Katia Lupinetti
email: katia.lupinetti@ge.imati.cnr.it

Franca Giannini
email: franca.giannini@ge.imati.cnr.itmarina.monti@ge.imati.cnr.it

Marina Monti

Jean-Philippe Pernot
email: jean-philippe.pernot@ensam.eu

CAD ASSEMBLY DESCRIPTORS FOR KNOWLEDGE CAPITALIZATION AND MODEL RETRIEVAL

Keywords: Assembly representation, assembly retrieval, multilayered search, multi-modal descriptors

Today, there exists a huge amount of digital data easily downloadable from Internet and/or simply accessible from large databases. Despite this rise, the methods to retrieve and search for specific data have not been sufficiently studied and developed, notably when considering 3D contents. Thus, it is sometime more efficient to define new 3D shapes starting from scratch rather than to try to make use of existing ones hardly identifiable within those databases. This is particularly true when considering CAD assembly models often resulting from a long and time-consuming modeling phase within the Product Development Process. Thus, having new methods, models and tools to capitalize, retrieve and reuse CAD assembly models would help saving a lot of time. This paper addresses such a difficult problem of finding a method to characterize and structure CAD assemblies so as to be able to search for similar ones. A framework has been designed for the retrieval of globally and/or partially similar assembly models according to different user-specified search criteria. It is based on an assembly descriptor, called the Enriched Assembly Model (EAM), which encodes all the required data automatically extracted from the geometry and structure of the CAD models. The data are organized in several layers thus enabling multilevel structuring and queries. It also allows fuzzy queries, which can be further refined.

INTRODUCTION

CAD models have become mainstream in many industrial applications. They can be considered as digital product reference models stored within a Digital Mock-Up used and shared all along the Product Development Process [START_REF] Falcidieno | Processing free form objects within a Product Development Process framework[END_REF]. Over the last few years, CAD model databases grew up so much that it has become challenging to develop new searching and browsing methods and tools. This is notably true for what concern the structuring, the access and the reuse of those databases. Actually, being able to reuse existing information and data, including existing 3D models, became a crucial factor for the competiveness of the industries whose behavior is often driven by the wellknown triptych cost-quality-delay. Being faster in developing or releasing new products is a key issue.

During the design process of a new product, the ability to retrieve existing models, either parts or assemblies, can be useful to reach several objectives as reusing an existing assembly in new configurations, or providing access to existing design knowledge (e.g. simulation results, manufacturing phases) related to similar products [START_REF] Gupta | Content-based search techniques for searching CAD databases[END_REF] or to identify similar configurations that could benefit from a standardization. However, such a retrieval process is not straightforward when considering CAD models potentially made of several hundreds of thousands of parts. Thus, specific search techniques have to be developed and optimized to be able to retrieve elements in a reasonable time.

In simple text-based retrievals, the user types a list of words or sentences that he/she wants to search in the database. However, with this technique, some models are not retrieved, even if they are semantically related with the query since they have not the same text in their annotations. These limitations may be overcome by using search methods based on thesauri, i.e. collections of controlled vocabulary terms that use associative relationships. However, these techniques are not sufficient since annotations may not be present and there is no guarantee of compliance to name conventions; moreover, they do not consider the shapes of the parts. Anyhow, in case of complex products made of several parts a method based only on the shape is not sufficient for retrieving the target assembly model. Actually, 3D models with similar shapes can be assembled in different ways, involving different kinematic characteristics and then different relationships between their parts. For example, an assembly of two parts with 5 screws can be considered similar to the same assembly with 6 screws. In this case, at a higher level, what is important is that the two parts have been screwed whatever the number of screws. Thus, an advanced search method has to incorporate mechanisms working at different levels (e.g. geometry, kinematic, annotation).

For the effective re-use of existing models, contentbased methods should also allow queries without the specification of a CAD model as input. This possibility is particularly important since at the early design stage, the designer would be interested in expressing vague queries, e.g. simply by specifying some attributes of the assembly model, just to take inspiration from the available models. Therefore, the challenge is to find an assembly representation able to support user requests at different level of specification details but also that is automatically defined. Indeed, to require the user to add manually some information may represent a heavy limitation.

In this context, it is crucial to provide a tool for the retrieval of assembly models which can be tailored to the user needs, and thus, that is able to consider the assembly and the part shapes, the interlinks between sub-assemblies and parts and other aspects that are implicitly stored in the 3D data.

In this paper, we propose a framework for the re-trieval of globally and/or partially similar assembly models according to different search criteria that can be convenient for designers. It is based on an assembly descriptor, called the Enriched Assembly Model (EAM), which encodes all the required data extracted by analyzing the geometry and structure of the CAD model without user intervention. It allows fuzzy queries, which can be further refined and applied on search results. Section 2 reviews the related works. The EAM model is introduced in section 3 and the complete framework in section 4. The last section ends this paper with some conclusions and perspectives.

RELATED WORKS

Gupta et al. have categorized the various use scenarios for content-based assembly search as follows [START_REF] Gupta | Content-based search techniques for searching CAD databases[END_REF]:

• Design reuse. The retrieving of an existing similar model allows the designer to make modifications to it and to obtain faster a new product. • Accessing the related existing knowledge. The availability of information related to similar models (e.g. costs, reliability and failure reports, shape adaptation processes for simulation and analysis results) makes easier the evaluation of the new product solutions.

In addition to these use cases, one can also imagine having mechanisms to identify similar CAD models in order to be able to standardize parts or assemblies.

In literature, a lot of works focuses on part retrieval. Tangelder and Veltkamp [START_REF] Tangelder | A survey of content based 3D shape retrieval methods[END_REF] and Iyer et al. [START_REF] Iyer | Three-dimensional shape searching: state-of-the-art review and future trends[END_REF] provide a complete state-of-the-art on 3D shape retrieval. However, these techniques focus on shape characteristics and do not take into account other important aspects for assembly models' similarity, such as the relationships between the parts and are not able to retrieve similar assembly structures or more simply a sub-assembly in another assembly.

There are various works addressing the analysis of assembly models for diverse objectives, such as the identification of assembly constraints or the assembly sequence planning; only a limited set of works directly addresses the search for assembly models similar to a given one.

To the purpose of our research, we identified some useful criteria for a comprehensive analysis of the works directly or indirectly addressing the identification of similarities in assembly models under different perspectives. The adopted criteria can be grouped into the following macro-categories:

• context,

• type of information exploited,

• type of assembly descriptors defined,

• query model. 1 summarizes the analyzed works directly aimed at the retrieval of similar assembly models according to the above criteria. Among these works, only two adopt a more comprehensive approach [START_REF] Chen | A flexible assembly retrieval approach for model reuse[END_REF][START_REF] Deshmukh | Content-based assembly search: A step towards assembly reuse[END_REF]. Deshmukh et al. take into account many different aspects that play a meaningful role in the description of an assembly product [START_REF] Deshmukh | Content-based assembly search: A step towards assembly reuse[END_REF]. The main limit of the presented work is in the required availability of several important information (such as the component orientation, component relationships and the joint constraints). A global approach, which aims to overcome this limitation, is proposed by Chen et al. [START_REF] Chen | A flexible assembly retrieval approach for model reuse[END_REF]. This work does not consider the overall assembly shape but focuses on the product structure and the relationships between the different parts of the assembly.

The assembly descriptor presented in this work takes into account different information levels. It includes topological structure, relationships between assembly components, and geometric information. It permits the use of rough and partial incomplete queries to allow a search completely adaptable to the designer requirements. The framework we propose is based on an assembly model that similarly to the ones presented in [START_REF] Chen | A flexible assembly retrieval approach for model reuse[END_REF] and [START_REF] Deshmukh | Content-based assembly search: A step towards assembly reuse[END_REF] is able to support user requests at different level of specification details but differently than [START_REF] Deshmukh | Content-based assembly search: A step towards assembly reuse[END_REF] does not require the user to add manually some information. Differently than [START_REF] Chen | A flexible assembly retrieval approach for model reuse[END_REF] the mapping algorithm is not limited to the identification of assembly models with the same structure in terms of sub-assemblies. Moreover, the framework does not rely on a specific CAD native file format, but requires as input a STEP file describing the assembly model, in which only geometric information is available and allows the retrieval of assembly models of which the query model is a sub-set.

