Louis Bigo

Laurent Feisthauer
email: laurent@algomus.fr

Mathieu Giraud
email: mathieu@algomus.fr

Florence Levé
email: florence@algomus.fr

RELEVANCE OF MUSICAL FEATURES FOR CADENCE DETECTION

Cadences, as breaths in music, are felt by the listener or studied by the theorist by combining harmony, melody, texture and possibly other musical aspects. We formalize and discuss the significance of 44 cadential features, correlated with the occurrence of cadences in scores. These features describe properties at the arrival beat of a cadence and its surroundings, but also at other onsets heuristically identified to pinpoint chords preparing the cadence. The representation of each beat of the score as a vector of cadential features makes it possible to reformulate cadence detection as a classification task. An SVM classifier was run on two corpora from Bach and Haydn totaling 162 perfect authentic cadences and 70 half cadences. In these corpora, the classifier correctly identified more than 75% of perfect authentic cadences and 50% of half cadences, with low false positive rates. The experiment results are consistent with common knowledge that classification is more complex for half cadences than for authentic cadences.

INTRODUCTION 1.1 Cadences

Music, like all languages, is organized into structural units. In Western tonal music, these units often end with strong harmonic formulas called cadences, from the Latin cadere, "to fall." Despite their structural function, cadences are hard to define. Based on a review of dozens of music theory papers, Blombach defined the cadence as "any musical element or combination of musical elements, including silence, that indicates relative relaxation or relative conclusion in music" [START_REF] Blombach | Phrase and cadence: A study of terminology and definition[END_REF]. This definition highlights the way a listener (whether musically trained or not) can hear the presence of a cadence by feeling that the music "breaths". A cadence is generally characterized by local musical elements, such as a specific harmonic progression and a falling melody. However, these elements do not necessarily imply a cadence. A global or high-level structure such as the sonata form may also induce the impression of a cadence [START_REF] Hepokoski | Elements of Sonata Theory: Norms, Types, and Deformations in the Late-Eighteenth-Century Sonata[END_REF].

c Louis Bigo, Laurent Feisthauer, Mathieu Giraud, Florence Levé. Licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Attribution: Louis Bigo, Laurent Feisthauer, Mathieu Giraud, Florence Levé. "Relevance of musical features for cadence detection", 19th International Society for Music Information Retrieval Conference, Paris, France, 2018.

Cadences are usually classified by harmonic progression. The authentic cadence is characterized by a dominant harmony (notated V) followed by a tonic harmony (notated I). In the American terminology, when both chords are in root position and the melody ends on the tonic, the authentic cadence is said to be perfect (PAC), otherwise it is imperfect (IAC). If the IAC is in root position (but the melody does not end on the tonic), it is said to be a rooted IAC (rIAC). The half cadence (HC) ends with a dominant harmony, generally in root position. The deceptive cadence (DC) is an authentic cadence where the expected final tonic is replaced by another harmony (often VI). Some authors theorize the evaded cadence as a particular IAC, while others see it as a DC-like progression but including a melodic break, for instance while repeating a phrase [START_REF] Schmalfeldt | Cadential processes: The evaded cadence and the "one more time" technique[END_REF]. Some scholars do not consider the plagal progression IV/I as a cadence but rather as a post-cadential prolongation [START_REF] Caplin | The classical cadence: Conceptions and misconceptions[END_REF].

Each cadence type provide a different feeling of closure. The strongest cadence is the PAC, followed in turn by the rIAC, IAC, HC, and the DC and related cadences [START_REF] Sears | Perceiving the classical cadence[END_REF]. Some traditions consider the rIAC to be very conclusive. For instance, French music teachers refer to both PAC and rIAC as cadence parfaite. Using a preparation chord before the dominant chord, generally a subdominant harmony (SD, that is II, IV, or V/V), strengthens the salience of a PAC/rIAC. In contrast, DC and related cadences renew tension, extending the musical phrase and delaying closure until a more conclusive cadence is used.

Cadences, Musicology and MIR

Modeling cadences is a current challenge in musicology [START_REF]What Is a Cadence? Theoretical and Analytical Perspectives on Cadences in the Classical Repertoire[END_REF]. Although cadence definitions found in music education textbooks are often quite short, music theorists agree on the difficulty to define cadences because of the variety of their realizations observed in the repertoire [START_REF] Caplin | The classical cadence: Conceptions and misconceptions[END_REF].

Cadences are therefore usually studied within the frame of one specific corpus -see for example Martin and Pedneault-Deslauriers's study of HC in Mozart's piano sonatas [START_REF] John | The Mozartean Half Cadence[END_REF]. However, more systematic analyses of large corpora would help to understand the evolution of compositional choices over time. Rohrmeier and Neuwirth suggested a first characterization using grammars, based on the degrees and the bass line [START_REF] Rohrmeier | Towards a Syntax of the Classical Cadence[END_REF].

Detecting cadences throughout the score requires a specific training to find clues pointing out to the breaths in music. Can we algorithmically detect cadences from a score encoded in symbolic notation? Some works in MIR have

X ⑤ ii ▼ Y V ④ ▼ (6) i ③ ③ ③ PAC ▲ i ② ① ▼ Z Figure 1.
Haydn, op. 17/4, iv, PAC at measure 8 (offset Z). Compare to Figure 1 of [START_REF] David | Modeling harmony with skip-grams[END_REF]. Features describe here the constitution of the Z chord (Z-in-perfect-triad, Zin-perfect-triad-or-sus4, Z-highest-is-1), voice leading to Z (Z-1-comes-from-7 1 , Z-3-comes-from-4 2 ), rests after Z (R-after-Z-rest-lowest, -middle 3 ) and the metric structure (R-Z-strong-beat). Features also describe relations with chord Y (Y-Z-bass-moves-compatible-V-I 4 , Y-Z-bass-same-voice), and cadence preparation (X-Y-bassmoves-2nd-Maj 5 ). Note that the heuristic choice of a single offset Y implies here that the features Y-in-V7-3 and Y-has-7 are not true, even if the dominant chord actually contain several pitches 3 and 7 (circled notes). Nevertheless, these pitches are caught by the tonality features (Z-bass-compatible-with-I, Z-bass-compatible-with-I-scale) and some of them are considered by the voice leading features ( 1 , 2 ).

vi ▼ X ii ③ ▼ Y ① ▼ Z HC ▲ V ① ① (7) ② ② Figure 2.
Haydn, op. 17/5, i, HC at measure 8. Features notably describe here a voice leading (Z-6-comesfrom-7, Z-4-comes-from-5, Z-1-comes-from-2 1 ) continued by a 6/4 suspension (Z-6-moves-to-5, Z-4-moves-to-3 2 ). Other features also describe the tonality compatibility of an HC (bass-Z-compatible-with-V, circled notes, but both Z-bass-compatible-with-I and Z-bass-compatible-with-I-scale are also true due to the squared c#) as well as bass movements (Y'-Y-bass-moves-chromatic, Y-Z-bass-moves-2nd-Maj 3 ) and the metric structure (R-Z-strong-beat).

