# Analytical Model for Angular Distortion in Multilayer Welding under Constraints
## 2 Methods
### Mechanics of Welding Distortion
When \(K\) equals 1, the constraints are removed, because \(M_{c}\) becomes zero and, when \(K\) is above 1, the degree of constraint increases. Equation (3) can be obtained by calculating Equations (1) and (2) for the elastic and plastic regions. \[ \begin{split}\sum M&=\int_{-\frac{1}{2}}^{\frac{1} {2}}EezdA+(K-1)\int_{-\frac{1}{2}}^{\frac{1}{2}}EezdA+\int_{\frac{1}{2}-a}^{ \frac{1}{2}}wYzdA\\ &=E\int_{-\frac{1}{2}}^{\frac{1}{2}}\left(\varepsilon_{N}-\frac {z}{\rho}\right)\!dz+\int_{\frac{1}{2}-a}^{\frac{1}{2}}wYzdz=0\end{split}  \tag{3}\]  
where \(\sigma_{z}=-\frac{Ez}{\rho},\ \varepsilon_{N}-\frac{z}{\rho}=\varepsilon,\ E:\) elastic modulus, \(\varepsilon:\) elastic strain, \(\varepsilon_{N}:\) elastic strain at neutral axis, \(\rho:\) curvature, \(w:\) weighting factor, \(Y:\) yield strength, and \(K:\) degree of constraint, \(K=\frac{M_{c}}{M_{r}}+1\). Solving the integral of Equation (3), we obtain  
\[\rho=\frac{KEt^{3}}{6 wYab} \tag{4}\]  
Geometrically, this can be expressed as the angular distortion that is given by  
\[\theta=\frac{L}{\rho-\frac{t}{2}}=\frac{3 we_{Y}bA_{p}}{Kt^{3}-3 we_{Y}abt} \tag{5}\]  
Figure 1: Free body diagram of welding distortion (\(t\): plate thickness, a: height of plasticity area, \(L\): width of plasticity area, \(\rho\): curvature, \(F\): force by weld shrinkage, \(F_{r}\): reaction force, \(M_{r}\): reaction moment of base plate, \(M_{c}\): reaction moment by surrounding structures, including external moment). where \(\varepsilon_{Y}\): strain at yield strength, \(Y/E\), \(A_{p}\): plasticity area. We assume that the plate is much thicker than the size of the plastic region. Thus, \(3 we_{Y}ab\) is much smaller than the square of thickness \(t^{2}\) and Equation (5) can be expressed as  
\[\theta\approx\frac{3 we_{Y}A_{p}}{Kt^{2}} \tag{6}\]  
where \(b\approx t\gg a\), \(Kt^{2}\gg 3 we_{Y}ab\), which indicates that the angular distortion is proportional to the area of the plastic region and inversely proportional to the square of thickness. The direct relationship between plastic region and angular distortion has been determined in studies on welding and laser forming [2; 18; 19]. The plasticity of a material is likely to occur at high temperatures between 500 and 800 \({}^{\circ}\)C, which produce a sharp stiffness drop. Therefore, the plastic region can be estimated from the distribution of the peak temperature field inside the material during heating and cooling. As we consider a thick plate, the equation of the peak temperature field by the point heat source on the semi-infinite plate is defined, as follows, according to the derivation in [20]:  
\[\pi r^{2}=\frac{2Q_{\text{net}}}{c_{p}\lambda e}\frac{1}{T_{\text{max}}} \propto A_{\text{p}} \tag{7}\]  
where \(r\): distance from point heat source \(Q_{\text{net}}\): effective heat input, \(c_{p}\): specific heat, \(\lambda\): density, \(T_{\text{max}}\): maximum temperature experienced, with \(r\) representing the area within an isotherm, because it is the distance from the center of the heat source to a specific temperature. If plasticity occurs at a specific temperature between 500 and 800 \({}^{\circ}\)C, the area of plastic region \(A_{\text{p}}\) is proportional to the heat input, according to Equation (7). The wire melting rate is expressed as the sum of the wire resistance heating and the arc heat, which also originate from electrical energy. Thus, for typical arc welding, the bead area is proportional to the heat input. As the plastic region and bead area are determined by the heat input, the angular distortion can be obtained from the bead area, as follows:  
\[\theta=\frac{C}{K}\frac{A}{t^{2}} \tag{8}\]  
where \(A\): bead area, \(C=3 we_{Y}A_{p}/A\). Equation (8) indicates that the angular distortion by welding is proportional to the bead area and inversely proportional to the square of thickness and degree of constraint only for materials with sufficient thickness when compared to the heated part.