Rodrigo Caye Daudt

Bertrand Le Saux

Alexandre Boulch

FULLY CONVOLUTIONAL SIAMESE NETWORKS FOR CHANGE DETECTION

Keywords: Change detection, supervised machine learning, fully convolutional networks, Earth observation

This paper presents three fully convolutional neural network architectures which perform change detection using a pair of coregistered images. Most notably, we propose two Siamese extensions of fully convolutional networks which use heuristics about the current problem to achieve the best results in our tests on two open change detection datasets, using both RGB and multispectral images. We show that our system is able to learn from scratch using annotated change detection images. Our architectures achieve better performance than previously proposed methods, while being at least 500 times faster than related systems. This work is a step towards efficient processing of data from large scale Earth observation systems such as Copernicus or Landsat.

INTRODUCTION

Change Detection (CD) is one of the main problems in the area of Earth observation image analysis. Its study has a long history, and it has evolved alongside the areas of image processing and computer vision [START_REF] Singh | Review article digital change detection techniques using remotely-sensed data[END_REF][START_REF] Hussain | Change detection from remotely sensed images: From pixel-based to objectbased approaches[END_REF]. Change detection systems aim to assign a binary label per pixel based on a pair or sequence of coregistered images of a given region taken at different times. A positive label indicates the area corresponding to that pixel has changed between the acquisitions. While the definition of "change" may vary between applications, CD is a well defined classification problem. Changes may refer, for example, to vegetation changes, urban expansion, polar ice melting, etc. Change detection is a powerful tool in the production of maps depicting the evolution of land use, urban coverage, deforestation, and other multi-temporal types of analysis.

Programs such as Copernicus and Landsat make available large amounts of Earth observation imagery. These can be used in conjunction with advanced supervised machine learning algorithms that have been on the rise for the past decade, especially in the area of image analysis. It is therefore of interest to find efficient ways to make use of the available data. In the context of change detection there is a lack of large annotated datasets, which limits the complexity of the models that can be used. Nevertheless, there are available pixelwise annotated change detection datasets available that can be used to train supervised machine learning systems that detect changes in image pairs, such as the Onera Satellite Change Detection dataset 1 presented in [START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF] and the Air Change dataset [START_REF] Benedek | Change detection in optical aerial images by a multilayer conditional mixed markov model[END_REF].

In this paper we present three Fully Convolutional Neural Network (FCNN) architectures that perform change detection on multi-temporal pairs of images Earth observation images. These architectures are trained end-to-end from scratch using only the available change detection datasets. The networks are tested in both the RGB and multispectral cases when possible. They are extensions of the ones presented in [START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF], which was the first CD method to be trained end-to-end, to a fully convolutional paradigm. This improves both the accuracy and the inference speed of the networks without increasing the training times. We present fully convolutional encoderdecoder networks that use the skip connections concept introduced in [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF]. We also propose for the first time two fully convolutional Siamese architectures using skip connections.

This paper is organized as follows. Section 2 explores other works that were used for inspiration or comparison during the development of this work. Section 3 describes in detail the three proposed FCNN architectures for change detection. Section 4 contains quantitative and qualitative comparisons with previous change detection methods. Finally, Section 5 contains this work's concluding remarks.

RELATED WORK

The history of change detection began not long after aerial images became possible [START_REF] Singh | Review article digital change detection techniques using remotely-sensed data[END_REF][START_REF] Hussain | Change detection from remotely sensed images: From pixel-based to objectbased approaches[END_REF]. The proposed techniques have followed the tendencies of computer vision and image analysis: at first, pixels were analyzed directly using manually crafted techniques; later on, descriptors began to be used in conjunction with simple machine learning techniques [START_REF] Le | Urban change detection in sar images by interactive learning[END_REF]; recently, more elaborate machine learning techniques (deep learning) are dominating most problems in the image analysis field, and this evolution is slowly reaching the problem of change detection [START_REF] Stent | Detecting change for multi-view, long-term surface inspection[END_REF][START_REF] Liu | A deep convolutional coupling network for change detection based on heterogeneous optical and radar images[END_REF][START_REF] Gong | Change detection in synthetic aperture radar images based on deep neural networks[END_REF][START_REF] Arabi | Zoom out cnns features for optical remote sensing change detection[END_REF][START_REF] Zhan | Change detection based on deep siamese convolutional network for optical aerial images[END_REF][START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF][START_REF] Mou | Multitemporal Very High Resolution From Space: Outcome of the 2016 IEEE GRSS Data Fusion Contest[END_REF].

Due to the limited amount of available data, most of these methods use various techniques based on transfer learning, 1 http://dase.grss-ieee.org/ taking as a starting point networks that have been trained on larger datasets for different problems. This is limiting in many ways, as it assumes similarities between these datasets and the relevant change detection data. For example, most of the networks have been trained on RGB images, and cannot be transfered for SAR or multispectral images, which is the case of the dataset presented in [START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF]. These methods also avoid endto-end training, which tends to have better results for successfully trained systems. For this reason, the focus of this paper is algorithms that are able to learn solely from the available change detection data, and can therefore be applied to any available datasets.

Using machine learning for comparing images is not a new idea. Convolutional Neural Networks (CNNs) are a family of algorithms especially suited for image analysis, and have been applied in different contexts for the comparison of image pairs [START_REF] Chopra | Learning a similarity metric discriminatively, with application to face verification[END_REF][START_REF] Zagoruyko | Learning to compare image patches via convolutional neural networks[END_REF][START_REF] Stent | Detecting change for multi-view, long-term surface inspection[END_REF]. Recently, fully convolutional architectures (FCNNs) have been proposed for problems that involve dense prediction, i.e. pixel-level prediction [START_REF] Long | Fully convolutional networks for semantic segmentation[END_REF][START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF][START_REF] Bertinetto | Fully-convolutional siamese networks for object tracking[END_REF]. Despite achieving state-of-the-art results in other Earth observation problems [START_REF] Audebert | Beyond rgb: Very high resolution urban remote sensing with multimodal deep networks[END_REF], these techniques have not yet been applied to CD to the best of our knowledge. The usefulness of such ideas in the context of Earth observation and their superiority over patch based, superpixel based and other approaches has already been studied [START_REF] Audebert | Segment-before-detect: Vehicle detection and classification through semantic segmentation of aerial images[END_REF]. Siamese architectures have also been proposed in different contexts with the aim of comparing images [START_REF] Bromley | Signature verification using a" siamese" time delay neural network[END_REF][START_REF] Zagoruyko | Learning to compare image patches via convolutional neural networks[END_REF]. To our knowledge, the only time a fully convolutional Siamese network was proposed previously was by Bertinetto et al. in [START_REF] Bertinetto | Fully-convolutional siamese networks for object tracking[END_REF] to tackle the problem of object tracking in videos. Despite achieving good results, the architecture that was proposed in that work is specific to that problem and was not used as an inspiration to this work.

PROPOSED APPROACH

The work presented in this paper aimed to propose FCNN architectures able to learn to perform change detection solely from change detection datasets without any sort of pretraining or transfer learning from other datasets. These architectures are able to be trained end-to-end, unlike the majority of recent works on change detection. These fully convolutional architectures are an evolution of the work presented in [START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF], where a patch based approach was used. Moving the patch-based architectures to a fully convolutional scheme improves accuracy and speed of inference without affecting significantly the training times. These fully convolutional networks are also able to process inputs of any sizes given enough memory is available.

Two CNN architectures were compared in [START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF]: Early Fusion (EF) and Siamese (Siam). The EF architecture concatenated the two patches before passing them through the network, treating them as different color channels. The Siamese architecture processed both images separately at first by identical branches of the network with shared structure and parameters, merging the two branches only after the convolutional layers of the network.

To extend these ideas we used the concept of skip connections that were used to build the U-Net, which aimed to perform semantic segmentation of images [START_REF] Ronneberger | U-net: Convolutional networks for biomedical image segmentation[END_REF]. In summary, skip connections are links between layers at the same subsampling scale before and after the encoding part of an encoder-decoder architecture. The motivation for this is to complement the more abstract and less localized information of the encoded information with the spatial details that are present in the earlier layers of the network to produce accurate class prediction with precise boundaries in the output image.

