Huy Duong Bui

Abdelbacet Oueslati
email: oueslati@lms.polytechnique.fr

Solutions exactes de fissure d'interface sous contact frottant avec un milieu indéformable

Keywords: Solides et structures, Frottement, Fissure d'interface Solids and structures, Friction, Interface crack

On étudie une fissure à l'interface d'un massif élastique semi-infini avec un demi-plan rigide, en présence de frottement de Coulomb. Il existe deux solutions singulières à la tête d'une fissure semi-infinie correspondant physiquement aux deux problèmes d'enfoncement et d'arrachement de la fibre indéformable dans une matrice élastique. Les solutions exactes montrent que seul le mode II est présent et qu'il n'y a pas de comportement oscillant près de la pointe de la fissure. La solution pour une fissure d'interface de longueur finie montre qu'il n'y a pas d'oscillations près des pointes de la fissure, mais qu'il y en a plus loin à la transition avec la zone décollée. Il y a quatre types de solutions singulières pour une fissure finie dans un solide borné, mais une seule solution dans un milieu infini.

Abridged English version

Let the elastic medium Ω + be bonded to the rigid body Ω -except on the crack. The general boundary conditions are given by Eqs. ( 4)-( 6) and [START_REF] Deng | An asymptotic analysis of stationary and moving cracks with frictional contact along bimaterial and in homogeneous solids[END_REF]. The method of solution is based on the use of the classical conjugate transform of function g(z) → ḡ(z) and the assumption that the function φ(z) defined in Ω + can be continued in Ω -along the bonded zone, which can be verified a posteriori on the solution. Then the conditions (4) of vanishing displacement suggest the definition of the function ψ(z) on the entire complex plane with the cut along the crack as Eq. ( 11) ψ(z) = κ φ(z)φ (z).

From the contact condition on the interface crack u 2 = 0, we show that for the finite interface crack φ (z) = -φ (z) + C 0 (14). We first derive the solution for the non-opening case. From Coulomb's law, we obtain Eq. ( 17) φ (t + ) = gφ (t -) + B 0 on the crack with g given by (18). The homogeneous equation possess two singular solutions φ (z) = C(za) -1/2±α . The sign (+) corresponds to the push-in type solution (A solution), the (-) sign corresponds to the pull-out type solution (B solution). The general feature of the solutions are: unbounded stresses at the crack tip; singular compressive stress σ 22 (x 1 < 0, x 2 = 0) in the crack, vanishing normal stress σ 22 ahead the crack tip, confirming Comninou's result on the boundedness of this component of stress, derived in the frictionless case. Therefore, from the classical definition of the stress intensity factor in mode I, one gets K I = lim r→0 √ (2πr)σ 22 (r > 0, θ = 0) = 0. The unusual definition of the mode I stress intensity factor using the compressive normal stress behind the tip is meaningless. There is no oscillatory behavior of stress and displacement at the crack tip, but only at the transition points with the stress free zone.

The interface crack under friction propagates only in mode II. Since there is friction in the crack, the energy release rate cannot be determined in the singular solution, but only in the case of finite geometry of interface crack. In the case of finite crack, four types of solutions are derived, The A-B, A-A, B-B, B-A solutions. Contrary to common belief, there exists an oscillatory behavior of the solution at the transition points with the stress free zone. Physically, only one type of solution of finite energy (the A-A solution) exists in an infinite medium, the other solutions are not acceptable for physical reasons unless one changes the friction law.

Introduction

On trouve les fissures d'interface dans le délaminage entre les couches qui constituent un composite, dans l'arrachement d'une fibre hors de la matrice ou encore dans l'enfoncement d'un pieu etc. Sur le sujet considéré, il y a déjà de nombreux travaux : citons parmi les premiers travaux ceux de Williams [START_REF] Williams | The stress around fault or crack in dissimilar media[END_REF], Rice et Sih [START_REF] Rice | Plane problems of cracks in dissimilar media[END_REF] et Willis [START_REF] Willis | Fracture mechanics of interfacial cracks[END_REF] et bien d'autres. La solution asymptotique des champs de déplacements et de contraintes diffère notablement de celle du milieu homogène par le fait que, d'après les travaux connus pour le cas sans frottement, les potentiels de Muskhelishvili ont des singularités de type puissances à exposants complexes z -1/2-i , avec la « constante bi-élastique » en déformation plane

= 1 2π Log{ (3-4ν 1 )µ 2 +µ 1 (3-4ν 2 )µ 1 +µ 2 }, cf. [4]
. Ce résultat a été obtenu avec les conditions aux limites de surface libre de contraintes sur la fissure, σ 12 = σ 22 = 0 sur R -. Deux conséquences résultent du modèle. La singularité des contraintes sur le ligament (z = r e iθ ∈ Ω + ou Ω -) est oscillante, de longueur d'onde de plus en plus courte à mesure que l'on se rapproche de la pointe de la fissure [START_REF] Wang | The interface crack between dissimilar anisotropic composite materials[END_REF] ont traité le problème de fissure finie, avec contact mais sans frottement. L'objet de cette Note est de réexaminer le problème d'une fissure d'interface R -, dans un cas très particulier, celui d'un demi-plan élastique collé à un substrat indéformable le long de R + en considérant le contact unilatéral avec frottement de Coulomb à coefficient f constant (f > 0). Notre analyse diffèrera notablement de celle de Comninou [START_REF] Comninou | The interface crack[END_REF] par la méthode de prolongement analytique utilisée ici, utilisée également dans Deng [START_REF] Deng | An asymptotic analysis of stationary and moving cracks with frictional contact along bimaterial and in homogeneous solids[END_REF], nous conduisant à une solution explicite, alors que celle de Comninou, basée sur l'étude des équations intégrales singulières couplées pour les dislocations inconnues, requiert une résolution numérique. En outre nous donnons la solution exacte de la fissure d'interface de longueur finie. Le cas de vrais bi-matériaux fera l'objet d'une autre publication.

(σ 22 + iσ 12 )(r, θ = 0) = Kr i √ 2πr = K √ 2πr cos( Log r) + i sin( Log

Équations du problème

Nous considérons un solide élastique occupant le demi-plan Ω + , soudé au demi-plan indéformable Ω -le long de l'axe réel, sauf sur la fissure d'interface de longueur 2a, cf. (

σ 22 + iσ 12 ) = φ (z) + φ (z) + zφ (z) + ψ (z) (3) u 1 = u 2 = 0 sur la partie collée (4) u 2 = 0, |σ 12 | = -f σ 22 (σ 22 0) sur la partie en contact frottant (5) (σ 22 -iσ 12 ) = 0 (u 2 > 0) sur la partie décollée (6) σ ik → t ∞ 11 , t ∞ 22 , t ∞ 12 quand |z| → ∞ dans Ω + (7) φ (z → ∞) = 1 4 t ∞ 11 + t ∞ 22 -i(k + 1) -1 t ∞ 12 (8) ψ (z → ∞) = 1 2 t ∞ 22 -t ∞ 11 + 2it ∞ 12 ( 9
)
où κ = 3 -4ν en déformation plane et z désigne la variable complexe définie par z = x 1 + ix 2 . Dans ( 5) et ( 6) les inégalités entre parenthèses ne sont pas imposées, mais sont à vérifier a posteriori pour la cohérence physique du modèle.

