Thermomechanically induced post-buckling analysis of functionally graded material plates with circular cut-outs resting on elastic foundations

Abstract

Post-buckling analysis of functionally graded material (FGM) plates resting on Winkler and Pasternak elastic foundations subjected to thermomechanical loadings with circular cut-outs at centre and random material properties is presented. The material properties of each constituent’s materials, volume fraction index, thermal expansion coefficients, foundation stiffness parameters and thermal conductivities are taken as independent basic random input variables. The basic formulation is based on applying Reddy’s higher order shear deformation theory, which requires C¹ continuous element approximation. A modified form C⁰ continuity is applied in the present investigation. A serum-free expansion medium with mean-centred first-order regular perturbation technique for composite plates is extended for FGM plates to solve the random eigenvalue problem. Typical numerical results are presented to examine the second-order statistics for effect of the volume fractions index, plate length-to-thickness ratios, plate aspect ratios, types of loadings, amplitude ratios, support conditions and various shape and size of holes with random thermomechanical properties. The results obtained by the present solution approach are validated with published papers and the robust method of simulation. It is found that the laminates with round cuts (FGM plates resting on Winkler and Pasternak elastic foundations) have a significant influence on the post-buckling response under Thermomechanical loading conditions. Present investigations are useful for the applications and further research.

Keywords

FGM laminates post-buckling response random properties cut-outs thermomechanically

Introduction

Functionally graded materials (FGMs) consisting of metal and ceramic possess some outstanding thermomechanical properties as fracture toughness and temperature resistance by maintaining the desired integrity. Laminates with circular, square and rectangle cuts are available as part members to further reduce the weight of machines. However, the sizing of structural members with various shaped cut-outs is often determined by stability (buckling) constraints. The buckling behaviour of geometrically nonlinear FGM plates resting on elastic foundations subjected to in-plane loadings with temperature dependent and independent properties is of utmost importance in the design and development of high-performance structural components from the stability point of view. To predict the structural response in terms of stability accurately and to enable a better understanding of the characterization of the actual behaviour under mechanical and thermal loading with various shaped cut-outs are important problems to be paid special attention for reliability of the design.

However, FGMs display inherent uncertainties due to lack of perfect knowledge for fabrication processes. This uncertainty is attributed to temperature variations. As a result, the buckling characteristics of FGM plates become stochastic variables and can be quantified probabilistically. The probabilistic method is a powerful mathematical tool used to incorporate and handle the structural system uncertainties, so that the structural response may not differ from the actual response and the final structures may be safe.

Earlier research works that applied the deterministic approach are available in the literature.^1–18

Relatively little efforts have been made in the past by researchers and investigators on the prediction of post-buckling response statistics of the structures made of laminated composites and FGMs having random system properties. Yang et al.¹⁹ and Onkar et al.²⁰ investigated the generalized buckling of the laminated composite circular cut-out plate with random material properties using classical plate theory combined with first order perturbation technique (FOPT). Singh et al.²¹ investigated the effect of uncertain parameters on the post-buckling of laminates with elastic foundation using higher order shear deformation theory (HSDT) and serum-free expansion medium (SFEM). Jagtap et al.²² presented uncertainties in natural frequency of FGMs laminates resting on supports with independent (TID) and dependent (TD) material properties based on HSDT with von-Karman strains using modified C⁰ continuity.

Kar and Panda^23–28 investigated the post-deformed response of FGM curved shell with edge compression.

Duc et al.^29–47 investigated buckling, post-buckling of plates, vibrations, static and dynamic response of cylindrical shell panels, curved panels, carbon nanotube (CNT)-reinforced panels and piezoelectric circular cylindrical shells with elastic foundations effects analytically.

It is evident from the published papers mentioned above that investigation of stochastic post-buckling response of FGM plates resting on elastic foundations subjected to temperature combined with mechanical force loadings involving randomness in material properties of constituent materials with circular cut-outs using computationally efficient direct iterative based C⁰ nonlinear finite-element method (FEM) in combination with mean-centred first-order regular perturbation technique (FORPT) is not dealt with by researchers to the best of the author’s knowledge.

The results are presented with a new concept in the form of tables, which can suit as a benchmark for future research.

Mathematical formulations

Consider a rectangular FGM plate with holes consisting of metal and ceramic at the top and bottom layers of length a, width b, and total thickness h, defined in (x, y, z) system with x- and –y-axes located in the middle plane and its origin placed at the corner of the plate. Let $(\bar{u}, \bar{v}, \bar{w})$ be the displacements parallel to the x, y, z axes, respectively as shown in Figure 1(a). In the present analysis, for plates with circular holes of various sizes one-quarter plate is used due to the symmetry of the FGM plate. The load–displacement relation between the plate and the supporting foundations is given as

Figure 1.

(a) Geometry of rectangular FGM plate with cut-outs. (b) Quarter of functionally graded laminates with circular cut-out.

P = K_{1} w - K_{2} \nabla^{2} w

where P is the foundation reaction per unit area, $\nabla^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}$ and K₁ and K₂ are linear Winkler (normal) foundation, linear Pasternak (shear layer) foundation parameters, respectively, and w is the transverse displacement of the plate. This model is simply known as Winkler type when K₂ = 0. The circular cut-outs geometry for one-quarter of the FGM plate is given in Figure 1(b). The hole size for circular holes are expressed, respectively, as 2r/a, where r is the radius of the circular holes.

The properties of the FGM plate are assumed to vary according to power law through the thickness of the plate only, such that the top surface z = h/2 is ceramic-rich and the bottom surface z = −h/2 is metal reach. The effective thermomechanical properties of the functionally graded plate within the plate domain are expressed as

\begin{array}{l} E (z, T) = E_{b} (T) + [E_{t} (T) - E_{b} (T)] V_{c} (z) \\ α (z, T) = α_{b} (T) + [α_{t} (T) - α_{b} (T)] V_{c} (z) \\ k (z, T) = k_{b} (T) + [k_{t} (T) - k_{b} (T)] V_{c} (z) \end{array}

where t and b represent the hybrid laminate, respectively, while E, α and k are the effective Young’s modulus, thermal expansion coefficient and thermal conductivity, respectively. The ceramic metal volume fraction V_C is the function of coordinate in the thickness direction, z and is expressed as

V_{C} (z) = {(0.5 + \frac{z}{h})}^{n}, - h / 2 \leq z \leq h / 2, 0 \leq n ≺ \infty

where n is the volume fraction index and is always positive. For n taken as zero, the plate is fully ceramic and for n taken as one, the composition of ceramic and metal is linear. Poisson’s ratio v depends weakly on temperature change and is assumed to be constant.⁴⁸

