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

13

60º Strain Rosette

120,60,0 cba

)(3

2

)22(3

1

cbxy

acbyy

axx

6. Strain Transformation 25

3