A trigonometric plate theory with 5-unknowns and stretching effect for advanced composite plates

Abstract

A simple but accurate trigonometric plate theory (TPT) for the bending analysis of functionally graded single-layer and sandwich plates is presented. The significant feature of this formulation is that, in addition to including the thickness stretching effect, it deals with only 5 unknowns as the first order shear deformation theory (FSDT), instead of 6 as in the well-known TPT. The TPT possesses in-plane and transverse shear strain shape functions (sin(z/m) and cos(z/n)) containing the parameters “m” and “n” that should be properly selected. The governing equations and boundary conditions are derived by employing the principle of virtual work. A Navier-type closed-form solution is obtained for functionally graded single-layer and sandwich plates subjected to bi-sinusoidal load for simply supported boundary conditions. Numerical results of the present TPT are compared with the FSDT, other quasi-3D higher order shear deformation theories (HSDTs), and 3D solutions. The important conclusions that emerge from the present numerical results suggest that: (a) for powerly graded plates the present TPT produces as good results as refined quasi-3D HSDTs, however (b) for exponentially graded plates the present TPT yields improved results; and (c) it is possible to gain accuracy keeping the unknowns’ number constant but by selecting properly the parameter “m” and “n”.

Introduction

Classical composite structures such as plates and beams made for example of carbon fiber and epoxy or E glass and Polyester are extensively used in several industries such as in the marine industry. However, despite the designers’ effort to tailor different laminate’s properties to suit a particular application, some classical composite structures still suffer from discontinuity of material properties at the interface of the layers. Therefore, the stress fields in these regions create interface problems and thermal stress concentrations under high temperature environments. Furthermore, large plastic deformation of the interface may trigger the initiation and propagation of cracks in the material [1]. In order to alleviate and perhaps solve this problem, functionally graded materials (FGMs) were proposed by Bever and Duwez [2], and then developed and successfully used in industrial applications since 1984 [3].

FGMs made possible to avoid abrupt changes in the stress and displacement distributions. Currently FGMs are alternative materials widely used in aerospace, nuclear reactor, energy sources, biomechanical, optical, civil, automotive, electronic, chemical, mechanical, and shipbuilding industries. Recently, the application of FGMs can be seen in micro and nano-devices. Then, FGMs provides a potential topic for further research.

In general, FGMs are both macroscopically and microscopically heterogeneous advanced composites which are made for example from a mixture of ceramics and metals with continuous composition gradation from pure ceramic on one surface to full metal on the other. In fact, FGM are materials with spatial variation of the material properties, see for example the paper related to beams by Giunta et al. [50] and further information in the book by Carrera et al. [51]. However, in most of the analytical and numerical methods to study FGMs available in the literature, as in the present work, such variation is through the plate thickness only. Therefore, it evidences the need of a lot of research on FGMs.

Recently, several researchers have provided results on functionally graded plates and shells. Both analytical and numerical solutions can be found in the literature. An interesting literature review also may be found in the paper of Birman and Byrd [4], complemented by Mantari and Guedes Soares [5], [6], [7], [8], [9]. An updated literature review of FGMs can be found in the work by Jha et al. [10]. Therefore, for completeness, in the present article, only the relevant and related work on FGMs performed during the last two years is described.

As one important conclusion of the literature review, it can be said that FGMs are important due to outstanding properties of being able to withstand high temperature gradients, strong mechanical performance and reduce the possibility of catastrophic fracture. Therefore, a lot attention was given to the thermal behavior of FGMs. However, even the recent effort to study the bending and vibration analysis of FGMs, this is an interesting research topic that still deserves attention.

Carrera et al. [11] studied the static analysis of functionally graded plates (FGPs) and shells. The stretching effect was included in the mathematical formulation and the importance of the transverse normal strain effects in the mechanical prediction of stresses of FGPs and shells was remarked. Recently, Neves et al. [12], [13] and Ferreira et al. [14] presented a quasi-3D hybrid type (polynomial and trigonometric) shear deformation theory for the static and free vibration analysis of functionally graded plates by using meshless numerical method. Their formulation can be seen as a generalization of the original Carrera’s Unified Formulation (CUF), by introducing different non-polynomial displacement fields for in-plane displacements, and polynomial displacement field for the out-of-plane displacement. Therefore, new non-polynomial shape strain functions, which can be adapted to this advanced generalized formulation for perhaps more accurate results, are needed.