THE ENRICHED ASSEMBLY MODEL

In order to allow an efficient and meaningful retrieval of CAD assembly models, a suitable description of the assembly should be provided. As previously described, an effective reuse of already existing models requires the possibility to perform search with queries using incomplete information and involving characteristics which might be not explicitly encoded in the CAD models, but highlighting features valuable for the search purposes. To this aim in the following, we propose a new shape descriptor called Enriched Assembly Model (EAM), which organizes information in a multi-layer structure aimed at guaranteeing the search flexibility. This permits search queries involving one or several layers of information in various combinations. The considered layers are: structural layer, statistic layer, shape layer and interface assembly layer.

Structural layer

The structural layer of the EAM encodes the hierarchical assembly structure of the CAD model as specified by the designer. At this level, information is stored as a tree graph in which arcs indicate the relation part-of and nodes correspond to the assembly elements. In particular, leaves represent the parts of the assembly model, the root the entire assembly model, while intermediate nodes represent subassemblies of the original model. Attributes are associated to leaf nodes to indicate the type of the corresponding component: fastener (e.g. bolts, screws) and principal. Here, the rational is that not always the geometry of fasteners is fully specified but they may be simply indicated as annotations. Therefore distinguishing between principal shape components and fasteners allows to possibly discard the second ones during the comparison, thus permitting to focus only on the most specific parts of the assembly.

Additionally, each component is classified as thin or normal part, such that if the comparison among the shape of the single components is required in the query, it can be performed only on compatible ones discarding the others. The thin part class includes different types of objects, such as those similar to thin plates, or having an arbitrary shape with almost constant thickness distribution, or presenting a large emptiness in its bounding box. Different criteria are used to distinguish the typology of thin objects as described in [START_REF] Pernot | Categorizatiotn of CAD models based on thin part identification[END_REF]. For subassembly nodes, we consider the shape resulting from the Boolean union operation performed on the subassembly components, such that a single object is obtained on which the classification into thin or normal part and the various shape descriptors are computed. Such an overall assembly shape is useful to guarantee the possibility of evaluating similar assemblies from the structural and shape point of view, such that subassembly can be treated as it were a single unique part. Such decomposition into parts, subassemblies or a single component resulting from the merging of all the parts and subassemblies also correspond to the way designers may focus on a product. Actually, designers can consider the product either as a whole object with its characteristics (e.g. volume, gravity center), or focusing on subassemblies (e.g. for kinematics purposes) or parts (e.g. for manufacturing issues). Figure 1 shows an example of the structural layer of an assembly model. The object is an engine formed by three sub-assemblies (S1, S2, S3): a piston, a crank shaft and a mass and two linking parts (P3, P10).

Example of structural layer of an assembly Figure 1 model

It is important to notice that the assembly decomposition depends on the context and, even if it represents a semantic organization, it is not unique. For example, the assembly can be organized in a way that the forthcoming assembly simulation steps are eased, or it can be organized with respect to visualization issues according to an octree-based decomposition, or it can be decomposed according to criteria based on the constitutive materials. Therefore, assembly similarity cannot strongly require same structures if not specifically requested by the user. However, we believe that there is a level of decomposition under which multiple decompositions of a same product will remain similar, and would in this case correspond to the smallest common denominators. Such an understanding can be performed at the level of the parts and subassemblies, but also at the level of the joints between the elements constituting the global assembly. This point is not further discussed in this paper.

Finally, all the elements in the tree are associated to data in the other information layer to fully characterize them.

Statistic layer

This layer collects several information that can be represented in form of numerical values for allowing a quick search and filtering. Statistics referring to the overall assembly or to sub-assembly nodes are:

• number of sub-assemblies,

• number of principal parts,

• number of fasteners,

• number of thin parts,

• number of linear patterns of repeated components,

• number of circular patterns of repeated components, • number of reflective patterns of repeated components.

While statistics referring to the parts include:

• percentages of specific type surfaces (i.e. planar, cylindrical, spherical, free form, toroidal) with respect to the overall area, • numbers of maximal faces (i.e. adjacent faces sharing the same underlying surface are considered as a single face) of a specific type surface (i.e. planar, cylindrical, spherical, free form, toroidal).

Statistics related to assembly interfaces are the number of elements in contact for each type (e.g. planar pair, planar-cylindrical pair).

The inclusion of information related to the presence of patterns can support the search of similar models having similar manufacturing or assembly processes.

Of course, based on this first list of statistics, additional descriptors can be used to enrich the characterization of the assembly model at different levels of the structural layer.

Shape layer

The shape layer includes the specification of the information related to the shape of the elements in the assembly. Useful for their comparison, this layer can also be interesting for the visualization. In particular, for each node of the structural layer (both parts and sub-assemblies), two mesh representations are associated together with several shape descriptors. The first is a rather precise model, the second corresponds to its rough representation (e.g. convex hull or more simply its bounding box). These double representations allow both a fast browsing of the objects and of the search results as well as the specification of both precise and rough queries with unprecise shapes. This is quite useful during product design when the shape is under development and it is quite reasonable to search for possibly re-usable products that share the main behavioral (e.g. degree of freedom) and overall shape characteristics. For each intermediate node (i.e. nodes associated to a sub-assembly), meshes and shape descriptors are provided and computed as previously described.

The choice of considering different shape descriptors associated to a single shape is due to the fact that it has been demonstrated that there is no best shape descriptor for the comparison of each shape type [START_REF] Iyer | Three-dimensional shape searching: state-of-the-art review and future trends[END_REF].

On the contrary, depending on the type of object a specific shape descriptor performs better. In our model, we intend to consider descriptors related to both the overall component and to its shape variation. Among the descriptors of the first type, we consider the volume and the surface area, which are size dependent. The other shape descriptors considered for the nodes in the structural graph are the spherical harmonics and shape distribution, which perform well with prismatic parts and shapes of revolution of which are mostly composed the mechanical products we are considering [START_REF] Iyer | Three-dimensional shape searching: state-of-the-art review and future trends[END_REF]. Therefore, depending on the values related to the percentages of surface type of the component (at the statistic layer), the comparison is activated to the concerned descriptor or to a weighted combination.

Interface assembly layer

This layer encodes the relationships between the different parts in an assembly model regardless the assembly organization. Therefore, it consists of arcs linking leaf nodes. Links between sub-assemblies and the rest of the assembly components can be obtained by simply considering the arcs that do not join the sub-assembly components. The possible relationships between two parts can be grouped into contact, interference and clearance [START_REF] Shahwan | Deriving Functional Properties of Components from the Analysis of Digital Mock-ups[END_REF] as shown in figure 2.

Two parts are in contact, if they touch along lowlevel geometric entities such as faces, edges or points without any shared volume. Two parts define an interference if a common volume exists between them. This relation cannot exist between two real objects, however during the modeling phase it may happen for instance after local shape simplification for simulation.

Possible relationships between parts Figure 2 Another possibility is due to some tolerances in the modelling, i.e. still at the intermediate design stage where dimensions are not fully tuned or there are tolerance issues. Since interference should not exist in a correct assembly model, they are not included in the EAM and, if present, are treated as contact. Clearance occurs when the distance between two surfaces of two parts is meaningful for the considered assembly, i.e. it is a small non-null distance between two parts in the assembly. This case is rather ambiguous since the design intent can correspond to both non-contact and contact configurations (i.e. due to tolerance problems). The interface layer is itself a multilayered one. Clearance and contacts are further detailed in terms of the types of kinematic pairs between parts, i.e. number and types of geometric elements, which are (almost) mating. At the lowest level, the couples of geometric elements (points, edges or faces) in contact of the two parts are specified, while at the highest level these are classified according to the resulting assembly feature [START_REF] Ullah | Assembly Features: definition, classification, and usefulness in sequence planning[END_REF] obtained combining the effect of kinematic pairs. In particular, we classify the contacts and clearance as positioning and interlocking types [START_REF] Chan | Generating Assembly Features Onto Split Solid Models[END_REF] depending on the final degree of freedom (DOF) they allow. In particular, interlocking configurations are those that prevent any direction movement, i.e. with zero DOF (see Figure 3). Finally, arcs have associated the attributes specifying the DOF between the linked parts in the assembly.

THE RETRIEVAL FRAMEWORK

CAD systems use different and proprietary file formats, able to store all the information directly inserted by the user during the design of the specific parts.

