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Thermal buckling analysis for hybrid and composite laminated plate by using new displacement function

  • Marwah G. Kareem EMAIL logo
Published/Copyright: April 9, 2024
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Open Engineering
From the journal Open Engineering Volume 14 Issue 1

Abstract

The analysis of a critical buckling temperature is dependent on the new displacement function field provided in the literature and uses third-order shear deformation theory. The value of the parameter “m,” which determines the new displacement function, closed with the three-dimensional (3D) elasticity results. Thermal critical buckling is analyzed for thick and thin laminated plates by using the Navier solution for symmetric and anti-symmetric cross-ply simply supported boundary conditions. Several design parameters are considered such as extension thermal coefficient ratio (α 1⁄α 2), number of schemes, thickness ratio (a/h), aspect ratio (a/b), and modular ratio (E 1/E 2) when analyzing the dynamic behavior of the critical buckling temperature for different materials, including composite (glass/epoxy) and hybrid (glass/carbon/epoxy), under uniformly distributed temperature load. The accuracy for theoretical results by using Matlab R2019b was checked with other results for different researchers and gave good agreement. Increasing orthotropic ratio and aspect ratio resulted in increasing critical buckling temperature in M1 than M2, whereas increasing thickness ratio, thermal coefficient expansion, and number of layers resulted in decrease in critical buckling temperature in M2 than M1.

Keywords: thermal critical buckling; thick plates; hybrid material; composite material

1 Introduction

Composite laminated plate and shell, one of the most basic structural components, is frequently used in a wide range of real-world applications, including bridges, aviation structures, naval vessels, and several technical sectors. The environmental circumstances in each of these applications are highly complicated, and the composite laminated plate and shell are subjected to numerous complex loads that have the potential to produce bigger acceleration shocks quickly. Therefore, it is of great significance to analyze the critical buckling temperature of the composite laminated plate under a variety of boundary conditions to take full advantage of them and ensure reliable structural performance. Thangaratnam and Ramachandran [1] improved critical buckling temperature for composite laminates by using finite element modeling (FEM) under thermal load. The set of total potential energy for the second variation is equal to zero in order to obtain the equation of motion for critical temperature and analyze angle-ply and cross-ply symmetric and anti-symmetric laminated plates under different boundary conditions. Chang and Leu [2] used higher order shear deformation for symmetric and anti-symmetric angle ply simply supported laminated plates to analyze thermal critical temperature under uniform thermal load. The study has taken into account the effects of transverse shear and normal strain to obtain a mathematical solution. The solutions have been validated with other theories such as higher and first-order shear deformation, and the comparison appeared very close with them. Chen et al. [3] presented the critical temperature buckling of laminated plates by using an FEM under constant and linear thermal loads. The effect of transverse shear and normal strain is examined to obtain an exact-closed solution by using the Mindli theory of laminated plates for thermal elastic. Meyers and Hyer [4] presented thermal buckling and post-buckling using the Rayleigh–Ritz theory for symmetrical laminated composite plates. Different parameters are taken into consideration including the boundary conditions, number of layers, thickness ratio, and modular ratio under a uniform temperature load. Noor and Burton [5] presented critical buckling temperature for double-layered angle-ply laminated plates by using a three-dimensional theory under thermal elastic properties. The surface axis temperature was assumed in the study to be independent, but it changes when the ply plate is symmetrical. Noor and Burton [6] studied combined thermal and axial loadings for laminated plate thermal buckling. The multi-parameter reduction approach is used to investigate the mixed FEM with a first-order shear deformation theory. The effect of some design conditions on the critical buckling temperature analysis of the plate was studied. Noor and Peters [7] analyzed thermal buckling of laminated plates by using three-dimensional elasticity theory and taking pre-buckling deformations into account in their calculations. Chen and Liu [8] presented critical buckling temperature for angle-ply composite plates by using first order under uniform thermal load combined with Levy-type approach boundary conditions. Prabhut and Dhanaraj [9] employed the FE approach with first order shear deformation theory to investigate the critical buckling temperature of laminated plates for symmetric and anti-symmetric (cross and angle) ply laminates under constant thermal load. Matsunaga [10] utilized higher-order shear deformation theory on critical buckling temperature to study sandwich plates and laminated angle-ply plate. The continuous displacement components for sandwich plates and simply supported laminated plate depend on power series expansion is used to solve equations of motion derived from the analysis of the virtual work principle. Shi et al. [11] used the FE approach to study the effect of mechanical and critical temperature loads for nonlinear thermal post-buckling on the angle ply laminated plate to anti-symmetric type. Enhanced (functionally graded materials) sandwich plate critical buckling temperature analysis was conducted by Bourada et al. [12] using a novel four-variable refined plate theory (RPT). The study considered different thermal load conditions such as constant and different linearity temperature increases in the ply plate direction. Abdul-Majeed et al. [13] used the Rayleigh–Ritz method of isotropic thermoelastic thin plates to investigate the governing differential equation of thermal buckling analysis. The effects of consistent temperature and irregular thermal dispersion cross-ply plates have been considered on the stability of the plate. Kreja [14] dealt with composite and sandwich panels and their mathematical processing. More than 200 texts on composite and sandwich panels are included in the theoretical FEM. In order to address deformation caused by the thickness of composite panels, the study demonstrated the impact of increasing the number of layers while decreasing the level of complexity and analysis. Jameel and Nsaif [15] examined symmetric, anti-symmetric, angle-ply, cross-ply of laminated plates by using classical theory and higher-order shear deformation plate theory on the critical buckling temperature analysis. The equations of motion were solved using Navier and Levy methods, Singh [16] analyzed the behavior of a laminated composite curved panel to study the critical buckling temperature using alloy fiber of shape memory. The responses have been obtained by depending on FEM with the vibration principle subjected to uniform thermal loading. Sayyad et al. [17] examined the flexibility of composite panels under the influence of sinusoidal thermal stress throughout thickness using three different types of theories: post shear deformation theory, third order shear deformation, and higher order shear deformation. In contrast, the exception of a little discrepancy between the elastic shear deformation theory and analytical shear deformation theory results for shear stress and investigation indicated some degree of agreement between the results of the theories. Cetkovic and Gyorgy [18] used generalized layer-wise theory to improve the angle ply of the thermal buckling laminates plate. The study dealt with a geometric stiffness matrix and an element stiffness matrix derived depending on FEM. Cetkovic [19] improved the analytical solution by using the new version of layer-wise to solve equations of motion and studied the thermal buckling of composite plates depending on Navier’s type. As the FE technique is used to analyze it for numerical solutions, Chen et al. [20] used the variation method to study free vibration and thermal buckling for the initially stressed composite plate. Two analyses of thermal load such as constant and linear distributed through the plate thickness were used in this study. Xing and Wang [21] focused on the functionally graded rectangular for examining critical buckling in thin plate applications. They used different parameters on laminated plates with variations in thermal loads (constant, linear, and non-uniform) by separating the variables and relying on closed-form solutions to investigate the thermal buckling [22]. Mohammed and Widad [23] used the general boundary conditions utilizing the classical theory with the Rayleigh–Ritz approach to examine the thermally critical buckling of laminated plates. The study has considered additional parameters such as the a/h ratio, number of schemes, modular ratio, and ply of orientation to improve its impact on the thermally critical buckling. Ahmed and Widad [24] applied the Levy-type solution and used the first-order shear deformation theory with different boundary conditions to laminated shallow shells for critical buckling temperature analysis. More design parameters have been considered in thermal buckling analysis, such as the number of layers, shallowness ratios (R/a), modular ratios (E 1/E 2), and thickness ratio (a/h), with change boundary conditions.

