E Simlissi

M Martiny
email: marion.martiny@univ-lorraine.fr

S Mercier

S Bahi

L Bodin

Elastic-plastic analysis of the peel test for ductile thin film presenting a saturation of the yield stress

Keywords: peel test, plastic dissipation, decohesion, Voce flow law, interface fracture energy

The paper investigates the peeling test of an elastic plastic film on an elastic substrate. The case of the film material presenting a saturation of the flow stress is considered. Based on earlier approaches of the literature, see for instance Kim and Aravas (1988) (Kim, K.S., Aravas, N., 1988, International Journal of Solids and Structures, 24, 417-435), a semianalytical expression for the plastic dissipation during bending/reverse bending is proposed. The validity of the present expression is established based on finite element calculations. It is shown that for the interpretation of results of peeling test at 90 degrees when the peeling force and the curvature have been measured, the present approach provides a precise interface fracture energy. The comparison is made with predictions based on an elastic perfectly plastic approach.

Introduction

Peeling is an affordable test conducted to determine the strength of the interface between two materials or between two layers. Depending on the mechanical behavior of the materials which form the interface, the prediction of the interface fracture energy from the measurement is not always straightforward. A large effort has been provided in the literature to analyse in details such experiments. For inextensible tape on a rigid substrate, [START_REF] Rivlin | The effective work of adhesion[END_REF] has observed that the peel force during steady state provides a direct measure of the interface fracture energy. From an energy balance, [START_REF] Kendall | Shrinkage and peel strength of adhesive joints[END_REF] and [START_REF] Kendall | Thin-film peeling-the elastic term[END_REF] have extended the Rivlin's approach to account for elastic contribution of the peel arm. Still in that configuration, the sole knowledge of the peel force provides an estimate of the interface fracture energy. The possibility to obtain this interface property when complex material behaviors are present needs the evaluation of the plastic dissipation during peeling. An important contribution has been proposed by [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF]. From the moment curvature relationship for the peel arm ( assuming pure bending under plane strain), the plastic dissipation is obtained. They investigate the case of materials presenting hardening (power law expression for the flow stress). Based on their work, the knowledge of the peel force and the slope of the film at the tip of the process zone is needed to predict the interface fracture energy. [START_REF] Kim | Mechanical effects in peel adhesion test[END_REF] and [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] consider next an elastic perfectly plastic material. With that specific behavior, an explicit moment curvature relation during the bending and the reverse bending is obtained. A closed form expression for the plastic dissipation is in addition proposed. As a consequence, from the energy balance of the peel test, the interface fracture energy can be derived from the knowledge of two experimental observations : force and angle at the tip. This model is efficient, has been adopted and discussed widely in the literature. [START_REF] Kinloch | The peeling of flexible laminates[END_REF] analyse the case where the film may develop also plasticity during the test. Instead of considering a powerlaw relation for the hardening behavior as in [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF], they consider that for a certain class of material, the non linear response of the material can be modelled by a bilinear relationship.

Therefore, an explicit expression for the plastic dissipation is also found. The model is shown to be salient for polyethylene film. Note that the configuration with a film material with a bilinear elastic plastic behavior was revisited by [START_REF] Moidu | On the determination of fracture energy using the peel test[END_REF] mostly for the dissipation during reverse plastic unloading stage. One may also cite [START_REF] Williams | The influence of peel angle on the mechanics of peeling flexible adherends with arbitrary load -extension characteristics[END_REF] for considering pre-strain and also non linear response. More recently, a large strain formalism has been proposed by [START_REF] Molinari | Peeling of elastic tapes: Effects of large deformations, pre-straining, and of a peel-zone model[END_REF] since during the decohesion process, the thin film may experience intense bending-reverse bending generating large strain. Close to the crack tip, the assumption of pure bending response may be questionable. Therefore, [START_REF] Wei | Interface strength, work of adhesion and plasticity in peel test[END_REF] adopt finite element calculations to analyse in details the mechanical response of the system in the vicinity of the process zone, the detached arm still being analyzed with the bending approach. The interface response is modelled by introducing cohesive element. They also propose a clear parametric analysis of the peel test based on non dimensional parameters, see also Kim and Kim (1988). They clearly show that the interface fracture energy is depending upon the material parameters (elastic and plastic responses) and also properties of the cohesive el-ement. [START_REF] Wei | Modeling non linear peeling of ductile thin films -critical assessment of analytical bending models using FE simulations[END_REF] compares the bending model of [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF] and [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] with the combined model of [START_REF] Wei | Interface strength, work of adhesion and plasticity in peel test[END_REF]. [START_REF] Wei | Modeling non linear peeling of ductile thin films -critical assessment of analytical bending models using FE simulations[END_REF] extends also the previous approach by accounting for the elastic compressibiliy. By still adopting a power law relationship for the hardening response of the film, a semi analytical expression is found for the link between moment and curvature during the bending, but it is not possible to provide a simple expression for the reverse bending phase. So clearly, as the material description becomes more complex, the possibility to derive analytical or semi analytical expression for the plastic dissipitation is reduced. [START_REF] Wei | Peeling experiments of ductile thin films along ceramic substrates -critical assessment of analytical models[END_REF] propose a critical assessment of the different methods (bending versus combined bending + FE approaches). They illustrate their findings on specific configurations for metallic films made of aluminium or copper on a ceramic. Peel tests are conducted at various angles (135 and 180) in addition to 90 degrees. The thickness of the Al film is varying from 20 to 200 microns.

They also consider ultra thin film of copper deposited on the ceramic (from 1 to 15 microns). Based on the previous works [START_REF] Wei | Modeling non linear peeling of ductile thin films -critical assessment of analytical bending models using FE simulations[END_REF]Hutchinson (1998), Wei (2004)), they conclude that the combined model (cohesive element and bending of the free arm) is valid for a weak interface strength. For a strong interface case, a full finite element model is preferable. Zhao and Wei (2007) confirm that a two parameters model is needed to obtain the interface fracture energy from a peel test. Instead of measuring the base angle at the crack tip, they measure the curvature of the film during the bending / reverse bending process. With these two informations (force + curvature), the strength of the interface can be determined precisely. They are also able to propose values for the cohesive law parameters (interface fracture energy and the separation strength). Experimental evidence based on copper thin fims is discussed. To analyse T-peel test on Cu/Cr/ polyimide system, [START_REF] Song | Analysis of T-peel strength in a Cu/Cr/Polyimide system[END_REF] determine the mechanical response in tension of both Cu/Cr and polyimide. The plastic response of Cu is shown to saturate at very small plastic strain so the authors consider that copper has an elastic perfectly plastic response. From the theoretical expression of the plastic dissipation proposed by [START_REF] Kinloch | The peeling of flexible laminates[END_REF], [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF] and [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF], the interface fracture energy is obtained.

From [START_REF] Wei | Interface strength, work of adhesion and plasticity in peel test[END_REF] and Zhao and Wei (2007), when the material response is known, the most important parameters controlling the decohesion are the interface fracture energy and the strength of the interface. They highlight that when the peel force and the curvature are measured in experiments, the interface fracture energy is theoretically found. The goal of the paper is to propose a contribution to the precise definition of the dissipated energy due to plasticity during bending -reverse bending for a particular class of material responses. Indeed, as shown in [START_REF] Song | Analysis of T-peel strength in a Cu/Cr/Polyimide system[END_REF], the flow stress of copper saturates rapidly. As a consequence, instead of adopting a perfectly plastic response as proposed in [START_REF] Song | Analysis of T-peel strength in a Cu/Cr/Polyimide system[END_REF], we propose to adopt a Voce type law to mimic such saturation like behavior. The paper is organized as follows.

First the geometrical configuration and the material response are presented. In a second part, a semi analytical expression for the bending -reverse bending of metallic material with a Voce law is proposed. Therefore, the plastic dissipation can be derived. Closed form expression is provided in some specific cases.

Finally, the proposed expressions are compared to the [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] approach and validated based on finite element simulations of peeling.

Configuration

A two layer woven composite / Cu sample is considered in this theoretical work. This configuration is representative of the one often present in printed circuit board. More generally, a system with a thick elastic substrate and a thin ductile film can also be investigated through the proposed approach. As shown in Fig. 1 a), the metallic film thickness t is usually small compared to the one of the substrate ( t between 17.5µm and 70µm). [START_REF] Song | Analysis of T-peel strength in a Cu/Cr/Polyimide system[END_REF] have characterized electroplated Cu layer under uniaxial tension. The work hardening is very large at early deformation stage and presents a saturation like behavior for large plastic strain. Based on this observation, they analysed T-peel test by adopting the perfectly plastic analysis proposed by [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] and [START_REF] Kinloch | The peeling of flexible laminates[END_REF]. Instead of considering the material as an elastic perfectly plastic one, we propose to model its plastic response by adopting a Voce flow stress, [START_REF] Voce | The relationship between stress and strain for homogeneous deformations[END_REF]. In that case, the mechanical response is written as :

σ = Eǭ for σ ≤ σ o (1) σ = σ o + Q(1 -exp(-γ(ǭ -ǫ o ))) for σ > σ o (2)
where E is the Young modulus, σ the equivalent von Mises stress, ǭ the accumulated strain and ǫ o = σo E . The material is assumed incompressible. In the following of the paper, when plastic yielding is triggered, the flow stress (2) is written in a different way :

σ = α -β exp(-γǭ) (3) with α = σ o + Q and β = Q exp (γǫ o ).
With the present modelling, the initial yield stress is σ o and the saturation value of the yield stress is α, see Section 3 and Fig. 2 for an illustration of the adopted response. The present work can be viewed as an application of the bending approach proposed by [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF] for a specific material response presenting a saturation of the flow stress.

The moment curvature relationship during peeling test is analysed assuming that 2D plane strain prevails, following the work of [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF], [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF], [START_REF] Kinloch | The peeling of flexible laminates[END_REF] or more recently [START_REF] Wei | Modeling non linear peeling of ductile thin films -critical assessment of analytical bending models using FE simulations[END_REF]. The main equation of the analysis is recalled below with emphasis of the novelty in the moment curvature relationship due to the Voce law for plastic yielding.

During the steady state phase of the test, any material point will experience a bending / reverse bending path, see Fig. 1 b). So the stress and strain tensors are (neglecting shear components) :

σ =       σ 11 0 0 0 σ11 2 0 0 0 0       ε =       ε 11 0 0 0 0 0 0 0 -ε 11       (4)
In the elastic domain, path (OA), from the Hooke's law and owing to incompressibility, one has σ 11 = 4E 3 ǫ 11 while during plastic loading, |σ 11 | = 2 √ 3 σ. Pure bending is supposed to prevail so as in [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF], ǫ 11 = -Kz where K is the curvature of the middle surface referred to the coordinate z = 0 and z is varying in the range [-t 2 , t 2 ]. In Section 3, it will be shown that for a certain range of material parameters, even if the pure bending approach is not fully justified, the plastic dissipation during the peeling process is still accuratly estimated. This aspect was already mentioned in Kim and Kim (1988).

