TP Goutte

TP Goutte

Si l’on dépose une goutte sur une surface lisse elle s’étale si les forces de pesanteur sont plus élevées que les forces capillaires (une toute petite goutte reste sphérique!!)

Si l’on arrive à mesurer la dynamique d'étalement dans le régime inertiel d’après [ ^Guyon2001 ] le rayon de la goute varie comme

où le coefficient k varie entre 1/8 pour une surface lisse et 1/4 pour une rugueuse.

La conservation de la masse peut nous donner une relation entre le volume de la goutte et le rayon final d’étalement

où l’épaisseur h dépend des caractéristiques du liquide (tension de surface) et du couple fluide-solide (angle de contact).

Goutte sur papier buvard

Si nous imaginons le papier buvard comme un milieu poreux

la goutte va s’étaler mais aussi pénétrer à l’intérieur de la structure

- Peut-on observer deux régimes différents inertiel et puis capillaire (si l’on suppose le milieu poreux comme un réseau imbriqué de tubes capillaires)? (la réponse peut être bien non!)

On rappelle ici que pour une montée capillaire la loi Washburn [ Washburn-Wikipedia] donne une loi en k=1/2

- le rayon final de la goutte étalée, peut-on nous donner une idée de l'épaisseur «effective» du papier?

r(t) ∼ t ^k

Matériel:

papier buvard, encre, seringue, caméra, billes Références :

[Guyon 2001] Hydrodynamique Physique, E. Guyon, J-P. Hulin, L. , EDP Sciences, pages 244-47.

[Washburn-Wikipedia] en.wikipedia.org/wiki/Washburn's_equation

V = hπr _f ²

vendredi 22 février 2013

!"#$%%&'()"*+,-)*%$.-$+/*+&)"*-)*0-1/*/)*2*

2)*%$.-$+/*3$"*/)*3(-4/3/)0*+&)"*-)*0-1/*/)*2*("#$%%/*)&0-5/%%/3/)06*7/*1-0*

+/*#/*89*/"0*+,:%-#$+/5*%,(5$;$)/*+/*#/"*("#$%%&'()"*/0*+/*#(3<5/)+5/*%/-5"*0/3<"*

#&5&#0:5$"'.-/"* =<:5$(+/>* &3(5'""/3/)0?6* @/* <A:)(3B)/* /"0* -'%$":* <(-5*

&3(5'5* %/"* #(-<"* +/* 1:%$/5* +&)"* %/"* $3<("&)0/"* #&)&%$"&'()"* +/"* #/)05&%/"*

