Document Type : Research Paper
Authors
Structural Engineering Department, Veermata Jijabai Technological Institute, Matunga, Mumbai 400 019, India.
Abstract
Keywords
Main Subjects
Quick Review and Assessment of Thermal Stress Analyses for Exact and Assumed Temperature Distribution for
Composite and Sandwich Laminates
Structural Engineering Department, Veermata Jijabai Technological Institute, Matunga, Mumbai 400 019, India.
KEYWORDS 

ABSTRACT 
Composites; Laminate; BVP; PDE; ODE. 
A simple semianalytical approach is used in the present studies to achieve a thermal response of composite and sandwich layered materials in stresses and displacements. Threedimensional (3D) heat conduction formulation has been formulated as a boundary value problem (BVP) to obtain accurate throughthickness temperature variation, which is further used for thermal stress analysis. Four layered domains of composite and/or sandwich laminate with variable degrees of orthotropic and several transverse and inplane aspect ratios have been used for numerical investigation. Exact solutions from past studies have been used for comparison with the outcomes received in the present study and proved the accuracy and efficiency of current developments. Additionally, simple constant and linear throughthickness temperature variation has been considered for stress analysis to highlight the importance and need for exact temperature distribution for thermal stress analysis of composite and sandwich laminates. The presented semianalytical approach achieves the benefit of exact analysis and numerical analysis and leads to accuracy and computational efficiency. 
Advancements in technology demanded qualitative materials rather than quantities. Quality and accuracy in composites increase their uses in most of the industry. These composites comprise more than single pure substances that are macroscopically combined. Most of these engineered materials show superior performance to conventional materials like metals. Nowadays, a composite material formed by two or more materials in a macroscopic form is commonly used. These are engineered materials with specific properties that prove better than conventional materials like metals.
During manufacturing processes and service life, these laminates formed with different materials are subject to thermal stresses at the adjacent layer due to their different thermal properties. External imposed mechanical and thermal loads cause interlaminar transverse stresses, delamination in laminae layers, and fractures in the matrix in the longitudinal direction. Therefore, a precise evaluation of transverse stresses and deformation leads to milestone achievement in this area.
Pell [1] first discovered the thermal deformation of anisotropic thin plates subjected to random loading. Classical plate theory (CLT) has been used by Timoshenko and WoinowskyKrieger [2], Boley and Weiner [3], Johns [4], Parkus [5], Burgreen [6] and Vinson [7] for the calculation of thermal stresses in beams, plates, and shell structures. Flexural response for various temperatures has been noted by Srinivas and Rao [8]. Boleys [9] has revealed the shortcoming in accessing thermal stresses in beams. Bapu Rao [10,11] developed a threedimensional (3D) formulation for a thick isotropic plate under thermal loading. Further, Tungikar and Rao [12] extended this study and developed an exact 3D solution for orthotropic laminate. Thermal stresses of symmetric and unsymmetric laminates in plane strain conditions have been highlighted by working on it by Wu and Tauchert [13]. Finite element formulation depends on the displacement model for laminate under thermal loading was developed by Reddy and Chao [14] and Reddy et al. [15]. Carrera [16] developed equivalentsinglelayer (ESL) and layerwise (LW) modeling for both classical and mixed approaches. Fourthorder displacement and stress field conditions are used to develop the thermomechanical governing equations. Several investigations [17, 18] have been carried out using four variable deformation theories for bending and vibration analysis of sandwich plates.
Burton and Noor [19] published a critical review of computational models for laminates under thermomechanical loading and showed the path for future studies for modeling highertemperature laminates. The elastic analysis based on displacementbased higherorder shear deformation (HOSD) theories has been detailed with the help of formulation for composite plates by Jonalagadda et al. [20]. Composite laminates are analyzed with HOSDbased C0 isoparametric FE formulation for the assumed thermal gradient. Four layered square laminates imposed to sudden uniform temperature change have been analyzed by presenting the transient heat conduction equation by Savoia and Reddy [21]. Bhaskar et al. [22] assumed a linear throughthickness thermal profile and formulated a 3Dimensional elasticity solution for cylindrical and bidirectional bending of laminate. Transverse shear stresses with the help of firstorder shear deformation (FOSD) based analytical model has been presented by Rolfes et al. [23]. Abualnour et al. [24] studied thermomechanical bending behavior using the fourvariable trigonometric plate theory. Bakoura et al. [25] studied the mechanical behavior of composite plates using three variable refined plate theory. Carrera [26,27] examined the thermomechanical response of layered, anisotropic plates by applying the temperature profile in the thickness directions. The impact of temperature profiles on the accuracy of plate models is then examined for a broad range of classical and advanced multilayered plate theories. Using the CUF and Reissner mixed variational theorem, Robaldo and Carrera [28] investigate the thermoelastic analysis of multilayered anisotropic plates (RMVT). To create the finite element (FE) matrices, the RMVT is used in conjunction with assumptions for the displacement fields in the thickness direction.
Composite and sandwich beam has been analyzed with the development of HOSD Formulation by Kapuria et al. [29] and showed their accuracy compared with the zigzag theory. Further, Kapuria and Achary [30] developed a higherorder zigzag theory to analyze laminated plates under thermal loading. A similar approach by Tungikar and Rao [12] has been used by Robaldo et al. [31], who developed an FE solution of heat conduction for stress analysis of anisotropic laminates. The sandwich plate and viscoelastic core have been analyzed to obtain buckling, damping behavior, and frequency with the help of FE analysis by incorporating CLT theory as carried out by Pradeep et al. [32]. Gherlone and Sciuva [33] have presented thermomechanical FE solutions using the Hermitian zigzag theory. A semianalytical approach has been used by Kant et al. [34,35] for layered composite and sandwich laminate analysis imposed by thermal loading. The vibration response of multilayered sandwich laminate has been calculated by Pradeep and Ganesan [36]. Pagani and Carrera [37] used a unified formulation for laminated composite beams. Large deflection and postbuckling analyses of laminated composite with the help of a dualphaselag model of generalized thermoelasticity interaction in a threedimensional homogeneous and isotropic sandwich structure have been investigated by Sur and Kanoria [38]. Garg and Chalak [39] review the analysis of the laminated composite and sandwich structures under hygrothermal conditions. Further, Garg and Chalak [40] analyze nonskew and skew laminated composite and sandwich plates subjected to hygrothermomechanical conditions. Naik and Sayyad [41] developed a higherorder displacement model for calculating the cylindrical bending of laminated and sandwich plates subjected to Environmental loads.
Kant and Shiyekar [42] considered linear throughthickness temperature variation and obtained an HOSDbased analytical formulation by incorporating the impact of shear deformation and transverse normal thermal strain. Trigonometric shear deformation theory has been used for the flexural analysis of crossply laminated plates subjected to nonlinear thermal loading by Ghugal and Kulkarni [43]. Further responses obtained from CPT, FOSD, and HOSD for symmetric crossply laminated plates acted linear and nonlinear thermal and mechanical loads simultaneously have also been compared by Ghugal and Kulkarni [44]. Sayyad et al. [45] outlined hyperbolic shear deformation theory for thermal response in stresses and displacement. Various combination of heat conduction, convection, and radiation has been considered by Norouzi et al. [46] to develop an exact solution for a multilayer spherical fiberreinforced composite. The optimal buckling temperature of the laminated composite skew plate has been obtained by the differential quadrature method with genetic algorithms by Parviz et al. [47] with FOSD formulation. Crossply laminated composite plate's behavior in terms of stress analysis has been derived by Sayyad et al. [48,49] by using an exponential shear deformation theory and four variables shear deformation theory, respectively. The effect of uniform heat flux supply on the circular and noncircular plate with a circular opening in the form of stresses and displacements has been noted by Jafari and Jafari [50].
Fazzolari and Carrera [51] developed quasi3D ESL and Zigzag theory using Carrera's unified formulation to examine the thermal stability of FGM isotropic and sandwich plates subjected to various throughthickness temperature distributions. Various researchers [52–54] used quasi3D HOST to analyze the bending, buckling, and vibration of FG plates resting on a viscoelastic foundation.
Researchers have put considerable effort into developing simplified and refined numerical and analytical models for thermal stress analyses of composite domains concerning the available literature. Mostly, simple linear and constant temperature gradients along the composite laminate depth were assumed and used for analyses. However, this rude assumption is not valid for composite laminates, which have been proven and reported in little research. Hence, efforts have been put into this paper to form simple and efficient semianalytical models for governing heat conduction equations to obtain real temperature gradients along with the depth of laminated composites. The prime object of this paper is the development of a semianalytical formulation for obtaining actual temperature variation along with stress analysis. Formulations include describing twopoint BVP ruled by a group of coupled firstorder ODEs (Eq. 1) within the depth of laminate.

