ANNÉE 2014
THÈSE / UNIVERSITÉ DE RENNES 1 sous le sceau de l’Université Européenne de Bretagne
pour le grade de
DOCTEUR DE L’UNIVERSITÉ DE RENNES 1 Mention : mécanique
Ecole doctorale SDLM présentée par
Yannick Laridon
Préparée à l’unité de recherche TERE
(Technologie des équipements agroalimentaires) Irstea
Modélisation et visualisation d’une croissance de bulle dans un milieu viscoélastique évolutif et hétérogène
Thèse soutenue à Rennes le 5 juin 2014
devant le jury composé de :
Valérie GUILLARD
Maître de conférence, UMR iAte Montpellier rapporteur
Sofiane GUESSASMA
Chargé de recherche, INRA Nantes rapporteur
Jean-Dominique DAUDIN
Directeur de recherche, INRA Theix-Clermont examinateur
Alain LE BAIL
Professeur, ONIRIS Nantes examinateur
Tiphaine LUCAS
Directrice adjointe scientifique, Irstea Rennes directeur de thèse
Denis FLICK
Professeur, UMR 1145, Paris co-directeur de thèse
La reconnaissance silencieuse ne sert à personne
— G. B. Stern
R E M E R C I E M E N T S
La première de mes pensées va à Tiphaine. Il est difficile de donner sur le papier la mesure ma gratitude à son égard, mais son implication sans faille à chaque étape, la force de travail qu’elle su (et pu) déployer, l’invariable méticulosité qui caractérise son travail, sans oublier sa sympathie ont été autant de facteurs clés pour l’aboutisse- ment de cette thèse. Un très grand merci à Denis pour sa patience, sa persévérence, la grande richesse scientifique de ses apports et la constante justesse de ses réflexions. Je tiens à remercier très chaleureusement David pour son accompagnement, son soutien sans faille, sa disponibilité ininterrompue, son amitié, son grand sens physique et la pertinence ses contributions. Je ne désespère pas de danser ensemble une bourrée en bal un de ces jours !
Je remercie très sincèrement Valérie Guillard et Sofiane Guessasma qui ont accepté la tâche de rapporteurs de ce mémoire, ainsi que Jean-Dominique Daudin et Alain Le Bail qui ont accepté de les rejoindre dans le jury.
Tous les travaux réalisés dans le cadre de cette thèse ont bénéficié de la contribution de beaucoup de personnes. Un grand merci à Christophe pour son aide à l’analyse nu- mérique des problèmes étudiés, et pour sa ténacité dans le débogage des programmes qui m’ont accompagné au quotidien. Merci aussi aux personnes de l’équipe IRMFood et de l’unité TERE avec qui j’ai pu travailler. Je remercie en particulier Sylvain pour sa grande disponibilité et sa bonne humeur, Dominique et Michel pour l’ingéniosité dont ils ont fait preuve et qui ont permis que la validation expérimentale se passe aussi bien que possible, Stéphane pour sa pédagogie dans ma découverte de l’IRM, Amina pour son soutien technique et, last but not least, un très grand merci à Yvonne, Julie et Marie-Christine pour la très grande qualité de leur accompagnement. Merci aussi à François de m’avoir accueilli, et d’avoir supporté mes velléités à l’encontre des logi- ciels propriétaites en général, et d’Office en particulier. Merci aussi à lui et à tous les partenaires du projet U 2 M-ChOp pour leurs retours sur le travail accompli pendant cette thèse. En particulier, je remercie Camille pour avoir contribué à lever le voile de brume qui entourait la rhéologie à la sortie de mon master, et tout le monde à Agro- ParisTech Massy pour m’avoir accueilli et avoir réussi en un temps très bref à me faire renouer avec un accessoire que je pensais avoir laissé au vestiaire il y a longtemps : la blouse. Merci aussi à Guy et Benoît pour leur recul très appréciable par rapport au produit.
Toute ma reconnaissance et ma sympathie pour Maëlle et David H., qui ont contri- bué de façon non-négligeable à l’aboutissement de cette thèse.
