1
Strain Transformation
Strain at an element on x-y-z coordinates
xzxy
xx
22
zzyzxz
yzyy
xy
xx
zzzyzx
yzyyyx
xzxyxx
22
22
22
ε
Strain at the same point at x’-y’-z’ coordinates
zxyxxx
zxyxxx
22
6. Strain Transformation 1
There should be some relationships between and ’The relationships are called strain transformation
zzzyzx
zyyy
yx
zzyzxz
zyyyxy
22
22ε
Plane-Strain Transformation
Plane-strain condition: strains related to z direction are all vanished
6. Strain Transformation 2
Only 3 independent strain components: xx, yy, xy (xy/2)
2
Plane-Strain Transformation
Plane-strain condition is not equal to plane-stress conditionstress condition
Plane stress produces zz
or
6. Strain Transformation 3
or 0zz
Plane-Strain Transformation
Strain at the same point but different coordinates
6. Strain Transformation 4
scoordinate - thein calculate want to
s,coordinate - thein known are If
'''''' yx, ,
yx, ,
yxyyxx
xyyyxx
3
Plane-Strain Transformation
Consider a line segment dx’
sin
cos
xddy
xddx
6. Strain Transformation 5
dydx
xd
and from
of elongation know the Want to
Plane-Strain Transformation
xx
6. Strain Transformation 6
jii ˆsinˆcosˆ dxdxdx xxxxxx
4
Plane-Strain Transformation
yy
6. Strain Transformation 7
jij ˆcosˆsinˆ dydydy yyyyyy
Plane-Strain Transformation
xy
6. Strain Transformation 8
jii ˆsinˆcosˆ dydydy xyxyxy
5
Plane-Strain Transformation
yy
xx
x
x
xd x
dxdy
cossincos xyyyxxxx xd
dydydx
xd
x
6. Strain Transformation 9
cossin2sincos
cossinsincos
cossincos
22
22
xyyyxx
xyyyxx
xyyyxx xd
dy
xd
dy
xd
dx
Plane-Strain Transformation
yxyx
x
xd
yyd
sincossintan xyyyxx dydydxy
6. Strain Transformation 10
2sincossinsincos
sincossin
tan
xyyyxx
xyyyxx xd
dy
xd
dy
xd
dxxdxd
6
Plane-Strain Transformation
yx
sin)
2cos( ydyddx
x
xd
yyd
cossincostan xyyyxx
yd
dydydx
ydx
x
y
yd
dx
2dy
cos)
2sin( ydyddy
6. Strain Transformation 11
2cossincoscossin
cossincos
xyyyxx
xyyyxx yddy
yddy
yddx
ydyd
Plane-Strain Transformation
yxyx
x
xd
yyd
)sin(coscossin)(2 22 xyxxyyyx
6. Strain Transformation 12
)sin(coscossin)(2
)sin(cos2cossin)(2
22
22
xyxxyy
yxyx
xyxxyy
7
Plane-Strain Transformation
Plane-strain transformation
)cossin2(sincos 22 xyyyxxxx
)sin(cos2
cossin)(2
or
)sin(coscossin)(
)cossin2(2
sincos
)(
22
22
22
xyxxyy
yx
xyxxyyyx
xyyyxx
xyyyxxxx
6. Strain Transformation 13
Similar formulas as plane-stress transformation
)sin(coscossin)(
)cossin2(sincos22
22
xyxxyyyx
xyyyxxxx
Plane-Strain Transformation
Conclusions:To obtain principal strains or maximum in-planeTo obtain principal strains or maximum in-plane shear strain, one can use similar methods as plane-stress transformation, such as Mohr’s circle method, eigenvalue method, etc.
6. Strain Transformation 14
cases all in or 2
use :Note xyxy εγ
8
Strain Measurement
Stress measurement: not possible, t li it d hexcept very limited cases, such as
contact stresses
Strain measurement: always can be done by several methods
6. Strain Transformation 15
Strain Measurement
Normal Strainy y
Strain gages1. Mechanical: extensometers
O ti l ti l diff ti
x0l
xxl
0
011
l
ll
x
u
x
u xxx
Undeform Deformed
6. Strain Transformation 16
2. Optical: optical diffraction3. Electrical: resistance, capacitance, inductance4. Acoustical5. Semiconductor: piezoresistive properties
9
Extensometer
MTS extensometer
6. Strain Transformation 17
Electrical-Resistance Strain Gages
ASTM E251: Standard Test Methods for Performance Characteristics of Metallic Bonded Resistance Strain Gages
Gage Factor:
// 000 RRllRRK
6. Strain Transformation 18
//0
0
0
0
0
0
RlRK
10
Electrical-Resistance Strain Gages
6. Strain Transformation 19
Electrical-Resistance Strain Gages
ASTM E1561: Standard Practice for Analysis of Strain Gage Rosette Data
Configurations of strain gages:
6. Strain Transformation 20
11
Electrical-Resistance Strain Gages
6. Strain Transformation 21
Electrical-Resistance Strain Gages
Normal strain measurement along x axis
y
measurement along x-axis
Off-angle normal strain measurement
x
cossinsincos 22
xyyyxx
xx
6. Strain Transformation 22
x
y
unknownunknownunknownmeasured
Hope to obtain xx, yy, xy
Need 3 strain gages
12
Strain Rosettes
aaxyayyaxxa cossinsincos 22
6. Strain Transformation 23
xyyyxx
ccxycyycxxc
bbxybyybxxb
,,
cossinsincos
cossinsincos22
22
45º Strain Rosette
90,45,0 cba
)(2 cabxy
cyy
axx
cba
6. Strain Transformation 24