1. Mechanical Properties of Metals
Chapter 6 Outline
Mechanical Properties of Metals
How do metals respond to external loads?
Stress and Strain
Tension
Compression
Shear
Torsion
Elastic deformation
Plastic Deformation
Yield Strength
Tensile Strength
Ductility
Toughness
Hardness
Not tested: true stress-true stain relationships, resilience, details of the different types of hardness tests, variability of material properties

2. How materials deform as a function of applied load 
Introduction
How materials deform as a function of applied load 
Testing methods and language for mechanical properties of materials.
Stress,  (MPa)
Strain,  (mm / mm)

3. Types of Loading Tensile Compressive Shear Torsion
Torsional forces (T) produce a rotational motion about the longitudinal axis of one end of the member relative to the other
Shear
Torsion

4. (For Tension and Compression)
Stress
(For Tension and Compression)
To compare specimens , the load is calculated per unit area.
Stress:  = F / Ao
F: is load
A0: cross-sectional area
A0 perpendicular to F before application of the load.

5. Strain (For Tension and Compression)
Strain:  = l / lo ( 100 %)
l: change in length
lo: original length.
Stress / strain = / 

6. Shear and Torsion Shear stress:  = F / Ao
F is applied parallel to upper and lower faces each having area A0.
Shear strain:  = tan ( 100 %)
 is strain angle
Shear
Torsion

7. Torsion Torsion: like shear. Load: applied torque, T
Strain: angle of twist, .
Torsion
Shear

8. Stress-Strain Behavior
(Tension)
Elastic Plastic
Elastic deformation
Reversible:
( For small strains)
Stress removed  material returns to original size
Plastic deformation
Irreversible:
Stress removed  material does not return to original dimensions.
Stress
Strain

9. Higher E  higher “stiffness”
Elastic deformation
Gives Hooke's law for Tensile Stress
 = E 
E = Young's modulus or modulus of elasticity (same units as , N/m2 or Pa)
Unload
Slope = modulus of
elasticity E
Stress
Load
Strain
Higher E  higher “stiffness”

10. Nonlinear elastic behavior
In some materials (many polymers, concrete...), elastic deformation is not linear, but it is still reversible.
/ = tangent modulus at 2
Definitions of E
/ = secant modulus between origin and 1

11. Elastic Deformation: Atomic scale
Chapter 2: Potentials and Force
Attractive is positive here
High modulus
Strongly
bonded
Force, F
Separation, r
Low modulus
Weakly
bonded
E ~ (dF/dr) at ro
F= (sign) dV/dr 
E~ curvature of potential
at equilibrium, r0

12. (time dependence of elastic deformation)
Anelasticity
(time dependence of elastic deformation)
Have assumed elastic deformation is time independent
(applied stress produces instantaneous strain)
Elastic deformation takes time; can continue even after load release.
This behavior is known as anelasticity.
Small effect in metals; can be significant for polymers (visco-elastic).

13. Poisson’s ratio Loaded Unloaded Tension  shrink laterally
Compression  bulge.
Ratio of lateral to axial strain called
Poisson's ratio .

14. Poisson’s ratio dimensionless. Sign:
lateral strain opposite to longitudinal strain
Theoretical value:
for isotropic material: 0.25
Maximum value: 0.50,
Typical value:

15. = G , Shear Modulus Unloaded Loaded Shear stress to shear strain:Zo y 
Unloaded
Loaded
Shear stress to shear strain:
= G ,
 = tan = y / zo
G is Shear Modulus (Units: N/m2)

16. Elastic Modulus Poisson’s Ratio and Shear Modulus
For isotropic material:
E = 2G(1+)  G ~ 0.4E
Single crystals are usually elastically anisotropic
Elastic behavior varies with crystallographic direction.

