R Rabah
email: rabah@emn.fr

G M Sklyar
email: sklar@univ.szczecin.pl

A V Rezounenko
email: rezounenko@univer.kharkov.ua

On pole assignment and stabilizability of neutral type systems

In this note we present a systematic approach to the stabilizability problem of linear infinite-dimensional dynamical systems whose infinitesimal generator has an infinite number of instable eigenvalues. We are interested in strong non-exponential stabilizability by a linear feed-back control. The study is based on our recent results on the Riesz basis property and a careful selection of the control laws which preserve this property. The investigation may be applied to wave equations and neutral type delay equations.

Introduction

For the linear finite-dimensional control system ẋ = Ax + Bu,

the problem of stabilizability is naturally connected to the possibility to move, eigenvalues of A, which are in the closed right-half plane by a linear feedback u = F x to eigenvalues of A + BF which are in the open left-half plane, such that the closed-loop system ẋ = (A + BF )x becomes asymptotically stable.

If we can do that with arbitrary given eigenvalues for A and A + BF , we say that the system is completely stabilizable (cf. [START_REF] Zabczyk | Mathematical Control Theory: an introduction[END_REF]) or that the pole assignment problem is solvable [START_REF] Wonham | Linear multivariable control. A geometric approach[END_REF]. The last property is connected with the complete controllability: the system is completely stabilizable if it is completely controllable. When the system is not completely controllable, the problem of stabilizability may be solvable if the unstable modes of the spectrum are controllable. One can then obtain asymptotic stability by feedback, which is in fact also exponential stability. It is well known that the situation in infinite dimensional spaces is much more complicated (see for example [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF][START_REF] Zabczyk | Mathematical Control Theory: an introduction[END_REF] and references therein). The possibility to move the spectrum is not so simply connected with the controllability property and the last one also is multiply defined (exact, approximative, spectral). The spectrum itself is not sufficient to describe the asymptotic behavior of the solution of the infinite dimensional system even in Hilbert spaces [START_REF] Van Neerven | The asymptotic behaviour of semigroups of linear operators[END_REF]. Our purpose is to analyze the pole assignment problem in the particular situation of a large class of linear neutral type systems. Our approach is based one the infinite dimensional framework. Let us first give some precision about the used notions.

Let us suppose that the system (1) is given in Hilbert spaces X, for the state, and U for the control. There are essentially two notions of stabilizability : exponential and strong (asymptotic, non-exponential).

Definition 1. The system is exponentially stabilizable if for some linear feedback F the semigroup of the closed loop system e (A+BF )t verifies:

∃M ω > 1, ∃ω > 0, ∀x, e (A+BF )t x ≤ M ω e -ωt x .
The system is strongly stabilizable if ∀x, e (A+BF )t x → 0, as t → ∞.

For some particular systems the two notions are equivalent. For example it is the case for linear parabolic partial differential equations with discrete spectrum or for linear retarded systems. For linear neutral type systems and hyperbolic partial differential equations they are different: such systems may be asymptotically stable but not exponentially stable and then the same situation occurs for stabilizability (see for example [START_REF] Brumley | On the asymptotic behavior of solutions of differentialdifference equations of neutral type[END_REF][START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF][START_REF] Sklyar | On Asymptotic Stability of Linear Differential Equation in Banach Space[END_REF][START_REF] Van Neerven | The asymptotic behaviour of semigroups of linear operators[END_REF]). This situation is related to the location of the spectrum near the imaginary axis (see our paper about the stability problem [START_REF] Rabah | On strong stability and stabilizability of systems of neutral type[END_REF]). For the neutral type systems, as for other infinite dimensional systems (for exemple hyperbolic partial differential systems), the spectrum may contain an infinite set close to the imaginary axis. In [START_REF] Rabah | On strong stability and stabilizability of systems of neutral type[END_REF][START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF] we gave an analysis of this situation on stability conditions. It is shown that even complete information on the location of the spectrum of the operator A (and A + BF ) does not provide the description of the cases when the system is stable or unstable. As clearly indicated in [18, Theorem 22, p.415], there is an example of two systems (of neutral type) which have the same spectrum in the open left-half plane but one of them is asymptotically stable while the other is unstable. The reason for this is that not only the location of the spectrum (eigenvalues in the case of discrete spectrum), but also the geometric characteristics (the structure of eigenspaces and generalized eigenspaces) are important. In this connection we notice that even for particular systems (except finite-dimensional) the complete description of the stability properties is not available at the present time. In such a situation, the stabilizability problem inherits many of the technical difficulties arisen in the study of the stability.

