N Perry

Keywords: Design for recycling, eco-design, organic matrix composites, composite recycling, knowledge exchange

The use of composites in industry is ever increasing. However, endof-life solutions for composites are still under development. In this paper, a solution linking design strategies with a recycling process for thermoset composite materials is proposed. The recovery solution for these materials is a supercritical water solvolysis process. The needs and multi-disciplinary skills required for taking recycling possibilities into account in the early stages of product design and the necessity to standardize product-recycling capabilities based on design requirements will be discussed. The paper highlights the need for designers to take a functional approach into consideration, including characterization of materials behavior, recycling process limits, constraints and opportunities. This paper will show the first lessons learned from experiments, using this technique.

Information exchanges between experts for thermoset composites design for recycling

Nicolas Perry * I2M -UMR 5295 -Arts et Métiers ParisTech, F-33400 Talence, France. E mail : n.perry@i2m.u-bordeaux1.fr

Stéphane Pompidou, Olivier Mantaux 1 Introduction

Today, the key focus for the transport industry is to make lighter vehicles. Reduction in weight leads to a decrease in energy consumption and CO 2 emissions. Reducing greenhouse gases and pollution is one of our society's main challenges as it strives for sustainable development. Composites provide good opportunities for combining high modulus materials with free definitions of geometry. As a result, their use in industry is increasing. The aerospace and aeronautics sectors have integrated composites at different levels for their products (organic matrix based for cold applications and metallic or ceramic-based composites for high temperature applications).

This study is based on European regulations and tend to anticipate their future evolutions (such as REACh or electronic wastes) [START_REF] Reach | Registration, Evaluation, Authorisation and Restriction of Chemicals (REACH)[END_REF] (WEEE, 2002). Today, in the automotive industry, the limits of the use for the composites are their potential recyclability. Indeed, in a global and eco-friendly approach, one must analyze and take end-of-life solutions for systems into account, at an early stage of their development process. Moreover, in the case of automotive, regulations impose a 95% ratio of recycling of an out-of-use vehicle.

The term de-manufacture has become more and more common, especially in the electronics industry. It characterizes the process of recycling materials and products, including end-of-life strategies and logistics in product development [START_REF] Berry | Design for recycling[END_REF]) [START_REF] Gaustad | Design for Recycling: Evaluation and Efficient Alloy Modification[END_REF]. Design engineers have to balance safety, energy efficiency and cost. Unfortunately, they rarely get to the point of thinking about what will happen to the product at the end of its useful life [START_REF] Kriwet | Systematic integration of design-for-recycling into product design[END_REF] [START_REF] Vallet | How ecodesign tools are really used -Requirements list for a context-related ecodesign tool[END_REF].

However, as new materials and technologies are developed, the challenge that recyclers face in safely and economically recycling those products grows ever more difficult [START_REF] Calcott | Waste, recycling and design for environment: Role of markets and policy instruments[END_REF]. Recycling a product means: (i) have a recycling technology available, (ii) get a dismantle solution and an access for the product, and (iii) dispose identification plus selection (maybe clean or pure) possibilities for the materials. Moreover, the recycling processes have to balance the technical, economical and environmental aspects of the end-of-life proposal. Composite applications have opened up a new field of research and development in the domain of new recycling processes.

Water under supercritical conditions gives the opportunity to recover composite reinforcements (carbon fibers) and open new opportunities for second-life composites.

We are focusing on research activities that aim at integrating recycling constraints in the design stage of composite parts. It is necessary to promote discussion between designers and recyclers in order to innovate in the definition of new recycled composite products. This means that information from both actors will be shared. However, it also implies that information from the material level should be enriched (for both designers and recyclers point of views). Therefore, it is necessary to include third parties in the discussion: experts in material and mechanical characterization. This expertise makes the link at the material level giving the expected information for designers and recyclers, enabling the material specification.

The second section of this paper presents an overview of composite recycling possibilities and the technical and economic reasons for its development. The third part will focus on the design for recovery issue and the specificities related to composite design. The fourth section will explain our understanding in terms of skills, needs and know-how required for addressing this issue. Before concluding, the last section will illustrate some feedback and the first lessons learned regarding eco-design for composites.

Composites recycling motivations and solutions

Future regulatory constraints are driving industries to develop efficient end-of-life alternatives based on technical and economic constraints. For carbon and aramid based products, high prices (e.g. carbon prepreg: approx. 180 €•kg-1; Kevlar©: approx. 150 €•kg-1) and the world shortage in raw materials production, are the leitmotiv for finding technical and cost-effective recycling solutions. In such cases, second use of composite fibers will be dedicated to the manufacture of medium or low loaded parts (non-structural in many cases). Indeed, the recycled fibers and reprocessed semi-products have to reach the full acceptance and the trust of users (designers), regarding their health and quality. In fact, many stakeholders upon seeing the "recycled" label still tend to think of low quality [START_REF] Piñero-Hernanz | Chemical recycling of carbon fibre reinforced composites in nearcritical and supercritical water[END_REF] [START_REF] Pimenta | Recycling carbon fibre reinforced polymers for structural applications: Technology review and market outlook[END_REF]. However, as far as recycled carbon fibers are concerned, this is far from the truth. Methods exist today by which carbon fibers and prepreg can be recycled, and the resulting recyclate retains up to 90 percent of the fiber's mechanical properties. In some cases, the method enhances the electrical properties of the carbon recyclate because the latter can deliver a performance close or superior to the initial material (Perry et al., 2010a). So it is necessary to create demand for recycled fiber, by packaging it in a form useful or attractive to end-users. For example, cheap materials with very good properties could find larger applications than composites today. Moreover, this material, resulting from a reprocessing path, is more environmental friendly, and has the potential of a new recycling loop. In this way this is a down-cycling solution as explained by [START_REF] Mcdonough | Cradle to Cradle: Remaking the Way We Make Things[END_REF]. We develop a remanufacturing solution in order to limit or master the properties losses.

To make such a recycled product cycle viable, the key factors are mainly the waste deposit quantity and its availability. In order to ensure the efficiency of the recycle path collecting and identifying infrastructure must be established in order to guaranty and improve the weighting of the recycled flux. Finally, the end-user must have confidence in the quality of the product in terms of robustness and value. Mechanical, physical and chemical studies must be purchased in order to enrich the recycled material data and confirm the properties of recycle fibers, semi-products and structures. These actions have been done successfully in the plastics industry. For example, forms of identification labels were added on plastic parts to facilitate collecting and sorting. Most composites manufacturers are already engaged in waste management procedures, pushed by the REACH regulation; as a result, they recycle waste materials on the workshop when collecting and processing solutions are proposed.

Thermoset matrix based composites designers and manufacturers are currently taking two different directions. On one hand, they are trying to increase the use of green composites or bio-composites, on the other hand, they develop or improve recycling technologies. In the first place, the interest of using bio-composites lies in their small environmental impact Indeed they are made of:

1) a bio-polymer matrix reinforced by synthetic fibers (glass, carbon, Kevlar©); 2) a petroleum-derived matrix reinforced by natural fibers (kenaf, flax, hemp, bamboo, coconut stems); 3) biopolymers (e.g. polylactic acid -PLA-or polyhydroxyalkanoates -PHA) reinforced with natural fibers. Natural fibers are being tested to estimate their reinforcement properties. They promise to be the future solution for organic matrix composites parts [START_REF] Feng | The biomaterial for green composites[END_REF]) [START_REF] Mohamad | Natural fibres for the 3rd millenium[END_REF]. For example, their mechanical properties can compete easily with glass fibers. Indeed, their specific density ranges from 20 to 45 and their tensile stress ranges from 400 to 1500 MPa. Furthermore, solutions already exist for high-performance mechanical composites, for example by improving the weaving processes (Weager, 2010). Biopolymers or bio-compound (a combination of bio-polymer and petroleum derived polymer) have also been studied [START_REF] Bourmaud | Rigidity analysis of polypropylen/vegetal fibre composites after recycling[END_REF]. For example, they come from PLA derived from cornstarch, or PA11 from castor seeds and its ricin protein.

On the one hand, their carbon footprint is reduced, by the use of a bio renewable material, and on the other hand by the increase of the recyclability potential.

In addition, the end-of-life impact awareness helps developing recycling technology. The thermoset matrix can be removed either by burning or grinding techniques. It's cheap but very aggressive for the carbon fibers [START_REF] Mantaux | Recycling study of end of life products made of ABS resin[END_REF]2009). Complex thermal, chemical and mechanical processes are needed to obtain high quality recycled carbon fibers. Pyrolysis or solvolysis are two of these very promising solutions. However, there are some limitations; for example, it is impossible to recycle different categories of matrix simultaneously. At the same time, specific coatings (e.g. metallic cladding for electric behavior) are not compatible with some processes. Thus, specific requirements must come from the recycling stages if they are to be efficient. The most obvious but also important thing is to extract and free all the metallic inserts, even before grinding. Moreover, before recycling, the products have to be dismantled and adapted to the recycling process reactor. These reactors are mainly cylindrical Cutting operations are compulsory.

In southwest France, composites recycling will increase in terms of quantity due to the creation of two dismantling platforms:

1) TARMAC platforms are dedicated to civil aircraft applications in collaboration with Airbus, EADS Sogerma. TARMAC first focuses on the re-use and the certification of replacement parts in aircraft maintenance, 2) P2P platform (close to Bordeaux) deals with the disassembly of ballistic weapons.

