internal stress measurement using xrd elasticity, for an isotropic elastic solid: the elastic...
TRANSCRIPT
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
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
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
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
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
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
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
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
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
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:
* 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
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
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
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