Nonlinear thermomechanical buckling of sandwich FGM oblique stiffened plates with nonlinear effect of elastic foundation

Abstract

In this article, the nonlinear thermomechanical buckling behaviors of sandwich functionally graded plates subjected to an axial compression and external pressure are analytically analyzed resting on nonlinear elastic foundation. Assuming that the plates are reinforced by oblique stiffeners and rested on nonlinear elastic foundation. The formulations are established using the higher-order shear deformation theory taking into account the geometrical nonlinearity of von Kármán. The Lekhnitskii’s smeared stiffener technique is developed for shear deformable oblique stiffener system using the coordinate transformation technique with both mechanical and thermal terms. The Galerkin method is utilized to obtain the nonlinear algebraically equation system, then, solve it to determine the explicit expressions of critical buckling loads and postbuckling load–deflection curves. Numerical results show the effects of temperature, nonlinear elastic foundation, stiffeners, and material and geometrical properties on nonlinear behaviors of plates.

Keywords

Oblique stiffeners nonlinear buckling sandwich functionally graded material higher-order shear deformation theory thermo-mechanical nonlinear elastic foundation

Introduction

Plates are the fundamental structural elements of engineering structures, such as aircraft, spacecraft, civil structures, and so on. Along with the appearance of advanced materials, the research and applications of composite materials have increased during the past decades.

The ultimate strength limit obtained through full nonlinear transient analysis and strengthening effects of stiffeners on regular and arbitrarily stiffened plates subjected to uniaxial stress and biaxial stress were discussed by Liu and Wang.¹ Breslavsky et al.² studied the static deflection, free and forced nonlinear vibration behavior of plates made from two types of hyperelastic materials, namely, rubber and soft biological tissues. Duc and Thu³ investigated the nonlinear buckling response of imperfect laminated three-phase polymer composite plates based on the classical plate theory with the effect of uniform temperature change. Huang et al.⁴ reported a finite-element model to predict the buckling behavior of stiffened laminated composite plates with a curved beam element for stiffeners compatible with the shell element. Bikakis et al.⁵ studied the linear buckling behavior of simply supported and clamped fiber-metal laminated plates under axial compressive loads using finite-element method and eigenvalue buckling analysis. Khazaeinejad et al.⁶ proposed a simple mathematical approach to determine the nonlinear response of unrestrained or restrained against lateral translation plates with a thickness-dependent temperature field and subjected to the transverse mechanical loads. A finite-element investigation of buckling behavior of stiffened and unstiffened composite plates with three types of composite materials based on the ANSYS 14.0 APDL software was studied by Kumar et al.⁷ Anish et al.⁸ discussed the effects of openings and additional mass on the linear free vibration behavior of higher-order shear deformable laminated sandwich rhombic plates.

Functionally graded material (FGM) and functionally graded carbon nanotubes reinforced composite (FG-CNTRC) are the advanced composite materials, which have the thermomechanical properties varying continuously from one surface to the other. FGM and FG-CNTRC plates have been widely utilized in many advanced engineering designs because of superior properties in thermomechanical behavior.^9–11

Zenkour^12,13 studied the static and vibration behavior of nonsymmetric sandwich FGM plates using the sinusoidal shear deformation plate theory. The buckling and postbuckling of rectangular, annular, and circular FGM plates under thermal and mechanical loads were reported.^14–19 Effects of elastic foundation interaction, shear deformation, and geometrical imperfection were numerically investigated. Using the two-step perturbation technique, Shen et al.^20,21 studied the postbuckling behavior of FGM and sandwich FGM plates subjected to thermal and mechanical loads. Sobhy²² proposed a four-variable shear deformation plate theory to investigate the hygrothermal vibration and buckling behavior of sandwich FGM plates on Pasternak’s elastic foundation. By developing the higher-order shear deformation theory (HSDT) and normal deformable plate theories, Mohammadi et al.²³ studied the bending, buckling, and free vibration responses of incompressible thick FGM plates. Using the quasi-3D shear deformation theories, Zarga et al.²⁴ and Mahmoudi et al.²⁵ discussed the thermomechanical behavior of sandwich FGM plates with and without the Pasternak’s elastic foundation. Effect of hygrothermal environment on the dynamic stability behavior of sandwich FGM plates was investigated by Hellal et al.²⁶ Kumar²⁷ presented the thermomechanical postbuckling behavior of FGM plates with random material properties and circular cutouts at center, taking into account the plate–foundation interaction modeled by Pasternak’s elastic foundation model.

In fact, the FGM plates and other complex structures usually reinforced by longitudinal and transversal (orthogonal) stiffeners to provide outstanding benefit for structures. Based on the first-order shear deformation theory (FSDT) and the element-free improved moving least-squares Ritz method, Zhang et al.^28–32 studied the postbuckling behavior of FG-CNTRC rectangular, polygonal, and arbitrarily straight-sided quadrilateral plates subjected to different load cases. Using Lekhnitskii’s smeared stiffener technique and Galerkin procedure, the nonlinear buckling behavior of eccentrically stiffened FGM double curved shallow shells with elastic foundation and initial imperfection effects was investigated by Dung and Dong.³³ Kim et al.³⁴ studied nonlinear buckling and vibration of eccentrically oblique stiffened FGM plates using the classical thin shells theories. Lekhnitskii’s smeared stiffeners technique was improved for classical oblique stiffeners with the assumption of ignoring the thermal stress in stiffeners.

The fully improved Lekhnitskii’s smeared stiffeners technique for classical shell theory and FSDT is developed^35–37 considering the thermal terms of stiffeners. Nam et al.³⁵ and Phuong et al.³⁶ studied nonlinear buckling of FGM and multilayer FGM cylindrical shell reinforced by spiral (oblique) stiffeners subjected to torsional loads with and without shell–foundation interaction effects. Nam et al.³⁷ improved the Lekhnitskii’s smeared stiffener technique for the oblique stiffeners in framework of the FSDT to investigate the nonlinear thermomechanical stability of stiffened multilayer FGM plates.

Effect of sandwich FGM structures on the vibration, bending, and buckling behavior of plates and shallow spherical caps were mentioned and discussed by Meksi et al.,³⁸ Rahmani et al.³⁹ and Phuong et al.⁴⁰ Ansari et al.⁴¹ developed a nonlinear microstructure-dependent third-order shear deformable beam model and investigated the nonlinear mechanical behavior of an FGM microbeam using the variational differential quadrature method.Zhang et al.⁴² used the element-free kp-Ritz method to investigate the postbuckling behavior of the glass-protection film exposed in the thermal environment. Effects of Pasternak foundation, Kerr foundation, and viscoelastic foundation on the mechanical behavior of FGM plates and single-layered graphene sheets were investigated and discussed by Addou et al.,⁴³ Boulefrakh et al.,⁴⁴ and Bellal et al.⁴⁵

Due to the large deflection of plates, the nonlinear effect of elastic foundation needs to be considered. According to the best of authors’ knowledge, there are no studies on the large deflection postbuckling of stiffened FGM plates resting on nonlinear elastic foundation. This article proposes a new analytical approach for nonlinear buckling and postbuckling of sandwich FGM plates under external pressure and axial compression in the thermal environment resting on three-parameter nonlinear elastic foundation. The plates are reinforced by higher-order shear deformable orthogonal or oblique stiffeners taking into account the thermal terms of stiffeners. The improved Lekhnitskii’s smeared stiffener technique is applied for oblique FGM stiffeners using the coordinate transformation technique within the framework of HSDT and Hooke’s law with both thermal effects of plate skin and stiffeners. The very large effects of oblique stiffeners and three-parameter nonlinear elastic foundation are obtained on the nonlinear buckling behavior of structures in numerical investigations.

Sandwich FGM plate reinforced by oblique stiffeners on nonlinear elastic foundation

Consider a sandwich FGM plate of length a, width b, and thickness h, which is subjected to axial compressive load and external pressure load in the thermal environment. The plate skin is assumed to consist of three layers: upper and lower FGM layers and isotropic core layer with the thickness denoted by $h_{t}, h_{c}$ , and $h_{b},$ respectively. The plate reinforced by lattice stiffeners included longitudinal, transversal, and oblique stiffeners with the thickness of stiffeners denoted by $h_{x}, h_{y}$ , and h_l, respectively. The width of longitudinal, transversal, and oblique stiffeners is $b_{x}, b_{y}$ , and b_l, and the distance between stiffeners is $d_{x}, d_{y}$ , and d_l, respectively. The midplane of plate skin is referred to the coordinate ( $x, y, z$ ), as shown in Figure 1.

