Sonia Cafieri
email: sonia.cafieri@enac.fr

Pierre Hansen
email: pierre.hansen@gerad.ca

Nenad Mladenović
email: nenad.mladenovic@brunel.ac.uk

Edge-Ratio Network Clustering by Variable Neighborhood Search

The analysis of networks and in particular the identification of communities, or clusters, is a topic of active research with application arising in many domains. Several models were proposed for its solution. In [Cafieri et al., Phys. Rev. E 81(2):026105, 2010], a criterion is proposed for a graph bipartition to be optimal: one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges (edge ratio), and it is used in a hierarchical divisive algorithm for community identification in networks. In this paper, we develop a VNS-based heuristic for hierarchical divisive edge ratio network clustering. A k-neighborhood is defined as move of k entities, i.e., k entities change their membership from one to another cluster. A local search is based on 1-changes and k-changes are used for shaking the incumbent solution. Computational results on datasets from the literature validate the proposed approach.

Résumé : L'analyse de réseaux et en particulier l'identification de communautés, ou classes, est un sujet de recherche très actif dont les applications sont nombreuses dans de multiples domaines. Plusieurs modèles ont été proposés pour sa résolution. Dans [Cafieri et al., Phys. Rev. E 81(2):026105, 2010], on propose un critère pour que la bipartition d'un graphe soit optimale : on cherche à maximiser le minimum pour les deux classes de la bipartition du rapport du nombre d'arêtes internes au nombre d'arêtes coupées (edgeratio). Ce critère est utilisé dans un algorithme hiérarchique divisif pour l'identification de communautés dans les réseaux. Dans le présent article, nous développons une heuristique pour la classification hiérarchique descendante basée sur la métaheuristique de recherche à voisinage variable. Un k-voisinage est défini comme le mouvement de k entités, c'est-à-dire que k entités changent leur appartenance d'une classe à l'autre. Une recherche locale est basée sur les 1-échanges et les k-échanges sont utilisés pour perturber la meilleure solution connue. Les résultats de calcul sur des données de la littérature valident l'approche proposée.

Introduction

Complex systems in a variety of domains are represented by networks. The most prominent examples include social networks, describing individuals and their interactions and relationships, telecommunication networks, transportation networks, biological networks, and many more. A modular structure characterizes many complex systems, which contain subgroups of entities sharing some common properties. A topic of particular interest in the study of complex networks is therefore the identification of modules, also called clusters or communities. This is very useful to identify some properties of the system described by the studied network starting from its structural features. The reader is referred to Newman [START_REF] Newman | Networks: An Introduction[END_REF] for an introduction to networks and to Fortunato [START_REF] Fortunato | [END_REF] for a recent and deep survey of network clustering.

Let us consider a graph G = (V, E), with V the set of vertices and E the set of edges, used to represent a network. Several models and clustering criteria have been proposed. One often maximizes or minimizes a criterion function. The most used is modularity [3], based on the idea of comparing the fraction of edges falling within communities to the expected fraction of such edges. An alternative approach to the maximization of a criterion function is based on the satisfaction of a priori conditions to have a community. Radicchi et al. [4] proposed two such conditions defining communities in a strong and a weak sense, respectively. Recall that the degree k i of a vertex i belonging to V is the number of its neighbors (or adjacent vertices). Let S ⊆ V be a subset of vertices. Then the degree k i can be separated into two components k in i (S) and k out i (S), i.e., the number of neighbors of i inside S and the number of neighbors of i outside S. A set of vertices S forms a community in the strong sense if and only if every one of its vertices has more neighbors within the community than outside:

k in i (S) > k out i (S), ∀i ∈ S.
A set of vertices S forms a community in the weak sense if and only if the sum of all degrees within S is larger than the sum of all degrees joining S to the rest of the network:

i∈S k in i (S) > i∈S k out i (S).
This is equivalent to the condition that the number of edges within S is at least half the number of edges in the cut of S.

The weak condition was used [4] as a local stopping criterion in hierarchical clustering; also, it led Wang et al. [5] to define a community S indivisible if there is no bipartition, (S 1 , S 2 ) of S, such that both S 1 and S 2 satisfy the weak condition.

These concepts inspired the definition of the edge-ratio criterion in [6]. More precisely, in [6] the definition of community in the weak sense is extended into a criterion for a bipartition to be optimal: one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges. Specifically, the ratio of the number of edges within a community to the number of cut edges which have one end point only within that community is considered:

r(S) = i∈S k in i (S)/ i∈S k out i (S).
When dividing S in S 1 and S 2 , this ratio for both communities S 1 and S 2 is considered and the smallest value is maximized:

f (S 1 , S 2 ) = max S1,S2⊂V
min (r(S 1 ), r(S 2 )) .

Solving sequentially this problem yields a hierarchical divisive clustering algorithm, with a well defined criterion for bipartitioning. To build the divisive algorithm, the weak definition is first strengthen by quantifying how much the number of inner edges is larger than the number of cut edges. This is done by introducing a parameter α in the weak condition:

i∈S k in i (S) ≥ α i∈S k out i (S). ( 2
)
The coefficient α, equal to the ratio of twice the number of edges within the community S divided by the number of edges within the cut of that community ((2) in case of equality), is called the edge ratio [6].

So, a divisive algorithm can be built performing each bipartition in such a way to seek the maximum value of α for which the current sub-network S is divisible (i.e., twice the ratio of the number of edges within S divided by the number of edges within the cut of S). Bipartitioning is the crucial point in a hierarchical divisive algorithm and usually gives rise to the most computationally demanding steps of the algorithm. The problem of maximizing the edge-ratio for bipartitioning is NP-hard [7]. This motivates studying heuristic methods as an alternative to exact methods, which combine computational efficiency with the ability to provide good quality results.

In this paper we propose an efficient Variable Neighborhood Search based heuristic for divisive hierarchical clustering with edge-ratio criterion. The paper is organized as follows. In Section 2 we present the Variable Neighborhood Search heuristic for the considered optimization problem. In Section 3 we validate our approach on a set of datasets from the literature and compare the obtained results with an exact algorithm for bipartition previously proposed. Section 4 concludes the paper.

2 Variable Neighbourhood Search for hierarchical divisive edge-ratio clustering

Variable Neighbourhood Search

Variable Neighbourhood Search (VNS) is a metaheuristic (i.e., a framework for building heuristics), aimed at solving combinatorial and global optimization problems. It relies on iteratively exploring neighbourhoods of the current (local) optimum to identify better local optima, the main idea consisting in a systematic change of neighbourhood combined with a local search [8,9,[START_REF] Hansen | Variable neighbourhood search[END_REF][START_REF] Hansen | Variable neighbourhood search[END_REF][START_REF] Hansen | [END_REF]. More precisely, VNS iteratively (until a given termination condition is satisfied) escapes from the current local optimum x * by initiating other local searches from starting points sampled from a neighbourhood of x * which increases its size iteratively until a local minimum better than the current one is found.

