Post on 05-Feb-2021
© Valerian Hirschberg, 2019
Fourier Transform Rheology as a Tool to Determine the Fatigue Behavior of Polymers
Thèse
Valerian Hirschberg
Doctorat en génie chimique
Philosophiæ doctor (Ph. D.)
Québec, Canada
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Fourier Transform Rheology as a Tool to
Determine the Fatigue Behavior of Polymers
Valerian Hirschberg
Doctorat en génie chimique
Philosophiae doctor (Ph.D.)
Directeur de recherche
Prof. Dr. Denis Rodrigue
Co-directeur de recherche
Prof. Dr. Manfred Wilhelm
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Résumé
Cette thèse propose un nouveau concept d'analyse, de quantification et de prédiction de la
fatigue mécanique d'un polymère amorphe à l'aide d'une méthode basée sur la
décomposition de la contrainte via la transformation de Fourier. En particulier, des essais
de fatigue ont été réalisés sous déformation contrôlée en torsion et en tension/tension. La
déformation, le couple et la force ont été enregistrés en fonction du temps et décomposés
en contributions linéaires et non-linéaires, quantifiés par des harmoniques plus élevées.
De plus, trois concepts ont été développés pour déterminer quantitativement le
comportement mécanique des échantillons en fonction du temps. Premièrement, il a été
établi que la génération de fissures macroscopiques était en corrélation avec l’augmentation
soudaine de l’intensité de I2/1. Deuxièmement, une méthode directe pour prédire la durée
de vie en fatigue a été développée, basée sur le taux de changement de I3/1 par rapport au
nombre de cycle N (dI3/1/dN) avant l'apparition de la rupture. Cette prédiction s'est avérée
beaucoup plus précise que les prédictions des courbes de Wöhler puisque les corrélations
présentent en moyenne des écarts-types beaucoup plus faibles (30 vs 60%). Troisièmement,
un critère de fatigue basé uniquement sur la non-linéarité mécanique a été développé,
appelé la non-linéarité cumulée Qf. Ce paramètre corrèle l'intégrale de la non-linéarité Q (Q
= I3/1/02) jusqu'à la rupture avec le nombre de cycles à la rupture Nf. L'écart-type de la
corrélation Qf vs Nf s'est avéré inférieur à 30%, indiquant que Qf est un critère de fatigue
plus précis que ceux couramment utilisés tels que la densité d'énergie dissipée cumulée ou
la contrainte cumulée (±50%). Enfin, ces trois concepts ont été appliqués avec succès dans
différentes conditions (type de déformation, plage de fréquence, amplitude de déformation)
et différents polymères tels que le polystyrène (PS), le polyméthylméthacrylate (PMMA), le
styrène acrylonitrile (SAN) et le polytertbutylméthylacrylate (PtBMA).
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Abstract
This thesis proposes a new framework to analyse, quantify and predict the mechanical
fatigue of amorphous polymer using a method based on the decomposition of the stress
response via Fourier transform. In particular, fatigue tests were performed under strain
controlled torsion and tension/tension deformation and the time data of the strain, torque
and force were recorded and decomposed into linear and nonlinear contributions via higher
harmonics. In particular, three concepts have been developed to quantitatively determine
the time behavior of the samples. Firstly, the generation of macroscopic cracks was found
to correlate with sudden increases in the I2/1 intensity. Secondly, an on-line method to
predict the fatigue lifetime was developed, based on the rate of change of I3/1 with respect
to the cycle number N (dI3/1/dN) before the onset of failure. This prediction was found to be
more precise than Wöhler curves predictions since the correlations have on average much
lower standard deviations (30 vs. 60%). Thirdly, a fatigue criterion solely based on
mechanical nonlinearity was developed: the cumulative nonlinearity Qf. This parameter
correlates the integral of the nonlinearity Q (Q = I3/1/02) until failure with the number of
cycles to failure Nf. The standard deviation of the Qf vs. Nf correlation was found to be less
than 30%, indicating that Qf is a more precise fatigue criterion than commonly used ones
such as the cumulative dissipated energy density or the cumulative stress (±50%). Finally,
these three concepts were successfully applied on different conditions (type of deformation,
range of frequency, deformation amplitude) and polymers such as polystyrene (PS),
polymethylmethacrylate (PMMA), styrene acrylonitrile (SAN) and
polytertbuthylmethacrylate (PtBMA).
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Table of contents
Résumé ........................................................................................................................ iii
Abstract ........................................................................................................................ iv
List of Tables ................................................................................................................. x
List of Figures ............................................................................................................... xi
Abbreviations .............................................................................................................. xvii
Symbols .................................................................................................................... xviii
Acknowledgement ....................................................................................................... xxii
Foreword ................................................................................................................... xxiii
General introduction ...................................................................................................... 1
Chapter 1 ...................................................................................................................... 5
1.1 Literature overview ............................................................................................ 5
Dynamic Rheology and Mechanics of Polymers in their Solid State ........................... 5
Résumé ................................................................................................................... 5
Abstract .................................................................................................................. 6
1.1.1 Introduction ................................................................................................... 7
1.1.2 Viscoelasticity of solid polymers ...................................................................... 8
1.1.2.1 Viscoelasticity of polymers ........................................................................ 8
1.1.2.2 Linear viscoelasticity ................................................................................ 8
1.1.2.3 Nonlinear viscoelasticity ........................................................................... 9
1.1.2.4 Large amplitude oscillatory shear (LAOS) ................................................ 10
1.1.2.4.1 Fourier Transform ........................................................................... 10
1.1.2.5 Fourier transform rheology ..................................................................... 11
1.1.2.5.1 Odd higher harmonics ..................................................................... 12
1.1.2.5.1.1 The intrinsic nonlinearity Q0 ...................................................... 13
1.1.2.5.2 Even higher harmonics .................................................................... 14
1.1.3 Mechanical fatigue ....................................................................................... 14
1.1.3.1 Different loading modes .......................................................................... 15
1.1.3.2 Wöhler curves ........................................................................................ 16
1.1.3.3 Approaches following the material properties .......................................... 17
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1.1.3.3.1 Concepts related to the dissipated energy ......................................... 17
1.1.3.3.1.1 Viscous heating ......................................................................... 18
1.1.3.3.2 Concepts related to nonlinear changes in the stress response ........... 19
1.1.3.3.2.1 Viscoelastic nonlinear parameter ............................................... 19
1.1.3.3.2.2 Analysis of higher harmonics in the stress response ................... 20
1.1.3.3.2.2.1 Thermoplastics ................................................................... 20
1.1.3.3.2.2.2 Elastomers .......................................................................... 22
1.1.4 Conclusion ................................................................................................... 23
Acknowledgments ................................................................................................. 24
1.2 Objective ......................................................................................................... 24
1.3 Application ...................................................................................................... 26
Chapter 2 Fatigue behavior of polystyrene (PS) analyzed from the Fourier transform (FT) of
stress response: First evidence of I2/1(N) and I3/1(N) as new fingerprints ......................... 28
Résumé ................................................................................................................. 29
Abstract ................................................................................................................ 30
2.1 Introduction .................................................................................................... 31
2.2 Theory ............................................................................................................. 32
2.2.1 Oscillatory Shear .......................................................................................... 32
2.3 Methods .......................................................................................................... 34
2.4 Material .......................................................................................................... 34
2.5 Experimental results ....................................................................................... 35
2.6 Discussion and analysis .................................................................................. 37
2.6.1 Analysis of the linear parameter: the storage modulus G’ ............................... 37
2.6.2 Analysis of the odd higher harmonic I3/1........................................................ 38
2.6.3 Even Higher Harmonics ................................................................................ 40
2.6.4 Notched samples .......................................................................................... 41
2.6.5 Comparison of G’ vs. I2/1 and I3/1 ................................................................... 42
2.7 Conclusion ...................................................................................................... 45
Acknowledgment ................................................................................................... 45
Supplementary material ............................................................................................... 46
Chapter 3 Influence of molecular properties on the mechanical fatigue of polystyrene (PS)
analyzed via Wöhler curves and Fourier Transform rheology ......................................... 47
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Résumé ................................................................................................................. 48
Abstract ................................................................................................................ 49