focused on melodic cadences [START_REF] Peter Van Kranenburg | Cadence detection in western traditional stanzaic songs using melodic and textual features[END_REF] and a few studies have tackled the problem of the identification of harmonic progressions [START_REF] Paiement | A probabilistic model for chord progressions[END_REF] or their representation as musical trajectories [START_REF] Bigo | Computation and visualization of musical structures in chord-based simplicial complexes[END_REF]. The authors of [START_REF] Bas De | Modeling harmonic similarity using a generative grammar of tonal harmony[END_REF] took Rohrmeier's works further, extending it into a system deriving harmonic relations between chords, where grammars rules were inferred for jazz harmony. Currently, only a few algorithms recognize simple cadences [START_REF] Plácido | Harmonic, melodic, and functional automatic analysis[END_REF]. We previously suggested a rule-based detection of PAC/rIAC in fugues [START_REF] Mathieu | Computational fugue analysis[END_REF] and used it in a study on the sonata form [START_REF] Bigo | Sketching sonata form structure in selected classical string quartets[END_REF]. Recently, Sears and colleagues [START_REF] David | Simulating melodic and harmonic expectations for tonal cadences using probabilistic models[END_REF] used the software IDyOM [START_REF] Thomas | The construction and evaluation of statistical models of melodic structure in music perception and composition[END_REF] on a corpus of Haydn string quartets to show that music predictability increases at cadential points and decreases on the following note.

Contents

Our goal is to identify binary, musical, and local features that coincide with cadences and that can be used to train a model that detects new cadences, either PAC/rIAC or HC. Rather than agnostically discovering cadential features on the musical surface, we intend here to confirm and study traditional music theory knowledge regarding cadences. The proposed strategy avoids chord segmentation, which is itself a difficult MIR problem. Section 2 details the selected features and Section 3 describes the learning process. Finally, Sections 4 and 5 discuss the application of the method on Bach and Haydn corpora.

MUSICAL FEATURES AT THREE ONSETS

Each beat Z of the score is considered as the potential arrival point of a cadence. A set of 44 binary features is computed at each beat. These features are then used to train a classifier whose aims to predict whether a beat corresponds to the arrival point of a cadence or not. The features aim at detecting cadences at a local level, i.e. the surroundings of the cadential beat including its immediate past, presumably corresponding to the preparation of the cadence. The idea is to try and detect SD-V-I progressions for a PAC/rIAC, and progressions ending with V for an HC.

What we propose here is a simple heuristic focusing on three specific onsets: Z, Y (Z) and X(Z), or for short Z, Y , and X. Most of the features describe sets of notes sounding at these onsets (even when they begin before), namely chord(Z), chord(Y), and chord(X). We therefore do not start from a complete harmony analysis nor a chord segmentation, that can be error-prone. Even when the methods finding Y (Z) and X(Z) return approximate onsets, the computed features may be relevant.

Features on the Arrival Point Z or around it

The arrival chord of a cadence is usually a perfect triad, possibly with some suspensions. A first set of features describes this chord and its immediate neighborhood:

• Z-in-perfect-major-triad (respectively Z-in-perfect- triad): chord(Z) is included in {1, 3M , 5} 1 (resp. {1, 3m, 3M , 5})
1 Pitches in underlined figures (i.e. 1, 3, etc.) are here computed by the interval modulo octave relative to the bass. As some chords are not in root

V I vi X ii ▼ V ▼ Y V/V 6 8 6 8 (81)
PAC

I ▼ Z ▲ Figure 3
. Bach, fugue #15 in G major BWV860, PAC at measure 83. Voice leading and highlighted notes as in the Figures 1 and2. To cope with the faster harmonic rhythm, every eighth before Z is considered as a potential Y .

• Z-in-perfect-triad-or-sus4: chord(Z) is included in {1, 3, 4, 5} • Z-is-sus4: chord(Z) is exactly {1, 4, 5} • Z-highest-is-1 (resp. Z-highest-is-3):
The highest note of chord(Z) is the tonic 1 (resp. the major or minor third 3), as expected for a PAC (rIAC)

Another set of features describes voice leadings from preceding notes (see list on Table 2). Z-β-comes-from-α means that the note β in chord(Z) is an "immediate resolution" of a note α (the interval being still relative to the bass of Z) that is exactly before β (even if this note is not at the onset Y that will be defined later). For example, Z-3comes-from-4 means that there is a 3 in chord(Z) that is an immediate resolution of a 4 (dominant seventh in the case of a PAC, see 2 on Figure 1).

There can also be a suspension at the arrival point, as on the HC on Figure 2. We thus add symmetrical features Z-αmoves-to-β (see list on Table 2). For example, Z-4-movesto-3 means that there is a suspended fourth 4 in chord(Z) that is immediately resolved to the third 3.

Finally, features try to grasp the tonality in the neighborhood of Z. We do not perform tonality estimation [START_REF] Krumhansl | Tracing the dynamic changes in perceived tonal organisation in a spatial representation of musical keys[END_REF][START_REF] Temperley | What's key for key ? the Krumhansl-Schmuckler key-finding algorithm reconsidered[END_REF] because of the usual difficulty of algorithms to disambiguate adjacent tonalities in the circle of fifths.

• Z-bass-compatible-with-I (resp. Z-bass-compatiblewith-V ): Both notes 4 and 7 of the tonality that would be implied if the bass of Z is I (resp. V) are present in the four beats before Z

• Z-bass-compatible-with-I-scale: The 8 previous beats exhibits the whole scale of the same implied tonality -Temperley suggesting that such PACs with SD before V-I feel more conclusive [START_REF] Temperley | Music and probability[END_REF].

For example, on the PAC of Figure 1, Z-basscompatible-with-I and -with-I-scale are true (and not -with-V ), and on the HC of Figure 2, Z-bass-compatible-with-V is true. However, these features may be triggered by position, these pitches may differ from the actual function. For example, the top voice d on offset X on Figure 1 is the sixth 6 of the chord II6 but is actually the tonic of the II chord.

other events close on the circle of fifths: Both Z-basscompatible-with-I and -with-I-scale may be triggered by a previous V/V (as on Figure 2) or, in minor, when Z is actually a III in root position.

Rhythmic Features around the Arrival Point Z

These textural features intend to detect either breaks or continuation in music.

• R-Z-strong-beat: Z is a strong beat (beat 1 and 3 for 4/4, and beat 1 for other time signatures)

• R-Z-same-rhythm-1 (resp.