Thus, the main ingredients of the VNS metaheuristic are:

-Definition of a neighbourhood of the current solution; -Neighbourhood changes; -Local search; -Shaking, i.e., a procedure to perturb the current solution; -Update of the current solution.

Divisive hierarchical clustering

Hierarchical divisive heuristics (see, e.g. [START_REF] Cafieri | [END_REF]) proceed from an initial partition containing all the n vertices of the graph and iteratively divide a community S into two (bipartitioning step) in such a way to improve as much as possible the objective function value. Bipartitions are iterated until a partition into n communities having each a single entity is obtained; in practice, bipartitions can be ended once they do not improve the objective function value anymore. Let ℓ be in {1, . . . , n} and S ℓ the community selected for bipartition at each iteration of the algorithm (at the first iteration, S ℓ = V ). The bipartition of S ℓ gives two new communities, S 1 and S 2 . We let S ℓ = S 1 and S ℓ+1 = S 2 , then ℓ is increased by one and all process repeated. In the present study, the objective function is based on the edge-ratio criterion (1) and bipartitioning is done by Variable Neighbourhood Search. A sketch of the divisive algorithm is given in Algorithm 1, where threshold is used to decide if the two subgraphs obtained by a bipartition have to be kept (i.e., the current community to be bipartitioned is divisible).

In the remainder of this section, we focus on the step 4 of Algorithm 1 and we present a VNS algorithm to perform community splitting.

a partition P of V 1 ℓ = 1, S ℓ = {v 1 , . . . ,

VNS for edge-ratio

Let us introduce binary variables x (a vector with components x i ) to denote to which of the two sets, obtained by the bipartition done at each hierarchy level, vertex v i belongs for all i. By convention, x i = 1 if v i belongs to S 1 and x i = 0 otherwise. To perform community splitting (step 4 of Algorithm 1), we solve an optimization problem with decision variables x and objective function defined in [START_REF] Newman | Networks: An Introduction[END_REF].

Neighborhoods. To build a VNS heuristic for this optimization problem, we first have to define a neighborhood N (x) of x. In discrete optimization problems with binary variables, neighborhoods of x are obtained from x by some simple modification, for instance complementing one of its components x i , or complementing two complementary components (i.e., setting one component from 1 to 0 and another from 0 to 1).

In a VNS framework, neighborhood changes typically correspond to various types of moves, or perturbations, of the current solution.

Thus, in our VNS algorithm, starting from a random allocation of entities x i to the two sub-communities of the current community S to be bipartitioned, we use a basic allocation move to change neighborhood of x. Allocation move works by taking some entity v i from S = S 1 ∪ S 2 and changing its allocation: if v i ∈ S 1 (i.e., x i = 1) then, after the allocation move, it will belong to S 2 (i.e., x i = 0) and vice-versa.

When performing such a change (move), we need to check if it leads to some improvement of the objective function value. Notice that, after the move, the new objective function to be maximized for bipartition is:

f new = min 2m new 1 c new 1 , 2m new 2 c new 2 , (3)
where m new

m new 1 = m 1 -k in v , m new 2 = m 2 + k cut v (4)
because the new number of inner edges m 1 of S 1 is reduced by k in v (the inner degree of v) and the number of edges in S 2 is augmented by the cut degree of v. The number of cut edges c 1 and c 2 are updated as follows:

c new 1 = c 1 + k in v -k cut v -k rest v ( 5
)
c new 2 = c 2 + k in v -k cut v + k rest v ( 6
)
where

k rest v = k v -k v (S)
(computed for each vertex at each hierarchy level). Indeed, since v moves to S 2 , the number of cut edges c new 1 of S 1 is increased by its inner degree in S 1 (+k in v ); also, it is reduced by k cut v , since its cut degree contributes to inner degree of S 2 ; finally, all vertices from S \ (S 1 ∪ S 2 ) connected with v after the move to S 2 do not produce cut edges of S 1 anymore (-k rest v

). So, the new objective function value is

f new = min 2 • (m 1 -k in v ) c 1 + k in v -k cut v -k rest v , 2 • (m 2 + k cut v ) c 2 + k in v -k cut v + k rest v (7)
and similarly for the opposite case of v moving from S 2 to S 1 .

Local search. A local search procedure is used in the VNS to improve the objective function value. It consists of starting from an initial solution vector x and iteratively moving to the best neighbor x ∈ N (x) such that the value of the objective function f is improved (f (x > f (x) in the case of maximization). The heuristic stops when no such neighbor exists anymore. In our VNS setting, a move is done changing vertex memberships. Thus, the local search is an iterative application of the above allocation move, which is repeated until there is no vertex whose membership change will give partition with larger value of f (S 1 , S 2 ). In that case, a local maximum has been found.

The following proposition holds.

Proposition 1 If a bipartition (S 1 , S 2 ) of a set S = V \ (S 1 ∪ S 2
) is locally optimal with respect to allocation neighborhood structure then the following holds:

[(∀v ∈ S 1 )(f new 1 < f 1 ∨ f new 1 < f 2 )] ∧ [(∀v ∈ S 2 )(f new 2 < f 1 ∨ f new 2 < f 2 )]
Proof. The fact that there is a move of an entity v from S 1 to S 2 , or from S 2 to S 1 such that f new is larger than both f 1 and f 2 , may be expressed in the following way:

[(∃v ∈ S 1 )(f new 1 > f 1 ∧ f new 1 > f 2 )] ∨ [(∃v ∈ S 2 )(f new 2 > f 1 ∧ f new 2 > f 2 )].
If there is no such improvement move, it can formally be presented as negation of the previous expression.

Then the statement of the proposition follows from De Morgan's laws.

Proposition 2 The time complexity of one iteration of the local search algorithm is O(n).

Proof. The proposed local search is based on the move procedure, which is repeated n times. The results follows from the observation that the procedure move is O(1).

Shaking. Shaking is a procedure used to generate a point x ′ at random from the k-th neighborhood N (x) of the current solution x. A new local search will then started at x ′ .

The natural way to introduce a distance in the solution space of the considered problem is to use Hamming distance. We say that two solutions x and y are at distance k if their Hamming distance is equal to k. As solution vectors are binary vectors, the Hamming distance between x and y measures the minimum number of changes of zeros to ones, or reciprocally, required to change x into y. We build the shaking procedure on this concept, perturbing the current solution x to a new vector which has Hamming distance k from it. Thus, k zeros or ones are switched to their opposite value, or, in other words, k entities change their original cluster.