Maharudra et al. [25] studied the effect of different thermal loads and design parameters such as trapezoidal shape, ply orientation, boundary conditions, and aspect ratio by using FEM on the buckling behavior of the trapezoidal panel. FEM used nine nodes for the trapezoidal panel in order to take account of the effect of rotary inertia and shear deformation on the thin plate configuration. Widad and Ibtehal [26] used the thermal buckling analysis of a simply support laminated plate using a new displacement function field with higher-order shear deformation theory under different loads distributed along the thickness. Different design characteristics are considered for the symmetric and anti-symmetric for thin and thick laminated plates such as E 1/E 2 ratio, aspect ratio (a/b), and (α 2/α 1) ratio. Aman and Chalak [27] analyzed thermal buckling of thick and angle-ply laminated plates by applying higher-order zigzag theory which considered quadratic transverse displacement laminated plates to include core schemes with constant for face layers. Rahul et al. [28] studied the thermal buckling analysis of laminated plates with different types of holes under uniform and non-uniform thermos mechanical loads that depend on FEM and first-order shear deformation theory. It improved uniaxial and biaxial load for all edges or two edges with additional characteristics taken into consideration such as aspect ratio, thickness ratio, shape of hole, ply orientation and distributed way of thermo mechanical loads. Hussein [29] investigated from uniform and non-uniform temperature distributions for critical buckling temperature of cross-ply and angle-ply composite plates by using RPT with five independent unknown factors. Theoretical analysis has been used to study the effect of design parameters such as thickness ratio (a/h), aspect ratio (a/b), orthogonality ratio (E 1/E 2), coefficient of thermal expansion ratio (α 2⁄α 1), and numbers of plies on critical buckling temperature. In this work, a new displacement function based on high order shear deformation theory is used to provide critical thermal buckling analysis for symmetric and anti-symmetric thin and thick plates. The effect of design parameters such as thickness ratio (a/h), aspect ratio (a/b), orthogonality ratio (E 1⁄E 2), coefficient of thermal expansion ratio (α 2⁄α 1), and numbers of layers on thermal buckling of laminated composite (M1) and hybrid (M2). The results indicate that the mechanical and thermal properties for hybrid materials are more efficient than composite materials, and it is very best to use in different applications of engineering.

2 Displacement functions

This study used the displacement field suggested by Mantari et al. [30] to investigate the thermal buckling of simply supported cross-ply laminated plates, as in Figure 1. In accordance with higher order theory:

(1) u ̅ ( x 1 · x 2 · z · t ) = 1 + z R 1 × u ( ( x 1 · x 2 · t ) − z × ∂ w ∂ x 1 + z × m − 2 × z h 2 × ∅ 1 ,

(2) v ̅ ( x 1 · x 2 · z · t ) = 1 + z R 2 × v ( ( x 1 · x 2 · t ) − z × ∂ w ∂ x 2 + z × m − 2 × z h 2 × ∅ 2 ,

(3) w ̅ ( x 1 · x 2 · z · t ) = w ( ( x 1 · x 2 · t ) .

Figure 1 Coordinate system of k-layers of laminated rectangular plates [31].
Figure 1

Coordinate system of k-layers of laminated rectangular plates [31].