The second main difficulty of the stabilization problem in infinite dimensional spaces is related to the action of the control, namely the dimension of the control variable and the quality of the feedback. So, it is known [START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF][START_REF] Sklyar | The problem of the perturbation of an element of a Banach algebra by a right ideal and its application to the question of the stabilization of linear systems in Banach spaces[END_REF][START_REF] Rabah | On exact controllablity and complete stabilizability for linear systems in Hilbert spaces[END_REF] that one can essentially change the spectrum of the system by use of a feedback only in the case when it is exactly controllable (this means, in particular, that the image of B is infinite dimensional). For example, it is possible to achieve for any µ > µ 0 :

σ(A + BF ) = σ(-A * -µI).
When the control space is finite dimensional our possibilities to influence spectrum by feedback is restricted. Only a finite number of eigenvalues may be assigned by a linear feedback (bounded or A-bounded, see [START_REF] Nefedov | A criterion for stabilizability of dynamic systems with finite-dimensional input[END_REF][START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF]). The neutral type systems may have an infinite number of unstable eigenvalues that must be moved from the right half plane by finite dimensional feedback.

In the present paper the pole assignment in this case is considered. After a preliminary section where the infinite dimensional model of the neutral type system is given, we consider in Section 3 the case of the abstract system (1) with a one-dimensional control and with an operator A having a Riesz basis of eigenvectors. The second part of Section 3 is concerned with a more general abstract case, when there is no Riesz basis of eigenvectors. Section 4 is devoted to the case of neutral type system, after that we give some concluding remarks.

The neutral type system and the infinite dimensional model

We consider the following neutral type system

ż(t) = A -1 ż(t -1) + 0 -1 A 2 (θ) ż(t + θ)dθ + 0 -1 A 3 (θ)z(t + θ)dθ (2)
where

A -1 is a constant n × n-matrix, det A -1 = 0, A 2 , A 3 are n × n-matrices whose elements belong to L 2 (-1, 0).
In our previous work [START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF] we analyzed asymptotic stability conditions of the system (2). One of the main point of the cited work is the fact that for (2) it may appear asymptotic non exponential stability (see also [START_REF] Brumley | On the asymptotic behavior of solutions of differentialdifference equations of neutral type[END_REF] for the behavior of solutions of a class of neutral type systems). We gave a detailed analysis of non exponential stability in terms of the spectral properties of the matrix A -1 . As a continuation of those results we consider in the present work the control system

ż(t) = A -1 ż(t -1) + 0 -1 A 2 (θ) ż(t + θ)dθ + 0 -1 A 3 (θ)z(t + θ)dθ + Bu, ( 3
)
where B is a n × p-matrix, and study the property for this system of being asymptotic stable after a choice of a feedback control law. Namely, we say that the system (2) is asymptotically stabilizable if there exists a linear feedback control u(t) = F (z t (•)) = F (z(t + •)) such that the system (2) becomes asymptotically stable.

It is obvious that for linear systems in finite dimensional spaces the linearity of the feedback implies that the control is bounded in every neighbourhood of the origin. For infinite dimensional spaces the situation is much more complicated. The boundedness of the feedback law u = F (z t (•)) depends on the topology of the state space.

When the asymptotic stabilizability is achieved by a feedback law which does not change the state space and is bounded with respect to the topology of the state space, then we call it regular asymptotic stabilizability. Under our assumption on the state space, namely

H 1 ([-1, 0], C n ), the natural linear feedback is F z(t + •) = 0 -1 F 2 (θ) ż(t + θ)dt + 0 -1 F 3 (θ)z(t + θ)dt, (4)
where

F 2 (•), F 3 (•) ∈ L 2 (-1, 0; C n ).
Several authors (see for example [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF][START_REF] Tarn | On stabilization by state feedback for neutral differential equations[END_REF][START_REF] Pandolfi | Stabilization of neutral functional differential equations[END_REF][START_REF] Dusser | On exponential stabilizability of linear neutral systems[END_REF]] and references therein) use feedback laws which for our system may take the form

k i=1 F i ż(t -h i ) + 0 -1 F 2 (θ) ż(t + θ)dt + 0 -1 F 3 (θ)z(t + θ)dt. (5)
This feedback law is not bounded in H 1 ([-1, 0], C n ) and then stabilizability is not regular. If the original system is not formally stable (see [START_REF] Loiseau | Neutral-type time-delay systems that are not formally stable are not BIBO stabilizable. Special issue on analysis and design of delay and propagation systems[END_REF]), i.e. the pure neutral part (when A 2 = A 3 = 0) is not stable, the non regular feedback (4) is necessary to stabilize. Later we shall return to this issue from an operator point of view.