17. Plastic deformation (Tension) Plastic deformation:
stress not proportional to strain
deformation is not reversible
deformation occurs by breaking and re-arrangement of atomic bonds (crystalline materials by motion of defects)

18. Tensile properties: Yielding
Elastic Plastic y
Yield point: P
Where strain deviates from being proportional to stress
(the proportional limit) P
Stress
Strain
Yield strength: y Permanent strain= 0.002
0.002
A measure of resistance to plastic deformation

19. Tensile properties: Yielding
Stress
Strain
For a low-carbon steel, the stress vs. strain curve includes both an upper and lower yield point.
The yield strength is defined in this case as the average stress at the lower yield point.

20. Tensile Strength Tensile strength = max. stress (~ 100 - 1000 MPa)
If stress maintained specimen will break
Fracture Strength
Stress, 
“Necking”
Tensile strength =
max. stress
(~ MPa)
Strain, 
Yield stress, y , usually more important than tensile strength. Once yield stress has been passed, structure has deformed beyond acceptable limits.

21. Tensile properties: Ductility Ductility  Deformation at Fracture
Strain offset method to define the yield strength
Ductility  Deformation at Fracture
percent elongation or
percent reduction in area

22. Mechanical Properties of Metals
Yield strength and tensile strength vary with thermal and mechanical treatment, impurity levels, etc.
Variability related to behavior of dislocations (Elastic moduli are relatively insensitive)
Yield and tensile strengths and modulus of elasticity: Decrease with increasing temperature.
Ductility increases with temperature.

23. Toughness
Toughness: ability to absorb energy up to fracture (Area under the strain-stress curve up to fracture)
Units: the energy per unit volume, e.g. J/m3
Can be measured by an impact test (Chapter 8).

24. True Stress and Strain True Stress True Strain
T = F/Ai T = ln(li/lo)
 = F/Ao  = (li-lo/lo)
True Strain
True stress: load divided by actual area in the necked-down region, continues to rise to the point of fracture, in contrast to the engineering stress.

25. Elastic Recovery During Plastic Deformation
Deformed plastically, stress released, material has permanent strain.
If stress is reapplied, material again responds elastically at the beginning up to a new yield point that is higher than the original yield point.
Elastic strain before reaching the yield point is called elastic strain recovery.

26. Hardness (I)
Hardness measure of material’s resistance to localized plastic deformation
(e.g. dent or scratch)
Moh’s scale  ability of a material to scratch another material: from 1 (softest = talc) to 10 (hardest = diamond).
Variety of hardness tests
(Rockwell, Brinell, Vickers, etc.). Small indenter (sphere, cone, or pyramid) forced into surface of material under controlled magnitude and rate of loading.
Depth or size of indentation is measured.
Tests are approximate, but popular because they are easy and non-destructive (except for the small dent).

27. Hardness proportional to tensile strength
Hardness (II)
Tensile strength (MPa)
Tensile strength (103 psi)
Brinell hardness number
Tensile strength and hardness  degree of resistance to plastic deformation.
Hardness proportional to tensile strength
Proportionality constant depends on material.

28. What are the limits of “safe” deformation?
Stress
For practical engineering design, the yield strength is usually the important parameter
Reliability is a big area… take into account economical, legal and other factors…
Strain
Design stress:
d = N’c : c = maximum anticipated stress,
N’ the “design factor” > 1.
Make sure d < y, safe or working stress:
w = y/N where N is “factor of safety” > 1.

29. Summary Make sure you understand language and concepts: Anelasticity
Ductility
Elastic deformation
Elastic recovery
Engineering strain
Engineering stress
Hardness
Modulus of elasticity
Plastic deformation
Poisson’s ratio
Proportional limit
Shear
Tensile strength
Toughness
Yielding
Yield strength

30. Reading for next class:
Chapter 7: Dislocations and Strengthening Mechanisms
Dislocations and Plastic Deformation
Motion of dislocations in response to stress
Slip Systems
Plastic deformation in
single crystals
polycrystalline materials
Strengthening mechanisms
Grain Size Reduction
Solid Solution Strengthening
Strain Hardening
Recovery, Recrystallization, and Grain Growth
Optional reading (Part that is not covered / not tested):
7.7 Deformation by twinning
In our discussion of slip systems, §7.4, we will not get into direction and plane nomenclature