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Constitutive Model

A failure analysis of quasi-brittle materials like concrete, rock, or ice 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 corresponding increasing principal strain). Standard local constitutive models are inappropriate for materials which exhibit strain-softening behaviour. It has been demonstrated [1], that this approach is not objective with regard to employed discretization. The strain-softening damage tends to localize into a zone, width of which depends on the element size. As the mesh 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.

The use of so-called localization limiter is therefore necessary to obtain objective results. A simple localization limiter, introduced by Bazant and Oh [1] is the crack band model. This model leads to proper energy dissipation, but it does not provide any information about the localization profile. More general localization limiters are based on high-order gradient models, viscoplastic regularization, or Cosserat continuum. A computationally efficient and widely used limiter is the nonlocal concept, introduced by Bazant [2]. It is based on 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.

It must be pointed out, that it is desirable to use weight function with a limited support (the closure of the set of points, where weight function is nonzero, is finite), in order to make the parallel strategy effective. In this paper, the following definition of nonlocal weight function has been employed




Parameter is related to the internal length. Since it corresponds to the largest distance of point that affects the nonlocal average at point , it is called interaction radius. The volume is defined as , ensuring that the normalizing condition holds at any point of interest.

The application of nonlocal constitutive models for concrete fracture is demonstrated using a nonlocal rotating crack model with transition to scalar damage (RC-SD, see [3] and [4]). It is based on the classical rotating crack model. The nonlocal version is described by the following equation


where is the secant constitutive stiffness matrix evaluated for the nonlocal strain . The resulting stress is the product of the ``nonlocal'' secant stiffness matrix with the local strain. This standard rotating crack procedure is applied only during the initial stage of the cracking process. Once cracking process reaches a certain critical state (identified by principal stress to tensile strength ratio and by current shear stiffness to shear modulus ratio), the procedure switches to a damage type formulation. The final stage is then described by the damage model, that uses the anisotropic stiffness multiplied by a scalar factor, that decays to zero value as the cracking continues.

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Daniel Rypl