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A fast efficient multi-scale approach to modelling the development of hydride microstructures at the component level in zirconium alloys Hydrogen Diffusion and Hydride Precipitation in Zirconium Mitesh Patel , Luca Reali , Adrian P Sutton,

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A fast efficient multi-scale approach to modelling the development of hydride microstructures at the component level in zirconium alloys

Hydrogen Diffusion and Hydride Precipitation in Zirconium

Mitesh Patel†, Luca Reali†, Adrian P Sutton, Daniel S Balint and Mark R Wenman*

†Theory and Simulation of Materials †Departments of Physics, Mechanical Engineering and Materials †Imperial College London ∗Email: m.wenman@imperial.ac.uk

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

CONTEXT

Figure : Cross-sectional schematic of a PHWR nuclear fuel pin clad.

Mitesh Patel 2

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

RESEARCH AIMS

Figure : A simplified stage-by-stage breakdown of DHC.

Mitesh Patel 3

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

HYDRIDES IN ZIRCONIUM

Source: [Carpenter (1973)] Source: [Patel (2017)] Hydrostatic stress/GPa Source: [Bailey (1963)], [Perovic (1983)]

Opening stress/GPa

Figure : Micro-hydride needles and macro-hydride structures in zirconium.

Mitesh Patel 4

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

HYDRIDES IN ZIRCONIUM

Source: [Patel (2019)]

Figure : The hydride represented as a continuous distribution of dislocations, leading to the dislocation dipole approximation.

Mitesh Patel 5 Source: [Carpenter (1973)]

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

ρ22

Mitesh Patel 6

THE DISLOCATION DIPOLE APPROXIMATION

Figure : Continuous distributions of dislocations and relative dipole approximation.

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

ρ22

Mitesh Patel 7

A DISCRETE DISLOCATION SUBMODEL

Figure : Once it has dissolved, the hydride leaves behind a tensile region that is likely to aid subsequent hydride nucleation ("memory effect"). Figure : The stress of a hydride can induce plasticity in its neighbourhood. The discrete dislocation simulation investigates effects of multiple dissolution-reprecipitation cycles.

Source: [Patel (2017)]

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

MODELS OF DELAYED HYDRIDE CRACKING

Source: [Cui (2009)] Source: [Cui (2009)]

Figure : Micrographs of notch-tip hydride patterns. Feature Finite Element Phase-Field Present Work Diffusion ✓ ✓ ✓ Precipitation ✓ ✓ ✓ Macro-Hydrides ✗ ✓ ✓ Micro-Hydrides ✗ ✓ Texture ✗ ✗ ✓ Simulation Speed Hours–Days Hours–Days Seconds–Minutes

Mitesh Patel 8

✓

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

MODELS OF DELAYED HYDRIDE CRACKING

Figure : Modelling techniques and typical lengthscales compared to the component size. Feature Finite Element Phase-Field Present Work Diffusion ✓ ✓ ✓ Precipitation ✓ ✓ ✓ Macro-Hydrides ✗ ✓ ✓ Micro-Hydrides ✗ ✓ Texture ✗ ✗ ✓ Simulation Speed Hours–Days Hours–Days Seconds–Minutes

Mitesh Patel 8 Adapted from: [Kim (2007)]

✓ DFT : Density Functional Theory MD : Molecular Dynamics QPF : Quantitative Phase Field FEM : Finite Element Method

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

MODELLING STRATEGY

Input Notch geometry Texture Thermomechanical cycle Methods Planar elasticity Voronoi tesselation Thermodynamics Classical nucleation Diffusion Precipitation Output Hydrogen profile Hydride distribution

Figure : Flowchart of simulation.

Mitesh Patel 9

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

STRESS-DRIVEN DIFFUSION

Figure : A tethrahedral interstice in Zr, wherein the dipole tensor ρij is defined.

Mitesh Patel 10

Figure : Stresses develop at notches. In the model, the theory of distributed dislocations yields analytical expressions for a variety of geometries.

Source: [El Chamaa (2018)]

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

ρ22

Mitesh Patel 11

DIFFUSION-CONTROLLED PRECIPITATION

Figure : Implementation of nucleation sites within a mass balance approach.