Let us consider know the operator model of the system (3) used in [START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF] (see also [START_REF] Burns | Linear functional-differential equations as semigroups on product spaces[END_REF]):

ẋ = Ax + Bu, x(t) = y(t) z t (•) , ( 6
)
where A is the generator of a C 0 -semigroup and is defined by

Ax(t) = A y(t) z t (•) = 0 -1 A 2 (θ) żt (θ)dθ + 0 -1 A 3 (θ)z t (θ)dθ dz t (θ)/dθ , (7)
with the domain

D(A) = {(y, z(•)) : z ∈ H 1 ([-1, 0]; C n ), y = z(0) -A -1 z(-1)} ⊂ M 2 , ( 8
)
where

M 2 def = C n × L 2 (-1, 0; C n ). The operator B : C p → M 2 is
(θ) = z(t + θ), θ ∈ [-1, 0].
This model was used in particular in [START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF] for the analysis of the stability of the system (2) and in [START_REF] Rabah | The analysis of exact controllability of neutral type systems by the moment problem approach[END_REF] for the analysis of the controllability problems (see also [START_REF] Burns | Linear functional-differential equations as semigroups on product spaces[END_REF][START_REF] Lunel | A functional model approach to linear neutral functional differential equations[END_REF]).

From the operator point of view, the regular feedback law (4) means a perturbation of the infinitesimal generator A by the operator BF which is relatively A-bounded (cf. [START_REF] Kato | Perturbation theory for linear operators[END_REF]) and verifies D(A) = D(A + BF). Such a perturbation does not mean, in general, that A + BF is the infinitesimal generator of a C 0 -semigroup. However, in our case, this fact is verified directly [START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF][START_REF] Rabah | The analysis of exact controllability of neutral type systems by the moment problem approach[END_REF] since after the feedback we get also a neutral type system like (2) with D(A) = D(A + BF) (see below for more details).

From a physical point of view, A-boundedness of the stabilizing feedback F means that the energy added by the feedback remains uniformly bounded in every neighbourhood of 0 (see also another point of view in [START_REF] Loiseau | Neutral-type time-delay systems that are not formally stable are not BIBO stabilizable. Special issue on analysis and design of delay and propagation systems[END_REF]). Hence the problem of regular asymptotic stabilizability for the systems (3),( 6) is to find a linear relatively A-bounded feedback u = Fx such that the operator A+BF generates a C 0 -semigroup e (A+BF )t with D(A + BF) = D(A) and for which e (A+BF )t x → 0, as t → ∞ for all x ∈ D(A).

Main approach to the problem of infinite pole assignment

This section presents the main methodology that we propose to solve the problem of infinite pole assignment. We consider first the case when there exists a Riesz basis of eigenvectors and the after that the case where there is only a basis of invariant subspaces.

Riesz basis of eigenvectors

This approach has been developed in [START_REF] Sklyar | A theorem on the strong asymptotic stability and determination of stabilizing control[END_REF] and then essentially extended to a more general systems as will be described in the next sections. We consider a system ẋ = Ax + Bu,

where A generates a contractive semigroup {e At } t≥0 in a Hilbert space H; B is a bounded operator from a Hilbert space U to H. We consider equation (9) under the assumptions: i) A is an unbounded operator with discrete spectrum consisting of simple eigenvalues {λ k } ∞ k=1 , ii) there exists a constant C σ ≡ 1 2 min i =j |λ i -λ j | > 0, i.e. the spectrum is separated, iii) the space U is one dimensional, so we associate B with a vector b ∈ H; besides, if {φ n } ∞ n=1 is an orthonormal eigenbasis Aφ n = λ n φ n , then b n = b, φ n = 0, n ∈ N, i.e. the system is approximatively controllable.

One of the key facts in our approach is the possibility to establish the existence of a special Riesz basis of the state space. We recall the definition. Definition 2. A basis {ψ j } of a Hilbert space H is called a Riesz basis if there are an orthonormal basis {φ j } of H and a linear bounded invertible operator R, such that Rψ j = φ j .

