Mechanical Properties

1
Section 3

• Mechanical Properties
• of Solids

2
Hookes Law - Mark One

• The restoring force of an ideal spring is-
• F -kx
• where k is the spring constant and x is the
  displacement from the unstrained length
• the minus sign is significant!

3
Restoring Force

• The restoring force is opposite to the applied
  force
• Minus sign - the restoring force always points
  in a direction opposite to the displacement of
  the spring

4
Short Springs

• The 10 coil spring has a spring constant k
• The 5 coil spring has a spring constant of 2k
• Spring sections behave like capacitors

5
Small Elastic Deformation

• Longitudinal deformation - stretching a rod
• Strain typically 0.01
• Youngs Modulus
• Nylon 3.7 x 109 Pa
• Steel 2 x 1011 Pa
• Empirical Law-
• F Y(DL/L0)A

6
Stress Strain

• Stress is the deforming force per unit area and
  is measured in pascals or N/m2
• Strain is the change in length per unit length
  and is dimensionless
• Youngs modulus (and other modulii) is a
  characteristic of the material
• Generally it depends on direction of applied
  stress relative to crystal plains

7
Shear Deformation

• Empirical Law-
• Steel 8.1 x 1010 Pa
• Stress/Strain defined as for long. defor.

8
Bulk Deformation

• Empirical formula-
• Minus sign - extra pressure results in a smaller
  volume

9
Hookes Law - Mark Two
10
Stress/Strain Characteristics

• Elastic region
• Hookes Law holds
• Elastic limit/Yield stress
• Plastic region
• larger strains (several 100 times)
• Fracture
• No plastic region for some materials
• ductile or brittle

11
Poissons Ratio

• Definition of Poissons Ratio-

12
Mechanical Modulii

• Three modulii are related
• Mechanical properties of the elastic region are
  characterised by-
• Y(E)
• S (G)
• B (K)
• Poissons Ratio (n)

13
Y versus G - Atomic Model
14
Stress/Strain Curve - Ductile Material
15
Microscopic Approach
16
Non-Linearity of Y - Whiskers
17
Elastic Behaviour of Rubber
18
Ductile Shear Deformation

• Theoretical yield stress is G/30 Pa for ductile
  materials
• This value is based on the atomic/crystal model
• Also based on inter-atomic force function shown
  earlier
• Poor agreement with expt. Why?

19
Theory Versus Experiment
20
Dislocations to Explain Discrepancies

• 1934 - Geoffrey Taylor introduces concept of
  dislocations to explain G/30 anomaly
• 1955 - dislocations first observed experimentally

21
Moving Dislocation - Small Stress
22
Impurities - The Other Factor
23
Yield Stress for Alloys

• Dislocation/impurities theory
• Alloys should have higher yield stress than pure
  material
• Mild steel (Fe/C) has yield stress 10 times that
  of pure iron

24
Plastic Deformation

• Beyond yield stress, the dislocations move more
  easily
• Small change in stress causes large change in
  strain
• easy glide region

25
Ductile Stress/Strain Curve
26
Work Hardening Stress/Strain Curve
27
Work Hardening Process
28
Improved Elastic Region
29
Ductile Fracture
30
Brittle Stress/Strain Curve
31
Brittle Fracture

• Atomic model predicts 0.1Y Pa
• Glass (brittle) - predicted fracture stress - 7.5
  x 109 Pa
• Glass - actual fracture stress - 0.6 x 109 Pa
• Why is the actual value much lower than expected?

32
Stress Concentration Factor

• SCF
• Atomic model -
• r - tip curvature l - depth of groove

33
Example SCF for Glass

• Y 75 x 109 Pa
• stheory 0.1Y 7.50 x 109 Pa
• SCF
• sactual 7.5 x 109/89 0.8 x 109 Pa
• This is close to the experimental value of 0.6 x
  109 Pa

34
Laminated Glass
35
Brittle or Ductile

• Whether brittle or ductile is a function of
  temperature
• Glass ductile at high temperature
• Iron brittle at 273 K
• Depends on bond type and arrangement of
  molecules/atoms e.g. polyethylene LDPE, MDPE, HDPE

36
Modulus Values
37
Stress-Strain Curves
38
Brittle Stress/Strain Curve
39
Ductile Stress/Strain Curve
40
Work Hardening Stress/Strain Curve