Conclusion

Il n’existe pas a l’heure actuelle de modele permettant de prevoir de maniere satisfaisante les

effets arrieres d’un impact non penetrant (( hautes )) vitesses sur le thorax. La question : (( Quelle

est, de l’onde de choc ou du cone dynamique, la partie du choc la plus dangereuse, celle vers

laquelle l’attention du concepteur de protection doit etre dirigee? )), posee il y a vingt ans, est

toujours d’actualite.

Notre objet n’etait pas de trancher cette question mais d’explorer sur le plan theorique la

partie (( onde de choc )) de l’impact. Nos resultats, associes a ceux obtenus par d’autres auteurs

concernant le cone dynamique, completent la caracterisation de la reponse du thorax. Bien

entendu, seules des etudes experimentales sur le vivant pourront, in fine, demontrer une letalite

plus importante de l’une ou l’autre des parties du choc ; mais la modelisation a un role d’autant

plus important a jouer que les experimentations sur le vivant sont difficiles a mettre en œuvre,

tant sur le plan ethique que sur le plan technique.

Nos motivations pour etudier specifiquement l’onde de choc tenaient en deux points :

i) cette partie du choc n’avait recu que tres peu d’attention (les modeles developpes actuellement,

utilisant la methode des elements finis, sont concus pour etudier les consequences lesionnelles de

l’enfoncement du cone dynamique et les maillages sont trop grossiers pour rendre compte de la

propagation de l’onde de choc) ;

ii) le poumon, organe vital, est repute tres sensible aux sollicitations (( rapides )).

La presente etude apporte, a notre connaissance, les premiers resultats theoriques sur la

transmission de l’energie dans le poumon dans le cas d’un chargement tres court du thorax

(d’une duree de l’ordre de 100 µs). Les apports du travail de these pour la comprehension de la

reponse du thorax sont synthetises dans le chapitre 9 ; nous y avons montre comment l’ensemble

de nos resultats peuvent s’articuler pour construire un mecanisme lesionnel.

Pour ce travail, nous nous sommes donne un modele idealise du thorax : un milieu stratifie

elastique, le modele (( couche sur substrat )). En outre, nous avons fait les hypotheses que la

reponse du thorax n’entraıne que de petites deformations et de petits deplacements (bien qu’une

premiere apprehension du phenomene physique de l’impact sur le thorax puisse laisser imaginer

le contraire, mais il faut se rappeler que la partie cone dynamique, associee, elle, a de grands

deplacements, est ignoree).

L’etude du modele (( couche sur substrat )) a clarifie l’analyse de la reponse en ce qu’elle a

136 Conclusion

permis de dissocier les phenomenes physiques locaux — sous le point d’impact — de la reponse

globale de la structure (mise en vibration des organes, influence de la rigidite de l’ensemble de

la cage thoracique, etc.). Ceci aux depens d’un fort realisme ; le modele est en effet incapable

de fournir des resultats quantitatifs dans des zones anatomiques bien determinees, il fournit

seulement des (( tendances )) de la reponse.

Sur le plan de la Mecanique, l’etude a consiste a calculer la reponse dynamique d’un milieu

stratifie. Nous avons principalement exploite des methodes analytiques pour la resolution des

equations de l’elastodynamique, plutot que des methodes purement numeriques de type elements

finis, differences finies ou elements de frontieres. Ceci a permis, en particulier,

i) de contourner certaines difficultes numeriques (comme celles mises en evidence dans le chap. 8)

et

ii) de calculer des expressions analytiques pour la reponse du thorax propices a une analyse

physique.

Une part de l’originalite de notre travail reside dans le choix de la methode de Cagniard-de

Hoop, associee a la theorie des rayons generalises, pour calculer des reponses. La methode est,

semble-t-il, la plus appropriee au calcul de fonctions de Green transitoires dans les milieux

stratifies elastiques. Dans de nombreux cas, elle seule fournit des solutions exactes, valables en

champ (( proche )) comme en champ (( lointain )) et ce quelle que soit la duree de la sollicitation.

Cette derniere caracteristique s’est revelee un atout majeur pour le traitement du probleme

physique car la (( duree )) de l’onde de choc est mal definie ; ainsi nous avons pu, avec la meme

methode, calculer des reponses pour toute la gamme de duree de chargement mesurees (0-300 µs).

Le present travail rassemble, au fil des chapitres de resultats, des exemples d’applications de

la methode de Cagniard-de Hoop. A plusieurs reprises, la mise en œuvre de la methode, tant

en ce qui concerne l’interaction des ondes avec les interfaces, le calcul de reponses en des points

particuliers que l’implementation des solutions, a donne lieu a des developpements originaux

dont l’interet depasse la cadre de l’application biomecanique.

Nous pensons avoir exploite le modele de thorax (( couche sur substrat )) autant que possible ;

le travail presente est une etude quasi-exhaustive du comportement du thorax vu comme une

telle structure.

Nous esperons que les fruits du present travail seront utiles, d’une part pour concevoir des

experiences a venir, et d’autre part pour guider en partie la construction des futurs modeles

numerique (( realiste )) du thorax, capables a la fois de prendre en compte la propagation de

l’onde de choc et de decrire avec precision la geometrie des organes.

L’avenir de la modelisation de la reponse du thorax a un impact non penetrant (( hautes ))

vitesses est la modelisation associee a la methode des elements finis, a l’image de ce qui est realise

dans l’industrie automobiles pour des chocs (( basses )) vitesses. Pour le calcul de la propagation de

l’onde de choc avec de telles methodes, il faudra au prealable s’efforcer de resoudre les difficultes

inherentes a la brievete de la sollicitation.

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[119] J.H. Yu et J.H. Stuhmiller. (( A surrogate model of thoracic response to blast loading )).Rapport Technique DAMD17-85-C-5238, JAYCOR, 11011 Toreyana Rd., San Diego. CA,92138, 1989.

References non citees

Lexique des regles typographiques en usage a l’imprimerie Nationale. Imprimerie Nationale, 2002.

Le Petit Robert, dictionnaire de la langue francaise. 2001.

Webster’s Students Dictionary. American Book Company, New-York, 1953.

M. Grevisse. Precis de grammaire francaise. DeBoeck Duculot, 2001.

Annexe A

Un modele pour la propagation des

ondes dans le poumon

Cette annexe est la retranscription d’un article publie dans Journal of Biomechanics (volume35, p. 1081-1089, 2002) intitule (( A one-dimensional model for the propagation of transientpressure waves through the lung )), par Quentin Grimal, Alexandre Watzky And Salah

Naıli.

Resume

Contexte. La propagation des ondes longitudinales dans le poumon a ete etudiee par denombreux auteurs dans le cadre de la physiologie respiratoire, de techniques ultrasonores ouencore de la biomecanique de l’impact. Dans la plupart des etudes theoriques, le poumon a etemodelise comme un materiau isotrope et homogene, et en utilisant la loi de Hooke pour lecomportement mecanique (voir par ex. [38,47,62]).

L’hypothese de milieu homogene peut etre mal appropriee pour certains problemes ou lesfrequences mises en jeu sont relativement elevees : a cause de sa structure proche de celle d’unemousse, le poumon est susceptible d’avoir un comportement qui depend de la frequence.

D’autre part, des lesions sont souvent observees dans les poumons quand de l’energie mecaniqueest delivree tres rapidement au thorax : impact non penetrant, corps a proximite d’une explosion,etc. Le poumon est susceptible d’etre endommage a la fois par une deformations globales (grandsdeplacements de l’organe) et par un phenomene (( hautes )) frequences associe a la propagationd’une onde.

Objectifs. L’etude presentee dans cette annexe a un double objectif :1) preciser, a l’aide d’un modele theorique, l’influence de la microstructure du poumon (alveoles)sur la propagation des ondes transitoires afin d’evaluer la fiabilite d’un modele de poumon ho-mogene ;2) caracteriser, a l’aide du meme modele theorique, la propagation d’une onde liee a un mecanismede blessure (( hautes )) frequences.

Modelisation. Le poumon est vu comme un milieu poreux a cellules fermees, c’est-a-dire quel’on suppose que les phenomenes etudies sont si courts que l’air n’a pas le temps de s’ecoulerpendant le passage d’une onde. La microstructure est representee comme un empilement 1D,

146 Un modele pour la propagation des ondes dans le poumon

bi-periodique, de strates elastiques d’air et de tissu mou : les strates A ont les proprietes de l’eau(paroi des alveoles), et les strates B celles de l’air. L’empilement ainsi constitue est faiblementcouple : les strates A sont (( dures )) et les strates B (( molles )).

Des resultats theoriques sur la propagation des ondes dans les milieux bi-periodiques [32,33] ont mis en evidence que dans le cas d’un faible couplage entre les strates, l’empilementa un comportement equivalent a celui d’une chaıne masses-ressorts : les parois des alveoles secomportent comme des masses rigides et l’air dans les alveoles joue le role de ressorts. Afin desimplifier le traitement mathematique du probleme, nous substituons donc au modele de milieucontinu par morceaux du poumon, un modele de milieu discret : une chaıne masses-ressorts.L’approximation qui est faite en identifiant l’empilement (milieu continu) a la chaıne masses-ressorts consiste a negliger la propagation des ondes de plus hautes frequences (( piegees )) al’interieur des strates.

Resultats. Des resultats sont presentes dans les domaines frequentiel et temporel. Le modelepredit une vitesse de propagation des ondes proche de celles mesurees experimentalement (del’ordre de 40 m.s−1), et une frequence de coupure voisine de 90 kHz. Nous trouvons que la taillemoyenne des alveoles joue beaucoup sur le comportement du modele vis-a-vis de la frequence(frequence de coupure, vitesse de phase, etc.).

Le modele est capable de quantifier les erreurs liees a l’utilisation d’un modele homogene, telque ceux utilises dans les logiciels de calcul par elements finis ou dans les modeles analytiquesetudies dans les chapitres 2 a 7. Une impulsion est plus ou moins perturbee par la microstructuresuivant son contenu frequentiel. Une impulsion qui a un contenu frequentiel (( suffisamment ))

bas, est peu perturbee : elle se propage dans le modele masses-ressorts comme dans un modele demateriau homogene. Une impulsion dont le contenu frequentiel est (( suffisamment )) eleve, subitde la dispersion. Dans la reference [50], nous avons montre que l’onde propagee dans le poumonsuite a une explosion type [61] ((( blast ))) a proximite du corps est susceptible d’etre assezperturbee par la microstructure tandis que celle due a un impact non penetrant type (projectileutilise avec les armes non-letales de 30 g et 60 m.s−1 [8]) n’est pratiquement pas perturbee.

Le differentiel de pression entre deux alveoles consecutives est un critere de lesion hy-pothetique [22] qui serait lie a un mecanisme de lesion par etirement excessif des parois alveolaires.Le calcul du differentiel pourrait permettre d’associer un risque de lesion a un impact donne.Nous montrons comment le modele permet de formaliser ce mecanisme de blessure et commentil peut fournir des valeurs du differentiel de pression engendre par la propagation d’une ondedonnee.

Abstract

The propagation of pressure waves in the lung has been investigated by many authors concer-ned with respiratory physiology, ultrasound medical techniques or thoracic impact injuries. Inmost of the theoretical studies, the lung has been modeled as an isotropic and homogeneous me-dium, and by using Hooke’s constitutive law (see, e.g., [38,47,62]), or more elaborated materiallaws (see, e.g., [14, 105, 117]). The hypothesis of homogeneous medium may be inappropriatefor certain problems. Because of its foam-like structure, the behavior of the lung — even if theair and the soft tissue are assumed to behave like linearly elastic materials — is susceptible tobe frequency dependant. In the present study, the lung is viewed as a one dimensional stack ofair and soft tissue layers; wave propagation in such a stack can be investigated in an equivalent

A.1. Introduction 147

mass-spring chain [32,33], where the masses and springs respectively represent the alveolar wallsand alveolar gas. Results are presented in the time and frequency domains. The frequency de-pendence (cutoff frequency, variations in phase velocity) of the lung model is found to be highlydependent on the mean alveolar size. We found that short pulses induced by high velocity im-pacts (bullet stopped by a bulletproof jacket) can be highly distorted during the propagation.The pressure differential between two alveoli is discussed as a possible injury criterion.

Keywords: wave, injury mechanism, lung, impact, ultrasound

A.1 Introduction

Severe lung contusions are often observed after blunt impacts on the thorax or exposure toexplosions. For “high” velocity impact — when the thoracic wall velocity at the impact point ismore than 30 m.s−1 — and blasts, many authors have suspected the propagation of a pressurewave through the thorax to be the main cause of lung injury [14, 23, 25, 37, 105, 106]. However,more investigation is needed to understand which are the principal features of the transientwave involved (duration, intensity or frequency content). In experimental works Yen et al. [118],Fung [36], Cooper and Maynard [23], Cooper et al. [24,25], Yu and Stuhmiller [119] have shownthat the velocity and acceleration of the thoracic wall at the impact point correlate well with theoccurence of lesions. The time history of the pressure wave generated in the thorax is dependanton the motion of the thoracic wall; it has been considered as an indicator for impact lethalityby Stuhmiller et al. [106] and Cooper et al. [23, 25]: they reported that waves that rise slowlyto their peak value have less lethality than those characterized by a fast rising shock front. Inother words, the authors suggest that lung injury is associated with a high frequency damagemechanism. Ultrasounds (US) techniques developed for medical applications use high-frequencycontinuous (images of internal organs) or pulsed sound waves (destruction of kidney stones);while they are considered safe for use in humans, small animals’ lung tissue can be damaged.The damage observed, called capillary lung bleeding, require more research before it can be fullyexplained; for some time, this injury was thought to be due to a cavitation phenomenon until newobservations made this explanation doubtful [30]. The susceptibility for capillary bleeding seemsto be species dependant (depending, among other things, on the alveolar size) and correlates withthe peak pressure and frequency of the US wave. In contrast, the risk of injury does not appearto depend on time-averaged intensity, exposure time or pulse repetition frequency [30,84]. Thereare similarities between capillary lung bleeding observed with US and lung damage followingblasts or high velocity blunt impacts — Pode et al. [87] have demonstrated that, since it inducessimilar injuries, a shock-wave lithotriptor may be used to study the physiological consequencesof blasts on rats. One may suspect that the same injury mechanisms are involved.

