### Engineering Equations for Strength and Modulus

```Engineering Equations for Strength
and Modulus of Particulate
Reinforced Composite Materials
M.E. 7501 – Reinforced Composite Materials
Lecture 3 – Part 2
Particulate Reinforcement
d
Example: idealized cubic
array of spherical particles
s
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 (1 1.21v )
2/3
p
(6.65)
where
Syc is the yield strength of the composite
Sym is the yield strength of the matrix material
vp is the volume fraction of particles
the coefficient 1.21 and the exponent 2/3 are selected so as to insure
that Syc decreases with increasing vp, that Syc = Sym when vp=0, and
that Syc=0 when vp=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
S yc  S ym (1 1.21sin2  v2/3
p )
where θ is the interfacial bonding angle,
θ = 0o corresponds to good adhesion, and
θ = 90o 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
Ec 1  ABv p

Em 1  B v p
where
A  kE  1
(6.67)
E

B
E
p
p

 A
/ Em  1
/ Em
 1  v p max
  1  2
 v p max


 vp


and where
is the Young’s modulus of the composite
Ec is the Young’s modulus of the particle
E p is the Young’s modulus of the matrix
Em is the Einstein coefficient
is the particle volume fraction
kE is the maximum particle packing fraction
vp
v p max
Semi empirical Models
Use empirical equations which have a
theoretical basis in mechanics
Halpin-Tsai Equations
Where
E2 1  v f

Em 1  v f

E

E
f
f
Em   1
Em   
(3.63)
(3.64)
And
 curve-fitting parameter
  2 for E2 of square array of
circular fibers
  1 for G12
As
As
   Rule of Mixtures
  0 Inverse Rule of
Mixtures
6
Young's Modulus (106 psi)
5
4
Experimental [62]
3
Eq. 3.27
Eq. 3.40
Eq. 6.67
2
1
0
0
0.1
0.2
0.3
0.4
0.5
Particle Volume Fraction
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.)
```