Benjamin Sala

Xavier Gabrion

Thomas Jeannin

Frédérique Trivaudey

Violaine Guicheret-Retel

Fabrizio Scarpa

Vincent Placet

Effect of hygrothermal ageing on the shear creep behaviour of eco-friendly sandwich cores

Keywords: sandwich core, eco-friendly, creep/recovery, hygrothermal environment

The development of sustainable sandwich materials is needed in the transportation sector to address environmental concerns related to the production and operation of vehicles. In addition to biobased composite skins, alternatives to classic synthetic core materials must be found to reduce the ecological footprint of whole sandwich-structured composites. This study focused on three eco-friendly lightweight core materials: balsa wood, paper honeycomb, and recycled PET foam. The effect of the hygrothermal ageing on their shear creep/recovery behaviour has been here investigated. Two different environmental conditions were tested: 23 °C-50% RH and 70 °C-65% RH. The results indicate that the maximum shear strain, the time-delayed strain and the residual strain increase for the three core materials with the severity of the hygrothermal conditions. This was attributed to the softening of the constitutive polymeric materials of the cell walls at temperatures close to 70 °C. The balsa wood exhibits the best creep resistance under the two environmental conditions. The identification of the viscoelastic properties highlights that the release times and the shear viscous parameter of the balsa wood and the PET foam depended on the stress level and the hygrothermal conditions.

INTRODUCTION

A sandwich-structured composite consists of two stiff composite skins bonded to a low-strength and lightweight core material [START_REF] Gay | Composite materials: design and applications[END_REF]. It is generally designed to produce panels with high bending stiffness with an overall low specific weight. Sandwich composites are commonly used in various applications, such as aeronautical, automotive, and civil engineering [START_REF] Castanie | Review of composite sandwich structure in aeronautic applications[END_REF][START_REF] Vijaya Ramnath | Review on Sandwich Composite and their Applications[END_REF]. Under bending, the core material is subjected to shear stress. Therefore, the selection of the core material is mainly based on lightness and transverse shear performance. The most common and commercially available cores are foams such as polyurethane (PU) and polyvinyl chloride (PVC), balsa wood (Ochroma pyramidale), and honeycombs (made of paper, aluminium or aramid) [START_REF] Manalo | State-of-the-Art Review on FRP Sandwich Systems for Lightweight Civil Infrastructure[END_REF][START_REF] Aktay | Numerical modelling of honeycomb core crush behaviour[END_REF]. To address global concerns over environmental issues and to ensure an ecological transition towards a sustainable development of various industrial sectors, more environmentally friendly sandwich materials are required. Recent studies have focused on plant fibre composites [START_REF] Baley | Eighty years of composites reinforced by flax fibres: A historical review[END_REF][START_REF] Moudood | Environmental effects on the durability and the mechanical performance of flax fiber/bio-epoxy composites[END_REF][START_REF] Alsubari | Potential of Natural Fiber Reinforced Polymer Composites in Sandwich Structures: A Review on Its Mechanical Properties[END_REF] and biobased and/or recycled materials for cores [START_REF] De Oliveira | A novel sandwich panel made of prepreg flax skins and bamboo core[END_REF][START_REF] Oliveira | Improved sustainable sandwich panels based on bottle caps core[END_REF][START_REF] Mancuso | Mechanical Behaviour of a Green Sandwich Made of Flax Reinforced Polymer Facings and Cork Core[END_REF][START_REF] Sergi | The Potential of Agglomerated Cork for Sandwich Structures: A Systematic Investigation of Physical, Thermal, and Mechanical Properties[END_REF][START_REF] Duigou | PLLA/Flax Mat/Balsa Bio-Sandwich-Environmental Impact and Simplified Life Cycle Analysis[END_REF][START_REF] Monti | Élaboration et caractérisation d'une structure composite sandwiche à base de constituants naturels[END_REF]. New classes of biobased materials can represent an alternative to traditional fossil constituents and reduce carbon and more general environmental footprints. The transition to more environmentally friendly sandwich materials begins with the selection of lightweight biobased or recycled materials with high shear properties. Most biobased materials are also more sensitive to temperature and moisture than the majority of the traditional synthetic ones, and also require a thorough investigation of their mechanical performance, in particular their creep behaviour in various and changing environments to which sandwich materials are often submitted in service conditions. Three eco-friendly materials have been selected in this study to evaluate their potential as cores in sustainable sandwich materials: reconstituted balsa wood panels, paper honeycombs, and recycled polyethylene terephthalate (rPET) foam. Balsa wood panels were selected because they are made from natural and renewable resources that are light and available. Balsa is a medium-sized fast-growing pioneer subtropical and tropical tree (typically reaching 25 m in height and 1 m in diameter) that is deciduous or evergreen and occurs in both pure and mixed stands in association with other pioneer species [START_REF] Fletcher | Balsa-Production and utilization[END_REF]. Balsa is not a tree in danger of extinction due to its fast growth and ease of reproduction. It is cultivated on plantations during 5-7-year rotations, primarily in the South American country of Ecuador. Its production, reaching 18 000 ha in 2008 in Ecuador, contributes to the economy of the country [START_REF] Midgley | Balsa: biology, production and economics in Papua New Guinea[END_REF]. Due to its rapid growth, balsa wood also has a relatively low density compared to other wood species [START_REF] Fletcher | Balsa-Production and utilization[END_REF]. The density of dry balsa wood ranges from 60 to 380 kg.m -3 [START_REF] Borrega | Composition and structure of balsa (Ochroma pyramidale) wood[END_REF]. Ecuador's environment, altitude, and climate offer stable wood in terms of density. Additionally, during photosynthesis, some atmospheric CO2 is stored in the wood, approximately 1.81 kg of CO2 per kg of balsa wood [START_REF] Duigou | PLLA/Flax Mat/Balsa Bio-Sandwich-Environmental Impact and Simplified Life Cycle Analysis[END_REF]. Balsa also improves soil fertility [START_REF] Martínez-Garza | Effect of initial soil properties on six-year growth of 15 tree species in tropical restoration plantings[END_REF]. For application as a core in sandwich panels, balsa wood often comes in the form of reconstituted wood blocks bonded with a polyvinyl alcohol (PVA) adhesive. To produce 1 kg of balsa wood panel, approximately 4 kg of balsa trunks are needed [START_REF] Duigou | PLLA/Flax Mat/Balsa Bio-Sandwich-Environmental Impact and Simplified Life Cycle Analysis[END_REF].

Paper honeycombs were also selected as a biobased material with a high mechanical performance to weight ratio, surpassing bulk wood cores [START_REF] Pflug | Continuously Produced Paper Honeycomb Sandwich Panels for Furniture Applications[END_REF]. Additionally, paper honeycombs are a low-cost material [START_REF] Bitzer | Honeycomb Technology[END_REF]. In the construction field, cardboard honeycombs are increasingly used as insulation materials, as a substitution for polyurethane foam or glass wool [START_REF] Salavatian | Assessment of Cardboard as an Environment-Friendly Wall Thermal Insulation for Low-Energy Prefabricated Buildings[END_REF]. Paper pulp is extracted from raw material using mechanical or chemical processes. To produce one tonne of pulp, 3 and 1.1 tonnes of wood are necessary using the abovementioned processes, respectively. Energy consumption accounts for approximately 16% of the total production costs of the pulp and paper [START_REF] Latka | Paper in architecture: Research by design, engineering and prototyping[END_REF].

Paper Industries [24], the energy consumed to produce pulp and paper is based on biomass. A total of 1.05 MWh are needed to produce 1 tonne of product. Additionally, the direct CO2 emissions per kilotonne of product are equal at 0.34 kt.

The last type of core considered in this work was a foam made of recycled polyethylene terephthalate (PET) from water bottles. The amount of plastic waste is worldwide increasing. Between 2006 and 2016 in Europe only, plastic waste collection has increased by 11% from 24.5 million tonnes to 27.1 million tonnes [START_REF]Plasticsthe Facts 2018: An analysis of European plastics production, demand and waste data[END_REF]. This waste can have a detrimental effect on the environment, hence the necessity to recycle this material [START_REF]Current research trends on plastic pollution and ecological impacts on the soil ecosystem: A review[END_REF]. The recycling rate for plastic packaging was 40.8% in Europe with a target of 50% by 2025 [START_REF]Plasticsthe Facts 2018: An analysis of European plastics production, demand and waste data[END_REF]. Therefore, recycling water bottles into a PET foam helps to reduce the impact of plastic waste on the environment. In addition, according to Armacell's life cycle assessment [START_REF]Life Cycle Assessment: PET foams[END_REF], the production of the recycled PET foam allows having an environmental carbon footprint 49% lower than producing a virgin PET foam. However, for a same density and according to the technical data sheets, the shear modulus of the recycled PET foam is approximatively 30% lower than that of the virgin PET foam (AIREX T92.80) [START_REF]Technical Datasheet Core GR[END_REF][START_REF]3A Core Materials[END_REF].