The bending moment per unit width of the metallic film is classicaly given by :

M = - t 2 -t 2 σ 11 zdz = -2 t 2 0 σ 11 zdz (5)
In the elastic phase of peeling (Path OA in Fig. 1 b), the moment curvature relationship is :

M = EKt 3 9 0 ≤ K ≤ K e (6)
with

K e = √ 3σo
Et . For a ductile thin metallic film, the maximum curvature in point B of Fig. 1 b), K B , is larger than K e so plasticity plays an important role and a precise calculation of the dissipation is needed to extract from the measurement of the peel force, the true (or objective as named in Kim and Kim (1988)) interface fracture energy Γ. The moment curvature relationship during plastic loading (path AB) is :

M = -2 h 0 σ 11 zdz -2 t 2 h σ 11 zdz (7)
where h = σo √ 3

2EK corresponds to the position of the elastic-plastic transition when the curvature K is between K e and K B . Note that during the plastic loading and due to 2D plane strain, the accumulated strain is ǭ = 2 √ 3 Kz. Owing to the behavior (2) of the film, the expression (7) of the moment becomes :

M = 8EKh 3 9 + 4 √ 3 t 2 h α -β exp(- 2γKz √ 3 ) zdz (8)
By integration by parts, the integral term on the right hand side of Eq. ( 8) can be evaluated since :

z2 z1 z exp(δz)dz = F (δ, z 2 ) -F (δ, z 1 ) (9)
with the function F defined as F (δ, z) = exp(δz)( z δ -1 δ 2 ). As a consequence, an explicit expression for Eq. ( 8) is found :

M = αt 2 2 √ 3 + σ 2 o E 2 K 2 ( σo √ 3 - √ 3α 2 ) - 4β √ 3 F (- 2γK √ 3 , t 2 ) -F (- 2γK √ 3 , h) (10)
From Eq. ( 2) when the material is perfectly plastic with Q = 0 ( i.e. α = σ o and β = 0), the relationship provided by [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] and Kim and Kim (1988) is recovered.

During the unloading phase, corresponding to path (BC) on Fig. 1 b),

(K B -K * B ≤ K ≤ K B )
, the moment is linked to the curvature, as in Kim and Kim (1988) :

M = M B + E 9 (K -K B )t 3 (11)
where M B = M (K = K B ) given in Eq. ( 10). In the following, we denote h B = σo √ 3

2EKB the position of the elastic plastic transition surface when the curvature is maximum, equal to K B . The unloading phase stops in C when reverse plasticity develops at the outer surface of the film z = t 2 . Let us denote

K C = K B -K * B ≤ K B the
curvature leading to this condition. Following [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF] or [START_REF] Kinloch | The peeling of flexible laminates[END_REF], the curvature K C is defined as follows :

2σ √ 3 = - 2σ √ 3 + 4E 3 (K B -K C ) t 2 (12)
where σ stands for the yield stress prevailing at z = t 2 at the end of plastic bending phase (for

K = K B ) : σ = α -β exp(-γKB t √ 3 ). As a consequence, K c = K B -K * B with K * B = 2 √ 3 Et (α -β exp(-γKB t √ 3
)) For the peeling test, the reverse plastic loading, path (CD) on 12), an implicit relationship

for h ′ is found α -β exp(- 2γK B h ′ √ 3 ) = E √ 3 (K B -K)h ′ (13)
In the theory as discussed by [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF], the reverse plastic loading can continue for negative curvature. During peeling test, according to Fig. 1 b), such possibility is not considered here. For a given curvature The peel angle is Φ and the force per unit width is P . The traction separation law is also presented where σ represents the peak stress and Γ the interface fracture energy. A local coordinate system is adopted so that the position z = 0 corresponds to the middle plane of the film. During peeling, the film is bended. Let us note R the radius of curvature R of the top surface z = t 2 . b) Schematic normalized moment versus normalized curvature under plane strain. Letters correspond to specific loading phase during the peeling test. cumulated some plasticity during the bending stage (AB). Finally, in the third part h ′ ≤ z ≤ t 2 the material point cumulates reverse plasticity. For the latter part of the film, the accumulated strain is, as given in [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF] and [START_REF] Kinloch | The peeling of flexible laminates[END_REF] :

K B -K * B ≥ K ≥ 0, the film thickness is divided in three parts. For 0 ≤ z ≤ h B , the material is elastic. For h B ≤ z ≤ h ′ ,
ǭ = 2 √ 3 (2K B -K)z - √ 3 E α -β exp(- 2γK B z √ 3 ) (14)
By considering the three above strain histories, the moment can be evaluated in a semi analytical manner:

M = 8EKBh 3 B 9 - 8E 9 (KB -K)(h ′ ) 3 + 2α √ 3 (2(h ′ ) 2 -h 2 B - t 2 4 ) - 4β √ 3 F (- 2γKB √ 3 , h ′ ) -F (- 2γKB √ 3 , hB) + 4β √ 3 t 2 h ′ z exp(-γǭ)dz (15)
with h ′ defined in Eq. ( 13) and ǭ in Eq. ( 14). Note that when the curvature K B and the parameter γ are large and the ratio σo+Q E is small (situation which is observed in peeling of copper film on an elastic substrate), the accumulated strain can be approximated by :

ǭ = 2 √ 3 (2K B -K)z (16)
In that case, a closed form expression for M defined in Eq. ( 15) is available.

Indeed the integral term of the right hand side of Eq. ( 15) becomes :

t 2 h ′ z exp(-γǭ)dz = F (- 2γ(2KB -K) √ 3 , t 2 ) -F (- 2γ(2KB -K) √ 3 , h ′ ) (17)
The domain of validity of this approximate solution will be discussed in Section 3.

As proposed by [START_REF] Kendall | Shrinkage and peel strength of adhesive joints[END_REF], [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] or [START_REF] Wei | Interface strength, work of adhesion and plasticity in peel test[END_REF] among others of the literature, the energy balance provides a link between the peeling force per unit width P (see Fig. 1 a), the interface fracture energy Γ and the plastic dissipation Ψ :

P (1 -cos Φ) = Γ + Ψ ⇒ Γ = P (1 -cos Φ) -Ψ (18)
Eq. ( 18) is valid assuming that the elastic stored energy of the system and other contributions are negligeable when compared to the contribution of plasticity during the bending, see for example Kim and Kim (1988) for further discussions.

As mentioned by [START_REF] Kim | Elastoplastic analysis of the peel test[END_REF], the plastic dissipation Ψ (or the work expenditure per unit film width) per unit crack advance is Ψ = L M dK where L is the loading path (OABCD) in the moment curvature Fig. 1 b) :

Ψ = Ke o M dK + K B Ke M dK + K B -K * B K B M dK + 0 K B -K * B M dK (19)
Interestingly, the moment curvature relationship is known for the whole path (OABCD). Owing to the explicit expression on the path (OABC), the first three terms on the right hand side of Eq. ( 19) can be evaluated analytically. More details can be found in Appendix A. Finally, we obtained :

Ψ = σ 2 o t 6E + αt 2 2 √ 3 (KB -Ke) + ( σo √ 3 - √ 3α 2 ) σ 2 o E 2 ( 1 Ke - 1 KB ) - √ 3β γ 2 G - γt √ 3 , KB -G - γt √ 3 , Ke + √ 3β γ exp(- γσo E ) ( σo E + 1 γ )( 1 KB - 1 Ke ) -MBK * B + Et 3 18 (K * B ) 2 + 8EKBh 3 B 9 - 2α √ 3 (h 2 B + t 2 4 ) + 4β √ 3 F (- 2γKB √ 3 , hB) (K * B -KB) + I (20)
where the function G(x, y) is defined as G(x, y) = exp(xy) y and I given by :

I = 0 K B -K * B 8E 9 (K -KB)(h ′ ) 3 + 4α √ 3 (h ′ ) 2 - 4β √ 3 F (- 2γKB √ 3 , h ′ ) + 4β √ 3 t 2 h ′ z exp(-γǭ)dz dK (21)
The expression ( 20) is semi-analytical. Based on the present derivation, the plastic dissipation has almost a closed form expression except for the last term.

Indeed, h ′ , defined by the implicit relationship (13), is a function of the current curvature K. Nevertheless, when the radius of curvature is small (K B large)

and the parameter γ has an important value, then the accumulated strain ( 14) can be simplified, see Eq. ( 16). In addition, from Eq. ( 13), we propose to

approximate h ′ by h ′ = √ 3α E(KB -K) .
Based on these two assumptions, the last integral term I can be evaluated in closed form :

I approx = 4α 3 √ 3E 2 ( 1 KB - 1 K * B ) - √ 3β γ 2 G(- γt √ 3 , 2KB) -G(- γt √ 3 , KB + K * B ) + √ 3β KBγ 2 G(-2 γα E , KB K * B ) -G(-2 γα E , 1) + G(-2 γα E , KB + K * B K * B ) -G(-2 γα E , 2) (22)
The present approach leading to the dissipation expression (20) contains by construction the perfectly plastic case. Indeed, when the parameter Q of Eq.

( 2) is set to zero, then obviously the material is elastic perfectly plastic. In that case, since α = σ o , one obtains

K * B = 2K e , h B = σo √ 3 2EKB .
It is easily shown that the expression of Ψ is identical to the expression of the dissipation provided by [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] or Kim and Kim (1988) for a perfectly plastic case :

Ψ P P = M o K e (2 K B K e -5 + 10 3 K e K B ) when K B ≥ 2K e (23)
where

M o = 1 2 √ 3 σ o t 2
is the fully plastic moment. Note also that when γ is becoming large, then the saturation occurs for low strain and therefore, it can be argued that the material is almost perfectly plastic with the yield stress being equal to the saturation stress Q + σ o (this assumption was adopted by [START_REF] Song | Analysis of T-peel strength in a Cu/Cr/Polyimide system[END_REF]). This case will be discussed in Section 3. One can also mention that the extension of the present approach to a constitutive behavior containing a finite number of Voce law :

σ = σ o + N i=1 Q i (1 -exp(-γ i (ǭ -ǫ o ))) ( 24
)
is straigthforward. This can be done by imposing α = σ o + N i=1 Q i , and replacing β by β i = Q i exp(γ i ǫ o ) and γ by γ i . For sake of brevity, the corresponding expressions with the summation are not reproduced here.

Results

The first part of this section is devoted to the evaluation of the moment curvature response of a thin film under bending / reverse bending. The difference between the present development based on a Voce flow stress with the one of Kim and Kim (1988) for elastic perfectly plastic response is enlighted. In a second part, the interface fracture energy estimated based on the plastic dissipation during the bending /reverse bending is compared to the one imposed in the finite element simulation of the peeling test at angle Φ = 90. Finally, the effect of the parameters of cohesive law on the peel force and curvature is investigated for peel tests at various angles Φ in the range [45; 135].

Moment curvature response and plastic dissipation

The effect of the parameter γ of Eq. ( 2) on the elastic-plastic response of the metallic film is first investigated in Fig. 2. The reference parameters for the elastic plastic behavior of the film are listed in Table 1. The parameter γ influences the deformation level for which the flow stress approaches the plateau σ o + Q. Indeed for a low value of γ (here γ = 5), a large strain is needed to reach the saturation regime ( around ǭ = 1), while for a large value of γ (here γ = 160), a small strain is suffisant for the yield stress to reach the plateau (around ǭ = 0.03). The initial plastic hardening modulus ∂ σ ∂ǭ (ǭ = ǫ o ) is highly affected by the value of γ. For a metal, it must be lower than the elastic Young's modulus, leading to the condition for γ : γ ≤ E Q . With the present material parameters of Table 1, one obtains : γ lim = 361. 2) is considered. Depending on the value of γ, the saturation state is approached for different accumulated strains.

