N Bhouri
email: neila.bhouri@inrets.fr

Constrained Optimal Control strategy for multimodal urban traffic network

Keywords: Optimal control, Receding horizon, Linear quadratic, Constrained control, Macroscopic traffic model, Urban traffic, Public Transport

We propose in this paper a receding horizon optimal control strategy for the bimodal urban traffic, cars and public transport. The aim of this strategy is to control the traffic lights durations in order to relieve the traffic congestions in the hole of the network making more for the arcs where the public transport vehicles are present. The bimodal traffic is represented by a linear model. The constraints on the state and control variables are considered within the optimization problem. The numerical resolution is carried out thanks to the active set method. Simulations results are given for a small urban network.

INTRODUCTION

Many efforts are carried out nowadays to improve public transport and to make their use competitive as compared to the cars use. Among the range of tools likely to improve the surface public transport (buses, tramways, etc.) performances, we can cote the control systems giving the priority to the public transport vehicles at the traffic lights.

Giving the priority to the public transport vehicles at traffic lights allows a good improvement of their quality of service. When giving them the priority one can improve, at the same time their speed, their frequency and their regularity and give by there a better comfort to the users. By reducing the number and the duration of their stops at the traffic lights, one can increase their speed and cause a drop in their total time of courses. The comfort of the users increases: faster and more regular vehicles, therefore also less charged, circulating with less accelerations and decelerations.

Several real-time urban traffic control systems are able to give the priority to public transport vehicles. We can cote for example CRONOS Boillot et al. (2000), PRODYN Henry and Farges (1994), SCATS Chen et al. (1998), SCOOT Hunt et al. (1982), TUC Diakaki et al. (2002) and UTOPIA [START_REF] Mauro | Utopia[END_REF]. However, these systems can be applied to small network zones. A state of the art of theses systems is given in [START_REF] Bhouri | Rgulation multimodale du trafic routier et des transports en commun de surface. une classification des mthodes[END_REF]. However, in urban areas where traffic is very dense and the number of public transport vehicles is very important, giving the priority according to local considerations of the intersection or of small zones is not sufficient. It can even imply twisted effects, since it can enable to feed road network sections or congested intersections, resulting in a deterioration of the general traffic conditions including buses traffic conditions themselves. That is why the objectif of our work is to build a global strategy for large scale networks. Its aim is to act on the intersection traffic lights in order to give the priority to the public transport vehicles and to regulate the trafic on the hole of the network.

A first strategy called NeTPrior (Network Transit Priority) has been developed [START_REF] Bhouri | An intermodal traffic control strategy for private vehicle and public transport[END_REF] and [START_REF] Bhouri | Regulation du trafic urbain multimodal avec priorite pour les transports en commun[END_REF]). It consists in a Linear Quadratic (LQ) optimal control strategy. Its objective is to regulate the traffic lights with regards to the global traffic situation and to favor the arcs where and when the buses are present. The resolution of the LQ problem however does not take into account the traffic constraints when computing the optimal control solution. To mitigate this problem we proceeded in NeTPrior to a projection of the optimal control values given by the LQ control stage onto the set of feasible values.

We propose in this paper another methodology consisting in a receding horizon optimisation problem with constraints. We are still using, like in NeTPrior, a linear model and a quadratic criterion. By this way, we can use the advantages of the particular structure of the LQ problem and use one of the powerful available optimization tools. We choose an active set method to resolve this LQ problem. Active set methods are not very perform in terms of speed as compared of other methods such the interiorpoint methods but they are more robust and better suited for warm starts.

The rest of the paper is organized as follows. In section 2 we present the the linear bi-modal urban traffic model. Section 3 gives the quadratic objectif criterion and section 4 states the constraints. Section 5 is concerned with a brief recall of the NeTPrior strategy. In section 6 we give a short presentation of the receding horizon optimization. In section 7 we present the constrained LQ problem we are concerned with and in section 8 we give a very short description of the active set method which we use to get the numerical resolution of the LQ problem. Finally, the section 9 gives the main simulation results of this strategy.

Arc b

Arc a

I M τ b,a q a u a x a N M Intersection Intersection
O N e a s a Fig. 1. Variables' definition

DYNAMIC MODEL

The network is represented by a directed graph composed of nodes and arcs. The nodes j ∈ J represent intersections and the arcs a ∈ A the unidirectional travel links. On every arc, the model consists in two equations, one of them modeling the progress of the total number of vehicles on the arc, expressed as private vehicle unit (PVU) (for example a bus equals 2,3 PVU). The second equation models the number of PT vehicles on the arc.

The general traffic dynamic equations

The traffic on each arc a is modeled using the continuity equation :

x a (k + 1) = x a (k) + T [q a (k) + d a (k) -u a (k) -s a (k)], (1)
where x a is the number of cars on the link expressed in PVU, q a and u a are the inflow and the outflow of link a during [kT, (k + 1)T ] where k is the discrete time step and T is the sampling time. d a and s a are respectively the demand and the exit flow within the link. d a and s a can be generated by the parking vehicles or due to noncontrolled intersections situated between intersections M and N . See figure 1. We We will neglect thereafter this generated and consumed flow (d a and s a ), it would be easy to include them without substantially changing the current development.

In order to explicit the equations for q and u we will consider the saturation flow of each link S a , that represents the maximum traffic flow that can exit the link, expressed in PVU/s. The Store and Forward model assumes that the vehicles reaching the arc's end are stored there and exit with rate S a during the green light. Hence, we can write :

u a (k) = S a .G a (k) C , ( 2
)
where C is the cycle time and G a (k) is the effective green time of link a, i.e., the green light duration attributed to arc a during the traffic light cycle C of the intersection situated at the arc exit, and will be the control variable in our approach.

If the green light periods are attributed to arc a during different phases (see figure 3), G a (k) is equal to the sum of all of these green light durations,

G a (k) = i∈P a N G N,i (k), (3)
where G N,i (k) is the green light duration for the phase i on the junction N , P a N is the set of phases of the intersection N, during which arc a has the right of way (green light). It also assumes that the outflow is distributed among the different following links according to the coefficients τ ab , called turning rates, that represent the proportion of outflow from a entering in arc b.

G 1,2 G 1,3 G 2,1 G 1,2 G 2,1 G 2,3 Phase 1 Phase 2 1 1 1 2 2 2 G 4,2 3 G 4,1 3 G 4,3 3
G 1 = G 1 1,2 + G 2 1,2 + G 2 1,3 ; G 2 = G 1 2,1 + G 1 2,3 + G 2 2,1 ; G 3 = G 3 4,1 + G 3 4,2 + G 3 4,3
If the link a originates at the junction M , the inflow traffic rate entering arc a can be written as the sum of the outflow traffic rates coming from the arcs entering junction M (other than a). If the arc b precedes arc a, the corresponding flow is τ ba u b , so the total flow entering arc a is

q a (k) = b∈I M τ b,a u b (k), (4)
where I M is the set of arcs entering junction M .

Replacing all the previous definitions in the equation (1), we obtain the following model :

x a (k+1) = x a (k)+ T C [ b∈I M τ b,a S b G M,i b (k)-S a j∈P a N G N,j (k)]
(5) or in matrix form :

X(k + 1) = X(k) + B.G(k), ( 6
) where B is a matrix of dimension N × M, N is the number of links and M is the total number of phases on the network.This modeling is possible under the following assumptions :

• the sampling time interval T is at least equal to the duration of the light cycle C, we will use T = C. item the gaps between the intersections are not taken into account, • variations in the queue are neglected, which means that the model considers that all of the input flows on the arc have the green phase at the same time.

The Public Transport Traffic Dynamic Equations

As we will be considering two kinds of traffic, the general one and the public transport one, we will distinguish the state variables as x v for the number of vehicles and x b for the number of PT vehicles (buses). Knowing the sequence of arcs which are used by each PT line, the progress of the PT vehicles is modeled by a delay equation :

x bi a (k) = x bi a (k -ζ i a ) (7)
where x bi a is the number of vehicles of the public transport line number i on arc a, a is the arc preceding a for the line i and ζ i a is a parameter which expresses the mean travel time of the vehicles on line b i to travel from arc a to arc a. These values should be real ones, however, in order to be able to write the precedent equation, we take ζ i a as integer, meaning that the travel time is a multiple of the sampling interval T . Thus we consider that ζ i a is equal to 1 if the bus line has no station on arc a, otherwise ζ i a is equal to 2 (for example). Substituting these values in equation ( 7), the model of the public transport becomes the following :

x bi a (k + 1) = x bi a (k -1), if line i stops on a x bi a (k), otherwise (8)
This simplification complies with the dynamical modeling of the PC, since it consists in assuming that both the private cars and public transport are "stored" during the red light period and then are "distributed" during the green light period, thus they spend a light cycle on the arc. However, the choice of the cycle duration should be done carefully.

The equation ( 8) written in vectorial form gives :

X b (k + 1) = A b 0 X b (k) + A b 1 X b (k -1), (9)
where the matrix A b 0 is the adjacency matrix corresponding to the bus line for the arcs without stops. A b 1 is the adjacency matrix corresponding to the bus line for the arcs with a stop, and X b (k) is the vector of numbers of buses at each traversed arc. It can be further simplified as :

X B (k + 1) = A b .X B (k) (10
) where X B is the vector obtained stacking X b (k) and X b (k -1), and matrix A b is given by :

A b = A b 0 A b 1 I 0 . ( 11
)

The total bimodal Model

The state variable of the whole system consists is a vector of dimension (N + 2N b ), where N is the number of arcs in the system, N b is the number of arcs crossed by the PT lines. The dynamics of the system thus is represented by the following equation

X(k + 1) = AX(k) + BG(k) (12) where A is a matrix of dimension (N + 2N b ) × (N + 2N b ).
The matrix B is composed of two stacked blocks the upper one is defined by the topology of the road network, i.e., the coefficient B aj when different from 0 means that phase j is found entering or leaving arc a and its value is defined according to (5). The lower block corresponds to the influence of the green lights on the bus, which, as it is neglected, has to be 0. We have then

A = I 0 0 A b , B = B 0 . ( 13
)
With these matrices, it is clear that it will not be possible to command the public transport because of the null block of matrix B. However, it doesn't set any problem because in the definition of the model, we suppose that the travel times of the public transport are fixed. What we want is to act in such way that buses can comply with their schedules.

OPTIMISATION CRITERIA

From the viewpoint of the traffic regulation, our objective is to improve the traffic conditions of the public transport on the network, relative to the cars flow, without deteriorating the global traffic conditions. The objective function need to be quadratic in terms of the state and control variables to rest in the LQ case. We propose the following objective function :

J(G) = ∞ k=0 (α(X(k) X b (k))+β X(k) 2 +γ G(k) 2 ) (14)
where α, β and γ are non-negative weighting parameters and the X are given by the dynamic equations ( 5) and ( 8).

The first term of the criteria, (X(k) X b (k)) puts forward the traffic conditions on the arcs crossed by the public transport at the time these public transport vehicles are present on it. The second member aims at reducing the number of vehicles on every arc on the network and thus to equalize the congestion on every arc. The role of this second term is mainly to not degrade too much the traffic in the other arcs. The last term is used in order to avoid large variations of the control (green light times).

