C S Unnikrishnan

Bose-Einstein Condensates as Universal Quantum Matter

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A derivation that was a quantum leap

Foundations of physics are often unsettled. Physicists have to make postu-lates, assumptions and various approximations to reach where intuitions and empirical data converge. In this process, it is not unusual that concepts with some mutual tension from previous theories and experience are mixed with new insights, as and when needed. Satyendra Nath Bose was looking at one such important instance in the course of his teaching in the university, when he decided that the derivation that changed and revolutionized physics, of the Planck's law of the spectral distribution of the black body radiation, was conceptually deficient and unsatisfactory. Planck's quanta was transformedto particles of light by Einstein and the experimental evidence for their par-ticle nature in photoelectric effect was convincing to the physics world (and to the Nobel committee). The quanta of light possessed energy and momentum

, where referred to the frequency of an underlying oscillator. Bose noticed that all previous derivations of the Planck's law treated the counting of quanta with a particular range of energy by treating them as standing waves in a confined space, familiar from classical electrodynamics, and not by treating them as particles. He was troubled by this inconsistency. He set out to correct that glaring conceptual flaw by calling a particle a particle. That exercise culminated in the "Indian Bose's beautiful derivation" of the Planck's radiation law, the details and history of which are discussed and documented widely. [START_REF] Ghose | Bose Statistics : a historical perspective[END_REF] As well known, some events related to Bose's attempt to publish his derivation finally brought the paper to Einstein's desk, with a daring and confidently humble request to help in the publication after translation to German. Einstein saw the importance of dealing with light quanta as 'atoms of light', translated the paper and helped Bose in its publication. [START_REF] Bose | Planck's Law and the Light-Quantum Hypothesis[END_REF] Meanwhile Einstein made the characteristic conceptual quantum leap.

Planck's formula for the spectral density of radiation consists of three factors. First is the number of 'modes of radiation' with frequency , considered as the standing waves in a containing box; this is the coefficient . Other two are the Planck's expression for the energy of a quantum, h , and the statistical factor of average occupation (probability) of the state of energy . The '-1' factor in the denominator is the signature of Planck's quantum revolution. Bose's immediate achievement, in his own words, was to "deduce the coefficient in Planck's law independent of classical electrodynamics".

What was striking in Bose's derivation was his treating the quanta of light as particles with energy and momentum . There is no mention of waves of radiation. Bose's friend and classmate, Meghnath Saha had already used (in 1919) the concept of the momentum of the quanta in the astrophysical context to deduce that absorbing a 'corpuscle of light' could significantly alter the trajectory of atoms. 3 Bose assumed that the light quanta occupied 'elementary cells' of volume in the phase space of coordinates and momenta. The volume of the phase space with momentum between and , occupying spatial volume V, would be . Accounting for the two polarization states one gets . (Apparently, Bose meant and wrote 'two states of spin directions' rather than 'polarization states', but Einstein changed it to a more conventional, though inconsistent, description). Dividing by the volume of each phase space cell, according to Bose, one gets the number of quantum states and thus the important coefficient in Planck's law correctly. Thus, the factor hitherto derived using classical electrodynamics and counting of standing waves in a box, Bose derived 'quantum theoretically', for the first time. This was the original and new contribution, as stressed by Bose himself, and highlighted by Einstein. [START_REF] Bose | Planck's Law and the Light-Quantum Hypothesis[END_REF] Bose's derivation marked an important conceptual advance. The other details of Bose's paper followed some earlier derivations, but now with the quanta occupying the cells of phase space in different ways (the indistinguishability of the quanta was already recognized in some earlier derivations).

Einstein applied the insight and inspiration gained from Bose's paper on the 'photon gas' to the quantum mechanics of a thermal atomic gas. [START_REF] Einstein | Quantum theory of a monoatomic ideal gas -I[END_REF] The momentum of a particle for a given energy is . Following Bose's steps, Einstein got the number density of phase space cells as . He deduced that the gas of atoms followed a quantum mechanical law very similar to Planck's law for their statistical distribution, with the average number of atoms per cell as . With their number conserved, unlike photons, atoms distribute with a 'chemical potential' , which approaches zero at low temperature.