AC+5(:%/#05$.-/"*/0*&*"/54$*D*:0-+$/5*%/"*<5(<5$:0:"*+/*%,A:%$-3*"-</5E-$+/*&-F*

05B"*1&""/"*0/3<:5&0-5/"*=#,/"0GDG+$5/*&-F*&%/)0(-5"*+/*GHIJK@?>*&-F.-/%%/"*%/"*

0/#A)$.-/"**#%&""$.-/"*+,:0-+/*+/"*E-$+/"*"()0*05B"*+$L#$%/"*D*3/M5/*/)*N-45/6*

O-/*4&-0*%/*)(315/*+/*P/C)(%+"*&""(#$:*&-F*("#$%%&'()"*(1"/54:/"*&4/#*+/*%,/&-*

+&)"*%/*0-1/*+/*<%-"*;5&)+*+$&3B05/*Q*7&*<:5$(+/*+:</)+G/%%/*+/*%,&3<%$0-+/*+/"*

("#$%%&'()"*Q*+-*5/3<%$""&;/*+-*0-1/*Q*R)*+:+-$5/*%/"*;5&)+/-5"*</5')/)0/"*D*

<5/)+5/*/)*#(3<0/*+&)"*%,&)&%C"/*+$3/)"$())/%%/*+-*<5(#/""-"6*R)*+:+-$5/*-)/*

S(53/*<(-5*%&*%($*+,:#A/%%/*<(-5*%&*<:5$(+/*+/"*("#$%%&'()"*!6*

R0&1%$""/T*/F<:5$3/)0&%/3/)0*#/M/*%($*+,:#A/%%/*/0*#(3<&5/T*4("*3/"-5/"*D*4("*<5:+$#'()"6**

!"#$%&'() *)+/"* 0-C&-F* +/* +$&3B05/"* 4&5$:">* +/* %,/&->* +-* ;%C#:5(%>* +-* "/%>* +/*

%,:0A&)(%>*-)*#A5()(3B05/6*U(#-3/)0"*+())&)0*%&*+/)"$0:*/0*%&*4$"#("$0:*+/*%,/&-*

"&%:/* /0* +-* 3:%&);/* /&-G;%C#:5(%* /)* S()#'()* +/* "&* #(3<("$'()6* U/)"$0:* +/*

%,:0A&)(%*V*J>IW6*X$"#("$0:*+/*%,:0A&)(%*V**Y>H*39&6"6**

+",-)./#%')0/12#'3%',45)**

•  -)/*<A(0(*+/*4(05/*3()0&;/*

•  %&*<5:"/)0&'()*+/"*;5&)+/-5"*</5')/)0/"*D*5/0/)$5*<(-5*%,&)&%C"/*+$3/)"$())/%%/*

•  %/*+:0&$%*+/*%,:0&1%$""/3/)0*+/*%&*%($*+/*<-$""&)#/*<(-5*!"

•  +/"*;5&<A$.-/"*<5:"/)0&)0*4("*3/"-5/">*&##(3<&;):"*+/*%/-5"*#()#%-"$()"*5/%&'4/"*D*%,&)&%C"/*

+$3/)"$())/%%/*

•  -)/*#()#%-"$()*"-5*%&*%($*+/*<-$""&)#/*D*%&.-/%%/*(1:$0*!*/0*"-5*%&*4&%$+$0:*+/*%&*3(+:%$"&'()>*"$*

4(-"*&4/T*5:-""$*D*%&*S&$5/6*

•  $+/3*<(-5*!*

X(-"*<(-55/T*&-""$*:0&1%$5*%,/F<5/""$()*+/*!*/)*/Z/#0-&)0*-)/*3(+:%$"&'()*+/*%,:#(-%/3/)06*

R0-+$/T*/)"-$0/*/F<:5$3/)0&%/3/)0*%/*0/3<"*#&5&#0:5$"'.-/*!*+,&3(5'""/3/)0*+/"*("#$%%&'()"6*

X(-"*<(-55/T*:0&1%$5*%&*%($*+,:#A/%%/*/0*%,(5+5/*+/*;5&)+/-5*+/*!**/)*/F<%($0&)0*%/*S&$0*.-/*V*

([*#*/"0*%,:)/5;$/*3:#&)$.-/*+-*%$.-$+/>*

********"&*4&%/-5*3(C/))/*"-5*-)/*<:5$(+/*+,("#$%%&'()>**

********/"0*%,:)/5;$/*3:#&)$.-/*+$""$<:/*+-5&)0*-)/*<:5$(+/*+,("#$%%&'()6***

!=!r∧ρv#t

! = 0

∆φ= k₁(e₁+e₂)−(k₁e₁ +k₂e₂) = (k₁ −k₂)e₂ χ = −1

V

∂V

∂P

!!

!!

!S

= 1 ρc² τ = −!E˙#

!E#

!E#

!E˙#

1

! =!r∧ρv#t

! = 0

∆φ=k₁(e₁+e₂)−(k₁e₁+k₂e₂) = (k₁−k₂)e₂ χ=−1

V

∂V

∂P

!!

!!

!_S = 1

ρc² τ = −!E˙#

!E#

!E#

!E˙#

1

95:+$"/T* +&)"* .-/%%/"* #$5#()"0&)#/"* 4(-"* 4(-"* &M/)+/T* D* %&* +$"<&5$'()* +/"* ("#$%%&'()"6* 8/"0/T*

/F<:5$3/)0&%/3/)0*#/M/*<5:+$#'()6*

! = ! r ∧ ρv #

! = 0

∆φ = k

(e

+ e

) − (k

e

+ k

e

) = (k

− k

)e

χ = − 1 V

∂ V

∂P

!!

!!