In addition to their eco-friendliness, materials selected as constituents for a sandwich panel must also possess high shear mechanical properties. To accurately predict and understand the mechanical response at the sandwich scale, in-depth knowledge of the mechanical behaviour of the composite skins and of the core materials is required. For this work realized in the framework of the European project SSUCHY (https://www.ssuchy.eu/), the targeted application concerns the floor and trim panel structures for automotive sector, which requires long-term durability, particularly under varying hygrothermal conditions.

The quasi-static shear behaviour of the three selected cores has already been investigated in open literature. The shear behaviour of balsa wood is particularly well documented. Concerning balsa samples extracted from blocks of the reconstituted balsa panel, studies have highlighted a linear dependence of the shear modulus and shear ultimate stress on the density of the wood [START_REF] Osei-Antwi | Shear mechanical characterization of balsa wood as core material of composite sandwich panels[END_REF][START_REF] Borrega | Mechanics of balsa (Ochroma pyramidale) wood[END_REF][START_REF] Silva | Compressive response and failure of balsa wood[END_REF]. The presence of the adhesive between the blocks of the reconstituted balsa core tends to stiffen the material and improve its strength [START_REF] Osei-Antwi | Shear mechanical characterization of balsa wood as core material of composite sandwich panels[END_REF]. Garrido et al. [START_REF] Garrido | Effects of elevated temperature on the shear response of PET and PUR foams used in composite sandwich panels[END_REF] carried out an experimental campaign on balsa samples for a range of temperatures between 23 °C and 240 °C. The authors observed a degradation of the mechanical properties with increasing temperature. The influence of water immersion on the mechanical properties of the balsa was only studied under compression [START_REF] Sadler | Water immersion effect on swelling and compression properties of Eco-Core, PVC foam and balsa wood[END_REF]. The experimental results highlight a decrease in the compression modulus, particularly during the first week of immersion. At the scale of the reconstituted balsa core, Monti [START_REF] Monti | Élaboration et caractérisation d'une structure composite sandwiche à base de constituants naturels[END_REF] showed that balsa wood had a higher specific shear modulus than PVC foam, PET foam and Kevlar honeycomb. The shear mechanical properties were also determined by Fathi et al. [START_REF] Fathi | Full-field shear analyses of sandwich core materials using Digital Image Correlation (DIC)[END_REF] using a direct shear test or by the inverse method from the strain field measured in the core material during a four-point bending test on a sandwich beam with glass fibre reinforced epoxy skin. The shear modulus and the strength were shown to be independent of the method to acquire the data. The strain at failure was however underestimated when using the direct shear measurement. The use of strain field measurements also highlighted the presence of a heterogeneous strain field in the balsa material due to the natural origin of the material. Recently, some authors have developed a veneered core made of a thin layer of reconstituted balsa [START_REF] Wu | Mechanical properties of a balsa wood veneer structural sandwich core material[END_REF][START_REF] Vahedi | Thermomechanical characterization of a balsa-wood-veneer structural sandwich core material at elevated temperatures[END_REF]. By turning the layer orientations, the idea was to decrease the scatter of the mechanical properties and to tailor these properties to specific applications. In particular, the work of Vahedi et al. [START_REF] Vahedi | Thermomechanical characterization of a balsa-wood-veneer structural sandwich core material at elevated temperatures[END_REF] on veneered balsa core showed a decrease in the shear modulus and strength with increasing temperature. In addition, the increase in the moisture content of the reconstituted balsa leads to a decrease in the mechanical properties such as bending or compressive modulus and ultimate stress [START_REF] Bhangu | Effect of Moisture Absorption on the Mechanical Properties of Balsa Wood[END_REF][START_REF] Stahlhut | Effect of Absorption of Liquids on the Mechanical Properties of Balsa Wood[END_REF]. The influence of water content on the shear behaviour of balsa has not yet been investigated in the literature. The quasi-static shear behaviour of balsa wood in a severe environment (high temperature and humidity level) and the time-delayed behaviour have not yet been addressed by the scientific community.

Concerning PET foams and paper honeycombs, the shear behaviour is less documented. Although some authors have used paper honeycombs as the core of sandwich panels [START_REF] Basf | Green & Light and BASF developed Elastoskin® based paper honeycomb car trunk floor. BASF We Create Chem[END_REF][START_REF] Chen | Investigation of flexural creep of kraft paper honeycomb core sandwich panels using the finite element method[END_REF][START_REF] Chen | Investigation of elastic moduli of Kraft paper honeycomb core sandwich panels[END_REF], the main contribution was made by Pohl [START_REF] Pohl | Strengthened corrugated paper honeycomb for application in structural elements[END_REF]. The author characterized the quasi-static shear behaviour of a paper honeycomb and tested paper honeycomb specimens for a long time under real environmental conditions. The measure of the relative humidity during the tests showed that summer was characterized by a mean level of relative humidity higher than that in winter.

Interestingly, the shear strain rate was greater in the summer than during winter. However, the influence of a severe environment on time-delayed behaviour has not been explored.

Fathi et al. [START_REF] Fathi | Full-field shear analyses of sandwich core materials using Digital Image Correlation (DIC)[END_REF] showed that, despite the higher density, the shear modulus of PET foam was lower than that of PVC foams. The use of strain field measurements allowed to identify a more uniform distribution of the shear strain in the PET foam compared to PVC foam, also due to a more homogenous repartition of the cell sizes. Garrido et al. [START_REF] Garrido | Effects of elevated temperature on the shear response of PET and PUR foams used in composite sandwich panels[END_REF] loaded PET foam specimens under shear over a large temperature range (between -20 °C and 120 °C).

The results showed a decrease in shear modulus with increasing temperature. Those authors demonstrated that different empirical models can accurately describe the variation in the shear modulus as a function of the temperature level. No studies however have been carried out so far on the creep behaviour of PET foams.

The aim of this study is to fill the gap in literature concerning the time-delayed shear behaviour of core materials and to evaluate the influence of hygrothermal ageing on this mechanical response. This study focuses on the quasistatic and creep/recovery shear behaviour of balsa wood, paper honeycomb, and recycled PET (rPET) foam panels.

They were tested under two hygrothermal conditions, namely, 23 °C-50% relative humidity (RH), and 70 °C-65% RH. Due to the expression of an irreversible mechanism during the creep stage, under a severe environment, the viscoelastic behaviour was analysed in the recovery stage.

MATERIALS AND METHODS

Materials

Materials presentation and microstructure

Three different materials were studied in this research. Figure 1 presents them and the associated coordinate systems. The first core was the balsa wood panel Baltek SB50 from 3A Core ® provided by Sicomin ® . It was a 25.4 mm thick rigid panel made of blocks assembled by adhesive bonding. The blocks were cut perpendicular to the tree growth direction (end grain balsa). The length and width of the reconstituted balsa wood panel were chosen as the two main material directions (directions L and T, respectively). The second material was an expanded 8 cell size paper honeycomb provided by Axxor ® company. This core is labelled as H-3D-140-1400-8.1-20.0-N-57 and a thickness of 20 mm. The L-direction was defined as the direction of the ribbon, and the T-direction was perpendicular to the L-direction. The third studied material was a recycled polyethylene terephthalate (rPET) foam ArmaPET Struct GR80 manufactured from PET bottles and provided by Armacell ® . This foam has the particularity of having darker colour lines corresponding to welding lines. The material coordinate system was therefore chosen so that the L-direction was parallel to the direction of these lines and the T-direction was perpendicular to these lines. For all the core materials, the T' direction was defined as the direction perpendicular to the plane (L,T).

Microscopic observations of the core materials, carried out using a Keyence VHX5000 numerical microscope, are presented in Figure 1. The balsa wood is composed of fibres and rays. According to Borrega et al. [START_REF] Borrega | Composition and structure of balsa (Ochroma pyramidale) wood[END_REF], it represents 66% to 76% and 20% to 25% of the volume of the microstructure, respectively. Prismatic in shape and composed of thick cell walls, the fibres enable the tree to be maintained. The cell wall of the fibres is divided into 3 sublayers labelled S1, S2 and S3. The S2 layer is the thickest and is composed by cellulose microfibrils of a matrix of hemicellulose and lignin [START_REF] Silva | Compressive response and failure of balsa wood[END_REF]. Microfibrils are made up of both amorphous and crystalline cellulose with a crystallinity index of approximately 40% [START_REF] Borrega | Composition and structure of balsa (Ochroma pyramidale) wood[END_REF].

The paper honeycomb consists of open hexagonal cells with a characteristic size of approximately 7 mm. Based on the microscopic observation, the thickness of the paper wall is comprised between 0.2 and 0.5 mm. The principal constituent of the paper is cellulose fibres with lengths between 1 and 3 mm [START_REF] Pohl | Strengthened corrugated paper honeycomb for application in structural elements[END_REF]. Some additives can also be used in the paper and can alter the mechanical properties of the paper.

The structure of the rPET foam made of closed hexagonal cells is also visible in Figure 1. The width of these cells is between 0.6 and 1 mm.

