Sylvain Lazard
email: sylvain.lazard@inria.fr

William Lenhart
email: wlenhart@williams.edu

Giuseppe Liotta
email: giuseppe.liotta@unipg.it

On the Edge-length Ratio of Outerplanar Graphs

We show that any outerplanar graph admits a planar straightline drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any > 0 there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than 2 -. We also show that every bipartite outerplanar graph has a planar straight-line drawing with edge-length ratio 1, and that, for any k ≥ 1, there exists an outerplanar graph with a given combinatorial embedding such that any planar straight-line drawing has edge-length ratio greater than k.

Introduction

The problem of computing a planar straight-line drawing with prescribed edge lengths has been addressed by several authors, partly for its theoretical interest and partly for its application in different areas, including VLSI, wireless sensor networks, and computational geometry (see for example [START_REF] Di | Angles of planar triangular graphs[END_REF][START_REF] Doherty | Convex optimization methods for sensor node position estimation[END_REF][START_REF] Held | Combinatorial optimization in VLSI design[END_REF][START_REF] Savarese | Robust positioning algorithms for distributed ad-hoc wireless sensor networks[END_REF]). Deciding whether a graph admits a straight-line planar drawing with prescribed edge lengths was shown to be NP-hard by Eades and Wormald for 3-connected planar graphs [START_REF] Eades | Fixed edge-length graph drawing is NPhard[END_REF]. In the same paper, the authors show that it is NP-hard to determine whether a 2-connected planar graph has a unit-length planar straightline drawing; that is, a drawing in which all edges have the same length. Cabello et al. extend this last result by showing that it is NP-hard to decide whether a 3-connected planar graph admits a unit-length planar straight-line drawing [START_REF] Cabello | Planar embeddings of graphs with specified edge lengths[END_REF]. In addition, Bhatt and Cosmadakis prove that deciding whether a degree-4 tree has a planar drawing such that all edges have the same length and the vertices are at integer grid points is also NP-hard [START_REF] Sandeep | The complexity of minimizing wire lengths in VLSI layouts[END_REF].

These hardness results have motivated the study of relaxations and variants of the problem of computing straight-line planar drawings with constraints on the edge lengths. For example, Aichholzer et al. [START_REF] Aichholzer | Graph drawings with relative edge length specifications[END_REF] study the problem of computing straight-line planar drawings where, for each pair of edges of the input graph G, it is specified which edge must be longer. They characterize families of graphs that are length universal, i.e. they admit a planar straight-line drawing for any given total order of their edge lengths.

Perhaps one of the most natural variants of the problem in the context of graph drawing is that where, instead of imposing constraints on the edge lengths, one aims at computing planar straight-line drawings where the variance of the lengths of the edges is minimized. See for example [START_REF] Di Battista | Graph Drawing: Algorithms for the Visualization of Graphs[END_REF], where this optimization goal is listed among the most relevant aesthetics that impact the readability of a drawing of a graph. Computing straight-line drawings where the ratio of the longest to the shortest edge is close to 1 also arises in the approximation of unit disk graph representations, a problem of interest in the area of wireless communication networks (see, e.g. [START_REF] Chen | Separability and topology control of quasi unit disk graphs[END_REF][START_REF] Kuhn | Unit disk graph approximation[END_REF]).

Discouragingly, Eades and Wormald observe in their seminal paper that the NP-hardness of computing 2-connected planar straight-line drawings with unit edge lengths persists even when a small tolerance (independent of the problem size) in the length of the edges is allowed. To our knowledge, little progress has been made on bounding the ratio between the longest and shortest edge lengths in planar straight-line drawings. We recall the work of Hoffmann et al. [START_REF] Hoffmann | Quality ratios of measures for graph drawing styles[END_REF], who compare different drawing styles according to different quality measures including the edge-length variance.

In this paper we study planar straight-line drawings of outerplanar graphs that bound the ratio of the longest to the shortest edge lengths from above by a constant. We define the planar edge-length ratio of a planar graph G as the smallest ratio between the longest and the shortest edge lengths over all planar straight-line drawings of G. The main result of the paper is the following.

Theorem 1. The planar edge-length ratio of an outerplanar graph is strictly less than 2. Also, for any given real positive number , there exists an outerplanar graph whose planar edge-length ratio is greater than 2 -.

Informally, Theorem 1 establishes that 2 is a tight bound for the planar edge-length ratio of outerplanar graphs. The upper bound is proved by using a suitable decomposition of an outerplanar graph into subgraphs called strips, then drawing the graph strip by strip. The lower bound is proved by taking into account all possible planar embeddings of a maximal outerplanar graph whose maximum vertex degree is a function of . Theorem 1 naturally suggests some interesting questions that are discussed in Section 3.

We shall assume familiarity with basic definitions of graph planarity and of graph drawing [START_REF] Di Battista | Graph Drawing: Algorithms for the Visualization of Graphs[END_REF] and introduce only the terminology and notation that is strictly needed for our proofs.

Proof of Theorem 1

It suffices to establish the result for maximal outerplanar graphs. To show that the edge-length ratio of a maximal outerplanar graph is always less than 2, we imagine decomposing the dual G * of G into a set of disjoint paths, which we call chains. Each chain corresponds to some sequence of pairwise-adjacent triangles of G. The set of chains inherits a tree structure from G * , and we use this structure to direct an algorithm that draws each of the chains proceeding from the root of this tree down to its leaves. We formally define a chain to be a sequence T s , T s+1 , . . . , T t of triangles of G where s ≤ 0 ≤ t, and such that (i) T 0 consists of an outer edge of G whose vertices are labeled with 0, along with a third vertex labeled 1, (ii) for each i : 1 ≤ i ≤ t, the vertices of T i are labeled by {i -1, i, i + 1} so that T i and T i-1 share the edge having vertices labeled i and i -1, and (iii) for each i : -1 ≥ i ≥ s, the vertices of T i are labeled by {i, i+1, i+2} so that T i and T i+1 share the edge having vertices labeled i+1 and i + 2. Note that this definition prohibits fans (consecutive triangles all sharing a common vertex) containing more than 3 triangles, except for the vertex labeled 1, which has four incident triangles on the chain.

The decomposition into chains is constructed by first selecting an edge e on the outer face of some outerplanar topological embedding of G. The edge e is incident with a unique triangle of G. Label each vertex of e with 0, and label the third vertex of the triangle with 1. There is now a unique maximal chain C e in G containing this labeled triangle. The edges of C e can be partitioned into two sets: S e and L e where S e consists of edges of C e whose vertex labels differ by 1 and L e consists of all edges of C e whose vertex labels differ by 2, along with e .

Removing the edges of S e from G produces a set of 2-connected components in 1-1 correspondence with the edges of L e : Each component contains exactly one element of L e which lies on its outer face. For each edge e ∈ L e , let G e be the component of G -S e containing e. We can then recursively decompose each G e by choosing the (unique) maximal chain C e in G e containing the one triangle (if any) of G e that is incident with e. We call the set of chains so constructed a chain decomposition of G. A chain decomposition produces a decomposition of the edges of G into sets L and S, where L is the union of the edges in each L e and S is the union of all of the edges in each S e . Note that there is a single chain for each edge in L, and that the collection of chains produced naturally form a tree: The root of the tree is the chain C e and its children are the chains {C e : e ∈ L e }; the chain decomposition is entirely determined by the choice of external edge e .

The drawing algorithm proceeds by first drawing the root chain C e of the chain decomposition tree of G and then recursively drawing the chain decomposition trees of each G e : e ∈ L e . The algorithm depends on a specific method for drawing a single chain. To describe it, we need a few definitions. First, given a line segment s in the plane and a direction (unit vector) d not parallel to s, denote by S(s, d) the half-infinite strip bounded by s and the two infinite rays in direction d that have their sources at the endpoints of s. Finally, given a chain C, the edges of C ∩ L are called external edges of C; note that each external edge is incident to exactly one triangle of C. (iii) each external edge has length 1 and all other edges have lengths greater than 1/2; (iv) each external edge forms an angle less than θ 0 with d. Moreover, such a drawing can be computed in O(n)-time in the real RAM model.

Proof. Let T 0 be the triangle of C containing e. T 0 is either adjacent to zero, one, or two triangles of C. We handle these three cases in turn. If T 0 is the only triangle in C, then we simply draw T 0 in S(s, d) as an isosceles triangle with e drawn as s and with its third vertex drawn so that its two edges have length l, where 1 2 < l < 1. Assume now that T 0 is adjacent to a triangle T 1 of C. Denote the vertices of C as follows:

T 0 = {v - 0 , v + 0 v 1 }
, where e = {v - 0 , v + 0 } and v - 0 is not incident with T 1 . The vertex of T 1 not in T 0 is denoted by v 2 , and, subsequently, the vertex of each T i not in T i-1 is denoted by v i+1 . We draw T 0 as previously, but with more careful positioning of v 1 . To determine where to position v 1 , we draw edge {v + 0 , v 2 } of T 1 as a unit-length segment in direction d. As long as v 1 is positioned within S(s, d) but outside of the disks of radius 1 2 centered at v - 0 , v + 0 , and v 2 , the edges from each of these vertices to v 1 will have length greater than 1 2 (see top half of Figure 1). By placing v 1 close to e, the edges {v 1 , v - 0 } and {v 1 , v + 0 } will have lengths less than 1. Also, since ∠v 2 v + 0 v 1 < θ 0 , edge {v 1 , v 2 } will have length less than 1.