!S

= 1 ρc

τ = ! E #

! E ˙ #

! E #

! E ˙ #

1

LESAMORTISSEURSLIQUIDES

Existant dans la plupart des villes nord-américaines et dans les grandes métropoles d’Asie, les gratte- ciel sont de plus en plus nombreux. L’étalement des personnes, la fonctionnalité ou le choix des matériaux sont autant de critères conduisant à des structures de plus en plus hautes, fines et souples.

De ce fait, elles répondent dynamiquement aux sollicitations extérieures (le vent essentiellement).

Afin d’entraver ces vibrations gênantes, voire dommageables, certains architectes ont inclus dans les étages supérieures des structures oscillantes de type pendule (ex: une boule de 660 tonnes arrimée au 92è d’une tour à Taipei). Plus récemment, il a été réalisé qu’une piscine située au sommet de l’immeuble pouvait, grâce au mouvement du fluide, produire le même effet amortisseur.

Dans le compte-rendu:

- Vue d’ensemble: une photo d’ensemble de votre manip

- Paramètres: La liste des paramètres physiques pertinents et ceux que vous allez faire varier

- Dans la boîte: des montages ImageJ qui montrent ce qui se passe et comment ça varie en fonction des paramètres que vous faites varier

- Modèle: Proposer une modélisation simple du système immeuble+piscine - Expé/théo: Un graphique comparant mesures et théorie pour différentes fréquences d’excitation

- Interprétation: Un paragraphe décrivant la physique: à quel archétype se rapporte votre phénomène, quelles sont les forces en jeu ?

Matériel: réglette, pot de yaourt, cellophane.

Commencez par observer les oscillations et l’amortissement naturel d’une structure élastique. Comment caractériser

l’amortissement ? Puis remplissez peu à peu la piscine, et quantifiez le rôle de

l’amortisseur. La piscine a-t-elle une fréquence propre ? Dépend-elle de son remplissage ? Peut-on faire varier la fréquence de vibration de l’immeuble ?

La tour One Rincon Hill à San Francisco Au cours de ce TP, vous démontrerez qu’il

est possible d’amortir les oscillations d’une

structure élastique grâce à un amortisseur

liquide

LESJETSDECHARGECREUSE

À partir du XVIIIè siècle, les ingénieurs des mines ont constaté qu’une charge explosive légèrement évidée pouvait causer beaucoup plus de dégâts qu’une charge classique. Ce principe de la charge creuse est par la suite devenu un principe militaire qui a été à la base de la création du bazooka.

Mais au-delà de ses nombreuses applications, l’effet de charge creuse est sans doute avant tout le prototype des effets de focalisation en mécanique des fluides.

Dans le compte-rendu:

- Vue d’ensemble: une photo d’ensemble de votre manip

- Paramètres: La liste des paramètres physiques pertinents et ceux que vous allez faire varier

- Dans la boîte: des montages ImageJ qui montrent ce qui se passe et comment ça varie en fonction des paramètres que vous faites varier

- Dimensions: Les étapes de votre analyse dimensionnelle

- Expé/théo: Un graphique comparant mesures et théorie

- Interprétation: Un paragraphe décrivant la physique: à quel archétype se rapporte votre phénomène, quelles sont les forces en jeu ?

Matériel: tubes à essai, éthanol

Faites chuter un tube empli d’un liquide mouillant et observez la formation d’un jet de charge creuse. Quelles sont les

paramètres pertinents dans ce problème ? Quelles courbes allez-vous tracer ?

Pourquoi y-a-t-il un creux dans le liquide initialement ?

Jet liquide formé par impact

Au cours de ce TP, vous reproduirez et quantifierez l’effet de charge creuse.

FIG. 6. Photograph of a disk of lead (8 in. diameter, 2 in. thick) which was placed 18 in. from a charge like that in Fig. 3). When this charge exploded, the jet made the hole near the bottom. The slug is embedded near the top of the picture. Obviously the slug and jet often do not travel the same path.

their deepest penetrations when they are placed a short distance away from the target surface, the unlined cavity charges produce their deepest penetrations when they are placed in direct contact with the target.

Figure 8 shows the effect produced by an

FIG. 7.^JPhotograph of a steel cylinder, like tha(in Fig.

