 1

(27a)

Order $\in 2 k$ (k = 1, 2, 3, etc.)

[τ_{13}^{(k)} τ_{23}^{(k)} σ_{3}^{(k)}] = [0 0 0] on x 3 = \pm 1

(27b)

Along the edges ( $x 1 = 0$ and $x 1 = L / \sqrt{Rh}$ ), the boundary conditions are given as follows

For the simply-supported edge and order $\in 2 k$ (k = 0, 1, 2, 3, etc.)

σ_{1}^{(k)} = u_{2}^{(k)} = u_{3}^{(k)} = 0

(28a)

For the clamped edge and order $\in 2 k$ (k = 0, 1, 2, 3, etc.)

u_{1}^{(k)} = u_{2}^{(k)} = u_{3}^{(k)} = 0 .

(28b)

Asymptotic integration

The leading-order problem

We examine the sets of asymptotic equations and find that the present analysis can be carried out by integrating these through the thickness direction. We thus integrate equations (17) and (18) to obtain

u_{3}^{(0)} = u_{3}^{0} (x 1, x 2)

(29)

u (0) = u 0 - x 3 D u_{3}^{0}

(30)

where

u_{3}^{0}

and

u 0 = [u_{1}^{0} (x 1, x 2) u_{2}^{0} (x 1, x 2)] T

represent the displacement components on the middle surface, and these are also of the Kirchhoff–Love type in CST.

With the lateral boundary conditions on x₃ = −1 given in equation (27a), we then proceed to integrate equations (19) and (20), which yields

σ_{s}^{(0)} = - \int_{- 1}^{x 3} [L 4 (u 0 - η D u_{3}^{0}) + L 5 u_{3}^{0}] d η

(31)

σ_{3}^{(0)} = \int_{- 1}^{x 3} [L 8 (u 0 - η D u_{3}^{0}) + \tilde{l} 63 u_{3}^{0}] d η + \int_{- 1}^{x 3} (x 3 - η) D T [L 4 (u 0 - η D u_{3}^{0}) + L 5 u_{3}^{0}] d η

(32)

Imposing the remaining lateral boundary conditions on x₃ = 1 given in equation (27a) in equations (31) and (32), we obtain

K 11 u_{1}^{0} + K 12 u_{2}^{0} + K 13 u_{3}^{0} = 0

(33)

K 21 u_{1}^{0} + K 22 u_{2}^{0} + K 23 u_{3}^{0} = 0

(34)

K 31 u_{1}^{0} + K 32 u_{2}^{0} + K 33 u_{3}^{0} = {\bar{q}}_{3}^{+} - {\bar{q}}_{3}^{-}

(35)

where

K ij

(i, j = 1–3) are the relevant differential operators and given in Appendix 4.

In this article, the edge boundary conditions of the cylinder are considered as combinations of the clamped and simply-supported edges. After the asymptotic process, we thus obtain the edge conditions for the leading-order problem, as follows.

Case 1

For simple-simple (SS) supports

u_{2}^{0} = 0, u_{3}^{0} = 0, N_{1}^{(0)} = 0 and M_{1}^{(0)} = 0, at x 1 = 0 and x 1 = L / \sqrt{Rh}

(36)

in which

N_{1}^{(0)} = \int_{- 1}^{1} σ_{1}^{(0)} λ θ d x 3

and

M_{1}^{(0)} = \int_{- 1}^{1} x 3 σ_{1}^{(0)} λ θ d x 3

Case 2

For simple-clamped (SC) supports

u_{2}^{0} = 0, u_{3}^{0} = 0, N_{1}^{(0)} = 0 and M_{1}^{(0)} = 0, at x 1 = 0

(37a)

u_{1}^{0} = 0, u_{2}^{0} = 0, u_{3}^{0} = 0 and u_{3}^{0}, 1 = 0, at x 1 = L / \sqrt{Rh}

(37b)

Case 3

For clamped-clamped (CC) supports

u_{1}^{0} = 0, u_{2}^{0} = 0, u_{3}^{0} = 0 and u_{3}^{0}, 1 = 0, at x 1 = 0 and x 1 = L / \sqrt{Rh}

(38)

It is noted that the CST governing equations are recovered from equations (33) to (35) by introducing a geometric assumption with regard to the thin shell: $x 3 / R << 1$ . CST has thus been derived as a first-order approximation of the 3D theory. Solutions of equations (33) to (35) must be supplemented with one of the appropriate edge boundary conditions given in equations (36) to (38) to constitute a well-posed boundary value problem. Once the variables of $u_{1}^{0}, u_{2}^{0}, and u_{3}^{0}$ are determined, the leading-order solutions of displacements are given by equations (29) and (30), the transverse shear and normal stresses by equations (31) and (32), and the in-surface stresses by equation (21).

Higher order problems

Proceeding to order $\in 2 k$ and following the same process as before, we readily obtain

u_{3}^{(k)} = u_{3}^{k} (x 1, x 2) + ϕ 3 k (x 1, x 2, x 3)

(39)

u (k) = u k (x 1, x 2) - x 3 D u_{3}^{k} + ϕ k (x 1, x 2, x 3)

(40)

σ_{s}^{(k)} = - \int_{- 1}^{x 3} [L 4 (u k - η D u_{3}^{k}) + L 5 u_{3}^{k}] d η - f k (x 1, x 2, x 3)

(41)

σ_{3}^{(k)} = \int_{- 1}^{x 3} [L 8 (u k - η D u_{3}^{k}) + \tilde{l} 63 u_{3}^{k}] d η + \int_{- 1}^{x 3} (x 3 - η) D T [L 4 (u k - η D u_{3}^{k}) + L 5 u_{3}^{k}] d η - f 3 k (x 1, x 2, x 3)

(42)

where

u k = [u_{1}^{k} (x 1, x 2) u_{2}^{k} (x 1, x 2)] T ϕ 3 k (x 1, x 2, x 3) = - \int_{0}^{x 3} [L 1 u (k - 1) + \tilde{l} 33 u_{3}^{(k - 1)} - ({Qc}_{33}^{- 1}) σ_{3}^{(k - 2)}] d η ϕ k = {ϕ 1 k (x 1, x 2,, x 3) ϕ 2 k (x 1, x 2, x 3)} = \int_{0}^{x 3} (L 2 u (k - 1) + S σ_{s}^{(k - 1)} + L 3 σ_{s}^{(k - 2)} - D ϕ 3 k) d η f k = {f 1 k (x 1, x 2, x 3,) f 2 k (x 1, x 2, x 3)} = \int_{- 1}^{x 3} [L 4 ϕ k + L 5 ϕ 3 k + L 6 σ_{s}^{(k - 1)} + L 7 σ_{3}^{(k - 1)}] d η f 3 k (x 1, x 2, x 3) = - \int_{- 1}^{x 3} [L 8 ϕ k + \tilde{l} 63 ϕ 3 k - L 9 σ_{s}^{(k - 1)} - \tilde{l} 64 σ_{3}^{(k - 1)} + D T f k] d η

u_{3}^{k}

and

u k

represent the kth-order modifications to the variables of the displacement components. By imposing the associated lateral boundary conditions (equation (27b)) on equations (41) and (42), we obtain again the CST governing equations, and the nonhomogeneous terms can be calculated using the lower order solution. The resulting equations are as follows

K 11 u_{1}^{k} + K 12 u_{2}^{k} + K 13 u_{3}^{k} = f 1 k (x 3 = 1)

(43)

K 21 u_{1}^{k} + K 22 u_{2}^{k} + K 23 u_{3}^{k} = f 2 k (x 3 = 1)

(44)

K 31 u_{1}^{k} + K 32 u_{2}^{k} + K 33 u_{3}^{k} = f 3 k (x 3 = 1) + \frac{\partial f 1 k (x 3 = 1)}{\partial x 1} + \frac{\partial f 2 k (x 3 = 1)}{\partial x 2}

(45)

The edge conditions for the higher order problems are given as follows.

Case 1

For simple–simple (SS) supports

u_{2}^{k} = 0, u_{3}^{k} = 0, N_{1}^{(k)} = 0 and M_{1}^{(k)} = 0, at x 1 = 0 and x 1 = L / \sqrt{R h}

(46)

in which

N_{1}^{(k)} = \int_{- 1}^{1} σ_{1}^{(k)} λ θ d x 3

and

M_{1}^{(k)} = \int_{- 1}^{1} x 3 σ_{1}^{(k)} λ θ dx 3

Case 2

For simple–clamped (SC) supports

u_{2}^{k} = 0, u_{3}^{k} = 0, N_{1}^{(k)} = 0 and M_{1}^{(k)} = 0, at x 1 = 0

(47a)

u_{1}^{k} = 0, u_{2}^{k} = 0, u_{3}^{k} = 0 and u_{3}^{k}, 1 = 0, at x 1 = L / \sqrt{Rh}

(47b)

Case 3

For clamped–clamped (CC) supports

u_{1}^{k} = 0, u_{2}^{k} = 0, u_{3}^{k} = 0, and u_{3, 1}^{k} = 0, at x 1 = 0 and x 1 = L / \sqrt{Rh}

(48)

The higher order modifications of mid-surface displacement components ( $u_{1}^{k}, u_{2}^{k}, and u_{3}^{k}$ ) can be obtained by solving equations (43) to (45) combined with one of the appropriate edge conditions given in equations (46) to (48), and once these are determined, the higher order modifications of displacement components are given by equations (39) and (40), the transverse stresses by equations (41) and (42) and the in-surface stresses by equation (26).