Displacement field model

In the present study, the assumed displacement field is based on Reddy’s⁴⁹ higher order shear deformation plate theory, which requires C¹ continuous element approximation. Singh et al.⁵⁰ slightly changed the model proposed by Shankara and Iyengar⁵¹ to make the suitability of C⁰. The parameters for the x-, y-, and z-axes for FGM laminates is now written as

\begin{array}{l} \bar{u} = u + f_{1} (z) ψ_{x} + f_{2} (z) ϕ_{x}; \\ \bar{v} = v + f_{1} (z) ψ_{y} + f_{2} (z) ϕ_{y}; \\ \bar{w} = w; \end{array}

where $\bar{u}$ , $\bar{v}$ and $\bar{w}$ denote with reference to the (x, y, z) coordinate axes; u, v and w are the corresponding displacements of a point on the mid plane. Here, $ϕ_{x} = w,_{x}$ and $ϕ_{y} = w,_{y}$ are the slopes along x and y coordinates, and $ψ_{x}$ , $ψ_{y}$ are the expression to the mid-plane of the y- and x-axes. The functions $f_{1} (z)$ and $f_{2} (z)$ given in equation (1) can be written as

f_{1} (z) = C_{1} z - C_{2} z^{3}; f_{2} (z) = - C_{4} z^{3} with C_{1} = 1, C_{2} = C_{4} = 4 h^{2} / 3.

The displacement vector for the modified C⁰ continuous model can be written as

{q} = {[\begin{matrix} u & v & w & ϕ_{y} & ϕ_{x} & ψ_{y} & ψ_{x} \end{matrix}]}^{T},

where comma (,) denotes partial differential.

For the FGM plate considered here, the vector consisting of (in term of midplane shape change, rotation of normal and higher terms) von-Karman type and thermal vectors associate with the displacement are expressed as

{{\bar{ε}}_{i j}} = {{\bar{ε}}_{i j}^{L}} + {{\bar{ε}}_{i j}^{N L}} - {{\bar{ε}}_{i j}^{T}} (i, j = 1, 2, ..., 6)

where ${{\bar{ε}}_{i j}^{L}}$ , ${{\bar{ε}}_{i j}^{N L}}$ and ${{\bar{ε}}_{i j}^{T}}$ are the linear, non-linear and thermal strain vectors, respectively.

From equation (6), this can be written as

{\bar{ε}}_{i j}^{L} = H_{m n} ε_{k l}^{L}

where H_mn is the related to z expressed in Appendix 1, equation (2), and $ε_{k l}^{L}$ is expressed as

ε_{k l}^{L} = {\begin{matrix} ε_{1}^{0} & ε_{2}^{0} & ε_{6}^{0} & k_{1}^{0} & k_{2}^{0} & k_{6}^{0} & k_{1}^{2} & k_{2}^{2} & k_{6}^{2} & ε_{4}^{0} & ε_{5}^{0} & k_{4}^{2} & k_{5}^{2} \end{matrix}}^{T}

Assuming that the strains are much smaller than the rotations (in the von-Karman sense), one can rewrite the non-linear part ${{\bar{ε}}_{i j}^{N L}}$ given in equation (6) following Jagtap et al.²²

The temperature field for non-uniform temperature change is expressed as in the literature.⁴⁸

Δ T = T (z) - T_{0}

where T₀ is initial temperature and can be expressed as

T (z) = T_{b} + (T_{t} - T_{b}) η (z)

where T(z) is the temperature distribution along z direction, t and b are referred as the top and bottom surfaces and parameter η(z) can be written as

η (Ζ) = \frac{1}{c} [\begin{array}{l} (0.5 + \frac{z}{h}) - \frac{k_{t b}}{(n + 1) k_{b}} {(0.5 + \frac{z}{h})}^{n + 1} + \frac{{k_{t b}}^{2}}{(2 n + 1) {k_{b}}^{2}} {(0.5 + \frac{z}{h})}^{2 n + 1} - \frac{{k_{t b}}^{3}}{(3 n + 1) {k_{b}}^{3}} {(0.5 + \frac{z}{h})}^{3 n + 1} \\ + \frac{{k_{t b}}^{4}}{(4 n + 1) {k_{b}}^{4}} {(0.5 + \frac{z}{h})}^{4 n + 1} - \frac{{k_{t b}}^{5}}{(5 n + 1) {k_{b}}^{5}} {(0.5 + \frac{z}{h})}^{5 n + 1} \end{array}]

where $C = 1 - \frac{k_{t b}}{(n + 1) k_{b}} + \frac{{k^{2}}_{t b}}{(2 n + 1) {k^{2}}_{b}} - \frac{{k^{3}}_{t b}}{(3 n + 1) {k^{3}}_{b}} + \frac{{k^{4}}_{t b}}{(4 n + 1) {k^{4}}_{b}} - \frac{{k^{5}}_{t b}}{(5 n + 1) {k^{5}}_{b}}$

Here k, z and n indicate thermal conductivity, distance from central axis and volume fraction, respectively, with k_tb= k_t − k_b.

For uniform temperature change, equation (10) can be written as follows:

T (z) = T_{0} + (T_{t} - T_{b})

For uniform and non-uniform temperature rise, the initial (T₀) and bottom (T_b) temperatures of the plate are assumed to be 300 K and T_c as 600 K, respectively, throughout the analysis, unless otherwise mentioned.

Constitutive relations

The constitutive law of thermoelasticity for material under consideration relates the stresses with strains in a plane stress state for anisotropic layer of a laminate is expressed as

{\bar{σ}}_{i j} = {\bar{C}}_{i j k l} {\bar{ε}}_{i j} (i, j = 1, 2, ...6)

where ${\bar{C}}_{i j k l}$ is the elastic material constant matrix as defined in Appendix 1, equation (2A).

The strength of the FGM laminate consisting of two types of strengths undergoing deformation is written as

Π_{1} = Π_{L} + Π_{N L}

where $Π_{L}$ and $Π_{N L}$ are different. From equation (14), ( $Π_{a}$ ) is denoted as

Π_{L} = \int_{Ω} \frac{1}{2} {\bar{C}}_{i j k l} {\bar{ε}}_{i j}^{L} {\bar{ε}}_{k l}^{L} d Ω = \int_{Δ} \frac{1}{2} ε_{i j} D_{m n} ε_{k l} d Δ

$D_{m n}$ is expressed separately.

