E Barthel

A Perriot

G Sperling

Adhesive Contact to a Coated Elastic Substrate

We show how the quasi-analytic method developed to solve linear elastic contacts to coated substrates (Perriot A. and Barthel E. J. Mat. Res., 2004, 19, 600) may be extended to adhesive contacts. Substrate inhomogeneity lifts accidental degeneracies and highlights the general structure of the adhesive contact theory. We explicit the variation of the contact variables due to substrate inhomogeneity. The relation to other approaches based on Finite Element analysis is discussed.

Introduction

In elastic contact problems, it is known that the homogeneous substrate is a special case which leads to counterintuitive results. As an example, for adhesionless contacts, the relation between penetration δ and contact radius a is independent from the mechanical parameters (Young's modulus E, Poisson ratio ν, reduced modulus E ⋆ = E/(1 -ν 2 )) and can be in all generality expressed as a function of the shape of the contacting bodies z(r) only.

For instance, for a rigid cone of half included angle β in frictionless contact with an elastic substrate:

δ = 2 π a tan β (1)
Obviously, the reduced modulus is absent and one might conclude that the relation has a geometrical origin. To show that this is not the case, one needs only remove the assumption of homogeneous substrate. Then, for a coated substrate, for instance, the penetration will again depend upon the mechanical parameters of the system, in a highly non trivial manner [1]. Overlooking the more complex penetration dependence upon contact radius might actually result in inaccuracies in the determination of thin film mechanical properties by nanoindentation [2,3,4,5].

Similarly, for adhesive contacts, a counterintuitive result is that the adhesion force or pull-out force F pullout for a sphere of radius R and adhesion energy w is independent from the mechanical parameters. It is valued

F pullout = nπwR (2)
where various n values have been proposed such as 1 [6], 3/2 [7] or 2 [8]. ‡ Corresponding author: etienne.barthel@saint-gobain.com

However, changing the geometry from sphere to cone [START_REF] Maugis | Contact, Adhesion and Rupture of Elastic Solids[END_REF] or removing the assumption of a homogeneous substrate, for example with a coated substrate [START_REF] Johnson | [END_REF]11], again results in reintroducing the dependence upon mechanical parameters.

With the idea that the real structure of the elastic contact models only appears when considering a non homogeneous substrate, the present contribution aims at: 1) showing how the recently developed quasi analytical method for contacts to coated substrates [1] may be used to account for adhesion as previously used by Mary et al. [12] 2) exploring how the different concepts developed for homogeneous adhesive contacts may be applied to a non homogeneous system, to wit, a coated substrate.

Section 2 reviews the various concepts which have been developed to establish the relation between macroscopic contact variables F , δ and a and the physical phenomena of adhesion. Section 3 describes the surface stress transform method pioneered by Sneddon. Section 4 explores the application of the method to the adhesive contact to a homogeneous substrate, while section 5 explores the application of the method to coated substrates.

Adhesive elastic contact concepts and previous results on coated substrates

Elastic adhesive contact -problem description

Contact problems are characterized by mixed boundary conditions: the surface displacement is specified inside the contact zone and the normal stress at the surface outside. Note that this normal stress outside the contact zone is zero, except at the edge of the contact zone where more or less long range adhesive surface forces develop in the so-called interaction zone (Fig. 1).

Three concepts have been developed to handle these complex boundary conditions, provide a solution and establish relations between the macroscopic variables F , δ and a.

Three concepts to solve the adhesive elastic contact problem

Energy minimization

The first method disregards the contact edge details, assumes a given form for the surface stresses inside the contact zone and calculates the solution parameters by energy minimization, taking into account the adhesion energy w as a πwa 2 term. This macroscopic approach was implemented as early as 1934, when Derjaguin assumed a Hertz-like stress distribution and combined it with an adhesion energy contribution proportional to the contact area, which resulted [6] in a pull-out force given by n = 1 in Eq. 2.

Based on elastomer adhesion experiments, a more realistic surface deformation built up from the addition of a flat punch displacement to the Hertz solution resulted in the JKR model [7] § where n = 3/2 in Eq. 2. This model, now generally accepted for soft solids, provides a more consistent description of the stresses inside the contact zone than the original 1934 result.

More generally, a formal method to minimize the total energy with regards to the contact radius a naturally leads to introducing the energy release rate, a concept borrowed from fracture mechanics : if E is the stored elastic energy, the energy release rate G can be defined in an axisymmetric geometry as

2πaG(a) = ∂E ∂a δ (3)
Such an approach was used for instance as an alternative derivation of the JKR model in [START_REF] Johnson | Contact Mechanics[END_REF].

Stress intensity factor

The second method is to focus on the crack-like stress singularity at the contact edge. It is described in terms of the stress intensity factor.

The relation between the energy release rate and the stress intensity factor was initially proposed by Irwin [START_REF] Irwin | [END_REF]. In the field of the adhesive contact, this second method was mainly propounded by Maugis [START_REF] Maugis | Contact, Adhesion and Rupture of Elastic Solids[END_REF]16,17]. It is particularly useful for direct extensions of the Linear Elastic Fracture Mechanics where a description in terms of stresses at the crack tip is necessary: for instance, it may be used to include crack tip viscoelasticity [16,17,18].

Exact solution

The third method is actually more general: it takes into account the details of the attractive stress distribution outside the contact zone. This idea was somehow initiated by the DMT model [8] (n = 2 in Eq. 2) but was brought to a new dimension by the introduction of a cohesive zone model by Maugis [19], which resulted in a clear exposition of the transition between the two limiting models JKR and DMT. It also led to the development of the fully viscoelastic adhesive contact models [20,21]. In these models, stress relaxation inside the contact zone -as in the adhesionless viscoelastic contact by Ting [22] -and creep inside the interaction zoneas in the viscoelastic crack models [16,17,18] -are taken into account simultaneously. Just as the viscoelastic case extends the adhesive contact model from instantaneous response to time dependent constitutive relation, the coated substrate generalizes the model to an inhomogeneous half space. Our interest here will however be limited to the small cohesive zone sizes, located on the JKR side of the transition, which connect this approach to the previous two. For an extension to the JKR-DMT transition with a coated substrate, when the interaction zone grows to extensions comparable or larger than the film thickness, in the spirit of the Maugis model, see [23].

where it stands as a curiosity, not as the "Theorie der Haftung" developped in the body of the text, which mainly deals with surface roughness and plastic deformation.

Review of previous results on the adhesive contact to coated substrates

On the theoretical side, the literature on the adhesive contact to coated substrates is scanty, although, in practice, thin soft adhesive layers are often used, as in many applications of pressure sensitive adhesives and functional or protective organic coatings. Actually only two series of papers by Shull and coworkers [24,25] on one side and Johnson and Sridhar [START_REF] Johnson | [END_REF]11] on the other side have been published.

Shull and coworkers calculate the energy release rate from the contact compliance (cf. 5.2.2). To complete the calculations, the compliance of coated sustrates are computed by Finite Element (FE) methods and phenomenological fits or numerical differentiation may be carried out.

Johnson and Sridhar's approach goes along the line of the stress intensity factor (section 2.2.2) approach: still using FE, they directly calculate the energy release rate G by a stiffness matrix variation method [26] and extract stress intensity factors from G. From this stress intensity factor, they build the adhesive contact relations.

Both methods converge on their ultimate use of Finite Element calculations to handle the complexities of the coated substrate response. Appearances to the contrary, these two series of papers seem actually to rely on the same method and an in depth investigation of the Parks method in the context of the frictionless contact would probably connect the stiffness matrix derivative used by Parks to the contact stiffness derivative method as developped by Shull.

In the present paper, we show that the quasi-analytical method introduced previously [1] may be used to efficiently compute all the necessary contact variables and in particular provide direct numerical evaluations for the stress intensity factor for coated substrates. It is also useful to expose the equivalence of the various concepts for the adhesive contact in the wider context of non-homogeneous substrates.

Sneddon and the surface stress transform

Adequate transforms facilitate the investigation of linear elastic axisymmetric contacts formulated as integral equations [START_REF] Gladwell | Contact Problems in the Classical Theory of Elasticity[END_REF]. It is ideally suited to the adhesive contacts because of the relevant boundary conditions including the interaction zone.

Following our previous studies [20,21,[START_REF] Barthel | [END_REF]29], we use g(r), which is a transform of the normal stress distribution at the surface σ z defined as

g(r) = +∞ r sσ z (s) √ s 2 -r 2 ds (4)
and θ(r) which is a transform of the normal surface displacement u z defined by

θ(r) = ∂ ∂r r 0 su z (s) √ r 2 -s 2 ds (5)
These relations are suited to the adhesive contact problem because g(r) is expressed as a function of normal surface stresses at radii values larger than r and θ(r) as a function of surface normal displacement at radii values smaller than r, in agreement with the adhesive contact boundary conditions (Fig. 1). In particular g is zero outside the interaction zone.

Similarly, inside the contact zone (r ≤ a), u z (r) = δ -z(r) where z(r) is the shape of the indenter. Integration by parts transforms Eq. 5 into

θ(r) = δ -δ 0 (r). ( 6
)
where δ is the penetration and δ 0 depends only upon the shape of the indenter z(r) through

δ 0 (r) = r r 0 z ′ (s)ds √ r 2 -s 2 (7)
The full power of these transforms appears when it is recognized [20,21] that in the linear elastic case and for a homogeneous substrate, mechanical equilibrium results in

g(r) = E ⋆ 2 θ(r) (8)
This provides the direct solution to the adhesive contact problem which we detail now.

Adhesive contacts on homogeneous substrates

Before tackling the problem of the adhesion to coated substrates we review our present understanding of elastic contacts to homogeneous substrates using the transform method.