If the operator A has a basis of eigenvectors, it is important also that there is a Riesz basis of eigenvectors of the operator A + bq * , where by q * we denote the functional defined by q * x = x, q . It is obvious that the vector q must verify some condition. It will be seen from the next sections that this key property could be extended to a more general situation (e.g. neutral type systems) where not only Riesz basis of eigenvectors, but Riesz basis of invariant subspaces should be investigated.

The following assertion is of prime importance for our considerations since it gives a simple characterization of controls which do not destroy the Riesz basis property. This result was first proved in [START_REF] Sklyar | A theorem on the strong asymptotic stability and determination of stabilizing control[END_REF] for the case of a skew-adjoint operator A. Using the Riesz basis property we have the following main result for the system (9) under the above assumptions i)-iii).

Theorem 2. [21] Let { λ n } ∞ n=1 be any set of complex numbers such that i) |λ n -λ n | < C σ , n ∈ N; ii) ∞ n=1 |λn-λn| 2 |bn| 2 < Cσ ||b|| 2
, where C σ , b n ≡ b, φ n and λ n are as in Theorem 1, the family of eigenvectors {φ n } of A constitute a Riesz basis of H. Then there exists a unique control u(x) = q * x such that the spectrum σ( A) of the operator A = A+bq * is { λ n } ∞ n=1 and, moreover, the corresponding eigenvectors Aψ n = λ n ψ n , constitute a Riesz basis.

This theorem gives the description how to move the (simple) eigenvalues λ n by using an one-dimensional bounded control (of the form u(x) = q * x) inside of circles of radii proportional to r n • b n , where b n ≡ b, φ n and {r n } ∞ n=1 ∈ 2 .

Riesz basis of invariant subspaces

In this section we present an abstract approach to the stabilization problem for a general operator model and this approach will be used in the next section to stabilize the neutral type system.

The approach presented here is a generalization of the idea proposed in [START_REF] Sklyar | A theorem on the strong asymptotic stability and determination of stabilizing control[END_REF] for the problem of infinite pole assignment for the case of the wave (partial differential) equation. In this case, the operator under consideration is skewadjoint with a simple spectrum while in the present case it does not satisfy neither first nor second assumption. Nevertheless, the main idea of [START_REF] Sklyar | A theorem on the strong asymptotic stability and determination of stabilizing control[END_REF], after necessary improvements, allows us to treat more general case including neutral type operator model.

Here we use the notation A for an operator satisfying the assumptions given below. As it will be shown in the next section, the operator defined in [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF] satisfies these assumptions, so the reader mainly interested in the neutral type system may simply look at A as at the operator [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF].

We denote the points

λ (k) m = ln |µ m | + i(arg µ m + 2πk), m = 1, .., ; k ∈ Z and the circles L (k) m (r (k) ) centered at λ (k) m with radii r (k) , satisfying k∈Z (r (k) ) 2 < ∞. (10)
Let H be a complex Hilbert space. We consider an infinitesimal generator A of a C 0 -semigroup in H with domain D(A) ⊂ H. We have the following assumptions:

H1) The spectrum of A consists of the eigenvalues only which are located in the circles

L (k) m (r (k)
), where radii r (k) satisfy [START_REF] Lunel | A functional model approach to linear neutral functional differential equations[END_REF]. Moreover, there exists N 1 such that for any k, satisfying |k| ≥ N 1 , the total multiplicity of the eigenvalues, contained in the circles L (k) m (r (k) ), equals p m ∈ N, i.e. the multiplicity is finite and does not depend on k.

We need the spectral projectors

P (k) m = 1 2πi L (k) m R(A, λ)dλ (11)
to define the subspaces V (k)

m = P (k)
m H H2) There exists a sequence of invariant for operator A finite-dimensional subspaces which constitute a Riesz basis in H. More precisely, there exists N 0 large enough, such that for any N ≥ N 0 , these subspaces are {V The main result of this section is the following Theorem (the proof may be found in [START_REF] Rabah | On strong regular stabilizability for linear neutral type systems[END_REF]).

and

lim k→∞ k • b, ϕ k m 0 = c m , ∈ R for all m ∈ I. (13)
Then there exists N 2 ≥ N such that for any family of complex numbers λ k m ∈ L The condition [START_REF] Tarn | On stabilization by state feedback for neutral differential equations[END_REF] means that the eigenvalues to be changed are controllable, and then may be moved. The condition (13) represents a certain boundedness of the control operator.

The result is that the controllable eigenvalues may be moved arbitrarily in some neighbourhoods of the initial eigenvalues.

