Horst Tempelmeier

Sascha Herpers

ABC β --A Heuristic for Dynamic Capacitated Lot Sizing with Random Demand under a Fillrate Constraint

Keywords: CAPACITATED LOT SIZING, HEURISTICS Random Demand lot-sizing, capacity, dynamic random demand, fillrate

  L'archive ouverte pluridisciplinaire

We consider the stochastic version of the dynamic multi-item capacitated lot sizing problem (CLSP). The problem is to determine production quantities to satisfy demands for multiple products over a finite discrete time horizon such that the sum of setup and holding costs is minimized, whereby a capacity constraint of a resource must be taken into consideration. Setup times are assumed to be zero. In contrast to the deterministic CLSP, it is assumed that for every period t and product k the demand D kt (k = 1, 2, . . . , K; t = 1, 2, . . . , T ) is specified by its mean and its coefficient of variation.

The period demand may be non-stationary (to permit dynamic effects such as seasonal variations, promotions, or general mixtures of known customer orders with random portions of period demands). Demand that cannot be filled immediately from stock on hand is backordered. As the precise quantification of shortage penalty costs which involve intangible factors such as loss of customer goodwill is very difficult, if not infeasible, we assume that management has set a target service level. In particular, we assume that the fillrate criterion (β service level) is in effect, as this criterion is very popular in industrial practice [see [START_REF] Tempelmeier | Inventory management in supply networks -problems, models solutions[END_REF]].

In industry, mainly two different planning approaches to dynamic production planning under uncertainty are used [START_REF] Boulaksil | Setting safety stocks in multi-stage inventory systems under rolling horizon mathematical programming models[END_REF]]. The first approach is to use results from stochastic inventory theory. In this case, a production order is triggered through the occurrence of a random demand that reduces the inventory position below a given reorder point which is computed based on assumptions concerning the demand during the replenishment lead time.

The second approach, which is implemented in many industrial material requirements planning (MRP) systems, uses a forecasting procedure that provides a deterministic time series of future demands. Uncertainty is taken into consideration by reserving a fixed amount of inventory as safety stock. The amount of this reserve stock is usually computed with simple rules of thumb, e. g. the standard deviation of the demand during the risk period is multiplied with a quantile of the standard normal distribution, or even a multiple of the average demand. In the MRP calculations the safety stock is added to the net demand which is then used in the deterministic lot size calculations performed with a dynamic uncapacitated single-item lot sizing procedure [see [START_REF] Wijngaard | Mrp and inventories[END_REF]]. Whether the safety stock is set once for the complete planning horizon or is adjusted from time to time, in both approaches the actual timing and size of the replenishments are the outcome of the observed demand process, which is random. This directly translates into random resource requirements which from a practical point of view is undesirable.

Literature

The deterministic multi-item capacitated lot sizing problem has been studied for a long time. For recent overviews see [START_REF] Karimi | The capacitated lot sizing problem: A review of models and algorithms[END_REF] and Buschkühl et al. (2009). However, only a relative limited number of researchers have considered the capacitated lot sizing problem under random demand. A literature overview is presented in [START_REF] Sox | A review of the stochastic lot scheduling problem[END_REF].

Depending on the modeling of the time axis, two groups of stochastic lot sizing models can be differentiated. Continuous time models are the stochastic counterpart of the Economic Lot Scheduling Problem (ELSP) which assumes stationary demands for all products. Discrete time models are extensions of the Capacitated Lot Sizing Problem (CLSP). These models assume time-varying dynamic demands. While some research has been done on continuous time models, only very few papers have been presented that treat capacitated lot sizing under uncertainty in discrete time. This is surprising, as many industrial planning environments such as Material Requirements Planning (MRP) systems or Advanced Planning Systems (APS) are based on a discrete time structure and randomness is a major issue in most practical planning situations. [START_REF] Sox | Optimization-based planning for the stochastic lotscheduling problem[END_REF] solve a variant of the stochastic dynamic CLSP, where item-and period-specific backorder costs as well as extendible production capacities are considered. The authors propose a Lagrangean heuristic to solve the resulting non-linear integer programming problem that is repeatedly applied in a dynamic planning environment. [START_REF] Martel | Multiple items procurement under stochastic nonstationary demands[END_REF] proposed a branch-andbound procedure for the solution of a similar problem formulation. [START_REF] Brandimarte | Multi-item capacitated lot-sizing with demand uncertainty[END_REF] considers the stochastic CLSP where the uncertainty of the demand is represented through a scenario tree and unfilled demand is lost. As to our knowledge, no results are available for the dynamic capacitated lot sizing problem under random demand, when the performance is measured in terms of a fillrate. Considering the popularity of the fillrate in industrial practice, this is an open problem that deserves being studied.

The majority of papers that consider dynamic lot sizing problems under random demand refer to single-item problems with unlimited capacity. For our work the most relevant is the paper of [START_REF] Bookbinder | Strategies for the probabilistic lot-sizing problem with service-level constraints[END_REF]. These authors studied several so-called strategies which define different modes of planning the timing and size of the production quantities. According to the " static-dynamic uncertainty strategy", at the beginning of the planning horizon all production periods are fixed in advance. The actual production quantities, however, are determined with the help of order-up-to levels that are computed based on a target service level. As the actual production quantities are the result of the observed demands, the " static-dynamic uncertainty strategy" leads to random resource requirements which impedes the precise consideration of capacity constraints.

A second strategy discussed by [START_REF] Bookbinder | Strategies for the probabilistic lot-sizing problem with service-level constraints[END_REF] is the so-called " static uncertainty strategy". Here, the timing and the size of all production quantities are fixed in advance for the complete planning horizon. This strategy, although being inferior as far as the costs are concerned, has the significant advantage that the production quantities are deterministic decision variables and it is therefore possible to respect limited capacities in the presence of dynamic random demands.

Compared to the rather limited amount of literature on dynamic stochastic capacitated lot sizing, there exists a large number of publications on the stochastic economic lot scheduling problem with stationary demands. These are reviewed in [START_REF] Sox | A review of the stochastic lot scheduling problem[END_REF] and [START_REF] Winands | The stochastic economic lot scheduling problem: A survey[END_REF].

In this paper we present a formulation of the stochastic dynamic capacitated lot sizing problem (SCLSP) that implements the " static uncertainty" strategy of [START_REF] Bookbinder | Strategies for the probabilistic lot-sizing problem with service-level constraints[END_REF] and takes into account random demand and a fillrate constraint. We propose a heuristic solution procedure to this problem. The result is a production plan for the entire planning horizon that respects the available production capacity in every period, despite the randomness of demands.