The content may vary according to the type of functionalities provided by the specific CAD system, therefore building a generic retrieval system cannot trust on the presence of specific data. Furthermore, especially when different companies are working together for the specification of complex products, standard formats for the CAD data exchange are generally used.

Thus, in our framework we adopted the STEP standard format to read the assembly models. In this format many information, such as the relationships between the components, are not stored, while some annotations may be included as functional information in the file. Anyway, we consider only the geometric data since the user provided annotations might be by their nature not accurate.

The framework considers both an off-line process and real-time processes. Off-line process is adopted for the evaluation of the EAM for all the models in the databases on which the search has to be performed. Real-time processes evaluate the descriptors on the query and accomplish the comparison with the stored models. It must be noted that the EAM is a very rich model, including many information on the represented assembly. Some information is apparently redundant, but it has the advantage to allow scalable queries. Therefore, a complete EAM version is computed only for the stored models, whereas for the query only those layer information assessable on the required detail are computed and exploited for the matching, thus reducing the complexity of the system.

Architecture

The architecture of the framework is illustrated in Figure 4 and it shows the different modules and how they communicate at the different levels. There are three main layers: the user interface layer, the functional layer and the data layer.

The user interface layer is responsible of the visualization tasks and user interaction activities. It provides support for i) the visualization of the EAM; ii) the search query specification; iii) the browsing of the search results. With the graphic user interface the designer may easily specify queries, search criteria, filters for the visualization of the EAM and the dataset to be used for the retrieval process. The EAM visualization module allows picturing an assembly model by means of a graph structure, whose nodes can be selected to enquire the content at the different levels of details. The geometry of the overall assembly, as well as the one of the intermediate nodes and leaves, can be previewed, thanks to stored mesh

The framework architecture Figure 4 models.

The functional layer contains the main modules of the framework. It manages the creation of an EAM from CAD assemblies in which explicit and implicit information present in the CAD model are made available. Moreover, it deals with the creation of the EAM of the query using either an existing model (i.e. a STEP file) or an abstract model specified by the user through the user interface layer.

The last layer is the data layer, which provides the input for the functional layer and contains the elaborated data.

At the operative level, as first step, it is necessary processing the databases used for the retrieval. Each file of the dataset is processed by the module "EAM creation". In this phase, a file is created where all the extracted information are archived.

Different possibilities are foreseen for the assembly search. If the user wishes to retrieve all the models similar to an existing CAD model, the module "EAM creation for query model" generates its assembly descriptor using the "EAM creation" as for the models in the database. Conversely, if the user is interested in finding models with some specific structural characteristics, the "abstract query model creation" module creates the corresponding parts of the assembly descriptor. This means that the user, through the interface, can build a graph specifying several attributes, such as a rough component shape. The user indicates if a node is a part or a subassembly and assigns its attributes. The same happens for the arcs, which can express parental or contact relationships. This operation is possible thanks to the interaction with user the graphic interface module, which allows the user to specify the abstract query in graph format.

When the dataset is entirely processed and the user specified his/her own search criteria, the search process is managed by the "matching" module. This module communicates with the user interface layer for displaying the obtained results with their associated measures.

Module descriptions

EAM creation

Since STEP file contains only part of the information required for creating the enriched assembly model, several geometric reasoning processes on the assembly model need to be performed to extract the missing information and generate the multilevel assembly descriptors. When reading the STEP file the compo-nents and links of the structure layer of the EAM are created from the hierarchy of the assembly model. For each leaf, statistics are computed. Then, for the whole assembly and the related sub-assemblies, patterns of repeated components are detected using the module described in [START_REF] Chiang | Identification of regularities in CAD part and assembly models[END_REF], which identifies linear, circular and reflective arrangements of parts. Generally, repeated components are explicitly indicated in the STEP file. In case no indication is present, components are considered repeated when presenting the same values for the associated statistics (i.e. number of faces of a specific type and related area percentage) and for the volumes and surface areas. Of course, such criteria do not fully characterize repeated components but represent necessary and easy to check conditions to identify them.

Then, for all the sub-assemblies, the resulting shapes are evaluated (see section 3.1). For each of them and for each leaf in the assembly, the corresponding tessellation (if not already present in the dataset) is computed together with the corresponding shape descriptors. Tessellation is also used to detect the typology of the type of the corresponding part (i.e. thin or normal) according to the methodology described in [START_REF] Pernot | Categorizatiotn of CAD models based on thin part identification[END_REF], which uses D2 shape distribution and ratios among the bounding box characteristics (e.g. diagonal, width) to distinguish parts, see figure 6. The interface assembly layer is then created by analyzing the reciprocal relationships between components. This kind of information is not stored in the STEP file, therefore we have to detect the possible part interactions only by exploiting the geometry data available in the STEP file. We use the functionalities provided by the API of the commercial system SolidWorks for the detection of the contact and the clearance. This functionality allows accessing to the faces (or edges or vertices) formed by the nonregularized intersection formed by two parts.

Examples of circular and linear patterns of

The module for detecting and evaluate the relationships between parts includes the following tasks: i) Detection of clearances. According to the description provided in the interface layer, we retain the clearances with distance lower than a given threshold.

ii) Identification of parts in contact. For each clearance, we identify the involved parts.

iii) Identification of contacts between parts. We compute the non-regularized intersection between the parts having null clearance. iv) Identification of kinematic pairs. The analysis of the typology of the elements involved in the non-regularized intersections allows the identification of kinematic pairs. For example, if a planar/cylindrical face is involved in the nonregularized intersection, it indicates that the two parts are in contact through two planar/cylindrical faces. In case of points and curves in the non-regularized intersection, additional geometric verifications are needed to identify the type of the faces. This additional checking is not presented in this paper.

v) Identification of the type of contact. Connected parts of the non-regularized intersection give rise to specific degree of freedom. They are further analyzed by combining the kinematic pairs' degree of freedom [START_REF] Chen | A flexible assembly retrieval approach for model reuse[END_REF] to detect weather they correspond to just a positioning relation or if the components are interlocked.

Finally, to detect if components correspond to fastener elements or principal ones, whenever they are not explicitly annotated, a software module is foreseen. It is worth to repeat that the aim of this classification is to guarantee that very similar products can be retrieved independently on the fact that specific standard fasteners are explicitly modelled or not. Therefore, this classification can be quite rough, just indicating negligible parts for the comparison.

Thus, only components having a volume within a certain range will be analyzed and compared to reference shapes of the most suitable fastener category. The choice of the fastener category will be based on the number of components in contacts. Components not satisfying the similarity threshold will be classified as principal.

Query EAM creation

As mentioned before, the graphical user interface provides various functionalities to specify the query, which include the creation of an abstract EAM model, in which the various layers can be fully or partially detailed. To specify the query, the user can select all the characteristics stored in the EAM. In this process, the interesting values can be specified by the user or automatically computed from the provided example, which can range from a precise CAD assembly model to an abstract assembly graph (Figure 7).

When the query corresponds to an existing CAD model, the EAM is created using the same "EAM creation" module previously described. If during the query specification, the user relaxes some characteristics, which he/she considers irrelevant, the corresponding evaluation procedures are ignored.

In case of an abstract query, the user provides its ideas without building an assembly model in a CAD system. For abstract queries, the mandatory information is the number of constituting principal components and related interface links. The abstract query creation is allowed thanks to the user interface in which the user can add some nodes specifying different attributes linking the nodes through arcs and specifying the type of the arc, i.e. a structural arc or an interface arc and in the last case with the label regarding the constraint. The attributes that the user can specify are those described in the different layers of the enriched assembly model descriptor. In this case the attributes specified by the user are used as search criteria during the matching process.

Optionally a percentage of allowed variation on the various elements can also be inserted to speed up the retrieval process. It allows a pre-filtering of the candidate most similar models through the verification of the concerned statistics values.

Matching

The matching problem is faced at the different levels of the EAM in a top down manner. If the user expresses ranges in which two assemblies are considered similar, e.g. allowed percentage of different components or relations, a filtering can be applied based on the concerned statistics to reduce the number of models to be compared.

The problem of finding the matching between two assembly models is translated into the problem of finding an isomorphism between two EAMs based on the specified criteria.

