Towards a notch-sensitivity strength test for irradiated nuclear graphite structural integrity

T h e 4 t h E D F E n e r g y N u c l e a r G r a p h i t e S y m p o s i u m . E n g i n e e r i n g

C h a l l e n g e s A s s o c i a t e d w i t h t h e L i f e o f G r a p h i t e R e a c t o r C o r e s

© EMAS Publishing 2014

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TOWARDS A NOTCH-SENSITIVITY STRENGTH TEST FOR IRRADIATED NUCLEAR GRAPHITE STRUCTURAL INTEGRITY

T.J. Marrow, M.S.L. Jordan and Y. Vertyagina

The effect of radiolytic oxidation on notch strength and its variability is an important element of methodologies to assess the probable development of cracking from stress concentrators such as keyway roots. At present, notch strength must be inferred from flexural tests on smooth specimens, with very limited data on irradiated graphite. In principal, strength measurements are feasible from small, notched specimens fabricated from trepanned material. To be representative, the effects of microstructure in the necessarily small test populations and differences in stress and strain gradients between specimens and components must be accounted for. This paper summarises progress in work to observe deformation and fracture initiation at stress concentrations, using X-ray tomography and digital volume correlation to measure three-dimensional strain fields. High precision synchrotron diffraction studies on strained samples provide new insights into the inelastic deformation of non-irradiated graphite, with implications for the behaviour of irradiated graphite and the effects of specimen size and stress gradients. Finally, novel modelling techniques are being developed to evaluate the sensitivity of small specimen fracture tests to microstructure. The aim of this work is to give confidence in whether such tests on radiolytically-oxidised graphites will be sufficiently representative to support structural integrity assessments.

INTRODUCTION

Structural integrity assessment models for AGR brick cracking have been developed to a significant level, primarily addressing the potential problem of late-life keyway initiated fracture, which is a predicted consequence of the turnaround in brick stresses during the life of the core. The keyways are expected to act as stress concentrators, such that fracture of moderator bricks may propagate from a keyway root under the action of internal and external loading. It is important that models should provide confidence in predictions of the onset and rate of development of keyway root cracking. This may be affected by variations in graphite microstructure between bricks and the response of the

Oxford Materials, The University of Oxford




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fast neutron irradiated and radiolytically-oxidised microstructure to stress concentrations. Such an approach requires models for the role of microstructure in the fracture process.

Nuclear graphite is a quasi-brittle material, at least in the virgin (non-irradiated) condition, and as such exhibits less sensitivity to the stress concentrating effect of notches compared to brittle elastic materials. Notch sensitivity varies between virgin nuclear graphite grades and data from other graphitic and porous materials suggests notch sensitivity may reduce further with increasing weight loss. The models for keyway fracture of AGR graphite have only been validated (some might say “tuned”) using virgin material. There are few data on the fracture behaviour of notched neutron irradiated and radiolytic-oxidised nuclear graphite specimens, and none for radiolytic-oxidised material at high weight loss.

At the Graphite Fracture workshop organised by EDF Energy UK (April 2011, Mansfield College, Oxford) there was a consensus that the effects of high levels of radiolytic oxidation on the notch sensitivity of nuclear graphite were not well understood. This has consequences for predicting the fracture of reactor components such as moderator bricks. Whilst an increased capacity for strain accommodation through microstructural damage might be expected to decrease notch sensitivity, heterogeneous oxidation of the microstructure could encourage damage coalescence and aid crack nucleation. It was recognised that to improve confidence in the prediction of component behaviour there was a need for better models for damage nucleation that were validated by experiment and could make better use of the available graphite property data.

A comprehensive model for graphite component behaviour should take into account the accumulation and distribution of damage in the microstructure, and the criteria for the unstable propagation of cracks that arise from the coalescence of damage. Both the effects of stress concentration and the degree of constraint on damage nucleation and crack propagation should also be understood. An aim of such a model would be to determine the envelope of conditions (component geometry and stress state) within which more simple models such as that currently used (i.e. maximum principle tensile stress) would be sufficient and conservative for the prediction of component failure. A key concern is whether the strength reduction due to the weight-loss associated with radiolytic oxidation affects the strength of notched components and test specimens with the same functional relationship that is observed in the smooth bend test specimens obtained from trepanned cores. The question is; how does the notch sensitivity of features such as keyways vary with radiolytic oxidation?




