Laminated composite plates subject to thermal load using trigonometrical theory based on Carrera Unified Formulation
Introduction
Nowadays, laminated composite materials are used in several construction fields, including aerospace, marine, civil, biomedical and other areas. Because of their attractive mechanical properties (high specific stiffness, excellent fatigue strength and resistance to corrosion) the demand of these kinds of materials in the industries is increasing. However, it is not simple to study the behavior of the laminated structures. Therefore, various theories and different variational statements were developed in order to study the mechanical behavior of the laminated composite materials under different loading conditions. In what follows some theoretical model are discussed.
In order to study composite materials, the classical plate theory (CPT) for metallic structures, based on the Kirchhoff’s assumptions, was extended to laminated plates. Nonetheless, only give acceptable results for thin plates, because the shear deformation effect is ignored. To overcome this problem the first order shear deformation theory (FSDT) based on Reissner [1] and Mindlin [2] was introduced. A constant transverse shear strain component is taken into account with their respectively shear correction factor. This value depends on the material coefficients, geometry, boundary conditions and loading conditions, which is difficult to calculate. Therefore, higher-order shear deformation theories (HSDTs) were introduced to avoid the shear deformation effect. The HSDTs can be developed using polynomial shape functions, for example Reddy [3], [4], Levinson [5] and Librescu [6]. Further, non-polynomial shape functions are developed by Touratier [7], Soldatos [8], Karama [9], Mantari and Guedes [10], Zenkour [11], [12], [13], [14] and Mantari et al. [15], [16], [17], [18].
The thermoelastic problem for laminated cross-ply shells were studied by Khdeir et al. [19] using a third-order HSDT which was compared with the CPT and FSDT. In the HSDT presented by Zhen and Wanji [20] the shear stress continuity conditions of composite laminated and sandwich plates under thermo-mechanical load are satisfied. Zenkour and Alghamdi [21], [22], [23] have presented the thermoelastic bending analysis of functionally graded sandwich plates. The thermoelastic bending response is presented for a simply-supported cross-ply composite laminated plate subjected to a thermal field by Zenkour and Maturi [24].
Carrera Unified Formulation (CUF) was developed for composite laminated plates and shells [25], [26], [27] using originally Taylor’s expansions of N-order. A sinusoidal shear deformation theory (SSDT) within the CUF framework was developed by Ferreira et al. [28] for static and free vibration analysis of laminated shells. In addition, a static analysis and free vibration analysis for several theories based on polynomial, trigonometric, hyperbolic, exponential and zig–zag function were developed by Carrera et al. [29], [30] for laminated beam. The effect of the normal strain effect was analyzed for static thermostatic bending problem for multilayer plates by Carrera [31]. ESL and Layerwise (LW) theories were presented based on the Principle of the Virtual Displacement (PVD) and Reissner’s Mixed Variational Theorem (RMVT) in Carrera et al. [32]. Free vibration analysis for laminated and sandwich plates using the Hierarchical Trigonometric Ritz Formulation (HTRF) were developed by Fazzolari and Carrera [33]. Furthermore, a coupled thermo-mechanical analysis for multilayer plates was studied by Brischetto and Carrera [34]. Thermal buckling analysis based on HTRF for multilayer plates was carried out by Fazzolari et al. [35], [36] and extended to study sandwich plates made of FGM by Fazzolari et al. [37].
This paper proposes a modified non-polynomial displacement field under CUF to study the thermoelastic analysis of simply-supported laminated plates under bisinusoidal thermal load. Normalized shear strain function (sin1) based on trigonometric theory are developed for several order of expansion (N = 4, 5, 6, 7, 8, 9, 10, 15, 30) and are compared with trigonometric functions in non-normalized form (sin2) and the polynomial kinematic (pol). A linear temperature field through the thickness is taken into account in this work. In addition, the temperature profile obtained by solving heat conduction problem was carried out in order to get proper results. The governing equations for the static analysis are obtained through PVD, and solved using Navier solution method.
The paper is organized as following: The analytical modeling is presented in Section 2. The geometry of the composite laminated plate, kinematic, the principle of virtual works and the governing equations under CUF platform is presented in this section. The analytical solution methodology is given in Section 3. The results are presented and discussed in Section 4. Finally, conclusions and the references list are given.
Section snippets
Plate geometry
This paper considers a simply-support cross-ply laminated plate with a uniform thickness , length , and width on the plane at in a Cartesian coordinate system as shown in Fig. 1. The relationship between the plate coordinates , , and the material coordinates 1, 2, 3 are represented in Fig. 2, where is the fiber orientation.
Model kinematics
CUF states that displacement field for plates, , is modeled in the following manner:
Analytical solution
Navier type closed form solution are possible for simply-supported cross-ply laminated plates such displacement variables and the transverse distributed load can be expressed in the following Fourier serieswhere , and , , and are amplitudes, and are the number of waves and and are the dimensions of the plate.