Non adhesive contact of smooth indenters

In the absence of adhesion, Eq. 4 shows that g(a) = 0. If the indenter shape z(r) is smooth at the contact edge, then Eqs. 6 and 8 imply θ(a) = 0 and

δ = δ 0 (a) (9)
Thus the indenter penetration in an adhesionless contact is the function δ 0 (a), which depends only upon the geometry of the indenter (Eq. 7) and not on the elastic parameters. This is the counterintuitive result stated in section 1.

Non adhesive contact of flat punches

A special case however is the flat punch for which δ 0 (r) = 0 for r < a and g(r) = 0 for r > a. The penetration is independent of the contact radius as expected since the contact radius is constant, equal to the punch radius.

Then θ(a -) = δ so that g(a + ) = 0 and g(a -) = 0. The exact meaning of this discontinuity in the g transform at a will be discussed below (section 4.4.2).

Note also that the force can be obtained through the simple but general expression [29]

F = 4 +∞ 0 g(s)ds (10)
Specifically here,

F 0 (a) = 4 a 0 g(s)ds (11)
For the flat punch the force is then directly calculated from Eqs. 6, 8 and 10 as

F 0 (s) = S(a)δ f p (12)
with

S(a) = 2aE ⋆ . ( 13
)

Adhesive contacts

In the adhesive case, however, although the indenter shape is smooth at the contact edges, the condition g(a) = 0 is relaxed due to the adhesive interactions. The core of the JKR approximation is to neglect the details of the stresses and deformations at the contact edge and encapsulate the adhesive contribution in an additional flat punch displacement. This amounts to a description where the attractive interaction stresses drop abruptly to zero outside the contact zone, leading to a stress singularity akin to those met in fracture mechanics. Then Eqs. 9 and 11 become

δ = δ 0 + δ f p (14)
F = F 0 + S(a)δ f p (15)
The question is to provide a relation between this additional flat punch displacement δ f p and the adhesion energy w for a given contact radius a.

Various concepts for the derivation of adhesive contact results

For future reference, we now apply the surface stress transform solution to the three methods developed to handle adhesive contacts (section 2.2).

Energy Release Rate

A very compact derivation is obtained when it is recognized that for a homogeneous substrate g et θ are independent of a inside the contact zone. In addition the total mechanical energy E as a function of the normal surface stress distribution is [29]

E = 4 E ⋆ +∞ 0 ds g 2 (s) = E ⋆ +∞ 0 ds θ 2 (s) (16)
Then for constant penetration, with g(a) = 0 outside the contact zone,

2πaG(a) = 4g(a) 2 E ⋆ = E ⋆ θ(a) 2 (17)
Equating G and the adhesion energy w results in

w = 2g(a) 2 πE ⋆ a (18)
The energy release rate G(a) can be expressed as a function of the local stress distribution at the crack tip because of Eq. 4 and g(r) = 0 for r > a.

Stress intensity factor

This expression together with the plain strain Irwin relation [START_REF] Maugis | Contact, Adhesion and Rupture of Elastic Solids[END_REF][START_REF] Irwin | [END_REF] between stress intensity factor and energy release rate

G = K I 2 /(2E ⋆ ) (19)
suggests that 2g(a)/ √ πa assumes the role of a stress intensity factor. A direct derivation is obtained [START_REF] Maugis | Contact, Adhesion and Rupture of Elastic Solids[END_REF] if we use the following expression for the stress intensity factor at the contact edge

K I = lim ǫ→0 (2πǫ) 1/2 σ(a -ǫ) (20)
Eq. 4 can be inverted [29] providing

σ z (s) = 2 π g(a)Θ(a -s) (a 2 -s 2 ) 1/2 + +∞ s dt g ′ (t) (t 2 -s 2 ) 1/2 (21)
where it has been assumed that g is smooth except for a discontinuity at a. We use the notation g(a -) = g(a) and assume g(a + ) = 0. Then

K I = 2g(a) √ πa (22)
which confirms that g(a) has the form and meaning of a stress intensity factor. As a stress intensity factor, g(a) is a measure of stress singularity at the contact edge. An ancillary property of the transform defined by Eq. 4 is that it regularizes the singular crack like stress distributions and Eq. 5 establishes a relation between the additional flat punch penetration and the stress singularity through

θ(a) = δ f p = 2g(a) E ⋆ (23)
4.4.3. Self-consistent method The self-consistent description of the interaction zone [19,[START_REF] Barthel | [END_REF]30] can be explored at the limit of negligible interaction zone extension, i.e. in the JKR limit.

One possible starting point of the self-consistent method [START_REF] Barthel | [END_REF]29] is to calculate the adhesion energy by

w = - +∞ a σ(s) dh ds ds (24)
Following [21](Eq 15) the gap between the surfaces h is split into the contributions of the contact stresses and the interaction stresses

h(r) = h Hertz (r, a) + h int (r, a) (25)
If the interaction range is small (which results in c -a << a), the radial extension of the interaction will also be small, g ′ is peaked around a, and the dominant term in Eq. 24 will be the ∂u z,int /∂s term. Thus [29],

w ≃ - c a σ s dh int ds ds ≃ - 4 πaE ⋆ +∞ a g ′ (t)g(t)dt (26)
As a result, one again recovers Eq. 18. Combined with Eq. 22, this method may be viewed as a direct derivation of the Irwin relation [START_REF] Irwin | [END_REF].

The 1/2 factor comes from the rigidity of the punch.

Normalized form

The adhesive contact equations are obtained by combining Eqs. 14-15 and Eq. 18. For later reference, we specialize the results to the spherical indenter and use the Maugis normalization [19]

P = F πwR (27)
D = δ π 2 w 2 R E ⋆2 1/3 (28)
ā = a

πwR 2 E ⋆ 1/3 (29)
One obtains

P = 4 3 ā3 -2 √ 2ā 3/2 (30)
D = ā2 - √ 2ā 1/2 (31)
These are the usual JKR equations which will be generalized in the next section. Their structure directly reflects the general structure of Eqs. 14-15 with the identifications δfp = -√ 2ā 1/2 and Sfp = 2ā.

Coated Substrates -Thin Films

Description of the contact

If the substrate is not homogeneous the simple diagonal equilibrium relation Eq. 8 between g and θ is lost. However, keeping the same transforms, we have shown [1] that a useful relation subsists (32) where K expresses the elastic response of the coated substrate [1], t stands for the coating thickness and [E] for the four mechanical parameters (film and substrate Young's moduli and Poisson ratios). Since g is zero outside the contact zone, the upper boundary will typically be the contact radius a or the interaction zone radius c if c = a. The finite integration interval facilitates the numerical inversion [1] of Eq. 32. Note that direct analytical inversion is impossible because of the complex expression for K.

θ(r; a, t, [E]) = 2g(r; a, t, [E]) E 1 ⋆ + 2 π +∞ 0 g(s; a, t, [E])K(r, s; t, [E])ds
We now split the stress distribution and the displacement field into their non adhesive δ H and flat punch δ f p components.

For the non adhesive contact of an indenter of regular shape, the penetration is given by

δ H (a, t, [E]) -δ 0 (r) = 2g H (r; a, t, [E]) E 1 ⋆ (33) + 2 π a 0 g H (s; a, t, [E])K(r, s; t, [E])ds (34)
where δ 0 (r) reflects the indenter shape and g H (a; a, t, [E]) = 0 in the absence of adhesion, ensuring a unique solution. Eq. 34 generalizes Eq. 15 and introduces the non-trivial relation mentioned in section 1.

For the contact of the flat punch the problem is linear with δ f p and

δ f p = 2g f p (r; a, t, [E]) E 1 ⋆ + 2 π a 0 g f p (s; a, t, [E])K(r, s; t, [E])ds (35)
with g f p (r; a, t, [E]) ∝ δ f p and g(a; a, t, [E]) = 0. Eq. 35 generalizes Eq. 23 into a non trivial proportionality relation between g(a) and the flat punch penetration δ f p .

Then the penetration for the adhesive contact of an indenter of arbitrary shape is

δ(a, t, [E]) = δ H (a, t, [E]) + δ f p (36)
With the following definition of the contact stiffness

S(a, t, [E]) = F f p (a, t, [E]) δ f p ( 37
)
the force is

F (a, t, [E]) = F H (a, t, [E]) + S(a, t, [E])δ f p (38)
Equations 36 and 38 generalize Eqs. 15 and 14. Here again, the question is to provide a relation between the additional flat punch displacement δ f p and the adhesive interaction.

Derivation

For coated substrates, the resolution benefits from more general expressions derived for viscoelastic adhesive contacts [31]. This is not fortuitous but results from a similar breakdown of the simplified relations derived through Eq. 8 when spatial heterogeneity or time dependence is introduced.

Energy Release Rate

because G(a) is calculated at constant displacement so that inside the contact zone, the surface displacement -and therefore θ(r) -are unaffected by the additional stress distribution g(a): there is only a stress rearrangement inside the contact zone which does affect g(r), r < a for a coated substrate: indeed the local relation Eq. 8 between g and θ breaks down in this case. Following Mary et al. [12], one can show that this non local contribution is actually canceled by the non local contribution included in the g(a)θ(a) term. For s < a

- π E 1 ⋆ dg(s) da = g(a)K(a, s) + a 0 dr dg(r) da K(r, s) (41)
and multiplying by g(s) and integrating between 0 and a, one obtains

g(a)θ(a) + a 0 dr dg(r) da θ(r) = 2g(a) 2 E 1 ⋆ (42)
The region affected by the crack tip stresses extend over a distance commensurate with the interaction zone size, i.e. it is smaller than the film thickness so that the energy release rate is controlled by the film compliance.

Compliance Method

The compliance formulation is at the core of the method used by Shull and coworkers but the derivation given in some of their earlier papers was obscured by unnecessary assumptions. This formulation emerges readily when E in Eq. 39 is expressed as a function of strain instead of stress ¶.