Update of the current solution. Update of the current solution vector leads to update a number of entities appearing in the problem description. We need indeed to update the number of vertices of S 1 and S 2 , their number of inner edges and cut edges, and correspondingly the degree of vertices of the two subgraphs obtained by bipartition.

The basic VNS algorithm for edge-ratio clustering bipartition can finally be summarized in Algorithm 2.

Computational results and analysis

In this section, we discuss the results obtained on 11 datasets from the literature, corresponding to various real-world applications and often used to test algorithms and heuristics for community detection. In particular, we consider Zachary's karate club dataset [14] describing friendship relationships in a karate club, Lusseau's dolphins dataset [15] describing communications between dolphins, Hugo's Les Misérables dataset describing characters in Victor Hugo's masterpiece and their interactions [START_REF] Knuth | The Stanford GraphBase: A Platform for Combinatorial Computing[END_REF], Krebs' political books dataset [START_REF] Krebs | [END_REF], a dataset representing the schedule of football games between American college teams [START_REF] Girvan | [END_REF], a network dealing with connections between US airports [19], a dataset on a coauthorship network of scientists working on network theory and experiment [20], a network describing electronic circuits [21], a network representing e-mail interchanges between members of a university [22], a network giving the topology of the Western States Power Grid of the United States [23] and a network of authors collaborations [19]. These datasets are all undirected and unweighted networks without loops. They are listed Table 1 together with their number of vertices n and number of edges m.

Comparison with modularity

In Table 1 we also report the number of communities M in partitions obtained by applying the proposed divisive heuristic with the VNS algorithm for bipartitioning. We compare this number with the number of communities obtained when the known criterion modularity [3] is used. An exact algorithm for modularity maximization is used [24] for the first 8 datasets, and a heuristic [25] refined using the procedure in [26] is used for the last three datasets, where an exact solution is not known. In the VNS, the value of parameter k max is set to min{50, n/2} in all test problems. We compare results obtained using two treshold values for deciding about indivisibility of a community. The default value for this threshold is 1 and, specially for small-scale networks, it allows us to obtain a number of communities that is very close to that obtained using modularity. In other cases, a threshold equal to 1 leads to consider a high number of communities in the final partition, often containing very small communities. In such cases, we experimentally find different values for the threshold which give results very close to results obtained with the modularity criterion. Notice that the impact of the threshold for indivisibility in edge-ratio clustering was not studied in previous work, as only small-scale networks were previously analyzed.

Comparison with a locally optimal hierarchical divisive algorithm

In [6] a mathematical programming formulation for identification of optimal communities according to the edge ratio criterion is proposed, which has a linear objective (i.e., maximization of α), but non linear and non convex constraints (products between α and binary variables arise). However, if α is fixed, a linear program in 0-1 variables is obtained. This suggests to solve the optimal bipartition problem with a dichotomous search on the values of α. An initial value α equal to 1 can first be chosen. If there is no feasible solution for that value, the network is indivisible. Otherwise, the value of α may be doubled and feasibility checked until a value is attained for which the weak condition cannot be satisfied. This gives an upper bound ᾱ and the previous value of α gives a lower bound α. Then the dichotomous search proceeds by considering the mid value of the interval [α, ᾱ]. The procedure stops when the length ᾱ -α of the current interval is smaller than some given tolerance ǫ. This basic procedure can be accelerated in several ways, including using an initial value of α corresponding to a solution obtained by some heuristic and removing symmetries by fixing a variable. The algorithm is locally optimal in the sense that an exact method is used to solve the bipartition problem. See [6] for more details.

In Table 2 we compare, in terms of execution time, the results obtained using VNS and the exact algorithm for bipartition in the splitting step ot the hierarchical divisive algorithm. Threshold 1 is used. Tests were run on AMD Opteron 2 GHz CPU, 128 GB RAM. We remark that VNS allows us to obtain results significantly faster than exact bipartitioning. The computational time is reduced up to 7 orders of magnitude. Thus, we are able to solve larger problems with respect to problems that we can solve using an exact method. Furthermore, using VNS does not significantly affect the quality of the results. In the most of the cases, we obtain the same partition using VNS and the exact bipartitioning method. Two examples of obtained partitions, and the corresponding dendrogram illustrating the hierarchical divisive algorithm, are shown in Figures 1 and2. These two partitions, obtained using VNS, are the same obtained using exact bipartitioning. In this paper we propose a heuristic based on (basic) Variable Neighborhood Search to perform the bipartitioning step in a hierarchical divisive algorithm based on the recent edge-ratio network clustering criterion, which was shown to be an alternative approach to modularity criterion. Neighborhoods to be used in the VNS are defined using the Hamming distance. Systematic perturbations of the current incumbent solution also use the Hamming distance between any two solution. We develop a local search for the addressed problem and evaluate its complexity.

Computational results show that VNS allows us to obtain good quality results significantly reducing the computational time needed to perform bipartitioning steps.

Algorithm 1 :

 1 Hierarchical divisive algorithm Input: graph G = (V, E), where |V | = n and |E| = m Output:

1 and m new 2 arenew 1 and c new 2 are

 212 the number of edges in the first and in the second subgraph obtained by bipartition, and c the number of cut edges with one end point in the first subgraph and the number of cut edges with one end point in the second subgraph respectively. If v ∈ S 1 changes its membership to S 2 , then we have:

Figure 1 :Figure 2 :

 12 Figure 1: (Color online) Partition and dendrogram obtained for dataset polbooks

  v n }, P = S ℓ 2 while ℓ ≤ |P | and ∃ S ℓ ∈ P not visited do

	3	select S ℓ ∈ P (not visited)
	4	find a bipartition of S ℓ into S 1 and S 2 according to the edge-ratio criterion
	5	if f ≥ threshold
	6	then
	7	P ← (P ∪ {S 1 } ∪ {S 2 }) \{S ℓ }
	8	S ℓ ← S 1 ; S ℓ+1 ← S 2
	9	end if
	10	ℓ ← ℓ + 1
	11 end while

Algorithm 2 :

 2 Basic VNS for edge-ratio clustering bipartitionInput: graph G S = (S, E S )

	Output: a bipartition of S
	1 Initialization
	2 while time ≤ t max do
	3	k ← 1
	4	while k ≤ k max do
	5	apply Shaking
	6	perform Local search
	7	k ← k + 1 (next neighborhood)
	8	if f improved
	9	then
	10	perform a move
	11	end if
	12	end while
	13 end while

Table 1 :

 1 Results obtained using VNS, on datasets from the literature

	dataset	n	m	M (threshold)	M (threshold)	M modularity
	karate	34	78	4	(1)	-	-	4
	dolphins	62	159	8	(1)	5	(1.5)	5
	les miserables	77	254	9	(1)	7	(1.5)	6
	political books	105	441	10	(1)	5	(2)	5
	football	115	613	12	(1)	10	(1.5)	10
	Usair97	332	2126	16	(1)	6	(2)	6
	netscience main	379	914	69	(1)	18	(6.5)	19
	s838	512	819	101	(1)	12	(5)	12
	email	1133	5452	39	(1)	12	(1.5)	11
	power	4941	6594	1322	(1)	48	(6)	46
	erdos02	6927	11850	268	(1)	32	(1.8)	34

