Arthur Levy
email: arthur.levy@mcgill.ca

Steven Le Corre
email: steven.lecorre@univ-nantes.fr

Irene Fernandez

Irene Fernandez Villegas
email: i.fernandezvillegas@tudelft.nl

Modeling of the heating phenomena in ultrasonic welding of thermoplastic composites with flat energy directors

Keywords: thermoplastic composite, energy directors, ultrasonic welding, viscoelastic heating, friction heating, multiphysical simulation

A model for the mechanics (oscillating deformation), heat transfer including viscoelastic heat generation and friction dissipation, and degree of adhesion (intimate contact and healing) is proposed for the initial transient heating phase.

Numerical resolution was performed using a multi-physical finite element code. The predicted dissipated power evolution exhibits a good correlation with previous experimental measurement of delivered power, and shows that the apparatus has a global efficiency of 13%. The predicted degree of adhesion also confirms the experimental observation that adhesion starts at the edge of the contact area, and progressively extends to the whole contact area.

The numerical model was further used to investigate the physical mechanisms occurring during the welding process. As suggested in the literature, the first heating mechanism is confirmed to be due to interfacial friction. Bulk viscoelastic dissipation becomes predominant when the interface reaches higher temperatures. The dissipated power is suddenly increased when the whole interface reaches the glass transition temperature.

A c c e p t e d M a n u s c r i p t

Introduction

Thermoplastic composites offer new possibilities for the aerospace industry. Huge structures (several meters) can be processed rapidly and more cost-effectively than when thermoset composites are used, since the latter need to undergo lengthy curing reactions. The ability to fuse thermoplastic resins gives new perspectives for processing and assembling as mentioned by [START_REF] Grewell | Welding of plastics: Fundamentals and new developments[END_REF] usually known as welding is, indeed, a group of joining techniques specific to thermoplastic composites, that offers a very interesting alternative to traditional assembling via adhesive bonding or mechanical fastening [START_REF] Yousefpour | Fusion Bonding/Welding of Thermoplastic Composites[END_REF].

Within this group, ultrasonic welding of thermoplastic composites is based on low-amplitude and highfrequency vibrations that cause surface friction and viscoelastic heating at the welding interface. Its main advantages are very high speed (welding times of a few seconds) and the fact that no foreign material, such as a metal mesh or metal particles, is needed at the interface, regardless the nature of the adherends. As first introduced by Fernandez Villegas (2013), localized heating in ultrasonic welding of thermoplastic composites can be achieved by placing a film of neat matrix at the welding interface (Fig. 1). This straightforward "flat energy director", as defined in this previous work [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF], is a solution that contrasts with more traditional and complex energy directors derived from the plastics industry, consisting of neat resin protrusions molded on the surfaces to be welded (Benatar et al., 1989). Flat energy directors concentrate heat dissipation at the welding interface as a result of their lower stiffness and hence their higher cyclic strain, as compared to the fibrous adherends. Since they cover the whole overlap, they provide 100% welded areas without the need for any shape or size optimization as suggested in previous work [START_REF] Levy | Ultrasonic welding of thermoplastic composites: a numerical analysis at the mesoscopic scale relating processing parameters, flow of polymer, and quality of adhesion[END_REF].

The experimental study [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF] coupled with the work of [START_REF] Zhang | Study on Heating Process of Ultrasonic Welding for Thermoplastics[END_REF] on heating of rectangular energy directors gave a relevant insight into the different heating mechanisms and their roles in this ultrasonic process. They particularly highlighted the significant effect of dry friction on the initiation of heating, which is combined to the usual self-heating as described by former author on ultrasonic welding such as [START_REF] Tolunay | Heating and bonding mechanisms in ultrasonic welding of thermoplastics[END_REF] or Benatar et al. (1989). Nevertheless, further modeling work is needed in order to fully understand and quantify the phenomena taking place in the flat-energy-director ultrasonic welding process. Of particular importance is the temperature developed at the welding interface and the extension of the heat affected zone (HAZ) in the adherends [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF], which are difficult to measure by using thermography or traditional thermocouples. Indeed, on the one hand, the HAZ is confined in the vicinity of the overlapping interface, thus not directly visible, and the other hand, introducing a thermocouple at the interface would be very intrusive and might even act as an energy director itself. In order to invetigate the heating mechanisms in the process and to predict the temperature at the welding interface and the extension of the HAZ, a multiphysical model was built and solved in COMSOL Multiphysics.

Experimental methods

Material

The material used for this research was T300 carbon-fiber-fabric (5HS) reinforced polyetherimide (CF/PEI) provided by Ten Cate Advanced Composites, The Netherlands. The modeling of such a welding process requires many thermo-physical parameters that are detailed below. First of all, it is important to note that thermal degradation can be a key problem in thermoplastic welding. It would limit the validity of the present approach. Nevertheless, according to [START_REF] Augh | Degradation of continuous carbon fiber reinforced polyetherimide composites during induction heating[END_REF], the values for degradation temperature for CF/PEI, obtained by thermogravimetric analysis, range from 450 • C (isothermal mode) to 500 • C (nonisothermal mode). As it will be shown later, the temperatures reached in the present process, predicted by the model, are far from this bounds.

The transverse k zz and in-plane k xx thermal conductivity of the composite and of the neat PEI matrix k were measured by DSM R&D Solutions, the Netherlands, using nanoflash tests. They are plotted versus temperature in Fig. 2. In spite of the visible thermo-dependence of the conductivity, the following constant approximations were adopted:

k zz = 0.61 W /mK k xx = 2.8 W /mK k = 0.25 W /mK (1)
The specific heat capacity c P EI of the neat PEI matrix was measured at TU Delft, using a differential scanning calorimeter (DSC). Its density ρ P EI was measured by the manufacturer Sabic Innovative Plastics using a PVT apparatus. As shown in Fig. 3 the volumetric heat capacity of the PEI ρ P EI c P EI was approximated by the linear relation:

ρ P EI c P EI = 4110 × T [ • C] + 1.24 × 10 6 J /m 3 .K.
(2)

The viscoelastic mechanical behavior of PEI was studied in Appendix A.

The specific heat capacity c comp of the CF/PEI composite was measured with a DSC. Considering a constant density ρ CF = 1760 kg /m 3 for the T300 carbon fiber and ρ P EI above, the law of mixtures gives the density of the composite ρ comp . The volumetric heat capacity is plotted versus temperature in Fig. 4. It is in good agreement with the volumetric heat capacity obtained using a law of mixtures:

ρ comp c comp = v f (ρ CF c CF ) + (1 -v f ) (ρ P EI c P EI ) (3)
where v f = 50% is the fiber volume fraction, and c CF = 800 J /kg.K the T300 carbon fiber specific heat.

Finally, the following linear dependence on temperature was assumed:

ρ comp c comp = 2700 × T [ • C] + 1.3 × 10 6 J /m 3 .K. (4)
A c c e p t e d M a n u s c r i p t The in-plane elastic modulus in the warp and weft directions was found by the manufacturer to vary very little (less than 3%) (Tencate Advanced Composites, 2008). Moreover it is rather constant (within 5%) between 23 • C and 80 • C. Nonetheless, the composite adherends are loaded transversely in compression.

Therefore, as a first simplified model, the composite is considered isotropic with a Young modulus equal to its transverse modulus. This transverse modulus E comp is evaluated using the rule of mixture proposed by [START_REF] Jacquet | Calculation of the transverse modulus of a unidirectional composite material and of the modulus of an aggregate. Application of the rule of mixtures[END_REF] which yields:

E comp = 10.3 GPa. ( 5
)
for a fiber modulus of 180 GPa and a resin modulus of ∼ 4 GPa (see Appendix A). A Poisson ratio of 0.4 was used.

Processing conditions

Six layers of five-harness satin CF reinforced prepreg were used to manufacture the base laminates in a hot platen press. The stacking sequence of the laminates was [0/90] 3s and the manufacturing conditions were 20 minutes at 320 • C and 20 bar. The nominal thickness of the consolidated laminates was h plates = 1.92 mm.

Rectangular samples (101.6 mm × 25.4 mm) were cut out of the CF/PEI laminates with an abrasive saw, so that the longer side of the samples coincided with the main apparent orientation of the fibers. These samples were degreased and welded in near-field (Benatar et al., 1989), into a single-lap configuration, as illustrated in Fig. 1, with an overlap area of 12.7 mm × 25.4 mm, as indicated by the ASTM D 1002 standard (ASTM International, 2010). A PEI film of thickness h f ilm = 0.25 mm was used as a flat energy director.

A c c e p t e d M a n u s c r i p t [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF].

For the welding of the samples a 20 kHz Rinco Dynamic 3000 ultrasonic welder with a maximum power input of 3000 W was used. This welding unit automatically adjusts the electrical power input in order to keep the amplitude of vibration constant. It allows for a certain range of amplitude values between 51.8 and 86.2 µm. The edge of the sonotrode in contact with the parts to be welded was cylindrical with a 0.40 mm diameter, much larger than the welding zone. The clamping device, shown in Fig. 5, was designed to allow for vertical motion of the upper adherend and to ensure parallelism between both composite samples. More detail on the apparatus and experimental conditions can be found [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF].

The processing parameters relevant to the analysis in this paper were:

• welding forces of 300, 1500, and 500 N, which are the minimum, maximum, and an intermediate values of the welder force capability

• vibration amplitudes of 51.8 and 86.2 µm which are the minimum and maximum capabilities of the welder.

