Keywords: metal forming, rigid-plastic FEM, BEM, rigid-plastic domain-BEM, hybrid element method, compatibility.

In this paper, a metal forming problem is analyzed by using the rigid-plastic hybrid element method. The rigid-plastic hybrid element method is a mixed approach of the rigid-plastic domain-BEM and the rigid-plastic FEM. For the rigid- plastic hybrid element method, the compatibilities of not only velocity but also velocity's derivative between the adjoining boundary elements and finite elements can be met, the velocity and the derivative of the velocity can be calculated with the same precision. While, for the conventional rigid-plastic FEMs, the velocity and the derivative of the velocity cannot be calculated with the same precision. Therefore, it may be considered that the rigid-plastic hybrid element method is a more precise method than the conventional rigid-plastic FEMs.

In simulations of metal forming, the conventional rigid-plastic FEMs [1,2] are formulated with single variable (velocity), and the compatibility of velocity's derivative cannot be met. The BEMs are formulated with mixed variables, and the compatibilities of function and function's derivative can be met, so that BEMs possess a merit in that the function and the function's derivative can be calculated with the same precision. While in metal forming problems, boundary conditions of friction and free surface are in common use, and these boundary conditions cannot be easily imposed in BEMs.

In this paper, to impose the boundary conditions of friction and free surface, the rigid-plastic hybrid element method is formulated. This method is a mixed approach of the rigid-plastic domain-BEM and the rigid-plastic FEM in which the boundary conditions of friction and free surface can be imposed easily. In the rigid-plastic hybrid element method, the division of the domain for the rigid-plastic domain-BEM is the same as that of the element for the rigid-plastic FEM, and the equations of the two methods are simultaneously used, so that the compatibilities of both velocity and velocity's derivative between the adjoining boundary elements and finite elements can be met, the velocity and the velocity's derivative can be calculated with the same precision for the rigid-plastic hybrid element method.

In the rigid-plastic hybrid element method, equations derived from the rigid- plastic FEM and equations derived from the rigid-plastic domain-BEM are used, simultaneously. The regular BEM [3,4,5] is employed to obtain required independent equations, which are the domain-boundary integral equations, in the rigid-plastic domain-BEM. Coefficient matrices of the domain-boundary integral equations in the domain-BEM are sparse ones, while the coefficient matrices of equations in ordinary BEMs are not sparse ones.

A plane strain forging problem is analyzed by using the rigid-plastic hybrid element method and the rigid-plastic FEM. For the rigid-plastic hybrid element method, the three-noded quadratic boundary element and the eight-noded quadratic finite element are adopted, respectively. For the rigid-plastic FEM, the eight-noded quadratic finite element is adopted. Comparisons between the calculated results of the two methods have been made, them are about the same, while somewhat differences between them have also been seen.

1

C.H. Lee and S. Kobayashi, "New solutions to rigid-plastic deformation problems using a matrix method", Transactions of the ASME, Journal of Engineering for Industry, 95(3), 865-873, 1973.

2

K. Osakada, J. Nakano and K. Mori, "Finite element method for rigid-plastic analysis of metal forming: Formulation for finite deformation", International Journal of Mechanical Sciences, 24, 459-468, 1982. doi:10.1016/0020-7403(82)90056-X

3

C. Patterson and M.A. Sheikh, "A regular boundary element method for fluid flow", International Journal for Numerical Methods in Fluids, 2, 239-251, 1982. doi:10.1002/fld.1650020303

4

C. Patterson and M.A. Sheikh, "Application of the direct regular methods to linear elastic fracture mechanics", Boundary Elements, ed. C.A. Brebbia, Springer-Verlag, 9-29, 1984.

5

T.A. Cruse, "An improved boundary integral equation method for three dimensional elastic stress analysis", Computers & Structures, 4, 741-754, 1974. doi:10.1016/0045-7949(74)90042-X

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