ε γ =5 γ =10 γ =40 γ =160
It is also interesting to observe the difference between the moment curvature evolution predicted by adopting a Voce law and by assuming that the material is perfectly plastic. For illustration, in Fig. 3, a curvature for the metallic film of K B = 4000m -1 is adopted, leading to a radius of 0.25mm. The film thickness is t = 35µm. Two perfectly plastic cases are considered. The first one assumes that the yield stress of the material is σ = σ o while in the second case σ = σ o +Q. As expected, the response based on the Voce law is located between the two curves corresponding to perfectly plastic materials. Since the plastic dissipation is the area formed by the curve along the path (OABCD), it is clear that assuming an elastic perfectly plastic behavior for the material is only valid for very large values of γ. It is interesting to notice that due to strain hardening, the moment curvature response during the reverse bending (path CD of It is observed that the plastic dissipation Ψ is increasing rapidly for small values of γ. As γ becomes larger, the response of the material becomes close to an elastic perfectly plastic response, see Fig. 2. As a consequence, the plastic dissipation saturates, reaching the value Ψ P P , predicted by the elastic perfectly plastic approach of [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF]; Kim and Kim (1988). Theses trends are confirmed for two values of Q : Q = 90 or 180M P a.

For material parameters of Table 1 and two values of γ : 40 and 160, the dissipation evaluated based on Eqs. ( 20) and ( 21) is presented in Fig. 4 a) for a curvature K in the range 1000m -1 to 10000m -1 leading to a radius of the film varying from 100µm to 1mm. As expected, the plastic dissipation Ψ is larger when the normalized curvature is increased. For large value of the normalized curvature, the response is seen to be almost linear for both values of γ. This could have been anticipated. Indeed, Fig. 3 a) has shown that for large γ, the moment versus curvature response is quite similar to the one estimated based on a perfectly plastic approach. Therefore, in that case, the plastic dissipation a) for a Voce or a perfectly plastic material (with σ = σ o + Q) must be quite similar. From Eq. ( 23), the increment of plastic dissipation ∆Ψ produced by an increment of the normalized curvature ∆K Ke is expressed as follows, when K Ke is large :

∆Ψ = σ o (σ o + Q)t E ∆K K e (25)
With the parameters of Table 1, we observed that ∆Ψ ≈ 15 ∆K Ke . This is consistent with the observation of Fig. 4 a). In addition, for the two values γ = 40 and γ = 160, we have observed that the approximate dissipation based on Eq. ( 22) almost coincides with the one obtained with Eq. (20) (results not presented here).

Next, a more precise comparison between the predicted dissipation ψ evaluated based on the Voce law and based on a perfectly plastic response ψ P P (with σ = σ o + Q) is presented in Fig. 4 b). For the value γ = 40, the normalized ratio ψ ψ P P is still below 0.9 when the normalized curvature K Ke is equal to 88 (K = 6676m -1 or a radius of 149µm). Adopting a perfectly plastic aproximation for this material when analyzing results of peeling test will lead to a 10% difference in the interface fracture energy Γ, see the energy balance equation ( 18) for a test at Φ = 90 o . When γ = 160, we clearly observed that for K Ke = 88, less than 3% difference remains. Note that Fig. 3 was constructed with a value of the curvature of K B = 4000m -1 ( KB Ke = 52). In that case, a difference of 15% exists between Ψ based on expression (20) and Ψ P P for γ = 40 while for γ = 160, the difference reduces to 4%.

Finite element validation based on peeling test

In this section, we propose to analyze in more details the peeling test on a two layer woven composite /Cu sample. The plastic behavior of copper is described by Eq. ( 2). The substrate, representative of a woven composite used in printed circuit board industry remains elastic. The elastic behavior has been identified based on the strategy proposed in [START_REF] Girard | Experimental and numerical characterization of thin woven composites used in printed circuit boards for high frequency applications[END_REF]. The in plane Young modulus is around 20.6 GPa, while the out of plane one is 16.4 GPa. The in plane Poisson ratio is 0.17 1.

A Finite Element model of peeling test under plane strain conditions for various angles Φ has been considered on ABAQUS Software, see Fig. 1 a). The interface is modeled with cohesive elements : a bilinear separation law in mixed mode is considered. For limited separation of the interface, the response is linear σ n = K nn δ n and σ t = K tt δ t , with K tt = K nn = 10 6 N/mm 3 the initial stiffness of the cohesive element. σ n , σ t are the normal and tangential component of the traction force. δ n and δ t are the corresponding separation displacement at the level of the cohesive element. The following quadratic failure initiation and linear failure criterions are adopted :

σ n σn,c 2 + σ t σt,c 2 = 1 G n G n,c + G t G t,c = 1 (26)
with G n = σ n dδ n and G t = σ t dδ t . Such cohesive law is implemented in the ABAQUS software, see ABAQUS (2013). More information can be found elsewhere. Note that other choices for the cohesive law could have been adopted.

Discussion about mixed mode ratio can be found in [START_REF] Tvergaard | The influence of plasticity on mixed mode interface toughness[END_REF], [START_REF] Kinloch | The peeling of flexible laminates[END_REF], [START_REF] Wei | Modeling non linear peeling of ductile thin films -critical assessment of analytical bending models using FE simulations[END_REF] and [START_REF] Martiny | Numerical analysis of the energy contributions in peel tests : a steady-state multilevel finite element approach[END_REF]. In our The goal of the finite simulations is to provide values of the peel force and the curvature for a given cohesive law and a given material behavior for the thin film. Knowing the curvature, the plastic dissipation can be evaluated by Eq.

(20). Subsequently, the interface fracture energy Γ is derived from the energy balance (18). The predicted value for Γ is compared to the one adopted in the FE simulation. The reference parameters for the cohesive law and for the elastic plastic behavior of the film are listed in Table 1.

Plane strain elements (CPE4R) are adopted with 9 elements in the film thickness for Φ < 90. / reverse bending process. The cohesive elements are removed for presentation purpose. The longitudinal strain E 11 is displayed. A local orientation is adopted so that the direction 1 is always parallel to the middle surface, as presented in Fig. 1. presented in the area where the thin film is detached from the substrate. It is observed in this figure, that in the case of a peel test with an angle of Φ = 90 o , the film is curved with a radius R, which is assumed to be representative of the curvature at point B of Fig. 1 b). To be consistent with the bending model where the curvature is measured from the middle surface of the film, the curvature K B is related to R by K B = (R + t

2 ) -1 . Fig. 6 presents the corresponding evolution of the curvature versus γ for the two different values of Q. The evolution of the curvature versus γ is not monotonic. For moderature values of γ, K B is first decreasing. Then K B reaches a minimum value and then subsequently increases slightly as γ becomes larger. Note also that depending on the respective value of Q which is scaling the saturation of the flow stress, the minimum of the curve is reached for larger γ value as Q is larger. The non monotonic trend can be understood as follows. On the one hand, from Fig. 5, it is seen that when γ is increasing from 5 to 40, the level of the force is slightly increased. From the balance of energy, the larger force level is linked to an increase in plastic dissipation. Nevertheless, in Fig. 3 b), it has been shown that, when γ is increasing from 5 to 40, the plastic dissipation is strongly enhanced, for a fixed curvature. As a consequence, to preserve the balance of energy and the slight increase of the force, the curvature is reduced so as to propose a moderate increase in plastic dissipation. On the other hand, when γ is large, the plastic dissipation saturates when assuming that the curvature remains constant, see Fig. 3 b). Therefore, the larger peeling force observed for γ=160 when compared to γ = 40 originates from the increase of the curvature, which produces larger dissipation.

The prediction of the plastic dissipation can be obtained via Eq. ( 20) based on the curvature K B captured by the finite element calculation (Fig. 6). In addition, from the numerical measurement of the peeling force (Fig. 5), and from Eq. ( 18), the interface fracture energy is estimated. The result of the dialog between the present model and the finite element calculation is illustrated in Table 2 where are presented the predicted values for Γ adopting the dissipation 2) on the curvature K B of the film during a peeling test at Φ = 90. The ductile film behavior is elastic plastic with a flow stress described by a Voce equation ( 2). Material parameters are listed in Table 1.

estimate provided by the current approach Eq. ( 20), the approximate evaluation of the plastic dissipation based on Eq. ( 22) and the perfectly plastic dissipation Eq. ( 23). The obtained values are compared to Γ = 1055N/m, which is the prescribed value in the finite element simulations. All the results concern a peel test at an angle Φ = 90. Our present approach is able to predict the Γ value with less than 1.5% in the whole range of γ for the two considered cases Q = 90M P a and Q = 180M P a, see Table 2. As expected, adopting the formula (23) proposed by [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] is only valid when γ is larger than 40. In that case, the stress strain response in tension is quite similar to one expected when the material is elastic perfectly plastic, see Fig. 2. Note also that estimation of the interface fracture energy based on the approximate solution Eq. ( 22) is also accurate for the values of the γ parameter considered in the present paper. This observation confirms a previous finding where very limited difference was observed between predictions of the interface fracture energy based on Eqs (20) and ( 22) for results of Fig. 4.

Table 2: Predicted interface fracture energy. From the force and curvature measured owing to finite element simulations (here Γ = 1055N/m, Φ = 90), the plastic dissipation is evaluated based on the present approach (20), on the approximate solution ( 22) and on the perfectly approach of [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF], see Eq. ( 23). From the energy conservation (18), the interface fracture energy is estimated. During the unloading stage (BC), the strain is slightly decreasing while in the reverse bending phase (CD), the strain magnitude is strongly reduced.

Nevertheless, due to plasticity development and presence of the peeling force, a residual strain still exists in the peel arm. The residual strain is homogeneous in the thickness, meaning that the curvature is close to K D = 0 (straight arm at the end). It is also interesting to mention that for the condition presented here (Φ = 90), the strain profile in the thickness is almost skew symmetric. This is in close agreement with the bending theory of the beam which is based on the assumption that any material point entering into the process zone will experience pure bending /reverse bending E 11 = -Kz. Fig. 7 b) presents the strain development inside the film thickness at the exact end of the bending process, condition B. Since K B can be evaluated from the finite simulations, the theoretical predictions of the strain E 11 = -K B z is also added to the FE strains. We observe that for a peel angle of Φ = 90 where K B = 3759m -1 and R = 248µm, it exists a very good agreement between the theory and the FE results. A similar comparison is made for two other angles : Φ = 60 and Φ = 135. For Φ = 135, it is seen that the strain distribution is non linear, with a larger absolute value of the strain at the bottom surface when compared to the top surface. In that configuration K B = 6468m -1 corresponding to a radius of R = 137µm. It will be shown next that the limited correlation of the present theory in reproducing the expected value of the interface fracture energy is partially due to the observed small radius. This conclusion was already drawn in [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF]. Indeed, the plane strain bending approach is considered by [START_REF] Hill | The mathematical theory of plasticity[END_REF] as a good approximation assuming that the minimum radius is larger than four to five times the film thickness. With t = 35µm and Φ = 135, R is smaller than 4t. For Φ = 60, K B = 2434m -1 and R = 393µm, the strain response is affine. The main difference comes from the fact that the condition E 11 = 0 is observed for z = 5µm while in pure bending as in the proposed theory, this condition is fulfilled for z = 0. Clearly, for Φ < 90, instead of pure bending, a more complex loading path prevail with bending and extension. 1 with γ = 40.