Typical lung injury consecutive to a high velocity impact on the thorax or a bomb explosionare edema and hemorrhage. Fung et al. [37] review the literature on alveolar trauma: edemais likely to be caused by a change of permeability of the epithelium induced by overstrechingof the alveolar walls while hemorrhage is a consequence of the fracture of blood vessels of thealveolar wall. Fung et al. [37] proposed and tested the hypothesis that lung injury is caused by an

148 Un modele pour la propagation des ondes dans le poumon

excess of tensile strain in alveolar walls that occurs during the re-expansion of collapsed alveolifollowing the passage of a pressure wave. However, the strain rates used in Fung’s experimentswere orders of magnitude slower than those generated in a lung subjected to the impact wavesof interest in the present study. Another hypothesis discussed by Cooper et al. [24] is that, dueto its high rate of pressure rise, the impact wave may generate important pressure jumps acrossalveoli walls, leading to overstreching and fracture.

The speed of sound in air and water (the main component of the parenchymal tissue) arerespectively about 340 m s−1 and 1500 m s−1. Pressure waves, which will propagate at the speedof sound, might therefore be expected to propagate in the lung at a speed between 340 m s−1

and 1500 m s−1. However, this is not the case: the measured wave speeds in the lung vary from 1m s−1 to 40 m s−1 [15,36,62]. Many authors have studied wave propagation in the lung with thehelp of the elastic wave theory in homogeneous media; it has allowed to understand some featuresof the propagation like wave speeds (see for instance [15, 62]) and cutoff frequencies for surfacewaves [38]. In the elastic wave theory, the wave speed c is calculated with the elastic modulusE and the density ρ of the media with the formula: c =

√E/ρ. Since the lung is made of two

media (air and tissue), several wave velocities can be calculated as combinations of the two elasticmoduli and the two densities; the low wave speeds measured correlate well with theoretical wavespeeds calculated by using the theory of mixtures, i.e., the composite elastic modulus and theaverage density given by E = [(1−α)/Eg +α/Et]−1 and ρ = (1−α)ρg +αρt respectively, wherethe subscripts g and t stand for the gas and the tissue respectively, and where α and (1−α) arethe volumetric ratios of tissue and gas. Using finite element methods (see [14,105,117]) require,in practice, to represent the lung as a homogeneous medium, so that the mesh does not have tobe too fine (and the computing time not too long); the underlying hypothesis is that the wavepropagation is not perturbed by the microstructure. However, it may not be appropriate forwaves of high frequency content generated by high rates of loading of the thoracic wall. Authorsof experimental works try to correlate measurements of displacements, velocities, accelerationsor pressures at the thoracic wall (or excised lung surface) to the occurrence of injury insidethe thorax of several mammal species. Their model generally use homogeneous media; one maywonder if the steep front measured before the pleural surface keeps its shape unchanged as itpropagates through the foam-like structure of the lung. Clearly, such a heterogeneous mediumis susceptible to show uncommon wave propagation patterns depending on the wavelengthsinvolved.

In this study, we investigate a one-dimensional model of the parenchyma that takes intoaccount its “bi-phasic” nature, that is, the succession of air pockets and tissue. As opposed tothe models used in former studies (Bush and Challener [14] in their study of wave propagationin lung sections and references [38, 47, 62, 105, 117]), the model introduced here can reveal afrequency-dependant behavior in relation with the structure. More precisely, the model is usedto discuss the following points: i) whether the foam-like structure of the lung is likely to distortthe pressure wave history, and when the homogeneous medium assumption is valid; ii) theinfluence of the dimensions of the lung structure (that vary between mammal species used inthe experiments) on injury thresholds; iii) the generation of transeptal pressure differences fordifferent impact characteristics.

Our model is based on the assumption that the lung behaves like a closed-cell foam, i.e.,during the passage of the wave, the air stays in the alveoli; on the basis of Butler et al. [15]work, this assumption is adequate for the study of frequencies of more than a few hundredsof Hertz. We assume that the parenchyma can locally be modeled as a one-dimensional bi-

A.1. Introduction 149

periodic stack of tissue and air layers as shown in Figure A.1. Each unit set is made of a “soft”layer (the gas) and “hard” layer (the tissue); due to the succession of soft and hard layers, thestratified medium is “weakly-coupled”. Very interesting results concerning the propagation oftransient waves in these kinds of media have been published recently [19, 32, 33]; it has beenshown by El-Raheb [32,33] that most of the transient waves propagation patterns are the samein a weakly-coupled bi-periodic stack of layers and in a mass-spring chain; this allows to describethe transient wave propagation with simple mathematical expressions. Following El-Raheb, thelung model presented in this paper describes the wave motion in the lung by the displacementof rigid bodies (tissue between alveoli), coupled with springs (the air in the alveoli), about anequilibrium position. The waves trapped in one layer — that are of negligible intensity in aweakly coupled stack [32] — are not taken into account in this model. This model is supportedby comments and results of many authors: among them, Fung et al. [37] mentioned that a lungmodel should recognize the compressibility of the lung gas and the mass of the lung tissue;Jahed et al. [62] showed experimentally that the velocities of waves transmitted through thelung are only weakly dependant on gas density; Bush and Challener [14] have already analyzeda multilayer sandwich of air and tissue (very similar to the stack represented in Fig. 1) with thefinite element method.

p1(t)

ρB , EB

ρB , EB (air) hB

p1(t)

m

Kunit set

ρA , EA (tissue) hA

Fig. A.1: One-dimensional weakly-coupled bi-periodic layered media (left) and the equivalentmass-spring chain (right). Modele 1D de milieu bi-periodique faiblement couple (gauche) et le modelemasses-ressorts equivalent (droite).

With this introduction as background, we present in section 2 the mass-spring chain modelof the lung and give the dispersion equation together with the solution for the transient pressureat each mass. In section 3, the model is investigated both in the frequency domain with thehelp of the dispersion equation, and in the time domain for typical impact wave histories. Insection 4, we give a physical meaning to our results and try to shed light on some experimentalfacts.

150 Un modele pour la propagation des ondes dans le poumon

A.2 Method

The spongy lung parenchyma is composed of microscopic air cells (alveoli), roughly polyhe-dral in shape, bounded by thin interfacial membranes (alveolar walls) composed of a fibrillatedconnective tissue surrounded by a few muscular and elastic fibers. In human, the air cells diame-ter vary from 75 µm to 360 µm [40], being largest on the surface of the lung, at the borders andat the apex, and smallest in the interior. At the alveolar surface the air is saturated with watervapor at body temperature — the relative partial pressure of water is 6.2% [21], the density isρg = 1.11 kg.m−3 [114] and the speed of sound is cg = 357 m.s−1 [65], for later use, we defineYoung’s modulus for the gas to be Eg = ρgc

2g. The thickness of the alveolar walls is about

5µm. Compared to human, mammals used in experiments have smaller alveoli diameters: theratios to the mean human alveolar diameter are 1/5 for the mouse, 2/5 for the rabbit and 2.3/5for the pig [85]. Besides alveoli, the parenchyma is pervaded by blood vessels, by bronchioleemanating from the main bronchial tubes, by fiber surrounding openings in the alveolar wallsand by fibrous tissues extending from the pleural membrane. On the whole, only about 10% ofthe lung is occupied by tissue, the remainder being filled with air. The parenchyma itself can beregarded as an aggregate of individual air cells separated by thin alveolar walls.

In the present paper, the propagation of transient pressure waves in the lung is investigatedin a one-dimensional model, illustrated in Figure A.1, by considering a bi-periodic stack of layersrepresenting the lung tissue (layers of type A) and the air pockets (layers of type B). Since weare not interested in studying the effects of various reflections that might occur at the lungboundaries, it is sufficient to consider a semi-infinite stack; the boundary condition is a givenpressure history on the first layer A (alternatively, a displacement history may be given forthe first layer A, the corresponding equations are found in reference [33]). We define a unitset to be an assemblage of a layer A and a layer B. In the stack, each layer is defined to belinearly elastic, homogeneous and isotropic; their Young’s modulus, density and thickness arerespectively denoted by E, ρ and h, subscripts A and B refer to the corresponding layers. As aconsequence of the succession of layers A and B, this lung model is not homogeneous. Each unitset is made of a “hard” layer A (the tissue) and a “soft” layer B (the gas) so that the layeredmedium is said to be weakly-coupled (it is made of two media with very different mechanicalproperties). Following El-Raheb [32], the propagation of transient waves in the weakly-coupledcontinuous bi-periodic medium is investigated by considering an equivalent discrete mass-springsystem consisting of masses m connected with springs of stiffness K [32, 33]. The equivalentparameters m and K of the discrete system are derived from the properties of the continuousmedia:

m = ρAhA + ρBhB

K = (hA/EA + hB/EB)−1 (A.1)

We also define hs = hA + hB to be the thickness of the unit set.The behavior of the continuous medium in the frequency domain gives insight into the ap-

proximation (a complete mathematical justification is found in reference [32]). The continuoussystem exhibits an infinite number of alternating propagation (PZ) and attenuation zones similarto pass and stop bands in a filter. Let ωc be the smallest cutoff frequency; PZ1 (the first propa-gation zone) extends from ω = 0 to ω = ωc. It can be shown that for weakly-coupled systems,PZ1 is paramount in propagation; in other words, only signals of (relatively) low frequency aremeaningful to describe wave propagation. Indeed, most of the energy carried by high frequencywaves is trapped in the first layers of the stack due to the small transmission coefficients betweenhard and soft layers. Furthermore, El-Raheb [32] shows that ωc for a continuous weakly-coupled

A.3. Results 151

bi-periodic media approximates the cutoff frequency of the equivalent mass-spring chain. In fact,it appears that motions of the continuous system in PZ1 are those of the mass-spring chain:each hard layer acts as a rigid mass and each soft layer acts as a massless spring. This way, thestudy of wave propagation in a continuous medium with a huge number of interfaces is reducedto the study of wave propagation in a simple equivalent mass-spring chain model. The model isuseful to study the propagation of signals of any frequency content at a scale larger than a fewalveolar diameters, that is, more than 1 mm.

Equations of motion for the semi-infinite mass-spring chain of Figure A.1 exited at its endare, in terms of axial displacement un of each mass n, for the mass n = 1 and masses n (n 6= 1)respectively

mu1 + K(u1 − u2) = p1(t),mun + K(un − un−1) + K(un − un+1) = 0,∀n = 2, 3, ...,

(A.2)

where un means twice differentiation with respect to time and p1(t) is the source history. Pressurein the n-th spring is given by

pn = K(un − un−1). (A.3)

In the following section, we will study the system both in the frequency and time domains.In the frequency domain, we will use the frequency-dependant phase velocity cp(ω). Its ex-pression is a dispersion equation found by introducing the harmonic progressive wave solutionun = exp i(ωt − kxn) in (A.2), where i2 = −1, ω is the angular frequency, k = ω/cp(ω) is thewavenumber and the abscissa of mass n by xn = (n− 1)hs. We have

cp(ω) =ω

k=

ωhs

2(arcsin(ω/2ω0))−1, (A.4)

where ω0 =√

K/m is the natural angular frequency of the mass-spring set. From (A.4), wecan show that a cutoff frequency occurs at 2ω0. For the study in the time domain, we need thetransient pressure; using the Fourier transform of (A.2) and (A.3) and after some algebra, thepressure for any mass n is found to be [33]

pn(t) = δn1p1(t) + 2n∫ t

0

J2n[2ω0(t− τ)]t− τ

p1(τ)dτ, (A.5)

where J is the Bessel function of the first kind and δij is the Kronecker symbol.

A.3 Results

EA (Pa) 2.2e9ρA (kg.m−3) 1000

hA (µm) 5 (Medium I) 3(Medium II)EB (Pa) 141.5e3

ρB (kg.m−3) 1.11hB (µm) 360 (Medium I) 100(Medium II)

Tab. A.1: Mechanical parameters of the continuous layered media represented in Figure A.1.Parametres mecaniques du milieu continu stratifie represente sur la figure A.1.

The mass-spring chain model is investigated with a set of parameters associated with thephenomenon under study. The densities and elastic moduli of the hard and soft layers (ρA, ρB,

152 Un modele pour la propagation des ondes dans le poumon

EA and EB) are kept constant (see Table A.3) while the geometric parameters hA and hB varyin order to investigate the influence of alveoli mean size on wave propagation — we supposethat Young’s moduli and densities of the alveoli walls and of the alveoli air are not subjectedto meaningful physiological variations.

A.3.1 Frequency domain analysis

0

20

40

60

80

100

120

0.00E+00 1.00E+05 2.00E+05 3.00E+05 4.00E+05

frequency (Hz)

phas

e ve

loci

ty (

m s

-1)

hB=400 µm

hB=300 µm

hB=200 µm

hB=100 µm

hB=50 µm

hB=20 µm

Fig. A.2: Phase velocity as a function of frequency for different alveolar diameters. hA = 5 µmis kept constant. Vitesse de phase en fonction de la frequence pour differents diametres des alveoles.hA = 5 µm est maintenu constant.

Figures A.2 and A.3 show the phase velocity (given by (A.4)) as a function of the frequencyfor various alveolar diameters hB and for two different alveolar wall thicknesses hA; values forYoung’s moduli and densities are collected in Table A.3. The cutoff angular frequency ωc

diminishes and the phase velocity increases as the air cells diameter becomes larger. Both themean value and the amplitude of the variations of the phase velocity in the angular frequencyrange (0, ωc) vary with the alveoli dimensions. For large diameters of air cells, the amplitude ofthe variations of the phase velocity is important: for example, when hA=5µm and hB=300µm

(see Fig. A.2), the amplitude is reduced by 35% in the frequency interval (0,ωc). On the whole,the dispersion curves show that i) signals of different frequencies travel at significantly differentspeeds; ii) frequencies above ωc do not propagate; iii) the cutoff frequency, the mean valueof the phase velocity and the amplitude of its variations highly depend on the characteristicdimensions hA and hB.