The first proposed architecture is directly based on the U-Net model, and was named Fully Convolutional Early Fusion (FC-EF). We adapted the U-Net model into the FC-EF taking into account the amount of available training data. The FC-EF (Fig. 1(a)) contains therefore only four max pooling and four upsampling layers, instead of the five present in the U-Net model. The layers in FC-EF are also shallower than their U-Net equivalents. As in the patch based EF model, the input of this network is the concatenation the two images in the pair that is to be compared.

The two other proposed architectures are Siamese extensions of the FC-EF model. To do so, the encoding layers of the network are separated into two streams of equal structure with shared weights as in a traditional Siamese network. Each image is given to one of these equal streams. The difference between the two architectures is only in how the skip connections are done. The first and more intuitive way of doing that is concatenating the two skip connections during the decoding steps, each one coming from one encoding stream. This approach was named Fully Convolutional Siamese -Concatenation (FC-Siam-conc, Fig. 1(b)). Since in CD we are trying to detect differences between the two images, this heuristic was used to combine the skip connections in a different way. Instead of concatenating both connections from the encoding streams, we instead concatenate the absolute value of their difference. This approach was named Fully Convolutional Siamese -Difference (FC-Siam-diff, Fig. 1(c)).

EXPERIMENTS

To evaluate the proposed methods we used two change detection datasets openly available. The first one is the Onera Satellite Change Detection dataset [START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF] (OSCD), and the second is the Air Change Dataset [START_REF] Benedek | Change detection in optical aerial images by a multilayer conditional mixed markov model[END_REF] (AC). AC contains RGB aerial images, while OSCD contains multispectral satellite images. The networks were also tested using only the RGB layers of the OSCD dataset. The two classes (change and no change) were assigned weights inversely proportional to the number of examples in each one. The available data was augmented by using all possible flips and rotations multiple of 90 degrees to the training patches. Dropout was used to help avoid overfitting during training. All experiments were done using PyTorch and with an Nvidia GTX1070 GPU.

On the OSCD dataset, we split the data in train and test groups as recommended by the dataset's creators: fourteen images were used for training and ten images were used for testing. For the AC dataset, we followed the data split that was proposed in [START_REF] Zhan | Change detection based on deep siamese convolutional network for optical aerial images[END_REF] Szada-1 and Tiszadob-3 images were extracted for testing, and the rest of the data for each location (another 6 and 4 images, respectively) was used for training. This allowed direct comparison to three CD algorithms. Each location (Szada and Tiszadob) was treated completely separately as two different datasets, and the images named "Archieve" were ignored, since it comprises only one image pair which is not enough data to train the models presented in this paper. Table 1 contains the quantitative evaluation of the proposed CD architectures, along the same measures of other state-of-the-art methods. The Early Fusion and Siamese methods presented in [START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF] were used for comparison for the OSCD dataset. For the AC dataset, the methods user for comparison were DSCN [START_REF] Zhan | Change detection based on deep siamese convolutional network for optical aerial images[END_REF], CXM [START_REF] Benedek | Change detection in optical aerial images by a multilayer conditional mixed markov model[END_REF], and SCCN [START_REF] Liu | A deep convolutional coupling network for change detection based on heterogeneous optical and radar images[END_REF], using the values claimed by Zhan et al. in [START_REF] Zhan | Change detection based on deep siamese convolutional network for optical aerial images[END_REF]. The table contains the precision, recall and F1 rate from the point of view of the "change" class, as well as the global precision when available.

The results relative to the OSCD dataset show that the methods proposed in this paper far outperform the ones proposed in [START_REF] Caye Daudt | Urban change detection for multispectral earth observation using convolutional neural networks[END_REF], which is the precursor of this work. While the patch based methods achieve good recall rates, they do very poorly on the precision metric, which also reduces the F1 rate. Inference time of the fully convolutional architectures was under 0.1 s per image for all our test cases, while the patch based voting approach took several minutes to predict a change map for each image pair. On this dataset, the FC-EF and FC-Siamdiff architectures obtained the best F1 scores, while the results obtained by FC-Siam-conc were still clearly superior to the ones obtained by the patch based approaches. An illustration of our results on this dataset can be found in Fig. 2.

The results obtained on the AC dataset also show the superiority of our method compared to other previous methods. For the Szada/1 case, all proposed architectures outperformed the other methods used for comparison in the F1 metric, the FC-Siam-diff being once again the best architecture. For the Tiszadob/3 case, the best F1 score was obtained by our FC-EF architecture, while the other architectures were outperformed by DSCN and SCCN. Once again the inference time of the fully convolutional architectures were below 0.1 s, which is a speedup of over 500x compared to the processing time of 50 s claimed by Zhan et al. [START_REF] Zhan | Change detection based on deep siamese convolutional network for optical aerial images[END_REF] for the SCCN method on a very similar setup. The results for these cases can be viewed in Fig. 3.

In these tests, the FC-Siam-diff architecture seems to be the most suited for change detection, followed closely by FC-EF. This is, we believe, due to three main factors which make this network especially suited for this problem. First, fully convolutional networks were developed with the express purpose of dealing with dense prediction problems, such as CD. Second, the Siamese architecture imbues into the system an explicit comparison between two images. Finally, the difference skip connections also explicitly guides the network to compare the differences between the images, in other words, to detect the changes between the two images. The FC-EF architecture, which is more generic, also achieved excellent results despite having to learn the heuristics used for developing the FC-Siam-diff architecture from the training data.

The significant speedup of these FCNNs with no loss in performance compared to previous CD methods is a step towards efficient processing of the massive streams of Earth observation data which are available through programs like Copernicus and Landsat. These programs monitor very large areas with a high revisit rate. Deployment of such systems in conjunction with the methods proposed in this paper would enable accurate and fast worldwide monitoring.

CONCLUSION

In this paper we presented three fully convolutional networks trained end-to-end from scratch that surpassed the state-ofthe-art in change detection, both in accuracy and in inference speed without the use of post-processing. Most notably, the fully convolutional encoder-decoder paradigm was modified into a Siamese architecture, using skip connections to improve the spatial accuracy of the results.

A natural extension of the work presented on this paper would be to evaluate how these networks perform at detecting semantic changes. It would also be interesting to test them with other image modalities (e.g. SAR), and to attempt to detect changes in image sequences. It is also likely that these networks would profit from training on larger datasets.

  FC-Siam-diff.

Fig. 1 .

 1 Fig. 1. Schematics of the three proposed architectures for change detection. Block color legend: blue is convolution, yellow is max pooling, red is concatenation, purple is transpose convolution. Red arrows illustrate shared weights.

  FC-EF. (e) FC-Siam-conc. (f) FC-Siam-diff.

Fig. 2 .

 2 Fig.2. Illustrative results on the montpellier test case of the OSCD dataset using all 13 color channels. In images (d), (e), and (f) white means true positive, black means true negative, green is false positive, and magenta is false negative.

Fig. 3 .

 3 Fig.3. Comparison between the results obtained by the method presented in[START_REF] Zhan | Change detection based on deep siamese convolutional network for optical aerial images[END_REF] and the ones described in this paper on the Air Change dataset. First row: Szada/1, second row: Tiszadob/3.

Table 1 .

 1 : the top left 748x448 rectangle of the Evaluation metrics on change detection datasets, in percent.

	Data	Network	Prec. Recall Global	F1
	OSCD-3 ch.	Siam. [3] EF [3] FC-EF FC-Siam-conc 42.89 47.77 21.57 79.40 21.56 82.14 44.72 53.92 FC-Siam-diff 49.81 47.94	76.76 33.85 83.63 34.15 94.23 48.89 94.07 45.20 94.86 48.86
	OSCD-13 ch.	Siam. [3] EF [3] FC-EF FC-Siam-conc 42.39 65.15 24.16 85.63 28.35 84.69 64.42 50.97 FC-Siam-diff 57.84 57.99	85.37 37.69 88.15 42.48 96.05 56.91 93.68 51.36 95.68 57.92
		DSCN [11]	41.2	57.4	NA	47.9
	Szada/1	CXM [4] SCCN [8] FC-EF FC-Siam-conc 40.93 65.61 36.5 58.4 24.4 34.7 43.57 62.65	NA NA 93.08 51.40 44.9 28.7 92.46 50.41
		FC-Siam-diff 41.38 72.38	92.40 52.66
		DSCN [11]	88.3	85.1	NA	86.7
	Tiszadob/3	CXM [4] SCCN [8] FC-EF FC-Siam-conc 72.07 96.87 61.7 93.4 92.7 79.8 90.28 96.74	NA NA 97.66 93.40 74.3 85.8 93.04 82.65
		FC-Siam-diff 69.51 88.29	91.37 77.78


