## 2 Numerical Modelling
However, in the radial direction of the wire, axial residual stress is non-uniformly distributed with tensile stress (T\({}^{+}\)) near the wire surface and compressive stress (C\({}^{-}\)) in the wire core (Figure 2c). Figure 2: (**a**) Scheme of the real manufacture process by multi-step wire drawing; (**b**) Residual stress field after the last step of cold drawing in the wires; and (**c**) Radial distribution of axial residual stress (T\({}^{+}\), tensile stress and C\({}^{-}\) compressive stress). Regarding the second numerical simulation, the same commercial FE code (MSC.Marc(r)) was used to carry out the simulation of the cyclic loading applied to the prestressing steel wires. Due to the axisymmetric geometry of the wire, the three-dimensional (3D) case is simplified to an equivalent two-dimensional (2D) problem by applying null displacement in the radial direction (\(\delta_{\mathrm{r}}\)) at the symmetry axis of the wire. Accordingly, a null displacement in the axial direction (\(\delta_{\mathrm{x}}\)) was imposed at the symmetry central transversal plane of the wire (Figure 3). In this way, an important saving in computing time is achieved in simulations. Finally, a cyclic loading (\(\sigma_{\mathrm{app}}\)) corresponding to each of the previously depicted cases of study (Table 1) was applied at one edge of the wire. In order to include the previously obtained residual stress profile in the numerical simulations of in-service cyclic loading, the same mesh used in the numerical simulation of wire drawing was considered: the FE size progressively decreases as the radial coordinate \(r\) increases. Thus, the mesh is more refined in the tensile zone (T\({}^{+}\) in Figure 2c), where the main variations in the residual stress were observed, whereas in the wire core (where less significant variations appear), a coarse mesh is suitable. Thus, an accurate approach to the residual stress profile (Figure 2c) can be achieved by assigning an initial stress value to each element (constant in each finite element). In wire drawing simulations, the base material at the beginning of the manufacturing chain is a hot-rolled pearlite steel whose chemical composition is: 0.740% C, 0.700% Mn, 0.200% Si, 0.015% P, 0.023% S. Experimental tensile tests up to fracture were carried out to obtain the stress-strain curve (blank points in Figure 4), according to which [PERSON]'s modulus \(E=199\) GPa and 0.2% offset yield strength \(\sigma_{\mathrm{Y}}=710\) MPa were obtained. In the case of the simulations of cyclic loading that were carried out, the constitutive model for the material was chosen to be elasto-plastic with isotropic strain hardening rule given by the master stress and strain curve obtained experimentally from a simple tensile test (full points in Figure 4). Results provide the following material properties: Young modulus, \(E=200\) GPa and 0.002% offset yield stress, \(\sigma_{\mathrm{Y}}=1300\) MPa. Figure 4: Master stress versus strain curves of a hot-rolled bar (initial product before cold drawing) and a prestressing steel wire (final commercial product after cold drawing in six steps). Figure 3: Scheme of the boundary conditions applied in the numerical simulation of the in-service loading considering residual stress induced by cold drawing of the prestressing steel wires.