A.3.2 Time domain analysis

Impact wave

A typical pressure impact wave has two phases: a fast rise of short duration T1 and anexponential decay of the form exp(−t/T2) (Fig. A.4) [23–25]. The frequency content of this

A.3. Results 153

0

20

40

60

80

100

120

140

160

0.00E+00 1.00E+05 2.00E+05 3.00E+05 4.00E+05 5.00E+05

frequency (Hz)

phas

e ve

loci

ty (

m s

-1)

hB=500 µm

hB=360 µm

hB=200 µm

hB=100 µm

hB=50 µm

hB=20 µm

Fig. A.3: Phase velocity as a function of frequency for different alveolar diameters. hA = 3 µmis kept constant. Vitesse de phase en fonction de la frequence pour differents diametres des alveoles.hA = 3 µm est maintenu constant.

Pulse 1 Pulse 2T1 (s) 1e-5 1e-6T2 (s) 5e-5 1e-5

highest frequency (MHz) 0.2 1

Tab. A.2: Characteristic times and frequency content of the pressure pulses. Temps caracteristiqueset contenu frequentiel des impulsions de pression.

signal depends on how fast the pressure rises: the faster the rise, the wider the spectrum. Inorder to test the influence of the characteristic times T1 and T2 of typical pressure pulses followingnon-penetrating impact, we have tried two different pressure histories p1(t) (see Table A.3.2).

Geometric and mechanical parameters are given in Table A.3; Medium I and Medium II areassociated with two different alveoli sizes and hence with two different cutoff frequencies (86 kHzand 215 kHz respectively). The analysis in the frequency domain of the previous section helps tounderstand the results associated with the transient phenomenon: from the dispersion equation(A.4), we have learned that each individual frequency travels at a specific velocity, therefore,the waveform in Figure A.4 cannot remain unchanged in form; it becomes distorted as shown inFigures A.5-A.6. Effects of the dispersion manifest themselves in the time domain as oscillationsoverlying the general pulse shape. The frequency range of the pulse affects the magnitude of theseoscillations: pulses with wide frequency content have large oscillations (Pulse 2 is more distortedthan Pulse 1); alternatively, we found that the dispersion for sufficiently slow pulses is negligible.Only plots for Medium I (Pulse 1 and Pulse 2) are represented (Fig. A.5 and A.6); plots forMedium II globally show less dispersion. Figures A.5 and A.6 show that: i) the maximum ofp(t) stays almost constant along the stack for Pulse 1 while it decreases significantly for Pulse 2;ii) the maximum slope of p(t) decreases significantly and the discontinuity in the derivative of

154 Un modele pour la propagation des ondes dans le poumon

0

0,2

0,4

0,6

0,8

1

1,2

time

amp

litu

de

T1 T2

Fig. A.4: Typical impact wave pressure history. Histoire typiques des ondes de pression induites pardes impacts.

the pulse at t = 0 disappears; iii) the oscillations increase with the distance from the top of thestack.

Pressure differentials

The hypothesis of Cooper et al. [22] that pressure jumps across alveolar walls may be relatedto lung injury is tested with the model. Pressure differentials across a set are defined to be thepressure differences between two consecutive masses of the chain: ∆p(t) = pn+1(t)− pn(t). Themass-spring chain model yields, for each mass, the pressure p(t) corresponding to the interfacialpressure between a soft layer and a hard layer (interface B-A). We have used the parameters andpulses respectively defined in Tables A.3 and A.3.2. Only ∆p(t) in Medium I for Pulse 2 is shown(Fig. A.7); the same shapes and orders of magnitude are observed for Pulse 1 and Medium II.For the typical pulses defined in Table A.3.2, we note that: i) pressure differentials lay betweenone half and one tenth of the initial pressure magnitude (Fig. A.7): the maximum pressure jumpbetween two adjacent alveoli is, at the most, of the order of magnitude of the pressure pulse; ii)the maximum is reached for the masses closest to impact; iii) the pressure gap diminishes as thewave propagates until it reaches a limit (approximatively 1/10 of the initial pulse amplitude).The shape and amplitude of ∆p(t) highly depend on the pulse history (duration and shape):computations (not represented) for “slow” pulses (more than a few milliseconds) have yieldedvery small values of ∆p(t) in comparison to those of Figure A.7.

A.4 Discussion and Conclusions

As shown in Figures A.2 and A.3, phase velocities obtained for characteristic dimensions ofhuman alveoli are of the same order of magnitude as those measured in the lungs (approximately40 m.s−1) although a little above the experimental values. A reason for this variation may be

A.4. Discussion and Conclusions 155

-0.2

0

0.2

0.4

0.6

0.8

1

0.00E+00 5.00E-04 1.00E-03 1.50E-03 2.00E-03

time

p(t)

d=0.365 d=36.5 d=73 d=109.5 d=146

Fig. A.5: Time histories of the pressure at several distances d (mm) from the impact pointcorresponding to Medium I and Pulse 1. Histoires temporelles de la pression a plusieurs distances d

(mm) du point d’impact pour Medium I et Pulse 1.

that wave speeds are measured across an entire lung (the wave not only travels through theparenchyma but also through large blood vessels, cartilage and air tubes) while our model onlydescribes local propagation. Our model is based on the assumption that the parenchyma can beregarded locally as an ordered bi-periodic stack of layers. Let us now consider the introduction ofdisorder to take into account tissues other that alveoli wall (bronchiole, cartilage...) or variationsof alveoli mean size from the surface (large diameter) to the interior (small diameter) of thelung. The first kind of disorder is likely to scatter the wave but not to change its history withdispersion; for the second kind of disorder, we may reason as follows: we have shown that thestructures the likeliest to distort the pulse (structures with the lowest cutoff frequencies) arethose with the largest alveolar diameter, hence, as the wave propagates from the surface to theinterior of the lung, it travels through a less and less dispersive medium, the influence of thisdisorder should thus be small.

The work presented in this paper is part of a project dedicated to the modeling of highvelocity nonpenetrating thoracic impact, and in particular to the propagation of steep frontpressure waves in the lung. The impact characteristics reported in Table A.3.2 are associatedwith the former problem. Cutoff frequencies for physiological alveolar dimensions may be belowthe highest frequencies of a short pulse spectrum. As a consequence, according to our lung model,the parenchyma can not be subjected to any rate of pressure rise, in fact there is an upper limitgiven by the cutoff frequency. If most of the energy of a pulse is associated with high frequenciesabove the cutoff frequency then little energy will propagate inside the lung. The distortion of thepulse depends on its frequency content; sufficiently “slow” pulses, for which the pressure rise lastsseveral milliseconds (like in car accidents), are not distorted. Pressure differentials across thealveolar septa calculated for long pulses (associated with low speed impacts) (not represented)are orders of magnitude smaller than the pressure peak of the pulse, while ∆p(t) for short pulsesmay be of its order of magnitude. This leads to the conclusion that if the pressure differential

156 Un modele pour la propagation des ondes dans le poumon

-0.2

0

0.2

0.4

0.6

0.8

1

0.00E+00 5.00E-04 1.00E-03 1.50E-03 2.00E-03

time

p(t)

d=0.365

d=36.5d=73

d=109.5d=146

d=18.25

Fig. A.6: Time histories of the pressure at several distances d (mm) from the impact pointcorresponding to Medium I and Pulse 2. Histoires temporelles de la pression a plusieurs distances d

(mm) du point d’impact pour Medium I et Pulse 2.

is the damage mechanism at work in “high” speed impacts then a different one is probablyinvolved in “low” speed impacts. It is to note that there is no way to derive the pressure (eventhough it varies continuously) at each point of a set from the knowledge of the pressure on eachside of a set; then ∆p(t) can only be regarded as an indicator of the pressure differential acrossan alveolar wall since it is the pressure differential across an entire unit set. Furthermore, it isobvious that ∆p(t) is closely related to the rate of pressure rise (the time derivative of the pulse,denoted by p

′1(t)). A simple reasoning leads to this results: for a weakly-distorted pulse (little

dispersion) the pressure p(t) along the stack may be approximated by p1(t); if the propagationtime across a set (T = hs/c) is small in comparison with the time for the change in curvatureof the pulse shape, then a good approximation of the differential is

∆p(t) = p′1(t)

hs

c, (A.6)

where c is some velocity subjected to an appropriate choice depending on the frequency range.Besides, in experimental works, Cooper et al. [24] and Stuhmiller et al. [106] found that the rateof pressure rise of the pulse is a good criterion for the risk of injury (even though we have noknowledge of a threshold); consequently, since ∆p(t) is a function of the derivative of the pulse(see Eq. A.6), then the pressure differential may be related to a damage mechanism.

The propagation of harmonic ultrasound signals in the lung have been investigated with themass-spring chain model. Computations in the time domain with (A.5) showed that each massof the mass-spring system oscillates at the input frequency after a transient stage. We couldcheck that frequencies above ωc do not propagate farther than a few alveolar diameters. Byusing (A.6), for a pressure signal given by p(t) = P sin(ωt), the maximum for ∆p(t) — aftera transient period during which the amplitude of the oscillations slowly rise to their steadystate value — is found to be Pωhs/c; we note that the pressure differential depends both on

A.4. Discussion and Conclusions 157

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

time

p(t)

d=1d=3.65

d=36.5

d=109.5

∆

Fig. A.7: Time histories of the pressure differential ∆p(t) at several distances d (mm) fromthe impact point corresponding to Medium I and Pulse 2. Plots at different distances have beentranslated in time for more clarity. Histoires du differentiel de pression ∆p(t) a plusieurs distancesd (mm) du point d’impact pour Medium I et Pulse 2. Les courbes pour les differentes distances ont etetranslatees dans un soucis de clarte.

the frequency and amplitude. This result might help to understand the connections betweenamplitude, frequency, alveolar size and occurence of capillary bleeding. Besides, the increase ofthe pressure threshold observed for high frequencies pressure signals (ECMUS, 1999) and thespecies dependence of the thresholds might be related to a “low pass filter” effect. For bothimpact waves and US waves, the results suggest that wave propagation patterns in the lung arespecies-dependent because of the variations in alveolar diameters from one species to another;the differences between species should increase as the frequency content of the pulse widens (seeFig. A.2 and Fig. A.3).

There is a need to choose a homogeneous material representative of the parenchyma for usein finite element models. Results derived from the mass-spring chain model indicate that thepulse is not distorted and is propagated as in a homogeneous medium if the pressure rise issufficiently slow (in this case, the use of a homogeneous material for the lung in a numericalmodel of the thorax is justified); alternatively, if the pressure rise is fast enough, the modelindicates that the distortion of the pulse might be important and should be considered. Theseresults are not surprising, they only are an illustration of the necessity to determine the scaleof the microstructure of the material to be taken into account in the model with respect to thewavelengths of interest; this is well-known to finite elements users. Because of the link between∆p(t) and the time derivative of the pulse, an injury criterion representative of the pressuredifferential across the alveolar septa in a numerical model of the lung can be chosen to be thederivative of the pressure.

In the present study, we have built a mass-spring chain model to investigate the propagationof transient waves in the foam-like parenchymal structure. The main results derived from the

158 Un modele pour la propagation des ondes dans le poumon

model are i) there exists a cutoff frequency for wave propagation in the lung; ii) a pressurepulse is distorted as it propagates through the lung depending on its frequency content; iii) fora sufficiently slow pressure rise, the pressure wave is not distorted and one can model the lungas an equivalent homogeneous material; iv) the pressure differential across the alveolar septais related to the rate of pressure rise of the pulse; v) the comparison of experimental data fordifferent mammal species (mouse, rabbit, pig, human) should consider the mean alveolar sizeand the frequency content of the pulse.

Annexe B

Coefficients de reflexion et de

transmission

Resume : Les coefficients de transmission et de reflexion sont au cœur de la methode

developpee au chapitre 3. Les coefficients qui interviennent dans la theorie des rayons

generalises ont des expressions identiques aux coefficients de transmission/reflexion

des ondes planes. Les coefficients a une interface avec des conditions de glisse-

ment sans frottement ne sont pas classiques et nous n’avons pu les trouver dans la

litterature ; il sont calcules et donnes pour reference dans cette annexe. Les coefficients

classiques a une interface avec des conditions de contact colle et a une surface libre

sont egalement donnes pour reference.

Introduction

Au chapitre 3, les coefficients de reflexion/transmission sur les surfaces I et II sont obte-nus (( naturellement )) au cours des manipulations algebriques des equations dans l’espace deLaplace-Fourier (voir les eqs. (3.34) et (3.41)).

Une maniere classique et elegante de determiner les expressions des coefficients de reflexionet de transmission des ondes planes est d’utiliser les (( matrices de diffraction )) ((( scatteringmatrix formalism )) en termes anglo-saxons, voir [4], p. 144).

Les operations algebriques — resolution de systemes lineaires d’equations — qui permettentd’obtenir les expressions des coefficients sont en fait les memes dans la methode du chapitre 3et avec les (( matrices de diffraction )).

Dans cette annexe, nous donnons1) les expressions des coefficients classiques pour une surface libre (§ B.1) et pour une interfaceavec des conditions de contact colle (§ B.2) ;2) les expressions des coefficients a une interface avec des conditions de contact glissant (sansfrottement) (§ B.3) [46]. Formellement un peu plus simple a obtenir que pour des conditionsde contact colle (voir, par ex. [4]), les expressions de ces coefficients sont toutefois absentes desreferences classiques.

Pour ces derniers, nous presentons la derivation des coefficients avec la methode des (( ma-trices de diffraction )). (Parmi les rares travaux avec la condition de contact glissant, on trouve [3]et [101], dans cette derniere reference, la transmission/reflexion a l’interface entre deux milieuxanisotropes est consideree).

160 Coefficients de reflexion et de transmission

B.1 Coefficients de reflexion a la surface libre

A partir de la premiere expression de (3.34),

RI = −S−11 G1

= −D(1)4

−1D

(1)3 ,

et en calculant formellement les matrices inverses, les coefficients de reflexion a la surface libresont obtenus

RPP = RSS =(sP ;13 sS;1

3 S2 − χ21

)∆−1

R;1 ;

RPS =(−2cP ;1sS;1s

P ;13 χ1S

)∆−1

R;1 ; RSP =(2sP ;1cS;1s

S;13 χ1S

)∆−1

R;1,(B.1)

ou ∆R;1 = sP ;13 sS

3;1S2 + χ2

1 est le denominateur de Rayleigh.

