Transcript Slide 1

Chapter 8
Flow and mechanical properties of polymers
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Concepts, coefficients,
definitions
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•
•
Fluid shear: the shear stress on a
fluid element is related to the
viscosity gradient by
Volume change on deformation:
some fluids (constant density
under shear) and solids (crosslinked elastomers) deform
isochorically. Poisson’s ratio, 0 <
n < 0.5.
Modulus of elasticity (Young’s
modulus). The strain in a solid is
related to the load by the modulus
of elasticity.
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 xy
dU x
  
    xy
dy
n  x   y   z
  E 
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Concepts, coefficients,
definitions, cont’d.
•
•
•
Shear modulus: the shear stress
of a solid is related to the strain by
The elastic and shear moduli are
related using the bulk modulus
(measures how the solid volume
changes with pressure) and
Poisson’s ratio. When Poisson’s
ratio = 0.5 (perfect elasticity), the
tensile modulus is three times the
shear modulus.
Compliance: the inverse of the
elastic modulus.
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  Gs
E  3  B1  2 n   2  1 n  G
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J
E
3
Concepts, coefficients,
definitions, cont’d.
•
•
Dynamic measurements of solids
and fluids yield two coefficients
(Young’s modulus used as the
example)
The dynamic modulus contains a
storage (or elastic) component
and a loss (or damping)
component
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E  E 'i  E ' '
*
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Rheology
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Fluid element under
simple shear
Newtonian fluid: the coefficient linking shear stress to shear rate
is constant over the entire range of the variable. Molecular
relaxations are much faster than the time scale of the shear force
or shear rate. Steady flows – velocity profile is constant;
oscillating flows – fluid responds instantly to forcing function.
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Defining relationship
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Non-Newtonian fluid
Viscosity changes with shear rate.
Apparent viscosity is always defined by
the relationship between shear stress
and shear rate.
Many polymeric fluids are shearthinning, i.e., their viscosities decrease
with shear rate or shear stress.
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Generalized Oswald
fluid
Pseudoplastic: shear thinning.
Shear thickening: viscosity increases with
shear stress.
Dilatant: shear thickening fluids that contain
suspended solids. Solids can become
close packed under shear.
Time-dependent: in many polymeric fluids,
the response time of the material may be
longer than response time of the
measurement system, so the viscosity
will change with time. Thixotropic: shear
thinning with time; antithixotropic: shear
thickening with time. Rheopectic:
thixotropic materials that can recover
original viscosity under low shear.
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Generalized Oswald
fluid
a. Shear rate vs. shear stress with high
and low stress limits on viscosity
b. Viscosity vs. shear rate. Zero shear
rate, 0, and infinite shear rate, ∞,
viscosities.
Pseudoplastic: shear thinning.
Shear thickening: viscosity increases
with shear stress.
Dilatant: shear thickening fluids that
contain suspended solids
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Pseudoplastics
Flow of pseudoplastics is consistent
with the random coil model of polymer
solutions and melts. At low stress, flow
occurs by random coils moving past
each other w/o coil deformation. At
moderate stress, the coils are
deformed and slip past each other
more easily. At high stress, the coils
are distorted as much as possible and
offer low resistance to flow.
Entanglements between chains and
the reptation model also are
consistent with the observed viscosity
changes.
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Viscometers
In order to get meaningful
(universal) values for the
viscosity, we need to use
geometries that give the
viscosity as a scalar
invariant of the shear
stress or shear rate.
Generalized Newtonian
models are good for these
steady flows: tubular, axial
annular, tangential annular,
helical annular, parallel
plates, rotating disks and
cone-and-plate flows.
Capillary, Couette and
cone-and-plate
viscometers are common.
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Power law
parameters
Material
k, Pa-sn
n
Shear rate range,
s-1
Ball point pen ink
10
0.85
1 – 1000
Fabric conditioner
10
0.6
1-100
Polymer melt
10000
0.6
100-10,000
Molten chocolate
50
0.5
0.1 – 10
Synovial fluid
0.5
0.4
0.1 – 100
Toothpaste
300
0.3
1- 1000
Skin cream
250
0.1
1 – 100
Lubricating grease
1000
0.1
0.1 - 100
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Generalized
Newtonian models
Power law model
n
 xy  k  xy ;   k  xy
 xy  k  xy ;  xy   k  xy
n
n 1
n 1
Ellis model
  xy 
0

