# Effect of Plate Curvature on Heat Source Distribution in Induction Line Heating for Plate Forming
## 2 Model and Method
### Heat Transfer and Mechanical Analysis  
In the heat transfer calculation, the heat generation distribution is the input. Solving the temperature field change follows the heat conduction equation for solids, Equation (8):  
\[\rho c\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\bigg{(}\lambda \frac{\partial T}{\partial x}\bigg{)}+\frac{\partial}{\partial y}\bigg{(} \lambda\frac{\partial T}{\partial y}\bigg{)}+\frac{\partial}{\partial z} \bigg{(}\lambda\frac{\partial T}{\partial z}\bigg{)}+q_{surf} \tag{8}\]  
where \(\rho\) is the density (kg/m\({}^{3}\)), \(c\) is the specific heat (J/ (kg\(\cdot\)\({}^{\circ}\)C)), \(T\) represents the temperature (\({}^{\circ}\)C), and \(\lambda\) represents the thermal conductivity (W/ (m\(\cdot\)\({}^{\circ}\)C)).  
The convection boundary condition can be described in Equation (9):  
\[\lambda\frac{\partial T}{\partial\overrightarrow{n}}=-h(T-T_{env}) \tag{9}\]  
where \(h\) refers to the convective heat transfer coefficient (W/(m\({}^{2}\)\(\cdot\)\({}^{\circ}\)C)), \(\overrightarrow{n}\) is the external normal direction of the steel plate surface, and \(T_{env}\) refers to the ambient temperature (\({}^{\circ}\)C).  
In the mechanical analysis, the basic equations are Equations (10) and (11):  
\[\sigma_{ij,j}+F_{bi}=0 \tag{10}\]  
where \(\sigma_{ij}\) refers to the stress tensor and \(F_{bi}\) refers to the body force.  
The stress-strain relationship can be expressed as:  
\[\{\mathrm{d}\sigma\}=[D]\{\mathrm{d}\epsilon\}-\{\mathrm{C}\}\mathrm{d}T \tag{11}\]  
where \(\{\mathrm{d}\sigma\}\) refers to the stress increment, \([D]\) is the elastoplastic matrix, \(\{\mathrm{d}\epsilon\}\) is the strain increment, \(\{\mathrm{C}\}\) is the thermal stiffness matrix, and \(\mathrm{d}T\) is the temperature increment.  
With regards to setting the boundary conditions of simple support on three corners, see Figure 4. The deformation and strain distribution of the plate can be obtained by solving the equations. Furthermore, the plastic strain can be decomposed into two parts--in-plane strain and out-of-plane strain, according to Equations (12) and (13):  
\[\varepsilon_{in\_plane}=\frac{1}{t_{plate}}\int_{z}\varepsilon\mathrm{d}z \tag{12}\]  
\[\varepsilon_{out\_of\_plane}=\varepsilon-\varepsilon_{in\_plane} \tag{13}\]  
where \(t_{plate}\) refers to the plate thickness, \(\varepsilon_{in\_plane}\) refers to the in-plane strain, and \(\varepsilon_{out\_of\_plane}\) refers to the out-of-plane strain.  
We performed the heat transfer calculation and mechanical analysis using commercial software ABAQUS/standard 6.14. Only a 1 m \(\times\) 1 m square plate was modeled in this calculation (see Figure 4). The mesh of the plate was a 10 mm \(\times\) 10 mm rectangular shape element using the 4-node heat transfer shell element (DS4 element) in the heat transfer calculation and the 4-node stress/displacement shell element with reduced integration (S4R element) in the mechanical analysis, with five integration points in thickness. There were 10,000 elements and 10,201 nodes in total. In the heat transfer calculation, the heating generation was applied as a surface heat flux density by defining a Fortran subroutine (DFLUX), in which the distribution of the heat source along the surface is defined. The boundary condition of natural convection heat dissipation was set on the surface of the plate, and the convective heat transfer coefficient was 15 W/(m\({}^{2}\)\(\cdot\)\({}^{\circ}\)C). In the mechanical analysis, we read the temperature distribution result obtained in the heat transfer calculation. The material of the plate was carbon steel, and the material properties refer to [PERSON]'s [11] data and are shown in Table 1. The hardening criteria for the material used a linear hardening model; a bilinear curve represents the strain-stress properties.