1 1 1 1 1 1 2 + ε 1 2 + ε 1 2 + ε 1 2 + ε 1 2 + ε 1 2 + ε 1 2 + ε 1 1 2 + ε T 0 T 1 1 1 2 + ε 1 1 2 + ε T 0 1 1 1 2 + ε 1 2 + ε 1 2 + ε 1 1 2 + ε 1 2 + ε T 2 T 3 T 4 T -1 T -2 T -3 T t T s v1 v - 0 v2 v3 v4 v5 vt-1 vt vt+1 v1 v + 0 v-1 v-2 v-3 vs+2 vs+1 vs d θ v + 0 v - 0 Fig. 1. Drawing a chain
Assuming that T 0 , . . . , T i-1 have been drawn for some i > 1, T i is drawn by positioning v i+1 one unit distant from v i-1 in direction d. The result is that each T i is congruent to T 1 and so the edge-length ratio of C is less than 2. At this point, all of the unit-length segments, except for e, lie on the two rays in direction d emanating from v + 0 and v 1 . By rotating these rays a very small amount towards one another, we can preserve the lengths of the unit-length segments while ensuring that all of the remaining segments have lengths in the range ( 12 , 1). See the top of Figure 1.

Finally, suppose that T 0 has two adjacent triangles. Starting with T 0 , label the other triangles in C so that the labels of adjacent triangles differ by 1. Thus, for example, T 0 is adjacent to T 1 and T -1 . The vertices in the T i , i ≥ 0 will be labeled as in the previous case: the unique vertex in T i not in T i-1 is labeled v i+1 . The vertices in the T i , i < 0 will be similarly labeled: the unique vertex in

T i not in T i+1 is labeled v i .
Draw each T i , i > 0 as in the previous case. Now draw the T i , i < 0 in a similar fashion: Place vertex v i one unit distant from v i+2 in direction d. Then, as above, all of the unit length edges of the triangles T i , i < 0 will lie on the two rays in direction d emanating from v 1 and v - 0 , and these two rays can be rotated slightly towards each other while maintaining the length of the unit-length edges and ensuring that the other edges still have lengths in the range ( 12 , 1). See the bottom of Figure 1. This can clearly be done so that all external edges form angles less than θ 0 with d.

However, we need to ensure that v 1 can be placed so that both triangle T 1 and triangle T -1 can simultaneously satisfy the required edge-length conditions: Namely, that edges {v - 0 , v + 0 }, {v 1 , v -1 }, and {v + 0 , v 2 } are all unit-length, while edges {v 1 , v 2 }, {v + 0 , v 1 }, {v 1 , v - 0 }, and {v - 0 , v -1 } all have lengths in the range ( 12 , 1). However, it is relatively simple to show that v 1 can always be successfully placed if the (smaller) angle between s and d is less than θ 0 = arccos(1/4).

[The angle θ 0 is the angle opposite an edge of length 2 in an isosceles triangle having side lengths 2, 2, and 1.] Finally, the computation of the locations of the vertices can each be computed in constant time in the real RAM model, giving a run-time linear in the size of the chain.

We are now ready to prove the following lemma. For a planar straight-line drawing Γ , we denote with ρ(Γ ) the ratio of a longest to a shortest edge in Γ . Lemma 2. A maximal outerplanar graph G with n vertices admits a planar straight-line drawing Γ with ρ(Γ ) < 2 that can be computed in O(n) time assuming the real RAM model of computation .

Proof. We call the drawing computed as in Lemma 1 a U-strip drawing of C and adopt the same notation as in Lemma 1. Recall that in a chain decomposition of a graph, the external edges of the chains are exactly the edges of L. A drawing of G is computed as follows.

1. Compute a chain decomposition tree for G; let C e be the root of the tree. 2. Select a line segment s of length 1 in the plane and an initial direction d not parallel to s such that ∠sd < θ 0 . 3. Apply Lemma 1 to compute a U-strip drawing of C e . 4. Each edge e ∈ L e is drawn as a segment s e of length 1, not parallel to d, that forms an angle with d that is less than θ 0 , so draw the subtree of C e rooted at C e in the empty strip S(s e , d).

The result is an outerplanar straight-line drawing in which all edges of L (long edges) have length 1 while all edges in S (short edges) have length strictly greater than 1 2 . If we assume that the input is provided to the algorithm in the form of a doubly-connected edge list [START_REF] Muller | Finding the intersection of two convex polyhedra[END_REF], then a chain decomposition tree for G can be computed in linear time. Also, since by Lemma 1 each chain can be drawn in time proportional to its length, the algorithm runs in O(n) time in the real RAM model.

The following lemma can be proved by means of a packing argument and elementary geometry. Lemma 3. For any > 0 there exists a maximal outerplanar graph whose planar edge length ration is greater than 2 -.

We conclude the section by observing that Lemmas 2 and 3 imply Theorem 1.

Additional Remarks and Open Problems

The upper and the lower bound of Theorem 1 suggest some questions that we find worth investigating. One question is whether better bounds on the planar edge-length ratio can be established for subfamilies of outerplanar graphs (for example, it is easy to show that trees have unit-length drawings). A second question is whether an edge length variance bounded by a constant can be guaranteed for drawings of outerplanar graphs where not all vertices lie in a common face. By a variant of the approach used to prove Lemma 2 and by using some simple geometric observations, the following results can be proved.

Theorem 2. The planar edge-length ratio of a bipartite outerplanar graph is 1.

The plane edge-length ratio of a planar embedding G of a graph G is the minimum edge-length ratio taken over all embedding-preserving planar straightline drawings of G. Theorem 3. For any given k ≥ 1, there exists an embedded outerplanar graph whose plane edge-length ratio is at least k.

We conclude this paper by listing some open questions that we find interesting to study: (i) Study the edge-length ratio of triangle-free outerplanar graphs. For example, it is not hard to see that if all faces of an outerplanar graph have five vertices, a unit edge length drawing may not exist; however, the planar edge length ratio for this family of graphs could be smaller than the one established in Theorem 1. (ii) Extend the result of Theorem 1 to families of non-outerplanar graphs. For example it would be interesting to study whether the planar edgelength ratio of 2-trees is bounded by a constant. (iii) Study the complexity of deciding whether an outerplanar graph admits a straight-line drawing where the ratio of the longest to the shortest edge is within a given constant. This problem is interesting also in the special case that we want all edges to be unit length.

Lemma 1 .

 1 Given a chain C with n vertices, an external edge e of C, a segment s of length 1 in the plane, and a direction d such that the (smaller) angle between s and d is θ < θ 0 = arccos(1/4) ≈ 75.5 • , there exists a planar straight-line drawing of C such that: (i) the drawing is completely contained within the strip S(s, d); (ii) no external edge e of C is parallel to d, and the strips S(e , d) are all empty;


Francisco Navas
email: francisco.navas-matos@inria.fr

Vicente Milanés
email: vicente.milanes@renault.com

Fawzi Nashashibi
email: fawzi.nashashibi@inria.fr

Youla-Kucera Based Online Closed-Loop Identification For Longitudinal Vehicle Dynamics

Keywords: Closed-Loop system identification, longitudinal vehicle dynamics, linear-parameter-varying system, Youla-Kucera parametrization

This paper deals with the identification of longitudinal dynamics of a cycab for subsequent control performance's improvement. Low-speed vehicle dynamic is used as physical model which non-linear behavior is highly complex to properly identify. Among several identification techniques, closed-loop identification gives better performance in a model-based control design. Here, the Hansen scheme is used to transform a closedloop identification problem in an open-loop-like. The algorihtm is tested in a string of two cycabs equipped with a proportionalderivative-based cooperative adaptive cruise controller, showing how the resulting model is improved in comparison with an openloop identification algorithm-ARX model.

I. INTRODUCTION

Intelligent vehicles field is rapidly growing worldwide [START_REF] Greenblatt | Automated vehicles, on-demand mobility, and environmental impacts[END_REF]. Autonomous cars promise to save time, improve traffic flow, reduce accidents, death and injuries, and make travels accessible and easier to everyone. Despite several demonstrations in recent years, there are still remaining challenges on the road to get fully autonomous vehicles. Manufacturers and academia focus in partially automated vehicles for aiding the driver. Antilock braking systems (ABS), emergency braking, lane changing systems, yaw stability control or adaptive cruise control (ACC) are some examples of already commercial applications. The design of each assistance system involves the specification of a model of vehicle dynamics and its use in the design of the corresponding control to satisfy good performance in different operation points.

Friction force between tire and road is the main reason why vehicle moves [START_REF] Pacejka | Tire and vehicle dynamics[END_REF]. It converts the motor torque to longitudinal force. The conversion depends on powertrain system, engine, aerodynamics resistance, tire/road surface conditions, tire pressure among others [START_REF] Hunt | Constructive empirical modelling of longitudinal vehicle dynamics using local model networks[END_REF]. The relation between them brings an insight into the understanding of vehicle dynamics. A complete physical model that really reflects the behavior of the vehicle results in a complex, non-linear dynamic model difficult to achieve. It motivates vehicle dynamics identification by using experimental data; the identified model should capture the nonlinearities of the vehicle while being suitable for control design. When it comes to identification methods from online data, linear parameter-varying (LPV) identification [START_REF] Bruzelius | Linear parameter varying systems[END_REF] offers a systematic way to obtain a non-linear model suitable for non-linear controllers. LPV system is a linear state-space representation whose dynamics vary as function of certain time-varying parameters called scheduling parameters. Two different approaches can be distinguished in LPV modeling: local and global. In local identification, the LPV model is composed by interpolated Linear-Time-Invariant (LTI) models in different operating points; while in global identification, the model changes continously with the scheduling parameter. The latter results in a more accurate model, as it changes with every scheduling parameter's modification [START_REF] Caigny | Interpolating model identification for siso linear parameter-varying systems[END_REF].

