Shubhendu Jena

Franck Multon

Adnane Boukhayma

Monocular Human Shape and Pose with Dense Mesh-borne Local Image Features

. Our quantitative and qualitative results on standard benchmarks show that using local features improves on global ones and leads to competitive performances with respect to the state-of-the-art.

I. INTRODUCTION

Reconstructing human bodies e.g. [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF], [START_REF] Lin | End-to-end human pose and mesh reconstruction with transformers[END_REF] and their parts (faces e.g. [START_REF] Feng | Joint 3d face reconstruction and dense alignment with position map regression network[END_REF], [START_REF] Abrevaya | Crossmodal deep face normals with deactivable skip connections[END_REF], hands e.g. [START_REF] Boukhayma | 3d hand shape and pose from images in the wild[END_REF], [START_REF] Ge | 3d hand shape and pose estimation from a single rgb image[END_REF]) from minimal, partial and noisy inputs is one of the most sought-after goals of human centered machine learning. In this regard, human shape and pose recovery from a single color image is a popular problem in computer vision and graphics spurring a vast research literature, with applications in various areas such as action recognition, avatarization, human machine interaction, etc. While earlier approaches to 3D reconstruction relied on multi-view triangulation or depth information, the recent surge of deep learning has allowed the reduction of acquisition constraints to as little as a single input RGB image. The ill-posedness of this monocular setting is alleviated through learning strong statistical priors from large training datatsets with deep CNNs, which has shown to be successful especially for single shape class settings such as humans. Such approaches also benefit from transfer learning techniques by leveraging networks pretrained on massive general datasets (e.g. ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF]). While several work [START_REF] Kanazawa | End-toend recovery of human shape and pose[END_REF], [START_REF] Kolotouros | Learning to reconstruct 3d human pose and shape via model-fitting in the loop[END_REF], [START_REF] Boukhayma | 3d hand shape and pose from images in the wild[END_REF] show that the learning can be further regularized by integrating differentiable parametric naked human body models (e.g. SMPL [START_REF] Loper | Smpl: A skinned multi-person linear model[END_REF], GHUM [START_REF] Xu | Ghum & ghuml: Generative 3d human shape and articulated pose models[END_REF], Frank [START_REF] Joo | Total capture: A 3d deformation model for tracking faces, hands, and bodies[END_REF]) within deep networks, other methods advocate predicting model-free 3D shape outputs (e.g. [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF], [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF]), with the prospect of more expressive results, whilst the parametric model is usually involved in generating training 3D pseudoground-truth in this case.

In this work, given a single color image, we tackle the problem of estimating human 3D shape and pose in the form of a fixed-topology triangle mesh, using a feed forward deep neural network. We specifically focus on improving on a class of model-free methods that propose to use GCNs for this task [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF], [START_REF] Ge | 3d hand shape and pose estimation from a single rgb image[END_REF], [START_REF] Kulon | Weakly-supervised mesh-convolutional hand reconstruction in the wild[END_REF], [START_REF] Kulon | Single image 3d hand reconstruction with mesh convolutions[END_REF], the graph's vertices and edges being defined as those of the 3D mesh representing the human shape. Traditionally, these methods extract a global latent feature vector from the image using a CNN, and this same vector is used as input feature to all the mesh vertices equally, as in [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF]. The GCN then starts from these mesh-borne features and deforms a T-posed template mesh towards the target posed mesh. A noteworthy variant of this strategy consists in predicting a global feature and mapping it subsequently to initial low resolution mesh vertex features [START_REF] Ge | 3d hand shape and pose estimation from a single rgb image[END_REF], [START_REF] Kulon | Weakly-supervised mesh-convolutional hand reconstruction in the wild[END_REF], [START_REF] Kulon | Single image 3d hand reconstruction with mesh convolutions[END_REF], and it was mostly explored for 3D hand prediction rather than full body partly due to the smaller mesh size.

In contrast to these methods, we propose here to use per vertex pixel-aligned local image features as initialization for the GCN stage, as illustrated in Fig. 1. We use a method for predicting dense pixel-to-surface correspondences (Dense-Pose [START_REF] Güler | Densepose: Dense human pose estimation in the wild[END_REF]) of humans to map each visible vertex in the template geometry to a pixel in the input image. Bi-linear interpolation is then used to build a different feature vector per vertex, by sampling and stacking local image features at the vertex's corresponding 2D location in the image space, at different CNN feature depths. Given a T-posed initial mesh appended with vertex specific local image features, a GCN regresses the final mesh vertex positions. The network composed of the image CNN and mesh GCN is trained endto-end using 3D supervision following the training scheme in [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF].

We evaluate our method on standard benchmarks for human mesh prediction from images, namely 3DPW [START_REF] Marcard | Recovering accurate 3d human pose in the wild using imus and a moving camera[END_REF] and Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF]. Our numerical and visual results demonstrate that using local features improves on using only global ones in GCN based human mesh recovery from single image. We also show that our method yields competitive results in comparison to the state-of-the-art methods.

II. RELATED WORK

There is a substantial body of work on the subject of human shape and pose estimation from a single image. We review in this section works that we deemed most relevant to the context of our contribution.

A. Optimization-based methods

Most current optimization-based methods rely on using 2D landmarks such as key-points and silhouettes [START_REF] Sigal | Combined discriminative and generative articulated pose and non-rigid shape estimation[END_REF], [START_REF] Bogo | Keep it smpl: Automatic estimation of 3d human pose and shape from a single image[END_REF], [START_REF] Guan | Estimating human shape and pose from a single image[END_REF] and optimize parametric models such as SCAPE [START_REF] Anguelov | Scape: shape completion and animation of people[END_REF] and SMPL [START_REF] Loper | Smpl: A skinned multi-person linear model[END_REF] to fit them to these landmarks. The optimization objective consists of mainly two kinds of loss terms. The first is prior terms designed to penalize unnatural human poses and shapes. The second is data terms minimizing the difference between the inferred 2D landmarks obtained by projecting the predicted mesh onto the image and the groundtruth body landmarks. Additionally, there have been other recent methods [START_REF] Zanfir | Monocular 3d pose and shape estimation of multiple people in natural scenes-the importance of multiple scene constraints[END_REF] which incorporated additional terms such as part segmentation, scene and temporal constraints in the optimization objective. Although optimization-based methods provide generally reliable solutions, they are notoriously slow and prone to getting stuck in local minima especially with challenging initializations. This incentivizes learning-based solutions such as ours, which offer faster inference and do not require initialization.

B. Learning-based methods

1) Model-based: Model-based methods make use of a parametric 3D human body model to perform 3D human pose and shape estimation. The learning problem is thus reduced to learning the parameters associated with the body model from images and other types of input. A notable example is the work of [START_REF] Kanazawa | End-toend recovery of human shape and pose[END_REF] which directly regresses the SMPL parameters from a single RGB image, by performing a weak supervision comprising of a 2D key-point reprojection error and an adversarially learnt pose prior. SPIN [START_REF] Kolotouros | Learning to reconstruct 3d human pose and shape via model-fitting in the loop[END_REF] improves on this with a self-improving framework that incorporates 3D human model parameter optimization into network training process. To deal with occlusions and noisy situations which make image-based methods more susceptible to failure, some approaches use alternative inputs such as 2D joint heatmaps [START_REF] Tung | Selfsupervised learning of motion capture[END_REF], silhouettes [START_REF] Pavlakos | Learning to estimate 3d human pose and shape from a single color image[END_REF], [START_REF] Varol | Bodynet: Volumetric inference of 3d human body shapes[END_REF] and semantic segmentation maps [START_REF] Omran | Neural body fitting: Unifying deep learning and model based human pose and shape estimation[END_REF]. In-spite of the aforementioned advantages behind model-based methods, such strategies can conversely also be somewhat restrictive. The tight relationship between the parameters and the model's output reduces the expressiveness of the generated meshes. Hence, as a model-free method, our framework focuses instead on directly regressing the 3D human body mesh vertices corresponding to an input image following seminal work.