It is observed that for the material points located close to the external surface of the film, the strain cumulated up to B is large enough so that the flow stress has already reached its saturation value σ o + Q. Therefore during the reverse bending process (CD), the stress level remains almost constant. In that stage, the material behavior can be approximated by an elastic perfectly plastic response. For Φ = 135 (see Fig. 8 b)), the same conclusion is made except for the maximum strain. Indeed, we observed that the maximum longitudinal strain at the top surface is 20% larger in absolute value than the strain experienced at the bottom surface. From the knowledge of the curvature K B , the theoretical prediction of the stress σ 11 versus strain E 11 response is also added to Figs 8

a) and b) for two points located at z = ±16.6µm (see dotted lines with squares or circles). It is clearly observed that some discrepancy is shown for the theory when compared to the finite element response. Of course, in the theory, a skew symmetric distribution of strain is assumed, which is not clearly the case in the Finite Element simulations. The bending assumption is better fulfilled for Φ = 90 than for Φ = 135.

At least for the parameters adopted as reference in Table 1, the proposed expression Eq. ( 20) for the plastic dissipation enables to recover precisely the interface fracture energy inserted in the numerical calculations when the peel angle is Φ = 90, as presented in Table 2. Peeling tests with an angle in the range [+45, +135] were also simulated by FE. The reference parameters of Table 1 are still adopted with γ = 40. Fig. 9 displays the estimated value of the fracture energy Γ assuming that the force per unit width P and the maximum curvature K B are known quantities (obtained from FE simulations). As already mentioned, the pure bending theory predicts a consistent interface fracture energy using Eqs ( 18) and (20). It is observed that for Φ = 120, 135, the estimate for Γ is 1160N/m, instead of 1055N/m. Clearly, the film radius is too small when compared to the film thickness for the bending theory to be accurate, see [START_REF] Hill | The mathematical theory of plasticity[END_REF]. For Φ = 45, 60, the strain map in the film thickness is neither fully consistent with a pure bending approach, see Fig. 7 b). Indeed, for Φ < 90, the stretching of the film is more intense, and the tensile contribution is noticeable.

It is observed that while for Φ = 60, the estimation of Γ is still accurate (less a)

- given by Eq. ( 20). The prediction of Γ based on the perfectly plastic approach Eq. ( 23) is also superimposed to Fig. 9. The perfectly approach is underestimating Γ for the considered material parameters, except for the large angle condition with Φ = 135 or Φ = 120. So from Fig. 9, it can be mentioned that the estimation of the interface fracture energy can be reasonably captured by the present work, from the knowledge of the experimental force and curvature for tests with peel angle around Φ = 90. Interestingly, it is seen that the closed form expression derived from Eq. ( 22) provides also consistent estimate and can be used instead of Eq. ( 20). 20) and on the equation ( 23) derived for perfectly plastic material are compared. The material parameters are listed in Table 1 with γ = 40.

From previous works of the literature [START_REF] Wei | Interface strength, work of adhesion and plasticity in peel test[END_REF] , Kim and Kim (1988), [START_REF] Wei | Modeling non linear peeling of ductile thin films -critical assessment of analytical bending models using FE simulations[END_REF]), the results of finite element simulation of peeling tests in presence of plasticity are highly influenced by the properties of the interface, i. e. the parameters of the cohesive law ( the interface fracture energy Γ and the peak stress σ). Next, we propose to investigate their relative influence on the force and curvature during a peel test. The material behavior is still described by a Voce law with parameters listed in Table 1 and γ = 40. We assume that for a given angle Φ = 90, the force P is known. For different values of Γ, the peak stress σ is adjusted so that the force P at Φ = 90 is reproduced, see [START_REF] Martiny | Numerical analysis of the energy contributions in peel tests : a steady-state multilevel finite element approach[END_REF]. In Fig. 10, the evolution of the peeling force and the curvature K B are displayed for 6 sets of parameters (Γ, σ) defined in Table 3.

Each of the set leads to the same force per unit width P for Φ = 90. As Γ is reduced, the peak stress has larger value leading to a larger curvature (small radius R) so that the plastic dissipation is enlarged. For Γ = 1055N/m and σ = 10M P a, finite element simulation of the peel test at Φ = 90 has provided a curvature of K B = 3759m -1 ( a radius R = 248µm) while for Γ = 527.5N/m and σ = 72M P a, the curvature is K B = 6042m -1 corresponding to a radius of R = 148µm. After this calibration, the force P is evaluated for four other angles Φ = 45, 60, 120 and 135. Fig. 10 shows that the force is quite insensitive to the set (Γ, σ) for Φ > 90. On the contrary, there is an important influence on the force prediction for angles lower than 90. This effect is larger for Φ = 45 than for Φ = 60. For instance, with Φ = 45, one has P = 3777N/m for Γ = 1055N/m and only P = 3300N/m when the interface fracture energy is reduced by a factor of two Γ = 527.5N/m. Remember that σ is adjusted so as to preserve the force level at Φ = 90 degree. Fig. 10 enligthens also that the curvature is the most sensitive feature to any variation of the set (Γ, σ). Therefore, to identify from peeling tests the interface fracture energy, one can act as follows. A peeling test carried out at Φ = 90 where the force and curvature are recorded may be sufficient to provide a consistent value for Γ. Nevertheless, it seems important also to validate this estimate by confirming the predictions of the force for another test; a peeling test at angle Φ = 45 seems to be a good candidate.

Indeed, the force P for low angle Φ is sensitive to the interface fracture energy.

In that case, a finite element simulation of the considered configuration must be carried out. Of course, all the present results are valid for the specific cohesive law (26) and for the given material behavior. 3 are adjusted so that the peeling force P (Φ = 90) is preserved for all angles.

Conclusion

The estimation of the plastic dissipation during bending / reverse bending for a Voce type behavior is the main goal of the present work. A semi-analytical expression is provided for the plastic dissipation. As expected, the plastic dissipation is highly sensitive to the curvature. As proposed previously in the literature for other material constitutive laws, the knowledge of the dissipation enables to evaluate the interface fracture energy from the peeling force measurement. By comparison with finite element calculations performed for various angles in the range [45,135], it is found that the present theory is accurate to analyse peeling test for Φ = 90 and provide a satisfactory estimate for other peel angles. For Φ = 90, at least for the material parameters adopted in the present paper, the assumption of pure bending in the process zone is a quite satisfactory hypothesis. It is also observed that depending on the material parameters, the perfectly plastic approach proposed by Kim and Kim (1988), [START_REF] Aravas | On the mechanics of adhesion testing of flexible films[END_REF] can also be justified even for a Voce type material when the saturation of the flow stress is observed for limited strain.

The cohesive law parameters ( interface fracture energy and peak stress) can be estimated by an inverse method, assuming that the force and curvature have been identified with recourse to finite element simulations or experiments.

The identification of the cohesive law parameters could benefit from a further validation based on peeling tests carried out mostly for low peel angle Φ < 90. obtains from Eq. ( 6):

Ke 0 M dK = σ 2 o t 6E (27)
Since in the peeling test of thin copper film, the curvature K B is large compared

to K e , one needs to evaluate the contribution to Ψ along (AB). The relationship between the moment M and the curvature K is given in that case by Eq. ( 10).

With the definition F (-2γK √ 3 , t 2 ) = -exp(-γKt √ 3 ) t √ 3 4γK + 3 4γ 2 K 2 , and with a change of variable u = γKt √ 3 , one easily gets :

KB Ke F (- 2γK √ 3 , t 2 )dK = 3 4γ 2 G - γt √ 3 , K B -G - γt √ 3 , K e (28)
where G(x, y) = exp(xy) y . The contribution linked to the term F (-2γK √ 3 , h) with h = σo √ 3

2EK is evaluated in a straightforward manner, so the plastic dissipation along the path (AB) is :

KB Ke = αt 2 2 √ 3 (K B -K e ) + ( σ o √ 3 - √ 3α 2 ) σ 2 o E 2 ( 1 K e - 1 K B ) - √ 3β γ 2 G - γt √ 3 , K B -G - γt √ 3 , K e + √ 3β γ exp(- γσ o E )( σ o E + 1 γ )( 1 K B - 1 K e ) (29)
For the unloading path (BC), the contribution to Ψ is readily :

KB -K * B KB M dK = -M B K * B + Et 3 18 (K * B ) 2 (30)
The last term corresponding to the reverse plastic loading phase (CD) is the most complex contribution and can not be estimated in closed form expression in the general case. Indeed, the expression of M is given by Eq. ( 15) where an integral term is already present. In addition, it is related to the position of the elastic-plastic transition h ′ defined by an implicit relation (13

  Fig. 1 b), exists, since the free arm is pulled into straight plane with a final curvature of K D = 0. Let us denote h ′ (K) the position of the elastic-plastic transition during the reverse plastic loading. By construction, h ′ is larger than h B , meaning that a material point with z > h ′ has already faced plasticity during the bending phase up to the curvature K B . Similarly to Eq. (

Figure 1

 1 Figure1: a) Peeling test configuration. The film thickness is t. The peel angle is Φ and the force per unit width is P . The traction separation law is also presented where σ represents the peak stress and Γ the interface fracture energy. A local coordinate system is adopted so that the position z = 0 corresponds to the middle plane of the film. During peeling, the film is bended. Let us note R the radius of curvature R of the top surface z = t 2 . b) Schematic normalized moment versus normalized curvature under plane strain. Letters correspond to specific loading phase during the peeling test.

Figure 2 :

 2 Figure 2: Elastic-plastic behavior of the metallic film. A Voce type behavior Eq. (2) is considered. Depending on the value of γ, the saturation state is approached for different accumulated strains.

  Fig. 1 b) for γ = 160 corresponds closely to the response of a perfectly plastic material with σ = σ o + Q. Nevertheless, during the bending (path AB of Fig. 1 b), some difference exists which affects the total dissipation. On the contrary, for γ = 40, the plastic dissipation during the bending phase is strongly overestimated by assuming a perfectly plastic response. with σ = σ o + Q. Fig. 3 b) displays the evolution of the plastic dissipation Ψ as γ is enlarged, keeping K B = 4000m -1 .

Figure 3

 3 Figure 3: a) Estimated moment curvature relationship for an elastic perfectly plastic material and a material with a Voce yield stress. Effect of the parameter γ controling the deformation for which the flow stress saturates, see Eq. (2). b) Plastic dissipation as a function of the parameter γ for two values of Q = 90M P a and Q = 180M P a.

Figure 4

 4 Figure 4: a) Plastic dissipation versus normalized curvature K/K e during bending and reverse bending for two values of γ : γ = 40 and γ = 160. b) Ratio between the plastic dissipation evaluated based on a Voce flow stress and a perfectly plastic response. Material parameters for the Voce law are given in Table1.

  FE simulations, as in the literature on that subject, only two parameters are kept for the definition of the cohesive law by imposing : σn,c = σt,c = σ and G n,c = G t,c = G c . σ represents the peak stress and G c the fracture energy, which corresponds here to the interface fracture energy Γ, see Fig. 1 a).

Figure 5

 5 Figure5: a) Force per unit width versus time. Different values of material parameters Q and γ are tested. b) View of the film curvature during the bending / reverse bending process. The cohesive elements are removed for presentation purpose. The longitudinal strain E 11 is displayed. A local orientation is adopted so that the direction 1 is always parallel to the middle surface, as presented in Fig.1.

Figure 6 :

 6 Figure 6: Effect of the γ parameter of Eq. (2) on the curvature K B of the film during a peeling test at Φ = 90. The ductile film behavior is elastic plastic with a flow stress described by a Voce equation (2). Material parameters are listed in Table1.