Figure 1.

Geometry and coordinate system of oblique stiffened plate rested on nonlinear elastic foundation.

The sandwich FGM plate skin are investigated in two cases (see Figure 2) of FGM—full metal—FGM plates and FGM—full ceramic—FGM plates.

Figure 2.

Material distribution of stiffened sandwich FGM plates. (a) First case and (b) second case.

In the first case, the volume fraction of materials of FGM’s upper layer varies from the ceramic-rich surface ( $z = - h / 2$ ) to the metal-rich interface ( $z = - h / 2 + h_{t}$ ), the volume fraction of materials of the FGM’s lower layer varies from metal-rich interface ( $z = h / 2 - h_{b}$ ) to ceramic-rich surface ( $z = h / 2$ ), and the isotropic core layer is a full metal layer. The volume fraction of materials of FGM stiffeners varies from the ceramic-rich interface ( $z = h / 2$ ) to the metal-rich surface ( $z = h / 2 + h_{i}, i = x, y, l$ ).

Elastic modulus E and thermal expansion coefficient α of sandwich FGM plate skin are considered in the form

\begin{array}{l} [E_{p} (z, T), α_{p} (z, T)] = \\ = \{\begin{cases} [E_{c} (T), α_{c} (T)] + [E_{m c} (T), α_{m c} (T)] {(\frac{2 z + h}{2 h_{t}})}^{k_{t}} for - \frac{h}{2} \leq z \leq - \frac{h}{2} + h_{t} \\ [E_{m} (T), α_{m} (T)] for - \frac{h}{2} + h_{t} \leq z \leq \frac{h}{2} - h_{b} \\ [E_{c} (T), α_{c} (T)] + [E_{m c} (T), α_{m c} (T)] {(\frac{- 2 z + h}{2 h_{b}})}^{k_{b}} for \frac{h}{2} - h_{b} \leq z \leq \frac{h}{2} \end{cases} \end{array}

where $E_{m}, E_{c}$ are Young’s moduli of metal and ceramic, respectively, and $α_{m}, α_{c}$ are thermal expansion coefficients of metal and ceramic, respectively; $E_{m c} = E_{m} - E_{c} = - E_{c m},$ $α_{m c} = α_{m} - α_{c} = - α_{c m}$ , $k_{t} \geq 0$ , and $k_{b} \geq 0$ are the volume fraction indices of the FGM’s upper layer, the FGM’s lower layer of the plate, and for the stiffeners³⁸

\{\begin{cases} [E_{s x} (z, T), α_{s x} (z, T)] = \\ = [E_{c} (T), α_{c} (T)] + [E_{m c} (T), α_{m c} (T)] {(\frac{2 z - h}{2 h_{x}})}^{k_{x}} for \frac{h}{2} \leq z \leq \frac{h}{2} + h_{x} \\ [E_{s y} (z, T), α_{s y} (z, T)] = \\ = [E_{c} (T), α_{c} (T)] + [E_{m c} (T), α_{m c} (T)] {(\frac{2 z - h}{2 h_{y}})}^{k_{y}} for \frac{h}{2} \leq z \leq \frac{h}{2} + h_{y} \\ [E_{s l} (z, T), α_{s l} (z, T)] = \\ = [E_{c} (T), α_{c} (T)] + [E_{m c} (T), α_{m c} (T)] {(\frac{2 z - h}{2 h_{l}})}^{k_{l}} for \frac{h}{2} \leq z \leq \frac{h}{2} + h_{l} \end{cases}

where $k_{x}, k_{y}$ , and k_l are the volume fraction indices of longitudinal, transversal, and oblique stiffeners, respectively.

In the second case, the volume fraction of materials of FGM’s upper layer varies from the metal-rich surface ( $z = - h / 2$ ) to the rich ceramic interface ( $z = - h / 2 + h_{t}$ ), while the volume fraction of materials of FGM’s lower layer varies from ceramic-rich interface ( $z = h / 2 - h_{b}$ ) to metal-rich surface ( $z = h / 2$ ), and the core layer is made of full ceramic. The volume fraction of materials of FGM stiffeners varies from the metal-rich interface ( $z = h / 2$ ) to the ceramic-rich surface ( $z = h / 2 + h_{i}, i = x, y, l$ ).

The elastic moduli and the thermal expansion coefficients of plate and stiffeners are considered as the first case by changing the c, m, and $m c$ by m, c, and $c m$ , respectively.

The plate–foundation interaction of three-parameter nonlinear elastic foundation model is assumed as follows⁴⁶

q_{f} = K_{1} w - K_{2} \nabla^{2} w + K_{3} w^{3}

where K₁ (N/m³) is the Winkler foundation modulus, K₂ (N/m) is the shear layer foundation stiffness of the Pasternak model, and K₃ is the nonlinear modulus (N/m⁵). $\nabla^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}$ is the Laplace operator.

Note that the nonlinear modulus may be positive or negative, it depends on the properties of foundation.

Governing equation establishment

By applying the HSDT, the strain components of plate at a distance z from the mid-surface are obtained as

\{\begin{cases} ε_{x} = ε_{x}^{0} + z ε_{x}^{(1)} + z^{3} ε_{x}^{(3)}, \\ ε_{y} = ε_{y}^{0} + z ε_{y}^{(1)} + z^{3} ε_{y}^{(3)}, \\ γ_{x y} = γ_{x y}^{0} + z γ_{x y}^{(1)} + z^{3} γ_{x y}^{(3)}, \end{cases} \{\begin{cases} γ_{x z} = γ_{x z}^{0} + z^{2} γ_{x z}^{(2)} \\ γ_{y z} = γ_{y z}^{0} + z^{2} γ_{y z}^{(2)} \end{cases}

in which the relations between the strain and displacement at the mid-surface taking into account the von Kármán’s geometrical nonlinearity are determined by

\{\begin{cases} ε_{x}^{0} = u_{, x} + \frac{1}{2} w_{, x}^{2} + w_{, x} w_{, x}^{*}, \\ ε_{y}^{0} = v_{, y} + \frac{1}{2} w_{, y}^{2} + w_{, y} w_{, y}^{*}, \\ γ_{x y}^{0} = u_{, y} + v_{, x} + w_{, x} w_{, y} + w_{, x} w_{, y}^{*} + w_{, y} w_{, x}^{*}, \end{cases} \{\begin{cases} γ_{x z}^{0} = ϕ_{x} + w_{, x} \\ γ_{y z}^{0} = ϕ_{y} + w_{, y} \end{cases}

and

\{\begin{cases} ε_{x}^{(1)} = ϕ_{x, x}, \\ ε_{y}^{(1)} = ϕ_{y, y}, \\ γ_{x y}^{(1)} = ϕ_{x, y} + ϕ_{y, x}, \end{cases} \{\begin{cases} γ_{x z}^{(2)} = - 3 λ (ϕ_{x} + w_{, x}), \\ γ_{y z}^{(2)} = - 3 λ (ϕ_{y} + w_{, y}), \end{cases} \{\begin{cases} ε_{x}^{(3)} = - λ (ϕ_{x, x} + w_{, x x}) \\ ε_{y}^{(3)} = - λ (ϕ_{y, y} + w_{, y y}) \\ γ_{x y}^{(3)} = - λ (ϕ_{x, y} + ϕ_{y, x} + 2 w_{, x y}) \end{cases}

where $ε_{x}^{0}$ and $ε_{y}^{0}$ are the normal strains, $γ_{x y}^{0}$ is the in-plane shear strain, $γ_{x z}^{0}$ and $γ_{y z}^{0}$ are the transversal strains at the $x z$ and $y z$ planes of plate. In addition, $w^{*} = w (x, y)$ is the constant initial geometrical imperfection of deflection and is assumed to be very small in comparison with the thickness of plate h, $λ = 4 / (3 h^{2})$ .

