Sophie Guillemet-Fritsch

Christophe Tenailleau
email: tenailleau@chimie.ups-tlse.fr

Helene Bordeneuve
email: cbordeneuve@laas.fr

Abel Rousset
email: drousset@chimie.ups-tlse.fr

Magnetic properties of cobalt and manganese oxide spinel ceramics

Keywords: magnetism, neutron diffraction, cobalt and manganese oxides, spinel, ceramics

Magnetic susceptibility measurements, magnetization and neutron diffraction results at low temperature for cobalt and manganese oxide spinel ceramics are presented. The Curie temperature varies similarly with the sample composition in ceramics and powders. The experimental molar Curie constant variation is explained by the presence of Co 2+ , Co III , Mn 3+ and Mn 4+ , and possibly Co 3+ in the octahedral sites for the cobalt rich phases. The magnetic moments of the cations in tetrahedral and octahedral sites are not collinear and the global magnetization is oriented in a third direction.

Introduction

Cobalt and manganese oxides Mn 3-x Co x O 4 (0≤x≤3), which crystallize in the spinel structure, are multiferroics that can be used in various technological applications (batteries, memory devices, NTC thermistors…) thanks to their thermal, optical, electrical and magnetic properties.

We have recently undertaken a detailed study of the structural properties of Mn 3-x Co x O 4 ceramics over the whole solid solution range in relationships with their electrical properties [1]. Such materials exhibit very interesting properties due to the multiplicity of each cation that can be present on the tetrahedral A and octahedral B sites, with various possible oxidation numbers, spin states and orbital moments. Magnetic measurements can help to describe the cation distribution of cobalt and manganese in spinels thanks to their different scattering factors and characterize their magnetically ordered state at low temperature.

Below T N = 42K, Mn 3 O 4 is ferrimagnetic [2]. Following Yafet-Kittel's theory, the haussmanite magnetic structure can be described by a double crystal structure along the b-axis. The Mn 2+ magnetic moments for the A sites are ordered in the [010] direction with a moment of 4.2µ B [3]. For the B sites, the magnetic moments are sub-divided in two non collinear sub-lattices in the (100) plane with a resulting moment of 1.96µ B oppositely aligned to the tetrahedral moment. At 33K, the magnetic and crystallographic structures become identical with antiferomagnetic couplings. The B-B interaction is predominant due to the low distances separating the Mn 3+ . This allows a direct interaction exchange by Mn 3+ ions orbital covering. For the A-A and A-B interactions, a superexchange can take place through the oxygen ions.

On the other end of the Mn 3 O 4 -Co 3 O 4 system, the cobalt oxide spinel is antiferromagnetic below T N = 40K [4]. Structural, thermodynamic and magnetic calculations usually agrees with a high spin Co 2+ on the A sites and low spin Co III on the B sites [5][6][START_REF] Griffith | The theory of transition-metal ions[END_REF]. For the A sites, the Co 2+ magnetic moments are spread out on two equal sub-lattices of opposite orientations. Each Co 2+ ion is surrounded by four neighbors in tetrahedral environment with anti-parallel moments compared to the central ion. The macroscopic global magnetization is then equal to zero. In comparison with other direct spinels, Co 3 O 4 exhibits strong interactions between the Co 2+ in the A sites that are probably due to Co III -O-Co III super-exchange interactions [START_REF] Roth | [END_REF]. When substituting Co for Mn, large ferromagnetic components appear and a material taken from the Mn 3-x Co x O 4 series (0<x≤3) is usually ferrimagnetic [9,10].

Sample preparation

Mixed manganese and cobalt oxalic powders were prepared by the co-precipitation method using ammonium oxalate and cobalt and manganese nitrates in stoichiometric proportions [11]. Oxalic powders were then heated for 4 hours in air at 800°C for sample decomposition. The resulting mixed manganese and cobalt oxide powders were either mixed with an organic binder before a 2×6mm (height×diameter) disc was put under 500MPa for 30s and left overnight at a sintering temperature 1160≤T 1 ≤1300°C (conventional sintering technique), or prepared as ceramics using a Spark Plasma Sintering apparatus in the P = 50MPa for 5min and 700≤T≤750°C conditions (SPS technique) [1]. The thermal cycles are schematized in Figure 1. The conventional sintering technique was applied to the samples for which x<1.78 and the SPS technique for x≥1.78 in order to prepare high density pellets of single phase ceramics. For the former technique, samples were slowly cooled down from T 1 to T 2 and left for 1 hour before quenching in water to prevent sample cracking (see Fig. 1), while for the latter technique a (Co,Mn)O thin layer formed on both sides of each pellet was removed by polishing before measurements.

Structural characterization

For x<1.78, a pure tetragonal phase was evidenced by XRD at room temperature for each oxide spinel ceramic while a pure cubic phase was characterized for any further x values.

Room temperature neutron diffraction measurements allowed us to determine the cation distributions for tetrahedral and octahedral sites of the manganese and cobalt oxide spinel ceramics [1]. For 0≤x≤1, Mn 2+ on the A sites are progressively replaced by Co 2+ while Mn 3+ remain on the B sites. For 1<x≤3, Co 2+ and low spin Co 3+ (Co III ) appear on the B sites in the presence of Mn 3+ and Mn 4+ . This cation distribution, confirmed by thermogravimetric analysis performed on the starting oxide powders [12], is original in ceramics and was, to our knowledge, only mentioned earlier for Mn x Co 3-x O 4 (0≤x≤1) thin films and CoMn 2 O 4 nanocrystals [13,14], while both manganese and cobalt oxidation states have recently been reported in RE-perovskites [15,16]. Our ceramics show the highest electrical conductivity in the region where a maximum of couples Co 2+ /Co III and Mn 3+ /Mn 4+ concentrations are present in the B sites [1]. Presence of two oxidation states for each element drives to specific conduction mechanisms and double-exchange magnetic interactions.

Magnetic susceptibility measurements

10 to 20 mg of each sample powder, obtained after grinding of the ceramic, was introduced into a transparent diamagnetic pharmaceutical holder for magnetic susceptibility measurements with a MPMS-XI Quantum Design magnetometer put under vacuum. A magnetic field of 3000Oe was applied during measurements in the [2-300K] temperature range. Except for Co 3 O 4 , which is antiferromagnetic below the magnetic ordering temperature T N , the thermal variation of the inverse molar magnetic susceptibility is associated with a ferrimagnetic behavior. The variation of the T C Curie temperature as a function of the cobalt amount in the manganite spinel ceramics (see Table 1) is very close to that previously observed for powders [17].

For our Co 3 O 4 ceramic, T N is equal to 33K while Roth found a value of 40K [4]. The variation of T C obtained for ceramics is very similar to those reported earlier for sample powders of same composition despite different thermal treatments, especially with slower cooling in the case of powders. For an increasing cobalt content up to x ~ 2, T C increases, then it decreases. Therefore, this is where the concentrations of manganese and cobalt couples are the highest that the magnetic interactions are the strongest. For all ceramics, the paramagnetic temperatures Θ p , which are characteristics of the strongest magnetic interactions, are negatives with relatively high values, in accordance with predominant antiferromagnetic interactions (see Table 1).

Experimental molar Curie constant C m extracted from the magnetic susceptibility variations versus temperature were compared to those reported earlier in the literature with various cation distributions (see Table 1). C m of Mn 3-x Co x O 4 (1.54≤x≤3) ceramics decreases when the cobalt content increases, in relation with the appearance of diamagnetic Co III in octahedral sites. In order to explain the experimental C m values, it is necessary to use the magnetic orbital contribution of Co 2+ as mentioned in previous works [START_REF] Michel | Phénomènes magnétiques et structure[END_REF][START_REF] Poix | [END_REF]. Therefore, C m is equal to 2.18 and 4.07emu.mol -1 .K -1 for Co 2+ in tetrahedral and octahedral environment, respectively. For the trivalent cobalt in octahedral sites, the structural and electrical properties relationships showed that high spin Co 3+ would exist in the structure, with a maximum concentration corresponding to the one for Co 2+ in the same sites, the main part being of low spin Co III . In Table 1, the C m theoretical values for only Co III , and (Co 3+ +Co III ) are given for comparison. The experimental C m values are closer from the theoretical ones in the latter case, which still remains to be confirmed by using spectroscopic techniques, for instance. However, if the usual cation distributions proposed in the literature are taken into account with either (Co III and Mn 3+ ) or (Co 2+ , Co III and Mn 4+ ) on the B sites, the experimental C m values can never be explained. Only, the presence of Co 2+ , Co III , Co 3+ , Mn 3+ and Mn 4+ can justify the values obtained in our ceramics for 1<x<3.

Table 1: Curie temperatures and molar Curie constants for Mn

3-x Co x O 4 ceramics x T C [K] θ P [K] C m exp [uem.mol -1 .K -1 ] C m theo (Co III B 100%) [uem.mol -1 .K -1 ] C m theo ((Co III +Co 3+ ) B ) [uem.mol -1 .K -1 ]

Magnetization at 5 K

Magnetization measurements were performed on each grinded ceramic with the MPMS-XI Quantum Design magnetometer at 5K with a magnetic field H varying from 500 to 70000Oe. First magnetization variation I vs field for Mn 0.78 Co 2.22 O 4 ceramic is represented in Fig. 2, for example. For all compositions under study, the variations become linear at high field (H>20000Oe) and the saturation could never be reached, even for the highest field applied. Therefore, our ceramics exhibit a strong magneto-crystalline anisotropy and the magnetic moments of cations in A and B sites are not collinear. The spontaneous magnetization I o was determined by extrapolating to H=0 the linear variation of the magnetization vs field. Fig. 2 shows the complex I o variation as a function of the cobalt content for our ceramics, which is very similar to the variations observed by previous authors on oxide powders of the same compositions [9,17]. The sample thermal treatment has therefore no real influence on the magnetic ordering. The spontaneous magnetizations are very small compared to the values determined for a collinear model with two sub-lattices of Néel by considering the average distributions usually given in the literature with the presence of Co III or (Co III and Co 3+ ) in B sites. This result was already observed in other manganese oxide spinels such as NiMn 2 O 4 and FeMn 2 O 4 with I o = 1.63 and 1.55µ B .mol -1 and M s = 3.83 and 3.82µ B .mol -1 , respectively [20,21]. The lack of saturation of the experimental magnetization and the complex magnetic structures of Yafet-Kittel type in the tetragonal phase or of 'star' type in the cubic phase can explain these differences. Thus, from a simple consideration of a saturation moment it seems difficult to have a complete cation distribution over the A and B sites in the Mn 3 O 4 -Co 3 O 4 system.

