Keywords: hybrid finite elements, composite materials, junctions, delamination cracks, stress intensity factors, energy release rate.

If the design of composite structures as laminates or sandwich plates can be considered as globally mastered, the one of junctions still holds unsolved difficulties. The junctions may be of various kinds, for instance inserts, rivets or sticking. We must face the complex 3D nature of these problems. Moreover, for brittle materials as composites, cracks can be initiated from regions where the stress field is singular, around lines at the junction between different anisotropic or, and isotropic materials. These lines are qualified as "junction lines". A junction with aluminium insert in a glass-epoxy sandwich plate is considered in Figure 1.

Let be the angular parameter along a junction line (or a crack front), and be the polar coordinates in the current plane orthogonal to the junction line (or the crack front). In the curvilinear coordinate system , the asymptotic singular part of the stress field near the junction line has the following form [1,2]:

The complex number a is the singularity exponent. Unlike for cracks in homogeneous materials, it is not generally equal to -1/2. The determination of its value is a first numerical difficulty to face. This computation already gives qualitative information, helpful to the design. This requires considering an eigenvalue problem in complex variable. A numerical algorithm was successfully implemented for solving it and obtaining the eigenstress field .

Next, accurate values of the associated stress intensity factors are computed. We implemented a 3D mongrel hybrid finite element, based on a unisolvent displacement field and an equilibrated stress field enhanced by the previously determined singular field. The Babuška-Brezzi inf-sup condition enables the selection the appropriate stress mode functions [3], as highlighted by a benchmark. The approach is illustrated for the previous sandwich-insert junction in Figure 2.

1

S.G. Lekhnitskii, "Theory of Elasticity of an Anisotropic Body", Holden-Day, San Francisco, USA, 1963.

2

D. Leguillon, É. Sanchez-Palencia, "Computation of Singular Solutions in Elliptic Problems and Elasticity", Masson, Paris, France, 1987.

3

T.H.H. Pian, D. Chen, "On the suppression of zero energy deformation modes", International Journal of Numerical Methods in Engineering, 19, 1743-1752, 1983. doi:10.1002/nme.1620191202

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