N Abbasnezhad
email: navideh.abbasnezhad@ensam.eu

A Khavandi
email: khavandi@iust.ac.ir

J Fitoussi
email: joseph.fitoussi@ensam.eu

H Arabi
email: arabi@iust.ac.ir

M Shirinbayan

A Tcharkhtchi
email: abbas.tcharkhtchi@ensam.eu

Influence of loading conditions on the overall mechanical behavior of polyether-ether-ketone (PEEK)

Keywords:

Testing methods have been developed to compare the mechanical responses and failure behavior of polyetherether-keton (PEEK) thermoplastic polymer; under quasi-static, high strain rate tensile tests and fatigue loading. Tensile tests were performed with the strain rates varying from 0.0003 s -1 to 60 s -1 and at different temperatures to compare the flow characteristics of the samples undergone various testing conditions. Fatigue tests at different amplitudes and frequencies were also performed to evaluate the temperature rise during cyclic loading and its effect on the fracture behavior. Results show that dynamic tension, in comparison with quasistatic behavior, causes brittle fracture; whereas under fatigue test at high frequencies and loading amplitudes the material behaves not only a more ductile behavior but also it clearly shows the influences of induced self-heating in the modulus and mechanical properties of the PEEK were significant. So the major aim of this article is to discuss about the induced temperature and its effect on the fracture surface. Thermal fatigue has a very significant role in increasing temperature and reducing fatigue life; from there it is necessary to know the conditions at which thermal fatigue happens and also the amount of energy which is consumed. Obtained equation from the experimental results and calculations can estimate the energy dissipation in the fatigue tests which is as a function of cycle and frequency.

Introduction

Polymeric materials have been used broadly in engineering applications due to their light weight. They are exposed to dynamic loading and high strain rate deformation in a variety of important applications such as aircraft and automotive components, as well as during highspeed processes such as extrusion and blow molding [START_REF] Chen | Tension and compression tests of two polymers under quasi-static and dynamic loading[END_REF]. Aircraft manufacturers plan to use linear aromatic polymers such as PEEK or as matrices of carbon fibers reinforced composite materials because of their high thermomechanical performances [START_REF] Rae | The mechanical properties of poly(ether-ether-ketone) (PEEK) with emphasis on the large compressive strain response[END_REF]. PEEK is a novel polymer with potential advantages for its use in demanding orthopedic applications (e.g. intervertebral cages) and it is considered to be one of the most important engineering polymers due to its properties such as: high mechanical strength, thermal stability (beyond 300 °C) and chemical resistance in corrosive environments [START_REF] Monich | Mechanical and biological behavior of biomedical PEEK matrix composites[END_REF]. Therefore, the influence of a physiological environment on the mechanical stability of PEEK is more requesting [START_REF] Ferguson | The long-term mechanical integrity of nonreinforced PEEK-OPTIMA polymer for demanding spinal applications: Experimental and finite-element analysis[END_REF]. In addition, PEEK is a thermoplastic which is considered as a high-performance biomaterial [START_REF] Chen | Fracture characteristics of PEEK at various stress triaxialities[END_REF].

Strain rate and temperature are two parameters that considerably influence the mechanical behavior of polymers [START_REF] Chen | Hot tensile fracture characteristics and constitutive modelling of polyether-ether-ketone (PEEK)[END_REF]. Since the beginning of polymer science, numerous experimental studies have been carried out on polymers to characterize the mechanical behavior as a function of temperature and strain rate. Among these studies, a great deal of attention has been given to the yield stress [START_REF] Richeton | Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress[END_REF]. The first work on the high speed response of plastics is due to Kolsky [START_REF] Kolsky | An investigation of the mechanical properties of materials at very high rates of loading[END_REF] in which he identified the importance of sample thickness on the measured response of the polymers. Walley et al. [START_REF] Richeton | Influence of temperature and strain rate on the mechanical behavior of three amorphous polymers: Characterization and modeling of the compressive yield stress[END_REF] accumulated reviews on the high strain rate studies of polymers, and they also expressed that the strain hardening behavior of glassy polymers is dependent on the strain rate and temperature.

Chou et al. [START_REF] Chou | The effect of strain rate and heat developed during deformation on the stress-strain curve of plastics[END_REF] analyzed the compressive behavior of several plastics and concluded that the temperature rise induced during deformation cannot be disregarded. They also detected that the yield strength increases with increasing strain rate. Similar results for the flow stress were also found by Briscoe and Nosker [START_REF] Briscoeb | The flow stress of high density polyethylene at high rates of strain[END_REF].

While numerous studies have investigated the influence of strain rate on the constitutive response for several polymers at room temperature, the influence of temperature at high strain rate has received much less attention. Among these studies, Rietsch and Bouette [START_REF] Rietsch | The compression yield behaviour of polycarbonate over a wide range of strain rates and temperatures[END_REF] studied the compression yield stress of PC over a wide variety of temperatures and strain rates, and exposed the importance of the secondary transition to account for the flow stress increases.

Cady et al. [START_REF] Cady | Determining the constitutive response of polymeric materials as a function of temperature and strain rate[END_REF] studied the mechanical response of PEEK and Polyamide-imides (PAI) under dynamic loading at high temperatures. Results of their study which have been conducted at strain rate of 0.001 s -1 and a temperature up to 125 °C showed a relatively constant apparent loading modulus occurs for PEEK material under the mentioned conditions. At higher temperatures, between 125 °C and 150 °C, the apparent loading modulus begins to gradually decrease and above 150 °C the modulus drops significantly. Similar behavior is seen at a strain rate of 1 s -1 where the apparent loading modulus had a drop from ∼ 3.1 GPa at 22 °C to ∼550 MPa at 175 °C. The compressive stress-strain response of PAI and PEEK were found to be depended on both the applied strain rate 0.001 s -1 ≤ έ ≤ 3000 s -1 , and the test temperature -55 °C ≤ T ≤ 300 °C, the apparent loading modulus of both materials drops dramatically above the glass transition temperature [START_REF] Cady | Determining the constitutive response of polymeric materials as a function of temperature and strain rate[END_REF].

High-strain rate studies on PEEK by Hamdan and Swallowe [START_REF] Hamdan | The strain-rate and temperature dependence of the mechanical properties of polyetherketone and polyetheretherketone[END_REF] showed that: (a) the yield strength increases by increasing strain rate or decreasing temperature, (b) the flow stress characteristics are related to the amount of crystallinity, and (c) samples held at temperatures above T g for long periods of time can show an increase in strength. A linear relationship between the compressive yield stress and logarithm of the strain rate has been observed for a large number of polymers [START_REF] Mulliken | Mechanics of the rate-dependent elastic-plastic deformation of glassy polymers from low to high strain rates[END_REF]. This observation supports the concept that a single deformation rate theory appears adequate to describe the plastic response of many polymers over a wide variety of imposed strain rates [START_REF] Mulliken | Mechanics of the rate-dependent elastic-plastic deformation of glassy polymers from low to high strain rates[END_REF].

Testing at temperatures above T g on thermoplastics and other polymers has shown that both the loading modulus and peak flow stress decrease with increasing temperature [START_REF] Cady | Determining the constitutive response of polymeric materials as a function of temperature and strain rate[END_REF].

In a study by Rakish Shrestha et al. [START_REF] Shrestha | Load history and sequence effects on cyclic deformation and fatigue behavior of a thermoplastic polymer[END_REF] about strain effect on fatigue behavior it can be observed that, the size of the crack initiation site in PEEK specimens generally decreased with increase in strain ratio, but not significantly. The crack initiation sites were observed to be subsurface, which is due to presence of a larger inclusion (particles or void-like defects). Furthermore, specimens with void-like defects, serving as the source of crack initiation, were found to have shorter fatigue lives as compared to the specimens containing particles of similar size.

When polymers are subjected to cyclic behavior a part of the energy which has been given to perform fatigue test, dissipates as thermal and mechanical energy, and because of low thermal conductivity this energy accumulates in the polymer and reduces the life time of the polymer. Thermal energy which is known as self-heating causes temperature increase and stabilized after some cycles. Having fracture while increasing the temperature is called thermal fracture, but if it is in the stabilized temperature it is controlled by crack initiation and propagation which is called mechanical fracture.