It should be noticed that the ultrasonic welder provided feedback on the power dissipated during the process and the displacement of the sonotrode. The dissipated power, as well as the transformations undergone by the flat energy director in the different phases of the process as described [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF], were used for the validation of the model.

A c c e p t e d

M a n u s c r i p t

Modeling

In this section, a modeling of heating phenomena that enable ultrasonic welding is proposed. Many authors studied the classical case where triangular shape energy directors are used [START_REF] Nonhof | Estimates for Process Conditions During the Ultrasonic Welding of Thermoplastics[END_REF]. [START_REF] Tolunay | Heating and bonding mechanisms in ultrasonic welding of thermoplastics[END_REF] suggested that the heating is solely due to dissipation of mechanical work through visco-elastic deformation. This suggested phenomenon has been widely assumed in the literature [START_REF] Benatar | Ultrasonic Welding of PEEK Graphite APC-2 Composites[END_REF]) and allowed to perform analysis of industrial energy director shapes [START_REF] Wang | FEM Investigation of the Temperature Field of Energy Director During Ultrasonic Welding of PEEK Composites[END_REF] or study the effect of design on the process [START_REF] Suresh | Modeling of temperature distribution in ultrasonic welding of thermoplastics for various joint designs[END_REF]. More recently, Levy et al. (2011b) confirmed the heating term due to visco-elastic work dissipation, using time homogenization technique based on asymptotic expansion. An integrated finite element code solving for the vibration, the heat transfer and the flow was developed (Levy et al., 2011a).

In the present study, because the energy director is not triangular but a film at the interface, different phenomena occur. The heating mechanism, as suggested by [START_REF] Zhang | Study on Heating Process of Ultrasonic Welding for Thermoplastics[END_REF] and Fernandez Villegas (2013) also consists in frictional dissipation at the interface which has to be incorporated as an additional interfacial heat source.

General macroscopic framework

Because of the sample geometry, a plane strain is assumed and the three domains shown in Fig. 1 are described in a two-dimensional framework.

The present study, focuses on the heating phase during ultrasonic welding with flat energy directors.

The frameworks is that of Levy et al. (2011a) except that we do not account for the flow, that mostly occur in the subsequent phases of the process Fernandez Villegas (2013).

The physics solved are therefore:

• An elastic problem that describes the vibration effect. The boundary condition being a sinusoidal displacement imposed by the tool. As noticed by [START_REF] Levy | Ultrasonic welding of thermoplastic composites: a numerical analysis at the mesoscopic scale relating processing parameters, flow of polymer, and quality of adhesion[END_REF], this approach neglects the hammering effect brought up by [START_REF] Tolunay | Heating and bonding mechanisms in ultrasonic welding of thermoplastics[END_REF], and considers contact between the sonotrode and the top composite during the whole process. In this study, the contact between the sonotrode and the composite assembly is considered to be lost during a part of the cycle.

• A heat transfer problem that describes temperature evolution in the setup. Following existing work by [START_REF] Tolunay | Heating and bonding mechanisms in ultrasonic welding of thermoplastics[END_REF], the viscoelastic dissipation appears as a thermal source of the form:

Qbulk (x, y) = α 2 h ωE ε * : ε * 2 (6)
E being the loss modulus of the material and ε * (x, y) the amplitude strain tensor obtained by solving one nominal static elastic problem with the amplitude of vibration as a load. [START_REF] Benatar | Ultrasonic Welding of PEEK Graphite APC-2 Composites[END_REF] used a similar approach for their carbon PEEK composite welding study. Levy et al. (2011b) A c c e p t e d M a n u s c r i p t recovered this term using a time homogenization technique and used it to simulate industrial cases [START_REF] Levy | Ultrasonic welding of thermoplastic composites: a numerical analysis at the mesoscopic scale relating processing parameters, flow of polymer, and quality of adhesion[END_REF]. [START_REF] Nonhof | Estimates for Process Conditions During the Ultrasonic Welding of Thermoplastics[END_REF] also retained this heat source but suggested that because of the hammering effect, during an ultrasonic cycle, contact is lost between the sonotrode and the sample. This phenomenon is accounted here by introducing the empirical hammering correction factor α h in equation ( 6). If no hammering occurs, α h = 1, and the classical dissipation term from the literature is recovered. In the case of hammering, the sinusoidal strain ε = ε * sin (ωt) occurs only during a fraction of the cycle, leading to a reduced viscoelastic dissipation. The data obtained with the nominal problem will be denoted with a star. Using the complex viscosity η = η + iη = Ē/iω instead ( Ē being the complex viscoelastic modulus), equation ( 6) becomes

Qbulk (x, y) = α 2 h ω 2 η ε * : ε * 2 . ( 7
)
It explicitly shows the quadratic dependency of the viscous dissipated power on the vibration frequency.

Microscopic analysis at the interface

In the following we call "interfaces" the lower and upper interface between the flat energy director and the lower, successively upper, composite adherend.

First the friction phenomenon, that induces an interfacial heat source, cannot be neglected. Then the contact evolution at the microscopic scale is modeled using an intimate contact model.

Friction

Mechanics. Because the stiffness of the neat matrix flat energy director at the interface is lower than that of the composite plates, its longitudinal deformation will be larger than that of the composite resulting in slippage and friction at the interfaces. In order to simulate this displacement discontinuity at the interface, a special connection was considered at the interfaces between the plates and the film. A "thin elastic film", as defined in COMSOL, was assumed with the following anisotropic properties:

• A very high normal stiffness K n = 10 18 N/m 3 constrains the normal displacement to be continuous, thus ensuring a contact condition.

• A very low initial tangential stiffness K 0 t ∼ 0 simulates a perfect slip with no friction. This first model neglects the effect of friction on the deformation and therefore probably over-estimates the slippage. This simple "perfect slip" approach allows one to keep a linear elastic problem, thus allowing to describe the sinusoidal response to the sinusoidal load by solving one single static problem (as discussed in section 3.1). As adhesion occurs at the interface, the tangential stiffness K t increases. An empirical quadratic dependency of K t on the degree of adhesion D a defined hereunder (section 3.2.3) is assumed:

K t = E comp h 2 × 10 -3 + D 2 a • (8)
A c c e p t e d M a n u s c r i p t

When D a reaches 1, full sticking occurs and K t reaches the very high value of E comp /h where h = 2 µm is a very thin equivalent layer thickness (∼ 1% of the film's thickness).

Normal stress. After solving the elastic problem, one gets the horizontal displacement discontinuity δu * (x) across the interface, for each position x along the interface. Back to the temporal space, the discontinuity writes:

δu (x) = δu * (x) sin (ωt) (9)
and the velocity discontinuity

δv (x) = δu * (x) ω cos (ωt) . ( 10
)
The tangential stress τ , even though it was neglected while computing the elastic problem, can be approximated assuming a Coulombic friction. It then depends on the normal stress magnitude N applied on the interface. This normal force is the sum of an oscillating term σ n linked to the vibration and a static term P app linked to the sonotrode static holding force:

N = P app + |σ n | (t) . (11)
For an order of magnitude approximation, P app can be considered uniform on the whole interface thus leading to

P app = F imp A , (12)
where F imp is the sonotrode holding force, of the order of 500 N, and A = 323 mm 2 is the overlap area, such that

P app ∼ 1.5 × 10 6 Pa. ( 13
)
On the other hand, a lower bound of σ n can be obtained by considering a uniform vertical sinusoidal deformation of amplitude a/h within the whole system thickness h = h f ilm + 2h plates , a ∼ 80 µm being the sonotrode displacement amplitude. Using the PEI Young's modulus E ∼ 4 GPa (see Appendix A) as a minimum bound for the system stiffness, the vertical stress |σ n | is therefore bounded by:

|σ n | > E a h f ilm + 2h plates ∼ 8 × 10 7 Pa. ( 14
)
This shows that the vibrating term in equation ( 11) is significantly higher than the static term. The normal stress N on the interface can therefore be reduced to the term σ n .

Using the Coulomb friction behaviour the tangent stress then writes:

τ (x) = µ. |σ n (x)| δv (x, y) |δv (x, y)| . (15)
A c c e p t e d M a n u s c r i p t

where µ is the friction coefficient, that depends on temperature, and was obtained using a friction setup, at

McGill University, as described in Appendix B. The parenthesis shows that the tangential force is always acting against slippage. σ n is obtained from the elastic problem as:

σ n (x) = σ * yy (x) sin (ωt) (16)
σ * yy (x) = σ * (x)•e y •e y being the vertical stress on the horizontal interface obtained by solving the normalized problem presented in section 3.1.

Dissipation. The average dissipated mechanical work associated to friction during an ultrasonic cycle writes

Qfric (x) = τ • δv ( 17
)
the operator • being the average operator over an ultrasonic vibration period 2π/ω. Noting that

δv |δv| • δv = |δv| (18) it gives Qfric (x) = α 2 h ω 2π × 2π ω 0 µ σ * yy (x) sin (ωt) • |δu * (x) ω cos (ωt)| dt (19)
where the hammering coefficient α h accounts for the part of the cycle when contact is lost between the sonotrode and the sample. Developing leads to:

Qfric (x) = α 2 h ω 2 2π µ σ * yy (x) δu * (x) × 2π ω 0 |sin (ωt) cos (ωt)| dt (20)
and finally

Qfric (x) = α 2 h ω π µ σ * yy (x) δu * (x) . (21)
Note that this friction dissipated power is proportional to the vibration frequency ω. This is related to the dry friction behavior assumed in eq. ( 15).