B.2 Coefficients de reflexion et de transmission a l’interface

collee

A partir de la seconde expression de (3.34),

RII = −S−12 G2

= −[D

(1)1 −D

(2)2

(D

(2)4

)−1D

(1)3

]−1 [D

(1)2 −D

(2)2

(D

(2)4

)−1D

(1)4

],

(B.2)

et

TII =[(

D(1)1

)−1D

(2)2 −

(D

(1)3

)−1D

(2)4

]−1 [(D

(1)1

)−1D

(1)2 −

(D

(1)3

)−1D

(1)4

].

et en calculant formellement les matrices inverses, les coefficients de reflexion et de transmissionl’interface (ecrit ci-dessous avec la factorisation utilisee par Aki et Richard [4]) sont obtenus

RPP =(F (bsP ;1

3 − csP ;23 )−H(a + dsP ;1

3 sS;23 )S2

)D−1 ;

RPS =(−2sP ;1

3 (ab + cdsP ;23 sS;2

3 ScP ;1

)(cS;1D)−1 ;

RSP =(2sS;1

3 (ab + cdsP ;23 sS;2

3 ScS;1

)(cP ;1D)−1 ;

RSS =(E(bsS;1

3 − csS;23 )−G(a + dsP ;2

3 sS;13 )S2

)D−1

(B.3)

TPP =(2ρ1s

P ;13 FcP ;1

)(DcP ;2)

−1 ; TPS =(−2ρ1s

P ;13 HScP ;1

)(DcS;2)

−1 ;

TSP =(2ρ1s

S;13 GScS;1

)(DcP ;2)

−1 ; TSS =(2ρ1s

S;13 EcS;1

)(DcS;2)

−1 , (B.4)

oua = ρ2(1− 2c2

S;2S2)− ρ1(1− 2c2

S;1S2) ; b = ρ2(1− 2c2

S;2S2) + 2ρ1c

2S;1S

2 ;c = ρ1(1− 2c2

S;1S2) + 2ρ2c

2S;2S

2 ; d = 2(ρ2c2S;2 − ρ1c

2S;1) ;

E = bsP ;13 + csP ;2

3 ; F = bsS;13 + csS;2

3 ;G = a− dsP ;1

3 sS;23 ; H = a− dsP ;2

3 sS;13 ; D = EF + GHS2.

B.3. Coefficients de reflexion et de transmission a l’interface glissante 161

B.3 Coefficients de reflexion et de transmission a l’interface glis-

sante

Bien que seuls les coefficients de transmission des ondes du milieu 1 vers le milieu 2 soientutiles pour notre etude, tous les coefficients sont calcules et donnes pour reference. Les notationssont celles du chapitre 2.

Chaque composante du vecteur w represente l’amplitude d’une onde determinee a la fois parune polarisation et une direction de propagation par rapport a x3. La methode pour obtenir lescoefficients consiste a exprimer, a une interface, les amplitudes des ondes diffractees (reflechieset transmises) en fonction des amplitudes des ondes incidentes — w(1)− et w(2)+ sont ecrites enfonction de w(1)+ et w(2)−

w(1)− = R+w(1)+ + T−w(2)−

w(2)+ = T+w(1)+ + R−w(2)−,(B.5)

ou T± et R± sont les matrices 3×3 des coefficients de reflexion/transmission. Les exposants’±’ de T et R sont relatifs aux directions de propagation des ondes incidentes. La methode estillustree sur la figure B.1.

medium 1

medium 2

x3

x1

o

w1+ w =R w

1 1- + +

w =T w2 1+ + +

w2-

w =R w2 2+ - -

w =T w1 2- - -

Fig. B.1: Illustration du formalisme des matrices de diffraction.

Les matrices de coefficients de reflexion/transmission ont la forme suivante

R± =

R±PP R±

SP 0R±

PS R±SS 0

0 0 R±SHSH

T± =

T±PP T±SP 0T±PS T±SS 00 0 T±SHSH

, (B.6)

ou la premiere des deux lettres (P ou S) en indice des coefficients est relative a la polarisationde l’onde incidente, et la seconde a celle de l’onde reflechie ou transmise. Remarquons que ledecouplage des mouvements P -SV et SH se traduit par les zeros dans la matrice des coefficients.

162 Coefficients de reflexion et de transmission

Les conditions de contact a l’interface glissante (2.7) sont recrites, dans le domaine destransformees, en termes des amplitudes des ondes w

6∑

n=1

D(1)3n w(1)

n =6∑

n=1

D(2)3n w(2)

n ,6∑

n=1

D(1)6n w(1)

n =6∑

n=1

D(2)6n w(2)

n ,

6∑

n=1

D(1)4n w(1)

n = 0,6∑

n=1

D(1)5n w(1)

n = 0,

6∑

n=1

D(2)4n w(2)

n = 0,6∑

n=1

D(2)5n w(2)

n = 0.

(B.7)

Pour simplifier les notations, nous introduisons Dk pour designer le vecteur correspondant a laligne k de la matrice D ; ce vecteur est compose de deux parties associees aux ondes montanteset descendantes comme indique ci-dessous.

Dk = (D−k , D+

k ).

En introduisant (B.5) dans (B.7), un systeme d’equations qui fait apparaıtre les coefficients dereflexion/transmission est obtenu

D3(1)−(R+w(1)+ + T−w(2)−) + D3

(1)+w(1)+ = D3(2)−w(2)− + D3

(2)+(T+w(1)+ + R−w(2)−)D6

(1)−(R+w(1)+ + T−w(2)−) + D6(1)+w(1)+ = D6

(2)−w(2)− + D6(2)+(T+w(1)+ + R−w(2)−)

D4(1)−(R+w(1)+ + T−w(2)−) + D4

(1)+w(1)+ = 0D5

(1)−(R+w(1)+ + T−w(2)−) + D5(1)+w(1)+ = 0

D4(2)−w(2)− + D4

(2)+(T+w(1)+ + R−w(2)−) = 0D5

(2)−w(2)− + D5(2)+(T+w(1)+ + R−w(2)−) = 0.

(B.8)Le systeme (B.8) est valable quelles que soient la valeurs des amplitudes w

(1)+1 , w

(1)+2 , w

(1)+3 ,

w(2)−1 , w

(2)−2 et w

(2)−3 . En choisissant w

(1)+1 = w

(1)+2 = w

(2)−1 = w

(2)−2 = 0, on montre directement

que les coefficients pour les ondes SH sont

T±SHSH = 0; R±SHSH = 1.

(Ces valeurs correspondent par ailleurs aux coefficients pour les ondes SH a une surface libre.)Les coefficients pour les ondes P et S sont obtenus en choisissant w

(1)+3 = w

(2)−3 = 0 ; dans

ce cas, les lignes (B.8)3 et (B.8)4 d’une part, et (B.8)5 et (B.8)6 d’autre part, se trouvent etreequivalentes, de sorte que le systeme (B.8) est recrit comme un systeme de quatre equationsseulement. En prenant, dans le systeme (B.8), successivement zero pour trois des quatre ampli-tudes w

(1)+1 , w

(1)+2 , w

(2)−1 et w

(2)−2 , quatre systemes de quatre equations lineaires sont obtenus ;

chacun de ces systemes donne quatre coefficients de reflexion/transmission. Ecrits sous la formematricielle Ma = y, les quatre systemes ont la meme matrice M . En appelant Y la matrice 4×4dont les colonnes sont les vecteurs y, les coefficients sont donnes par

(R′+ T

′−

T′+ R

′−

)= M−1Y, (B.9)

ou R′± et T

′± sont les matrices 2×2 qui contiennent les coefficients du probleme P/SV (voir(B.6)). Les matrices Y et M sont donnees respectivement par

Y =

−D(1)34 −D

(1)35 D

(2)31 D

(2)32

−D(1)64 −D

(1)65 D

(2)61 D

(2)62

−D(1)44 −D

(1)45 0 0

0 0 −D(2)41 −D

(2)42

,

B.3. Coefficients de reflexion et de transmission a l’interface glissante 163

M =

D(1)31 D

(1)32 −D

(2)34 −D

(2)35

D(1)61 D

(1)62 −D

(2)64 −D

(2)65

D(1)41 D

(1)42 0 0

0 0 D(2)44 D

(2)45

.

L’ordre des colonnes de Y est obtenu en prenant successivement des valeurs non nulles de w(1)+1 ,

w(1)+2 , w

(2)−1 et w

(2)−2 dans (B.8), dans cet ordre.

Les expressions explicites des seize coefficients qui apparaissent dans (B.9) ont ete obtenuesen resolvant separement les quatre systemes d’equations. Les manipulations algebriques sontgrandement simplifiees par le fait que tous les systemes ont la meme matrice M . Grace auxnombreuses symetries, les formes finales des expressions sont relativement simples. En utilisantles quantites

Em = 2µmS−1∆R;m ; Fm = 0.5S−1sP ;m3 s2

S;m ;Gm = 2µm(sP ;m

3 sS;m3 S − χm

2S−1) ; H = (F1E2 + F2E1),

ou ∆R;m = sP ;m3 sS;m

3 S2 + χm2, m = 1, 2 est le coefficient de Rayleigh, les coefficients sont

R+P→P = (E2F1 + F2G1)H−1 R+

S→P = (−4F2µ1cS;1

cP ;1sS;13 χ1)H−1

R+P→S = (4F2µ1

cP ;1

cS;1sP ;13 χ1)H−1 R+

S→S = (−E2F1 + F2G1)H−1

T+P→P = (4F1µ1

cP ;1

cP ;2χ1χ2S

−1)H−1 T+S→P = (4F1µ1

cS;1

cP ;2sS;13 χ2)H−1

T+P→S = (4F1µ1

cP ;1

cS;2χ1s

P ;23 )H−1 T+

S→S = (4F1µ1cS;1

cS;2sS;13 sP ;2

3 S)H−1

R−P→P = (G2F1 + F2E1)H−1 R−

S→P = (4F1µ2cS;2

cP ;2sS;23 χ2)H−1

R−P→S = (−4F1µ2

cP ;2

cS;2sP ;23 χ2)H−1 R−

S→S = (G2F1 − F2E1)H−1

T−P→P = (4F2µ2cP ;2

cP ;1χ1χ2S

−1)H−1 T−S→P = (−4F2µ2cS;2

cP ;1sS;23 χ1)H−1

T−P→S = (−4F2µ2cP ;2

cS;1sP ;13 χ2)H−1 T−S→S = (4F2µ2

cS;2

cS;1sS;23 sP ;1

3 S)H−1,

(B.10)

ou sP,S;m3 = (s2

P,S;m−S2)1/2, et χm = 0.5s2S;m − S2. Les expressions donnees dans (B.10) ont ete

verifiees avec un logiciel de calcul symbolique.

En partant de (B.9), les coefficients peuvent etre evalues numeriquement ; il suffit pour celade calculer M−1. Cette facon de faire est bien sur utile pour verifier l’implementation des co-efficients ; mais on peut egalement envisager, dans un programme de calcul, de se passer del’implementation des expressions explicites, en effet toutes les quantites apparaissant dans lesmatrices M et Y sont par ailleurs requises par le programme — c’est le choix fait par van der Hi-

jden [55] pour les coefficients a une interface collee. Nous avons observe des instabilites dans lecalcul numerique des coefficients pour des milieux de proprietes trop eloignees, dans ce cas, lamatrice M est mal conditionnee et les expressions explicites doivent etre utilisees. Pour nombrede problemes, le calcul numerique des coefficients peut cependant etre la meilleure alternative,d’autant que la forme sous laquelle les matrices sont donnees pour l’interface glissante est in-terchangeable avec celle donnee dans [55] pour l’interface collee, de sorte qu’un programme decalcul peut aisement utiliser, alternativement, l’une ou l’autre des conditions de contact.

Tous les coefficients de reflexion/transmission ont le meme denominateur H. L’equationobtenue par Achenbach et Epstein [3] qui determine la vitesse de propagation des ondes

164 Coefficients de reflexion et de transmission

d’interface est equivalente a l’expression obtenue pour H. Ainsi, de meme que pour les ondes deScholte, les ondes de Rayleigh ou les ondes de Stoneley a une interface collee, la vitessedes ondes de Stoneley a une interface glissante est determinee par les poles de H.

Pour reference, nous introduisons la quantite

∆S = S2H = sP ;13 s2

S;1µ2∆R;2 + sP ;23 s2

S;2µ1∆R;1. (B.11)

ou l’on retrouve l’expression du denominateur de Rayleigh

∆R;m = sP ;m3 sS;m

3 S2 + χm2 m = 1, 2. (B.12)

Annexe C

Recherche des poles des coefficients

de transmission/reflexion

Resume : Cette annexe est une illustration de l’application du principe de l’argument

pour compter le nombre de poles des coefficients de transmission/reflexion.

Introduction. Les poles des coefficients de transmission/reflexion sont des singularites dontil faut tenir compte dans la methode de Cagniard-de Hoop. Il se trouve que dans les casetudies, les poles n’ont jamais du etre pris en compte : ils ne se trouvaient pas dans la zone duplan complexe dans laquelle le theoreme de Cauchy etait applique. Nous nous sommes assurede l’absence de pole dans cette zone en utilisant le (( principe de l’argument )). Un exempled’application du principe est detaille dans cette annexe.

Le principe de l’argument. C’est une consequence directe du theoreme des residus de Cau-

chy (voir par ex. [29] p. 244 ou [116] p. 211) qui peut etre enonce comme suit. Soit L une courbefermee, tracee dans un plan complexe, parcourue une fois dans le sens positif (antihoraire) (Lest une courbe de Jordan) ; soit f(z) une fonction holomorphe partout a l’interieur de L saufa un nombre fini de poles, et qui n’a ni pole ni zero sur L. Si Z et N sont respectivement lesnombres de zeros et de poles de la fonction (comptes avec leurs multiplicites) a l’interieur de L

alors

Z − P =1

2πi

∫

L

f′(ξ)

f(ξ)dξ.

