In this paper, we aim at better understanding how human mental representations are structured in the specific case of the perception of urban soundscapes. This task is traditionally studied using questionnaires, surveys or categorization tasks followed by a lexical analysis. In contrast, we propose a new experimental approach to tackle this aim. In this approach, the subject is asked to manipulate sound events and textures within a dedicated computer environment in order to recreate two complex urban soundscapes, one ideal and the other not ideal. Subjects have access to a sound data set which has been designed and structured based upon perceptual considerations, and may alter the physical parameters of the selected sound samples. In order to achieve this, we use an audio-digital environment and a web audio interface for sound mining developed for the purpose of this study. The latter allows subjects to explore a sound database without resorting to text. By focusing on the auditory modality during the experimental process, this new paradigm potentially allows the subject to be better put in context and provides a more detailed description of the actual mental representations. In the light of the results presented in this paper, it seems that it also reduces the potential bias of only using verbalization during the experiment.

Introduction

Even if the city has always been a noisy environment, whatever the times, our perception of this noise has evolved. It is in the 80s that the association between noise and pollution has been the strongest. Noise was considered as an overall degradation of the quality of life [START_REF] Raimbault | Simulation des ambiances sonores urbaines: intégration des aspects qualitatifs[END_REF]. In response, " anti-noise " legislation took place and planed to fight the noise by reducing its intensity level. But the problem remains, and for good reason, noise is a subjective phenomenon, that depends on the " listener appreciation ". The noise is a matter of context. It may entertain as well as disturb or annoy. Improving the sound environment by only focusing on acoustical parameters, which are by definition objective, is not enough. Besides, this acoustical approach does not tackle directly the issue of improving the soundscape, as it is focused on isolated " negative sounds ", and not on interconnected " positive sounds ". To summarize, a pleasant city is not a silent city.

Understanding the noise requires a methodology that differs from the more traditional approach of psychophysics, as noise is more " a cognitive object than a physical object " [START_REF] Guastavino | Etude sémantique et acoustique de la perception des basses fréquences dans l'environnement sonore urbain[END_REF]. For those reasons a new concept of soundscape had been introduced by R.M. Schafer [START_REF] Schafer | The Tuning of the World[END_REF]. A commonly agreed definition of the " soundscape " has been given by Truax who worked into the World Soundscape Project: " an environment of sound (sonic environment) with emphasis on the way it is perceived and understood by the individual, or by a society " (Truax quoted in [START_REF] Niessen | Categories for soundscape: toward a hybrid classification[END_REF]). Soundscape is a subject centered approach that considers a sound environment from the listener point a view. It assumes that the qualitative evaluation of a soundscape mainly depends on the context, and the knowledge of the community or the subject that experiment the environment. The question is no longer when the noise is annoying, but why the sound is annoying.

Since its introduction, this approach has been widely used to study urban sonic environment [START_REF] Davies | Perception of soundscapes: An interdisciplinary approach[END_REF]. It allowed the development of a database of useful qualitative descriptors and sounds to better understand the way humans perceive their sound environments. One major today's challenge is to connect these perceptual data derived from questionnaires, categorization tasks and psycho-linguistic studies to acoustic measurements in order to establish an effective policy of noise reduction adapted to each situation [START_REF] Schulte-Fortkamp | Soundscape-focusing on resources[END_REF]. To this end, we propose a new experimental approach to study soundscape perception. In order to investigate what could be the nature of an ideal urban environment, we ask subjects to simulate two soundscapes, one qualified as " ideal " (in which you would like to live) the others " not ideal " (in which you would not live), from a dedicated urban sound data set. By analyzing the reconstruction process, we are able to objectify the mental representations of an ideal (not ideal resp.) sound environment.

We will first introduce the experimental paradigm as well as the technological devices on which it relies, then we will detail the creation of our sound data set, and finally we will present the results of a pilot study.

Experimental Protocol

Paradigm of the proposed approach

Traditional psycho-cognitive studies addressing mental representations of sound environments use linguistic resources to objectify mental categories. Two types of approach has been widely used: The sorting tasks, where subjects are asked to sort sounds and label the classes (categorization tasks), and the describing tasks, where subjects are asked to describe a sound via a adapted questionnaire (usually free or semi structured questionnaire). Both approaches consider verbal data as input, and thus rely on psycho-linguistic analysis to objectify generic and meaningful mental categories from names or labels.

In this paper, the proposed approach is also a psychocognitive one, as we also consider subjective literal data. But these data are related to quantitative data that are assumed to be directly linked to the " reality of the word ". To do so, we adopt a simulation paradigm. The subject is asked to recreate a complex sound environment, by choosing sound classes in a sound data set, and modifying their physical properties (sound levels, time positioning). The selection process is made without any text-written help (see Section 2.3 for more details). The subject must name each selected sound. At the end of the creation process, the subject has to give a global name (title) to the simulated scene, and freely comment its creation. Doing so, we obtain 2 types of data: the objective data and the subjective data. The subjective data depend on the subject. They are composed of the names given by the subjects to the sound classes they selected, the titles of the simulated scenes, and the free comments. The objective data depend on the sound data set. They are non ambiguous and may be controlled by the experimenters. The objective data are composed of 1) the numerical data that are the values of the audio control parameters set by the subject to simulate the scene, and 2) non ambiguous verbal data called tag. A tag is the " real name " of a sound class of the data set. It is given by the experimenter who recorded it. The tags of the sound classes act as a ground-truth compared to the names given by the subjects.

Mixing both objective and subjective data allows us to obtain subject-centered results that are both meaningful and easily analyzable. These two outputs may always be weighted relative to each other in order to give meaningful results.

Creation of the sound data set

Special care has been taken to build the sound data set as it represents the " sonic world " upon which subjects will rely to simulate a desired soundscape. The goal was to propose a sound data set that 1) is representative of the diversity of all the sounds that populate the urban environment, 2) is able to offer several variants of a same sound class, and 3) could be quickly explored in order to supplies to the time constraints of the experiment. Moreover, as our tags entirely depend on the sound data set nomenclature, a consistent and, as much as possible, generic typology of the different classes of urban sounds has been designed. To this sense, the sound data set structure was motivated by practical and perceptual considerations.

We decided to divide the sound data set in two subsets, each corresponding to a specific type of sounds: the sound events and the sound textures. Several studies have pointed out that this two types of sounds provoke two distinct cognitive processes. Maffiolo [START_REF] Maffiolo | De la caractérisation sémantique et acoustique de la qualité sonore de l'environnement urbain[END_REF] showed that event sequences lead to a descriptive (semantic) analysis, which mainly rely on the identification of the sound events, whereas amorphous sequences, which are sequences without salient events that can be regarded as textures (" traffic hubbub ", " street hubbub "), lead to a holistic analysis which depends on the acoustical properties of the sequence. McDermott and Simoncelli [START_REF] Mcdermott | Sound texture perception via statistics of the auditory periphery: Evidence from sound synthesis[END_REF] recently showed that sound textures perception mainly depends on the analysis of simple statistical properties of the acoustical signal. For those reasons we consider a soundscape as " a skeleton of events on a bed of textures " [START_REF] Nelken | An ear for statistics[END_REF].