Bien que φ (z) soit seulement définie dans Ω + nous supposons qu'elle soit prolongeable dans Ω -, à travers la zone collée (point à vérifier a posteriori). Donc φ (z) est holomorphe dans tout le plan complexe avec la coupure suivant la fissure. Soit la transformation de conjugaison définie par g(z) → ḡ(z) := g(z) qui à toute fonction holomorphe dans Ω + (resp. Ω -) fait correspondre la fonction dite conjuguée, obtenue en remplaçant z par z puis en prenant le conjugué du résultat, qui est holomorphe dans Ω -(resp. Ω + ). La propriété essentielle est que sur l'axe réel z = z, là où la fonction g(z) est continue, on a l'égalité ḡ(z) = g(z) = g(z).

Méthode de résolution du problème. Cas sans décollement

Sur la zone soudée de l'axe réel, remplaçons dans (1) φ(z) par φ(z) et z par z, conformément à la remarque du paragraphe précédent. Nous obtenons de la condition d'adhérence (1)

2µ(u 1 -iu 2 ) = κ φ(z) -zφ (z) -ψ(z) = 0 (10) d'où ψ(z) = κ φ(z) -zφ (z) (11)
Le prolongement analytique utilisé ici, obtenu avec (4) diffère de celui fait classiquement à partir de [START_REF] Comninou | The interface crack[END_REF]. Nous définissons la fonction ψ(z) donnée par le second membre de (11) et la prolongeons non seulement dans Ω + mais dans tout le plan complexe coupé suivant la fissure. En dérivant (11) on obtient ψ (z) que l'on peut reporter dans l'expression de la contrainte sur la fissure (σ 22iσ 12 ) pour z = z + , z = z -ainsi que dans celle de la dérivée partielle ∂u 2 /∂x 1 qui est nulle dans la fissure, car le contact frottant est supposé maintenu partout. Nous trouvons

σ 22 -iσ 12 = φ (z) + κφ (z) sur l'axe réel (12) La nullité de ∂u 2 /∂x 1 implique que 4iµ ∂u 2 ∂x 1 = κ φ (z) -φ (z) -φ (z) + φ (z) = 0 (
φ (z) = -φ (z) + C 0 C 0 = φ 0 + φ 0 ( 14
)
Pour achever la détermination de φ (z), nous explicitons la condition de frottement en écrivant σ 12 = f σ 22 en présupposant σ 12 < 0 et σ 22 < 0, avec

2iσ 12 = -(κ + 1) φ (z) + φ (z) -C 0 , ( 15
)
2σ 22 = (κ -1) φ (z) -φ (z) + C 0 (1 + κ) ( 16
)
Nous obtenons l'équation suivante

φ (t + ) = gφ (t -) + B 0 sur [-a, a], ( 17
) où g = - (κ + 1) + if (κ -1) (κ + 1) -if (κ -1) (18) avec B 0 = -C 0 (κ+1)(if -1) (κ+1)-if (κ-1) et |g| = 1.
En prenant les mêmes notations que celle du livre de Muskhelishvili [START_REF] Muskhelishvili | Some Basic Problems of the Mathematical Theory of Elasticity[END_REF], nous posons tg(πα

) = f κ-1 κ+1 , d'où 0 α 1 2 .
Les deux bornes de α réel proviennent de f 0 et de 17) est une variante d'un problème de Hilbert, dont la solution est donnée dans Muskhelishvili [START_REF] Muskhelishvili | Some Basic Problems of the Mathematical Theory of Elasticity[END_REF]. La solution singulière de l'équation homogène est

κ -1 = 3 -4ν -1 = 2(1 -2ν) > 0. Par conséquent g = -e 2iπα = e 2iπ(α+ 1 2 ) . Posons γ = 1 2iπ Log(g) = α + 1 2 . L'Éq. (
X 0 (z) = (z + a) -1/2-α (a -z) -1/2+α (19)
σ 12 = - b(κ + 1) 2 r -1/2+α , σ 22 = 0 et u 1 = u 2 = 0 (|x 1 | > a) ( 20
)
σ 12 = - b(κ + 1) 2 sin(πα)r -1/2+α , σ 22 = - b(κ -1) 2 cos(πα)r -1/2+α (|x 1 | < a) (21) u 1 = - bκ µ(1 + 2α) cos(πα)r 1/2+α , u 2 = 0 (|x 1 | < a) (22)
La condition de contact est bien respectée. La constante b est ici positive pour que σ 22 et σ 12 soient toutes les deux négatives sur la fissure, comme il a été supposé. Dans ce cas, la face de la fissure se déplace par rapport au substrat vers la gauche (u 1 < 0) ce qui explique le cisaillement négatif. En l'abscence de frottement (α = 0), nous retrouvons la singularité de Comninou [START_REF] Comninou | The interface crack[END_REF], avec son surprenant résultat sur la finitude de σ 22 sur le ligament. Physiquement, la première singularité φ (z) = C(za) -1/2+α correspond cinématiquement à l'enfoncement de la fibre rigide dans le corps élastique (en anglais push-in), tandis que la deuxième singularité φ (z) = C(za) -1/2-α correspond à l'arrachement de la fibre (en anglais pull-out), cf. Fig. 1. Une solution de (17) est une combinaison des deux types de singularités, singularité arrachement ou enfoncement à gauche ou à droite.

φ (z) = 1 2πi B 0 X 0 (z) a -a 1 X 0 (t + ) dt t -z + C 1 B 0 X 0 (z) (23) = 1 2 B 0 1 -i tg(πα) + 1 2 B 0 i cos(πα) (z + 2aα) + 2C 1 X 0 (24)
La fonction X 0 (z), donnée par (19), dans le cas enfoncement-arrachement prend une valeur réelle pour z = z + , mais il y a d'autres types de fonctions X 0 (z) correspondant au type enfoncement-enfoncement etc. La constante complexe C 1 = C + iC est déterminée par deux conditions de fermeture pour assurer la continuité de u 1 (a) = u 2 (a) = 0, obtenues intégrant 2µ(u 1 + iu 2 ) sur la fissure

z = t + i0 2µ(u 1 + iu 2 )(x 1 , 0 + ) = x 1 -a κφ (z) -zφ (z) -φ (z) -ψ (z) dt (25)
En résumé, la solution sans décollement est déterminée entièrement, pourvu que l'on ait

σ 12 x 1 + i0; t ∞ 11 , t ∞ 22 , t ∞ 22 < 0 et σ 22 x 1 + i0; t ∞ 11 , t ∞ 22 , t ∞ 22 < 0, ∀x 1 ∈ [-a, +a] (26

Cas du décollement

Supposons qu'il y a un seul décollement sur l'intervalle [c, d] (-a < c < d < a) et que les zones de contact frottant sont les intervalles [-a, c] et [d, a]. Le cisaillement sur l'interface avant le glissement étant négatif, le mouvement de glissement de la face de la fissure va vers la gauche. On a une singularité de type enfoncement en z = +a et une singularité de type arrachement en z = -a. Nous cherchons s'il y a une solution sous la forme 11) demeure valable car elle résulte de la condition d'adhérence en dehors de la fissure. Les contraintes au niveau de la fissure sont données par

