macrochemical analysis of lumina

1. Chap 2 Macromechanical
Analysis of a Lamina
• Laminate is constructed by stacking a
number of laminas in the direction of the
lamina thickness)
• Limina is the building block of a laminate

2. 2.1 Brief Review
•Macromechanics means that the average mechanical
properties are focused, ignoring the interaction of
components in composites.
•Generally, a single-layer board is analyzed in terms
of a plane stress problem since its thickness is much
smaller than its length and width. Only the in-plane
stresses exist.

3. 












































































12
31
23
3
2
1
66
65
64
63
62
61
56
55
54
53
52
51
46
45
44
43
42
41
36
35
34
33
32
31
26
25
24
23
22
21
16
15
14
13
12
11
12
31
23
3
2
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C

4. 












































































12
31
23
3
2
1
66
56
46
36
26
16
56
55
45
35
25
15
46
45
44
34
24
14
36
35
34
33
23
13
26
25
24
23
22
12
16
15
14
13
12
11
12
31
23
3
2
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
• Constants Cij=Cji
• C is symmetrical, including 21 independent constants.
For anisotropic elastomers:

5. Material with one plane of symmetry
• If the material has one symmetry plane, the number of elastic
parameters is reduced to 13.













































































12
31
23
3
2
1
66
36
26
16
55
45
45
44
36
33
23
13
26
23
22
12
16
13
12
11
12
31
23
3
2
1
C
0
0
C
C
C
0
C
C
0
0
0
0
C
C
0
0
0
C
0
0
C
C
C
C
0
0
C
C
C
C
0
0
C
C
C
单对称材料

6. Orthotropic Material
•More symmetry planes lead to the reduction of
independent constants.
-- If there are two symmetry planes -- 9 independent
constants.













































































12
31
23
3
2
1
66
55
44
33
23
31
23
22
21
13
12
11
12
31
23
3
2
1
C
0
0
0
0
0
0
C
0
0
0
0
0
0
C
0
0
0
0
0
0
C
C
C
0
0
0
C
C
C
0
0
0
C
C
C
No coupling!

7. Transversely Isotropic Material
•Five independent constants














































































12
31
23
3
2
1
12
11
44
44
33
13
13
13
11
12
13
12
11
12
31
23
3
2
1
2
C
C
0
0
0
0
0
0
C
0
0
0
0
0
0
C
0
0
0
0
0
0
C
C
C
0
0
0
C
C
C
0
0
0
C
C
C
2
C
C
C 12
11
66


Subscripts 1 and 2 can exchange

8. •Two independent constants
2
/
)
C
C
(
C
C
C
C
C
C
C
C
C
12
11
66
55
44
31
23
12
33
22
11




























































































12
31
23
3
2
1
12
11
12
11
12
11
11
12
12
12
11
12
12
12
11
12
31
23
3
2
1
2
C
C
0
0
0
0
0
0
2
C
C
0
0
0
0
0
0
2
C
C
0
0
0
0
0
0
C
C
C
0
0
0
C
C
C
0
0
0
C
C
C
Isotropic Material

9. Constitutive equation（flexibility matrix）













































































12
31
23
3
2
1
66
56
46
36
26
16
56
55
45
35
25
15
46
45
44
34
24
14
36
35
34
33
23
13
26
25
24
23
22
12
16
15
14
13
12
11
12
31
23
3
2
1
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
S
• It is inverse to stiffness matrix
• The both matrices has the same features

10. Flexibility matrix expressed by engineering constants
 















































12
31
23
3
2
23
1
13
3
32
2
1
12
3
31
2
21
1
ij
G
1
0
0
0
0
0
0
G
1
0
0
0
0
0
0
G
1
0
0
0
0
0
0
E
1
E
E
0
0
0
E
E
1
E
0
0
0
E
E
E
1
S
(Orthotropic material)

11. 




































