Closure to “Discussion of ‘Damage Modeling in Random Short Glass Fiber Reinforced Composites Including Permanent Strain and Unilateral Effect’ ” (2006, ASME J. Appl. Mech., 73, pp. 347–348)

Abstract

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]12Eσii−σii−≠12Etr[(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++σii−σii−+σijσij∣i≠j)−υE(σiiσjj−σijσij)=12E[σ]:[σ]−υE((tr[σ])2−tr([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).