3.1 Introduction .................................................................................................... 50
3.2 Materials ......................................................................................................... 51
3.2.1 PS with low PDI ............................................................................................ 51
3.2.2 PS with broad MWD ..................................................................................... 52
3.3 Fatigue Testing ................................................................................................ 53
3.4 Rheological Measurements .............................................................................. 53
3.4.1 Strain Sweep Tests ....................................................................................... 53
3.5 Fatigue Testing ................................................................................................ 55
3.5.1 Wöhler curves of low PDI polystyrene ............................................................ 55
3.5.2 Wöhler curves for broad PDI PS .................................................................... 57
3.5.3 Analysis of the Time Dependent Stress Response via Fourier Transform ........ 58
3.5.3.1 Polystyrene with low PDI ........................................................................ 58
3.5.3.2 PS with a broad PDI ............................................................................... 59
3.5.3.3 Detection of macroscopic cracks ............................................................. 60
3.5.3.4 Number of cycles to the occurrence of a macroscopic crack Nc as a function
of the molecular weight ...................................................................................... 61
3.6 Conclusion ...................................................................................................... 62
Acknowledgements ................................................................................................ 63
Chapter 4 Fatigue life prediction via the time-dependent evolution of linear and nonlinear
mechanical parameters determined via Fourier transform of the stress ......................... 64
Résumé ................................................................................................................. 65
Abstract ................................................................................................................ 66
4.1 Introduction .................................................................................................... 67
4.2 Fatigue testing ................................................................................................ 68
4.3 Results and Discussion ................................................................................... 69
4.3.1 Polystyrene ................................................................................................... 69
4.3.2 Validation with polymethylmethacrylate and styrene-acrylonitrile .................. 75
4.4 Safety limits via the time evolution of the stress ............................................... 78
4.5 Conclusion ...................................................................................................... 79
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Acknowledgements ................................................................................................ 80
Supplementary material ........................................................................................ 81
Chapter 5 Cumulative nonlinearity as a criterion to quantify mechanical fatigue ........... 83
Résumé ................................................................................................................. 84
Abstract ................................................................................................................ 85
5.1 Introduction .................................................................................................... 86
5.2 Materials and Methods .................................................................................... 89
5.3 Experimental results and discussion ................................................................ 90
5.3.1 Cumulative Nonlinearity ............................................................................... 91
5.3.1.1 Effect of frequency and molecular weight ................................................ 93
5.3.2 Comparison with the cumulative dissipated energy and stress density ........... 95
5.4 Conclusion .................................................................................................... 100
Acknowledgement ............................................................................................... 101
Declarations of interest: ...................................................................................... 101
Chapter 6 Fatigue and fracture analysis via Fourier transform of the stress of brittle
polymers in tension/tension ....................................................................................... 102
Résumé ............................................................................................................... 103
Abstract .............................................................................................................. 104
6.1 Introduction .................................................................................................. 105
6.2 Experimental setup ....................................................................................... 109
6.3 Materials ....................................................................................................... 109
6.4 Results and discussion .................................................................................. 110
6.4.1 Fourier spectra of solid polymers in tension ................................................ 110
6.4.2 Dumbbell sample ....................................................................................... 112
6.4.2.1 Evolution of I2/1 and I3/1 ....................................................................... 112
6.4.3 Investigation of notched members ............................................................... 116
6.4.3.1 Crack detection and propagation .......................................................... 116
6.4.3.2 Correlation of dI3/1/dN and Qf with Nf ................................................... 121
6.5 Conclusion .................................................................................................... 123
Acknowledgement ............................................................................................... 124
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Conclusion ................................................................................................................ 125
Outlook ..................................................................................................................... 127
Literature .................................................................................................................. 130
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List of Tables
Table 1: Molecular properties of the synthetized PS. ...................................................................... 52
Table 2: Molecular characteristics of the PS with a broad and bimodal MWD. ..................... 52
Table 3: The measured number of cycles to failure (Nf) for tests at 0 = 0.7% and ω1/2π =
1 Hz, the rate of change of G', G'' and I3/1 and the calculated fatigue lifetimes via Eq. (41) -
(43), as well as their average value. The closest prediction to the experimental value is
underlined. ................................................................................................................................................... 72
Table 4: The coefficient of determination r2, the pre-factors AW (Eq. (57)) and AQ (Eq. (62))
as well as their standard deviation for the correlations of Wf and Qf with Nf. ........................ 99
Table 5: The proportionality factors AWöhler, AI3/1 and AQ (Eq. (53), (68) and (70)) for the
investigated polymers and geometries, as well as their standard deviations in absolute
value and percent for easier comparison. The lowest standard deviations are marked in
bold. .............................................................................................................................................................. 122
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List of Figures
Figure 1: Spectacular cases of failure: the Tacoma Narrows bridge due to strong winds (left)
and the break-up of a newly-constructed ship after releasing from the shipyard (right)
[Elliot, 1940; Parker, 1957]. ........................................................................................... 2
Figure 2: Typical G' and G'' behavior during a strain sweep test in the linear (SOAS) and
nonlinear (LAOS) regime of a polylactic acid/nanocrystal composite (PLA2002D, Mn =
12.5 kg/mol) at ω1/2π = 1 Hz and T = 165°C, according to [Hyun, 2002]. ...................... 10
Figure 3: Fourier spectrum of the stress response of a polylactic acid/nanocrystal
composite (PLA2002D, Mn = 12.5 kg/mol) at γ0 = 268%, ω1/2π = 1 Hz and T = 165°C. The
signal to noise (S/N) ratio is in the range of 100000:1. .................................................. 11
Figure 4: Typical response for the third harmonic (I3/1) intensity as a function of the applied
strain amplitude (γ0), showing the behavior in SAOS and LAOS of a polylactic acid/(treated)
nanocrystal composite (PLA2002D, Mn = 12.5 kg/mol) at ω1/2π = 1 Hz and T = 165°C. . 13
Figure 5: Typical behavior of the stress response of a strain controlled fatigue test and its
corresponding hysteresis loops/Lissajous curves. ......................................................... 15
Figure 6: Schematic representation of a Wöhler curve with the three typical regimes: low
and high cycle fatigue, followed by the fatigue limit. ...................................................... 16
Figure 7: Storage modulus (E') and Nonlinear Viscoelastic Parameter (NVP) as a function of
time for three different loading modes of a glass fiber/Nylon6 (GF/Ny6) sample:
tension/tension (empty circle), tension/compression (full circle) and
compression/compression (empty square) at ω1/2π = 11 Hz and T = 303 K [Liang, 1996].
................................................................................................................................... 19
Figure 8: The (normalized) decrease of G' against the (normalized) increase in I3/1 until a
macroscopic crack occurred in polystyrene [Hirschberg, 2017]. ..................................... 21
Figure 9: Storage modulus (G’) as well as I2/1 and I3/1 intensity as a function of the number
of cycles (N). The I2/1 intensity increases substantially (over a factor of ten) when a
macroscopic crack occurs in the sample (between N = 40 and N = 60 cycles) [Hirschberg,
2017]. .......................................................................................................................... 21
Figure 10: The second (I2) and third harmonic (I3) as a function of the number of cycles for
a rubber at different strain amplitude (15-35%). The values are not normalized to the
fundamental harmonic (I1). The author labeled I2 as the first and I3 as the second (higher)
harmonics. Adapted from [Lacroix, 2004]. .................................................................... 22
Figure 11: Reproducibility of fatigue experiments at γ0 = 15% showing the absolute values
of the higher harmonics at the beginning of the test and their trends as a function of the
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number of cycles. The author labeled I2 as the first and I3 as the second (higher) harmonics.
