Elsevier

Composite Structures

Volume 92, Issue 3, February 2010, Pages 653-661
Composite Structures

A C0-type higher-order theory for bending analysis of laminated composite and sandwich plates

https://doi.org/10.1016/j.compstruct.2009.09.032Get rights and content

Abstract

In this paper, a C0-type higher-order theory is developed for bending analysis of laminated composite and sandwich plates subjected to thermal/mechanical loads. The total number of unknowns in the present theory is independent of number of layers. The continuity conditions of transverse shear stresses at interfaces are a priori enforced. Moreover, the conditions of zero transverse shear stresses on the upper and lower surfaces are also considered. Based on the developed higher order theory, the typical solutions are presented for comparison. It is very important that the first derivatives of transverse displacement w have been taken out from the in-plane displacement fields of the proposed model, so that its finite element counterparts may avoid using the C1 interpolation functions. To assess the developed theory, the C1-type higher-order theory is chosen for comparison. Numerical results show that the present model can accurately predict the thermal/mechanical response of laminated composite and sandwich plates. Moreover, the present model is able to accurately calculated transverse shear stresses directly from constitutive equations without any postprocessing methods.

Introduction

Laminated composite and sandwich plates/shells are being widely used in aerospace, automobile and other industries due to their light weight, higher stiffness and good tailoring capability. Laminated composite structures generally possess different material constants along the thickness directions at each ply. Due to the discontinuity of the mechanical characteristics through the thickness directions, some technique has to be required to satisfy the continuity of interlaminar stresses for equilibrium reasons. Once the continuity conditions of transverse shear stresses at interfaces are a priori enforced, however, the first derivatives of transverse displacements will be involved into the in-plane displacement fields, so that such model’s finite element counterparts require C1 interpolation functions. Therefore, the models, in which the first derivatives are involved into in-plane displacement fields, are called as the C1-type laminated plate theories. Reddy’s theory [1], [2], the zig-zag theories [3], [4], [5], [6], [7] and the global–local higher order theory [8] belong to the C1-type laminated plate theories. Although the continuity conditions of transverse shear stresses at interfaces are unable to be satisfied, the conditions of zero transverse shear stresses on the upper and lower surfaces were enforced in Reddy’s theory. The zig-zag theories above mentioned are able to a priori satisfy the continuity conditions of transverse shear stress as well as the free conditions of transverse shear stresses on the upper and lower surfaces of laminates. However, the zig-zag theories are still unable to accurately compute transverse shear stresses directly from constitutive equations. To obtain accurately transverse shear stresses, the three-dimensional equilibrium equation methods have to be employed. However, the equilibrium equation approach requires the higher order derivatives of transverse displacement that result in difficulties of finite element implementation. Moreover, the global equilibrium equations related to full domain [9] have to be adopted, which is rather expensive. To obtain accurately transverse shear stresses directly from constitutive equations, a global–local higher order theory has been developed by Li and Liu [8]. The number of unknowns in Li and Liu’s model is independent of the number of layers, whereas their model’s finite element counterparts will require C1 interpolation functions.

To avoid using C1 interpolation functions during finite element implementation, the C0-type laminated plate theories, in which the first derivatives of transverse displacements were not included in the in-plane displacement fields, are widely used. The first order shear deformation theories [10], [11] and the global higher order theories [12], [13], [14], [15], [16], [17], [18], [19] are of the C0-type laminated plate theories. Without using the shear correction factors, the global higher order theories can predict more accurately static and dynamic response of laminated composite and sandwich structures in comparison with the first order theories. Moreover, for the global higher order theories, the C0 interpolation functions are only required during their finite element implementation. Therefore, they are suitable for implementation in commercial finite element codes to support the more general design practices. However, due to violation of continuity conditions of interlaminar stresses, the global higher order theories can not predict accurately transverse shear stresses directly from constitutive equations. Moreover, the global higher order theories [20], [21] will also encounter the difficulties for dynamic and stability problems of soft-core sandwich plates.

It can be deducted from the literature review that although their finite element counterparts only require C0 interpolation functions, the published C0-type higher order theories generally violate the continuity conditions of transverse shear stress at interfaces, and are less accurate in comparison with the C1-type laminated plate theories. In view of this situation, a C0-type higher order theory, which can accurately predict in-plane and transverse shear stresses directly from constitutive equations, ought to be expected. To this end, a C0-type higher-order theory including interlaminar stress continuity is developed in this paper. The total number of unknowns in the developed theory is independent of the number of layers, so that present theory belongs to the equivalent single layer theories. Moreover, to the best knowledge of the authors, it cannot be found that the published C0-type equivalent single layer plate displacement-based theories in the literature are more accurate than the present model [22]. To assess the developed theory, the analytical formulations and solutions are also presented for thermo-mechanical analysis of laminated composite and sandwich plates with simply-supported boundary conditions.

Section snippets

Theoretical formulation

To clearly compare the C0-type higher order theory with the C1-type higher order theory, the expressions of two theories including interlaminar stress continuity will be detailedly presented in this section. Initial displacement fields can be given byuk(x,y,z)=uG(x,y,z)+u¯Lk(x,y,z)+uˆLk(x,y,z)vk(x,y,z)=vG(x,y,z)+v¯Lk(x,y,z)+vˆLk(x,y,z)wk(x,y,z)=wG(x,y,z)in which, uG, vG and wG denote global displacement components; u¯Lk and v¯Lk are of two-term local groups; uˆLk and vˆLk are of one-term local

Analytical formulation

In this section, the analytical formulations of the C0-type higher order theory are presented in detail. Following the same procedure, analytical solutions of another higher-order theory are also included in this paper for comparison.

Numerical examples

In this section, the thermo-mechanical problems of laminated composite and sandwich plates are considered to verify the accuracy of the C0-type higher order theory. The material constants used are given as follows:

Material 1 laminated plates [26]E1=172.4GPa,E2=E3=6.89GPa,G12=G13=3.45GPa,G23=1.378GPa,v12=v13=v23=0.25.Material 2 sandwich plate [23]

Face sheets (0.1h × 2):E1=172.4GPa,E2=E3=6.89GPa,G12=G13=3.45GPa,G23=1.378GPa,v12=v13=v23=0.25.Core material (0.8h):E1c=E2c=0.276GPa,E3c=3.45GPa,G13c=G23c

Conclusions

This paper develops a C0-type higher order theory for bending analysis of laminated composite and sandwich plates subjected to thermal/mechanical loads. The total number of unknowns in the present theory is independent of the number of layer, so that present theory belongs to the equivalent single layer theories. To assess the present theory, the C1-type higher order theory proposed by Li and Liu is chosen for comparison. The first derivatives of transverse displacements are included in the

Acknowledgements

The work described in this paper was supported by the National Natural Sciences Foundation of China (No. 10802052, 10672032), Aeronautical Science Foundation of China (No. 2008ZA54003) and Liaoning Province Science Foundation for Doctors (No. 20081004).

References (26)

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