Intimate contact evolution

During the welding, the quality of the contact between the adherends and the flat energy director will evolve as the surface asperities get squeezed. During the last couple of decades, several models have been proposed to predict the evolution of this contact.

In the present study the [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF] model is used to predict the intimate contact evolution.

The degree of intimate contact D ic is a scalar defined as the contact area ratio, and reaches 1 when full contact is obtained. The Lee and Springer model, followed by [START_REF] Mantell | Manufacturing process models for thermoplastic composites[END_REF] has been widely used in the literature [START_REF] Ageorges | Avdances in Fusion Bonding Techniques for Joining Thermoplastic Matrix Composites: a Review[END_REF], and proved its efficiency. The model for D ic writes: Using contact surface profile measurements presented in Fig. 6, [START_REF] Yang | A fractal Cantor set based description of interlaminar contact evolution during thermoplastic composites processing[END_REF] proposed to directly obtain a 0 as the standard deviation of the surface function. This was also the approach followed by [START_REF] Levy | Inter-layer Thermal Contact Resistance Evolution with the Degree of Intimate Contact in the Processing of Thermoplastic Composite Laminates[END_REF]. Nonetheless the other parameters b 0 and w 0 cannot be obtained. In this study the parameters used are simply adapted from the literature and are given in table 1. P app is the static load applied on the interface, i.e. the sonotrode holding force divided by the overlap area, and η (T (x)) is the viscosity of the roughness, that is taken as the pure matrix viscosity [START_REF] Levy | Inter-layer Thermal Contact Resistance Evolution with the Degree of Intimate Contact in the Processing of Thermoplastic Composite Laminates[END_REF]. It follows an Arrhenius law

D ic (x) = 1 w * × 1 + a * t 0 P app η (T (x))
η (T ) = A × exp E a RT [K] ( 23
)
where R is the gas constant. The pre-exponential factor A and the free energy E a are determined experimentally using rheometry measurements (see Fig. 7).

Equation ( 22) and ( 23) give the evolution of D ic as an ordinary differential equation:

∂D ic ∂t = D -4 ic a 0 b 0 2 × 1 1 + w0 b0 4 P app A exp - E a T (x) . (25)
A c c e p t e d M a n u s c r i p t The bad contact also results in a temperature gap at the interface. The thermal contact resistance associated with this gap decreases as the contact increases (as D ic increases). In this study this resistance evolves according to the empirical relation proposed by [START_REF] Levy | Inter-layer Thermal Contact Resistance Evolution with the Degree of Intimate Contact in the Processing of Thermoplastic Composite Laminates[END_REF]:

R c (D ic (x)) = D ic (t = 0) × a 0 1 kD 2 ic (x) + 1 -D ic (x) k air D ic (x) + 1 k , (26)
k being the thermal conductivity of the matrix and k air of the air. k air = 0.03 W /mK shall be used.

Note that because the temperature is not continuous across the interface, the interfacial source Qfric (eq. 21) is equally split on each side of the interface.

Adhesion evolution

In order to get a good adhesion, once intimate contact is achieved, the bond must be kept hot enough, and for a time long enough, to ensure diffusion of the polymer macromolecules across the interface. Following the reptation theory by [START_REF] De -Gennes | Reptation of a polymer chain in the presence of fixed obstacles[END_REF], the reptation time t r represents the time the macromolecule needs to fully change its configuration. t r can be modeled using an Arrhenius law:

t r = A r exp E a R.T [K] . ( 27
)
The complex viscoelastic moduli at high temperature were measured at f = 10 Hz with a rheometer as shown in Fig. 8. Using the Cox-Merck principle [START_REF] Ferry | Viscolelastic Properties of Polymers[END_REF], the characteristic time t r = 1 2πf = 15.9 ms (28)

A c c e p t e d M a n u s c r i p t is associated to this test. Following [START_REF] Whorlow | Reological Techniques[END_REF] this is the relaxation time at 281.5 • C where the two curves cross.which is the relaxation temperature for this test. Using eq. ( 27), A r is identified at this temperature as:

A r = 4.71 × 10 -13 s (29)

Once the macromolecule has fully diffused across the interface, healing is complete. Therefore, the degree of healing D h is defined such that [START_REF] Regnier | A Multi-physic and Multi-scale Approach to Model the Continuous Welding of Thermoplastic Matrix Composites[END_REF]:

∂D h ∂t = 1 t r (T )
.

(30) [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF] proposed a coupled model to ensure that only the surface already in intimate contact starts healing. [START_REF] Butler | An analysis of mechanisms governing fusion bonding of thermoplastic composites[END_REF] extended the idea and investigated which of the intimate contact or healing is the limiting phenomenon. [START_REF] Yang | Nonisothermal healing and interlaminar bond strength evolution during thermoplastic matrix composites processing[END_REF] suggested an improvement of the intimate contact model but kept the coupled model idea. Based on these models, the degree of adhesion

D a = D ic × D h ( 31
)
is defined. It ensures that only the fraction in intimate contact heals. D a (x) is computed for each position

x of the interface using D ic and D h obtained by solving the ordinary differential equations ( 25) and (30).

Implementation

The model is solved with finite element methods using the commercial software COMSOL multiphysics.

Thanks to the symmetries of the system, a quarter of the two dimensional geometry is simulated (see Fig. 9).

13

A c c e p t e d M a n u s c r i p t Sym.

Sym. It consists of half the upper composite adherents and a quarter of the flat energy director. The quarter geometry is meshed with 5063 unstructured triangles with refinement along the interface. The unknowns are: (i) the displacement vector and (ii) the temperature in the whole domain, obtained by solving the elastic problem and the heat transfer using a quadratic interpolation; and (iii) the degree of intimate contact and (iv) the degree of healing at each node of the interface. The couplings are summarized in Fig. 10.

Adherend Energy director

A backward Euler solver is used for the time integration. The resolution is stopped when the minimum temperature in the film exceeds T g + 20 • C, T g = 215 • C being the glass transition temperature of the PEI. This temperature is reached before 0.5 s. The computation is performed on a desktop computer in less than 5 min. Following the experimental procedure, at initial time t = 0, the sonotrode holding force F imp is applied.

Then, the amplitude a (t) of the displacement imposed by the sonotrode on the upper part of the assembly starts increasing. It is ramped from 0 at t = 0 to the nominal amplitude at time t = t ramp . The reference case presented [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF]

Results and Discussion

Dissipated powers

Fig. 11 shows the different powers dissipated in the system. It was obtained by integrating successively the visco-elastic dissipation Qbulk (eq. ( 6)), the friction dissipation Qfric (eq. ( 21)) and the sum of both (the total dissipation) over the domain.

Fernandez Villegas (2013) measured the experimental electric power delivered to the ultrasonic apparatus P rig . Because of several losses, amongst which:

• the generator efficiency itself

• acoustic losses

• dissipation in the composite plates only a fraction of this delivered power was dissipated in the system. Considering a constant efficiency χ = 13%, the experimental dissipated power P exp = χ • P rig fitted with the predicted total dissipation.

In Fig. 11 after the initial ramp that corresponds to the sonotrode amplitude increase, a first peak around t ramp = 0.05 s is associated with the highest friction dissipation. Despite the good qualitative representativity of the obtained power curves, there is a systematic discrepancy in this initial stage, which is also visible for other processing conditions (see Figs. 17 and18). This effect is mainly due to the modeling of the sonotrode transient regime. In practice, just after the contact with the plates, the vibration amplitude starts from zero to the desired value within a certain duration. Only after this stage, a stationary regime of the loading is reached. In order to simulate this initial step, a linear variation of the vibration amplitude a was imposed, which led to a somewhat quadratic evolution of the dissipated energy, as it can be expected from Eqs. ( 7) and ( 21).

Following that peak, adhesion started, such that the friction dissipation quickly decreased, leading to a slight decrease of the total power. Around t min = 0.38 s, accurately reproducing the experimental trend, the power rose.

Temperature predictions

Fig. 12 presents the temperature along the center of the energy director film at three different characteristic times. Note that because of symmetry only a half of the interface is plotted and position 0 corresponds to the center of the film. At t ramp = 0.05 s, the temperature showed a slight increase at the edge of the energy directors. This is explained by the slippage at the interface that is higher at the edge than at the Page 18 of 28 A c c e p t e d M a n u s c r i p t center (as shown in Fig. 13). It induces a higher friction dissipation. Then, the temperature increased until t = 0.15 s, when the hottest point (the edge) reached the PEI glass transition temperature T g = 215 • C.

Then, the visco-elastic dissipated power was steady (as shown in Fig. 11), as a larger part of the film reached the glass transition temperature. Eventually, at time t = 0.4 s, the whole film almost reached the glass transition temperature. The system stiffness dropped and allowed for higher deformation amplitudes in the film.

In addition with the increased loss modulus of the polymer around T g , the total visco-elastic dissipation quickly increased. This was observed experimentally [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF] and corresponds to the beginning of the flow.

Adhesion evolution

Fig. 15 shows the degree of adhesion along the energy director/adherends interface at different times.

At time t = 0.1 s, because of the higher temperatures, adhesion started at the edge (around x = 6 mm).