 1  

  1/ 2 
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n 1
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Example 8.2
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Dependence of viscosity
on molecular weight
Branched polymers have different
rheology. Melt viscosities of LMW
materials are lower than those of
linear polymers because the volume
occupied by a branch unit is smaller
than that of a chain element.
Melt viscosities of high molecular
weight materials have the reverse
trend. Branched polymers have a
higher zero shear viscosity. Usually,
linear polymers are preferred for
processing.
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Effects of variables
on polymer viscosity
 Ei 
  A  exp 

 RT 
The Arrhenius equation can be
used to scale the viscosity. This
can be applied to constant shear
rate or constant shear stress
values over moderate ranges of
temperature.
Plasticizers tend to reduce melt viscosities while fillers tend to increase melt
viscosity.
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Molecular weight
effects
For M < Mc;  = k * M
For M > Mc;  = k * M3.4
The critical molecular weight is the
point at which molecular
entanglements restrict the
movement of polymer molecules
relative to each other.
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Free volume model
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Shift factors
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Modulus vs. t
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Failure pressure
scaled with t, T
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Extensional flow
geometry
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Normal stress
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Elongational,
extensional,
shear-free flows
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Sheet die
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Elastic State
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Unique conditions of
polymer elasticity
• Elastomers are used above Tg; the temperature range for elastic
performance increases with molecular weight
• At low stress, there is no visible elongation of the elastomer
• Crystallization can occur in the stretched state, and increases the
tensile strength
• Deformation of elastomers (noncrystalline segments) stores energy
in changed conformations (entropic), meaning that the modulus
increases with temperature
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Volume vs. P and T
 V 
 V 
dV  

dT



  dP
 T  p
 P T
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Total derivative of volume
•
Fractional volume change
•
Term for temperature derivative is
the volume expansivity, b, and that
for the pressure derivative is the
isothermal compressibility, k.
These coefficients are relatively
independent of temperature and
pressure for moderate ranges.
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dV 1  V 
1  V 
 
  dT   
  dP
V V  T  p
V  P T
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dV
 b  dT  k  dP
V
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elongation vs. T & F
 L 
 L 
dL  
  dT  
  dF
 T  F
 F T
•
Total derivative of length
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Fractional length change
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Term for temperature derivative is
linear expansivity, a, and that for
the force derivative is the Young’s
modulus, E.
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The fractional change in length is:
•
This is a mechanical equation of
state for elastomers
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dL 1  L 
1  L 
 
  dT   
  dF
L L  T  F
L  F T
L  F 
1  L 
E  
 ;a   

A  L T
L  T  F
1
  a  dT 
 dF
A E
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In-class exercise
• A butyl rubber part is being used to suspend a motor. As the motor
is used, the temperature of the part increases by 25 C. Estimate the
change in force exerted by the butyl rubber mount when this occurs.
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In-class exercise:
solution
•
A butyl rubber part is being used
to suspend a motor. As the motor
is used, the temperature of the
part increases by 25 C. Estimate
the change in force exerted by the
butyl rubber mount when this
occurs.
1
  a  dT 
 dF
A E
Suppose that the elongation
does not change so  ~ 0.
1
0  a  dT 
 dF
A E
1
 a  dT 
 dF
A E
dF
 a  A  E
dT
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Mechanical performance
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Tensile test
•
A0 – initial cross-sectional area
•
L0 – initial length
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F – force, L – length, A – crosssectional area
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Elastic deformation, a constant
volume process for small
deformations
 eng = engineering stress =
load/initial area
 eng = engineering strain = length
change/initial length
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Definition of yield
Test equipment has some “slack” in it.
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Additional definitions
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True stress and strain
At high strains, many polymers
crystallize so that DV is not zero
and this analysis is not correct
True stress and true strain are
always larger than the engineering
values
When the volume is constant on
strain:
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 true 
load
F