With m = 0.05, equations of strain depend on displacement functions:

(4) ε 1 = ε 1 0 + z × ε 1 1 + z × m − 2 × z h 2 × ε 1 2 ,

(5) ε 2 = ε 2 0 + z × ε 2 1 + z × m − 2 × z h 2 × ε 2 2 ,

(6) ε 4 = 1 − 4 × log ( m ) × z h 2 × m − 2 × z h 2 × ε 4 3 ,

(7) ε 5 = 1 − 4 × log ( m ) × z h 2 × m − 2 × z h 2 × ε 5 3 ,

(8) ε 1 = ε 6 0 + z × ε 6 1 + z × m − 2 × z h 2 × ε 6 2 ,

where

(9) ε 1 0 = ∂ u ∂ x 1 ε 1 1 = ∂ ∅ 1 ∂ x 1 − ∂ 2 w ∂ x 1 2 ε 1 2 = ∂ ∅ 1 ∂ x 1 , ε 2 0 = ∂ v ∂ x 2 ε 2 1 = ∂ ∅ 2 ∂ x 2 − ∂ 2 w ∂ x 2 2 ε 2 2 = ∂ ∅ 2 ∂ x 2 , ε 4 3 = ∅ 2 ε 5 3 = ∅ 1 , ε 6 0 = ∂ v ∂ x 1 + ∂ u ∂ x 2 ε 6 1 = ∂ ∅ 1 ∂ x 1 + ∂ ∅ 2 ∂ x 2 − 2 × ∂ 2 w ∂ x 1 ∂ x 2 ε 6 2 = ∂ ∅ 1 ∂ x 2 + ∂ ∅ 2 ∂ x 1 .

According to Hamilton’s principles:

(10) ∫ t 2 t 1 ( δ U + δ V ) ∂ t = 0 ,

where

(11) δ U = ∫ A ∫ z ( σ 11 × δ ε 1 + σ 22 × δ ε 2 + σ 44 δ ε 4 + σ 55 × δ ε 5 + σ 66 × δ ε 6 ) × A 1 × A 2 × d z ) d x 1 d x 2 ,  

(12) δ U = ∫ A ( N 1 × δ ε 1 0 + M 1 × δ ε 1 1 + P 1 × δ ε 1 2 + N 2 × δ ε 2 0 + M 2 × δ ε 2 1 + P 2 × δ ε 2 2 + N 6 × δ ε 6 0 + M 6 × δ ε 6 1 + P 6 × δ ε 6 2 + K 1 × δ ∅ 1 + K 2 × δ ∅ 2 ) d x 1 d x 2 ,

where

(13) { N i · M i · P i } = ∫ − h 2 h 2 σ i × { 1 · z · z × m − 2 × z h 2 × A 1 × A 2 × d z . ( i = 1 , 2 , 6 ) ,

(14) { Q i · K j } = ∫ − ( h 2 ) ( h 2 ) σ i 1 . 1 − 4 log ( m ) z h 2 m − 2 × z h 2 A 1 A 2 d z ( i = 4 , 5 ) , ( j = 1 , 2 ) ,

(15) δ V = − ∫ { N x 1 T δ ∂ 2 w ∂ x 1 2 + N x 2 T δ ∂ 2 w ∂ x 2 2 } d x 1 d x 2 .

Working on Hamilton’s equation gives triple and double integral parts; by substituting equations (9)–(15) in equation (11), this gives five equations of motion as follows:

δ u : ∂ N 1 ∂ x 1 + ∂ N 6 ∂ x 2 = 0 ,

δ v : ∂ N 2 ∂ x 2 + ∂ N 6 ∂ x 1 = 0 ,

(16) δ w : 2 π m h ∂ 2 M 6 ∂ x 1 ∂ x 2 + π m h ∂ 2 M 1 ∂ x 1 2 + ∂ 2 M 2 ∂ x 2 2 + N x 1 T ∂ 2 w ∂ x 1 2 + N x 2 T ∂ 2 w ∂ x 2 2 = 0 ,

δ ∅ 1 : ∂ P 1 ∂ x 1 + ∂ P 6 ∂ x 2 + ∂ 2 M 6 ∂ x 1 + ∂ 2 M 1 ∂ x 1 2 − K 1 + π m h Q 1 = 0 ,

δ ∅ 2 : ∂ P 2 ∂ x 2 + ∂ P 6 ∂ x 1 + 2 π m h ∂ 2 M 6 ∂ x 2 + π m h ∂ 2 M 2 ∂ x 2 2 − K 2 + π m h Q 2 = 0 .

The stress–strain relation for the kth lamine is given by

(17) σ 11 σ 22 σ 66 σ 44 σ 55 = Q 11 Q 12 Q 16 0 0 Q 12 Q 22 Q 26 0 0 Q 16 Q 26 Q 66 0 0 0 0 0 Q 44 0 0 0 0 0 Q 55 ε 1 − α 11 ∆ T ε 2 − α 22 ∆ T ε 6 − α 66 ∆ T ε 4 ε 5 .

The force results-strain related as follows:

(18) N 1 N 2 N 6 M 1 M 2 M 6 P 1 P 2 P 6 = A 11 A 12 A 16 A 12 A 22 A 26 A 16 A 26 A 66 B 11 B 12 B 16 B 12 B 22 B 26 B 16 B 26 B 66 B 11 B 12 B 16 B 12 B 22 B 26 B 16 B 26 B 66 E 11 E 12 E 16 E 12 E 22 E 26 E 16 E 26 E 66 E 11 E 12 E 16 E 12 E 22 E 26 E 16 E 26 E 66 F 11 F 12 F 16 F 12 F 22 F 26 F 16 F 26 F 66 E 11 E 12 E 16 E 12 E 22 E 26 E 16 E 26 E 66 F 11 F 12 F 16 F 12 F 22 F 26 F 16 F 26 F 66 H 11 H 12 H 16 H 12 H 22 H 26 H 16 H 26 H 66 * ε 1 0 ε 2 0 ε 6 0 ε 1 1 ε 2 1 ε 6 1 ε 1 2 ε 2 2 ε 2 6 ,