φ (z) = F (z)X 0 (z) avec X 0 (z) = (z + a) -1/2-α (a -z) -1/2+α (27) où F (z) est holomorphe dans C -[c, d]. L'Éq. (
σ 22 -iσ 12 = φ (z + ) + κφ (z -) (28)
Si F (z) est continue, alors les deux conditions de contact frottant u 2 = 0 et σ 12 = f σ 22 , en présupposant σ 12 et σ 12 négatifs, sont remplies par la fonction discontinue X 0 (z). Maintenant, la condition de surface libre dans la zone décollée est

(σ 22 -iσ 12 = 0) sur [c, d], c'est à dire φ (z + ) + κφ (z -) = 0 sur [c, d] (29)
En tenant compte de X 0 (t + ) = gX 0 (t -) nous obtenons l'équation de discontinuité

F (t + ) + κ g F (t -) = 0 ⇒ F (t + ) = g * F (t -) dans [c, d] (30)
En posant g * = -κ g = e 2iπβ avec β = β 1 + iβ 2 complexe, on obtient β = -α -i 2π Log(κ). On pose β = -i 2π Log(κ). Une fonction satisfaisant à (30) est

F (z) = (z -c) α+(i/(2π)) Log(κ) (d -z) 1-α-(i/(2π)) Log(κ) (31)
Remarquons que la coupure, pour la fonction F (z), se trouve maintenant sur [c, d]. Nous allons combiner les singularités pour respecter plusieurs conditions : les branches des fonctions φ (z) singulières de type (z -a i ) -γ (zb i ) γ -1 à choisir sont celles pour lesquelles elles tendent à l'infini vers ( 1 z ). Les fonctions φ (z) sont régulières en z = c et z = d, holomorphes dans tout le plan complexe coupé suivant la fissure, tendent vers une constante à l'infini et vérifient les conditions de discontinuités (contact frottant et surface libre). Pour l'holomorphie, il est nécessaire que la somme des exposants des fonctions puissances dans les solutions ci-dessous soit égale à un entier positif, négatif ou nul.

Solution de type (B-A)

. 2-α . Les zones d'interpénétration hors de la zone de contact sont de plusieurs ordre de grandeur plus petits (10 -6 a) par rapport aux zones oscillantes connues dans le cas des solutions sans contact ( de l'ordre de 10 -4 a). La deuxième condition (6) est vérifiée pratiquement sur la surface libre, exceptée deux petites zones.

φ BA = C(z + a) -1/2-α (z -c) -β +α (d -z) 1+β -α (a -z) -1/2+α (32)
Ainsi, s'il n'y a pas d'oscillations à la pointe de la fissure, grâce au contact unilatéral, les oscillations se déplacent plus loin, aux points de décollement c et d, contrairement aux idées reçues reportées dans la littérature (sans frottement) soit sur des solutions asymptotiques avec frottement de Deng [START_REF] Deng | An asymptotic analysis of stationary and moving cracks with frictional contact along bimaterial and in homogeneous solids[END_REF], soit sur des calculs numériques sans frottement de Gautesen et Dundurs [START_REF] Gautesen | The interface crack under combined loading[END_REF]. Ceci tient à la nature mathématique des solutions des problèmes aux limites mixtes, conditions de Dirichlet et de Neumann, de part et d'autre part d'un point ou conditions mélangées (c'est notre cas). Dans Muskhelishvili [START_REF] Muskhelishvili | Some Basic Problems of the Mathematical Theory of Elasticity[END_REF] (page 486), un problème de poinçonnement avec les données de (u 1 + iu 2 ) d'un côté et (σ 22 + iσ 12 ) de l'autre, étudié par Abramov, donne un déplacement oscillatoire en dehors de la zone de contact.

Pour obtenir les résultats ci-dessus, il reste à préciser les constantes inconnues c et d, déterminées par les conditions de continuité du déplacement sur la fissure sachant que u 1 (-a, 0) = 0 et que u 2 (c, 0) = 0, à savoir les deux conditions de fermeture U(a) = 0 et V (d) = 0, avec les parties réelle et imaginaire de (25)

U(x 1 ) = x 1 -a 2µu 1 dt et V (x 1 ) = x 1 c 2µu 2 dt ( 33
)
Convexes admissibles. En nous limittant à deux paramètres de chargement, on obtient finalement quatre types de convexes des contraintes définissant le convexe admissible par leur intersection

σ 12 x 1 + i0; t ∞ 12 , t ∞ 22 < 0 ∀x 1 ∈ [-a, c] ∪ [d, a] σ 22 x 1 + i0; t ∞ 12 , t ∞ 22 < 0 ∀x 1 ∈ [-a, c] ∪ [d, a] u 1 x 1 + i0; t ∞ 12 , t ∞ 22 < 0 ∀x 1 ∈ [-a, c] ∪ [d, a] u 2 x 1 + i0; t ∞ 12 , t ∞ 22 > 0 ∀x 1 ∈ [c, d] (u 1 et σ 12 sont de même signe sur [-a, c] et [d, a])
Remarque 2. Nous avons indiqué en détail la solution de type B-A en tête de fissure. La solution de type A-B en tête de fissure est obtenue par symétrie miroir. Ces deux solutions ont des singularités des contraintes ou de l'énergie non intégrable à une tête de fissure et donc ne sont pas physiques, comme l'a remarqué aussi Deng [START_REF] Deng | An asymptotic analysis of stationary and moving cracks with frictional contact along bimaterial and in homogeneous solids[END_REF]. Ce sont des solutions mathématiques, aussi non intégrables en énergie que les fonctions de Bueckner.

Fig. 1 . 2 )Fig. 1 .

 121 Fig. 1. Délaminage par cisaillement et tension.

Fig. 1 .

 1 Fig. 1. Delamination by shearing and tension.

  13) sur la fissure, d'où en séparant les variables z d'un côté et z de l'autre on voit que la fonction φ (z) + φ (z) déja holomorphe dans C -[-a, +a] est continue à travers la fissure et donc holomorphe dans C. Cette fonction est une constante égale à sa valeur C 0 à l'infini. Par conséquent nous avons la relation qui définit la fonction conjuguée φ (z) dans tout C -[-a, a] en fonction de φ (z)

  Étudions d'abord la nature de la singularité à droite, en posant r = |z -a|, φ (z) = C(za) -1/2+α . Prenons C = i b 2 imaginaire pur pour que le cisaillement soit réel sur le ligament, d'où

  Elle est de type arrachement en z = -a et z = d et de type enfoncementen z = c et z = a. Cette fonction est régulière en z = c car α > 0 et en z = d car (1α) > 0. Enfin la constante C est déterminée par la condition à l'infini, d'où C = 1 4 (t ∞ 11 + t ∞ 22 )i(k + 1) -1 t ∞ 12 . Au voisinage de z = c et z = d, la solution B-A présente un comportement non singulier mais oscillatoire (η = 1 2π Log(κ)), φ (zc) ≈ r α r iη et φ (zd) ≈ r 1-α r -iη . Ceci n'entraine pas d'interpénétration sensible pour le déplacement normal, car au voisinage de z = c et z = d, le déplacement u 2 oscille entre les deux courbes ±Br 1+α et ±Br

Remarque 3 .Remarque 4 .Remarque 5 .Remarque 6 .

