Multi-scale characterization and modelling of damage evolution in nuclear Gilsocarbon graphite
Dong Liu1, Peter Heard1, Branko Šavija2, Gillian Smith1, Erik Schlangen2, and Peter Flewitt1,3 1 Interface Analysis Centre, School of Physics, University of Bristol, Bristol UK
2 Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, Netherlands
3 HH Wills Physics Laboratory, University of Bristol, Bristol UK
ABSTRACT
In the present work, the microstructure and mechanical properties of Gilsocarbon graphite have been characterized over a range of length-scales. Optical imaging, combined with 3D X-ray computed tomography and 3D high-resolution tomography based on focus ion beam milling has been adopted for microstructural characterization. A range of small-scale mechanical testing approaches are applied including an in situ micro-cantilever technique based in a
Dualbeam workstation. It was found that pores ranging in size from nanometers to tens of
micrometers in diameter are present which modify the deformation and fracture characteristics of the material. This multi-scale mechanical testing approach revealed the significant change of mechanical properties, for example flexural strength, of this graphite over the length-scale from a micrometer to tens of centimeters. Such differences emphasize why input parameters to
numerical models have to be undertaken at the appropriate length-scale to allow predictions of the deformation, fracture and the stochastic features of the strength of the graphite with the required confidence. Finally, the results from a multi-scale model demonstrated that these data derived from the micro-scale tests can be extrapolated, with high confidence, to large
components with realistic dimensions.
INTRODUCTION
Gilsocarbon graphite is used as moderating and structural components in the reactor core of currently operating gas-cooled nuclear reactors in the UK. However, there remains a need to improve the confidence for the evaluation of the overall durability and integrity of these core bricks despite the wide range of work on the understanding of the fracture of both virgin and irradiated material. Fracture criteria such as maximum stress [1], maximum strain [2] and
maximum energy [3] have been proposed; the maximum stress approach has been adopted by the UK community but it is considered to be conservative. Where the complexity lies is that these polygranular aggregate graphites show quasi-brittle characteristics when deform and fracture [4], i.e. there is a non-linear stage prior to final failure. Unlike ductile materials, this non-linear stress-strain behavior in graphite is attributed to the formation of high-strain zones or distributed micro-cracks prior to the formation of macro-cracks. Post the peak load, the material does not fail in an abrupt manner as linear-elastic brittle materials, but rather follows a ‘graceful’ failure with a rising R-curve due to various toughening mechanisms such as the formation of bridges and deflected crack path. As a consequence, conventional linear-elastic fracture mechanisms do not fully apply to Gilsocarbon graphite.
For these quasi-brittle materials, the change of test specimen size often leads to variations in the measured properties. Conventionally, a Weibull distribution [5] has been mathematically formulated for the evaluation of the randomness in the strength of a brittle material. But, this
approach is considered to be insufficient when addressing graphite-like quasi-brittle materials [6]. The limitation in nuclear community, however, is the available volume of irradiated material that can be removed from plant and tested. Cylindrical samples are trepanned from graphite bricks during inspection and the test data are embedded in computer modelling to predict the service life of bricks within the reactor cores.
For nuclear graphites, quite a few models have been developed with representation of the deformation and fracture at different length-scales [7-9]. One example of numerical models is that proposed by Rose and Tucker [10] which combines weakest link theory from Weibull analysis and a fracture mechanics approach. For these models, there are usually two important inputs: (i) a simplified, experimental-based characterization of the microstructure to represent the main features of the material such as the size and distribution of the filler particles and pores over the service time; and (ii) the mechanical properties for the constituent elements. Both of these must consider the scales in terms of the 2D length or 3D volume of the model that is adopted. It has been recognised that the data produced by conventional macro-tests are not sufficient in the latter case. Therefore, there is a need to develop micro-scale techniques that could provide required information related to the constituent element of the microstructure [11].
