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

11

pc

m p

ABvEE 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

cEpEmEEk

pvmaxpv

Semi empirical ModelsUse empirical equations which have a theoretical basis in mechanics

Halpin-Tsai Equations

f

f

m vv

EE

1

12 (3.63)

Where

mf

mf

EEEE 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. 3.27Eq. 3.40Eq. 6.67

Particle Volume Fraction

You

ng's

Mod

ulus

(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