By observation of the governing equations of the leading-order problem (equations (33) to (35)) and the higher order problems (equations (40) to (42)), we find that the differential operators among the various order problems remain identical, and the nonhomogeneous terms of higher order problems can be calculated from the lower order solution. It is thus shown that the solution process of the leading-order problem can be repeatedly applied to the higher order problems. The present asymptotic solutions can be determined order-by-order in a hierarchical and consistent manner.

DRK interpolation

In this article, the DRK interpolation functions [44] are used to construct the shape functions of the primary field variables of this problem in the axial (x₁) direction, and the DRK interpolation functions and their relevant derivatives are briefly described, as follows.

It is assumed that there are n_p discrete nodes randomly selected and located at $x 1 = (x 1) 1, (x 1) 2, \dots, (x 1) n p$ , respectively, in the x direction, in which a function $F (x 1)$ is interpolated as $F a (x 1)$ and defined as

F a (x 1) = \sum_{l = 1}^{n p} ψ l (x 1) F l = \sum_{l = 1}^{n p} [\bar{φ} l (x 1) + \overset{\land}{φ} l (x 1)] F l

(49)

where

\bar{φ} l (x 1)

(l = 1, 2, …, n_p) denote the enrichment functions, which are determined by imposing the nth-order reproducing conditions and are given by

\bar{φ} l (x 1) = w a (x 1 - (x 1) l) P T (x 1 - (x 1) l) \bar{b} (x 1)

, in which

P T (x 1 - (x 1) l) = [1 (x 1 - (x 1) l) (x 1 - (x 1) l) 2 \dots (x 1 - (x 1) l) n]

, n is the highest order of the base functions,

\bar{b} (x 1)

is the undetermined function vector, and

w a (x 1 - (x 1) l)

is the weight function centered at the node,

x 1 = (x 1) l

, with a support size a;

\overset{\land}{φ} l (x 1)

(l = 1, 2,…, n_p) denote the primitive functions, which are used to introduce the Kronecker delta properties;

ψ l (x 1)

is the shape function of

F a (x 1)

at the sampling node,

x 1 = (x 1) l

; and F_l is the nodal function of

F a (x 1)

x 1 = (x 1) l

By selecting the complete nth-order polynomials as the basis functions to be reproduced, we obtain a set of reproducing conditions to determine the undetermined functions of $\bar{b} i (x 1) (i = 1, 2, \dots, n + 1)$ . These conditions are given as

\sum_{l = 1}^{n p} [\bar{φ} l (x 1) + \overset{\land}{φ} l (x 1)] (x 1) l r = (x 1) r r \leq n .

(50)

Equation (50) represents (n+1) reproducing conditions, and the matrix form of these is given as

\sum_{l = 1}^{n p} P (x 1 - (x 1) l) \bar{φ} l (x 1) = \sum_{l = 1}^{n p} P (x 1 - (x 1) l) w a (x 1 - (x 1) l) P T (x 1 - (x 1) l) \bar{b} (x 1) = P (0) - \sum_{l = 1}^{n p} P (x 1 - (x 1) l) \overset{\land}{φ} l (x 1),,

(51)

where

P (0) = [100 \dots 0] T

According to these conditions, we may obtain the undetermined function vector $\bar{b} (x 1)$ in the following form

\bar{b} (x 1) = A - 1 (x 1) [P (0) - \sum_{l = 1}^{n p} P (x 1 - (x 1) l) \overset{\land}{φ} l (x 1)],

(52)

where

A (x 1) = \sum_{l = 1}^{n p} P (x 1 - (x 1) l) w a (x 1 - (x 1) l) P T (x 1 - (x 1) l)

Substituting equation (52) into equation (49) yields the shape functions of $F a (x 1)$ in the form of

ψ l (x 1) = \bar{φ} l (x 1) + \overset{\land}{φ} l (x 1) (l = 1, 2, \dots, n p),

(53)

where

\bar{φ} l (x 1) = w a (x 1 - (x 1) l) P T (x 1 - (x 1) l) A - 1 (x 1) [P (0) - \sum_{l = 1}^{n p} P (x 1 - (x 1) l) \overset{\land}{φ} (x 1)] .

It is noted that if we select a set of primitive functions satisfying the Kronecker delta properties (i.e. $\overset{\land}{φ} l ((x 1) k) = δ lk$ ), a priori, then a set of the shape functions with these properties will be obtained (i.e. $ψ l ((x 1) k) = δ lk$ ), due to the fact that the enrichment functions vanish at all the nodes (i.e. $\bar{φ} l ((x 1) k) = 0$ ).

In implementing the present scheme, the weight and primitive functions (i.e. $w (s)$ and $\overset{\land}{φ} (s)$ ) must be selected in advance. Following Wang et al. [44], the normalized Gaussian function is selected as the weight and primitive functions at each sampling node, and this is given as

w (s) = {\frac{e - (s / α) 2 - e - (1 / α) 2}{1 - e - (1 / α) 2} for s \leq 1 0 for s > 1

(54)

where

w a (x - x l) = w (s)

s = | x - x l | / a

, and a denotes the radius of the influence zone, which is assigned not to cover any neighboring node for the primitive function, and its optimal value for the weight function will be discussed later in this work. Wang et al. [44] suggests α = 3 for the analysis of an elastic solid, and this is also used in this article. Moreover, the derivatives of these DRK interpolation functions are given in Wu and Jiang [53,54] and not repeated here.

Applications

Leading-order solution

The bending problem of functionally graded elastic circular hollow cylinders with clamped and simply-supported edges is studied using the asymptotic DRK-based meshless method, in which the variables of displacement and stress components are expanded as the Fourier series functions in the circumferential coordinate, and then the DRK interpolation functions are used to interpolate the variables in the axial coordinate. For this problem the governing equations of the leading-order problem can thus be solved by letting

u_{1}^{0} (x 1, x 2) = \sum_{\overset{\land}{n} = 0}^{\infty} {\tilde{u}}_{1}^{0} (x 1) \cos \tilde{n} x 2

(55)

u_{2}^{0} (x 1, x 2) = \sum_{\overset{\land}{n} = 0}^{\infty} {\tilde{u}}_{2}^{0} (x 1) \sin \tilde{n} x 2

(56)

u_{3}^{0} (x 1, x 2) = \sum_{\overset{\land}{n} = 0}^{\infty} {\tilde{u}}_{3}^{0} (x 1) \cos \tilde{n} x 2

(57)

where

\tilde{n} = \overset{\land}{n} \sqrt{h / R}

\overset{\land}{n}

is a positive integer or zero, and the summation sign will not be shown in the following derivation for brevity.

Substituting equations (55) to (57) into equations (33) to (35) gives

[k 11 k 12 k 13 k 21 k 22 k 23 k 31 k 32 k 33] {{\tilde{u}}_{1}^{0} {\tilde{u}}_{2}^{0} {\tilde{u}}_{3}^{0}} = {00 {\tilde{q}}_{3}^{+} - {\tilde{q}}_{3}^{-}}

(58)

in which the applied external loads on the lateral surfaces,

{\bar{q}}_{3}^{\pm} (x 1, x 2)

, are expressed as

{\bar{q}}_{3}^{\pm} (x 1, x 2) = \sum_{\overset{\land}{n} = 0}^{\infty} {\tilde{q}}_{3}^{\pm} (x 1) \cos \tilde{n} x 2

, and the expressions of

k ij (i, j = 1 - 3)

are given in Appendix 5.

Using the DRK interpolation, we express the field variables in the form of equation (49) and their higher order derivatives, and then rewrite the edge boundary conditions and governing equations of the leading order, as follows.

Appling the governing equations (equation (58)) to the ith-sampling node ( $x 1 = (x 1) i i = 1 - n p$ ) and using the DRK interpolation leads to

d 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{1}^{0}) j + d 12 ({\tilde{u}}_{1}^{0}) i + d 13 \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) ({\tilde{u}}_{2}^{0}) j + d 14 \sum_{j = 1}^{n p} ψ_{j}^{(3)} ((x 1) i) ({\tilde{u}}_{3}^{0}) j + d 15 \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) ({\tilde{u}}_{3}^{0}) j = 0

(59)

d 21 \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) ({\tilde{u}}_{1}^{0}) j + d 22 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{2}^{0}) j + d 23 ({\tilde{u}}_{2}^{0}) i + d 24 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{3}^{0}) j + d 25 ({\tilde{u}}_{3}^{0}) i = 0

(60)

d 31 \sum_{j = 1}^{n p} ψ_{j}^{(3)} ((x 1) i) ({\tilde{u}}_{1}^{0}) j + d 32 \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) ({\tilde{u}}_{1}^{0}) j + d 33 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{2}^{0}) j + d 34 ({\tilde{u}}_{2}^{0}) i + d 35 \sum_{j = 1}^{n p} ψ_{j}^{(4)} ((x 1) i) ({\tilde{u}}_{3}^{0}) j + d 36 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{3}^{0}) j + + d 37 ({\tilde{u}}_{3}^{0}) i = {\tilde{q}}_{3}^{+} ((x 1) i) - {\tilde{q}}_{3}^{-} ((x 1) i)

(61)

in which

i = 1 - n p

, and the expressions of

d ij

are given in Appendix 6.

