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.