In Section 3 the mathematical model is formulated. In Section 4 we present a solution approach that is based on the A/B/C-Heuristic originally introduced by [START_REF] Maes | A simple heuristic for the multi item single level capacitated lotsizing problem[END_REF] for the solution of the deterministic CLSP. In Section 5 the results of a numerical experiment are given. The last section contains some conclusions. These moments, which may vary over time, are the outcome of a forecasting procedure.

Unfilled demands are backordered and the amount of backorders is controlled by imposing a fillrate (β service level) constraint. We define the fillrate as the ratio of the expected demand observed during the coverage time of a production order that is routinely filled from stock on hand and the actual lot size. More precisely, let τ be a production period of product k, let t be the period immediately before the next production of product k and let q kτ t be the lot size produced in τ covering the demand up to t. Finally, let F kt (q kτ t ) be the product k backorders that occur in period t. Then for a target service level

β ⋆ k it is required, that 1 - E t i=τ F ki (q kτ t ) E t i=τ D ki ≥ β ⋆ k k = 1, 2, . . . , K (1)
This constraint is equivalent to the fillrate definition under stationary conditions which relates the average backorders per cycle to the average replenishment quantity. At the beginning of the planning horizon there is a known initial inventory I k0 (k = 1, 2, . . . , K)

which may be zero.

The mathematical formulation of the considered dynamic multi-item Stochastic Capacitated

Lot-Sizing Problem with fixed production quantities and fillrate constraint is as follows: 1 -

Model SCLSP q β Minimiere Z = K k=1 T t=1 s k • γ kt + h k • E [I kt ] + (
I k,t-1 + q kt -D kt = I kt k = 1, 2, ..., K; t = 1, 2, ..., T (3)
q kt -M • γ kt ≤ 0 k = 1, 2, ..., K; t = 1, 2, ..., T (4)
K k=1 tb k • q kt ≤ b t t = 1, 2, ..., T (5)
I f,prod kt = -[I k,t-1 + q kt ] - k = 1, 2, ..., K; t = 1, 2, ..., T (6)
I f,end kt = -[I kt ] - k = 1, 2, ..., K; t = 1, 2, ..., T (7)
F kt = I f,end kt -I f,prod kt k = 1, 2, ..., K; t = 1, 2, ..., T (8)
l kt = (l k,t-1 + 1) • (1 -γ kt ) k = 1, 2, ..., K; t = 1, 2, ..., T (9)
l k,0 = -1 k = 1, 2, ..., K (10)
ω kt = γ k,t+1 k = 1, 2, ..., K; t = 1, 2, ..., T -1 (11) ω kT = 1 k = 1, 2, ..., K (12)
E t j=t-l kt F kj E t j=t-l kt D kj ≥ β * k k = 1, 2, ..., K; t ∈ {t | ω kt = 1} (13) q kt ≥ 0 k = 1, 2, ..., K; t = 1, 2, ..., T (14)
γ kt ∈ {0, 1} k = 1, 2, ..., K; t = 1, 2, ..., T (15)
Legend: (4) forces the setup indicator γ kt to 1, whenever there is a positive production quantity q kt , according to the assumptions of a big-bucket lot sizing model. Constraint (5) requires that the available capacity b t per period must not be exceeded. Equation ( 6) defines the backlog in period t immediately after a production has taken place and all outstanding backorders, if any, have have been filled as much as possible before the new demand of period t is filled. Equation ( 6) describes the backlog at the end of period t. Equation ( 8)

defines the backorders that newly occurred in period t.

The remaining equations are use for book-keeping. In order to calculate the average fillrate during an order cycle, we count the number of periods covered by a production quantity with the help of variable l kt . Equations ( 11) and ( 12) set the indicator variable ω kt to 1, if either period t + 1 is a setup period or the planning horizon ends in period t. In addition, the length of an order cycle (the number of periods between two consecutive setups) must be known. This is computed with the help of equations ( 9) and ( 10). l kt is reset to zero whenever γ kt = 1, i. e., when t is a setup period for product k. Otherwise, l kt is incremented by one to (l k,t-1 + 1). Equation ( 13) defines the expected fillrate within the actual production cycle since the last production of product k.

The expected inventory of product k at the end of period t is calculated as follows. Let

Q (t)
k be the sum of the inital inventory I k0 and the production quantities produced up to period t and let Y (t) k denote the cumulated demands from period 1 to t. Then the expected inventory at the end of period t is equal to E{[Q

(t) k -Y (t)
k ]} + . This can be written as

E{I p kt } = Q (t) k 0 (Q (t) k -y) • f Y (t) k (y) • dy = Q (t) k -E{Y (t) k } + G 1 Y (t) k (Q (t) k ) k = 1, 2, . . . , K; t = 1, 2, . . . , T (16)
where G 1 Y (Q) is the first-order-loss function with respect to the random variable Y and the quantity Q.

Note that with model SCLSP q βc the timing and the size of replenishments are determined in advance. The model and the underlying " static uncertainty" planning strategy (see [START_REF] Bookbinder | Strategies for the probabilistic lot-sizing problem with service-level constraints[END_REF]) are easily applicable within the context of a forecast-driven planning system such as an MRP system. It is noteworthy that there is no need to differentiate between cycle stock and safety stock, as with the replenishment quantity there is only one decision variable that ensures the fulfillment of the expected period demands as

Solution Approach

Model SCLSP q β describes a stochastic dynamic optimization problem. Up to now, there is no exact solution procedure available for this model. In the sequel we propose a heuristic solution approach which is based on the A/B/C heuristic presented by [START_REF] Maes | A simple heuristic for the multi item single level capacitated lotsizing problem[END_REF] for the solution of the deterministic CLSP. The A/B/C heuristic is a period-by-period procedure which transforms a matrix of demands d kt into a matrix of production quantities q kt (k = 1, 2, . . . , K; t = 1, 2, . . . , T ). It consists of three parts, namely the lot sizing step, a feasibility routine and an improvement step.