A consortium of rocket manufacturers (EADS Astrium Space Transportation, Snecma Propulsion Solide, etc.) are working together on the RECCO project (RECCO is a french acronym for composites recycling). The final goal is to validate an industrial solution and an industrial demonstrator for composite recycling. The solvolysis process is chosen for removing the thermoset matrix [START_REF] Aymonier | Review of supercritical fluids in inorganic materials science[END_REF] [START_REF] Cunliffe | Pyrolysis of composite plastic waste[END_REF]. In this technological and industrial background, we promote an early integration of the recycling constraints and possibilities in the design process of carbon composite parts. The next sessions will explain the integration levels we face in order to develop a design for recovery approach.

Composite design for recycling / recovery

In order to take the end-of-life information into consideration in the product design phase, we naturally opted for design for x approaches [START_REF] Wallace | High-performance biocomposites: novel aligned natural fibre reinforcments[END_REF] [START_REF] Hauschild | Design for Environment approaches? Do We Get the Focus Right[END_REF]. In our case, we worked on design for recovery (instead of recycling) taking common consideration (disassembly in this case) that should lead designers to propose solutions for products. In this perspective, all the recycling requirements are considered as input (data) that must be taken into account in a product's functional specifications. In addition, from a semantic point of view, we promote the use of the term recovery instead of recycling, in order to emphasize the second life and second use of the product or material after the recycling phase. Besides, we shift from the cradle to grave cycle to the spirit of the cradle to cradle [START_REF] Mcdonough | Cradle to Cradle: Remaking the Way We Make Things[END_REF][START_REF] Mcdonough | Peer Reviewed: Applying the Principles of Green Engineering to Cradle-to-Cradle Design[END_REF] [START_REF] Braungart | Cradle-to-cradle design: creating healthy emissions -a strategy for eco-effective product and system design[END_REF]. Thus, even if we are using design for recycling methodology, or using the term recycling, we take it as a dynamic state, as a rebirth for future use and not as a static and final goal. That means considering future product design or future second life material at the recycling level. It also means that the material down-cycling effect have to be taken into account in order to improve material data base with its different evolutions and recycling solutions. In other words, we are in two dual areas of research. Firstly, design for the end of life can be summarized as design for recovery. Secondly, from recycling to design, the research deals with robust material recycled for design.

Design for recovery challenges aim to protect the environment and create sustainable means for preserving our resources and reducing energy loss and pollution. It seeks to achieve two very basic goals. The first one is to eliminate or reduce the use of hazardous or toxic materials which may present a serious threat to the environment, or put a recycler's workforce in jeopardy. The second is to discourage the use of materials that are not recyclable or manufacturing techniques that make a product non-recyclable using current technologies. The best time to address these issues is during the design stage [START_REF] Ferro | Design for recycling in the automobile industry: new approaches and new tools[END_REF]. Addressing a product's end of life is essential at the very beginning. Adopting this premise helps to ensure an efficient recycling chain, this goes well beyond the scrap processor to the mill, smelter, or extruder that will take the recycled materials and make them into new materials. Design for recycling is a mindset that all design engineers must embrace if they hope to have their products considered as environmentally friendly. As mentioned before, design for recycling is driven by governmental mandates like the European Union's waste electrical and electronic equipment directive (WEEE) or end-of-life vehicle (ELV) directive (in Europe, the rate of re-use and recovery should reach 95% in 2015, and 85% for re-use and recycling, in average weight per vehicle and per year).

There is more than environmental compliance at stake here. As new materials are developed, such as carbon-based composites, they bring about a new threat in terms of recycling. As these new materials are introduced into products and are replacing materials that have been recyclable for generations, they affect recyclability negatively both practically and financially, and can have a devastating impact. Even materials that are recyclable can pose a problem when used in combination. Take for instance a product that is made from many different types of plastics. Today's recycling technology can only sort two or three different types of polymer materials using a mechanical solution, at best. Composites effectively become non-recyclable, or at least the plastics fraction of that product will be non-recyclable [START_REF] Seager | A taxonomy of metrics for testing the industrial ecology hypotheses and application to design of freezer insulation[END_REF] (Perry et al., 2010b).

Therefore, in order to address the global problem, we are working on different levels: (i) design teams, (ii) design methods and (iii) design tools, so as to reduce the gap between the existing recycling solutions or bio-composites possibilities, and designer's today solutions. Not only engineers have new materials and new product design solutions for eco-responsible products, but there are now also different tools available to help model and evaluate the solutions and the impact of the product life. This end-of-life requirements definition, also points out the needs, in terms of maintenance or parts fixing.

Unfortunately, the results of these rating tools depend on the information available. Most of the time, little information is available about the life and end of life of a product in the design stage. Furthermore, in the case of recycling processes under development, it is necessary to anticipate the potentiality of the technologies and their applications. The uncertainty of decisions will therefore be increased. Indeed, design decisions, in terms of end-of-life consequences, will undoubtedly appear 5 to 20 years later.

Before giving further explanation, it is important to remember that composite design is complex due to the fact that it is necessary to simultaneously define (i) geometry and shape, (ii) materials (fiber and matrix) and the reinforcement orientations, and (iii) manufacturing process. These three topics are linked and inter-dependent in the design and optimization process. For example, laminates selections limit shapes possibilities and depend on manufacturing capability. Consequently, as illustrated in figure 1, the ecodesigner has to juggle with constraints from various sources, in addition to internal constraints and relation-ships of the composite design. Up to now, these additional constraints are not taken into consideration, except for the consequences of the European regulation (REACh) which focuses on manufacturing aspects.

Levels of complexity for the integrated design

Integrated design means addressing the recycling process development issue carefully while at the same time, proposing the possibility of including such evolutions in design methodology. We identified three main problems:

• the first is linked with recycling physics and scheduling to link and reach design methodologies; • the second is dedicated to the uncertain and non-complete nature of the available information;

• the third problem has to do with competencies and skills needed to for the design of robust recovery problems. We subscribe to the proposal of Nonaka who explains that information becomes knowledge when it is used by a human in a cognitive process. It becomes know-how and even skill when this knowledge is efficiently used and applied with others [START_REF] Nonaka | SECI, Ba and Leadership: a Unified Model of Dynamic Knowledge Creation, California Management Review[END_REF].

Physics and timetable

This part deals with the knowledge of integration in design of the recycling process. These processes are most of the time under development, searching for breakthroughs, innovations and applications. The recycling rules that must be included in the design process are still under formalization while designers must take decisions. Furthermore, the decisions taken today affect the product much later. Figure 2 shows the core elements that build the skeleton of a product's end of life. Different stakeholders are included in the loop and should point out information, expected data and decision rules they apply to switch from one stage to another. The different schedule between real recycling process and part design increases the integration difficulties. For existing and robust end-of-life paths, constraints and material re-process and re-use are well known. For new recycling processes, robust validation can take time while designers have to take decisions now. An extreme example is nuclear plants. Built 40 years ago, there are still no efficient end-oflife solutions as their dismantling will soon begin. Research and development teams are banking on some kind of technological breakthrough to guide developments in recycling. Innovation is needed not only in the recycling phase, but also for all the key stages. It is important to improve disassembly techniques (see design for assembly-disassembly approaches) or selection efficiency [START_REF] Boothroyd | Design for Assembly and Disassembly[END_REF]) [START_REF] Aymonier | Review of supercritical fluids in inorganic materials science[END_REF]) (Lopinet et al., 2010). At the end of the process, it is essential to develop innovative and valuable uses to compete with virgin raw materials (for similar characteristics) or to find new opportunities, at the very early design of the recycling path.

As far as composites are concerned, new processes enable fibers to be recovered with very little distortion and fracture from the initial disposal of the composite part. Supercritical fluids (id. es. solvolysis) give such opportunities (Loppinet-Serani et al., 2010) [START_REF] Kromm | Tensile and creep properties of ultra high molecular weight PE fibres[END_REF]. However, the problem of alignment and restructuration of the fibers still remains. Competences (knowledge and know-how) and skills on fibers spinning and weaving have been integrated by recycling teams. Other alternatives consist in reprocessing medium size flat rectangular pieces of 1D or 2D carbon fibers. The innovation consists in proposing a patchwork's part design approach. Moreover, specific works must identify the mechanical characteristics and efficient strategy for material characterization according to the product design development phase, from the recycled fibers to the final structure [START_REF] Laurin | Approche multiéchelle des mécanismes de ruine progressive des matériaux stratifiés et analyse de la tenue de structures composites[END_REF] [START_REF] Rollet | Vers une maitrise des incertitudes en calculs des structures composites[END_REF]. This pyramidal testing problematic, from the micro to macro behavior, at all the stages of the product (from the fiber to the structure), must integrate the uncertainty of complex but in real case tests [START_REF] Dennison | The Test Pyramid: A Framework for consistent evaluation of RFID tags from design and manufacture to end use[END_REF] [START_REF] Ladevèze | Lack of knowledge in structural model validation[END_REF].

Information access and result trusts

Figure 2 is a starting point for identifying the existing information available at each stage of the process. This map integrates the different stakeholders, and identify which kind of information is needed or available, who is the owner or requires this information [START_REF] Bernard | Virtual Engineering based on knowledge integration[END_REF]. This map is the base of the information system (PLM based) in order to link information and engineering tools for the product development phase and the material recycling and characterization. Rosemann proposed an engineering platform in order to optimize disassembly and recycling strategy at the early stage of the product design. It is based on 3 tools: i) ReGrEd: a graph interface-modeling tool of products, ii) RecyKon: a recycling-engineering-kit to help the functionality analysis, and iii) DisPlay: a disassembly Planning System. This platform was the premise of the today PLM system that can be adapted to early end of life integration [START_REF] Rosemann | Design for recycling, recycling data management and optimal end-of-life planning based on recycling-graphs[END_REF].