Table 2 :

 2 Comparison of results obtained using VNS and an exact algorithm for bipartition, on datasets from the literature

	dataset	n	m	Time VNS (sec.)	Time exact (sec.)
	karate	34	78	0.0	62.10
	dolphins	62	159	9.99e-3	172.2
	les miserables	77	254	1.99e-2	283.49
	political books	105	441	1.99e-2	716.45
	football	115	613	3.99e-2	11780.79
	Usair97	332	2126	0.62	3752906.94
	netscience main	379	914	1.14	2347.24
	s838	512	819	0.64	12612.53
	email	1133	5452	6.84	-
	power	4941	6594	29.80	-
	erdos02	6927	11850	35.50	-


Cunsheng Zhang
email: zhangcs@sdu.edu.cn

Guoqun Zhao

Hao Chen

Lionel Leotoing

Numerical investigation of temperature and forming rate effect on AA5086 warm formability

Keywords: Forming Limit Diagrams (FLDs), Marciniak test, Warm formability, UHARD subroutine

The poor formability of aluminum alloy at room temperature limits its use in some products with complex shapes, hence the warm forming process is intended to overcome this problem by using an elevated temperature. Now, the warm formability of AA5086 has not previously been well investigated in the literatures, especially at a rapid forming rate. In this paper, a numerical method has been developed to investigate the warm formability of an AA5086 sheet. Firstly, the dynamic tensile test was carried out under different forming temperatures and forming rates to identify an appropriate constitutive law for the sheet. The inverse analysis was performed to identify the parameter values in the constitutive law. Then based on the commercial finite element program ABAQUS, the Marciniak test was simulated to evaluate the sheet formability of by implementing a user-defined material subroutine UHARD. The effects of forming temperature and forming rate on sheet formability were investigated and it is shown that the formability of AA5086 seems to be insensitive to the forming temperature and forming rate.

Introduction

Due to its advantage of high-strength to weight ratio, high strength and corrosion resistance, aluminum alloy is becoming one of the main materials to replace steel in order to reduce weight. It is reported that between 1995 and 2000, the use of aluminum has increased by over 80% in automotive applications [1]. However, aluminum alloy shows the poor formability at room temperature compared to steels. Hence, the warm forming process is intended to overcome this problem by using an elevated forming temperature which is below the recrystallization temperature [2][3].

For sheet metal forming, FLDs, which proposed by Keeler and Backofen in 1960s [4], are an efficient diagnostic tool for evaluating sheet formability and many methods have been developed to determine the FLDs [5][6][7]. For experimental predictions of FLDs, two main kinds of forming tests have been developed, the so-called out-of-plane stretching and the in-plane stretching. However, it is a very timeconsuming procedure to establish FLDs and the scatter in experimental data for a given sheet is usually large [8]. As a result, significant efforts have been made on developing more analytical or numerical models for construction of FLDs. Using the LDH (Limiting Dome Height) test, Narasimhan [START_REF] Narasimhan | Proceedings of the 8th International Conference on Numerical Methods in Industrial Forming Processes[END_REF] has predicted the onset of localized necking by calculating the thickness strain gradient across neighboring regions. With the Marciniak test, Petek et al. [5] proposed that the maximum of the second temporal derivative of the thickness strain corresponds to the onset of localized necking. Volk [START_REF] Volk | Proceedings of the FLC-Zurich 06[END_REF] identified the onset of localized necking by the two following main effects: increase of point number with high strain rate (in the localization region) and decrease of the strain rate outside the localization bands.

Aluminum alloy 5086 (AA5086) is a medium to high strength alloy, which is commonly used in the manufacture of unfired, welded pressure vessels, marine, auto aircraft cryogenics, etc. However, little attention has been put on the warm forming behavior and sheet formability for this alloy, especially for a rapid forming rate. Hence, the paper begins with the dynamic tensile test for an AA5086 sheet. An inverse analysis will be applied to identify warm forming behaviors of this aluminum sheet and the corresponding parameters in the constitutive law will be determined. Furthermore, with commercially available finite-element program ABAQUS, the FE model based on the Marciniak test will be developed and simulated to investigate numerically the warm formability of AA5086. And the effects of forming temperature and forming rate on the sheet formability will be investigated.

Determination of material constitutive laws

The tensile test is carried out on a computer-controlled servo-hydraulic testing machine under different forming temperatures (20, 230, 290, 350°C) and forming rates (10, 750, 1000mm/s). At high temperatures, the mechanical behavior of aluminum alloys results from a balance between workhardening and thermal softening. This behavior can be represented by the use of a modified Voce's hardening law ( [START_REF] Lukac | [END_REF][12]) (Eq.1)

( ) ( ) 4 0 1 1 C n T m T T C e e ε σ σ ε     -   = + - (1)
Where σ and ε are the equivalent plastic stress and strain, respectively, and 0 σ the yield strength. The indexes n and m are the function of forming temperature T :

( )

( ) 3 2 C T n T C e = and ( ) 5 6 m T C C T = + (2)
The coefficients i C ( i =1~6) are constants, they are identified by the inverse analysis procedure, which has been described in detail in author's previous paper [13]. It is supposed that the value of the yield strength depends only on temperature but not on strain rate level, which is commonly verified for aluminum alloys [14]. The yield strengths and the identified constitutive parameters i C at 20, 230, 290 and 350°C are given in Table 1.

The comparison between experimental and identified loads versus displacement at 10mm/s is illustrated in Fig. 1. The agreement between the experimental curves and the ones identified by inverse analysis is not perfect but the Voce's constitutive law and the identified parameters permit to well model the global effect of temperature and strain rate on the behavior of this aluminum alloy. Considering self-heating, thermal radiation and conductivity, not presented in this work, should improve the identification of the parameters. ( )

Numerical determination of FLDs with the Marciniak test

1 3 C K - ( ) 4 C K 5 C ( )

FE model

In this part, the Marciniak test is modeled with the ABAQUS to evaluate the AA5086 sheet formability. The FE model of the Marciniak test consists of three parts: a rigid cylindrical punch with a flat bottom, a die and a deformable sheet. In addition, a pressure load is directly applied on the sheet as the blank-holder force (Fig. 2). Keeping the blank in place by means of pressure and friction will be used also in the future experiments.