Several studies address the difficulty of obtaining a generic classification of all the elements that populated the sound environment [START_REF] Niessen | Categories for soundscape: toward a hybrid classification[END_REF][START_REF] Brown | Towards standardization in soundscape preference assessment[END_REF], as it is difficult to regroup them under consistent and non ambiguous classes. Indeed, sound classes as " traffic sound ", " human voice " or more specific classes as " cars " or " sounds of a man " may regroup a large variety of sounds, depending on the context of the study. Considering these remarks, we design a hierarchical sound data set of urban environmental sounds adapted for our approach. In this hierarchy, top level classes represent large categories of concept such as " sounds of nature " or " motorized transportation sounds ". Deeper the class is in the hierarchy, and lesser is the diversity between the sounds that belong to it. Each class of the leaf level (" car passing ", " man yelling "), regroup several recording of sound samples that are perceptively close. In order to choose relevant class name (tag) for the typology, we did a bibliographical review of several papers addressing mental categories of urban environmental sounds [START_REF] Guastavino | Etude sémantique et acoustique de la perception des basses fréquences dans l'environnement sonore urbain[END_REF][START_REF] Niessen | Categories for soundscape: toward a hybrid classification[END_REF][START_REF] Maffiolo | De la caractérisation sémantique et acoustique de la qualité sonore de l'environnement urbain[END_REF][START_REF] Brown | Towards standardization in soundscape preference assessment[END_REF][START_REF] Raimbault | Urban soundscapes: Experiences and knowledge[END_REF][START_REF] Guastavino | The ideal urban soundscape: Investigating the sound quality of french cities[END_REF][START_REF] Guastavino | Categorization of environmental sounds[END_REF][START_REF] Al Brown | Advancing the concepts of soundscapes and soundscape planning[END_REF].

It has to be noted that subjects only interact with the leaf classes to simulate the soundscape, and not with the recorded samples.

Human voice

Man voice

Man yelling

Simulating a soundscape

To simulate a soundscape we use a web software called Simscene. Simscene allows the user to manipulate sound classes of events and textures (" man yelling ", " car passing ", " rain ", " parc hubbub ") in order to simulate a soundscape. To free the subjects from the influence of the pre-established nomenclature of the sound data set, the selection interface of Simscene has been designed not to use a keyword based search. The selection interface allows the subject to explore the data set by listening to the sounds. To do so, the leaf classes of the typology are distributed in a 2D space, each leaf class being represented by a circle (see figure 1). The space configuration of the circles depends on the hierarchical structure of the typology. Circles representing leaf classes belonging to the same class are packed together, and so on until the top-classes are reached. Relative positions of the top-classes are meaningless.

Subjects may explore the data set by clicking on the circles. When a circle is clicked, the subject hears the sound prototype of the leaf class represented by the circle. By listening to the different prototypes, the subject may select a sound class.

When a sound class is selected, a time sequence populated with the samples (event or texture) belonging to the selected leaf class is displayed.

The subject may then adjust the control parameters of the sequence being the average/variance of time intervals between samples, or average/variance of sound levels. As subjects interact with sound classes, control parameters have been designed to manipulate sequences of sounds and not unique samples. Simscene can be accessed via the link http://soundthings.org/simScene/.

3 The Pilot study

Task and collected data

Subjects are asked to successively create two urban soundscapes. The first must be ideal (i.e. the favorite urban soundscape of the subject), the second not ideal (ie. the worst urban soundscape of the subject). Subject are not restricted o, their design choices. The only constraint is to simulate a physically plausible urban environment. In other words, unrealistic situations as " a dog barking every 10 milliseconds " are forbidden.

Two types of data are collected: The verbal data and the numerical data. For the verbal data, we consider the title of the soundscape, the names given by the subject to each selected sound class, the free comment describing the simulation process, and the missing sounds, that is, the sounds that were not found by the subject in the sound data set. The numerical data are the audio control parameters from which we analyze the sound levels of each samples, and the time intervals between sound events.

Participant

Participants are 10 french subjects individually selected from volunteers. They are 7 males and 3 females and are about the same age (M : 24.1, STD 1.7). All of them are urban dwellers.

Apparatus

Subjects have access to a sound data set of 483 urban environmental sounds including 381 sound events and 102 textures. Among them, 260 events and 72 textures were recorded. 121 sound events and 30 textures which proved to be particularly difficult to record came from existing sound banks. All recordings were performed using a shotgun microphone audio technica AT8035 connected to a ZOOM H4n recorder. We chose to use a shotgun microphone (highly directive) in order to isolate sound recordings from undesired events. All the sounds were normalized to the same peak level.

The experiment is run with one subject at a time, in the audiometric test booth of the IRCAM french institute 1 . Audio is presented in monophonic to each participant via Yamaha MSP 5 speakers (active speaker), on Macintosh Mac Book Pro type computer, connected to a RME FireFace 800 sound card. The software (SimScene) is located on a distant server and loaded via Google-Chrome navigator in a Linux operating system. At the end of each task, data is automatically collected server side. One experimenter is always present to give instructions and answer queries if needed. Experiment lasts about one hour and a half for each subject.

4 Results of the Pilot study

Objective and Subjective verbal data analysis

Selected sound classes analysis

For the ideal scenes, subjects used 51 event classes and 27 texture classes. For not-ideal scenes, they used 96 event classes and 32 texture classes. This result indicates that not-ideal scene are composed of a larger number of sound sources than ideal scenes. Table 1 shows the number of texture and event classes used, averaged over the subjects.

The tags (see section 3.1) of the sounds chosen by the subjects are now analyzed. Tags were regrouped based on their position in the typology. For ease of reading, We chose the semantic hierarchical levels that are the most meaningful. Figure 2 and 3 display the results for the ideal and not ideal scenes. The dominant event classes for the not-ideal scenes are: " alarm " and " horn " (35% of occurrences), " construction work " (22% of occurrences) and " traffic " (13.5% of occurrences). The majority of the texture classes belong to the classes " construction work " (construction work hubbub : 28% of occurrences , work vehicles : 19% of occurrences) and " human voice " ( 25% of occurrences). For the ideal scenes, event classes are mainly sounds of humans (footsteps : 21.6% of the occurrences, and human voices 19.6% of occurrences), sounds of " birds " (11.8% of occurrences), " sounds of bicycle " (11.8% of the occurrences) and " bells " ( 11.8% of occurrences). For the textures, sounds of humans are well represented (voice: 33.3% of occurrences). Some results may be considered as counter-intuitive. For the ideals scenes, we observe the presence of the classes " alarm" , " construction work " and " traffic ". Traffic textures (" traffic hubbub ") are also well represented (25.9% of occurrences). The same applies to the not-ideal scenes where textures of " human voices " and " fountains " are observed.