(19) K 1 K 2 = J 44 J 45 J 45 J 55 ε 5 3 ε 4 3 + L 44 L 45 L 45 L 55 ε 5 3 ε 4 3 ,

(20) Q 1 Q 2 = J 44 J 45 J 45 J 55 ε 5 3 ε 4 3 + L 44 L 45 L 45 L 55 ,

(21) N x 1 T N X 2 T = ∑ K = 1 N ∫ k k + 1 Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 α 11 α 22 2 α 66 ∆ T d z ,

(22) M x 1 T M X 2 T = ∑ K = 1 N ∫ k k + 1 Q 11 Q 12 Q 16 Q 12 Q 22 Q 26 α 11 α 22 2 α 66 ∆ Tz d z ,

where

A ij = ∫ − h / 2 h / 2 Q ij d z , i = 1 , 2 , 4 , 5 , 6 ,

B ij · D ij · E · F ij · H ij = ∫ − h / 2 h / 2 Q ij z · z 2 · z m − 2 z h 2 · z 2 m − 2 z h 2 · z 2 m − 4 z h 2 d z , i = 1 , 2 , 6 ,

(23) L ij = ∫ − h / 2 h / 2 Q ij × 1 − log ( m ) × z h 2 × m − 2 × z h 2 2 d z ,

j ij = ∫ − h / 2 h / 2 Q ij × 1 − log ( m ) × z h 2 × m − 2 × z h 2 2 d z .

While ∆ T = T f − T i , the above equations are solved using Navier’s solution for cross-ply, which is presented as follows [24]:

u ( x 1 · x 2 ) = ∑ m = 1 ∞ ∑ n = 1 ∞ A mn cos ( α x 1 ) sin ⁡ ( β x 2 ) v ( x 1 · x 2 ) = ∑ m = 1 ∞ ∑ n = 1 ∞ B mn sin ( α x 1 ) cos ⁡ ( β x 2 ) ,

(24) w ( x 1 · x 2 ) = ∑ m = 1 ∞ ∑ n = 1 ∞ C mn sin ( α x 1 ) sin ( β x 2 ) ,

∅ 1 ( x 1 ⸱ x 2 ) = ∑ m = 1 ∞ ∑ n = 1 ∞ D mn cos ( α x 1 ) sin ⁡ ( β x 2 ) ) ,

∅ 2 ( x 1 ⸱ x 2 ) = ∑ m = 1 ∞ ∑ n = 1 ∞ E mn sin ( α x 1 ) cos ⁡ ( β x 2 ) ) ,

where α = n × π a β = m × π b A mn , B mn , C mn , D mn , and E mn are arbitrary constants. The moment resultants and force from (18) and (22) can be applied to equations of motion (16); after that, we will apply equations for cross-ply to produce

(25) k 11 k 12 k 13 k 14 k 15 k 22 k 23 k 24 k 25 k 33 − α 2 ( N X T + M X T ) − ( N Y T + M Y T ) − α β N XY T k 34 k 35 k 44 k 45 k 55 u v w ∅ 1 ∅ 2 = 0 .

The procedure of analytical solution to find thermal critical buckling in this present work by using MATLAB 19, as shown in Figure 2.

Figure 2 The typical flow chart of solving critical buckling thermal problem derived by using new higher order shear deformation.
Figure 2

The typical flow chart of solving critical buckling thermal problem derived by using new higher order shear deformation.

3 Validation

New higher-order shear deformation is employed to investigate its capability level for dynamic analysis of thermal critical buckling for the symmetric and anti-symmetric cross-ply laminated plate and compared with other theories used by other researchers such as HOSD. Table 1 shows a validation for the critical buckling temperature obtained from the present work by applying MATLAB (R2019b) with results produced by HOSD in the study of Cetkovic [20] for change width to length ratio (a/h) of the isotropic plate under uniform temperature. Also, the critical buckling temperature results are obtained from the present work for symmetric and anti-symmetric cross-ply thin and thick plates under uniform temperature compared with the study of Cetkovic [20], as shown in Table 2. The maximum discrepancy between the results obtained by the present theory and other studies is within 6.68%. The mechanical and thermal properties used in Tables 1 and 2, respectively, E 1 = E 2 = 380 GPa, ν = 0.3, α = 7.4 × 10−6, and E 1/E 2 = 15, G 12 = G 13 = 0.5, E 2 E 2 ν12 = 0.3, G 23 = 0.3356, α 1/α 0 = 0.015, α 2/α 0 = 1, α 0 = 10−6 [5], dimensions are a = b and open value of width-to-thickness ratio.

(26) Discrepancy ( % ) = ∣ Present work results – Reference result / Reference results ∣ × 100 .

