M2794.007700 Smart Materials and Design
Stress – Strain in Anisotropic Material
April 11, 2017
Prof. Sung-Hoon Ahn (安成勳)
STRESS-STRAIN IN
ANISOTROPIC MATERIAL
2
General anisotropic materials
▪ Recap - isotropic stress, strain
- generalized Hooke’s law ; 81 elements
▪ Special material
Work per unit volume
j i ij i
i
C
2 1 2
W 1
i Cijj
j ij i
i
W C
ij j
i
W C
2
also
j i ij 2C
W 1 j iji
W C
ji
ij C
C
ji
ij S
S
© Sung-Hoon Ahn
Orthotropic material
▪ 3 mutually perpendicular planes of material symmetry
4
3 2 1 3 2 1
66 55
44 33
23 13
23 22
12
13 12
11
3 2 1 3 2 1
C C
C C
C C
C C
C
C C
C
9 elements
3 2
1
3
2
Transversely isotropic material
- One of principal plane is isotropic
3 2 1 3 2 1
55 23
22 22
23 12
23 22
12
12 12
11
3 2 1 3 2 1
C C
2 C C
C C
C
C C
C
C C
C
2 1
3
© Sung-Hoon Ahn
Isotropic material
6
3 2 1 3 2 1
12 11
12 11
12 11
11 12
12
12 11
12
12 12
11
3 2 1 3 2 1
2 C C
2 C C
2 C C
C C
C
C C
C
C C
C
2 constants
3 2 1 3 2 1
66 55
23 22
22 23
12
23 22
12
12 12
11
3 2 1 3 2 1
C C
2 C C
C C
C
C C
C
C C
C
12 31 23
τ τ τ
More on the orthotropic material
▪ No interaction between
▪ No interaction between
▪ No interaction between
3 2 1 3
2
1
, , & , ,
3 2 1 3
2
1
, & , ,
2 1
&
eg.
plane different
in '
&
'
s s
Under plane stress
1 13
20
23
3
When stresses in thickness direction (3) assumed to zero eg. Thin plate, (pressure vessel)
Normal component very small
2
© Sung-Hoon Ahn
More on the orthotropic material
8
3 2 1 3 2 1
66 55
44 33
23 13
23 22
12
13 12
11
3 2 1
C C
C C
C C
C C
C
C C
C
0 0 0
3 66 3
2 1
33 2 23 33
1 13 3
3 33 2
23 1
13
3 23 2
22 1
12 2
3 13 2
12 1
11 1
C
0
C C C
C C C
C 0
C C
C
C C
C
2 22 1
12 2
33 23 23 22
1 33
23 13 12
2
2 12 1
11 2
33 13 23 12
1 33
13 13 11
1
Q C Q
C C C
C C C C
Q C Q
C C C
C C C C
More on the orthotropic material
S 0
0
0 0
3 2 1
66 22
21
12 11
3 2 1
Q Q
Q
Q Q
eq.1
1
2 fiber
direction
© Sung-Hoon Ahn
How to relate S ij to engineering constants
▪ Orthotropic (general case)
10
Test 1.
11 1 12 1
12 2
1 1 1
E E
2Test 2.
Test 3.
2 2 21 2
21 1
2 2 2
E E
12 12
12
G
1
1
E1
1 2
fiber direction
▪ In matrix form
How to relate S ij to engineering constants
1 11 12 1
2 21 22 2
3 66 3
12
1 1
1 21
2
2 2
3
12
0 0
0 0
1
E E
1
E E
1 G S S
S S
S
2 21 1
12
E E
21 12
2 22
21 12
1 11
1 1 Q E Q E
© Sung-Hoon Ahn
Transformation of stress & strain
▪ Two coordinate systems
▪ Loading axes (x,y)
▪ Principal material axes (1,2)
12
2 1
x y
s y x
T
3 2
1
s y x
T
2 1 2
1
3 2 1
2 2 2
2
2 2
2 2
n m mn mn
mn m
n
mn n
m T
2 2 2
2
2 2
1 2
2
n m mn mn
mn m
n
mn n
m T
T
sin
cos
n m
Transformation of elastic parameters
▪ recap – simplification of material constants General anisotropic (81)
orthotropic (9)
Transverseley isotropic (5)
Plane stress (4)
isotropic (2) – (E,υ)
231 2133 0
66 12 22
11,Q ,Q ,Q Q
© Sung-Hoon Ahn
Transformation of elastic parameters
▪ In simple form
14
x,y Q
x,y
x,yTo obtain [Q]
x,yin loading axes (off-axes) using [Q]
1,2in material
axes & transformation
x y y
x
, 2
, 1 1
-
2 , 1 2
, 1 1
-
2 , 1 -1
,
T Q
T
Q T
T
x yy x
, 2
, 1
2 , 1 2
, 1 2
, 1
, 2
, 1
T Q T
Known value
sin , cos T
, T
G , ,
E , E Q
1 -
12 12
2 2 1
, 1
f
f
Transformation of elastic parameters
▪ Actual matrix form
▪ Stress - Strain
Eg. (3.31) & (3.35) are for ten/comp along principal axes – no shear strain Similarly, under pure shear, τ3, along principal axes(1,2), only a pure strain γ3
No coupling between normal stresses & shear deformation and between shear stress
& normal strains
Not true if along arbitrary axes(x,y)
) 63 . 3 ( .