In the lot sizing step the demands are accumulated to lots. Thereby three parameters are used the govern the sequence in which future demands are considered for inclusion into a production lot:

A With parameter A the selection sequence of the products is influenced. Here, criteria like the average time-between-orders (TBO), the ratio of setup and holding costs, and four other criteria are used. B Once a product has been selected, parameter B defines the criterion for deciding whether the extension of the current production lot is economically favorable. Here, the criteria known from the dynamic single-item lot sizing heuristics such as the Silver-Meal criterion, the least unit cost criterion, the least total cost criterion or the absolute cost criterion are used.

C Finally, parameter C defines how the search through the demand matrix is performed, for example, product by product (east), period by period (south) or a combination thereof.

s k h k •tb k •d k

EC

expected average costs per period:

s k TBO k + h k •d k •TBO k 2 ES
expected savings when combining demand over TBO periods:

(s k -h k • d k ) + (s k -2 • h k • d k ) + ... + (s k -(TBO k -1) • h k • d k ) ESC ES divided by average capacity requirement tb k • d k Parameter B (Cost criterion) SM Silver-Meal criterion s k +h k • j t=τ (t-τ )d kt j-τ +1

LUC

Least unit cost criterion:

s k +h k • j t=τ (t-τ )•d kt j t=1 d kt LTC Least total cost criterion: s k -h k • j t=τ (t -τ ) • d kt AC Absolute cost criterion: h k • j t=τ +1 (t -τ ) • d kt ≤ s k h k Parameter C (Direction) E East S South SE South-east
Table 1: Abbreviations Whenever a production lot is about to be fixed and before the actual planning period is increased, a look-ahead routine is used to ensure feasibility. In particular, the amount of capacity caused by future demands (from periods t + 1, t + 2, . . . , T ) still to be met in the actual planning period t is computed as

CF t = max 0, K k=1 tb k • d k,t+1 -b t+1 + CF t+1 t = T -1, T -2, . . . , 1 (17)
with CF T = 0.

A specific heuristic is defined through the combination of the parameters A, B and C. As computation times are very small, [START_REF] Maes | A simple heuristic for the multi item single level capacitated lotsizing problem[END_REF] propose to select the best production plan found after applying all 72 possible combinations of the parameters.

The reported computational results show that for the deterministic CLSP the A/B/C heuristic can be expected to find high-quality solutions with low computational effort [START_REF] Maes | A simple heuristic for the multi item single level capacitated lotsizing problem[END_REF]; [START_REF] Maes | Multi-item single-level capacitated dynamic lotsizing heuristics: A general review[END_REF]). Moreover, due to its structural flexibility, the A/B/C heuristic can be used as the basis for solving the stochastic version of the CLSP, as stated by model SCLSP q β . In the following we propose a heuristic called ABC β that includes the adjustments required to use the principle structure of the A/B/C heuristic for the solution of model SCLSP q β . These adjustments refer to

• the calculation of the lot sizes,

• the cost criteria used, and

• the feasibility routine.

Calculation of the lot sizes. While in the deterministic case the lot sizes result from the cumulation of consecutive period demands, with random demand the target fillrates

β ⋆
k must be taken into consideration. If the production lot for product k in period τ is extended to cover the demand of an additional period t, then the lot size is recalculated according to the fillrate constraint as follows:

q * kτ t = min q kτ t 1 - E t i=τ F ki (q kτ t ) E t i=τ D ki ≥ β * k t = τ, τ + 1, τ + 2 . . . (18)
The optimum lot size q * kτ t can be found with a standard search procedure. Cost criteria. With deterministic demands, a typical dynamic lot sizing rule increases the production quantity in period τ until C τ,t+1 > C τ t , where C τ t denotes the 'costs' that result if the quantity produced in period τ covers the demands from period τ up to period t.

As with random demand the development of the inventory over time is random, too, the cost criteria known from the single-item dynamic lot sizing heuristics are adjusted as follows [see [START_REF] Tempelmeier | Dynamic uncapacitated lot sizing with random demand under a fillrate constraint[END_REF]]. For simplicity, in the following the product index k is omitted.

If the Silver-Meal criterion is used as the basis, we obtain

E{C τ t } = s + h • t ℓ=τ E    I τ -1 (P τ -1 ) + q * τ t - ℓ i=τ D i +    t -τ + 1 τ = 1, 2, . . . ; t = τ + 1, τ + 2, . . . ( 19
)
where P τ -1 is the actual production plan (sequence of lots) from period 1 up to period (τ -1) and τ denotes the current production period for which the lot sizes are determined.

For the adjusted Least unit cost criterion the criterion is

E{C τ t } = E              s + h • t ℓ=τ I τ -1 (P τ -1 ) + q * τ t - ℓ i=τ D i + t i=τ D i              τ = 1, 2, . . . ; t = τ + 1, τ + 2, . . . (20)
For the adjusted Least total cost criterion we get

E{C τ t } = E    s + h • t ℓ=τ I τ -1 (P τ -1 ) + q * τ t - ℓ i=τ D i +    τ = 1, 2, . . . ; t = τ + 1, τ + 2, . . . (21)
For the adjusted Absolute cost criterion we obtain max

   t h • t ℓ=τ +1 E    I τ -1 (P τ -1 ) + q * τ t - ℓ i=τ D i +    ≤ s    τ = 1, 2, . . . ; t = τ, τ + 1, . . . (22)
Feasibility routine. Under deterministic demand equation ( 17) is sufficient to ensure feasibility when proceeding from period τ to τ + 1, as the total capacity requirements in the future are constant. In the case of random demand, however, the total production quantities depend on the lot sizes which are influenced by the target fillrates. Therefore, the total capacity requirements in periods τ +1, τ +2, . . . , T and consequently the feasibility depend on the lot sizes which are determined only in a later planning step. To circumvent this problem, we take a conservative view and compute the capacity requirements based on a lot-for-lot policy for the periods (t+1, t+2, . . . , T ), where each production lot (q ⋆ k,t+1,t+1 ) in the future is determined with equation ( 18). Thus, equation ( 17) is modified as follows:

CF t = max 0, K k=1 tb k • q ⋆ k,t+1,t+1 -b t+1 + CF t+1 (23)
In any planning period τ the feasibility check may require that future demand must be produced in period τ in order to avoid a capacity overload in future periods τ < t. If this infeasibility is avoided through a new setup for a product that is not produced in period τ , then the length of the last order cycle before period τ of this product decreases.

This in turn requires the adjustment of the corresponding lot size which now covers a smaller number of period demands. In our implementation, this adjustment is made in a post-processing step at the end of the heuristic.

