[CdG [RT [Cd[R

Where[Cd]G is material matrix in the global axes; and [R] is transformation matrix defined by:

and ¡3 is angle of principal strain direction. If di = d2 = d, the following relation is obtained:

which represents the damaged isotropic model. This model differs from equation 5.12 by a square factor because of the energy equivalence.


Now we try to relate the damage variable to the state of an element using uniaxial behaviour of a concrete specimen.. Consider the elastic brittle uniaxial behavior as shown in Figure 5.9. If a point At on the stress strain curve (a, e) moves to A2, because of damage, a certain amount of energy is dissipated (dWd). Under conditions of infinitesimal deformation and negligible thermal effects, the first law of thermodynamics requires:

where dWe is elastic energy variation; and dWd is energy dissipated by damage. It can be seen that after damage the strain will reach to its original strain at zero. The total dissipated energy is calculated as:

— 2 sin ßcosß 2 sin ßcosß cos2 ß — sin2 ß

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