In this paper, an experimental study is carried-out to investigate the influence of loading conditions on the overall mechanical behavior of a PEEK. As mentioned, this polymer is a high performance thermoplastic material having good combination of properties such as: high melting temperature, high toughness, low moisture absorption, high fatigue capabilities, good resistance to dynamic cycles, suitable hydrolytic stability and high thermal properties [START_REF] Rakish | Cyclic deformation and fatigue behavior of polyether ether ketone (PEEK)[END_REF]. The organization of this work is as follow: after a presentation of the main physical-chemical characteristics and microstructure of PEEK, the results of the quasi static and high strain rate tensile tests up to failure are presented. Furthermore, fatigue tests with SEM fractography analysis emphasize the effect of loading conditions and a special attention is given to the analysis of the temperature rise.

Material description and methods

Polyether-ether-ketone (PEEK)

The material used in this study was plates of a PEEK grade 90G supplied by Victrex, unreinforced PEEK in the form of semi crystalline granules, is suitable for injection molding, and can easily flow under suitable conditions. It has complex geometries with thinner cross sections or longer flow lengths for high strength and stiffness as well as good ductility. Its cross-linking mechanism is also shown in Fig. 1.

Specimen geometry

An optimum geometry has been chosen in order to get a good quality of high speed tensile tests. Indeed, geometrical parameters were considered in order to reduce the stress wave perturbations due to inertial effects during high speed loading. In this work, all mechanical characterizations have been performed on the ISO-527-2-5A specimen geometry (see Fig. 2). In fact, its small dimension of the active length allows generating homogeneous stress/strain field from the early stages of the high strain rate tensile tests.

Methods

Physical-chemical characterization

• Microscopic observations: Scanning Electronic Microscope (HITACHI 4800 SEM), was used to investigate qualitatively the material microstructure and especially the fractured surfaces of the tensile specimens.

• DSC analysis: DSC Q10 V9.0 was utilized to find out the tempera- ture revolution near the glassy temperature up to the melting point. The sample used, was 6.2 mg which was first heated from 100 °C up to 350 °C; from this temperature, it was then cooled to 100 °C, always with the rate of 2 °C/min.

• DMA analysis: Dynamic mechanical analysis (DMA) testing was performed using a Q800 V21.2 Dynamic Mechanical Analyzer. Rectangular PEEK specimens with approximate dimensions of 25 × 4 × 3 mm 3 were tested over the entire temperature range of the DMA instrument from 100 °C to 300 °C at 1 Hz with constant amplitude. The storage modulus and loss modulus were measured as a function of temperature, and the corresponding parameter was calculated.

Mechanical characterization

• Quasi static tensile tests: Quasi static tensile tests by velocity of 2 mm/min were performed with an Instron 5881 machine in the low and high temperatures to see the effects of strain rate and temperature on mechanical behavior. • High-speed tensile tests: High-speed tensile tests were performed in a servo-hydraulic test machine. As specified by the manufacturer (Schenk Hydropuls VHS 5020), the test machine can reach a crosshead speed range from 10 -4 m/s (quasi-static) to 20 m/s. Moreover, the load level was measured by a piezoelectric crystal load cell having a capacity of a 50 KN. High-strain rate tensile tests were performed at different strain-rates until the polymer specimen failure. The testing machine was equipped with a launching system. The specimen was positioned between the load cell (upper extremity) and the moving device (lower extremity) as sketched in Fig. 3. A contactless technique was used to measure the local deformation using a high speed camera (FASTCAM-APX RS).

• Fatigue test: Tension-tension fatigue tests were performed at dif- ferent applied maximum stress on MTS 830 hydraulic fatigue machine. The minimum applied stress was always chosen to be equal to 10% of the maximum applied stress. In this paper, the results of experiments performed at different frequencies and different amplitudes at the constant temperature of 23 °C have been presented. In order to measure precisely the stiffness reduction due to the first loading stage, each fatigue test was preceded by a quasi-static tensile loading-unloading-reloading stage. Fig. 4 shows the loading-unloading-reloading and cyclic loading stages before cyclic loading.

During cyclic loading, the temperature rise (due to self-heating) was measured using an infrared camera (Raynger-MX4) in specific area (maximum temperature). DSC was used to investigate the response of PEEK as temperature variations. The result presented in Fig. 5 shows PEEK grade 90 has a glass transition temperature of 155 °C, crystallization temperature of 317 °C and melting point about 345 °C.

Experimental results and discussion

Dynamic mechanical analysis

DMA results (Fig. 6) show that the increasing temperature, in the zone of glassy state, until 135 °C has no significant effect on the storage modulus; whereas from this temperature it sharply decreases. The glass transition is relatively large from 130 °C to 180 °C. The value of α transition temperature is equal to 155 °C.

Tensile behavior of PEEK

Effect of temperature

In order to show the effect of temperature on the mechanical properties of PEEK, tensile tests were conducted at three different temperatures; environment temperature, below and above glass transition temperature. The results have been presented in (Fig. 7) and Table 1. For PEEK samples, the effect of temperature on Young's modulus as well as on yield stress and strain at break was significant. Variation of the temperature from 24 °C to 180 °C, lead to an abrupt decrease of the Young's modulus from 3600 MPa to 280 MPa (nearly 12 times), while stress at break decreased by 65%. This effect on strain at break is non-significant up to 110 °C but strain at break increased by 450% at 180 °C when the amorphous region of the polymer is at rubbery state. In this state the molecular mobility in the structure of the polymer is very high and the polymer shows ductile behavior however at 110 °C the polymer is always in glassy state and its failure was characterized as a quasi-brittle failure.

Effects of strain-rate on the material characteristics

The stress-strain curves illustrate three stages:

(i) Linear elastic behavior, (ii) Damage initiation corresponding to a knee point (apparition of a non-linear behavior), (iii) Damage propagation corresponding to a second quasi-linear stage and failure.

These stages are different for medium and high strain rate tensile tests. Stress-strain (σ-ε) tensile curves plotted for several strain rates are presented in Fig. 8. For each strain rate about 5 samples were tested and the stress-strain curves obtained for each strain rate were covered by the repeated tests especially for low strain rates (Fig. 8). For high strain rate tensile test also some tests were conducted. Here although there are differences at the values of failure strain but it is not very significant and it may occur due to existence of some faults like inclusions or hollows, so the highest strain at break can well express the polymer's behavior under high strain rate tensile test.

Tensile curves in Fig. 9(a) clearly indicate that the overall behavior of PEEK was influenced by loading-rate. Indeed, under rapid tensile loading (i.e. 60 s -1 ), PEEK exhibit with the high yield point and material failure happened rapidly after the first stage. Comparing the curve of quasi-static with high strain rates it can be notice that by increasing strain rate a delay in commencing the second stage (delay of the kneepoint) and ultimate stress appeared.

One can note that in Fig. 9(b) by increasing strain rate from quasistatic to 60 s -1 failure strain decreases about 75% which is an indication that the material alters from viscous behavior to a brittle one.

Material characteristics, namely Young's modulus (E), damage thresholds corresponding to the first non-linearity (σ threshold ; ε threshold ) and ultimate stress and strain (σ ultimate ; ε ultimate ) as a function of strain rate are shown in Figs. 10 and11. Note that the ultimate characteristics correspond to the maximum stress level. Fig. 10 enables us to emphasize that the material elastic modulus measured in the first stage of the stress-strain curve remains insensitive to the strain rate.

It is noticeable from the Fig. 11(a) and (b) that damage threshold in term of stress and strain increase by increasing strain rate. Fig. 11(c) shows that by increasing strain rate from quasi-static to 60 s -1 maximum stress is increased about 20% whereas threshold stress is increased about 100% and PEEK behave like a brittle polymer in high strain rates.

Fracture surface observation after tensile tests

Fig. 12(a) shows clearly the effect of cooling rate on the morphology of the polymer. As it can be seen a thin layer (skin) is formed at the surface of the sample because of high cooling rate. The morphology of this region is different from internal part of the specimen; it is more amorphous. It is not so clear on the picture but the reality is that the spherolits have more important diameter in internal part of the sample; the degree of crystallinity of the polymer in this region is higher. This difference of morphology gives the sample a non-isotropic behavior. The skin part of the specimen is more ductile as was shown in Fig. 12(b).