 3456 Il n'y a pas de solutions des deux types A-A et B-B en l'abscence de décollement parce que les potentiels définis comme précédemment ne sont pas holomorphes dans C -[-a, +a]. Il existe une seule solution physique à énergie intégrable de type A-A aux extrémités de fissure, avec décollement en c et dφ AA = C(z + a) -1/2+α (zc) 1-β -α (dz) 1+β -α (az) -1/2+α(34)Elle est de type enfoncement en z = -a et z = a et arrachement en z = c et z = d. Cette solution correspond, pour certain choix de C, à un état de contraintes de flexion t ∞ 22 = 0 et t ∞ 11 = λx 2 . La singularité de (34) est faible aux pointes de fissure, le taux de restitution d'énergie, défini par une intégrale J[START_REF] Deng | An asymptotic analysis of stationary and moving cracks with frictional contact along bimaterial and in homogeneous solids[END_REF] (limitée ici à un demi-cercle de rayon tendant vers 0) est nul. L'intégrale J dépend du contour parce qu'elle exprime la dissipation par frottement sur les lèvres de la fissure situées à l'intérieur du contour considéré. La solution de type B-B aux extrémités de fissure et décollement en c et d n'est pas physique (énergie non intégrable). Elle est de typearrachement en z = -a et z = a et enfoncement en z = c et z = d. φ BB = C(z + a) -1/2-α (zc) 1-β +α (dz) 1+β +α (az) -1/2-α(35)On a donné une solution pour le milieu infini. Pour un milieu borné, pour des singularités de type arrachement comme en (32) et (35) on obtient une énergie finie en multipliant respectivement les potentiels par des polynômes, par exemple on multiplie (32) par (z + a) et (35) par (z 2a 2 ). Les singularités dans ce cas (milieu borné) ont un comportement ϕ BA (z + a) 1/2-α (zc) -β +α (dz) 1+β -α (az) -1/2+α ϕ BB (z + a) 1/2-α (zc) 1-β +α (dz) 1+β +α (az) 1/2-α

)

  Dans l'espace des chargements, ces inégalités définissent deux familles de convexes (des demi-espaces), indexées par x 1 , dont l'intersection commune définit le convexe admissible pour le chargement donnant une solution sans décollement. Prenons le cas de chargement à deux paramètres (t ∞ 22 et t ∞ 12 ) avec une compression forte (tension négative) et un cisaillement faible ou nul, il est clair que le point (t ∞ 22 = -∞ et t ∞ 12 = 0) fait partie du convexe admissible. Par contre, le point (t ∞ 22 = +∞ et t ∞ 12 < 0) ne fait partie de ce convexe, d'après les travaux antérieurs sur les solutions existantes dans le cas sans frottement. Le convexe admissible n'est pas vide. Notons que la contrainte σ 22iσ 12 = φ (z) + κφ (z) au niveau de la fissure σ 22 qui contribue au délaminage dépend uniquement de C 0 = (φ 0 + φ 0 ). Or d'après (8) on voit que la contrainte normale de tension horizontale t ∞ 11 > 0 joue le même rôle vis à vis du délaminage que la tension verticale t ∞ 22 > 0. Intuitivement, sous la contrainte par exemple résiduelle de traction horizontale t ∞ 11 > 0 la matière s'allonge horizontalement, mais son déplacement horizontal est empêché au niveau de l'interface adhérente (|x 1 | > a), d'où un mouvement de flexion vers le haut qui peut ouvrir la fissure, cela même si la contrainte t ∞ 22 est compressive. En outre, une tension t ∞ 11 > 0 entraîne par effet de Poisson une contraction dans le sens vertical, empêchée par l'adhérence avec le substrat. Ceci tend aussi à ouvrir la fissure.

	Remarque 1.


G Gary

H Zhao

Dynamic testing of fibre polymer matrix composite plates under in-plane compression

Keywords:

Experimental investigations of the failure strength of fibre reinforced polymer matrix composite plates under compressive impact loading is presented in this paper. A split Hopkinson pressure bar (SHPB) is used to measure these properties. The specimen being a plate, its crosssectional area is small compared with the area of the bars and the failure strength is weak. One has to then use low impedance bars made of a viscoelastic material. Subsequent experimental problems, such as dispersion corrections in viscoelastic bars, are analysed. One also has to use a special anti-buckling device to prevent the overall buckling of the specimen. It is shown that the presented SHPB system provides a precise measurement of forces and displacements at both ends of the specimen. A special attention is then given to the analysis of the test, especially in situations where a non-homogeneous state of stress in the specimen in observed.

Introduction

Compressive failure is a design limiting feature of fibre polymer matrix composite plates as it is always much less than the tensile one. Recent research indicates that this decrease is mostly due to a localised compressive plastic buckling. A review of these problems is given by Budiansky and Fleck [START_REF] Budiansky | Compressive failure of fibre composites[END_REF], mainly concerning studies on the compressive failure strength of composite plates under quasi-static loading.

As the fibre polymer matrix composite is going to be involved in dynamic loading situations, compressive failure strengths under impact loading are required. In the crash situation of automobiles, composite plates, when used, are indeed submitted to an in-plane compressive impact loading.

For fibre reinforced composite plates, most studies at high strain rates report the strength of the composite plates under transverse localised impact or piercing behaviour [START_REF] Davies | Flexed plate impact testing of carbon fibre-reinforced polymer composite[END_REF][START_REF] Liu | Matrix cracking in impacted glass/epoxy plate[END_REF][START_REF] Glinicki | Plate impact testing method for GRC materials[END_REF]. It corresponds to the most frequently found situations. Meanwhile, other tests to determine the energy absorbing capacity or the tensile impact behaviour are also reported [START_REF] Harding | A tensile testing technique for fibre-reinforced composites at impact rates of strain[END_REF][START_REF] Harding | Impact damage in composite materials[END_REF]. A review of works in this field has been written by Cantwell and Morton [START_REF] Cantwell | The impact resistance of composite materials-a review[END_REF]. As indicated by Cantwell and Morton, there are no standard impact tests for composites because it is hardly possible to predict their behaviour from one type of loading to another.

The plates to be studied were provided by a car company. The aim of our study was to evaluate their capability to absorb energy in a crash situation where they were loaded in compression. Therefore, the experiment is performed in a condition similar to the real application.

This paper presents a testing method specially developed to measure the compressive failure strength of composite plates using a viscoelastic split Hopkinson pressure bar (SHPB). Some experimental results for glass fibre polymer matrix composites are shown to illustrate the validity of the used technique.

Impact test measurement techniques

To test composite plates under impact, there are various testing configurations [START_REF] Cantwell | The impact resistance of composite materials-a review[END_REF]. For relative low strain rates, Charpy or Izod pendulum tests, drop-weight test and rapid servo-hydraulic machine can be used. On the contrary, in the range of high strain rates, SHPB or gas gun impact tests are employed. To perform a compressive failure test, all those configurations could be used. The pendulum test is significantly accurate mainly for the measurement of the energy absorption. Signals obtained with drop-weight impact tests or servo-hydraulic machine tests often contain perturbations due to the vibration of the testing machine [START_REF] Beguelin | caracte ´risation me ´canique des polyme `res et composites a `l'aide d'une machine d'essai rapide[END_REF]. It has been proved that, for instance, those measurements depend on the location of the sensors [START_REF] Tan | Use of statical indentation laws in the impact analysis of laminated composite plates[END_REF][START_REF] Genin | Endommagement de plaques composites sous sollicitation de choc[END_REF].

