Embed Size (px)
Citation preview
Practice Theory Standards Research
A.C. Pronk
3rd 4PB Workshop
Davis, CA, 17/18 September 2012
Conflicts & Compromises
30 0
4
k + 1
24k = 1 0
2 2 k - 1
L 2 sin ( t )F LV ( , t) =
2 E I
Asin ( 2 k 1 )
L ( )
( 2 k 1 )
. . . .
. .
. . .
1 .
. . 1
Centre Deflection (no extra masses etc.)
1st order approximation 3
00
4 20
21
x A2 sin ( ) sin ( )F L sin ( t )L LV ( x , t ) =
E I 1
3 2
2
3/ 2
12 41
00
20
21
LF A A sin ( t )V ( L , t )
E I L L -
Modified 1st order approximation
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60
Hz
No Mass Correction
"Exact" Mass Correction
Virtual Extra Mass Correction
Influence of Moving Masses
• In general !
• 0 – 10 Hz: Moving masses can be ignored
• 10 – 30 Hz: It’s sufficient to know the moving masses
• > 30 Hz: A virtual mass has to be added in order to
use the 1st order approximation only.
The value has to be obtained in calibration tests.
Consequences for neglecting Shear
0
2
4
6
8
10
0.00 0.05 0.10 0.15 0.20
VS/V
B[%
]
H/L
Ratio VS/VB 0,15 0,25 0,35a = 0,85
From comparison with 1D & 3D finite element results it was found that a = 0,85 is a more appropriate value. The differences, if a value 2/3 is used, are small
Shear Force • If the ratio of the height H over the length L is
not small ( H/L > 0.05) the shear force in the cross section of the beam will attribute to the total deflection Vt = Vbending + Vshear.
HL L
Vs VbL A
2
2 2
4 1
2 23 4
a
a 0.85 0.667 is valid for a plate)
Homogeneity
• From a research point of view the
smallest dimension of a specimen should
be at least 10 times the maximum grain
size.
• Compromise in the Euiropean standard:
3 times
Greatly exaggerated bending of the
beam due to its own weight
Supports
“”Creep””
• Due to its own weight it is possible that, specially at high temperature and low frequencies, permanent deformation will occur in the beam between the supports.
• For such conditions a shorter beam has to be used but than deflection due to shear forces must be taken into account.
4PB device
Boundary conditions • When the beam is bended upwards the “neutral” line
(halfway the beam) ought to stay constant in length. This means that the 4 supports have to move inwards to the centre.
• However, the same yields when the beam is bended downwards.
• So, (in theory) the horizontal movement of a support will be a repeated half sine which from a mechanical point of view is not attractive at all.
Horizontal translations of the supports
Horizontal Translation
Tim
e
Clamping
• In order to avoid “rattling” of the beams in the clamps, a clamping force is applied.
• Nevertheless, the clamping force should be as low as possible.
Order of 10 mm
Frame stiffness Kf [N/m]
Beam stiffness Kb [N/m]
Total Deflection = Vf + Vb = Vt =
Force/Kf + Force/Kb >
1/Kf = Vt/Force – 1/Kb
Increasing phase lag with frequency
3 Aluminum Reference Beams
Standarization - Research
• CEN Standard > National Standard > Type testing : Characterization of a material property
• At the moment some properties, which are determined and defined according to the CEN standard but with different devices, are different
• Changes in CEN Standard take a long time because consensus has to be reached?
CEN Smix
• The CEN standard describes the procedure and
protocols for the determination of a stiffness
modulus. So, you can perform frequency sweeps at
different temperatures, leading to a mastercurve.
• However, for the national application, type testing of
a specific asphalt mix, only one value at a certain
frequency and temperature is used for the
characterization of the mix.
- k - h
2i + . ( i ) )
o
o
1 1 2
- E EE ( ) = + E
1 + . (
The Huet-Sayegh model consists out 4 elements: two springs and two parabolic dashpots. A linear dashpot can be added in series for the simulation of linear creep (used in VEROAD) Also often the spring Eo can be deleted and stil a good comparison between data and model can be found.
S.sin(j)
j = /2
j = a./2
S.cos(j) s h.a – 1.Wa{e(t)}
The Parabolic Dashpot is a rheological element between a linear spring (complete elastic: X axis) and a linear dashpot (complete viscous: Y axis)
The response on a block load is given by:
.
. ( / ) ( / ). { }
1 a
a a0
0 0t t 2 t t 2
1 a
s e
h
CEN Fatigue
• The CEN standard gives the procedures and protocols for the determination of fatigue lives (at the moment Nf,50 is still used for the definition of fatigue life) including Wöhler curve etc.
