Keywords: finite element, interlaminar stresses, free edge, multiparticle model.

This paper presents a finite element model based on the Multiparticle Model of Multilayered Materials (M4) [1,2,3] developed in Institut Navier-LAMI during the last years. The laminated plate is considered as a superposition of Reissner-Mindlin plates, in taking the shear and moment resultant, in each layer into account, along with interlaminar shear and normal stresses at the interface ( equilibrium equations). The various studies of Caron and Ehrlacher [1], Naciri et al. [2], Carreira et al. [3] are inspired from the work of Pagano [4] with more simplified approaches. These works validate the multiparticle approach by comparison with 3D finite element (FE) and experiments. The proposed finite element model is capable of computing interlaminar stresses and other localized effect wich is impossible with classical 2D finite element model. This multiparticle finite element presents a 2D type data structure that provides several advantages over a conventional 3D finite element model (simplified input data, ease of mesh alteration). Moreover, the present model provides finite values of interlaminar stresses, even on the free edges of a structure. In fact, it is well know that 3D FE results present a stress singularity. In the 3D FE analysis, the magnitude of the interlaminar stresses depends strongly on the refinement of the mesh (see [5]). On the contrary, according to [3] stress singularity at the free edge does not exist with the proposed model. The physical meaning of these finite interlaminar stresses has been justified in [6]: a limit stress criterion using these finite values on the edge predicts the delamination onset with accuracy.

We developped here an eight-node isoparametric multiparticle element. This element has d.o.f per node ( is the layer's number of the laminate), thus giving a total d.o.f per element. The interpolated displacement variables of this element are biquadratic in and . The interlaminar stresses are given directly in a straight forward manner using constitutive equations without postprocessing works. A finite element program called MPFEAP (MultiParticle Finite Element Analysis Program) has been developped for the implementation of the multiparticle element.

In order to illustrate the convergence characteristics and the accuracy of the proposed multiparticle element, the classical exemple of straight free edge problem has been carried out. The finite width laminates under uniaxial tension has been studied with different stacking sequences: and . The analytical solution [3,7] is reached even using a coarse mesh. The MPFEAP results are also compared with M4 analytical solution [7], with 3D FE calculations [3] and with Pagano's local model [4]. A high level of correlation between all the curves are shown. In the second set of exemples, an analysis of laminates with a central circular hole is realized. The tangential interlaminar shear stress distributions around the hole is presented and is compared with the 3D FE calculations in [8,9,10]. In spite of the difference in magnitude, all the models provide similar curve shapes and have their largest values at about from the load axis. The different values is surely due to the singularities at the edge for the 3D solution fields.

1

J.F. Caron, A. Ehrlacher "Modelling the kinetics of transverse cracking in composite laminates", Compos. Sci. Technol., 57, 1261, 1997. doi:10.1016/S0266-3538(97)00059-6

2

T. Naciri, A. Ehrlacher, A. Chabot "Interlaminar stress analysis with a new multiparticle modelization of multilayered materials (M4)", Compos. Sci. Technol., 58, 337, 1998. doi:10.1016/S0266-3538(97)00085-7

3

R.P. Carreira, J.F. Caron, A. Diaz Diaz "Model of multilayered materials for interface stresses estimation and validation by finite element calculations", Mechanics of Materials, 34, 217-230, 2002. doi:10.1016/S0167-6636(02)00102-3

4

N.J. Pagano, "Stresses fields in composite laminates", Int. J. Solids Struct., 14, 385-400, 1978. doi:10.1016/0020-7683(78)90020-3

5

I.S. Raju, J.H. Crews "Interlaminar stress singularities at a straight free edge in composite laminates", Comput. Struct., 14, 21, 1981. doi:10.1016/0045-7949(81)90079-1

6

J.F. Caron, R.P. Carreira, A. Diaz Diaz "Critèeres d'initiation de délaminage dans les tratifiées", C. R. Acad. Sci. Paris, t. 327 (séerie II b), 1291, 1999.

7

A. Diaz Diaz, J.F. Caron, R.P. Carreira "Software application for evaluating interfacial stresses in inelastic symmetrical laminates with free edges", Composite Structures, 58, 195-208, 2002. doi:10.1016/S0263-8223(02)00123-X

8

I.S. Raju, J.H. Crews "Three-dimensional analysis of (0/90)s and (90/0)s laminates with a central circular hole", Composites Technology Review, 4, 116-124, 1982.

9

R.P. Carreira "Validation par éléments finis des modèles multiparticulaires des matériaux multicouches", PhD Thesis, Ecole Nationale des Ponts et Chaussées, Marne la Vallée, France, 1998.

10

W.M. Lucking, S.V. Hoa, T.S. Sankar "The effect of geometry on interlaminar stresses of (0/90)s composite laminates with circular holes", Journal of Composite Materials, 17, 188, 1984. doi:10.1177/002199838401800207

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