In the perspective of a classical V cycle of product development, two elements arise. Firstly, the kinds of data and their accuracy depend on the end-of-life solution chosen. Indeed, as illustrated in Figure 3, in the case of a re-use of product, the product life information might need to be certified and guaranteed for compliance. The materials database will store all this information, structured for each recycling stage. The designer will use the data stored. Nevertheless, in many cases, the kind of data required is known, but the real value of the data is either unknown, or fuzzy, or ranges out of the limits. As a result, design evaluation becomes uncertain. Therefore, end-of-life solutions are not fully defined at an early design stage. Nevertheless, these initial decisions will impact the environmental footprint. In addition, as illustrated in part 4.1, an efficient end-of-life solution might not have been developed yet. Consequently, end-of-life evaluations must be used with care, dividing the trustable results with the uncertain ones. Similar levels of information completeness can compare different solutions. Otherwise, the results should be taken as trends or qualitative comparisons. It is necessary to formalize and map the design process cycle, with each key decision, regarding the life and end-of-life impact. The decisions will need data, decision rules, etc. In addition, as started in figure 2, all the information must be captured in detail (requirements and constraints) in order to generate recycling rules. The connection link between the two aspects, identified with the use or the generation of the same data, will link designer to recycler.

Multidisciplinary needs

The previous section explains how we will create the link between recyclers and designers. However, in many cases, designers need data regarding a specific characteristic (e.g. maximum tensile stress) and the recyclers are not able to give such information. Inversely, the recyclers have to know about life damage, but the designer can only inform on the use cases employed for the design. Consequently, complementary information arises in this dual relationship. Material and product characterization is compulsory at different levels. People coming from material, chemical and mechanical fields will be able to provide a way of translating the requests or the requirements into real data. Indeed, many different characterizations must be carried out before and after the recycling process.

Figure 4 Summary of the interrelation of the three skills

For example, assembly consequences on the material, end of life and the possible disassembly damage must be identified. These data guide the recycling process in order to minimize the variability incidence. In a second stage, the recycled material or reprocessed semi-product has to be tested in order to assess its quality and enrich the designer possibility (with a second life or reprocessed material). Figure 4 sums up the interaction between these three required skills. This picture gives an idea of the cyclonic cycle of the material properties moving from one field of application (high mechanical constrains) to other field (low mechanical constrains) [START_REF] Perry | Improving design for recycling -Application to composites[END_REF]. This downcycling view gives the possible use of the recycled material to the recycler and the designer. It allows developing final products (like sport devices with initial aeronautic material) or semi products (like patch). Mechanical models simulate the material behavior, including the size stochastic evolution of the recycle fibers, the random defects repartition (such as orientation, density) and the mix of initial material. These models help improving the recycling process, and optimize the design of the second life materials.

From the characterization point of view, the key problem is adaptation of scale. The testing pyramid strategy helps to identify sufficient and necessary tests from the elementary sample level to the full system level. In most cases, specific tests must be developed in order to guarantee rel also applied to the development of n 5 Solvolysis and recycled fib

Recycled fibers level

RECCO project (in which we are recycle and re-process) carbon com recycle thermoset matrix composite removal with a little carbon fibers fibers remain, oriented according to Solvolysis process heat the wat under super-critical conditions an compound (even metallic). Using matrix is removed in an average of layers).The solvolysis has minim solvolysis vessel has size limitation cutting process impact the fiber rep

Figure 5 EBM pictures of carbon fib

Nevertheless, this process is v material such as metal. For examp inserts, and even parts of the solvo particles in and on the carbon fiber uni-directional or woven orientatio flow concentrates this pollution on This pollution has little influen bridges and an upper skin that red When sizing and imbue with matrix on mechanical properties. Regardin behavior locally. It results a metal have to be reduced and mastered b in the dismantling process in order with the cutting step in order to cut relevant and reliable results. This multi level approach is of new recycling processes and their industrialization.

fibers results

re involved) search industrial solutions to recover (i.e. omposite wastes. The solvolysis method is developed to sites. The specificity of this process is its maximal matrix ers degradation. After the solvolysis phase, dry carbon to the product composite sequence. ater at 400°C under a 200bars of pressure. The water is and become a very efficient solvent for any organic ng a flow in the High pressure vessel, the thermo-set of 30 minutes (for around 10 centimeters of composites imal water consumption and pollution. Nevertheless ion. It implies cutting composite parts. Consequently, the eprocessing as illustrated in figure 5. fibers free (left) and glue at an edge due to cutting (right) s very sensitive to the presence of other non-organic mple, the solvolysis process oxidized rivets, screws or lvolysis reactor in some cases. It spreads small metallic bers as illustrated in figure 6. Depending on the whether ation that was used for the fibers, the solvolysis liquid n the outer edge of the fibers or at the cross of the tows. uence on the dry fibers mechanical strength. It creates reduces the re-process-ability (spinning or re-weaving). trix polymer it eases mechanical grip and has little effect ding the physical properties, this pollution modified the tallic fraction limit in the global composite volume that d before solvolysis. This point included a specific phase er remove all the metallic elements. It could be coupled cut efficiently both composite parts and fixing elements.

When the complete removal is not ensure a limitation of the metallic p Figure 6 EBM pictures of clean and Recycled fiber with small si Recycled fibers with consider overall pollution from steel (F Another feedback has resulted the parts, the cutting processes cha from the adjustments of the parts to the cutting burns the matrix. This c fibers together at their edges. This when hard condition cutting param the recyclate. To spin a long tow w shreds (good for mechanical handle time it reduces the spinning effici solutions such as water jet, mach impact. These results give best pr locally does not heat the material fi Up to now, we have integrated to develop the remanufacturing end development and we are able to tra carbon fibers. The mechanical char fibers, except for their length. Fibers with slight nickel pollution derable and Fibers with edge pollution l (Fe + Cr) ed from the analysis of these pollutions. At the edge of change the properties of the matrix. This cutting comes s to the solvolysis reactors sizes. Indeed, the local heat of s carbon based layer (as illustrated in figure 5) glues the his thickness of this layer can reach up to 200 microns ameters are used. This property can ease the handling of with short or medium size fibers, this glue reduces the dle and for keeping fibers aligned. However, at the same iciency of inter fibers grip. We tested different cutting chining and grinding in order to measure their edge's practices for dismantling solution. All the process that l fit with the best handling fibers' properties. ed design competences in textile manufacturing in order nd of the recovery loop. The pilot demonstrator is under transform composite parts to tows or woven of recycled aracteristics of these fibers are nearly the same as virgin ped based on the recycled and re-aligned carbon fibers. t due to the early emergence of this material. It is mainly sportive application due to the ne quality of the recycled carbon fib general public target, and submitte product. Natec manufactures one o crank has been processed with recy This product shows a very problematic of alignment of fibe directional semi product (if needed parameter. In this case, the non ali the nearly +/-30° comparatively to gives the possibility to have only n final crank mass is 187g versus 16 fiber placement and displacement defects are identified. This validate compression molding.

Recycled product level

Conclusions

We were able to recycle carbon-b development highlights the fact th included in the early design stages from materials characterization and provide information about the par products derived from recycling. Fr skills that must interweave in or objectives and to develop robust re order to facilitate discussion and c have to work on the definitions of must also take the recycler's poin requirements and semantic align communities has to be implemented need of mechanical properties expected. To proof the fiber, an application has been developed with a more itted to mechanical stresses to fit to a high-performance e of the lightest racing bike pedal board in the world. A cycled carbon fibers (see Figure 7). ed characteristics: stiffness and equivalent mass to the initial y acceptable aspect. The zoom illustrates the today ibers that must be improved to propose nearly unied). This is a key point of the re-manufacturing process alignment of some fibers gives mechanical properties in to the main 0° orientation in the length of the crank. It y nearly 0° layer instead of woven or 0/90° layers. The 163g for the initial mass-product with virgin fibers. The nt has been observed during the process. No specific ates the possibility of using second generation fibers by based composites and recover the carbon fibers. This t that recycling possibilities and constraints should be es. Skills and competences (knowledge and know-how) and mechanical characterization are required in order to parts to be recycled, and about the materials or semi-. From our point of view, the three communities have the order to reach the design under recycling constraint t recycling processes for second design perspectives. In communication within the three areas of expertise, we of the requirements for and from the designer, but we oint of view into account. Systematic formalization of ignment between the different (yet not so distant) ted.

An environmental assessment of the solvolysis recycling process still has to be carried out in order to compare energy and material consumption, pollution, cost, etc. for the same quantity of material (new or recycled). The increasing use of composites in industry has brought about the development of end-of-life solutions. Regulatory constraints and financial perspectives (cost-cutting) will give rise to design for recovery or eco-design approaches.

