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

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