Free and Forced Vibration of Laminated and Sandwich Plates by Zig-Zag Theories Differently Accounting for Transverse Shear and Normal Deformability

A number of mixed and displacement-based zig-zag theories are derived from the zig-zag adaptive theory (ZZA). As a consequence of their different assumptions on displacement, strain, and stress fields, and layerwise functions, these theories account for the transverse shear and normal deformability in different ways, but their unknowns are independent of the number of layers. Some have features that are reminiscent of ones that have been published in the literature for the sake of comparison. Benchmarks with different length-to-thickness ratios, lay-ups, material properties, and simply supported or clamped edges are studied with the intended aim of contributing toward better understanding the influence of transverse anisotropy on free vibration and the response of blastloaded, multilayered, and sandwich plates, as well as enhancing the existing database. The results show that only theories whose layerwise contributions identically satisfy interfacial stress constrains and whose displacement fields are redefined for each layer provide results that are in agreement with elasticity solutions and three-dimensional (3D) finite element analysis (FEA) (mixed solid elements with displacements and out-of-plane stresses as nodal degrees of freedom (d.o.f.)) with a low expansion order of polynomials in the in-plane and out-of-plane directions. The choice of their layerwise functions is shown to be immaterial, while theories with fixed kinematics are shown to be strongly case-sensitive and often inadequate (even for slender components)

Zig-zag theories differently accounting for layerwise effects of multilayered composites

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-to-thickness 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

Approximate 3-D model for analysis of laminated plates with arbitrary lay-ups, loading and boundary conditions

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 physically-based 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

New 3-D zig zag theories: elastostatic assessment of strategies differently accounting for layerwise effects of laminated and sandwich composites

New physically based 3-D, fixed d.o.f., theories which enable to analyze cases with general loading and boundary conditions efficiently are proposed. Here the aim is to study the effects of an arbitrary choice of through-thickness representation of kinematic/stress variables and of zig-zag functions. The same trial functions and expansion order of analytical solutions are assumed to assess theories under the same conditions. Comparisons are carried out with exact and/or 3-D FEA solutions. Their computational burden is still comparable to that of classical plate models. The results show that whenever coefficients of representation are recalculated across the thickness by enforcing the fulfillment of all constraints prescribed by the elasticity theory, the choice of the representation form and of zig-zag functions is immaterial. In this way, a high order of generalization is allowed because the representation of one single displacement can be freely varied across the thickness and be completely different from that of other displacements. Moreover, zig-zag functions can be arbitrarily chosen or even omitted without any accuracy loss. Instead, accuracy is shown to be strongly dependent upon the assumptions made for theories only partially satisfying constraints

Free Vibration of flexible soft-core sandwiches according to layerwise theories differently accounting for the transverse normal deformability

Abstract This study aims to generalize a previously developed accurate and inexpensive 3-D zig-zag theory up to an arbitrary representation form and to determine which simplifications are yet accurate in determining transverse shear and normal stress/deformation effects on vibrations of soft-core sandwiches with not moving middle/neutral plane (pumping). Natural frequencies are calculated using displacements assumed differently across the thickness, having fixed d.o.f., not yet explored forms of representation and zig-zag functions differently accounting for the transverse normal deformability and that partially or fully fulfill physical constraints. Applications are presented for sandwich plates and beams with length-to-thickness ratios and material properties of faces and core varying within an industrial range, for which layerwise effects are very important and so suited to the evaluation of theories. Analytical solutions are found using the same trial functions and expansion order for all theories, so to evaluate their accuracy under the same conditions. The choice of the representation form and of zig-zag functions is shown immaterial if displacement field coefficients are recomputed across the thickness by enforcing the fulfillment of all physical constraints (using symbolic calculus). Furthermore, it is shown that assigning a specific role to each coefficient is immaterial, as well as exchanging the order of representation of in-plane and transverse displacement components and even that zig-zag functions could be omitted. This no longer occurs for lower-order theories with only a partial fulfillment of constraints. Pumping motions are highlighted as the first modes, which require the theories much accurately accounting for transverse normal deformability

Novel HW mixed zig-zag theory accounting for transverse normal deformability and lower-order counterparts assessed by old and new elastostatic benchmarks

