## 2 Problem and Equations
### Equations  
The problem of instantaneous thermoelasticity of a plate with a crack during welding heating can be represented in Figure 3.  
A cracked body with an open initial domain \(\Omega\in R_{3}\) and a piecewise smooth boundary \(\Gamma\) is considered. \(\Gamma_{u}\cup\Gamma_{t}\cup\Gamma_{c}\cup\Gamma_{T}\cup\Gamma_{q}=\Gamma\). \(t\) represents stress, \(u\) represents displacement, \(q\) represents heat flux, \(\Gamma_{t}\) represents the stress boundary, \(\Gamma_{u}\) represents the displacement boundary, \(\Gamma_{q}\) represents the  
Figure 3: The thermo-elastic model of the structure caused by a crack during welding heat. Figure 2: Gauss heat flux distribution (The colors represent the intensity of the heat flux distribution, the red represents the largest, the blue represents the smallest). heat flux boundary, \(\Gamma_{T}\) represents the temperature boundary. \(T(X,t)\) is the transient temperature field throughout the structure produced by the \(q\). \(q\) contains a heating source \(q_{s}\) and a heat flux toward the element surface \(q_{f}\). The problem of instantaneous thermoelasticity consists in the governing equations written by  
\[\
abla\cdot\sigma-\rho b=0 \tag{3}\]  
\[k\cdot\
abla^{2}T=\rho c_{E}\frac{\partial T}{\partial t}-q_{s} \tag{4}\]  
\[\varepsilon=\
abla_{s}u \tag{5}\]  
\[\varepsilon_{T}=\alpha(T-T_{0})I \tag{6}\]  
\[\sigma=D:(\varepsilon-\varepsilon_{T}) \tag{7}\]  
where the symbol \(\
abla\) is the spatial gradient operator, \(\rho\) is the total density, \(q_{s}\) is the heat source externally into the body per unit volume, \(T\) is the temperature, \(k\) is the thermal conductivity of the material. The displacement \(u\), the strain tensor \(\varepsilon\), the stress tensor \(\sigma\) and the thermal expansion \(\varepsilon_{T}\) are defined with respect to a reference temperature \(T_{0}\); the material properties are the expansion coefficient \(\alpha\) and the isotropic fourth-order Hooke tensor \(D\), \(I\) is the identity second-order tensor, and \(\
abla_{s}\) is the symmetric gradient operator on a vector field. \(c_{E}\) is the specific heat at constant strain. The temperature of the structure is \(T(X,t)\) at a given time, the prescribed displacements \(\overline{u}\) are imposed on \(\Gamma_{u}\), the prescribed stresses \(\overline{t}\) are imposed on \(\Gamma_{t}\), \(q\) is imposed on \(\Gamma_{q}\), \(\overline{T}\) is imposed on \(\Gamma_{T}\). The crack forms or propagates, \(\Gamma_{c}\) should be accounted as a propagation of the already existing \(\Gamma_{c}\) in the body. For this case, the solution for the problem of instantaneous thermoelasticity of the governing Equations (3)-(7) is subjected to the following boundary and initial conditions  
\[\text{Temperatureboundary condition}\;\Gamma_{T}:T=\overline{T} \tag{8}\]  
\[\text{Heat flux boundary condition}\;\Gamma_{q}:\;q\cdot n=\overline{q} \tag{9}\]  
Figure 3:0  
Figure 3:1  
Figure 3:2  
Figure 3:3  
where \(\overline{q}=a_{k}(T-T_{0})+\theta\Big{[}(T-T_{0}^{\prime})^{4}-(T_{0}-T_{0}^ {\prime})^{4}\Big{]}\), \(a_{k}\) is the coefficient of heat transfer, \(T_{0}\) is the sink temperature, \(T_{0}^{\prime}\) is the value of absolute zero of the temperature scale. \(\theta\) is the radiation coefficient. **3. The XFEM Formulation of Governing Equations**  
The XFEM is employed to numerically model an elasticity medium with discontinuities by adding appropriate enrichment functions to the standard finite element (FE) approximation. In order to derive the weak form of the governing Equations (3) and (4), the trial functions \(u(X,t)\), \(T(X,t)\) and the test functions \(\delta u(X,t)\), \(\delta T(X,t)\) are required to be smooth enough to satisfy all essential boundary conditions.