We split the total stress and strain fields in their non adhesive (g H and θ H ) and flat punch components (g f p and θ f p ). This allows for easy integration because θ f p (r; a, t, [E]) = δ f p is independent of r. Using Betti's theorem to calculate the cross terms

2 +∞ 0 dsg H (s; a, t, [E])θ f p (s; a, t, [E]) (43) = 2 +∞ 0 dsg f p (s; a, t, [E])θ H (s; a, t, [E]) (44) = 1 2 δ f p P H (a, t, [E]) (45)
and similarly using Eq. 37

2 +∞ 0 dsg f p (s; a, t, [E])θ f p (s; a, t, [E]) = 1 2 S(a, t, [E])δ 2 f p (46)
to express the flat punch elastic energy, one obtains the total energy

E(a, δ f p ) = U H (a, t, [E]) + 1 2 S(a, t, [E])δ 2 f p + P H (a, t, [E])δ f p (47)
where δ f p is negative.

A graphic illustration of this result is given in Fig. 2. The energy release rate G is the differential of the total energy E(a, δ f p ) with respect to contact area (Eq. 3) at constant total penetration δ where

δ = δ H (a) + δ f p ( 48
)
dE da δ = ∂E ∂a + ∂E ∂δ f p dδ f p da (49)
Now

dU H = P H (a, t, [E])dδ H ( 50
)
dP H = S(a, t, [E]
)dδ H (51) ¶ Such a strain energy release rate has also been introduced previously in viscoelastic crack problems by Schapery [32] and Greenwood and Johnson [18]. so that finally all terms cancel + except the differential of the flat punch elastic energy and

2πaG(a) = 1 2 δ 2 f p dS da (52)
This differentiation process is also illustrated on Fig. 2.

Stress Intensity Factor

The derivation of the stress intensity factor expression in section 4.4.2 is indendent of the material properties and only results from the definition of the transform g. It is therefore unchanged in the more general case and the pending problem is actually the relation between g(a) and the energy release rate G.

Self-consistency

In Eq. 25,

h int (r, a, t, [E]) = 2 π r a θ(s) -θ(a) √ r 2 -s 2 ds (53)
If at the crack tip the deformation predominantly results from the adhesive interactions -and this is the essence of the JKR limit -then Eq. 24 becomes [20]

w = - 2 πa c a θ ′ (r)g(r)dr (54)
From Eq. 32 we obtain

θ ′ (r) = 2g ′ (r) E 1 ⋆ + 2 π +∞ 0 g(s) dK(r, s) dr (55)
The second term is well behaved when c → a since K is the elastic response of the coated substrate: the contact edge singularity results from the boundary conditions, not the response function. Thus it is g ′ which is peaked around a [START_REF] Barthel | [END_REF] so that in the end for c < r < a

θ ′ (r) ≃ 2g ′ (r) E 1 ⋆ ( 56
)
and again

w = 2g(a) 2 πE ⋆ a (57)
even for a non-homogeneous substrate. The local response is indeed dominated by the local stress distribution and the film compliance: the same relation Eq. 17 between adhesion energy and stress intensity factor applies.

+ Due to the counterintuitive results obtained for the homogeneous substrate, one may develop doubts about the identity of the stiffness defined by Eq. 51 and the stiffness defined by Eq. 37. That this identity does hold for a non homogeneous substrate is shown in the appendix (section 8. It is due to the absence of adhesion: the normal stress at the edge of the contact is zero so that contact radius variation does not result in force variation, to first order.

Relation between g(a) and S

Comparing Eqs. 52 and 57, one infers

4g(a) 2 E 1 ⋆ = 1 2 δ 2 f p dS da (58)
This is a remarkable result which one should in principle be able to derive from Eq. 35.

Note however that the present quasi-analytical method [1] simultaneously provides the flat punch stiffness and and contact edge stress singularity g(a) (or the stiffness derivative through Eq. 35) by the simple resolution of a linear system.

Normalized Solution

The procedure to compute a full force curve for the adhesive contact to coated substrates is therefore: for a given value of the contact radius a (i) compute the adhesionless penetration and force for the given indenter shape (ii) compute the force and stress intensity factor (or g(a)) for the flat punch for a unit value of the penetration (iii) compute actual value of g(a) from Eq. 57 and rescale flat punch force and penetration (iv) compute the solution from Eqs. 36 and 38

This provides the solution under the form of two relations between the local deformation (stress intensity factor) and the local response on the one hand and the remote loading and the macroscopic response (contact stiffness) on the other hand.

In normalised form, with

r = r t (59)
one has *

F 0,s = a 3 E 1 ⋆ 2R Π s (ã, t, [E]) (60)
δ 0,s = a 2 R ∆ s (ã, t, [E]) (61)
S = 2aE 1 ⋆ E eq (ã, t, [E]) (62) g(r; a, t, [E]) = δ f p E 1 ⋆ 2 Γ(r; ã, t, [E]) (63)
where the normalized variables can be numerically calculated by the simple algorithm presented in [1]. * For a cone:

F 0,c = πa 2 E 1 ⋆ 4 tan β Π c (ã, t, [E]) δ 0,c = πa 2 tan β ∆ c (ã, t, [E])
All variables equal 1 for the homogeneous substrate except Π s = 8/3 for the sphere♯. Γ(r; ã, t, [E]) is the surface stress transform normalized to penetration δ. In particular, Γ(ã; ã, t, [E]), denoted Γ(1) below for brevity, is the contact edge singularity g(a; a, t, [E]) incurred for a coated system -normalized to a homogeneous material with the film elastic properties -at identical δ f p value. The variable Γ(1), which is positive since both δ f p and g(a) are negative in Eq. 63, is a function of ã, and depends upon the mechanical parameters of the system. From Eq. 52 the following identity holds:

Γ(1) = 1 2E 1 ⋆ dS da = d(aE eq ) da (64)
.

Then for the sphere, keeping the Maugis normalization, one introduces the film thickness normalized to the typical adhesive contact radius

η = t πwR 2 E 1 ⋆ 1/3 (65)
and with ā = ηã one obtains

Π s = (ηã) 3 Π s,0 2 -2 √ 2 (ηã) 3/2 E eq Γ(1) (66)
D s = (ηã) 2 ∆ s,0 - √ 2 (ηã) 1/2 1 Γ(1) (67)
These equations generalize Eqs. 30-31. The homogeneous substrate force and penetration terms are corrected by the coated system factors Π s,0 and ∆ s,0 , the homogeneous contact stiffness by E eq and, for identical stress intensity factor, the homogeneous penetration is corrected by 1/Γ(1).

Examples of Numerical Results

Reduced variables

For given mechanical parameters E 1

⋆ /E 0 ⋆ , ν 0 and ν 1 one may calculate the four variables Π s,0 ,∆ s,0 ,E eq and Γ(1) as a function of ã = a/t. Typical results are illustrated in Fig. 3 for E 1

⋆ /E 0 ⋆ = 10 and Fig. 4 for E 1 ⋆ /E 0 ⋆ = 0.1. The results for these reduced variables compare well with the FE calculations by Sridhar and Johnson [11].

The thin film contact problem exhibits a transition between film dominated to substrate dominated behaviour. In each limit cases, the contact behaves like a contact to a homogeneous system. The non trivial behaviour is apparent in the transition which, roughly speaking, occurs for ã ≃ 1, but is shifted to higher values for compliant films (Fig. 3) and to lower values for stiff films (Fig. 4) [1].

E eq exhibits a transition between film to substrate reduced modulus as the contact radius increases is consistent with numerous previous works [33]. The behaviour of ∆ s,0 , which deviates from 1 in the midst of the transition [1,5] has been much less studied, as ♯ For a cone Π c = 2 mentioned previously (section 1). Similarly for the flat punch we note that Γ(1) tracks E eq at small contact radius values, up to about the middle of the transition. This does result from Eq. 64 as a linear expansion of E eq shows (note that E eq (0) = 1). For larger contact radius values, higher order terms come into play and Γ(1) starts to deviate from E eq .

Adhesive contact solutions

The reduced variables Π s (ã, t, [E]), ∆ s (ã, t, [E]), E eq (ã, t, [E]) and Γ(1) = Γ(ã; ã, t, [E]) provide the normalized solution (Eqs. 66 and 67) as a function of η for a set of mechanical parameters. It is worth emphasizing that -provided the reduced variables have been calculated on a wide enough range of values of ã = a/t -the solution for any values of contact radius, film thickness and adhesion energy can be calculated from the reduced variables through the elementary arithmetics of the normalization Eqs. 66 and 67. This should provide for an easy algorithm to fit experimental data for adhesive contact on thin films.

Results for various values of η are displayed on Fig. 5 and6 for E 1 ⋆ /E 0 ⋆ = 10 and Fig. 7 for

E 1 ⋆ /E 0 ⋆ = 0.1 with ν 0 = ν 1 = 0.25.
For a coated substrate in the presence of adhesion, the system undergoes two transitions. The first one is is the transition between coating and substrate response [1], when ã ≃ 1 as described in the previous section. The second one is the transition from adhesion dominated to non adhesive contact as the load and therefore the contact radius increase [34]. This transition occurs when ā ≃ 1. The deviation from the JKR results occur when the two transitions are simultaneous, i.e. when η ≃ 1.

Explicitly, for large values of η (i.e. thick films), the adhesive stage of the contact will occur for small ã and the contact is film dominated and behaves as a homogeneous material with the mechanical parameters of the film.

For small values of η (i.e. thick films), the adhesive stage occurs for large ã and the contact is substrate dominated: the contact behaves as a homogeneous material with the mechanical parameters of the substrate. However, at the edge of the contact, this contact is not equivalent to an adhesive contact to a homogeneous material with substrate properties. Indeed, the stress intensity factor is dominated by the film properties (Eq. 57).

As a result, moderate deviations on the pull-out force are evidenced. For soft layers, the adhesion force is enhanced [11]. For rigid layers there is a small reduction of the adhesion force † †.

