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Model Regularization

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


where is the domain of interest and is a nonlocal weight function. For efficiency reasons, it is desirable to use the weight function with a limited support (the closure of the set of points, where the weight function is nonzero, is finite). In this paper, the following definition of nonlocal weight function has been employed




is a bell-shaped function. Parameter , called interaction radius, corresponds to the largest distance of point that affects the nonlocal average at point and is related to the material internal length. The volume is defined as


ensuring that the normalizing condition


holds at any point of interest.

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 [15] 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


where is the damage (ranging from 0 for virgin material to 1 for fully degraded material), is a state variable taken as maximum equivalent strain ever reached at a particular integration point calculated from the nonlocal strain and is the elastic constitutive stiffness matrix. The damage evolution is controlled by an exponential law given by


The damage starts above a threshold that defines the upper bound of the elastic branch of stress strain relationship. The parameter then controls the slope of the softening branch at the peak and indirectly the fracture energy.

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Daniel Rypl