Among open-loop (OL) and closed-loop (CL) identification methods, it is well-known that for model-based control design, CL identification gives better performance [START_REF] Hjalmarsson | For model-based control design, closed-loop identification gives better performance[END_REF]. There are several techniques in the literature for LPV identification, but most does not take into account CL operation of the system because its associated difficulties [START_REF] Sznaier | An lmi approach to control-oriented identification and model (in) validation of lpv systems[END_REF] [8] [START_REF] Santos | Identification of linear parameter varying systems using an iterative deterministicstochastic subspace approach[END_REF]. Using the Hansen scheme proposed in [START_REF] Hansen | Closed-loop identification via the fractional representation: Experiment design[END_REF], one can transform the CL problem into an OL-like identification problem. The same has been used and extended to LPV systems [START_REF] Bendtsen | Closed?loop identification for control of linear parameter varying systems[END_REF] inside of the Danish research program entitled Plug and Play Process Control [START_REF] Stoustrup | Plug & play control: Control technology towards new challenges[END_REF]. Identification of coupled dynamics in heat distribution systems [START_REF] Trangbaek | Lpv identification of a heat distribution system[END_REF], and identification of actuators/sensors in the latter [START_REF] Knudsen | Plug and play process control applied to a district heating system[END_REF] were carried out into the project, showing promising model results.

II. PROBLEM FORMULATION

One of the objectives in the development of the Intelligent Transportation Systems (ITS) is to reduce the use of private vehicles in urban areas, by offering new modern and public transportation systems. In order to fulfil this objective, INRIA has developed the cybercar concept (cycab and cybus) as mobile platforms for urban aplications. Cycab has a kinematic structure that differs from that of a car-like: it turns its rear wheels as a linear function of the steering angle of the front wheels [START_REF] Sekhavat | The cycab robot: a differentially flat system. in intelligent robots and systems[END_REF]. Cycab is commanded through velocity. This velocity is limited to 5m/s.

The design of different urban applications pass through the correct characterization of the vehicle. When doing cycab characterization (see Fig. 1), a non-linear longitudinal behavior is detected with different time responses and damping factors depending on the control velocity v c . A non-linear model suitable for non-linear control is needed to satisfy good performance in different operation points. LPV identification fits the needs.

The longitudinal model of cycab G β can be represented as a minimal state space representation of a LPV system:

G β = A β B β C β D β (1)
where β is the scheduling parameter, which corresponds to the control velocity v c . This system could be stabilized by any appropriate LPV controller K β , also represented in state-space as:

K β = A k β B k β C k β D k β (2)
Here, the global LPV CL identification algorithm provided by the dual Youla-Kucera (YK) parameterization (so-called Hansen scheme) is implemented for identification of longitudinal cycab dynamics. The same is compared with a wellknown OL identification algorithm-auto regressive model with external inputs (ARX) [START_REF] Karaboyas | Efficient adaptive algorithms for arx identification[END_REF]. Then, the performance of the algorithm is tested.

The rest of the paper is structured as follows. Section III introduces the mathematical basis used for OL-like identification of a general LPV system. How this theory is applied to CL cycab dynamics identification is presented in Section IV. Comparison results between CL and OL algorithms are obtained both in simulation and experimental environments. Finally, some concluding remarks are given in Section V.

III. OL-LIKE IDENTIFICATION OF LPV SYSTEMS

This section introduces the mathematical basis in which Hansen scheme relies on for global LPV CL identification. As already mentioned, Hansen scheme allows OL-like identification through the dual YK parameterization (see [START_REF] Dasgupta | A parametrization for the closed-loop identification of nonlinear time-varying systems[END_REF]). This parameterization is based on the doubly coprime factorization described in [START_REF] Pommaret | Generalized bezout identity[END_REF].

A. LPV doubly coprime factorization

Any controller and model set that satisfies both Eqs. ( 1) and ( 2) for any change in the scheduling parameter β , can be rewritten as double coprime factors according to [START_REF] Ishihara | Doubly coprime factorizations related to any stabilizing controllers in state space[END_REF]. These coprime factorizations should be such that G β and K β are:

G β = N β M -1 β = Mβ -1 Ñβ K β = U β V -1 β = Ṽβ -1 Ũβ (3)
These right and left coprime factorizations should be stable and satisfy the double Bézout's identity [START_REF] Pommaret | Generalized bezout identity[END_REF]:

Ṽβ -Ũβ -Ñβ Mβ M β U β N β V β = = M β U β N β V β Ṽβ -Ũβ -Ñβ Mβ = I 0 0 I (4)
According to [START_REF] Ishihara | Doubly coprime factorizations related to any stabilizing controllers in state space[END_REF], coprime factorization to any stabilizing controller in state space can be obtained if Theorem 1 is satisfied.

Theorem 1: Consider a LPV plant in state space representation as

G β = C β (sI -A β ) -1 B β with A β , B β ,C β sta- bilizable and detectable and a stabilizing controller K β = C k β (sI -A k β ) -1 B k β + D k β . F k β and F k β should be chosen such that A β + B β F β and A k β + B k β F k β ∈ RH pxm ∞ .
Then the matrices given by 3) and (4).

M β U β N β V β =      A β + B β F β 0 -B β 0 0 A k β + B k β F k β 0 B k β -F β C k β + D k β F k β I D k β -C β F k β 0 I      (5) Ṽβ -Ũβ -Ñβ Mβ =      A β + B β D k β C β B β C k β -B β B β D k β B k β C β A k β 0 B k β F β -D k β C β -C k β I -D k β C β -F k β 0 I      (6) satisfy (

B. Hansen scheme for LPV systems

Dual YK parameterization can be used to recast the CL identification into a OL-like problem. The original method in [START_REF] Hansen | Closed-loop identification via the fractional representation: Experiment design[END_REF] for LTI systems is extended for LPV systems in [START_REF] Bendtsen | Closed?loop identification for control of linear parameter varying systems[END_REF]. This method is used here once double coprime factorization of the controller/model set is obtained. Thus, all the models stabilized by a controller K β are represented by:

G β = (N β +V β S β )(M β +U β S β ) -1 (7)
From the general description of any model stabilized by K β in Eq. ( 7), it is possible to identify new dynamics from an initial model G 0 β to another one G 1 β just by obtaining the dual YK parameter S β . Then the set of all plants stabilized by K β can be represented by the scheme in Fig. 2. Here, n = ( Mβ + Sβ Ũβ )n y is the measurement noise relocated to affect the output of S β ; and r 1 and r 2 are external excitation signals. From Fig. 2 and Bezout's identity in (4) the following expressions yields:

ζ = Ũβ r 1 + Ṽβ r 2 (8) z = Mβ y -Ñβ u (9)
Even when control signal u and output y are measured in CL, the identification of S β is OL-like by using filtered signals ζ and z.

OL identification algorithms like ARX [START_REF] Karaboyas | Efficient adaptive algorithms for arx identification[END_REF] or PBSIDopt [START_REF] Van Wingerden | The asymptotic variance of the pbsidopt algorithm[END_REF] can be used for obtaining S β . By identifying S β in OL, advantages of CL identification are preserved with a simpler method. The CL equivalent model results:

G 1 β = (N 0 β +V 0 β S β )(M 0 β +U 0 β S β ) -1 (10)

IV. LONGITUDINAL DYNAMICS IDENTIFICATION

In this section, the described CL LPV identification is applied to the longitudinal dynamics model of the experimental platform already described in Section II. The vehicle is controlled in CL to compare OL and CL identification algorithms.

Here, the controller K (see Fig. 3) is a PD-based cooperative adaptive cruise controller (CACC) as the one in [START_REF] Naus | String-stable cacc design and experimental validation: A frequency-domain approach[END_REF]. Its aim is to reduce the response time to different speed changes by using the communication link with its preceding vehicle. Thus, intervehicle distance can be reduced, improving highway capacity. A scheme of the vehicle with the controller in place is shown in Fig. 3; where i is the vehicle index; s d is a constant distance between consecutive vehicles at standstill; h d is the constant head time, v i is the vehicle velocity, d r is the distance between vehicles obtained by laser, and F a feedforward filter depending on the communication link with the preceding vehicle. Notice how the resulting control signal u will be equivalent to the commanded velocity v c , while the output y corresponds to the ego-vehicle velocity v i . The controller K has been designed for a point of operation which is not necessarily the same in all operating points. A better performance could be achieved by designing a LPV controller in terms of the scheduling parameter v c . The design should be done according to the identified model.

First, an OL LPV model of the vehicle is obtained by using v c and v i directly in the ARX model (see [START_REF] Karaboyas | Efficient adaptive algorithms for arx identification[END_REF]). Its basis is equivalent to the following equation:

y k+1 = Zθ k + v k ( 11
)
where y is the output vector, Z the matrix with old inputs and output values, θ k the model parameters and v k white noise. Then, a CL LPV identification is carried out applying the Hansen scheme presented in Section III. Note that the output of block F (see Fig. 3) and the desired distance relative coming from s d + h d v i serve as external excitation signals r 1 and r 2 in the CL identification. Instead of using v c and v i directly in the ARX model, filtered signals ζ and z are employed to get the dual YK parameter S β . Once S β is known, the CL vehicle model is updated by using Eq. [START_REF] Hansen | Closed-loop identification via the fractional representation: Experiment design[END_REF].

Mβ -1 Nβ

In order to evaluate the performance between both algorithms, the metric called Vinnicombe ν-gap in [START_REF] Vinnicombe | Uncertainty and Feedback: H inf loop-shaping and the v-gap metric[END_REF] is used. ν-gap goes from 0 to 1 and expresses the difference between two models in reference to similarity in its CL operation. The closer its value to 0, the better the performance. It is been only defined for linear time invariant (LTI) system; its use into LPV systems cames with the definition of a set of LTI systems that composes the LPV model. The ν-gap metric is used during the simulation step with two different LTI models of the cycab. Experimental results regarding real, OL estimated and CL estimated outputs are provided to support the simulation results.

