Engineering Equations for Strength and Modulus of Particulate Reinforced Composite
Materials
M.E. 7501 – Reinforced Composite MaterialsLecture 3 – Part 2
Particulate Reinforcement
d
s
s
s
Example: idealized cubicarray of spherical particles
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.)
Experiments show that, for typical micron-sized particulatereinforcement, as the particle volume fraction increases, themodulus increases but strengthand elongation decrease
Experimental observations on effects ofparticulate reinforcement
2/3(1 1.21 )yc ym pS S v (6.65)
Yield strength of particulate composites
Nicolais-Narkis semi-empirical equation for casewith no bonding between particles and matrix
where Syc is the yield strength of the composite Sym is the yield strength of the matrix material vp is the volume fraction of particlesthe 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, andthat 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 – matrixinterfacial adhesion
2 2/3(1 1.21sin )yc ym pS S v (6.66)
where θ is the interfacial bonding angle, θ = 0o corresponds to good adhesion, andθ = 90o corresponds to poor adhesion
(a)
(b)
Finite element models for particulate composites
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)
development of axisymmetric RVE
axisymmetric finite element models of RVE
Modulus of particulate composites
Katz -Milewski and Nielsen-Landel generalizations of the Halpin-Tsai equations
1
1pc
m p
ABvE
E B v
(6.67)
where 1EA k
/ 1
/
p m
p m
E EB
E E A
max
2max
11 p
pp
vv
v
and where
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 is the maximum particle packing fraction
cEpE
mE
Ek
pvmaxpv
Semi empirical Models
Use empirical equations which have a theoretical basis in mechanics
Halpin-Tsai Equations
f
f
m v
v
E
E
1
12 (3.63)
Where
mf
mf
EE
EE 1(3.64)
And curve-fitting parameter
2 for E2 of square array of circular fibers
1 for G12
As Rule of Mixtures
As Inverse Rule of Mixtures
0
Comparison of predicted and measured values of Young’s modulus for glass microsphere-reinforced polyester composites of various particle volume fractions.
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
1
2
3
4
5
6
Experimental [62]
Eq. 3.27
Eq. 3.40
Eq. 6.67
Particle Volume Fraction
You
ng'
s M
odu
lus
(106
psi
)
(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.)
Hybrid multiscale reinforcements
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