This criteria can be written in the matrix form :

J(G) = ∞ k=0 x 2 Q + u 2 R ( 15
)
where

Q = β * I N xN α/2 * I N b xN b α/2 * I N b xN b β * I N xN , R = (γ * I M xM ); I M xM is the M-dimension identity matrix.
The choice of the values of these parameters enables to modify the objective of the regulation. For example, for α = 0, β = γ = 1 the strategy is equivalent to TUC, which doesn't take into account the presence of the public transport. On the other hand a significant parameter α (α >> β) will strongly penalize the arcs which don't support the public transport.

THE CONSTRAINTS

For operative needs, at every intersection j, the durations of green lights should comply with a certain number of constraints :

• the cycle duration (C),

• the phase diagram : all of phases P j should have their green light within the cycle, • the clearance times between phases R j , which implies :

i∈P j G j,i + R j = C. ( 16
)
On the other hand, for safety reasons, the traffic lights have to respect some constraints. The duration of green lights is limited by a minimum and a maximum value. Indeed, a too short green light can not allow pedestrians to cross the road safely and a too long implies a long red light duration on the antagonistic road which can be interpreted by users as a malfunction of the intersection lights and imply their non-compliance :

G j,i,min ≤ G j,i ≤ G j,i,max . (17)
Also the state variable X, the number of vehicles on each arc must respect the capacity of the road and can not be negative :

0 ≤ X a ≤ X a,max .

(18)

THE NETPRIOR STRATEGY

In the already published strategy NeTprior [START_REF] Bhouri | An intermodal traffic control strategy for private vehicle and public transport[END_REF], we resolve a non-constrained LQ optimal control problem over an infinite horizon. The advantage of that relies in the fact that the optimal solution can be written as a linear feedback law and the matrix that defines this law is the solution of a matrix equation (Ricatti equation) stated in terms of the given data.

Control Law

The problem of optimal control consists in minimizing the criteria given by equation ( 14) respecting the dynamics of the system given by the equations ( 12).

Using the LQ optimisation method, the applied command law is given by the following equation

G(k) = G N -F.X(k) (19)
where F is the Feedback matrix defined as F = (R + B T P B) -1 B T P A and the matrix P solves the Riccati matrix equation P = Q+A T P A-A T P BF which depends on the coefficients α, β, and γ of the objective function through matrices Q ad R.

Constraints

The solution of the optimal control problem by the LQ methodology doesn't enable us to take the constraints into account because the Ricatti equation will no longer be valid. We solved this problem in NeTPrior, through a projection of the obtained control values onto the set of feasible values defined by the above constraints. It means to obtain the closest (for some distance) values to the optimal but not feasible ones.

RECEDING HORIZON OPTIMAL CONTROL WITH CONSTRAINTS

The Receding Horizon Optimal Control consists in starting with a fixed optimization horizon, of length K, using the current state of the plant as the initial state, we then optimize the objective function over this fixed interval accounting to constraints, obtain an optimal sequence of K control moves {u i , ..., u i+K-1 }. Apply only the first control move{u i } to the plant. Time than advances one step and the same K-step optimization problem is considered using the new state of the plant as initial state as shown on the Figure 4. Thus one continuously revises the current control action based on the current state and accounting for the constraints over an optimisation horizon of length K [START_REF] Goodwin | Constrained Control and Estimation, an optimisation Approch[END_REF].

k 0 1 2 3 k 0 1 2 3 k 0 1 2 3 4 k 0 1 2 3 4 k=0; k= 1
Prediction Horizon Prediction Horizon

Fig. 4. Receding horizon optimisation principle

As the traffic conditions in the dense urban areas are very changeable it is important that control solution can be applied in a feedback loop, this is possible with the receding horizon optimal control as shown on the diagram 5. In the general case the algorithms for the receding horizon optimal control remain relatively complex, however remarkable simplifications occur for the particular case of linear systems subject to linear inequality constraints. Thats why we will place our selves in this particular case. The fixed horizon optimal control problem for linear system with a quadratic objective function and linear constraints can be set up as a quadratic program, written for the general case as follows :

P N (x) :                        J (x k ),(u k ) = M in 1 2 x T K P x K + 1 2 K-1 k=0 (x T k Qx k + u T k Ru k ) x k+1 = Ax k + Bu k x 0 = x u k ∈ U k ∀k = 0, 1, . . . , N -1, u ∈ R m x k ∈ X k ∀k = 0, . . . , N, x ∈ R n
The control and state variables u k and x k are subject to the inequality :

U k = {u k : Ku k ≤ v u } X k = {x k : Dx k ≤ v x }

OUR CONSTRAINED LQ OPTIMAL CONTROL

Given the time-invariant model of the system (12) the fixed horizon optimization problem ( 14) can be transformed into a Quadratic Problem (QP).

we denote by u s and x s the desired steady state values for u k and x k . We have then :

u s = [(I -A) -1 B] -1 x s x s = (I -A) -1 Bu s ( 20
)
our aim is to find, for the system (12) the K control sequence {u 0 , • • • , u K-1 } and corresponding state sequence {x 0 , • • • , x K } that minimize the finite horizon objective function :

J (x k ),(u k ) = M in 1 2 x T K P x K + 1 2 N -1 k=0 (x T k Qx k + u T k Ru k ) (21)
where P > 0, Q ≥ 0 and R > 0, K is the prediction horizon, K ≤ K is the control horizon, and u k , x k are the input and state setpoints given by ( 20). The control is set equal to its steady state setpoint after M steps, that is u k = u s for all k ≥ K. The above fixed horizon minimization problem is solved at each time step for the current state and disturbance values. Then the first move of the resulting control sequence is used as the current control, and procedure is repeated at the next time step in a RHC fashion, as described in section (6)

Objective function handling

Writing from ( 12)with x 0 = x, and using the constraint that

u k = u s for all k ≥ M .                  x 1 = Ax 0 + Bu 0 x 2 = Ax 1 + Bu 1 = A 2 x 0 + ABu 0 + Bu 1 x 3 = Ax 2 + Bu 2 = A 3 x 0 + A 2 Bu 0 + ABU 1 + Bu 2 . . . x N = A N x 0 + A N -1 Bu 0 + • • • • • • + A 1 Bu N -2 + Bu N -1 let us note : U = (u 0 , u 1 , • • • , u N -1 ) ; X = (x 1 , x 2 , • • • , x N )
the state equation ((12) became :

X = S.U + A.x (22)
where S is an (N.n) * (N.m) and A an N.n matrices given by :

S =     B 0 • • • • • • 0 AB B 0 • • • 0 . . . . . . A N -1 B A N -2 B • • • AB B     ; A =      A A 2 . . . A N     
using this vector notation ( 22) and rewriting the objective function ( 21) to yield :

J = 1 2 x 0 Qx 0 + 1 2 (Ax) Q(Ax) constante + 1 2 U (S QS) H 1 U +U (S QAx) F ⇔ J = 1 2 U (S QS + R) H U + U (S QAx) F ( 23
)
This criterion is a function only of the control vector U ; matrices Q and R are given by:

Q =      Q • • • 0 0 . . . . . . . . . . . . 0 • • • Q . . . 0 • • • 0 P      R =    R • • • 0 . . . . . . . . . 0 • • • R   
We can notice that to have a positive definite matrix H it is sufficient to have matrices Q and R positive definite independently of the system dynamic. However, due to the non controllability of the model, this H matrix is independent of the parameter "α" of the Q matrix. Only the second term of the objective function (matrix F ) is sensitive to transit vehicles.

Contraints Handling

In the same way, we use the vector notation to write the constraints. Let us note

V u = (v u , v u , • • • , v u ) N times ; So U k = {u k : Ku k ≤ v u } ⇔ KU ≤ V u ; and V x = (v x , v x , • • • , v x ) N times ; ; so X k = {x k : Dx k ≤ v x } ⇔ DX ≤ V x ; where K =    K • • • 0 . . . . . . . . . 0 • • • K    and D =    D • • • 0 . . . . . . . . . 0 • • • D   
State constraints can be expressed as a function of the only control variables :

DX ≤ V x ⇔ DS.U ≤ V x -DAx (24)
The QP problem is than formulated :

(QP )

         J = 1 2 U HU + U F subject to : DS.U ≤ V x -DAx KU ≤ V u (25)
8. NUMERICAL SOLUTION There are several approaches for solving the constrained QP problem given by ( 25). One could use, for example, methods of feasible directions, which solve the problem by moving from a feasible point to an improved feasible point. Methods of primal-dual interior point algorithms have also been proposed, such the infeasible interior point methods proposed by [START_REF] Wright | A path-following interior-point algorithm for linear and quadratic problems[END_REF]. Another used algorithm is the active set method, in which at each step certain constraints, indexed by the active set, are regarded as equalities whilst the rest are temporarily disregarded. The method then sequentially solves an equality-constrained QP and adjusts the active set in order to identify the correct active constraints at the optimal solution. It is this method which is used in this work (for more details see [START_REF] Bonnans | Numerical Optimization, theoretical and practical aspects[END_REF]).

active set method

To give a clear presentation of the method, let us simplifying the notation by writing the constraints in the form of A i u ≤ b i ∀i ∈ I which means that A = [DS; K] and the constraints b = [(V x -DAx); V u ]. The problem can be written in a classical form :

     M inimize{f u 1 2 u Hu + u b} subject to : A i u ≤ b i ∀i ∈ I (26)
where H est positive definite, and where the index set I correspond to inequality contraints. Let u k be a feasible solution for (26) and let I k = {i ∈ I : A i u k = b i } the set that records the active constraints at u k . Because the constraints are linear, and since the the optimization problem is strictly convex, a necessary and sufficient condition for a feasible solution to be unique optimal solution of ( 26) is that there exist Lagrange multipliers λ i for i ∈ I k , such that the following Karush-Kuhn-Tucker (KKT) conditions holds :

Hu k + c + i∈I k λ i A i = 0 (27) λ i ≥ 0, i ∈ I k
The active set method iterates to find the optimal solution of (26) in the following way. At the kth iteration we assume that a feasible solution u k is available. The method now seeks to minimize the objective function subject only to equality constraints recorded in I k and ignoring the remaining inequality constraints. If we parameterize u as u = u k + d, so : We use a feasible direction method.

f (u) = f (u k + d) = f (u k ) + 1 2 d Hd + d (b + Hu k )

SIMULATION RESULTS

In order to evaluate the receding horizon strategy we performed several simulation tests on a small urban traffic network (Figure 6). This network has two inputs (E1) and (E2), and two outputs (S1) and (S2). It has seven intersections and fourteen arcs numerated according to the notation of section (6) : J = {1, 2, ...7} and A = {1, 2, ...14}. Arcs related to exits are not considered. The transit vehicles follow a fixed line which is given by the arcs (1), ( 4) and ( 6) (the green lanes on (Figure 6). Vehicles has a commercial stop at arc (2).

For example we show here the traffic state on the arcs respectively 1 and 4 on which circulate the public transport vehicles (figures 7 and 8). The first parts of these figures give the state of the traffic optimized without favoring the public transport (α = 0). In the second part, the blue curves give the traffic state when the public transport vehicles are given a favor (α = 10 4 ) and the green color curves gives the moments of presence of the public transport vehicles on the arcs. the arcs when the buses are present. For the arc 4 (Fig. 8), where a bus vehicle is always present, there is no-vehicle on the queue since the fifth step of the regulation. On the other hand, as we can see on the table (1) we can notice that the total congestion is more important when the parameter alpha is important. A finer analysis of the results shows than the improvement of the arcs supporting public transport is done by retaining the traffic on the entries of the network and not on its totality. It is the price to pay to improve the trajectory of public transport.