Figure 1

 1 Figure 1 Composite design constraints.

Figure 2

 2 Figure2Recycling process analyses in the perspective of the design product specification and recycling process information

Figure 3

 3 Figure 3 V cycle for information exchange

  Different applications are develope The application domains are tight d ot possible, a metallic fraction measurement is needed to c pollution. d polluted recycled carbon fibers l silicates

Figure 7

 7 Figure 7 Natec pedal crank : desired one (approx. 185 g)


Martin Costabel
email: martin.costabel@univ-rennes1.fr

Michel Crouzeix
email: michel.crouzeix@univ-rennes1.fr

Monique Dauge
email: monique.dauge@univ-rennes1.fr

Yvon Lafranche
email: yvon.lafranche@univ-rennes1.fr

The inf-sup constant for the divergence on corner domains

Keywords: Mathematics Subject Classification. 30A10, 35Q35. Key words and phrases. LBB condition, inf-sup constant, essential spectrum of the Cosserat operator

The inf-sup constant for the divergence, or LBB constant, is related to the Cosserat spectrum. It has been known for a long time that on non-smooth domains the Cosserat operator has a non-trivial essential spectrum, which can be used to bound the LBB constant from above. We prove that the essential spectrum on a plane polygon consists of an interval related to the corner angles and that on three-dimensional domains with edges, the essential spectrum contains such an interval. We obtain some numerical evidence for the extent of the essential spectrum on domains with axisymmetric conical points by computing the roots of explicitly given holomorphic functions related to the corner Mellin symbol. Using finite element discretizations of the Stokes problem, we present numerical results pertaining to the question of the existence of eigenvalues below the essential spectrum on rectangles and cuboids.

INTRODUCTION

The inf-sup constant of the divergence β(Ω) is defined for a domain Ω ⊂ R d as (1.1)

β(Ω) = inf q ∈ L 2 • (Ω) sup v ∈ H 1 0 (Ω) d div v, q Ω |v| 1,Ω q 0,Ω .
Here L 2 • (Ω) denotes the space of square integrable functions with mean value zero, with norm • 0,Ω and scalar product •, • Ω , and H 1 0 (Ω) is the usual Sobolev space, closure of the space of smooth functions of compact support in Ω with respect to the H 1 seminorm, which for vector-valued functions is defined by

|v | 1,Ω = grad v 0,Ω = d k=1 d j=1 ∂ x j v k 2 0,Ω 1/2 .
The inf-sup condition β(Ω) > 0 is also called LBB condition [START_REF] Malkus | Eigenproblems associated with the discrete LBB condition for incompressible finite elements[END_REF] or Ladyzhenskaya-Babuška-Brezzi condition (a term coined ca. 1980 by T. J. Oden, on suggestion by J. L. Lions [START_REF] Oden | Mafelap conference Uxbridge[END_REF]), and β(Ω) is therefore also known as LBB constant [START_REF] Dobrowolski | On the LBB constant on stretched domains[END_REF][START_REF]On the LBB condition in the numerical analysis of the Stokes equations[END_REF]. It plays an important role in the discussion of the stability of solutions and numerical approximations of the equations of hydrodynamics. The exact value of the constant is known only for a small class of domains, first for balls and ellipsoids (derived from the Cosserat spectrum [START_REF] Cosserat | Sur les équations de la théorie de l'élasticité[END_REF][START_REF] Cosserat | Sur la déformation infiniment petite d'un ellipsoïde élastique[END_REF]), then in three dimensions for spherical shells [START_REF] Cosserat | Sur la déformation infiniment petite d'une enveloppe sphérique élastique[END_REF][START_REF] Liu | The discrete Cosserat eigenfunctions for a spherical shell[END_REF], and finally in two dimensions for annular domains [START_REF] Chizhonkov | On the domain geometry dependence of the LBB condition[END_REF] and some domains defined by simple conformal images of a disk [13, §5], [START_REF] Horgan | On inequalities of Korn, Friedrichs and Babuška-Aziz[END_REF][START_REF] Án | On the spectrum of the Schur complement of the Stokes operator via conformal mapping[END_REF]. Its precise value remains, however, unknown for such simple domains as a square.

Finding estimates, from below and from above, for β(Ω) has therefore been an important subject for many years. The question can be rephrased in terms of the Cosserat eigenvalue problem, see relation (1.5) below, and it is this problem that we will study in the present paper. In particular, any number known to belong to the nontrivial part of the spectrum of the Cosserat operator will imply an upper bound for the LBB constant.

Other inequalities and eigenvalue problems are known to be related to the Cosserat eigenvalue problem and the LBB condition, namely the Korn and Friedrichs inequalities. In two dimensions in particular, such relations have been investigated since the paper [START_REF] Friedrichs | On certain inequalities and characteristic value problems for analytic functions and for functions of two variables[END_REF] by Friedrichs, and the equivalence between these inequalities and equations between the corresponding constants have been studied in the classical paper [START_REF] Horgan | On inequalities of Korn, Friedrichs and Babuška-Aziz[END_REF] by Horgan and Payne, see [START_REF]On the inequalities[END_REF] for improved versions of some of their results.

For the Cosserat eigenvalue problem, the values 0 and 1 are eigenvalues of infinite multiplicity, and 1/2 is an accumulation point of eigenvalues. On smooth domains, these are the only points in the essential spectrum, as shown by Mikhlin [START_REF] Mihlin | The spectrum of the pencil of operators of elasticity theory[END_REF], see [START_REF] Crouzeix | On an operator related to the convergence of Uzawa's algorithm for the Stokes equation[END_REF] for a complete proof for C 3 domains. On non-smooth domains, a non-trivial essential spectrum is present, as was already pointed out in [START_REF] Friedrichs | On certain inequalities and characteristic value problems for analytic functions and for functions of two variables[END_REF] for the equivalent Friedrichs problem.

We show in Theorem 3.3 for two-dimensional domains that a polygonal corner of opening ω ∈ (0, 2π) contributes an interval [ 1 2 -sin ω 2ω , 1 2 + sin ω 2ω ] to the essential spectrum of the Cosserat operator. This has been known for a while already [START_REF] Costabel | On the Cosserat spectrum in polygons and polyhedra[END_REF], but the proof has not yet been published. A corollary is an upper bound for the LBB constant, see also [START_REF] Stoyan | Towards discrete Velte decompositions and narrow bounds for inf-sup constants[END_REF][START_REF]Iterative Stokes solvers in the harmonic Velte subspace[END_REF] (1.2)

β(Ω) ≤ 1 2 - sin ω 2ω .
For the square Ω = , in [14, (6.39)] a conjecture was offered that amounts to β( ) = 2/7 = 0.53 . . . , which is obviously incompatible with the upper bound β ≤ 1 2 -1 π = 0.42 . . . from (1.2). Although already conjectured in [START_REF] Stoyan | Towards discrete Velte decompositions and narrow bounds for inf-sup constants[END_REF], it is still unknown whether this latter inequality is an equality or whether it is strict, that is, whether for the square there exist Cosserat eigenvalues below the minimum of the essential spectrum. In Section 5 below, we present numerical evidence suggesting that there are no such eigenvalues. But this is not yet proven.

For three-dimensional domains with edges, the two-dimensional corner domain transversal to the edge implies an inclusion of the corresponding interval in the essential spectrum, which therefore always contains such an interval symmetric with respect to the point 1/2 (see Section 4.1). This is very different from the case of smooth domains, where there exists the known example of the ball that has a Cosserat spectrum consisting (apart from the trivial point 1) of a sequence of eigenvalues σ k = k 2k+1 , k ≥ 1, converging to 1/2 from below and therefore has no spectrum in the interval (1/2, 1) [START_REF] Cosserat | Sur la déformation infiniment petite d'un ellipsoïde élastique[END_REF][START_REF] Crouzeix | On an operator related to the convergence of Uzawa's algorithm for the Stokes equation[END_REF][START_REF] Simader | Introduction to the Cosserat problem[END_REF].

For three-dimensional bounded domains having conical boundary points with tangent cones of revolution, we present in Section 4.2 numerical results showing that there are also intervals contained in the essential spectrum of the Cosserat operator.

The Cosserat eigenvalue problem as originally formulated by E. and F. Cosserat [START_REF] Cosserat | Sur les équations de la théorie de l'élasticité[END_REF] can be written as follows: Find nontrivial u ∈ H 1 0 (Ω) d and σ ∈ C such that

(1.3) σ∆u -∇ div u = 0 .
This is the spectral problem of the bounded selfadjoint operator ∆ -1 ∇ div on H 1 0 (Ω) d , where ∆ -1 is the inverse of the Laplace operator with Dirichlet conditions ∆ :

H 1 0 (Ω) → H -1 (Ω).
On the orthogonal complement of the kernel of div, which is the eigenspace for the trivial eigenvalue σ = 0, this operator is equivalent to the operator

(1.4) S = div ∆ -1 ∇ : L 2 • (Ω) → L 2 • (Ω)
, the Schur complement operator of the Stokes system.

We define the Cosserat constant σ(Ω) of the domain Ω as the minimum of the spectrum of the operator S.

It is then an exercise in elementary Hilbert space theory to show that there holds

(1.5) σ(Ω) = β(Ω) 2 .
To see this, write the LBB constant using the definition of the H -1 norm and the fact that ∆ is an isometry from H 1 0 (Ω) to H -1 (Ω):

β(Ω) 2 = inf q ∈ L 2 • (Ω) |∇q| 2 -1,Ω q 2 0,Ω = inf q ∈ L 2 • (Ω) -∆ -1 ∇q, ∇q Ω q 2 0,Ω = inf q ∈ L 2 • (Ω) Sq, q Ω q 2 0,Ω = σ(Ω) .
For numerical approximations of the Cosserat eigenvalue problem, we will use the equivalent formulation as an eigenvalue problem for the Stokes system:

Find u ∈ H 1 0 (Ω) d , p ∈ L 2 • (Ω) \ {0}, σ ∈ C such that -∆u + ∇p = 0 div u = σp . (1.6)
In the following, we will mainly discuss the essential spectrum of the Cosserat problem. This is the set of σ ∈ C such that the operator

(1.7) L σ = σ∆ -∇ div : H 1 0 (Ω) d → H -1 (Ω) d is not a Fredholm operator.
From the variational formulation of the operator

L σ for u, v ∈ H 1 0 (Ω) 2 (1.8) L σ u, v = -σ Ω ∇u : ∇v + Ω div u div v
it is not hard to see that such σ have to be real, between 0 and 1, and that L σ is Fredholm if and only if it is Fredholm of index 0 and if and only if it is semi-Fredholm.