Fig. 12(b) shows enormous amount of plastic flow of the polymer prior to final failure in the quasi static tensile test. Besides the polymer bonding after tensile test was stretched compared to the virgin material and caused formation of gaps between the bonds which makes it easy to failure [START_REF] Rae | The mechanical properties of poly(ether-ether-ketone) (PEEK) with emphasis on the large compressive strain response[END_REF][START_REF] Bassett | On crystallization phenomena in PEEK *[END_REF].

Besides, the effect of loading rate on the failure behavior of the polymer must not be neglected. Comparing the curves presented in Fig. 9(a), and the macroscopic images of the fractured samples presented in Fig. 13, one can notice that as the strain rate increased failure mode changed from quasi-ductile to brittle and the failure of the sample subjected to high strain rate tensile test was accompanied by fragmentation.

In order to emphasize on the change of fracture mode (i.e. the ductile and brittle fracture), fracture surfaces of the samples were investigated also at microscopic scale. Micrographs presented in Fig. 14 show the microscopic surfaces obtained from the unloaded sample and the fractured surface of the sample after quasi-static tensile test and also after high strain rate tensile test (60 s -1 ). These micrographs indicate that in contrary to quasi-static loading, high strain rate loading lead to a more brittle fracture.

By comparing Fig. 14(b) & (c), one can observe that there is a large amount of crack initiation and plastic deformations which is occurred in Fig. 14(b). The distances between the river like features increased by increasing the strain rate and changes of fracture mode to brittle, so that the cracks were aligned in the same direction. It can be noticed that for an applied stress equal to 65 MPa, the fatigue life is about 1.5 × 10 5 cycles whereas the fatigue life is about 10 3 cycles for an applied stress of 85 MPa. This indicates that a variation of 23% stress can result to a fatigue life about 100 times higher. It is depicted in Fig. 15 that a decrease in the life time is very significant above stresses of 70 MPa.

Effect of loading amplitude

Depending on the loading conditions in terms of frequency and amplitude, the fatigue behavior can be in such a way that induces selfheating [START_REF] Medalia | Heat generation in elastomer compounds: Causes and effects[END_REF]. It has been reported [START_REF] Medalia | Heat generation in elastomer compounds: causes and effects[END_REF] that high frequency and applied stress can lead to an increase of the self-heating phenomenon and cause an "Induced Thermal Fatigue" (ITF). For low frequency, the fatigue behavior is mainly driven by the damage mechanisms occurring at the microscopic scale such as micro-cracking of the polymer. This kind of phenomenon has been introduced as "Mechanical Fatigue" (MF) [START_REF] Mortazavian | Fatigue behavior and modeling of short fiber reinforced polymer composites: A literature review[END_REF].

Coupled effect of loading amplitude and frequency can lead to a competition between induced self-heating and damage phenomenon [START_REF] Mortazavian | Effect of cycling frequency and self-heating on fatigue behavior of reinforced and unreinforced thermoplastic polymers[END_REF]. In order to illustrate the effect of loading amplitude and frequency, different loading amplitude at the same frequency and different frequencies at the same loading amplitude were carried out.

It is clearly indicated in Fig. 16 that the temperature rises first up to a certain cycle; then it decreases. At the same time, the relative modulus (E/E 0 ) changes inversely; it decreases first, before increasing after. The increase of temperature in the first step is due to energy dissipation and self-heating. The decrease of relative modulus in this step is due to selfheating. The increase of relative modulus in the second step is related to the decrease of the temperature. Also this decrease in the temperature which is followed after increasing the temperature can be the result of heat convection which occurs after a while with the environment to equilibrate with it.

Fig. 16 also shows the effect of loading amplitude. The results indicate that by increasing the loading amplitude (at the same frequency); the temperature rise becomes more significant. However, after defined cycles the relative modulus increases due to decreasing of temperature.

Effect of frequency

The results of Wöhler curves obtained by fatigue tests for different frequencies of 10, 30, 50, and 100 Hz and the same loading amplitude of 75 MPa for PEEK samples are shown in Fig. 17. This figure indicates that there is a difference between the fatigue lifetime related to various frequencies; at medium frequency the differences is small while at high frequency fatigue lifetime reduced and at low frequency it increased substantially.

Fig. 17 also shows for an applied stress equal to 75 MPa, a variation of the frequency from 10 Hz to 50 Hz lead to a fatigue life 2.5 times smaller (10 4 and 4 × 10 3 respectively) and it became 3.5 times smaller when the frequency raised to 100 Hz. These ratios become more significant when the fatigue stress increased (observed in Fig. 19). Thus it can be concluded, independent of the loading amplitudes, for frequencies of 30 Hz and above, there is a determinant influence of the frequency on the fatigue lifetime. This phenomenon, in fact is due to the self-heating during fatigue test pointed out in the next section.

Fig. 18 illustrates the variation of temperature during fatigue tests of PEEK samples at different frequencies and loading amplitudes. It is represented clearly in Fig. 18 that the amplitude and frequency effects are not very noticeable up to 200 cycles, but at higher cycles, the effects of these parameters on the temperature is considerable.

For the same amplitude it is clear that by increasing frequency, temperature increased accordingly. This increase is very noticeable for the frequency above 30 Hz.

Concerning the applied stress and frequency: as shown in Fig. 18(a) and (b), the temperature increases more rapidly until fracture for high applied stresses and frequencies. Moreover, at low applied stresses and frequencies, a stabilization of the temperature can be observed until specific number of cycles which highlights the effect of heat exchange made possible by a lower rate of self-heating.

Indeed, in the range of temperature corresponds to the beginning of the modification of the physical state of the PEEK.

It should be noted that the couple frequency-amplitude affects the nature of the fatigue overall response which can be governed by the damage mechanisms accumulation (Mechanical Fatigue) and/or by the self-heating (Induced Thermal Fatigue). For fatigue loading at 100 Hz, self-heating has been observed and yielded to a temperature rise up to 55 °C.

Fracture surface observation after fatigue tests

Fig. 19 shows the SEM analysis on fracture surfaces of the sample after fatigue tests.

As it was discussed before, by changing strain rate, frequency or loading amplitude for PEEK, the mode of rupture of fracture surface also changes. So it is important to emphasize that the material behaves differently under different conditions of fatigue tests. As it is shown in Fig. 19(a) for PEEK in 10 Hz ductile fracture appears. For the frequency of 100 Hz fracture surface consist of both ductile and brittle sections, shell layer faces ductile fracture (more amorphous) and the core layer (more crystalline) exhibits brittle fracture.

Calculation of energy dissipation

The samples used in this section are in the shape of rectangles with the dimensions of 45 × 4 × 3 mm which was subjected to the cyclic behavior by the MTS 830 hydraulic fatigue machine. The samples were tested in different frequencies of 10, 30 and 80 Hz and at constant amplitude of 68 MPa. In this amplitude we don't have the failure of sample rapidly and also the fatigue test doesn't take a long time. At the same time the evaluation of the temperature was measured by infrared camera (Raynger-MX4) in specific area (maximum temperature).

Fig. 20 is the different cycles of fatigue test in the frequency of 30 Hz and amplitude of 68 MPa in which we have the hysteresis loops so we can get the energy dissipation and follow the modulus evaluation.

For this reason in this section, samples in three different frequencies and constant amplitude of 68 MPa are tested to analyze this phenomenon.

As it is clearly comprehensible energy dissipation in the fatigue test refers to the energy which is not gain back from the energy which is given for applying the cyclic test, this energy is assumed as dissipated energy which is consumed as the work for plastic deformation or thermal energy to increase the heat or to form physical transformation.

Dissipated energy is equal to the area inside the hysteresis loop, which is a kind of energy in Joule so for calculating it, curves of force-displacement for each cycle which is obtained by the help of OriginLab 2016 software, is used. By subtracting these two areas shown in the Fig. 21, energy dissipation in each cycle is obtained. For 10 Hz there is a failure of sample after 500,000 cycles, for 30 Hz 100,000 cycles and for 80 Hz it is about 2000 cycles but in this analysis for better comparing we have chosen the optimized cycles.

By calculating energy dissipation of some cycles we can trace energy dissipation -cycle curve (see Fig. 22) in which we can use for correlation. As usual the premier cycles of the fatigue test are not reliable especially for the high frequency fatigue test because it takes a little time for the test to arrive to the amplitude which we want to perform the test. So it is because of that we haven't considered some premier cycles of the test.

