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ELASTICITY. Elasticity Elasticity of Composites Viscoelasticity. Elasticity: Theory, Applications and Numerics Martin H. Sadd Elsevier Butterworth-Heinemann, Oxford (2005). Revise modes of deformation, stress, strain and other basics (click here) before starting this chapter etc. .

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slide1

ELASTICITY

  • Elasticity
  • Elasticity of Composites
  • Viscoelasticity

Elasticity: Theory, Applications and Numerics

Martin H. Sadd

Elsevier Butterworth-Heinemann, Oxford (2005)

slide2

Revise modes of deformation, stress, strain and other basics (click here) before starting this chapter etc.

slide3

What kind of mechanical behaviour phenomena does one have to understand?

  • Phenomenologically mechanical behaviour can be understood as in the flow diagram below.
  • Multiple mechanisms may be associated with these phenomena (e.g. creep can occur by diffusion, grain boundary sliding etc.).
  • These phenomena may lead to the failure of a material.

Mechanical Behaviour

Elasticity

Recoverable deformation

Permanent deformation

Plasticity

Fracture

Propagation of cracks in a material

Fatigue

Oscillatory loading

Creep

Elongation at constant load at High temperatures

Note: above is a ‘broad’ classification for ‘convenience’. E.g. Creep is also leads to plastic deformation!

slide4

Recoverable

Instantaneous

Elastic

Time dependent

Anelasticity

Deformation

Instantaneous

Plastic

Time dependent

Permanent

Viscoelasticity

slide5

Elasticity

  • Elastic deformation is reversible deformation- i.e. when load/forces/constraints are released the body returns to its original configuration (shape and size).
  • Elastic deformation can be caused by tension/compression or shear forces.
  • Usually in metals and ceramics elastic deformation is seen at low strains (less than ~10–3).
  • The elastic behaviour of metals and ceramics is usually linear.

Linear

E.g. Al deformed at small strains

Elasticity

Non-linear

E.g. deformation of an elastomer like rubber

slide6

Atomic model for elasticity

  • Let us consider the stretching of bonds (leading to elastic deformation).
  • Atoms in a solid feel an attractive force at larger atomic separations and feel a repulsive force (when electron clouds ‘overlap too much’) at shorter separations. (At very large separations there is no force felt).
  • The energy and the force (which is a gradient of the energy field) display functional behaviour as below.

Energy

Force

Repulsive

Attractive

The plots of these functions is shown in the next slide

A,B,m,n → constantsm > n

slide7

Repulsive

Repulsive

r0

Force (F) →

Potential energy (U) →

r →

r →

r0

Attractive

Attractive

Equilibrium separation

r0

slide8

Elastic modulus is the slope of the Force-interatomic spacing curve (F-r curve), at the

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

slide9

Young’s modulus (Y / E)**

  • Young’s modulus is :: to the –ve slope of the F-r curve at r = r0

Stress →

Tension

strain →

Compression

** Young’s modulus is not an elastic modulus but an elastic constant

slide10

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

>

slide11

Other elastic moduli

  •  = E. E → Young’s modulus
  •  = G. G → Shear modulus
  • hydrodynami = K.volumetric strain K → Bulk modulus
slide12

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
slide13

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]
slide14

Material dependence

Elastic modulus

Property

Geometry dependence

Elastic modulus in design

  • Stiffnessof 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
slide15

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

slide16

Increasing the modulus of a material

  • METALS

► By suitably alloying the Young’s 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
slide17

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

slide18

Composite modulus in isostress and isostrain conditions

Voigt averaging

  • 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

Reuss averaging

Ef

Isostrain

Ec→

Isostress

For a given fiber fraction f, the modulii of various conceivable composites lie between an upper bound given by isostrain conditionand a lower bound given by isostress condition

Em

f

A

B

Volume fraction →