Determination of the density and moisture content at equilibrium

The equilibrium moisture content (EMC) of the three core materials was determined following the ASTM C272 standard [START_REF]C272/Test Method for Water Absorption of Core Materials for Structural Sandwich Constructions[END_REF] under two different environmental conditions: 23 °C-50% RH, labelled EC1, and 70 °C-65% RH, labelled EC2. A balance with an accuracy of 0.001 g was used to weight the specimens. Square section samples with 150 mm side and 25 mm thick were used for the balsa wood and square section samples with 120 mm side and 20 mm thick were used for the two other materials. The edges of the samples were not protected. All materials are first stored for a minimum of one and a half months in a room with a temperature and humidity of 23±3 °C-50±10% RH. In order to control the hygrothermal conditions more finely, the specimens were then conditioned in environment EC1 using a Memmert HPP 108L climatic chamber. Then, the climatic chamber is set to the environmental condition corresponding to the environment EC2. Finally, the cores were removed from the climatic chamber and putted into an oven at 103 °C to dry the specimens in accordance with the ASTM D4442 standard [START_REF]D4442 -Test Methods for Direct Moisture Content Measurement of Wood and Wood-Based Materials[END_REF]. The mass of the different samples was measured after at least 20 days in EC1 (𝑚 𝐸𝐶1 ), then 7 days in EC2 (𝑚 𝐸𝐶2 ) and finally after drying for 7 days at 103 °C (𝑚 𝑑𝑟𝑦 ). For each condition, the equilibrium was considered to be reached when the variation of mass was infinitesimal between two measuring points with time interval of 24h.

The equilibrium moisture content (𝑀𝐶 𝐸𝐶𝑖 ) under the environmental condition ECi was calculated as follows:

𝑀𝐶 𝐸𝐶𝑖 = 100 × 𝑚 𝐸𝐶𝑖 -𝑚 𝑑𝑟𝑦 𝑚 𝑑𝑟𝑦 (1)
The density of the core materials was measured under environment EC1. At least 10 samples with a length of 300 mm and a width of 50 mm were tested. The thicknesses of the samples were 25 mm for balsa wood and 20 mm for the other core materials.

Mechanical testing

Monotonic shear tests

Testing parameters

The core materials were tested under shear solicitation using an MTS ® Criterion 45 machine instrumented with a 100 kN load cell. For this purpose, a shear assembly was specially designed in accordance with the ASTM C273 standard [START_REF]C273/Test Method for Shear Properties of Sandwich Core Materials[END_REF]. The samples are first bonded to two metal parts. One of these metal parts was then embedded in a mobile plate, while the other part was embedded in a fixed plate. Monotonic shear tests were realized in EC1 (23±3 °C-50±10% RH) and EC2 (70±3 °C-65±5% RH). To control the environment during shear testing and then maintain hygrothermal conditions similar those used during sample conditioning, a polymethyl methacrylate (PMMA) box, connected to an Inec70-90 relative humidity generator from Ineltec ® , was installed on the tensile machine. Figure 2 shows the set-up. The displacement between the two metal parts was measured using an LVDT sensor with a measurement range of 10 mm. The solicitation speed was fixed at 1 mm/min. The dimensions of the samples were 50 mm in width and 300 mm in length. The thicknesses of the samples were equal to the dimensions of the initial core material panels. The shear planes (L,T') and (T,T') were subjected to loading and at least 4 samples were tested in each direction and environmental condition. The shear strain and stress according to the direction (i,T'), where the direction i corresponds to the L or T material direction, were calculated as follows:

𝜀 𝑖𝑇′ = atan ( 𝑢 𝑖 𝑡 ) 2 = 𝛾 𝑖𝑇′ 2 (2)
𝜎 𝑖𝑇′ = 𝑃 𝑖 𝐿𝑏 (3)
where 𝑢 𝑖 is the displacement of the material oriented along direction i and measured by the LVDT sensor, t is the thickness of the sample, 𝛾 𝑖𝑇′ is the angular distortion in plan (i,T'), 𝑃 𝑖 is the load on the material oriented along direction i, L is the length of the sample and b is the width of the sample.

The shear moduli in each material direction were calculated using a linear regression on the shear strain/stress curve for a strain range between 0 and 1.5.10 -3 . The test was stopped once the specimen has broken slowly or the measuring range of the LVDT was reached.

Bonding conditions

The conditions for bonding the core materials to the metal parts as a function of the tested environmental conditions are summarized in Table 1. Concerning the thermocompression process, an Agila ® Presse 100 kN machine was used. For the paper honeycomb, expanding/foaming adhesives have been used to provide a larger bonding surface.

The bonding of the materials for mechanical testing under the EC2 environment was more difficult. It was necessary to ensure a high interlaminar strength between the metal plates and the core materials, while at the same time ensuring that no damage was caused by water desorption into the material itself.

The balsa wood was therefore bonded directly at condition EC2 using clamps. The rPET foam was first conditioned under condition EC2, then bonded and finally conditioned again under condition EC2. The honeycomb was bonded using thermocompression under condition EC2. Just before the beginning of the mechanical test, the paper-based material was conditioned under the environment EC2 until the EMC was reached.

Creep shear tests

The time-delayed behaviour of the core materials was investigated under condition EC1 for different stress levels generator were also used to maintain the two studied environmental conditions with temperature and relative humidity variations of ±3 °C and ±5%, respectively, during the creep/recovery test. The bonding conditions of the core materials were the same as those for the monotonic tests. Materials were solicited along the stiffer material direction: L-direction for the balsa wood and the paper honeycomb, T-direction concerning the rPET foam. For each material and environmental condition, the creep/recovery tests were realized on 3 samples. Some parameters were extracted from the time/strain and strain/stress curves (Figure 3):

the instantaneous shear strain ε ins obtained when the nominal stress is reached for the first time, the instantaneous shear modulus Gins measured between zero strain and 1.5.10 -3 strain, the maximum shear strain ε max , the secondary creep stage strain rate 𝜀̇𝑐 𝑟𝑒𝑒𝑝 determined in a range of times between 2000 s and 3500 s, the time delayed strain during the creep stage ε delc defined as the difference between ε max and ε ins , the residual shear strain ε res corresponding to the shear strain at the end of the creep/recovery test, the time delayed strain during the recovery stage ε delr defined as the difference between the shear strain once the recovery stage begins and ε res .

Identification of the viscoelastic properties

Viscoelastic model

A 3D model was used to study the viscoelastic behaviour of the core materials based on the formulation of Boubakar et al. [START_REF] Boubakar | A meso-macro finite element modelling of laminate structures[END_REF]. The viscoelastic flow 𝜀̇𝑣 𝑒 is defined as the sum of the elementary viscoelastic flows 𝜉 𝑖 ̇ (see Eq. 4). The number of viscoelastic mechanisms (labelled N) is chosen empirically at 31. It allows to accurately describe the viscoelastic behaviour of cellulose based materials as shown in [START_REF] Sala | Influence of the stress level and hygrothermal conditions on the creep/recovery behaviour of high-grade flax and hemp fibre reinforced GreenPoxy matrix composites[END_REF].

𝜀̇𝑣 𝑒 = ∑ 𝜉 𝑖 Ṅ 𝑖=1 = ∑ 1 𝜏 𝑖 (𝜇 𝑖 𝑆 𝑣𝑒 𝜎 -𝜉 𝑖 ) 𝑁 𝑖=1 , (4)
The expression of the i th elementary viscoelastic flow is given by the following equation:

𝜉 𝑖 ̇= 1 𝜏 𝑖 (𝜇 𝑖 𝑆 𝑣𝑒 𝜎 -𝜉 𝑖 ), (5)
where 𝜏 𝑖 and µ 𝑖 are the release time and the weighting coefficient of the i th elementary viscous mechanism, respectively. The viscoelastic compliance tensor is 𝑆 𝑣𝑒 and 𝜎 is the Cauchy stress tensor.

Considering an orthotropic material, the viscoelastic compliance can be written as follows:

𝑆 𝑣𝑒 = [ 𝛽 𝐿 𝐸 𝐿 -𝛽 𝐿𝑇 * 𝜈 𝐿𝑇 𝐸 𝐿 -𝛽 𝐿𝑇′ * 𝜈 𝐿𝑇′ 𝐸 𝐿 0 0 0 -𝛽 𝐿𝑇 * 𝜈 𝐿𝑇 𝐸 𝐿 𝛽 𝑇 𝐸 𝑇 -𝛽 𝑇𝑇′ * 𝜈 𝑇𝑇 ′ 𝐸 𝑇 0 0 0 -𝛽 𝐿𝑇′ * 𝜈 𝐿𝑇′ 𝐸 𝐿 -𝛽 𝑇𝑇′ * 𝜈 𝑇𝑇 ′ 𝐸 𝑇 𝛽 𝑇′ 𝐸 𝑇′ 0 0 0 0 0 0 𝛽 𝐿𝑇 𝐺 𝐿𝑇 0 0 0 0 0 0 𝛽 𝐿𝑇′ 𝐺 𝐿𝑇′ 0 0 0 0 0 0 𝛽 𝑇𝑇′ 𝐺 𝑇𝑇′ ]
where 𝛽 𝐿 , 𝛽 𝑇 , 𝛽 𝑇′ , 𝛽 𝐿𝑇 * , 𝛽 𝐿𝑇′ * , 𝛽 𝑇𝑇′ * , 𝛽 𝐿𝑇 , , 𝛽 𝐿𝑇 and 𝛽 𝑇𝑇′ are the viscous parameters that characterize the viscosity of the material, 𝐸 𝑖 is the elastic moduli in each material direction; and 𝜈 𝑖𝑗 𝑎𝑛𝑑 𝐺 𝑖𝑖 are the Poison's ratios and shear moduli within the plane (i,j).

