Page 76 - 3D Fibre Reinforced Polymer Composites

Micromechanics Models for Mechanical Properties 65
where [SI is the inverse matrix of [C] and is given by:

'11 '12 'I3 'I6
'21 '22 '23 '24 '25 '26
['I = '31 '32 '33 '34 '35 '36 (4.4)
'41 '42 '43 '44 '45 '46
'51 '52 '53 '54 '55 '56
'61 s62 '63 '65 '66
where S, are the elastic compliances. For small deformation in the Cartesian coordinate
system, the strains can be defined as:

I au. auj
E.. = -(--'- +-1 (4.5)
2 ani ani
I~

where ui (i=1,2,3) are the displacements in the directions of the three Cartesian
coordinates, and xi (i=1,2,3) are the three coordinates in the Cartesian system.
For an orthotropic material, in which there are three orthogonal symmetrical planes,
we have the following Hooke's law:
-
I
I
0, '11 '12 '13 "11
022 CI2 c22 c23 0 0 0 E22
033 '13 ' 2 3 '33 ~ '33
0 0 0 c4, 0 0
c23 2E2,
0 0 0 0 C5, 0
O3 I 2~~~
0 0 0 0 0 C6,
,012 -. 2EI2
in which there are only nine independent elastic stiffness constants. Similarly, there are
only nine independent elastic compliance constants, and the compliance matrix for a
unidirectional fibre reinforced composite material is given by:

I
[si = [c]-l= o - v23/E2 0 0 0 0
- V13/EI
0
0
0

where El, Ez, E3, G12, G23, GN, v12, v23 and v3l are engineering constants.