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The main objectives of the research at Oxford University are (i) to develop a strength test, with a sharp corner, that could be used on specimens of radiolyically-oxidised graphite, machined from the small samples extracted from the reactor cores, and (ii) to develop a modelling framework for the prediction of strength for quasi-brittle material components with stress concentrations. These objectives are being addressed through two EDF-supported projects, and also via participation in the EPSRC-funded project QUBE (Quasi-brittle fracture: an experimentally validated approach, EP/J019992/1). QUBE aims in particular to develop microstructure-dependent damage models for the general class of quasi-brittle materials that include graphite, concrete and ceramic matrix composites, and to implement these within a component model. The damage model will be calibrated using the experimental data on specimen strength and deformation, including the development of the size and shape of the fracture process zone. This paper presents an interim description of the results and findings of this work.

TEST SPECIMEN DEVELOPMENT

The load-displacement trace obtained in a flexural bending fracture test should provide data on the critical stress for nucleation of a crack, thus flexural tests with notched specimens may investigate the effects of irradiation and oxidation on notch sensitivity. However, some test specimen geometries may not be ideal for measurement of the properties for graphite fracture, particularly at high weight loss. Such tests need to performed on samples extracted from the graphite core, which are necessarily small. Small specimen tests might be differently affected than larger components by changes in material properties. In particular, pre-fracture inelastic deformation at the notch and geometric constraints will influence the nominally elastic strain field in the test specimen; this needs to be understood as it may affect the onset of unstable fracture. These effects are analogous to the well-known size-effect in ductile metals that is due to plasticity; in a quasi-brittle material such as graphite such effects will be quite smaller, but need to be quantified in order to have high confidence in the application of small specimen test data to component integrity assessment. To study this, novel methods are being investigated that can measure deformation on the surface and also within small test specimens.

Digital image correlation (DIC) is a powerful tool to measure the full field displacement distribution on the surface of specimens. The strain field may be derived from the displacement field and employed for the early detection of fracture nuclei, to which the strain field is very sensitive. The surface length of the observed cracks can be measured directly from the DIC data and the depth of the cracks may be estimated from the




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opening displacements. DIC has thus been applied to study crack behaviour in a broad range of materials, including nuclear graphite; e.g. (1)-(3). Three-dimensional in-situ observation of damage within materials has become possible through high-resolution X-ray computed micro-tomography (µXCT), e.g. (4)-(6). Digital volume correlation (DVC) can map relative changes in displacements between tomographic datasets (7), allowing quantitative observations of the three-dimensional deformations that occur within materials when they respond to loads. In high quality tomographs of suitable microstructures, the displacement resolution is sub-voxel (a voxel is the three-dimensional equivalent of a pixel), and both elastic and plastic deformations can be studied; e.g.(8)-(10). Crack propagation in virgin, non-irradiated polygranular graphite is accompanied by the development of a micro-cracked fracture process zone (FPZ), and through the combined use of µXCT and DVC, the deformation of the fracture process zone has been measured, and described by cohesive zone modelling (11), (12).

To observe the fracture process zone developing at a stress-concentrating notch, an experiment was performed at the Diamond Light Source (DLS) (experiment EE8519). The experiment involved the driving, and subsequent extraction, of a loading wedge into notched specimens of virgin GCMB grade Gilsocarbon graphite, simulating blunt features with radii of 2 and 4 mm. The samples were cyclically loaded under quasi-static conditions to progressively higher peak loads until a drop in the load with increasing displacement was observed: a ‘pop-in’ of a crack from the notch. During loading and unloading, radiographs were collected at intervals, and X-ray computed tomographs collected in the loaded and unloaded states (voxel size: 1.8 µm). DIC performed on the radiographs was used to measure the Notch Mouth Opening Displacement (NMOD). DVC performed on the tomographs calculated the three-dimensional displacement field and these data were used to investigate the development of the fracture process zone prior to and following crack initiation.

