Eranezhuth Wasan Awin

Abhijeet Lale

K C Hari Kumar

Samuel Bernard

Ravi Kumar

Disordered mesoporous polymer derived N-doped TiO 2 /Si-O-C-N nanocomposites with nanoscaled heterojunctions towards enhanced adsorption and harnessing of visible light

The mesoporous N-doped TiO 2 /Si-O-C-N ceramic nanocomposites has been revealed to be a potential candidate Mesoporous towards visible light photocatalytic degradation of organic dyes. The polymer-derived ceramic route was im-Nanocomposite plemented to prepare uniformly distributed in-situ crystallized N-doped TiO 2 nanocrystals in a mesoporous Photocatalysis amorphous siliconoxycarbonitride matrix. This chemical approach assisted by the hard template pathway re-Adsorption sulted in a high surface area (186 m 2 /g) nanocomposite exhibiting predominantly mesoporous structure with an Heterojunction average pore size of 11 nm. The two-step process involved pyrolysis of the polyhydridomethyilsiloxane impregnated in CMK3 (hard template) under argon generating SiOC-C composites and functionalizing it with titanium n-tetrabutoxide to be pyrolyzed under ammonia to form the titled nanocomposite. Interestingly, pyrolysis in a reactive ammonia atmosphere resulted in the incorporation of nitrogen in the titania lattice while decomposing the template. The Si-O-C-N support on which N-doped TiO 2 exhibited superior adsorption of organic dye molecules and photocatalytically active in the visible wavelength. The nanoscaled heterojunctions reduced the recombination rate and the presence of superoxide anions/hydroxyl radicals was found to be responsible for the dye degradation.

Introduction

Titania (TiO 2 ), known for its photocatalytic applications ever since its discovery by Fujishima and Honda [START_REF] Fujishima | Electrochemical photolysis of water at a semiconductor electrode[END_REF] has attracted a lot of interest in the research community ranging its application from hydrogen pro-duction and water purification to solar cells [START_REF] Anpo | The design and development of highly reactive titanium oxide photocatalysts operating under visible light irradiation[END_REF][START_REF] Tang | Mechanism of photocatalytic water splitting in TiO2. Reaction of water with photoholes, importance of charge carrier dynamics, and evidence for four-hole chemistry[END_REF][START_REF] Pavasupree | Synthesis, characterization, photocatalytic activity and dye-sensitized solar cell performance of nanorods/nanoparticles TiO2 with mesoporous structure[END_REF][START_REF] Houas | Photocatalytic degradation pathway of methylene blue in water[END_REF]. The low cost, oxi-dizing power and durability of TiO 2 has made it a material of interest. However, the band gap of TiO 2 of 3.2 eV restricts itself from harnessing of light in the visible wavelength thereby limiting its photocatalytic activity in the UV region [START_REF] Zhang | New understanding of the diff erence of photo-catalytic activity among anatase, rutile and brookite TiO2[END_REF]. Several methods have been explored to-wards the reduction of band gap of TiO 2 which includes doping of TiO 2 with non-metals such as C, N, F, P and S [START_REF] Luo | Photo-catalytic conversion of oxygenated hydrocarbons to hydrogen over heteroatom-doped TiO2 catalysts[END_REF][START_REF] Xu | Simple fabrication of twist-like helix N, S-codoped titania photocatalyst with visible-light response[END_REF][START_REF] Yu | Eff ects of F-doping on the photocatalytic activity and microstructures of nanocrystalline TiO2 powders[END_REF][START_REF] Chen | The electronic origin of the visible-light absorption properties of C-, N-and S-doped TiO2 nanomaterials[END_REF]. The introduction of dopants generates an impurity state above the valence band of TiO 2 , hence raising the valence band edge, resulting in the absorption of visible light. Ever since Asahi et al. reported the reduction of band gap in nitrogen-doped TiO 2 (N-TiO 2 ) and improved absorption of solar light [START_REF] Asahi | Visible-light photocatalysis in nitrogen-doped titanium oxides[END_REF], various approaches have been adopted towards synthesis of Ndoped TiO 2 photocatalysts. This includes sol-gel reaction [START_REF] Li | Palladium oxide nanoparticles on nitrogendoped titanium oxide: accelerated photocatalytic disinfection and post-illumination catalytic "memory[END_REF],

Corresponding author. E-mail address: nvrk@iitm.ac.in (R. Kumar).

solvothermal reaction [START_REF] Yin | Synthesis of excellent visiblelight responsive TiO2-xNy photocatalyst by a homogeneous precipitation-sol-vothermal process[END_REF][START_REF] Yin | Visible-light-induced photocatalytic activity of TiO2-xNy prepared by solvothermal process in urea-alcohol system[END_REF] and hydrothermal reaction [START_REF] Yin | Visible-light induced photocatalytic activity of TiO2-xAy (A = N, S) prepared by precipitation route[END_REF]. Re-cently, titanium oxynitride (Ti-O-N) microspheres were synthesized by ammonolysis of TiO 2 microspheres and were reported to exhibit visible light photocatalytic activity [START_REF] Yin | Visible-light induced photocatalytic activity of TiO2-xAy (A = N, S) prepared by precipitation route[END_REF]. Visible light photocatalytic activity of mesoporous N-TiO 2 possessing high surface area was also reported to absorb light in the visible light regime [START_REF] Yin | Visible-light induced photocatalytic activity of TiO2-xAy (A = N, S) prepared by precipitation route[END_REF].

The retrieval of dispersed photocatalysts from aqueous solution as well as the agglomeration of TiO 2 nanoparticles hinders the im-plementation of the material for water purification applications. Anchoring/immobilizing the photocatalysts on a suitable substrate is an alternative way to overcome the above mentioned disadvantage. TiO 2 immobilized on high surface area mesoporous SBA-15 was shown to act as an active photocatalyst [START_REF] Tseng | Synthesis of TiO2/SBA-15 photocatalyst for the azo dye decolorization through the polyol method[END_REF]. The mesoporous substrate with high surface area plays a prominent role in determining the adsorption characteristics, thereby influencing the overall reaction rate. Precursor routes such as the Polymer-Derived Ceramics (PDCs) approach enable controlling the size, composition, porosity and shape of the final pro-duct [START_REF] Ionescu | Silicon-containing polymer-derived ceramic nanocomposites (PDC-NCs): preparative approaches and properties[END_REF][START_REF] Zaheer | Polymer derived non-oxide ceramics modified with late transition metals[END_REF][START_REF] Bernard | Ordered mesoporous polymer-derived ceramics and their processing into hierarchically porous boron nitride and silicoboron carbonitride monoliths[END_REF]. The amorphous Si-O-C-N possess enhanced properties such as thermal stability, creep resistance, chemical stability and oxi-dation resistance [START_REF] Ionescu | Silicon-containing polymer-derived ceramic nanocomposites (PDC-NCs): Preparative approaches and properties[END_REF][START_REF] Colombo | Polymer-derived ceramics: 40 years of research and innovation in advanced ceramics[END_REF][START_REF] Riedel | Silicon-based polymer-derived ceramics: synthesis properties and applications-a review dedicated to Prof. Dr. Fritz Aldinger on the occasion of his 65th birthday[END_REF]. Hence, Si-O-C-N ceramics in the form of composites, fibers, coatings and membranes off ers to be a potential candidate in extreme/harsh operating conditions. For instance, ex-cellent adsorption capacity was shown by hierarchically porous silicon-carbon-nitrogen (Si-C-N) ceramics prepared from preceramic polymers in contrast to its bulk counterparts [START_REF] Meng | Hierarchically porous silicon-carbonnitrogen hybrid materials towards highly efficient and selective adsorption of or-ganic dyes[END_REF]. Such materials were shown to be highly efficient in removing organic pollutants. The template as-sisted PDCs approach results in porous structures with high stability, surface area and tailorable pore size [START_REF] Shi | Ordered mesoporous SiOC and SiCN ceramics from atmosphere-assisted in situ transformation[END_REF]. In the current study, a two-step template assisted PDCs approach has been adopted to synthesize uniformly distributed N-doped TiO 2 anchored on an amorphous Si-O-C-N matrix possessing high surface area and disordered mesoporous structure, especially by tailoring the chemistry of precursors and using their reactivity with ammonia as a reactive atmosphere. The resulting material was evaluated for its photocatalytic activity towards de-gradation of methylene blue in visible light.