Conclusion

The surface stress transform is adequate to handle the adhesive contact to coated substrates. It provides a numerically simple method to compute the four quantities necessary for the actual description of such adhesive contacts. For a given layer and substrate mechanical properties, fits to data with free adhesion energies and coating thicknesses may be performed easily. It also allows a consistent description of the crack tip which connects the various concepts developed in adhesive contact problems.

The method also allows the inclusion of finite indenter stiffness, a question which arises in practice and which will be studied in more details in a subsequent paper.

Appendix: contact stiffness and flat punch stiffness

The identity of the contact stiffnesses defined by Eq. 37 and by Eq. 51 may look questionable: is the contact stiffness for an adhesionless curved indenter identical with the flat punch stiffness for an identical contact radius, even for an inhomogeneous substrate ?

Since the curved indenter is adhesionless, g(a) = 0. Then from Eq. 10 we have

dP dδ = 4 a 0 dg(r) dδ dr (68)
From the differentiation of Eq. 32 with δ together with Eq. 5,

1 = 2 π a 0 dg(r) dδ K(r, s; t, [E])dr (69)
This is the flat punch equilibrium equation for unit penetration. Then the force

F f p = 4 a 0 dg(r) dδ dr (70)
is the flat punch force for unit penetration, i.e. the flat punch stiffness S(a). Therefore for an adhesionless curved indenter

dP dδ = S(a) (71)
Note however that in general this stiffness is not the hertzian 3P R/4a 3 . Captions Fig. 1: schematics of an adhesive contact: the contact zone (radius a) is surrounded by the interaction zone. The gap between surfaces outside of the contact zone is h(r). The normal stress distribution turns from compressive at the center of the contact to tensile on both sides of the contact edge. The transform θ(r 1 ) is calculated from the normal surface displacement at r < r 1 . The transform g(r 2 ) is calculated from the normal surface stress for r > r 2 .

Fig. 2: Adhesive contact configuration obtained from the adhesionless contact by a flat punch displacement δ f p along the tangent to the adhesionless force F -penetration δ curve. Integration gives the stored elastic energy. The energy release rate is obtained as the variation with contact radius a of the stored elastic energy at constant total penetration δ (shaded area).

Fig. 3: Normalized force Π s (left) and penetration ∆ s (right) for an adhesionless sphere as a function of contact radius normalized to film thickness ã = a/t. The film to substrate modulus ratio is 0.1 and Poisson ratio 0.25 for both materials. The contact stiffness E eq (left) and contact edge stress intensity factor Γ(1) (left) are calculated from the flat punch boundary conditions. For ã ≃ 1 the transition from film dominated to substrate dominated contact occurs.

Fig. 4: Same plot as Fig. 3 for a film to substrate modulus ratio equal to 10. The transition occurs earlier due to film rigidity. Fig. 5: Normalized force versus normalized contact radius as a function of η, the ratio of the film thickness to the typical zero load adhesive contact radius. When η ≃ 1, adhesive effects are significant in the transition from film to substrate dominated regimes: deviations from the JKR model are observed and increased adhesion forces are observed.

Fig. 6: Normalized force versus normalized penetration as a function of η. Identical parameters as in Fig. 5. Fig. 7: Similar plot as in Fig. 5 with a film to substrate modulus ratio of 10. In the intermediate regime η ≃ 1, adhesion forces are reduced.

  The total energy is[31]

  Figure 1.

  Figure 3.

  Figure 4.

  Figure 5.

  Figure 6.

Figure 7 .

 7 Figure 7.

† † This is not, however, the reason why on soft materials a rigid layer may drastically reduce the adhesion, which is often due to the suppression of non elastic additional dissipation phenomena. Note that here the adhesive force reduction is accompanied by an increase of the stress intensity factor.

Acknowledgments

The authors thank A. Chateauminois and C. Frétigny for several enlightening discussions on this and other related topics.


apparent rhythm

5 syl 5 syl real rhythm 4 6 (or 5 in discontinuous rhythm)

1 Rough draft of a lecture at Charles University in Prague (nov. 14). Thank you to the CEFRES, to Jakub Rihà, Josef Hrdlicka, and other colleagues and students at Charles University for their welcome and suggestions, and particularly to Anthony McKenna for many stylistic improvements of

Benoît de Cornulier, Laboratoire de Linguistique de Nantes / Centre d'Études Métriques novembre-décembre 2014 (mise en ligne décembre 2015)

On the use of the pantoum in Baudelaire's Harmonie du soir

From metrical analysis to interpretation 1

What is the use of metrical studies? Among other things : 1) Metrics being concerned with systematic rhythmic regularities, looking for such regularities may help to reveal relevant aspects of speech which otherwise would not be obvious. One example concerning prosody will be briefly evoked in §1.

2) In France at least, metrical studies are not always well-received among literary scholars and teachers, one objection being that they are useless as regards the interpretation of texts. I will try to show by an example in Part 2 that, on the contrary, careful examination of the metrical structure of a poem may help to understand its meaning or symbolism.

I. Adapting notions

First example, extracted from a French traditional song, considered as verse (when read without music) ; while we would be « intellectually » tempted to analyse it as indicated on the left side of the display below, the observed regularities are rather those shown on the right In such tralalas as shown below (from traditional song4 ), the observed regularity is such that the frontier between the final contrasting part and the identical beginning part occurs precisely at the beginning of the catatonic form (thus non-concomitant with a syllabic frontier) :

Total phonemic form Anaton. phon. form Catat. form Flon, flon, Larira, dondaine, Flon, flon, Larira, dond--aine Fon, flon, Larira, dondon Flon, flon, Larira, dond--on

II. On the metrics and meaning of Harmonie du soir

Study of metrical form often provides a useful line of attack for the interpretation of a poem.

I try to illustrate this method in a book De la métrique à l'interprétation (2009) about poems by Rimbaud (about 1870-1872). For instance, the interpretation of his Chanson de la plus haute Tour can be improved by recognizing that, underlying this highly literary poem, is a traditional, rural song for children, about agricultural work and food. -More recently, this method helped me to (try to) show that two of his most famous (quasi)-metrical poems, Mémoire and « Qu'est-ce pour nous, mon Coeur », apparently semantically unrelated (the former being melancholic, the latter, as revolutionary), are semantically related, and seem to form a sort of diptych5 .

First approach of Baudelaire's pantoum

I present below a very famous poem by Baudelaire, Harmonie du soir6 . All its lines, being alexandrine, normally begin with the same left margin; but a particular lay-out is used herewith two different left margins and a few words in bold type -to help understand the analysis I will propose… and the poem itself.

HARMONIE DU SOIR

Module-distichs A Module-distichs B

Let us begin with a fairly traditional metrical description of the « pantoum ».

Rhyme : Stanza rhyme schema abab, or sometimes abba traditionally often analysed as including a distich bb surrounded by a pair of a…a rhymes, whence the name rimes embrassées (dd lines embraced by both a…al ines). Repetition : Lines 2 and 4 of each stanza become the lines 1 and 3 of the following stanza (line-repetition schema xAxB AxBx). This very particular repetition schema is the most obvious, if not main, characteristic of the pantoum7 . Semantic Alternance. -Two distinct semantic sequences are supposed to alternate in a regular pantoum : a sequence A constituted by « the first two lines » of each stanza (here A1 to A4), and a sequence B constituted by its « last two lines » (here B1 to B4)8 . Global Line-Repetition Closure. -In a regular pantoum, the last line repeats (comes back to) the first line.

In such a description, the semantic alternance and the repetition schema are unrelated with the supposed « embraced » rhyme schema.

It is generally admitted as obvious that the Semantic Alternance principle is not observed in Baudelaire's pantoum, and obvious that he did not observe the Line-repetition closure.

I will try to show that9 :

1) The three rhyme, repetition and semantic schemas are related ; 2) Baudelaire observed the semantic alternance principle, and the semantic spirit, if not the letter of the global closure rule.

Useful metrical notions

For a better understanding of the « pantoum », the following notions may be useful.

Rhyme structure.

In French poetry, most « classical » stanzas are constituted of one or two rhyme equivalence groups (r.e.g.), each of which is a pair of rhyming modules of one to three lines, such as:

Bi-phonic alternance (intertwining).

In various oral traditions and in a variety of ways, a systematic alternance between two speakers A and B may be « metrified » in an AB-AB-AB… binary periodic sequence, resulting from the intertwining of two unary periodic sequences, one of A's and one of B's; this alternance is metrically realised in a bi-phonic or quasi-bi-phonic alternance in some literary traditions (e.g. some antique amoebean verses). Such a binary periodicity departs from the unilinear tendancy of pure literary metrics.

An alternating, bi-phonic structure is obvious in Baudelaire's Abel et Caïn (1861 Fleurs du Mal n° 119), where ab ab rhyme equivalence groups are divided in a sequence of initial ab modules, let us say the A's, adressed to the « Race d'Abel » and a sequence of concluding ab modules, let us say the B's, addressed to the « Race de Caïn ». -In the following poem Les Litanies de Satan, a series of aa distichs is intertwined with a series of repetitions of a unique line-sentence (refrain) ; the combination of each distich with the following refrain occurrence constitutes one sentence in which the distich is a vocative, for exemple the first two : Ô toi, le plus savant et le plus beau des Anges, Dieu trahi par le sort et privé de louanges, Ô Satan, prends pitié de ma longue misère Ô Prince de l'exil, à qui l'on a fait tort, Et qui, vaincu, toujours te redresses plus fort, Ô Satan, prends pitié de ma longue misère Here the bi-phonic structure has a metrical reflex : two metrical rules, the Cadenza Alternance and the 2-colour rule, are violated in a purely uni-linear reading of the poem, but are observed in each A or B uni-linear sequence considered in itself : the successive rhyme-words « louanges » and « misère » have feminine, double endings, althoug they do not rhyme together ; and the line ending in « misère » will really rhyme (not by mere repetition) only with the last distich of the poem (« colère » and « Père »), though they are separated by (much) more than one rhyme-colour.

