Fig. 2. 3D Thermal softening function
Assuming now a multiplicative hardening with the coupled effects of strain and temperature included within the softening function, the general Considère condition of eq. (3) must be updated; for the quasistatic case with temperature effect (\(\frac{\partial{\dot{\varepsilon}}_{\text{True}}}{\partial\varepsilon_{\text{True}}}=0\),\({\dot{\varepsilon}}_{\text{True}}=0\) and \(R=1\)) it takes the following form:
\(\sigma_{Eq-S}\bullet S-\frac{\partial\sigma_{Eq-S}}{\partial\varepsilon_{\text{True}}}\bullet S-\sigma_{Eq-S}\bullet\frac{\partial S}{\partial T}\bullet\frac{\partial T}{\partial\varepsilon_{\text{True}}}-\sigma_{Eq-S}\bullet\frac{\partial S}{\partial\varepsilon_{\text{True}}}=0\)\(\rightarrow\ \varepsilon_{N-DT}\) (9)
then eq. (5) only valid if \(S=S\left(T\right)\), is substituted by the more realistic eq. (10) which is based on\(S=S\left(T,\varepsilon\right)\).
\(\varepsilon_{N-T}=\frac{n}{1-\frac{1}{S}\ \bullet\ \left(\frac{\partial S}{\partial\varepsilon_{\text{True}}}+\frac{\partial S}{\partial T}\ \bullet\ \frac{\partial T}{\partial\varepsilon_{\text{True}}}\right)}\)(10)
The significant improvement of eq. (10) with respect to eq. (5) is that, thanks to the incorporation of the strain-temperature coupling within\(S\left(T,\varepsilon\right)\) and to the related term\(\frac{\partial S}{\partial\varepsilon_{\text{True}}}\), the former equation recognizes the necking onset anticipation due to constant temperatures higher than \(T_{\text{Room}}\), which eq.(5) was not able to capture.
The obtained quasi-static thermal softening is now applied to the dynamic tests, for also predicting the corresponding changes of the necking strain. The anticipation effect due to the temperature will be in contrast to the delay effect caused by the strain rate variation, and the analysis of dynamic experiments will show which one of the two effects will prevail.
Analysis of the necking phenomenon in dynamic tests with combined strain rate and temperature effects
During dynamic SHTB tests, both temperature and strain rate undergo significant variations. In fact the temperature greatly increases due to the plastic work generated in almost adiabatic conditions, while the plastic strain rate evolves from zero at first yield up to the regime value at plateau.
Then, also the necking inception strain can remarkably change with respect to \(\varepsilon_{N}\) from static tests at \(T_{\text{Room}}\), because of the combined effect from the thermal softening and from the dynamic amplification of the stress.
Such combined effect is very clear in the left part of Fig. 3, where the dynamic true curves are compared to the quasi-static ones at room temperature.