1. Pendahuluan Material Komposit
BAB 2 Macromechanical Analysis of a Lamina
Invariant Form of Transformed Matrices
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

2. Invariant Form of Stiffness Matrix

3. Invariant Form of Compliance Matrix

4. Pendahuluan Material Komposit
BAB 2 Macromechanical Analysis of a Lamina
Maximum Stress Failure Theory
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

5. Strength Failure Theories for an Angle Lamina
The failure theories are generally based on the normal and shear strengths of a unidirectional lamina.
In the case of a unidirectional lamina, the five strength parameters are:
Longitudinal tensile strength
Longitudinal compressive strength
Transverse tensile strength
Transverse compressive strength
In-plane shear strength

6. Maximum Stress Failure Theory
The lamina is considered to be failed if:
is violated.
Note that all five strength parameters are positive numbers.
Each component of stress does not interact with each other.

7. Example Find the maximum value of S>0 if a stress of
is applied to a 60o lamina of Graphite/Epoxy. Use Maximum Stress failure theory. Use properties of a unidirectional Graphite/Epoxy lamina given in Table 2.1 of the textbook Mechanics of Composite Materials by Autar Kaw.
Gambar 2.33
Off-axis loading in the x-direction

8. Solution
The stresses in the local axes are

9. Solution
The ultimate strengths of a unidirectional Graphite/Epoxy lamina are:

10. Solution
Then using the inequalities of the Maximum Stress Failure Theory:
or,
All the inequality conditions (and
)are satisfied if
The above inequalities also show that the angle lamina will fail in shear. The maximum stress that can be applied before failure is:

11. Pendahuluan Material Komposit
BAB 2 Macromechanical Analysis of a Lamina
Maximum Strain Failure Theory
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

12. Strength Failure Theories for an Angle Lamina
The failure theories are generally based on the normal and shear strengths of a unidirectional lamina.
In the case of a unidirectional lamina, the five strength parameters are:
Longitudinal tensile strength
Longitudinal compressive strength
Transverse tensile strength
Transverse compressive strength
In-plane shear strength

13. Maximum Strain Theory The lamina is considered to be failed if:
is violated, where:
= Ultimate longitudinal tensile strain (in direction 1),
= Ultimate longitudinal compressive strain (in direction 1),
= Ultimate transverse tensile strain (in direction 2),
= Ultimate transverse compressive strain (in direction 2),
= Ultimate in-plane shear strain (in plane 1-2).

14. Example Find the maximum value of S>0 if a stress of
is applied to a 60o lamina of Graphite/Epoxy. Use Maximum Strain Failure Theory. Use properties of a unidirectional Graphite/Epoxy lamina given in Table 2.1 of the textbook Mechanics of Composite Materials by Autar Kaw.
Gambar 2.33
Off-axis loading in the x-direction

15. Solution
The stresses in the local axes are:

16. Example
The strains in the local axes are:

17. Example
Assume there is a linear relationship between all the stresses and strains till failure, then the ultimate failure strains are:

18. Example
or:
which give:

19. Observations
The strength ratio from Maximum Stress Failure Theory and Maximum Strain Failure Theory is MPa. There is no difference between the two values because the mode of failure is shear.
However, if the mode of failure were other than shear, there would have been a difference between the strength ratios due to the Poisson's ratio effect, which couples the normal strains and stresses in the local axes.
Neither, the Maximum Stress Failure Theory nor the Maximum Strain Failure Theory have any coupling between the three possible modes of failure.

20. Pendahuluan Material Komposit
BAB 2 Macromechanical Analysis of a Lamina
Tsai-Hill Failure Theory
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

21. Strength Failure Theories for an Angle Lamina
The failure theories are generally based on the normal and shear strengths of a unidirectional lamina.
In the case of a unidirectional lamina, the five strength parameters are:
Longitudinal tensile strength
Longitudinal compressive strength
Transverse tensile strength
Transverse compressive strength
In-plane shear strength

22. Tsai-Hill Failure Theory
Based on the distortion energy theory, Tsai and Hill proposed that a lamina has failed if:
This theory is based on the interaction failure theory.
The components G1 thru G6 of the strength criteria depend on the strengths of a unidirectional lamina.

