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damage to fracture (formation and propagation of the discontinuity)


I am working on modeling of Continuum damage in isotropic materials. I have written a Matlab code for modeling of a continuous damage and I have verified it with several examples. In this modeling, I have mesh-size independent through the smeared damage model. Now, my aim is to present a continous-discontinous model.

In other words, my aim is transition from Continuum Damage to macro-cracks. In the first step, I need the time for forming a crack and its angle. For doing this, I want to apply the bifurcation analysis. My question is that how can I model the starting mechanism and discontinuity propagation?  

In fact, I have used the localization tensor(acoustic tensor) (Q = n.Cd.n) and I have considered the direction of discontinuity in the where the determinant of this matrix is zero (det(Q)=0). As we know, for calculation of this determinant, we need the tangent constitutive operator(Cd) where it denotes the relationship between the d(sigma) and d(epsilon). d(sigma) and d(epsilon) are stress increment and strain increment, respectively ( d(sigma)=Cd x d(epsilon) ).

My question: I have constructed the tangent constitutive operator(Cd) in my work, but one its elements in the main diagonal is negative. Considering that our phenomenon is a softening one, is it logical to have a negative element in Cd matrix? Can we apply the method of localization tensor in this case?

I am wondering that if I can find a person who has some experiences related applying this localization tensor.


Lihua Jin's picture

By sigma and epsilon, do you mean true stress and true strain? To me it's reasonable that d(nominal stress)/d(nominal strain) can be negative. If d(true stress)/d(true strain) is negative, there must be some intrinsic mechanism for this instability. What's that soften mechanism?

Hello Sir,


Do you think you can give me a very basic matlab code for CDM please, so I know how you do implementation.  I need this for my MS project.  Or you can tell me where to get it.


Thank you

N. Situ 


Yes, negative eigenvalues often exhibit in material model related to strain softening. Therefore it's usual if you have negative diagonals in your tangent operator. It indicates the material point pass to some instable state.

 By the way, i also implemented a code for continuum isotropic damage model. What i observered is that the convergence property is very poor. Especially for the later stage of damage. For most of examples i tried it always failed after some steps. The algorithm only converge if i combine with line-search technique. Did you observe the same behaviour?


Giang Bui



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