3, cut in half after being shot with a charge from"which the steel cone had been removed. This unlined charge was detonated in contact with the cylinder because it produces its deepest holes at zero standoff, whereas linedfcharges produce their deepest holes at a larger standoff.

568

ordinary charge with no cavity. This charge obviously contained more explosive than the others, since it had the same dimensions but con- tained no cavity.

These phenomena can be readily understood by applying some very simple theoretical con- siderations which will now be presented.

THEORY OF JET FORMATION WITH CONICAL AND WEDGE-SHAPED LINERS An elementary mathematical discussion can be given of the formation of jets from conical and wedge-shaped liners. Consider a charge having a cross section as shown in Fig. 9. In the conical case the charge is obtained by rotat- ing the cross section about the dashed axis of symmetry. The wedge-shaped case is obtained by supposing the cross section to be the same for an indefinite distance perpendicular to the cross section-or, equivalently, to be confined be- tween rigid walls at both ends. \Ve first consider the case of a wedge.

The mechanism of liner collapse is illustrated in Fig. 10. Starting from the booster a detona-

FIG. 8. Photograph of a steel cylinder that has been cut in half after being blasted by a solid charge. This charge contained more explosive than the others, since it' was cast in the same mold without a cavity.

JOURNAL OF ApPLIED PHYSICS

Downloaded 22 Nov 2011 to 134.157.34.225. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

FIG. 3. Photograph of a solid steel cylinder (3.25 in. in diameter, 7 in. in length) which has been shot and then sawed in half to show the nature of the hole produced. A cross-sectioned replica of the charge that produced this hole is shown in the position that the real charge occupied before it was detonated. The real charge contained 0.25 lb. of Pentolite. The cavity liner was a steel cone (0.025 in. thick, 1.63-in. base diameter).

which they have been used. Consequently, fur- ther discussion will be confined to charges with these two shapes of liners. Since most of the re- liable data have been obtained with cone-lined charges similar to that shown in Fig. 2, much of the following discussion applies only to them.

The basic principles in the phenomenon> are fairly simple. The mechanism of cone collapse and the resulting formation of high speed jets have been revealed independently in the United States, England, and Germany by means of high speed radiographs⁶ made of charges during the explosion process. Referring to Fig. 2, the detonation of the booster starts an explosive wave down the charge. When this wave reaches the apex of the thin-walled steel cone it suddenly produces very high pressures on the outside of

• Much pioneer work toward understanding this phe- nomenon was done by W. M. Evans and A. R. Ubbelohde under the auspices of the British Ministry of Supply.

6 The clearest and most revealing photographs the writers have seen were taken by L. B. Seely and J. C.

Clark at the Aberdeen Ballistics Research Laboratory; the earliest photographs were taken by J. L. Tuck in England.

566

FIG. 4. Photograph of a solid lead cylinder (4 in. in diameter, 9.5 in. in length) which has been treated the same as the steel cylinder in Fig. 3. The steel slug from the liner can be seen embedded in the lead about 5 in. from the bottom of the cylinder.

this cone that cause its walls to collapse. The forces are so great that the strength of the steel has a negligible effect on the process, and the steel can be treated as though it were a perfect fluid. The explosive pressures on the outside cause the thin walls of the cone to move inward nearly perpendicular to their surfaces at high velocities. The moving steel retains a conical shape with the apex moving to the right along the axis. To the left behind the moving apex is found a section of thoroughly collapsed cone which contains metal only from the outer part of the cone. The inner part of the cone forms a jet which is squeezed out from the inner apex of the lining and travels at high speeds along the axis, to the right in Fig. 2. In other words, the metal in the cone lining divides into two parts with the dividing surface between these two parts being a cone lying somewhere between the inner and outer surfaces of the original hollow cone. The metal from the outer cone forms into a slug that travels to the right in Fig. 2 at rela- tively slow speeds (500 to 1000 meters/sec.). The metal from the inner cone forms into a jet that JOURNAL OF ApPLIED PHYSICS

Downloaded 22 Nov 2011 to 134.157.34.225. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Birkhoff et al., J. Appl. Phys (1948)

Explosive Ogive

Conioal Steel tiner

FIG. 1. Cross section of the head of a U. S. Army bazooka, showing conical steel liner in the shaped charge.

weapon, called the Panzerfaust, was launched from a small light steel pipe employing the recoilless mortar principle. It is a rather large shaped-charge projectile stabilized in flight by tail fins. All of these weapons were briefly de- scribed in the Popular Science of February, 1945. They made it possible for an infantryman operating from ambush to defeat the largest tank. The Germans also had a shaped-charge weapon for hand placing which was equipped with permanent magnets to hold it to a tank until its time fuse set it off. Characteristically, the Japanese developed a shaped charge (Lunge mine) fastened to the end of a long wooden pole.

