Peridynamic modelling of cracking in TRISO particles for high temperature reactors

A linear-elastic computer simulation (model) for a single particle of TRISO fuel has been built using a bond-based peridynamic technique implemented in the ﬁnite element code ‘Abaqus’. The model is able to consider the elastic and thermal strains in each layer of the particle and to simulate potential fracture both within and between layers. The 2D cylindrical model makes use of a plane stress approximation perpendicular to the plane modelled. The choice of plane stress was made by comparison of 2D and 3D ﬁnite element models. During an idealised ramp to normal operating power for a kernel of 0.267 W and a bulk fuel temperature of 1305 K, cracks initiate in the buffer near to the kernel-buffer interface and propagate towards the buffer-iPyC coating interface, but do not penetrate the iPyC and containment of the ﬁssion products is maintained. In extreme accident conditions, at around 600% (1.60 W) power during a power ramp at 100% power (0.267 W) per second, cracks were predicted to form on the kernel side of the kernel-buffer interface, opposite existing cracks in the buffer. These were predicted to then only grow further with further increases in power. The SiC coating was predicted to subsequently fail at a power of 940% (2.51 W), with cracks formed rapidly at the iPyC-SiC interface and propagating in both directions. These would overcome the containment to ﬁssion gas release offered by the SiC ‘pressure vessel’. The extremely high power at which failure was predicted indicates the potential safety beneﬁts of the proposed high temperature reactor design based on TRISO fuel.

Following a small number of accidents in which nuclear fuel has been severely damaged, there is a desire to move towards fuel cladding technology that is more inherently safe.A number of socalled accident tolerant fuels (ATF) have been proposed for current light water reactor (LWR) reactor designs.These include coated zirconium alloy claddings; FeCrAl claddings; novel pellet materials and SiC-SiC composite cladding [1][2][3][4][5][6][7][8][9][10][11][12][13][14] .Moving to the longer term, advanced technology fuels (also termed ATF) offer the possibility of not only enhanced safety, but the opportunity to move towards higher burnups.Small modular reactors (SMRs) built on a factory-based production line could offer reduced build times and hence financial risk prior to commencement of initial power sup-ply.In addition, SMRs offer the potential for load-following capability as well as higher temperatures giving better thermodynamic efficiency and improved economics [15][16][17][18][19][20][21][22][23][24] .They also offer the opportunity for nuclear to move beyond merely providing a baseload generating capacity in the drive for 'net zero'.By providing high temperature heat, the reactors could supply district heating, heat for industrial uses, and heat for hydrogen co-generation [25] .
One example of an advanced technology fuel is tristructuralisotropic (TRISO) particle fuel, a micrograph of which is shown in Fig. 1 .The TRISO particle consists of a uranium-based fuel kernel surrounded by four layers of materials designed to offer fission product retention and structural support [26] with a total diameter of around 1 mm.The fuel kernels have historically been formed of uranium dioxide (UO 2 ) [27] or uranium oxycarbide (UCO) [ 15 , 28 ], although uranium nitride (UN) [22] has been suggested as a means of increasing the density of fissile material.The layers encasing the fuel kernel consist of: a porous buffer layer of carbon in-Fig.1.A micrograph of a TRISO particle.Reproduced from Fig. 1 in [30] .
tended to absorb fission product recoil and accommodate dimensional change; a inner dense pyrolytic carbon (iPyC) layer; a SiC layer to provide strength and act as a barrier to gaseous and metallic fission products [16] ; and, a dense outer pyrolytic carbon (oPyC) layer.The iPyC and oPyC layers, at around 90% theoretical density, act as a barrier to species migration and protect the SiC from chemical attack [23] .The three outer layers (iPyC-SiC-oPyC) are termed the TRISO coating [29] .The TRISO particles are then embedded into surrounding graphite / SiC matrix 'compacts' to provide moderation and structural integrity.
Given that the SiC layer forms the primary barrier against fission product release, most models have been focused on this layer, frequently treating it as a pressure vessel.Li et al. considered the SiC as a rigid pressure vessel, perfectly bonded to the PyC layers [18] .A number of models have been developed that treat the carbon layers as visco-elastic and the SiC layer as purely elastic material [23] .The stress in the SiC is designed to be zero at the start of life.It then becomes compressive due to densification of the PyC layers, reaching the order of 100 MPa at a fast fluence of approximately 0.5 × 10 25 n m −2 [23] .Due to PyC swelling and the buildup of carbon monoxide and dioxide (CO and CO 2 ), the stress increases (becomes more tensile) towards the end of life.This has the potential to cause failure if it becomes tensile and greater than the fracture stress [23] .The historic approach is to assume spherical symmetry and solve the mechanics of the particle using closed form 1D analytical solutions [23] .This is analogous to conventional 2D( r -z ) axisymmetric models for conventional pellet fuel in tubular cladding such as TRANSURANUS and ENIGMA [ 31 , 32 ].
The most complete 1D model for TRISO fuel performance is PARFUME ( par ticle fu el m od e l) [ 19 , 33 , 34 ].In common with other fuel performance codes, PARFUME uses stress concentration factors derived from finite element models to account for a pre-imposed crack.The effect of a crack in the iPyC layer was to change the maximum principal stress in the SiC layer from a compressive stress of around 700 MPa (in the uncracked case) to a tensile stress of 440 MPa (in the cracked case) [33] .A further evolution is to analyse the statistical prediction of failure due to of a number of interacting failure mechanisms which are computationally demanding to capture [35] .
The introduction of complicated fracture patterns such as crack coalescence or branching is challenging using techniques such as the extended finite element method (XFEM) [36] .In contrast, peridynamics allows complicated crack patterns to develop naturally.Phenomena such crack initiation, branching and retardation can be predicted as a result of the underlying geometry, material properties and loading.This paper investigates the behaviour of a TRISO fuel particle during early service life using a bond-based peridynamics model implemented in Abaqus.This enables the prediction of the crack patterns within each layer, together with the interaction between cracks in each layer.