An adequate measurement technique for impact loading is the Hopkinson bar, which is a widely used experimental technique to study constitutive laws of materials at high strain rates [START_REF] Hopkinson | A method of measuring the pressure in the deformation of high explosives by the impact of bullets[END_REF][START_REF] Kolsky | An investigation of mechanical properties of materials at very high rates of loading[END_REF]. The SHPB test has gained a great popularity in the past decades and many works have contributed to improve the accuracy of the set-up [START_REF] Kolsky | Stress waves in solids[END_REF][START_REF] Lindholm | Some experiments with the split Hopkinson pressure bar[END_REF] and to extend the technique to the tensile (by Harding et al. [START_REF] Harding | Tensile testing of materials at impact rate of strain[END_REF]) and to the torsional loading (by Duffy et al. [START_REF] Duffy | On the use of a torsional split Hopkinson bar to study rate effects in 1100-0 aluminium[END_REF]). The three-dimensional effects in the specimen such as radial inertia and friction have also been studied by Davies and Hunter [START_REF] Davies | The dynamic compression testing of solids by the method of the split Hopkinson pressure bar[END_REF], and more recently by Malinowski and Klepaczko [START_REF] Malinowski | A unified analytic and numerical approach to specimen behavior in the split Hopkinson pressure bar[END_REF]. These corrections are in agreement with a two-dimensional numerical simulation by Bertholf and Karnes [START_REF] Bertholf | Two-dimensional analysis of the split Hopkinson pressure bar system[END_REF]. The assumption of the homogeneous state in the specimen has also been critically analysed by means of the transient wave simulation by Jahsman [START_REF] Jahsman | Reexamination of the Kolsky technique for measuring dynamic material behavior[END_REF] and by Gary and Zhao [START_REF] Zhao | On the use of SHPB techniques to determine the dynamic behavior of materials in the range of small strains[END_REF].

For composite specimens having a cylindrical form, the SHPB technique has already been used in tension or in compression [START_REF] Griffiths | A study of the dynamic behavior of carbonfibre composite using the split Hopkinson pressure bar[END_REF][START_REF] Sierakowski | Dynamic compressive strength and failure of steel reinforced epoxy composites[END_REF]. To perform a compressive impact test on plates with SHPB, some specific difficulties such as low impedance specimen effects or prevention of buckling effects will be solved.

Use of SHPB technique

Classical SHPB analysis

A SHPB set-up is composed of long input and output bars with a short specimen placed between them (Fig. 1). By the impact of a projectile at the free end of the input bar, a compressive longitudinal incident wave e i t in the input bar is developed. This incident wave will induce a reflected wave e r t in the input bar and a transmitted wave e t t in the output one. With strain gages cemented on the two bars, one can record those three waves, which allow for the determination of the forces and particle velocities at both faces of the specimen.

Assuming that F input , F output , V input ,V output denote forces and particle velocities at the specimen-bar interfaces and S B , E and C 0 are, respectively, the cross-section of the bars, Young's modulus, and the longitudinal wave speed, the following equations are used to obtain forces and velocities at both ends of specimen:

F input t S B Ee i t ϩ e r t V input t C 0 e i t Ϫ e r t F output t S B Ee t t V output t C 0 e t t 1
In order to obtain the stress-strain curve of the specimen from the forces and the velocities at both specimen faces, the classical analysis is based on the hypothesis of the homogeneity of stresses and strains within the specimen. The result is in the form of the following equations:

_ e s V output t Ϫ V input t l s Ϫ 2C 0 e r t l s 2a e s Ϫ t 0 2C 0 e r t l s dt 2b s s t F output S s 2c
where l s is the length of the specimen and S s is the cross sectional area of the specimen.

Limitations of the classical analysis

The assumption of the homogeneity of stresses and strains within the specimen is obviously not correct at the early stage of the test due to the transient effects: the loading starts at one face of the specimen whereas the other face remains at rest. When low impedance materials are tested, the time for the waves to travel through the specimen becomes longer. Moreover, when viscoelastic materials are tested, this effect is increased and the use of an inverse method can be necessary to find a constitutive model for the material tested [START_REF] Zhao | On the use of SHPB techniques to determine the dynamic behavior of materials in the range of small strains[END_REF][START_REF] Gary | Inverse methods for the dynamic study of non-linear material with a Split Hopkinson Bar[END_REF]. As it will be seen further in Fig. 5, such a situation occurs for the testing of fibre reinforced polymer plates. The independent measurement of the input and output forces is then needed.

Viscoelastic SHPB set-up

The tested plates have a small cross sectional area compared with the area of the bars because of their small thickness. As the material strength is also weak, the compressive crushing failure strength is small. To use the SHPB technique to measure it, one must overcome a major difficulty. Indeed, when the force in the specimen is small compared to the force associated with the incident wave, the transmitted wave will be weak, and one has to use a more sensitive measuring device to amplify the signal. On the contrary, e r t is almost equal to Ϫe i t according to Eq. (1). The input force then cannot be precisely measured because the difference of two almost equal values is nearly zero. The use of common metallic bars to test composite specimens is then difficult, even when the specimen is cylindrical and has the same cross sectional surface as the bar, as shown by Griffiths and Martin [START_REF] Griffiths | A study of the dynamic behavior of carbonfibre composite using the split Hopkinson pressure bar[END_REF][START_REF] Sierakowski | Dynamic compressive strength and failure of steel reinforced epoxy composites[END_REF]. Thus, an SHPB set-up made of a low impedance material is needed, for example, PMMA or nylon bars. These polymeric low impedance bars always demonstrate a viscoelastic behaviour. As compared with a conventional SHPB, the use of viscoelastic bars in a SHPB set-up introduces the following complications. The first is due to the wave dispersion in a viscoelastic bar. The second concerns the calculation of stress and particle velocity from the measured strain.

The wave dispersion in an elastic bar has been studied by Davies [START_REF] Davies | A critical study of Hopkinson pressure bar[END_REF]. On the basis of the Pochhammer and Chree's longitudinal wave solution for an infinite cylindrical elastic bar [START_REF] Pochhammer | Uber die Fortpflanzungsgeschwindigkeiten Kleiner Schwingungen in einem Unbergrenzten Isotropen Kreiszylinder[END_REF][START_REF] Chree | The equations of an isotropic elastic solid in polar and cylindrical co-ords., their solutions and applications[END_REF], a dispersion correction has been proposed. Even though the Pochhammer-Chree solution is not exact for a finite bar, it is easily applicable and sufficiently accurate for long bars. Such a correction is then accepted and applied by many authors [START_REF] Follansbee | Wave propagation in the split Hopkinson pressure bar[END_REF][START_REF] Gong | Dispersion investigation in the split Hopkinson pressure bar[END_REF].

This approach is extended to the case of viscoelastic bars [START_REF] Zhao | A three dimensional analytical solution of longitudinal wave propagation in an infinite linear viscoelastic cylindrical bar. Application to experimental techniques[END_REF][START_REF] Zhao | On the use of a viscoelastic split Hopkinson pressure bar[END_REF]. The solution of this equation gives a dispersive relation between wave number j and frequency v. Once the dispersive relation is known, one can calculate the wave propagated at a distance Dz; u p z t; from the measured wave u m z t as follows

u p z t FFT Ϫ1 {e ijvDz FFT u m z t } 3
Such a correction is very important, especially when dealing with viscoelastic bars. In our case, the input bar is 40 mm in diameter and 3 m long. Fig. 2a andb illustrates this point in the case of a test without specimen where the equality of input and output forces has to be checked. The result without dispersion correction shows a significant gap between both forces.

