Elastoplastic deformation of ferritic steel and cementite studied by neutron diffraction and self-consistent modelling
Introduction
Internal and residual stresses in materials have a considerable effect on material properties, including fatigue resistance, fracture toughness and strength. These stresses can vary greatly as a function of position within engineering components, due to manufacturing processes during the production route. Their measurement and interpretation is thus of considerable interest to the engineer. This paper will address a number of issues relating to mechanisms of materials deformation, and to the practical measurement of stress in components by diffraction from the Fe/Fe3C system. The paper also describes the new in situ loading facilities on the ENGIN diffractometer at ISIS.
The strength of a two phase material depends both on the level of in-situ matrix strengthening (for example due to an increased dislocation density from the presence of the reinforcing phase), as well as on the efficiency of load transfer between the matrix and reinforcing phase. If a high proportion of the applied load is carried by the reinforcing phase, the composite is efficient. The level of load sharing is dependent on reinforcement volume fraction, shape and orientation and on the relative elastic properties of the phases [1]. The partitioning ratio of an applied load between matrix and reinforcement will remain constant with increasing applied load provided that both phases remain elastic. However, once the stress levels in one of the phases are high enough for relaxation or inelastic deformation processes to occur, the load partitioning ratio changes.
One experimental technique ideal both for profiling macro-strain variations as a function of position, and for measuring phase specific load transfer, is neutron diffraction. Relying on the same physics describing the analagous measurement of stress using X-ray diffraction [2], neutrons have one principal benefit compared to traditional X-rays for the engineer or materials scientist. Since neutrons interact primarily with the nucleus, rather than the electron cloud as X-rays do, the penetration depth of neutrons is very large compared to X-rays. For example, the penetration depth (1/e) for steel is ∼1 cm for thermal neutrons, but less than 10 μm for X-rays of comparable wavelength [3]. This results in neutrons being a probe suitable for bulk average measurements of material properties. For many practical applications, it is bulk averaged properties that the engineer or material scientist is primarily interested in.
Diffraction measurement of strain involves the monitoring of changes in separation of one or more suitably orientated crystallographic lattice planes. It is thus a direct measure of the elastic strain in the material. In a polycrystal a diffraction peak represents the average lattice separation over all the grains in the irradiated volume which are suitably oriented to diffract. In the presence of an applied stress, the responses of individual diffraction peaks may differ in magnitude with respect to each other, and to the continuum macrostrain (e.g. [4]). The importance of understanding the mechanisms of interaction between differently orientated crystallites is highlighted when one considers that not only do individual grains exhibit elastic stiffness anisotropy, but also that plastic relaxation occurs preferentially on certain slip systems (plastic anisotropy). Despite increasingly sophisticated models (e.g. [5]) the current state of the art is far from quantitatively predicting the evolution of hkl dependent strains or the implications of their spread on failure. The use of self-consistent models does however give a direct insight into plasticity in polycrystals; the onset of deformation in differently oriented crystallites and the importance of hardening mechanisms. Systems where an elastic phase interacts with a ductile phase have been treated extensively using continuum mechanics arguments [1], but are relatively unexplored using crystal plasticity arguments. This paper applies a self consistent model to the deformation of bcc iron with a small volume fraction of cementite (Fe3C) present, and compares the results with lattice reflection data from neutron diffraction experiments. Wilson and Konnan [6] first measured lattice strains in the phases of a spheroidised high carbon steel using X-ray diffraction in 1964; such results however suffer from near surface effects due to the low penetration depth of traditional X-rays. More recently, residual stresses have been studied in both phases of a similar material in studies on fatigue crack propagation [7]. Oliver et al. [8] have reported interphase and intergranular measurements on a high carbon steel (19% cementite); these measurements were complicated by the presence of significant Lüders straining which caused very rapid transfer of load from the ferrite to cementite phases.
