Emir Bouarroudj Erredir Chahrazad
email: cerredir@yahoo.fr

Mohamed Lahdi Riabi

Swarm Intelligence Algorithm for Microwave Filter Optimization

Keywords: Swarm Intelligence, Spider Monkey Optimization, Social Spider Optimization, Teaching Learning Based Optimization, Particle Swarm Optimization, Microwave Filter. 1

In this paper, three recent swarm intelligence algorithms (spider monkey optimization (SMO), social spider optimization (SSO) and teaching learning based optimization (TLBO)) are proposed to the optimization of microwave filter (H-plane three-cavity filter). The results of convergence and optimization use of these algorithms are compared with the results of the most popular swarm intelligences algorithm, namely particle swarm optimization (PSO) for different common parameters (population size and maximum number of iteration). The results showed validation of the proposed algorithms.

Introduction

Swarm intelligence (SI) algorithms are natural inspired techniques that involve the study of collective behavior of decentralized, self-organized systems [START_REF] Yang | Nature-inspired metaheuristic algorithms[END_REF][START_REF] Lim | Innovations in Swarm Intelligence[END_REF]. The swarm intelligence systems contain a set of particles (or agents) that interact locally with one another and with their environment. Swarm Intelligence techniques can be used in several engineering applications where the SI algorithms have been successfully applied to solve complex optimization problems including continuous optimization, constrained optimization and combinatorial optimization.

To date, several swarm intelligence models based on different natural swarm systems have been proposed in the literature, and successfully applied in many real-life applications. Examples of swarm intelligence models are: Ant Colony Optimization [START_REF] Dorigo | Ant Colony Optimization[END_REF], Particle Swarm Optimization [START_REF] Kennedy | Particle Swarm Optimization[END_REF], Artificial Bee Colony [START_REF] Karaboga | An Idea Based On Honey Bee Swarm for Numerical Optimization[END_REF], Bacterial Foraging [START_REF] Passino | Biomimicry of Bacteria Foraging for Distributed Optimization and Control[END_REF], Cat Swarm Optimization [START_REF] Chu | Cat swarm optimization[END_REF], Artificial Immune System [START_REF] Bakhouya | An Immune Inspired-based Optimization Algorithm: Application to the Traveling Salesman Problem[END_REF], and Glowworm Swarm Optimization [START_REF] Krishnanand | Glowworm swarm optimization for searching higher dimensional spaces[END_REF].

In this paper, we will primarily focus on three of the recent swarm intelligences models, namely, spider monkey optimization (SMO), social spider optimization (SSO) and teaching learning based optimization (TLBO) to optimize the waveguide microwave filter (H-plane three-cavity filter). The results of optimization obtained are validated by comparing them with those obtained using the optimization algorithm available in the literature Particle swarm optimization (PSO). The details of each algorithm are presented in the next section.

2

Swarm Intelligence Algorithm

Particle Swarm Optimization

Particle swarm optimization (PSO) is a stochastic method of optimization based on the reproduction of a social behavior. It was invented by Eberhart, R. C, and Kennedy, J [START_REF] Kennedy | Particle Swarm Optimization[END_REF] in 1995. They tried to simulate the ability of animal societies that don't have any leader in their group or swarm (bird flocking and fish schooling) to move synchronously and their ability to change direction suddenly while remaining in an optimal formation (food source). PSO consists of a swarm of particles, where particle represent a potential solution. The particles of the swarm fly through hyperspace and have two essential reasoning capabilities: their memory of their own best position local best (LB) and knowledge of the global or their neighborhood's best global best (GB). The essential steps of particle swarm optimization are presented by the following algorithm:

Step.1. Initialize the optimization parameters (population size, number of generations, design variables of the optimization problem and the specific parameters of algorithm) and define the optimization problem (minimization or maximization of fitness function).

Step.2. Generate a random population (position and velocities), according to the population size and the limits of the design variables.

Step.3. Evaluate each initialized particle's fitness value, then calculate LB the positions of the current particles, and GB the best position of the particles.

Step.4. the best particle of the current particles is stored. The positions and velocities of all the particles are updated according to (1) and ( 2), then a group of new particles are generated.

𝑋 𝑖 (𝑡 + 1) = 𝑋 𝑖 (𝑡) + 𝑉 𝑖 (𝑡 + 1) (1)
𝑉 𝑖 (𝑡 + 1) = 𝑤 * 𝑉 𝑖 (𝑡) + 𝑐 1 𝑟 1 [𝐿𝐵 𝑖 (𝑡) -𝑋 𝑖 (𝑡)] + 𝑐 2 𝑟 2 [𝐺𝐵(𝑡) -𝑋 𝑖 (𝑡)] (2)
V i (t), X i (t) are the velocity and the position for particle i at time t. w is the inertia weight, at each iteration update with the following equation [START_REF] Singh | A New Modified Approach of Mean Particle Swarm Optimization Algorithm[END_REF] 𝑤 𝑡 = 𝑤 𝑚𝑎𝑥 -

(𝑤 𝑚𝑎𝑥 -𝑤 𝑚𝑖𝑛 ) * 𝑡 𝑚𝑎𝑥𝑖𝑡 (3)
The parameters w max , w min , maxit, c 1 and c 2 are constant coefficients determined by the user. r 1 and r 2 are random numbers between 0 and 1.

Step.5. Repeat the procedure from step 3 until the maximal iteration is met.

Spider Monkey Optimization Algorithm

Spider Monkey Optimization (SMO) algorithm is a new swarm intelligence algorithm based on the foraging behavior of spider monkeys, proposed by J. C. Bansal et al in 2014 [START_REF] Bansal | Spider Monkey Optimization Algorithm for Numerical Optimization[END_REF]. There are four important control parameters necessary for this algorithm: perturbation rate (Pr), local leader limit (LLL), global leader limit (GLL) and maximum number of groups (MG). The SMO process consists of six phases:

Local Leader Phase (LLP). In this phase, Spider Monkey SM of each group updates its position based on the experience of the local leader and local group members as following expression:

𝑆𝑀 𝑖 𝑗 = 𝑆𝑀 𝑖 𝑗 + 𝑟 1 𝐿𝐿 𝐾 𝑗 -𝑆𝑀 𝑖 𝑗 + 𝑟 2 𝑆𝑀 𝑟 𝑗 -𝑆𝑀 𝑖 𝑗 (4)
Where SM i j is the j th dimension of the i th SM, LL k j represents the j th dimension of the k th local group leader position. SM r j is the j th dimension of the r th SM which is chosen randomly within k th groupsuchthatr≠i,r 1 is a random number between 0 and 1 and r 2 isarandomnumberbetween-1 and 1.

Global Leader Phase (GLP

Where GL j represents the j th dimension of the global leader position and j ∈ {1, 2... D} is the randomly chosen index; D is the number of design variables. The positions are updated based upon some probability given by the following formula.

𝑃 𝑖 = 0.9

𝐹 𝑖 𝑚𝑎𝑥 _𝐹 + 0.1 (6)
Where P i is the probability, F i is the fitness of i th SM, and max_F is the maximum fitness of the group.

Local Leader Learning Phase (LLLP).

In this phase, the position of the local leader isupdatedbyapplyingthegreedyselectioninthatgroup.IftheLL'spositionremains same as before, then the Local Limit Count is increased by 1.

Global Leader Learning Phase (GLLP).

In this phase, the position of the global leaderisupdatedbyapplyingthegreedyselectioninthepopulation.IftheGL'sposition remains same as before, then the Global Limit Count is increased by 1.

Local Leader Decision Phase (LLDP).

If a LL position is not updated for a predetermined number of iterations Local Leader Limit (LLL), then the positions of the spider monkeys are updated either by random initialization or by using information from both LL and GL through equation ( 7) based on the perturbation rate (pr)

𝑆𝑀 𝑖 𝑗 = 𝑆𝑀 𝑖 𝑗 + 𝑟 1 𝐺𝐿 𝑗 -𝑆𝑀 𝑖 𝑗 + 𝑟 1 𝑆𝑀 𝑖 𝑗 -𝐿𝐿 𝐾 𝑗 (7)
Global Leader Decision Phase (GLDP). If the position of GL is not updated in predetermined number of iterations Global Leader Limit, then the population is split into subgroups. The groups are split till the number of groups reaches to maximum allowed groups (MG), then they are combined to form a single group again.

The details of each step of SMO implementation are explained below:

Step.1. Initialize the optimization parameters and define the optimization problem. Control parameters necessary for this phases; perturbation rate (Pr), local leader limit (LLL), global leader limit (GLL) and maximum number of groups (MG). Some settings of control parameters are suggested as follows:

-MG = N/10, i.e., it is chosen such that minimum numberofSM'sinagroupshould be 10.

-

Global Leader Limit ∈ [N/2, 2 × N].
-Local Leader Limit should be D × N.

Pr 𝑡 + 1 = Pr 𝑡 + (0.4-0.1) 𝑚𝑎𝑥𝑖𝑡 , Pr 1 = 0.1 (8)
Step.2. Initialize the population and evaluate the corresponding objective function value.

Step.

Locate global and local leaders

Step.4. The local leader phase starts by update the position of all group members' equation [START_REF] Kennedy | Particle Swarm Optimization[END_REF]. Accept of a new solution if it gives better function value. All the accepted function values at the end of this phase are maintained and these values become the input to the global leader phase.

Step.5. Produce new positions for all the group members, selected by probability (P i ), by using self experience, global leader experience and group members' experiences equation [START_REF] Karaboga | An Idea Based On Honey Bee Swarm for Numerical Optimization[END_REF].

Step.6. Update the position of local and global leaders, by applying the greedy selection process on all the groups (see. LLLP, GLLP)

Step.7. if any Local group leader is not updating her position after a specified number of times (Local Leader Limit) then redirect all members of that particular group for foraging (LLDP)

Step.8. If Global Leader is not updating her position for a specified number of times (Global Leader Limit) then she divides the group into smaller groups (see. GLDP).

Step.9. Repeat the procedure from step 3 to until the termination criterion is met.