Application to neutral type systems

The main contribution of this paper is that under some controllability conditions on the unstable poles of the system, we can assign arbitrarily the eigenvalues of the closed loop system into circles centered at the unstable eigenvalues of the operator A with radii r k such that r 2 k < ∞. This is, in some sense, a generalization of the classical pole assignment problem in finite dimensional space. Precisely we have the following In fact the controllable eigenvalues may be arbitrarily assigned in some neighbourhoods of the initial eigenvalues. Let us also precise that the neutral part of the system (namely here, the matrix A -1 ) is not modified).

Conclusion

Under some controllability condition we obtain that some infinite part of the spectrum of a neutral type system may be moved arbitrarily by a finite dimensional regular feedback. The counterpart is that it may be made only in some neighbourhoods of the original eigenvalues. But for neutral type systems it is sufficient to insure asymtotic stability provided that the spectrum close to the imaginary axis is simple.

Bu 0 .

 0 defined by the n × p-matrix B as follows Bu def = The relation between the solution of the delay system (3) and the system (6) is z t

Theorem 1 .

 1 Let ||b|| • ||q|| < C σ /2, where C σ ≡ 1 2 min |λ i -λ j | > 0 and the family of eigenvectors {φ n } of A constitute a Riesz basis of H. Then the eigenvectors ψ k of the operator A ≡ A + bq * constitute a Riesz basis of H as well.

  and W N , where the last one is the 2(N + 1)n-dimensional subspace spanned by all eigen-and rootvectors, corresponding to all eigenvalues of A, which are outside of all circlesL (k) m , |k| ≥ N, m = 1, .., .The scalar product and the norm in which all the finite-dimensional subspaces V (k) m and W N are orthogonal and form a Riesz basis of subspaces are denoted by •, • 0 and • 0 . H3) The system is of single input, i.e. the operator B : C → H is the operator of multiplication by b ∈ H.

Theorem 3 .

 3 [START_REF] Rabah | On strong regular stabilizability for linear neutral type systems[END_REF] (On infinite pole assignment). Assume the assumptions H1)-H3) are satisfied. Consider an infinite set of circlesL (k) m (r (k) ) such that each L (k) m (r (k)) contains only one simple eigenvalue of A, i.e. p m = 1. We denote the set of indexes of these circles by m ∈ I.We assume that b ∈ H is not orthogonal to eigenvectors ϕ k m , of A * for m ∈ I i.e. b, ϕ k m 0 = 0 for all |k| ≥ N, m ∈ I

  (k) m (r (k) ), m ∈ I, |k| ≥ N 2 there exists a linear control F : D(A) → C, such that 1) the complex numbers λ k m are eigenvalues of the operator A + BF; 2) the operator BF : D(A) → H is relatively A-bounded.

Theorem 4 .

 4 Consider the system (3) under the following assumptions: 1) All the eigenvalues of the matrix A -1 satisfy |µ| ≤ 1.

2 ) 1 e 1 e

 211 All the eigenvalues µ j ∈ σ 1 def = σ(A -1 ) ∩ {z : |z| = 1} are simple (we denote their index j ∈ I).Then the system (3) is regularly asymptotic stabilizable if 3) rank ( ∆ A (λ) B ) = n for all Re λ ≥ 0, where∆ A (λ) = -λI + λe -λ A -1 + λ 0 -λs A 2 (s)ds + 0 -λs A 3 (s)ds, 4) rank ( µI -A -1 B ) = n for all |µ| = 1.


Mahadev Bar

R Alagirusamy

Apurba Das

Pierre Ouagne

Emmanuel De Luycker

Influence of composite architecture and interfacial bonding on their impact properties

Resistance to impact thrust is one of the most important requirement along with many others for defining suitable applications of any composite materials. In this present work, the influence of composite architecture and composite interfacial bonding strength on the impact performance of flax reinforced PP composites have been investigated. In this regard two different composite architecture through conventional film stacking method and through DREF (DR. E. Fehrer open-end friction spinning process) yarn consolidation method have been produced and to alter the fibre-matrix interfacial bonding the flax fibres are treated with MAgPP (Maleic Anhydride grafted Polypropylene). Finally, the notch Izod impact behaviour of all composite samples are evaluated and compared.