(1) 
Further, stress analysis has been performed and reported for exact as well as for simple assumed constant and linear temperature gradients along with the laminate depth. Numerical studies have been reported and discussed for a wide range of lamination schemes and inplane and transverse aspect ratios.
A layered laminate of depth ‘d’ consisting of perfectly bonded isotropic/orthotropic, linearly elastic laminae of constant depth and having length and breadth dimension (l x b) as considered in Fig. 1. All peripheral edges are simple diaphragm supported as in Table 1. The plate has been imposed by thermal load and/or mechanical load on its top surface. Any other external stresses on all surfaces are neglected here. The fibers are aligned in the lamina along the laminate reference axis ‘X1’ and perpendicular to it.
Fig. 1. 3D composite laminate
This section is devoted to the formation of a twopoint boundary value problem (BVP) governed by a set of coupled ordinary differential equations (ODEs) along with the laminate thickness from the basic governing differential heat conduction Equation.
The differential heat conduction equation for a homogenous isotropic/orthotropic solid for steadystate conditions in 3D, without internal heat generation, is

(2) 
Table 1. Boundary conditions (BCs)
Locations 
BC imposed on the 
BC imposed on the 
Face 

_ 
Face 


Face 

_ 
Face 

. 
Top face 
 

Bottom face 
 

According to Fourier’s theory of heat conduction, heat flux in direction , and is given by,

(3) 
where,
= coefficient of thermal conductivity in
= heat flux in and
= temperature in Kelvin .
By considering total heat remains zero due to heat flow, the equilibrium equation in 3D is written as,

(4) 
For exact analysis in laminated layers, the continuity of heat flux and temperature at the interface must be maintained and taken care of in the present development. By using the mathematical conversion of the Equations. (2), (3), and (4), a set of partial differential equations (PDEs) includes dependent variables and , only two in quantity are derived as follows,

(5) 
The above PDEs stated by Equation (5) can be converted to coupled firstorder ODEs with the help of Fourier trigonometric series expansion for primary variables. These equations need to be

(6) 
satisfied simply supported boundary conditions at = 0, l and = 0, b as follows,
Substituting Equation (6) and its derivatives into Equation (5), the following set of firstorder ODEs is obtained.