A. Simulation results

As mentioned aboved, two different LTI models are considered for identification comparison. An initial LTI model G 0 of the cycab was obtained when the control velocity changes from 2m/s to 4m/s, resulting in the model shown in the Eq. ( 12). The PD-based CACC controller K is designed under consideration of this LTI model. Controller parameters are found in the Table I. A different LTI model G 1 is also considered, corresponding to the braking phase between 3m/s and 1m/s (see Eq. ( 13)). The model G 1 is used together with the controller K. White noise with a signal-noise-ratio (SNR) of 42.42dB is added to the output. The modification from G 0 to G 1 emulates the case in which an erroneus model has been identified, being the real one G 1 . As the real model is known, ν-gap metric comparison results reliable among OL and CL identified models. An OL identification has been carried out for obtaining the model G 1 . Signals v c and v i shown in Fig. 4 are used into an ARX model of third order, with one sample delay. The identification is performed using a sliding window approach of 400 samples with a sample time of 0.1s. The resulting model is compared with G 1 by calculating the corresponding ν-gap.

G 0 = A 0 B 0 C 0 D 0 =   1.856 -0.867 0.125 1 0 0 0.0818 0.00282 0.0024   (12) G 1 = A 1 B 1 C 1 D 1 =   1.
The proposed identification system computes ζ and z by using Eqs. ( 8) and ( 9) with signals r 1 , r 2 , v c and v i in Fig. 4. Notice that these signals are now input and output of the same ARX model. Same order and delay are chosen, so advantages of CL identification can be studied under same conditions. Results of the OL identification of the dual YK parameter S is provided in Fig. 5. The identified S is used in Eq. [START_REF] Hansen | Closed-loop identification via the fractional representation: Experiment design[END_REF], for obtaining the CL model of the vehicle. The identified model is compared with G 1 through the ν-gap metric.

ν-gap results from OL and CL identification are compared in Fig. 6. A better model is obtained when using ARX model together with the Hansen scheme. Under same conditions, νgap results closer to zero. CL nature of the data affects the ARX model, and the Hansen scheme helps to mitigate these effects; obtaining a model closer to the real one. of the string, so the algorithm can be tested with slower and faster velocity changes. The signals from Fig. 7 corresponds to the slower velocity changes case. An OL identification through ARX model is carried out with v c and v i ; while the same is also used with ζ ans z, for the OL identification of S. Its goodness is proved through results in Fig. 8, where real and identified output of S are compared. Identified S and coprime factors from K and G 0 are used in Eq. [START_REF] Hansen | Closed-loop identification via the fractional representation: Experiment design[END_REF] to get the CL model of a cycab with a CACC controller. As a priori knowledge of the real model is missing, ν-gap metric is not employed. Instead, estimated output from the model obtained by Eq. ( 10), OL model and real output are compared in Fig. 9. On the other hand, Fig. 10 shows the signals used in the identification for the faster velocity changes case. The procces is exactly the same than in previous cases. It is important to note that goodness of the OL identification for S in Fig. 11 is poorer than in the case of Fig. 8, fact related to the faster signal changes. Even with that, the CL identified model provided by the Hansen scheme is able to follow non-linearities, while the direct ARX model is not. Comparison of identified outputs is shown in Fig. 12. Differences between OL and CL algorithm are visible when faster changes are carried out; demonstrating the good performance of the algorithm previously tested in simulation.

In the latter case, the importance of modeling for controller design is noticeable. The peformance of the controller is quite oscillating, except for the velocity change from 2m/s to 4m/s, which coincides with the identified model G 0 used for the design of K. A LPV controller could provide a better performance in all system operation conditions.

V. CONCLUSION

In this paper, Hansen scheme is used for CL identification of longitudinal dynamics of a cycab. The good performance of the algorithm is tested when connected to a PD-based CACC controller. The CL identification provided by the Hansen scheme is based in the OL identification of the dual YK parameter S. Thus, the same OL algorithm-ARX model-is used for performance comparison of CL and OL identifications. Simulation and experimental results are conducted to validate the proposed identification algorithm. Results show the feasiblity of the proposed scheme, improving ARX model identification. Future work will be focused on controller tuning depending on the identification to yield better overall system performance.

Fig. 1 .

 1 Fig. 1. Nonlinear velocity cycab response.

Fig. 2 .

 2 Fig. 2. Hansen scheme for CL identification.

Fig. 3 .

 3 Fig. 3. CACC control structure.

θ

  k is found by minimizing the least squared prediction error between real and estimated output.

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig. 4. Signals for identification. Simulation results.

BFig. 7 .

 7 Fig. 7. Signals for identification. Experimental results with soft speed profile.

Fig. 8 .

 8 Fig. 8. OL identification of dual Youla-Kucera parameter S. Experimental results with soft speed profile.

Fig. 9 .Fig. 10 .

 910 Fig. 9. CL/OL comparison through estimated output. Experimental results with soft speed profile.

Fig. 11 .Fig. 12 .

 1112 Fig. 11. OL identification of dual Youla-Kucera parameter S. Experimental results with abrupt speed profile.

ACKNOWLEDGMENTS

Authors express their gratitude to the french project VALET and the RITS Team for its support in the development of this work.


Michael Callahan
email: michael.callahan@centralesupelec.fr

Olivier Hubert

François Hild

Astrid Perlade

Jean-Hubert Schmitt

Coincidence of Strain-Induced TRIP and Propagative PLC Bands in Medium Mn Steels

Keywords: Digital image correlation, Magnetization measurement, Medium Mn steel, Portevin-le-Châtelier bands, TRIP

An Fe-0.2C-5Mn-2.5Al medium manganese steel is studied for three different intercritical annealing temperatures T IA to observe the effects of variations of retained austenite stability on the mechanical work hardening in uniaxial tension. Digital image correlation (DIC) showed that both Lüders bands and type A Portevin-le-Châtelier (PLC) bands occurred in samples with a lower T IA , but both disappeared when T IA was increased. In-situ measurements of sample magnetization with a correction for stress effects on the magnetic properties of the sample allowed for continuous measurement of the retained austenite volume fraction. This method provided a complete characterization of the kinetics of transformation-induced plasticity (TRIP) in the sample over the course of tensile testing. It is shown that the martensite transformation was purely strain-induced and coincided with the passage of both Lüders and PLC bands. An Olson-Cohen model was applied to magnetic data to attempt to explain the differences in TRIP kinetics with varying T IA .

Introduction

Increasingly stringent vehicle emissions standards and a high consumer demand for fuel efficient vehicles have led to the development of a third generation of advanced high strength steels (AHSS) with complex multiphase microstructures. One such alloy family is Medium Manganese steels.

Medium Mn steels are composed of an ultra-fine-grained (UFG) mixture of ferrite, retained austenite, and in some cases martensite. They can achieve high ductility and high work hardening due to transformation-induced plasticity (TRIP) or twinning-induced plasticity (TWIP) in the austenite phase, which locally harden the steel and delay mechanical instability [START_REF] Pardoen | On the Role of Martensitic Transformation on Damage and Cracking Resistance in TRIP-Assisted Multiphase Steels[END_REF][START_REF] Chiang | Effect of microstructure on retained austenite stability and work hardening of TRIP steels[END_REF][START_REF] Park | Tensile properties of cold-rolled TWIP-cored three-layer steel sheets[END_REF]. Previous studies on Medium Mn steels have shown total elongation values of up to 70% and ultimate tensile strengths (UTS) ranging from 800 to 1500MPa [START_REF] Arlazarov | Evolution of microstructure and mechanical properties of medium Mn steels during double annealing[END_REF][START_REF] Cai | Austenite stability and deformation behavior in a cold-rolled transformation-induced plasticity steel with medium manganese content[END_REF][START_REF] Chen | Correlation between mechanical properties and retained austenite characteristics in a low-carbon medium manganese alloyed steel plate[END_REF][START_REF] Gibbs | Austenite stability effects on tensile behavior of manganeseenriched-austenite transformation-induced plasticity steel[END_REF][START_REF] Lee | Tensile Behavior of Intercritically Annealed Ultra-Fine Grained 8% Mn Multi-Phase Steel[END_REF]. These steels are intended for high formability in that an initially ductile steel can be hardened significantly during the forming process, thereby achieving mechanical strength criteria without complicating the forming process. Despite this potential, these steels present some challenges for forming operations because the adiabatic heat generated at high strain rates could suppress the TRIP effect, resulting in a loss of hardening capacity.

In order to create an UFG microstructure, the steel is generally homogenized and quenched before cold-rolling to obtain a large fraction of martensite. If the cold-rolled steel is then annealed in the intercritical ferrite + austenite domain, austenite will nucleate at the intersections of martensite laths. The resulting sub-micrometer austenite grain size makes it possible to thermodynamically stabilize the austenite at room temperature with Mn additions of only 5-10 wt% rather than the 20 wt% or more needed in TWIP steels [START_REF] Tang | Deformation twinning and martensitic transformation and dynamic mechanical properties in Fe0.07C23Mn3.1Si2.8Al TRIP/TWIP steel[END_REF][START_REF] Rahman | The effect of grain size on the twin initiation stress in a TWIP steel[END_REF]. The retained austenite thus obtained is metastable at room temperature and can undergo TRIP if sufficient mechanical energy is applied.

However, the high strain rates used in forming operations can stabilize the retained austenite via adiabatic heating and result in a significant loss in hardening capacity. This illustrates the need to understand how the stability of the austenite is related to the kinetics of the martensite transformation during TRIP. This study will seek to establish a method of characterizing the TRIP kinetics that could be used in future work to optimize compositions and high strain rate performance of TRIP steels.