Einstein noticed a peculiar feature occurring at low temperature. [START_REF] Einstein | Quantum theory of a monoatomic ideal gas -2[END_REF] The total number of atoms is limited, at temperature T in volume V as (The numerical factor is over all positive integers). Thus there is a maximum density (or minimum temperature ), beyond which the added particles have to go to a state where their presence would not be felt! Einstein wrote [START_REF] Einstein | Quantum theory of a monoatomic ideal gas -2[END_REF] , "So what happens when we increase the density of the substance, N/V, at constant temperature (for example by isothermal compression)? I suggest that in this case, as you increase the overall density, an ever increasing number of molecules drop into the ground state (i.e. the first quantum state, state of no kinetic energy)...My prediction is the emergence of something very similar to what occurs when vapour is isothermally compressed beyond the saturation volume. A separation will take place; one part will condense, the rest will remain as a 'saturated ideal gas' ...When exceeding this density the surplus molecules will drop out as immobile ('condensing' without forces of attraction)."

I think that decades of intimacy with thermodynamics and statistical physics helped Einstein to see this subtle result transparently. Bose-Einstein Condensation is the most spectacular prediction from the thermal statistical distribution of gas atoms, called the Bose-Einstein distribution.

Bose-Einstein Condensates in the laboratory

The prediction from the 'most famous physicist' remained undiscussed for years, until Fritz London's calculations and identification, in 1938, of the superfluid Helium as a genuine Bose-Einstein condensation transition. 6 Despite the binding interactions in the liquid phase, the light atoms of Helium has significant zero-point energy that makes them loosely bound, almost like the atoms of a gas, a fraction of which becomes a Bose-Einstein Condensate (BEC). That superfluid Helium is a BEC was evident from its physical behaviour and the closeness of the condensation temperature to the one calculated from Einstein's theory.

The relation between superconductivity and Bose-Einstein Condensation is not straightforward because the electrons with half-integer intrinsic spin are involved. It was known that the statistical behaviour of particles with integer values for their intrinsic spin (like photons) is drastically different from those with half-integer spin. However, the mysterious fact that only the total spin of a composite entity matters for determining its quantum statistical behaviour gave clarity for the identification of superconductivity as involving the BEC of the bound 'Cooper pairs' of electrons, with total zero spin. 7 Attempts to produce a near ideal BEC in atomic gases were pursued slowly, starting in late 1970s with Hydrogen, notably by D. Kleppner in the USA. This turned out to be extremely difficult with many technical hurdles. However, a variety of experimental techniques and insights were developed on the way to reaching low effective temperatures. The thermodynamic temperature in this case refers to the average kinetic energy in a thermalised sample of atoms, as . Most important of these techniques was perhaps the method of evaporative cooling, to be discussed later.

An explosive development in atomic physics and in the field of atom-light interactions, which happened in a relatively short time of a decade, changed the entire scenario. [START_REF] Metcalf | Laser cooling and trapping[END_REF] First was the rapid progress in the precision spectroscopy of atomic transitions using lasers. [START_REF] Demtröder | Laser Spectroscopy: Basic Principles[END_REF] The invention of the remarkably simple pump-probe saturation spectroscopy enabled identifying the transitions without Doppler broadening and tuning of the laser frequencies to these transitions with a precision of a part in . Meanwhile, several types of new laser sources, including many semiconductor lasers, became available. Radiation pressure from photons on atoms, 'out of proportion to the sizes of the atoms' as M. N. Saha remarked, became the tool for slowing down and stopping atoms, and holding them in the diffusive viscous state, called the optical molasses. Ideas and techniques in the new field of laser cooling of atoms grew quickly, adapting from the extensive theoretical knowledge developed in the interaction of light with two level and three level atoms, especially optical pumping and coherent state manipulations. At the low kinetic energy corresponding to the temperature below a milli-Kelvin and speed of a few m/s speed, atoms can be held with very small forces, available from their interaction with magnetic fields and light. A suggestion by J. Dalibard in Paris to combine a small quadrupole magnetic field with appropriately polarized light beams to create a magneto-optical trap (MOT) to spatially confine the laser cooled slow atoms enabled much of the subsequent experimental activity. Realizing the BEC of nearly non-interacting atoms in the gas phase was one primary goal. However, other areas like precision spectroscopy, metrology, matter-wave and atomic state interferometry and detailed study of many features of atomic physics itself, flourished as well.

