Pierre Biasutti
email: pierre.biasutti@labri.fr

Vincent Lepetit

Mathieu Brédif

Jean-François Aujol

Aurélie Bugeau

Jean-Franc ¸ois Aujol

LU-Net: An Efficient Network for 3D LiDAR Point Cloud Semantic Segmentation Based on End-to-End-Learned 3D Features and U-Net

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Introduction

The recent interest for autonomous systems has motivated many computer vision works over the past years. The importance of accurate perception models is a crucial step towards system automation, especially for mobile robots and autonomous driving. Modern systems are equipped with both optical cameras and 3D sensors, mostly LiDAR sensors. These sensors are now essential components of perception systems as they enable direct space measurements, providing an accurate 3D representation of the scene. However, for most automation-related tasks, raw Li-DAR point clouds require further processing in order to be used. In particular, point clouds with accurate semantic segmentation provide a higher level of representation of the scene that can be used in various applications such as obstacle avoiding, road inventory, or object manipulation.

This paper focuses on the semantic segmentation of 3D LiDAR point clouds. Given a point cloud acquired with a LiDAR sensor, we aim at estimating a label for each point that belongs to objects of interest in urban environments (such as cars, pedestrians and cyclists). The tradi-tional pipelines used to tackle this problem consider ground removal, clustering of remaining structures, and classification based on handcrafted features extracted on each clusters [START_REF] Himmelsbach | Lidar-Based 3D Object Perception[END_REF][START_REF] Feng | Fast Plane Extraction in Organized Point Clouds Using Agglomerative Hierarchical Clustering[END_REF]. The segmentation can be improved with variational models [START_REF] Landrieu | Large-Scale Point Cloud Semantic Segmentation with Superpoint Graphs[END_REF]. These methods are often hard to tune as handcrafted features usually require tuning many parameters, which is likely to be data dependent and therefore hard to use in a general scenario. Finally, although the use of regularization can lead to visual and qualitative improvements, it often leads to a large increase of the computational time.

Recently, deep-learning approaches have been proposed to overcome the difficulty of tuning handcrafted features. This has become possible with the arrival of large 3D annotated datasets such as the KITTI 3D object detection dataset [START_REF] Geiger | Are We Ready for Autonomous Driving? the KITTI Vision Benchmark Suite[END_REF]. Many methods have been proposed to segment the point cloud by directly operating in 3D [START_REF] Qi | PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation[END_REF] or on a voxel-based representation of the point cloud [START_REF] Zhou | VoxelNet: End-To-End Learning for Point Cloud Based 3D Object Detection[END_REF]. However, this type of methods either needs very high computational power, or are not able to process the amount of points acquired in a single rotation of a sensor. Even more recently, faster approaches have been proposed [START_REF] Wu | Squeezeseg: Convolutional Neural Nets with Recurrent CRF for Real-Time Road-Object Segmentation from 3D Li-DAR Point Cloud[END_REF][START_REF] Wang | Pointseg: Real-Time Semantic Segmentation Based on 3D Li-DAR Point Cloud[END_REF]. They rely on a 2D representation of the point cloud, called rangeimage [START_REF] Biasutti | Range-Image: Incorporating Sensor Topology for Li-DAR Point Cloud Processing[END_REF], which can be used as the input of a convolutional neural network. Thus, the processing time as well as the required computational power can be kept low, as these range-images consist in low resolution, multichannel images. Unfortunately, the choice of input channels, as well as the difficulty of processing geo-spatial information using only 2D convolutions have limited the results of such approaches, which have not yet achieved good enough scores for practical use, especially on small objects classes such as cyclists or pedestrians.

In this paper, we propose LU-Net-for LiDAR U-Netan end-to-end model for the semantic segmentation of 3D LiDAR point clouds. LU-Net benefits from a high-level 3D feature extraction module that can embed 3D local features in 2D range-images, which can later be efficiently used in a U-Net segmentation network. We demonstrate that, beside being a simple and efficient method, LU-Net largely outperforms state-of-the-art range-image methods, as shown in Figure 1.

The rest of the paper is organized as follows: We first discuss previous works on point cloud semantic segmentation, including methods designed for processing LiDAR data. We then detail our approach, and evaluate it on the KITTI dataset against state-of-the-art methods and discuss the results.

Related Work

In this section, we discuss previous works on image semantic segmentation as well as 3D point cloud semantic segmentation below.

Semantic Segmentation for Images

Semantic segmentation of images has been the subject of many works in the past years. Recently, deep learning methods have largely outperformed previous ones. The method presented in [START_REF] Long | Fully Convolutional Networks for Semantic Segmentation[END_REF] was the first to propose an accurate end-to-end network for semantic segmentation. This method is based on an encoder in which each scale is used to compute the final segmentation. Only a few month later, the U-Net architecture [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] was proposed for the semantic segmentation of medical images. This method is an encoder-decoder able to provide highly precise segmentation. These two methods have largely influenced recent works such as DeeplabV3+ [START_REF] Chen | Encoder-Decoder with Atrous Separable Convolution for Semantic Image Segmentation[END_REF] that uses dilated convolutional layers and spatial pyramid pooling modules in an encoder-decoder structure to improve the quality of the prediction. Other approaches explore multi-scale architectures to produce and fuse segmentations performed at different scales [START_REF] Lin | RefineNet: Multi-Path Refinement Networks for High-Resolution Semantic Segmentation[END_REF][START_REF] Zhao | IC-Net for Real-Time Semantic Segmentation on High-Resolution Images[END_REF]. Most of these methods are able to produce very accurate results, on various types of images (medical, outdoor, indoor). The survey [START_REF] Briot | Analysis of Efficient CNN Design Techniques for Semantic Segmentation[END_REF] of CNNs methods for semantic segmentation provides a deep analysis of some recent techniques. This work demonstrates that a combination of various components would most likely improve segmentation results on wider classes of objects.

Semantic Segmentation of Point Clouds

3D-based methods. As mentioned above, the first approaches for point cloud semantic segmentation were done using heavy pipelines, composed of many successive steps such as: ground removal, point cloud clustering, feature extraction as presented in [START_REF] Himmelsbach | Lidar-Based 3D Object Perception[END_REF][START_REF] Feng | Fast Plane Extraction in Organized Point Clouds Using Agglomerative Hierarchical Clustering[END_REF]. However, as mentioned above, these methods often require many parameters and they are therefore hard to tune. In [START_REF] Landrieu | Point Cloud Oversegmentation with Graph-Structured Deep Metric Learning[END_REF], a deep-learning approach is used to extract features from the point cloud. Then, the segmentation is done using a variational regularization. Another approach presented in [START_REF] Qi | PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation[END_REF] proposes to directly input the raw 3D LiDAR point cloud to a network composed of a succession of fully-connected layers to classify or segment the point cloud. However, due to the heavy structure of this architecture, it is only suitable for small point clouds. Moreover, processing 3D data often increases the computational time due to the dimension of the data (number of points, number of voxels), and the absence of spatial correlation. To overcome these limitations, the methods presented in [START_REF] Li | 3D Fully Convolutional Network for Vehicle Detection in Point Cloud[END_REF] and [START_REF] Zhou | VoxelNet: End-To-End Learning for Point Cloud Based 3D Object Detection[END_REF] propose to represent the point cloud as a voxel-grid which can be used as the input of a 3D CNN. These methods achieve satisfying results for 3D detection. However, semantic segmentation would require a voxel-grid of very high resolution, which would increase the computational cost as well as the memory usage. Range-image based methods. Recently, SqueezeSeg, a novel approach for the semantic segmentation of a LiDAR point cloud represented as a spherical range-image [START_REF] Biasutti | Range-Image: Incorporating Sensor Topology for Li-DAR Point Cloud Processing[END_REF], was proposed. This representation allows to perform the segmentation by using simple 2D convolutions, which lowers the computational cost while keeping good accuracy. The architecture is derived from the SqueezeNet image segmentation method [START_REF] Iandola | Squeezenet: AlexNet-Level Accuracy with 50x Fewer Parameters and < 0.5 MB Model Size[END_REF]. The intermediate layers are "fire layers", i.e. layers made of one squeeze module and one expansion module. Later on, the same authors improved this method in [START_REF] Wu | SqueezesegV2: Improved Model Structure and Unsupervised Domain Adaptation for Road-Object Segmentation from a LiDAR Point Cloud[END_REF] by adding a context aggregation module and by considering focal loss and batch normalization to improve the quality of the segmentation. A similar rangeimage approach was proposed in [START_REF] Wang | Pointseg: Real-Time Semantic Segmentation Based on 3D Li-DAR Point Cloud[END_REF], where a Atrous Spatial Pyramid Pooling [START_REF] Chen | Deeplab: Semantic Image Segmentation with Deep Convolutional Nets, Atrous Convolution, and Fully Connected CRFs[END_REF] and squeeze reweighting layer [START_REF] Hu | Squeeze-And-Excitation Networks[END_REF] are added. Finally, in [START_REF] Biasutti | RIU-Net: Embarrassingly simple semantic segmentation of 3D LiDAR point cloud[END_REF], the authors offer to input a range-image directly to the U-Net architecture described in [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF]. This method achieves results that are comparable to the state of the art of range-image methods with a much simpler and more intuitive architecture. All these range-image methods succeed in real-time computation. However, their results often lack of accuracy which limits their usage in real scenarios.