Neutron diffraction at 10 K

The macroscopic measurement of magnetization is not the only technique that can show evidence for the presence of a spontaneous magnetic order in ferromagnetic samples. Neutron diffraction can also be a really interesting technique thanks to the neutron spin that can interact with the electron spin. Besides the structural information given by neutron diffraction measurements, the location and orientation of the magnetic moments for each A and B network can be determined below the T C magnetic ordering temperature, in relation with the cation distributions and oxidation states.

Four samples of Mn 3-x Co x O 4 ceramics, with x = 1.54, 1.78, 2.22 and 2.77, were measured by neutron diffraction at 10 K with a Helium cryostat on the Super-D2B instrument (wavelength of 1.594Å) at the ILL in Grenoble, France. Fig. 3 shows the neutron diffraction patterns of Mn 1.46 Co 1.54 O 4 (tetragonal phase) and Mn 0.78 Co 2.22 O 4 (cubic phase) ceramics. There is a good Wickham and Croft [ref] Exp. Boucher et al. [ref] agreement between the observed and calculated intensities obtained by Rietveld refinements with the FULLPROF program. The magnetic and structural cells are always identical. The magnetic moments for each crystallographic site were determined and compared with values reported elsewhere for manganites of spinel structure, namely NiMn 2 O 4 and FeMn 2 O 4 (see Table 2). The spontaneous magnetization values determined at 5 K and previously discussed in this paper are also presented for comparison. For the sample compositions where x is closer to 2, the magnetic moments for the A and B sites are close to the values obtained by Boucher et al. [9]. In particular, M A ~ 3µ B .mol -1 is to be related to the magnetic moment of Co 2+ . For those compositions, a ferromagnetic A sub-lattice can be envisaged with an almost complete alignment of the Co 2+ magnetic moments. Whereas the resulting magnetic moments for the B sites are lower than those calculated with all Co 2+ , Co III , Co 3+ , Mn 3+ and Mn 4+ ions supposed to be aligned in the same direction. As a matter of fact, the M B resulting moment is an average value of non-aligned magnetic moments usually described in the manganese oxide spinels (see [22], for instance). The magnetic structure is thus composed of two sub-lattices of magnetization M A and M B forming an angle lower than 180° between each other. The resulting magnetization is then situated in a third direction. For the other two ceramic compositions (x =1.54 and 1.77) studied by neutron diffraction at very low temperature, the magnetic structures are more complex according to the low magnetization values obtained for the Co 2+ in the tetrahedral sites (see Table 2). This result can be interpreted by the occurrence of two sub-lattices of Co 2+ ions non-collinear in the A sites and at least two sublattices of manganese and cobalt ions in the B sites. The resulting magnetizations for each site will then form an angle lower than 180° and the global magnetization will be oriented in a third direction with non-collinear structures.

Conclusion

Magnetic properties of dense and single phase Mn 3-x Co x O 4 (0≤x≤3) ceramics were characterised. These properties are strongly related to the cation distribution of the material which depends upon the preparation method and thermal treatment. Our ceramics present similar magnetic properties to those reported for powders of the same compositions. The Curie constant decrease as the cobalt content increase is in agreement with a larger proportion of trivalent cobalt ion on the octahedral sites. The magnetic structure of Mn 3-x Co x O 4 (0≤x<3) is ferrimagnetic. At low temperature, the magnetic ordering is characterised by a non-collinear arrangement of spins, as described by Yafet-Kittel for the tetragonal phases or the 'star' type for the cubic phases.

Figure 1 :

 1 Figure 1: Phase diagram for Mn 3-x Co x O 4 with 0<x≤3. TS and CS correspond to Tetragonal and Cubic Spinel structures, respectively. Open squares symbolize the T 1 temperature and close squares T 2 (conventional sintering) or T (SPS) used in the preparation of ceramics.

Figure 2 :

 2 Figure 2: Spontaneous magnetization variation at 5 K with 0≤x≤3 in Mn 3-x Co x O 4 ceramics, in comparison with values reported for powders. Inset shows the magnetization variation versus field at 5 K for a Mn 0.78 Co 2.22 O 4 ceramic.

Figure 3 :

 3 Figure 3: Neutron diffraction patterns observed (dots), calculated (lines) and difference for tetragonal (left, x = 1.54) and cubic (right, x = 2.22) phases of Mn 3-x Co x O 4 ceramics at 10 K. Vertical lines indicate nuclear (top) and magnetic (bottom) peaks.

Table 2 :

 2 Magnetic moments on the A and B sites of some Mn 3-x Co x O 4 ceramics determined using neutron diffraction measurements at 10 K. Spontaneous magnetizations determined at 5 K and literature data for NiMn 2 O 4 and FeMn 2 O 4 are given for comparison.

  Sample x M A [µ B .mol -1 ] M B [µ B .mol -1 ] I o exp [µ B .mol -1 ] Mn 3-x Co x O 4


Désiré Saba

Gilbert Bayili

Serge Nicaise

Polynomial stabilization of the wave equation with a time varying delay term in the dynamical control

Keywords: 2010 Mathematics Subject Classification. 35B35, 93D15 Dynamical control, stability, time varying delay

We consider the one-dimensional wave equation with a time-varying delay term in the dynamical control. Under suitable assumptions, we show the well posedness of the problem. These results are obtained by using semi-group theory. Combining the multiplier method with a non linear integral inequality, a rational energy decay result of the system is established. A fundamental aspect of this paper is that our involved operator is time dependent, therefore the standard frequential method cannot be invoked.

Introduction

In this paper, we are interested in the effect of a time-varying delay term in boundary stabilization of the wave equation in one-dimensional case with dynamical control. Delay effects arise in many physical, electrical engineering systems or modelling of biological problems , refer [START_REF] Anikushyn | Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell's equations under a nonlinear boundary feedback with delay[END_REF][START_REF] Hadeler | Delay equations in biology[END_REF][START_REF] Peralta | Analysis of a nonlinear fluid-structure interaction model with mechanical dissipation and delay[END_REF][START_REF] Peralta | Stabilization of the wave equation with acoustic and delay boundary conditions[END_REF][START_REF] Datko | Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks[END_REF][START_REF] Nicaise | Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks[END_REF][START_REF] Rebarber | Robustness with respect to delays for exponential stability of distributed parameters systems[END_REF] and it is well known that they can induce some unstabilities, and that sometimes, an arbitrarly small delay in the feedback can destabilise or improve the stability of the system, see [START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Abdallah | Delayed positive feedback can stabilize oscillatory systems[END_REF][START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedback[END_REF].

There is a huge number of papers that deal with stabilitity properties of the damped wave equation and many others dealing wave propagation with delay, see for instance [START_REF] Akil | Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay[END_REF][START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Nicaise | Stability and instability results of the wave equation with a delay term in the boundary or internal feedback[END_REF][START_REF] Peralta | Stabilization of the wave equation with acoustic and delay boundary conditions[END_REF][START_REF] Nicaise | Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks[END_REF][START_REF] Bayili | Rational energy decay rate for the wave equation with delay term on the dynamical control[END_REF][START_REF] Nicaise | Exponential stability of the wave equation with boundary timevarying delay[END_REF]. Time delay, in a system or evolution equation, is a property of the system for which the response to an applied action is delayed. For some systems, the delay is assumed to be constant over the course of the evolution, but in practice this assumption may seem unrealistic. The goal of this paper is to consider the case when a varying delay term appears in a dynamical control. More precisely, on the interval (0, 1), we look for the displacement u, solution of the next wave equation and the dynamical control η solution of the next linear differential equation: u tt (x, t) -u xx (x, t) = 0 on (0, 1) × (0, ∞) , [START_REF] Abdallah | Delayed positive feedback can stabilize oscillatory systems[END_REF] with boundary conditions u (0, t) = 0 ;

u x (1, t) + η(t) = 0 ∀ t ∈ (0, ∞) , ( 2
)
η t -u t (1, t) + β 1 η(t) + β 2 η(t -τ (t)) = 0 ∀ t ∈ (0, ∞) , (3)
where, as usual the subindex t (resp. x) means the time (resp. space) derivative of a function.

In [START_REF] Anikushyn | Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell's equations under a nonlinear boundary feedback with delay[END_REF], τ (t) is the time varying delay that is supposed to be non-negative (and smooth enough, see below), β 1 > 0 is a positive real number and β 2 ∈ R is a real number.

In [START_REF] Anikushyn | Global well-posedness and exponential stability for heterogeneous anisotropic Maxwell's equations under a nonlinear boundary feedback with delay[END_REF], the term η(t -τ (t)) is not defined if t -τ (t) < 0, therefore we impose the relation

η(t -τ (0)) = f 0 (t -τ (0)), ∀ t ∈ (0, τ (0)) . ( 4
)
Assuming that τ ′ (t) < 1, we will always have τ (t) -τ (0) < t and if moreover t -τ (t) < 0, then t -τ (t) will be in (-τ (0), 0), and there exists a unique s t ∈ (0, τ (0)) such that t -τ (t) = s t -τ (0). Therefore (4) allows to give a meaning to (3) for t -τ (t) < 0.

Finally the system is closed with an initial condition on (u, u t , η) = (u 0 , u 1 , η 0 ) . Alltogether we obtain the system.