The Japanese solider was expected to charge out from ambush and jam this charge against the side of a tank. Obviously it was another suicide weapon designed on the principle that the life of one or two Japanese soliders was not too high a price to pay for the destruction of a tank.

Since the ability of these weapons to pierce targets is independent of the striking velocity, much of the experimental work has been done with statically fired charges. Figure 2 shows a typical experimental hollow charge lined with a steel cone. When the booster is detonated, it starts an explosive wave which travels parallel with the axis of the charge and collapses the steel cone starting from its apex. The collapsing cone forms a long thin jet of steel that travels to the right in the figure along the extension of the cone axis at speeds up to 30,000 ft./sec.^l

1 Rifle bullets ordinarily have muzzle velocities around 2000-3000 ft./sec., though with special devices projectiles have been shot from gun barrels up to 5000 ft./sec.

F. Zwicky and F. Whipple have suggested using these jets as artificial meteors for controlled studies, since their velocities are about the same as the velocities of the slower meteors. Shaped charges are carried to the upper atmos-

564

These high speed jets force their way through armor plate in much the same way as a stream of water from a powerful fire hose would force² its way through a bank of soft mud. The pres- sures produced by the jet traveling at these high velocities are so much greater than the ultimate strength of the armor plate that the strength of the armor plays a negligible role in retarding the penetration. In fact, mild steel provides almost as much protection as hard-steel armor against these weapons.

The lined hollow-charge principle is such a unique and surprising phenomenon that a great deal of thought has been given to possible peace- time Of the many suggestions that have been advanced, only a few have proved to be practical, since the desired results can often be accomplished as well or better in other ways.

Shaped charges have frequently been used for the rapid boring of holes in demolition work.

After a hole has been blown with a shaped charge, the cavity can be filled with explosive for further blasting. Such a procedure is quick and easy, but because of the relatively high cost of the explo- sives used in shaped charges it can seldom be justified on a cost basis. High explosives of the type used in military operations are needed to obtain satisfactory performance with shaped charges. The performance with the cheaper dyn<).mites and other low pressure explosives is definitely inferior.

phere by V-2 rockets and exploded when the rocket has reached its highest point. Calculations indicate that these

"meteors" should be observable from the ground.

2 Like most analogies this is far from perfect, for while a stream of water washes mud out of a mud bank the jet of metal does not wash or erode metal out of the target.

Careful weighings have shown that a metal jet is captured by a metal target, which loses no weight except a very small amount at the front surface. The hole is produced by plastic flow of the target material in a radial direction.

JOURNAL OF ApPLmD PHYSICS

Downloaded 22 Nov 2011 to 134.157.34.225. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions

Schéma de principe d’un bazooka

TP séance Dans le compte-rendu:

- Vue d’ensemble: une photo d’ensemble de votre manipe

- Paramètres: La liste des paramètres physiques pertinents et ceux que vous allez faire varier

- Dans la boîte: des montages ImageJ qui montrent ce qui se passe et comment ça varie en fonction des paramètres que vous faites varier

- Dimensions: Les étapes de votre analyse dim (nombres sans dim...), et la loi de puissance

- Expé/théo: Un graphique comparant mesures et théorie avec plusieurs valeurs du préfacteur

- Interprétation: Un paragraphe décrivant la physique: à quel archétype se rapporte votre phénomène, quelles sont les forces en jeu.

La conservation de l’énergie nous dit que l'énergie mécanique totale soit l'énergie potentielle plus l’énergie cinétique se conserve

Roulement (encore) des billes/cylindre sur un plan incliné.

Prenez différentes billes : l’énergie totale est la même? Les vitesses? Quels sont les paramètres qui varient? Il y a-t-il un «défaut» d’énergie?