12
2
1
66
22
12
12
11
12
2
1
S
0
0
0
S
S
0
S
S
2.2 Relationship in Plane stress State
(Orthotropic)
1
2
3
Only three stresses 1, 2,12
are non-zero.
The flexibility matrix is
simplified as

12. 12
12
12
2
2
)
2
(
2
2
1
12
)
2
(
1
1
1
12
)
1
(
2
1
1
)
1
(
1
G
1
E
1
E
E
E
1



















1
2
1

2

12

1

2

12

Deducing…
by
2.2 Relationship in Plane stress State
(Orthotropic)
by
by
No coupling!

13. Based on the superposition principle：
12
12
12
2
2
1
1
12
)
2
(
2
)
1
(
2
2
2
1
12
1
1
)
2
(
1
)
1
(
1
1
G
1
E
1
E
E
E
1









































































12
2
1
12
2
1
12
2
21
1
12
2
1
G
1
0
0
0
E
1
E
0
E
E
1





































12
2
1
66
22
12
12
11
12
2
1
S
0
0
0
S
S
0
S
S
12
66
2
22
2
21
1
12
12
1
11
G
1
S
E
1
S
E
E
S
E
1
S










14. 




































12
2
1
66
22
12
12
11
12
2
1
Q
0
0
0
Q
Q
0
Q
Q
66
66
2
12
22
11
11
22
2
12
22
11
12
12
2
12
22
11
22
11
S
1
Q
S
S
S
S
Q
S
S
S
S
Q
S
S
S
S
Q







12
66
21
12
2
22
21
12
1
21
21
12
2
12
12
21
12
1
11
G
Q
1
E
Q
1
E
1
E
Q
1
E
Q



















Stiffness constants in plane stress state

15. 2
21
1
12
E
E



E
S
E
1
S
)
S
S
(
2
0
0
0
S
S
0
S
S
12
11
12
2
1
12
11
11
12
12
11
12
2
1















































































12
2
1
66
11
12
12
11
12
2
1
Q
0
0
0
Q
Q
0
Q
Q
G
)
1
(
2
E
Q
1
E
Q
1
E
Q
66
2
12
2
11











four independent constants:
E1,E2,12 & G12
For isotropic materials

16. 34
.
0
,
GPa
80
.
5
G
,
GPa
50
.
8
E
,
GPa
3
.
134
E 12
12
2
1 




Try to calculate the stiffness constants and flexibility constants.
GPa
80
.
5
G
Q
GPa
91
.
2
E
m
Q
Q
GPa
56
.
8
mE
Q
,
GPa
3
.
135
mE
Q
0074
.
1
)
E
E
1
(
m
,
TPa
4
.
172
G
/
1
S
TPa
53
.
2
E
/
S
S
TPa
6
.
117
E
/
1
S
,
TPa
45
.
7
E
/
1
S
12
66
2
12
21
12
2
22
1
11
1
1
2
2
12
1
12
66
1
1
12
21
12
1
2
22
1
1
11































1
1
2
2
12
1
21
12 )
E
E
1
(
)
1
(
m 
 






Let
Eg. The engineering constants for a lamina (T300/648)
are

17. 2.3 Constitutive eq of single-layer
board in arbitrary direction
Coordinate Transformation is necessary when the principle
directions are inconsistent with the geometric directions.
Oblique to the shaft
1
2
y
x
+

18. Using the stresses σ1, σ2 and σ3 in 1-2 coordinate
system to express σx, σy and σz in x-y coordinate
system:
2.3 Constitutive eq of single-layer
board in arbitrary direction
     
T 
 
 
     