CONCLUSION

The simulation shows that the strategy makes a good improvement to the arcs on which public transport circulates exactly when they are present but by retaining the traffic on the entries of the network Future work should include a nonlinear model of the traffic which will give a better precision on the progression of the buses and will thus allow a better precision at their time of presence in the arcs.

FlowFig. 2 .

 2 Fig. 2. The store and forward modeling

Fig. 3 .

 3 Fig. 3. Traffic light periods are attributed during different phases

Fig. 5 .

 5 Fig. 5. Receding horizon optimal control operating in a feedback loop

Fig. 6 .

 6 Fig. 6. The simulation urban network Hence instead of solving (26), we solve the equivalent direction-finding problem : 1 2 d Hd + d (b + Hu k ) subject to A i d = 0 for all i ∈ I k

Fig. 8 .

 8 Fig. 7. Traffic state on the arc 1

Table 1 .

 1 Number of vehicles on the networkAs we can see on the second curves: when the value of (α) is very large, the strategy manages to cancel the queue on


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email: sabine.cantournet@ensmp.fr

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Micromechanics and Macromechanics of Carbon Nanotube Enhanced Elastomers

Keywords: Elastomers, nanotubes, anisotropic, nonlinear behavior PACS:

The effects of carbon nanotubes on the mechanical behavior of elastomeric materials is investigated. The large deformation uniaxial tension and uniaxial compression stress-strain behaviors of a representative elastomer are first presented. This elastomer is then reinforced with multi-wall carbon nanotubes (MWNTs) and the influence of weight fraction of MWNTs on the large deformation behavior of the resulting composite is quantified. The initial stiffness and subsequent strain-induced stiffening at large strains are both found to increase with MWNT content. The MWNTs are also found to increase both the tensile strength and the tensile stretch at break. A systematic approach for reducing the experimental data to isolate the MWNT contribution to the strain energy of the composite is presented. A constitutive model for the large strain deformation behavior of MWNT-elastomer composites is then developed. The effects of carbon nanotubes are modeled via a constitutive element which tracks the stretching and rotation of a distribution of wavy carbon nanotubes. The MWNT strain energy contribution is due to the bending/unbending of the initial waviness and provides the increase in initial stiffness as well as the retention and further enhancement of the increase in stiffness with large strains. The model is shown to track the stretching and rotation of the CNTs with macroscopic strain as well as predict the dependence of the macroscopic stress-strain behavior on the MWNT content for both uniaxial tension and uniaxial compression.

Introduction

Elastomeric materials have long been reinforced by stiff fillers such as carbon black and silica particles in order to provide increased stiffness and strength, hysteresis, as well as electrical conductivity (e.g., [START_REF] Guth | Theory of filler reinforcement[END_REF], [START_REF] Harwood | Stress-softening in natural rubber vulcanizates. part iii: Carbon black-filled vulcanizates[END_REF], [START_REF] Dannenberg | The effet of surface chemical interactions on the properties of filler reinforced rubbers[END_REF], [START_REF] Kilian | Universal properties in filler loaded rubbers[END_REF], Bergstrom andBoyce (1999, 2000)). Typical volume fractions lie in the range of 0.10 to 0.40. These property enhancements are invariably accompanied by a decrease in extensibility and tensile strain to failure.

The introduction of carbon nanotubes to the library of potential fillers offers new possibilities to further tailor the properties of elastomers via blending with relatively low volume fractions of fillers. Here, we explore the influence of multi-wall carbon nanotubes (MWNTs) on the mechanical behavior of a representative elastomer.

Over the past decade, there has been an explosion in research investigating the stiffness and strength enhancements that carbon nanotubes potentially offer as fillers in polymers (e.g. [START_REF] Baughman | Carbon nanotubes the route toward applications[END_REF], [START_REF] Coleman | Small but strong: A review of the mechanical properties of carbon nanotube polymer composites[END_REF]). For example, the elastic modulus of single and multi-wall carbon nanotubes is approximately 1TPa (e.g., [START_REF] Wong | Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes[END_REF]) and, hence, relatively small volume fractions could potentially provide tremendous increases in stiffness over that of the polymer matrix (see, [START_REF] Shaffer | Fabrication and characterization of carbon nanotube/poly(vinyl alcohol) composites[END_REF], [START_REF] Qian | Load transfer and deformation mechanisms in carbon nanotubepolystyrene composites[END_REF], [START_REF] Safadi | Multiwalled carbon nanotube polymer composites: synthesis and characterization of thin films[END_REF], [START_REF] Dalmas | Multiwalled carbon nanotube/polymer nanocomposites: processing and properties[END_REF]. Such improvements depend on good dispersion and alignment of the nanotubes as well as excellent bonding between the nanotube and the matrix (e.g., [START_REF] Chen | Field emission of different oriented carbon nanotubes[END_REF] and [START_REF] Cooper | Distribution and alignment of carbon nanotubes and nanofibrils in a polymer matrix[END_REF]). Although there has been some success in dispersing and aligning nanotubes, these issues have not been universally resolved and bonding is still a limiting issue. The research has also primarily focused on the stiffness and strength enhancement of thermoplastics with limited attention to the influence of carbon nanotubes on elastomeric behavior. Aspects of alignment and bonding which may be beneficial to optimally enhancing thermoplastic properties may actually be detrimental to enhancing elastomeric mechanical behavior.

Elastomeric applications generally require and make use of the large deformation extensibility and resilience of the elastomer. Upon incorporation of stiff fillers into elastomers, it is generally desired to enhance the stiffness (i.e., enhance the initial stiffness and retain this stiffness enhancement for overall large strain deformation behavior) while also retaining the important attributes of large strain resilient behavior and large strain-to-break. The alignment (in terms of both overall alignment as well as waviness of the nanotubes) and bonding detriments encountered when incorporating MWNTs into thermoplastics may be used to advantage in their incorporation into elastomers. The waviness that leads to minimal stiffness enhancement in glassy and/or semicrystalline polymers (see, for example, [START_REF] Bradshaw | Fiber waviness in nanotube-reinforced polymer compositesii modeling via numerical approximation of the dilute strain concentration tensor[END_REF]) can still provide sufficient enhancement to the stiffness and strength of the comparatively very low modulus elastomer. The property enhancement can be achieved through the unbending (straightening out) of the MWNTs with deformation such that the extensibility of the elastomer can be retained. Furthermore, if stiffness enhancement primarily results from unbending of the waviness of CNTs as opposed to axial straining of the CNTs (which relies on good bonding and shear lag load transfer from the matrix to the CNT), the stiffness enhancement will not be lost with large strains. Experiments on elastomeric-like matrices enhanced with CNTs have been reported, for example, by [START_REF] Frogley | Mechanical properties of carbon nanoparticlereinforced elastomers[END_REF], [START_REF] Dufresne | Processing and characterization of carbon nanotube/poly(styren-co-butyl acry-late) nanocomposites[END_REF], [START_REF] Koerner | Deformation-morphology correlations in electrically conductive carbon nanotube-thermoplastic polyurethane nanocomposites[END_REF] ,and very recently [START_REF] Chen | Carbon nanotube-reinforced polyurethane composite fibers[END_REF].The Frogley study reports on silicone-based elastomers filled with single wall CNTs (SWNTs) and carbon nanofibers (CNFs). The Frogley study reveals a dramatic increase in initial stiffness with small fractions of SWNTs, however , this stiffness enhancement is lost after only 10-20 percent strain whereupon the tangent stiffness of the nanocomposites returns to that of the parent elastomer due to debonding of the CNTs from the matrix; the tensile strain-to-break was found to decrease significantly with increase in volume fraction of either CNTs or CNFs. The Dufresne study report on a cast latex-MWNT composite and found increased stiffness with increasing MWNT content, retained stiffness to large strains, but reduced failure strains with increase in MWNT content. The Koerner and the Chen studies report on thermoplastic elastomer polyurethanes (TPUs) enhanced with CNTs where the Koerner study produced thin sheets using a casting process and the Chen study produced extruded, melt-drawn fibers thereby using processing to help achieve axial alignment of the CNTs. It is important to note that TPUs exhibit elastomeric-like behavior where, due to their copolymer structure, the TPU mechanical behavior characteristically exhibits a relatively stiff small strain region followed by a rollover at a relatively low stress level (typical of order of a few MPa to tens of MPa) to a more compliant elastomeric-like stress-strain behavior. Both the Koerner study and the Chen study report an increase in the very initial modulus with increasing CNT weight fraction, as well as an increase in the rollover flow stress; the post-rollover tangent stiffness is found to parallel that of the neat polymer, losing the dramatic stiffness enhancement seen at small strains. The Koerner study reports maintaining the same magnitude of tensile strain-to-break as the parent TPU (with a modest decrease in strain-to-break with increasing weight fraction) and the Chen study reports modest increase in strain-to-break for the lower weight fraction cases and a decrease in strain-to-break for the highest weight fraction of 17.7wt percent. These studies demonstrate aspects of the enhancements that can be achieved when CNTs are incorporated into elastomeric materials. However, the loss in stiffness seen at larger strains suggests that CNT/matrix configurations which do not rely on straight well-bonded CNTS may be more beneficial for elastomeric applications than for thermoplastic applications. This is demonstrated in some of the very recent work of [START_REF] Ahir | Photomechanical actuation in polymer nanotube composites[END_REF] and [START_REF] Ahir | Infrared actuation in aligned polymer-nanotube composites[END_REF] where MWNTs are incorporated in a silicone-based elastomer to produce an IR-induced actuating polymer. In the Ahir studies, the CNTs are in a wavy configuration. While only limited mechanical data is presented, the stiffness is found to increase with increasing volume fraction of CNT and this stiffness enhancement is maintained with increasing strain. This aspect of the mechanical behavior thus contributes to the resilient and reversible behavior needed for actuation mechanisms. There is great potential for enhancing the mechanical performance of elastomers via incorporation of small weight fractions of CNTs; the multifunctional possibilities of further using such enhancements to produce actuators is enormous (note the Koerner study was directed at creating CNT-TPU actuating materials). Hence the need for quantifying and predicting the large strain multiaxial stress-strain behavior of CNT-enhanced elastomers with microstructurally-informed constitutive models will provide a tool for further designing and optimizing materials for a wide range of applications (mechanical and otherwise).

In this paper, we study the mechanics of MWNT-elastomer composites. A series of material compositions with a range in weight fraction of MWNTs are produced and evaluated in large deformation uniaxial tension and compression testing. The contribution of the MWNTs to the deformation is then evaluated in a systematic reduction of the strain energy contributions of the elastomer matrix and the MWNTs to the overall composite strain energy. A three-dimensional constitutive model is then developed and shown to capture the dependence of the large deformation stress -strain behavior of the elastomer nanocomposites subjected to different types of deformation (uniaxial tension and uniaxial compression).

