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DISCUSSION

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) OPEN ACCESS

[+] Author and Article Information
Noël Challamel

Laboratoire de Génie Civil et Génie Mécanique (LGCGM),  INSA de Rennes, 20, avenue des Buttes de Coësmes, 35043 Rennes cedex, Francenoel.challamel@insa-rennes.fr

Christophe Lanos

Laboratoire de Génie Civil et Génie Mécanique (LGCGM),  INSA de Rennes, 20, avenue des Buttes de Coësmes, 35043 Rennes cedex, Francechristophe.lanos@insa-rennes.fr

Charles Casandjian

Laboratoire de Génie Civil et Génie Mécanique (LGCGM),  INSA de Rennes, 20, avenue des Buttes de Coësmes, 35043 Rennes cedex, Francecharles.casandjian@insa-rennes.fr

J. Appl. Mech 73(2), 347-348 (Mar 01, 2006) (2 pages) doi:10.1115/1.2126694 History:

The paper of Mir [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 still raises some questions.

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)]2tr[(σ̱̱̃)2]}
The projector tensors are introduced as in Eq. (3) of (1):Display Formula
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 asDisplay Formula
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=0̱̱M̱̱̱̱(Ḏ̱)=1̱̱̱̱σ̱̱̃=σ̱̱
The complementary elastic energy is then reduced to
Ue(σ̱̱)=12Etr[(σ̱̱)2]+12E{tr[(σ̱̱)2]σii2}ν2E{[tr(σ̱̱)]2tr[(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)
Q̱̱f(Q̱̱σ̱̱Q̱̱T)f(σ̱̱)
In this simplified case, the elastic constitutive law is given byDisplay Formula
ϵ̱̱=Ue(σ̱̱)σ̱̱ϵ̱̱=1+νEσ̱̱νE(trσ̱̱)1̱̱+1E(σ̱̱σiiei̱ei̱)
(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.

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+andσ̱̱σ̱̱: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+hIσIhIIσII2cos2θ
σ22+=hIσI+hIIσII2hIσIhIIσII2cos2θ
τ12+=(hIσIhIIσII)sin2θ
with
θθ0=12arctanτ12σ11σ22andτ12=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+σ11σ222cos(2θ2θ0))
hII=H(σ11+σ222σ11σ222cos(2θ2θ0))
The Jacobian matrix (J) can be introduced as
(J(σ11,σ22,τ12))=(σ11+σ11σ22+σ11τ12+σ11σ11+σ22σ22+σ22τ12+σ22σ11+τ12σ22+τ12τ12+τ12)
In this case, the Jacobian can be more easily evaluated for θ=θ0:
σ̱̱+σ̱̱:σ̱̱+θ=θ0=(σ11+σ11σ22+σ11τ12+σ11σ11+σ22σ22+σ22τ12+σ22σ11+τ12σ22+τ12τ12+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+andσ̱̱σ̱̱: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:σ̱̱̃σ̱̱̃
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).

Copyright © 2006 by American Society of Mechanical Engineers
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