**Appendix C**

For the validation of the neglected elastic and thermal strains in the plastic region for the analytical model, the elastic and total strain resulting from the numerical simulation are evaluated (Figure A2). For room temperature, the error resulting from neglecting the elastic strains is 6%. From 200 ◦C, each increase of 100 ◦C increases this error by approximately 1% as thermal strain increases. The maximum error from excluding thermal and elastic strains occurs at 600 ◦C partial heating temperature. The resulting error for the strains is 10%.

**Figure A2.** Difference between plastic strain and total strain. (**a**) evaluated position, (**b**) total strain and plastic strain evaluated through FE simulation for different partial heating temperatures.

## **Appendix D**

The slope of a curve *f* (*x*) at a position *x* is defined using the relations

$$m = \frac{df(x)}{dx}.\tag{A1}$$

 <sup>௫</sup> As the slope of *f* (*x*) can be dependent on the position *x* a mean value of the slope can be calculated with

()

$$
\overline{m} = \frac{\int\_{\mathcal{X}} m d\mathbf{x}}{\Delta \mathbf{x}}.\tag{A2}
$$

୮,୰൫୮୪̅ , ൯

Plastic strain and strain rates are dependent on the *y*-axis position in the cross-section. Consequently, to calculate a mean slope of the flow curve, a mean value over strains and strain

, )<sup>୮୪</sup> ሶ̅

.

.

+ න න ୷,୦

୷,ୖ ୷,୦

ଶ

(ଶ + (−)

4( + (−))

ቀ ଶ ି௭బି௧ቁ

ିቀ ଶ ି௭బቁ

− ൰ ൬

 ଶ ି௬బ

ିቀ ଶ ା௬బቁ

+ (−)

ଶ) ଶ

2( + (−))

ଶ

− 2 ൰

 ୮୪̅ (୮୪̅ , <sup>୮୪</sup> ሶ̅

∆<sup>୮୪</sup> ሶ̅

∆୮୪̅

୮,୰൫୮୪̅ , ൯ቇ ୮୪̅

ቆ

ఌത౦ౢ

=න න ୷,ୖ

 ଶ ି௬బ

 ଶ ି௬బି௧

∙ ൬ −

ଶ

<sup>ଷ</sup> 3 +

+ (−)

( − )<sup>ଷ</sup> 3

2( + (−))

ଶ

−

ఌതሶ౦ౢ

୮,୰൫୮୪̅ , ൯ =

୮ () =

> ଶ ା௭బ

ିቀ ଶ ି௭బି௧ቁ

௬௭,ୖ =

rates is necessary to approximate the hardening behavior. Using Equations (A1) and (A2), the mean slope of the flow curve over the strain rate *<sup>E</sup>*p,r *ε*pl, *T* can be calculated to:

$$E\_{\rm p.r} \left( \overline{\varepsilon}\_{\rm pl.}, T \right) = \frac{\int\_{\dot{\overline{\varepsilon}}\_{\rm pl.} \, \frac{d}{d\overline{\varepsilon}\_{\rm pl.}} k\_{\rm f} (\overline{\varepsilon}\_{\rm pl.} \dot{\overline{\varepsilon}}\_{\rm pl.} \, T) \, d\dot{\overline{\varepsilon}}\_{\rm pl.}}{\Delta \dot{\overline{\varepsilon}}\_{\rm pl.}}. \tag{A3}$$

Using Equation (A3), the mean slope of the flow curve over strain and strain rate is

$$E\_{\rm p}(T) = \frac{\int\_{\overline{\varepsilon}\_{\rm pl}} \left( \frac{d}{d\overline{\varepsilon}\_{\rm pl}} E\_{\rm p,r} \left( \overline{\varepsilon}\_{\rm pl}, T \right) \right) d\overline{\varepsilon}\_{\rm pl}}{\Delta \overline{\varepsilon}\_{\rm pl}}. \tag{A4}$$

With the mean slope of the flow curve, the relation between shear stresses and shear strain rate can be approximated.