In the next section, we propose LU-Net: an end-toend model for the accurate semantic segmentation of point clouds represented as range-images. We will show that it outperforms all other range-image methods by a large margin on the KITTI dataset, while offering a robust methodology for bridging between 3D LiDAR point cloud processing and 2D image processing.

Methodology

In this section, we present our end-to-end model for the semantic segmentation of LiDAR point clouds inspired by the U-Net architecture [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF]. An overview of the proposed method is available in Figure 2.

Network input

As mentioned above, processing raw LiDAR point clouds is computationally expensive. Indeed, these 3D point clouds are stored as unorganized lists of (x, y, z) Cartesian coordinates. Therefore processing such data often involves preprocessing steps to bring spatial structure to the data. To that end, alternative representations, such as voxel grids or 2D pinhole projections in 2D images, are sometimes used, as discussed in the Related Work section. However, high resolution is often needed in order to represent enough details, which involves heavy memory costs. Modern LiDAR sensors often acquire 3D points, following a strict sensor topology, from which we can build a dense 2D image [START_REF] Biasutti | Range-Image: Incorporating Sensor Topology for Li-DAR Point Cloud Processing[END_REF], the so-called range-image. The range-image offers a lightweight, structured and dense representation of the point cloud.

Range-images

Whenever the raw LiDAR data (with beam number) is not available, the point cloud has to be processed to extract the corresponding range-image. As 3D LiDAR sensors acquire 3D points with a sampling pattern of a few number of scan lines and quasi uniform angular steps between samples, the acquisition follows a grid pattern that can be used to create a 2D image. Indeed, each point is defined by two angles and a depth, (θ, φ, d) respectively, with steps of (∆θ, ∆φ) between two consecutive positions. Each point p i of the LiDAR point cloud P can be mapped to the coordinates (x, y) with x = θ ∆θ , y = φ ∆φ of a 2D range-image u of resolution H × W = Card(P ), where each channel represents a modality of the measured point. A range-image is presented on Figure 3.

In perfect conditions, the resulting image is completely dense, without any missing data. However, due to the nature of the acquisition, some measurements are considered invalid by the sensor and they lead to empty pixels (no-data). This happens when the laser beam is highly deviated (e.g. when going through a transparent material) or when it does not create any echo (e.g. when the beam points in the sky direction). We propose to identify such pixels using a binary mask m equal to 0 for empty pixels and to 1 otherwise. The analysis of multi-echo LiDAR scans is subject to future work.

High-level 3D feature extraction module

In [START_REF] Wang | Pointseg: Real-Time Semantic Segmentation Based on 3D Li-DAR Point Cloud[END_REF], [START_REF] Wu | Squeezeseg: Convolutional Neural Nets with Recurrent CRF for Real-Time Road-Object Segmentation from 3D Li-DAR Point Cloud[END_REF] and [START_REF] Wu | SqueezesegV2: Improved Model Structure and Unsupervised Domain Adaptation for Road-Object Segmentation from a LiDAR Point Cloud[END_REF], the authors use a 5-channel rangeimage as input of their network. These 5 channels are made of the 3D coordinates (x, y, z), the reflectance (r) and the spherical depth (d). However, the analysis presented in [START_REF] Biasutti | RIU-Net: Embarrassingly simple semantic segmentation of 3D LiDAR point cloud[END_REF] showed that feeding a 2-channel range-image with only the reflectance and depth information to a U-Net architecture achieves comparable results to the state of the art.

In all these previous works, the choice of the number of channels of the range-image appears to be empirical. For each application, a complete study or a large set of experiments must be conducted to choose the best within all the possible combinations of channels. This is tedious and time consuming. To bypass such an expensive study, we propose in this paper a feature extraction module that is able to directly learn meaningful features adapted to the target application-here, semantic segmentation.

Moreover, processing geo-spatial information using 2D convolutional layers can cause issues in terms of data normalization as LiDAR sensors sampling typically decreases when acquiring farther points.

Inspired by the Local Point Embedder presented in [START_REF] Landrieu | Point Cloud Oversegmentation with Graph-Structured Deep Metric Learning[END_REF], we propose a high-level 3D feature extraction module that is able to learn N meaningful high-level 3D features for each point and to output a range-image with N channels.

Contrary to [START_REF] Landrieu | Point Cloud Oversegmentation with Graph-Structured Deep Metric Learning[END_REF], our module exploits the range-image to directly estimate the neighbors of each points instead of using a pre-processing step. Moreover, our module outputs a range-image, instead of a point cloud, which can be used as input to a CNN. Given a point p i = (x, y, z), and r i its associated reflectance, we define N (p i ) the set of neighboring points of p i in the range-image (e.g. the points that correspond to the 8-connected neighborhood of p i in the range-image). This set is illustrated Figure 4. We also define N (p i ) = {q -p i | q ∈ N (p i )} the set of neighbors in coordinates relative to p i . Note that if either p i or q is an empty pixel, then q -p i = (0, 0, 0).

Similarly to [START_REF] Landrieu | Point Cloud Oversegmentation with Graph-Structured Deep Metric Learning[END_REF], the set of neighbors N (p i ) is first processed by a multi-layer perceptron (MLP), which consists of a succession of linear, ReLU and batch normalization layers. The resulting set is then maxpooled to a point feature set, which is concatenated with p i and r i . The resulting vector is processed through another MLP that outputs a vector of N 3D features for each p i . This module is illustrated in Figure 5.

As linear layers can be done using 1 × 1 convolutional layers, the whole P point cloud can be processed at once. In this case, the output of the 3D feature extraction module is a Card(P ) × N matrix, which can then be reshaped to a H × W × N range-image.

Semantic segmentation

Architecture. The U-Net architecture [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF] is an encoderdecoder. As illustrated in Figure 6, the first half consists in the repeated application of two 3 × 3 convolutions followed by a rectified linear unit (ReLU) and a 2 × 2 max-pooling layer that downsamples the input by a factor 2. Each time a downsampling is done, the number of features is doubled to compensate for the loss of resolution. The second half of the network consists of upsampling blocks where the input is upsampled using 2 × 2 up-convolutions. Then, concatenation is done between the upsampled feature map and the corresponding feature map of the first half. This allows the network to capture global details while keeping fine details. After that, two 3 × 3 convolutions are applied followed by a ReLU. This block is repeated until the output of the network matches the dimension of the input. Finally, the last layer consists in a 1x1 convolution that outputs as many features as the wanted number of possible labels i.e. K 1-hot encoded.

Loss function. The loss function of our model is defined as a variation of the focal loss presented in [START_REF] Lin | Focal loss for dense object detection[END_REF]. Indeed, our model is trained on a dataset in which the number of example for each class is largely unbalanced. Using the focal loss approach helps improving the average score by few percents, as discussed later in Section 4. First, we define the pixel-wise softmax for each label k:

p k (x) = exp(a k (x)) K k =0 exp(a k (x))
where a k (x) is the activation for feature k at the pixel position x. After that, we define l(x) the groundtruth label of pixel x. We then compute the weighted focal loss as fol-lows:

E = x∈Ω -1 {m(x)>0} w(x)(1 -p l(x) (x)) γ log(p l(x) (x))
where Ω is the domain of definition of u, m(x) > 0 are the valid pixels, γ = 2 is the focusing parameter and w(x) is a weighting function introduced to give more importance to pixels that are close to a separation between two labels, as defined in [START_REF] Ronneberger | U-Net: Convolutional Networks for Biomedical Image Segmentation[END_REF].

Training We train the network with the Adam stochastic gradient optimizer and a learning rate set to 0.001. We also use batch normalization with a momentum of 0.99 to ensure good convergence of the model. Finally, the batch size is set to 4 and the training is stopped after 10 epochs.

Experiments

We trained and evaluated LU-Net using the same experimental setup as the one presented in SqueezeSeg [START_REF] Wu | Squeezeseg: Convolutional Neural Nets with Recurrent CRF for Real-Time Road-Object Segmentation from 3D Li-DAR Point Cloud[END_REF] as they provide range-images with segmentation labels exported from the 3D object detection challenge of the KITTI dataset [START_REF] Geiger | Are We Ready for Autonomous Driving? the KITTI Vision Benchmark Suite[END_REF]. They also provide the training / validation split that they used for their experiments, which contains 8057 samples for training and 2791 for validation and which can be used for a fair comparison between each result of each method.

We have manually tuned the number of layers N , i.e. the number of 3D features learned for each points. On all our experiments, best semantic segmentation results were obtained by setting N = 3. This small amount of channels is enough to highlight the structure of the objects that are latter used in the U-Net in charge of the segmentation task. All results reported in this section are with this value. Nevertheless, if using the high-level 3D feature extraction module for other applications, one should consider adapting this value.