To quantify the deformation behaviour, comparison with a 3-D Finite Element (FE) simulation (Figure 1a) was made. In this analysis a linear elastic FE simulation has been developed in Abaqus/Standard; as in the experiment, the model deformation is driven by the insertion of a wedge, opening the notch. The simulation is composed of quadratic hex elements of approximately 1 mm dimension, with refinement around the wedge-contact area and notch. The wedge contact interaction is defined as surface-to-surface with finite sliding, “Hard” Contact pressure behaviour and the Abaqus Default constraint enforcement method, with no correction made for frictional sliding. The graphite properties used were Young’s modulus, E = 10 GPa, Poisson ratio, ν = 0.2. Figure 1b&c shows a qualitative comparison of the experimental and theoretical displacements in a section across the 4 mm radius notch, measured under an 85 N load; the pop-in load was




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approximately 95 N. The magnitude of the simulation displacements (the maximum is < 5 µm) agrees well with the DVC measured values. The shape of the contours also demonstrates similar trends in the deformations. Mapping the 3D residual error between the data and model can be used to emphasise and quantify their differences; this is shown in Figure 1d at loads of 50, 75 and 85 N, in which it can be seen that the disagreement increases with increasing load. This is partly due the linear elastic model with constant properties; it has been shown previously that the elastic properties of virgin graphite depend on strain (13). Improved fitting with non-linear properties will be used to investigate this in more detail. The contribution of inelastic deformation will also be examined using the data obtained in the unloaded state. The analysis will also investigate the development of damage at the notch tip; stable crack development in the FPZ should be revealed as a discontinuity in the displacement-error field.

Figure 1: Simulation and observation of deformation at a notch; a) FE linear elastic simulation showing the maximum principal strain, b) FE simulated horizontal displacements, Ux, at the notch, c) measured horizontal displacements at the notch, d) error maps of the difference between the measured and simulated Ux displacements at 50, 75 and 85N.

More recently, as part of experiment EE9478 at the Diamond Light Source, several small (6 mm x 6 mm x 19 mm) of virgin Gilsocarbon notched beams were tested in four point flexure (Figure 2a); the material was from the same billet as experiment EE8519. Two geometries of notch were tested: a 2 mm diameter ‘blunt’ U notch, and a ‘sharp’ 90° V notch. X-ray computed tomographs (voxel size:~3.3 µm) were collected at intervals as the samples were loaded under quasi-static conditions to progressively higher peak loads (no unloading) until a drop in the load with increasing displacement was observed. An example is presented in Figure 2b, showing the crack that propagated from the notch. DVC is currently being performed on these tomographs, as above, to examine the effects

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of the reduced constraint and notch geometry on the development of the fracture process zone in comparison to the higher constraint, larger specimens of EE8519.

Figure 2: In situ tomography study of 4-point bend flexure of a small notched graphite test specimen; a) specimen prior to testing, b) tomographic visualization showing a crack propagated from the notch.

A recent experiment (EE9036 at the Diamond Light Source) has made novel measurements of elastic strains in the FPZ developed for a propagating crack in virgin Gilsocarbon graphite; the material is from the same billet as the notched specimens described above. This required synchrotron X-ray diffraction mapping, supported by DVC applied to µXCT. Prior to loading, the region ahead of a notch was mapped by diffraction by rastering a beam that had been reduced via slits to 1.5 x 1.5 mm (the area mapped was 9.5 mm x 5 mm, with step size of 0.75 mm). The (00.2) diffraction ring produced by the graphite crystals in the path of the beam was observed at a distance of 2.55 m (beam energy 80 keV) and a novel cross-correlation method was applied to measure the sub-pixel changes in the ring diameter caused by the change in Bragg diffraction angle with the crystal strains. The same region was also mapped by overlapping tomographs (at 1.8 µm voxel size). Radiography during loading was used to control the experiment by observing crack initiation and propagation until a 4 mm crack length had been developed. Diffraction mapping and µXCT were then repeated with the crack under load. Unfortunately, due to a major power failure that terminated the experiment, only the crack wake could be tomographed. However, the experiment was repeated with a new specimen of the same graphite in April 2014 (experiment EE9478), successfully collecting diffraction and tomography data of the crack wake and fracture process zone as the crack was propagated in several stages up to 10 mm from the notch.

