Thermomechanical simulation of shape memory alloys structures: variational methods and associated numerical tools

Shape Memory Alloys (SMA) offer new perspectives in various fields such as aeronautics, robotics, biomedicals, or civil engineering. Efficient design of such innovative systems requires both adequate material models and numerical methods for simulating the response of SMA structures. Whereas much effort has been devoted to developing constitutive laws for describing the behaviour of SMAs, the structural problem (i.e. the simulation of a three-dimensional SMA structure) has received far less attention, in spite of substantial difficulties notably due to the strong thermomechanical coupling and the presence of physical constraints on the internal variables. The time-discretization of the evolution problem obtained is not obvious, and special care must be taken to avoid convergence difficulties and ensure robustness of the numerical schemes. Computation time and ease of implementation (for instance in an existing finite element code) also are major issues that need to be addressed. In this communication are presented some recent results in that direction. A central result is a recent time-discretization scheme for the thermomechanical problem. A variational formulation is attached to the corresponding incremental problem, allowing one to prove the existence of solutions for a large class of usual SMA models. The variational nature of the problem at hand also calls for an easy implementation in an existing finite element code, building on well-established descend algorithms. Using that approach, the solution of the thermomechanical incremental problem is typically obtained by solving a sequence of linear thermal problems and purely mechanical (i.e. at prescribed temperature) nonlinear problems. That approach is fairly general and applies for a wide range of SMA models. The numerical scheme for solving the purely mechanical problem, however, strongly depends on the particular model that is used. In a micromechanical modelling of SMAs, the phase transformation is described locally by an internal vectorial variable which is physically constrained to satisfy a set of inequalities at each point. We show that the corresponding incremental problem can be recast as a linear complementarity problem, for which efficient algorithms (such as interior-point methods) are available. That reformulation essentially consists in a change of variables. In terms of variational formulation, that approach amounts to replace a convex but non-quadratic minimization problem with an equivalent quadratic minimization problem

A Study of the Formation of Single- and Double-Walled Carbon Nanotubes by a CVD Method

The reduction in H2/CH4 atmosphere of aluminum-iron oxides produces metal particles small enough to catalyze the formation of single-walled carbon nanotubes. Several experiments have been made using the same temperature profile and changing only the maximum temperature (800-1070 °C). Characterizations of the catalyst materials are performed using notably 57Fe Mo¨ssbauer spectroscopy. Electron microscopy and a macroscopical method are used to characterize the nanotubes. The nature of the iron species (Fe3+, R-Fe, ç-Fe-C, Fe3C) is correlated to their location in the material. The nature of the particles responsible for the high-temperature formation of the nanotubes is probably an Fe-C alloy which is, however, found as Fe3C by postreaction analysis. Increasing the reduction temperature increases the reduction yield and thus favors the formation of surface-metal particles, thus producing more nanotubes. The obtained carbon nanotubes are mostly single-walled and double-walled with an average diameter close to 2.5 nm. Several formation mechanisms are thought to be active. In particular, it is shown that the second wall can grow inside the first one but that subsequent ones are formed outside. It is also possible that under given experimental conditions, the smallest (<2 nm) catalyst particles preferentially produce double-walled rather than single-walled carbon nanotubes

Carbon Nanotubes by a CVD Method. Part II: Formation of Nanotubes from (Mg, Fe)O Catalysts

The aim of this paper is to study the formation of carbon nanotubes (CNTs) from different Fe/MgO oxide powders that were prepared by combustion synthesis and characterized in detail in a companion paper. Depending on the synthesis conditions, several iron species are present in the starting oxides including Fe2+ ions, octahedral Fe3+ ions, Fe3+ clusters, and MgFe2O4-like nanoparticles. Upon reduction during heating at 5 °C/min up to 1000 °C in H2/CH4 of the oxide powders, the octahedral Fe3+ ions tend to form Fe2+ ions, which are not likely to be reduced to metallic iron whereas the MgFe2O4-like particles are directly reduced to metallic iron. The reduced phases are R-Fe, Fe3C, and ç-Fe-C. Fe3C appears as the postreaction phase involved in the formation of carbon filaments (CNTs and thick carbon nanofibers). Thick carbon nanofibers are formed from catalyst particles originating from poorly dispersed species (Fe3+ clusters and MgFe2O4-like particles). The nanofiber outer diameter is determined by the particle size. The reduction of the iron ions and clusters that are well dispersed in the MgO lattice leads to small catalytic particles (<5 nm), which tend to form SWNTS and DWNTs with an inner diameter close to 2 nm. Well-dispersed MgFe2O4-like particles can also be reduced to small metal particles with a narrow size distribution, producing SWNTs and DWNTs. The present results will help in tailoring oxide precursors for the controlled formation of CNTs