Adapted from [Lacroix, 2004]. ...................................................................................... 23
Figure 12: Schematic overview of the objectives of the thesis. ........................................ 25
Figure 13: Typical results for solid samples in oscillatory shear tests [Hirschberg, 2015]. a)
Lissajous (stress-strain) curve of a solid PS-PI-PS triblock copolymer sample (MW = 153
kg/mol, polydispersity index (PDI) = 1.06) at ω1/2π = 1 Hz and room temperature in the
linear regime (γ0 = 0.0027) and b) the FT spectrum of the stress. c) Lissajous (stress-strain)
curve of the same PS-PI-PS triblock copolymer in the nonlinear regime (γ0 = 0.05). The
corresponding FT spectrum of the stress in d) contains odd higher harmonics; the even
harmonics are within the noise level. In the linear regime, only the peak at the fundamental
frequency appears, whereas in the nonlinear regime higher harmonics up to the 13th can
be detected. ................................................................................................................. 33
Figure 14: Typical FT spectrum of the stress response of the first 15 cycles after adjusting
of a fatigue test with constant strain amplitude γ0 = 0.012. The excitation frequency is ω1/2π
= 0.5 Hz. The signal to noise ratio (S/N ratio) is about 1x105......................................... 35
Figure 15: Storage modulus (G’) and the intensities of the higher harmonics (I2/1 and I3/1)
during a fatigue test at ω1/2π = 1 Hz and γ0 = 0.012 (left) or 0.014 (right), and room
temperature (RT). The pictures below are taken from a video of the fatigue test and failure
of the sample as labelled in the plot a) above. ............................................................... 36
Figure 16: The first derivative of the storage modulus G’ as a function of the number of
cycles N (dG’/dN) for the PS data of Figure 15 (γ0 = 0.012, ω1/2π = 1 Hz). The dashed line
indicates the occurrence of the first macroscopic crack, corresponding to pictures a) and b)
in Figure 15. ................................................................................................................ 37
Figure 17: The Lissajous (strain/stress) curves of two nonlinear stress signals. The first
Lissajous curve (I3/1 = 0.022) represents the beginning of a fatigue test on PS as reported
in Figure 15 (dashed line), while for the second one, I3/1 increased by 45% (I3/1 = 0.0319,
full line) which was found as a typical increase during a fatigue test until a macroscopic
crack occurs. ............................................................................................................... 39
Figure 18: The intensity of I2/1 as a function of the number of cycles N at ω1/2π = 1 Hz and
a) γ0 = 0.01 and b) γ0 = 0.012. Three tests were conducted at both strain amplitudes. In a)
macroscopic cracks occur after about 2200 cycles (black line), 1900 (red) and 3100 (blue),
in b) after 1600 cycles (black line), 1400 (red) and 2800 (blue). At about the same time than
the macroscopic cracks occur, the intensity of I2/1 increases drastically. ........................ 40
Figure 19: Typical behavior of the storage modulus (G’) and the intensities of the higher
harmonics (I2/1 and I3/1) during a fatigue test of a notched PS sample at ω1/2π = 2 Hz and
γ0 = 0.012. The images taken from the video belong to the label in the plot above to show
when changes occur in the sample. .............................................................................. 42
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Figure 20: The values of G’ and I3/1 (red line) normalized by their initial values at the
beginning of the test as a function of the number of cycles. The box indicates the region
where no macroscopic crack can be seen. ..................................................................... 43
Figure 21: Relative values of the elastic shear modulus as a function of the relative values
of the third harmonic when the first macroscopic crack occurs. .................................... 44
Figure 22: Experimental setup used for the fatigue measurements. ............................... 46
Figure 23: G’ (a) and G’’ (b) as a function of the strain amplitude (ω1/2π = 1 Hz, RT) for the
PS listed in Table 1 (low PDI). ....................................................................................... 53
Figure 24: I3/1 as a function of strain amplitude (ω1/2π = 1 Hz, RT) for the PS samples listed
in Table 1 (low PDI). For each strain amplitude, 15 cycles were performed. .................... 54
Figure 25: Number of cycles to failure (Nf) as a function of the strain amplitude (γ0) for low
PDI polystyrene. ........................................................................................................... 55
Figure 26: Value of the parameter A in Eq. (35) as a function of the weight average molecular
weight for low PDI PS (see Table 1). The lines represent the correlations given by Eq. (36)
and (37). ...................................................................................................................... 56
Figure 27: Values of the parameter A in Eq. (35) as a function of: a) Mw and b) Mn. The lines
represent the correlations given by Eq. (36) and (37). .................................................... 57
Figure 28: Normalized elastic modulus (G’/G’0) a) and relative third harmonics intensity
(I3/1/(I3/1)0) b) as a function of the number of cycles for the different low PDI PS investigated
(γ0 = 2%, ω1/2π = 1 Hz, RT). ......................................................................................... 58
Figure 29: Normalized elastic modulus (G’/G’0) a) and relative third harmonics intensity
(I3/1/(I3/1)0)) b) as a function of the number of cycles for the different broad and bimodal
MWD PS (γ0 = 2%, ω1/2π = 1 Hz, RT). ........................................................................... 59
Figure 30: The linear parameters (G’ and G’’) as well as the nonlinear parameters (I2/1 and
I3/1) during a fatigue test of PS200k (γ0 = 2%, ω1/2π = 1 Hz, RT). An increase of I2/1 is found
while the crack propagates. .......................................................................................... 61
Figure 31: The number of cycles to the occurrence of a macroscopic crack as a function of:
a) Mw and b) Mn. (γ0 = 2%, ω1/2π = 1 Hz, RT). The lines represent the correlations given by
Eq. (36) and (39). ......................................................................................................... 62
Figure 32: The Wöhler curve of the tests at ω1/2π = 0.5 Hz (green triangle), 1 Hz (black
star), 2 Hz (pink circle) and 5 Hz (orange square). ......................................................... 69
Figure 33: Typical evolution of the three mechanical parameters (G’, G’’ and I3/1) as a
function of the number of cycles (N). The curves can be divided into three regimes: after an
initial phase (regime I, up to NII cycles), the three parameters linearly de/increase with the
number of cycles (regime II, up to NIII cycles), before failure onset occurs (regime III). In
regime II, the slope of the three parameters were calculated (dG’/dN, dG’’/dN and d3/1/dN).
The tests were performed at ω1/2π=1 Hz, γ0 = 0.5% and RT. .......................................... 70
xiv
Figure 34: Fatigue lifetime (Nf) of the PS samples as a function of the absolute values of the
rates of change of: a) G’, b) G’’ and c) I3/1 calculated in regime II. The full lines represent the
correlations of Eq. (41) - (43), while the dashed lines represent a 30% deviation. ........... 74
Figure 35: Typical evolution of the three mechanical parameters (G’, G’’ and I3/1) as a
function of the number of cycles (N) for PMMA (a) and SAN (b) samples. The curves can be
divided into three regimes: after an initial phase (regime I, up to NII cycles), the three
parameters de-/increase linearly with the number of cycles (regime II, up to NIII cycles),
before failure onset occurs (regime III). In regime II, the slopes of the three parameters were
calculated (dG’/dN, dG’’/dN and d3/1/dN). The tests were performed at ω1/2π = 1 Hz, γ0 =
0.5% and RT. ............................................................................................................... 76
Figure 36: Fatigue lifetime (Nf) of PMMA samples as a function of the absolute values of the
rates of change of G’, G’’ and I3/1 calculated in regime II. The full lines represent the
correlations of Eq. (44)- (46), while the dashed lines represent a 30% deviation. ............ 77
Figure 37: Fatigue lifetime (Nf) of SAN samples as a function of the absolute values of the
rates of change of G’, G’’ and I3/1 calculated in regime II. The full lines represent the
correlations of Eq. (47) - (49), while the dashed lines represent a 30% deviation. ........... 78
Figure 38: The linear parameters G' and G'' as well as the nonlinear ones I3/1 and I2/1 as a
function of the number of cycles during a fatigue test of PS (ω1/2π = 1 Hz, γ0 = 0.8%). At
the transition from regime II to regime III, the I2/1 intensity first decreases before it increases
drastically by nearly two orders in magnitude until failure. To better see these changes, a
logarithmic axis is employed, in comparison to linear ones for I3/1 above. ...................... 81
Figure 39: Wöhler curves of PMMA (left) and SAN (right) at ω1/2π = 1 Hz. ..................... 81
Figure 40: NII/Nf (black square) and NIII/Nf (red circle) as a function of the applied strain
amplitude for: a) PS, b) PMMA and c) SAN. ................................................................... 82
Figure 41: Normalized number of cycles to the de-/increase of G'00, G’’00 and I3/1,00 by the
corresponding prefactor AF of Eq. (41) - (49). ................................................................. 82
Figure 42: The parameter Q as a function of cycle number for PS98k (full red line). The
dashed surface represents the cumulative nonlinearity Qf. ............................................ 90
Figure 43: The cumulative nonlinearity until failure as a function of the number of cycles
to failure for selected polymers: PS98k (black square), PMMA (red circle), SAN (green
diamond) and PtBMA (blue triangle). ............................................................................ 92
Figure 44: Cumulative nonlinearity as a function of the number of cycles to failure for
different frequencies (0.5, 1, 2 and 5 Hz) for different PS molecular weights (Mn = 82, 98,
125, 190, 380 and 840 kg/mol). ................................................................................... 94
Figure 45: Prefactor AQ of the correlations between the cumulative nonlinearity with the
number of cycles (Eq.(62)) as a function of the number average molecular weight Mn (a))
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and the prefactor of the Wöhler curves (fatigue resistance, Eq.(53), AWöhler) for PS samples
with different molecular weight. ................................................................................... 95
Figure 46: Cumulative stress density σf and dissipated energy density Wf as a function of
the number of cycles to failure for the investigated polymers PS98k, PMMA, SAN and PtBMA
a) and b), for PS98k at different frequencies c) and d), and for PS with different molecular
weights e) and f).This analysis is analog to Figure 43 and Figure 44 for Qf. .................... 96
Figure 47: The different pre-factors (AW and Aσ) for the correlations between the cumulative
stress and dissipated energy density with the number of cycle to failure as a function of the
number average molecular weight Mn (a) and of the pre-factor of the Wöhler curves (b),
fatigue ressistance, AWöhler) for PS samples with different molecular weights. ................. 97
Figure 48: Stress controlled cycling of PMMA, PS200k and PS400k as a function of the
deformation a) and a magnification of the small deformation zone is given in b). The curves
reveal that the three materials deform plastically, as even after the first cycle a zero stress
does not correspond to a zero strain. .......................................................................... 110
Figure 49: Fourier spectra of the force response of dumbbell samples of PMMA on the
Electro Force 3300-AT from BOSE (a), ω1/2π = 5 Hz, ε0 = 0.8%, R = 0.1) and on the Acumen
3 from MTS (b), ω1/2π = 5 Hz, ε0 = 0.8%, R = 0.1). For the signal from the BOSE, beside the
fundamental harmonic at ω1/2π = 5 Hz, especially odd higher harmonics at ω1/2π = 15,
25, 35 Hz, etc. can be seen. The signal to noise ratio is in the range of 1:104 to 1:105 for
both machines. .......................................................................................................... 110
Figure 50: The normalized complex modulus ׀E*׀ and the higher harmonics intensities (I2/1
and I3/1) as a function of the applied strain amplitude ε0 as precited by the Neo-Hooke law.
