1. Micromechanics
Macromechanics
Fibers
Lamina
Laminate
Structure
Matrix

2. Macromechanics
Study of stress-strain behavior of composites using effective properties of an equivalent homogeneous material. Only the globally averaged stresses and strains are considered, not the local fiber and matrix values.

3. Stress-Strain Relationships for Anisotropic Materials
First, we discuss the form of the stress-strain relationships at a point within the material, then discuss the concept of effective moduli for heterogeneous materials where properties may vary from point-to-point.

4. General Form of Elastic - Relationships for Constant Environmental Conditions
Each component of stress, ij, is related to each of nine strain components, ij
(Note: These relationships may be nonlinear)

5. Expanding Fij in a Taylor’s series and Retaining only the first order terms,
for a linear elastic material

6. 3D state of stress

7. Generalized Hooke’s Law for Anisotropic Material=
(2.2)

8. Symmetry Simplifies the Generalized Hooke’s Law
Symmetry of shear stresses and strains:
Same condition for shear strains,
Material property symmetry – several types will be discussed. 2 O 1

9. Symmetry of shear stresses and shear strains:
Thus, only 6 components of ij are independent, and likewise for ij.
This leads to a contracted notation.

10. Stresses
Tensor Notation
Contracted Notation
11 1
22 2
33 3
23= 32 4
13= 31 5
12= 21 6

11. Strains
Tensor Notation
Contracted Notation
 11
 1
 22
 2
 33
 3
2 23= 2 32=  23=  32
 4
2 13= 2 31=  13=  31
 5
2 12= 2 21=  12= 21
 6

12. Geometry of Shear Strain
 xy = Engineering Strain
 xy = Tensor Strain
Total change in original angle =  xy
Amount each edge rotates =  xy/2 =  xy

13. Using contracted notation
or in matrix form
where and are column vectors and [C] is a 6x6 matrix (the stiffness matrix)
(2.3)
(2.4)

14. where [S] = compliance matrix and
Alternatively, or
where [S] = compliance matrix
and
(2.5)
(2.6)

15. Expanding:

16. Up to now, we only considered the stresses and strains at a point within the material, and the corresponding elastic constants at a point.
What do we do in the case of a composite material, where the properties may vary from point to point?
Use the concept of effective moduli of an equivalent homogeneous material.

17. Concept of an Effective Modulus of an Equivalent Homogeneous Material
Heterogeneous composite under varying stresses and strains
Stress,
Strain,
Equivalent homogeneous material under average stresses and strains
Stress
Strain

18. Effective moduli, Cij
(2.9)
where,
(2.7)
(2.8)

19. 3-D Case General Anisotropic Material
[C] and [S] each have 36 coefficients, but only 21 are independent due to symmetry.
Symmetry shown by consideration of strain energy.
Proof of symmetry:
Define strain energy density

20. (2.12)
Now, differentiate:
but
if i=j
if i j

21. But if the order of differentiation is reversed,
(show)
(2.11)
(2.13)
But if the order of differentiation is reversed,
(2.14)

22. Since order of differentiation is immaterial,
(Symmetry)
Similarly,
and
Only 21 of 36 coefficients are independent for anisotropic material.

23. Stiffness matrix for linear elastic anisotropic material with no material property symmetry
(2.15)

24. 3-D Case, Specially Orthotropic
Fiber direction 2
1, 2 , 3 principal material coordinates 1

25. (a)
(b)
(c)
Simple states of stress used to define lamina engineering constants for specially orthotropic lamina.

26. Consider normal stress 1 alone:3 1 2
Resulting strains,
(2.19)

27. Typical stress-strain curves from
ASTM D3039 tensile tests
Stress-strain data from longitudinal tensile test of carbon/epoxy composite.
Reprinted from ref. [8] with permission from CRC Press.

28. Similarly,
where E1 = longitudinal modulus
ij = Poisson’s ratio for strain along j direction due to loading along i direction

29. Now consider normal stress 2 alone:
Strains: 3 1 2
(2.20)
Where E2 = transverse modulus
Similar result for 3 alone

30. Observation:
All shear strains are zero under pure normal stress (no shear coupling).
For
alone

31. Now, consider shear stress alone,3 1 2
Strain
Where G12 = Shear modulus in 1-2 plane
(2.21)
(No shear coupling)

32. Similarly, for alone
and for alone
Now add strains due to all stresses using superposition

33. Specially Orthotropic 3D Case
(2.22)
12 coefficients, but only are 9 independent

34. Symmetry:
Only 9 independent coefficients.
Generally orthotropic 3-D case – similar to anisotropic with 36 nonzero coefficients, but 9 are independent as with specially orthotropic case

35. Specially Orthotropic – Transversely Isotropic3
Reinforcement 2
Fibers randomly packed in 2-3 plane, so properties are invariant to rotation about 1-axis (2 same as 3) 1

36. Now, only 5 coefficients are independent.
Specially orthotropic, transversely isotropic (2 and 3 interchangeable)
(2.23)
Now, only 5 coefficients are independent.

