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残余应力测试


BSSM Workshop PART II 2ψ Method Using The sin Laboratory X-Rays XJudith 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”

The sin2ψ Method How Does it Work?
We measure STRAIN (ε) not STRESS (σ)
? 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

? Strain = ε = ?d/d

Changes in d-spacing dwith Stress
Consider a bar which is in tension

? The d-spacings of the planes normal to the applied stress increase, as the stress is tensile ? The d-spacings of the planes parallel to the applied stress decrease, due to Poisson strain

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 2θ, of the reflection & we can measure this 2θ (Only Just!!)

? 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 polygives a high angle Bragg reflection
? Metals ? Ceramics (not easy!) ? Multi-phase materials Multi-

– Not usually applied to polymers, as no suitable reflections, can add a metallic powder, reported in the literature

Why use the sin2Ψ Method The Advantages
? Most Important
– A stress free d-spacing is NOT required for the bi-axial case which is almost always used bi– Other advantages
? Low cost (compared with neutrons & synchrotrons, but not hole drilling) ? Non-destructive, unlike hole drilling Non? Easy to do & fairly fool proof (if you are careful!!)

Disadvantages
? Most Important
– Surface method only, X-ray beam penetration Xdepth 10 to 20 microns, at best – For depth profiling must electro-polish, gives 1electro11.5mm – Other Disadvantages
? 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

? Notation, σ(ij) σ the stress component acting on face i in direction (parallel to axis) j

Basic Theory

? The normal stresses act normal to the cube faces & the two subscripts are the same – e.g.. σ (22) ? The shear stresses (twisting forces) act parallel to the cube faces & the two subscripts are different – e.g. σ (31) or in the general case σ (ij) ? We measure normal stresses & shear stresses, but that’s not what we want, we don’t get all of the information! Why??

Basic Theory Normal Stresses
? From elastic theory of isotropic materials, the 3 normal strains are given by,
ε11 = 1 [σ11 - ν(σ22 + σ33)] E ε22 = 1 [σ22 - ν(σ33 + σ11)] E ε33 = 1 [σ33 - ν(σ11 + σ22)] E ? The strain in any direction is a function of the stress in the others!!. Ideally, we should measure more than one direction

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 φ, in its own plane, exact details & diagrams later

How the Sin2Ψ Method Works Sample in “Bragg Condition”
Diffraction vector, normal to sample surface

θ
dn

θ

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

Also called focussed geometry

How the Sin2Ψ Method Works
Ψ
θ

Diffraction vector, titled with respect to sample surface

Defocused geometry

Tilt the sample through an angle Ψ and measure the dspacing again. These planes are not parallel to the free surface. Their d-spacing is changed by the stress in the sample.

How the Sin2Ψ Method Works
? We tilt the sample through an angle psi,Ψ to measure magnitude the normal & shear stresses
– We use a range of values of Ψ (called offsets) for example, from 0 to 45° in steps of 5° 45° 5° – NEVER use the “Double Exposure Method” which uses just one Ψ offsets. Not enough data points! offsets.

? We rotate the the sample through an angle, φ to determine the directions of the principle stresses

No Stress Free d-Spacing dNeeded The Approximation
? The depth of penetration of the X-ray beam in the sample Xis small, typically < 20? 20? ? We can say that there is no stress component perpendicular to the sample surface, that is σ33 = 0 ? We can use the d-spacing measured at ψ = 0 as the stress free d-spacing
– This is the d-spacing of the planes parallel to the sample surface d-

? A reasonable approximation!! The error is <2%, certainly less than trying to make a stress free standard!!!

The Equation for the sin2Ψ Method
? The simplest form of the equation is, σφ = E (1 + ν) sin2Ψ Were
– – – – – –

dΨ - dn dn

Strai n Term

σφ = Stress in direction φ E = Young’s modulus (GPa) ν = Poisson’s ratio Ψ = Tilt angle (degrees) dΨ = d-spacing measured at tilt angle, Ψ (?) dn = The “stress free d-spacing” from our approximation measured at Ψ = 0 (?)

The sin2Ψ Plot : The Results!