Using the DRK interpolation, we rewrite the appropriate edge conditions of Cases 1 to 3 for the leading-order problem, as follows.

Case 1

For the simple–simple (S-S) supports

({\tilde{u}}_{2}^{0}) 1 = 0, ({\tilde{u}}_{3}^{0}) 1 = 0 \overset{\land}{A} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{1}^{0}) j + \tilde{n} \tilde{A} 12 ({\tilde{u}}_{2}^{0}) 1 - \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (0) ({\tilde{u}}_{3}^{0}) j + (\tilde{A} 12 + \tilde{n} 2 \tilde{B} 12) ({\tilde{u}}_{3}^{0}) 1 = 0 \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{1}^{0}) j + \tilde{n} \tilde{B} 12 ({\tilde{u}}_{2}^{0}) 1 - \overset{\land}{D} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (0) ({\tilde{u}}_{3}^{0}) j + (\tilde{B} 12 + \tilde{n} 2 \tilde{D} 12) \times ({\tilde{u}}_{3}^{0}) 1 = 0, at x 1 = 0

(62a-62d)

({\tilde{u}}_{2}^{0}) n p = 0, ({\tilde{u}}_{3}^{0}) n p = 0 \overset{\land}{A} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) ({\tilde{u}}_{1}^{0}) j + \tilde{n} \tilde{A} 12 ({\tilde{u}}_{2}^{0}) n p - \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (L / \sqrt{Rh}) ({\tilde{u}}_{3}^{0}) j + (\tilde{A} 12 + \tilde{n} 2 \tilde{B} 12) ({\tilde{u}}_{3}^{0}) n p = 0 \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) ({\tilde{u}}_{1}^{0}) j + \tilde{n} \tilde{B} 12 ({\tilde{u}}_{2}^{0}) n p - \overset{\land}{D} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (L / \sqrt{Rh}) ({\tilde{u}}_{3}^{0}) j + (\tilde{B} 12 + \tilde{n} 2 \tilde{D} 12) ({\tilde{u}}_{3}^{0}) n p = 0, at x 1 = L / \sqrt{Rh}

(63a-63d)

Case 2

For the simple–clamped (SC) supports

({\tilde{u}}_{2}^{0}) 1 = 0, ({\tilde{u}}_{3}^{0}) 1 = 0 \overset{\land}{A} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{1}^{0}) j + \tilde{n} \tilde{A} 12 ({\tilde{u}}_{2}^{0}) 1 - \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (0) ({\tilde{u}}_{3}^{0}) j + (\tilde{A} 12 + \tilde{n} 2 \tilde{B} 12) ({\tilde{u}}_{3}^{0}) 1 = 0 \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{1}^{0}) j + \tilde{n} \tilde{B} 12 ({\tilde{u}}_{2}^{0}) 1 - \overset{\land}{D} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (0) ({\tilde{u}}_{3}^{0}) j + (\tilde{B} 12 + \tilde{n} 2 \tilde{D} 12) \times ({\tilde{u}}_{3}^{0}) 1 = 0, at x 1 = 0

(64a-64d)

({\tilde{u}}_{1}^{0}) n p = 0, ({\tilde{u}}_{2}^{0}) n p = 0, ({\tilde{u}}_{3}^{0}) n p = 0, \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) ({\tilde{u}}_{3}^{0}) j = 0, at x 1 = L / \sqrt{Rh}

(65a-65d)

Case 3

For the clamped–clamped (CC) supports

({\tilde{u}}_{1}^{0}) 1 = 0, ({\tilde{u}}_{2}^{0}) 1 = 0, ({\tilde{u}}_{3}^{0}) 1 = 0, \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{3}^{0}) j = 0, at x 1 = 0

(66a-66d)

({\tilde{u}}_{1}^{0}) n p = 0, ({\tilde{u}}_{2}^{0}) n p = 0, ({\tilde{u}}_{3}^{0}) n p = 0, \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) ({\tilde{u}}_{3}^{0}) j = 0, at x 1 = L / \sqrt{Rh}

(67a-67d)

The governing equations (59) to (61) associated with one set of the appropriate boundary conditions (equations (62) to (67)) represents a set of ( $3 n p + 8$ ) simultaneously algebraic equations in terms of $3 n p$ unknowns. The leading-order solutions of the mid-surface displacement components at each sampling node can then be obtained using a weighted least square method with the weight number 10,000 for the essential boundary conditions, 100 for the natural boundary conditions, and 1 for the governing equations, as suggested by Wang et al. [44]. As mentioned above, once the variables of $({\tilde{u}}_{1}^{0}) j, ({\tilde{u}}_{2}^{0}) j, and ({\tilde{u}}_{3}^{0}) j, j = 1 - n p$ , are determined, the leading-order solutions of displacements are given by equations (29) and (30), the transverse shear and normal stresses by equations (31) and (32), and the in-surface stresses by equation (21).

Higher order modifications

Carrying on the solution to order $\in 2 k (k = 1, 2, \dots)$ , we find that the nonhomogeneous terms ( $f ik (x 1, x 2, 1), i = 1 - 3$ ) and the revelant functions $φ ik (x 1, x 2, x 3)$ (i = 1–3) for fixed values of $\overset{\land}{m} and \overset{\land}{n}$ in the $\in 2 k$ -order equations are

f 1 k (x 1, x 2, 1) = \tilde{f} 1 k (x 1, 1) \cos \tilde{n} x 2

(68)

f 2 k (x 1, x 2, 1) = \tilde{f} 2 k (x 1, 1) \sin \tilde{n} x 2

(69)

f 3 k (x 1, x 2, 1) = \tilde{f} 3 k (x 1, 1) \cos \tilde{n} x 2

(70)

and ϕ 1 k (x 1, x 2, x 3) = \tilde{ϕ} 1 k (x 1, x 3) \cos \tilde{n} x 2

(71)

ϕ 2 k (x 1, x 2, x 3) = \tilde{ϕ} 2 k (x 1, x 3) \sin \tilde{n} x 2

(72)

ϕ 3 k (x 1, x 2, x 3) = \tilde{ϕ} 3 k (x 1, x 3) \cos \tilde{n} x 2

(73)

where,

\tilde{f} 1 k, \tilde{f} 2 k, \tilde{f} 3 k

and

\tilde{ϕ} 1 k, \tilde{ϕ} 2 k, \tilde{ϕ} 3 k

are the relevant coefficients.

In view of the recurrence of the equations, the $\in 2 k$ -order solution can be obtained by letting

u_{1}^{k} (x 1, x 2) = {\tilde{u}}_{1}^{k} (x 1) \cos \tilde{n} x 2

(74)

u_{2}^{k} (x 1, x 2) = {\tilde{u}}_{2}^{k} (x 1) \sin \tilde{n} x 2

(75)

u_{3}^{k} (x 1, x 2) = {\tilde{u}}_{3}^{k} (x 1) \cos \tilde{n} x 2

(76)

Substituting equations (68) to (76) into equations (43) to (45) gives

[k 11 k 12 k 13 k 21 k 22 k 23 k 31 k 32 k 33] {{\tilde{u}}_{1}^{k} {\tilde{u}}_{2}^{k} {\tilde{u}}_{3}^{k}} = {\tilde{f} 1 k (x 1, 1) \tilde{f} 2 k (x 1, 1) \tilde{f} 3 k (x 1, 1) + \tilde{f} 1 k (x 1, 1), 1 + \tilde{n} \tilde{f} 2 k (x 1, 1)} .