The EAM can be seen as an attributed graph structure. Thus, a partial shape correspondence between two graphs corresponds to the problem of finding their maximum common sub graph (MCS). Among the various techniques proposed for the identification of the MCS [START_REF] Bunke | A comparison of algorithms for maximum common subgraph on randomly connected graphs[END_REF], the chosen strategy is to solve the problem of the identification of the maximum clique (MC) [START_REF] Pelillo | Replicator equations, maximal cliques, and graph isomorphism[END_REF]. To compute it, an association graph is constructed in which nodes equivalent in the two attributed graphs to be compared are mapped into a single node. Similarly, arcs in the associated graph are present when the corresponding nodes in the attributed graphs are connected in the same way. The maximum clique thus corresponds to the maximum set of nodes all connected together of this newly defined association graph. In our system, the maximum clique finding problem is solved using an existing method exploiting the simulated annealing technique and described in [7][17].Two nodes are considered equivalent if they have the same attributes specified by the user through the search criteria accessible from the interface. To limit unnecessary comparisons, if the search is looking for assemblies similar in all the aspects, at first the comparison is performed on the structural nodes and assembly interface layers, then the geometric matching on the part geometry is performed only on the returned corresponding candidates. In this case, nodes are then considered equivalent if they are associated to the same type (principal or fastener) of components presenting the same type of shape (thin or normal). Similarly, arcs are considered equivalent when at the interface layer, corresponding arcs have the same classification and DOF. Statistics information at the node and interface layer are then used to adjust the similarity ranking. Being the assembly comparison important at the various information layers, the matching process provides as result a vector of measures: the first measure refers to the identified cliques, the second to the interface statistics, the third is related to the component shape similarity.

Figure 7 illustrates an example of assembly retrieval results obtained applying various criteria at different stages, thus allowing a refined browsing of the data set. The model used as query has two main parts (one is the reflection of the other) and a set of screws and a bolts (both arranged in a circular pattern). At the first stage, the search is looking for models in which there are two parts in contact with the same number of possible rotations and translations as in the query model. The assemblies that satisfy these requirements locally are shown in the second row in Figure 7.

In the second stage the retrieved set of assemblies is filtered according to the pattern type of the query model. Thus, the system retrieves the models with a reflective pattern and two circular patterns. This step allows reducing the number of false positive retrieved before.

In the last stage, dimension information are used to filters the achieved results such that the returned assemblies (fourth row in Figure 7) present the same volume with 30% of tolerance. Last row in the figure presents some of the models not retrieved at the first stage.

This example shows that for searching assemblies a single key is not sufficient to have adequate results, while combining several queries the results improve. In this sense, the proposed approach helps finding similar assembly while being closer to the user intent. If the user really wants an assembly made of 4 screws, only such assemblies will be retrieved. However, if the user is interested in screwed assemblies, they will all be retrieved whatever the number of screws.

CONCLUSIONS AND FUTURE WORKS

In this paper, a framework for the retrieval of similar CAD assemblies has been proposed. It is based on a three layers architecture. CAD models available in the database are pre-processed off-line and enriched with data extracted from them. Once enriched, the so-called Enriched Assembly Models are available and ready for the matching step. Then, a query model is created by the user and compared to each EAM of the database using specific criteria. Both the creation of the query model and the criteria used for the comparison can follow different scenarios.

The proposed Enriched Assembly Model is a first step toward the definition of a unified multi-level structure for CAD assembly description. Together with the definition of an advanced hierarchical matching process, it helps retrieving CAD assemblies in huge databases.

Our approach has the following advantages. It allows the automatic computation of the joints between the components, as well as a pre-classification of the parts so as to speed up the matching step. Moreover, query models should not be fully specified thus enabling fuzzy queries very useful in the early design phases where the product specifications are not fully known.

The proposed EAM is open so as to incorporate more advanced shape descriptors enabling more accurate retrieval of CAD assemblies defined by such complex features. The framework is under development and we are currently completing the module for the automatic identification and interpretation of the assembly component interface.

  Examples of thin (a, b) and normal parts (c) Figure6

Figure 7

 7 Figure 7 Example of assembly retrieval

Paper Input model Work objective Geometric/ topological infor- mation used Assembly relation- ships Assembly descriptor level Descriptor represen- tation Query model

					The type of assembly descriptors includes indications
					on the assembly description level, i.e. (i) assembly
					level: the assembly is described by its parts and/or
	The context includes the work objectives and the required type of input data (i.e. B-Rep, 3D mesh or point cloud). The type of information refers to the information (i.e. geometric and/or topological in-formation) the authors use to characterize the as-sembly model. Either this information can be implic-itly coded in the input data or it can be explicitly associated to it. Moreover, it specifies if the assem-bly relationships (i.e. the relationships between the assembly components) are explicit in the CAD native models, automatically derived from assembly geome-try or specified by the user.	their relationships; (ii) part level: the assembly is described by its parts; (iii) feature level: the assembly is described through shape portions having specific assembly meaning. Moreover, it indicates how the assembly descriptors are implemented, i.e. the de-scriptor representation used to store the infor-mation, such as graph or vector. The query model indicates the type of data required to express the query (i.e. a single CAD assembly model, a set of CAD assembly models, a set of CAD part models such as a sub-assembly without the con-straints, a 3D mesh, or an abstract assembly de-
	[20]	B-Rep	Search for frequent simi-lar sub-assembly models	Curvature, Model components, Mating constraints	model Already existing in the CAD	Assembly Part	Face Adjacency Graph	Assembly model
			Search for
	[10]	3D mesh	globally simi-lar assembly	Model components	none	Part	Component Vector	Assembly model
			models
	[3]	B-Rep	Search for globally or partially simi-lar assembly models	Model elements Mating constraints Degree of freedom annotation	Partially extracted	Assembly Part	Hierar-chical graph	Assembly model
	[18]	B-Rep	Search for globally simi-lar assembly models	Model elements Surface properties Contact relations	Manually inserted	Assembly Part	Component attributed graph relation	Assembly model
	[12]	B-Rep	Search for globally simi-lar assembly models	Shape characteris-tics Mating constraints	Already existing in the CAD	Assembly Part	Attributed Assembly graph	Assembly model
					model
	[5]	B-Rep	Search for globally or partially simi-lar assembly models	Model elements Mating constraints Degree of freedom Annotation	Already model existing in the CAD	Assembly Part	Mating graph	Set of statistics Mating graph

Table 1

 1 Summary of works directly addressing assembly models' retrieval scriptor).

	Table


Amélie Morançais

Mathieu Fèvre

Manuel François

Nicolas Guel

Serge Kruch

Pascale Kanouté

Arnaud Longuet

FATIGUE LIFE OF A SHOT-PEENED NICKEL-BASED SINGLE CRYSTAL SUPERALLOY : FROM MEASUREMENTS TO MODELING

Keywords: Residual stress, Single crystal, Superalloy, X-ray diffraction, Lifetime prediction

In this work, X-ray diffraction measurements and finite elements calculations are combined to investigate the effect of the shot-peening process on the fatigue lifetime of the AM1 nickel-based single crystal superalloy. The Ortner method is used to determine residual elastic stress depth profiles in plane-parallel samples. They exhibit a 130-160µm thick hardened layer where compressive stresses up to 1000-1400 MPa take place. The tensile stresses which ensure the mechanical equilibrium of the samples are not localized in a specific layer but rather distributed in a few millimeters thick layer. The eigenstrain theory is then used to incorporate measured stresses in the elastoviscoplatic modelling of shot-peened fatigue test specimens. A numerical method is proposed to initialize hardening variables in the shot-peened layer independently of the complexity of the constitutive law or measurements in calibration samples. Finally, a fatigue analysis at 650 • C is performed in samples with a stress-concentration. The effect of shot-peening on the fatigue lifetime is studied using both modelling and measurements. Results are in good agreement in the investigated range of applied stresses. However, measurements show that the shotpeening operation is not always beneficial.