The forces and the particle velocities associated with a viscoelastic wave can be calculated after the strain using the Fourier analysis. When the constitutive relation (complex modulus) of the bar is known, the stress and the particle velocity are derived from the strain in the frequency domain by using the Fourier coefficients [START_REF] Zhao | On the use of a viscoelastic split Hopkinson pressure bar[END_REF].

Validation of the method

In order to give an experimental demonstration of the accuracy of the method, we put a Nylon bar and an aluminium bar in contact so that the force and the particle velocity at the interface measured by those two bars should be the same. Since the measurement by an aluminium bar is considered reliable, the precision of measurement with the Nylon bar can be then evaluated [START_REF] Zhao | On the use of a viscoelastic split Hopkinson pressure bar[END_REF]. The comparison of the two measurements is shown in Fig. 3a andb. This experiment provides also a global verification of the validity of the use of viscoelastic bars.

Anti buckling device

As mentioned by Budiansky and Fleck [START_REF] Budiansky | Compressive failure of fibre composites[END_REF], the most frequent mode of compression failure for composite plates is the localised plastic buckling. The experimental aim is then to determine the failure strength of the studied compo-site plate in this mode of failure. Therefore, other modes of failure should be avoided in the test.

The composite plate is not attached to the bar in order to prevent the perturbations of measurements due to additional interfaces [START_REF] Harding | A tensile testing technique for fibre-reinforced composites at impact rates of strain[END_REF][START_REF] Harding | Impact damage in composite materials[END_REF][START_REF] Griffiths | A study of the dynamic behavior of carbonfibre composite using the split Hopkinson pressure bar[END_REF][START_REF] Sierakowski | Dynamic compressive strength and failure of steel reinforced epoxy composites[END_REF]. It is then just in contact with the bars which actually increase the probability of elastic buckling. To eliminate these effects, honeycomb structures are used as complementary supports (Fig. 4). These structures are just placed in contact with the plates so that they can induce a reaction to prevent overall elastic buckling. Since their stiffness is very weak in the direction of the bars, they cannot perturb the axial measurements made with the SHPB set-up.

Experimental results

Imperfect equilibrium

The material to be tested is an industrially made glass/ epoxy composite plate of 3 mm thickness. The details of the manufacturing process and the exact fibre volume fraction have not been given by the providers. Square specimens 36mm × 36mm are cut from it. Using the viscoelastic SHPB made of PMMA (Young's modulus being about 6 × 10 9 MPa; forces and velocities at both ends of the specimen are measured. A typical result is illustrated in Fig. 5.

The hypothesis of homogeneous stress and strain fields along the specimen is not exactly verified. Indeed, as the input force and the output force are not equal (Fig. 5) the stress field cannot be uniform along the specimen length. Such an observation has been confirmed by high-speed photography in a similar situation [START_REF] Zhao | On the use of a viscoelastic split Hopkinson pressure bar[END_REF]. To obtain an accu-rate and detailed interpretation of the measurements (forces and velocities), a transient calculation using a failure model must be performed following the inverse calculation technique [START_REF] Zhao | On the use of SHPB techniques to determine the dynamic behavior of materials in the range of small strains[END_REF][START_REF] Gary | Inverse methods for the dynamic study of non-linear material with a Split Hopkinson Bar[END_REF] used for concrete [START_REF] Zhao | A study on testing techniques for concrete-like materials under impact loading[END_REF][START_REF] Gary | Measurement of dynamic behaviour of concrete under impact loading[END_REF] and polymers [START_REF] Zhao | A study of specimen thickness effects in the impact tests on polymers by numeric simulations[END_REF]. Such a calculation could explain, for instance, the unusual observation of the output force becoming greater than the input one after a short time (at point A, Fig. 5). This point could be related to the damage initiation in the composite plate.

Simplified analysis

For an industrial application, and considering that the deviation from equilibrium is not very important, using the conventional analysis (Eqs. (2a), (2b) and (2c)where the average force is set equal to the output forceis sufficient to study the influence of the strain-rate on the mean response of the plates. A typical result is shown in Fig. 6. It is seen on this figure that the strain rate is not constant during the test. It is assumed that failure occurs when the force starts decreasing. Then, the reference strain-rate for the test will be conventionally taken as the maximum value of the strain rate obtained before failure. In the example of Fig. 6, this value equals 282/s.

The composite plates exhibit a non-linear behaviour with a strain-rate sensitivity (Fig. 7). To confirm the strain rate effect, quasi-static tests in compression have been done. The tests have been performed with a "Instron" standard testing machine using the same anti-buckling device and same specimens as for dynamic tests. The loading rate, equal to 0.5 mm/min, corresponds to a strain rate of 2.3 10 Ϫ4 /s. It is observed, in Fig. 8, that the influence of the fibre direction does not depend on the loading rate and that the strain-rate effect increases more rapidly when the loading is in the fibre direction (0Њ).

Conclusions

Experimental results on the compressive failure strength of glass fibre epoxy matrix composite plates under dynamic loading are analysed. It is shown that the use of a viscoelastic SHPB set-up is an indispensable technique to obtain accurate measurement under an in-plane impact loading when low impedance specimens are used. The situation is especially critical for reinforced polymer matrix composite showing a relatively low failure strength. In particular, the measurement of both input and output force is made possible.

It is observed that the failure strength is rate sensitive and that the fibre orientation has an influence on the failure strength.

Fig. 1 .

 1 Fig. 1. SHPB set-up.

Fig. 2 .

 2 Fig. 2. (a) Input and output forces without dispersion correction. (b) Input and output forces with dispersion correction.

Fig. 4 .

 4 Fig. 4. Detail of the experimental anti-buckling device.

Fig. 5 .

 5 Fig. 5. Input and output forces of a SHPB test on a composite plate.

Fig. 3 .

 3 Fig. 3. (a) Impact Nylon bar-Aluminium bar: comparison between force measurements of Nylon bar and that of aluminium bar. (b) Impact Nylon bar-Aluminium bar: comparison between velocity measurements of Nylon bar and that of aluminium bar; the precision of the measured velocity is less good than that of the force because it is proportional to the difference between incident wave and reflected wave (which are two great values of same sign in this configuration) whereas the force is the sum.

Fig. 7 .

 7 Fig. 7. Stress-strain responses of composite plates under various loading rates. The failure strength increases with the strain rate.

Fig. 6 .

 6 Fig.6. Stress and strain rate vs. strain for a SHPB test on a composite plate.

Fig. 8 .

