Philippe Borianne
email: philippe.borianne@cirad.fr

Frédéric Borne
email: frederic.borne@cirad.fr

Julien Sarron
email: julien.sarron@cirad.fr

Émile Faye
email: emile.faye@cirad.fr

Deep Mangoes: from fruit detection to cultivar identification in colour images of mango trees

Keywords: Faster R-CNN, mango fruit detection, mango cultivar identification, neural network

This paper presents results on the detection and identification mango fruits from colour images of trees. We evaluate the behaviour and the performances of the Faster R-CNN network to determine whether it is robust enough to "detect and classify" fruits under particularly heterogeneous conditions in terms of plant cultivars, plantation scheme, and visual information acquisition contexts. The network is trained to distinguish the 'Kent', 'Keitt', and 'Boucodiekhal' mango cultivars from 3,000 representative labelled fruit annotations. The validation set composed of about 7,000 annotations was then tested with a confidence threshold of 0.7 and a Non-Maximal-Suppression threshold of 0.25. With a F1-score of 0.90, the Faster R-CNN is well suitable to the simple fruit detection in tiles of 500x500 pixels. We then combine a multi-tiling approach with a Jaccard matrix to merge the different parts of objects detected several times, and thus report the detections made at the tile scale to the native 6,000x4,000 pixel size images. Nonetheless with a F1score of 0.56, the cultivar identification Faster R-CNN network presents some limitations for simultaneously detecting the mango fruits and identifying their respective cultivars. Despite the proven errors in fruit detection, the cultivar identification rates of the detected mango fruits are in the order of 80%. The ideal solution could combine a Mask R-CNN for the image pre-segmentation of trees and a double-stream Faster R-CNN for detecting the mango fruits and identifying their respective cultivar to provide predictions more relevant to users' expectations.

INTRODUCTION

The estimation of pre-harvest agricultural production is essential to meet development challenges and reduce the vulnerability of populations to global changes. Indeed, one of the main issues hindering the development of perennial crops is the impossibility of early, easily, and accurately estimating crop yields in order to guide farm management effectively. To date, yield estimation in tropical orchards is still based on a visual inspection of a limited sample of trees, a tedious, time-and cost-consuming method that depends on the reliability and accuracy of the observer [START_REF] Linker | Determination of the number of green apples in RGB images recorded in orchards[END_REF]. The cultivation of mango (Mangifera indica L.) in West Africa, and more particularly in Senegal, needs particular attention because of the physiological specificities of trees, especially the reproductive asynchronism and the inter-tree heterogeneity [START_REF] Dambreville | Deciphering structural and temporal interplays during the architectural development of mango trees[END_REF]. Moreover, West African farmers require tools adapted to the conditions of their family-sized and diversified cropping systems that can provide them with production information in a simple, inexpensive and convenient way.

Over the last few years, several studies described efficient machine vision systems for fruit detection and fruit yield estimation [START_REF] Gongal | Sensors and systems for fruit detection and localization: A review[END_REF]. These systems ranged from simple colour pixel segmentation to more advanced machine learning method using combination of colour, shape and texture features acquired by multiple sensors. Moreover, they were usually developed and evaluated in orchards under homogeneous conditions in terms of plant cultivars, plantation schemes (density, row, etc.), cultivation practices and visual information acquisition methods [START_REF] Payne | Estimating mango crop yield using image analysis using fruit at 'stone hardening' stage and night time imaging[END_REF][START_REF] Hung | A feature learning based approach for automated fruit yield estimation[END_REF][START_REF] Qureshi | Machine vision for counting fruit on mango tree canopies[END_REF]. Under heterogeneous conditions, the performance of machine learning approaches drops significantly [START_REF] Sarron | Is machine learning efficient for mango yield estimation when used under heterogeneous field conditions[END_REF].

One of the major advances in the object localization and detection was the multi-scale sliding window algorithm [START_REF] Sermanet | Overfeat: Inte-grated recognition, localization and detection using convolutional networks[END_REF][START_REF] Guo | Deep learning for visual understanding: A review[END_REF] for using Convolutional Neural Networks (CNNs). Regions with CNN features or R-CNN [START_REF] Girshick | Rich feature hierarchies for accurate object detection and semantic segmentation[END_REF] improved by almost 50% the detection performance by combining the object extraction using a Selective Search [START_REF] Uijlings | Selective search for object recognition[END_REF] and the region classification by SVMs [START_REF] Adankon | Support vector machine[END_REF]. This key step was published under the name Fast R-CNN [START_REF] Girshick | Fast r-cnn[END_REF]. As the R-CNN, it used Selective Search to extract possible objects, but instead of typing them using SVM classifiers, it applied the CNN on all the image and then used both Region of Interest (RoI) and Pooling on the feature map with a final feed forward network for classification and regression. The RoI Pooling layer and the fully connected layers allowed the model to be end-to-end differentiable and easier to train. Faster R-CNN [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF][START_REF] Chen | An implementation of faster RCNN with study for region sampling[END_REF] Different works used deep CNN for yield estimation [START_REF] Hou | Fruit recognition based on convolution neural network[END_REF] and/or fruit counting [START_REF] Rahnemoonfar | Deep count: fruit counting based on deep simulated learning[END_REF], and especially based on the Faster R-CNN [START_REF] Sa | Deepfruits: A fruit detection system using deep neural networks[END_REF][START_REF] Dore | Détection d'objets en milieu naturel: application à l'arboriculture[END_REF]. Despite the asynchronous production cycles of mango trees, this neural network obtained high performances for fruit detection and counting at plot scale in homogeneous field conditions [START_REF] Stein | Image based mango fruit detection, localisation and yield estimation using multiple view geometry[END_REF][START_REF] Bargoti | Deep fruit detection in orchards[END_REF].

To our knowledge, no work has been done to identify the stage of development of the fruits and their cultivar, which is assumed to be known. However, these two aspects are essential for estimating economic yields and forecasts of fruit farms, particularly on family farms that are increasingly using multiple grafts: several fruit cultivars are simultaneously grafted on the same tree. One of the major interests of this type of practice is to be able to diversify the production on a small area in order to make the farmer less dependent on the fluctuations of economic markets while preserving biodiversity and the health balance of the crops.

Besides mango detection in heterogeneous mango orchards, this study presents the results of fruits classification among mango cultivars. Indeed, visual images of mango trees often encompassed fruits carried by other nearby trees (adjacent or in background), and fruits too heavily occluded so that their status is difficult to assess. In order to provide farmers with the most accurate results of the number of fruits that will actually be produced per tree, we investigated the capacities of a Faster R-CNN network to classify the fruit condition.

The evaluation of the fruit detection and identification is usually done using the standard Mean Average Precision [START_REF] Powers | Evaluation: from precision, recall and Fmeasure to ROC, informedness, markedness and correlation[END_REF]. Among these indicators, we used the F1-score well suited to the statistical comparison with a real "image truth". Different notions were introduced, for example the global Jaccard indicator for qualifying the geometric fitting between expert annotations and network predictions or the multi-tiling for aggregating multiple detections.

The detection reporting from tiles to images is also described and discussed.

MODEL AND METHODS

2.1

The Faster R-CNN

Faster R-CNN started with 500x500 pixel pre-cut images called tiles and provided a list of labelled bounding boxes with prediction probabilities. Its architecture is based on three components (see Figure 1).

First, a Region Proposal Network (RPN) is used to find up to a predefined number of regions that may contain objects. Fixed sized reference bounding boxes are placed uniformly throughout the original image. They give a list of possible relevant objects and their locations in the original image.

Second, a Region of Interest Pooling (RoIP) is applied using the features extracted by the CNN in the previously given bounding boxes to extract those features which would correspond to the relevant objects. This step significantly accelerates the identification of the interest areas.

Third, the R-CNN module uses that information to classify the content in the bounding box (or discard it using the "background" label) and adjust the bounding box coordinates (so it better fits the object).

This method involves using a CNN pre-trained for the task of classification and then re-trained for the recognition of mangoes.

