1. Elastic properties Young’s moduli Poisson’s ratios Shear moduli
Bulk modulus
John Summerscales

2. Elastic properties Young’s moduli Poisson’s ratio Shear moduli
uniaxial stress/unixaial strain
Poisson’s ratio
- transverse strain/strain parallel to the load
Shear moduli
biaxial stress/biaxial strain
Bulk modulus
triaxial stress (pressure)/triaxial strain

3. Terminology: ------- as subscripts -------
transverse Y 2
through-thickness 3 Z X 1
axial, or
longitudinal
as subscripts
single subscript for linear load (e.g. tension)
double subscript for planar load (e.g. shear)
triple subscript for volume (e.g. pressure)

4. Young’s modulus (E) Strain = elongation (ε)/original length (l)
Stiffness = force to produce unit deformation
Stress = force (F)/area (A)
Strength = stress at failure
E = Fl/εA but E may vary with direction in composites
< carbon composite
< glass composite
stress
strain

5. Variation of E with angle: fibre orientation distribution factor ηo

6. Load sharing models Reuss model: Voigt model
up to 0.5% strain, equal stress in both the fibres and the matrix.
Voigt model
above 0.5% strain, equal increases in strain in both fibre and matrix.

7. Variation of E with fibre length: fibre length distribution factor ηl
Cox shear-lag
depends on
Gm: matrix modulus
Af: fibre CSA
Ef: fibre modulus
L: fibre length
R: fibre separation
Rf: fibre radius
< Shear
< Tension

8. Variation of E with fibre length: fibre length distribution factor ηl
Cox shear-lag equation:
where
Critical length:

9. Poisson’s ratio (isotropic: ν)
 = -(strain normal to the applied stress)
(strain parallel to the applied stress).
thermodynamic constraint restricts the values to -1 <  < 1/2

10. Poisson’s ratio (orthotropic: νij)
Maxwell’s reciprocal theorem
ν12E2 = ν21E1
Lemprière constraint restricts the values of ν to
(1-ν23ν32), (1-ν13ν31), (1-ν12ν21), (1-ν12ν21-ν13ν31-ν23ν32-2ν21ν32ν13) > 0
hence νij ≤ (Ei/Ej)1/2 and ν21ν23ν13 < 1/2.

11. Poisson’s ratios for GRP
Peter Craig measured νij for C1: 13 layers F&H Y119 unidirectional rovings A2: 12 layers TBA ECK25 woven rovings
confirmed Lemprière criteria were valid for both materials        
UD C1
WR A2
E1 (GPa)
20.3
15.5
E2 (GPa)
7.9
17.5
E3 (GPa)
7.1
9.4
G12 (GPa)
3.45
3.0

12. Poisson’s ratio: beware !!
For orthotropic materials, not all authors use the same notation
subscripts may be stimulus then response
subscripts may be response then stimulus
The following page uses stimulus then response:
1= fibres
2 = resin (UD) or fibre (WR)
3 = resin

13. Poisson’s ratios for GRP
UD C1
WR A2
νij
√Ei/Ej
ν12
0.308
1.606
0.140
0.942
ν21
0.123
0.623
0.109
1.061
ν13
0.354
1.687
0.408
1.285
ν31
0.124
0.593
0.247
0.778
ν23
0.417
1.051
0.380
1.364
ν32
0.414
0.952
0.297
0.733
high values low values

14. Extreme values of νij Dickerson and Di Martino (1966):
orthotropic (cross-plied) boron/epoxy composites Poisson's ratios range from to 0.878
±25º laminate boron/epoxy composites Poisson's ratios range from to 1.97

15. Shear moduli Isotropic case Orthotropic case (Huber’s equation, 1923)
Pure Simple

16. Bulk modulus
Isotropic case
Orthotropic case

17. Negative Poisson’s ratio (auxetic) materials
Re-entrant or chiral structures

18. Summary Young’s moduli Poisson’s ratios, Shear moduli Bulk modulus
including reentrant/chiral auxetics
Shear moduli
Bulk modulus