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

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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

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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) Z Y X 2 transverse 3 through- thickness 1 axial, or longitudinal

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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

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Variation of E with angle: fibre orientation distribution factor η o

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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.

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Variation of E with fibre length: fibre length distribution factor η l Cox shear-lag depends on o G m : matrix modulus o A f : fibre CSA o E f : fibre modulus o L: fibre length o R: fibre separation o R f : fibre radius < Shear < Tension

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Cox shear-lag equation: where Critical length: Variation of E with fibre length: fibre length distribution factor η l

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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

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Poisson’s ratio (orthotropic: ν ij ) Maxwell’s reciprocal theorem o ν 12 E 2 = ν 21 E 1 Lemprière constraint restricts the values of ν to (1- ν 23 ν 32 ), (1- ν 13 ν 31 ), (1- ν 12 ν 21 ), (1- ν 12 ν 21 - ν 13 ν 31 - ν 23 ν ν 21 ν 32 ν 13 ) > 0 hence ν ij ≤ (E i /E j ) 1/2 and ν 21 ν 23 ν 13 < 1/2.

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Poisson’s ratios for GRP Peter Craig measured ν ij 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 C1WR A2 E 1 (GPa) E 2 (GPa) E 3 (GPa) G 12 (GPa)

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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

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Poisson’s ratios for GRP UD C1 WR A2 ν ij √E i /E j ν ij √E i /E j ν ν ν ν ν ν high values low values

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

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

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Bulk modulus Isotropic case Orthotropic case

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

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

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