Peridynamic modelling in abaqus
In peridynamics, objects are decomposed into a number of 'material points'.The behaviour of each material point is dependant upon those in a horizon surrounding the material point, not just its nearest neighbours [37] .An example of this is shown Fig. 2 .
'State-based' peridynamics uses information about the surrounding material points to determine the response of a material point.This approach has been applied by a number of authors [38] , as well as to nuclear fuel more specifically [39][40][41][42] .'Bond-based' peridynamics considers a number of bonds connecting a material point to surrounding material points.It can be readily introduced to a finite element code [ 37 , 43-48 ] and the full range of material properties available in a finite element model employed.For this reason, a bond-based implementation of peridynamics in Abaqus was chosen for this work.
This work broadly follows the theoretical background, described by Le and Bobaru [49] .In the Abaqus implementation presented here, each 'material point' is represented by a node, and truss elements represent the 'bonds' linking the material points.This approach is discussed in more detail in [37] .

Peridynamic model for TRISO particles
Fig. 3 shows the model for TRISO particles used to investigate cracking during an idealised power history.Seventy rings of nodes were used across the radius of the particle, giving a nominal nodal separation of 5.6 μm.This gives a total of 15,373 nodes and 211,108 trusses at a horizon ratio of three.At the top, bottom, left and right of the model, the particle is constrained to not move in the az-Fig.3. A schematic for the peridynamic model for TRISO particles used to investigate cracking during an idealised power history.Red 'dots; indicate nodes; green lines indicated bonds; squares represent boundary conditions; and the circle and arrow represents a plane polar co-ordinate system.For clarity, the nodal separation shown in this figure has been increased by a factor of 10 over that used in the simulations, making the mesh 10 times coarser.

