A numerical approach for modelling thin cracked plates with XFEM

The modelization of bending plates with through the thickness cracks is investigated. We consider the Kirchhoff-Love plate model which is valid for very thin plates. We apply the eXtended Finite Element Method (XFEM) strategy: enrichment of the finite element space with the asymptotic bending and with the discontinuity across the crack. We present two variants and their numerical validations and also a numerical computation of the stress intensity factors. Résumé. Cet article traite de l’approximation numérique des plaques en flexion en présence d’une fissure traversante. Nous considérons un modèle de Kirchhoff-Love, valide pour les plaques très minces, et des éléments finis de type triangles HCT et quadrangles FVS réduits. La stratégie de la méthode des éléments finis étendue (XFEM) est alors appliquée : enrichissement de l’espace élément fini par le déplacement de flexion asymptotique en fond de fissure et par la discontinuité à la traversée de la fissure. Nous présentons deux variantes que nous validons numériquement ainsi que des calculs de facteurs d’intensité de contrainte. INTRODUCTION Very thin plates are widely used for instance in aircraft structures and some of them may present through the thickness cracks. We describe in this paper an adaptation of XFEM (eXtended Finite Element Method) to the computation of thin cracked plates. XFEM is a strategy initially developed for plane elasticity cracked problems (see [16,17]) and is now the subject of a wide literature (among many others, see [2–4,12,18,20] and references therein). It mainly consists in the introduction of the discontinuity across the crack and of the asymptotic displacements into the finite element space. As far as we know, the unique attempt to adapt XFEM to plate models is presented in [10] in which a Mindlin-Reissner model is considered. However, in this reference, the numerical tests are degraded by an important shear locking effect for very thin plates, despite the use of some classical locking-free elements. This suggests that the locking effect is due to the XFEM enrichment. Even though most of the finite element codes are based on the Mindlin-Reissner plate model, the socalled Kirchhoff-Love model provides also a realistic description of the displacement for a thin plate since it is the limit model of the three-dimensional elasticity model when the thickness vanishes (see [7]). For instance, the panels used in aeronautic structures can be about one millimeter thin, for several meters long. On this kind of plate, the shear effect can generally be neglected and consequently the Kirchhoff-Love model is mechanically appropriate. Furthermore, for through the thickness cracks, the limit of the energy release rate of the three-dimensional model can be expressed with the Kirchhoff-Love model solution (see [8] and [9]). 1 MIP-IMT, CNRS UMR 5219, INSAT, Complexe scientifique de Rangueil, 31077 Toulouse, France, [email protected] 2 Université de Lyon, CNRS INSA-Lyon, ICJ UMR5208, LaMCoS UMR5259, F-69621, Villeurbanne, France, [email protected] 3 Université de Toulouse, ISAE, 10 av. Edouard Belin, 31055 Toulouse cedex, France, [email protected] The Kirchhoff-Love model is not submitted to the so-called "shear locking" phenomenon for the finite element discretization, even for a very small thickness. So, the reliability of the numerical results is potentially better compared to the use of the Mindlin-Reissner model, which needs some specific treatments to avoid the locking phenomenon such as a reduced integration (QUAD 4 element, see [15]). Moreover, these treatments are not straightforwardly adaptable to an enriched finite element space. Since the Kirchhoff-Love model corresponds to a fourth order partial differential equation, a conformal finite element method needs the use of C 1 (continuously differentiable) elements. We consider the reduced Hsieh-Clough-Tocher triangle (reduced HCT) and the reduced Fraejis de Veubeke-Sanders quadrilateral (reduced FVS) because they are the less costly conformal C 1 elements (piecewise cubic elements), see [5]. In the XFEM framework, the knowledge of the asymptotic crack tip displacement is required. For a Kirchhoff-Love isotropic plate, it corresponds to the bilaplacian singularities (see [11]). Now, we briefly describe the features of the specific XFEM enrichment. In order to represent the discontinuity across the crack, a “jump function” (or Heaviside function) is used. Then two strategies for the crack tip enrichment are proposed, following the ideas already presented in [12]. In both of them, an enrichment area of fixed size is defined, centered on the crack tip. In the first strategy, the crack tip asymptotic displacements are added to the finite element space multiplied by the shape functions corresponding to the nodes which are in the enrichment area. In the second one, the asymptotic displacements are added without being multiplied by some shape functions. In order to limit its influence, this enrichement is only considered inside the enrichment area. A matching condition is prescribed in order to ensure the (weak) continuity of the displacement and its derivatives across the interface between the enrichment area and the rest of the domain. The paper is organized as follows. Section 1 describes the model problem. Section 2 is devoted to the finite element discretization of the Kirchhoff-Love model. In Section 3, the two enrichment strategies are detailed. In section 4, some numerical results are presented, which illustrate the capabilities of these methods. The last section deals with the computation of the Stress Intensity Factors: these values are used in the crack propagation criteria by mechanical engineers. 1. THE MODEL PROBLEM 1.1. Notations and variational formulation Let us consider a thin plate, i.e. a plane structure for which one dimension, called the thickness, is very small compared to the others. For this kind of structures, starting from a priori hypotheses on the expression of the displacement fields, a two-dimensional problem is usually derived from the three-dimensional elasticity formulation by means of integration along the thickness. Then, the unknown variables are set down on the mid-plane of the plate, denoted by ω. The mid-plane ω is an open subset of R. In the three-dimensional cartesian referential, the plate occupy the space { (x1, x2, x3) ∈ R , (x1, x2) ∈ ω and x3 ∈ ]−ε ; ε[ } . So, the x3 coordinate corresponds to the transverse direction, and all the mid-plane points have their third coordinate equal to 0. The thickness is 2ε (see Fig. 1). Finally, we assume that the plate has a through the thickness crack and that the material is isotropic, of Young’s modulus E and Poisson’s ratio ν. In plate theory, the following approximation of the three-dimensional displacements is usually considered: