The concepts of damage effective stress and damage equivalence I sis play an important role in the development of continuum damage mechanics. Based on a generalization of the damage equivalence hypothesis, the so-called damage isotropy principle, it is found that the effective stress as a second-order tensor-valued function of the usual stress tensor and the damage tensor (s) has to be isotropic. Particularly. this prop-erty is regardless of the initial material symmetry (isotropy or anisotropy) and the type of damage variables; and thus. it allows general invariant modeling of the effective stress by the use of theory of tensor function representations. Damage material constants are then consistently introduced to the invariant models oftha effective stress and the damage effect tensors. Three new models of the damage effect tensor capable of including realistic dimensionless damage material constants are proposed. The significance of the damage material constants is examined by micromechanical analysis and computer experiments on effective elastic moduli.