This viscoelastic model has the peculiarity of linking the logarithm of the release times to the weighting coefficients of the viscous mechanisms by a Gaussian distribution defined by its mean noted ln (𝜏 1 ) and its standard deviation (SD).

Identification of the model parameters

The identification of the viscoelastic parameters was carried out on the recovery part of the test. Although the viscoelastic behaviour can be affected by the creep stage, the method used in this work however excludes the irreversible mechanisms that can be expressed in this phase of the loading, particularly during tests in harsh environments. The recovery stage was considered to start once the load is null.

Only three parameters remain to be identified when one considers a one-dimensional shear loading: the viscous parameter 𝛽 𝐿𝑇 ′ (𝛽 𝑇𝑇 ′ for rPET foam), the mean and the standard deviation of the Gaussian distribution of the elementary viscoelastic components. The two previous parameters and the term

𝛽 𝐿𝑇′ 𝐺 𝐿𝑇′ ( 𝛽 𝑇𝑇′ 𝐺 𝑇𝑇′
for rPET foam) have been identified using an inverse method applied to the viscoelastic compliance during the recovery stage. The viscous parameter was then calculated by measuring the modulus during the unloading between ε max and ε max -1.5. 10 -2 . The optimization algorithm used in this work adopts a genetic algorithm and a Levenberg-Marquardt algorithm to avoid early convergence to a local minimum. All these tools are already implemented in the free software MIC2M [START_REF] Richard | Identification du comportement et évaluation de la fiabilité des composites stratifiés[END_REF] used in this study.

RESULTS AND DISCUSSION

Density and moisture content

The density of the cores at 23 °C-50% RH and the EMC under conditions EC1 and EC2 are shown in Table 2. The densities of the balsa wood panel, the paper honeycomb and the rPET foam are 97 kg.m -3 , 47 kg.m -3 and 80 kg.m - 3 , respectively. These values are in accordance with the data provided by the suppliers. The distribution of the density values of the balsa wood is the most scattered. This can be explained by the fact that the latter is composed of an assembly of balsa blocks. Balsa wood and paper honeycomb have a larger water content under ambient conditions than rPET. The EMC under environment EC1 is equal to 9.2%, 7.3% and 0.66% for the balsa wood panel, paper honeycomb and rPET foam, respectively. The three materials present different moisture sorption mechanisms. In the paper honeycomb, several moisture diffusion mechanisms occur: (i) diffusion of water vapour through the empty space between the fibres, (ii) diffusion into small pores, (iii) diffusion within the wood fibres [START_REF] Ramarao | Moisture sorption and transport processes in paper materials[END_REF]. The sorption of water in the balsa wood panel is principally due to the hydrophilic constituents of the fibre cell wall [START_REF] Jin | Creep Behavior of Wood Plasticized by Moisture and Temperature[END_REF]. In the case of the petroleum-based material, water penetrates the PET by forming hydrogen bonds with polar ester groups. The number of hydrogen bonds are then reduced and the relaxation of the main chain become so easier [START_REF] Chen | Plasticization and Crystallization of Poly(ethylene Terephthalate) Induced by Water[END_REF]. The direct consequence is the glass transition temperature decreases so with the increase of water content [START_REF] Chen | Glass transition temperature of copolyesters of PET, PEN and PHB as determined by dynamic mechanical analysis[END_REF][START_REF] Wang | Study of the glass transition temperature and the mechanical properties of PET/modified silica nanocomposite by molecular dynamics simulation[END_REF]. For a very long time of exposure, the water can cause the hydrolysis of the material by cutting the chains of ester functions [START_REF] Van Schoors | Vieillissement hydrolytique des géotextiles polyester (polyéthylène téréphtalate): Etat de l'art[END_REF][START_REF] Jabarin | Effects of water absorption on physical properties and degree of molecular orientation of poly (ethylene terephthalate)[END_REF]. The diffusion kinetics is influenced by the average molecular weight of the polyester chains, by the orientation of these chains and also by the crystallinity of the material [START_REF] Van Schoors | Vieillissement hydrolytique des géotextiles polyester (polyéthylène téréphtalate): Etat de l'art[END_REF][START_REF] Jabarin | Effects of water absorption on physical properties and degree of molecular orientation of poly (ethylene terephthalate)[END_REF].

Under the environmental conditions EC2, a small decrease in the EMC is observed for the balsa wood and the paper honeycomb with mean values of 7.4% and 6.0% respectively. These results are in accordance with the sorption curves of wood [START_REF] Straube | Methodology and Design of Field Experiments for Monitoring the Hygrothermal Performance of Wood Frame Enclosures[END_REF]. For the rPET, the EMC increases to a mean value of 0.76%. Therefore, the main difference induced by the two conditioning conditions in the materials is primarily due to the temperature, since the moisture content remains slightly lower or higher, depending on the core material considered.

Monotonic and creep behaviour under ambient conditions

Monotonic behaviour

The monotonic shear behaviour of the core materials and the associated mechanical properties are presented in Figure 4 and in Figure 5 for the (L,T') and (T,T') shear planes under environment EC1. The mean values of the shear modulus according to the (L,T') shear plane are measured at 123 MPa, 49 MPa and 18 MPa for the balsa wood panel, paper honeycomb and rPET foam, respectively. These values are in accordance with those provided in the technical datasheet [START_REF]Technical Datasheet Core GR[END_REF][START_REF]A Composites Core Materials[END_REF]. The specific shear modulus of the balsa wood panel, paper honeycomb and rPET foam are measured at 1,27 MPa.kg -1 .m 3 , 1,04 MPa.kg -1 .m 3 and 0,23 MPa.kg -1 .m 3 respectively. In comparison, the specific moduli of PVC and PUR foam are equal to 0,37 MPa.kg -1 .m 3 and 0,09 MPa.kg -1 .m 3 , respectively [START_REF] Fathi | Full-field shear analyses of sandwich core materials using Digital Image Correlation (DIC)[END_REF][START_REF] Garrido | Effect of service temperature on the shear creep response of rigid polyurethane foam used in composite sandwich floor panels[END_REF].

The values of the shear modulus of the balsa wood panel and the paper honeycomb are 1.5 and 3.8 times higher according to the (L,T') shear plane than to the (T,T') plane, respectively. This difference was already observed on balsa wood in the literature [START_REF] Osei-Antwi | Shear mechanical characterization of balsa wood as core material of composite sandwich panels[END_REF]. The maximum shear stress also depends on the loading planes, with a strength 1.6 times higher in the (L,T') shear plane than in the (T,T') plane for balsa wood. Concerning the paper honeycomb, the strength is 2.2 times higher in the (L,T') shear plane than in the (T,T') plane. The shear strength of the balsa is significantly higher than that of the other core materials. For the rPET foam, the shear modulus and strength are similar in the two planes.

Figure 5 presents the evolution of the apparent shear modulus as a function of the stress for each material according to the two studied shear planes. The behaviour of the materials is nonlinear. After a first phase in which the apparent rigidity is almost constant or marked by a slight decrease, a clear break in slope is observed. The yield stresses according to the shear planes (L,T') and (T,T') are measured at approximately 1 and 0.5 MPa for the balsa wood panel, respectively, 0.3 and 0.15 MPa for the paper honeycomb and 0.3 MPa in both plane for the rPET foam.

After the yield point, the apparent shear modulus decreases progressively due to damage propagation. Figure 4 shows damage phenomena such as shear buckles or deflection of honeycomb cells as already seen in the work of Pohl [START_REF] Pohl | Strengthened corrugated paper honeycomb for application in structural elements[END_REF]. The failure of the rPET and balsa panel is initiated by the appearance of cracks at 45°.

Time-delayed behaviour

The time-delayed behaviour of the core materials is evaluated by using creep/recovery tests under ambient environment (23 °C-50% RH). Stress levels below and above (or close) the yield points have been here considered.

