# Finite Element Analysis of Thermal Stress and Thermal Deformation in Typical Part during SLM
## 2 Mobile Heat Source and Thermal-Mechanical Coupling Algorithm
### Stress-Strain Equations  
The material is in an elastic state as each layer of powder begins to melt with the laser beam. For simplified isotropic materials, the thermal strain equation can be obtained through the constitutive equations of elasticity theory [17]:  
\[\varepsilon_{ii}=\begin{array}{c}\frac{1}{E}\Big{[}\sigma_{ii}-\mu(\sigma_{ jj}+\sigma_{kk})\Big{]}+a_{e}\Delta T,\varepsilon_{jj}...,\varepsilon_{kk}...j \gamma_{ij}=\frac{\tau_{ij}}{G},\gamma_{jk}...,\gamma_{ik}...\\ (i,j,k\in x,y,z)\end{array} \tag{13}\]  
where \(\gamma\) is the shearing strain; \(E\), \(\mu\), and \(G\) are the modulus of elasticity, Poisson's ratio, and shear modulus, respectively.  
The thermal stress equation can be obtained through the thermal strain equation solution:  
\[\begin{array}{c}\sigma_{ii}=-E\frac{(1-\mu)\varepsilon_{jj}+\mu(\varepsilon_ {ii}+\varepsilon_{kk})-a_{e}T(1+\mu)}{(1+\mu)(-1+2\mu)},\sigma_{jj}...,\sigma_{ kk}...j\tau_{ij}=\gamma_{ij}G,\tau_{IK}...,\tau_{ik}...\\ (i,j,k\in x,y,z).\end{array} \tag{14}\]  
As the melting temperature increases, the stress-strain increases, and the material enters a plastic state. According to the Mises yield criterion and the Prandtl-Resus flow criterion, the incremental constitutive equation is used to calculate the elastic-plastic strain:  
\[\begin{array}{c}d\varepsilon_{ii}=d\varepsilon_{ii}{}^{e}+d\varepsilon_{ii}{ }^{p},d\varepsilon_{jj}...,d\varepsilon_{kk}...j\gamma_{ij}=d\gamma_{ij}{}^{e }+d\gamma_{ij}{}^{p},d\gamma_{jk}...,d\gamma_{ki}...\\ (i,j,k\in x,y,z)\end{array} \tag{15}\]  
\[\begin{array}{c}d\varepsilon_{ii}=\frac{1}{2G}d\sigma_{ii}+d\lambda\sigma_{ ii},d\varepsilon_{jj}...,d\varepsilon_{kk}...;\;d\gamma_{ij}=\frac{1}{G}d \tau_{ij}+2d\lambda\tau_{ij},d\gamma_{jk}...,d\gamma_{ik}...\\ d\gamma_{ij}=\frac{1}{G}\lambda\tau_{ij}+2d\lambda\tau_{ij},d\gamma_{jk}...,d \gamma_{ik}...\\ (i,j,k\in x,y,z)\end{array} \tag{16}\]  
where \(d\lambda\) is the positive instantaneous constant  
\[d\lambda=\frac{3d\varepsilon_{ii}{}^{p}}{2\sigma_{s}},(\sigma_{s}=\frac{1}{ \sqrt{2}}\sqrt{\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2}-\sigma _{3}\right)^{2}  
Meanwhile, as the natural convection and surrounding radiation with the gases were considered, the boundary conditions of the simulation model can be defined. The heat flux at the boundary is equal to the laser heat flux at the boundary unit.  
\[-k(\frac{\partial T}{\partial n})_{w}=q_{(r)w} \tag{19}\]  
where n is the normal direction of the model surface, and w is the model border.  
\[-k(\frac{\partial T}{\partial n})_{w}=h(T_{w}-T_{f})+\varepsilon\sigma(T_{w} ^{4}-T_{f}^{4}) \tag{20}\]  
where \(h\) is the heat transfer coefficient at the model surface; \(T_{w}\) is the temperature of the model surface; \(T_{f}\) is the temperature of the surrounding gas. \(\varepsilon\) is the radiation coefficient of the black body, and \(\sigma\) is the Stefan-Boltzmann constant.  
Next, according to the input laser parameters such as P, r, A, and v as well as the property parameters of the material at different temperatures T, such as k, c, p, \(\alpha\), E, u, and G, the software established the stiffness matrix to solve the stress balance equation (Navier equation), which can be obtained through the displacement of any unit \([u_{i},v_{i},w_{i}]\). Through the Cauchy equation, the stress-strain value of any unit can be solved by Equations (13)-(18).