The cooling of gases to the low temperature required for the transition to a BEC is set by the condition of quantum degeneracy. The maximum density in Einstein's formula happens when a fundamental quantum cell of size of the order of de Broglie wave length contains more than one particle, or equivalently the inter-particle separation is smaller than the quantum wavelength. This amounts to slowing the atoms down to an average velocity of a few cm/s, keeping the density low enough to avoid other types of conventional condensations. The thermal de Broglie wavelength at temperature T is . Then the degeneracy condition is approximately . At a gas density of about atoms/cm 3 , the required refrigeration is to a temperature of around a micro-Kelvin.

In 1995, the first atomic gas BEC was created in a spectacular fashion by researchers from the National Institute of Standards and Technology and University of Colorado, USA, with Rubidium atoms, which have mild repulsive interactions. [START_REF] Anderson | [END_REF] To be fair, it was like a marathon race ran by several competent groups, and a few getting to the podium eventually, with others finishing close. This, as well as the techniques for laser cooling, were recognized as major contributions, with Nobel prizes in 1997 [START_REF] Chu | Phillips won the Nobel prize in 1997 'for development of methods to cool and trap atoms with laser light[END_REF] and 2001. [START_REF] Cornell | for the achievement of Bose-Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates[END_REF] Crucial to this breakthrough was the solution to the problem of holding the atoms in a 'trap' (or a cage) at high density while progressively cooling them to the required ultra-low temperature below a micro-Kelvin. The proven technique for progressive and continuous cooling was called 'forced evaporative cooling' in a magnetic trap. The method involves two important stages. One is to trap the atoms in a three-dimensional magnetic trap. This is possible because the interaction of the magnetic field and the magnetic moment of the atom is 'attractive' for one orientation of the spin and repulsive for the other. Atoms can be spin polarized with light in the favourable orientation. The magnetic field direction changes in the trap, but if the atoms are moving slowly, their spin direction follows the direction of the field and the atoms are stable trapped, except if the atoms pass through the centre of the trap where the magnetic field is zero. Then the spin can flip and the atoms can be lost from the trap. This problem was solved by adding a small rotating magnetic field to the strong quadrupole field of two coils with opposite currents. The next step is the evaporative cooling, allowing the 'hottest' atoms with the highest energies in the trap to escape while the rest of the atoms collide and thermalise. This can be forced by selectively spin flipping atoms above a certain energy, because in a magnetic trap the Zeeman energy and the spin flip energy gap increase outwards from the trap centre. Starting with a radio-frequency field at a resonant frequency corresponding to some peripheral circle in the trap, one progressively reduces the frequency to eject the highest energy atoms, while allowing the rest to thermalise. This way, the average energy and temperature come down continuously, while the hot atoms are lost from the trap, as in the familiar cooling of liquids by evaporation.

From the early days of BEC of alkali atoms like Rubidium and Lithium, the atomic gas BEC list has now grown to span a wide range of properties in the atomic table. The list now includes 1 H, [START_REF] Einstein | Quantum theory of a monoatomic ideal gas -I[END_REF] He*, 7 Li, [START_REF] Robins | Atom lasers: Production, properties and prospects for precision inertial measurement[END_REF] Na, 39 K, 41 K, 40 Ca, 52 Cr, 85 Rb, 87 Rb, 84 Sr, 86 Sr, 88 Sr, 133 Cs, 164 Dy, 168 Er, and 174 Yb (numbers specify the isotope mass number). All the atoms of a Bose-Einstein condensate are in the ground state, occupying the same quantum state. This is reflected in the 'wave nature' of the entire BEC, with the de Broglie wavelength corresponding to the uncertainty momentum of a single atom. This phase coherence of the matter-wave is impressively seen in the direct interference of the condensate released from two traps 13 , much like the double slit interference (Figure 1).