No real consensus exists on the best network architecture for the pre-training step, i.e. between different networks with a varying number of weights. Even if the ZF [START_REF] Zeiler | Visualizing and understanding convolutional networks[END_REF], MobileNet [START_REF] Howard | Mobilenets: Efficient convolutional neural networks for mobile vision applications[END_REF] and DenseNet [START_REF] Huang | Densely connected convolutional networks[END_REF] pre-training seem attractive, we used the VGG [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] pre-trained on ImageNet [START_REF] Deng | Imagenet: A large-scale hierarchical image database[END_REF] because it was dedicated to the large-scale image recognition. The network has been fine-tuned from a learning set of annotated tiles, where the annotations are bounding boxes surrounding visible (part of) mango fruits and labelled according to the defined classes.

Key network parameters

The confidence score. The network gives each box the probability of containing an object. The confidence-threshold is set to detect only boxes with high certainty.

The non-maximum suppression [28]

The Non-Maximum Suppression (NMS) pre-vents multiple detection of the same object. The NMS-threshold is set to define the non-significant intersections of boxes.

The iteration number

The main mechanism of networks, called gradient descent, is based on an iterative minimum search algorithm: the algorithm needs to be repeated to converge to the best possible solution.

The number of iterations (given as a train-ing parameter) indirectly defines the number of times the network observes the data.

Matching Function

Matching function consists in associating two by two the expert annotation and network prediction boxes. The optimum matching between the expert annotations Ai and the network predictions P j was given by the Jaccard matrix [START_REF] Bouchard | A proof for the positive definiteness of the Jaccard index matrix[END_REF]; the J i,j Jaccard index was defined as the ratio of areas between both the intersection and the reunion of two boxes (1). Matches were made in descending order of Jaccard indices as long as greater than 0.25. Once fitted, the boxes were removed from the list of possible candidates and the process was repeated until all the boxes have been matched or no box could be matched.

(

) 1
The algorithmic complexity of the matching was in O(IJ) where I and J respectively represented the maximum numbers of annotations and predictions per tile (which visibly does not exceed 20 objects). Image tiling, i.e. partitioning the image into tiles which can be processed by the neuronal network, is necessary for the detection of "small" objects in "large" images. Without tiling, small objects "disappear" during the geometric transformations applied for data formatting. If the recomposition of the image is simply achieved by juxtaposing the tiles, the aggregation of detected objects is potentially more complex, especially for objects "cut" into several parts by the tiling. When sufficiently significant i.e. quite close to the examples given in the training data set, these different parts are respectively detected, and the object is counted several times (see Figure 2). When not significant enough, no part of the object is detected and the object is not recorded. The multiple detections of the same object appearing on several tiles must be merged to adjust as well as possible the count and the shape of the boxes characterizing this last one.

Merging Function

We proposed an approach based on the following assumption: "The size of the objects to be detected is significantly smaller than those of the tiles used. Consequently, there should be at least one tile in which the object appears integrally". We used so different tiling, i.e. ways to partition the native image. For a given tiling, the objects validated as detected (or detections) were those whose bounding box was not incident at any edges of the tile that contained it. Objects not checking this condition were not retained: they were detected in another tiling, more exactly in a more appropriate tile containing them integrally. The problem was thus to merge detections coming from different tiling while taking care not to count more than once a given object.

The matching function previously described, based on the use of a Jaccard matrix, was used here to identify matches between objects detected in two different tiling. In case of matching, the smaller bounding box was removed. Objects that find no homologue correspond to the identification of "integral" objects in one of the tiling and "cut" objects in the other.

Merging Function

The statistical indicator

We used the F1-score (2) as test's accuracy between 'image truth' and network predictions; it is defined as the best compromise between precision (3) and recall (4), two statistical indicators taking into account respectively the numbers of "True Positive", "False Positive" and "False Negative" classes where  "True Positives" (TP) are mangoes annotated by the expert and correctly detected by the network,  "False Negatives" (FN) are mangoes annotated by the expert and not detected by the network,  "False Positives" (FP) are mangoes detected by the network but not considered as such by the expert.

(

(3) (4)

The geometric fitting

A global Jaccard index (5) was introduced to for quantifying the difference between the geometry of the A i expert annotations and the P i network predictions, i.e. the precision of the geometric fitting of the bounding boxes of mangoes.

(

) 5
An index of 1 will be the perfect superposition of annotations and prediction boxes.

RESULTS AND DISCUSSION

The main question is whether the network is sensitive enough to identifying the cultivar of the mango fruits it detects. We used 150 native colour images of 'Kent', 'Keitt', and 'Boucodiekhal' (Bdh) mango cultivars taken at a distance of 5 meters by a Sony Nex-7 RGB camera with a fixed focal length of 18 mm. The 4000x6000 pixel native images were cut in 500x500 pixel tiles which were manually annotated and labelled under ImageJ [START_REF] Schneider | NIH Image to ImageJ: 25 years of image analysis[END_REF] to form a representative set of about 10,000 annotations in terms of shape, colour, sunlight conditions or occlusions. The labelled annotations were rectangular boxes, from 10 to 80 pixels side, including as closely as possible the visible fruits on the images: labels specified mango cultivar. These annotated data have been distributed as follows: 3,000 heterogeneous annotations for network training, 7,000 for cross validation.

Parameter setting

Various experiments were conducted to determine the best settings, i.e. the ones that maximizes the performances of the network in detection mango fruit. The same set-tings were used for fruit cultivar identification.

The confidence and NMS levels

The fruit detection network was trained with an arbitrarily number of 50,000 iterations. The validation set was then tested by simultaneously varying the confidence and NMS thresholds from 0.35 to 0.9 and 0.005 to 0.5 respectively in steps of 0.05 (see Figure 3). The F1-score (2) ranged from 0.66 for the thresholdcouple (0.35, 0.5) to 0.90in fact 0.899687for (0.7, 0.25). Different settings led to relatively close performances. We chose the setting resulting in the highest performances, acknowledging that this result certainly depended on both the training and the validation sets used. Experimentation could have been further developed by studying the performance of the cultivar identification network to find the optimum threshold pair values, but we considered that the optimum parameter setting of the fruit detection network could be transposed to the cultivar identification network.

The number of iteration

The network was trained with numbers of iterations ranging from 500 to 30,000 in steps of 500. The validation set was then tested with a confidence threshold of 0.7 and a NMS threshold of 0.25 (see Figure 4).

Error and accuracy were indicative: they represented respectively the percentage of 'correct' and 'incorrect' predictions by the network. The functions characterizing the network behaviour were strongly correlated and were stabilized from 15,000 iterations.

The size of the training data set

The network was trained with an iteration number of 15,000 and a size of the training data set ranging from 200 to 5,000 by steps of 200. The validation set was then tested with a confidence threshold of 0.7 and a NMS threshold of 0.25 (see Figure 5).

The network performance reached its maximum after 2,650 training annotations: the accuracy of the geometric fitting did not exceed 77%, even when significantly increasing the size of the training data set. A training set of 3,000 annotations reinforced the stability of the detection process.

Localization and identification of fruits

In the orchards of Western Africa, fruit plots often mix different cultivars of fruit for a better use of soils and to spread the fruit production along the year. Estimating mango yield at the orchard scale requires assessing the production of each cultivar grown, i.e. the number of fruits produced (or to be produced) per cultivar and tree. For all the experiments below, the network was fine-tuned with 3,000 annotations and 15,000 iterations. The validation set was then tested with a confidence threshold of 0.7 and a NMS threshold of 0.25. The F1-score on the validation set was 0.90 for a J0 fitting precision of 0.77 (see Figure 4): the geometric offset between annotations and predictions could appear high, but it was a very satisfactory result considering the heterogeneity of the data set.

The mango detection at the tile scale

In Figure 6, the expert annotations (in red) were almost superimposed on the network predictions (in blue).

The in-depth study of False Negatives pointed out the network's difficulties in detecting partially occulted fruits, essentially in fruit clusters (see Figure 7.A) and " fruits inside canopy (see Figure 7.B). In the first case, the NMS threshold prevented 'correct' detection of background fruits partially masked by foreground fruits. In the second case, fruits were rather hidden by leaves and wood, causing a significant change in the signature of the mango.

On the False Positive side, most of the residual confusion concerned leaf arrangements with curvatures similar to those of mangoes (see Figure 7.C). Interestingly, the network detected fruits not annotated by the expert (see Figure 6.B and 7.D): in absolute terms, they were false False-Positives which penalized the network performance.

The mango detection at the full image scale

Four tiling were used to cover a maxi-mum of cutting configurations, without however being able to guarantee that they are all well considered: the 1st is a partition in tiles of 500x500 pixels starting from the image point coordinates (0,0), the second from the point (0,250), the third from the point (250,0) and the fourth from the point (250,250).