Il est pratique de rechercher le nombre de zeros de ∆S = ∆S/sP ;13 plutot que ceux de la

quantite ∆S definie par (B.11) ; en procedant de la sorte, les coupures dans le plan complexe,introduites pour rendre la quantite etudiee analytique, restent bornees. En utilisant la conventionque le symbole racine carree (√ ) designe une quantite de partie reelle positive, ∆S est donnepar

∆S(S) = s2S;1µ2

[(0.5s2

S;2 − S2)2− S2

√s2P ;2 − S2

√s2S;2 − S2

]

+s2S;2µ1

qs2P ;2−S2q

s2P ;1−S2

[(0.5s2

S;1 − S2)2− S2

√s2P ;1 − S2

√s2S;1 − S2

].

(C.1)

Nous restreignons l’analyse au cas ou sP ;1 < sS;1 < sP ;2 < sS;2. La discussion qui suit estvalable pour les recepteurs et les proprietes de materiaux consideres dans les chapitres 5 a 7. Enprenant Re[sP,S;1,2

3 ] ≥ 0, la fonction ∆S est analytique dans le plan coupe de S represente sur la

166 Recherche des poles des coefficients de transmission/reflexion

L ¥

Re[S]

Im[S]

L+

L-

Fig. C.1: Contour dans le plan complexe de S pour l’application du Principe de l’Argument. Lescoupures sont representees en trait gras discontinu.

figure C.1. Huit points de branchement sont places a ±sP,S;1,2, les coupures associees s’etendentle long de l’axe des reels entre sP ;1 et sS;2, et entre −sP ;1 et −sS;2. Notons que ∆S(S) a ununique pole au point de branchement S = sP ;1 qui se trouve en dehors du plan coupe de S.

En choisissant L dans le plan coupe de S, le principe de l’argument, avec f = ∆S , permetde calculer le nombre de zeros (le domaine a l’interieur de L ne contient pas de pole).

En utilisant

v = ∆S(S),

une image du plan de S est obtenue dans le plan de la variable complexe v. L’image de L dansle plan de v est notee Lv. Le principe de l’argument dans le plan de v s’ecrit

Z =1

2πi

∫

Lv

dv

v. (C.2)

L’integrande 1/v a un unique pole en v = 0, donc Z est le nombre de fois que Lv entoure l’originedu plan de v. La courbe de Jordan L, dans le plan de S, utilisee pour l’application du principede l’argument est montree sur la figure C.1 ; L est construite comme suit : L∞ est un cercle,parcouru dans la direction positive, centre a l’origine et de rayon arbitrairement grand ; L+ etL− sont deux chemins, parcourus dans la direction negative, arbitrairement proches des coupuresle long des parties positive et negative de l’axe des reels respectivement, L+ et L− incluent desboucles circulaires de rayons arbitrairement petits autour de chaque point de branchement (L+

et L− sont representes fig. C.1 a une certaine distance de l’axe des reels mais, dans les calculs,

167

les quantites sont evaluees sur l’axe des reels). Pour que le principe de l’argument puisse etreapplique, L doit etre une courbe fermee ; deux chemins, arbitrairement choisis, devraient etretraces pour relier L+ et L− a L∞ ; si il n’y a pas de zero sur ces chemins, ni dans le domainequ’ils delimitent (il est toujours possible de trouver de tels chemins), l’integration (C.2) le longde ces chemins est nulle. Finalement, le contour pour l’application du principe de l’argument estsimplement L = L∞ ∪ L+ ∪ L−.

Contour L∞. Pour des valeurs de |S| grandes (en utilisant (ε− 1)1/2 = i(1− ε/2) + O(ε2) et(1− ε/2)2 = 1− ε + O(ε2), ou ε est un petit parametre), l’equation (C.1) devient

∆S(S) = −0.5S2[s2S;1µ2(s2

P ;2 − s2S;2) + s2

S;2µ1(s2P ;1 − s2

S;1)]. (C.3)

L’expression (C.3) indique que l’image de L∞ dans le plan de v entoure l’origine deux fois dansla direction positive ; donc, en notant Lv,∞ l’image de L∞

12πi

∫

Lv,∞

dv

v= 2. (C.4)

Contours L±. Puisque ∆S est seulement fonction de S2, les calculs pour L+ et L− sontidentiques. De plus, ∆S satisfait le principe de reflexion de Schwarz (∆S(S∗) = ∆S(S)∗, ou S∗

est le complexe conjugue de S), de sorte que les images des parties des chemins au-dessous etau-dessus des coupures sont symetriques par rapport a l’axe des reels. Pour le calcul des imagesdes chemins autour des coupures, nous pouvons donc nous restreindre a l’analyse de la partie deL+ au-dessus de la coupure. En ce qui concerne les arcs circulaires, seulement l’image du cercleautour du point de branchement (et pole) sP ;1 contribue a l’integrale. En prenant S = sP ;1+εeiθ

et en faisant tendre ε vers zero, nous trouvons que l’image dans le plan de v est un demi-cerclesuivi dans la direction positive.

En suivant le chemin sur l’axe des reels pour sP ;1 < S < sS;2, et en utilisant la notation“RP” pour (( reel positif )) et “IN” pour (( imaginaire negatif )), les radicaux prennent les valeurs

sP ;1 < S < sS;1

sP ;13 , IN

sS;13 ,RP

sP ;23 , RP

sS;23 ,RP

; sS;1 < S < sP ;2

sP ;13 , IN

sS;13 , IN

sP ;23 , RP

sS;23 , RP

; sP ;2 < S < sS;2

sP ;13 , IN

sS;13 , IN

sP ;23 , IN

sS;23 , RP

.

(C.5)En rassemblant les resultats (C.5) et (C.1), l’image du chemin L+ peut etre tracee. Un exemplede l’image d’une boucle autour d’une coupure est montre sur la figure C.2.

Les images des contours L+ et L− (suivis dans la direction negative) entourent chacunl’origine du plan de v une fois dans la direction positive, c’est-a-dire en notant Lv,± les imagesde L±

12πi

∫

Lv,±

dv

v= −1. (C.6)

Contour L = L∞ ∪ L+ ∪ L−. En rassemblant les resultats (C.4) et (C.6), le nombre de zerosde ∆S est Z = 2− 1− 1 = 0.

Dans le cas etudie, les coefficients de reflexion/transmission n’ont donc pas de pole.

168 Recherche des poles des coefficients de transmission/reflexion

Im[ ]v

Re[ ]v

Fig. C.2: Un exemple d’une image dans le plan complexe de la variable v d’une des boucle L+

ou L− (voir fig. C.1) autour d’une coupure du plan complexe de S. Le contour — qui se refermeen dehors de la zone de trace en haut et en bas a droite — encercle l’origine une fois dansle sens negatif. Les lignes fines et grasses correspondent respectivement aux chemins parcourusau-dessus et en dessous de la coupure ; le demi-cercle sur la gauche est l’image du cercle autourdu pole sP ;1 de ∆S.

Annexe D

Transient elastic wave propagation in

a spherically symmetric bimaterial

medium modeling the thorax

Cette annexe est la retranscription d’un article publie dans International Journal of Solidsand Structures (volume 39, p. 5345-5369, 2002) intitule (( Transient elastic wave propagationin a spherically symmetric bimaterial medium modeling the thorax )), par Quentin Grimal,

Salah Naıli Et Alexandre Watzky [44]. Dans un soucis de coherence, quelques notationsont ete adaptees.

Abstract The transient response resulting from an impact wave on an elastic bimaterial,made out of a “hard” medium and a “soft” medium, welded at a spherical interface, have beeninvestigated by using an integral transform technique. This technique permits isolation of thepressure and shear waves contributions to the wave-field. The method of solution makes useof the generalized ray/Cagniard-de Hoop method (GR/CdH) associated with a “flattening ap-proximation” (FA) technique, similar to the Earth flattening transformation used in geophysics.The GR/CdH method and the FA technique are briefly presented, together with their numericalimplementations. The FA has proved to be useful in geophysical application, however, as far asthe authors know, it has never been investigated for other applications. For the purpose of thispaper, numerous tests of the method have been performed in order to check that the FA is ap-propriate to compute transient responses in the special case presented here. We could determineappropriate values for some parameters involved in the FA. This paper follows [43] in which weinvestigated the same bimaterial with a plane — instead of spherical — interface. Numericalexamples are concerned with the propagation of an impact wave in the thorax modeled as a bi-material (thoracic wall - lung). In addition to the effects of the weak coupling of the two mediaalready observed in our previous study, we found that, for interface curvatures characteristic ofthose measured in the thorax, focalization of energy is manifest.

D.1 Introduction

The study of transient elastic waves in layered media is of interest for many applications. Inthe past ten decades, geophysicians have developed powerful analytical and numerical methods

170Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

to solve problems derived from seismology; these methods have then been exported to otherfields of mechanical engineering such as non-destructive evaluation of composites. Seismology isconcerned with wave propagation in Earth. Layered models with spherical symmetry (concentriclayers) are a good representation of the Earth geometry but they lead to complicated equationsas compared to those ruling planarly layered media. Seismologists have been looking for trans-formations — Earth Flattening Approximations (FA) —, acting not only upon the coordinatesystem but also on the media properties, to allow for the study of wave propagation in a modelof spherical Earth by using existing analytical and numerical methods for “flat” Earth models.Thinking in terms of geometrical rays, the idea is to study curved rays in a flat structure ins-tead of straight rays in a spherical structure. This is achieved through the introduction in theflat representation of depth-dependence media parameters, so that the rays are continuouslyrefracted.

The main reasons and advantages for using the FA are i) most of the formalisms and ana-lytical methods dedicated to layered media (scattering matrix formalism, exact generalized raytheory...), cannot be used in spherical geometry (i.e., with a spherical system of coordinates);ii) FA avoid some algebraic and numerical difficulties appearing when manoeuvring with Bes-sel functions that occur in the spherical formulation [5]; iii) minimal changes are required tointroduce the FA in the numerical programs dedicated to planarly layered media.

Unlike conformal transformations used for the resolution of the Laplace’s equation withcomplex boundary shapes, FA are not exact — except for the study of Love waves [10]. As amatter of consequence, several FA have been investigated for different wave propagation studies.However, most of the authors (see Aki p. 463 [4] and [5,20,79]) have used the same set of trans-formations — derived from the kinematic properties of geometrical rays — for the coordinatesand the wave velocities. On the other hand, many transformations have been proposed for themechanical parameters (Young’s modulus and Poisson’s ratio or, alternatively, Lame’s coeffi-cients), for different wave problems (study of body or surface waves, SH or P-SV problems...).

The present paper follows a study [43] in which the wave propagation in a bimaterial medium,with a plane interface, modeling the thorax was investigated; the bimaterial represented thethoracic wall (medium 1) and the lung (medium 2); a point source generated a wave motion inmedium 1 and the response was computed in medium 2.

Our work is part of a preliminary study aiming at characterizing the strains and stressesundergone by thoracic tissues during a non-penetrating impact. This is of interest to the defenseindustry concerned with high velocity impacts (design of bulletproof jackets) and to the auto-motive industry concerned with lower velocity impacts (crash safety systems). The necessity forinvestigations on the interaction of stress waves with thoracic tissues has been underlined byFung et al. [37] and Yen et al. [118]. Indeed, the impact wave is supposed to play an importantrole in the occurrence of lung injuries.

In our previous works [42,43], the interface between the two media was plane; we could showthat, due to the weak acoustic coupling between the two media, the energy was propagated ina relatively narrow zone in medium 2 (a few centimeters around an axis passing through thesource and perpendicular to the interface); this has been recognized as a focalization effect. Inthe present study, the wave propagation in the same bimaterial model, but with a sphericalinterface, is investigated (Fig. D.1); our purpose is to quantify the additional focalization dueto the curvature of the interface.

In this paper, we use the flattening technique together with the exact three-dimensionalgeneralized ray/Cagniard-de Hoop (GR/CdH) method. The GR/CdH method is a very powerful

D.2. Model and method of solution 171

tool for the study of transient wave propagation in planarly layered media. It allows to computeexact solutions directly in the time domain; relevant references for these methods are [4, 66,86]. The generalized ray method splits the solution for the wave motion at some point of themedia (called the receiver) in a sum of contributions. Each contribution is associated with a“generalized ray path”, i.e., with a succession of transmissions and reflections (with or withoutmode conversion) at each interface between the source and the receiver. Finally, a generalizedray (GR) is defined both by a path and by a sequence of modes of propagation (longitudinalor transverse polarization), i.e., one mode per layer crossed. Formally, each GR is an integraltransform representation of a contribution to the solution; they are obtained in a Fourier-

Laplace domain dual of the space-time domain. The Cagniard-de Hoop method applies to eachGR integral; in essence, it is a mathematical trick to avoid integrations over the frequency, orover the wavenumber, required to transform the solutions back to the space-time domain. Gilbertand Helmberger [39] has shown that the GR/CdH method can be used in an approximate way inspherical geometry. However, in our study, we choose to apply the exact GR/CdH method to aflat representation, equivalent to a spherically symmetric model, obtained with an approximateflattening transformation.

In a previous paper [43], the GR/CdH method and its numerical implementation in a bima-terial medium with a plane interface have been described. The aim of this paper is to presentthe flattening transformation procedure, associated with the exact three-dimensional GR/CdHmethod. We discuss the choice of some parameters involved in the transformation for the lengthand time scales related to our biomechanical application. We not only give the transformationsfor the coordinates and mechanical properties as used in geophysics, but also we propose someprocedures to validate the method before it can be used with length and times scales differentto those used in geophysical applications. As far as we know, the FA have only been used forgeophysical applications; however, they are of interest to other fields of mechanical engineeringconcerned with transient wave propagation in elastic media.

With this introduction as background, we present in section 2 the basic set of transformationsused to derive the plane representation (Figure D.2) equivalent to the spherical representationillustrated in Figure D.1. For the sake of completeness, the main steps of the GR/CdH method forplane-layered media are given. Then we expose the method used to validate the transformationand to choose the parameters. Finally, section 3 is devoted to the biomechanical application.

