Keywords: finite element method, geometrical non-linearity, fibre-reinforced solids, curved fibre finite elements, tailoring, fibre-matrix conform coupling.

Fibre reinforced solids are usually analysed using a homogeneous analogue which makes it difficult to identify the contact stresses between fibres and matrix. Alternatively, interesting techniques proposing fibre-matrix coupling are present in the literature. Some of them introduce fibres into the continuum by direct mathematical considerations generating solid finite elements that consider fibre properties [1,2,3]. Others introduce fibres as a point to point bar discretization which leads to a difficult mesh generation process. Another strategy is to write fibre node coordinates as functions of solid finite element nodes by means of shape functions and to introduce the strain energy of fibres into the solution process [4]. The last strategy makes it possible to consider fibres at any place in the continuum without increasing the amount of unknowns in the solution procedure. However, in general, it does not guarantee the continuity among the continuum and fibre material between nodes and, as a consequence, this technique is called a tailoring process [1]. This paper is devoted to introducing curved fibres with high order approximations in a continuum media using a tailoring process demonstrating that, if the order of the fibre approximation is at least the same as the solid finite element, the coupling is confirmed and the deficiency of the technique disappears. Moreover the spreading strategy is presented that makes it possible to make a complete analysis of the randomly fibre reinforced materials without increasing the number of variables. Applications for two-dimensional large deformation analysis of elastic bodies using the Green-Lagrange strain and Saint-Venant-Kirchhoff strain energy function show the potential of the proposed methodology.

1

F.K.F. Radtke, A. Simone, L.J. Sluys, "A partition of unity finite element method for simulating non-linear debonding and matrix failure in thin fiber composites", International Journal for Numerical Methods in Engineering, 86(4-5), 453-476, 2011.

2

F.K.F. Radtke, A. Simone, L.J. Sluys, "A partition of unity finite element method for obtaining elastic properties of continua with embedded thin fibres", International Journal for Numerical Methods in Engineering, 84(6), 708-732, 2010. doi:10.1002/nme.2916

3

F.K.F. Radtke, A. Simone, L.J. Sluys, "A computational model for failure analysis of fibre reinforced concrete with discrete treatment of fibres", Engineering Fracture Mechanics, 77(4), 597-620, 2010. doi:10.1016/j.engfracmech.2009.11.014

4

L. Vanalli, R.R. Paccola, H.B. Coda, "A simple way to introduce fibers into FEM models", Communication in numerical methods in Engineering, 24, 585-603, 2008.

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