The F1-score was 0.71 for a J 0 fitting precision of 0.67. This drop in performance is explained by the significant increase in False Positives (see Figure 8) compared to the previous experiment: the expert limited his count to the foreground tree while the network processed all the trees in the image. Previously, only tiles in which the expert had observed a fruit were used, which partially excluded fruits from adjacent trees; here, all tiles in the images were used.

The algorithmic complexity of the detection merging is in O(n 2 ) where n represents the number of fruits visible in the image: it is fixed by the calculation of the Jaccard matrix. It can look high even if it only takes 30'' to process a native image on a Dell Studio XPS 8 x i7 3 GHz with a Nvidia Quadro M2000 graphics card.

Further study to determine the optimal tiles to use for the fusion of detected fruits would be of definite theoretical interest, even if the best way to reduce computation times would be to use adaptive tiles, i.e. tiles centred on "clipped" objects: but such an approach would require two instantiations of the neural network. The major problem is to reduce the exploration area of the network to the foreground tree. The YOLO network [START_REF] Redmon | You only look once: Unified, real-time object detection[END_REF] is currently under study to pre-zone the image and exclude background trees: to refine the results, it will probably be necessary to use (in addition or not) segmentation networks such as SegNet [START_REF] Badrinarayanan | Segnet: A deep convolutional encoder-decoder architecture for image segmentation[END_REF] or similar [START_REF] Garcia-Garcia | A review on deep learning techniques applied to semantic segmentation[END_REF]. For example, Mask-RCNN [START_REF] He | Mask r-cnn[END_REF] customizes the Faster-RCNN with an additional parallel branch that makes it powerful for both the bounding-box object detection and the object segmentation. For this experiment, the network was trained to identify the 'Kent', 'Keitt' and 'Bdh' cultivars composing the data set (see Figure 9). For the learning step, each cultivar class was described by about 1,000 labelled annotations.

Identification of mango cultivars

In identifying cultivars, the F1-score was to 0.56 for a J 0 fitting precision of 0.71. This significant drop in performances is explained by the high increase in False Negatives, i.e. mangoes not detected: barely 4,000 mangoes were detected by the network during the cross-validation while the data set contains nearly 7,000 expert annotations.

When the tested fruit looks alike fruits of at least two variety classes, the network assigns predictive probabilities (by class) to it that are quite close and finally lower than the confidence threshold used by the network for the detection phase. Consequently, the object was not considered as a fruit and is therefore not detected. (see Figure 10, blue rectangles). Lowering the confidence threshold may in some cases be an option, but it might significantly increase the number of False Positives. The best option would probably be to use a two-stream faster R-CNN [START_REF] Peng | Multi-region two-stream R-CNN for action detection[END_REF] to firstly detected all the fruits of the image and then qualify them in classes.

Figure 10 shown the behaviour of the identification network: mangoes correctly detected and identified by the cultivar identification network were represented by green rectangles, mangoes correctly detected but incorrectly identified by red rectangles, and mangoes not detected by the identification network although counted by a simple detection network by blue rectangles.

Errors in identification probably have several causes. Among them, the physiological features are not negligible. Since the expression of the really discriminating features of fruits only appears after the juvenile stages, there would therefore be a link between the cultivar identification certainty and the fruit development stages. It is therefore necessary to know / be able to identify the stage of development of the fruits to certify the detected fruit cultivars. This is all the more important in the case of grafted trees that carry several fruit cultivars. Moreover, angles of view or partial occlusions of fruits probably have a greater impact on the responses of an identification network than on those of a detection network.

Beyond this observation, the question was how many correctly detected mangoes were well identified (classified). Table 1 gave the identification percentages, class by class, of correct detected fruits (900 'Bdh' mangoes detected on 2,000 annotated, 1,200 'Keitt' on 2,500 and 1,600 'Kent' on 2,500). The identification rates without error; i.e. the detected fruits attributed to the right cultivar class, ranged from 80 to 90% (see bold values of the Table 1). The identification errors were caused by visual similarities in mangoes, especially when partially masked, viewed from irrelevant angles or at too early stages of development.

Table 1. Multi-class identification rate. Each column indicates how the neural network identifies the fruits of a given cultivar.

Some questions remain open: for example, why is there 3 times more confusion between the Keitt and Kent fruits than between Kent and Keitt fruits: a simple conjuncture or a deeper network learning problem? It is obvious that, unlike adult or mature fruits, young fruits have very similar aspects, perhaps too similar, so the network can unambiguously identified their respective cultivar. But if that were possible, the solution would surely based on a Ensemble Neuronal Network [START_REF] Kumar | An ensemble of fine-tuned convolutional neural networks for medical image classification[END_REF][START_REF] Zameer | Intelligent and robust prediction of short term wind power using genetic programming based ensemble of neural networks[END_REF] i.e. on the simultaneous use of different networks fine-tuned on a same dataset or on several identical networks trained on different data for significantly increasing the relevance of fruit cultivar identification unless this qualification is limited to sufficiently mature fruit.

CONCLUSION

We evaluated the behaviour of the Faster R-CNN network to determine whether it was robust enough to "detect and identify" fruits under particularly heterogeneous conditions in terms of tree cultivars, plantation scheme, and visual information acquisition contexts.

The network was trained using 3,000 representative labelled annotations of 'Kent', 'Keitt', and 'Boucodiekhal' mango cultivars.

The validation set composed of about 7,000 labelled annotations was tested with a confidence threshold of 0.7 and a Non-Maximal-Suppression threshold of 0.25.

The network accuracy (F1-score) was 90% for fruit detection, but fell to 56% for fruit cultivar identification. When the tested fruit looks alike fruits of at least two variety classes, the network assigns predictive probabilities (by class) to it that are quite close and finally lower than the confidence threshold used by the network for the detection phase. Consequently, the object was not considered as a fruit and is therefore not detected. The cultivar identification rates of the detected mango fruits were in the order of 80%. Errors were caused by visual similarities in mangoes, especially when partially masked, viewed from irrelevant angles or at too early stages of development.

Image tiling, i.e. partitioning the image into tiles which can be processed by the neuronal network, is necessary for the detection of "small" objects in "large" images. Tiling may lead to "artificially" clipped objects, which will result in multiple detection of these objects. We combined a multi-tiling approach with a Jaccard matrix to identify and merge multiple detections and thus report the detections made at the tile scale to the native image.

The cross validations showed the need to undertake additional reflexion to make the predictions of the R-CNN Faster network more relevant to users' expectations. Future works could focused on (i) the image pre-processing to separate neighbouring trees from each other (using convolutional segmentation networks such as SegNet or Mask R-CNN) and, (ii) the two-stream Faster R-CNN developing to identify the cultivar of all detected fruits on each segmented tree.

5.

Figure 1 .

 1 Figure 1. Complete Faster R-CNN architecture..

Figure 2 .

 2 Figure 2. Image tiling and object detection. A-The green tiling splits the object into 4 parts which are each detected by the network; in the blue tiling, the object is detected only once in its entirety. B-The blue tiling breaks 3 objects into 4 parts leading to 5 detections; in the green tiling, the 3 objects are correctly detected.

Figure 4 .

 4 Figure 4. Variation in the network performances according to the number of iterations.

Figure 6 .

 6 Figure 6. Detection of mangoes in tree tiles. Red boxes were expert annotated fruits, and blue boxes network detected fruits. The network detected visible mangoes well, regardless of light exposure or fruit growing stage. The pictures shown tiles of mango trees seen at 5 meters distance for 'Keitt' (A, B) and 'Kent' (C, D) cultivars.

Figure 3 .Figure 5 .

 35 Figure 3. Variation in the network performance according to the confidence and NMS thresholds..

Figure 7 .Figure 8 .

 78 Figure 7. Detection errors. False Negatives are red boxes, False Positives blue boxes. A-a background mango was not detected due to the application of the NMS threshold; B-mango partially hidden by foliage was not detected because this situation is not sufficiently represented in the training data set. C-a leaf was confused with a mango by the network. D-a mango inside the tree was detected by the network even though it had not been annotated by expert.

Figure 9 .