Comparison with the state of the art

We compare the proposed method to 4 range-image based methods of the state of the art: PointSeg [START_REF] Wang | Pointseg: Real-Time Semantic Segmentation Based on 3D Li-DAR Point Cloud[END_REF], SqueezeSeg [START_REF] Wu | Squeezeseg: Convolutional Neural Nets with Recurrent CRF for Real-Time Road-Object Segmentation from 3D Li-DAR Point Cloud[END_REF], SqueezeSegV2 [START_REF] Wu | SqueezesegV2: Improved Model Structure and Unsupervised Domain Adaptation for Road-Object Segmentation from a LiDAR Point Cloud[END_REF], and RIU-Net [START_REF] Biasutti | RIU-Net: Embarrassingly simple semantic segmentation of 3D LiDAR point cloud[END_REF]. RIU-Net is a previous version of LU-Net we developed and was solely based on the raw reflectance and depth features instead of the 3D features learned in the end-to-end network of LU-Net. Similarly to [START_REF] Wu | Squeezeseg: Convolutional Neural Nets with Recurrent CRF for Real-Time Road-Object Segmentation from 3D Li-DAR Point Cloud[END_REF] and [START_REF] Wu | SqueezesegV2: Improved Model Structure and Unsupervised Domain Adaptation for Road-Object Segmentation from a LiDAR Point Cloud[END_REF], the comparison is done based on the Intersection-over-Union score:

IoU l = |ρ l G l | |ρ l G l |
where ρ l and G l denote the predicted and groundtruth sets of points that belongs to label l respectively. The performance comparisons between LU-Net and state-of-the-art methods are displayed Table 1. The first observation is that the proposed model outperforms existing methods in terms of average IoU by over 10%. In particular, the proposed model achieves better results on each of the classes compared to PointSeg, SqueezeSeg and RIU-Net. Our method also largely outperforms SqueezeSegV2 for both pedestrians and cyclists.

Our method is very similar to RIU-Net as both methods use a U-Net architecture with a range-image as input. While RIU-Net uses 2 channels-the reflectance and depth-LU-Net automatically extracts a N-dimensional high-level features per point thanks to the 3D feature extraction module. Table 1 demonstrates that using an additional network to automatically learn high-level features from the 3D point cloud largely improves the results, especially on classes that are less represented in the dataset.

Figure 7 presents visual results for SqueezeSegV2 and LU-Net. We here observe that visually, the results for cars are comparable. Nevertheless, by looking closer at the results, we observe that SqueezeSegV2 is more subject to false positives (Figure 7, orange rectangle). Moreover, our method provides a better segmentation of the cars in the back of the scene, compared to SqueezeSegV2 (Figure 7, purple rectangle). First, we analyse the influence of relative coordinates N as input to the 3D feature extraction module (Figure 5). We trained and tested the model using absolute coordinates N . We name this version LU-Net w/o relative. As Table 2 shows, relative candidates provide better results than neighbors in absolute coordinates. We believe that by reading relative coordinates as input, the network learns high-level features characterizing the local 3D geometry of the point cloud, independently of its absolute position in the 3D environment. These absolute positions are re-introduced once this geometry is learned, i.e. before the second multi-layer LU-Net Zooms in the following order Groundtruth, SqueezeSegV2, LU-Net For fair comparison, we also experimented using absolute coordinates N and adding a supplementary convolutional layer as the first layer. Indeed, we could expect this additional layer to characterize the transformation from absolute and local coordinates. Nevertheless, this architecture brought numerical instability while not managing to learn such transform, as it ended up with an average IoU of 30.6%.

Ablation study

Next, we analyse the influence of the focal-loss. As seen in Table 2, the use of focal-loss largely improves the scores on both cyclists and pedestrians. This is related to the imbalance between each class in the dataset, where there are 10 times more car examples than cyclists or pedestrians.

Additional results

Apart from being convincing in terms of IoUs, the results produced by our method are also very convincing visually, as it is demonstrated Figure 1 and9. Our segmentations are very close to those of the groundtruth. In Figure 9d), one of the pedestrians was not detected. When looking closely at the depth values in the range-image, this pedestrian is in fact hardly visible. It is also the case in the reflectance image. This is also related to the resolution of the sensor as only few points fall on the pedestrian, and could probably be solved by adding an external modality such as an optical image.

In Figure 9e), a car in the foreground is missing from the groundtruth, this causes the IoU to drop from 89.7% when ignoring this region of the image, down to 36.4%. Thus, removing examples with wrong or missing annotations in the dataset could lead to better results on LU-Net as well as on other methods. However, due to the amount of examples in the dataset, having a perfect annotation is practically very difficult.

Finally, LU-Net is able to operate at 24 frames per second on a single GPU. This is a lower frequency compared to other systems, yet still above the frame rate of the LiDAR sensor (10fps for the Velodyne HDL-64e). Moreover, our system uses only a few more parameters than RIU-Net for a significant improvement in terms of IoU scores.

Conclusion

In this paper, we have presented LU-Net, an end-toend model for the semantic segmentation of 3D LiDAR point clouds. Our method efficiently creates a multi-channel range-image using a learned 3D feature module. This range-image later serves as the input of a U-Net architecture. We show that this methodology efficiently bridges between 3D point cloud processing and image processing. The resulting method is simple, but yet provides very high quality results far beyond existing state-of-the-art methods.

The current method relies on the focal loss function. We plan to study possible spatial regularization schemes within this loss function. Finally, fusion of LiDAR and optical data would probably enable reaching a higher level of accuracy.

Figure 1 .

 1 Figure 1. The top two images show the segmentation of LiDAR data obtained with our method, and the groundtruth segmentation, seen in the sensor topology. The bottom two images show the same segmentations from a different point of view.

Figure 2 .

 2 Figure 2. Proposed pipeline for 3D LiDAR point cloud semantic segmentation. First, the topology of the sensor is used to estimate the 8-connected neighborhood of each point. Then, each point and its neighbors are fed to the high-level 3D feature extraction module, which outputs a multichannel 2D range-image. The range-image is finally used as the input of a U-Net segmentation network.

Figure 3 .

 3 Figure 3. Turning a point cloud into a range-image. (a) A point cloud from the KITTI database [7], (b) the same point cloud as a a range-image. Note that the dark area in (b) corresponds to pulses with no returns. Colors correspond to groundtruth annotation, for better understanding.

Figure 4 .

 4 Figure 4. Illustration of the notation of the input of the feature extraction module. pi is the point, N (pi) is the set of neighbors of pi.

Figure 5 .

 5 Figure 5. Architecture of the 3D feature extraction module. The output is an 1 × N feature vector for each LiDAR point.

Figure 6 .

 6 Figure 6. LU-Net architecture with the output of the 3D feature extraction module as the input (top) and the output segmented rangeimage (bottom).

Figure 7 .

 7 Figure 7. Visual comparison of the proposed model against SqueezeSegV2 [21] and the ground truth. Results are shown on the range-image where depth values are encoded with a grayscale map. Both SqueezeSegV2 and LU-Net globally achieve very satisfying results. Nervertheless, LU-Net is less subject to false positives than SqueezeSegV2, as can be seen in the orange areas and corresponding zooms. It also better segments farther objects such as the cars on the back of the scene in the purple rectangle, which reduces the amount of false-negatives, which are crucial for autonomous driving applications.

Figure 8 .

 8 Figure 8. Visual results of the ablation study. The use of neighbors in absolute coordinates results in incomplete segmentations of the objects compared to neighbors in relative coordinates. Moreover, the use of the focal loss (FL) helps the network to better distinguish classes that have similar aspects, here, cyclists and pedestrians.

Figure 9 .

 9 Figure 9. Results of the semantic segmentation of the proposed method (bottom) and groundtruth (top). Results are shown on the rangeimage where depth values are encoded with a grayscale map. Labels are associated to colors as follows: blue for cars, red for cyclists and lime for pedestrians.

Table 2

 2 presents intermediate scores in order to highlight the contribution of some model components.

Table 1 .

 1 Comparison (IoUs, %) of LU-Net with the state of the art for the semantic segmentation of the KITTI dataset.

		C a r s	P e d e s t r i a n s	C y c l i s t s	A v e r a g e
	SqueezeSeg	[20] 64.6	21.8	25.1	37.2
	PointSeg	[19] 67.4	19.2	32.7	39.8
	RIU-Net	[2] 62.5	22.5	36.8	40.6
	SqueezeSegv2 [21] 73.2	27.8	33.6	44.9
	LU-Net	72.7	46.9	46.5	55.4

Table 2 .

 2 Ablation study for the semantic segmentation of the KITTI dataset. Results in terms of (IoUs, %) for LU-Net w/o relative: which uses absolute coordinates N instead of relative N as input to the feature extraction module; LU-Net w/o FL : proposed model without focal-loss; LU-Net: proposed model with relative coordinates and focal-loss.

		C a r s	P e d e s t r i a n s	C y c l i s t s	A v e r a g e
	LU-Net w/o relative 62.8	39.6	37.5	46.6
	LU-Net w/o FL	73.8	42.7	32.9	49.8
	LU-Net	72.7	46.9	46.5	55.4

Acknowledgement

The authors thank GEOSAT for funding part of this work. This project has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 777826.


Nicolas Raimbault
email: nicolas.raimbault@univ-nantes.fr

Adeline Heitz
email: adeline.heitz@ifsttar.fr

Adrien Beziat
email: adrien.beziat@entpe.fr

Nathan Bounie
email: nathan.bounie@ifsttar.fr

Can logistics compensate for the local effects of deindustrialisation? The situation in

    Can logistics compensate for the local effects of deindustrialisation? The situation in the Ile-de-France region between 1982 and 2012

Over the last few years, logistics have emerged as an economic activity offering an alternative to the decline of industrial employment [START_REF] Husing | Logistics & Distribution: An Answer to Regional Upward Social Mobility[END_REF][START_REF] Lara | Goods movement and metropolitan inequality[END_REF]. But what about in Ile-de-France, a region marked by a long history of industrialisation that still widely structures its urbanisation [START_REF] Mischi | The world of the blue-collar worker: changed, but not gone[END_REF] and which is still one of the main consumer markets and Europe's logistic gateway ? 1 Employment generated by logistic activities is yet to be analysed or compared with industrial jobs. In terms of social geography, it is not only a matter of evaluating the number of jobs logistic development is able to promote in the region in order to reduce the impact of deindustrialisation processes, such as job losses among workers with few skills [START_REF] Husing | Logistics & Distribution: An Answer to Regional Upward Social Mobility[END_REF], but also of examining the location of these jobs at local level which may well reduce or increase spatial mismatch 2 for such workers [START_REF] Korsu | Job accessibility, residential segregation and risk of long-term unemployment in the Paris region[END_REF].

