Engineering Transactions, 49, 4, pp. 637–661, 2001
10.24423/engtrans.546.2001

Nonlinear Regression Problem of Material Functions Identification for Porous Media Plastic Flow

Z. Nowak
Institute of Fundamental Technological Research
Poland

A. Stachurski
Institute of Control and Computation Engineering
Poland

In the paper we present the identification problem arising in modelling the processes of nucleation and growth of voids in the elastic–plastic media. Identification is carried out on the basis of Fisher's data measured on the cylindrical steel specimens subjected to the uniaxial tension. The identification problem is formulated as the standard nonlinear regression problem. Our aim was to select appropriate formulae of the material functions appearing in the porosity model in the right-hand side of the differential equation, and to identify their unknown parameters. The resulting nonlinear regression problem was solved by means of the global optimization method of Boender et al. As the local minimizer we have implemented the modified famous BFGS quasi-Newton method. Modifications were necessary to take into account box constraints posed on the parameters. As the directional minimizer we have prepared a special procedure joining quadratic and cubic approximations and including a new switching condition. We have tested two variants of the porosity model; in the first one with variable shape of the material function $g$, and the second one – with constant $g$. The results suggest that the model with material function $g \equiv 1$ describes well the nucleation and growth of voids. However, our attempt to identify that constant has brought an unexpected value smaller than 1, and approximately equal to 0.84.
Keywords: plastic flow of porous media; material functions identification; global optimization; nonlinear regression; nonlinear programming
Full Text: PDF
Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN).

References

A.A. AFIFI, S.P. AZEN, Statistical analysis. A computer oriented approach, Academic Press, New York 1979.

C.G. BOENDER, A.H.G. RINNOOY KAN, L. STROUGIE, G.T. TIMMER, A Stochastic method for global optimization, Mathematical Programming, 22, 125–140, 1982.

P.W. BRIDGMAN, Studies in large plastic flow and fracture, McGraw-Hill, 1952.

C.C. CHU, A. NEEDLEMAN, Void nucleation effects in biaxially stretched sheets, Trans. ASME, J. Engng. Materials and Technology, 102, 249–256, July 1980.

W. FINDEISEN, J. SZYMANOWSKI, A. WIERZBICKI, Theory and computational methods of optimization [in Polish], PWN, Warszawa 1977.

J.R. FISHER, Void nucleation in spheroidized steels during tensile deformation, Ph.D. Thesis, Brown University, June 1980.

R. FLETCHER, Practical methods of optimization, second edition, John Wiley & Sons, Chichester 1987.

A.A. GOLDSTEIN, On steepest descent, SIAM J. Control, 3, 147–151, 1965.

A.L. GURSON, Continuum theory of ductile rupture by void nucleation and growth. Part 1. Yield criteria and flow rules for porous ductile media, J. Engng. Materials and Technology, Trans. of the ASME, 99, 2–15, 1977.

R. HILL, Mathematical plasticity, Oxford Press, 1950.

K. LEVENBERG, A Method for the solution of certain nonlinear problems in least squares, Quart. Appl. Math., 2, 164–168, 1944.

D.W. MARQUARDT, An algorithm for least squares estimation of nonlinear parameters, SIAM Journal on Applied Mathematics, 11, 431–441, 1963.

A. NEEDLEMAN, J.R. RICE, Limits to ductility set by plastic flow localization, D.P. KOISTINEN, N. M. WANG, [Eds.], Mechanics of Sheet Metal Forming, 237–267, Plenum, New York 1978.

Z. NOWAK, A. STACHURSKI, Global optimization in material functions identification for voided media plastic flow, submitted for publication in CAMES, 2002.

P. PERZYNA, Internal state variable description of dynamic fracture of ductile solids, Int. J. Solids Structure, 22, 797–818, 1986.

P. PERZYNA, Z. NOWAK, Evolution equation for the void fraction parameter in necking region, Arch. Mech., 39, No. 1–2, 73–84, 1987.

W.H. PRESS, S.A. TEUKOLSKY, W.T. VETTERLING, B.P. FLANNERY, Numerical recipes in C: The art of scientific computing, Cambridge University Press, Cambridge 1993.

J.W. RUDNICKI, J.R. RICE, Conditions for the localization of deformation in pressure–sensitive delatant materials, Journal Mech. Phys. Solids, 23, 371–394, 1975.

M. SAJE, J. PAN, A. NEEDLEMAN, Void nucleation effects on shear localization in porous plastic solids, Int. J. Fracture, 19, 163, 1982.

A. STACHURSKI, A.P. WIERZBICKI, Introduction to optimization [in Polish], Publishing House of the Warsaw University of Technology, Warsaw 1999.

A. TÖRN, A. ŽILINSKAS, Global optimization, Springer Verlag, Berlin, Heidelberg 1989. Received December 29, 2000.




DOI: 10.24423/engtrans.546.2001