 9 Figure 9. Mango cultivars at the pre-harvest stage. A-'Bdh' mango fruit is ovoid-flat with a skin col-our ranging from red to green and an average weight of 450 grams. B-'Keitt' mango fruit is ovoid-oblong; skin color is pink with less than 30% red and the average weight is 500 to 600 grams. C-'Kent' mango fruit is rather ovoid-wide with yellow skin and red spot; the average weight does not exceed 550 grams.A -

Figure 10 .

 10 Figure 10. Identification errors of Bdh (A), Keitt (B) and Kent (C) cultivars.

ACKNOWLEDGMENTS

This work was supported by the French National Research Agency under the Invest-ments for the Future Program (ANR-16-CONV-0004) and by the PixYield Creativity and Scientific Innovation action funded by CIRAD, the French agricultural research and international cooperation organization. The authors wish to thank Dr. Sané C.A.B for its expert work and active participation in the annotation of visual data.

6.


Benjamin Gerin
email: benjamin.gerin@ensam.eu

Etienne Pessard

Franck Morel

Catherine Verdu

A non-local approach to model the combined effects of forging defects and shot-peening on the fatigue strength of a pearlitic steel

Keywords: Steel, Forging, Shot-peening, High Cycle Fatigue, Surface Defects, Non-local Approach

come

A non-local approach to model the combined effects of forging defects and shot-peening on the fatigue strength of a pearlitic steel

Introduction

The high cycle fatigue behaviour of industrial components can be affected by material characteristics but also by the production process. These elements must be taken into account during the design process.

Because crack initiation is usually located at the surface, a large number of articles in the literature focus on how surface integrity affects fatigue behaviour. The term "surface integrity" includes various aspects that can all affect fatigue behaviour:

• Surface roughness

• Surface defects • Residual stresses • Hardening • Microstructure
Surface integrity is very dependant on the process being used, so studies on this subject usually focus on a specific process: machining [START_REF] Shahzad | Influence of anodizing process on fatigue life of machined aluminium alloy[END_REF][START_REF] Chomienne | Étude de l'influence de l'intégrité de surface en tournage de l'acier 15-5PH sur la tenue en fatigue en flexion rotative[END_REF][START_REF] Souto-Lebel | Characterization and influence of defect size distribution induced by ball-end finishing milling on fatigue life[END_REF], stamping [START_REF] Shang | Research on the stamping residual stress of steel wheel disc and its effect on the fatigue life of wheel[END_REF], punching [START_REF] Dehmani | Experimental study of the impact of punching operations on the high cycle fatigue strength of Fe-Si thin sheets[END_REF]. A large number of papers also deal with the impact of localised natural or artificial defects on fatigue strength. A classic example is Murakami's work [START_REF] Murakami | Effects of hardness and crack geometries on ∆K th of small cracks emanating from small defects[END_REF][START_REF] Murakami | Metal fatigue: effects of small defects and non-metallics inclusion[END_REF]. More recently, studies have been performed on steel [START_REF] Billaudeau | Multiaxial fatigue limit for defective materials: mechanisms and experiments[END_REF][START_REF] Pessard | The effect of quenching and defects size on the hcf behaviour of boron steel[END_REF][START_REF] Guerchais | Micromechanical investigation of the influence of defects in high cycle fatigue[END_REF], on cast iron [START_REF] Chantier | Probabilistic approach to predict the very high-cycle fatigue behaviour of spheroidal graphite cast iron structures Fatigue and Fracture of Engineering Materials and Structures[END_REF][START_REF] Nadot | Influence of casting defects on the fatigue limit of nodular cast iron[END_REF][START_REF] Morel | Comparison between defects and micro-notches in multiaxial fatigue -The size effect and the gradient effect[END_REF], on cast aluminium alloys [START_REF] Koutiri | High cycle fatigue damage mechanisms in cast aluminium subject to complex loads[END_REF][START_REF] Morel | Simulation of the Kitagawa-Takahashi diagram using a probabilistic approach for cast Al-Si alloys under different multiaxial loads[END_REF][START_REF] Serrano-Munoz | Influence of surface and internal casting defects on the fatigue behaviour of A357-T6 cast aluminium alloy[END_REF] and cast titanium alloys [START_REF] Léopold | Influence of artificial and casting defects on fatigue strength of moulded components in Ti-6Al-4V alloy, Fatigue and Fracture of Engineering Materials and Structures[END_REF]. Most investigations on defects are performed on homogeneous materials, without residual stresses. However, in the case of hot-forged steel, shot-blasting is generally used to clean off the scale. Shot-blasting is akin to shot-peening and affects surface integrity by introducing residual stresses, hardening and surface damage, and changing the microstructure. Numerous articles have been published on how shot-peening affects surface integrity and fatigue behaviour. Gariépy et al. [START_REF] Gariépy | Experimental and numerical investigation of material heterogeneity in shot peened aluminium alloy AA2024-T351[END_REF] conducted a thorough characterisation of the surface of a shot-peened aluminium.

Bhuiyan et al. [START_REF] Bhuiyan | The influence of mechanical surface treatments on fatigue behavior of extruded az61 magnesium alloy[END_REF] investigated shot-peened magnesium alloy specimens, some of which were stress-relieved after peening. McKelvey & Fatemi [START_REF] Mckelvey | Surface finish effect on fatigue behavior of forged steel[END_REF] studied the effects of decarburisation and shot-cleaning of the surface of steel specimens. Sakamoto et al. [START_REF] Sakamoto | Effect of fibprocessed sharp flaw on fatigue limit of shot peened medium carbon steel[END_REF][START_REF] Sakamoto | Effect of surface flaw on fatigue strength of shot-peened medium-carbon steel[END_REF] studied the influence of surface cracks introduced by shot-peening on steel specimens Kim et al. [START_REF] Kim | Residual stress relaxation and low-and high-cycle fatigue behavior of shot-peened medium-carbon steel[END_REF] showed that in high cycle fatigue, surface residual stresses show little stress-relief during fatigue cycling.

In order to take into account the influence of the various aspects of the surface integrity, different approaches have been suggested. The majority of the studies on the effect of surface integrity on fatigue behaviour are performed on machined components. Machined surfaces typically have a periodic roughness profile. Arola & Ramulu [START_REF] Arola | An examination of the effects from surface texture on the strength of fiber reinforced plastics[END_REF] have therefore suggested a model based on the roughness values and on the estimation of the stress concentration factor suggested by Peterson [START_REF] Peterson | Stress Concentration Factors[END_REF]:

K f = 1 + q(K t -1) with K t = 1 + n R a ρ R y R z and q = 1 1 + γ/ρ ( 1
)
where ρ is the notch curvature radius and (n, γ) are material parameters. Extending this model, Suraratchai et al. [START_REF] Suraratchai | Modelling the influence of machined surface roughness on the fatigue life of aluminium alloy[END_REF] suggested estimating the stress concentration with 2D simulations of the roughness profiles. Using 3D simulations from 3D surface scans to estimate the stress distribution was first performed by Shahzad and then used on surface states from other processes like forging or punching [START_REF] Dehmani | Experimental study of the impact of punching operations on the high cycle fatigue strength of Fe-Si thin sheets[END_REF]. Guillemot and Souto Lebel suggest taking into account the stress concentration using a statistical model which considers micro-defects as pre-existing population of defects. Also worth mentioning is the original work of Le Pecheur [START_REF] Pécheur | Polycrystal modelling of fatigue: Pre-hardening and surface roughness effects on damage initiation for 304L stainless steel[END_REF] and Guilhem [START_REF] Guilhem | Effects of surface roughness on plastic strain localization in polycrystalline aggregates[END_REF] based on evaluating the surface effects using crystal plasticity. When studying defects, the Kitagawa-Takahashi diagram [START_REF] Kitagawa | Applicability of fracture mechanics to very small cracks or cracks in the early stage[END_REF] is a very useful tool, as it allows to illustrate the effect of defect size on the fatigue strength.