Proper characterization of the kinetics of the TRIP effect in a particular alloy requires the volume fraction of retained austenite to be measured as a function of strain. This is notoriously difficult to do, however, as dif-ferent retained austenite volume fraction measurement techniques can give very different values for similar alloys [START_REF] Zhao | Magnetic and X-ray diffraction measurements for the determination of retained austenite in TRIP steels[END_REF][START_REF] Jacques | On measurement of retained austenite in multiphase TRIP steels results of blind round robin test involving six different techniques[END_REF]. Determination of the austenite content is nonetheless critical in alloy design, as it would provide the metallurgist control over the strain hardening contributions from TRIP and TWIP. Attempts have been made to measure the austenite volume fraction as a function of strain by X-ray diffraction (XRD) [START_REF] Moallemi | Deformation-induced martensitic transformation in a 201 austenitic steel: The synergy of stacking fault energy and chemical driving force[END_REF] and magnetization measurements using a ferritescope [START_REF] Moallemi | Deformation-induced martensitic transformation in a 201 austenitic steel: The synergy of stacking fault energy and chemical driving force[END_REF][START_REF] Beese | Identification of the Direction-Dependency of the Martensitic Transformation in Stainless Steel Using In Situ Magnetic Permeability Measurements[END_REF][START_REF] Radu | Continuous magnetic method for quantitative monitoring of martensitic transformation in steels containing metastable austenite[END_REF] in order to quantify TRIP kinetics. While in terms of the accuracy of the measurement method, magnetic measurements provide no advantage over other methods [START_REF] Jacques | On measurement of retained austenite in multiphase TRIP steels results of blind round robin test involving six different techniques[END_REF], the primary interest of the method as employed herein is the extremely high sampling rate compared to, for example, XRD or EBSD. It enables one to measure the retained austenite volume fraction in-situ without having to pause the mechanical test, which would not be possible if one wishes to perform the test at the very high strain rates used during stamping operations for example. Sampling rates this high are not normally obtainable by other in-situ methods, making magnetic saturation measurements the unique choice for such an application if the experimenter wants intermediate points between the unstrained and ruptured states. Moreover, a more complete description of the kinetics of TRIP using several hundreds or thousands of experimental data points would make it possible to refine existing TRIP models used in simulations of mechanical behavior.

Recent studies on Medium Mn steels have noted that strain hardening is sometimes unstable and that long Lüders plateaus are often observed in alloys containing low levels of aluminum [START_REF] Arlazarov | Evolution of microstructure and mechanical properties of medium Mn steels during double annealing[END_REF][START_REF] Gibbs | Strain partitioning in ultra-fine grained medium-manganese transformation induced plasticity steel[END_REF]. Additionally, for certain compositions and heat treatments, serrations have been observed in plots of dσ dε versus either true stress or true strain. Gibbs et al. [START_REF] Gibbs | Austenite stability effects on tensile behavior of manganeseenriched-austenite transformation-induced plasticity steel[END_REF] suggested that these serrations are due to a variable martensite transformation rate resulting from the competition between the consumption and creation of martensite nucleation sites during plastic deformation. Lee et al. [START_REF] Lee | Effect of the Intercritical Annealing Temperature on the Mechanical Properties of 10Pct Mn Multi-phase Steel[END_REF] attributed the serrations to dynamic strain aging (DSA) leading to the formation of Portevin Le-Châtelier (PLC) bands. Cai et al. [START_REF] Cai | Austenite stability and deformation behavior in a cold-rolled transformation-induced plasticity steel with medium manganese content[END_REF] instead proposed that the stability of the austenite phase itself varied from one grain to another and thus could transform to martensite in bursts.

Digital image correlation (DIC) has historically been a common method for characterizing strain localizations such as PLC [START_REF] Besnard | Finite-Element" Displacement Fields Analysis from Digital Images: Application to Portevin-Le Chatelier Bands[END_REF][START_REF] Kang | On the sequence of inhomogeneous deformation processes occurring during tensile deformation of strip cast AA5754[END_REF][START_REF] Halim | The Portevin-Le Chatelier (PLC) effect and shear band formation in an AA5754 alloy[END_REF][START_REF] Benallal | An experimental and numerical investigation of the behaviour 32 of AA5083 aluminium alloy in presence of the Portevin-Le Chatelier effect[END_REF][START_REF] Réthoré | Experimental investigation of localized phenomena using digital image correlation[END_REF][START_REF] Zdunek | Digital Image Correlation investigation of Portevin-Le Chatelier effect in an aluminium alloy[END_REF][START_REF] Zavattieri | Spatio-temporal characteristics of the Portevin-Le Chatelier effect in austenitic steel with twinning induced plasticity[END_REF][START_REF] Coër | Piobert-Lüders plateau and Portevin-Le Chatelier effect in an Al-Mg alloy in simple shear[END_REF][START_REF] Min | Effect of strain rate on spatio-temporal behavior of portevin-le chatelier bands in a twinning induced plasticity steel[END_REF][START_REF] Cheng | Investigation of Portevin-Le Chatelier effect in 5456 Al-based alloy using digital image correlation[END_REF]. As such, it is an ideal tool for investigating the jerky flow present in Medium Mn steels [START_REF] Sun | Discontinuous strain-induced martensite transformation related to the Portevin-Le Chatelier effect in a medium manganese steel[END_REF][START_REF] Wang | Kinematic and thermal characteristics of Lüders and Portevin-Le Châtelier bands in a medium Mn transformation-induced plasticity steel[END_REF].

This paper will seek to use a combination of DIC and in-situ measurements of the retained austenite content via the sample's saturated magnetization to demonstrate that the observed strain instabilities coincide with the TRIP effect.

Materials and Methods

Fabrication and Microstructural Characterization

A Medium Mn steel containing 0.2 wt% C, 5 wt% Mn and 2. XRD was used to determine the initial austenite volume fraction for each intercritical annealing temperature. X-rays were produced with a Co-K α source with an energy of 6.915 keV. As XRD is a surface measurement, the sample was oscillated in order to increase the volume of measurement (still at the surface) by scanning over a larger area. This provides better statistical sample size in terms of the number of grains analyzed. Spectra were obtained for several orientations of the sample relative to the incident beam. The inclination angle of the sample with respect to the incident beam, ψ, was varied from 0 • to 70 • with a step size of 5 • . The rotation of the sample around its normal to the sample surface, Φ, varied from 0 • to 355 • with a step of 5 • . These spectra were summed together in order to minimize texture effects on the relative peak intensities of the bcc and fcc phases, which can vary significantly depending on the orientation of the sample. The volume fraction of retained austenite was calculated using the method presented in

Ref. [START_REF] Bach | Diffraction X en traitements thermiques: dosage de l'austénite résiduelle par diffraction des rayons X[END_REF] and the phase volume fractions for each annealing temperature are summarized in Figure 2.

Tensile Testing and DIC

Tensile dog bone specimens were cut along the rolling direction from 1.25mm thick sheet. The specimens had an active gauge length 60mm long and 10mm wide. Unidirectional tensile tests were performed using an Instron extensometer with a 12.5mm initial length. The samples were strained until rupture using an Instron 4430 electromechanical tensile testing machine with a strain rate in the sample of 5 • 10 -4 s -1 .

Images for DIC analyses were obtained by painting the polished sample surface with a black and white speckle pattern and taking pictures every 3s during testing. A single Canon 60D camera with a 105mm macro objective lens was used to acquire the images with an exposure time of 1 320 s -1 , ISO grade of 800, and an aperture f-stop of f/9. Two halogen lamps were used to illuminate the specimen surface. The images recorded were 3465×5202 pixels in definition with 16bit encoding and were cropped to remove excess empty space and converted to .tif format for registration purposes. Correlation of the images was done using 3-node triangular elements and regularization using RT3-DIC [START_REF] Tomicevc | Mechanics-aided digital image correlation[END_REF]. Regularization, as used herein, is a means of reducing uncertainties in the displacement fields measured by DIC. It does this through the use of an elastic constitutive law that aids the convergence of the displacement field to mechanically admissible values. A 10pixel element size and regularization length of 100pixels were used for the calculations.

Pixel size varied between 0.025 and 0.05mm from one test to another and this scale was determined using the measured width of the sample. The displacement fields in photo n were calculated with respect to the reference image using the results from photo n -1 to initialize the calculation. That is, Figures 8 and 9a report the nominal strain fields calculated using the first image as the reference image. Figures 8 and9b show the strain rate fields computed with the (Lagrangian) velocity field.

When Lüders and/or PLC bands were observed, the width and angle of strain bands were characterized using ImageJ [START_REF] Abramoff | Image processing with ImageJ[END_REF]. Images of the longitudinal strain field where the bands were clearly defined were selected and 8 line plots of the strain level were analyzed along the tensile direction in each image.

The resulting peaks in the line profiles were fitted with Gaussian functions.

The full width mid height (FWMH) was used to calculate band width and the peak locations in each of the 8 profiles were used to calculate the band angle. Band width measurements are corrected for the displacement of the band during the 3s interval between images. The band velocity is calculated using the shift in position of the strain peaks from one image to another.

Magnetic Measurements of Austenite Volume Fraction

A measurement system was developed based on Ref. [START_REF] Bormio-Nunes | Piezomagnetic behavior of FeAlB alloys[END_REF] in order to per-form in-situ measurements of the retained austenite volume fraction during tensile testing. While the setup used is applied to flat tensile samples, it could easily be adapted to other geometries for either uniaxial or biaxial tests [START_REF] Hubert | Energetical and multiscale approaches for the definition of an equivalent stress for magneto-elastic couplings[END_REF].

The sizes of the coils can be made larger or smaller as needed (as long as one is careful to note the number of loops in the coil as that will be needed for magnetostatic calculations and to calibrate the applied field using, for example, a Hall effect H-coil). The primary consideration when changing sample geometries is that of the measurement depth, or "skin depth", e, which is related to the frequency in the driving coil by

e = 1 2 √ πf σµ ( 1
)
where σ is the electrical conductivity of the sample and µ its magnetic permeability. In the current case, this provides an estimation of the skin depth of 8mm which is several times greater than the thickness of the sample and ensures that the measure is volumetric. For this particular sample geometry, the frequency could be increased to 200-300Hz while still maintaining a volumetric measurement. It should be noted that this would affect the measurement in other ways, however, such as increasing the noise in the signal.