• But again in the national application for type testing of an asphalt mix, only the strain value is required for which 106 repetitions can be performed at 30 Hz and 20 oC.
Wöhler curve
Log(e)
Log(N) Low Endurance Limit
Lissajous Force
Deflection
Total Dissipated
Energy per cycle
Dissipated Energy
per cycle used for
Fatigue Damage
Nevertheless the deviations between fatigue lives determined in different bending tests become smaller if the definition N1
0
2000
4000
6000
8000
10000
12000
0 20000 40000 60000 80000 100000 120000 140000 160000
N [-]
Ed
yn
[M
Pa
]
Change in curvature
Nf,50
Traditional determination of fatigue life (Nf,50)
The first part of the curve
can be fitted very well by:
DW=a.Nb
Original definition
Deviation of straight line
Possible definition for standard
Crossing of two lines
Strain/Deflection Controlled Fatigue Tests
N
SDWi / DWn
Stress/Force Controlled Fatigue Tests
A C
B
Force
Deflection
Fatigue Damage
Lissajous figures Loop B will become loop C
Fatigue damage for loop A and loop C are the same
1. Fatigue damage is related to the distortion
energy (deformation of material)
DD :: C.Dwdistortion
2. If the dilatation per cycle is positive (DV > 0)
than C > 1; For DV < 0 than C = 1.
3. Both at the top and bottom of an asphalt layer
the distortion energy per load passage will be
high and of the same order.
4. At the bottom of the asphalt layer DV > 0
At the top of the asphalt layer DV < 0
5. Thin asphalt pavement: Cracks bottom up
Thick asphalt pavement: Cracks top down
A
B W p
O r i g i n a l n e u t r a l p o s i t i o n
N e w n e u t r a l p o s i t i o n a f t e r b e n d i n g
HEALING
Healing is defined as the ratio of the number of
load repetitions in a discontinuous fatigue test
and a continuous fatigue test.
There is no consensus how the discontinuous test
has to be performed. 1 load repetition followed
by 10 rest periods or should it be 1000 and
10.000 cycles.
It’s difficult to apply just one sine
Future
Development of Fatigue Models like
the Modified Partial Healing Model
Application of Finite Element Models
Fatigue Life Definition leading to
comparable values for 2PB, 3PB, 4PB
and UPP or T/C tests
1 1
{ } ( ){ }
0
t dQ tF t F e d
o d
a
2 2
{ } ( ){ }
0
t dQ tG t G e d
o d
a
“Healing” during testing
Probably a good
mathematical expression
for the thixotropic
behaviour
“Permanent” damage
during continuous
testing
0
2000
4000
6000
8000
10000
0 80000 160000 240000
Number of Load cycles
Sti
ffn
ess
mo
du
lus
[MP
a]
Discontuous Continuous
a 1 = 500
a 2 = 1000
1 = 0
2 = 0
= 5000
If there is no permanent fatigue damage (=0) an equilibrium
will be reached in a continuous test and complete repair takes
place after a rest period (if long enough; in this example N
loading is 40.000 and N rest is 400.000)
0
2000
4000
6000
8000
10000
12000
14000
0 200000 400000 600000
Cycles
Sm
ix [
MP
a]
15
20
25
30
Ph
ase
lag
[o
]
0
2000
4000
6000
8000
10000
12000
14000
0 200000 400000 600000
CyclesS
mix
[M
Pa]
15
20
25
30
Ph
ase l
ag
[o
]
Left: No permanent damage Equilibrium
Right: No (Partial) Healing Monotonic change
1* = 0.46 106 (e - 88 10-6 )
R2 = 0.85
2* = 1.365 106 (e - 74 10-6 )
R2 = 0.998
0
20
40
60
80
100
120
140
50 70 90 110 130 150 170 190
PH
param
eters
1*
&
2*
Strain amplitude [m/m]
1* 2
*
Indication of Low Endurance Limit
* 2
1,2 1,2 0
0( )
it
f e
e e endurance lim
Fictive fatigue life N1 as a function of the applied strain and
a power fit on the interval N1 = 100 to N1 = 1,000,000 cycles.
N = 2.34 1012 e -3.22
R2 = 0.999
N .e2.(e-eendurance)=C
R2 = 1
All 9 data points
were generated with
• This will lead to a Wöhler curve fit with
an exponent m > 2+k > 3+
, endurance limit
,
constant . .
constant . .
k2
1 2
2 l
1 2
with k 1 and l 1
e e e
a e e
• Healing
• How to determine it?
