On the crucial role of imperfections in quasi-static viscoplastic solutions

The stability and structure of shear bands and how they relate to initial imperfections is studied within the framework of a one-dimensional boundary value problem. It is shown that in strain-softening viscoplasticity the structure of the band depends on the structure of the imperfection. A Fourier analysis shows that the width of the shear band depends directly on the width of the imperfection, suggesting that the imperfection scales the response of the viscoplastic material. For continuously differentiable imperfections, the shear band is continuously differentiable, whereas when the imperfection is C^o at the maximum, the shear band is C^o, and cusp-shaped. For step function imperfections, the shear band is shown to be a step function, but it is shown that this solution is unstable.