Electro-Thermo-Elastic Simulation of Graphite Tools Used in SPS Processes

Abstract

In the range of field-assisted sintering technology or spark plasma sintering all materials in the testing machine undergo very large temperature changes. The powder material, which has to be sintered, is filled into a graphite die and mechanically loaded by a graphite punch. The heat is produced by electrical induction and the cooling process is performed by conduction and radiation. Both the heating and the cooling process are very fast. In order to understand the process of the highly loaded graphite parts, experiments, modeling and computations have to be carried out. On the thermal side the temperature-dependent material properties such as heat capacity and heat conductivity have to be modeled. Since the heat capacity is not independent of the Helmholtz free-energy a particular consideration of the free-energy is carried out. On the other hand, the temperature changes of the electrical resistivity and the material properties of the graphite tool must be taken into considerations. Accordingly, the material properties of “Ohm’s law” must be modeled as well. The fully coupled system comprising the electrical, thermal and mechanical field are solved numerically by a monolithic finite element approach. After the spatial discretization using finite elements one arrives at a system of differential-algebraic equations which is solved by means of diagonally implicit Runge-Kutta methods. Issues and open questions in the numerics are addressed and problems in modeling a real application are discussed.

Appendix

In the numerical studies the copper is used in the die system. For this the temperature-dependent heat capacity and conductivity are measured, see Fig. 8. The heat capacity at constant pressure of copper is measured with a Netzsch DSC 204 F1 Phoenix apparatus, which uses the Differential Scanning Calorimetry, see Fig. 8b. The thermal diffusivity $\displaystyle a(\Theta )$ of copper is measured with a Netzsch Laserflash LFA 457 and subsequently the thermal conductivity is computed by $\displaystyle \kappa {}_{\Theta }(\Theta )=a(\Theta )\rho c{}_{\Theta }(\Theta )$, see Fig. 8a. For both quantities a linear temperature dependence is assumed,

$$\begin{aligned} \kappa _{\Theta ,\text{ Cu }}(\Theta )&= -78.3347 \times {10^{-3}}\;{\mathrm{{W}}} /({\mathrm{{mK^2}}}) \Theta + {433.173}{\mathrm{{W}}}/({\mathrm{{mK}}}) \end{aligned}$$

(38)

$$\begin{aligned} c_{\Theta ,\text{ Cu }}(\Theta )&= 82.2141 \times {10^{-3}}\;{\mathrm{{J}}}/({\mathrm{{kgK}}}^2) \Theta + {373.728}{\mathrm{{J}}} /({\mathrm{{kgK}}}^2) \end{aligned}$$

(39)

which fit very well with experimental data. The properties of alumina and the electrical conductivity of copper are taken from the literature, see [30]. The investigated temperature range in this publication is between $\displaystyle {300}\,{\mathrm{{K}}}$ and $\displaystyle {1300}\,{\mathrm{{K}}}$.

$$\begin{aligned} \kappa _{\Theta ,\text{ Al }}(\Theta )&= \frac{65181330.4+\Theta }{-669628.8+8175.85\Theta } \quad \text{ in }\quad {\mathrm{{W}}}/({\mathrm{{mK}}}),\end{aligned}$$

(40)

$$\begin{aligned} c_{\Theta ,\text{ Al }}(\Theta )&= \frac{7770.25{\Theta }}{249.4+{\Theta }} + \frac{790.15}{249+{\Theta }} + 0.008\Theta \quad \text{ in }\quad {\mathrm{{J}}}/({\mathrm{{kgK}}}) ,\end{aligned}$$

(41)

$$\begin{aligned} \kappa _{\varphi ,\text{ Al }}&= {10^{-8}}\mathrm{{A}}/(\mathrm{{Vm}})\end{aligned}$$

(42)

$$\begin{aligned} \kappa _{\varphi ,\text{ Cu }}(\Theta )&= (5.5+0.038\Theta )\times 10^{9}\quad {\text{ in }}\quad {\mathrm{{A}}}/({\mathrm{{Vm}}}) \end{aligned}$$

(43)