Source: [Patel (2019)]

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

ρ22

Mitesh Patel 11

DIFFUSION-CONTROLLED PRECIPITATION

Figure : Implementation of nucleation sites within a mass balance approach.

Source: [Patel (2019)]

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

DIFFUSION-CONTROLLED PRECIPITATION

Figure : Model formulation for diffusion-controlled hydride precipitation.

Mitesh Patel 12 Source: [Patel (2019)]

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

THE AUTOCATALYTIC MECHANISM

Figure : The autocatalytic nucleation: nucleation sites are added so that the global elastic energy is minimised with respect to ϕ.

Mitesh Patel 13

Figure : The resulting macro- hydide and its deck-of-cards

structure. Flexibility of ϕ leads to

local stress reorientation.

Source: [Patel (2019)]

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

ρ22

Mitesh Patel 14

COMPUTATIONAL GEOMETETRY

Figure : Hydride colonies reaching grain boundaries and allowed stacking configurations and are dealt with using computational geometry: the polygon-line segment intersection (PLSI) and the line segment intersection (LSI) algorithms.

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

SIMULATION METHODOLOGY

Figure : The architecture of the code.

Mitesh Patel 15 Source: [Patel (2019)]

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

THE REORIENTATION EFFECT

Figure : Elastochemical and mechanical equilibrium state.

Mitesh Patel 16 5 10 15 20 2 4 6 8 10 c(x1, x2)/c0 5 10 15 20

20 40 60 80 100 σyy(x1, x2)/GPa

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

HYDRIDE MORPHOLOGICAL EVOLUTION

Figure : Simulating micro-hydride needles and macro-hydride structures.

Mitesh Patel 17 5 10 15 20

20 40 60 80 100 σyy(x1, x2)/GPa 5 10 15 20 2 4 6 8 10 c(x1, x2)/c0

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

MICRO-HYDRIDE STACKING

Figure : Formation of macoscopic hydride features.

Mitesh Patel 10 November 2017 500 1000 2 4 6 8 10 c(x1, x2)/c0 Initial Concentration: 100.00 ppm

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

NOTCH-TIP MACRO-HYDRIDE FEATURES

500 1000 2 4 6 8 10 c(x1, x2)/c0 Initial Concentration: 100.00 ppm 500 1000 2 4 6 8 10 c(x1, x2)/c0 Initial Concentration: 100.00 ppm

Figure : Formation of macoscopic hydride features for different notch geometries.

Mitesh Patel 11 November 2017

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

SIMULATED MICROGRAPHS

Source: [Cui (2009)] Source: [Cui (2009)]

Figure : Formation of macoscopic hydride features with more realistic grain sizes.

Mitesh Patel 12 November 2017

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

SUMMARY AND FUTURE DIRECTIONS

Further Considerations :

Grain boundary effects: intergranular nucleation sites & compatibility

stresses (PhD starting in Oct 2019)

Thermal rachetting: featuring a discrete dislocation plasticity submodel to

include the "memory effect" in hydride precipitation.

Mitesh Patel 21

Goal : Fast & efficient (i.e. seconds on a laptop) framework to simulate H profile and hydrided micrographs. At component scale, while retaining full detail of the micro-hydride substructure. Strategy : Physically based multiscale model, not encumbered by unnecessary details, rooted in analytical expressions and optimised computational geometry algorithms. Parametric Study : Effect of texture, notch geometry and loading conditions is possible after calibration of the parameters through experiments.

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EXPERIMENTAL SCALE

Component Scale Macroscopic Scale Mesoscopic Scale Microscopic Scale

Multiscale Modelling of Delayed Hydride Cracking Rolls-Royce plc

ACKNOWLEDGEMENTS

PhD Study

◮ EPSRC Centre for Doctoral Training on Theory and Simulation of Materials

Supervisors

◮ Dr Mark R Wenman

Centre for Nuclear Engineering, Materials

◮ Prof. Adrian P Sutton

Condensed Matter Theory, Physics

◮ Dr Daniel S Balint

Mechanics of Materials, Mechanical Engineering Imperial College London

◮ Luca Reali, Saïd El Chamaa, Sana Waheed

Rolls-Royce plc.

◮ Mike Martin, Salah Muflahi, Dave Rugg, Rob Bentley, Chas Gee, Ted Darby, Robb

Marshall

Mitesh Patel 22