Social Spider Optimization Algorithm

Social-spiders optimization (SSO) [START_REF] Cuevas | A swarm optimization algorithm inspired in the behavior of the social-spider[END_REF][START_REF] Cuevas | A new algorithm inspired in the behavior of the social-spider for constrained optimization[END_REF] is a new proposed swarm optimization algorithm; it is based on the natural spider's colony behavior. An interesting characteristic of social-spiders is the highly female-biased populations, where the number of females N f is randomly selected within the range of 65-90% of the entire population NP and the rest is the number of male Nm. Therefore, N f and N m are calculated by the following equations:

𝑁 𝑓 = 𝑓𝑙𝑜𝑜𝑟 0.9 -𝑟𝑎𝑛𝑑 * 0.25 * 𝑁𝑃 (9)
𝑁 𝑚 = 𝑁𝑃 -𝑁 𝑓 [START_REF] Singh | A New Modified Approach of Mean Particle Swarm Optimization Algorithm[END_REF] Where floor rounds each element to the nearest integer, and rand is a random number in the unitary range [0, 1].

After the initialization process the algorithm starts the searching loop that only ends when the maximum number of function evaluations or the target function value is reached. The first step in the searching loop is to calculate the spider's weight. This calculation is done according to:

𝑊 𝑖 = 𝑊𝑜𝑟𝑠𝑡 -𝑓 𝑥 𝑖 𝑊𝑜𝑟𝑠𝑡 -𝐵𝑒𝑠𝑡 (11)
Where W i is the weight of the i th spider, f (x i ) is the fitness value of the spider x i . The values Worst and Best are defined as follows (considering a minimization problem):

𝐵𝑒𝑠𝑡 = 𝑚𝑖𝑛 𝑖=1…𝑁𝑃 𝑓 𝑥 𝑖 (12)
𝑊𝑜𝑟𝑠𝑡 = 𝑚𝑎𝑥 𝑖=1…𝑁𝑃 𝑓 𝑥 𝑖 [START_REF] Cuevas | A new algorithm inspired in the behavior of the social-spider for constrained optimization[END_REF] In the colony, the spiders communicate with each other directly by mating or indirect by a small vibration to determine the potential direction of a food source, this vibration depend on the weight and distance of the spider which has generated them.

𝑉𝑖𝑏 𝑖𝑗 = 𝑤 𝑗 * exp -𝑑 𝑖𝑗 2 (14)
Where w j indicates the weight of the j th spider, and d i j is the euclidean distance between i th and j th spiders. Every spider is able to consider three vibrations from other spiders as:

-Vibrations Vib ci are perceived by the individual i (X i ) as a result of the information transmitted by the member c (X c ) who is an individual that has two important characteristics: it is the nearest member to i and possesses a higher weight in comparison to i

(W c > W i ).
-The vibrations Vib bi perceived by the individual i as a result of the information transmitted by the member b (X b ), with b being the individual holding the best weight (best fitness value) of the entire population NP, such that :

𝑊𝑏 = max 𝑘=1..𝑁 (𝑊 𝑘 ). (15)
-The vibrations Vib fi perceived by the individual i (X i ) as a result of the information transmitted by the member f (X f ), with f being the nearest female individual to i. Depending on gender, each individual is updating their position according to three operations (female operation, male operation and mating operation). In female operation, the female individuals are updating as follow equation: 16)

X i = X i + α Vib ci (X c -X i ) + β Vib bi (X b -X i ) + δ (rand -0.5)withprobabilityPF (
X i = X i -α Vib ci (X c -X i )-β Vib bi (X b -X i ) + δ (rand -0.5)withprobability1-PF (17)
WherePFisthresholdparameter,α,β,δandrand are random numbers between [0, 1] The male spiders are divided into two different groups (dominant members D and nondominant members ND) according to their position with regard to the median member.

According to this, change of positions for the male spider can be modeled as follows:

𝑋 𝑖 =
Where the individual (X f ) represent the nearest female individual to the male member. After all males and females spiders are update, the last operator is representing the mating behavior where only dominant males will participate with females who are within a certain radius called mating radius given by

𝑅 = 𝑋 𝑑 ℎ -𝑋 𝑑 𝑙 𝑑 =1…𝑛 2 𝐷 (20)
Where X h and X l are respectively the upper and lower bound for a given dimension and n is the problem dimension. Males and females which are under the mating radius generate new candidate spiders according to the roulette method. Each candidate spider is evaluated in the objective function and the result is tested against all the actual population members. If any member is worse than a new candidate, the new candidate will taketheactualindividualpositionassumingactualindividual'sgender.

Teaching Learning Based Optimization

Rao et al. [START_REF] Rao | Teaching-Learning-Based Optimization: A Novel Method for Constrained Mechanical Design Optimization Problems[END_REF][START_REF] Rao | Teaching-Learning-Based Optimization: An Optimization Method for Continuous Non-Linear Large Scale Problems[END_REF][START_REF] Rao | Optimization of Fused Deposition Modeling Process Using Teaching-Learning-Based Optimization Algorithm[END_REF] proposed an algorithm, called Teaching-Learning-Based Optimization (TLBO), based on the traditional teaching learning phenomenon of a classroom. TLBO is a population based algorithm, where a group of students (i.e. learner) is considered as population and the different subjects offered to the learners are analogous with the different design variables of the optimization problem. The results of the learner are analogous to the fitness value of the optimization problem. The best solution in the entire population is considered as the teacher. Teacher and learners are the two vital components of the algorithm, so there are two modes of learning; through the teacher (known as the teacher phase) and interacting with other learners (known as the learner phase).

Teacher Phase. In this part, learners take their knowledge directly through the teacher, where a teacher tries to increase the mean result value of the classroom to another value, which is better than, depending on his or her capability. This follows a random process depending on many factors. In this work, the value of solution is represented as X j, k, i , where j means the jth design variable (i.e. subject taken by the learners), j= 1, 2, ..., m; k represents the kth population member (i.e. learner), k = 1, 2, ..., N ; and i represents the ith iteration, i = 1, 2, ...,maxit , where maxit is the number of maximum generations (iterations). The existing solution is updated according to the following expression

X ' j, k, i = X j, k, i + DM j, k, i (21)
DM j, k, i the difference between the existing mean and the new mean of each subject is given by

DM j, k, i =r * (X j, kbest, i -TF * M j, i ) (22)
M j, i the mean result of the learners in a particular subject j, X j, kbest, i the new mean and is the result of the best learner (i.e. teacher) in subject j. r is the random number in the range [0, 1]. TF The teaching factor is generated randomly during the algorithm in the range of [START_REF] Yang | Nature-inspired metaheuristic algorithms[END_REF][START_REF] Lim | Innovations in Swarm Intelligence[END_REF], in which 1 corresponds to no increase in the knowledge level and 2 corresponds to complete transfer of knowledge. The in between values indicates the amount of transfer level of knowledge. The value of TF is not given as an input to the algorithm its value is randomly decided by the algorithm

TF = round [1+rand (0, 1) {2-1}] (23)
Learner Phase. In this part, learners increase their knowledge by interaction among themselves. A learner interacts randomly with other learners for enhancing his or her knowledge. A learner learns new things if the other learner has more knowledge than him or her. At any iteration i, each learner is compared with the other learners randomly. For comparison, randomly select two learners P and Q such that

X ' P, i ≠X ' Q, i
(where X ' P, i and X ' Q, i are the updated values at the end of the teacher phase).

X " j, P, i =X' j, P, i +r * ( X' j, P, i

+ X' j, Q, i ), f (X' P, i ) < f (X' Q, i ) (24)
X " j, P, i =X' j, P, i +r * ( X' j, Q, i + X' j, P, i ), f (X'

P, i ) > f (X' Q, i ) (25)
Accept X " j, P, i if it gives a better function value.

Comparison of Optimization Techniques

The swarm intelligence algorithms have been widely used to solve complex optimization problems. These methods are more powerful than conventional methods based on formal logic or mathematical programming. In terms of comparison of intelligence algorithms of the swarm we have:

-The four algorithms studied in this paper are population-based techniques that implement a group of solutions to achieve the optimal solution. -The PSO and TLBO algorithms use the best solution of the iteration to change the existing solution in the population, which increases the rate of convergence. -TLBO and PSO do not divide the population unlike SSO and SMO -TLBO, SSO and SMO implement greed to accept the right solution -Each method requires parameters that affect the performance of the algorithm.

•PSOrequirescoefficientsofconfidenceandinertia •SSOrequiresthethresholdsetting •SMOrequirestheperturbationrate(Pr),thelocalleaderlimit(LLL), the global leader limit (LGL) and the maximum number of groups (MG).

• In contrary, TLBO does not require any parameters, which simplifies the implementation of TLBO.

Application Example and Results

In this section, the application of a proposed algorithm is presented for the optimization of rectangular waveguide H-plane three-cavity filter [START_REF] Yang | Investigation of Multiple Rectangular Aperture Irises in Rectangular Waveguide Using TE Modes[END_REF] Fig. 1. When the main guide is WR28, four parameters are to be optimized W 1 and W 2 (the opening of the iris), l 1 and l 2 (the distance between the iris). The thicknesses of the iris are fixed to t 1 =1.45mm, t 2 =1.1 mm. Table 1 contains the geometric variables of the structure and the corresponding ranges. As for the frequency range, it was chosen to be f ∈ (34, 35.5 GHz). 2 with the population size of 50 and the maximum number of iterations take the following values (30, 50, 300). Table 3 shows the convergence of the fitness functions for the number of iterations is 50 and the population size (30, 70 and 100). Every algorithm is run 10 independent times. The other specific parameters of algorithms are given below: -PSO Settings: c 1 and c 2 are constant coefficients c 1 =c 2 =2, the inertia weight decreased linearly from 0.9 to 0.2 -SSO Settings: the threshold parameter PF=0.7 -TLBO Settings: for TLBO there is no such constant to set -SMO Settings: the parameter of SMO depends on the population size where: (N=30: MG=3, GLL= 30 and LLL=120); (N=70: MG=7, GLL= 70 and LLL=280); (N=100: MG=10, GLL= 100 and LLL=400). Fig. 2 shows the convergence of the fitness function of the best individual of each algorithm. The results of the optimization are presented in the Table 4. It is observed from Table 2, Table 3 and Fig. 2 that, the SMO and TLBO algorithms performs better in terms of convergence than the PSO and SSO algorithms. In which SMO and TLBO algorithms converge to the minimum optimal for the first's iterations maxit=30, on the other hand PSO and SSO algorithms converge to the minimum optimal in 300 iterations.

