Elsevier

Composite Structures

Volume 244, 15 July 2020, 112233
Composite Structures

Considerations about the choice of layerwise and through-thickness global functions of 3-D physically-based zig-zag theories

https://doi.org/10.1016/j.compstruct.2020.112233Get rights and content

Abstract

A generalization of physically-based fixed degrees of freedom 3-D zig-zag theories is developed, which allows for any arbitrary choice of layerwise and representation functions. Thereby users can choose arbitrarily the representation case-by-case. This paper aims to prove that the choice of global and layerwise functions is immaterial whenever coefficients are recalculated exactly (via symbolic calculus) by the enforcement of interfacial stress continuity, boundary conditions and equilibrium in point form, as prescribed by the elasticity theory. Vice versa, accuracy of theories partially fulfilling constraints will prove to largely depend on the assumptions made and to be inadequate when strong layerwise effects rise, under distributed/localized step loading and boundary conditions other than simply supported edges (tests carried out in closed-form). The present and previous authors’ theories are tested with the aim to understand in which cases an adequate level of accuracy is still achieved by lower-order theories that are derived as particularizations.

Introduction

As well known, laminated fibre-reinforced and sandwich composites offer excellent specific strength and stiffness, fatigue and energy absorption properties, better resistance to corrosion than metals and greater design flexibility. To ward off any possible catastrophic failure or intolerable loss of performance of these materials, an accurate prediction of through-thickness displacement, strain and stress fields is crucial [1]. Displacements have to be C° continuous at interfaces and with the suited slope (zig-zag effect) so that out-of-plane stresses are continuous and satisfy local equilibrium equations.

So far many laminated plate and shell theories with varying degrees of accuracy, computational costs and memory storage occupation have been developed, as explained in detail in the book by Reddy [2] and in the papers by Reddy and co-workers [1], [3], Vasilive and Lur’e [4], Lur’e, and Shumova [5], Noor et al. [6], Altenbach [7], Carrera [8], [9], [10], [11], [12], Qatu [13], Qatu et al. [14], Wanji and Zhen [15], Khandan et al. [16] and Kapuria and Nath [17].

Equivalent single-layer [18], [19], [20], [21], discrete-layer [22], [23], zig-zag [11], [24], hierarchical [25], [26] and axiomatic/asymptotic [27], [28] theories can be categorized, which further subdivide into displacement-based and mixed formulations (displacements, strains and stress fields chosen separately).

Equivalents single layer theories are of limited validity, not even being able to predict overall behaviour quantities, but even this limited goal could be disregarded, as shown among many others by Icardi and Sola [29], Icardi and Urraci [30], [31], Kapuria et al. [32], Zhen and Wanji [33], Burlayenko et al. [34] and Jun et al. [35] for cases with a strong transverse anisotropy. A forcibly schematic description like that of this article leads to summarize discrete-layer theories as potentially accurate irrespective of lay-up, property variation across the thickness, loading and boundary conditions, but also problematic because they could overwhelm the computational capacity when structures of industrial interest are analyzed, owing to their too much number of variables. Recent accurate, versatile and efficient theories like Carrera’s unified formulation [8] and refined zig-zag theories [29], [30], [31], [36] are instead suitable for analysis of industrial structures. Theory [8] allows displacements to take arbitrary forms that can be chosen by the user as an input of the analysis and it is able to get existing theories as particularizations. This paper intends to pursue the same purpose with a new generalized zig-zag theory.

Zig-zag theories subdivide into: (i) Di Sciuva’s like physically-based theories (see [37], [38], [39] as examples), or (ii) Murakami’s like kinematic-based mixed theories, (see, e.g. [11], [40], [41]), given the different type of zig.zag functions used. Both strike the right balance between accuracy and cost saving, allowing designers’ demand of theories in a simple already accurate form to be met.

Theories (i) generally have a fixed number of unknowns irrespective of the number of constituent layers. Their layerwise contributions are the product of linear [37] (or nonlinear [42]) zig-zag functions and unknown zig-zag amplitudes, inferred by enforcing the fulfilment of stress continuity conditions at layer interfaces.

