Cohesive Failure Law

Hi Ruchika,

I am also interested in traction-separation laws (TSLs). I do not know much but there are something that I can tell you. 

Basically, a  TSL expresses the relation between the traction and the displacement jump on the crack surfaces. There are mode I STLs and mixed mode STLs. In the former, only the normal traction and the normal displacement jump delta u_n matter. Whereas, in the latter both normal and tangential are involved.

There are rigid STLs and non-rigid STLs. For rigid STLs, the traction starts from the tensile strength of the material while for non-ridig STLs there is a linear elastic stiffness. Often, the former is employed in XFEM where the crack is introduced whenever a failure criterion is met. The latter is used in finite elements with interface elements since you need the linear elastic stiffness to model the perfect bond before fracturing.

About numerical methods used for STLs, there are XFEM where cracks can be located arbitrarily on the mesh, or interface elements (also called cohesive zone elements) where cracks are along the interboundaries of the elements (hence it limits the application of the method). There is also the embeded strong discontinuity method where the enhanced strain method is used to represent the displacement discontinuity inside the element. 

By the way, I do not well know how to do the stress update for STLs. I mean how to handle loading, unloading, penetration (negative displacement jump) ... If you know how to do, please show me since my current implementation is not robust for STLs for brittle materials although they are just simple laws like linear STL or exponential STL for mode I problems.

 Regards,

Phu

There is a large body of work on cohesive laws.  Here is a review I wrote a while ago:

G. Bao and Z. Suo, "Remarks on crack-bridging concepts," Applied Mechanics Review. 45, 355-366 (1992).

This review should give you a basic idea of cohesive-zone models.