Chams Lahlou
email: chams.lahlou@emn.fr

Stéphane Dauzère-Pérès
email: stephane.dauzere-peres@emn.fr

Single machine scheduling with time window dependent processing times

Keywords: scheduling, one-machine, time dependent processing time, time windows, computational complexity

In the one-machine scheduling problems analysed in this paper, the processing time of a job depends on the time at which the job is started. More precisely, the horizon is divided into time windows and with each one a coecient is associated that is used to determine the actual processing time of a job starting in it. Two models are introduced, and one of them has direct connections with models considered in previous papers on scheduling problems with time dependent processing times. Various computational complexity results are presented for the makespan criterion, which show that the problem is NP-hard, even with 2 time windows. Solving procedures are also proposed for some special cases.

Introduction

One of the most classic problems in scheduling theory consists in scheduling a set J = {1, . . . , n} of independent jobs on a single machine. Preemption is not allowed (a job cannot be interrupted while being processed) and no more than one job can be processed at a time.

Processing times of the jobs are usually considered to be given and constant. However, 1 in practical settings, the processing time of a job may depend on the time at which an operation starts on a resource. Looking at the vast body of literature in scheduling, only a few papers have considered time dependent processing times. The rst papers on the subject are cited in Gawiejnowicz 1 . A very extensive survey can be found in Alidaee and Womer 2 , and a more recent survey is provided in Cheng et al. 3 .

From now on, we only consider problems with the makespan criterion, i.e for the minimisation of the schedule length. When processing times are linear, p j = x j + α j t j , where α j is the rate at which the processing time of the job increases (or decreases, if α j < 0). The problem can be solved in O(n log n) time. This result was proved independently by Tanaev et al. 4 , Gupta and Gupta 5 , Browne and Yechali 6 and Gawiejnowicz and Pandowska 7 .

In the case of piecewise linear processing times, three models have been studied. In the model proposed by Sundararaghavan and Kunnathur 8 , p j = x j if t j ≤ d, where d is a given date, and p j = x j + α j else. The problem has been shown to be binary NP-hard by Mosheiov 9 , whereas Alidaee and Womer 2 have shown that it can be solved in time O(n log n) if the x j 's are identical. In a more general model introduced by Mosheiov 9 , the processing times are subject to multi-step deteriorations, i.e each job j has several dates d 1,j < d 2,j < • • • < d w,j and coecients α 1,j < α 2,j < • • • < α w,j such that p j = x j + α i,j if d i-1,j < t j ≤ d i,j . In the third model, there are two given dates, d and D, and p j = x j if t j ≤ d, p j = x j + α j (t j -d) if d < t j < D, and p j = α j (D -d) otherwise. The problem has been shown to be binary NP-hard by Kononov 10 Since the problems studied in this paper have few connections with the nonlinear pro-2 cessing time case, the reader is referred to the paper of Alidae and Womer 2 .

In our problem, a sequence of dates (d 1 , . . . , d w+1 ) denes w time windows [d i , d i+1 [ where d i < d i+1 , d 1 = 0 and d w+1 = +∞. With each job j is associated a normal processing time x j and w coecients α ij , one for each time window. The actual processing time p j of a job j depends on the coecient of the time window in which it starts. For a job j starting at time t j , we dene the two following models:

• Model M + : p j = x j + α ij if d i ≤ t j < d i+1 . • Model M × : p j = α ij x j if d i ≤ t j < d i+1 ,
The objective is to nd a schedule that minimises the makespan C max = max j∈J C j , where C j = t j + p j is the completion time of job j.

In model M + , the coecient α ij models a waste of or a saving on the normal processing time of job j when it starts in time window [d i , d i+1 [: job j can take more time (i.e α ij > 0), less time (i.e α ij < 0) or the same time (i.e α ij = 0) than the normal processing time. This model has some connections with the two rst piecewise linear processing time models we have presented. First, it is a generalisation of the one proposed by Sundaraghavan and Kunnathur 8 , where p j = x j if t j ≤ d and p j = x j + α j otherwise. Indeed, to get their model we just have to consider two time windows such that α 1j = 0, α 2j = α j and d 2 = d + (with being strictly positive and smaller than the smallest processing time in a window). First, the case of M + where α i < α i+1 is equivalent to a special case of the model proposed by Mosheiov 9 where d i,j = d i , for all j. There are actually two dierences between Mosheiov's model and M + : (i) contrary to Mosheiov's model where only deteriorating processing times are considered, any kind of step function can be used in M + to model the processing times; (ii) in Mosheiov's model, each job has its own sequence of time windows whereas in our model the jobs share the same sequence of time windows.

The model M + can be useful to solve practical scheduling problems when the processing times depend on time periods. For instance, consider a workshop where the processing time of job j is x j when all the workers are present (e.g. there are k workers), and suppose there is a delay p - j (a saving p + j ) to process job j if the number of workers is smaller (larger) than k. The problem of scheduling the jobs in minimum time is modelled by associating with each day a time window, and with each pair (day i, job j) a coecient α ij according to the number of workers present in day i : we set α ij = p - j if the number of workers is smaller than k, α ij = -p + j if the number of workers is larger than k, and α ij = 0 otherwise.

Contrary to M + , the model M × has no direct link to previous models. However, it deserves to be studied since in some practical cases the processing time of a job is proportional to the availability of a resource. For instance, if in a network of processors the speed of a task depends linearly on the number of available processors, we set

p j = x j /n i , that is α ij = 1 n i
, where n i is the number of available processors in time period i. Moreover, as we shall see, there exist interesting dierences between M + and M * from a complexity point of view.

Finally, another application of M + and M × could be the approximation of non linear processing times (see the survey of Alidaee and Womer 2 for examples of this type of processing time), by using time windows as intervals for the discretisation of the non linear function.

In the next section, we present an optimal algorithm for both models when the sequence of jobs is given. Then, the M × model with two time windows is studied: It is shown that the problem is NP-hard in the general case and can be solved using a pseudo-polynomial time algorithm when the coecient of a time window is the same for each job, i.e α ij = α i ∀j.

Next, the M + model is analysed: The problem is shown to be polynomial when α ij = α i ∀j and the coecients are increasing or decreasing in the order of the time windows; complexity results for the remaining cases are also discussed. Finally, some conclusions and perspectives for future research are given.

Scheduling in M + and M × according to a given sequence

We introduce an algorithm which solves the problem in both models. The following dominance property is used.

Theorem 1 In models M + and M × , there exists an optimum schedule such that each job starts either at the beginning of a time window or at the completion time of the previously scheduled job.