                                   u tt (x, t) -u xx (x, t) = 0 on (0, 1) × (0, ∞) , u (0, t) = 0 ∀ t ∈ (0, ∞) , u x (1, t) + η(t) = 0 ∀ t ∈ (0, ∞) , η t -u t (1, t) + β 1 η(t) + β 2 η(t -τ (t)) = 0 ∀ t ∈ (0, ∞) , u(•, 0) = u 0 , u t (•, 0) = u 1 on (0, 1) ; η(0) = η 0 , η(t -τ (0)) = f 0 (t -τ (0)) ∀ t ∈ (0, τ (0)) .
(

Notice that in [START_REF] Wehbe | Rational energy decay rate for a wave equation with dynamical control[END_REF], A. Wehbe has established an optimal polynomial decay rate in 1 t of the system [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF] without delay (namely β 2 = 0). More recently in [START_REF] Bayili | Rational energy decay rate for the wave equation with delay term on the dynamical control[END_REF], assuming 0 ≤ β 2 < β 1 , in general space dimension with constant delay, G. Bayili and all, have obtained on one hand, a rational decay of energy and have established on the other hand in [START_REF] Silga | Polynomial stability of the wave equation with distributed delay term on the dynamical control[END_REF] a polynomial stability of the system with distributive delay. From the previous results, we cannot expect to have an exponential decay of the energy of our system. Furthermore, since our operator is time-dependent, we cannot use a frequency domain approach to obtain some polynomial decay results (see for instance [START_REF] Borichev | Optimal polynomial decay of functions and operator semigroups[END_REF]), therefore we use an argument combining the multiplier method with a nonlinear integral inequality. The present paper is organized as follows. In section 2, we prove the well posedness of the system (5) for any coefficients β 1 > 0 and β 2 ∈ R and with strictly positive time varying delay. Moreover for smooth enough initial data, we establish in section 3 the stability of the system with an energy decay rate of order 1 t .

Then the initial system ( 5) is equivalent to the following one

                                                   u tt (x, t) -u xx (x, t) = 0 on (0, 1) × (0, ∞) , u (0, t) = 0 ∀ t ∈ (0, ∞) , u x (1, t) + η(t) = 0 ∀ t ∈ (0, ∞) , η t -u t (1, t) + β 1 η(t) + β 2 w(1, t) = 0 ∀ t ∈ (0, ∞) , u(•, 0) = u 0 , u t (•, 0) = u 1 on (0, 1) ; η(0) = η 0 , (1 -rτ ′ (t))w r + τ (t)w t = 0 ∀ (r, t) ∈ (0, 1) × (0, ∞) , w(0, t) = η(t), ∀ t ∈ (0, ∞) , w(r, 0) = f 0 (-rτ (0)) ∀ r ∈ (0, 1) . ( 7
)
Now, by the reduction of order method, setting U = (u, v, η, w) T , the system ( 5) is equivalent to the following Cauchy problem

U ′ (t) = A(t)U (t) in H, ∀t > 0, U (0) = U 0 = (u 0 , u 1 , η 0 , f 0 (-• τ (0))) ⊤ in H, (8)
where

V = {u ∈ H 1 (0, 1) : u(0) = 0},
and H is defined by

H = V × L 2 (0, 1) × R × L 2 (0, 1) , ( 9
)
which is a Hilbert space with the inner product

U, Ũ H = 1 0 (u x ũx + vṽ) dx + 1 0 w(r) w(r)dr + η η, ∀ U = (u, v, η, w) ⊤ , Ũ = (ũ, ṽ, η, w) ⊤ ∈ H. (10)
In [START_REF] Hadeler | Delay equations in biology[END_REF], the linear operator A(t) (depending on the time parameter t) is defined by

A(t)     u v η w     =       v u xx v(1) -β 1 w(0) -β 2 w(1) rτ ′ (t) -1 τ (t) w r       , ( 11
)
with domain D (A(t)) given by

D (A(t)) = (u, v, η, w) ∈ V ∩ H 2 (0, 1) × V × R × H 1 (0, 1) : η + u x (1) = 0; w(0) = η. . ( 12
)
Let us directly notice that the domain D (A(t)) is time independent, i.e., ∀t > 0, D(A(t)) = D(A(0)), [START_REF] Nicaise | Exponential stability of the wave equation with boundary timevarying delay[END_REF] and that

A(t) is well-defined provided τ ∈ W 1,∞ ([0, ∞)) satisfies    τ ′ (t) ≤ d < 1, ∀t ≥ 0, 0 < τ 0 ≤ τ (t) ≤ R, ∀ t ≥ 0, (14)
for two positive real numbers τ 0 , R and a non negative real number d such that d < 1. For future purposes, let us set

q = sup t≥0 |τ ′ (t)|. (15)
A general existence theory for equations of type (8) was developed using semigroup theory in [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF][START_REF] Kato | Abstract differential equations and nonlinear mixed problems[END_REF]. Indeed, in reference to [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF][START_REF] Kato | Abstract differential equations and nonlinear mixed problems[END_REF], (precisely, Theorem 1.9 of [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF] or Theorem 1.2 of [START_REF] Kato | Abstract differential equations and nonlinear mixed problems[END_REF]) if the triplet {A, H, Y } (with Y = D(A(0)), A = {A(t), t ∈ [0, T ]} for some fixed T > 0) forms a CD-system (or constant domain system), then an existence and uniqueness result follows. For completeness, let us recall this result. Theorem 2.1. Let T be a fixed real number. Let H be a Hilbert space and A(t) : D(A(t)) ⊂ H -→ H a linear operator satisfying the following hypotheses:

(H 1 ) : D(A(t)) = D(A(0)) ∀ t ∈ [0, T ]; (H 2 ) : D(A(0)) is a dense subset of H; (H 3 ) : ∀ t ∈ [0, T ] , A(t)
S tj (s j )u ∥ H ≤ Ce ms ∥ u ∥ H , f or all u ∈ H, s j ≥ 0, 0 ≤ t 1 ≤ . . . t k ≤ T, k ∈ N); (H 4 ) : ∂ ∂t A ∈ L ∞ * ([0, T ] ; B (D(A(0)), H)) which is the space of equivalent classes of essentially bounded, strongly measurable f unctions f rom [0, T ] into the set B (D(A(0)), H) of bounded operators f rom D(A(0)) into H. Then problem (8) has a unique solution U ∈ C ([0, T ] ; H)) for any initial datum U 0 ∈ H. Moreover if U 0 ∈ D(A(0)), then U ∈ C ([0, T ] ; D(A(0))) ∩ C 1 ([0, T ] ; H) .
Note that this Theorem also furnish a unique solution to our problem on (0, ∞) if we check that (H 1 )-(H 4 ) are valid for all T > 0. As mentionned before the assumption (H 1 ) always holds. To prove that (H 2 ) and (H 3 ) hold, we use a time variable-norm technique, namely for all t ≥ 0, we introduce the following time dependent inner product

U, Ũ t = 1 0 (u x ũx + vṽ) dx + β 1 τ (t) 1 0 w(r) w(r)dr + η η, ∀U = (u, v, η, w) ⊤ ∈ H, Ũ = (ũ, ṽ, η, w) ⊤ ∈ H. ( 16
)
By our assumption on τ , the norm ∥ • ∥ t induced by this new inner product is equivalent to the use one ∥ • ∥ H 1 induced by [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF], namely for all t ≥ 0, min{1,

β 1 τ 0 }∥U ∥ H ≤ ∥U ∥ t ≤ max{1, β 1 R}∥U ∥ H , ∀U ∈ H. (17)
For all t ≥ 0 let us now set

φ(t) = 1 + τ ′2 (t) 2τ (t) + β 2 2 2β 1 (1 -d) , ( 18
)
and

K(t) = A(t) -φ(t)I, ( 19
)
where I means here the identity operator from H into itself.

Theorem 2.2. For all fixed t ≥ 0, the operator K(t) is maximal dissipative in H for the inner product defined by [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF].

1 i.e., ∥U ∥ 2 t = U, U t , and ∥U ∥ 2 H = U, U H , for all U ∈ H
Proof. We first prove that for all fixed t ≥ 0, K(t) is dissipative in H endowed with inner product <, > t . Fix U = (u, v, η, w) ⊤ ∈ D(A( 0)) (that is clearly the domain of K(t)), then by the definition of A(t), we have

< A(t)U, U > t =       v u xx v(1) -β 1 w(0) -β 2 w(1) rτ ′ (t) -1 τ (t) w r       ,     u v η w     t = 1 0 (v x u x + u xx v) dx + β 1 1 0 (rτ ′ (t) -1)ww r dr + η (v(1) -β 1 w(0) -β 2 w(1)) .
Integrating by parts in the first and second terms of this right-hand side, it follows

< A(t)U, U > t = [u x v] 1 0 + β 1 2 (rτ ′ (t) -1)w 2 1 0 - β 1 2 τ ′ (t) 1 0 w 2 dr + η (v(1) -β 1 w(0) -β 2 w(1)) . Since u x (1) = -η, v(0) = 0, w(0) = η, we get < A(t)U, U > t = - β 1 2 η 2 -β 2 ηw(1) + β 1 2 (τ ′ (t) -1) w 2 (1) - β 1 τ ′ (t) 2 1 0 w 2 dr. ( 20
)
Since here we have no smallness assumption on β 2 , we cannot absorb the second term of this right-hand side by the first and third ones, that is the reason of the use of the operator K(t). Indeed recalling that

< K(t)U, U > t =< A(t)U, U > t -φ(t) < U, U > t ,
we then have

< K(t)U, U > t = - β 1 2 -φ(t) η 2 -β 2 ηw(1) + β 1 2 (τ ′ (t) -1) w 2 (1) + - β 1 2 1 + τ ′2 (t) + τ ′ (t) + β 2 2 τ (t) β 1 (1 -d) 1 0 w 2 dr -φ(t) 1 0 u 2 x + v 2 dx.
By Young's inequality, for all α > 0, we get

< K(t)U, U > t ≤ - β 1 2 -φ(t) + |β 2 | 2α η 2 + α |β 2 | 2 + β 1 2 (τ ′ (t) -1) w 2 (1) + -φ(t) 1 0 u 2 x + v 2 dx - β 1 2 1 + τ ′2 (t) + τ ′ (t) + β 2 2 τ (t) β 1 (1 -d) 1 0 w 2 dr. ≤ -β 1 2 -φ(t) + |β 2 | 2α η 2 + α |β 2 | 2 + β 1 2 (τ ′ (t) -1) w 2 (1).
Note that if β 2 = 0 (that is no delay occurs in the system), then

< K(t)U, U > t ≤ 0. But for β 2 ̸ = 0, choosing α = (1 -d)β 1 |β 2 | , we obtain < K(t)U, U > t ≤ - β 1 2 -φ(t) + β 2 2 2β 1 (1 -d) η 2 + β 1 2 (τ ′ (t) -d) w 2 (1).
But the definition (18) of φ(t) and the estimate τ

′ (t) ≤ d < 1 in (14) imply that - β 1 2 -φ(t) + β 2 2 2β 1 (1 -d) ≤ - β 1 2 + 1 + τ ′2 (t) 2τ (t) < 0, τ ′ (t) -d ≤ 0, therefore ⟨K(t)U, U ⟩ t ≤ 0, ∀U ∈ D(A(0)). ( 21
)
Now we prove that there exists λ > 0 such that λI -A(t) : D -→ H is surjective. In other words, for

