A new generalized hypothesis of elastic energy equivalence is proposed. The proposed generalized hypothesis is inclusive of all the existing different hypotheses of equivalence in continuum damage mechanics and all are obtained as special cases. Specifically, the hypothesis of elastic strain equivalence and the hypothesis of elastic energy equivalence are obtained as special cases of the generalized hypothesis proposed here. In addition, the generalized hypothesis has some unusual properties when the integer exponent n approaches infinity. In particular, it turns out that the strain energy density function is a vector for even values of the integer exponent. This conclusion is totally unexpected but an attempt is made to explain this result based on geometry.