? ?

dn is obtained by extrapolating a plot of dΨ (or strain) against sin2Ψ to Ψ = 0 Stress is obtained from the gradient, m of the sin2Ψ plot σφ = E (1 + ν) m

If the d-spacing decreases, the stress is compressive (planes pushed together) If the d-spacing increases the stress is tensile (planes pulled apart)

The sin2Ψ Plot : Example
We can plot STRAIN against sin2ψ & obtain the STRESS from the gradient
Shot Peened Steel
y = -0.0028x - 2E-05
0.00000 0.0000 -0.00010 -0.00020 -0.00030 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000

Strain

-0.00040 -0.00050 -0.00060 -0.00070 -0.00080 -0.00090

Sin psi

2

The sin2Ψ Plot : Example
? Also, we can plot d{hkl} against sin2ψ & obtain the stress from the gradient, which is the same on both plots
Shot Peened Steel
y = -0.00278x + 1.17038
1.17050 1.17040 1.17030 1.17020 1.17010

d{hkl}

1.17000 1.16990 1.16980 1.16970 1.16960 1.16950 0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000

sin2 psi

General: Stress Diffractometers ? Basically adapted powder diffractometers ? Can accommodate larger, heavier samples
– Maximum accessible 2θ angle is larger 2θ ? Usually about 165° 2θ (check this if you buy 165° one!!!) – More axes of rotation than a standard powder diffractometer, omega and 2θ can move independently 2θ

There are Two Basic Types
? Laboratory Based Systems
– Fixed location – Can usually be used for other applications, for example phase identification

? Portable systems
– Designed specifically for residual stress measurements – Can carried and fixed to a large component (aircraft!)

Diffraction Angles used in Stress Analysis
Omega ω Goniometer circle θ ψ Sample 2θ Incident X-ray beam Phi φ ω Normal to sample surface 2θ = 0°

Chi χ

Diffracted X-ray beam

?

Diffraction Geometry Summary of the Angles Used in Residual Stress Analysis TwoTwo-theta (2θ)
The Bragg angle, angle between the incident (transmitted) and diffracted Xray beams. beams.

? Omega (ω) (ω
– The angle between the incidence X-ray beam and the sample surface. Both Xomega and two-theta rotate in the same plane. two-

? Phi (φ)
– The angle of rotation of the sample about it’s surface normal. normal.

? Psi (ψ) (ψ
– Angles through which the sample is rotated, in the sin2ψ method. We start at psi=0, where omega is half of two-theta and add (or subtract) successive twopsi offsets, for example, 10, 20, 30 and 40° 40°

? Chi (χ) (χ
? Angle of rotation about the axis of the incident beam. Chi rotates in the plane normal to that containing omega and two-theta. This angle is also twosometimes (confusingly) referred to as ψ

Instrumentation The Omega Method Portable Systems

Instrumentation an Example an of a Portable System, Manchester’s Proto i-XRD i-

Deciding What to do?
? We need to decide how to make our measurements, we need to make some choices, ? Which X-ray tube to use? X? Which crystallographic plane do we choose? – The best thing to do is copy what someone else has done!
? Your results will be comparable with those made by other workers ? Many Industries have “set” methods

Radiation Selection
Choice of X-Ray Tube X(Wavelength!) ? ALWAYS check what other people have done in the past as, generally measurements on different planes with different wavelengths are not comparable

? 3 Considerations
(1) Dispersion (2) Fluorescence (3) Choice of crystallographic plane

Radiation Selection
Choice of X-Ray Tube X(Wavelength!)
? We can measure the stress in a variety of materials (i.e. ferrite, austenite, nickel, aluminium, corundum etc) using the same diffractometer, by changing the X-ray tube & consequently the Xwavelength of the X-rays. X– Most residual stress diffractometers will have a selection of X-ray tubes available X-

– How do we choose?????

Choice of X-Ray Tube X(1) Dispersion ? We need a 2θ angle, ideally > 140° 2θ 2θ 140°
– The change in d-spacing, due to strain, is very small, typically in the third decimal place – The dispersion of the diffraction pattern is much greater at high 2θ angles. The small changes in d-spacing 2θ d-

can only be detected at angles > 125° 2θ 125°

Choice of X-Ray Tube X(1) Dispersion An Example ? If we have a reflection from ferrite {211} at 156° 156° 2θ. Using radiation from a chromium anode X-ray tube of wavelength 2.2897? X? If we introduce a stress of 200 MPa, given Young’s modulus of 220 GPa, what is the change in the 2θ angle? 2θ
? Answer, the new 2θ angleis155.51° 2θ angleis155.51° ? The difference is 0.48° NOT MUCH!!! 0.48°

Choice of X-Ray Tube X(2) Sample Fluorescence ? If the K-α1 component of the incident XKXray beam causes the sample emit its own fluorescent X-rays, DO NOT USE IT X– X-ray penetration depth will be very small <5 microns & ∴ NOT representative of the bulk – Peak to background ratio will be terrible – May damage sensitive detectors

Choice of X-Ray Tube X(3) Choice of Crystallographic Plane
– For accurate comparison with other peoples data CHECK which planes have been used historically!!
? Measurements made on planes with different Miller {hkl} indices are not usually comparable.

