Layerwise zig‐zag model with selective refinement across the thickness

Abstract

SUMMARYIn this paper a multilayered model with a hierarchic representation of displacements is developed. The representation, which is based on Jacobi polynomials, can be different for any computational layer, from point to point across the thickness and for any functional d.o.f., aimed at adapting the model to the local variations of solutions. This refinement is obtained by keeping the number of d.o.f. unchanged. The displacements are described through a zig-zag representation in any computational layer, in order to fulfil the stress contact conditions at the inner interfaces. A displacement-based version of the model with the six displacement components at the upper and lower faces as functional d.o.f. and a mixed version with the six interlayer stress components are developed. The numerical results mainly concern simply supported thick sandwich beams under sinusoidal loading, but results are also presented for thin cantilevered sandwich beams and laminated plates. The model accurately predicts displacements and stress fields using a single computational layer, even when material properties abruptly change. The computational effort for the adaptive representation is advantageous by the viewpoint with respect to a fixed representation or with the use of post-processing methods.

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

“…Any other condition of interest can be satisfied in this way; in particular, the consistency with a state of zero transverse shear stress in presence of nonzero bending strain in the thin regime and with a state of nonzero transverse normal stress with a nonzero bending strain in the thick regime can be achieved contemporaneously [37], nevertheless their requirements seems in contrast. Indeed, the former condition implies that the transverse displacement should have a lower order than the in-plane displacements, while the latter one requires the opposite with a fixed representation.…”

“…Obviously, lower order models can be obtained as particular cases; for example, the classical (CLT) and first order shear deformation (FSDT) plate models can be particularized from the displacement fields assuming a constant transverse displacement and the shear rotations as vanishing or being linear. The present representation of displacements is chosen since the same accuracy of DL models can be achieved with a lower computational effort, as shown in [37,38], because a refinement across the thickness is obtained without increasing the number of primary variables. The high-order "adaptive" terms O 4 that appear in the expressions of the inplane displacements (1), (??…”

“…Because the derivatives of displacements are unpractical by the computational viewpoint, they can be converted into their primitive functions in a way that preserves their contribution to the strain energy, as shown in Ref. [37].…”

“…Using this representation, the interlaminar stresses can be always accurately predicted from constitutive equations since the displacement and stress fields can adapt to the variation of solutions across the thickness. In the mixed version [37], the d.o.f. are the displacement components and the interlaminar stresses at the upper and lower faces of the computational layers, while in the displacement-based version [38], they are the mid-plane displacements and the shear rotations as in ESL and ZZ classical plate models.…”

“…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.…”