Conclusion

In this works, the study and application of the three recent swarm intelligence algorithms in literature (spider monkey optimization (SMO), social spider optimization (SSO) and teaching learning based optimization (TLBO)) are presented. These three algorithms are used for the optimization of microwave filter (H-plane three-cavity filter). The results of convergence and optimization are compared with the results of

Fig. 1 .

 1 Fig. 1. Rectangular waveguide H-plane three-cavity filter

Fig. 2 .

 2 Fig. 2. The convergences of the fitness function of the best individual of each algorithm.

Table 1 .

 1 Geometric variables of the structure and the corresponding rangesThe objective is to minimize the fitness function in the frequency range, where the fitness functions is the mean value of the coefficient of reflection S11.With PT is the number of points in the interval[f 1 , f 2 ].the convergence of the fitness functions of each algorithm (Best, Worst, Mean) is presented in Table

	Variables	Min	Max
	W 1 (mm)	1.80	5.40
	W 2 (mm)	1.20	3.60
	l 1 (mm)	2.07	6.22
	l 2 (mm)	2.35	7.05
	𝑓𝑖𝑡𝑛𝑒𝑠𝑠 =	𝑓 2 𝑓 1	𝑆 11 𝑓 𝑃𝑇	(26)

Table 2 .

 2 The fitness functions for population size=50

	Maxit	Algorithm Best	Worst	Mean
		PSO	0.0193	0.1588	0.0907
	30	SSO TLBO	0.0191 0.0082	0.1405 0.0100	0.0680 0.0089
		SMO	0.0082	0.0092	0.0088
		PSO	0.0126	0.0945	0.0791
	50	SSO TLBO	0.0110 0.0081	0.0353 0.0089	0.0190 0.0082
		SMO	0.0081	0.0081	0.0081
		PSO	0.0084	0.0210	0.0108
	300	SSO TLBO	0.0081 0.0081	0.0121 0.0081	0.0091 0.0081
		SMO	0.0081	0.0081	0.0081

Table 3 .

 3 The fitness functions for Maxit=50

	Population	Algorithm Best	Worst	Mean
		PSO	0.0197	0.0849	0.0411
	30	SSO TLBO	0.0120 0.0081	0.0536 0.0140	0.0409 0.0089
		SMO	0.0081	0.0081	0.0081
		PSO	0.0126	0.0634	0.0326
	70	SSO TLBO	0.0119 0.0081	0.0501 0.0082	0.0272 0.0081
		SMO	0.0081	0.0081	0.0081
		PSO	0.0120	0.0352	0.0255
	100	SSO TLBO	0.0103 0.0081	0.0311 0.0081	0.0162 0.0081
		SMO	0.0081	0.0081	0.0081

Table 3 .

 3 The geometrical parameters optimized

Variables

[START_REF] Yang | Investigation of Multiple Rectangular Aperture Irises in Rectangular Waveguide Using TE Modes[END_REF]

PSO SSO TLBO SMO W 1 (mm) 3.60 4.705 4.694 4.709 4.078 W 2 (mm) 2.40 3.598 3.585 3.600 3.599 l 1 (mm) 4.15 2.936 2.917 2.915 2.910 l 2 (mm) 4.70 3.633 3.613 3.613 3.608

the most popular swarm intelligences algorithm, particle swarm optimization (PSO). The results showed validation of the proposed algorithms.


Christine Boher
email: christine.boher@enstimac.fr

D Attaf

Luc Penazzi

Christophe Levaillant

Wear behaviour on the radius portion of a die in deep-drawing: Identification, localisation and evolution of the surface damage

Keywords: Wear process-simulator, Deep-drawing, Wear mechanisms, Simulation, Friction coefficient

published or not. The documents may come

Wear behaviour on the radius portion of a die in deep-drawing: Identification, localisation and evolution of the surface damage

Introduction

Deep-drawing is the forming process of a blank by plastic deformation. The problems related to deep-drawing can be grouped into three categories [START_REF] Ronde-Oustau | Conception et mise au point d'une gamme d'emboutissage[END_REF][START_REF] Samper-Mangin | Etude théorique et expérimentale du frottement et des forces de retenue en emboutissage des tôles d'acier nues et galvanisées[END_REF]:

• Problems in maintaining tolerance of the workpiece dimensions. • Sheet constriction and anisotropy problems. Before rupture, constriction can produce a sheet surface aspect like orange skin, which makes it useless. • Wear problems. Wear affects both the aspect of the workpiece and, in addition, the die radius is the area of the tool which undergoes the most thermo-mechanical stress.

The general aim of this work is to focus on the third category by studying the degradation mechanisms of the die radius used in deep-drawing. To do so, a new tribological test has been designed: the wear deep-drawing process-simulator. It is concerned with the die radius wear rather than metal sheet wear.

The bibliography covers various tests for studying friction and wear in relation to the deep-drawing process. Most of these tests study sheet wear rather than tool wear [START_REF] Felder | Tribologie de l'emboutissage[END_REF][START_REF] Kim | An experimental study on forming characteristics of pre-coated sheet metals[END_REF].

For the plane-drawing test, a strip slides between two plane tools that generate a normal clamping force on it. The objective of this test is to simulate friction conditions operating on the part of the workpiece that is the least deformable but where contact exists over a great sliding length. This test reveals the evolution of asperity deformation.

The Inland test has an identical principle to the above test. The strip slides against a cylindrical tool, and it is plastically deformed in this way. Even though this test does not represent real deep-drawing conditions, it enables the study of possible seizure between the strip and the tool.

The Bending test, is more representative of what occurs under a drawing press [START_REF] Felder | Tribologie de l'emboutissage[END_REF][START_REF] Kim | An experimental study on forming characteristics of pre-coated sheet metals[END_REF][START_REF] Coubrough | Angle contact between sheet and die during stretch-bend deformation as determined on the bending-under-tension friction test system[END_REF]. The friction coefficient can be measured. A strip is plastically curved on a radius cylinder by the application of a traction force. By modifying the pulling forces as well as the restraining forces, the tool radius contact pressure can be changed.

The test called "Restrained Ring" allows the curving of the strip by a ring indentation between two edges up to a specified value [START_REF] Felder | Tribologie de l'emboutissage[END_REF][START_REF] Coubrough | Angle contact between sheet and die during stretch-bend deformation as determined on the bending-under-tension friction test system[END_REF]. When this value is reached, the strip slides at a constant speed. The traction and clamping forces are measured. This test is used to investigate the metal strip formability and its roughness through the restrained ring passage.

The Erichsen test of strip expansion on a hemispherical punch is used to study strip formability and ductility under non-axisymmetric expansion conditions [START_REF] Ericksen | The influence of die geometry on tool wear in deepdrawing[END_REF]. A round punch is pressed on a strip clamped by a blankholder. The metal strip is plastically distorted by expansion with bilateral stretching around the punch bottom making the metal strip thinner. Then the maximum level before sheet tearing is recorded using predefined conditions of punch speed and blankholder clamping force.

Finally, The Swift test is a deep-drawing test using a flat bottom punch that has been codified by the International Deep-Drawing Research Group (IDDRG) [START_REF] Samper-Mangin | Etude théorique et expérimentale du frottement et des forces de retenue en emboutissage des tôles d'acier nues et galvanisées[END_REF]. Mechanical analyses of this test allow a correlation of the friction coefficients of the blankholder and the tool radius using pre-defined conditions of pressure and speed. The reach limiting drawing ratio can be obtained with this test.

In the current study, a new tribological test to study friction and wear of a die radius in the deep-drawing process was used. We focused our study on the degradation of the die radius; the strip surface degradation is not investigated. After the presentation of the deep-drawing process-simulator, the results will demonstrate that there are always two mechanisms of degradation on the die radius. These two mechanisms are not localised in the same areas on the die radius and they have a different evolution according to the cycle number.

Experimental friction equipment: the deep-drawing process-simulator (DDPS)

The aim of the deep-drawing process-simulator is to study the tribological interaction between the metal strip and the tool (Fig. 1). The strip slides over the die radius to simulate drawing conditions. With this process-simulator, laboratory tests were performed under quasi-industrial conditions (materials, loading, evolution over time, sliding velocity, lubrication, etc.).

A steel strip, unrolled directly from a coil, was in contact with a portion of the radius tool. The flat blankholder and the die radius constituted the working system of our DDPS. A rolling up engine pulled the strip through this working system (Fig. 2). The loading of the die radius was a result of the restraining forces (H) and the pulling forces (T). The blankholder forces were controlled by a hydraulic cylinder.

The sliding of the steel strip over the die radius varied in accordance with a defined angle (α) which simulates the running of the strip steel on the tool (Figs. 3 and4). We fixed the strip exit angle (α) in relation to the angular position of the reversing cylinder.

With the rolling up and unrolling engines and with a programmable logic control unit, two user modes can be programmed: either continuous or repeated sliding of the strip over the tool. In the continuous case, the mechanical load- ing conditions are constants, so the closed friction contact is not refreshed by air convection the contact temperature increases and produces thermal friction. In the repeated sliding cycle, we load and unload the contact, and thus refreshing contact conditions are produced. However, the effects of loading and unloading produce mechanical fatigue of the die radius.

Materials and samples

The die radius steel grade is X160CrMoV12 (AISI D2 or DIN 1.2379). The chemical composition (in wt.%) is given in Table 1. Steel AISI D2 is commonly used in the deep-drawing process. It contains 12% chromium with a large quantity of free carbides, which confer great wear resistance (Fig. 5). The steel is quenched at 1080 • C for 30 min and tempered twice for 1 h at 200 • C to a hardness of 60 HRC. The die radius is R m = 6 mm (Fig. 6). The arithmetic roughness (Ra) of the die radius is 1.02 m.

The strip is made of low skin-passed steel used in cold forming, DC04 grade steel (DIN 1.0338) (Table 1). The strip is covered with a thin, protective oil film. The metal strip is 50 mm wide, 1 mm thick and the coil length is 150 m. The strip mechanical characteristics are detailed in Table 2. These values will be required to calculate the friction coefficient between the die radius and the sheet.