(7) 
Equation (7) shows the governing twopoint BVP in ODEs in a domain with known temperatures at the top and bottom surfaces of a beam.

(8) 



(9) 





(10) 


in which and are the stress and strain vector correspondingly and are free thermal strains concerning laminate axes
( , , ). is the transformed elasticity constants matrix of the i^{th} lamina regarding the laminate axes. On performing a few algebraic operations on Equations (8)(10) a set of partial differential equations (PDEs) that includes only eight chosen main variables and are obtained as,



(11) 
The above PDEs stated by Equation (11) can be converted to coupled firstorder ODEs by using Fourier trigonometric series expansion for primary variables satisfying the simple (diaphragm) support end conditions at = 0, l and = 0, b as shown below,

(12) 
where m, n=1, 3, 5, …….
Next, temperature changes along the inplane directions are also shown in sinusoidal form as,

(13) 
Substituting Equations (12) and (13) and their differentiation into Equation (11), the below 6 coupled firstorder Ordinary Differential Equations (ODEs) are derived.













(14) 
Equation (14) represents the governing twopoint BVP in ODEs in the domain with stress components known at the upper and bottom surfaces of a laminate.
The solution of BVP obtained in Equations (7) and (14) are obtained by converting BVP into initial value problems (IVPs) (Tables 2 and 3). Finally, one nonhomogeneous and n/2 homogeneous solution has been formed with the satisfaction of BCs at = + , as detailed by Kant and Ramesh [55]. Changes in the material properties (Table 4) are inserted by the change in the coefficient of the material matrix appropriately during numerical integration performed by the fourthorder RungeKutta Gill algorithm.
Table 2. Transformation of BVP into IVPs for thermal analysis
Integration No. 
Bottom edge

Top edge 






1 
known 
0 
M11 
M21 
2 
0 
1 
M12 
M22 
3 

K1 


Table 3. Transformation of BVP into IVPs for stress analysis
Integration No. 
Starting edge; 
Final Edge; 
Load term 














1 
0 
0 
0 
0 
0 
0 
Y_{11} 
Y_{21} 
Y_{31} 
Y_{41} 
Y_{51} 
Y_{61} 
Include 
2 
1 
0 
0 
0 
0 
0 
Y_{12} 
Y_{22} 
Y_{32} 
Y_{42} 
Y_{52} 
Y_{62} 
Delete 
3 
0 
1 
0 
0 
0 
0 
Y_{13} 
Y_{23} 
Y_{33} 
Y_{43} 
Y_{53} 
Y_{63} 
Delete 
4 
0 
0 
1 
0 
0 
0 
Y_{14} 
Y_{24} 
Y_{34} 
Y_{44} 
Y_{54} 
Y_{64} 
Delete 
Final 
X_{a} 
X_{b} 
X_{c} 
known 
known 
known 



0 
0 
0 
Include 
Table 4. Material Properties
Set 
Material Property 

I 





II 





III 





IV 





Face 





Core 





= thermal conductivity coefficient = coefficient of thermal expansion
Ref: Kapuria and Acharya [30]
Four, allaround simply supported composite laminates, (A), (B), (C), and (D) having different material schemes and lamination properties (Table 4) showing high inhomogeneous characteristics and variable degrees of orthotropic have been assessed for various transverse and inplane aspect ratios. These are correlated with the real solution presented by Kapuria and Achary [30] wherever available.
A fivelayered composite laminate (A) has been considered with variable inhomogenous properties for tension and shear stiffness as well as thermal coefficients and conductivity. It has individual layer thickness as 0.1d/0.25d/0.15d/0.2d/0.3d of material sets
I/ II/ III/ III/ III with orientations .
Laminates (B) and (C) are symmetric and antisymmetric crossply composites of material set IV consisting of four equal thicknesses (0.25d) of the lamina.
Further, fivelayered sandwich laminate (D) has graphiteepoxy faces and softcore with a thickness of each lamina as 0.05d/0.05d/0.8d/0.05d/0.05d has been considered.
Two different temperature cases, viz, constant temperature rise [Equation 15] along the laminate thickness simulating thermal stretching and equal rise (laminate top surface) and fall (laminate bottom surface) simulating thermal bending [Equation 16] have been considered here whereas along with inplane directions, bidirectional sinusoidal variation is assumed [Equation 13].

(15) 

(16) 
For maintaining the consistency in numerical values comparison, the coefficients presented in Equation 17 have been used here for normalization.

