An analytical solution with local elastoplastic models for the evolution of dynamic softening

Strain softening is associated with the evolution of localization. With the use of the similarity method, a closed-form solution for wave propagation in a strain softening bar is derived in the paper through a partitioned-modeling procedure with local elastoplastic constitutive models. The initial point of the localization is taken as the point at which the type of governing di€erential equation transforms from a hyperbolic one to an elliptic one due to material softening. The evolution of localization is then represented by a moving material surface between the softening domain and non-softening domain. The motion of the material surface is of di€usion type, representing macroscopically the progressive percolation of heterogeneous ¯ow or microdamage. The evolution of relevant ®eld variables along the bar is shown, and the e€ects of the model parameters on the solution are discussed to demonstrate the proposed procedure. The analytical solution is unique and stable for the given set of boundary and initial data, and material properties, based on the theory of di€erential equation