Analysis of the second Piola‐Kirchhoff stress in nonlinear thick and thin structures using exact geometry SaS solid‐shell elements

Abstract

This paper presents the finite rotation exact geometry four-node solid-shell element using the sampling surfaces (SaS) method. The SaS formulation is based on choosing inside the shell N SaS parallel to the middle surface to introduce the displacements of these surfaces as basic shell unknowns. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N-1 in the through-thickness distributions of displacements, strains and stresses leads to a robust higher-order shell formulation. The SaS are located at only Chebyshev polynomial nodes that make possible to minimize uniformly the error due to Lagrange interpolation. The proposed hybrid-mixed four-node solid-shell element is based on the Hu-Washizu variational principle and is completely free of shear and membrane locking. The tangent stiffness matrix is evaluated through efficient 3D analytical integration and its explicit form is given. As a result, the proposed exact geometry solid-shell element exhibits a superior performance in the case of coarse meshes and allows the use of load increments, which are much larger than possible with existing displacement-based solid-shell elements.KEYWORDS exact geometry solid-shell element, finite rotation, hybrid-mixed method, sampling surfaces method, second Piola-Kirchhoff stress tensorUsing (4), (16), (23), and (27) and considering identities ∑ J M J ( 3 ) = 0, ∑ J J 3 M J ( 3 ) = 1, (30) which in turn follow from trivial identities ∑ J L J ( 3 ) = 1, ∑ J J

Nonlinear analysis of laminated composite shells with piezoelectric patches under displacement‐dependent pressure loads

“…The proposed hybrid-mixed solid-shell element is free from shear and membrane locking and passes the three-dimensional bending patch test for homogeneous, laminated and FGM plates [43,54,55], the inf-sup test for structures discretized by distorted meshes [38], the zero energy mode test for elastic and piezoelectric structures [56,57]. It is insensitive to the number of load increments [31,58,59] and can be recommended for three-dimensional stress analysis of laminated composite structures with piezoelectric patches subjected to displacement-dependent loads in a range of arbitrarily large displacements and rotations. The developed laminated piezoelectric solid-shell element demonstrates superior performance in the case of coarse meshes and allows the use of a single load step in most numerical examples.…”

An efficient isostatic mixed shell element for coarse mesh solution

“…Finally, by recalling Equations (12) and (15), the mixed strain energy is conveniently evaluated as line integral evaluated along Γ.…”

“…Mixed elements are also developed in the framework of the virtual element method by Artioli et al [9][10][11] Furthermore, mixed FE are largely used for the geometrically nonlinear analysis of thin-walled structures. [12][13][14][15][16] Recent formulations of mixed FE can be found in References 17,18. Moreover, mixed FE are diffusely employed for the analysis of structures undergoing physical nonlinearities.…”

“…Then, Zucco et al 8 formulated a mixed FE for the linear static and buckling analyses of variable‐angle tow composite plates. Mixed elements are also developed in the framework of the virtual element method by Artioli et al 9‐11 Furthermore, mixed FE are largely used for the geometrically nonlinear analysis of thin‐walled structures 12‐16 . Recent formulations of mixed FE can be found in References 17,18.…”