Using equation (5), the compatibility equation is obtained as

ε_{x, y y}^{0} + ε_{y, x x}^{0} - γ_{x y, x y}^{0} = w_{, x y}^{2} - w_{, x x} w_{, y y} - w_{, x x} w_{y y}^{*} + 2 w_{, x y} w_{x y}^{*} - w_{x x}^{*} w_{, y y}

Hooke’s law for the sandwich FGM plate skin taking into account the thermal effects is written as

\{\begin{cases} σ_{x}^{​} = \frac{E_{p} (z, T)}{1 - ν^{2}} [ε_{x} + (1 + ν) ε_{y}] - \frac{E_{p} (z, T) α_{p} (z, T)}{1 - ν} Δ T \\ σ_{y}^{​} = \frac{E_{p} (z, T)}{1 - ν^{2}} [(1 + ν) ε_{x} + ε_{y}] - \frac{E_{p} (z, T) α_{p} (z, T)}{1 - ν} Δ T \\ σ_{x y}^{​} = \frac{E_{p} (z, T)}{2 (1 + ν)} γ_{x y}, \\ σ_{x z}^{​} = \frac{E_{p} (z, T)}{2 (1 + ν)} γ_{x z}, \\ σ_{y z}^{​} = \frac{E_{p} (z, T)}{2 (1 + ν)} γ_{y z}, \end{cases}

with $Δ T = T - T_{0}$ is the uniform temperature change of environment.

Hooke’s law for shear deformable FGM stiffeners (in the local coordinate system $η_{1}$ or $η_{2}$ ) is expressed by

\begin{array}{l} σ_{η}^{s} = E_{s} (z, T) ε_{η} - E_{s} (z, T) α_{s} (z, T) Δ T \\ σ_{η z}^{s} = \frac{E_{s} (z, T)}{2 (1 + ν)} γ_{η z} \end{array}

Using the developed Lekhnitskii’s smeared stiffener technique, the internal force and moment expressions of the sandwich FGM plate reinforced by FGM stiffeners are obtained based on the coordinate transformation technique for oblique stiffeners as follows

\{\begin{cases} N_{x} = A_{11} ε_{x}^{0} + A_{12} ε_{y}^{0} + B_{11} ϕ_{x, x} + B_{12} ϕ_{y, y} - \\ - λ C_{11} (ϕ_{x, x} + w_{, x x}) - λ C_{12} (ϕ_{y, y} + w_{, y y}) - φ_{1}^{p} - φ_{1}^{s x} \\ N_{y} = A_{12} ε_{x}^{0} + A_{22} ε_{y}^{0} + B_{12} ϕ_{x, x} + B_{22} ϕ_{y, y} - \\ - λ C_{12} (ϕ_{x, x} + w_{, x x}) - λ C_{22} (ϕ_{y, y} + w_{, y y}) - φ_{1}^{p} - φ_{1}^{s y} \\ N_{x y} = A_{66} γ_{x y}^{0} + B_{66} (ϕ_{x, y} + ϕ_{y, x}) - λ C_{66} (ϕ_{x, y} + ϕ_{y, x} + 2 w, x y) \end{cases}

\{\begin{cases} M_{x} = B_{11} ε_{x}^{0} + B_{12} ε_{y}^{0} + D_{11} ϕ_{x, x} + D_{12} ϕ_{y, y} - \\ - λ E_{11} (ϕ_{x, x} + w_{, x x}) - λ E_{12} (ϕ_{y, y} + w_{, y y}) - φ_{2}^{p} - φ_{2}^{s x} \\ M_{y} = B_{12} ε_{x}^{0} + B_{22} ε_{y}^{0} + D_{12} ϕ_{x, x} + D_{22} ϕ_{y, y} - \\ - λ E_{12} (ϕ_{x, x} + w_{, x x}) - λ E_{22} (ϕ_{y, y} + w_{, y y}) - φ_{2}^{p} - φ_{2}^{s y} \\ M_{x y} = B_{66} γ_{x y}^{0} + D_{66} (ϕ_{x, y} + ϕ_{y, x}) - λ E_{66} (ϕ_{x, y} + ϕ_{y, x} + 2 w, x y) \end{cases}

\{\begin{cases} T_{x} = C_{11} ε_{x}^{0} + C_{12} ε_{y}^{0} + E_{11} ϕ_{x, x} + E_{12} ϕ_{y, y} - \\ - λ L_{11} (ϕ_{x, x} + w_{, x x}) - λ L_{12} (ϕ_{y, y} + w_{, y y}) - φ_{4}^{p} - φ_{4}^{s x} \\ T_{y} = C_{12} ε_{x}^{0} + C_{22} ε_{y}^{0} + E_{12} ϕ_{x, x} + E_{22} ϕ_{y, y} - \\ - λ L_{12} (ϕ_{x, x} + w_{, x x}) - λ L_{22} (ϕ_{y, y} + w_{, y y}) - φ_{4}^{p} - φ_{4}^{s y} \\ T_{x y} = C_{66} γ_{x y}^{0} + E_{66} (ϕ_{x, y} + ϕ_{y, x}) - λ L_{66} (ϕ_{x, y} + ϕ_{y, x} + 2 w, x y) \end{cases}

where

\begin{array}{l} φ_{1}^{p} = \frac{1}{1 - v} \int_{- h / 2}^{h / 2} E_{p} (z, T) α_{p} (z, T) Δ T d z \\ φ_{1}^{s x} = μ_{1} \frac{b_{x}}{d_{x}} \int_{h / 2}^{h / 2 + h_{x}} E_{s x} (z, T) α_{s x} (z, T) Δ T d z + \\ + μ_{2} \frac{b_{l}}{d_{l}} \int_{h / 2}^{h / 2 + h_{l}} 2 E_{s l} (z, T) α_{s l} (z, T) ({cos}^{6} θ + {sin}^{2} θ {cos}^{2} θ) Δ T d z \\ φ_{1}^{s y} = μ_{1} \frac{b_{y}}{d_{y}} \int_{h / 2}^{h / 2 + h_{y}} E_{s y} (z, T) α_{s y} (z, T) Δ T d z + \\ + μ_{2} \frac{b_{l}}{d_{l}} \int_{h / 2}^{h / 2 + h_{l}} 2 E_{s l} (z, T) α_{s l} (z, T) ({sin}^{6} θ + {sin}^{2} θ {cos}^{2} θ) Δ T d z \end{array}

with $Δ T = const$ .

\begin{array}{l} φ_{1}^{p} = φ_{10}^{p} Δ T, φ_{10}^{p} = \frac{1}{1 - v} \int_{- h / 2}^{h / 2} E_{p} (z, T) α_{p} (z, T) d z \\ φ_{1}^{s x} = φ_{10}^{s x} Δ T \\ φ_{10}^{s x} = μ_{1} \frac{b_{x}}{d_{x}} \int_{h / 2}^{h / 2 + h_{x}} E_{s x} (z, T) α_{s x} (z, T) d z + \\ + μ_{2} \frac{b_{l}}{d_{l}} \int_{h / 2}^{h / 2 + h_{l}} 2 E_{s l} (z, T) α_{s l} (z, T) ({cos}^{6} θ + {sin}^{2} θ {cos}^{2} θ) d z \\ φ_{1}^{s y} = φ_{10}^{s y} Δ T, φ_{10}^{s y} = μ_{1} \frac{b_{y}}{d_{y}} \int_{h / 2}^{h / 2 + h_{y}} E_{s y} (z, T) α_{s y} (z, T) d z + \\ + μ_{2} \frac{b_{l}}{d_{l}} \int_{h / 2}^{h / 2 + h_{l}} 2 E_{s l} (z, T) α_{s l} (z, T) ({sin}^{6} θ + {sin}^{2} θ {cos}^{2} θ) d z \end{array}

where

if $µ_{1} = 1, µ_{2} = 0$ , the plate is stiffened by longitudinal, transversal stiffeners;

if $µ_{1} = 0, µ_{2} = 1$ , the plate is stiffened by oblique stiffener system; and

if $µ_{1} = 1, µ_{2} = 1$ , the plate is stiffened by longitudinal, transversal, and oblique stiffeners.

Transverse shear forces and the higher-order shear forces are expressed as

\{\begin{cases} Q_{x} = {\bar{H}}_{44} w_{, x} + {\bar{H}}_{44} ϕ_{x}, \\ Q_{y} = {\bar{H}}_{55} w_{, y} + {\bar{H}}_{55} ϕ_{y}, \end{cases} \{\begin{cases} S_{x} = H_{66} w_{, x} + H_{66} ϕ_{x} \\ S_{y} = H_{77} w_{, y} + H_{77} ϕ_{y} \end{cases}

where the stiffness parameters $A_{ij}, B_{ij}, C_{ij}, D_{ij}, E_{ij}, L_{ij}, H_{i j}, {\bar{H}}_{i j}$ are presented in Online Appendix A1.