Figure 6 presents the viscoelastic compliance (defined as the ratio between the time-delayed angular distortion and the nominal shear stress) during the creep stage. Under constant loading, the deformation of the material increases rapidly, at first. This phase is generally called the primary creep stage for viscoelastic materials. This is followed by a secondary creep phase characterized by an almost constant strain rate. For the rPET and balsa wood panels, the viscoelastic compliance varies as a function of the stress level. Regardless of the shear stress level, the balsa wood panel presents the lowest time-delayed strain levels with viscoelastic compliance at the end of the creep stage equal to 1.1.10 -3 and 2.4.10 -3 MPa -1 for a shear stress of 0.5 MPa and 1 MPa, respectively. rPET exhibits a less pronounced time-delayed behaviour when subjected to a load lower than the yield stress (0.3 MPa) with a viscoelastic compliance equal to 5.1.10 -3 MPa -1 at 0.2 MPa compared to 20.4.10 -3 MPa -1 at 0.4 MPa. The timedelayed behaviour of the paper honeycomb is quite similar regardless of the level of solicitation. Although lighter than the other core materials (see Table 2), the paper honeycomb has the highest time-delayed strain under creep solicitation. The viscoelastic compliance at the end of the creep stage is equal to 34.8.10 -3 MPa -1 and 28.9.10 -3

MPa -1 at 0.27 and 0.15 MPa, respectively.

Other features were also extracted from the creep/recovery tests. They are synthetized in Table 4, Table 5 and Table 6 for the balsa wood, the paper honeycomb and the rPET foam, respectively. The irreversible nature of the behaviour is more pronounced in the paper honeycomb samples. For a nominal stress corresponding to 38% of the quasi-static shear strength, the residual strain is equal to 1.1.10 -3 against 1.2.10 -4 for the rPET and the balsa wood panel, respectively, both subjected to a 36% of the quasi-static shear strength. For the wood and foam samples, the instantaneous shear modulus is of the same order of magnitude as the one determined during the quasi-static tests.

In contrast, for paper honeycomb samples, the instantaneous modulus (measured during the loading phase of the creep stage) is approximately 22% higher than the elastic modulus measured during the quasi-static tests. This is due to the difference in loading rate between the two experiments (0.03 MPa.s -1 for quasi-static tests against 0.14 MPa.s -1 for creep/recovery test). This feature also highlights the strong dependence of the paper honeycomb on the loading rate.

The results also show that the strain rate in the secondary creep stage (ε̇c reep ) increases with the stress level. For paper honeycomb, it is twice as high at 0.27 MPa than it is at 0.15 MPa. Regardless of the material, the time delayed strain during the creep stage is higher than that measured during the recovery stage, therefore indicating the presence of an irreversible mechanism, such as damage and deformation (collapse and buckles of cell units), during the creep part. For balsa, the time delayed strain is twice larger during creep than under recovery.

Impact of severe environmental conditions on the monotonic shear behaviour

The monotonic shear behaviour of the core materials is also studied under the more severe hygrothermal condition, EC2. The stress-strain curves and the failure profiles along the shear planes (L,T') and (T,T') are presented in Figure 7. The evolution of the tangent shear modulus as a function of the stress is given in Figure 8. The quasistatic behaviour of all the core materials is first linear until a yield stress, whose values are shown in Table 7. The tangent shear modulus then decreases until the failure of the specimen. Compared to environmental conditions EC1, a decrease in the yield stress is also observed. In particular, the yield stress of the balsa core is 40% and 30% lower under the severe environment in the shear planes (L,T') and (T,T'), respectively. The evolution of the stress as a function of the strain is like the one observed under ambient conditions. The failure profiles are presented in Figure 7. The shear failures of the paper honeycomb samples are identical to those obtained at room temperature.

Although cracks at 45° appear, the failure of the balsa is also due to the detachment of the balsa blocks glued together by a polyvinyl acetate base adhesive [START_REF] Osei-Antwi | Shear mechanical characterization of balsa wood as core material of composite sandwich panels[END_REF]. Concerning the rPET foam, the samples tested in the plan (L,T')

do not present cracks as under ambient conditions. The tests are stopped due to the detachment of the material from the metal parts and, for one sample, due to the limit measure of the LVDT sensor. The detachment of the sample is also observed in the other shear plane. For the sample with the highest maximum stress, a crack at 45° is responsible for the failure. On the basis of these observations, the measured maximum stress thus gives the load threshold that should not be exceeded before skin/core separation and/or core failure in a sandwich structure consisting of an rPET foam core. According to the shear plane (L,T'), the shear modulus of balsa, paper honeycomb and rPET is measured at 88 MPa, 31 MPa and 10 MPa respectively. The shear stiffness is therefore 28%, 37% and 50% lower than that in the ambient environment. For the other shear planes, this decrease is 38%, 86% and 37% for the wood, honeycomb and foam, respectively. Considering that the moisture content in the core materials (see Table 2) is not very different under environments EC2 and EC1, the decrease in the shear stiffness is assumed to be mainly related to the temperature variation. For balsa wood, the mechanical properties of the material are mainly driven by the S2 layer of the cell wall, which is made up of cellulose microfibrils embedded in a matrix of lignin and hemicelluloses [START_REF] Borrega | Mechanics of balsa (Ochroma pyramidale) wood[END_REF]. The decrease in the shear properties of the balsa wood panel under environment EC2 is attributed to the softening of the hemicellulose at 70°C due to the changes from a rigid to a viscous state [START_REF] Goodrich | High-temperature mechanical properties and thermal recovery of balsa wood[END_REF]. The decrease in stiffness of the paper honeycomb is principally due to the activation of the viscoelastic properties of the paper constituents by temperature. The values of modulus measured for the rPET are in accordance with those measured by Garrido et al. [START_REF] Garrido | Effects of elevated temperature on the shear response of PET and PUR foams used in composite sandwich panels[END_REF]. Additionally, the decrease in this property, compared to ambient conditions, is due to (i) the plasticization of the polymer by the water molecule causing a decrease in the glass temperature transition [START_REF] Jabarin | Effects of water absorption on physical properties and degree of molecular orientation of poly (ethylene terephthalate)[END_REF][START_REF] Kamiya | Sorptive dilation of polysulfone and poly(ethylene terephthalate) films by high-pressure carbon dioxide[END_REF] and (ii) a glass transition temperature of PET close to 70 °C [START_REF] Chen | Glass transition temperature of copolyesters of PET, PEN and PHB as determined by dynamic mechanical analysis[END_REF][START_REF] Wang | Study of the glass transition temperature and the mechanical properties of PET/modified silica nanocomposite by molecular dynamics simulation[END_REF]. A decrease in the maximum shear stress is also observed for the core materials under a more severe environment. In particular, the shear stress for the balsa samples on the (L,T) plane is 30% lower at 23 °C than at 70 °C. Conversely, for this material, the strain at failure is twice as high

Impact of severe environmental conditions on the creep shear behaviour

The time-delayed behaviour of the three core materials is also investigated under the environmental conditions EC2. Figure 9 shows the evolution of the shear strain of the materials as a function of time during the creep/recovery test for the two studied environmental conditions. The mechanical characteristics are extracted from the raw experimental data and are shown in Table 4, Table 5 and Table 6 (balsa wood panel, paper honeycomb and rPET foam, respectively). As observed at ambient environment, the instantaneous shear modulus of the balsa wood panel and rPET foam is on the same order of magnitude as the one obtained from the quasi-static test. For the paper honeycomb, this engineering constant is higher than the one measured by the monotonic test. With the increase in severity of the environment, the irreversible character of the material is more pronounced. The residual shear strain is 5.3, 4.6 and 72 times higher than that at 23 °C for the balsa, paper honeycomb and rPET, respectively.

Additionally, the time-delayed strains of the balsa wood panel, paper honeycomb and rPET foam determined during the recovery stage are 41%, 69% and 25% lower than those obtained during creep, respectively. For the same stress level, the maximum and the time-delayed strains during creep are larger under severe environment, regardless of the type of material considered. Specifically, for the balsa wood, these two mechanical parameters are twice and five times larger than those measured in ambient conditions, respectively. The strain rate related to the paper honeycomb during the secondary creep stage also appears to be higher at 70 °C, with a value equal to 3.1.10 -7 s -1 at ambient environment and 7.1.10 -7 s -1 in a severe environment. For the rPET foam, the decrease in the mechanical properties is explained by the fact that the temperature is close to the glass transition temperature.

In view of the EMC decrease of the paper honeycomb and balsa wood under the EC2 environmental conditions (see Table 2), the decrease of the material the time-delayed behaviour is largely due to the higher temperature.

Thus, the increase in the time-delayed strain can be explained by the activation of the viscoelastic properties of wood and paper with temperature [START_REF] Placet | Viscoelastic properties of green wood across the grain measured by harmonic tests in the range 0-95°C: Hardwood vs. softwood and normal wood vs. reaction wood[END_REF]. Figure 10 represents the viscoelastic compliance of the core materials tested under the severe environment. The balsa wood allows a lower time-delayed strain compared to the other materials with a mean compliance value of 5.9.10 -3 MPa -1 . The increase in the temperature on the rPET drastically increases the viscoelastic compliance with a value of 5.5.10 -1 MPa -1 at 70 °C compared to 5.1.10 -3 MPa -1 at 23 °C.