(77)

Appling the governing equations (equation (77)) to the ith-sampling node ( $x 1 = (x 1) i i = 1 - n p$ ) and using the DRK interpolation leads to

d 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{1}^{k}) j + d 12 ({\tilde{u}}_{1}^{k}) i + d 13 \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) ({\tilde{u}}_{2}^{k}) j + d 14 \sum_{j = 1}^{n p} ψ_{j}^{(3)} ((x 1) i) ({\tilde{u}}_{3}^{k}) j + d 15 \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) ({\tilde{u}}_{3}^{k}) j = \tilde{f} 1 k ((x 1) i, 1)

(78)

d 21 \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) ({\tilde{u}}_{1}^{k}) j + d 22 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{2}^{k}) j + d 23 ({\tilde{u}}_{2}^{k}) i + d 24 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{3}^{k}) j + d 25 ({\tilde{u}}_{3}^{k}) i = \tilde{f} 2 k ((x 1) i, 1)

(79)

d 31 \sum_{j = 1}^{n p} ψ_{j}^{(3)} ((x 1) i) ({\tilde{u}}_{1}^{k}) j + + d 32 \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) ({\tilde{u}}_{1}^{k}) j + d 33 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{2}^{k}) j + d 34 ({\tilde{u}}_{2}^{k}) i + d 35 \sum_{j = 1}^{n p} ψ_{j}^{(4)} ((x 1) i) ({\tilde{u}}_{3}^{k}) j + d 36 \sum_{j = 1}^{n p} ψ_{j}^{(2)} ((x 1) i) ({\tilde{u}}_{3}^{k}) j + + d 37 ({\tilde{u}}_{3}^{k}) i = \tilde{f} 3 k ((x 1) i, 1) + \sum_{j = 1}^{n p} ψ_{j}^{(1)} ((x 1) i) \tilde{f} 1 k ((x 1) j, 1) + \tilde{n} \tilde{f} 2 k ((x 1) i, 1)

(80)

in which i = 1 − n_p.

The appropriate edge conditions of Cases 1 to 3 for the $\in 2 k$ -order problem can be rewritten, as follows.

Case 1

For the simple–simple (S-S) supports

({\tilde{u}}_{2}^{k}) 1 = 0, ({\tilde{u}}_{3}^{k}) 1 = 0 \overset{\land}{A} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{1}^{k}) j + \tilde{n} \tilde{A} 12 ({\tilde{u}}_{2}^{k}) 1 - \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (0) ({\tilde{u}}_{3}^{k}) j + (\tilde{A} 12 + \tilde{n} 2 \tilde{B} 12) ({\tilde{u}}_{3}^{k}) 1 = - \int_{- 1}^{1} [Q 13 (σ_{3}^{(k - 1)}) 1] d x 3 - \int_{- 1}^{1} [\tilde{Q} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) (\tilde{ϕ} 1 k) j + \tilde{n} \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 2 k) 1 + \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 3 k) 1] d x 3 \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{1}^{k}) j + \tilde{n} \tilde{B} 12 ({\tilde{u}}_{2}^{k}) 1 - \overset{\land}{D} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (0) ({\tilde{u}}_{3}^{k}) j + (\tilde{B} 12 + \tilde{n} 2 \tilde{D} 12) ({\tilde{u}}_{3}^{k}) 1 = - \int_{- 1}^{1} [Q 13 x 3 (σ_{3}^{(k - 1)}) 1] d x 3 - \int_{- 1}^{1} (x 3) [\tilde{Q} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) (\tilde{ϕ} 1 k) j + \tilde{n} \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 2 k) 1 + \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 3 k) 1] d x 3, at x 1 = 0

(81a-81d)

({\tilde{u}}_{2}^{k}) n p = 0, ({\tilde{u}}_{3}^{k}) n p = 0 \overset{\land}{A} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) ({\tilde{u}}_{1}^{k}) j + \tilde{n} \tilde{A} 12 ({\tilde{u}}_{2}^{k}) n p - \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (L / \sqrt{Rh}) ({\tilde{u}}_{3}^{k}) j + (\tilde{A} 12 + \tilde{n} 2 \tilde{B} 12) ({\tilde{u}}_{3}^{k}) n p = - \int_{- 1}^{1} [Q 13 (σ_{3}^{(k - 1)}) n p] d x 3 - \int_{- 1}^{1} [\tilde{Q} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) (\tilde{ϕ} 1 k) j + \tilde{n} \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 2 k) n p + \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 3 k) n p] d x 3 \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) ({\tilde{u}}_{1}^{k}) j + \tilde{n} \tilde{B} 12 ({\tilde{u}}_{2}^{k}) n p - \overset{\land}{D} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (L / \sqrt{Rh}) ({\tilde{u}}_{3}^{k}) j + (\tilde{B} 12 + \tilde{n} 2 \tilde{D} 12) ({\tilde{u}}_{3}^{k}) n p = - \int_{- 1}^{1} [Q 13 x 3 (σ_{3}^{(k - 1)}) n p] d x 3 - \int_{- 1}^{1} (x 3) [\tilde{Q} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) (\tilde{ϕ} 1 k) j + \tilde{n} \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 2 k) n p + \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 3 k) n p] d x 3, at x 1 = L / \sqrt{Rh}

(82a-82d)

Case 2

For the simple–clamped (SC) supports

({\tilde{u}}_{2}^{k}) 1 = 0, ({\tilde{u}}_{3}^{k}) 1 = 0 \overset{\land}{A} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{1}^{k}) j + \tilde{n} \tilde{A} 12 ({\tilde{u}}_{2}^{k}) 1 - \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (0) ({\tilde{u}}_{3}^{k}) j + (\tilde{A} 12 + \tilde{n} 2 \tilde{B} 12) ({\tilde{u}}_{3}^{k}) 1 = - \int_{- 1}^{1} [Q 13 (σ_{3}^{(k - 1)}) 1] d x 3 - \int_{- 1}^{1} [\tilde{Q} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) (\tilde{ϕ} 1 k) j + \tilde{n} \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 2 k) 1 + \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 3 k) 1] d x 3 \overset{\land}{B} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{1}^{k}) j + \tilde{n} \tilde{B} 12 ({\tilde{u}}_{2}^{k}) 1 - \overset{\land}{D} 11 \sum_{j = 1}^{n p} ψ_{j}^{(2)} (0) ({\tilde{u}}_{3}^{k}) j + (\tilde{B} 12 + \tilde{n} 2 \tilde{D} 12) ({\tilde{u}}_{3}^{k}) 1 = - \int_{- 1}^{1} [Q 13 x 3 (σ_{3}^{(k - 1)}) 1] d x 3 - \int_{- 1}^{1} (x 3) [\tilde{Q} 11 \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) (\tilde{ϕ} 1 k) j + \tilde{n} \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 2 k) 1 + \tilde{Q} 12 λ_{θ}^{- 1} (\tilde{ϕ} 3 k) 1] d x 3, at x 1 = 0

(83a-83d)

({\tilde{u}}_{1}^{k}) n p = 0, ({\tilde{u}}_{2}^{k}) n p = 0, ({\tilde{u}}_{3}^{k}) n p = 0, \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) ({\tilde{u}}_{3}^{k}) j = 0, at x 1 = L / \sqrt{Rh}

(84a-84d)

Case 3

For the clamped–clamped (CC) supports

({\tilde{u}}_{1}^{k}) 1 = 0, ({\tilde{u}}_{2}^{k}) 1 = 0, ({\tilde{u}}_{3}^{k}) 1 = 0, \sum_{j = 1}^{n p} ψ_{j}^{(1)} (0) ({\tilde{u}}_{3}^{k}) j = 0, at x 1 = 0

(85a-85d)

({\tilde{u}}_{1}^{k}) n p = 0, ({\tilde{u}}_{2}^{k}) n p = 0, ({\tilde{u}}_{3}^{k}) n p = 0, \sum_{j = 1}^{n p} ψ_{j}^{(1)} (L / \sqrt{Rh}) ({\tilde{u}}_{3}^{k}) j = 0, at x 1 = L / \sqrt{Rh}

(86a-86d)

Again, the governing equations (78) to (80) associated with one set of the appropriate boundary conditions (equations (81) to (86)) represent a set of ( $3 n p + 8$ ) simultaneously algebraic equations in terms of $3 n p$ unknowns. The $\in 2 k$ -order modifications of the mid-surface displacement components at each sampling node can then be obtained using a weighted least square method, with the weight number 10,000 for the essential boundary conditions, 100 for the natural boundary conditions, and 1 for the governing equatons. Once the variables of $({\tilde{u}}_{1}^{k}) j, ({\tilde{u}}_{2}^{k}) j and ({\tilde{u}}_{3}^{k}) j, j = 1 - n p$ , are determined, the higher order modifications of displacement components are given by equations (39) and (40), the transverse stresses by equations (41) and (42), and the in-surface stresses by equation (26).

Illustrative examples

Laminated composite hollow cylinders

A benchmark problem with regard to the static behaviors of simply supported, multilayered orthotropic circular hollow cylinders under a sinusoidally distributed load applied at the inner surface (

{\bar{q}}_{r}^{+} = 0

and

{\bar{q}}_{r}^{-} = - q 0 \sin (π x / L) \cos 4 θ

) was investigated by Varadan and Bhaskar [55], and the benchmark solutions are used in this work to validate the accuracy and convergence of the asymptotic DRK-based meshless method in Tables 1 and 2, in which the material properties of the cylinders are given as

E L = 25 \times 106 psi (172.4 GPa), E T = 1.0 \times 106 psi (6.89 GPa) G LT = 0.5 \times 106 psi (3.45 GPa), G TT = 0.2 \times 106 psi (1.378 GPa), ε LL = ε LT = 0.25

(87a-87e)

where the subscripts of L and T denote the directions parallel and transverse to the fiber directions, respectively.

