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Unformatted text preview: Lecture 14
Viscoelasticity 1
Wednesday, November 17, 2010 Last time…
•
•
• •
• We showed the development of the affine, phantom and MooneyRivlin
models of network deformation
We derived an expression for modulus, G=nkT/V; this is (related) to the
slope of stress strain curves
The classical models fail to capture the effect of entanglements, which are
particularly significant if the strand length between crosslinks is large; there
is some dependence on preparation condition
We discussed viscosity of polymers, introducing a time dependence
We discussed how polymers entangle to form temporary crosslinks, and
how the time scale of their movement (“reptation”) influences the viscous
and elastic properties of the material 2
Wednesday, November 17, 2010 Entropy driven elasticity
•
• Flory constructed a simple method to separate energetic from entropic
contribution to the elastic force
Consider a typical temperature dependence of a retractive force, f, for a
network of constant volume V at constant elongation L. ⎛ ∂f ⎞
slope = ⎜ ⎟
⎝ ∂T ⎠V ,L
⎛ ∂E ⎞
fE = ⎜ ⎟
⎝ ∂L ⎠T ,V
⎛ ∂f ⎞
⎛ ∂S ⎞
f S = T⎜ ⎟ = −T⎜ ⎟
⎝ ∂T ⎠V ,L
⎝ ∂L ⎠T ,V ⎛ ∂F ⎞
⎛ ∂E ⎞
⎛ ∂S ⎞
f = ⎜ ⎟ = ⎜ ⎟ − T⎜ ⎟
⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T
⎝ ∂L ⎠V ,T
(in ideal
networks, this
is
approximately
zero) (in crystalline
materials, this is
low or
approximately
zero) 3
Wednesday, November 17, 2010 This time
•
•
• We’ll combine our discussions of viscous and elastic behavior
We’ll discuss how to characterize, model and utilize viscoelastic behavior
If there’s time we’ll talk about some other polymer properties and then
about polymer processing. 4
Wednesday, November 17, 2010 Time dependent Deformation
Deformation occurs via motion of polymer chains relative each other
Sliding polymer chains past each other takes some time
if we apply the load slowly, the chains will slide easily
if the load is applied quickly, the chains do not have time to slide
This is VISCOELASTIC behavior
At high temperatures or low strain rates, polymers have more
ductility
At low temperatures or high strain rates, polymers have less ductility 5
Wednesday, November 17, 2010 Viscous versus elastic response to deformation 6
Wednesday, November 17, 2010 Viscoelasticity
Creep: Permanent deformation to relieve stress. At constant stress,
the strain increases with time.
Stress Relaxation: Under (sudden) deformation to strain ε0, the stress
decays with time, as the force required to maintain the deformation
decreases. The amount of stress in the material as a result of a
constant deformation decreases with time.
Hystersis: nonconservative force; energy loss upon applied stress
Creep Stress Relaxation Hysteresis deformation deformation
strain spring (Hookean) creep
(viscoelastic) stress stress relaxation
(viscoelastic) force lost energy spring (Hookean) time Wednesday, November 17, 2010 time deformation 7 Creep
• Creep: Permanent deformation to relieve stress. At constant stress, the
strain increases with time. 8
Wednesday, November 17, 2010 Example: Creep in Sutures 9
Wednesday, November 17, 2010 Stress Relaxation
• Stress Relaxation: Under (sudden) deformation to strain ε0, the stress
decays with time, as the force required to maintain the deformation