• How to characterize it?
Nevertheless the deviations between fatigue lives determined in different bending tests become smaller if the definition N1
Beam 26-02
2000
3000
4000
5000
6000
7000
8000
0 40000 80000 120000 160000
Number of Load cycles
Sti
ffn
ess
mo
du
lus
[MP
a]
Measured "Predicted"
Load periods: 40.000 Rest periods: 400.000
Not explained by MPH model
“Real Healing” ??
Smix (UPV) = 0.37 Smix (4PB) + 20,000 [MPa]R2 = 0,96
21000
22000
23000
2000 3000 4000 5000 6000 7000 8000
Sti
ffn
ers
s m
od
ulu
s U
PV
[M
Pa]
Stiffness modulus 4PB [MPa]
Beam 03026-02
B
Receiver Sender Beam
B
Receiver Sender Beam
The first resonance frequency is determined by the
dimensions of the beam and the stiffness modulus
Discontinuous Fatigue tests but instead of
real Rest Periods (no load), Pseudo-Rest
Periods are applied using a strain level
below the low endurance limit.
This allows the measurement of the
recovery in the complex stiffness modulus
during a pseudo rest period . Using the PH
model the contribution due to thixotropic
behaviour can be determined
Evolution of Smix in “Rest” Blocks
Strain = 80 m/m
Evolution of Smix in “Rest” Blocks
Strain = 40 m/m
NPH = 8.36 1013
e- 3.9
R2 = 0.75 (UPP + 4PB)
10
100
1000
1.00E+04 1.00E+05 1.00E+06 1.00E+07Cycles
Str
ain
[
m/m
]
4PB data UPP data
NPH = 8.36 1013
e- 3.9
R2 = 0.75 (UPP + 4PB)
10
100
1000
1.00E+04 1.00E+05 1.00E+06 1.00E+07Cycles
Str
ain [
m/m
]
4PB data UPP data
Dissipated Energy Ratio
N
NPH N1
Data
Straight Line N1
MPH model curve
E13D 180-17
0
2000
4000
6000
8000
10000
12000
14000
0 20000 40000 60000 80000 100000 120000
N
Sti
ffn
es
s [
Mp
a]
0,0
0,5
1,0
1,5
2,0
2,5
Te
mp
era
ture
Inc
rea
se
[o
C]
Measured Smix Calculated Temperature increase
Calculated decrease in Smix due to
Temperature
• Thank you for your attention and
patience
• Questions?
Load Period: 10.000 cycles
Pseudo-Rest Period: 2.000 cycles
Measure in Pseudo-Rest Period
First
Last
First Measure in Load Period
“Normal” Behaviour
Surface Damage ?
Controlled Deflection Mode
Strain: 160 m/m
Strain: 40 m/m
a2* = 6,15 10
-6 e
R2 = 0,83
400
600
800
1000
1200
50 70 90 110 130 150 170 190
Strain amplitude [m/m]
PH
para
mete
r a
2*
A.C. Pronk & A.A.A. Molenaar
11th ISAP Conference, Nagoya, Japan, 1-6 August 2010
* *
1 1
0
* *
2 2
0
2 2
* 2
1,2 1,2 0 1,2
0
0
;
;
t
t
mix
t
t
mix
dis dis
dis
dis mix
dQS t Sin t F t F e d
d
dQS t Cos t G t G e d
d
dQ dW Wf W
d d T
W S Sin F
f f
j a
j a
e j e
a a e
DD
D
D
Controlled strain mode:
* 2
1,2 0e
Basic equations of the (modified) Partial Healing model
Thank you John,
Dear audience,
Slide 1: Title
The title of my keynote is: “Theory-Practice &
Standardization – Research”: Conflicts and
Compromises. In this keynote I will give examples of
conflicts between Theory and Practice and the chosen
compromises. There are also conflicts between the
protocols and procedures used in standards and the
results obtained from research. It takes a long time
before research results are embedded in the standards
and often compromises have to be made. I will not state
conclusions etc, my intention is just to show items and
problems which can be discussed during this workshop.
Slide 2: 4PB logo
The 4PB test looks very simple and easy to handle but
in my view there are many conflicts between the
underlying theory for a test method like the 4PB test and
the realisation of the test method in practice. I will
enlighten some examples and how one can deal with it
in practice. As I said before you may have conflicts
between the procedures and protocols needed for
standardization and harmonization of the test at one
hand and research findings at the other hand. It takes a
long time before research results are implemented in
standards which are used in daily practice for
characterizing asphalt mixes. You have to make
compromises in order to minimize the gap between
research findings and daily practice using standard
protocols.