Introduction

The fatigue analysis of blades in the hot stage of aeronautical gas turbines is extensively investigated to improve the engine reliability. Life prediction of these components under complex thermomechanical loading and environmental conditions is nowadays associated with the design process by the industry. Finite element calculations at the level of the component are able to identify fatigue critical zones and crack initiation time depending on its thermal and mechanical history [START_REF] Géradin | Three dimensional turbine blade analysis in thermo-viscoplasticity[END_REF]. The development of a consistent approach connecting the physical variables required to describe the mechanical behavior of the material to those required to describe the fatigue process is still a challenging task because damage is closely connected to the microstructure of the material. For blades made of AM1 nickel-based single crystal superalloy, a lifetime analysis has been developed since the 80's in the framework of continuum damage mechanics. The elastoviscoplastic model which enables to reproduce the anisotropic behavior of the single crystal at different temperatures is coupled to a creep-fatigue damage model [START_REF] Chaboche | Application of a Kinematic Hardening Viscoplasticity Model with Thresholds to the Residual Stress Relaxation[END_REF][START_REF] Gallerneau | Fatigue Life Prediction of Single Crystals for Turbine Blade Applications[END_REF][START_REF] Chaboche | Cyclic Inelastic Constitutive Equations and their Impact on the Fatigue Life Predictions[END_REF].

To delay crack initiation and propagation and thus improve fatigue lifetime, turbine blade roots are shotpeened. The compressive residual stress layer and the plastic deformation created at the surface of the component contribute to relax stress concentration caused by the fir-tree geometry. Due to the complex thermomechanical loading endured by blades during service, one may wonder in which circumstances the shot-peening treatment is really beneficial.

The aim of this study is to take into account stress fields and work hardening generated by shot-peening in the finite element code Zset/Zebulon used by SNECMA (SAFRAN group) and ONERA for modeling cyclic fatigue in the AM1 single-crystal superalloy. The ultimate goal of this work is to get efficient estimations of the benefit of the shot-peening treatment regarding the fatigue lifetime.

The paper is organized as follows. The first section is devoted to the determination of elastic stresses using the Ortner method [START_REF] Ortner | Röntgenographische Spannungsmessung an einkristallinen Proben[END_REF]. After a description of the formalism and technical details, stress measurements on a specimen with shot-peened flat surfaces oriented along the <100> and <110> crystallographic directions are presented. In the second section, the procedure used to incorporate stresses in the finite element calculations is described and applied to generate the initial mechanical state of a shot-peened single crystal fatigue test specimen. In the last section, the low cycle fatigue lifetime of specimens exhibiting a stress concentration is calculated and compared to measurements.

Residual stress determination in single crystals using X-ray diffraction

Formalism and methodology

The strain of the crystal lattice is determined by X-ray diffraction measurements. The elastic stress tensor T is then deduced with the generalized Hooke law:

T = C : E (1)
where C and E are the elastic stiffness and strain tensors of the crystal lattice. In the case of single crystals, the determination of strains with the sin 2 ψ method is not possible. When a monochromatic wavelength is used, the Ortner method can be employed [START_REF] Ortner | Röntgenographische Spannungsmessung an einkristallinen Proben[END_REF]. The Green-Lagrange strain tensor components E ij are related to the metric tensor components g ij and g 0 ij of the deformed and undeformed crystal lattices by:

E ij = 1 2 (g ij -g 0 ij ) (2)
where g ij = a i • a j with a i (i=1,2,3) the crystal lattice basis vectors. Diffraction techniques involve measurements in the dual (reciprocal) space of the crystal lattice. By definition of the reciprocal space basis {a * i } (i=1,2,3), the components of its metric tensor g * ij = a * i •a * j are obtained by inverting the g ij matrix :

[g * ij ] = [g ij ] -1 (3)
and are related to diffraction angles by :

3 i,j=1 h i h j g * ij = d -2 h = 4 sin 2 θ h m 2 λ 2 (4)
where the coefficients {h i } correspond to the Miller indices of the lattice plane associated with the node of the reciprocal lattice designated by the vector h, θ h is the angle between the incident beam and the scattering planes, m is an integer corresponding to the order of the reflection and λ is the wavelength of the incident wave. In principle, six Bragg angles corresponding to non colinear h vectors have to be measured in order to fully determine the metric tensor. However, in order to reduce the effect of uncertainties on the accuracy of the solution, the number of lattice planes N is usually taken larger [START_REF] Ortner | The choice of lattice planes in X-ray strain measurements of single crystals[END_REF][START_REF] François | Analyse des contraintes résiduelles dans les monocristaux et les matériaux gros grains par diffraction des rayons X[END_REF]. Therefore, the system which must be solved is made of N linear equations with six unknown components (g * 11 , g * 22 , g * 33 , g * 23 , g * 13 , g * 12 ) since metric tensors are symmetric. The best solution in the sense of leastsquare minimization is obtained by solving the normal equations. Thus, the six independent coefficients of the reciprocal space metric tensor g * ij are given by :

[g * ] = [h] T [h] -1 [h] T [d -2 ] (5) with [h] =      h 2 1 k 2 1 l 2 1 2k 1 l 1 2h 1 l 1 2h 1 k 1 h 2 2 k 2 2 l 2 2 2k 2 l 2 2h 2 l 2 2h 2 k 2 . . . . . . h 2 N k 2 N l 2 N 2k N l N 2h N l N 2h N k N      (6) [g * ] = g * 11 , g * 22 , g * 33 , g * 23 , g * 13 , g * 12 T (7) [d -2 ] = 4 sin 2 θ 1 m 2 λ 2 , • • • , 4 sin 2 θ N m 2 λ 2 T ( 8
)
where h i , k i and l i are the Miller indices of the ith lattice plane used for the measurement. To minimize numerical errors, the number of lattice planes and their orientation relationship must be chosen such that the condition number associated with the linear equation 5 is small [START_REF] Ortner | The choice of lattice planes in X-ray strain measurements of single crystals[END_REF][START_REF] François | Analyse des contraintes résiduelles dans les monocristaux et les matériaux gros grains par diffraction des rayons X[END_REF][START_REF] Sen | Regression analysis: theory, methods, and applications[END_REF]. The methodology applied in the following to determine residual stress depth profiles can be summarized as follows :

• measurement of N > 6 Bragg angles • calculation of g * ij using Eq. 5

• calculation of [g ij ] = [g * ij ] -1
• calculation of the crystal lattice strain tensor E c ij using Eq. ??

• calculation of the crystal lattice stress tensor T c ij using Eq. 1

• calculation of the stress tensor in the reference system of the specimen

• material removal by electropolishing and iteration of the procedure.

Material and technical details

Samples Electro Discharge Machining (EDM) is used to obtain 15 mm x 6 mm x 6mm plane-parallel samples from a bar of the AM1 nickel-based single crystal produced by directional solidification at SNECMA. For all samples, the 6 mm x 6mm surfaces are oriented along the [010] crystallographic direction which corresponds to the solidification axis and the primary arms axis of dendrites. Other surfaces have <100> (Fig. 1a) or <110> (Fig. 1b) crystallographic orientations depending on the batch of samples. Optical microscopy observations reveal dendrites with a primary arms separation in order of 500µm (Fig. 1c and Fig. 1d). Porosity due to the solidication process and coarse γ particles which have not been entirely removed by the solution heat treatment are also present (black and white spots in (Fig. 1c-d). Shot-peening and diffraction measurements are realized on one of the 15 x 6 mm 2 surfaces.

To remove the oxidation layer and the residual stresses which may have been introduced by the EDM, this surface is subjected to a soft mechanical polishing. Solution and ageing heat treatments are then realized to obtain an austenitic nickel rich γ matrix strengthened by about 70% volume fraction of γ -Ni 3 Al ordered precipitates with a cubic L1 2 structure. γ precipitates have cuboidal shapes with sizes lower than 1 µm and are homogeneously distributed in the γ matrix ((Fig. 1e). The lattice mismatch between the two phases is smaller than 0.05%. The last step of samples preparation is the shotpeening which is is realized with 1 mm diameter 100Cr6 steel shots at 100% coverage by an ultrasonic technique.

