Peter Stockinger

Problems and challenges

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

Introduction

This paper presents : the audiovisual archive programme for humanities (social and human sciences) launched in 2001 by the Maison des Sciences de l'Homme (MSH) in Paris; a working (indexing and annotation) environment for individual or collective users ("communities") in order to extract audiovisual segments, to classify them in personal archives, to describe them following the particular user point of views and to reuse them in given academic and/or educational, viz. professional activities.

The presentation and discussion of this example is undertaken in the perspective to better understand the interests and challenges of audiovisual archives for public research and educational institutions.

1) The audiovisual archives of the MSH

The Maison des Sciences de l'Homme (MSH) is a public research foundation, located in Paris (France) 1 .

Since 2001, it has started to build an audiovisual archive in social and human sciences. The principal -institutional -goals of this programme are the following ones: a) production of scientific knowledge in humanities:

constitute an (audiovisual, textual, …) content base that could play the role of an "incubator" for the emergence of novel knowledge as well as for collectively shared knowledge standards,

1 Official web site of the Maison des Sciences de l'Homme : http://www.msh-paris.fr

create an observatory for ongoing research and research trends. b) conservation of scientific knowledge: preserve and conserve the scientific contributions of researchers especially in humanities create a public and open archive of scientific heritage in humanities. c) diffusion: create a central access to sometimes highly dispersed information and knowledge especially in humanities create a base of reusable elements for multisupport and "personalised" publishing of scientific knowledge. d) exploitation: create a sharable knowledge base for research and teaching communities, create high graded knowledge references for professional uses (media, policy makers, …), create a "competence pool" for young researchers.

The principal content of the audiovisual archives of the MSH are footages of: 1. research seminars; 2. symposia, workshops; 3. the "life" in research labs; 4. and especially of in-depth interviews with researchers (average time of one interview session: 2, 5 hours; a researcher may repeat as often as he wants an interview session; average time of one interview: 3,5 hours)

The actual state of the art of the archive can be summarized as follows:

simple access to this archives via the web site of the MSH (http://www.msh-paris.fr -section "Manifestations En Ligne"; cf. figure 1); -rich access, via the portal of the archive -the "Médiathèque Sciences Humaines et Sociales"-(http://e-msha.msh-paris.fr/Mediatheque_SHS; cf. figure 2); -450 hours of scientific video on-line (some 300

hours have yet to be digitized till the end of august 2003); most of the videos are in three formats: 56k; 256k and MPEG 1 or 2; the major part of the content is in French (but there does not exist any limitation for the use of other languages …).

The principal information unit of the audiovisual archive is the so-called "event" (scientific, educational, ….) event: it

Equipe Sémiotique Cognitive et Nouveaux Médias (ESCoM)

Maison des sciences de l'Homme (MSH) 54, Bd. Raspail -F-75006 Paris

2

is an interview (one session of an interview), a seminar, a workshop, a round table, …; -Is generated as a dynamic web site composed by a set of data registered in a relational database (MS SQL server); -Is the base indexing and description unit in the media portal (implemented on MS Sharepoint server; cf.figure 3).

The dynamic web site visualizing a scientific event is organized in : a "home page" (cf.figure 4) describing the event and the actors of it and giving access to the other pages; a "video page" which is the principal interface for consulting an event (cf.figure 5); a "thematic index page" with an access to the principal scientific notions and their definitions as well as to the videos where they are developed (cf.figure 6); several other pages containing supplementary information of the event (related documents, …).

The "video page" (cf.figure 5) contains : 1. the embedded MS media player; 2. the principal thematic sections following which an event can be explored; 3. supplementary information sources (as pdf, ppt, image, other video… files) that accompany an event; 4. several information concerning a selected video file (title, duration, format, …).

Finally, the "canonical" production and editing chain adopted for realizing an event can be summarized as follows: 1. Shooting of an event (with DV cameras); 2. Digitizing of footages; 3. Cutting of a digitized footage in several avi files; 4. Post-editing of each file; 5. Thematic description and indexing of the files; 6. Exporting of avi files in asf and mpeg formats; 7. Registration of asf and mpeg files as well other files and information in the database; 8. Generation of a dynamic web site of the event; 9. "Broadcasting" of the web site within the MSH archive.

2) Users, objectives and needs

The principal users of the MSH audiovisual archive in humanities are:

-Researchers and research teams; -Teachers and graduate students; -Professional user groups (journalists, policy makers, …).

As already mentioned above (cf. 1 st chapter) the archive is supposed to fulfil: -A heritage function -An observatory function, -A transfer (teaching and training) function, -An incubating function for research and knowledge production.

In order to satisfy genuinely these functions, the principal needs may be identified as follows. Around the MSH audiovisual archive in social and human sciences:

-Make it possible to create work spaces for individual and collective users ("communities) attuned to the specific (scientific, educational, professional) usages of a set of -scientificevents; -Make possible -for authorised users -to identify and select, within the existing audiovisual data pool, (audiovisual and/or textual) segments which are relevant with respect to the users knowledge and information requirements; -Allow individual or collective users to specify their own view points following which the selected segments should be classified and described; -Make possible for a user, a user group to create his/their "personalised" ontologies (thematic grids, thesaurus, …) and annotation categories in order to index and annotate (describe) the selected segments following a chosen and specified viewpoint; -Make possible the sharing and updating of viewpoints, ontologies and annotation categories; -Make possible to create personal archives (or again archives attuned to group specific interests) of scientific events out of the common MSH research archive; -Make possible to research and explore video and textual segments indexed and described following a specific view point; -Make possible for an individual or collective user to produce a new "montage" (a new "hyperdocument") on the basis of the selected and described segments -Make possible to edit such a montage either as a web site, a CD-ROM, a DVD, a VCD, etc. The « output » of the project is the prototype of an integrated environment (the Opalesenvironment) for describing (indexing, annotating,…) selected audiovisual and textual segments following a specific viewpoint as well as for producing a « new montage » of described segments.

Equipe

The project started in september 1999 and finished in march 2003. A second phase of this project is planned in order to overcome the limitations of the actually existing prototype (cf. chapter 6).

4) The Opales indexing and annotation environment

The general architecture of the Opales environnement (cf.figure 9; figure 10) is composed of:

1. the Opales client; 2. the video server and 3. the Opales environment server (cf.figure 7).