(17) 

Firstly, heat conduction solutions have determined the exact thermal gradient for both load Case I and II, and the same has been used for thermal stress analysis (termed as Present Model 1).
Thermal stress analyses have also been performed for simple constant and linear (generally cited in literature) thermal gradient approximation (Present Model 2).
These parametric studies mainly aimed to show and compare structural responses for the exact and approximate thermal gradient. Transverse aspect ratios ranging from thick to thin and inplane aspect ratios 1.5, 2, and 3 have also been considered for parametric studies, both for thermal and stress analyses.
Through thickness, thermal gradients have been presented in Figures 2 to 4 for composite laminates (A), (B), and a sandwich (D), respectively, and compared with available exact solutions (Kapuria and Achary [30]) to prove its accuracy. From Figures 2 to 4, it has been observed that thermal variation along with the depth of all composite laminates, (A), (B), and (D) remains constant for a transverse aspect ratio of more than 20 when the domain is exposed to the thermal load case I. However, for thick and moderately thick laminates, these variations showed a considerable reduction of temperature value when compared to reference constant temperature. Nearly 60% reduction in reference temperature has been noted for all cases. In symmetric laminates, the maximum reduction is noted at the half depth of laminates (Figures 4 and 5).
In contrast, for composite laminate (A), maximum reduction (more than 70%) has been located at the height of 0.3d from the bottom surface. Moreover, when the domain is exposed to thermal load case II, the depth distribution of temperature depends on the lamination configuration. For symmetric composite laminate (B), thermal variation is observed to be linear, whereas, for laminates (A) and (D), a change in gradient has been observed at the lamina interfaces. However, the transverse aspect ratio effect is absent in this case.
Figures 5 to 7 depicted through depth thermal gradient for rectangular laminates. Inplane ratios affected temperature distribution along with the depth of laminates when the domain is subjected to thermal load Case I and not to thermal load Case II.
Further, obtained thermal gradient from heat conduction solutions has been used for stress analyses. Structural responses for inplane stresses, inplane and transverse shear stresses , and transverse displacement at silent locations for thermal stress analysis have been presented in Tables 5 to 8 for laminates A, B, C, and D, respectively.
Comparisons have been made against exact solutions presented by Kapuria and Acharya [30] and noted to match them exactly here. Moreover, an additional comparison has also been made with a Model 2 solution obtained for simple constant (Case I) and linear (Case II) thermal gradient along with the laminate depth.


Fig. 2. Through thickness temperature variation in a fivelayered symmetric composite laminate
for the thermal load; (a) Case I and (b) Case II


Fig. 3. Through thickness temperature variation in a fourlayered symmetric composite laminate
for the thermal load (a) Case I and (b) Case II


Fig. 4. Through thickness temperature variation in the fivelayered sandwich composite plate
for thermal load; (a) Case I and (b) Case II


Fig. 5. Through thickness temperature variation in a fivelayered rectangular symmetric
composite plate for thermal load; (a) Case I and (b) Case II


Fig. 6. Through thickness temperature variation in the fourlayered rectangular symmetric composite plate
for thermal load; (a) Case I and (b) Case II