The strain–force relations are reversely written from equation (10)

\{\begin{cases} ε_{x}^{0} = X_{11} N_{x} - X_{12} N_{y} - X_{13} ϕ_{x, x} - X_{14} ϕ_{y, y} + λ λ X_{15} (ϕ_{x, x} + w_{, x x}) + \\ + λ X_{16} (ϕ_{y, y} + w_{, y y}) + X_{17} φ_{1}^{p} + X_{18} φ_{1}^{s x} + X_{19} φ_{1}^{s y} \\ ε_{y}^{0} = - X_{12} N_{x} + X_{21} N_{y} - X_{22} ϕ_{x, x} - X_{23} ϕ_{y, y} + λ X_{24} (ϕ_{x, x} + w_{, x x}) + \\ + λ X_{25} (ϕ_{y, y} + w_{, y y}) + X_{26} φ_{1}^{p} + X_{27} φ_{1}^{s x} + X_{28} φ_{1}^{s y} \\ γ_{x y}^{0} = X_{31} N_{x y} - X_{32} (ϕ_{x, y} + ϕ_{y, x}) + λ X_{33} (ϕ_{x, y} + ϕ_{y, x} + 2 w_{, x y}) \end{cases}

Substituting equation (14) into equations (11) and (12), the moment–force relations are determined as

\{\begin{cases} M_{x} = X_{13} N_{x} + X_{22} N_{y} + Y_{11} ϕ_{x, x} + Y_{12} ϕ_{y, y} - λ Y_{13} (ϕ_{x, x} + w_{, x x}) - \\ - λ Y_{14} (ϕ_{y, y} + w_{, y y}) + Y_{15} φ_{1}^{p} + Y_{16} φ_{1}^{s x} + Y_{17} φ_{1}^{s y} - φ_{2}^{p} - φ_{2}^{s x} \\ M_{y} = X_{14} N_{x} + X_{23} N_{y} + Y_{21} ϕ_{x, x} + Y_{22} ϕ_{y, y} - λ Y_{23} (ϕ_{x, x} + w_{, x x}) - \\ - λ Y_{24} (ϕ_{y, y} + w_{, y y}) + Y_{25} φ_{1}^{p} + Y_{26} φ_{1}^{s x} + Y_{27} φ_{1}^{s y} - φ_{2}^{p} - φ_{2}^{s y} \\ M_{x y} = X_{32} N_{x y} + Y_{31} (ϕ_{x, y} + ϕ_{y, x}) - λ Y_{32} (ϕ_{x, y} + ϕ_{y, x} + 2 w, x y) \end{cases}

\{\begin{cases} T_{x} = X_{15} N_{x} + X_{24} N_{y} + Y_{13} ϕ_{x, x} + Y_{14} ϕ_{y, y} - λ Z_{11} (ϕ_{x, x} + w_{, x x}) - \\ - λ Z_{12} (ϕ_{y, y} + w_{, y y}) + Z_{13} φ_{1}^{p} + Z_{14} φ_{1}^{s x} + Z_{15} φ_{1}^{s y} - φ_{4}^{p} - φ_{4}^{s x} \\ T_{y} = X_{16} N_{x} + X_{25} N_{y} + Y_{23} ϕ_{x, x} + Y_{24} ϕ_{y, y} - λ Z_{21} (ϕ_{x, x} + w_{, x x}) - \\ - λ Z_{22} (ϕ_{y, y} + w_{, y y}) + Z_{23} φ_{1}^{p} + Z_{24} φ_{1}^{s x} + Z^{25} φ_{1}^{s y} - φ_{4}^{p} - φ_{4}^{s y} \\ T_{x y} = X_{33} N_{x y} + Y_{32} (ϕ_{x, y} + ϕ_{y, x}) - λ Z_{31} (ϕ_{x, y} + ϕ_{y, x} + 2 w, x y) \end{cases}

in which $X_{i j}, Y_{i j}, Z_{i j}$ are observed in Online Appendix A2.

The equilibrium equation system of the stiffened sandwich FGM plates based on the HSDT is applied as

\{\begin{cases} N_{x, x} + N_{x y, y} = 0 \\ N_{x y, x} + N_{y, y} = 0 \\ Q_{x, x} + Q_{y, y} - 3 λ (S_{x, x} + S_{y, y}) + λ (T_{x, x x} + 2 T_{x y, x y} + T_{y, y y}) + N_{x} (w_{, x x} + w_{x x}^{*}) + \\ + 2 N_{x y} (w_{, x y} + w_{x y}^{*}) + N_{y} (w_{, y y} + w_{y y}^{*}) + q - q_{f} = 0 \\ M_{x, x} + M_{x y, y} - Q_{x} + 3 λ S_{x} - λ (T_{x, x} + T_{x y, y}) = 0 \\ M_{x y, x} + M_{y, y} - Q_{y} + 3 λ S_{y} - λ (T_{x y, x} + T_{y, y}) = 0 \end{cases}

Boundary condition and solution method

In this article, two types of boundary condition are considered

For the first type, four edges of the stiffened sandwich FGM plate are simply supported and freely movable (FM), the boundary conditions are written as

\begin{array}{l} w = 0, M_{x} = 0, N_{x y} = 0, ϕ_{y} = 0, T_{x} = 0, N_{x} = N_{x 0} = - h P_{x}, at x = 0, x = a \\ w = 0, M_{y} = 0, N_{x y} = 0, ϕ_{x} = 0, T_{y} = 0, N_{y} = N_{y 0} = - h P_{y}, at y = 0, y = b \end{array}

For the second type, four edges of the plate are simply supported. The edges $x = 0, x = a$ are assumed to be FM, and the two extant edges are immovable. The axial compressive loads are loaded in the x direction. The boundary conditions are obtained as

\begin{array}{l} w = 0, M_{x} = 0, N_{x y} = 0, ϕ_{y} = 0, T_{x} = 0, N_{x} = N_{x 0} = - h P_{x}, at x = 0, x = a \\ w = 0, v = 0, ϕ_{x} = 0, M_{y} = 0, T_{y} = 0, N_{y} = N_{y 0}, at y = 0, y = b \end{array}

where $N_{x 0}$ is the prestress in the x direction, $N_{y 0}$ is the prestress in the y direction for the first type and it is axial compressive edge load of immovable edges for the second type.

Assuming that a stress function $f (x, y)$ may be found, which satisfies

N_{x} = f_{, y y}, N_{y} = f_{, x x}, N_{x y} = - f_{, x y}

As can be observed, the first two equations of (17) are automatically satisfied.

Substituting equation (20) into equation (14) and then substituting into equation (7), the compatibility equation is rewritten as

\begin{array}{l} X_{21} f_{, x x x x} + (X_{31} - 2 X_{12}) f_{, x x y y} + X_{11} f_{, y y y y} - (X_{22} - λ X_{24}) ϕ_{x, x x x} + \\ + [X_{32} - X_{13} - λ (X_{33} - X_{15})] ϕ_{x, x y y} + [X_{32} - X_{23} - λ (X_{33} - X_{25})] ϕ_{y, x x y} + \\ - (X_{14} - λ X_{16}) ϕ_{y, y y y} + λ X_{24} w_{, x x x x} + λ (X_{15} + X_{25} - 2 X_{33}) w_{, x x y y} + \\ + λ X_{16} w_{, y y y y} - w_{x y}^{2} + w_{, x x} w_{, y y} + w_{, x x} w_{, y y}^{*} - 2 w_{, x y} w_{, x y}^{*} + w_{, x x}^{*} w_{, y y} = 0 \end{array}