  Fig.7a) presents the time evolution of the strain component E 11 for three points located on the top of the film (z = 16.6, 14.6 and 12.6µm) and 3 on the bottom of the film (z = -16.6, -14.6 and -12.6µm). As expected, the points located at the bottom (resp. top) of the film z < 0 (resp. z > 0) experience positive (resp. negative) value for the strain component E 11 . The maximum strain is reached when the curvature is maximum, loading condition B of Fig.1 b). During the unloading stage (BC), the strain is slightly decreasing while

Fig. 8 =Figure 7

 87 Fig.8presents the stress-strain loading path faced by 6 material points inside the thickness during the peeling process for two angles : Φ = 90 and Φ = 135. The loading stages (OABCD) are also added in Fig.8 a). The material parameters are listed in Table1with γ = 40. In Fig.8 a), for an angle Φ = 90, we clearly observed that the assumption of pure bending / reserve bending is mostly satisfied. During the loading path (OAB), the trajectory of all points in the stress-strain plane is identical up to a maximum strain E max 11 = -K B z.

Figure 8 :

 8 Figure 8: Loading path faced by material points along the thickness during the peeling process. a) Peel angle of 90 o b) Peel angle of 135 0 . A local orientation is adopted in the Finite element simulations. The reference parameters are those of Table1with γ = 40.

Figure 9 :

 9 Figure 9: Effect of the angle on the estimated interface fracture energy. The predictions based on the present work Eq. (20) and on the equation (23) derived for perfectly plastic material are compared. The material parameters are listed in Table1with γ = 40.

Table 3 :

 3 Couples (Γ, σ) leading to the same force per unit width for the peel angle Φ

Figure 10 :

 10 Figure10: Force per unit width and curvature for various peeling angles Φ. Note that the cohesive law parameters listed in Table3are adjusted so that the peeling force P (Φ = 90) is preserved for all angles.

Table 1 :

 1 Reference parameters for the cohesive law and the elastic plastic behavior of the metallic film

	t	E	ν	σo	Q		γ	Gc	σ
	µm GP a		M P a M P a			N/m M P a
	35	65	0.26	100	180	5-160 1055	10
		300
		250
	-σ, MPa	200
	Yield stress	100 150
		50
		0
		0	0.2	0.4		0.6		0.8	1
				Accumulated strain	-

  ). Therefore one Zhao H, Wei Y (2007) Determination of interface properties between micronthick metal film and ceramic substrate using peel test. International Journal of Fracture 144:103-112

Acknowledgements

The authors acknowledge the support of Agence Nationale de Recherche through the program Labcom LEMCI ANR-14-LAB7-0003-01.

The research leading to these results has received funding from the European Union's Horizon 2020 research and innovation programme (Excellent Science, Marie Sklodowska-Curie Actions) under REA grant agreement 675602 (OUT-

the predictions in terms of peel force and curvature are preserved. The cohesive element size in the 1 direction is five times lower than the element size in the film. The height of the cohesive element is vanishing small. In all calculations, the velocity for the peeled arm is V = 0.5mm/s. To ensure that the peel angle Φ remains constant during the FE simulation, the substrate is moving in the horizontal direction with the same velocity V = 0.5mm/s. Note that the length of the substrate is 50 mm long. It has been checked that a stationary process prevails after a quite short time period.

First we present a comparison of the model capability based on a peeling test with Φ = 90. This working condition is the most often used in the literature and for practical applications. For instance, a peeling test at Φ = 90 is recommended for the printed circuit board industry to evaluated the interface strength, see the IPC-TM-650 2.4.8 standard. The force versus time is presented in Fig. 5 for three values of the γ parameter : 5, 40 and 160 and for two values of Q : 90 and 180 MPa. A stationary process for the peeling force is established after 3s. We clearly observed that for the same interface fracture energy Γ = 1055N/m(orJ/m 2 ), the force is larger when γ increases. Indeed, for Q = 180M P a, the level of the force is evolving from 1531N/m with γ = 5 to 1707N/m for γ = 160. As Q is larger, meaning that the plateau value for the yield stress is larger, the peeling force is also enhanced significantly for large values of γ. Remember that as the parameter γ is large, the saturation stress is reached for lower strain, see Fig. 2. From the energy balance (18), and since Γ is fixed, this enhancement of the force means that the plastic dissipation is larger. A key factor controling the plastic dissipation is the maximum curvature K B , existing at the end of the loading path (AB).

During the stationary process in the FE simulations, the radius of curvature R of the top surface z = t 2 can be measured, see Fig. 5 b) where a zoom is COME project).

A Derivation of the expression for the plastic dissipation

The present appendix provides some details concerning the derivation for the expression of the plastic dissipation Ψ along the loading path (OABCD). For the elastic response of the film along path (OA), as depicted in Fig. 1 b), one can simply noticed that :

where the term I is provided in Eq. ( 21). As mentioned in Section 2, for metallic film when K B and γ have large values, the accumulated strain (14) (resp. h ′ ) can be approximated as proposed in Eq. ( 16) (resp. by h ′ =

√ 3α E(KB -K) ). After some classical mathematical development, the integral term I is evaluated and the approximate closed form expression ( 22) is found.


Meghyn Bienvenu

Pierre Bourhis

Mixed-World Reasoning with Existential Rules under Active-Domain Semantics

In this paper, we study reasoning with existential rules in a setting where some of the predicates may be closed (i.e., their content is fully specified by the data instance) and the remaining open predicates are interpreted under active-domain semantics. We show, unsurprisingly, that the main reasoning tasks (satisfiability and certainty / possibility of Boolean queries) are all intractable in data complexity in the general case. However, several positive (PTIME data) results are obtained for the linear fragment, and interestingly, these tractability results hold also for various extensions, e.g., with negated closed atoms and disjunctive rule heads. This motivates us to take a closer look at the linear fragment, exploring its expressivity and defining a fixpoint extension to approximate non-linear rules.

Introduction

There has been significant interest in recent years in ontologymediated query answering (OMQA) [START_REF] Bienvenu | Meghyn Bienvenu and Magdalena Ortiz. Ontology-mediated query answering with data-tractable description logics[END_REF], in which data is enriched with an ontology that provides general domain knowledge. Ontologies are typically expressed using decidable fragments of first-order logic. While description logics (DLs) [START_REF] Baader | [END_REF] remain the most widely-studied formalism, existential rules (aka Datalog +/-) [START_REF] Baget | [END_REF][START_REF] Baget | [END_REF][START_REF] Krötzsch | Extending decidable existential rules by joining acyclicity and guardedness[END_REF][START_REF] Calì | [END_REF], which we adopt in this paper, have been drawing increasing interest, as they allow relations of arbitrary arity (unlike DLs which are restricted to unary and binary predicates) and generalize both Datalog [Abiteboul et al., 1995] and tractable DLs like DL-Lite and EL [Calvanese et al., 2007;[START_REF] Baader | [END_REF].

The vast majority of work on OMQA adopts the open world assumption, whereby facts that are not present in the data instance are treated as unknown. Formally, each knowledge base (KB) (R, I), consisting of an ontology R and instance I, gives rise to a set of models, defined as the first-order structures that make true all facts in I and satisfy all rules (or axioms) in R. When querying a KB, we are interested in obtaining the certain answers, i.e. the answers that hold for every model of the KB. By contrast, relational databases make the closed-world assumption, where each instance I is interpreted as the finite structure whose domain is equal to the active domain of I (i.e., the constants explicitly mentioned in I) and which makes true precisely the facts in I.

In practice, it seems natural to assume that applications may involve some predicates whose contents are fully known and others for which we have only partial information. This motivates the consideration of mixed-world semantics, where the set of predicates is partitioned into closed and open predicates, and models are required to coincide with the instance on the closed predicates. Mixed-world OMQA was first explored for DL ontologies [START_REF] Lutz | [END_REF][START_REF] Lutz | [END_REF][START_REF] Ahmetaj | Shqiponja Ahmetaj, Magdalena Ortiz, and Mantas Simkus. Polynomial datalog rewritings for expressive description logics with closed predicates[END_REF][START_REF] Ngo | [END_REF] and has only recently been studied for existential rules [Benedikt et al., 2016]. Further variants of traditional OMQA semantics can be obtained by placing additional restrictions on the models. In particular, as the classical semantics considers arbitrary models with possibly infinite domains, it is interesting to consider the restriction to finite models [START_REF] Ibáñez-García | [END_REF][START_REF] Gogacz | [END_REF] , or models with fixed or bounded-size domains [Gaggl et al., 2016;[START_REF] Rudolph | Not too big, not too small... complexities of fixed-domain reasoning in first-order and description logics[END_REF].

The present paper combines elements of these different lines of research by exploring OMQA with existential rules under a hybrid mixed-world active-domain semantics, in which we can use both closed and open predicates, and the semantics is based upon active-domain models whose domains are equal to the active domain of the instance. Such a semantics is appropriate for scenarios in which all relevant constants are made available in the instance. A possible use case (formalized in Example 1 in simplified form), which served as a motivation for this work, is in analyzing the trajectories of people circulating in a geographically restricted area (e.g., industrial facility, closed medical facility) that is equipped with sensors and secure entry and exit (so that the set of people in the area is known at each timepoint).

The paper is structured as follows. In Section 3, we formally introduce our framework for mixed-world reasoning with existential rules under active-domain semantics, in particular, defining the certain and possible answers to conjunctive queries (CQs). In the following section, we analyze the data complexity of the three central reasoning tasks which are to determine whether a mixed-world knowledge base (MWKB) is satisfiable, and whether a Boolean CQ is certain or possible w.r.t. a MWKB. We show that all three tasks are intractable (NP-or coNP-complete in data complexity) for MWKBs based upon arbitrary existential rules. However, for the linear fragment of our language (where rule bodies may contain at most one atom with an open predicate), we are able to establish several tractability results. Indeed, we can show that satisfiability, possibility of BCQs, and certainty are all PTIME-complete in data complexity, and moreover, the PTIME upper bounds remain valid in the presence of useful extensions, like (in)equality atoms, negated closed atoms, and disjunction in ruleheads. Motivated by these encouraging results, we investigate the linear fragment in more detail. In Section 4, we explore the expressive power of the linear fragment and prove our most technically challenging result, namely, that atomic queries coupled with mixed-world linear existential rules, extended with either closed negated atoms or disjunctive ruleheads and interpreted under either certain or possible active-domain semantics, can express all PTIME queries over ordered databases. Section 5 provides a general method of approximating unrestricted existential rules by means of linear rules augmented with a fixpoint semantics. We conclude the paper with a discussion of related and future work. Due to space restrictions, some proof details have been deferred to the appendix of the long version [START_REF] Bienvenu | Long version of this paper[END_REF].

Preliminaries

We assume the reader is familiar with standard notions from first-order logic. A term is either a constant or variable. An atom takes the form P (t 1 , . . . , t k ), where P is a predicate (or relation) symbol of arity k ≥ 0, and each t i is a term. A fact is an atom that does not contain any variables, and an instance is a finite set of facts. The active domain of an instance I, denoted adom(I), is the set of constants occurring in I.

A signature Σ is a finite set of predicate symbols. A Σinstance (resp. Σ-atom, Σ-fact) is an instance (resp. atom, fact) that only mentions predicates from Σ. The projection of an instance I onto Σ, denoted I| Σ , is the set of Σ-facts in I. When S is a set of instances, we let S| Σ = {I| Σ | I ∈ S}.