Mixed zig-zag plate theories are derived from a recently developed 3-D five d.o.f. zig-zag “adaptive” theory ZZA (a priori fulfillment of interfacial stress constraints) under steadily growing limiting assumptions on displacement, strain and stress fields. The intended aim is trying to save computational costs simultaneously preserving accuracy. Lower-order theories assume a uniform or a polynomial transverse displacement, out-of-plane stresses are derived from local equilibrium equations or retaken from higher-order theories. The focus of this study is twofold: (i) to assess whether and for which cases a higher-order through-thickness zig-zag transverse displacement representation is essential, or vice versa a simpler kinematics can be assumed; (ii) to compare accuracy of theories based on Murakami’s and Di Sciuva’s zig-zag functions with the same expansion order across the thickness. A number of challenging benchmarks are retaken from literature and new benchmarks with a strong variation of material properties (damaged layers), distributed or localized step loading and some different boundary conditions are considered to assess accuracy of theories and of FEA 3-D (constituting the reference solution in lack of exact results). For these benchmarks, closed-form solutions are obtained assuming the same expansion order across the thickness and the same in-plane trial functions for all theories. The numerical illustrations show that lower order and Murakami’s based theories fail for cases having the strongest layerwise and transverse anisotropy effects, or a marked through-thickness asymmetry. The Hu–Washizu highest-order theory HWZZ is shown to be always the only as accurate as ZZA, despite it reduces the computational effort

Impact Damage Analysis with Stress Continuity Constraints Fulfilment at Damaged-Undamaged Regions and at Layer Interfaces

Low-velocity impacts have a relevant importance for safety of laminated and sandwich composite structures, because they are highly susceptible to damage. In this paper, a 3D cost effective zigzag model is developed in order to efficiently simulate such impacts. It a priori fulfills the continuity of out-of-plane stresses at layer interfaces, the continuity of stresses under in-plane variation of properties across undamaged and damaged regions and it is suitable for general boundary conditions. Its main advantage is its capability to accurately predict stresses from constitutive equations at a low cost, along with being refined across the thickness keeping fixed the number of unknowns. A modified Hertzian contact law that forces the target to conform to the shape of the impactor and the Newmark's implicit time integration scheme are used for computing the contact force time history. The progressive damage analysis is carried out using stress-based criteria. Non-classical feature, a continuum damage mesomechanic model is employed that accounts for the effects of local failures in homogenized form by modifying the strain energy expression. Fiber, matrix and delamination failures are predicted using stress-based criteria, and then the modified strain energy expression is employed for computing stresses. Such modeling options enable to account for the residual properties as they are in the reality, as shown by the comparison with the damage experimentally detected. As shown by the comparison with experiments, a closed-form solution by the Galerkin's method, obtained as a series expansion of trial displacement functions, accurately simulates the contact force and the damage progressively accumulated. The results show the importance of in-plane stress continuity for obtaining accurate prediction

Impact Damage Analysis with Stress Continuity Constraints Fulfilment at Damaged-Undamaged Regions and at Layer Interfaces

Abstract Low-velocity impacts have a relevant importance for safety of laminated and sandwich composite structures, because they are highly susceptible to damage. In this paper, a 3D cost effective zig-zag model is developed in order to efficiently simulate such impacts. It a priori fulfills the continuity of out-of-plane stresses at layer interfaces, the continuity of stresses under in-plane variation of properties across undamaged and damaged regions and it is suitable for general boundary conditions. Its main advantage is its capability to accurately predict stresses from constitutive equations at a low cost, along with being refined across the thickness keeping fixed the number of unknowns. A modified Hertzian contact law that forces the target to conform to the shape of the impactor and the Newmark’s implicit time integration scheme are used for computing the contact force time history. The progressive damage analysis is carried out using stress-based criteria. Non-classical feature, a continuum damage mesomechanic model is employed that accounts for the effects of local failures in homogenized form by modifying the strain energy expression. Fiber, matrix and delamination failures are predicted using stress-based criteria, and then the modified strain energy expression is employed for computing stresses. Such modeling options enable to account for the residual properties as they are in the reality, as shown by the comparison with the damage experimentally detected. As shown by the comparison with experiments, a closed-form solution by the Galerkin’s method, obtained as a series expansion of trial displacement functions, accurately simulates the contact force and the damage progressively accumulated. The results show the importance of in-plane stress continuity for obtaining accurate predictions