Tests on the wear deep-drawing process-simulator

Introduction

In the real deep-drawing process, the punch displacement involves a strip exit angle (α) scanning the die radius from 0 to 90 • . In our process-simulator, varying the strip exit angle (α) allows us to understand at which angle (α) position, the die radius is the most worn by the strip sliding.

To evaluate the influence of this exit angle (α) on the repartition and amplitude of the die radius contact pressure, a finite element model was designed (Fig. 7). This model reproduces the geometry of the die radius and the test conditions of the DDPS [START_REF] Attaf | Etude et analyse de la dégradation des rayons de matrice en emboutissage[END_REF][START_REF] Attaf | Mechanical study of a sheet metal forming dies wear[END_REF]. The pressure distribution is variable with two zones of strong pressures: one at the entry of the die radius and the other at the exit of the contact between the strip and the die radius. There is, therefore, a variation in the localisation and the amplitude of the two high-pressure peaks according to the strip exit angle (α). In this paper, the strip exit angles studied (α) are 70 • , 80 • and 90 • . Our experimental results were divided into three parts.

1. Identification of mechanisms of the die radius degradation. 2. Location of these surface degradations on the die radius. 3. Evolution of these surface degradations on the die radius in function of the cycle number.

Test parameters

The parameters (sliding speed, blankholder pressure, sliding distance, etc.) are selected to obtain a degradation of the die radius after only 1200 cycles (one coil). The selected parameters are given in Table 3:

Observation techniques for wear on the die radius

Tools were observed by scanning electron microscope (SEM) at each 100 cycles to track die radius surface damage. Only the die radius central section was observed which was then divided into five sub-sections (Fig. 8). The chemical elements of the die radius surface were analysed by electron dispersive spectroscopy. This analysis demonstrated that the only perceptible difference in the tool and sheet iron grades is the relative value of chromium content. The metal strip has no chromium content. For this reason, the weight percent of chromium evolution will be a solution to reinforce our findings concerning the degradation behaviour of the surface.

Results

Identification of degradation mechanisms on the die radius

Two mechanisms of surface degradation were identified on the die radius through the micrographs: adhesion and ploughing. Adhesion was in evidence from transferred particles and ploughing by the formation of grooves by plastic deformation.

The strip transfer on the die radius consisted of very small particles about 10 m diameter (Fig. 9). Transferred particles were identified by energy dispersive spectroscopy analysis. With the same analysis conditions and the same specific volume of interaction, the chromium content decreased (4% instead of 12.7 wt.%) when a transferred particle of the strip material was identified (Fig. 10).

The strip particles are essentially embedded in the tool grinding grooves (Fig. 11). Transfer is essentially due to mechanical interlocking of strip asperities on the die roughness. Some researchers have noticed that transfer is initiated with small particles "lumps" that grow by propagation to form lamellar structures [START_REF] Van Der | Lubricant failure in sheet metal forming processes[END_REF][START_REF] Rooij | Tribological aspects of unlubricated deepdrawing porocesses[END_REF].

The formation of grooves (or ploughing) results in fine scratches on the die radius (Fig. 12). The carbides, M 7 C 3 , contained in tool steel, play an important role in the formation of these scratches. The die radius matrix plastically flows around these carbides. Like the particle transfer mechanism, the ploughing damage rises as the strip exit angle (α) increases.

Location of degradations on the die radius

The observations with the scanning electron microscope highlight two zones of the die radius where degradations are present.

In order to make very precise die radius surface observations, with the motorised SEM stage, the die radii have been scanned each 350 m for a circumference from 0 • to 90 • . These investigations revealed that the transfer is localised in two definite zones: the entry of die radius (zone E) and a zone located between 40 • and 50 • according to the strip exit angle (α) (zone A). The plastic deformations induced by the ploughing mechanism are only observed at the entry of the die radius (zone E).

The equivalence between the damage observations observed by SEM stage and the die radius angle (β) is given in Fig. 13 and in Table 4.

The location of the first damage zone on the die radius is the same for the strip exit angles 70 • and 80 • but is quite different for 90 • . For the second zone, the damage locations are quite similar.

The contact between the strip and the die is not the same in function of the exit angle. It is more difficult to curve a Area width (d): the degradation area width is calculated from the distance "d" obtained from the origin of the die radius.

the strip with an exit angle of 90 • . The strip thickness and its mechanical properties play a role in the strip bending stiffness and the contact zones are different from the others angles.

To understand the areas of the degradations observed, the results of our numerical simulation (Fig. 7) were used. The numerical simulation model has shown that the pressure distribution is not constant on the die radius and it varies according to the test parameters (blankholder pressure, strip exit angle (α), die radius, etc.), the strip properties and design as well as the friction coefficient [START_REF] Coubrough | Angle contact between sheet and die during stretch-bend deformation as determined on the bending-under-tension friction test system[END_REF][START_REF] Ericksen | The influence of die geometry on tool wear in deepdrawing[END_REF][START_REF] Jensen | Applying the finite element model for determination of tool wear in conventionnal deep-drawing[END_REF][START_REF] Mortesen | A combined physical and numerical simulation of tool performance in conventionnal deepdrawing operations[END_REF]. One previous study pointed out that the areas of high pressures obtained by numerical simulation are the same as the areas of the degradations obtained from the wear process simulator tests [START_REF] Attaf | Etude et analyse de la dégradation des rayons de matrice en emboutissage[END_REF][START_REF] Attaf | Mechanical study of a sheet metal forming dies wear[END_REF]. Correlation between surface damage of the die radius and the pressure level can be made.

Evolution of the wear damage in function of the cycle number

Microscopic observations of the die radius surface are made at different cycle number (100, 300, 500, 700, 900 and 1200 cycles). For a strip exit angle (α) of 70 • , the degradation of the die radius occurs preferentially by the adhesion mechanism. Strip particle transfer is located in zones E and A on the die radius and starts from 100 cycles. For this cycle number, the ploughing scratches in zone E are not observed. Between 100 and 700 cycles, there is no significant evolution. From 700 cycles, in zone A, larger transferred particles are present on the die radius surface (Fig. 14) and some scratches begins to appear. These scratches affect the transferred particles surface but not really the die surface.

The degradation evolution for a strip exit angle (α) of 80 • , looks like the 70 • angle one. The die radius is little marked by the ploughing mechanism and the transfer is visible in zone A.

For the 90 • angle, the adhesion mechanism, in zone A, is observed from 100 cycles (Fig. 15) and some scratches of ploughing are also observed in zone E (Fig. 16). These scratches affect the die surface and not only the transferred particles like previously. From 700 cycles, there is a real increase of the wear damage in zone E (entry of the contact). The waves of plastic deformation are significant and they go round the carbides (Fig. 17). In zone A (exit of the contact), surface damage is always due to transferred particles (Fig. 18).

Friction coefficient results

The friction coefficient (f m ) between the strip and the die radius is calculated according to a model given in Eq. (1) [START_REF] Attaf | Etude et analyse de la dégradation des rayons de matrice en emboutissage[END_REF]. This model is based on the bibliography [START_REF] Ghosh | A method for determining the coefficient of friction in punch streching of sheet metals[END_REF][START_REF] Terreaux | Emboutissage: Simulation du frottement de la tôle sur l'arrondi de la matrice[END_REF] and the mechanical analysis of our DDPS (Fig. 2) (Appendix A).

f m = R m + e/2 θR m ln T ′ -F u H + F b (1)
where R m is radius of the die, e the strip thickness, T ′ the tensile force applied on the die radius, H the restrained force and θ is the contact angle between the strip and the die radius. F u and F b are the unbending and bending forces acting on the strip. The different parameters necessary to calculate the friction coefficient were obtained from the variations in the rolling up engine electrical currents (Appendix A). We established For each strip exit angle (α), three tests of 1200 cycles were carried out with many interruptions to measure the friction coefficient (Fig. 19). To compare the friction curves obtained for the three strip exit angles (α), we calculate polynomial curves from the average tendency for a given exit angle. The friction coefficient evolution results are presented in Fig. 20.

We notice a great similarity for the three angles, especially for the strip exit angle (α) of 70 • and 80 • . For these two angles, the values of friction coefficient vary from 0.65 to 0.5 at the end of the test. For the angle 90 • , the values are weaker and limited to 0.5 and 0.4.

The evolution of the friction coefficient presents three fields. The friction coefficient is high and quite constant in the first field. It decreases in the second one and in the last field, it is low and quite constant. The fields I and II correspond to the running-in period of the test for the three angles and the stabilisation of the friction coefficient is observed after 600 cycles. The friction coefficient evolution could be correlated with the evolution of the wear damage. The initial lower values obtained for a strip exit angle of 90 • indicate that the abrasion mechanism is more significant. In opposite, the high value for 70 • and 80 • could indicate that the adhesion mechanism is more important. After 600 cycles for all the angles, abrasion mechanism at the entry of the contact is observed and there is no evolution of the wear damage at the surface die radius.

Discussion

Wear surface damages are localised in two specific zones on the die radius and these locations are confirmed by a finite element model which gives mechanical response of a deep-drawing die radius. More specifically, there is a good agreement between high-pressure contact areas (FE model) and localised transfer zones (experiments). Two mechanisms of surface degradation are identified on the die radius through adhesion (or adherence) and ploughing. Adhesion is identified by transferred particles and ploughing by the formation of grooves by plastic deformation. Adherence could be also used because the strip asperities (named particles with global observations) are essentially embedded in the tool grinding grooves.

The transfer and ploughing mechanisms are thus correlated with the pressure distribution on the die radius. The model argue that the transfer mechanism can occur for val-ues of pressure weaker than those necessary to generate the ploughing mechanism.

When the ploughing mechanism is present, it occurs only on the entry of the contact (zone E), where the pressures are higher, whereas the transfer mechanism exists in zones A and E.

In these test conditions, the initial lubrication of the sheet is not sufficient to avoid asperity contact with the die and the die radius friction is carried out under boundary or mixed lubrication.

For a strip exit angle of 70 • and 80 • , the main wear damage at the surface of the die radius is adhesion or adherence. The transferred particles are observed at the entry of the contact (zone E) but also at the exit of the contact in zone A. The quantity of the transferred particles increases essentially in this last zone with the cycle number. For these angles, the abrasion mechanism is not really observed. For a strip exit angle of 90 • the main wear damage at the surface of the die radius is ploughing. This degradation is located at the entry of the contact and it increases in function of the cycle number.