The Marciniak test requires the application of different specimen geometries to cover different strain states, ranging from uniaxial through plane strain to equi-biaxial stretching. In this work, all test samples are shaped by cutting strips of different widths W in a circular flange, according to Fig. 3. To assure the occurrence of the maximal strains (to trigger localization) on the central part of the blank, the specimens are designed with a reduced central thickness (0.8 mm) compared to the thickness of adjoining part (1.5 mm) and the clamping part with a thickness of 2.0 mm.

The model is meshed with shell linear elements S4R. The elasticity of the specimen is defined with a Young's modulus of 67000 MPa and a Poisson's ratio of 0.3. The constitutive law (1) is implemented into ABAQUS by using the UHARD user subroutine. During the simulation, the die is fixed and the punch moves downwards at different speeds, where the forming rates correspond to the above dynamic tensile test. The contact interactions between the surfaces of the FE model are defined with Coulomb's friction law. The friction coefficients between punch and parts with thicknesses of1.5 mm, and 2.0 mm are 0.05 and 0.1, respectively. There is no friction between the punch and the central part of specimen, which is under the identical condition to the future experimental procedure.

Determination of localized necking

Here, a failure criterion, widely used in the M-K model, is applied to determine the onset of localized necking. When the ratio of equivalent plastic strain increment in localized element and its adjacent one on the blank reaches 7, the major and minor strains in the adjacent element are noted as limit strains. The same procedure is repeated for all specimens and the corresponding limit strains describe the forming limit curve. With the above failure criterion, the FLDs under different temperatures (20, 230, 290°C) and forming rates (10, 750, 1000mm/s) are determined. The comparisons of FLDs at a given forming rate of 10mm/s and those at a given forming temperature of 230°C are shown in Fig. 4 and Fig. 5, respectively. From these figures, the general shape of FLDs is observed, and the minimal formability occurs under the plane strain condition. In addition, the numerical results show that the sheet formability of AA5086 seems to be insensitive to forming temperature and forming rate, which do not completely agree with the conclusion for other aluminum alloys in the literatures [2][3]. However, the method proposed in this work permits to numerically evaluate the sheet formability. These results must be experimentally validated in order to determine accurate FLDs necessary and optimize the warm forming processes. This validation is in progress by means of a dynamic Marciniak test associated with a high-speed camera and the digital image correlation (DIC) technique to detect the onset of localized necking and construct FLDs.

Conclusion

In this paper, the warm forming behavior of AA5086 is investigated by experimental and numerical methods. A modified Voce's law was identified by the inverse analysis to describe its high temperature forming behavior. By means of a user-defined material subroutine UHARD, the material's constitutive law was implanted into ABAQUS and the Marciniak test was simulated to evaluate the sheet formability. The effects of forming temperature and forming rate on AA5086 sheet formability were also investigated. Numerical results show that the formability of this alloy seems to be insensitive to forming temperature and forming rate. This experimental validation of these results is in progress by means of a dynamic Marciniak test.

Fig. 1

 1 Fig.1 Comparison of experimental and identified load-displacement curves at 10 mm/s

Fig. 2 FEFig. 3

 23 Fig.2 FE model of the Marciniak test

Table 1 :

 1 Yield strengths and constitutive parameter values.

			27.41
			1.76
			3 2.08 10 -×
			606.20
			6 1 10 --×
	6 C K -	1	4 1.54 10 -×


Keywords: Forming Limit Diagrams (FLDs), Dynamic tensile test, Digital Image Correlation (DIC), Inverse analysis

M a n u s c r i p t into account the temperature and strain rate is proposed. An inverse analysis is carried out to identify the material parameters of the law for the tested aluminum alloy. Finally, with the above identified law, tensile tests are simulated. The experimental and numerical results show that the testing temperature and forming rate have a great influence on sheet formability. At high forming rates, the sheet formability of AA5086 is lowered up to a certain temperature, above this temperature, the formability is greatly enhanced. Furthermore, the agreement between experimental and numerical results indicates that the proposed constitutive law and the identified material parameters can be appropriate to model the sensitivity of AA5086 sheet towards strain rate and temperature.

A c c e p t e d M a n u s c r i p t

Experimental and numerical study on effect of forming rate on AA5086 sheet formability

Cunsheng ZHANG

Abstract

With increasing application of aluminum alloys in automotive or aeronautic industries, it is necessary to characterize their deformation behaviors at large strains, high strain rates and elevated temperatures, which is relatively lacking today. The aim of this paper is to experimentally and numerically investigate the influence of forming rate and temperature on formability of an AA5086 sheet. Firstly, tensile tests are carried out at different temperatures (20, 230, 290 and 350°C) and at different forming rates (10, 750 and 1000 mm/s). A technique of digital image correlation (DIC) associated with a high-speed camera is applied to evaluate the surface strains and a complete procedure is built to detect the onset of localized necking during the experiments. The influences of initial testing temperature and forming rate on the sheet formability are analyzed. Then in order to numerically determine the formability of this sheet, a form of Voce's constitutive law taking

Preprint submitted to Materials Science and Engineering: A September 4, 2009

Introduction

Sheet metal forming is a very commonly used method for producing various components in automotive or aeronautic industries. Especially with the innovative techniques, such as hydroforming and incremental forming, the manufacture of complex parts with low cost can be realized. However, in sheet metal forming operations, the sheet can be deformed only to a certain limit. This limit is usually imposed by the onset of localized necking, which may lead to early failure. The ability of sheet metal to deform into desired shape without local necking or fracture is defined as its formability. Formability depends on many factors like material properties (e.g. strain hardening coefficient, strain rate sensitivity, anisotropy ratio) or process parameters

2

A c c e p t e d M a n u s c r i p t (e.g. strain rate, temperature) [START_REF] Seth | High velocity formability and factors affecting it[END_REF]. Thus, understanding and characterizing the formability of metal sheets are crucial for controlling final product quality and then the success of the sheet forming operation, especially with the increasing use of aluminum alloys that exhibit low formability compared with typical mild steels [START_REF] Miller | Recent development in aluminium alloys for the automotive industry[END_REF][START_REF] Heinz | Recent development in aluminum alloys for aerospace applications[END_REF].

