Analytical estimation of thermomechanical distortion and interface layer thickness for gas metal arc lap joining of dissimilar sheets

Abstract

The thermomechanical distortion and the evolution of an interface layer with intermetallic phases are the two critical challenges for gas metal arc overlap joining of multimaterial sheets. Two novel analytical methods are proposed following mechanistic principles to estimate the thermomechanical distortion and the interface layer thickness. The analytically estimated results are tested rigorously with the corresponding experimentally measured results for gas metal arc joining of aluminium and steel sheets for different process conditions. Both the thermomechanical distortion and the interface layer thickness are influenced predominantly by the wire feed rate and the resulting heat input. The interface layer thickness and the thermal distortion are found to be the minimum for a heat input of 42.4 J/mm corresponding to the lowest wire feed rate of 4 m/min and the highest travel speed of 10 mm/s. The proposed analytical methods can serve as practical easy-to-use design tools for appropriate selection of process variables in gas metal arc overlapped joining of dissimilar sheets to mitigate the joint distortions and restrict excessive growth of the interface layer.

Appendices

Appendix 1

Estimation of peak temperature at the thermocouple monitoring location and the joint interface

Rosenthal’s pseudo-steady state moving heat surface solution for thin-plate geometry is used to calculate the temperature fields in dissimilar assembly of aluminium and steel sheets as

$${T}_{P}-{T}_{0}=\left(\frac{{~}^{{\eta I}_{\mathrm{a}}{V}_{\mathrm{a}}}\!\left/ \!{~}_{{\mathrm{th}}_{\mathrm{T}}}\right.}{2\pi {k}_{\mathrm{a}}}\right)\mathrm{exp}(-\mathrm{TS}\frac{\zeta }{2{\gamma }_{\mathrm{a}}}){K}_{0}\left(\mathrm{TS}\frac{{r}_{1}}{2{\gamma }_{\mathrm{a}}}\right)$$

(1.1)

where T_P and T₀ are peak and ambient temperatures, respectively. V_a and I_a are average arc voltage and current, η depicts the process efficiency ~ 0.75, TS is the travel speed and ${K}_{0}\left(\mathrm{TS}\frac{{r}_{1}}{2{\gamma }_{\mathrm{a}}}\right)$ represents the modified Bessel function of the second kind with order zero. k_a is the average thermal conductivity of aluminium and steel, and γ_a is the average diffusivity of aluminium and steel. r₁ is the distance between temperature measuring point and the arc centre, ${r}_{1}=\sqrt{{\upzeta }^{2}+{\uplambda }^{2}}$.

ζ depicts the distance between the monitoring location and the arc centre along the direction of the travel speed and is considered (100-vt), where t is the time interval, λ is the transverse distance from the monitoring location and 100 indicates half-length of the joint. th_T signifies sheet thickness, the values of th_T are taken as 1.8 mm and 1 mm in estimating temperature at the thermocouple monitoring location and at the joint interface, respectively.

A sample estimation of peak temperature at a thermocouple measuring location for TS of 10 mm/s and WFR of 5 m/min is shown in Table 6. The peak temperature is estimated 3 mm away from thermocouple location in transverse direction due to the limitation of Rosenthal’s solution, which estimates exaggerate values of temperature exactly under the arc. For a given process condition, T_P is estimated at several values of ζ under different time interval along joining direction and represented as a thermal cycle in Fig. 4.

Table 6 Values of variables in Eq. (1.1) for TS of 10 mm/s and WFR of 5 m/min

Appendix 2

Estimation of thermomechanical distortion

As per compatibility norm, Fig. 3(b), final contraction of the steel sheet will be equal to the final contraction of the aluminium sheet, that can be expressed as

$$\frac{L}{2}{\alpha }_{\mathrm{Fe}}\left({T}_{\mathrm{P}}-{T}_{0}\right)+\frac{L}{2}\left(\frac{{\sigma }_{\mathrm{Fe}}}{{E}_{\mathrm{Fe}}}\right)=\frac{L}{2}{\alpha }_{\mathrm{Al}}\left({T}_{\mathrm{P}}-{T}_{0}\right)-\frac{L}{2}\left(\frac{{\sigma }_{\mathrm{Al}}}{{E}_{\mathrm{Al}}}\right)$$