Table 1
The outer radius of each region of material of the TRISO fuel particles modelled.imuthal direction -this prevents spurious rotation.A temperature boundary condition is applied to the whole of the model and the temperature at individual nodes determined using the analytical solution described by Eq. (17) .It should be noted that whilst most finite element models for TRISO particles assume a buffer-iPyC gap, in this work we have not.This is because we wished to predict the debonding and cracking during routine early life.
The material properties of a bond depend upon the distance of the connected material points from the centre of the fuel kernel.Bonds which cross material interfaces are addressed in Section 2.3 .The radius of each region of material are given in Table 1 ; this was based upon a 'standard' particle with a 350 μm kernel diameter and the layer thicknesses applied in the US AGR-1 and AGR-2 experiments [20] .

Bonds crossing material interfaces
For trusses crossing an interface, the assignment of material properties such as the elastic modulus to bonds is somewhat complicated.For example, the macroscopic elastic modulus differs at each end of the bond.In addition, one should bear in mind that the bonds or trusses are modelling constructs and do not exist in the physical sense and are simply carrying forces between material points.In contrast, the nodes represent the material of a given volume surrounding themselves.The material exists and therefore the nodes have a physical meaning.
In this work, the strength of a truss crossing an interface is set to the assumed strength of the interface, given in Table 4 .There are however several different ways of assigning an elastic modulus to the trusses.In this work, four approaches were considered: 1. Taking the average elastic modulus of the nodes at each end of the truss.2. Modelling the truss itself as consisting of two materials with an interface at the midpoint of the truss.3. Assuming the maximum elastic modulus of the nodes at each end of the truss.4. Assuming the minimum elastic modulus of the nodes at each end of the truss.
By testing each approach, it was found that using the minimum elastic modulus of the nodes at either end of an interface truss gave the best overall behaviour for all of the interfaces.Employing the minimum elastic modulus also represents the minimum stress.In addition, using the minimum stress would be sensible in the light of Weibull's weakest link theory for brittle fracture [ 44 , 50 , 51 ].

Material properties
The approach taken by this work was to use start of life material properties.This involved neglecting the impact of burnup and changes in porosity.

Porosity
The initial fractional porosities were assumed to be 0.010 for the UO 2 kernel; 0.560 for the buffer; 0.160 for iPyC; 0.006 for SiC and 0.166 for oPyC [52] .

Elastic modulus
The elastic modulus of UO 2 , E UO2 , as described by MATPRO [53] , is given by (1) ; the units for the elastic modulus are GPa; P is the fractional porosity and T, the absolute temperature in Kelvin.It should be noted that these values were derived for conventional LWR fuel.The elastic modulus of the buffer, E BUF , is given [54] by (2) .
E BUF = 34 .5 e −2 . 03P (2) A number of expressions are available for the elastic modulus of PyC, E PyC , [ 18 , 54 , 55 ].The simplest, given in (3) and described by Park et al. [55] , uses a standard correction term for porosity, originally derived for alumina [56] , hence its representativeness of PyC is questionable and thus an area where new data should be sought.Further, it should be noted that this expression does not take into account the anisotropy of PyC; an alternative expression that includes the impact of this is available in [54] .
A number of correlations for the elastic modulus of SiC are available in the literature [57] .The relationship recommended by Powers [23] , for use in TRISO fuel performance codes, is given by ( 4) .

Poisson's ratio
In bond-based peridynamics, Poisson's ratio is constrained to 1/3 in plane stress [58] and ¼ in 3D [59] or plane strain [46] .Poisson's ratio was therefore constrained to 1/3 or 1/4, as appropriate in the bond-based peridynamics models; state-based models have no such constraint.For the finite element models, the values given in Table 2 were used.

