Geometrically exact beam theory with embedded strong discontinuities for the modeling of failure in structures. Part I: Formulation and finite element implementation

We introduce a three-dimensional geometrically nonlinear Reissner beam theory with embedded strong discontinuities for the modeling of failure in structures and discuss its finite element implementation. Existing embedded beam theories are geometrically linear or two-dimensional, motivating the need for the present work. We propose a geometrically nonlinear beam theory that accounts for cracks through displacement discontinuities in 3D. To represent the three modes of fracture inside an element, we enrich each displacement field with an incompatible mode parameter in the form of a jump, an additional degree of freedom. We then eliminate these nodeless degrees of freedom through static condensation and evaluate them within the framework of inelasticity by utilizing an operator split solution procedure. Seeing that the coupled strain-softening problem is non-convex, we present an alternating minimization (staggered) algorithm, thus retaining a positive definite stiffness matrix. Finally, the four-parameter representation by quaternions describes a three-dimensional finite rotation. We demonstrate a very satisfying and robust performance of these new finite elements in several numerical examples, including the fracture of random lattice structures with application to fibrous materials. We show that accounting for geometrical nonlinearity in the beam formulation is necessary for direct numerical simulations of fiber networks regardless of the density.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Automated summary

This study introduces a three-dimensional geometrically nonlinear Reissner beam theory for modeling failure in structures and discusses its finite element implementation. The theory considers cracks through displacement discontinuities and incorporates three modes of fracture inside an element. By eliminating nodeless degrees of freedom and using an alternating minimization algorithm, a positive definite stiffness matrix is obtained. The performance of these new finite elements in numerical examples, including the fracture of random lattice structures, is demonstrated. Geometrical nonlinearity in the beam formulation is shown to be necessary for direct numerical simulations of fiber networks.