Viscoelastic behaviour

The previous paragraphs have described the mechanical behaviour of the different materials observed during the creep/recovery tests, which involve the presence of elasticity, viscoelasticity and irreversible mechanisms within the cores considered in this work. The present section focuses on the specific viscoelastic behaviour from the recovery phase. The identification of the 3D viscoelastic model parameters allows evaluating in details the influence of the stress level and the hygrothermal conditions on the viscoelastic behaviour of the cores. The identified parameters of the viscoelastic model are presented in Table 8 for all the core materials and tested stress levels in an ambient environment. The values of the cost function are comprised between 1.10 -8 and 5.10 -5 . The associated Gaussian distribution of the weighting parameters as a function of the release times for each sample of core materials is presented in Figure 11. For the balsa wood panel, the results highlight that two families of Gaussian distributions can be distinguished for the tested loading levels. The two populations are clearly linked to the stress level. The mixing of some curves owing to the same stress level between the two clusters is likely due to the variation of the yield stress as a function of the specimens cut in the balsa wood panel. Although a significant variation was measured from one sample to one another, the results show the robust dependence of the viscoelastic behaviour on the stress level [START_REF] Reichel | Hygro-mechanically coupled modelling of creep in wooden structures, Part I: Mechanics[END_REF]. The mean value of the logarithm of the release time is higher at 1 MPa than at 0.5 MPa. This means that the primary creep phase lasts longer at 1 MPa than at 0.5 MPa. Additionally, the shear viscous parameter is twice larger at 1 MPa than at 0.5 MPa.

Two different types of Gaussian distributions can also be observed in the case of the rPET foam with a mean value of the parameter ln (𝜏 1 ) equal to 3.9 and 5.6 at 0.4 MPa and 0.2 MPa, respectively. As for balsa, the recovery viscoelastic behaviour of the rPET is characterised by one type of continuum spectrum for a stress level below the yield stress and another for a loading above this threshold. The viscous parameter is also more than twice higher for the stress level above the yield stress, than below. Opposite to the two other core materials, the viscoelastic behaviour of the paper honeycomb does not depend on the stress level. The values of the release time logarithm, rigidities and viscous parameter are similar for the different stress levels adopted. Moreover, all these parameters are larger than those of the rPET foam and the balsa wood panel. In particular, the shear viscous parameter is equal to 2.3 and 2.4 at 0.27 MPa and 0.15 MPa, respectively. Thus, the paper honeycomb is characterized by a higher viscoelastic strain than the other materials. The viscoelastic behaviour is also evaluated under the environmental conditions EC2. The identified model parameters are given in Table 9. The associated Gaussian distribution of the weighting parameters as a function of the release times for each sample of core materials is given in Figure 12.

For all the materials, the value of the shear viscous parameter increases with the severity of the environmental conditions. This material parameter is 5.4 higher than the one at ambient environment for the balsa wood panel, and even goes from a value of 0.06 to 3.28 in the case of the rPET foam. The Gaussian spectra are plotted in Figure 11 and compared with those obtained at EC1. Interestingly, when the wood is solicited at 0.5 MPa and the foam at 0.2 MPa, the Gaussian distributions at EC2 are similar to those at EC1 at 1 MPa for the balsa and 0.4 MPa for the rPET. The two studied stress levels under the severe environment are above the yield stress, and it seems that the viscoelastic behaviour of the rPET and the balsa is characterised by one distribution of weighting coefficients as a function of relaxation times below the yield stress and another after.

In the case of paper honeycombs, the Gaussian distribution appears to be not dependent on the environmental conditions.

CONCLUSIONS

The present paper focuses on the influence of hygrothermal conditions on the creep/recovery shear behaviour of three eco-friendly core materials: balsa wood, paper honeycomb and rPET foam. The monotonic and time-delayed behaviour are first studied under an ambient environment (23 °C-50% RH) and then under a more severe environment (70 °C-65% RH). Finally, the viscoelastic behaviour of the materials is identified using a 3D anisotropic model from the recovery phase under the two studied hygrothermal conditions.

This study highlighted the following points:

▪ A change of the environment only leads to small water uptake in the rPET foam and a small decrease of the moisture content in the balsa wood and the paper honeycomb. The decrease in the mechanical properties is therefore attributed to the higher temperature associated to those environements.

▪ Under ambient conditions, the balsa wood panel is the core material with the highest shear modulus and strength. When the loading plane is the one with the largest stiffness, the paper honeycomb performs better than the rPET foam.

▪ At room temperature, the balsa wood panels exhibits the lower time-delayed shear strain. The creep behaviour of the balsa wood panel and the rPET foam is dependent on the stress level. For a solicitation below the yield stress, the viscoelastic compliance of the rPET foam is lower than that of the paper honeycomb.

▪ A decrease in shear modulus, yield stress and shear strength is observed under the more severe environment for the three different core materials. This decrease is more pronounced in the case of the rPET foam.

▪ The maximum shear strain, the time delayed strain and the residual strain increase for all the core materials considered with the severity of the hygrothermal conditions. In the case of the rPET, this decrease is attributed to the PET glass transition. For the case of the two biobased materials, the temperature increase induces the softening of the cell wall constituents, in particular the hemicelluloses.

▪ The balsa wood exhibits the best creep resistance under the two tested hygrothermal conditions. The level of residual strain of this material in particular is lower than the one of the other core materials.

▪ The viscoelastic material release times and the shear viscous parameter of the balsa wood and the rPET foam depend on the stress level and the hygrothermal conditions. In particular, the post-creep viscoelastic behaviour of these two materials is characterized by one distribution of the release times for stress levels below the yield stress, and another distribution for those above. Unlike the viscous parameter, the release times of the viscoelastic behaviour of the paper honeycomb are not dependent on the test stress levels and the environmental conditions.

▪ The post-creep viscoelastic behaviour of the paper honeycomb is characterised by a higher viscoelastic strain than the two other materials, due to the larger values of viscous parameters related to this biobased honeycomb.

  below and above or close to the yield stress: (i) 1 and 0.5 MPa for the balsa wood, (ii) 0.27 and 0.15 MPa concerning the paper honeycomb and (iii) 0.4 and 0.2 MPa about rPET foam. This behaviour was also studied under environment EC2 and compared to that obtained at EC1 only for the stress at 0.5 MPa, 0.15 MPa and 0.2 MPa. The same machine and shear assembly as for the monotonic tests was used. The load path was divided into two parts: one hour of creep and one hour of recovery. The PMMA box and the Inec 70-90 relative humidity

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 7 .Figure 8 .Figure 9 .Figure 10 .Figure 11 .Figure 12 .

 123456789101112 Figure 1. Images of the balsa wood a), paper honeycomb b) and rPET foam c) at a macroscopic scale (left column), with their associated coordinate systems, and at microscopic scale following the plane (L,T') or (L,T) (right column)

Table 1 .

 1 Bonding conditions and adhesives selected for the different materials and environmental conditions

		Tests
	Material	environmental	Adhesives	Type of bonding
		conditions
		EC1	Sicomin ® GreenPoxy 56	Themocompression 1 h/130 °C/0.25 bar
	Balsa wood	EC2	Epoxy ® Loctite EA 9492	Bonding with clamps at least 24 h at 70 °C-65% RH
	Paper honeycomb	EC1 EC2	Sader ® expanding polyurethane glue Sicomin ® PB170	Bonding with clamps at least 48 h Themocompression 1 h/130 °C/0.25 bar
	rPET foam	EC1 EC2	Sicomin ® GreenPoxy 56	Themocompression 1 h/130 °C/0.25 bar Themocompression 1 h/80 °C/0.25 bar

Table 2 .

 2 Density at 23 °C-50%RH and equilibrium moisture content under environmental conditions EC1 and EC2

	Material	Density [kg.m -3 ] at 23 °C -50% RH	𝑀𝐶 𝐸𝐶1 [%]	𝑀𝐶 𝐸𝐶2 [%]
	Balsa wood	97±6	9.2±0.1	7.4±0.1
	honeycomb	47±2	7.3±0.2	6.0±0.2
	rPET foam	80±1	0.66±0.03	0.76±0.06

Table 3 .

 3 Mechanical properties of core materials for shear solicitation according to the shear planes (L,T') and (T,T') under the EC1 environment (mean ± standard deviation)

	Core material	Shear plane	Shear modulus [MPa]	Max. shear stress [MPa]	Shear strain at failure/at maximum [10 -3 .mm/mm] stress	Yield stress [MPa]
	Balsa wood	(L,T') (T,T')	123±6 81±9	1.4±0.2 0.9±0.3	11±1/8.2±1.0 14±2/8.3±1.5	1 0.5
	Paper	(L,T')	49±8	0.4±0.1	-/15±6	0.3
	honeycomb	(T,T')	13±1	0.18±0.02	-/21±5	0.15
	rPET foam	(L,T') (T,T')	18±1 19±0.1	0.56±0.01 0.56±0.03	65±13/65±13 43±12/43±12	0.3 0.3

Table 4 .

