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Elastic properties

Explore the key terminology, equations, and models related to elastic properties, such as Young’s moduli, Poisson’s ratios, and shear moduli. Learn about load sharing models, variations with fiber length, and the behavior of isotropic and orthotropic materials.

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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. Stress component notation • first letter in the suffix for planesecond letter for direction of stress • normal stress is positive if outwardi.e. producing tension • shear stress is positive if it hassame sense as corresponding normal stress. • Hooke’s law: σii = Eiii

  5. Young’s modulus (E) stress < carboncomposite < glasscomposite strain • Strain (ε) = elongation (e)/original length (l) • Stiffness = force to produce unit deformation • Stress = force (F)/area (A) • Strength = stress at failure

  6. Young’s moduli (Ei) E = Fl/eA … but E in compositesmay vary with direction 3 principal axes (x, y, z)

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

  8. 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.

  9. 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

  10. Variation of E with fibre length:fibre length distribution factor ηl • Cox shear-lag equation: where • critical length:

  11. Poisson’s ratio (isotropic: ν) •  = -(strain normal to the applied stress) (strain parallel to the applied stress). • thermodynamic constraintrestricts the values to -1 <  < ½for isotropic materials

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

  13. 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.

  14. Poisson’s ratios for GRP • Peter Craig measured νij forC1: 13 layers F&H Y119 unidirectional (UD) rovingsA2: 12 layers TBA ECK25 woven rovings (WR) • confirmed Lemprière criteriawere valid for both materials        

  15. Poisson’s ratios for GRP high values low values

  16. 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

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

  18. Bulk modulus • Isotropic case • Orthotropic case

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

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

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