Finite element methods for the multi-scale modeling of softening hinge lines in plates at failure

This paper presents a multi-scale framework for the modeling of softening hinge lines in inelastic plates and its finite element implementation. The proposed framework considers the Reissner–Mindlin problem of thick plates as the large-scale problem with the hinge introduced locally in small neighborhoods modeling the small scales of the material response. The hinges are defined as strong discontinuities of the generalized displacements describing the plate deformations. The consideration of the large-scale limit of vanishing small scales leads to a large-scale model of these localized failures of plates, involving the solution of a problem in terms of those large-scale generalized displacements with the standard regularity conditions. Enhanced finite elements define the natural framework for the numerical implementation of these considerations. The new finite elements are enhanced with the singular strain fields associated with the discontinuities or hinge lines, which given their local character are statically condensed at the element level. We present a general strategy for the definition of these local strain fields so the kinematics of the hinge lines are captured correctly, allowing in particular their fully softened state with no stresses and hence avoiding the so-called stress locking. We present several particular finite elements formulated in this framework, including triangular and quadrilateral finite elements. Several numerical simulations are presented illustrating their performance.