Engineering StressStrain Curve

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Engineering Stress-Strain Curve
Elongation
Sy
0.2 offset yield stress
Engineering Stress, SP/Ao
(Ultimate)
E
Su
E
Proportional Limit
Engineering Strain, e DL/Lo)
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Post Yield Stress-Strain Material Behaviour
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Quick Tip!

• Express Load in Newtons (N) and Area in mm2 to
  get Stress in MPa.
• Mechanical properties of engineering materials
  are almost always given in MPa or ksi.
• Imperial units Load in kips (1000 lbf) Area as
  in2 gives Stress in ksi (kips/in2)
• 1000 psi 1 ksi 6.89 MPa

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Hookes Law Elastic Deformation

• Elastic deformation is not permanent it means
  that when the load is removed, the part returns
  to its original shape and dimensions.
• For most metals, the elastic region is linear.
  For some materials, including metals such as cast
  iron, polymers, and concrete, the elastic region
  is non-linear.
• If the behavior is linear elastic, or nearly
  linear-elastic, Hookes Law may be applied
• Where E is the modulus of elasticity (MPa)

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Modulus of Elasticity - Stiffness
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Atomic Origin of Stiffness
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Shear Stress and Strain
Shear Stress
Shear Strain
shear stress, t Shear Load / Area shear strain,
g angle of deformation (radians) shear modulus,
G t /g (elastic region)
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Elastic Properties of Materials

• Poissons ratio When a metal is strained in one
  direction, there are corresponding strains in all
  other directions.
• For a uniaxial tension strain, the lateral
  strains are constrictive.
• Conversely, for a uniaxial compressive strain,
  the lateral strains are expansive.
• i.e. the lateral strains are opposite in sign to
  the axial strain.
• The ratio of lateral to axial strains is known as
  Poissons ratio, n.

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Poissons Ratio, n
For most metals, 0.25 lt n lt 0.35 in the elastic
range
Furthermore
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Plastic Deformation
Sy
Sy
Sy
Stress
Strain
0.002
0.002
0.002
Most Metals - Al, Cu
Clad Al-Alloys
Low carbon Steel
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Microstructural Origins of Plasticity

• Slip, Climb and Slide of atoms in the crystal
  structure.
• Slip and Climb occur at Dislocations and Slide
  occurs at Grain Boundaries.

t
t
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Elastic and Plastic Strain
P
(e,S)
Stress
Total Strain
The 0.2 offset yield stress is the stress that
gives a plastic (permanent) strain of 0.002.
Strain
Plastic
Elastic
ee
ep
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Elastic Recovery
Loading
Loading
Reloading
Stress
Unloading
Unloading
Strain
Strain
elastic strain
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Ductile Vs Brittle Materials

• Only Ductile materials will exhibit necking.
• Ductile if ELgt8 (approximately)
• Brittle if EL lt 5 (approximately)

Engineering Stress
Engineering Strain
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Typical Mechanical Properties
Metals in annealed (soft) condition
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True Stress and True Strain

• True stress, s, is the load, P, divided by the
  instantaneous area of the specimen, Ai.
• True strain, e, is determined from the rate of
  change in gauge length with respect to the
  instantaneous gauge length, Li.
• When strains are elastic, Ss, and ee.
• Up to strain where necking begins, specimen
  deforms with a constant volume in gauge section.
• Constant Volume gives

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True Stress
Assumes constant volume. Valid for all strains up
to point where necking begins Hence, valid for
S lt Su.
Special Case, True Fracture Stress
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True Strain
Assumes constant volume. Valid for all strains up
to point where necking begins Hence, valid for
S lt Su.
Special Case, True Fracture Strain
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Engineering Vs. True Stress-Strain Curves
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Equation of s-e curve
K
fracture
Log(true stress, s) MPa
n
necking
1.0
0.10
0.010
0.0010
Log(true plastic strain, ep)
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Ramberg-Osgood Equation
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Typical K and n values for metals
K Strain hardening coefficient n Strain
hardening exponent