T 
 
 
1
2 y
x
 1 cos( , )
l x x

 2 cos( , )
l y x

 3 cos( , )
l z x


1 cos( , )
m x y

 2 cos( , )
m y y

 3 cos( , )
m z y


1 cos( , )
n x z

 2 cos( , )
n y z

 3 cos( , )
n z z



19. 2 2 2
1 1 1 1 1 1 1 1 1
2 2 2
2 2 2 2 2 2 2 2 2
2 2 2
3 3 3 3 3 3 3 3 3
2 3 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 2
3 1 3 1 3 1 3 1 1 3 3 1 1 3 3 1 1 3
1 2 1 2 1
2 2 2
2 2 2
2 2 2
x
y
z
yz
zx
xy
l m n m n n l l m
l m n m n n l l m
l m n m n n l l m
l l m m n n m n m n n l n l l m l m
l l m m n n m n m n n l n l l m l m
l l m m n n









 
 
 
 

 
  
 
    
 
 
  2 1 2 2 1 1 2 2 1 1 2 2 1
x
y
z
yz
zx
xy
m n m n n l n l l m l m






 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
   
   
2 2 2
1 1 1 1 1 1 1 1 1
2 2 2
2 2 2 2 2 2 2 2 2
2 2 2
3 3 3 3 3 3 3 3 3
2 3 2 3 2 3 2 3 3 2 2 3 3 2 2 3 3 2
3 1 3 1 3 1 3 1 1 3 3 1 1 3 3 1 1 3
1 2 1 2 1
2 2 2
2 2 2
2 2 2
x
y
z
yz
zx
xy
l m n m n nl l m
l m n m n n l l m
l m n m n n l l m
l l m m n n m n m n n l n l l m l m
l l m m n n m n m n n l nl l m l m
l l m m n n









 
 
 
 

 
  
 
    
 
 
  2 1 2 2 1 1 2 2 1 1 2 2 1
x
y
z
yz
zx
xy
m n m n nl n l l m l m






 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
  
  

20. 





















































12
2
1
2
2
2
2
2
2
xy
y
x
sin
cos
cos
sin
cos
sin
cos
sin
2
cos
sin
cos
sin
2
sin
cos
Similarly,
Coordinate transformation matrix
𝜀𝑥
𝜀𝑦
𝜏𝑥𝑦
=
cos2𝜃 sin2𝜃 −sin𝜃cos𝜃
sin2
𝜃 cos2
𝜃 sin𝜃cos𝜃
2sin𝜃cos𝜃 −2sin𝜃cos𝜃 cos2𝜃 − sin2𝜃
𝜀1
𝜀2
𝜏12
     
T 
 
 
     
T 
 
 

21. 


















































12
2
1
66
11
12
12
11
12
2
1
xy
y
x
Q
0
0
0
Q
Q
0
Q
Q
 



























12
2
1
12
2
1
Q
When the principle axises of material are consistent with
the coordinate system:
If inconsistent:
2.3 Constitutive eq of single-layer
board in arbitrary direction
         
1
T S T S
 
  

  
 
   
         
1
T C T C
 
  

  
 
   
   
T
S T S T
 
  
     
T
C T C T
 
  
 

22.  



















































xy
y
x
66
26
16
26
22
12
16
12
11
xy
y
x
xy
y
x
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
)
sin
(cos
Q
cos
sin
)
Q
2
Q
2
Q
Q
(
Q
cos
sin
)
Q
2
Q
Q
(
cos
sin
)
Q
2
Q
Q
(
Q
cos
sin
)
Q
2
Q
Q
(
cos
sin
)
Q
2
Q
Q
(
Q
cos
Q
cos
sin
)
Q
2
Q
(
2
sin
Q
Q
)
sin
(cos
Q
cos
sin
)
Q
4
Q
Q
(
Q
sin
Q
cos
sin
)
Q
2
Q
(
2
cos
Q
Q
4
4
66
2
2
66
12
22
11
66
3
66
22
12
3
66
12
11
26
3
66
22
12
3
66
12
11
16
4
22
2
2
66
12
4
11
22
4
4
12
2
2
66
22
11
12
4
22
2
2
66
12
4
11
11























































• Nine non-zero constants.
• Coupling phenomena occur in this generalized constitutive eq. for
orthotropic materials. Be quite different from Q!
Odd
Even functions

23. 




