2) Model-free: As the name implies, model-free methods do not rely on parametric models for 3D human body reconstruction. Instead, they directly regress an explicit body shape representation from the input images. Some of the earlier work uses a volumetric reconstruction approach with a voxel output [START_REF] Varol | Bodynet: Volumetric inference of 3d human body shapes[END_REF], [START_REF] Zheng | Deephuman: 3d human reconstruction from a single image[END_REF]. The main drawback of voxelbased methods is their inability to represent detailed surfaces due to memory limitations. Other methods use different representations such as depth maps, point clouds, etc. [START_REF] Gabeur | Moulding humans: Non-parametric 3d human shape estimation from single images[END_REF]. However, these suffer mostly from lacking surface continuity and neighborhood connectivity. To deal with these problems, recent methods propose to directly regress 3D SMPL meshes representing the output human body shape. To the best of our knowledge, this line of work started with CMR [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF] which used a GCN to directly regress 3D coordinates of vertices from a global image feature. Pose2Mesh [START_REF] Choi | Pose2mesh: Graph convolutional network for 3d human pose and mesh recovery from a 2d human pose[END_REF] also did the same but from 2D joints as input instead. The work of Lin et al. [START_REF] Lin | End-to-end human pose and mesh reconstruction with transformers[END_REF] yields arguably state-of-theart performances through the use of Transformers [START_REF] Vaswani | Attention is all you need[END_REF] but requires considerable training time and data. Furthermore, there has been another class of methods focusing on learning dense correspondence between 2D images and 3D shapes. The seminal work of DensePose [START_REF] Güler | Densepose: Dense human pose estimation in the wild[END_REF] which provides dense mapping from images to a human body model by regressing 2D correspondence maps has led arguably to the advent of much similar work focusing on dense correspondence. [START_REF] Xu | Denserac: Joint 3d pose and shape estimation by dense render-and-compare[END_REF], [START_REF] Rong | Delving deep into hybrid annotations for 3d human recovery in the wild[END_REF] utilized DensePose correspondence maps for 3D human model recovery. However, they only leverage them as input images. Zhang et al. [START_REF] Zhang | Learning 3d human shape and pose from dense body parts[END_REF] predict and use local and global correspondence maps as input to further CNN stages for pose and shape prediction in a similar fashion. In contrast, we propose here to use correspondence maps to provide vertices with semantically meaningful image-aligned local features. In fact, our strategy is similar to DecoMR [START_REF] Zeng | 3d human mesh regression with dense correspondence[END_REF], where the authors establish dense correspondences between the surface and the input image, which is subsequently used to transfer image features to the UV-map domain and thereafter perform 3D coordinate regression with a 2D CNN. Differently, we explore here the idea of performing vertex position regression from local image features using a convolutional network defined naturally on the same data representation as the output (i.e. GCN on the mesh as opposed to CNN on the UV-map).

III. METHOD

We describe in this section the components of our method, which follows the illustration in Fig. 1. Our input is a single RGB image and our output is a triangle human mesh with the SMPL [START_REF] Loper | Smpl: A skinned multi-person linear model[END_REF] template topology. Given the input image, a convolutional neural network (CNN) is tasked with extracting 2D image features. These features are fed to a graph convolutional network (GCN) that transforms a template mesh to the final output mesh. The graph's topology is defined as per that of the mesh. Each vertex in the graph is initialized with a local feature extracted from the image encoder feature maps. This extraction is performed using a DensePose [START_REF] Güler | Densepose: Dense human pose estimation in the wild[END_REF] correspondence map that maps pixel locations to the mesh's visible surface.

We use a ResNet50 [START_REF] He | Deep residual learning for image recognition[END_REF] network that we denote f to build convolutional image feature maps from an input image I, extracted in a coarse-to-fine fashion at 5 network stages:

{F l I } 1≤l≤5 = f (I), F l
I being the feature map at stage l. The feature maps' spatial dimensions decrease gradually from the input image dimensions (H,W ) downwards and their respective feature dimensions are 64, 256, 512, 1024 and 2048, summing up to D = 3904. The final feature is a spatially globally pooled one.

In parallel, the DensePose [START_REF] Güler | Densepose: Dense human pose estimation in the wild[END_REF] network, denoted by h, predicts a dense mapping from the image pixels to the template mesh, in the form of a 3-channel image IUV:

IUV = h(I),
where IUV ∈ 0, 24 × [0, 1] × [0, 1] H×W . The first channel indicates which one of 24 pre-defined body parts the pixel belongs to, 0 being the background label. The second and third channels indicate the UV-coordinate of the pixel in a pre-defined UV-map of that body part's template mesh (See Fig. 2).

Given a vertex k in the template mesh, the corresponding pixel location c k ∈ 1, H × 1,W in the input image is obtained using the following thresholded nearest-neighbor strategy:

ĉk = arg min (i, j) ||IUV(i, j) -IUV k || 2 , c k = ĉk , if ||IUV( ĉk ) -IUV k || 2 ≤ δ k / 0, otherwise
where IUV k is the k th vertex IUV coordinates in the predefined template geometry body partitioning and per part UV-map (Fig. 2). Threshold δ k is defined as the k th vertex distance to its closest adjacent neighbor in the template UVmap space. This thresholding insures no pixels are assigned to occluded surface regions. We construct then the input local mesh-borne feature F M (k) for the k th vertex as follows: Finally, a GCN g, who's topology is defined by the template mesh connectivity, takes as input the mesh local features F M ∈ R N V ×(D+5) and predicts the final mesh vertex coordinates M ∈ R N V ×3 : M = g(F M ), N V being the total number of vertices. g uses the formulation from [START_REF] Kipf | Semi-supervised classification with graph convolutional networks[END_REF] and follows the architecture described in [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF], where regular graph convolutions are substituted by the semantic graph convolutions introduced in [START_REF] Zhao | Semantic graph convolutional networks for 3d human pose regression[END_REF]. We note that body joints J can be obtained from meshes using a fixed linear regressor W : J = W M.

F M (k) =
We train the parameters of the convolutional networks f and g following the losses, training scheme, data augmentation and 3D supervision introduced in [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF]. Our loss combines L1 reconstruction errors between the prediction and the ground-truth for mesh vertices L vertex , joints L joint , normals L normal and edges L edge with the following weighting scheme:

L = L vertex + L joint + 0.1L normal + 0.1L edge .
The normal and edge losses are used to ensure smoother and more visually pleasing results.

IV. RESULTS

We present in this section our experimental setup in addition to our results. We train our network on datasets Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] and MSCOCO [START_REF] Lin | Microsoft coco: Common objects in context[END_REF] following the data augmentation and 3D supervision described in [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF]. We use the Adam optimizer [START_REF] Kingma | A method for stochastic optimization[END_REF] in PyTorch on a Quadro RTX 5000 GPU for 12 epochs with a learning rate of 10 -4 , followed by another 2 epochs using a learning rate of 10 -5 and finally 1 extra epoch using a learning rate of 10 -6 . The image feature extraction network f is initialized with the ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] pretrained weights.

