A Diffusion Model for the Effect of Applied Stress on Void and Loop Growth

Abstract

The effect of a general state of stress on swelling has been considered in a theoretical void growth model. Angular-dependent loop growth has been incorporated into the model in order to include the effects of shear stress. It has been found possible to consider the effects of stress on void and loop growth as a relatively simple deformation process. We define the stress effect as follows: where (dϵi/dt)σ=0 is the strain rate associated with the isotropic swelling that would occur in stress-free conditions. For an applied stress having both hydrostatic and shear components, the hydrostatic component provides a driving force for isotropic volume swelling and the shear component causes a shape change. The total deformation process (stress-induced volume increase and shear) may be represented by the relation: where the σ's are applied tensile stresses and the i, j, K are the principal axes. K and ν are two material parameters that depend upon microstructure (dislocation density, void number density, and void size) but do not depend upon flux. K depends strongly on temperature whereas ν is independent of temperature. The extent to which plastic flow must accompany stress-assisted swelling is governed by the parameter ν. As ν approaches 1/2, the deformation process becomes pure shear. It is found that the value of ν is bounded and lies between -1/3 and +1/2. Thus, except under the application of purely hydrostatic stress, stress-assisted swelling cannot be completely shear-free. When transmission electron microscopy (TEM) data for neutron-irradiated solution-treated Type 316 stainless steel were used, the values of ν were found to lie between 0 and 0.5. For cold-worked steels, ν is expected to lie in the upper part of the range just given. The significance of the present results becomes evident in the analysis of fuel pin profiles. For isotropic swelling with no plastic flow, the fractional change in fuel pin diameter is 1/3 the fractional isotropic volume change. However, for stress-assisted swelling, the relation may be much different and depends upon the parameter ν. For a biaxial stress state of the type that occurs in pressurized cylindrical tube, for example, Δ D/D ≈ 0.7 when ν = 0.1, whereas ΔD/D≈ 1.4 ΔV/V when ν = 0.3.

Stress state dependence of in-reactor creep and swelling

“…Finally, note that when m p ¼ 1=2, which is the case for creep of void free material, Eq. (8) shows that there is no stress-affected swelling component, as _ S r ¼ 0.…”

“…In this expression m p is a material parameter that, by analogy to linear elasticity, is identified [8,10] as a plastic Poisson's ratio, which is defined as a ratio of the stress-dependent plastic increments of strain in a tensile test, m p À_ e 2 = _ e 1 . The third term on the right hand side of Eq.…”

“…With these assumptions, SGW [8] developed approximate expressions for the plastic Poisson's ratio: m p À_ e 2 = _…”

“…The effect of a shear (deviatoric) stress on swelling was considered by Straalsund, Guthrie and Wolfer (SGW) [8] in an early void growth model, leading to a general conclusion ''that both stress-accelerated void growth and dislocation loop creep are manifestations of the same mechanism, and must therefore occur simultaneously in the presence of a stress state that contains both deviatoric and hydrostatic components". Their model, however, related stress effect on void growth entirely to the hydrostatic stress component and, therefore, did not include the possibility that deviatoric stress may affect void growth in the absence of a hydrostatic component.…”

“…Using microstructural modeling estimates of equation parameters, obtained from the work of SGW [8], the predicted effects of stress state triaxiality, stress exponent and plastic Poisson's ratio on deviatoric strain rate were explored parametrically. Although not emphasized in this early paper, Hall's equations also predict that swelling is a function of deviatoric stress, but only for non-linear creep and a for mixed deviatoric plus hydrostatic stress state.…”