Our analyses mobilise data on municipal employment as estimated by the successive population censuses of 1982, 1990, 1999, 2006-2007 and 2011-2012. They also make use of various socioeconomic data, once again at municipal level.

INDUSTRY AND LOGISTICS: SIMILARITIES IN THE NATURE OF BLUE COLLAR JOBS AND LOCATION FACTORS

Definitions of logistics include strategic management of movement and the storage of flows of goods across production and distribution chains [START_REF] Christopher | Marketing Logistics[END_REF], and physical distribution activities, i.e. production of such flows including transport, handling and packaging, warehousing, and trans-shipment [START_REF] Mckinnon | Transport challenges and opportunities: briefing paper on the freight transport sector, Report for the UK Commission for Integrated Transport Raimbault N[END_REF].

The first activity field involves employment of middle and upper managers and office employees. These "white collars" are mainly located in offices in major tertiary centres. The second field involves jobs located in warehouses (Savy, 2015), which involve tasks such as handling, preparing orders, sorting or loading-unloading (Gaborieau, 2017), as well as employment of heavy goods vehicle drivers and delivery drivers. According to the French Nomenclature of Socioprofessional Categories and Professions (PCS 2003) established by INSEE, these are "blue-collar" workers: employees performing routine or semi-routine manual work as part of a hierarchical organisation of labour offering very little autonomy.

According to a study carried out using population census data and published by the Afilog association (2016) -which, since 2001, has brought together professionals from the logistics field and territorial representativesin 2012, workers accounted for 80% of logistic employee workforces, compared with 14% for technicians and 6% for managerial employees. Statistics are similar in the industry sector, where generally low-skilled workers are required for physical and routine operations and where, since the 1960s, the volume of qualified employment, and the number of managerial employees in particular 3 , has increased considerably [START_REF] Choffel | Adaptation des structures d'emploi et modernisation des entreprises[END_REF] outside factories, usually in offices located in major tertiary centres.

In terms of working conditions, logistical workers' jobs mainly present the same characteristics as manufacturing jobs [START_REF] Benvegnù | Produire le flux. L'entrepôt c o m m e p r o l o n g e m e n t d ' u n monde industriel sous une forme logistique[END_REF]. In warehouses, where the main activities are 3. The total number of senior management, intermediate profession and office employee jobs accounts for over 50% of manufacturing sector jobs, whilst workers account for only 44% of such jobs. preparation of orders, packaging, labelling and the organisation of consumer goods deliveries, operations performed are very similar to assembly-line work in factories, including, for example, the presence of conveyor belts and repetitive sorting tasks. The evolution of techniques and technologies has not brought an end to the Taylorist approach, but has transformed it instead. Thus, traceability of flows has been accompanied by such new operators' tools as vocal guidance, which has drastically reduced workers' autonomy and intensified the physical tasks performed, and whose introduction causes significant health issues [START_REF] Benvegnù | Produire le flux. L'entrepôt c o m m e p r o l o n g e m e n t d ' u n monde industriel sous une forme logistique[END_REF].

The blue-collar nature of these occupations also matches the low level of qualifications required to perform warehouse operations. In this sense, such operations are therefore very close to tasks performed by factory workers.

A comparison between location criteria for logistical activities and industrial activities highlights the similarity of these two sectors' spatial rationales. Excluding the criterion of proximity to consumption areas, factories and warehouses generally meet the same criteria: search for large land plots, close to transport hubs and within areas with a large catchment area of a low-skilled workforce. However, reconfiguration of logistical chains has caused a decrease of the number of warehouses on production sites -in immediate proximity to factories -and a sharp increase outside such sites, closer to consumption areas. This is the beginning of the construction of so-called "latest-generation" warehouses (DTZ, 2011). Logistical geography is therefore in the process of emancipating itself from industrial geography. At national level, distribution centres have increasingly focused on large urban agglomerations since the 1980s (Hesse, 2008[START_REF] Cidell | Concentration and decentralisation: the new geography of freight distribution in US Metropolitan areas[END_REF][START_REF] Dablanc | L a m é t r o p o l e l o g i s t i q u e , L e transport de marchandises et le territoire desgrandes villes[END_REF]. At regional level, the search for available, inexpensive land has led to logistical sprawl [START_REF] Dablanc | The impacts of logistic sprawl: How does the location of parcel transport terminals affect the energy efficiency of goods' movements in Paris and what can we do about it?[END_REF][START_REF] Dablanc | Atlanta: A Mega Logistics Center in the Piedmont Atlantic Megaregion (PAM)[END_REF], Dablanc et al, 2014). The emergence of a financialised property market tends to emphasize this pattern [START_REF] Hesse | Logistics and freight transport policy in urban areas: a case study of Berlin-Brandenburg/Germany[END_REF][START_REF] Raimbault | Ancrer le capital dans les flux logistiques : la financiarisation de l'immobilier logistique[END_REF].

IN ILE-DE-FRANCE, INDUSTRIAL JOB LOSSES PARTIALLY COM-PENSATED FOR BY LOGISTIC JOBS

Between 1982 and 2012, the number of blue collar jobs dropped by 62.7% in the industrial sector and by 11% in the logistics sector in Ilede-France. Three periods can be identified. Between 1982 and 1999, an all-time high of 275,849 jobs in logistics was reached whilst, the number of jobs in the industry sector fell (-43.6%) over the same period. In 2000, with job numbers continuing to plummet in the industrial sector (-21.3%), production employment in the logistics sector also started to decline (25%). The logistics sector had entered a maturation and rationalisation phase that was at the origin of the decline. It was also the very start of robotisation and automation in warehouses, likely to be expressed by a decreasing number of jobs for workers. Widespread recourse to interim workers (who are most certainly underestimated here) provides an explanation, at least in part, for the decline in logistic production employment.

Proximity to consumption areas

Need to be closer to the consumer, particularly to facilitate delivery and "last-mile" logistics.

No benefit in being closer to the consumer.

Proximity to suppliers

Only for some logistics activities (industrial logistics). Very important for manufacturing firms, in order to secure supply.

Land availability

Significant requirements in terms of land, search for large, inexpensive plots, leading to such activities being moved progressively away from agglomerations, starting in the 1950s for the industrial sector and in the 1990s for the logistics sector.

Accessibility of transport networks

Importance of proximity to motorway hubs. Sometimes to rail and inland waterway modes of transport.

Workforce availability

Need to be relatively close to an area with a low-skilled workforce.

Public policies

Severe regulatory constraints for the development of these activities in densely populated areas, context quite favourable to their development in agglomeration peripheries.

Regulations on establishments classified for environmental protection (ICPE) favourable to development of these activities in periurban areas.

Facilitation of development of industrial and logistical activities in economic activity zones (ZAE).

Activities perceived as generating nuisance and pollution.

Real-estate offers

Obsolescence of old industrial and logistical buildings due to sector evolution towards larger, more modern buildings that meet new standards.

Economic environment (Cluster)

Location in former industrial territories and new logistical and industrial clusters.

According to Mérenne-Schoumaker, 2007;Masson & Petiot, 2015;Hayter 1998;Aydalot, 1985;and Sergeot, 2004 Physical and routine operations are thus carried out in factories and warehouses alike, in the majority of cases by typically low-skilled workers. The dynamics of industrial office jobs are very different, while logistical "white collar workers" are few and far between. If we are to analyse the effects of deindustrialisation, we therefore need to gain a clearer idea of the spatial dynamics specific to factories and warehouses and precisely define those jobs located, on the one hand, on industrial sites and, on the other, on logistical sites. We propose to call these jobs (which, though primarily blue collar, also include technicians and associate professionals) industrial and logistical blue collar jobs.

Graph 1. Evolution of the number of blue collar jobs in the industry and logistics sectors in Ile- de-France (1982de-France ( -2012) ) 4. Number of inhabitants, number of jobs and average per capita income a year in 1990 and 2012

More recently, the decrease in blue collar jobs in the logistics sector stabilised (-5.3%) and the fall in this type of employment in the industrial sector slowed (-12.1%). 2012 marked the beginning of the fourth phase, in which both sectors stabilised. It may therefore be noted that logistics had a compensating effect on the industry sector until the end of the 1990s, which continues to operate today to a lesser extent.

LOCATION OF INDUSTRIAL AND LOGISTICS WORKER JOBS IN ILE-DE-FRANCE: BETWEEN CONTINUITY AND DIVERGENCE

Nearly three quarters of Ile-de-France's municipalities accommodate industrial and logistics jobs. This proportion is stable, reaching 72.1% in 1982 and 71.1% in 2012. On a finer scale, this period shows a strong correlation between the decrease in the number of jobs in industry and logistics in Paris and its inner suburbs (-69.5% and -33.8% respectively). However, in its outer suburbs, non-skilled employment in the industrial sector is falling (-47.5%) whilst it is rising at the same rate in the logistics sector (+ 47.9%). Transition from industrial territory to logistical territory therefore seems to be at work in the outer suburbs of Paris.

