A4
Beyond elasticity: plasticity, yielding and
ductility
Elastic project – which avoids plastic
deformation, ensure that the car cabin does not
deform in the collision. The plasticiy absorbs the
impact energy, and allows forming metals and
polymers.
Strength, plastic work and ductility: definition and
measurement
yield properties and ductility are measured using the standard tensile tests with the
materials taken to failure
σy – yield strength (tensão de cedência ) or
elastic limit (MPa) is the tension beyond
which the material is no longer elastic;
εpl – plastic strain (extensão plástica) is the
permanent strain resulting from plasticity
εpl= εtot - σ/E;
ductilidade – is a measure of how much
plastic strain a material can tolerate. It is
measured by the tensile strain at break, εf.
Plastic work
The plastic work per unit volume at fracture, important in energy-absorbind
applications, is
Note: just the area under the stress-strain curve
Hardness test (teste de dureza)
tensile and compression tests need large samples and are destructive
(although they are more complete and accurate
There are numerous hardness scales which are
empirically defined, and all are a measure of σy
Hardness H is the load F
divided by the area A of the
indent projected onto a plane
perpedicular to the load. The
indent means that plasticity has
occurred, and H is the
resistence to it.
Hv≈σy/3
The big picture: charts for yield strength
The strength range for a given class of materials can span a factor of 10 or more, while the spread in
stiffness is at most 10%.
The modulus-strength chart
yield strain: σy/E is the strain at which the material ceases to be linearly elastic
The origins of strength and ductility
- perfection: the ideal strength
The distance over which interatomic forces act is
small – a bond is broken if it is stretched to more than
10% of its original length. So the force needed to
break a bond is roughly:
On this basis the ideal strength of a solid should
therefore be roughly (since E=S0/a0):
The stress-strain curve for a single
atomic bond (it is assumed that each
atom occupies a cube of side a0)
The origins of strength and ductility
- perfection: the ideal strength
None of the materials (metals, polymers, ceramics) achieve the ideal value
σy/E de 1/10; most don’t even come close. Why not?
→ because materials are imperfect!
Crystalline imperfection:
defects in metals and ceramics
Defects are
always
present in
crystals
point defects
0D
1D
Dislocations make
metals soft and
ductile
2D
grain
boundaries
form in pure
metals and
alloys
Defects explain diffusion, strength, ductility, electrical resistance, thermal
conductivity, and much more.
Dislocations and plastic flow
The strength (resistência mecânica) of
engineering materials is much smaller
that that calculated from the interatomic
forces in a perfect crystal (~E/10). In the
40s and 50s of the 20th century it was
understood that a crystal with
dislocations can deform at significantly
lower tensions than those predicted for
an ideal crystal.
atom configuration at an edge
dislocation in a simple cubic crystal
It is far easier to move a dislocation
through a crystal, breaking and
remaking bonds only along its line as it
moves, than it is to simultaneously
break all the bonds in the plane before
remaking them. It is like moving a
heavy carpet by pushing a fold across
it rather than sliding the whole thing at
one go.
Dislocations and plastic flow
A screw dislocation. The slip vector b
is parallel to the dislocation line S-S)
Dislocation motion causes extension. The
slip displacements are tiny (one dislocation
produces a displacement of ~10-10 m). But if
large numbers of dislocations traverse a
crystal, moving on many different planes, the
shape of the material changes at the
macroscopid length scale.
Manipulating strength – strengthening metals
The way to make crystalline materials stronger is to make it harder for
dislocations to move
A “dislocation-eye” view of the slip plane across which it must move: each
strengthening mechanism presents a new obstacle course
Solution hardening
Endurecimento por introdução
de impurezas
τss ∝ c1/2
c: solute
concentration
Precipitation hardening
Endurecimento por formação
de precipitados
τss ∝ L-1
L: particle
spacing
Work hardening
encruamento
τss ∝ ρd1/2
ρd: dislocation
density
Grain boundary hardening
Solid solution hardening is strengthening by
deliberate additions of impurities or, more properly
said, by alloying. The addition of Zn to Cu makes
the alloy brass. The Zn atoms replace Cu atoms
to form a random substitutional solid solution. The
Zn atoms are bigger than those of Cu and, in
squeezing into the Cu lattice, they distort it. This
roughens the slip plane, making it harder for
dislocations to move. Brass, bronze and stainless
steels dervie their strength in this way.
