Discussion: “Damage Modeling in Random Short Glass Fiber Reinforced Composites Including Permanent Strain and Unilateral Effect” (Mir, H., Fafard, M., Bissonnette, B., and Dano, M. L., 2005, ASME J. Appl. Mech., 72, pp. 249–258)

The complementary elastic energy is assumed by authors to be $($see Eq. (2) of (1)$)$The projector tensors are introduced as in Eq. (3) of (1):It can be interesting to simplify the model in order to check some properties in simple cases. One can assume, for instance, that the damage tensor and the stress tensor have the same principal directions. In this case, the projection tensor only depends on the stress tensor asThis is the classical projection tensor introduced by Ortiz (2). One can, moreover, assume that the material in undamaged:The complementary elastic energy is then reduced toIt can be noticed that the function $Ue$ is not an isotropic tensorial function. It is clear, for instance, that the functionis not an isotropic tensorial function. It is sufficient to show that an orthogonal tensor can be found such as (3)In this simplified case, the elastic constitutive law is given byThe authors assume that this relation is only true in the principal damage directions which are not necessarily the same as the one of the stress tensor (in the general case). For instance, for the undamaged material, this relation should be verified whatever the directions, and the classical elastic linear isotropic relation is not found.

We think that there are some difficulties in the derivation of the projection tensor. It has been shown, for instance, thatwhen the damage principal axes coincide with the one of the stress tensor, as it is the case in Eq. (A2) (4). This relation is implicitly used in most stress-based anisotropic damage modeling including unilateral effects (see, for instance, (5)). However, this relation is no more true in the case considered by authors, and given by Eq. (A1), when the principal axes of the damage tensor do not coincide with the one of the stress tensor. Following the reasoning of (4), it can be shown for a two-dimensional problem thatwith$θ0$ is the rotation value between the principal axes of the stress tensor and the one of the damage tensor. $hI$ and $hII$ are two Heaviside functions of the principal stresses:The Jacobian matrix $(J)$ can be introduced asIn this case, the Jacobian can be more easily evaluated for $θ=θ0$⁠:It is easy to calculate the first line:andIt appears thatAs a conclusion, the relationis not true when the principal axes of the damage tensor and the ones of the stress tensor do not coincide. As a consequence, the calculation of the strain variable from the derivation of the complementary elastic energy is difficult to achieve with the presented model. This is furthermore more difficult to achieve when considering the fourth-order damage operator:These remarks concern the thermodynamical background of the developed model. Similar remarks can be formulated for the derivation of the damage strain energy release rate as the projective tensor also depends on the damage tensor (see (6) without introducing this coupling).