Mechanical Properties Polymer_141013
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Transcript of Mechanical Properties Polymer_141013
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Mechanical Properties ofMechanical Properties of
PolymersPolymers
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Deformation of Materialsplastic rubber
Lo
L
L
Effective spring constant of the material: LFk = /Material dependent property (modulus):
0/
/
LL
AFM
=
Youngs modulus (E) specifies resistance of the material to elongations at smalldeformations.
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Dependence of the Modulus vs
Temperature
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Dependence of the Modulus vsTime
Schematic modulus-time curve for a polymer at constant temperature
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Specific-volume vs Temperature
Specific-volume data for poly(vinyl acetate) used to determine
glass transition temperature Tg.
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Phenomenological TreatmentPhenomenological Treatmentofof ViscoelasticityViscoelasticity
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Elastic ModulusStress force per unit area
YZ
F
A
FE ==
Units: dynes/cm2 (dynes per square centimeter), or N/m2 (Newton per square meter)
FF
Y
ZX X
Tensile strain resulted form applicationof uniaxial stress
XX=
Tensile modulus and the tensile compliance:
DE E
1==
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Elastic Modulus
F
FY
Z
X
XXShear deformation
Shear stress:
ZX
F=
Shear strain:
tan=
= YX
Shear modulus G and shear complianceJare defined as
JG
1==
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Deformation at Constant Volume
Initial state
Deformed stateF F
Material stress ij is a stress produced by the deformed material.
x
xxx
AF=
Fx x
y
zAx
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Deformation at Constant VolumeForce balance at the boundary
xxAxpAx
xxAx
pxxxx +=
or rearranging
pxxxx =
pAy
yyAypyy =
yyxxxx =
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Deformation at Constant VolumeHooks law
ijij G =
where ij is a deformation tensor
The definition of the deformation should be consistent with
the shear experiment
i
j
j
iij
xu
xu
+
=
where x1
=x, x2
=y and x3
=z
2=
+
=
x
u
x
u xxxx
Tensile experiment
=
=
+
=
y
u
y
u
y
u yyyyy 2
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Deformation at Constant Volume
Relation between Shear and Youngs modulus
( ) GGG yyxxyyxxxx 3)(2 ====
xxE
E ==
GE 3=
Using the definition of the Youngs modulus
Sample does not change volume!!!
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Poissons Ratio1
1
1
1+ 1-
1-
Volume change upon deformation
)21(1)1)(1( 2 +=VNo volume change when the Poissons ratio =1/2
General relation between Shear and Youngs modulus
)1(2 +
=E
G
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Elastic Properties of Polymers as Compared to Other
Materials (T=300K)
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Understanding the mechanical
response on the molecular levelThe elastic properties of solids (glasses) is a result of the
Intermolecular forces between the atoms.
a
Deformed sample
x
The force acting between atoms at small
deformations is equal to
)( axkF =The area per atom is equal to a2 and thetensile stress is written as
Ea
ax
a
k
a
axk
=
=
=)()(
2
E=k/aYoungs Modulus
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Understanding the mechanical
response on the molecular level
The constant k is obtained by expanding intermolecular potential
around the equilibrium separation a
2
)(...
)()(
2
1)()(
2
2
22 axkconst
dx
xdUaxaUxU
ax
+=++=
=
Thus, we find
axdx
xdUk
=
=2
2)(
For a potential U(x)=f(x/a) which has a minimum value atx=a
)1()( ''
2
int
2
2
fadx
xdUk
ax
==
=
)1(''3
int fa
E
=
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Molecular Rearrangements
Liquid t~ 10-10 sec
Solids t>>100 sec
Polymer
chaint~100 sec
t~ 10-9 sec
T>>Tg
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Linear Viscoelasticity
v
The Newtonian liquid obeys the equation
dt
d
dt
d
==
where is the viscosity of liquid
Viscose Response of a Liquid
Linear viscoelasticity = elastic + viscosev
v
e
e
v
e
e
dt
dG
+=
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Readings:
Ch. 1,2 p.p.1-33, M. T. Swah, W. J. MacKnight, Introduction toPolymer Viscoelasticity.
Ch. 6. p.p. 162-174 D. I. Bower, An Introduction to Polymer Physics
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Creep response of material to a constant stress, .
Strain
Time
Solid
/Geq
Liquid.
Jeq
In these experiments the strain is monitored
as function of time, (t).
The creep complianceJ(t) is defined as
the ratio of the time dependent strain (t)
and the applied stress
)()(
ttJ
Stress-relaxation - response of materials to a constant strain, .
Stres
s
Time
Solid Geq
Liquid
In these experiments the stress is monitored
as function of time, (t).