 4 Mechanical parameters identified from the creep/recovery tests at different stress levels on the balsa wood in the environmental conditions EC1 and EC2 (mean ± standard-deviation)

	Stress level [MPa]	1	0.5
	Environmental condition	23 °C-50% RH (EC1)	23 °C-50% RH (EC1)	70 °C-65% RH (EC2)
	Gins [MPa]	119±8	126±11	92±13
	𝛆 𝐧𝐬 .10 -3 [-]	4.5±0.3	2.0±0.2	3.0±0.4
	𝛆 𝐝𝐞𝐥𝐜 .10 -3 [-]	1.2±0.3	0.3±0.2	1.5±0.2
	𝛆 𝐦𝐚𝐱 .10 -3 [-]	5.8±0.6	2.3±0.3	4.6±0.6
	𝜺̇𝒄 𝒑 .10 -7 [s -1 ]	0.92±0.08	0.31±0.23	1.43±0.32
	𝛆 𝐝𝐞𝐥𝐫 .10 -3 [-]	0.63 ± 0.10	0.14 ± 0.02	0.88 ± 0.17
	𝛆 𝐫𝐞𝐬 .10 -3 [-]	0.27 ± 0.19	0.12 ± 0.10	0.63 ± 0.18

Table 5 .

 5 Mechanical parameters identified from the creep/recovery tests at different stress levels on the paper honeycomb in the environmental conditions EC1 and EC2 (mean ± standard-deviation)

	Stress level [MPa]	0.27	0.15
	Environmental condition	23 °C-50% RH (EC1)	23 °C-50% RH (EC1)	70 °C-65% RH (EC2)
	Gins [MPa]	60±6	54±10	41±5
	𝛆 𝐧𝐬 .10 -3 [-]	2.4±0.2	1.5±0.2	2.1±0.3
	𝛆 𝐝𝐞𝐥𝐜 .10 -3 [-]	4.8±2.9	2.2±1.1	7.4±2.0
	𝛆 𝐦𝐚𝐱 .10 -3 [-]	7.2±3.0	3.7±1.0	9.5±0.3
	𝜺̇𝒄 𝒑 .10 -7 [s -1 ]	6.2±4.4	3.1±1.5	7.1±2.1
	𝛆 𝐝𝐞𝐥𝐫 .10 -3 [-]	1.9 ±0.7	1.1±0.7	2.3±0.3
	𝛆 𝐫𝐞𝐬 .10 -3 [-]	2.7 ± 2.0	1.1 ± 1.3	5.1 ± 1.4

Table 8 .

 8 Values of the viscoelastic model parameters identified from tests realized under environmental condition EC1 (mean ± standard deviation)

	Core material	Stress level [MPa]	𝐥𝐧 (𝝉 𝟏 )	SD	Shear viscous parameter (𝛽 𝐿𝑇′ or 𝛽 𝑇𝑇′ )
	Balsa wood	1 0.5	5.1±0.5 3.8±1.6	3.4±0.8 2.9±1.8	0.15±0.03 0.07±0.01
	Paper honeycomb	0.27 0.15	12.2±1.9 13.0±3.8	6.4±0.3 6.0±1.7	2.3±0.3 2.4±0.3
	rPET foam	0.4 0.2	3.9±0.3 5.6±0.6	3.3±0.4 1.9±0.2	0.14±0.02 0.06±0.01

Table 9 .

 9 Values of the viscoelastic model parameters identified from tests realized under environmental condition EC2 (mean ± standard deviation)

	Core material	Stress level [MPa]	𝐥𝐧 (𝝉 𝟏 )	SD	Shear viscous parameter (𝛽 𝐿𝑇′ or 𝛽 𝑇𝑇′ )
	Balsa wood	0.5	5.6±0.3	4.0±0.4	0.38±0.02
	Paper honeycomb	0.15	10.6±0.6	5.3±0.5	3.18±0.42
	rPET Foam	0.2	4.3±0.1	2.73±0.02	3.48±0.21

Acknowledgements

The authors would like to acknowledge the funding received from the Bio Based Industries Joint Undertaking under the European Union's Horizon 2020 research and innovation program under grant agreement No 744349-SSUCHY project. The authors would also like to thank Armacell Company for providing the rPET foam. This work has also been supported by the EIPHI Graduate school (contract "ANR-17-EURE-0002").


Maxime Savary

Svenja Hermann

Christophe Espanet

Valentin Préault

Gaël Chevallier

Pauline Butaud

Jean-François Manceau

Development of a magneto-mechanical bench and experimental characterization of magneto-rheological elastomers

Keywords: Magneto-Rheological Elastomers, Magneto-mechanical coupling, Test bench

Magneto-rheological elastomers belong to the class of smart materials whose mechanical properties can be controlled by an external magnetic field. These materials can be integrated into mechatronic systems and submitted to multiple loadings such as temperature, mechanical stress and magnetic field. Thus, the present work is dedicated to the development of a magneto-mechanical bench and on first experimental characterizations of hard magneto-rheological elastomers taking multiphysics coupling into account. Regarding the mechanical loading, the experimental setup is able to create a uniaxial tensile stress in case of low strain (< 1%) without friction effect. In regards to the magnetic loading, a magnetic circuit made of a strong permanent magnet has been designed to impose a variable and a homogeneous magnetic field strength up to 41 kA/m. Experimental analysis has been performed on silicone rubber filled with 36%vol. of NdFeB particles. The purpose was first to investigate the evolution of the Young modulus with or without magnetic field. Results obtained from measurements show that the developed test bench is able to depict the mechanical behavior and phenomena linked to rubber-like material.

I. INTRODUCTION

Magneto-rheological elastomers (MREs) are composites consisting of hard or soft magnetic particles embedded in a cross-linked elastomer with various filling ratios. They belong to the class of smart materials whose stiffness and damping can be controlled by an external stimuli as magnetic field. Due to their great flexibility and strain amplitude of the order of 10 -3 -10 -2 under a magnetic field, they can find an interest in the field of mechatronic systems such as soft-robotic or medical technologies [START_REF] Bira | A review of magnetic elastomers and their role in soft robotics[END_REF][START_REF] Ze | Magnetic memory polymers with integrated multifunctional shape manipulation[END_REF].

Magneto-mechanical coupling of MREs needs to be clearly understood for the development of mechatronic applications where MREs will be submitted to magnetic and mechanical loadings at the same time. Intense researches have been performed on the magnetic field-induced stiffening effect as well-known magneto-rheological effect (MR effect) [START_REF] Han | Field-stiffening effect of magnetorheological elastomers[END_REF][START_REF] Winger | Influence of the particle size on the magnetorheological effect of magnetorheological elastomers[END_REF][START_REF] Jolly | A model of the behaviour of magnetorheological materials[END_REF]. This phenomenon is defined as a variation of the Young modulus due to magnetic particles interaction and is strongly correlated to the material's microstructure [START_REF] Han | Field-stiffening effect of magnetorheological elastomers[END_REF][START_REF] Winger | Influence of the particle size on the magnetorheological effect of magnetorheological elastomers[END_REF]. The investigation of the MR effect of MREs has been largely studied in the presence and in the absence of an external magnetic field with additional mechanical stress [START_REF] Schubert | Large-strain behaviour of Magneto-Rheological Elastomers tested under uniaxial compression and tension, and pure shear deformations[END_REF][START_REF] Varga | Magnetic field sensitive functional elastomers with tunable elastic modulus[END_REF][START_REF] Bodelot | Experimental investigation of the coupled magneto-mechanical response in magnetorheological elastomers[END_REF][START_REF] Stepanov | Effect of a homogeneous magnetic field on the viscoelastic behavior of magnetic elastomers[END_REF]. These researches are mainly focused on soft MRE (soft magnetic particles embedded in the elastomeric matrix) and in the case of large strain. The widely test method used consists of a pure shear stress mode [START_REF] Schubert | Large-strain behaviour of Magneto-Rheological Elastomers tested under uniaxial compression and tension, and pure shear deformations[END_REF][START_REF] Stepanov | Effect of a homogeneous magnetic field on the viscoelastic behavior of magnetic elastomers[END_REF]. Other modes, as well as tensile stress [START_REF] Schubert | Large-strain behaviour of Magneto-Rheological Elastomers tested under uniaxial compression and tension, and pure shear deformations[END_REF][START_REF] Bodelot | Experimental investigation of the coupled magneto-mechanical response in magnetorheological elastomers[END_REF][START_REF] Stepanov | Effect of a homogeneous magnetic field on the viscoelastic behavior of magnetic elastomers[END_REF] and compression stress test [START_REF] Schubert | Large-strain behaviour of Magneto-Rheological Elastomers tested under uniaxial compression and tension, and pure shear deformations[END_REF][START_REF] Varga | Magnetic field sensitive functional elastomers with tunable elastic modulus[END_REF] are less widespread. Regarding the magnetic loading, the use of a pair of strong permanent magnet is not suitable due to non-uniform magnetic field [START_REF] Schubert | Large-strain behaviour of Magneto-Rheological Elastomers tested under uniaxial compression and tension, and pure shear deformations[END_REF]. Other solutions consist of using an electromagnet [START_REF] Varga | Magnetic field sensitive functional elastomers with tunable elastic modulus[END_REF][START_REF] Bodelot | Experimental investigation of the coupled magneto-mechanical response in magnetorheological elastomers[END_REF] or a coil [START_REF] Varga | Magnetic field sensitive functional elastomers with tunable elastic modulus[END_REF], but a cooling system is required to limit overheating. The present work deals with the development of an innovative magnetomechanical test bench and first experimental characterization performed on a hard MRE in the case of low uniaxial strain (<1%) and for a high and homogeneous magnetic field strength. We describe the experimental setup in the next section and then, in section III, we present the first experimental results.