Table 1.

Convergence studies for asymptotic meshless DRK solutions of the stress and displacement components at the crucial positions of the $[90 \circ / 0 \circ / 90 \circ]$ laminated composite cylinder with fully simple supports and under a sinusoidally distributed load (L/R=4 and R/(2h)=10).

η	a	Theories	n_p	$\in 2 k$ order	$\bar{σ} x (\frac{L}{2}, 0, h)$	$\bar{σ} θ (\frac{L}{2}, 0, h)$	$\bar{σ} r (\frac{L}{2}, 0, - \frac{h}{3})$	$\bar{τ} θ r (\frac{L}{2}, \frac{π}{8}, - \frac{h}{3})$	$\bar{u} r (\frac{L}{2}, 0, 0)$
4	4.1Δx	Present	17	$\in 0$	0.04793	3.77170	−1.58833	−3.44525	0.48200
				$\in 2$	0.07283	4.77697	−1.50821	−3.34939	1.18270
				$\in 4$	0.07230	4.66031	−1.51426	−3.36662	1.21976
				$\in 6$	0.07258	4.67854	−1.51381	−3.36455	1.22448
			21	$\in 0$	0.04793	3.77170	−1.58833	−3.44525	0.48200
				$\in 2$	0.07284	4.77699	−1.50821	−3.34941	1.18270
				$\in 4$	0.07230	4.66033	−1.51426	−3.36664	1.21976
				$\in 6$	0.07258	4.67856	−1.51381	−3.36456	1.22449
			25	$\in 0$	0.04793	3.77171	−1.58833	−3.44525	0.48200
				$\in 2$	0.07284	4.77699	−1.50821	−3.34941	1.18270
				$\in 4$	0.07230	4.66034	−1.51426	−3.36665	1.21976
				$\in 6$	0.07258	4.67857	−1.51381	−3.36457	1.22449
4	4.6Δx	Present	17	$\in 0$	0.04793	3.77169	−1.58833	−3.44524	0.48200
				$\in 2$	0.07283	4.77694	−1.50820	−3.34936	1.18269
				$\in 4$	0.07229	4.66028	−1.51425	−3.36659	1.21975
				$\in 6$	0.07257	4.67851	−1.51380	−3.36452	1.22447
			21	$\in 0$	0.04793	3.77170	−1.58833	−3.44525	0.48200
				$\in 2$	0.07284	4.77697	−1.50821	−3.34939	1.18270
				$\in 4$	0.07230	4.66032	−1.51426	−3.36662	1.21976
				$\in 6$	0.07258	4.67854	−1.51381	−3.36455	1.22448
			25	$\in 0$	0.04793	3.77170	−1.58833	−3.44525	0.48200
				$\in 2$	0.07284	4.77698	−1.50821	−3.34940	1.18270
				$\in 4$	0.07230	4.66033	−1.51426	−3.36664	1.21976
				$\in 6$	0.07258	4.67856	−1.51381	−3.36456	1.22448
5	5.1Δx	Present	21	$\in 0$	0.04793	3.77169	−1.58833	−3.44524	0.48200
				$\in 2$	0.07284	4.77696	−1.50821	−3.34939	1.18270
				$\in 4$	0.07230	4.66031	−1.51426	−3.36662	1.21975
				$\in 6$	0.07258	4.67853	−1.51381	−3.36454	1.22448
			25	$\in 0$	0.04793	3.77170	−1.58833	−3.44525	0.48200
				$\in 2$	0.07284	4.77698	−1.50821	−3.34940	1.18270
				$\in 4$	0.07230	4.66032	−1.51426	−3.36663	1.21976
				$\in 6$	0.07258	4.67855	−1.51381	−3.36456	1.22448
			31	$\in 0$	0.04793	3.77170	−1.58833	−3.44525	0.48200
				$\in 2$	0.07284	4.77699	−1.50821	−3.34941	1.18270
				$\in 4$	0.07230	4.66034	−1.51426	−3.36664	1.21976
				$\in 6$	0.07258	4.67857	−1.51381	−3.36457	1.22449
6	6.1Δx	Present	25	$\in 0$	0.04793	3.77171	−1.58833	−3.44526	0.48200
				$\in 2$	0.07284	4.77701	−1.50821	−3.34942	1.18271
				$\in 4$	0.07230	4.66036	−1.51427	−3.36666	1.21977
				$\in 6$	0.07258	4.67859	−1.51381	−3.36458	1.22449
			31	$\in 0$	0.04793	3.77171	−1.58833	−3.44526	0.48200
				$\in 2$	0.07284	4.77701	−1.50821	−3.34942	1.18271
				$\in 4$	0.07230	4.66036	−1.51427	−3.36666	1.21977
				$\in 6$	0.07258	4.67859	−1.51381	−3.36458	1.22449
			37	$\in 0$	0.04793	3.77171	−1.58833	−3.44526	0.48200
				$\in 2$	0.07284	4.77701	−1.50821	−3.34942	1.18271
				$\in 4$	0.07230	4.66036	−1.51427	−3.36666	1.21977
				$\in 6$	0.07258	4.67859	−1.51381	−3.36458	1.22449
Exact 3D elasticity method					0.0739	4.683	NA	NA	1.223
Modified Pagano method					0.07392	4.68272	−1.51082	−3.34769	1.22329
State space DRK method (n=4 and n_p=21)					0.07387	4.68342	−1.51073	−3.3474	1.22359
Finite prism method (Q9, 48×12)					0.07379	4.68268	−1.51598	−3.36169	1.22329

DRK: differential reproducing kernel.

Table 2.

The asymptotic meshless DRK solutions of the stress and displacement components at the crucial positions of $[90 \circ / 0 \circ / 90 \circ / 0 \circ / 90 \circ] s$ laminated composite cylinders with fully simple supports and under a sinusoidally distributed load (L/R=4).

S (R/(2h))	Theories	$\bar{σ} x (\frac{L}{2}, 0, h)$	$\bar{σ} θ (\frac{L}{2}, 0, h)$	$\bar{σ} r (\frac{L}{2}, 0, 0)$	$\bar{τ} θ r (\frac{L}{2}, \frac{π}{8}, 0)$	$\bar{u} r (\frac{L}{2}, 0, 0)$
5	Present $\in 0$	0.0641	5.2829	−0.9178	−3.4713	0.6990
	Present $\in 2$	0.1131	6.6126	−0.8045	−3.2204	2.9974
	Present $\in 4$	0.1079	6.4180	−0.8355	−3.2736	3.0597
	Present $\in 6$	0.1071	6.2569	−0.8100	−3.2481	3.0678
Exact 3D elasticity method		NA	NA	NA	NA	NA
Modified Pagano method		0.1106	6.3588	−0.8259	−3.2624	3.0561
State space DRK method (n=4 and η_p₌₂₁)		0.1106	6.3588	−0.8222	−3.2597	3.0553
Finite prism method (Q9, 48×12)		NA	NA	NA	NA	NA
10	Present $\in 0$	0.0688	5.4430	−1.3573	−3.4677	0.6994
	Present $\in 2$	0.0854	5.8666	−1.3203	−3.4744	1.3205
	Present $\in 4$	0.0865	5.8821	−1.3235	−3.4748	1.3776
	Present $\in 6$	0.0865	5.8758	−1.3230	−3.4753	1.3812
Exact 3D elasticity method		0.0877	5.8750	−1.32	−3.479	1.38
Modified Pagano method		0.0876	5.8750	−1.3275	−3.4788	1.3800
State space DRK method (n=4 and n_p=21)		0.0876	5.8749	−1.3237	−3.4789	1.3800
Finite prism method (Q9, 48×12)		0.0876	5.8750	−1.32	−3.478	1.38
50	Present $\in 0$	0.0890	5.2867	−4.5853	−3.2773	0.6614
	Present $\in 2$	0.0962	5.5154	−4.7525	−3.4176	0.7527
	Present $\in 4$	0.0968	5.5260	−4.7600	−3.4236	0.7611
	Present $\in 6$	0.0968	5.5263	−4.7602	−3.4238	0.7617
Exact 3D elasticity method		0.0971	5.529	−4.76	−3.425	0.7622
Modified Pagano method		0.0971	0.5285	−4.7620	−3.4253	0.7622
State space DRK method (n=4 and n_p=21)		0.0970	0.5279	−4.7620	−3.4250	0.7622
Finite prism method (Q9, 48×12)		0.0971	5.528	−4.76	−3.425	0.7622
100	Present $\in 0$	0.1000	4.5363	−7.4716	−2.7956	0.5642
	Present $\in 2$	0.1070	4.6748	−7.6841	−2.8825	0.6215
	Present $\in 4$	0.1074	4.6759	−7.6856	−2.8831	0.6255
	Present $\in 6$	0.1074	4.6756	−7.6850	−2.8828	0.6257
Exact 3D elasticity method		0.1076	4.677	−7.69	−2.884	0.6261
Modified Pagano method		0.1075	4.6770	−7.6870	−2.8840	0.6261
State space DRK method (n=4 and n_p=21)		0.1075	4.6756	−7.6850	−2.8829	0.6259
Finite prism method (Q9, 48×12)		0.1076	4.677	−7.69	−2.884	0.6261
500	Present $\in 0$	0.0512	0.8145	−6.6124	−0.4902	0.0989
	Present $\in 2$	0.0516	0.7779	−6.3296	−0.4676	0.1006
	Present $\in 4$	0.0516	0.7769	−6.3217	−0.4669	0.1005
	Present $\in 6$	0.0516	0.7769	−6.3217	−0.4669	0.1005
Exact 3D elasticity method		0.0516	0.777	−6.32	−0.467	0.1006
Modified Pagano method		0.0516	0.7768	−6.3210	−0.4669	0.1006
State space DRK method (n=4 and n_p=21)		0.0515	0.7760	−6.3140	−0.4663	0.1004
Finite prism method (Q9, 48×12)		0.0516	0.777	−6.32	−0.467	0.1006

DRK: differential reproducing kernel.

