effect of contiguity on shear elastic modulus of fibre reinforced ...

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14 Raluca Hohan, Liliana Bejan and Nicolae Ţăranu The shear stresses are equal in fibres, matrix and composite and the compatibility <strong>of</strong> shear deformations is assured (Gibson, 2012). The in-plane shear modulus, G12, determined on the model is defined by relation G 12 τ = γ where τ12 is average composite shear stress in the (1,2) plane and γ12 is the average engineering shear strain in the same plane. Using the model given in Fig. 5, a formula based on the inverse rule <strong>of</strong> mixtures has been deduced G 12 12 GG f m 12 = , GV m f + GV f m Fig. 6 – Variation <strong>of</strong> G12 by inverse rule <strong>of</strong> mixtures compared to extreme <strong>contiguity</strong>. and the corresponding shear modulus values are illustrated in Fig. 6, the bottom curve, where G12 is the in-plane shear modulus <strong>of</strong> the composite, Gf – the shear modulus <strong>of</strong> fibres, Gm – the shear modulus <strong>of</strong> matrix; Vf and Vm – the fibres and matrix volume fractions, respectively. , (5) (6)
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 15 The utilized model for G12 does not give precise results because it is based on many simplifications. In addition, the shear modulus values determined experimentally are significantly higher than those determined with eq. (6). To obtain more accurate results complex analyses have been performed to give convenient predictions (Mathews et al., 2008). Some attempts have focussed on improving the inverse rule <strong>of</strong> mixture formula for shear loading, one <strong>of</strong> them based on <strong>contiguity</strong>. The approach that considers the <strong>contiguity</strong> has been analysed by Tsai (Jones, 1999) who has developed the following formulas for the shear modulus: ( ) ( ) ( ) ( ) ( ) ( ) 2Gf− Gf − Gm Vm Gf + Gm − Gf −Gm Vm G12 = ( 1− C) Gm + CG f . 2G + G − G V G + G + G − G V m f m m f m f m m The <strong>contiguity</strong> factor, C, takes values between 0 and 1 (0 in case <strong>of</strong> totally separated fibres (Fig. 4 a) and 1 for total fibres <strong>contiguity</strong> (Fig. 4 c)). An averaging factor has been used by the authors to determine the intermediate stiffness values, considering the square array <strong>of</strong> reinforcing fibres (C0.785) and the triangular array (C0.907) respectively. These averaging factors take into account the <strong>effect</strong>ive fibre volume fractions C V 0785 . f 0. 785 = , C 0907 . V f = . 0. 907 Figs. 7 a and 7b represent the influence <strong>of</strong> <strong>contiguity</strong> on shear modulus, considering the averaging factor <strong>effect</strong>s (C0.785 and C0.907) versus extreme values <strong>of</strong> <strong>contiguity</strong> (C = 0 and C = 1). The specialized literature (Cooke, 1995) provides the upper (superior limit) and lower (inferior limit) bounds <strong>of</strong> shear modulus; the numerical results, calculated with these formulas, coincide with those obtained from formula (7) in which the extreme <strong>contiguity</strong> values have been introduced (Fig. 8) V f G12 − = Gm+ 1 Vm + G − G 2G f m m , Vm G12 + Gf 1 V f G G 2G = + + − m f f . (7) (8) (9)
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