To check if those results are due to a misidentification of the sounds by the subjects, we look at the names (see section 3.1) given by the subjects to each selected sounds. To achieve the linguistic analysis, we rely on the following rules: We connect a name to a tag of our typology when the former makes explicit reference to an event class (" foot step ", " man calling someone "), or a texture class (" courtyard ", " street atmosphere "). For the names that cannot be explicitly connected to a tag, we check if some of them belong to the same lexical field.

If we detect a common lexical field along the names,

we regroup them under the same designation 2. If we do not detect a common lexical field, we eliminate the isolated name of the analysis. These sounds are then referred to as " untreated "

We show the results of the linguistic analysis in the Figures 4 and5 for respectively the ideal scenes and the not-ideal scenes. In general, the events were well identified with 90% of non ambiguous associations between names and tags for the ideal scenes, and 92% for the not-ideal scenes. The identification is less obvious for the textures with only 48% of non ambiguous associations between names and tags for the ideal scenes, and 47% for the not-ideal scenes. We observe that more than 25% of the textures names do not refer to a sound source, but rather to a global description of the all texture. These observations are in line with those obtained by Maffiolo [START_REF] Maffiolo | De la caractérisation sémantique et acoustique de la qualité sonore de l'environnement urbain[END_REF] which states that sequences of events trigger a semantic analysis (i.e. identification of sound sources), while amorphous sequences are subject to a holistic treatment, which depends on the acoustical properties of the signal, and thus does not promote the identification of sound sources. Several names given to the textures for the ideal as well as the not-ideal scenes directly refer to what could be considered as sound events (birds, public transport). This is most probably due to the fact that it is very difficult to record a scene without any sound events. Some of them, specially birds sounds, occur in the texture sequences. It is interesting to note that it is those unwanted events that drive the identification process of the textures, even if they are poorly represented in the textures. This underlines the fact that semantic value of a soundscape mainly depends on the recalled sound events.

Considering the textures of the ideal scenes, we see a significant decrease of the number of sounds related to " traffic hubbub " between the tags and the names. Traffic sounds are designated by the subject with generic names: " street atmosphere " (french: " ambiance de rue "), or " background " (french: " bruit de fond "), indicating that sounds related to urban traffic are accepted only if they are part of the background of the scene. There is a notable difference between the tags and the names for the not-ideal textures. Many sound tagged " construction work " has been identified as urban traffic sounds.

Tags analysis reveals the presence of sound events belonging to the " alarm/horn " category (sounds used by one Figure 5: Names given by the subjects for the not-ideal scenes (percentage of the total amount of names used) subject) and construction work (sounds used by 2 subjects) in the ideal scenes. By analyzing the free comments given by the subjects, we note that the alarm sounds have been chosen conscientiously, and are described as " discreet and nice to hear ". According to the subjects, " construction work " sounds have been chosen for realistic considerations.

We propose to compare our results to those obtained with a similar study conducted by Guastavino [START_REF] Guastavino | The ideal urban soundscape: Investigating the sound quality of french cities[END_REF], in which she asked subjects to describe their ideal urban sound environment. Thanks to a psycho-linguistic analysis, she derived categories of positive and negative sounds. For the ideal scenes, our results are consistent with those of Guastavino. In both cases the main represented categories are: human sounds (footsteps and voice) and sounds of nature (animals, weather). The difference lays in the appreciation of some categories. In both studies, sound categories of " traffic ", " construction work" , " alarm/horn " are used to characterize ideal cities. In the case of Guastavino, terms relative to those categories are found in negative construction (" with no cars " or " with less traffic " ), as they give rise to negative judgments. In our case, the analysis of the free comments made by the subjects of their scenes showed that these sounds were used to make the simulated environment plausible. Several subjects clearly indicated that they found distant sounds of " traffic " and " construction work " pleasant.

Missing sounds analysis

We identify 23 references to missing sounds, 14 for the ideal scenes and 9 for the not-ideal scenes. Among these sounds, 3 refer to musical sounds and 10 refer to sounds that are present in the data set, and were not found by the subjects. As musical sounds are deliberately not integrate in the typology, this leave us with 10 well missing sounds. This small number tends to show that the proposed corpus is representative of the diversity of an urban sound environment. 50% of the subjects spontaneously specified in the free comments that the diversity of the corpus was sufficient for the ideal scenes (20% subjects), the not-ideal scenes (20% subjects), or both (10% subject).

Title analysis

We perform the analysis of the titles given by the subjects to the simulated scenes. The goal is to identify categories of geographic location. As a title may contains several distinct semantic entities, one title may be put in several geographical categories. From the lexical analysis of the titles of the 10 ideal scenes, 4 categories were derived: " park " (4 subjects) " pedestrian space " (3 subjects), " courtyard " (2 subjects) and " quiet street " (2 subjects). Titles categories refer to urban area where traffic noise is absent or limited, but where human sounds are well represented. From the analysis of the not-ideal titles, 4 geographical categories were identified: " street " (4 subjects) , " boulevard " (2 subjects), " avenue " (2 subjects) and " crossroads " (2 subjects). 3 subjects titles clearly suggests the presence of construction work sounds, and 3 others the presence of traffic sounds.

Numerical data analysis

Relative sound levels and time intervals

In this section the sound levels and the time intervals set by the subjects via the control parameters are investigated. Table 2 displays the means and standard deviations of the events and textures sound levels for the ideal and the notideal scenes. It should be noted that all sounds have an initial relative level fixed to 0 dB SPL. Subjects may only reduce the sound level. Results show that the mean sound levels for the ideal scenes is lower than for the not-ideal scenes. Similarly textures levels are lower than events levels (t-test: between ideal events and textures: p = 0.02; between not ideal events and textures: p = 5.10 -4 ).

Table 3 shows the averages and standard deviations of the time intervals between events. This parameters apply only to the event classes. Given the length of a simulated scene (60 seconds), we note that the average time intervals between events are not significantly different for the ideal and the not ideal scenes (t-test: p = 0.59).

Results tend to indicate that if the overall sound level of an ideal city is lower than that of a not-ideal city, the ideal city remains an active environment in term of sound events activity.

Discussion and perspective

In this paper, a new experimental protocol used to objectify mental representations of urban soundscapes has been introduced. To perform the experiment, we created 3: Average and standard deviation of the time intervals between events for the ideal and the not-ideal scenes Events Ideal scenes 17;16 Not-ideal scenes 18.5;16.5 a corpus of environmental sounds based on a typology established from a dedicated literature review. To free the subjects from the influence of an a priori nomenclature, as well as the lack of lexical words to describe adequately the acoustic phenomena [START_REF] Dubois | A cognitive approach to urban soundscapes: Using verbal data to access everyday life auditory categories[END_REF][START_REF] Guastavino | The ideal urban soundscape: Investigating the sound quality of french cities[END_REF], we have developed an interface which allows subjects to explore a sound data set by listening to the sounds themselves.