Table 1

Validation of non-dimensionless thermal buckling (T cr) of isotropic plate

a/h 10 20 40 60 80 100
Present work 2948.127 817.7786 205.8765 91.36183 50.95799 32.54329
Cetkovic [20] 3266.311 845.027 213.113 94.871 53.395 34.182
CLPT 3409.821 844.955 203.738 84.995 43.434 24.198
HOSD 3224.968 833.032 202.984 84.484 43.387 24.177
Table 2

Non-dimensionless thermal buckling for square laminated plate (T cr = 0.1000 × Tα 0)

Nub. layers References T cr
a/h
2 10/3 4 5 20/3 10 20 100
(0/90) Cetkovic [20] 0.3695 0.2391 0.1926 0.1419 0.09052 0.04449 0.01188 4.86 × 10−4
Present 0.369465 0.239077 0.192581 0.141886 0.090511 0.044486 0.01191 4.90 × 10−4
Discrepancy% 4.25 3.05 2.43 1.78 1.12 0.41 0.068 0.058
(0/90/0) Cetkovic [20] 0.3595 0.2625 0.2272 0.1848 0.134 0.07628 0.02316 9.96 × 10−4
Present 0.33546 0.259163 0.226327 0.184185 0.13393 0.075976 0.023423 9.89 × 10−4
Discrepancy% 6.68 1.27 0.38 0.3329 0.015418 0.04331 0.074993 0.088593

4 Results and discussion

The critical temperature buckling is studied for symmetric and anti-symmetric cross-ply simply supported laminated plates under uniformly distributed temperature and different parameters (i.e., thickness ratio a/h, modulus ration E 1/E 2, aspect ratio a/b, thermal expansion ratio α 2/α 1, and change in the number of layers) on M1and M2. The properties of the material used in the present work are given in Table 3 [32].

Table 3

Mechanical and thermal properties for M1 and M2

Composite material (M1) Hybrid material (M2)
Glass/epoxy Glass/carbon/epoxy
Young’s modulus, E 1 = 44.8 GPa E 1 = 102.1 GPa
E 2 = 12.1 GPa E 2 = 13.1 GPa
Shear modulus, G 12 = G 13 = 4.47 GPa, G 23 = 4.35 GPa G 12 = G 13 = 4.44 GPa, G 23 = 4.34 GPa
Passion ratio, µ 12 µ 12 = µ 13 = µ 23 = 0.26 = µ 13 = µ 23 = 0.3
Thermal expansion, α 1 = 6.13 × α 0, α 2 = 24.87 × α 0, α 0 = 1 × 10−6 α 1 = 4.714 × α 0, α 2 = 301 × α 0, α 0 = 1 × 10−6
Density, ρ = 2,060 kg/m3 ρ = 1,712 kg/m3
Volume fractions: glass = 0.6 and epoxy = 0.4 Volume fractions: glass = 0.3, carbon = 0.3, and epoxy = 0.4

Table 4 shows the effect of coupling between bending and extension on the dimensionless critical thermal buckling (T cr = 1,000 × Tα 0) of cross-ply anti-symmetric (0/90) simply supported plates with an increase in the number of layers, the bending–stretching coupling has the effect of lowering on the critical thermal buckling. For example, the four-layer plate has critical thermal buckling lower than those of the eight-layer anti-symmetric laminate for (M1 and M2), respectively, at the same thickness ratio and aspect ratio, while the value of thickness ratio of the laminated plate at the same number of layers increases from 10 to 100 the magnitude of dimensionless critical buckling decreases for M2 more than in M1, due to the stiffness increase of the plate in M2 (i.e., with an increases number of layers) because improved mechanical properties for M2 (i.e., larger orthotropic ratio E 1/E 2).

Table 4

Non-dimensionless thermal buckling (T cr = 1,000 × Tα 0) with different schemes of cross-ply (0/90) with change thickness ratio (a/h) to square plate between M1 and M2

Type of materials M1 M2
Layers
a/h (0/90)2 (0/90)3 (0/90)4 (0/90)2 (0/90)3 (0/90)4
10 0.815 0.83 0.841 0.1661 0.2319 0.304
20 0.228 0.231 0.2327 0.04736 0.07769 0.0975
50 0.03636 0.03793 0.03882 0.007933 0.01367 0.0224
100 0.00823 0.00962 0.00981 0.00231 0.00356 0.0054

Figure 3 shows that the critical thermal buckling load is achieved by using thickness ratios (a/h = 10, 20, 50, and 100) and modular ratios (E 1/E 2 = 5,10, 20, and 30). In all cases, the thermal buckling load obeys the same behavior, which decreases when the thickness ratio and modular ratio increase. This is because the plate stiffness becomes weak with a rise of the E 1/E 2 ratio and a/h ratio with respect to the effect of the stretching–bending energy increases, which causes decreasing stiffness of the plate.

Figure 3 Normalized thermal critical buckling (T cr = T × 1,000 × α 0) with change (E 1/E 2) ratio and thickness ratio (a/h) of cross-ply [0/90] (M = 1, N = 1) for a-M2 and b-M1.
Figure 3

Normalized thermal critical buckling (T cr = T × 1,000 × α 0) with change (E 1/E 2) ratio and thickness ratio (a/h) of cross-ply [0/90] (M = 1, N = 1) for a-M2 and b-M1.

Figure 4 shows the change value of the (α 2/α 1) ratio of thermal expansion with the critical thermal buckling of anti-symmetric [0/90]. The dimensionless critical buckling temperature (T cr = T × α 0 × 1,000) decreases when the thermal expansion ratio (α 2/α 1) increases from 2 to 10 because the stiffness of the plate decreases in composite material (M1) more than hybrid material (M2) at thickness ratio (a/h = 100) and square plate (a/b = 1).

Figure 4 Variation of (α 2/α 1) ratio on normalized thermal buckling (T cr × 1,000 × α 0) for [0/90] cross-ply square plate between M1 and M2.
Figure 4

Variation of (α 2/α 1) ratio on normalized thermal buckling (T cr × 1,000 × α 0) for [0/90] cross-ply square plate between M1 and M2.

Figure 5 shows the variation of critical thermal buckling between materials (M1 and M2) laminated cross-ply plates of various aspect ratios.

