1. Composite Materials Chapter 10. Micromechanics of Composites

2. 10.1 Mechanical Properties
Fig 10.1 Unidirectional composite: a. isostrain or action in parallel, b. isostress or action in series
Krishan K. Chawla, Composite materials science and engineering, Springer-Verlag, (1998)

3. 10.1 Mechanical Properties
Density
mc = mf + mm (1)
Where mc : mass of composite, mf : mass of fiber,
mm : mass of matrix
vc = vf + vm + vv (2)
Where vv : volume of voids
Mf + Mm = (3)
vf + vm + vv = (4)
Composite density, ρc(= m/v) is given by
ρc = mc/ vc = mf + mm / vc = ρfvf + ρmvm / vc
and

4. 10.1 Mechanical Properties
or ρc = ρfvf + ρmvm (5)
We can also derive an expression for ρc in terms of mass fraction. Thus,
ρc = mc / vc = mc / vf+vm+Vv = mc / (mf / ρf + mm / ρm + Vv)
= 1 / ( Mf / ρf + Mm / ρm + Vv / mc )
= 1 / ( Mf / ρf + Mm / ρm + vv / ρcvc )
= 1 / ( Mf / ρf + Mm / ρm + Vv / ρc ) (6)
Rewriting eq.(6), we obtain
ρc = ρc / ρc [ Mf / ρf + Mm / ρm ] + Vv
Vv = 1 – ρc ( Mf / ρf + Mm / ρm ) -----(7) or

5. 10.1 Mechanical Properties
Fig 10.2 Variation of longitudinal modulus (E11) and transverse modulus (E22) with fiber volume fraction (Vf)

6. 10.1 Mechanical Properties
Isostrain or action in parallel situation
If two components adhere perfectly and if they
have the same poisson ratio two components will
under go the same longitudinal at elongation, ∆l.
εf = εm + εcl = ∆l / l (8)
Where εcl is the strain in the composite in longitudinal direction by Voigt[1910]
σf = Efεf , σm = Emεm

7. 10.1 Mechanical Properties
Let Ac, Am, Af be the cross-sectional area of the
composite, matrix, and all the fiber respectively.
From the equilibrium of for in fiber direction,
Pc = Pf + Pm or σclAc = σfAf + σmAm ---(9)
From eqs. (8) and (9),
σclAc = (EfAf + EmAm) εcl
or Ecl = σcl / εcl = Ef ( Af / Ac) + Em ( Am / Ac)
for a given composites length, Af / Ac = Vf,
Am / Ac =Vm, then the above expression can be simplifield
to Ecl = EfVf + Em Vm = E (10) : rule of mixture for E
Similar expression can be obtained for composite longitudinal strength from eq(9), namely
σcl = σf Vf + σm Vm –-(11) : rule of mixture for strenth

8. 10.1 Mechanical Properties
Isostress or Action in series
Equal stress in twocomponents [by Reuss, (1929)]
for loading transverse to the fiber direction,
We have
For total displacements of the composite, tc is the sum of displacements of the fibers and matrix.

9. 10.1 Mechanical Properties
Diving through by tc, composite gage length, we obtain
Now, , composite strain in transverse direction,
while and equal the strains in matrix and fiber times their respective gage length :
and Then
or (12)

10. 10.1 Mechanical Properties
Volume fraction of fiber and matrix can be written
and
This simplifies eq(12) to (13)
In elastic regime, , Eq(13) becomes
or (14)

11. 10.1 Mechanical Properties
Rules of Mixture
(Density)
▣ Iso-strain, action in Parallel
▣ Iso-stress condition, action in series

12. 10.1 Mechanical Properties
Fig Various fiber arrays in a matrix : (A) square (b) Hexagonal (c) random
Krishan K. Chawla, Composite materials science and engineering, Springer-Verlag, (1998)

13. 10.1 Mechanical Properties
Fig 10.4 Transversely isotropic fiber composites : plane transverse to fibers (2-3 plane) is isotropic
Krishan K. Chawla, Composite materials science and engineering, Springer-Verlag, (1998)

14. 10.1 Mechanical Properties
Fig 10.5 A Single Fiber sum owened by its matrix shell
Krishan K. Chawla, Composite materials science and engineering, Springer-Verlag, (1998)

16. 10.1 Mechanical Properties
Fig 10.6 Three dimensional stress distribution in the unit composite. Transverse stress (σr and σθ) result from the differences in the Poisson ratios of the fiber and matrix.
Krishan K. Chawla, Composite materials science and engineering, Springer-Verlag, (1998)

17. 10.1 Mechanical Properties
Stress Distribution in a fiber composite in the above Fig 10.6
Fallowing Inferences are drawn at below
1. Axial stress is uniform in component 1 and 2,stress magnitude depends on the respective elastic constants.
2. In central component 1, σr1 and σθ1 are equal in magnitude and sense. In the sleeve 2, σθ2 vary as 1-b2/r2 and 1+b2/r2, respectively.
3. When the Poisson ratio difference ( V2 – V1 ) goes to zero, σr and σθ go to zero ; the rheological interaction will vanish.
4. Small difference in Poisson ratio of components of metallic composite.
→Transverse stress developing in elastic regime will be small.

18. 10.2 Thermal Properties
• Thermal expansion coefficients of are less than the value expressed by rule of mixture (=αfνf+ αmνm)
← mechanical constraint on the matrix.
A fiber results in the a greater constraint than a partical.

19. 10.2 Thermal Properties
Volumetric expansion coefficient of a composite
-spherical particles dispersed in a matrix-
By Kerner, proc. Phys. Soc. London, B69, 808(1956)
Where subscripts m and p denote the matrix and particle respectively ; αc,αm and αp are the volumetric thermal expansion coefficients of composite, matrix and the particle, respectively ; km and kp are the bulk moduli of matrix and particle.
αc does not significantly differ from the value obtained from the rule of mixture.

20. 10.2 Thermal Properties Expansion coefficient of a fiber composite
by Schapery, J.composite Mater., 2, 311(1969)
Assuming the Poisson ratios of components are not very different. Longitudinal
--Eq(10.34)
Transverse
--Eq(10.35)
When v is Poisson ratio given by v = vfVf + vmVm
For high fiber volume fraction, Vf ≥ 0.2
--Eq(10.36)
Expansion Coefficient for a Composite conTaining
Randomly Oriented fibers in three dimension

21. 10.2 Thermal Properties
Thermal Conductivity ( ) of a fiber composite in fiber direction.
by E. Behrens, J. Composite Mater., 2, 2(1968)

22. 10.2 Thermal Properties
Fig 10.7 Longitudinal and transverse linear thermal expansion coefficients versus fiber volum fraction for alumina fiber in an aluminum matrix.
Krishan K. Chawla, Composite materials science and engineering, Springer-Verlag, (1998)

23. 10.2 Thermal Properties
Table 10.4 Thermal properties of transversely isotropic composite (matrix isotropic, fiber anisotropic)
Krishan K. Chawla, Composite materials science and engineering, Springer-Verlag, (1998)

24. 10.2 Thermal Properties
Fig 10.8 Strain-free reference state ; b thermal strain ( eT ) ; c hygral ( eH ) and thermal ( eT ) strains ; d hygral ( eH ), thermal ( eT ), mechanical strain ( e ) – final state.
Krishan K. Chawla, Composite materials science and engineering, Springer-Verlag, (1998)