Finally, in the computation of the TBO criterion (parameter A) used for product selection, the average period demand is replaced by the average lot size that would result for a lotfor-lot policy.

After the lot sizing step is completed, in the same way as for the deterministic CLSP an improvement step tries to eliminate lots by combining them with earlier production quantities [START_REF] Maes | A simple heuristic for the multi item single level capacitated lotsizing problem[END_REF]; [START_REF] Dixon | A heuristic solution procedure for the multi-item singlelevel limited capacity lot-sizing problem[END_REF]).

Numerical Results

In order to test the quality of the proposed ABC β heuristic we conducted a numerical experiment including a large number of invented problem instances. Unfortunately, at present for the considered stochastic lot sizing problem there are no benchmark solutions available. Therefore, the objective of the study is to find out which parameter combination is likely to perform best and which type of parameter (A -sorting, B -cost criterion, C -search direction) has the dominating influence on the solution quality.

We constructed a set of test instances based on deterministic problems from the literature.

The first set of 360 problem instances including 12 products and 12 periods was created with an implementation of the problem generator described by [START_REF] Maes | A simple heuristic for the multi item single level capacitated lotsizing problem[END_REF]. The problem instances were generated through the variation of the five control factors studied in the experimental design used by these authors, namely variability of average demands, capacity absorption, average time between orders, tightness of capacity As noted above, no benchmark solutions are available for the problem at hand. Therefore, we cannot make precise statements concerning the solution quality of the proposed heuristic. Instead, we study the relative performance of the different parameter combinations.

In order to accomplish this, for each problem instance we proceed as follows. Let w i be the highest (worst) solution value found with any of the 72 parameter combinations. Let x ij be the solution value of parameter combination j (j = 1, 2, . . . , 72) for problem instance i.

Then the average relative deviation of parameter combination j over all 13307 problem instances is ∆ j = 1 13307 • 13307 i=1 w i -x ij w i . ∆ j can be interpreted as a relative performance measure for the solution quality of parameter combination j.

Figure 1 shows the ∆ j -values for all parameter combinations. In each triple group of vertical bars, the left bar denotes the east direction (parameter C), while the other bars stand for the south and the south-east direction, respectively. 6 Conclusions and directions for future research

In this paper we introduced a model for the single level capacitated lot sizing problem with stochastic demand and a fillrate constraint. The underlying strategy that fixes the timing and the sizes of all production quantities in advance has the positive characteristic, that limited capacities can be respected with certainty. Alternative strategies that wait until demand is observed and react on the demand do not have this advantage.

We proposed a heuristic procedure that is based on the A/B/C heuristic published by [START_REF] Maes | A simple heuristic for the multi item single level capacitated lotsizing problem[END_REF]. Our heuristic is as flexible as the deterministic variant. While our numerical study shows that the Silver-Meal criterion dominates other cost criteria in the same way as under deterministic demand conditions, the least unit cost criterion does not.

Further research will include other service levels (e. g. the probability distribution of the customer waiting time) and the consideration of setup times. In addition, different solution techniques for the current problem formulation SCLSP q β are required so that the absolute performance of the heuristic could be judged. This is the subject of our future research agenda. Further, another direction of research could be the introduction of the " static uncertainty strategy" into other dynamic lotsizing models.

  We consider K products that are produced to stock on a single resource. The planning situation is completely identical with the classical dynamic capacitated lot sizing problem without setup times (CLSP) with one exception: For each product k, the period demands D kt are random variables with given expected values E{D kt } and variances V {D kt }.

  , whereby the experimental values given in[START_REF] Maes | A simple heuristic for the multi item single level capacitated lotsizing problem[END_REF] were used. The second set of problem instances included 540 instances with 20 periods and 4, 6 and 8 products taken from[START_REF] Sürie | Description of clspl test instances[END_REF], which summed up to a total of 900 deterministic instances. For both problem sets, we added the control factors 'standard deviation of period demand' and 'fillrate'. We considered four different standard deviations of the period demands (σ ∈ {0.1, 0.2, 0.3, 0.4}) and four different target fillrates (β * ∈ {0.875, 0.925, 0.95, 0.98}) resulting in 14400 problem instances. For the problem instances with σ < 0.3 we assumed normal demands while for larger values of σ gammadistributed demands were used. Each instance was solved using all 72 combinations of the heuristic's parameters. 1093 problem instances were skipped as no feasible solution was found. The remaining 13307 problem instances are the basis for the following analysis.

Figure 2 :Page

 2 Figure 2: Average relative improvement per parameter type

Table 1

 1 provides an overview over the abbreviations used in the sequel.

	9

14 Page 14 of 18 http://mc.manuscriptcentral.com/tprs Email: ijpr@lboro.ac.uk International Journal of Production Research

		12% 12% 12% 12% 12% 12% 12% 12%	Absolute cost Sorting	Least total cost	Least unit cost Cost Criterion	Silver-Meal Direction
			E E E E E E	S S S S S S	SE SE SE SE SE SE
	Average Improvement Average Improvement	F o r 10% 6% 8% 10% 4% 6% 8% 10% 0% 2% 4% 6% 8% 10% 0% 2% 4% 6% 8% 10% 0% 2% 4% 6% 8% 10% 10% 10% Figure 1: Average relative improvements for all parameter combinations r 2% o F 8% 4% 6% 8%
		0%	EC	P P SH ES SHC TBO ESC	SM LTC LUC AC	E	S	SE
	e e r It appears that on the average the parameter combinations using the Silver-Meal criterion (parameter B) perform best, while the specification of the other parameters (A and C) plays only a minor role. Nevertheless, the 'east' search direction outperforms 'south' and r e e Parameter Parameter
	R e v i e w The most effective parameter combinations are EC/SM/E, ES/SM/E and SH/SM/E, 'south-east'. whereas many combinations based on the AC criterion and the LUC criterion show inferior results. R e v i e w
							O n O n
								l y l y
						15

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A Bartók

L Daniel

A Razek

Micro-macro modeling of stress-dependent anisotropic magnetoresistance

Keywords: numbers: 73.43.Qt, 75.80.+q, 75.30.Gw, 75.60.-d

Anisotropic magnetoresistance (AMR) is the basic phenomenon of a spread class of sensors. AMR effect has a strong mechanical stress dependence. Micromagnetic simulations are often used for modeling the magnetoresistance of ferromagnetic materials, but these approaches do not allow to investigate macroscopic effects (for example behavior of a polycrystal under stress) due to the high number of interactions and degrees of freedom. On the other hand macroscopic phenomenological approaches fail in describing the main role of microstructure on the effective behavior. In this work a micro-macro model is proposed to describe the effect of stress on the AMR in ferromagnetic polycrystals. Results are discussed and compared to experimental data from the literature.

