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

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

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)

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

Variation of E with angle: fibre orientation distribution factor ηo

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.

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

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 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) 20.3 15.5 E2 (GPa) 7.9 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 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

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

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 ±25º laminate boron/epoxy composites Poisson's ratios range from -0.414 to 1.97

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

Bulk modulus Isotropic case Orthotropic case

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

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