The comparison between our results and those of Guastavino [START_REF] Guastavino | The ideal urban soundscape: Investigating the sound quality of french cities[END_REF] shows that the proposed approach provides consistent outputs.

However, while the approach of Guastavino allows to consider the mental representations of an urban sound environment as a whole, the proposed paradigm allows to refine the analysis of the categories, thanks to the pre-established typology.

The pilot study shows that the proposed protocol provides us with ecologically viable data. Indeed, motivated by a realism concern, some subjects have voluntarily placed in their ideal scenes sounds that could be a priori considered as unpleasant events. The same tendency is observed for the textures of the ideal scenes from which 25.9% are sounds relative to " traffic hubbub ". Several subjects stated in their free comments that they deliberately deleted sound elements of their soundscape in order not to overload the final sound environment, particularly for the ideal scenes. This fact reinforces the ecological validity of the experimental protocol. This deletion is in fact the result of an awareness, by the subject of the sonic context of an urban environment. This awareness is encouraged by the experimental paradigm. We believe that the simulation process acts as a mirror, allowing the subject to adjust its responses during the experiment, and thus provides meaningful data. Subjects do not manipulate isolated sounds, but interconnected sounds as they are perceived in the " real word ". The analysis of the numerical data shows a significant difference between the average sound levels of an ideal urban environment and that of a not-ideal environment. Nevertheless, both environments present the same activity in terms of occurrences of sound sources.

To conclude, the pilot study described in this paper validates the potential of this new experimental paradigm, as well as many of the design choices that were made. In order to reach conclusive results about the points discussed in this section, future work will focus on the development of this experiment at a larger scale.

Figure 1 :

 1 Figure 1: Sound selection Interface of Simscene with illustration of the urban sound data set hierarchy

Figure 2 :Figure 3 :

 23 Figure 2: Tags chosen by the subjects for the ideal scenes (percentage of the total amount tags used)

Figure 4 :

 4 Figure 4: Names given by the subjects for the ideal scenes (percentage of the total amount of names used)

Table 1 :

 1 Average and standard deviation of the numbers of event and texture classes used by each subject

		Events Textures
	Ideal scenes	5.1;2.4	2.7;1.1
	not-ideal scenes 9.6;3.1	3.2;1.8

Table 2 :

 2 Average and standard deviation of the events and textures sound levels for the ideal and the not-ideal scenes

		Events(dB) Textures(dB)
	Ideal scenes	-6.1;7	-10.3;8.1
	Not-ideal scenes	-1.2;3.2	-4.1;5.5
	Table

Ircam: (Institut de Recherche et Coordination Acoustique/Musique) www.ircam.fr/

Acknowledgments

Research project partly funded by ANR-11-JS03-005-01.


Jean-Yvon Faou

Guillaume Parry

Sergey Grachev

Etienne Barthel
email: etienne.barthel@saint-gobain.com

Telephone cord buckles -a relation between wavelength and adhesion

Thin films with low adhesion and large residual stresses may buckle. The resulting morphologies are varied, but one of the most commonly observed is an intriguing oscillating pattern -the so-called "telephone cord" -which has been extensively investigated in the recent years. We have studied the kinematics of formation of telephone cords using a geometrically non-linear plate model and mode dependent interfacial toughness, captured via a cohesive zone. Through extensive Finite Element Simulations, we have demonstrated a simple, non trivial relation between telephone cord wavelength and interfacial toughness. To validate this prediction, highly stressed Mo thin films where deposited on Si wafers, with a well defined interface and very reproducible adhesion. Studying the morphology of the resulting buckles for different film thicknesses and stresses, we observed a trend which was fully consistent with our simulation results. From the data fit, an adhesion energy of 0.58 ± 0.04 Jm -2 for the SiO 2 /Ag interface was inferred, which compares well with literature estimates.

Introduction

Coatings are widely used for surface functionalization so that the mechanical stability of coatings is of critical importance in many applications. An elastic film subject to in-plane compression may partially delaminate and form blisters (Fig. 1) depending on factors such as thickness, stress and adhesion. Indeed buckling driven delamination is a frequent failure mode when the coating is subject to external or residual compressive stresses and adhesion is low. This subject has been investigated for many years and a variety of blister morphologies have been studied [START_REF] Gilles | Buckling instability and adhesion of carbon layers[END_REF][START_REF] Wang | Effects of strain cycling on buckling, cracking and spalling of a thermally grown alumina on a nickel-based bond coat[END_REF][START_REF] Thouless | Combined buckling and cracking of films[END_REF][START_REF] Evans | On the mechanics of delamination and spalling in compressed films[END_REF][START_REF] Hutchinson | Delamination of compressed films on curved substrates[END_REF][START_REF] Moon | Buckle delamination on patterned substrates[END_REF][START_REF] Chen | Spatial and kinetic evolutions of telephone cord buckles[END_REF]. The problem is to couple plate non-linearity with adhesion. For simple buckle morphologies such as straight sided and circular blisters, relations between buckle size and adhesion have been obtained (Hutchinson and Suo, 1992a;Hutchinson et al., 1992b;[START_REF] Jensen | Energy release rates and stability of straight-sided, thin-film delaminations[END_REF]. From these relations one would be able to derive film adhesion from the measurement of the buckle width (straight sided) or buckle diameter (circular). Unfortunately both straight and circular blisters are seldom found in practice. In the ubiquitous case of equi-biaxial compression, by far the most usual morphology is an undulating blister called telephone cord (TC) buckle (Moon et al., 2002a). However, the morphology of TCs is poorly understood. As an approximation, it was proposed to map the buckle morphology to simpler blister geometries, namely a circular or a pinned circular blister (Moon et al., 2002b). With this method the energy release rate can be estimated from the width of the blister [START_REF] Cordill | Adhesion measurements using telephone cord buckles[END_REF][START_REF] Lee | Comparison of the telephone cord delamination method for measuring interfacial adhesion with the four-point bending method[END_REF]. Along a different route, [START_REF] Gioia | Determination of thin-film debonding parameters from telephone-cord measurements[END_REF] have proposed an approximate plate theory to model the morphology of undulating buckles, from which they derived a relation between adhesion and TC buckle period. However, these methods rely on approximate descriptions so that, to date, no clear relation between adhesion and morphological parameters of TC buckles is available.

In a previous work, we have explored the kinematics of TC propagation and demonstrated how it results in the TC morphology: mode dependent adhesion conspires with local sagging of the film to form effective pinning points, from which the oscillating propagation of TCs results [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF]. Here we present a more quantitative investigation of the impact of TC formation kinematics on morphology. Rationalizing the morphologies of simulated buckles for various film thicknesses, residual stresses and adhesion, we find an unexpectedly simple relation between TC wavelength (not width) and adhesion. To test this prediction, we have studied the morphology of a variety of Mo buckles with different thicknesses and stresses and constant adhesion. We find that a reasonable value of film adhesion can indeed be derived from this relation, which compares favorably with literature values.