12
2
1
66
22
12
12
11
12
2
1
S
0
0
0
S
S
0
S
S
    



















































xy
y
x
66
26
16
26
22
12
16
12
11
xy
y
x
T
xy
y
x
S
S
S
S
S
S
S
S
S
T
S
T
)
sin
(cos
S
cos
sin
)
S
S
4
S
2
S
2
(
2
S
cos
sin
)
S
S
2
S
2
(
cos
sin
)
S
S
2
S
2
(
S
cos
sin
)
S
S
2
S
2
(
cos
sin
)
S
S
2
S
2
(
S
cos
S
cos
sin
)
S
2
S
2
(
sin
S
S
)
sin
(cos
S
cos
sin
)
S
S
S
(
S
sin
S
cos
sin
)
S
S
2
(
cos
S
S
4
4
66
2
2
66
12
22
11
66
3
66
12
22
3
66
12
11
26
3
66
12
22
3
66
12
11
16
4
22
2
2
66
12
4
11
22
4
4
12
2
2
66
22
11
12
4
22
2
2
66
12
4
11
11























































Similarly, the strain can be
expressed by stress
in x-y system:

24. 




































12
2
1
66
26
16
26
11
12
16
12
11
12
2
1
Q
Q
Q
Q
Q
Q
Q
Q
Q





































12
2
1
66
26
16
26
22
12
16
12
11
12
2
1
S
S
S
S
S
S
S
S
S
12
12
,
2
2
3
,
12
26
12
12
,
1
1
1
,
12
16
12
66
2
22
2
21
1
12
12
1
11
G
E
S
G
E
S
G
1
S
E
1
S
E
E
S
E
1
S





















i
ij
i
,
ij
ij
i
ij
,
i










The generalized constitutive eq. of a common anisotropic
elastomer can be deduced in a similar way.
Hint: The off-axis tension in composites
leads to axial extension and shear strain!
(New engineering constants
— interaction coefficient )

25. i
ij
i
,
ij
ij
i
ij
,
i










New engineering constants — interaction coefficient
The first kind: the extension in i-direction caused by
the shear stress in i-j plane
Off-axis tension in composites
The second kind: the shear stress in i-j plane caused
by the normal stress in i-direction

26. 2
22
2
21
1
12
12
1
11
E
1
S
E
E
S
E
1
S








)
sin
(cos
S
cos
sin
)
S
S
4
S
2
S
2
(
2
S
cos
sin
)
S
S
2
S
2
(
cos
sin
)
S
S
2
S
2
(
S
cos
sin
)
S
S
2
S
2
(
cos
sin
)
S
S
2
S
2
(
S
cos
S
cos
sin
)
S
2
S
2
(
sin
S
S
)
sin
(cos
S
cos
sin
)
S
S
S
(
S
sin
S
cos
sin
)
S
S
2
(
cos
S
S
4
4
66
2
2
66
12
22
11
66
3
66
12
22
3
66
12
11
26
3
66
12
22
3
66
12
11
16
4
22
2
2
66
12
4
11
22
4
4
12
2
2
66
22
11
12
4
22
2
2
66
12
4
11
11























































12
12
,
2
2
3
,
12
26
12
12
,
1
1
1
,
12
16
12
66
G
E
S
G
E
S
G
1
S














27. 



































































































 








































 




3
12
1
12
2
3
12
1
12
1
y
y
,
xy
3
12
1
12
2
3
12
1
12
1
x
x
,
xy
4
4
12
2
2
12
1
12
2
1
xy
4
2
2
2
1
12
12
4
1
y
2
2
12
2
1
4
4
1
12
x
xy
4
2
2
2
1
12
12
4
1
x
cos
sin
G
1
E
2
E
2
cos
sin
G
1
E
2
E
2
E
cos
sin
G
1
E
2
E
2
cos
sin
G
1
E
2
E
2
E
)
cos
(sin
G
1
cos
sin
G
1
E
2
E
2
E
2
2
G
1
cos
E
1
cos
sin
E
2
G
1
sin
E
1
E
1
cos
sin
G
1
E
1
E
1
)
cos
(sin
E
E
sin
E
1
cos
sin
E
2
G
1
cos
E
1
E
1
Engineering constants with off-axis loading is expressed by