37. Isotropic 2 independent coefficients
Usually measure E, υ – calculate G

38. Isotropic – 3D case
Same form for any set of coordinate axes

39. 3-D Isotropic – stresses in terms of strains

41. 3-D Case, Generally Orthotropicz
Reinforcement direction y
Material is still orthotropic, but stress-strain relations are expressed in terms of non-principal xyz axes x

42. Generally Orthotropic
Same form as anisotropic, with 36 coefficients, but 9 are independent as with specially orthotropic case

43. Elastic coefficients in the stress-strain relationship for different materials and coordinate systems
Material and coordinate system
Number of nonzero coefficients
Number of independent coefficients
Three – dimensional case
Anisotropic 36 21
Generally Orthotropic
(nonprincipal coordinates) 9
Specially Orthotropic (Principal coordinates) 12
Specially Orthotropic, transversely isotropic 5
Isotropic 2
Two – dimensional case (lamina) 6
(nonprincipal coordinates) 4
Balanced orthotropic, or square symmetric
(principal coordinates) 3

44. 2-D Cases
Use 3-D equations with,
Plane stress, Or

45. Specially Orthotropic Lamina
(2.24) Or
(2.26) 2 1
5 Coefficients - 4 independent

46. 5 nonzero coefficients 4 independent coefficients
Specially Orthotropic Lamina in Plane Stress
(2.24)
5 nonzero coefficients independent coefficients

47. Or in terms of ‘engineering constants’
(2.25)

49. Experimental Characterization of Orthotropic Lamina
Need to measure 4 independent elastic constants
Usually measure E1, E2, υ12, G12
(see ASTM test standards later in Chap. 10)

50. Stresses in terms of tensor strains,
(2.26)

51. Inverting [S]:

52. Off – Axis Compliances:
Off – Axis Stiffnesses:
Where fij and fij’ are found from transformations of stress and strain components from 1,2 axes to x, y axes

53. Sign convention for lamina orientationy y 1 2 2 x x 1

54. Stress Transformation:Y 2 1 X dA
and

55. Equations used to generate Mohr’s circle.
(2.29)
Equations used to generate Mohr’s circle.

56. Resulting stress Transformation:
(2.30)
Where or
(2.31)

57. Strain Transformation:
Where
(2.32)
Strain Transformation:
(2.33)

58. Recall: Tensor shear strain
Where xy = engineering shear strain or

59. Substituting (2.33) into (2.26), then substituting the resulting equations into (2.30)
(2.34)
Carrying out matrix multiplications and converting back to engineering strains,
(2.35)

60. Where
(2.36)
Alternatively
(2.37)
Where

61. Generally Orthotropic Lamina (Off Axis)
(2.37) Or x 2 y 1
9 Coefficients - 6 independent

62. In expanded form:
(2.38)

63. Off-axis lamina engineering constants
Young’s modulus, Ex 2
When y
(2.39) or 1 x
(2.40)

64. Complete set of transformation equations for lamina
engineering constants
(2.40)

65. Variations of off-axis engineering constants with lamina orientation for
unidirectional carbon/epoxy, boron/aluminum and glass/epoxy composites.
(From Sun, C.T Mechanics of Aircraft Structures. John Wiley & Sons, New York.
With permission.)

66. Shear Coupling Ratios, or Mutual Influence Coefficients
Quantitative measures of interaction between normal and shear response.
Example: when
Shear Coupling Ratio
Analogous to Poisson’s Ratio
(2.41)

67. Example of off-axis strain in terms of off-axis engineering constants
(2.43)

68. Compliance matrix is still symmetric for
off-axis case, so that, for example
and

69. Balanced Orthotropic Lamina (Ex: Woven cloth, cross-ply)2 1
Only 3 independent coefficients

70. Lamina Stiffness Transformations

71. The lamina stiffness transformations can be written as:
Use of Invariants
The lamina stiffness transformations can be written as:
(2.44)

72. Where the invariants are
(2.45)

73. Alternatively, the off-axis compliances can be expressed as
(2.46)

74. where the invariants are
(2.47)

75. Example: decomposition of using invariants
References: Mechanics of Composite Materials, Jones
Introduction to Composite Materials, Tsai & Hahn

76. Invariants in transformation of stresses: Mohr’s circle
Where
(2.48)
Invariant I
Invariant

77. Similar graphical interpretation of stiffness transformations
Ex:
(2.49)
Orthotropic Part
Isotropic Part

78. U1 and U4: First order invariants
U2 and U3: Second order invariants
Radii of circles indicates degree of orthotropy.
(i.e., if U2=U3=0, we have isotropic material)