Experimental Program

Materials

The elastomer is a brominated polymer derived from a copolymer of isobutylene and paramethylstyrene (PMS) with a trademark name of EXXPRO (or BIMSM in accordance to ASTM) [START_REF] Powers | Patent : Para-alkylstyrene/isolefin copolymers and functionalized copolymer thereof[END_REF]). Specifically, EXXPRO 3745 from ExxonMobil Chemical Company with 1.2 mole % bromine was the BIMSM elastomer used. This BIMSM elastomer has a specific density of 0.92 and is known to possess a good combination of properties including temperature resistance, heat stability, low permeability, high damping and ozone/weathering resistance [START_REF] Kresge | Butyl Rubbers[END_REF]), Multi-wall carbon nanotubes (MWNTs) of type PR19-PS MWNT were obtained from Applied Science. The MWNTs had a diameter range of approximately ¢¡ ¤£ ¦¥ to § ©¤¡ ¢£ ¦¥ as estimated from SEM and TEM micrographs.

Blends of BIMSM and MWNTs were prepared using a Banbury internal mixer directly operating at 100 RPM under ¡ RAM pressure. The polymer was added first whereas MWNTs were added ¤¡ later. All mixes were removed from the internal mixer at a dump temperature of § ! "# %$ or lower with a total mix time of 3 to 5 minutes. Curatives of 1 phr (parts per hundred of polymer) zinc oxide and 1.5 phr stearic acid were subsequently added using a roller mill operating at & ¡ '# ($ to § !¡ ¡ # $ . These blends were then compression molded and cured to form 2 ¥ ¥ thick plaques in a curing press. The cure temperature was set at § ! ¡ # $ and cure time applied was t90 + 5 (5 minutes plus the time required to reach 90 % cure as measured by an oscillating disk cure meter). BIMSM/MWNT nanocomposites with a range in weight fraction of MWNT were produced. Thermogravimetric analysis (TGA) was conducted on all samples to verify the final material composition and found the prepared samples to possess a weight fraction close to that blended. Table 1 summarizes the compositions.

Scanning electron microscopy on freeze-fractured surfaces as well as transmission electron microscopy (TEM) (Figure 1) revealed good dispersion with random distribution and orientation of MWNTs within the elastomeric matrix.

Mechanical Testing

Uniaxial tension and uniaxial compression tests were conducted on all material compositions. Tests were conducted to large strains at constant engineering strain rate (displacement rate divided by specimen gage length) on a Zwick/Roell Z010 single axis screw machine. Specimens were cut from the ¡ ¥ ¥ thick molded plaques. Tensile specimens were rectangular of dimension ¡ ¢¥ ¥ *¢¥ ¥ . Two types of grips were used. For moderate strains (strains less than 0.70) tensile grips with surfaces designed to test relatively thin compliant polymers were used; one surface is aluminum and the mating surface is a polyurethane. For larger strains (to final failure), Zwick grip 8122 with a spring-loaded grip action was used to avoid any sliding at very large deformation. The transverse displacement was monitored optically during extension and the axial strain was determined assuming incompressibility. Load-unload-reload tests were conducted to quantify any hysteresis and/or straininduced softening of the materials. Tests were also conducted at different engineering strain rates to quantify any rate dependence. Some samples were also extended until failure to examine the effect of MWNTs on the tensile stress and strain at break. Compression specimens were cylinders of nominally ¢ ¤¥ ¥ in diameter and ¤¥ ¥ in height (accomplished by stacking three ¡ ¥ ¥ thick cylinders). Thin Teflon sheets were placed between the specimens and the compression platens to reduce friction; additionally, a powder composed of PTFE and PE was placed between the specimens and the Teflon sheets. Load-unload-reload tests were conducted. Tests were also performed at different engineering strain rates. Tensile and compression data are reported in terms of true stress (load divided by current deformed crosssectional area) and true strain (the natural logarithm of the current axial gage length divided by the initial gage length). Since the bulk modulus of rubber is substantially (i.e. 2-3 orders of magnitude greater) than the shear modulus of rubber, the deformed cross-sectional area during uniaxial tension and compression can be estimated from the measured axial strain assuming incompressibility. A minimum of at least three tests were conducted for each loading situation; stress-strain curves were repeatable in all cases and , in the case of uniaxial tension, strain-to-break measurements were also repeatable for all data sets.

Experimental results

Uniaxial tension

Uniaxial tension true stress-strain data are shown in Figure 2. Figure 2a reveals the elastomer matrix material to exhibit a small level of hysteresis (tests at different strain rates, not shown, revealed negligible strain rate sensitivity); the stiffness and the hysteresis are both observed to increase with an increase in MWNT content. The hysteresis was further quantified through cyclic load-unload-reload tests (representative data are provided in Appendix A) and was found to be a Mullins effect (see, [START_REF] Mullins | Effect of stretching on the properties of rubber[END_REF][START_REF] Mullins | Softening of rubber by deformation[END_REF], Mullins andTobin (1954, 1965)) whereby the hysteresis was nearly absent after the first cycle. Figure 2b depicts the uniaxial tensile data to large strain up to final failure where the stiffening with strain is observed to increase with increase in MWNT content and the strain to failure was also found to increase with increase in MWNT content (failure occurred in the gage length of the specimen and was not a grip-induced failure). The inset of Figure 2b shows the initial modulus as a function of wt% MWNT where the 12.2wt% MWNT-elastomer composite has a modulus 2.5 times that of the neat elastomer.Comparatively, a similar weight fraction of carbon black filler would provide a modulus 1.3 times that of the neat elastomer (e.g., [START_REF] Bergstrom | Mechanical behavior of particle filled elastomers[END_REF]). These data show an ability to substantially increase the tensile stiffness and retain the stiffness enhancement during large strain deformation as well as the ability to increase the tensile strength and the tensile strain-to-break.

Uniaxial Compression Results

Uniaxial compressive true stress-true strain data are shown in Figure 3a revealing the same basic trends as observed in the tensile data. The overall stiffness and the stiffening with strain are seen to increase with MWNT content. The neat elastomer exhibits a low level of hysteresis; the nanocomposite hysteresis increases with an increase in MWNT content. The compressive stress-strain behavior is compared to the tensile stress-strain behavior in Figure 3b for the neat elastomer and for the 6.1wt% and 12.2wt% MWNT-elastomers. These data show the classic differences seen in elastomer compression vs tension stress-strain data. In an elastomer, this difference is attributable to the evolution in alignment of the underlying molecular network with different states of strain. The network aligns in a biaxial orientation when subjected to uniaxial compression and aligns in a uniaxial orientation when subjected to uniaxial tension. Here, the MWNTs do not alter this compression vs tension dependence of the stress-strain behavior and, indeed, must also orient with deformation as discussed further in the modeling section.

Strain Energy

The tensile and compression stress vs strain data were further reduced to plots of strain energy density vs axial strain. Figure 4 shows the strain energy density results for all material compositions for uniaxial tension (Figure 4a) and uniaxial compression (Figure 4b). This data is then once again reduced to provide a plot of the strain energy contribution of the MWNTs, ¢¡ ¤£ ¦¥ ¨ § , to the composite behavior.

Here, we assume a simple rule of mixtures approach giving:

© ¡ ¤£ ¥ § § "! # %$ & (' (1) or, ¡ ¤£ ¥ § ) 0 1 (© 2! 3 § 4! # 5$ 6 (' 7$ 98 @ A 5$ (2)
where (© is the strain energy density of the composite, ¢' is the strain energy den- sity of the elastomer, and is the volume fraction of the MWNTs. Figure 5 depicts ¡ ¤£ ¦¥ ¨ § vs. axial strain for uniaxial tension (Figure 5a) and for uniaxial compression (Figure 5b) as obtained by reducing the data for all compositions. Note that the curves obtained from each composition are nearly coincident for the tensile data as well as for the compression data and thus reflect the unit contribution of the MWNTs to the deformation behavior of the composite. The coincidence of these data supports the use of equation 2 as a surprisingly good first approximation to assess and model the contribution of MWNTs to the composite behavior. This form of the reduced data will be shown to be instrumental in constructing the microstructurally-informed constitutive model for these nanocomposites B .

B

The loading curves were used to construct the strain energy density curves and, hence, contain the hysteresis contibutions seen in the elastomer compositions which were found to increase with an increase in MWNT content. As a first-order approach we are neglecting hysteresis effects.

Constitutive model

Model Development

Our experimental results indicate that the strain energy density, © , of the MWNTelastomer composites under consideration can be decomposed into a contribution from the elastomer, ' , and a contribution from the MWNTs, ¢¡ ¤£ ¥ § , using the simple rule of mixtures approach given earlier in equation ( 1). This representation is physically realistic for this case of a low to modest volume fraction of wavy (bent) fibers distributed within a compliant elastomeric matrix, where the fibers will essentially deform with the matrix, unbending and rotating with the imposed deformation. The wavy fibers offer little disturbance to the basic deformation field of the elastomer (in other words, there is not a significant amplification of strain in the matrix) and hence the elastomer deformation can be directly described by the macroscopic deformation gradient.

Here, the elastomer matrix contribution will be modeled using the Arruda-Boyce (A-B) eight-chain network model of rubber elasticity [START_REF] Arruda | A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials[END_REF]) and the MWNT contribution will be captured using the framework for anisotropic hyperelastic materials as originally developed by [START_REF] Spencer | Constitutive Theory for strongly Anisotropic Solids[END_REF]. The strain energy contributions will be described below and each will be a function of properties of the constituent material and the imposed deformation. The macroscopic deformation is described by the deformation gradient, ¡ £¢ ¥¤ 28 ¦¢ ¨ § , with polar decompositiom ¡ © © , § denotes the position of a material point in the initial, reference configuration and ¤ denotes the the position vector of that same material point in current, deformed configuration. The corresponding right Cauchy-Green tensor is given by ¡ ¡ "! and the left Cauchy-Green tensor is given by # ¡ ¡ ! . The invariants of and # are given by: $ B &% (' ¦) 10 32 # $ ¢ 4 65 ¨7 B 5 ¨7 7 5 87 9 A@ $ 7 § ¡ CB $ 7 B ! D% (' E) 10 F2 # 7 $ HG 4 5 ¨7 B 5 87 7 5 ¨7 B 5 87 9 5 ¨7 7 5 87 9 $ 9 PI Q2 R% # $ ¢ 5 7 B 5 7 7 5 7 9

(3)

where 5 ¨S are the principal stretches.

Elastomer Matrix Representation

The BIMSM elastomer is a crosslinked molecular network and was found to exhibit little time-dependence. The Mullins effect (quantified in the data presented in Appendix A) is small and can be modelled using the [START_REF] Qi | Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials[END_REF] model, but this effect is neglected in this paper in order to more clearly focus on the primary contributions of the MWNTs to the mechanical behavior. Hence, the matrix is approximated as hyperelastic and modeled using the A-B eight-chain network model of rubber elasticity. The A-B model represents a random isotropic molecular network as eight chains emanating from a central junction point to the corners of a cube; the cube is taken to be aligned with the principal axes of stretch. In this representation, each chain stretches by the root mean square of the imposed stretch, 5 © ¡ £¢ S ¤ ¥ 5 7 B 5 7 7 5 7 9

$ 78 ¤ ¥ $ B 8 ¢ and also rotates towards the axis(es) of maximum principal stretch(es) (Figure 6). Thus, the eight-chain model captures the essence of the physics of deformation of an isotropic random network as verified by its ability to predict the stress-strain behavior of an elastomer under different states of strain. The strain energy function for a compressible version of the eight-chain model (see, [START_REF] Bischoff | A new constitutive model for the compressibility of elastomers at finite deformation[END_REF]) is given by:

(' §¦ © 5 © ¡ ¢ S ¤ £ ! #" %$ '& ! # £ )(

@

; where 6 98 B is the inverse Langevin function with 6 " $ CB ED GF & $ ! § 8 , ( $ 9

and 2 H3 is the bulk modulus.

