Antoine Pirrone
email: antoine.pirrone@labri.fr

Marie Beurton Aimar

Nicholas Journet
email: journet@labri.fr

Papy-S-Net : A Siamese Network to match papyrus fragments

Keywords: Applied computing → Document Analysis;, Computing methodologies → Neural networks Historical documents, Deep Learning, Siamese network

Like all heritage documents, papyri are the subject of an in-depth study by scientists. While large volumes of papyri have been digitized and indexed, many are still waiting to be so. It takes time to study a papyrus mainly because they are rarely available in one piece. Papyrologists must review a large number of fragments, find those that go together and then assemble them to finally analyze the text. Unfortunately, some fragments no longer exist. It is then a time consuming puzzle to solve, where not all the pieces are available and where fragments boundaries are not perfectly matching.

This article describes a method to help Papyrologists save time by helping them to solve this complex puzzle. We provide a solution where an expert use a fragment as a request element and get fragments that belong to the same papyrus. The main contribution is the proposal of a deep siamese network architecture, called Papy-S-Net for Papyrus-Siamese-Network, designed for papyri fragment matching. This network is trained and validated on 500 papyrus fragments approx. We compare the results of Papy-S-Net with a previous work of Koch et al. [14] which proposes a siamese network to match written symbols. In order to train and validate the network, we proceed to the extraction of patches from the papyrus fragments to create our ground truth. Papy-S-Net outperforms Koch et al. 's network. We also evaluate our approach on a real use case on which Papy-S-Net achieves 79% of correct matches.

INTRODUCTION

The papyri studied here are coming from Egyptian mummies. These documents were written around the third century BCE and their supports were afterwards reused to make mummy casings called "Cartonnages". These casings were painted funeral ornaments decorating the mummies. A funeral mask made up of several dozen papyri fragments can be seen on the left of Figure 3. Papyrus being an expensive and rare resource at that time, these ornaments were made with documents that did not bear interest anymore since they were already covered with writing. On each fragment we extract patches that will be used to train our network. For example on fragment 10 we extract 7 patches Modern Papyrologists are interested in studying the content of these papyri in order to obtain information on the economic activity of the region from which they come : agricultural production, tax policy . . . In order to analyse their textual content, the ornaments were in the past torn apart but are nowadays carefully separated. Figure 3 shows on the right an example of fragments extracted from an ornament. For several decades, hundreds of fragments coming from a large number of administrative documents have been extracted. This results in a complex puzzle in which some pieces are missing and most are very damaged. Papyrologists have no choice but to resolve this complex puzzle before being able to analyse the writings. The amount of fragments to be analysed makes the assembly task beyond the scope of manual work only. They need to be assisted with a digital process for reconstructing the papyri. Methods already exist to solve puzzle-type problems. They mainly tackle the cases in which the documents are contemporary and all the pieces are available, which is not the case with papyrus fragments. In addition, most of them are hand-crafted methods. A strong knowledge of the content of the image is necessary in order to be able to adjust all the parameters, making these methods difficult to generalize.

In this article, we describe the first version of a tool that aims to help the Papyrologists to sort the fragments according to an image similarity criterion. Our main contribution is the creation of a process providing a list of similar fragments for a requested fragment. For that purpose, we tuned a deep siamese neural network architecture. As presented in figure 1 1 , a papyrus can be composed of many fragments on which we extract patches. Figure 2 describes how our network is trained. We extract several patches from each fragment. We train the network on carefully selected pairs of patches belonging to the same papyrus and patches of fragments coming from different papyri.

RELATED WORK

The problem addressed in this work is a first step towards solving lots of puzzles of which all the pieces are mixed together. It is first necessary to identify which pieces come together in the same puzzle before tackling the assembly process. In the context of our work, we have multiple papyri (our puzzles) that have been torn apart in several fragments (our pieces) that are all mixed up together. The puzzle problem has mainly been studied in the context of natural images. This objective of reconstructing an image from many fragments is addressed as a np-hard optimization problem where, like in [START_REF] Liu | Automated assembly of shredded pieces from multiple photos[END_REF], pieces of fragments of photos are matched then arranged using shape and image content information. Recently, methods based on deep neural networks [START_REF] Sholomon | DNN-Buddies: A deep neural network-based estimation metric for the jigsaw puzzle problem[END_REF] open the way to other approaches.

In the specific context of document image reconstruction, some researchers have been interested in torn newspapers [START_REF] Richter | Partial contour matching for document pieces with content-based prior[END_REF] but most focused on the reconstruction of typewritten documents shredded by a machine [START_REF] Ahmed S Atallah | A step toward speeding up cross-cut shredded document reconstruction[END_REF]. Indeed, in the past many documents (military archives, intelligence services) have been shredded to make information disappear. These fragments were digitized for historical purposes and must be reassembled so that historians can analyze them. This research area was very active in the early 2010's, a competition has even been initiated. It has been created by the U.S. Defence Advanced Research Projects Agency (DARPA) [START_REF] Butler | The deshredder: A visual analytic approach to reconstructing shredded documents[END_REF].

The puzzle problem can be addressed in different ways. One is to consider it as an optimization problem. In [START_REF] Chen | A Solution to Reconstruct Cross-Cut Shredded Text Documents Based on Constrained Seed K-means Algorithm and Ant Colony Algorithm[END_REF][START_REF] Justino | Reconstructing shredded documents through feature matching[END_REF], the authors propose relatively similar approaches to automatically resolve the puzzle. They first extract and approximate the contours. As they are in a context where all the fragments are available with contours that perfectly match from one fragment to another, they propose to model the problem as a cost function to minimize, like a similarity measure between puzzle pieces contours. 1 Image taken from the University of Michigan Library Digital Collections Another way is to apply greedy methods to solve the puzzle. In [START_REF] Richter | Learning to reassemble shredded documents[END_REF], features are computed on contours of each fragment : contour length, color histogram . . . these features are used to train a SVM (Support-Vector Machine) to find matching pairs of points of adjacent fragments. The alignment between fragments is done using a graphical modeling (vertex=fragment, edge=alignment) by trying to find a spanning tree in it. The greedy methods assume that the dataset size remains reasonable.

As far as we know, most of computer science papers dealing with papyrus image analysis like [START_REF] Athina A Alexopoulou | Multispectral documentation and image processing analysis of the papyrus of tomb II at Daphne, Greece[END_REF][START_REF] Gargano | A perceptual approach to the fusion of visible and NIR images in the examination of ancient documents[END_REF] aim to improve the digitalization process of physical documents and then to carry out image processing algorithm in order to be able to clearly distinguish the ink from the letters and the background pixels. Only few recent papers like [START_REF] Levi | A Method for Segmentation, Matching and Alignment of Dead Sea Scrolls[END_REF] detail methods which are going further than just enhancing the image readability. Indeed, they present a deep architecture to precisely segment the fragment without being disturbed by the presence of the ruler, color charts and the scotch tape that hold the fragment on the support. They also propose to use a classical image retrieval process where features like : the size, the shape and SIFT descriptors are used to compare the fragments with each other. This fragment comparison allows to find the exact copies of previously digitized fragments (with a different acquisition protocol).

