Variational formulation of a three-dimensional surface-related solid-shell finite element

The solution of structural analysis problems, especially of shell structures, demands an efficient numerical solution strategy. Since unilateral contact problems are investigated, the shell model is formulated with respect to one of the outer surfaces, i.e., the shell formulation is surface-related. In particular, the investigation of textile reinforced strengthening layers (Brameshuber (ed.) in State-of-the-Art Report of RILEM Technical Commitee 201—TRC, 2006) will be carried out by this approach. Since shells are three-dimensional structures, i.e., bodies, the field equations of continuum mechanics are the starting point. This set of partial differential equations with pertinent boundary conditions has to be solved. An efficient numerical solution of this problem becomes easier, if the problem is reformulated using variational formalism. A corresponding mathematically abstract formulation of the underlying variational principle of the three-dimensional surface-related solid-shell finite element is stated. The discretization of the mathematically abstract principle is, among others, the source of several locking phenomena. The presented shell formulation assumes linear shell kinematics with six displacement parameters, circumventing a rotation formulation. This low-order shell kinematics produces parasitical strains and stresses, leading to poor approximations of the solution or even useless results. Therewith, extensions and/or adjustments of well-known techniques to prevent or at least reduce locking like the assumed natural strain method (Simo and Hughes in J Appl Mech 53:52–54, 1986) and the enhanced assumed strain method (Simo and Rifai in Int J Numer Methods Eng 29:1595–1638, 1990) have to be carried out. Using these adapted methods, a reliable and efficient solid-shell element with tremendously reduced locking properties is obtained. This concept comprises the utilization of unmodified three-dimensional constitutive relations by a minimal number of kinematical parameters. Finally, two nonlinear examples illustrate the reliability and the efficiency of the new solid-shell element.