Figure 5 Variation of normalized thermal critical buckling (T cr × 1,000 × α 0) for [0/90] cross-ply plate with aspect ratio (a/b) and different (a/h) ratio between M1 and M2.
Figure 5

Variation of normalized thermal critical buckling (T cr × 1,000 × α 0) for [0/90] cross-ply plate with aspect ratio (a/b) and different (a/h) ratio between M1 and M2.

It can be seen that, in both M1 and M2, the critical buckling temperature decreases initially and increases for a thickness ratio from 10 to 100. It is thus concluded that a plate with a large thickness ratio undergoes a small critical buckling temperature and that the buckling temperature decrease of an M1 is much greater than that of an M2. The value of thermal critical buckling is highly dependent on aspect ratio (a/b) and decreases when the ratio increases from 1 to 4. The decreasing ratio from thermal critical buckling with aspect ratio (1) to that with aspect ratio (4) is equal to (27.87%) for M1 and (24.54%) for M2 at thickness ratio (a/h = 10). Figures 6 and 7 show the first three mode shapes of M1 and M2 materials for plate anti-symmetric cross-ply (0/90) simply supported boundary conditions.

Figure 6 Normalized thermal critical buckling mode for cross-ply (0/90) plate with different modes, Nub. of schemes = 2, a/h = 10, a/b = 1.
Figure 6

Normalized thermal critical buckling mode for cross-ply (0/90) plate with different modes, Nub. of schemes = 2, a/h = 10, a/b = 1.

Figure 7 Normalized thermal critical buckling modes for anti-symmetric cross-ply (0/90) squared plate (a/b = 1), a/h = 10.
Figure 7

Normalized thermal critical buckling modes for anti-symmetric cross-ply (0/90) squared plate (a/b = 1), a/h = 10.

From the figures, the thermal critical buckling behaviors of the plate can be seen vividly of hybrid (M2) are less than composite materials (M1), because in the equation of dimensionless critical buckling as T cr = T × ( a 2 / π 2 ) ρ h / D 22 , the bending-twisted stiffness (D22) is in the denominator, and the magnitude of (D22) for hybrid (M2) is more than composite (M1). If the value of the (D22) gets increased, dimensionless critical buckling will decrease. Also, the effect of the density is greater than that of the effect of (D22) on stiffness.

5 Conclusions

The results are derived from this mathematical modeling [Matlab 2019] and validated using different ways. The main conclusions from the obtained results are as follows:

  1. The dynamic behavior for critical buckling temperature of anti-symmetric cross-ply plates under uniform temperature load is studied between composite and hybrid materials by applying a new displacement function, and it gave results closed to another result obtained by another theory.

  2. The effect of twisted curvature coupling with bending stretching is very large on the dimensionless critical thermal buckling when raising the modular ratio (E 1/E 2).

  3. Dimensionless critical thermal buckling of cross-ply anti-symmetric simply supported plate increases with an increase in the number of layers because the bending–stretching coupling has a lower effect on it.

  4. The effect of change thickness ratio (a/h) and aspect ratio (a/b) on the composite material was more than on hybrid materials because the enhanced mechanical properties’ effect increased the stiffness and reliability in resisting buckle thermal load on laminate plates.

  5. The value of the dimensionless thermal critical buckling hybrid (M2) is less than composite materials (M1), because, in the equation of the thermal critical backing parameter, the bending-twisted stiffness (D22) is in the denominator and the magnitude of (D22) for hybrid (M2) more than composite (M1).

  6. As the ratio of the thermal expansion coefficient increases from 2 to 10, the critical thermal buckling load decreases by a percentage like an exponential decay.

  7. The present work studied the engineering applications under different working conditions and different materials and reinforcement in order to understand the mechanical behavior of composite and hybrid materials during different loads.

  1. Funding information: Author declares that the manuscript was done depending on the personal effort of the author, and there is no funding effort from any side or organization.

  2. Conflict of interest: Author states no conflict of interest.

  3. Data availability statement: Most datasets generated and analyzed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.

References

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Received: 2023-12-22
Revised: 2024-02-06
Accepted: 2024-02-07
Published Online: 2024-04-09

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/eng-2022-0597
Keywords for this article
thermal critical buckling; thick plates; hybrid material; composite material
Creative Commons
BY 4.0