A DEEP SIAMESE NEURAL NETWORK TO MATCH PAPYRUS FRAGMENTS 3.1 Siamese architectures

Siamese networks are a special kind of neural networks having the particularity to contain two (or more) identical sub-networks (sometimes called branches). In a siamese network the layers arrangement is the same across the sub-networks, all their parameters are constrained to the same values. The goal is to define a similarity metric to decide if the inputs are similar or not. Siamese networks originally have been designed to compare handwritten signatures [START_REF] Bromley | Signature verification using a "siamese" time delay neural network[END_REF] with great success. More recently, they have been getting increasingly popular with applications in face recognition [START_REF] Wu | Face Recognition based on Convolution Siamese Networks[END_REF], object tracking [START_REF] Bertinetto | Fully-Convolutional Siamese Networks for Object Tracking[END_REF] or analysis of medical images [START_REF] Bhagwat | Modeling and prediction of clinical symptom trajectories in AlzheimerâĂŹs disease using longitudinal data[END_REF].

More recently, siamese networks have been used by Kassis [START_REF] Kassis | Alignment of Historical Handwritten Manuscripts Using Siamese Neural Network[END_REF] to align portions of the same text coming from two different documents and to find matching words pairs. Zhong et al. [START_REF] Zhong | SpottingNet: Learning the Similarity of Word Images with Convolutional Neural Network for Word Spotting in Handwritten Historical Documents[END_REF] use siamese and pseudo-siamese networks to perform word spotting in handwritten historical documents. We can also note [START_REF] Ahrabian | On Usage of Autoencoders and Siamese Networks for Online Handwritten Signature Verification[END_REF] which uses autoencoders together with siamese networks for automatic signature verification.

Architecture of Papy-S-Net

The proposed siamese architecture was inspired by the architecture presented by Koch et al. in [START_REF] Gregory | Siamese Neural Networks for One-Shot Image Recognition[END_REF], where a deep siamese network is used for one-shot learning of handwritten characters from a variety of languages (see the Omniglot dataset, introduced in [START_REF] Brenden | Humanlevel concept learning through probabilistic program induction[END_REF]). This architecture is designed to work on dataset which's structure resembles the Omniglot dataset images, that is black on white images containing characters. Our dataset is very different. Although also containing characters, they are written on papyri, which contain a lot of texture information (see Figure 2) that can be exploited for better recognition. We describe here the architecture of Papy-S-Net and how and why it differs from Koch et al. 's.

Papy-S-Net (a diagram of the architecture can be seen on Figure 4 2 ) consists of two identical convolution / pooling branches with shared weights, meaning that at any time, every corresponding neuron weights are the same between the two branches. Contrary to Koch et al.'s architecture, we choose to have two consecutive convolutional layers before applying the max pooling operation every time. Indeed, each pooling operation reduces the size of the image (by a factor of 2 in our case), which reduces the amount of details available. Thus, multiplying sequential convolution operations before pooling allows for a finer representation of the data at each scale. This is useful to capture local details, such as texture, as well as global information on the image, such as writing style.

After flattening the outputs of the convolutional parts of the network, we compute the element-wise absolute difference of the two feature vectors. Some siamese networks perform a concatenation of these vectors instead of this absolute difference (i.e. [START_REF] Zagoruyko | Learning to Compare Image Patches via Convolutional Neural Networks[END_REF][START_REF] Zhong | SpottingNet: Learning the Similarity of Word Images with Convolutional Neural Network for Word Spotting in Handwritten Historical Documents[END_REF]), but trying it out, we noticed it created an asymmetry in the results : permuting the input branches on the same pair of patches outputted different results.

As we do not really want a similarity score as in most siamese networks, but rather a 2-by-2 classification of which patches come from which fragments, we choose the output of Papy-S-Net to be a 2 classes Softmax function ([dissimilar, similar]) after the two fullyconnected layers. This, combined with the use of the cross-entropy loss function, tends to push the classification scores to extremes (close to 0 and close to 1) which makes the decision process easy (no need for an expertly tuned threshold), but removes the possibility of manually adjusting the threshold to filter false positives.

For training, we use a batch size of 64, a learning rate of 0.0001 and the AdamOptimizer [START_REF] Diederik | Adam: A Method for Stochastic Optimization[END_REF] Tensorflow implementation. We train for 80 epochs as accuracy and loss values stabilize around this number and the performance do not improve by training for longer. At each epoch during training, the batches are recomputed, randomizing the order in which the network sees all the positive and negative pairs available in the training set. The validation set is fixed at the beginning of training, and the validation accuracy is computed on all the set at the end of each epoch. All the experiments presented in the next sections are done on an Nvidia Titan X (Pascal) with 12GB of video memory and using Tensorflow version 1.13.

Dataset creation process

In the context of deep learning, a large dataset is needed for correctly training a model. A model is considered good if it can generalize well to new data it has never seen before. It means that it has learned the underlying concept represented in the data it was trained on, and did not just memorize the data, which would be Overfitting.

Our dataset consists of just over 500 papyrus fragments (from now on, this dataset will be called B500). The papyri are from the University of Michigan Library Digital Collections. These papyri have very varied contents since they come from different periods (from 600 years BCE to 400 years CE), they have been written in different languages (Coptic, Greek, Arabic, Hebrew, Hieratic, Latin and Demotic) and are of different sizes. They have also been digitized at different times with different settings. Because real ground truth dataset is very difficult to obtain in our context, we choose to create our ground truth automatically by extracting patches from the fragments. As the fragments differ in size, we obtain a different number of patches for each of them. In order to be accepted as part of the dataset, a patch needs to contain at least 80% of material (i.e. less than 20% of background). This approach allows us to generate enough ground truth to train the deep siamese neural network. The goal of the network is now to decide if two different patches come from the same fragment. In order to test if the results are influenced by the presence or absence of text in the patches, we conduct the experiments with three sets created with different patch extraction approaches : a set of patches all containing text, a set of patches containing only texture (no text) and a set with randomly selected patches (no constrains on the presence or not of text).