Page 19 of 28 This was confirmed experimentally [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF]. As shown in Fig. 16, weak adhesion starts at the edges. Right after that, heating and adhesion increased everywhere in the interface. At time t = 0.4 s, the degree of adhesion was non negligible everywhere along the interface, which was also observed experimentally.

Parametric study

A second simulation was performed with a higher sonotrode holding force F imp = 1500 N to reproduce the case 1500 N/86.2 µm investigated by Fernandez Villegas (2013). Because a higher holding force results in a reduced hammering effect, the hammering coefficient was increased to α h = 0.38. Using the same efficiency χ = 13%, the simulated power curve accurately predicts the plateau, followed by the increase around 0.22 s, as shown in Fig. 17.

Note that the the friction dissipation Qfric (eq. ( 21)) does not depend directly on the sonotrode holding force because the effect of P app was assumed to be negligible. A higher holding force will only result in a decreased hammering effect and thus, a higher energy transfer, and faster heating (term α h in eq. ( 21)).

Indeed, the tangential vibrating interfacial force τ (eq. 15) is significantly higher than the static tangential force µP app acting at the interface as discussed in section 3.2.1.

The higher holding force also resulted in a faster intimate contact and thus a faster adhesion. In that case, where F imp = 1500 N, the maximum degree of adhesion at the end of the heating phase (at t = 0.25 s) reached 0.34. As visible in figure 15, it is comparable to the maximum obtained in the reference case, where F imp = 500 N, after a shorter heating phase (t = 0.4 s).

A third simulation was performed with a lower holding force F imp = 300 N and a smaller sonotrode amplitude a = 51.8 µm. The lower sonotrode amplitude led to a reduced hammering effect even though the holding force was reduced. Indeed, using a hammering coefficient α h = 0.39 led to an acceptable recovery of the dissipated power, as shown in Fig. 18.

A c c e p t e d M a n u s c r i p t The developed numerical tool consistently approximated the experimental measurements, for different holding forces and sonotrode amplitudes. The only parameter to fit was the hammering coefficient, which was shown to control the transmitted power in the process and the magnitude of the total dissipation. In the presented simulations, α h was set to a constant empirical value, but more efforts are necessary to better model the evolution of this parameter with the sonotrode amplitude and holding force.

Conclusions

Ultrasonic welding of Carbon/PEI composite was investigated. A neat PEI film at the interface between the two plates to be welded acts as a flat energy director. It allows a local heating and ensures a progressive adhesion of the two adherends to be welded. The main contributions of the present work are the following:

• A numerical multiphysical model of the leading physical phenomena occurring during the heating phase was developed. The model consists of:

an elastic problem that predicts the effects of the ultrasonic vibration a heat transfer problem, that predict the temperature increase and a coupled bonding evolution problem at the interface, that predicts the evolution of the adhesion.

• The simulations aimed at replicating the experiments given [START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF].

-The results confirm that the heating is mostly initiated by friction dissipation between the flat energy director and the composite adherends.

-Once adhesion occurs, slippage is reduced and friction drops while viscoelastic dissipation takes over.

-The temperature at the interface progressively increases until it reaches the glass transition temperature. Afterwards, the flow of the resin can start, and the heating phase is finished.

• The predicted dissipated powers were compared with the experimental electric power delivered to the ultrasonic unit. A transmitted power of 13% of the apparatus power was found to accurately fit the predicted dissipated power in the bond for three different combinations of welding parameters.

A c c e p t e d M a n u s c r i p t where T is the tangent load and N is the normal load applied between the adherend and the energy director. µ was determined using a friction setup based on the ASTM method D1894 (ASTM International, 1995), initially developed for measuring thin plastic sheet friction coefficient. The setup was similar to the one proposed by [START_REF] Thije | Design of an experimental setup to measure tool-ply and ply-ply friction in thermoplastic laminates[END_REF], [START_REF] Lebrun | Thermoforming-Stamping of Continuous Glass Fiber/Polypropylene Composites: Interlaminar and Tool-Laminate Shear Properties[END_REF] or [START_REF] Sun | Interply friction of carbon fiber/epoxy prepreg stacks under different processing conditions[END_REF]. It consisted of two platens that applied a controlled normal load and was installed in an MTS tensile machine (depicted in Fig. B.21).

Two PEI films similar to those used as flat energy directors were attached on each platen, and a CF/PEI plate, similar to the adherends used in the study, was compressed between those wrapped platens. The contact area was that of the platens: 50.8 mm × 101.6 mm. Normal loads corresponding respectively to 86 and 129 kPa were applied and the plate was pulled through, using the MTS machine, at a constant velocity of successively 1.17 mm /s and 2.33 mm /s. The measured extraction force was averaged over steady state phase of the test, that was reached after about 20 mm . Using the two normal loads, values of the friction coefficient were obtained. The test was performed at room temperature. Results are given in table B.3. The four friction coefficient obtained using the two different normal and two different velocities presented less than 10% dispersion and showed the adequacy of the Coulomb law. An average of these four values was used in the simulation. Note that only the room temperature behavior was studied since the friction will mostly occur at the initial stage of the process, at low temperature.

Figure 1 :

 1 Figure 1: Principle of the ultrasonic welding.

Figure 2 :

 2 Figure 2: Thermal conductivity of the CF/PEI composite and the neat PEI matrix versus temperature.

Figure 3 :

 3 Figure 3: Volumetric heat capacity of the neat PEI matrix ρ P EI c P EI p versus temperature. It is approximated by a linear relation.

Figure 4 :

 4 Figure 4: Heat capacity of the CF/PEI composite c comp p

Figure 5 :

 5 Figure 5: Ultrasonic welding device. 1: sonotrode, 2: sliding platform, 3: clamp for upper sample, 4: clamp for lower sample[START_REF] Fernandez | Situ Monitoring of Ultrasonic Welding of Thermoplastic Composites Through Power and Displacement Data[END_REF].

Figure 6 :

 6 Figure 6: Contact surface profile of the composite.

  w * = 1 + w 0 /b 0 and a * = 5w * (a 0 /b 0 ) 2 are geometric parameters related to the interface roughness.

A

  = 7.23 × 10 -7 Pa.s E a = 111.7 kJ/mol .

Figure 7 :

 7 Figure 7: Logarithm of viscosity versus inverse of temperature for PEI resin. The linear fit gives the Arrhenius parameters.

Figure 8 :

 8 Figure 8: Elastic and Loss Moduli versus temperature for PEI measured at 10 Hz. The relaxation temperature of 281.5 • C is obtained.

Figure 9 :

 9 Figure 9: Mesh over the computation domain.

Figure 10 :

 10 Figure 10: The three physical problems to be solved and their couplings.

Figure 11 :

 11 Figure 11: Total simulated dissipated powers and measured transmitted power Fernandez Villegas (2013) (assuming an efficiency χ = 13%) versus time. Reference case with a sonotrode amplitude of 86.2 µm and a holding force of 500 N.

Figure 12 :

 12 Figure 12: Temperature predictions along the interface at three characteristic times.

Figure 13 :

 13 Figure 13: Temperature and displacement fields at time tramp = 0.05 s. General view and closeup.

Fig. 14

 14 Fig.14presents the temperature profile across the adherends and the flat energy director at three different locations: in the middle of the sample, at three quarters and at the edge of the film. Once again, for symmetry reasons, only a quarter of the domain is considered. The temperatures are plotted at time t = 0.4 s. During this initial heating phase, the heating of the bond was very local. That is very encouraging for an efficient welding without deconsolidation of the adherends. This was, once again, experimentallyobserved Fernandez Villegas (2013), during this initial heating phase. One should finally notice that the temperature in the middle plane of the film (y = 0) is higher than the temperature at the interfaces energy director/adherends (y = 0.125 mm), which shows that flow might start in the flat energy director centerplane before the interfaces reach T g .

Figure 14 :

 14 Figure 14: Predicted temperature profile at t = 0.4 s across the assembly thickness at three locations: bond centerline (--), bond quarter (••) and bond edge (-). Heating is localized in the energy director.

Figure 15 :

 15 Figure 15: Predicted degree of adhesion at different times along the interface. Adhesion starts at 0.05 s at the edge of the overlap.

Figure 16 :

 16 Figure 16: First physical transformations occur at the edges of the energy director and leave no significant traces on the composite adherends (100 ms vibration time at 500 N and 86.2 µm) (Fernandez Villegas, 2013).

Figure 17 :

 17 Figure 17: Total simulated dissipated powers and measured transmitted power Fernandez Villegas (2013) (assuming an efficiency χ = 13%) versus time. Sonotrode amplitude of 86.2 µm and holding force of 1500 N.

Figure 18 :

 18 Figure 18: Total simulated dissipated powers and measured transmitted power Fernandez Villegas (2013) (assuming an efficiency χ = 13%) versus time. Sonotrode amplitude of 51.8 µm and holding force of 300 N.

Figure A. 19 :

 19 Figure A.19: Obtained shift factor and its extrapolation to 20 kHz.

  Figure A.20: Elastic and loss moduli of PEI at 20 kHz.

Figure

  Figure B.21: Friction coefficient measurement setup comprising two platens and the tensile machine grip for tangent load.

  ultrasonic welding. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 1 :

 1 Intimate contact parameters.

  is replicated. The parameters that were not given in the text are in table 2.