A conjunctive query (CQ) is a conjunction of atoms in which some of the variables may be existentially quantified. Atomic CQs consists of a single atom, and Boolean conjunctive queries (BCQs) are CQs whose variables are all existentially quantified. We use vars(Q) to denote the set of variables in a CQ Q. We sometimes treat (B)CQs as sets of atoms, e.g., using α ∈ Q to say that α occurs as a conjunct of Q.

We recall that an existential rule (sometimes abbreviated to rule) is a first-order sentence of the form ρ = ∀ xϕ( x) → ∃ yψ( x, y), where ϕ( x) and ∃ yψ( x, y) are CQs (called respectively the body and head of ρ), and all variables in x occur in ϕ( x). An (existential) ruleset (over Σ) is a finite set of existential rules (whose predicates are drawn from Σ). For brevity, we omit the initial universal quantifiers in rules, use commas in place of ∧, and sometimes simplify to just ϕ → ψ (in which case the variables in vars(ψ) \ vars(ϕ) are implicitly existentially quantified). As is common, we shall assume that ruleheads always consist of a single atom. This is w.l.o.g. since we can simulate conjunctive ruleheads by means of fresh predicates, obtaining a ruleset that is a conservative extension of the original one.

Existential rules are classically interpreted under first-order logic semantics. An existential rule ϕ → ψ is satisfied in a first-order structure J with domain D if every variable assignment h : vars(ϕ) → D that makes ϕ hold in J can be extended to an assignment h : vars(ϕ) ∪ vars(ψ) → D such that ψ holds in J. We can extend the notion of satisfaction to instances as follows. With every instance I, we associate a finite first-order structure J I whose domain is adom(I) and such that every predicate P is interpreted as the set of tuples c such that P ( c) ∈ I. We then say that rule ρ is satisfied in an instance I (or that

I satisfies ρ) if ρ is satisfied in J I . Likewise, a BCQ Q holds in instance I if Q holds in J I .
Datalog rules are existential rules with no existentially quantified variables, and a Datalog program is a finite set of Datalog rules. It is typical to partition the signature into extensional predicates that occur only in rule bodies, and intensional predicates that may occur both in bodies and heads. Given a Datalog program Π and an instance I over its extensional predicates, we denote by Π(I) the minimal instance that contains I and satisfies all rules in Π. A Datalog query takes the form (Π, Goal), where Π a Datalog program and Goal an intensional predicate. When evaluated on instance I, such a query returns those tuples a such that Goal( a) ∈ Π(I). The preceding notions naturally extend to semi-positive Datalog, which allows negated extensional atoms in rule bodies.

Existential Rules with Closed Predicates and Active-Domain Semantics

This section introduces a framework for reasoning with existential rules in a setting where some of the predicates are declared as closed, and for the remaining open predicates, an active domain semantics is adopted, whereby only facts over the explicitly named constants are considered. Formally, we will consider mixed-world signatures Σ = (Σ c , Σ o ), whose predicates are partitioned into a set Σ c of closed predicates, and a set Σ o of open predicates. Often we will only define Σ c , leaving it implicit that all other predicates in Σ belong to Σ o . A mixed-world knowledge base (MWKB) is a triple (Σ, R, I), where Σ is a mixed-world signature, R is an existential ruleset over Σ, and I is a Σ-instance. Note that R and I may use both open and closed predicates. Later in the paper, we will be especially interested in linear existential rules, which we define in our setting as existential rules whose bodies contain at most one Σ o -atom (but can contain any number of Σ c -atoms1 ).

We now define the active-domain (AD) semantics of MWKBs

. A Σ-instance M is an active-domain model of a MWKB (Σ, R, I) if adom(M ) = adom(I), M | Σc = I| Σc , I| Σo ⊆ M ,
and M satisfies every rule in R. The set of active-domain models (henceforth abbreviated to models) of (Σ, R,

I) is denoted M ods AD (Σ, R, I). A MWKB (Σ, R, I) is satisfiable under AD semantics if M ods AD (Σ, R, I) = ∅.
There are two natural ways of interpreting queries in our setting:

• a BCQ Q is certain w.r.t. (Σ, R, I) (under AD seman- tics) if Q holds in every M ∈ M ods AD (Σ, R, I); • a BCQ Q is possible w.r.t. (Σ, R, I) (under AD seman- tics) if Q holds in some M ∈ M ods AD (Σ, R, I).
We extend these notions to arbitrary CQs in the obvious way: a tuple a of constants from adom(I) is a certain (resp. possible) answer to a CQ Q( x) with free variables x iff the BCQ Q( a) obtained by replacing x by a is certain (resp. possible). Observe that a BCQ Q is not possible just in the case that its negation ¬Q holds in every model, so the reasoning task of determining whether a BCQ is possible can be rephrased as determining whether a BCQ is certainly false. Note that because open predicates can be constrained by the closed predicates, not all open facts are possible (hence, some queries involving open predicates may be certainly false).

We will also consider a more general form of existential rule that allows for the following useful features: (in)equality atoms (t = t , t = t ) in rule bodies and heads, negated Σ catoms in rule bodies, and rule heads that take the form of a (possibly empty) disjunction of existentially quantified atoms. We will call these extended (existential) rules. An extended linear rule is an extended rule with at most one Σ o -atom in its body, where (in)equality atoms are viewed as Σ c -atoms. The notion of an extended rule holding in an instance is defined in the obvious way, following first-order logic semantics.

We conclude this section by an example that illustrates the use of our logic to reconstruct trajectories of users whose positions are recorded by different sensors in an area. Example 1. The goal is to reconstruct paths of people circulating in a circumscribed area based upon sensor events. Every such event has an identifier e and provides the position p of a (possibly unidentified) user u at a certain time t. Nominative events are stored as facts N (e, p, t, u), and all events (both nominative and anonymous) are stored as facts E(e, p, t) in a generic event relation E. For the purposes of the example, we assume events occur at regular intervals (every ∆ t time units), and every user participates in some event (e.g., is detected by some sensor) at each timepoint (if this event identifies the user, then it is nominative, else anonymous). We further assume that N ext contains all pairs of events (e, e ) that can feasibly occur in sequence i.e., the timepoint of e directly follows that of e, and it is physically possible to move between their respective positions in ∆ t . Finally, we assume special nominative events containing the starting and ending positions of a user in the considered area, which can be found in relations S and F . Based upon the preceding information (stored in closed predicates), we construct possible trajectories by means of an open predicate T r, where T r(u, e, e ) means that user u transitions from event e to event e ; a second open predicate Loc(u, p, t) states that u is at position p at time t. Our first two rules guess a subsequent event for every event up to the final event (note that the variable e with the guessed event is existentially quantified):

S(u, e, p, t), F (u, e , p , t ), t < t → T r(u, e, e )

T r(u, e 0 , e), E(e, p, t), F (u, e , p , t ), t < t → T r(u, e, e )

We use N ext to constrain the pairs of events in T r, T r(u, e, e ) → N ext(e, e )

and enforce that if the true position of a user is known at a given timepoint t (via a nominative event), then any selected event for that user at time t gives the same position:

T r(u, e 0 , e), E(e, p, t), N (e , p , t, u) → p = p Loc-facts record the times and position of events in the guessed trajectories given by relation T r:

T r(u, e, e ), E(e , p, t) → Loc(u, p, t)

and the final two rules forbid a user from being in more than one position at a time, and for distinct users being associated to the same position at the same time:

Loc(u, p, t), Loc(u, p , t) → p = p Loc(u, p, t), Loc(u , p, t) → u = u
The first five rules are linear, while the last two are not. The certain (resp. possible) Loc-facts provide us with the sure (resp. potential) location of users at different timepoints, which could be used for visualizing and analyzing user trajectories. For example, displaying (or aggregating) the certain and possible locations of users (given by Loc) could help with understanding which sectors are the most frequented.

Complexity Analysis

We study the complexity of the three main reasoning tasks in our setting: satisfiability, certainty, and possibility. We focus on data complexity, which is measured in terms of the size of the instance. Our results are formulated for BCQs, but they apply equally well to the decision problem of testing whether a given tuple is a certain / possible answer.

Our first result shows that reasoning in the full language is intractable in data complexity, which is not surprising in light of prior negative results for mixed-world reasoning. Theorem 1. For (possibly extended) MWKBs with AD semantics, deciding if a BCQ is certain is coNP-complete in data complexity, while KB satisfiability and BCQ possibility are NP-complete in data complexity. The lower bounds hold for binary signatures and when the BCQ is a fact.

Proof sketch. We focus on certainty, but the proofs are easily adapted for satisfiability and possibility. To show q is not certain w.r.t. (Σ, R, I), it suffices to guess an instance J with adom(J) = adom(I), and verify that I| Σo ⊆ J, J coincides with I on Σ c , all rules in R are satisfied in J, and q does not hold in J. As J is of polynomial size in |I|, and we can verify rule satisfaction in PTIME w.r.t. |J|, we get an NP upper bound (data complexity) for non-certainty.

For the lower bound, we adapt a reduction from 2+2UN-SAT sketched in [START_REF] Lutz | [END_REF]. Let R 2+2 consist of:

• for i ∈ {1, 2}: P i (x, y) → ∃zR(y, z), P i (x, y), R(y, z), F (z) → U i (x)

• for i ∈ {3, 4}: N i (x, y) → ∃zR(y, z), N i (x, y), R(y, z), T (z) → U i (x) • R(y, x) → B(x) • S(w, x), 1≤i≤4 U i (x) → Q(w) Given a CNF ϕ = λ 1 , . . . , λ n with λ i = v i 1 ∨ v i 2 ∨ ¬v i 3 ∨ ¬v i 4 , we construct the following instance I 2+2 : {P (c i , v i 1 ), P (c i , v i 2 ), N (c i , v i 3 ), N (c i , v i 4 ) | 1 ≤ i ≤ n} ∪ {B(t), T (t), B(f ), F (f )} ∪ {S(a, c i ) | 1 ≤ i ≤ n} It is easily verified that, letting Σ c = {B}, the fact Q(a) is certain for (Σ, R 2+2 , I 2+2 ) iff ϕ is unsatisfiable.
We next turn to linear rulesets. A straightforward adaptation of Theorem 1 shows that linearity does not always yield tractability, even for normally well-behaved classes of BCQs: Theorem 2. For linear MWKBs with AD semantics, deciding if an acyclic 2 BCQ is certain is coNP-hard in data complexity.

Proof sketch. We adapt the reduction from Theorem 1 by (i) using the acyclic BCQ obtained by 'unfolding' Q(a) w.r.t. the rules with head predicates Q and U i , and then (ii) removing all non-linear rules from the ruleset R 2+2 .

Nevertheless, we can show several positive results for linear rules, exploiting the following maximal model property: Lemma 1. Consider a signature Σ = (Σ c , Σ o ), an extended linear ruleset R over Σ, and Σ-instances I, N with adom(N ) ⊆ adom(I). If {M ∈ M ods AD (Σ, R, I) | M ∩ N = ∅} is non-empty, then it has a maximal element w.r.t. set inclusion, called the maximal model of (Σ, R, I) that omits N . Moreover, computing this model (or determining its non-existence) can be done in polynomial time in |I|.

