## 2 Numerical Modelling
It is a common practice to obtain those states from numerical simulation of the cold drawing process by FE [6, 8, 10], and experimentally by means of synchrotron and neutron diffraction techniques [25], and more recently by laser ultrasonic [26]. In both procedures, similar values were obtained for conventional prestressing steels. In this paper, to achieve an estimation of those states, a numerical simulation of a real drawing chain was carried out by means of a commercial FE code (MSC.Marc(r)). Accordingly, the wire section is progressively reduced from an initial diameter of 12 mm to a final diameter of 7 mm during seven drawing steps, according to the following sequence (cf. Figure 2a): \(d_{0}\) = 12.00 mm, \(d_{1}\) = 10.80 mm, \(d_{2}\) = 9.75 mm, \(d_{3}\) = 8.90 mm, \(d_{4}\) = 8.15 mm, \(d_{5}\) = 7.50 mm, \(d_{6}\) = 7.00 mm. The revolute symmetry allows for simplification of the geometry to an axisymmetric case, with the consequent savings in computing time. Four-node quadrilateral elements were selected for meshing the wire due to its simple geometry. Diverse meshes were tested until the required convergence in results was achieved. It is worth remarking that the results of this simulation reveal the residual stress state (Figure 2b) that will be used later as an input in the numerical simulation of the in-service loading considering both stress states: the residual and that caused by in-service loading. The same mesh was used in both simulations. Figure 1: Scheme of the cyclic loading: (**a**) Without overload; (**b**) Overload at the first loading cycle; (**c**) Overload at an intermediate loading cycle; and (**d**) Overload at the final loading cycle. To achieve the required high accuracy in the radial dimensions of the prestressing steel wire after cold drawing process, the dimensions of the drawing dies at the contact zone must remain the same during the plastic straining of the cold drawn wire. Thus, the drawing dies must not undergo plastic strains or wear, and the dies must also undergo low elastic strains. Taking these design requirements into account, the drawing dies are commonly made of high strength and stiffness ceramics materials such as tungsten carbides. Accordingly, in this study, the drawing dies of the drawing chain shown in Figure 2a are modelled as rigid bodies [17]. Two parameters are needed to define the geometry of the drawing die: the inlet die angle (\(\alpha\)) and the bearing length (\(l_{\mathrm{z}}\)) [10]. Common values used in industry for both parameters (\(\alpha=7^{\circ}\) and \(l_{\mathrm{z}}=d_{0}\)) were chosen for each of the drawing steps of the considered drawing chain (labeled as A in Figure 2). The boundary conditions applied in the FE modelling are the following: a prescribed axial displacement imposed on the front extreme of the rod, and null displacement imposed to those nodes placed at the symmetry axis of the wire. Elastoplastic analysis was performed using large deformations and large strains, with updated Lagrangian formulation. The simple geometry of the wire in the 2D axisymmetric approach (a rectangle) allows the consideration of a simple mesh of four-node quadrilateral elements. During cold drawing, the outer surface of the wire is in contact with the die, thereby undergoing plastic strains. In this way, the meshing of the wire was locally refined at the contact surface. Thus, on one hand, the size of the quadrilateral elements is progressively reduced in the radial direction, and on the other hand, the same size of the elements in the axial direction was used. Thus, the elements of the zone where the main variations are expected to appear (near the contact zone between the wire and the drawing die) have a lower size. According to the results, the wire at the end of the process exhibits a uniform distribution of axial stress in the axial direction (Figure 2b).