material functions part 3 introduction to the rheology of complex fluids dr aldo acevedo - erc sops1
TRANSCRIPT
Material Functions
Part 3
Introduction to the Rheology of Complex Fluids
Dr Aldo Acevedo - ERC SOPS 1
To find constitutive equations, experiments are performed on materials using standard flows.
Numerous standard flows may be
constructed from the two sets of flows, by varying the functions σ(t) and ε(t) (and b)
stress responses → materials & type of flow
timestrain
strain rate (or other kinematic parameters)chemical nature of the material
functions of the kinematic parameters that characterize the rheological behavior are
material functionsDr Aldo Acevedo - ERC SOPS 2
Material Functions
Definitions of material functions consist of three parts:
1. Choice of flow type
2. Details of the σ(t) and ε(t) (and b) that appear in the definition of the flows.
3. Material function definitions – based on the measured stress quantities
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Material Functions
To predict them: Use the kinematics and const. eq. to predict the stress
components Calculate the material functions
To measure them:1. Impose the kinematics on material in a flow cell2. Measure the stress components
To choose a constitutive equation to describe a material, we need both to measure the material function and to predict it.
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Steady Shear
Kinematics for steady shear 0)( t
constant
Produced in a rheometer by:1. Forcing the fluid through a capillary at a constant rate and
the steady pressure required to maintain the flow is measured.
2. Using cone-and-plate and parallel-plate geometry, rotate at constant angular velocity while measuring the torque generated by the fluid.
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Steady-State Shear
0
21)(
20
332220
22
)()(
N
20
221120
11
)()(
N
Viscosity
For S.S., the stress tensor is constant in time, and the three stress quantities are measured.
The three material functions that are defined are:
First normal-stress coefficient
Second normal stress coefficient
Either + or -, depending on the flow direction and the choice of coordinate system.
Zero-shear viscosity 00 )(lim
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Steady-State Shear
Either + or -, depending on the flow direction and the choice of coordinate system.
shear rate strain stress
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Unsteady Shear
Made in the same geometries as steady shear.
Measured pressures and torques are functions of time.
Many types of time-dependent shear flows: Shear-stress growth Shear-stress relaxation/decay Shear creep Step shear strain Small-amplitude oscillatory shear
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Shear-Stress Growth
Before steady-state is reached, there is a start-up portion of the experiment in which the stress grows from its zero at-rest value to the steady-state value.
This start-up experiment is one time-dependent shear flow experiment
0
00)(
0 t
tt
may be positive or negative
Kinematics for shear-stress growth
no flow initially
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Shear-Stress Growth
0
21),(
t
20
332220
22
)(),(
Nt
Viscosity
The three material functions that are defined are:
20
221120
11
)(),(
NtFirst normal-
stress growth coefficient
Second normal-stress growth
coefficient
Dr Aldo Acevedo - ERC SOPS 11
Shear-Stress Growth
)(),(lim tt
)(),(lim 22 tt
Viscosity
At steady-state these material functions become steady-state functions:
)(),(lim 11 tt
First normal-stress growth
coefficient
Second normal-stress growth
coefficient
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Shear-Stress Decay
Relaxation properties of non-Newtonian fluids may be obtained by observing how the steady-state stresses in shear flow relax when the flow is stopped.
00
0)( 0
t
tt
may be positive or negative
Kinematics for shear-stress decay
no flow initially
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Shear-Stress Decay
0
21),(
t
20
332220
22
)(),(
Nt
Viscosity
The three material functions are analogous to stress growth and are defined as:
20
221120
11
)(),(
NtFirst normal-
stress decay coefficient
Second normal-stress decay
coefficient
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Shear-Stress Decay
Newtonian fluids relax instantaneously when the flows stops.
For many Non-Newtonian fluids relaxation takes a finite amount of time.
The time that characterizes a material’s stress relaxation after deformation is called the relaxation time, λ.
A dimensionless number that is used to characterize the importance of λ is the Deborah number De
flowtDe
material relaxation time
flow time scale
Deborah number can help predict the response of a system to a particular deformation.