One important technique to evaluate the formability of sheet metals is the use of forming limit diagrams (FLDs) developed by Keeler and Backofen in the 1960s [START_REF] Keeler | Plastic instability and fracture in sheets stretched over rigid punches[END_REF]. For sheet metal forming, FLDs are realistic and efficient diagnostic tools for evaluating formability. In addition, due to the improvement of production rates, strain rates in sheet forming processes can be located in the range of intermediate strain rates (10 -2 to 500s -1 ). Nevertheless, the mechanical behavior of materials at intermediate and high strain rates can be considerably different from that observed with quasi-static loading because of the strain-rate sensitivity of the material and propagation of stress waves [START_REF] Tarigopula | A study of localisation in dual-phase high-strength steels under dynamic loading using digital image correlation and FE analysis[END_REF]. Further, strain-rate sensitivity has been identified as an important factor determining formability of sheet metal and can alter substantially the level and the shape of FLCs. Of particular interest here is the influence of strain rate on the forming capacity of metals during the forming process. A review of currently available literature shows that relatively little attention has been paid to models for FLDs taking strain rate into account. model. One of the first studies taking into account this effect was proposed by [START_REF] Hutchinson | Mechanics of sheet metal forming[END_REF]. Considering a von Mises' yield function, the effect of strain rate sensitivity have been evaluated. Afterwards, anisotropy of sheet metals was introduced in this classical model by using different yield function, like the Hill's criterion [START_REF] Barata Da Rocha | Prediction of the forming limit diagrams of anisotropic sheets in linear and non-linear loading[END_REF] or the Logan's and Hosford's criterion [START_REF] Graf | Calculations of forming limit diagrams[END_REF]. In these analytical models, the strain hardening behavior and the strain rate sensitivity of materials are generally represented by simplistic power laws. For some materials, especially for aluminium alloys, this formulation is not well adapted to model the strain rate effects and then the onset of local necking [START_REF] Diot | Forming process of a 5083 aluminium alloy. constitutive model covering a large range of temperature[END_REF]. Moreover, it has been shown that these simplistic laws only permit the study of the influence of the strain rate sensitivity index but not the influence of the value of the strain rate and temperature [START_REF] Zhang | Theoretical and numerical study of strain rate influence on AA5083 formability[END_REF].

Experimentally, the Nakazima [START_REF] Nakazima | Study on the formability of steel sheet[END_REF] and Marciniak tests [START_REF] Marciniak | Influence of the plastic properties of the material on the forming limit diagram for sheet metal tension[END_REF] are widely used to study sheet formability at quasi-static loading. Under dynamic conditions, it is relatively difficult to adapt these conventional tests. Percy [START_REF] Percy | The effect of strain rate on the FLD for sheet metal[END_REF] analyzed the influence of strain rate on FLDs by explosive forming and concluded that FLDs level was dependent on the strain state and forming rates.

Broomhead et al. [START_REF] Broomhead | The effect of strain rate on the strain to fracture of a sheet steel under biaxial tensile stress conditions[END_REF] performed bulge forming over a range of strain rates from 10 -3 to 70s -1 and concluded that the position of FLDs for biaxial tensile conditions was lowered with increasing strain rate. The effects of temperature and forming speed on FLCs of an Al-Mg alloy (5083-O) sheet have been investigated by Naka and Yoshida [START_REF] Naka | The effects of temperature and forming speed on the forming limit diagram for type 5083 aluminummagnesium alloy sheet[END_REF] for different speeds (0.2-200 mm/min) and at temperatures of 20-180 °C. Experimental results showed that the level of FLC increases with increasing temperature and decreasing forming speed.

By deep drawing of rectangular parts at a certain strain rate (1 s -1 ) con-A c c e p t e d M a n u s c r i p t ducted by Li and Ghosh [10], FLCs under warm forming conditions (250, 300 and 350 °C) were also determined for three different aluminum alloys (AA5754, AA5182 and AA6111-T4). They all exhibit a significant improvement in their formability with increasing temperature. Moreover, Ghosh [START_REF] Ghosh | Sheet metal stamping die design for warm forming[END_REF] has pointed out that warm formability drops with increasing forming speed.

However, no correlations have been presented between these experimental results and those of numerical or analytical predictive model of FLCs.

With the increasing application of computational techniques, numerical predictions of FLDs have become more attractive and the finite element method (FEM) has been selected to analyze the necking process ( [START_REF] Petek | An improved method for determining forming limit diagram in the digital enviroment[END_REF], [START_REF] Volk | New experimental and numerical approach in the evaluation of the FLD with the FE-method[END_REF]).

In these models, the choice of an appropriate failure criterion is a key point for the determination of accurate FLCs [START_REF] Situ | A suitable criterion for precise determination of incipient necking in sheet materials[END_REF]. Moreover, it is found that the hardening law implemented in finite element codes influence greatly the level and shape of FLCs [START_REF] Zhang | Investigations on the effect of strain rate sensitivity on formability of aluminium alloy sheets[END_REF]. Therefore, an appropriate viscoplastic constitutive model and a reliable identification of its parameters are primordial for the formulation of an efficient formability predictive tool.

By use of the uniaxial tensile test, one point of the FLC can be obtained.

From the previous analytical studies of present authors, this point seems to be very sensitive to the strain rate sensitivity of the material [START_REF] Zhang | Investigations on the effect of strain rate sensitivity on formability of aluminium alloy sheets[END_REF]. Moreover, the uniaxial tensile test is well controlled under dynamic loading. Hence, a tensile test performed at various forming rates can firstly be used to analyze the sensitivity of forming behavior to strain rate.

Therefore, it is necessary to establish a rigorous procedure and carry out image correlation associated with a high-speed camera is used to evaluate the strains on the specimen surface and a complete procedure is built to detect the onset of localized necking during the experiments. Then a form of Voce's constitutive law is proposed and an inverse analysis is applied to identify the material parameters of this aluminum sheet for the different investigated conditions. Finally, with the above identified laws, the dynamic tensile test is simulated with ABAQUS and the comparison of experimental and numerical results is given.

Experimental procedure and results

The dynamic tensile tests are carried out on a computer-controlled servo-hydraulic testing machine. The actuator accelerates upward with the crosshead to achieve a desired velocity before loading the test sample and this velocity can be considered to remain constant during the test. To augment the sensitivity of this alloy to strain rate, tensile tests are conducted at different temperatures (20, 230, 290 and 350 °C). The displacements are controlled at constant rate and increased from 10 to 1000 mm/s, prescribing average strain rates of order 10 -1 to 10 2 s -1 in the defined specimen.

Experimental device

In this work, a tensile specimen is designed with a notch which results in necking initiation in the central region of the specimen and then facilitates To satisfy the requirement that the longitudinal axis of the test specimen coincides with tensile direction of the testing machine, a recently designed grip system is used during the experiments, as shown in Fig. 2. The heels are used to clamp the test specimen at its two ends. They are mounted on the testing machine with the upper and lower grips. More importantly, under dynamic conditions, the device allows a predetermined run where the sample is accelerated to a desired test velocity. Here, to capture the consecutive frames during the experiments, a highspeed camera is used (FASTCAM ultima APX-RS digital CMOS camera) associated with a macro lens, which is ideally suited for high magnification close range applications. Additional light is required to illuminate the sample during testing. A high frequency induction heating system is applied to heat the specimen. Once the specimen is heated to the target temperature, tensile test is performed. The camera and the load acquisition are triggered just before the test is started to capture a picture of undeformed specimen.