ILE-DE-FRANCE'S INDUSTRIAL AND LOGISTICAL EMPLOYMENT IS LOCATED IN WORKING-CLASS AREAS

By cross-referencing the employment landscape with Ile-de-France municipalities' socio-economic characteristics 4 , we observe that it is the most working-class areas that accommodate industrial and logistics activities. Municipalities with average to low population density, with employment concentration average or under the regional average and the lowest per capita income are those that appear to be most attractive for logistics development ("Municipalities with logistics employment growth" and "Municipalities with logistics and industrial employment growth" categories). These municipalities form a "productive belt" that brings together a poorer population, less-skilled jobs and less-desirable activities due to the nuisances that they generate and the low employment density they provide in relation to the space used.

Between 1982 and 2012, the decline in the number of industrial blue-collar jobs in Ilede-France was only partly compensated by blue-collar jobs in the logistics sector. Rationalisation of work, productivity gains and recourse to interim work all contribute to explaining this low compensation. In addition, municipalities that had lost industrial jobs were not necessarily winners during the rise of logistics.

Hence, municipalities at the heart of the Parisian agglomeration and residential municipalities in the outer suburbs that were widely affected by deindustrialisation are only seeing low-levels of logistics development today. These municipalities are marked by a gentrification process, in particular in the centre of the Parisian region. However, a partial deindustrialisation compensation effect can be observed in a productive belt farther away, which has witnessed mild growth in industrial employment and strong growth in logistics employment. These municipalities have relatively working-class social profiles with average income below the regional average.

Hence, there is a form of partial compensation of industrial jobs by logistics jobs, with location of logistics blue-collar jobs mapping a relatively new productive geography which is progressively emancipating itself from the industrial geography it inherited.

The fringes of the Parisian agglomeration, between the inner and outer suburbs, now constitute a new industrial and logistics belt.

Table 1 .

 1 Main factors of factory and warehouse location

	LOCATION DETERMINANTS	LOGISTICS	INDUSTRY

Savy M. (2015). La logistique en France. Etat des lieux et pistes de progrès. Rapport du comité scientifique ministère de l'Écologie, du Développement durable et de l'Énergie, du ministère de l'Économie, de l'Industrie et du Numérique, et de la Commission nationale des services. 103 p.


A Berthier

J Nicolas

J Verdun

J.-P Boy

MSSA Analysis Of Seasonal Loading-Induced Deformation From GNSS Time Series

Keywords:

Singular Spectrum Analysis (SSA) is a statistical method that allows to decompose time series of scalar quantities into oscillatory components and noises. Multi Singular Spectrum Analysis (MSSA) does the same for time series of vector quantities.

)

Introduction and study objectives

• Loading crustal deformations induced by ocean, atmosphere and continental water mass redistributions can be monitored by space geodesy (e.g. GNSS and GRACE) • Usually surface waters (rivers, lakes) are not included in hydrological loading models considering only snow and soil-moisture • Huge discrepancies between GNSS observations and models arise around large rivers such as the Amazon where nearly half of the vertical signal cannot be explained by the combination of loading models • In this study, Multi Singular Spectrum Analysis (MSSA) and Taylor Diagram Analysis are used together to compare GNSS observations and loading models to numerically evaluate the part of crustal deformation not yet explained by the loading models • Studied area is located in South America (Figure 1) where the hydrological effects are among the largest in the world mainly due to the river level variations can be determined from one of the radial graduations (in red in Fig. 3) plotted on the external semicircle • The corresponding centered Root Mean Square difference is given by one semicircle (in green in Fig. 3) from a second set of dashed semicircles

Conclusion and prospects

• These results are preliminary and only concern the vertical component and the first RC for the moment. Our method for estimating RC period still needs to be refined • The case of POVE (Figure 5) clearly suggests that the annual vertical displacement is best explained by models that account for rivers (better correlation MERRA: 72 % -> 79 %; GLDAS: 76% -> 85 %; lesser RMS differences MERRA: 3.8 mm -> 1.6 mm; GLDAS: 3.5 mm -> 1.8 mm)

• The UNPA case (Figure 6) indicates that despite a very low correlation between GNSS observations and loading models at annual period (< 30 %), the RMS differences remains moderate (< 1.3 mm) • The case of MAPA (Figure 7) is typical of an overestimation of the annual vertical movement by models including rivers (MERRA: 2.4 mm -> 13 mm; GLDAS: 2.5 mm -> 14 mm) • Such analyses have now to be carried out for the 3D position vector components (East, North, Up) of large number of GNSS sites (Figure 8) at different periods (few hours to seasonal) in order to confirm the results and to infer geophysical interpretations. For the loading models please refer to the EOST loading service (http://loading.u-strasbg.fr)

First promising results

Figure 2 :

 2 Figure 2: MSSA applied to GNSS-determined position of POVE station

Figure 3 :

 3 Figure 3: Example of Taylor diagram

Figure 1 :•Figure 6 :Figure

 16 Figure 1: Network of GNSS stations and vertical time series examples


Djaffar Boussaa
email: boussaa@lma.cnrs-mrs.fr

Coupled heat equation in thermoelasticity with temperature dependent moduli

Keywords: generalized thermoelasticity, temperature dependent properties, Cattaneo-Vernotte heat conduction equation, Lord-Shulman model, thermoelastic stress analysis 1

New and consistent expressions for the coupled heat equation are developed within the framework of small-strain thermoelasticity for both the Fourier and Cattaneo-Vernotte conduction models. These expressions place no restrictions on the changes in temperature, allow for the temperature dependence of the thermoelastic moduli, and include all the coupling terms as functions of the thermoelastic moduli and their derivatives. As applications, (i) an extended Lord-Shulman-type model is derived that takes into account the temperature dependence of the thermoelastic moduli, and (ii) the equations underpinning the experimental technique of thermoelastic stress analysis are revisited.

Introduction

The thermoelastic coupling terms occurring in the heat conduction equation are usually small and neglected in practice [START_REF] Boley | Survey of recent developments in the fields of heat conduction in solids and thermo-elasticity[END_REF][START_REF] Allen | Thermomechanical coupling in inelastic solids[END_REF][START_REF] Barron | Design for Thermal Stresses[END_REF]. Taking them into account, however, has been found important in many applications, such as the modeling of the vibration of resonant microelectromechanical systems (MEMS) [START_REF] Prabhakar | Analysis of frequency shifts due to thermoelastic coupling in flexural-mode micromechanical and nanomechanical resonators[END_REF][START_REF]Resonant MEMS. Advanced Micro and Nanosystems[END_REF][START_REF] Li | Study of intrinsic dissipation due to thermoelastic coupling in gyroscope resonators[END_REF], dynamic crack propagation [START_REF] Kokini | Thermal shock of a cracked strip: Effect of temperature-dependent material properties[END_REF][START_REF] Atkinson | Fracture in fully coupled dynamic thermoelasticity[END_REF][START_REF] Zamani | Implementation of the extended finite element method for dynamic thermoelastic fracture initiation[END_REF], thermal shocks [START_REF] Shindo | Thermal shock of cracked composite materials with temperature dependent properties[END_REF][START_REF] Ekhlakov | A BDEM for transient thermoelastic crack problems in functionally graded materials under thermal shock[END_REF][START_REF] Zhang | Thermomechanical coupling of non-Fourier heat conduction with the C-V model: Thermal propagation in coating systems[END_REF][START_REF] Wang | Thermoelastic response of thin plate with variable material properties under transient thermal shock[END_REF][START_REF] Wang | Thermoelastic behavior of elastic media with temperature-dependent properties under transient thermal shock[END_REF], ultrafast laser heating in thermal processing of materials [START_REF] Tzou | Recent development of ultrafast thermoelasticity[END_REF][START_REF] Tsai | Analysis of microscale heat transfer and ultrafast thermoelasticity in a multi-layered metal film with nonlinear thermal boundary resistance[END_REF][START_REF] Yu | Macro-to microscale heat transfer: the lagging behavior[END_REF][START_REF] Entezari | 3D dynamic coupled thermoelastic solution for constant thickness disks using refined 1D finite element models[END_REF], and wave propagation [START_REF] Sinha | Reflection of thermoelastic waves at a solid half-space with thermal relaxation[END_REF][START_REF] Sharma | Generalized thermoelastic wave propagation in circumferential direction of transversely isotropic cylindrical curved plates[END_REF][START_REF] Jiangong | Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation[END_REF][START_REF] Dobovšek | Structure and intrinsic properties of dispersion relation in hyperbolic thermoelasticity[END_REF]. In some of these applications, using the Cattaneo-Vernotte conduction model instead of Fourier's law and/or accounting for the temperature dependence of the material moduli have resulted in better predictions [START_REF] Wang | Thermoelastic behavior of elastic media with temperature-dependent properties under transient thermal shock[END_REF][START_REF] Tay | Finite element analysis of thermoelastic coupling in composites[END_REF][START_REF] Palumbo | Data correction for thermoelastic stress analysis on titanium components[END_REF].

