Continuum theory for strain-softening

In heterogeneous materials such as concretes or rocks, failure occurs by progressive distributed damage during which the material exhibits strainsoftening, i.e., a gradual decline of stress at increasing strain. It is shown that strain-softening which is stable within finite-size regions and leads to a nonzero energy dissipation by failure can be achieved by a new type of nonlocal continuum called the imbricate continuum. Its theory is based on the hypothesis that the stress depends on the change of distance between two points lying a finite distance apart. This continuum is a limit of a discrete system of imbricated (regularly overlapping) elements which have a fixed length, l, and a cross-section area that tends to zero as the discretization is refined. The principal difference from the existing nonlocal continuum theory is that the equation of motion involves not only the averaging of strains but also the averaging of stress gradients. This assures that the finite element stiffness matrices are symmetric, while those obtained for the existing nonlocal continuum theory are not. Broad-range stresses are distinguished from local stresses and a different stress-strain relation is used for each—the broad range one with strain-softening, the local one without it. Stability of the material is analyzed, and an explicit time-step algorithm is presented. Finally, convergence and stability are numerically demonstrated by analyzing wave propagation in a one-dimensional bar.