In real dialogical pantun (in oral tradition), B1 can appear as a variant response to A1, and so forth each Bi as a response to its corresponding Ai ; this dialogical correspondance can be transposed, in literary pantun, into a parallelism (possibly symbolic) between the A and B sequences.

« Pantoum » structure

These notions allow a coherent formulation of the metrics and semantics of the « pantoum » :

1) Modular structure of stanzas : Each stanza is a rhyme equivalence group constituted of twho (distich) modules, with the rhyme schema ab ab (or sometimes ab ba). Combinations of the « pantoum » (French term), or pantun, can be observed in different, apparently independent, oral traditions in the world (for instance in the 13th century Galicean cantigas d'amigo by Martin Codax). This quasi bi-phonic structure can arise in oral traditions where two persons improvise alternately11 .

Irregular appearance of Baudelaire's pantoum.

Baudelaire was an intellectual , literary poet, an admirer of Edgar Poe, and a careful metrician and poet. It seems rather paradoxical that he should have chosen the (rather difficult) form of a « pantoum » while neglecting its fundamental principle (semantic alternance with parallelism), and without seriously exploiting it. And, in this respect, it seems that it would not make much sense simply to claim that his poem is « musical » (without analysis of that very vague notion), or even that its form is related to « valse » rhythm (how ???). -In what follows, rather than systematically investigating from metrical form to meaning, I will try to relate these two aspects informally.

A remarkable lexical/rhyme set.

It is often admitted that, in good poetry, the most efficient word in a line is its last, rhyming word, or, as Banville puts it in his Petit Traité (1872 : 42) :

On n'entend dans un vers que le mot qui est à la rime […]. Le rôle des autres mots contenus dans le vers se borne donc à ne pas contrarier l'effet de celui-là et à bien s'harmoniser avec lui […] This suggests giving a stylistic importance not only to the rhyming word which concludes a line, but also to the rhyming-words which conclude larger metrical units, for instance : Line rhyme-word, concluding any line ; a lower degree could be a hemistich word even if, at the caesura, it does not even rhyme.

Module rhyme-word : rhyming word concluding not only a line, but a module (as b 1 and a 2 in ab ba).

Stanza rhyme-word (or at leat rhyme-equivalence-group word) : rhyming word concluding not only a module, but a stanza (as a 2 in ab ba) (or at least an r. e. g.).

And even poem rhyme-word, which concludes a poem (as « ostensoir » in Harmonie du soir).

In Baudelaire's pantoum, there are four modules ending in « -ige » and four modules ending in « -oir », and as many module-rhyming-words. The « -oir » series is particularly important because :

-Three of the four « -oir » module-rhyming-words, namely « encensoir », « reposoir », and « ostensoir », share an ostentatious similarity:

• Not only are they 3-syllable words (as could also be « pas-soi-re »), but, more relevantly, they have the same anatonic rhythm (3 anatonic vowels), and the same cadenza or catatonic rhythm (1 catatonic vowel, i. e. they all have a simple, prosodically « masculine » ending). Moreover (unlike in « soir » [= evening]), this common phonic ending is associated to one and same nominal suffix « -oir » meaning something like a functional object.

• In these three words, while this common suffixal and catatonic ending already contains three phonemes (/war/), it is (graphically) preceded by the same letter « s », and (phonetically) by the same ± voiced consonant /S/ ; this /S/ is similarly un-voiced in the first and last words « encen-soir » and « osten-soir », in which, morever, the longest common ending includes the same vowel spelled, with the same spelling « en »12 .

• These last two words freely « rhyme » with the title Harmonie du soir, whose last word, or last syllable, is included in the end of « encen-soir » and « osten-soir », and graphically in « repo-soir » (same last syllable).

• These three word belong to a very specific common lexical field : they are liturgical instruments (in the catholic religion dominant in Baudelaire's culture), the use of which can be related in certain rites as will be specified later.

-The module-rhyming-words « encensoir » and « ostensoir » occupy key positions of the poem. This is most obvious for « ostensoir » which, together with the last line (and last sentence), obviously concludes the last module, the last stanza, the whole poem. « Encensoir », its most similar rhyming word, concludes the first module, and sentence, of the poem. Now it happens that the first and last sentences are semantically more or less clearly detached: the first sentence, as an announcement of something important about to happen, in an already religious tone (« Voici venir les temps où… »), where definite, plural « les temps » has a more serious resonance than simply « season » or « part of the day » (cf. the eschatological notion of the end of the world, when God will manifests himself to the whole Mankind ; the last sentence, being detached by the five suspension points, and introducing the adressee (« toi ») implied in « ton souvenir » = « le souvenir de toi ») and the subject « moi ». This final appearance of the subject is all the more striking since it follows the indefinite « Un coeur tendre… », revealing that this heart was the heart of the subject.

-It might then be relevant that « reposoir », the other element of this lexical and liturgical trio, concludes the first half of the poem, so that the two halves of the poem are respectively concluded by a « reposoir » and an « ostensoir ».

Liturgical symbolism

Do the « encensoir », « reposoir » and « ostensoir » simply constitute a sort of bric-à-brac borrowed from « la religion la plus fadasse » (Étiemble13 ) and belonging to « un fonds romantique assez commun » (Crépet et Blin approuved by Pichois 1975 : 919).

Let us rather question this set of liturgical instruments, such as framed within the architecture of poem, and only then examine whether the semantic alternance principle of the pantoum is really neglected.

Encensoir, first module-rhyme-word:

The « encensoir » (Engl. « censer » 14 ) , swayed by humans -here below, on earth -, could send perfumes (« encens ») in the air towards the divinity above. For instance, during the « Offertory » part of a catholic solemn mass (eucharistic service), before being consecrated, the bread and wine were offered to the divinity and incensed with codified gestures, with the following words (italics mine) :

Incensum istud […] ascendat ad te, Domine, et descendat super nos misericordia tua (= Let the incense ascend towards you, Lord, et let your pity descend on us).

Note the double direction, human offerings with incense upwards towards heaven, calling for divine pity downwards towards the Earth.

Only in the following part of the service (« Canon »), were the bread and wine transformed by « consecration » into the blood and body of Jesus (Corpus Christi) who thus descended from Sky in the consecrated wine and bread.

Ostensoir, last module-rhyme words. After the mass in which it was consecrated (« transsubstantiated » into the body or flesh of Jesus), the white host could be exposed to worship in an « ostensoir » (Eng. « monstrance »), an instrument generally made of gold throwing out beams around the white host, in such a way that the now Holy Host (Corpus Christi, divine body) looked like the sun15 ; as Giles Dimock ([Dominican] recalls in Questions and Answers on the Eucharist, New Jersey, 2006 : 86) :

At the end of Vespers or Compline (Evening Prayer and Night Prayer respectively), which were well attended in religious-order and even parish churches, the priest at the end of the service would take out the Blessed Sacrament [the consecrated host] in the monstrance and the congregation would salute it with a hymn (hence the French term salut for Benediction of the Blessed Sacrament) and often salute the Blessed Virgin Mary with the « Salve Regina » or other Marian hymn. The priest would then bless the people with the Blessed Sacrament […] In the course of time Benediction became an independent ceremony […] and […] became very popular in itself.

Thus, in the preparation of the holy sacrifice (offertory), the « encensoir » sent a message upwards from the humans on earth towards God in heaven ; and by the consecration, the divinity descended on earth into the host, in which he could be « received » (ingested) in Communion, or exposed and worshipped as, for instance, in the evening Benediction of the Blessed Sacrament, during the « Vêpres », commonly attended by laymen in churches where the Holy Sacrament was exposed ; « vêpres », that is, the evening office, « (office du) soir », where the etymological notion, soir, is part of the title « Harmonie du soir », and a rhyming-word in the first stanza, related to the long-awaited eschatological « fin des temps » (cf. « Voici venir les temps »).

Reposoir, middle module-rhyme word. The most common English translation of this word occurring in the middle of the poem, « altar », is too general, not specific enough in this poem. It is aptly defined in Nodier & Verger's 1832 Dictionnaire as (italics mine): terme de culte catholique: sorte d'autel qu'on élève et qu'on prépare dans les lieux où la procession passe le jour de la Fête-Dieu [Corpus Christi], pour y faire reposer le Saint Sacrement16 .

Interestingly, in several similar definitions, the reposoir is not directly characterized as a place in which the monstrance with the holy sacrament is already exposed, but a place prepared to receive it at the moment the procession will make a momentary halt there.

The specific meaning of the Corpus Christi procession, happening once a year, was, as the archbishop of Lyon (de Bonald, 1841: 4) explained: … le mémorial […] du mystère d'un Dieu qui est descendu parmi les hommes […]. Ainsi quand vous verrez Jésus, caché sous les espèces eucharistiques, porté par nos mains à travers les places de la cité, ne vous semblera-t-il pas que le Fils de Dieu […] passe encore sur la terre en faisant le bien […]? (= the memorial of a God who has descended among the humans? Thus, when you see Jesus, hidden in the eucharistic species [?], carried by our hands through the squares and streets of the city, will it not seem to you that the Son of God still passes on earth en faisant le bien [= English "doing good"??).

In such processions, whereas the consecrated host was normally kept inside a sacred place (church), it was carried and exposed in different places of non-consecrated roads or streets, in the ordinary world, among the people, believers or not; and at several stage-points of the procession, the monstrance was laid down on a reposoir prepared for that purpose, and was there exposed to worship. Before that, the reposoir was ready to receive it (waiting for it), and people could wait for it there. It can thus make sense that the heaven-« reposoir » is evoked halfway between the « encensoir » which can be used before the consecration and the « ostensoir » which exposes the result of consecration to worship.

