Keywords: fracture mechanics, cohesive zone model, extrinsic law.

This paper deals with the numerical implementation of cohesive laws with an infinite initial stiffness, usually called extrinsic cohesive laws. Two formulations are proposed for the implementation of these kinds of cohesive zone models in the case where the discontinuity propagates between the continuous elements of the mesh. An equivalent cohesive stress and an equivalent displacement jump are defined to express the cohesive law as a function of those two quantities as it is done in [1], and the relation between the normal and the tangential behaviour is obtained from an analogy with a crack band model [2].

A first formulation which uses the nodal displacements as discrete variables for the continuous elements of the bulk and the cohesive stresses for the discontinuous elements of the interface is presented. With this formulation, a compliance matrix is calculated for the cohesive zone in replacement of the usual stiffness matrix. Consequently, there is no need to penalize the stiffness of the interface before the initiation of the cohesive zone. A drawback of this formulation is that it cannot deal with cohesive elements that have completely failed because the compliance of these elements is infinite. A solution could be to suppress failed elements from the discretization, but this can lead to numerical errors if the cohesive stresses are not exactly equal to zero in an element when it is suppressed.

To tackle this problem, a second formulation has been developed for which a change of variable permits the replacement of the equivalent cohesive stress with a modified equivalent cohesive stress. In this way, tangent operators are calculated using a modified cohesive law that gives the modified equivalent cohesive stress as an increasing function of the equivalent displacement jump. This allows the implementation of an extrinsic cohesive law in a similar way to the first formulation without changing the discretization during the calculation.

Both formulations have been tested on a bi-dimensional problem with linear elements. An arc-length method has been used for the control of the loading factor. We have observed that the use of two Newton-Cotes integration points for the interface element leads to cohesive stress fields with no oscillations. A convergence study shows that the global behaviour of the problem converges when the mesh is refined.

1

G.T. Camacho, M. Ortiz, "Computational modelling of impact damage in brittle materials", International Journal of Solids and Structures, 33(20), 2899-2938, 1996. doi:10.1016/0020-7683(95)00255-3

2

Z.P. Bazant, B.H. Oh, "Crack band theory for fracture of concrete", Materials and Structures, 16(3), 155-177, 1983. doi:10.1007/BF02486267

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