For comparison purposes, the set of normalized variables used in Varadan and Bhaskar [55] is defined as

[\bar{u} x \bar{u} θ] = \frac{10 E L}{q 0 (2 h) S 3} [u x u θ], \bar{u} r = \frac{10 E L}{q 0 (2 h) S 4} u r [\bar{σ} x \bar{σ} θ \bar{τ} x θ] = \frac{10}{q 0 S 2} [σ x σ θ τ x θ] [\bar{τ} x r \bar{τ} θ r] = \frac{10}{q 0 S} [τ x r τ θ r], \bar{σ} r = σ r / q 0, S = R / (2 h)

(88a-88e)

Table 1 shows the convergence studies for the asymptotic DRK solutions of the stress and displacement components induced at the crucial positions of the simply-supported, $[90 \circ / 0 \circ / 90 \circ]$ laminated composite cylinders, in which L/R = 4 and R/(2h) = 10. In the table, a uniform distribution of nodes (n_p) along the axial direction of the mid-surface of the cylinder is adopted, n_p = 17, 21, 25, 31, and 37, the highest order of the base functions (n) is n = 4, 5, and 6, and the radius of the influence zone for each sampling node (a) is a = 4.1 $Δ x$ and 4.6 $Δ x$ in the cases of n = 4, a = 5.1 $Δ x$ in the cases of n = 5, and a = 6.1 $Δ x$ in the cases of n = 6, in which $Δ x = L / (n p - 1)$ . It can be seen in Table 1 that the asymptotic DRK solutions converge rapidly, and the solutions obtained using (n = 6, a = 6.1 $Δ x$ ) are slightly more accurate than those using (n = 4, a = 4.1 $Δ x$ ) and (n = 5, a = 5.1 $Δ x$ ). The convergent solutions are yielded at the $\in 4$ -order for this moderately thick cylinder when (n = 4, a = 4.1 $Δ x$ ), (n = 5, a = 5.1 $Δ x$ ) and (n = 6, a = 6.1 $Δ x$ ) are used, and these are in excellent agreement with the exact 3D solutions [55], modified Pagano solutions [9], state space DRK method [53], and FPM solutions [41].

Table 2 shows the asymptotic DRK solutions of the displacement and stress components induced at the crucial positions of simply-supported, laminated $[90 \circ / 0 \circ / 90 \circ / 0 \circ / 90 \circ] s$ composite cylinders with different mid-surface radius-to-thickness ratios S, S = R/(2h), and under the sinusoidally distributed load mentioned above, in which n_p = 21, n = 4, a = 4.1 $Δ x$ , L/R = 4, and S = 5, 10, 50, 100, and 500. It is seen in Table 2 that convergence of the asymptotic DRK methods occurs the $\in 4$ -order for the moderately thick cylinders (S = 10), and $\in 2$ -order for the thin cylinders (S>50). These convergent solutions are also shown to be in excellent agreement with the exact 3D [55], modified Pagano [9], state space DRK [53], and FPM solutions [41]. The convergence rate for the very thick cylinders (S = 5) is, however, slower than those for the moderately thick (S = 10) and thin (S>50) cylinders. The relative errors between the present $\in 6$ -oder solutions and the solutions obtained using the approximate and exact 3D methods are less than 2% and 3% for the primary variables (i.e. the displacement components and the transverse stress ones) and the secondary ones (i.e. in-surface stress components), respectively. If one wishes to have more accurate solutions than those shown in Table 2 for the very thick cylinders, one or two more orders modifications need to be performed, although the very thick cylinders are not commonly used in practical industries.

Table 3 shows the asymptotic DRK solutions of the displacement and stress components induced at a particular position (L/2, 0, 0) on [0°/90°/0°] laminated cylinders with three different edges (SS, SC, and CC edges) and under a uniformly distributed load (i.e.

{\bar{q}}_{r}^{+} = 0

and

{\bar{q}}_{r}^{-} = - q 0

), in which L/R = 5, R/2h = 10, n = 4,

a = 4.1 Δ x

, and n_p = 21. It can be seen in Table 3 that the convergence rate of these solutions for different edge conditions is SS>SC>CC, in which “>” represents faster, even though the deviations among them are insignificant. The convergent solutions were obtained at the

\in 4

-order, and these were compared with those obtained using the modified Pagano method [9], state space DRK method [53], and ANSYS commercial software for comparison purposes, in which a 20-node cylindrical brick element with (25×8×9), (50×16×9), and (100×32×9) meshes for the ANSYS analysis was used. The results were shown to be in excellent agreement with the accurate solutions obtained using the above-mentioned methods.

Table 3.

The asymptotic DRK solutions of the displacement and in- and out-of-surface stress components of $[00 / 900 / 00]$ laminated composite cylinders with various boundary conditions and under a uniformly distributed load (L/R=5, R/(2h)=10).

Variables	Theories	SS	SC	CC
$\bar{u} r (L / 2, 0, 0)$	Present $\in 0$	0.27717	0.27717	0.27706
	Present $\in 2$	0.26330	0.26331	0.26320
	Present $\in 4$	0.26282	0.26283	0.26274
	Present $\in 6$	0.26276	0.26271	0.26255
	State space DRK method (N_l=9, n_p=41)	0.26642	0.26641	0.26635
	ANSYS (1/4 model, 25 × 8 × 9)	0.26641	0.26639	0.26634
	ANSYS (1/4 model, 50 × 16 × 9)	0.26643	0.26643	0.26638
	ANSYS (1/4 model, 100 × 32 × 9)	0.26643	0.26643	0.26641
	Modified Pagano method	0.26250	NA	NA
$\bar{σ} x (L / 2, 0, 0)$	Present $\in 0$	0.02619	0.02617	0.02777
	Present $\in 2$	0.01235	0.01233	0.01384
	Present $\in 4$	0.01291	0.01291	0.01444
	Present $\in 6$	0.01302	0.01296	0.01437
	State space DRK method (N_l=9, n_p=41)	0.01398	0.01398	0.01468
	ANSYS (1/4 model, 25 × 8 × 9)	0.01403	0.01403	0.01472
	ANSYS (1/4 model, 50 × 16 × 9)	0.01395	0.01395	0.01458
	ANSYS (1/4 model, 100 × 32 × 9)	0.01398	0.01398	0.01464
	Modified Pagano method	0.01399	NA	NA
$\bar{σ} r (L / 2, 0, 0)$	Present $\in 0$	−0.49490	−0.49490	−0.49490
	Present $\in 2$	−0.47016	−0.47016	−0.47015
	Present $\in 4$	−0.46801	−0.46800	−0.46800
	Present $\in 6$	−0.46819	−0.46819	−0.46820
	State space DRK method (N_l=9, n_p=41)	−0.47174	−0.47174	−0.47173
	ANSYS (1/4 model, 25 × 8 × 9)	−0.41321	−0.41322	−0.42298
	ANSYS (1/4 model, 50 × 16 × 9)	−0.45491	−0.45477	−0.45276
	ANSYS (1/4 model, 100 × 32 × 9)	−0.46647	−0.46647	−0.46647
	Modified Pagano method	−0.47295	NA	NA

DRK: differential reproducing kernel.

FGM sandwich hollow cylinders

The static behaviors of FGM sandwich circular hollow cylinders, consisting of two homogeneous face-sheets and an embedded FGM core, with combinations of simply-supported and clamped edges and under either a sinusoidally or a uniformly distributed load (i.e. either ${\bar{q}}_{r}^{+} = 0$ , ${\bar{q}}_{r}^{-} = - q 0 \sin (π x / L) \cos 4 θ$ or ${\bar{q}}_{r}^{+} = 0$ , ${\bar{q}}_{r}^{-} = - q 0$ ) are investigated. The configuration, cylindrical coordinate systems and layer-up sequence of the cylinder are shown in Figure 1(a) and (b). The thickness ratio for each layer of the sandwich cylinder is $h 1 : h 2 : h 3$ , while $\sum_{m = 1}^{3} h m = 2 h$ , and the Young’s modulus of the material at the mid-surface of the core-layer and that of the face-sheets are denoted as E₀ and E_f, respectively, for which E₀ = 70 GPa (aluminum) and E_f = 380 GPa (alumina) are used in this example, Poisson’s ratios $ε (m)$ (m = 1^_3) are taken to be 0.3, and the corresponding bulk moduli (K) and shear ones (G) are K₀ = 58.333 GPa, K_f = 316.666 GPa, G₀ = 26.923 GPa, G_f = 146.154 GPa.