The Opales client interface is composed of two components (cf.figure 8):

1. A research and exploration interface; 2. A working interface.

The working interface is composed by a set of specialised tools such as: 1. A video explorer tool (an enhanced MS media player for selecting segments and magnifying zones in an audiovisual stream); 2. An ontology builder (of concepts or themes and relations between concepts and themes representing the knowledge of a given domain of expertise); 3. A conceptual graph editor (which takes as the input an ontology of themes and relations and which produces as the output a particular conceptual or thematic configuration, i.e. a set of selected themes and selected relations between themes representing specific scenes in a audiovisual segment); 4. A point of view editor (which is a very generic "formulary" composed by a set of features and which can be attuned to the requirements of a user or user group); 5. A video segment indexing and annotation interface (by the means of which themes, conceptual graphs, viewpoints or simply free annotations are "attached" to a video segment or a particular zone within a segment); 6. A segment montage editor (allowing mainly to create oriented navigation paths through a set of selected and described video segments); 7. A tool for creating a personal or collective work space.

Note: the working interface is an open oneother services can be added to the already existing ones.

The research and exploration interface offers the user different possibilities to access the content of a (personal and/or collective) archive: -Direct access to the personal library of selected video segments; -Access to specific scenes developed in selected segments via a request by the means of conceptual graphs; -Access to the content of a personal library of video segments via defined themes in an ontology; -Access via key words to the personal library of selected segments;

Equipe

4

-Access to the hyper-documents composed out of selected and described (indexed and annotated) video segments.

5) A typical user scenario

A (not fictive) case: A researcher in social history more specially concerned by the processes of western industrialisation wants to build his/her personal library of audiovisual and textual documents for different purposes:

evaluation and critical comparison of existing knowledge with respect to further researches; identification of specific research topics; preparation of a series of specialised workshops, research seminars or publications; -….

In order to assist the researcher in these and other comparable activities, the Opales environment has been conceived and developed (as a prototype) with reference to 8 very typical types of steps (or types of "tasks") that able to cover -as it seems -most of the above quoted researcher's activities. Naturally, in speaking of "types of steps" or "tasks", the Opales environment refers to an ideal scenario : for his/her concrete needs and goals, a researcher may only use very partially the possibilities offered by the Opales environment.

Step 1) Once connected to the Opales interface, the researcher opens the Opales work space tool and creates his own (personal) work space in which he will have : the Opales video description environment as well as more or less publicly resources such as already existing ontologies, viewpoints, commented video segments, etc.

Step 2) The researcher explores -via the MSH audiovisual archive portal 2 -the for his research topic relevant scientific events (seminars, interviews,…) and creates (in the form of "annotated bookmarks"), progressively, his personal base library of relevant scientific events (this base can be publicly accessible or accessible only for a particular user group).

Step 3) Then, the researcher starts in identifying, within the (audiovisual, but also textual) data composing an event these segments which 2 cf. http://e-msha.msh-paris.fr/Mediatheque_SHS are of particular interest for him and creates his personal library ("list") of (audiovisual, textual, …) segments (this activity can, naturally, occur several times, …). This activity of selecting specific segments, the researcher can perform with the help of the enhanced MS Media Player (cf.figure 12) that possesses several options such as the identification of start and end points of a segment, the magnifying of specific zones of a single frame, etc.

Step 4) The researcher starts in specifying his "ontology". This -crucial -step can be, normally, decomposed in several tasks: specification of a hierarchy of themes (or concepts) representing the knowledge of a given domain (i.e. in our case, modern industrialisation processes; specification of typical relationships between themes (other than the is-a relationship which is the default relationship for building an ontology of themes and relationships); elaboration of definitions for each theme and each relationship previously identified; building the ontology wih the help of the Opales ontology builder (cf.figure 13, figure 14, figure 15); assessment of the ontology.

Once the ontology established, the researcher can then start to index his segments (parts of his segments). For this: -He opens a segment with the enhanced MS media player; -Identifies the temporal positions of the parts within the segment; -"Attaches" one or more themes to these parts.

Step 5) After having built his ontology, the researcher, normally will have to specify typical scenes that are developed in the previously selected audiovisual segments.

As far as the interviews with researchers are concerned, such typical scenes are, for instance: In order to capture and describe (index) such typical scenes in his personal library of selected audiovisual segments, the researcher: -Opens the conceptual graph editor; -Imports themes and relations from his ontology in the conceptual graph editor; -Builds conceptual graphs (i.e. configurations) of a given set of themes and relations in qualifying the particular position between two themes within a given configuration by the means of a specific, oriented and named relation (cf.figure 16, figure 17) .

Each conceptual graph represents, by definition, a type of scenes (cf. supra) in the audiovisual segments.

After having finished the building of conceptual graphs, the researcher opens (with the enhanced MS media player) an audiovisual segment and describes the scenes developed in this segment with the help of a conceptual graph: -He identifies the temporal positions of a given scene; -Selects a conceptual graph that represent the type of this scene; -"Attaches" the conceptual graph to this scene; -Adds key words or textual descriptions to the instanciated graph (i.e. attached to a given scene) in using, for this, the referent field of a conceptual graph (remember, al graph has a conceptual field and a referent field, i.e. a field to which a concept refers: [CONCEPT: REFERENT]).

Note: Another -complementary -exploitation of a conceptual graph built on the basis of an already existing ontology is to improve the search mechanism of the ontology based search and exploration interface of Opales.

Step 6) In order to adapt ("personalize") even more the indexing and description interface, the researcher will open the View point editor (cf.figure 18, figure 19, figure 20). This editor is a highly generic formulary provided by a set of already given features that the researcher restricts to his special case. He also may define new features, not yet provided by the formulary.

For instance, the researcher would like to include in his description of audiovisual segments specific moments highly important for his own research such as:

-The rhetorical character of an interview -The relative importance of a theme developed within a segment or one of its parts; -The credibility of information communicated within a segment; -The "added value" of a communicated information; -The specialisation degree of a communicated information; etc. All these items, the researcher can specify in the view point editor and elaborate a highly specialised, "personal" description model which allows him to enhance a description of a video segment done by the sole means of an ontology and a set of conceptual graphs.