DOMAINS WITH CONICAL POINTS

We will show how Kondrat'ev's classical theory [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF] applies to the Cosserat operator L σ when Ω is a domain with conical points in R d . This means that the boundary of Ω is smooth except in a finite set C of points c, called the corners of Ω, and in the neighborhood of each corner c the domain Ω is locally diffeomorphic to a regular cone Γ c , i.e., the section

G c = Γ c ∩ S d-1 is a smooth domain in S d-1 .
From the discussion in [24, §2], we know that the system σ∆ -∇ div is elliptic at any point of Ω if (and only if) σ ∈ {0, 1}, and that the Dirichlet boundary condition covers σ∆ -∇ div at any smooth point of ∂Ω if, moreover, σ = 1 2 .

In contrast with the smooth case when 0, 1 2 , 1 are the only values for which L σ is not Fredholm, the corners of Ω cause an enlargement of this exceptional set, in general.

2.1. The Mellin symbol of the Cosserat operator. For determining the values σ such that L σ in (1.7) is not Fredholm, we use Kondrat'ev's [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF] technique of corner localization and Mellin transformation.

Let us choose a corner c and write (r, ϑ) ∈ R + × G c for the polar coordinates in the tangent cone Γ c . The homogeneous Lamé system σ∆ -∇ div can be written as

σ∆ -∇ div = r -2 L σ (ϑ; r∂ r , ∂ ϑ )
where the d × d system L σ has coefficients independent of r. The Mellin transformation u → ∞ 0 r -λ-1 u(r, ϑ) dr transforms r∂ r into the multiplication by λ. The Mellin symbol A c σ at the corner c is the operator valued function (known as operator pencil in the Russian literature) defined as (2.1)

A c σ (λ) = L σ (ϑ; λ, ∂ ϑ ) : H 1 0 (G c ) d → H -1 (G c ) d , λ ∈ C.
2.2. The Fredholm theorem. In Kondrat'ev's theory, the Fredholm property of L σ is studied in the framework of weighted Sobolev spaces [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF]. In the book [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF] by Kozlov-Maz'ya-Rossmann, these spaces are defined as follows: For ℓ ∈ N and β ∈ R

(2.2) V ℓ β (Ω) = {u ∈ L 2 loc (Ω) : |x -c| β+|α|-ℓ ∂ α x u ∈ L 2 (Ω) ∀c ∈ C, ∀α ∈ N d , |α| ≤ ℓ}. Lemma 2.1.
Let Ω ⊂ R d be a domain with conical points. The space H 1 0 (Ω) coincides with the subspace of V 1 0 (Ω) consisting of functions with zero boundary traces. This lemma is proved using a Poincaré inequality in angular variables, see [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]Lemma 6.6.1]. Then [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]Theorem 6.3.3] implies the following result.

Theorem 2.2. Let Ω ⊂ R d be a domain with conical points. Assume that σ does not belong to {0, 1 2 , 1} and that for each corner c and all complex numbers λ with real part 1 -d 2 , the Mellin symbol A c σ (λ) is invertible. Then L σ is Fredholm. Note that we cannot apply [START_REF] Kondrat'ev | Boundary-value problems for elliptic equations in domains with conical or angular points[END_REF]Theorem 3.1] right away because it is assumed there that the Sobolev exponent ℓ is at least 2. For this reason we have to use the extension performed in [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]. The assumption σ ∈ {0, 1 2 , 1} ensures that the boundary value problem L σ is elliptic. Concerning the critical abscissa 1 -d 2 , there is a simple rule of thumb for determining it: A corner c being chosen together with a non-zero angular function W ∈ H 1 0 (G c ) and a compactly supported cut-off function χ equal to 1 near c, we find that 1 -d 2 is the smallest real number η such that all functions of the form χ r λ W (ϑ) belong to H 1 0 (Γ c ), whenever Re λ > η.

PLANE POLYGONS

Let Ω ⊂ R 2 be a polygon, that is, a bounded Lipschitz domain the boundary ∂Ω of which consists of a finite set of straight segments. This set being chosen in a minimal way, the corners c of Ω are the ends of these segments. Each corner c belongs to two neighboring segments and the tangent cone Γ c is a plane sector, the opening of which is denoted by ω c . Thus the section G c can be identified with the interval (-ωc 2 , ωc 2 ).

3.1. The Mellin determinant. In two-dimension, the invertibility of the Mellin symbol A c σ (λ), which is a 2 × 2 Sturm-Liouville system on an interval with polynomial dependence on λ, can be further reduced to the non-vanishing of a scalar holomorphic function, which we may call Mellin determinant M c σ (λ). This Mellin determinant is well known for the case of the Dirichlet problem of the Lamé system of linear elasticity. It is constructed and described in detail in the book [START_REF]Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations[END_REF]Chapter 3], where also references to earlier works can be found. We can use the results of these calculations, simply by noticing that the operator L σ is the Lamé operator, if we set σ = 2ν -1

with the Poisson ratio ν. What is non-standard here, compared to the discussion in [START_REF]Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations[END_REF], is first that the σ considered here correspond to the "non-physical" range 1 2 < ν < 1, and second that we consider the question of loss of H 1 regularity, that is, zeros of the Mellin determinant on the critical line {Re λ = 1 -d 2 = 0}. In [17, (3.1.22/23)], the Mellin determinant for a corner of opening angle ω is given as

M(λ) = λ -2 (3 -4ν) 2 sin 2 λω -λ 2 sin 2 ω .
The characteristic equation M(λ) = 0 can therefore be written as the two equations

(3.1) (1 -2σ) sin λω λ = ± sin ω.
In the following paragraph 3.2 we present a new and straightforward way to calculate M together with the singular functions associated with the roots λ of M, i.e. the solutions w λ of the equation L σ w = 0 that are homogeneous of degree λ. Remark 3.1. These singular functions play a double role here: If Re λ = 0, they do not belong to H 1 0 near the corner, which is the reason why L σ is then not a Fredholm operator and σ belongs to the essential spectrum, see Theorems 3.2 and 3.3. Figure 1 shows this case: As a function of σ and ω, the imaginary part of λ is plotted. On the other hand, if Re λ > 0, then the singular functions w λ belong to H 1 0 near the corner and they describe the corner asymptotics of any solution in H 1 0 (Ω) of L σ u = f with smooth f , in particular of Cosserat eigenfunctions for such eigenvalues σ that are not in the essential spectrum:

(3.2) u ∈ H 1+s (Ω) 2 , ∀s < min{Re λ | Re λ > 0, λ root of (3.1) }
If Re λ is close to zero, the derivatives of u (thus the "pressure" part p of the Stokes solution in (1.6)) will go to infinity quickly at the corner, which poses problems for the numerical approximation of the Cosserat eigenvalue problem, see the discussion in Section 5. In Figure 2, we plot λ as function of σ and ω in this case. Note that for 0 ≤ σ ≤ 1, λ is either real or purely imaginary.

Singular functions.

Let us choose a corner c and drop the reference to that corner in the notation. We assume for simplicity that c is the origin. The symbol A σ (λ) is not invertible iff it has a nonzero kernel. We note that for W ∈ H 1 0 (-ω 2 , ω 2 ) 2 we have the equivalence So we first solve

(3.3) W ∈ ker A σ (λ) ⇐⇒ (σ∆ -∇ div)(r λ W (θ)) = 0.
(3.4) (σ∆ -∇ div)(r λ W (θ)) = 0
without boundary conditions and in a second step find conditions on λ so that there exist nonzero solutions satisfying the Dirichlet conditions at θ = ± ω 2 .

STEP 1. We know from [5, Theorem 2.1] that for any λ ∈ C the solutions of (3.4) form a space of dimension 4, which we denote by W σ (λ). Moreover [5, §2.b(iii)] tells us that if λ ∈ N, it is sufficient to look for each component of the homogeneous function r λ W (θ) in the space generated by z λ , zλ , z λ-1 z, and zλ-1 z. Here we have identified R 2 with the complex plane C by the formula z = x 1 + ix 2 . So the Ansatz space for w = r λ W (θ) has the basis

w (1) = 1 i z λ , w (2) = 1 i zλ , w (3) = 1 i z λ-1 z, w (4) = 1 i zλ-1 z, w (1) = 1 -i z λ , w (2) = 1 -i zλ , w (3) = 1 -i z λ-1 z, w (4) = 1 -i zλ-1 z.
Applying L σ w = 0 to these vector functions is an easy computation using complex derivatives for functions of z and z and writing for a scalar function f the formulas ∇f = ∂f ∂z

1 i + ∂f ∂ z 1 -i , div f if = 2 ∂f ∂ z , div f -if = 2 ∂f ∂z ,
and

∆f = 4 ∂ 2 f ∂z ∂ z .
In this way we find that for any λ ∈ C \ {0, 1}, the following four vectors w k λ given by

w 1 λ = w (1) , w 2 λ = w (2) , w 3 λ = w (3) + 2σ -1 λ w (1) , w 4 λ = w (4) + 2σ -1 λ w (2) ,
form a basis 1 for the space W σ (λ).