Thus the lexical and rhyming set encensoir -reposoir -ostensoir located at key-positions in the poem rhythmically correspond to a ritual eucharistic progression.

Communion.

All this eucharistic symbolism ends in the mention of the sun dying « dans son sang » and the last line « Ton souvenir en moi luit comme un ostensoir ».

The formula which performatively transforms (« trans-substantiates ») bread (host) and wine into divine flesh and blood (Corpus Christi) repeats words pronounced by Jesus offering bread and wine to his disciples at the Last Supper before his death17 (italics mine) :

Take, and eat ye all of this ; for this is my body […] For this is the chalice of my blood […], which shall be shed for you, and for many, for the remission of sins. As often as ye do these things, ye shall do them in remembrance of me.

After this consecration, the host(s) ultimately could be received 18 and ingested by the believers in the so-called Communion sacrament. In receiving communion, the believer was supposed, according to a well-known expression, to receive Jesus in his heart (« recevoir Jésus dans son coeur ») and to be intimately united with him.

In this context, after the action of the « coeur » which « recueille » the « vestiges » of the sun drowned in his blood, and after the lapse of time suggested by the five suspension points which succeeds to this statement just before the last line of the pantoum, the present tense of « luit » (glowing in the dark) may suggest the state in which all this ends. Thus the result of the poem, namely the presence of « ton souvenir » in my heart, is analogous to the real presence of God in the heart of a believer after Communion ; and the « harmonie » announced in the title can be analogous to the intimate union of the believer's heart with God, or « ton souvenir » in « moi » 19 .

This mystical symbol makes Harmonie du soir a poem of worship… of the lost person, who, by means of its « sorcellerie évocatoire », can still be present in the memory of the subject 20 .

Semantic alternance in Harmonie du soir

By the Eucharist, a God, invoked by a mystical human rite, comes back from heaven down to Earth where he can be worhipped and received in the heart of believers. The double direction thus implied seems to be reflected in the two series A and B at least throughout the first three stanzas of Harmonie du soir, making it a real pantoum21 : First stanza. A1, by the rhyme-word « tige », first of the A series and of the poem, introduces the stems of the flowers ; the stem is not the most commonly poetical part of a flower, but it is relevant as evoking the ground (on earth) to which each flower is attached, as opposed to the above air and sky. In the second line, the comparison of a flower with a censer may concern not only the fragrance (not yet named) which emanates from both (« évapore »), but also the « vibr[ation] » of both : during a liturgical service, the censer was permanently swayed, either sofly just to maintain its burning, either more strongly to activate the emanation of its fragrance during certain symbolical, liturgical gestures ; its movement was provoked by the chain held by a person (standing on earth), comparable to the stem which ties the flower to the ground. In a solemn mass, during the Offertory (where the « holy sacrifice » of divine Jesus to his Father is prepared), while the prayer « Incensum istud a te benedictum ascendat… » (Let this incense ascend towards you) was pronounced, the censer was swayed, then rotated, above the offered bread (host) and wine22 before they were transformed by the formulae of « Consecration » into the very flesh and blood of Jesus (Corpus Christi). -But by evaporation the flowers and censer send perfumes from earth towards heaven. The ascending fragance was interpreted as a symbol of the prayers hich, from the poor humans on Earth, should ascend ut to the high-above divinity (« le Très-haut » [« the Highest »]).

B1, Line 1, explicitly introduces « les parfums » (fragrance) only implied in A1. While the first rhyme-word in A1 tied the flowers to the grounds (earth), the first rhyme-expression in B1 situates them (with sounds) in the « air » (in French, « en l'air » means : above the ground, not supported by it).

The sounds (« Les sons… ») in air also come from an instrument on the ground, yet to be mentioned Line 2 introduces a « valse », which can be a dance (by couples of persons on the ground) or a music (therefore in the air) 23 . The dance is characterized by a turning movement of the dancers ; but then, no person has been introduced, the turning movement is ascribed to the sounds and perfumes which « tournent dans l'air », before this movement is qualified in the second line as a walz : a waltz of sounds and perfumes in the air.

Thus the explicit space of B1 is the air, even though the melancholy and « langoureux vertige » attributed to the waltz may evoke a human being24 ; the epithet « langoureux » could apply to music (Littré).

Second stanza. A2. Line 2, mentions the musical instrument which, handled by some human on the ground, can send music in the air. By comparison with this violin is introduced a yet undefined « coeur qu'on afflige » (possibly subject to melancholy and languor).

The notion « un coeur qu'on afflige » is much less commonplace than « un coeur affligé » (which would not rhyme in the poem such as it is). While it could allude to some sentimental relation where a loved person does harm to a loving person, the following lines do not seem to give a clear confirmation of such an interpretation ; but the end of the stanza may suggest that the heart is afflicted by the loss of a « passé lumineux » (for instance, in a masculine poem, loss of a loved woman).

The heart « qu'on afflige » is compared with a violin which « frémit », where the quivering of the violin is analogous to the vibration of the parallel flowers and censer. The quivering of the violin corresponds to the vibration of the instrument being played, where, as a dictionary puts it, « Toutes les parties d'un violon qu'on joue entrent en vibration » (p. 667 of Nouveau dictionnaire universel des arts et métiers, Francoeur & al., Bruxelles, Jamar, 1839, italics mine). Thus the heart « qu'on afflige » vibrates or quivers like violin « qu'on joue ». This heart obviously is, or belongs to, a person on earth, unlike a dead person who might already be in heaven. But the comparison of the heart « qu'on afflige » with a violin being played which sends its music in the air may suggest that the quivering (vibration) of the heart produces something analogous to the music sent in the air, or to the perfumes sent towards the divinity.

The strong liturgical framework of the poem provides religious connotations which give sense to this precise analogy. Human life was most commonly depicted by preachers as being, since Original Sin, a miserable life on earth, during which humans, waiting for a happy life above after death. prayed heaven for consolation. The most famous and popular prayer embodying this view was the Salve Regina ; in singing this hymn, addressed to the mother of God, « mother of mercy », who is above us in heaven, the faithful sing : « To thee, […], we banished children of Eve […] send forth our sighs, mourning and weeping in this vale of tears » (« ad te suspiramus, gementes in flentes in hac lacrymarum valle »), and ask her to show them Jesus « after this exile » in the vale of tears (« Jesum … nobis ostende »). The terms « affligé », « afflictions », were the most commonly used to designate the sufferings of human life (see examples in Appendix). Through this catholic ideology, in the same way as, from a vibrating censer or violin, perfumes and music can ascend in the air, mourning and sighs or prayers for pity can ascend in the air from an afflicted heart. However strange such a combination of images may seem nowadays, it was far from exotic during the catholic Second Empire, and it is clear that on some other occasions Baudelaire made an ostentatious, complex use of Catholic (and liturgical) symbolism (for instance the Franciscae meae laudes linguistically and rhythmically transpose a hymn to Mary, mother of Jesus, into a hymn to a beloved woman).

B2 : Line 2 explicitly mentions the sky / heaven (« Le ciel) ») and immediately associates it with a religious, eucharistic comparison to a « reposoir », the place where the host could be located (as it will be later at the very end of the poem). This symbol of expectation of the divine host is coherent with the prayers of an afflicted heart. The sadness of the sky / heaven (« triste ») is not incoherent with the fact that the host is not there. the officiating priest ingested both, but generally, in the Catholic church, the laymen received only the bread (hosts) which thus, alone, could represent the whole body of Jesus. In the last module, the blood and bread (host) respectively conclude the two lines (« son sang qui se fige », « un ostensoir »). But in the last line, the host alone (in the monstrance) can represent the divinity.

This double revelation, linked by « comme », makes explicit the analogy between the (let's say) sentimental or lyrical and the religious aspects of the poem. The presence of the lost loved person (through « ton souvenir ») in my heart is analogous to the presence of the Jesus (dead and ressuscitated) in the host ; the relation you-in-me (« ton souvenir en moi »), following the idea that « un coeur » (me) « recueille tout vestige » of the sin (you), may evoke, more specifically, the communion rite.

The five suspension-point before the last line, and its present tense (« luit ») after « s'est noyé », suggest that the gleaming of your memory in my heart is the present result of this « sorcellerie évocatoire ».

Relevance of the notion « harmonie » in the pantoum.

[To be completed]. *

  same regularities, observed in such corpuses as Dante's Divine Comedy or French poetry, lead us to associate to the two following sub-lines, rather than an « intellectually » apparent 5-5 rythm, a regular rhythm 4-6 rhythm : O Vierge mère, fille de ton fils 3 (cf. Dante : Vergine madre, figlia del tuo figl[io])

3 )

 3 Semantic coherence of module sequences : Let A be the sequence of initial modules (ab…), B the sequence of terminal modules (…ab) ; each sequence, A or B, tends to be semantically coherent, resulting in a semantic alternance of the distichs along the poem. 2) Repetition-chaining of each module sequence : In each sequence A or B, any two successive modules are chained according to the repetition schema …a a…, the first line of a module repeating the last line of the preceding one in its sequence. As a result, any two successive stanzas have the repetition schema [xa xb] [ax bx]. 4) Semantic parallelism and final convergence of the chains : Chains A and B are supposed to be semantically parallel in some poetical way, and they are supposed to converge in the last line of the « pantoum » by the global repetition schema [A………A] (the last line of the « pantoum » repeating its first line).

Example from a 1722 vaudeville. Same catatonic, non syllabic contrast, in "mironton, mirontaine", "tralala, tralalaire", etc.

In the journals Cahiers du Centre d'Études Métriques [on line] and Parade sauvage.

Number 47 in the 1861 edition of the Fleurs du Mal (text chosen here), first published in 1857.

