Wave propagation in a strain-softening bar: Exact solution

A closed-form solution is given for a one-dimensional bar which undergqgs strain softening,(i.e a gradual decline of stress to zero at increasing strain).it is shown that-strain softening can occur in the interior of a body, and thafThe length of the strain-softening region tends to localize into a point, which agrees with what was previously shown by stability analysis for static situations. The stress in the strain-softening cross section drops to zero' in stantly, regardless of the shape of the strain-softening diagram, and the total energy dissipated in the strain-softening domain of the bar is found to vanish. Despite these unpleasant features, the problem apparently possesses a solution for certain boundary and initial conditionsHowever, the fact that the energy dissipation in the strain-softening process vanishes is not representative of the experimentally observed behavior of real strain-softening materials such as concrete or geomaterials.