Zig-zag theories differently accounting for layerwise effects of multilayered composites
Abstract
This paper essays the effects of the choice of through-thickness representation of variables and of zig-zag functions within a general theory by the authors from which the theories considered are particularized. Characteristic feature, coefficients are calculated using symbolic calculus, so to enable an arbitrary choice of the representation. Such choice and that of zig-zag functions is shown to be always immaterial whenever coefficients are recalculated across the thickness by enforcing the fulfillment of elasticity theory constraints. Assigning a specific role to each coefficient is shown immaterial. Moreover, the order of representation of displacements can be freely exchanged with one another and, most important, zig-zag functions can be omitted if part of coefficients are calculated enforcing the interfacial stress field compatibility. Vice versa, accuracy of theories that only partially satisfy constraints, is shown to be strongly dependent upon the assumptions made. Applications to laminated and soft-core sandwich plates and beams having different length-tothickness ratios, different material properties and thickness of constituent layers, various boundary conditions and distributed or localized loading are presented. Solutions are found in analytic form assuming the same trial functions and expansion order for all theories. Numerical results show which simplifications are yet accurate and therefore admissible.
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Introduction
Laminated and sandwich composites, which find continuously increasing applications because of their superior specific strength and stiffness, better energy absorption, fatigue properties and corrosion resistance than traditional materials, need to be analyzed with specific structural models.
Indeed, differently from non-layered materials their displacement field can no longer be C1-continuous, but instead has to be C°-continuous, i.e. slope discontinuities must occurs at the interfaces of layers with different properties as only in this way local equilibrium equations can be satisfied, which means that out-of-plane shear and normal stresses and the transverse normal stress gradient must be continuous at interfaces (zig-zag effect).
As a result of the strong differences between in-plane and transversal properties, 3-D stress fields arise whose out-of-plane components can assume the same importance as those in the plane and which play a fundamental role in the formation and growth mechanisms of damage and for failure. Considered that multilayer composites are used for the construction of primary structures, this being the only way to fully exploit their advantages, so far many multilayered theories of various order and degree have been developed, wherein sandwiches are described as multilayered structures whenever cell scale effect of honeycomb core aren’t the object of the analysis. Sandwiches are often described as three-layer laminates where the core is assumed as the intermediate layer being shear resistant in the transverse direction, free of in-plane normal and shear stresses and deformable in the thickness direction (see, e.g. Frostig and Thomsen [1]). But often higher-order sandwich theories are considered wherein in-plane and transverse displacements vary nonlinearly across the thickness, taking different forms in the faces and the core (see, Rao and Desai [2] and Yang et al. [3]), or a separate representation is used for each of them (see, Cho et al. [4]).
A broad discussion of this matter is found, among many others, in the papers by Carrera and co-workers [5-9], Demasi [10], Vasilive and Lur’e [11], Reddy and Robbins [12], Lur’e, and Shumova [13], Noor et al. [14], Altenbach [15], Khandan et al. [16] and Kapuria and Nath [17] and the book by Reddy [18]. As shown by the quoted contributions, theories can be categorized into equivalent single-layer (ESL) formulations borrowed from those for isotropic materials, which completely disregard layerwise effects and therefore they are only suitable for predicting overall response quantities (but not even for all loading, material properties and stack-up and certainly not for sandwiches as shown e.g. [19] to [25]) and layerwise formulations which differently account for layerwise and zig-zag effects, presenting a different degree of accuracy in predicting through-thickness displacement and stress fields and a different computational burden.
Conclusion
Various new zig-zag theories in displacement-based or mixed form, with a different representation of variables across the thickness and differently assumed zig-zag functions, so ultimately differently accounting for layerwise effects, or retaken from previous papers by the authors have been compared. Challenging elastostatic benchmarks with strong layerwise effects, under distributed or localized loading and simply-supported and clamped edges and with distinctly different material properties and thickness of layers, mainly in form of sandwich structures have been considered.
All zig-zag theories have the same five functional degrees of freedom like FSDT and HSDT (that in the cases here examined are very inaccurate), so the number of unknowns is independent from the number of constituent layers. To compare theories under the same conditions, the same trial functions and expansion order are used to obtain solutions in closed form.
The prefixed purpose was to show on a broader series of theories and benchmarks than in the former papers by the authors that whenever the expressions of coefficients of displacements are determined a priori by enforcing the fulfillment of the full set of interfacial stress compatibility conditions, of stress boundary conditions and of local equilibrium equations at a number of selected point sufficient to determine the expressions of all coefficients, the choice of the representation form and of zigzag functions can be arbitrary without the results changing. When all these conditions are mutually occurring, it has been shown that zig-zag functions can even be omitted, with self-evident advantages from the computational standpoint.
Higher-order zig-zag theories, whose coefficients are redefined layer-by-layer by imposing the fulfillment of interfacial displacement and stress compatibility conditions, stresses boundary conditions at upper and lower bounding faces and local equilibrium equations at different points across the thickness proved to be always those most accurate and efficient, as a computational burden still comparable to that of ESL was required for all benchmarks. In particular, it was demonstrated that zig-zag functions can be freely chosen and variables can be assumed in an arbitrary form, i.e. different form one to another and from region to region across the thickness, without the results changing. According, a specific role does not need to be assigned to individual coefficients of displacements, being sufficient that the total number of coefficients to be determined corresponds to the number of conditions to be imposed. Consequently the expansion order of displacements can be freely chosen if this condition is met and at least it cubic/quartic.
In fact theories ZZA, ZZA*, HWZZ, HWZZ_RDF, ZZA*_43, HSDT_34, HWZZM, ZZA-X1 to ZZA-X4 and ZZA-X1* to ZZA-X4* based on totally different forms of representation but satisfying the conditions mentioned above show indistinguishable results from each other and always prove to be the most accurate and efficient. The most efficient of all are ZZA*, ZZA*_43, HSDT_34, ZZA-X1 to ZZA-X4 and ZZA-X1* to ZZA-X4* that omit the explicit presence of zig-zag functions, therefore they constitute a convenient option to much expensive 3-D finite element methods and discrete-layer models.
A partial fulfillment of above mentioned constraints implies instead that the accuracy decreases and becomes strongly depending on assumptions made. Lower-order theories HRZZ, HRZZ4, MHWZZA, MHWZZA4 and in particular ones that incorporate Murakami’s zig-zag function MHR and MHR4 belong to this category. However in some cases they provide quite accurate results, but in general are rather inaccurate so, it is not possible to deduce any general rule about their usability. The only rule that can be drawn is that the higher-order zigzag theories of this paper with a redefinition of the coefficients obtained through the enforcement of the complete set of physical constraints are always accurate and efficient.