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The data from that experiment are currently being analysed and the results will be published in due course.

The DVC analysis of the tomographs from EE9036 measured the crack opening displacements in the crack wake; the displacement gradients can be used to visualise the crack shape. In Figure 3, the crack opening profile is superposed on a map of the [00.2] crystal strains (i.e. the relative change in the (002) separation). The preliminary data, presented here only for strains perpendicular to the crack plane, show the elastic strain concentration measured around a crack tip in polygranular graphite for the first time. An interesting observation is that a compressive strain change occurs in the crack wake; this is attributed to relaxation via microcracking of the significant thermal strains that exist after graphitization at >2000°C.

Figure 3: A notched graphite sample (top-right) is wedge loaded until a crack is observed via radiography of the notch (bottom-left). DVC of µXCT data (centre) measures the crack opening displacement profile, which is compared with the diffraction-measured crystal strain map obtained from the change in basal (00.2) interplanar distance (left). The rectangle indicates the same region in each image (laser illuminated in top-right).

Work is now in progress to use a calibration of the crystal strains, as a function of bulk engineering strain applied in tension, to quantify the bulk elastic strain field in the FPZ and its surroundings. These data will be compared with FE simulations with the aim of using reverse-modelling to obtain the strain/damage relations that are needed to predict the deformation fields in specimens of different geometry. These will be evaluated using




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the DVC measurements from the small notched bend specimens. Ultimately, it is expected that these methods will be applied to radiolytically-oxidised graphite test specimens in order to determine their suitability as a method of assessing the notch-sensitivity of components of different geometry.

MODELLING

A fully developed model should take into account the accumulation and distribution of damage in the microstructure, and the criteria for the unstable propagation of cracks that arise from the coalescence of damage. Both the effects of stress concentration and the degree of constraint or stress state on damage nucleation and crack propagation should also be understood. An aim of such a model would be to determine the envelope of conditions (component geometry, stress state and oxidation) within which more simple models (i.e. maximum principle tensile stress, such as currently used by EDF) would be sufficient and conservative for the prediction of component failure.

The working hypothesis for model development is that a short crack-like defect develops locally in response to the deformation of the microstructure, propagating stably before its instability leads to fracture of the specimen. This interaction with microstructure contributes to the variability in component strength. The heterogeneous porous structure of polygranular nuclear graphite comprises regions of different elastic moduli, due to local variations in crystal orientation and porosity. The strain to failure and the work of fracture may also vary with microstructure, and should not be expected to correlate directly to elastic modulus. There is no expectation that irradiation and oxidation would affect the local properties of microstructure similarly. A general modelling framework is needed that can represent the key microstructural parameters, and so assess the sensitivity of fracture resistance to variations in microstructure. It is also important that the model is computationally efficient.

For a computationally efficient and flexible modelling approach, we are using cellular automata (CA) Error! Reference source not found., which represents a discrete system consisting of a limited number of cells. The states of these cells change synchronously at every time step and the cell behaviour is determined by the state of this cell and the states of the current environment. Advantages of CA include: cellular automata represents a mathematical alternative for differential equations, which can work in those areas where differential equations cannot; flexible rules can be given for evolution of the system; local interactions in the model will lead to global behaviour of the entire system; local interactions allow the use of parallel high performance computations; the topology of the object under study is reproduced by the model itself; any boundary conditions can be




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applied to the model; there is a possibility to observe development of the process step-by-step; and CA can be combined with other modelling techniques such as the Monte-Carlo technique or Finite Element Analysis. There are three main disadvantages of CA: the discrete nature of CA can affect the reproducibility of certain physical phenomena; if the rules for system evolution are chosen incorrectly, this will lead to unphysical results; increasing the model’s size leads to increased calculation time, though this problem is partially solved by using parallel computations.