 8 Fig.8. Influence of strain rate on the failure strength of composite plates under dynamic compression.


Peixi Li

Yannick Benezeth

Keisuke Nakamura

Randy Gomez

Chao Li

Fan Yang

Comparison of region of interest segmentation methods for video-based heart rate measurements

Keywords: remote photoplethysmography (rPPG), Heart Rate (HR), Region of Interest (ROI)

Conventional contact photoplethysmography (PPG) sensors are not suitable in situations of skin damage or when unconstrained movement is required. As a consequence, remote photoplethysmography (rPPG) has recently emerged because it provides remote physiological measurements without expensive hardware and improves comfort for longterm monitoring. RPPG estimation methods use the spatially averaged RGB values of pixels in a Region Of Interest (ROI) to generate a temporal RGB signal. The selection of ROI is a critical first step to obtain reliable pulse signals and must contain as many skin pixels as possible with a low percentage of non-skin pixels. In this paper, we experimentally compare seven ROI segmentation methods in the perspective of heart rate (HR) measurements with dedicated metrics. The algorithms are compared using our in-house database UBFC-RPPG, comprising of 53 videos specifically geared towards rPPG analysis.

I. INTRODUCTION

T HE principle of photoplethysmography (PPG) is actually very simple as it only requires a light source and a photodetector. The light source illuminates the tissue and the photodetector measures the small variations in transmitted or reflected light associated with changes in perfusion in the tissue [START_REF] Sun | Photoplethysmography revisited: from contact to noncontact, from point to imaging[END_REF]. However, conventional contact PPG sensor is not suitable in situations of skin damage or when unconstrained movement is required. With the emergence of videobased health care monitoring, remote photoplethysmography (rPPG) has recently been developed [START_REF] Verkruysse | Remote plethysmographic imaging using ambient light[END_REF] as it allows remote physiological measurements only based on the ambient light and a video camera, hence reducing user constraint and without any expensive and specialist hardware requirement. The very low signal-to-noise ratio (SNR), precisely the ratio between the rPPG signal and all possible noises, represents the main difficulty of this methodology. Some research teams have proposed computer vision or signal processing techniques to improve the robustness of the original method [START_REF] Verkruysse | Remote plethysmographic imaging using ambient light[END_REF]. Current rPPG methods are not yet as accurate as the ECG's measurements nevertheless it seems obvious that, in a close future, they will enable a flexible monitoring of human vital signs (e.g. the heart rate, breathing rate) with a significant reduction of user's constraints or eventually extending the duration of the monitoring.

As discussed in different review papers [START_REF] Sun | Photoplethysmography revisited: from contact to noncontact, from point to imaging[END_REF], [START_REF] Mcduff | A Survey of Remote Optical Photoplethysmographic Imaging Methods[END_REF], many rPPG methods implement a general pipeline-based framework: regions of interest (ROI) are first detected and tracked over frames, RGB channels are then combined to estimate the pulse signal, which is filtered and analyzed to extract physiological parameters such as heart rate or respiration rate. This pipeline-based framework emphasizes the importance of the common first step of ROI segmentation. Several approaches have been proposed for ROI selection in the video stream. In earlier studies, manual selection of the ROI have been used [START_REF] Verkruysse | Remote plethysmographic imaging using ambient light[END_REF]. ROI segmentation can also be based on the result of classical face detection [START_REF] Poh | Advancements in noncontact, multiparameter physiological measurements using a webcam[END_REF] and tracking algorithms and possibly refined with skin pixel classification [START_REF] Wang | A novel algorithm for remote photoplethysmography: Spatial subspace rotation[END_REF].

Pixels in the ROI are then usually spatially averaged and the process is repeated in each video frame. The result of this process is a time series, that is later used to obtain rPPG signal. It has been shown in several studies (e.g. in [START_REF] Bousefsaf | Continuous wavelet filtering on webcam photoplethysmographic signals to remotely assess the instantaneous heart rate[END_REF]) that the quality of the ROI has a direct impact on the quality of the rPPG signal. First, because a smaller number of skin pixels leads to larger quantized RGB errors, it can be observed that the quality of rPPG signal deteriorates while down-sampling the ROI. This may be understood as the reduction of the sensor noise amplitude by a factor equals to the square root of the number of pixels used in the averaging process [START_REF] Guazzi | Non-contact Measurement of Oxygen Saturation With an RGB Camera[END_REF]. Second, the quality is also affected by the percentage of non-skin pixels in the ROI. All rPPG algorithms suffer from performance degradation when the ROI is not properly selected.

From these observations, we propose to quantitatively evaluate several ROI segmentation methods in the perspective of HR measurements with dedicated metrics. The algorithms are compared using the two datasets of our inhouse database UBFC-RPPG [START_REF] Bobbia | Unsupervised skin tissue segmentation for remote photoplethysmography[END_REF], comprising of 53 videos specifically geared towards rPPG analysis.

In section 2, seven commonly-implemented ROI segmentation methods are described. The video dataset used to compare those methods is described in section 3 while results and conclusion are presented in sections 4 and 5.

II. ROI SEGMENTATION ALGORITHMS

Although different, most ROI segmentation techniques are based on the result of classical face detection and tracking algorithms and possibly refined with skin pixel classification or more precise ROI definition based on a set of landmarks. In this section, we present seven implemented ROI segmentation methods.

Face detection (face)

The easiest way to segment the ROI is to use directly the detected face. It is important to note that all other methods presented in this paper are based on the face detection result. In this experiment, we have used the well-known Viola-Jones face detector [START_REF] Viola | Rapid object detection using a boosted cascade of simple features[END_REF]. In order to avoid spurious movements of the detected face, we also use Kanade-Lucas-Tomasi tracking [START_REF] Lucas | An iterative image registration technique with an application to stereo vision[END_REF] algorithm (cf. Fig. 1(a)).

Face cropped (crop)

The face ROI contains a significant amount of non-skin pixels from the background, the hair or the clothes (cf. Fig. 1(b)). As suggested by [START_REF] Poh | Advancements in noncontact, multiparameter physiological measurements using a webcam[END_REF], it is possible to simply crop the ROI selecting the center 60% width and height of the box as ROI (cf. Fig. 1(b)).

Rule-based skin detection (rule)

Because in the context of remote heart rate measurements, we are only interested in skin pixels, another obvious refinement of ROI segmentation method is to perform a pixelbased skin classification on the detected face. Many review papers have been published presenting various color-based skin detection rules (e.g. in [START_REF] Kakumanu | A survey of skin-color modeling and detection methods[END_REF]). It is clearly impossible to implement all possible methods so we have selected a fairly common method [START_REF] Kovac | Human skin color clustering for face detection[END_REF] in this experiment (cf. Fig. 1(c)). Histogram-thresholding skin detection (Conaire) Macwan et al. [START_REF] Macwan | Remote photoplethysmography measurement using constrained ica[END_REF] uses Conaire et al. [START_REF] Conaire | Detector adaptation by maximising agreement between independent data sources[END_REF] in their rPPG algorithm. This skin detector is based on a thresholding of a non-parametric histogram trained using manually annotated skin and non-skin pixels. This method is eventually based on a Look-Up-Table (LUT) and is thus very fast (cf. Fig. 1(d)). Adaptive-range skin detection (adaptive) Instead of using fixed thresholds for detecting skin pixels, it is possible to derive the range of skin pixel values using specific ROI of the face. For example, in this experiment, a small ROI in the center of the face bounding box is used to create a reference skin color specific to the detected face. Similar pixels in the detected face ROI are then segmented to create the ROI mask. The main advantage of this method is that it is person-specific and does not rely on global threshold values (cf. Fig. 1(e)). Graph-cut based skin segmentation (Graph-cut) All previous methods perform pixel based skin detection.