The complex modulus ׀E*׀ decreases as a function of the strain amplitude, whereas the I2/1
intensity increases and the I3/1 intensity increases at strain amplitudes above 0.05 for the
resolution used. The fatigue tests were performed at strain amplitudes smaller 0.01,
indicated by the dashed purple line. ........................................................................... 111
Figure 51: Typical behavior of the complex modulus ׀E*׀ (black line), the average stress E0
(green line) and the nonlinear parameters I3/1 (red line) and I2/1 (blue line) as a function of
the cycle number for a dumbbell sample of PMMA (a), ω1/2π = 5 Hz, ε0 = 0.3%, R = 0.1 and
PS200k (b), ω1/2π = 5 Hz, ε0 = 0.3%, R = 0.1. The solid black line through the I3/1 data
represents the rate of change of I3/1 (dI3/1/dN), the E0 data of the PMMA tests shown is the
fitted exponential decrease to smoothen the data. ....................................................... 113
Figure 52: The logarithm of the I3/1 increase as a function of the E0 decrease in regime I
((I3/1)II/(I3/1)0 vs. E0/EII) for PMMA (a) and PS200k (b). For PMMA different loading ratios R
= 0.1 (black square), 0.3 (red circle) and 0.5 (blue triangle) were investigated and for PS200k
R = 0.1. The tests were performed at ω1/2π = 5 Hz. The black lines represent the best fits
xvi
with a slope of -2.7 for PMMA and -9.3 for PS200k, showing much higher changes in I3/1
for PS200k than for PMMA for similar changes in E0. .................................................. 114
Figure 53: The number of cycles to failure (Nf) as a function of the rate of change of I3/1
(dI3/1/dN) in regime II, for dumbbell samples of PMMA (a) and PS200k (b) at ω1/2π = 5 Hz.
For PMMA different deformation ratios were investigated: R = 0.1 (black square), 0.3 (red
circle) and 0.5 (blue triangle) and for PS200k: R = 0.1. The black lines represent the best fit
for R = 0.1 with slopes of -1.3. .................................................................................... 115
Figure 54: The cumulative nonlinearity (Qf) as a function of Nf for dumbbell samples of
PMMA (a) and PS200k (b) at ω1/2π = 5 Hz. For PMMA different deformation ratios were
investigated: R = 0.1 (black square), 0.3 (red circle) and 0.5 (blue triangle), for PS200k: R =
0.1. The black lines represent the best fit for R = 0.1, with slopes of 1.75. ................... 115
Figure 55: The evolution of ׀E*׀, I2/1 and I3/1 as a function of the cycle number for notched
PMMA samples (ω1/2π = 5 Hz, ɛ0 = 0.35% and 0.2%) at different deformation ratios: a),b) R
= 0.1, c),d) R = 0.3, e) R = 0.5 and f) R = 0.7. ............................................................... 117
Figure 56: The evolution of ׀E*׀, I2/1 and I3/1 as a function of the cycle number for a notched
PS200k (ε0 = 0.3%) and PS400k (ε0 = 0.35%) sample (ω1/2π = 5 Hz, R = 0.3). .............. 118
Figure 57: Micrographs of PMMA, PS200k and PS400k (ω1/2π = 5 Hz, ε0 = 0.2%, 0.3% and
0.35%, R = 0.3) broken surfaces revealing two zones of different crack propagation rates via
a rough (the river patterns, fast crack growth with plastic deformation) and a smooth
surface area, containing (for PMMA) striations (slow crack growth, brittle failure). ....... 119
Figure 58: Correlations between Qf, dI3/1/dN and Nf for notched rectangular PS200k (black
square) and PS400k (red circle) samples. The black lines represent the best fit. ........... 121
xvii
Abbreviations
DE Dissipated Energy during one cycle
FT Fourier transform
LAOS Large amplitude oscillatory shear
MAOS Medium amplitude oscillatory shear
PDI Polydispersity index
PI Polyisoprene
PLA Polylactic acid
PMMA Polymethylmethacrylate
PS Polystyrene
PtBMA Polytertbuthylmethacrylate
S-N curve Stress-number of cycles to failure curve
S/N ratio Signal to noise ratio
SAN Styrene-acrylonitrile
SAOS Small amplitude oscillatory shear
SI Polystyrene-polyisoprene diblock copolymer
SIS Polystyrene-polyisoprene-polystyrene triblock copolymer
xviii
Symbols
A Surface
AQ Proportionality constant of the cumulative nonlinearity vs. number of cycles
correlation
AI3/1 Proportionality constant of the dI3/1/dN vs. number of cycles correlation
AWöhler Proportionality constant of the Wöhler curve
B Prefactor of the Wöhler correlation
C Exponent of the Wöhler correlation
δ1 Phase angle of the fundamental harmonic
δ2 Phase angle of the second harmonic
δ3 Phase angle of the third harmonic
δ4 Phase angle of the fourth harmonic
δ5 Phase angle of the fifth harmonic
dI3/1/dN Rate of change of I3/1 in regime II
abs(dG’/dN) Rate of change of G’ in regime II
abs(dG’’/dN) Rate of change of G” in regime II
EDiss Dissipated energy during one cycle
E’ Storage modulus (elongation)
E’’ Loss modulus (elongation)
(Complex modulus (elongation ׀*E׀
E0 Relaxation modulus (elongation)
ε Strain (elongation)
ε0 Strain amplitude (elongation)
F Force
xix
Strain (torsion)
0 Strain amplitude (torsion)
+ Clockwise strain
- Counter-clockwise strain
G’ Storage modulus (shear)
G’’ Loss modulus (shear)
G* Complex modulus
I2/1 Ratio of the second harmonic normalized by the fundamental
I3/1 Ratio of the third harmonic normalized by the fundamental
I4/1 Ratio of the fourth harmonic normalized by the fundamental
I5/1 Ratio of the fifth harmonic normalized by the fundamental
Mn Number average molecular weight
Mw Weight average molecular weight
η Viscosity
NI Cycle number at the transition from regime I to regime II
NII Cycle number at the transition from regime II to regime III
Nf Number of cycles to failure
NVP Nonlinear viscoelastic parameter
ω Angular frequency
ω1 Angular frequency of the deformation
Q Ratio of the intensity of the third harmonic over the square of the strain
amplitude
Q0 Parameter of the intrinsic nonlinearity
Qf Cumulative nonlinearity
R Stress ratio
xx
RDEC Ratio of Dissipated Energy Change
σ Stress
s(t) Time signal
S(ω) Signal in the frequency domain after FT
t Time
Tg Glass transition temperature
T/T Tension/tension
Wdiss Dissipated Energy Density
Wf Cumulative Dissipated Energy Density
xxi
To my family
xxii
Acknowledgement
Most importantly, I would like to thank everyone who helped me to complete my Ph.D.
thesis successfully. Firstly, I would like to thank my supervisor Prof. Denis Rodrigue for
the highly interesting and scientifically challenging subject of my thesis, his guidance and
support, the very fruitful and challenging discussions over the past three years, and last
but not least the financial support which allowed me to focus on my research. Secondly, I
would like to thank my co-supervisor Prof. Manfred Wilhelm for the time I could spend in
his laboratories, the fruitful and challenging discussions and the financial support. Thirdly,
I would like to thank all my friends and colleagues at ULaval and KIT, with special thanks
to Jian Zhang (ULaval), Dr. Dimitri Merger (KIT), Lorenz Faust (KIT) and Dr. Mahdi Abbasi
(KIT), as well as Mr. Yann Giroux (ULaval) and Mr. Daniel Zimmermann (KIT) for the
excellent technical support. Fourthly, I would like to thank Prof. Florian Lacroix and Prof.
Stéphane Méo for the opportunity to stay in their Institute at the Université de Tours in
summer 2018, the support over there and the discussions. Furthermore, I would like to
thank NSERC, CREPEC, CQMF and CERMA for financial and technical support, as well as
providing the equipment.