Thermal conductivity
The thermal conductivity of nuclear materials is a function of their burnup, temperature, porosity and level of fission product contamination.A number of correlations for the thermal conductivity of UO 2 are available in the literature and have been discussed in depth elsewhere [63] .The starting point is normally setting the thermal conductivity of irradiated fuel to be the product of the thermal conductivity of unirradiated fuel, k 0 , and a number of correction factors.The thermal conductivity of unirradiated fuel [in J K −1 m −1 ] is usually expressed, using (5) , as the sum of a phonon term, k ph , [typically a power series in the absolute temperature, T ] and an electronic term, k el , [typically a negative exponential of the temperature].
Lucuta's model [64][65][66] is widely applied in the literature and so is given in more detail as an example.Lucuta's phonon and electronic terms, given by ( 6) and (7) , are based upon the work of Harding & Martin [67] .
T 2 e −1 . 6361×10 4 /T (7) A multiplicative porosity correction factor is given by ( 8) , in which P is the fractional porosity and σ S a shape function [set to 1.5 for spherical pores].In this work, it was assumed that the fuel porosity was 1.0%.In the peridynamic models, the average thermal conductivity was determined in a user-defined sub-routine and used to determine the temperature profile in the kernel.More details on this approach are given in Section 2.5 .

Thermal expansion
Table 3 gives the coefficient of thermal expansion, α, assumed for each layer in the TRISO particle; all materials were assumed to be isotropic.In the peridynamics model, the total expansion of a truss is determined in the user-defined field subroutine and passed to Abaqus through a field variable.The code was amended to set the expansion as the average of that of the nodes at each end of the truss.

Density
A volume-averaged density was determined to be 2.72 ×10 3 kg m −3 and this was applied to the models.Assigning an averaged density to nodes is a simplification and current a limitation of the current pre-processor.The approximation was judged to not have a great influence upon the results given the constraint upon the crack fragments.For instance, in the kernel the mass applied to each material point will be a third of that which it should be.Upon bond failure, elastic energy is converted into kinetic energy in the node.The velocity of the nodes will therefore be underestimated by a factor of √ 3. Work is ongoing to overcome this underestimate.

Fracture stress
For a bond entirely within a material, i.e. not near a free surface or material interface, the fracture strength of the bond, σ F,Bond , will be given by the fracture strength of that material.This is passed to Abaqus through the user-defined field sub-routine.The fracture stresses of each material, or pair of materials in the case of an interface are given in Table 4 .The available data is sparse, consisting predominantly of strengths obtained at room temperature.The values used are given in Table 4 .Of particular significance is that no data is available on the strength of either the kernel-buffer or buffer-PyC interface.In this work, we have set the strength of these two interfaces to be equal to the average of the materials on  either side of the interface.Future work could investigate the appropriateness of this assumption against the backdrop of validation against experimental data.

Thermal model for a TRISO fuel particle
A temperature boundary condition was applied to all nodes using the user-defined displacement sub-routine; in Abaqus the displacement boundary condition is used to define all degrees of freedom, including temperature.The temperature profile was based upon a simple pseudo-steady-state temperature solution.This relied upon the following two approximations: • That all regions except for the kernel were held at the same bulk temperature.Effectively the kernel is embedded in an infinite thermal sink.This approximation was made as particles will reside in regions with different temperature gradients across them due to their comparative position in the pellet/pebble/matrix.In addition, the TRISO particles are so small that no significant temperature gradients are expected.• The thermal conductivity of the kernel varied with temperature, as shown and discussed in Section 2.3 .The average thermal conductivity, k av , was then determined This was justified by the comparatively low temperature drop across the kernel and significant computational saving.
Poisson's equation for heat flow in the presence of a volumetric heat flux in W m −3 , q , is given by ( 9) , in which k is the thermal conductivity in W m −1 K −1 , T the temperature in K and ∇ 2 the Fig. 6.Fracture patterns in the TRISO particle at the end of the rise to 100% power over 24 h.Red lines define the layer boundaries and white donates bond failure, a surrogate for damage / cracking.Laplace operator with units m −2 [70] .
At the origin ( r = 0), the heat flux is a minimum in order to maintain symmetry.This necessitates setting the integration constant, A , to zero.Eq. ( 11) therefore becomes (12) .Integrating once more gives (13) , in which B is a second integration constant.
Substituting the outer radius of the kernel, R , and the bulk temperature T Bulk into ( 13) gives (14) .
This allows B to be determined.