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Il serait injuste de laisser penser que ces années se résument uniquement aux inter- actions professionnelles. Aussi, je tiens à remercier très chaleureusement tous les gens qui rendent plaisante la vie à l’I rstea de Rennes. Je pense aux différentes équipes de Mosaïque, mais aussi à ceux que j’ai eu le plaisir de rencontrer et avec qui j’ai partagé des moments conviviaux pendant ces quelques années. Une pensée amicale à tous les doctorants du centre. Tout spécialement, je voudrais remercier Cécile, Clément, Del- phine et Faustine de s’être trouvés là au même moment. L’aventure n’aurait pas été aussi agréable sans eux.
Naturellement, une pensée aux membres de ma famille qui se demandent bien com- ment on peut faire des trucs et des machins sur du fromage, mais qui n’ont jamais remis en question mon choix de me lancer dans cette entreprise et m’ont toujours soutenu.
Enfin, un merci tout particulier à Riwalenn pour son soutien infaillible.
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À mon grand-père.
TA B L E D E S M AT I È R E S
Introduction 1
Chapitre 1 14
1 n e w a s s e s s m e n t m e t h o d o f t h e v i s c o e l a s t i c pa r a m e t e r s o f m a - t e r i a l s . a p p l i c at i o n t o s e m i - h a r d c h e e s e . 15
1 i n t r o d u c t i o n 17
2 m at e r i a l s a n d m e t h o d 21
2 . 1 Modelling of the compression-relaxation test . . . . 21
2 . 1 . 1 Modelling hypotheses . . . . 21
2 . 1 . 2 Governing laws . . . . 22
2 . 2 Identification of the Maxwell model parameters . . . . 24
2 . 2 . 1 Instantaneous compression . . . . 25
2 . 2 . 2 Linear compression . . . . 25
2 . 2 . 3 Non-linear compression . . . . 27
2 . 3 Implementation of the proposed method . . . . 27
2 . 4 Implementation of the reference method . . . . 29
2 . 5 3 -D models . . . . 29
2 . 6 Simulated data sets . . . . 30
2 . 6 . 1 Reference material . . . . 30
2 . 6 . 2 Application of the methods to replicates of simulated solicitation trials and estimation of their effectiveness . . . . 32
2 . 7 Compression-relaxation tests of semi-hard cheese . . . . 32
2 . 7 . 1 Experimental procedure . . . . 32
2 . 7 . 2 Sample preparation . . . . 33
2 . 7 . 3 Measurement of force-position-time data: compression- relaxation experiment . . . . 33
2 . 7 . 4 Validity of the modelling hypotheses and numerical applications 34 3 r e s u lt s a n d d i s c u s s i o n 37 3 . 1 Optimisation of the conditions for identification of parameters of the last Maxwell element with the proposed method . . . . 37
3 . 2 Estimation of the uncertainty associated with the proposed method . . 38
3 . 2 . 1 Effects of experimental noise . . . . 38
3 . 2 . 2 Effects of cross-head speed of the upper plate of the rheometer . 41 3 . 2 . 3 Effects of non-parallelity between the upper and lower surfaces of the sample . . . . 43
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3 . 2 . 4 Estimation of the overall uncertainty in the determination of the
Maxwell model parameters . . . . 46
3 . 3 Application to semi-hard cheese . . . . 47
3 . 3 . 1 Effects of ripening on the viscoelastic parameters of the Maxwell model . . . . 47
3 . 3 . 2 Effects of location in the cheese block on the Maxwell model vis- coelastic parameters . . . . 52
4 c o n c l u s i o n 55 Chapitre 2 61 2 m o n i t o r i n g a s i n g l e e y e g r o w t h u n d e r k n o w n g a s p r e s s u r e : m r i m e a s u r e m e n t s a n d t e a c h i n g s a b o u t t h e m e c h a n i c a l b e - h av i o u r o f a s e m i - h a r d c h e e s e 61 1 i n t r o d u c t i o n 63 2 m at e r i a l a n d m e t h o d s 65 2 . 1 Cheese . . . . 65
2 . 2 Sample preparation . . . . 65
2 . 3 Experimental procedure . . . . 67