Previous results

We first summarize previous results concerning the relations between blister geometry, loading and adhesion. We consider an elastic film with Young's modulus E, Poisson's ratio ν, thickness h, in-plane equi-biaxial compressive stress σ 0 and adhesion (or rather interfacial toughness) G c .

Straight sided blisters

Although straight sided delamination is rarely observed, we briefly summarize the main results here to introduce the key concepts. Hutchinson and Suo (1992a) modeled the straight sided blister as a clamped Euler column. Bending of thin plates is described by the non-linear Foppl-Von Kármán equations. For a straight sided blister the strain component along the blister direction is unaffected by buckling whereas the stress is partially relaxed in the perpendicular direction. It is found that the blister profile is sinusoidal. Denoting 2b the blister width, the critical buckling stress σ c for buckling is given by

σ c = π 2 12 E 1 -ν 2 ( h b ) 2 (1)
For a straight sided blister, the loading ratio σ 0 /σ c is the sole parameter governing the buckle shape. For instance the buckle height is

δ h = √ 4 3 ( σ 0 σ c -1
) a form which highlights the non linearity of the problem. Similarly the energy release rate depends only on the loading ratio.

Adhesion is introduced when this solution is coupled to an edge crack. The total elastic energy stored in the film is G 0 = ((1 -ν)/E)σ 2 0 h. A fraction of this energy can be released during crack propagation to overcome adhesion. But this energy release rate G is mainly an increasing function of the width 2b so that in this simpler picture, it is expected that buckles are unstable. However it has been evidenced experimentally [START_REF] Liechti | Asymmetric shielding in interfacial fracture under in-plane shear[END_REF]) that crack face separation in shear (mode II) requires more energy than pure opening (mode I). Hutchinson and Suo (1992a) pointed out that as the straight sided buckles becomes wider, the shear contribution increases, so that G c may eventually reach G whereupon the edge crack stops.

In brief, in the buckle problem, the main stabilizing force is provided by the mode dependence of interfacial toughness so that the morphology results from the interplay between plate geometric non-linearities and mode dependent adhesion.

Telephone cord blisters

Unlike the straight sided blister, undulating patterns such as TCs partially relax film stresses along both transverse and axial directions. As a result, the energy release rate G is larger.

In their detailed investigation of TC morphologies, [START_REF] Moon | Buckle delamination on patterned substrates[END_REF] pointed out that the precise experimental determination of TC width 2b is by itself a complex problem, as several candidate measures arise. In practice 2b is often defined as the width within which the TC buckle is confined [START_REF] Moon | Buckle delamination on patterned substrates[END_REF]. Besides the width, they also focussed on the wavelength λ and drew two conclusions about the aspect ratio λ/2b: 1) the aspect ratio is not a constant, but varies from about 0.8 to 1.5; 2) the aspect ratio is not a function of the loading ratio σ 0 /σ c only.

In an attempt to better understand the wavelength, Moon et al. (2002b) built upon the idea of secondary buckling. Indeed the TC morphology can be approximately reproduced assuming that there is no adhesion over a strip of width 2b: when the loading ratio exceeds σ 0 /σ c = 6.5 [START_REF] Audoly | Stability of straight delamination blisters[END_REF][START_REF] Audoly | Secondary buckling patterns of a thin plate under in-plane compression[END_REF][START_REF] Parry | Stability diagram of unilateral buckling patterns of strip-delaminated films[END_REF] (for ν = 0.3) the plate destabilizes from straight sided into wavy bukles, a morphology quite similar to TCs. However, using Finite Element Analysis of these wavy buckles, [START_REF] Moon | Buckle delamination on patterned substrates[END_REF] found that the energy release rate is independent of the wavelength. In brief, secondary buckling does not predict the aspect ratio.

FEM model for buckling driven delamination

To analyze the morphology of TC buckles, we use our recent model, which has been briefly described earlier [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF]. The thin film is modeled as a geometrically non-linear plate, as depicted in Fig. 2. The midplane surface is defined by the (O, x, y) plane, with (u, v, w) the respective displacement components along the axes (Ox, Oy, Oz) for any point of the midplane. The various buckling patterns are characterized by large values of the out-of-plane displacement w(x, y), and the calculation must be carried out within the framework of large displacements using the Green Lagrange strain tensor.

Modeling of the film/substrate interface

For the interface debonding process, a mixed-mode cohesive zone model is used. Cohesive zone models have been widely used and implemented for frac-ture mechanics studies for the last two decades [START_REF] Tvergaard | The relation between crack growth resistance and fracture process parameters in elastic-plastic solids[END_REF][START_REF] Xu | Void nucleation by inclusion debonding in a crystal matrix[END_REF][START_REF] Barthel | On the description of the adhesive contact of spheres with arbitrary interaction potentials[END_REF]. The constitutive behavior of the interface consists of a traction vs. separation law. Due to interaction between the two separating faces, the interface traction T depends upon the separation vector δ, which is the relative displacement between opposite crack faces at a point initially joined on the interface. The separation and traction vectors can be resolved into their normal components (δ n , T n ), the opening or mode I contribution, and their tangential components resolved in the direction normal to the crack front, (δ t , T t ), the shearing or mode II contribution. The tangential component resolved tangent to the crack front, the mode III contribution, is not expected to play an important role and is not taken into account in the traction-separation law.

As described in the previous section, an essential aspect of the buckling delamination phenomenon is mixed mode behavior with higher resistance to delamination under mode II than mode I. The following mixed mode tractionseparation law embodies this type of dependency, and it can be implemented as a standard ABAQUS cohesive element (COH) (ABAQUS, 2010). The law is a linear/softening model depicted in Fig. 3. In Stage 1, prior to attaining the peak traction, the traction-separation relation is linear and reversible:

T = Kδ (2)
with stiffness K. For any combination of traction components, (T n , T t ), peak values (T n i , T t i ) are reached when (c.f., Fig. 3)

( T n T n 0 ) 2 + ( T t T t 0 ) 2 = 1 (3)
where T 0 n is the maximum traction in mode I and T 0 t is the maximum traction in mode II. The corresponding separations at the peak are δ i = (δ i n , δ i t ). Once peak traction is attained, softening sets in (Stage 2), resulting in decreasing traction, as in Fig. 3.

The two stages are combined such that the separation energy/area, G c (ψ), depends on the mode mix ψ in a specific manner, as now described. The measure of the mode mix in the cohesive law is defined in terms of the combination of tractions attained at the peak, as the angle where tan(ψ) = T i t /T i n . This definition differs from that employed in linear fracture mechanics in that it is defined point-wise on the interface in the vicinity of crack front (i.e., from element to element), whereas the conventional definition of mode mix, ψ, is defined in terms of the mode I and II stress intensity factors at the crack front as tan(ψ) = K II /K I . The definition of ψ used here is natural within the context of a cohesive law employed in a numerical finite element simulation wherein a softening traction behavior is assigned for the remaining interface resistance after the peak is attained in a manner depending on the mode experienced at that point on the interface.