From equation (15) the nonlinear strain energy ( $Π_{N L}$ ) of the plate can be rewritten as

\begin{array}{l} Π_{N L} = \int_{Ω} \frac{1}{2} {\bar{ε}}_{i j}^{L} D_{3} {\bar{ε}}_{i j}^{N L} d Ω + \frac{1}{2} \int_{Ω} {\bar{ε}}_{k l}^{N L} D_{4} {\bar{ε}}_{i j}^{L} d Ω \\ + \frac{1}{2} {\int_{Ω} {\bar{ε}}_{k l}}^{N L} D_{5} {\bar{ε}}_{i j}^{N L} d Ω (i,j,k,l = 1, 2, 3) \end{array}

where Ω, ${\bar{ε}}^{L}_{i j}$ and ${\bar{ε}}^{N L}$ denotes un-deformed configuration of plate and linear and non-linear strain tensors, respectively.

Equation (16) is expressed as

\begin{array}{l} Π_{N L} = \frac{1}{2} \int_{Ω} A_{n i j} ϕ_{n i j} D_{3} {\bar{ε}}_{i j}^{L} d Ω + \frac{1}{2} \int_{Ω} {\bar{ε}}_{i j}^{L} D_{4} A_{n k l} ϕ_{n k l} d Ω \\ + \frac{1}{2} \int_{Ω} A_{n i j} ϕ_{n i j} D_{5} A_{n k l} ϕ_{n k l} d Ω, (i, j, k, l = 1, 2, 3) \end{array}

where D₃, D₄ and D₅ are the laminate stiffness matrices of the plate expressed in Appendix 1, equation (4A). The ( $Π_{2}$ ) storage by heat load (change in temperature across the thickness) due to change in temperature, pre-buckling forces, that is, in-plane heat effected compressive internal resisting forces resultants in the plate are generated. These stress resultants are the reason for the buckling. The potential energy due to the in-plane thermal stress resultants is expressed as

\begin{array}{l} \prod_{2} = \frac{1}{2} \int_{A} [N_{x t m} {(w,_{x})}^{2} + N_{y t m} {(w,_{y})}^{2} + 2 N_{x y t m} (w,_{x}) (w,_{y})] d A \\ = \frac{1}{2} {\int_{A} {\begin{matrix} w,_{x} \\ w,_{y} \end{matrix}}}^{T} [\begin{matrix} N_{x t m} & N_{x y t m} \\ N_{x y t m} & N_{y t m} \end{matrix}] {\begin{matrix} w,_{x} \\ w,_{y} \end{matrix}} d A \end{array}

where N_xtm, N_ytm and are written as N_xm − N_xt, N_ym, − N_yt and N_xym − N_xyt respectively. For mechanical application, in the above expression, N_xt, N_yt, N_xyt are assumed as zero.

Forces causing change in dimensions due to temperature can be expressed as

\begin{array}{l} Π_{3} = \frac{1}{2} \int_{A} {K_{1} w^{2} + K_{2} [{(w_{, x})}^{2} + {(w_{, y})}^{2}]} d A, \\ Π_{3} = \frac{1}{2} \int_{A} {\begin{matrix} w \\ w_{, x} \\ w_{, y} \end{matrix}}^{T} [\begin{matrix} K_{1} & 0 & 0 \\ 0 & K_{2} & 0 \\ 0 & 0 & K_{2} \end{matrix}] {\begin{matrix} w \\ w_{, x} \\ w_{, y} \end{matrix}} d A \end{array}

Finite element models

Strain energy of the plate element

In the present study, a C⁰ nine-noded isoparametric FE with 7 DOFs per node is employed.

{q} = \sum_{i = 1}^{N N} ϕ_{i} {q}_{i}; x = \sum_{i = 1}^{N N} ϕ_{i} x_{i}; and y = \sum_{i = 1}^{N N} ϕ_{i} y_{i}

Following this and applying FEM model this can be written as

Π_{1} = \sum_{e = 1}^{N E} {Π_{1}}^{(e)}

\begin{array}{l} Π_{1} = \frac{1}{2} \sum_{e = 1}^{N E} [{q_{i}}^{T (e)} {[K_{i j}]}^{(e)} {q_{j}}^{(e)} - {q_{i}}^{T (e)} {[{F_{i}}^{T}]}^{(e)}] = \frac{1}{2} {q_{i}}^{T} [K_{i j}] {q_{j}} - {q_{i}}^{T} [{F_{i}}^{T}] \end{array}

where $[K_{i j}] = [K_{L_{i j}} + K_{N L_{i j}}]$ with $[K_{N L_{i j}}] = \frac{1}{2} [K_{N L_{1 i j}}] + [K_{N L_{2 i j}}] + \frac{1}{2} [K_{N L_{3 i j}}]$ and where $[K_{L_{i j}}] and [K_{N L_{i j}}]$ are expressed as global matrix of the plate and are expressed in Appendix 1, equation (5A).

FE model is written as

Π_{2} = \sum_{e = 1}^{N E} {Π_{2}}^{(e)} = \frac{1}{2} \sum_{e = 1}^{N E} {q_{i}}^{T (e)} λ {[K_{(G) i j}]}^{(e)} {q_{j}}^{(e)} d A = \frac{1}{2} λ {q_{i}}^{T} [K_{(G) i j}] {q_{j}}

where $λ$ and $[K_{{(G)}_{i j}}]$ are expressed as the deformation load due to temperature parameters and the global matrix, respectively.

The solution procedure adopted for post-deformed investigations of FGM laminates resting on supports and undergoing thermomechanical loadings can be derived using mathematical principle, which is generalization of the principle of virtual displacement. For the displacement field of the buckling, the minimization of π(π₁+π₂+π₃) with respect to generalized displacement vector and after simplification, using this, equations (22) and (23) can be represented following Reddy JN, as⁵²

[K_{i j}] {q_{i}} = [{F_{i}}^{T}]

For the critical buckling state corresponding to the neutral equilibrium condition, the second variation of total potential energy (π) must be zero. Following this condition, one obtains the standard eigenvalue problem

| {[K_{i j}] + λ_{i} [K_{{(G)}_{i j}}]} | {q_{i}} = 0

Using this, equation (24) can be rewritten as

[K_{i j}] {q_{i}} = λ_{i} [K_{{(G)}_{i j}}] {q_{i}}

The plate stiffness matrix $[K_{i j}]$ consisting of linear and nonlinear stiffness matrices and geometric stiffness matrix are random in nature, being dependent on the system properties of the structure.

Direct iterative-based SFEM application

The full procedure is explained in Lal et al.⁵³

The detailed DISFEM solution procedure for post-buckling analysis is shown in the flowchart of Figure 4.