Finally, I would like to thank my family, my parents, my brother Leander and my girlfriend
Shan Lyu for their unconditional love and support.
xxiii
Foreword
The work presented hare was done under the supervision of Prof. Denis Rodrigue at
Université Laval in Quebec City/Canada and the co-supervision of Prof. Manfred Wilhelm
at the Karlsruhe Institute of Technology (KIT) in Karlsruhe/Germany. The mechanical
testing in torsion was mostly done at ULaval, whereas the synthesis of polymer model
systems was done in the laboratories at KIT. The tests in tension/tension where mostly
done during a research stay at the Université de Tours in France, funded by a “MITACS
Globalink Award” scholarship.
This thesis contains nine chapters, one is a book chapter, three are published articles and
two are submitted articles.
Chapter 2
Hirschberg V, Wilhelm M, Rodrigue D. Fatigue behavior of polystyrene (PS) analyzed
from the Fourier transform (FT) of stress response: First evidence of I2/1(N) and I3/1(N)
as new fingerprints. Polym Testing 2017, 60, 343-350.
Chapter 3
Hirschberg V, Schwab L, Cziep M, Wilhelm M, Rodrigue D. Influence of molecular
properties on the mechanical fatigue of polystyrene (PS) analyzed via Wöhler curves
and Fourier Transform rheology. Polymer 2018, 138, 1-7.
Chapter 4
Hirschberg V, Wilhelm M, Rodrigue D. Fatigue life prediction via the time-dependent
evolution of linear and nonlinear mechanical parameters determined via Fourier
transform of the stress. J Appl Polym Sci 2018, 46634.
Chapter 5
Hirschberg V, Wilhelm M, Rodrigue D. Cumulative nonlinearity as a criterion to
quantify mechanical fatigue. Subm. to Int J Fatigue, 2018.
Chapter 6
Hirschberg V, Lacroix F, Wilhelm M, Rodrigue D. Fatigue and fracture analysis via
Fourier transform of the stress of brittle polymers in tension/tension. Subm. to Mech
Mater, 2019.
xxiv
My contribution as the first author includes the planning, the measurements, the analysis
and the interpretation of the data obtained, as well as the writing of the first version of the
articles. My supervisors, Prof. Rodrigue and Prof. Wilhelm, as co-authors of these articles,
supervised the work, helped with highly fruitful and challenging discussions and revised
the written articles.
The results were published in scientific journals, but also presented at several conferences
as oral presentation and as poster, as well as institute/group/center seminars at ULaval
and KIT. Two invited seminars were also presented, the references are listed below. The
poster presented at the “90th Annual Meeting of the Society of Rheology” in Houston (USA)
won the 2nd prize in the Ph.D. student/Post-doc poster competition.
Conference Contributions:
1.
V. Hirschberg, M. Wilhelm, D. Rodrigue.
“Fatigue fingerprints via Fourier transform of the stress.”
90th Annual Meeting of the Society of Rheology, Houston, USA, 14-18.10.2018, paper SG17.
2.
V. Hirschberg, M. Wilhelm, D. Rodrigue.
“Fatigue analysis via Fourier transform of the stress.”
90th Annual Meeting of the Society of Rheology, Houston, USA, 14-18.10.2018, poster
PO102.
3.
V. Hirschberg, M. Wilhelm, D. Rodrigue.
“Fatigue Analysis and Prediction via Fourier Transform of the Stress. “
12th International Fatigue Congress, Poitiers (France), 27.05.-01.06.2018, paper PS18,
#00375.
4.
V. Hirschberg, M. Wilhelm, D. Rodrigue.
“Fingerprints of mechanical fatigue via Fourier transform of the stress response with
application on polystyrene (PS)”.
29th Canadian Material Science Conference. Ottawa (CAN), 20-23.06.2017.
xxv
5.
M. Wilhelm, S. Nie, L. Schwab, V. Hirschberg, M. Cziep, J. Lacayo-Pineda.
“FT-Rheology on Rubber Materials”.
TA-Instruments Conference. Würzburg (Ger), 28.04.2017.
6.
D. Rodrigue, V. Hirschberg, M. Wilhelm.
“The effect of testing conditions on the mechanical properties of polymers during fatigue
testing”.
88th Annual meeting of the Society of Rheology. Tampa (USA), 12-16.02.2017.
7.
V. Hirschberg, M. Wilhelm, D. Rodrigue.
“Fatigue Behavior of Polylactic Acid (PLA) analyzed from the Fourier Transform (FT) of their
Stress Response”. Proceedings of the XVIIth International Congress on Rheology (ICR2016).
Kyoto (JP), 8-13.08.2016.
8.
V. Hirschberg, M. Wilhelm, D. Rodrigue.
“New Fingerprints of Fatigue: Nonlinear Parameters I2/1 and I3/1(t) of Fourier Transform (FT)
of the Stress Response with Application on Polystyrene (PS) and High Impact Polystyrene
(HIPS)”.
PolyMerTec Conference. Merseburg (Ger), 15-17.07.2016.
Invited Seminars
1.
Valerian Hirschberg, Manfred Wilhelm, Denis Rodrigue.
“Fatigue Analysis of Polymers via Fourier Transform of the Stress”
Seminar of ‘GRK 2078’, Institute of Engineering Mechanics, Karlsruhe Institute of
Technology, invited by Prof. Thomas Böhlke, 31.07.2018.
2.
Valerian Hirschberg, Manfred Wilhelm, Denis Rodrigue.
“Fatigue Analysis and Prediction via Fourier Transform of the Stress”
Seminar CERMEL, Université Tours, invited by Prof. Florian Lacroix, 20.06.2018.
http://www.itm.kit.edu/english/index.php
xxvi
Furthermore, during my stays in the laboratories of Prof. Wilhelm at the KIT, the effect of
PS/PI copolymerisation on the fatigue properties and the evolution of the higher harmonics
was systematically investigated. Therefore, model systems of PS-PI-PS (SIS) and PS-PI (SI)
with different molecular weight and PI content were synthesized via anionic polymerisation.
In total over 25 samples were synthesized, but as the characterisation is not yet finished
and published, this work is not further discussed here.
1
General introduction
The world plastic production has risen rapidly since the 1950s from about 1.5 Mt/a to 335
Mt/a in 2016 and is expected to grow to over 1800 Mt/y in 2050, the so-called success
story of plastics. About one third of the world production of plastics is used for packaging,
another third in buildings such as pavements, windows and piping. The last third includes
industrial components such as cars, toys and furniture [Andrady, 2009]. In several
applications, plastics are replacing common materials. Plastics offer various advantages
against common materials such as metals, wood and ceramics: low density, resistance
again corrosion, wide range of processing possibilities and low costs (~2 US$/kg) for the
commodity plastics such as polyethylene (PE), polyvinyl chloride (PVC) or polypropylene
(PP). Furthermore, polymers are viscoelastic. As a result, the stress response (force
normalized by the loaded area) contains both an elastic and a viscous part, characterized
as the storage modulus (E’) and loss (E’’) moduli in tension or G’ and G’’ in shear/torsion.
This viscoelastic behavior results in a nonlinear stress response under large
load/deformation. Nonlinearity means in this context that the stress response is no longer
directly proportional to the applied load/deformation.
Due to the lower mechanical strength of neat polymers compared with metals (in machines)
or concrete (in construction), polymers are seldom used as load bearing components.
Polymers are typically used for all kind of applications where the properties of polymers are
superior to other materials but are still exposed to a mechanical stress. For example, in
pavement, floor-covering, windows or piping, often also in combination with physical aging
(influence of temperature/weather).
With important advancements in the technology of 3D printing not only small and low-cost
objects, but also complex structures up to complete houses, can be printed with high
quality and speed. Polymers and polymer-based materials can be used due to their
rheological and mechanical properties. The increase of high-quality polymer-based
products also underlines the demand for an in-depth understanding of the mechanical
long-term properties during application, thus mechanical fatigue.
The failure of a material under a repetitive load or deformation is called mechanical fatigue
[Andrew, 1995]. The phenomenon of mechanical fatigue is a long-term process due to a
recurring deformation or load and is in general hardly detectable unless visible cracks or
the catastrophic event itself (failure) occurs. Fatigue is one of the main limitations with
respect to lifetime of the application of every solid part and responsible for about 90% of all
mechanical service failures [Campbell, 2008]. One of the underlying processes resulting in
2
accidents such as the collapse of bridges or dams, train accidents, plane crashes or the
failure of machines and devices is mechanical fatigue. Prominent accidents due to fatigue
are the collapse of the Tacoma Narrows bridge or the Liberty ships (Figure 1). In fact, most
accidents and catastrophic events, which are not directly related to human failure, are
related to mechanical fatigue. For our daily life, the safe use without the fear of failure is of
fundamental interest and consequently attracted the attention of scientists and engineers
from the beginning of the industrial revolution. Additionally, the economical losses due to
mechanical fatigue are reported to be around 4% of the Gross Domestic Product (GDP) in
the United States [Stephens, 2000], corresponding (for 2016) to a loss of 742 billion US$.