Experimental section

2.1. Synthesis of the mesoporous nanocomposite 0.2 g of mesoporous carbon (Sigma-Aldrich, Bangalore, India) was kept at 100 °C overnight to remove the moisture present in the pores. Subsequently 1 ml of polyhydridomethyilsiloxane (PHMS, Sigma-Aldrich, Bangalore, India) was impregnated into mesoporous carbon by stirring the mixture in vacuum for 24 h. The mixture was then trans-ferred to an alumina crucible and pyrolyzed at 1000 °C in argon at-mosphere at a heating rate of 2 °C/min and held for 2 h. The pyrolyzed product, SiOC-C composite was then mixed with 1 ml of titanium n-tetrabutoxide (TB, Sigma-Aldrich, Bangalore, India) stirred overnight and subsequently pyrolyzed in ammonia atmosphere at 500 °C for 10 h maintaining a heating rate of 5 °C/min.

Characterization

The powder X-ray diff raction (XRD, D8 Discover, Bruker AXS X-ray diff ractometer, USA) was carried out to understand the phase evolution. The Raman spectrum (Labram HR800, Horiba, Japan) was recorded in the range of 100 and 2000 cm -1 using a He-Ne laser source. The sur-face morphology as well as the size distribution of the particles was studied in assistance with scanning electron microscopy (SEM, Hitachi S-4800, Japan) and transmission electron microscopy (TEM, JEOL 3010, Japan). The sample was sputter coated for 80 s prior to SEM analysis so as to avoid the charging eff ects. To perform the TEM ana-lysis, the sample was dissolved in acetone, ultrasonicated for 15 min and a couple of drops were poured on the copper grid. In order to quantify the porosity, the sample was degassed at 150 °C for 6 h and adsorption-desorption isotherm was obtained at -196 °C using BET analyzer (TriStar 3020, Micromeritics, USA). The pore size distribution was determined using Barrett-Joyner-Halenda (BJH) method. The XPS measurements (Omnicron nanotechnology, ESCA-14, Germany) were carried out using Al K α (1486.6 eV) as radiation source. The deconvo-lution of the spectra was done using CASA XPS software. The calibra-tion of the spectra was done with respect to C1s peak at a binding energy value of 284.8 eV. The bonding characteristics were studied with the aid of FTIR spectroscopy (Perkin Elmer Spectrum, USA). The solid state 29 Si NMR was recorded using Bruker AV III, Germany spectrometer operating at 500 MHz.

Photocatalytic studies

To evaluate the photocatalytic activity of the nanocomposite, me-thylene blue (0.06 and 0.12 mM) was chosen as the representative dye and 50 mg of the catalyst was added to 100 ml of the solution. The mixed solution was then kept for 12 h in dark to achieve ad-sorption-desorption equilibrium. Subsequently, the solution was stirred and subjected to visible light exposure (500 W tungsten halogen lamp). At regular time intervals, 2 ml of the solution was taken out and cen-trifuged, to remove the catalyst. The concentration of the MB was measured using UV-vis spectrophotometer (ThermoFisher Scientific, Evolution 220, USA).

Results and discussion

PHMS, with a chemical structure of (CH 3 ) 3 SiO[(CH 3 )HSiO] n Si (CH 3 ) 3 was used as the molecular precursor for the synthesis of silicon oxycarbide (SiOC) matrix. The pyrolysis of PHMS in an inert atmo-sphere leads to amorphous SiOC. Herein, PHMS was infiltrated into CMK3 to replicate the nanostructured mesoporous structure by nano-casting. The functionalization of the composite with titanium was done using TB (Ti(OCH 2 CH 2 CH 2 CH 3 ) 4 ) and the template was reduced by pyrolyzing in a reactive ammonia atmosphere. The overall reaction procedure is shown in Fig. 1. The XRD pattern of SiOC-C composite before functionalization with TB was characteristic of an amorphous compound with broad peaks at 2θ ~ 23° and 43.8°. The broad hump at 23° corresponds to either amorphous SiO 2 [START_REF] Bréquel | Systematic structural characterization of the high-temperature behavior of nearly stoichiometric silicon oxycarbide glasses[END_REF] or the presence of SiOC [START_REF] Fukui | Structures and lithium storage performance of Si-O-C composite materials depending on pyrolysis tem-peratures[END_REF]. The peak at 2θ ~ 43.8° corresponds to the characteristic amorphous disordered carbons [START_REF] Subramanian | Nitrogen-and oxygen-containing activated car-bons from sucrose for electrochemical supercapacitor applications[END_REF]. The XRD of N-TiO 2 /Si-O-C-N nanocomposite (after functionalization with TB and ammonia treatment) clearly reveals the crystallization of anatase phase with major peaks at 2θ = 25.3°, 37.8° and 48.0° corresponding to (1 0 1), (0 0 4) and (2 0 0) planes of anatase, respectively (Fig. 2a). Interestingly, an additional peak was observed at 2θ ~ 43° and has been assigned to the (2 0 0) plane of titanium oxy-nitride (TiO x N y ) phase (Fig. 2b). The initiation of the crystallization of TiO x N y phase is an indication of gradual transformation of anatase phase with prolonged holding hours in ammonia atmosphere. The Scherrer equation was used to calculate the crystallite size of anatase and was found to have an average size of 5 nm.

The analysis of Raman spectra (Fig. 3) indicated the presence of six bands at 151 cm -1 (E g ), 201 cm -1 (E g ), 399 cm -1 (B 1g ), 505 cm -1 (A 1g ), 519 cm -1 (B 1g ) and 630 cm -1 (E g ) confirming the presence of anatase phase. This is in good agreement with the XRD result. The inset image shows the presence of D and G peaks at 1348 cm -1 and 1588 cm -1 respectively attributing to the existence of free carbon. The pyrolysis in reactive atmosphere such as ammonia is known to reduce the carbon entities via. transamination reactions during the initial stages of polymer to ceramic conversion [START_REF] Bechelany | Nanocomposites through the chemistry of single-source precursors: understanding the role of chemistry behind the design of monolith-type nanostructured titanium nitride/silicon nitride[END_REF][START_REF] Bechelany | In situ controlled growth of titanium nitride in amorphous silicon nitride: a general route toward bulk nitride nanocomposites with very high hardness[END_REF]. However, the high affinity of titanium atoms towards nitrogen could hinder this reaction. This could result in the formation of free carbon by the phase separation of SiOC at higher temperatures. The Raman spectra of SiOC-C compo-site (i.e., before functionalization with TB) clearly revealed the for-mation of free carbon as a result of the phase separation of SiOC. Nevertheless, the carbon template is believed to be reduced because of its decomposition under ammonia.

The scanning electron micrographs of mesoporous carbon (CMK-3) and the TiO x N y-derived materials are shown in Fig. 4a and b respec-tively. This clearly indicates that the structure of the hard template is indeed retained in the nanocomposite after nitridation process which highlight Raman discussion on the identification of carbon in our ma-terials. This also displays the fact that the polymer impregnated well in to the pores of the carbon template. The EDS spectrum of the nano-composite is shown in Fig. 4c. The EDS spectra clearly reveals the presence of silicon, oxygen, carbon, titanium and nitrogen.

The structural organization of the sample was analyzed using transmission electron microscopy (TEM). The micrograph clearly re-veals the presence of well-defined crystals that are uniformly dis-tributed on a support matrix (Fig. 5a). At higher magnification, the lattice fringe interplanar spacing of 0.35 nm represents the (1 0 1) plane of anatase (Fig. 5b). The crystals exhibit a spherical morphology with an average crystal size of 5-10 nm, which substantiated the XRD re-sults.

The nitrogen adsorption-desorption isotherm exhibit a typical type IV curve with a H2 hysteresis loop revealing the mesoporous nature of the nanocomposite as exemplified in Fig. 6. The impregnation/pyrolysis process resulted in the reduction of BET surface area from 478 m 2 /g for CMK-3 to 186 m 2 /g for the nanocomposite. The Barrett-Joyner-Halenda (BJH) pore size distribution profile is presented as an inset image and the mean pore size was found to be 11 nm. The pore volume was found to be 0.29 cm 3 /g. The surface area, pore volume and pore size of SiOC-C composite was revealed to be 350 m 2 /g, 0.45 cm 3 /g and 19 nm re-spectively. The decrease in pore volume and surface area of the me-soporous SiOC-C composite after functionalization with TB then an-nealing under ammonia confirms SEM observation and most probably the presence of the TiO 2 phase as nanocrystals identified by TEM in the porous structure of the pristine SiOC-C composite.

