Reduced representations of assumed fields for Hu–Washizu solid-shell element

Abstract

Mixed eight-node (hexahedron) solid-shell elements based on the standard or partial version of the three-field Hu–Washizu (HW) functionals are developed for Green strain. Three reduced representations of the assumed stress/strain fields are selected. They improve effectiveness, yet retaining good accuracy and convergence properties. At the outset, the standard HW functional and the assumed stress/strain representations of the 3D solid element B8-15P (Weissman in Int J Numer Methods Eng 39:2337–2361, 1996) are used to derive a solid-shell element with 51 parameters. To eliminate locking, the ANS method is applied to the thickness strain (Betsch and Stein in Commun Numer Methods Eng 11:899–909, 1995) and to the transverse shear strain (Dvorkin and Bathe in Eng Comput 1:77–88, 1984). It is a correct element which, however, yields too large displacements for coarse meshes and trapezoidal through-thickness shapes. To improve the above formulation, the $$\zeta $$ ζ -independent reduced representations of the assumed stress/ strain fields are selected and the transformations to Cartesian components are modified. The thickness strain is enhanced by the EAS method. The element with 35 parameters is derived from the standard/enhanced HW functional, but, to further reduce the assumed fields, partial/enhanced HW functionals are constructed from the 3D potential energy by applying the Lagrange multiplier method only to selected strain components. In the element with 27 parameters, this is applied to the constant in-plane strain and to the transverse shear strain while in the element with 19 parameters, to the constant in-plane strain only.Two other modifications are implemented to enhance the behavior of these elements: (A) the skew coordinates are used in the reduced representations of the in-plane stress/strain (Wisniewski and Turska in Int J Numer Methods Eng 90:506–536, 2012), and (B) the Residual Bending Flexibility correction of the transverse shear stiffness (MacNeal in Comput Struct 8(2):175–183, 1978) is adapted. Finally, the performance of the proposed solid-shell HW elements is demonstrated on several linear and non-linear examples for the linear elastic material and the hyper-elastic material. The proposed elements are compared to each other and to the best existing elements of this class.