Theories (ii) assume zig-zag functions featuring a periodic change of sign of the slope of displacements at interfaces, as occurring for periodic laminations, therefore without taking into account orientation angle, material properties and thickness of constituent layers. Their number of variables (kinematic and stress quantities) depends upon the number of constituent/computational layers. They can be inaccurate [30], [31], [43], [44], but they enable a more easily obtainment of C° formulations of plate theories for the development of efficient finite elements. Mixed multilayered theories accurately predict stresses, with the merit of keeping simpler kinematics as is evident from the papers by Tessler et al. [36], Kim and Cho [45], Barut et al. [46], Iurlaro et al. [47] and Zhen and Wanji [48]. When strong layerwise effects rise, (ii) require many more degrees of freedom and/or a higher expansion order than (i), as it can be seen, for example, comparing the results of [30], [49].

Efficient and accurate theories where zig-zag functions are not explicitly incorporated based on a global-local superposition of displacement fields were proposed by Li and Liu [50] and refined by Zhen and Wanji, e.g. [51], which in the version of Shariyat [52] consider a non-uniform transverse displacement across the thickness.

Theories with a hierarchical set of locally defined polynomials where neither zig-zag contributions are incorporated in the kinematics, nor post-processing steps are required (see, [25], [26], Catapano et al. [53] and de Miguel et al. [54]) have been developed, for favouring numerical efficiency. Anyway, they require a quite large number of variables to mitigate the effects of the omitted enforcement of interfacial stress contact conditions, so to avoid small interfacial stress jumps.

Physically-based 3D zig-zag theories like [29], [30], [31] also allow a variable representation arbitrarily chosen by the user, without increasing their only 5 d.o.f. , while Strain Energy Updating Technique [55] can be used to obtain a C° formulation. Because their coefficients are redefined across the thickness, they can be counted among those with variable-kinematics (see, e.g. Vescovini and Dozio [56]).

So far, power series expansion, hierarchic polynomials, Taylor’s series, trigonometric and exponential functions, a combination of both and radial basis functions (see [57], [58], [59], [60], [61], [62], [63], [64], [65], [66]) have been used to represent variables across the thickness and a high sensitivity to the different assumptions made was shown.

Going into more detail regarding physically-based 3D zig-zag theories, in Icardi and Sola [29] a displacement-based laminated plate theory with piecewise cubic in-plane displacements, a fourth-order transverse displacement and only 5 d.o.f. (mid-plane displacements and rotations), referred as ZZA, was developed, whose coefficients are redefined across the thickness by imposing the fulfilment of elasticity theory constraints (interfacial stress compatibility conditions, stress boundary conditions and local equilibrium equations at arbitrary selected points across the thickness).

Variants of ZZA were developed in [30], assuming different layerwise functions, in order to evaluate how the different choices affect accuracy. For the same purpose, different forms of representation of global functions and various layerwise functions (as well as without them) were assumed in [67], [68], [69], [70], [71]. In an attempt of lowering the computational cost, a mixed variant of the already very efficient ZZA theory was developed in [31] within the framework of Hu-Washizu variational theorem, so to retain separately only the essential contributions of displacement, strain and stress fields. Theories [29], [30], [67], [68], [69], [70], [71], all developed by using symbolic calculus to exactly fulfil the elasticity theory constraints, were proven to be efficient and suitable to analyse challenging elastostatic and dynamic cases, including pumping modes which are usually solved through FEM. In effects, their accuracy was proven to be similar to that of discrete layer and layerwise models with a very high through-thickness expansion order of variables and so a larger computational burden.

A not negligible advantage of these theories is that there is no need to approximate loading with a series expansion, because its mathematical expression is used to exactly compute the work of external forces via symbolic calculus. Another advantage is that also in-plane displacement and stress continuity can be enforced, with the aim to analyse also structures with step variable properties along in-plane directions and not only across the thickness. Moreover, stresses of theories [29], [30], [67], [68], [69], [70], [71] are obtained from constitutive relations and there is no need to post-processing.

Summing up the features of theories [29], [30], [67], [68], [69], [70], [71], similarly to [50], [51], [52], it is not necessary to include specific zigzag functions to respect the interfacial out-of-plane stress compatibility, because a number of coefficients can be redefined for each layer for this purpose. However, unlike [50], [51], [52] and all conventional plate theories, in [29], [30], [67], [68], [69], [70], [71] all coefficients can be redefined across the thickness so that stress-boundary conditions are met at the outer layers and equilibrium can be satisfied in a strong point form within inner layers. Similarly to hierarchical and asymptotic theories, the representation can be enriched in order to achieve a better accuracy, but the expansion order and the number of variables do not grow once the piecewise cubic-quartic representation adopted is fragmented into computational layers and/or changing the type of representation from layer to layer and differently for each displacement. Unlike the theories that today are very popular for their accuracy and versatility (e.g. [8] and subsequent developments), [29], [30], [67], [68], [69], [70], [71] are formulated by respecting physical constraints exactly (from which the appellation of physically-based theories follows) so as to limit the computational burden, rather respecting them in a limit sense increasing the expansion order and / or the number of variables.

