Issue 39

M. Romano et alii, Frattura ed Integrità Strutturale, 39 (2016) 226-247; DOI: 10.3221/IGF-ESIS.39.22 230 Parametric FE-calculations allow the consideration of elastic parts. The numerical investigations focus on plain representative sequences as a two-dimensional verification of the analytic model. Degree of ondulation In order to definitely describe the differently shaped ondulations regarding the respective intensity of ondulation the specific value O ~ is introduced. The non-dimensional value is based on the afore mentioned characteristic and variable geometric parameters amplitude A and lengths L F and L R , respectively. The definition of the degree of ondulation is based on a purely analytical sine        L x A y  2 sin (2) For modeling more sequences in a row it can be repeated in series as often as required, and still yields a continuously differentiable function over the whole domain of definition [27], [28]. For the introduction of the degree of ondulation O ~ the wave steepness S used in nautics, as exemplarily defined in Büsching 2001 [29] L A L H S 2   (3) is modified. In Eq. (3) the geometric parameters are the (absolute) wave height A H 2  and the wave length L . In contrast, following the outlook of Ottawa et al. 2012 [23] and the ideas described in Romano et al. 2014 [24, 25] and elaborated in Romano 2016 [26], the degree of ondulation in fabric reinforced plastics is defined by R F ~ L A L A L A O     (4) where the geometric parameters are the amplitude A , the lengths of the ondulation in the fabric L F and the length of the cross-section of a roving as a fill yarn L R , respectively, and  is a characteristic factor characterizing the type of the fabric. It is for example 2   for a plain weave fabric and 4   for a twill weave 2/2 fabric and can be varied for different fabric constructions. In case of a plain weave fabric it is F R R PL 2 L L L L     (5) Comparison of the structural mechanical motivations The reason for the modification of the wave steepness S in Eq. (3) to the degree of ondulation O ~ in Eq. (4) is based on the different motivations, necessities and intentions of the respective subject area. In nautics it is important to characterize the absolute load a structure undergoes during its life-cycle, e. g. regarding fatigue strength issues. Therefore, twice the amplitude A corresponding to the absolute wave height H is considered in the calculation of the specific value S [29]. In contrast, when fabric reinforced composites are considered, the characterization of the deviation from the ideally orientated unidirectionally reinforced single layer caused by the ondulation in the fabric is focused. The modification towards considering the single value of the amplitude A can also be interpreted as a degree of eccentricity compared to an unidirectionally reinforced single layer that exhibits an ideally orientated reinforcement without ondulations. The degree of ondulation O ~ defined in Eq. (4) is a non-dimensional specific value. It corresponds to the rate of intensity of the continuously differentiable geometric direction change of the ondulation. The value enables the comparability of the representative sequences of the analytic model and the ones of the numeric investigations by FE-analyzes. This is at the same time the requirement for the later described verification. The geometric are chosen in selected and at the same time realistic steps. Whereas the amplitude A is varied from 0.05 mm to 0.25 mm in five substeps of 0.05 mm (0.05 mm; 0.10 mm; 0.15 mm; 0.20 mm; 0.25 mm) the length of the cross- section of a roving as a fill yarn L R is varied from 2.5 mm to 7.5 mm in five substeps of 1.25 mm (2.5 mm; 3.75 mm; 5.0 mm; 6.25 mm; 7.5 mm). According to Eq. (5) for a plain weave fabric this yields lengths of the ondulation in the fabric L F from 5.0 mm to 15.0 mm.