The polygranular Gilsocarbon graphite is described, at this time, in terms of filler particles and pores distributed in a matrix. The general algorithm for graphite microstructure modelling consists of two main stages. The configuration of the filler particles in graphite is formed at the first stage. Two probabilistic procedures, namely modelling of the centres of nucleation and particle development, have been run simultaneously for this purpose. The nucleation process is regulated by the probability of the formation of filler particles centres. Particle development is regulated by the growth probability. The second stage of the modelling allows the formation of the porosity in the graphite’s microstructure. This simulation is also regulated by two probabilities. The probability of nucleation controls the number of pore centres in the microstructure. The probability of growth is responsible for the average size of the pores. As a result we can obtain a model of graphite microstructure and size distributions of filler particles and pores, in which fracture can be simulated. An example is shown in Figure 4, which compares a tomographic characterization of the Gilsocarbon microstructure with a simulation; random microstructures can thus be generated using statistical data from experimental observations.

Using the cellular automata technique, we consider the cells corresponding to the pores and matrix as local elements with different properties; the cells corresponding to the matrix are inactive elements of the structure with state 1, and the cells of pores are active elements with state 2. The stress field generated by each active element (i.e. by the pore) in the matrix is computed according to the elasticity theory formulas for calculation of the stresses around holes. Linear elastic fields have been assumed at this stage, but the model is applicable to non-linear elastic and elastic-plastic fields. The total stress field is obtained as a superposition of elastic stress fields, generated by all cells with the active state 2 in each cell with the inactive state 1; the fields are currently calibrated with the analytical solutions for an infinite volume with a central spherical void, but calibrations for the more general case, such as elliptical pores, are being developed. The calculated local stresses in each cell of the matrix allow calculation of the local strain and strain energy density values in each cell, and the criterion of crack initiation is checked in each cell. If the criterion is satisfied in any cell, it is marked as cracked. Recalculation of the




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stress distribution, taking into account the cracked cells, is then performed for a damaged microstructure. The process is iterated to simulate crack propagation; an example is shown in Figure 5.

Although qualitatively correct in that it describes the mechanical interactions of pores in a complex arrangement, and can simulate the nucleation and growth of damage within a porous microstructure, the CA model requires considerable further development, which will be done as the project continues. In particular, approximations are currently made in the fracture criterion (a critical stress is assumed in individual cells) and in the treatment of complex pore shapes, which need refinement and validation. Nonetheless the CA method offers significant enhancements in computational efficiency compared to the finite element simulation methods of equivalent microstructural fidelity. This spatial complexity is necessary to examine the stochastic behaviour of microstructures and also to assess the sensitivity to the effects of irradiation and oxidation on the microstructure and its local properties.

Despite the computational efficiency of CA, it is not practical to simulate mechanical components, nor small test specimens. In order to extend the CA model to the component scale, we aim to use the CAFE (Cellular Automata Finite Element) modelling framework that is being developed in the EPSRC QUBE project (15). This method provides two sets of elements representing the finite element model of the component and the microstructure. The first is used to link the engineering scale problem with the microstructure, obtaining the stress and strain fields of the macro-mechanical problem. With those, we compute the micro-mechanical fields via the second set of elements, which describe explicitly the heterogeneity of properties from the microstructure; these elements have stochastic properties that will be defined by CA model described above. The model uses the Meshfree approach to simulate the damage development through the microstructure. The material properties of the finite elements are recomputed according to the microstructural damage and the fracture path is completely free with respect to the finite element mesh, which can be very coarse. By this method quasi-brittle fracture can develop freely through the microstructure, improving the accuracy and computational cost of the calculations at engineering length-scales in complex microstructures.

CONCLUDING SUMMARY

The combined use of X-ray tomography and digital volume correlation to measure three-dimensional displacement fields can be used to study the deformation associated with stress concentrations, in particular the non-linear strains within in the fracture process zone are measurable. The intention is that such data, combined with diffraction-




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based measurements of elastic strains in the graphite crystals, will be used to validate microstructure-based models for damage development. Computationally efficient microstructure-based models are being developed, using cellular automata and meshfree methods, with the aim of simulating small specimen test behaviour to measure the notch sensitivity of radiolytically oxidised graphite.

ACKNOWLEDGEMENT

This work is supported by EDF Energy Generation (GRA/GNSR 6057 GRA/GNSR 6061) and also EPSRC (EP/J019992/1). The opinions expressed are the authors' and not necessarily those of EDF Energy.