It is apparent that for the fatigue test of 80 Hz fatigue is dominated by thermal fatigue and the hysteresis loop is getting larger as it is in accordance with the temperature-cycle curve and its modulus is sharply decreasing the other frequencies which are lower frequencies (30 & 10 Hz) process is dominated by both mechanical and thermal fatigue. Hereby in Fig. 22 related curve fits of energy dissipation versus cycles for three different frequencies are presented; by comparing the results and equations one can note that there is a relation between energy dissipation which is in the function of frequency and cycle. For generalizing the equation we can reach to the following equation with a good approximation:

= × E a N b ( )
where "E" is energy dissipation; "N" is the number of cycles and "a" is constant. One can note the relation between the frequency and the parameter "b". So, on the whole obtained equation for calculating the energy dissipation which is as a function of frequency and cycle is To verify this equation the fatigue tests were performed repeatedly for the frequency of 40 Hz and the results which are driven from the experimental and from the above equation (Fig. 23).

So it is apparent from the curve that the dispersion is negligible so as a result the equation is covering the energy dissipation per cycle for different frequencies. As it is known the accumulation of the energy in the certain cycles is obtained from the integration of the area under the energy dissipationcycle so one can calculate the energy dissipation for cycles.

Conclusions

In this paper, tensile tests were performed with the strain rates varying from quasi-static to 60 s -1 and at different temperatures to compare the flow characteristics of PEEK undergone various testing conditions. An overall analysis of the fatigue behavior of PEEK has been performed through the analysis of the coupled effect of loading amplitude and frequency. In order to highlight the induced thermal fatigue, the effect of frequency has been studied. At the same time, the effect of the loading amplitude allowed emphasizing the effect of damage kinetic on the nature of fatigue behavior of PEEK. In addition, the conditions at which thermal fatigue happens and also the amount of energy which is consumed are discussed. Based on the experimental results, the following conclusions may be drawn:

• By increasing the strain rate from quasi-static tensile test to high strain rate tensile test, the mechanism of fracture changes from ductile to brittle mode. This change doesn't influence the Young's modulus which can be influenced by frequency and amplitude. As the amplitude and frequency increased Young's modulus decreased.

• Different behavior for the fatigue fracture was observed at high and low frequency and amplitude. By increasing frequency or loading amplitude, the fracture surface changed from ductile to mixed ductile and brittle mode and this was interpreted to be due to the induced self-heating phenomenon.

• The core layer failed in brittle mode while the shell layer failed in ductile mode.

• Thermal fatigue has a very significant role in increasing temperature and reducing fatigue life; from there it is necessary to know the conditions which thermal fatigue happens and also the amount of energy which is consumed. Obtained equation from the experimental results and calculations can estimate the energy dissipation in the fatigue tests which is as a function of cycle and frequency.

Fig. 1 .

 1 Fig. 1. PEEK cross-linking mechanism.

Fig. 2 .

 2 Fig.2. Specimen dimensions obtained from optimized procedure results.

3. 1 .

 1 Physical-chemical characterization 3.1.1. DSC analysis

Fig. 3 .

 3 Fig. 3. Experimental device used for high-speed tensile tests.

Fig. 4 .

 4 Fig. 4. Loading-unloading-reloading stages before cyclic loading.

Fig. 5 .

 5 Fig. 5. DSC results for PEEK.

Fig. 6 .

 6 Fig. 6. DMA curve of PEEK material (frequency = 0.1 Hz).

3. 3 .

 3 Fatigue behavior of PEEK 3.3.1. Wöhler curve Fig. 15 shows the Wöhler curve obtained in tension-tension mode of stress-controlled fatigue tests at a frequency of 30 Hz for PEEK samples.

Fig. 7 .

 7 Fig. 7. Tensile response of PEEK samples at 24 °C, 110 °C and 180 °C at 2 mm/min.

Fig. 8 .

 8 Fig. 8. Representative curves of quasi-static tensile tests (2 mm/min).

Fig. 9 .

 9 Fig. 9. (a) Representative curves of different strain rates tensile tests and (b) Influence of strain rate on the failure strain.

Fig. 10 .

 10 Fig. 10. Influence of strain rate on the elastic modulus.

Fig. 11 .

 11 Fig. 11. Influence of strain rate on the: (a) threshold stress, (b) threshold strain, (c) maximum stress and (d) maximum strain.

Fig. 12 .

 12 Fig. 12. (a) Cut section of virgin material, (b) plastic deformation after quasi-static tensile test.

Fig. 13 .

 13 Fig. 13. Macroscopic observation of fractured samples at different strain rates; (a) quasistatic test and (b) strain rate of 60 s -1 .

Fig. 14 .

 14 Fig. 14. (a) Microscopic observation of unloaded sample and fractured surfaces of the samples: (b) after quasi-static tensile test and (c) after high strain rate tensile tests (60 s -1 ).

Fig. 15 .

 15 Fig. 15. Wöhler curves for PEEK samples at 30 Hz.

Fig. 16 .

 16 Fig. 16. Evolutions of the relative Young's modulus (E/E 0 ) and the self-heating phenomenon during fatigue tests: (a) f = 30 Hz and σ max = 60 MPa, (b) f = 30 Hz and σ max = 65 MPa, (c) f = 30 Hz and σ max = 70 MPa and (d) f = 30 Hz and σ max = 78 MPa.

Fig. 17 .

 17 Fig. 17. Applied frequency versus fatigue lifetime for PEEK samples for σ = 75 MPa.

Fig. 18 .

 18 Fig. 18. Temperature variation at different frequencies and amplitudes for PEEK samples: (a) f = 30 Hz and different amplitudes, (b) σ max = 75 MPa and different frequencies.

Fig. 19 .

 19 Fig. 19. Fracture surfaces at different loading conditions: (a) σ max = 75 MPa and f = 10 Hz, (b and c) σ max = 75 MPa and f = 100 Hz.

Fig. 20 .

 20 Fig. 20. Stress-strain fatigue curve of PEEK sample in the σ = 68 MPa and f = 30 Hz.

Fig. 21 .

 21 Fig. 21. Force -Displacement curve of fatigue test for calculating energy dissipation.

Fig. 22 .

 22 Fig. 22. Energy dissipation-cycle curve of fatigue test for PEEK at f = 10, 20, 30, 80 Hz & σ = 68 MPa.

Table 1

 1 Mechanical characters at different temperatures.

		E (MPa)	σ max (MPa)	ɛ r (%)
	24 °C	3600	83	5
	110 °C	2100	42	5.3
	180 °C	280	28	27.5

Conflicts of interest disclosure

This study has no funding and the authors declare that they have no conflict of interest.


Jacky Cresson

Aris Daniilidis

Masahiro Shiota

On the First Integral Conjecture of René Thom

Keywords: Structural stability, first integral, o-minimal structure, Sard theorem AMS Subject Classification. Primary 37C20 ; Secondary 34D30, 14P10

More than half a century ago R. Thom asserted in an unpublished manuscript that, generically, vector fields on compact connected smooth manifolds without boundary can admit only trivial continuous first integrals. Though somehow unprecise for what concerns the interpretation of the word "generically", this statement is ostensibly true and is nowadays commonly accepted. On the other hand, the (few) known formal proofs of Thom's conjecture are all relying to the classical Sard theorem and are thus requiring the technical assumption that first integrals should be of class C k with k ≥ d, where d is the dimension of the manifold. In this work, using a recent nonsmooth extension of Sard theorem we establish the validity of Thom's conjecture for locally Lipschitz first integrals, interpreting genericity in the C 1 sense.

Introduction

The purpose of this paper is to discuss the following conjecture attributed to René Thom (see [START_REF] Thom | F-64000 Pau[END_REF], [START_REF] Franc ¸oise | Géométrie analytique et systèmes dynamiques, Cours de troisième cycle[END_REF], [START_REF] Peixoto | Qualitative theory of differential equations and structural stability[END_REF], for example) which is part of the folklore in the dynamical systems community:

Thom conjecture: For 1 ≤ r ≤ ∞, C r -generically vector fields on d-dimensional compact, smooth, connected manifolds without boundary do not admit nontrivial continuous first integrals.