(2.1)

$$\frac{{\sigma }_{\mathrm{Al}}}{{E}_{\mathrm{Al}}}+\frac{{\sigma }_{\mathrm{Fe}}}{{E}_{\mathrm{Fe}}}=\left({\alpha }_{\mathrm{Al}}-{\alpha }_{\mathrm{Fe}}\right)\left({T}_{\mathrm{P}}-{T}_{0}\right)$$

(2.2)

The resistance against contraction will develop a pair of equal and opposite forces on the transverse cross-sections of both sheets. The force equilibrium can be expressed as

$${F=\sigma }_{\mathrm{Al}}{A}_{\mathrm{Al}}={\sigma }_{\mathrm{Fe}}{A}_{\mathrm{Fe}}$$

(2.3)

Equating Eqs. (B.2) and (B.3) will provide Eq. (1). The equivalent Young’s modulus in Eq. (3) is estimated as

$${E}_{\mathrm{eqv}}=\frac{{E}_{\mathrm{Al}}{\rho }_{\mathrm{Al}}+{E}_{\mathrm{Fe}}{\rho }_{\mathrm{Fe}}}{{\rho }_{\mathrm{Al}}+{\rho }_{\mathrm{Fe}}}$$

(2.4)

where ρ_Fe and ρ_Al are density of steel and aluminium sheets, respectively.

The position of the neutral plane is located considering concept of pure bending with resultant axial force acting along the cross-section is zero; therefore,

$$\begin{array}{cc}{\int }_{\mathrm{Al}}{\sigma }_{\mathrm{Al}}dA+{\int }_{\mathrm{Fe}}{\sigma }_{\mathrm{Fe}}dA=0& -{\int }_{\mathrm{Al}}\frac{{yE}_{\mathrm{Al}}}{r}dA-{\int }_{\mathrm{Fe}}\frac{{yE}_{\mathrm{Fe}}}{r}dA=0\end{array}$$

(2.5, 2.6)

where r is the radius of curvature and y is the distance from the neutral plane. The first integral in Eq. (2.6) is evaluated over the cross-sectional area of aluminium sheet and filler deposit, and the second is evaluated over the area of the steel sheet. Since the radius of curvature is constant at any given cross-section; therefore, it can be eliminated and the equation to locate neutral plane is

$$-{E}_{\mathrm{Al}}{\int }_{\mathrm{Al}}ydA-{E}_{\mathrm{Fe}}{\int }_{\mathrm{Fe}}ydA=0$$

(2.7)

The integral of the above equation depicts the first moment of aluminium and steel sheets cross-section of the overlapped assembly with respect to the neutral plane.

The longitudinal distortion at any position is deduced from the pure bending analysis of the distorted bi-metallic assembly as shown in Fig. 3(c). An independent point P is chosen at the neutral plane M₁N₁ which is always at the distance of r from the origin. Therefore, the vertical location of point P can be expressed as

$$O{Q}_{1}=\sqrt{\left({OP}^{2}-{{Q}_{1}P}^{2}\right)}=\sqrt{\left({r}^{2}-{z}^{2}\right)}$$

(2.8)

Since the length SP and h₁ are very small as compare to the radius of curvature of the joint assembly, the arc SV can be considered a tangent at point V indicating the ∆PVS as a right-angle triangle. Therefore, a comparison of ∆OQ₁P and ∆PVS expresses

$$\mathrm{PS}=\frac{\left(OP\right)\left(PV\right)}{O{Q}_{1}}=\frac{r{h}_{1}}{O{Q}_{1}}=\frac{r{h}_{1}}{\sqrt{\left({r}^{2}-{z}^{2}\right)}}$$

(2.9)

The distance PG and GH are given as

$$\mathrm{PG}=\left(r-O{Q}_{1}\right)= \left(r-\sqrt{\left({r}^{2}-{z}^{2}\right)}\right); \mathrm{GH}=h2$$

(2.10, 2.11)

Eventually, the distortion of any point on the top of the assembly along longitudinal axis A′A′ can be estimated by adding Eqs. (2.9) to (2.11).