## 3 Method
### Calculation Process
#### 3.2.2 Residual Stress Model  
During arc welding, the electrical power source produces a voltage (\(U\)) between the electrode and base metal, resulting in the flow current. In this process, energy loss is due to the convection and radiation within the electric arc. Consequently, only a tiny fraction of energy is employed for melting materials. To account for this, a variable, namely power efficiency (\(\eta\)), was introduced. The effective welding heat input can be calculated as:  
\[Q=\eta U\Pi \tag{3}\]  
where \(Q\) is the heat input, \(\eta\) is the power efficiency, \(U\) is the voltage, and \(I\) is the current.  
Considering the thermal model, the thermal gradient can be evaluated by establishing an energy balance [40]:  
\[\rho(T)c(T)\frac{\partial T}{\partial t}=Q+\frac{\partial}{\partial x}[Kx(T) \frac{\partial T}{\partial x}]+\frac{\partial}{\partial y}[Ky(T)\frac{ \partial T}{\partial y}]+\frac{\partial}{\partial z}[Kz(T)\frac{\partial T}{ \partial z}] \tag{4}\]  
where \(\rho\) is the material density, \(c\) is the specific heat, \(Q\) is the heat input, \(T\) is the temperature, \(t\) is the time, and \(K_{x}\), \(K_{y}\), and \(K_{z}\) are the thermal conductivity coefficients in their respective directions. The heat flow involved is non-linear, and the material's thermophysical properties are strongly temperature-dependent.  
Using the thermal gradients from Equation (4), residual stress can be calculated by evaluating the gradient deformation:  
\[\sigma_{v}=\delta_{v}\lambda_{eik}+2\mu\varepsilon_{v}-\delta_{v}(3\lambda+2 \mu)\alpha T \tag{5}\]  
where \(\lambda\) and \(\mu\) represent the Lame constants, which are associated with the elastic modulus (\(E\)) and Poisson's ratio (\(v\)). Variable \(\varepsilon_{v}\) is further related to the deformation and displacement:  
\[\varepsilon_{v} = \frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right) \tag{6}\]  
Accordingly, the residual stress calculation method involves elastoplastic behavior, including isotropic hardening, and the relevant data were extracted from the strain during the welding process. As strain can be categorized by elastic, plastic, and thermal properties, the formula of the total strain can be given as:  
\[\varepsilon_{ij}=\ \varepsilon_{ij}^{e}+\varepsilon_{ij}^{p}+\varepsilon_{ij}^{ th} \tag{7}\]  
where \(\varepsilon_{ij}\) is the total deformation, \(\varepsilon_{ij}^{e}\) is the elastic deformation, \(\varepsilon_{ij}^{p}\) is the plastic deformation, and \(\varepsilon_{ij}^{th}\) is the thermal deformation.  
Furthermore, according to [PERSON]'s law, the elastic deformation can be expressed as:  
\[\varepsilon_{kl}^{e}=\sigma_{ij}^{e}\cdot E(T)^{-1} \tag{8}\]  
The thermal-induced deformations are as follows:  
\[\varepsilon_{kl}^{th}=\alpha_{ij}\cdot(T-T_{\infty}) \tag{9}\]  
where \(\alpha_{ij}\) is the linear thermal coefficient and \(T_{\infty}\) is the reference temperature. Plasticity theory characterizes the elastic-plastic response of materials using mathematical relationships that rely on specific restrictive assumptions. Among these assumptions, plastic deformation is the result of transient stress and is independent of time [41].  
Finally, according to the flow rule model that determines the plastic direction of metals under minor displacements, the plastic potential function (\(g\)) is equivalent to the flow area capability (\(f\)). This relationship is known as the associated flow rule, where the plasticdirection is considered to be perpendicular to the flow surface. The plastic strain ratio is determined as:  
\[\mathrm{d}\epsilon_{kl}^{pl}=\mathrm{d}\lambda\frac{\partial f}{\partial\sigma_{ ij}} \tag{10}\]  
where \(\lambda\) is a positive constant dependent on the properties of the material. For a perfect elastic-plastic material and for the cases in which a Von Mises surface is used as the flow area capability criterion, parameter \(\lambda\) can be described as:  
\[\lambda=\frac{3G\mathrm{s}_{ij}\mathrm{s}_{kl}}{\sigma_{e}{}^{2}} \tag{11}\]