Transcript Training - Plymouth University

Elastic properties
Young’s moduli
Poisson’s ratios
Shear moduli
Bulk modulus
John Summerscales
Elastic properties
• Young’s moduli
o uniaxial stress/unixaial strain
• Poisson’s ratio
o
- transverse strain/strain parallel to the load
• Shear moduli
o
biaxial stress/biaxial strain
• Bulk modulus
o
triaxial stress (pressure)/triaxial strain
Terminology:
Y
transverse
2
throughthickness
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)
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
stress
< carbon composite
< glass composite
strain
Variation of E with angle:
fibre orientation distribution factor ηo
Load sharing models
• Reuss model:
o
up to 0.5% strain,
equal stress
in both the fibres and the matrix.
• Voigt model
o
above 0.5% strain,
equal increases in strain
in both fibre and matrix.
Variation of E with fibre length:
fibre length distribution factor ηl
• Cox shear-lag
• depends on
o
o
o
o
o
o
Gm: matrix modulus
Af: fibre CSA
Ef: fibre modulus
L: fibre length
R: fibre separation
Rf: fibre radius
< Tension
< Shear
Variation of E with fibre length:
fibre length distribution factor ηl
• Cox shear-lag equation:
where
• Critical length:
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
Poisson’s ratio (orthotropic: νij)
• Maxwell’s reciprocal theorem
o 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.
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)
E2 (GPa)
20.3
7.9
15.5
17.5
E3 (GPa)
7.1
9.4
G12 (GPa)
3.45
3.0
Poisson’s ratio: beware !!
• For orthotropic materials,
not all authors use the same notation
a. subscripts may be stimulus then response
b. subscripts may be response then stimulus
The following page uses stimulus then response:
• 1= fibres
• 2 = resin (UD) or fibre (WR)
• 3 = resin
Poisson’s ratios for GRP
UD C1
ν12
ν21
ν13
ν31
ν23
ν32
high values
WR A2
νij
UD C1
√Ei/Ej
νij
WR A2
√Ei/Ej
0.308
1.606
0.140
0.942
0.123
0.623
0.109
1.061
0.354
1.687
0.408
1.285
0.124
0.593
0.247
0.778
0.417
1.051
0.380
1.364
0.414
0.952
0.297
0.733
low values
Extreme values of νij
• Dickerson and Di Martino (1966):
orthotropic (cross-plied) boron/epoxy composites
Poisson's ratios range from 0.024 to 0.878
o ±25º laminate boron/epoxy composites
Poisson's ratios range from -0.414 to 1.97
o
Shear moduli
• Isotropic case
E
G
2(1  )
Pure
Simple
• Orthotropic case (Huber’s equation, 1923)
G12 

Ex E y
2 1   xy yx

Bulk modulus
• Isotropic case
E
K
3(1  2 )
• Orthotropic case
K

3
Ex E y Ez
3 1  2.3  xy zx yz

Negative Poisson’s ratio
(auxetic) materials
• Re-entrant
or
chiral structures
Summary
• Young’s moduli
• Poisson’s ratios,
o
including reentrant/chiral auxetics
• Shear moduli
• Bulk modulus