An analysis of strain localization and wave propagation in plastic models of beams at failure Original Research Article

This paper presents an analysis of localized failures in Timoshenko beams/rods. Standard elastoplastic models are considered first. A spectral analysis of the linearized problem shows the instability of the viscous problem and the ill-posedness of the inviscid rate-independent problem in the presence of strain softening. The consequences of the ill-posedness are illustrated by obtaining the exact solution of wave propagation in a simply supported beam and observing its lack of physical significance: the failure of the beam occurs with no energy dissipation. Finite element solutions of the problem are presented, exhibiting the characteristic pathological mesh-size dependence in these conditions. The lack of a localized dissipative mechanism in these models is identified as the main cause of these inconsistencies, regardless of the length scale present in the problem (namely, the thickness of the beam). To solve these difficulties, we formulate the so-called localized models, incorporating these localized dissipative mechanisms in the form of strong discontinuities in the generalized displacements, leading naturally to the classical notion of plastic hinges. These models consider a localized softening law between the bending moment and the rotation jump. The model problem of wave propagation in a simply supported beam is then reexamined in this context, arriving again at the exact modal solution. This solution recovers its physical significance (the breaking of the beam is modeled with a non-zero energy dissipation), illustrating the well-posedness of the problem and the regularizing effects introduced by the localized models. The comparison of this exact solution with finite element solutions of the problem allows to evaluate the approximation properties of the numerical techniques, showing now the required objective energy dissipation with respect to the mesh size.