y x ys
yy yx
xs xy
xx y
x
Q Q
Q
Q Q
Q
Q Q
Q e
i
x,y Q
x,y
x,yMatch with previous page’s eq
© Sung-Hoon Ahn
Transformation of elastic parameters
▪ rewrite
16
) 64 . 3 ( 2
2 1 2 2
s y x
ss sy
sx
ys yy
yx
xs xy
xx
s y x
Q Q
Q
Q Q
Q
Q Q
Q
3 2 1
66 22
12
12 11
1
3 2 1 1
s y
Q 0
0
0 Q
Q
0 Q
Q
T T
x
on axis
s y x
T T
T
2 2Q 1
0 0
0 Q
Q
0 Q
Q
2 2Q 1
0 0
0 Q
Q
0 Q
Q
66 22
21
12 11
1
3 2 1
66 22
21
12 11
1
to use eq.60. (3.65)
B
Transformation of elastic parameters
▪ Result
T
2 0
0
0 0 T
2 2 2
66 22
12
12 11
1 -
Q Q
Q
Q Q
Q Q
Q
Q Q
Q
Q Q
Q
ss sy
sx
ys yy
yx
xs xy
xx
Different from [Q]
1,2Matrix
3 3
3 3
3 3
66 3
3 12
3 3
22 3 11
3
66 2 2 12
4 4
22 2 2 11
2 2
66 2 2 12
2 2 22
4 11
4
66 2 2 12
2 2 22
4 11
4
) (
2 )
(
) (
2 )
(
4 )
(
4 2
4 2
Q mn n
m Q
mn n
m nQ
m Q
mn Q
Q n m mn
Q n m mn
Q mn nQ
m Q
Q n m Q
n m
Q n m Q
n m Q
Q n m Q
n m Q
m Q
n Q
Q n m Q
n m Q
n Q
m Q
xs xy yy xx
A=B
© Sung-Hoon Ahn
Transformation of elastic parameters
▪ Strain - Stress
18
x,y S
x,y
x,y
s y x
ss sy
sx
ys yy
yx
sx xy
xx x
S S
S
S S
S
S S
S
2 1 2
1 2
1 2
1
s
y
Similarly
x yy x y
x y
x
, 1
2 , 1 2
, 1 1
2 , 1 1
, ,
,
T S T
S T
T S
CTransformation of elastic parameters
3 2 1
66 22
12
12 11
1
3 2 1 1
s y
2 0 1
0
0 S
S
0 S
S
2 1
2
1
S T
T
x
) 61 . 3
( (3.33)
11 12
1
12 22
66
S S 0
S S 0
0 0 1
2
x y s
T T
S
(3.57)C=D D
S S
S S
S
xx xy xs
12
11
0
© Sung-Hoon Ahn
Transformation of elastic parameters
▪ Result
20
66 2 2 2
12 2 2 22
2 2 11
2 2
66 3
3 12
3 3
22 3
11 3
66 3
3 12
3 3
22 3 11
3
66 2 2 12
4 4
22 2 2 11
2 2
66 2 2 12
2 2 22
4 11
4
66 2 2 12
2 2 22
4 11
4
) (
8 4
4
) (
) (
2 2
2
) (
) (
2 2
2
) (
2 2
S n
m S
n m S
n m S
n m S
S mn n
m S
mn n
m nS
m S
mn S
S n m mn
S n m mn
S mn nS
m S
S n m S
n m
S n m S
n m S
S n m S
n m S
m S
n S
S n m S
n m S
n S
m S
ss ys xs xy yy xx
Transformation of elastic parameters
S
x,yin terms of engineerin g constant
“ shear coupling coefficient”
shear strain
s y x
xy y
ys x
xs
xy sy xy
xy yx sx
x
G E
n E
n
G n G
n
1 E
1 E
E E
1
y x
y
s y
x s
nxs
Normal loading
Shear strain
) 77 . 3 . (eq
© Sung-Hoon Ahn
Transformation of elastic parameters
▪ Micromechanics
▪ Volume
▪ Mass
22
E E V V S A
f
Cij f , m,f ,m, f , m, ,
v m
f V V
V
V
Volume
Volume fraction
1 , ,
v m
f
i i
V V
V
v m f V i
V V
How to get ?
1 ,
0
when Vv Vf Vm Vv
void,
Vm
matrix, Vf
fiber,
m m f
f
m m f
f
m m f
f
v v
m f
V V
V V V
V V
M
V V
M
M M
M M
M
0
Transformation of elastic parameters
▪ Representative elements for elastic constants
▪ Longitudinal Young’s modulus
Longitudinal (ROM)
Transverse (IROM)
L
L
L
Af
Am L
L L
Flip
Lf
F1
m f m
f f m
m f
A A A F A
A A F
A F A
F A
F
F F
F
1 1
© Sung-Hoon Ahn
Rule of mixture
▪ Transverse Young’s modulus
24
1 1 1
1 1 1
f f f
m m m
E E
m m f
f
m m m f
f f
V E V
E E
V E V
E E
1 1
1
1 1 1
1 1
1
ROM
L ∆L
m m F2
L ΔL
A E
A F
2
2 2 2
2
2
2Lmεm2 Lf f
ΔL
Rule of mixture
(IROM-continued)
1 IROM
) Assume
(
,
2
2 2
2
2 2 2
2 2
2
2 2
2
2 2 2
2 2 2
2 2
2 2
2 2
f f m
m
f f f m
m m
m f
f f f
m m m
f f m
m
E V E
V E
E V σ E
V σ E
E σ E
σ
L L L
E L E
out hand
, ,
, 23 12 23
12 G
G
Halpin-Tsai
- Semi empirical model
E
ROM
Test data
© Sung-Hoon Ahn
Rule of mixture
26
psi E
V
psi E
psi E
f m
f
6
6 6
1
6 6
10 16
. 18
) 4 . 0 ( 10 4
. 0 ) 6 . 0 ( 10 30
6 . 0
10 4
. 0
10 30
psi E
E
6 2
6 6
2
10 98
. 0
10 4
. 0
4 . 0 10
30 6 . 0 1
What about particulate composites
ROM, IROM 1)
2)
