## Nonlinear Modelling Of Concrete Dams Using Damage Mechanics

Micro cracking in concrete is believed to occur at relatively low levels of loading. Therefore, cracking progresses in a heterogeneous medium because of an increase in micro cracking, and because of the linking of various micro cracked zones. Experiments performed on cement paste as well as concrete show that micro cracking has an arbitrary orientation. When the load is increased, macroscopic cracks develop and the crack orientations follow the principal stress directions in the material.

Proper understanding and mechanical modeling of the damage process of concrete, brought about by the internal defects, is of vital importance in discussing the mechanical effects of the material deterioration on macroscopic behavior. Modeling of this phenomenon has triggered intensive research activities over the past 20 years or so.

The main concept of this theory is to represent the damage state of material by an internal variable, which directly characterizes the distribution of micro cracks formed during the loading process. Each damage model established mechanical equations to describe the evolution of the internal variables and the mechanical behavior of damaged material. The damage mechanic model can be divided into isotropic and anisotropic damage models. The isotropic damage mechanics model uses a single scalar parameter and is based on Lenlaitre's hypothesis of strain equivalence. It was used with some success to describe the damage of concrete. Yet, few practical applications to real structures were conducted using this approach. However, it was experimentally observed that crack growth in concrete structures significantly depends on the direction of the applied stress and strain. Hence, the damage process in concrete is essentially orthotropic, the isotropic description is a mere simplification.

Compared with the fracture mechanics theory used in the context of discrete cracks the continuum models with fixed mesh have the advantage of avoiding remeshing when finite elements are adopted. However, when compared with the smeared crack approach there seems to be a small advantage. Researchers have proved that different mechanical phenomena can be formulated within the same framework of damage mechanics. The swelling problem is investigated in concrete dams using this concept. Finally, damage mechanics permits the easy implementation of any initial damage due to thermal stresses or any other phenomena, such as alkali-aggregate reactions, acting on an existing dam.

5.6.1 NUMERICAL PROBLEMS RELATED TO STRAIN SOFTENING

Damage process is associated with strain softening, which is usually accompanied by sudden transition from a smoothly varying deformation field into a localized band. The treatment of strain softening and localization by finite elements could lead to serious difficulties, as reported by Simo (1989), Bazant and Belytschko (1985), and Wu and Freund (1984).

Some of these difficulties are:

(1) cracks tend to localize in a band that is generally the size of the element used in the discretization;

(2) the fracture energy dissipated decreases as the mesh is refined, in the limit it tends towards zero;

(3) the solution obtained is extremely dependent on mesh size and orientation.

To overcome these difficulties, the following solutions were proposed:

1 . The strain softening is related to the element size (Pietruszczak and Mroz 1981). which leads to the mesh-dependent hardening modulus technique (Bazant and Oh 1983. Pramono and Wiliam 1989; Simo 1989).

2. Nonlocal damage is formulated as proposed by Bazant and Pijaudier-Cabot (1988), Mazars et at. (1991), Saouridis and Mazars (1992).

3. Introduction of viscoplasticity as suggested by Needleman (1988), Loret and Prevost (1991), Sluys and de Borst (1992).

4. The use of the second gradient theory in the definition of the strain tensor (sluys et al. 1993).

5. Cosserat continuum approach (de Borst 1991; Vardoulakis 1989). From a purely mathematical viewpoint Simo (1989) showed that the first three solutions can be interpreted as regularization procedures of the dissipation function. Pijaudier-Cabot et al. (1988) performed a comparative study of these techniques for the propagation of waves in a longitudinal bar.

### 5.6.2 FUNDAMENTAL EQUATIONS OF DAMAGE MECHANICS

The concept of the damage mechanics model is based on the dissipation of energy by means of cracking and loss of rigidity of the material. Any induced plastic behavior is ignored.

Equilibrium of the system is defined by the thermodynamic potential, which is considered here as the strain energy W(e, d) and is a function of the strain vector {e} and the damage variable d. Thus, W(e, d) can form the basis to formulate the following expressions:

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