The stress relaxation modulus G(t) is definedas the ratio of the time dependent stress (t)
and the applied strain
)(
)(
t
tG
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Mechanical Models
of Viscoelastic Materials
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The Maxwell ModelThis model consists of a spring and dashpot in series
s, s, G
d, d, For elements connected in series
For springss
G =
For the dashpotdt
d dd
=
ds == and ds +=
+=
dt
d
Gdt
d 1
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The Maxwell Model
Creep experiment: deformation at constant stress d/dt=0
=+= dtdGdtd 1
t
GtJ
ttt +==+=
1)(
)()(
0
00
Solving the equation with initial conditions at time t=0
This illustrates that the compliance of the Maxwell model increases
without limit, a behavior characteristic for viscoelastic fluids.
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The Maxwell Model
Creep experiment:
(t)
(t)
0
0
Time
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The Maxwell ModelStress relaxation experiment: deformation at constant strain d/dt=0
=+= dt
d
Gdt
d
Gdt
d 11
Solving the equation
G
dtdt
Gd=== where,
==
t
tdt
t exp)(ln))(ln( 00
The stress decays to zero at infinite time!!!
=
==
tG
tttG expexp
)()(
0
0
0
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The Maxwell ModelThe response of the Maxwell model in a stress relaxation
experiments corresponds to an elastic solid at t.
>>
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The Maxwell ModelStress-relaxation experiment:
(t)
(t)
0
G0
Time
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The Voigt ModelThis model consists of a spring and dashpot in parallel
s, s, G
d, d,
For elements connected in parallels
For springss G =
For the dashpotdt
d dd
=
ds += and ds ==
dt
dG
+=
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The Voigt ModelCreep experiment: deformation at constant stress =0
dt
dG
+=0
0 Solution of this equation is
Gwhere,exp1)( 0 =
=t
Gt is retardation time
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The Voigt ModelCreep experiment:
(t)
(t)
0/G
0
Time
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The Voigt ModelStress relaxation experiment: deformation at constant strain d/dt=0
dtdG +=
G=
This model can not describe stress relaxation !!!!
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More Complicated Models
Generalized Maxwell Model Voigt-Kelvin Model
..
{Gi, i}
{Gi, i}
These models describe systems with
multiple relaxation times.
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Generalized Maxwell Can
Describe Relaxation Process in Real Polymers
Th B lt S iti
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The Boltzmann Superposition
PrincipleThe strain from any combination of small step stresses is the linear
combination of the strains resulting from each individual step
i
applied at time ti
(t)
(t)
1
2
2J(t-t2)t1 t2
1J(t-t1)
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The Boltzmann Superposition
Principle
For the multi-step loading program:
..)()()()( 332211 +++= ttJttJttJt
For continuously changing stress:
s
s
s
t
ss
t
s dtdt
tdttJtdttJt )()()()()(
==
For the strain experiments
ss
s
t
ss
t
s
dtdt
tdttGtdttGt
)()()()()(
==
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Creep and Recovery
(t)
(t)
0
t1 t2
r
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Dynamic Measurements
Elastic solid:
A sinusoidal strain with angular frequency
( )tt sin)( 0=
)sin()()( 0 tGtGt ==
The stress is perfectly in phase with the strain for Hookean solid.
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Dynamic Measurements
Newtonian liquid:
)2sin()cos(
)(
)( 00
+=== ttdttd
t
The stress is out of phase with the strain for Newtonian liquid.
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Dynamic MeasurementsIn general, the linear response of the viscoelastic materials always
has stress oscillate at the same frequency as the applied strain, but
the stress leads strain by aphase angle
)sin()( 0 += tt
The stress can be separated into two orthogonal functions that
oscillate with the same frequency, one in phase with the strain and
the other out-of-phase with the strain by /2
)cos()()sin()()( "'0 tGtGt +=
G() is the storage modulus G() is the loss modulusExamples:
Elastic solids: G() =G and G()=0
Newtonian liquids: G() =0 and G()=
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Dynamic MeasurementsWe can rewrite storage and loss modulus in terms of phase angle
)sin()cos()sin()cos()sin( ttt +=+
)cos()(0
0'
=G
Using
and
[ ])sin()cos()sin()cos()sin()( 00 tttt +=+=
One obtains
)sin()(0
0"
=G
)(
)(
)tan( '
"
G
G
=
)cos()()sin()()( "'0 tGtGt +=
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Application to the Maxwell Model
+=
dt
d
Gdt
d 1
)sin()( 0 += tt( )tt sin)( 0=For the time dependent strain and stress
Substitution to the Maxwell model
results in
)sin()cos(
1)cos( 000
+++=
tt
Gt
Collecting terms at cos(t) and sin (t) one has
)sin()cos(
1 000 +=
G
)cos()sin(
10 00 +=
G
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Application to the Maxwell Model
)()(
"' GG
G+=
)()(0
'" GG
G+=
Rewriting the last equations in terms of loss and storage modulus
Solving this system of equations for G and G one arrives at
22
22'
1)(
+= GG
22
"
1
)(
+
= GG
1
)(
)()(tan
'
"
==G
G
G
G
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Readings:
Ch. 3 p.p.51-66, M. T. Swah, W. J. MacKnight, Introduction toPolymer Viscoelasticity.
Ch. 7. p.p. 187-204 D. I. Bower, An Introduction to Polymer Physics