Dynamic analysis of laminated composite plates using a layer-wise mixed finite element model
Introduction
Advanced composite materials are finding increasing application in aircraft, automobiles, marine and submarine vehicles besides other engineering applications owing to their high specific strength and stiffness. Consequently, these applications have stimulated interest in the development of mathematical models for prediction of the dynamic behaviour of the physical models with sufficient accuracy. Moreover, all these applications of composite materials in advanced technology areas, where precision and reliability play a paramount role, demand clear understanding of their behaviour and performance under severe operating environments.
The analysis of the multi-layered structures is a complex task, compared with the conventional homogeneous isotropic metallic structures, due to the exhibition of coupling among membrane, torsion and bending strains; weak transverse shear rigidities; and discontinuity of the mechanical characteristics along the thickness of the laminates. A theory that can address these complexities and also remain general in its approach, is required for an accurate analysis of multi-layered composite structures under general condition of loading and supports. Present work is an attempt to develop a three-dimensional mixed FE model consisting of displacement and transverse stress degrees-of-freedom in order to incorporate most of the behavioural requirements of multi-layered laminated composites.
Numerous higher order shear deformation theories have been proposed to analyze the response characteristics of laminated plates. Excellent reviews of these theories have been presented by Noor and Burton [1], Kapania and Raciti [2], Reddy [3], Leissa [4], Mallikarjuna and Kant [5], Varadan and Bhaskar [6] and recently by Kant and Swaminathan [7]. These shear deformation plate theories can be grouped as (i) the equivalent single-layer plate theories, (ii) the layer-wise plate theories and (iii) the individual-layer plate theories.
Elastic solutions of layered plates [8], [9], [10] indicate that interlaminar continuity of transverse normal and shear stresses as well as the layer-wise continuous displacement field through the thickness of the laminated plates are the essential requirements in their analysis. Thus, a layer-wise analysis, which can ensure the continuity of both, the transverse stresses as well as those of displacement fields through the thickness, is often required for the laminated composite structures. A layer-wise mixed FE model with displacement and transverse stresses as nodal degrees-of-freedom satisfy the continuity requirements of transverse stress components as well as those of displacement fields through the thickness of the laminated composite structures. Further, through such FE model the transverse stress components are evaluated directly, thereby integration of equilibrium equations are advantageously avoided, which otherwise introduce error in their evaluation. Wu and Lin [11], for example, presented a two-dimensional mixed FE scheme based on a local high order displacement model for the analysis of sandwich structure, where displacement continuity conditions at the interface between layers were regarded as the constraints and the interlaminar stresses were introduced as the Lagrange multiplier. Shi and Chen [12] also developed a mixed FE model based on global–local laminate three-dimensional variational model. The model proposed a mixed use of a hybrid stress element within a high precision stress solution region in the thickness direction of the laminate and a conventional displacement FE in the remaining domain. Carrera [13], [14], [15] has developed a layer-wise mixed FE model based on Rieissner’s mixed variational principle. Because the stress fields are assumed independent of the displacement fields in mixed FE model developed by using Reissner’s variational principle, the fundamental elasticity relation cannot be satisfied exactly.
An 18-node three-dimensional mixed FE model has been developed in the present formulation, by using the Hamilton’s principle. The transverse stress quantities (τxz, τyz and σz where z is the thickness direction) have been invoked from the assumed displacement fields by using fundamental elastic relations. This ensures the satisfaction of elastic equations throughout an elastic continuum, which is lacking in numerical models based on other mixed variational principles. Because the transverse stress components are the nodal degrees-of-freedom in the present FE model, their computations do not require the integration of equilibrium equations which otherwise reduce the accuracy in determination of these stresses. Moreover, it can appropriately model a composite laminated structural member of any number of lay-ups of different materials as it satisfies exactly, the requirements of through thickness continuity of transverse stress and displacement fields.
Drawback of such three-dimensional mixed FE models is often attributed to their being computationally expensive. However, due to the advent of high-speed computers and also with the use of parallel computing, this drawback can be minimized to a great extent. Further, the difficulty like non-positive definiteness of matrices [16] is advantageously avoided because the present formulation uses the fundamental elasticity relations for incorporating transverse stress variables as nodal degrees-of-freedom in conjunction with an extremum, minimum energy principle.
Section snippets
Formulation
An anisotropic composite laminated plate consisting of N-layers of orthotropic lamina has been considered for FE analysis as shown in Fig. 1(a) and (b). The plate has been discretized in to a number of three-dimensional elements. Each element lies completely within a lamina and no element crosses the interface between any two successive laminae.
Illustrative examples
A computer program incorporating the present three-dimensional mixed theory has been developed in FORTRAN-90 to analyze natural vibrations of symmetric/unsymmetric composite laminated plates. Numerical computations have been performed for various examples. Results have been compared with the available elastic, analytical and FE solutions wherever these are available in the literature. Illustrative examples encompassing a couple of symmetric and unsymmetric cross-ply laminated plates under
Conclusions
A novel methodology for the formulation of mixed FE model has been presented in this paper. The FE model has been developed by maintaining the fundamental elastic relationship between constituents of the stress, strain and displacement fields within the elastic continuum. Because it is a layer wise FE formulation with the transverse stress components as the nodal degrees-of-freedom, both the primary requirements of continuity of displacement fields, as well as those of transverse stress
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