Analysis of viscosity function models used in friction stir welding

Abstract

Friction stir welding (FSW) is a solid-state process, where a tool that consists of a shoulder and a pin rotates between the plates to be welded by plastic deformation. This process involves several physical phenomena. To better understand this complex phenomenon, simulations have been performed for a range of maximum viscosity values. The viscosity model used in friction stir welding simulations depends on two variables, the temperature and strain rate; however, the viscosity goes toward infinity for low values of temperature and strain rates. This study analyzed two different friction stir welding simulation, by observing how the viscosity functions behave for low values of temperature and strain rates. The maximum viscosity value was shown to be restricted to values, where the velocity field tends to zero. Furthermore, when the viscosity value exceeds the maximum value, the temperature and viscosity field is significantly impaired. In the correct results, the change in viscosity is restricted to stir zone.

Appendix

1.1 1 AISI 304 stainless steel constants

$$\begin{aligned} \begin{array}{ll}C_{{\rm P-workpiece}}\,({\rm J/kg\, K})&276 + 0.851\cdot T - 0.000851\cdot T^2 + 3\times 10^{-7}\cdot T^3\\ k_{{\rm workpiece}}\, ({\rm W/m\, K})&14.3 - 0.00902\cdot T + 4.52\times 10^{-5} \cdot T^2 - 2.49\times 10^{-8} \cdot T^3\\ \rho _{{\rm workpiece}}\,({\rm kg/m}^3)&7200\\ P_N\,({\rm MPa})&109\\ R_{\rm s}\,({\rm mm})&9.53\\ R_{\rm p}\,({\rm mm})&3.17\\ U_1\,({\rm mm/s})&1.693\\ C_{{\rm P-tool}} \,({\rm J/kg\, K})&158 + 1.06\cdot T - 1.63\cdot T^2\\ \eta&0.5\\ k_{{\rm tool}} ({\rm W/m\, K})&0.367 - 2.29\cdot T + 1.25\times 10^{-7} \cdot T^2\\ \rho _{{\rm tool}}\,({\rm kg/m}^3)&19400\\ \delta&0.7\\ \mu&0.4\\ \omega \,({\rm RPM})&300\\ a_0 \,({\rm s}^{-1})&1.36\times 10^{35}\\ b_0 \,({\rm s}^{-1})&8.03\times 10^{26}\\ G\, ({\rm Pa})&73.1\times 10^{9}\\ k_0 \, ({\rm Pa})&150\times 10^{6}\\ Q \,({\rm J/mol})&410\times 10^{3}\\ Q_0 \,({\rm J/mol})&91\times 10^3\\ \lambda&0.15\\ M&7.8\\ N&5\\ C \, ({\rm Pa})&132\times 10^6\\ D_0 \,({\rm s}^{-1})&10^8\\ m_0&2.148\\ n_0&6\\ R\, ({\rm J/mol\, K})&8.3144621 \end{array}\end{aligned}$$

1.2 2 Ti–6Al–4V alloy constants

$$\begin{aligned} \begin{array}{ll} C_{{\rm P-workpiece}}\,({\rm J/kg\, K})&622 - 0.367\cdot T - 0.000545\cdot T^2 + 2.39\times 10^{-8}\cdot T^3\\ k_{{\rm workpiece}} ({\rm W/m\, K})&19.2 + 0.0189\cdot T - 1.53\times 10^{-5} \cdot T^2 + 1.41 \times 10^{-8} \cdot T^3\\ \rho _{{\rm workpiece}}\,({\rm kg/m}^3)&7200\\ \theta \,({\rm MPa})&9.09\times 10^2 + 1.11\cdot T - 3.05\times 10^{-3} \cdot T^2 + 1.26\times 10^{-6}\cdot T^3 \\ P_N\,({\rm MPa})&60.0\\ R_{\rm s}\,({\rm mm})&9.5\\ R_{\rm p}\,({\rm mm})&3.95\\ U_1\,({\rm mm/s})&1.6\\ C_{{\rm P-tool}} \,({\rm J/kg\, K})&128.3 - 3.279\times 10^{-2}\cdot T + 3.41\times 10^{-6}\cdot T^2\\ k_{{\rm tool}} ({\rm W/m\, K})&153.5 - 9.56\times 10^{-2}\cdot T + 5.23\times 10^{-5} \cdot T^2\\ \rho _{{\rm tool}}\,({\rm kg/m}^3)&19400\\ \delta _0&0.7\\ \mu _0&0.4\\ \eta&0.7\\ \omega \,({\rm RPM})&275\\ A \,({\rm s}^{-1})&229.725\\ \alpha \, ({\rm MPa}^{-1})&0.0066\\ Q\,({\rm J/mol})&501000\\ n&5\\ P&0.08\end{array} \end{aligned}$$