Principal Stresses
• We should measure more than one direction to get a complete picture of the stress in the component • If we measure 3 directions or more we can calculate the PRINCIPAL STRESSESS, these are the directions on which no shear stress acts • We do this by rotating the sample through an angle , in its own plane, exact details & diagrams later
Defocused geometry
How the Sin2 Method Works
• We tilt the sample through an angle psi, to measure magnitude the normal & shear stresses
– We use a range of values of (called offsets) for example, from 0 to 45 in steps of 5 – NEVER use the “Double Exposure Method” which uses just one offsets. Not enough data points!
Disadvantages
• Most Important
– Surface method only, X-ray beam penetration depth 10 to 20 microns, at best – For depth profiling must electro-polish, gives 11.5mm – Other Disadvantages
Basic Theory Normal Stresses
• From elastic theory of isotropic materials, the 3 normal strains are given by,
11 = 1 [11 - (22 + 33)] E 22 = 1 [22 - (33 + 11)] E 33 = 1 [33 - (11 + 22)] E • The strain in any direction is a function of the stress in the others!!. Ideally, we should measure more than one direction
• Low cost (compared with neutrons & synchrotrons, but not hole drilling) • Non-destructive, unlike hole drilling • Easy to do & fairly fool proof (if you are careful!!)
• Affected by grain size, texture (preferred orientation) & surface roughness • Doesn’t work on amorphous materials (obviously!!)
Basic Theory
• Consider a unit cube (quite a big one!) embedded in a component
• Metals • Ceramics (not easy!) • Multi-phase materials
– Not usually applied to polymers, as no suitable reflections, can add a metallic powder, reported in the literature
Measuring Elastic & Inelastic Strain
• Primarily we are measuring macro stresses
– This is a uniform displacement of the lattice planes – These cause a VERY SMALL shift in the position, the Bragg angle 2, of the reflection & we can measure this (Only Just!!)
BSSM Workshop PART II The sin2ψ Method Using Laboratory X-Rays Judith Shackleton School of Materials, University of Manchester
The sin2ψ Method What are We Measuring?
• We measure the ELASTIC Strain. – We can determine – Magnitude of the stress, – Its direction – Its nature
• Compressive or tensile
– We use the planes of the crystal lattice as an atomic scale “strain gauge”
• We rotate the the sample through an angle, to determine the directions of the principle stresses
No Stress Free d-Spacing Needed The Approximation
Also called focussed geometry
How the Sin2 Method Works


d
Diffraction vector, titled with respect to sample surface
Tilt the sample through an angle and measure the d-spacБайду номын сангаасng again. These planes are not parallel to the free surface. Their dspacing is changed by the stress in the sample.
The sin2ψ Method How Does it Work?
We measure STRAIN () not STRESS ()
• We CALCULTE STRESS from the STRAIN & the ELASTIC CONSTANTS • We use the planes d{hkl} , of the crystal lattice as a strain gauge • We can measure the change in d-spacing, d
• Most Important
Why use the sin2 Method The Advantages
– A stress free d-spacing is NOT required for the bi-axial case which is almost always used – Other advantages
• Inelastic stresses cause peak broadening, which can be measured. This is an extensive subject, not covered here.
Which Materials Can We Measure?
– Works on any poly-crystalline solid which gives a high angle Bragg reflection
• The depth of penetration of the X-ray beam in the sample is small, typically < 20 • We can say that there is no stress component perpendicular to the sample surface, that is 33 = 0 • We can use the d-spacing measured at = 0 as the stress free d-spacing
– This is the d-spacing of the planes parallel to the sample surface
• A reasonable approximation!! The error is <2%, certainly less than trying to make a stress free standard!!!
How the Sin2 Method Works Sample in “Bragg Condition”
Diffraction vector, normal to sample surface

dn

We measure the dspacing with the angle of incidence () & the angle of reflection of the Xray beam (with respect to the sample surface) equal. These planes are parallel to the free surface & unstressed, but not unstrained