The use of conventional X-ray diffraction is limited to near-surface residual stress measurements due to its low penetration depth. For stress measurements deep inside bulk components, more specialised techniques, requiring dedicated large-scale facilities, using either neutrons or high energy X-rays are required. Neutrons have the advantage over X-rays that for wavelengths comparable to the atomic spacing, their penetration into engineering materials is typically several centimetres with good spatial resolution.
There are two sources for the production of intense neutron beams: reactor sources and accelerator-based spallation sources. In reactor sources, neutrons are generated by the fission of uranium whilst in spallation sources, neutrons are obtained by the bombardment of protons into a heavy nucleus (e.g. Tungsten, Uranium, Tantalum). Examples of reactor sources are: <ILL> in France, <ANSTO> in Australia and examples of spallation sources are: <ISIS> in UK and <PSI> in Switzerland. Spallation sources are also sometimes referred to as pulsed sources or time-of-flight diffractometer.
The measurement of residual stresses using neutron diffraction is based on Bragg’s law of diffraction. If neutrons with a wavelength λ are incident on a crystal and are diffracted from lattice planes of spacing dhkl, diffraction peaks will be observed at an angle 2θ relative to the incident beam. Under an applied tensile (or compressive) stress, the lattice spacing in individual crystallite grains expands (or contracts). This change in the lattice spacing can be detected, at constant wavelength, as a shift (Δθhkl) in diffraction peaks.
Principle of neutron diffraction showing a Bragg reflection from favourably aligned crystal planes. There are normally a large number of grains in the gauge volume, but only a fraction of these will be in the correct orientation to meet the diffraction condition at any time.
In a time-of-flight (TOF) diffractometer, the neutrons originating from the moderator have a wide energy range from a few meV up to many eV, which corresponds to wavelengths in the range from 0.1–10 Å. When a pulse of neutrons is directed onto a sample, those with different wavelengths will require different times to reach the sample at a fixed distance from the moderator. Thus if the detected neutron count is plotted as a function of time it will exhibit a series of peaks corresponding to the different dhkllattice planes in the material.
Typical TOF diffraction spectrum on ENGIN-X, in this case for a stainless-steel specimen.
If one can measure the d-spacing in a stress-free material and a stressed material, the strain can be easily calculated from the change in lattice spacing:
where εxx is the strain in the xx direction, dxx is the stressed lattice spacing and d0xx is the stress-free lattice parameter.
The stress in a given direction (e.g.) can then be calculated using the generalised Hooke’s law equations:
Where E is the bulk elastic modulus for the material, v is Poisson’s ratio, and ex, ey and ez are the measured strains in three orthogonal orientations x, y and z, respectively.
Stress-free lattice spacing
The stress-free lattice spacing, d0, is a critical measured parameter required for determining residual stresses as very small changes in its value can result in large uncertainties. Changes in composition, martensitic phase transformations, changes in temperature and intergranular strains are some of the factors that can affect the stress-free lattice parameter. Recommended methods for determining a representative stress-free reference value are , :
- Measurement at a position or time known to contain negligible stress;
- Measurement on a stress-free powder or filings that are representative of the material being examined;
- Measurements on stress-free reference cubes or comb specimens extracted from the sample;
- Application of force/moment balance
- Imposing a zero stress condition perpendicular to a free surface
Generally cubes are used, but these are difficult to handle and position accurately on the instrument. Furthermore, in some components (for example weldments), there may be gradual changes in stress-free lattice spacings, which calls for an extensive process of extracting cubes accurately from different locations. Combs and matchsticks are becoming increasingly common for measurements of stress-free lattice parameters in such cases.