For inverse analysis, an objective function to be minimized should be defined. For the inverse heat transfer analysis, the objective function should be defined in terms of the temperature difference between the FEM prediction and experimental measurement. Therefore, we assumed the objective function f to be minimized as follows:

where t denotes time, tc denotes the thermocouple number, FEM Ttct denotes the temperature at time t obtained from the FEM simulation for thermocouple locations, and FEM Ttct denotes the temperature at time t measured at the same thermocouple locations.

Before conducting the FEM forming simulation coupled with heat conduction, it is necessary to assume the form of the interface heat transfer coefficient. In the present investigation, we assumed the interface heat transfer coefficient to be

That is, hint was assumed to be linearly proportional for contact pressures up to 0.1 MPa and constant for contact pressures equal to or greater than 0.1 MPa. Although the interface heat transfer coefficient was regarded as a function of the contact pressure or process time in previous investigations (Lee et al., 2012; Kim and Kang, 2010), we

assumed that the heat transfer coefficient depends linearly on the contact pressure in the range of small pressures and that it is a constant above the critical contact pressure, which was assumed to be 0.1 MPa in the present investigation.

Various interface heat transfer coefficients for dies were applied in the FEM simulation of the hot press forming process. The objective function in equation (1) was minimized using search optimization techniques to determine the interface heat transfer coefficient of the die surface. As the interface heat transfer coefficient hint could be very small under the low contact pressure condition based on equation

(2), considerable values of hint were mainly concentrated around the corners of the top and bottom dies. Finally, the

value of hint was determined to be 8.5 kW/m2K around the die corners where the die came into contact with the

workpiece. The heat transfer coefficient at the regions where there was no contact between the die and the workpiece, i.e., 0.02 kW/m2K, was approximated as the value for the natural cooling condition.

3. EVALUATION OF DIE LIFE BASED ON STRESS ANALYSIS

Using the interface heat transfer coefficient obtained by the inverse approach, a coupled FEM analysis of the hot press forming process was performed using the commercial finite element program DEFORM-2D. The analysis led to the estimation of the temperature and stress states of the top and bottom dies. In the FEM analysis, the top and bottom dies were assumed to be an elastic body made of STD61, and the workpiece was assumed to be a plastic material of boron steel 22MnB5, which is commercially used for hot press forming. The flow stress of the workpiece played a crucial role in determining the elastic stress state of the die.

In the case of diffusion-controlled phase transformation, the volume fraction of transformed phase i from austenite is calculated based on the Avrami type equation, which is expressed by (Lee et al., 2009).

Here, Xieq is the thermo-dynamical equilibrium fraction of phase i determined from the equilibrium phase diagram, and Bi and ni are empirical material constants defining transformation for phase i.

The diffusionless transformation of austenite to martensite is obtained based on the empirical equation proposed by Koistinen and Marburger (1959). The volume fraction of martensite Xm at the current temperature T below Ms is defined as follows:

where Xa and Ms are the retained austenite volume fraction and martensite start temperature, respectively.

For hot press forming, the strain-rate   is defined as

288 D. Y. KIM, H. Y. KIM, S. H. LEE and H. K. KIM

where  ,  ,  ,  , and   represent the strain-rate components corresponding to elastic, plastic, thermal, phase transformation, and transformation plasticity, respectively.