Issue 39

M. Romano et alii, Frattura ed Integrità Strutturale, 39 (2016) 226-247; DOI: 10.3221/IGF-ESIS.39.22 231 The selected values can be considered realistic, as described in detail for the comparable sets of specimens in the experimental validations carried out in Ottawa et al. 2012 [23], Romano et al. 2014 [24, 25] and Romano 2016 [26] or reported by Ballhause 2007 [11], Matsuda et al. 2007 [12] and Kreikmeier et al. 2011 [20]. For a plain weave fabric according to Eq. (5) with 2   and the afore listed geometric parameters yield values of the degree of ondulation O ~ according to Eq. (4) lying in the range of 0.0033 and 0.0500. Fig. 1 right illustrates the degree of ondulation O ~ over a plane spanned by the axes for amplitude A and length of the ondulation in a plain weave fabric L PL with isolines. 0,000 0,010 0,020 0,030 0,040 0,050 5,0 10,0 15,0 Degree of ondulation Õ = A / L PL h SL,n =4 A L R L PL =2 L R h R =2 A A A 0 L /4 L /2 3 L /4 L h SL,a =2 A . . . . . . . . . Figure 1 : Representative sequences of one complete ondulation based on the characteristic geometric parameters amplitude A and length L of a sine in the analytic model (left top) and for the numerical investigations with FE-calculations (left bottom). Graphical illustration of the degree of ondulation O ~ over a plane spanned by the axes of the geometric parameters amplitude A and length of the ondulation in a plain weave fabric L PL with isolines (right). A NALYTICAL MODEL AND PROCESSING he investigations of the purely analytic model are described with its presumptions, processing and evaluation. Mathematical model and presumptions The mathematical model for the analytical description of ondulations in fabric reinforced composites reduces the complexity down to a one-dimensional problem. Therefore, only the centerline of an ondulated yarn is considered. The obtained mesoscopic geometry is thus an analytical sine. For a representative sequence of one complete ondulation according to Eq. (2) the argument of the sine is presumed in the interval     x L 2 0, 2    that leads to the domain of definition   x L 0,  . The advantages of a trigonometric function are steadiness, differentiability and integrability of its shape on the whole domain of definition [27, 28]. Furthermore the representative domain of definition can be reduced to the wavelength of one complete ondulation, i. e.   D L 0,  because of symmetry properties. With arbitrary amplitude A and   x L 0,  the representative domain of the argument in the trigonometric function   D 0, 2   is obtained. In the analytical approach the ondulated yarn is assumed not to lengthen or shorten due to strain or compression, but remaining constant in length. Mechanically the presumption of an ideally stiff and at the same time ideally flexible yarn can be stated in terms of an infinite high Young’s modulus in longitudinal direction and a negligible flexural modulus transverse to it, E E L 1   and E E T 2 0   (6) T