F = (f, g, h, y) ⊤ ∈ H, we look for U = (u, v, η, w) ⊤ ∈ D(A(0)) such that (λI -A(t)) U = F , or equivalently                                    λu -v = f, λv -u xx = g, λη -v(1) + β 1 w(0) + β 2 w(1) = h, λw + 1-rτ ′ (t) τ (t) w r = y, w(0) = η, u(0) = η + u x (1) = 0. ( 22
)
Assuming that a solution exists, then eliminating

v = λu -f, ( 23
)
by the first identity, replacing w(0) by η, and noticing that

1 -rτ ′ (t) ≥ 1 -d > 0, ∀t ≥ 0, r ∈ (0, 1), (24)
we find that u is solution of the boundary value problem

λ 2 u -u xx = λf + g, u(0) = η + u x (1) = 0, ( 25
)
while w is solution of the following first order differential equation with initial condition at 0:

       λτ (t) 1 -rτ ′ (t) w + w r = y τ (t) 1 -rτ ′ (t) , w(0) = η, ( 26
)
where η is given by

(λ + β 1 )η + β 2 w(1) = h -f (1) + λu(1). ( 27
)
Since w(1) depends on η via (26), we then first solve this differential system. As 1 -rτ ′ (t) is independent of r if τ ′ (t) = 0, its resolution is made according to the case τ ′ (t) = 0 and τ ′ (t) ̸ = 0. In both cases, by the method of variation of the constant, the solution of (26) is

w(r) = e -λτ (t)r η + τ (t) r 0 e λτ (t)s y(s)ds , ∀r ∈ (0, 1) , ( 28
) if τ ′ (t) = 0, while w(r) = exp λτ (t) τ ′ (t) ln (1 -rτ ′ (t)) η + r 0 τ (t)y(s) 1 -sτ ′ (t) exp - λτ (t) τ ′ (t) ln (1 -sτ ′ (t)) ds , ∀r ∈ (0, 1) (29)
if τ ′ (t) ̸ = 0. Note that the expression (28) can be recovered from (29) with the convention that

ln (1 -rτ ′ (t)) rτ ′ (t) = -1, ∀r ∈ (0, 1] if τ ′ (t) = 0.
From now on, using this convention, using (29) we find

w(1) = b(t)η + a(t), ( 30
)
where

a(t) = b(t) 1 0 τ (t)y(s) 1 -sτ ′ (t) exp - λτ (t) τ ′ (t) ln (1 -sτ ′ (t)) ds, b(t) = exp λτ (t) τ ′ (t) ln (1 -τ ′ (t)) .
Inserting the expression (30) in ( 27), we obtain

η = h + λu(1) -f (1) -β 2 a(t) λ + β 1 + β 2 b(t) , ( 31
)
as soon as λ is large enough so that the denominator is positive, in other words for

λ > -β 1 -β 2 b(t). ( 32
)
Coming back to (25) and using (31), we deduce that u is solution of

       λ 2 u -u xx = λf + g in (0, 1), u(0) = 0, λ λ + β 1 + β 2 b(t) u(1) + u x (1) = - h -f (1) -β 2 a(t) λ + β 1 + β 2 b(t) .
Note that the boundary condition at 1 is of Robin type and that a(t) depends only on the datum y (and not on u). This boundary value problem has a unique solution in H 2 (0, 1), because the coefficient

λ λ + β 1 + β 2 b(t) is non-negative.
With this solution u, we define v by (23), η by (31) and w by (29), that by construction satisfies (22). In other words, for λ ≥ 0 satisfying (32), λI -A(t) is surjective.

Finally as λI -K(t) = (λ + φ(t)) I -A(t), we deduce that λI -K(t) is also surjective if λ ≥ 0 satisfies (32) (reminding that φ(t) ≥ 0). This surjectivity results and ( 21) complete the proof of Theorem 2.2. ■ Before going on, let us notice that since we are in a Hilbert setting, the above Theorem implies that D(K(t)) is dense in H, consequently assumption (H 2 ) holds for A(t) recalling that D

(K(t)) = D(A(t)) = D(A(0)).

We are further ready to check that the operators A(t) satisfy the assumption (H 3 ) from Theorem 2.1.

≤ φ(t) ≤ m := 1 + q 2 2τ 0 + β 2 2 2β 1 (1 -d) ,
we deduce that ∥e sA(t) U ∥ t ≤ e ms ∥U ∥ t , ∀U ∈ H, s ≥ 0, for any t ≥ 0. Finally in order to have the stability properties, we use the smoothness property of the inner product ( 16) with respect to t. Indeed for U = (u, v, η, w) ⊤ ∈ H, we see that

d dt ∥U ∥ 2 t = β 1 τ ′ (t) 1 0
w(r) 2 dr, hence using the assumption ( 14), we obtain

d dt ∥U ∥ 2 t ≤ β 1 1 0 w(r) 2 dr ≤ 1 τ 0 ∥U ∥ 2 t .
This estimate and Gronwall's inequality imply that

∥U ∥ t ≤ e 1 2τ 0 |t-s| ∥U ∥ s , ∀s, t ∈ [0, ∞).
This proves the Proposition with the help of Proposition 1.1 of [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF]. ■

Finally, under the assumption

τ ∈ W 2,∞ ([0, T ]) ∀ T > 0, (33)
we readily check that the hypothesis (H 4 ) holds. Indeed under this assumption for U = (u, v, η, w) ⊤ ∈ D(A(0)) we have

∂A(t) ∂t U =       0 0 0 rτ ′′ (t)τ (t) -τ ′ (t)(rτ ′ (t) -1) τ 2 (t) w r       . ( 34
)
Since 0 < τ 0 ≤ τ, and by (33), we see that

S := sup t∈[0,T ] rτ ′′ (t)τ (t) -τ ′ (t)(rτ ′ (t) -1) τ (t) < ∞,
which directly leads to

∥ ∂A(t) ∂t U ∥ H ≤ S∥ 1 τ (t) w r ∥ L 2 (0,1) ≤ S 1 -d ∥A(0)U ∥ H , which shows that ∂A(t) ∂t ∈ L ∞ * ([0, T ] ; L (D(A(0)), H)).
Owing to Theorem 2.1, we can state the following existence and uniqueness theorem.

Theorem 2.3. Assume that ( 14) and (33) hold, then for any initial datum U 0 ∈ H, there exists a unique solution U ∈ C ([0, ∞) ; D(A(0))) of the system [START_REF] Hadeler | Delay equations in biology[END_REF].

Furthermore, if U 0 ∈ D(A(0)), then U ∈ C ([0, ∞) ; D(A(0)))∩ C 1 ([0, ∞) ; H) .
In the setting of the previous theorem, since w satisfies the transport equation

           (1 -rτ ′ (t))w r + τ (t)w t = 0 ∀ (r, t) ∈ (0, 1) × (0, ∞) , w(0, t) = η(t), ∀ t ∈ (0, ∞) , w(r, 0) = f 0 (-rτ (0)) ∀ r ∈ (0, 1) , we have w(r, t) =    η(t -rτ (t)), ∀t -rτ (t) > 0, f 0 (t -rτ (t)) ∀t -rτ (t) < 0. Hence if we set T 0 = inf t≥0 {t : t -τ (t) ≥ 0}, ( 35
) then w(r, t) = η(t -rτ (t)), ∀t ≥ T 0 , r ∈ (0, 1). ( 36
)
Let us finally give an improved regularity result for smoother initial data.

Theorem 2.4. Assume ( 14) hold with the additional regularity

τ ∈ W 3,∞ ([0, T ]) ∀ T > 0, ( 37
)
then for any initial datum U 0 ∈ D(A(0) 2 ), the unique solution

U ∈ C ([0, ∞) ; D(A(0))) of the system (8) satisfies U ∈ C 1 ([0, ∞) ; D(A(0)))) ∩ C 2 ([0, ∞) ; H) .
Proof. We apply Theorem 2.13 of [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF] with n = 2 and X j = Y j = D(A(0)). In such a case, the assumption (P1) is satisfied due to Proposition 2.1, (P2) follows from the assumption (37), while (P3) is here automatically satisfied. Finally, we notice that D 2 (0)) = D(A(0) 2 ). Indeed by section 2.4 of [START_REF] Kato | Linear and quasi-linear equations of evolution of hyperbolic type[END_REF], we have

D 1 (t) = D(A(0)) and D 2 (t) = ϕ ∈ D 1 (t) : S k (t)ϕ ∈ D(A(0)), ∀ 0 ≤ k ≤ 1 ,
where S 0 (t) = I, and S 1 (t) = A(t). Therefore

D 2 (t) = {ϕ ∈ D 1 (t) : A(t)ϕ ∈ D(A(0))} , which gives D 2 (0)) = D(A(0) 2 ). ■ From the previous theorem, if U 0 ∈ D(A(0) 2 ), we deduce that U = (u, u t , η, w) satisfies u ∈ C 3 ([0, ∞), L 2 (0, 1)), u t ∈ C 1 ([0, ∞), H 2 (0, 1)), η ∈ C 2 ([0, ∞)).
In particular, it holds

u ttt = u xxt in L 2 (0, 1), ∀t ≥ 0.

Rational decay of Energy

We have shown in the previous section that the system ( 5) is well posed for any β 1 > 0, β 2 ∈ R under the assumptions ( 14) and ( 33). Here, we aim to establish a rational energy decay result of the system under the additional assumption

|β 2 | < ρβ 1 . ( 38
)
where

ρ = min √ 1 -d ; 1 √ 1 + q
with d ≥ 0 from the assumption ( 14) and q was defined in [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary time-varying delays[END_REF]. Let U be the unique solution of the system (5) with an initial datum U 0 ∈ H, then we define its associated energy E(t) by

E(t) = 1 2 ∥ U ∥ 2 t = 1 2 1 0 u 2 x + u 2 t dx + β 1 τ (t) 1 0 η 2 (t -rτ (t)) dr + η 2 (t) , ∀t ≥ 0. (39)
If morever U 0 ∈ D(A(0)), then we can define its energy E 1 (t) of higher order for t ≥ T 0 by

E 1 (t) = 1 2 1 0 u 2 tx + u 2 tt dx + β 1 τ (t) 1 0 η 2 t (t -rτ (t)) dr + η 2 t , ∀t ≥ T 0 . ( 40
)
Note these two energies are well-defined owing to Theorem 2.3 due to (24) and (36).