The origin of the « pantoum » tradition in French is a translation of a Malayan poem published in a note of Victor Hugo's Orientales (1829), where « pantoum » was a misprint for « pantoun » (= pantun), so that the « pantoum » can be considered a particular modern form, first codified in French literary tradition, of a more general sung or oral tradition.

For instance in Banville, Petit Traité de versification française (1872). In such descriptions, the numerical notion « the two [first or last] lines » superficially compensates the lack of a corresponding metrical unit (module) in the metrical theory.

A first sketch of this analysis was published in L'Atelier de Baudelaire: "Les Fleurs du Mal" (Cl.Pichois & J. Dupont, Slatkine, 2005, vol. 4: 3559-3560). I did not then know that David[START_REF] Scott | Translating Baudelaire[END_REF], the first to my knowledge, had already proposed a distinct interpretation of the two series of distichs (although substantially different from the interpretation proposed here).

Chaining or concatenation (Latin concatenatio), a notion familiar to medievists and folklorists. Child folklore quasi-syllabic example of chaining : « Trois p'tits chats -Chapeau d'paill' -Paillasson… ».

Rip Cohen (2013) tries to explain the structure of a Galician-Portuguese cantiga d'amigo.

The longest common ending (plus grande commune terminaison or PGCT in French), is not to be confused with the minimal regular common ending or rhyme (= catatonic form).

Quoted by G.Voisset (1997 : 99).

The beaming sun was a very common, traditional, analogy of God in his glory, in religious texts, painting, &c., but rather like a rising sun than like a setting sun as in Harmonie du soir.

= A sort of altar which is raised and prepared in the places where the procession passes on Corpus-Christi day, to place the Holy Sacrament.

The sacramental formulae had to be uttered in Latin, « Accipite, et manducate ex hos omnes ; Hoc est enim Corpus meum […], Hic est enim calix Sanguinis mei […] qui pro vobis et pro multis effundetur in remissionem peccatorum. […]. Haec quotiescumque feceretis, in mei memoriam facietis » (quoted from a

Paroissien Romain complet selon le rit romain (Paris, Leroux, p. 85)). English translation borrowed from a 1850 Golden Manual being a Guide to Catholic devotion (Londres, p. 234).18 As in the common expression « recevoir la sainte communion » (to receive holy communion).19 Jesus, who had returned to the Sky after his death and resurrection, was supposed to be substantially contained, at every mass, « under the species » of consecrated bread and wine.20 This interpretation was ony sketched in my contribution to Cl. Pichois & J. Dupont's Atelier deBaudelaire (2005 : 3561). The only precise interpretation of Harmonie du soir as eucharistic I know of is that of FrançoiseMetzer (2011 : 231-234) which I found on Internet after my Prague lecture ; she mentions the formula « Faites ceci en mémoire de moi » and the notion of « sorcellerie évocatoire ».

The semantic alternance of Baudelaire' pantoum was sketched by CliveScott (2000 : 114) : « Baudelaire observes the pattern of repetitions and, more unusually, incorporates that convention of pantoums, insisted on Banville (19809 : 243-8), whereby each pair of lines of the quatrain (i.e. first and second, third and fourth), pursues, through the length of the poem, a different thematic thread : the first two lines of each stanza are devoted to the vicissitudes of the heart, while lines three and four trace the fortunes of the dying light, the poem's final line reaching across into the world of the heart » (I did not know this study when I wrote my 2005 interpretation).

The priest « encense le pain et le vain, dirigeant son encensoir par trois fois au-dessus du calice et de l'hostie », and « Ensuite il fait tourner son encensoir autour du calice et de l'hostie » [italics mine ; in…].

As in : « Je le priai de me jouer une valse. Il en joua deux avec assez de verve » (dans Scènes et Proverbes de Octave Feuillet, 1855, Paris, Michel Lévy, p. 326).

In classical French, the meaning of « vertige », far from being reduced to the notion of dizziness, still was related to the latin root of « vertere » (= to turn). The Littré Dictionnaire defines it as a state in which « il semble que tous les objets tournent et que l'on tourne soi-même » ; if the sounds turn in the air, the « vertige » can apply to the sounds. The languor could also be provoked by incense.

The fact that the blood « se fige » (clots) is a manifestation of death (the Mass represented the sacrifice of Jesus).

 observes the convergence of the last line with Lines 1-2.

Third stanza.

A3 qualifies the afflicted heart (on earth), and perhaps his very affliction, hating « le néant vaste et noir », anologous to the feared, expected end of the evening, which will be complete darkness. -That this heart is « tendre » may reinforce the sentimental interpretation of the afflicted heart, but at the same time is compatible with the religious evocation of an afflicted life in the valley or tears, taking into account a classical meaning of tendre (sensitive -thus sensitive to suffering).

B3 introduces the sun, which is in the sky, and, through an allusion to the reddening colour of the lower sky in the dusk, the idea of the death of the sun (« noyé ») and of its blood, both in precise correspondence with the eucharistic symbol introduced by the « reposoir » in the preceding B-module. The « holy sacrifice » of Jesus not only symbolized, but supposedly really reproduced by Catholic mass, is his death, even if associated with his resurrection 25 .

Thus, so far, the A-modules concerned earthly elements -« fleurs », « encensoir », « violon » -and the B-modules, elements in the air or sky (« sons », « parfum », « air du soir », « ciel », « soleil »).

Last stanza. A4, Line 2, introduces « tout vestige […] du passé lumineux », where the notion « passé lumineux » seems to be ambivalent. Considered in the sole context of the A series, it seems to refer to the bright, happy past of the heart (on Earth), where the notion « vestige » implies that this happy time is finished (« passé »).

But at the same time the image of a heart which hates the « black » void and « recueille tout vestige » « du passé lumineux », in the prevailing context of « soir » (title, « air du soir » and rhymes), alludes to the day, with its light, which is passed when the « black » night is there.

By this ambivalence, the series A and B, parallel in the other stanzas, converge in the first module of the last one. This convergence is pointed out, as it were, and concentrated, in the word : « recueille », since this word designates an action which unites (one into the other) the sun (or « tout vestige » of its light) and the heart.

The last line of the poem will restrospectively specify the symbolism of « recueille ». One may observe that the heart is central in the the A-series : after it has been symbolically prepared by the flowers, censer and violin as its analogues, his evocation extends over the three other stanzas : « (A2) Un coeur qu'on afflige … (A3) un coeur tendre, qui hait le passé vaste et noir… (A4) du passé lumineux recueille tout vestige ».

B4. Line 2, last line of the poem, introduces three entities : In h1 (first hemistich), you, or at least the memory of you (« ton souvenir ») ; me ; and, in h2, a monstrance. This final, explicit reference to « you » and « me » had been carefully delayed by the indefinite notion « un coeur ». The « ostensoir » (monstrance) has the shape of a beaming sun (and had sometimes been named « soleil ») ; the comparison of the last line equates you (« ton souvenir ») and the monstrance sun. It follows that an A-entity « un coeur » (heart of the subject on earth, « moi ») and a Bentity, the sun meet together in the last line of the poem 26 .

Thus, A and B entities meet together in each module of the last stanza. In this way, the semantic principle of final convergence in the pantoum, if not its metrical form (repetition closure), is observed.

In all the poems of the Fleurs du Mal, when the sex of the subject is relevant, it is masculine, and it is often equated with the poet. Several clues in the poem suggest that the melancholy and sadness are associated with the loss of a lost loved person, plausibly a woman.

The concluding, rhyming-word of the poem, « ostensoir », prepared by the « encensoir » and the « reposoir », makes explicit the eucharistic meaning, as the ostensoir contains the consecrated host, believed to really be the divine body of Jesus. It was sometimes specified that the consecrated wine becomes the blood, and the consecrated bread (host), the flesh of Jesus ;


Camille Schreck

Damien Rohmer

Stefanie Hahmann

Marie-Paule Cani

Interactively animating crumpling paper

Keywords: I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism Computer Animation, Geometric Modeling, Interactive Deformation, Paper Crumpling, Developable Surface

We present the first method in computer graphics to animate sheets of paper at interactive rates while automatically generating a plausible set of sharp features when the sheet is crumpled. Our hybrid, geometric and physical, model is based on a high-level understanding of the physical constraints that act on real sheets of paper, and of their geometric counterparts. This understanding enables us to use an adaptive mesh carefully representing the main geometric features of paper in terms of singular points and developability.

INTRODUCTION

Computer animations of crumbled paper sheets are almost absent from virtual environments although it is a very common material in everyday life. This is due to the extreme difficulty of efficiently capturing the dynamic, non-smooth developable geometry of paper sheets. Indeed the microscopic fibers structure significantly affects the macroscopic behavior and makes it challenging to reproduce. The goal of this work is to achieve the animation of paper crumpling processes at interactive rates. This may greatly simplify the work of artists who need to model this phenomenon. Let us list the macroscopic features that need to be captured in order to crumple a sheet of paper in a visually plausible way:

. Length preservation. The assumption can be made that the surface of a sheet of paper preserves lengths with respect to an original 2D pattern. It can be flattened onto a plane without distortion (stretching or compressing). This is a property which is called developability. Sharp features. Crumpled sheets of paper exhibit singular points, and possibly some sharp creases. Plastic behavior and shape memory. In real life, the microscopic fibers of paper or more precisely the bonds between them may break upon deformation, causing irreversible damages to the structure.

