Table 3 Exemplary evolution of reference strain for repeated cooling and heating assuming constant strains during solidification
Time | Temperature | Reference strain | \(r_p\) | \(r_s\) | \(r_m\) |
|---|---|---|---|---|---|
\(t^0\) | \(T < T_s\) | \(\varvec{\varepsilon }_\text {ref}^0 = 0\) | 1.0 | 0.0 | 0.0 |
\(t^1\) | \(T = 0.5(T_l+T_s)\) | \(\varvec{\varepsilon }_\text {ref}^1 = 0\) | 0.5 | 0.0 | 0.5 |
\(t^2\) | \(T < T_s\) | \(\varvec{\varepsilon }_\text {ref}^2 = \frac{1}{0.5} 0.5 (\varvec{\varepsilon }^{1} - \varvec{\varepsilon }_T^{1})\) | 0.5 | 0.5 | 0.0 |
\(t^3\) | \(T = 0.25T_l +0.75T_s\) | \(\varvec{\varepsilon }_\text {ref}^3 = \frac{1}{0.25} 0.25 \varvec{\varepsilon }_\text {ref}^2 = \varvec{\varepsilon }_\text {ref}^2\) | 0.5 | 0.25 | 0.25 |
\(t^4\) | \(T < T_s\) | \(\varvec{\varepsilon }_\text {ref}^4 = \frac{1}{0.5} (0.25 \varvec{\varepsilon }_\text {ref}^3 + 0.25(\varvec{\varepsilon }^3 - \varvec{\varepsilon }_T^3)) =\) \( \frac{1}{2}\varvec{\varepsilon }_\text {ref}^3 +\frac{1}{2}(\varvec{\varepsilon }^3 - \varvec{\varepsilon }_T^3) \) | 0.5 | 0.5 | 0.0 |
\(t^5\) | \(T = 0.75T_l +0.25T_s\) | \(\varvec{\varepsilon }_\text {ref}^5 = \lim _{r_s \rightarrow 0}\frac{r_s}{r_s}\varvec{\varepsilon }_\text {ref}^4 = \varvec{\varepsilon }_\text {ref}^4\) | 0.25 | 0.0 | 0.75 |
\(t^6\) | \(T < T_s\) | \(\varvec{\varepsilon }_\text {ref}^6 = \frac{1}{0.75}(0\cdot \varvec{\varepsilon }_\text {ref}^5 + 0.75(\varvec{\varepsilon }^5 - \varvec{\varepsilon }_T^5)) = \varvec{\varepsilon }^5 - \varvec{\varepsilon }_T^5\) | 0.25 | 0.75 | 0.0 |