The effective material properties of the FGM core are evaluated using the Mori–Tanaka scheme [52], and are written as follows

K (ζ) = \frac{V f (ζ) (K f - K 0)}{[1 + (1 - V f (ζ)) \frac{K f - K 0}{K 0 + (4 / 3) G 0}]} + K 0

(89a)

G (ζ) = \frac{V f (ζ) (G f - G 0)}{[1 + (1 - V f (ζ)) \frac{G f - G 0}{G 0 + f 0}]} + G 0

(89b)

In which $f 0 = G 0 \frac{9 K 0 + 8 G 0}{6 (K 0 + 2 G 0)}$ . $V f (ζ)$ are the volume fractions of the face sheet material of the cylinder, and are given by

V f (ζ) = [| ζ | / (h 2 / 2)] κ p when (- h 2 / 2) ⩽ ζ ⩽ (h 2 / 2)

(90)

It is apparent that when k_p = 0, $V f (ζ)$ = 1, this FGM sandwich cylinder is reduced to a single-layered homogeneous cylinder with material properties E_f = 380 GPa and $ε f$ = 0.3; while when k_p = ∞, $V f (ζ)$ = 0, it is reduced to a homogeneous sandwich cylinder with material properties $E (1) = E (3)$ = 380 GPa, $E (2)$ = 70 GPa, and $ε (m)$ = 0.3 (m = 1^_3). In addition, the set of dimensionless field variables, used in Example 8.1 and given in equation (88), is also used in this example, except that E_L is replaced by E₀.

Figure 2 shows the asymptotic DRK solutions for the through-thickness distributions of assorted variables induced at the sections, $(x, θ) = (L / 5, 0) and (4 L / 5, 0)$ , of a simply supported, FGM sandwich cylinder subjected to the mechanical loads, ${\bar{q}}_{r}^{+} = 0$ and ${\bar{q}}_{r}^{-} = - q 0 \sin (π x / L) \cos 4 θ$ , in which L/R = 5, R/(2h) = 5, $h 1 : h 2 : h 3 = 0.2 h : 1.6 h : 0.2 h$ , 2h = 0.1m, $κ p = 3$ , n = 4, $a = 4.1 Δ x$ and n_p = 21. Again, it is shown in Figure 2 that the current $\in 4$ -order solutions are in excellent agreement with the exact 3D elasticity solutions obtained using the modified Pagano method [9]. The asymptotic DRK solutions for the through-thickness distributions of various variables at these two positions, $(x, θ) = (L / 5, 0) and (4 L / 5, 0)$ , are symmetric with each other, because the configuration and loading conditions of the cylinder are symmetric with respect to the position of $(x, θ) = (L / 2, 0)$ .

Figure 2.

Convergence studies for the asymptotic DRK solutions of the through-thickness distributions of various field variables at the (x = L/5, $θ = 0$ ) and (x = 4L/5, $θ = 0$ ) sections of a simply supported, FGM sandwich cylinder with $κ p$ =3.

Figure 3 shows the $\in 4$ -order DRK solutions for the through-thickness distributions of various variables induced at the section, $(x, θ) = (L / 2, 0)$ , of the FGM sandwich cylinder with SS, SC, and CC edge conditions and under a uniformly distributed load, ${\bar{q}}_{r}^{+} = 0$ and ${\bar{q}}_{r}^{-} = - q 0$ , in which L/R = 10, R/(2h) = 10, $h 1 : h 2 : h 3 = 0.2 h : 1.6 h : 0.2 h$ , 2h = 0.1m, $κ p = 3$ , n = 4, $a = 4.1 Δ x$ , and n_p = 21. It can be seen in Figure 4 that the effect of different boundary conditions on the variables, $\bar{u} r (ζ)$ and $\bar{σ} x (ζ)$ , is more significant than the variable $\bar{σ} r (ζ)$ . The magnitudes of $\bar{u} r (ζ)$ and $\bar{σ} x (ζ)$ for different edge conditions are SS>SC>CC and CC>SC>SS, respectively, and the deviation among the solutions of $\bar{σ} r (ζ)$ for the cases of SS, SC, and CC edge conditions is very insignificant, and cannot be distinguished in the figure.

Figure 3.

The $\in 4$ -order DRK solutions for the through-thickness distributions of various field variables at the (x=L/2, $θ = 0$ ) section of an FGM sandwich cylinder with SS, SC, and CC supports.

Figure 4.

The $\in 4$ -order DRK solutions for the through-thickness distributions of various field variables at the (x=L/4, $θ = 0$ ) or (x=L/4, $θ = π / 8$ ) section of a simply supported, FGM sandwich cylinder with different material-property gradient index, in which $κ p$ =∞, 5, and 0.

Figure 4 shows the $\in 4$ -order DRK solutions for the through-thickness distributions of various variables induced at the section, $(x, θ) = (L / 4, 0)$ , of simply supported, FGM sandwich cylinders with different material-property gradient indices, $κ p$ , and under a sinusoidally distributed load, ${\bar{q}}_{r}^{+} = 0$ and ${\bar{q}}_{r}^{-} = - q 0 \sin (π x / L) \cos 4 θ$ , in which L/R = 5, R/(2h) = 10, $h 1 : h 2 : h 3 = 0.2 h : 1.6 h : 0.2 h$ , 2h = 0.1m, $κ p = \infty$ , 5 and 0, n = 4, $a = 4.1 Δ x$ , and n_p = 21. As mentioned above, when k_p = 0 and ∞, the FGM cylinder is reduced to a single-layered homogeneous cylinder and a homogeneous sandwich one, respectively. It is shown that the distributions of in- and out-of surface variables appear to be globally linear and globally higher order polynomial variations through the thickness coordinate, respectively, for the single-layered homogeneous cylinder (i.e. k_p = 0), while they appear to be layerwise linear and layerwise higher order polynomial variations, respectively, for the homogeneous sandwich cylinders (i.e. k_p = ∞). As for the FGM sandwich cylinders (k_p = 5), the through-thickness distributions of assorted field variables induced in the FGM cylinders differ significantly from those induced in both the homogeneous single-layered and sandwich cylinders. Moreover, the in-surface stresses are continuously distributed through the interfaces between the face-sheet and core layers for FGM sandwich cylinders (k_p = 5), but change abruptly for the homogeneous sandwich cylinders (k_p = ∞), and the transverse shear stresses induced at the interfaces between the face-sheet and core layers for FGM sandwich cylinders are reduced in comparison with those for homogeneous sandwich cylinders, and this can prevent the delamination failure that often occurs at the interfaces between adjacent layers of homogeneous sandwich cylinders.

Conclusions

On the basis of the RMVT, in this article we first developed an asymptotic DRK-based meshless method for the 3D bending analysis of FGM sandwich circular hollow cylinders and laminated composite ones with combinations of simply-supported and clamped edges, which are relatively few in the open literature. It is shown in the examples that in the cases of moderately and thin cylinders these asymptotic DRK solutions converge rapidly, and are in excellent agreement with the 3D exact solutions of simply-supported, laminated composite cylinders and FGM ones available in the literature, and those for the cases of SC and CC edge conditions can be used as a reference for assessing the solutions obtained using other 2D and 3D methodologies and numerical techniques. The convergence rate decreases when the cylinder becomes thick, while in the cases of thick cylinders, which are seldom used in the practical industries, the relative errors between the sixth-order asymptotic solutions and the available exact 3D ones are still less than 3%. It is also seen in the examples that the through-thickness distributions of the in- and out-of-surface variables of the FGM sandwich cylinders appear to have layerwise higher order polynomial variations. Moreover, the transverse shear stresses induced at the interfaces between the face-sheet and core layers for sandwich FGM cylinders are reduced in comparison with those for homogeneous sandwich cylinders, which may prevent the delamination failure that often occurs at the interfaces between adjacent layers of homogeneous sandwich cylinders.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Science and Technology of Republic of China through Grant MOST 103-2221-E-006-064-MY3.

Appendix 1

Appendix 2

Appendix 3

Appendix 4

Appendix 5

Appendix 6

References

Chiu

Wang

. A review on the three-dimensional analytical approaches of multilayered and functionally graded piezoelectric plates and shells. CMC-Comput Mater Con 2008; 8: 93–132.

Brischetto

. Exact elasticity solution for natural frequencies of functionally graded simply-supported structures. CMES-Comput Model Eng Sci 2013; 95: 391–430.