Energy decay

Theorem 3.1. Assuming that [START_REF] Nicaise | Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks[END_REF] and (38) hold, then for any 5) is nonincreasing and there exists C 1 > 0 such that

U 0 ∈ D(A(0)), the energy E of the unique solution U ∈ C ([0, ∞) ; D(A(0))) ∩ C 1 ([0, ∞) ; H) of (
d dt E(t) ≤ -C 1 η 2 (t) + η 2 (t -τ (t)) ≤ 0, ∀t ≥ 0. ( 41
)
If additionally we assume that (37) holds, then the energy E 1 is nonincreasing and there exists C 2 > 0 such that

d dt E 1 (t) ≤ -C 2 η 2 t + η 2 t (t -τ (t)) ≤ 0, ∀t ≥ T 0 . ( 42
)
Proof. Let us start with the first assertion. By its definition, we have

d dt E(t) =< U t , U > t + β 1 τ ′ (t) 2 1 0 w(r, t) 2 dr =< A(t)U, U > t + β 1 τ ′ (t) 2 1 0 w(r, t) 2 dr
where U = (u, u t , η, w) ⊤ . Due to [START_REF] Silga | Polynomial stability of the wave equation with distributed delay term on the dynamical control[END_REF], we then have

d dt E(t) = - β 1 2 η 2 (t) -β 2 η(t)η (t -τ (t)) + β 1 2 (τ ′ (t) -1) η 2 (t -τ (t)) .
Young's inequality then yields

d dt E(t) ≤ - β 1 2 + |β 2 | 2α η 2 (t) + α|β 2 | 2 + β 1 2 (d -1) η 2 (t -τ (t)) ∀ α > 0. ≤ - β 1 2 + |β 2 | 2 √ 1 -d η 2 (t) + √ 1 -d|β 2 | 2 + β 1 2 (d -1) η 2 (t -τ (t)) f or α = √ 1 -d ≤ |β 2 | -β 1 √ 1 -d 2 √ 1 -d η 2 (t) + √ 1 -d |β 2 | -β 1 √ 1 -d 2 η 2 (t -τ (t))
Thanks to the hypothesis (38),

|β 2 | -β 1 √ 1 -d 2 < 0 and taking C 1 = - √ 1 -d |β 2 | -β 1 √ 1 -d 2
we get (41). On the other hand for U 0 ∈ D(A(0) 2 ), using the additional regularity

U ∈ C 1 ([0, ∞) ; D(A(0)))) ∩ C 2 (
(u ttx u tx + u ttt u tt ) dx + η t η tt + β 1 τ (t) 1 0 η t (t -rτ (t)) η tt (t -rτ (t)) (1 -rτ ′ (t))dr + β 1 τ ′ (t) 2 1 0 η 2 t (t -rτ (t)) dr. = 1 0 (u ttx u tx + u txx u tt ) dx + η t η tt -β 1 1 0 η t (t -rτ (t)) ∂ ∂r (η t (t -rτ (t))) (1 -rτ ′ (t))dr + β 1 τ ′ (t) 2 1 0 η 2 t (t -rτ (t)) dr, because u ttt = u txx = [u tt u tx ] 1 0 + η t η tt - β 1 2 (1 -rτ ′ (t))η 2 t (t -rτ (t)) 1 0 . = u tt (1, t)u tx (1, t) + η t η tt + β 1 2 η 2 t + β 1 2 (τ ′ (t) -1) η 2 t (t -τ (t)) . = - β 1 2 η 2 t -β 2 (1 -τ ′ (t))η t η t (t -τ (t)) + β 1 2 (τ ′ (t) -1) η 2 t (t -τ (t)) .
Again thanks to Young's inequality, we obtain

d dt E 1 (t) = - β 1 2 η 2 t -β 2 (1 -τ ′ (t))η t η t (t -τ (t)) + β 1 2 (τ ′ (t) -1) η 2 t (t -τ (t)) . ≤ - β 1 2 + |β 2 |(1 -τ ′ (t)) 2α η 2 t + (1 -τ ′ (t)) α|β 2 | 2 - β 1 2 η 2 t (t -τ (t)) ∀ α > 0. ≤ - β 1 2 + |β 2 |(1 + q) 2α η 2 t + (1 -τ ′ (t)) α|β 2 | 2 - β 1 2 η 2 t (t -τ (t)) ∀ α > 0. ≤ - β 1 2 + |β 2 | √ 1 + q 2 η 2 t + (1 -τ ′ (t)) |β 2 | √ 1 + q 2 - β 1 2 η 2 t (t -τ (t)) , for α = √ 1 + q. Thanks to (38), - β 1 2 + |β 2 | √ 1 + q 2 <
0, hence, by (24), taking

C 2 = β 1 2 - |β 2 | √ 1 + q 2 (1 -d),
we obtain (42). ■

Polynomial stability of the system

In order to establish the rational decay result of the energy, we first prove the following lemmas.

Lemma 3.1. Assume that ( 14) and ( 33) hold. With T 0 ≥ 0 defined by (35), we have

T S 1 0 η 2 (t -rτ (t))drdt ≤ E(S -τ (S)) (1 -d)C 1 , ∀T 0 ≤ S ≤ T. ( 43
)
Proof. Fix S and T such that T 0 ≤ S ≤ T . By Fubini's theorem, we may write

T S 1 0 (1 -rτ ′ (t))η 2 (t -rτ (t))drdt = 1 0 T S (1 -rτ ′ (t))η 2 (t -rτ (t))dtdr.
By the change of variable s = t -rτ (t) it holds

1 0 T S (1 -rτ ′ (t))η 2 (t -rτ (t))dtdr = 1 0 T -rτ (T ) S-rτ (S) η 2 (s)dsdr. ( 44
)
Thanks to Theorem 3.1, we have η

2 (t) ≤ - E ′ (t) C 1 for all t ≥ 0. Therefore, T -rτ (T ) S-rτ (S) η 2 (s)ds ≤ - T -rτ (T ) S-rτ (S) E ′ (s) C 1 ds ≤ E (S -rτ (S)) C 1 ≤ E (S -τ (S)) C 1 . ( 45
)
Since by (24), we have

(1 -d) 1 0 T S η 2 (t -rτ (t))dtdr ≤ 1 0 T S (1 -rτ ′ (t))η
E(S) + 3 2C 2 E 1 (S) + β 1 R 2(1 -d)C 1 E (S -τ (S)) , ∀ T 0 ≤ S < T. ( 46
)
Proof. Integrating by parts, we have

1 0 u 2 x + u 2 t dx = x u 2 x + u 2 t 1 0 -2 1 0 x (u x u xx + u t u tx ) dx. = u 2 x (1, t) + u 2 t (1, t) -2 1 0 x (u x u tt + u t u tx ) dx. = u 2 x (1, t) + u 2 t (1, t) -2 d dt 1 0 xu x u t dx . ( 47
)
The boundary conditions give

u 2 x (1, t) + u 2 t (1, t) = η 2 (t) + (η t (t) + β 1 η(t) + β 2 η(t -τ (t))) 2 ≤ (1 + 3β 2 1 )η 2 (t) + 3β 2 2 η 2 (t -τ (t)) + 3η 2 t (t). ( 48
)
From ( 47) and (48), it follows

1 0 u 2 x + u 2 t dx ≤ (1 + 3β 2 1 )η 2 (t) + 3β 2 2 η 2 (t -τ (t)) + 3η 2 t (t) -2 d dt 1 0 xu x u t dx .
This estimate and the definition of the energy E(t) (and recalling that τ (t) ≤ R) yield

E(t) ≤ 2 + 3β 2 1 + 3β 2 2 2 η 2 + η 2 (t -τ (t)) + 3 2 η 2 t - d dt 1 0 xu x u t dx + β 1 R 2 1 0 η 2 (t -rτ (t))dr.
Thanks to Theorem 3.1, it follows that

E(t) ≤ - 2 + 3β 2 1 + 3β 2 2 2C 1 E ′ (t) - 3 2C 2 E ′ 1 (t) - d dt 1 0 xu x u t dx + β 1 R 2 1 0 η 2 (t -rτ (t))dr, ∀t ≥ T 0 . (49)
Integrating the left-hand side between S and T , with T 0 ≤ S < T , we get

T S E(t)dt ≤ 2 + 3β 2 1 + 3β 2 2 2C 1 E(S) + 3 2C 2 E 1 (S) + β 1 R 2 T S 1 0 η 2 (t -rτ (t))drdt + 1 0 xu x (x, S)u t (x, S)dx - 1 0 xu x (x, T )u t (x, T )dx. ( 50
)
By Young's inequality and thanks to the non increasing property of the energy E, we obtain

1 0 xu x (x, S)u t (x, S)dx - 1 0 xu x (x, T )u t (x, T )dx ≤ 2E(S). (51)
This estimate and (43) in (50) imply that (46) holds. ■

We conclude this section by the following rational energy decay result of the initial value problem.

Theorem 3.2. Assume that ( 14) and (37), and let U ∈ C ([0, ∞) ; D(A(0))) ∩ C 1 ([0, ∞) ; H) be the unique solution of ( 5) with U 0 ∈ D(A(0)) different from 0. Then there exists a positive constant M (E(0), E 1 (T 0 )) depending on E(0) and E 1 (T 0 ) such that

E(t) ≤ E(0) 2M t + M , ∀ t ≥ 0. ( 52
)
Proof. In a first step, we assume that U 0 ∈ D(A(0) 2 ) (and is different from 0). Then thanks to Lemma 3.2 and to the decrease of the energy E, for all T 0 ≤ S < T we have

m = 2 + 2 + 3β 2 1 + 3β 2 2 C 2 + 3 E1(T0) E(0) C 1 2C 1 C 2 + β 1 R 2(1 -d)C 1 • (55)
Taking M = max{m; T 0 }, the estimate (54) implies that Therefore if we denote by U (n) the solution of ( 5) with initial datum U (n) 0 , it holds U (n) (t) → U (t) in D(A(0)), ∀t ≥ 0.