BEC with Bosonic and Fermionic Atoms

Einstein's quantum theory of mono-atomic gases assumed that all atoms behave similar to Bose's light quanta. Starting with the Pauli exclusion principle, physicists learned that fundamental particles with their spin angular momentum quantized as (half-integer spin) behave very differently from particles that have integer spin, like Bose's photons. Formal developments completed the classification into two --the Bose-Einstein statistics to describe the collective behaviour of particles of integer spins and the Fermi-Dirac statistics to describe those of half-integer spins. The former class of particles came to be known as Bosons and the latter class, as Fermions.

All the fundamental particles that make the atom -electron, proton and neutron -are Fermions with spin . An atom is not a fundamental particle. Then how do we decide whether a given atom is a Boson or a Fermion? Most interestingly, and equally enigmatically, what matters physically is the 'external appearance', in that the total spin of the different particles together determines the spinorial nature of the composite. In fact, the phenomenon of superconductivity is a remarkable consequence of this feature. The conducting charge carriers in materials are electrons, which are Fermions. However, two electrons can pair up through a weak attractive interaction available in the material lattice, and then the pairs with the integer spin (0 or 1) behave like Bosons and condense into a free flowing BEC at low temperature. Thus, atoms come in both varieties -Bosons and Fermions. In fact, while one isotope of a species of atom, like Potassium-39 with 19 protons and 20 neutrons can behave as a Boson, another isotope Potassium-40 with one more neutron is a Fermion. Therefore, 39 K can be made into a BEC, but not 40 K. Another such pair is Lithium isotopes of 7 Li and 6 Li. The drastic difference is seen when one cools the two species in similar experiments (figure 2). The quantum pressure of Fermions that resist compression is the reason for the stability, up to a maximum density, of the compact stars like while dwarfs and neutron stars, as first discovered by S. Chandrasekhar. The experimenters were very innovative in designing and executing experiments with Fermionic atoms. Since Fermions in identical states obey the Pauli exclusion principle, spinpolarized cold Fermions in a magnetic trap do not collide and thermalise. Hence, evaporative cooling is not an option to cool them to quantum degeneracy. A clever way is to arrange for 'sympathetic cooling' where a evaporatively cooled Bosonic species is used to simultaneously cool the Fermions in the same trap. [START_REF] Ketterle | Making, probing and understanding ultra-cold Fermi gases[END_REF] If the Fermionic atoms are made to pair up into composite dimers first, they can subsequently become a BEC. Since this route is similar to the pairing up of two electrons to become a 'Cooper pair' that subsequently pass to the condensed phase BEC, as described by the BCS theory of superconductivity, this is refereed to as BCS-BEC route. In the case of the atoms, the strength and sign of the pair interactions are tunable with what is called the Feshbach resonance, in the presence of a magnetic field. Working with Fermionic isotopes of atoms like Lithium and Potassium enabled unprecedented studies that are inaccessible in condensed matter systems. [START_REF] Ketterle | Making, probing and understanding ultra-cold Fermi gases[END_REF]

BEC at home and the world

My involvement with Bose-Einstein Condensates has a home story as well as some prior history abroad. Fascinated with the possibilities of laser cooled atoms, especially in metrology and research in gravity and foundational issues in quantum physics, I sought to learn the techniques from the experts. At the Laboratoire Kastler Brossel (LKB) of the École Normale Supérieure, Paris, I joined a determined group, and a complicated experiment, trying to condense Helium atoms.

The Helium atoms are peculiar in that it is not the atoms in the ground states that have to be cooled. The spectral transitions in Helium suitable for laser cooling is accessible only when one of the electrons is excited, using a discharge. This excited state with the enormous energy of 19.2 eV is 'metastable', with a very long lifetime of about 8000 seconds, as long as the atoms are protected from collisionseven collisions among atoms can violently deexcite the metastable He* atoms.