Measurements conditions and data analysis The diffraction measurements are realized using a diffractometer composed of a Rigaku microfocus X-ray generator equiped with parabolic optics (Cu wavelengths), a general purpose Huber 6-circles goniometer and a NaI(Tl) scintillation detector located at 700 mm from the sample position. The opening of the divergence and detector crossed slits is set to 1 mm × 1 mm in order to have a good signal to noise ratio when high angles diffraction peaks are investigated in shot-peened samples. Once the diffraction condition has been found for a given (hkl) reflection, all the goniometer motor positions are recorded and a θ-2θ scan is realized to obtain the diffraction peak. Data are then corrected by the Lorentz-polarisation factor and scans are analyzed with a least-square fitting procedure using a Rachinger doublet [START_REF] Rachinger | A Correction for the α1 α2 Doublet in the Measurement of Widths of X-ray Diffraction Lines[END_REF]. As explained above, the knowledge of the Miller indices and the Bragg angle of at least six reflections enables to determine the metric tensor of the crystal unit cell, thus the six parameters of the Bravais lattice. Due to the small lattice mismatch between the γ and γ phases, the contribution of the two phases in (hkl) diffraction peaks with h,k,l having the same parity are not differentiated using our experimental setup. In principle, additional monochromators would probably help but with higher X-ray beam fluxes. Also, the large peak broadening caused by shot-peening would not guaranty a quantitative deconvolution of the two contributions. Therefore, our analysis of fundamental reflections provide the average strain between the γ and γ phases. Best strain sensitivity and a small conditions numbers are obtained with reflections diffracting at high Bragg angles. In this work, {420}, {331}, {004} and {222} reflections are considered. This corresponds to diffraction angles close to 147 • , 138 • , 118 • and 96 • respectively. 21 Bragg angles are used in the calculations when the sample surface is oriented along the [100] crystallographic direction, 20 when the surface is oriented along the [110] direction. To determine the residual stress tensor components, a macroscopic stiffness tensor determined from mechanical testing measurements at room temperature is used : C 1111 =296 GPa, C 1122 =204 GPa and C 2323 =125 GPa [START_REF] Espié | Étude expérimentale et modélisation numérique du comportement de monocristaux de superalliages[END_REF]. Superstructure reflections are related to the γ phase only and have intensities more than 1000 times smaller than fundamental reflections [START_REF] Morançais | Residual stress determination in a shot-peened nickel-based singlecrystal superalloy using X-ray diffraction[END_REF]. For the shot-peened specimen, crystal misorientations near sample surface reduce significantly the maximum intensity of Bragg peaks resulting in intensities comparable to the noise intensity. Thus, our setup is not able to provide a complete residual strain depth profile related to the γ phase.

The evaluation of uncertainties is performed through a Monte-Carlo approach, assuming random fluctuation around the measured diffraction angle. Uncertainties include lattice parameter variations due to composition fluctuations, mosaic spread introduced by the dendrites and errors arising from the least-squares fitting procedure of the Bragg peaks. Details are provided in [START_REF] Morançais | Residual stress determination in a shot-peened nickel-based singlecrystal superalloy using X-ray diffraction[END_REF]. Two methods are commonly employed to determine the metric tensor g 0 of the undeformed state: measurements of the unit cell parameters in a reference sample or, for low penetrating radiations, the plane stress approximation, where the lattice parameter of the refer-ence state is taken such that the component T 33 of the stress tensor related to the deformed state is zero (with ê3 direction along the surface normal of the sample). For the Cu wavelength, the X-ray attenuation length in the superalloy is close to 10 µm. Depending on the orientation of the sample with respect to the incoming and diffracted beams, the characteristic depth probed ranges from 2µm for grazing angles to 10 µm close to normal angles. Both methods are assessed in this work.

Results and discussion

Marty et al. and Brückner et al. have shown that in the diffraction condition, several local maxima are visible due to the misorientation between dendrites [START_REF] Marty | X-ray study on single crystal superalloy SRR99: Mismatch γ/γ', mosaicity and internal stress[END_REF][START_REF] Brückner | The influence of the dendritic structure on the γ/γ'-lattice misfit in the single-crystal nickel-base superalloy CMSX-4[END_REF]. This is illustrated in Fig. 2. In such case, θ-2θ scans are recorded for n local maxima, with n ≤ 4 in order to quantify the variations of the lattice plane spacing. The strain is calculated from the middle of the variation range of peak positions, while the error bar is computed from half the range. After an optimization work detailed in Ref. [START_REF] Morançais | Residual stress determination in a shot-peened nickel-based singlecrystal superalloy using X-ray diffraction[END_REF], we show that the lattice parameter fluctuations related to the dendritic microstructure and the choice of reflections in the calculations result in 30 MPa uncertainties on the calculated stress tensor components.

Reference sample Figure 3 represents the six unit cell parameters (a i , α i ) as a function of depth in a heat treated reference sample with its surface normal oriented along the {100} crystallographic directions. Each unit cell length oscillates as a function of depth with an average of 3.5897 Å or 3.5900 Å (dashed lines) with maximum deviations of ±4. 10 -4 Å. At the sample surface, lattice constants differ significantly probably due to chemical inhomogeneities and/or remaining stresses caused by the heat treatments and the mechanical polishing respectively. The reference state can be assimilated to a cubic crystal with a lattice parameter a 0 = 3.58985 Å which leads to :

g 0 ij = (a 0 ) 2 δ ij = 3.58985 2 δ ij (9)
where δ ij is the Kronecker delta.

Sample with a {100} shot-peened surface Stress calculations in samples with a shot-peened surface oriented along the [001] crystallographic direction are determined from 21 Bragg angles. Residual elastic stress depth profiles resulting from sets of measurements and layer removals are represented in Fig 4 using the metric tensor of Eq. 9 (Fig. 4a) and the plane stress approximation (Fig. 4b). The evolution of T 11 and T 22 which differ by less than 30 MPa, shows compressive stresses taking place in the 150µm thick surface layer and almost zero residual stress deeper. The shear components (T ij with i = j) are close to zero for all depths. Although Dashed line corresponds to the ultimate stress of the AM1 superalloy determined from tensile or fatigue tests at room temperature [START_REF] Caron | Influence of Chemistry on the Tensile Yield Strength of Nickel-Based Single Crystal Superalloys[END_REF].

each measurement is realized on a free surface, in the layer affected by the shot-peening process, T 33 significantly differs from zero when the metric tensor of the reference sample is used (Fig. 4a). This is probably due to the modification of the material properties (solid solution gradients, loss of coherency strains) in this layer because of the strong hardening caused by shot-peening and the lattice parameter of the reference state should not be taken like the measured one in the reference sample. In figure 4, the dashed line represents the ultimate stress limit of the AM1 material [START_REF] Caron | Influence of Chemistry on the Tensile Yield Strength of Nickel-Based Single Crystal Superalloys[END_REF]. In Fig. 4a stresses close to the sample surface are larger than this value. The maximum difference with this limit is about 300 MPa at the 20 µm depth. This is not incompatible since the material is strongly hardened in this area and if all the tensor components are considered, the Von Mises stress is only 60 MPa above the ultimate stress limit. Two additional corrections must be done to the stress profile calculated from Bragg peak positions : the first one to account for the stress relaxations resulting from material removals and the second to account for the fact that the X-ray beam probes a stress gradient. In the latter case, it can be shown that the measured value of the T 33 component close to the surface plane is twice its true value providing that the stress gradient at the surface is zero (see discussion in [START_REF] Morançais | Residual stress determination in a shot-peened nickel-based singlecrystal superalloy using X-ray diffraction[END_REF]). The Moore and Evans correction for stress relaxations due to material removals is applied [START_REF] Moore | Mathematical Correction for Stress in Removed Layers in X-ray Diffraction Residual Stress Analysis[END_REF]. The resulting stress profiles are represented in Fig. 5. The residual stress depth profile exhibit a 130-160µm thick hardened layer where T 11 and T 22 compressive stresses up to 1000-1400 MPa take place depending on the assumption used to describe the initial state. The tensile stresses which ensure the mechanical equilibrium of the sample are not localized in a specific layer with high levels of stresses but rather distributed in a few millimeters thick layer.

Effects of the crystal anisotropy

To quantify the effect of the crystalline anisotropy on the stress field caused by the shot-peening process, samples with surfaces oriented along the < 100 > and < 110 > crystallographic directions have been shot-peened in the same conditions. The relationship between the sample surface and the crystal unit cell orientations is represented in Fig. 6. For both orientations, shear components of stress tensors are almost zero (within the ± 30 MPa uncertainty) with re- Table 1: Diagonal components of the T stress tensor at a 20µm depth when the shot-peened sample surface is oriented along the [001] or the [101] crystallographic direction. Values are given in MPa in the (ê 1 ,ê 2 ,ê 3 ) sample basis (see Fig. 6). The reference state corresponds to a cubic lattice with a 0 = 3.58985 Å or to the plane stress hypothesis. 6).