Fig. 7. Through thickness temperature variation in a fivelayered rectangular sandwich composite plate
for thermal load; (a) Case I and (b) Case II
For load Case I, it is worth to be noted that simple constant and linear thermal gradient assumptions and overestimate responses for all parameters are considered here. As expected, these overestimations are maximum for thick laminates and minimum for thin laminates. (transverse aspect ratio, > 30) as expected.
The approximate percentage differences between Model 1 and Model 2 for composite laminate (A) are 30% (inplane stresses), 45% (transverse shear stresses), 38% (inplane shear stress) and 45% (transverse displacement) have noted whereas, for symmetric and antisymmetric laminates (B, C), these differences are 20% (inplane stresses), 19% (transverse shear stresses), 12% (inplane shear stress) and 12% (transverse displacement). For load Case II, the marginal difference between numerical values obtained from Model 1 and Model 2 analyses expects few locations to be noticed for composite laminates A, B, and C.
However, the underestimate structural response has been observed for inplane normal stresses and transverse shear stresses for thick laminates A. In contrast, overestimation has been observed for transverse displacement and inplane shear stresses.
Further, these responses have been reversed for thin composite laminates. For crossply symmetric and antisymmetric composite laminates (B and C), the variation between Model 1 and Model 2 has a uniform from thick to thin laminates.
All displacements and stress parameters have been overestimated marginally for thick laminates, whereas for thin laminates, the difference between Model 1 and Model 2 analyses has almost been nullified. Moreover, for sandwich laminate (D), the comparison between Model 1 and Model 2 numerical values showed very large differences even for thin laminates in both Cases I and II. It is established that considered temperature variation overpredicts the structural response as much as 300% for thick plates, which is minimized to approximately 135% for a thin plate with an aspect ratio greater than 30.
Through depth, variations of all displacements and stresses were available in the paper presented by Kapuria and Achary [30] and compared in present studies.
However, concerning the presented tables in the previous section, the accuracy of the current development has been very well proven. Therefore, only a few throughthickness variations showing the major discrepancy between Model 1 and Model 2 analyses have been presented in this paper for the sake of brevity. Throughthickness variations of inplane displacement , transverse displacement , transverse shear stress , and inplane normal stress have been depicted in Figures 8 and 9 for sandwich laminate (D) subjected to Case I and II thermal loadings. Through thickness distribution of transverse stress and inplane normal stress indicates a very large discrepancy and different variation patterns for the assumed thermal gradient (Model 2) when compared with Model 1. Within the core section of sandwich laminates, transverse shear stress shows linear and parabolic patterns for Case I and Case II when the sandwich plate is subjected to an assumed thermal gradient which is supposed to be constant as observed in Model 1 analyses. Moreover, the analysis response has shifted from positive to negative ordinates between Model 1 and Model 2.
For inplane and transverse displacements, variation showed a similar pattern; however, major differences have been noticed between numerical values. These observations have also been noticed in other examples; however, fewer differences have been noted than sandwich laminates.
Therefore, it needs to determine the exact thermal gradient when laminates are subjected to temperature loading and mechanical loading to capture proper behavior. Simple assumptions, either constant, linear or any other patterns, will lead to wrong determination and interpretation of structural responses in laminated composites, especially composite with a high degree of orthotropic like sandwich laminates and lower transverse aspect ratios (a/h) that is thick laminates. Further, a selective result for rectangular plates has been presented here for futureready reference.
Normalized inplane normal stresses , transverse shear stresses and the transverse displacement of fourlayered symmetric composite rectangular ( = 2) laminate have depicted in Table 9.
Through depth, changes of inplane displacement , transverse displacement transverse shear stress , and inplane normal stress had shown in Figures 10 and 11 for fivelayered homogeneous orthotropic laminates (A) and sandwich laminates (D) when laminates were subjected to thermal load Case I.
Additionally, changes of normalized inplane displacement transverse displacement , transverse shear stress , and inplane normal stress along with the thickness for fourlayered symmetric (B) and antisymmetric (C) laminates have presented in Figures 12 and 13, respectively under the applied thermal load Case II. A mixed kind of observations has been drawn from these parametric studies.
The effect of inplane aspect ratios (b/a) has been observed for some parameters [Figure 10 (a, b, c), Figure 11 (a), Figure 12 (b, c, d), and Figure 12 (b)]. In contrast, in a few parameters, partial parts of laminate along the depth remain independent to inplane aspect ratios, viz Figure 10 (d), Figure 11 (b, c, d), Figure 12 (a), and Figure 13 (a, c, d).
Efforts have been made to determine the exact thermal gradient for laminated plates by solving heatconduction equations with the help of a semianalytical approach followed by stress analyses under applied two thermal loading cases defining stretching and bending effects. The formulation depends on the formation of BVP along with the depth of laminate, and then the solution is attempted by adopting a shooting approach. Obtained thermal gradients and stress analyses result from extensive numerical studies have been compared with available solutions, and the accuracy of current development has been proved.
Further, comparisons have also been documented for simple through thickness constant and linear thermal profiles. It is observed and noted here that it is needed to capture the exact thermal profile when laminate components are subjected to thermal loading with/without other kinds of loading.
Fig. 8. Through thickness variation of normalized (a) inplane displacement (b) transverse displacement (c) transverse shear stress (d) inplane normal stress of fivelayered sandwich composite laminate subjected to the thermal load, (Case I)
Fig. 9. Through thickness variation of normalized (a) inplane displacement (b) transverse displacement (c) transverse shear stress
(d) inplane normal stress of fivelayered sandwich composite laminate subjected to the thermal load, (Case II)
Fig. 10. Through thickness variation of normalized (a) inplane displacement (b) transverse displacement (c) transverse shear stress
(d) inplane normal stress of fivelayered homogeneous orthotropic laminate subjected
to thermal load (Case II) for different inplane (s_{2}) aspect ratio
The degree of orthotropic, lamination scheme, and transverse and inplane aspect ratios are sensitive parameters toward the correct response prediction for laminated structures when subjected to thermal loading. The present formulation uses analytical and numerical approaches, achieving accuracy and simplicity.
It also helps to reduce the dimension of the elasticity problem and helps to avoid a complex 3D stress analysis. This unique formulation can analyze composite material subjected to environmental/externally imposed loads such as thermal, hygrothermal, piezoelectric, mechanical, and its combinations with little modification in the future.
Moreover, the application of present semianalytical models for multidirectional FG materials can also be tested in the future. However, the presented formulation is restricted to simply supported boundary conditions and distributed transverse loading in bidirectional sinusoidal and uniform loading conditions.
Fig. 11. Through thickness variation of normalized (a) inplane displacement (b) transverse displacement (c) transverse shear stress (d) inplane normal stress of fivelayered sandwich composite laminate subjected
to the thermal load, (Case I) for different inplane aspect ratio
Fig. 12. Through thickness variation of normalized (a) inplane displacement (b) transverse displacement
(c) transverse shear stress (d) inplane normal stress of fourlayered symmetric laminate subjected
to the thermal load, (Case II) for different inplane aspect ratio
Fig. 13. Through thickness variation of normalized (a) inplane displacement (b) transverse displacement (c) transverse
shear stress (d) inplane normal stress of fourlayered antisymmetric composite laminate subjected
to the thermal load, (Case II) for different inplane aspect ratio.
Table 5. Inplane normal stresses , transverse shear stresses , and the transverse displacement
of fivelayered orthotropic composite square laminates
Case I 
s 
Source 