Therefore, the governing equations
Numerical results and discussions
The thermoelastic bending analysis of simply-supported laminated plate is presented in what follows. Trigonometric theory within the CUF framework is developed and compared with the unified polynomial theory for several orders of expansion. Three layers symmetric cross-ply plate (0°/90°/0°) under a bisinusoidal thermal load according to Eq. (9). Linear temperature profile with and nonlinear temperature profile, , obtained by solving heat conduction
Conclusions
This paper presents an analytical solution for the thermoelastic bending analysis of simply-supported laminated plates using Carrera’s Unified Formulation with trigonometric shear strain functions. A classical PVD and ESL model without interlaminar continuity are used in the present work.
These new trigonometric displacement fields have the capability to reproduce the same results of the classical Taylor expansion for thin and thick laminated cross-layer plates. Furthermore, good results
Acknowledgement
The authors want to dedicate this work to Professor J.N. Reddy for his 70 anniversary.
References (43)
- et al.
A higher-order shear of deformation theory of laminated elastic shells
Int J Eng Sci
(1985) An accurate, simple theory of the statics and dynamics of elastic plates
Mech Res Commun
(1980)On the theory of anisotropic elastic shells and plates
Int J Solids Struct
(1967)An efficient standard plate theory
Int J Eng Sci
(1991)Mechanical behavior or laminated composite beam by the new multilayer laminated composite structures model with transverse shear stress continuity
Acta Mech
(2003)- et al.
A trigonometric plate theory with 5-unknowns and stretching effect for advanced composite plates
Compos Struct
(2014) Generalized shear deformation theory for bending analysis of functionally graded plates
Appl Math Modell
(2006)Hygro-thermo-mechanical effects on FGM plates resting on elastic foundations
Compos Struct
(2010)- et al.
Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higher-order shear deformation theory
Compos Struct
(2011) - et al.
A new trigonometric deformation theory for isotropic, laminated composite and sandwich plates
Int J Solids Struct
(2012)
Vibrational analysis of advanced composite plates resting on elastic foundation
Compos Part B Eng
A new tangential-exponential higher order shear deformation theory for advanced composite plates
Compos B Eng
Thermal effects on the response of cross-ply laminated shallow shells
Int J Solids Struct
A quadrilateral element based on refined global–local higher-order theory for coupling bending and extension thermo-elastic multilayered plates
Int J Solids Struct
Analysis of laminated shells by a sinusoidal shear deformation theory and radial basis functions collocation, accounting for through-the-thickness deformations
Compos B Eng
Laminated beam analysis by polynomial, trigonometric, exponential and zig–zag theories
Eur J Mech A/Solids
Free vibration analysis of sandwich plates with anisotropic face sheets in thermal environment by using the Hierarchical trigonometric Ritz formulation
Compos B Eng
Coupled thermo-mechanical analysis of one-layered and multilayered plates
Compos Struct
Advanced variable kinematics Ritz and Galerkin formulations for accurate buckling and vibration analysis of laminated composite plates
Compos Struct
A refined sinus plate finite element for laminated and sandwich structures under mechanical and thermomechanical loads
Comput Methods Appl Mech Eng
Thermoelastic solutions for orthotropic and anisotropic composite laminates
Compos B
Cited by (33)
New quasi-three-, and two-dimensional trigonometric-cubic monomial HSDT for thermal buckling and thermo-mechanical bending analyses of FGM symmetrical/non-symmetrical sandwich plates with hard/soft core
2023, Composite StructuresCitation Excerpt :The shear strain shape function is an odd one, satisfying the zero transverse shear stresses on the top and bottom edge surfaces of the plate. This important feature allows the researchers to analyze several kinds of HSDTs, such as the third-order shear deformation theory (TSDT) [32–43], the trigonometric shear deformation theory, or the sinusoidal shear deformation theory (SSDT) [44–77], the hyperbolic shear deformation theory [78–94], and the exponential shear deformation theory [95–97] based on it, searching for more accurate solutions to the global responses as well as the thermal and mechanical behaviors analyses of structural elements, as this paper will show. This paper makes use of shear deformation plate theory using a new trigonometric-cubic monomial shape function of shear strain to build refined plate, theory models.
A neural network-assisted 3D theoretical thermoelastic solution for laminated liquid crystal elastomer plate used in restoring cardiac mechanical function
2022, Journal of the Mechanical Behavior of Biomedical MaterialsThermo-mechanical coupling analysis of laminated composite plates using wavelet finite element method
2022, Thin-Walled StructuresCitation Excerpt :The proposed finite element involving eleven unknown parameters was constructed based on sinus theory with considering layer refinement and the transverse normal effect. Then Zenkour [20] presented analytical solutions for thermal bending analysis of laminated plates using Carrera’s unified formulation with trigonometric shear strain functions. In the framework of inverse hyperbolic shear deformation theory (IHSDT), Grover [21,22] investigated the thermo-mechanical response characteristics of cross-ply and angle-ply laminated composite plates.