II. EXPERIMENTAL SETUP

An overview of the test rig is presented in Fig. 1. Concerning the mechanical part, the method used consists of a uniaxial tension setup. Due to non-linearities of the mechanical behavior in case of large deformation, strain is limited to a maximum value of 1%. Thus, the challenge is to generate a mechanical loading without friction effect while controlling the stress induced by imposed displacement. The system assembly is composed by a fixed clamp (part 1) and a mobile clamp (part 3) mounted in series with a force sensor (part 6) and a micrometric movable stage (part 5compact 5-axis Stage TM ).

The displacement of mobile clamp, is performed by the micrometric movable platform on which a load cell sensor (part 6 -S100-005 TM ) is fixed by an aluminum holder (part 8). Thus, the beam of the stress controller (part 6), due to its high rigidity (E ~ 200 GPa), is able to impose the displacement of the movable clamp and thus the one of the MRE. An optical position sensor (part 7 -optoNCDT-1420 TM ) has been added into the experimental setup to monitor the displacement of the tested sample from the intermediate stage (part 8). All mechanical parts mentioned are in a POM-C copolymer for parts 2, 3, 6 and 10, in brass for rods assembly (part 9) and in a stainless steel for the force sensor (part 6). The fact that those components are non-magnetic ones helps to improve the magnetic field uniformity in the sample and to reduce the magnetic flux dispersion. Measurement of the displacement (u) and the force (F) are achieved by an oscilloscope. Afterwards, datas are analyzed with using a Python code. Concerning the magnetic part (Fig. 2), loading is performed by a magnetic circuit composed of a static part made of pure iron (part 1) in which a strong permanent Neodymium Iron Boron magnet (Br = 1.3 T) is enclosed (part 3). To apply a variable magnetic field strength, the strategy consists of a variation of the air gap by the displacement of the movable part (part 2) along the vertical axis. During the magneto-mechanical test, the magnetic field close to the sample can be measured by inserting a Hall effect sensor in the test bench.

The composite, used for the characterization of the magnetomechanical coupling, is composed of a high consistent silicone rubber (Nusil TM Med 4014 -E ~ 10 6 Pa) mixed by a rolling process with a Neodymium Iron Boron powder (MQFP TM -14-12), provided by Magnequench with a particle size distribution D50 around 5µm. The magneto-mechanical behavior has been characterized for an isotropic MRE with a volumetric filling ratio of 36%. After curing process under a temperature of 116 °C during 10 minutes, samples with a parallelepipedal shape have been magnetized in a strong external magnetic field of 4 T generated by a special magnetization head. Thus, the obtained remanent flux density is 0.3 T and the magnetization direction is along the length of the sample corresponding to the direction of the mechanical and magnetic loadings.

III. EXPERIMENTAL INVESTIGATIONS

A. Mechanical characterization

The sample's compression during clamping creates a strained state due to a lateral extension and a buckling of the sample. In order to eliminate this deformation, an initial displacement is imposed to the sample. Mechanical loading is performed step by step using the linear displacement platform. A standby time of 2 minutes is required between step loading to reach a stable state due to stress relaxation. The mechanical quantities such as nominal stress (σ) and nominal strain (ε) have The results concerning mechanical loading cycles is depicted in Fig. 3 without any external magnetic field. As it can be seen, a hysteretic mechanical behavior is observed with a remanent strain around 0.2% -0.3%. This effect is well known as Mullins effect with rubber-like materials and corresponds to a cyclic stress softening [START_REF] Mullins | Stress softening in rubber vulcanizates. Part 1. Use of a strain amplification factor to describe the elastic behavior of filler-reinforced vulcanized rubber[END_REF]. Results presented here was also observed for additional tests after demounting and remounting sample in the test rig (Fig. 3b) and for additional specimen. Hence, the setup allows to perform a repeatable and reproductible mechanical loading for MRE samples.

B. Magnetic characterization

The magnetic flux density generated from the magnetic circuit in the air gap has been investigated. This measurement consists of the displacement of a Hall effect sensor in the midof the air gap by using a characterizing tool developed by the company MMT. This characterization has been performed for different air gaps controlled by the length w (Fig. 2). The magnetic field strengths, calculated from the measurement, are between 29 kA/m to 41 kA/m in the central region (Fig. 4). It can be also observed that the magnetic flux density is relatively homogenous in the central area (with an estimated variation around 1.5%) and a non symmetric magnetic field for the outer regions along y-axis due to the proximity of the permanent magnet.

C. Magneto-mechanical characterization

The first experimental analysis concerning the magnetomechanical coupling consists of the study of the elastic modulus both with and without an external magnetic field. Displacement and force signals show a strong variation during the time acquisition in which the magnetic circuit (H = 41 kA/m) is added into the setup (from 5 s to 15 s -Fig. 5). Thus, a difference of the force and the displacement of 0.07N and 5µm respectively from the stationary values depicted by green and red dots (Fig. 5) is observed. For a better understanding, the variations of the stress and the strain (respectively ∆σ and ∆ε) have been compared due to the offset induced by the magnetic field. magnetization are oriented toward the same direction. A comparison of the mechanical behavior both in the absence and in the presence of the magnetic field is made for the first cycle loading (Fig. 6a) and the second cycle loading (Fig. 6b). It can be observed that the Young modulus and its standard deviation calculated from slopes between consecutive measurement point (Table 1) are not affected by the magnetic loading. According to Borin et al. [START_REF] Borin | Tuning the tensile modulus of magnetorheological elastomers with magnetically hard powder[END_REF], a Young modulus of 200 kPa is the limit for a significant MR effect. Thus, due to the high stiffness of the elastomeric matrix used for the experiments (E ~ 10 6 Pa), the magnetic field has no significant influence neither on the particles motion nor on the Young modulus.

TABLE I COMPARISON OF THE YOUNG MODULUS FOR DIFFERENT MAGNETIC LOADING

IV. CONCLUSION Magneto-rheological elastomers exhibit a variation of their stiffness under the application of an external magnetic field due to magnetic particle interaction. A supplementary loading such as a mechanical stress can act on this magneto-rheological effect. This class of smart materials can be integrated into mechatronic systems whose magneto-mechanical properties must be clearly understood.

The present work proposed an innovative characterizing tool for the investigation of the magneto-mechanical coupling in magnetically hard MRE. The setup presented here is composed of an apparatus able to generate a mechanical strain up to 1% without friction effect. In addition, a reluctant circuit has been designed as a magnetic loading. The magnetic circuit is composed of a strong permanent magnet and a movable part which is able to reach a homogenous magnetic field in the sample area between 29kA/m and 41kA/m. The first experimental characterization has been presented in this work. Uniaxial tension tests were conducted on hard MRE in the presence and in the absence of an external magnetic field. The experimental data have shown the ability of the setup to characterize the mechanical behavior of the magnetic elastomer and common phenomena of cross-link elastomers such as Mullin's effect. The magnetic field had no significant influence on the Young modulus due to the stiffness of the matrix. Thus, these results are very promising for the design of mechatronic application for which magnetic and mechanical loading can be considered separately. Further tests must be performed on softer elastomeric matrices, for different filling ratios and polarization orientation for isotropic and anisotropic MRE to study the influence of these parameters on the magneto-mechanical coupling.

Fig. 1 :

 1 Fig. 1: (a) Overview of the experimental setup; (b) Schematic view of the functional parts of the mechanical loading. 1fixed clamp, 2 -MRE sample, 3movable clamp, 4aluminum holder, 5micrometric movable platform, 6force sensor, 7optical displacement sensor, 8intermediate platform, 9brass rods, 10mechanical frame, 11magnetic circuit

Fig. 2 :

 2 Fig. 2: (a) Magnetic reluctant circuit. 1static part, 2-mobile part, 3 -NdFeB permanent magnet; (b) MRE sample

  datas such as σ = F/S and ε = u/L0 where S and L0 refer to the cross section area (S = 8 mm 2 ) and the free length of sample respectively (L0 = 17 mm).

Fig. 3 :

 3 Fig. 3: Mechanical behavior of an MRE filled with 36%vol of NdFeB particles in absence of an external magnetic field for two consecutive mountings of the specimen (a) and (b)

Fig. 4 :

 4 Fig. 4: Cartography of the magnetic field in the mid plane for different air gaps.

Fig. 5 :

 5 Fig. 5: Evolution of (a) the displacement and (b) the force due to the addition of the magnetic circuit to the setup.The magneto-mechanical coupling experiments have been conducted only with an external magnetic field of 41 kA/m (w = 0 mm). The direction of the magnetic field and sample's

Fig. 6 :

 6 Fig. 6: Comparison of the mechanical behavior in the presence and in the absence of an external magnetic field (a) first loading cycle; (b) second loading cycle.