STEP 2. We look for conditions on λ so that there exists a nonzero w ∈ W σ (λ) which satisfies the Dirichlet conditions on θ = ± ω 2 . We note that, setting a = sin(λ-1)ω sin(λω) and b = sin ω sin(λω) ,

for any ε ∈ R the function

w = w (3) -aw (1) -bw (2) + ε w (4) -a w (2) -b w (1) is zero on θ = ± ω 2 . It is clear that w belongs to W σ (λ) iff ε = ±1 and -εb = 2σ-1 λ , i.e. (3.5) σ = 1 2 1 -ε λ sin ω sin(λω) , ε = ±1.
Note that this is the same equation as (3.1). The associated singular function is

(3.6) w λ = w 3 λ + εw 4 λ -a(w 1 λ + εw 2 λ ).
1 To obtain a basis valid for λ = 0, we could define w 3 λ by w

(3) + w (4) + 2σ-1 λ ( w (1) + w (2) -w (1) -w (2) ) and w 4 λ by w (3) -w (4) + 2σ-1 λ ( w (1) -w (2) + w (1) -w (2) ).
3.3. Singular sequences at corners and essential spectrum. The result of the Kondrat'ev theory for our polygon can be written as follows.

Theorem 3.2. Let Ω be a polygon in the plane. Assume that σ does not belong to {0, 1 2 , 1}. Then the Cosserat operator L σ is Fredholm from H 1 0 (Ω) 2 to H -1 (Ω) 2 if and only if for each corner c ∈ C and ω = ω c the characteristic equations (3.1) have no solution on the line Re λ = 0.

The "if" part is a consequence of Theorem 2.2. For the "only if" part, we could simply quote [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]Remark 6.3.4], but we prefer to present a rather simple explicit proof in the spirit of [9, §9.D], namely the construction of a singular sequence approximating the non-H 1 corner singularity.

Choose a corner c, which we assume to sit at the origin, set ω c = ω. Let R > 0 be such that Ω coincides with the plane sector Γ c in the ball of center c and radius 2R. Now choose a cut-off function χ ∈ C 1 (R 2 ) such that χ(x) = 1 for |x| ≤ R and χ(x) = 0 for |x| > 2R, and for any complex λ set u λ = χ w λ with w λ the singular function defined in (3.6). Then u λ = 0 on ∂Ω. Therefore u λ ∈ H 1 0 (Ω) 2 if and only if Re λ > 0. Let σ be such that the equation (3.1) has a solution λ ∈ iR, and take ε ∈ {±1} so that σ = 1 2 (1-ε λ sin ω sin(λω) ). Define (3.7)

λ n = λ + 1 n and σ n = 1 2 1-ε λ n sin ω sin(λ n ω) . Then (3.8) L σ u λn = (σ-σ n )∆u λn + f n with f n = L σn u λn .
From L σn w λn = 0 follows that f n = 0 for |x| ∈ [R, 2R] and that f n 0,Ω can be estimated by w λn H 1 ({R<|x|<2R}) , which remains bounded as n → ∞. Note that for n → ∞, σ n -σ → 0 and u λn 0,Ω remains bounded, while |u λn | 1,Ω → ∞. Using the fact that ∆ is an isometry from H 1 0 (Ω) to H -1 (Ω), we obtain with (3.8)

(3.9) lim n→∞ |L σ u λn | -1,Ω |u λn | 1,Ω = 0.
Altogether, this shows that L σ cannot satisfy an a-priori estimate of the form

|u| 1,Ω ≤ α|L σ u| -1,Ω + β u 0,Ω with constants α, β ≥ 0.
Hence L σ is not Fredholm and σ belongs to the essential spectrum.

We can now conclude in the case of polygonal domains.

Theorem 3.3. Let Ω ⊂ R 2 be a polygon with corner angles ω c ∈ (0, 2π), c ∈ C. Then the essential spectrum of the Cosserat problem in Ω is given by

(3.10) σ ess (S) = {1} ∪ c∈C 1 2 - sin ω c 2ω c , 1 2 + sin ω c 2ω c .
The LBB constant satisfies

(3.11) 0 < β(Ω) ≤ min c∈C 1 2 - sin ω c 2ω c 1 2 .
Proof. As Ω is Lipschitz, β(Ω) > 0 [START_REF] Horgan | On inequalities of Korn, Friedrichs and Babuška-Aziz[END_REF]. For σ ess (S), we have to find the σ for which (3.1) has solutions λ with Re λ = 0. The function λ → sin λω λω maps iR to [1, ∞). For σ = 1 2 , the necessary and sufficient condition is therefore

- sin ω ω ≤ 1 -2σ ≤ sin ω ω .
This proves (3.10).

Remark 3.4. The result of Theorem 3.3 remains true if Ω is a curved polygon, that is a bounded Lipschitz domain with a piecewise C 2 boundary. This follows from general perturbation techniques that are part of the Kondrat'ev theory [15, §2], see also [START_REF] Kozlov | Elliptic boundary value problems in domains with point singularities[END_REF]Ch. 6].

If the corner angles have cusps, Theorem 3.3 extends as follows: An inward cusp or crack (ω = 2π) does not contribute to the nontrivial essential spectrum [START_REF] Friedrichs | On certain inequalities and characteristic value problems for analytic functions and for functions of two variables[END_REF], whereas in the presence of an outward cusp (ω = 0), the essential spectrum of the Cosserat problem fills the entire interval [0, 1] [START_REF] Friedrichs | On certain inequalities and characteristic value problems for analytic functions and for functions of two variables[END_REF][START_REF] Tartar | An introduction to Navier-Stokes equation and oceanography[END_REF]. The estimate (3.11) for β(Ω) was proved by Stoyan in [START_REF] Stoyan | Towards discrete Velte decompositions and narrow bounds for inf-sup constants[END_REF], using the equivalence between the Friedrichs constant and the LBB constant [START_REF] Horgan | On inequalities of Korn, Friedrichs and Babuška-Aziz[END_REF][START_REF]On the inequalities[END_REF].

THREE-DIMENSIONAL DOMAINS WITH EDGES OR CORNERS

Let Ω ⊂ R 3 . We consider domains with edges and corners. For the case when Ω contains edges in its boundary, we prove that the essential spectrum of the Cosserat problem contains an interval that depends on the opening angle of the edges. For the case when Ω has a conical point with an axisymmetric tangent cone, we describe how to reduce the question to the determination of the roots of a sequence of holomorphic functions and we show the result of numerical computations. Polyhedral corners are the subject of some comments at the end of this section.

4.1.

Edges. In a first step, we assume that the boundary of Ω contains a piece of a straight edge of opening ω ∈ (0, 2π) in the sense that for some R > 0

(4.1) x ∈ Ω | x 2 1 +x 2 2 ≤ 2R, |x 3 | ≤ 2R = x ∈ R 3 | x 2 1 +x 2 2 ≤ 2R, |x 3 | ≤ 2R, | arg(x 1 +ix 2 )| < ω 2 .
Theorem 4.1. Under the hypothesis (4.1), the interval

[ 1 2 -sin ω 2ω , 1 2 + sin ω 2ω ] is contained in the essential spectrum of the Cosserat problem. Proof. Let σ ∈ [ 1 2 -sin ω 2ω , 1 2 + sin ω 2ω ] \ { 1 2 }.
We construct a singular sequence based on the one that we have used for a two-dimensional corner in Section 3.3: Like there, we set z = x 1 + ix 2 , and for λ ∈ C we define w λ = (w λ,1 , w λ,2 ) ⊤ as in equation (3.6) and use the same cut-off function χ. Now we choose another cut-off function θ

∈ C 2 (R) that satisfies θ(x 3 ) = 1 if |x 3 | ≤ R and θ(x 3 ) = 0 if |x 3 | ≥ 2R. Then we define u λ as u λ (x) = θ(x 3 ) χ(x 1 , x 2 )   w λ,1 (z) w λ,2 (z) 0   .
For λ, λ n , σ n chosen as before, see (3.7), the vector functions u λn belong to H 1 0 (Ω) 3 and L σ u λn has still the expression (3.8) with f n = (f n,1 , f n,2 , f n,3 ) ⊤ given by

f n,1 f n,2 = θ(x 3 ) L 2D σn (χw λn ) + σ n θ ′′ (x 3 ) χw λn f n,3 = -θ ′ (x 3 ) div(χw λn ) .
We conclude as before that u λn 0,Ω and f n,1 f n,2 0,Ω remain bounded as n → ∞ whereas |u λn | 1,Ω → ∞. As for f n,3 , we see that |f n,3 | -1,Ω remains bounded, so that we can again conclude that u λn is a singular sequence satisfying (3.9), and that σ therefore belongs to the essential spectrum.

Remark 4.2. Yet another contribution to the essential spectrum may come from the edges, this time not from the essential spectrum of the transversal Cosserat problem L 2D σ as described in Theorem 4.1, but from eigenvalues of the "edge symbol", which is the boundary value problem defined on the sector

Γ = {(x 1 , x 2 ) ∈ R 2 | | arg(x 1 +ix 2 )| < ω 2 }, cf. (4.1), (4.2) R ∋ ξ → L σ (ξ) = σ(∂ 2 1 +∂ 2 2 -ξ 2 )I 3 -   ∂ 2 1 ∂ 1 ∂ 2 iξ∂ 1 ∂ 1 ∂ 2 ∂ 2 2 iξ∂ 2 iξ∂ 1 iξ∂ 2 -ξ 2   : H 1 0 (Γ) 3 → H -1 (Γ) 3 ,
obtained by partial Fourier transformation along the edge. Indeed, it is known that for an elliptic boundary value problem on a domain with edges to be Fredholm, it is not sufficient that its edge symbol is Fredholm (this latter condition is satisfied as soon as the transversal Mellin symbol is invertible on a certain line Re λ = const), but it has to be invertible for all ξ = 0 [23, Theorem 10.1], see also [START_REF] Rossmann | Elliptic equations in polyhedral domains[END_REF]. For our Cosserat problem, if such eigenvalues exist below the essential spectrum of the transversal problem L 2D σ , then the lower bound of the essential spectrum on Ω, and therefore the LBB constant, may be smaller than what is described by Theorem 4.1. Whether this actually happens is unknown, and some numerical experiments we did rather seem to indicate that it does not.

