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Table 4 Exemplary evolution of reference strain for a cyclic, partial remelting of solid phase assuming \(\varvec{\varepsilon }- \varvec{\varepsilon }_T\) is constant in all phase change intervals

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 \(r_s\) \(r_m\)
\(t^0\) \(T < T_s\) \(\varvec{\varepsilon }_\text {ref}^0 = 0\) 1.0 0.0
\(t^1\) \(T = 0.5(T_l+T_s)\) \(\varvec{\varepsilon }_\text {ref}^1 = 0\) 0.5 0.5
\(t^2\) \(T < T_s\) \(\varvec{\varepsilon }_\text {ref}^2 = \frac{1}{1} (0.5 \varvec{\varepsilon }_\text {ref}^1 + 0.5 (\varvec{\varepsilon }- \varvec{\varepsilon }_T)) = \frac{1}{2} (\varvec{\varepsilon }- \varvec{\varepsilon }_T)\) 1.0 0.0
\(t^3\) \(T = 0.5(T_l+T_s)\) \(\varvec{\varepsilon }_\text {ref}^3 = \frac{1}{0.5} 0.5 \varvec{\varepsilon }_\text {ref}^2 = \varvec{\varepsilon }_\text {ref}^2\) 0.5 0.5
\(t^4\) \(T < T_s\) \(\varvec{\varepsilon }_\text {ref}^4 = \frac{1}{1} (0.5 \varvec{\varepsilon }_\text {ref}^3 + 0.5(\varvec{\varepsilon }- \varvec{\varepsilon }_T)) = \frac{3}{4}(\varvec{\varepsilon }- \varvec{\varepsilon }_T)\) 1.0 0.0
\(t^5\) \(T = 0.5(T_l+T_s)\) \(\varvec{\varepsilon }_\text {ref}^5 = \frac{1}{0.5} 0.5 \varvec{\varepsilon }_\text {ref}^4 = \varvec{\varepsilon }_\text {ref}^4\) 0.5 0.5
\(t^6\) \(T < T_s\) \(\varvec{\varepsilon }_\text {ref}^6 = \frac{1}{1} (0.5 \varvec{\varepsilon }_\text {ref}^5 + 0.5(\varvec{\varepsilon }- \varvec{\varepsilon }_T)) = \frac{7}{8}(\varvec{\varepsilon }- \varvec{\varepsilon }_T)\) 1.0 0.0