Table 2 :

 2 Parameters used in the reference simulation.

	α h	hammering coefficient	0.32
	ω	sonotrode pulsation	125000 rad.s -1
	a	sonotrode amplitude	86.2 µm
	t ramp time to establish vibration	50 ms
	P app sonotrode holding pressure	1.55 MPa

Table B .

 B 3: Friction coefficient measurements.

	Velocity Normal Force Friction coefficient µ
	mm/s	kPa
	1.17	86	0.189
	2.33	86	0.196
	1.17	129	0.201
	2.33	129	0.196
			0.195 ± 3%

• dissipation/damping in the rig

Acknowledgement

The authors would like to thank professor Pascal Hubert without whom this collaboration would not have been possible. The authors also acknowledge the help of Marcus Scaramanga with the design and manufacturing of the friction measurement apparatus.

Appendix A. Determination of the viscoelastic modulus at high frequency

The elastic and loss moduli E and E of the neat PEI were measured using a DMA apparatus between room temperature and 235 • C for five frequencies f : 0.1, 1, 10 and 100 Hz. Using a time temperature superposition principle, the obtained data were shifted onto the f = 1 Hz master curve, assuming that:

and

The obtained shift factors a (f → 1) are plotted for each frequency in

Appendix B. Measurement of the friction coefficient

The classical coulomb law has been widely used to characterize composite friction behavior. It consists in considering a behavior such that a constant friction coefficient µ can be determined as:


Arthur Levy
email: levy@udel.edu

Dirk Heider

John Tierney

John Gillespie

Inter-layer Thermal Contact Resistance Evolution with the Degree of Intimate Contact in the Processing of Thermoplastic Composite Laminates

Keywords: Forming process, consolidation, thermal modeling, bonding mechanism, through thickness properties, on-line quality control, Carbon PEEK APC2

This paper focuses on the contact between layers in forming processes of composite laminate. The link between the degree of intimate contact and the consequent thermal contact resistance between layers is investigated. A hot plate forming process experiment allows to propose a relation and determine the missing parameter for APC2 thermoplastic prepreg composite. Beside the new proposed relation, this work showed that the internal thermal contact resistances in the laminate is significant. Therefore, thermal modeling of forming processes of composite laminate (such as automatic tape placement) should account for this phenomenon.

INTRODUCTION

For some decades now, composite materials tend to replace traditional metallic structures in the industry. Their specific properties make them very com-1 petitive compared to metals. Although thermoset matrix composites are traditionally used in the industry, for several years now, thermoplastic composites open new prospects.

Indeed, the ability to melt the matrix and the non-necessity to cure it open the way to new forming processes. One may give as an example the hot press forming [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF][START_REF] Mantell | Manufacturing process models for thermoplastic composites[END_REF], automatic prepreg tape placement [START_REF] Grove | Thermal modelling of tape laying with continuous carbon fibre-reinforced thermoplastic[END_REF][START_REF] Sonmez | Analysis of the on-line consolidation process in thermoplastic composite tape placement[END_REF][START_REF] Nicodeau | Modélisation du Soudage en Continu des composites à Matrice Thermoplastique[END_REF][START_REF] Tierney | Modeling of in situ strength development for the thermoplastic composite tow placement process[END_REF][START_REF] Khan | Identification of some optimal parameters to achieve higher laminate quality through tape placement process[END_REF], or new welding techniques [START_REF] Ageorges | Avdances in Fusion Bonding Techniques for Joining Thermoplastic Matrix Composites: a Review[END_REF][START_REF] Levy | Ultrasonic Welding of Thermoplastic Composites, Modeling of the Process Using Time Homogenization[END_REF]. All those processes consist in heating the part to melt the matrix and apply pressure to bond the layers together.

In order to better monitor such processes, a good understanding of the physical phenomena involved in bonding is required. In addition to experimental campaign, numerical simulation appears to be a great tool to determine the optimal range for the process parameters. One of the very influent parameter in the process of thermoplastic is the temperature. It underlines the need for an accurate thermal modeling when simulating those processes.

Beside the temperature field, a typical objective in the simulation of composite laminate processing is to predict the quality of the bonding between the layers. With this aim, the first step is to qualify the contact between the layers at the microscopic scale. Lee and Springer [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF] define a variable called degree of intimate contact that reaches 1 when contact is perfect. They propose a model of the layer surface roughness as a succession of rectangular asperities and give the degree of intimate contact as a function of temperature and pressure.

Because modeling the surface as a succession of rectangular asperities is a very basic approach Yang and Pitchumani [START_REF] Yang | A fractal Cantor set based description of interlaminar contact evolution during thermoplastic composites processing[END_REF] proposed an improvement of the geometrical description. After reviewing the different existing approaches for modeling the degree of intimate contact evolution, we make use of the Lee and Springer model that was widely used in the literature [START_REF] Mantell | Manufacturing process models for thermoplastic composites[END_REF][START_REF] Sonmez | Analysis of the on-line consolidation process in thermoplastic composite tape placement[END_REF][START_REF] Ageorges | Avdances in Fusion Bonding Techniques for Joining Thermoplastic Matrix Composites: a Review[END_REF][START_REF] Nicodeau | Modélisation du Soudage en Continu des composites à Matrice Thermoplastique[END_REF][START_REF] Lamethe | Polymer dynamics applied to PEEK matrix composite welding[END_REF][START_REF] Tierney | Modeling of in situ strength development for the thermoplastic composite tow placement process[END_REF][START_REF] Khan | Identification of some optimal parameters to achieve higher laminate quality through tape placement process[END_REF].

The presents work aims at showing that the thermal properties of the laminate is highly dependent on this degree of intimate contact. Indeed, as suggested by Barasinski et al. [START_REF] Barasinski | An Improvement in Thermal Modeling of Automated Tape Placement Process[END_REF][START_REF] Barasinski | In situ consolidation for thermoplastic tape placement process is not obvious[END_REF], a badly bonded laminate will result in air gaps between the layers and internal thermal contact resistance. This thermal contact resistance phenomenon is observed in other processes involving bad contact. For instance, in the injection molding process, the contact between the part and the mold is not perfect, and gives rise to such a thermal contact resistance [START_REF] Bendada | Analysis of thermal contact resistance between polymer and mold in injection molding[END_REF][START_REF] Masse | Heat transfer with mechanically driven thermal contact resistance at the polymer/mold interface in injection molding of polymers[END_REF].

The thermal contact resistance at a metal interface has been studied in the past. Thomas and Probert [START_REF] Thomas | Thermal contact resistance: The directional effect and other problems[END_REF] or Sridhar and Narh [START_REF] Sridhar | Measurement and Modeling of Thermal Contact Resistance at a Plastic-Metal Interface[END_REF] used a modeling of the surface geometry as a succession of asperities. Nonetheless no author related the thermal contact resistance to the degree of intimate contact, especially when they are evolving, which is likely the case during the processing of a composite laminate. This paper aims at relating internal thermal contact resistance and degree of intimate contact between the layers of a laminate.

In the first part of this paper, the existing models for the degree of intimate contact are reviewed. Then, focusing on the interface, homogenization consideration allows to identify two bounds for the thermal contact resistance as a function of the degree of intimate contact. In the second part, the experimental setup is presented. It consists in a specific hot plate processing, a nanoflash apparatus for measuring thermal properties, and an optical analysis of cross sections of the laminate for measuring degree of intimate contact. The results are finally presented and discussed in section 4 and 5.

Based on experimental results, an empirical relation between thermal contact resistance and degree of intimate contact is proposed and the importance of the phenomenon is highlighted.

ANALYTICAL MODEL

Intimate contact models

When processing prepreg composite using tape such as the one shown on figure 1, the quality of the contact between the tapes will evolve as the surface asperities get squeezed. During the last couple of decades, several models have been proposed to predict the evolution of this contact. This first section aims at reviewing those different existing models.

Because of the unidirectional fiber orientation, the roughness of the surface is suppose to be an extrusion in the direction of the fibers. The raw tape micrography 1 shows surface asperities of few tens of microns containing fibers. It corroborates this assumption. Therefore, the study is usually restricted to two dimensions with plain strain assumption. While processing, temperature and pressure applied on the tape will allow a squeezing of the asperities. Therefore, the degree of intimate contact, defined as the contact area ratio by Lee and Springer [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF] will evolve. Depending on the description of the initial surface roughness, different models have been proposed. [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF] modeled the initial tape roughness as a succession of rectangles as shown on Fig. 2a. The degree of intimate contact is classically defined as:

Rectangle model. Lee and Springer

D ic = b b 0 + w 0 (1)
where the geometric parameters are given on Figure 2a. The material being incompressible, the volume of the squeezed rectangles remains constant and writes:

V 0 = a 0 b 0 = ab, (2)
which allows to express the degree of intimate contact as a function of a only:

D ic = a 0 /a 1 + w 0 /b 0 . ( 3
)
Then, Lee and Springer consider a Newtonian flow of the rectangles in a lubrication framework. It is assumed that the asperity consists of an homogeneous material that behaves as a Newtonian fluid of equivalent viscosity µ (T ). The micrography 1 shows, indeed, that the asperities contain many fibers. At this point, improvements of the Lee and Springer model could be proposed to account for non-Newtonian behavior of the resin or better handling of the heterogeneous behavior. The following Arrhenius law

µ (T ) = A × exp E a T [K] ( 4
)
is proposed for the Newtonian viscosity thermal dependency, with the preexponential factor A and the free energy E a : The rectangle height a is found to be governed by an ordinary differential equation involving the applied pressure P app and the temperature T . Under constant pressure and isothermal conditions, an analytical solution is obtained. Using relation (3), the model for D ic then writes [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF]:

A = 1.14 × 10 -12 Pa.s E a = 26300 K -1 . ( 5
D ic = 1 1 + w 0 b 0 × 1 + 5P app µ (T ) 1 + w 0 b 0 a 0 b 0 2 t 1 5 . ( 6
)
Mantell and Springer [START_REF] Mantell | Manufacturing process models for thermoplastic composites[END_REF] extended this model to account for non-isothermal conditions few years later.

