1. Assist.Prof.Dr. Ahmet Erklig
Ch3 Micromechanics
Assist.Prof.Dr. Ahmet Erklig

2. Objectives Find the nine mechanical: four elastic moduli,
five strength parameters
Four hygrothermal constants:
two coefficients of thermal expansion, and
two coefficients of moisture expansion
of a unidirectional lamina

3. Micromechanics
Determining unknown properties of the composite based on known properties of the fiber and matrix

4. Micromechanics

5. Uses of Micromechanics
Predict composite properties from fiber and matrix data
Extrapolate existing composite property data to different fiber volume fraction or void content
Check experimental data for errors
Determine required fiber and matrix properties to produce a desired composite material .

6. Limitations of Micromechanics
Predicted composite properties are only as good as fiber and matrix properties used
Simple theories assume isotropic fibers many fiber reinforcements are orthotropic
Some properties are not predicted well by simple theories
more accurate analyses are time consuming and expensive
Predicted strengths are upper bounds

7. Notations
Subscript f, m, c refer to fiber, matrix, composite ply, respectively
v volume
V volume fraction
w weight
W weigth fractions
ρ density

8. Terminology Used in Micromechanics
Ef, Em – Young’s modulus of fiber and matrix
Gf, Gm – Shear modulus of fiber and matrix
υf, υm – Poisson’s ratio of fiber and matrix
Vf, Vm – Volume fraction of fiber and matrix

9. Micromechanics and Assumptions
Approaches:
Mechanics of materials approach,
Semi-empirical approach; Involves rigorous mathematical solutions.
Assumption: the lamina is looked at as a material whose properties are different in various directions, but not different from one location to another.

10. Volume Fractions Fiber Volume Fraction 𝑉 𝑓 = 𝑣 𝑓 𝑣 𝑐
𝑉 𝑓 = 𝑣 𝑓 𝑣 𝑐
Matrix Volume Fraction
𝑉 𝑚 = 𝑣 𝑚 𝑣 𝑐
𝑉 𝑓 + 𝑉 𝑚 =1

11. Mass Fractions Fiber Mass Fraction 𝑊 𝑓 = 𝜌 𝑓 𝜌 𝑚 𝜌 𝑓 𝜌 𝑚 𝑉 𝑓 + 𝑉 𝑚 𝑉 𝑓
𝑊 𝑓 = 𝜌 𝑓 𝜌 𝑚 𝜌 𝑓 𝜌 𝑚 𝑉 𝑓 + 𝑉 𝑚 𝑉 𝑓
Matrix Mass Fraction
𝑊 𝑚 = 1 𝜌 𝑓 𝜌 𝑚 (1−𝑉 𝑚 )+ 𝑉 𝑚 𝑉 𝑚
𝑊 𝑓 + 𝑊 𝑚 =1

12. Density
Total composite weigth: wc = wf + wm Substituting for weights in terms of volumes and densities 𝜌 𝑐 𝑣 𝑐 = 𝜌 𝑓 𝑣 𝑓 + 𝜌 𝑚 𝑣 𝑚 Dividing through by vc gives, 𝜌 𝑐 = 𝜌 𝑓 𝑣 𝑓 𝑣 𝑐 + 𝜌 𝑚 𝑣 𝑚 𝑣 𝑐 𝜌 𝑐 = 𝜌 𝑓 𝑉 𝑓 + 𝜌 𝑚 𝑉 𝑚 1 𝜌 𝑐 = 𝑊 𝑓 𝜌 𝑓 + 𝑊 𝑚 𝜌 𝑚

13. Density
When more than two constituents enter in the composition of the composite material
where n is the number of constituent.

14. Void Content
During the manufacture of a composite, voids are introduced in the composite.
This causes the theoretical density of the composite to be higher than the actual density.
Also, the void content of a composite is detrimental to its mechanical properties. These detriments
include lower
• Shear stiffness and strength
• Compressive strengths
• Transverse tensile strengths
• Fatigue resistance
• Moisture resistance

15. Effects of Voids on Mechanical Properties
†Lower stiffness and strength
†Lower compressive strengths
†Lower transverse tensile strengths
†Lower fatigue resistance
†Lower moisture resistance
†A decrease of 2-10% in the preceding matrix- dominated properties generally takes place with every 1% increase in void content .