Fathallah et al. [START_REF] Fathallah | High cycle fatigue behavior prediction of shot-peened parts[END_REF] studied the effect on fatigue behaviour of shot-peening coverage (100% or 1000% surface coverage). To model the fatigue behaviour, they performed 2D finite element simulations, taking into account the specimen geometry, the surface defects, the surface damage, the residual stresses and the hardening introduced by shot-peening. To study the influence of surface defects, other authors have used non-local approaches based on the theory of critical distances defined by Taylor [START_REF] Taylor | Geometrical effects in fatigue : a unifying theoretical model[END_REF][START_REF] Taylor | The theory of critical distances: a new perspective in fracture mechanics[END_REF]. This method has the advantage of using the elastic stress distribution to predict fatigue strength. It has been used in a number of applications [START_REF] Susmel | The theory of critical distances: a review of its applications in fatigue[END_REF][START_REF] Taylor | Applications of the theory of critical distances in failure analysis[END_REF], such as welded joints [START_REF] Taylor | Some new methods for predicting fatigue in welded joints[END_REF][START_REF] Crupi | Fatigue assessment of welded joints using critical distance and other methods[END_REF], fretting [START_REF] Araújo | A multiaxial stress-based critical distance methodology to estimate fretting fatigue life[END_REF], cast alumium [START_REF] Zheng | Numerical modeling of fatigue crack propagation based on the theory of critical distances[END_REF], punching defects [START_REF] Dehmani | Experimental study of the impact of punching operations on the high cycle fatigue strength of Fe-Si thin sheets[END_REF], corrosion defects [START_REF] May | Non-local high cycle fatigue strength criterion for metallic materials with corrosion defects[END_REF] and vapour deposition defects [START_REF] Herasymchuk | Fatigue life calculation for titanium alloys considering the influence of microstructure and manufacturing defects[END_REF]. This approach has been combined with a multiaxial fatigue criterion and applied to shot-peened specimens [START_REF] Benedetti | On the combination of the critical distance theory with a multiaxial fatigue criterion for predicting the fatigue strength of notched and plain shot-peened parts[END_REF]. It has also been used to develop a non-local damage model for shot-peened turbine blades [START_REF] Morançais | Effet du grenaillage sur la durée de vie des aubes monocristallines de turbine[END_REF].

The component studied in this paper is a hot-forged connecting rod, which is shot-blasted after forging. The connecting rods therefore exhibit localised surface defects introduced during forging in addition to hardening and residual stress gradients generated by the shot-blasting. The goal of this paper is to develop a fatigue approach which can account for the specific surface integrity of hot-forged and shot-peened components. This work is based on the results of a large experimental campaign, detailed in a previous article [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF]. The various studied surface states and the fatigue results will first be quickly summed up. These results will then be used to model the fatigue behaviour of the specimens. Two approaches will be detailed in this article. The first uses ellipsoids to fit the shape and size of the forging defects present on the fatigue specimens. The defect dimensions are then used to try and detect the critical defect and its fatigue strength. In addition, the statistics of extremes are used to predict defect sizes on larger sample sizes. The second approach is based on finite element (FE) simulations using the geometry of real defects.

The theory of critical distances combined with the residual stress profiles is used to predict the specimen's fatigue strength.

Fatigue testing results

In the previous study, fatigue specimens were machined out of connecting rods (figure 1a).

Several different surface states were analysed in order to quantify the influence of the various aspects of surface integrity. The various surface states used in this study are as follows:

• As-forged surface, manually cleaned of scale

• Shot-blasted, the surface state of industrially produced connecting rods

• Shot-peened with shot diameter 800 µm, Almen intensity 30-60 A with 200% coverage.

This shot-peening was performed on as-forged specimens and was chosen so as to be as close as possible to the shot-blasted surface.

• Shot-peening with shot diameter 400 µm, Almen intensity 20-30 A with 200% coverage.

This shot-peening was performed on as-forged specimens and was chosen so as to have a lower roughness and a shallower residual stress profile.

• Polished, used as the reference in fatigue The surface of the specimens was scanned using a profilometer prior to fatigue testing. This allowed the identification of the critical defect (the defect where crack initiation occurred) for each specimen. Figure 1b shows the crack path and the critical defect for an as-forged specimen.

In addition to the surface topography, the hardness, residual stresses and microstructure gradients were analysed. Figure 2 shows the residual stress profiles of all batches. Additional information on the preparation and the analysis of the specimens can be found in the article [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF].

Fatigue test were performed in plane bending with a load ratio R = -1. Some specimens were stress-relieved in tension prior to fatigue testing, in order to quantify the influence of the residual stresses. Fatigue tests were also performed in tension R = -1. However, not all specimens were valid as crack initiation sometimes occurred in the corner. Only one of the tension tests was valid, performed on a stress-relieved shot-peened ∅800 µm shot, all the others had a corner crack initiation.

Fig. 2: Residual stress profiles for the various batches, in the direction of the specimen axis (long.), and the transverse direction (tvs.). [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF] Table 1 1: Overview of the number of valid tests and fatigue strength range for each batch. [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF] reference fatigue strength of the material: σ D 0 = 424 MPa. Crack initiation for valid specimens was always located on a large forging defect. The fatigue results for all specimens are represented in a Kitagawa diagram (figure 3). Defect size is represented using the square root of their surface area, projected along the loading direction [START_REF] Murakami | Effects of hardness and crack geometries on ∆K th of small cracks emanating from small defects[END_REF].

The fracture surfaces can be used to calculate the projected surface area of each critical defect, [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF] which is generally used to represent defect size. All the defects have a projected area of similar shape, with the width much bigger than the depth (defects are typically 500-2000 µm wide and 50-200 µm deep). Calculating the projected area is done using the SEM fracture surface images: the total width and maximum depth of the defects are measured and their projected surface area is calculated by supposing that they have a semi-elliptical shape. As suggested by Murakami, the defect width is limited by a threshold of 10 times the depth when calculating the projected area.

The fatigue results show that:

• Forging defects play a significant role in fatigue. Crack initiation was always located on a forging defect. The as-forged specimens all have a lower fatigue strength than the reference fatigue strength because of the defects. Furthermore, all batches are influenced by defect size: larger defects lead to lower fatigue strength.

• Shot-blasting and shot-peening improve fatigue strength significantly. Shot-blasting and shot-peening increase the fatigue strength by approximately 150 MPa compared to the as-forged specimens (for similar defect sizes). This increases the fatigue strength much higher than the reference, despite the influence of defects. The two shot-peened batches do not have quite the same fatigue strength, with the ∅400 µm shot being around 50 MPa lower.

• Residual stresses have a major influence. The stress-relieved specimen shows a significant drop (70 MPa) in fatigue strength compared to the other specimens of the same batch. This drop is only due to the absence of the residual stresses in the loading direction, as no other aspect of the surface integrity was changed.

The fatigue results show that the forging defects and the residual stresses are the two features of the surface integrity that have the most influence on fatigue behaviour.

The experimental part of the study has already been performed and published [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF]. This article will therefore focus on establishing a fatigue model taking into account influence of the surface integrity.

Geometrical analysis of the surface defects

The specimen's surface contains numerous forging defects, all with a unique shape and size. These defects do not all have the same influence in fatigue and it is therefore of interest to be able to determine the most harmful defect (the critical defect) a priori. The fatigue results show that the critical defect is not necessarily the deepest or the largest defect. Furthermore, the forging defects usually form complex networks in which individual defect are not easily isolated.

In order to simplify the analysis of the defects, their geometry will be approximated using an ellipsoid. The ellipsoid shape is chosen because most defects are much larger than they are deep and are very smooth, with the deepest point near the center of the defect. Each defect is therefore approximated by a half-ellipsoid defined by three parameters (a, b, c), the semi-axes of the ellipsoid, which represent the length, width, and depth of the defect. The dimensions of the ellipsoid are then used to determine the defect's influence in fatigue. To simplify the fitting of the ellipsoid, the center of the ellipsoid and its semi-axes a and b are locked parallel to the surface.

The semi-axis c is therefore parallel to the z axis and the orientation of the ellipsoid is defined by a single angle. In order for the ellipsoid to correctly represent the depth of the defect, the value of the semi-axis c is set to the maximum depth of the defect. Each defect can thus be represented by an ellipsoid using only four parameters: the three semi-axes (a, b, c) and θ, the angle between the major semi-axis a and the loading direction x (figure 4). To limit the total number of defects per Fig. 4: Diagram of a surface with a defect (green) and the corresponding ellipsoid (blue). The grid is the reference plane. specimen, only defects deeper than 15 µm are analysed. Using this method, approximately 100 defects were detected per specimen, on average.