Thermal boundary conditions & loads
The temperature assumed for all materials except the kernel was set to 1305 K at full power.This was based upon the fuel volume averaged temperature reported for AGR-2 [52] .During a single 24-hour step, the temperature of the outside of the kernel was increased linearly from 293 K to 1305 K.This temperature was chosen the reflect the fuel volume averaged temperature in the AGR-2 experiment.
The volumetric heat flux applied to the model was taken from Hu and Uddin [71] and gives a power of 0.267 W (1.19 ×10 10 W m −3 ) for a 350 μm diameter kernel and agrees with the value of 0.3 W used by Rodríguez Garcia et al. [72] .
To model extreme events, separate simulations were run in which the kernel was increased to 10 0 0% power (2.267 W) to assess the impact of a severe ramp upon cracking.

Benchmarking simulations
In previous work, a 2D( r -θ ) slice through a cylindrical pellet waist or section of cladding was used as an approximation, normally under generalised plane strain, to a 3D cylinder [ 37 , 43 , 44 , 63 , 73 ].In this work, a 2D( r -θ ) model is used to approximate a 3D sphere.Given that an axisymmetric model is not available in the current bond-based peridynamics implementation, it was necessary to pick either a plane strain or plane stress approximation.
Our current implementation of bond-based peridynamics does however give considerable flexibility in that it allows the user to leverage much of the materials modelling capability available within Abaqus.Utilising Abaqus also offers a lower barrier to technique adoption than a bespoke code.The ability to combine finite element and peridynamic regions in order to capture phenomena such as contact is another advantage.It is hoped that by applying peridynamics in 'industry standard' simulation software, it will accelerate commercial acceptance of the technique.Whilst the approach taken here compares favourably with others [46] , one of the downsides of peridynamics compared to other techniques is the considerable computational cost.To avoid this cost, to date, we have applied the technique to 2D models under conditions of both plane strain and plane strain.This is accompanied a restriction in Poisson's ratio to 1/4 or 1/3 respectively.Clearly, the 3D stress state in a TRISO particle would be best captured by an axisymmetric model.This was not an option due to the truss elements available.
To make the most appropriate choice, between plane strain and plane stress models, a number of equivalent finite element models were run using the same boundary conditions and loads as the peridynamics models detailed above.The finite element models run included: • An axisymmetric model to capture the correct stress state in the particle.• A plane stress model.
• A plane strain model.
• A plane stress model with the limitation of Poisson's ratio to 1/3, as per bond-based peridynamics.

Stress state in TRISO particles during idealised power histories
Fig. 4 gives the maximum principal stress at the end of the ramp to 100% in each of the finite element simulations described in Section 2.3 .Use of a plane strain approximation can be discounted due to the significant stress in the SiC layer compared to the surrounding PyC layers; this is not seen in the axisymmetric or plane stress models.The plane stress models can be seen to give more plausible stress results.For example, the peak stress on the inner surface of the buffer was 27 MPa in the axisymmetric model and 21 MPa in the plane stress model, 22% lower.Using a Poisson's ratio of 1/3 in the model, as used in bond-based peridynamics under conditions of plane stress, makes only a small difference.The maximum hoop stress on the inner surface of the buffer was around 4% higher using a uniform Poisson's ratio than a materialspecific Poisson's ratio.For context, it should be noted that for an 'engineering' fuel performance code, errors of up to 10% are more than likely to below those due to uncertainties in other parameters such as many of the material properties.Overall, the use of plane stress model with a uniform Poisson's ratio can be argued to be fair, given the absence of a better solution in the peridynamics implementation detailed here.Work is ongoing on developing the implementation to accelerate 3D modelling for future work.