2 . 4 Methods of measurement . . . . 68
2 . 4 . 1 Water content . . . . 68
2 . 4 . 2 Gas pressure measurements within the bubble . . . . 68
2 . 4 . 3 MRI measurements . . . . 69
2 . 5 Image and data analysis . . . . 70
2 . 5 . 1 Bubble volume . . . . 70
2 . 5 . 2 Radii . . . . 71
2 . 5 . 3 Upper surface deflected shape . . . . 73
2 . 5 . 4 Bi-extensional strain rate and strain within the cheese cylinder . 74 3 r e s u lt s 77 3 . 1 Water content . . . . 77
3 . 2 Bubble volume . . . . 77
3 . 3 Bubble radius and upper deflected shape . . . . 79
4 d i s c u s s i o n 81 4 . 1 Ability of MRI and associated image analysis to describe the geometry of bubble and cheese dynamically . . . . 81
4 . 2 Low pressure was found to be able to inflate the cheese bubble . . . . . 81
4 . 3 Quasi-absence of time-independant elasticity and small elastic strain in cheese during bubble growth . . . . 82
4 . 4 Linear response of the cheese and low bi-extensional strain rate . . . . . 83
5 c o n c l u s i o n 85
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Chapitre 3 87
3 m o d e l l i n g o f t h e m e c h a n i c a l d e f o r m at i o n o f a s i n g l e b u b - b l e i n s e m i - h a r d c h e e s e , w i t h e x p e r i m e n ta l v e r i f i c at i o n a n d
s e n s i t i v i t y a na ly s i s 87
1 i n t r o d u c t i o n 89
2 m at e r i a l s a n d m e t h o d s 93
2 . 1 Experimental procedure . . . . 93
2 . 1 . 1 Sample preparation . . . . 94
2 . 1 . 2 Measurements . . . . 94
2 . 1 . 3 Data analysis and uncertainties . . . . 95
2 . 2 Modelling . . . . 96
2 . 2 . 1 Model assumptions . . . . 96
2 . 2 . 2 Governing equations . . . . 96
2 . 2 . 3 Boundary conditions . . . . 97
2 . 2 . 4 Evaluation of model parameters . . . . 98
2 . 3 Numerical implementation . . . . 99
2 . 4 Sensitivity analysis . . . . 100
2 . 4 . 1 Screening method . . . . 101
2 . 4 . 2 Statistical method . . . . 101
3 r e s u lt s a n d d i s c u s s i o n 103 3 . 1 Sensitivity analyses . . . . 103
3 . 2 Experimental validation . . . . 104
4 c o n c l u s i o n s 111 Chapitre 4 115 4 s i m u l at i o n o f b u b b l e g r o w t h i n s e m i - h a r d c h e e s e w i t h m a s s a n d m o m e n t u m t r a n s p o r t : c o m pa r i s o n w i t h e x p e r i m e n t a n d s e n s i t i v i t y a na ly s i s 115 1 i n t r o d u c t i o n 117 2 e x p e r i m e n ta l p r o c e d u r e a n d d ata a na ly s i s 121 3 m o d e l d e s c r i p t i o n 123 3 . 1 Geometry . . . . 123
3 . 2 Hypotheses . . . . 124
3 . 3 Mechanical behaviour . . . . 124
3 . 4 Mass transport . . . . 124
3 . 5 Coupling of transport phenomena . . . . 125
3 . 6 Boundary conditions . . . . 126
3 . 7 Initial conditions . . . . 126
3 . 8 Numerical implementation and calculations . . . . 127
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4 e s t i m at i o n o f va l u e s f o r i n p u t pa r a m e t e r s 129
4 . 1 Carbon dioxide production rate . . . . 129
4 . 2 Carbon dioxide diffusivity . . . . 130
4 . 3 Carbon dioxide concentration in cheese and saturation . . . . 131
4 . 3 . 1 Estimation of CO
2saturation . . . . 131
4 . 3 . 2 Estimation of CO
2concentration directly on cheese material . . . 131
4 . 3 . 3 Comparison between experimental CO
2concentration and ex- pected concentration at saturation . . . . 132
4 . 4 Mechanical properties of cheese . . . . 134
5 r e s u lt s a n d d i s c u s s i o n 137 6 c o n c l u s i o n s 143 Chapitre 5 145 5 f i r s t s t e p s t o wa r d s a b e t t e r u n d e r s ta n d i n g o f t h e g r o w t h o f m u lt i p l e n e i g h b o u r i n g b u b b l e s i n c h e e s e 145 1 i n t r o d u c t i o n 147 2 m at e r i a l a n d m e t h o d s 149 2 . 1 Experimental set-up . . . . 149
2 . 2 Cheese sampling . . . . 151
