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.