This implies that the energy E (n) (t) (resp. E

(n) 1 (t)) associated with U (n) (t) converges to E (n) (t), for all t ≥ 0 (resp. E (n) 1 (t), for all t ≥ T 0 ). As the estimate (52) is valid for U (n) , namely

E (n) (t) ≤ E (n) (0) 2M (n) t + M (n) , ∀ t ≥ 0.
with M (n) = max{m (n) ; T 0 }, and m (n) given by (55) replacing E(0) (resp. E 1 (T 0 )) by E (n) (0)) (resp. by E (n) 1 (T 0 )), we can pass to the limit in n, and conclude that (52) remains valid. ■

∞SE 2

 2 (t)dt ≤ M E(S)E(0), ∀ S ≥ M. (56)In application of a classical result of Komornik[START_REF] Komornik | Exact controllability and stabilization[END_REF] Theorem 9.1], we conclude from (56) that (52) holds.For U 0 ∈ D(A(0)) different from 0, we use a density argument. Namely, as D(A(0) 2 ) is dense in D(A(0)), there exists a sequence of U (n) 0 ∈ D(A(0) 2 ) (that, without loss of generality, we can suppose to be different from 0) such that U(n) 0 → U 0 in D(A(0)).

  [0, ∞) ; H) from Theorem 2.4, using the same argument than in the proof of Theorem 2.2 and noticing that η t (t -rτ (t)) satisfies the same transport equation than η (t -rτ (t)), we have for any t ≥ T 0

	d dt	E 1 (t) =	0	1

The Author was supported by (PAES) the Higher Education Support Program of Burkina Faso.


Daniel Monceau
email: daniel.monceau@ensiacet.fr

Djar Oquab
email: bdjar.oquab@ensiacet.fr

Claude Estournes
email: cestourne@chimie.ups-tlse.fr

Mathieu Boidot
email: mathieu.boidot@ensiacet.fr

Serge Selezneff
email: eserge.selezneff@ensiacet.fr

Nicolas Ratel-Ramond
email: fnicolas.ratel-ramond@cemes.fr

Thermal Barrier Systems and Multi-Layered Coatings Fabricated by Spark Plasma Sintering for the Protection of Ni-Base Superalloys

Keywords: Coatings, TBC, spark plasma sintering SPS, superalloys, high temperature oxidation

Aeronautic gas turbine blades, vanes and combustion chambers are protected against high temperature oxidation and corrosion by single or multilayered coatings. These include environmental coatings, generally based on Pt-modified Ni aluminides or MCrAlY overlays (where M = Ni and/or Co), thermal barrier coating (TBC) systems including a ceramic thermally insulating layer, and abradable seals. The present work shows the ability of the Spark Plasma Sintering technique to rapidly develop new coatings compositions and microstructures. This technique allows combining powders and metallic foils on a superalloy substrate in order to obtain multilayered coatings in a single short production step. Fabrication of MCrAlY overlays with local Pt and/or Al enrichments is shown, as well as fabrication of coatings made of ζ-PtAl 2 , ε-PtAl, α-AlNiPt 2 , martensitic and β-(Ni,Pt)Al or Pt-rich γ/γ' phases, including their doping with reactive elements. The fabrication of a complete TBC system with a porous and adherent Yttria Stabilized Zirconia (YSZ) layer on a bond-coating is also demonstrated, as well as the fabrication of a CoNiCrAlY-based cermet coating for abradable seal application. Difficulties of fabrication are reviewed, such as Y segregation, risks of carburization, local over-heating, or difficulty to coat complex shaped parts. Solutions are given to overcome these difficulties.

Introduction

Powders which are difficult to sinter by other techniques, such as SiC or WC, have been densified by Spark Plasma Sintering (SPS) without any additives [1]. The reasons for this success are not clearly identified [2]. It was thought initially that plasma was created between grains by the pulse current, but this was not proven. When comparing the technique with other sintering processes, one can notice that 1/ the electric current causes the fast heating of a graphite die by Joule effect and the apparatus arrangement allows a fast cooling [3]; 2/ the sintering is performed in a vacuum chamber with graphite die and punch which impose a very low oxygen partial pressure [2] and 3/ an uniaxial load is applied by the graphite punch.

To the authors' knowledge, the possibility to coat structural alloys using SPS has never been reported in review papers on coatings fabrication [4], nor in reviews on the SPS technique [3] including a recent review on patents [5]. Only the extensive paper by Orru et al. [1] reports few works on ceramic coatings for structural metallic alloys. Indeed, questions arise about the possibility to industrialise such SPS coating process. Nevertheless, uncoated parts of complex shape as large as 500 mm, are currently in production [5]. In addition, pseudo-isostatic pressure can be obtained by embedding the part to sinter in a not sintering powder [5]. Then, it is reasonable to think that SPS coatings could be produced in a close future as it has been demonstrated for Functionally Graded Materials (FGM) [6]. If so, this process has several advantages including no pollution by chemicals, no dangerous gases, a minimum waste of materials, and a very fast processing.

Moreover, it was shown recently that SPS allows rapid development of new coatings compositions and microstructures, not with the direct objective to industrialize the SPS process but with the goal of fabricating laboratory samples on which properties, such as the resistance to high temperature oxidation, can be studied. It was demonstrated that a fully dense and unoxidized MCrAlY and Al-Pt surface modified NiCoCrAlYTa coating could be prepared in less than 1 h using this process [7][8]. These results were completed by the development of several kinds of Ni aluminides diffusion coatings, including reactive elements (RE) doped Pt-rich γ-Ni/γ'-Ni 3 Al bond coatings, and the possibility to process a full TBC system in a one step process [2].

The development of such coatings is driven by the objectives of increasing the operating temperature and the durability of gas turbines. Multi-layered coatings are developed to protect single crystal Ni-base superalloy blades against several types of attacks. An ideal coating should be a thermal barrier reducing the operating temperature of the superalloy blade, it should be resistant to erosion and to CMAS corrosion, and should prevent high temperature oxidation and hot corrosion of the superalloy, and finally it should not degrade the mechanical properties of the superalloy [2]. Development of such multi-layered coatings requires numerous fabrication techniques [4] and the cost, complexity and delay to obtain test coupons certainly slow down and limit research works. Then, any shortcut in the development process should allow to test innovative solutions or to optimize existing compositions.

Materials and Processing

The substrates used for this study were 8, 9 or 24.5 mm diameters discs prepared from single crystal rods made of first generation Ni-base superalloys AM1® or AM3® [9]. Before use, the substrate surface was mechanically polished down to 0.25 µm diamond grit for MCrAlY-base coatings or ground to #600 SiC grit for Ni aluminide-base coatings. NiCrAlY, CoNiCrAlY and NiCoCrAlYTa commercial powders (resp. Sultzer-Metco AMDRY® 962, 995 and 997, grain size of 35 µm), pure Pt and Al foils (GoodFellow) with thickness between 2 and 10 µm, Yttria Stabilized Zirconia powder (Tosoh Corporation, with either 8 or 10 mol%Y 2 O 3 , grain size of 50 to 200 nm) were used for the fabrication of the systems. For the making of CoNiCrAlY-talc -base cermet, the CoNiCrAlY powder was mixed with 1:20 weight ratio of natural talc particles with an average size around 1.8µm. All samples presented in this study were prepared by SPS (Dr Sinter 2080, SPS Syntex Inc., Japan). This SPS equipment, consists of a pulsed direct current (3.3 ms of duration) generator, a hydraulic press system, a vacuum and water cooled chamber, two upper and lower punch electrodes graphite spacers and finally a control system for the temperature (thermocouples and optical pyrometer), for the environment pressure inside the chamber and for the positioning of the lower electrode on z-axis which allows to follow the shrinkage of the sample. The materials were loaded onto 8, 15 or 24.5 mm inner diameter cylindrical graphite dies. The pulse pattern 12-2 (i.e. 12 pulses followed by 2 rest intervals) recommended by the system manufacturer was used. The temperature was monitored by a thermocouple placed in a small hole located at the surface of the die. A heating rate of 100°C/min was used to reach the final temperature of 950, 1000°C or 1150°C depending of the samples. Dwell time at the desired temperature was between 0 and 30 min. Loads of several kN depending on the systems and corresponding to uniaxial stresses between 16 and 100 MPa were applied during the cycle [2,7,8,[START_REF] Ratel | Surf and Coat Tech[END_REF][START_REF] Boidot | Surf and Coat Tech[END_REF]. In order to add controlled quantities of reactive elements in their metallic form (Y, Hf, Si) in Ni aluminide coatings, some of the AM1® superalloy substrates were PVD coated by RF-sputtering before the SPS process [2]. Some of the samples where annealed in secondary vacuum in order to obtain the desired microstructure. For example, it was shown that the combination of Pt and Al foils can be used to obtain ζ-PtAl 2 , ε-PtAl, α-AlNiPt 2 , martensitic and β-(Ni,Pt)Al or Pt-rich γ/γ' phases, depending of the thickness of the foils, the temperature and duration of the SPS processing, and the temperature and duration of a post-SPS annealing at 1100°C [2].

Most of the coatings were tested under a cyclic oxidation test. An automatic rig was used. One cycle consists in 4 min heating and 56 min dwell at 1100°C under laboratory air followed by a 15 min cooling to room temperature under a high pressure and filtered air flow. Samples surfaces and polished cross sections were caracterized by metallography, SEM and TEM when necessary, after 0, 300 and sometimes 1000 cycles. Thermogravimetric experiments were also ran at 1050°C under synthetic air to determine the oxidation kinetics of bulk CoNiCrAlY-talc mixture after SPS sintering.