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Deborah Number – An unusual interpretation
The prophetess Deborah said:
“The mountains flowed before the Lord” (Judges 5:5)
The interpretation of Prof. Markus Reiner:
Deborah knew two things:
1. Mountains flow as everything flows.
2. But they flowed before the Lord, not before man
Reiner’s interpretation: “Man in his short lifetime cannot see them flowing, while the time of observation of God is infinite.”
Thus, even some solids “flow” if they are observed long enough.
Reiner, M. “The Deborah Number,” Physics Today, pp 62, January, 1964.
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Shear Creep
An alternative way of producing steady shear flow is to drive the flow at constant stress.
Constant driving pressure in a capillary flow, or by driving the fixtures with a constant-torque motor.
The unsteady response to shear flow when a constant stress is imposed is necessarily different from the response when a constant strain rate is imposed.
In the constant-stress experiment, the time-dependent deformation of the sample is measured during the transient flow.
The unsteady shear experiment where the stress is held constant is called creep.
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Shear Creep
The material function will prescribe the stress:
0constant
00)(
021 t
tt
Prescribed stress
function for creep
In creep deformation of a sample is measured, that is how the sample changes shape over some time interval as a result of the imposition of the stress.
To do that the concept of strain must be defined.
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Shear Creep
To measure deformation we use the shear strain.
Strain is a measure of the change of the shape of a fluid particle, that is, how much stretching or contracting a fluid experiences.
Shear strain is denoted by γ21(tref, t) Refers to the strain at time t with respect to the shape of the fluid
particle at some other time (i.e. tref) may be abbreviated as γ21(t), where tref = 0
For short time intervals:
where u1 is the displacement function in the 1-direction2
121 ),(
x
uttref
Shear Strain
(small deformations)
21
Shear Creep
Then the displacement function is:
1233
2
1
1233
2
1
)(
)(
)(
)(
)(
)(
)(
)(
tx
tx
tx
tr
tx
tx
tx
tr
ref
ref
ref
ref
)()(),( refref trtrttu
u1 is just the 1-direction component
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Shear Creep
Physical Interpretation:
2
1
2
121 ),(
x
u
x
uttref
)()(),( refref trtrttu
is the slope of the side deformed particle
Thus, strain is related to the change in shape of a fluid particle in the vicinity of points P1 and P2.
23
Shear Creep
where r is the initial particle position and the velocity is 120 exv
1233
2
1
1233
2
201
1233
2
1
)(
)(
)(
)(
)(
)(
)()(
)(
)(
)(
tx
tx
tx
tr
tx
tx
xtttx
tx
tx
tx
r
ref
ref
refref
For steady shear flow over short time intervals, the particle position vector is:
For steady simple shear flow over a short time interval from 0 to t then, we calculate the displacement function and strain:
tx
ut
xttttu refref
02
121
201
),0(
)(),(
Strain in steady-shear over short interval 24
Shear Creep
where tp = pΔt and Δt = t/N.
),)1((...),(...),(),0(),0( 121121 ttNtttttt pp
The deformation in the creep experiment occurs over a long time interval, and the previous equation is for small deformations is not sufficient for calculating strain in this flow.
However, we can break a large strain into a sequence of N smaller strains:
The steady shear-flow displacement function is given for short time intervals by:
tx
utt
xtttu
pp
pp
02
11
2011
),(
),(
Therefore, for each small-strain interval,
independent of timeDr Aldo Acevedo - ERC SOPS 25
For creep, an unsteady flow, the relationship between γ21(0,t) and the measured shear rate is a bit more complicated since the shear rate varies with time.
The displacement function is the same, however is replaced by the
measured time-dependent shear rate function
Shear Creep
Same results as for short time intervals and it is valid in steady-shear flows.
)(21 t
00
1
012121 ),(),0( ttNttt
N
ppp
The total strain over the entire interval from 0 to t is given by:
0
Now we will consider the general case of strain between two times t1 and t2.