Step 7) The researcher may want to produce a "complex" hyper-document out of all selected and described video segments (as a ppt presentation, for instance). For this he uses the video montage editor (cf.figure 21).

Step 8) Finally the researcher certainly wants to decide which of his resources (ontologies, conceptual graphs and view points) could be used by other researchers or not. He also will decide which descriptions and annotations are completely personal (only for him), which others are accessible for identified user groups and which are publicly accessible. This, the user can perform with the already mentioned (cf. step 1) personal work space tool.

Note: there are again some other tools included in the Opales working environment but the above mentioned can be considered to be the most crucial ones for a scientific (pedagogical or professional) work on the data of audiovisual archives.

6) Difficulties, limitations and further orientations

The Opales project has been an extremely important one for the development of tools and services allowing specific user groups to exploit but also to enhance, to enrich the audiovisual archives of the MSH in the humanities.

It also has been interesting to better identify difficulties in the use of such rather complex and sophisticated tools and services.

Indeed, an important number of these difficulties concern especially the working habitudes and traditions of researchers (teachers, professional users,…). Besides the -well known -fact that researchers are not always disposed to share not-definitive results of their ongoing researches, many of them are indeed not well prepared to use Opales like tools in services because of methodological reasons. Indeed, even if a researcher always uses -practically -some classification and description schemas, he/she is not necessarily prepared to make explicit these practical and intuitively used schemas. Working with an ontology or one or several view points, nevertheless, presupposes this elicitation process of practical classification and description schemas, i.e. of schemas grounded on the experience of the researcher and tacitly used by him and/or his community. In other words: such schemas constitute a part of the scientific culture of a research community and their elicitation is a work that is rather similar to the analysis and the interpretation of texts or other documents for which particular methodologies (such as semiotics or rhetoric) are needed in which the concerned research community not always may possess. This first type of difficulties explains also an other type of difficulties due to the fact that the Opales working interface is provided with to rich and to general tools and services: the "common" researcher simply get lost in front of such an environment (and one can guess that this should be also the case for teachers and other categories of professional users). This means that in a second phase of the Opales project, an important effort will have to be done: -To provide simplified and "standard" working interfaces attuned to more specific but already prior identified types or profiles of working and exploitation contexts (such as "realise a course"; "build a personal archive", "create a web portal", …) -To provide especially the ontology editor, the conceptual graph editor and the view point editor with already pre-defined models (pre-defined ontologies, pre-defined typical scenes developed in a video of an interview or a seminar, predefined view points for analysing a segment of a scientific event).

Other difficulties concern more particularly actual limitations of the Opales working environment. The plan is to remove these limitations in the second phase of the Opales project (and in other, related projects): -Transformation of the Opales tools in web based tools, viz. in genuine "web services"; -Introduction of a genuine multilingual dimension in these tools;

-Enhancement of specific tools of the Opales environment (such as, for instance, the "montage editor for producing hyper-documents out of selected and described audiovisual segments); -Development of genuine multi-support publishing tool (for CD-ROM, DVD, web sites but also traditional VHS cassettes); -And, especially, integration of principal standards in an Opales description and annotation of audiovisual segments (standards such as DC, OAI, MPEG 7, TVA, etc.).

Annexe : Illustrations

  Sémiotique Cognitive et Nouveaux Médias (ESCoM) Maison des sciences de l'Homme (MSH) 54, Bd. Raspail -F-75006 Paris 3 3) The OPALES Project These needs constitute the "input" and the principal motivation of the French OPALES project. Opales (« Outils pour des Portails Audiovisuels Educatif et Scientifique ») has been a French R&DT project, financed by the Ministery of Industry within a specific national programme (the RIAM -Recherche et Innovation en Audiovisuel et Multimedia -programme) aiming at the development of innovative technological, economic and socio-cultural models in the digital audiovisual sectors. The Opales project has been leaded by the INA (Institut National de l'Audiovisuel). The principal partners have been: -Université de Montpellier -LIRMM (Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier) ; -Université d'Angers ; -Ministère de l'Education National -CNDP (Centre National de Documentation Pédagogique) ; -Cité des Sciences de la Villette à Paris ; -France 5 ; -Maison des Sciences de l'Homme (MSH) -Equipe Sémiotique Cognitive et Nouveaux Medias (ESCoM) ; -Etc.

  Sémiotique Cognitive et Nouveaux Médias (ESCoM) Maison des sciences de l'Homme (MSH) 54, Bd. Raspail -F-75006 Paris

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  The discussion of given problem (object, historical data, …) by the interviewed researcher; -The presentation of the historical filiation (genealogy) of a given (problem, object, …); -The qualification of already existing researches; -The qualification of the achievement of a personal research; -The argumentation of a scientific affirmation (hypothesis, theory, …); -The confrontation (critical comparison) of two or more research results; -Etc.

  : a part of the "scientific events" library -the "Interviews")Equipe Sémiotique Cognitive et Nouveaux Médias (ESCoM)Maison des sciences de l'Homme (


S R X Lopes

P B Gonçalves

D C Pamplona

Influence of initial geometric imperfections on the stability of thick cylindrical shells under internal pressure

This paper investigates numerically and experimentally the influence of initial geometric imperfections on the critical loads of initially stretched thick hyperelastic cylindrical shells under increasing uniform internal pressure. Imperfections in shells can have a global or local character. First, two types of local imperfections are considered: (1) a local axially symmetric imperfection in the form of a ring and (2) a small rectangular imperfection. The influence of the imperfection thickness, position and size are analysed in detail. Results show that the critical load decreases as the imperfections increase in size or thickness and as they move from the boundaries to the centre of the shell. The influence of multiple local imperfections is also studied in the present paper. Finally, the influence of global imperfections is considered with the imperfections described as a variation of the shell curvature in the axial direction. The results show that thick hyperelastic shells may be sensitive to local and global imperfections. In all cases the experimental results are in good agreement with the numerical ones, corroborating the conclusions.

these shells. As the imperfections of these test specimens were small, a good agreement between the experimental and numerical results was obtained. However, the initial geometric imperfections may have a marked influence on the behaviour of this type of structure, as it happens in the analysis of thin metal cylindrical shells. Koiter [START_REF] Koiter | On the stability of elastic equilibrium[END_REF] was the first one to develop a theory that explains in a rational form, based on the influence of imperfections, the discrepancy between theoretical and experimental results in the buckling analysis of certain structures. Amongst them, axially compressed cylindrical shells and spherical shells submitted to external pressure stand out. The theory of Koiter allows the determination of the type of bifurcation that occurs at the first bifurcation point along an equilibrium path and the verification of its sensitivity to initial geometric imperfections.