5 
^{1}Exact solutions 
0.6799 
0.2610 
1.5802 2.3311 
1.3365 
1.1570 
0.3155 

Present Model 1 
0.6803 
0.2611 
1.5804 2.3319 
1.3370 
1.1573 
0.3156 

Present Model 2 
0.8918 
0.3281 
2.3291 2.6962 
1.6647 
1.5978 
0.4619 

10 
^{1}Exact solutions 
0.3441 
0.2083 
0.3989 0.7718 
1.8825 
1.0761 
0.5273 

Present Model 1 
0.3442 
0.2083 
0.3989 0.7719 
1.8827 
1.0762 
0.5274 

Present Model 2 
0.3712 
0.2159 
0.4805 0.7802 
2.0083 
1.1836 
0.5874 

20 
^{1}Exact solutions 
0.1996 
0.1595 
0.0259 0.3351 
2.0871 
1.0060 
0.6292 

Present Model 1 
0.1996 
0.1595 
0.0259 0.3351 
2.0871 
1.0060 
0.6292 

Present Model 2 
0.2032 
0.1597 
0.0159 0.3312 
2.1229 
1.0313 
0.6471 

40 
^{1}Exact solutions 
0.1564 
0.1422 
0.1461 0.2245 
2.1452 
0.9812 
0.6608 

Present Model 1 
0.1564 
0.1422 
0.1461 0.2245 
2.1452 
0.9812 
0.6608 

Present Model 2 
0.1571 
0.1421 
0.1444 0.2232 
2.1545 
0.9874 
0.6655 

Case II 
s 
Source 







5 
^{1}Exact solutions 
0.8914 
0.3787 
1.3061 2.8454 
1.3531 
0.4574 
0.5237 

Present Model 1 
0.8918 
0.3788 
1.3064 2.8461 
1.3536 
0.4574 
0.5239 

Present Model 2 
0.8368 
0.3702 
1.5743 2.7791 
0.9534 
1.2766 
0.4846 

10 
^{1}Exact solutions 
0.8037 
0.5004 
1.0784 1.4850 
1.1968 
0.2924 
0.6036 

Present Model 1 
0.8038 
0.5004 
1.0784 1.4886 
1.1969 
0.2924 
0.6037 

Present Model 2 
0.8877 
0.5676 
1.3723 1.6761 
0.6143 
0.9427 
0.5425 

20 
^{1}Exact solutions 
0.7849 
0.5501 
0.9827 1.0872 
1.1241 
0.2237 
0.6291 

Present Model 1 
0.7849 
0.5501 
0.9827 1.0872 
1.1241 
0.2236 
0.6291 

Present Model 2 
0.9318 
0.6595 
1.2562 1.3324 
0.4502 
0.7967 
0.5629 

40 
^{1}Exact solutions 
0.7810 
0.5647 
0.9547 0.9810 
1.1020 
0.2032 
0.6360 

Present Model 1 
0.7810 
0.5648 
0.9547 0.9810 
1.1020 
0.2032 
0.6360 

Present Model 2 
0.9469 
0.6877 
1.2195 1.2385 
0.3993 
0.7525 
0.5686 
^{1}Exact solutions = Kapuria and Achary [30]
Table 6. Inplane normal stresses , transverse shear stresses , and the transverse displacement
of fourlayered symmetric composite square laminates
Case I 
s 
Source 






5 
^{1}Exact solutions 
0.9134 
0.5603 
2.2777 2.2777 
0.9186 
0.1426 
0.5553 

Present Model 1 
0.9138 
0.5605 
2.2784 2.2784 
0.9190 
0.1427 
0.5554 

Present Model 2 
1.1023 
0.6678 
2.5672 2.5673 
1.0011 
0.1598 
0.5916 

10 
^{1}Exact solutions 
0.8075 
0.6356 
0.6256 0.6256 
0.8108 
0.0996 
0.6338 

Present Model 1 
0.8076 
0.6356 
0.6256 0.6256 
0.8109 
0.0996 
0.6338 

Present Model 2 
0.8520 
0.6657 
0.6460 0.6460 
0.8279 
0.1027 
0.6452 

20 
^{1}Exact solutions 
0.7753 
0.6572 
0.1604 0.1604 
0.7763 
0.0868 
0.6567 

Present Model 1 
0.7753 
0.6572 
0.1604 0.1604 
0.7763 
0.0868 
0.6567 

Present Model 2 
0.7862 
0.6649 
0.1617 0.1617 
0.7803 
0.0875 
0.6597 

40 
^{1}Exact solutions 
0.7668 
0.6628 
0.0404 0.0404 
0.7671 
0.0835 
0.6627 

Present Model 1 
0.7668 
0.6628 
0.0404 0.0404 
0.7671 
0.0835 
0.6627 

Present Model 2 
0.7695 
0.6648 
0.0404 0.0404 
0.7680 
0.0837 
0.6634 

Case II 
s 
Source 






5 
^{1}Exact solutions 
0.7463 
0.1933 
2.7529 1.5126 
0.7510 
0.1316 
0.1403 

Present Model 1 
0.7466 
0.1933 
2.7537 1.5131 
0.7513 
0.1317 
0.1403 

Present Model 2 
0.7745 
0.1934 
2.8210 1.5399 
0.7646 
0.1348 
0.1463 

10 
^{1}Exact solutions 
0.7524 
0.2850 
1.6449 1.3250 
0.5152 
0.0975 
0.2632 

Present Model 1 
0.7525 
0.2850 
1.6450 1.3251 
0.5152 
0.0975 
0.2632 

Present Model 2 
0.7593 
0.2855 
1.6554 1.3329 
0.5175 
0.0981 
0.2652 

20 
^{1}Exact solutions 
0.7603 
0.3193 
1.3155 1.2349 
0.4192 
0.0864 
0.3130 

Present Model 1 
0.7603 
0.3193 
1.3155 1.2349 
0.4192 
0.0864 
0.3130 

Present Model 2 
0.7621 
0.3195 
1.3177 1.2368 
0.4196 
0.0865 
0.3136 

40 
^{1}Exact solutions 
0.7630 
0.3290 
1.2283 1.2081 
0.3916 
0.0834 
0.3274 

Present Model 1 
0.7630 
0.3290 
1.2284 1.2081 
0.3915 
0.0834 
0.3274 

Present Model 2 
0.7634 
0.3291 
1.2289 1.2086 
0.3917 
0.0834 
0.3275 
^{1}Exact solutions = Kapuria and Achary [30]
Table 7. Inplane normal stresses , transverse shear stresses , and the transverse displacement
of fourlayered antisymmetric composite square laminates
Case I 
s 
Source 







