Constitutive behaviour under hot stamping conditions
Introduction
Nowadays direct hot stamping is common practice in automotive manufacturing to produce parts with complex shapes and high strength. The hot stamping process starts with a heating step to austenize the blanks in a furnace, typically at 900 °C. When the blank is released from the furnace, it is quickly transferred to a press. During this transport the blank remains in the austenitic phase by virtue of its slow transformation kinetics. In a quick press stroke, the product gets its final shape, and is quenched to a hardening structure. The final in-press transformation ensures that no residual stresses are present in the product, and springback is minimized. For FE analysis of the hot stamping process, an accurate description of the material model is essential. Therefore, the hardening behaviour as a function of temperature and strain rate must be described for the austenitic phase. Åkerström (2004) developed a method to determine the mechanical response (flow stress) for the austenite, based on multiple overlapping continuous cooling and compression tests in combination with inverse modelling. Hein (2005) described a global approach for FE analysis of hot stamping where all parameters needed for a good simulation are described. Flow curves for the austenitic state between 650 and 900 °C for different strain rates are recommended. It is also stated that at temperatures lower than 600 °C, problems can occur due to phase transformations. Merklein et al. (2006) described the effect of strain rate on 22MnB5 tensile tests. Turetta (2008) showed hot tensile tests on standard tensile specimens according to ISO 10130. Hardening curves for different temperatures and strain rates were measured. Lechler et al. (2008) used a phenomenological model for the generation of hardening curves as a function of strain rate and temperature. This model is a multiplicative model: it assumes that the work hardening rate of the flow stress is a product of the strain rate dependency and the temperature sensitivity.
An improved sample geometry and test set-up for use in a Gleeble tensile testing machine is given to avoid the inhomogeneous heating as is observed in the standard sample. Also a physical modelling approach for the dependence from strain rate and temperature is proposed since a correct description of strain-rate/temperature sensitivity is crucial for predicting strain non-uniformities. The model recognizes two distinct strain-rate/temperature effects on the flow stress, the first being due to dislocation glide resistance which pertains to dislocation propagation, the second due to dynamic recovery and which pertains to dislocation multiplication (work hardening). They can be separated experimentally by first validating the dislocation glide resistance at the yield point, where work hardening is absent, followed by a fit of the work hardening function to the post-yield part of the hardening curve. The model is based on dislocation theory by the principle of additive contributions of yield stress, glide resistance and work hardening to the flow stress, described e.g. by Klepaczko and Chiem (1986), who discuss the fundamentals of how to construct constitutive relations on the basis that the total flow stress is the sum of the effective stress (or glide resistance due to surmounting local obstacles by dislocations) and the internal stress (which pertains to dislocation multiplication). They show the validity of this concept from rate jump experiments, and recognize that these contributions cause, respectively an instantaneous and a strain dependent strain rate sensitivity. van Liempt et al. (2002) published a flow stress model based on this principle of additive flow stress contributions. Also Sarkar and Militzer (2009) propose a similar approach of flow stress modelling at elevated temperatures. The individual flow stress contributions are described in several papers. Bailey and Hirsch (1960) experimentally validated the correlation between dislocation density and flow stress that was first proposed by Taylor (1934). Krabiell and Dahl (1981) described a function for the dislocation glide resistance which is the stress to move mobile dislocations at the required velocity at a specific temperature.
The work hardening theory used is based on theories using dislocation density and dynamic recovery and annihilation or alternatively remobilization of dislocations. Kocks (1976) characterized recovery as dislocations getting “annihilated or becomes ineffective in some other way at each potential recovery site”, where Bergström (1969–1970), interpreted it as remobilization of stored dislocations. The Bergström model was later developed further by Vetter and van den Beukel (1977) by incorporating the effect of dislocation density in the storage term, yielding the Bergström model mathematically identical to the Kocks–Mecking theory. Kocks and Mecking (2003) later revisited Kocks’ model, discussing further implications of the theory.
With the parameters found for the equations, a strain rate and temperature dependent hardening description can be made. The parameters were derived from a limited dataset within the hot stamping regime. To verify the accuracy of the predicted curves, they were compared with the measurements and simulations with these curves were compared with thickness measurements on a hot formed part.
Section snippets
Basic principle
The goal for this testing programme is to have accurate measurements, which requires tensile tests with a homogeneous temperature along the deformation area of the sample, including the shoulders, for which special sample shape and set-up have been designed. Another goal is to cover a wide range of temperatures and strain rates with a limited amount of tests. Therefore the tests are parameterized with a model based on physical parameters. The use of a physical model increases the understanding
Workflow and boundary conditions
For the input of the finite element code, a dataset of hardening curves spanning a broad range of forming temperatures and strain rates is necessary. To limit the amount of tests, and also find a solution for some difficult combinations of strain rate and temperature where phase transformation occurs, a parameterized model is a useful method to determine the hardening curves. For hot stamping simulations the hardening description must be valid in a range of temperatures and strain rates. The
Discussion
With all parameters and constants known, curves can be calculated for all temperature and strain rate combinations for the austenitic condition. Note that the dynamic strain rate influence cannot be negative and needs to be cut off at zero for negative values. The hardening curves from the model were compared with the direct fits on the measured data. In Fig. 14, the comparison for the measurements at different strain rates and temperatures is shown.
To quantify the accuracy of the model, the
Conclusions
The heat distribution in the tensile bar for tensile tests under hot conditions is very important. It is possible to create a homogeneous temperature distribution in the tensile bar by using shunts. These shunts allow for an equal current density in the complete sample. Samples without these shunts have the risk of an inhomogeneous temperature distribution resulting in non-austenized areas of that sample. Considering that for high temperatures ferrite behaves softer than austenite, necking can
Acknowledgements
We would like to thank Delft University of Technology for facilitating the Gleeble test machine, Nick den Uijl, Roy Frinking and Hans Hofman for their contribution to the testing work on the Gleeble test machine.
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