We evaluate our contribution numerically using the 3DPW [START_REF] Marcard | Recovering accurate 3d human pose in the wild using imus and a moving camera[END_REF] and Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] datasets. We report the MPJPE and PA-MPJPE metrics of our method in Table I when using only one global feature for all vertices (F 5 I ) Global (Ours), and also when using vertex specific local features (full F M ) Local (Ours). For a fair comparison with the competition, we show other methods ( [START_REF] Kanazawa | End-toend recovery of human shape and pose[END_REF], [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF], [START_REF] Kolotouros | Learning to reconstruct 3d human pose and shape via model-fitting in the loop[END_REF], [START_REF] Choi | Pose2mesh: Graph convolutional network for 3d human pose and mesh recovery from a 2d human pose[END_REF], [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF]) trained

Methods

Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] 3DPW [34] MPJPE PA-MPJPE MPJPE PA-MPJPE HMR [START_REF] Kanazawa | End-toend recovery of human shape and pose[END_REF] 153.2 85.5 300.4 137.2 GraphCMR [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF] 78.3 51.9 126.5 80.1 SPIN [START_REF] Kolotouros | Learning to reconstruct 3d human pose and shape via model-fitting in the loop[END_REF] 72.9 85.5 113.1 71.7 Pose2Mesh [START_REF] Choi | Pose2mesh: Graph convolutional network for 3d human pose and mesh recovery from a 2d human pose[END_REF] 67.9 49.9 91.4 60.1 I2L-MeshNet [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF] 55 3DPW [START_REF] Marcard | Recovering accurate 3d human pose in the wild using imus and a moving camera[END_REF]. ALL METHODS ARE TRAINED ON HUMAN3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] AND MSCOCO [START_REF] Lin | Microsoft coco: Common objects in context[END_REF].

on the same data as us and we relay their performances as they were reported in [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF].

We firstly showcase the effect of using local features on the convergence of our model in Fig. 3. For both of the PA-MPJPE and MPJPE metrics, our pixel-aligned local mesh features enable the model to reach significantly lower generalization errors on Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] at the same training epoch compared to our global feature baseline. Furthermore, we note that our Global baseline shares the same network design as GraphCMR [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF]. However, by substituting regular convolutions with learnable adjacency matrix ones [START_REF] Zhao | Semantic graph convolutional networks for 3d human pose regression[END_REF] and training with the same supervision and training scheme as in [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF], we manage to improve its performance by roughly 8mm in MPJPE for Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF], 14mm in MPJPE and 13mm in PA-MPJPE for 3DPW [START_REF] Marcard | Recovering accurate 3d human pose in the wild using imus and a moving camera[END_REF]. Our proposed Local method improves on our Global baseline substantially in almost all figures, by roughly 5mm in MPJPE and 8mm in PA-MPJPE for Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF], and 2mm in MPJPE for 3DPW [START_REF] Marcard | Recovering accurate 3d human pose in the wild using imus and a moving camera[END_REF]. It is noteworthy that our Local version also achieves competitive results in comparison to the state-ofthe-art, as it outperforms all methods presented in the table on Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] except for I2L-MeshNet [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF], while ranking 3 rd on 3DPW [START_REF] Marcard | Recovering accurate 3d human pose in the wild using imus and a moving camera[END_REF] closely behind I2L-MeshNet [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF] and Pose2Mesh [START_REF] Choi | Pose2mesh: Graph convolutional network for 3d human pose and mesh recovery from a 2d human pose[END_REF], which uses 2D joints (from HRNet [START_REF] Sun | Deep high-resolution representation learning for human pose estimation[END_REF]) as input rather than a RGB image. While [START_REF] Choi | Pose2mesh: Graph convolutional network for 3d human pose and mesh recovery from a 2d human pose[END_REF] is advantaged by the 2D pose input, we believe the performance of [START_REF] Moon | I2l-meshnet: Image-to-lixel prediction network for accurate 3d human pose and mesh estimation from a single rgb image[END_REF] is by virtue of their Lixel architecture which is not readily applicable to irregular graphs.

The numerical superiority of our contribution compared to our baseline is also confirmed with qualitative results. As shown in Fig. 4, we notice that using local image features in the GCN stage yields improved visual results, as witnessed by these examples from the 3DPW [START_REF] Marcard | Recovering accurate 3d human pose in the wild using imus and a moving camera[END_REF] and Human3.6M [START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] datasets. The red boxes in the figure illustrate in

Local Global

Local Global particular the better positioning of the body limbs with our Local method compared to our Global baseline. We note that following [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF], we show the results after an additional linear layer trained to predict SMPL instances from the previous output for smoother visual results.

V. CONCLUSION

We presented a method for 3D human shape and pose estimation from a single RGB image. The method is modelfree and relies on a GCN that starts from a template T-posed mesh and regresses the final vertex coordinates. Contrarily to seminal work [START_REF] Kolotouros | Convolutional mesh regression for single-image human shape reconstruction[END_REF], we propose to initialize the graph convolutions with pixel-aligned vertex-specific features instead of only one global feature. These features are extracted at multiple feature map stages of an image CNN, and mapped subsequently to the graph vertices using a pixel-tosurface correspondence map [START_REF] Güler | Densepose: Dense human pose estimation in the wild[END_REF]. Our results demonstrated the benefit of using local features in GCN based human 3D shape and pose estimation. Next, we will attempt to make the entire pipeline fully differentiable, by including the correspondence estimation network h training in the end-toend learning framework alongside the image feature network f and graph network g.

Fig. 1 .

 1 Fig. 1. Overview. Given an input color image, DensePose [11] (Bottom left) produces dense pixel-to-surface correspondences. Meanwhile, an image convolutional neural network (CNN) (Top) builds feature maps at multiple depths (Shades of red). The correspondences are then used to sample (Dashed lines) local image features from the CNN feature maps for each template surface vertex at its corresponding image location (Red Circle). Next, we use a graph convolutional network (GCN) (Right) to map the template surface with vertex specific local image features to the final posed surface.

Fig. 2 .

 2 Fig. 2. Illustration of the template geometry IUV mapping.

F 1 I

 1 (c k ), . . . , F 4 I (c k ), F 5 I , c k , x k , y k , z k where F l I ( / 0) = 0 for 1 ≤ l ≤ 4 so the non-visible vertices are assigned null local features. We note that the last feature F 5 I is a global one and hence does not depend on the spatial 2D sampling. (x k , y k , z k ) are the k th vertex coordinates in the T-pose initial mesh, hence F M (k) ∈ R D+5 .

Fig. 3 .

 3 Fig.3. MPJPE (left) and PA-MPJPE (right) testing losses on Human3.6M[START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] for our baseline using only a global image feature (Global) and our proposed approach using per vertex pixel-aligned local features (Local).

Fig. 4 .

 4 Fig. 4. Comparison of our baseline using only a global image feature (Global) and our proposed approach using per vertex pixel-aligned local features (Local) on the 3DPW [34] (left) and Human3.6M [13] (right) datasets.

TABLE I COMPARISON

 I OF MPJPE AND PA-MPJPE ON HUMAN3.6M[START_REF] Ionescu | Human3. 6m: Large scale datasets and predictive methods for 3d human sensing in natural environments[END_REF] AND

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F Dau

M.-L Dano

B Vérone

J Girardot

Z Aboura

J.-M Morvan

In-plane and out-of-plane characterization of a 3D angle interlock textile composite

The present work focuses on the characterization of the in-plane and out-of-plane behavior of a threedimensional angle-interlock woven composite. In-plane tensile and compression tests, short beam shear tests and through-the-thickness compressive tests were carried out to assess the mechanical behavior of the composite material. The results are presented in terms of stress-strain responses, elastic moduli, failure strains and strengths. In addition, the material elastic properties were measured using an ultrasonic method. Results are compared with the values of the elastic properties determined by mechanical testing. A good correlation is observed which demonstrates the pertinence of this technique to measure in particular the through-thickness elastic properties of the material.

