Theoretical background
• Khachaturyan developed theory on elastic strain energy due to inhomogeneous solids which gives approach to calculate the energy due to precipitates configuration; The equilibrium state of inhomogeneous solids with coherent lattice should correspond to minimal elastic strain energy
Strain energy under uniform stress
When the system of multicomponent with coherent interaction, the external stress could affect the configurational energy, hence the arrangement of precipitates due to minimal B(n)
1 Gil ijkl n j nk
The use of above equation should be proceeded with caution since no previous results have been found as benchmark. Despite the definition of ijkl is also called elastic average by arbitrary P M ijkl ijkl (1 )ijkl
Reference: Mishima, Y., S. Ishino, and H. Kawnishi, Some observations on the dissolution and precipitation of zirconium hydrides in a-zirconium by electron microscopy. Journal Name: pp 489-496 of International Congress on Hydrogen in Metals. Vols. 1 and 2. Paris Editions Science et Industrie, 1972.
0 vol% hydride,0.075 with stress oriented at 0
1 1 1 d 3k d 3k 0 * 0 0 E Vincl Cijkl ij ij kl kl (V Vincl )Cijkl ij kl s n G [n (q )]s 0 jl nl 3 3 ik k ij 2 2 2 (2 ) (2 )
* * Therefore, B( n ) s ik nk Gij ( n )s jl nl
Is modified with the elastic tensor of precipitates when w approach zero this is compared to the B function using elastic tensor of average
Université Scientifique et Technologique de Lille
Laboratoire de Métallurgie Physique et Génie des Matériaux USTL Bât C6 – 2e étage 59655 Villeneuve d’Ascq
28th of September 2011, Les Renardières EDF R&D
Experimental review
Re-orientation: time-dependent, concentration dependent
Hoop stress definition sh=pr/t, t:thickness, p: internal pressure, r:radius
Literatures milestones
f: the angle between unit vector and referential axis q: the angle between applied stress and referential axis
The other documents that well studied the hydrides distribution free or under stress is found only in 1970’s by Mishima et al in which the massive hydrides appears to be less affected by the stress and the dissolution instead of re-orientation was found on the basal plane. This means reorientation could occur in prismatic planes that has to be studied theoretically.
3D modeling of minimum strain energy surface
Conclusion and future works Acknowledgement
Literatures and theoretical background
literature milestones Microelasticity of Khachaturyan’s inhomogeneous solid Limits of microelastic theory
* kl
* * Therefore, B( n ) s ik nk Gij ( n )s jl nl
n is unit vector of normal to habit plane and varies from 0-p to calculate B
Limit of microelastic theory
P M ijkl ,ijkl Elastic tensor of precipitates and matrix respectively

is volume fraction
1 P Gil ijkl n j nk
Therefore the Green function is modified to
Higher the hydrides concentration higher stress level is needed to trigger re-orientation
tline
Literature and theoretical background 2D modeling on basal plane: minimium strain energy and phase field method 2D modeling on prismatic plane: minimum strain energy and phase field method
One of the key tasks of the B function calculation lying with the determination of the Green function in 2D or 3D. Khachaturyan et al has developed the theoretical approach that inverse Green function is related to the elastic tensor of matrix and precipitates in coherent case by spacial unit vector xi x j therefore the inverse G-1
2D modeling of microstructure dependent strain energy vs. phase field modeling
Strain energy for Hydride volume fraction approach zero Phase field modeling for single hydride Phase field modeling for a population of hydrides Strain energy for Hydride volume fraction >>0
* 0 * s ij Sijkl kl
0 Sijkl
is average compliance tensor dependent on elastic stiffness and volume fraction of each materials component is effective strain due to applied uniform stress for instance along axis
0 B( n ) s ik nk Gij ( n )s 0 jl nl
The stress term in above equation is the stress free stress; In order to calculate the mimimal B(n) as function of angel respective to the reference axis, one should replace stress-free stress by effective stress.
0 Configurational B(n ) s ik nk ij (n )s 0 jl nl energy