Due to inherent hardware limitations of the camera, the higher the resolution, the lower the possible sample rate. In our experiments, the tests are performed at different crosshead speeds. Therefore, in order to get a good representation of the central zone deformation of the specimen, the testing speed and view resolution should be optimized for each speed, the chosen balance is listed in Table 1.

8

A c c e p t e d M a n u s c r i p t For strain measurement using the DIC technique, the sample is prepared by the application of a random dot pattern to its surface. In this work, to avoid burning paints at elevated temperature, shotblasting treatment is applied to replace white paint and produce white basecoat on specimen surface. The zone of interest on the specimen surface is divided into small subsets (small grid in Fig. 3(a)). Then the system tries to match the grey level distribution of these subsets on the deformed images. This matching operation is carried out throughout a displacement transformation applied to the subsets.

To perform correlation analysis in this work, the commercial digital imag- shows a saturation in the equivalent plastic strain. The necking ultimately ends with the onset of a rupture in the localized zone. The growth of this localized necking can also be illustrated in Fig. 4, which plots the ratio of average equivalent plastic strain increments in the localized (Zone b) and nonlocalized zones (Zone a) throughout time. One can observe that this ratio keeps relatively stable (close to 1) until about 0.52 s. After this moment, the ratio increases suddenly and localized necking, i.e. plastic instability, begins to occur. In this work, a failure criterion widely used for the M-K model is chosen: when this ratio reaches 7, localized necking is assumed to occur. At this moment, the principal strains (-0.067, 0.30) in Zone b are retained as limit strains. As illustrated in Fig. 4, the equivalent plastic strain increment ratio changes so suddenly, hence the critical value in the failure criterion has no great influence on the determination of the limit strains. Therefore, the value of 7 is reasonable as a failure criterion for predicting the onset of the localized necking.

ing

A c c e p t e d M a n u s c r i p t

Experimental results

Figure 5 represents the effect of temperature on the evolution of the forcedisplacement curves measured during the plastic stage of the specimen deformation. For a quasi-static test speed (10mm/s), a thermal softening is clearly observed on these curves, when the test temperature increases from 20°C to 350°C the maximum tensile force is decreased by half. On Figure 6, the temperature is fixed at 350°C and the evolutions of the force-displacement curves are plotted for three different forming speeds. In order to properly analyze the mechanical behavior of the material under dynamic conditions (750mm/s and 1000mm/s), the oscillations due to impact-induced stress wave travelling through the specimen have been filtered out. An increasing load level is observed with increasing forming speed, the magnitude of this Page 14 of 26 A c c e p t e d M a n u s c r i p t effect depends on the level of the test temperature. As a result, for high temperatures, the effect of strain rate on the material behavior should be taken into account for accurate rheological predictions. Figure 7 shows the evolution of the major strains in terms of the temperature (20, 230, 290 and 350 °C) and testing speeds (10, 750 and 1000 mm/s). The minor strains are not represented here but the same tendencies are observed, due to the constant strain path for all the specimens. Three specimens are tested for each condition, and the different curves are plotted by using the average major strain for each condition. In all cases, the scatter from test to test is relatively low.

As the figure 7 major strain can be observed, while at 350 °C, a relative decrease of 51 % is exhibited. It is true that there is no significant differences between strains at 750 and 1000 mm/s, because these two forming rates are relatively close. In conclusion, at elevated temperature, forming rate plays a prominent role in determining sheet formability. At low forming rate (10 mm/s), the testing temperature always plays a positive effect on sheet formability. It may be concluded that at 350 °C, an increasing forming rate compensates the positive effect of temperature on the formability, and this leads to an insignificant effect of temperature on the formability for this range of rate (major strain remains close to 30 %). At 230 • C, the compensating effect of forming rate is more pronounced, consequently, we observe the lowest major strains at this temperature for the two high speeds tested. Differently from the conclusion Page 17 of 26

A c c e p t e d M a n u s c r i p t in [10, [START_REF] Naka | The effects of temperature and forming speed on the forming limit diagram for type 5083 aluminummagnesium alloy sheet[END_REF][START_REF] Ghosh | Sheet metal stamping die design for warm forming[END_REF], with increasing temperature, there does not exhibit a single trend in major strains at different forming rates. The aim of the following parts is to find numerically this notable evolution thanks to an appropriate constitutive law.

Identification of material's constitutive laws

Based on the above tensile test with a notched specimen, it is difficult to identify the material's flow behavior with conventional analytical methods due to the irregular specimen cross-section. Nowadays, the inverse methods are intensively used to adjust the material parameters for more and more complex constitutive laws or irregular specimen geometry. The basic concept of an inverse analysis for parameter identification is to find out a set of unknown material parameters in constitutive equation thanks to a Finite Element simulation of the test.

Constitutive law

At high temperatures, the mechanical behavior of aluminium alloys results from a balance between work-hardening and thermal softening. This behavior can be represented by the use of a Voce's hardening law ( [START_REF] Lukac | Characterisation and modelling of the plastic behaviour of aluminium alloys[END_REF], [START_REF] Diot | Forming process of a 5083 aluminium alloy. constitutive model covering a large range of temperature[END_REF]) (Eq. 1)

σ = σ 0 + C 1 1 -e -n(T )ε e ( C 4 T ) εm(T) (1)
where σ and ε are the equivalent plastic stress and strain, respectively, and σ 0 the yield strength. In order to model the influence of temperature on For the strain rate sensitivity index m(T ), a linear evolution with temperature seems to be appropriate :

m(T ) = C 5 + C 6 T
The coefficients C i (i = 1..6) are constants, they will be identified through the inverse analysis procedure. Due to the sheet plane quasi-isotropy of AA5086, a von Mises's yield function is used in the FE model.

Results

The identification procedure (OPTPAR) used in the present work, developed by Gavrus [START_REF] Gavrus | Identification automatique des paramètres rhéologiques par analyse inverse[END_REF], is based on an iterative Gauss-Newton's algorithm, using numerical computation of the first order derivatives of the cost function with respect to the parameters to be identified. The identification of the constitutive parameters is performed with a FE simulation of the tensile test using the ABAQUS software until the minimum of the cost function is reached. The FE model for inverse analysis is shown in Fig. 8, which is meshed with triangle elements. By means of the user-defined FORTRAN subroutine UHARD, the constitutive law (Eq. 1) can be implemented into the finite element code.

In this work, we suppose that the value of the yield strength depends only on temperature but not on strain rate level, which is commonly verified for aluminum alloys [START_REF] Pedersen | Strength and ductility of aluminium alloy AA7030[END_REF]. The yield strengths and the identified constitutive parameters C i are given in table 2. The comparison between experimental and calculated loads versus displacement at 10mm/s is illustrated in Fig. 9. There is much discrepancy between experimental and identified curves of load vs. displacement for higher temperatures, but these curves curves are obtained under the same low forming rate (10 mm/s), therefore, the influence of strain rate on this discrepancy is probably less. The agreement between the experimental curves and the ones identified by inverse analysis is not perfect but the Voce's constitutive law and the identified parameters permit to well model the global effect of temperature and strain rate on the behavior of this aluminium alloy, on the wide range of tested temperatures and forming rates.