The setup consists of a primary coil placed around a tensile sample that creates a magnetic field in the sample. A schematic of the setup is provided in Figure 3. The magnetic response of the sample is then measured using a pick-up coil wound around the sample surface. The field is enclosed by two high-permeability ferrimagnetic yokes used to contain the magnetic field and concentrate it in the sample. The pick-up coil measures the average magnetic flux in the portion of the sample around which the driving coil is placed. In the current setup, this corresponds to a measurement over 37mm of the 60mm active tensile length of the sample. The measurement is performed at low frequency (here 2Hz) in order to ensure that no eddy currents are present and that the penetration depth of the measurement is sufficient to include the whole thickness of the sample. It must be noted that because the driving coil encloses the sample, it is not possible to perform DIC and magnetic measurements simultaneously on the same sample with this method. Instead, the plastic strain in tensile experiments is compared with the plastic strain from magnetic measurement experiments and the presence of bands is indicated by plateaus that manifest in the extensometer data.

While specific bands cannot be correlated to specific increases in martensite fraction, the existence of bands in both tensile and magnetic data over the same ranges would strongly indicate their coincidence. By cycling the current applied to the driving coil at a frequency of 2Hz, hystereses can be obtained using the readings from the pick-up coil with which the magnetic properties of the sample are analyzed. In this method, as it is typically employed for retained austenite measurements, the saturation value of the magnetization, M s , is used to determine the combined volume fraction of the ferromagnetic phases (here ferrite and martensite) using a simple mixture rule

M s = f α+α M s α+α + f γ M s γ (2)
where ferrite, martensite, and austenite are represented by the subscripts α, α , and γ respectively. The value of M s γ is assumed to be 0 due to the austenite's negligible magnetic response compared to ferrite or martensite.

Assuming that M s α = M s α = M s α+α , one can express the ferromagnetic volume fraction as

f α+α = M s M s α (3)
where M s is the measured saturated magnetization for a given strain level, the subscript α denotes a value measured in a 100% ferromagnetic sample, and f α+α the combined volume fraction of ferrite and martensite. Figure 4 shows the M -H hystereses thus obtained over 37mm of the 60mm tensile length for increasing levels of plastic strain for a sample annealed at 780 • C. It can been seen that as the plastic strain increases, the magnetostatic energy in the sample also increases (visible in its integral and in the value of M s ) due to the martensitic transformation.

Equation ( 3) is frequently used to calculate the retained austenite volume fraction from ferritescope measurements [START_REF] Beese | Identification of the Direction-Dependency of the Martensitic Transformation in Stainless Steel Using In Situ Magnetic Permeability Measurements[END_REF][START_REF] Vinogradov | Kinetics of deformation processes in high-alloyed cast transformationinduced plasticity/twinning-induced plasticity steels determined by acoustic emission and scanning electron microscopy: Influence of austenite stability on deformation mechanisms[END_REF][START_REF] Hecker | Effects of Strain State and Strain Rate on Deformation-Induced Transformation in 304 Stainless Steel: Part I[END_REF], but it is not suitable for in-situ measurements at stresses exceeding 100-200 MPa due to stress effects on the sample's magnetic properties [START_REF] Perevertov | Influence of the applied elastic tensile and compressive stress on the hysteresis curves of Fe-3%Si non-oriented steel[END_REF]. The value of M s α usually comes from a reference measurement and is not called into question. M s is more questionable. Figure 5 shows, for example, a clear sensitivity of the magnetization to an applied stress in unloading during which no phase change occurs. This phenomenon, which is known as the Villari effect [START_REF] Hubert | Two phase modeling of the influence of plastic strain on the magnetic and magnetostrictive behaviors of ferromagnetic materials[END_REF], must be taken into account. Equation (3) assumes that the true saturation is reached during the experiments. This is not achieved in Figure 4. This unsaturated situation leads to some localization effects (i.e. the magnetic field in the ferromagnetic phase is different from the applied field). Two kinds of corrections must consequently be considered as discussed in the following sections. ), yet the magnetization increases. This is a result of the Villari effect. The magneto-mechanical coupling demonstrated herein must be corrected if measurements are to be made under an applied stress.

Stress Correction

If measurements of the sample magnetization are to be made in-situ, then the effect of the applied stress on the magnetic properties must be taken into account since it has been shown that even modest loads can have a non-negligible impact on magnetization [START_REF] Vanoost | Magnetic hysteresis at the domain scale of a multi-scale material model for magneto-elastic behaviour[END_REF]. A model has been recently developed, which was used to predict the magneto-mechanical coupling in dual phase steels for stresses up to approximately 600 MPa by relating kinematic hardening to magneto-elastic effects occurring within each phase at the grain scale [START_REF] Hubert | Two phase modeling of the influence of plastic strain on the magnetic and magnetostrictive behaviors of ferromagnetic materials[END_REF]. This approach has been adapted to the current study to correct for the Villari effect.

The multidomain model used here was first introduced in Ref. [START_REF] Lazreg | A multidomain modeling of the magnetoelastic behavior for nondestructive evaluation[END_REF]. It is able to model the magnetization and the magnetostrictive strain of an isotropic polycrystalline ferromagnetic material submitted to a uniaxial applied stress. Two scales are involved: the domain scale (6 domains are considered) and the grain scale taken as an equivalent polycrystalline medium.

Macroscopic and crystallographic frames of reference are coincident (i.e.

(x, y, z)=([100],[010],[001])). In the simplified description we propose here, the rotation of the magnetization is not considered. The stress tensor σ and magnetic field H are assumed to be homogeneous. The free energy of a domain φ then becomes a sum of the magnetostatic (Zeeman) and magnetoelastic energies:

W φ = -µ 0 H • M φ -σ : µ φ ( 4
)
where µ 0 is the magnetic permeability of vacuum, M φ is the magnetization vector inside a domain such that |M φ |= M s α , and µ φ the magnetostrictive strain tensor (see details in Ref. [START_REF] Lazreg | A multidomain modeling of the magnetoelastic behavior for nondestructive evaluation[END_REF]). The volume fraction of a domain f φ is calculated from Boltzmann statistics:

f φ = exp(-A s W φ ) φ exp(-A s W φ ) (5)
where A s is a fitted parameter proportional to the magnetic initial susceptibility χ 0 without an applied stress (A s = 3χ 0

µ 0 M s α 2
). The magnetization of the single crystal is obtained from an averaging operation

M = φ f φ M φ (6)
when the magnetic field and uniaxial stress σ are applied along x. After simplifying and applying a second order Taylor expansion of the exponential, one obtains

M = M • x = χ 0 H 1 + 6χ 0 µ 0 M s α 2 λσ ( 7
)
where λ is a global magnetostriction constant. This relationship permits a new definition of the initial susceptibility in which it is linearly related to the magnitude of stress as

χ 0 σ = M H = χ 0 1 + 6χ 0 µ 0 • M s α 2 λσ (8)
This equation applies for the initial susceptibility but is assumed to also be applicable for the secant susceptibility at any magnetic field level and particularly at the magnetic field level at which the experimental saturation of magnetization is measured. A linear correction of the stress effect is then obtained

χ σ = χ(1 + ησ) (9)
where η is a constant that can be identified using magnetization measurements obtained during elastic unloading (see Figure 5) without any phase transformation. The constant η is typically on the order of -10 -5 MPa -1 .

This relationship between susceptibility and stress applies to the magnetization level as well, so a stress-sensitive "saturated" magnetization is defined as

M s σ = f α+α χ σ H = f α+α M s α (1 + ησ) = M s (1 + ησ) (10)
Equation ( 3) is thus modified to become

f α+α = M s σ (1 + ησ)M s α ( 11
)
which provides a correction to the ferromagnetic volume fraction f α+α for a magnetic saturation value M s obtained with applied stress σ.

The effect of the stress correction is shown in Figure 6 for a sample annealed at 780 • C. This linear correction of the M s value improves the accuracy of retained austenite measurements at high stress where differences of up to 0.05 in f α+α were observed between calculations using M s and M s σ . It also makes it possible to determine whether or not stress-induced martensite transformation occurs.

Localization Correction

The ratio presented in Equation ( 3) is the most commonly used method of calculating the ferrite + martensite volume fraction using magnetic measurements. However, these equations require the assumption that the field H is uniform throughout the microstructure, which is not true as the negligible magnetic response of the austenite creates a demagnetizing field in the ferromagnetic phase [START_REF] Hao | Modelling the electromagnetic response of two-phase steel microstructures[END_REF], which changes the magnetic properties of the macroscopic sample. An Eshelby model for a spherical inclusion embedded in a homogeneous matrix can be used to localize the field H within each phase and thus correct for the effect of the demagnetizing field on the magnetic susceptibility χ of the sample [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF][START_REF] Mballa-Mballa | Multidomain modelling of the magneto-mechanical behaviour of dual-phase steels[END_REF]. If the magnetic behavior in a phase i is described by

M i = χ i H i (12)
then the solution to the Eshelby problem will give

H i -H ∞ = 1 3 + 2χ ∞ (M ∞ -M i ) ( 13
)
where H ∞ and M ∞ denote the macroscopic (average) magnetic field and magnetization and χ ∞ is the macroscopic susceptibility defined as χ ∞ = M ∞ /H ∞ . Applying Equation ( 13) to the current problem without an applied stress, the field along the magnetization axis in the ferromagnetic phase α becomes

H α = H + 1 3 + 2χ (M -M α ) (14)
with χ = M /H. With the magnetic susceptibility of the ferromagnetic phase defined as χ α = Mα /Hα, it follows that

H H α = 3 + 2χ + χ α 3 + 3χ (15)
where the volume fraction of the ferromagnetic phase is defined as

f α = M M α (16)
It should be noted that this definition differs from the classical one given in Equation (3) since true saturation is not achieved. This relationship is most appropriate for low magnetic field levels where the magnetic behavior can be considered as linear. Equation ( 16) is transformed by introducing the corresponding magnetic fields and susceptibilities

f α = χH χ α H α (17)
Finally, by combining Equations ( 15) and ( 17), f α reads

f α = χ(3 + 2χ + χ α ) χ α (3 + 3χ) (18)
However, this expression is only valid considering the initial susceptibility (or linear behavior). By recognizing that χ = 1 3 A s µ 0 M s2 and χ α =

f α ≈ M s M s α 2 (19)
The volume fraction of ferromagnetic phase thus obtained provides a second estimation of f α+α with the hypothesis that the ferromagnetic regions do not interact closely. This description is only partially true and a more appropriate estimation may be defined between expressions (3) and [START_REF] Kang | On the sequence of inhomogeneous deformation processes occurring during tensile deformation of strip cast AA5754[END_REF]. We propose the use of a combination of the two estimations using a ratio parameter κ, which leads to

f α+α = M s M s α + κ M s M s α 2 - M s M s α ( 20
)
for the case without applied stress and

f α+α = M s σ (1 + ησ)M s α 1 + κ M s (1 + ησ)M s α -1 (21)
to account for the Villari effect and measured saturation M s σ . The κ parameter is identified via a magnetic measurement on a partially transformed material (in the unstrained state) with a known volume fraction of ferromagnetic material and was determined to be 2 3 in the current case.