Fe/Co Alloys for the Catalytic Chemical Vapor Deposition Synthesis of Single- and Double-Walled Carbon Nanotubes (CNTs). 1. The CNT−Fe/Co−MgO System

Mg0.90FexCoyO (x + y ) 0.1) solid solutions were synthesized by the ureic combustion route. Upon reduction at 1000 °C in H2-CH4 of these powders, Fe/Co alloy nanoparticles are formed, which are involved in the formation of carbon nanotubes, which are mostly single and double walled, with an average diameter close to 2.5 nm. Characterizations of the materials are performed using 57Fe Mo¨ssbauer spectroscopy and electron microscopy, and a well-established macroscopic method, based on specific-surface-area measurements, was applied to quantify the carbon quality and the nanotubes quantity. A detailed investigation of the Fe/Co alloys’ formation and composition is reported. An increasing fraction of Co2+ ions hinders the dissolution of iron in the MgO lattice and favors the formation of MgFe2O4-like particles in the oxide powders. Upon reduction, these particles form R-Fe/Co particles with a size and composition (close to Fe0.50Co0.50) adequate for the increased production of carbon nanotubes. However, larger particles are also produced resulting in the formation of undesirable carbon species. The highest CNT quantity and carbon quality are eventually obtained upon reduction of the iron-free Mg0.90Co0.10O solid solution, in the absence of clusters of metal ions in the starting material. Introduction Catalyti

Carbon Nanotubes by a CVD Method. Part I: Synthesis and Characterization of the (Mg, Fe)O Catalysts

The controlled synthesis of carbon nanotubes by chemical vapor deposition requires tailored and wellcharacterized catalyst materials. We attempted to synthesize Mg1-xFexO oxide solid solutions by the combustion route, with the aim of performing a detailed investigation of the influence of the synthesis conditions (nitrate/urea ratio and the iron content) on the valency and distribution of the iron ions and phases. Notably, characterization of the catalyst materials is performed using 57Fe Mo¨ssbauer spectroscopy, X-ray diffraction, and electron microscopy. Several iron species are detected including Fe2+ ions substituting for Mg2+ in the MgO lattice, Fe3+ ions dispersed in the octahedral sites of MgO, different clusters of Fe3+ ions, and MgFe2O4-like nanoparticles. The dispersion of these species and the microstructure of the oxides are discussed. Powders markedly different from one another that may serve as model systems for further study are identified. The formation of carbon nanotubes upon reduction in a H2/CH4 gas atmosphere of the selected powders is reported in a companion paper

Fe-substituted mullite powders for the in situ synthesis of carbon nanotubes by catalytic chemical vapor deposition

Powders of iron-substituted mullite were prepared by combustion and further calcination in air at different temperatures. A detailed study involving notably Mo¨ssbauer spectroscopy showed that the Fe3+ ions are distributed between the mullite phase and a corundum phase that progressively dissolves into mullite upon the increase in calcination temperature. Carbon nanotube-Fe-mullite nanocomposites were prepared for the first time by a direct method involving a reduction of these powders in H2-CH4 and without any mechanical mixing step. The carbon nanotubes formed by the catalytic decomposition of CH4 on the smallest metal particles are mostly double-walled and multiwalled, although some carbon nanofibers are also observed

Surface Composition of Carbon Nanotubes-Fe-Alumina Nanocomposite Powders: An Integral Low-Energy Electron Mo1ssbauer Spectroscopic Study