28. •Generalized elasticity modulus of anisotropic
elastomers varies with the angle between the
loading direction and the principle direction of
composite material under off-axis loading.
•The ultimate value of material performance (max.
or min.) may not appear in the principle directions
of material.
•Performance of composites can be designed and
optimized according to the above features.
Summary

29. 2.4 Strength of Orthotropic Laminae
Strength -- an important concept
•Practically, a lamina is hardly used due to its low
strength and stiffness in transverse direction.
However, multi-layer composite board is often
used as structural materials.
•Purpose of this section is to express the strength
in arbitrary direction by the properties in the
principle directions of material. (differing from the
traditional isotropic materials)

30. •Basic strength – in the principle of material
Xt -- longitudinal tensile strength
Xc -- longitudinal compressive stength
Yt -- transverse tensile strength
Yc -- transverse compressive strength
S -- in-plane shear strength
•They, together with E1,E2,12 and G12, are called
engineering constants. (9 constants)
•Strength is dependent of direction.

31. The strength of isotropic material reflects the material
strength in simple loading state.
•Plastic – yield stress or constraint yield stress
•Brittle – ultimate stress
•Others: shear yield stress, fatigue, etc.
For orthotropic materials
•Strength varies with direction
•The mechanism of tensile failure differs from
that of compressive failure
•In-plane shear strength is independent.

32. Eg. A unidirectional fibre reinforced lamina is shown in
Fig.1.
1
2
X
Y
S 2
2
2
cm
/
N
2000
S
cm
/
N
1000
Y
cm
/
N
50000
X



Stress field
2
12
2
2
2
1
cm
/
N
1000
cm
/
N
2000
cm
/
N
45000






Fig. 1
The maximum principle stress σ1 is lower than the maximum
material strength X, but the board will fail in 2-direction,
resulting from the higher stress σ2 when compared to the
strength in 2-direction.
Conclusions
Strength

33. •The shear strength in the principle direction is
independent of tension and compression. The
maximum value of shear stress
-- constant regardless of the performance difference
between compression and tension.
-- invariable in spite of a positive shear stress or a
positive shear stress.
•The maximum value of the off-axis shear strength
is dependent of the positive/negative sign of shear
stress.
Eg.

34. Different off-axis shear strengths
1
2
1
2 1
2
1
2
Shear in principle directions Shear at 45֠ to the principle axises
complicate
（identical max. ）
(different shear strength)
+ -
=
=

35. Experimentally Determining Strength and Stiffness
•Strength constants
Xt Xc Yt Yc S
•Stiffness constants
E1 – elasticity modulus in 1-direction
E2 – elasticity modulus in 1-direction
12 = 2/1, when 1=  and other stress =0
21 = -1/2, when 2=  and other stress = 0
G12 – shear modulus in 1-2 plane

36. •Basic principle in tests:
The stress-strain relationship is linear with the increase of
the load until failure.
• Generally, the tensile curve exactly shows linear feature
while the compressive curve and shear curve are nonlinear.
•The key point in the tests is to load specimens
evenly.
Experimentally Determining Strength and Stiffness

37. Normal stress and shear strain
Shear strain and normal stress
Normal stress and bending curvature
Bending stress and normal strain
Coupling effect
for orthotropic materials, the orthotropy
often leads to coupling phenomena when the
load is off-axis.
Experimentally Determining Strength and Stiffness

38.  Unidirectional tensile test in 1-direction
1
2
P
P
×
1
1
1
E1
1 max=X
1、2 can be determined by measurement
A
/
P
X
E
A
/
P
max
1
2
12
1
1
1
1











