Quality prediction of friction stir welded joint based on multiple regression: entropy generation analysis

Abstract

Due to the outstanding advantages in light alloy material processing, friction stir welding technology is of great significance for obtaining high-quality welding products and accelerating aerospace lightweight. At present, most related studies predict joint quality before welding through computer technology. However, due to the complex thermo-mechanical coupling in the welding process, there may be a large deviation between the predicted results and the actual results, resulting in energy waste. In addition, these prediction models have poor real-time and versatility. Therefore, a new joint quality prediction method based on entropy production analysis is proposed in this paper. Firstly, based on non-equilibrium thermodynamics and extrusion theory, the entropy generation analysis model of the friction stir welding system is deduced. Using Liouvile’s Formula, the analytic solution of the entropy generation analysis model with unknown parameters is obtained. Secondly, combined with numerical simulation and multiple regression, the unknown parameters of the entropy generation analysis model are determined. Finally, multiple sets of welding experiments are designed to verify the effectiveness of the entropy generation analysis model. The welding process is analyzed by the proposed entropy generation analysis model to achieve the quality prediction of friction stir welded joint.

Appendix

1.1 Appendix A. Derivation of Q ₁’

According to reference [46], the nominal contact area between the shoulder with workpieces (A₁), and the side of the welding pin with workpieces (A₂) per unit time respectively are

$${\mathrm{d}A}_{1}=2\pi r\mathrm{d}r$$

(40)

$${\mathrm{d}A}_{2}=2\pi \frac{\sqrt{{H}^{2}+{\left({R}_{2}-{R}_{3}\right)}^{2}}}{H}\sqrt{{x}^{2}+{y}^{2}\mathrm{d}z}$$

(41)

So, for a period, the contact area between the welding tool and workpieces

$${A}_{\alpha }=\left[\pi \left({x}^{2}+{y}^{2}\right)+2\pi \frac{\sqrt{{H}^{2}+{\left({R}_{2}-{R}_{3}\right)}^{2}}}{H}\sqrt{{x}^{2}+{y}^{2}}z\right]t$$

(42)

Based on reference [47], friction force between the shoulder and the workpieces can be expressed as

$${F}_{f}=c{\mathrm{F}}_{\mathrm{N}}+\left(1-{c}^{2}\right)\phi \left\{{\left(\frac{D}{2-D}\right)}^{\frac{2-D}{2}}{\left(1.72{D}^{2}-6.2075D+7.4476\right)}^{{\left(\frac{2-D}{2}\right)}^{2}}{S}^\frac{D}{2}{G}^{2-D}{\left(\frac{\pi {\mathrm{E}}^{2}}{225{\sigma }_{\mathrm{y}}^{2}}\right)}^{\frac{2-D}{2\left(D-1\right)}}\right\}$$

(43)

where c is area coefficient, and c = 0.93; F_N is normal load; ϕ is adhesive shear strength; D is fractal dimension, and D = 1.63; G is the scale factor, and G = 1E-6; σ_y is the yield strength of aluminum; E is comprehensive elastic modulus of aluminum alloy materials, and E = 73 GPa. Thus, the frictional heat between welding tool and workpieces.

$$\begin{array}{ll}Q_1'=2\pi nF_frS\\=2\pi^2nt{(x_1^2+x_2^2)}^\frac32\left\{ {\text{F}}_\text{N} +\phi(1-c^2){(\frac D{2-D})}^\frac{2-D}2{(1.72D^2-6.2075D+7.4476)}^{\left(\frac{2-D}2\right)^2}{\lbrack\pi(x_1^2+x_2^2)t\rbrack}^\frac D2G^{2-D}{(\frac{\pi\text{E}^2}{225\sigma_\text{y}^2})}^\frac{2-D}{2(D-1)}\right\}\end{array}$$

(44)

Where n is rotation rate. According to Hilbert Space Projection Theorem, the above equation (A5) is simplified to the following equation (A6).

$$Q_1'=0.6430\left(\frac\chi2-3.8580\right)exp\left(-0.6430\chi\right)$$

(45)

Where

$$\chi ={\tau }^{-\frac{1}{2}}\left({x}_{1}+{x}_{2}-2{x}_{3}\right)$$

(46)

1.2 Appendix B. Derivation of Q ₆’

According to reference [48], the plastic deformation heat during FSW

$$\begin{array}{l}Q_6'=2\pi ntM\\=\frac{16\pi^3\lambda\Pr n^2R_1^2x^2(x^2+y^2+z^2)zt}{c_v(x^2+y^2)\lbrack R_1^2-(x^2+y^2+z^2)\rbrack}\langle a_0+b_1\frac1{1+\exp\lbrack-(n-c_1)/d_1\rbrack}\frac1{1+\exp\{-\lbrack(\sqrt{x^2+y^2+z^2}-\sqrt{x^2+y^2})-c_2\rbrack/d_2\}}\rangle-4\pi^2\tau_0R_1^2nz\ln\vert\frac{x^2+y^2+z^2}{R_1^2}\vert\frac{x^2+y^2+z^2}{R_1^2-(x^2+y^2+z^2)}\end{array}$$

(47)

According to Hilbert Space Projection Theorem, the above equation (B1) is simplified to the following equation (B2).

$$Q_6'=0.8129\left(\frac\chi2-5.7744\right)exp\left(-0.8129\chi\right)$$

(48)