The incorporation of nitrogen in the material was confirmed by XPS as observed in Fig. 7. The deconvolution of N1s peak (Fig. 7b) resulted in three peaks at 397.2, 399.4 and 402.2 eV. The peak at 397.2 eV could be assigned to the NeTieN bonds implying the nitrogen ions occupying the substitutional sites and is close enough to the reported values of TiN [START_REF] Asahi | Visible-light photocatalysis in nitrogen-doped titanium oxides[END_REF][START_REF] Valentin | N-doped TiO2: theory and experiment[END_REF]. The peak at 399.4 eV corresponds to the TieNeO and TieOeN linkage which could be ascribed to the positively charged N + ions lo-cated at the interstitial site of TiO 2 lattice [START_REF] Kusano | Influence of electronic structure on visible light photocatalytic activity of nitrogen-doped TiO2[END_REF]. The substitution of N 2 in OeTieO structure results in reduced electron density around N and increase in binding energy of N1s peak in OeTieN environment in contrast to that of NeTieN. Reports suggest that both substitutional and interstitial type N evokes visible light photocatalytic activity. The presence of signal at 402 eV could be attributed to the NH 3 chemi-sorbed or molecular N 2 onto TiO 2 [START_REF] Kusano | Influence of electronic structure on visible light photocatalytic activity of nitrogen-doped TiO2[END_REF].

The XPS spectra of Ti 2p spectra resulted in peaks at binding en-ergies of 459.8 eV and 465.5 eV corresponding to Ti 2p 3/2 and Ti 2p 1/2 respectively (Fig. 7a). The diff erence in binding energy of 5.7 eV be-tween the doublets confirmed that titanium exists in Ti 4+ oxidation state. This indicates that there is negligible amount of transfer of electron from oxygen vacancies to titanium atoms [START_REF] Chen | Reaction of NH3 with titania: N-doping of the oxide and TiN formation[END_REF]. The Si 2p 1/2 peak (Fig. 7d) exhibited a signal at binding energy 103.5 eV indicating the presence of SiO 4 units on the surface of the nanocomposite [START_REF] Abass | Synthesis and electrochemical performance of a polymer-derived silicon oxycarbide/boron nitride nanotube compo-site[END_REF]. Fig. 7c shows the C1s spectra of the mesoporous nanocomposite. The main peak at 284.8 eV corresponds to CeC and the three peaks at 285.8, 287.5 and 289.4 eV corresponds to C]N, CeN and CeO bonds respectively [START_REF] Yan | Titanium oxynitride nanoparticles anchored on carbon nanotubes as energy storage mate-rials[END_REF][START_REF] Wang | One-step hydrothermal synthesis of N-doped TiO2/C nanocomposites with high visible light photocatalytic activity[END_REF]. This in turn suggests that the amorphous matrix to be composed of CeNeO bonds.

XPS clearly highlights the presence of a TiO x N y most probably supported by a SieOeC matrix . FTIR of the SiOCeC composite and mesoporous nanocomposite were also performed to understand the chemical bonding and is exemplified in Fig. 8. The FTIR spectrum of SiOCeC composite shows a peak at 1630 cm -1 corresponding to C]C bond [START_REF] Meng | Hierarchically porous silicon-carbonnitrogen hybrid materials towards highly efficient and selective adsorption of or-ganic dyes[END_REF], indicating the presence of mesoporous carbon. In the case of mesoporous nanocomposite, the broad peak at around 1000-1100 cm -1 and 600-800 cm -1 corresponds to the presence of SieOeSi bonds and TieOeTi stretching modes respectively [START_REF] Wang | Nitrogen-doped TiO2 microspheres with hierarchical micro/nanostructures and rich dual-phase junctions for enhanced photocatalytic activity[END_REF].

The peak at around 1420 cm -1 corresponds to the CeN bond [START_REF] Yan | Titanium oxynitride nanoparticles anchored on carbon nanotubes as energy storage mate-rials[END_REF]. This further confirms the incorporation of nitrogen into the amorphous matrix. Apart from this, the peak at 1210 cm -1 corresponding to NeC]O was also observed [START_REF] Wang | Preparation of nitrogen-doped TiO2/graphene nanohybrids and application as counter electrode for dyesensitized solar cells[END_REF]. Hence, it is understood that pyr-olyzing in ammonia atmosphere not only leads to the doping of ni-trogen into TiO 2 lattice but also modifies the amorphous matrix.

The local environment around silicon atoms was further in-vestigated by analyzing the NMR spectrum shown in Fig. 9. The peaks present at -100 ppm and -109 ppm could be attributed to SiO 3 (OH) and SiO 4 units, respectively. The shoulder peak at -90 ppm could be assigned to SiO 3 N units, where Si atoms share bonds with oxygen and nitrogen atoms [START_REF] Kohn | Evidence for the formation of SiON glasses[END_REF][START_REF] Hayashi | Fast and almost complete nitridation of mesoporous silica MCM-41 with ammonia in a plug-flow reactor[END_REF]. The occurrence of SiO 3 N peak implies the fact that the pyrolysis of the molecular precursor under ammonia atmo-sphere has led to the incorporation of nitrogen atoms in the Si-based network. The signal at -18 ppm corresponds to SiC 4 . In general, be-sides the presence of SiC 4 and SiO 4 signals, the silicon oxycarbide based ceramics exhibits additional resonance peaks at SiO 3 C (-70 ppm), SiO 2 C 2 (-35 ppm) and SiOC 3 (6 ppm) [START_REF] Kalfat | 29Si MAS NMR investigation of the pyrolysis process of cross-linked polysiloxanes prepared from poly-methylhydrosiloxane[END_REF]. The absence of these mixed bonds implies that the ammonolyzed sample has undergone phase se-paration at higher temperatures resulting in the formation of multi-phasic system and free carbon. The XRD and Raman spectra (Figs. 2 and3) also substantiate the aforementioned fact. Fig. 10a shows the absorption spectra of MB solution of 0.06 mM kept in dark for 12 h. It is clearly observed that due to the ultra-high adsorption capacity exhibited by the nanocomposite, most of the MB molecules gets adsorbed on to the material system which results in complete decolourization of the solution. Hence, in order to compre-hend the kinetics of photocatalytic degradation in visible light, the concentration of MB solution was doubled to 0.12 mM. The absorption spectrum of MB solution (0.12 mM) before and after exposure of visible light is exemplified in Fig. 10b. As seen from the spectrum, the con-centration of the MB solution gradually decreases with increase in irradiation time indicating the reaction of catalyst with the solution in the presence of light. The photocatalytic activity of the nanocomposite in visible light could be ascribed to the doping of nitrogen in the TiO 2 lattice.

In order to determine the photodegradation constant, Langmuir -Hinshelwood kinetic model was used which determines the photo-catalytic activity independent of the amount of MB concentration left in the solution [START_REF] Ao | A simple route for the preparation of anatase titania-coated magnetic porous carbons with enhanced photocatalytic activity[END_REF]. The experimental data was fitted using first order kinetic equation, ln(C/C o ) = -kt where C is the concentration of the MB at time t, C o is the initial concentration and k is the photo-degradation constant. The plot of ln C/C o vs time follows a first order reaction in which ln C/C o linearly decreases with time as depicted in Fig. 10c. The photocatalytic degradation constant, k(×10 -3 min -1

) was calculated to be 8.3. The UV-vis diff use reflectance spectrum of the N-TiO 2 /SiOCN nanocomposite exhibited a very strong absorption in the visible to near infra-red region as shown in Fig. 10d. However, no specific band edge was observed in the UV-DRS spectrum. This could be due to the presence of amorphous Si-O-C-N matrix present in the nanocomposite, since the Bloch theorem is no longer applicable in amorphous materials due to the lack of lattice periodicity, leaving the crystalline hk momentum unidentified [START_REF] Adachi | Optical Properties of Crystalline and Amorphous Semiconductors: Materials and Fundamental Principles[END_REF].

The BET surface area of the N-TiO 2 /SiOCN nanocomposite was found to be 130 m 2 /g after the adsorption/catalysis experiments (Fig. 11) in contrast to 186 m 2 /g for the nanocomposite before the experiments. This further proves the fact that pores play a major role in the adsorption process.