The separation energy/area, G c (ψ), is taken to have a dependence on mode mix characterized by the relation

G c (ψ) = G Ic (1 + tan 2 (ηψ)) (4)
with G Ic as the mode I separation energy, or toughness, and a dependence on the parameter η seen in Fig. 4. This is a phenomenological relation that has been successfully fit to experimental data sets for various interfaces (Hutchinson and Suo, 1992a). Values of η close to zero imply a weak dependence on the mode mix, while values close to unity have a strong dependence. With Eq. 4 as the total separation energy and ψ determined by the traction combination attained at the peak, a (scalar) effective relative interface displacement at complete separation δ f is defined from the relation

G c (ψ) = 1 2 δ f T i (5)
where

T i = √ T i n 2 + T i t 2
is the peak stress magnitude. The softening is implemented as a reduction of the stiffness by the factor (1 -d) where d can be regarded as a damage variable. The definition of the cohesive traction Eq. 2 becomes

T = K(1 -d)δ (6)
The variable d is defined as

d = δ f δ δ -δ i δ f -δ i (7)
where δ is the separation magnitude

δ = √ δ 2 n + δ 2 t (8)
and similarly δ i is the peak separation magnitude. Inserting Eq. 7 into Eq. 6 we obtain

T = T i δ f -δ δ f -δ i δ δ (9)
which shows that T , which is colinear with the current separation δ, has a magnitude proportionally reduced from T i . Integration of the work of the cohesive tractions using Eq. 2 in Stage 1 to (T i n , T i t ) and then, in Stage 2, to full separation using Eq. 9 gives 1/2δ f T i . Thus, the definition of δ f in Eq. 5 provides the desired mode-dependent separation energy.

In the simulations of buckling delamination some regions of the crack front are observed to undergo softening separations beyond the peak and then, prior to complete separation, the separation reverses, or unloads, as depicted in Fig. 3. Beyond the peak, we assume that the interface has undergone irreversible damage and the unloading response is a linear connection to the origin of the traction-separation relation (Fig. 3), as results when in Eq. 7 δ is fixed at the maximum value attained δ max .

Loading and boundary conditions

The loading consists of an eigenstrain ϵ 0 > 0 applied uniformly to the plate (ϵ xx = ϵ yy = ϵ 0 , ϵ xy = 0) at time t = 0. An equi-biaxial compressive stress state is generated in the flat adherent parts of the film:

σ xx = σ yy = -E/(1 -ν)ϵ 0 = -σ 0 , σ xy = 0.
Nucleation is controlled with a small rectangular strip located near an edge of the domain. This area remains adhesion-free so that it can buckle and nucleate a delamination front that will subsequently propagate along the plate-substrate interface.

The calculations are carried out with the Finite Element Method, using the software ABAQUS (ABAQUS, 2010). Triangular shell elements are used for the film, within a large displacement framework. The element size is chosen so that there are about 40 elements in the buckle width. Quadrilateral cohesive elements are used in order to model the interface behavior. The upper surface is tied to the shell elements, whereas the lower surface is fixed since the substrate is rigid (Fig. 2). Because during subsequent evolution of the plate, the gap may close back after cohesive elements have been broken, a unilateral contact condition w(x, y) ≥ 0 is added to the shell elements. It is not implemented in the cohesive element, but is managed separately using a classical hard/frictionless contact algorithm. With this unilateral contact conditions plate-substrate interpenetration is avoided.

A quasi-static analysis is carried out using the explicit solver of ABAQUS. Due to the presence of sources of non-linearity (geometrical non-linearity and cohesive law), an explicit algorithm is more efficient even if the analysis is quasi-static. This approach is commonly adopted, requiring the dynamic parameters to be chosen carefully so that the kinetic energy always remains very small compared to the strain energy in the elements. A number of implicit calculations have also been carried out and very good agreement with explicit results is obtained.

For all simulations, Poisson's ratio was kept equal to 0.31 and a strong mode II dependence was chosen with η = 0.95. K is chosen so that δ i ≪ δ f , while T 0 n =17 MPa and T 0 t =200 MPa. It would have been useful to choose the variable simulation parameters so as to keep the loading ratio σ 0 /σ c constant but it cannot be estimated prior to the simulation since the width 2b is a result of the calculations. More directly, material parameters were varied according to two normalized ratios. σ 0 / Ē was varied from 1 × 10 -3 to 1 × 10 -2 and G 0 /G Ic from 0.1 to 0.6.

Numerical results

The results evidence a wide variety of morphologies (Fig. 5).

Relation between loading ratio and aspect ratio

Following [START_REF] Moon | Buckle delamination on patterned substrates[END_REF] we have inspected the aspect ratio and found values ranging from λ/2b = 0.8 up to λ/2b = 1.5. We have also evaluated the loading ratio σ 0 /σ c from Eq. 1 taking the full width of the TC buckle for 2b (Fig. 5). Loading ratios vs. aspect ratios are plotted in Fig. 6 left. A dispersion similar to [START_REF] Moon | Buckle delamination on patterned substrates[END_REF] is observed.

To go further, we color-coded the simulation points with the energy ratio G 0 /G Ic and found that smaller values of energy ratio correlate with larger aspect ratios. In fact iso-values of energy ratio define a family of parallel curves. We also found that normalizing the loading ratio by √ G 0 G Ic -1 collapses all the points on a single master curve (Fig. 6 right). This normalized loading ratio is now a strictly decreasing function of the aspect ratio. (left) loading ratio σ 0 /σ c as a function of the aspect ratio λ/2b, color coded with G 0 /G Ic . Also shown is the critical loading ratio at which the transition from straight sided (Euler) to wavy (TC) buckle is expected to occur from linear stability analysis [START_REF] Audoly | Stability of straight delamination blisters[END_REF], and the expected wavelength; (right) normalized loading ratio (Eq. 10) as a function of aspect ratio, and resulting linear relation for λ (inset, Eq. 11). The dots are experimental data points from section 5.

Wavelength of the delamination pattern

Fitting the master curve connecting aspect ratio and normalized loading ratio gives a quadratic relation

σ 0 σ c √ G 0 G Ic -1 ∝ ( 2b λ ) 2 (10)
Since the loading ratio itself depends quadratically upon the width 2b through Eq. 1, the width actually drops out of Eq. 10. As a result we find that the quantity which is directly linked to interfacial toughness is the wavelength of the TC buckle, not the width. In summary the relation between λ and interfacial toughness can be expressed as:

λ = kL = kh √ E σ 0 ( G 0 G Ic -1 ) -1/4 (11)
where all the other variables are in fact known material parameters or geometrical quantities. Relation 11 is plotted in Fig. 6 (right, inset). The slope k = 14.5.

Experiments -Relation between wavelength and adhesion

With Eq. 11 we can consider measuring adhesion from buckle wavelength. To experiment on the morphologies of TC buckles and test this prediction, we turned to the well calibrated two-layer thin films we have used previously [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF]. In this system a 15nm thick silver layer is deposited onto the native oxide of a two-inch (1 0 0) oriented silicon wafer. A cap layer of compressively stressed molybdenum is added to provide elastic energy for buckling driven delamination. The sample structure is illustrated in Fig. 1, where the Ag layer is a much thicker than usual for clarity.

