Numerical fracture experiments using nonlinear nonlocal models

Abstract

In this talk I will focus on the modelling of fracture in the continuum solids using nonlocal interaction. This is referred to as Peridynamics. Peridynamics proposes an alternative framework from that of a classical elastic theory and is shown to be very suitable in modelling fracture and the material breaking. The key ingredient in the theory is how the forces at the material points are computed. In the classical theory of continuum mechanics, the force at a point is expressed as a divergence of stress at that point. Instead, in Peridynamics, the force at a point is given by the summation of the pairwise (bond) forces between this point and the neighboring points. The size of the neighborhood sets the nonlocal length scale. The pairwise force is assumed to be the function of a relative stretch (pairwise strain) between the two points. Different models in the Peridynamics differ in how the force behaves with the pairwise strain, however, they all have one thing in common: If the stretch between the two points is beyond a threshold value (critical strain), the two points no longer interact, i.e. the bond between two points is severed. This behavior of the pairwise force enables the material to fracture under the appropriate boundary conditions. The fracture zone is defined as a collection of the material points which have bonds with strain above the critical strain. It is shown theoretically and also through the numerical simulations that the fracture zone localizes to a very small region encapsulating the crack. The width of the region depends on the nonlocal length scale and in the limit of the nonlocal length scale going to zero the fracture zone collapses to the crack line. At any stage in the theory, the derivatives of the displacement field are not computed which results in advantage in situations where the displacement field is not well behaved. My goal will be to give a brief overview of the Peridynamics theory through the results of my work with Dr. Robert Lipton at the Louisiana State University.