Substituting equations (13), (15), (16), and (20) into three end equations of equation (17) leads to

\begin{array}{l} λ X_{24} f_{, x x x x} + λ (X_{15} + X_{25} - 2 X_{33}) f_{, x x y y} + λ X_{16} f_{, y y y y} + λ (Y_{13} - λ Z_{11}) ϕ_{x, x x x} + \\ + λ [Y_{23} + 2 Y_{32} - λ (Z_{21} + 2 Z_{31})] ϕ_{x, x y y} + ({\bar{H}}_{44} - 3 λ H_{66}) ϕ_{x, x} + \\ + λ (Y_{24} - λ Z_{22}) ϕ_{y, y y y} + λ [Y_{14} + 2 Y_{32} - λ (Z_{12} + 2 Z_{31})] ϕ_{y, x x y} + \\ + ({\bar{H}}_{55} - 3 λ H_{77}) ϕ_{y, y} - λ^{2} Z_{11} w_{, x x x x} - λ^{2} (Z_{12} + Z_{21} + 4 Z_{31}) w_{, x x y y} - \\ - λ^{2} Z_{22} w_{, y y y y} + ({\bar{H}}_{44} - 3 λ H_{66}) w_{, x x} + ({\bar{H}}_{55} - 3 λ H_{77}) w_{, y y} + f_{, y y} (w_{, x x} + w_{, x x}^{*}) \\ - 2 f_{, x y} (w_{, x y} + w_{, x y}^{*}) + f_{, x x} (w_{, y y} + w_{, y y}^{*}) + q - K_{1} w + K_{2} \nabla^{2} w - K_{3} w^{3} = 0 \end{array}

22a

\begin{array}{l} (X_{13} - X_{32}) f_{, x y y} + X_{22} f_{, x x x} + (Y_{11} - λ Y_{13}) ϕ_{x, x x} + \\ + [Y_{12} + Y_{31} - λ (Y_{14} + Y_{32})] ϕ_{y, x y} + (Y_{31} - λ Y_{32}) ϕ_{x, y y} - \\ - λ Y_{13} w_{, x x x} - λ (Y_{14} + 2 Y_{32}) w_{, x y y} - λ {(X_{15} - X_{33}) f_{, x y y} + X_{24} f_{, x x x} + \\ + (Y_{13} - λ Z_{11}) ϕ_{x, x x} + [Y_{14} + Y_{32} - λ (Z_{12} + Z_{31})] ϕ_{y, x y} + \\ + (Y_{32} - λ Z_{31}) ϕ_{x, y y} - λ Z_{11} w_{, x x x} - λ (Z_{12} + 2 Z_{31}) w_{, x y y}\} - \\ - {\bar{H}}_{44} ϕ_{x} - {\bar{H}}_{44} w_{, x} + 3 λ (H_{66} ϕ_{x} + H_{66} w_{, x}) = 0 \end{array}

22b

\begin{array}{l} (X_{23} - X_{32}) f_{, x x y} + X_{14} f_{, y y y} + (Y_{22} - λ Y_{24}) ϕ_{y, y y} + \\ + [Y_{21} + Y_{31} - λ (Y_{23} + Y_{32})] ϕ_{x, x y} + (Y_{31} - λ Y_{32}) ϕ_{y, x x} - \\ - λ Y_{24} w_{, y y y} - λ (Y_{23} + 2 Y_{32}) w_{, x x y} - λ {(X_{25} - X_{33}) f_{, x x y} + X_{16} f_{, y y y} + \\ + (Y_{24} - λ Z_{22}) ϕ_{y, y y} + [Y_{23} + Y_{32} - λ (Z_{21} + Z_{31})] ϕ_{x, x y} + \\ + (Y_{32} - λ Z_{31}) ϕ_{y, x x} - λ Z_{22} w_{, y y y} - λ (Z_{21} + 2 Z_{31}) w_{, x x y}\} - \\ - {\bar{H}}_{55} ϕ_{y} - {\bar{H}}_{55} w_{, y} + 3 λ (H_{77} ϕ_{y} + H_{77} w_{, y}) = 0 \end{array}

22c

The approximate solutions of equations (22a) to (22c) satisfying the boundary conditions (18) and (19) are assumed in the trigonometric functions as

\begin{array}{l} w = W sin α x sin β y \\ ϕ_{x} = Φ_{x} co s α x sin β y \\ ϕ_{y} = Φ_{y} sin α x cos β y \end{array}

The geometrical imperfection $w^{*}$ is assumed to have the same form of deflection of plate, represented as

w^{*} = ξ h sin α x sin β y

with $\begin{array}{l} α = m π / a, β = n π / b \end{array}$ ; m and n are numbers of half-waves in x and y directions; and ξ is the geometrical imperfection size.

Substituting equations (23) and (24) into the compatibility equation (21), the stress function expression is obtained as follows

f = f_{1} cos 2 α x + f_{2} cos 2 β y + f_{3} sin α x sin β y + \frac{1}{2} N_{0 x} y^{2} + \frac{1}{2} N_{0 y} x^{2}

where

\begin{array}{l} f_{1} = \frac{β^{2}}{32 α^{2} X_{21}} W (W + 2 ξ h), f_{2} = \frac{α^{2}}{32 β^{2} X_{11}} W (W + 2 ξ h) \\ f_{3} = M_{1} Φ_{x} + M_{2} Φ_{y} + M_{3} W \end{array}

with $D = X_{21} α^{4} + (X_{31} - 2 X_{12}) α^{2} β^{2} + X_{11} β^{4}$

\begin{array}{l} M_{1} = \frac{(X_{14} - λ X_{22}) α^{3} + [X_{13} - X_{32} - λ (X_{15} - X_{33})] α β^{2}}{D} \\ M_{2} = \frac{[X_{23} - X_{32} - λ (X_{25} - X_{33})] α^{2} β + (X_{14} - λ X_{16}) β^{3}}{D} \\ M_{3} = \frac{- λ [X_{24} α^{4} + (X_{15} + X_{25} - 2 X_{33}) α^{2} β^{2} + X_{16} β^{4}]}{D} \end{array}

Substituting equations (23), (24), and (25) into equations (22a), (22b), (22c), then applying the Galerkin method, leads to

\begin{array}{l} κ_{11} W + κ_{12} Φ_{x} + κ_{13} Φ_{y} + κ_{14} Φ_{x} (W + ξ h) + κ_{15} Φ_{y} (W + ξ h) + \\ + ς_{1} W (W + ξ h) + ς_{2} W (W + 2 ξ h) + ς_{3} W (W + ξ h) (W + 2 ξ h) + \\ + ς_{7} W^{3} + ς_{4} q - (N_{x 0} α^{2} + N_{y 0} β^{2}) (W + ξ h) = 0 \\ κ_{21} W + κ_{22} Φ_{x} + κ_{23} Φ_{y} + ς_{5} W (W + 2 ξ h) = 0 \\ κ_{31} W + κ_{32} Φ_{x} + κ_{33} Φ_{y} + ς_{6} W (W + 2 ξ h) = 0 \end{array}

where the expressions of coefficients $κ_{i j}$ and $ς_{i}$ are presented in Online Appendix B.

Clearly, the Galerkin method is applied to reduce the nonlinear partial differential equation system (22a), (22b), (22c) to the solution of nonlinear algebraic equation system (26).

Nonlinear buckling analysis

From the two end equations of equation (26), these equations can be expressed by $Φ_{x}, Φ_{y}$ as

\begin{array}{l} Φ_{x} = \frac{(κ_{23} κ_{31} - κ_{21} κ_{33}) W + (κ_{23} ς_{6} - κ_{33} ς_{5}) W (W + 2 ξ h)}{κ_{22} κ_{33} - κ_{23} κ_{32}} \\ Φ_{y} = \frac{(κ_{32} κ_{21} - κ_{22} κ_{31}) W + (κ_{32} ς_{5} - κ_{22} ς_{6}) W (W + 2 ξ h)}{κ_{22} κ_{33} - κ_{23} κ_{32}} \end{array}

Substituting equation (27) into the first equations of equation (26), the load–deflection relation is received by

\begin{array}{l} a_{1} \bar{W} + a_{2} h \bar{W} (\bar{W} + ξ) + a_{3} h \bar{W} (\bar{W} + 2 ξ) + \\ + a_{4} h^{2} \bar{W} (\bar{W} + ξ) (\bar{W} + 2 ξ) + \frac{ς_{4}}{h} q - (N_{x 0} α^{2} + N_{y 0} β^{2}) (\bar{W} + ξ) + ς_{7} h^{2} {\bar{W}}^{3} = 0 \end{array}

in which $\bar{W} = \frac{W}{h}$ and

\begin{array}{l} a_{1} = κ_{11} + \frac{κ_{12} (κ_{23} κ_{31} - κ_{21} κ_{33}) + κ_{13} (κ_{32} κ_{21} - κ_{22} κ_{31})}{κ_{22} κ_{33} - κ_{23} κ_{32}} \\ a_{2} = ς_{1} + \frac{κ_{14} (κ_{23} κ_{31} - κ_{21} κ_{33}) + κ_{15} (κ_{32} κ_{21} - κ_{22} κ_{31})}{κ_{22} κ_{33} - κ_{23} κ_{32}} \\ a_{3} = ς_{2} + \frac{κ_{12} (κ_{23} ς_{6} - κ_{33} ς_{5}) + κ_{13} (κ_{32} ς_{5} - κ_{22} ς_{6})}{κ_{22} κ_{33} - κ_{23} κ_{32}} \\ a_{4} = ς_{3} + κ_{14} \frac{κ_{23} ς_{6} - κ_{33} ς_{5}}{κ_{22} κ_{33} - κ_{23} κ_{32}} + κ_{15} \frac{κ_{32} ς_{5} - κ_{22} ς_{6}}{κ_{22} κ_{33} - κ_{23} κ_{32}} \end{array}

Equation (28) is the governing equation, which is utilized to analyze the buckling and postbuckling of sandwich FGM plates reinforced by oblique FGM stiffeners resting on nonlinear elastic foundation based on HSDT.

