Thermal Stress Analysis of Composite Laminates using Trigonometric Shear Deformation Theory and Finite Element Method

Laminated composite materials offer a versatile design approach for achieving the desired levels of stiffness and strength by selecting specific lamination schemes. The Trigonometric Shear Deformation Theory (TrSDT) effectively addresses the appropriate distribution of transverse shear strains throughout the plate thickness while maintaining stress-free boundary conditions on the plate's top surfaces. Consequently, there is no need for a shear correction factor. In this research paper, we use the Trigonometric Shear Deformation Theory (TrSDT) that takes into account the influence of transverse shear deformation. The in-plane displacement field incorporates a sinusoidal function with respect to the thickness coordinate to accommodate the effects of shear deformation. Theories that involve trigonometric functions based on the thickness coordinate in the displacement fields are collectively referred to as Trigonometric Shear Deformation Theories (TrSDTs). In the present study, we conduct a thermal stress analysis of Laminated Composite Plates using the TrSDT. This theory eliminates the need for shear correction factors and provides a more accurate distribution of interlaminar stresses compared to other methods like CPT and FOST. We assess deflection and stress at various locations and for different aspect ratios under thermal loads using TrSDT. Stress evaluations are carried out analytically, and the results are validated by comparing them with existing findings from the literature. To further verify our findings, we model a composite laminate under thermal loads using the commercial Finite Element Method tool ABAQUS, and our results are validated against those obtained with the TrSDT for plates with simply supported boundary conditions.


INTRODUCTION
Plate structures play a pivotal role as primary load-bearing elements in the realm of Structural Mechanics, finding widespread applications in terrestrial, naval, and aeronautical engineering.These plates often endure substantial in-plane compression forces and/or shear loads.What makes them particularly versatile is their mechanical properties in various directions and their exceptional strength-to-weight ratios, which allow for tailored configurations.Furthermore, they exhibit a range of unique characteristics, encompassing resistance to corrosion, high damping capacity, temperature resilience, and a low coefficient of thermal expansion.
These exceptional attributes have led to an expanded utilization of advanced composite materials in structures exposed to thermal environments.The use of highly fiber-reinforced materials empowers designers to finely control the structure's stiffness and strength.Composite laminates are formed by stacking layers of different composite materials and/or adjusting fiber orientations, with their planar dimensions being significantly larger than their thickness.In many cases, laminates are employed in applications that demand both axial and bending strength, treating them effectively as plates.
Numerous plate theories are accessible to describe the static and dynamic behavior of such plates.The choice of a specific plate theory depends on factors like plate geometry and material properties, with one theory being more suitable than another in various scenarios.Understanding the distinctions between these theories and their practical application is of great significance to engineers involved in plate structures and researchers engaged in advancing our understanding of plate behavior.
Shimpi and Ghugal [1] introduced a novel layerwise trigonometric shear deformation theory for the analysis of two-layered cross-ply laminated beams.This theory not only streamlines the number of primary variables, reducing them to even fewer than those in the first-order shear deformation theory, but also eliminates the need for a shear correction factor.Ghugal and Shimpi [2] provided an overview of displacement and stress based refined theories for isotropic and anisotropic laminated plates.They discussed various equivalent single-layer and layerwise theories for laminated plates, highlighting their advantages and disadvantages.The paper also referenced exact elasticity solutions for plate problems