DETC03/VIB-48324 BIAXIAL VIBRATIONS OF AN ELASTO-PLASTIC BEAM WITH A PRESCRIBED RIGID-BODY ROTATION

Abstract

ABSTRACT In the present work, the development of plastic strains in a flexural beam is studied. The beam is modeled as a BernoulliEuler beam, where large rigid-body rotations and biaxial bending in the small strain regime are studied. The deformation is split into the spatial deformation of a hinged-hinged beam and the movement of the second support. Neglecting axial displacements of the beam, this support moves on a sphere. In the present paper, the latter motion is considered as prescribed. The beam thus is assumed to possess only flexural degrees-offreedom. Such a problem is frequently to be encountered in machine dynamics or robotics. We assume the stiffness of the beam to be considerably lowered due to catastrophic environmental influences, such that the deformations relative to the rigid-body motion, albeit small, reach the plastic regime. The equations of motion are derived by Hamilton's principle. The potential energy follows from the internal energy due to the elastic part of the deformation and the potential due to gravity. Plastic strains are treated according to the theory of eigenstrains, which act as sources of self-stress upon the linear elastic beam. The biaxial deflections are discretized in space by means of Legendre polynomials. The plastic strains are discretized over length, height and width of the beam by small plastic cells. The plastic strains are computed in every time-step by a suitable iterative procedure. An implicit midpoint rule, which preserves the total energy of the system, is used for integration of the equations of motion. Linear elastic/ perfectly plastic behavior is exemplarily treated in a numerical study. INTRODUCTION Heavily loaded deforming structures performing rigid body motions may exhibit elastic as well as plastic strains, particularly when the elastic stiffness and the yield stress is considerably lowered by environmental influences such as a high temperature. In order to consider the occurrence of plastic strains properly, it is useful to retain and extend the essential assumptions of structural dynamics. The present paper has been written under the point of view of the implementability of an elastoplastic constitutive model in a respective dynamic structural beam theory. We restrict ourselves to small deformations superimposed upon a given large three-dimensional rigid-body rotation, extending previous papers of our group on the plane motion of elasto-plastic beams, see [2] and [3] for reference. In these latter formulations, a linear elastic background structure has been introduced, where plasticity acts as eigenstrains analogous to a temperature loading. This point of view is motivated by the fact that plasticity is connected to microstructural changes in the virgin (elastic) material, and it opens a consistent way of introducing plasticity in structural theories. In the present contribution we present an extension of the eigenstrain method with respect to the spatial development of plastic zones in a beam. As a benchmark problem, a large three-dimensiona