Additional comments on discrete Cohesive Elements

This post is in response to Yuval Freed of May 13, 2009 (see below).

I checked with some of my MSC colleagues related to the question of off-diagonal terms in the stiffness matrix for cohesive elements.

It was pointed out to me that we have 2 quadrature options for Cohesive Elements: Gauss quadrature and Lobatto quadrature

When Lobatto quadrature is applied, the relative displacements at the integration points are directly equal to the relative nodal displacements. For some cases, it is known that the Lobatto scheme behaves better than the Gauss scheme.

We took a look at the Cohesive Element stiffness matrix generated from the 2 quadrature options. It appears that the Lobatto quadrature produces a stiffness matrix with only a single off-diagonal term in each row, so definitely sparse.

Kim Parnell
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From Yuval Freed of May 13, 2009:

I would like to add a comment on the interesting lecture of Dr. Kim Parnell from yesterday:

If we are looking at the stiffness matrix of cohesive elements, off-diagonal terms may appear in it. These terms represent the interaction between the different degrees of freedom of the interface element and are the outcome of its continuum nature. However, the off-diagonal terms sometimes result in numerical difficulties. To overcome this problem, the group of Prof. Tony Waas (from Univ. of Michigan) introduced a discrete cohesive zone model (DCZM) and a suitable element in which the stiffness matrix is sparse. This results from the fact that in the DCZM elements, the direct nodal displacement values are used in the traction separation laws, rather than the interpolated values as in the continuum framework used here.

I recommand to look at the following references (among many others of this group):

Xie D., Salvi AG., Sun C., Waas AM. and Caliskan A. Discrete cohesive zone model to simulate static fracture in 2D triaxially braided carbon fiber composites. Journal of Composite Materials,
40: 2025-2046, 2006.

Xie D. and Waas AM. Discrete cohesive zone model for mixed-mode fracture using finite element analysis. Engineering Fracture Mechanics, 73: 1783-1796, 2006.