If Equation ( 21) is used instead of the ratio given in Equation (3), this correction improves measurement accuracy when high volume fractions of retained austenite are present.

In the current study, this method is implemented for in-situ tensile testing at a strain rate of 5 • 10 -4 s -1 on flat tensile samples for each intercritical annealing temperature T IA . While the quasi-static strain rate used herein is hardly representative of the strain rates used in forming operations, the intention is to demonstrate the utility of the method in performing high-frequency measurements of the retained austenite volume fraction. A successful implementation of this method would make it possible to perform further studies at much higher strain rates for which no method of measuring the retained austenite fraction during straining exists.

Results and Discussion

Tensile Characterization

The samples' mechanical behavior in uniaxial tension varied significantly with intercritical annealing temperature. Such a variation of mechanical behavior with respect to intercritical annealing temperature is frequently observed in Medium Mn steels even for differences in T IA of only 5-10 • C [START_REF] Arlazarov | Evolution of microstructure and mechanical properties of medium Mn steels during double annealing[END_REF][START_REF] Cai | Austenite stability and deformation behavior in a cold-rolled transformation-induced plasticity steel with medium manganese content[END_REF][START_REF] Chen | Correlation between mechanical properties and retained austenite characteristics in a low-carbon medium manganese alloyed steel plate[END_REF][START_REF] Gibbs | Austenite stability effects on tensile behavior of manganeseenriched-austenite transformation-induced plasticity steel[END_REF][START_REF] Lee | Tensile Behavior of Intercritically Annealed Ultra-Fine Grained 8% Mn Multi-Phase Steel[END_REF]. As seen in Figure 7(a), the samples annealed at 740 • C and 760 • C showed significant yield point elongation in the form of a Lüders plateau.

The samples annealed at 780 • C did not show a traditional Lüders band, but rather a sort of pseudo-plateau during which stress is not constant. Tests using DIC later confirmed that there is no strain localization in samples annealed at 780 • C, so the pseudo-plateau observed on its tensile curve is not a Lüders effect. The strain hardening of the samples, shown in Figure 7(b) was characterized using the derivative of the tensile curve, dσ /d , plotted against the true stress. Approximating dσ /d with a trapezoidal method will naturally create a very noisy signal, so a Savitsky-Golay filter [START_REF] Savitsky | Smoothing and Differentiation of Data by Simplified Least Squares Procedures[END_REF] was applied to smooth out the data. It can be seen that the samples annealed at 740 • C and 760 • C exhibited significant serrations in the strain hardening curves that were not observed in the sample annealed at 780 • C. These serrations are very similar to those previously observed [START_REF] Cai | Austenite stability and deformation behavior in a cold-rolled transformation-induced plasticity steel with medium manganese content[END_REF][START_REF] Gibbs | Austenite stability effects on tensile behavior of manganeseenriched-austenite transformation-induced plasticity steel[END_REF][START_REF] Lee | Effect of the Intercritical Annealing Temperature on the Mechanical Properties of 10Pct Mn Multi-phase Steel[END_REF] and are expected to come from the same phenomenon-strain bands due to a sort of dynamic strain aging effectobserved in other Medium Mn alloys. Such serrations are undesirable for forming operations because passing below the Considére criterion can lead to premature necking if the relaxation associated with a given strain band is significant. The jerky flow in samples annealed at 740 • C and 760 • C was further characterized using DIC.

Digital Image Correlation

Strain maps obtained by DIC showed two distinct phenomena, namely, a clear Lüders band at yielding, followed by a series of other strained bands that were primarily propagative. It can be seen in Figure 8 that for the sample annealed at 740 • C, several propagative PLC-type bands move across the gauge length of the specimen and always in the opposite direction of the initial Lüders band. The strain increment everywhere outside the band is very small. In the sample annealed at 760 • C, a Lüders band is once again observed, but the subsequent bands are much less continuous. Bands frequently will stop suddenly and another band appears elsewhere along the length of the sample, as seen in Figure 9. The bands themselves were characterized by using ImageJ to analyze images of the strain increment field for the component along the loading direction ∆ε yy and plotting several line profiles along the loading direction.

Using the peaks in the line profiles, the band width and inclination angle from the loading direction were calculated for both the Lüders bands and the subsequent PLC-type bands. The PLC-type bands that appeared in the sample annealed at 760 • C were too random to obtain reliable measurements of their propagation speed and width, so values are only presented for the sample annealed at 740 • C. Table 1 summarizes the results of the band characterization (note that the velocities given for PLC bands is the average of each PLC band for that T IA ). Figure 10 shows the strain magnitude in each band for both intercritical annealing temperatures. The strain in the first band is significantly higher than in subsequent band, suggesting that two different phenomena are responsible for the first band and those that follow. As can be seen in Figure 11, the velocity of the PLC-type bands decreased roughly linearly with total strain while the strain rate in the band remained constant. Both this trend and the observed velocities are in agreement with previous experiments [START_REF] Wang | Kinematic and thermal characteristics of Lüders and Portevin-Le Châtelier bands in a medium Mn transformation-induced plasticity steel[END_REF]. Comparing these results to those published for aluminum alloys or TWIP steels exhibiting the PLC effect, it can be seen that for propagative type-A PLC bands the band velocity also linearly decreases with strain at a rate roughly one order of magnitude slower than observed here [START_REF] Renard | Characterisation of the Portevin-Le Châtelier effect affecting an austenitic TWIP steel based on digital image correlation[END_REF][START_REF] Ait-Amokhtar | Kinematics fields and spatial activity of Portevin-Le Chatelier bands using the digital image correlation method[END_REF][START_REF] Ait-Amokhtar | Crossover from continuous to discontinuous propagation in the Portevin-Le Chatelier effect[END_REF]. It is thus proposed that the bands observed in Medium Mn steels are likely type-A PLC bands. The difference in rate of band velocity decrease with strain could be due to chemical composition (via the solute concentrations and concentration gradients), and interaction with the martensitic transformation, or both. PLC resulting from an interaction be-

Magnetic Measurements

The in-situ measurement of sample magnetization throughout tensile testing allowed the retained austenite volume fraction (via the combined volume fractions of ferrite and martensite) to be calculated. This method provided continuous phase volume fraction measurements over the 37mm of the sample's gauge length that is contained within the driving coil, thereby ensuring a very large measurement volume that is statistically representative of the steel's microstructure. Additionally, because of the difference in the magnetic measurement length and the tensile length of the sample (60mm), if any strain localizations are present that induce TRIP their effect should produce plateaus in the measurement of the austenite volume fraction as a function of strain. Figure 12 presents the true stress and retained austenite volume fraction as a function of true plastic strain as calculated from the crosshead displacement for each intercritical annealing temperature. The plastic strain is calculated by assuming that the tensile machine deforms purely elastically, making it possible to simply remove the elastic strain e = σ/E. An important clarification must be made. Because the magnetic measure-428 ment system completely encloses the tensile sample, the strain in the sample 429 was not directly measured during tensile testing. Instead, the true plastic strain is used in order to eliminate the elastic strain in the tensile machine as recorded by the crosshead displacement. The curves presented for each intercritical annealing temperature in Figure 12 are thus from two different samples and, while the dispersion in tensile data was small, the indications of banding will not be identical in two curves for the same annealing temperature. This effect aside, it is clear that in the magnetic data the retained austenite volume fraction decreases sharply at several points, creating the same "steps" that were previously seen in tensile data using an extensometer. This strongly suggests that the passage of strain bands and the TRIP effect are, in fact, coincident.

Previous work by Wang et al. [START_REF] Wang | Kinematic and thermal characteristics of Lüders and Portevin-Le Châtelier bands in a medium Mn transformation-induced plasticity steel[END_REF] showed that the TRIP effect occurred ≈1-2%), the heat given off by the reaction is "drowned out" by the adiabatic heating (even at quasi-static strain rates) in the band and thus difficult to distinguish by infrared imagery. It is also possible that the retained austenite in Wang et al.'s steel was simply stable enough to not transform noticeably.

This point is important to address because the origin of the PLC-type localizations is, as mentioned in Section 3.2, a point of contention currently and if one is able to have PLC without a phase transformation in these steels it would indicate that the PLC effect is not initiated by the phase transformation.

The correction of the stress effect on magnetization allowed stress-induced and strain-induced martensite transformations to be separated and showed that no stress-induced transformation occurred during elastic deformation.