In order to investigate the active species involved in dye degrada-tion, reaction species trapping experiments were carried out (Fig. 12). The isopropanol (0.02 M), benzoquinone (0.001 M) and triethanola-mine (0.01 M) were used as sacrificial agents for scavenging hydroxyl radicals ( % OH), superoxide anion radicals ( % O 2 -) and photogenerated holes (h + ) respectively.

Fig. 12 shows the degradation efficiency of methylene blue with and without the presence of scavengers. It could be clearly understood that the methylene blue without the presence of catalyst showed highest degradation efficiency. The introduction of isopropanol drastically reduced the degradation efficiency, implying that % OH plays a significant role towards the degradation of methylene blue dye. A decrease in degradation efficiency was also observed with the addition of triethanolamine, indicating that h + also has a minor role to play in the degradation process. However, the catalyst in the presence of benzoquinone had the least eff ect on the degradation effi-ciency, manifesting that % O 2 -has minimal role to play in the catalytic degradation process.

The plausible mechanism for the visible light photocatalytic activity of N-TiO 2 /Si-O-C-N nanocomposite is shown in Fig. 13a. The crystalline form of anatase transfers the electron from the valence band to the conduction band up on irradiation of an UV light, thereby favoring the photocatalytic reaction. However, since the band gap of TiO 2 being high, the electrons fails to excite in the visible light regime. The in-corporation of nitrogen in the TiO 2 lattice, substantiated by the XPS analysis, is believed to alter the electronic band structure of TiO 2 . The doping of nitrogen atoms results in the creation of an additional N2p band above the O2p band, which results in the reduction of bang gap. Thus, instead of electron being excited from the O2p band (in pure TiO 2 ), diff usion of charge carriers occurs from the N2p band to Ti3d conduction band. The resulting electrons in the conduction band reduce with the oxygen molecules to produce superoxide ions, O 2 -. Con-comitantly, the holes in the valence band react with the water mole-cules to produce the hydroxyl radical, OH % . The aforementioned su-peroxide anions and the hydroxyl radical are believed to be responsible for the degradation of MB molecules. The heterojunction formation between amorphous Si-O-C-N and crystalline N-TiO 2 is expected to reduce the recombination rate as evident from HR-TEM micrograph (Fig. 13b). The heterojunction formation between the crystalline N-TiO 2 and amorphous Si-O-C-N matrix results in the migration of pho-togenerated electrons from the conduction band of amorphous Si-O-C-N A hard template assisted precursor derived approach was adopted to synthesize mesoporous, in-situ crystallized nitrogen doped TiO 2 in an amorphous Si-O-C-N matrix. The pyrolysis of pre-ceramic polymer in Ar/NH 3 atmosphere resulted in complete polymer to ceramic trans-formation leading to the formation of uniformly distributed nitrogen doped TiO 2 in an amorphous Si-O-C-N matrix. Pyrolysis in a reactive ammonia atmosphere has led to the inclusion of nitrogen atoms in the TiO2 lattice as well as in the amorphous matrix. The ammonia assisted pyrolysis treatment has also aided in the reduction of CMK-3 template, to the conduction band of N-TiO 2 . Similarly, the photogenerated holes will migrate from the valence band of N-TiO 2 to the valence band of amorphous Si-O-C-N.

The reactions occurring during dye degradation could be general-ized as follows:

The methylene blue molecules gets adsorbed on to the surface which turns out to be the first step (Eq. ( 1)) in the catalytic process. Up on exposure to visible light, the charges are excited (Eq. ( 2)) to form % O 2 -and % OH radicals.

Fig. 1 .

 1 Fig. 1. Schematic representing the generalized processing route for the synthesis of the mesoporous nanocomposite.

Fig. 2 .

 2 Fig. 2. (a) X-ray diff ractrograms revealing the amorphous and crystalline phases of the nanocomposite before and after functionalization with TB and (b) The slow scan XRD of N-TiO 2 /Si-O-C-N exemplifing the TiO x N y phase.

Fig. 3 .

 3 Fig. 3. Raman spectra of the nanocomposite before and after functionalization with TB implying the presence of anatase phase and free carbon.

Fig. 4 .

 4 Fig. 4. Scanning electron micrograph exhibiting the porous nature of (a) CMK-3 (b) the TiO x N y-derived materials and (c) EDS spectrum of the nanocomposite.

Fig. 5 .

 5 Fig. 5. Transmission electron micrograph illustrating (a) uniform distribution of the TiO x N y-derived materials and (b) Lattice fringes unveiling a spacing of 0.35 nm consigned to (1 0 1) plane of the anatase phase in the TiO x N y-derived materials.

Fig. 6 .

 6 Fig. 6. N 2 adsorption-desorption isotherm indicating the presence of mesopores in SiOC-C composite and N-TiO 2 /Si-O-C-N nanocomposite and the inset image shows the BJH pore size distribution curve.

Fig. 8 .Fig. 7 .

 87 Fig. 8. FTIR spectrum showing the bonding characteristics of the SiOC-C composite and N-TiO 2 /Si-O-C-N nanocomposite.

Fig. 9 .

 9 Fig. 9. 29 Si MAS NMR spectra of N-TiO 2 /Si-O-C-N nanocomposite implying the incorporation of N atoms in the Si-based network.

Fig. 10 .

 10 Fig. 10. UV-vis absorption spectra of (a) 0.06 mM of MB with 50 mg of the catalyst kept in dark for 12 h (b) 0.12 mM of MB with 50 mg of the catalyst exposed to visible light after keeping it in dark for 12 h (c) photocatalytic activity of the mesoporous nanocomposite and (d) UV-visible absorption spectrum of the mesoporous nanocomposite.

Fig. 11 .

 11 Fig. 11. . N 2 adsorption-desorption isotherm of the nanocomposite after the Fig. 12. . Catalytic degradation efficiency of methylene blue solution with and adsorption/catalysis.without the presence of presence of scavengers.

Fig. 13 .

 13 Fig. 13. (a) Schematic representing the visible light photocatalytic activity of the nanocomposite and (b) TEM micrograph exemplifying the heterojunction for-mation.

  -radicals are formed when photogenerated electrons reacts with the oxygen molecules as shown in Eq. (3) H 2 O 2 are formed when % O 2 -radicals gets disproportionated or reduced by an electron (Eq. (4))and the formed H 2 O 2 subsequently reacts with electrons to from % OH radicals as expressed in Eq. (5).

				1)	resulting in a high surface area (186 m	2 /g) mesoporous nanocomposite.
	Methylene blue -adsorbed N-TiO 2 + hν Methylene blue-adsorbed N-TiO 2 (e -, h + )	(2)	The mesoporous nanocomoposite exhibited high adsorption capacity and visible light photocatalytic activity towards degradation of me-
				However, as the reaction trapping experiments suggests, the	% OH	thylene blue. The doping of nitrogen in to the TiO 2 lattice is expected to
	radicals are the major contributing species when compared to indicating that % O 2 -are unstable in aqueous solution resulting in im-% O 2 -, mediate conversion to % OH radicals. The plausible degradation me-	reduce the band gap, thereby harnessing light in the visible light wa-velength regime. The hydroxyl radicals and the superoxide anions formed are believed to degrade the organic molecules. The re-
				combination rate is expected to be lowered due to the formation of
	chanism is explained in the below mentioned reactions.
				The	% O 2	heterojunctions between the amorphous and crystalline phases.
	O 2 + e -→	% O 2 -
	% O 2	-+ e -+ 2H + → H 2 O 2
	H 2 O 2 + e -→	% OH + OH -
	The h	+ also reacts with the water molecules resulting in the for-
	mation of	% OH radicals Eqs. (6) and (7).
	H 2 O → H + + OH -	7)
	OH -+ h	+ →	% OH
	Finally, the

% OH and % O 2 -reacts with the dye methylene blue molecules decomposing into water, carbon dioxide (CO 2 ) and other inorganic compounds (Eqns. (8) and (9)). % OH + dye → water + CO 2 + inorganic compounds % O 2 -+ dye → water + CO 2 + inorganic compounds 8


Hocine Chalal
email: hocine.chalal@ensam.eu

Farid Abed-Meraim
email: farid.abed-meraim@ensam.eu

SIMULATION OF STRUCTURAL APPLICATIONS AND SHEET METAL FORMING PROCESSES BASED ON QUADRATIC SOLID-SHELL ELEMENTS WITH EXPLICIT DYNAMIC FORMULATION