Introduction

Composite materials are increasingly being used for primary aircraft components because of their superior structural performances such as high strength, high stiffness, long fatigue life, and lightweight. These structures are prone to impact loading during their service life. Conventional laminated composites have fibers oriented only in the plane of the laminate and therefore exhibit poor through-thickness mechanical performances. Three-dimensional (3D) woven composites in which through-the-thickness yarns are introduced offer significant advantages over conventional laminates. In particular, 3D composites do not exhibit delamination, prove better damage tolerance and superior impact performances [START_REF] Potluri | Influence of fiber architecture on impact damage tolerance in 3D woven composites[END_REF]. Therefore, interest for using 3D woven composites for aircraft applications is growing.

In general, 3D woven composites consist of warp yarns that run in the machine direction (1-direction), weft yarns that run perpendicular to the machine direction (2-direction) and binder yarns that go through the thickness (3-direction). 3D woven composites can be divided in four types depending on the angle of the binder yarns and how deep the binder yarns penetrate in the fabric as shown in Fig. 1. If the angle is equal to 90 • , the fabric is called orthogonal interlock (ORT) (Fig. 1a and Fig. 1b) whereas if the angle is less than 90 • , the fabric is referred to angle interlock (AI) (Fig. 1c and Fig. 1d). If the binder yarns go through the entire fabric thickness, it is classified as through-thickness interlock (TT) (Fig. 1a and Fig. 1c) whereas if the binder yarns interlace with only a few layers, it is called layer-to-layer interlock (LTL) (Fig. 1b and Fig. 1d). The type of interlock fabric affects directly the unit-cell size and degree of crimp.

Much work has been done to characterize the mechanical behavior of various types of 3D woven composites along the warp and weft directions in tension [START_REF] Potluri | Influence of fiber architecture on impact damage tolerance in 3D woven composites[END_REF][START_REF] Cox | Failure mechanisms of 3D woven composites in tension, compression, and bending[END_REF][START_REF] Pochiraju | Three-dimensionally woven and braided composites. II: an experimental characterization[END_REF][START_REF] Leong | The effect of binder path on the tensile properties and failure of multilayer woven CFRP composites[END_REF][START_REF] Tan | Behavior of 3D orthogonal woven CFRP composites. Part I. Experimental investigation[END_REF][START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF][START_REF] Ivanov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 2: comprehensive experimental results[END_REF][START_REF] Gerlach | In-plane and through-thickness properties, failure modes, damage and delamination in 3D woven carbon fibre composites subjected to impact loading[END_REF][START_REF] Dai | Influence of fibre architecture on the tensile, compressive and flexural behavior of 3D woven composites[END_REF][START_REF] Cox | On the tensile failure of 3D woven composites[END_REF][START_REF] Warren | Experimental investigation of threedimensional woven composites[END_REF][START_REF] Castaneda | Investigation of the Z-binder role in progressive damage of 3D woven composites[END_REF][START_REF] Saleh | Characterizing the loading direction sensitivity of 3D woven composites: Effect of z-binder architecture[END_REF][START_REF] Wang | Effect of fabric structures on the mechanical properties of 3-D textile composites[END_REF], compression [START_REF] Cox | Failure mechanisms of 3D woven composites in tension, compression, and bending[END_REF][START_REF] Pochiraju | Three-dimensionally woven and braided composites. II: an experimental characterization[END_REF][START_REF] Dai | Influence of fibre architecture on the tensile, compressive and flexural behavior of 3D woven composites[END_REF][START_REF] Warren | Experimental investigation of threedimensional woven composites[END_REF][START_REF] Wang | Effect of fabric structures on the mechanical properties of 3-D textile composites[END_REF] or bending [START_REF] Cox | Failure mechanisms of 3D woven composites in tension, compression, and bending[END_REF][START_REF] Pochiraju | Three-dimensionally woven and braided composites. II: an experimental characterization[END_REF][START_REF] Dai | Influence of fibre architecture on the tensile, compressive and flexural behavior of 3D woven composites[END_REF][START_REF] Wang | Effect of fabric structures on the mechanical properties of 3-D textile composites[END_REF]. A few studies have also been reported on the off-axis (45 • ) behavior of 3D woven composite [START_REF] Pochiraju | Three-dimensionally woven and braided composites. II: an experimental characterization[END_REF][START_REF] Lomov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 1: materials, methods and principal results[END_REF][START_REF] Ivanov | A comparative study of tensile properties of non-crimp 3D orthogonal weave and multi-layer plain weave E-glass composites. Part 2: comprehensive experimental results[END_REF][START_REF] Gerlach | In-plane and through-thickness properties, failure modes, damage and delamination in 3D woven carbon fibre composites subjected to impact loading[END_REF][START_REF] Warren | Experimental investigation of threedimensional woven composites[END_REF][START_REF] Saleh | Characterizing the loading direction sensitivity of 3D woven composites: Effect of z-binder architecture[END_REF]. However, experimental studies to characterize the through-thickness properties of 3D woven composites are very scarce [START_REF] Gerlach | In-plane and through-thickness properties, failure modes, damage and delamination in 3D woven carbon fibre composites subjected to impact loading[END_REF][START_REF] Wang | Effect of fabric structures on the mechanical properties of 3-D textile composites[END_REF][START_REF] Gras | Identification of the Outof-Plane Shear Modulus of a 3D Woven Composite[END_REF][START_REF] Fishpool | Interlaminar toughness characterisation of 3D woven carbon fibre composites[END_REF]. This is due to the lack of suitable and standardized test methods to investigate the throughthickness response of this kind of materials. 3D woven composites are usually thin. Increasing the thickness would require increasing the thickness of the preform, which would change the yarn architecture.

A comprehensive review of the available literature on the mechanical characterization of 3D woven composites has been published [START_REF] Saleh | Recent advancements in mechanical characterisation of 3D woven composites[END_REF]. It highlights very clearly the knowledge gaps still present in the literature. The paper mentions in particular that the behavior of ORT interlock composites has been the most studied especially under in-plane tension and compression. Through-thickness angle-interlock (TTAI) composites have received much less attention although they were shown to outperform ORT woven composites [START_REF] Dai | Influence of fibre architecture on the tensile, compressive and flexural behavior of 3D woven composites[END_REF]. Therefore, there is a need to characterize the mechanical behavior of TTAI architecture 3D composites subjected to warp, weft, 45 • directions in-plane loading as well as through-thickness loading.