The friction coefficient of the die radius is calculated with a specific model using the mechanical characteristics of the DDPS and the mechanical properties of the strip. The calculated coefficient is a global friction coefficient on the die radius. It integrates all the contact phenomena occurring on all the zones of contact of the die radius. The correlation of the wear surface evolution with the friction coefficient evolution could explain the three fields of the friction coefficient. In field I the strip transfer mechanism or adhesion mechanism on the die radius is dominating. The ploughing mechanism is then dominating starting from field III. Field No. II corresponds to the progressive transition of the two mechanisms. The high values of the friction coefficient are associated with adhesion and the lower value with ploughing. The ploughing mechanism is more important for a strip exit angle (α) of 90 • . In this case, the friction coefficient is less high than the 70 • and 80 • one. For the strip exit angle (α) of 90 • , the die radius contact pressure seems to be more important.

The ploughing mechanism could be induced by a mechanical fatigue phenomena, because it is not well observed at the beginning of cycles like transfer phenomena. Degradation evolution reveals that the transfer mechanism occurs after the first cycles whereas the mechanism of ploughing is visualised after 500 or 700 cycles, under our conditions.

To understand the transfer mechanism it is necessary to consider the mechanical behaviour of the strip asperity. For an elastic deformation of the asperity, the contact model is based on Hertz theory. With transferred particles, the deformation mode is based on plastic or visco-plastic behaviour of the strip asperity. Sliding could induce large deformations in one of the two antagonists. The plastic deformation thickness may reach a few micrometers in function of tribological conditions. These tribological transformations corroborate that the material has got a ductile behaviour [START_REF] Suh | The delamination theory of wear[END_REF][START_REF] Rigney | Wear process in sliding systems[END_REF][START_REF] Rigney | Comments on the sliding wear of metals[END_REF][START_REF] Kapoor | Wear by plastic ratchetting[END_REF]. Others authors confirm the assumption of ductile deformation with the mechanism of transfer and adhesion [START_REF] Rooij | Tribological aspects of unlubricated deepdrawing porocesses[END_REF][START_REF] Chowdhury | Adhesion between metal surfaces: the effect of surface roughness[END_REF].

In our present study, to explain the transfer particles, we consider three steps in the asperity behaviour. For instancethe thermo-mechanical aspect is not yet study -the transfer assumptions are only based on a mechanical theory.

The contact leads to plastic deformation conditions. The strip asperities, sliding on the die radius, have got a ductile behaviour. The notion of a low cycle fracture model is of prime necessity to understand the particle transfer.

Under contact pressure and shearing stresses they plastically flow with ductile behaviour. Under high-contact pressure, low cycle number is necessary to break of the asperity and the opposite for low contact pressure. Theses assumptions could explain the difference of transferred particles evolution between the entry and the exit of the contact on the die radius.

Conclusion

The deep-drawing process-simulator, designed in our laboratory is interesting because it is possible to simulate quasiindustrial conditions and to study the influence of the strip exit angle on the die radius degradation. It also allows a great length of strip to study the wear behaviour of the die surface. For most of the tribological tests used in the deep-drawing field, the strip sliding length is rather short and the wear behaviour is studied on a few cycles. The die radius surface observations confirm that wear industrial phenomena can be reproduced on the process simulator: strip particle transfer is the main wear damage and it is located on two specific areas of the die radius. Ploughing or abrasion is essentially observed at the entry of the die radius. There is a good agreement between high-pressure contact areas calculated by a finite element model and localised wear zones. The tribological behaviour of the die radius is quite different in function of the strip exit angle. For low strip exit angles, particle transfer on the die radius is important and for high strip exit angle, the main damage is abrasion. The friction coefficient may also give informations about the contact evolution. Its modelling depends on the specificities of the DDPS. This wear processsimulator can be used to study different grades of die radius or strip.

Appendix A. Measurement of the friction coefficient of a die radius portion on our wear DDPS

The calculation of the die radius friction coefficient, on our DDPS, is based on an analytical theory obtained with the bibliography for the expression of a strip friction coefficient on a cylinder. The second part of this calculation is based the engine outputs and required to interpret the strip traction on this equipment.

A.1. General model of the strip friction coefficient on a cylinder

Authors [START_REF] Ghosh | A method for determining the coefficient of friction in punch streching of sheet metals[END_REF][START_REF] Terreaux | Emboutissage: Simulation du frottement de la tôle sur l'arrondi de la matrice[END_REF]

A.2. Model of the forces acting on the strip when it slides around several rolls or cylinders

The general equation given the force action with several cylinders is detailed in

F = T - 4 i=2 2lK (n + 1)(n + 2)R i (n+1) e 2 (n+2) (2 -3a ci ) (A.1)
where F is restrained force (N), T is tensile force (N), l is strip width (mm), n is strip hardening coefficient, K is consistancy (MPa), R i is cylinder radii (mm), e is strip thickness (mm), a ci is coefficient of the dissipated energy by plastic deformation of the strip bending. This value is estimated with the fractional part of the contact surface of strip bending (a mp ) and unbending (a md ) on the die radius. This value is assessed with the calculation of contact surface fraction due to bending.

The specific equations given the force action with a blankholder and a portion of a die radius (DDPS) are detailed in bending forces acting on the strip (N), F u is unbending forces acting on the strip (N).

A.3. Model of the die radius friction coefficient on our DDPS

To determine the value F t , and T ′ , the engines characteristics are necessary (evolution of the engine electrical current during strip sliding). They are detailed in Table A.1. The main effects acting on the engine current are the decurler effect, the blankholder pressure, the blankholder friction, the die radius friction, the strip exit angle, the unrolling and rolling up effects and the coil diameter.with

Fig. 1 .

 1 Fig. 1. Global view of the deep-drawing process-simulator.

Fig. 2 .

 2 Fig. 2. Diagram of the DDPS.

Fig. 3 .

 3 Fig. 3. Magnification of the active part of the process simulator: visualisation of the samples.

Fig. 4 .

 4 Fig. 4. Diagram of the active part of the process simulator: visualisation of the strip exit angle (α).

Fig. 5 .

 5 Fig. 5. Die radius without degradation: visualisation of carbides in dark zones (SEM observations with back scattering electrons).

Fig. 6 .

 6 Fig. 6. X160CrMoV12 (AISI D2 or DIN 1.2379) die sample of the DDPS.

Fig. 7 .

 7 Fig. 7. Numerical model of our DDPS: meshing of the die radius and observations of the two pressure peaks at the entry of the contact (E) and at its exit (A).

Fig. 8 .

 8 Fig. 8. Schema of the die radius and localisation -on the central part -of the micrographs for SEM observations.

Fig. 9 .

 9 Fig. 9. Example of die radius wear by transfer particles after 100 cycles (on the left the real micrograph, on the right in red lines, the transferred particles).

Fig. 10 .

 10 Fig. 10. Energy dispersive spectroscopy analysis of die radius: (a) before wear and (b) analyses of a transfer zone: diminution of the chromium content.

Fig. 11 .

 11 Fig. 11. Transferred particles embedded in the die grinding grooves. Example of die radius wear (transfer) after 700 cycles (α = 90 • ) in zone A (in white lines, the transferred particles).

Fig. 12 .

 12 Fig. 12. Visualisation of the ploughing mechanism on the die radius. The plastic flow occurs around the carbides (angle = 90 • ).

Fig. 13 .

 13 Fig. 13. Diagram of the equivalence between the damage observations localised by SEM stage and the die radius angle (β).

Fig. 14 .

 14 Fig. 14. Die radius degradation for α = 70 • in zone E after 100 cycles on the left and after 700 cycles on the right (in black lines, the transferred particles).

Fig. 15 .

 15 Fig. 15. Die radius degradation for α = 90 • , at the exit of the contact (zone A) after 100 cycles (on the left the real micrograph, on the right, in black lines, the transferred particles). a measurement routine of these parameters according to various test configurations. This routine requires test interruptions. Stopping every 100 cycles, we measured the different value of engine electrical currents for each of 10 cycles. This was necessary to calculate friction-under three test conditions: (1) with a strip exit angle (α) of 0 • and without load (blankholder); (2) with a strip exit angle (α) of 0 • and under load and (3) finally under the test conditions.

Fig. 16 .

 16 Fig. 16. Die radius degradation for α = 90 • , at the entry of the contact (zone E) after 100 cycles.

Fig. 17 .

 17 Fig. 17. Die radius degradation for α = 90 • , at the entry of the contact (zone E) after 700 cycles: ploughing wear mechanism.

Fig. 18 .

 18 Fig. 18. Die radius degradation for α = 90 • , at the exit of the contact (zone A) after 700 cycles: transferred particles.

Fig. 19 .

 19 Fig. 19. Example of the friction coefficient evolution in function of the cycle number for the strip exit angle (α) 70 • .

Fig. A. 1 .

 1 Fig. A.1. Diagram to model the friction coefficient on a radius portion.

  have established a relation for the friction coefficient (f m ) of a strip on a cylinder -with a radius (R)in function of the tensile force (T) and the restrained force (H) (Fig. A.1 and Eq. (1)). F u and F b are the unbending and bending forces acting on the strip. Strip thickness is the value (e).

Fig. A. 2 .

 2 Fig. A.2. Diagram to model the strip traction forces with several cylinders.

  Fig. A.2 and Eq. (A.1)

  Fig. A.3. Model of blankholder friction coefficient on our DDPS.

T( 2 -( 2 -

 22 ′ -F u = √ 3U cos(φ)η m η ener K r (I -I empty ) ν + F t -2lK (n + 1)(n + 2) R (n+2) 3a mp )a md H + F b = √ 3U cos(φ)η m η ener K r (I load -I empty ) ν + F t + 2lK (n + 1)(n + 2) R (n+2) 3a mp )a mp F t = √ 3U cos(φ)η m η ener K r (I decurler -I 0 )ν All the forces are detailed to calculate the friction coefficient (Eq. (1)) of the die radius on our DDPS.