R-Z-same-rhythm-2):

There is exactly the same sequence of durations in the one (resp. the two) beat(s) preceding Z than on the one (resp. the two) beat(s) at onset Z

• R-Z-sustained-note: At least one note sounding at Z started before Z

• R-after-Z-rest-highest, R-after-Z-rest-lowest, R-after-Z-rest-middle: There is a rest in some voice right after the note at onset Z (see Figure 1)

• R-after-Z-one-voice-ends: Z is the last onset in at least one voice (end of the piece)

Features on the Point Y or around it

For each arrival beat Z, we identify a point Y prior to Z supposed to pinpoint the chord "preceding" Z. For example, identifying chord(Y) as a dominant chord is a sign indicating a potential PAC at Z. Although V chords generally span over more than one beat, associating Y with a single beat eases the computation of features.

We thus propose to identify the point Y as the latest beat preceding Z for which the bass voice includes a sounding note, limited to one measure in the past. If the bass includes a rest just before Z, we look just before. The usual time span corresponding to the preparation of a cadence depends on the harmonic rhythm and varies among musical styles. The beat resolution to search the Y point should therefore depend on the corpus.

• Y-Z-offsets-at-most-1: Y is at most one quarter note before Z Some features are concerned with chord(Y) :

• Y-has-7 (resp. Y-has-9): chord(Y) contains 7 (resp. 9), that is the leading tone (resp. the dominant seventh or the dominant ninth) in the case of a candidate PAC

• Y-in-V7 : chord(Y) is included within a dominant seventh chord • Y-in-V7-3 : chord(Y)
Table 1. Corpora with manual annotations of cadences. Cadences are labeled at about 2% of the beats. We narrow to sets with significant number of cadences (PAC and HC for the Haydn corpus, PAC and PAC+rIAC for the Bach corpus).

• Y'-Y-bass-moves-chromatic: The bass note preceding Y is at a distance of one semitone (HC)

• Y-Z-bass-moves-2nd-min (resp. Y-Z-bass-moves-2nd-Maj)

• Y-Z-bass-same-voice: Bass notes of both chords are on the same voice

• Y-Z-bass-moves-compatible-V-I (resp. Y-Z-bass- moves-compatible-I-V ):
The bass moves by an ascending fourth or descending fifth (PAC) (resp. ascending fifth or descending fourth, HC I-V)

Features on the Cadence Preparation (Point X)

We identify the onset X as the latest beat before Y whose lowest sounding note has a different pitch (modulo octave) than the lowest note of Y. Features focus on this bass move:

• X-Y-bass-moves-2nd-min (V/V-V-I) • X-Y-bass-moves-2nd-Maj (IV-V-I or II6-V-I)
• X-Y-bass-moves-4th (expected in II-V-I)

CLASSIFICATION PROCESS

A model is built in order to reflect the correlation between the features listed in Section 2 and the occurrences of cadences in corpora. These corpora bear manual annotations indicating the position of PAC, rIAC and HC. Assuming that the arrival points of cadences do not fall between beats, each beat (quarter note, or three eights depending on the time signature) of each piece is described by:

• a vector of boolean values corresponding to the set of features and computed from the musical score,

• a boolean class specifying whether the beat is annotated in the reference as a PAC/rIAC/HC or not.

This way of representing data enables us to reformulate cadence detection as a classification task. To avoid overfitting, each dataset is randomly divided into two subsets: a training set used to train a classifier and a test set left to evaluate the classifier performance at the end. The classifier and the value of its hyper-parameters have been selected by performing Leave-One-Piece-Out crossvalidation over the training set. This is done by evaluating the classification on each piece of the training set by a classifier trained on the remaining pieces of the training set. Indeed, the traditional Leave-One-Out (LOO) crossvalidation approach that would consist in leaving only one beat of one piece out of the training set would result here in overfitting due to intra-piece musical repetitions.

EXPERIMENTS AND DISCUSSION

Corpora and Implementations

Table 1 shows the corpora which was used in this study. The corpus bach-wtc-i includes the 24 fugues of the first book of the Well-Tempered-Clavier by J.-S. Bach. Cadence annotations were taken from our previous work [START_REF] Mathieu | Computational fugue analysis[END_REF]. The corpus haydn-quartets includes 42 expositions from movements of Haydn string quartets in sonata form, annotated with cadences by Sears and colleagues [START_REF] David | Simulating melodic and harmonic expectations for tonal cadences using probabilistic models[END_REF]. Even if these annotated corpora model cadences in the light of a global analysis of the form, we have used them as a benchmark on our local feature-based detection. Only a minority of annotated PAC are final in the sense that they are included in the last four measures of the piece (or of the exposition).

Pieces were downloaded as voice-separated .krn files from kern.ccarh.org [START_REF] Huron | Music information processing using the Humdrum toolkit: Concepts, examples, and lessons[END_REF]. Note that the features proposed here could also apply to non-separated files, except for after-Z-rest-* and Y-Z-bass-same-voice. In this case, features on voice leading would only check that the coming note or the suspended note is found at the right place in the polyphonic texture.

For each beat Z (and their related onsets X, Y ), the features described in Section 2 are extracted using code based on the Python framework music21 [START_REF] Scott | music21: A toolkit for computer-aided musicology and symbolic music data[END_REF]. Points Y and X are searched at a beat resolution of a quarter note (Haydn) or eight note (Bach, see Figure 3). Classifiers were computed thanks to the scikit-learn framework [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF].

Discussion on Feature Statistics

Table 2 shows tallies of features, their correlation with cadences as well as an estimation of their significance. Many features are significant in both corpora, despite differences in musical style. Unsurprisingly, features R-Z-strong-beat, Y-Z-bass-moves-compatible-V-I, Z-perfecttriad-or-sus4 and Z-highest-note-is-1 are activated nearly for every PAC. Note that PAC lacking the fifth leap are the ones where the bass passes by another note before tonic resolution. Rhythmic and break features are also quite significant. Some features differ between corpora. For example, R-Z-sustained-note is absent in nearly all PACs in the Haydn corpus, whereas it can be found in some PACs in Bach fugues due to the contrapuntal writing.

We were expecting to find more suspensions for both PAC and HC as a way to retain tension before the ultimate resolution but they do not appear significantly in these corpora. We also notably lack strong significant features for HC. Indeed, the Y-Z bass move in a HC is variable (it is typically similar to X-Y moves in PAC). Z-6-comes-from-7

200 • / 2 • / 1 130 • / 1 • / 1 Z-2-moves-to-1 57 • / 0 • / 0 90 2 / 1 1 / 0 Z-4-moves-to-3 160 2 / 2 1 / 0 340 2 / 4 11 * / 3
Z-6-moves-to-5 Table 2. Feature tallies for PAC (both corpora), rIAC (Bach corpus) and HC (Haydn corpus). The table shows, for each feature, the number of beats where this feature occurs (all beats, cadential points or not), followed by its number of occurrences on beats labeled as cadences in the reference annotation, as well as, in small, its expected number should the feature be random and uniformly distributed across the beats. (• means 0, and not significant). For example, there are 70 HC out of 7173 beats in the Haydn quartets corpus. There are 35 beats corresponding to a HC with the feature Y-Z-bass-2nd-min, out of 1328 beats with this feature, and compared to only 12 beats should this feature be random.