The surface state of carbon nanotubes-Fe-alumina nanocomposite powders was studied by transmission and integral low-energy electron Mo¨ssbauer spectroscopy. Several samples, prepared under reduction of the R-Al1.8-Fe0.2O3 precursor in a H2-CH4 atmosphere applying the same heating and cooling rate and changing only the maximum temperature (800-1070 °C) were investigated, demonstrating that integral low-energy electron Mo¨ssbauer spectroscopy is a promising tool complementing transmission Mössbauer spectroscopy for the investigation of the location of the metal Fe and iron-carbide particles in the different carbon nanotubenanocomposite systems containing iron. The nature of the iron species (Fe3+, Fe3C, R-Fe, ç-Fe-C) is correlated to their location in the material. In particular, much information was derived for the powders prepared by using a moderate reduction temperature (800, 850, and 910 °C), for which the transmission and integral low-energy electron Mössbauer spectra are markedly different. Indeed, R-Fe and Fe3C were not observed as surface species, while ç-Fe-C is present at the surface and in the bulk in the same proportion independent of the temperature of preparation. This could show that most of the nanoparticles (detected as Fe3C and/or ç-Fe-C) that contribute to the formation of carbon nanotubes are located in the outer porosity of the material, as opposed to the topmost (ca. 5 nm) surface. For the higher reduction temperatures Tr of 990 °C and 1070 °C, all Fe and Fe-carbide particles formed during the reduction are distributed evenly in the bulk and the surface of the matrix grains. The integral low-energy electron Mo¨ssbauer spectroscopic study of a powder oxidized in air at 600 °C suggests that all Fe3C particles oxidize to R-Fe2O3, while the R-Fe and/or ç-Fe-C are partly transformed to Fe1-xO and R-Fe2O3, the latter phase forming a protecting layer that prevents total oxidation

Chirality of internal metallic and semiconducting carbon nanotubes

We have assigned the chirality of the internal tubes of double walled carbon nanotubes grown by catalytic chemical vapor deposition using the high sensitivity of the radial breathing ~RB! mode in inelastic lightscattering experiments. The deduced chirality corresponds to several semiconducting and only two metallic internal tubes. The RB modes are systematically shifted to higher energies when compared to theoretical values. The difference between experimental and theoretical energies of the RB modes of metallic tubes and semiconducting tubes are discussed in terms of the reduced interlayer distance between the internal and the external tube and electronic resonance effects. We find several pairs of RB modes corresponding to different diameters of internal and external tubes

Synthesis of γ-(Al1-xFex)2O3 solid solutions from oxinate precursors and formation of carbon nanotubes from the solid solutions using methane or ethylene as carbon source

This work reports for the first time the synthesis of ?-(Al1-xFex)2O3 solid solutions with a high specific surface area (200-230 m2/g) by the decomposition of metal oxinate [(Al1-xFex)(C9H6ON)3] and investigated the potential of these materials as catalysts for the synthesis of carbon nanotubes by catalytic chemical vapor deposition using methane or ethylene as carbon the source. The nanocomposite powders prepared by reduction in H2-CH4 contain carbon nanotubes (CNTs), which are mostly double-walled but also contain a fair amount of undesirable carbon nanofibers, hollow carbon particles, and metal particles covered by carbon layers. Moreover, abundant metallic particles are observed to cover the surfaces of the matrix grains. By contrast, the nanocomposite powders prepared by reduction in N2-C2H4 are not fully reduced, and the CNTs are much more abundant and homogeneous. However, they are multiwalled CNTs with a significant proportion of defects. The powders were studied by several techniques including Mössbauer spectroscopy and electron microscopy

Presence of Metallic Fe Nanoclusters in r-(Al,Fe)2O3 Solid Solutions

Powders of R-(Al1-xFex)2O3 solid solutions prepared by the calcination in air of the corresponding γ-(Al1- xFex)2O3 powders were studied by several techniques including X-ray diffraction, field-emission-gun scanning electron microscopy, transmission Mössbauer spectroscopy, integral low-energy electron Mössbauer spectroscopy (ILEEMS), and Fe K-edge X-ray absorption near-edge structure (XANES) measurements. The asymmetry of the characteristic Mo¨ssbauer doublet representing Fe3+ ions substituting for Al3+ ions in the corundum lattice of R-(Al1-xFex)2O3 solid solutions was resolved and explained for the first time by using two additional subspectra, i.e., a broad second doublet characteristic of a very distorted octahedral site for Fe3+ and a singlet attributable to R-Fe, suggesting the presence of metallic iron nanoclusters consisting of only a few number of atoms within the solid solution grains. ILEEMS studies showed that the Fe nanoclusters are evenly distributed among the surface layers and the cores of the grains. Fe K-edge XANES measurements further confirmed the occurrence of metallic iron. The proportion of Fe nanoclusters increases when the total iron content is decreased, as does the proportion of distorted octahedral site, suggesting that they are located around the iron nanoclusters. The formation of the metallic Fe nanoclusters in the R-(Al1-xFex)2O3 grains is thought to be a consequence of the γ f R phase transition which implies structural rearrangement on both the cationic and anionic sublattices