Residual Stress analysis

Informations: strains

Fe Cu• Macro elastic strain tensor (I kind)• Crystal anisotropic strains (II kind)

Macro and micro stresses

C

Applied macro stresses

Residual Stress/Strain definition

!'''

!'!''

!

x

!' : Macrostress

!'' : Microstress

!''' : r.m.s. Microstress

Experimental setting

• Measurement of a high 2theta peak position for different tilting of the sample

• The sample can be tilted in omega or psi (chi)

• Changing phi we will scan different direction for the stress in the sample

• The simple behavior is a linear relationship between the measured d-

spacing and sin2psi . The slope is proportional to the macrostrain.

Strain measurement

Strain measurements

eRN

' e33% [e

11cos2N % e

12sin2N % e

22sin2N & e

33] sin2R

% [e13

cosN % e23

sinN] sin2R

x3C

[001]

[100]

[010]

x1C

x2C

(hkl)

Hhkl

x3C

[001]

[100]

[010]

x1C

x2C

(hkl)

Hhkl

x3S

x1S

x2S

C S x3L=M!

"

C S La(#1,#2,#3) $(!,")

6 unknowns: over-determined system, least-squares

xL

i' (ij x

S

j, (

ij' R

ikNkj

'

cosN cosR sinN cosR &sinR

&sinN cosN 0

cosN sinR sinN sinR cosR

; Hhkl2x

L3 .

eL

33 / eRN

' dRN

/ d0 & 1

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Isotropic planar stresses:

The full formula:

Shear stresses

H. DOLLE 493

0 0

(~1 0 0 t / 0 " 1 1 0 " 1 2 0"13~ 0.2 0 , ~0.x2 0.E2 0.23~. (21a-d) 0 0"3/ \0"13 0"23 0"33 ]

While the tensors (2 la, b) represent (residual or applied) stresses in free surfaces, the tensors (21c, d) may represent residual stress states in the interior of a material or stress states caused by multiaxial loading. It is obvious that for the tensors a and c, the principal axes of the stress state coincide with the sample system Pi. The 'classical' d-sin 2 ~k law (1) (Macherauch & Mfiller, 1961) results when (21a) is substituted into (20). For the tensors (21b, c) there are additional terms, but d vs sinE~

is still linear. For the most general stress state (21d), d is no longer linear vs sin 2 0. The terms 013 and 0"23 in (20) have a sin 2qJ dependence. This results in different d vs sin z q/relationships for ~ > 0 and ¢ < 0 as illustrated in Fig. 2(a). This effect has been termed 'lp splitting' (D611e & Hauk, 1977) and has actually been detected by Walburger (1973) in ground steel. In recent years, these lattice-strain distributions were found after grinding or milling (Faninger & Walburger, 1976; Wolfstieg & Macherauch, 1976; Macherauch & Wolfstieg, 1977; D611e & Cohen, 1979), or on the surfaces of wheels and wear (Christ & Krause, 1975; Krause & Jiihe, 1976, 1977).

2.3 The evaluation of the stress tensor for isotropic

materials

The lattice strain for the direction L3 is (Evenschor & Hauk, 1975b)

e~3=ell cos E tp sin 2 tP+ex2 sin 2~o sin E

-[- el 3 COS ~O sin 2~ +/322 s ine q~ sin E

~-/323 sin q~ sin 2~-be33 COS 2 ~ . (22)

Introducing the average strain, a l, and the deviation, aE, from this average strain ('~ splitting'), we find

1 a 1 = :,:[eq,~, + +/3~oq, - ] = (d0,q, + + dq, q, _ )/Edo - 1

=/333 + [/311 cos 2 q~ +/31E sin 2q~

+/322 sin 2 (P--/333] s inE Ip, (23a)

1 a 2 = 7[e~, + -/3,p~, _ ] = (d,p~, + - d~o~, - )/2do

"-" [~13 COS (p -JI- e23 sin ~0] sin[2~9[. (23b)

Thus,/333 can be determined from the intercept of al vs sin 2 ~, if do is known. As a check on this result, note that this value is independent of q~. The tensor components el 1/312/322 can be obtained from (Oal/8 sin E ~). For q~ =0, (el l -e33) is obtained, whereas for q)= 90 °, (e22-/333) is evaluated. The tensor component /312 can then be evaluated from (eal/esinZO) at ~o=45 °. From

(Oaz/t~ sin IE~P[),/3~3 results when q)=0, and e23 when ~0 =90 ° .

Taking the crystallographic anisotropy into account, the stress components a u can be calculated from

1 ~ 5 Sl(hkI)

au= !s2(hkI) L e q - q !SE(hk--~-~sl(hkl)

X (ell nt-E22-b" e33)]. (24)

This method, developed by D611e & Hauk (1976), can also be used when there is no ~b splitting (which implies that 0.13=O'E3=0). It may be a useful procedure to determine whether or not the surface stress condition

0"33-" 0 is fulfilled.