In order for the ellipsoid to reflect the defect geometry as accurately as possible, the ellipsoids are fitted using the least squares method. A second ellipsoid is fitted on an area restricted to a 100 µm radius around the defect's deepest point. This second ellipsoid is mostly useful in the case of larger defects, for which the general shape does not necessarily reflect the geometry at the bottom of the defect. Values for the radius ranging from 50 to 200 µm were tested without any noticeable improvement of the results. The intermediate value of 100 µm was therefore chosen. Figure 5 shows a defect from an as-forged surface with the two fitted ellipsoids.

The method used to detect the defects supposes that the crack initiation point is located in the deepest part of the defects. However, for two of the 34 as-forged specimens, crack initiation was located in a shallow area connected to a larger and deeper defect. For these two specimens (n°13 and 21), the algorithm was not able to correctly fit the critical area as it ignored the shallow crack initiation area in favour of the deeper, non-critical part of the defect.

Detecting the critical defect

This method analyses both critical and non-critical defects, which means it is possible to compare them in order to detect the critical defects on the specimens' surface. The classical criterion used to compare defects is the projected area, as used in Murakami's model. An ellipsoid's maximum cross-section contains its center and has the shape of an ellipse, making it easy to calculate its projected area. The projected area is calculated using the total length and the depth of the large ellipsoid. As before, a length threshold of 10 times the depth is applied when determining the projected area. Figure 6 compares, for each as-forged specimen, the projected area of the critical defect with that of every other defect. According to Murakami's model, defects with the greatest projected area are the most influential in fatigue. The critical defects of the as-forged specimens have mostly high projected areas compared to the other defects. However, the critical defect has the highest area for only six of the 32 as-forged specimens. The same results are observed for the shot-peened specimens. This confirms prior observations on as-forged specimens where the critical defect is smaller in size to other defects [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF]. This means that the square root of the projected area cannot be used to accurately detect the critical defect.

Peterson's criterion [START_REF] Peterson | Stress Concentration Factors[END_REF] gives a defect's stress concentration factor, K t , using the defect's depth c and its curvature radius ρ in the loading direction:

K t = 1 + 2 c ρ (2)
An ellipse's curvature radius is easily calculated. In the case of the ellipse along the loading direction:

ρ x = (semi-axis x ) 2 c (3)
The curvature radius at the bottom of the defects can thus be calculated using the semi-axes of the small ellipsoid.

Figure 7 compares the K t of all defects for every specimen. For the as-forged and shot-peened batches, the critical defect has the highest K t of all defects for roughly a third of all specimens. In the case of the shot-blasted specimens, the critical defects' large size make them easily identified (this batch has large uniformly large defects because of choices made during fatigue testing [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF]).

The K t is therefore a better criterion for detecting the critical defects than the projected area, but is not able to detect all critical defects. There are several possible explanations for this:

• The K t is calculated using the curvature radius of the small ellipsoid fitted at the bottom of the defects. This means that most of the defect geometry is ignored in favour of the deepest part.

• The 100 µm radius value might not be the best choice, however, tests with different values have shown no improvement of the results.

• Defects can be close together and thus interact with each other. Each defect is analysed independently and thus potential interactions are ignored.

In order to predict the fatigue limit, Peterson gives an empirical relationship using the K t to calculate a K f (defined as

K f = σ D 0 /σ D defect ): K f = 1 + q(K t -1) with q = 1 1 + γ/ρ (4)
γ is a material parameter. γ can be fitted with the least squares method, using the experimental results. The value thus obtained is γ = 1.01 (with ρ in mm).

When compared to the fatigue results, the K f values of the critical defects of the as-forged specimens have an average error of 8% with a maximum error of 19%. However, when comparing all defects together in order to detect the critical defects, the K f is a less effective criterion than K t .

As-forged

Predicting the maximum defect size on a larger sample

The present study is performed on less than a hundred specimens when the industrial production of forged components can be in the millions. The number of analysed critical defects is too small to be a representative sample. Furthermore, the critical defects have the most impact in fatigue and are therefore in the tail of the defect distribution. The statistics of extremes, defined by Gumbel [START_REF] Gumbel | Statistics of Extremes[END_REF], are used to study statistically rare events. This approach has been used by other authors to study defects in fatigue, for example Murakami [START_REF] Murakami | Quantitative evaluation of effects of non-metallic inclusions on fatigue strength of high strength steels. ii : Fatigue limit evaluation based on statistics for extreme values of inclusion size[END_REF]. The method is applied with the following steps:

• Define a reference area S 0 and determine the studied event's distribution by identifying the most extreme event on N samples of size S 0

• Verify that the distribution of the studied events has indeed the same form as Gumbel's distribution • If it is the case, determine the (λ, δ) parameters of the distribution • Use the return period to predict the extreme events on a larger sample Gumbel's cumulative distribution function is:

F(A max ) = exp exp -(A max -λ) δ (5)
where A max is the studied event (for example, the critical defect's projected area or its K t ), and (λ, δ) are the parameters for the distribution. In order to compare the distributions, the standardised variable y is introduced:

y = A max -λ δ (6)
The cumulative distribution function can then be expressed as F(y) = exp(exp(-y)) and the standardised variable as y =ln(-ln(F(y))). The linear relationship between y and A max in equation ( 6) means that Gumbel's distribution is a line in a y versus A max plot.

Each event for the N observations are sorted and numbered in increasing order: A max, j with j from 1 to N. For each numbered event, the corresponding cumulative probability is:

F j = j N + 1 (7)
the corresponding standardised variable is y j =ln(-ln(F j )). If the extreme events follow Gumbel's distribution, then plotting the (A max, j , y j ) points will show a linear relationship. Using equation [START_REF] Murakami | Effects of hardness and crack geometries on ∆K th of small cracks emanating from small defects[END_REF], it is therefore possible to determine the distribution's (λ, δ) parameters. The distribution can then be used to predict extreme events on larger sample sizes.

To estimate the extreme event on a surface S > S 0 , Gumbel uses the concept of return period, defined as T = S /S 0 , the associated cumulative probability is:

F S = T -1 T (8)
The most extreme event on the surface S is then estimated with:

A max,S = λ + δ y S (9)
where y S =ln(-ln(F S )).

In our case, the samples are the fatigue specimens, the reference area S 0 is therefore the surface area of the central part of the specimens (twice 20 × 6.5 mm, top and bottom). The extreme event is the critical defect, there is only one per specimen and it is determined by the fatigue tests. The defects can be quantified using different parameters (depth, projected area, K t , etc.), however not all parameters follow Gumbel's distribution. Figure 8 shows the standardised variable plots for the K t and the depth of the critical defects of the as-forged specimens. The K t follows Gumbel's distribution, but the depth does not. The parameters for the K t distribution can be calculated with equation ( 6), and the maximum K t on a larger sample can be predicted with equation ( 9). The reference surface being one specimen, the return period T can be considered to be the total number of specimens in the larger sample. The predicted maximum K t for 10 6 is: K t,10 6 = 6.1.

This method can be used to predict defect size on entire industrial productions, without having to individually check every component.

Comparing the K t to the projected area, not only is the K t a better criterion for detecting the critical defect, it also follows Gumbel's distribution. In an industrial context, where fatigue tests cannot be performed on all components, the K t is therefore the best criterion for the analysis in fatigue of forging defects.

Finite element simulations of the critical defects

Fitting the defects with ellipsoids allows to model the effect of their geometry in fatigue but ignores other aspects of the surface integrity, specifically those of shot-peening. The fatigue tests have shown that the two most important aspects of surface integrity are the defects and the residual stresses introduced during shot-peening and shot-blasting. Finite element simulations allow to more accurately model their influence on fatigue behaviour:

• The geometry of the defect is directly used in the FE simulation, with no simplification or approximation.

• Interactions between defects can be taken into account if the simulated area is large enough.

• Integrating the residual stress profile into the FE simulations allows the accurate analysis of its influence.

FE modelling of the forging defects

The simulations are performed using the surface scans of the defects. The surfaces of the specimens are too large to be simulated in their entirety, therefore the defects are simulated independently. Because of the large number of defects, the simulations are restricted to only the critical defects. If large defects are close to the critical defect, these are included in the simulation in order to take into account eventual interactions.

The scans are simulated with a thickness of 2 mm (the specimens have a total thickness of around 6 mm). The elements are linear tetrahedra, meshing is performed using the GMSH software [START_REF] Geuzaine | Gmsh : a threedimensional finite element mesh generator with built-in preand post-processing facilities[END_REF], with a element size gradient in order to reduce the number of elements. Mesh size at the surface (top) is 7 µm, gradually increasing to a size of 1 mm at the bottom (figure 9). With these meshing parameters, a 2 × 2 mm scanned surface results in a mesh with approximately 400 000 elements.