In order to extract only patches containing text or patches not containing text, we need a way to locate the lines of text in the papyrus fragments. We used the first part of ARU-Net [START_REF] Grüning | A two-stage method for text line detection in historical documents[END_REF] to do so. Results can be seen in Figure 5. With this information, we can now extract three different kinds of (non overlapping) patches on which to perform the experiments. Random patches are extracted without any constraints. Without text patches are selected so that they do not contain text, only background (papyrus texture). With text patches are selected so that they all do contain text. All the extracted patches do not contain borders because we do not want to learn the shape of the fragments but their contents. We extract as many non overlapping patches which satisfy their respective constraints as possible for each fragment. Naturally, we get a lot more of Random patches than of Without text or With text. To be able to compare the results, we need to have roughly the same amount of data for the three approaches. We therefore randomly discard patches in the Random class until getting the same number of patches in the three classes. It is important to note that the number of patches extracted per fragment can vary quite a lot depending on the size of the given fragment, ranging from 2, for the smallest ones, to 123 for the biggest ones.

We then prepare our datasets by following the classic Train / Validation / Test split. We first split the dataset into [Train / Validation], 90% and [Test], 10%. We get about 1000 patches coming from about 50 fragments for the Test set. The [Train / Validation] part is then split in Train, 80% and Validation, 20%, giving us about 8500 patches for the Train set and about 2000 patches for the Validation set. For these sets, we create the same number of similar (patches coming from the same fragment) and dissimilar (patches coming from different fragments) pairs. Batches for training are pre-computed at each new epoch, we ensure that all examples of the Train set are seen at each epoch in a random order by associating two patches for each patch in the set, one similar and one dissimilar. We also add the permutation of each created pair to the dataset. Finally, statistics on the Test set are computed on all the possible pairs permutations on the set.

EXPERIMENTS AND RESULTS

We compare the results obtained on Papy-S-Net and Koch et al. 's architectures with our three patches extraction approaches on two test datasets. As can be seen on Figure 9, the best accuracy performance is achieved with our method on patches containing text after 70 epochs. This seems to be confirmed by Figures 6 and7.

It is interesting to note that we respectively obtain globally better results when trained with Random patches than with Without text patches, our model being slightly better or very close most of the time. However, we can notice an improvement on Papy-S-Net when trained on patches With text. As can be seen on Figures 6 and7, Papy-S-Net outperforms Koch et al. 's for all metrics in this case, and actually almost outperforms every other approaches (see bold numbers). We tested the different approaches on two datasets.

Random

Without While the performance difference between these two examples is quite significant (between 8 and 10 percentage points), the respective performance differences between the Random and With text approaches exist, but is smaller. Taking into account the overall pre-processing cost to detect the baselines in order to extract patches all containing text, which in itself can probably be improved and brings its own uncertainties, the Random approach seems to bring acceptable results. However, random being what it is, this approach may not cope well with papyri containing little text, as it would statistically tend to extract lots of patches only containing texture, which is the worst performing approach with our model. In this case, we would recommend the With text approach. The second one is a subset of the aforementioned Test part on which we know that different fragments actually belong to the same papyrus, let's call it RealUseCase. This dataset contains 60 fragments belonging to 19 papyri. This is closer to a real world case, where we have different fragments and we try to find which belong to the same papyrus. The B500 dataset was used to train the model, this dataset is emulating the real work of Papyrologists. Knowing the ground truth information of the correct belongings of the fragments, we can see if patches are correctly classified in their corresponding papyrus, which is composed of two or more fragments. Again, as can be seen on Figure 7 Figure 8 shows correct and incorrect pairings on the RealUseCase dataset in the form of a confusion matrix. For each papyrus, we compare the extracted patches to the other patches of all other papyri and register which pairs were considered as "similar" and increment the number in the corresponding case of the matrix, which is afterwards normalized. We can see on the diagonal that the patches are mostly classified in the correct papyri. Interestingly, the papyrus "greek012" seems to attract a lot of false positives, especially on other papyrus containing greek language. One reason for that could be because 96 patches were extracted on this papyrus, which is a lot more than most of the others, making this class statistically more likely to be classified incorrectly. Another reason could be that the network implicitly learns some language classification skills.

Random

CONCLUSION AND FUTURE WORK

With the introduction of Papy-S-Net, we have proposed a promising method for sorting papyri fragments in terms of their belonging to the same papyrus. Along with the siamese network itself, we proposed a patch based methodology in order to prepare the datasets, and concluded that training the network using patches all containing text yielded the best matching results. The pre-processing phase needed for this approach can be very much improved, for example, by accurately segmenting the papyri, we could be more precise in selecting patches without background.

Papy-S-Net is a good first step towards the final goal of our research, which is actually reconstructing the puzzle. It can already be used to perform a significant sorting process on existing fragments databases, however we still suffer from a too high false negative rate for the needs of the Papyrologists, it is indeed worst to incorrectly discard fragments than to accept wrong fragments which could be further filtered by other means. This issue could be addressed by a more linear output of the network, giving the ability to express doubt about the result instead of classifying binarily. We also want to continue this work by experimenting with patch level data augmentation which, combined with better pre-processing, may greatly improve the current results.

Figure 1 :

 1 Figure 1: Definition of terms related to the study of papyri. This image shows a papyrus composed of 10 fragments.On each fragment we extract patches that will be used to train our network. For example on fragment 10 we extract 7 patches

Figure 2 :

 2 Figure 2: Overview of Papy-S-Net. The network is trained with similar and dissimilar pairs as input. In our case, the pairs are patches extracted from papyrus fragments. The network decides if 2 patches belong to the same fragment.

Figure 3 :

 3 Figure 3: Left image : a funeral mask made from papyrus fragments. Right image : several fragments extracted from a decoration

Figure 4 :

 4 Figure 4: The architecture of Papy-S-Net. The input images are of size 128x128x3, each pooling operation (red squares) divides the dimension of the images by 2, each convolution operation uses a kernel of size 3x3. We use ReLU activation functions everywhere except for the last layer, which uses Softmax for binary classification. The weights of the two branches are shared and their outputs are combined by taking the absolute value of the element wise difference of the flattened outputs of each branch. This vector is then given to a Softmax two layers fully-connected network.

Figure 5 :

 5 Figure 5: Typical line detection results obtained with ARU-Net on our dataset. Green, red and blue pixels respectively denote correct, false positives and false negatives relative to manually labeled ground truth.

Figure 6 :

 6 Figure 6: Comparing the performance on B500 obtained for our architecture and that of Koch, et al. The best-performing method for each metric is listed in bold-faced type, and the worst method is underlined The first one is the Test part of each training for each patch extraction method, as explained in section 3.3. We can see on Figure 6 that the best True positive, False Positive, True Negative and False Negative rates are obtained when training Papy-S-Net on patches With text, and most of the worst rates are obtained when training Koch et al.'s architecture on patches With text.While the performance difference between these two examples is quite significant (between 8 and 10 percentage points), the respective performance differences between the Random and With text approaches exist, but is smaller. Taking into account the overall pre-processing cost to detect the baselines in order to extract patches all containing text, which in itself can probably be improved and brings its own uncertainties, the Random approach seems to bring acceptable results. However, random being what it is, this approach may not cope well with papyri containing little text, as it would statistically tend to extract lots of patches only containing texture, which is the worst performing approach with our model. In this case, we would recommend the With text approach.