Table 1

 1 Chemical composition (in wt.%) of the two materials

		C	Cr	Mo	Mn	V	Si	S	Pb	Fe
	DIN 1.2379	1.45-1.6	11-13	0.7-1	0.2-0.6	0.7-1	0.1-0.6	0.03	0.03	Other
	DIN 1.0338	0.08	-	-	0.4	-	-	0.03	0.03	Other

Table 2

 2

	Strip mechanical characteristics
	Hardening coefficient, n	0.225
	Consistancy, K (MPa)	530
	Poisson coefficient, ν	0.3
	Young modulus, E (GPa)	210
	Shear modulus, G (GPa)	80
	Hardness (MPa)	320
	Thickness, e (mm)	1
	Width, l (mm)	50

Table 3 Test

 3

parameters with the DDPS Sliding speed, V (mm/s) 50 Sliding distance, L (mm) 100 Blankholder pressure, P (MPa) 4 Strip exit angle, α ( • ) 70, 80 and 90

Table 4

 4 Correspondence between SEM observations on the die radius and die radius angle (β)

	Strip exit anglem,	First zone transfer + ploughing mechanisms		Second zone transfer mechanism
	α ( • )	Area width, d (m) a	Area width (2β • )	Median position (2β • )	Area width, d (m) a	Area width (2β • )	Median position
							(2β • )
	70	275-420	3.6-5.4	4.5	3970-4054	42.4-43.2	42.9
	80	150-450	2.0-5.8	3.9	4100-5000	43.6-52.2	47.8
	90	720-1030	9.1-12.6	10.8	4460-4850	47.1-50.8	48.9


A Orcesi
email: andre.orcesi@ifsttar.fr

A Feraille

S Chataigner

A LIFE CYCLE ANALYSIS APPROACH APPLIED TO THE STRENGTHENING OF STEEL BRIDGES

Keywords: Strengthening and repair, CFRP, Life Cycle Cost, Life Cycle Analysis

This paper presents some results obtained in the Infravation research project called "Fast and effective solution for steel bridges life-time extension" (FASSTbridge). The aim of this project is to develop and demonstrate a reliable, preventive, cost-effective and sustainable solution for steel bridges fatigue life-time extension. Original sustainable preventive strengthening techniques will be tested to help postpone difficult, resourceconsuming and costly retrofitting and repair interventions and demolitions of the steel and composite steel bridges stock. This paper addresses the definition of the needed data and the methodology for the realization of an actual cost-benefit analysis and a life cycle analysis of the CFRP solution compared with traditional techniques (use of steel plates) and considering the economic and environmental profit of the preventing policy.

INTRODUCTION

Various methods exist to strengthen steel bridges once fatigue damage have been identified. The most popular among engineers is the attachment of steel plates to the tension flange of the girders (Figure 1). However, this method has several disadvantages: plates are usually bulky, heavy, difficult to fix and prone to corrosion and fatigue. In general, conventional strengthening techniques are labor intensive and disruptive to traffic flows, thus limiting their application in the wide network of European and American steel bridges. Carbon Fibre Reinforced Polymers (CFRP) composites, although more expensive than steel plates in terms of price per unit/m2, have several and relevant advantages that make them suitable and cost effective for retrofitting of steel bridges: their application is less time consuming than traditional solutions (i.e. from one month to a few days, therefore less traffic disruptive), have a high strength to weight ratio, excellent fatigue properties, low space disruption, high durability and versatility, and are easy to transport, handle and apply (without heavy equipment) (Figure 2).

Figure 1. Additional riveted plate

Figure 2. Adhesively bonded composite reinforcement The use of CFRP strips for fatigue strengthening is not only justified by its optimum mechanical performance, proven in several research studies and projects [START_REF] Fawzia | Bond-slip models for double strap joints strengthened by CFRP[END_REF][START_REF] Chataigner | Use of a single lap shear test to characterize composite-to-concrete or composite-to-steel bonded interfaces[END_REF][START_REF] Ghiassi | Numerical analysis of bond behaviour between masonry bricks and composite materials[END_REF][START_REF] Jiao | A comparative study on fatigue behaviour of steel beams retrofitted with welding, pultruded CFRP plates and wet layup CFRP sheets[END_REF][START_REF] Lepretre | Application of Fibre Reinforced Polymer Composites to the Highway Infrastructure[END_REF]. It is also cost-effective in terms of the entire execution process. The swiftness of the intervention, compared to the traditional attachment of steel plates, reduces considerably traffic disruptions: reducing the costs of auxiliary means (platform, scaffolds), making easier the transport (steel plates are heavy and difficult to handle in the worksite) and the labour (welding requires qualified and experienced workers). In a report from the National Cooperative Highway Research Program (NCHRP 503, 2003), the findings of a study to develop a strategic plan for guiding the application of fibre reinforced polymer composites in the highway infrastructure are presented and discussed. Among all the elements in the proposed strategic plan to create widespread understanding and application of FRP materials in highway infrastructure, it was recommended to perform a study of the relative costs of FRP versus traditional materials. Some of the key issues regarding the use of FRP composites for rehabilitation of steel components the authors mentioned in the report were:

 costs of strengthening deteriorated steel components are largely labor and societal costs,  costs of materials in rehabilitation projects are usually less significant,  higher initial costs of FRP composites can be offset by reduced labor, reduced construction time, and reduced traffic disruptions during construction,  cost-effectiveness of rehabilitation of steel components with FRP composites would be mainly due to durability of FRP, reductions in construction time and traffic delays.  Based on the strategic plan outlined in the NCHRP 503, this report not only aims to look at the material/labor costs and environmental impact but also at the costs associated with traffic disruptions/delays during the repair or retrofit of the bridge. These aspects are considered in this paper which introduces a cost-benefit and life-cycle analysis to perform the comparison of costs of implementing traditional repair strategies and the use of FRP composites. Such analysis should make a link between the variables in the design of the strengthening action, the cost and the gain on performance and life-time extension. The data needed were introduced and a database with unit cost was elaborated in the realm of the project FASSTbridge (2017a,b). This project aims at developing and demonstrating a reliable, preventive, cost-effective and sustainable solution for steel bridge fatigue life-time extension. FASSTbridge is co-funded by Funding Partners of the ERA-NET Plus Infravation and the European Commission.

INFORMATION USED IN LIFE CYCLE ANALYSES

Input data

This section provides impacts information when using welded steel plate (Table 1), bolted steel plate (Table 2) or CFRP (Table 3) to increase the fatigue remaining life. These estimations are based on actual interventions made by the partners of FASSTbridge consortium. To the unit costs in Tables 1 to 3 should be added the costs of scaffolding and rental equipment such as forklift in work site, when necessary (Table 4). The average work duration for such a strengthening project is assessed at 12 days for welded and bolted solutions and 5 days for CFRP. It is noted that there is no forklift location in work site when using CFRP. Despite the fact that the unitary cost of material is higher for CFRP than for steel plates (welded or bolted), the former solution seems cost-effective in terms of the entire execution process. The swiftness of the intervention, compared to the traditional attachment of steel plates can reduce the costs of auxiliary means (platform, scaffolds), making easier the transport (steel plates are heavy and difficult to handle in the worksite) and the labour (welding requires qualified and experienced workers, see Table 1).

Performance analysis

The estimation of the remaining fatigue life time is done according to the regulation of Eurocode EN 1993-1-9 7.1 (3) with following formulae (analogue formulae for shear):

For nominal stress spectra with stress ranges above and below the constant amplitude fatigue limit (2)

The fatigue detail category 50 (reference value C  =50N/mm 2 for the fatigue strength at 2 million cycles) is considered for illustration in the following. The properties of the steel section and fatigue design life configuration are provided in Table 5.

       C  =50N/mm 2 D  =36.85N/mm 2
To simplify the illustration of the methodology, the strengthening project is based on this fatigue assessment configuration, extended to the scale of the structure. The design life in number of cycles is then R N  6,439,900 if stress relieved and R N  5,313,961 if not stress relieved. As mentioned previously, three options are taken into account in this report for fatigue service life increase:

 welded steel plates,  bolted steel plates,  CFRP strips. In order to make a comparison based on the function or performance equivalent the solutions are compared keeping the product of the Young modulus (E) by the section constant. The characteristics associated with each of the three solutions are provided in Tables 6, 7 and8 To these costs above need to be added the additional costs due to the scaffold and platform rent, installation/dismantling, transport, and to the use of forklift in work sites (rental equipment and transport). The corresponding unit costs for the welded/bolted steel plates and use of CFRP are provided in Table 4.

COST ANALYSIS

Materials

The proposed methodology assesses the NPV at ,0 s T of all future costs including strengthening ones that can occur at different years. Doing this, the "Do Nothing" strategy (DN) is characterized only by the discounted demolition/reconstruction costs assumed to occur at 0 f T (year 100 in this example):

    0 ,0 0 1 fs f s TT C NPV T r    (3)
The alternative strategies with some strengthening actions (use of welded or bolted steel plates or use of CFRP) are applied at 0 ,0 ss f T T T  and added to the end of life costs which occur at 1 2 3 0 ,,

f f f f T T T T 
in case of maintenance (when s G  14% or 30% or 46%, respectively) since the residual fatigue remaining life is increased (the effect being dependent on s G and the effective year of strengthening action). The total cost for each type of repair project can be calculated with information previously provided on unit cost , j su c , additional cost j w C and length s L of strengthening. The net present value at ,0 s T for strengthening solution j and gain of fatigue service life i applied at year s T can be obtained as follows:

      ,0 ,0 , , 1 1 i ss fs jj f s u s w j i s TT TT C c L C NPV T r r       with 0 ,0 ss f T T T  (4)
Such an approach reflects a decision making process where one investigates at a specific year (here ,0 s T ) the interest to postpone some strengthening action and to find the optimal year for it in the future. In this approach, 0 ,0 ss f T T T  but the decision is always considered at year ,0 s T . Figure 3 NPV follows the same trend with the same inflection point at year 69. The inflection point for 3,1

NPV is reached sooner at year 52. Tthe "Do nothing" cost ( 0 NPV ) is fixed for any s T as it is always expressed in reference to year ,0 s T and is equal to 325,000€. NPV T 325000€, the use of welded and bolted steel plates ( j  1 and 2) is more interesting than DN option until year 81. It is more interesting until year 85 for CFRP ( j  3). In Figure 5 when 46% s G  , it is more interesting not to wait at year 50 to strengthen the bridge and that for any of strengthening options. The net present values for each option are very close. Indeed, additional costs j w C are more interesting when using CFRP. However, this advantage is neutralized when 46% s G  as the unit costs for CFRP are larger than those for steel plates and necessary quantities of materials are more significant than when 14% s G  and 30% (see Tables 6,7 and 8).