The interest in retaining thermoelastic coupling terms is not limited to extreme applications. For instance, these terms are the foundation of the experimental technique known as thermoelastic stress analysis (TSA), which does not involve severe conditions. In a typical TSA test, temperature changes of an order of magnitude of one thousandth of a Kelvin are induced by sinusoidal reversible straining at a frequency of the order of magnitude of ten Hertz [START_REF] Greene | Thermoelastic stress analysis[END_REF][START_REF] Stanley | Beginnings and early development of thermoelastic stress analysis[END_REF]. Interestingly, despite the smallness of the temperature changes in TSA tests, models that (i) take into account the temperature dependence of the material moduli and (ii) involve their temperature-derivatives have been found to explain experimental observations more satisfactorily than linear thermoelasticity [START_REF] Belgen | Infrared radiometric stress instrumentation application range study[END_REF][START_REF] Wong | Thermoelastic constant or thermoelastic parameter?[END_REF][START_REF] Pitarresi | A review of the general theory of thermoelastic stress analysis[END_REF], in which the elasticities and the coefficients of thermal expansion are assumed temperature-independent.

In the case of small strains, small temperature changes, and temperatureindependent thermoelastic moduli, there is a consensus on the explicit form of the coupled heat equation for both the Fourier and Cattaneo-Vernotte conduction models [START_REF] Hetnarski | Thermal Stresses -Advanced Theory and Applications[END_REF]. In the case of large temperature changes and temperaturedependent thermoelastic moduli, there currently seems to be no consensus on the explicit form of the coupled heat equation, even for the Fourier conduction model. So far, several distinct expressions have been used that (i) involve different, often implicit, assumptions, and (ii) generally lack some of the coupling terms. Thus, the expressions obtained by letting the material moduli depend on temperature in the coupled heat equation obtained in linear thermoelasticity [START_REF] Wang | Thermoelastic behavior of elastic media with temperature-dependent properties under transient thermal shock[END_REF][START_REF] Ezzat | The dependence of the modulus of elasticity on the reference temperature in generalized thermoelasticity[END_REF][START_REF] Mohamed | Lord-Shulman theory under the dependence of the modulus of elasticity on the reference temperature in twodimensional generalized thermoelasticity[END_REF][START_REF] Hamdy | Thermal shock problem of generalized thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity[END_REF][START_REF] Ashraf | A generalized thermoelasticity problem of an annular cylinder with temperature-dependent density and material properties[END_REF][START_REF] Kumar | Investigation on the effects of temperature dependency of material parameters on a thermoelastic loading problem[END_REF] implicitly neglect, among other things, the coupling terms involving the temperature derivatives of the elasticity tensor. Similarly, the expressions stemming from polynomial expansions of the Helmholtz energy to an order higher than second in the temperature change [START_REF] Wang | Thermoelastic behavior of elastic media with temperature-dependent properties under transient thermal shock[END_REF][START_REF] Dillon | A nonlinear thermoelasticity theory[END_REF][START_REF] Lord | A generalized dynamical theory of thermoelasticity[END_REF][START_REF] Wang | The expression of free energy for thermoelastic material and its relation to the variable material coefficients[END_REF] also lack some of the coupling terms because of the truncation. Moreover, such expansions result in forms for the thermoelastic moduli, as functions of temperature, that do not fit well the actual temperature dependence of these moduli in many instances, for example, between room and cryogenic temperatures [START_REF] Marquardt | Cryogenic material properties database[END_REF].

The purpose of this study is to provide expressions for the coupled heat equation that include all the coupling terms for both the Fourier and Cattaneo-Vernotte conduction models while taking into account large temperature changes, defined as temperature changes large enough to require consideration of the temperature dependence of the thermoelastic moduli. The thermoelastic framework used here assumes the strain to be small but does not place the usual restrictions imposed in linear thermoelasticity on the temperature changes [START_REF] Kovalenko | Osnovy termouprugosti[END_REF][START_REF] Kovalenko | The current theory of thermoelasticity[END_REF][START_REF] Carlson | Linear thermoelasticity[END_REF]. The approach is based on the expressions of the thermodynamic potentials obtained in [START_REF] Boussaa | On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions[END_REF].

The rest of the paper is organized as follows. Section 2 reviews some basic thermodynamic results of thermoelasticity with temperature-dependent material moduli. Section 3 introduces thermoelastic moduli that enter the expressions of the coupled heat equations to be developed in later sections. The presentation emphasizes the dependence of these moduli on the mechanical variables in addition to temperature. Section 4 develops the coupled heat equation for both the Fourier and the Cattaneo-Vernotte conduction models, with strain and temperature as independent variables. Section 5 does the same with stress and temperature as independent variables. Section 6 applies the results of Sections 4 and 5 to obtain (i) approximate expressions for the coupled heat equation, including a Lord-Shulman-type model that accounts for the temperature dependence of the thermoelastic moduli and (ii) a specialization of the coupled heat equations to the adiabatic case, which is the framework underpinning the TSA. Section 7 offers conclusions.

2 Framework, energy balance, potentials and heat conduction laws

Framework

The framework of this study is small-strain thermoelasticity with allowance made for large changes in temperature, as defined in the introduction, and temperature dependence of the material moduli [START_REF] Kovalenko | Osnovy termouprugosti[END_REF][START_REF] Kovalenko | The current theory of thermoelasticity[END_REF][START_REF] Boussaa | On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions[END_REF]. The basic assumption of the framework is that the stress-strain relation is an affine function with temperature-dependent coefficients. This amounts to assuming that the elasticity tensor and, as to be expected from a thermoelasticity model, the thermal strain tensor are temperature dependent. This assumption fixes the form of the thermodynamic potentials as functions of temperature and either stress or strain [START_REF] Boussaa | Effects of superimposed eigenstrains on the overall thermoelastic moduli of composites[END_REF], and therefore that of the remaining thermoelastic moduli, i.e., the heat capacities, the coefficient of thermal expansion (CTE) tensor, and the stress-temperature tensor.

Local energy balance equation

In small deformation, the local form of the energy balance equation reads [45, p. 42]

u = σ • ˙ -div q + r, (1)
where the central dot denotes the inner product between two second-order tensors, the overdot denotes differentiation with respect to time, u is the internal energy per unit reference volume, σ is the stress tensor, is the (small) strain tensor, q is the heat flux vector, and r is the heat supply per unit reference volume.

Let h denote the enthalpy per unit reference volume. The identity

u = h + σ • (2)
allows rewriting the energy balance equation ( 1) in terms of the enthalpy as

ḣ = -• σ -div q + r. (3)

Helmholtz potential in the case of temperature-dependent moduli

The Helmholtz potential per unit reference volume, expressed in terms of its natural variables, is given by [START_REF] Boussaa | Effects of superimposed eigenstrains on the overall thermoelastic moduli of composites[END_REF] f

( , T ) = 1 2 • L(T ) + l(T ) • - T T0 ξ T0 C (0, ν) ν dν dξ -η 0 (T -T 0 ) + f 0 . ( 4
)
where T is the temperature; T 0 is the reference temperature; L is the isothermal elasticity tensor; l is the thermal stress, i.e., the stress that develops in the material when the temperature is made to vary under zero-strain conditions; C ( , T ) is the heat capacity at constant strain per unit reference volume at a state ( , T ) and C (0, T ) its value at the state (0, T ); ν and ξ are dummy variables; and η 0 and f 0 are the values of the entropy and the Helmholtz potential at the reference state, respectively. The elasticity tensor L is assumed to be symmetric, and the thermal stress l is assumed to vanish at the reference state (i.e., l (T 0 ) = 0). From

σ = ∂f ∂ T , (5)
η = - ∂f ∂T , (6)
it follows that the state equations associated with the Helmholtz potential ( 4) are

σ = L(T ) + l(T ), (7)
η = - 1 2 • dL dT - dl dT • + T T0 C (0, ν) ν dν + η 0 , ( 8
)
where η is the entropy.

Combining the Gibbs-Helmholtz equation

u = f -T ∂f ∂T (9)
with the Helmholtz potential (4) yields the following expression for the internal energy:

u ( , T ) = 1 2 • L -T dL dT + l -T dl dT • + T T0 C (0, ν) dν + f 0 + η 0 T 0 . (10)

Gibbs potential in the case of temperature-dependent moduli

The Gibbs potential per unit reference volume, expressed in terms of its natural variables, is given by [START_REF] Boussaa | Effects of superimposed eigenstrains on the overall thermoelastic moduli of composites[END_REF] g(σ, T ) = -

1 2 σ • M (T ) σ -m(T ) • σ - T T0 ξ T0 C σ (0, ν) ν dν dξ + g 0 -η 0 (T -T 0 ) . ( 11
)
where M = L -1 is the compliance tensor; m is the thermal strain, i.e., the strain, measured from the reference state, that develops in the material when the temperature is made to vary under zero-stress conditions; C σ (σ, T ) is the heat capacity at constant stress per unit reference volume at a state (σ, T ) and C σ (0, T ) is its value at the state (0, T ); and g 0 is the value of the Gibbs potential at the reference state. By definition, m (T 0 ) = 0. From

= - ∂g ∂σ T , (12)
η = - ∂g ∂T σ , (13)
it follows that the state equations associated with the Gibbs potential [START_REF] Ekhlakov | A BDEM for transient thermoelastic crack problems in functionally graded materials under thermal shock[END_REF] are

= M (T ) σ + m(T ), (14)
η = 1 2 σ • dM dT σ + dm dT • σ + T T0 C σ (0, ν) ν dν + η 0 . ( 15
)
Combining the Gibbs-Helmholtz equation

h = g -T ∂g ∂T σ ( 16
)
with the Gibbs potential [START_REF] Ekhlakov | A BDEM for transient thermoelastic crack problems in functionally graded materials under thermal shock[END_REF] gives

h(σ, T ) = - 1 2 σ • M -T dM dT σ -m -T dm dT • σ + T T0 C σ (0, ν) dν + g 0 + η 0 T 0 . (17)

Heat conduction laws

Two heat conduction laws are used below. The first is Fourier's law, i.e.,

q = -k grad T, ( 18
)
where k is the heat conductivity tensor, which, in general, is a function of the thermodynamic state.