2 . 3 Experimental procedure . . . . 152
2 . 4 Measurements . . . . 153
2 . 4 . 1 Pressure and temperature . . . . 153
2 . 4 . 2 MRI sequences . . . . 153
2 . 5 Image analysis for indirect assessment of bubble volume . . . . 154
2 . 5 . 1 Determination of the volume of other cavities . . . . 154
2 . 6 Model of bubble growth and carbon dioxide transport . . . . 155
3 p r e l i m i na r y e x p e r i m e n ta l r e s u lt s 157 3 . 1 Experiment 1 . . . . 158
3 . 2 Experiment 2 . . . . 160
3 . 3 Experiment 3 . . . . 163
3 . 4 General discussion of the experimental results . . . . 165
4 s i m u l at i o n r e s u lt s 167 5 c o n c l u s i o n s 173 Conclusions et perspectives 175 Annexes 181 a f a b r i c at i o n d ’ u n f r o m a g e à pât e p r e s s é e n o n - c u i t e 183 a . 1 Définition . . . . 183
a . 2 Étapes de fabrication . . . . 184
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a . 2 . 1 Préparation du lait . . . . 184
a . 2 . 2 Fabrication du fromage . . . . 184
a . 2 . 3 Affinage . . . . 185
b p r i n c i p e s d ’ i m a g e r i e pa r r é s o na n c e m a g n é t i q u e ( i r m ) 187 b . 1 Résonance magnétique nucléaire ( rmn ) . . . . 187
b . 1 . 1 Spin, moment magnétique . . . . 187
b . 1 . 2 Principe de mesure rmn — T
1et T
2. . . . 188
b . 2 Imagerie . . . . 189
b . 2 . 1 Appareillage . . . . 189
b . 2 . 2 Séquences . . . . 191
b . 2 . 3 Obtention des images — Pondération . . . . 192
b . 3 Apports et contraintes . . . . 192
Bibliographie 195
b i b l i o g r a p h i e 197
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TA B L E D E S F I G U R E S
Figure 1 . 1 Experimental relaxation curve and step responses calculated with 3 , 4 , and 5 Maxwell elements. . . . 31 Figure 1 . 2 Simulated force-time step responses generated from the refer-
ence material (see Table 1 . 1 ) before the addition of noise, and its comparison with the best adjustment provided by the pro- posed and the reference methods performed on the force-time simulated dataset for the reference material, with and without noise added. . . . 39 Figure 1 . 3 Effects of cross-head speed of the upper plate of the rheometer
on the determination of Young’s modulus with the reference and the proposed methods; errors calculated from Eq. 1 . 44 us- ing simulated sets of force-time data generated from the refer- ence material without noise (see Table 1 . 1 ). . . . 42 Figure 1 . 4 Effects of cross-head speed of the upper plate of the rheometer
on the relaxation parameters with the reference method (+) and the proposed method (o); errors calculated from Eq. 1 . 44 using simulated sets of force-time data generated from the reference material without noise (see Table 1 . 1 ). Vertical line at 3 mm/s indicates the value retained for the experimental study. . . . . 42 Figure 1 . 5 Stress-strain curves computed with the reference material (see
Table 1 . 1 ) with the angle β ( 0 °, 1 ° and 2 °) formed by the upper surface of the sample with the horizontal line; comparison with a typical experimental stress-strain curve from a compression- relaxation test performed on semi-hard cheese. . . . 44 Figure 1 . 6 Effects of the angle β ( 0 °, 1 ° and 2 °) formed by the upper surface
of the sample with the horizontal line,on the determination of the Young’s modulus by the proposed and reference methods applied to simulated sets of force-time data generated from the reference material without noise (see Table 1 . 2 ). The horizontal line indicates the Young’s modulus of the reference material. . 45 Figure 1 . 7 Step responses obtained with the proposed and reference meth-