138 1 / 1 • / 0 180 • / 2 8 * / 1 Z-7-moves-to-1 7 • / 0 • / 0 105 2 / 1 1 / 1 Point Y Y-in-V7
For each feature and each cadence type, p-values are estimated by an exact Fisher test computed by the Python scipy package. Fisher tests are computed independently. To account for the large number of tests, only features with p-values under .001 (bold, * ) can be considered as significant, either by their absence (italic) or their presence. For example, the feature Y-Z-bass-2nd-min is significantly absent in PACs of both corpora (p < 10 -7 ) and significantly present in HCs of the Haydn corpus (p < 10 -8 ). . The PAC at m67 is hard to detect with the silence at the bass. In their global analysis of the form, Sears et. al see the end of the secondary theme (called the EEC, for Essential Expositional Closure by [START_REF] Hepokoski | Elements of Sonata Theory: Norms, Types, and Deformations in the Late-Eighteenth-Century Sonata[END_REF]) at m67 and discard any further PAC in the following concluding section [START_REF] David | Simulating melodic and harmonic expectations for tonal cadences using probabilistic models[END_REF]:

The PAC candidate at m71, found by the proposed strategy, is thus counted here as a FP. It could be debated whether the EEC is indeed at m67 (first cadential I, but weakened by the bass rest) or rather at m71 (cadential feeling augmented by the following rests and bass note on upbeat, m67 considered as an evaded cadence).

Learning Process

A linear Support Vector Machine (SVM) classifier was trained on each training set as explained in Section 3, splitting the feature space with a hyperplane [START_REF] Cortes | Support-vector networks[END_REF]. As datasets are unbalanced (about 98% of the beats are "noncadential"), we assigned stronger weights to data belonging to the under-represented class, here the cadential beats. Other classifying algorithms such as k-nearest-neighbor or decision trees were tested and turned out to provide comparable or inferior results.

Discussion on Detection Results

Table 3 shows the comparison between the predictions of each classifier on the test set of each corpus and the reference annotations. The detection of PAC is good, with more than 75% PAC detected and a low false positive rate (< 1%). Note that we previously reported 82% of PAC detection in fugues [START_REF] Mathieu | Computational fugue analysis[END_REF] but with manual hard-coded rules that may have resulted in overfitting.

False positives (FP) beats may still have many cadential features. An inspection of the 28 PAC reported as FP in the Haydn corpus shows that at least 5 FP can be seen as actual cadences, for example measure 71 in Haydn op. 55/3, i, shown on Figure 4. Other notable sources of FP are tonic chords following actual HC cadences activating significant features for PAC. The same Figure 4 shows an example of FN, where a silence in the bass makes the computation of many features fail.

Adding rIAC (Bach corpus) lowers the results, but there may be too few such cadences to efficiently build the model. The detection of HC is difficult (Haydn corpus), as there is not a single feature applicable to every case. Half of them are detected, with about 2% FP. Table 4 further studies these results while varying the set of considered features. Some features in Z already consider the past. Nevertheless, the features around Y are essential to improve the overall detection. Features on X bring a small but significant gain for PAC. Rhythmic features (R) bring an improvement especially for HC, in particular with R-Z-strong-beat that correctly filters out more than half of the beats.

CONCLUSION

Different musical clues give the cadential impression of a "breath in music". We evaluated cadential features on and around three onsets at the arrival and in the preparation of cadences. Without performing any chord segmentation, these features describe the underlying harmony, the voice leading as well as structural aspects of the music.

These features reflect common knowledge of music: we have shown that some of them are specific to cadential points. They make it possible to learn how to predict cadences -PAC/rIAC, and, to a lesser extent, HC -in corpora with reference annotations. Such features may also be used in other systematic musicology approaches.

Perspectives include the extension of our set of features to cadential and non-cadential positions. Some features could be not necessarily theory driven and could possibly have metric values. Coupled with automatic selection, this could lead to the discovery of significant but unexpected features. More generally, the method used to identify points X and Y could be compared to other heuristics. Cadence preparations could for example be described by features regarding contiguous "spans" of onsets rather than single onsets X and Y, in order to improve the harmony relevance of the model. Research along these lines could significantly improve HC detection.

Figure 4 .

 4 Figure 4. Haydn, op. 55/3, i, potential PACs at m67 and m71. The PAC at m67 is hard to detect with the silence at the bass. In their global analysis of the form, Sears et. al see the end of the secondary theme (called the EEC, for Essential Expositional Closure by[START_REF] Hepokoski | Elements of Sonata Theory: Norms, Types, and Deformations in the Late-Eighteenth-Century Sonata[END_REF]) at m67 and discard any further PAC in the following concluding section[START_REF] David | Simulating melodic and harmonic expectations for tonal cadences using probabilistic models[END_REF]:The PAC candidate at m71, found by the proposed strategy, is thus counted here as a FP. It could be debated whether the EEC is indeed at m67 (first cadential I, but weakened by the bass rest) or rather at m71 (cadential feeling augmented by the following rests and bass note on upbeat, m67 considered as an evaded cadence).

Table 3 .

 3 Detection of cadences on the test sets of both corpora using all features: Number of beats annotated in the reference annotation (ref), true positives (TP), false positives (FP), false negatives (FN), and F 1 measure (harmonic mean of the recall and the precision).

				beats ref TP FP FN	F 1
	haydn-quartets PAC		3583 51 42 28	9	0.69
	(21 quatuors)	HC		3583 32 18 73 14 0.29
	bach-wtc-i	PAC		2357 36 26	3	10 0.80
	(12 fugues)	PAC+rIAC 2357 46 30 12 16 0.68
		haydn-quartets		bach-wtc-i
			PAC	HC	PAC PAC+rIAC
	All features XYZR	0.69	0.29	0.80	0.68
	Features YZR	0.69	0.27	0.71	0.68
	Features ZR	0.59	0.24	0.52	0.34
	Features XYZ	0.72	0.25	0.74	0.54

Table 4 .

 4 F 1 measure while detecting cadences on the test sets of both corpora with different sets of features.

Acknowledgements. The authors are grateful to Robin Alais, Mikaela Keller, Marie Perrier, and the Algomus team for fruitful discussions and proofreading of the manuscript. We also thank the anonymous reviewers for their insightful comments. This work is partially funded by French CPER MAuVE (ERDF, Région Hauts-de-France) and by a grant from the French Research Agency (ANR-11-EQPX-0023 IRDIVE).


Giuseppe Geymonat
email: giuseppe.geymonat@polytechnique.edu

Françoise Krasucki
email: francoise.krasucki@umontpellier.fr

Michele Serpilli
email: m.serpilli@univpm.it

ASYMPTOTIC DERIVATION OF A LINEAR PLATE MODEL FOR SOFT FERROMAGNETIC MATERIALS

Keywords: Asymptotic methods, Plates, Magnetoelasticity 2010 MR Subject Classification 74F15, 74K20

We apply the asymptotic expansion method by P.G. Ciarlet to obtain a Kirchhoff-Love-type plate model for a linear soft ferromagnetic material. We also give a mathematical justification of the obtained model by means of a strong convergence result.