Proof. We outline a procedure for testing whether there is a model of (Σ, R, I) that omits N , and constructing the maximal such model when it exists. If N ∩I = ∅, then we immediately fail. Otherwise, we set J equal to the (unique) maximal Σ-instance with adom(J) = adom(I) and J ∩ N = ∅ that coincides with I on Σ c . If J satisfies R, then we have found the desired maximal model. Otherwise, repeat the following while J does not satisfy R:

1. Pick a rule γ → ψ that is not satisfied in J, together with a variable assignment h : vars(γ) → adom(J) such that h(γ) is satisfied in J but there is no extension h of h to vars(γ) ∪ vars(ψ) such that h (ψ) holds in J.

2. If γ contains an atom P ( x) with P ∈ Σ o and P (h( x)) ∈ I, then remove P (h( x)) from J.

3. Else, halt and return 'no such model'.

If at some point J satisfies R, halt and return J.

Correctness of this procedure can be established by an inductive argument, which shows that the current set J is always a superset of the desired maximal model if it exists (intuitively because every fact removal is 'forced'). Note that the initial J can be constructed in polynomial time in |I|, and there at most polynomially many iterations of the above steps, so the procedure runs in PTIME w.r.t. data complexity.

As an immediate consequence of Lemma 1, we have: 2 We recall that a BCQ Q is acyclic if the undirected graph (VQ, EQ) is acyclic, where VQ contains the terms of Q, and EQ contains (t, t ) iff there is an atom of Q containing t and t .

Theorem 3. Satisfiability of extended linear MWKBs with AD semantics is in PTIME for data complexity.

We can also use Lemma 1 to identify classes of queries for which the certainty problem is tractable. Theorem 4. For extended linear MWKBs with AD semantics, deciding if a fact is certain is in PTIME for data complexity. The same holds for ground CQs and for disjunctions of facts.

Proof sketch. To test whether fact α is not certain, it suffices to determine the existence of a model that omits N = {α}, a PTIME-checkable condition by Lemma 1. For a ground CQ α 1 ∧ . . . ∧ α n , we just perform this test for each α i separately. Finally, note that a disjunction of facts α 1 ∨ . . . ∨ α n is certain iff there is no model that omits {α 1 , . . . , α n }.

We now provide matching PTIME lower bounds for satisfiability and fact certainty. Recall that the latter problem is in AC 0 PTIME for linear existential rules under classical semantics without closed predicates [START_REF] Calì | [END_REF]. Theorem 5. For linear MWKBs with AD semantics, satisfiability and fact certainty are PTIME-hard in data complexity.

Proof sketch. The proof is by reduction from the PTIMEcomplete Path Systems Accessibility (PSA) problem [START_REF] Garey | [END_REF]. A path system is a tuple (U, E, S, t) where U is a set of nodes, E ⊆ U × U × U is an accessibility relation, S ⊆ U is the set of source nodes, and t ∈ U is the distinguished terminal node. The problem is then to determine whether t ∈ access(S), where access(S) is the least subset of U such that (i) S ⊆ access(S), and (ii) if u 1 , u 2 ∈ access(S) and (u 1 , u 2 , u 3 ) ∈ E, then u 3 ∈ access(S).

Observe that u ∈ access(S) iff there is a labelled binary tree such that (i) the root is labelled u, (ii) every leaf is labelled by some u ∈ S, and (iii) for every non-leaf node with label u 3 , there is (u 1 , u 2 , u 3 ) ∈ E such that the node's children are labelled u 1 and u 2 . We remark that if such a witness tree exists, then there is one with depth is at most |E|. The idea for the reduction is to try to build such a witness tree. Given a path system P S = (U, E, S, t) with |E| = m, we construct (by a logspace transducer) the instance I P S with:

• Init(t, 0), with t the unique terminal node

• R(u, u , u , k, k + 1), for (u, u , u ) ∈ E, 0 ≤ k < m • R(u, u, u, k, k + 1), for each u ∈ S and 0 ≤ k < m • R(u, u, u, m, m), for each u ∈ S
We set Σ c = {Init, R}, and let R P S consist of the rules:

• Init(x, z) → ∃y 1 , y 2 , z T (y 1 , y 2 , x, z, z ) • T (y 1 , y 2 , x, z, z ) → R(y 1 , y 2 , x, z, z ) • T (y 1 , y 2 , x, z, z ) → ∃v 1 , v 2 , z T (v 1 , v 2 , y 1 , z , z ) • T (y 1 , y 2 , x, z, z ) → ∃v 1 , v 2 , z T (v 1 , v 2 , y 2 , z , z )
The rough idea is that we guess a witness tree topdown, from the root t to the leaves, using the T -facts to record which triples from E were used. Correctness follows from the fact that every model of (Σ, R P S , I P S ) gives rise to a witness tree for t ∈ access(S) (constructed from the T -facts), and conversely, every such tree can be used to build a model. The numbers occurring in the last two positions of T play a crucial role, by ordering the T -facts (allowing them to be assembled into a witness tree) and forcing each 'branch' to eventually stabilize in S (as once value m has been reached, we can only use T -facts of the form T (u, u, u, m, m) with u ∈ S).

It follows that (Σ, R P S , I P S ) is satisfiable iff t ∈ access(S).

For certainty, we pick a fact α = Init(u, v) ∈ I P S and note that α is certain iff (Σ, R P S , I P S ) is unsatisfiable.

For possibility, we obtain tractability for arbitrary BCQs: Theorem 6. For (extended) linear MWKBs with AD semantics, BCQ possibility is PTIME-complete for data complexity. The lower bound holds when the BCQ is a fact.

Proof sketch. To decide if q is possible, it suffices to check whether q holds in the unique maximal model, which can be done in PTIME w.r.t. data complexity by Theorem 1.

The lower bound is by reduction from the PSA problem introduced earlier. The ruleset will consist of the following four linear rules, with

Σ c = {N R 0 , B, In, Out}: N R(x) → N R 0 (x) N R(x), Out(y, x) → ∃zN RIn(y, z) N RIn(y, z) → B(z) In(y, z, x), N RIn(y, z) → N R(x)
The input (U, E, S, t) is encoded by the following instance:

• B(0), B(1), and N R 0 (u) for every u ∈ U \ S • for every e i = (u i,0 , u i,1 , u i,2 ) ∈ E, the facts In(e i , 0, u i,0 ), In(e i , 1, u i,1 ), and Out(e i , u i,2 ) It can be verified that u ∈ access(S) iff N R(u) is not possible for every u ∈ U , and in particular, when u = t, yielding a reduction from PSA to non-possibility.

Expressive Power of Linear Fragment

We now study mixed-world linear existential rules under AD semantics from the perspective of query expressivity.

We recall that from an abstract perspective, a k-ary query Q over a signature Σ is a function that maps every Σinstance I to a finite k-ary relation Q(I) (containing the query answers). A query is called generic if it is invariant under renaming of constants 3 , which implies in particular that adom(Q(I)) ⊆ adom(I). With every k-ary query Q over Σ, we can associate the following recognition problem: given a Σ-instance I and k-tuple a of constants from adom(I), decide whether a ∈ Q(I). The class QPTIME consists of all generic queries whose recognition problem is computable in polynomial time w.r.t. data complexity.

Finding a logical language that precisely captures QPTIME is a major open problem in descriptive complexity. The following well-known result shows that the class of semipositive Datalog queries captures QPTIME over so-called ordered instances with min and max, i.e., instances I with a binary predicate Succ providing a successor relation among all constants in adom(I), and unary predicates M in and M ax containing the smallest and largest constants. Theorem 7 ( [Papadimitriou, 1985]). Over ordered instances with min and max, semi-positive Datalog captures QPTIME. 3 Queries that mention constants are not generic, but can be simulated using generic queries, see [Abiteboul et al., 1995] for details.

Interestingly, we can show that our linear fragment, extended with either closed negated atoms or disjunctive ruleheads and interpreted with either certain or possible semantics, captures QPTIME over ordered instances. Formally, the next theorem concerns the classes of queries obtained by associating with every constant-free atomic query A( x) and extended linear existential ruleset R over Σ the (generic) queries Q A,R,c and Q A,R,p that map I respectively to the certain and possible answers of A( x) w.r.t. (Σ, R, I). Theorem 8. Over ordered instances, atomic queries coupled with mixed-world linear rulesets extended with either closed negated atoms or disjunctive ruleheads and interpreted under either the certain or possible AD semantics capture QPTIME.

In the remainder of this section, we outline the key steps in the proof of Theorem 8. We start by noting that differently from Theorem 7, our result does not require the instance to provide the values min and max. This is because we can show how to compute these values from Succ using our rules.

Suppose we are given a semi-positive Datalog program Π with extensional predicates Σ Π ⊇ {Succ, M in, M ax} and k-ary output relation Goal. The core of the proof will be concerned with constructing a set of extended linear existential rules R Π over Σ with Σ c = Σ Π \ {M in, M ax} such that for every ordered Σ c -instance I and k-tuple of constants a, the following statements are equivalent:

1. Goal( a) ∈ Π(I ∪{M in(min), M ax(max)}, with min and max the smallest and largest constants in

I 2. Ans( a) is certain w.r.t. (Σ, R Π , I) 3. Ans( a) is possible w.r.t. (Σ, R Π , I)
To simplify the construction, we initially let R Π consist of linear rules extended with both negated closed predicates and disjunctive ruleheads. Then in a final step, we show how we can get rid either of negated atoms or disjunctions.

Let us describe at a high level the ruleset R Π . We include rules that populate an open predicate T uple with all k-ary tuples of constants from the active domain and then guess for each tuple, whether it is an answer to the Datalog query:

T uple( x) → Ans( x) ∨ N otAns( x)
To verify that these guesses are correct, we rely upon the fact that α ∈ Π(J) iff there is a proof tree for α [Abiteboul et al., 1995], i.e., a tree whose root is labelled α, whose every leaf node is labelled with either an extensional fact β with β ∈ J or a negated extensional fact ¬β with β ∈ J, and such that for each non-leaf node with label β and whose children have labels γ 1 , . . . , γ k , the ground rule γ 1 , . . . , γ k → β can be obtained from a rule in Π by instantiating its variables with constants from adom(J). It is known that the depth of a minimal proof tree is bounded by p(|I|) for some polynomial function p. To verify Ans( a), we follow a similar approach to the proof of Theorem 5 and guess a proof tree for Goal( a) by working from root to leaves (failing if no such proof tree exists). To verify a fact N otAns( a), we use a second set of rules to enforce that there is no proof tree for Goal( a) of depth at most p(n). Essentially, we guess topdown a 'noproof' tree whose root is labelled Ans( a), whose leaves are labelled by extensional facts not in J (here we used closed negated atoms) or negations of extensional facts from J, and such that for every non-leaf node ν with label β and every ground instantiation γ 1 , . . . , γ k → β of a rule in Π, there is a child of ν that is labelled with some γ i . We implement counters that keep track of the depth of the guessed trees and fail if we exceed p(n). By construction, (Σ, R Π , I) is satisfiable. Moreover, while the models may differ on which trees are guessed, they always coincide on Ans and N otAns facts. It follows that Ans( a) is certain iff it is possible iff Goal( a) ∈ Π(I ∪ {M in(min), M ax(max)}.

The rules implementing the preceding construction presume the existence of at least two values (min = max), which are used to implement binary counters. Thus, we must identify instances containing a single constant (min = max) and create a separate set of rules to compute the answers in this limit case. Briefly, the idea is to create a dedicated rule for each single-constant instance, using negated closed atoms to identify the contents of the input single-constant instance.