Solution approach: Perturbation technique

In the present analysis, the material properties of constitute materials such as $E_{c}, E_{m}, υ_{c}, υ_{m}, n, α_{c}, α_{m}, k_{1}, k_{2}, k_{c} and k_{m}$ volume fraction index are treated as independent random variables (RVs).

For formulation, $[K_{i j}]$ $λ_{N L_{i}},$ can be easily extended for even dependent random variables.

In general, without any loss of generality, any arbitrary random variable can be represented as the sum of its mean and zero mean random part, denoted by superscripts ‘d’ and ‘r’, respectively asin Zhang et al.⁵⁴

[{K_{i}}_{j}] = [K_{i j}^{d}] + [K_{i j}^{r}], [K_{{(G)}_{i j}}] = [K_{​_{{(G)}_{i j}}}^{d}] + [K_{​_{{(G)}_{i j}}}^{r}], λ_{i} = {λ_{i}}^{d} + {λ_{i}}^{r}, {q_{i}} = {{q_{i}}^{d}} + {{q_{i}}^{r}}

where $[K_{i j}^{d}]$ and $[K_{{(G)}_{i j}}^{d}]$ are the mean elastic (linear and nonlinear) and geometric stiffness matrices of the structures, respectively. Correspondingly, $[K_{i j}^{r}]$ and $[K_{_{{(G)}_{i j}}}^{r}]$ are the first-order derivatives that are sufficient to yield results with the desired accuracy having low variability as in the sensitive applications. After comparing the zeroth- and first-order terms one obtains the following.

Zeroth-order perturbation equation

([{K_{i j}}^{d}] + {λ_{i}}^{d} [K_{{(G)}_{i j}}^{d}]) {{q_{i}}^{d}} = 0 (i = 1, 2, 3, ..., n : no sum over k)

First-order perturbation equation

[K_{i j}^{d}] {{q_{i}}^{r}} + [K_{i j}^{r}] {{q_{i}}^{d}} = {λ_{i}}^{d} [K_{{(G)}_{i j}}^{r}] {{q_{i}}^{d}} + {λ_{i}}^{d} [K_{{(G)}_{i j}}^{d}] {{q_{i}}^{r}} + {λ_{i}}^{r} [K_{{(G)}_{i j}}^{d}] {{q_{i}}^{d}} (i = 1, 2, 3, ..., n : no sum over k)

Here, equation (28) is the deterministic equation relating to the mean eigenvalues and the corresponding mean eigenvectors, which can be determined by conventional eigen-solution procedures. Equation (29) is the random equation, defining the stochastic nature of the mechanical and thermal buckling, which cannot be solved using the conventional method. For this, further analysis is required.

Variance of post-buckling load

The above expression is used to obtain the eigenvalue with respect to the basic random variables, which are then used to the post-deformation due to force covariance.

To obtain the statistics of critical post-buckling load, multiply both sides of equation (29) by mean eigenvector ${q_{i}^{d}}$ computed from equation (28) for minimum mean eigenvalue ${λ_{c r}^{d}}$ . This gives

{q_{i}^{d}} ([K_{i j}^{d}] + λ_{c r}^{d} [K_{{(G)}_{i j}}^{d}]) {{q_{i}}^{r}} = - {λ_{c r}}^{r} ({q_{i}^{d}} [K_{{(G)}_{i j}}^{d}] {q_{j}^{d}}) - {q_{i}^{d}} ([K_{i j}^{r}] + λ_{c r}^{d} [K_{{(G)}_{i j}}^{r}]) {q_{j}^{d}}

Since both $[K_{i j}^{d}]$ and $[K_{{(G)}_{i j}}^{d}]$ are symmetric, therefore the left-hand side of equation (30) equals zero by definition of the zeroth-order equation. By employing $[K_{{(G)}_{i j}}^{d}]$ orthonormal conditions (using orthogonal properties), the first term on the right-hand side equation reduces to ${λ_{c r}}^{r}$ .

The expression for the first-order derivative of the eigenvalue is then written as

{λ_{c r}}^{r} = - {q_{i}^{d}} ([K_{i j}^{r}] + [K_{{(G)}_{i j}}^{r}]) {q_{j}^{d}} (i, j = 1, 2, ..., n)

Using equation (22), the variances of the eigenvalue can now be expressed as in the work by Singh et al.²¹

Var (λ_{c r}) = \sum_{j = 1}^{p} \sum_{k = 1}^{p} λ_{i, j}^{d} λ_{i, k}^{d} Cov ({b_{j}}^{r}, {b_{k}}^{r})

where $Cov ({b_{j}}^{r}, {b_{k}}^{r})$ is the cross-variance between ${b_{j}}^{r}$ and ${b_{k}}^{r}$ . The systematic overview of the present study is shown in Figure 3.

Numerical results and explanation

A computer program has been developed in MATLAB [R2010a] environment to compute the second-order statistics of post-buckling response of the FGM plates resting on Winkler and Pasternak elastic foundation subjected to thermomechanical loading with circular cut-outs using the proposed DISFEM probabilistic method. The validity and efficacy of the proposed algorithm is examined by comparing the results with those available in the literature and the independent method of simulation (MCS). A 567-noded laminate is used for the HSDT study for discretizing the laminate. For the computation of results, non-uniform temperature with TID and TD material properties have been considered.

For the computation of results, full integration schemes (3 × 3) are used for thick plate and selective integration scheme (2 × 2) for thin plate. In the present analysis, foundation stiffness parameters, various support conditions, volume fraction index and hole sizes are used to check the efficacy of the present model. However, the formulation and code are useful tools.

Throughout the analysis, it is assumed that the materials are perfectly elastic during the deformation.

In the present study, the following sets of boundary conditions are used.

\begin{array}{l} SSSS: \\ u = v = w = θ_{y} = ψ_{y} = 0, at x = 0, a; u = v = w = θ_{x} = ψ_{x} = 0 at y = 0, b \\ CCCC: \\ u = v = w = ψ_{x} = ψ_{y} = θ_{x} = θ_{y} = 0, at x = 0, a and y = 0, b; \\ CSCS: \\ u = v = w = ψ_{x} = ψ_{y} = θ_{x} = θ_{y} = 0, at x = 0 and y = 0; \\ v = w = θ_{y} = ψ_{y} = 0, a t x = a u = w = θ_{x} = ψ_{x} = 0, at y = b \end{array}

[TD] FGM laminates are investigated. COV is defined as the coefficient of variation. The dimensional mean is represented as dimensional post-buckling thermomechanical critical load. Being the linear nature of variations of COV as mentioned earlier and passing through origin, the results are only presented by coefficient of correlation (COC) of system properties equal to 0.1. However, the investigated outcomes for standard deviation revealed that the DISFEM approach is of 0.2.^54,55

Ti-6Al-4V/ZrO₂ and SUS304/ Si₃N₄ types of FGM properties for TID and TD material properties are used for computation.¹⁶ The assumed basic random input variables (b_i) are sequenced.

b_{1} = E_{c}, b_{2} = E_{m}, b_{3} = υ_{c}, b_{4} = υ_{m}, b_{5} = α_{c}, b_{6} = α_{m}, b_{7} = k_{1}, b_{8} = k_{2}, b_{9} = k_{c}, b_{10} = k_{m} and b_{11} = n

where $E_{c}, E_{m}, υ_{c}, υ_{m}, α_{c}, α_{m}, k_{1}, k_{2}, k_{c}, k_{m} and n$ are Young’s moduli, Poisson’s ratios and volume fraction index with other notations.