In the EXPERIMENT section we describe the approaches that have been adopted for the characterisation of the microstructure and mechanical properties of Gilsocarbon graphite over multiple length-scales. Results are presented and discussed with respect to (i) the challenges of measurements and (ii) the suitability for input data into multi-scale microstructure-based numerical models.
EXPERIMENT
Characterizations of the 3D microstructure
Two approaches have been adopted in the present work for the characterisation of the microstructure of the unirradiated Gilsocarbon graphite: (1) X-ray computed tomography (CT) scan using the facility at Manchester University - see Table I for the experimental parameters for the measurements; (2) High-resolution focus ion beam (FIB) slice and view with Amira
reconstruction in a Helios Nanolab 600i Dualbeam workstation (resolution ~10 nm). In addition, 2D optical imaging has been undertaken to extend the range of observation.
Table I Experimental parameters adopted for the X-ray CT scans Nikon Metrology 225/320 kV Custom Bay
Target Cu Voxel size 15.8 μm
Voltage 65 kV Filter none
Current 140 μA Exposure time 1415 ms
Number of projections 3142 Acquisition time 1h15min
Specimen size 20 mm x 20 mm x 20 mm
Characterization of the mechanical properties
A range of conventional mechanical testing approaches are adopted including three and four point bending, and Brazilian disc compression. At the micro-scale, a technique for in situ testing cantilever beam specimens has been used [12, 13], Fig. 1. This approach combines a Dualbeam FEI Helios NanoLab 600i Workstation and a customized force measurement system (FMS) from Kleindiek Nanotechnik. The procedure of creating the micro-cantilever specimens has been described by Liu et al [14]. The specimens are usually created at the edge of the bulk
material to allow access for the loading probe, Fig. 1. For all the tests the system is calibrated against single crystal silicon. The measurement system instantly outputs the load (with the resolution of 0.01 μN) applied to the cantilever, whereas the displacement at the loading position is measured on the SEM images recorded during the test (with the resolution of 0.5 pixel).
Fig. 1 A scanning electron microscope (SEM) image showing the in situ loading of micro-cantilever by force measurement probe in a Dualbeam workstation chamber.
RESULTS AND DISCUSSION Multi-scale microstructure
The X-ray CT analysis revealed that the size of the pores in the samples cover a range from the resolution limit of 15.8 µm to 0.3 mm dia.; the diameter is an equivalent diameter assuming the pores are spheres, Fig. 2a. The 3D FIB tomography investigated the pores at nano- to micro-scale, Fig. 2b. Combining the two approaches, it was found that the pores in this graphite range in size from 40 nm to 0.3 mm dia. Further, the material comprises matrix, filler particles and macro-pores, Fig. 3a, and localized FIB milling showed that a large amount of nano-scale porosity exists in the solid phase of the particle, Figs. 3b and c, and in the matrix, Figs. 3d and e. The larger pores (≥ 10 µm) occupy about 5-6 vol.% and smaller pores ~15% of the total volume. This is an indication that when seeking to model the mechanical behavior of this material, the role of small pores should be taken into account.
Fig. 2 The distribution of pores obtained by (a) X-ray CT and (b) high-resolution FIB reconstruction.
Fig. 3 (a) Optical image of Gilsocarbon graphite (4 mm x 4 mm field of view) showing the matrix (light grey color), macro-pores (dark grey color) and filler particles; (b) SEM image of a typical particle; (c) a 2D section of the particle in (b) created by ion-milling; (d) SEM image of a typical area of the matrix; (e) a 2D section showing the pores from a ion-milled section of the matrix in (d).
Multi-scale mechanical properties
The mechanical properties of the Gilsocarbon graphite have been obtained over a size range from micrometer to centimeter. At the micro-scale range, the response of the cantilevers are shown in Fig. 4a for the elastic range and in Fig. 4b for the full range to failure. The flexural strength varies from zero when the cantilever samples a large pore and up to 979 MPa. The larger tests for specimens of the size of 20 mmx10 mmx150 mm tested by four point bending with an inner span of 50 mm and outer span 138 mm and Brazilian disc compression (12 mm dia. 4 mm thickness) give a flexural strength of 26.67±4.04 MPa and a tensile strength of 15.70±1.80 MPa, respectively.