decreases. The amount of stress in the material as a result of a constant
deformation decreases with time. 10
Wednesday, November 17, 2010 Example: Stress Relaxation in Bone Cement 11
Wednesday, November 17, 2010 Viscoelasticity
In stress relaxation, the modulus is time dependent. The time dependent
modulus is simply the time dependent stress divided by the (constant) strain: E (t) = σ (t )
ε0 12
Wednesday, November 17, 2010 Relaxation Modulus
The viscoelastic relaxation modulus characterizes the stress relaxation
behavior of a polymer
Stress relaxation measures stress versus time at constant strain
The relaxation modulus is
G( t ) ≡ σ (t )
γ With time, the apparent stiffness of the
material changes 13
Wednesday, November 17, 2010 How do we characterize this
timedependent behavior? 14
Wednesday, November 17, 2010 Dynamic Mechanical Testing Response for Classical
Extremes
Purely Elastic Response
(Hookean Solid) Purely Viscous Response
(Newtonian Liquid) δ = 90° δ = 0°
Stress
Stress
Strain Strain γ ( t ) = γ 0 sin 2πft = γ 0 sin ωt
τ ( t ) = Gγ 0 sin ωt d
(γ 0 sinωt )
dt
τ ( t ) = ηγ 0 cosωt
. τ (t) = η γ = η Courtesy of TA Instruments
Wednesday, November 17, 2010 15 Dynamic Mechanical Testing: Viscoelastic Material Response
γ ( t ) = τ 0 sin(ωt + δ )
τ ( t ) = (τ 0 cosδ ) sinωt + (τ 0 sinδ ) cosωt
Phase angle 0° < δ < 90° Strain Stress Courtesy of TA Instruments
Wednesday, November 17, 2010 16 DMA Viscoelastic Parameters The Complex, Elastic, &
Viscous Stress
The stress in a dynamic experiment is referred to as the complex stress τ*
The complex stress can be separated into two components:
1) An elastic stress in phase with the strain. τ' = τ*cosδ
τ' is the degree to which material behaves like an elastic solid.
2) A viscous stress in phase with the strain rate. τ" = τ*sinδ τ" is the degree to which material behaves like an ideal liquid. Phase angle δ τ* = τ' + i τ" Complex Stress, τ* Strain, ε
Wednesday, November 17, 2010 Courtesy: TA Instruments
17 DMA
•
• τ ( t ) = (τ 0 cosδ ) sin ωt + (τ 0 sin δ ) cosωt Clearly, there is an inphase and outofphase component to this shear
stress
Since we know modulus is stress divided by strain, we can rewrite the
equation: ⎡⎛ τ
⎤
⎞
⎛τ0
⎞
0
τ ( t ) = γ 0 ⎢⎜ cosδ ⎟ sin ωt + ⎜ sin δ ⎟ cosωt⎥
⎠
⎝ γ0
⎠
⎣⎝ γ 0
⎦ • This yields expressions for the inphase and outof phase modulus
(moduli): τ0
cos δ
γ0
τ
G" (ω ) = 0 sin δ
γ0
G' (ω ) = τ ( t ) = γ 0 [G' (ω ) sinωt + G" (ω ) cosωt ]
Courtesy of TA Instruments
Wednesday, November 17, 2010 18 DMA Viscoelastic Parameters
τ ( t ) = γ 0 [G' (ω ) sinωt + G" (ω ) cosωt ]
The Complex Modulus: Measure of
materials overall resistance to
deformation.
The Elastic (Storage) Modulus:
Measure of elasticity of material. The
ability of the material to store energy.
The Viscous (Loss) Modulus:
The ability of the material to dissipate
energy. Energy lost as heat.
Tan Delta:
Measure of material damping  such
as vibration or sound damping. G* = Stress*/Strain
G* = G’ + iG” G' (ω ) = τ0
cos δ
γ0 τ0
G" (ω ) = sin δ
γ0 tan δ = G"/G'
Courtesy of TA Instruments Wednesday, November 17, 2010 19 Complex and Dynamic Viscosity
The viscosity measured in an oscillatory experiment is a Complex Viscosity much
the way the modulus can be expressed as the complex modulus. The complex
viscosity contains an elastic component and a term similar to the steady state
viscosity.
