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Table 3 Exemplary evolution of reference strain for repeated cooling and heating assuming constant strains during solidification

From: A simple yet consistent constitutive law and mortar-based layer coupling schemes for thermomechanical macroscale simulations of metal additive manufacturing processes

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