Surface

Results also show that the T 11 component is 265 MPa higher when the surface is oriented along the [101] direction whereas the T 22 component is 53 MPa lower.

When the lattice parameter of the reference sample is used in the calculations, T 11 and T 22 differ by 10% and 14% between the two sample orientations.

Finite element calculations incorporating residual stresses resulting from shot-peening

The initial state introduced in the calculations must be as close as possible to the state generated by the shot-peening process and corresponds to a mechanical equilibrium. To reproduce stress redistributions dur-ing cyclic loading conditions, internal variables such as eigenstrain, kinematic hardening and accumulated plastic deformation must be carefully initialized in addition to the choice of their governing equations. In the case of single crystals, data obtained from measurements are not able to provide such values for every Gauss point. Therefore assumptions and interpolation schemes are required. In the following, we present a generic approach to initialize internal variables in the layer affected by the shot-peening in the case of single crystals. The method is then applied to investigate the lifetime of a fatigue test specimen made of the AM1 suerpalloy.

Internal variables initialization

Eigenstrain. The eigenstrain framework enable to account for strains caused by a wide range of phenomena such as thermal expansion mismatch, phase transformation or plastic deformation. The formalism can be easily implemented in finite element calculations and a detailed modeling of the process generating residual stresses is not required when measurements are available. The method is extensively used to investigate stresses in shot-peened components mainly for the following reasons : (1) for a given eigenstrain distribution, an equilibrated residual stress distribution can be calculated for any geometry. (2) the residual stress state evolution during an additional plastic activity in the material is able to be modelled through the modification of the eigenstrain distribution (3) the eigenstrain distribution related to the shot-peening process can be determined from residual stress measurements on samples with simple geometries and then be incorporated into the modeling of components with more complex geometries.

In the small strain approximation, the mechanical equilibrium of a system with stress free boundary conditions at the surfaces is obtained by solving the following set of equations :

       Inc = 0 ∇ • σ = 0 σ • n = 0 = S : σ + * (10)
where σ is the residual stress tensor, is the total strain tensor, S =C -1 is the compliance tensor, * is the eigentrain tensor and n correponds to the surface normal. The shot peening being operated in normal conditions, we assume that only the stress tensor components in the surface plane are non zero. The solution of Eqs. 10 for a plate-like sample which has been shot-peened on a face oriented along the ê3 direction has the following

   * 11 (z) = A 1 z + B 1 -[S 11 σ 11 (z) + S 12 σ 22 (z)] * 22 (z) = A 2 z + B 2 -[S 12 σ 11 (z) + S 22 σ 22 (z)] * 33 (z) = -[ * 11 (z) + * 22 (z)] (11)
with

A i = 12 h h/2 -h/2
z * ii (z)dz ( 12)

B i = 1 h h/2 -h/2 * ii (z)dz =< * ii > z ( 13
)
where h is the sample thickness. As illustrated in Fig. 7, the residual elastic stress depth profile of Fig. 5 is easily modelled if a Gaussian function is assumed for the z dependence of * 11 (z) and * 22 (z) (dashed lines in the figure). Kinematic hardening and accumulated plasticity. The method is inspired by the work of Grasty and Andrew [START_REF] Grasty | Shot peen forming sheet metal: finite element prediction of deformed shape[END_REF] where the upper layers of the system are subjected to a squeeze pressure such that a small plastic deformation is generated. This process was iterated by the authors until a known curvature caused by the indentation process is retrieved. Here, we use a similar strategy in the sense that a pressure is applied at different depths in such a way that after release eigenstrain values determined previously are found. The pressure is applied using a serrated function of time until the mechanical state has reached the equilibrium and the accumulated plastic deformation a target value, which is material history dependent. Here, as a rough approximation, diffraction peak widths recorded in shotpeened sample are compared to peak widths evolution determined in calibration samples with a known amount of macroscopic plastic deformation. To reduce calculation costs, this procedure is applied only to a finite set of representative volume elements (RVE) chosen at different locations in the system of interest. The values of variables resulting from independent calculations are then associated with the considered Gauss point and values related to other points are determined by interpolation. In practice, due to the symmetry of the eigenstrain tensor determined previously, the pressure is applied only to the element surface which is parallel to the shot-peened surface, all other boundaries being fixed. Crystal anisotropy. To take into account the effect of the crystal anisotropy, the procedure developed above can be applied to residual elastic stress measurements in samples with different surface orientations. However, if data are not available, the pressure caused by multiple shots during the peening operation can be supposed independent of the crystallographic orientation of the sample surface. This implies that the pressure value and the number of cycles determined for the [100] crystallographic direction can be used to determine eigenstrain, kinematic hardening, accumulated plastic deformation and residual elastic stress values for all crystal orientations. This methodology is applied to a fatigue test specimen with a cylindrical geometry (3.11mm radius). Calculations are realized at room temperature with the anisotropic elastoviscoplatic model developed for the AM1 superalloy [START_REF] Fournier Dit Chabert | Calibration and validation of a constitutive model for single crystal nickel based superalloys[END_REF]. Fig. 8 represents the hoop (σ θθ ) and axial (σ zz ) stresses in a quarter of cross-section. The hoop stress is significantly affected by the crystal anisotropy since in Fig. 8a, stresses differ by about 500 MPa between the [100] and [110] crystal directions in the shot-peened layer. In the case of the [100] direction, the depth profile exhibit compressive stresses up to 1000 MPa in a 150µm thick layer and tensile stresses are smoothly distributed with depth (solid line in Fig. 8b). In the case of the [110] direction, compressive stresses up to 1500 MPa take place in a 100µm thick layer and a tensile stress layer is clearly visible. The axial stress behavior is less affected by the crystal anisotropy as shown in Fig. 8c and Fig. 8d (dashed line). Calculations only show that the layer where compressive stresses take place is thinner in the [110] direction than in the [100] direction. At a 20 µm depth, the calculated hoop component is 34% lower in the [110] direction than in the [100] direction. This value has to be compared with 28% in the case of measurements in flat samples (see Tab. 1). The axial stress is 10% lower in the calculations whereas it is 5% higher in experiments. X-ray measurements in a shot-peened fatigue specimen are in progress to assess the relevance of trends observed in calculations.

Towards fatigue lifetime assessment

The ultimate goal of this work is to get efficient estimations of the benefit of the shot-peening treatment regarding the fatigue lifetime. The generated residual stresses that influence the mechanical state at the surface of the part may evolve with the development of the plasticity in the component during cyclic loading conditions and work hardening will also play a significant role on this stress relaxation. Residual stresses and work hardening need then both to be taken into account in the fatigue analysis. The method proposed in the previous section to introduce theses quantities in a finite element mechanical analysis in fatigue allows for an assessment of their combined and separated contribution. In the following, the method is used to estimate the low cycle fatigue lifetime to crack initiation at 650 • C on notched specimens made of the AM1 single crystal superalloy. The accuracy of the proposed approach to account for the influence of residual stresses and work hardening on lifetime prediction is assessed through comparisons with experimental results.