It was a massive effort, led by the senior colleagues Claude Cohen Tannoudji (Nobel laureate, 1997, for developing ideas and theory of laser cooling of atoms) and Michèle Leduc, along with several of us with experience in different aspects of experimental physics. My association was intermittent, during the regular visits of several months a year. We built up an entirely new experiment with an efficient discharge source of He* atoms and Zeeman slower that slowed down metastable Helium atoms from over a km/s to below 50 m/s, suitable for trapping in the 3D MOT. A large low density MOT received and held these atoms which were subsequently transferred to a magnetic trap, after spin polarizing the atoms. Spin conservation prohibits spin-changing collisional de-excitation in spin-polarized gas. The evaporative cooling was not easy, made more difficult for monitoring by the low sensitivity of camera at these wavelengths. Finally in January 2001, simultaneously with another group led by Alain Aspect in the neighbouring Institute d'Optique [START_REF] Robert | [END_REF] , BEC of metastable He atoms was achieved 16 , thanks to the persistence and courage of the then graduate student Franck Pereira dos Santos.

While at LKB, I started formulating a sure and reliable way of achieving an 'alloptical BEC' with the simpler Rubidium atoms ( 87 Rb), after a discussion with M. S. Chapman from Georgia Institute of Technology, who made a BEC in a purely optical trap, instead of in a magnetic trap. He remained the only one who managed to do so, despite several others trying the method, even years after his feat in 2001. Atoms experience a pure trapping force without any scattering and radiative force, towards the high intensity region of a focused laser beam, if the optical frequency is much lower than the transition frequencies of the atom. While the magnetic trap BEC needed holding cold atoms for over 50 seconds in the trap, optical route needed only a few seconds of evaporation. In such optical dipole traps, evaporative cooling amounted to just slowly reducing the intensity of the trapping laser beam. The novelty was tempting. It was obvious to many that the key to success was in getting a starting sample of large number of cold atoms at high density.

I designed a 10 cm long 2-dimensional MOT as the source of pre-cooled atoms to load a dense 3D MOT in another chamber with lower pressure. It took a few years to arrange the finances and a laboratory at TIFR. But, the asset any funding cannot get was ready at handhard working and enthusiastic graduate students. While the first student Ashok Mohapatra built and established the basic techniques, the next two -Saptarishi Choudhury and Sanjukta Roy -working as an untiring pair, set up the whole scheme for BEC within three years. The experiment consisted of several elements -Lasers for cooling, 2D+MOT, 3D MOT, CO 2 laser for the optical trap, modulators and countless optical elements around a ultra-high vacuum set up at mbar. It involved complicated optical set up and alignment, electronics and programming of pulse sequences for automatic controls, sensitive CCD camera etc. Working with the powerful and invisible infrared beams of the CO 2 laser, at about 50 Watts, had to be done carefully. We innovated resources and methods for a large number of hitherto unfamiliar tasks. Once we were ready, it was only a matter of months to see the first glimpses of the Bose-Einstein condensation. The dipole optical trap was formed with the stable beam of a CO 2 laser, with reduced intensity fluctuations and pointing jitter. The tight trap of about 40 micron size was formed by crossing two focused beams and we had over 10 8 atoms at the start of the evaporative cooling. Within about 1 s, the evaporation takes the sample to a few hundred nano Kelvin and high density exceeding 10 14 atoms/cm 3 , when the condensation happens. 17 We also did some experiments where the condensation was performed in even tighter traps formed as an optical lattice. This is a series of periodic potential wells formed in the standing waves of light, when the focussed CO 2 laser beam is retro-reflected symmetrically and overlapped on itself. The news of this first home-BEC, and one of the very few all optical BEC then, was enthusiastically carried by the Indian press in 2007, especially in Kolkata and Mumbai. The then prime minister of India, Dr. Manmohan Singh, who was also at one time in the governing council of TIFR, was very attentive of the modest yet significant development and sent individual letters of appreciation to the three researchers involved.

It was amusing that many senior theoretical physicists who showed great enthusiasm and activism about making a BEC in Bose's home country remained lukewarm and uninterested when the first Indian BEC was made by our small group at TIFR. Most physicists working on theoretical aspects here have never stepped into the laboratories were things happen. Due to this noninteracting or even mildly repulsive nature, experimenters are rarely motivated by the work of theorists here, which is an unfortunate situation.