Proof Suppose there is a job j scheduled just after a job k. If t j = C k the property is true, otherwise let i be the window in which j starts. If t j = d i , the property is true, else by setting t j = max{C k , d i }, the actual processing time of j does not change, neither in

M + nor in M × , since j still starts in the same time window [d i , d i+1 [. 2
Without loss of generality we suppose now that the sequence is (1, . . . , n). Let C(j, t) be the completion time of job j if it starts at time t (note that C j = C(j, t j )). The algorithm is the following:

Algorithm 1:

1. t 1 = d k with C(1, d k ) = min d i C(1, d i ) 2. for j = 2 to n t j = C j-1 if C(j, C j-1 ) ≤ min {d i :C j-1 ≤d i } C(j, d i ) d k if C(j, C j-1 ) > C(j, d k ) = min {d i :C j-1 ≤d i } C(j, d i )
Let t * j and C * j be the starting time and completion time of job j in an optimum schedule.

Theorem 2 In models M + and M × , when the sequence of jobs is given, there exists an optimum schedule such that C * j = C j for every job j.

Proof Consider the rst job. By Theorem 1, t * 1 = d i for some i. So it cannot be completed before time min The algorithm also solves the following special case, in both models.

d i C(1, d i ) = C 1 . Therefore if C * 1 > C
Corollary 1 When the sequence is not given, but the normal processing times are identical and the time window coecients do not depend on the jobs, the minimum makespan problem can be solved in models M + and M × in time O(nw)

Proof Since x j = x, for some x, and α ij = α i , we have p j = α i x in M × and p j = x + α i in M + : The actual processing time of a job does not depend on the job. Consequently, Algorithm 1 can be used with any sequence to get an optimum schedule. 2

Scheduling in M × with two time windows

NP-hardness

We rst prove that minimising the makespan is NP-hard if either α ij = α i or x j = 1, by reductions to the following problems: Proof With an instance of PARTITION is associated an instance of Π 1 as follows :

-J = {1, . . . , n} A pseudo-polynomial time algorithm for the α ij = α i case Lemma 1 If α 1 < α 2 , there exists an optimum schedule such that the last job to start in the rst time window has the largest normal processing time.

-d 1 = 0, d 2 = 2B and d 3 = +∞ -x j = b j for 1 ≤ j ≤ n -α 1j = 2 and α 2j = 1, for 1 ≤ j ≤ n -T = 3B
Proof Let m be a job with the largest normal processing time. Suppose we have an optimum schedule (without idle times since α 1 < α 2 ) such that the last job to start in the rst time window is not m but a job j. We have two cases to consider according to the time window in which m starts:

1. If m starts in the rst time window, the job can be interchanged with j since jobs starting in the same time window can be sequenced according to any order.

2. Otherwise, m starts in the second time window. As jobs starting in the same time window can be sequenced according to any order, m can be interchanged with the j is completed (remember that the schedule has no idle time). The completion time C m is equal to t j + α 1 x j + α 2 x m . If we interchange m and j, C j (the new completion time of j) is equal to t m + α 1 x m + α 2 x j = t j + α 1 x m + α 2 x j . Let us now compare C m and C j . We have C j -C m = (α 1 -α 2 )(x m -x j ). Since α 1 < α 2 and x m ≥ x j , we get C j ≤ C m and, as a result, the schedule remains optimum if the two jobs are interchanged. 2

Theorem 6 If there are two time windows and α ij = α i , the makespan minimisation problem, for model M × , can be solved in pseudo-polynomial time.

Proof We shall prove that the problem can be modeled as a knapsack problem (see Garey and Johnson 13 for example), which is a well-known problem that can be solved in pseudo- polynomial time with very ecient algorithms (for up to several thousands of items, see

Martello and Toth 14

). The knapsack problem with n items can be written as follows:

   max j a j y j j b j y j ≤ d y j ∈ {0, 1}, ∀j ∈ {1, . . . , n} where a j and b j are respectively the cost and the weight of item j, and d is the capacity of the knapsack. The binary variable y j is equal to 1 if item j is placed in the knapsack, and is equal to 0 otherwise. In our case, with each job j is associated a weight b j equal to α 1 x j , and a boolean variable y j such that y j = 1 if and only if job j starts in the rst time window. The values of a j and d are dened as follows.

1. If α 1 < α 2 , there is an optimum schedule without idle times and such that (by Lemma 1) a job with the largest normal processing time is the last one to start in the rst time window. Let m be that job. The makespan of the schedule veries C max = j =m α 1 x j y j + α 1 x m + j =m α 2 x j (1 -y j ). Moreover, jobs starting in the rst time window must complete before d 2 -1 since since m is the last job to start in the rst time window (recall that d 2 is the start time of the second time window).

Hence we must have j =m α 1 x j y j ≤ d 2 -1 . Therefore, the makespan minimisation problem can be modeled as:

   min C max = j =m α 1 x j y j + α 1 x m + j =m α 2 x j (1 -y j ) j =m α 1 x j y j ≤ d 2 -1 y j ∈ {0, 1}, ∀j ∈ J Minimising C max is equivalent to maximising j =m (α 2 -α 1 )
x j y j since α 1 x m + j =m α 2 x j is a constant. Hence, we have to solve a knapsack problem where a j = (α 2 -α 1 )x j and d = d 2 -1:

   max j =m (α 2 -α 1 )x j y j j =m α 1 x j y j ≤ a 2 -1 y j ∈ {0, 1}, ∀j ∈ J 2. If α 1 > α 2 ,
there may be an overlapping job in the optimum schedule. Hence, there are two cases:

• If no job overlaps the two windows, the makespan depend on the jobs that start in the second time window, that is C max = d 2 + j α 2 x j (1 -y j ). Moreover j α 1 x j y j ≤ d 2 since jobs starting in the rst time window cannot be completed after time d 2 (otherwise there would be an overlapping job). Since d 2 + j α 2 x j is a constant, the minimisation of the makespan is equivalent to the following knapsack problem where a j = α 2 x j and d = d 2 :

   max j α 2 x j y j j α 1 x j y j ≤ d 2 y j ∈ {0, 1}, ∀j ∈ J • If one job (say m) overlaps the two windows, the makespan depends on the starting time t m of the overlapping job m. It veries C max = t m + α 1 x m + j =m α 2 x j (1 -y j ) since job m is only followed by jobs that start in the second time window. Since job m starts in the rst time window, it is completed before time d 2 + α 2 x m : Otherwise, we would get a schedule with no overlapping job by starting m in the second time window, at time d 2 exactly. Hence, t m + α 1 x m ≤

d 2 + α 2 x m , that is t m ≤ d 2 + (α 2 -α 1 )
x m . Since the jobs that start in the rst time window are completed before time t m , j =m α 1 x j y j ≤ d 2 + (α 2 -α 1 )x m .

