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Elastic properties

Elastic properties. Young’s moduli Poisson’s ratios Shear moduli Bulk modulus John Summerscales. Elastic properties. Young’s moduli uniaxial stress/unixaial strain Poisson’s ratio - transverse strain/strain parallel to the load Shear moduli biaxial stress/biaxial strain

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Elastic properties

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

  2. Elastic properties Young’s 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 ------- • single subscript for linear load (e.g. tension) • double subscript for planar load (e.g. shear) • triple subscript for volume (e.g. pressure) transverse Y 2 through-thickness 3 Z X 1 axial, or longitudinal

  4. Young’s modulus (E) stress < carboncomposite < glasscomposite strain • Strain = elongation (ε)/original length (l) • Stiffness = force to produce unit deformation • Stress = force (F)/area (A) • Strength = stress at failure • E = Fl/εAbut E may vary with direction incomposites

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

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

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

  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 constraintrestricts the values to -1 <  < 1/2

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

  11. Poisson’s ratios for GRP • Peter Craig measured νij forC1: 13 layers F&H Y119 unidirectional rovingsA2: 12 layers TBA ECK25 woven rovings • confirmed Lemprière criteria were validfor both materials        

  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 high valueslow values

  14. Extreme values of νij • Dickerson and Di Martino (1966): • orthotropic (cross-plied) boron/epoxy compositesPoisson's ratios range from 0.024 to 0.878 • ±25º laminate boron/epoxy compositesPoisson's ratios range from -0.414 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, • including reentrant/chiral auxetics • Shear moduli • Bulk modulus

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