René Thom [START_REF] Thom | F-64000 Pau[END_REF] proposes a scheme for a formal proof relying on the assumption that a C r closing lemma (r ≥ 1) is true [START_REF] Franc ¸oise | Géométrie analytique et systèmes dynamiques, Cours de troisième cycle[END_REF]. The C 1 -closing lemma (case r = 1) has indeed been proved by Pugh ([14], [START_REF] Pugh | An improved closing lemma and a generic density theorem[END_REF]). Nevertheless, very little is known for a C r -closing lemma with r ≥ 2 (see also [START_REF] Gutierrez | A counter-example to a C 2 -closing lemma[END_REF], [START_REF] Pugh | Against the C 2 -closing lemma[END_REF]) so that Thom's strategy should be revised.

In [START_REF] Peixoto | Qualitative theory of differential equations and structural stability[END_REF], Peixoto proves Thom conjecture for r = 1 assuming that first integrals are of class C k with k ≥ d, i.e. a relation between the regularity class of the first integrals to be considered and the dimension of the underlying compact manifold. As pointed out by Peixoto, this condition is only of technical nature and relates to the use of the classical Sard theorem in a crucial part of the proof. More precisely, denoting by X M the set of C 1 -vector fields on the d-dimensional compact manifold M , Peixoto's proof is divided in three steps:

Let X ∈ X M and f : M → R be a first integral of class C d .
-By Sard's lemma, there exists in f (M ) an interval ]a, b[ made up of regular values. For any y ∈]a, b[, f -1 (y) is an (d -1)-dimensional, compact, differentiable manifold, invariant under X.

-By Pugh's general density theorem for each y ∈]a, b[, f -1 (y) does not contain singularities or closed orbits of X since they are generic. As a consequence, singularities and closed orbits are all located at the critical levels of f . 1 -Any trajectory γ in f -1 (y) is such that ω(γ) ⊂ f -1 (y) and cannot be contained in the closure of the set of singularities or closed orbits, in contradiction to Pugh's general density theorem.

T. Bewley [START_REF] Bewley | Non genericity of first integrals[END_REF] extends Peixoto's theorem for 1 ≤ r ≤ ∞. The proof has then been simplified by R. Mañe [START_REF] Mañe | A note on first integrals of vector fields and endomorphisms[END_REF]. Mañe's proof seems to have been rediscovered by M. Hurley [START_REF] Hurley | On the generic nonexistence of first integrals[END_REF]. However, the technical assumption of Peixoto (C d -regularity of the first integrals) stays behind all these works, because of the use of Sard's theorem. Nevertheless, Thom conjecture seems to be true in general.

In this paper, we cancel the aforementioned regularity condition by interpreting the word "nontrivial" as "being essentially definable with respect to an o-minimal approximation" (see Definition 2.6 below). In this context we prove the validity of Thom's conjecture for r = 1 and for Lipschitz continuous first integrals.

The technic of the proof follows the strategy used by M. Artin and B. Mazur ( [START_REF] Artin | On Periodic Points[END_REF]) to prove that generically the number of isolated periodic points of a diffeomorphism grows at most exponentially. Indeed, using a recently established nonsmooth version of Sard theorem (see [START_REF] Bolte | Clarke critical values of subanalytic Lipschitz continuous functions[END_REF]Theorem 7] or [START_REF] Bolte | Clarke subgradients of stratifiable functions[END_REF]Corollary 9]) and Peixoto's scheme of proof ( [START_REF] Peixoto | Qualitative theory of differential equations and structural stability[END_REF]) we first show that in an o-minimal manifold (that is, a manifold that is an o-minimal set), generically, o-minimal first integrals are constant. Then by approximating every compact differentiable manifold by a Nash manifold we derive a general statement.

• Preliminaries in dynamical systems.

Given a C 1 manifold M we denote by X M the space of all C 1 -vector fields on M equipped with the C 1 topology. Let φ t : M → M be the one-parameter group of diffeomorphisms generated by a vector field X on M . A point p ∈ M is called nonwandering, if given any neighborhood U of p, there are arbitrarily large values of t for which U ∩ φ t (U ) = ∅. Denoting by Ω the set of all nonwandering points we have the following genericity result due to C. Pugh ([14], [START_REF] Pugh | An improved closing lemma and a generic density theorem[END_REF]).

General density theorem (GDT):

The set G M of vector fields X ∈ X M such that properties (G 1 )-(G 4 ) below hold is residual in X M .

(G 1 ) X has only a finite number of singularities, all generic ;

(G 2 ) Closed orbits of X are generic ;

(G 3 ) The stable and unstable manifolds associated to the singularities and the closed orbits of X are transversal ;

(G 4 ) Ω = Γ, where Γ stands for the union of all singular points and closed orbits of X.

We use the following definition of a first integral: Definition 1.1 (First integral). A first integral of a vector field X on a compact connected manifold M of dimension d is a continuous function f : M → R which is constant on the orbits of the flow generated by X but it is not constant on any nonempty open set of M .

As mentioned in the introduction, Peixoto [START_REF] Peixoto | Qualitative theory of differential equations and structural stability[END_REF] only considers C k -first integrals with k ≥ d.

• Preliminaries in o-minimal geometry.

Let us recall the definition of an o-minimal structure (see [START_REF] Van Den Dries | Geometric categories and o-minimal structures[END_REF] for example).

Definition 1.2 (o-minimal structure). An o-minimal structure on the ordered field R is a sequence of Boolean algebras

O = {O n } n≥1 such that for each n ∈ N (i) A ∈ O n =⇒ A × R ∈ O n+1 and R × A ∈ O n+1 ; (ii) A ∈ O n+1 =⇒ Π(A) ∈ O n (Π : R n+1 → R n denotes the canonical projection onto R n ) ;
(iii) O n contains the family of algebraic subsets of R n , that is, the sets of the form

{x ∈ R n : p(x) = 0},
where p : R n → R is a polynomial function ;

(iv) O 1 consists exactly of the finite unions of intervals and points.

An important example of o-minimal structure is the collection of semialgebraic sets (see [START_REF] Coste | An introduction to semi-algebraic geometry[END_REF] for example), that is, sets that can be obtained by Boolean combinations of sets of the form {x ∈ R n : p(x) = 0, q 1 (x) < 0, . . . , q m (x) < 0} , where p, q 1 , . . . , q m are polynomial functions in R n . Indeed, properties (i), (iii) and (iv) of Definition 1.2 are straightforward, while (ii) is a consequence of the Tarski-Seidenberg principle.

A subset

A of R n is called definable (in the o-minimal structure O) if it belongs to O n . Given any S ⊂ R n a mapping F : S → R is called definable in O (respectively, semialgebraic) if its graph is a definable (respectively, semialgebraic) subset of R n × R.
• Preliminaries in variational analysis.

Let g : U → R be a Lipschitz continuous function where U is a nonempty open subset of R d . The generalized derivative of g at x 0 in the direction v ∈ R n is defined as follows (see [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]Section 2] for example):

g o (x 0 , e) = lim sup x→x 0 tց0 + g(x + te) -g(x) t (1)
where t ց 0 + indicates the fact that t > 0 and t → 0. It turns out that the function v → g o (x 0 , v) is positively homogeneous and convex, giving rise to the Clarke subdifferential of g at x 0 defined as follows:

∂g(x 0 ) = { x * ∈ R d : g o (x 0 , v) ≥ x * , v , ∀v ∈ R d }. (2)
In case that g is of class C 1 (or more generally, strictly differentiable at x 0 ) it follows that ∂g(x 0 ) = {∇g(x 0 )}.

A point x 0 ∈ U is called Clarke critical, if 0 ∈ ∂g(x 0 ). We say that y 0 ∈ g(U ) is a Clarke critical value if the level set g -1 (y 0 ) contains at least one Clarke critical point. Given a Lipschitz continuous function f : M → R defined on a C 1 manifold M we give the following definition of (nonsmooth) critical value. Definition 1.3 (Clarke critical value). We say that y 0 ∈ f (M ) is a Clarke critical value of the function f : M → R, if there exists p ∈ f -1 (y 0 ) and a local chart (ϕ, U ) around p such that 0 ∈ ∂(f • ϕ -1 )(ϕ(p)). In this case, p ∈ M is a Clarke critical point for f . It can be easily shown (see [START_REF] Rockafellar | Variational Analysis, Grundlehren der Mathematischen[END_REF]Exercise 10.7], for example) that the above definition does not depend on the choice of the chart.