Introduction

Since last few decades with rising ecological concerns and public awareness related to sustainability, a growing interest has been observed in the use of bio-based, environment-friendly, sustainable materials. Development of natural fibre reinforced thermoplastic composites (NFRTC) fits well with the above scenario [1]. Several studies on NFRTC materials have already been carried out and these studies have concluded that NFRTC can successfully replace the glass-reinforced polymer composites in the areas of automobiles, constructions, households, packaging, etc. [2]. However, these reported studies mainly focus on the mechanical (tensile and flexural) properties of the NFRTC materials and only a few of them have investigated the impact responses of the NFRTC materials [3,4]. There is a high chance that during their service life, NFRTC materials can be subjected to accidental impact forces which may result from tool drop, stone strike, crash impacts, etc. Hence, it is necessary to understand the impact responses of the composite laminates.

In general, the impact thrust are of two types namely low velocity impact (LVI) by a large mass (dropped tool) and high velocity impact (HVI) by a small mass (small arms fire). Natural fibres have very low modulus, strength and work of rupture than the high performance synthetic fibres (HVI-resistant materials) such as aramid, ultra-high-molecular-weight polyethylene, etc. Hence, the composite made of natural fibres are suitable for the application where the materials are subjected to LVI only. Generally, the LVI performance of a NFC is studied in three different ways, through the Charpy impact test method, Izod impact test method or through the drop-weight impact test method. Among these test methods Izod method of impact testing is very popular for evaluating the impact behavior of natural fibre reinforced polymer composites and all composite samples in this present study are tested by Izod impact test method.

Two different composite structures have been developed to investigate the influence of composite architectures on their impact behavior. The first one through DREF yarn consolidation method and the second one through conventional film stacking method. Reinforcing flax yarns are treated with MAgPP in order to alter the composite's interfacial bonding. The fibre to matrix ratios in each composite Structure structures are varied at 40 wt %, 50 wt% and 60 wt% levels respectively. In this way, in total nine sets of composite samples have been produced to investigate the effect of composite architecture and interface on their impact properties.

Experiment

MAgPP treatment of flax yarns

In the present study, MAgPP treatment of the flax yarns are carried out in a boiling solvent medium which is made of 80% xylene and 20% toluene. The process is conducted in a reaction kettle which is fitted with a flowing water condenser to avoid the solvent lost during the treatment. 5 wt% MAgPP is dissolved in the boiling solvent (T = 130 ± 5 °C) with a yarn to solvent ratio of 1:25. The yarns in hank form are then socked in the boiling solvent for 10-15 min. After socking, the hanks are washed several times with the xylene heated just above 80 °C to remove the un-bonded MAgPP from the yarn surface. Finally, the washed samples are dried in an air oven at 105 °C for overnight. Exposure to xylene and toluene can cause acute and chronic effects (such as mild headache, fatigue etc.) in human body systems. Hence proper protection such as use of hand gloves, face mask, safety goggle etc. is taken care and the whole experiment is conducted inside a hood.

Manufacturing of DREF yarn

The DREF or friction spinning system is an open-end spinning process was developed by Dr. E. Fehrer, in 1973. This process produces core-sheath structure hybrid yarn at very high delivery rate and without altering the feed core yarn twist. In the present work, a flax yarn is used as core whereas PP is used as sheath. During DREF spinning, the flax yarn is fed as core to the yarn formation region while the PP slivers (sheath components) are fed from the back side of the machine. After feeding, the PP slivers are opened up into individual fibre entity by a pinned beater and are transferred to the yarn forming zone to wrap over the core flax yarn. DREF yarn having different flax content i.e. 40 wt%, 50 wt% and 60 wt% respectively are manufactured after varying the PP sliver feed rate with respect to the constant core yarn feed rate of 40 m/min. An image of produced DREF yarn having 50% flax and 50% PP is shown in Fig. 1.

Preparation of composite laminates

In this present study, two different unidirectional composite structures through DREF yarn consolidation method and through conventional film stacking method have been manufactured. In case of DREF yarn consolidations, the hybrid yarns are wound over a spring-loaded rectangular metallic frame in a parallel configuration in layer-by-layer manner. While in case of film stacking method, pre-weighed and predetermined (based on fibre to matrix ratios) number of PP strips having 20 mm width are placed evenly between the parallel wound flax yarn layers. All stacked yarn layers are then wrapped with a thin PP film and subsequently tied with a PP filament. The metallic frame having stacked yarn sample is then placed in the groove of female mold half and subjected to consolidation after closing the mold. Consolidation is carried out at 190 °C temperature, with 8 bar consolidation pressure and 8 bar vacuum pressure for 5 min. After curing, whole set-up is cooled down to below 100 °C without releasing the consolidation pressure. The images of the produced DREF yarn consolidated composite and film stacked composites are shown in Fig. 2.