Slide 3: Solution as an infinite series
Let’s have a closer look at the underlying theory for the
4PB test. This slide shows the complete solution for
cycling and pure bending of a slender beam. As you can
imagine this is not a practical expression. So, here we
have to search for an alternative.
The basic theory is partly based on the pseudo-static
bending of a thin slender beam which is characterized
by the product of moment I times the modulus E. In
order to incorporate cyclic bending at low frequencies
and an easy to handle back calculation formula a first
order approximation is adopted.
Slide 4: First order approximation
It’s a first order approximation because only the first
term is taken of the exact solution which consists out of
an infinite sum of terms. For clarity I have left out the
terms connected to mass inertia forces. So here already
a compromise is made because an infinite series for
daily practice is not done. Furthermore instead of the
coefficient for the first term a modification is used
based on the solution for the pseudo static test. In that
case the modified first order approximation will become
equal to the pseudo-static solution for a frequency of nil
Hz.
Slide 5: Virtual masses
The approximation contains also a mass inertia term due
to the moving masses between load cell and beam
during cyclic bending.
Fortunately in 4PB tests moving masses have only a
small effect for frequencies below 10 Hz. But above 10
Hz the influence increases fast. The effect is only
partially corrected for by the weighted sum of all real
moving masses. Especially above 30 Hz one has to use
an extra virtual mass in order to back calculate the
correct modulus using the modified first order
approximation. This can only be achieved by carrying
out calibration tests with elastic beam.
Slide 6: Moving mass effects
So, in general no mass correction is needed for
frequencies up to 10 Hz. By taking into account the real
value for the moving masses no extra corrections are
needed up to 30 Hz. Above 30 Hz a virtual mass has to
be introduced in order to back calculate the right E
value with the first order approximation.
Slide 7: Shear force.
Of course in our tests we don’t have real slender beams.
The slenderness of a beam is characterized by the ratio
of the height H over the length L. If this ratio is below
0.05 the contribution of the deflection due to shear can
be neglected. However, in practice the ratio will be
higher and as a consequence the shear deflection can’t
be neglected. The contribution is still small in the order
of 3-5 percent. But this deflection doesn’t contribute to
the horizontal strain. So, without this correction you will
overestimate the horizontal strain with 3-5 percent.
Slide 8: Shear formula
Unfortunately the formula for the shear deflection
contains an unknown factor alpha which is related to the
unknown distribution of the shear force over a cross
section. Several expressions are known from literature.
Based on finite element calculations Rien Huurman and
I showed at the 2nd 4PB workshop in Guimares that the
correct value is around 0.85. The value of 0.85 is also
used in the 2D version of the Finite Element Program
ABAQUS. The value of 0.85 is in contrast with the
value of 0.67 used in some programs but the difference
is small.
Slide 9: Slenderness
There are also conflicts or problems related to the
geometrical dimension of the beam. At one hand you
have the required geometrical slenderness for a beam
leading to a ratio of H/L < 0.05 but at the other hand
you have to deal with the non homogeneity of the
material. Based on experience we know that from a
research point of view the smallest geometrical distance
(height H and width B) should be at least 10 times
bigger than the maximum grain size in the asphalt mix.
Given the fact that the maximum grain size is
sometimes in the order of 20 mm or even more, this
means a height and width of 200 mm leading to a length
of 4m for the beam. In Europe we decided that for
practical reasons this ratio has to be 3.
Slide 10: Homogenuity
So with a height and width of 50 mm, asphalt mixes
with a maximum grain size of 16 mm can be tested
according to the standard. If one would like to test
mixes with a maximum grain size of for example 25
mm he/she has to perform tests with beams of 75 mm in
height and width and he/she might encounter problems
like creep. Specially in the case of a small value for the
ratio H over L.
Slide 11: Creep
It’s tempting to use long beams with a length of 1 m. Of
course you can in general neglect shear deflection for
that length. However, you might hit from the rain in the
drops. Especially at low frequencies and high
temperatures the beam might deform between the 4
supports due to its own weight. And your formulas for
bending of a straight beam are no longer valid.
Slide 12: Creep 2
If you encounter such a problem (large maximum grain
size, low frequency and high temperatures) it will be
better to use short beams and take into account the
deflection due to shear.
Slide 13: Boundary conditions
In the solution of the differential equations it’s assumed
that all supports should have free rotation freedom and
horizontal translation freedom. In the different devices
this is achieved in different ways.