The Cauchy stress is then obtained by differentiating ¢' , giving: The material properties were fit to the BIMSM tensile data, giving ¦ )¨ ¡ SV ẌW `Y ) and ¡ ¡ ¡ ; the bulk modulus was taken to be ba cY ) . The model fit is shown to capture the tensile behavior in Figure 7. As is well known, elastomeric stress-strain behavior is strongly dependent on the state of imposed deformation (see, for example, [START_REF] Treloar | The Physics of Rubber Elasticity[END_REF], [START_REF] Arruda | A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials[END_REF]), where the greatest contrast is seen in comparisons of uniaxial tension to uniaxial compression (or, alternatively, uniaxial tension and equibiaxial tension). The model prediction of the uniaxial compression behavior based on its fit to the tensile data is also shown in Figure 7 , demonstrating the ability of the model to capture the matrix behavior. The small discrepancy between model prediction and experiment is most likely due to our neglecting the small amount of hysteresis and Mullins effect (see Appendix A) exhibited by this material. However, for purposes of examining the effect of CNTs on the uniaxial tension and compression behavior , this discrepancy is small.

I ' ¡ ( ¢ % ' ¢ $ B # ¢ % ' ¢ ©( QP (

MWNT Representation

The contribution to the composite strain energy emanating from the deformation of the constituent MWNTs is modeled by first building from the basic framework presented in [START_REF] Spencer | Constitutive Theory for strongly Anisotropic Solids[END_REF] for anisotropic fiber-reinforced hyperelastic materials. This framework has been followed and/or built upon in recent years by a number of investigators including, for example, Bischoff et al. (2002b,a) who established an anisotropic molecular network model for skin tissue; [START_REF] Holzapfel | Structural and numerical models for the viscoelastic response of arterial walls with residual stresses[END_REF][START_REF] Holzapfel | Nonlinear solid mechanics, a continuum approach for engineering[END_REF] in developments of models of arterial wall tissue, and [START_REF] Reese | Finite element modelling of orthotropic material behavior in pneumatic membranes[END_REF] in models of fabric reinforced elastomers and [START_REF] Horgan | A new consititutive theory for fiberreinforced incompressible nonlinearly elastic solids[END_REF] in a model of an incompressible elastomer reinforced with a single family of fibers. The framework begins with the recognition of fiber direction, , as a building block of needed invariants for constructing the anisotropic strain energy function. Using a notation ¢¡ as representing the direction of the ith family of fibers, several additional invari- ants (beyond $ B , $ 7 , and $ 9

introduced previously) become relevant. The primary invariants of interest being:

$ ¤£ ¦¥ § S V 4V ¨ S $ © ¥ § S V 7 V S (6)
where additional invariants which capture coupling between fiber families constitute the remaining invariants (for an overview, see, for example, [START_REF] Holzapfel | Nonlinear solid mechanics, a continuum approach for engineering[END_REF]).

In our representation, based on the experimental data, we first take the simplest approach and capture the influence of the wavy MWNTs (i.e., the MWNTs are not straight nor are they aligned, instead they are randomly dispersed and in an undulated or bent configuration within the matrix) by simply building a strain enregy function based on the $ £ ¦¥ , where $ ¤£ ¦¥ physically represents the square of the stretch of the fiber family in the ith direction: 5 7

¡ ¤£ ¥ § ¥ $ ¤£ ¦¥
. The MWNT strain energy contribution can then be expressed as:

¡ ¤£ ¥ § S S ¡ ¤£ ¥ § ( 5 87 ¡ ¤£ ¥ § ¥ $ (7)
where S

is the volume fraction of MWNTs whose end-to-end vector is aligned in the ith direction and the sum is over all MWNT end-to-end directions present in the material; the "fiber stretch" here refers to the extension of the end-to-end distance of this wavy MWNT where this extension results in the unbending or straightening out of the MWNT waviness.

For the material under consideration, we note that the orientation distribution is isotropic and we thus take a leap similar to that of the eight-chain network model of rubber elasticity: we represent the contribution of the random distribution of nanotubes by the average initial orientation direction. The average orientation is approximately given by the azimuthal angle of 55 degrees (see, for example, [START_REF] Bergstrom | Deformation of elastomeric networks: relation between molecular level deformation and classical statistical mechanics models of rubber elasticity[END_REF]) corresponding to the fibers associated with the % § § ! § family (i.e. (1,1,1), (1,1,-1), (-1,1,1),(1,-1,1)... ). This isotropic distribution is more optimally captured by considering the average orientation given with respect to the principal stretch directions, i.e. taking the azimuthal angle of # with respect to the maximum principal stretch direction. This is thus the "8-chain" network approach and gives: 5 ¡ ¤£ ¥ § ¥ $ B 8 ¤ . We thus have "isotropized" the anisotropic strain energy function for a random isotropic distribution of MWNTs which gives the MWNT contribution to be:

) ¡ ¤£ ¥ § ) ¡ ¡ ¤£ ¦¥ ¨ § 5 7 ¡ ¤£ ¦¥ ¨ § $ (8) where 5 7 ¡ ¤£ ¦¥ ¨ § $ B 8 ¤ .
This isotropized contribution also provides a framework for further reduction of our experimental data of Figures 5a and5b which we can now represent as ¡ ¤£ ¦¥ ¨ § vs 5 7

¡ ¤£ ¦¥ ¨ § ! § $ as shown in Figure 8. Fitting the data as represented in Figure 8 provides the MWNT strain energy function:

¡ ¤£ ¦¥ ¨ § ¢ B 5 ¡ ¤£ ¦¥ ¨ § 7 ! § $ ¨ ¢ 7 5 ¡ ¤£ ¡ § 7 ! § $ 7 ! ¡ ¢ B £ $ ( (9) or, ¡ ¤£ ¥ § ) ¢ B $ B ! $ ¨ ¢ 7 & $ B ! $ 7 ! ¡ ¢ B £ $ (
where the model curve fit, with ¢

B 'V ¢ W `Y ) and ¢ 7 ¡ V ¡ & bW `Y ) , is shown in
Figure 9. The model for the more general case of a distribution in MWNT orientations is given in Appendix B. We further emphasize that the strain energy contribution from the MWNTs is a result of the energy from bending and/or unbending of undulations in the MWNTs during the imposed macroscopic deformation, where the MWNTs stretch (extending their end-to-end distance via unbending) and rotate with the imposed deformation such that they deform with the matrix.

Composite Constitutive Model

The composite strain energy function is simply the sum of the contributions from the elastomer and from the MWNT as given earlier in equation 1 and now using equations 4 and 9 for the elastomer and MWNT contributions, respectively. Differentiation of the strain energy function gives the Cauchy stress:

I § ! # 5$ ¦ R S( T 5 © ¡ £¢ S ¤ # ! # P 0 ¡ ¤ ¢ B ¡ ¢ 7 $ B ! $ ¦¥ B ! ¡ ¢ B 1 2 H3 5( ! § $ P (10)

Comparison of model and experimental results

The constitutive model results are compared to data in Figures 10 and11. Figure 10 shows the model results for uniaxial tension of the neat elastomer and the MWNT-elastomer composites plotted together with the data. The model is found to quantitatively capture the features of the uniaxial tension stress-strain behavior and its dependence on MWNT content. The inset of Figure 10 shows the model results for initial modulus as a function of wt% MWNT. The model results for uniaxial tension and compression to strains of 1.0 and -1.0, respectively, together with the corresponding data are shown in Figure 11. The model is also found to be in good agreement with experimental data for uniaxial compression. The model is found to capture the data to large strain and to quantitatively predict the uniaxial compression behavior showing the distinct differences in tension and compression data as well as the dependence of the curves on MWNT content.

Conclusions

The influence of multi-wall carbon nanotubes on the mechanical behavior of elastomeric materials was investigated. The BIMSM elastomer was blended with multiwall carbon nanotubes (MWNTs) and the influence of weight fraction of MWNTs on the large deformation tensile and compressive behavior was quantifed. The initial stiffness and subsequent stretch-induced stiffening at large strains were both found to increase with MWNT content. The MWNTs were also found to increase the tensile strength and the tensile strain-to-break. The strain energy density of the composite materials as obtained experimentally was reduced into two contributions: a matrix contribution and a MWNT contribution. The ability to decompose the experimentally obtained strain energy guided the development of a hyperelastic constitutive model for the composite materials based on the volume fraction of matrix and MWNT. The model was found to predict the effects of MWNT content on the large deformation stress strain behavior of the composite materials in tension and compression, showing excellent agreement with the experimental data. Formulation ( ¢¡ ¤£ ) 2.43/.0113 6.97/.0372 9.09/.049 13.04/.0718 TGA ( ¢¡ ¤£ ) 2.60/.0136 6.07/.0323 8.8/.0474 12.16/.0667 =weight percent; £ =volume fraction

A Mullins effect

The elastomer and MWNT-elastomer composites of this study were found to exhibit a Mullins effect (a stretch-induced softening of the stress-strain behavior commonly observed in elastomeric material, see, for example, [START_REF] Mullins | Effect of stretching on the properties of rubber[END_REF][START_REF] Mullins | Softening of rubber by deformation[END_REF], Mullins andTobin (1954, 1965), [START_REF] Harwood | Stress-softening in natural rubber vulcanizates. part iii: Carbon black-filled vulcanizates[END_REF]) as evidenced by a more compliant response on reloading of the material after having been subjected to a prior load-unload cycle. For completeness, Figures A.1 and A.2 show loadunload-reload data for uniaxial tension (A.1) and compression (A.2) of the neat elastomer and shows the uniaxial tensile loadunload-reload behavior to/from strains of 0.40 and 0.69 for the neat elastomer and the 15phr MWNT-elastomer composite. Figure A.2 shows the load-unload-reload behavior of the neat elastomer and the 15phr MWNT-elastomer nanocomposite in compression to/from a strain of -1.0.

B Anisotropic Strain Energy for Composite Containing Initial Preferential Orientation of MWNTs

The materials under consideration were found to contain a randomly oriented distribution of MWNTs. However, the modeling presented is also applicable to the case of an initially preferentially oriented distribution of MWNTs as is presented in this appendix for purposes of both completeness and comparison. The fibers direction are given by unit vectors S S ¢¡ B ¤£ S ¥¡ 7 §¦ S ¢¡ 9

in the initial configuration and 5 ©¨ ¡ B V 4V ¡ B , 5 ©¨ ¡ 7 V HV ¡ 7 , 5 ©¨ ¡ 9 V 4V ¡ 9

represent the stretches along the principales fiber axes. We take the MWNT contribution to be given by:

¡ ¤£ ¦¥ ¨ § S ¡ ¤£ ¦¥ ¨ § ¥ ¡ ¤£ ¦¥ ¨ § ( 5 ¨7 ¡ ¤£ ¥ § ¥ $ (B.1) where 5 ¡ ¤£ ¦¥ ¨ § ¥ 7 $ £ S S S
giving the composite strain energy function: In tension simulations, the nanocomposite is observed to stiffen with strain and the strain stiffening is observed to increase with the initial fiber alignment, where nanocomposites with fibers initially more axially oriented stiffen more with strain. In compression simulations, the strain stiffening is also found to depend upon initial fiber orientation, where initial alignments with the fibers more biaxially oriented (in the plane with normal coinciding with the compression axis) are observed to strain stiffen more with strain as the fibers become more biaxially aligned in the plane normal to the compression axis thus resisting the compression.