Le cylindre peut être «rempli» de pâte à modeler : à quantité égale étudiez l’effet de la répartition de la masse (par exemple symétrique ou a-symétrique).

Pourriez vous calculer le moment cinétique?

Matériel: plan incliné, billes, cylindres, pâte à modeler

Théorème de Koening

(dus -- il y en a 2 -- à Johann Samuel König (allemand né à Büdinger en 1712, mort à Amerongen aux Pays-Bas en 1757). Ils permettent d'exprimer le moment cinétique (ou angulaire) et l'énergie cinétique d'un système de points matériels sous des formes plus facilement interprétables physiquement.)

«L’énergie cinétique d’un solide est égale à la somme de l’énergie cinétique du centre de masse et de l’énergie cinétique du solide correspondant à son mouvement dans le référentiel barycentrique.»

Bref

g

La petite bille rattrape-t-elle la grande?

E _p + E _c = cte

Course à l'échalote

Ceci n’est pas un Koening

E _c = 1

2 M v _G + 1

2 I ω ²

jeudi 21 février 2013

!"#$%&'()*+,(+-(./0$.'1+$)+.(+20&%'+,(#+3$)'(4..(#+

5)4+ %6&+ 7&#+ 8&4'+ 20&%'(*+ )%(+ 3$)'(4..(+ ,(+ 349*(+ .(+ #$4*+ &)+ 8$%,+ ,(#+