The second is the Cattaneo-Vernotte heat conduction law, which is characterized by the following equation:

q + τ q = -k grad T, ( 19
)
where τ is the relaxation time, assumed to be a nonnegative constant. This equation is postulated to hold as given for general anisotropy [START_REF] Chandrasekharaiah | Thermoelasticity with second sound: A review[END_REF][START_REF] Chandrasekharaiah | Hyperbolic thermoelasticity: A review of recent literature[END_REF].

3 Thermoelastic moduli

Coefficient of thermal expansion tensor

The CTE tensor, α, is defined by

α = ∂ ∂T σ . (20)
From ( 14) it follows that

α (σ, T ) = dM dT σ + dm dT . (21)
Unlike in linear thermoelasticity, where M is independent of T and m is linearly dependent on T so that α is independent of the state, equation [START_REF] Jiangong | Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation[END_REF] shows that the CTE tensor depends on σ in addition to T . The free thermal expansion coefficient (CFTE) tensor, α 0 (T ), is defined as α (0, T ) in ( 21), i.e.,

α 0 (T ) = dm dT , (22)
whence

m(T ) = T T0 α 0 (ν) dν, (23)
where m (T 0 ) = 0 was used. The CFTE tensor defined by ( 22) is referred to as the tangent or instantaneous CFTE tensor. Another CFTE tensor is used in the literature, namely, the secant CFTE tensor, which is defined as

α sec 0 (ν) = 1 T -T 0 m (T ) = 1 T -T 0 T T0 α 0 (ν) dν. ( 24
)
In terms of this quantity, the stress-strain relation reads

= M (T )σ + α sec 0 (T ) (T -T 0 ) . (25)
In linear thermoelasticity, α, α 0 and α sec 0 coincide and the fact that they are not distinguished from each other has no consequences. This is not the case if the temperature dependence of the elasticities is taken into consideration.

Stress-temperature tensor

The stress-temperature tensor β is defined by

β = ∂σ ∂T . (26)
From ( 7) it follows that

β ( , T ) = dL dT + dl dT . (27)
As with α, the dependence of the elasticities on T causes a dependence of β on the mechanical variable, here . The discussion in Subsection 3.1 on α can be repeated for β in an obvious manner, yielding

β 0 (T ) = β(0, T ), (28)
l(T ) = T T0 β 0 (ν) dν, (29)
and

β sec 0 (ν) = 1 T -T 0 l (T ) = 1 T -T 0 T T0 β 0 (ν) dν, (30)
as well as

σ = L(T ) σ + β sec 0 (T ) (T -T 0 ) . ( 31
)

Heat capacities

The heat capacity per unit reference volume at constant strain, C , is defined by

C = T ∂η ∂T . (32)
It follows from (8) that

C ( , T ) = C (0, T ) -T 1 2 • d 2 L dT 2 + d 2 l dT 2 • . ( 33
)
In view of [START_REF] Shindo | Thermal shock of cracked composite materials with temperature dependent properties[END_REF], it is possible to check that, as was to be expected,

∂u ∂T = C ( , T ) . ( 34
)
The heat capacity per unit reference volume at constant stress, C σ , is defined by

C σ = T ∂η ∂T σ , (35)
which, combined with (8), gives

C σ (σ, T ) = C σ (0, T ) + T 1 2 σ • d 2 M dT 2 σ + d 2 m dT 2 • σ . ( 36
)
In view of [START_REF] Yu | Macro-to microscale heat transfer: the lagging behavior[END_REF], one can check that, as was to be expected,

∂h ∂T σ = C σ (σ, T ) . ( 37
)
Within the thermoelasticity framework used here, the thermoelastic moduli discussed in this subsection depend not only on the temperature but also on the mechanical variable, which will increase the number of coupling terms.

Remark

If expressions in terms of the thermal strain, m, its first T -derivative, α 0 , and its second T -derivative , dα 0 /dT , are preferred to those involving the thermal stress, l, then the following equations can be used:

l = -Lm, ( 38
) dl dT = - dL dT m -Lα 0 , (39)
d 2 l dT 2 = - d 2 L dT 2 m -2 dL dT α 0 -L dα 0 dT . (40)
Equation ( 38) can be derived from [START_REF] Kokini | Thermal shock of a cracked strip: Effect of temperature-dependent material properties[END_REF] by noting that during a free thermal expansion σ = 0 and = m.

4 Coupled heat equation with strain and temperature as independent variables

General form

From (1), ( 7), ( 10), ( 26), ( 27), [START_REF] Hamdy | Thermal shock problem of generalized thermoelasticity for an infinitely long annular cylinder with variable thermal conductivity[END_REF] and [START_REF] Ashraf | A generalized thermoelasticity problem of an annular cylinder with temperature-dependent density and material properties[END_REF] it follows that

C ( , T ) Ṫ = -div q + T β • ˙ + r. (41)

Case of Fourier's law

Inserting ( 18) into [START_REF] Kovalenko | The current theory of thermoelasticity[END_REF] gives

div (k grad T ) + r = C ( , T ) Ṫ -T β( , T ) • ˙ . (42)
The thermoelastic coupling occurs through not only ˙ terms as in linear thermoelasticity but also through terms, in view of the expressions of β( , T ) and C ( , T ) as given by ( 27) and ( 33), respectively. Another potential source of coupling is the possible dependence of k on . A thermoelastic model where the two-way coupling is caused by the dependence of k on is described in [START_REF] Fu | Thermo-Mechanical Coupling for Ablation[END_REF].

The coupling in ( 42) is now discussed according to whether or not the conductivity depends on the strain.

Case of -dependent conductivity If the conductivity depends on the strain, then the two-way coupling is operative whether or not the problem depends on time.

Case of -independent conductivity If the conductivity is independent of the strain, then, in general, the two-way coupling is inoperative in timeindependent problems and operative in time-dependent problems. For the twoway coupling to be inoperative and the heat equation ( 42) uncoupled from the mechanical fields in the time-dependent case, it is necessary that the stresstemperature tensor β be zero. The vanishing of β implies that L is independent of T and l is zero. In turn, this implies that C is independent of the strain. The vanishing of the stress-temperature tensor β is therefore also a sufficient condition for the heat equation to be uncoupled from the mechanical fields.

Case of the Cattaneo-Vernotte law

Combining [START_REF] Kovalenko | The current theory of thermoelasticity[END_REF] and its time derivative with [START_REF] Sinha | Reflection of thermoelastic waves at a solid half-space with thermal relaxation[END_REF] gives

div (k grad T ) + r + τ ṙ = τ C T + τ ∂C ∂T Ṫ 2 + C + τ ∂C ∂ T -β -T ∂β ∂T • ˙ Ṫ -β • ˙ + τ β • ¨ + ˙ • ∂β ∂ T ˙ T, (43)
where

∂C ∂ T = -T d 2 L dT 2 + d 2 l dT 2 , ∂C ∂T = ∂C ∂T (0, T ) - 1 2 • d 2 L dT 2 - d 2 l dT 2 • -T 1 2 • d 3 L dT 3 + d 3 l dT 3 • , and
∂β ∂ T = dL dT , ∂β ∂T = d 2 L dT 2 + d 2 l dT 2 .
The following higher order Maxwell relation can be used in ( 43)

∂C ∂ T = -T ∂β ∂T . (44)
Similar conclusions to those reached in subsection 4.2 on the thermoelastic coupling hold true in this subsection. In particular, for the thermoelastic coupling to be inoperative, it suffices that k be independent of and β be zero.

Equation [START_REF] Boussaa | On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions[END_REF] represents the general coupled heat equation in the case of the Cattaneo-Vernotte law. It contains all the coupling terms in explicit form while being more convenient and more accurate than the equations based on the expansion of the Helmholtz potential in power series of the temperature change.

5 Coupled heat equation with stress and temperature as independent variables

General form

From ( 3), ( 14), ( 17) , ( 20), ( 21), ( 36) and ( 37) it follows that

-div q + r = C σ Ṫ + T α • σ. ( 45
)

Case of Fourier's law

Combining ( 18) and [START_REF] Lubliner | Plasticity theory[END_REF] gives

div (k(σ, T ) grad T ) + r = C σ (σ, T ) Ṫ + T α(σ, T ) • σ, (46)
where α(σ, T ) and C σ (σ, T ) are given by ( 21) and [START_REF] Dillon | A nonlinear thermoelasticity theory[END_REF], respectively.

It can be shown, as in Subsection 4.2, that in the case of σ-independent conductivity the vanishing of the CTE tensor α is a necessary and sufficient condition for the heat equation to be uncoupled from the mechanical fields. The vanishing of only the CFTE tensor α 0 does not suffice to uncouple the heat equation from the mechanical fields.