Thick shells submitted to large elastic deformations under the action of increasing internal pressure can display global and local instabilities. Initially, when submitted to internal pressure, the shell deforms almost uniformly throughout its length (except near the boundaries). After a certain critical pressure is reached, a bulb appears in the shell. This instability is followed by a sudden decrease of the internal pressure and a decrease in the radial deformation of the other parts of the structure. This instability corresponds to a limit point along the non-linear equilibrium path, that is, a turning point of the surface tension that corresponds to a bifurcation point. In a perfect shell, the bulb has an axi-symmetrical form and occurs in the middle of the shell. This behaviour is known in literature as localization [START_REF] Tvergaard | On the localization of buckling patterns[END_REF] and was observed in thick hyperelastic membranes by Kyriakides and Chang [START_REF] Kyriakides | The initiation and propagation of a localized instability in an inflated elastic tube[END_REF], who analysed this problem both theoretically and experimentally. However, due to imperfections, the bulb can occur in any place along the length of the shell and may become completely asymmetrical. This behaviour can be observed in different areas of study, with the formation of aneurisms being one of the most important (Figure 1).

The linear constitutive laws valid for the small strain regime are sometimes extended to the large displacement, small strain, which can lead to unrealistic results when large deformations are

INTRODUCTION

In previous works [START_REF] Pamplona | Finite axisymmetric deformations of an initially stressed fluid-filled cylindrical membrane[END_REF][START_REF] Pamplona | Finite deformations of cylindrical membrane under internal pressure[END_REF] the authors studied the non-linear behaviour and instability of thin cylindrical shells and carried out a detailed numerical and experimental parametric analysis of present. In such cases, elasticity in the fully non-linear range must be employed. The pioneering work of Green and Adkins [START_REF] Green | Large Elastic Deformations and Non-Linear Continuum Mechanics[END_REF] on non-linear elasticity set up the basis for the analysis of structures under large deformations. Historically, the large deformations observed in rubber-like materials under several loading conditions motivated researchers to express the associated non-linear elastic behaviour through hyperelastic models. Among the hyperelastic models, strain invariant based Mooney-Rivlin model [START_REF] Mooney | A theory of large elastic deformations[END_REF][START_REF] Rivlin | Large elastic deformations of isotropic materials: I. Fundamental concepts. II. Some uniqueness theorems for pure homogeneous deformation[END_REF] is the oldest one. The simplest constitutive model is the Neo-Hookean model, which can be viewed as a simplification of the Mooney-Rivlin law. The analysis of thin hyperelastic cylindrical shells has been a problem of continued interest. The investigation of the equilibrium and stability of thin cylindrical tubes under uniform pressure loading or loads acting along the boundaries has been examined theoretically, for instance, by Alexander [START_REF] Alexander | Tensile instability of initially spherical balloons[END_REF], Ratner [START_REF] Ratner | Tensile stability of cylindrical membranes[END_REF], Haseganu and Steigmann [START_REF] Haseganu | Theoretical flexure of a pressurized cylindrical membrane[END_REF], Haughton [START_REF] Haughton | Axially elastic membranes subjected to fluid loads[END_REF] and Gent [START_REF] Gent | Elastic instabilities in rubber[END_REF]. Experimental investigations were carried out by Pamplona and Bevilacqua [START_REF] Pamplona | Large deformations under axial force and moment load of initially flat membranes[END_REF] and Pamplona et al. [START_REF] Pamplona | Finite axisymmetric deformations of an initially stressed fluid-filled cylindrical membrane[END_REF][START_REF] Pamplona | Finite deformations of cylindrical membrane under internal pressure[END_REF], among others. When the thickness to the outer radius ratio is small, it can be modelled as an ideal membrane subjected only to in-plane normal and shearing forces. When the membrane becomes thicker, flexural effects cannot be disregarded in the analysis and the structure must be modelled as a shell or a solid. The analysis of large deformations of hyperelastic thick shells is not so popular, nevertheless there are some important publications such as the ones by Kyriakides and Chang [START_REF] Kyriakides | The initiation and propagation of a localized instability in an inflated elastic tube[END_REF] and, more recently, the works by Tang et al. [START_REF] Tang | Wall stress and strain analysis using three-dimensional thick-wall model with fluid-structure interactions will be blood flow in carotid arteries with stenoses[END_REF] and Haussy and Ganghoffer [START_REF] Haussy | An orthotropic hyperelastic model of cylindrical thick shells under pressure: application to the modeling of aneurysm[END_REF] where the theory of thick hyperelastic shells was used for modelling carotid arteries and aneurysms, respectively. An analytical solution for a perfect elastic thick tube under axial load was presented by Haughton and Ogden [START_REF] Haughton | Bifurcation of inflated circular cylinders of elastic material under axial loading-exact theory for thick-walled tubes[END_REF].

In this work the imperfection sensitivity of thick hyperelastic shells under internal pressure is analysed numerically and experimentally. To the authors knowledge this is the first work where the effect of large local imperfection in thick hyperelastic shells is analysed in detail. Non-symmetrical and axi-symmetrical local and global imperfections are considered in the analysis. The effect of multiple local imperfections is also investigated. Variations of the imperfection thickness, length and position along the shell length are considered. In the theoretical formularization of the problem, the material of the shell is considered to be incompressible, homogeneous and isotropic and modelled as a Mooney-Rivlin material, which is described by two elastic constants. These assumptions are sufficiently reasonable when compared with the physical behaviour of the shell used in the experimental analysis (a latex shell). In the analysis of the large deformations of the shell, the non-linear shell response is obtained by the finite element code ABAQUS [START_REF] Hibbit | ABAQUS Standard Theory Manual[END_REF][START_REF] Hibbit | ABAQUS Standard User's Manual[END_REF]. This code offers a good library of non-linear finite elements and constitutive laws for analyses of shells under large deformations. Newton-Raphson algorithm is used together with Riks arc-length method to obtain the post-critical behaviour of the shell under traction and internal pressure. In the experimental analysis, two types of imperfections are considered: the first one, an axi-symmetrical imperfection in the form of a ring, and then in the form of a small square. In all the cases a good qualitative agreement is observed between the numerical and experimental analysis. Since one of the main influences on the formation of aneurysms is the presence of local imperfections due to variations in the wall thickness, we feel that the present work is a useful contribution in this area.

