The paper of Mir et al. [ASME J. Appl. Mech., 72, pp. 249–258 (2005)] analyzes the deformation behavior of random short glass fiber composites using a continuum damage mechanics model incorporating permanent strain and unilateral effect. Unilateral effect is a difficult topic [see, for instance, J. L. Chaboche, Int. J. Damage Mech., 2, pp. 311–329 (1993)] and it seems to us that the model of Mir et al. still raises some questions.

## Analysis of the Complementary Elastic Energy

The complementary elastic energy is assumed by authors to be $($see Eq. (2) of (1)$)$
$Ue(σ̱̱,Ḏ̱)=12Etr[(σ̱̱̃)+2]+12Etr[(σ̱̱̃)−2]+12E{tr[(σ̱̱̃)2]−σ̃ii2}−ν2E{[tr(σ̱̱̃)]2−tr[(σ̱̱̃)2]}$
The projector tensors are introduced as in Eq. (3) of (1):
$σ̱̱+=P̱̱̱̱+(σ̱̱,Ḏ̱):σ̱̱andσ̱̱−=P̱̱̱̱−(σ̱̱,Ḏ̱):σ̱̱$
A1
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 as
$σ̱̱+=P̱̱̱̱+(σ̱̱):σ̱̱andσ̱̱−=P̱̱̱̱−(σ̱̱):σ̱̱$
A2
This is the classical projection tensor introduced by Ortiz (2). One can, moreover, assume that the material in undamaged:
$Ḏ̱=0̱̱⇒M̱̱̱̱(Ḏ̱)=1̱̱̱̱⇒σ̱̱̃=σ̱̱$
The complementary elastic energy is then reduced to
$Ue(σ̱̱)=12Etr[(σ̱̱)2]+12E{tr[(σ̱̱)2]−σii2}−ν2E{[tr(σ̱̱)]2−tr[(σ̱̱)2]}withσ̱̱++σ̱̱−=σ̱̱$
It can be noticed that the function $Ue$ is not an isotropic tensorial function. It is clear, for instance, that the function
$f(σ̱̱)=σii2$
is not an isotropic tensorial function. It is sufficient to show that an orthogonal tensor can be found such as (3)
$∃Q̱̱∕f(Q̱̱σ̱̱Q̱̱T)≠f(σ̱̱)$
In this simplified case, the elastic constitutive law is given by
$ϵ̱̱=∂Ue(σ̱̱)∂σ̱̱⇒ϵ̱̱=1+νEσ̱̱−νE(trσ̱̱)1̱̱+1E(σ̱̱−σiiei̱⊗ei̱)$
A3
The 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.

## On the Projection Derivative Operator

We think that there are some difficulties in the derivation of the projection tensor. It has been shown, for instance, that
$∂σ̱̱+∂σ̱̱:σ̱̱+=σ̱̱+and∂σ̱̱−∂σ̱̱:σ̱̱−=σ̱̱−$
when 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 that
$σ11+=hIσI+hIIσII2+hIσI−hIIσII2cos2θ$
$σ22+=hIσI+hIIσII2−hIσI−hIIσII2cos2θ$
$τ12+=(hIσI−hIIσII)sin2θ$
with
$θ−θ0=12arctanτ12σ11−σ22andτ12=σ12+σ21$
$θ0$ 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:
$hI=H(σ11+σ222+σ11−σ222cos(2θ−2θ0))$
$hII=H(σ11+σ222−σ11−σ222cos(2θ−2θ0))$
The Jacobian matrix $(J)$ can be introduced as
$(J(σ11,σ22,τ12))=(∂σ11+∂σ11∂σ22+∂σ11∂τ12+∂σ11∂σ11+∂σ22∂σ22+∂σ22∂τ12+∂σ22∂σ11+∂τ12∂σ22+∂τ12∂τ12+∂τ12)$
In this case, the Jacobian can be more easily evaluated for $θ=θ0$:
$∣∂σ̱̱+∂σ̱̱:σ̱̱+∣θ=θ0=(∂σ11+∂σ11∂σ22+∂σ11∂τ12+∂σ11∂σ11+∂σ22∂σ22+∂σ22∂τ12+∂σ22∂σ11+∂τ12∂σ22+∂τ12∂τ12+∂τ12)(θ=θ0)(σ11+σ22+σ12+)$
It is easy to calculate the first line:
$∣∂σ11+∂σ11∣(θ=θ0)=hI+hI2(1+cos2θ0);∣∂σ22+∂σ11∣(θ=θ0)=hI2(1+cos2θ0)$
and
$∣∂τ12+∂σ11∣(θ=θ0)=hIsin2θ0$
It appears that
$∂σ11+∂σ11σ11++∂σ22+∂σ11σ22++∂τ12+∂σ11σ12+≠σ11+exceptforθ0=0$
As a conclusion, the relation
$∂σ̱̱+∂σ̱̱:σ̱̱+=σ̱̱+and∂σ̱̱−∂σ̱̱:σ̱̱−=σ̱̱−$
is 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:
$∂[P̱̱̱̱+:M̱̱̱̱:σ̱̱]∂σ̱̱:σ̱̱̃+≠σ̱̱̃+and∂[P̱̱̱̱−:M̱̱̱̱:σ̱̱]∂σ̱̱:σ̱̱̃−≠σ̱̱̃−$
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).
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