¡ ¤£ ¦¥ ¨ § ¢ B 5 ¡ ¤£ ¦¥ ¨ § ¥ 7 ! § $ ¢ 7 5 ¡ ¤£ ¦¥ ¨ § ¥ 7 ! § $ 7 ! ¡ ¢ B £ $ 5 ¥ ¨5 ¥ ¨5 ¥ ¨ ¨$ (B.

Fig. 1 .

 1 Fig. 1. (a) Low magnification Scanning Electron micrographs of a freeze fractured surface and (b) TEM of MWCNT/elastomer nanocomposite of 15 wt¥ , showing uniformity of MWCNT dispersion

Fig. 2 .Fig. 3 .

 23 Fig. 2. (a) Uniaxial tension load-unload true stress vs. true strain at an engineering strain rate of 0.01 /s to a strain of 0.4 for the elastomer and the elastomer-MWNT nanocomposites; (b)Uniaxial tension true stress vs. true strain at a strain rate of 0.01/s taken to failure; the inset plot reports the initial axial modulus as a function of weight fraction MWNT.

Fig. 4 .

 4 Fig. 4. Strain energy density vs true axial strain for all compositions for (a) uniaxial tension, and (b) uniaxial compression

Fig. 5 .

 5 Fig. 5. Strain energy density of the MWNTs vs true axial strain for all compositions for (a) uniaxial tension and (b) uniaxial compression.

Fig. 6 .

 6 Fig. 6. Eight-chain network model is (a) undeformed, (b) uniaxial tension, and (c) uniaxial compression,[START_REF] Arruda | A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials[END_REF]

Fig. 11 .

 11 Fig. 11. Constitutive model results for uniaxial tension and uniaxial compression compared to data for MWNT weight percents (a) 12.2% and (b) 2.6%.

  Fig. A.1.Differentiation of strain energy function gives:

  Fig. B.1. Constitutive model results for composite containing initial preferential orientation of MWNTs for uniaxial tension (a) and uniaxial compression (b). The initial orientation of MWNTs are given by the table B.1 and represented by a "cone" (a).

  Fig. B.2. Small strain behavior as a function of the initial alignment of the fibers for uniaxial tension and uniaxial compression.

Table 1 :

 1 Material Compositions

	Nanocomposite No.	1	2	3	4
	Elastomer (phr)	100	100	100	100
	MWNT (phr)	2.5	7.5	10	15

Acknowledgements

This research was sponsored, in part, by USAFOSR through the DURINT Polymer Nanocomposites, Grant No. F49620-01-1-0447, and, in part, by ExxonMobil Chemical Company.


Jumana A Karasneh
email: jumana2003@yahoo.com

Rola A Al Habashneh

Nour Aldain

S Marzouka

Martin H Thornhill

Effect of cigarette smoking on subgingival bacteria in healthy subjects and patients with chronic periodontitis

Keywords: Bacteria, Cigarette smoking, Chronic periodontitis, Polymerase chain reaction, 16S ribosomal RNA

Background: Cigarette smoking is known to increase the risk of periodontal destruction and developing chronic periodontitis (CP). It is also reported to affect the subgingival bacterial profile among CP patients. However, studies on the effect of smoking on the bacterial profile among healthy subjects are still limited. Therefore, the aim of this study was to investigate the impact of smoking on the subgingival bacterial profile in both healthy adults and CP patients. Methods: Subgingival plaque samples were collected from CP patients (30 nonsmokers and 9 smokers) and healthy subjects (37 non-smokers and 18 smokers). Genomic DNA was extracted and 25 bacterial species were detected using PCR of 16S rRNA. Comparing smokers to non-smokers from each group was conducted using chi2 and binary logistic regression analysis. Results: After correcting for confounding factors, the odds of having Slackia exigua, Selenomonas sputigena and Campylobacter rectus was higher among healthy smokers (ORadj = 10.1, 6.62 and 5.62 respectively). While for CP group, the highest odds were observed for Treponema amylovorum, Treponema medium, Slackia exigua and Treponema vincentii (ORadj = 20.7, 7.97, 6.37 and 5.37 respectively) and the increase in Treponema amylovorum was statistically significant (p = 0.05). Conclusion: Smoking affects the subgingival bacterial profile in healthy individuals and is responsible for the depletion of beneficial bacteria and the increase in periodontopathogenic bacteria. In the CP patient group, our study suggests that subgingival bacteria (particularly Treponema species) make a more substantial contribution in the etiology of CP among non-smokers. Further studies using a larger sample set and more sensitive and quantitative techniques (such as real -time PCR) are needed to enhance our understanding of the exact effect of smoking on subgingival biofilm.

Background

Periodontitis is a group of inflammatory diseases affecting the supporting tissues of the teeth. Periodontal diseases are classified as complex disorders where environmental, lifestyle and genetic factors contribute to their etiology [START_REF] Hassell | Genetic influences in caries and periodontal diseases[END_REF]. The most important environmental factor is the diverse microbiota arranged in a biofilm [START_REF] Filoche | Oral biofilms: emerging concepts in microbial ecology[END_REF] which acts as an initiator of periodontal destruction, while lifestyle factors, such as smoking and oral hygiene habits, act as modifiers of disease presentation [START_REF] Stabholz | Genetic and environmental risk factors for chronic periodontitis and aggressive periodontitis[END_REF]. Cigarette smoking has been shown to be responsible for accelerated periodontal destruction and increased risk of periodontitis in young adults [START_REF] Linden | Cigarette smoking and periodontal destruction in young adults[END_REF]. Based on epidemiological studies, smokers have more severe chronic periodontitis (CP) than non-smokers [START_REF] Tomar | Smoking-attributable periodontitis in the United States: findings from NHANES III. National Health and Nutrition Examination Survey[END_REF] and the relationship is dose-dependent [START_REF] Bergström | Exposure to tobacco smoking and periodontal health[END_REF], both in terms of number of cigarettes smoked and number of years smoking [START_REF] Jette | Tobacco use: a modifiable risk factor for dental disease among the elderly[END_REF]. It has been suggested that increased severity of periodontitis among smokers compared to non-smokers is, in part, induced by differences in the subgingival bacterial profile [START_REF] Haffajee | Relationship of cigarette smoking to the subgingival microbiota[END_REF].

Reports on the effect of smoking on the composition of the subgingival microbiota are inconclusive; some studies report a minor effect [START_REF] Boström | Smoking and subgingival microflora in periodontal disease[END_REF] and some report no effect of smoking on the subgingival microbiota [START_REF] Apatzidou | Impact of smoking on the clinical, microbiological and immunological parameters of adult patients with periodontitis[END_REF]. However, most studies report a significant difference in the subgingival microbiota between smokers and nonsmokers [START_REF] Haffajee | Relationship of cigarette smoking to the subgingival microbiota[END_REF][START_REF] Gomes | Periodontal status in smokers and never-smokers: clinical findings and real-time polymerase chain reaction quantification of putative periodontal pathogens[END_REF][START_REF] Kamma | Clinical and microbiological characteristics of smokers with early onset periodontitis[END_REF][START_REF] Eggert | Effects of smoking and treatment status on periodontal bacteria: evidence that smoking influences control of periodontal bacteria at the mucosal surface of the gingival crevice[END_REF][START_REF] Van Winkelhoff | Smoking affects the subgingival microflora in periodontitis[END_REF]. These studies were all conducted on patients with periodontal disease. In contrast, studies on healthy individuals are limited with only one study investigating the effect of smoking on the subgingival microbiota of healthy young adults [START_REF] Shiloah | The prevalence of pathogenic periodontal microflora in healthy young adult smokers[END_REF]. In order to determine the effect of smoking on subgingival bacteria, other confounding factors that would dilute or interfere with the effect of smoking, particularly those associated with periodontal disease, should be eliminated.

Therefore, the aim of the present study was to determine whether smoking affects the subgingival bacteria in healthy individuals. It also aimed to investigate if there is a difference in the subgingival microbiota between smoking induced periodontitis and periodontitis induced by factors other than smoking.

Methods

Study groups

The study was independently reviewed and approved by the Institutional Research Board (IRB) of Jordan University of Science and Technology (JUST). The purpose and methodology of the study, including possible future publication of clinical datasets, were fully explained to all participants or their parents if they were under the age of 18 years and their written consent was obtained before interviews, examinations and sample collection, complying with the tenets of the revised Declaration of Helsinki.

In total, 150 participants were enrolled in the study at the initial screening unit in the Dental Teaching Centre (DTC) of Jordan University of Science & Technology (JUST). Demographic data and medical history were collected, and brief extraoral and intraoral clinical examinations were performed. Patients with clinically healthy gingiva were referred to the oral diagnosis clinic and those with suspected periodontal disease were referred to the periodontics clinic, both located in the DTC. Selection criteria were: systemically healthy; Jordanian; no periodontal treatment or antimicrobial therapy for at least 6 months before the examination; at least seven natural teeth/quadrant with no fixed or removable prosthesis. Any participants using long-term medications, receiving orthodontic treatment, under the age of 16, or pregnant were excluded. Once in the designated clinic, supragingival plaque was scored and, when present, it was carefully removed. Teeth were isolated using cotton rolls, and subgingival plaque was collected prior to any periodontal examination by inserting a paper point for 30 s into the mesial sulcus of each posterior tooth apart from the third molar if present (16 sites in total). The clinical diagnosis was based on the 1999 Consensus Classification of Periodontal Diseases [START_REF] Armitage | Development of a classification system for periodontal disease and conditions[END_REF]. Four clinical variables were recorded for all participants: the periodontal pocket depth (PPD), clinical attachment loss (CAL), plaque index (PI) and gingival index (GI) [START_REF] Löe | The gingival index, the plaque index and the retention index system[END_REF]. The absence of periodontal disease was defined as the absence of sites with PPD of ≥4 mm in any tooth and confirmed by radiographic examination. Paper points contaminated with blood were not included as this might affect DNA quality, only points from non-bleeding sites were pooled together, placed in Eppendorf tubes and stored at -20 °C for extraction of genomic DNA and analysis. Only participants who satisfied the inclusion criteria and had a clear-cut diagnosis were included in the study and DNA was extracted from plaque samples. In total a study population of 94 unrelated individuals: 30 nonsmoker CP cases (18-50 years), 9 current smoker CP cases (28-48 years), 37 non-smoker periodontally healthy subjects (17-53 years), and 18 current smoker periodontally healthy subjects (17-43 years).

Molecular technique

Samples were placed in 2 ml phosphate-buffered saline (PBS) and incubated for 3 h at room temperature. Each sample was centrifuged at 5000 × g for 10 min and the supernatants removed. Genomic DNA was then extracted from the pellets using QIAamp DNA Mini Kit (Qiagen GmbH, Hilden, Germany) according to the manufacturer's instructions.

