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

Surface science: physical chemistry of surfaces

Massimiliano Bestetti

Lesson N° 9 - 10 November 2011

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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.

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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.

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X-ray diffraction residual stress techniques

Microstrain: average value of lattice fluctuation in diffracting volume, e = <d/d>,

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.

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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 dhkl value.

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X-ray diffraction residual stress techniques

Microstress relief in brass

sample by annealing

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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

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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.

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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.

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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.

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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.

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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 σ1

and σ2 in the plane of the surface; no stress perpendicular to the surface, σ3 = 0.

However, a strain component perpendicular to the surface ε3 (ε3 0) exists as a result

of the Poisson's ratio contractions caused by the two principal stresses.

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X-ray diffraction residual stress techniques

Strain εφψ in the direction defined by the angles φ and ψ

E modulus of elasticity

Poisson's ratio

α1, α2 angle cosines of the strain vector

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X-ray diffraction residual stress techniques

If ψ = 90°, the strain vector lies in the plane of the surface, and the

surface stress component, σφ is

The strain in the sample surface at an angle φ from the principal stress σ1

Equation relates the surface stress σφ, in any direction defined by the angle ψ, to the strain, , in the direction (φ, ψ) and the principal ∈stresses in the surface

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X-ray diffraction residual stress techniques

dφψ is the spacing between the lattice planes measured in the

direction defined by φ and ψ.

The strain can be expressed in terms of:

d0 is the stress-free lattice spacing

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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

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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 dφψ is a linear function of sin2ψ.

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X-ray diffraction residual stress techniques

d(311) versus sin2ψ plot for a shot peened 5056-O aluminum alloy

having a surface stress of -148 MPa.

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X-ray diffraction residual stress techniques

The intercept of the plot at

sin2ψ = 0 is

unstressed lattice spacing, d0, minus the Poisson's ratio contraction caused by the sum of the principal stresses

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X-ray diffraction residual stress techniques

The slope of the plot is

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X-ray diffraction residual stress techniques

The x-ray elastic constants can be determined empirically.

The unstressed lattice spacing d0 is generally unknown.

Because E » (σ1 + σ2) dφ0 differs from d0 by not more than ± 1%,

and σφ may be approximated to

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X-ray diffraction residual stress techniques

2

22

1 sencos

21000 Eddd

2

22

1 sencos

By rotating the sample in the plane of an angle w1 the measurements are

repeated. We will obtain s + 1f w .

The unknowns are f, s1 e s2.

System of equations

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X-ray diffraction residual stress techniques