About linear and quadratic “Solid-Shell” elements at large deformations

An efficient analysis for many shell problems is obtained only by using shell elements based on classical shell theory instead of fully 3D solid elements. However, for many investigations of e.g. shell stretching and shell connections, strains and stresses in thickness direction as well as general 3D material laws would be required but cannot be provided by `classicalʹ shell formulations. Therefore in Hauptmann [Int J Numer Meth Engng 42 (1998) 49] among others as Schoop [Ingenieur-Archiv 56 (1986) 427] Parisch [Int J Numer Meth Engng 38 (1995) 1855] Miehe [Comput Meth Appl Mech Engng 155 (1998) 193] a so-called “Solid-Shell” formulation which is described briefly in the following section, was proposed. The “Solid-Shell” formulation is based on displacement degrees of freedom belonging to the upper and lower shell surfaces and thus the use of rotational degrees of freedom can be avoided. Thus also a simple combination of these elements with solid elements is possible and in particular no problems concerning large rigid body motions and rotational updates appear. The typical locking phenomena mostly obtained for shell formulations as well as the additional locking effects belonging to 3D effects of the presented “Solid-Shell” elements have to be discussed in detail. A particular focus is on numerical instabilities occurring with the many approaches to avoid locking for linear and quadratic shape functions at large deformations.