Slide 14: Movement text
However, no one, including me, really took a look at the
required horizontal movements of the supports during
cyclic bending. The theory of a slender bending beam is
based on the assumption that the neutral line will stay
constant during cyclic bending.
Slide 15: Movement supports plaatje
This can only be achieved if all supports go inwards
during bending of the beam upwards but also during
bending of the beam downwards. From a mechanical
point of view this is terrible. You approach the zero
position at the highest speed and have to turn
immediately back ward with the same speed. Given this
engineering conflict I changed my mind and decided to
favour the Haversine mode in which this conflict
doesn’t appear. However, for me a Haversine test will
be still a pure sine test but on a pre-bended beam. So, in
principle you are not testing a straight beam for which
the formulas are valid. However, I think the introduced
errors are very small. It could be checked with FE
calculations. By the way Haversine testing and the
interpretation will be one of the items for discussion
tomorrow.
It might also be of interest if you want to increase the
range of frequencies. I think it would be easier to
increase the allowable frequencies if a Haversine mode
is used.
Slide 16: Clamping
In theory the beam rest on supports with an infinite
small contact area. In practice the contact area have
substantial dimensions. The width is equal to the width
of the beam and the length is in the order of 10 mm.
Because the boundary condition between clamp and
beam is not known this effect is uncertain but will not
be large. The deviation by the outer support on the
central deflection can be neglected but this not the case
for the inner support. The same yields for the clamping
force. In theory you don’t have a clamping force but in
practice the beam has to be clamped for example by
springs or servo motors.
In general the clamping forces are taken as minimal as
possible just to avoid the rattle of the beam in the
clamps. However, this force level might depend on the
modulus/stiffness of the beam. For the moment I think
that with FE modelling it might be possible to obtain an
answer.
Slide 17: Finite frame stiffness
In the theory it is assumed that no vertical deflections
occur at the two outer supports. It’s assumed that the
main frame has an infinite stiffness. Well in practice this
is not true and due to the load the outer supports may
deform a bit. If you use a relative deflection measure, so
measuring the deflection with reference to supports
resting on the beam than in principle the non infinite
stiffness of the main frame will not influence the test.
But if you measure the deflection absolutely, so
measuring with reference to a move less point you have
to carry out a correction procedure if the frame is not
stiff enough. The simplest one is to simulate the whole
device by two springs in series and use a heavy stiff
elastic steel beam. Sometimes it appears to be that the
substituting spring for the main frame depends on the
applied force.
Slide 18: Phase lag
In case of an elastic material the phase lag between
force and deflection ought to be nil. But in practice, in
spite of all efforts and electronic tools often a linear
increasing or decreasing phase lag with frequency is
obtained. In my view this has to be corrected for
example by software routines before you enter the back
calculation procedures with the measured data. If the
increase or decrease is not linear and also the equivalent
frame spring is not a constant it might be worthwhile to
stiffen the frame or to use a spring/dashpot simulation
for the frame stiffness.
Slide 19 Standards - Research
Now I will talk about the conflicts between standards
and research. Standards are needed for the
characterization of the tested material. In the standards
protocols, procedures and tests are given for the
determination of the parameter or value which
describes a certain property of the material. As a
researcher I like the standards because it is the reference
for exchange of data. However, it takes a long time
before changes in procedures, protocols and definitions
are implemented in the standard. But the European
standards are also used for the national standards and
the type testing of materials. Here I will only talk about
the standards as used in Europe for the stiffness
modulus and fatigue properties. Starting with the
stiffness modulus.
Slide 20 Text van uit CEN
In Europe only the stiffness modulus at a certain
temperature and at a certain frequency is prescribed in
the national standards. For research one will be more
interested in the master curve which contains much
more information. However, in the national standards
the master curve is only optional. It will take a long time
before a procedure will be adopted leading to a property
which contains the whole stiffness information.
Slide 21: MHS
In my view a nice compromise would be the
application of a modified Huet-Sayegh model which has
proven to be an excellent rheological model for
describing the master curve. In contrast with other
rheological models, the Huet-Sayegh model has two
parabolic dashpots of which the response is not well
known.
Slide 22: Parabolic dashpot
The parabolic dashpot can be seen as an rheological
element between the elastic spring and the linear
dashpot. You have to do the mathematics in Fourier
space but for simple load signals the response in time
domain can be calculated directly.
But instead of one Smix value you get 6 parameters
when the Huet-Sayegh model is used. For the standard,
which describes the procedures and protocols this is not
an essential problem but the standards are afterwards
used for the type testing of the materials. And in type
testing you want to have only 1 or 2 parameters for the
characterization of a mix property.