Scanning electron microscopy (SEM)

Effect of MAgPP treatment on flax fibres are studied through SEM image analysis. This study is carried out in the following manner. Initially, a small amount of MAgPP treated and untreated flax fibres are placed over a metallic sample holder with the help of a double-sided carbon tape and are subjected to a gold coating. After coating, the samples are examined using a scanning electron microscope (SEM ZEISS EVO 50) at an accelerating 5-30 AC voltage.

Testing of composite laminates

In the present study, the composite laminates are tested for their tensile and Izod Impact behaviors. The tensile properties of the MAgPP treated and untreated flax reinforce composite samples are tested according to ASTM D638 test method in order to assure the effectiveness of MAgPP treatment on Flax fibres. Tensile testing is carried out at 2 mm/min test speed after clamping the composite sample over an area of 40 mm  20 mm at both ends leaving 100 mm gauge length. The thickness of all tensile test specimen varies between 3.3 and 3.7 mm. The notch Izod impact behavior of the composite samples is tested in an impact strength tester, according to ASTM D256-97 test method. The specimens for notch Izod impact test have a dimension of 64 mm  12.7 mm  3.5 (±0.2) mm. Before testing a V-shaped notch is engraved in all specimens using a notch cutting machine on the shoulder side of 12.7 mm dimension of the specimen or on the side parallel to the direction of the application of molding pressure. The notch etched on the test specimens has a notch angle of 45 ± 1°with a radius of curvature at the apex of 0.25 (±0.05) mm. The depth under the notch of the specimen is 10.2 mm. All the tests are carried out at 27 ± 2 °C temperature and 65% RH. Each reported value represents the average of five samples.

Experimental design

The main aim of this present work is to study the influence of composite architecture and composite interfacial bonding strength on the impact performance of flax reinforced PP composites. DREF yarn having three cores to sheath ratios are manufactured after using MAgPP treated and untreated flax yarn as core to study the influence of interface on their impact properties. In order to study the influence of composite architecture on composite's impact properties, two different composite architectures with three different flax to PP ratios are manufactured through conventional film stacking method and through DREF yarn consolidation method. The details of all composite sample produced in this study are tabulated in Table 1.

Effect of composite's interface on their impact behavior

Photographs of the MAgPP treated and untreated flax reinforced Izod impact tested composite samples are shown in Fig. 5. It is observed that all composite samples fail with a valid fracture mode i.e. no longitudinal cracking reaches to the remote sample end. However, lots of pulled out fibres are observed at the impact fracture end of the untreated flax reinforced composite samples while a sharp failure is observed in case of MAgPP treated flax reinforced composite samples.

The Izod impact strength values of MAgPP treated and untreated composites are shown in Fig. 6. It is observed that the impact strength of the composite samples decrease up to 33% as the flax fibre content in the composite structure increases from 40% to 60%. A further 30% decrement in the impact strength of the unidirectional composite samples is observed after the MAgPP treatment of core flax yarn. Untreated flax yarn offers a poor fibre matrix interfacial bonding, on the other hand, MAgPP treatment of the reinforcing yarn improves the fibre-matrix interaction. A strong flax-PP interface is beneficial to the tensile and flexural properties of the composite samples but, it has a detrimental effect on the impact strength. Because the energy absorption during impact testing in the form of fibre-matrix de-bonding, sample deformation and fibre pull out is suppressed by the strong interface [7,8]. Hence, untreated flax yarn reinforced composites show higher impact strength than the MAgPP treated flax reinforced composites.

Effect of composite architecture on their impact properties

The photographs of Izod impact tested DREF yarn compressed and film stacked composite specimens are shown in Fig. 7. It is observed that both the composite samples break completely and the line of fracture along the sample width lies almost in a plane. Moreover, no fibre pull out is observed at the impact failure ends of both composite samples.