ods applied to the step response simulated for the reference material, with noise added. Effects of the angle β ( 0 °, 1 ° and 2 °) formed by the upper surface of the sample with the horizontal line. Solid red line indicates the step response obtained with the reference material parameters. . . . 45
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Figure 1 . 8 Evolution of Young’s modulus at core of cheese block during ripening. The error bar is the overall uncertainty due to the proposed method. Vertical lines at d = 0 denote the change in room temperature. The horizontal plain and dotted lines rep- resent the mean value calculated over all values and the overall uncertainty (see values in Table 1 . 4 ). . . . 50 Figure 1 . 9 Evolution of relaxation times (a) and proportions (b) of core
samples of cheese material during ripening (vertical line at d = 0 refers to change from cold to warm storage rooms). The horizontal plain and dotted lines represent the mean value cal- culated over all values and the overall uncertainty (see values in Table 1 . 4 ). . . . 51 Figure 1 . 10 Effects of x-position in the cheese block on (a) Young’s modulus,
and (b) step responses. The error bar in (a) is the overall uncer- tainty due to the proposed method. The horizontal plain and dotted lines in (a) represent the mean value calculated over all values for 4 ≤ x ≤ 9 cm and the overall uncertainty (experimen- tal variability and uncertainty produced by the identification method). . . . 53 Figure 1 . 11
Experimental set-up for cutting the samples. The boring cylinder was placedinside the cylinder to avoid strain when lowering the blades.
. . . . 59 Figure 1 . 12
Typical experimental force-time curve from a compression-relaxation test per-formed on a semi-hard cheese sample.
. . . . 59 Figure 1 . 13
Location of samples in the cheese block. The XY plane is located at the middleof the cheese block height.
. . . . 60 Figure 2 . 1 Experimental device for the bubble growth experiment. (a) Lo-
cations of the cylinders of cheese sampled in the cheese block and used for bubble growth experiment and for water content analysis. The 32 samples used for the analysis of the gradi- ent of water content can be seen using the left and right hand side drawings (the lattices) (b) Cylinder of cheese in its sustain- ing apparatus. The height of the cylinder of cheese is in the ( O, depth ) direction of the larger cheese block and the width of the cylinder of cheese is in the ( O, x ) direction. . . . 66
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Figure 2 . 2 Typical MRI image and associated processed inages. (a) Raw MRI image containing the largest section of the bubble (largest horizontal lines constituted by the voxels in the bubble) amongst the 88 images of the 3 D reference sequence ( 300 ) be- fore the creep. (b) Thresheld image of the image presented in (a). (c) A mask was applied in order to make the thread rod disappear in the image (b). (d) Masked and thresheld image at the end of the creep. . . . 71 Figure 2 . 3 Dimensional calculations applied to a typical bubble in cheese,
with their uncertainties. Calculations were applied onto the im- age presenting the largest dimensions for the bubble, and after thresholding and labeling of the MRI image. The bubble pre- sented here was acquired with the 3 D ( 500 ) sequence, at the end of the creep. The pixel indetermination on x
iand y
iare presented. y
band y
tare the lowest and highest positions of the largest diameters in the image. . . . 73 Figure 2 . 4 Comparison between the MRI method and a comparator gauge
applied to a reference upside surface of a cheese cylinder roughly cut with a knife. The vertical bar depicted the com- bined standard deviation on the comparator measurement (in- cluding three measurements at each of the 18 points) and the horizontal bar, the standard uncertainty on the horizontal posi- tion of the comparator gauge. . . . 75 Figure 2 . 5 Contours were obtained with the 3 D ( 300 ) on the cylinder of