Conclusions

This paper investigates experimentally and numerically the effect of forming rate and temperature on sheet formability of an AA5086 sheet. By carrying out a dynamic experimental tensile test and its associated FE model, the following conclusions can be drawn.

• The technique of digital image correlation associated with a high-speed camera has been successfully applied to evaluate the strains at high forming rates and at high temperatures. With the time-sequence of equivalent plastic strain profiles along the specimen's longitudinal axis, a complete procedure is built to detect the onset of localized necking during the experiments.

• To describe the forming behavior of this AA5086 sheet, a form of Voce's hardening law taking into account the temperature and strain rate is proposed. The inverse method is used to identify the material parameters for this law. The agreement between experimental and numerical results indicates that this constitutive law and identified material parameters can model the high temperature deformation behavior of the sheet.

• The comparison of experimental and numerical results shows that for high strain rates (around 10 2 s -1 ), the sheet formability seems to be lowered up to a certain temperature (between 200 • C and 250 • C), above this temperature, the formability is enhanced and the negative effect of strain rate on formability is compensated.

Page 24 of 26

  Different analytical models have been developed that focus either on diffuse or localized necking. These models not only help to understand the necking phenomenon but they represent also useful tools to successfully and rapidly predict the formability of sheet metals in industrial practice. The strain rate sensitivity has been studied mainly through the classical M-K Page 4 of 26 A c c e p t e d M a n u s c r i p t

  more experimental investigations about forming behavior in a large range of strain rates. One of the aims of this work is to propose an experimental Page 6 of 26 A c c e p t e d M a n u s c r i p t procedure to characterize the effect of initial testing temperature and forming rate on sheet formability. Firstly, a dynamic tensile test is conducted for an AA5086 sheet at different temperatures (20, 230, 290 and 350 °C) and different forming rates (10, 750 and 1000 mm/s). A technique of digital

6

 6

  A c c e p t e d M a n u s c r i p t the capture of consecutive frames. The geometry and dimensions of the specimens are shown in Fig. 1. The thickness of the AA5086 sheet is 2.0 mm.

Figure 1 :

 1 Figure 1: Geometry and dimensions of the notched specimen (lengths in mm).

Page 8

 8 of 26 A c c e p t e d M a n u s c r i p t

Figure 2 :

 2 Figure 2: Grip system used in the dynamic tensile test.

A

  Fig. 3(b) displays the time-sequence of these equivalent plastic strain profiles along the axis x with a regular time step.

Figure 3 :

 3 Figure 3: Experimental analysis and determination of onset of localized necking.

From the figure 3

 3 (b), it is observed that at early stage of the test, a quasi-homogeneous deformation is distributed along the axis x. The subsequent development of deformation is concentrated to the central part of the Page 12 of 26 A c c e p t e d M a n u s c r i p t specimen notch, and the plastic strain is localized to a smaller and smaller region. After a certain moment, the equivalent plastic strain in the localized zone (Zone b) rises extremely rapidly while the non-localized zone (Zone a)

Figure 4 :

 4 Figure 4: Variation of equivalent plastic strain increment ratio in the localized and nonlocalized zones.

Figure 5 : 14 AFigure 6 :

 5146 Figure 5: Influence of temperature on force-displacement evolutions at 10mm/s and for the plastic stage.

Figure 7 :

 7 Figure 7: Limit strains under different forming rates and temperatures.

  the wide range tested (from 20 to 350 °C), an exponential form of the strain hardening index n(T ) is proposed in this work : Page 18 of 26 A c c e p t e d M a n u s c r i p t n(T ) = C 2 e (C 3 T )

Figure 8 :

 8 Figure 8: FE model used for inverse analysis.

  Considering self-heating, thermal radiation and conductivity, not presented in this work, should improve the identification of the parameters. Page 20 of 26 A c c e p t e d M a n u s c r i p t

Figure 9 :Figure 10 :

 910 Figure 9: Experimental and identified curves of load versus displacement at 10mm/s

A c c e p t e d M a n u s c r i p t

  1 * , Lionel LEOTOING 2 , Dominique GUINES 2 , Eric RAGNEAU 2 1 Shandong University, Key Laboratory of Liquid Structure and Heredity of Materials (Ministry of Education) 73 Jingshi Road, Jinan, Shandong Province, P.R.CHINA, 250061 Université Européenne de Bretagne, France, INSA-LGCGM -EA 3913 20, avenue des Buttes de Coësmes 35043 RENNES Cédex

	Email: zhangcs@sdu.edu.cn
	Tel : +86(0)53181696577 Fax : +86(0)53188395811
	Tel : +33(0)2 23 23 86 64 Fax : +33(0)2 23 23 87 26

2

Table 1 :

 1 Camera settingsDigital image correlation (DIC) is a technique for non-contact strain measurement and surface profilometry, which is nowadays widely used for forming analysis of metal sheets as well as the determination of material properties.

	Testing speed (mm/s) Camera resolution (Pixels) Camera speed (images/s)
	10	512×758	1500
	750	256×656	12000
	1000	128×352	18000
	2.2. Strain evaluation and determination of the onset of localized necking

Table 2 :

 2 Yield strengths and constitutive parameter values.

	20°C 230°C 290°C 350°C
	σ 0 (MPa) 162.7 147.2 131.7 116.2
	C 1 (MPa)	27.41
	C 2	1.76
	C 3 (K -1 )	2.08 10 -3
	C 4 (K)	606.20
	C 5	-1 10 -6
	C 6 (K -1 )	1.54 10 -4
	4. Numerical study and discussion
	With the above identified constitutive laws, the tensile test is simulated
	at different temperatures and forming speeds. The onset of localized necking
	is analyzed with the identical procedure to experiments. Figure 10 shows
	the comparison of evolution of the limit major strains in terms of the tem-
	perature (20, 230, 290 and 350 °C) and high tensile speeds (750 and 1000
	mm/s) by experimental and numerical methods. From this figure, one can
	observe the same phenomenon on numerical and experimental results, i.e.
	the negative effect of strain rate on formability up to 230 °C. The experi-
	mental limit major strains are slightly lower than numerical ones (except for
	350 °C). Especially at 230 °C, a relative difference of 20 % in major strains
	exists. One reason for this divergence might results from the discrepancies
	observed between experimental load-displacement curves and the identified
	ones. Even So, the compensating effect of temperature and forming rate is
	clearly observed and the same tendencies of major strain versus temperature
	20