23. Components of Tsai-Hill Failure Theory
Apply to a unidirectional lamina, then the lamina will fail. Hence, Equation reduces to:

24. Components of Tsai-Hill Failure Theory
Apply to a unidirectional lamina, then the lamina will fail. Hence, Equation reduces to:

25. Components of Tsai-Hill Failure Theory
Apply to a unidirectional lamina, and assuming that the normal tensile failure strength is the same in direction (2) and (3), then the lamina will fail. Hence, Equation reduces to:

26. Components of Tsai-Hill Failure Theory
Apply to a unidirectional lamina, then the lamina will fail. Hence, Equation reduces to

27. Components of Tsai-Hill Failure Theory

28. Tsai-Hill Failure Theory – Plane Stress
Because the unidirectional lamina is assumed to be under plane stress - that is,

29. Tsai-Hill Failure Theory
Unlike the Maximum Strain and Maximum Stress Failure Theories, the Tsai- Hill failure theory considers the interaction among the three unidirectional lamina strength parameter.
The Tsai-Hill Failure Theory does not distinguish between the compressive and tensile strengths in its equation. This can result in underestimation of the maximum loads that can be applied when compared to other failure theories.
Tsai-Hill Failure Theory underestimates the failure stress because the transverse strength of a unidirectional lamina is generally much less than its transverse compressive strength.

30. Example Find the maximum value of S>0 if a stress of
is applied to a 60o lamina of Graphite/Epoxy. Use Tsai-Hill Failure Theory. Use properties of a unidirectional Graphite/Epoxy lamina given in Table 2.1 of the textbook Mechanics of Composite Materials by Autar Kaw.
Gambar 2.33
Off-axis loading in the x-direction

31. Example

32. Example 2.18

33. Example 2.18

34. Modified Tsai-Hill Failure Theory

35. Pendahuluan Material Komposit
BAB 2 Macromechanical Analysis of a Lamina
Tsai-Wu Failure Theory
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

36. Tsai-Wu Failure Theory
H1σ1+H2σ2+H6τ12+H11
Tsai-Wu Failure Theory
+H22
+H66
+2H12σ1σ2< 1
Tsai-Wu applied the failure theory to a lamina in plane stress. A lamina is considered to be failed if:
is violated. This failure theory is more general than the Tsai-Hill failure theory because it distinguishes between the compressive and tensile strengths of a lamina.
The components H1 – H66 of the failure theory are found using the five strength parameters of a unidirectional lamina.

37. Components of Tsai-Wu Fail
a) Apply
to a unidirectional lamina, the lamina will fail. Equation
(2.152) reduces to:
b) Apply
to a unidirectional lamina, the lamina will fail. Equation
(2.152) reduces to:
From Equations (2.153) and (2.154),

38. Components of Tsai-Wu Fail
c) Apply
to a unidirectional lamina, the lamina will fail. Equation
(2.152) reduces to
d) Apply
to a unidirectional lamina, the lamina will fail. Equation
(2.152) reduces to:
From Equations (2.157) and (2.158):

39. Components of Tsai-Wu Fail
e) Apply
to a unidirectional lamina, the lamina will fail. Equation
(2.152) reduces to:
f) Apply
to a unidirectional lamina, the lamina will fail. Equation
(2.152) reduces to:
From Equations (2.157) and (2.158),

40. Apply equal tensile loads along the two material axes in a unidirectional composite. If
is the load at which the lamina fails, then:
The solution of the Equation (2.165) gives:

41. Take a 450 lamina under uniaxial tension . The stress
Take a 450 lamina under uniaxial tension
. The stress
at failure is noted.
If this stress is
then using Equation (2.94), the local stresses at failure are:
Substituting the above local stresses in Equation (2.152):

42. as per Tsai-Hill failure theory8 as per Hoffman criterion10
as per Tsai-Hill failure theory8
as per Hoffman criterion10
as per Mises-Hencky criterion11

43. Example 2.19 and Find the maximum value of if a stress
are applied to a 600 lamina of Graphite/Epoxy. Use Tsai-Wu failure theory. Use the properties of a unidirectional Graphite/Epoxy lamina from Table 2.1.