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Shear Creep
The strain for each interval is:
)(),( 1212
1121
ppp ttx
utt
212111
1
)(),(
1....2,1,0
xttttu
Nptptt
ppp
p
Break the interval into N pieces of duration Δt:
With varies with time. Thus, for unsteady shear flow, a large strain between times t1 and t2 is given by:
1
0121
1
01212121 )(),(),(
N
pp
N
ppp tttttt
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Shear Creep
This expression for strain is valid in unsteady shear flows such as creep.
In the limit Δt goes to zero
Shear strain in the creep experiment may be obtained by measuring the instantaneous shear rate as a function of time and integrating it over the time interval.
2
1
')'()(lim),( 21
1
012102121
t
t
N
ppt dtttttt
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Shear Creep
0
210
),0(),(
ttJ
In creep because the stress is prescribed rather than measured, the material functions relate the measured sample deformation (strain) to the prescribed stress.
The creep compliance is:
The creep compliance curve has many features, and several other material functions related to it can be defined.
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Shear Creep
)(),()(
state-steady00
t
tJJ s
Steady-state compliance is defined as the difference between the compliance function at a particular time at steady-state and t/η, the steady-flow contribution to the compliance function at that time:
Creep recovery – when the driving stress is removed, elastic and viscoelastic materials will spring back in the opposite direction to the initial flow direction, and the amount of strain that is recovered is called the steady-state recoverable shear strain or recoil strain.
Sample is constrained such that no recovery takes place in the 2-direction.
t
tr tdtt2
)()(
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Shear Creep
Recoil Function
000 ),(lim)(
)(lim
tRR
t
t
rt
),(),(
)(),(
00
00
tJtR
ttJ
r
rr
Recoverable Creep Compliance
Recoverable Shear
Ultimate recoil function
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Shear Creep
Nonrecoverable shear strain due to steady shear flow
For small stresses (i.e. linear viscoelastic limit) the strain at all times is just the sum of the strain that is recoverable and the strain that is not recoverable (due to steady viscous flow at infinite time)
the shear rate attained at s.s. in creep experiment
t
Advantages to creep flow:• more rapid approach to steady-state• Creep-recovery gives important insight into elastic memory effects• Sometimes materials are sensitive to applied stress levels rather than
shear-rate levels• It is straightforward to determine critical stresses
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Step Shear Strain
One of the interesting properties of polymers and other viscoelastic materials is that they have partial memory (stresses that do not relax immediately but rather decay over time) .
The decay is a kind of memory time or relaxation time for the fluid.
To investigate relaxation time, one of the most commonly employed experiments is the step-strain experiment in shear flow.
constant
00
0
00
lim)(
0
00
t
t
t
tKinematics of step shear strain
The limit expresses that the shearing should occur as rapidly as possible. The condition __ relates to the magnitude of the shear strain imposed.
33
Step Shear Strain
Taking the time derivative and applying Leibnitz rule:
As previously discussed:
For step-strain experiment:
0021
0 0
0
21
21
212121
)(
'0''0)(
)(),(
')'(),(2
1
t
dtdtdtt
tdt
ttd
dtttt
t
ref
t
t
It is called the step-strain experiment: this flow involves a fixed strain applied rapidly to a test sample at time t=0.
Dr Aldo Acevedo - ERC SOPS 34
Step Shear Strain
Where the function multiplying the strain is an asymmetric impulse or delta function
The prescribed shear-rate function in terms of the strain is:
)()(
1)(
and
0
01
00
lim)(
0
01
00
lim)(
0
-
0
00
tt
dtt
t
t
t
t
t
t
t
t
Thus we can write:
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Step Shear Strain
Heaviside step function
The strain can be expressed:
01
00)(
where
)(),(
0
00)(),(
021
0021
t
ttH
tHt
t
ttdtt
t
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Step Shear Strain
Relaxation modulus
The response of a non-Newtonian fluid to the imposition of a step strain is a rapid increase in shear and normal stresses followed by a relaxation of these stresses.
The material functions are based on the idea of modulus rather than viscosity. Modulus is the ratio of stress to strain and is a concept that is quite useful for elastic materials.