– If the sample is textured (preferred orientation) select a set of planes with a high multiplicity

Choice of Measurement Conditions: Summary
– Ask someone who has experience with that particular material
? Don’t re-invent the wheel ? Choose radiation type carefully
– Avoid X-ray tubes which cause K α-1 fluorescence

– Lot’s of “tricks of the trade” see the NPL Good Practice Guide for Residual Stress Measurements using the sin2ψ Method

Data Collection Positioning the Sample
– Sample must be centre of rotation of the goniometer, most instruments have depth gauge or a pointer – Be careful that the sample is as flat as possible, bent samples will give artificial shear stresses – For curved and uneven samples restrict the irradiated area
? Hoop direction, Spot size < R/4, where R= radius of curvature ? Axial direction, Spot size < R/2

Data Collection
– Make sure that you collect data over a sufficient 2θ range! Include the background on both sides of the peak. Can be difficult as inelastic is usually present & this causes peak broadening. Peaks can be up to 10° 2θ 10° – Count for a sufficient time to ensure adequate statistics, need > 1000 count at the top of the peak if possible

Data Processing
– ALWAYS CHECK THIS STAGE – Need a program with good graphics – Stages in the data processing
? ? ? ? Background stripping K-α2 stripping (only if K-α2 peak is visible) KLorentz Polarisation Correction Peak fitting to locate maximum
– Critical Stage, check the results on the screen. – A variety of peak models are available most of which will work. Usually use Gaussian, don’t use parabola

? Good quality data can be fitted with most models, this is a good test!

How Precise are the Results
– Generally there’s a lot of scatter on sin2ψ plots!
? The error bars printed out by most PC’s are just the standard deviation of the points from the fitted line and tend under estimate the errors

– Large error bars are not necessarily unacceptable and are due to,
? Texture, large grain size, poor peak fitting etc ? For example, 200 ± 50MPa is quite normal ? Check the peaks on the screen!

– Values of less than ± 50MPa, can usually be thought of as zero, this depends on the instrument
? To confirm such low readings make several measurements & see if they all come out with the same sign (i.e. all compressive)

Instrument Misalignment
- Omega-2θ misalignments

-Omega-ψ misalignments (side inclination method) Instrument misalignment causes,
? Shifts in the positions of the reflections and incorrect stress values ? The positive and negative ψ measurements give different peak positions, this is called ψ splitting

We must measure at least two standards to verify that the machine is working correctly

Instrument Misalignment
- Recommended Standards ? A stress free powder
– Not an easy thing to make – Beware stresses due to filing and oxidation – Can be combined with resin for ease of use

? A stressed standard
– – – – Be careful, always measure in the same direction Shot peened samples are good Usually verified by Round-Robin tests No certified standards (???)

– One set for each tube anode

A typical Example of a Stress Profile in a Shot Peened Sample
– A shot peened surface, depth profiled by ElectroElectropolishing
400 200 0

Residual Stress (MPa)

-200 -400 -600 -800 -1000 -1200 -1400 0 40 80 120 160 200 240 280 320
1.2% Strain, R=0 o 350 C / 1 cycle, Longitudianl o 350 C / 1 cycle, Transverse As-peened

Depth (?m)

Good one, the material has a small grain size (< ≈ 100?) 100? is isotropic, rather than textured & there’s no shear stress. Ideal!

Problems!!!
Shear Stress, the positive & negative ψ plots split

Texture, the “wiggle” Our sample is not isotropic

Shear Stresses sin2Ψ Splitting
– Positive and negative Ψ give different results when a shear stress is present (or sample is not correctly positioned, always check!) – Function of the direction of the measurement

Conclusions
The sin2Ψ method works well if you are careful – Check to see what’s been done by others
? Don’t reinvent the wheel

– Choose you X-ray tube with care X– Position the sample carefully
? Think about the directions you wish to measure

– Measure a sufficient range of 2θ & count for a 2θ sufficient time – Check you peak fitting – Do the results make sense??????

Thank You & Happy Landings!


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