To complete the proof, we show how to simulate negated closed atoms using disjunction in ruleheads, and conversely, how to simulate disjunctive ruleheads using negated closed atoms. The latter translation only works for instances with at least two constants, but this is sufficient for our purposes, as disjunction only occurs in the rules that fire on instances with multiple constants. We point out that if we were to restrict to the class of instances with at least two constants, then both transformations are applicable and yield (plain) linear rules.

Fixpoint Extension of Linear Fragment

While the (extended) linear has desirable computational properties, it cannot express some useful constructs, such as functionality (cf. final rule from Example 1). In this section we show how arbitrary rulesets can be approximated via a fixpoint-style extension of the linear fragment.

The idea is as follows. Given an arbitrary ruleset R over Σ = (Σ c , Σ o ), we can first compute the certain and possible facts using only the linear rules in R, which yields a subset of the 'true' certain facts and a superset of the possible ones. We store these facts in new closed predicates R lb and R ub , use rules to link them to the original open predicates, create new linear rules from the non-linear rules by replacing all open atoms but one with closed R lb -atoms, then recompute the certain and possible facts. Iterating this process until fixpoint allows us to obtain more refined approximations of the certain and possible facts of the original ruleset.

Formally, an (extended) linear f p ruleset over

Σ = (Σ c , Σ o ) is an (extended) linear ruleset over Σ + = (Σ + c , Σ o ) where Σ + c = Σ c ∪ {R ub , R lb | R ∈ Σ o } and
R lb -atoms (resp. R ub -atoms) may only occur positively and in rule bodies (resp. ruleheads). Given such a ruleset R, we denote by R + the ruleset

R ∪ {R lb ( x) → R( x), R( x) → R ub ( x) | R ∈ Σ o }
(with x a tuple of distinct variables of the same arity as R). These rules formalize that R lb (resp. R ub ) places a lower (resp. upper) bound on the set of R-facts.

Given a linear f p ruleset R over Σ = (Σ c , Σ o ) and a Σinstance I, we define inductively the set M ods k (Σ, R, I) of models of (Σ, R, I) at each stage k ≥ 1:

• M ods 1 (Σ, R, I) = M ods AD (Σ + , R + , I 0 )| Σ , where

I 0 = I ∪ {R ub ( a) | R ∈ Σ o of arity n, a ∈ adom(I) n } • M ods k+1 (Σ, R, I) = M ods AD (Σ + , R + , I k )| Σ where I k = I ∪ {R lb ( a) | R( a) ∈ Cert k (Σ, R, I)} ∪ {R ub ( a) | R( a) ∈ P oss k (Σ, R, I)}
where, for every k ≥ 1, the set Cert k (Σ, R, I) (resp. P oss k (Σ, R, I)) contains the Σ-facts over adom(I) that appear in every (resp. some) M ∈ M ods k (Σ, R, I). The next lemma shows that these sets eventually converge to a fixpoint. Lemma 2. For every linear f p ruleset R over Σ and Σinstance I, M ods j+1 (Σ, R, I) ⊆ M ods j (Σ, R, I) for every j ≥ 1, and there exists k * ≥ 1 such that M ods j (Σ, R, I) = M ods k * (Σ, R, I) for every j ≥ k * .

Proof Sketch. We can show by simultaneous induction that (i) Cert j-1 (Σ, R, I) ⊆ Cert j (Σ, R, I), (ii) P oss j (Σ, R, I) ⊆ P oss j-1 (Σ, R, I), and (iii) M ods j+1 (Σ, R, I) ⊆ M ods j (Σ, R, I) for every j ≥ 1.

In what follows, we'll use M ods ∞ (Σ, R, I) to refer to the fixpoint M ods k * (Σ, R, I) from Lemma 2, and similarly for Cert ∞ (Σ, R, I) and P oss ∞ (Σ, R, I). We observe that the stage k * where the fixpoint is reached is bounded polynomially in |I|. When combined with Theorems 4 and 6, this yields the following tractability result: Theorem 9. The problems of deciding whether a fact belongs to Cert ∞ (Σ, R, I) or to P oss ∞ (Σ, R, I) are both in PTIME for data complexity.

Next consider an extended existential ruleset R over Σ = (Σ c , Σ o ). The linearization of R, denoted R lin , is obtained by replacing each rule ε ∧ β 1 ∧ . . . ∧ β m → γ ∈ R, whose body Σ o -atoms are β 1 , . . . , β m , with the set of rules:

ε ∧ 1≤j≤m,j =i * β lb j ∧ β i * → γ (1 ≤ i * ≤ m)
where

β lb i = R lb ( t) when β i = R( t).
The following theorem shows that R lin , under the above fixpoint semantics, provides an under-approximation of the certain facts and an over-approximation of the possible facts. Theorem 10. Let R be an extended existential ruleset over Σ, and let R lin be its linearization. For every Σ-instance I: M ods AD (Σ, R, I) ⊆ M ods ∞ (Σ, R lin , I). In particular, this implies that for every fact α:

• α ∈ Cert ∞ (Σ, R lin , I) ⇒ α is certain w.r.t. (Σ, R, I); • α is possible w.r.t. (Σ, R, I) ⇒ α ∈ P oss ∞ (Σ, R lin , I).
The next result shows that linearization provides a nontrivial approximation in the sense that in general there is no instance-independent bound on the number of iterations needed to reach the fixpoint. instances. To obtain a more widely applicable PTIME result, we provided a method of approximating non-linear rules by means of linear rules equipped with a fixpoint semantics.

We point out that a PTIME upper bound for possible BCQs with mixed-world semantics has been shown in [Benedikt et al., 2016;[START_REF] Benedikt | [END_REF] for a more restricted notion of linear rule (and without PTIME-hardness and QPTIME results). The key difference with [START_REF] Benedikt | [END_REF] and other works on mixed-world reasoning [START_REF] Lutz | [END_REF][START_REF] Lutz | [END_REF][START_REF] Ahmetaj | Shqiponja Ahmetaj, Magdalena Ortiz, and Mantas Simkus. Polynomial datalog rewritings for expressive description logics with closed predicates[END_REF][START_REF] Ngo | [END_REF] is that we adopt active-domain semantics, whereas these other works use classical semantics under which new values can appear in the open predicates. This has a very significant impact on the decidability and complexity of reasoning. For example, certainty of BCQs is decidable in our setting even for arbitrary existential rules, while it is undecidable under classical semantics; for frontier-guarded rules, certainty of BCQs was shown to be EXPTIME-complete in data complexity [Benedikt et al., 2016], while it is coNP-complete in data complexity for our setting. A notable similarity is that our PTIME membership results employ a greatest fixpoint construction, which is in the same spirit as techniques used in [START_REF] Benedikt | [END_REF]Benedikt et al., 2016]. Other non-trivial data tractability results have been obtained for fact certainty in the presence of linear disjunctive Datalog rules [START_REF] Kaminski | [END_REF], and for exact views given as LAV source-to-target rules [START_REF] Benedikt | [END_REF].

The present work can be continued in several directions. First, we can study the combined complexity of reasoning in our setting for different classes of existential rules, and the impact of using a fixed but succinctly represented domain, in the spirit of [START_REF] Rudolph | Not too big, not too small... complexities of fixed-domain reasoning in first-order and description logics[END_REF]. There are also many interesting questions related to expressive power and the (non)existence of translations to various Datalog extensions (e.g, disjunctive Datalog [Eiter et al., 1997], or Datalog with non-deterministic choice [START_REF] Abiteboul | Non-determinism in logic-based languages[END_REF]). We note that polynomial translations to disjunctive Datalog with negation have been proposed to compute certain facts w.r.t. mixed-world KBs in expressive description logics [START_REF] Ahmetaj | Shqiponja Ahmetaj, Magdalena Ortiz, and Mantas Simkus. Polynomial datalog rewritings for expressive description logics with closed predicates[END_REF], and a similar approach may be worth exploring in our setting. On the practical side, we have already begun applying our framework to integrating geolocalization data as illustrated by our example. Here the main challenge is to develop more efficient PTIME algorithms that do not require the construction of full maximal models, which although polynomial may be impractical for large domains. We would also like to allow for closed relations that are implicitly defined by PTIME-computable functions (e.g., N ext from Example 1 that could be defined using distances between locations) that are not computed and stored in advance but rather produced as needed.

Linear existential rules[Calì et al.,

2012] are usually restricted to a single body atom. Our definition is more general and in the spirit of linear Datalog rules[Abiteboul et al., 1995].

We have used an equality atom for convenience, but it is possible to modify the proof so that it works in the basic formalism.

Acknowledgements

We thank the reviewers for their detailed and constructive feedback. This work was partially supported by CNRS Momentum project "Managing Data without Leaks" and ANR project CQFD (ANR-18-CE23-0003).

Lemma 3. There exists a Σ-ruleset R such for every j ≥ 0, there is a Σ-instance I such that M ods j+1 (Σ, R, I) = M ods j (Σ, R, I).

Proof. Consider the following ruleset 4 R sat :

• Clause(u) → SatLit(u, x, y)

• SatLit(u, x, y) → ClauseLit(u, x, y)

• SatLit(u, x, y) → P ickV al(x, y)

• P ickV al(x, y) → B(y)

• P ickV al(x, y) ∧ P ickV al(x, y ) → y = y

With every CNF ϕ = c 1 ∧ . . . ∧ c n over v 1 , . . . , v m , we associate the instance I ϕ consisting of the following facts:

We let Σ sat contain all of the preceding predicates, where Clause, ClauseLit, B are the only closed predicates. It can be verified that (Σ sat , R sat , I ϕ ) iff ϕ is satisfiable. Moreover, P ickV al(p i , 1) (resp. P ickV al(p , 0)) is certain iff every satisfying valuation of ϕ sets p i to true (resp. false).

The linearization (R lin sat ) + retains the first four rules of R sat and adds new rules, the most relevant being:

• P ickV al lb (x, y) ∧ P ickV al(x, y ) → y = y • P ickV al lb (x, y) → P ickV al(x, y),

• P ickV al(x, y) → P ickV al ub (x, y)

Consider the sequence of propositional CNF ξ i (i ≥ 1), where ξ i = c 1 ∧ . . . ∧ c i , c 1 = p 1 , and c +1 = ¬p ∨ p +1 for 1 ≤ < i. As every ξ i is satisfiable, we know that (Σ sat , R sat , I ϕ ) is satisfiable. By Theorem 10, the same is true of (Σ sat , R lin sat , I ξi ). Moreover, ξ i |= p for every 1 ≤ ≤ i, so by the above, we have P ickV al(p , 1) ∈ Cert ∞ (Σ sat , R lin sat , I ξi ) for all 1 ≤ ≤ i. A simple inductive argument shows that for every i ≥ 1 and 1 ≤ k ≤ i:

, then k ≥ i. Thus, by considering ξ i for increasing values of i, we can delay the achievement of the fixpoint, meaning it cannot be bounded independently of the instance.

A relevant question for future work is to analyze, either formally or experimentally, the quality of our approximation.

Concluding Remarks

In this paper, we investigated the problem of reasoning with existential rules under a hybrid mixed-world active-domain semantics. While querying in our setting is intractable in the general case, we identified an interesting and non-trivial fragment based upon linear rules that is PTIME-complete in data complexity, and in fact, captures all of QPTIME over ordered