Finally, minimising the makespan is equivalent to maximising j =m α 2 x j y j because t m +α 1 x m + j =m α 2 x j is a constant, and we get the following knapsack problem where a j = α 2 x j and

d = d 2 + (α 2 -α 1 )x m :    max j =m α 2 x j y j j =m α 1 x j y j ≤ d 2 + (α 2 -α 1 )x m y j ∈ {0, 1}, ∀j ∈ J
Therefore, we get a solution to the problem by solving n + 1 knapsack problems.

Indeed, we just have to solve the case with no overlapping job, and n cases with an overlapping one (one problem for each possible overlapping job). Then, the optimum schedule is the best schedule among these n + 1 schedules. 2

Scheduling in M +

The α ij = α i and increasing (or decreasing) α i case Lemma 2 If α i < α i+1 , there exists an optimum schedule such that the jobs are scheduled according to the non-decreasing normal processing time order.

Proof Consider a schedule with two jobs j and j such that t j < t j and x j ≤ x j . Let α and α be the coecients associated with the time windows in which j and j start respectively. Let us denote by S the sequence of jobs scheduled during the time interval [C j , t j [, i.e after j completes and before j starts.

If j and j are interchanged, the new schedule is such that j starts at time t j = t j , followed by the jobs in S and then by job j, without idle times. To show that this interchange is always possible, we are going to prove that C j ≤ C j , where C j is the completion time of j in the new schedule.

First, C j = t j + p j + k∈S p k + p j = t j + x j + α + k∈S p k + x j + α , where α is the coecient of the time window in which j starts. Then, because there may be idle times in the former schedule, C j ≥ t j + x j + α + k∈S p k + x j + α . Hence, the following inequality holds:

C j -C j ≤ k∈S (p k -p k ) + α -α
Note that C j ≤ C j because x j ≤ x j . It follows that jobs in S start earlier in the new schedule and thus k∈S p k ≤ k∈S p k . Therefore, α ≤ α since job j also starts earlier in the new schedule which implies that C j ≤ C j . 2 Lemma 3 If α i > α i+1 , there exists an optimum schedule such that the jobs are scheduled according to the non-increasing normal processing time order.

Proof By using a similar interchange argument. 2

Theorem 7 If the model is M + and α ij = α i , the makespan minimisation problem can be solved in time:

• O(n log n), if α i < α i+1 , • O(n log n + nw), if α i > α i+1 .
Proof -If α i < α i+1 , there exists an optimum schedule without idle times (it is always possible to shift a job to the left). Thus, to get an optimum schedule, it is enough to know how to sequence the jobs, that is, by Lemma 2, to sort the jobs according to the time to sort n jobs, that is O(n log n).

If α i > α i+1 , there may be idle times in an optimum schedule. By Lemma 3 we know how to sequence the jobs. To get an optimum schedule, Algorithm 1 can be applied, which is dened for the case of a given sequence in Section . Sorting the jobs and applying the algorithm takes time O(n log n) and O(nw) respectively, that is time O(log n + nw). 2

Complexity of the case with a xed number of time windows

Let us rst consider the case with two time windows. Alidaee and Womer 2 have shown that the makespan minimisation problem, for the piecewise linear model where p j = x j if t j ≤ d and p j = x j + α j otherwise, can be solved in time O(n log n) if the x j 's are identical.

Mosheiov 9 proved that it is a binary NP-hard problem if we have arbitrary x j 's. We deduce from these two results that our problem (in which d 2 corresponds to d + , where can be chosen as small as we want) can be solved in polynomial time if α 1j = 0 and the x j 's are identical, but is NP-hard if α 1j = 0 and the x j 's are arbitrary. Moreover, a consequence of Theorem 7 is that nding an optimum schedule if α ij = α i is a polynomial time problem (with two time windows, either α

1 < α 2 or α 1 > α 2 ).
Therefore two open questions are (i) the complexity of the case where the α ij 's are arbitrary but the x j 's are identical, and (ii) the complexity of the case where α ij = α i and there are three time windows. In order to prove the NP-hardness of these two cases, the two following problems are considered: Theorem 8 Π 3 is NP-complete.

Proof The transformation is the following (see Figure 2 for an illustration):

-J = {1, . . . , n + 1}

-x j = b j , for 1 ≤ j ≤ n -x n+1 = n 2 B + 1 -d 1 = 0, d 2 = B + n 2 B + 1, d 3 = 2B + nB + 1, and d 4 = +∞ -T = 3B + 3 2 nB + 1 -α 1j = α 3j = B, and α 2j = T , for 1 ≤ j ≤ n + 1
The construction can be done in polynomial time. Now, suppose there is a yes answer for PARTITION. Jobs associated with S are rst scheduled, followed by job n + 1, The second model handles the case of jobs with processing times that are proportional to the availability of a ressource. Both models can also be used to approximate nonlinear

  , Cai et al. 11 and Kubiak and van de Velde 12 , even if D = ∞. For the case where D = ∞, a pseudopolynomial time algorithm that runs in O(nd x j ) and a branch-and-bound algorithm have been proposed by Kubiak and van de Velde 12 . In the same paper, two pseudopolynomial time algorithms are proposed for the case where D < ∞.

  polynomial transformations from the PARTITION problem (Garey and Johnson 13 ): INSTANCE: Integers b 1 , . . . , b n such that n i=1 b i = 2B. QUESTION: Is there a subset S ⊂ {1, . . . , n} such that i∈S b i = B? Theorem 4 Problem Π 1 is NP-complete.

Figure 1 1 and J 2 2 Theorem 5 2 j∈J 1 α

 1122521 Figure 1 illustrates the transformation. The construction can be done in polynomial

• Problem Π 3 :

 3 INSTANCE: A set of jobs J , a sequence of three time windows W = (d 1 , d 2 , d 3 , d 4 ), normal processing times x j , coecients α ij such that α ij = α i and an integer T . QUESTION: is there a schedule with makespan less than or equal to T ? • Problem Π 4 : INSTANCE: A set of jobs J , a sequence of two time windows W = (d 1 , d 2 , d 3 ), normal processing times x j = 1, coecients α ij and an integer T . QUESTION: is there a schedule with makespan less than or equal to T ? It can be checked in polynomial time if a schedule has a makespan less than or equal to T , so the two problems are in NP. The NP-completeness of Π 3 and Π 4 is proved by polynomial transformations from a subcase of the PARTITION problem (Garey and Johnson 13 ): INSTANCE: Integers b 1 , . . . , b n such that n i=1 b i = 2B. QUESTION: Is there a subset S ⊂ {1, . . . , n} such that i∈S b i = B and |S| = n 2 ?