Fractal model. Because describing the surface as a succession of rectangles is not realistic, Yang and Pitchumani [START_REF] Yang | A fractal Cantor set based description of interlaminar contact evolution during thermoplastic composites processing[END_REF] proposed to improve the Lee and Springer model. Indeed, the initial rectangles dimensions a 0 , b 0 and w 0 cannot be determined physically and must therefore be fitted using experimental results. Their approach consists in describing the surface morphology as a Cantor set fractal surface (cf. fig 2b), where each geometric parameter can be obtained from a surface profile measurement. To model the squeezing of this morphology and the evolution of the degree of intimate contact Yang and Pitchumani propose the following assumption. The highest order generation of rectangle is squeezed first without any deformation of the lower generation (bigger rectangles). Once this n-th generation troughs is filled, squeezing of the next generation n -1 starts, and so on. The degree of intimate contact finally reaches 1 when the biggest troughs of the first generation are filled. For each successive generation n the squeezing of the rectangle is modeled using a similar method as that of Lee and Springer.

Finite element model.

A model using an even more realistic description of the initial surface has been lately proposed [START_REF] Gruber | Thermoplatic In Situ Placement Requires Better Impregnated Tapes and Tows[END_REF]. The real surface geometry measured with a profilometer, such as the one shown on figure 2c, is meshed. The squeezing of this geometry is then solved using a finite element method (polyflow software). The authors showed a good correlation of this new model with the Lee and Springer rectangle model. Finally, different model exist to describe the initial geometry of the tape surface. One should notice that all of these models consider the degree of intimate contact D ic as an accurate scalar to describe the contact at the interface. All these models also consider a Newtonian flow of the asperities and allow to relate the degree of intimate contact with the processing parameters (applied pressure, temperature). Further improvements of these model could account for a better description of the asperities behavior.

In this paper, for the sake of simplicity, we use the rectangle model, that has been largely used [START_REF] Mantell | Manufacturing process models for thermoplastic composites[END_REF][START_REF] Sonmez | Analysis of the on-line consolidation process in thermoplastic composite tape placement[END_REF][START_REF] Ageorges | Avdances in Fusion Bonding Techniques for Joining Thermoplastic Matrix Composites: a Review[END_REF][START_REF] Nicodeau | Modélisation du Soudage en Continu des composites à Matrice Thermoplastique[END_REF][START_REF] Lamethe | Polymer dynamics applied to PEEK matrix composite welding[END_REF][START_REF] Tierney | Modeling of in situ strength development for the thermoplastic composite tow placement process[END_REF], and proved its efficiency. The model chosen for D ic has no influence on the results shown. The presented review of the existing models for describing the evolution of the degree of intimate contact provides guideline for the reader. In this work, the degree of intimate contact is kept as a good candidate to describe the contact at the interface. Thus, this study aims at relating the thermal contact resistance and the degree of intimate contact regardless of the way the latter is obtained.

Microscopic modeling of the thermal contact resistance

The bad contact between the layers during the processing of a laminate results in an internal thermal contact resistance. This section aims at linking the degree of intimate contact D ic described in section 2.1 and the thermal contact resistance between layers. We remind that the thermal contact resistance R c is defined as:

Φ z = 1 R c T + -T - ( 7
)
where Φ z is the heat flux across the contacting interface, and T + and T -are the temperature on each side of the interface. Some author [START_REF] Thomas | Thermal contact resistance: The directional effect and other problems[END_REF] instead define the thermal contact conductance C c as the inverse of the thermal contact resistance. The thermal contact resistance is linked to the morphology of the contact at the microscopic scale [START_REF] Thomas | Thermal contact resistance: The directional effect and other problems[END_REF][START_REF] Salgon | A mechanical and geometrical approach to thermal contact resistance[END_REF].

Thin Layer Model

Let us consider the surface asperities of a tape as a thin resistive layer of thickness a, (as in figure 3) . This thin layer is composed of two phases: a composite phase and an air phase. The degree of intimate contact is assumed to be tractable to the whole thin layer thickness a, so that the volume fraction of the composite phase is considered to be D ic in the thin layer. The two dimensional framework adopted in the intimate contact modeling is retained here. Characterizing the thin layer properties is a spatial homogenization problem in heterogeneous media. The homogenized thermal conductivity λ * of the thin layer is defined such that the average flux φ writes

φ = λ * . ∇T . ( 8
)
Where ∇T is the temperature gradient and • is the spatial averaging operator in the thin layer. Let us consider the particular case where the temperature gradient is normal to the interface. For symmetry reasons, the heat flux is normal to the interface as well. The vertical direction is therefore a principal direction of λ * . This results in the following zero values in the representation of λ * in the (x, y, z) base:

λ * =    λ * xx λ * xy 0 λ * xy λ * yy 0 0 0 λ * z    (x,y,z) (9)
z being the direction normal to the interface.

Projecting equation ( 8) on this vertical direction of normal vector e z , one obtains:

φ z = λ * z ∇T z (10)
Now, one wants to identify with the interface magnitudes of equation [START_REF] Beran | Statistical Continuum Theories[END_REF].

We should notice that the interface is composed of two tapes in contact. Therefore there is two contacting thin layers. The interface then has a thickness 2a and is comprised between z = -a and z = +a, the origin being the contacting points. This assumption is enforced in the following experimental setup by using a cross ply stacking that prevents asperity conformation. The average values then write:

∇T z = 1 2a +a -a ∇T z dz = T + -T - 2a (11)
and

φ z = Φ z ( 12
)
So that:

Φ z = λ * z T + -T - 2a . ( 13
)
Finally, using equation ( 7), R c is identified as:

R c = 2a λ * z . ( 14
)
This equation shows that the thermal contact resistance behaves as a resistive thin layer of thickness 2a and of thermal conductivity λ * z

In order to obtain the evolution of the thermal contact resistance R c with D ic , one has to propose an homogenization law for λ * z . In the following paragraphs we remind the two bounds for the homogenized thermal conductivity λ * z of this thin layer in this homogenization framework. Those bounds are classical in the elastic theory (for instance [START_REF] Hill | The Elastic Behaviour of a Crystalline Aggregate[END_REF][START_REF] Kanit | Determination of the size of the representative volume element for random composites: statistical and numerical approach[END_REF]) and are extended without loss of generality to the heat transfer problem. One should notice that those two bounds are valid whatever the representation of the the surface asperity is. Therefore, the following analysis stands for any degree of intimate contact model chosen among the one presented in section 2.1.

Voigt Bound

Considering the vertical slab geometry shown on figure (4a), Beran [START_REF] Beran | Statistical Continuum Theories[END_REF] obtains the upper Voigt bound. In this case, the temperature gradient ∇T is the same in both phase and is therefore homogeneous in the thin layer

∇T = T + -T - 2a .e z ( 15
)
According to the Fourier law, the respective vertical flux φ i z in each phase simply writes: λ i z being the respective thermal conductivities of the phases in the z direction. Averaging the flux gives

φ i z = λ i z ∇T.e z ( 16
)
φ i z = λ i z ∇T.e z ( 17
)
which allows to identify

λ * z = λ i z = D ic λ + (1 -D ic ) λ air , ( 18
)
λ air being the conductivity of the isotropic air phase and λ that of the com-215 posite in the transverse direction z. If we neglect the thermal conductivity of the air versus that of the composite, λ * z finally writes

λ * z = λD ic . ( 19
)
Reminding that

D ic = a 0 a 1 + w 0 b 0 = D ic (t = 0) a 0 a ( 20
)
we can substitute the value of a in equation ( 14) and obtain:

R c = D ic (t = 0) D 2 ic 2a 0 λ (21) 10
In terms of conductance, it writes:

C c = D 2 ic λ D ic (t = 0) .2a 0 (22)
In this case the thermal contact conductance C c is proportional to the square of the degree of intimate contact. It is explained by the fact that, when D ic increases, both the composite phase volume fraction increases and the thickness 2a of the thin layers decreases.

Reuss bound

Considering the other extreme geometry for which the interface is the stacking of alternate air and composite horizontal slabs (like one figure 4b), one obtains the lower Reuss bound [START_REF] Beran | Statistical Continuum Theories[END_REF]. In this case, the vertical flux φ is the same in both phases. The Fourier law ( 16) on the z direction then rewrites:

φ z = λ i z ∇T i .e z ( 23
)
dividing by λ i z and averaging, one obtains

1 λ i φ z = ∇T i .e z ( 24
)
It gives, by identifying with equation ( 10):

1 λ * z = 1 λ i (25) or 1 λ * z = 1 -D ic λ air + D ic λ ( 26
)
Substituting in equation ( 14) and using the definition of D ic , one obtains

R c = D ic (t = 0) .2a 0 × 1 -D ic D ic 1 λ air + 1 λ . ( 27
)
3 EXPERIMENTAL METHOD

Material

The material used in this study is APC2 prepreg unidirectional composite provided by Cytec (IPS05-09-001). It is made of AS4 carbon fibers and PEEK thermoplastic polymer. The fiber volume content is 64%. It was shipped by Cytec in 12" width tape that was cut into 3" by 3" (76.2 mm by 76.2 mm) squares.

