The Absolute Coordinate Formulation with Elasto-Plastic Deformations

Abstract

The present work contributes to the field of multibody systems with respect to the absolute coordinate formulation with a reduced expression of the strain energy and a non-linear constitutive model. Standard methods for multibody systems lead to highly non-linear terms either in the mass matrix or in the stiffness matrix and the most expensive part in the solution of the equations of motion is the assembling of these matrices, the computation of the Jacobian of the non-linear system and the solution of a linear system with the system matrices. In the present work, a consistent simplification of the equations of motion with respect to small deformations but large rigid-body motions is performed. The absolute coordinate formulation is used, therefore the total displacements are the unknowns. This formulation leads to a constant mass matrix while the non-linear stiffness matrix is composed of the constant small strain stiffness matrix rotated by the underlying rigid body rotation. Plastic strains are introduced by an additive split of the strain into an elastic and a plastic part, a yield condition and an associative flow rule. The decomposition of strain has to be performed carefully in order to obey the principle of objectivity for plasticity under large rigid body rotations. As an example, a two-dimensional plate which is hinged at one side and driven by a harmonic force at the opposite side is considered. Plastic deformation is assumed to occur due to extreme environmental influences or due to failure of some attached parts like a defect bearing.