2 KB + Kx + n 2 B

 22 then by the remaining jobs. Jobs associated with S are processed from time 0 to time j∈S (b j + B) = B + n 2 B, since there are exactly n 2 jobs in S. Hence, job n + 1 starts at time B + n 2 B = d 2 -1 and is completed at time d 2 -1 + x n+1 + α 1n+1 = d 3 . Finally, Suppose there is a yes answer for PARTITION. Jobs in S are rst scheduled from time 0 to time j∈S p j = j∈S (1 + K(b j + B) -1) = K( n 2 + 1)B = d 2 . The remaining jobs are then processed in the second time window and are completed by time d 2 + j∈{1,...,n}-S (1+ b j + B -1) = d 2 + ( n 2 + 1)B = T : The schedule denes a yes answer for PARTITION. Now, suppose we have a yes answer for Π 4 . First, note that no more than n 2 jobs can start in the second time window. Otherwise, as p j = b j + B and b j ≥ 1, these jobs would be processed during at least ( n 2 + 1)(1 + B) time units, which is not possible since the time window is of length ( n 2 + 1)B time units. Similarly, no more than n 2 jobs can start in the rst time window. Otherwise, as p j = b j + B and b j ≥ 1, these jobs would be processed during at least ( n 2 + 1)(K + KB) = K( n 2 + 1)B + ( n 2 + 1)K > T , since K > B. Hence n 2 jobs start in the rst time window and n 2 in the second. Now, let J 1 be the set of jobs starting before time d 2 , J 2 the set of jobs starting at or after time d 2 , x = j∈J 1 b j and y = j∈J 2 b j . First, note that x + y = 2B. Since j∈J 2 p j ≤ T -d 2 , we also have y ≤ B, and consequently x ≥ B. Finally, because j∈J 1 p j + j∈J 2 p j ≤ T , we have n + y ≤ T , i.e Kx + y ≤ KB + B, which implies x ≤ B. Therefore, x = y = B and the sets J 1 and J 2 dene a yes answer for PARTITION. 2 Conclusion This paper has introduced a new type of scheduling problems with time dependent processing times, where the time horizon is divided into time windows and the processing time of a job is associated with the time window in which the job starts. Two models are investigated. The rst one is related to two other models proposed in the literature.

Figure 2 :

 2 Figure 2: polynomial transformation for Theorem 8.

  1 , Job 1 is shifted to the left in the optimum schedule, i.e t * 1 = t 1 and thus C * 1 = C 1 . Now, suppose C * j = C j for 1 ≤ j ≤ k and k < n. By Theorem 1, which applies to both models, t * k+1 = C * k or t * k+1 = d i for some i such that d i ≥ C * k . Therefore job k + 1 cannot be completed before time min{C(k + 1, C * k ), min {d i :C * k ≤d i } C(k + 1, d i )} = C k+1 . Again, if C * k+1 > C k+1 , the job is shifted to the left in the optimum schedule by setting t * k+1 = t k+1 . So C * k+1 = C k+1 and, by induction, the proof is completed. 2 Theorem 3 When the sequence of jobs is given, the minimum makespan problem can be solved in models M + and M × in time O(nw) by Algorithm 1.

Proof -From Theorem 2, which applies to both models, it follows that C max = C * max .

Finally, because Step 2 of the algorithm is run n -1 times and there are O(w) comparisons each time, the total computation time is O(nw). 2

•

  Problem Π 1 : INSTANCE: A set of jobs J , a sequence of two time windows W = (d 1 , d 2 , d 3 ), normal processing times x j , coecients α ij such that α ij = α i , and an integer T . QUESTION: is there a schedule with makespan less than or equal to T ? • Problem Π 2 : INSTANCE: A set of jobs J , a sequence of two time windows W = (d 1 , d 2 , d 3 ), normal processing times x j = 1, coecients α ij , and an integer T .

QUESTION: is there a schedule with makespan less than or equal to T ? These problems are in NP since one can check in polynomial time if a schedule has a makespan less than or equal to T . We prove the NP-completeness of Π 1 and Π 2 by

Suppose now that at least n 2 + 1 jobs start before job n + 1. Since their normal processing times are greater than or equal to 1 (recall the transformation), the last scheduled one is completed at least at time

Similarly, there cannot be more than n 2 jobs starting after job n + 1. Therefore, there are exactly n 2 jobs processed before job n + 1 and n 2 jobs processed after. Let J 1 and J 2 be the set of jobs starting before and after job n + 1, respectively.

We know that

As a result, we get j∈J 1 x j = j∈J 2 x j = B, and the schedule denes a yes answer for PARTITION. 2

Theorem 9 Π 4 is NP-complete.

Proof The following polynomial time transformation is used:

-J = {1, . . . , n}

time dependent processing times by discretising the time horizon. The results presented in this paper are summarised in Table 1. Our current research aims at developing heuristic and exact procedures. In particular, two original integer programming models have been proposed, whose tight linear relaxations allow rather large instances to be solved using standard solvers.


N Despringre

Y Chemisky

F Meraghni
email: fodil.meraghni@ensam.eu

J Fitoussi

G Robert

FATIGUE DAMAGE IN SHORT GLASS FIBER REINFORCED PA66: MICROMECHANICAL MODELING AND MULTISCALE IDENTIFICATION APPROACH

Keywords: Polymer-matrix composite, Short glass fiber, Multi-scale modeling, Viscoelastic-Damage behavior

The paper presents a new micromechanical high cycle fatigue visco-damage model for short glass fiber reinforced thermoplastic composites, namely: PA66/GF30. This material, extensively used for automotive applications, has a specific microstructure which is induced by the injection process. The multi-scale developed approach is a modified Mori-Tanaka method that includes coated reinforcements and the evolution of micro-scale damage processes. The description of the damage processes is based on the experimental investigations of damage mechanisms previously performed by the authors and presented elsewhere [M.F. Arif et al. "In situ damage mechanisms investigation of PA66/GF30 composite: Effect of relative humidity." Composites Part B: Engineering, Volume 61: 55-65, 2014]. Damage chronologies have been proposed involving three different local degradation processes: fiber-matrix interface debonding/coating degradation, matrix microcracking and fiber breakage. Their occurrence strongly depends on the microstructure and the moisture content. The developed model integrates these damage kinetics and accounts for the complex matrix viscoelasticity and the reinforcement orientation distributions induced by the process. Each damage mechanism is introduced through an evolution law involving local stress fields computed at the microscale. The developed constitutive law at the representative volume element scale is implemented into the finite element code Abaqus using a User MATerial subroutine. The model identification is performed via reverse engineering, taking advantage of the multiscale experimental results: in-situ SEM tests as well as quantitative and qualitative µCT investigations. Experimental validation is achieved using high cycle strain controlled fatigue tests.