Mechanical buckling analysis

Consider a sandwich FGM plate reinforced by FGM stiffeners resting on nonlinear elastic foundation, subjected to axial compressive load P_x uniformly distributed on two edges $x = 0$ and $x = a$ (first type of boundary condition). In this case, $q = 0, P_{y} = 0$ .

From equation (28), the relation between P_x and $\bar{W}$ is written by

P_{x} = \frac{- 1}{h α^{2}} [a_{1} \frac{\bar{W}}{\bar{W} + ξ} + a_{2} h \bar{W} + a_{3} h \frac{\bar{W} (\bar{W} + 2 ξ)}{\bar{W} + ξ} + a_{4} h^{2} \bar{W} (\bar{W} + 2 ξ) + ς_{7} h^{2} \frac{{\bar{W}}^{3}}{\bar{W + ξ}}]

From equation (29), when $\bar{W} \to 0$ and $ξ = 0$ , the upper axial compressive buckling load expression is obtained

P_{x u p p e r} = \frac{- a_{1}}{h α^{2}}

The critical axial compressive buckling load is determined by condition $P_{x}^{c r} = min P_{x u p p e r} vs. (m, n)$

Consider a sandwich FGM plate subjected to uniform external pressure with $N_{x 0} = N_{y 0} = 0$ (first type of boundary condition).

From equation (28), the q and $\bar{W}$ relation is written by

q = \frac{- h}{ς_{4}} [\begin{array}{l} a_{1} \bar{W} + a_{2} \bar{W} h (\bar{W} + ξ) + a_{3} \bar{W} h (\bar{W} + 2 ξ) + \\ + a_{4} \bar{W} h^{2} (\bar{W} + ξ) (\bar{W} + 2 ξ) + ς_{7} h^{2} {\bar{W}}^{3} \end{array}]

Equation (31) is utilized to investigate the postbuckling behavior of sandwich FGM plate reinforced by orthogonal and/or oblique stiffeners.

Thermomechanical buckling analysis

The plate is simply supported with FM edges $x = 0, x = a$ and immovable edges $y = 0, y = b$ (second type of boundary condition) with uniform temperature change $Δ T = const$ . The conditions for end-shortening displacements at $y = 0$ and $y = b$ are calculated in average sense

\int_{0}^{a} \int_{0}^{b} v_{, y} d y d x = 0

Consider an imperfect sandwich FGM plate subjected to axial compressive load P_x uniformly distributed at $x = 0$ and $x = a$ in the thermal environment ( $q = 0$ ).

By employing the condition $N_{x 0} = - P_{x} h$ and equation (32), the compressive edge load at immovable edges is determined

N_{y 0} = - \frac{X_{12}}{X_{21}} P_{x} h + τ_{1} W + τ_{2} W (W + 2 ξ h) - \frac{X_{26}}{X_{21}} Δ T φ_{10}^{p} - \frac{X_{27}}{X_{21}} Δ T φ_{10}^{s x} - \frac{X_{28}}{X_{21}} Δ T φ_{10}^{s y}

in which $τ_{1}, τ_{2}$ are defined in Online Appendix C.

Substituting equation (33) into equation (28) yields

\begin{array}{l} P_{x} = \frac{a_{1}}{a_{5}} \frac{\bar{W}}{(\bar{W} + ξ) h} + \frac{a_{2} - τ_{1} β^{2}}{a_{5}} \bar{W} + \frac{a_{3}}{a_{5}} \frac{\bar{W} (\bar{W} + 2 ξ)}{(\bar{W} + ξ)} + \\ + \frac{ς_{7} h}{a_{5}} \frac{{\bar{W}}^{3}}{\bar{W} + ξ} + \frac{a_{4} - τ_{2} β^{2}}{a_{5}} h \bar{W} (\bar{W} + 2 ξ) - \frac{a_{6}}{a_{5} h} Δ T \end{array}

Equation (34) is utilized to determine the axial compressive load–deflection postbuckling curves of imperfect sandwich FGM plate in the thermal environment.

When $\bar{W} \to 0$ and $ξ = 0$ , equation (34) becomes

P_{x u p p e r} = \frac{a_{1}}{a_{5} h} - \frac{a_{6}}{a_{5} h} Δ T

in which $a_{5} = - (α^{2} + \frac{X_{12}}{X_{21}} β^{2})$ and $a_{6} = - β^{2} (\frac{X_{26}}{X_{21}} φ_{10}^{p} + \frac{X_{27}}{X_{21}} φ_{10}^{s x} + \frac{X_{28}}{X_{21}} φ_{10}^{s y})$ .

The critical thermo-axial compressive buckling load is determined by condition $P_{x}^{c r} = min P_{x u p p e r} vs. (m, n)$

Consider a stiffened imperfect sandwich FGM plate subjected to external pressure q in the thermal environment ( $P_{x} = 0$ ).

From equation (28), the external pressure load–deflection postbuckling curve expression can be obtained in the explicit form

\begin{array}{l} q = - \frac{a_{1} h}{ς_{4}} \bar{W} - \frac{(a_{2} - τ_{1} β^{2}) h^{2}}{ς_{4}} \bar{W} (\bar{W} + ξ) - \frac{a_{3} h^{2}}{ς_{4}} \bar{W} (\bar{W} + 2 ξ) + \\ - \frac{ς_{7} h^{3}}{ς_{4}} {\bar{W}}^{3} - \frac{(a_{4} - τ_{2} β^{2}) h^{3}}{ς_{4}} \bar{W} (\bar{W} + ξ) (\bar{W} + 2 ξ) + \frac{a_{6}}{ς_{4}} Δ T (\bar{W} + ξ) \end{array}

Numerical results and discussion

Validation results

In this article, to validate the present analysis, the nondimensional critical axial compressive buckling loads $\bar{P} = \frac{P_{x}^{c r} a^{2}}{100 E_{0} h^{2}}$ with $E_{0} = 1 GPa$ of unstiffened sandwich FGM plates of the present approach are compared with the ones of Zenkour¹³ (Table 1). Results of Zenkour¹³ are obtained according to the classical theory, FSDT, and HSDT.

As can be observed, the perfect agreements are obtained, especially, the results of Zenkour¹³ which are utilized HSDT coincide with the present results.

Table 1.

Comparison of nondimensional critical axial compressive buckling load $\bar{P}$ with the results of Zenkour.¹³

k_t	Theory	$\bar{P}$
k_t	Theory	(I)	(II)	(III)	(IV)
1	CLT¹³	5.3325	6.0273	6.1946	7.7841
	HSDT¹³	5.1671	5.8401	6.4647	7.5066
	FSDT¹³	5.1424	5.8138	6.4389	7.4836
	HSDT (present)	5.1671	5.8401	6.4647	7.5066
5	CLT¹³	2.7308	3.1070	3.6573	4.8572
	HSDT¹³	2.6582	3.0426	3.5796	4.7347
	FSDT¹³	2.6384	3.0225	3.5596	4.7147
	HSDT (present)	2.6582	3.0426	3.5796	4.7347

(I): $h_{t} = h_{b} = 0.5 h$ ; (II): $h_{t} = h_{b} = 0.4 h$ ; (III): $h_{t} = h_{b} = h / 3$ ; (IV): $h_{t} = h_{b} = 0.25 h$ ; CLT: classical theory.

