Layerwise mixed element with sublaminates approximation and 3D zig‐zag field, for analysis of local effects in laminated and sandwich composites

Abstract

SUMMARYThe analysis of damaged sandwich composites and thick laminates remains a still challenging task, owing to their intricate distributions of displacements and stresses across the thickness. The arduous task is capturing these distributions with an affordable computational effort. The use of sublaminate zig-zag models appears promising since they allow to group several physical layers into a sublaminate without violating the continuity of interlaminar stresses. To contribute to research in this field, a mixed, sublaminate element is developed in which displacements and stresses are approximated by two independent zig-zag models. Like for any zig-zag model, the interfacial contact conditions are fulfilled a priori suitably choosing the continuity functions and constants. In addition to the contact conditions on interlaminar shears, here those on the transverse normal stress and stress gradient are also fulfilled, as required by the elasticity theory. This uncustomary feature implies the presence of unwise terms which need C 2 continuous shape functions. To overcome this drawback, C 1 and C 2 terms are eliminated from the displacement field, as allowed by the mixed approach, then the interlaminar stresses obtained by the C 0 -made displacement model are cast into a separate zig-zag representation which restores their continuity at the interfaces. As nodal d.o.f. the three displacements and the three interlaminar stresses at the upper and lower faces are assumed, in order to fulfil the contact conditions assembling the sublaminates. Standard, C 0 , serendipity interpolation polynomials are used to represent both internal displacements and stress fields. As a consequence of this choice, the intra-element stress equilibria are fulfilled in an approximate integral form. This makes easier the development of the element, but does not compromise its accuracy and convergence rate, as shown by former applications. A postprocessing procedure is developed which improves the estimation of interlaminar stresses and helps to reduce the number of sublaminates. The result is that the analysis can be carried out using a single computational layer across the thickness and a reasonably fine in-plane discretization, even with distinctly different material properties of layers and in presence of damage. The overall computational time is reduced to an half of that required by a conventional sublaminate model, like present one deprived of zig-zag terms and postprocessing procedure.

Layerwise zig‐zag model with selective refinement across the thickness

“…As shown in Reference [31], these approximate continuity functions result in negligible slope errors at the interfaces for the displacements and in small jumps of the interlaminar stresses. The approximate expressions of (k) u1 to (k) 9 are then used for determining approximate expressions for…”

“…In these models, here referred as zig-zag sublaminate (SZZ) models, the required accuracy can be obtained either by subdividing into sublaminates or by increasing the order of representation of the displacements within the sublaminates. The senior author contributed towards developing SZZ models along with the transverse normal stresses and stress gradient continuous at the interfaces (see References [28][29][30][31]), as prescribed by the elasticity theory, disregarded in the previous SZZ models.…”

“…In general, SZZ models are able to obtain a degree of accuracy that is comparable to DL models, but with a lower effort. However, as shown in Reference [31], cases exist for which this advantage is only apparent, a long post-processing time or even the subdivision of physical layers into one or more computational layers being required. In addition, examples exist for which the stresses computed by post-processing methods can be still not accurate enough in comparison with three-dimensional elasticity solutions, as shown by Cho et al [32].…”

Multilayered Plate Model with “Adaptive” Representation of Displacements and Temperature Across the Thickness and Fixed D.O.F.

“…As SBB fails to be accurate using a reasonable number of computational layers when the properties abruptly change [36], Icardi and Ferrero developed a ZZ model with a hierarchical representation of the displacements across the thickness, either in mixed [37] and displacement-based [38] version, that does not need postprocessing or staking of computational layers. These models, which are constructed incorporating hierarchic high-order terms into the ZZ model [39], have a fixed number of functional d.o.f., since the coefficients of these terms are determined enforcing equilibrium conditions at intermediate points across the thickness.…”

First-Order Displacement-Based Zigzag Theories for Composite Laminates and Sandwich Structures: A Review

“…Extensions of the kinematics of the Di Sciuva's First-order zigzag theory (linear zigzag model) include, among other: (i) extension to the linear [41] and nonlinear [48] multilayered shell theory; (ii) general lamination lay-up [42,47,49]; (iii) satisfaction of the shear stress-free boundary conditions on the top and bottom surfaces [44,46,50]; (iv) polynomial expansion of the global part to any degree [47,49]; (v) extension to the dynamics, buckling [16,47,49,48]; (vi) thermal effects [68], [69]; (vii) sublaminates approach [68,69,70,71,72,73,74]; (viii) inclusion of von Kármán geometrically nonlinear effects [16,48,49]; (ix) extension to the nonlinear theory of laminated composites with damaged interfaces [49,68,69,74,75,76]; (x) inclusion of the transverse normal strain and stress [74,77,78]; (xi) visco-elastic effects [79]; (xii) active control of beams, plates and shells [80].…”