Keywords: Finite elements, quadratic solid-shell elements, explicit dynamic analysis, 3D simulations, thin structures, sheet metal forming

In this work, nonlinear dynamic analysis of thin structures is investigated using quadratic solid-shell (SHB-EXP) elements. The proposed SHB-EXP elements are based on a fully three-dimensional formulation using an in-plane reduced-integration scheme along with the assumed-strain method in order to alleviate most locking phenomena. These developments consist of a twenty-node hexahedral element, denoted SHB20-EXP, and its fifteen-node prismatic counterpart, denoted SHB15-EXP. The formulation of these elements is combined with fully three-dimensional behavior models, including elastic behavior as well as anisotropic plastic behavior for metallic materials. The resulting formulations are implemented into the ABAQUS explicit/dynamic software package in the framework of large displacements and rotations. First, to assess the performance of the SHB-EXP elements, four representative nonlinear dynamic benchmark tests have been conducted. Then, impact / crash problem and deep drawing of cylindrical cup have been performed to demonstrate the capabilities of the SHB-EXP elements in handling various types of nonlinearities (large strains, anisotropic plasticity, and double-sided contact). Comparisons with results obtained by ABAQUS elements as well as with reference solutions taken from the literature show the good capabilities of the developed quadratic SHB-EXP elements for the explicit dynamic simulation of thin structures.

Introduction

Thin structures are increasingly used in many engineering applications, and especially in automotive industries. These structures are usually modeled by the finite element method using conventional shell elements for both linear and nonlinear problems (see, e.g., [START_REF] Berg | Cyclic plasticity modelling for ANDES thin shell and line-spring finite elements[END_REF], [START_REF] Lampeas | Interlaminar stresses calculation using a stacked-shell finite element modeling approach[END_REF], [START_REF] Cui | A central point-based discrete shear gap method for plates and shells analysis using triangular elements[END_REF] and [START_REF] Lei | Modal synthesis with the isogeometric Kirchhoff-Love shell elements[END_REF]). However, for the three-dimensional (3D) simulation of sheet metal forming processes, shell elements have some drawbacks associated with their formulations: planestress assumptions; no thickness variations since only the mid-plane of the sheet is modeled; difficulty in handling double-sided contact, etc. To overcome the latter issues associated with shell elements, continuum solid elements are alternatively used to allow more realistic modeling of structural applications thanks to their 3D formulation, thus avoiding geometric (mid-plane) or kinematics assumptions, as well as constitutive (planestress) restrictions. However, the use of solid elements for the simulation of thin structures requires very fine meshes to obtain accurate solutions due to the various locking phenomena that are inherent to these elements (see, e.g., [Feng et al., 2012] and [START_REF] Wang | Simple and accurate eight-node and six-node solid-shell elements with explicit element stiffness matrix based on quasi-conforming element technique[END_REF]). Moreover, in explicit dynamic simulations of thin structures using solid elements, the time step depends on the smallest element size, the latter often being in the thickness direction, which may lead to very high computational costs.

In order to obtain accurate and reliable numerical results, with respect to traditional shell and solid elements, the solid-shell elements have been developed during the last decades. They are based on a fully 3D formulation with only displacements as degrees of freedom. Combined with the reduced-integration technique, various methods have been proposed in the literature to eliminate most locking phenomena (see, e.g., [START_REF] Cho | Stability analysis using a geometrically nonlinear assumed strain solid shell element model[END_REF], [START_REF] Hauptmann | Extension of the 'solid-shell' concept for application to large elastic and large elastoplastic deformations[END_REF]], [Abed-Meraim andCombescure, 2002, 2009], [START_REF] Xie | Drape simulation using solid-shell elements and adaptive mesh subdivision[END_REF] and [START_REF] Wang | Simple and accurate eight-node and six-node solid-shell elements with explicit element stiffness matrix based on quasi-conforming element technique[END_REF]), among which the assumed-strain method (ASM), the enhanced assumed strain (EAS) formulation, and the assumed natural strain (ANS) approach.

In the context of dynamic and vibration analyses of thin structures with solid-shell elements, [START_REF] Pagani | Explicit simulation of forming processes using a novel solid-shell concept based on reduced integration[END_REF][START_REF] Pagani | Computationally efficient explicit nonlinear analyses using reduced integration-based solid-shell finite elements[END_REF] and [START_REF] Cocchetti | Selective mass scaling and critical time-step estimate for explicit dynamics analyses with solid-shell elements[END_REF] have developed a loworder solid-shell element, in which an efficient selective mass scaling method has been introduced in order to control the critical time step. [START_REF] Hajlaoui | Nonlinear dynamics analysis of FGM shell structures with a higher order shear strain enhanced solid-shell element[END_REF] have proposed an 8-node hexahedral solid-shell element for the nonlinear dynamic analysis of functionally graded materials (FGM). With this element, a quadratic distribution of the shear stress through the thickness is considered, which allows enhancing the dynamic behavior of FGM shell structures. Based on the method of incompatible modes for conventional solid elements, [START_REF] Mattern | Highly efficient solid and solidshell finite elements with mixed strain-displacement assumptions specifically set up for explicit dynamic simulations using symbolic programming[END_REF] have proposed a solid-shell element with linear interpolation, as an alternative to the well-known EAS technique, for explicit dynamic simulations using symbolic programming.

In this work, prismatic and hexahedral solid-shell elements with quadratic interpolation of displacements, denoted SHB15-EXP and SHB20-EXP, respectively, are proposed for the nonlinear dynamic simulation of 3D thin structures. These two formulations are developed starting from earlier works on the family of solid-shell (SHB) elements, which includes linear solid-shell elements and their quadratic counterparts (see [START_REF] Abed-Meraim | SHB8PS--a new adaptive, assumedstrain continuum mechanics shell element for impact analysis[END_REF], [START_REF] Abed-Meraim | An improved assumed strain solid-shell element formulation with physical stabilization for geometric non-linear applications and elastic-plastic stability analysis[END_REF], [START_REF] Trinh | A new assumed strain solid-shell formulation "SHB6" for the six-node prismatic finite element[END_REF], [START_REF] Salahouelhadj | Application of the continuum shell finite element SHB8PS to sheet forming simulation using an extended large strain anisotropic elastic-plastic formulation[END_REF], [START_REF] Abed-Meraim | New quadratic solid-shell elements and their evaluation on linear benchmark problems[END_REF], [Wang et al., 2017a[Wang et al., , 2017b]]). The quadratic versions of the SHB elements, which have been first developed in the framework of small strain and quasi-static analysis, are extended in the current work to the explicit dynamic framework. The motivation behind this extension is to allow analyzing all types of structural problems (e.g., impact/crash, complex sheet metal forming processes), which cannot be modeled with the previously developed quasi-static/implicit versions of the SHB elements, due to convergence issues of implicit solvers.

The remainder of the paper is organized as follows. The explicit dynamic formulation of the quadratic SHB-EXP solid-shell elements is first presented in Section 2. In Section 3, the performance of the quadratic SHB-EXP elements is first evaluated through the simulation of nonlinear dynamic benchmark problems. Then, more complex and challenging applications are considered to assess the performance of the SHB-EXP elements in severe dynamic elasto-plastic problems. Finally, the main conclusions and remarks are drawn in Section 4.

Formulation of the SHB-EXP Elements

Reference geometry and location of integration points

The proposed SHB-EXP elements are based on a fully 3D formulation using an in-plane reduced-integration scheme. Figure 1 shows the reference geometry of the quadratic prismatic SHB15-EXP and hexahedral SHB20-EXP solid-shell elements as well as the location of their integration points. The local direction ζ is considered as the thickness direction, where an arbitrary number of integration points are distributed.

Discrete gradient operator and internal forces

Using standard interpolation functions for traditional continuum prismatic and hexahedral elements, the complete expression of the velocity field i v for the SHB-EXP elements can be written as follows:

0 1 1 2 2 3 3 1 1 i i i i i i i v a a x a x a x c h c h α α = + + + + + + ⋯ 1, 2, 3 , i = with (1)
where h α are functions of the local coordinates , , ξ η ζ , in the reference coordinate system, and α varies from 1 to 11 for the SHB15-EXP element, and from 1 to 16 for the SHB20-EXP element. Note that the expressions of the h α functions can be found in [Wang et al., 2017b], where the quasi-static/implicit formulations of the quadratic SHB elements have been detailed.