Cracking during rise to 100% power
The temperature profile in the fuel at the end of the raise to power is shown in Fig. 5 .The temperature drop across the kernel of 18 K is much lower than seen across the pellets of conventional LWR fuel at the start of life, which is approximately 550 K [68] .The temperature gradient is however similar, at 100 K mm −1 , compared to the typical gradient of 140 K mm −1 seen in conventional LWR fuel at the same point Fig. 6 shows the fracture pattern in the TRISO particle at the end of a rise to 100% power i.e. normal operation (a kernel power of 0.267 W and a temperature of 1305 K, as described in Section 2.5 ) over 24 h.Twenty cracks initiate in the buffer, close to the kernel-buffer interface and around half of these extend outwards radially.They do not however initiate damage in the overlying iPyC or any of the other layers further towards the outer surface.In addition, circumferential damage is seen in the buffer, close to the kernel-buffer interface.Here, a fundamental difference is seen between TRISO and conventional LWR during the initial rise to power.Namely, that the UO 2 was not predicted to crack in the TRISO fuel, whilst it does in conventional LWR fuel [74] .This might offer potential thermal and fission gas release benefits.Fig. 7 shows the percentage power at the end of each increment and demonstrates one of the advantages of implementing bond-based peridynamics in Abaqus.Namely, that time steps similar to those required for an explicitly integrated solution can be used during periods of rapid crack propagation and implicit time steps in between, when there is no rapid change in the model.The average time increment during the simulation was 37 s, the minimum 4.3 ns and the maximum 1.8 h.The simulation took 9.5 h to run on a workstation computer.
Fig. 8 shows the development of cracking in the buffer at a number of time points indicated in Fig. 7 .Between 62 and 65% power, there is a period of crack initiation at the kernel-buffer interface (points A to D).This is followed by a period of slow crack growth towards the buffer-iPyC interface (points D to E) until 89-91% power, when additional cracks initiate at the kernel-buffer interface (points E to H).This second period of initiation is followed by a period of slow crack growth towards the buffer-iPyC interface in a smaller subset of cracks (points H to I).

Cracking during power ramp
Fig. 9 shows the power level reached during the hypothetical extreme power ramp to 10 0 0% power by increment number.In contrast to Fig. 7 , there are no intermediate stages of periods of rapid cracking, merely the onset of rapid crack growth at 940% power.The very high power required to give rise to cracking in the SiC layer confirms the potential safety advantage of TRISO particle fuel.
Fig. 10 shows the crack patterns at the points A to H shown in Fig. 9 .At 610% of full power (point A), cracks initiate in the fuel kernel, close to the kernel-buffer interface.These cracks initiate opposite existing cracks in the overlaying buffer.The cracks in the kernel grow slowly (points A to C) until 940% power (point D), when cracks initiate in the SiC at the SiC-PyC interface (points D and E).These cracks increase in number (points F and G) and cracking of the TRISO particle proceeds rapidly towards the outer surface of the SiC layer (point H).It should be noted at point F, the iPyC layer is breached, allowing potential chemical attack of the SiC 'pressure vessel' by fission products.In addition, at point H, the SiC pressure vessel is breached, allowing the egress of fission products, especially gases, from the particle.

Kernel-Buffer interf ace
In this work, we have assumed that the kernel and buffer were initially bonded.There is however evidence that immediately following manufacture, the kernel and buffer are de-bonded [75] .This is supported by the observation of debonding post-irradiation [52] .Fig. 11(a) is an optical micrograph shows debonding between the buffer and a ZrO 2 kernel manufactured from zirconia, a common surrogate for UO 2 .The clean debonding seen experimentally can be reproduced by our model, as shown in Fig. 11(b) , which shows clean cracking between the buffer and kernel due to a uniaxial load applied in the horizontal direction.The branching crack is likely to be due to post-initiation twisting leading to increasingly mixed mode loading as the crack grows.The impact of residual stresses due to manufacture and the associated cracking of the kernel-buffer interface is clearly something which deserves further research.
The formation of cracks in SiC opposite those in the underlying iPyC show the value of employing more advanced explicit crack modelling techniques to predict the failure of TRISO particle fuel than those which do not take into account cracks in the underlying layers.The necessity to model cracks in underlying layers is likely to result from the relatively thin nature of the coating layers in a TRISO fuel particle.