Numerical investigations

Numerical results are presented in this section for orthogonal or oblique stiffened FGM plates. Silicon nitride and stainless steel are applied for FGM, referred to as ${Si}_{3} N_{4} / SUS 304$ . The material properties ${Pr}_{i}$ , such as Young’s modulus E_i and thermal expansion coefficient $α_{i}$ , are expressed as a nonlinear function of temperature

{Pr}_{i} = P_{0} (P_{- 1} T^{- 1} + 1 + P_{1} T + P_{2} T^{2} + P_{3} T^{3}); i = c, m

where P₀, $P_{- 1}$ , P₁, P₂, and P₃ are the coefficients of temperature $T (K)$ and are unique to the constituent materials. The material properties of constituent materials of considered FGM are presented in Table 2.

Table 2.

Material properties of constituent materials of the considered FGM plate.

Material	Properties	P ₀	P ₁	P ₂	P ₃
Ceramic ( ${Si}_{3} N_{4}$ )	E (Pa)	3.48 × 10¹¹	–3.07 × 10⁻⁴	2.16 × 10⁻⁷	−8.95 × 10⁻¹¹
Ceramic ( ${Si}_{3} N_{4}$ )	α ( $K^{- 1}$ )	5.87 × 10⁻⁶	9.10 × 10⁻⁴	0	0
Metal ( $SUS 304$ )	E (Pa)	2.01 × 10¹¹	3.08 × 10⁻⁴	6.53 × 10⁻⁷	0
Metal ( $SUS 304$ )	α ( $K^{- 1}$ )	1.23 × 10⁻⁵	8.09 × 10⁻⁴	0	0

FGM: functionally graded material.

Consider a sandwich FGM plate reinforced by FGM orthogonal or oblique stiffeners system rested on the nonlinear elastic foundation with temperature dependence (T-D) of material properties. The geometrical and material properties of plate and stiffeners are $a = b = 0.4 m$ $b / h = 20$ , $h_{t} = h_{b} = 0.1 h (m)$ $k_{t} = k_{b} = k$ , $k_{x} = k_{y} = k_{l} = 1 / k$ , $b_{x} = b_{y} = b_{l} = 0.01 m$ , $h_{x} = h_{y} = h_{l} = 0.02 m$ , and $d_{x} = d_{y} = d_{l} = 0.005 m$ The plate is rested on nonlinear elastic foundation with the parameters as $K_{1} = 5 \times 10^{7} {N/m}^{3}$ , $K_{2} = 2 \times 10^{5} N/m$ , and $K_{3} = 10^{12} {N/m}^{5}$ . In this article, the comparisons of buckling behavior between orthogonal and oblique stiffeners are investigated with the same material quantity of stiffeners.

The effects of stiffener and boundary condition types on the critical axial compressive buckling load of perfect sandwich FGM plates ( $ξ = 0$ ) are presented in Tables 3 and 4. In these tables, the oblique stiffeners are investigated with the stiffener angle $θ = π / 4$ . As can be observed, critical axial compressive buckling load of stiffened sandwich FGM plate is larger than the one of unstiffened sandwich FGM plate, respectively. Especially, the superior effects of oblique stiffener in comparison with orthogonal stiffener are obtained.

Table 3.

Effects of stiffener and boundary condition types on the critical axial compressive buckling load $P_{x}^{c r} (GPa)$ in the first case ( $m, n$ ) = (1, 1).

K	Unstiffener		Orthogonal stiffeners		Oblique stiffeners
K	FM	IM	FM	IM	FM	IM
0.2	2.0023	1.5402	4.1821	3.3430	6.0080	4.4105
1	2.1755	1.6734	4.6115	3.7013	6.6148	4.8306
5	2.3330	1.7947	5.1539	4.1518	7.4323	5.4019
10	2.3654	1.8195	5.3091	4.2803	7.6778	5.5748

FM: freely movable; IM: immovable.

Table 4.

Effects of stiffener and the boundary condition types on the critical axial compressive buckling load $P_{x}^{c r} (GPa)$ in the second case ( $m, n$ ) = (1, 1).

k	Unstiffener		Orthogonal stiffeners		Oblique stiffeners
k	FM	IM	FM	IM	FM	IM
0.2	2.8390	2.1838	5.9517	4.7398	8.5993	6.3425
1	2.6653	2.0502	5.5162	4.3791	7.9743	5.9050
5	2.5067	1.9282	4.9556	3.9205	7.1001	5.2802
10	2.4741	1.9031	4.7928	3.7886	6.8310	5.0852

FM: freely movable; IM: immovable.

Figures 3 and 4 show the effects of stiffener type on $P_{x} - W / h$ postbuckling curve for the first case and second case FGM plates, respectively. The results show that the postbuckling curves of stiffened sandwich FGM plates are greater than ones of unstiffened sandwich FGM plates, respectively. Bifurcation phenomenon is observed in the case of perfect plates. On the contrary, there is no bifurcation point for imperfect plates. Investigated curves also show that the postbuckling curves of oblique stiffened plates are upper than the ones of orthogonal stiffened plates, respectively.

Figure 3.

Effect of stiffener type on $P_{x} - W / h$ postbuckling curve (first case).

Figure 4.

Effect of stiffener type on $P_{x} - W / h$ postbuckling curve (second case).

Effects of stiffener type on $q - W / h$ postbuckling curve of sandwich FGM plates are considered in Figures 5 and 6. Clearly, there is no bifurcation phenomenon with sandwich FGM plates subjected to external pressure. Similar to the case of axial compressive load, in the case of external pressure load, the postbuckling curves of oblique stiffened plates are upper than the ones of orthogonal stiffened plates.

Figure 5.

Effect of stiffener type on $q - W / h$ postbuckling curve (first case).

Figure 6.

Effect of stiffener type on $q - W / h$ postbuckling curve (second case).

Table 5 and Figures 7 and 8 show that the angle of stiffener θ strongly influence the critical axial compressive buckling load and postbuckling curves of plates. For investigated results, the critical axial compressive buckling load and load carrying capacity in the postbuckling state of square plate is the largest with $θ = \frac{π}{4}$ .

Table 5.

Effect of angle of stiffeners θ on the critical axial compressive buckling loads $P_{x}^{c r} (GPa)$ (k = 1, ( $m, n$ ) = (1,1)).

Model	$θ (rad)$
Model	$\frac{π}{18}$	$\frac{π}{9}$	$\frac{π}{6}$	$\frac{π}{4}$	$\frac{π}{3}$	$\frac{7 π}{18}$	$\frac{4 π}{9}$
First case	4.4245	5.1858	6.0167	6.6148	6.0217	5.1858	4.4245
Second case	5.4070	6.2935	7.2689	7.9743	7.2689	6.2935	5.4070

Figure 7.

Effect of angle of stiffeners θ on $P_{x} - W / h$ postbuckling curve (first case).

Figure 8.

Effect of angle of stiffeners θ on $q - W / h$ postbuckling curve (second case).

Effects of volume fraction index k of FGM layers on the critical axial compressive buckling loads and postbuckling curves of plates are considered in Tables 3, 4 and Figures 9, 10. The obtained results show that, for the first case, the critical buckling load increases when the volume fraction index k increases and the postbuckling curve is upper with the higher value of k, the opposite phenomena are obtained for the second case.

Figure 9.

Effect of k on $P_{x} - W / h$ postbuckling curve (second case).

Figure 10.

Effect of k on $q - W / h$ postbuckling curve (first case).

Consider an oblique stiffened sandwich FGM plate in the first case subjected to axial compressive load in thermal environment. Effects of temperature change on the critical axial compressive buckling loads and postbuckling load–deflection curves of sandwich FGM plate are shown in Tables 6, 7 and Figures 11, 12. As can be seen, critical axial compressive buckling load decreases when the thermal environment increases in two cases of T-D of material properties and temperature independence (T-ID) of material properties (Tables 6 and 7).

Table 6.

Effect of temperature change $Δ T$ and volume fraction index k on the critical axial compressive buckling loads $P_{x}^{c r} (GPa)$ in the first case ( $m, n$ ) = (1,1).