Interface formation and thin film deposition

The films are sputtered in a high vacuum sputter deposition chamber equipped with two DC magnetron sputtering guns. Argon was used as the sputtering gas for both silver and molybdenum. Molybdenum thin films were sputtered at 2×10 -6 bar of Ar, lowest pressure for which stable plasma could be maintained. Film stresses were measured in-situ by curvature measurement for each sample. A detailed study of residual stress tuning can be found in a previous publication [START_REF] Faou | Stress tuning in sputter-deposited MoOx films[END_REF]. In brief, low deposition pressure results in high compression in molybdenum thin films. This is due to the large mean free path of argon neutrals, thus enhancing energies at which they bombard the sample. Higher energies result in atomic peening, causing compressive stress. Another smaller contribution to the compressive stress is ion peening. It remains small due to the low number of ionized species reaching the sample. The contribution can be enhanced by accelerating positive ions with the help of negative bias on the sample. Control over the negative bias provides a good command of the residual compressive stress in the molybdenum top layer.

Deposition time for molybdenum ranges from 3 to 8 minutes resulting in film thickness 80 to 180 nm having a residual compression between 1.7 and 2.3 GPa in equi-biaxial state.

Delamination was observed when the sample was brought to atmospheric pressure. Direct observation of the interface separation was made under scanning electron microscope. For all samples, delamination occurred at the interface between the native oxide of the substrate and silver (Fig. 1). The topography of the buckles was determined by atomic force microscopy. As the wavelength is very sensitive to adhesion, the λ values are averages over ten periods. The width 2b is the total width within which the buckle is contained.

We observed TC buckles for 0.89 < G 0 < 1.5 J.m -2 . TCs do not form if the elastic energy per unit area is too small compared to adhesion, and if it is too large, the film completely delaminates without forming TC patterns. Three examples of TCs with different morphologies are presented in Fig. 7. Morphological parameters and mechanical loading for all the buckles are summarized in Table 1.

Experimental relation between loading ratio and aspect ratio

The aspect ratios λ/2b of the TC buckles we have obtained range from 0.8 to 1.5, which is similar to the observations reported by [START_REF] Moon | Buckle delamination on patterned substrates[END_REF] and to the values obtained in the previous section. The loading ratio σ 0 /σ c 1.74 29.9 24.9 1.12 calculated from residual stress and total width (Eq. 1 and Table 1) is plotted against aspect ratio λ/2b in Fig. 8. We observe a general trend for the loading ratio to decrease as aspect ratio increases but there is a significant dispersion, similar to [START_REF] Moon | Buckle delamination on patterned substrates[END_REF]'s. However, color coding the points with elastic energy stored in the film (Fig. 8) suggests that the results can indeed be rationalized following Eq. 10. To do so, we need to identify the optimal value of interfacial toughness G Ic for the SiO 2 /Ag interface. Equivalently, this can be done by a least square fit based on Eq. 11 with G Ic and k as free variables. In Fig. 8 we plot the non-dimensional loading parameter (λ/h) 2 σ/E as a function of the total elastic energy in the film G 0 along with the best fit (k = 12.3 ± 0.6 and G Ic = 0.58 ± 0.04 Jm -2 ). This value of mode I interfacial toughness is consistent with the values measured by cleavage test (0.8 Jm -2 [START_REF] Barthel | Asymmetric silver to oxide adhesion in multilayers deposited on glass by sputtering[END_REF] and 0.5 Jm -2 [START_REF] Patinet | Finite size effects on crack front pinning at heterogeneous planar interfaces: Experimental,finite elements and perturbation approaches[END_REF] for oxidized Si 3 N 4 /Ag). Using this value for G Ic we also plotted the data on Fig. 6.

Discussion

Kinematics and wavelength

Dimensional analysis of the Von Karman plate equations for in-plane compressive loading σ 0 shows that there is a non-dimensional loading parameter σ 0 Ē ( a h

) 2 where Ē = E/(1 -ν 2 ) and the length a is characteristic of the buckle equilibrium morphology [START_REF] Parry | Stability diagram of unilateral buckling patterns of strip-delaminated films[END_REF]. The length a can be the half-with b of a straight-sided blister or the radius R of a circular blister (Hutchinson and Suo, 1992a). For these highly symmetric geometries the buckle edge is everywhere critical, i.e. the energy release rate is everywhere equal to the critical energy release rate. For TCs however, this is no longer the case. The morphology is defined by the propagation kinematics, and it may (and does) happen that this kinematics leaves behind portions of buckle edge where the energy release rate lies below the critical value. This is precisely the case for the so-called pinning point we identified earlier [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF], where plate sagging develops strong mode II. As a result, approximations of the TC buckle morphology by models assuming critical loading all along the edges are prone to errors in estimating both the energy release rate and the phase angle. In contrast, for TCs our experiments and simulations point to the wavelength λ as the primary morphological parameter, in relation to adhesion. Rewriting Eq. 11 we obtain σ 0 E ( λ h

) 2 = k 2 √ G 0 G Ic -1 (12)
The wavelength depends on the position of the pinning point (or sag) forming along the front of the buckle, which is controlled by the propagation kinematics of the buckle, impacted by both plate response and adhesion.

For TCs, we conclude that it is the wavelength which directly depends on adhesion. For simpler buckles, it is the width 2b. In many aspects, in TC buckles, the wavelength behaves differently from -even sometimes contrary to -the width. For instance, let us imagine a slight increase of adhesion while keeping other properties (material, stress, thickness) unchanged. For a straight-sided blister, increased adhesion would result in smaller width. However, for the TC blisters, following from Eqs. 11 or 12, stronger adhesion results in longer wavelength. In fact, this behavior is due to the narrower front (smaller width), which is more difficult to destabilize, delaying the formation of the pinning point. Conversely increasing the residual stress facilitates destabilization, resulting in smaller TC wavelength, while for a straight-sided buckle, it results in increased width.

Total elastic energy and energy release rate in TC buckles

It is somewhat unexpected that the normalized loading ratio involves the energy ratio G 0 /G Ic (Eq. 10) and not the energy release rate ratio G/G Ic .

To better understand this difference, we have calculated the average energy release rate G during propagation for several TC buckle morphologies and plotted G/G 0 as a function of σ 0 /σ c in Fig. 9. We also plotted the results obtained by [START_REF] Moon | Buckle delamination on patterned substrates[END_REF] for the delamination of a straight-sided blister. Altogether G = 0.65 ± 0.05 G 0 , i.e. there is a very limited variation of G/G 0 within the family of buckles we have investigated. This observation points to the fact that G/G 0 is almost insensitive to both loading ratio and aspect ratio. Indeed [START_REF] Moon | Buckle delamination on patterned substrates[END_REF] commented that the maximum dependence of G/G 0 on loading ratio is observed for σ 0 /σ c < 20: precisely the range where no TC buckle is observed experimentally or numerically! Note however that the average energy release rates found here (for full buckle kinematics) are lower by about 15% than [START_REF] Moon | Buckle delamination on patterned substrates[END_REF]'s values (for secondary buckling). Although these two varieties of TCs look alike, the exact morphologies differ. In fact, when going through the full kinematics, the film does not completely delaminate over the strip of width 2b: it is still attached to the substrate close to the pinning points [START_REF] Faou | How does adhesion induce the formation of telephone cord buckles?[END_REF].