In the interpretation of many strain mapping measurements the assumption is made that macrostrain effects outweigh microstrain effects in importance, at least as far the stress measurement engineer is concerned, and despite the atypical responses for specific reflections associated with intergranular effects. The practical outcome of this assumption is that a suitable (individual) plane for macrostrain profiling is defined as one that is little affected by intergranular strains. More specifically it is one whose response is linear with applied or with residual stress. Indeed, it is on the selection of a suitable reflection that the efficiency of measurements at constant wavelength sources relies since it is often impractical to measure multiple reflections. At pulsed neutron or energy dispersive X-ray sources however, a diffraction pattern is recorded over a range of lattice spacings (or wavelengths) and thus many diffraction peaks are recorded (in the same sample direction). This has been used to provide information about intergranular interactions, which are fundamental to an understanding of polycrystalline plasticity [9], [10], [11], [12]. However we can also analyse the data from the viewpoint of identifying a suitable parameter for macrostrain profiling. In previous papers [13], [14] we analysed diffraction data obtained during a uniaxial loading test on face centred cubic steel and hexagonal close packed beryllium by Rietveld refinement. In those cases a “macrostrain” response was obtained from the lattice parameters which was comparable in linearity with the “best” single peak fits, with stiffness very close to the macroscopic modulus. The result was obtained despite the imperfection of the Rietveld fit, which assumed an “undeformed” crystal structure. This paper extends the discussion to a body centred cubic crystal structure.
Section snippets
Uniaxial tensile test on iron
A series of uniaxial tensile loads was applied sequentially to a specimen in situ on the ENGIN instrument of the PEARL beamline at the ISIS pulsed neutron facility, Rutherford Appleton Laboratory. The load frame is newly commissioned, and purpose built for use within the neutron beam. Fig. 1 shows the rig in place on the beamline. The loading axis is horizontal and typically at 45° to the incident beam, allowing simultaneous measurement of lattice plane spacings both parallel and perpendicular
Self-consistent modelling
The elastic–plastic properties of a polycrystalline aggregate have been described using the Hill self-consistent approach [10], which was first implemented by Hutchinson [18]. A population of grains is chosen with a distribution of orientations and volume fractions that match the measured texture. Each grain in the model is treated as an ellipsoidal inclusion and is attributed anisotropic elastic constants and slip mechanisms characteristic of a single crystal of the material under study.
Rietveld refinement of diffraction data
At time-of-flight sources, such as ISIS, neutron pulses, each with a continuous range of velocities and therefore wavelengths, are directed at a specimen. The flight times of detected (diffracted) neutrons are measured, allowing calculation of wavelengths, and the recording of diffraction spectra. The incident spectra are polychromatic, thus all possible lattice planes are recorded in each measurement. The scattering vectors for all reflections recorded in one detector lie in the same
Macroscopic response
The agreement achieved between the model and experimental macroscopic curves is reasonable (Fig. 3). However, two important discrepancies exist, which further modifications to the hardening parameters do not remove. Firstly a reverse yield can be seen at the base of the experimental unload, which is not captured in the model. Secondly, the measured macroscopic response immediately after yield is somewhat softer than shown in the model; the plastic hardening curve is initially shallower in the
Conclusions
Good agreement has been achieved between experimental and model data for uniaxial loading of a simple two phase iron–iron carbide system, in which the elastic properties of the two phases are the same, while only one phase has been modelled as exhibiting plastic slip.
The use of Rietveld refinement, either with or without the anisotropy strain, has been shown to provide a good description of the behaviour of the iron. The importance of considering the influence of even a small volume fraction of
Acknowledgments
Diffraction experiments were carried out at the ISIS pulsed neutron facility at the Rutherford Appleton Laboratory, Oxon., UK, under support from the Engineering and Physical Sciences Research Council. The authors would like to acknowledge Daimler–Chrysler for supplying the samples, and Lyndon Edwards for considerable aid in the design and specification of the stress rig. The stress rig was designed and built in a collaboration between ISIS, the Open University and Instron Plc and was funded by
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