Solopko I.V.   Lubanova A.S.  

The mathematical modeling of the heat transfer during continuous extrusion

Reporter: Solopko I.V.

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\title{}{\bf THE MATHEMATICAL MODELING OF THE HEAT TRANSFER DURING CONTINUOUS EXTRUSION}

\author{}{Solopko I.V., Lyubanova A.Sh.}

{\it Siberian federal university, Krasnoyarsk}

{\it isolopko@sfu-kras.ru, lubanova@mail.ru}

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The research is devoted to the mathematical modeling of the heat transfer during continuous extrusion by the Conform Method. This method is based on the use of the contact friction between a billet and the mobile part of the split-face
container to extrude a metal into the die hole \cite{LGSZ}. The process is supposed to be steady-state and the metal is considered as an incompressible rigid visco-plastic homogeneous medium or the Bingham liquid \cite{K}. The heat exchange between metal and the pressing tool is caused by the heat release as a result of work of plastic deformation forces and heat output into the tool. The cooling system takes the heat from the tool.

The heat flux through the tool is directed in the normal to the surface of the contact between the tool and metal.This enabled to find the explicit temperature distribution in the tool as a solution of Cauchy problem for appropriate ordinary differential equations and to get the temperature dependence on the coolant temperature on the surface between the tool and metal. Hereafter, the dependence will be used in the investigation of the inverse problem on the maintenance of the required temperature mode by controlling the coolant temperature.

Authors stated the boundary value problem of the steady-state heat exchange at the zone of rigid visco-plastic deformation and devised an algorithm for numerical solution of this problem by the finite elements method. The method choice is governed by the fact that the derivatives of the components of the metal velocity may be discontinuous on certain slide lines. The linear rectangle functions of element shape were taken as finite elements. The shape functions are formed on the eight mesh points neighboring \cite{Fl}. The non-uniform mesh consisting of four parts was used for forming the basis functions. The choice of the mesh is due to the geometrical features of the deformation zone.


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\bibitem{LGSZ}
{\it Lyubanova A. Sh., Gorokhov Yu. V., Solopko I. V., Ziborov A. Yu.} Optimization of the Uniformity of a Metal Flow
during Continuous Extrusion by the Conform Method // Russian Metallurgy (Metally), Vol. 2010, No. 3, P. 178ֱ82. Original Russian Text published in Metally, 2010, No. 2, pp.~28-33.
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\bibitem{K}
{\it Kolmogorov V.} Mekhanika obrabotki metallov davleniem. Moscow: Metallurgiya, 1986. 320 p. (Russian)
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\bibitem{Fl}
{\it Fletcher C. A. J.} Computational Galerkin  Methods. Berlin: Springer-Verlag, 1984. 352 p.

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Abstracts file: SolopkoIV.zip


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