A failure analysis of quasi-brittle materials requires the evaluation of progressive damage due to distributed cracking. The cracking is characterized by a fracture process zone, distributed over a finite size volume, which exhibits so-called strain softening (the stress strain relation, in which the maximum principal stress decreases with increasing corresponding principal strain). Standard local constitutive models are inappropriate for materials which exhibit strain-softening behaviour. They are not objective with respect to employed discretization because the strain softening damage tends to localize into a zone, width of which depends on the element size. As the element size is refined, the size of the localization zone converges to zero and the total energy consumed by the fracture process converges to zero as well, which makes the local constitutive models unacceptable.
In this study, a computationally efficient and widely used localization limiter based on the nonlocal concept of integral type is adopted. It consists in replacement of a suitable locally defined quantity by its nonlocal counterpart , obtained by weighted averaging of the local quantity over a certain representative volume of the material
The application of nonlocal constitutive approach for concrete fracture is demonstrated on damage model. A number of nonlocal damage formulations have been published in the literature. However, not all of them can provide a clean description of the complete degradation process up to the final failure. It was shown  on a simple uniaxial localization test that the complete fracture is correctly reproduced by models that average the equivalent strain, the energy release rate, or the compliance variable. In contrast to that, models based on the evaluation of the inelastic stress from the nonlocal strain and models averaging the damage variable, inelastic stress, or inelastic stress increment lead to spurious residual stresses and to a pathological expansion of the softening zone.
In the present study, isotropic damage model based on the nonlocal equivalent strain is employed. It is described by the following constitutive equation