Properties of hemp fibre polymer composites -An optimisation of ...

Appendix D: Density and mechanical properties
The composite can be investigated relative to the composite density as a function of the
fibre weight fraction using the porosity constants for fibres, matrix, strength and
stiffness. At first an expression for the composite density is derived:
m = V ρ v = ρ W v
( ) ( )
f f f c f
c
ρ f
ρc
= Vf W
=
W ρ α
Wf
ρm W ρ α
ρ f
×
W
( 1+ ) + ( 1− ) ( 1+
)
f f m f f f m f
c
ρc
=
W ρ α
ρmρ f
W ρ α
( 1+ ) + ( 1− ) ( 1+
)
f m f f f m
in which v is composite volume. The next point is to get an expression for the composite
stiffness Ecp:
E = V E + V E
c
E
c
E
cp f f m m
cp
cp
( ) ( ) ( )
f ρm f + ( 1−
f ) ρ f m
( 1+ ) + ( 1− ) ( 1+
)
( 1−
Wf ) ρ fEm ( ) ( ) ( α )
Wf ρmEf
= +
W ρ 1+ α + 1− W ρ 1+ α W ρ 1+ α + 1− W ρ 1+
f m f f f m f m f f f m
W E W E
=
W ρ α W ρ α
f m f f f m
The next point is to combine the expressions for ρc and Ecp:
( )
( 1+ ) + ( 1− ) ( 1+
)
( 1 ) ( 1 )
( ) ( ) ( 1 α )
E W ρ E + 1− W ρ E W ρ 1+ α + 1−
W ρ +
cp
= ×
ρ W ρ α W ρ α
ρ ρ
c
f m f f f m f m f f f m
c f m f f f m
m f
E W ρ E + −W ρ E W E −W
E
= = +
ρ ρ ρ ρ ρ
cp f m f f f m f f
f m
c m f f m
by substituting Ecp with the real composite stiffness Ec, the following expression is
derived:
E ⎛WE ( 1−
Wf ) E ⎞
n
c f f
m
E
= ⎜ + ⎟(
1−Vp)
ρ ⎜ c ρ f ρ ⎟ m
⎝ ⎠
c
nE
E ⎛WE ( 1− Wf ) E ⎞⎛ m Wf ρm + ( 1−W
f f
f ) ρ ⎞
c
f
= ⎜ +
⎟⎜
⎟
ρ ⎜ c ρ f ρ ⎟⎜ m Wf ρm( 1+ α f ) + ( 1− Wf
) ρ f ( 1+
αm)
⎟
⎝ ⎠⎝
⎠
c
nE
E ⎛WE ( 1−W
f f
f ) E ⎞ ⎛ m Wf
( ρm − ρ f ) + ρ ⎞
c
f
= ⎜ + ⎟× ⎜
⎟
ρ ⎜ c ρ f ρ ⎟ ⎜ m ⎝ ⎠ ⎝
Wf ρm( 1+ α f ) + ( 1− Wf
) ρ f ( 1+
αm)
⎟
⎠
Risø-PhD-11 77