Abstract
The properties of hydrogen atoms dissolved in α and β zirconium are modeled treating the atoms as misfitting defects. Relationships are derived on this basis for their interactions with each other, with dislocations and with other external and internal sources of stress. Experimental results are given showing that hydrogen is more soluble in the β compared to in the α zirconium phase. The variation of this solubility difference, expressed as a partitioning ratio, with Nb concentration in the β zirconium phase is given. Following Kirchheim, relationships are given for the trapping of hydrogen (as hydrides) at dislocations, using Fermi–Dirac statistics with a density-of-sites-energy approach for hydrogen site occupancy.
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- 1.
Strictly speaking, this relationship determines only the extremum condition. Whether this extremum is a stable (equilibrium) one requires taking the second derivative of this function to determine whether the extremum represents a maximum, minimum or saddle point condition.
- 2.
Note that this expression is often given in the literature with the opposite sign. This follows a sign convention where pressure is given as negative when tensile while the opposite sign convention is sometimes used for stress components. In this text, both pressure and individual stress components are taken as positive when tensile. Since the sign convention used is not always indicated in a given text, it can be indirectly inferred from the results for the interaction energy, which should yield a negative (attractive) interaction energy when the misfit strain (volume) is positive and tensile stresses are applied.
- 3.
Hence, the components of this strain dipole are equivalent to the stress-free misfit or transformation strains of the point defect.
- 4.
In the literature it is sometimes said that the reason that defects with positive misfit strains preferentially locate to regions of elevated tensile stress is because the enhanced lattice dilation produced by the elevated tensile stress makes it easier for the defect (which also includes hydrides) to be accommodated there. The foregoing treatment shows that this is not the reason for the attractive interaction energy. In fact, in linear elastic solids there is no interaction between the strains of the lattice produced by a stressed solid and the misfit strains of the defect.
- 5.
The interaction energy terms given in this section are generally of second order and not required in DHC theory. However, the existence of these interactions is sometimes exploited in experimental techniques to determine the defect’s stress-free misfit strain. For this reason, a description of their derivation is given here.
- 6.
- 7.
When dealing with the stress field produced by dislocations most authors have used the sign convention in which compressive stresses have positive signs and tensile stresses negative signs. The opposite sign convention has mostly been used in describing the stress state in front of cracks or for externally applied stresses. In the following, to be consistent with the predominant usage in the literature, we use the sign convention usually employed for describing the stress field of dislocations.
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Puls, M.P. (2012). Solubility of Hydrogen. In: The Effect of Hydrogen and Hydrides on the Integrity of Zirconium Alloy Components. Engineering Materials. Springer, London. https://doi.org/10.1007/978-1-4471-4195-2_4
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