ANALYSIS OF AXI-SYMMETRICAL IMPERFECTIONS

Experimental analysis

The numerical results are conducted considering latex shells with undeformed radius R, length L and thickness H . The material of the shell is considered, based on experimental results, as homogeneous, isotropic and incompressible [START_REF] Xavier | Nonlinear behaviour and instability of hyperelastic membranes and shells[END_REF]. Its geometric characteristics were measured by the Institute of Metrology of the Catholic University of Rio de Janeiro (ITUC-PUC-Rio) and the results are presented in Table I.

The local imperfection has the form of a ring, as illustrated in Figure 2. It is generated, with the aid of an electric sander, by the reduction of the thickness of the shell. It must be pointed out that it is very difficult to control the thickness of the shell in the imperfection region. In the tested specimens, the wall thickness was reduced between 40 and 50%. The imperfect shell initially is under traction through an imposed axial displacement L with a relation L/L = 0.16. After the initial elongation, the shell is fixed in its two extremities. Then it is gradually filled with air. The imperfection has length L imp , thickness H imp and is located at a distance Z i from the upper edge. Five different values of Z i are considered, as shown in Table II.

The results of these five experiments are shown in Figure 3, where, for each imperfection, the critical configuration (formation of the bulb or aneurysm) and the respective value of the measured critical pressure P cr (10 -2 MPa) is shown.

For imperfections located in positions Z 1 and Z 2 , the bulb does not occur in the place of the local imperfection. This shows that even large imperfections near the supports do not interfere with the localization of the bulb. However, for imperfections far from the supports (Z 3 to Z 5 ) the bulb appears exactly in the place where the imperfection was generated, as observed in the photographs. When the imperfection is near the support, the place of the buckling localization is influenced by other small imperfections that may exist in the specimen. As observed, even for the so-called perfect shell, the bulb does not occur in the middle, as expected from symmetry conditions for a perfect shell. The theory can, of course, be rewritten to force the bulb to appear anywhere along the shell. Figure 4 shows the variation of the critical load with the distance of the imperfection from the upper support, Z i . Initially, a small increase in the critical pressure occurs (position Z 1 ), followed by a decrease of the critical pressure for the other four imperfection positions. This is also observed numerically, as it will be shown in the next section.

Z 1 0.02L Z 2 0.05L Z 3 0.20L Z 4 0.30L Z 5 0.45L (

Numerical analysis

Analytical studies of thin hyperelastic shell can be conducted only for simple geometries and loading cases [START_REF] Alexander | Tensile instability of initially spherical balloons[END_REF][START_REF] Ratner | Tensile stability of cylindrical membranes[END_REF][START_REF] Haseganu | Theoretical flexure of a pressurized cylindrical membrane[END_REF][START_REF] Haughton | Axially elastic membranes subjected to fluid loads[END_REF][START_REF] Gent | Elastic instabilities in rubber[END_REF][START_REF] Pamplona | Large deformations under axial force and moment load of initially flat membranes[END_REF]. For thick hyperelastic shells even simple problems can rarely be treated theoretically [START_REF] Haughton | Bifurcation of inflated circular cylinders of elastic material under axial loading-exact theory for thick-walled tubes[END_REF]. So, for most of the problems related to real hyperelastic shells, the solutions are usually obtained through numerical techniques. The most used methodology is the finite element method. However, for a correct analysis a shell element appropriate for large deformation analysis and a proper constitutive law for hyperelastic materials are required. Here, the numerical study is carried out using the finite element program ABAQUS. The shell has the same geometry as the one used in the experiments (Table I). Isotropic hyperelastic materials are conveniently represented in terms of a strain energy density w. Assuming the complete recoverability after deformation, the strain energy density depends only on the final state of strain and not on the loading. Thus given an undeformed reference state, the strain is characterized by the principal stretches 1 (meridional), 2 (azimuthal) and 3 (transversal) or, alternatively, by the strain invariants I 1 , I 2 and I 3 , that is

w = w( 1 , 2 , 3 ) = w(I 1 , I 2 , I 3 ) (1)
The three strain invariants of the deformation field are given by

I 1 = g i j G i j I 2 = G i j g i j I 3 I 3 = G g G = |G i j |, g = |g i j | (2a)
where G i j and g i j are the contravariant components of the metric tensors of the undeformed and deformed states, respectively, and G i j and g i j are the corresponding components of the covariant metric tensor [START_REF] Green | Large Elastic Deformations and Non-Linear Continuum Mechanics[END_REF].

The three deformation invariants can be rewritten in terms of the principal stretches i (i = 1, 2, 3) as

I 1 = 2 1 + 2 2 + 2 3 I 2 = ( 1 2 ) 2 + ( 2 3 ) 2 + ( 1 3 ) 2 I 3 = ( 1 2 3 ) 2 (2b)
Volume changes in rubber-like materials are very small, as shown by Treolar [START_REF] Treolar | The Physics of Rubber Elasticity[END_REF], and the simplifying assumption of incompressibility is usually adopted. Thus the constraint

I 3 = 1 (3)
is identically satisfied throughout the material. The strain energy density is then considered as a function of I 1 and I 2 only. Hence, the theoretical representation of the behaviour of a hyperelastic incompressible material is given by the definition of the function w(I 1 , I 2 ).

There are several constitutive laws in literature particularly adapted to the representation of elastomers. Rivlin [START_REF] Rivlin | Large elastic deformations of isotropic materials: I. Fundamental concepts. II. Some uniqueness theorems for pure homogeneous deformation[END_REF] has proposed the following polynomial form for the energy density function:

w = m,n C mn (I 1 -3) m (I 2 -3) n (4)
also known as the Mooney-Rivlin strain energy density function because the first-order polynomial function

w = C1(I 1 -3) + C2(I 2 -3) (5)
has first been introduced by Mooney [START_REF] Rivlin | Large elastic deformations of isotropic materials: I. Fundamental concepts. II. Some uniqueness theorems for pure homogeneous deformation[END_REF]. Equation ( 5) is a function of two constants, C1 and C2. This is probably one of the most used density functions in finite element codes.