In the first amplification, the 16S rRNA genes were amplified by PCR with universal primers 27 F and 1492R [START_REF] Lane | 16S/23S rRNA sequencing[END_REF]. The primer sequences were: 27 F, 5′-AGA GTT TGA TCC TGG CTC AG-3′; and 1492R, 5′-TAC GGG TAC CTT GTT ACG ACT T-3′. PCR was performed in 25 μl volumes using 12.5 μl of PCR master mix (Promega, USA), 6.5 μl of nuclease free water, 1 μl of forward and reverse primers and 5 μl of DNA. Amplifications were performed using a PCR Thermal Cycler (Bio-RAD, USA) programed for 10 min at 95 °C for initial heat activation, 35 cycles of 45 s at 95 °C for denaturation, 45 s at 60 °C for annealing, 45 s at 72 °C for extension, and 7 min at 72 °C for final extension. The size of the PCR product using the universal primers was 1505 bp.

The presence or absence of the bacteria was determined by the presence or absence of the PCR band resulting from using previously reported species-specific primers [START_REF] Mayanagi | Detection frequency of periodontitis-associated bacteria by polymerase chain reaction in subgingival and supragingival plaque of periodontitis and healthy subjects[END_REF] on the first universal PCR product as a template after being checked on Primer-BLAST (www.ncbi.nlm.nih.gov/tools/primer-blast/). Primer sequences can be found in additional word document file (see Additional file 1). PCR mixtures were prepared using 5 μl of the first PCR amplification mixture and 20 μl of the reaction mixture containing 0.5 μl of species-specific primer, 13 μl of PCR master mix (Promega, USA) and 6.5 μl of nuclease free water. PCR amplification was performed in a PCR Thermal Cycler (Bio-RAD, USA) programed for 10 min at 95 °C for initial heat activation, 35 cycles of 1 min at 95 °C for denaturation, 45 s at 55 °C for annealing, 45 s at 72 °C for extension, and 7 min at 72 °C for final extension.

The PCR products were separated on 2% agarose gels stained with ethidium bromide and visualized under a UV light transilluminator. With each gel run, negative and positive control samples were run simultaneously.

In any case of uncertainty about the presence or absence of a bacteria-specific band, the PCR was repeated.

Statistical analysis

Continuous variables were presented using mean ± SD and compared using the independent sample t-test, categorical variables were presented using frequencies and percentages, and were compared using Chi squared analysis. Binary logistic regression analysis was used to calculate the odds of having each bacterium in smokers compared to non-smokers for both healthy and CP Both crude and OR corrected for age, gender, education level and PPD (OR adj ) were presented. All statistical analyses were conducted using IBM SPSS 24 (SPSS Inc., Chicago, USA). The adjusted OR (OR adj ) was adopted for interpreting the results and a p-value ≤ 0.05 was considered to be statistically significant. SPSS could not calculate the OR for the bacteria if the prevalence was 100% or 0% in any of the compared groups; in that case, only the p-value for χ 2 analysis was shown. A power calculation was conducted to determine the statistical power of our samples to detect an association using the G*Power program [START_REF] Faul | G*Power 3: A flexible statistical power analysis program for the social, behavioral, and biomedical sciences[END_REF]. The input parameters were: effect size = 0.3 (small size = 0.1, medium size = 0.3), statistical significance level α = 0.05 and two tailed.

Results

The sociodemographic and clinical data for the participants are summarized in Table 1.

In our sample set, a significant difference in gender was observed between smokers and non-smokers in the two groups, where males dominated the smokers groups (p < 0.0001). The PPD was significantly higher among the healthy group compared to healthy smokers. No significant difference was observed in the oral hygiene habits between the four investigated groups (data not shown).

The frequency of bacteria and the odds of having each of the 25 bacteria among healthy smokers compared to healthy non-smokers are presented in Fig. 1 and Table 2, respectively. Both crude and adjusted regression analyses are presented in Table 2. The odds of having 15 bacteria was increased in healthy smokers compared to healthy non-smokers, and the highest increases were detected in Slackia exigua, Selenomonas sputigena, and Campylobacter rectus (OR adj =10.1, 6.62 and 5.62, respectively). The odds of having eight bacteria was decreased among healthy smokers, and the decrease was mostly noted in Prevotella nigrescens, Capnocytophaga ochracea, and Eikenella corrodens (OR adj = 0.31, 0.32 and 0.35, respectively). None of these differences were statistically significant. Eubacterium saphenum was not present among nonsmokers.

Within the CP group, when smokers were compared to non-smokers, the prevalence of six bacteria was increased among smokers (Fig. 2). When correcting for confounders, Treponema amylovorum, Treponema medium, Slackia exigua and Treponema vincentii had the highest odds ratios (OR adj = 20.7, 7.97, 6.37 and 5.37, respectively) and the increase in Treponema amylovorum was statistically significant (p = 0.05) (Table 3). Treponema denticola, Treponema socranskii and Tannerella forsythia were present in all CP smokers. The remaining bacteria were decreased among CP smokers (Table 3).

Power calculations revealed that the sample set had 62.9 and 48.1% power to detect an association for healthy and CP groups, respectively, when the significance level was set at 0.05.

Discussion

This study reports the effect of smoking on the bacterial profile of healthy and CP subjects by comparing the bacterial profile of 25 bacterial species among smokers and non-smokers. These particular bacterial species were analyzed because there is previous evidence of their association with periodontal diseases [START_REF] Mayanagi | Detection frequency of periodontitis-associated bacteria by polymerase chain reaction in subgingival and supragingival plaque of periodontitis and healthy subjects[END_REF][START_REF] Flemmig | Periodontitis[END_REF]. Among healthy individuals, the bacteria present among smokers differ from those present among non-smokers although both are periodontitis-free groups (CAL = 0.00). A moderate increase in the red complex bacteria was observed among smokers compared to non-smokers, which agrees with previously reported results [START_REF] Shiloah | The prevalence of pathogenic periodontal microflora in healthy young adult smokers[END_REF]. However, smoking was responsible for increasing the prevalence of other periodontopathogenic bacteria in subjects with healthy gingiva that were also prevalent in CP patients. After ruling out the effects of other confounding factors, smoking was responsible for increasing the prevalence of Slackia exigua, Selenomonas sputigena, and Campylobacter rectus by 910%, 562%, and 462% times, respectively. Furthermore, smoking was responsible for decreasing the prevalence of bacteria that were considered to be less periodontopathogenic, for instance, Eikenella corrodens was 65% less likely to be present among healthy smokers. These findings indicate that smoking is responsible for the depletion of beneficial bacteria and increasing the abundance of periodontopathogenic bacteria among healthy individuals.

When comparing the bacterial profiles of smoker and non-smoker CP patients, an unexpected observation was the increased prevalence and the odds of having many of the investigated bacterial species among the non-smokers. Exceptions were limited to Treponema species and Slackia exigua which were increased among CP smokers. Previous reports concluded that smoking is responsible for enriching the subgingival microbiota with pathogenic species [START_REF] Van Winkelhoff | Smoking affects the subgingival microflora in periodontitis[END_REF][START_REF] Zambon | Cigarette smoking increases the risk for subgingival infection with periodontal pathogens[END_REF]. This effect was supported by more recent studies conducted on CP patients, which concluded that smoking has a detrimental impact on the periodontal status and the microbial profile [START_REF] Gomes | Periodontal status in smokers and never-smokers: clinical findings and real-time polymerase chain reaction quantification of putative periodontal pathogens[END_REF][START_REF] Shchipkova | Subgingival microbial profiles of smokers with periodontitis[END_REF]. This was further supported by a longitudinal study investigating the effect on bacteria of smoking cessation after 6 and 12 months among CP patients [START_REF] Fullmer | Smoking cessation alters subgingival microbial recolonization[END_REF]. Our results support the view that smoking affects the subgingival microbiota in CP patients and that it is responsible for the increase in the Treponema species and Slackia exigua. They also suggest that, in non-smoker CP patients, bacteria tend to play a more crucial role in pathogenesis, where a larger number of bacterial species will be involved in constructing the subgingival plaque, while in smoker CP patients, smoking is more likely to contribute to the disease by inducing changes within gingival tissues and the gingival sulcus. Smoking has been reported to affect the immune response against bacteria [START_REF] Palmer | Mechanisms of action of environmental factorstobacco smoking[END_REF] by lowering gingival crevicular fluid (GCF) volume [START_REF] Holmes | Effects of smoking and/or vitamin C on crevicular fluid flow in clinically healthy gingiva[END_REF][START_REF] Persson | Tobacco smoking and gingival neutrophil activity in young adults[END_REF] or by affecting the host antibody response [START_REF] Haber | Cigarette smoking: a major risk factor for periodontitis[END_REF], through reducing sera immunoglobulin G antibody titers against the subgingival bacteria [START_REF] Apatzidou | Impact of smoking on the clinical, microbiological and immunological parameters of adult patients with periodontitis[END_REF]. When looking at the whole sample set, smoking had two distinct effects on subgingival bacteria: some bacteria were more prevalent among smokers as observed with Tannerella forsythia, Treponema denticola and Treponema medium, while other bacteria were less prevalent among smokers as observed with Aggregatibacter actinomycetemcomitans, Fusobacterium nucleatum, Prevotella tannerae, Capnocytophaga ochracea and Mogibacterium timidum. Based on these observations, smoking might induce changes in the oral and gingival ecosystem that make the environment more favorable for some bacterial species to grow, while for others, these changes make the environment less favorable for growth.

Gender was found to be significantly different between smokers and non-smokers of both groups. This was mainly attributed to the fact that the prevalence of smoking is much higher in males compared to females in the Jordanian population [START_REF] Khader | Smoking habits among university students in Jordan: prevalence and associated factors[END_REF][START_REF] Khabour | Waterpipe tobacco and cigarette smoking among university students in Jordan[END_REF]. The prevalence of smoking in Jordanian men is reported to be in the range of 56.9-62% compared to 11.0-11.4% in women [START_REF] Khader | Smoking habits among university students in Jordan: prevalence and associated factors[END_REF][START_REF] Khabour | Waterpipe tobacco and cigarette smoking among university students in Jordan[END_REF]. The effect of gender on the subgingival bacterial flora is controversial; several studies conducted on periodontitis patients have reported a difference in the bacterial profile between males and females [START_REF] Slots | Age and sex relationships of superinfecting microorganisms in periodontitis patients[END_REF][START_REF] Umeda | Risk indicators for harboring periodontal pathogens[END_REF], while other studies have reported that gender has no effect on the subgingival bacterial profile in CP patients or in healthy controls [START_REF] Schenkein | The influence of race and gender on periodontal microflora[END_REF][START_REF] Fischer | Influence of the menstrual cycle on the oral microbial flora in women: a case-control study including men as control subjects[END_REF]. The PPD was significantly higher in healthy non-smokers compared to smokers; this increase is most likely related to increased age among the nonsmokers.

This study investigated the effect of smoking on the prevalence of bacterial species; however, no changes in the serotype induced by smoking could be determined. Furthermore, no quantitative judgments on the amount of bacteria could be made as this requires using a realtime PCR technique, which was not available to us. An effect of smoking on the immune response [START_REF] Apatzidou | Impact of smoking on the clinical, microbiological and immunological parameters of adult patients with periodontitis[END_REF][START_REF] Palmer | Mechanisms of action of environmental factorstobacco smoking[END_REF][START_REF] Holmes | Effects of smoking and/or vitamin C on crevicular fluid flow in clinically healthy gingiva[END_REF][START_REF] Persson | Tobacco smoking and gingival neutrophil activity in young adults[END_REF][START_REF] Haber | Cigarette smoking: a major risk factor for periodontitis[END_REF] could not be ruled out, and this would provide another mechanism of action in inducing periodontal disease. Finally, although differences in the bacterial profile could be detected in this sample set, only the increase in Treponema amylovorum in CP smokers was statistically significant (p = 0.05); other differences were not statistically significant (p > 0.05) due to limited power related to the small sample number, which is one of the limitations of the study; hence, conclusions based on these results should be interpreted with caution.