5 
^{1}Exact solutions 
0.8549 
0.5091 
2.2801 2.2801 
0.8549 
0.1429 
0.5091 

Present Model 1 
0.8553 
0.5093 
2.2808 2.2808 
0.8553 
0.1429 
0.5093 

Present Model 2 
1.0176 
0.5936 
2.5707 2.5707 
1.0176 
0.1601 
0.5936 

10 
^{1}Exact solutions 
0.7771 
0.6110 
0.6257 0.6257 
0.7771 
0.0996 
0.6110 

Present Model 1 
0.7771 
0.6111 
0.6257 0.6257 
0.7772 
0.0996 
0.6111 

Present Model 2 
0.8137 
0.6347 
0.6461 0.6461 
0.8137 
0.1028 
0.6347 

20 
^{1}Exact solutions 
0.7657 
0.6496 
0.1604 0.1604 
0.7657 
0.0868 
0.6497 

Present Model 1 
0.7657 
0.6496 
0.1604 0.1604 
0.7657 
0.0868 
0.6496 

Present Model 2 
0.7745 
0.6557 
0.1617 0.1617 
0.7745 
0.0875 
0.6557 

40 
^{1}Exact solutions 
0.7643 
0.6608 
0.0404 0.0404 
0.7643 
0.0835 
0.6608 

Present Model 1 
0.7643 
0.6608 
0.0404 0.0404 
0.7643 
0.0835 
0.6608 

Present Model 2 
0.7664 
0.6623 
0.0404 0.0404 
0.7664 
0.0837 
0.6623 

Case II 
s 
Source 







5 
^{1}Exact solutions 
0.6473 
0.2648 
2.7869 1.5492 
0.8426 
0.1380 
0.4145 

Present Model 1 
0.6476 
0.2649 
2.7878 1.5497 
0.8426 
0.1380 
0.4146 

Present Model 2 
0.6737 
0.2735 
2.8557 1.5772 
0.8389 
0.1413 
0.4248 

10 
^{1}Exact solutions 
0.5917 
0.2942 
1.8455 1.5276 
0.8768 
0.1130 
0.4471 

Present Model 1 
0.5918 
0.2942 
1.8457 1.5277 
0.8768 
0.1130 
0.4471 

Present Model 2 
0.5979 
0.2966 
1.8569 1.5364 
0.8760 
0.1137 
0.4499 

20 
^{1}Exact solutions 
0.5759 
0.3025 
1.5937 1.5136 
0.8863 
0.1060 
0.4561 

Present Model 1 
0.5759 
0.3025 
1.5937 1.5137 
0.8863 
0.1060 
0.4561 

Present Model 2 
0.5774 
0.3031 
1.5961 1.5159 
0.8861 
0.1062 
0.4568 

40 
^{1}Exact solutions 
0.5718 
0.3046 
1.5296 1.5095 
0.8888 
0.1042 
0.4584 

Present Model 1 
0.5718 
0.3046 
1.5296 1.5095 
0.8888 
0.1042 
0.4584 

Present Model 2 
0.5722 
0.3047 
1.5302 1.5101 
0.8887 
0.1043 
0.4586 

^{1}Exact solutions = Kapuria and Achary [30]
Table 8. Inplane normal stresses , transverse shear stresses , and the transverse displacement
of fivelayered sandwich composite square laminates
Case I 
s 
Source 