Case of the Cattaneo-Vernotte law

Combining [START_REF] Lubliner | Plasticity theory[END_REF] and its time derivative with [START_REF] Sinha | Reflection of thermoelastic waves at a solid half-space with thermal relaxation[END_REF] gives

div (k grad T ) + r + τ ṙ = τ C σ T + τ ∂C σ ∂T σ Ṫ 2 + C σ + τ ∂C σ ∂σ T + α + T ∂α ∂T σ • σ Ṫ + α • σ + τ α • σ + σ • ∂α ∂σ T σ T, (47)
where

∂C σ ∂σ T = T d 2 M dT 2 σ + d 2 m dT 2 , ∂C σ ∂T σ = ∂C σ ∂T (0, T ) + 1 2 σ • d 2 M dT 2 σ + d 2 m dT 2 • σ + T 1 2 σ • d 3 M dT 3 σ + d 3 m dT 3 • σ , and
∂α ∂σ T = dM dT , ∂α ∂T σ = d 2 M dT 2 σ + d 2 m dT 2 .
Note that in (47)

∂C σ ∂σ T = T ∂α ∂T σ . (48)
Equations ( 43) and ( 47) are of course equivalent. It is keeping all the coupling terms that makes it possible to pass from one equation to the other without unnecessary additional assumptions. The appendix gives relations that can be used to move from [START_REF] Boussaa | On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions[END_REF] to [START_REF] Chandrasekharaiah | Hyperbolic thermoelasticity: A review of recent literature[END_REF].

The conclusions reached in Subsection 5.2 on the thermoelastic coupling hold true in this subsection. In particular, for the thermoelastic coupling to be inoperative, it suffices that k be independent of σ and α (and not α 0 ) be zero.

Applications

Some simplified expressions

Equations ( 42), ( 43), [START_REF] Chandrasekharaiah | Thermoelasticity with second sound: A review[END_REF], and (47) are numerically challenging and are not meant to be used as they are in all circumstances. They are rather intended to build justified approximate models. As they do not involve any approximations other than those defining their theoretical framework (small strains and linear stress-strain relation with temperature-dependent coefficients) and contain all the corresponding coupling terms, they make it possible to estimate on a caseby-case basis the order of magnitude of each of the coupling terms and decide which ones to keep. This subsection discusses some such approximate models.

An approximate heat equation within Fourier conduction

If the temperature gradient, temperature rate, strain rate, heat supply, together with strain, are assumed to be small of the same order, then, up to first-order, the heat equation for Fourier's law [START_REF] Carlson | Linear thermoelasticity[END_REF]

reduces to div (k(0, T ) grad T ) + r = C (0, T ) Ṫ -T dl dT (T ) • ˙ . ( 49
)
This equation is valid even for large temperature changes. If in addition to the above assumptions the relative change in temperature, (T -T 0 )/T 0 , is also assumed to be small, then the foregoing equation reduces to div (k (0, T 0 ) grad

T ) + r = C (0, T 0 ) Ṫ -T 0 dl dT (T 0 ) • ˙ . ( 50
)
Since dl/dT (T 0 ) = -L (T 0 ) α 0 (T 0 ), (50) can be recognized as the heat conduction equation used in linear thermoelasticity, namely, div (k (0, T 0 ) grad

T ) + r = C (0, T 0 ) Ṫ + T 0 L (T 0 ) α 0 (T 0 ) • ˙ . (51)
It is worth noting that it does not suffice to replace T 0 by T in the foregoing equation to retrieve the coupled heat equation in the case of temperature-dependent material moduli [START_REF] Carlson | Linear thermoelasticity[END_REF], or even the approximate version (49).

An extended Lord-Shulman model

In the case of the Cattaneo-Vernotte conduction also, if the temperature gradient, temperature rates, strain rates, heat supply, and heat supply rate, together with strain, are assumed to be small, then [START_REF] Boussaa | On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions[END_REF] reduces to div (k(0, T ) grad

T ) + r + τ ṙ = C (0, T ) Ṫ + τ T -T dl dT (T ) • ( ˙ + τ ¨ ) . ( 52
)
Equation ( 52) takes into account large temperature changes and the temperature dependence of the moduli. It therefore represents an extended version of the classical Lord-Shulman equation.

Again, if in addition to the above assumptions the temperature change is also assumed to be small, then the above equation reduces to

div (k (0, T 0 ) grad T ) + r = C (0, T 0 ) Ṫ + τ T + T 0 L (T 0 ) α 0 (T 0 ) • ( ˙ + τ ¨ ) .
(53) This is the anisotropic version of of the coupled heat equation proposed by Lord and Shulman [START_REF] Lord | A generalized dynamical theory of thermoelasticity[END_REF]. Replacing T 0 by T in this equation gives back neither (43) nor (52).

Application to TSA

The independent variables commonly used in TSA are the temperature and stress and the conduction model is Fourier's law. This subsection will therefore focus on [START_REF] Chandrasekharaiah | Thermoelasticity with second sound: A review[END_REF]. This equation is first particularized to the adiabatic case, which is the usual case considered in TSA studies. The results obtained are then compared with those given in [START_REF] Wong | Thermoelastic constant or thermoelastic parameter?[END_REF], which are presently the most widely used [START_REF] Pitarresi | A review of the general theory of thermoelastic stress analysis[END_REF][START_REF] A F Robinson | A review of residual stress analysis using thermoelastic techniques[END_REF][START_REF] A F Robinson | Paint coating characterization for thermoelastic stress analysis of metallic materials[END_REF][START_REF] Garinei | Constant stress field measurements through thermoelasticity[END_REF][START_REF] Howell | A stress-free model for residual stress assessment using thermoelastic stress analysis[END_REF][START_REF] Howell | Identification of Plastic Strain using Thermoelastic Stress Analysis[END_REF] if there is a need to take into account the experimentally observed dependence of the thermoelastic parameter on both temperature and stress [START_REF] Belgen | Infrared radiometric stress instrumentation application range study[END_REF][START_REF] Machin | Mean stress dependence of the thermoelastic constant[END_REF].

In the adiabatic case, i.e., q = 0 and r = 0, (46) reduces to

C σ (σ, T ) Ṫ T = -α(σ, T ) • σ, (54)
or equivalently, by [START_REF] Jiangong | Circumferential thermoelastic waves in orthotropic cylindrical curved plates without energy dissipation[END_REF], to

C σ (σ, T ) Ṫ T = - dM dT σ + α 0 • σ. (55)
Specializing equation (55) to the isotropic case gives

C σ (σ, T ) Ṫ T = -α 0 + ν E 2 dE dT - 1 E dν dT tr σ tr σ + 1 + ν E 2 dE dT - 1 E dν dT σ • σ, (56)
where α 0 is the coefficient of free thermal expansion (α 0 = α 0 i, with i being the second-order identity tensor), E is Young's modulus and ν is Poisson's ratio. Equation ( 56) is now compared with equation (2.23) given in [START_REF] Wong | Thermoelastic constant or thermoelastic parameter?[END_REF], which is often referred to as "revised higher order theory equation." The general form of the two equations is the same. The terms involving the mechanical moduli are identical in the two equations. However, those involving the heat capacities and those involving the coefficients of free thermal expansion differ. First, the heat capacity at constant strain is used in [28, equation (2.23)], whereas the heat capacity at constant stress is used in (56). Second, the secant CFTE is used in [28, equation (2.23)], whereas the tangent CFTE is used in (56). The two equations are therefore different for general adiabatic transformations.

The difference can be explained by approximations used in [START_REF] Wong | Thermoelastic constant or thermoelastic parameter?[END_REF] but not used here. The derivation of the stress-based equations [28, equation (2.22)] and [28, equation (2.23)] from the strain-based equation [28, equation (2.20)] and [28, equation (2.21)] has required approximations, such as "omitting higher order terms". By contrast, (56) is free of such approximations. Now, as far as TSA is concerned, the new equations should not a priori question most existing results, as C σ and C are close to each other, as are, even more so, the tangent and secant CFTE tensors in view of the smallness of the change in temperature in typical TSA tests.

Conclusions

This study has developed expressions for the coupled heat equation within the framework of small-strain thermoelasticity with temperature-dependent thermoelastic moduli and without the restrictions on temperature changes imposed in linear thermoelasticity. The expressions, developed for the Fourier and Cattaneo-Vernotte conduction laws, contain all the coupling terms, which involve not only the thermoelastic moduli but also their derivatives with respect to the independent state variables. Retaining all the coupling terms has made it possible to present a coherent framework without unnecessary assumptions and to obtain equations that are more accurate and more convenient than those obtained by polynomial expansion. As applications, the expressions obtained were used (i) to derive a new Lord-Shulman-type model that accounts for the temperature dependence of the thermoelastic moduli and large changes in temperature, and (2) to revisit the theory underpinning TSA.

C = C σ -T α • M -1 α ∂C ∂T = ∂ ∂T C σ -T α • M -1 α σ - ∂ ∂σ C σ -T α • M -1 α T • M -1 α = ∂C σ ∂T σ - ∂C σ ∂σ T • M -1 α -α • M -1 α -2T α • M -1 ∂α ∂T σ + 3T α • M -1 dM dT M -1 α ∂C ∂ T = T -2M -1 dM dT M -1 α + M -1 ∂α ∂T σ • Terms in β β = -M -1 α ∂β ∂T = 2M -1 dM dT M -1 α -M -1 ∂α ∂T σ ∂β ∂ T = -M -1 dM dT M -1
• Terms in

= M σ + m ˙ = α Ṫ + M σ ¨ = M σ + α T + 2 dM dT σ Ṫ + ∂α ∂T σ Ṫ 2

Appendix

The heat equations [START_REF] Boussaa | On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions[END_REF] and ( 47) are of course equivalent. One can use the following relations to derive (47) from ( 43):

• Terms in C