$Δ T$	k	Unstiffener		Orthogonal stiffeners		Oblique stiffeners
$Δ T$	k	T-ID	T-D	T-ID	T-D	T-ID	T-D
$Δ T = 100 K$	0.2	1.2963	1.2615	3.0340	2.9685	4.1743	4.0965
	1	1.4328	1.3940	3.3952	3.3187	4.5988	4.5082
	5	1.5592	1.5168	3.8528	3.7626	5.1765	5.0687
	10	1.5854	1.5423	3.9833	3.8891	5.3509	5.2379
$Δ T = 200 K$	0.2	1.0524	0.9502	2.7250	2.5378	3.9381	3.7190
	1	1.1922	1.0857	3.0892	2.8898	4.3669	4.1343
	5	1.3238	1.2138	3.5538	3.3399	4.9510	4.7001
	10	1.3513	1.2406	3.6862	3.4682	5.1270	4.8705
$Δ T = 300 K$	0.2	0.8084	0.6192	2.4160	2.0666	3.7019	3.2903
	1	0.9515	0.7603	2.7831	2.4275	4.1350	3.7199
	5	1.0883	0.8962	3.2548	2.8942	4.7255	4.3056
	10	1.1172	0.9250	3.3892	3.0274	4.9031	4.4816

T-D: temperature dependence; T-ID: temperature independence.

Table 7.

Effect of temperature change $Δ T$ and volume fraction index k on the critical buckling loads $P_{x}^{c r} (GPa)$ in the second case ( $m, n$ ) = (1,1).

$Δ T$	k	Unstiffener		Orthogonal stiffeners		Oblique stiffeners
$Δ T$	k	T-ID	T-D	T-ID	T-D	T-ID	T-D
$Δ T = 100 K$	0.2	1.9952	1.9430	4.5008	4.3950	6.1571	6.0214
	1	1.8570	1.8087	4.1336	4.0388	5.7108	5.5885
	5	1.7323	1.6877	3.6724	3.5917	5.0793	4.9767
	10	1.7067	1.6629	3.5402	3.4637	4.8831	4.7866
$Δ T = 200 K$	0.2	1.8066	1.6869	4.2619	4.0337	5.9717	5.6919
	1	1.6637	1.5481	3.8882	3.6716	5.5165	5.2499
	5	1.5363	1.4247	3.4242	3.2231	4.8785	4.6315
	10	1.5104	1.3995	3.2918	3.0952	4.6809	4.4402
$Δ T = 300 K$	0.2	1.6180	1.4149	4.0230	3.6543	5.7863	5.3507
	1	1.4705	1.2687	3.6427	3.2783	5.3222	4.8868
	5	1.3404	1.1407	3.1761	2.8184	4.6777	4.2438
	10	1.3140	1.1148	3.0434	2.6876	4.4787	4.0454

T-D: temperature dependence; T-ID: temperature independence.

Figure 11.

Effect of temperature change $Δ T$ on $P_{x} - W / h$ postbuckling curve (second case).

Figure 12.

Effect of temperature change $Δ T$ on $q - W / h$ postbuckling curve (first case).

Figures 13 and 14 show the effects of temperature-dependence properties on the postbuckling curve of oblique stiffened sandwich FGM plates. The investigated results show that the postbuckling curve of the case of T-ID material properties is upper than the one of T-D material property case.

Figure 13.

Effect of temperature dependence on $P_{x} - W / h$ postbuckling curve (first case).

Figure 14.

Effect of temperature dependence on $q - W / h$ postbuckling curve (second case).

Effects of nonlinear elastic foundation on the critical axial compressive buckling loads and postbuckling curves are presented in Tables 8 and 9. It can be seen that when the foundation moduli increase, the critical compression load also increases. The tables also show that the nonlinear modulus of foundation K₃ does not affect the critical compressive buckling loads that it only affects the postbucking load–deflection behavior of plates.

Table 8.

Effect of foundation moduli on the critical axial compressive buckling loads, $P_{x}^{c r} (GPa)$ (first case) (( $m, n$ ) = (1,1), $Δ T = 300 K$ , $k = 1$ ).

K₁ (N/m³), K₂ (N/m), K₃ (N/m⁵)	Unstiffener		Orthogonal stiffeners		Oblique stiffeners
K₁ (N/m³), K₂ (N/m), K₃ (N/m⁵)	T-ID	T-D	T-ID	T-D	T-ID	T-D
$0, 0, 0$	0.9050	0.7138	2.7345	2.3789	4.0908	3.6758
$5 \times 10^{7}, 0, 0$	0.9361	0.7449	2.7671	2.4114	4.1204	3.7053
$5 \times 10^{7}, 2 \times 10^{5}, 0$	0.9515	0.7603	2.7831	2.4275	4.1350	3.7199
$5 \times 10^{7}, 2 \times 10^{5}, 10^{14}$	0.9515	0.7603	2.7831	2.4275	4.1350	3.7199
$0, 2 \times 10^{5}, 0$	0.9203	0.7291	2.7506	2.3949	4.1054	3.6904
$0, 2 \times 10^{5}, 10^{14}$	0.9203	0.7291	2.7506	2.3949	4.1054	3.6904

T-D: temperature dependence; T-ID: temperature independence.

Table 9.

Effect of foundation moduli on the critical axial compressive buckling loads $P_{x}^{c r} (GPa)$ (second case) (( $m, n$ ) = (1,1), $Δ T = 300 K$ , $k = 1$ ).

K₁ (N/m³), K₂ (N/m), K₃ (N/m⁵)	Unstiffener		Orthogonal stiffeners		Oblique stiffeners
K₁ (N/m³), K₂ (N/m), K₃ (N/m⁵)	T-ID	T-D	T-ID	T-D	T-ID	T-D
$0, 0, 0$	1.4240	1.2221	3.5947	3.2303	5.2774	4.8420
$5 \times 10^{7}, 0, 0$	1.4551	1.2533	3.6269	3.2625	5.3074	4.8720
$5 \times 10^{7}, 2 \times 10^{5}, 0$	1.4705	1.2687	3.6427	3.2783	5.3222	4.8868
$5 \times 10^{7}, 2 \times 10^{5}, 10^{14}$	1.4705	1.2687	3.6427	3.2783	5.3222	4.8868
$0, 2 \times 10^{5}, 0$	1.4393	1.2375	3.6106	3.2462	5.2922	4.8568
$0, 2 \times 10^{5}, 10^{14}$	1.4393	1.2375	3.6106	3.2462	5.2922	4.8568

T-D: temperature dependence; T-ID: temperature independence.

Figures 15 and 16 show the effects of nonlinear modulus of foundation K₃ on the postbuckling curve of oblique stiffened sandwich FGM plates. As can be observed, the large effects of nonlinear modulus of foundation are obtained. The postbuckling curve of plate is upper with a larger value of K₃. Especially, an irregular tendency of the postbuckling curve is obtained when K₃ is negative.

Figure 15.

Effect of nonlinear modulus of elastic foundation on $P_{x} - W / h$ postbuckling curve (first case).

Figure 16.

Effect of nonlinear modulus of elastic foundation on $q - W / h$ postbuckling curve (second case).

Conclusions

Using the coordinate transformation technique, Lekhnitskii’s smeared stiffener technique is developed for oblique shear deformable stiffeners taking into account the thermal effects for plate and stiffeners in this article. Governing equations of orthogonal/oblique stiffened sandwich FGM plate resting on nonlinear elastic foundation is established according to the HSDT combined with von Kármán geometrical nonlinearity. Nonlinear thermomechanical buckling behavior of stiffened plates is obtained using the Galerkin method. The most significant remark is that the effects of oblique stiffeners on the critical buckling load are greater than the one of orthogonal stiffeners.

Some other important remarks are obtained from numerical investigated as follows

Material sandwich model strongly influences the nonlinear buckling behavior of the oblique stiffened FGM plate.

Volume fraction index, nonlinear elastic foundation, thermal environment, and geometrical parameters significantly influence on the nonlinear buckling behavior of plates.

Critical buckling loads of temperature-independent material FGM plates are larger than ones of temperature-dependent material FGM plates.

Nonlinearity of elastic foundation does not influence to the linear critical buckling, reversely, it strongly influences to the nonlinear postbuckling behavior of plates.

Supplemental material

Supplemental Material, Appendix - Nonlinear thermomechanical buckling of sandwich FGM oblique stiffened plates with nonlinear effect of elastic foundation

Supplemental Material, Appendix for Nonlinear thermomechanical buckling of sandwich FGM oblique stiffened plates with nonlinear effect of elastic foundation by Dang Thuy Dong, Vu Hoai Nam, Nguyen Thoi Trung, Nguyen Thi Phuong and Vu Tho Hung in Journal of Thermoplastic Composite Materials

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

ORCID iD

Nguyen Thi Phuong

Supplemental material

Supplemental material is available online.

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