The final case study deals with understanding the effect of a multi-axial state of stress on creep life predictions. All models discussed inadvertently pre­dict the deformation rates under a uniaxial state of stress. However in real life situations, the engineering structures experience a multi-axial state of stress as well as variations of loading with time. Here we present an example of how a multi-axial state of stress would affect diffusion creep. For a more detailed account of the effects of multi-axial state of stress and loading his­tory, we refer the reader to Penny and Marriott.82

Raj109 proposed relationships to understand the strain rates of deforma­tion during diffusion creep under a multi-axial state of stress. Since strain rate in diffusion creep has a linear correlation with applied stress, Raj sug­gested that under a multi-axial state of stress the strain rates of deformation can be represented in terms of principal stresses and strain rates,

[3.63 ]

These relationships are borrowed from equations of elasticity and Poisson’s ratio has been assumed to be 0.5 for constancy of volume. Here k1, k2 and k3 are creep constants that correspond to the appropriate diffusion creep mechanism. These strain rates of deformation will be identical for an iso­tropic or equiaxed microstructure but would be different for non-isotropic grain configuration. Also the diffusional creep should be independent of the hydrostatic state of stress which has an effect only on the diffusivity term.

Materials such as Zr and other hexagonal close packed metals and alloys exhibit distinct crystallographic textures leading to anisotropic mechanical and creep properties. Zr-alloys are commonly used as thin-walled tubes to clad radioactive fuel such as UO2 and tube reduction processes render them highly textured during the manufacturing of cladding tubes. Prediction of dimensional changes in-reactor of the cladding along both the diametral and axial directions requires consideration of multi-axial loading particularly with axial load superimposed with internal/external pressure. A modified

Hill formulation was shown to be convenient wherein the generalized stress, ag, is defined as the uniaxial stress along the tube axial direction,110

_2 _ R (( — ъ?)2+RP ( — ъ )2+P ( — ъ )2 [3.64]

g P(R +1) ‘

The parameters, R and P, are the mechanical anisotropy parameters. Using the Prandtl-Reuss energy balance in conjunction with the above yield crite­rion, the strain increments or strain rates along the three orthogonal direc­tions are related to the respective stresses

where £g is the generalized strain rate corresponding to the generalized stress ag. These equations define the yield and flow loci, and the corre­sponding creep locus is referred to at a constant energy dissipation-rate, W (= atj etj = ag eg).111 It is clear from the above equations that the mechan­ical anisotropy parameters, R and P, are the transverse contractile strain (-rate) ratios in uniaxial tests. The contractile strain ratios (CSRs), R and P, define the resistance to wall thinning of an anisotropic material and thus control the formability which is of importance to the material manufac­turers, in tube reduction processes, sheet drawing and forming, etc.112 The mechanical anisotropy parameters are directly related to the preferred ori­entations of the grains. Thus obtained creep loci for cold-worked stress — relieved annealed and following complete recrystallization are shown in Fig. 3.28.113 Crystallite orientation distribution function (CODF) creep model predictions113 are shown as solid lines along with the experimental results.

These types of creep studies are not commonly found albeit materials in real engineering structures experience such complex multi-axial loadings. Additional complexity is encountered in attempting to predict transients in creep under such multi-axial loading.