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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