Presented by: Nassia Tzelepi

Progress on the Graphite Crystal Plasticity Finite Element Model (CPFEM)

J F B PayneL Delannay, P Yan (University of Louvaine)

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Overview

• Background• Basic principles• Inputs and assumptions • Model description• Results• Future work

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• It is clear that the graphite microstructure is extremely important to its behaviour under irradiation.

• The mechanical properties and the response to irradiation of polycrystalline graphites differ greatly from single graphite crystals • Most notably polycrystalline graphite is much more nearly isotropic than the highly anisotropic graphite crystal• Strains well beyond the elastic limit of the crystal are accommodated in the polycrystalline material.

• The anisotropy of individual crystals and void formation affects the mechanical behaviour of many polycrystalline materials• The effect of crystal anisotropy has been studied in these materials.

Background

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1. The first piece of physics in the model is that the crystallites making up the polycrystalline agglomerate are anisotropic and that the crystal symmetry axes of adjacent crystallites are not aligned.• As pointed out by Mrozowski this misalignment can cause cracking because of the mismatch between thermal expansion of adjacent crystallites along the crystallite boundary. Therefore…

2. The second piece of physics in the model is the possibility of cracking when a threshold strength is exceeded.

3. The final piece of physics in the model is the behaviour of isolated graphite crystals. • Elasticity, thermal and dimensional change data for an isolated crystal are taken from published experimental (MTR) results.

Basic principles

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Model inputs

Constitutive behaviour of single graphite crystals

Total strain:

The only inputs to the model are:• Single crystal elasticity• Single crystal CTE• Single crystal response to irradiation induced dimensional

change• Irradiation creep only by shear between basal planes

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stiff in-plane directions :

compliant along

the c-axis :

Isotropic in the basal plane

Model Inputs: Elastic Anisotropy

‘3’

‘1’

‘2’ direction taken to coincide with Z

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Contraction of basal plane is negligible

Other

Model inputs: Anisotropic thermal expansion

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Volume is preserved

per neutron

Model inputs: Irradiation induced dimensional change

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At the crystal level:

Creep occurs by shear between basal

planes only:

Macroscopically:

Model inputs: Irradiation induced creep

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Cooling (manufacturing): The anisotropy of the graphite crystal is such that it shrinks almost exclusively in the direction of the c-axis

Irradiation: After the 4th step the c-axis recovers its initial length Dimensional change preserves the volume

Simulation of manufacturing followed by irradiation

Manufacturing

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Model description (1)

• 100 crystallites included as Voronoï cells• Their seeds are positioned randomly inside a square

section and then periodically reproduced outside the square

• Min distance between seeds: dmin = 0.08L

• Uniform crystallite sizes and realistic crystallite shapes

• The crystal c-axis lies within the 2D plane• Random orientation of crystallites in 2D

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Model description (2)

• Periodicity of the model microstructure • Modified UMAT used for elastic-plastic deformation of individual crystallites in metals.

x

y

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Model description (3)

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• Background• Basic principles• Model description• Inputs and assumptions• Results• Future work

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Preliminary investigations

What is the influence of the internal stresses and cracks

due to cooling from manufacturing temperatures on the

CTE and dimensional change of the polycrystal under

irradiation?

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After cooling After Step 4

Crack closure midway through irradiation

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After cooling After full irradiation

Opening of new cracks upon further irradiation

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Preliminary findings

Manufacturing

-4

-3

-2

-1

0

1

2

3

4

2000-1000C 1000C-RT Irrad 2 Irrad 3

d(ar

ea) b

etw

een

FE s

teps

(%)

Single crystal

Polycrystal (no cracks)

Polycrystal (cracks, no creep)

Polycrystal (cracks+creep)

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Without creep With creep

Colored according to the shear stress on the basal plane

Preliminary findings

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Results

1° Calculate the elastic modulus of the polycrystal without

cracks

• Which crystallite configuration is more likely to lead

to cracking at the boundary?

2° CTE change of the polycrystal with cooling from

manufacturing temperatures and irradiation

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•Pure elastic modelling (no cracking)

•Average elastic modulus of the polycrystal

Results – Pure elastic modelling

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•Sensitivity of the macroscopic YM to the crystal stiffness

Results – Pure elastic modelling

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•Stress distribution at the crystal boundary

•Which crystallite configuration is most likely to lead to cracking?

•The interface stresses (normal and shear) depend on the two angles between the interface and the two c-axes of the twograins connected by the interface

θ1 varies from -90˚to 90 ˚, while θ2 varies from 0˚to 90˚

Results – Pure elastic modelling

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Results – Pure elastic modelling

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Results – Pure elastic modelling

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•Evolution of cracking

Results – Model with cracks

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•Variation of shear strain (in local coordinates of individual crystals)

Results – Model with cracks

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Results – Model with cracks

Manufacturing Irradiation

2000˚C1000˚C

ambient

c-axis recovery

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•Model input data are crystal properties only. 

•Crystal orientation of each crystallite is chosen randomly in modelling plane.

•The model predicts that:

•cracks occur on cooling from graphitisation temperature

•cracks close with irradiation

•new cracks open as irradiation proceeds.

•The model shows that this cracking behaviour explains

•the reduction of CTE of as-manufactured graphite

•the irradiation induced dimensional change.

In summary

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•The model without cracks has provided some useful insights on:

•the modulus of elasticity of the polycrystal

•the crystallite configurations that are most likely to lead to cracking.

•The model predicts CTE change during manufacturing and during irradiation.

In summary

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Further work

• The existing 2D model will be refined and its application to thermal and irradiation conditions expanded.

• A 3D model is being developed, based upon the refined 2D model.

• Sensitivity of the models to known features of manufactured nuclear graphite, including raw materials, crystallite size, porosity and degree of anisotropy.

• Both models will be compared with historical graphite irradiation data.

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Thank you for your attentionThank you for your attention

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dArea dVoid

Isolated -3 % 0 %

FE (no cracks) -2.4 % N/A

FE -2.4 % 0.1 %

Macroscopic dimensional changes during cooling: 2300 K 1300 K

dLin = -1.2 x 10-2

Average CTE over cooling step = -1.2 x 10-5/K

2300-1300 = 1000K

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dArea dVoid

Isolated -3 % 0 %

FE (no cracks) -2.3 %

FE -1.5 % 1.6 %

•The model predicts that the presence of cracks reduces CTE of as-manufactured graphite and affects the temperature dependence of CTE

•CTE is lower at lower T, where more cracks are open

Second cooling step: 1300 K 300 K

Average CTE over cooling step= -0.75 x 10-5/K

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- Irradiation preceded by 2000 K cooling during manufacturing.

- Two irradiation “steps” needed to recover the c-axis length

Estimated fluence at crack closure

Manufacturing