M.E. 7501 – Reinforced Composite Materials
Lecture 3 – Part 2

s
d
Example: idealized cubic array of spherical particles s s

Experimental observations on effects of particulate reinforcement
Experiments show that, for typical micron-sized particulate reinforcement, as the particle volume fraction increases, the modulus increases but strength and elongation decrease
Flexural stress-strain curves for 30 µm glass bead-reinforced epoxy composites of various bead volume fractions. (From Sahu, S., and Broutman, L. J. 1972. Polymer Engineering
and Science, 12(2), 91-100. With permission.)

Yield strength of particulate composites
Nicolais-Narkis semi-empirical equation for case with no bonding between particles and matrix
S yc

S ym
 v
2/3 p
)
(6.65) where
S yc
S ym v p is the yield strength of the composite is the yield strength of the matrix material is the volume fraction of particles the coefficient 1.21 and the exponent 2/3 are selected so as to insure that S yc decreases with increasing v p
, that S yc
= S ym when v p
=0, and that S yc
=0 when v p
=0.74 , the particle volume fraction corresponding to the maximum packing fraction for spherical particles of the same size in a hexagonal close packed arrangement

Liang – Li equation includes particle – matrix interfacial adhesion
S yc

S ym
2
 v
2/3 p
) where θ is the interfacial bonding angle,
θ = 0 o corresponds to good adhesion, and
θ = 90 o corresponds to poor adhesion
(6.66)

Finite element models for particulate composites
(a) development of axisymmetric RVE
(b) axisymmetric finite element models of RVE
Finite element models for spherical particle reinforced composite.
(From Cho, J., Joshi, M. S., and Sun, C. T. 2006. Composites Science and Technology,
66, 1941-1952. With permission)

Modulus of particulate composites
Katz -Milewski and Nielsen-Landel generalizations of the Halpin-Tsai equations where
E c
E m
A k

1

1

ABv p p
1 B



E / E p m
E p
/ E m




1
A
(6.67)


1
 v p max v
2 p max

 v p

and where
E c
E p
E m is the Young’s modulus of the composite is the Young’s modulus of the particle is the Young’s modulus of the matrix is the Einstein coefficient is the particle volume fraction k
E is the maximum particle packing fraction v p v p max

Use empirical equations which have a theoretical basis in mechanics
Halpin-Tsai Equations
Where
 
E
2
E m

1
  v f
1
  v f


E
E f f
E m
E m




1

(3.63)
(3.64)





2 for E
2 of square array of circular fibers

1 for G
12

 
0
Mixtures

6
5
4
3
2
1
0
0
Experimental [62]
Eq. 3.27
Eq. 3.40
Eq. 6.67
0,1 0,2 0,3
Particle Volume Fraction
0,4 0,5
Comparison of predicted and measured values of Young’s modulus for glass microsphere-reinforced polyester composites of various particle volume fractions.

Hybrid multiscale reinforcements
(a)
(b)
Improvement of mechanical properties of conventional unidirectional E-glass/epoxy composites by using silica nanoparticle-enhanced epoxy matrix. (a) off-axis compressive strength. (b) transverse tensile strength and transverse modulus. (From Uddin, M. F., and Sun, C. T. 2008. Composites
Science and Technology, 68(7-8), 1637-1643. With permission.)