1. Surface science: physical chemistry of surfaces Massimiliano Bestetti Lesson N° 9 - 10 November 2011

2. Nome relatore 2 X-ray diffraction residual stress techniques The term residual stress measurement is frequently employed to refer to the experimental determination of the residual stress field in a component. Most analytical techniques measure strain rather stress. Stress can be obtained from strain data through a suitable mechanical model.

3. Nome relatore 3 X-ray diffraction residual stress techniques Residual stress or macrostress: cause a strain which is reflected in an average change of lattice spacing with respect to the stress free state of the surface layer. According to the Bragg law such a change in the average lattice spacing value is visualized by a shift in peakposition as a function of orientation of diffracting planes with respect to the substrate surface. Strained (dry) and unstrained (wet) AgBr in a photographic film.

4. Nome relatore 4 X-ray diffraction residual stress techniques Microstrain: average value of lattice fluctuation in diffracting volume, e =, where d is the interplanar spacing of crystallographic planes (hlk). These fluctuations are inside individual grains and/or as fluctuation from grain to grain. These two origins of microstrain are indistinguishable by a sole XRD. According to Bragg law, 2( d±d)sin( ± )=. The fluctuations of d cause line broadening. Microstrain e can be determined from Williamson-Hall plot cos / = 1/D + (4e/ )sin where is the line broadening and D is an unknown grain size in the direction normal to diffracting planes (peaks = Cauchy functions). The instrumental broadening is subtrated from the measured total broadening to get the physical broadening.

5. Nome relatore 5 X-ray diffraction residual stress techniques Both size and microstrain broadening effect produce a symmetric broadening. Microstrains in crystallites can come from a number sources: dislocations, vacancies, defects, shear planes, thermal expansion and contractions, etc.. Whatever the cause of the residual stress in a crystallite, the effect will cause a distribution of d- values about the normal, unstrained or macrostrained d hkl value.

6. Nome relatore 6 X-ray diffraction residual stress techniques Microstress relief in brass sample by annealing

7. Nome relatore 7 X-ray diffraction residual stress techniques In x-ray diffraction residual stress measurement, the strain in the crystal lattice is measured, and the residual stress producing the strain is calculated, assuming a linear elastic distortion of the crystal lattice. Although the term stress measurement has come into common usage, stress is an extrinsic property that is not directly measurable. All methods of stress determination require measurement of some intrinsic property, such as strain or force and area, and the calculation of the associated stress. http://www.lambdatechs.com/publications/diffraction-notes.html

8. Nome relatore 8 X-ray diffraction residual stress techniques A monochromatic beam of x-rays at a high diffraction angle (2θ) from the surface of a stressed sample for two orientations of the sample relative to the x-ray beam. The angle ψ, defining the orientation of the sample surface, is the angle between the normal of the surface and the incident and diffracted beam bisector, which is also the angle between the normal to the diffracting lattice planes and the sample surface.

9. Nome relatore 9 X-ray diffraction residual stress techniques Sample in the ψ = 0 orientation. The presence of a tensile stress in the sample results in a Poisson's ratio contraction, reducing the lattice spacing and slightly increasing the diffraction angle, 2θ.

10. Nome relatore 10 X-ray diffraction residual stress techniques Sample rotated through some angle ψ. The tensile stress present in the surface increases the lattice spacing over the stress-free state and decreases 2θ.

11. Nome relatore 11 X-ray diffraction residual stress techniques a) Measuring the change in the angular position of the diffraction peak for at least two angles ψ enables calculation of the stress present in the sample surface lying in the plane of diffraction, which contains the incident and diffracted x-ray beams. b) To measure the stress in different directions at the same point, the sample is rotated about its surface normal to coincide the direction of interest with the diffraction plane.

12. Nome relatore 12 X-ray diffraction residual stress techniques X-ray diffraction stress measurement is confined to the surface of the sample. Plane stress is assumed to exist: the stress distribution is described by principal stresses σ 1 and σ 2 in the plane of the surface; no stress perpendicular to the surface, σ 3 = 0. However, a strain component perpendicular to the surface ε 3 (ε 3 0) exists as a result of the Poisson's ratio contractions caused by the two principal stresses.

13. Nome relatore 13 X-ray diffraction residual stress techniques Strain ε φψ in the direction defined by the angles φ and ψ E modulus of elasticity Poisson's ratio α 1, α 2 angle cosines of the strain vector

14. Nome relatore 14 X-ray diffraction residual stress techniques If ψ = 90°, the strain vector lies in the plane of the surface, and the surface stress component, σ φ is The strain in the sample surface at an angle φ from the principal stress σ 1 Equation relates the surface stress σ φ, in any direction defined by the angle ψ, to the strain,, in the direction (φ, ψ) and the principal stresses in the surface

15. Nome relatore 15 X-ray diffraction residual stress techniques d φψ is the spacing between the lattice planes measured in the direction defined by φ and ψ. The strain can be expressed in terms of: d 0 is the stress-free lattice spacing

16. Nome relatore 16 X-ray diffraction residual stress techniques the elastic constants (1 + /E)(hkl) and ( /E)(hkl) are not the bulk values but the values for the crystallographic direction normal to the lattice planes in which the strain is measured as specified by the Miller indices (hkl). Elastic anisotropy

17. Nome relatore 17 X-ray diffraction residual stress techniques The lattice spacing for any orientation is Fundamental relationship between lattice spacing and the biaxial stresses in the surface of the sample. The lattice spacing d φψ is a linear function of sin 2 ψ.

18. Nome relatore 18 X-ray diffraction residual stress techniques d(311) versus sin 2 ψ plot for a shot peened 5056-O aluminum alloy having a surface stress of -148 MPa.

19. Nome relatore 19 X-ray diffraction residual stress techniques The intercept of the plot at sin 2 ψ = 0 is unstressed lattice spacing, d0, minus the Poisson's ratio contraction caused by the sum of the principal stresses

20. Nome relatore 20 X-ray diffraction residual stress techniques The slope of the plot is

21. Nome relatore 21 X-ray diffraction residual stress techniques The x-ray elastic constants can be determined empirically. The unstressed lattice spacing d 0 is generally unknown. Because E » (σ 1 + σ 2 ) d φ0 differs from d 0 by not more than ± 1%, and σ φ may be approximated to

22. Nome relatore 22 X-ray diffraction residual stress techniques By rotating the sample in the plane of an angle the measurements are repeated. We will obtain. The unknowns are, 1 e 2. System of equations

23. Nome relatore 23 X-ray diffraction residual stress techniques