A simplified form of Equation ( 5)

w = C1(I 1 -3) (6)
known as the Neo-Hookean strain density function, has also been extensively used in literature.

Here, the shell material is modelled as a hyperelastic material of the Mooney-Rivlin type with elastic constants C1 = 0.100398 MPa and C2 = 0.150843 MPa. These constants were obtained experimentally for this material by Xavier [START_REF] Xavier | Nonlinear behaviour and instability of hyperelastic membranes and shells[END_REF], and a good agreement was observed between experimental and numerical tests. The experimental methodology can also be found in Pamplona et al. [START_REF] Pamplona | Finite axisymmetric deformations of an initially stressed fluid-filled cylindrical membrane[END_REF][START_REF] Pamplona | Finite deformations of cylindrical membrane under internal pressure[END_REF].

Influence of the position of the local imperfection-Z i .

First, a convergence analysis of the mesh used with elements of type S4R (quadrilateral shell elements with reduced integration) is conducted. It is observed that a good convergence is attained for the critical load for a number of elements more than 1340. Therefore, all analyses are carried out using 1992 shell elements of type S4R.

The local imperfection position (Z i ) varies from 0.02L to 0.45L in relation to the lower support of the shell (see Table II). The ring imperfection has thicknesses that vary from H imp = 1.5 to 3.0 mm, that is, from H imp = 0.40H to 0.85H , with H being the thickness of the undeformed shell. For the imperfection length two values are considered: L imp = 5 mm and L imp = 0.025L, where L is the undeformed length of the shell.

Figure 5 shows the variation of the internal pressure as a function of the maximum radial displacement, U r , for different positions of the ring imperfection, as well as the response of the perfect shell. The influence of the imperfection on the bifurcation load is small and decreases as the imperfection approaches the support, as shown in Figure 5(b).

Figure 6 shows the critical configuration for each case. It is observed that, for an imperfection of thickness H imp = 0.85H , in positions Z 1 and Z 2 (Figure 6(a) and (b)), the formation of the bulb occurs, as for the perfect shell, in the middle of the shell. However, for positions Z 3 , Z 4 and Z 5 (Figures 6(c)-(e)), the bulb appears at the position of the imperfection ring (the imperfection in the middle of the bulb). This agrees with the general behaviour observed in the experimental analysis.

Figure 7 shows the variation of the critical pressure with the position of the imperfection for different imperfection thicknesses. There is a small increase in the critical pressure for positions Z 1 and Z 2 when compared with the perfect shell when the thickness of the imperfection is between zero and 0.85H (see detail in Figure 7(b)). These imperfections are close to one of the supports. However, for smaller imperfection thicknesses, this increase occurs solely for position Z 1 . As the distance of the imperfection from the support increases, the critical pressure decreases. This decrease is also followed by a change in the localization of the bulb, as observed previously (Figure 6).

Effect of the variation of the thickness and length of the local imperfection.

Consider again the same shell submitted to the same traction level and under increasing internal pressure.

Here the influence of the variation of thickness and length of the imperfection ring are analysed for an imperfection located in position Z 5 (Table II). This is the most sensitive location for the imperfection, as shown previously. Two lengths of the imperfection are considered: L imp = 0.0125L and L imp = 0.0250L. The thickness of the imperfection varies from 0.45H to 0.85H . Figure 8 shows the variation of the critical pressure with the thickness of the shell for the two distinct lengths of imperfection. One notices that, as expected, the critical pressure of the shell is directly proportional to the thickness of the imperfection, but inversely proportional to its length of the imperfection. Now the effect of length variation for imperfections located in positions Z 2 and Z 5 is analysed, considering H imp = 0.85H . The length of the imperfection varies from 0.0125L to 0.05L. Figure 9 9.5 presents the variation of the critical load with the increase of the length of the imperfection. As the imperfection length increases, the critical load decreases and approaches a lower bound. For position Z 2 , the bulb appears initially in the central region of the shell. However a change in the localization of the bulb occurs for imperfection lengths of 0.025L or higher (Figure 10). However, for position Z 5 , the bulb appears at the position of the imperfection for all the considered imperfection lengths.

STUDY OF THE LOCAL GEOMETRIC IMPERFECTIONS IN A SQUARE SHAPED REGION

Experimental analysis

Again, for the same shell geometry experiments considering shells with imperfections in the form of a small square along the length of the shell are carried out. These imperfections also are generated with the aid of an electric sander, through the reduction of the thickness of the shell, with dimensions of 0.025L × 0.025L. It must be pointed out that it is very difficult to control the thickness of the shell in the imperfection region. The critical configurations for imperfections at positions Z 1 -Z 4 are presented in Figure 11.

Figure 12 shows the variation of the critical pressure with the position of the imperfection. One notices that, for all the imperfection positions, a small decrease of the critical pressure occurs. However, the critical load for position Z 2 is slightly higher than that of position Z 1 . After this local maximum, the pressure decreases as the distance of the imperfection from the support increases. This fact, although puzzling, since the imperfection is actually strengthening the tube, was observed both numerically and experimentally. For positions Z 1 and Z 2 the bulb position is not affected by the imperfection, while for positions Z 3 and Z 4 the bulb appears exactly at the position of the imperfection. A behaviour similar to that observed experimentally for the axi-symmetric imperfection.

Numerical analysis

Now a numerical investigation of the behaviour of the imperfect shell is conducted. The local imperfection is represented by the reduction of the thickness of the shell in a square shaped region with dimensions of 0.025 × 0.025L and variable thickness. The variation of the position of the imperfection along the length of the shell is the same one adopted in the previous item (Z 1 , Z 2 , Z 3 and Z 4 ). The convergence analysis carried out in the previous item also applies to this analysis. Thus, as in the previous analysis, a mesh with 1992 S4R elements and 2016 nodes is considered. The results, shown in Figure 13, exhibit a small variation of the critical pressure with the position of the imperfection. In this case the imperfection thickness is H imp = 0.85H . Figure 14 illustrates the critical configurations obtained numerically. As in the experimental analysis, for imperfection Z 1 and Z 2 the bulb position is not affected by the imperfection, while for positions Z 3 and Z 4 the bulb appears exactly at the position of the imperfection. Also it is observed for positions Z 3 and Z 4 a loss of symmetry that affects the global behaviour of the shell. In addition to the local instability, a global, column type instability occurs. Now the same shell with an imperfection of thickness H imp = 0.75H is studied. The variation of the critical pressure as a function of the imperfection position is shown in Figure 15. For this imperfection thickness the critical pressure is different from the previous case (Figure 13). Here a small increase of the critical pressure in relation to the perfect shell occurs for Z 2 . However, for the other positions, the critical pressure decreases, being this decrease more significant in position Z 1 . As illustrated in Figure 16, for an imperfection in positions Z 1 , the bulb develops in the middle of the shell. For the other imperfection positions the bulb appears in the place of the imperfection. Again, a loss of symmetry of the deformed shell for these imperfection positions is observed.