Figure 13 shows the sample magnetization vs. true stress for each intercritical annealing temperature. In all samples the magnetization remains constant in the elastic domain, proving that no stress-induced transformation occurs during elasticity and that the first martensitic transformation occurs only after the onset of plasticity. The changes of the austenite volume fraction (f γ = 1-f α+α ) as calculated from magnetization measurements were used to fit the parameters of an Olson-Cohen model [START_REF] Olson | Kinetics of Strain-Induced Martensitic Nucleation[END_REF]

f α+α = 1 -exp(-β(1 -exp(-α p ) n ) ( 22
)
where the α parameter is related to both the number of shear bands and the stacking fault energy of the austenite, β accounts for the probability of a martensite nucleus to form, which is in turn defined by the driving force for the phase transformation. The exponent n is usually considered a material constant, but is not fixed in the current case due to the varying composition of the austenite for each T IA . Perhaps surprisingly, the model can be fit to the measured austenite volume fraction change fairly well even when strain localizations are present as can be seen in Figure 14. This provides a set of effective Olson-Cohen parameters, given in Table 2, that are valid in a macroscopic sense for a given microstructure and austenite stability. Estimations of the martensite start temperature, M S , are also provided in Table 2. Alternatively, it is also possible that the trends in the model parameters are not the same when varying test temperature (as Olson and Cohen did) or austenite composition/volume fraction (as in the current study). In any case, one could further investigate the interactions between phases using numerical models parameterized using the kinetics obtained for each intercritical annealing temperature in these experiments. Doing so would allow for a better description of the true kinetics since microstructure-level strains and stresses would be available.

Conclusions

A system for measuring the retained austenite volume fraction via sample magnetization has been developed and used in-situ during tensile testing.

This method provides very rapid measurements of the retained austenite fraction (here 2Hz) relative to other methods such as XRD or EBSD and enables measurements to be conducted without pausing or otherwise interrupting the tensile test. However, the measurement system encloses the sample, so the strain must either be calculated from the crosshead displacement or measured with a strain gauge (extensometers or DIC methods cannot be used on a sample upon which these magnetic measurements are made).

This system was employed to study the effects of intercritical annealing temperature and strain instabilities on the TRIP effect present in samples of a Fe-0.2C-5Mn-2.5Al Medium Mn steel that had been intercritically annealed The experimental procedures established were intended to study the effects of T IA and strain instabilities on TRIP, but provide a tool that can be extended to use in the optimization of Medium Mn TRIP steels in general. The magnetic measurement system is capable of measuring the retained austenite fraction in-situ even for the high strain rates used in forming operations or for applied loads other than unidirectional tension. The data obtained could be used to calibrate a micromechanical model of TRIP kinetics and thus enable the metallurgist to predict the martensite transformation rates for a given microstructure and strain rate. The extension of this method to such a purpose is underway as well as studies of the effect of the applied strain rate on transformation kinetics.

Figure 1 :

 1 Figure 1b after straining until rupture. The variations in intercritical annealing temperature result in samples with varying retained austenite stability and initial phase volume fractions. As the retained austenite is metastable, it can transform to α (bct) martensite when mechanically loaded, thereby resulting in high strain hardening 1 .

Figure 2 :

 2 Figure 2: Initial phase volume fractions in the unstrained state after annealing and quenching as measured via XRD (α and γ) and as estimated from SEM images (α ) as a function of the intercritical annealing temperature used.

Figure 3 :

 3 Figure 3: Schematic of the magnetic measurement system. A primary coil (H-field) placed around the tensile sample induced a magnetic H field in the sample, and the current thereby induced in the sample is measured by the probe coil (B-field) and converted to magnetization, M . The coils are enclosed in two ferrimagnetic yokes to contain the magnetic fields.

Figure 4 :

 4 Figure 4: M -H hystereses obtained during unloading of a sample annealed at 780 • C after several levels of plastic strain (denoted by the color bar) where M is the sample magnetization and H the applied magnetic field. The saturation value of magnetization, M s , increases with increasing plastic strain due to the martensite transformation.

Figure 5 :

 5 Figure 5: In-situ magnetization measurements of a sample annealed at 760 • C during unloading after straining until a point just before necking. During unloading, the microstructure does not change (i.e. no TRIP occurs), yet the magnetization increases. This is a result of the Villari effect. The magneto-mechanical coupling demonstrated herein must be corrected if measurements are to be made under an applied stress.

Figure 6 :

 6 Figure 6: Magnetization and volume fraction of ferromagnetic phases (ferrite + martensite) for T IA = 780 • C as functions of plastic strain as calculated from in-situ data with and without the stress correction developed in this study. High levels of stress reduce the measured magnetization and result in several percent difference in calculated volume fractions.

1 3 A s µ 0 M s α 2

 12 following the theoretical development in Section 2.3.1 and considering χ 1, the volume fraction of ferromagnetic phase is expressed as

Figure 7 :

 7 Figure 7: (a) Representative true stress -true strain curves for each intercritical annealing temperature employed. (b) Strain hardening rate dσ/d plotted as a function of the true stress σ for each intercritical annealing temperature T IA . The Considère criterion, wherein the strain hardening rate is equal to the applied true stress, is included as well to show that localizations can bring the hardening rate below this criterion and lead to premature necking.

Figure 8 :Figure 9 :

 89 Figure 8: Spatiotemporal DIC measurements of (a) nominal longitudinal strain and (b) longitudinal strain rate measurements for annealing at 740 • C. The horizontal axis is the time in seconds, the vertical axis represents the position along the tensile direction, and the color bar represents either the total strain in the tensile direction or the strain rate (1/s) in the tensile direction.

Figure 10 :

 10 Figure 10: Plot of the longitudinal strain increment in each band as a function of time for samples annealed at 740 • C and 760 • C. The stark difference in magnitude between the first band and those that follow suggests two different underlying mechanisms. It is proposed that there is an initial Lüders band followed by PLC-type bands.

  tween dislocations and Mn-C clusters has been previously proposed by Jung and DeCooman[START_REF] Jung | Temperature dependence of the flow stress of Fe-18Mn-0.6C-xAl twinning-induced plasticity steel[END_REF], who suggested that the movement of C atoms in C-Mn clusters between octahedral and tetrahedral interstices could lock partial dislocations in stacking faults. Hickel et al.[START_REF] Hickel | Impact of nanodiffusion on the stacking fault energy in high-strength steels[END_REF] also confirmed a strong effect of C-Mn clusters on SFE and showed that the active deformation mechanism can locally redistribute solutes and thus significantly change the local SFE.

Figure 11 :

 11 Figure 11: Calculated band propagation velocities and the strain rate within each band plotted as a function of the nominal strain for the sample annealed at 740 • C. The velocities of the PLC-type bands decrease roughly linearly with strain while the strain rate in the band remains more or less constant for bands occurring after the Lüders band.

Figure 12 :

 12 Figure 12: Plots of (a) true stress and (b) retained austenite volume fraction vs. true plastic strain for each intercritical annealing temperature. The same steps previously observed in tensile data are again seen in the retained austenite volume fraction, suggesting that the TRIP effect coincides with the passage of strain bands. Both plots share identical strain axes.

  during the Lüders plateau, but that little or no transformation occurred during PLC. It has also been shown by Sun et al. [28] that some transformation does occur during PLC. The current work then confirms Sun et al.'s result that TRIP occurs both during Lüders deformation and PLC. It is possible that when the amount of austenite transformed by a single band is low (i.e.

Figure 13 :

 13 Figure13: Sample magnetization (with stress correction) plotted as a function of true stress shows that during elasticity, the magnetization remained constant meaning that there is no stress-induced transformation before yielding.

Figure 14 :

 14 Figure 14: Experimental transformation rates of retained austenite in alloys annealed at 740 • C, 760 • C, 780 • C (symbols), fit with a standard Olsen-Cohen model (dashed lines).

  at 740 • C, 760 • C, and 780 • C. There was a clear difference in the rate of martensitic transformation for different intercritical annealing temperatures, though it is not understood exactly how the micromechanics of the TRIP effect are altered by varying T IA . Plateaus corresponding to the passage of strain bands appear both in tensile data measured with an extensometer and in magnetization data. This indicates that the martensitic transformation and the passage of strain bands occur simultaneously. DIC analyses of the strain bands present in samples annealed at 740 • C and 760 • C showed that the Lüders and post-Lüders bands are from two separate phenomena and that, because the band velocity in the sample annealed at 740 • C decreased linearly with strain at a rate similar to that observed in Al and TWIP steels exhibiting PLC[START_REF] Zdunek | Digital Image Correlation investigation of Portevin-Le Chatelier effect in an aluminium alloy[END_REF][START_REF] Zavattieri | Spatio-temporal characteristics of the Portevin-Le Chatelier effect in austenitic steel with twinning induced plasticity[END_REF][START_REF] Coër | Piobert-Lüders plateau and Portevin-Le Chatelier effect in an Al-Mg alloy in simple shear[END_REF], the post-Lüders bands are likely to be propagative type-A PLC bands. The micromechanical origin of the PLC effect is unknown and has been hypothesized to be related to either C-Mn clusters or the martensite transformation itself[START_REF] Cai | Austenite stability and deformation behavior in a cold-rolled transformation-induced plasticity steel with medium manganese content[END_REF][START_REF] Gibbs | Austenite stability effects on tensile behavior of manganeseenriched-austenite transformation-induced plasticity steel[END_REF][START_REF] Lee | Effect of the Intercritical Annealing Temperature on the Mechanical Properties of 10Pct Mn Multi-phase Steel[END_REF].

Table 1 :

 1 DIC strain band characterization

		Velocity (mm/s) Band Width (mm) Inclination θ( • )
	740 • C Lüders	0.381 ± 0.008	0.605 ± 0.033	66.4 ± 0.496
	PLC	0.583 ± 0.084	0.770 ± 0.042	69.8 ± 0.605
	760 • C Lüders	0.623 ± 0.003	0.246 ± 0.044	69.3 ± 0.800

Sometimes (hcp) martensite appears as an intermediary phase between γ and α . In the material studied, no martensite was observed in TEM or XRD so it is assumed that the retained austenite transformed directly to α martensite when plastically strained. If martensite were present, the magnetic measurements would include it within the ferromagnetic volume fraction (not with the retained austenite).

Acknowledgements

The authors would like to thank Dr. Denis Solas from Université Paris Sud for his expertise with XRD measurements. This study has been performed within the framework of the ANR 13-RMNP-0002 MedMnAl Steels funded by the Agence Nationale de la Recherche (ANR). This project is supported by the competitive cluster "Materalia".