  , Papy-S-Net outperforms Koch et al.'s architecture in nearly all experiments.

Figure 8 :

 8 Figure 8: Confusion matrix on the RealUseCase dataset for the With text patch extraction approach. The names (such as "greek016") represent a papyrus composed of at least two different fragments.Figure8shows correct and incorrect pairings on the RealUseCase dataset in the form of a confusion matrix. For each papyrus, we compare the extracted patches to the other patches of all other papyri and register which pairs were considered as "similar" and increment the number in the corresponding case of the matrix, which is afterwards normalized. We can see on the diagonal that the patches are mostly classified in the correct papyri. Interestingly, the papyrus "greek012" seems to attract a lot of false positives, especially on other papyrus containing greek language. One reason for that could be because 96 patches were extracted on this papyrus, which is a lot more than most of the others, making this class statistically more likely to be classified incorrectly. Another reason could be that the network implicitly learns some language classification skills.

Figure 9 :

 9 Figure 9: Train and Validation accuracy plots and table for the different experiments

This diagram was generated using github.com/HarisIqbal88/PlotNeuralNet

ACKNOWLEDGMENTS

The research leading to this results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 758907 and is part of the GESHAEM project, hosted by the Ausonius Institute. The tests were carried out on images from the Image Bank, University of Michigan Library Digital Collections. https://quod.lib.umich.edu/a/apis Accessed: June 04, 2019. we also thank Pierre Biasutti for his advice and Pierre Célor for his work during his master's internship.


Eduardo García

Damien Méresse

Iñigo Pombo

Souad Harmand

José Antonio Sánchez

Stefan-Boltzmann Constant

Identification of heat partition in grinding related to process parameters, using the inverse heat flux conduction model

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Grinding is an abrasive machining process characterized by producing high quality components for high added-value industries (aerospace, energy, tooling…) in terms of tight dimensional tolerances and smooth surface roughness. By contrast, grinding is also characterized by requiring a high amount of energy input per unit volume of material removed. This energy is turned into heat in the contact zone and can cause excessive heating of the workpiece leading to thermal damage on its surface [START_REF] Malkin | Grinding Technology[END_REF]. Thus, the occurrence of thermal damage becomes one of the limiting constraints of productivity in grinding technology.

Increasing material removal rate is limited by the apparition of workpiece burn. The classic solution for burning problems is the use of cooling fluids but their effect is limited. Hence researches try to improve cooling effect by refrigerating the coolant [START_REF] Gao | An active coolant cooling system for applications in surface grinding[END_REF] or improving their convective effect [START_REF] Zhang | Investigation for convective heat transfer on grinding work-piece surface subjected to an impinging jet[END_REF] . However, these advances are not enough for avoiding burning in some cases. Therefore, understanding the mechanisms that govern workpiece temperature increase and relating them to industrial process parameters become a key factor for grinding process control and optimization.

It is assumed that the power consumed by the grinding wheel spindle is transformed into heat in the contact zone. This heat is evacuated through four ways: workpiece, wheel, ground chips and cooling fluid. The partition of this heat that goes into the workpiece, Rw, is the cause of workpiece temperature increase and thermal damage [START_REF] Malkin | Thermal Aspects of Grinding: Part 1-Energy Partition[END_REF]. Therefore its determination is very important for process optimization. However, it is a parameter difficult to assess since it depends on a large number of parameters: tribological, mechanical and geometrical. In fact, on shallow grinding with Al2O3 wheels scientific literature give values between 0.25 and 0.85.

Usually, authors have identified Rw by matching temperature data experimentally obtained with calculated temperature from theoretical models. One of the main limitations of this methodology, and one of the causes of the wide dispersion of Rw above, is the steep thermal gradients (up to 1000K/s [START_REF] Kohli | Energy Partition to the Workpiece for Grinding with Aluminium Oxide and CBN Abrasive Wheels[END_REF]) found in the contact zone, that limit the reliability of the measured temperatures.

For obtaining the theoretical temperature modelization of the process is necessary. Initially, the grinding wheel heating was modeled as a rectangular moving heat source along a semi-infinite body [START_REF] Kohli | Energy Partition to the Workpiece for Grinding with Aluminium Oxide and CBN Abrasive Wheels[END_REF]. Analytical models [START_REF] Guo | Analysis of energy partition in grinding[END_REF], [START_REF] Lavine | An exact solution for surface temperature in down grinding[END_REF], [START_REF] Rowe | Thermal analysis of high efficiency deep grinding[END_REF] have allowed to highlight some behaviors of the rubbing contact like the triangular shape of the generated heat flux. These were progressively replaced by numerical models with the rapid increase of calculation power. As it is gathered in [START_REF] Doman | Finite element modeling approaches in grinding[END_REF], numerical models are used following similar approaches, workpiece discretization, effect of grinding wheel as a moving heat source and consideration of the convective effect of cooling fluid; but differ in four main aspects: geometry of heat source (rectangular vs triangular), contact length estimation, material properties (constant or not with temperatures), and 2D or 3D models. This way, Brosse [START_REF] Brosse | Temperature measurement and heat flux characterization in grinding using thermography[END_REF] used finite element methods and thermography to characterize the distribution of Rw in the contact zone finding the triangular or the parabolic shape of the heat fluxes provided more accurate results. Following the approach above described, Kohli found values in the range 0.6-0.75 for shallow grinding with Alumina wheels [START_REF] Kohli | Energy Partition to the Workpiece for Grinding with Aluminium Oxide and CBN Abrasive Wheels[END_REF]. He used the Jaegers [START_REF] Jaeger | Moving sources of heat and the temperature of sliding contacts[END_REF] solution with a triangular heat flux shape. Hadad studied grinding with Minimum Quantity of Lubrication-MQL [START_REF] Hadad | Thermal analysis of minimum quantity lubrication-MQL grinding process[END_REF]. Based on analytical models found in [START_REF] Rowe | Thermal analysis of high efficiency deep grinding[END_REF], his experiments concluded that Rw varies between 0.73 and 0.77 for grinding with MQL, 0.82 for dry grinding and the use of cooling fluid reduced Rw to 0.36.