Slide 23 CEN Fatigue
The next item is the protocol for the characterization of
the fatigue properties. In the European standards again
only a single value is chosen for the characterization of
the fatigue properties. This value is the strain value for
which the material can wither stand one million of load
repetitions in a cyclic fatigue tests in constant deflection
mode at a frequency of 30 Hz and a temperature of 20
oC.
Slide 24: Wohler
For a contractor this is an easy way to handle procedure
for the type testing of his product. But for me as a
researcher and also for pavement designers it is not
sufficient enough. I want to know the slope of the
fatigue curve, the existence of a possible low endurance
limit and the Wohler curves for at least 2 temperatures.
Slide 25: Energy loops
In the last decade it became clear that fatigue is more
related to the dissipated energy than to only the strain
amplitude. On this slide you see the Lissajous figure of
the applied force and measured deflection during one
cycle. The area within the loop is the dissipated energy
per cycle. The vast majority of this dissipated energy is
transformed into heat, which by the way will also lower
the stiffness modulus during a fatigue test. Many
researchers now assume that a small part of this energy
is used for fatigue damage. From this point of view the
fatigue characteristics ought to be related to dissipated
energy.
Slide 26: Fatigue life definitions
The first step that had to be taken for such a change in
thinking is the fatigue life definition. Traditional the
fatigue life of an asphalt mix was defined as the number
of load repetitions at which the stiffness modulus was
decreased to half its initial value. The initial stiffness
modulus was defined as the modulus value for the 100th
cycle. However, it is found in many research projects
that the fatigue lives defined in this way differ for the
same mix if the mix was tested using different devices.
So, in principle this definition is not very useful for
standardization. In the past Piet Hopman and I proposed
a new definition which is based on a change in the
dissipated energy per cycle.
Slide 27: N1
We plotted the ratio of the dissipated energy in cycle n
and the accumulated dissipated energy up to cycle n.
The underlying thought was that the change or deviation
from a straight line reflected the point at which
somewhere in the beam micro defects start to change in
macro cracks. So, the end of the crack initiation phase.
This concept is nowadays used by many researchers.
However, I want to emphasize that the evolution of this
dissipated energy ratio is just a visualization of the
process. You get a straight line if the dissipated energy
can be fit by a power function. Nevertheless, the
deviations between fatigue lives determined in different
bending tests become smaller if the definition N1 is
used.
It might well be that the “”correct”” fatigue life is
greater than N1 because the correct function which
describes the evolution in the dissipated energy is not
known yet. Moreover it is rather very subjective where
the straight line starts to deviate from the measured data.
So it looks attractive to define N1 in the following way:
As the intercept or crossing of two straight lines for
strain controlled tests and force controlled tests as
indicated in this slide for strain controlled tests by point
A. From the point of view with respect to
standardization the procedure to determine the fatigue
life is more objective and less user dependent.
Slide 28 N1 Force controlled
For stress controlled tests the inverse ratio has to be
used and the second line becomes horizontal just hitting
the maximum ratio.
Slide 29: Loop dissipated energy
I like to go back to the dissipated energy per cycle in
relation to the application of a Haversine deflection
signal. On this slide loop A is the dissipated energy per
cycle of pure sine signals for the load and the
deflection. It is symmetrical around the origin. When
you start a fatigue test with a Haversine signal for both
the force and deflection, in this slide loop B, you will
find that very, very soon after the start the force signal
changes into a pure sine signal. So, loop B is transferred
to the position of loop C. It is often assumed that the
part of the dissipated energy per cycle which is related
to fatigue damage depends on the strain amplitude. How
higher the strain, how more fatigue damage. So, one
could argue that the fatigue damage for loop C has to be
bigger than the fatigue damage for loop A. However,
fatigue tests showed that the fatigue lives in both tests
are the same. This leads in my view to the conclusion
that in case of loop C one is testing directly from the
start on a pre bended beam as indicated on this slide.
Before I will deal with this, I want to make another
remark which maybe will become an issue for
discussion for the sessions of tomorrow.
Slide 30 Hypothese
I think that fatigue damage is related to the distortion of
the material during a loading. So, the shear modulus G
becomes important. If the dilatation is positive (volume
increase) the damage might be higher.
In my view this already can explain the phenomenon of
surface cracking for thick asphalt layers. The material is
also distorted at the surface but the dilatation is
negative. But still the material gets fatigued without
cracking. For crack growth etc you need tensile stresses
which can occur in the contact area between wheel and
pavement or by temperature induced stresses. I think
this might be a nice topic for tomorrow or at the dinner
tonight.