In the present analysis, mean-centred first-order perturbation technique has been used to compute the numerical results, keeping in mind the complicity and difficulty of using higher order perturbation at the cost of very little improvement in results, especially for the non-linear problem. It is also noted that the results are presented for lower amplitude ratios (W_max/h = 0.2 to 0.8) due to convergence and computational time for holes. For the computation of results, the following dimensionless post-buckling parameters are used and expressed:

λ_{c r} = \frac{N_{c r} a^{2}}{π^{2} D_{0}}

where $D_{0} = \frac{E_{c} h^{3}}{12 (1 - {ν_{c}}^{2})}$ and $λ_{T c r} = α {(Δ T)}_{c r} \times 10^{3}$ ; $λ_{c r}$ and $λ_{T c r}$ are the dimensionless mean post-buckling load and temperature parameters, respectively. In the above expression, $N_{c r}$ and ${(Δ T)}_{c r}$ are the dimensional exceptional deformed loading force and express thermal condition, respectively. Also in the expression ${(Δ T)}_{c r}$ is expressed as $λ_{T} Δ T$ , where $λ_{T}$ is the critical thermal buckling force parameter and $Δ T$ is defined as T_c − T_m. For the mean dimensional post-buckling analysis of FGM plate, the dimensionless parameters $λ_{c r}$ and $λ_{T c r}$ are used. While for the calculation of COV, the dimensional mean $N_{c r}$ and ${(Δ T)}_{c r}$ are used.

It is noted that the values given in parenthesis in the tables are the dimensionless mean values of thermomechanical post-buckling load and temperature. Throughout the analysis, both type of loadings, uniaxial and biaxial compression without considering shear effect, have been used. For numerical illustration, the following parameters are taken as plate thickness ratio (a/h = 10, 15, 40, 50, 60, and 100), plate aspect ratio (a/b = 1 and 1.5), amplitude ratio (W_max/h = 0.2, 0.4, 0.6 and 0.8) and volume fraction index (n = 0.5, 1, 5 and 10).

Validation study for mean post-deformed behaviour

To assure the accuracy and proficiency of the present outlined stochastic formulation, three test examples are analyzed for post-deformed behaviour of FGM plates resting on elastic foundations. For validation purposes, the plates studied are subjected to uniaxial force application with uniform or non-uniform thermal condition distribution. The characteristics of constituents (metal and ceramic) are studied at 300 K.

We first consider the accuracy of the present deterministic FEM by comparing the results with those available in the literature. The results of initial buckling load and temperature with and without cut-out are validated through numerical examples.

Validation study of buckling load parameter (Ń_cr = N_cra²/π²D₀) of the SSSS and CCCC supported square Al/ZrO₂ plate with circular cut-out under uniaxial compression with a/h = 100, with those available in the study by Zhao et al.¹⁶ and Lal et al.⁵³ as shown in Tables 1 and 2 and Figure 2. From these tables and Figure 2, it can be seen that the SFEM applied is in very good agreement.

Figure 2.

Validation of mean post-buckling loading forces of clamped square FGM plates subjected to thermomechanical loading, simple support and biaxial compression.

Table 1.

Validation study of buckling load parameter Ń_CR = N_cra²/π²D₀ of the clamped square Al/ZrO₂ plate, under uniaxial compression with a/h = 100, mode = 1 and hole size = 2r/a circular hole; SSSS support.

Type of hole	Hole size		Ń_cr = N_cra²/π²D₀
Type of hole	Hole size		n = 0	n = 0.5	n = 1	n = 2	n = 5
Circular	0.1	Present HSDT	8.0675	6.4396	5.8303	5.3500	4.8598
		Lal et al.⁵³	8.0675	6.4396	5.8303	5.3500	4.8598
		Zhao et al.¹⁶	8.0873	6.4749	5.5973	5.1270	4.6325
	0.2	Present HSDT	6.7773	5.3994	4.8902	4.4963	4.0969
		Lal et al.⁵³	6.9279	5.5193	4.9989	4.5963	4.1880
		Zhao et al.¹⁶	6.9711	5.4112	4.6858	4.0808	4.0609
	0.4	Present HSDT	5.2575	4.1788	3.7855	3.4882	3.1898
		Lal et al.⁵³	5.8417	4.6431	4.2062	3.8757	3.5442
		Zhao et al.⁶	5.2611	4.1152	3.6761	3.3672	3.1238

Al/ZrO2: HSDT: higher order shear deformation theory.

Table 2.

Validation study of buckling load parameter Ń_CR = N_cra²/π²D₀ of the simply supported square Al/ZrO₂ plate with circular hole under uniaxial compression, a/h = 100, mode = 1, hole size = 2r/a circular hole; CCCC support.

Type of hole	Hole size		Ń_cr = N_cra²/π²D₀
Type of hole	Hole size		n = 0	n = 0.5	n = 1	n = 2	n = 5
Circular	0.1	Present DT	18.3866	14.4852	12.9406	11.7850	10.8136
		Lal et al.⁵³	18.4292	14.5493	12.9960	11.8131	10.8065
		Zhao et al.⁶	18.3518	14.3866	14.2659	11.4694	10.1226
	0.2	Present HSDT	15.6728	12.3126	11.0015	10.0440	9.2557
		Lal et al.⁵³	15.6843	12.3485	11.0320	10.0523	9.2338
		Zhao et al.¹⁶	15.1882	12.5212	11.2438	10.3533	8.3753

SDT: shear deformation theory; HSDT: higher order shear deformation theory.

