Chapter 2 - Nonlocal continuum plasticity

Abstract

In classical continuum theory, the constitutive model of a material point can be solely defined from certain characteristic properties at that point. The coupling to the rest of continuum body is governed by the standard balanced equations and through the stress tensor. These families of continuum models are categorized as the local models. Local continuum models incorporate no length scale, and scaling of the specimen size does not change the governing equations. In reality, however, due to the material heterogeneities, the material behavior is size dependent. In fact, as the sample size becomes comparable to the length of the nature of heterogeneities, such as aggregate size in concrete, the governing mechanism of deformation becomes a function of sample length. At least one length scale should be incorporated into the material model to capture the size effects in different materials which leads to the nonlocal continuum models. Various methods have been developed to incorporate the size effects in plasticity through different procedures. The nonlocal plasticity models can be categorized as two families. First, the models which addresses size effects by incorporating additional kinematic variables in addition to strain including higher order gradient of displacement. Furthermore, the weighted spatial averages of strain can be also included in this category. Second, the nonlocal models which modify the constitutive equation by adding the gradients of the internal variables or their thermodynamic conjugates. Again, the weighted spatial averages of internal variables and their thermodynamic conjugates are accounted as the same group. Another way to categorize the nonlocal plasticity models is based on an operator used to capture size effects which includes the gradient and integral-type models. In this chapter, local and nonlocal plasticity models will be discussed in both frameworks of small strain and finite strain continuum mechanics.