Introduction

The anisotropic magnetoresistance (AMR) effect in ferromagnetic materials (FM) was first discovered by William Thomson in 1857. Due to its high sensitivity and flexibility of design it is still used in a wide array of sensors for measurement of Earth's magnetic field (electronic compass) [START_REF] Vcelak | AMR navigation systems and methods of their calibration[END_REF], for electrical current measuring (by measuring the magnetic field created around the conductor) [START_REF] Mlejnek | AMR current measurement device[END_REF], for traffic detection [START_REF]Vehicle detection and compass applications using amr magnetic sensors[END_REF] and for linear position and angle sensing [START_REF] Adelerhof | New position detectors based on amr sensors[END_REF].

The electrical resistivity in FM depends on the angle between the direction of electrical current and orientation of the magnetization in the material. This anisotropic dependence lies in spin-orbit coupling. It is fundamentally related to the larger probability of s-d scattering for electrons traveling parallel to the magnetization [START_REF] Smit | Magnetoresistance of ferromagnetic metals and alloys at low temperatures[END_REF]. As the magnetization rotates, the electron cloud about each nucleus deforms slightly and this deformation changes the amount of scattering undergone by the conduction electrons in their passage through the lattice [START_REF] Cullity | Introduction to Magnetic Materials[END_REF]. Thus the AMR effect is strongly dependent on the local magnetization in the material. In that sense it has some similarities with the magnetostriction effect as discussed later in this paper. On the other hand, owing to the magnetic domain structure of FM, the distribution of magnetization within FM is very heterogeneous. This is the reason why numerical models for AMR effect are mostly based on micromagnetic calculations [START_REF] Adeyeyea | Magnetoresistance behavior of single castellated ni 80 fe 20 nanowires[END_REF][START_REF] Hafner | Theory of anisotropic magnetoresistance in atomic-sized ferromagnetic metal contacts[END_REF][START_REF] Serrano-Guisan | Inductive and magneto-resistive measurements of gilbert damping in ni 81 fe 19 thin films and microstructures[END_REF]. In these approaches the number of degrees of freedom and interactions are growing quickly with the number of magnetic moments, so that these simulations can only address small volumes corresponding to a limited number of domains. If the effect of the microstructure on the overall AMR properties is to be investigated -for example the effect of crystallographic texture in polycrystalline media -these methods are not relevant.

It is also known that the AMR effect has a strong mechanical stress dependence [START_REF] Bozorth | Ferromagnetism[END_REF]. Indeed mechanical stress applied to a magnetic media changes the distribution of domain orientations. As a consequence it modifies the local resistivity, and thus the overall resistivity. This effect of stress on magnetoresistance is for instance used as the basis for a particular type of strain measurement gauges [START_REF] Sonehara | Strain sensor using stress-magnetoresistance effect of nife/mnir exchange-coupled magnetic film[END_REF].

Some macroscopic phenomenological models for AMR effect have also been proposed [START_REF] Kwiatkowski | Numerical analysis of the shape anisotropy and anisotropy dispersion in thin film permalloy magnetoresistors[END_REF][START_REF] Mcguire | Anisotropic magnetoresistance in ferromagnetic 3d alloys[END_REF][START_REF] Li | A phenomenological approach to the anisotropic magnetoresistance and planar hall effect in tetragonal la 2/3 ca 1/3 mno 3 thin films[END_REF] but they cannot account for microstructure or composition related effects. They have to be identified on macroscopic measurements and cannot be used as predictive tools for material design.

An intermediate approach, standing between micromagnetic and macroscopic modeling, would be useful in order to provide a design tool sensitive to complex microstructural effects, and notably accounting for the crystallographic texture. Indeed this microstructural parameter can be controlled and significantly modifies the overall AMR properties of FM. In this work a micro-macro model for stress-dependent AMR is proposed. This model is based on a magneto-elastic multiscale model allowing the definition of the local stress and magnetic field in heterogeneous materials from the knowledge of the macroscopic loading. A phenomenological law for AMR effect is then used at the local scale. A homogenization step is finally performed to define the macroscopic change in resistivity. This model can naturally account for the effect of stress on the overall AMR effect, and includes the influence of microstructural parameters such as the crystallographic texture.

The paper is divided into three parts. In the first part the micro-macro model is detailed. In the second part the approach is validated thanks to experimental measurements on iron, nickel and permalloy polycrystals taken from the literature. The approach is finally applied in the third part to the investigation of the effect of crystallographic texture on AMR.

Micro-macro modeling

AMR effect depends on the local magnetization orientation, itself depending on local magnetic field and stress (at the magnetic domain scale). Owing to the heterogeneity of materials, stress and magnetic field are not uniform within the material. Their local values have to be determined through an appropriate micro-macro scheme. Once these values for the local loading are known, the magnetic domain structure has to be determined, in order to define the local magnetization (at the domain scale). A model for the AMR effect can then be applied. The overall response of the material (polycrystalline scale) is then obtained through appropriate averaging operations. Thus, the model proposed in this paper is based on a three-scale description (polycrystal, single crystal, magnetic domain) and follows the scheme presented in Figure 1. The uppermost calculation scale in the model -called macroscopic scale -is the polycrystalline representative volume element (RVE) which is seen as an assembly of single crystals or grains (g) with respect to a given orientation function. The intermediate scale -called mesoscopic scale -is the single crystal or grain, that is seen as a collection of magnetic domains (α) with given magnetization orientation. The lowest scale -called microscopic scale -is the magnetic domain, that is an area with uniform magnetostriction strain ‡ and magnetization.

The main steps of this model are divided as follows.

(i) The localization steps aims at defining the local loading (magnetic field H α , stress σ α and current i α ) at the microscopic scale as a function of the macroscopic loading (magnetic field H, stress σ and current I). The loading at the mesoscopic scale (magnetic field H g , stress σ g and current i g ) is calculated as an intermediate step. These localization steps highly depend on the microstructure of the material. ‡ magnetostriction strain is the spontaneous strain undertaken by magnetic materials.