Table 3 explains using the following data: *D = Eh³/12(1 − υ), E₁ = 130.0 GPa, E₂ = E₃ = 10.0 GPa, G₁₂ = G₁₃ = 5.0 GPa, ν₁₂ = ν₁₃ = ν₂₃ = ν₃₂ = 0.35. The plate dimension considered is 2000 × 2000 mm². The thickness of each layer of this eight-layer laminates is 1.5 mm. It is clear from the table that the present FEM [HSDT] results are in good agreement with those of the theoretical and ABAQUS results of Al Qablan et al.²⁹

Table 3.

Comparison of buckling load between theoretical method and finite element model for isotropic plate without cut-outs.

Theoretical method	E (Pa)	υ	b (m)	h (m)	Non-dimensional buckling load (theoretical) N_crb^2/Eh3	Non-dimensional buckling load (ABAQUS) N_crb2/Eh3	Non-dimensional buckling load (FE model) N_crb2/Eh3
Theoretical method	E (Pa)	υ	b (m)	h (m)	Al Qablan et al.²⁹		Present HSDT
Uniaxial case N_cr = 4π²D/b²	2×10¹¹	0.3	2.0	0.02	3.615	3.623	3.6794
Biaxial case N_cr = 2π²D/b²	2×10¹¹	0.3	2.0	0.02	1.807	1.811	1.8398

FE: finite element; HSDT: higher order shear deformation theory.

^aD=Eh³/12(1 − υ), E1 = 130.0 GPA, E2 = E3 = 10.0 GPA, G12 = G13=5.0 GPA, ν12 = ν13 = ν23 = ν32 = 0.35. The plate dimension considered is 2000 × 2000 mm². The thickness of each layer of this eight-layer laminate is 1.5 mm.

Validation study for stochastic post-buckling response

Figure 3 shows the compared results of DISFEM with independent MCS. For the MCS approach, the sample values are generated using MATLAB software to fit the desired mean and standard deviation (SD) using Gaussian probabilistic distribution function. The applicability of MCS outcomes is utilised by taking the different numbers. Material properties E_c, which are given as input to the present deterministic equation (28) post-buckling load is calculated. The MCS approach is used only for validation purposes.

Figure 3.

Validation for the coefficient of variations due to uncertainties in properties (b_i, I = 1,…, 4) for CCCC support, biaxial compression, Al/ZrO₂ laminate with n = 0.2 and 0.8 and a/h = 10.

Figure 4 shows the validation of dimensionless mean post-deforming loading force (Ń_cr = N_cra²/π²D₀) of square Al/ZrO₂ plate with circular cut-out under bixial compression with a/h = 100 of clamped square FGMs circular cut-out laminate under combined temperature and mechanical loading, clamped support, biaxial compression with those available in the literature.^16,53

Figure 4.

Validation of dimensionless mean post-deforming loading force of clamped square FGM circular cut-out laminates under combined temperature and mechanical loading, clamped support and biaxial compression.

Flow chart of the solution expression of stochastic procedure is given in Figure 5.

Figure 5.

Flow diagram of stochastic procedure for post-buckling.

Parametric study for second-order statistics (dimensionless mean and COV) of post-buckling response

It is observed that plate with TID and TD material properties and Pasternak elastic foundations increase the mean and COV compared to Winkler elastic foundations and plates without foundation (Table 4). TD conditions have significant effects when the volume fraction index is increased.

Table 4.

Effect of thermomechanical loading, volume fraction index with random material properties {bi (i = 1,…, 7) = 0.1} on the dimensionless mean and COV of post-buckling load and temperature of SSSS-supported FGM square plate without and with Winkler (k₁ = 100, k₂ =0) and Pasternak (k₁ = 100, k₂ = 10), elastic foundations, circular hole (hole size = 0.1) having TID and TD material properties, a/h = 50, and W_max/h = 0.4.^a

	Hole type	Uniaxial				Biaxial
Loading	Circular	TID		TD		TID		TD
Loading	Circular	n = 1	n = 5	n = 1	n = 5	n = 1	n = 5	n = 1	n = 5
Thermomechanical	(k₁ = 000, k₂ = 0)	0.0610 (6.8064)	0.0590 (5.5428)	0.0613 (4.9475)	0.0613 (4.2657)	(3.5390) 0.0608	(2.8799) 0.0585	0.0610 (2.5723)	(2.2166) 0.0607
	Linear	5.1313	4.3022	3.8366	3.3896	2.6614	2.2294	1.9895	1.7567
	(k₁ = 100, k₂ = 0)	0.0659 (8.2108)	0.0618 (6.9502)	0.0658 (5.9165)	0.0626 (5.2354)	0.0664 (4.3088)	0.0617 (3.6503)	(3.1022) 0.0659	(2.7468) 0.0620
	Linear	6.5449	5.7133	4.8137	4.3662	3.4334	3.0020	2.5209	2.2884
	(k₁ = 100, k₂ = 10)	0.0721 (10.9067)	0.0671 (9.5809)	0.0718 (7.8064)	0.0670 (7.1031)	0.0743 (5.8575)	0.0697 (5.1982)	0.0733 (4.1678)	(3.8121) 0.0682
	Linear	9.2744	8.3882	6.7151	6.2478	4.9818	4.5496	3.5867	3.3539

COV: coefficient of variance; TID: independent material properties; TD: dependent material properties; FGM: functionally graded material.

^a The dimensionless mean mechanical post-buckling load and temperature are given in brackets. Uniaxial and biaxial compression.

The dimensionless mean thermomechanical post-buckling load and temperature are given in brackets (Table 5). It is seen that there is significant biaxial compression on the pates without support and with support when there is increase in the volume fraction index. The mean and COV values for uniaxial and Pasternak supports are more differentiated with biaxially compressed laminates. It is observed that the laminate with circular hole shows appreciable increase in COV and the corresponding higher average post-deformed force and temperature. This is because the stress concentration in the circular holed plate is lower.

Table 5.

