An Efficient Inverse Method for Determining the Material Parameters and Coefficient of Friction in Warm Rolling Process

Abstract

In the present work, the material parameters for power law and the coefficient of friction are obtained using inverse analysis by measuring exit strip temperature and slip. The procedure makes use of a finite element model for deformation and an analytical method for the estimation of temperature. A heuristic optimization algorithm is used for this purpose that minimizes the error between the measured and estimated flow stresses. The method is verified by conducting some numerical experiments.

Estimation of Temperature Distribution in the Strip at Exit Side

The temperature distribution of strip at the exit side is obtained as (Kim et al. 2009)

$$T_{e} \left( {y,t} \right) = T_{0} + \sum_{n = 1}^{\infty } {{ \exp }\left( { - \frac{{k_{s} }}{{\rho c_{p} }}\lambda_{n}^{2} t} \right)\left( {\frac{{4\lambda_{n} \int\nolimits_{0}^{{\frac{{h_{2} }}{2}}} {T\left( {y,0} \right)} { \cos }\left( {\lambda_{n} y} \right){\text{d}}y - 4T_{0} { \sin }\left( {\lambda_{n} \frac{{h_{2} }}{2}} \right)}}{{\lambda_{n} h_{2} + { \sin }\left( {\lambda_{n} h_{2} } \right)}}} \right)} { \cos }\left( {\lambda_{n} y} \right),$$

(27)

where T(y, 0) is the temperature at the exit of deformation zone that corresponds to time t = 0, T ₀ is the temperature of the coolant, h _a is the convective heat transfer coefficient at the strip surface, k _s is the thermal conductivity of strip, ρ is the density of strip, c _p is the specific heat, and λ _n are obtained by solving the following equation:

$$k_{s} \lambda_{n} { \sin }\left( {\lambda_{n} \frac{{h_{2} }}{2}} \right) - h_{a} { \cos }\left( {\lambda_{n} \frac{{h_{2} }}{2}} \right) = 0.$$

(28)

The temperature below the sensor is obtained by substituting

$$y = \frac{{h_{2} }}{2}\quad {\text{and}}\quad t = \frac{\text{distance from the exit of roll bite to sensor}}{\text{exit velocity of the strip}} \,$$

(29)

in Eq. (27).