(ii) The microscopic magneto-elastic model allows to define in a statistical way the domain configuration, introducing as an internal variable the volumetric fraction f α of domains with orientation α in a grain g. In each domain, depending on the considered orientation α, the magnetization M α and magnetostriction strain ε µ α are known. (iii) The microscopic AMR model allows to define the local resistivity (ρ α ) depending on the magnetization orientation α in the considered domain. (iv) The homogenization step allows to retrieve the overall response of the material at the polycrystal scale (magnetization at the macroscopic scale M, macroscopic magnetostriction strain ε µ and the variation of the macroscopic resistivity δρ) depending on the local values of the response at the microscopic scale.

These steps are detailed hereafter. The magneto-elastic part of the model is based on a reversible magneto-elastic model previously published [START_REF] Daniel | A constitutive law for magnetostrictive materials and its application to terfenol-d single and polycrystals[END_REF][START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF].

Localization step

The simplest assumption to define the microscopic loading (H α , σ α , i α ) as a function of the macroscopic loading (H, σ, I) would be to consider uniform field hypotheses. Under such hypotheses the localization rules are very simple (H α = H, σ α = σ, i α = I). However due to the heterogeneity of the materials, these assumptions are often inappropriate.

1.1.1. Macro-meso scale transition In a polycrystal the susceptibility from one grain to another can vary very significantly. For instance in pure iron the permeability of a grain can vary up to 70% at 400 A/m or or 60% at 2000 A/m depending on its relative orientation with respect to the magnetic field [START_REF] Daniel | A note on the effective magnetic permeability of polycrystals[END_REF]. This heterogeneity results in a significant heterogeneity of the magnetic field within the material. In the case of polycrystals, the self-consistent scheme is known to provide satisfying results. The macro-meso localization rule is written as follows [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF]:

H g = H + 1 3 + 2χ m (M -M g ) (1)
M and M g are the magnetization respectively at the macroscopic and mesoscopic scales. In the case of self-consistent hypothesis χ m is the overall magnetic susceptibility of the material.

The elastic response to a given mechanical loading also significantly differs from one grain to another in a polycrystal. As an example in the case of pure iron the Young's modulus can vary up to 115% depending on the crystallographic orientation (see the elastic constants in Table 1). The self-consistent localization scheme is known to provide satisfying estimates for polycrystalline media [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF]. Under such hypotheses, the macro-meso localization rule can be written in the following form [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF]:

σ g = B σ : σ + L inc : (ε µ -ε µ g ) (2)
ε µ and ε µ g are the magnetostriction strain respectively at the macroscopic and mesoscopic scale. B σ denotes the so-called concentration tensor and L inc is a tensor accounting for elastic incompatibilities due to magnetostriction. The way to calculate these fourth order tensor can be found in [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF][START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF] and is briefly re-called in Appendix A.

In the case of the electrical resistivity, and as will be shown in the following, the heterogeneity is weak. Depending on the orientation of the single crystal, the electrical resistivity does not vary more than a few percent. This is why we applied uniform electric current conditions.

i g = I (3)
1.1.2. Meso-micro scale transition In the case of the localization rules from the grain to the domain scale, an accurate definition of the microstructure, namely the magnetic domain structure, would be requisite §. This information is unknown. For the sake of simplicity we assumed uniform magnetic field, uniform strain and uniform current within the single crystal. However in the proposed microscopic magneto-elastic model, the mean values at the single crystal scale are often sufficient data.

Microscopic magneto-elastic model

The magneto-elastic model for the single crystal is derived from [START_REF] Daniel | A constitutive law for magnetostrictive materials and its application to terfenol-d single and polycrystals[END_REF]. The single crystal is seen as an assembly of magnetic domains. The potential energy of a domain α is written:

W α = W K α + W σ α + W H α ( 4
)
where W K α denotes the magneto-crystalline energy, W σ α denotes the elastic energy and W H α denotes the magneto-static energy.

In the case of cubic crystallographic structure the magneto-crystalline energy can be written

W K α = K 1 (α 2 1 α 2 2 + α 2 2 α 2 3 + α 2 3 α 2 1 ) + K 2 (α 2 1 α 2 2 α 2 3 ) (5)
where K 1 and K 2 denote the magneto-crystalline anisotropy constants of the cubic crystal and α = t [α 1 α 2 α 3 ] the direction cosines of the magnetization (M α = M s α with M s the saturation magnetization of the material).

Under uniform strain hypotheses the elastic energy can be written [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF]:

W σ α = -σ g : ε µ α (6)
In the case of cubic crystallographic symmetry, the magnetostriction strain tensor ε µ α can be written as

ε µ α = 3 2   λ 100 (α 2 1 -1 3 ) λ 111 α 1 α 2 λ 111 α 1 α 3 λ 111 α 1 α 2 λ 100 (α 2 2 -1 3 ) λ 111 α 2 α 3 λ 111 α 1 α 3 λ 111 α 2 α 3 λ 100 (α 2 3 -1 3 )   (7)
where λ 100 and λ 111 are the magnetostrictive constants of the single crystal.

The magneto-static energy of a domain is written as

W H α = -µ 0 M α .H α ( 8
)
where µ 0 is the vacuum permeability. § except for the electric current since the hypotheses of weak heterogeneity of the resistivity is still valid.

We then introduce the volumetric fractions f α of domains with magnetization orientation α [START_REF] Néel | Les lois de l'aimantation et de la subdivision en domaines élémentaires d'un monocristal de fer[END_REF][START_REF] Chikazumi | Introduction to ferromagnetic materials[END_REF][START_REF] Buiron | A multiscale model for magneto-elastic couplings[END_REF][START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF]. These internal variables are obtained through the numerical integration of the following Boltzmann-type relation [START_REF] Daniel | A constitutive law for magnetostrictive materials and its application to terfenol-d single and polycrystals[END_REF]:

f α = exp(-A s .W α ) α exp(-A s .W α )dα (9)
where A s is an adjustable parameter that can be deduced from low field measurement of the anhysteretic magnetization curve [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF] (A s = 3χ0 µ0M 2 s where χ 0 is the initial anhysteretic susceptibility of the material).

The magnetostriction strain and the magnetization over the single crystal are defined by an averaging operation over the single crystal (volume V g ):

ε µ g = ε µ g = 1 V g α ε µ dV = α f α ε µ α ( 10
)
M g = M g = 1 V g α M dV = α f α M α . ( 11
)
If needed, the elastic strain ε e g in the single crystal can be easily calculated from Hooke's law (Equation ( 12)) using the single crystal stiffness tensor C g . This elastic strain is superimposed to the magnetostriction strain to obtain the total strain of the single crystal (ε = ε e + ε µ ).