Effect of thermomechanical loading, volume fraction index and various hole sizes with random material properties {b_i (i = 1,…, 7) = 0.1} on the dimensionless mean and COV of post-buckling load and temperature of SSSS-supported FGM square plate without and with Winkler (k₁ = 100, k₂ = 0) and Pasternak (k₁ = 100, k₂ = 10) elastic foundations, circular hole having TD material properties, a/h = 10, and W_max/h = 0.4.^a

	Hole type	Uniaxial				Biaxial
Loading	Circular	n = 0.5		n = 10		n = 0.5		n = 10
Loading	Circular	Hole size = 0.1	Hole size = 0.2	Hole size = 0.1	Hole size = 0.2	Hole size = 0.1	Hole size = 0.2	Hole size = 0.1	Hole size = 0.2
Thermomechanical	(k₁ = 00, k₂ = 0)	(4.4605) 0.0664	(4.1746) 0.0664	(3.3686) 0.0646	(3.1500) 0.0651	(2.3072) 0.0658	(2.1958) 0.0660	(1.7415) 0.0637	1.6592) 0.0641
	Linear	3.3802	3.1392	2.6301	2.4439	1.7421	1.6622	1.3560	1.2949
	(k₁ = 100, k₂ = 0)	(5.0811) 0.0761	(5.0692) 0.0733	(4.2441) 0.0661	(4.0460) 0.0645	(2.8515) 0.0714	(2.5093) 0.0718	(2.2839) 0.0617	(2.0160) 0.0621
	Linear	4.3757	4.0611	3.6057	3.3367	2.2823	2.0191	1.8958	1.6811
	(k₁ = 100, k₂ = 10)	(7.0145) 0.0886	(6.5583) 0.0776	(6.0506) 0.0717	(5.5000) 0.0677	(3.9190) 0.0786	(3.4730) 0.0789	(3.3492) 0.0678	(2.9769) 0.0681
	Linear	6.1754	5.6282	5.2378	4.8000	3.3527	2.9780	2.9637	2.6363

COV: coefficient of variance; TD: dependent material properties; FGM: functionally graded material.

^a The dimensionless mean thermomechanical post-buckling load and temperature are given in brackets.

It is noticed that when the amplitude ratios are increased, the mean values and COV decrease for uniaxially and biaxially compressed plates without elastic foundations (Table 6). The plates resting on Pasternak elastic foundations have significant effects on mean and COV when there is increase in amplitude ratios; however, these values decrease for biaxially compressed plates.

Table 6.

Effect of thermomechanical loading and amplitude ratios with random material properties {b_i (i = 1,…, 7) = 0.1} on the dimensionless mean and COV of post-buckling load and temperature of SSSS supported FGM square plate without and with Winkler (k₁ = 100, k₂ = 0) and Pasternak (k₁ = 100, k₂ = 10) elastic foundations, circular holes having TD material properties, a/h = 60, hole size = 0.2 and n = 5.^a

Loading	Hole type	Uniaxial				Biaxial
	Hole type	W_max/h = 0.2	W_max/h = 0.4	W_max/h = 0.6	W_max/h = 0.8	W_max/h = 0.2	W_max/h = 0.4	W_max/h = 0.6	W_max/h = 0.8
	Circular
Thermomechanical	(k₁ = 000, k₂ = 0)	(3.5287) 0.0688	(4.1301) 0.0613	(5.0287) 0.0540	(5.1812) 0.0529	(1.8748) 0.0686	(2.1857) 0.0607	(2.6561) 0.0528	(3.2861) 0.0460
	Linear	3.2409	3.2409	3.2409	3.2409	1.7239	1.7239	1.7239	1.7239
	(k₁ = 100, k₂ = 0)	(4.4952) 0.0710	(5.0820) 0.0651	(5.9671) 0.0596	(5.8031) 0.0684	(2.4291) 0.0717	(2.7414) 0.0652	(3.2132) 0.0580	(3.8428) 0.0512
	Linear	4.2101	4.2101	4.2101	4.2101	2.2775	2.2775	2.2775	2.2775
	(k₁ = 100, k₂ = 10)	(6.2760) 0.1715	(6.8858) 0.1591	(7.8644) 0.1517	(7.8835) 0.1466	(3.5186) 0.1896	(3.8345) 0.1980	(4.3103) 0.1993	(4.9417) 0.1945
	Linear	5.9907	5.9907	5.9907	5.9907	3.3649	3.3649	3.3649	3.3649

COV: coefficient of variance; TD: dependent material properties; FGM: functionally graded material.

^a The dimensionless mean thermomechanical post-buckling load and temperature are given in brackets.

It is seen that on varying the plate thickness ratios and volume fraction index with increase in hole size the mean value decreases, while COV increases for plates without foundations (Table 7). When plates are resting on Winkler and Pasternak elastic foundations, the mean values increase on increasing the volume fraction index. COV is significant for thin plates compared to thick plates. It is also expected that laminates with holes show less dimensionless average and the corresponding COV of deformed load and thermal condition. This is because post-buckling stiffness of the plate decreases for holed plates. It is concluded that from the reliability point of view, circular holed plates are preferred for small hole sizes.

Table 7.

Effect of combined temperature and mechanical loading, n, a/h and various hole sizes with uncertain characteristics {b_i (i = 1,…, 7) = 0.1} on the dimensionless mean and COV of post-deforming force and temperature of SSSS-supported, biaxial compressed FGM circular plate without and with Winkler (k₁ = 100, k₂ = 0) and Pasternak (k₁ = 100, k₂ = 10) elastic foundations circular hole, TD, W_max/h = 0.4.^a

	Hole type	a/h =20				a/h =100
Loading	Circular	n = 0.5		n = 10		n = 0.5		n = 10
Loading	Circular	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3
Thermomechanical	(k₁ = 000, k₂ = 0)	(2.3069) 0.0661	(2.2086) 0.0669	(1.750) 0.0651	(1.675) 0.0654	(2.7428) 0.0677	(2.5956) 0.0666	(2.0798) 0.0670	(1.9704) 0.0656
	Linear	1.7787	1.7006	1.3900	1.3297	2.1705	2.0182	1.6918	1.5770
	(k₁ = 100, k₂ = 0)	(2.783) 0.0738	(2.679) 0.0737	(2.2242) 0.0666	(2.1364) 0.0669	(3.2659) 0.0737	(3.1410) 0.0736	(2.6031) 0.0668	(2.5163) 0.0665
	Linear	2.2614	2.1675	1.8693	1.7921	2.6887	2.5650	2.2108	2.1241
	(k₁ = 100, k₂ = 10)	(3.6901) 0.3163	(3.5406) 0.3265	(3.1044) 0.1420	(2.9766) 0.1420	(4.3036) 0.3196	4.1572) 0.3210	(3.6396) 0.1577	(3.5313) 0.1592
	Linear	3.1848	3.0430	2.7562	2.6268	3.7191	3.5763	3.2404	3.1344

COV: coefficient of variance; TD: dependent material properties; FGM: functionally graded material.

^a The dimensionless mean thermomechanical post-buckling load and temperature are given in brackets.

For a/b = 2, n = 10), and cut diameter of 0.3, the COV varies for Pasternak support changes because of stress concentration near the round cut, which is responsible for decrease of post-buckling internal resisting force, with increase in round cut size (Table 8). Here the round cut size, aspect ratio, fraction index and supports matter much.

Table 8.