From the literature, it appears that the determination of the elastic properties has been performed through destructive mechanical tests. Researchers usually used strain gages [START_REF] Tan | Behavior of 3D orthogonal woven CFRP composites. Part I. Experimental investigation[END_REF][START_REF] Gerlach | In-plane and through-thickness properties, failure modes, damage and delamination in 3D woven carbon fibre composites subjected to impact loading[END_REF][START_REF] Dai | Influence of fibre architecture on the tensile, compressive and flexural behavior of 3D woven composites[END_REF], extensometers or digital image correlation (DIC) to measure the strains [START_REF] Gerlach | In-plane and through-thickness properties, failure modes, damage and delamination in 3D woven carbon fibre composites subjected to impact loading[END_REF][START_REF] Dai | Influence of fibre architecture on the tensile, compressive and flexural behavior of 3D woven composites[END_REF][START_REF] Warren | Experimental investigation of threedimensional woven composites[END_REF][START_REF] Castaneda | Investigation of the Z-binder role in progressive damage of 3D woven composites[END_REF][START_REF] Saleh | Characterizing the loading direction sensitivity of 3D woven composites: Effect of z-binder architecture[END_REF]. Although ultrasonic (US) methods have been used to characterize conventional laminates [START_REF] Baste | Evaluation de la matrice d'élasticité des composites orthotropes par propagation ultrasonore en dehors des plans principaux de symétrie[END_REF][START_REF] Hosten | Stiffness matrix invariants to validate the characterization of composite materials with ultrasonic methods[END_REF][START_REF] Hood | Determination of elastic moduli in anisotropic media from ultrasonic contact measurements[END_REF][START_REF] Audoin | Ultrasonic Evaluation of Stiffness Tensor Changes and Associated Anisotropic Damage in a Ceramic Matrix Composite[END_REF], it has so far not been applied to 3D woven composites. Therefore, there is a need to investigate the use of this technique to evaluate the elastic properties of 3D composites.

This paper focuses on the static characterization of a 3D angleinterlock woven composite. The objectives of the present work are to (1) characterize the in-plane and out-of-plane mechanical behavior of a 3D through-thickness angle-interlock composite (TTAI); (2) provide reliable and extensive data on a 3D angle-interlock composite to be used in numerical models; (3) assess the accuracy of ultrasonic methods to evaluate the elastic properties of 3D composites.

Experimental investigations are conducted to determine in-plane and out-of-plane elastic and failure properties of a 3D angle-interlock composite. No standard test methods have been specifically developed for 3D textile composites. In many cases, the usual standard test methods for traditional high performance composites can be used if proper specimen geometries and gage sizes are selected [START_REF] Astm Standard | Standard Guide for Testing Fabric-Reinforced[END_REF]. Depending upon the size of the yarns and the pattern of the weave, the inhomogeneity within a 3D textile composite can be quite large. In the case of 3D angle-interlock composites, the representative unit cell can be several dozen millimeter long. This implies, if the recommendations given in [START_REF] Astm Standard | Standard Guide for Testing Fabric-Reinforced[END_REF] are followed, to use very large specimens. In this work, various specimen sizes will be considered for the tensile tests to assess scale factors effect.

After describing the 3D angle-interlock composite studied in this work, in-plane tensile and shear tests conducted to determine the elastic modulus in both weft and warp directions and the shear modulus are first presented. Then, the results of in-plane compressive tests in the warp and weft directions are discussed. Next, the 3D angle-interlock composite was subjected to through-the-thickness compression and short-beam shear (SBS) tests. Full-field surface strain distributions obtained using a digital image correlation system are presented and discussed. Finally, the determination of the through-thickness elastic properties of the 3D angle-interlock composite using an ultrasonic method is presented. The results are then compared and discussed. Finally, general conclusions are drawn and prospects for future research conclude this work.

Material description

The present study focuses on a 3D angle-interlock woven composite already investigated by the authors for low velocity impact loading [START_REF] Dau | Experimental investigations and variability considerations on 3D interlock textile composites used in low velocity soft impact loading[END_REF]. The fabric is composed of binding (or 3D) warp yarns, stuffer (or straight warp) yarns and filler (or weft) yarns as depicted in Fig. 2a. The ratio of 3D (binding) warp yarns to straight warp yarns is equal to 55%. The fabric has the same number of warp and weft yarns. The present fabric can be defined as a 3D warp interlock A-T 9-5 {5-4} according to the general definition of 3D warp interlock fabric referenced in [START_REF] Boussu | General definition of 3D warp interlock fabric architecture[END_REF]. Both the warp and the weft yarns were manufactured with carbon HR Tenax-E HTS40 F13 12 K 10Z yarns. The surface density is 2720 g/m 2 . The fiber volume fraction is about 58.5 %. Fig. 2b shows a micro-computertomography (micro-CT) of the dry 3D angle-interlock woven fabric before being injected with resin.

The 3D angle-interlock woven fabric is then processed by Resin Transfer Moulding (RTM) using a RTM6 epoxy resin. The resin is injected at 120 • C and the composite plate is cured at 160 • C. The thickness of the dry interlock fabric is 10 mm before injection whereas the final thickness of the plates is 2.8 mm. The useful dimensions of the plates issued from the RTM process are 500 mm × 500 mm. The microstructure of the composite material was studied by performing micrographs of the plate cross sections. Fig. 3a and Fig. 3b illustrate the cross section of the composite along the warp and the weft directions, respectively. A binder warp yarn, running through the entire thickness is clearly visible in Fig. 3a. Fig. 3b shows that the weft yarns do no remain straight but wave significantly in a repetitive manner. The representative unit cell (RUC) could be determined from the micrographs and is represented by the red rectangles in Fig. 3. One can estimate that the unit cell has a thickness of 2.8 mm, a length of about 50 mm in the warp direction, and a width of about 10 mm in the weft direction.

Weft yarns

Warp yarns Binder yarns Matrice Fig. 4 shows a close-up of the interlaced weft and warp yarns. One can see that the fiber distribution within a yarn is uneven which is caused by compaction during the RTM process. Some resin rich areas are clearly visible and can take the form of paths within the yarn. As a consequence, the mechanical properties will probably exhibit some variations.

Characterization of the in-plane properties

Tensile tests along the weft and warp directions and in-plane shear tests were performed to determine the tensile Young modulus in the weft and warp directions, the in-plane Poisson ratio and the in-plane shear modulus, respectively. Compressive tests were also performed to measure the compressive Young modulus in the weft and warp directions.

Tensile tests along the weft and warp directions 3.1.1. Device and measurements

A Zwick tensile machine equipped with self-locking grips was used. The crosshead speed of the machine was set to 2 mm/min. The crosshead displacement of the machine and the loading force were recorded. A laser extensometer was used to measure the longitudinal strain. Video cameras were added to allow post-processing using digital image correlation. The images frequency was 1 Hz.

Scale effect study

As mentioned in section 2, the representative unit cell of the 3D angle-interlock composite has a length of about 50 mm in the warp direction, and a width of about 10 mm in the weft direction. In the ASTM Standard Guide for Testing Fabric-Reinforced "Textile" Composite Materials [START_REF] Astm Standard | Standard Guide for Testing Fabric-Reinforced[END_REF]. it is recommended to use a specimen geometry that contains at least two unit cells along the width. This implies that the width of the specimen should be 100 mm for the tensile tests in the weft direction. Since it is inconvenient to use such large specimens, a scale effect study is conducted to evaluate if it would be possible to use smaller specimens without compromising the results.

To assess the scale effect, tensile tests have been performed in the weft direction on specimens with two different geometries as shown in Fig. 5. The first one called "narrow" is of thickness 2.8 mm, out-of-grip length 180 mm and width 30 mm. The second one referred as "wide" has the same thickness but has an out-of-grip length of 340 mm and a width of 78 mm. It was not possible to test a specimen with a 100 mm width due to width limitation of the grips.

As depicted in Fig. 5, the representative unit cell is repeated many times along the length for the two specimens (weft direction). However, along the width (warp direction), the narrow specimen contains slightly over a half unit cell and the wide specimen about one and a half unit cell.