Chen

Ding

Liang

. The exact elasto-electric field of a rotating piezoelectric spherical shell with a functionally graded property. Int J Solids Struct 2001; 38: 7015–7027.

Han

Pan

Roy

. Responses of piezoelectric, transversely isotropic, functionally graded, and multilayered half spaces to uniform circular surface loadings. CMES-Comput Model Eng Sci 2006; 14: 15–29.

Pan

. Exact solution for functionally graded anisotropic elastic composite laminates. J Compos Mater 2003; 37: 1903–1920.

Dumir

Dube

Kapuria

. Exact piezoelectric solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending. Int J Solids Struct 1997; 34: 685–702.

Heyliger

. A note on the static behavior of simply-supported laminated piezoelectric cylinders. Int J Solids Struct 1997; 34: 3781–3794.

Chen

. Exact solutions of free vibration of simply-supported functionally graded piezoelectric sandwich cylinders using a modified Pagano method. J Sandwich Struct Mater 2013; 15: 229–257.

Tsai

. Exact solutions of functionally graded piezoelectric material sandwich cylinders by a modified Pagano method. Appl Math Model 2012; 36: 1910–1930.

10.

Chiu

. Thermoelastic buckling of laminated composite conical shells. J Therm Stress 2001; 24: 881–901.

11.

Chiu

. Thermally induced dynamic instability of laminated composite conical shells. Int J Solids Struct 2002; 39: 3001–3021.

12.

Carrera

. An assessment of mixed and classical theories on global and local response of multilayered orthotropic plates. Compos Struct 2000; 50: 183–198.

13.

Carrera

. An assessment of mixed and classical theories for the thermal stress analysis of orthotropic multilayered plates. J Therm Stress 2000; 23: 797–831.

14.

Carrera

. Developments, ideas, and evaluations based upon Reissner’s mixed variational theorem in the modeling of multilayered plates and shells. Appl Mech Rev 2001; 54: 301–329.

15.

Carrera

. Historical review of zig-zag theories for multilayered plates and shells. Appl Mech Rev 2003; 56: 287–308.

16.

Carrera

Brischetto

. Reissner mixed theorem applied to static analysis of piezoelectric shells. J Intell Mater Syst Struct 2007; 18: 1083–1107.

17.

Zenkour

. The effect of transverse shear and normal deformations on the thermomechanical bending of functionally graded sandwich plates. Int J Appl Mech 2009; 1: 667–707.

18.

Zenkour

. Bending analysis of functionally graded sandwich plates using a simple four-unknown shear and normal deformations theory. J Sandwich Struct Mater 2013; 15: 629–656.

19.

Zenkour

Alghamdi

. Thermomechanical bending response of functionally graded nonsymmetric sandwich plates. J Sandwich Struct Mater 2010; 12: 7–46.

20.

Lurlaro

Gherlone

Sciuva

. Bending and free vibration analysis of functionally graded sandwich plates using the refined zigzag theory. J Sandwich Struct Mater 2014. DOI: 10.1177/1099636214548618.

21.

Bourada

Tounsi

Houari

MSA

. A new four-variable refined plate theory for thermal buckling analysis of functionally graded sandwich plates. J Sandwich Struct Mater 2012; 14: 5–33.

22.

Sadighi

Benvidi

Eslami

. Improvement of thermo-mechanical properties of transversely flexible sandwich panels by functionally graded skins. J Sandwich Struct Mater 2011; 13: 539–577.

23.

Bessaim

Houari

MSA

Tounsi

. A new higher-order shear and normal deformation theory for the static and free vibration analysis of sandwich plates with functionally graded isotropic face sheets. J Sandwich Struct Mater 2013; 15: 671–703.

24.

Meziane

MAA

Abdelaziz

Tounsi

. An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions. J Sandwich Struct Mater 2014; 16: 293–318.

25.

Ramirez

Heyliger

Pan

. Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach. Compos Part B: Eng 2006; 37: 10–20.

26.

Ramirez

Heyliger

Pan

. Discrete layer solution to free vibrations of functionally graded magneto-electro-elastic plates. Mech Adv Mater Struct 2006; 13: 249–266.

27.

Zhou

FTK

Cheung

. Three-dimensional vibration analysis of circular and annular plates via the Chebyshev-Ritz method. Int J Solids Struct 2003; 40: 3089–3105.

28.

Zhou

Cheung

FTK

. Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method. Int J Solids Struct 2002; 39: 6339–6353.

29.

Dong

. Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev-Ritz method. Mater Des 2008; 29: 1518–1525.

30.

Uymaz

Aydodgu

. Three-dimensional vibration analyses of functionally graded plates under various boundary conditions. J Reinf Plast Compos 2007; 26: 1847–1864.

31.

Bert

Malik

. Differential quadrature method in computational mechanics: A review. Appl Mech Rev 1996; 49: 1–27.

32.

Lim

Lin

. Application of generalized differential quadrature method to structural problems. Int J Numer Meth Eng 1994; 37: 1881–1896.

33.

Liew

Teo

. Modeling via differential quadrature method: three-dimensional solutions for rectangular plates. Comput Meth Appl Mech Eng 1998; 159: 369–381.

34.

Liew

Teo

Han

. Three-dimensional static solutions of rectangular plates by variant differential quadrature method. Int J Mech Sci 2001; 43: 1611–1628.

35.

Liew

Zhang

. Differential quadrature-layerwise modeling technique for three-dimensional analysis of cross-ply laminated plates of various edge-supports. Comput Meth Appl Mech Eng 2002; 191: 3811–3832.

36.

Lü

Chen

Shao

. Semi-analytical three-dimensional elasticity solutions for generally laminated composite plates. Eur J Mech A/Solids 2008; 27: 899–917.

37.

Lü

Huang

Chen

. Semi-analytical solutions for free vibration of anisotropic laminated plates in cylindrical bending. J Sound Vib 2007; 304: 987–995.

38.

Lü

Lim

Chen

. Semi-analytical analysis for multi-directional functionally graded plates: 3D elasticity solutions. Int J Numer Meth Eng 2009; 79: 25–44.

39.

Reissner

. On a certain mixed variational theory and a proposed application. Int J Numer Meth Eng 1984; 20: 1366–1368.

40.

Reissner

. On a mixed variational theorem and on a shear deformable plate theory. Int J Numer Meth Eng 1986; 23: 193–198.

41.

. RMVT-based finite cylindrical prism methods for multilayered functionally graded circular hollow cylinders with various boundary conditions. Compos Struct 2013; 100: 592–608.

42.

. An RMVT-based finite rectangular prism method for the 3D analysis of sandwich FGM plates with various boundary conditions. CMC-Comput Mater Con 2013; 34: 27–62.

43.

Chen

Wang

. A Hermite DRK interpolation-based collocation method for the analysis of Bernoulli-Euler beams and Kirchhoff-Love plates. Comput Mech 2011; 47: 425–453.

44.

Wang

Chen

. A meshless collocation method based on the differential reproducing kernel interpolation. Comput Mech 2010; 45: 585–606.

45.

Ferreira

AJM

Roque

CMC

Martins

PALS

. Radial basis functions and higher-order shear deformation theories in the analysis of laminated composite beams and plates. Compos Struct 2004; 66: 287–293.

46.

Ferreira

AJM

Carrera

Cinefra

. Analysis of laminated doubly-curved shells by a layerwise theory and radial basis functions collocation, accounting for through-the-thickness deformations. Comput Mech 2011; 48: 13–25.

47.

Sladek

Atluri

. Meshless local Petro-Galerkin method in anisotropic elasticity. CMES-Comput Model Eng Sci 2004; 6: 477–489.

48.

Sladek

Zhang

. Static and dynamic analysis of shallow shells with functionally graded and orthotropic material properties. Mech Adv Mater Struct 2008; 15: 142–156.

49.

Sladek

Stanak

. Analysis of the bending of circular piezoelectric plates with functionally graded material properties by a MLPG method. Eng Struct 2013; 47: 81–89.

50.

Zhang

Lei

Liew

. Large deflection geometrically nonlinear analysis of carbon nanotube-reinforced functionally graded cylindrical panels. Comput Meth Appl Mech Eng 2014; 273: 1–18.

51.

Liew

Lei

. Postbuckling of carbon nanotube-reinforced functionally graded cylindrical panels under axial compression using a meshless approach. Comput Meth Appl Mech Eng 2014; 268: 1–17.

52.

Mori

Tanaka

. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 1973; 21: 571–574.

53.

Jiang

. A state space differential reproducing kernel method for the 3D analysis of FGM sandwich circular hollow cylinders with combinations of simply-supported and clamped edges. Compos Struct 2012; 94: 3401–3420.

54.

Jiang

. A state space differential reproducing kernel method for the buckling analysis of carbon nanotube-reinforced composite circular hollow cylinders. CMES-Comput Model Eng Sci 2014; 97: 239–279.

55.

Varadan

Bhaskar

. Bending of laminated orthotropic cylindrical shells-an elasticity approach. Compos Struct 1991; 17: 141–156.