cheese. The initial contour at t0 was the contour obtained just before the creep and the contour at the End of the creep (at t1) was obtained just after the beginning of recovery. dR
han dR
vwere variations in horizontal radius between t0 and t1. L was the distance between the right-hand side of the bubble and the right-hand side of the cylinder. . . . 76 Figure 2 . 6 (a) Volume and air pressure according to time; creep, P > 0 ,
recovery, P = 0 , lowering in pressure, P < 0 . The natural pres- sure in semi-hard cheese (Grenier et al. 37 ) is indicated by the grey area in (a). Bars are the expanded standard uncertainties on the pressure ( 0 . 3 kPa) and on the volume (as a function of the volume see section 3 . 1 ). (b) Details of the volume and pres- sure according to time for the stages of creep, (c) the recovery and (d) the lowering in pressure. . . . 78
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Figure 2 . 7 (a) Horizontal and vertical radii according to time (creep); ( 1 ) Instantaneous increase in horizontal radius, ( 2 ) 3.7 · 10
−7m/s, ( 3 ) linear increase in horizontal radius, 3.2 · 10
−8m/s deflected shapes according to time (c) creep (d) recovery. The expanded standard uncertainty (not shown) on the upper surface de- flected shape was evaluated at 0 . 20 mm (see Table 2 . 1 and sec- tion 3 . 1 ). . . . 80 Figure 3 . 1 Sustaining apparatus (the values given for the heights of the
cheese cylinder, the bubble and the radius of the bubble are specific to Experiment 2 ). . . . 94 Figure 3 . 2 (a) geometry and boundary conditions of the domain and (b)
the associated mesh retained after mesh sensitivity analysis (see section 2 . 3 for more details).For each experiment, r
cwas set at 27 mm. . . . 98 Figure 3 . 3 Sensitivity indices for the model parameters selected, at dif-
ferent simulation times. (a) sensitivity indices on the bubble volume, (b) sensitivity indices on the upside surface deflected shape. The higher the indices, the greater the influence of the parameter on the model output. . . . 106 Figure 3 . 4 Selection of second order sensitivity indices for bubble volume,
at t = 100 ks. . . . 107 Figure 3 . 5 MRI cross-sections of the experimental set-up at the beginning
(left) and end (right) of the experiment. . . . 107 Figure 3 . 6 (a) Experimental deflected shape of the upside surface of cheese
over time, with no apparent initial strain (Experiment 2 ); (b) Ex- perimental deflected shape over time, with initial strain (Exper- iment 3 ). Uncertainties on the upside surface are not shown for clarity, but were estimated at ± 0 . 2 mm. Plain horizontal line denotes the expected height of the samples. . . . 108 Figure 3 . 7 Experimental and simulated bubble volumes and headspace
pressures (a, c, e) displacement of the middle of the upside surface (b, d, f) for Experiments 1 , 2 , 3 respectively. Due to combinatory effects, the propagation of uncertainties did not necessarily yield output curves that were centered on the refer- ence ouput, especially for (b, d, f). . . . 109 Figure 4 . 1 Axisymmetrical geometry used for modelling (left), and do-
main meshing (right) . . . . 123 Figure 4 . 2 Carbon dioxide production under rind (top) and at core (bot-
tom) at 293 K. Two replications are shown for each graph, i.e.
one bottle for each replication. . . . 135
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Figure 4 . 3 Experimental and simulated bubble volumes at the core of the cheese block. Effect of the variations of input parameters of the model all other parameters being set at fixed value: (a) t
sat, (b) r
CO2, (c) D
chCO2