Introduction

The use of the framework of continuum mechanics for the study of the influence of electromagnetic effects on solids has been largely stimulated by Truesdell and Toupin [START_REF] Truesdell | The classical field theories[END_REF]. Their research on the coupling between the mechanical and magnetic responses of magnetoelastic solids was the first of a long list (without any attempt to be exhaustive, see, e.g. Brown [START_REF] Brown | Magnetoelastic Interactions[END_REF] Tiersten [START_REF] Tiersten | Coupled magnetomechanical equations for magnetically saturated insulators[END_REF], Maugin and Eringen [START_REF] Maugin | Deformable magnetically saturated media I. Field equations[END_REF], Pao and Yeh [START_REF] Pao | A Linear Theory for Soft Ferromagnetic Elastic Solids[END_REF], Moon [START_REF] Moon | Magneto-solid mechanics[END_REF], Maugin [START_REF] Maugin | Continuum Mechanics of Electromagnetic Solids[END_REF], Dorfmann and Ogden [START_REF] Dorfmann | Nonlinear magnetoelastic deformations of elastomers[END_REF], Kankanala and Triantafyllidis [START_REF] Kankanala | Magnetoelastic buckling of a rectangular block in plane strain[END_REF], ...). An important stimulus for the development of these researches was the study of the magnetoelastic buckling problem. Indeed a plate, made of a magnetoelastic material, subject to a transverse magnetic field, buckles when the magnetic field attains a critical value; see also [START_REF] Kankanala | Magnetoelastic buckling of a rectangular block in plane strain[END_REF][START_REF] Danas | Instability of a magnetoelastic layer resting on a non-magnetic substrate[END_REF], for a general analysis of the buckling of some magnetoelastic structures. Following a pioneering experimental and theoretical research of Moon and Pao, the first rigorous attempt to analyze this problem is due to Pao and Yeh [START_REF] Pao | A Linear Theory for Soft Ferromagnetic Elastic Solids[END_REF]. Maugin and Goudjo [START_REF] Maugin | The equations of soft ferro-magnetic elastic plates[END_REF] have considered a plate model with particular attention on the regularity of the boundary. More recently, in the case linear soft magnetoelastic materials, Zhou and Zheng [START_REF] Zhou | A theoretical model of magnetoelastic buckling for soft ferromagnetic thin plates[END_REF][START_REF] Zhou | General expression of magnetic force for soft ferromagnetic plates in complex magnetic fields[END_REF][START_REF] Zhou | Magnetoelastic bending and snapping of ferromagnetic plates in oblique magnetic fields[END_REF] have revisited the subject by adapting the usual Kirchhoff-Love and von Kármán models only modifying the equivalent transverse force with the addition of the magnetic effects.

The paper is organized as follows. In Sect. 2, we briefly recall the governing equilibrium equations of magnetoelasticity and, then, in Sect. 3, we state the problem on a variable domain assuming that the magnetic forces are given. In order to apply the Ciarlet's method (see, e.g., [START_REF] Ciarlet | Mathematical Elasticity[END_REF]), we must at first prove that the magnetic forces give rise to a linear and continuous form. This can be achieved under suitable assumptions on the magnetic forces, see lemma 3.1. In Sect. 4, we introduce the usual scaling on the mechanical quantities and we scale the magnetic quantities in such a way that the Gauss' Law (2.1) is conserved. Using the classical change of variables, we deduce the scaled equations. Then we can apply the asymptotic methods, following [START_REF] Ciarlet | Mathematical Elasticity[END_REF], to obtain the limit problem and the strong convergence result. It is interesting to remark that in the simplest situation of a transversal magnetic field, we recover the Kirchhoff-Love model of Zhou and Zheng, which is, hence, completely justified.

Governing equations of magnetoelasticity for linear soft ferromagnetic materials

In the sequel, Greek indices range in the set t1, 2u, Latin indices range in the set t1, 2, 3u, and the Einstein's summation convention with respect to the repeated indices is adopted. Let us consider a three-dimensional Euclidean space identified by R 3 and such that the three vectors e i form an orthonormal basis. We introduce the following notations for the vector product: a ^b " a i e i ^bj e j " a i b j ijk e k , for all vectors a " pa i q P R 3 and b " pb i q P R 3 , where ijk denotes the alternator Ricci's symbol.

When a magnetizable, deformable elastic solid Ω is placed in a magnetic field, magnetic moments are induced inside the body. The action of the external magnetic induction B 0 manifests itself in magnetization M (magnetic moment per unit volume). Within the body, the magnetic induction B is not necessarily equal to B 0 . The induced magnetization M " pM i q is related to B " pB i q by B " µ 0 pH `Mq, where H " pH i q is called magnetic intensity and µ 0 is the magnetic permeability of vacuum. Generally, we have H " HpMq, but, in the sequel, we restrict our attention to a class of linear isotropic magnetoelastic materials called soft ferromagnetic materials, which are characterized by the fact that their local average magnetization becomes zero when the external field is set to zero. In this particular case, the hysteresis loops are narrow and the influence of induced currents is small in comparison with the effect of magnetization. Therefore, it is possible to use the quasi-static approximation, i.e., the equations of magneto-statics:

" B i B i " 0 in Ω, (Gauss' Law) kij B j H i " 0 in Ω. (Ampère's Law) (2.1)
The magnetic constitutive law takes the following linear form

M " χH or B " µ 0 µ r H, (2.2)
where χ represents the magnetic susceptibility and µ r :" χ `1 is the relative magnetic permeability. For linear soft ferromagnetic materials, such as steels, iron, cobalt and various alloys, the relative permeability is very large, µ r or χ " 10 2 " 10 5 .

In this work we use the model proposed by Brown [START_REF] Brown | Magnetoelastic Interactions[END_REF] where the action of the magnetic field is given by a magnetic body force (per unit volume) f m " pf m i q and a magnetic body couple (per unit volume) I m " pI m i q:

f m " µ 0 p∇HqM and I m " µ 0 M ^H Title 3

i.e. component-wise, f m i " µ 0 M j B j H i and I m i " µ 0 kji M k H j , This choice of the action of the magnetic field is sometimes called the dipole model of microcurrents and has been used in particular by Pao-Yeh for soft ferromagnetic elastic solids [START_REF] Pao | A Linear Theory for Soft Ferromagnetic Elastic Solids[END_REF]. Using the Gauss' law and the Ampère's law, the magnetic body force f m can also be written as the divergence of a second order tensor T m " pT m ij q, the so-called Maxwell's stress tensor

f m " divT m , (2.3)
or, component-wise,

f m j " B i T m ij with T m ij :" B i H j ´1 2 µ 0 H 2 δ ij and H 2 :" H k H k .
As pointed out by [START_REF] Brown | Magnetoelastic Interactions[END_REF], other different choices of the Maxwell's stress tensor are possible and, indeed, they depend on the choice of the Helmomtz free energy; see in particular [START_REF] Kankanala | Magnetoelastic buckling of a rectangular block in plane strain[END_REF], for a clear explanation of the influence of the choice of the arguments in the free energy on the Maxwell stress, magnetic body forces and traction boundary conditions.