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Figure 4: Gilsocarbon graphite microstructures and their pore size distributions: a) Reconstructed tomographic data; total pore fraction: 15.89%, number of pores: 717, volume size: 500x500x500 voxels. Resolution: 0.8 µm/voxel, average pore radius: 1.096 µm; b) Simulated microstructure; total pore fraction: 16.04%, number of pores: 791, volume size: 100x100x100 cells, average pore radius: 2.08 cells.

Figure 5: CA simulation of the crack development in inhomogeneous microstructure at uniaxial tension along green axis. The simulated volume is 50x50x50 cells, with an initial porosity of 7.72%.

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REFERENCE LIST

(1) Li, H., Duff, J. and Marrow T.J, In-situ observation of crack nucleation in nuclear graphite by digital image correlation (2008) Proceedings of PVP2008 ASME Pressure Vessels and Piping Division Conference, Chicago (USA). Paper PVP2008-61136.

(2) Mostafavi, M. and Marrow, T.J. (2011) In situ observation of crack nuclei in polygranular graphite under ring-on-ring equi-biaxial and flexural loading, Engineering Fracture Mechanics 78 pp 1756-1770

(3) Aswad, M.A. and Marrow, T.J. (2010) Intergranular crack nucleation in polycrystalline alumina, 18th European Conference on Fracture, ECF18, Dresden (Germany).

(4) Stock, S. R., (1999) X-ray microtomography of materials, International Materials Review 44 pp 141–164.

(5) Maire, E., Buffière, J.-Y., Salvo, L., Blandin, J. J., Ludwig, W., and Letang, J. M. (2001) On the application of X-ray microtomography in the field of materials science, Adv. Eng. Mater. 3 pp 539–546.

(6) Marrow, T. J., Buffière, J.-Y., Withers, P. J., Johnson, G. and Engelberg, D. (2004) High resolution X-ray tomography of short fatigue crack nucleation in austempered ductile cast iron, Int. J. Fatigue 26, pp 717-725.

(7) Bay, B. K., Smith, T. S., Fyhire, D. P. and Saad, M. (1999) Digital volume correlation: three-dimensional strain mapping using X-ray tomography. Exp. Mech. 39 pp 217-226.

(8) Forsberg, F. and Siviour, C. R. (2009) 3D deformation and strain analysis in compacted sugar using x-ray microtomography and digital volume correlation. Meas. Sci. Tech., 20 art. no. 095703.

(9) Barranger, Y., Doumalin, P., Dupre, J.-C., Germaneau, A., Hedan, S. and Valle, V. (2009) Evaluation of three-dimensional and two-dimensional full displacement fields of a single edge notch fracture mechanics specimen, in light of experimental data using X-ray tomography. Eng. Fract. Mech. 76 pp 2371-2383.

(10) Mostafavi, M., Vertyagina, Y., Reinhard, C., Bradley, R., Jiang, X., Galano and M., Marrow, J. (2014) 3D studies of indentation by combined X-ray




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tomography and digital volume correlation. Key Eng. Mater. 592-593 pp 14-21.

(11) Mostafavi, M., Baimpas, N., Tarleton, E., Atwood, R. C., McDonald, S. A., Korsunsky, A. M., & Marrow, T. J. (2013). Three-dimensional crack observation, quantification and simulation in a quasi-brittle material. Acta Materialia, 61(16), pp. 6276–6289

(12) Mostafavi, M., McDonald, S. A., Mummery, P. M., & Marrow, T. J. (2013). Observation and quantification of three-dimensional crack propagation in poly-granular graphite. Engineering Fracture Mechanics, 110, pp. 410–420

(13) Mostafavi, M., & Marrow T. J. (2012). Quantitative in situ study of short crack propagation in polygranular graphite by digital image correlation. Fatigue & Fracture of Engineering Materials & Structures, 35(8), 695–707

(14) Wolfram, S., (2002). A new kind of science. IL: Wolfram Media.

(15) Saucedo-Mora, L., & Marrow, T. J. (2014). 3D Cellular Automata Finite Element Method with Explicit Microstructure: Modeling Quasi-brittle Fracture using Meshfree Damage Propagation. Procedia Materials Science Vol. 3, pp. 1143–1148

Towards a notch-sensitivity strength test for irradiated nuclear graphite structural integrity

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