ENVIRONMENTAL ANALYSIS

LCA methodology

LCA methodology enables the evaluation of the product on its entire life cycle, that is from cradle to grave, calculating different impact indicators, such as climate change, eutrophication… It is important to note that the fact that LCA is multicriteria allows to avoid a pollution transfer from one environment to another. LCA method has been undertaken complying with the standards 14040 at 14044 of the lnternational Organization for Standardization (lSO). The indicators in Table 9 are assessed according to the European standard EN 15804 regarding the construction products. Impact-oriented characterisation (CML 2001) Table 10 present the environmental results obtained for the solution with for a life cycle increase of 14%, 30% and 46%. Similar tables can be obtained for the two other solutions. It's important to note that these results have been obtained with the hypothesis described and with generic environmental data which might not be representative of the type of materials used in the FASSTbridge project. If more representative data were obtained in the near future, the methodology to be applied would be the same, but the results of the calculations of impact indicators at material scale should be updated. In order to be able to compare different reinforcement solutions, we propose to focus on the climate change just to be more readable. Concerning CFRP and under the assumptions of this study, the climate change impact has been the most important in all cases. Welded structure is always the one with the less impact comparing with the two other solutions. It is noted that uncertainties have not been calculated in this case but may be important in LCA, given the uncertainty of the data but also the methods of calculating the impact. Thus CFRP could be competitive regarding bolted structure.

An LCA study has been performed on a bridge by Cimbeton (CT 87). It is a concrete bridge so it is not the same case as ours, but it takes into account the whole life cycle or this bridge for a lifetime of 100 years. As there is not any other bridge case LCA studied we will consider this as a first approximation in order to compare different reinforcement solutions for this bridge along all its life cycle. In the publication of Cimbeton for the whole life cycle, climate change is equal to 677,000 eq CO2. Figure 7 shows the representative fraction of the strengthening action on the whole life cycle considering climate change indicator. It is observed that even in the worst case this fraction does not exceed 1.7% of the total life cycle of the bridge chosen as a reference.

CONCLUSIONS

This paper proposes economic and environmental analyses for highlighting the benefits of different strategies for fatigue strengthening. The aim is to provide a general framework that gathers precise information on unit costs to assess the overall cost of traditional strengthening projects (use of welded and bolted steel plates) and also original repair techniques (use of CFRP). Impacts include the cost of the strengthening action for the bridge owner and the corresponding impact on the end of the service life, and consequently the effective year for demolition/reconstruction. The general framework is highly dependent on input parameters and enables the comparison of several strengthening strategies. In particular, it is shown how several parameters can significantly influence the comparison results both for direct costs (yearly discount rate of money, costs of maintenance works, area of the bridge deck concerned by strengthening, end-of-life costs). The perspectives for future work is to apply this framework to the Jarama Bridge located in the city of Madrid in Spain over Jarama river and to analyze the effect of uncertainties on input parameters. 0,00% 0,20% 0,40% 0,60% 0,80% 1,00% 1,20% 1,40% 1,60% 1,80% 14% 30% 46%

D

   the fatigue strength is based on the extended fatigue strength curve as follows:

Figure

  Figure 3. Net present value

  Figure 4. Net present value

Figure

  Figure 5. Net present value

Figure 6 :

 6 Figure 6: Climate change in kg eq CO 2 for each solution of reinforcement while reparing 50m, 100m or 150m.

Figure 7 :

 7 Figure 7: Part of the reinforcing solution compared to the total bridge for climate change.

Table 1 :

 1 Steel S275JR, welded structure (1 kg).

	Item	Unit Quantity	Ecoinvent Process	Unitary Cost Cost
	Workmanship	h	0.018		18.96	0.34
	Workmanship (auxiliary)	h	0.018		17.83	0.32
	Qualified welders	h	0.018		24.5	0.44
	Laminated steel S275 JR	kg	1.05	steel. low-alloyed. hot rolled	0.99	1.04
	Electrolytic coating	l	0.01	Zinc coating	7.59	0.08
	Crane 30m. span. 750 kg capacity	h	0.005		18.92	0.09
	Diesel Welder	h	0.018	Diesel burned in building machine	3.62	0.07
	Fungible material	%	0.019		5	0.1
	Table 2: Steel S275JR, Bolted structure (1 kg).
	Item	Unit Quantity	Ecoinvent Process	Unitary Cost Cost
	Workmanship	h	0.028		18.96	0.53
	Workmanship (auxiliary)	h	0.028		17.83	0.5
	Laminated steel S275 JR	kg	1.05	steel, low-alloyed, hot rolled	0.99	1.04
	Electrolytic coating	l	0.01	Zinc coating	7.59	0.08
	Crane 30m. span. 750 kg capacity	h	0.01	Diesel burned in building machine	18.92	0.19
	Fungible material	%	0.023		5	0.12

Table 3 :

 3 CFRP laminate strengthening (1m).

	Item	Unit Quantity	Ecoinvent Process	Unitary Cost	Cost
	MasterBrace P 3500. 5 kg (primer)	kg	0.048	epoxy resin production. liquid	MasterBrace P 3500. 5 kg (primer)	kg
	MasterBrace ADH 4000 (adhesive)	kg	0.374	epoxy resin production. liquid	MasterBrace ADH 4000 (adhesive)	kg

Table 4 .

 4 Additional costs/m2.

	Item	Description	Welded solution Bolted solution CFRP
	Scaffold and platforms Rental equipment -400€/day 4800	4800	2000
		Installation + dismantling	350	350	350
		Installation certificate	300	300	300
		Transport	400	400	400
	Forklift in work site	Rental equipment -80€/day	960	960	-
		Transport	1200	1200	-

Table 5 .

 5 Properties of steel section and axial stress considered for illustration.

	Cross section

Table 6 .

 6 . Steel S275JR, welded structure, cost/m for several design life increase configurations.

	n a	b	E	Al	Al/A	Stress reduction	Stress range	Design life	Design life increase	Weight	Cost/m
	mm mm GPa mm²						Kg
	1 145 6	210 870.00 0.05 0.96	32.4	7353107 1.14	6.699	13.7
	2 145 6	210 1740.00 0.09 0.92	31.1	8348804 1.30		27.3
	3 145 6	210 2610.00 0.14 0.88	29.8	9430559 1.46	20.097	41.0

Table 7 .

 7 Steel S275JR. bolted structure. cost/m for several design life increase configurations.

	n a	b	E	Al	Al/A	Stress reduction	Stress range	Design life	Design life increase	Weight	Cost/m
	mm mm GPa mm²						Kg
	1 145 6	210 870.00 0.05 0.96	32.4	7353107 1.14	6.699	16.5
	2 145 6	210 1740.00 0.09 0.92	31.1	8348804 1.30	13.398	33.0
	3 145 6	210 2610.00 0.14 0.88	29.8	9430559 1.46	20.097	49.4

Table 8 .

 8 CFRP. cost/m for several design life increase configurations.

	n a	b	E	Al	Al/A

Stress reduction Stress range Design life

							Design
							life	Cost/m
							increase
	mm	mm GPa mm²
	1 100	4	460 876.19	0.05 0.96	32.4	7359896	1.14	233.8
	2 100	4	460 1752.38 0.09 0.92	31.1	8363586	1.30	466.7
	3 100	4	460 2628.57 0.14 0.88	29.8	9454615	1.47	699.5

  illustrates the cost

		NPV obtained with different solution (	j 	1, 2,3	for welded, bolted steel plates and
		j
	CFRP, respectively), with	50 Lm  s	,	s G 	14%	( 1 i  ), 0 100 f T 	and ,0 50 s T  . It is observed that	1,1 NPV
	gradually decreases until year 69, then grows again over time until	0 f TT  . s	2,1

Table 9 :

 9 Environmental impact indicators calculated in the following study and calculation methods used (CML, EDIP and CED); these methods have been chosen for their international representativeness.

	Environmental indicator	Unit	Calculation method used
	Consumption of energy resources
	Renewable energy	MJ	Cumulative Energy Demand
	Non renewable energy
	Depletion of abiotic resources	kg eq antimony Impact-oriented characterisation (CML 2001)
	Wastes	kg	Environmental Design of Industrial Products (EDIP)
	Radioactive wastes	kg	Environmental Design of Industrial Products (EDIP)

Table 10 :

 10 Environmental impacts for 1m of CFRP.

	Indicator impact	Unit	14%	30%	46%
	Acidification potential	Kg eq SO 2	0.30079	0.398780726 0.496767694
	Climate change	Kg eq CO 2	39.44833	57.52334768 75.59836625
	Depletion of abiotic resources Kg eq Sb	0.32011	0.495088248 0.670066297
	Dangerous wastes	kg	0.41127	0.65697567	0.902684559
	Non dangerous wastes	kg	3.56786	5.9002119	8.232565633
	Eutrophication	kg eq PO 4 3-	0.06949	0.104183964 0.13888189
	Non renewable energy	MJ	590.35404 1143.250925 1696.147813
	Photochemical oxydation	kg eq ethylene 0.00808	0.01194248	0.015804783
	Radioactive waste	kg	0.00723	0.012315639 0.017397316
	renewable energy	MJ	22.72660	43.72423107 64.72186321
	Stratospheric ozone depletion Kg eq CFC-11-0.00001	1.13534E-05 1.5118E-05


Yuxin Liu

Abdelkrim Talbi

Bahram Djafari-Rouhani

Lucie Drbohlavová

Vincent Mortet

Houssaine El El

Boudouti

Olivier Bou Matar

Philippe Pernod

Interaction of Love waves with coupled cavity modes in a 2D holey phononic crystal

Keywords: numbers: 68, 60, Bs

The interaction of Love waves with square array of pillars deposited on a cavity defined in a 2D holey phononic crystal is numerically investigated using Finite Element Method. First, the existence of SH surface modes is demonstrated separately for phononic crystals that consist of square arrayed holes, or Ni pillars, respectively in, or on, a SiO 2 film deposited on a ST-cut quartz substrate. The coupling between SH modes and torsional mode in pillars induces a transmission dip that occurs at a frequency located in the range of the band-gap of the holey phononic crystal. Second, a cavity is constructed by removing lines of holes in the holey phononic crystal and results in a transmission peak that matches the dip. The optimal geometrical parameters enable us to create a coupling of the cavity mode and the pillar mode by introducing lines of pillars into the cavity, which significantly improved the efficiency of the cavity. The obtained results will pave the way to implement advanced designs of high-performance electroacoustic sensors based on coupling modes in phononic crystals.