ε e g = C -1 g : σ g (12)
At this stage, the macroscopic magnetization and magnetostriction strain could also be calculated thanks to an averaging operation over the whole volume of the RVE.

Single-domain model of AMR

Let β = t [β 1 β 2 β 3 ] be the direction cosines determining the orientation of the current used for measuring the electrical resistance (α = t [α 1 α 2 α 3 ] are still the direction cosines of the magnetization in the considered domain). The general expression for the magnetoresistance in any direction of a cubic crystal can be written in a series form of α and β [START_REF] Mcguire | Anisotropic magnetoresistance in ferromagnetic 3d alloys[END_REF]. Döring used the following form [START_REF] Bozorth | Ferromagnetism[END_REF] for cubic crystals with negative magnetocrystalline anisotropy constant K 1 (such as nickel and Fe 11 Ni 89 permalloy):

ρ α = ρ 0 1 + k 1 (α 2 1 β 2 1 + α 2 2 β 2 2 + α 2 3 β 2 3 -1 3 ) +2k 2 (α 1 α 2 β 1 β 2 + α 2 α 3 β 2 β 3 + α 3 α 1 β 3 β 1 ) +k 3 (s -1 3 ) +k 4 (α 4 1 β 2 1 + α 4 2 β 2 2 + α 4 3 β 2 3 + 2s 3 -1 3 ) +2k 5 (α 1 α 2 β 1 β 2 α 2 3 + α 2 α 3 β 2 β 3 α 2 1 + α 3 α 1 β 3 β 1 α 2 2 ) (13) in wich s = α 2 1 α 2 2 + α 2 2 α 2 3 + α 2 3 α 2 1
, ρ 0 is the resistivity in the demagnetized state and k 1 ,k 2 ,k 3 ,k 4 ,k 5 are material constants. For a crystal with positive magnetocrystalline anisotropy constant ¶ K 1 (such as iron) the expression is the same except that the term k 3 /3 is absent.

in that case easy magnetization directions are 111 directions. ¶ in that case easy magnetization directions are 100 directions.

Effective properties

The macroscopic magnetization and strain are obtained through an averaging operation over the whole volume V of the RVE.

M = M V = M g V (14
)

ε = ε V = ε e + ε µ V = ε g V (15)
If needed, the macroscopic magnetostriction strain can be obtained using the following relation [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF]:

ε µ = t B σ : ε µ V = t B σ : ε µ g V (16)
Since the local electric conductivity ς α in a domain is known (ς α = 1/ρ α ), the effective macroscopic conductivity ς can be obtained through a self-consistent approach, applying the classical Bruggeman relation [START_REF] Bruggeman | Berechnung verschiedener physikalischer konstanten von heterogenen substanzen[END_REF][START_REF] Stroud | Generalized effective-medium approach to the conductivity of an inhomogeneous material[END_REF][START_REF] Daniel | A note on the effective magnetic permeability of polycrystals[END_REF]. ς is solution of Equation ( 17) that can be solved easily using a fixed point method.

ς = ς α 2ς + ς α V 1 2ς + ς α V (17)
where the operation . V is an averaging operation over the whole volume of the RVE. The effective resistivity ρ is deduced from the effective conductivity (ρ = 1/ς). In the following the variation of the macroscopic resistivity δρ (Equation ( 18)) will be plotted. It can be noticed from Equations ( 13) and ( 17) that δρ does not depend on the value of ρ 0 . δρ = ρ -ρ 0 ρ 0 (18)

Validation on isotropic polycrystals

The proposed micro-macro approach has been validated on isotropic iron, nickel and permalloy (Fe 11 Ni 89 ) polycrystals.

Model parameters

The mechanical and magnetic characteristics of iron, nickel and a permalloy (Fe 11 Ni 89 ) single crystal were used + for the calculations. At the single crystal scale, for the calculation of the volumetric fractions f α , a 10242 orientation (α) data file was used [START_REF] Daniel | A constitutive law for magnetostrictive materials and its application to terfenol-d single and polycrystals[END_REF]. The material constants are defined in Table 1 and Table 2. The distribution function for crystal orientation in the case of isotropic polycrystals has been obtained by a regular zoning of the crystallographic space [START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF]. The corresponding pole figures are given in Figure 2.

+ The accurate data of the constants of Döring expression of Fe

Prediction of the AMR effect

The similarity in the phenomena of magnetostriction and magnetoresistance has been known for a long time. This similarity is linked to the strong dependence of both phenomena on the local magnetization state and thus on the magnetic domain configuration. Our model is based on a micro-macro approach of the magnetoelastic behavior of the materials which can give the magnetostriction (λ(H)) and the magnetoresistive curves (δρ(H)). From these results the relation between the change of resistivity and the magnetostriction strain can be easily obtained. It is illustrated in the case of isotropic nickel and pure iron polycrystals without external stress and compared with experimental data from the literature [START_REF] Bozorth | Ferromagnetism[END_REF] in Figure 3 and4. These figures plot the effective magnetostriction strain as a function of the effective change in resistivity for parallel configuration (magnetic field and electrical current are parallel). Experimental observations show that magnetostriction strain first increases with magnetoresistance and then decreases in the case of iron, and magnetostriction decreases continuously in the case of nickel. The different behavior of these materials results from the different sign in their material constants (magnetostriction and Döring expression). The experimental curve is accurately predicted by the model in the case of nickel. The experimental curve for iron is qualitatively predicted, but quantitatively overestimated. It is shown by the dot plot in Figure 4 that a macroscopic tension of amplitude 25 MPa applied in the direction of the magnetic field provides a numerical result closer to the experimental observation. This point shows that a residual stress in the material could explain the discrepancies in the case of the iron specimen.

Prediction of the effect of stress on the AMR

In order to study the effect of an applied uniaxial stress on the magnetoresistance, modeling results were compared to experimental results [START_REF] Bozorth | Ferromagnetism[END_REF] on a permalloy (Fe 11 Ni 89 ) polycrystal. In this case the δρ(B) curves were calculated from the combination of B(H) and δρ(H) modeling results. This permalloy has negative magnetostriction so that the tension orients its domains perpendicularly to the direction of tension and this effect decreases the initial resistivity. The comparison between numerical and experimental results (Figure 5) gives very satisfying results. It shows the nonlinear stress-dependence as well. The stress decreases the initial resistivity and increases the slope of the curves which is important in the sensor application (higher sensibility in low field measurement). The model can predict the effect of a multiaxial stress as well but these results are not presented in this paper due to the lack of experimental results for validation.