44. Example 2.19
Using Equation (2.94), the stresses in the local axes are:

45. H12=
Example 2.19

46. Example 2.19 Substituting these values in Equation (2.152), we obtain:or

47. Example 2.19
If one uses the other two empirical criteria for H12 as per Equation (2.171), one obtains:
Summarizing the four failure theories for the same stress-state, the value of S obtained is:
S = (Maximum Stress failure theory),
= (Maximum Strain failure theory),
= (Tsai-Hill failure theory),
= (Modified Tsai-Hill failure theory),
= (Tsai-Wu failure theory).

48. Pendahuluan Material Komposit
BAB 2 Macromechanical Analysis of a Lamina
Comparison of Failure Theories
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

49. Strength Failure Theories of an Angle Lamina
The failure theories are generally based on the normal and shear strengths of a unidirectional lamina.
An isotropic material generally has two strength parameters:
normal strength and shear strength.
In the case of a unidirectional lamina, the five strength parameters are:
Longitudinal tensile strength
Longitudinal compressive strength
Transverse tensile strength
Transverse compressive strength
In-plane shear strength

50. Experimental Results and Failure Theories
Tsai and Wu compared the results from various failure theories to some experimental results. He considered an angle lamina subjected to a uniaxial load in the x-direction.
Gambar 2.33
Off-axis loading in the x-direction

51. Experimental Results and Maximum Stress Failure Theory
Gambar 2.34
Maximum normal tensile stress in x-direction
as function of angle of lamina using maximum
stress failure theory

52. Experimental Results and Maximum Strain Failure Theory
Gambar 2.35
Maximum normal tensile stress in x-direction
as function of angle of lamina using maximum
Strain failure theory

53. Experimental Results and Tsai-Hill Failure Theory
Gambar 2.36
Maximum normal tensile stress in
x-direction as function of angle of
lamina using Tsai-Hill failure theory

54. Experimental Results and Tsai-Wu Failure Theory
Gambar 2.37
Maximum normal tensile stress in
x-direction as function of angle of
lamina using Tsai-Wu failure theory

55. Comparison of Strength Ratios
S = (Maximum Stress failure theory),
= (Maximum Strain failure theory),
= (Tsai-Hill failure theory),
= (Modified Tsai-Hill failure theory),
= (Tsai-Wu failure theory)

56. END

57. Pendahuluan Material Komposit
BAB 3 Micromechanical Analysis of a Lamina
Volume Fractions, Weight Fractions,
Density, and Void Content
Qomarul Hadi, ST,MT
Teknik Mesin
Universitas Sriwijaya
Sumber Bacaan
Mechanics of Composite Materials by Kaw

58. Volume Fractions

59. Weight Fractions

60. Volume and Weight Fractions

61. Volume and Weight Fractions

62. Volume and Weight Fractions

63. Density

64. Example
Example 3.1
A Glass/Epoxy lamina consists of a 70% fiber volume fraction. Use properties of glass and epoxy from Tables 3.1 and 3.2, respectively to determine the
density of lamina
b) mass fractions of the glass and epoxy
c) volume of composite lamina if the mass of the lamina is 4 kg.
d) volume and mass of glass and epoxy in part (c).

