computation G

[Van Dung Nguyen]

Dear all,

I have questions about the MoFEM code. In the function CPSolvers::calculateGc in the  crack propagation module of MoFEM, to estimate G, 

 - first both the the material force and Griffith force vectors are projected following a surface matrix estimated by the function cP.calculateSurfaceProjectionMatrix(...) (using SurfaceSlidingConstrains); 

 - then the both vectors are projected to the crack front following a matrix estimated by cP.calculateFrontProjectionMatrix (using ConstantArea) to estimate G1, and G3.

Can anyone help me to explain why two opeations must be used? It seems not to be the same procedure as described in https://onlinelibrary.wiley.com/doi/10.1002/nme.4603 (I did not see informations about the first projection).

And why the value G3 is estimated but not G2?

Best regards,

Van Dung 

[Lukasz Kaczmraczyk]

Hello,

Note, that,

||G|| = sqrt(g1^2+g2^2+g3^2), 

where  

• g1 - is a force component in the crack surface orthogonal to crack front,
  
• g2 - is a force component orthogonal to crack surface and orthogonal to crack front,
  
• g3 - is a force component in the crack surface tangent to crack front,
  
• G - is a material force at the crack front.
  

In the case of K1 fracture mode, straight crack front g1 is equal to ||G|| and has an interpretation of J-integral. 

Note that G work adjoint to displacements at the crack front, and work is interpreted as work done at the crack front on topological changes, i.e. crack propagation. Since we consider crack propagation,  the body geometry constrains crack displacements, i.e. crack front cannot go beyond the shape of the body. Thus, if the crack front node is on the boundary of the body, is constrained by the shape (cP.calculateSurfaceProjectionMatrix(...)). Note that

G * W_(constrained_by_the_body) = G_(projected_on_body_surface) * W

where W is material displacement at crack front.  Then to calculate g1 and g3 we using (cP.calculateFrontProjectionMatrix). Once we know g1 and g3, we can work out what is g2 form equation at the top.

Kind regards,

Lukasz

[Van Dung Nguyen]

Thanks for your response!

As you described, the surface contraint is applied on the crack front node of it is on the boundary of the body. It means that if the crack does not intersect with the boundary of the body (e.g ellipsoidal crack inside the body), these kind of constraints do not need to be applied?

Regards,

Van Dung

[Lukasz Kaczmraczyk]

Yes, if the ellipsoidal crack is in the body, not crossing the boundary, surface constrains does not apply. 

[Van Dung Nguyen]

It is clear for me, thanks a lot!