Inverse methods are numerous and widely used to study various heat transfer phenomena [START_REF]Thermal measurements and inverse techniques[END_REF]. Recently, the function specification method was used by Meresse to get heat flux repartition on a disc in braking conditions [START_REF] Meresse | Experimental disc heat flux identification on a reduced scale braking system using the inverse heat conduction method[END_REF]. Ludowski used the Levenberg-Marquardt method [START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF] to get the thermal boundary conditions in heat exchangers [START_REF] Ludowski | Identification of thermal boundary conditions in heat exchangers of fluidized bed boilers[END_REF]. The conjugated gradient method with adjoint problem has been used by Luchesi to identify a moving heat source in machining conditions [START_REF] Luchesi | An inverse method to estimate the moving heat source in machining process[END_REF]. Inverse methodology has already been used in grinding to determine Rw by matching experimental temperature with and an output numerical data, the nodal temperature at the sensor location, and minimizing the error. Three inverses analyses are used in [START_REF] Guo | Inverse heat transfer analysis of grinding. PartI: Methods[END_REF] to obtain Rw, the geometry of the heat source, and the convection coefficient hf from temperature data. Experiments are described in [START_REF] Guo | Inverse heat transfer analysis of grinding. PartII: Applications[END_REF] and authors have found Rw values in the range 0.70 and 0.74 for grinding with Al2O3 wheels using hardened steel and plain carbon steel. More recently, Hong has presented the assessment of Rw using a finite element method as direct model but results have not been validated by experimental data [START_REF] Hong | An inverse analysis for the heat conduction during a grinding process[END_REF]. Anderson performed this work for shallow and deep grinding. For the assessment of workpiece temperature distribution they used the finite elements method, and took into account the material removed by deleting elements in the model for the case of deep grinding. They estimated heat partition ratios to the workpiece by matching numerical results to those obtained with an infrared camera measurements obtaining Rw in the range 0.8-0.85. Results match those obtained by the model developed by Rowe [START_REF] Rowe | Thermal analysis of high efficiency deep grinding[END_REF].

The objective of this paper is to provide an original methodology and experimental setup that contribute to a better understanding of the influence of grinding variables on the heat partition ratio Rw. With respect to existing literature, the new methodology eliminates the errors induced by the steep thermal gradients that occur in grinding operations. Uncertainty related to the actual area of contact between both bodies (wheel and workpiece) is also suppressed by using a rigid and controlled geometrical configuration. Assessment of Rw in shallow grinding of hardened steel with Al2O3 wheels, and its relation with actual grinding parameters is presented in this paper using an inverse heat conduction model. Grinding tests are developed in the so-called On-Machine Test Bench, an alternative experimental set-up that allows reproducing grinding conditions while accurate temperature measurements and control of contact area are possible. The transient heat conduction problem is solved by a finite element model which takes into account the workpiece material removal during the process. The heat partition to the workpiece is identified by matching temperature data from thermocouple measurement to nodal temperature of the finite element model. The Rw parameter is identified with the Levenberg-Marquardt algorithm. Validation of temperature measurements is indirectly carried out by metallurgical analyses, which consists of measuring the micro-hardness of the tested samples and comparing it to the analytical model proposed in [START_REF] Fedoseev | Analysis of Tempering and Rehardening for Grinding of Hardened Steels[END_REF]. Results have led to a time-dependent definition of Rw which had not been previously proposed in literature; in addition it has been possible to relate the observed variations of Rw to the physical material removing mechanisms of grinding.

GRINDING PROCESS

As it has been mentioned, in grinding, for practical issues it is accepted that all the energy consumed in the grinding wheel spindle is transformed into heat in the contact zone. Therefore, the total heat flux generated in the contact zone, qt, can be obtained from equation 1, where P is the power consumption in the grinding wheel spindle. The partition of this heat that is evacuated through the workpiece, Rw, and the heat flux directed to the workpiece can be obtained from Equation 2.

= •

Eq.1

= . •

Eq.2

An experimental test bench is developed to reproduce this machining process and to get temperature data close to the contact zone. A thermal finite element model describing the heat conduction through the workpiece is then defined to allow the identification of the heat flux boundary condition by an inverse method. The experimental work was developed in the new test bench configuration shown in Figure 1. A horizontal-spindle CNC surface grinding machine is used for the study. A cup wheel, with a constant spinning speed, carries out grinding of a static workpiece plate. The wheel removes material by moving towards the workpiece at a constant infeed speed, vf. Insulating PTFE (Teflon) plates are placed at both sides of the steel plate to minimize heat losses to the clamps. The actual relative velocity between abrasive and workpiece is equal to the wheel spinning speed. This disposition presents some advantages compared to traditional grinding tests developed in grinding machines. The area of contact between wheel and workpiece, which is the area whereby heat enters the workpiece, is constant and known at every moment. In the direction of the spindle axis the configuration is very rigid, so that the effect of machine deformations can be neglected and the actual equivalent chip thickness, effectively controlled. Finally, continuous heat flux to the workpiece in the contact zone produces progressive heating of it, so that high temporal thermal-gradients are avoided and temperature can be accurately measured with thermocouples.

Experimental Test Bench

A Power meter device Load Controls Inc. UPC-FR installed on the wheel spindle and a Kistler 9257B dynamometer are also used for power and force measurement. Here experimental tests are performed with AISI4140 hardened steel (630HV). The workpiece is a rectangular prism of 100mm length x 5mm width x 45mm height. A hole is machined by EDM on the side of the workpiece and a K-type thermocouple of 200µm diameter is welded inside by capacitive discharge to track the temperature evolution. The hole is 1mm in diameter and is placed 3.5mm far from the edge of the workpiece, and 2.5mm in depth. Due to machining and welding alignment errors, the positioning of thermocouples is submitted to an uncertainty that has been estimated to be under 0.4mm.

Finite element model

Heat generation on the ground surface and heat diffusivity in the workpiece are modeled using the finite element method. ANSYS 13.0 is used for its ability to solve the heat equation for 3D transient problems and to take into account the steel removal during the process. The insulating PTFE plates are also included in the model in order to take into account non negligible heat losses. Material properties of the two materials have been considered constant and are listed in Table 1. To simulate the heat generation on the ground workpiece surface, a moving heat source is used. For a correct definition of this boundary condition the amount of heat flux entering the workpiece and its distribution over the contact area is needed. This area is theoretically defined by the product of the contact length lc and the wheel width b. Due to uncertainties related to the nature of deformations that must still be tackled, the value of lc considered in grinding varies from the geometrical contact length lg [4][11], up to values that can be 2 to 3 times higher [START_REF] Marinescu | Tribology of abrasive machining processes[END_REF].