Main results

Throughout this section M will be a C 1 compact connected submanifold of R n (without boundary), T M its corresponding tangent bundle and X M the space of C 1 vector fields on M equipped with the C 1 topology. Let us recall that submanifolds of R n admit ε-tubular neighborhoods for all ε > 0 sufficiently small. We further consider a C 1 -submanifold N of R n , a C 1 -diffeomorphism F : M → N and ε > 0.

Definition 2.1 (Approximation of a manifold). (i) We say that (N, F ) is a C 1 -approximation of M (of precision ǫ), if N belongs to an ǫ-tubular neighborhood U ǫ of M and F can be extended to a C 1 -diffeomorphism F : R n → R n , which is isotopic to the identity id, satisfies F | R n Ur ≡ id and max

x∈R n || F (x) -x|| + || d F (x) -id || < ǫ.
(We shall use the notation F ∼ ǫ id to indicate that F is ǫ-C 1 -closed to the identity mapping.) (ii) A C 1 -approximation (N, F ) of M is called semialgebraic (respectively, definable) if the manifold N is a semialgebraic subset of R n (respectively, a definable set in an o-minimal structure).

In the sequel, we shall need the following approximation result.

Lemma 2.2 (Semialgebraic approximation).

Let M be a C 1 compact submanifold of R n . Then for every ε > 0, there exists a semialgebraic ε-approximation of M .

Proof. Fix ε > 0 and let U be an open ǫ-tubular neighborhood of M in R n for some ǫ ∈ (0, ε). Applying [18, Theorem I.3.6] (for A = R n and B = C 1 ), we deduce the existence of a C 1embedding F of M into U which is ǫ-close to the identity map id in the C 1 topology such that F (M ) = N is a Nash manifold (that is, N is a C ∞ -manifold and a semialgebraic set). Then F can be extended to a C 1 diffeomorphism F of R n by a partition of unity of class C 1 such that F = id on R n \ U . Moreover there exists a C 1 isotopy {F t } t∈[0,1] , such that F t = id on R n \ U , F 0 ≡ id and F 1 = F and the map

F t : R n × [0, 1] → R n is ǫ-close to the projection to R n in the C 1 topology.
Given a C 1 -manifold M and a C 1 -vector field X ∈ X M , the following result relates generic singularities of hyperbolic type of X with Clarke critical values of Lipschitz continuous first integrals of X.

Lemma 2.3 (Location of singularities).

Assume f : M → R is a Lipschitz continuous first integral for the vector field X ∈ X M . Then all generic singularities and all closed orbits of hyperbolic type are located at the Clarke-critical level sets.

Proof. Let p 0 be either a singular point of hyperbolic type or any point of a closed orbit of hyperbolic type and let π : U → M be the exponential mapping around p 0 = π(0), where U is an open neighborhood of 0 ∈ T p 0 M ∼ = R d (d denoting the dimension of M ). It follows that the function g = f • π is Lipschitz continuous. Since the stable and unstable manifolds of the flaw of the field X at p 0 are transversal and since f is a first integral, it follows that for some basis {e i } i∈{1,...,d} of T p 0 M one has

g o (0, ±e i ) ≥ g ′ (0, ±e i ) := lim tց0+ g(±te i ) -g(0) t = 0,
where g 0 (0, •) is given by [START_REF] Artin | On Periodic Points[END_REF]. In view of (2) we deduce that 0 ∈ ∂g(0), hence f (p 0 ) is a critical value of f . Lemma 2.4 (Density of critical values for GDT fields). In the situation of Lemma 2.3, let us further assume that X ∈ G M . Then the Clarke critical values of f are dense in f (M ).

Proof. Let Γ denote the union of all singular points and closed orbits of the field X. Since X ∈ G M , it follows from Lemma 2.3 that the set f (Γ) is included to the Clarke critical values. Continuity of f and compactness of

M yield f (Γ) = f (Γ) = f (Ω).
Since Ω contains all ω-limits of orbits of X, taking any y ∈ f (M ) and any x ∈ f -1 (y) Γ we denote by γ the orbit passing through x and by γ ∞ the set of ω-limits of γ. Then by continuity

y = f (γ) = f (γ ∞ ) ⊂ f (Ω).
This proves the assertion. If the manifold M is not a definable subset of R n the above result holds vacuously and gives no information. To deal with this case, the forthcoming notion of essential o-minimality with respect to a given o-minimal approximation turns out to be a useful substitute for our purposes. Let us fix ǫ > 0 and a definable ǫ-approximation (N, F ) of M .

Definition 2.6 (Essential o-minimality with respect to a definable approximation).

A mapping f : M → R is called essentially o-minimal with respect to a definable approximation

(N, F ) of M if the mapping f • F -1 : N → R is o-minimal.
Note that every o-minimal function on M is essentially o-minimal with respect to any approximation (N, F ) of M for which the diffeomorphism F is o-minimal. Setting

M = {p ∈ R 2 : p ∈ Graph(h)}
where h(t) = t 3 sin(x -1 ), if t = 0 and 0 if t = 0, we obtain a nondefinable C 1 -submanifold of R 2 . Thus the (projection) function f : M → R with f ((t, h(t)) = t is not o-minimal. It can be easily seen that for every ǫ > 0 there exists an ǫ-approximation (N, F ) with respect to which f is essentially o-minimal. On the other hand, if χ K is the characteristic function of the Cantor set K of (0, 1), the function g : R → R defined by g(x) = x 0 χ K (t) dt for all x ∈ R is not essentially o-minimal with respect to any approximation. Roughly speaking, a function that is not essentially o-minimal contains intrinsic irreparable oscillations.

In view of Lemma 2.2, for every ǫ > 0 there exists a C 1 definable manifold N and a diffeomorphism F : M → N such that F ∼ ǫ id . Fixing the approximation, we associate to every vector field X : M → T M on M the conjugate C 1 -vector field X : N → T N on N defined as follows:

X(q) = dF (F -1 (q), X(F -1 (q)).

Note that X is uniquely determined by X. Let us further denote by G N the vector fields of N that satisfy the generic GDT assumptions. We are ready to state our main result.

Theorem 2.7 (Genericity of non-existence of first integrals). Let M be a C 1 compact submanifold of R n and ǫ > 0. For the C 1 topology, the set of vector fields in M that do not admit Lipschitz continuous first integrals which are essentially o-minimal with respect to a given definable ǫ-approximation of M is generic.

Proof. Let us fix any definable ǫ-approximation of M and let us denote by G N the vector fields of N that satisfy the generic GDT assumptions. By Pugh's density theorem G N is a C 1 -residual subset of X N and by Corollary 2.5, if Y ∈ G N then Y does not possess any o-minimal Lipschitz continuous first integral. Let G denote the set of vector fields of X that conjugate inside G N , that is, G = {X ∈ X M : X ∈ G N }.

Then G is residual in X M . Pick any X ∈ G and assume that f : M → R is a Lipschitz continuous first integral of X. Since the trajectories of X are transported to the trajectories of X ∈ G N through the mapping F, it follows that f = f • F is a first integral of X in N. This shows that f cannot be essentially o-minimal with respect to (N, F ).

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Corollary 2 . 5 (

 25 Thom conjecture -definable version). Let X ∈ G M . Then X does not admit any Lipschitz continuous definable first integral.Proof. Assume f is a Lipschitz continuous first integral of X on M and denote by S the set of its Clarke critical points. If f is o-minimal, then so is M (cf. property (ii) of Definition 1.2), the tangential mappings π : U ⊂ T p 0 M → M (around any point p 0 ∈ M ) and the composite functions of the form g = f • π (notation according to the proof of Lemma 2.3). Note that p is a critical point of f if and only if π -1 (p) is a critical point of g = f • π where π is any tangential mapping with p ∈ π(U ). Applying [4,Corollary 8] we deduce that the set of Clarke critical values of each function g is of measure zero. Using a standard compactness argument we deduce that f (S) is of measure zero, thus in particular f (M ) \ f (S) contains an interval (y 1 , y 2 ). But this contradicts the density result of Lemma 2.4.

Acknowledgment. Part of this work has been made during a research visit of the second author to the University of Pau (June 2007). The second author wishes to thank his hosts for hospitality and Olivier Ruatta (University of Limoges) for useful discussions. The first author thanks Jean-Pierre Bourguignon for his help with R. Thom's archive and J-P. Françoise for useful references. J.C. has been supported by the French projet de l'Agence Nationale de la Recherche "Intégrabilité réelle et complexe en Mécanique Hamiltonienne", N.JC05-41465