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

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.