The Complex viscosity is defined as: η* = η’  iη”
or
η* = G*/ω 20
Wednesday, November 17, 2010 DMA 21
Wednesday, November 17, 2010 Frequency Sweep: Material Response Transition
Region log G'and G" Terminal
Region Rubbery
Plateau
Region Glassy Region Storage Modulus (E' or G')
Loss Modulus (E" or G") log Frequency (rad/s or Hz)
Wednesday, November 17, 2010 Courtesy: TA Instruments 22 Dynamic Temperature Ramp or Step and Hold: Material
Response Log G' and G" Glassy Region Transition
Region Rubbery Plateau
Region Terminal Region Storage Modulus (E' or G')
Loss Modulus (E" or G") Temperature
Wednesday, November 17, 2010 Courtesy: TA Instruments 23 Viscoelastic Behavior of Amorphous Materials
• Polymer properties are time and temperature dependent 24
Wednesday, November 17, 2010 Viscoelastic Behavior of Amorphous Materials
• Polymer properties are time and temperature dependent 25
Wednesday, November 17, 2010 ModulusTemperature Curve from: Dynamic Mechanical Analysis: A Practical Introduction 2nd Edition by Kevin Menard
26
Wednesday, November 17, 2010 How do we describe
viscoelastic behaviour? 27
Wednesday, November 17, 2010 Viscoelasticity
Material has a viscous component and elastic component; typical of polymers Models of viscoelasticity separate the viscosity component from the elasticity
component
ε
Parallel:
Voigt model ∂ε( t )
σ ( t ) = Eε( t ) + η
∂(t) η 1 ∂σ σ ∂ε( t )
+=
E ∂t η
∂t Serial:
Maxwell model
η ε (η is viscosity)
28 Wednesday, November 17, 2010 Higher Order Models
•
• Real materials are characterized by a spectrum of relaxation times
More complicated models have been developed to try to describe this
behaviour 29
Wednesday, November 17, 2010 Can we predict properties over
a broad time range? 30
Wednesday, November 17, 2010 Temperature Dependence on Deformation
Increasing T
decreases E
decreases yield strength
increases ductility 31
Wednesday, November 17, 2010 Williams, Landel and Ferry (WLF) model and the Time
Temperature SuperPosition Principle (TTS)
Used for predicting polymer behavior over time, when measurements would
otherwise take years. Can only be used for materials above their Tg; below Tg,
Arrheniustype dependence.
log aτ = C1 (T − Tg )
17.44 (T − Tg )
log τ (T )
=−
=−
log τ (T0 )
51.6 + T − Tg
C2 + (T − Tg )
10 4 T1 (90 C) log a( !) T2 (85 C) T Tg T3 (75 C) 3 10
E(t) T 4 ( 70 C)
T 5 ( 65 C)
T6 (60 C) 2 10 1 10 !ref ! 10 T7 (50 C)
T8 (45 C)
T9 (35 C) 0 10 log t 5 4 10 10 3 10 2 1 10 10 0 10 1 2 10 3 10 10 4 5 10 6 10 10 7 time Free volume justification:
Wednesday, November 17, 2010 f = f g + (T − Tg )α f aτ = τ (T )
η(T )
=−
τ (Tg )
η(Tg ) 32 Other Properties to Consider 33
Wednesday, November 17, 2010 Other Physical Properties
Brittleness  materials that are brittle experience little or no plastic
deformation upon fracture (~5% or less fracture strain)
Ductility a measure of the degree/extent of plastic deformation upon
fracture
Resilience  the capacity of a material to absorb energy on elastic
deformation and then allow this energy to be recovered on unloading
(energy storage)
Hardness  resistance to localized plastic deformation. Rockwell,
Shore (Durometer) tests
Toughness  energy to break a unit volume of material, approximately
the area under the stressstrain curve 34
Wednesday, November 17, 2010 Ductility
• Amount of plastic deformation prior to failure
– % elongation
brittle – % reduction in area ⎛ A0 − A f
% RA = ⎜
⎜A
0
⎝ ductile
Stress ⎛ l f − l0 ⎞
% EL = ⎜
⎜ l ⎟ ×100
⎟
⎝0⎠ ⎞
⎟ ×100
⎟
⎠
strain Temperature dependent!
Wednesday, November 17, 2010 l0
35 Resilience
•
• Capacity to absorb and release energy during
elastic deformation
Modulus of resilience Ur σy = strain energy per unit volume
εy U r = ∫ σ dε
for linear elastic materials
1
U r = σ yε y
2 Stress 0 0.002 εy strain
36 Wednesday, November 17, 2010 Toughness • Capacity to absorb and release energy prior to
fracture
Low strain rate: toughness = ∫ εf
0 σ dε Stress • σy 0.002 εf strain
37
Wednesday, November 17, 2010 Hardness test simple (easy prep)
– need smooth, flat surface
– stay away from edge & from
adjacent indentations
inexpensive
lessdestructive
can estimate Tensile Strength from
hardness
– both measure resistance to plastic
deformation TS [MPa ]= 3.45 × HB
TS [psi ]= 500 × HB
38
Wednesday, November 17, 2010 Polymer Processing 39
Wednesday, November 17, 2010 Polymer Additives
Improve mechanical properties, processability, durability, etc.