20
33220
20
22110
0
0210
)(),(
)(),(
),(),(
2
1
tG
tG
ttG
First normal-stress step shear relaxation modulus
Second normal-stress step shear relaxation modulus
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Step Shear Strain
The second normal stress … modulus is seldom measured since it is small and requires specialized equipment.
For small strain, the moduli are found to be independent of strain, this limit is called the linear viscoelastic regime.
In the linear viscoelastic regime G(t,strain) is written as G(t), and often high strain data are reported relative to G(t) through the use of a material function called the damping function:
)(
),()( 00 tG
tGh
only reported when it is independent of time.
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Small-Amplitude Oscillatory Shear
One of the most common material functions set.
The flow is again shear flow, and the time-dependent shear-rate function used for this flow is periodic (a cosine function).
tt
xt
v
cos)(
0
0
)(
0
123
2
Kinematics for SAOS
Usually done (but not limited to) in parallel plate or cone-and-plate.
frequency (rad/s)constant amplitude of the shear rate function
39
Small-Amplitude Oscillatory Shear
From the strain, the wall motion required to produce SAOS can be calculated.
Small shear strains can be written as
ttt
tdttdtt
h
tbt
x
u
tt
sinsin),0(
cos)(),0(
)(),0(
00
21
0 00 2121
21
2
121
If b(t) is the time-dependent displacement of the upper plate (for example) and h the gap between the plates
And strain can be calculated from the strain rate:
strain amplitudeDr Aldo Acevedo - ERC SOPS 40
Small-Amplitude Oscillatory Shear
Thus the motion of the wall is:
Moving the wall of a shear cell in a sinusoidal manner does not guarantee that the shear-flow velocity profile will be produced, but one can show that a linear velocity profile will be produced for sufficiently low frequencies or high viscosities.
thtb sin)( 0
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Small-Amplitude Oscillatory Shear
ttt
ttt
tt
cos)sin(sin)cos()(
)cossincos(sin)(
)sin()(
0021
021
021
Expanding using trigonometric identities:
When a sample is strained at low strain amplitudes, the shear stress that is produced will be a sine wave of the same frequency as the input strain wave.
The shear stress usually will not be in phase with the input strain.It can be expressed as:
there is a portion of the stress wave that is in phase with the imposed strain (sen) and a portion of the stress wave that is in phase with the strain rate (cos)
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Newtonian fluids: stress is proportional to shear rate.
For elastic materials: •shear stress is proportional to the imposed strain, that is to the deformation
•similar to mechanical springs (which generate stress that is proportional to the change in length)
Small-Amplitude Oscillatory Shear - Significance
2121 GHooke’s law(shear only)
The stress response generated in SAOS has both a Newtonian-like and an elastic part. Thus, SAOS is ideal for probing viscoelastic materials (i.e. materials that show both viscous and elastic properties) 43
Small-Amplitude Oscillatory Shear
SAOS material functions
sin)(
cos)(
cossin
0
0
0
0
0
21
G
G
tGtG
storage modulus
loss modulus
portion of the stress wave that is in phase with the strain wave divided by the amplitude of the strain wave
portion of the stress wave that is out of phase with the strain wave divided by the amplitude of the strain wave
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Small-Amplitude Oscillatory Shear – The Limits
For a Newtonian fluid, the response is completely in phase with the strain rate:
For an elastic solid that follows Hooke’s law (a Hookean solid) , the shear stress response is completely in phase with the strain.
0
0
GGG
GG
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Small-Amplitude Oscillatory Shear
Several other material functions related to G’ and G” are also used by the rheological community, although they contain no information not already present in the two dynamic moduli already defined.
Table 5.1
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Material Functions for Elongational Flow
Based on the velocity field:
Only the stress differences can be measured.
Stress measurements are very challenging to make in elongational geometries.
In many experiments flow birefringence is used. Flow birefringence is an optical property that is proportional to stress.
Measurements of strain are sometimes made by videotaping a marker particle in the flow and analyzing the images using computer software.
123
3
1
1
)(
)1)((2
1
)1)((2
1
xt
xbt
xbt
v
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Steady Elongation
Steady-state elongational flow is produced by choosing the following
kinematics:
For these flows, constant stress differences are measured. The material functions defined are two elongational viscosities based on the measured normal stress-differences.
