Keywords: second moment reliability methods, laminates, meso-macro modelling, damage, (visco)elasticity, (visco)plasticity, incremental laws, failure probability.

Because of the lack of knowledge of the long-term composite materials behaviour, the building of sufficiently reliable composite structures needs designers to use very high safety coefficients. As a result, the composite solution in leading industries, aeronautics for instance, looses a large part of its interest because of over sizing. This article proposes an objective reliability index for damaged composites with strongly non-linear behaviour. To calculate this index, a reliability computation algorithm is coupled with a finite element code in which the integration of the finite strain laminate behaviour is perfectly assessed, to minimize the errors and uncertainties of the modelling response.

Within the context of a mesoscopic approach to model the non-linear behaviour of laminate composites made of a polymer matrix reinforced with long fibres, the simple ply constitutive law, including damage, (visco)plasticity and (visco)elasticity by Boubakar et al. [1,2] is recalled. Its integration within a context of space-shell finite elements has been realised using a multi-step prediction-correction algorithm. Modelling the identification parameters has been executed using the method of Carbillet [3] with a hybrid algorithm and the experimental data is obtained using cylindrical tubes subject to traction and internal pressures.

To evaluate laminate reliability, one can calculate its failure probability, defined as the joint probability density integral on the failure domain in the X-space (physical parameters space), by taking into account the failure scenarios and the nominal parameter uncertainties. An estimation of the failure probability can be obtained using a second-moment method such as the first order reliability method (FORM) due to Breitung [4], which determines the first order reliability index due to Hasofer et al. [5]. In order to compute , the Y-space of the reduced centered Gaussian variables is employed. To assess the Y-space from the X-space, the Nataf [6] transformation is used. The value of and the conception point (failure surface point) are classically obtained by solving a minimisation problem using the Rackwitz-Fiessler algorithm [7]. To introduce the space conception parameters' variability, an exponential type autocorrelation is applied.

The numerical integration parameters' influence, on the reliability index, is studied in the case of a laminated squared tube [+55, -55, +55, -55, 90] clamped at one of its edges and loaded by a deviated force acting on the other edge. The geometric, loading and material data are defined as being Gaussian uncorrelated random variables.

1

M.L. Boubakar, F. Trivaudey, D. Perreux, L. Vang, "A meso-macro finite element modelling of laminate structures, Part I : Time-independent behaviour", Composite Structure, 58, 271-286, 2002. doi:10.1016/S0263-8223(02)00049-1

2

M.L. Boubakar, L. Vang, F. Trivaudey, D. Perreux, "A meso-macro finite element modelling of laminate structures, Part II : Time-dependent behaviour", Composite Structure, 60, 275-305, 2003. doi:10.1016/S0263-8223(03)00012-6

3

S. Carbillet, "Contribution aux calculs fiabilistes sur des structures composites", Thèse de l'Université de Franche Comté no1063, 2005.

4

K. Breitung, "Asymptotic approximation for multinormal integrals", Journal of Engineering Mechanics, 110(3), 1984.

5

A.M. Hasofer, N.C. Lind, "Exact and Invariant Second Moment Code Format", Journal of Engineering Mechanics Division, ASCE, 100, 111-121, 1974.

6

A. Nataf, "Détermination des distributions dont les marges sont données", Comptes rendus académie des Sciences, 225, 42-43, 1962.

7

R. Rackwitz, B. Fiessler, "Structural reliability under combined load sequences", Computer. Struct, 9(5), 489-494, 1978. doi:10.1016/0045-7949(78)90046-9

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