65. Example Table 3.1 Typical Properties of Fibers (SI system of units)
Property
Units
Graphite
Glass
Aramid
Axial modulus
Transverse modulus
Axial Poisson's ratio
Transverse Poisson's ratio
Axial shear modulus
Axial coefficient of thermal expansion
Transverse coefficient of thermal expansion
Axial tensile strength
Axial compressive strength
Transverse tensile strength
Transverse compressive strength
Shear strength
Specific gravity
GPa
- - -
μm/m/0C
MPa
230 22
0.30
0.35
-1.3
7.0
2067
1999 77 42 36
1.8 85
0.20
35.42 5
1550 35
2.5
124 8
0.36
0.37 3
-5.0
4.1
1379
276 7 21
1.4

66. Example Property Units Epoxy Aluminum Polyamide
Table 3.2 Typical Properties of Matrices (SI system of units)
Property
Units
Epoxy
Aluminum
Polyamide
Axial modulus
Transverse modulus
Axial Poisson's ratio
Transverse Poisson's ratio
Axial shear modulus
Coefficient of thermal expansion
Coefficient of moisture expansion
Axial tensile strength
Axial compressive strength
Transverse tensile strength
Transverse compressive strength
Shear strength
Specific gravity
GPa
- - -
μm/m/0C
m/m/kg/kg
MPa
3.4
0.3
1.308 63
0.33 72
102 34
1.2 71
0.30 27 23
0.00
276
138
2.7
3.5
0.35
1.3 90 54
108

67. Example Property Units Graphite Glass Aramid
Table 3.3 Typical Properties of Fibers (USCS system of units)
Property
Units
Graphite
Glass
Aramid
Axial Modulus
Transverse modulus
Axial Poisson's ratio
Transverse Poisson's ratio
Axial shear modulus
Axial coefficient of thermal expansion
Transverse coefficient of thermal expansion
Axial tensile strength
Axial compressive strength
Transverse tensile strength
Transverse compressive strength
Shear strength
Specific gravity
Msi
- - -
μin/in/0F
ksi
33.35
3.19
0.30
0.35
3.889
299.7
289.8
11.16
6.09
5.22
1.8
12.33
0.20
5.136
2.778
224.8
5.08
2.5
17.98
1.16
0.36
0.37
0.435
-2.778
2.278
200.0
40.02
1.015
3.045
1.4

68. Example Property Units Epoxy Aluminum Polyamide
Table 3.4 Typical Properties of Matrices (USCS system of units)
Property
Units
Epoxy
Aluminum
Polyamide
Axial modulus
Transverse modulus
Axial Poisson's ratio
Transverse Poisson's ratio
Axial shear modulus
Coefficient of thermal expansion
Coefficient of moisture expansion
Axial tensile strength
Axial compressive strength
Transverse tensile strength
Transverse compressive strength
Shear strength
Specific gravity
Msi
- - -
μin/in/0F
in/in/lb/lb
ksi
0.493
0.3
0.1897 35
0.33
10.44
14.79
4.93
1.2
10.30
0.30
3.915
12.78
0.00
40.02
20.01
2.7
0.5075
0.35
0.1885 50
7.83
15.66

69. Example
Example 3.1 a)

70. Example
Example 3.1 b)

71. Example
Example 3.1 c)

72. Example
Example 3.1 d)

73. Example
Example 3.1 d)

74. Example
Example 3.1 d)

75. Example
Example 3.1 d)

76. Void Content
Gambar 3.2 Photomicrographs of cross-section of a lamina with voids.

77. Void Content

78. Void Content

79. Void Content

80. Example
Example 3.2
A Graphite/Epoxy cuboid specimen with voids has dimensions of and its mass is Mc. After putting it in a mixture of sulphuric acid and hydrogen peroxide, the remaining graphite fibers have a mass Mf. From independent tests, the densities of graphite and epoxy are ρf and ρm, respectively. Find the volume fraction of the voids in terms of a, b, c, Mf, Mc, ρf, and ρm.

81. Example

82. Example

83. Example

84. Example

85. Example
Alternative Solution

86. Example
Alternative Solution

87. Example
Alternative Solution

88. Example
Alternative Solution

89. Example
Alternative Solution

90. END