Issue 39

M. Romano et alii, Frattura ed Integrità Strutturale, 39 (2016) 226-247; DOI: 10.3221/IGF-ESIS.39.22

deformations x n ,  positions x n , ,FE 

(17) and the evaluated or calculated ones by FE-analyzes in longitudinal direction at the relevant

for every substep n 1, , 39   is



     



(23)

x n,

x n ,

x n x n, ,

, kin

, FE

It serves for the identification of the elastic deformation of the plain representative sequence and the longitudinally cut ondulated warp yarn in the FE-model. Results of the analytical model Tab. 3 lists the evaluated results according to Eq. (18) in the w rel - u rel -diagram as absolute values y M ,a  over the selected degrees of ondulation O  of the analytical model. The results are listed in ascending order regarding the corresponding degree of ondulation O  . Fig. 5 graphically illustrates the correlations of the selected degrees of ondulation and the corresponding sensitivity to the mesomechanic kinematic. The consideration of the absolute values of the slopes in the w rel - u rel -diagram y M ,a  allows a graphical illustration of the correlation between degree of ondulation O  and the sensitivity to the mesomechanic kinematic in a linear equidistant scaled diagram and in a double logarithmic scaled diagram. In both cases the results are illustrated separately for the large and for the small interval, and the equations of a hyperbolic approximation of the correlation for the respective interval are indicated. Additionally, the upper bound that can be identified for the exponent -1.5 is visualized by a bold dashed line.

~ M

Slope

in the w rel

- u rel

-diagram

Geometric parameters of the ondulation

a,

y

Analytically for 3 rel 101     u in steps of 4 101  

Analytically for 4 rel 101  

Degree of ondulation O ~

  u in steps of

Amplitude A

Length of ondulation L

5 101  

0.05 0.05 0.05 0.10 0.05 0.10 0.15 0.10 0.05 0.15 0.20 0.10 0.15 0.20 0.25 0.25 0.20 0.15 0.10 0.25 0.20 0.15 0.25 0.20 0.25

15.0 12.5 10.0 15.0 12.5 15.0 10.0 12.5 15.0 10.0 12.5 15.0 12.5 10.0 7.5 5.0 7.5

-2448.20 -1886.70 -1369.90 -914.03 -914.59 -729.59 -540.08 -540.13 -540.14 -369.62 -293.84 -293.67 -229.49 -200.78 -184.74 -127.63 -127.54 -127.64 -127.50

-5040.10 -3288.40 -2047.40 -1143.40 -1141.30 -793.20 -507.26 -507.13 -505.79 -352.36 -285.42 -285.23 -225.43 -198.17 -182.66 -126.94 -126.90 -126.92 -126.90

0.00333 0.00400 0.00500 0.00667 0.00667 0.00800 0.01000 0.01000 0.01000 0.01200 0.01333 0.01333 0.01500 0.01600 0.01667 0.02000 0.02000 0.02000 0.02000 0.02500 0.02667 0.03000 0.03333 0.04000 0.05000

7.5 5.0 7.5 5.0 7.5 5.0 5.0

10.0

-81.45 -71.61 -56.60 -45.86 -31.92 -20.50

-81.28 -71.49 -56.54 -45.83 -31.91 -20.50

~ M of the selected degrees of ondulation O ~ for the two considered intervals of u rel .

Table 3 : Slopes in the w rel

- u rel

-diagram, i. e.

a,

y

240