3$):(#;+ <(#+ 3)'+ ,(+ 2(+ =>+ (#'+ ,6".)24,(*+ .6$*4:4%(+ ,(+ 2(+ #$%+

2&*&2'"*4#?@)(+ ,(#+ 3$)'(4..(#+ (%'&/"(#+ ('+ ,(+ .(+ 2&*&2'"*4#(*A+

-4#'$*4@)(/(%'+)?.4#"+7&*+B$(%4:+7$)*+8&3*4@)(*+.(+7*(/4(*+&%&.C#()*+

,(+#7(2'*(+,(+D$)*4(*+,)+#$%E+.(+F+*"#$%&'()*+,(+-(./0$.'1+GE+,$%'+.&+

3$)'(4..(+ (#'+ )%+ (H(/7.(E+ (#'+ )?.4#"+ 7$)*+ .64%#$%$*4#&?$%+ ,(#+ 7$'#+

,6"20&77(/(%'+('+(%+&2$)#?@)(+&*204'(2')*&.(A+

>&*+&4..()*#E+26(#'+)%+(H(/7.(+#4/7.(+,(+,4#7$#4?8+&2$)#?@)(+'(.+@)6I+*"#$%&%2(E+.&+.$%:)()*+,6$%,(+

,)+#$%+(#'+:*&%,(+,(J&%'+.&+'&4..(+,)+*"#$%&'()*A+<&+J4'(##(+('+.&+7*(##4$%+#$%'+,$%2+0$/$:9%(#+,&%#+

20&@)(+ 7&*?(+ ,(+ .&+ 3$)'(4..(+ K:$).$'E+ 2$*7#LA+ M%+ 2$%#'&'(+ (H7"*4/(%'&.(/(%'+ ('+ $%+ 7()'+ N)#?O(*+

*4:$)*()#(/(%'+ @)6I+ 2&)#(+ ,(+ 2(.&E+ .&+ 3$)'(4..(+ (H24'"(+ 7&*+ .(+ #$)P(+ #(+ 2$/7$*'(+ 2$//(+ )%+

$#24..&'()*+0&*/$%4@)(+,$%'+.&+*&4,()*+(#'+,)(+I+.&+*"#4#'&%2(+I+.&+2$/7*(##4$%+,(+.6&4*+2$%'(%)+,&%#+

.(+ 2$*7#+ ,(+ .&+ 3$)'(4..(+ ,(+ J$.)/(+ Q+ ('+ .&+ /&##(+ (#'+ &77*$H4/&?J(/(%'+ 2(..(+ ,(+ .6&4*+ 2$%'(%)+ ('+

$#24..&%'+(%+3.$2+,&%#+.(+:$).$'+,(+.$%:)()*+<+('+,(+#(2?$%+,6&4*(+RA+

S%+ J$)#+ #$)J(%&%'+ @)(+ .&+ 2$/7*(##434.4'"+!+ ,6)%+ /4.4()+ K2$/7*(##434.4'"+ 4#(%'*$74@)(E+ 26(#'TIT,4*(+

2(..(+2&*&2'"*4#&%'+.(#+2$/7*(##4$%#T,"'(%'(#+*&74,(#+,)+/4.4()+&##$24"(#+I+.&+7*$7&:&?$%+,)+#$%L+

(#'+.4"(+I+#&+,(%#4'"+!+('+I+.&+J4'(##(+,)+#$%+,&%#+2(+/4.4()+!+7&*+.&+*(.&?$%+U+

! =!r∧ρv#t

! = 0

∆φ = k₁(e₁+e₂)−(k₁e₁+k₂e₂) = (k₁−k₂)e₂ χ =−1

V

∂V

∂P

!!

!!

!_S = 1

ρc²

1

7*$7$#(1+)%(+8$*/(+7$)*+.&+.$4+,6"20(..(+I+.&@)(..(+$3"4'+.&+8*"@)(%2(+,(+*"#$%&%2(+"+,(+.&+3$)'(4..(A+

Q$)#+7$)**(1+(%#)4'(+"J(%')(..(/(%'+7*"24#(*+2(V(+.$4+,6"20(..(+(%+(H7*4/&%'+.&+/&##(+('+.&+*&4,()*+

,(+ .6$#24..&'()*+ (%+ 8$%2?$%+ ,(#+ :*&%,()*#+ 2&*&2'"*4#?@)(#+ ,)+ #C#'9/(+ ('+ (%+ )?.4#&%'+ .6(H7*(##4$%+

2.&##4@)(+,(+.&+8*"@)(%2(+,(+*"#$%&%2(+,6)%+$#24..&'()*+0&*/$%4@)(+(%+8$%2?$%+,(+#&+/&##(+('+,(+#&+

*&4,()*A+

!"#$%&'()*))%(+3$)'(4..(E+)%+,4#7$#4?8+,6&..$%:(/(%'+,)+:$).$'E+)%+/42*$E+)%+$#24..$#2$7(A+

S%+ 8&4#&%'+ J&*4(*+ .(+ J$.)/(+ ,)+ 2$*7#+ ,(+ .&+ 3$)'(4..(+ K2$//(%'+ WL+ ('+ .&+ .$%:)()*+ ,)+ :$).$'E+ ('+ (%+

/(#)*&%'+2$//(%'+.&+8*"@)(%2(+"+,)+20&%'+,(+.&+3$)'(4..(+J&*4(E+"'&3.4##(1+(H7"*4/(%'&.(/(%'+.&+.$4+

,6"20(..(+ I+ .&@)(..(+ $3"4'+"+ ('+ '(#'(1+ .&+ J&.4,4'"+ ,(+ .&+ /$,".4#&?$%+ 7*$7$#"(A+ X$//(%'+ /$%'*(*+

#4/7.(/(%'+&J(2+J$'*(+,4#7$#4?8+@)(+#().+4%'(*J4(%'+.(+J$.)/(+,)+2$*7#E+('+7&#+#&+8$*/(+W++

+",-)./#%')0/12#'3%',45)*)

•  .&+7*"#(%'&?$%+,(#+:*&%,()*#+7(*?%(%'(#+I+*('(%4*+7$)*+.6&%&.C#(+,4/(%#4$%%(..(+

•  .(+,"'&4.+,(+.6"'&3.4##(/(%'+,(+.&+.$4+,(+7)4##&%2(+7$)*+"#

•  ,(#+:*&704@)(#+7*"#(%'&%'+J$#+/(#)*(#E+&22$/7&:%"#+,(+.()*#+2$%2.)#4$%#+*(.&?J(#+I+.6&%&.C#(+

,4/(%#4$%%(..(+

•  )%(+2$%2.)#4$%+#)*+.&+.$4+,(+7)4##&%2(+I+.&@)(..(+$3"4'+"+('+#)*+.&+J&.4,4'"+,(+.&+/$,".4#&?$%E+#4+J$)#+

&J(1+*")##4+I+.&+8&4*(A+