The modeling of paper as been studied in numerous previous works that can be divided into two main approaches : Geometric modeling. Bending and creasing virtual paper can be expressed through geometric deformation [START_REF] Kergosien | Bending and creasing virtual paper[END_REF]. An interactive tool is presented in [START_REF] Zhu | Soft folding[END_REF] to fold thin sheets of materials. Other methods addresse interactive origami modeling [START_REF] Tachi | Origamizing polyhedral surfaces[END_REF][START_REF] Mitani | Interactive design of planar curved folding by reflection[END_REF][START_REF] Solomon | Flexible developable surfaces. Computer Graphics Forum (CGF)[END_REF] where the folds are explicitly defined by the user. A geometric way to create a virtual sheet of paper with sharp features from a 3D boundary curves and a 2D pattern is proposed in [START_REF] Rohmer | Folded paper geometry from 2d pattern and 3d contour[END_REF] but cannot be extended to animation. Physically based modeling. Thin sheet materials are typically modeled in computer graphics using 2D finite elements with a linear elastic behavior. This approach has been used to bend sheets of paper using the thin plate theory [START_REF] Grinspun | Discrete shells[END_REF][START_REF] Burgoon | Discrete shells origami[END_REF]. However, this method cannot handle the dynamic formation of singular points and sharp edges. Kang et al. [START_REF] Kang | Plausible virtual paper for real-time application[END_REF] propose to model the plastic behavior of the paper using breakable springs, but the approach suffers from visual artifacts as edge-split and singularities can only occur along preexisting coarse triangle edges. Recently Narain et al. [START_REF] Narain | Folding and crumpling adaptive sheets[END_REF] obtained high-quality visual simulation of paper by combining this inelastic deformation model with an adaptive remeshing method. Yet, this approach is computationally expensive due to the large number of triangles in the folded regions, so it cannot be used in interactive applications.

The key idea of this work is to obtain interactive rates and automatic generation of sharp feature by interleaving standard physically-based simulation steps with procedural generation of a piecewise continuous developable surface. Our contributions include a hybrid physically-based and geometric model, a realistic modeling of singular points, an optimal adaptive isometric meshing and a validation through comparison with real paper.

As already mentioned, a sheet of paper can be considered as a developable surface, which can be defined as a ruled surface with zero Gaussian curvature (product of the principal curvatures) everywhere. As a consequence, if a smooth part of the paper is bent such that one principal curvature is not zero, then the other principal curvature in the direction of the fold is necessarily zero. Thus the surface becomes rigid in the direction of zero-curvature, leading discontinuities to appear instead of having the surface bend if a compression is applied in the orthogonal direction. The appearance of singularities has also been studied in the mechanical literature: due to the fact that the bending rigidity of thin plates is much smaller than their stretching rigidity, the paper mechanical transformations are favorable to bending [START_REF] Cambou | Three-dimensional structure of a sheet crumpled into a ball[END_REF]. In addition, thin sheets have the specific behavior of concentrating energy when being constrained [START_REF] Witten | Stress focusing in elastic sheets[END_REF]. When the thickness tends to 0, this energy concentrates into singular points, which are called d-cones [START_REF] Amar | Proceedings of the Royal Society[END_REF][START_REF] Mahadevan | Conical surfaces and crescent singularities in crumpled sheets[END_REF][START_REF] Cerda | Conical surfaces and crescent singularities in crumpled sheets[END_REF]. These d-cones denote developable generalized cones defined by a fixed point called the apex and a one-parameter family of straight lines (rulings) passing through the apex. Outside of the influence of the d-cones, the surface is smooth and exhibits a low distribution of elastic energy [START_REF] Schroll | Elastic building blocks for confined sheets[END_REF]: it can be deformed according to standard linear models for continuous mechanics laws. We choose thus to have the motion of the surface guided by an usual physic simulation based on continuous mechanics. The surface model used to represent the shape of a sheet of paper must be highly deformable, as sharp features may appear anywhere during deformation. This usually requires to use dense meshes, which provide a large number of degrees of freedom and reduce visual artifacts due to badly oriented edges. Instead we choose to dynamically adapt the mesh to the surface using a dedicated geometric model in order to have the necessary degrees of freedom for surface deformation while keeping a coarse mesh. This reduces most of the time usually spent in physically-based simulation. The observations made above lead us to use the following model to get an optimized mesh (represented in Figure 1). The surface of the paper is approximated by a set a curved regions (in magenta), represented by generalized cones, and planar regions (in green). The mesh used by the physicallybased simulation is composed of the triangles and quadrangles formed by the rulings of generalized cones and of a very coarse triangulation of the planar regions. In this way, the curved regions are rigid along the direction of the rulings, as for real paper, while planar regions are still able to bend in any direction. The sharp creases of the paper are modeled by singular points (red dots) which can only be apices of d-cones. They are generated in the regions where conflicts between two bending directions arise. Their positions are computed using a probability law in the conflict's region. As the apparition of sharp points is a plastic event, we record the position of these vertices on the 2D-pattern. We interleave the updating of the mesh with the simulation steps in order to obtain a simplified, yet visually realistic paper behavior. The optimized mesh stays coarse such that the animation can be achieved in interactive time. Moreover, during the whole animation process, we are able to maintain an approximative isometric mapping between the paper surface S and its pattern S defined in the 2D parameter domain. We propose the following algorithm, a more detailed description can be found in [START_REF] Schreck | Non-smooth developable geometry for interactively animating paper crumpling[END_REF]. Initialization. The input is a developable surface S and its isometric mapping to a 2D pattern S. Additionally, some handles (black dots in Figure 1), i.e. a finite set of points that the user can manipulate, are defined. They act as hard constraints and are used to govern the deformation. Animation loop. (see Figure 2) • Step 1: Elastic deformation. A state-of-the-art physicallybased simulation is applied for efficiently deforming the surface mesh in a plausible way but without taking care of paper plasticity at this stage. Isometry to S, and therefore developability, may thus be lost.

• Step 2: Modeling bending and crumpling. The connectivity of the mesh is modified to update the position of the curved regions and the flat regions. Notably the conflicts that may appear between two regions bent in different directions are solved by introducing new singular vertices.

• Step 3: Developable and isometric tracking. The curved regions are segmented into quasi developable regions by locally computing the best approximation by a set of generalized cones. Developability is then geometrically enforced by aligning the mesh edges along these rulings. The isometry preservation is then optimized by constraining edge lengths using a method similar to [START_REF] Wang | Towards flattenable mesh surfaces[END_REF]. Finally the mesh is displayed.

RESULTS AND VALIDATION

Validation. We used experiments involving real pieces of paper to validate our virtual model. Note that contrary to previous approaches, we specifically conduct comparisons in the critical situation when the first few singular vertices appear. Indeed, when and where these first sharp features appear is very noticeable and capturing it accurately is a key element towards realism. The first experiment, shown in Figure 3, was used to validate the way a new singular vertex appears: in real, we moved Then we applied an orthogonal constraint on one of the borders until a singular point appeared (see Figure 3-left). We repeated the experiment 20 times with other sheets of the same thickness and then 20 times using our method. The resulting distributions of the position of the first singular point are depicted in Figure 3 (center and right), for two different types of paper. Our results show a good accordance between real measurements and our virtual model. Visual comparison between real paper and our results are also provided in Figure 4. The top row illustrates the deformation of a smooth paper strip continuously twisted into a Moebius strip. The second row compares the behavior of a sheet of paper after a singular vertex is inserted: we get the same effect of the sheet straightening when the constrained points at the back are Comparison with other methods. Figure 5 compares our method with a model using a standard finite element method applied to a fine mesh of fixed connectivity, and with one using adaptive remeshing from Narain et al. [START_REF] Narain | Folding and crumpling adaptive sheets[END_REF]. All three methods use the same physical FEM solver, and are able to obtain comparable visual results. Our method requires approximately ten times less triangles than the adap- tive remeshing method in Narain et al. and twenty times less than the standard FEM approach with fixed connectivity for similar visual result, leading to far better time rates (see Table 1). Other results. Table 2 provides the average length distortion over all the animation frames and the computational times for the different examples. Figure 6 gathers other re- Computed for the example presented in Fig. 5. sults of our method.

CONCLUSION

We get a variety of results that qualitatively look as real paper, indicating that our method has reached most of its goals.

We would like to extend our work to have creased curves and not only singular points as sharp features, as shown in mechanical studies. Also, although it is much faster than state-of-the-art methods, our prototype currently runs at 1 to 10 fps. As future work, we would like to improve this method to reach real-time performances.

Figure 1 :

 1 Figure 1: Photograph of a sheet of paper bent in several directions (top left). Our geometric surface representation (pattern on the bottom left, and 3D surface on the bottom right) is divided into curved regions (magenta) and planar regions (green). Singular vertices (red dots) are apices of generalized cones.

Figure 2 :

 2 Figure 2: Overview of the algorithm

Figure 3 :

 3 Figure 3: Comparison of the positions of the first appearing singular vertex. Left: deformations applied to the sheet of paper. Right: the measurements on real paper (black crosses) and the positions generated by our method (red dots) for a 70 g/m 2 paper (center) and a 90 g/m 2 paper (right). We use different standard deviation for computing the position of the singular points in order to take into account the different type of paper.

Figure 4 :

 4 Figure 4: Comparing our results with real paper deformations.

Figure 5 :

 5 Figure 5: Comparison with other methods. The Standard FEM (left) and Narain et al. [10] method (middle) required a very dense mesh in order to get a result of similar visual quality as our method (right).

Figure 6 :

 6 Figure 6: Some other results.

Table 1 :

 1 Average computation times.

		Our	standard Narain
		method	FEM	et al.
	Remeshing time (ms)	1	-	450
	Simulation time (ms)	100	14,700	4,640
	Nb. of triangles	89	1668	788

Table 2 :

 2 Measures of error in length and Computational time analysis. cm is the mean percentage of compression over all the model edges and Nt is the average number of triangles of the mesh through the animation. t simu and t geom are the respective times spent in average in the simulation steps versus in the geometrical steps.

	Models	cm	Nt t sim (ms) tgeom (ms)
	Fig. 4 (top)	0.17 117	294	3.3
	Fig. 4 (bottom) 0.19	87	60	1.4
	Fig. 6 (left)	1.47 156	247	2.8

Acknowledgements: This research was partially funded by the ERC advanced grant no. 291184 EXPRESSIVE.