The impact strength of DREF yarn compressed and Film Stacked composite samples are reported in Fig. 8. The impact strength of both the types of composite specimens show around 20% decre- ment when the flax content in the respective composite structure increases from 40% to 60%. The impact strength of the composite samples having non-uniform fibre-matrix distribution is mainly determined by the matrix deformation [9]. The matrix deformation during impact testing of the composite samples having 40% flax content is more than those of the composites having 60% flax. Cabral et al. [10] have observed a similar effect on jute-PP composite system when the jute content goes beyond 20%. However, at constant fibre content the hybrid yarn reinforced composites exhibit around 10% higher impact strength than the film stacked composites. There is no fibre pull out observed in the impact fracture surfaces of both composite samples. Hence the impact strength of the composite samples is mainly dependent on the amount of energy absorbed to break the flax fibres, matrix and the fibre matrix interphase. The DREF yarn composites have higher fibre bundle-matrix interface area than those of film stacked composites, thus the DREF yarn compressed composites exhibit better impact strength than film stacked composites.

Conclusions

In the present work, the influence of composite architectures and composite interphase on their impact behavior have been investigated. Two different composite structure, first one through DREF yarn consolidation method and the second one through conventional film stacking method have been manufactured. Reinforcing flax yarns are treated with MAgPP in order to alter the composite's interfacial bonding. The tensile strength of the composite laminates improves when the flax fibres are treated with MAgPP which confirms the improved fibre-matrix interfacial bonding after MAgPP treatment. A negative effect on the impact properties of the composite samples is observed when the reinforcing flax fibres are treated with MAgPP. The DREF yarn compressed composites have better fibre matrix distribution in its structure which results in better impact performance than the film stacked composites.

CRediT authorship contribution statement

Fig. 1 .

 1 Fig. 1. Flax-PP (50:50) based DREF yarn cross-section.

3. Results and discussions 3 . 1 .

 31 Effect of MAgPP treatment on flax fibre SEM images of the MAgPP treated and untreated flax fibres are reported in Fig. 3. It is observed that after the treatment MAgPP forms a continuous protecting layer around the core flax. Beside this continuous layer, some microspheres are also observed on the MAgPP treated flax fibre surface. Literature reveals that MAgPP reacts with the various functional groups of flax fibre through ester linkage formation and enhances the fibre-matrix bonding of the resultant composite system [5,6]. Tensile strength of the MAgPP treated and untreated flax core based DREF yarn compressed composites are shown in Fig. 4. As expected, the mechanical properties of both types MAgPP treated and untreated flax reinforced PP com-posites increases with increasing flax content which is mainly due to improved fibre matrix interaction.

Fig. 2 .

 2 Fig. 2. The cross-sectional images of hybrid yarn (DREF yarn) reinforced (DYC) composite and film stacked (FSC) composite.

Fig. 3 .

 3 Fig. 3. SEM images of (a) untreated flax fibres (b) MAgPP treated flax fibres.

Fig. 4 .

 4 Fig. 4. Tensile strength of the MAgPP treated and untreated flax core based DREF yarn compressed composites.

Fig. 5 .

 5 Fig. 5. Photographs of the MAgPP treated and untreated flax reinforced Izod impact tested composites.

Fig. 6 .

 6 Fig. 6. Izod impact strength of the MAgPP treated and untreated flax-PP composites.

Fig. 7 .

 7 Fig. 7. Impact fractured composites having different composite architecture.

Mahadev Bar :

 Bar Conceptualization, Data curation, Formal analysis, Methodology, Writing -original draft. R. Alagirusamy: Supervision, Validation, Visualization, Writing -review & editing. Apurba Das: Supervision, Validation, Visualization, Writingreview & editing. Pierre Ouagne: Conceptualization, Funding acquisition, Investigation, Methodology, Project administration, Resources, Writing -review & editing. Emmanuel De Luycker:

Table 1

 1 Details of produced composite samples.

	Sample No.	Flax cont. (%)	PP cont.(%)	Surface treatment of flax fibre	Method of composite fabrication
	1	60	40	Untreated	DREF yarn consolidation
	2	50	50	Untreated	DREF yarn consolidation
	3	40	60	Untreated	DREF yarn consolidation
	4	60	40	Treated with MAgPP	DREF yarn consolidation
	5	50	50	Treated with MAgPP	DREF yarn consolidation
	6	40	60	Treated with MAgPP	DREF yarn consolidation
	7	60	40	Treated with MAgPP	Film stacking method
	8	50	50	Treated with MAgPP	Film stacking method
	9	40	60	Treated with MAgPP	Film stacking method

Funding acquisition, Investigation, Project administration, Resources, Writing -review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Andréa Julien

Charline Giguet-Covex

Erwan Messager

Laurent Marquer

Anne-Lise Develle

Fabien Arnaud

Pierre Sabatier

A fast, non-destructive and low-cost dating approach of sediment cores over the last century

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