# Characteristics of GMAW Narrow Gap Welding on the Armor Steel of Combat Vehicles
## 3 Measurement and Prediction of Residual Stress Distribution
### FEM Prediction of Welding Process  
In this study, a coupled thermo-mechanical analysis is used to calculate the welding residual stress by the multi-pass welding process. The shape of the groove and the weld passes of each model are shown in Figure 6a,b. MSC Marc software is used to conduct the thermo-mechanical finite element analyses.  
Thermal Analysis  
Analyses of the welding heat transfer for a given welding condition are performed in the 3D finite element analysis (FEA) models. During welding, the governing equation for transient non-linear heat transfer is:  
\[\frac{\partial}{\partial x}(\lambda\frac{\partial T}{\partial x})+\frac{ \partial}{\partial y}(\lambda\frac{\partial T}{\partial y})+\frac{\partial}{ \partial z}(\lambda\frac{\partial T}{\partial z})+Q=\rho c\frac{\partial T}{ \partial t} \tag{1}\]  
The thermal effects due to solidification of the weld pool are modeled by considering the latent heat for fusion. The value of the latent heat is 270 J/g [10]. The liquidus temperature, \(T_{L}\), and the solidus temperature, \(T_{S}\), are assumed to be 1500 \({}^{\circ}\)C and 1450 \({}^{\circ}\)C, respectively. The liquidus temperature and solidus temperatures were typed into the solidus and liquidus temperature module of MSC Marc software.  
Figure 6: (**a**) Cross-section of the FEM model: NGW. (**b**) Cross-section of the FEM model: X-groove.  
Heat losses due to convection and radiation are also taken into account in the finite element models. All surfaces of the model are assumed to lose heat by convection to the surrounding air. Heat loss due to convection is accounted for by using [PERSON]'s law [11]:  
\[q_{r}=-h_{c}(T_{S}-T_{0}) \tag{2}\]  
Here, \(h_{c}\) is the heat transfer coefficient, \(T_{S}\) is the surface temperature of the workpieces, and \(T_{0}\) is the ambient temperature. For a surface element of a solid body contacted by flowing gas or liquid, the heat flow density \(q_{r}\) is, according to [PERSON]'s law, proportional to the difference between surface temperature, \(T_{S}\), and gas or liquid temperature, \(T_{0}\), through the coefficient of convective heat transfer \(h_{c}\).  
In this study, a temperature-dependent film coefficient [12] is used and the ambient temperature is assumed to be 25 \({}^{\circ}\)C.  
The heat loss due to radiation is modeled using the Stefan-Boltzmann law [13]:  
\[q_{r}=-\varepsilon\sigma[(T_{S}+273)^{4}-(T_{0}+273)^{4}] \tag{3}\]  
Here, \(\varepsilon\) is the emissivity and \(\sigma\) is the Stefan-Boltzmann constant. In this study, the emissivity is assumed to be 0.2 [14].  
In welding technology, heat dissipation of the cooling of a relatively small body (body temperature, \(T_{S}\)) in relatively extensive surroundings (body temperature, \(T_{0}\)) occurs by means of radiation in accordance with Equation (3). Thermal conductivity and specific heat are used as the temperature-dependent thermal properties as shown in Figure 7. The element activation and deactivation methods are used to model the deposition of weld metal during multi-pass welding. The elements of the weld deposition are activated and then heated to model the moving heat source.