Now returning to the Haversine and the pre-bended
beam
Slide 31: Pre-bended beam.
Because in the start you have a Haversine load creep or
permanent deformation will occur leading to a new
position of the neutral line. When the load is removed
you will end up with a bended beam. However, ending
the test should be performed in such away that the net
force is nil. This is hard to realize in practice. In my
view adopting viscous permanent bending of the beam
explains why the force in such a test changes
immediately from a Haversine signal into a pure
sinusoidal signal.
Slide 32 Healing
Another important aspect is the healing capacity of a
mix. At the moment this is not in the European
standards and it will take a long time because there is no
consensus how healing has to be measured and how to
characterize it. In The Netherlands a contractor can
develop a new mix but he has to show/proof what the
healing capacity is otherwise this value will be only 1 in
contrast with accepted mixes from the past for which
this factor is 4. But there is no agreement on how you
should determine the healing factor.
This might be also a good issue for tomorrow because
everyone want to avoid the time consuming
discontinuous tests.
Slide 33 Future
On this slide I have put three items but there are many
more such as Healing. I will not talk today on the item
of Finite Element modelling because later on I will
present a paper aimed at this application. The third item
is a good discussion item for the workshops on Tuesday.
Instead I will according to a Dutch saying, but maybe it
is the same in English, preach to the converted. I hope
this is the correct expression. I want to end my keynote
with a short overview of the possibilities of the
modified partial healing model which I have developed
in the last years.
Slide 34: MPH model
Don’t be afraid of the expressions. The first expression
describes the evolution of the loss modulus and the
second one the evolution of the storage modulus in a
continuous fatigue tests. The expression contains a
reversible part indicated by the parameters alpha and
beta and an irreversible part indicated by the parameter
gamma.
In the past I thought the reversible part could be seen as
partial healing which already takes place during a
continuous test. But now I’m sure that it is more a rather
good description for the thixotropic behaviour of
asphalt. You can compare it with the viscosity of
yoghurt. When you stir the yoghurt the viscosity will
drop but when you stop stirring the viscosity will in the
end return to its original value.
Slide 35: partial healing
On this slide you can see that in case the parameter
gamma is nil, an equilibrium will be reached in a
continuous fatigue test and that complete recovery of
stiffness and phase lag occur if the rest periods in a
discontinuous test are big enough.
Slide 36; Verschil in damage
This slide shows at the left the evolution in stiffness
modulus and phase lag if no permanent fatigue damage
occur. It means also that if a strain level is applied
below the low endurance limit, the stiffness and phase
lag may decrease and increase due to thixotropy but
fatigue will not occur (in theory). The figure at the right
show the decrease in stiffness and increase in phase lag
if no thixotropy is present.
Slide 37: Parameters gamma
For real fatigue damage the parameters of the
irreversible part are more important. Because it is
related to energy the parameters are at least depending
on the square of the applied strain.
It turned out that these two parameters can be written as
shown on this slide
Slide 38: Gamma = Strain square times (Strain –
Endurance limit)
So, I think it will be possible to determine the endurance
limit (the strain value for which no fatigue damage
occurs) from continuous fatigue tests at normal strain
levels. In this way you can avoid the time consuming
fatigue tests at low strain levels.
Another in my view very interesting point will be the
accumulated summation of the irreversible fatigue
damage up to the fatigue life. I think, or better I hope,
that this summation turns out to be a constant which can
be used as a characterization for fatigue: Fatigue
strength.
By the way the introduction of the endurance limit will
not lead to a conflict with the commonly accepted
Wöhler curves, as I will demonstrate with the two
following slides.
Slide 39: Wohler Endurance 1A
On this slide I have plotted a few virtual test data. You
see that it can be well fitted by the so called Wohler
curve in the interval from 100 to 1 million load
repetitions . So, I only used the 5 data points on this
interval. The power exponent is 3.22 what is not an
unusual value for these curves.
But in fact these data were calculated using the
following expression:
Slide 40: Wohler Endurance 1B
So, I think that the accumulation of the irreversible
damage par might be a constant. You might call it the
fatigue strength. You may argue that often you find
power coefficients above 4 for fitting fatigue data. But
given the limited data I had I could only drawn a linear
relationship for the gamma coefficients.. It might well
be that in general the expression for the irreversible
damage would be:
Slide 41: e2*(e-elim)k
The same yields for the reversible part. Also in many
procedures the fatigue life is directly related to the
decrease in stiffness modulus. And although the
reversible part doesn’t contribute to real fatigue damage
it will lower the stiffness modulus. I’m retired now but I
think this model has a big potential and is worthwhile
for further investigation
Slide 42: Healing
Finally I want to talk a bit on the possibilities to
determine and characterize healing
In my view real healing, so repair of fatigue damage,
will only occur if real rest periods or periods without
fatigue damage are applied.