Experimentally Determining Strength and Stiffness

39. Unidirectional tensile test in 2-direction
2
1
P P
×
2
2
1
E2
2 max=Y
Measure 1、2
A
/
P
Y
E
A
/
P max
2
1
21
2
2
2
2 










Experimentally Determining Strength and Stiffness

40. 2
21
1
12
E
E



Stiffness coefficients
must meet the equation:
 The test devices are not accurate;
 Miscalculation;
 The stress-strain relationship of the tested
material is nonlinear.
If not
Experimentally Determining Strength and Stiffness

41. Unidirectional test at 45 degrees to 1-axis
450
2
y
1
x
P
P
x
x
1
Ex
2
1
1
12
x
12
2
12
1
12
1
x
x
x
E
1
E
1
E
2
E
4
1
G
E
1
G
1
E
2
E
1
4
1
E
1
A
/
P
E




















Measure x
（G12 is determined）
Based on
Experimentally Determining Strength and Stiffness

42. 2.5 Bidirectional strength theory of
anisotropic laminae
• The above methods often used in unidirectional stress
state.
• Practically, an elastomer may be loaded in two or three
directions.
• A new failure criterion should be identified when loading
in multi-directions.
• Failure criterion is used for predicting failures. It is not
the model of true failures and consequently the failure
mechanism cannot be uncovered by it.

43. x
y
Failure Data
×
Failure
×
Yield
2.5 Bidirectional strength theory of
anisotropic laminae
Failure envelope -- combinations of the normal and shear
stresses.
Two-dimensional failure envelope with a given shear stress

44. 1. Maximum Stress Failure Theory
In plane stress state, a failure is predicted in a
unidirectional lamina if any stress component in the
local axes equals or exceeds the corresponding
ultimate strength.
This criterion includes 3 independent expressions
or 3 sub-criteria.
The stress should be expressed in the principle
axises of material.
Description

45. S
Y
X 12
t
2
t
1 





S
Y
X 12
c
2
c
1 





























cos
sin
S
sin
Y
cos
X
cos
sin
sin
cos
x
2
x
2
t
x
x
12
2
x
2
2
x
1
Criterion
Tensile
Compressive
Theoretical and experimental
results are often conflicted

46. 2. Maximum Strain Failure Theory
The criterion also includes 3 independent expressions
or sub-criterions.
Strain should be expressed in the material axes.
Especially, a term incorporating Poisson’s ratio is involved
in the expression, which indicates the effect of bidirectional
stress is considered. The effect will be small if the Poisson’s
ratio is small.
Description
In plane stress state, a failure is predicted if any
strain component in the material axes exceeds the
maximum strain.

47. Expressions


 




 S
Y
X 12
2
1 t
t
12
12
12
1
21
2
2
2
2
12
1
1
1
G
)
(
E
1
)
(
E
1















In 2-D state (plane stress state):

48. x
12
12
x
2
21
2
2
2
x
2
12
2
1
1
)
cos
(sin
G
1
)
cos
(sin
E
1
)
sin
(cos
E
1




















2
c
1
c
12
2
t
1
t
E
Y
Y
E
X
X
G
S
S
E
Y
Y
E
X
X
c
c
t
t


























cos
sin
S
cos
sin
Y
sin
cos
X
x
2
21
2
x
2
12
2
t
x














cos
sin
sin
cos
x
12
2
x
2
2
x
1
Maximum Strain theory
Big difference between
theoretical and
experimental results
Unidirectional：
Let

49. 3. Tsai-Hill Failure Theory
1
N
2
M
2
L
2
F
2
G
2
H
2
)
G
F
(
)
H
F
(
)
H
G
(
2
12
2
13
2
23
3
2
3
1
2
1
2
3
2
2
2
1
























Yield criterion
where F, G, H, L, M, N are constants related to failure
load.
This theory is based on the Von-Mises’ distortional energy
yield criterion.