INTRODUCTION

Thermoplastic composites are especially used for their thermomechanical properties along with their high strength to weight ratio and their ability to be injection molded. These characteristics make them a good choice to replace metallic structural components in automotive industry. This study focuses on polyamide-66 reinforced by 30%wt short glass fiber (PA66/GF30). Its behavior is governed by several coupled phenomena: complex damage mechanisms, matrix viscosity influenced by environmental conditions such as temperature and relative humidity, applied loading and the microstructure induced by the manufacturing process. Design of structural part, which includes fatigue design, thus needs advanced modeling tools that account for these coupled effects Damage mechanisms have been previously studied by Sato [START_REF] Sato | Microfailure behaviour of randomly dispersed short fibre reinforced thermoplastic composites obtained by direct SEM observation[END_REF], Horst, [START_REF] Horst | Fatigue fracture mechanisms and fractography of short glass fibre reinforced polyamide 6[END_REF], Arif et al. [START_REF] Arif | Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66[END_REF][START_REF] Arif | In situ damage mechanisms investigation of PA66/GF30 composite: Effect of relative humidity[END_REF] and Cosmi and Bernasconi [START_REF] Cosmi | Micro-CT investigation on fatigue damage evolution in short fibre reinforced polymers[END_REF], using in-situ Scanning Electron Microscopy (SEM) tests as well as X-ray micro-computed tomography observations. Damage scenarii have thus been established, according to relative humidity (RH). Four damage mechanisms arise during fatigue life involving low frequencies (< 5Hz): fiber breakage, matrix micro-cracking, interfacial debonding fiber/matrix and matrix deformation band [START_REF] Fitoussi | Effect of the matrix behavior on the damage of ethylene-propylene glass fiber reinforced composite subjected to high strain rate tension[END_REF]. Interface degradation and related micro-cracks are proven to drive the whole fatigue damage evolution.

Phenomenological approaches have been frequently used to reproduce the composite cyclic behavior. A constitutive model has been recently published by Launay et al. [START_REF] Launay | Multiaxial fatigue models for short glass fiber reinforced polyamide -Part I: Nonlinear anisotropic constitutive behavior for cyclic response[END_REF][START_REF] Launay | Multiaxial fatigue models for short glass fibre reinforced polyamide. Part II: Fatigue life estimation[END_REF] to handle the matrix elasto-viscoplastic aspect but it lacks damage consideration. Moreover, Nouri's et al. model [START_REF] Nouri | Fatigue damage model for injection-molded short glass fibre reinforced thermoplastics[END_REF] accounts for the anisotropic damage evolution but does not handle the specific composite microstructure. Micromechanical models have also been developed to take into account inclusion shape and orientation. Doghri et al. [START_REF] Doghri | Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method[END_REF][START_REF] Kammoun | First pseudo-grain failure model for inelastic composites with misaligned short fibers[END_REF] indeed worked on a multi-scale approach based on a mean field homogenization for an elasto-viscoplastic matrix reinforced by elastic fiber.

As far as PA66/GF30 is concerned, damage mechanisms deeply depend on microstructure and environmental conditions [START_REF] Arif | In situ damage mechanisms investigation of PA66/GF30 composite: Effect of relative humidity[END_REF]. They thus need to be included in a multi-scale model to be linked to microscale fields. A new micromechanical model is here developed to handle the coupled phenomena, based on a incremental Mori-Tanaka approach [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF]. It especially accounts for the matrix viscoelastic rheology. Each phase has its own constitutive equations which can be influenced by local damage evolution laws. The effective constitutive law is then implemented in a Finite Element (FE) code (Abaqus) via User MATerial subroutine.

Damage at interface plays a crucial role within composite fatigue life [START_REF] Sato | Microfailure behaviour of randomly dispersed short fibre reinforced thermoplastic composites obtained by direct SEM observation[END_REF][START_REF] Horst | Fatigue fracture mechanisms and fractography of short glass fibre reinforced polyamide 6[END_REF][START_REF] Arif | Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66[END_REF][START_REF] Arif | In situ damage mechanisms investigation of PA66/GF30 composite: Effect of relative humidity[END_REF][START_REF] Cosmi | Micro-CT investigation on fatigue damage evolution in short fibre reinforced polymers[END_REF]. To model its evolution, a coating is set around fibers. When damage occurs, the coating mechanical properties as well as the load transfer between coating and fiber both decrease. The load transfer ratio is computed from Shear Lag Model (SLM) equations, initially developed by Cox [START_REF] Cox | The elasticity and strength of paper on other fibrous materials[END_REF].

Parameter identification is done by means of inverse methods, using the multiscale data from matrix and composite tensile tests as well as in-situ SEM and X-ray micro-computed tomography (µCT) analyses.

In this paper, the second section presents the developed multiscale model through its homogenization method and coating description. The third one focuses on damage mechanisms and load transfer. The final part introduces the identification strategy and some of its results.