An experimental example of this kind of study (D611e & Cohen, 1979) is discussed below. On ground steel, the e-sin E ~b curves shown in Fig. 4 were measured at the

211 reflection; the evaluation of stresses resulted in the residual stress tensor

/390 306 14 6~/

- 1 9 2 /

(components in MPa). (25)

It is remarkable how well (20) with the stress tensor (25) substituted fits the data points. The principal axes of the stress tensor evaluated by principal axes transformation were tilted about the transverse direction of the sample

P2 by about 11 °.

3. S tres s m e a s u r e m e n t on t e x t u r e d m a t e r i a l s

3.1 X-ray elastic constants of textured materials

For textured materials, the X-ray elastic constants defined by (5), even for the same hkl reflection, depend on the direction ~0@ of the measurement (D611e & Hauk, 1978). Neglecting the interaction of crystallites (t'33ij

=0), we will calculate the anisotropic X-ray elastic constants for a sharp texture in cold-rolled e-iron, as an example of the procedures. Since the averages will be taken over single-crystal compliances s'33u, the X-ray elastic constants will result in the Reuss limit. It will be assumed that the orientation distribution of the crystallites can be idealized by some combination of

?

0 . 5 t--

- ~ 0 .

- .5 '

o/ . /

L..S"

tp=45 ° tO = 90~" [

I

o. .5 ol .5 o. .5

sin2qu Fig. 4. Lattice strain vs sin 2 ~O measured at the 211 reflection of ground

steel (D611e & Cohen, 1979). Measured values: • ~b>0; © ~O <0. ( - - ) Calculated from stress tensor (25). (---) Average strain at (23a).

Non linear behavior

• In many cases oscillation of d vs. sin2psi are observed; some possible

causes:

Textured sample -> the elastic tensor is anisotropic.

Plastic deformation: anisotropy of the plasticity behaviour and elastic

tensor results in anisotropy of the residual stresses/strains

Thermal expansion anisotropy

Shear stresses normal to the surface

Coherent and semicoherent interfaces (in thin film….)

…………

Dolle in 1979 (J. Appl. Cryst., 12, 489) analyzed the problem in general and

was followed by other authors: Noyan and Nguyen for the plastic

deformation, Barral et al. for the texture connection.

Texture-Stress

Procedure:

Measurement of the texture ODF by traditional pole figures

Measurement of the d-spacing vs. sin2psi for high angle reflections

Computation of the effective macro-elastic tensor using single crystal elastic constants and the ODF

Different theories can be used to average the elastic tensor over the ODF:

Voigt (stress compatibility)

Reuss (strain compatibility)

Hill (mean value between Reuss and Voigt)

Self Consistent, FEA.... (costly)

Geometrical mean

Analysis of the d-spacing vs. sin2psi using the averaged elastic tensor

Pro:

You control the entire process

Cons:

Lengthly procedure, two measurements, two analyses

Does not work (very difficult) for highly stressed or strongly textured materials

Traditional methods for the ZrO2 films

1.545

1.550

1.555

1.560

1.565

1.570

1.575

0 0.2 0.4 0.6 0.8 1

Voigt model (no texture)Reuss model (texture)

d 113 [

Å]

sin2(!)

160

170

180

190

200

210

0 20 40 60 80

ZS3

ZS1

ZS2E

last

ic m

od

ulu

s [G

Pa

]

! [degrees]

<C> vs. psi

ZrO2 thin films

0

500

1000

1500

2000

25 35 45 55 65

Inte

nsi

ty [

cou

nts

]

2!

Macro residual stress on the ZrO2 serie

-5

-4

-3

-2

-1

0

0.3 0.5 0.7 1 1.2

Whole pattern analysis

sin2! method

sin2! with texture

Res

idual

str

ess

[GP

a]

Thickness [µm]

Voigt model

Reuss model

F&L model

Measuring the stress also by the curvature

0

2

4

6

8

10

12

14

16

0 4000 8000 12000 16000 20000

[µm

]

scan length [µm]

!2y/!x2=-2.9272e-7 µm-1

---- data

___ fit

!" = K #d s2

d f#a11

2

Comparison of results

method: XRD: sin2! curvature method XRD

(220) plane (200) plane(113) plane Stoney's

formula

modified

formula

Ferrari and

Lutterotti

"11 = "22

[GPa]

-2.73 -1.66 -3.06 -1.36 -1.48 -2.92

"c

* , i = j

"c

* , i # j

[GPa]

0.631

0.043

Stress-texture for a zirconia film (WIMV)

Residual Stresses/Texture analysis

• Voigt model + WIMV

ZrO2 film: results

Very high in plane residual stresses (compression):

Reuss model: 3.6 GPa

Bulk Path GEO: 3.47(5) GPa

Curvature method: > 10 Gpa !?

Reconstructed pole figures

Thickness: 320 Nanometer

Experimental errors

• Example: the CPT film shows big shift of the peaks increasing !.

• The shift is not smaller at low 2theta angle.

• In the fitting was perfectly reproduced by a beam 0.59 mm higher than the goniometer center.

• Using the Rietveld method peak shifts from low angle positions are also used normally -> good sample positioning required, perfect alignment of the instrument also.