The world loss is consequently about of 3.22 trillion USD (2017, using the world gross
product), which is roughly double the GNP of Canada in 2017, underlining the whole extent
of damage of mechanical fatigue.
Figure 1: Spectacular cases of failure: the Tacoma Narrows bridge due to strong winds (left) and the
break-up of a newly-constructed ship after releasing from the shipyard (right) [Elliot, 1940; Parker,
1957].
Since the beginning of the industrial revolution around 1800, the then-new phenomenon
of cyclic mechanical fatigue of metal in construction and machines, such as bridges and
railways, has been observed and investigated. Firstly, publications on the systematic
analysis of fatigue can be found from the 1830th on by English, French and German
engineers. The most well-known name related to fatigue at that time is August Wöhler who
first published the concept of the analysis of the number of (sinusoidal) cycles (N) to failure
of metal dumbbell samples as a function of the applied stress amplitude (S), nowadays
known as Wöhler or S-N curve (“Über Versuche zur Ermittlung der Festigkeit von Achsen”,
1867, in (more ancient) German, “On fatigue experiments of axel”) [Wöhler, 1867]. Wöhler
curves are still the state-of-the-art approach to characterize and quantify the fatigue
resistance of a material, connecting the fundamental parameters in mechanical fatigue
3
testing: the number of cycles to failure (N) and the stress (S) or deformation amplitude. The
main challenge of Wöhler curves and mechanical fatigue testing in general is the
probabilistic nature of fatigue, thus the poor reproducibility of the number of cycles to
failure for the same material under the exact same testing conditions. The standard
deviations of Wöhler curves are typically far above 100% for metals and typically above 50%
for polymers. The range of lifetime for metallic samples under the same measurement
conditions is reported as up to several decades and in an older extreme case, with up to
four orders of magnitude. Due to the highly probabilistic nature of mechanical fatigue,
intensive research in this field is necessary to improve the understanding of fatigue and to
protect human and environment from the catastrophic event of failure.
An in-depth understanding of fatigue needs to investigate the different stages of fatigue and
fracture. In general, three different regimes can be observed: crack initiation, crack
propagation and fast crack growth/catastrophic failure. Before crack initiation, no changes
due to damage are visible. So, several experimental, empirical and analytical approaches
were developed to better detect, quantify and predict those evolutions and events during
mechanical fatigue. Methods based on external devices such as video or thermal cameras,
or on the material parameters themselves, for example via an analysis of the dissipated
energy during one cycle or in total up to failure, were proposed. The fatigue analysis in
deformation/strain controlled tests, providing more information about the current state of
fatigue, without using additional devices, as e.g. a camera, are based on the analysis of the
stress response of the material.
From rheology, different techniques and frameworks exist to investigate and quantify
nonlinear contributions in the stress, the strain or strain rate in the time domain. In the
strain/strain rate domain, hysteresis loops or so-called Lissajous curves, can be used to
quantify changes from the linear response via Chebyshev coefficients for example. In the
time domain, the stress can be decomposed into linear and nonlinear contributions via
Fourier transform with high sensitivity; i.e. the so-called FT-rheology. The excitation mode
of choice is nowadays sinusoidal and can be performed on the experimental devices with a
high signal to noise (S/N) ratio, typically in the range of 105:1 and nonlinear contributions
in the rage of 10-3 to 10-4 can be measured and reproducible at small strain amplitudes.
Rheology investigates the linear and nonlinear mechanical material properties as a function
of the deformation or deformation rate, but mechanical fatigue of solids is typically not
investigated. However, in mechanical fatigue testing, only few papers can be found dealing
with the analysis of the stress and none applying the technique of FT -rheology. An example
from fatigue testing is the so-called nonlinear viscoelastic parameter (NVP), calculating the
intensity of the sum of higher stress harmonics over the fundamental, analog to the total
4
harmonic distortion for sound systems. This parameter was investigated for fatigue tests in
tension and compression and found to increase at failure. With the knowledge and
understanding from FT-rheology, the drawbacks of the NVP as well as its limited predictive
power are easily understandable. Firstly, it does not analyze odd and even higher harmonics
separately, so no difference is made between the higher harmonics representing the
geometric symmetry (odd higher harmonics) and the asymmetric contributions (even higher
harmonics) of the stress response. For an undamaged sample, noise is summed up via the
even or the even and odd higher harmonics, depending on the material, the deformation
direction and the applied stress/strain amplitude.
This thesis investigates a multidisciplinary topic at the interface between polymer science,
rheology and mechanical engineering. The work deals with the investigation, quantification
and prediction of mechanical fatigue and failure via the linear and nonlinear contributions
of the materials stress response. Mechanical fatigue and fracture itself are nonlinear
processes, so the expectation is high to find typical patterns and fingerprints in the stress.
5
Chapter 1
1.1 Literature overview
Dynamic Rheology and Mechanics of Polymers in their Solid
State
Hirschberg V, Wilhelm M, Rodrigue D. Book chapter submitted to: Llewellyn, P. Rheology
of Polymer Blends and Nanocomposites: Theory, Modelling and Applications. Elsevier.
Résumé
Ce chapitre présente une revue de littérature sur les propriétés rhéologiques et mécaniques
dynamiques des polymères à l'état solide. En présence de grandes déformations, la
contrainte des polymères n’est plus linéaire et différentes techniques ont été proposées pour
analyser le comportement non-linéaire de ces matériaux. Les techniques et théories
actuelles sont présentées, mais la rhéologie à transformation de Fourier (FT) est mise en
détail.
De plus, ce chapitre traite de la fatigue mécanique des polymères. Premièrement, les
concepts de base sont présentés en utilisant la représentation bien connue des courbes de
Wöhler du nombre de cycles à la rupture en fonction de l'amplitude de la contrainte
(contrainte imposée) ou de la déformation (déformation imposée). Deuxièmement, une
discussion est faite sur les différentes approches pour suivre les changements des
propriétés des matériaux au cours des essais, par exemple: l'énergie dissipée, le paramètre
viscoélastique non-linéaire et l'analyse des harmoniques supérieures de la réponse aux
contraintes via FT. Enfin, les tendances actuelles des recherches sont présentées.
6
Abstract
This chapter presents an overview of the dynamic rheological and mechanical properties of
polymers in their solid state. Under large deformation, the material response of polymers is
no longer linear viscoelastic and different techniques have been proposed to analyze the
nonlinear behavior of these materials. The current techniques and theories are presented,
but Fourier transform (FT) rheology is more detailed.
Furthermore, this chapter deals with the mechanical fatigue of polymers. The basic
concepts are presented first using the well-known Wöhler curves representation of the
number of cycles to failure as a function of the applied stress (stress controlled) or strain
(strain controlled) amplitude. Secondly, a discussion is made on the different approaches
to follow the material properties changes during testing/application such as: the dissipated
energy function, the nonlinear viscoelastic parameter and the analysis of higher harmonics
of the stress response via FT. Finally, the current investigation trends are presented.
Keywords: Dynamic rheology, Nonlinearity, Fourier transform rheology, mechanical fatigue.
7
1.1.1 Introduction
The world plastic production rapidly increased since the 1950s from 1.5 to more than 300
Mt/y today (2018). About one third of the world plastics production is used for packaging,
another third in buildings (pavements, windows and piping), while the last third includes
industrial components (automotive, toys and furniture) [Andrady, 2009]. In several
applications, plastic parts replace common materials (metals, wood and ceramics) since
they provide various advantages such as low density, corrosion resistance, a wide range of
processing possibilities to allow basically all shapes of the final part in combination with
low costs, especially for the so-called commodity plastics like polyethylene and
polypropylene.
But polymers are viscoelastic materials and show, in the melt and solid state, a nonlinear
response when the deformation exceeds a certain threshold. Viscoelasticity implies that the
stress response contains both an elastic and a viscous part characterized by the storage E’
and loss E’’ moduli in tension, or G’ and G’’ in shear/torsion. Nonlinearity means that the
stress response is no longer proportional to the load/deformation. Under simple shear, the
viscosity might de-/increase as a function of the applied shear rate after a constant plateau
value at small deformations/rates [Bird, 1987; Carreau, 1997]. Under oscillatory shear, the
stress response becomes nonlinear, so the parameters G, E, and tan(δ) are not constant
anymore, but depend on the applied conditions. During oscillatory shear higher harmonics
(through FT) can be detected, see below [Giacomin, 1993; Wilhelm, 2002].
The rheological properties of melts are of high importance for polymer processing. So, it is
important to know the viscosity at a given temperature and shear rate, as well as how it is
influenced by their changes in processes like extrusion, injection molding, etc.
One of the main limitations with respect to the application of every solid part is its lifetime.