For both wide and narrow specimens, strains were obtained by digital image correlation. For the narrow specimens, only one camera was used whereas a second camera was necessary to capture the strain all over the surface for the wide specimens.

Fig. 6 presents the stress-strain curves obtained for the narrow and wide specimens. For the wide specimen, there are two curves: one was plotted using the strain measured on the lower section of the specimen and the other the strain measured on the upper section. As can be observed, up to 100 MPa, the stress-strain curves obtained for the two sections are similar. Then, the curve for the upper section slightly deviates probably due to damage occurrence in the top part of the specimen. The stress-strain curve for the narrow specimen is very similar to the one obtained from the lower section of the wide specimen. Therefore, the narrow specimen does not seem to be strongly affected by edge effect as it could have been expected. Based on these results, narrow specimens will be used for the remaining tests as it allows material saving.

Digital image correlation was also used in this test to capture damage mechanisms. Fig. 7 shows (a) the strain field issued from image correlation post-processing and (b) the speckle applied on the specimen. The strain field exhibits high strain concentrations at a few locations. By superposing the strain field with the portion of the specimen where the speckle pattern has been removed, it is possible to identify that the strain concentrations appear where the binding warp tows intersect with the weft ones. They are located on the picture by the identical small areas depicted on the two pictures (a) and (b).

Test results

Five specimens were initially tested in both the warp and the weft directions to characterize the material behavior. Unfortunately, one of the tensile tests in the warp direction could not be exploited. Fig. 8 presents the stress-strain curves issued from these tests.

As shown in Fig. 8, the 3D angle-interlock woven composite exhibits a rather linear elastic behavior up to 0.6% strain for both the warp and the weft directions. Above 0.6 % strain, the behavior in the warp direction becomes slightly nonlinear up to final failure. This phenomenon has been previously observed for 3D orthogonal and angle-interlock composites [START_REF] Dai | Influence of fibre architecture on the tensile, compressive and flexural behavior of 3D woven composites[END_REF][START_REF] Cox | On the tensile failure of 3D woven composites[END_REF][START_REF] Warren | Experimental investigation of threedimensional woven composites[END_REF]. The non-linear behavior is due to damage occurring in the specimens such as matrix cracking. The nonlinearity is much more significant for specimens loaded in the weft direction. As can be observed in Fig. 8b, the stress-strain curve is first linear and then reaches a plateau as progressive failure occurs. These results are in agreement with results obtained in [START_REF] Warren | Experimental investigation of threedimensional woven composites[END_REF] where it was observed that the behavior of a 3D angle-interlock composite in the weft direction was nonlinear with softening damage. The observed differences in the behavior between the warp and the weft directions may be attributed to more important Fig. 5. Specimens geometry for the scale effect study. interactions between wavy yarns and the matrix when loaded in the weft direction.

Mean values of tensile modulus and strength determined from the experimental tests are presented in Table 1. The 1-, 2-and 3-directions are the warp, weft and through-the-thickness directions, respectively.

In-plane shear tests 3.2.1. Device and measurements

A MTS tensile machine equipped with self-locking grips was used. The crosshead speed of the machine was set to 4 mm/min. The crosshead displacement of the machine and the loading force were recorded. A digital image correlation system was again used to measure the strain field on the specimen surface. Five 300 mm-long and 24.5 mm-wide specimens were tested.

Test results

Fig. 9 presents the stress-strain curves obtained from the tensile tests performed in the 45 • -direction. As shown in the figure, the in-plane shear behavior of the 3D angle-interlock woven composite is nonlinear. The initial linear portion of the curve was used to determine the shear modulus. The strength was determined from the ultimate applied force. The mean values of the shear modulus and strength are presented in Table 1.

Compressive tests along weft and warp directions

Device and measurements

Compressive tests were performed using a combined loading compression fixture according to the ASTM standard D6641 [START_REF]Standard Test Method for Compressive Properties of Polymer Matrix Composite Materials Using a Combined Loading Compression (CLC) Test Fixture[END_REF]. The fixture was placed between two compressive platens attached to a MTS universal testing machine. The crosshead speed of the machine was set to 1.3 mm/min. The compressive tests were conducted on 140 mm-long by 12 mm-wide specimens. Back-to-back strain gages (CEA-06-125UW-350) were bonded on the specimen surface along the longitudinal direction as shown in Fig. 10. Scale effect was not studied due to lack of suitable testing device and material. A total of five tests were conducted in the warp and in the weft directions. The percentage of bending in the specimen was determined during testing using:

%bending = ⃒ ⃒ ⃒ ⃒ ε 1 -ε 2 ε 1 + ε 2 ⃒ ⃒ ⃒ ⃒ × 100 (1)
where ε 1 and ε 2 are the strains measured from the two strain gages. The test results were discarded if a bending percentage over 10% was obtained.

Test results

Fig. 11 presents the stress-strain curves obtained from the in-plane compressive tests on the 3D angle-interlock woven composite. The strain is the average of the two strain gages measurements. As can be observed, the material presents a rather brittle elastic behavior in compression in both the warp and weft directions with no sign of progressive damage.

Compressive modulus as well as failure stress and strain were determined from the experimental tests and are presented in Table 1. The results of Tests 1 and 5 for the weft direction and Test 5 for the warp direction were not considered in the average computation since the bending percentage was over 10%.

It is interesting to note that the compressive modulus is about 20% less than in tension for both directions. The compressive strength is smaller in the warp direction than in the weft direction, which is probably caused by the important waviness of the through-thickness yarns in this direction. Fig. 9. Stress-strain curve for the tensile test in the 45 • -direction.

Through-the-thickness compression tests

Device and measurements

Conducting the through-the-thickness compression tests on the 3D angle-interlock woven composite was challenging due to the small thickness of the plate (2.8 mm). To overcome this difficulty, the tests were performed on a stack of three 15 mm × 15 mm square samples as shown in Fig. 12. This configuration allows having enough space between the two compression platens to use a digital image correlation system and prevents any barrel effect to occur. Moreover, the samples were extracted from the initial plate at different locations as shown in Fig. 13. Samples G were cut on a same weft axis than samples D but for two different warp axes. Similarly, samples D and H are cut on the same warp axis but for different weft axis. Therefore, by using samples G i , D i and H i (i = 1 to 9) for the same compression test will allow studying material variability.

The through-the-thickness compression tests were conducted on a MTS universal test machine using compression platens and the compression fixture of the ASTM standard D6641 [START_REF]Standard Test Method for Compressive Properties of Polymer Matrix Composite Materials Using a Combined Loading Compression (CLC) Test Fixture[END_REF] as shown in Fig. 14a. The compressive fixture was diverted from its initial function and used as a convenient means to apply a uniform compressive force on the stack of samples. This set-up allowed determining the throughthickness elastic modulus of the three samples at the same time using DIC (Fig. 14b). The areas used to average the strains are depicted in Fig. 14c. The areas were selected to be sufficiently away from the sample edges and top and bottom surfaces to obtain a representative average strain.

The crosshead speed of the machine was set to 1.3 mm/min. A total of nine compression tests were conducted using stacked samples H i /D i / G i (i = 1 to 9). Only the elastic behavior was investigated and failure was not reached during testing.

Test results

Fig. 15 shows the results obtained for one test. After an initial nonlinearity for one of the sample probably due to a non-uniform load introduction, it appears that the 3D angle-interlock woven composite exhibits a linear elastic behavior through-the-thickness direction. Since the compression moduli showed negligible variability between samples H, D and G the compressive modulus was taken as the average of the values obtained for each sample. The through-thickness elastic modulus obtained from these tests is indicated in Table 1.