Considering the expressions above, in the absence of electric field, charge distribution and conduction current, the mechanical governing equations defined in a magnetized body Ω can be expressed by

$ & % B i t ij `f m j " 0 in Ω, t ij ´tji `µ0 pM i H j ´Mj H i q " 0 in Ω, t ij n i " 1 2 µ 0 pM n q 2 n j in Ξ, (2.4)
where t ij is the non-symmetric total stress tensor, pn i q represents the unit normal vector to the boundary Ξ Ă BΩ and M n :" M i n i is the normal surface boundary magnetization. The nonsymmetry of the total stress tensor is due to the presence of a magnetic body couple. In order to simplify the model, we neglect magnetostriction and piezostrictive terms in the constitutive laws and we consider an isotropic linear elastic material. Thus, we obtain that

t ij " σ ij `µ0 M i H j , with σ ij " λe pp δ ij `2µe ij , (2.5)
where σ ij denotes the symmetric Cauchy stress tensor, associated with the linearized strain tensor e ij " 1 2 pB i u j `Bj u i q, being pu i q the displacement field, through the classical Lamé's constitutive equations. The presence of the term µ 0 M i H j in the decomposition of the total stress always follows from the form of the Helmoltz free energy (see [START_REF] Brown | Magnetoelastic Interactions[END_REF] and [START_REF] Kankanala | Magnetoelastic buckling of a rectangular block in plane strain[END_REF]).

In the Pao-Yeh's case of soft ferromagnetic materials, thanks to (2.2), the magnetic body couple I m " µ 0 χH ^H " 0, and, hence, the stress tensor t ij becomes symmetric, i.e., t ij " t ji . Moreover, substituting the expression of the Maxwell's stress tensor into the divergence relation (2.3) and using (2.1), we can find an alternative form of the magnetic body force:

f m " µ 0 p∇HqM " µ 0 χ 2 ∇pH 2 q.
In the sequel, we will focus our attention to the reduced mechanical model arising from the use of the asymptotic methods, assuming that the magnetization M and the magnetic intensity H are a given external magnetic source.

Position of the problem

Let ω Ă R 2 denote a smooth domain in the plane spanned by vectors e α , with boundary γ; γ 0 Ă γ is a measurable subset of γ with strictly positive length measure; γ 1 :" γzγ 0 is the complement of γ 0 with respect to γ; finally, 0 ă ε ă 1 is a dimensionless small real parameter which shall tend to zero. For each ε, we define

Ω ε :" ω ˆp´h ε , h ε q, Γ ε :" γ ˆp´h ε , h ε q Γ ε 0 :" γ 0 ˆp´h ε , h ε q, Γ ε ˘:" ω ˆt˘h ε u,
with h ε ą 0. Hence the boundary BΩ ε of Ω ε is partitioned into the lateral surface Γ ε and the upper and lower faces Γ ε `and Γ ε ´; the lateral surface is itself partitioned as

Γ ε " Γ ε 0 Y Γ ε 1 , with Γ ε 1 :" γ 1 ˆp´h ε , h ε q. Moreover, we let p Γ ε :" Γ ε ˘Y Γ ε 1 " BΩ ε zΓ ε 0 , the complement of Γ ε 0 with respect to BΩ ε .
We assume that Ω ε is constituted by a homogeneous isotropic linear soft ferromagnetic material, whose constitutive law is given in (2.5). We suppose that the Lamé's coefficients satisfy the classical positivity properties. The plate is clamped on Γ ε 0 , so that u ε " 0, and, for simplicity, we consider that no mechanical charges are applied to the body. The only source terms are given by M ε i and

H ε i . Let V pΩ ε q :" v ε P H 1 pΩ ε ; R 3 q; v ε " 0 on Γ ε 0 (
be the functional space of admissible displacements. The variational formulation of problem (2.4), defined on the variable domain Ω ε , takes the following form

" Find u ε " pu ε i q P V pΩ ε q such that A ε pu ε , v ε q " L ε pv ε q, for all v ε " pv ε i q P V pΩ ε q, (3.1)
where the bilinear form A ε p¨, ¨q and the linear form L ε p¨q are, respectively, defined by

A ε pu ε , v ε q :" ż Ω ε t ε ij e ε ij pv ε qdx ε , L ε pv ε q :" ż Ω ε µ 0 M ε j B ε j H ε i v ε i dx ε `żp Γ ε 1 2 µ 0 pM ε n q 2 n ε i v ε i dΓ ε , with t ε ij :" λe ε pp pu ε qδ ij `2µe ε ij pu ε q `µ0 M ε i H ε j .
In order to prove the wellposedness of the problem, by virtue of the Lax-Milgram's lemma, we rewrite (3.1) in an alternative form:

"

Find u ε " pu ε i q P V pΩ ε q such that Āε pu ε , v ε q " Lε pv ε q, for all v ε " pv ε i q P V pΩ ε q,

where

Āε pu ε , v ε q :" ż Ω ε λe ε pp pu ε qe ε qq pv ε q `2µe ε ij pu ε qe ε ij pv ε q ( dx ε , Lε pv ε q :" ż Ω ε µ 0 M ε j B ε j H ε i v ε i ´M ε i H ε j e ε ij pv ε q ( dx ε `żp Γ ε 1 2 µ 0 pM ε n q 2 n ε i v ε i dΓ ε .
Since the bilinear form Āε p¨, ¨q is V pΩ ε q-coercive, in order to apply the Lax-Milgram's lemma, we have only to prove that the linear form Lε p¨q is continuous on V pΩ ε q. For this when 1 ă p ă `8 we denote W 1 p pΩ ε q the Banach space of v P L p pΩ ε q whose first order derivatives (in the distribution sense) also belong to L p pΩ ε q. The continuity of the linear form Lε p¨q is the object of the following lemma. Lemma 3.1 Let us assume:

M ε i , H ε i P W 1 12{5 pΩ ε q (3.3)
then Lε p¨q is continuous on V pΩ ε q. Proof. (i) Since v ε i P H 1 pΩ ε q, thanks to the Sobolev imbedding theorem, (see e.g. Nečas [START_REF] Nečas | Direct Methods in the Theory of Elliptic Equations[END_REF], Chap. 2, Theor. 3.4) we obtain that v ε i P L 6 pΩ ε q and that f m,ε i P L 6{5 pΩ ε q; hence, by means of Hölder's inequality, we can infer that

| ż Ω ε µ 0 M ε j B ε j H ε i v ε i ( dx ε | ď C}v i } 1,Ω ε (3.4) (ii) Since M ε i , H ε i P W 1
12{5 pΩ ε q, thanks to the Sobolev imbedding theorem, we get that M ε i , H ε i P L 12 pΩ ε q Ă L 4 pΩ ε q and, thus,

