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Phase Field Modelling of Fracture Mechanics

Submitted by PrajwalSabnis on

Hello To one and All,

 

I am currently working on my Master's thesis in Computational
Mechanics and as the Title suggests, I am working on using phase field
theories to model fracture mechanics. The whole idea /concept of phase
field modelling in itself has been a very complex topic for me, as it
originates in superconductivity theories of which I have but a
superficial understanding. I am ok with fracture mechanics terminology
as I have worked on it before.

 

The problem I am currently facing is in understanding the theory
behind computational modelling of fracture using these phase field
theories. The PFT use the "order variable/order parameter" which is a
scalar quantity in itself to model phase transitions, but  quite a few
papers that I have come across use the terms "Conserved Order
Parameters/ Non Conserved Order Parameters". Can anyone explain these
terms to me please?

 

I am guessing itis based on the model being used, i.e. to say, if
the model is based on the Balnce Laws such as the Momentum Balance and
Mass Balance/Volume Balance which are basically conservation laws, in
other words, if the evoliution off the parameter is given on basis of
one of the conservation laws(mass balance law in e.g. Phase Field
modelling of Mode I fracture by L. O. Eastgate et. al.)then it is
called a "conserved order parameter". And on the other hand if the
evolution for the parameter is given by a constitutive relation , then
it is called a "non conserved parameter".

 

Is my Guess correct? Can someone please help me out with this or provide a more reasonable explanation for it?

 

Regards

Prajwal

Forums
Fracture Mechanics Forum
Mogadalai Gururajan

Dear Prajwal,

If there is a constraint on your order parameter (namely, that the integral of the order parameter over the system volume is constant as a function of time), then the order parameter is a conserved quantity; for example, composition is a conserved order parameter. On the other hand, if there is no such constraint, then the order parameter is non-conserved; for example, if you use an order parameter in a phase field model for solidification which indicates whether the system is in solid phase or liquid phase, it is non-conserved since at the beginning of the simulation the system was almost entirely liquid and at the end of the simulation it is almost entirely solid.

The evolution, on the other hand, in both cases is postulated using some kind of kinetic law; in the case of conserved order parameters, however, there is an additional constraint of equation of continuity that needs to be imposed, which is what makes the evolution laws looks different. For example, in the case of composition, it should not only be conserved, but it should also evolve in such a fashion that the equation of continuity holds. One can, of course, try and impose the constancy of integrated composition over the system volume using a Lagrange multiplier; however, the evolution of microstructure that one obtains using such an apporach need not be physically meaningful; hence, only if, your interest is in obtaining the final equlibrium microstructure for a given system, one can (and people do) use such a model for a conserved order parameter.

Guru 

Fri, 04/10/2009 - 05:54 Permalink