In fact, Mantari et al. [15], [16], [17], [18], [19] and Mantari and Guedes Soares [20] presented several new non-polynomial shear strain shape functions, which were found to be important when they were used to solve classical and advanced composites due to accuracy of the achieved results. For example, Mantari and Guedes Soares [5], [6], [7], [8], [9] presented bending results of FG plates by using new non-polynomial HSDTs. The authors used 5-unknowns HSDT without including the stretching effect. In [7], [8], the stretching effect was included by adding 1-unknown in the displacement field (6-unknowns HSDT) and improved results of displacements and in-plane normal stresses compared with [5], [6] were obtained.

On the other hand, based on the interesting work on the 2-unknowns plate theory for isotropic and orthotropic plates by Shimpi [21] and Shimpi and Patel [22], [23], respectively, Mechab et al. [24] studied the static behavior of FGPs by using 4-unknowns plate theory and polynomial shear strain shape function. Then, Abdelaziz et al. [25] studied the static analysis of functionally graded sandwich plates with the same plate theory. Houari et al. [26] and Hamidi et al. [27] analyzed the thermoelastic behavior of FGPs by the 4-unknowns plate theory and polynomial shear strain function. Mechab et al. [28] studied the behavior of classical composites under thermomechanical loading by mean of the same plate theory with non-polynomial shear strain function (sinusoidal). Consequently, Mechab et al. [29] considered the static and dynamic analysis of FGPs with new non-polynomial shear strain shape function (hyperbolic). Recently, a 5-unknowns TPT with stretching effect was developed by Thai and Kim [30] with good accuracy respect its counterpart the TPT with 6-unknowns.

The paper deals with the TPT (also called sinusoidal HSDT), and it is important to mention that the TPT was originally developed by Levy [31], corroborated and improved by Stein [32], extensively used by Touratier [33] and recently by Vidal and Polit [34], [35], [36], [37] and adapted to powerly graded plates (PGPs) and exponentially graded paltes (EGPs) by Zenkour [38], [39].

The present paper mainly uses the ideas behind the 2-unknowns [21], [22], [23], 4-unknowns without thickness stretching effect [24], [25], [26], [27], [28], [29], 5-unknowns with thickness stretching effect [30], and the 6-unknowns with thickness stretching effect plate theory [7], [8], [40] which are described in what follow.

In [21], [22], [23], in a remarkable work, the authors include w_s and w_b (bending and shear transverse displacement) to model the transverse displacement of the shear deformation theories (in many theories assumed constant and called w₀), in fact, the authors adopt the ideas belonging to Huffington [46]. In [24], [25], [26], [27], [28], [29], the authors used the previous contribution to develop HSDTs with 4 unknowns without including the thickness stretching effect. In [30], the authors introduce a TPT including the thickness stretching effect by adding 1-unknown to similar HSDT [24], [25], [26], [27], [28], [29]. In [7], [8] the authors developed a novel trigonometric quasi-3D HSDT and a generalized hybrid type quasi-3D HSDT with 6-unknowns. In [40] an optimized sinusoidal HSDT for the analysis of functionally graded shells with thickness stretching effect and 6-unknowns was developed. In the present paper the platform to develop quasi-3D hybrid type generalized HSDTs is presented, but the paper focus their studies on the TPT with modified displacement field to its optimization.

Complementarily, the present theory complies with the tangential stress-free boundary conditions on the plate boundary surface, and thus a shear correction factor is not required. The plate governing equations and their boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for plates subjected to bi-sinusoidal transverse load for simply supported boundary conditions. The results of present optimized HSDTs are compared with 3D exact, quasi-exact, and with other closed-form solution published in the literature.