Shear softening and localization: Modelling the evolution of the width of the shear zone

In this paper, the question of shear localization inside a band is revisited. Using simplified constitutive equations we look at a 1D layer in a classical medium. Even though the example is simple, we show once more the loss of uniqueness of the solution. By studying the initial boundary value problem we also show that the width of the localization zone stays constant or decreases. A second gradient model is then introduced and the evolution of the width of the localization zone is studied. Analytical solutions are derived for a quasi-brittle material and estimations made for a ductile constitutive law. It is shown that the width of the localization zone stays constant for the quasi-brittle material and decreases for the ductile one. Numerical simulations are finally provided to validate the approach and to get more precise numerical results for the ductile case.