ELASTICITY

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ELASTICITY

• ELASTICITY
• COMPOSITES

Elasticity Theory, Applications and
Numerics Martin H. Sadd Elsevier
Butterworth-Heinemann, Oxford (2005)
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Recoverable
Instantaneous
Elastic
Time dependent
Anelasticity
Deformation
Instantaneous
Plastic
Time dependent
Permanent
Viscoelasticity
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Elasticity
Linear
Elasticity
Non-linear
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Atomic model for elasticity
Force
Energy
Attractive
Repulsive
A,B,m,n ? constantsm gt n
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Repulsive
Repulsive
r0
Force (F) ?
Potential energy (U) ?
r ?
r ?
r0
Attractive
Attractive
r0
Equilibrium separation
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Near r0 the red line (tangent to the F-r curve at
r r0) coincides with the blue line (F-r) curve
r ?
Force ?
r0
For displacements around r0 ? Force
displacement curve is approximately linear? THE
LINEAR ELASTIC REGION
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Youngs modulus (Y / E)

• Youngs modulus is to the ve slope of the
  F-r curve at r r0

Stress ?
Tension
strain ?
Compression
Youngs modulus is not an elastic modulus but
an elastic constant
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Stress-strain curve for an elastomer
Tension
?T due to uncoilingof polymer chains
Stress ?
?C
strain ?
?T
Due to efficient filling of space
Compression
?C
?T
gt
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Other elastic modulii

• ? E.? E ? Youngs modulus
• ? G.? G ? Shear modulus
• ?hydrodynami K.volumetric strain K ? Bulk
  modulus

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Bonding and Elastic modulus

• Materials with strong bonds have a deep
  potential energy well with a high curvature ?
  high elastic modulus
• Along the period of a periodic table the
  covalent character of the bond and its strength
  increase ? systematic increase in elastic modulus
• Down a period the covalent character of the
  bonding ? ? ? in Y
• On heating the elastic modulus decrease 0 K ?
  M.P, 10-20 ? in modulus

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Anisotropy in the Elastic modulus

• In a crystal the interatomic distance varies
  with direction ? elastic anisotropy
• Elastic anisotropy is especially pronounced in
  materials with ? two kinds of bonds E.g. in
  graphite E 10?10 950 GPa, E 0001 8 GPa
  ? Two kinds of ordering along two
  directions E.g. Decagonal QC E 100000 ?? E
  000001

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Material dependence
Elastic modulus
Property
Geometry dependence
Elastic modulus in design

• Stiffness of a material is its ability to resist
  elastic deformation of deflection on loading ?
  depends on the geometry of the component.
• High modulus in conjunction with good ductility
  should be chosen (good ductility avoids
  catastrophic failure in case of accidental
  overloading)
• Covalently bonded materials- e.g. diamond have
  high E (1140 GPa) BUT brittle
• Ionic solids are also very brittle

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• METALS
• ? First transition series ? good combination
  of ductility modulus (200 GPa) ? Second
  third transition series ? even higher modulus,
  but higher density
• POLYMERS ? Polymers can have good plasticity
  ? but low modulus dependent on
• ? the nature of secondary bonds- Van der Walls /
  hydrogen ? presence of bulky side groups ?
  branching in the chains ? Unbranched
  polyethylene E 0.2 GPa, ? Polystyrene
  with large phenyl side group E 3 GPa, ?
  3D network polymer phenol formaldehyde E 3-5
  GPa ? cross-linking

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Increasing the modulus of a material

• METALS
• ? By suitably alloying the Youngs modulus can be
  increased
• ? But E is a structure (microstructure)
  insensitive property ? the increase is ?
  fraction added? TiB2 ( spherical, in
  equilibrium with matrix) added to Fe to increase
  E
• COMPOSITES? A second phase (reinforcement) can
  be added to a low E material to ? E (particles,
  fibres, laminates)? The second phase can be
  brittle and the ductility is provided by the
  matrix ? if reinforcement fractures the crack is
  stopped by the matrix

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COMPOSITES
Laminate composite
Alignedfiber composite
Particulate composite
Modulus parallel to the direction of the fiberes

• Under iso-strain conditions
• I.e. parallel configuration
• m-matrix, f-fibre, c-composite

Volume fractions
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Composite modulus in isostress and isostrain
conditions

• Under iso-strain conditions ?m ?f ?c
• I.e. resistances in series configuration

• Under iso-stress conditions ?m ?f ?c
• I.e. resistances in parallel configuration
• Usually not found in practice

Ef
Isostrain
Ec ?
Isostress
For a given fiber fraction f, the modulii of
various conceivable composites lie between an
upperbound given by isostrain conditionand a
lower bound given byisostress condition
Em
f
A
B
Volume fraction ?