Elastic Deformation

1
Elastic Deformation

• Mechanical behaviour and properties
• response of materials to applied force
• deformation change of shape
• fracture
• properties such as stiffness, strength, ductility
  and toughness

• Phenomenon
• spring
• rubber
• deformation is only temporary (gone with the
  release of force)
• Quantitative description
• stress
• strain
• elastic modulus
• Physical process
• stretching of bonds

Reading 6.1-6.5 (6th ed)
2
Elastic Deformation

• Stress
• Force is not important unless we know on how big
  an area it is acting on
• definition force per area
• types of stresses
• normal stress force normal to the area divided
  by the area
• s Fn/Ao

• shear stress force parallel to the area divided
  by the area
• t Fs/Ao
• General case

Fs
Ao
This is called engineering stress
Fn
Ao
Fz
F
Unit of F N Unit of area m2 Unit of stress
N/m2 (Pa - Pascal) or more commonly, MN/m2 106
N/m2 (MPa)
Fy
Fx
3
Elastic Deformation

• Strain
• response of material to applied stress (measure
  of deformation)
• types of strain
• normal strain

• lateral strain
• Poisson's ratio
• minus sign as e and ew usually have opposite
  signs and we want a positive value
• shear strain
• g tan q

n around 0.3 for many engineering materials
This is called engineering strain
wo
wo
w
w
Unit of strain dimensionless
4
Elastic Deformation

• Relationship between stress and strain
• Tensile (compressive) testing
• recording load
• recording displacement or extension

• Tensile stress-strain curve
• convert load into stress
• convert extension into strain
• plot stress vs strain
• Elastic deformation
• all deformation is recovered upon the removal of
  stress

5
Elastic Deformation

• Elastic modulus
• relationship is linear for many materials slope
  elastic modulus
• a measure of the resistance of material to
  elastic deformation

• for materials that show a non-linear relationship
  during elastic deformation (e.g. rubber)

Hooke's Law
limited to e lt 0.1
s E e E Young's modulus
t G g G shear modulus
unit GPa (109 N/m2)
E values Al 70 GPa Steel 200 GPa Diamond
1000 GPa
The higher the elastic modulus, the
stiffer Stiffness how resistant to elastic
deformation
For isotropic materials E 2G (1 n)
G 0.4E
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Elastic Deformation

• Physical meaning of E
• depending on the stiffness of the bonds

• Determination of E
• direct
• by definition
• E s / e
• indirect
• vibration measure resonance frequency, f
• E ? f2
• velocity of sound, v
• E r v2

F
F
r density
7
Elastic Deformation

• Elastic deformation behaviour
• Linear (e.g most metals and ceramics)
• elastic energy stored in the deformed material
  loaded to s

• non-linear (e.g. rubber and other polymers)
• energy dissipation
• for some materials, a significant amount of
  energy cannot be recovered during unloading

loading
s
unloading
loading
unloading
e
energy stored upon loading
energy spent per cycle (e.g. as heat)
energy recovered upon unloading
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Elastic Deformation

• Anelasticity
• time dependent elastic deformation

• Summary
• Concepts of stress and strain
• normal
• shear
• Poisson's ratio
• Relationship between stress and strain
• linear Hooke's law
• elastic modulus
• Young's
• shear
• Stress vs strain curves characterising elastic
  behaviour

anelastic strain
instant recovery
(instant) elastic strain
loading time
unloading time
anelasticity is usually neglegible for metals and
ceramics, but may be significant for polymers
9
Elastic Deformation - Example

• Example Problem 6.2
• Known
• do 10 mm
• Dd di do 2.5 µm 2.5 x 10-3 mm
• E 97 GPa
• n 0.34
• Question F ?
• Solution

Tensile stress Elastic deformation