They are usually very fast but are somehow limited because it does not use any spatial information. Another strategy is to formulate the skin segmentation problem as a segmentation problem using for example Graph-cut [START_REF] Hu | Skin segmentation based on graph cuts[END_REF] (cf. Fig. 1(f)).

Landmark based skin segmentation (landmarks)

Another strategy to select the ROI is to define a polygon based on a set of landmarks. In this experiment, we used the algorithm proposed by Kazemi et al. [START_REF] Kazemi | One millisecond face alignment with an ensemble of regression trees[END_REF]. We have selected the contour of the face using the detected landmarks as illustrated in Fig. 1(g).

III. VIDEO DATASET

The ROI segmentation algorithms have been evaluated in the framework of remote heart rate measurements. To this end, we have used the two datasets of our in-house database UBFC-rPPG [START_REF] Bobbia | Unsupervised skin tissue segmentation for remote photoplethysmography[END_REF] comprising of 53 videos. In the first dataset (7 videos), the volunteers were asked to sit still while in the second dataset (46 videos) the subjects were required to play a time sensitive mathematical game that aimed at augmenting their heart rate while simultaneously emulating a normal human-computer interaction scenario. The database which is focused specifically on rPPG analysis was created using a custom C++ application for video acquisition with a Logitech C920 web camera placed at a distance of about 1 meter from the subject. The video was recorded with a frame resolution of 640x480 in 8-bit uncompressed RGB format at 30 frames per second. A CMS50E transmissive pulse oximeter was used to obtain the ground truth PPG data comprising of the PPG waveform as well as the PPG heart rates. The experimental setup with sample images is depicted in Fig. 2. Video frames synchronized with PPG sensor data can be downloaded from our project page 1 .

IV. EXPERIMENTS

A. System framework

For each video frame, the segmented ROI is spatial averaged to obtain the RGB values. The result of this process is a RGB time series. The RGB temporal traces are then pre-processed by zero-mean and unit variance normalization, detrended using smoothness priors approach and bandpass filtered with Butterworth filter. The rPPG signal is then extracted using the chrominance-based method (later called CHROM) [START_REF] De Haan | Robust pulse rate from chrominance-based rppg[END_REF]. This method applies simple linear combinations of RGB channels and obtains very interesting performance with low computational complexity. Let y c (t) be the RGB time series obtained after pre-processing, where c ∈ {R, G, B} is the color channel, CHROM method projects RGB values onto two orthogonal chrominance vectors X and Y :

X(t) = 3y R (t) -2y G (t), Y (t) = 1.5y R (t) + y G (t) -1.5y B (t).
( The pulse signal S is finally calculated with S(t) = X(t) -αY (t) where α = σ(X)/σ(Y ). Because X and Y are two orthogonal chrominance signals, PPG-induced variations will likely be different in X and Y , while motion affects both chrominance signals identically.

) (a) (b) (c) (d) (e) (f) (g) 2
For each video, we estimate heart rate in a sliding window framework. Welch's method is used to obtain the periodogram over a 20 seconds moving window, with a step size of one second. Heart rate is given by the position of the peaks on the frequency axis. The same heart rate estimation procedure was used on the PPG signal recorded with the contact sensor and on the rPPG signal given by each evaluated ROI segmentation methods.

B. The Evaluation Metrics

To evaluate the performance of the ROI segmentation method in the perspective of heart rate measurements, the following metrics are used: and HR P P G . • Signal-to-Noise Ratio (SNR) is calculated as the ratio of the power of the main pulsatile component and the power of background noise, computed in dB due to the wide dynamic range. • Precision at 5 bpm. This metric represents the percentage of estimations where the absolute error is under a threshold (2.5 or 5 bpm).

C. Results

Averaged results are presented in Table I. First, it is possible to observe that the ROI selection has indeed a large impact on the final heart rate measurement precision. One example for this idea is that the SNR obtained with face is only 2.11 dB while adaptive has a SNR of 4.15 dB. Second, it is also possible to observe that the overall ranking changes from one metric to another one. Graph-cut for example is the best method according to the MAE but performs poorly according to the SNR. From this observation, it is very difficult to give a final ranking of ROI segmentation method. However, we can observe that face obtains consistently the worst result. Then, more complex methods such as Graph-cut or landmarks obtain interesting results but are significantly computationally more expensive than pixelbased methods. crop obtains surprisingly good results in this experiment. One possible explanation is that in all videos, the volunteers remained silent and never opened their mouths. The mouth area is very important in the crop ROI (cf. Figure 1(b)). As a consequence, even if it is not the best for all metrics, the histogram thresholding method proposed by Conaire et al. [START_REF] Conaire | Detector adaptation by maximising agreement between independent data sources[END_REF] is an interesting ROI selection. It obtains consistently good performance with all metrics and interestingly, this method is very fast because it is based eventually on a simple LUT. V. CONCLUSION ROI segmentation is a critical first step in all remote PPG algorithms to obtain a reliable pulse signal. The ROI must contain as many skin pixels as possible with a low percentage of non-skin pixels. From this observation, we present in this paper a comparative study of seven different ROI selection methods. These methods were implemented and evaluated in the perspective of heart rate measurements. The experiments were done with a low cost camera and a contact PPG as the ground truth. 53 videos were recorded and the averaged results of each method were calculated. The results show that the ROI selection has indeed a large impact on the final heart rate measurement precision and that the histogram thresholding method proposed by Conaire et al. [START_REF] Conaire | Detector adaptation by maximising agreement between independent data sources[END_REF] obtains in average very good performance with all metrics. Moreover, this method is interestingly very fast because it is based eventually on a simple LUT.

Even if color-based ROI selection method are fast and reliable, these methods do not consider the distinct pulsatility feature of informative ROI, i.e. only skin tissue generates pulsatility. In future work, this particular property may help in improving ROI segmentation methods. Second, it has been shown that the rPPG signal is not distributed homogeneously across the skin, as a consequence the regular spatial averaging of the segmented ROI may not be optimal and offer good opportunities to improve current ROI segmentation techniques.

  (R,G,B) is classified as skin if R > 95 and G > 40 and B > 20 and max{R, G, B} -min{R, G, B} > 15 and |R -G| > 15 and R > G and R > B (1)

Figure 1 .

 1 Figure 1. ROI segmentation result examples for (a) face (b) crop (c) rule (d) Conaire (e) adaptive (f) Graph-cut (g) Landmarks.

Figure 2 .

 2 Figure 2. Experimental Setup (top) and sample images from the UBFC-RPPG database.

•

  Mean Absolute Error (MAE) in beats per minute (bpm) is calculated between heart rate estimated from rPPG signals HR rP P G and heart rate estimated from PPG signals HR P P G with |HR rP P G -HR P P G |, averaged per video. • Root mean square error (RMSE) between HR rP P G

Table I THE

 I AVERAGE EVALUATION VALUES FOR ROI DETECTION

		Precision5 MAE	RMSE	SNR
	face	0.883	4.43	7.19	2.11
	crop	0.938	2.35	3.86	2.98
	rule	0.909	3.91	5.34	3.11
	Conaire	0.931	3.02	4.99	3.02
	adaptive	0.913	3.63	5.50	4.15
	Graph-cut	0.941	2.34	3.82	2.95
	landmarks	0.922	3.21	4.95	2.87

https://sites.google.com/view/ybenezeth/ubfcrppg