1. Plasticity, Ductility 4/15/2017 4/15/2017 1
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2. Yield strength is the stress beyond which a material becomes plastic – deformation is permanent
Determined by standard tensile testing procedures
Units: Mpa - MN/m2 or psi - lb/in2
1 Mpa = psi

3. Maximum stress is defined as the tensile strength σts
Figure 6.1
Yield strength σy is defined by a 0.2% offset from the linear elastic region
When strained beyond σy, most metals work harden, causing the rising part of the curve
Maximum stress is defined as the tensile strength σts

4. Figure 6.2
σy is identified as the stress at which the stress-strain curve becomes markedly non-linear, typically around a strain value of 1%
The behavior of the polymer beyond the yield point depends on its temperature relative to the materials glass transition temperature

5. Figure 6.3
Glasses and ceramics have a yield strength; however, it is so large that it is never reached during a tensile test – the material fractures first
The elastic limit σel is defined by the end of the elastic region of the stress-strain curve – this is the value generally used to compare the strength of ceramics with other materials

6. Tensile and compression tests require a large sample and are destructive – hardness tests require only a small volume and are non-destructive
Figure 6.4
In a hardness test, a diamond or ball shaped indenter is pressed into the surface of a material
The hardness of the material is determined by its resistance to the indentation

7. Most common are Brinell and Rockwell
Units for hardness change based on which scale is used
Figure 6.5

8. Figure 6.7

9. Figure 6.8
Yield strain is the strain at which the material ceases to be linearly elastic – polymers have large yield strain (0.01 – 0.1) while the value for metals is at least a factor of 10 smaller

10. Stress – strain curve for a single atomic bond
Ideally, the strength of a material is the force necessary to break inter-atomic bonds
A bond is broken if it is stretched beyond about 10% of its original length – therefore the force needed to break a bond is roughly:
Figure 6.9

11. Figure 6.10
Further calculations that account for the curvature of the force-distance curve predict a ratio of 1/15

12. Defects in metals and ceramics prevent materials from achieving their ideal strength
Figure 6.11
Common Defects:
Vacancies
Solute atoms on interstitial and substitutional sites
Dislocations
Grain boundaries

13. Vacancies Solute Atoms
A vacancy is a site at which an atom is missing – while vacancies play a role in diffusion, creep, and sintering, they do not influence strength
Solute Atoms
Substitutional solid solution – dissolved atoms replace those of the host
Interstitial solid solution – dissolved atoms squeeze into spaces or “interstices” between the host atoms
Dissolved atoms rarely have the same size as the host material, so the surrounding lattice is distorted

14. Dislocations Grain Boundaries
A dislocation is an extra half-plane of atoms in the crystal – in the figure, the upper part of the crystal has one more double-layer of atoms than the lower part – dislocations distort the lattice and make metals soft and ductile
Grain Boundaries
Grain boundaries form when differently oriented crystals meet – the individual crystals are called grains, the meeting surfaces are grain boundaries

15. Figure 6.12
The edge dislocation is made by cutting, slipping, and rejoining bonds across a slip plane
The dislocation line separates the part of the plane that has slipped from the part that has not
(b) represents the resulting atomic configuration – called an edge dislocation because it is formed by the edge of the extra half-plan

16. When a dislocation moves it makes the material above the slip plane slide relative to that below
Figure 6.13
Initially perfect crystal
(b) – (d) the passage of the dislocation across the slip plane shears the upper part of the crystal over the lower part by the slip vector b; when it leaves, the crystal has suffered a shear strain

17. In a screw dislocation, the upper part of the crystal is displaced
Figure 6.14
In a screw dislocation, the upper
part of the crystal is displaced
parallel to the edge of the cut
rather than normal to it
All dislocation are either edge or
screw or mixed, meaning they
are made up of little steps of
edge and screw
In the screw dislocation, the slip vector b is parallel
to the dislocation line S – S

18. Figure 6.15
For a dislocation to move, only bonds along the line it moves must be broken – this is significantly easier than breaking all of the bonds in the plane
In crystals there are preferred planes and directions for which dislocation movement is easier – these are called the slip planes and slip directions
Slip displacements are tiny – however, if a large number of dislocations travers a crystal, moving on many slip planes, the material deforms at a macroscopic level

19. Crystals resist the motion of dislocations with a friction-like
resistance f per unit length
Dislocations move from an applied shear stress τ – as they move
the upper half of the crystal shifts relative to the lower half by
a distance b
Dislocations move if
τ exceeds f/b
Figure 6.16

20. Drawing aligns polymer chains in the direction in which the material
Figure 6.17
Drawing aligns polymer chains
in the direction in which the material
is stretched – this can increase
strength and stiffness by a factor of 8
Polymers with high Tg cannot be drawn
at room temperature – they craze, forming
small crack-shaped regions within the
polymer
When crazing limits ductility in tension,
large plastic strains may still be possible
in compression by shear banding

21. The way to strengthen crystalline materials is to make it
harder for dislocations to move
Figure 6.18

22. Two factors determine the influence of obstacles
on dislocation movement:
Spacing
Strength
L: distance between obstacle and the slip plane
NL: number of obstacles touching unit length of dislocation line
p: pinning force exerted by obstacle on dislocation line
α: dimensionless constant characterizing the strength of obstacle
The shear stress needed to force
the dislocation through a field
of obstacles

23. Strengthening of a metal by alloying – deliberate additions of impurities
Alloying elements are generally bigger than those of the host material, making it harder for dislocations to move

24. Dispersion: disperse small, strong particles into a liquid metal, trapping the particles when it is cast in to shape
Precipitation: solute dissolved in a metal while bother are molten; precipitates as small particles when cooled
Dispersed and precipitated particles obstruct dislocation movement

25. Caused by the accumulation of dislocations generated by plastic deformation
Dislocation density is defined as the length of dislocation line per unit volume

26. If a dislocation advances, it shears the material above
the slip plane relative to that
below, creating a little step
called a jog
Contribution of work hardening
to the shear stress required to
move the dislocation
Pinning force exerted
on dislocations by jogs

27. Figure 6.20
Successive positions of a dislocation as it bypasses particles that obstruct its motion.

28. Grain size D is typically 10-100 μm
Dislocations cannot simply slide from one grain to the next because the slip planes do not line up
Effect of grain boundaries on shear strength required for a dislocation to move

29. Approximation of shear yield strength based on
strengthening mechanisms
This describes the yield strength of a singled crystal
loaded in shear – we need the yield strength of
a polycrystalline material loaded in tension

30. Many engineering materials can be strengthened
through various hardening mechanisms – however,
an increase in strength almost always results in a
decrease in ductility

31. Figure 6.22
The figure shows strengthening mechanisms and their effect on ductility – mechanisms are frequently combined, but an increase in strength will generally lower the ductility

32. Dislocations do not play a role in the strength of non-crystalline solids – instead, the relative slippage of two segments of a polymer chain must be considered
Impeding the slippage of molecular chains can be done through blending, drawing, cross- linking, and by reinforcement with particles, fiber, and fabrics

33. Stronger alloys tend to have lower ductility
Figure 6.23
Stronger alloys tend to have lower ductility

34. SUMMARY and CONCLUSIONS
Load Bearing Structures require materials with reliable, known strength.
There are several measures of strength
Elastic Design requires:
NO PART of the structure suffers plastic deformation
No stresses should exceed the yield strength
Plastic Design requires:
Some parts of the structure can deform plastically but the whole structure must survive.
Ductility and Tensile Strength become important properties
Materials can be made stronger
Metals, Ceramics: stopping dislocations from moving
Polymers: increasing bond strength between chains