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  69. Numerical and experimental comparison study of piled raft foundation
  70. Effect of asphalt modified with waste engine oil on the durability properties of hot asphalt mixtures with reclaimed asphalt pavement
  71. Hydraulic model for flood inundation in Diyala River Basin using HEC-RAS, PMP, and neural network
  72. Numerical study on discharge capacity of piano key side weir with various ratios of the crest length to the width
  73. The optimal allocation of thyristor-controlled series compensators for enhancement HVAC transmission lines Iraqi super grid by using seeker optimization algorithm
  74. Numerical and experimental study of the impact on aerodynamic characteristics of the NACA0012 airfoil
  75. Effect of nano-TiO2 on physical and rheological properties of asphalt cement
  76. Performance evolution of novel palm leaf powder used for enhancing hot mix asphalt
  77. Performance analysis, evaluation, and improvement of selected unsignalized intersection using SIDRA software – Case study
  78. Flexural behavior of RC beams externally reinforced with CFRP composites using various strategies
  79. Influence of fiber types on the properties of the artificial cold-bonded lightweight aggregates
  80. Experimental investigation of RC beams strengthened with externally bonded BFRP composites
  81. Generalized RKM methods for solving fifth-order quasi-linear fractional partial differential equation
  82. An experimental and numerical study investigating sediment transport position in the bed of sewer pipes in Karbala
  83. Role of individual component failure in the performance of a 1-out-of-3 cold standby system: A Markov model approach
  84. Implementation for the cases (5, 4) and (5, 4)/(2, 0)
  85. Center group actions and related concepts
  86. Experimental investigation of the effect of horizontal construction joints on the behavior of deep beams
  87. Deletion of a vertex in even sum domination
  88. Deep learning techniques in concrete powder mix designing
  89. Effect of loading type in concrete deep beam with strut reinforcement
  90. Studying the effect of using CFRP warping on strength of husk rice concrete columns
  91. Parametric analysis of the influence of climatic factors on the formation of traditional buildings in the city of Al Najaf
  92. Suitability location for landfill using a fuzzy-GIS model: A case study in Hillah, Iraq
  93. Hybrid approach for cost estimation of sustainable building projects using artificial neural networks
  94. Assessment of indirect tensile stress and tensile–strength ratio and creep compliance in HMA mixes with micro-silica and PMB
  95. Density functional theory to study stopping power of proton in water, lung, bladder, and intestine
  96. A review of single flow, flow boiling, and coating microchannel studies
  97. Effect of GFRP bar length on the flexural behavior of hybrid concrete beams strengthened with NSM bars
  98. Exploring the impact of parameters on flow boiling heat transfer in microchannels and coated microtubes: A comprehensive review
  99. Crumb rubber modification for enhanced rutting resistance in asphalt mixtures
  100. Special Issue: AESMT-6
  101. Design of a new sorting colors system based on PLC, TIA portal, and factory I/O programs
  102. Forecasting empirical formula for suspended sediment load prediction at upstream of Al-Kufa barrage, Kufa City, Iraq
  103. Optimization and characterization of sustainable geopolymer mortars based on palygorskite clay, water glass, and sodium hydroxide
  104. Sediment transport modelling upstream of Al Kufa Barrage
  105. Study of energy loss, range, and stopping time for proton in germanium and copper materials
  106. Effect of internal and external recycle ratios on the nutrient removal efficiency of anaerobic/anoxic/oxic (VIP) wastewater treatment plant
  107. Enhancing structural behaviour of polypropylene fibre concrete columns longitudinally reinforced with fibreglass bars
  108. Sustainable road paving: Enhancing concrete paver blocks with zeolite-enhanced cement
  109. Evaluation of the operational performance of Karbala waste water treatment plant under variable flow using GPS-X model
  110. Design and simulation of photonic crystal fiber for highly sensitive chemical sensing applications
  111. Optimization and design of a new column sequencing for crude oil distillation at Basrah refinery
  112. Inductive 3D numerical modelling of the tibia bone using MRI to examine von Mises stress and overall deformation
  113. An image encryption method based on modified elliptic curve Diffie-Hellman key exchange protocol and Hill Cipher
  114. Experimental investigation of generating superheated steam using a parabolic dish with a cylindrical cavity receiver: A case study
  115. Effect of surface roughness on the interface behavior of clayey soils
  116. Investigated of the optical properties for SiO2 by using Lorentz model
  117. Measurements of induced vibrations due to steel pipe pile driving in Al-Fao soil: Effect of partial end closure
  118. Experimental and numerical studies of ballistic resistance of hybrid sandwich composite body armor
  119. Evaluation of clay layer presence on shallow foundation settlement in dry sand under an earthquake
  120. Optimal design of mechanical performances of asphalt mixtures comprising nano-clay additives
  121. Advancing seismic performance: Isolators, TMDs, and multi-level strategies in reinforced concrete buildings
  122. Predicted evaporation in Basrah using artificial neural networks
  123. Energy management system for a small town to enhance quality of life
  124. Numerical study on entropy minimization in pipes with helical airfoil and CuO nanoparticle integration
  125. Equations and methodologies of inlet drainage system discharge coefficients: A review
  126. Thermal buckling analysis for hybrid and composite laminated plate by using new displacement function
  127. Investigation into the mechanical and thermal properties of lightweight mortar using commercial beads or recycled expanded polystyrene
  128. Experimental and theoretical analysis of single-jet column and concrete column using double-jet grouting technique applied at Al-Rashdia site
  129. The impact of incorporating waste materials on the mechanical and physical characteristics of tile adhesive materials
  130. Seismic resilience: Innovations in structural engineering for earthquake-prone areas
  131. Automatic human identification using fingerprint images based on Gabor filter and SIFT features fusion
  132. Performance of GRKM-method for solving classes of ordinary and partial differential equations of sixth-orders
  133. Visible light-boosted photodegradation activity of Ag–AgVO3/Zn0.5Mn0.5Fe2O4 supported heterojunctions for effective degradation of organic contaminates
  134. Production of sustainable concrete with treated cement kiln dust and iron slag waste aggregate
  135. Key effects on the structural behavior of fiber-reinforced lightweight concrete-ribbed slabs: A review
  136. A comparative analysis of the energy dissipation efficiency of various piano key weir types
  137. Special Issue: Transport 2022 - Part II