Fig. 4 Typical loading curve from micro-scale cantilever beam tests: (a) the linear response of a cantilever (1.9x1.9x15 µm) loaded and unloaded within the elastic range; and (b) a cantilever (2.25x2.25x11µm) loaded to failure.
In addition, Gilsocarbon graphite beams (60 specimens of 6.35mmx6.35mmx200mm,; 60 specimens of 19.05mmx19.05mm x200 mm and 60 specimens of the size of 70mmx70mmx200 mm) were tested under four point bending by Novovic et al [15]; the flexural strengths are found to be 24.68 ± 1.26 MPa, 26.96 ± 1.68 MPa, 22.94 ± 2.37 MPa respectively. These data are consistent with the results produced in the present work for the larger test specimens. Compared with the scatter of the data obtained at the micro-scale, the change of strength with sample size is significantly small. This is an interesting result as it demonstrates that the current Weibull
approach adopted in industry practice to extrapolate data obtained at cm-scale to a size comparable to a reactor core brick is not convincing. This is consistent with the proposal by Mitchell et al [16] that Weibull theory does not predict the failure well for four-point bending and L-shaped bending specimens. Novovic et al [15] pointed out, however, that the scatter of the measured strength seems to change with test specimen size: a smaller scatter for larger
specimens. This applies for graphite as for a smaller sample volume there would be a greater potential for microstructural variation.
Modelling
Fig. 5 (a) a 2D slice through a 3D model cm cube with filler particles in black and pores in white; (b) Simulated microstructures with 20 vol.% porosity (Red - filler particles; White - matrix; Blue - pores); (c) the scatter and average flexural strength decrease as a function of sample size (the solid circle represents the average value).
To extend the understanding of the extrapolation of data obtained from a small test specimen to a large component, a multi-scale microstructure-based lattice-type model is adopted. This approach provides a basis for evaluating the strength of Gilsocarbon graphite at a range of specimen size, Fig. 5. Two important inputs are necessary for the model: one is a representation of the Gilsocarbon graphite microstructure which is shown in Figs. 5a and 5b; the other is the mechanical response obtained at micro-scale. Modelling results have shown that the average mechanical properties (flexural strength) decrease monotonically with the increase in specimen size, Fig. 5c. However, the scatter of simulated mechanical properties decreases for larger specimens. These are consistent with experimental observations.
CONCLUDING COMMENTS
The results indicate to several important factors that should be taken into account when seeking to model the deformation and fracture of Gilsocarbon graphite. (i) The multi-scale microstructure should be considered and the complexity of the constituent elements to be represented by the models, especially the role of small pores as they are about 15% of the total volume. (ii) The micro-scale properties, such as the flexural strength, can be an order of
magnitude larger than those values obtained at millimeter scale from trepanned from reactor bricks. It is important to have the correct length-scale input parameters when modelling complex materials such as Gilsocarbon graphite. To conclude, it is proposed here that the mechanical properties obtained via the cantilever beams at micro-scale are more appropriate as inputs to microstructure-based multi-scale models. However, it worth pointing out that these test
specimens sample nano-scale pores which affect the strength of the cantilevers and the scatter of the data. Ideally, the ‘true’ properties of the constituent elements of Gilsocarbon graphite should be obtained from defect-free test specimens sampled from selected locations. However, this is not practical since it will reduce the size of the cantilevers to a scale at which the measured mechanical properties will be influenced significantly by other factors so that the data are not a true representation.
ACKNOWLEDGMENTS
The authors acknowledge the financial support from the EPSRC grants EP/J019801/1 (Bristol): QUBE: QUasi-Brittle fracture: a 3D Experimentally-validated approach.
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