Solving was performed using a software developed at LAMPA. More information about the solver can be found in the article by Robert & Mareau [START_REF] Robert | A comparison between different numerical methods for the modeling of polycrystalline materials with an elastic-viscoplastic behavior[END_REF].

The loading conditions are given in figure 10.

Previously performed analysis of the residual stresses during fatigue cycling have shown that there is no residual stress relaxation, even in the forging defects [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF]. This indicates that there is very little plastic strain at the surface, an elastic behaviour is thus used for the simulations. The cyclic fatigue loading is simulated using a single step of equal load.

Figure 11 shows the various steps used to perform FE simulations of the critical defects.

Figure 12 compares the surface scan of a critical defect with the surface stress values obtained through FE simulation. The simulation correctly identifies the critical defect and, in this particular case, the K t values calculated with the ellipsoids are very close to the results of the FE simulation. Loading direction (x) ----Surface scan FE simulation Fig. 12: FE simulation of a critical defect compared to its surface scan. Indicated on the scan are the defects' K t value, calculated using the geometrical analysis.

FE fatigue model

The surface K t values calculated through the FE simulations cannot be directly used to predict fatigue strength. Furthermore, due to notch effect, these values are heavily dependent on the surface mesh size. Another method is therefore required to accurately predict fatigue strength.

In order to model the shot-peened and shot-blasted specimens, the residual stresses must be taken into account. The Dang Van criterion [48] can be used to integrate the residual stresses in the stress value obtained with the point or volume methods. In the case of a proportional uniaxial loading, the criterion is written as:

τ a + α σ H,max β (10)
where τ a is the shear stress amplitude, σ H,max is the maximum hydrostatic stress and (α, β) are material parameters. The residual stresses are constant during cycling and can therefore be integrated into the hydrostatic stress. The defect geometry will locally affect both τ a and σ H,max , proportionately to the loading stress. In this way, the model takes into account both the defect's geometry and the residual stresses. In the case of the polished specimens, there are no defects and no residual stresses, the local stress tensor is therefore equal to the loading stress: σ D 0 , and equation ( 10) can be written as:

σ D 0 2 + α σ D 0 3 = β ⇒ σ D 0 = β 1/2 + α/3 (11)
For the shot-peened and shot-blasted specimens, the residual stresses along the vertical axis are considered negligible, the residual stress tensor is therefore:

σ RS =                  σ RS 11 0 0 0 σ RS 22 0 0 0 0                  (12)
The residual stress analysis has shown that they are equi-biaxial, as shot-peening and shot-blasting do not favour a particular direction.

Different criteria were tested and the chosen model is based on the theory of critical distance [START_REF] Taylor | Geometrical effects in fatigue : a unifying theoretical model[END_REF]. Instead of using the surface stress, this model uses the stress a depth d below the defect and compares it directly to the reference fatigue stress σ D 0 . The critical distance d is a material parameter. In the case of an elastic simulation, this means that a K f value can be directly determined by comparing the stress at the critical distance with the loading stress. Several methods are available for calculating the stress at the critical distance. The simplest is the point method, where the stress is simply the stress value at a single point located at the distance d/2

beneath the tip of the defect. The volume method requires averaging the stress values in a sphere of radius d centred on the tip of the defect. Figure 13 shows an example of both methods being applied to a critical defect. As the point and volume methods give very similar results, only point method results will be presented in this article. For the point method, the contribution of the residual stresses is obtained simply by using the residual stress values at the depth d/2 = 40 µm. These values are:

• Shot-blasting: 500 MPa

• Shot-peening ∅800 µm shot: 525 MPa

• Shot-peening ∅400 µm shot: 445 MPa

Determining the critical distance and the (α, β) parameters

The critical distance is a material parameter, and therefore should change during shot-peening or shot-blasting. However, the most important aspects in fatigue being the defects and the residual stresses, the critical distance and other material parameters are considered equal to the as-forged values for the sake of simplicity.

The critical distance d is determined using the critical defects of the as-forged specimens. Each critical defect is loaded with its respective fatigue limit, ranging from 286 to 400 MPa, and the stress gradient below the maximum surface stress value is plotted. Because the defects have different depths, varying from 31 to 171 µm, the gradients are plotted in relative depth, with 0 depth being the surface. The resulting stress gradients are plotted in figure 14. According to the In order to apply the Dang Van model, the (α, β) parameters must be determined. If we assume that the (α, β) values are independent of the hardening, then all batches have identical parameters.

Using the FE simulations of all specimens, it is therefore possible to determine the (α, β) pair that gives the overall best fit of the experimental results. The values obtained this way are: α = 0.27 and β = 250 MPa.

Results and discussion

Figure 15 plots the predicted versus experimental fatigue strengths for all critical defects of all batches (point method). The simulations correctly predict the fatigue limit, with only a few points having an error greater than 15%. This means that taking into account only the defect geometry and the residual stresses is enough to accurately predict the fatigue behaviour of a hot-forged and shot-blasted (or shot-peened) surface. It is important to note that the point for each batch are centred around the 1:1 line. This indicates that their dispersion is not due to the choice of parameters used for the simulations. Changing the (α, β) parameters to other values would not have improved the dispersion, as the parameters affect all points similarly. This is also true for the critical distance (figure 14): selecting a slightly different value would not have improved the dispersion and only changed the average error.

The model used here ignores the hardening introduced by shot-blasting and shot-peening. Taking this effect into account is possible and would mean changing the β value for each batch, leading to new α values. However, as stated previously, these new values would not improve the results, as the dispersion is due to the specimens themselves (defect geometry, surface integrity, etc.).

The simulations correctly reproduce the roughly 150 MPa difference in fatigue strength between the as-forged specimens and the shot-blasted (or shot-peened) specimens. This difference is due to the effect of the residual stresses, their effective contribution to the Dang Van stress value is:

∆σ RS eff = α 3 σ RS 11 + σ RS 22 1/2 + α/3 (13)
The values for the effective contribution of the residual stresses for each batch are given in table table contributions for all batches. However, the 50 MPa difference between the two shot-peened batches seen in the experimental results (figure 3) is also present in the simulation results (figure 15), but is not matched by the difference in the contribution of the residual stresses. This means that the defect geometry is also contributing to the difference between the two batches. This is confirmed by calculating the K f values of the defect geometry without the residual stresses: all batches have an average defect K f value between 1.22 and 1.25, except for the shot-peened ∅800 µm shot which has an average defect K f of 1.12. This difference is not due to defect size, as all batches have defects of similar sizes (figure 3). Furthermore, all shot-peened specimens were originally from the as-forged batch, and no particular selection was made when choosing which specimens were to be shot-peened. All batches should then have similar defects distributions, which means that the difference comes from the shot-peening itself. The large shot diameter could have a smoothing effect on defect geometry. This is not the case for the shot-peening with ∅400 µm shot, the shot diameter being twice as small means that a much smaller surface area is affected by each impact. The effect is also not observed for the shot-blasting despite using ∅1 mm shot. This is because the shots are not often changed and break or wear down to much smaller effective diameters. Furthermore, the critical defects of the shot-blasted specimens are extremely large, and the smoothing effect might not be as pronounced for defects of such size.

To illustrate this difference, simulations were performed with defects scanned before and after shot-peening. This analysis was not performed for shot-blasting as is was not possible to scan connecting rods before shot-blasting. Two defects of similar size and orientation were specifically chosen amongst the scanned surfaces. These are not critical defects, as the critical defects of these specimens were not similar enough for accurate comparison. The surface scans (before and after shot-peening) and the corresponding simulations are given in figure 16. Comparing the Before shot-peening ∅800 µm shot After shot-peening ∅800 µm shot Before shot-peening ∅400 µm shot After shot-peening ∅400 µm shot Fig. 16: Comparison of two defects, before and after shot-peening, for both shot-peened batches.

The scanned surface and the corresponding surface K t (FE simulation) are shown.

surface scans shows how the shot impacts have affected the defects: the general shape and the depth are unchanged but the edges are less clearly defined. The shot impacts also smooth out the local geometry: the maximum surface K t is lower after shot-peening, for both batches. Figure 16 give the K f values without residual stresses of the defects, before and after shot-peening.

