*2.3. Numerical Model and Material Parameters*

The material in use for the profiles is S500MC. Delivery conditions are according to EN 10140-2 (Table 2).

**Table 2.** Mechanical properties and chemical composition of steel S500MC in delivery condition according to EN 10140-2.


The temperature-dependent Young's modulus and temperature- and strain ratedependent flow curves were characterized through isothermal tensile tests on a Z250 tensile testing machine built by ZwickRoell, Ulm, Germany with induction heating. For the tensile tests, specimens with a parallel length of 30 mm and width of 10 mm are used to achieve a homogenous heating. The specimens are produced through laser cutting of the profile material in delivery condition. The strains are measured using a Maytec PMA-12/1N7-1 Extensometer produced by Maytec, Singen, Germany. Logarithmic strain rates 0.0003, 0.003, 0.03, 0.3, and 0.1 1/s as well as temperatures from 25 ◦C and between 200 and 600 ◦C in 100 ◦C steps were investigated (Figure 5). The specimens are evaluated according to ISO 6892. The software Abaqus 2018 with explicit global time incrementation is used for the numerical FE simulation. The profile is discretized with tri-linear hexahedral elements with full integration (C3D8T). The simulations employ full thermo-mechanical coupling. The deformable profile is meshed with 16 elements over the profile width (2.5 mm thick elements). Isotropic, linear, temperature-dependent elasticity is assumed for the workpiece material. The plastic behavior is modelled as isotropic according to von Mises with temperature and strain-rate dependent hardening. The rolls are modelled as rigid bodies with temperature degrees of freedom. The flow curves for each temperature set are extrapolated using a Tanimura–Voce model (see Appendix A for the model parameters)

$$k\_{\rm f} = \text{C}\_{1} + (\text{C}\_{2} - \text{C}\_{1}) \exp\left(-\text{C}\_{3} \cdot \overline{\varepsilon}\_{\rm pl}\right) + \left(\text{C}\_{4} - \text{C}\_{5} \cdot \overline{\varepsilon}\_{\rm pl} \overset{\leftarrow}{\text{C}}\_{\rm pl}\right) \log\left(\frac{\dot{\overline{\varepsilon}}\_{\rm pl}}{\dot{\overline{\varepsilon}}\_{\rm pl,0}}\right) + \text{C}\_{7} \dot{\overline{\varepsilon}}\_{\rm pl} \overset{\leftarrow}{\text{C}}\_{8} \tag{1}$$

with the flow stress *k*<sup>f</sup> , the material constants *C*<sup>1</sup> to *C*8, the equivalent plastic strain *ε*pl, the equivalent plastic strain rate . *ε*pl, and the initial equivalent plastic strain rate . *ε*pl,0. The temperature dependency of Young's modulus is approximated by () ൎ −0.1579 Kଶ ∙ ∆<sup>ଶ</sup> − 9.5596 K ∙ ∆ + 180098 MPa, ∆

̅ሶ

MPa

MPa

$$E(T) \approx -0.1579 \frac{\text{MPa}}{\text{K}^2} \cdot \Delta T^2 - 9.5596 \frac{\text{MPa}}{\text{K}} \cdot \Delta T + 180,098 \text{ MPa},\tag{2}$$

<sup>ଵ</sup> ଼ ̅<sup>୮୪</sup>

<sup>୮୪</sup> ̅ሶ

୮୪,

where ∆*T* is the difference between room temperature and heating temperature. The maximum error for this approximation is 6%. The Poisson's ratio is set as 0.3.

**Figure 5.** Simulation set-up with geometrical- and material parameters.

Continuous heating and cooling are realized through isothermal rigid strips with high conductivity. This heating method assumes that the temperature is evenly distributed after the material reaches the heating or the cooling zone. The strip for the heated area is at maximum temperature while the strip for the cooling area is at room temperature. The simulation is divided into three steps comparable to the bending tests. These steps are pre-bending, temperature-assisted continuous push bending, and unloading.

#### **3. Analytical Modelling of the Kinematic Push-Bending Phase**

In this segment, the developed analytical procedure is explained. First, the process is abstracted for the simplification of the calculations. Then, the assumptions for the model are displayed. After that, the resulting equations for strains and strain rates are defined. Using these definitions, the methods to evaluate warping angle and bending moment are shown.
