# Geometrical and Mechanical Modeling of Polymeric Multi-Ply Yarns
## 3 Results
### Tensile Tests on Two-Ply Griege and Dipped Cords  
To simulate the behavior of cords, we first compute the distribution of the mean fiber inclination on an equivalent cylinder, following the procedure described in Section 2.1. This mean inclination is then assigned to each Gauss point of the mesh of the equivalent cylinder employed in numerical analyses. The material parameters previously identified using results on single-ply yarns (as in Table 1) are adopted. Figure 7 shows the comparison of experimental (dashed lines) and numerical (solid lines) results in terms of global force vs strain curves for uniaxial tensile tests conducted at different strain rates. A good agreement is observed for strain below 0.1. The model is able to predict the correct stiffness of the cord both in the initial linear elastic part and in the following non-linear elasto-plastic one. For very large strains, the experimental curves show a significant locking effect due to the variation in fiber inclination occurring at large imposed cord elongation which cannot be predicted in the small strain frame adopted in our modeling. Figure 8a shows the contour plot of the axial stress on the cross-section for \(\varepsilon=0.1\) obtained with an axial-symmetric coarse mesh and a structured refined mesh. One can observe a good agreement between the results obtained with the two meshes. It is worth noting that a small central region of the cord is subject to compressive stresses. This counter-intuitive effect is the result of the fiber inclination, which is higher in the central part of the cord than in the external one. The variation of the axial stress along the radius for different imposed strains (marked by dots in Figure 7) is shown in Figure 8b. The effect of the reduced stiffness of the fibers in compression, which limits the compressive stress value, is visible. The boundary between the compression and the tension regions evolves during the cord stretching due to the progressive inception of plasticity (from the external layers to the internal ones). This progressive development of plastic strain can be seen in Figure 9 which shows the contour plots of the plastic strain in the fiber direction at different imposed strains marked by the letters A, B, C and D in Figure 7.  
The effect of dipping on the uniaxial mechanical response of cords has been also considered. During the dipping process, the adhesive material penetrates the external layers of fibers for a thickness of two or three filaments, visible in the microscopic image  
Figure 7: Uniaxial tensile test of greige cords C2, comparison between experimental data (dashed lines) and numerical results (solid lines) at strain rates of 0.017 s\({}^{-1}\), 0.003 s\({}^{-1}\) and 0.0003 s\({}^{-1}\). of Figure 10a as darker filaments. This treatment increases the stiffness of the cord and reduces the failure strain. Figure 10b compares the experimental uniaxial response of greige (in blue, dashed) and dipped (in orange, dashed) for a test at \(\dot{\epsilon}=0.003\) s\({}^{-1}\). Figure 8: (a) Axial stress distribution on the cross-section of cord C2 corresponding to \(\epsilon=0.1\), point D in Figure 7 (tension test at \(\dot{\epsilon}=0.003\) s\({}^{-1}\)) with refined and coarse mesh; (b) axial stress versus radial position at different strain levels, points A–D in Figure 7, obtained with the refined mesh. Figure 10: (a) Microscopy image of the cross-section of dipped cord DC2 inside rubber matrix; (b) uniaxial tensile test on greige and dipped cords, C2 and DC2, at a strain rate of 0.003 s\({}^{-1}\): experimental data (dashed lines) and numerical results (solid lines). Figure 9: Evolution of the plastic strain in the fiber direction in tension test of cord C2 at \(\dot{\epsilon}=0.003\) s\({}^{-1}\): contour plots at strain levels of points A, B, C and D in Figure 7.  
To simulate the effect of dipping we consider enhanced material properties in the external part of a cylinder, in a layer of 36 um thickness, in agreement with the microscopy image in Figure 10a.