MULTISCALE MODELING

Homogenization

The injection molding process induced a non-homogeneous microstructure in PA66/GF30 samples. Fiber orientation are here taken into account using N orientation families, defined by two Euler angles. The Representative Volume Element (RVE) thus regroups the matrix, voids from micro-cracks, N oriented fiber and their N coating, both taken as ellipsoids. The fiber breakage mechanism will create two new oriented phases: the broken fiber and the related coating. The volume fraction of each orientation is computed thanks to a Simpson numerical integration of experimental Orientation Density Functions (ODF). The effective macroscopic behavior is computed using the Mori-Tanaka homogenization [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF] (1). r is a phase, c r its volume fraction, L r its stiffness tensor and A r its strain localization tensor according the chosen micromechanical method, here Mori-Tanaka. This approach has already been widely used to get the effective behavior of a composite material [START_REF] Launay | Multiaxial fatigue models for short glass fibre reinforced polyamide. Part II: Fatigue life estimation[END_REF][START_REF] Desrumaux | Generalized Mori-Tanaka Scheme to Model Anisotropic Damage Using Numerical Eshelby Tensor[END_REF][START_REF] Jendli | Multi-scales modelling of dynamic behaviour for discontinuous fibre SMC composites[END_REF][START_REF] Lagoudas | Elastoplastic behavior of metal matrix composites based on incremental plasticity and the Mori-Tanaka averaging scheme[END_REF]. The Eshelby tensor is numerically computed based on Lagoudas and Gavazzi's method [START_REF] Gavazzi | On the numerical evaluation of Eshelby's tensor and its application to elastoplastic fibrous composites[END_REF], which accounts for the anisotropy due to a non-linear tangent modulus of the matrix. A r are also adapted to coated reinforcements thanks to Cherkaoui et al. [START_REF] Cherkaoui | Elastic composites with coated reinforcements: A micromechanical approach for nonhomothetic topology[END_REF]. This approach is validated by the analytic equations derived by Hervé and Zaoui [START_REF] Hervé | n-Layered inclusion-based micromechanical modelling[END_REF]. Homogenization is finally suited for the non-linear matrix rheology, the composite microstructure and the damage mechanisms. The coating is at the heart of the damage at interface and its role is explained in sections 3.3 to 3.5. In order to couple this phenomena, the loading path is divided into small time increments, thus defining an incremental Mori-Tanaka scheme.

(1)

Phase constitutive laws

Voids follow linear isotropic elasticity, with a null Young's modulus and a Poisson's coefficient equal to 0.5. Fibers follow the same law and are defined by their two Euler angles, their aspect ratio, their Young's modulus and their Poisson's coefficient.

DMTA analyses in both tensile and shear loadings reveal that the PA66 is viscoelastic. The Cole-Cole diagram of its tensile modulus (Fig. 1) shows three phase transitions. The first peak, corresponding to the glass transition, is the lone one of practical interest, due to its range of time and temperature. Inverse methods are applied to find the most adapted viscoelastic model for PA66. A comparative study is done between different models: Zener and its generalized version, Maxwell and its, and other methods accounting for peak asymmetry as the one of Havriliak and Negami [START_REF] Havriliak | A complex plane representation of dielectric and mechanical relaxation processes in some polymers[END_REF]. Generalized Zener ends up to be the best model to fit the experimental data and to capture the matrix viscoelasticity. A simple Zener model would thus be enough if one neglects the dissipation aspect or if the related parameters depends on the environmental conditions (temperature and RH).

DAMAGE MECHANISMS

Previous work on damage mechanisms [START_REF] Sato | Microfailure behaviour of randomly dispersed short fibre reinforced thermoplastic composites obtained by direct SEM observation[END_REF][START_REF] Horst | Fatigue fracture mechanisms and fractography of short glass fibre reinforced polyamide 6[END_REF][START_REF] Arif | Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66[END_REF][START_REF] Arif | In situ damage mechanisms investigation of PA66/GF30 composite: Effect of relative humidity[END_REF][START_REF] Cosmi | Micro-CT investigation on fatigue damage evolution in short fibre reinforced polymers[END_REF][START_REF] Fitoussi | Effect of the matrix behavior on the damage of ethylene-propylene glass fiber reinforced composite subjected to high strain rate tension[END_REF] lead to the consideration of three damage mechanisms within our modeling philosophy: fiber breakage, matrix microcracking and interfacial debonding (fiber/matrix). These processes all occur at microscale but there is still a scale difference between them. Matrix micro-cracking and fiber breakage occur at fiber level, while coating degradation (interfacial debonding) takes place at a relatively smaller scale. This justifies applying continuum damage theory to the mechanism associated with the coating [START_REF] Chaboche | Continuous damage mechanics: A tool to describe phenomena before crack initiation[END_REF][START_REF] Krajcinovic | Continuum damage mechanics[END_REF]. The three damage mechanisms are separately implemented.

Fiber breakage

When a fiber breaks, the related volume fraction (as well as the one of its coating) decreases in favor of the broken fiber phase with the same orientation (and respectively its coating). Fiber breakage was modeled with a Weibull-like law which computes the damage rate at each time step. This phenomenological approach is based on the work of Meraghni and Desrumaux [START_REF] Meraghni | Implementation of a constitutive micromechanical model for damage analysis in glass mat reinforced composite structures[END_REF]. This damage mechanisms is experimentally proven to occur for low moisture content and at start of fatigue life. The hypothesis is here that a few specific fiber setup will induce local stresses high enough to break some fibers. Such specific configuration are then supposed to be rarer as the loading goes on. The modified Weibull law accounts for this effect (2). (m) is a shape coefficient,  0 a stress threshold,  a time constant,  et  the parameters linked to damage saturation.  eq is the inclusion Von Mises stress.

Matrix micro-cracking

Matrix micro-cracks mainly come from two sources. On one hand, the stress concentration of local fiber setup will induce crack initiation and propagation, especially if there is already a fiber breakage. On the other hand, the damage accumulation inside the composite leads the loading to be more and more transferred to the matrix, until material rupture. The first effect is modeled in a similar way than fiber breakage. The second one is expressed as a damage dependant exponential phenomenon [START_REF] Arif | Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66[END_REF]. When the matrix damage parameter rises, the matrix volume fraction decreases in favor of void one. The void aspect ratio is set to 1 (spherical), but further analyses would lead the void shape to be orthogonal to the main loading direction.

(3) (m) is a shape coefficient,  0 a stress threshold,  a time constant,  et  the parameters linked to damage saturation.  eq is the matrix Von Mises stress. () and () are the parameters linked to void coalescence and its exponential propagation near the end of fatigue life.

Damage at interface

Interfacial debonding, mainly initiating at fiber ends, plays a crucial role in material life. This damage mechanism occurs at relatively small stresses and continually modifies the composite behavior. The chosen approach is based on Jendli's et al. work [START_REF] Jendli | Multi-scales modelling of dynamic behaviour for discontinuous fibre SMC composites[END_REF]. The developed damage law derived from a quadratic rupture criterion of the interface (4). The latter depends on shear  and normal  N interface stresses, which can be obtained by partially computing the stress jump at interface, as done by Meraghni et al. [START_REF] Meraghni | Effect of interfacial decohesion on stiffness reduction in a random discontinuous-fibre composite containing matrix microcracks[END_REF]. These two stress components can then be computed at each ellipsoid point.  rupture and  rupture are respectively the maximum normal et tangential stresses prior to debonding.