Kernel-iPyC interface
In our work, we did not predict a iPyC-buffer crack, possibly as a result of not modelling irradiation-induced shrinkage and possibly a result of not modelling residual stresses from manufacture.If the buffer and kernel are de-bonded, as has been assumed elsewhere [76] , the stress concentration in the kernel might not be as large and so the crack might not 'jump' from one material to the other.This in turn, might reduce the stress concentration in the SiC, again analogous to the situation observed in conventional pellet-clad interaction [77] .We are currently working to implement both residual stress and irradiation-induced shrinkage in our model.

Conclusions
In this work, we have developed a 2D linear-elastic peridynamic model for TRISO particle fuel during the initial rise to power.This has shown that: In the absence of the capability to carry out an axisymmetric simulation, the most appropriate 2D approximation to the 3D solution for a volumetrically heated sphere is plane stress.
During an idealised raise to power for a fuel kernel power of 0.267 W and a bulk fuel temperature of 1305 K, cracks initiate in the buffer near to the kernel-buffer interface and propagate radially outwards towards the buffer-iPyC interface.The cracks do not fully penetrate the buffer layer during normal operation.
During a hypothetical accident involving a power ramp from 100% power (0.267 W) at 0.267 W s −1 , cracks formed on the kernel side of the kernel-buffer interface at 610% power.These were opposite existing cracks in the buffer.The cracks in the kernel then propagated slowly.
The TRISO particle failed at a power of 940%, with numerous cracks rapidly formed at the iPyC-SiC interface, propagating in both directions.During this hypothetical accident, these would overcome the containment to fission gas release offered by the SiC 'pressure vessel'.However, this very high resistance to an extreme power ramp compared to other fuel forms, confirms the potential inherent accident tolerance of coated particle fuel.

Fig. 2 .
Fig. 2. A material point surrounded by a horizon of other material points.In bondbased peridynamics, bonds connect all of the material points within the horizon.Adapted from Fig. 1 in [37] .

Fig. 4 .
Fig.4.The maximum principal stress in each of the finite element models described in Section 2.7 at the end of the rise to 100% power.

Fig. 5 .
Fig. 5.The temperature in the TRISO particle at the end of the rise to 100% power over 24 h.

Fig. 7 .
Fig. 7. Incrementation during the simulated rise to 100% power (0.267 W kernel power) over 24 h.Increment duration ranged from 4.3 ns to 1.8 h.The letters A-I are points of interest, used in Fig. 8 .It should be noted that the fuel power increases linearly with time.

Fig. 8 .
Fig. 8. Crack formation during the rise to full power.The letters A-I are points of interest shown in Fig. 7 .For clarity, only the buffer region is shown.

Fig. 9 .
Fig. 9. Incrementation during the simulated power ramp from 100% to 10 0 0% power.Crack patterns at points A to H are shown in Fig. 10 .

Fig. 10 .
Fig.10.Crack patterns at the points A to H in the rapid power ramp to 10 0 0% power shown in Fig.9.

Fig. 11 .
Fig. 11.(a) -An optical micrograph shows debonding between the buffer and a kernel of zirconia in.(b) -Cracking between the buffer and kernel due to a uniaxial load applied in the horizontal direction.

Table 2
Poisson's ratio for each material in the finite element model.

Table 3
Coefficient of thermal expansion for each material.

Table 4
Material fracture stresses assumed.