A generalization of previous 3-D zig-zag theories is developed in this paper, whose formulation allows user to arbitrarily choose layerwise and representation functions, also differently for each displacement and for each layer. Nevertheless any number of d.o.f. could be assumed (not necessarily just mid-plane displacements and rotations), it is here limited to five to carry out comparisons with ZZA and other previous theories under the same conditions. From the present generalized theory an approximate non-plate theory (number of d.o.f. not fixed that depends from the expansion order imposed) AT-3D is derived, which can be used as reference solution when exact one is unavailable. Given its generality, also hierarchical theories could be obtained by the present theory, which however won’t be considered in this research.

Six theories different from those of [29], [30], [31], [67], [68], [69], [70], [71] are particularized in (3.1) assuming sinusoidal, exponential, power series or a combination of them (which can be different for each displacement and from layer to layer) to describe the variation of displacements across the thickness. Fourteen additional lower order theories are developed in (3.2)-(3.8) to test along with the former six if a plate theory with fixed d.o.f. can enable a kinematic-variable representation across the thickness requiring a lower computational burden than existing theories with same features, and to understand in which cases it is still allowed to reach an adequate level of accuracy. The results will show that a piecewise cubic and a fourth-order polynomial for in-plane and transverse displacements respectively achieve this goal. In one theory, equilibrium equations are enforced in weak form, instead than in a point form as for all others theories mentioned, in order to verify the degree of accuracy achievable. Results will confirm on a broader basis what preliminarily demonstrated in [29], [30], [31], [67], [68], [69], [70], [71], that the choice of zig-zag and representation functions is immaterial if coefficients are redefined across the thickness and calculated by imposing full set of physical constraints.

Table 1 provides a quick reference pattern of all cases considered in the numerical applications and lists details on length-to-thickness ratio, lay-up, loading and boundary conditions. Mechanical properties of materials are in Table 2, Table 3 reports trial functions, expansion order and equilibrium points position across the thickness for each case, Table 4 contains normalizations, Table 5 records a brief description of theories, while the processing time is listed in Table 6.

Section snippets

Theoretical framework

Notations and basic assumptions, that are common to all the theories, are discussed in the following section. Features of the new theories introduced in this paper are examined in details. Readers are referred to the literature quoted for previously developed theories.

New theories of this paper

New theories, which constitutes the main theoretical contribution of this paper, are proposed as a generalization of ZZA and all previously developed theories by the authors [29], [30], [31], [42], [55], [68], [69], [70], [71], in order to prove the objective set in the introductory section. Initially, the displacement field is thought in the following form:uαj=i=0nα=3jCαiα,βFαi(ς)uςj=i=0nς=4jCςiα,βGi(ς)that does not contain zig-zag functions because coefficients are redefined across the

Numerical assessments and discussion

Accuracy of previous theories is assessed considering different benchmarks, loading and boundary conditions, some of which exhibit strong layerwise effects. The aim is to evaluate whether accuracy of results is independent on the choice of global and layerwise functions. Conversely, it will be shown that accuracy of theories only partially satisfy physical constraints, is highly dependent on the choices made.

Low length-to-thickness ratios with strong layerwise effects in most cases, but also

Concluding remarks

This study illustrated numerically that the degree of accuracy of higher-order physically-based zig-zag theories (in displacement-based and mixed form) is independent on the choice of global and layerwise functions, once all stress continuity, boundary and equilibrium conditions are enforced at the same time.

On the contrary, if coefficients are not redefined or physical constraints are partially satisfied, results are strongly dependent by choices made, as demonstrated by assessments of the

Data availability

All data generated or analyse during the study are included in the manuscript.

CRediT authorship contribution statement

Ugo Icardi: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization, Supervision, Project administration. Andrea Urraci: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - original draft, Writing - review & editing, Visualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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