Results and Discussion

Fig. 1 shows some architectures which have been fabricated using the SPS process. The process consists in stacking up several layers in the die : the superalloy substrate with or without PVD thin films is covered with metallic foils (Al or Pt) or MCrAlY powder or a mixture of CoNiCrAlY and talc powders and finally a ceramic YSZ powder to realize a TBC system. Several difficulties were encountered during the development of SPS coatings. They were reviewed in [2] and include the measurement of the temperature during the SPS process, the carburization and the Y segregation. Satisfactory solutions were found concerning the measurement of temperature and the carburization [2]. Using this "know how", the SPS processing of these assemblies at high temperature under vacuum and mechanical load, results in smooth and homogeneous metallic bond coatings without pores or internal oxidation [2,7,8] with a porous YSZ top coat of desired porosity (between 15 and 35 vol%) in order to ensure a low thermal conductivity [START_REF] Boidot | Surf and Coat Tech[END_REF]. It is noticeable that fully dense metallic coatings can be obtained even in the presence of a hole in the substrate when using a powder [7] and without Kirkendall pores when combining pure Al and Pt foils [2]. This is certainly a beneficial effect of the combination of the uniaxial load in a confined die which leads to a pseudo-isostatic stress state, and to the very low oxygen partial pressure in the apparatus which limits the oxidation of the foils and powders and then facilitates their sintering. -4 present three examples of SPS-processed coatings. In Fig. 2, the coating consists in an overlay NiCoCrAlYTa dense coating modified by an overlay of Pt and Al. Such a coating would be generally fabricated using vacuum plasma spaying of the MCrAlY powder on the superalloy followed by a Pt electrolytic deposition, annealing at 1050-1100°C during several hours, and a vapor phase aluminisation during 1-3 h at about 1100°C. Pt and Al enrichments are realized to improve the oxidation and corrosion resistance of the MCrAlY coating [4]. With the SPS process, only one step of less than 1h is necessary to obtain a sample. Nevertheless, it was shown that this one-step process leads to a high level of Y and Pt segregation at the grain boundaries of the β-(Ni,Pt)Al top layer. This resulted in Y-rich oxide growth creating heterogeneities in the α-alumina protective oxide layer. This could be detrimental during cyclic oxidation, but it has not yet been tested. Only isothermal tests at 1100°C were performed, and it was shown that the oxidation kinetics after 500 h at 1100°C were low [8], see Fig 5.

Fig. 3 shows an example of a TBC system including an AM1® superalloy substrate, a dense Pt-rich γ/γ' bond coating and a YSZ porous top layer. It was obtained from the following stacking : AM1/ 5µm Pt / 2µm Al / Tosoh TZ8Y powder. An uniaxial force was applied, corresponding to a pressure of 50MPa. The stack was first heated up at 700°C during 10 min and then at 1150°C during 10 min. Heating rates were 100°C/min and cooling rate was 20°C/min. Only one SPS step was necessary to fabricate this system, nevertheless one additional step would be required to obtain RE-doped bond coating. In [2], the fabrication of a full TBC system including a Pt-rich γ/γ' bond coating doped with a controlled quantity of metallic Y, Hf and Si was reported. The doped bond coating was obtained when the superalloy substrate was first coated with PVD thin layers of controlled thicknesses of metallic Y, Hf and/or Si before the SPS process. It can be noted also that for the thickest bond coatings, it is also necessary to perform a final annealing of a few hours at 1100°C after SPS in order to obtain the Pt-rich γ/γ'microstructure. It is remarkable here that SPS process can be conducted without oxidation of the RE elements (no interfacial RE oxide were observed). It is also shown that it is possible to obtain a fully dense bond coating together with a porous and adherent YSZ top coat. This was achieved using nanometric YSZ powder in order to enhance its sintering ability. Finally, it is shown that an interfacial TGO (thermally grown oxide) of alumina can be formed between the bond coating and the ceramic top coat during the SPS process [2,[START_REF] Boidot | Surf and Coat Tech[END_REF].

Another application of high temperature coatings on Ni-base superalloys concerns the abradable seals. The evaluation of the high temperature abradability of coated materials is complex and expensive. Then, it is interesting to test first the high temperature stability, oxidation and corrosion of new candidate materials. Fig. 4 reports a coating fabricated by SPS for this purpose. It consists in a mixture of natural talc and CoNiCrAlY powders. A perfect mixture of the powders could be obtained and a fully dense coating was produced with the SPS process, after 10 min at 1000°C under 100 MPa [START_REF] Ratel | Surf and Coat Tech[END_REF]. Then, the reaction between the talc and the CoNiCrAlY during SPS and during further annealing at high temperature could be studied, as reported in [START_REF] Ratel | Surf and Coat Tech[END_REF]. For example, Fig. 4 shows the external surface of the cermet coating after 50 h at 1050°C in laboratory air. Experimental observations by optical and electron microscopy showed that the microstructure of the bulk specimen was constituted with CoNiCrAlY grains, showing a β/γ structure, covered with fine layers of oxides resulting of the interaction between CoNiCrAlY and talc which decomposed at 1050°C. An external duplex protective oxide scale was also formed consisting in an external continuous layer of spinel MgAl 2 O 4 lying on an internal continuous layer of Al 2 O 3 . The underlying CoNiCrAlY grains were depleted in Al leading to the β-phase dissolution.

After such rapid fabrication of these complex multi-layered systems, large amount of work is necessary to study the properties of these materials. Fig. 5 gathers some preliminary results on the oxidation resistance of the SPS fabricated coatings. The oxidation kinetics of the bulk CoNiCrAlY-talc based cermet was measured using thermogravimetry, but all the multi-layered systems were coated on one side only. Then, oxidation kinetics were evaluated by the measurement of the TGO oxide layer thickness, measured using the SEM on polished cross sections, as a function of time and assuming a parabolic behavior. When several oxidation durations were available, the local calculation of the parabolic constant k p .[mg²/cm 4 /s] was performed using [START_REF] Monceau | [END_REF]:

e 2 ² -e 1 ² =F².k p .(t 2 -t 1 )
where e 2 and e 1 are the oxide scale thicknesses at time t 2 and t 1 , and where F is a factor allowing the conversion from the scale thickness to the oxygen mass gain by unit surface area (F=0.185 mg/cm² for a 1µm thick α-Al 2 O 3 oxide layer). These evaluations were done after 500 h isothermal oxidation at 1100°C in laboratory air (MCrAlY-Pt-Al) or after 300x1h cycles at 1100°C (TBC systems with RE-doped Pt-rich γ/γ' bond coatings). Results given in Fig. 5 show the good quality of the fabricated coatings. Indeed, they all demonstrate oxidation kinetics typical of α-alumina formers, as it can be seen from the comparison with alumina growth kinetics measured on pure β-NiAl [13]. It can be seen also that very low oxidation kinetics can be obtained through Y doping of alumina formed on a NiCoCrAlYTa coating modified by Pt and Al which allows the formation of a continuous and protective alumina layer (and despite Y-rich pegs). Thanks to this coating processing, the reactive element co-doping (Y, Hf, Si) effect on high temperature oxidation of Ni-aluminides, including γ/γ' bond coatings, can be studied in a systematic and reliable manner since the RE elements are added in controlled quantities and at their unoxidized state. This work is in progress.

Figure 1 :

 1 Figure 1: Schematic of several stacking used to fabricated coated superalloy systems before introduction in the spark plasma sintering (SPS) apparatus.

Figure 2 :

 2 Figure 2: NiCoCrAlY coating on AM3 superalloy modified by Pt and Al using the stacking: AM3/NiCoCrAlYTa powder/10µm Pt/12µm Al After 500 h oxidation at 1100°C in air.

Figure 3 :

 3 Figure 3: Thermal Barrier Coating system fabricated in one step by SPS from the stacking: AM1/ 5µm Pt / 2µm Al / Tosoh TZ8Y powder

Figure 4 :

 4 Figure 4: Cross section of a CoNiCrAlY+talcbased coating on AM3 superalloy forming an external duplex oxide scale of MgAl 2 O 4 /Al 2 O 3 . after 50h oxidation at 1050°C in laboratory air.

Figure 5 :

 5 Figure 5: Parabolic rate constants of various coatings fabricated by SPS (see text). "85Y" stands γ/γ' bond coating-base TBC system including a Y PVD coating of 85 nm. α, θ, γ-Al 2 O 3 lines refer to oxidation kinetics of pure β-NiAl [13].

Acknowledgements

This work was partly performed thanks to the financial support of DGA, SNECMA-SAFRAN and TURBOMECA-SAFRAN. RF-sputtering was done at ICMCB, Bordeaux, France.


C Garnier

S Mistou

O Pantalé
email: cpantale@enit.fr

Influence of process and material parameters on impact response in composite structure: methodology using Design of Experiments

Keywords: impact response, RTM process, LRI process, composite structure, design of experiments

Even if the mechanical performances of composite materials give new perspectives for the aircraft and space design, the variability of their behavior, linked to the presence of initial microscopic defects or led in service, constitute however a still important brake in their development. As regards particularly the response to fatigue loads or ageing, the behavior of these materials is affected by several sources of uncertainties, notably on the nature of the physical mechanisms of degradation, which are translated by a strong dispersion in life time.

In aerospace industry, low energy impact phenomenon is not well known concerning composite materials and composite structures. Many manufacturers use important safety factors to design structures. The aim of this work is to define the most predominant parameters which permit a good response of damage using experiences plans. The differences of these parameters by using Resin Transfer Molding (RTM) or Liquid Resin Infusion (LRI) process than prepreg one is also studied in this work.

Introduction

Nowadays, in aerospace industry, composite materials are used but their behavior is not well known. During their life in aerostructure, these materials are subjected to many different loads as maintainability impacts, hail, birds, and others. These loads can be classified according to their energies or strain speed, as shown on figure 1 and2.

Under dynamics loads or medium loads, the structure is affected and many damages are visible. In this case, this structure is automatically replaced. In case of quasi-static loads, the surface of the structure is few affected but damage can occur in it. The detailed inspection (DET) and General Visual Inspection (GVI) inspection do not reveal systematically these defects.

One of the main problems is to know how the structure responds on impact in terms of damage produced inside and residual strength after impact. The knowledge of the entire phenomenon which acts in this response is a good way to control it. First, we have to define these phenomena and all the parameters linked to them. Then, we have to understand how they act to choose the best configuration. To understand all the effects, we have to make experiments using a special test apparatus. For the low energy impacts, a drop weight machine is used, and for upper energies, a gas gun is used. Furthermore, we decided to know how the sample, impacted or not impacted, behave under fatigue loads. The aim is to understand how the damage will initiate for no impacted samples and how it will propagate for others.

State of the art.

Parameters acting on impact response.

In its final report in 1998, the FAA [1] does a state of the art about parameters which can modify the impact response of composite structures. In its report they classify these parameters in three parts. One part of these reported to the impactor, another one of these reported to the target and a third one for parameters reported to the material and its manufacturing process.

In 1992, Demuts and al. [2] compare 2 resin/epoxy systems: graphite / epoxy and graphite / BMI. They note that using the resin BMI doesn't give any advantage concerning residual strength in compression after impact.

In 1994, Hitchen and al. [3] study the influence of the lay up on carbon/epoxy samples impacted by 7 J. They denote that the layup modifies the shape and the size of the delamination. It also modifies the energy level to initiate delamination. Besides, the layup acts on the structure strength before and after impact.