50. Tsai-Hill Theory
2
2
2
2
Z
1
G
F
Y
1
H
F
X
1
H
G
S
1
N
2







If only 12 is applied
If only 1 is applied
If only 2 is applied
If only 3 is applied
2
2
2
2
2
2
2
2
2
Z
1
Y
1
X
1
F
2
Z
1
Y
1
X
1
G
2
Z
1
Y
1
X
1
H
2










According to the criterion
( X, Y, Z are failure strengths in the three
orthotropic directions, respectively )

51. 1
S
Y
X
X 2
2
12
2
2
2
2
2
1
2
2
1









0
23
13
3 



















cos
sin
sin
cos
x
12
2
x
2
2
x
1
2
x
2
4
2
2
2
2
2
4
1
Y
sin
sin
cos
X
1
S
1
X
cos















Based on transformation eq.
Hill-Tsai Theory
Unidirectional
loading
For a unidimentional lamina under plane stress:

52. Features of Tsai-Hill Theory
 It is a failure criterion.
 Strength curve varying with direction angle is smooth without
sharp point.
 Unidirectional strength decreases with respect to the direction
angle (different from Max. Stress Theory and Max Strain
Theory)
 The results are more consistent with experimental results.
 In Hill-Tsai theory, the interaction among the failure strengths
X, Y and S is significantly considered, which is ignored by the
other strength theory.
 The result for isotropic elastomers can be obtained by
simplification.

53. Disadvantages
 It is not suitable for all materials.
The function should be applied for compressive loading
differs from the function for tensile loading.
Hoffman suggested a adapted failure criterion for the
materials with different behavior under compressive loading
when compared to tensile loading.
1
S
Y
Y
Y
Y
X
X
X
X
Y
Y
X
X 2
2
12
2
c
t
t
C
1
c
t
t
C
c
t
2
2
c
t
2
1
2
1















-- Hoffman failure criterion

54. 4. Tsai-Wu Failure Theory
6
,
2
,
1
j
,
i
1
F
F j
i
ij
i
i 






1
F
2
F
F
F
F
F
F 2
1
12
2
6
66
2
2
22
2
1
11
6
6
2
2
1
1 














Assume that the failure surface in stress space is
( Fi and Fij are coefficient tensors)
12
6
13
5
23
4 








For orthotropic elastomers in plane stress state,
This failure theory is based on the total strain energy failure
theory of Beltrami.

55. Tsai-Wu Failure Theory
1
X
F
X
F 2
t
11
t
1 

c
t
22
c
t
2
Y
Y
1
F
Y
1
Y
1
F




1
X
F
X
F 2
c
11
c
1 

Apply 1=Xt, 2= 0=12:
c
t
11
c
t
1
X
X
1
F
X
1
X
1
F




Similarly
in plane stress state
Some components of the strength tensor can be determined by
engineering parameters:
The shear strength in the principle directions is independent of
the positive/ negative sign of shear stress.
Apply 1=Xc, 2= 0=12:
Apply 1=0, 2= 0,12=S(-S)

56. Tsai-Wu Failure Theory
1
)
F
2
F
F
(
)
F
F
( 2
12
22
11
2
1 






The tensor coefficient Fij indicates the interaction of the stresses
in two orthotropic directions (1 and 2).




 2
1































 2
c
t
c
t
c
t
c
t
2
12
Y
Y
1
X
X
1
Y
1
Y
1
X
1
X
1
1
2
1
F
Substituting the stresses into the tensor eq. in plane stress
state yields
If 2F12=-F11, the eq. is as same as Hoffman.
If compressive strengths are identical, 2F12=-1/X2.
(as same as Hill-Tsai eq.)
Design a bidirectional tensile test:

57. Features of Tsai-Wu failure theory
• The linear terms Fi describe the effect of unidirectional
stress;
• The quadratic terms Fij describe the effect of the space
stresses, dependent of the interaction of the normal
stressesi and j.
• The quadratic terms Fij are symmetric.
6
,
2
,
1
j
,
i
1
F
F j
i
ij
i
i 