Characterization of transverse shear moduli

Since the 3D angle-interlock composite studied is thick, transverse shear effects may be significant, especially in the case of impact loading. Therefore, the determination of the transverse shear moduli G 13 and G 23 is of great interest. In this section, Short Beam Shear (SBS) tests are conducted to identify these properties.

SBS tests consist in subjecting a short composite specimen to threepoint bending until failure. The test method has been standardized [START_REF] Astm Standard | Standard Test Method for Short-Beam Strength of Polymer Matrix Composite Materials and Their Laminates[END_REF] for continuous or discontinuous fiber -reinforced composite materials. It was developed originally to determine the interlaminar shear strength. In this work, the test method is used in conjunction with a Tabs Strain gage no 2 Strain gage no 1 digital image correlation system to measure the transverse shear strains across the thickness. Therefore, transverse stress-strain curves can be obtained from which transverse shear moduli are determined.

Test methodology

Device and specimen geometry

Without standard test method for the present 3D angle-interlock woven composite, an existing fixture has been adapted for this study. The fixture configuration is presented in Fig. 16. The specimens were 15 mm wide (out-of-plane dimension in Fig. 16), 2.8 mm thick and the span was 20 mm. This choice follows [START_REF] Astm Standard | Standard Test Method for Short-Beam Strength of Polymer Matrix Composite Materials and Their Laminates[END_REF] in which a span-over-thickness ratio ranging between 4 and 10 is recommended to promote shear failure. The specimens were oriented along the warp direction.

Two specimens were tested. For each specimen, cycling loading has been performed until rupture. Table 2 indicates the load level for each cycle. Conducting such a cycling test allows studying the evolution of the transverse shear modulus. The loading speed was 0.2 mm/min. Pictures were taken every two seconds by the digital image correlation system.

Transverse shear stress and strain assessment

From classical beam theory, maximum transverse shear stress can be assessed using:

Table 2

Bending load cycles for the two specimens.

τ 13 = 3F 4bh ( 2
)
where F is the applied force on the loading nose, b is the specimen width and h the specimen thickness. An example of the results obtained for the shear strain measurements is shown in Fig. 17. The image corresponds to the right half of the specimen. As expected, the strain field displays concentration areas near the loading nose and support. To properly evaluate the transverse shear modulus, shear strain will have to be assessed in an area where it is the most constant. To study the overall strain field distribution beyond the supports, a first area (rectangular zone in Fig. 17) is selected for postprocessing. It corresponds to 60% of the entire thickness where the shear strain level is maximum and to the entire half specimen length (respectively H and L in Fig. 17). The analysis of the shear strain field evolution will allow identifying the area where the shear strain is uniform to subsequently reduce the area length for the final postprocessing.

Transverse shear strain evolution on the selected area

To study the evolution of the shear strain field along the length (L) during testing, the shear strain is averaged over the area height (H) for each image. Fig. 18a shows the shear strain field of Fig. 17 in the vertical position. By averaging the shear strain over the area height (H), a line is obtained for each image. The juxtaposition of the lines obtained for each image allows constructing Fig. 18b. Therefore Fig. 18b represents the resulting average along the area length (in pixels) as a function of image numbers. As can be observed, for each cycle, the shear strain increases, reaches a maximum value and decreases to zero. The transverse shear strain distribution seems to be constant over a region comprised between 550 and 1550 pixels. Based on these results, the retained area for averaging the shear strain field γ 13 is shown in Fig. 19.

Test results

Determination of transverse shear modulus G 13

Fig. 20 shows the transverse shear stress-strain curves obtained for the two specimens during the cycling bending loading. The curves do not exhibit too much hysteretic behavior except for the last cycle for which damage probably occurred. The transverse shear modulus is evaluated from the slope of the linear part of the curve during loading and unloading.

Evolution of transverse shear modulus G 13 during cycling

Tables 3 and4 present the values of G 13 extracted from the curves of Fig. 20. The values are obtained from both the loading and unloading phases starting at cycle 3 for specimen 1 and cycle 4 for specimen 2. The first cycles for each specimen were not considered because the linear part for these cycles was too small to obtain a reliable value for the slope.

From the tables, it can be noticed that:

-The shear modulus does not change significantly with the number of cycles. -The G 13 values extracted from the unloading phases are slightly greater than the ones obtained from the loading phases except for the very last cycle for which damage has occurred. -The G 13 values for specimen 2 are slightly smaller than for specimen 1.

-Globally, all G 13 values are very close.

For specimen 1, the values obtained for the loading and unloading phases of cycles 3, 4, 5 and 6 are averaged to 3.04 GPa. For specimen 2, the values obtained for the loading and unloading phases of cycles 4, 5, and 6 are averaged to 2.89 GPa. The average of these two values is reported in Table 1 as well as the average strength and strain to failure.

The same test method has been applied to determine G 23 . The final value is reported in Table 1. One can notice that the value for the transverse shear modulus and strength are close to the ones for the 1-3 plane.

Assessment of the elastic properties by ultrasonic method

Principle of measurement

An ultrasonic immersion method is used to identify the full set of elastic stiffness tensor C ij of the material with a single sample and experiment by performing an angular investigation [START_REF] Baste | Evaluation de la matrice d'élasticité des composites orthotropes par propagation ultrasonore en dehors des plans principaux de symétrie[END_REF][START_REF] Audoin | Ultrasonic Evaluation of Stiffness Tensor Changes and Associated Anisotropic Damage in a Ceramic Matrix Composite[END_REF]. Although for a different kind of materials, the main principles are explained in the international standard ISO 18610 [27]. By inverting the elastic stiffness tensor C ij , (in contracted notation) obtained by ultrasonic propagation, the elastic fourth order compliance tensor, S ij is obtained. The compliance tensor, S ij in contracted notation, then allows to calculate the more commonly used engineering values of elastic properties given by:

E 1 = 1/S 11 E 2 = 1/S 22 G 12 = 1/S 66 G 13 = 1/S 55 G 23 = 1/S 44 (3)
The main principles of ultrasonic evaluation have been explained by Roux [START_REF] Roux | Elastic Wave Propagation in Anisotropic Materials[END_REF] for the evaluation of the elastic coefficients of homogeneous anisotropic materials. Here after and following convention used in the ultrasonic method, 2 and 3 denote the in-plane directions while 1 indicates the through-the-thickness direction. In order to identify the nine elastic constants C ij which fully describe the elastic behavior of an orthotropic material, the wave propagation velocities are collected in the two accessible principal planes (1-2 and 1-3 planes in Fig. 21) and in a non-principal plane (1-45 • plane), described by the bisectrix of axes 2 and 3. The identification in the 1-2 plane gives four elastic constants C 11 , C 22 , C 66 and C 12 . Three other elastic constants, C 33 , C 55 and C 66 , are obtained in the 1-3 plane. The two remaining coefficients, C 44 and C 23 , are identified by propagation in the non-principal 1-45 • plane. However, when the material exhibits a tetragonal symmetry, the 1-45 • plane becomes a principal plane. It is thus impossible to measure independently the two stiffness coefficients. The value of the in-plane shear modulus C 44 is obtained from the phase velocity of a shear wave generated with a pair of contact transverse transducers. This value is used together with the 1-45 • plane data to simplify and to improve optimization [START_REF] Hosten | Stiffness matrix invariants to validate the characterization of composite materials with ultrasonic methods[END_REF]. The dimensions of the tested specimen were 2.8 mm × 98.5 mm × 95.1 mm in the 1-, 2-, and 3-directions, respectively. Its density was 1486 kg.m -3 .