In order to understand the effect of the imperfection thickness on the critical load and mode, a shell with H imp = 0.55H is now analysed. Again, as shown in Figure 17, a small variation of the critical pressure in relation to the one of the perfect shell is observed. From these results, one can conclude that the imperfection thickness and position has only a small influence on the critical pressure. However the variation of this critical pressure, although small, is different for different values of the imperfection thickness. On the other hand, the imperfections have an important influence on the local and global instability mode of the shell.

STUDY OF MULTIPLE LOCAL GEOMETRIC IMPERFECTIONS

Here two imperfections, equidistant from the supports and forming an angle imp between them, are considered. The angle imp varies from 0 • to 180 • , as illustrated in Figure 18. As in the previous analysis, the imperfections have the form of a small square and are considered to have the same geometry. The analysed shells have 3 mm thicknesses, 400 mm length, and 11 mm diameter (H/R = 0.55 and L/R = 72.7). The imperfections are located 0.05L from the supports (position Z 2 , as presented in Table II) and two imperfection thicknesses are considered, namely H imp = 0.55 and 0.75H .

Again a convergence analysis was carried out and it was observed that 1500 S4R elements are sufficient to obtain a good convergence for all configurations. Based on this analysis, a mesh of 1688 S4R elements and 1656 nodes is adopted.

Figure 19 shows the obtained critical configurations for imp = 0 • . The imperfections have thicknesses of H imp = 0.55 and 0.75H and are located at a distance of 0.05L from the supports. It is observed in the Figure 19(a) that, for an imperfection thickness 0.55H , the shell does not exhibit a global instability, with instabilities concentrated in the regions of the imperfections. However, as shown in the Figure 19(b), for an imperfection thickness of 0.75H , besides the two local instabilities, a global instability mode is observed. Figure 19(c) shows the typical non-linear behaviour of a hyperelastic thick shell under traction and increasing internal pressure. As expected, the critical load decreases as the thickness of the imperfection decreases.

The influence of the angle between the two imperfections is studied by varying imp from 0 • to 180 • and maintaining the geometry of both imperfections constant. The results are shown in

INFLUENCE OF THE INITIAL CURVATURE OF THE SHELL

Finally the effect of a global imperfection on the non-linear behaviour and load carrying capacity of the shell is studied. The imperfection takes the form of a non-zero curvature in the axial direction of the shell. Positive and negative initial curvatures are considered, as illustrated in Figure 22. The positive and negative curvature is generated through ABAQUS by using a spline through three given points, namely the extremities of the shell, (R, L/2) and (R, -L/2), and the point located at the middle height of the shell (x = 0) and radial co-ordinate r = R ± R, and R equals to 0.5, 1.0 and 1.5 mm. The inner and outer surface of the shell are subjected to the same curvature.

Initially, the behaviour of three shells of distinct thicknesses is analysed, considering two positive radius of initial curvature. Later two negative radius of curvature for the same shells are considered.

Influence of positive initial curvature

A numerical study of the behaviour of shells with positive initial radius of curvature of r 1 = +0.5, +1.0+2.0 × 10 4 mm is carried out considering shells with undeformed thicknesses H = 1.0, 2.0, and 3.0 mm. The cylindrical shell of reference (without imperfection) has a radius of R = 9.0 mm and an undeformed length of L = 300 mm and is submitted to an initial traction of L/L = 0.16.

Figure 23 shows the variation of the internal pressure with the maximum radial displacement for shells with initial thicknesses of 2.0 and 3.0 mm. A considerable reduction of the critical load of the shell with the increase of the initial curvature is observed for both shell thicknesses. The deformation pattern is the same found for the perfect shell, as shown in Figure 24, with one bulb in the middle of the shell.

Influence of negative initial curvature

The same study is carried out for the geometries presented in the previous item considering now a negative initial curvature of r 1 = -0.5, -1.0 and -2.0 × 10 4 mm. Figure 25 shows the variation of the internal pressure with the maximum radial displacement for two shells. In this case, the critical load increases with increasing imperfection. Also, the critical configuration is completely different from that observed for the perfect shell, with the formation of two bulbs near the supports, as illustrated in Figure 26.

Figure 27 shows the variation of the critical pressure of the shell with the radius of curvature r 1 .

CONCLUSIONS

The non-linear behaviour and stability of a thick hyperelastic shell under constant traction and increasing internal pressure was analysed both numerically and experimentally. In the numerical analysis, the material of the shell was modelled as a Mooney-Rivlin material with two elastic constants and the shell was discretized by the finite element method using shell elements for large strain analysis. The objective was to study the effects of local imperfection on the critical load and particularly on the local buckling of the shell. First a local axi-symmetric imperfection in the form of a ring was generated through the reduction of the shell thickness. A parametric analysis of this type of imperfection was carried out varying the position, thickness and length of the imperfection. It was observed that the dimensions of the imperfection have a small influence on the critical pressure. The critical load decreases as the thickness of the shell in the region of the imperfection decreases and as the local imperfection moves away from the supports. It is also observed that the critical load is inversely proportional to the length of the imperfection. A good agreement in terms of the global behaviour was verified between the obtained numerical results and the experimental results for imperfect shells.

Another type of local imperfection in the form of a small square was analysed. A behaviour similar to the one observed previously was obtained.