In particular, the failure of a solid material under a repetitive load is called mechanical
fatigue [Andrew, 1995], which is responsible for about 90% of all the mechanical service
failures [Campbell, 2008]. For our daily life, safe use without the fear of failure is of
fundamental interest.
It is a challenge to predict the long-term performance of a material under a load. The first
publications on the systematic analysis of fatigue can be found from the 1830s by English,
French and German engineers, investigating the then-new phenomenon of cyclic fatigue of
metal in construction and machines like railways. The most well-known name related to
fatigue at that time is August Wöhler (“Versuche über die Festigkeit der
Eisenbahnwagenachsen” (“Study on the Stability of Railroad Vehicle Axes”) [Wöhler, 1867]),
who analyzed the number of (sinusoidal) cycles (N) to failure of a metal dumbbell sample
as a function of the applied stress amplitude (S), nowadays known as Wöhler or S-N curve.
However, Wöhler curves is an old characterization tool for mechanical fatigue under
8
dynamic loading but is still state-of-the-art since it represents the simplest way to
characterize and quantify the fatigue resistance of a material. The main drawback of Wöhler
curves are their typically large standard deviations [Bathias, 2013], and the resulting
limited meaning for an individual sample. The deformation mode of choice is nowadays
sinusoidal and can be performed with a low signal to noise (S/N) ratio.
The mechanical fatigue testing of polymers must also take a polymer specific property into
account: their viscoelasticity and their nonlinearity at large strain amplitudes [Sauer,
1980]. Recent methods of fatigue analysis in deformation/strain controlled tests, providing
more information about the current state of fatigue, without using additional devices
(camera), are based on the stress response analysis of the material. The stress response
can be analyzed in strain/strain rate or time domains. In the strain domain (hysteresis
loops or so-called Lissajous curves), although (large) changes can be seen by the naked eye,
they can hardly be quantified. In the time domain, different parameters and analysis
methods exist based on a FT of the stress. As an example, the nonlinear viscoelastic
parameter (NVP) was proposed which represents the sum of the stress higher harmonics
over the fundamental, analog to the total harmonic distortion for sound systems. Another
approach is to analyze the higher harmonics separately which has the advantage, especially
in torsion/shear, of separating between the material dependent parameters (higher odd
harmonics) and parameters related to asymmetry in the stress response (higher even
harmonics). These concepts are described next.
1.1.2 Viscoelasticity of solid polymers
1.1.2.1 Viscoelasticity of polymers
Rheology investigates the flow behavior of materials under deformation. The basic
parameters are therefore the normalized deformation, the strain () and the resistance force
(F) of the material, typically normalized by the loaded surface (A) to get a system
independent quantity, which is the stress (σ):
𝜎 =𝐹
𝐴
(1)
Static and dynamic deformations can be applied to a material, but the focus here is on
dynamic (sinusoidal) deformations.
1.1.2.2 Linear viscoelasticity
The typical models to describe elasticity and viscosity are the ideal elastic solid and the
ideal viscous fluid, represented by a spring and dashpot (mechanical analogy). The ideal
elastic solid can be described by Hooke’s law with the modulus G as the proportional
constant between strain and stress. Newton’s law describes the stress for an ideal viscous
fluid, with the viscosity η as the proportional factor between shear rate and stress.
9
𝜎 = 𝐺𝛾 (2)
𝜎 = 𝜂�̇� (3)
Models to describe the rheological and mechanical behavior of viscoelastic materials
combine typically spring and dashpot (Hooke’s and Newton’s law) respectively, in different
numbers and ways. The simplest viscoelastic solid model is the Kelvin-Voigt model,
combining spring and dashpot in parallel. On the other hand, the Maxwell model combines
spring and dashpot in series for a viscoelastic liquid. More advanced models are also
available (Burgers, Jeffreys, etc.).
Under small amplitude oscillatory shear (SAOS), the stress response is linear with the
strain amplitude; i.e. if the strain is sinusoidal, the stress is sinusoidal as well but with a
lag. A complex notation can be used, so the strain is written as (according to Euler’s rule):
𝛾 = 𝑖𝛾0𝑒𝑖𝜔1𝑡 (4)
𝜎 = 𝑖𝐼1𝑒𝑖(𝜔1𝑡+𝛿1) (5)
Consequently, the stress response under oscillatory shear can be completely
mathematically described by a complex function with the magnitude I1 and phase angle δ1.
The magnitude I1 divided by the strain amplitude is the complex modulus (G*), and together
with the phase angle, form a set to describe in a complex plane the real (G’) and imaginary
part (G’’). This regime is called the linear or SAOS regime. In the linear regime, the storage
and loss moduli are independent of the applied strain amplitude and provide rheological
parameters to completely characterize a material in a meaningful way.
1.1.2.3 Nonlinear viscoelasticity
In the nonlinear regime, under the conditions of large amplitude oscillatory shear (LAOS),
the above statements do not hold anymore. Consequently, G’ and G’’ depend on the applied
strain amplitude. Figure 2 presents an example for G’ and G’’ as a function of the strain
amplitude. For small strain amplitudes, G’ and G’’ are constant and independent of the
strain amplitude, but they can increase/decrease for larger strain amplitudes, so:
𝐺 = 𝐺(𝛾0) ≠ constant (6)
For fluids, four main types of G’ and G’’ behavior as a function of strain amplitude were
described [Hyun, 2002]. When G’ and G’’ decrease with increasing strain, strain softening
(Type I) occurs, while the reverse is called strain hardening (Type II). If G’ decreases but G’’
increases before decreasing, leading to a weak strain overshoot (“Payne effect” for filled
rubbers) this is associated to Type III. Finally, when both parameters increase before they
decrease with a strong strain overshoot, Type IV is obtained. However, mixed behaviors out
of these four main types have been reported.
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Figure 2: Typical G' and G'' behavior during a strain sweep test in the linear (SOAS) and nonlinear
(LAOS) regime of a polylactic acid/nanocrystal composite (PLA2002D, Mn = 12.5 kg/mol) at ω1/2π =
1 Hz and T = 165°C, according to [Hyun, 2002].
A stress versus strain curve (so-called Lissajous curve) can be plotted for each cycle. The
shape of a Lissajous curve is a straight line for an ideal elastic body and a circle for an ideal
viscous one [Larson, 2006]. Consequently, the shape of the Lissajous curve can be
correlated to the material viscoelasticity. It shows if a viscous or elastic behavior dominates
and if the stress response is linear or not. However, Lissajous curves can hardly detect
small nonlinear contributions and are unable to quantify them.
1.1.2.4 Large amplitude oscillatory shear (LAOS)
1.1.2.4.1 Fourier Transform
The FT is a reversible mathematical function decomposing a signal into a sum of sine and
cosine functions or (using Euler’s approach) a sum of exponentials [Brigham, 1974]. It
calculates the periodic contributions of a time signal, displaying their amplitudes and
phases (or in a complex plane as the real and imaginary parts) as a function of frequency.
The FT of any real- or complex-time signal s(t), or frequency dependent spectrum S(ω)
(inverse FT), are typically defined as:
𝑆(𝜔) = ∫ 𝑠(𝑡) 𝑒−𝑖𝜔𝑡 𝑑𝑡∾
−∾
(7)
𝑠(𝑡) =1
2𝜋∫ 𝑆(𝜔)𝑒𝑖𝜔𝑡𝑑𝜔
∞
−∞
(8)
11
Moreover, the FT is an orthogonal (linear) transform. Consequently, the sum of two
functions in the time domain correspond to their sum in the frequency domain [Bracewell,
1986].
𝑎𝑠(𝑡) + 𝑏𝑔(𝑡) ↔ 𝑎𝑆(𝜔) + 𝑏𝐺(𝜔) (9)
Since the FT transposes a real (time) signal into a complex (frequency) signal, a FT spectrum
contains real and imaginary parts. In the complex plane, each signal can be considered as
a vector from the origin with a magnitude Iω and a phase φ(ω).
1.1.2.5 Fourier transform rheology
The stress response becomes nonlinear under LAOS conditions [Giacomin, 1993].
Nonlinearity means in this context that the stress and strain/strain rate are not directly
proportional to each other.
𝜎 ≠ 𝛾, �̇� (10)
Figure 3: Fourier spectrum of the stress response of a polylactic acid/nanocrystal composite (PLA2002D,
Mn = 12.5 kg/mol) at 0 = 268%, ω1/2π = 1 Hz and T = 165°C. The signal to noise (S/N) ratio is in the
range of 100000:1.
From an experimental point of view, this means that the modulus and viscosity are affected
by the testing conditions (strain/strain rate, see Figure 2). The linear parameters G’ and G’’
are sufficient to describe the stress response completely in the linear case (Eq. (2-3)).
