3.2.4 Modeling Constitutive Behavior of Steels

Kevin J. Coakley

Dom Vecchia

Statistical Engineering Division, ITL

Yi-Wen Cheng

Materials Reliability Division, MSEL

Scientists in the Materials Reliability Division seek to improve the quality of sheet metal products manufactured by hot-strip rolling. The project is funded by the American Iron and Steel Institute and the Department of Energy. To achieve this goal, it is necessary to understand how a metal deforms under high stress. Based on experimental data collected at NIST, SED is developing a statistical model for predicting stress-strain behavior of metals as a function of chemistry, grain size, temperature and initial strain rate. The current model is an improved version of an earlier model developed at NIST.

The prediction variables in the new model are normalized so that the relative contribution of the different sources of variability are apparent. In the model, there are two terms in the prediction for stress. The first prediction term is a monotonically increasing function of strain. The second term represents a correction due to the dynamic recrystallization of the material. Due to this effect, stress is not necessarily a monotonically increasing function of strain.

Due to the high number of parameters in the model, the estimated parameters were obtained using a regularization approach. The model parameters are determined by minimizing a loss function which is the weighted sum of two terms. The first term is the sum of squared residuals. The second term is a penalty function. The model predicts the asymptotic value of stress for large values of strain. The penalty function is large when the predicted asymptotic value of stress is far from a prior estimate of the aysmptotic value. A weighting factor determines how much influence the penalty function has, relative to the sum of squared residuals term, in determining the parameter values. Estimates were obtained for various values of the weighting factor. Scientific judgement was used to select the best value of the weighting factor.

The new model has better theoretical properties than does the earlier model. For certain choices of initial strain rate and temperature, the stress-strain curves should satisfy monotonicity constraints. For low to moderate strains, as strain is increased, the predicted stress curves for different grain sizes should not cross. However, the predicted stress curves from the older model did cross. In contrast, the predicted stress curves from the improved model do not cross.

In coming months, data collection and validation studies will be based on formal statistical planning to further refine or replace stress-strain models.

Figure 14: The measured stress-strain curves and a statistical model for the curves are compared for different experimental conditions. Due to dynamic recrystallization, strain is not a monotonic function of stress.

Date created: 7/20/2001
Last updated: 7/20/2001