(4)

To be accounted for the micromechanical approach, this criterion has to be expressed in a probabilistic way. This allows the damage at interface to be computed with any stress tensor. This is ensured by a normal distribution N (5) which adds two parameters : a shape one µ and a damage threshold (between 0 and 100%). The latter adjusts the mean and the standard deviation sd of the normal distribution in order to get 99% damage within the rupture criterion [START_REF] Fitoussi | Effect of the matrix behavior on the damage of ethylene-propylene glass fiber reinforced composite subjected to high strain rate tension[END_REF]. sd 0 is the standard deviation when there is 99% damage on the rupture criterion for  equal to 0. The incremental model requires a consistent damage rate to be computed, and a normalization is thus done with a time constant  and the damage d from the previous time step. ( A last normalization has to be made due to the normal density function having strictly positive value on ]-∞; +∞[. The interesting values of the rupture criterion are between 0 and 1. Interval [1; +∞[ is inherently taken into account, in contrary to ]-∞; 0] which is then removed [START_REF] Launay | Multiaxial fatigue models for short glass fiber reinforced polyamide -Part I: Nonlinear anisotropic constitutive behavior for cyclic response[END_REF]. [START_REF] Launay | Multiaxial fatigue models for short glass fiber reinforced polyamide -Part I: Nonlinear anisotropic constitutive behavior for cyclic response[END_REF] In literature, the damage at interface is often only computed along one fiber equator [START_REF] Meraghni | Effect of interfacial decohesion on stiffness reduction in a random discontinuous-fibre composite containing matrix microcracks[END_REF]. Figure 2 here shows that this limits the ability to find the maximum damage value. The ellipsoid has thus to be meshed and the damage parameter computed on each point of this mesh. This is also valid in pure tensile setup: the shear stress at fiber extremities can shift the point of maximum damage.

In a continuum damage mechanics framework, this law will reduce the mechanical properties of the related coating. Its tangent modulus will then proportionally decreases with the damage parameter.

Influence on transfer load

Reducing the mechanical properties of the coating is not enough to capture the influence of damage at interface on load transfer. A specific low has thus to implemented and is here inspired from the Shear Lag Model (SLM). A brief reminder of the method is here done. Further details about the equations and hypothesises of the SLM can be found in Nairn's paper [START_REF] Nairn | On the use of shear-lag methods for analysis of stress transfer in unidirectional composites[END_REF].

The fiber is here assumed cylindrical. The SLM ends up with a differential equation ( 8) involving the stress in a fiber section according to its abscissa z (axis origin is set at centre of the fiber of length l).  SL is the shear lag parameter as defined by McCartney [START_REF] Mccartney | Stress Transfer for Multiple Perfectly Bonded Concentric Cylinder Models of Unidirectional Composites[END_REF] and Nayfeh [START_REF] Nayfeh | Thermodynamically induced interfacial stresses in fibrous composites[END_REF]. It only depends on material properties, which constitutes one of the strength of this approach.  f ∞ is the theoretical stress in a fiber section if the fiber is infinitely long. The differential equation is then integrated, assuming no stress at fiber extremities [START_REF] Nouri | Fatigue damage model for injection-molded short glass fibre reinforced thermoplastics[END_REF]. This is equivalent to debonding at fiber ends and represents the most common case of damage initiation at interface [START_REF] Sato | Microfailure behaviour of randomly dispersed short fibre reinforced thermoplastic composites obtained by direct SEM observation[END_REF][START_REF] Horst | Fatigue fracture mechanisms and fractography of short glass fibre reinforced polyamide 6[END_REF][START_REF] Arif | Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66[END_REF][START_REF] Arif | In situ damage mechanisms investigation of PA66/GF30 composite: Effect of relative humidity[END_REF][START_REF] Cosmi | Micro-CT investigation on fatigue damage evolution in short fibre reinforced polymers[END_REF]. This equation in fact defines the effective load transfer in each fiber section. A new integration on half the fiber is then done to get the average load transfer Ψ  between the fiber and the coating [START_REF] Doghri | Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method[END_REF]. Damage can now be added to the law through the following hypothesis: it decreases the effective fiber length involved in the real transfer load . This length is indeed reduced by the damage percentage [START_REF] Kammoun | First pseudo-grain failure model for inelastic composites with misaligned short fibers[END_REF]. A normalization is finally required to ensure a load transfer equal to 100% at virgin state. This law is shown in figure 3. At the loading beginning, the load transfer does not decrease a lot with damage. When a critical value is reached, it decreases faster until interface rupture. N. Despringre, Y. Chemisky, F. Meraghni, J. Fitoussi, G. Robert [START_REF] Doghri | Mean-field homogenization of elasto-viscoplastic composites based on a general incrementally affine linearization method[END_REF] (11) Figure 3: Load transfer between fiber and coating according to damage at interface

Effective behavior

The implementation of the previous mechanisms follows the literature philosophy [START_REF] Sato | Microfailure behaviour of randomly dispersed short fibre reinforced thermoplastic composites obtained by direct SEM observation[END_REF][START_REF] Horst | Fatigue fracture mechanisms and fractography of short glass fibre reinforced polyamide 6[END_REF][START_REF] Arif | Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66[END_REF][START_REF] Arif | In situ damage mechanisms investigation of PA66/GF30 composite: Effect of relative humidity[END_REF][START_REF] Cosmi | Micro-CT investigation on fatigue damage evolution in short fibre reinforced polymers[END_REF]. What's more is that they are each implemented with an independent law, and are thus competing with each other. Damage scenario is then only driven by the evolution law parameters, despite being designed from experimental observations. At virgin state of the composite, most of the loading is transferred to fibers. Stress at interface raises, leading to a gradual degradation of the coating; especially for the orientation families parallel to the loading direction. The coating tangent modulus proportionally decreases with damage at interface. The effective behavior is not impacted as long as the load transfer between fiber and coating is not reduced too much by the relative law [START_REF] Kammoun | First pseudo-grain failure model for inelastic composites with misaligned short fibers[END_REF]. When the latter is compromised, the matrix is under heavier loading and thus starts to crack. Its volume fraction decreases in favor of the void one. In the meantime, local and specific fiber setups are leading to stress concentration which eventually leads to some fibre breakages and matrix micro-cracks. The fiber volume fraction (and the one of the related coating) thus decreases in favor of the one of broken fiber phase (and the one of their coating), with respect to orientation families. The matrix is finally damaged enough to make the voids coalesce and propagate in a brittle way. The mesoscale crack rapidly grows and leads to material ruin. The nonlinear visco-damage behavior of the composite does not allow for a direct access to a macroscopic damage parameter. The effective modulus of the RVE can however be written based on equation 1 [START_REF] Mori | Average stress in matrix and average elastic energy of materials with misfitting inclusions[END_REF]. All the properties but damage parameters are the initial ones. Strain localization tensors A are obtained with the Mori-Tanaka equations adapted to coated reinforcements. Indices 0, v, f, c, bf, et bc respectively stand for matrix, voids, fibers, coatings, broken fibers and their coatings. Θ is an orientation family and L a tangent modulus (L v is thus a null tensor but is still written for sake of understanding).