D.2 Model and method of solution

D.2.1 Configuration and definitions

The model geometry is illustrated in Figure D.1; it consists of two media with concentricspherical boundaries, the center of the spheres is the point O. With this model geometry, wewill use both spherical and Cartesian coordinates systems, together with their associated basis.The spherical coordinates are denoted by (ϕ,ψ,R) and the associated orthonormal sphericalbasis is denoted by (eϕ, eψ, eR). The Cartesian coordinates originating at point O are denotedby (d1, d2, d3) and the associated orthonormal basis is denoted by (d1,d2,d3). Let I be a pointon axis d3. The spherical coordinates are defined with respect to the Cartesian basis as follows.For a given point M , R is taken to be the distance OM ; ϕ is taken to be the angle between d3

and eR; finally, ψ is the angle between the projection of OM in the plane defined by (d1,d2) andd1. Due to symmetry, the problem is ψ-independent; consequently, in the Cartesian coordinatesystem, we will only use the distances d =

√d2

1 + d22 and d3.

172Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

a

Rs

Ri

I

M(ϕ,

ϕ

medium 2: lung�2 �2 !2

medium 1:thoracic wall�1 �1 !1

R

P

PP PS

P

d1

d3

O

eϕϕ

eR

ψ, R)

Fig. D.1: Configuration and coordinate systems in the spherical representation.

medium 1�

1(x3) � 1(x3) � 1(x3)

x3

I

M(x1,x3) medium 2�

2(x3) � 2(x3) � 2(x3)

PPPS

P

P

x1

Fig. D.2: Configuration and coordinate system in the equivalent plane representation.

The media are assumed homogeneous, isotropic and linearly elastic; we will use the Lame’selastic parameters λ and µ, and the mass density ρ (see Table D.2.1). At the — spherical— interface, the media are welded. Pressure and shear wave speeds are respectively definedby cP =

√(λ + 2µ)/ρ and cS =

√µ/ρ. Wave slownesses are defined as sP,S = 1/cP,S where

P or S must be used for pressure and shear waves respectively. All through the paper, eachtime a comma appears between P and S means that the quantities relative to P-waves or

D.2. Model and method of solution 173

λ× 106 (Pa) µ× 106 (Pa) ρ (kg.m−3) cS (m.s−1) cP (m.s−1)medium A 1126 562 1000 750 1500medium B 0.034 0.008 500 4 10

Tab. D.1: Lame’s coefficients, densities and wave speeds for layers A and B. cS and cP arethe pressure and shear waves speeds. Coefficients de Lame, masses volumiques et vitesses des ondesdans les milieux A et B. cS et cP designent les vitesses des ondes de cisaillement et longitudinales,respectivement.

SV-waves respectively must be used. The particle velocity and the Cauchy stress tensor arerespectively denoted by v and σ. The elastodynamics problem will be solved for the six com-ponents of the motion-stress vector whose expression in the spherical system of coordinates is:b = (vϕ, vψ, vR,−σRϕ,−σRψ,−σRR)T (here T means transpose). The welded-interface conditionrequires the continuity of b at the interface.

The wave motion is generated by a point source, located in medium 1 at point I (Fig. D.1) ofcoordinate (0, 0, d3;I) in the Cartesian frame, or alternatively, of coordinate (0, 0, RI) in sphericalframe. Computations have been made for two types of sources (see [43]): a buried point forcewith direction d3 (in this case, medium 1 is unbounded), and a point force with direction d3

at a free spherical surface defined by R = RI (in this case, medium 1 is bounded). For lateruse, we define z = RI − d3, so that the source is placed at z = 0. Now, by using the flatteningapproximation (see next subsection), we are going to achieve a model configuration similar tothat investigated in [43].

D.2.2 The flattening transformations (FA)

In order to compute the transient response resulting from an impact wave on a layered me-dium, we use a set of transformations — similar the Earth flattening approximation used ingeophysics — that defines a “flat” representation equivalent to the “spherical” representationdescribed above; in the “flat” representation, the spherical interface is replaced by a plane in-terface (Fig. D.2). In what follows, subscripts s and f will respectively refer to the sphericalrepresentation and its equivalent flat representation. Another Cartesian reference frame is asso-ciated with the flat representation (Fig. D.2): the coordinates originating at point I (as definedin the spherical configuration above) are denoted by (x1, x2, x3), and the associated orthonormalbasis by (x1,x2,x3), where x3 (corresponding to −d3) is perpendicular to the — plane — inter-face. (Note that the vectors of this basis have the same directions as the vectors of the Cartesianbasis (d1,d2,d3) defined in the model with the spherical geometry.) In the flat representation,media 1 and 2 are unbounded in directions x1 and x2. Making use of the axisymmetry of theproblem, we will, without loss in generality, make all the computations for points in the (x1,x3)plane.

The flattening transformations leading to the “flat” representation involve both geometri-cal transformations (acting on the coordinate system) and transformations of the mechanicalparameters. The set of FA may be split into two groups. First, transformations for the coordinatesand the wave speeds are defined by the kinematic properties of the geometrical ray (far field)theory [20, 79], i.e., these FA introduce an approximation that is optimum for a description ofwave propagation with the aid of geometrical rays – we will assume, following Muller [79], thatthe approximation is also valid for calculations with the exact theory. Coordinates and wave

174Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

speeds in the equivalent flat representation are given by

x1 = aϕ; x3 = a ln(

R

a

); cP ;f (x3) =

a

RcP ;s; cS;f (x3) =

a

RcS;s, (D.1)

where a is a reference radius to be discussed below, and “ln” denotes the Neperian logarithm.Second, the computation of stresses — and not only displacements and travel times, as for mostgeophysical applications — requires some additional transformations given by Chapman [20].The mechanical parameters in the flat representation are given by

λf (x3) = (R/a)mλs; µf (x3) = (R/a)mµs; ρf (x3) = (R/a)m+2ρs. (D.2)

Once solutions have been computed in the flat representation, the motion-stress vector bcorresponding to the spherically symmetric problem is calculated by using the following trans-formations

vf 7−→ vs = (a/R)lvf ; σf 7−→ σs = (a/R)l+m+1σf , (D.3)

where vectors and tensors with subscripts f and s are expressed in the Cartesian basis (x1,x2,x3)and in the spherical basis (eϕ, eψ, eR) respectively. Thus, for instance, the first transformationgiven by Eqs. (D.3) allows to obtain the components of the vector vs in the basis (eϕ, eψ, eR)from the components of the vector vf calculated in the basis (x1,x2,x3). The parameters m andl will be discussed in the next section. We could not find any study that establishes the degree ofvalidity of these last five transformations [20] pointed out the difficulty of doing this), however,similar forms of the transformations have yielded exact solutions in some specific cases [10] andmany authors have used them with success (see, e.g., references [5, 7, 9].

In this study, transient responses in the equivalent flat representation, illustrated in Fi-gure D.2, are calculated with the exact three-dimensional GR/CdH method. In the followingsubsections, we write down the main steps of the method; more details on the GR/CdH methodare given in [43] and a thorough presentation can be found in [55]. In what follows, we willomit the subscript f since all the equations will be written for the flat representation. Beforewe can use the GR/CdH method, media 1 and 2 of the flat representation (which density andLame coefficients, given by Eq. (D.2), are continuous functions of x3) must be sliced into thinhomogeneous layers, so that λ, µ and ρ are constant in each thin layer. This last approximation— substituting to a media with continuously varying properties, a stack of layers with constantproperties — has been extensively studied by geophysicians (see reference [4], p. 385) and it hasbeen shown that doing a sufficiently fine discretization yields accurate results; the influence ofthe spatial discretization will be discussed in the following section.

D.2.3 Governing equations, boundary and initial conditions in the flat confi-

guration

In each homogeneous layer illustrated in Figure D.2, the equation of motion is

∂jσij − ρ∂tvi = −fi, (D.4)

where ∂j and ∂t denote partial derivatives with respect to xj and to time respectively. Einstein’ssummation convention is used. The term fi stands for a volume density of force. The timederivative of Hooke’s constitutive law for an elastic isotropic medium can take the form

∂tσij − λδijδpq∂pvq − µ(∂ivj + ∂jvi) = 0, , (D.5)

D.2. Model and method of solution 175

where δij is the Kronecker symbol.The initial condition is that all the layers are at rest for t < 0. For the purposes of the present

study, we have made calculations with two types of sources: (i) for a buried point force we takef = (0, 0, f0φ(t)δ(x1, x2, x3)), where δ(x1, x2, x3) is the three-dimensional Dirac function, f0 isthe amplitude of the source and φ(t) defines the temporal shape of the source; (ii) for a pointforce at the free surface, we take f = (0, 0, 0) in (D.4) and, in addition to (D.4) and (D.5), theboundary conditions at the free surface (defined by x3 = 0) require

σ33(x1, x2, 0, t) = f0φ(t)δ(x1, x2),σ13(x1, x2, 0, t) = σ23(x1, x2, 0, t) = 0,

(D.6)

where δ(x1, x2) is the two-dimensional Dirac function. In addition, the welded contact conditionat the interfaces requires that the motion-stress vector is continuous at the interfaces betweentwo adjacent media.

D.2.4 The method of solution in the flat representation

First, the solution of the elastodynamics problem for the model configuration illustrated inFigure D.2 is found in a Laplace-Fourier transform-domain after some algebraic manipu-lations; then the Cagniard-de Hoop method is used to obtain the solution in the space-timedomain. In what follows, we recall the solution for an infinite, or semi-infnite medium, as pre-sented in [43]. Then, we present the formalism for the derivation of the solution in a layeredmedium.

Solution in a homogeneous medium

The first step of the method consists in writing Eqs. (D.4)-(D.6) in a transformed domainin which first order ordinary differential equations are solved. For this purpose, the one-sidedLaplace transform with respect to time and the Fourier transform with respect to the x1

and x2 coordinates are used. The Laplace and Fourier transform parameters are respectivelyp and pkl (l = 1, 2). Quantities in the transformed domain are indicated with . and Laplace-transformed quantities by . .

The solution in the transformed domain is obtained as a sum of six wave amplitudes. For-mally, the solution for any component of the motion-stress vectorb = (v1, v2, v3,−σ13,−σ23,−σ33)T takes the form

bi = φ(p)×6∑

n=1

{DinWn exp[± psP,S

3 x3]}

=6∑

n=1

bni . (D.7)

In the formalism used, the terms Din are stocked in a matrix whose lines and columns areassociated with components bi and with the polarization of a wave respectively; and Wn is asource term standing for the amplitude of a wave of a given polarization emitted by the source.For the purposes of the present paper, we have used source terms corresponding to three typesof sources: buried source of strain rate (explosion) [55], buried point force [55], and point forceacting at a free surface [43]. The exponential term is a phase term accounting for the propagationof a wave. The term sP,S

3 = (s2P,S +k1

2+k22)1/2 is the slowness along the x3-axis for P or S-waves.

Each term bni of the sum in Eq. (D.7) is called a Generalized Ray (GR).

176Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

The solution in the Laplace domain for one of the six GR is obtained by applying theinverse Fourier transform

bnj (x, p) = φ(p)× (p/2π)2

∫ ∞

−∞

∫ ∞

−∞Djn(k1, k2)Wn(k1, k2)

exp[−p(ik1x1 + ik2x2 ∓ sP,S3 x3)]dk1dk2.

(D.8)

The Laplace domain solution for a component of the motion-stress vector in an infinitemedium or in a semi-inifnite medium (with the source acting at the free surface), for a pointforce source of direction x3, takes the form

bj(x, p) = b4j (x, p) + b5

j (x, p), (D.9)

where b4j (x, p) and b5

j (x, p) are the P and SV-wave contributions to the solution respectively(SH-wave contribution is zero because the loading is along the x3-axis). (The numbering, inrelation with the polarizations, is consistent with the formalisms used in [55] and [43].)

Solution for the layered medium

Let us write the solution for waves emitted in medium 1 and transmitted in medium 2; thissolution has been used to obtain the numerical results presented in section 3. In reference [43],we have been concerned with a bimaterial, hence few GR were involved (we computed up tofour); in the present paper, the flat representation consists in a layered medium with numerousinterfaces. The computation of the exact solution requires, in theory, to compute all the GR;however, since we are interested in the transmission of energy from medium 1 to medium 2, asubstantial simplification can be made. Among the many interfaces, only the one that separatesthe two media and corresponds to a physical interface, at which there is a significative jump(discontinuity) in the mechanical properties. Except for this interface, almost all the energy istransmitted between two sub-layers, i.e., there is neither reflexion nor conversion of the wave(the effect of the additional interfaces within media 1 and 2, introduced by the FA, is basicallyto deviate the energy, that is, to “bend the rays”). Thus, very much like in referenve [43], wewill compute only four GR; the solution for a motion-stress vector component in the Laplace-domain is, for both types of sources (see Eq. (D.9))

bj(x, p) = bP→Pj (x, p) + bP→S

j (x, p) + bS→Pj (x, p) + bS→S

j (x, p), (D.10)

where the meaning of the notations is as follows. Let the Greek letters α and β stand for P or S

symbols in media 1 and 2 respectively, then the notation α → β is read: the GR correspondingto the wave of polarization α emitted in medium 1, propagated — with no loss of energy — ineach sub-layer of medium 1, and transmitted in medium 2 as a wave of polarization β up to thereceiver (in medium 2, this wave is also transmitted with no loss of energy at each sub-interface).In the case of a force at the free surface, we do not include in the computation of the solution thewaves reflected at the free surface and transmitted in medium 2; hence the solution computedwill be valid up to the arrival time of the first wave that has undergone a reflection at the freesurface.

Each of the terms in (D.10) is deduced from the solution (D.8) in the single-medium confi-guration. Many authors have shown with different formalisms [4, 55, 66] that the interaction ofwaves with interfaces can be accounted for by introducing specific terms called “generalizedtransmission and reflection coefficients”. Aside from the introduction of a transmission coeffi-cient, the phase term sP,S

3 x3 appearing in Eq. (D.8) is changed to account for the travel times

D.2. Model and method of solution 177

in each homogeneous layer. Let n1 and n2 be the number of sub-layers introduced by the FA inmedia 1 and 2 respectively, with n = n1 +n2. The solution for each GR in the Laplace-domain,corresponding to a wave emitted in layer 1 and a receiver in layer n, takes the form

bα→βj (x, p) = φ(p)×

( p

2π

)2∫ ∞

−∞

∫ ∞

−∞Bα→β

j (k1, k2) Tα→β

exp[−p(ik1x1 + ik2x2 +n1∑

y=1

sα;y3 hy +

n∑

y=n1+1

sβ;y3 hy)] dk1dk2, (D.11)

where sP,S;y3 = (s2

P,S;y + k12 + k2

2)1/2; hy is the thickness of a sub-layer and Tα→β is the trans-mission coefficient at the “physical interface” between media 1 and 2 (all the other transmissioncoefficients are set equal to one; there is neither reflexion nor conversion at these sub-interfaces).The term Bα→β

j (k1, k2) accounts both for the coupling of the source to the medium, the com-ponent of b, and the type GR; it may also be written as

Bα→βi (k1, k2) = D

(n)im W

(1)l , (D.12)

where superscripts (n) and (1) stand for the layer number in which the quantities are evaluated.Subscripts l and m are associated with polarizations α and β respectively; like in reference [43],l and m take the values 4 or 5 depending on the generalized ray under study; for example, theray denoted by P → S is associated with l = 4 and m = 5.