Conclusions

Smoking affects the subgingival bacterial profile in healthy individuals and is responsible for the depletion of beneficial bacteria and the increase in periodontopathogenic bacteria. In the CP patient group, our study suggests that subgingival bacteria (particularly Treponema species) might make a more substantial contribution in the etiology of CP among non-smokers. Further studies using a larger sample set and more sensitive and quantitative techniques (such as real-time PCR) are needed to enhance our understanding of the exact effect of smoking on subgingival biofilm.

Fig. 2

 2 Fig. 2 Prevalence of bacterial species in CP patients who were smokers or non-smokers

Table 1

 1 Demographic, socioeconomic and clinical data for the study groups

		Healthy		Chronic periodontitis
		Non-smoker	Smoker	Non-smoker	Smoker
		N = 37	N = 18	N = 30	N = 9
	Gender
	Male (%)	14 (37.8)	16 (88.9)	5 (16.7)	7 (77.8)
	Female (%)	23 (62.2)	2 (11.1)	25 (83.3)	2 (22.2)
	P value	<0.0001		<0.0001
	Age (mean ± SD), years	31.84 ± 10.10	26.94 ± 8.39	38.96 ± 10.65	40.63 ± 6.72
	P value	0.065		0.600
	Education level
	≥ University (%)	15 (40.5)	10 (55.6)	13 (46.4)	7 (87.5)
	< University (%)	22 (59.5)	8 (44.4)	15 (53.6)	1 (12.5)
	P value	0.294		0.039
	Family income
	Low income (%)	17 (45.9)	10 (55.6)	19 (65.5)	5 (55.6)
	Middle income (%)	12 (32.4)	4 (22.2)	10 (34.5)	4 (44.4)
	High income (%)	7 (18.9)	4 (22.2)	0 (0.0)	(0.0)
	P value	0.750		0.699
	PPD (mean ± SD), mm	2.41 ± 0.82	1.87 ± 0.64	5.14 ± 0.74	4.75 ± 0.89
	P value	0.017		0.284
	CAL (mean ± SD), mm	0.0 ± 0.0	0.0 ± 0.0	5.14 ± 1.06	4.88 ± 0.84
	P value	1.000		0.471
	PI (mean ± SD)	1.41 ± 0.50	1.60 ± 0.51	2.31 ± 0.54	2.38 ± 0.74
	P value	0.240		0.823
	GI (mean ± SD)	1.47 ± 0.62	1.60 ± 0.63	2.31 ± 0.54	2.25 ± 0.46
	P value	0.511		0.758
	Smoking quantitiy
	< 10 cig/day (%)	-	3 (16.7)	-	0 (0.0)
	10-20 cig/day (%)	-	11 (61.1)	-	7 (77.8)
	> 20 cig/day (%)	-	4 (22.2)	-	2 (22.2)
	Smoking duration
	< 5 years (%)	-

Fig. 1 Prevalence of bacterial species in healthy smokers healthy non-smokers

Table 2

 2 Odds of having each bacterium detected in plaque samples of healthy smokers compared to healthy non-smokers Smoking was the dependent variable and independent variables (covariates) were: age, gender, education level, PPD and type of bacteria. Gender, education level and type of bacteria were set as categorical variables, while age and PPD were set as continuous variables (default).

	Bacterial species	Crude OR (95% CI)	P	Adjusted a OR (95% CI)	P-value
	Aggregatibacter actinomycetemcomitans	0.57 (0.2-2.1)	0.40	0.76 (0.1-4.1)	0.74
	Campylobacter rectus	2.67 (0.5-13.9)	0.25	5.62 (0.4-73.3)	0.19
	Capnocytophaga ochracea	0.57 (0.1-2.4)	0.44	0.32 (0.1-2.4)	0.26
	Capnocytophaga sputigena	0.57 (0.2-1.8)	0.33	0.41 (0.1-2.3)	0.31
	Centipeda periodontii	1.14 (0.3-4.0)	0.83	0.60 (0.1-3.8)	0.58
	Dialister pneumosintes	1.75 (0.5-6.5)	0.40	2.10 (0.3-14.0)	0.44
	Eikenella corrodens	0.56 (0.2-1.8)	0.33	0.35 (0.1-1.8)	0.21
	Eubacterium saphenum	-	-	-	-
	Fusobacterium nucleatum	0.36 (0.1-1.2)	0.09	0.57 (0.1-2.7)	0.48
	Mogibacterium timidum	0.67 (0.2-2.3)	0.52	0.41 (0.1-3.0)	0.38
	Parvimonas micra	2.20 (0.6-9.2)	0.28	3.22 (0.2-48.8)	0.40
	Porphyromonas endodontalis	0.25 (0.0-2.3)	0.22	1.54 (0.3-7.3)	0.58
	Porphyromonas gingivalis	1.26 (0.4-4.0)	0.70	4.37 (0.7-26.5)	0.11
	Prevotella intermedia	1.40 (0.5-4.4)	0.56	3.95 (0.6-24.1)	0.14
	Prevotella nigrescens	0.14 (0.0-1.1)	0.06	0.31 (0.0-4.7)	0.40
	Prevotella tannerae	0.51 (0.2-1.6)	0.25	0.41 (0.1-2.1)	0.28
	Selenomonas sputigena	4.52 (0.9-22.8)	0.07	6.62 (0.9-50.5)	0.07
	Slackia exigua	3.08 (0.7-13.3)	0.13	10.1 (0.7-146.1)	0.09
	Tannerella forsythia	1.00 (0.2-6.1)	1.00	2.68 (0.2-29.9)	0.42
	Treponema amylovorum	6.05 (1.1-34.4)	0.04	4.38 (0.5-33.1)	0.15
	Treponema denticola	2.50 (0.7-9.1)	0.17	1.87 (0.3-12.4)	0.51
	Treponema maltophilum	1.12 (0.4-3.5)	0.85	1.26 (0.2-6.8)	0.79
	Treponema medium	1.57 (0.5-4.9)	0.44	1.1 (0.2-5.6)	0.91
	Treponema socranskii	1.00 (0.3-3.9)	1.00	0.92 (0.1-6.7)	0.93
	Treponema vincentii	8.29 (0.7-104.9)	0.10	1.41 (0.2-9.2)	0.72

a Adjusted for gender and age patients.

Table 3

 3 Odds of having each bacterium detected in plaque samples of CP smokers compared to CP non-smokers

	Bacterial species	Crude OR (95% CI)	P-value	Adjusted a OR (95% CI)	P-value
	Aggregatibacter actinomycetemcomitans	0.40 (0.1-1.8)	0.24	0.53 (0.07-4.4)	0.57
	Campylobacter rectus	-		-	0.01*
	Capnocytophaga ochracea	-		-	0.06*
	Capnocytophaga sputigena	1.27 (0.2-7.5)	0.79	4.8 (0.2-108.1)	0.32
	Centipeda periodontii	0.50 (0.1-2.6)	0.41	0.88 (0.1-8.4)	0.91
	Dialister pneumosintes	0.40 (0.1-2.2)	0.29	0.76 (0.07-8.2)	0.82
	Eikenella corrodens	-		-	-
	Eubacterium saphenum	-		-	-
	Fusobacterium nucleatum	0.12 (0.0-1.5)	0.10	0.52 (0.0-44.3)	0.77
	Mogibacterium timidum	0.57 (0.1-7.1)	0.66	0.09 (0.0-2.0)	0.13
	Parvimonas micra	-		-	0.06*
	Porphyromonas endodontalis	0.25 (0.0-2.3)	0.22	0.25 (0.02-3.5)	0.31
	Porphyromonas gingivalis	0.28 (0.0-4.9)	0.38	0.19 (0.01-5.3)	0.32
	Prevotella intermedia	0.70 (0.1-4.4)	0.70	0.78 (0.05-13.3)	0.87
	Prevotella nigrescens	0.14 (0.0-1.0)	0.06	0.66 (0.04-10.3)	0.77
	Prevotella tannerae	0.17 (0.0-0.9)	0.04	0.57 (0.1-5.7)	0.63
	Selenomonas sputigena	0.12 (0.0-1.5)	0.10	0.34 (0.0-46.5)	0.66
	Slackia exigua	1.07 (0.2-6.4)	0.95	6.37 (0.3-146.1)	0.25
	Tannerella forsythia	-		-	*0.58
	Treponema amylovorum	6.05 (1.1-34.4)	0.04	20.7 (0.9-466.0)	0.05
	Treponema denticola	-		-	*0.58
	Treponema maltophilum	0.73 (0.2-3.6)	0.70	0.79 (0.1-7.2)	0.84
	Treponema medium	1.50 (0.3-8.7)	0.65	7.97 (0.4-67.1)	0.23
	Treponema socranskii	-		-	1.00*
	Treponema vincentii	8.29 (0.7-104.9)	0.001	5.37 (0.2-151.8)	0.32
	*P value based on χ 2 analysis
	a Adjusted for gender and education level
	Significant P values ≤ 0.05 are shown in bold

(22.2) -2 (22.2)

5-10 years (%) -10 (65.6) -4 (44.5) > 10 years (%) -4 (22.2) -3 (33.3) Significant P values ≤ 0.05 are shown in bold

Acknowledgements

Not applicable.

Funding

This study was fully funded by the Deanship of Scientific Research, Jordan University of Science and Technology (JUST), Grant number 84/2008.

Availability of data and materials

The data will not be made available in order to protect the participants' identity.

Additional file

Additional file 1: Target Bacteria and Their Species-Specific Primers. A table including names of bacteria investigated and the species-specific primers (forward and reverse) used for the PCR. (DOCX 16 kb) Abbreviations bp: Base pair; CAL: Clinical attachment loss; CP: Chronic periodontitis; DTC: Dental teaching centre; F: Forward; g: Relative centrifugal force; GCF: Gingival crevicular fluid; GI: Gingival index; IRB: Institutional Research Board; JUST: Jordan University of Science and Technology; min: Minute; OR: Odds ratio; OR adj : Adjusted odds ratio; PBS: Phosphate-buffered saline; PCR: Polymerase chain reaction; PI: Plaque index; PPD: Periodontal pocket depth; R: Reverse; rRNA: Ribosomal ribonucleic acid; s: Seconds; SD: Standard deviation.

Authors' contributions

JK was responsible for the study design, preparing the questionnaires used for the study, statistical analysis and writing the paper. RA was responsible for evaluating the patients for a final diagnosis, collecting plaque samples and preparing the questionnaires used in the study. NAM was responsible for conducting the laboratory work including bacterial DNA extraction and PCR. MT commented on the methodology of the study and was responsible for editing the paper and final drafting. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Consent for publication

Not applicable.

Ethics approval and consent to participate

The study was independently reviewed and approved by the Institutional Research Board (IRB) of Jordan University of Science and Technology (JUST). Written consent was obtained before interview and after explaining the study purposes, methodology and possible future publication of clinical datasets according to the Declaration of Helsinki.

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