Analysis on the Complementary Elastic Energy

If we compare the first equation of the Discussion to Eq. (2) of the paper, we can remark that the authors of the Discussion have misunderstood the former equation, 
12Eσ̃ii+σ̃ii+12Etr[(σ̃+)2]12Eσiiσii12Etr[(σ̃)2]
Considering this, their remark is irrelevant. Nevertheless, we can observe easily that in the particular case [D]=0, Eq. (2) is simplified to Eq. (1) of the paper, 
Ue(σ,[D]=0)=12E(σii+σii++σiiσii+σijσijij)υE(σiiσjjσijσij)=12E[σ]:[σ]υE((tr[σ])2tr([σ]2))
Furthermore, we can observe that if we put [D]=0, the fourth-order damage operator defined in Eq. (6) becomes [M]=[I4]. Thus, in this particular case, Eq. (9) is exactly equal to the classical complementary energy presented at Eq. (1).

On the Projection Derivative Operator

The authors of the Discussion mentioned that “when the principal axes of the damage tensor do not coincide with the one of the stress tensor,” some relations presented in the paper are not applicable. We agree with this affirmation because we mentioned it in Sec. 2.3 of the paper and in the Conclusion. When we spoke about proportional loading, this means that the principal stress directions, and thus, those of damage tensor, are identical and do not change during loading. We apologize not having mentioned it more clearly at the beginning of the paper.