Stress–displacement stabilized finite element analysis of thin structures using solid-shell elements, Part I: On the need of interpolating the stresses

This work studies the solid-shell finite element approach to approximate thin structures using a stabilized mixed displacement–stress formulation based on the Variational Multiscale framework. The work is divided in two parts. In Part I, the numerical locking effects inherent to the solid-shell approach are characterized using a variety of benchmark problems in the infinitesimal strain approximation. In Part II, the results are extended to formulate the mixed approach in finite strain hyperelastic problems. In the present work, the stabilized mixed displacement–stress formulation is proven to be adequate to deal with all kinds of numerical locking. Additionally, a more comprehensive analysis of each individual type of numerical locking, how it is triggered and how it is overcome is also provided. The numerical locking usually occurs when parasitic strains overtake the system of equations through specific components of the stress tensor. To properly analyze them, the direction of each component of the stress tensor has been defined with respect to the shell directors. Therefore, it becomes necessary to formulate the solid-shell problem in curvilinear coordinates, allowing to give mechanical meaning to the stress components (shear, twisting, membrane and thickness stresses) independently of the global frame of reference. The conditions in which numerical locking is triggered as well as the stress tensor component responsible of correcting the locking behavior have been identified individually by characterizing the numerical response of a set of different benchmark problems.