In the remainder of this subsection, we will describe the method used to transform eachgeneralized ray contribution (D.11) in the Laplace-transform domain back to the space-timedomain, by applying the Cagniard-de Hoop (CdH) method.

Upon introducing the change of variables

ik1 = s cos θ − iq sin θ, ik2 = s sin θ + iq cos θ, (D.13)

where θ and r (0 ≤ θ < 2π, 0 ≤ r < ∞) are the polar coordinates in the (x1, x2) plane; q is areal number and s is complex. Noting that ik1x1 + ik2x2 = sr, Eq. (D.11) becomes

bα→βj (x, p) = φ(p)× (p2/4iπ2)

∫ ∞

−∞dq

∫ i∞

−i∞Bα→β

j (s, q)Tα→β

exp[−p(sr +n1∑

y=1

sα;y3 hy +

n∑

y=n1+1

sβ;y3 hy)]ds. (D.14)

In Eq. (D.14), the integration over the variable s lies along the imaginary axis. The CdH methodconsists in a deformation of the contour of integration away from this axis; this requires toextend the definition of the integrand in the complex s plane by analytic continuation. This cannot be achieved without a detailed analysis of the analyticity of the integrand (for a detaileddiscussion see [55]): if during its deformation, the contour crosses a singularity, another integralcorresponding to a head wave must be evaluated by using residue theorems. Whether or not thishappens depends on the location of the receiver with respect to the source and on the materialproperties; in the numerical examples presented in this paper the contour does not cross anysingularity during its deformation (see [43]).

The next step of the method is to take τ to be the real variable — with the dimension oftime — defined by

τ = sr +n1∑

y=1

sα;y3 hy +

n∑

y=n1+1

sβ;y3 hy. (D.15)

178Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

The solution of this equation for s(q, τ) is the CdH contour for the GR under consideration.Considering the symmetry properties of both Bα→β

j (s, q) and the contour, Eq. (D.14) can berewritten as an integral over τ

bα→βi (x, p) = φ(p)× (p2/2π2)

∫ ∞

−∞dq

∫ ∞

T (q)=[Bα→β

i (s, q)Tα→β(s, q)∂τs] exp[−pτ ]dτ, (D.16)

where = denotes the imaginary part and T (q) is the minimum of τ on the contour; it correspondsto the point at which the contour intersects the real axis and is given by Van der Hijden [55]

T (q) =n1∑

y=1

(s2α;y + q2)1/2

hy

cosΘy(q)+

n∑

y=n1+1

(s2β;y + q2)1/2

hy

cosΘy(q); (D.17)

the angles Θy(q) being defined through the equations

(s2P,S;y + q2)1/2 sinΘy(q) = s0(q), and (D.18)

r =n1∑

y=1

hy tan Θy(q) +n∑

y=n1+1

hy tan Θy(q), (D.19)

where s0(q) is given by the intersection of the Cagniard-de Hoop contour with the real axis.Next, we must interchange the order of integration in Eq. (D.16), i.e., the integration over q

must be performed first. The new limits of integration are −Q(τ) and Q(τ); they are solutionsfor q of the equation τ = T (q). Eq. (D.16) becomes

bα→βi (x, p) = φ(p)×(p2/2π2)

∫ ∞

Ta

{∫ Q(τ)

−Q(τ)=[Bα→β

i (s, q)Tα→β(s, q)∂τs]dq

}exp[−pτ ]dτ, (D.20)

where Ta = T (q = 0) appears to be the arrival time of the wave associated with the GR denotedby α → β. In Eq. (3.58), the integration over τ has the form of a forward Laplace transform,the transformation back to the time domain can thus be done by inspection. Eventually, thespace-time domain solution for the contribution of one GR is given by

bα→βj (x, t) = 0 for 0 ≤ t ≤ Ta

bα→βj (x, t) = (1/2π2) ∂ttφ(t) ∗

∫ Q(t)

−Q(t)=[Bα→β

j (s, q)Tα→β(s, q)∂ts]dq for Ta < t,

(D.21)where ∗ denotes a convolution product.

Numerical computations For each component bj(x, t) of the motion-stress vector, one mustevaluate the Green’s function, that is: i) the integral appearing in (D.21) and its limits; andii) the Cagniard-de Hoop contour s(τ, q), given by (D.15), for each value of q under the integral. Itis to note that for receivers on the x3-axis (corresponding to axis d3), the integration over q can beevaluated analytically, see [43]. In this former study, a method and its numerical implementationhave been developed to compute expressions like (D.21) with n1 = n2 = 1; for the purposes ofthe present study, the program has been modified: the flattening transformations (D.1)-(D.3)have been implemented together with a procedure to generate the sub-layers accounting for thecontinuous variation of the mechanical parameters.

An inner product of the components of the motion-stress vector b yields the projection ofthe Poynting vector in direction eR: PR = vϕσϕR +vφσφR +vRσRR. Since we will only consider

D.2. Model and method of solution 179

receivers close to axis d3 for which the angle ϕ is small, PR is a very good approximation ofthe projection of the Poynting vector on d3, which we denote by P3 (for the receiver locationsused in this paper, the error introduced is less than 0.4 %). In what follows, we will considerP3 = PR. Some results presented in this paper are given in term of P3; they shall be comparedto the results — also given in terms of P3 — presented in reference [43]. This quantity is anindicator of the energy flux propagated.

The time history of the source is, for all the computations, a Blackman window (Fig. D.3)of 0.3 ms duration.

In what follows, we present: i) displacement responses u1(t) and u3(t) in the Cartesian referenceframe (I,x1,x2,x3), computed in the equivalent flat representation; and ii) P3(t) responses inthe spherical representation (vs and σs have been obtained from vf and σf , computed in theequivalent flat representation, by using Eq. (D.3)).

D.2.5 Influence of the parameters; settings

Choice of the parameters

Muller [79] has compared the displacements amplitudes calculated with the geometricalray theory in a medium with spherical symmetry to the corresponding displacements in a flatgeometry with depth dependant properties; he has shown that the FA (D.1) for the coordinatesand velocities are optimal for the displacements amplitudes. These transformations have thenbeen used for computations with exact and approximate theories; in this study, we have assumedthat the approximation is valid for computations with the exact theory. In addition to (D.1),the power-law transformations defined through (D.2) have been used by various authors; theyintroduce parameters m and l. Biswas showed, by using power-laws with specific values of m

and l, that the equations for Love waves — in an exact way [10] — and for Rayleigh waves —in an approximate way [9] — in flat geometry can be derived from the corresponding equationsin spherical geometry. More recently, Arora et al. [5] and Battacharya [7] investigated power-laws with application to specific computational methods. Chapman [20] attempted to find theoptimum power-law parameters m and l for different wave propagation problems. It appearedthat, unlike for SH problems (the study of horizontally polarized shear waves) or for wavepropagation in a fluid, there are no optimal values for the P-SV case (the coupled study oflongitudinally and vertically polarized shear waves). In addition, for a given P-SV problem, thechoice of m and l is not obvious and the estimation of the error is a difficult task. However, theFA have proved, in geophysical applications, to yield realistic results.

Before the method may be used to study a physical phenomenon, the influence of the para-meters m, l and a must be investigated; this is the aim this subsection.

Determination of the parameter m

Chapman [20] found, after some algebraic manipulations of the basic equations (equationof motion and constitutive law) in spherical and Cartesian coordinates, that m and l must belinked by

l =1−m

2, (D.22)

hence we will only investigate the influence of m. The discussion of the parameters used inthe present study is related to a biomechanical application and it probably depends on the

180Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

0

0.5

1

0.E+00 1.E-04 2.E-04 3.E-04

t (s)

Am

plit

ude

Fig. D.3: Normalized time pulse shape φ(t) (Blackman window [55]) used in the computations.The total duration of the pulse is 0.3 ms.

mechanical parameters used (that is, on the length and time scales). However, the method forthe determination of the parameters may be used for other cases.

Many different values of m have been used by various authors, e.g., m=3 [9] for an exact FAfor Love waves, m=0 [9] for an approximate FA for Rayleigh waves, m = −3 [52] and m = −2 [7]for an approximate FA for body waves.

Taking φ(t) = H(t) in (D.21), where H(t) is the Heaviside step function, the displacementin the medium is given by (u1, u2, u3) = (b1, b2, b3), and no convolution is required. From aphysical point of view, the displacement response to a step of force should show a transientphase before the static response displacement value is attained. Some responses to such a stepof force in the bimaterial (whose properties are collected in Table D.2.1) for a receiver locatedat d = 0.01 m and z = 0.025 m (see Fig. D.1) are presented in Figures D.4(a) and D.4(b).They correspond to a spherical interface of radius Ri = 0.018 m; the source is located atpoint I. The responses are given for three different values of the parameter m; these figuresalso show for comparison the response obtained for the plane interface investigated in [43]for the equivalent receiver (r = 0.01 m and x3 = 0.025 m in the configuration and with thenotations used in our aforementioned paper), source and interface locations. The responses arecomputed close to the interface, hence the effect of focalization is expected to be small, andthe amplitudes corresponding to the plane and spherical interface cases should be comparable:this can be checked on the figures. The arrival times for the three plots corresponding to thespherical interface are smaller than those for the plane interface case; this is a consequence ofthe differences in the location of the receiver with respect to the interface. (The jagged shape ofthe plots for large times is due to numerical errors; those are enhanced in the computation withthe FA but they have also been observed with the program dedicated to the plane interface.)The results illustrated in Figures D.4(a) and D.4(b) enforced our choice of a value of m: theplots for m = −3 and m = 3 diverge from the plane case plot (bold line) and the correspondingresponses u1(t) and u3(t) are not physically acceptable; in contrast, taking m = 0 seems to be a

D.2. Model and method of solution 181

-1.5E-08

-1.0E-08

-5.0E-09

0.0E+00

5.0E-09

1.0E-08

1.5E-08

2.0E-08

0.0004 0.0009 0.0014 0.0019 0.0024

time

u1

(t)

m=3

m=0

m=-3

(a) u1(t)

-1.20E-08

-2.00E-09

8.00E-09

1.80E-08

2.80E-08

0.0004 0.0009 0.0014 0.0019 0.0024

time

u3

(t)

m=-3

m=3

m=0

(b) u3(t)

Fig. D.4: Displacement responses versus time to a step of force in the spherical representation(thin line) for the receiver location d = 0.01 m and z = 0.025 m; the interface is at Ri = 0.18 m;the mechanical parameters are those collected in Table D.2.1. The response obtained in the samemodel with a plane interface (see [43]), for a receiver location r = 0.01 m and x3 = 0.025 m in theconfiguration and with the notations used in our previous paper, is represented for comparisonin bold line. Among the different values considered for the parameter m, only m = 0 gives aphysically acceptable result. (a) u1(t); (b) u3(t). Time is in seconds.

reasonable choice. However, the optimum value of m may depend on the mechanical parametersof the model.

The value of the reference radius a is chosen so that the amplitudes computed close to thespherical interface are the same as those computed close to the plane interface. The justificationfor this procedure is that the focalization increases with the distance from the interface; close

182Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

enough to the interface, the effect of focalization should not be seen, hence the numerical calcu-lations for the flat and spherical interfaces must give the same amplitudes. We found that thereference radius a must be equal to the distance between the source and the center of curvatureO of the spherical surface.

The number of homogeneous sub-layers to introduce in the flat representation to accountfor the x3-dependence of the mechanical properties is determined by testing several spatialdiscretizations. Figure D.5 shows the evolution of u3(t) with the number of sub-layers, for φ(t) =H(t), for a receiver located at d = 0.005 m and z = 0.06 m and for m = 0. It can be seen onthe plots that — for the length scales of the problem under study —, it is useless to take layersthiner than 0.5 mm (the mean relative error is less than 5 %).

0.0E+00

5.0E-09

1.0E-08

1.5E-08

2.0E-08

0.0038 0.0043 0.0048time

u3

(t)

Fig. D.5: Displacement responses u3(t) to a step of force, in the spherical representation for thereceiver location d = 0.005 m and z = 0.06 m; the interface is at Ri = 0.18 m. The four plotseach correspond to a spatial discretization (a given sub-layer thickness), from left to right thecomputation were made with: h = 0.01 m, h = 0.005 m, h = 0.0005 m and h = 0.0002 m, whereh is the thickness of sub-layer. Time is in seconds.

D.2.6 Numerical validation

The computations whose results are illustrated in Figures D.4(a) and D.4(b), have provedthat the method and its numerical implementation can yield physically acceptable results (form = 0). Other tests have been performed to validate the method in a few simple cases.(i) The two layers in the spherical geometry have been given the same mechanical propertiesso that the receiver and the buried point force are placed in a homogeneous medium. Thedisplacement responses for a buried point force, with φ(t) = H(t), have been computed andcompared with the results obtained for the model with a plane interface investigated in [43].Displacements u1(t) and u3(t) are shown in Figures D.6(a)-D.6(d) for m = 0. Since the twomedia have the same properties, there is no wave conversion or reflection at the interface andthe transmission coefficients are equal to one. A small error is introduced in the computationof the displacement responses u1(t), at a receiver in a homogeneous medium with properties of

D.3. Numerical example: repartition of energy in a model of the thorax 183

medium 1 (Fig. D.6(a)) or medium 2 (Fig. D.6(b)) (see Table D.2.1) : the approximation usedfor the spherical model lead to a small overestimation of the displacement (less than 10%). Incontrast, responses u3(t) (at a receiver in a homogeneous medium with properties of medium 1(Fig. D.6(c)) or medium 2 (Fig. D.6(d))), for the plane and spherical interfaces, are almostindistinguishable. (It is to note that the global contribution of u1(t) to the response in termsof the Poynting vector, is — at the receivers of interest in the present study — less thanu3(t).) With this comparison of the displacement responses, we conclude that the change ofcoordinate system and of mechanical properties introduced by the flattening approximationdoes not perturbate the wave propagation pattern.(ii) The computed travel times for receivers on axis d3 have shown good agreement with thosecalculated with known analytical expressions; for a sub-layer thickness of 0.5 mm, the error isless than 0.1%.(iii) All the calculations of interest in this paper are concerned with a wave traveling in thedirection of decreasing radius: in this case, we expect to observe focalization of energy (this isthe case, as shown in the next section). Likewise, we have checked that waves propagating inthe direction of increasing radius, decrease in amplitude with the distance from the interface,more rapidly in a model with a spherical interface than in a model with a plane interface.(iv) Letting the radius of curvature become very large, the results found with the methodincluding the FA converge towards the results obtained in the model with the plane interface.