In this case, thanks to our experimental configuration, the lc is equal to the width of the workpiece, b, and thus the contact area can be determined as the product of the workpiece width b and the depth of cut ae (eq.3):

= • Eq.3
The rate at which the energy is generated is approximately proportional to the rate of material removal. Taking into account the triangular geometry of the undeformed chip thickness, a triangular distribution has been chosen for the heat source [7][9]. This way, eq. 4 expresses the heat flux to the workpiece as a function of the distance to the left face of the workpiece, x:

= • • ( -) Eq.4
A perfect contact is considered between the workpiece and the insulating material as described by the equation eq. 5. On workpiece free areas, heat losses by convection and radiation are modeled corresponding to the equation eq.6. The heat convection parameter h depends on the workpiece nodal temperature and on the room temperature Tamb. A correlation of natural convection for vertical plate in laminar flow is used as described by the equation eq.7. Within the working conditions the heat transfer coefficient has been estimated to vary between 10 and 40W/m 2 . A sensitivity study was developed using the limit values of h. Results show that maximum workpiece temperature, whose value is close to 1300K, only decreases 11K when increasing h from 10 to 40W/m 2 . The low impact of convective parameter h in maximum workpiece temperature has led to consider a constant h value of 20W/m 2 . With regard to radiation losses, it has to be mentioned that usually thermal grinding models do not include radiation because of its negligible influence [START_REF] Doman | Finite element modeling approaches in grinding[END_REF]. However, due to the singularities of this test configuration, high temperatures were encountered in the vicinity of the contact zone (over 1300K in some cases), which motivated taking radiation heat losses into account. The value of the emissivity is set to 0.6 commonly used for medium carbon steel. A low influence of the emissivity has also been highlighted by simulations.

The model was meshed using eight nodes prism-shaped elements with a single degree of freedom, temperature, at each node. It can be seen on figure 2 that the mesh was refined near the contact zone. For the selection of the smallest element size in this area a sensitivity study was developed. 0.2mm, 0.05mm and 0.01mm element sizes were tested in this study. The maximum temperature difference between the 0.2mm model and the 0.01mm was 6%, however, the calculation time was increased notoriously, from 7min of the 0.02mm elements model to more than 7h for the 0.01mm elements model. Thus, 0.02mm was consider accurate enough for the issue of the study Finally, since the depth of cut during the test is not negligible, material removal was simulated using the element death feature built into ANSYS. In [START_REF] Anderson | Experimental validation of numerical thermal models for dry grinding[END_REF] this tool was used for modeling deep grinding. This feature does not actually remove the elements, but deactivate them by applying a severe reduction factor to their conductivity.

The solution of a test is obtained by incrementally stepping the heat flux along the workpiece. For each step the heat source moves the length of an element (0.2mm), and at the same time the precedent element is eliminated. Taking into account that the ground length is 2mm each test simulation consists of 11 steps. The time of each step is determined by the infeed speed of the grinding wheel vf.

The FEM has been completed with a model for simulating tempering. As it has already been explained, measuring temperatures in grinding is a very difficult task. From the previous explanation it becomes evident that the experimental set-up is characterized by low thermal gradients, which reduces errors in temperature measurement. Even though, in order to increase soundness of results, it is proposed here to use redundant data from indirect temperature measurement, as that provided by hardness alterations.

Taking into account the high temperatures suffered by the workpiece, metallurgical transformations (such as tempering) are expected to occur during the test.

For its simulation a tempering model based on the formulation of Fedosheev [START_REF] Fedoseev | Analysis of Tempering and Rehardening for Grinding of Hardened Steels[END_REF] has been implemented. The input of the model is the temperatures of the FEM and it calculates the loss of hardness due to tempering by using the concept of probability of tempering ψ:

= ℎ • (1 -)
Eq. 8

with h1 given by: ℎ ≡ exp -•

Eq. 9

The differential equation [eq. 8] has been transformed in a discrete equation [eq. 10] taking advantage of the step by step solution of the transient thermal problem:

= + ∆ • ℎ • (1 -) Eq.10
The final hardness H of the steel after tempering will be given by the following equation:

= -( -) •

Eq.11

INVERSE METHOD FOR THE HEAT PARTITION PARAMETER IDENTIFICATION

The identification of the heat partition to the workpiece, Rw, is possible thanks to reliable experimental temperature measurements and to an accurate thermal model. Knowing all the boundary conditions except Rw, an inverse method is used to evaluate this parameter value. The Levenberg-Marquardt algorithm is used to match the experimental value of the temperature with the numerical estimation at the sensor location. This iterative method has been largely applied to inverse heat transfer problems because of its stability and its accuracy [START_REF] Ludowski | Identification of thermal boundary conditions in heat exchangers of fluidized bed boilers[END_REF]. It consists of the minimization of quadratic functional S(Rw) where Rw can be a scalar or a vector. Minimizing S(Rw) is performed by equating to zero the derivative of S(Rw) with respect to Rw (eq.12).

∇ (

) = 2 * ( ) * - ( )
Eq.12

( ) = ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎡ … … … … … … … … … … … … … … … … ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎤ Eq.13
The Jacobian J(Rw), also called the sensitivity matrix, defines the sensitivity of the temperature at time step i to a variation of the parameter Rw at the time step j (eq.13). This is a N*N matrix for time-dependent Rw parameter. For constant parameter Rw, J is a vector and its length corresponds to the number of simulated time step N. In case of time-dependent Rw value with one value per time step, the Jacobian J becomes a N*N matrix.

The technique consists of correcting the values of the unknown vector Rw by solving the direct finite element model and applying a variation on values according to the sensitivity coefficients. This is an iterative method as the algorithm converges progressively on the optimal solution with several iterations. The variation on Rw values is applied thanks to the following formula (eq.14):

= + * - * * * (1 + μ ) Eq.14
The damping factor µ k is useful to allow the convergence on the optimal Rw value because it reduces oscillations and instabilities for an ill-conditioned problem. However the damping factor introduces a bias on the parameter estimation. This is progressively decreased with the iterations when the convergence is obtained on the optimal parameter value so as to be negligible on the final iteration. More details on the Levenberg-Marquardt method can be found in the literature [START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF]. Two stopping criteria have been defined. The iterative algorithm is stopped when the calculated Rw variation is lower than ΔRw =10 -3 between two iterations. Otherwise, the identification is stopped if the number of iterations has reached nbmax-it=8.

Numerical case was developed to assess the feasibility of the Rw identification.

Experimental temperature data are simulated by setting heat flux boundary condition in the range of the dissipated power in a real grinding test. Nodal temperature data is exported at the sensor location and the identified Rw value is compared to the previously imposed in the direct model. This study was performed with a constant Rw value (Rw =0.65) and time-dependent Rw. Results are presented for this second case on the figure 3. The convergence is rapidly obtained for the constant Rw value (2-4 iterations) depending on the initial guess. For time-dependent Rw, 6-7 iterations are necessary to get negligible temperature residuals. For the presented case, a good agreement can be observed between the imposed and identified Rw values. The temperature residual is lower than 1K on the entire temperature profile.