  138. Variability in road surface temperature in urban road network – A case study making use of mobile measurements
  139. Special Issue: BCEE5-2023
  140. Evaluation of reclaimed asphalt mixtures rejuvenated with waste engine oil to resist rutting deformation
  141. Assessment of potential resistance to moisture damage and fatigue cracks of asphalt mixture modified with ground granulated blast furnace slag
  142. Investigating seismic response in adjacent structures: A study on the impact of buildings’ orientation and distance considering soil–structure interaction
  143. Improvement of porosity of mortar using polyethylene glycol pre-polymer-impregnated mortar
  144. Three-dimensional analysis of steel beam-column bolted connections
  145. Assessment of agricultural drought in Iraq employing Landsat and MODIS imagery
  146. Performance evaluation of grouted porous asphalt concrete
  147. Optimization of local modified metakaolin-based geopolymer concrete by Taguchi method
  148. Effect of waste tire products on some characteristics of roller-compacted concrete
  149. Studying the lateral displacement of retaining wall supporting sandy soil under dynamic loads
  150. Seismic performance evaluation of concrete buttress dram (Dynamic linear analysis)
  151. Behavior of soil reinforced with micropiles
  152. Possibility of production high strength lightweight concrete containing organic waste aggregate and recycled steel fibers
  153. An investigation of self-sensing and mechanical properties of smart engineered cementitious composites reinforced with functional materials
  154. Forecasting changes in precipitation and temperatures of a regional watershed in Northern Iraq using LARS-WG model
  155. Experimental investigation of dynamic soil properties for modeling energy-absorbing layers
  156. Numerical investigation of the effect of longitudinal steel reinforcement ratio on the ductility of concrete beams
  157. An experimental study on the tensile properties of reinforced asphalt pavement
  158. Self-sensing behavior of hot asphalt mixture with steel fiber-based additive
  159. Behavior of ultra-high-performance concrete deep beams reinforced by basalt fibers
  160. Optimizing asphalt binder performance with various PET types
  161. Investigation of the hydraulic characteristics and homogeneity of the microstructure of the air voids in the sustainable rigid pavement
  162. Enhanced biogas production from municipal solid waste via digestion with cow manure: A case study
  163. Special Issue: AESMT-7 - Part I
  164. Preparation and investigation of cobalt nanoparticles by laser ablation: Structure, linear, and nonlinear optical properties
  165. Seismic analysis of RC building with plan irregularity in Baghdad/Iraq to obtain the optimal behavior
  166. The effect of urban environment on large-scale path loss model’s main parameters for mmWave 5G mobile network in Iraq
  167. Formatting a questionnaire for the quality control of river bank roads
  168. Vibration suppression of smart composite beam using model predictive controller
  169. Machine learning-based compressive strength estimation in nanomaterial-modified lightweight concrete
  170. In-depth analysis of critical factors affecting Iraqi construction projects performance
  171. Behavior of container berth structure under the influence of environmental and operational loads
  172. Energy absorption and impact response of ballistic resistance laminate
  173. Effect of water-absorbent polymer balls in internal curing on punching shear behavior of bubble slabs
  174. Effect of surface roughness on interface shear strength parameters of sandy soils
  175. Evaluating the interaction for embedded H-steel section in normal concrete under monotonic and repeated loads
  176. Estimation of the settlement of pile head using ANN and multivariate linear regression based on the results of load transfer method
  177. Enhancing communication: Deep learning for Arabic sign language translation
  178. A review of recent studies of both heat pipe and evaporative cooling in passive heat recovery
  179. Effect of nano-silica on the mechanical properties of LWC
  180. An experimental study of some mechanical properties and absorption for polymer-modified cement mortar modified with superplasticizer
  181. Digital beamforming enhancement with LSTM-based deep learning for millimeter wave transmission
  182. Developing an efficient planning process for heritage buildings maintenance in Iraq
  183. Design and optimization of two-stage controller for three-phase multi-converter/multi-machine electric vehicle
  184. Evaluation of microstructure and mechanical properties of Al1050/Al2O3/Gr composite processed by forming operation ECAP
  185. Calculations of mass stopping power and range of protons in organic compounds (CH3OH, CH2O, and CO2) at energy range of 0.01–1,000 MeV
  186. Investigation of in vitro behavior of composite coating hydroxyapatite-nano silver on 316L stainless steel substrate by electrophoretic technic for biomedical tools
  187. A review: Enhancing tribological properties of journal bearings composite materials
  188. Improvements in the randomness and security of digital currency using the photon sponge hash function through Maiorana–McFarland S-box replacement
  189. Design a new scheme for image security using a deep learning technique of hierarchical parameters
  190. Special Issue: ICES 2023
  191. Comparative geotechnical analysis for ultimate bearing capacity of precast concrete piles using cone resistance measurements
  192. Visualizing sustainable rainwater harvesting: A case study of Karbala Province
  193. Geogrid reinforcement for improving bearing capacity and stability of square foundations
  194. Evaluation of the effluent concentrations of Karbala wastewater treatment plant using reliability analysis
  195. Adsorbent made with inexpensive, local resources
  196. Effect of drain pipes on seepage and slope stability through a zoned earth dam
  197. Sediment accumulation in an 8 inch sewer pipe for a sample of various particles obtained from the streets of Karbala city, Iraq
  198. Special Issue: IETAS 2024 - Part I
  199. Analyzing the impact of transfer learning on explanation accuracy in deep learning-based ECG recognition systems
  200. Effect of scale factor on the dynamic response of frame foundations
  201. Improving multi-object detection and tracking with deep learning, DeepSORT, and frame cancellation techniques
  202. The impact of using prestressed CFRP bars on the development of flexural strength
  203. Assessment of surface hardness and impact strength of denture base resins reinforced with silver–titanium dioxide and silver–zirconium dioxide nanoparticles: In vitro study
  204. A data augmentation approach to enhance breast cancer detection using generative adversarial and artificial neural networks
  205. Modification of the 5D Lorenz chaotic map with fuzzy numbers for video encryption in cloud computing
  206. Special Issue: 51st KKBN - Part I
  207. Evaluation of static bending caused damage of glass-fiber composite structure using terahertz inspection
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