However, this is not enough in the nonlinear regime, so they partially lose their (physical)
meaning. Therefore, it is necessary to use a more advanced approach to completely describe
12
the stress response. Currently, the nonlinear stress response was analyzed quantitatively
in the strain/strain rate [Cho, 2005; Ewoldt, 2008] or the time domain [Wilhelm, 2002;
Hyun, 2011]. In the strain/strain rate domain, the corresponding Lissajous curves are
decomposed into elastic and viscous moduli, by using Chebyshev polynomials for example.
In the time domain, the stress signal is decomposed into a Taylor series of odd higher
harmonics of the deformation frequency, using FT. This decomposition is better done in the
time domain since it is easier to analyze for a time dependent process like fatigue. However,
Fourier series can be converted into Chebyshev coefficients.
1.1.2.5.1 Odd higher harmonics
The FT of the nonlinear stress response only shows odd higher harmonics, as shown in a
typical FT spectrum presented in Figure 3. This can be explained using the simplest
nonlinear 1D scalar model for an elastic solid body, a nonlinear spring [Doetsch, 2003]. In
such a model, the spring constant (which corresponds to the modulus) depends on the
strain:
𝜎 = 𝐺(𝛾)𝛾 (11)
Additionally, it is assumed that the modulus of the clockwise half cycle G(+) is equal to the
counterclockwise cycle G(-), which is a reasonable assumption for isotropic materials. As a
result, the (complex) modulus depends only on the absolute values of the strain:
𝐺(𝛾+) = −𝐺(𝛾−) = 𝐺(|𝛾|) (12)
𝜎 = 𝐺(|𝛾|)𝛾 (13)
Consequently, if the absolute value of the modulus is expanded in a Taylor series of the
strain, only even powers are obtained:
𝐺(|𝛾|) = 𝐺1 + 𝑎1𝛾2 + 𝑏1𝛾
4+. .. (14)
Inserting Eq. (4) and Eq. (14) in Eq. (13) gives:
𝜎 = 𝐺(|𝛾|)𝛾 = (𝐺1 + 𝑎1𝛾2 + 𝑏1𝛾
4+. . . )𝛾0𝑒𝑖𝜔1𝑡 (15)
𝜎 = 𝐺1𝛾0𝑒𝑖𝜔1𝑡 + 𝑎1𝛾0
3𝑒3𝑖𝜔1𝑡 + 𝑏1𝛾05𝑒5𝑖𝜔1𝑡+. .. (16)
which can be rewritten as:
𝜎 = 𝐼1𝑒𝑖𝜔1𝑡 + 𝐼3𝑒
3𝑖𝜔1𝑡 + 𝐼5𝑒5𝑖𝜔1𝑡+. .. (17)
To better quantify and compare a nonlinear behavior, mostly the third harmonic (I3ω1) is
considered and normalized to the fundamental harmonic (I1ω1) typically written as (I3/1).
From Eq. (16) and (17) it can be seen that the nth harmonic depends on the n power of the
strain amplitude and consequently the In/1 intensity depends on (0)n-1. The I3/1 intensity in
a strain sweep test, including linear and nonlinear regime, typically follow the behavior
described in Figure 4 (log-log plot). Firstly, the I3/1 intensity decreases with a slope of -1,
before it increases with a slope of 2 and levels off to a constant value at very large strain
amplitudes. The decrease of I3/1 with a slope of -1 (mostly in the SAOS regime) is caused by
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the sensitivity limit of the torque transducer (Eq. (18)) [Reinheimer, 2012b]. Below the
minimum instrument torque resolution, In cannot be detected and can be assumed to be
equal to noise N:
𝐼𝑛/1 ∼𝑁
𝐼1∼
𝑁
𝛾0∼ 𝛾0
−1 (18)
Figure 4: Typical response for the third harmonic (I3/1) intensity as a function of the applied strain
amplitude (0), showing the behavior in SAOS and LAOS of a polylactic acid/(treated) nanocrystal
composite (PLA2002D, Mn = 12.5 kg/mol) at ω1/2π = 1 Hz and T = 165°C.
The increasing part of the curve with a slope of 2 is also theoretically expected (Eq. (17)).
However, theoretical calculations predict that I3/1 quadratically depends on the strain
amplitude even in SAOS, but its values become negligibly small (
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intensity scales quadratically with 0 before leveling off, following a power-law relation
[Hyun, 2009; Cziep, 2016]:
𝑄 =𝐼3/1
𝛾02
(19)
For each material (and temperature/frequency) Q is constant over a range of strain
amplitudes and a parameter can be defined characterizing the material which is
independent of the applied strain amplitude:
𝑄0 = lim𝛾0→0
𝑄 =𝐼3/1
𝛾02
(20)
1.1.2.5.2 Even higher harmonics
As described in section 1.1.2.5.1 for isotropic materials (G(+) = G(-)), only odd higher
harmonics should be obtained in the stress FT spectrum under LAOS conditions. However,
higher even harmonics, typically with a lower intensity than the odd ones (I2/1 < I3/1, I4/1 <
I5/1, etc.), can be observed. For isotropic materials, these even higher harmonics are mostly
related to measurement artefacts on the macroscopic level, besides the error range of the
experiment (lack of sensitivity), for example wall slip [Hatzikiriakos, 1991] or inertia [Atalık,
2004]. However, Sagis et al. [Sagis, 2001] theoretically predicted and experimentally showed
the occurrence of even harmonics in the stress response caused by changes on a
microscopic level. For demonstration purpose, they used a viscoelastic material with
anisotropic rigid particles. Their main statement is that for broken shear symmetry to occur
(G(+) ≠ G(-)), odd and even harmonics are needed to completely describe the stress
response. Moreover, Sagis et al. [Sagis, 2001] were able to theoretically and experimentally
show that higher even harmonics first increase with increasing deformation, go to a
maximum, and then decrease at very large strain amplitudes.
1.1.3 Mechanical fatigue
This section describes the most important approaches related to the methodology and
analysis of dynamic fatigue testing. The experiments can be performed under controlled
(constant) strain or stress. The main focus here is on a strain controlled approach, since it
is the most used one towards continuous signal analysis. In strain controlled fatigue
testing, the strain amplitude is constant, typically following a sine function. The stress
response as a function of the number of cycles, can be presented by hysteresis
loops/Lissajous curves as described in Figure 5. Furthermore, the strain amplitude, as well
as the experimental setup/conditions (sampling rate), must be carefully chosen to
completely analyze the stress signal higher harmonics during a fatigue test.
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Figure 5: Typical behavior of the stress response of a strain controlled fatigue test and its corresponding
hysteresis loops/Lissajous curves.
1.1.3.1 Different loading modes
Fatigue tests can be performed under different loading conditions like shear or torsion,
tension/tension, tension/compression or three-point bending/dual cantilever. An
important characteristic for the test results comparability and the higher harmonics
contribution prediction is the stress ratio (R), which is defined as ratio between the
minimum and maximum stress:
𝑅 =minimum stress
maximum stress
(21)
Under oscillatory shear and three-point bending/dual cantilever geometries, the stress ratio
(R) is usually equal to -1, since it is not necessary to use a preload. This simplifies the
higher harmonics analysis and interpretation since the same higher harmonics are
expected as in the rheology of polymer melts under LAOS conditions. Consequently, only
odd higher harmonics are expected, representing the material viscoelastic nonlinearity. In
cases when the even harmonics intensity increases above the noise level during a test, this
indicates that the stress signal is not symmetric anymore and could be related to the
appearance of macroscopic cracks [Hirschberg, 2017].
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However, a preload results in higher even harmonics above the noise level making the
higher harmonics interpretation more complex. But in tension/tension, the deformation
oscillates around a preload value, larger than the deformation amplitude to avoid
compression. At large oscillatory strain amplitudes and consequently large preloads, the
stress response during the half cycle to the stress maximum is different than the one back
to the stress minimum. Consequently, the stress signal is not symmetric and contains even
and odd higher harmonics. This makes more difficult the interpretation of these higher
harmonics since it is not clear which harmonics are caused by the material nonlinearity
compared to the stress signal asymmetry due to the loading conditions. The same comment
is also valid for tension/compression, compression/compression and three-point bending
tests.
1.1.3.2 Wöhler curves
The first systematic approach for testing and analyzing dynamic mechanical fatigue data
can be traced back to the 19th century with the work of August Wöhler [Wöhler, 1867].
Figure 6: Schematic representation of a Wöhler curve with the three typical regimes: low and high cycle
fatigue, followed by the fatigue limit.
Wöhler studied the dynamic mechanical fatigue of metal specimen under a constant load
(stress controlled). The number of cycles to failure were analyzed as a function of the applied
load/stress, resulting in Stress - Number of cycles to failure (S-N) or in so-called Wöhler
17
curves. Wöhler curves have typically three different regimes (Figure 6). Firstly, the low cycle
fatigue regime is associated to high stress amplitudes. In this case, the stress amplitude is
very high and the samples fail within a few cycles (Nf), typically less than 103-104. Secondly,
for lifetime between 104 to 106 cycles,