IDENTIFICATION STRATEGY

The developed micromechanical model involves a lot of material parameters, ranging from composite and phase properties to damage law coefficients. A specific identification strategy is thus required to uncouple these parameters as far as possible. Their interdependency indeed prevents a rigorous identification from macroscopic data only. Firstly, the microstructure properties can be determined with the µCT and the in-situ SEM test. Moreover, the matrix rheology is the lone phase with a behavior complex enough to require a dedicated identification. Last but not least, damage parameters are assumed identical in both quasi-static and fatigue [START_REF] Sato | Microfailure behaviour of randomly dispersed short fibre reinforced thermoplastic composites obtained by direct SEM observation[END_REF][START_REF] Arif | Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66[END_REF]. Their identification is first approximately done with experimental observations and then more accurately with specific optimization tools. Reverse engineering algorithm here combines a genetic approach with a Levenberg-Marquardt method [START_REF] Meraghni | Parameter identification of a thermodynamic model for superelastic shape memory alloys using analytical calculation of the sensitivity matrix[END_REF].

Microstructure identification

The most important aspect of the composite microstructure is the fiber orientation distribution within the material. One can here make the most of µCT analysis. Each element is differentiated by its aspect ratio and its 3D volume. If it is a fiber, the 3D volume is added to the fiber volume fraction of the relative orientation family, as µCT gives oriented elements. ODF can thus be computed in every composite layer or within the whole sample. Figure 4 shows experimental data obtained on half the thickness of two samples of injection molded PA66/GF30. Optimization algorithm modeled the two curves with a chosen analytical function. A Pearson VII function with two peaks here fits the two datasets. The resulting ODF can be send as input in the micromechanical model, dividing the fiber/coating phases into different orientations with appropriate volume fractions. Furthermore, coating characteristics are determined thanks to in-situ SEM tests. The damage area around fibers is estimated during several loading phases and an average coating semi-axe is found. The fiber and its coating are not homothetic but their border are assumed equidistant. The coating semi-axes are then the fiber one plus a constant, given by the previous experimental observations. The coating aspect ratio can then be defined. Its volume fraction is consequently set by the one of fibers and by their own aspect ratio.

Matrix properties

Dissipative aspect is here neglected. According to section 2.2, the polyamide-66 can then be approximated by a Zener model, here in 3D, whose equations are inspired from the ones of Richter [START_REF] Richter | Upsetting and Viscoelasticity of Vitreous SiO2: Experiments, Interpretation and Simulation[END_REF]. The identification of this isotropic rheological model is done with stress/strain curves from both quasi-static and cyclic tests. The optimization algorithm gives consistent results whatever the applied frequency or the hydrothermal conditions. Figure 5 shows the result identified on both a quasi-static and a cyclic test at 1 Hz frequency with a loading ratio equal to 0.5. The obtained curves perfectly fit the data but the identified parameters depend on test conditions.

Damage mechanisms identification strategy

Identification of the different damage mechanisms is on-going and will be presented during the conference. Fiber breakage parameters can be estimated with µCT analyses. The number of broken fibers is the number of new fiber elements as recognized by Aviso, as the loading goes along. An appropriate thresholding has to be applied to distinguish matrix, fibers and voids; while knowing the stress state at observation areas. As far as matrix micro-cracking is concerned, it should not be a governing process until late fatigue life, where there is damage accumulation and interface saturation. Coating degradation can thus be identified with macroscopic data while considering the first part of material life. The second one will allow to calibrate the RVE meso-crack evolution. The parameters of those two damage mechanisms can finally be refined together.

CONCLUSIONS

A incremental micromechanical model with Mori-Tanaka homogenization has been developed to compute the fatigue behavior of polyamide-66 reinforced by 30%wt short glass fibers. The model accounts for the composite microstructure, the matrix viscoelasticity and the damage evolution. The damage mechanisms, previously identified by many authors [START_REF] Sato | Microfailure behaviour of randomly dispersed short fibre reinforced thermoplastic composites obtained by direct SEM observation[END_REF][START_REF] Horst | Fatigue fracture mechanisms and fractography of short glass fibre reinforced polyamide 6[END_REF][START_REF] Arif | Multiscale fatigue damage characterization in short glass fiber reinforced polyamide-66[END_REF][START_REF] Arif | In situ damage mechanisms investigation of PA66/GF30 composite: Effect of relative humidity[END_REF][START_REF] Cosmi | Micro-CT investigation on fatigue damage evolution in short fibre reinforced polymers[END_REF], have been implemented through evolution laws depending on microscale variable such as local stress. A specific law, inspired by the shear lag model, has been derived to take into account the load transfer evolution according to the damage at interface. The identification strategy takes advantage of the multiscale experimental data, using reverse engineering methods and allowing to uncouple the different phenomena. Its validation is on-going and will be presented during the conference. b.

Figure 1 :

 1 Figure 1: Normalized Cole-Cole diagram of the PA66 tensile modulus. Two dataset from DMTA are fitted with the same generalized Zener model.

  , Y. Chemisky, F. Meraghni, J. Fitoussi, G. Robert[START_REF] Horst | Fatigue fracture mechanisms and fractography of short glass fibre reinforced polyamide 6[END_REF]

Figure 2 :

 2 Figure 2: Ellipsoidal fiber under a. traction b. a random loading. µ = ½ and  = 0

∞

Figure 4 :

 4 Figure 4: ODF identification for two PA66/GF30 datasets obtained by µCT.

  , Y. Chemisky, F. Meraghni, J. Fitoussi, G. Robert

Figure 5 :

 5 Figure 5: 3D Zener parameter identification based on both a. an strain controlled oligo-cyclic test and b. a quasi-static test to get the other component of the isotropic stiffness tensor