area @ load A

length

L
  ln 
 true  ln
 initiallength 
 L0 
 true   eng  1   eng 
 true  ln  eng  1
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Additional definitions
• When a material is deformed, it absorbs energy as the force acts
over the distance, L-L0.
• Ductility – the amount of permanent strain prior to fracture failure
• Toughness – amount of energy absorbed by the material during
fracture failure, i.e., the area under the stress-strain curve.
• Initial yield – stress/strain to which deformations are elastic
• Maximum tensile strength – highest load the material can take prior
to fracture
• Resiliency – amount of energy absorbed elastically and completely
recoverable. Resilience = ½*max*max.
• At higher stresses, the sample has permanent strain.
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Other notes
• Cold drawing of fibers: stress above the yield point crystallizes the
material.
• Product failure can occur at the yield point as the original
dimensions are not recovered.
• In some cases, product failure occurs when the part breaks
• Toughness is a measure of energy needed to break the part.
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Effect of T on stressstrain curves
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Summary table
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End-use properties
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Failure mechanisms
polymers
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Failure mechanisms
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Elastic deformation
Brittle fracture initiated by shear banding or crazing
Plasticity terminating in ductile fracture
Cold drawing
Rubbery and viscous flow
Adiabatic heating
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Brittle fracture,
T< 0.8 Tg
• Material fails by brittle fracture; stress-strain is nearly linear to break
point. Fracture may be initiated by shear yielding or crazing.
• Elongation may be less than 5%
• Brittle fracture can also occur in ductile materials if the strain rate is
very high (projectile speeds)
• Failure in tension is initiated at cracks or flaws in the sample.
Polymers have a limiting critical flaw size, below which fracture
stress is independent of the flaws (fillers?)
• PMMA critical flaw size is 0.05 mm.
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Internal defect
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Compression
• Failure strength in compression may be an order of magnitude
greater than that in tension
• Crack growth is more difficult in compression – perhaps failure
occurs by plastic flow
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Crazing, T~0.8Tg
• Crazes are cracks that fill in with oriented, load-bearing material
• Usually initiated at free surfaces
• Crazing is thought to be a microdrawing process that results in
fibrillation of the polymer in the craze
• Crazes may thicken by pulling more material into the fibrils
• The thickening process stops when the local stress decreases due
to deformation
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Plasticity/ductile
failure, T> 0.8Tg
• Shear banding is observed as “kink bands” – local changes in
orientation often at an angle to the tensile or compressive force
• Shear yielding modes are common under compression
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Cold drawing
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•
Non-crystalline polymers
– yield point followed by
constant stress region
and then break
Semi-crystalline
polymers – yield point,
load drop, high
elongation with material
necking and
crystallization in this
region. The neck has a
nearly constant crosssectional area and pulls
in material from each
end. The chain
alignment gives materials
with much higher tensile
strength than the original
sample.
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Viscous flow
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•
•
•
T > 1.1 Tg
WLF equation
Polymer deform via viscous flow
Upper temperature limit is usually determined by degradation
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Adiabatic heating
• At high deformation rates, the heat generated in deformation may
not have time to be conducted away, and the local temperature can
increase significantly.
• Heating usually occurs in the craze and shear banding regions
• As the temperature increases, the local elastic modulus decreases
and the material can undergo strain softening.
• Necking then occurs
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Other factors
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•
•
•
•
T
P
Strain rate
Annealing
Cold drawing
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Time to failure
HDPE water pipes at 4
temperatures. Failure
modes: 1) ductile failure,
2) creep crazing.
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Composites
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Composites
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Impact failure
Izod test
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Notch tip radius,
material effects
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Impact speed effects
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