Results

The stiffness coefficients obtained by the ultrasonic characterisation with their 90% confidence interval are given under a matrix form by: Fig. 17. Transverse shear strain field.

After the compliances calculation, the elastic properties are obtained according to Eq. ( 3) and are presented in Table 5. Note that the 1-, 2-, and 3-axes correspond to the thickness, warp and weft directions, respectively.

Discussion

Hereafter standard composite notations are used for discussion. The results obtained by the US technique are very close to the ones obtained from mechanical tests (Table 5). The values of E 3 and G 23 determined by US technique show very low scatter and very good correlation with the mechanical tests measurements. Indeed, the difference for E 3 and G 23 is around 1.8% and 0.3%, respectively. The value of G 13 shows also very low scatter and a close correlation with the mechanical test measurement. However, the value of G 13 measured by US technique is slightly higher than G 23 whereas it is the opposite in the mechanical test 3.3 ± 0.6 0 3.9 ± 0.5 Although other measurements would be necessary to consolidate the tendency, US technique turns out to provide very good estimations of the elastic properties of a 3D angle-interlock textile composite. The technique is particular relevant and convenient to evaluate throughthickness elastic properties that can be difficult to determine using classical mechanical tests due to the small thickness of composite materials. The determination of these through-thickness elastic properties combined with in-plane properties allows identifying the complete elastic stiffness matrix, which is crucial for modeling and evaluating the mechanical behaviour of a composite structure subjected to different loading conditions.

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (GPa) (4)
Moreover, the US technique offers the advantage of using only one specimen compared to several specimens of different shapes and sizes for classic mechanical testing and is a non-destructive test method. Such a technique is also suitable for future variability studies.

Conclusions and research perspectives

This paper presented the static characterization of a 3D through-thethickness angle-interlock (or TTAI) woven composite. The objectives were to characterize the full set of mechanical properties (in-plane and out-of-plane), to provide reliable and extensive data to be used in numerical models, and to assess the accuracy of ultrasonic technique to evaluate the elastic properties of 3D composites.

Experimental investigations were conducted to determine in-plane and out-of-plane elastic and failure properties of a 3D angle-interlock composite. In the absence of standard test for such an interlock composite material with a large representative unit cell, various specimen sizes have been considered at first for the tensile tests in the weft direction to assess scale factor effect. No significant effects have been revealed. It would be very interesting to extend this study and investigate scale factor effect for other loading modes. This was not done in this research work due to limited material availability and lack of large fixtures required for this kind of investigation.

Then, in-plane tensile, compression and shear tests have been conducted to determine the elastic modulus in both weft and warp directions and the shear modulus. Compression tests revealed that the compressive modulus was about 20% less than in tension for both warp and weft directions. Besides, the compressive strength was smaller in the warp direction than in the weft direction probably due to the important waviness of the through-thickness yarns in this direction.

Next, the 3D angle-interlock composite was subjected to an original through-the-thickness compression test using a stack of small material pieces taken from different places in a composite plate to take into account variability. DIC was used to assess the through-thickness compressive Young modulus. Then, short-beam shear tests have been developed to assess transverse shear moduli. DIC was used to evaluate the transverse shear strain.

Finally, the through-thickness elastic properties of the 3D angleinterlock composite have been determined using an ultrasonic method. The technique turned out to be relevant to easily obtain good estimates for the through-thickness elastic modulus and transverse shear moduli. The results globally correlate very well with the ones obtained from the mechanical tests. In addition, the ultrasonic method is able to determine through-thickness Poisson ratios. Therefore, US characterization allows identifying the complete elastic stiffness matrix, which is crucial to develop constitutive models aimed at structural design. In addition, it offers the advantages of using only a single specimen and avoiding destructive tests.

To complete this work, variability studies could be done by taking advantage of the US technique. Furthermore, additional tests should be carried out to investigate the damage behavior of such TTAI materials.

Fig. 1 .

 1 Fig. 1. Types of 3D woven composites: (a) Through-thickness orthogonal, (b) Layer-to-layer orthogonal, (c) Through-thickness interlock, (d) Layer-to-layer interlock.

  (a) Diagram of the cross section unit cell (b) Micro-CT scan

Fig. 2 .

 2 Fig. 2. 3D angle-interlock woven composite.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Micrographs of the 3D angle-interlock woven composite cross section.

Fig. 6 .

 6 Fig. 6. Scale effect study for the tensile behavior of the 3D angle-interlock woven composite in the weft direction.

Fig. 7 .

 7 Fig. 7. Correlation between the weaving pattern and the high strain concentration areas.

Fig. 8 .

 8 Fig. 8. In-plane tensile stress-strain curves.

Fig. 10 .Fig. 11 .Fig. 12 .

 101112 Fig. 10. Specimen configuration used for the in-plane compressive tests.

Fig. 13 .

 13 Fig. 13. Sample distribution across the composite plate.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Experimental set-up for the through-the-thickness compressive tests.

3 Fig. 16 .

 316 Fig. 16. Three-point bending test configuration.

Fig. 18 .Fig. 19 .Fig. 20 .

 181920 Fig. 18. (a) Shear strain field image in vertical position, (b) Shear strain field evolution along the length L.

Fig. 21 .

 21 Fig. 21. Specimen instrumented for ultrasonic characterization.

Table 1

 1 Mechanical properties of the 3D angle-interlock woven composite.

	Test	Direction	Elastic	Poisson	Failure	Failure
			modulus	ratio	stress	strain (%)
			(GPa)		(MPa)
	Tension	1	58.0	-	527	1.05
	Tension	2	60.0	0.0688	400	1.4
	In-plane	1-2	4.61	-	64.6	2.9
	shear
	Compression	1	44.7	-	397	0.913
	Compression	2	47.2	-	443	0.944
	Compression	3	11.2	-	-	-
	Transverse	1-3	2.97	-	40.5	1.4
	shear
	Transverse	2-3	3.29	-	39.7	1.3
	shear

Table 3

 3 Evolution of the transverse shear modulus G 13 for specimen 1. Further in-depth investigation would be necessary to explain this trend difference but it is beyond the scope of this study. Finally, it should be underlined that only the US technique is able to estimate values for Poisson ratios v 31 and v 32 .

	Cycle number	Loading phase	Unloading phase	Difference (%)
	3	3.10	3.13	0.967
	4	3.09	3.12	0.708
	5	3.06	3.06	0
	6	2.91	2.90	-0.436

Table 5

 5 Through-thickness elastic properties determined by US technique.

	US notation	Standard composite notation	Ultrasounds	Mechanical tests
	E 1 (GPa)	E 3 (GPa)	10.8 ± 0.5	11.2 ± 1.14
	v 13	v 32	0.11	-
	v 12	v 31	0.12	-
	G 13 (GPa)	G

[START_REF] Dau | Experimental investigations and variability considerations on 3D interlock textile composites used in low velocity soft impact loading[END_REF]

(GPa) 3.3 ± 0.6 3.29 ± 0.45 G 12 (GPa) G 13 (GPa) 3.9 ± 0.5 2.97 ± 0.11

Table 4

 4 Evolution of the transverse shear modulus G 13 for specimen 2.

	Cycle number	Loading phase	Unloading phase	Difference (%)
	4	2.91	2.94	1.03
	5	2.88	2.91	1.04
	6	2.78	2.93	5.39
	7	2.90	2.79	-3.79

Funding

This work was partially funded by the Natural Sciences and Engineering Research Council of Canada.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.