Remark 4.3. The result of Theorem 4.1 remains true for some curved edges, too, similar to what we mentioned in two dimensions in Remark 3.4. In particular, if Ω is a finite straight cylinder with a smooth base, then the interval corresponding to the angle ω = π 2 , namely

[ 1 2 -1 π , 1 2 + 1 π ],
will belong to the essential spectrum.

Conical points.

We assume now that Ω is a domain in R 3 with conical points. Theorem 2.2 applies with the critical abscissa Re λ = -1 2 . We can prove exactly like in dimension 2 that the sufficient condition for Fredholmness is also necessary. Now the question is: Does a 3D cone produce a full interval of nonzero length inside the essential spectrum? Do we have symmetry with respect to σ = 1 2 like in 2D? We do not have a general answer to the first question, but a partial answer in the case of axisymmetric cones. This example shows that the symmetry property with respect to σ = 1 2 is lost. We learn from [START_REF] Kozlov | On singularities of solutions to the boundary value problems near the vertex of a rotational cone[END_REF][START_REF]On singularities of solutions of the displacement problem of linear elasticity near the vertex of a cone[END_REF] that semi-analytical calculation of the spectrum of the Mellin symbol A σ associated with an axisymmetric cone Γ is made possible by the use of the Boussinesq representation and separation of variables in spherical coordinates. Then, relying on the equivalence (3.3), we can follow the same track as in 2D, looking for homogeneous solutions w of the equation

(4.3) (σ∆ -∇ div)w = 0
without boundary conditions in a first step, and imposing the homogeneous Dirichlet conditions in a second step. We give a rapid overview of this procedure (see [17, §3.7] for details). The Boussinesq representation for solutions of the equation (4.3) states that w can be found as a linear combination of the three following particular solutions of the same equation (4.4)

w 1 = ∇Ψ, w 2 = curl(Θ e 3 ), w 3 = ∇(x 3 Λ) + 2(σ -1)Λ e 3 ,
where the scalar functions Ψ, Θ, and Λ are harmonic functions. Here we choose e 3 as the director of the cone axis. Then homogeneous w's of degree λ are given by finding Ψ and Θ homogeneous of degree λ + 1, and Λ of degree λ. Using spherical coordinates (r, θ, ϕ) such that

x 1 = r sin θ cos ϕ, x 2 = r sin θ sin ϕ, x 3 = r cos θ we split the 3D problem into an infinite sequence of 2D problems in (r, θ) parametrized by the azimuthal frequency m ∈ Z. The harmonic generating function can be written in separated variables as (4.5) Ψ = r λ+1 P -m λ+1 (cos θ) cos(mϕ), Θ = r λ+1 P -m λ+1 (cos θ) sin(mϕ), and Λ = r λ P -m λ (cos θ) sin(mϕ),

where P µ ν is the associated Legendre function of the first kind of order µ and degree ν. Combining (4.4) and (4.5), we find for each m ∈ Z three independent solutions w m,1 λ , w m,2 λ , and w m,3 λ of degree λ for equation (4.3). Denote by ω the opening of the cone Γ, which means that Γ is defined by the condition θ ∈ [0, ω) in spherical coordinates. The exponents λ we are looking for are those for which there exists an integer m such that the following 3 × 3 matrix is singular for all r > 0 and ϕ  

w m,1 λ,1 (r, ω, ϕ) w m,2 λ,1 (r, ω, ϕ) w m,3 λ,1 (r, ω, ϕ) w m,1 λ,2 (r, ω, ϕ) w m,2 λ,2 (r, ω, ϕ) w m,3 λ,2 (r, ω, ϕ) w m,1 λ,3 (r, ω, ϕ) w m,2 λ,3 (r, ω, ϕ) w m,3 λ,3 (r, ω, ϕ)  
Using the special forms (4.4) and (4.5), the variables r and ϕ disappear and after some obvious simplification we are left with the matrix 2

(λ + 1) cos ωP -m λ+1 (cos ω) -(λ + 1 -m)P -m λ (cos ω) m cos ωP -m λ+1 (cos ω) (λ+1+m) cos ω P -m λ+1 (cos ω) +(1-2σ) sin 2 ω P -m λ (cos ω) -(λ + 1) cos 2 ω P -m λ (cos ω) -mP -m λ+1 (cos ω) (λ+1-m) cos ω P -m λ (cos ω) -(λ + 1) P -m λ+1 (cos ω) -m cos ω P -m λ (cos ω)          
With this matrix at hand, it is possible to compute its determinant M m σ (λ) and find, for any chosen m, couples (ω, σ) for which the equation M m σ (λ) = 0 has roots on the line Re λ = -1 2 . We present in Fig. 3 the region R 0 of the (ω, σ) plane where such roots can be found when m = 0. This region is clearly non-symmetric with respect to σ = 1 2 . The region R 1 that we have calculated for m = 1 (which is the same for m = -1) is strictly contained in the region R 0 associated with m = 0, see Fig. 4. We observe that when |m| is increasing, the region R m is shrinking. 4.3. Polyhedra. By polyhedron we understand a bounded Lipschitz domain Ω ⊂ R 3 the boundary of which consists of a finite set of plane polygons, the faces. This set being chosen in a minimal way, the edges e of Ω are the segments which form the boundaries of the faces, and the corners c are the corners of the faces. Let E and C be the sets of edges and corners, respectively. To each edge e is associated a transversal sector Γ e and its opening ω e . To each corner c is associated the tangent infinite polyhedral cone Γ c and its section G c = Γ c ∩ S 2 .

Let σ ∈ C be the Cosserat spectral parameter. For each edge e, the edge symbol L e σ (ξ) is defined by (4.2). For each corner c, the corner Mellin symbol A c σ (λ) is defined by (2.1). If σ ∈ {0, 1 2 , 1}, and if the following two conditions are satisfied ∀e ∈ E, L e σ (ξ) is invertible for all ξ = 0 (4.7)

∀c ∈ C, A c σ (λ) is invertible for all λ, Re λ = -1 2 . (4.8)
then L σ is Fredholm on Ω. These conditions are also necessary and determine the essential spectrum of the operator S (1.4). We recall that, cf. Remark 4.2, the condition

σ ∈ 1 2 - sin ω e 2ω e , 1 2 +
sin ω e 2ω e is necessary for (4.7) to hold but is, possibly, not sufficient.

FINITE ELEMENT COMPUTATIONS FOR RECTANGLES AND CUBOIDS

5.1. Rectangles. For any rectangle Ω ⊂ R 2 , Theorem 3.3 shows that the essential spectrum of the Cosserat problem is the interval

[ 1 2 -1 π , 1 2 + 1 π ].
σ(Ω) ≤ 1 2 - 1 π ∼ 0.18169 .
In contrast to the essential spectrum, the discrete spectrum depends on the actual shape of the rectangle, namely of its aspect ratio (but not on its width and height separately). Our convention is to characterize a rectangle by the number a ∈ (0, 1] such that its aspect ratio is 1:a or a:1. The square has its factor a equal to 1, and small a corresponds to elongated rectangles (large aspect ratio).

To determine σ(Ω) and therefore the LBB constant, one can solve the Cosserat eigenvalue problem numerically and look for the smallest eigenvalue. A conforming discretization based on the Stokes eigenvalue problem (1.6) consists in choosing a pair of finite dimensional spaces U ⊂ H 1 0 (Ω) and P ⊂ L 2 (Ω) and constructing the following discrete version of the Schur complement

(5.2) S(U, P) = B 1 R -1 B ⊤ 1 + B 2 R -1 B ⊤ 2
where R is the stiffness matrix associated with the ∇ : ∇ bilinear form on U × U and B k is associated with the bilinear form (u, p) → Ω ∂ k u p on U × P, k = 1, 2. The discrete Cosserat eigenvalues σj , j ≥ 1, are the non-zero eigenvalues of S(U, P) ordered increasingly. Two main difficulties are encountered:

(i) The discrete pair (U, P) constructed from finite element spaces may have a behavior of its own, somewhat independent of the continuous pair (H 1 0 (Ω), L 2 (Ω)). This may give rise to spurious eigenvalues. (ii) The regularity of the eigenvectors p of S depends on the eigenvalue σ and gets worse as σ is closer to the essential spectrum, cf Remark 3.1. In Figure 5 we plot the supremum of exponents s such that p belongs to H s (Ω): This is the minimal Re λ for λ solution of (1 -2σ) sin There exist estimates for σ(Ω) from above and from below which prove that σ(Ω) → 0 like O(a 2 ) as a → 0:

(5.3) sin 2 1 2 arctan a ≤ σ(Ω) ≤ 1 - sinh ρ ρ cosh ρ , with ρ = aπ 2 .
The lower bound is deduced from the Horgan-Payne estimate [START_REF] Horgan | On inequalities of Korn, Friedrichs and Babuška-Aziz[END_REF][START_REF]On the inequalities[END_REF] and is equal to a 2 4 modulo O(a 4 ). The upper bound is obtained by plugging the quasimode p(x 1 , x 2 ) = cos x 1 on the rectangle (0, π) × (-ρ, ρ) into the Rayleigh quotient of the Schur complement S (see Lemma 5.1 below for a more general estimate of this type). It is an improvement at the order O(a 4 ) of the upper bound π 2 a 2 12 proven in [START_REF] Chizhonkov | On the domain geometry dependence of the LBB condition[END_REF] by Chizhonkov and Olshanskii. Note that the upper bound in (5.3) proves that for any a ≤ 0.53127, the bottom σ(Ω) of the spectrum of S is an eigenvalue.

