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Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic...
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Transcript of Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic...
![Page 1: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/1.jpg)
Internal stress measurement using XRD
Elasticity, for an isotropic elastic solid: the elastic constant E and v
kkijijij E
v
E
v
1 : Kroenecker’s delta
332211 kk
ij
Written explicitly:
)]([1
)(1
3322113322111111
vEE
v
E
v
)]([1
33112222 vE
)]([1
22113333 vE
121212 2
11
E
v
)1(2 v
E
Shear modulus
31312323 2
1
2
1
![Page 2: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/2.jpg)
Stress normal to a free surface ( ) must be zero at the surface, i.e.,
jn0 jij n
Equation of equilibrium (satisfied at each point of thematerial):
03
1
j j
ij
x
Transformation of the strain tensor (from one coordination system to another: ijnjmimn aa '
where defines the cosine of the angle between in the old coordinate system and in the new coordinate system.
mia ix
mx
![Page 3: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/3.jpg)
Supplement
Vector transformation from one (X) to another (X’) coordinationsystem:
X system:
X’ system:
332211 iiiA AAA
332211 iiiA AAA
332211332211 iiiiii AAAAAA
)()( 332211332211 iiiiiiiiAi AAAAAA jjj
jA)( 332211 iiiiii jjjj AAAA
3
2
1
332313
322212
312111
3
2
1
A
A
A
A
A
A
iiiiii
iiiiii
iiiiii)cos( jkkj ii
![Page 4: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/4.jpg)
1S
2S
3S 3L
Consider the transformation of the sample coordinate system to the laboratory coordinate system .
iS
iL
Find out the transformation matrix for the above case:1. Rotate along the axis by an angle ;2. rotating an angle along the
3S '
2S
100
0cossin
0sincos
y
x
yx
![Page 5: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/5.jpg)
100
0cossin
0sincos
cos0sin
010
sin0cos
transformation matrix for the coordinate system
cos0sin
010
sin0cos
z
y
x
'
'
'
z
y
x
z
y
x
cossinsinsincos
0cossin
sincossincoscos
z
x
z
x
![Page 6: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/6.jpg)
cossinsinsincos
0cossin
sincossincoscos
333231
232221
131211
aaa
aaa
aaa
ijnjmimn aa '
cossinsinsincos
0cossin
sincossincoscos
'33
'32
'31
'23
'22
'21
'13
'12
'11
333231
232221
131211
cossinsinsincos
0cossin
sincossincoscos
![Page 7: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/7.jpg)
Interested in ijjiaa 33'33
13122
1122'
33 cossincos2sincossin2sincos
332
232222 coscossinsin2sinsin
13 and 31
13122
1122'
33 2sincossin2sinsincos
332
232222 cos2sinsinsinsin
212
22332211
'33 sin2sin
1sincos)]([
1
E
vv
E
2233112213 sinsin)]([
12sincos
1
v
EE
v
222113323 cos)]([
12sinsin
1
v
EE
v
Change strain to stress
![Page 8: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/8.jpg)
Look at the 11 term, there are
211
2211
2211 cossinsinsincos
1
E
v
E
v
E
2211
2211 sincossincos
E
v
E
v Add and subtract one term
We get 1122
11 sincos1
E
v
E
v
Similar for 22 term
2333333
23333 sin
11cos
1
E
v
E
v
E
v
E
v
E
v
2222
22 sinsin1
E
v
E
v
For 33 term
![Page 9: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/9.jpg)
Let’s group the sin2 into one term, and the rest …
332
332
22122
11'33
1sin]sin2sincos[
1 E
v
E
v
2sin)sincos(1
)( 2313332211
E
v
E
v
The quantity measured at angles and . '33
: d-spacing in the stresses sample (measured for the plane whose normal is at angles , from the sample coordinate system); : d-spacing for the unstressed state is related
0
0'33 d
dd
d
0d
'33
![Page 10: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/10.jpg)
Three stress states of interests are: uniaxial, biaxial, and hydrostatic states.
000
000
0011 ij
1122
110
0'33 sin]cos[
1 E
v
E
v
d
dd
112sin
1 E
v
E
v
* uniaxial stress state:
![Page 11: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/11.jpg)
* biaxial stress state:
000
0
0
2221
1211
ij
)(sin]sin2sincos[1
221122
22122
11'33
E
v
E
v
)(sin1
22112
E
v
E
v
2
22122
11 sin2sincos
)(1
)]([1
2211333322113333 E
v
Ev
E
33332'
33
1sin
1 EE
v
033
332'
33 sin1
E
v
![Page 12: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/12.jpg)
0
0
33
33
0
33
0
033
0
033
'33
d
dd
d
dd
d
dd
d
dd
d
dd
2
0
0 sin1
E
v
d
dd
2
0
0
sin)1( v
E
d
dd
![Page 13: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/13.jpg)
volumetric strain (hydrostatic stress): H
)21(3 ;
v
EKKH K: bulk modulus
* Hydrostatic stress state:
00
00
00
ij
E
v
E
v
E
v 2131'33
volumetric strain : 332211
![Page 14: Internal stress measurement using XRD Elasticity, for an isotropic elastic solid: the elastic constant E and v : Kroenecker’s delta Written explicitly:](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649f045503460f94c1907c/html5/thumbnails/14.jpg)
2sin
Slope ~ E
vd
133
Linear relation when the sampleis in the biaxial stress state.
dWhen the sample is in the triaxialstate -splitting
2sin
d
2sin
d
2sin)sincos(1
...... 2313'33
E
v
asymmetric
The shear stress can lead tocompression of some plane spacingand expansion of others
Presence of stress gradient, textureand/or elastic and plastic anisotropic