*3.1. Abstraction of the Process Geometry Regarding the Profile Load*

Plasticity in the analytical model is assumed to start at a position *x = l<sup>p</sup>* to reach a set bending radius at the center fiber *r*mC for the profile (see Figure 6a). The counter roll is simplified as fixed support, at which the origin of the coordinate system is located in the profile cross-sectional center of gravity. Resulting from the bending, an angle *θ* is generated between the forming force *F* and the position of plasticity onset *lp*. The forming force *F* induces a bending moment *M*b. The partial heating on the profile is assumed to shift the position of the stress-free fiber *y*m (*T*). Additionally, through the forming force *F*, a section force *Q* on the cross-section (Figure 6b) is implied, which results in shear stresses *τ*yz due to the difference between the force application axis position and shear center position. The shear stresses result in a torsion moment *M*<sup>T</sup> leading to a warping deformation with the warping angle α. The alteration of the stress-free fiber position *y*<sup>m</sup> (*T*) is assumed to influence the position of the shear center. *θ τ* α

**Figure 6.** Mechanical model for analytical calculation. (**a**) length direction, (**b**) corresponding cross-sectional cut A-A.

#### *3.2. Assumptions*

For the analytical investigation, it is assumed that a modified version of the elementary bending theory is applicable. The assumptions used from elementary bending theory are [18]


#### *3.3. Calculation of Strains and Strain Rates*

In the continuous push bending process, the material behaves elastically at *x* < *l*<sup>p</sup> and the material can behave partially elastic or fully plastic over the cross-section. The temperature dependent elastic bending strain *ε*el,x can be described as

$$
\varepsilon\_{\rm el,x} = \frac{-y}{r\_{\rm mc}} + a\_{\rm T} \Delta T,\tag{3}
$$

with the loaded bending radius to the center of gravity fiber *r*mC, the thermal expansion coefficient *a*<sup>T</sup> and the difference between room temperature and heating temperature ∆*T* as the compressive zone is in the positive *y*-segment. Elastic shear deformation is neglected (*ε*el,yz = 0) Using Equation (3), the strain free fiber position *y*<sup>f</sup> can be calculated through *ε*el,x(*y*f) = 0. For the plastic area of the profile cross-section, the plastic strains are expressed as

$$
\varepsilon\_{\rm pl,x} = \ln\left(1 + \frac{-y}{r\_{\rm mC}}\right),
\tag{4}
$$

$$
\varepsilon\_{\rm pl,yz} = \tan(\alpha). \tag{5}
$$

Elastic and thermal strain components are neglected if the material behaves plastic as they are considered small compared to the plastic mechanical strain (see Appendix C).

The bending strain rate in the continuous push-bending phase can be expressed as

$$
\dot{\varepsilon}\_{\rm pl,x} = \frac{v\_{\rm f}}{\chi - l\_{\rm p}} \ln \left( 1 + \frac{-y}{r\_{\rm mC}} \right),
\tag{6}
$$

with the profile feed velocity *v*<sup>f</sup> and the *x* position of plasticity onset *l*p. Assuming that strain rates and strains at the position *x* remain constant over time, during the bending, due to constant feed velocity, the Levy–Mises flow rule can be used to calculate the shear strain rate in the plastic segment as

$$
\dot{\varepsilon}\_{\rm pl,yz} = \varepsilon\_{\rm pl,yz} \frac{\dot{\varepsilon}\_{\rm pl,x}}{\varepsilon\_{\rm pl,x}}.\tag{7}
$$