Before shot-peening, both defects have similar K f values. However, after shot-peening, the K f value is noticeably lower for the ∅800 µm shot while it remains unchanged for the ∅400 µm shot. This shows how the ∅800 µm shot seems to increase fatigue strength by reducing the impact of defect geometry. points follow the curve σ D = σ D 0 /K f , which validates the simulations' K f values.

Conclusion

The goal of this paper is to develop a fatigue approach which can take into account the combined effects of surface defects and shot-peening on fatigue behaviour. The components chosen for this study, hot-forged connecting rods, exhibit surface defects after forging and are subsequently shot-blasted. This study was performed on fatigue specimens machined out of hot-forged connecting rods. The specimens were thoroughly characterized and fatigue tests were then performed to quantify the effect of the surface integrity on fatigue behaviour. The fatigue results

show that the surface integrity has a major impact on fatigue behaviour: large forging defects and shot-blasting both affect fatigue strength. The two major aspects of the surface integrity are the forging defects and the residual stresses. The specimens were scanned before fatigue testing, so that the critical defects could be identified.

The fatigue results were analysed in order to develop a fatigue model, taking into account the surface integrity. Two approaches were used. The first is based on the fitting of the surface defects with an ellipsoid. This gives access to the defects' size and shape, allowing to quickly analyse a large number of defects. Combined with Peterson's K t model, this method was able to detect the critical defect for roughly a third of all specimens.

The second model used FE simulations of the defects, in order to more accurately analyse the effect of their geometry. The theory of critical distance is used to quantify the influence of the defects and the residual stresses are taken into account with Dang Van's model. The simulations correctly predict the fatigue strength with at most 15% error. These results show that it is possible to accurately model the effect of the surface integrity using the following parameters:

• The material's reference fatigue strength

• The critical distance of the material

• The surface topography of the defects

• The residual stress profile Despite the different profiles, the contribution of the residual stresses is almost the same for the shot-blasted and shot-peened batches. Additionally, the FE simulation approach showed that shot-peening with a large diameter shot can reduce the impact on fatigue of the forging defects.

List of figure captions

List of table captions

Table . 1: Overview of the number of valid tests and fatigue strength range for each batch. [START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF] Table . 2: Effective contribution of the residual stresses to the fatigue strength, for each batch.

Fig. 1 :

 1 Fig.1: a) Connecting rod with spark-machined specimen. b) Example of a critical defect on an as-forged specimen.[START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF]

  µm stress-rel.

Fig. 3 :

 3 Fig.3: Kitagawa diagram of the fatigue tests, with fatigue strength and critical defect size of each valid specimen.[START_REF] Gerin | Influence of surface integrity on the fatigue behaviour of a hot-forged and shot-peened C70 steel component[END_REF]

Fig. 5 :

 5 Fig. 5: Example of a surface defect (green) with the fitted ellipsoids (blue and red). The values for the semi-axes a and b are given in pixels (1 pixel = 3.5 µm) and the values for c are in µm.

Fig. 6 :

 6 Fig. 6:√ area of the defects from the as-forged specimens. For each specimen, the blue point are the defects and the critical defect is marked in red.

Fig. 7 :

 7 Fig. 7: Peterson's K t of each defect for all specimens. For each specimen, the blue point are the defects and the critical defect is marked in red.

Fig. 8 :

 8 Fig. 8: Plots versus the standardised variable of the critical defects' K t and depth values. The K t values follow a Gumbel distribution (straight line), but the depth does not.

Fig. 9 :

 9 Fig. 9: Close-up of the mesh showing the element size gradient. a) Mesh gradient along the entire thickness. b) Mesh gradient in the first 100 µm, surface is marked in orange.

Fig. 10 :Fig. 11 :

 1011 Fig. 10: Loading conditions of the FE simulations.

Fig. 13 :

 13 Fig. 13: Cross-section of the FE simulation of the critical defect in figure 12. Stress values are in MPa, the white dot locates the maximum stress. The point method (red dot) and volume method (sphere centred on the white dot) are indicated.

Fig. 14 :

 14 Fig.14: Stress gradients below the maximum stress for all critical defects of the as-forged specimens. Depth is relative: for each gradient, 0 depth is the tip of the defect.

Fig. 15 :

 15 Fig. 15: Simulated fatigue strength versus experimental fatigue strength, for all critical defects.

Figure 17 plotsFig. 17 :

 1717 Figure 17 plots the experimental fatigue limit versus the K f value obtained with the FE simulation, taking into account both the defect geometry and the residual stresses. The data

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :

 1234 Fig. 1: a) Connecting rod with spark-machined specimen. b) Example of a critical defect on an as-forged specimen. [43] Fig. 2: Residual stress profiles for the various batches, in the direction of the specimen axis (long.), and the transverse direction (tvs.). [43] Fig. 3: Kitagawa diagram of the fatigue tests, with fatigue strength and critical defect size of each valid specimen. [43] Fig. 4: Diagram of a surface with a defect (green) and the corresponding ellipsoid (blue). The grid is the reference plane.

Fig. 5 :

 5 Fig. 5: Example of a surface defect (green) with the fitted ellipsoids (blue and red). The values for the semi-axes a and b are given in pixels (1 pixel = 3.5 µm) and the values for c are in µm.

Fig. 6 :

 6 Fig. 6:√ area of the defects from the as-forged specimens. For each specimen, the blue point are the defects and the critical defect is marked in red.

Fig. 7 :

 7 Fig. 7: Peterson's K t of each defect for all specimens. For each specimen, the blue point are the defects and the critical defect is marked in red.

Fig. 8 :

 8 Fig. 8: Plots versus the standardised variable of the critical defects' K t and depth values. The K t values follow a Gumbel distribution (straight line), but the depth does not.

Fig. 9 :

 9 Fig. 9: Close-up of the mesh showing the element size gradient. a) Mesh gradient along the entire thickness. b) Mesh gradient in the first 100 µm, surface is marked in orange.

Fig. 10 :

 10 Fig. 10: Loading conditions of the FE simulations.

Fig. 11 :

 11 Fig. 11: Process for performing the FE simulations of the critical defects.

Fig. 12 :

 12 Fig.12: FE simulation of a critical defect compared to its surface scan. Indicated on the scan are the defects' K t value, calculated using the geometrical analysis.

Fig. 13 :Fig. 14 :

 1314 Fig.13: Cross-section of the FE simulation of the critical defect in figure12. Stress values are in MPa, the white dot locates the maximum stress. The point method (red dot) and volume method (sphere centred on the white dot) are indicated. Fig.14: Stress gradients below the maximum stress for all critical defects of the as-forged specimens. Depth is relative: for each gradient, 0 depth is the tip of the defect.

Fig. 15 :

 15 Fig. 15: Simulated fatigue strength versus experimental fatigue strength, for all critical defects.

Fig. 16 :

 16 Fig.16: Comparison of two defects, before and after shot-peening, for both shot-peened batches.The scanned surface and the corresponding surface K t (FE simulation) are shown.

Fig. 17 :

 17 Fig. 17: Simulated fatigue strength versus experimental fatigue strength, for all critical defects.

  details the number of valid specimens for each batch, with their respective fatigue strength range. The average value from the staircase method on the polished specimens is the

	Batch	Test type Nbr. of specimens σ D (MPa)
	Polished	Staircase	15	424
	As-forged	Locati	34	280-400
	Shot-blasting	Locati	8	420-500
	Shot-peening ∅800 µm shot	Locati	10	500-580
	Shot-peening ∅400 µm shot	Locati	18	400-500
	Tension SP ∅800 µm shot stress-relieved	Locati	1	431
	Table

Table 2 :

 2 2. Despite the different residual stress profiles, the residual stresses have similar Effective contribution of the residual stresses to the fatigue strength, for each batch.

	Stress in MPa	Shot-blasting SP ∅800 µm shot SP ∅400 µm shot
	σ RS at 40 µm depth	500	525	455
	∆σ RS eff	152.5	160.2	135.8

Acknowledgements

This work has been performed within the ANR (National Research Agency) DEFISURF project, in a partnership including several industrial partners (Ascometal, CETIM, Renault, Transvalor, Atelier des Janves, Gévelot Extrusion) and academic institutions (INSA Lyon MATEIS, ENSMP-CEMEF, Arts et Métiers ParisTech LAMPA).