Because of the moderate variation of G/G 0 within the TC buckle family, it is tempting to use G instead of G 0 in Eq. 11. We find that this is possible but less accurate in predicting the resulting wavelength. showing: 1) previous results for buckle delamination along a straight-sided strip, with straight sided (or Euler) blister (dashed line, valid for σ 0 /σ c < 6.5) and TC blister from secondary buckling (full line with markers, σ 0 /σ c > 6.5, from Moon et al. ( 2004)); 2) present results for full buckling driven delamination with mode dependent interfacial toughness (triangles, color coded for energy ratio).

Impact of mode II on wavelength

The linear relation Eq. 11 was obtained for the specific value η = 0.95. Changing the mode mixity dependence by varying η from 0.6 to 0.97, similar linear relations were found for λ as a function of loading, interface and material properties. It was found that the slope k decreases with η (Fig. 10). This variation is very slow however and k changes by only 10% when G IIc /G Ic drops from 450 to 2.9. We conclude that even though mode dependent interfacial toughness controls stability of the buckle edges and kinematics of buckle formation, the resulting wavelength is only weakly impacted by the mode II dependence. A possible reason is the range of mode mixity impacting TC buckle formation, which is quite remote from the limit mixity angle -π/2, where the maximum dependence of G c upon mode mixity is found.

Note that η is not the only parameter influencing k. The cohesive tractions T 0 n and T 0 t defining the cohesive zone also impact k and a moderate increase of k was observed when the cohesive traction was increased. It is expected that k converges to a finite value for large cohesive tractions. In addition, Eq. 4 is an idealized representation of the mode dependence of interfacial toughness. Real systems could possess a different dependence [START_REF] Strohband | Interface separation in residually-stressed thin-film structures[END_REF] also affecting the value of the k parameter. In brief, although our experimental value k = 12.2 lies close to the simulated values, we find it difficult to conclude on η and the mode mixity dependence based on this value.

Conclusion

We have studied the kinematics of formation of telephone cord buckles duly taking into account the details of film adhesion to the substrate, and especially the mode mixity dependence. Through extensive Finite Element Simulations, we have demonstrated that the aspect ratio λ/2b of the telephone cord is rationalized if the loading ratio, defined from the buckle width 2b, is normalized by a factor involving the energy ratio G 0 /G Ic . As a result, we find that it is the wavelength λ, and not the width, of the TC buckle which directly contains information about film adhesion, and a relation between λ and interfacial toughness in mode I G Ic has been identified. Experiments have also been carried out to confirm this relation, which has been tested against buckle morphology data. From the trends observed for various thicknesses and stresses, an estimate of the adhesion energy for an SiO 2 /Ag interface has been inferred, which compares well with literature values.

Acknowledgements

We thank John Hutchinson for very useful discussions.

Figure 1 :

 1 Figure 1: Scanning electron micrograph of a buckled thin film (cross-section). The stack consists of 80 nm of molybdenum and 80 nm of silver on a silicon wafer.

Figure 2 :

 2 Figure 2: Schematic sketch of the model.

Figure 3 :

 3 Figure 3: Traction vs separation law for interfacial toughness at the film/substrate interface including mixed mode dependence. Inset: schematic of the linear/softening traction separation law, in one dimension, also showing interfacial toughness as the total work done by the cohesive stress upon complete opening of the cohesive element (shaded area).

Figure 4 :

 4 Figure 4: Mode dependence of interfacial toughness, plotted from Eq. 4. The amplitude is directly set by the mode mixity parameter η.

Figure 5 :

 5 Figure 5: Out of plane displacement for two different TC morphologies obtained by numerical simulations, with E = 329 GPa: (left) h = 100 nm, σ 0 = 2.5 GPa, G Ic = 0.477 J.m -2 resulting in an aspect ration of 1.0; (right) h = 40 nm, σ 0 = 3.1 GPa, G Ic = 0.4 J.m -2 with an aspect ratio 1.35.

Figure 6 :

 6 Figure6: Numerical results: (left) loading ratio σ 0 /σ c as a function of the aspect ratio λ/2b, color coded with G 0 /G Ic . Also shown is the critical loading ratio at which the transition from straight sided (Euler) to wavy (TC) buckle is expected to occur from linear stability analysis[START_REF] Audoly | Stability of straight delamination blisters[END_REF], and the expected wavelength; (right) normalized loading ratio (Eq. 10) as a function of aspect ratio, and resulting linear relation for λ (inset, Eq. 11). The dots are experimental data points from section 5.

Figure 7 :

 7 Figure 7: Microscopy photographs (a,b) and AFM phase image (c) for different values of G 0 for the same interface, and hence same adhesion G c1 . This series of images illustrate the variation of the aspect ratio λ/2b with G 0 /G c 1 .

Figure 8 :

 8 Figure 8: (left) measured loading ratio σ 0 /σ c as a function of aspect ratio λ/2b for a series of films with identical SiO 2 /Ag interface; (right) non-dimensional loading parameter for the experimental TC buckle as a function of total elastic energy prior rupture.

Figure 9 :

 9 Figure9: Average elastic energy release rate G/G 0 as a function of loading ratio σ 0 /σ c , showing: 1) previous results for buckle delamination along a straight-sided strip, with straight sided (or Euler) blister (dashed line, valid for σ 0 /σ c < 6.5) and TC blister from secondary buckling (full line with markers, σ 0 /σ c > 6.5, from Moon et al. (2004)); 2) present results for full buckling driven delamination with mode dependent interfacial toughness (triangles, color coded for energy ratio).

Figure 10 :

 10 Figure 10: Computed prefactor k for the period of TC blisters as a function of the mode mixity parameter η.

Table 1 :

 1 Experiments: thickness, in-plane residual stress, wavelength of the TC blister, total elastic energy for Mo films on Ag/SiO 2 interface.

	h (nm) σ 0 (GPa) λ (µm) 2b (µm) G 0 (J/m 2 )
	25	4.00	3.1	2.1	0.84
	81	2.29	14.4	12.6	0.89
	94	2.15	17.5	15.4	0.91
	95	2.06	17.8	16.6	0.85
	101	2.25	16.4	13.5	1.08
	106	2.34	14.4	15.4	1.22
	108	1.98	19.4	20.2	0.89
	120	1.72	27.9	29.0	0.74
	137	1.81	24.0	21.2	0.94
	156	2.10	20.5	16.8	1.44
	177