The results of the aforementioned tests of the method, allow us to suppose — althoughthere is no complete mathematical validation for this — that the set of FA can yield physicallymeaningful results.

D.3 Numerical example: repartition of energy in a model of the

thorax

The impact wave generated by a “high” velocity non-penetrating impact on the thoracicwall (like when a projectile is stopped by a bulletproof jacket) is supposed to induce severe lunginjuries. The detailed study of the propagation of an impact wave in the thorax is a complicatedmatter; the present work, which is an extension of our work presented in [43], is part of apreliminary study whose objectives are to describe the wave propagation in a model of thethorax and to identify some phenomena involved in the mechanisms of lung injury. The specificmotivation for the work presented here is to evaluate the influence of the curvature of the thoraxon the repartition of energy. In the model, the impact wave is generated by a point force at thefree — spherical — surface of medium 1, the direction of the force is perpendicular to the surface(in the equivalent flat representation, the force is defined by (D.6)). (The way this point forceaccounts for the real loading of the thoracic wall will be investigated in another study.) As before,the force source is applied, normal to the free surface at 2 cm from the interface; the time historyof the source is a Blackman window (Fig. D.3) of 0.3 ms duration. In all the computations, wehave used m = 0.

As for the plane interface case investigated in [43], the wave fronts of the pressure and shearwaves are found to be almost plane in medium 2; obviously, this is due to the weak acousticcoupling between media 1 and 2.

Figure D.7 shows typical plots of P3(t) for three receivers placed in medium 2 on axis d3;the radius of curvature of the interface is Ri = 28 cm (these plots may be compared to thoseobtained with the plane interface in [43]). Again, the distortion of the input pulse with the

184Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

0.0E+00

1.0E-10

2.0E-10

3.0E-10

4.0E-10

5.0E-10

6.0E-10

7.0E-10

8.0E-10

9.0E-10

1.0E-09

0.00E+00 2.00E-04 4.00E-04 6.00E-04t

u1

(t)

(a) u1(t) by using the properties of medium 1

given in Table D.2.1

0.0E+00

5.0E-05

1.0E-04

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014t

u1

(t)

(b) u1(t) by using the properties of medium 2

given in Table D.2.1

0.0E+00

3.0E-09

6.0E-09

0.00E+00 2.00E-04 4.00E-04 6.00E-04t

u3

(t)

(c) u3(t) by using the properties of medium 1 gi-

ven in Table D.2.1

0.0E+00

1.3E-04

2.5E-04

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014t

u3

(t)

(d) u3(t) by using the properties of medium 2

given in Table D.2.1

Fig. D.6: Displacement responses versus time to a step of force in the spherical representationare plotted for m = −3, 3 and 0 in thin line, when the receiver location is given by d = 0.01 mand z = 0.04 m. The responses obtained in the model with a plane interface (see [43]), for anequivalent receiver location, are plotted for comparison in bold line. The mechanical parametersare the same in media 1 and 2. Time is in seconds.

distance from the source — due to the coupling of far and near-field terms — is manifest. It isto note that for receivers located near axis d3, the contribution of the shear waves is negligiblewith respect to the pressure waves contributions. The increase in amplitude with the distancefrom the spherical interface demonstrates the focalization of energy (this amplitude decreaseswith a plane interface).

Figure D.8 illustrates the influence of the radius of curvature of the interface on the degree offocalization. It shows the evolution of the maxima of the P3(t) (read on plots like Fig. D.7) versusthe distance from the source, at receivers in medium 2, for various interface’s curvatures andfor the buried explosion source used in [43] (using a force source instead of an explosion sourcehas little influence on the focalization pattern on axis d3). From the biomechanical applicationpoint of view, it is interesting to note, as displayed in Figure D.8, that P3 weakly decreases withz for Ri < 30 cm. So, the focalization is pronounced for radius of curvatures of less than 30 cm,

D.3. Numerical example: repartition of energy in a model of the thorax 185

0.0E+00

5.0E-04

1.0E-03

1.5E-03

2.0E-03

2.5E-03

0 0.5 1normalized time

P3

(t)

Fig. D.7: P3(t) responses to a point force with the time history illustrated in Figure D.3, forthree receivers on axis d3; the radius of curvature of the interface is Ri = 0.28 m; the threevalues of z are: 0.025 m (thin line), 0.04 m (intermediate line) and 0.1 m (bold line). The timescale is normalized with respect to the duration of the pulse and the plots are offset of the arrivaltime. The amplitude of P3(t) increases with the distance from the source; this is a focalizationeffect due to the interface’s curvature.

which is a value characteristic of the thoracic geometry.

In reference [43], we had represented the repartition of the P and S-waves transient strainenergy contributions, denoted by EP,S , in medium 2; the strain energy was calculated by usingthe formula EP,S = Max[PP,S

3 (t)](2cP,S)−1, where Max[PP,S3 (t)] is extracted from plots like

those shown in Fig. D.7 (P and S-waves contributions are well separated in time so that P3(t)can be split, without ambiguity, in two parts: PP

3 (t) and PS3 (t)). Figure D.9 shows the maximum

transient strain energies in medium 2 for a model with a spherical interface. Results shown inFigure D.9 correspond to a plane interface, and to an interface’s curvatures of Ri = 28 cm;for each case, three plots corresponding to three distances z from the source are represented;the values in abscissa correspond to distances d (see Fig. D.1) from the source (the choice ofthe coordinates (d, z) in the spherical geometry allow for comparison with the values compu-ted for the plane interface case). In the plane interface case, the amplitude of P and S-wavescontributions weakly decreases with the distance from the source; in contrast, in the sphericalinterface case, the amplitudes increase with the distance from the interface. Close to the inter-face (z=0.025 m), there are little differences in the amplitudes of the P-waves contributions, forthe plane and spherical interfaces, this illustrates that the focalization is negligible close to theinterface. S-waves contribution amplitudes are smaller for the spherical case; this is probablydue to differences in the receivers’ locations with respect to the interface. For a point source atthe free spherical surface, the focalization is shown to be important mostly close to axis d3, ford > 2 cm the plots corresponding to the two cases get superimposed.

186Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

0

1

2

3

4

5

6

7

-0.01 0.01 0.03 0.05 0.07 0.09 0.11 0.13 0.15 0.17

z (m)

Nor

mal

ized

max

imum

of

P3

(t)

Ri= 0.13

Plane interface

Ri = 0.33

Ri = 0.28

Ri = 0.23

Ri = 0.18

Fig. D.8: Normalized maximum of P3(t) for receivers on axis d3 for various interface curvatures.Distances are in meters.

0.0

0.5

1.0

0 0.01 0.02 0.03 0.04 0.05d

Max

[P3(t

)]/2

CP

,S

z=0.1

z=0.04

z=0.025

Pressure waves contribution

z=0.04 z=0.025z=0.1

Shear waves contribution

Fig. D.9: Maximum of strain energy contributions (a quantity derived from P3(t)) in medium 2of P-waves (rays PP and SP ) and S-waves (rays PS and SS). The source is a force at the freesurface of medium 1. Continuous lines correspond to the spherical interface case (Ri =0.28 m)and dotted lines to the plane interface case [43]. Distances are in meters.

D.4 Conclusion

Three-dimensional transient elastic wave propagation in a spherically symmetric bimaterialmedium has been investigated. The method of solution makes use of a set of flattening trans-formations (FA) to build a “flat” representation equivalent — with an approximation — to the

D.4. Conclusion 187

spherically symmetric representation; wave propagation is then computed with the exact three-dimensional generalized ray/Cagniard-de Hoop method in the flat representation. Chapman [20]has shown that FA for the coupled P-SV problems are not exact and his work proved that itis delicate to evaluate the validity of the approximation. Hence, the solution obtained in thisstudy is only an approximation to the exact solution. However, we have checked that the resultsare physically acceptable and are coherent with results obtained for a bimaterial with a planeinterface in a former work [43]. Except for the work of Chapman [20], we could not find anywork discussing specifications for the choice of the parameters (m and l) of the power-law trans-formations given by Eq. (D.2). We have found optimal values by comparing the displacementresponses to a step of force, for a spherical interface and a plane interface — for which the exactsolution could be computed.

Flattening transformations have been widely used in geophysics to study surface waves (Loveand Rayleigh waves), spheroidal modes of vibration, transient propagation of SH or P-SV bodywaves with many different analytical methods; however, the application of the FA to other fieldsof mechanical engineering is not known to the authors. We have used the FA to investigate thewave propagation in a simplified model of the thorax subjected to an impact wave loading. Wehave been interested in responses at receivers close to axis d3 in a weakly coupled bimaterial;these are the two main differences with the cases investigated by geophysicians: they rathercompute responses at large horizontal distances from the source with asymptotic approximations(that are not valid in the case investigated here), and in layered media that are not weaklycoupled. From the application’s point of view, the results presented in this paper yield qualitativeinformations on the propagation of an impact wave in the thorax; in particular the resultsindicate that the curvature of the thoracic wall-lung interface may contribute to focalize energy:the substitution of a spherical interface of radius Ri = 28 cm, to a plane one, increases theenergy transmitted, in the medium representing the lung, of about 50% at some points. Thismay play a role in the occurrence of injuries in certain curved regions of the lung.

188Transient elastic wave propagation in a spherically symmetric bimaterial medium

modeling the thorax

Annexe E

Resultats de calculs supplementaires

E.1 Reponses en termes de deplacement

Les figures E.1 a E.3 presentent les reponses u1(t) et u3(t) pour differents recepteurs dansle Cas III et dans la Configuration 2a (force ponctuelle). Les contributions des integrales debranches (ondes de tete), associees a l’indice IB, sont reproduites dans la courbe inferieure dechaque figure, on peut ainsi evaluer l’amplitude de leur contribution par rapport a celle desondes de volume. Les reponses correspondant a l’interface collee et glissante sont en traits fin etgras respectivement. Toutes les dimensions indiquees sont en centimetres.

E.2 Calcul par elements finis

Les figures E.4 a E.10 sont des cartes d’isovaleurs dans le modele elements finis (voir § 8.4.2,p. 125).

190 Resultats de calculs supplementaires

1 2 3 4 5 6 7 8 9

x 10−4

0

2

4

6

8x 10

−9

t

u1(t

)

x1=0.005 ; x

3=0.025

1 2 3 4 5 6 7 8 9

x 10−4

−1

−0.5

0

0.5

1

t

u1(t

) (I

B)

(a) u1(t)

1 2 3 4 5 6 7 8 9

x 10−4

0

0.5

1

1.5

2

2.5x 10

−8

t

u3(t

)

x1=0.005 ; x

3=0.025

1 2 3 4 5 6 7 8 9

x 10−4

−1

−0.5

0

0.5

1

t

u3(t

) (I

B)

(b) u3(t)

Fig. E.1: Deplacement pour un recepteur place a (x1 = 0.005, x3 = 0.025).

E.2. Calcul par elements finis 191

1 2 3 4 5 6 7 8 9

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2x 10

−8

t

u1(t

) x1=0.025 ; x

3=0.025

1 2 3 4 5 6 7 8 9

x 10−4

−0.5

0

0.5

1

1.5

2

2.5

3x 10

−9

t

u1(t

) (I

B)

(a) u1(t)

1 2 3 4 5 6 7 8 9

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2x 10

−8

t

u3(t

)

x1=0.025 ; x

3=0.025

1 2 3 4 5 6 7 8 9

x 10−4

−5

0

5

10

15

20x 10

−10

t

u3(t

) (I

B)

(b) u3(t)

Fig. E.2: Deplacement pour un recepteur place a (x1 = 0.025, x3 = 0.025).

192 Resultats de calculs supplementaires

1 2 3 4 5 6 7 8 9 10

x 10−4

−1

0

1

2

3

4

5x 10

−9

t

u1(t

)

x1=0.05 ; x

3=0.025

1 2 3 4 5 6 7 8 9 10

x 10−4

−4

−2

0

2

4

6x 10

−10

t

u1(t

) (I

B)

(a) u1(t)

1 2 3 4 5 6 7 8 9 10

x 10−4

−2

0

2

4

6x 10

−9

t

u3(t

)

x1=0.05 ; x

3=0.025

1 2 3 4 5 6 7 8 9 10

x 10−4

−2

0

2

4

6

8

10x 10

−10

t

u3(t

) (I

B)

(b) u3(t)

Fig. E.3: Deplacement pour un recepteur place a (x1 = 0.05, x3 = 0.025).

E.2. Calcul par elements finis 193

Fig. E.4: Isovaleurs des vitesses resultantes a T = 10 µs.

Fig. E.5: Isovaleurs des vitesses resultantes a T = 30 µs.

194 Resultats de calculs supplementaires

Fig. E.6: Isovaleurs des vitesses resultantes a T = 60 µs.

Fig. E.7: Isovaleurs des vitesses resultantes a T = 80 µs.

E.2. Calcul par elements finis 195

Fig. E.8: Isovaleurs des vitesses resultantes a T = 110 µs.

Fig. E.9: Isovaleurs des vitesses resultantes a T = 210 µs.

196 Resultats de calculs supplementaires

Fig. E.10: Isovaleurs des vitesses resultantes a T = 410 µs.