M i d[CoM
Since Y is a quadratic function of strain, it is positive and, thus, d is always increasing as shown in equation 5.9. This is a characterization of the irreversibility of damage.
Different hypothesis have been proposed for isotropic damage model. The basic assumption in behaviour of the damaged and equivalent undamaged element will results in different model. Among several models, special attention will be given to the two models. They are mainly based on two concepts of the modelling. In the first model, EnergyBased Damage Model, a single damage variable independent of directions of stresses describes the behavior of concrete. The damage evolution is defined as a function of the elastic strain energy of an undamaged equivalent material. To ensure mesh objectivity of the finite element solution, the softening parameters are made meshdependent using the energy equivalence concept. This technique leads to reasonable mesh size and the model is suitable for the analysis of large concrete structures. While in the second concept, StrainBased Damage Model, the damage is described by coupling the compression and tension effects to define a single damage variable d. This variable is calculated based on a certain measure of the material's strain field. The objectivity of the finiteelement solution is ensured by introducing the nonlocal description of this measure by averaging it within an influence area in the finiteelement mesh. The size of the influence area is of the order of three times the aggregate size. In this case an extremely refined mesh is required. If the aggregate size varies between 100 and 300 mm, it becomes impractical to model real dams with a reasonable size mesh since at least three elements are required within the influence area of about 300900 mm.
5.6.4 ANISOTROPIC DAMAGE MODEL FOR CONCRETE
A Gftype anisotropic damage model is described in this part for different reasons: when roller compacted concrete is used in dams, the material itself is initially orthotropic. Therefore, the development of an anisotropic damage model is essential. In addition, orthotropic damage models allow the modeling of joints in the dam and at the damfoundation interface, even though these problems are not addressed in this paper. It will be shown that by developing an orthotropic damage model, the isotropic model becomes a special case (Chow and Wang 1987; Ju 1989; Chow and Lu 1991).
Figure 5.8: Material model in the damage mechanics concept; A)effective areas for isotropic and anisotropic damages; B)characteristic length; C)strain equivalence hypothesis; D)stressstrain curve for equivaalence hypothesis; E)closingopening criterion; F)initial damage formulation
When isotropic damage is considered, the effective stress {a} (Lemaitre and Chaboche 1978) and the elastic stress {a} are related by 5.11. If damage is no longer isotropic, because of cracking, material anisotropy introduces different terms on the diagonal of [M (d)]. If the concept of net area is still considered, the definition for d becomes:
a where Q is tributary area of the surface in direction i; and Q* is lost area resulting from damage, as shown in Figure 5.8A. The index i(1, 2,3) corresponds with the Cartesian axes x,y and z. In this case the ratio of the net area over the geometrical area may be different for each direction.
The relation between the effective stresses {a} and the elastic equivalent stresses {a} becomes:
<1  
*  
<2  
* 
—  
<12  
*  
_ <21 _ 
In this case the effective stress tensor is no longer symmetric and an anisotropic damage model, based on equivalent strains, results in a nonsymmetric effective stress vector. Various attempts to restore symmetry were proposed by, Chow and Lu (1991) and Valliappan et al. (1990). They are based on the principle of elastic energy equivalence. This principle postulated that the elastic energy in the damaged material is equal to the energy of an equivalent undamaged material except that the stresses are replaced by effective stresses. If the symmetrized effective stress vector is defined as: {a} = {<T1<72<T12} = {<T1<T2 It can be related to the real stresses by:

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