Stiffness modelling method of inclusions

Due to the inherent symmetry of as-produced textile fabrics, woven composite plates are orthotropic. Their stiffness can be represented by engineering constants Young s modules shear modules Gy and Poisson coefficients py, ij= 1.3. If the reinforcement is deformed during production of the composite, or if the preform is net shaped, or for some knitted performs, then the assumption of orthotropy does not necessarily apply and the full stiffness matrix has to be introduced. 

Impregnated yams are considered as composites with unidirectional reinforcement (twist of the yarns used in composite reinforcement is normally negligible) with the given fibre volume fraction Vj this value is determined by the dimensions of the cross-section of the yam (which can differ from point to point in the unit cell, but most often assumed to be constant) and the amount of fibres inside the yam, given by the fibre count or yam linear density  

With all the simplicity of the OA method it produces reasonable predictions for the fibre-dominated properties and low crimp, when deviations from the iso-strain assumption play a minor role. For off-axis properties, or for cases when transverse parts of the reinforcement (e.g. weft yams for warp-direction loading) play an important role, or in the presence of high-crimp yams (e.g. 3D reinforcements with significant fraction of binder), the iso-strain assumption is not valid any more, and quality of the OA predictions is decreased. There are approaches that combine iso-strain and iso-stress formulations [3], but these approaches leave open the choice of the combination mles open, which makes their predictive abilities limited. 

This chapter describes the method of the stiffness homogenisation for textile composites based on Eshelby solution of the elastic problem for an ellipsoidal inclusion and Mori-Tanaka homogenisation scheme. The approach was proposed by Huysmans et al. [91—94] and is successfully applied to very different textile composites, woven [95], braided [96] and knitted [91,92]. In short, in the following discussion the approach will be called method of inclusions (Mol). 

Mol homogenisation of a heterogeneous medium, which should have a homogeneous matrix with a second (discontinuous) phase, or more phases of reinforcement embedded in it, is done via the following steps  

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