Effect of combined temperature and mechanical loading, n, a/b and various hole sizes with uncertain properties {b_i (i = 1,…, 7) = 0.1} on the dimensionless mean and COV of post-buckling load and temperature of SSSS, uniaxial compressed, functionally graded laminate (k₁ = 100, k₂ = 0) and (k₁ = 100, k₂ = 10) elastic foundations, circular hole, TD properties, a/h = 50, W_max/h = 0.4.^a

	Hole type	a/b = 1				a/b = 2.0
Loading	Circular	n = 0.5		n = 10		n = 0.5		n = 10
Loading	Circular	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3
Thermomechanical	(k₁ = 000, k₂ = 0)	(4.8274) 0.0675	(4.7154) 0.0684	(3.6665) 0.0671	(3.5803) 0.0693	(4.9620) 0.0768	(1.2315) 0.0793	(3.4341) 0.0943	(0.9339) 0.0954
	Linear	3.7646	3.6555	2.9408	2.8582	4.3187	0.5386	2.9229	0.3587
	(k₁ = 100, k₂ = 0)	(5.6754) 0.0736	(5.4605) 0.0745	(4.5128) 0.0676	(4.3250) 0.0703	(1.1576) 0.2859	(1.2976) 0.0788	(3.3782) 0.0959	(1.1388) 0.0695
	Linear	4.6538	4.4613	3.8238	3.6508	4.3579	0.5987	2.9720	0.4249
	(k₁ = 100, k₂ = 10)	(7.2863) 0.3081	(6.7957) 0.2020	(6.0843) 0.1276	(5.5891) 0.0775	(2.2005) 2.9312	(1.8627) 0.2017	4.0663) 0.1231	(1.6552) 0.1239
	Linear	6.3014	5.8344	5.4123	4.9506	0.1041	1.1228	3.3938	0.9700

COV: coefficient of variance; TD: dependent material properties.

^a The dimensionless mean thermomechanical post-buckling load and temperature are given in brackets.

It is observed that Pasternak support clamped with fraction index 0.5 and round cut of 0.3 have significance on the laminates compared to SSSS (Table 9). Average and coefficients of variations rise for laminates on supports. It is noticed that for the same compressed laminate and hole, the dimensionless average post-deformation force raised the corresponding COV. This is because of the amplitude ratio, and because of increase in boundary constraints that significantly increases the internal properties of the laminate.

Table 9.

Effect of thermomechanical loading, volume fraction index, support conditions and various hole sizes with random material properties {b_i(i = 1,…, 7) = 0.1} on the dimensionless mean and COV of post-buckling load and temperature of uniaxial compressed FGM square plate without and with Winkler (k₁=100, k₂=0) and Pasternak (k₁=100, k₂=10) elastic foundations, circular hole having TD material properties, a/h = 15, W_max/h = 0.4.^a

	Hole type	SSSS				CCCC
Loading	Circular	n = 0.5		n = 10		n = 0.5		n = 10
Loading	Circular	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3	Hole size = 0.2	Hole size = 0.3
Thermomechanical	(k₁ = 000, k₂ = 0)	(4.4618) 0.0666	(4.2898) 0.0657	(3.3769) 0.0651	(3.2469) 0.0644	(5.6623) 0.0918	(4.3549) 0.0893	(4.3549) 0.0893	(5.7563) 0.1323
	Linear	3.3921	3.2177	2.6499	2.5146	5.4569	4.2107	4.2107	5.4259
	(k₁ = 100, k₂ = 0)	(5.4092) 0.0740	(5.1904) 0.0738	(4.3159) 0.0672	(4.1372) 0.0670	(6.0077) 0.0960	(7.4467) 0.0980	(4.6900) 0.0886	(5.7559) 0.0912
	Linear	4.3624	4.1583	3.6046	3.4296	5.8723	7.3188	4.6013	5.6676
	(k₁ = 100, k₂ = 10)	(7.1836) 0.3218	6.7199) 0.2478	(5.9430) 0.1348	(5.5364) 0.1055	(7.5936) 0.1461	8.7810) 0.1936	(6.2131) 0.1220	(7.0642) 0.1362
	Linear	6.0912	5.6484	5.2177	4.8006	7.2715	8.6568	5.9779	6.9745

COV: coefficient of variance; TD: dependent material properties; FGM: functionally graded material.

^a The dimensionless mean thermomechanical post-buckling load and temperature are given in brackets.

Findings and concluding remarks

The mean dimensionless thermomechanical post-buckling load and temperature and the corresponding COVs of FGM plate subjected to uniaxial and biaxial compression decrease with increase in cut-out size. The dimensionless mean post-buckling load and temperature and corresponding COV are lower for cut-out laminates without support compared to laminates resting on supports. Laminate is most sensitive with uncertain change in E_c and E_m of functionally graded laminate with Pasternak support circular cut-out.

In general, as W_max/h increases the dimensionless average deforming force increases, while COV becomes less. However, increase in the volume fraction index results in decrease in dimensionless mean post-buckling load. The mean and COV of post-deforming force and thermal condition of FGM laminate changes with the volume fraction index, a/h, a/b, types of force application, supports, stiffness and types of materials used where stability is of utmost importance.

Thin plate with circular cut-out is more sensitive than thick plate with respect to dimensionless mean and COV of post-buckling load and temperature subjected to uniaxial and biaxial compression. Therefore, for stability and reliability point of view, rectangular plate with circular cut-out having low volume fraction and Pasternak elastic foundation should be considered. For stability and reliability point of view, clamped supported plates with various shapes and cut-outs would be desirable.

Supplemental material

Supplemental Material, Highlights - Thermomechanically induced post-buckling analysis of functionally graded material plates with circular cut-outs resting on elastic foundations

Supplemental Material, Highlights for Thermomechanically induced post-buckling analysis of functionally graded material plates with circular cut-outs resting on elastic foundations by Rajesh Kumar in Journal of Thermoplastic Composite Materials

Footnotes

Authors’ note

Rajesh Kumar is now affiliated with Department of Mechanical Engineering, SET, IIMT University, Meerut, U. P., India.

Funding

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

ORCID iD

Rajesh Kumar

Supplemental material

Supplemental material for this article is available online.

Nomenclature

Appendix 1

where

where,

with

\begin{array}{l} B_{i j} = \int_{- h / 2}^{h / 2} {Q_{i j}^{m} + (Q_{i j}^{c} - Q_{i j}^{m}) {[\frac{z}{h} + \frac{1}{2}]}^{n}} z d z = Q_{i j}^{m} h^{2} + (Q_{i j}^{c} - Q_{i j}^{m}) h^{2} [\frac{1}{n + 2} - \frac{1}{2 (n + 1)}] \end{array}

where

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