The influence of multiple imperfections was also analysed. For this, two local imperfections of the same form and thickness located symmetrically in relation to the middle of the shell were considered and the angle between the two imperfections in the circumferential direction was varied between 0 • to 180 • . When the angle between the imperfections varied between 0 • and 120 • and 165 • and 180 • two bulbs were observed in the numerical analysis. For angles between 120 • and 165 • , only one bulb was observed. However, in all experiments involving two imperfections only one bulb can be observed. This can be explained by the fact that in the experimental analysis is practically impossible to generate two identical local imperfections. Even small variations in the local imperfection make the bulb to form in only one location.

Finally, a global imperfection was generated through modifications in the initial curvature of the shell in the axial direction. It was observed that, for negative initial curvatures, the critical pressure suffers a small increase in relation to the one obtained for the cylindrical shell with constant radius. On the other hand, for positive initial curvatures the critical pressure decreases as the curvature increases.

This work is part of an ongoing research on the stability analysis of thick hyperelastic shells under various loadings with emphasis on the local and global instability phenomena. The ultimate goal of this research is to model and understand the formation of aneurysm in arteries. At the moment more refined constitutive laws are being tested both numerically and experimentally [START_REF] Holzapfel | A new constitutive framework for arterial wall mechanics and a comparative study of material models[END_REF][START_REF] Sokolisa | Assessment of the aortic stress-strain relation in uniaxial tension[END_REF].

Figure 1 .

 1 Figure 1. Cerebral aneurisms (Barrow Neurological Institute Web site).

3 Figure 2 .

 32 Figure 2. Geometry of the axi-symmetric imperfection.

Figure 3 .

 3 Figure 3. Critical configurations, obtained experimentally, and the corresponding critical pressures: (a) perfect shell and shell with imperfection at (b) Z 1 = 0.02L; (c) Z 2 = 0.05L; (d) Z 3 = 0.20L; (e) Z 4 = 0.30L; and (f) Z 5 = 0.45L.

Figure 4 .

 4 Figure 4. Variation of the critical pressure with the localization of the imperfection ring.

Figure 5 .

 5 Figure 5. (a) Internal pressure as a function of the maximum radial displacement for different positions of the imperfection (Z i ); and (b) the detail of the region of the maximum pressures.

Figure 6 .

 6 Figure 6. Deformed cylindrical shell with imperfection thickness H imp = 0.85H , for imperfection in positions Z 1 -Z 5 : (a) Z 1 = 0.02L; (b) Z 2 = 0.05L; (c) Z 3 = 0.20L; (d) Z 4 = 0.30L; and (e) Z 5 = 0.45L.

Figure 7 .Figure 8 .

 78 Figure 7. (a) Variation of the critical pressure with the position of the imperfection for different values of the imperfection thickness; and (b) detail near position Z 1 .

Figure 9 .Figure 10 .

 910 Figure 9. Variation of the critical load with the increase of the length of the ring of imperfection for 2 positions Z 2 and Z 5 .

Figure 11 .Figure 12 .

 1112 Figure 11. Experimental results. Critical configurations for shells with initial imperfection in a square shaped region in positions: (a) Z 1 = 0.02L; (b) Z 2 = 0.05L; (c) Z 3 = 0.20L; and (d) Z 4 = 0.30L.

Figure 13 .

 13 Figure 13. Variation of the critical pressure with the position of the imperfection (H imp = 0.85H ).

Figure 14 .

 14 Figure 14. Deformed configurations of shells with local imperfections of dimensions 0.025L × 0.025L × 0.85H for positions Z 1 -Z 4 : (a) Z 1 = 0.02L; (b) Z 2 = 0.05L; (c) Z 3 = 0.20L; and (d) Z 4 = 0.30L.

Figure 15 .

 15 Figure 15. Variation of the critical pressure with the position of the imperfection of dimensions 0.025L × 0.025L × 0.75H .

Figure 16 .

 16 Figure 16. Configurations of the shell at the critical points with an imperfection of dimensions 0.025L × 0.025L × 0.55H in positions Z 1 -Z 4 : (a) Z 1 = 0.02L; (b) Z 2 = 0.05L; (c) Z 3 = 0.20L; and (d) Z 4 = 0.30L.

Figure 17 .

 17 Figure 17. Variation of the critical pressure with the position of the imperfection for H imp = 0.55H .

Figure 18 .

 18 Figure 18. Position of the two local imperfections.

Figure 19 .

 19 Figure 19. (a), (b) Deformed configurations of shells with two local imperfections forming an angle of 0 • between them; and (c) respective load-displacement response.

Figures 20 and 21 .

 21 Figures 20 and 21. imp has a small influence on the non-linear response of the shell. However, with regard to the deformation pattern of the shell at the critical point, a puzzling fact is observed: for angles of 130 • , 150 • and 165 • , only one bulb appears, Figures21(e)-(g); for all other angles two bulbs are observed. It must be pointed out that the formation of two bulbs was not observed in the experiments. In the laboratory it is impossible to generate two exactly equal imperfections. Even small variations in the position and size of the imperfections, apart from other inherent model imperfections, are sufficient to localize the instability at only one position.

Figure 20 .

 20 Figure 20. Variation of the internal pressure as a function of the maximum radial displacement for different values of imp . H imp = 0.75H .

Figure 21 .

 21 Figure 21. Deformed configurations considering two local imperfections with imp varying from 0 • to 180 • : (a) imp = 30 • ; (b) imp =60 • ; (c) imp = 90 • ; (d) imp = 120 • ; (e) imp = 130 • ; (f) imp = 150 • ; (g) imp = 165 • ; and (h) imp = 180 • .

Figure 22 .Figure 23 .Figure 25 .

 222325 Figure 22. Representation of the initial curvature of the shell: (a) positive; (b) null; and(c) negative and its respective radius of curvature.

Figure 27 .

 27 Figure 27. Variation of the internal critical pressure of the shell with initial radius of curvature (r 1 ) for shells with undeformed thicknesses of 2.0 and 3.0 mm.

Table I .

 I Geometric characteristics of the latex shell used in the experimental analysis.

					Geometric
		External radius	Thickness	Length	relations
	Rubber	R(mm)	H (mm)	L(mm)	R/H	L/R
	Latex	9.38	3.58	200.00	2.63	22

Table II .

 II Position of the ring imperfection.

	Position of the	Position from the superior support
	imperfection	(%L)

ACKNOWLEDGEMENTS

This work was sponsored by the Brazilian research agencies CNPq and CAPES.