RESULTS

Rw values of tested configurations

Experimental tests and then Rw parameter estimations were performed for two types of abrasive wheels (2A46I8VWEB2 and 30SA46I8VWEB2) and five values of specific material removal rates Qw' by varying the infeed speed vf (eq.15).

′ = • eq. 15

The grinding wheels selected have a similar configuration but differ in the type of alumina used. The 2A46I8VWEB2 wheel is composed of fused alumina grits, while in the case of 30SA46I8VWEB2 the Sol-Gel 3M TM Cubitron TM 321 alumina type is used. For facilitating identification of the wheels through the text the following nomenclature is used 2 using Wheel 1 and Wheel 2, respectively. The evolutions of both, experimental and simulated temperature, as well as the value of the heat partition to the workpiece are plot. Regarding the experimental temperature, it must be noted that the measured signal is very stable and its increase is gradual (gradients below 100K/s). This is an important advantage for the accuracy of the results face to the conventional grinding tests. One of the problems of the thermal test in grinding is that the temperature varies so fast (10000K/s, in some cases [START_REF] Kohli | Energy Partition to the Workpiece for Grinding with Aluminium Oxide and CBN Abrasive Wheels[END_REF]) that the thermocouple is not able to accurately measure it. As it can be noticed from Figures 4 and5, errors related to this effect are minimized.

-Wheel 1 = 2A46I8VWEB2 (fused alumina) -Wheel 2 = 30SA46I8VWEB2 (Sol-Gel 3M
For the assessment of Rw, two approaches were followed. Initially, Rw was assumed to be constant along the test, but as it can be seen in Figures 4 and5 the numerical (circular dots) and the experimental temperatures (continuous line) do not match properly. After 3-4 iterations, the algorithm converges to a solution but residuals are not negligible and the difference between the numerical and the experimental temperature can reach 100K. As a consequence, a second approach was followed, which consisted of considering Rw as a time-dependent parameter.

The results obtained with the time-dependent Rw, matched almost perfectly with the experimental temperature being the residuals negligible. Paying attention to the evolution of Rw along the test it can be seen that it follows a similar behavior in both figures. The value of Rw is higher at the beginning of the test and decreases to a stabilized value as the test goes on. For Test 2, heq =7nm, its value starts at 0.72 and rapidly decreases to 0.55 (Figure 4). After 20sec, Rw looks to be stable until the end of the test (54sec). In Test 4, heq=35nm, the initial value is higher (0.86) and tends to stabilize around 0.66. This behavior has been observed in all tested conditions. This effect has not been previously described in scientific literature. The hypothesis for its explanation relies on the transient behavior of the abrasive grits at the beginning of use after the dressing operation. The dressing operation sharpens the abrasive grits, so that they recover their cutting ability. Sharp corners on the surface of the grits involve much reduced contact surfaces between abrasive grains and workpiece. Due to the extreme brittleness of alumina (which is a ceramic material) these sharp edges suffer a very fast microscopic wear shortly after the grinding operation starts. This effect is wellknown in grinding practice [START_REF] Malkin | Grinding Technology[END_REF]. After this short transient period, wear stabilizes and the operation progresses with a much lower wear rate. Therefore, during the transient period the actual area of contact for heat conduction increases until the regime of stable wear is reached. The growth of a metallic third body on the contact interface during the transient period [START_REF] Godet | The third body approach: mechanical view of wear[END_REF] may also have an influence on heat partition. Future work will include microscopic observation of the surface of the abrasive grits at different instants of the transient period, in order to validate the above hypotheses. Taking the above assumption into account the stabilized phase of the test is more representative of the grinding conditions found in an industrial process, due to the shortness of the unsteady part.

Figure 6 shows the values of Rw at the stable phase as a function of heq for all tested conditions. Clear differences can be observed between Test 1 and 2 (low values of heq, 4-7nm) and the others. For low values of heq (4-7nm), the part of the consumed energy directed to the workpiece is around 53% of the total. By contrast, for higher heq (18-53nm), Rw values are identified in the range of 0.7-0.8. This trend has been identified for the two abrasive materials tested. This difference is related to the different predominant mechanism of material removing (rubbing, ploughing and shearing) [START_REF] Rosenthal | Discussion: "On the Mechanics of the Grinding Process Under Plunge Cut Conditions[END_REF] as a function of heq. At low depths of cut (heq = 4 and 7nm), the abrasive grits hardly penetrate the workpiece. This leads to high plastic deformations (ploughing is predominant) and, consequently, a non-negligible part of the power is consumed mechanically, the assumption that all consumed energy is transformed into heat is not fulfilled. Increasing the grit penetration with a higher equivalent chip thickness implies that grits are able to remove the material by shearing it. The plastic strains become less important, all energy can be assumed to be transformed into heat and consequently the part of the consumed energy directed as heat to the workpiece is higher. It has been possible to determine this effect thanks to the tight control of heq (not achievable in conventional grinding tests) and to the accurate temperature measurements. Separation of ploughing and shearing effects in real grinding is not possible since the grit passes through both phases during its contact with workpiece material. Thus, the separated contribution of ploughing and shearing has not been addressed before.

During the contact of the grit with the workpiece chip thickness, hcu, varies, from a minimum (ploughing predominant) to a maximum (shearing predominant) in up grinding, or vice versa in down grinding. This assumption would imply that as long as the chip thickness varies Rw so should vary too, as it is shown in Figure 7 4.2. Uncertainty on the sensor location An analysis is realized to determine the error on the Rw value regarding the uncertainty on the sensor location. Because of closeness of the thermocouple on high thermal gradient area, an error on the position could induce an important error on Rw value. The inverse method is then performed by choosing another node in the model close the initial node. The location is varying in the three directions for a value of +/-0.4mm. This procedure allows simulating a different sensor location unless another experimental test. For each condition, it gives eight optimization calculations. Results are showed on Figure 7 for 3 experimental tests. The uncertainty study is only performed for timedependent Rw and is plotted regarding the material removal volume. For the 3 configurations, we observe a more significant uncertainty span at the beginning. This span reaches +/-0.05 on the first time step but rapidly decreases with the material removal. After a removal of 10 mm 3 , it becomes under +/-0.02. Besides it can be noticed that the tendency of Rw to decrease with material removal for each condition. The validation consists of comparing the hardness of the workpiece near the contact zone after the test, to the theoretical hardness obtained from the tempering model, described in Section 2.2. Since tempering is temperature and time dependant phenomena, if the final hardness is similar it can be deduced that the temperature history is the same. For the assessment of the theoretical hardness the temperature evolution of the FEM with the identified time-dependant Rw is used.