For both uniaxial and biaxial extension, the elongational viscosity base on 22-11 is zero for all fluids.
0
11330
0
11330
0
)()(
)()(
constant)(
B
t
uniaxial elongational viscosity
biaxial elongational viscosity
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Steady Elongation
Steady-state elongational flow is difficult to achieve because of the rapid
Rate of particle deformation that is required.
Very few reliable data are available for this important flow.
00 ln
)(),(
l
lt
tdtttt
trefref
The strain in elongational flow is defined as:
Integrating:
Hencky strain
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Elongation Stress Growth
Start-up elongational flow has the same problems as steady elongational flow, but some start-up curves have been reported.
The material functions for the startup of steady elongational analogously to shear flow.
Material for stress decay could be defined, but steady-state is seldom reached. Thus it is not very useful.
0
00)(
)(
)1)((2
1
)1)((2
1
0
123
3
2
1
t
tt
xt
xbt
xbt
v
Kinematics of startup of steady uniaxial elongation
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Elongational Creep
If instead of a constant elongational rate, a constant stress is applied to drive the flow, the flow is called elongational creep.
May be obtained by hanging a weight in a cylindrical sample. The deformation of the length (expressed as strain) measured quantity.
The material function is:
00
01133
),0(),(
0constant
00
ttD
t
tKinematics of elongational creep
Elongational creep compliance
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Elongational Creep
An experiment that gives some information about relaxation after elongational deformation is the unconstrained or free recoil experiment.
The material is able to relax in all three directions. The amount contraction that occurs can be expressed as an amount of recoil strain and is an indication of the amount elasticity in the material:
)0(
)(lnl
tlrUltimate recoverable
elongational strain
• L(tinf) is the length of the sample at the time at which the sample is cut free after it has had a chance to relax completely
• L(0) is the length of the sample at the time at which the sample is cut free of the driving mechanism
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Step Elongational Strain
The kinematics are:
constant
0
0
00
lim)(
)(
)1)((2
1
)1)((2
1
00
00
123
3
2
1
e
et
et
t
t
xt
xbt
xbt
v
e
Kinematics of step elongational strain
0 is the magnitude of the elongational strain imposed on the fluid.
short time interval
Dr Aldo Acevedo - ERC SOPS 55
Step Elongational Strain
Uniaxial and biaxial each have one non-zero step elongational modulus,
Planar has two moduli.
By convention , the strain measure is not the simple elongational strain e0, but rather the difference between two components of a strain tensor called the Finger strain tensor C-1.
02
001
00
00
21122
111
122
11220
21133
111
133
11330
21133
111
133
11330
21133
111
133
11330
1
)()(),(
)()(),(
)()(),(
)()(),(
eCCtE
eeCCtE
eeCCtE
eeCCtE
P
P
B
Uniaxial step elongational relaxation modulus
Biaxial step elongational relaxation modulus
Planar step elongational relaxation moduli
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Small-Amplitude Oscillatory Elongation
The kinematics are:
tt
xt
xt
xt
v
cos)(
)(
)(2
1
)(2
1
0
123
3
2
1
Kinematics of SAOE
Similar to SAOS, the deformation rate can be calculated as:
tt
ttdttt
sin),0(
sincos),0(
0
0
0
0
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Small-Amplitude Oscillatory Elongation
The stress tensor is assumed to be of the form:
12311
11
11
200
00
00
For small deformations and small deformation rates, the stresses generated in SAOE flow will be oscillatory functions of time with the same frequency as the input deformation wave.
The stress will be in general out of phase with respect to both deformation and deformation rate.
If we designate d as the phase difference between stress and strain, the 11-component of the stress as:
tt sin)( 011Dr Aldo Acevedo - ERC SOPS 58
Small-Amplitude Oscillatory Elongation
The stress difference on which material functions will be based::
1111111133 32
The material functions for SAOE:
sin3
)(
cos3
)(
cossin
0
0
0
0
0
1133
E
E
tEtESAOE material functions
Elongational storage modulus
Elongational loss modulus
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