The eigenfunctions are not very singular at the corners if the first eigenvalue is well below the minimum (5.1) of the essential spectrum (Figure 5), which is the situation for rectangles of large aspect ratio according to (5.3). We have quantified this by evaluating convergence rates of the first and second eigenvalues when a = 0.2 (aspect ratio 5:1), using uniform square meshes with 5 • 2 n • 2 n elements (n = 1, . . . , 7) and Q 2 -Q 1 polynomial spaces, see Table 1 where a convergence rate equal to twice the regularity exponent is observed. However, as the aspect ratio approaches 1, the first Cosserat eigenvalue approaches the essential spectrum and the numerical results become less reliable.

We present in Figure 6 computations done with the finite element library Mélina++ with different choices of quadrilateral meshes (uniform 12×12 for (a) and (c), strongly geometrically refined at corners with 144 elements for (b) and (d)) and different choices of polynomial degrees for u and p (tensor spaces of degree 8 and 6 for (a) and (b), 8 and 7 for (c) and (d)). Note that the meshes follow the elongation of the rectangles.

From these four discretizations, we observe a relative stability of the results below 1 2 -1 π , if we except a quadruple eigenvalue that appears for the degrees [START_REF] Crouzeix | On an operator related to the convergence of Uzawa's algorithm for the Stokes equation[END_REF][START_REF]On the inequalities[END_REF] and is rather insensitive to the mesh and the aspect ratio. Moreover, with the same restrictions, the first eigenvalue sits between the explicit lower and upper bounds (5.3), and further eigenvalues satisfy the estimate of [START_REF]On the LBB condition in the numerical analysis of the Stokes equations[END_REF]Theorem 5]. In contrast, the part of the numerical eigenvalues appearing above 1 2 -1 π is very sensitive to the mesh and flattens in a very spectacular way when a refined mesh is used.

As one can see from the graphs in Figure 6, at an aspect ratio of about 1:0.6 (golden ratio ?), the lowest computed eigenvalue crosses over into the interval occupied by the essential spectrum. This is observed rather stably in similar computations, and if true, it would mean that for all rectangles of smaller aspect ratio, in particular for the square, one would have the LBB constant corresponding to (5.1), as discussed in the introduction, see (1.2). But due to the large numerical errors arising from spurious numerical eigenvalues and from the strong corner singularities near the essential spectrum of the continuous operator, the numerical evidence is not as convincing as one would wish.

In Figure 7, we show the first 6 computed eigenfunctions for an aspect ratio of 10:1. One can see that the first eigenfunctions are almost independent of the transversal variable and look like the corresponding quasimodes cos x 1 , cos 2x 1 , cos 3x 1 ,. . . , although on close inspection, even the first one shows corner singularities. As the eigenvalue grows, the unboundedness at the corners becomes more pronounced, and when the essential spectrum is attained, the corner singularities completely dominate the behavior of the eigenfunction. Thus we see the behavior that was discussed above in Remark 3.1. 5.2. Cuboids. We show results of computations for domains Ω ⊂ R 3 , along the same lines as in two dimensions. We choose scalar finite dimensional spaces U ⊂ H 1 0 (Ω) and P ⊂ L 2 (Ω). The 3D discrete version of the Schur complement is (5.4) S(U,

P) = B 1 R -1 B ⊤ 1 + B 2 R -1 B ⊤ 2 + B 3 R -1 B ⊤ 3
where R is still the stiffness matrix associated with the ∇ : ∇ bilinear form on U × U and B k is the matrix associated with the bilinear form (u, p) → Ω ∂ k u p on U × P, now for k = 1, 2, 3.

We present computations on elongated cuboids of the form 1 a ×1×1 with a ranging from 0.05 to 1, see Figure 8, compare with [START_REF]On the LBB condition in the numerical analysis of the Stokes equations[END_REF]. From [START_REF] Dobrowolski | On the LBB constant on stretched domains[END_REF] we know that for such a "channel domain", an upper bound is valid: σ(Ω) ≤ γa 2 where the constant γ depends on the cross section of the channel. Here we prove an improvement of this estimate. ψ a ∈ H 1 0 (ω), (-∆ ′ + a 2 )ψ a = 1 in ω. Then, with µ(ω) the measure of ω, there holds (5.6) σ(Ω a ) ≤ a 2 ψ a , 1 ω µ(ω) .

Proof. We denote by (x 1 , x ′ ) ∈ (0, π a ) × ω the coordinates in Ω a and consider the quasimode p(x) = cos ax 1 . Then ∇p = (-a sin ax 1 , 0 . . . 0) ⊤ and we check that ∆ -1 ∇p = (a sin ax 1 ψ a (x ′ ), 0 . . . 0) ⊤ .

Hence Sp = a 2 cos ax 1 ψ a (x ′ ). It is easy to see that the Rayleigh quotient satisfies Sp, p p, p = a 2 ψ a , 1 ω µ(ω) , which ends the proof of (5.6). To obtain an upper bound for the cuboids Ω a of dimensions 1 a ×1×1, we take = (0, π)× (0, π) in Lemma 5.1. We can calculate ψ a by Fourier expansion in ω: Starting from

1 k 2 1 + k 2 2 .
We observe in Figure 8 that our computations are in good agreement with the bound (5.7). Nevertheless, they are more difficult to interpret than in 2D. The possible presence of spurious eigenvalues is not easy to distinguish from the manifestation of the essential spectrum. Moreover, at this stage, it is an open question whether the bottom of the essential spectrum comes from edges or from corners. Numerical experiments on cylinders with circular or annular sections, which are compatible with the upper bound (5.6), tend to suggest that the essential spectrum coming from the edges is restricted to [ 1 2 -1 π , 1 2 + 1 π ]. Therefore we may conjecture that we see in Figure 8 a manifestation of the essential spectrum coming from the corners of the cuboids.

FIGURE 1 .

 1 FIGURE 1. Color plot of the decimal logarithm of | Im λ| for purely imaginary roots λ of the characteristic equation (3.1) as a function of the opening ω/π ∈ (0, 2) (abscissa) and the Cosserat spectral parameter σ ∈ [0, 1] (ordinate).

FIGURE 2 .

 2 FIGURE 2. Color plot of smallest positive real roots λ of the characteristic equation (3.1) as a function of the opening ω/π ∈ (0, 2) (abscissa) and the Cosserat spectral parameter σ ∈ [0, 1] (ordinate).

FIGURE 3 .

 3 FIGURE 3. m = 0: Color plot of the decimal logarithm of Im λ of roots λ with real part -1 2 of characteristic equation M 0 σ (λ) = 0 as a function of the opening ω (in degrees, abscissa) and the parameter σ ∈ [0, 1] (ordinate).

  1)P -m λ+1 (cos ω) mP -m λ+1 (cos ω) (λ + 2σ -1) cos ω P -m λ (cos ω)

1 FIGURE 4 .

 14 FIGURE 4. m = 0, 1: Boundary of regions R 0 (solid line) and R 1 (dashed line) in the plane (ω, σ): Opening ω (in degrees, abscissa) and σ ∈ [0, 1] (ordinate).

1 FIGURE 5 .

 15 FIGURE 5. Regularity exponent s (ordinate) function of σ (abscissa) for rectangles. The vertical asymptote is σ = 1 2 -1 π .

FirstFirstFIGURE 6 .

 6 FIGURE 6. First 12 computed Cosserat eigenvalues on rectangles vs parameter a ∈ (0, 1] (abscissa). The solid horizontal line is the infimum1 2 -1 π of the essential spectrum. Solid curves are the lower and upper bounds (5.3).

σ 1 ≃FIGURE 7 .

 17 FIGURE 7. First six computed Cosserat eigenvectors on the rectangle with aspect ratio a = 0.1. Same mesh and polynomial degrees as in Fig. 6 (a).

Lemma 5 . 1 .

 51 Let ω be a Lipschitz domain in R d-1 (d ≥ 2) and for a > 0 set Ω a = (0, π a )×ω. Denote by ∆ ′ the Laplacian in ω and consider the solution ψ a of the Dirichlet problem(5.5)

FirstFIGURE 8 .

 8 FIGURE 8. First 16 computed Cosserat eigenvalues on cuboids 1a ×1×1 vs parameter a ∈ (0, 1] (abscissa). The solid curve is the upper bound (5.7).

TABLE 1 .

 1 λπ 2 = ±λ with positive real part, cf (3.2). Convergence rates for Cosserat eigenvalues on the rectangle a = 0.2

		Extrapolated value Regularity exponent s Convergence rate (n = 5, 6, 7)
	σ1	0.031375609	0.93189	1.85618
	σ2	0.109538571	0.69036	1.37814

The matrix (4.6) that we reproduce here is equivalent to the one we find in the preprint[START_REF] Kozlov | On singularities of solutions to the boundary value problems near the vertex of a rotational cone[END_REF]. Its correctness can be checked. Unfortunately a misprint appeared in further references[START_REF]On singularities of solutions of the displacement problem of linear elasticity near the vertex of a cone[END_REF][START_REF]Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations[END_REF]: The factor -m in (λ+1-m) is missing in the last term of the second column. Nevertheless, the numerical computations presented in these latter references where made with the correct formulas.