Slide 43: Data Discontinue
Here you see the data of a discontinuous fatigue test.
The red curves are the predictions of the PH model for
which the parameters are fitted using the data in the first
load period. As you see the PH model explains only a
bit of the stiffness recovery. In my view this is the
thixotropic effect. So, there is recovery with respect to
the so called permanent fatigue damage parameter
gamma. But now you face the problem how to measure
the healing during a rest period? If determination of the
stiffness modulus during a rest period is possible, you
get information for making a healing model which can
be validated.
In my view you have to look to non-destructive tests. In
cement concrete research two test methods are used.
These two methods can also be applied in discontinuous
fatigue tests.
Slide 44 UPV
The first one is the ultra soon velocity tester. This
device sends acoustic signals with a high frequency (50
kHz) through a specimen. The pulse velocity is
measured which depend on the modulus and Poisson
ratio of the material. Before you start a fatigue test on a
beam you measure using low strain levels at different
temperatures and frequencies the stiffness modulus and
the pulse velocity, just at the start and at the end of one
frequency. This gives you a kind of calibration line,
that’s to say a relation between stiffness modulus and
pulse velocity. So you will measure the pulse velocity
during a rest period in a discontinuous fatigue test and
you can relate it to a stiffness modulus based on the
calibration. If the temperature cabinet is large enough it
can be done within the cabinet otherwise you have to
take the beam out for the application of the UPV test.
Andrea Cocurello performed this test during his stage at
the Technical university of Delft and reported it at the
RILEM conference at Sardinie in 2009.
Slide 45: Resonance test
A similar acoustic test is the resonance test. Instead of
applying pulses the resonance test uses a frequency
sweep and the first resonance frequency is determined.
In contrast with the UPV test the resonance test will
give you directly the stiffness modulus. Unfortunately
this modulus value is determined at high frequencies in
the order of 2 kHz. So, again a kind of calibration curve
as done for the UPV test has to be established. Both
acoustic methods are useful but have their limits. I like
to mention that the resonance test can also be used very
well in real calibration tests of 4PB devices using real
reference beams (e.g. aluminium beams) in orcer to
obtain the E value for the reference beam more
precisely.
Slide 46: Endurance
A third option might be the application of a strain below
the endurance limit. In theory no irreversible fatigue
damage should occur at this strain level. Using the MPH
model the effect of thixotropy during the rest period can
be predicted and so the remaining recovery in stiffness
modulus had to be real healing. In this way it would be
possible to determine the healing in stiffness modulus
(and fatigue strength) during the pseudo rest periods in a
discontinuous fatigue test. It can even lead to shorter
fatigue tests but that is one step more.
In the past I used this method in several projects. I will
show you two examples
Slide 47: Examples
The strain level in the load periods was 160 micro
strain. This slide shows the evolution in a pseudo-rest
period if a strain of 80 micro strain was applied. Mark
that after a while the stiffness modulus decreases again.
So, the strain of 80 micro strain is above the endurance
limit for this mix.
Slide 48: Example
Here the strain level during the pseudo-rest period was
only 40 micro strain and it seems that the stiffness
modulus reaches an equilibrium or will slowly heal
further on.
Slide 49 UPP-4PB
Finally I like to show you that with the MPH model it is
possible to compare fatigue data obtained with cyclic
tension/compression tests and 4 PB tests. The fatigue
lives are defined as the number of load periods where
the MPH model starts to deviate from the measured
data.
Slide 50 Determination Nph
As you can see the fatigue life is a bit bigger than the
fatigue life N1 using a straight line.
Slide 51
At the end of this keynote I would like to make a remark
on one aspect that is still a pain in the ass. As I showed
you most of the dissipated energy will be transformed
into heat. So, during the fatigue test the temperature will
increase and therefore the stiffness modulus will
decrease. In the past I made two Excel programs for the
calculation of this temperature increase. If everything
went well the programs will be available on the
conference web site.
For low frequencies and moderate strain amplitudes the
increase in temperature is limited but will play a role for
higher values.
SLIDE 52 Questions
Well this is the end of my keynote lecturer. I hope I
didn’t lose you during all the info I spread out over you.