Using the same expansion as for the velocity field (1), the complete expression of the nodal velocity vectors i d ɺ can be expressed by the following compact form:

0 1 1 2 2 3 3 i i i i i i a a a a c α α α = + + + +  d s x x x h ɺ 1, 2, 3 , i = with (2)
where ( )

1 2 3
, , , ,

T i i i i iK x x x x = x
⋯ are the nodal coordinate vectors. In Eq. ( 2), index α ranges from 1 to 11 for the SHB15-EXP element, and from 1 to 16 for the SHB20-EXP element. Also, vector ( )

1, 1, , 1 T = s ⋯
has fifteen constant components in the case of the SHB15-EXP element, and twenty constant components for the SHB20-EXP element.

With the help of some well-known orthogonality conditions and of the [START_REF] Hallquist | Theoretical Manual for DYNA3D; Report UC1D-19041[END_REF] vectors

| 0 i i x ξ η ζ = = = ∂ = ∂ N b
, where vector N contains the expressions of the interpolation functions I N , the unknown constants ji a and i c α in Eq. ( 2) can be derived as:

, ,

T T ji j i i i a c α α = ⋅ = ⋅ b d d γ (3)
where the complete details on the expressions of vectors α γ can be found in [Wang et al., 2017b]. By introducing the discrete gradient operator B , the vector form of the velocity gradient operator can be expressed as follows:

( ) , , , , , , , , , , x x y
y x z z s y x y y x z y z z y x z z x v v v v v v v v v               ∇ = = ⋅ = ⋅   +         +     +   d v B d B d d ɺ ɺ ɺ ɺ (4)
where the expression of the discrete gradient operator B is:

, , , , , , , , ,
.

T T x x T T y y T T z z T T T T y y x x T T T T z z y y T T T T z z x x h h h h h h h h h α α α α α α α α α α α α α α α α α α   +   +     +     + +   + +     + +   b 0 0 0 b 0 0 0 b B = b b 0 0 b b b 0 b γ γ γ γ γ γ γ γ γ (5)
From the simplified form of the Hu-Washizu variational principle (see [START_REF] Simo | On the variational foundations of assumed strain methods[END_REF]), the expression of the internal force vector of the proposed SHB-EXP elements can be derived as follows:

int , e T d Ω = ⋅ Ω  f B σ (6)
where σ is the Cauchy stress tensor.

To solve explicit dynamic problems with the proposed SHB-EXP elements, a diagonal lumped element mass matrix is added to the above formulations for the prismatic and hexahedral SHB-EXP elements. These lumped mass matrices are defined in the reference coordinate system (see [START_REF] Zienkiewicz | The Finite Element Method[END_REF] and [Wang et al., 2017a] for more details).

Local coordinate frames

In the framework of large strains, the formulation of the SHB-EXP elements requires the definition of two local frames with respect to the global coordinate system for the calculation of the internal forces associated with the constitutive law, as illustrated in Fig. 2. The first type of local frame, which is denoted as the "element frame", is attached to the element mid-plane associated with each integration point. In these local physical coordinate systems, where the ζ -coordinate represents the thickness direction, the fourth-order elasticity tensor is specified. The second local frame is called the "material frame", in which the anisotropic plastic behavior of the material is defined. This material frame is also used to integrate the resulting constitutive equations in order to ensure the incremental objectivity of the model.

Numerical examples

The resulting quadratic SHB-EXP elements have been implemented into the finite element code ABAQUS using the explicit dynamic solver. First, four representative nonlinear dynamic problems have been conducted in order to assess the performance of the SHB-EXP elements. Then, impact / crash problem and deep drawing of cylindrical cup have been performed to demonstrate the capabilities of the SHB-EXP elements in handling various types of nonlinearities (large displacements and rotations, anisotropic plasticity, and double-sided contact).

For comparison purposes, all numerical results obtained with the SHB-EXP elements are compared with those given by ABAQUS linear elements, using the same in-plane meshes, along with reference solutions taken from the literature. Note that the following simulations are achieved using only a single element layer with two integration points through the thickness in the case of SHB-EXP elements. Furthermore, since no quadratic elements are available in the ABAQUS/Explicit software package, standard quadratic prismatic and hexahedral solid elements (i.e., 15-node solid element and 20-node solid element, respectively) have been implemented into the finite element code ABAQUS/Explicit for consistent comparisons with the proposed quadratic SHB-EXP elements. It is worth noting that the ABAQUS linear prismatic solid element (i.e., C3D6) is provided with a single integration point in the explicit dynamic code ABAQUS (see Table 1 for the description of all finite elements used for comparison purposes). Therefore, in what follows, several element layers are required for the ABAQUS linear prismatic solid element C3D6 in order to have the same number of integration points in the thickness direction as the other elements used for comparison.

In the following benchmark problems, the structures are meshed using a specific nomenclature for each type of finite element. This nomenclature is detailed in Table 2.

Cantilever beam subjected to concentrated force

The first explicit/dynamic problem consists of a simple cantilever beam subjected to a concentrated force at its free end. The geometry and the elastic properties of the beam are summarized in Fig. 3(a), while Fig. 3(b) illustrates the deformed shape of the cantilever beam with respect to the underformed shape. The dynamic response in terms of deflection history at the free edge of the beam, obtained using the explicit quadratic elements (i.e., standard quadratic solid elements and SHB-EXP elements) as well as ABAQUS explicit linear elements, is shown in Fig. 4 along with the reference solution given by [START_REF] Olovsson | Selective mass scaling for thin walled structures modeled with tri-linear solid elements[END_REF]. The results reveal that the solution yielded by the proposed SHB-EXP elements is in excellent agreement with the reference solution, which is also the case of the S3R, SC6R, S4R, SC8R and C3D8I ABAQUS elements as well as explicit quadratic PRI15 and HEX20 elements. For the linear prismatic ABAQUS solid element (i.e., C3D6), although requiring twice more elements (i.e., two element layers in the thickness direction), it is found that the solution it provides is far from the reference solution, which suggests resorting to very fine meshes for this element to obtain an accurate solution, while only five SHB20-EXP elements or ten SHB15-EXP are sufficient.

Spherical cap under uniform pressure

A dynamic benchmark test of a clamped spherical cap, subjected to a uniform pressure over its top surface, is analyzed in this section. Both elastic and elasto-plastic behavior models are considered, with the material properties given in Fig. 5(a). Figure 5(b) provides an illustration of the deformed shape of the spherical cap with respect to the underformed shape. Owing to the problem symmetry, only one quarter of the structure is modeled. In addition to the mesh nomenclature previously defined, three partitions for the quarter of the spherical cap are created in order to achieve a relatively regular mesh. This leads to the following new mesh nomenclature for this test: 3×(N1×N1)×N3 for hexahedral elements, where N1 indicates the number of elements along each edge and N3 the number of elements in the thickness direction (see Fig. 5(c)). For the prismatic elements, the total number of elements is twice that corresponding to hexahedral elements, which leads to 3×(N1×N1×2)×N3 elements. For the quadrilateral shell elements, the nomenclature for discretizing the quarter model is 3×(N1×N1), while this nomenclature is 3×(N1×N1×2) when triangular shell elements are used.

In Figs. 6 and7, the histories of the central deflection of the spherical cap, in the case of elastic and elasto-plastic materials, respectively, are shown. In these figures, the results obtained with the explicit quadratic elements (i.e., PRI15, HEX20 and SHB-EXP elements) are compared with those given by ABAQUS explicit linear elements as well as with the reference solutions given by [START_REF] Bathe | Finite element formulations for large deformation dynamic analysis[END_REF] and [START_REF] Belytschko | Explicit algorithms for the nonlinear dynamics of shells[END_REF]. From these figures, it can be seen that the results obtained with the proposed quadratic SHB-EXP elements are the closest to the reference solutions along the entire deflection history, while ABAQUS shell elements and the explicit quadratic HEX20 element provide results that deviate from the reference solutions when elastic material and elastoplastic material are considered, respectively (see Figs.