The fatigue life analysis of structures more often is based on the use of a damage model to describe the initiation and coalescence of micro-cracks leading to an observable macroscopic crack. The damage evolution usually expressed in terms of damage increment per cycle is then introduced into a lifetime workflow. As admitted generally for metallic materials, this workflow can be build assuming an uncoupled formalism between the descriptions of the viscoplasticity and hardening mechanisms occurring in the materials and the damage evolution [START_REF] Kruch | ONERAs Multiaxial and Anisothermal Lifetime Assessment for Engine Components[END_REF]. However, interaction between damage mechanisms induced by fatigue and creep is often considered eventually coupled with oxidation effects [START_REF] Gallerneau | Fatigue Life Prediction of Single Crystals for Turbine Blade Applications[END_REF]. The lifetime assessment of the fatigue samples considered in this study is then performed following several steps briefly presented hereafter. Firstly, the procedure described before is used to introduce in the finite element model the initial profiles of the residual stresses and all the hardening variables considered in the constitutive model, chosen to simulate the mechanical behavior of the material. Then, the fatigue loading sustained by the specimens presenting a stress concentration (K t =1.6 in the gauge length) is simulated to account for the stress redistributions due to cyclic plasticity namely in the affected layer by the shot-peening process. At this stage it is particularly important that the constitutive model chosen to describe the behavior of the material allows for an accurate simulation of both the residual stress relaxation and the work hardening evolution. Determination of their profile after interrupted fatigue tests may be necessary and a specific formulation of the kinematic hardening variables describing Bauschinger effects can be used [START_REF] Chaboche | Cyclic Inelastic Constitutive Equations and their Impact on the Fatigue Life Predictions[END_REF]. Finally, the damage model is applied as a post-processing on mechanical quantities obtained at the stabilized cycle such as for instance the octahedral shear stress amplitude, the mean value of the hydrostatic stress and the maximum stress eigen to compute the fatigue life and to get the crack localization in the structure. For the analysis performed in this study, a multiaxial creepfatigue damage model, recently improved, has been used to compare the estimated lifetime between smooth and shot-peened specimens. A zoom of the local damage map is represented in Fig. 9 for both type of samples. Calculations correspond to a cyclic fatigue test at 650 • C and a 800 MPa applied stress. In the case of the smooth specimen, the maximum of damage is localized at the sample surface and corresponds to 3000 cycles before crack initiation (Fig. 9a). The effect of shot-peening is clearly visible in Figs. 9c since the maximum of damage is localized about 200µm beneath the surface and corresponds to a number of cycles one order of magnitude higher. These trends are corroborated by the scanning electron micrographs which show that cracks initiate at the sample surface for the smooth specimen and at a 200µm depth for the shot-peened specimen (arrows in Fig. 9b and Fig. 9d). Numerical and experimental results obtained for different levels of nominal applied stress are represented in Fig. 10. Calculated and experimental lifetimes are in good agreement when the shot-peening process has beneficial effects. The lifetime is 10(12) times higher at 800MPa and 4(4) times higher at 635 MPa in the modelling (measurements) respectively. However, in measurements a crossover is observed between 585 MPa and 635 MPa since the lifetime is lower for shot-peened samples. This tends to show that the sample surface roughness is the limiting factor for the fatigue lifetime in the small applied stress regime.

Conclusion

In this paper, X-ray diffraction measurements and finite element calculations are performed to study the effect of shot-peening on the low cycle fatigue of a nickel-based single crystal superalloy (AM1). In the modelling, the mechanical state associated with the shot-peened operation is incorporated using the results of the residual elastic stress measurements, the eigenstrain framework and a generic procedure which enables to initialize the kinematic hardening and accumulated plastic deformation variables independently of the constitutive equation complexity. A specific approach is also implemented to take into account crystal anisotropy effects when residual stress depth profiles are not available for all surface orientations. These developments are applied to the lifetime prediction of notched fatigue test samples with a stress-concentration factor K t = 1.6.

The residual stress depth profile is determined using the Ortner method in a plane-parallel sample with a shot-peened surface oriented along the [100] crystallographic direction. The profile exhibits a 160 µm-thick hardened layer, where compressive in-plane stresses up to 1000-1400 MPa take place. The tensile stresses which ensure the mechanical equilibrium are smoothly distributed in the sample thickness. Measurements realized in a shot-peened sample with a surface oriented along the [110] crystallographic direction reveal a 30% increase of compressive stresses for the component which is not crystallographically equivalent with respect to the [100] surface orientation. The other in-plane component of the stress tensor is weakly affected by the crystal anisotropy. The eigenstrain distribution associated with the residual (elastic) stress profile is then calculated and elastoplastic calculations are performed to initialize the internal variables of the model in the surface layer affected by the shot-peening process. With this approach, a physically justified equilibrated mechanical initial state is generated and the study of stress redistributions during cyclic thermal and mechanical load-ings is possible in samples with specific geometries. Finally, the ability of the proposed approach to account for the influence of residual stresses and work hardening on lifetime prediction is assessed in the case of fatigue tests at 650 • C on samples having a stress-concentration (K t =1.6). The results of modelling are in good agreement with measurements in the 635-800 MPa range of applied stress. The increase of the fatigue lifetime due to shot-peening is between a factor 4 and 10. For smaller applied stress, a deleterious effect of shot-peening is measured probably due to a higher sensibility of the damage to the surface roughness of samples.

Figure 1 :

 1 Figure 1: Schematics of the relationship between sample geometry, dendrites and crystallographic orientations of investigated samples . (ê 1 ,ê 2 ,ê 3 ) designates the Cartesian coordinate system used to express tensor components. Shot-peening and X-ray diffraction measurements are realized on the dotted surface which is oriented along (a) the [001] crystallographic direction or (b) the [101] crystallographic direction. (c) and (d) are optical micrographs of sample sides where "PDAS" denotes the Primary Dendrite Arm Spacing. Black and white spots represent porosity and eutectic phases respectively. (e) is a Scanning Electron Microscopy micrograph of the cuboidal γ precipitates (dark grey) surrounded by the γ matrix (light grey).

Figure 2 :

 2 Figure 2: (ω, η) intensity maps (counts per second) around the (133) reflexion in the reference sample at at depth of 202 µm.

Figure 3 :

 3 Figure 3: Unit cell lattice parameters determined in a sample which has not been subjected to shot-peening. (a) cell lengths (b) cell angles α i = ( a j , a k )

Figure 4 :

 4 Figure4: Independent components of the residual stress tensor calculated from measured crystal lattice parameters a i and α i (i=1,2,3) as function of depth in the shot-peened sample whose surface is oriented along a {100} crystallographic direction. The Cartesian basis is oriented such as ê2 and ê3 are colinear with the surface normal and the long edge of the sample (schematics). Dashed line corresponds to the ultimate stress of the AM1 superalloy determined from tensile or fatigue tests at room temperature[START_REF] Caron | Influence of Chemistry on the Tensile Yield Strength of Nickel-Based Single Crystal Superalloys[END_REF].

Fig. 4b represents

  the residual stress profile determined from the same measurement but with the plane stress approximation (T 33 = 0). The evolution of the lattice parameter of the reference state (cubic symmetry) as a function of depth is represented in the inset. Up to 200 µm, its value differs from the lattice parameter of the reference sample (dashed line). As in figure Fig.4b, T 11 ans T 22 components are almost equal ans shear components close to zero. The highest absolute value is about 1000 MPa close to the surface.

Figure 5 :

 5 Figure 5: Evolution of residual stresses as a function of depth. Filled symbols correspond to T 11 = T 22 = T and open symbols to T 33 = T ⊥ . The profile with square symbols is determined with respect to the lattice parameter of a reference sample and circle with respect to the plane stress hypothesis.

Figure 6 :

 6 Figure 6: Relationship between the sample surface and the crystal unit cell orientations when the surface normal of the sample is aligned with (a) the [001] crystallographic direction and (b) with the [101] direction. (ê 1 ,ê 2 ,ê 3 ) and (a 1 ,a 2 ,a 3 ) are the sample and crystal coordinate systems respectively.

Figure 7 :

 7 Figure 7: Evolution of residual elastic stresses (left axis) and eigenstrain (right axis) as a function of depth. Circles corresponds to X-ray diffraction measurements and lines to modeling. X and X ⊥ denotes in-plane and normal components of X tensor.

Figure 8 :

 8 Figure 8: Calculated hoop (a) and axial (c) residual stress maps in a shot-peened cylindrical fatigue test specimen made of the AM1 single-crystal superalloy. Only a quarter of cross-section is shown. (b) and (d) graphs correspond to the stress depth profiles along the [100] and [110] crystallographic directions.

Figure 9 :

 9 Figure 9: Left -local damage maps calculated model in notched cylindrical fatigue test samples with a stressconcentration factor K t =1.6 subjected to an applied stress of 800 MPa at 650 • C (f=15Hz, R σ =0). Zoom in the crack initiation area in (a) a smooth sample (c) a shot-peened sample. Views are mirrored for sake of comparison. Right -fracture SEM micrographs in corresponding fatigue test specimens.

Figure 10 :

 10 Figure 10: S/N curve of K t =1.6 fatigue test specimen made of the AM1 single crystal superalloy (T=650 • C, f=15Hz, R σ =0). Circles corresponds to measurements and lines to modelling. Arrows indicate measurements stopped before failure.