Nonlinear displacement-based and hybrid-mixed quadrilaterals for three-dimensional stress analysis through sampling surfaces formulation

The finite deformation displacement-based and hybrid-mixed four-node quadrilateral elements using the sampling surfaces (SaS) technique are developed. The SaS formulation is based on choosing inside the plate N not equally spaced SaS parallel to the middle surface to introduce the displacements of these surfaces as basic plate unknowns. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the thickness direction permits the presentation of the plate formulation in a very compact form. The SaS are located at only Chebyshev polynomial nodes that allows one to minimize uniformly the error due to the Lagrange interpolation. To circumvent shear locking and have no spurious zero energy modes, the assumed transverse shear strain field is employed. Both plate quadrilaterals pass patch tests and exhibit superior performance in the case of coarse distorted mesh configurations. However, the hybrid-mixed element allows the use of load increments, which are much larger than possible with existing displacement-based elements. The developed finite elements can be useful for the 3D stress analysis of thin and thick plates, since the SaS solutions asymptotically approach the solutions of elasticity as N→∞.