4. Conclusion#

A finite element interface model is proposed in order to model the evolution of cohesive cracks along a predefined path. The latter is compatible with the use of conventional solid finite elements. Its unknowns are the nodal displacements on the lips of the cohesive crack as well as the nodal Lagrange multipliers, corresponding to the surface density of the cohesive forces. The Lagrangian of the problem is increased, this makes it possible to ensure a convexity condition (via the penalization parameter) which guarantees the uniqueness of the solution during the local integration of the law.

However, this approach has the following limitations and disadvantages:

  • The potential trajectories of the crack must be postulated a priory.

  • Additional degrees of freedom, corresponding to cohesive forces, are introduced. However, their number remains low since they are limited to the potential trajectories of the crack.

  • The introduction of Lagrange multipliers leads to a mixed formulation: solving the problem therefore amounts to a search for a saddle point and more to that of a minimum as was the case in the initial energy formulation 5.

  • It is necessary to increase the Lagrangian to have a local convexity property. This involves the introduction of a penalization parameter, which has no influence on the ongoing problem, but which could affect the results of the discretized problem. However, the numerical examples in 5 show that this sensitivity remains low and disappears with the refinement of the mesh.

  • The local integration of the cohesive law is based on the calculation of displacement discontinuities based on cohesive forces. A reverse approach is generally adopted in the literature for interface elements.

However, a certain number of interesting properties stand out:

  • No regularization of cohesive law is necessary, especially with respect to initial adherence or contact condition.

  • The choice of a quadratic discretization for the movements and linear for the Lagrange multipliers makes it possible to satisfy condition LBB. The latter ensures the convergence of the solution with the refinement of the mesh in terms of movements and cohesive forces (see numerical example in 5).

  • The convergence rate that would be obtained without an interface is not disturbed by the presence of interface elements (see 5).

  • The search for a saddle point leads to a symmetric tangent matrix.

This interface model is compatible with the usual*Code_Aster resolution algorithms such as the Newton method, linear search or load control. This is illustrated by 2D and 3D calculations in 5, thus demonstrating the robustness and reliability of such a model.