Investigation of crystallographic texture effect

The magnetoresistance in a single crystal is strongly anisotropic. Figure 6 shows the change of resistivity in a pure iron single crystal as a function of the angle between its easy magnetization direction 100 and the applied magnetic field in the {011} crystallographic plane. The change in resistivity (under no applied stress) can vary up to several hundred percents depending on the orientation of the magnetic field with respect to the crystal orientation, both in parallel and perpendicular configurations. As a consequence, the AMR effect can be expected to be very sensitive to crystallographic texture.

<100> <111> <011>

[100]

[011]

[111] Figure 6. Change of resistivity as a function of the angle between the 100 direction and the applied magnetic field (10 5 A/m) in the {011} crystallographic plane -pure iron single crystal in parallel (parallel magnetic field and electrical current) and perpendicular (perpendicular magnetic field and electrical current) configurations

In order to investigate this effect, the crystallographic texture of an Armco specimen (pure iron), known as isotropic, has been obtained from electron backscattered diffraction (EBSD) measurement. The corresponding pole figures are given in Figure 7. It shows a weak texture compared with the calculated isotropic crystal orientation distribution used previously and showed in Figure 2. The prediction of the AMR effect for pure iron using this latter crystallographic texture has been compared to the prediction using the isotropic orientation data file. Figure 8 shows the change of resistivity in the parallel current/applied magnetic field configuration as a function of current orientation in the polycrystal. In the case of Armco specimen two planes (XY and YZ) have been investigated.

It is shown that even for this very weakly textured material the magnetoresistance can vary up to 10 % depending on the orientation of the sollicitation (parallel configuration). The magnetoresistance is confirmed to be strongly dependent on crystallographic texture.

Conclusion

A micro-macro model for the effect of stress on the anisotropic magnetoresistance has been presented. It is based on a description of the magneto-mechanical coupling at several scales (domain, single crystal, polycrystal). The behavior of iron, nickel and a permalloy (Fe 11 Ni 89 ) polycrystal has been calculated. Numerical results have been compared with experimental results from the literature with very satisfying agreement. This model enables to investigate the effect of crystallographic texture on AMR effect and it highlights the strong influence of crystallographic texture on it. It also enables to investigate the effect of stress, and notably multiaxial stress on magnetoresistance. The development of micro-macro models accounting for microstructure and stress dependence of effective magnetoresistive properties should allow the design of high precision AMR devices.

H g σ g i g ρ α M g ε µ g Figure 1 .

 g1 Figure 1. Modeling strategy

Figure 2 .

 2 Figure 2. Pole figures for an isotropic polycrystal obtained from a regular zoning of the crystallographic space

0 0 . 3 -Figure 3 .

 033 Figure 3. Magnetostriction strain of nickel polycrystal as a function of the change of resistivity (current and applied magnetic field are parallel) -experimental data[START_REF] Bozorth | Ferromagnetism[END_REF] and obtained numerical results

  Numerical results (no applied stress) Numerical results (under applied stress) Experimental results

Figure 4 .Figure 5 .

 45 Figure 4. Magnetostriction strain of iron polycrystal as a function of the change of resistivity (current and applied magnetic field are parallel) -experimental data[START_REF] Bozorth | Ferromagnetism[END_REF] and obtained numerical results (dashed line: under 25 MPa tension applied in the direction of magnetic field)

  between the <100> direction and the applied magnetic eld (in degrees)

Figure 7 .

 7 Figure 7. Pole figures of an Armco steel obtained from EBSD measurement

Figure 8 .

 8 Figure 8. Change of resistivity (current and applied magnetic field are parallel) with change of the orientation of the applied magnetic field (10 5 A/m) for pure iron polycrystal using an isotropic texture (dotted line) and texture data from EBSD measurement in different planes (lines)

Table 1 .

 1 [START_REF] Sonehara | Strain sensor using stress-magnetoresistance effect of nife/mnir exchange-coupled magnetic film[END_REF] Ni 89 was not found. The constants of the Fe 15 Ni 85 permalloy were used instead. Physical constants used for the modeling

	Coefficient	M S	K 1 , K 2	λ 100 , λ 111		A s	C 11 , C 12 , C 44
	Unit	A/m	kJ/m 3	-		m 3 /J	GP a
	Iron [10, 6, 24]	1.71x10 6 42.7, 15	21, -21 (x10 -6 )	0.0020 238, 142, 232
	Nickel [10, 25]	4.91x10 5 -5.7, -2.3 -45.9, -24.3 (x10 -6 ) 0.016	250, 160, 118
	Fe 11 Ni 89 [10, 26] 7.50x10 5 -1, -2	-15, -10 (x10 -6 )	0.032	243, 148, 122
	Coefficient	k 1	k 2	k 3	k 4	k 5
	Iron [10]	0.00153 0.00593 0.00194 0.00053 0.00269
	Nickel [10]	0.0654	0.0266	-0.032	-0.054	0.020
	Fe 15 Ni 85 [27] 0.0518	0.0478	-0.0243 -0.0139 0.0259

Table 2 .

 2 Constants of Döring expression

Acknowledgments

The authors are greatly indebted to Dr. Anne-Laure Helbert (ICMMO, Univ Paris-Sud) for the EBSD measurement of the Amrco steel.

Appendix A: Calculation of mechanical localization operators

We briefly give hereafter the way to obtain the fourth order tensors B σ and L inc appearing in section 1.1. More detailed explanations can be found in [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF][START_REF] Daniel | Reversible magneto-elastic behavior: A multiscale approach[END_REF].

B σ is defined by Equation ( 19) that introduces the single crystal stiffness tensor C g , the polycrystal effective stiffness tensor C and the strain localization tensor A σ .

A σ is defined by Equation ( 20) where C * is the so-called Hill constraint tensor [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF] that can be obtained from the Eshelby tensor S E according to Equation ( 21), I being the fourth order identity tensor.

L inc is defined by Equation [START_REF] Bruggeman | Berechnung verschiedener physikalischer konstanten von heterogenen substanzen[END_REF].