5 
^{1}Exact solutions 
0.7973 
0.1215 
2.2860 2.2860 
0.8580 
0.0679 
0.1285 

Present Model 1 
0.7974 
0.1215 
2.2867 2.2867 
0.8580 
0.0679 
0.1285 

Present Model 2 
2.5120 
0.3997 
2.7229 2.7271 
2.1455 
0.7503 
0.1070 

10 
^{1}Exact solutions 
0.8045 
0.1322 
0.6544 0.6544 
0.8220 
0.0633 
0.1338 

Present Model 1 
0.8045 
0.1321 
0.6545 0.6545 
0.8220 
0.0633 
0.1338 

Present Model 2 
2.0543 
0.3372 
0.6792 0.6871 
1.7637 
0.1222 
0.1155 

20 
^{1}Exact solutions 
0.8068 
0.1354 
0.1699 0.1699 
0.8112 
0.0620 
0.1354 

Present Model 1 
0.8068 
0.1354 
0.1699 0.1699 
0.8112 
0.0620 
0.1354 

Present Model 2 
1.9337 
0.3207 
0.1665 0.1753 
1.6676 
0.1151 
0.1171 

40 
^{1}Exact solutions 
0.8075 
0.1362 
0.0429 0.0429 
0.8084 
0.0616 
0.1359 

Present Model 1 
0.8075 
0.1362 
0.0429 0.0429 
0.8084 
0.0616 
0.1359 

Present Model 2 
1.9032 
0.3165 
0.0382 0.0472 
1.6435 
0.1133 
0.1182 

Case II 
s 
Source 






5 
^{1}Exact solutions 
0.6455 
0.0817 
1.6070 0.7887 
0.9357 
0.0513 
0.1161 

Present Model 1 
0.6455 
0.0817 
1.6071 0.7887 
0.9357 
0.0513 
0.1161 

Present Model 2 
1.3497 
0.1978 
2.8026 1.4228 
1.2760 
0.4752 
0.1177 

10 
^{1}Exact solutions 
0.6486 
0.0890 
1.1075 0.8991 
0.9411 
0.0491 
0.1173 

Present Model 1 
0.6485 
0.0890 
1.1075 0.8991 
0.9411 
0.0491 
0.1173 

Present Model 2 
1.2902 
0.1919 
1.9804 1.6352 
1.2116 
0.8947 
0.1192 

20 
^{1}Exact solutions 
0.6487 
0.0938 
0.9821 0.9298 
0.9432 
0.0485 
0.1177 

Present Model 1 
0.6487 
0.0938 
0.9821 0.9298 
0.9432 
0.0485 
0.1177 

Present Model 2 
1.2752 
0.1904 
1.7749 1.6886 
1.1953 
1.7611 
0.1196 

40 
^{1}Exact solutions 
0.6486 
0.0956 
0.9510 0.9379 
0.9439 
0.0483 
0.1178 

Present Model 1 
0.6486 
0.0956 
0.9510 0.9379 
0.9439 
0.0483 
0.1178 

Present Model 2 
1.2714 
0.1900 
1.7235 1.7020 
1.1912 
3.5082 
0.1197 
^{1}Exact solutions = Kapuria and Achary [30]
Table 9. Inplane normal stresses, transverse shear stresses , and the transverse displacement
of fourlayered symmetric composite rectangular (b /l = 2) laminates
Case I 
s 
Source 






5 
^{1}Exact solutions 
1.0756 
0.6300 
2.3984 2.3984 
0.4826 
0.1197 
0.2685 

Present Model 1 
1.0758 
0.6301 
2.3989 2.3989 
0.4828 
0.1197 
0.2686 

Present Model 2 
1.1930 
0.6942 
2.5710 2.5710 
0.4948 
0.1273 
0.2794 

10 
^{1}Exact solutions 
0.9038 
0.6764 
0.6354 0.6354 
0.5529 
0.0979 
0.2900 

Present Model 1 
0.9039 
0.6764 
0.6355 0.6355 
0.5530 
0.0979 
0.2900 

Present Model 2 
0.9301 
0.6935 
0.6470 0.6470 
0.5572 
0.0995 
0.2931 

20 
^{1}Exact solutions 
0.8546 
0.6888 
0.1613 0.1613 
0.5723 
0.0919 
0.2959 

Present Model 1 
0.8546 
0.6888 
0.1613 0.1613 
0.5723 
0.0919 
0.2959 

Present Model 2 
0.8609 
0.6931 
0.1620 0.1620 
0.5735 
0.0922 
0.2966 

40 
^{1}Exact solutions 
0.8418 
0.6920 
0.0405 0.0405 
0.5773 
0.0903 
0.2974 

Present Model 1 
0.8418 
0.6920 
0.0405 0.0405 
0.5773 
0.0903 
0.2974 

Present Model 2 
0.8434 
0.6931 
0.0405 0.0405 
0.5776 
0.0904 
0.2976 

Case II 
s 
Source 






5 
^{1}Exact solutions 
0.4641 
0.0101 
1.9951 0.7105 
0.9653 
0.0529 
0.2371 

Present Model 1 
0.4643 
0.0101 
1.9956 0.7107 
0.9652 
0.0530 
0.2371 

Present Model 2 
0.4796 
0.0077 
2.0316 0.7176 
0.9645 
0.0539 
0.2419 

10 
^{1}Exact solutions 
0.3879 
0.0259 
1.0780 0.7504 
0.9796 
0.0332 
0.2693 

Present Model 1 
0.3879 
0.0259 
1.0781 0.7504 
0.9796 
0.0332 
0.2693 

Present Model 2 
0.3914 
0.0253 
1.0833 0.7538 
0.9795 
0.0334 
0.2707 

20 
^{1}Exact solutions 
0.3664 
0.0305 
0.8333 0.7509 
0.9836 
0.0277 
0.2787 

Present Model 1 
0.3664 
0.0305 
0.8333 0.7509 
0.9836 
0.0277 
0.2787 

Present Model 2 
0.3673 
0.0304 
0.8343 0.7519 
0.9836 
0.0277 
0.2790 

40 
^{1}Exact solutions 
0.3609 
0.0318 
0.7710 0.7504 
0.9847 
0.0263 
0.2812 

Present Model 1 
0.3609 
0.0317 
0.7710 0.7504 
0.9847 
0.0262 
0.2811 

Present Model 2 
0.3611 
0.0317 
0.7712 0.7506 
0.9847 
0.0263 
0.2812 
^{1}Exact solutions = Kapuria and Achary [30]
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this manuscript. In addition, the authors have entirely observed the ethical issues, including plagiarism, informed consent, misconduct, data fabrication and/or falsification, double publication and/or submission, and redundancy.
References