16. Void Content

17. Evaluation of Four Elastic Moduli
There are four elastic moduli of a unidirectional lamina:
Longitudinal Young’s modulus, E1
Transverse Young’s modulus, E2
Major Poisson’s ratio, υ12
In-plane shear modulus, G12

18. Strength of Materials Approach
Assumptions are made in the strength of materials approach
The bond between fibers and matrix is perfect.
The elastic moduli, diameters, and space between fibers are uniform.
The fibers are continuous and parallel.
The fiber and matrix follow Hooke’s law (linearly elastic).
The fibers possess uniform strength.
The composites is free of voids.

19. Representative Volume Element (RVE)
This is the smallest ply region over which the stresses and strains behave in a macroscopically homogeneous behavior. Microscopically, RVE is of a heterogeneous behavior. Generally, single force is considered in the RVE.

20. RVE
RVE
matrix
fibre

23. Longitudinal Modulus, E1
Total force is shared by fiber and matrix

24. Longitudinal Modulus, E1
Assuming that the fibers, matrix, and composite follow Hooke’s law and that the fibers and the matrix are isotropic, the stress–strain relationship for each component and the composite is
The strains in the composite, fiber, and matrix are equal (εc = εf = εm);

25. Longitudinal Modulus, E1
The ratio of the load taken by the fibers to the load taken by the composite is a measure of the load shared by the fibers.

26. Longitudinal Modulus, E1
Predictions agree well with experimental data

27. Transverse Young’s Modulus, E2

28. Transverse Young’s Modulus, E2
The fiber, the matrix, and composite stresses are equal.
σc = σf = σm
the transverse extension in the composite Δc is the sum of the transverse extension in the fiber Δf , and that is the matrix, Δm.
Δc = Δf + Δm
Δc = tc εc
Δf = tf εf
Δm = tm εm
tc,f,m = thickness of the composite, fiber and matrix, respectively
εc,f,m = normal transverse strain in the composite, fiber, and matrix, respectively

29. Transverse Young’s Modulus, E2
By using Hooke’s law for the fiber, matrix, and composite, the normal strains in the composite, fiber, and matrix are

30. Transverse Young’s Modulus, E2

31. Transverse Young’s Modulus, E2

32. Major Poisson’s Ratio, ν12

33. Major Poisson’s Ratio, ν12

34. Major Poisson’s Ratio, ν12

35. Major Poisson’s Ratio, ν12

36. In-Plane Shear Modulus, G12
Apply a pure shear stress τc to a lamina

37. In-Plane Shear Modulus, G12

38. In-Plane Shear Modulus, G12

39. In-Plane Shear Modulus, G12
FIGURE 3.13
Theoretical values of in-plane shear modulus as a function of fiber volume fraction and com-
parison with experimental values for a unidirectional glass/epoxy lamina

41. Halphin-Tsai Equation
Longitudinal Young’s Modulus
Major Poisson’s Ratio

42. Transverse Young’s Modulus, E2
For a fiber geometry of circular fibers in a packing geometry of a square array, ξ = 2. For a rectangular fiber cross-section of length a and width b in a hexagonal array, ξ = 2(a/b), where b is in the direction of loading.

43. Transverse Young’s Modulus, E2

44. In-Plane Shear Modulus, G12
For circular fibers in a square array, ξ = 1. For a rectangular fiber cross-sectional area of length a and width b in a hexagonal array,
ξ = 𝑙𝑜𝑔 𝑒 ( 𝑎 𝑏 ), where a is the direction of loading.
Hewitt and Malherbe suggested choosing a function

45. In-Plane Shear Modulus, G12

46. Elasticity Approach
Elasticity accounts for equilibrium of forces, compatibility, and Hooke’s law relationships in three dimensions. The elasticity models described here are called composite cylinder assemblage (CCA) models. In a CCA model, one assumes the fibers are circular in cross-section, spread in a periodic arrangement, and continuous.

47. Composite Cylinder Assemblage (CCA) Model

48. CCA Model

49. Longitudinal Young’s Modulus, E1

50. Major Poisson’s Ratio

51. Transverse Young’s Modulus, E2
The CCA model only gives lower and upper bounds of the transverse Young’s modulus of the composite.

52. Transverse Young’s Modulus, E2

53. Transverse Young’s Modulus, E2

54. Transverse Young’s Modulus, E2

55. Transverse Young’s Modulus, E2
FIGURE 3.21
Theoretical values of transverse Young’s modulus as a function of fiber volume fraction and comparison with experimental values for boron/epoxy unidirectional lamina

56. In-Plane Shear Modulus, G12

57. In-Plane Shear Modulus, G12