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PhD Defence Hans Mulder | Differential hardening of low carbon steel in sheet metal forming

Differential hardening of low carbon steel in sheet metal forming

The PhD Defence of Hans Mulder will take place in the Waaier building of the University of Twente and can be followed by a live stream.
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Hans Mulder is a PhD student in the department Nonlinear Solid Mechanics. (Co)Supervisors are prof.dr.ir. A.H. van den Boogaard from the faculty of Engineering Technology and dr.ir. H. Vegter from Tata Steel.

A significant deviation from the isotropic hardening concept is observed for the equibiaxial stress state of ferritic steel grades at the start of deformation. An observation that is generally found for all ferritic steel grades and which appears to be more apparent for strongly textured materials, like extra-deep-drawing steel grades. This differential hardening behaviour is not caused by predeformation due to temper rolling. Temper rolling is a required processing step for commercially supplied sheet steel. An identical differential hardening behaviour was observed prior to temper rolling for an interstitial-free steel grade.

Work hardening and anisotropic behaviour of sheet metals are well described by current constitutive models that are available for finite-element simulations, but the observed differential hardening phenomenon is not captured by any of the current models. There are two principle directions to model differential hardening as an extension to the current constitutive models:

  • Model one reference hardening curve, similar to the isotropic hardening concept, in combination with a yield locus that changes in shape with plastic work. The shape of the yield locus reflects the ratio between the yield stresses at the relevant level of plastic work.
  • Model multiple hardening curves for different load cases. It is convenient to specify stress as a function of plastic work instead of strain. To establish the yield point at an arbitrary stress state, interpolation between neighbouring load cases at equal plastic work is required. The flexibility in specifying hardening implies a separate plastic potential to establish the plastic strain-increment tensor.

The equibiaxial stress-strain response that is used for the observed differential hardening phenomenon, is established using the stack compression test. In this test a specimen of stacked sheets is compressed in the through-thickness direction. To ensure that the observed stress-strain response is the true material response and not an artefact of the test, two equibiaxial stress tests are comprehensively analysed. Next to the stack compression test, that is the hydraulic bulge test. In this test a clamped sheet metal is stretched by an applied hydraulic pressure. Improvements to the evaluation procedure of the hydraulic bulge test are proposed to obtain more accurate results. With the improvements the bulge test is accurate above 0.03 strain. Test results from both tests are compared, as well as with biaxial tensile test results using a cruciform specimen. Biaxial tensile tests using a cruciform specimen are reliable, but without special sample preparations the test is limited in applicability to the start of deformation. The three tests show a very good agreement, which indicates a high reliability of the equibiaxial stress-strain response.

The observed stress-strain response in a test does not only depend on the underlying work hardening behaviour of the steel grade, but also on the strain rate and temperature during the test. Based on an available temperature model for the uniaxial tensile test, temperature models for the stack compression test and the hydraulic bulge test have been developed, which allows compensation of the stress-strain responses for strain rate and temperature. After compensation, the results of the three equibiaxial tests for the observed steel grade are nearly identical for the overlapping strain ranges. With the compensated uniaxial stress-strain curve, the observed differential hardening phenomenon is confirmed.

Crystal plasticity simulations are used to evaluate possible root causes for the differential hardening phenomenon. In crystal plasticity simulations the grain structure and orientation distribution of the steel are modelled. Deformation is simulated according to the underlying mechanism of shear on slip systems in the crystal lattice.

Statistical crystal plasticity simulations use the volumetric distribution of crystal orientations in the metal to predict the stress-strain behaviour, without considering the grain structure. With this simulation method differential hardening could only be simulated as a result of texture developments due to deformation. That is a gradual process, which doesn’t explain the observed significant deviation from isotropic hardening for the equibiaxial stress-strain response at the start of deformation.

Full field crystal plasticity simulations consider the grain structure as well as the crystal orientation within the grains. These simulations therefore include the influence of grain interactions on the macroscopic behaviour. The expected isotropic hardening behaviour for a polycrystalline metal is accurately predicted for the observed steel grade after the initial deviation from isotropic hardening in the first 5 to 10% of deformation. Next to the gradual change of texture as a result of deformation, full field crystal plasticity simulations have shown that there is another mechanism that may cause differential hardening. Different, independent hardening behaviour on individual (families of) slip systems will influence the slip system partitioning and thereby cause differential hardening. The possibilities to differentiate hardening on individual slip systems are limited for the full field crystal plasticity software that was used. The observed differential hardening phenomenon could not be simulated with this model. Most of the possible explanations for this phenomenon could be rejected based on the simulation results. One possible explanation that could not be verified or rejected with the available crystal plasticity software is the non-Schmid behaviour of ferritic steel.

Deformation in crystal plasticity simulations is modelled as shear on a fixed number of slip systems in the crystal lattice. Shear always occurs in a direction of dense stacking of atoms because the lattice resistance in that direction is lowest. The Schmid behaviour states that deformation only occurs when the projection of the external stress tensor on a shear stress for a slip system, the resolved shear stress, exceeds a minimal value, the critical resolved shear stress. For body centered cubic lattices, which includes ferritic steel grades, deviations on this behaviour have been observed already from the moment that Schmid formulated his law. There are two known deviations for ferritic steel at the start of deformation. The first is that sometimes shear is observed in a different crystallographic orientation than predicted by the designated slip systems. The second is that shear on a designated slip system may occur at a resolved shear stress below the critical resolved shear stress. Both deviations occur at the start of deformation and are temperature dependent. This non-Schmid behaviour is a possible and plausible cause for the observed differential hardening behaviour. Modelling of non-Schmid behaviour is outside the context of the current analysis.

The isotropic hardening model is used in most forming simulations. The yield locus for steel in this case is being established on average stress ratio between the load cases for the established stress-strain curves at equal plastic work. The accuracy of forming simulations, using a yield locus that is established in this way, is remarkable in view of the observed deviation. That is confirmed in a hemispherical punch stretching example where equibiaxial hardening is important for the accuracy of the simulation. It is a possible explanation why the observed differential hardening phenomenon didn’t get more attention in nresearch as well as modelling. It is recommended to critically review forming simulations where biaxial stretching is limited to a relatively low level. If isotropic hardening simulations are less accurate in these situations, differential hardening may improve the prediction.