where available, addressing various critical issues related to plate theories based on the literature review.Matsunaga [3] described a two-dimensional global higher-order deformation theory for evaluating inter-laminar stresses and displacements in cross-ply multilayered composite and sandwich plates subjected to  [8] presented a review of the recent developments in finite element analysis for laminated composite plates from 1990 onwards.The review covered finite elements based on various laminated plate theories for free vibration, static analysis, buckling, postbuckling analysis, geometric nonlinearity, large deformation analysis, and failure and damage analysis of composite laminated plates.Roylance [9] outlined the mechanics of fiber-reinforced laminated plates, elucidating a computational approach that connects in-plane strain and curvature with tractions and bending moments imposed on the laminate.The paper reviewed constitutive relations for isotropic materials and demonstrated the straightforward extension to transversely isotropic composite laminae.Kant and Shiyekar [10] presented a comprehensive analytical model that accounted for shear deformation and transverse normal thermal strains in the thermal stress analysis of cross-ply laminates subjected to linear or gradient thermal profiles across the laminate's thickness.The model used twelve degrees of freedom to expand the primary displacement field in the thickness direction.The resulting equilibrium equations, based on higher-order shear and normal deformation theory, were variationally consistent and derived using the principle of virtual work.Numerical results for displacements and stresses were compared with classical plate theory, firstorder shear deformation theory, and higher-order shear deformation theory.Mantari et al. [11] developed a trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates, introducing a parameter 'm' to align results with three-dimensional elasticity bending solutions.This theory effectively distributes transverse shear strains across the plate thickness, eliminating the need for a shear correction factor.Tornabene et al. [12] employed the Generalized Differential Quadrature (GDQ) method to study laminated composite degenerate shell panels, including rectangular and annular plates.The method enabled the determination of stress profiles through the thickness of plates without specifying specific equations for these plate types, making the theoretical treatment general.Thai et al. [13] introduced an inverse tangent shear deformation theory (ITSDT) for the dynamic, free vibration, and buckling analysis of laminated composite and sandwich plates.Ghugal and Kulkarni [14] addressed thermal stresses and displacements in orthotropic, twolayer antisymmetric, and three-layer symmetric square cross-ply laminated plates subjected to nonlinear thermal loads through the thickness of the plates.They employed trigonometric shear deformation theory.Srinivasan and Rao [15] presented Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates.Pagano [16] developed exact Solutions for Rectangular Bidirectional Composites and Sandwich Plates.Karama [17] presented A new theory for laminated composite plates.Reddy [18] developed third order shear deformation theory.Kulkarni and Kapuria [19] developed A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates.Shaikh and Chakrabarti [20] presented a New Plate Bending Element Based on Higher Order Shear Deformation Theory for the Analysis of Composite Plates.Chen and Wu [21] presented a new higher-order shear deformation theory and refined beam element of composite laminates.Sahooand and Singh [23] developed a new shear deformation theory for the static analysis of laminated composite and sandwich plates.Kant and Swaminathan [24] developed analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory.Reddy and Liu [25] presented a higher-order shear deformation theory of laminated elastic shells.
In this paper, to confirm our results, we conduct a thermal analysis of a composite laminate in ABAQUS [22], a commercial Finite Element Method software.Subsequently, we compare our findings with those obtained through the application of the Trigonometric Shear Deformation Theory (TrSDT) to plates with boundary conditions characterized as simply supported.