In 1995, Hinrich and al. [4] work on the impact in composite panels, made by RFI (Resin Film Infused) process, under compression load. They denote that thinner is the sample, higher is the damage area. They also note that the contact force is affected by the shape and the speed of the impactor. The prediction of the contact force shall be done separating the kinetic energy in two energies : elastic energy and Hertz contact energy.

In 1998, Fuoss and al. [START_REF] Fuoss | [END_REF] decide to study the influence of the stacking sequence on the impact resistance of composites laminates varying three parameters: interface angle, ply orientation relative to a fixed axis and ply grouping. Concerning angle interface, they denote that increasing the number of angle interface reduces the damage area. For the ply orientation to a fixed axis, they note that changing the angle influences the impact response. Changing the stacking sequence changes the impact behaviour but each configuration is specific. Modifying specimen geometry or boundary conditions will change the impact response. It is noted that the ply grouping reduces the damage resistance and so, increases the damage area.

In 1998, Ambur and Starnes [START_REF] Ambur | In structures, structural dynamics, and material conference[END_REF] work on plane and curvative samples. They show that the radius of curvature doesn't act on the contact force but acts on the compression after impact residual strength.

In 1999, Tai and al. [START_REF] Tai | [END_REF] work on the influence of the thickness of composite on the response of impact and on its fatigue behaviour. They predict that a linear relation exists between the stress level and the life duration under fatigue load.

In 2002, Cartié et al. [START_REF] Cartié | [END_REF] study the influence of resin and fiber properties. They use six carbon / epoxy composites: four different resins and two different fibers. Impact tests were done using a drop weight machine with a speed which varies between 1 and 3 m.s -1 . They denote that, for CAI tests, resin toughness is the most significant parameter. Changing type of fiber does not alter or few alter compression after impact residual strength. Moreover, the type of resin acts on the level of damage initiation load after impact.

In 2004, Mitrevski and al. [9] want to determine the influence of the impactor shape using three different shapes: flat, hemispheric and conic and two impact energies. They also use two different lay up for carbon/epoxy prepreg samples. They conclude that coarser is the indentation, higher is the absorbed energy by the coupon. The hemispheric impactor increases the peak contact force, decreases the contact duration and creates a smaller indentation.

Currently, many approaches around materials are made. Instead of using only thermosetting resin, two components resins are used: thermosetting one with thermoplastic interleaving or particulate thermoplastic polymer are locally located in the resin [10,11,12,13].

Many parameters act on the response of impact of composite structures. In most of cases, authors use prepreg to make their samples. To confirm that the same parameters act on RTM, as shown on figure 3, or LRI processes, as shown on figure 4, a decision to make a design of experiments with many parameters was done.

Fatigue

Many investigations were carried out from fatigue loading on composite structure and their behaviour [14][START_REF] Talreja | Transeverse cracking and stiffness reduction in cross-ply laminates of different matrix toughness[END_REF][START_REF] Gathercole | Life prediction for fatigue of T800/5245 carbonfibre composite : I. Constant-amplitude loading[END_REF][START_REF] Mao | Fatigue damage modelling of composite materials[END_REF][START_REF] Dzenis | Cycle-based analysis of damage and failure in advanced composites under fatigue: 1. Experimental observation of damage development within loading cycles[END_REF][START_REF] Gros | NDT data fusion at pixel level[END_REF][START_REF] Vary | Nondestructive testing of fiber reinforced plastics[END_REF]. Most of these investigations show that fatigue behaviour of composite structure is a complex phenomenon. Many factors act on this behaviour, such as the loading level, the frequency, etc. However, few studies take into account the worsening character of compression during fatigue loading. Most of the pre-cited works deal with tension-tension fatigue loading. But, when a fatigue cycle takes into account compression, local or global buckling shall appear [21] and initiate in a premature way damages. Concerning the damage development, many properties of the materials act on it. The inhomogeneous microstructure and the variable quality of many commercial composite materials [START_REF] Smith | proceedings of the sixth International reinforced plastics Congress[END_REF], where there are small disparate porosities or cavities on it, the stiffness or strength of the composite lead to complex damage development and to different modes of failure like delaminations, disbondings, fibre breakage, etc [START_REF] Harris | Fatigue in Composites[END_REF]. About the experimental procedure, Bathias, Behesty and Tai [START_REF] Bathias | [END_REF]25,[START_REF] Tai | [END_REF] consider that the most penalizing cyclic damage is fatigue under traction/compression. Moreover, they found that the component in compression of a loading in traction/compression must be important to show its effect on the life reduction after impact. Consequently they use a loading ratio of -1 between the minimum and maximum stress.

In their works, Minnestyan, Philippidis and Post [START_REF] Minnetyan | Computational simulation of composite structural fatigue[END_REF][START_REF] Philippidis | [END_REF][28] make a non exhaustive list of models which predict residual strength. Existing models as Sarkani et al., Broutman and Sahu, Schaff and Davidson, Hahn and Kim, Rotem, Yang et al.,interaction model, Sendeckyj, modified Broutman and Sahu model, modified Hahn and Kim, modified interaction model and non linear model were compared. Different types of coupons were tested and compared with numerical models. They note that most of them are not completely reliable concerning residual strength after fatigue. Some models correlate well with the experiment curves of life at the beginning while others correlate near fracture. The experimental results show that the dispersion of residual static strength increases with the accumulation of the damage during the increase in the number of endured cycles. Most of the cited models follow this trend.

All the precited models and comparisons of them with experiments were carried out from prepreg Fiber Reinforced Plactics. None of them were compared with RTM-FRP or LRI-FRP.

Testing methodology

In order to do design of experiments, it has been decided to classify the parameters in 3 classes: process, materials and testing parameters. The first class includes the entire parameters which act during the manufacturing of the structure. The temperature of the resin, the temperature of the mold, the room temperature, the pressure, the time of polymerization, the time to reach the temperature of polymerization can be cited. All the data which can change or variable during the manufacturing can be integrated on the design of experiments. The second class involves the nature of the fibre, the tow, the density and the thickness of a ply, the basis weight, the different mechanical and chemical treatment of the fiber, the stacking sequence, the nature of the resin, the number of components of the resin. And for the third class, the weight and the shape of the impactor, the boundary conditions of the sample, the impact energy shall be parameters. The aim of the design of experiments (DoE) is to know the most important parameters acting on the impact response for two different processes: LRI (Liquid Resin Infusion) and RTM (Resin Transfer Molding). Two considerations in the building of the DoE are the fact that some parameters of these processes are different and that the direction of the resin injection is different. So it has been decided to build two DoE: one for LRI process and other one for the RTM one, keeping the same parameters for the materials (tow, basis weight, ...) and for the impact testing ( impactor weight, radius of impactor tap, ...) .

As seen before, many parameters should be used in our Design of Experiments. In order to predict the influence of each parameter, the coupling between them and the curvature of the DoE, it has been decided to reduce the number of factors and so, the number of specimen to produce. Here is the general response of the DoE:

2 0 i ii j i ij i i X B X X B X B B Y (Eq.1)
where Xi is the parameter i, Bi the influence of the parameter Xi, Bij the interaction between Xi and Xj, Bii the curvature of the plan and Y is the response. First, the way to reduce the high number of parameters for the material class has to be chosen. Nowadays, most of composite aerostructures are made with epoxy resin and carbon fiber. Moreover, only few resins are qualified for aircraft. Deciding to use only epoxy resin seems to be a good choice to reduce the number of parameters. Concerning fibers, we decided to choose carbon.

Secondly, concerning intrinsic fiber parameter, the study shall be done varying parameters which are easy to change. Some parameters shall be changed in concert with suppliers on basis of the existing fibers. Moreover these parameters must be, as far as possible, independent from one to other. So we choose these four parameters: tow, basis weight, type of fiber, type of fabric. One more chosen parameter for the experiments plan is the powdering of the fabric. The aim of this parameter is to predict the level of joining between the powder and the resin for LRI and RTM process. As we note in [Fuos98], the stacking sequence works on the impact response of composite structure. So it will be added in the design of experiments.

On the implementation of composite materials, we will take as the basic the polimerization cycle of epoxy resin. This cycle is composed of 3 stages: temperature rise, polymerization step and decreasing temperature. For the first one and the third one, the speed will be introduced as parameter. But for the polymerization step, we add time and temperature.

The main difference between the two processes is the pressure. LRI used a vacuum pressure while RTM use pressure. Each one will be a parameter for the concerned process.

For each parameter, a field variation will be chosen using for each one 2 or 3 levels.

To know how the structure reacts on impact, we have to make experiments. Low energy impact testing will be made on samples. We choose energy which will approximate the Barely Visible Impact Damage (BVID). So we will impact the samples using a drop weight impact tester (fig. 1) able to keep the drop weight after the first impact (brake). According to the AITM 1.0010, the measurements of samples are 100 mm * 150 mm with a thickness of 4 mm. Concerning the impactor, we will use one, made in steel, with hemispheric tap which weight 1 to 3 kg. This testing will take place on new samples and also on old samples, ageing under fatigue loading, impacted or not impacted.

To define the influence of each parameter, we have to measure responses of the experiments plans. We decided to measure the health of the samples before impact, three-dimension geometry of the damage produced by impact, and residual mechanical properties.

Figure 1 :Figure 2 :

 12 Figure 1 : Principle of Resin Transfer Molding

Figure 5 :

 5 Figure 5 : Drop weight test machine

Conclusions and perspectives

The response of the DoE is a good way to understand low energy impact phenomenon. The 3D geometry and the reduction in mechanical properties will be a help in the design of composite aerostructures. This study will give a safety ratio taking into account the impact phenomenon.

Also, this study is good start to evaluate the mechanical behavior of impacted composite structures under fatigue loading.

Summary

This study concerns composite structures and how they response to a low energy impact. The methodology used is based on a design of experiments (DoE). First, studying done works, a list of parameters acting on impact structure is made. These parameters should be classified on three groups: material, process and testing parameters. Three responses of the DoE are measured: material health, before and after impact loading, geometry of the produced damage and residual mechanical properties. The aim of this study is to understand the low energy impact phenomenon and how it act on impact repose of composite structure.

The second work is to understand the behavior of the structure before and after low energy impact loading under fatigue loads. Different medium are used to follow the damage growth.