M ε i H ε i P L 2 pΩ ε q. It then follows from Korn's inequality | ż Ω ε µ 0 M ε i H ε j e ε ij pv ε q ( dx ε | ď C}e ε ij pv ε q} L 2 pΩ ε q ď C}v} V pΩ ε q (3.5)
(iii) By virtue of a trace imbedding theorem (see e.g. Nečas [START_REF] Nečas | Direct Methods in the Theory of Elliptic Equations[END_REF], Chap. 2, Theor. 4.2), we have that M ε i | BΩ ε P L 8 pBΩ ε q. Besides, being v P V pΩ ε q, then the same trace imbedding theorem imply that v i | BΩ ε P L 4 pBΩ ε q, and so

| ż p Γ ε 1 2 µ 0 pM ε n q 2 n ε i v ε i dΓ ε | ď C}pM ε n q 2 } L 4 pBΩ ε q }v ε i } L 4 pBΩ ε q ď C}v} V pΩ ε q (3.6)
Collecting (3.4), (3.5) and (3.6), we obtain the desired result.

Thanks to the V pΩ ε q-coercivity of the bilinear form Āε p¨, ¨q and the continuity of the linear form Lε p¨q we deduce, using the Lax-Milgram's lemma, that the variational problem (3.1) admits one and only one solution.

The asymptotic expansion

In order to perform an asymptotic analysis, we need to transform problem (3.1), posed on a variable domain Ω ε , onto a problem posed on a fixed domain Ω (independent of ε). We suppose that the thickness of the plate h ε depends linearly on ε, so that h ε " εh. Accordingly, we let Ω :" ω ˆp´h, hq, Γ 0 :" γ 0 ˆp´h, hq, Γ 1 :" γ 1 ˆp´h, hq,

Γ ˘:" ω ˆt˘hu, p Γ :" Γ ˘Y Γ 1 .
and we define the following change of variables (see [START_REF] Ciarlet | Mathematical Elasticity[END_REF]):

π ε : x " px, x 3 q P Ω Þ Ñ x ε " px, εx 3 q P Ω ε , with x " px α q.
By using the bijection π ε , one has B ε α " B α and B ε 3 " 1 ε B 3 . Moreover, we define the following functional spaces V pΩq :" v " pv i q P H 1 pΩ; R 3 q; v " 0 on Γ 0 ( , W 1 12{5 pΩq.

In order to write the expression of the scaled problem, we need first to make some assumptions on the data which will define their dependences with respect to the small parameter ε. In our case, the only external data are represented by the magnetic field pB ε i q, the magnetic intensity field pH ε i q and the magnetization field pM ε i q. By virtue of the constitutive equations B ε i " µ 0 µ r H ε i and M ε i " χH ε i , we assume that B ε i , H ε i and M ε i will share the same dependence on ε. The scaling of B ε i must reflect the fact that the magnetic field is a solenoidal field, meaning that B ε i B ε i " 0 in Ω ε . This property must be satisfied also on the fixed domain Ω. Let us suppose that B ε α px ε q " ε q B α pxq, B ε 3 px ε q " ε p B 3 pxq, x P Ω, with B i independent of ε. By applying the change of variables π ε , we can write that the scaled divergence vanishes in Ω, so that ε q B α B α `εp´1 B 3 B 3 " 0 in Ω. In order to guarantee the consistency of this equation, we ask that p " q `1, finding a relation between the two exponents p and q. In the sequel, we choose q " 0 and, hence,

B ε α px ε q " B α pxq, B ε 3 px ε q " εB 3 pxq, x P Ω, H ε α px ε q " H α pxq, H ε 3 px ε q " εH 3 pxq, x P Ω, M ε α px ε q " M α pxq, M ε 3 px ε q " εM 3 pxq, x P Ω.
With the unknown displacement field u ε , we associate the scaled unknown displacement field upεq defined by u ε α px ε q " u α pεqpxq for all x ε " π

ε x P Ω ε , u ε 3 px ε q " 1 ε u 3 pεqpxq for all x ε " π ε x P Ω ε .
We likewise associate with any test function v ε , the scaled test function v, defined by the scalings:

v ε α px ε q " v α pxq for all x ε " π ε x P Ω ε , v ε 3 px ε q " 1 ε v 3 pxq for all x ε " π ε x P Ω ε .
According to the previous hypothesis, problem (3.1) can be reformulated on a fixed domain Ω independent of ε. Thus we obtain the following scaled variational problem:

"

Find upεq " pu i pεqq P V pΩq such that Apupεq, vq " Lpvq, for all v " pv i q P V pΩq,

where the scaled bilinear form Ap¨, ¨q and the scaled linear form Lp¨q are, respectively, defined by

Apupεq
ż Ω µ 0 M j B j H i v i dx `1 2 ż Γ`µ 0 pM 3 q 2 v 3 dΓ ´1 2 ż Γ´µ 0 pM 3 q 2 v 3 dΓ1 2 ż Γ1 µ 0 pM β n β q 2 n α v α dΓ,
where φ ˘:" φpx, ˘hq denotes the restriction of φ on Γ ˘. Since H i , M i P W 1 12{5 pΩq, thanks to lemma 3.1 and Lax-Milgram's lemma, we can prove that the scaled problem admits one and only one solution.

We are now in position to perform an asymptotic analysis of the scaled problem (4.1). Since the scaled problem (4.1) has a polynomial structure with respect to the small parameter ε, we can look for the solution of the problem as a formal series of powers of ε:

upεq " u 0 `ε2 u 2 `ε4 u 4 `. . . (4.2)
Hence, by substituting expressions (4.2) in (4.1) and by identifying the terms with identical power of ε, we can write the following sequence of variational subproblems:

P ´4 : a ´4pu 0 , vq " 0, P ´2 : a ´4pu 2 , vq `a´2 pu 0 , vq " 0, P 0 : a ´4pu 4 , vq `a´2 pu 2 , vq `a0 pu 0 , vq " Lpvq, . . . (4.3) By solving the above variational problems, we can characterize the leading term of the asymptotic expansion u 0 , the so-called the limit displacement field, and its associated limit problem.

The limit problem

We define the usual functional space of Kirchhoff-Love admissible displacements:

V KL pΩq :" tv P V pΩq; e i3 pvq " 0u, and V H pωq :" tv H " pv α q P H 1 pω; R 2 q; v H " 0 on γ 0 u, V 3 pωq :" tv 3 P H 2 pωq; v 3 " 0 and B ν v 3 " 0 on γ 0 u, where ν " pν α q is the outer unit normal vector to γ.

Theorem 5.1 a) The leading term u 0 of the asymptotic expansion (4.2) satisfies the following variational problem:

#

Find u 0 P V KL pΩq such that Apu 0 , vq " Lpvq for all v P V KL pΩq,

where Apu 0 , vq :"

ż Ω " 2µλ λ `2µ
e σσ pu 0 qe τ τ pvq `2µe αβ pu 0 qe αβ pvq `µ0 M α H β e αβ pvq * dx.