• Fillers
– Added to improve tensile strength & abrasion resistance,
toughness & decrease cost
– ex: carbon black, silica gel, wood flour, glass, limestone, talc,
etc.
• Plasticizers
– Added to reduce the glass transition
temperature Tg
– commonly added to PVC  otherwise it is brittle 40
Wednesday, November 17, 2010 Polymer Additives
• Stabilizers
– Antioxidants
– UV protectants • Lubricants
– Added to allow easier processing
– “slides” through dies easier – ex: Na stearate
• Colorants
– Dyes or pigments
• Flame Retardants
– Cl/F & B 41
Wednesday, November 17, 2010 Bioactive Contaminants Leach from Disposable
Laboratory Plasticware G. Reid McDonald, Alan L. Hudson, Susan
M. J. Dunn, Haitao You, Glen B. Baker, Randy M. Whittal, Jonathan W.
Martin, Amitabh Jha, Dale E. Edmondson, Andrew Holt
Science 2008, 322, 917 42
Wednesday, November 17, 2010 BisphenolA as endocrine disruptor • BPA is used as a monomer in some polycarbonates, as well as a
plasticizer for many plastics • BPA can mimic effect of estrogen? 43
Wednesday, November 17, 2010 Processing of Plastics
• Thermoplastic –
– can be reversibly cooled & reheated, i.e. recycled
– heat till soft, shape as desired, then cool
– ex: polyethylene, polypropylene, polystyrene, etc. • Thermoset
– when heated forms a network
– degrades (not melts) when heated
– mold the prepolymer then allow further reaction
– ex: urethane, epoxy 44
Wednesday, November 17, 2010 Processing Plastics  Molding
• Compression and transfer molding
– thermoplastic or thermoset Adapted from Fig. 15.23,
Callister 7e. (Fig. 15.23 is from
F.W. Billmeyer, Jr., Textbook of
Polymer Science, 3rd ed.,
John Wiley & Sons, 1984. ) 45
Wednesday, November 17, 2010 Processing Plastics  Molding
• Injection molding
– thermoplastic & some thermosets Adapted from Fig. 15.24,
Callister 7e. (Fig. 15.24 is from
F.W. Billmeyer, Jr., Textbook of
Polymer Science, 2nd edition,
John Wiley & Sons, 1971. ) 46
Wednesday, November 17, 2010 Processing Plastics – Extrusion Adapted from Fig. 15.25,
Callister 7e. (Fig. 15.25 is from
Encyclopædia Britannica, 1997.) 47
Wednesday, November 17, 2010 Polymer Types: Fibers Fibers  length/diameter >100
• Textiles are main use
– Must have high tensile strength
– Usually highly crystalline & highly polar
• Formed by spinning
– ex: extrude polymer through a spinnerette
• Pt plate with 1000’s of holes for nylon
• ex: rayon – dissolved in solvent then pumped through
die head to make fibers
– the fibers are drawn
– leads to highly aligned chains fibrillar structure 48
Wednesday, November 17, 2010 Polymer Types
• Coatings – thin film on surface – i.e. paint, varnish
– To protect item
– Improve appearance
– Electrical insulation • Adhesives – produce bond between two adherands
– Usually bonded by:
1. Secondary bonds
2. Mechanical bonding • Films – blown film extrusion • Foams – gas bubbles in plastic 49
Wednesday, November 17, 2010 BlownFilm Extrusion Adapted from Fig. 15.26, Callister 7e.
(Fig. 15.26 is from Encyclopædia
Britannica, 1997.) 50
Wednesday, November 17, 2010 Next time...
• An important class of viscoelastic
polymeric materials: hydrogels! 51
Wednesday, November 17, 2010 ...
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This note was uploaded on 11/29/2010 for the course BME 104 taught by Professor Kasko during the Fall '10 term at UCLA.
 Fall '10
 KASKO

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