The melting temperature T m and the glassy temperature T g of PEEK are known to be [START_REF] Cogswell | Thermoplastic aromatic polymer composites[END_REF][START_REF] Lamethe | Polymer dynamics applied to PEEK matrix composite welding[END_REF]:

T m = 334 • C T g = 143 • C (28)
The heat capacity ρc of APC2 at room temperature has been measured with a differential scanning calorimeter (DSC):

ρc = 2.2 × 10 6 J /m 3 K (29)
and is in agreement with the values found in the literature [START_REF] Cogswell | Thermoplastic aromatic polymer composites[END_REF][START_REF] Ageorges | Characteristics of resistance welding of lap shear coupons : Part I : Heat transfer[END_REF][START_REF] Nicodeau | Modélisation du Soudage en Continu des composites à Matrice Thermoplastique[END_REF][START_REF] Levy | A levelset based approach for the finite element simulation of a forming process involving multiphysics coupling : ultrasonic welding of thermoplastic composites[END_REF].

The transverse thermal conductivity λ of the APC2 at room temperature can be found in the literature [START_REF] Grove | Thermal modelling of tape laying with continuous carbon fibre-reinforced thermoplastic[END_REF][START_REF] Ageorges | Characteristics of resistance welding of lap shear coupons : Part I : Heat transfer[END_REF][START_REF] Nicodeau | Modélisation du Soudage en Continu des composites à Matrice Thermoplastique[END_REF] and is taken equal to:

λ = 0.63 W /m.K. ( 30
)

Surface Analysis

The surface of a raw APC2 tape is analyed using cross section micrograph such as the one shown on figure 1. A simple thresholding allows to extract the profile function. The magnification is ×5 and the resolution is about 2 pix /µm. Four profiles were measured, each one on a length of about 600 µm.

A typical profile is shown on figure 2c. The depth of the asperities a 0 is obtained as twice the standard deviation of the scanned function (similar to Yang and Pitchumani [START_REF] Yang | A fractal Cantor set based description of interlaminar contact evolution during thermoplastic composites processing[END_REF] approach):

a 0 = 21.1 µm. ( 31
)

Processing

In order to measure the thermal contact resistance between each layer of a laminate, badly bonded samples were manufactured. A specific hot plate processing was designed. Preparation Three layers of 3 in by 3 in (area S = 5806 mm 2 ) of prepreg are laid up on a 0 • /90 • /0 • sequence. This allows the degree of intimate contact to be minimized by avoiding fiber conformation that would arise in a unidirectional stacking [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF]. It will also help to determine the interface position accurately in the micrographs, even for perfectly bonded samples. The stacking is positioned in a folded film of Kapton coated with releasing interface (Frekote 55-NC).

Consolidation Seven different plates are processed. Each one are kept between the hot plate and the weight (see fig. 5) for different times, denoted hereunder as "consolidation times". The seven consolidation times are respectively 5 s, 10 s, 15 s, 20 s, 30 s, 1 min and 3 min. During these consolidations, the weight applies a compaction pressure P app . It is assumed homogeneous on the whole sample area, so that: The composite thermal diffusivity is λ/ (ρc). The characteristic heat diffusion time t c through thickness of the plates is therefore classically expressed as [START_REF] Carslaw | Conduction of Heat in Solids[END_REF]:

P app = wg S = 1340
t c ∼ (3e) 2 ρc λ = 0.7 s. ( 34
)
This value is obtained using the thermal conductivity and specific heat of the composite at room temperature (equations ( 29) and ( 30)). Even if consolidation occurs at high temperature, those properties will not vary over more than an order of magnitude. Therefore, a characteristic diffusion time around 1 s is reasonable. During consolidation heating occurs on both sides of the sample since the hot plate and the weight were preheated. The consolidation times being way higher than t c , the process can be considered as isothermal at temperature:

T app = 400 • C. ( 35
)

Measurement of Thermal Contact Resistance

The thermal contact resistance is computed using through thickness thermal diffusivity measurements. Nanoflash equipment (LFA 447 Nanoflash, Netzsch Instruments, Inc.) is used according to ASTM standard E1461 [START_REF]ASTM E1461 -07 Standard Test Method for Thermal Diffusivity by the Flash Method[END_REF]. It consists in applying a flash of light on one side of the sample and measuring the increase in temperature on the other side using infrared measurement. The tests are performed at 25 • C. The flash light emitted with a Xenon light has a broadband between visible and near infrared. The pulse is set to "long" which correspond to a rectangular pulse of 700 µs. The sensor used to measure temperature on the opposite face is an InSb infrared detector cooled with liquid nitrogen.

Sample preparation Square samples of dimension 27 mm by 27 mm were cut in the middle of the processed plate. Those samples were then coated with graphite to ensure that they are black bodies. A metal mask used in the nanoflash equipment restricts the measured area to a disk of diameter 12.7 mm. Each sample was exposed to 10 shots at 25 • C. Through thickness diffusivity D of the samples is then computed using the temperature measurement on the opposite side of the sample. The Cowan method [START_REF] Cowan | Pulse Method of Measuring Thermal Diffusivity at High Temperatures[END_REF], that accounts for heat loss in the air, is used. Notice that the heating resulting from the flash is lower than 0.1 • C so that the material properties at room temperature can be used in the analysis.

Thermal Contact Resistance Computation

Knowing the heat capacity ρc of the sample, we can obtain the equivalent through thickness thermal conductivity λ eq of each sample as:

λ eq = Dρc.
(36)

The sample being modeled as a series of three layers separated by two interfaces, we can relate this equivalent conductivity to the properties of each component. Following Maillet et al. [START_REF] Maillet | Non-destructive thermal evaluation of delaminations in a laminate: Part I -Identification by measurement of thermal contrast[END_REF] work, the air gap specific heat can be neglected versus that of the composite. The contact is therefore purely resistive. Thus, considering the steady state configuration will inform us on this resistive property. The heat flux φ in each of those components would be equal and would write successively:

φ = λ e T sup -T + 2 R c .φ = T + 2 -T - 2 φ = λ e T - 2 -T + 1 R c .φ = T + 1 -T - 1 φ = λ e T - 1 -T inf ( 37
)
where λ is the composite through thickness conductivity, e is the tape thickness, R c is the thermal contact resistance searched for, and the T i are the temperatures defined on Figure 6. The whole sample thickness being 3e, one can also write in term of equivalent conductivity:

φ = λ eq 3e (T sup -T inf ) . ( 38
)
Figure 6: Unidimensional through thickness thermal modeling of the laminate.

Equations ( 37) and (38) finally lead to the relationship between the thermal contact resistance and the equivalent conductivity:

R c = 3e 2 1 λ eq - 1 λ (39)
The results obtained for each consolidated plate are given on Figure 9.

In the presented analysis the bulk property λ of the tape is considered constant. One may argue that the processing might change it because of internal evolution of the matter, in particular intra-layer voids content. Nonetheless, the pressure P app = 1340 Pa applied is far from a real industrial pressure that might be encountered in a hot press or an autoclave. It does not likely influence the internal voids in the material. As a confirmation, the micrography corresponding to the 5 s and 180 s consolidated plates were analyzed optically and their intra-layer void content were found similar (10.0% for the 5 s sample and 10.2% for the 180 s one). Therefore, the term consolidation, used in this paper, refers to the contact evolution between the layers, but not to void reduction that the matter would undergo in an industrial process.

One should finally notice that the intra-layer voids are distributed slightly non-homogeneously over the layer thickness (as shown on figure 7). The material conductivity λ therefore represents the thermal conductivity of these composite tapes homogenized over the thickness.

Measurement of Intimate Contact

In order to relate the thermal contact resistance measured in the previous section to the intimate contact, a measurement of the degree of intimate contact D ic has been done on each consolidated plate.

Sample preparation

Samples were cut out of the remains of each plate. The four sections facing the sample used in the nanoflash were analyzed for each plate. They were embedded in epoxy resin and polished down to 3 µm. Gray scale pictures such as the one shown on Figure 7 were then taken with a microscope set on a magnification of ×5. The resolution is about 2 pixels per microns.

Intimate Contact Computation

In order to quantify the degree of intimate contact out of those micrographs, the gray scale value was plotted along the two interfaces upper/middle and middle/lower layer. These lines are taken in the center of the interface region which is easily determined optically thanks to the fiber orientation change. The darker region are assumed to be voids whereas lighter regions can be either matrix (light gray) or fibers (white). A simple thresholding allows to obtain the contact length along these lines. The gray scale threshold value is determined independently for each picture by checking that it is in the range between the lowest gray scale value for matrix and the highest value for voids [START_REF] Santulli | Void content measurements in commingled E-glass/polypropylene composites using image analysis from optical micrographs[END_REF]. Because of the two dimensional plane strain assumption, the contact length ratio is constant in the third dimension. The degree of intimate contact is therefore directly the contact length ratio. The four values obtained for each section surrounding the nanoflash sample are then averaged successively for the upper and lower interfaces.