Phononic crystals (PnCs) have received increasing attention in the last two decades as an analogue of photonic crystals (PtCs). [1][2][3][4][5][6][7][8] Coupling modes have been investigated in both PnCs [9][10][11][12][13][14] and PtCs [15][16][17][18][19], exhibiting features such as energy transferring [10,11,16], absorption [9,12], wave confinement [16] and frequency modulation [13]. Most of the research is based on the cavity/waveguide [10,13,16], cavity/cavity [13,18] or waveguide/ waveguide [11] systems. Wang et al. [12] demonstrated the transmission cancellations due to the coupling modes of a 1D locally resonant PnC. Ohta et al. [17] studied the coupling between a PtC nanobeam cavity and a quantum dot. The emission peak of the quantum dot could be tuned by temperature to cross the cavity mode which splits into two peaks. In recent years, peak splitting phenomena have also been discussed in the coupling of two PnC cavities [13], giving rise to a capacity of frequency modulation as well as improved quality factors. We believe that it is conceivable to investigate similar effects in different PnC devices.

Recently, cavity modes for Love waves, which is a shear horizontal (SH) surface acoustic waves (SAW), are demonstrated in a holey PnC (h-PnC) that consist of square arrayed holes in a silica (SiO 2 ) film deposited on a ST-cut quartz [20]. This letter investigates the interaction of Love waves with a Ni pillar-introduced cavity in the h-PnC. Love waves are analyzed separately in the h-PnC and the pillared PnC (p-PnC). A mode-coupling induced transmission dip for the p-PnC is found in the a) Electronic mail: yuxin.liu@phd.ec-lille.fr band-gap region of the h-PnC. An optimally sized cavity in the h-PnC gives rise to a cavity mode that matches the transmission dip. The cavity mode is coupled to the pillar mode by introducing the pillars into the cavity, which significantly improved the sharpness of the transmission peak. These effects are used to design the microelectromechanical resonators with highly confined cavity modes. The band structures and the transmission spectra are calculated with the finite element method(FEM, COMSOL Multiphysics ). , where Rayleigh waves cannot be generated [21] due to a zero electromechanical coupling factor in the substrate. The surfaces of the PnCs coincide with the plane z = 0. The wavelength normalized energy depth (NED) [20] is calculated to select the surface modes:

NED = D 1 2 T ij S * ij (-z)dxdydz nλ D 1 2 T ij S * ij dxdydz , ( 1
)
where T ij is the stress and S ij the strain. The star symbol (*) means the complex conjugate. -z denotes that we integrate in the whole domain of the unit cell D. λ is the wavelength. For a relatively small k (with wave speed greater than the SH waves velocity of the substrate), λ is fixed to 2a that is resulting from k = π a and k = 2π λ . n = 1 for the h-PnC and n = 1.3 for the p-PnC because of the pillar height, which equals to 0.6a = 0.3λ. Surface modes have a NED < 1. The NED can filter out the bulk modes as well as the plate modes that appear in our finite-depth substrate which is supposed to be semiinfinite for Love waves.

SAW include SH type SAW and Rayleigh type SAW. The SH polarization ratio is calculated to distinguish these two polarizations:

SH ratio = D u SH u * SH dxdydz D (u x u * x + u y u * y + u z u * z )dxdydz , (2)
where u x , u y and u z are respectively the displacement along the x, y, z directions. u SH is the SH displacement component that can be expressed as u x cos θ -u y sin θ, which is perpendicular to the wave vector k. θ is the The transmission spectra is calculated by simulating the same SAW device using our preceding approaches [20,22], which has been validated by experiments. The model consists of two parts of aluminum inter-digital transducers (IDTs) with the PnC located between IDTs. The input IDT is given a V 0 = 1V harmonic voltage signal. The output is measured by averaging the voltage difference between odd and even electrodes. The width of the electrodes of IDTs L IDT is updated for every frequency in the spectrum according to the relation The coupling between the torsional mode p2 and SH waves has induced a sharp transmission dip at 508 MHz, denoted as χ. The origin of this dip is a narrow band gap between 507.6 and 510.2 MHz, formed by the coupling of p2 and p3 [12]. The slight frequency shift is mainly due to the mesh construction of the FEM.

L IDT = λ 4 = v 4f . v
By removing lines of holes along the y direction in the h-PnC, a cavity is formed perpendicularly to the waves propagation direction. Here we set the cavity width W to 4.5a h , since a cavity mode is predicted [20] at almost the same resonant frequency as f χ . The resonator is realized by introducing the p-PnC into the cavity. A supercell containing 1×(6 + W a h ) unit cells is constructed, as shown in Fig 4, with periodic boundary conditions applied along the y direction. Since the crystal size only slightly affects the cavity mode frequencies [13,20], the transmissions are calculated with 4 holes on each side of the cavity.

The Love waves normalized transmission spectra and band structures for the cavity without and with pillars are shown in Fig 5(a) and (b), respectively. On the bare cavity (without pillars), two cavity modes C1 and C2 appear inside the band-gap of the perfect h-PnC, giving rise to two transmission peaks at 438.9 MHz and 507.5 MHz, respectively. The 2 nd peak matches the transmission dip χ at 508 MHz that is denoted by the dotted line (f C2 ∼ = f χ ). After introducing the pillars, the fundamental cavity modes C1 and C2 are coupled to the pillars and become two new modes Cp1 and Cp2, while new pillar modes ∀p appear above. Due to the injected pillars mass, Cp1 at 433.4 MHz has a slight down frequency shift compared with C1, whereas Cp2 has shifted down to 490.5 MHz to show a well separated peak from ∀p. The frequency shift of Cp2 results from both the mass loading and the peak splitting effect that is attributed to a degeneracy of the cavity mode. [13] Meanwhile, the transmission peak of Cp2 is significantly sharper than that of the mode C2, as a result of the matched frequencies between f C2 and f χ . The pillar modes ∀p are different combinations of pillars vibrating in mode p2 (since f p2 f χ ), with the maximum displacements existing in different pillars. The number of pillar modes equals to the number of pillars, i.e., 18 pillar modes with adjacent frequencies around f p2 , which can be adjusted by the p-PnC parameters. The dispersion curves of ∀p are zoomed on the right of Fig 5(b). The first pillar mode gives rise to a sharp transmission peak at 507.3MHz. The last mode, with a less smoothing and homogeneous dispersion curve, only arouses a small resonance at 517.5 MHz. Other pillar modes disappear in the transmission spectrum because they are not coupled to the SH waves. This transmission spectrum results from a coupling between the p-PnC and the bare cavity-containing h-PnC. The differences in sharpness of the transmission peaks for modes C2, Cp2 and the 1 st of ∀p could be explained after observing their displacement fields shown in Fig 5(c). When Love waves propagate through the cavity, the displacement of C2 is concentrated in the center of the model and is confined in the silica guiding film. The mode Cp2 is actually a coupling of p2, p3 and C2 modes. The pillars vibrating in mode p2 and p3 are alternated along the propagating axis. This pillar-coupling mode is simultaneously coupled to the fundamental cavity mode C2 guiding in the film. Nevertheless, the maximum displacement amplitude is in the pillars, leading to an energy more confined to the surface (a smaller NED). This is proved by the dispersion curve of Cp2 whose red color is more intense than that of C2. As for the 1 st collective pillar mode of ∀p, only tiny displacement exists in the guiding layer, leading to the most confined energy and the sharpest transmission peak. The magnification of these three peaks is shown aside. The quality factor (Q) of the mode Cp2 is 6 times greater than that of the mode C2. The Q of the sharpest peak is 80 times greater than that of C2, with a transmission peak at almost the same frequency. This means that the efficiency of the cavity could be significantly enhanced by introducing the optimally sized pillars.

In summary, we investigated the SH surface modes in two phononic crystals that consist of square arrayed holes/Ni pillars in/on a silica film deposited on a 90STcut quartz substrate. The h-PnC gives rise to a band gap within which two pillar modes of the p-PnC are coupled, leading to a sharp transmission dip. An optimally sized cavity is introduced into the h-PnC by removing lines of holes and arouses a cavity mode that matches the pillar-coupling frequency. Pillar-containing cavity for Love waves is first demonstrated by introducing the pillars into the cavity. A coupling of the cavity mode to the pillar mode is created with significant improvement on the cavity efficiency. The sharpest transmission peak observed at the original cavity mode frequency has a 80 times greater Q. This study could be used for potential PnC applications such as filters and bio-sensors.

FIG. 1 :FIG. 2 :FIG. 3 :

 123 FIG.1: Unit cells of the PnCs with square arrayed cylindrical holes, or Ni pillars, respectively in, or on, the silica film deposited on a 90ST-cut quartz. r h =0.2a h , r p =0.2a p , H h =H p =2.4µm, h p = 0.6a p , a h = 4µm, a p = 2µm.

FIG. 4 :FIG. 5 :

 45 FIG. 4: Supercell of pillar-introduced cavity in the h-PnC. W = 4.5a h , a h = 4µm, a p = 2µm.

  is the velocity of Love waves for H h = 0.6a h , resulting from the basic dispersion relation of Love waves. That is, each frequency corresponds to a single wavelength, and this model simulates the dispersive SAW devices. The frequency responses are then nor-malized by that of the matrix, referred to as normalized or relative transmissions ∆S 21 . Fig 3(b) shows the normalized transmission spectrum of the h-PnC around the band-gap region. The attenuation appears clear and is consistent with the band structure prediction. Fig 3(d) is the normalized transmission spectrum of the p-PnC.

f (MHz) Ni 0 1 SH polarisation ratio

ACKNOWLEDGMENT

The authors thank RENATECH, the French national nanofabrication network, the J.E. Purkyn fellowship awarded to V. Mortet by the Czech Academy of Sciences and Centrale Initiative Foundation for their financials supports.