ν = 0.3 ρ = 2.45 ×10 -4 σ 0 = 24000 N 1 N 1 N 1 N 1 N 1 N 1 N 1 N 1 N 1 N 3 N 3 N 3 (

Rectangular plate subjected to triangular pulse

The dynamic response of an elastic rectangular plate is considered here. The rectangular plate is simply supported at each edge, and subjected to a uniform pressure having triangular time variation, as shown in Fig. 8(a). Figure 8(b) gives an illustration of the deformed shape of the plate with respect to the underformed shape. Due to the problem symmetry, only one quarter of the plate is analyzed. It is worth noting that only the midsurface of the lateral edges of the plate is simply supported, which requires meshing the plate with two element layers in the case of ABAQUS linear solid and solid-shell elements (i.e., SC6R, C3D6, SC8R and C3D8I elements). As to the proposed quadratic SHB-EXP elements and standard quadratic PRI15 and HEX20 solid elements, only one element layer is needed to model this benchmark problem, thanks to the existing midsurface nodes in traditional continuum quadratic elements.

The deflection of the central point of the plate, obtained with the explicit quadratic elements, is depicted in Fig. 9 and compared with the results provided by ABAQUS explicit linear elements as well as with the reference solution given by [START_REF] Sheikh | Linear and nonlinear transient vibration analysis of stiffened plate structures[END_REF]. It can be observed once again that the SHB-EXP elements perform very well with respect to the reference solution, which is also the case for ABAQUS shell, solid-shell and C3D8I elements as well as the explicit quadratic PRI15 and HEX20 elements. However, as pointed out in the previous dynamic benchmark problems, finer mesh is required for the ABAQUS linear prismatic solid element in order to obtain an accurate solution.

Pinched hemispherical shell

From the above benchmark problems, it can be observed that the simulation results obtained with the SHB15-EXP are comparable to those provided by the SHB20-EXP element. Despite this, the prismatic SHB15-EXP element has been mainly developed for use in mesh discretization of complex geometries, for which regular mesh with hexahedral elements cannot be achieved. To this end, we consider in this section the popular benchmark test of a hemispherical shell that is loaded by alternating radial forces as shown in Fig. 10(a). By considering the symmetry, only a quarter of the structure is discretized. The hemispherical shell is meshed with a mixture of prismatic and hexahedral elements, or triangular and quadrilateral shell elements, which consists of 90 prismatic or triangular shell elements located at the top of the hemisphere (far from the load points, see Fig. 10(b)), and 110 hexahedral or quadrilateral shell elements for the remaining area. The simulation results obtained with the mixture of SHB-EXP elements, in terms of history of radial deflection at point A, are plotted in Fig. 11, and compared with those given by a mixture of standard explicit quadratic solid elements and a mixture of ABAQUS explicit linear elements, along with the reference solution given by [START_REF] Wu | A DKT shell element for dynamic large deformation analysis[END_REF]. Compared to the reference solution, it can be observed that the SHB-EXP elements successfully pass this benchmark test, based on a combination of prismatic and hexahedral solid-shell elements, which is also the case of ABAQUS linear shell and solid-shell elements. However, the results obtained by the combination of standard explicit quadratic solid elements (i.e., PRI15 and HEX20) and a mixture of ABAQUS linear solid elements (i.e., C3D6 and C3D8I) reveal the poor performance of these elements in this test, even when using two element layers in the thickness direction, due to their sensitivity to locking effects.

Impact of a boxbeam

In this complex benchmark problem, a steel boxbeam, clamped at one end as illustrated in Fig. 12(a), is impacted by an infinite mass with a constant velocity of 1.27 m/s. The geometry of the cross-section of the boxbeam is an empty rectangle with a thickness of 0.914 mm. The following Ludwig isotropic hardening law is used in the simulations for the elasto-plastic behavior of the boxbeam:

0 = + , N Y Y kε (7)
where ε and 0 Y are the equivalent plastic strain and the initial yield stress, respectively, while k and N are the hardening parameters. The elasto-plastic material parameters of the boxbeam are reported in Table 3. During the simulations, frictionless contact is considered between the boxbeam and the rigid impactor.

Due to the symmetry, one quarter of the boxbeam is modeled. In the case of hexahedral solid, solid-shell and quadrilateral shell elements, a mesh of 896 elements with a single element layer is adopted, while 1792 elements with a single element layer are used in the case of quadratic prismatic solid and solid-shell elements as well as triangular shell elements. For the ABAQUS explicit linear solid element C3D6, the same in-plane mesh as the prismatic solid-shell element is used, with however two element layers through the thickness.

Figure 13 shows the predicted reaction force-displacement curves for the impactor, as obtained using the SHB-EXP elements, which are compared with the results given by ABAQUS elements as well as explicit quadratic solid elements (i.e., PRI15 and HEX20), along with the experimental results provided by [START_REF] Zeng | A new one-point quadrature, general non-linear quadrilateral shell element with physical stabilization[END_REF]. The results predicted with the quadratic SHB-EXP elements reveal that the three impact force peaks in the reaction force-displacement curve, which are typical in such impact problems, are in good agreement with experiments. More specifically, the quadratic SHB-EXP elements provide the closest results to the experimental curve, while ABAQUS shell and solid-shell elements provide results that slightly deviate from the experimental curve. As to the explicit quadratic solid elements (i.e., PRI15 and HEX20), although the reaction force is in the order of magnitude of the experimental one, the force peaks are not well reproduced with these elements. Regarding ABAQUS linear solid elements (i.e., C3D6 and C3D8I), the latter provide the farthest results with respect to the experimental curve, and appear to be less suitable for this type of crashworthiness analysis.

Table 3. Elasto-plastic material parameters used in the simulation for the boxbeam.

Deep drawing of a cylindrical cup

This subsection is devoted to the simulation of deep drawing of a cylindrical cup in order to evaluate the capabilities of the quadratic SHB-EXP elements in modeling sheet metal forming processes. This sheet forming process is commonly used to study the earing profile of the cylindrical cup when anisotropic plastic behavior of sheet metals is considered. The geometry and dimensions of the forming setup are illustrated in Fig. 14. The sheet is made of an Aluminum AA2090-T3 material. The following Swift isotropic hardening law is used in the simulations for the elasto-plastic behavior of the sheet:

( ) The proposed SHB-EXP elements have been combined with the Hill'48 yield surface to account for the initial anisotropy of the sheet. The corresponding Hill'48 anisotropy coefficients are provided in Table 5 (see [START_REF] Yoon | Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function[END_REF]).

0 = . N Y k ε ε + (8)
Owing to the problem symmetry, one quarter of the initial sheet is modeled. In the case of hexahedral and quadrilateral shell elements, the quarter of the circular sheet is meshed with 255 elements, while 510 elements are used in the case of prismatic and triangular shell elements. The simulations are performed using only a single element layer for the quadratic solid and solid-shell elements as well as ABAQUS linear solid-shell elements. Note that only two integration points in the thickness direction are sufficient for the quadratic SHB-EXP elements to obtain converged results, while three integration points are considered for the explicit quadratic solid elements. As to the explicit ABAQUS linear solid elements, three integration points are required to obtain converged result, which involves three element layers in the case of the ABAQUS linear prismatic solid element, and two element layers in the case of the ABAQUS linear hexahedral solid element. Figure 15 shows the deformed meshes of the sheet, as obtained with the quadratic SHB-EXP elements, which correspond to a completely drawn cup. It can be seen that the proposed SHB-EXP elements predict four ears for the cylindrical cup, which is consistent with the use of the quadratic Hill'48 yield surface for the description of the material planar anisotropy. The final height profiles for the cylindrical cup, as obtained with the SHB-EXP elements, are reported in Fig. 16 and compared with those provided by the explicit quadratic solid elements as well as ABAQUS explicit elements, along with the experimental ones given by [START_REF] Yoon | Prediction of six or eight ears in a drawn cup based on a new anisotropic yield function[END_REF]. Overall, the shape of the earing profiles predicted with the SHB-EXP elements is in good agreement with the experiments, with the cup heights slightly underestimated at 0° and 90° from the rolling direction, while they are closer to the experiments in the range around the experimental peak value at 50° from the rolling direction. Note that the ABAQUS explicit linear solid element C3D8I as well as the ABAQUS shell and solid-shell elements provide the farthest results with respect to the experimental cup heights in the range around 0°, which emphasizes the limitations of these elements in handling double-sided contact in sheet metal forming processes. The thickness strain distribution along the rolling and transverse directions at the end of the forming process is also analyzed in this problem. Figure 17 compares the predicted and measured thickness strain in both directions. In the rolling direction (see Figs. 17(a) and (b)), the thickness strain distributions predicted with the proposed SHB-EXP elements and quadratic solid elements are in good agreement with the experiments from the center of the sheet to the initial position of 50 mm, which is not the case of ABAQUS solid and solid-shell elements. Beyond this position, the predicted results differ from the experiments for all elements except for the proposed SHB20-EXP, for which the predicted results are the closest to the experimental ones. In the transverse direction (see Figs. 17(c) and(d)), the thickness strain distributions predicted by the proposed SHB15-EXP and PRI15 as well as SHB20-EXP and HEX20 are very close to each other. More specifically, from the center of the sheet to the initial position of 50 mm, the results obtained with the prismatic quadratic solid and solid-shell elements are in good agreement with the experiments. However, the thickness strain distributions differ from the experiments beyond the initial position of 50 mm for all elements.