(5.2)

b) The sequence tupεqu εą0 strongly converges in H 1 pΩ; R 3 q to u 0 , the solution of the limit problem (5.1).

Proof. The proof is straightforward, following the approach by [START_REF] Ciarlet | Mathematical Elasticity[END_REF].

Let us focus our attention to the expression of the magnetic force work Lp¨q of the limit problem. By choosing a test function v P V KL pΩq, namely v α px, x 3 q :" η α pxq ´x3 B α η 3 pxq and v 3 px, x 3 q :" η 3 pxq, with η α P H 1 pωq and η 3 P H 2 pωq, after an integration along x 3 and by applying the Gauss-Green's formula, we get that

Lpηq " ż Ω µ 0 tpM 3 B 3 H 3 `Mα B α H 3 qη 3 `pM β B β H α `M3 B 3 H α qpη α ´x3 B α η 3 qu dx1 2 ż Γ`µ 0 pM 3 q 2 η 3 dΓ ´1 2 ż Γ´µ 0 pM 3 q 2 η 3 dΓ `1 2 ż Γ1 µ 0 pM β n β q 2 n α pη α ´x3 B α η 3 qdΓ " " ż ω f m i η i dx `żγ1 gm i η i dx `żγ1 hm 3 B ν η 3 dx.
where the reduced magnetic forces f m i , gm i and hm 3 have the following form

f m α :" µ 0 χxH i B i H α y " µ 0 xf m α y, f m 3 :" µ o χ " 1 2 pχ `1qppH 3 q 2 ´pH 3 q 2 q `xH β B β H 3 y `Bα xxH i B i H α yy * " " µ 0 χ " 1 2 pχ `1qppH 3 q 2 ´pH 3 q 2 q `xH β B β H 3 y * `µ0 xxB α f m α yy, gm α :" 1 2 µ 0 χ 2 xH 2 ν yν α , gm 3 :" ´µ0 χxxH i B i H α yyν α " ´µ0 xxf m α yyν α , hm 3 :" ´1 2 µ 0 χ 2 xxH 2 ν yy,
where H ν :" H α ν α , and xφypxq :"

ż h ´h φpx, x 3 qdx 3 , xxφyypxq :" ż h ´h x 3 φpx, x 3 qdx 3 .
It easy to verify that if the induced magnetic intensity field is normal to the middle plane of the plate, with H α " 0, the form of the limit magnetic force acting on a plate depends just on the jump of the square of magnetic intensities evaluated at the top and bottom faces of the plate. Indeed, since f m α " gm α " hm 3 " 0, one has f m 3 :" 1 2 µ 0 χ p1 `χq pH 3 q 2 ´pH 3 q 2 ( « 1 2 µ 0 χ 2 pH 3 q 2 ´pH 3 q 2 ( ,

χ is very large for soft ferromagnetic materials. Equation (5.3) is analogue to the one presented in [START_REF] Zhou | General expression of magnetic force for soft ferromagnetic plates in complex magnetic fields[END_REF] and it can be considered as a mathematical justification of the magnetic force acting on a plate, which is usually employed in magnetic instability problems. The limit problem (5.1) can be decoupled into a membrane and a bending problem, by virtue of the Kirchhoff-Love limit displacement field. The membrane problem reads as follows:

$ ' & ' % Find u 0
H " pu 0 α q P V H pωq such that ż ω n αβ pu 0 H qe αβ pη H qdx " ż ω f m α η α dx `żγ1 gm α η α dx for all η H " pη α q P V H pωq, where n αβ pu 0 H q :" 4hλµ λ `2µ e σσ pu 0 H qδ αβ `4hµe αβ pu 0 H q `µ0 xM α H β y, represents the ferromagnetic membrane stress tensor. After an integration by parts, we find

Title

The present work represents a first step on the asymptotic modeling of soft ferromagnetic plates. Indeed, we do not consider the coupling between the mechanical and magnetic behaviors within the equations of magnetostatics. We assume the magnetic charges as external loads without investigating the asymptotic behavior of the magnetostatic equations for what concerns with the magnetic field, the magnetic intensity and the magnetization. Therefore, the limit problem becomes linear and we cannot see, at first glance, the so-called magnetic buckling phenomenon.

Title 5

 5

  t4µe α3 pupεqqe α3 pvq `λe σσ pupεqqe 33 pvq `λe 33 pupεqqe σσ pvqu dx, a 0 pupεq, vq :" ż Ω tλe σσ pupεqqe τ τ pvq `2µe αβ pupεqqe αβ pvq `µo M i H j e ij pvqu dx,

	, vq :"	1 ε 4 a ´4pupεq, vq	`1 ε 2 a ´2pupεq, vq `a0 pupεq, vq,
	with
	ż
	a ´4pupεq, vq :"	pλ `2µqe 33 pupεqqe 33 pvqdx,
	Ω
	ż
	a ´2pupεq, vq :"
	Ω
	Lpvq :"

Acknowledgement We thank K. Danas and N. Triantafyllidis for many useful discussions on magnetoelastic materials.

that the membrane displacements u 0 α solve the following membrane differential problem:

Field equation:

in ω, Boundary conditions:

The bending problem takes the following form

where

represents the ferromagnetic moment stress tensor. After an integration by parts, we find that the transversal displacement u 0 3 solves the following bending differential problem:

Field equation:

Boundary conditions:

where τ " p´ν 2 , ν 1 q represents the unit tangent vector to γ.

Considering the case of an induced magnetic intensity field, normal to the middle plane of the plate, with H α " 0, n αβ and m αβ reduce to the classical elastic membrane stress tensor and moment stress tensor. Besides, since f m α " gm α " 0, we can infer that the membrane problem admits the only zero solution, so that n αβ " 0, and thus, in this case, the plate equilibrium problem takes just into account the bending behavior.

Concluding remarks

In this work we derive a model of a soft ferromagnetic isotropic linear plate by means of an asymptotic analysis. In the absence of mechanical loading, thanks to the particular scaling of the magnetic charges, we obtain a complex expression of the reduced magnetic forces acting on the plate. The problem can be decoupled as usual in a membrane problem and in a flexural problem. It is important to notice that in the simple case in which the magnetic charges are normal to the middle plane of the plate, we formally obtain an expression of the magnetic force (5.3), acting on the plate, which is equivalent to the one used in classical literature, see, e.g., [START_REF] Pao | A Linear Theory for Soft Ferromagnetic Elastic Solids[END_REF][START_REF] Zhou | A theoretical model of magnetoelastic buckling for soft ferromagnetic thin plates[END_REF][START_REF] Zhou | General expression of magnetic force for soft ferromagnetic plates in complex magnetic fields[END_REF]. Moreover, by virtue of the strong convergence result, we also give a mathematical justification to the limit model.