Y d MT IV

The formulation of a damage model first requires the definition of threshold of damage. which is the conditions that initiates the propagation of damage. Secondly, the evolution of the damage with loading must also be defined, and it is a function of a measure of strains, stresses or energy.

5.6.3 ISOTROPIC DAMAGE MODEL FOR CONCRETE

To establish the damage constitutive equation, it is necessary to relate the damage variable d to the other internal variables by some physical hypothesis. Here we try to briefly describe an isotropic damage mechanics model. The hypothesis of strain equivalence of Lemaitre and Chaboche (1978) is empirical in nature. It states that any constitutive equation for a damaged material can be derived from the same potentials as that for a virgin material by for a damaged material by replacing the stresses by effective stresses. The effective stresses are defined as:

where[M(d)] is in general a symmetric matrix of rank four. For isotropic damage model equation 5.10 is valid when the strains in the damaged material are assumed to be equivalent to strains in the virgin material, but possess a reduced modulus by a factor (1 — d). Figure 5.8C;D shows the basis of this assumption. In equation 5.10, [M] is a diagonal matrix equal to/; and / is identity matrix. [M] represents the damage matrix. Thus, in the context of strain equivalence, equation 5.10 reduces to:

The strain energy for a damaged material is W = 1/2{e}T[C(d)]{e}, where [C(d)] is a second order matrix of material properties. In such a case, the evolution of damage decreases the material stiffness and the following expression can be written where [Co] is initial matrix of undamaged material properties. The stress vector and the rate of energy dissipation can be easily obtained:

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