Approximate 3-D model for analysis of laminated plates with arbitrary lay-ups, loading and boundary conditions
Abstract
Available exact solution techniques of elasto-static problems entail limitations on the choice of lay-ups, loading and boundary conditions and impose restrictions on strain and stress fields as well, to overcome algebraic difficulties inherent to modeling of laminated and sandwich composites. Therefore in fact they become unsuitable for testing accuracy of modern laminated plate theories aiming to very accurately describing 3-D stress fields in real conditions of use of multilayered composites, nowadays widespread in engineering applications. To overcome the assumption of too restrictive hypotheses, an approximate 3-D solution technique is proposed and assessed that is able to automatically solve problems which due to the lay-ups, loading and boundary conditions assumed would not be solved with the exact techniques available. A quite general, accurate structural model is developed that comes to constitute a generalization of available physicallybased zig-zag theories, being free from through-thickness assumptions and because zig-zag functions are not explicitly contained, the layerwise contributions being represented by the redefinition of coefficients of the through-thickness series expansion. It is based solely on the prescriptions of the theory of elasticity, i.e., displacement and stress compatibility at interfaces, fulfillment of local equilibrium equations at points across the thickness and of stress boundary constraints. A truncated expansion series of trial functions and unknown amplitudes is used to represent variables, whose coefficients are determined in exact form using a symbolic calculus tool that enforces all elasticity constraints and in conjunction with Rayleigh-Ritz and Lagrange multipliers methods.
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Introduction
Laminated and sandwich composites are progressively replacing traditional metallic materials in various engineering applications, thanks to their superior properties in terms of specific strength and stiffness, energy absorption, fatigue and corrosion properties.
Due to strongly different in-plane and transversal elastic properties that distinguish these materials, 3-D stress fields take place that condition formation and growth of local damage and failure mechanisms and that require specific structural models. The main aspect to highlight is that their displacement field must be C°-continuous (zig-zag effect), so that appropriate slope discontinuities occurs and consequently local equilibrium equations can be satisfied at layer interfaces. A multitude of theories is to date available that differently account for the mentioned layerwise effects and which so have a different order of accuracy and different computational costs. A broad discussion of this matter is given by Carrera and coworkers [1-4], Vasilive and Lur’e [5], Reddy and Robbins [6], Lur’e, and Shumova [7], Noor et al. [8], Altenbach [9], Khandan et al. [10] and Kapuria and Nath [11] and in the book by Reddy [12].
Due to the inevitable limiting hypotheses a priori formulated, laminated plate theories may no longer be accurate for arbitrary lay-ups, loading and boundary conditions and strong transverse anisotropy that are typical of some practical applications. As most of the assessments have been carried out just considering cross-ply lay-ups, sinusoidal heap loading and simplysupported edges, since exact solutions can be determined and used for comparisons, and unfortunately usually a mild variation of thickness and elastic properties of layers, what the limits of available theories may be is not well known in general.
Exact solutions, of which by way of example the papers by Brischetto [13] (static analysis of multi-layered plates and shells), Yang et al. [14] (free vibration analysis of laminated, box and sandwich Icardi [15] (exact solution for a damaged sandwich beam with laminated faces) and by Ren [16] (laminated shell in cylindrical bending) are cited, usually assume thin constituent layers in order plane strain conditions can be postulated. In addition, they assume symmetric lay-ups, simplysupported conditions and sinusoidal loading, because solutions can be assumed in trigonometric form.
Conclusion
A 3-D approximate solution procedure has been proposed in this paper which avoids assuming any restrictive hypotheses across the thickness about through-thickness kinematics, strain and stress fields, like on the contrary usually done for developing laminated plate theories so to overcome algebraic difficulties.
A general representation of variables is assumed, which is based on a fixed number of unknowns irrespective for the number of constituent layers, whose intended aim is to capture all the salient three-dimensional effects with a low computational cost, so to provide a quick and accurate tool that provides a reference results for cases where exact solutions cannot be found with the known techniques, due to the considered lay-up, loading and boundary conditions.
It comes to constitute a generalization of zig-zag theories [19-21], which likewise uses a fixed number of d.o.f. irrespective of the number of layers, for the ideas on the basis of which it is developed. The fundamental aspect to underline is the lack of zig-zag layerwise functions, as the coefficients of the displacement field play themselves the role of layerwise functions being redefined across the thickness through the enforcement of physical constraints related to out-of-plane compatibility of stress components, displacements and fulfillment of stress-boundary conditions.
The characteristic feature that enables the development of such a general physically-based theory is that all constraints are satisfied in exact form through use of a symbolic calculus tool that provides once and for all expressions deriving from the enforcement of constraints. An analytical solution is searched as a truncated expansion series of trial functions and unknown amplitudes using Rayleigh-Ritz and Lagrange multipliers methods.
To assess its accuracy, a number of challenging benchmarks with strong layerwise effects have been considered, along with applications with less demanding characteristics from the standpoint of modeling, for which exact solutions are available.
The results of all these applications show that invariably the present approximate 3-D solution turns out to be accurate and of low cost, as shown by the comparison with the exact results when available and the results of 3-D FEA and by recent laminated plate theories by the authors, which although presenting similar characteristics as the lack of explicit zigzag functions and the imposition of strict constraints across the thickness are however based on a priori assumptions