The dispersion of small, strong particles is an
effective way to impede dislocations. For
example, an alloy of Al with 4% Cu, appropriately
treated, gives very small, closely spaced
precipitates of the hard compound CuAl2. Most
steels are strengthened by precipitates of
carbides.
Plastic deformation itself causes the accumulation
of dislocations which are themselves obstacles to
dislocation motion
Finally, grain boundaries obstruct dislocation
motion.
τgb ∝ D-1/2; D: grain size
Strength and ductility of allows
To a first approximation the strengthening mechanisms add up
Shear yield strength
Shear stresses make dislocations move
Microscopically, beyond a certain shear stress, dislocations move in the
material. But we want to know the yield stregth of a material in a tensile test!
When this shear stress acts upon an aggregate of microcrystals, some will have slip
planes oriented favorably with respect to the shear stress, and others will not. From
this combinations, it results
Mechanisms of increase mechanical resistance
in Cu alloys
Strengthening mechanisms and the consequent drop in ductility, here shown for Cu alloys.
The mechanisms are frequently combined. The greater the strength, the lower the ductility.
The bronze age
historical alloyCu-10%Sn
Sn
Hardening of Cu by a solid solution of Sn
The airplane age
The mechanical resistance of the
light Al alloys was significantly
improved in 1910 by the addition of
4%.
The solid solution is preserved if the
cooling is fast, forming a supersaturated solution.
A final treatment consists in
annealing the materials at specific
temperatures and times to form
uniformly dispersed very thin
precipitates (<10 nm) which very
efficienly block dislocation motion.
Plastic flow in polymers
At low temperatures, below 0.75
Tg , polymers are brittle
Above 0.75 Tg,
polymers
become plastic
craze (esfarrapamento) occurs
for polymers with higher Tg
light dispersion
causes whitening
drawn material is
stronger and stiffer, by a
factor of about 8, giving
drawn polymers
exceptional properties
(but you can only draw
fibers or sheet)
An example of the search for new materials
- metallic glasses
σy
Materials Today Jan-Feb 09 (www.materialstoday.com)
Material indices for yield-limited design
Minimizing weight: a light, strong tie-rod
m=ALρ
eliminating A
• high-strength end of several major alloy systems – Ni alloys, high-strength steels, and Al
and Mg alloys;
• Ti alloys are significantly better than the other metals
• CFRP is better still
• ceramics and glasess have high values of Mt but are impractical as structural ties
because of their britlleness
Material indices for yield-limited design
minimizing weight:
light, strong panels
Mp = σy1/2/ρ
Now all the light alloys (Mg,
Al, and Ti) outperform stell,
as do GFRP and wood.
CFRP still leads the way.
Summary and conclusions - I
•
Load-bearing structures require materials with reliable, reproducible strength.
•
Elastic design requires that no part of the structure suffers plastic deformation, and
this means that the stresses in it must nowhere exceed the yield strength, σy, of
ductile materials or the elastic limit of those that are not ductile
•
Plastic design, by contrast, allows some parts of the structure to deform plastically so
long as the structure as a whole does not collapse. Then tow further properties
become relevant: the ductility, εf, and the tensile strenght, σts, which are the
maximum strain and the maximum stress the material can tolerate before fracture.
•
Charts plotting strength show that material families occupy different areas of material
property space, depending on the strengthening mechanisms on which they rely.
•
Crystal defects – particularly dislocations – are central to understand the
strengthening mechanisms. It is the motion of dislocations that gives plastic flow in
crystalline solids, giving them unexpectadly low strengths.
•
When strength is needed it has to be provided by the strengthening mechanism that
impedes dislocations motion.
Summary and conclusions - II
•
Non-crystalline solids – particularly polymers – deform in a less organized way by the
pulling of the tangled polymer chains into alignment with the direction of deformation.
•
This leads to cold drawing with substantial plastic strain and, at lower temperatures,
to crazing. The stress required to do this is significant, giving polymers a considerable
intrinsic strength. This can be enhanced by blending, cross-linking, and reinforcement
with particles or fibers.
•
Sometimes, however, controlled plasticity is the aim.Then the requirement is that the
stress must exceed the yield strength over the entire section of the component.
Next classes
HW1 due Sunday March 1 by e-mail
Next week’s classes:
1) A5 – Fracture
2) A6 – Fatigue, friction and wear, and creep
3) A7 – Thermal, electrical, magnetic and optical
properties
Essential materials science classes (A2 to A11) are based on the textbook “Materials:
engineering, science, processing and design” by M. Ashby, H. Shercliff, D. Cebon,
Butterworth-Heinemann, Oxford, UK