Conclusions

In this paper, quadratic prismatic and hexahedral solid-shell elements, denoted SHB-EXP, have been proposed for the explicit dynamic modeling of 3D thin structures. These SHB-EXP elements are based on a purely three-dimensional approach, with displacements as the only degrees of freedom. Furthermore, for efficiency requirements and for alleviating locking phenomena, an in-plane reduced-integration scheme along with the assumed-strain method is adopted in the formulations. The resulting formulations are computationally efficient, since only a single layer of elements with an arbitrary number of through-thickness integration points is required to model 3D thin structures. The quadratic SHB-EXP elements have been implemented into the ABAQUS explicit software package, in the framework of large displacements and rotations, for the dynamic simulations of structural applications. Four representative dynamic benchmark tests, involving geometric and material nonlinearities, have been first conducted in order to assess the performance of the SHB-EXP elements. For each benchmark problem, the results given by the proposed SHB-EXP elements have been compared with those yielded by ABAQUS explicit linear elements as well as with reference solutions taken from the literature. In addition, since no quadratic elements are available in the ABAQUS/Explicit software package, standard quadratic prismatic and hexahedral solid elements (i.e., 15-node solid element and 20-node solid element, respectively) have been implemented in this work into the ABAQUS/Explicit software package for consistent comparison with the quadratic SHB-EXP elements. From these first dynamic analyses, the numerical results obtained with the SHB-EXP elements showed excellent agreement with the available reference solutions. Compared to the implemented standard explicit quadratic solid elements as well as ABAQUS explicit linear shell and solid-shell elements, the performance of the SHB-EXP elements is often comparable, and even better in some cases. However, in most dynamic benchmark tests, the numerical results obtained with the ABAQUS linear prismatic solid element (i.e., C3D6) reveal that this element requires finer mesh (in the plane and through the thickness) to provide accurate solutions.

Then, the performance of the SHB-EXP elements has been assessed via the simulation of more complex and challenging structural applications, involving large strains, anisotropic plasticity and double-sided contact. These applications consist of the simulation of impact of a boxbeam as well as deep drawing of a cylindrical cup made of an aluminum alloy with anisotropic plastic behavior. Comparisons between the simulation results given by the SHB-EXP elements and the experimental ones revealed that the proposed explicit solid-shell elements are able to successfully model such complex problems, using only a single element layer with few through-thickness integration points. Furthermore, the SHB-EXP elements provided the closest results to the experimental data, when compared to the standard explicit quadratic solid elements as well as ABAQUS explicit elements, which makes the proposed SHB-EXP solid-shell elements very competitive in such complex applications.

  Fig.1. SHB-EXP solid-shell elements: reference geometry and location of the associated integration points.

Fig. 2 .

 2 Fig. 2. Definition of the local frames with respect to the global coordinate system for the proposed SHB-EXP elements.

  N1: Number of elements along the length N2: Number of elements along the width N3: Number of elements along the thickness Triangular shell elements Prismatic elements Quadrilateral elements Hexahedral elements N1×N2×2 (N1×N2×2)×N3 N1×N2 (N1×N2)×N3

F

  Fig. 3. Cantilever beam.

  6(a) and 6(b) for shell elements, and Fig.7(b) for the HEX20 element). Similar to the previous benchmark test, the results obtained with the ABAQUS linear prismatic solid element (i.e., C3D6) are far from the reference solutions during the second stage of loading (from T = 0.2 s) for both elastic and elasto-plastic materials.

Fig. 5 .

 5 Fig. 5. Clamped spherical cap under uniform pressure.

  Fig. 7. Vertical displacement history for the point located at the apex of the clamped spherical cap in the case of elasto-plastic material.

Fig. 9 .

 9 Fig. 8. Simply supported rectangular plate subjected to triangular pulse.

Fig. 11 .

 11 Fig. 10. Pinched hemispherical shell.

  Fig. 12. Schematic representation of a boxbeam impacted by an infinite mass.

Fig. 14 .

 14 Fig. 14. Geometry and dimensions (in mm) of the cylindrical cup drawing setup.

Fig. 15 .

 15 Fig. 15. Final deformed shape for a completely drawn cylindrical cup.

  Fig. 16. Prediction of cup height profiles.

Fig. 17 .

 17 Fig. 17. Comparison of predicted and measured thickness strain distributions.

Table 1 .

 1 Prismatic, hexahedral as well as shell finite elements used in the simulations.

		SHB15-EXP	15-node solid-shell element with two integration points through the thickness
	Prismatic elements / Triangular shell element	PRI15 S3R SC6R	15-node solid element with three integration points through the thickness 3-node shell element with two integration points through the thickness 6-node solid-shell element with two integration points through the thickness
		C3D6	6-node solid element with one integration point through the thickness
		SHB20-EXP	20-node solid-shell element with two integration points through the thickness
	Hexahedral	HEX20	20-node solid element with three integration points through the thickness
	elements / Quadrilateral shell	S4R	4-node shell element with two integration points through the thickness
	element	SC8R	8-node solid-shell element with two integration points through the thickness
		C3D8I	8-node incompatible mode solid element with two integration points through the thickness

Table 2 .

 2 Description of the mesh strategy used in the simulations.

  Simulation of Structural Applications and Sheet Metal Forming Processes Based on Quadratic Solid-ShellElements with Explicit Dynamic Formulation 9 Vertical displacement history for the point located at the apex of the clamped spherical cap in the case of elastic material.

		0.08								0.08
				Bathe et al. [1975]							Bathe et al. [1975]
				Belytschko et al. [1984]						Belytschko et al. [1984]
		0.04		SHB15-EXP 3×(4×4×2)×1				0.04		SHB20-EXP 3×(4×4)×1
	Central displacement	-0.08 -0.04 0.00	PRI15 3×(4×4×2)×1 S3R 3×(4×4×2) SC6R 3×(4×4×2)×1 C3D6 3×(4×4×2)×2			Central displacement	-0.08 -0.04 0.00	HEX20 3×(4×4)×1 C3D8I 3×(4×4)×1 S4R 3×(4×4) SC8R 3×(4×4)×1
		0.0 -0.12	0.1	0.2	0.3	0.4	0.5	0.6		0.0 -0.12	0.1	0.2	0.3	0.4	0.5	0.6
					Time (s)								Time (×10 -3 s)
		(a) Triangular shell / prismatic elements			(b) Quadrilateral shell / hexahedral elements
	-0.12 0.08 Fig. 6. 0.0 -0.08 -0.04 0.00 0.04 Central displacement	0.1	0.2 Bathe et al. [1975] 0.3 0.4 Belytschko et al. [1984] SHB15-EXP 3×(4×4×2)×1 0.5 S3R 3×(4×4×2) SC6R 3×(4×4×2)×1 C3D6 3×(4×4×2)×2 PRI15 3×(4×4×2)×1	0.6	Central displacement	0.0 -0.12 0.08 0.04 -0.08 -0.04 0.00	0.1	0.2	0.3 Bathe et al. [1975] 0.4 Belytschko et al. [1984] 0.5 SHB20-EXP 3×(4×4)×1 S4R 3×(4×4) SC8R 3×(4×4)×1 C3D8I 3×(4×4)×1 HEX20 3×(4×4)×1	0.6
					Time (s)								Time (s)

Table 4

 4

	. Elasto-plastic material parameters used in the simulation for the Aluminum
			AA2090-T3.
	Material	E (GPa)	ν	ε	0	k (MPa)	N
	AA2090-T3	70.5	0.34	0.025	646	0.227

Table 5 .

 5 Hill'48 anisotropy coefficients for AA2090-T3 aluminum sheet.

	Material	F	G	H	L	M	N
	Aluminum	0. 2521	0. 8254	0. 1745	1.5	1.5	2.2380