The parameters used in equations 9-11 for the calculation of hardness are: U1 is in the range of 80 -90kJ/mol according to tempering curves in [START_REF]Carbon and alloy steels[END_REF]. The value of the attempt frequency A1 is set to 10000s -1 . The fully-tempered H1 and the fully quenched materials H3 have a hardness of 300 and 730HV respectively. The initial hardness value H0 has been measured on each workpiece before the experimental test.

Figure 8 shows the comparison for the Test 3 using Wheel 1 along the length direction. Two zones can be clearly seen: the rehardened zone close to the friction zone and the tempered zone. In this case, the deviation between the simulated and the measured curves of micro-hardness is below 5%. Although it is evident, it must be reminded that in the region on the left hand side of the re-hardening curve has occurred due to very high temperatures. Therefore the model for hardness prediction used at this point is no longer valid. Table 3 gathers the maximum and mean deviations between the measured hardness and the theoretical hardness obtained with the model using the variable Rw. In the worst case, the upper limit of the deviation punctually reaches 20% and the maximum difference is under 10% for mean values. This is an acceptable result taking into account the dispersion in micro-hardness measurements. This metallurgical study tends to confirm the results obtained for the Rw values and the efficiency of the methodology.

CONCLUSIONS

In this paper an original methodology and experimental set-up that contribute to a better understanding of the influence of grinding variables on the heat partition ratio Rw has been introduced. From the work carried out, the following conclusions can be drawn:

 With respect to existing literature, the new methodology eliminates the errors induced by the steep thermal gradients that occur in grinding operations. Experimental results showed that heating of the workpiece is progressive, and Exp.

Num

Re-Hardened material

Re-Hardened material the temperature increase in the workpiece is not so steep (100K/s), compared to the 10000K/s that can be found in conventional grinding tests. In addition, uncertainty related to the actual area of contact between both bodies (wheel and workpiece) is also suppressed by using a rigid and controlled geometrical configuration.  Results have led to a time-dependent definition of Rw which had not been previously proposed in literature. In all tested conditions a similar behavior is observed: high Rw initial values that decrease to stabilized smaller values. Initial values range from 0.9 to 0.75 depending on heq, descending to stabilized values between 0.7 and 0.5, respectively. This behaviour could be due to local variations on the contact surfaces as a consequence of abrasive grits wear and to the apparition of a third body.  For cutting conditions were the chip thickness (heq) is very small (4-7nm), the predominant material removing mechanism is ploughing, and results show that Rw varies between 0.50 and 0.55. By contrast, when the depth of cut is increased (chip thickness > 15nm) the prevailing material removing mechanism is shearing. Under these conditions the Rw is between 0.7 and 0.8. In the bibliography, there is no mention to the effect of material removing mechanism in Rw. The justification is that from all the energy available in the contact when ploughing is predominant a non negligible part of it is consumed in plastic deformations, whereby less amount of the consumed energy is transformed into heat and directed to the workpiece.  A metallurgical study offers the possibility of comparing the numerical temperature with the experimentally reached temperature, by means of a indirect parameter: hardness. The theoretical hardness profile along the workpiece thickness matches micro-hardness measurements (mean deviations between predicted and measured hardness on the range 9.5-1.5%). This implies that the temperature field obtained numerically after Rw identification is accurate enough and is able to predict the subsurface steel hardness after a grinding operation.

 Future work will include microscopic observation of the surface of the abrasive grits at different instants of the transient period, in order to optically identify the growth of the flat wear areas and the appearance of a third body that validates the above hypotheses of the time-dependant Rw.

Figure 1 -

 1 Figure 1 -Experimental Test Bench description

1 -Figure 2 -

 12 Figure 2 -Boundary conditions and meshing of the 3D finite element thermal model

Figure 3 -

 3 Figure 3 -Inverse method numerical validation

Figure 4

 4 Figure4and 5 shows the results obtained for test 2 and 4 of Table2using Wheel 1 and Wheel 2, respectively. The evolutions of both, experimental and simulated temperature, as well as the value of the heat partition to the workpiece are plot. Regarding the experimental temperature, it must be noted that the measured signal is very stable and its increase is gradual (gradients below 100K/s). This is an important advantage for the accuracy of the results face to the conventional grinding tests. One of the problems of the thermal test in grinding is that the temperature varies so fast (10000K/s, in some cases[START_REF] Kohli | Energy Partition to the Workpiece for Grinding with Aluminium Oxide and CBN Abrasive Wheels[END_REF]) that the thermocouple is not able to accurately measure it. As it can be noticed from Figures4 and 5, errors related to this effect are minimized.

Figure 4 -Figure 5 -

 45 Figure 4 -Temperature and estimated Rw value for heq=7nm -Wheel 1

Figure 6 -

 6 Figure 6-Stabilized Rw value as the function of the equivalent chip thickness heq

Figure 7 -

 7 Figure 7 -Error on Rw values regarding uncertainties on thermocouple location

Figure 8 -

 8 Figure 8 -Experimental vs simulated hardness for heq=18nm -Wheel 1

Table 2 -

 2 Qw' by the grinding wheel speed vs. The process parameters values are gathered in Table2. The low Qw' values of the Test 1 and 2 correspond to very low values of heq where ploughing is predominant. These values have been selected in order to assess the influence of the different material removing mechanisms in heat partition. The specific volume of part material removed, Vw', is small: 10mm 3 /mm, in order to avoid the effect of wear of the abrasive grits. Process parameters values

	Test	Qw'	heq	vf	vs	Vw'
		[mm 3 /mm s]	[nm]	[mm/min]	[m/s]	[mm 3 /mm]
	1	0.083	4	1	24	10
	2	0.17	7	2	24	10
	3	0.42	18	5	24	10
	4	0.83	35	10	24	10
	5	1.25	53	15	24	10

TM

Cubitron TM 321 alumina)

Qw' varies from a very low value (Qw'=0.083mm 3 /mm•s) to typically met value in grinding finishing operations (Qw'=1.25mm 3 /mm•s). The equivalent chip thickness heq is deduced by dividing

Table 3 -

 3 Maximum and mean deviations between theoretical and measured hardness values

	Qw'	heq	2A46I8VWEB2	30SA46I8VWEB2
	[mm 3 /mm s]	[nm]	Mean	Max.	Mean	Max.
	0.083	4	5.39%	13.43%	9.37%	20.05%
	0.17	7	4.95%	9.35%	5.44%	13.26%
	0.42	18	2.38%	5.05%	3.82%	13.67%
	0.83	35	1.70%	6.42%	3.59%	6.78%
	1.25	53	9.58%	18.23%	3.54%	11%