II. FORMULATION OF TRIGONOMETRIC SHEAR DEFORMATION THEORY (TrSDT)
We use a Trigonometric Shear Deformation Theory (TrSDT) that not only considers the impact of transverse shear deformation but also accounts for the influence of transverse normal strain.In this theory, the in-plane displacement field is defined using a sinusoidal function with respect to the thickness coordinate to accommodate the effects of shear deformation.Additionally, a cosine function in the thickness coordinate is employed in the transverse displacement to address the impact of transverse normal strain.The governing equations and boundary conditions for this This work is licensed under Creative Commons Attribution 4.0 International License.theory are derived through the application of the principle of virtual work.
Consider a rectangular cross-ply laminated plate with dimensions length a, width b, and total thickness h, consisting of orthotropic layers as depicted in Fig. 1.Each layer's material is assumed to possess a plane of material property parallel to the x-y plane.The coordinate system is aligned so that the plate's mid-plane coincides with the x-y plane, while the z-axis is perpendicular to the middle plane.The upper surface of the plate, situated at z = -h / 2, experiences a constant thermal load denoted as T (x, y, z).The plate's region is defined within the righthanded Cartesian coordinate system of (x, y, z).

The Displacement Field
In accordance with plate theory, the displacement field for a plate with a global thickness denoted as h can be defined as follows: (1) where, u and v are the in-plane displacements at any point (x, y, z).u 0 and v 0 signify the in-plane displacement of the point (x, y, 0) on the mid-plane, w is the transverse deflection, f x and f y are the rotations of the normal to the mid-plane about the y and x axes respectively.The strain-displacement relationships is given as,  Neglecting s z for each layer, the stress-strain relations in the orthotropic local coordinate system can be expressed as (5) This equation is referred to as the Constitutive Relationship for the material.In this equation, subscripts 1 and 2 represent the fiber and the direction normal to the fiber directions, 3 denotes the direction normal to the plate.Through appropriate coordinate transformations, the stress-strain relationships for the k th layer in the global x-y-z coordinate system can be derived as follows: (6)

Governing Equations
The governing equations are derived from the principle of virtual displacements.This work is licensed under Creative Commons Attribution 4.0 International License.(7) Substituting stress resultants in terms of displacements in the governing equilibrium equations, we can obtain the displacements, strains and stresses induced.

III. RESULTS AND DISCUSSION
In this section, examples of composite laminated plate are presented to show accuracy and applicability of ABAQUS under static loading.The results obtained are compared with published results.

Example 1: Numerical investigation of square composite laminated plates under mechanical loading (b=a)
A simply supported square laminated composite plate having dimension 1mx1m of side "a" and "b" thickness "h" for various aspect ratios is composed of three equally layers oriented at [0 0 /90 0 /0 0 ] subjected to doubly sinusoidal load 10 kN/m 2 as amplitude for m = 1, n = 1.The result of displacement and stress using ABAQUS software and with the literature are verified.The material properties are Formulae used for dimensionless maximum transverse deflection and stress are as follows.
The results and the contour plots for transverse deflection, normal stresses and in-plane shear stress are shown in below Table 1 and Fig. 2.   A simply supported rectangular composite laminated plate having dimension 1mx1m of side "a" and "b" thickness "h" for various aspect ratios is composed of three equally layers oriented at [0 0 /90 0 /0 0 ] subjected to doubly sinusoidal load 10 kN/m 2 as amplitude for m=1, n=1.The result of displacement and stress using ABAQUS software and with the literature are verified.The material properties are Formulae used for non-dimensionalsed maximum transverse deflection and stress are as per previous problem.The results as shown following Table 2.A simply supported laminated composite plate having dimension 1mx1m of side "a" and thickness "h" is composed of two and three equally layers oriented at [0 0 /90 0 ] and [0 0 /90 0 /0 0 ] respectively subjected to doubly sinusoidal temperature load of 30 0 c as amplitude for m=1, n=1.The result of displacement and stress are derived using ABAQUS software and with the literature are verified.Following are the material properties, Results are presented in Tables 3 and 4

IV. CONCLUSIONS
The conclusions drawn from the analysis in the present study are as follows: 1.When analyzing square and rectangular composite laminated plates with the [0 0 /90 0 /0 0 ] orientation under mechanical doubly sinusoidal loading, we calculated non-dimensional transverse displacement, normal stresses, and shear stress at their maximum absolute values.The results obtained using ABAQUS and TrSDT for non-dimensional transverse displacement were found to closely match with analytical results, for various aspect ratios under simply supported loading.ABAQUS provided better results for both displacement and stresses.
2. When analyzing square and rectangular composite laminated plates with the [0 0 /90 0 ] and [0 0 /90 0 /0 0 ] orientations under thermal doubly sinusoidal loading, we calculated non-dimensional transverse displacement, normal stresses, and shear stress at their maximum absolute values.Like the mechanical loading analysis, the results from ABAQUS and TrSDT for non-dimensional transverse displacement were in good agreement with Exact results for various aspect ratios under simply supported loading.Once again, ABAQUS produced more accurate results for displacement and stresses.

Example 2 :
Numerical investigation of rectangular composite laminated plates under mechanical loading (b=3a)

Table 3
In-plane displacement and normal stresses in Composite Laminated Square Plate for all edges simply supported composite laminated square plate [0 0 /90 0 /0 0 ] under sinusoidal temperature loading for a/h = 10 [ ] % error w.r.t.Exact

Table 4
In-plane displacement and normal stresses in Composite Laminated Square Plate for all edges simply supported composite laminated square plate [0 0 /90 0 ] under sinusoidal temperature loading for a/h = 10 [ ] % error w.r.t.Exact 3. In all the analyzed problems of composite laminated plates, ABAQUS provided increasingly accurate