RESULTS

Intimate contact

The measured degree of intimate contact for each consolidation times are shown on Figure 8. One can notice the difference between the degree measured on the upper and lower interfaces, especially for the short time consolidation plates. This is due to the operating conditions. Their is a short lag between the positioning of the sample on the hot plate, and the closing of the mold with the hot weight. This lag (lasting usually less than a second) results in non uniformity in the heating. The heating starts earlier from the bottom face. The lower interface will reach processing temperature slightly earlier. It also explains that the longer the consolidation time, the smaller the difference. Hereunder the upper and lower degree of intimate contact are averaged.

The parameters a 0 /b 0 and w 0 /b 0 of the Lee and Springer model (eq. ( 6))

are obtained using a least square method with this average value for each plate. It consists in minimizing the norm 2 error err (a 0 /b 0 , w 0 /b 0 ) between the model and the experiment:

err (a 0 /b 0 , w 0 /b 0 ) = n [D ic (a 0 /b 0 , w 0 /b 0 , t n ) -D n ic ] 2 (40)
where the summation is done over each experimental value (each consolidation time), D ic is the relation given in equation ( 6), and D n ic are the average value of degree of intimate contact measured experimentally for each consolidation time t n . The minimization problem is solved numerically using a direct simplex method in Matlab (fminsearch function). One obtains

a 0 b 0 = 1.237 w 0 b 0 = 0.743 (41)
which correspond to an initial degree of intimate contact

D ic (t = 0) = 0.574. ( 42
)
These values are in agreement with the previous identification by Lee and Springer [START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF] who found an initial degree of intimate contact D ic = 0.5. The difference is due to difference in the material which surface roughness has evolved with the production lines in 20 years.

One should finally notice that the high dispersion of measured degree of intimate contact is related to the experimental method used. For each sample, the degrees of intimate contact were measured out of four cross sections. This corresponds to about 40 pictures for each sample. Depending on the position, the intimate contact can highly vary. As in the optical determination of void content [START_REF] Santulli | Void content measurements in commingled E-glass/polypropylene composites using image analysis from optical micrographs[END_REF], the higher the number of sections analyzed, the more accurate the degree of intimate contact measurement. Nonetheless, one should notice that Santulli et al. [START_REF] Santulli | Void content measurements in commingled E-glass/polypropylene composites using image analysis from optical micrographs[END_REF] gave a void fraction error of 10% using no more than 20 images, which suggests that our measurement is accurate within less that 10%.

Figure 9: Thermal contact resistance vs. degree of intimate contact.

Thermal contact resistance

The measured thermal contact resistance are plotted on Figure 9. Once again, the relative error bars are very wide. It represents the maximum error within the 40 pictures analyzed for each point. Nevertheless, following Santulli et al. [START_REF] Santulli | Void content measurements in commingled E-glass/polypropylene composites using image analysis from optical micrographs[END_REF] approach, the error on the deduced void content is lower than 10%.

The Reuss and Voigt bounds (eq. ( 21) and ( 27)) are plotted on the graph using the value of a 0 obtained in section 3.2. One can notice that the experimental values lie within the bounds.

Following the Voigt-Reuss-Hill [START_REF] Hill | The Elastic Behaviour of a Crystalline Aggregate[END_REF] principle, one may suggests a basic arithmetic average of the Voigt and Reuss bound as an empirical relation between thermal contact resistance and degree of intimate contact:

R c (D ic ) = D ic (t = 0) a 0 1 λD 2 ic + 1 -D ic λ air D ic + 1 λ . ( 43
)
Note that this relation stands for the experimental points on figure 9 and does not depend on the intimate contact model used.

Actually, implementation of a nanoflash device on a production site to perform in-situ control is possible, and even easier, than an ultrasonic technique because it is a non contacting method. A similar non destructive thermal investigation method was suggested by Maillet et al. [START_REF] Maillet | Non-destructive thermal evaluation of delaminations in a laminate: Part I -Identification by measurement of thermal contrast[END_REF] to predict delamination on composite part during their lifetime.

Importance of the Phenomenon

In a forming process, the intimate contact and heat transfer phenomena are fully coupled. The degree of intimate contact depends on the temperature history and the temperature depends on the thermal contact resistance that depends on the degree of intimate contact. Accurately modeling an industrial forming process in order to predict the degree of intimate contact therefore requires the simultaneous solving of the heat transfer and the degree of intimate contact evolution. With this aim, relation (43) should be used in conjunction with an appropriate degree of intimate contact model (as reviewed in section 2.1). Moreover the thermal contact resistance values obtained are ranging between 5 × 10 -5 m 2 K /W and 5 × 10 -4 m 2 K /W. Those values are comparable with the ones obtained in other forming processes, such as the thermal contact resistance between mold and part in injection molding [START_REF] Masse | Heat transfer with mechanically driven thermal contact resistance at the polymer/mold interface in injection molding of polymers[END_REF][START_REF] Bendada | Analysis of thermal contact resistance between polymer and mold in injection molding[END_REF]. This confirms the importance of the phenomenon that will affect the heat transfer during processing of composite laminate.

As a matter of fact, the thermal contact resistance highly modifies the equivalent through thickness thermal conductivity. As shown on Figure 10, neglecting the phenomenon would be similar to making a mistake of more than 50% on the through thickness thermal conductivity.

Finally one should notice that the present study focused on the intimate contact, which is an interlayer void. To this end, the samples studied exhibited similar intra-layer void content. In a real industrial process, the intra-layer void content is likely to evolve too, resulting in an evolution on the bulk composite property. This issue might be addressed in future studies.

CONCLUSION

A relation between the degree of intimate contact and the internal thermal contact resistance between the layers was obtained with an experimental This thermal contact resistance should definitely be taken into account in the thermal simulation of composite laminate forming processes. This is the aim of the future development of a simulation tool accounting for the full coupling between intimate contact and heat transfer in the automatic tape placement process.

Figure 1 :

 1 Figure 1: Micrograph of a raw APC2 tape.

  As a succession of rectangles[START_REF] Lee | A model of the manufacturing process of thermoplastic matrix composites[END_REF].

  As a fractal Cantor set[START_REF] Yang | A fractal Cantor set based description of interlaminar contact evolution during thermoplastic composites processing[END_REF].(c) Using a real measured profile.

Figure 2 :

 2 Figure 2: Modeling of the initial surface roughness.

Figure 3 :

 3 Figure 3: Representation of the interface as an heterogeneous two media thin layer.

( a )

 a Vertical slabs, homogeneous temperature gradient, Voigt bound. (b) Horizontal slabs, homogeneous heat flux, Reuss bound.

Figure 4 :

 4 Figure 4: The two extreme morphologies for the thin layer leading to the two bounds.

Figure 5 :

 5 Figure 5: Experimental setup.

  Pa (32) g = 9.81 m.s -2 being the gravity acceleration. The three layers of thickness e give a total plate thickness 3 × e = 0.48 mm (33)

  (a) General view. The arrows materialize the lines along which the intimate contact analysis is performed.(b) Detail view with the line along which the intimate contact analysis is performed.

Figure 7 :

 7 Figure 7: Micrograph of the 10 s consolidated plate (magnification ×5 ).

Figure 8 :

 8 Figure 8: Measured degrees of intimate contact.

Figure 10 :

 10 Figure 10: Equivalent thermal conductivity through thickness for different consolidation times.

ACKNOWLEDGEMENT

The authors would like to thank EADS CTO North America and especially Marc Fraser for funding this research. Our thanks also go to Patrice Lefebure, from EADS IW, France who allowed this collaboration.

DISCUSSION

In the forming process of thermoplastic composite, predicting the quality of the bonding between layers requires a good modeling of the intimate contact evolution. The presented results focused on two coupled phenomena: the intimate contact evolution and the thermal contact resistance at the interfaces. Those two properties are shown to be related.

Linking Thermal Contact Resistance and Degree of Intimate Contact

Equation (43) proposes a relation between the thermal contact resistance and the degree of intimate contact. Note that all the parameters in the relation are physical. Even a 0 , the thickness of the initial thin layer, can be obtained as twice the standard deviation of the initial surface profile function, as described in section 3.2. This relation can be used as is, regardless of the modeling of the degree of intimate contact D ic .

Because this relation uses only physical parameters, its temperature dependency is straightforward: λ and λ air only need to be adapted to reflect their variation with temperature. Literature values such as Cogswell [START_REF] Cogswell | Thermoplastic aromatic polymer composites[END_REF] or Ageorges et al. [START_REF] Ageorges | Characteristics of resistance welding of lap shear coupons : Part I : Heat transfer[END_REF] might be used. The relation can therefore be extended to high temperature and used for in situ modeling of thermal contact resistance while processing.

The experimental work consisted in analyzing the produced samples by two means:

1. The nanoflash experiment allows to quickly obtain the average thermal contact resistance over the area of the flash beam.

2. The optical analysis requires a large number of image processing to obtain an accurate spatial average value of the degree of intimate contact. Given the sample preparation requirement (embedding and polishing), this is a time-consuming measurement.

The results showed a clear link between intimate contact and thermal resistance. Therefore, in a production framework, one could consider the thermal contact resistance to be the discriminant parameter for qualifying the quality of the intimate contact. Instead of requiring a minimum degree of intimate contact, one may require a maximum thermal contact resistance per layer.