It took another 8 years for the second BEC to be formed in an Indian laboratory, of Umakant Rapol at the IISER, Pune. 18 This was in a robust set up with a Zeeman slower as the source of the pre-cooled atoms and the BEC was formed in a quadrupole magnetic trap. This seems to be the only operational facility for BEC based experiments at present in India.

Atoms in a BEC have zero kinetic energy. But they have the quantum zero point energy. From the uncertainty principle we know that . Hence, if the condensed atoms are released from an anisotropic trap, the momentum and the rate of expansion are the largest in the direction in which the atoms are more tightly confined; then the initial anisotropy in space inverts after some expansion! The images in figure 4 show this phenomenon in our Rb BEC.

Figure 4: Expansion of the BEC with the quantum zero point energy of the atoms. The momentum is the largest in the direction of the tightest confinement. This is seen as the inversion of the initial anisotropy, with hardly any expansion in the vertical direction in this case.

BEC in fundamental physics and applications

The Bose-Einstein Condensate is a quantum phase transition, quite unlike the phase transitions we are familiar with, like steam to water, or water to ice etc. Though it involves going below a critical temperature, BEC does not involve any interactions. In fact it happens even with mildly repulsive interaction. It is a new state of matter, where the wave-particle duality manifests in a sample wide coherence. One may say that the quantum wavelength becomes larger than the typical inter-particle separation. But, the effect is more subtle; once the condensation happens, the state remains robust in spite of splitting the condensate or the expansion of the sample that increases the inter-particle separation beyond the quantum wavelength. BEC based experiments and theory have become a new culture (or cult) in physics, with a large number of researchers pursuing a still growing terrain of ideas and techniques. [START_REF] Anglin | Bose-Einstein condensation of atomic gases[END_REF]

Atomic physics

Except for the fact that all atoms in a BEC are in identical quantum state, the lowest energy state, there is no change in the atomic properties. Many atomic species spanning a wide range of atomic properties have been condensed, enabling very focused study of electric structure based atomic interactions. The interest in atoms like Cr and Dy is that they have very large magnetic moments and can be used to study entirely new features of atomic magnetism.

The quantum theory of BEC of weakly interacting atomic gases is described by a multi-particle Schroedinger equation which has a density dependent potentials, called the Gross-Pitaevskii equation [START_REF] Pethick | Bose-Einstein Condensation in Dilute Gases[END_REF] . Since the density is essentially the square of the wavefunction, , this is a nonlinear Schroedinger equation. Superfluidity and collective modes of oscillations, vortex generation in rotating condensates, etc. were the early studies. The enormous amount of work done in the context of atomic gases [START_REF] Pethick | Bose-Einstein Condensation in Dilute Gases[END_REF] is difficult to summarize. I will not attempt the impossible here.

Condensed matter physics

The atomic BEC has contributed most to the study of simple phenomena in condensed matter physics, involving multiparticle collective effects. The relevant phenomena are transport and conductivity, superfluidity and quantum tunneling. Some initial experiments focused on elementary phenomena of vortex formation and their dynamics, by rotating and imparting angular momentum to the atomic BEC with the help of focussed light fields (tweezers). The scene changed drastically by the demonstration of a spectacular transition from a superfluid phase to an insulating phase in an optical lattice, analogous to what might happen for electron conduction in a crystal lattice. The lattice in this case was optical, periodic potential wells made by standing waves of light in 1,2 and 3 dimensions. [START_REF] Greiner | Optical lattices[END_REF] The optical lattice can be easily created, with widely tunable trapping depth and spatial period, by interfering optical beams. The transport of the atoms loaded in the lattice is then controlled by just two parameters; the mild repulsive interaction between the atoms, and inter-site tunneling or 'hoping', related to the depth of the potential wells. This allows the realization of the versatile condensed matter model Hamiltonian systems called the Bose-Hubbard model [START_REF] Bloch | Many body physics with ultra-cold gases[END_REF] and variations (there is also the equivalent for Fermionic cold atoms called the Fermi-Hubbard model). In the real condensed matter situation, electrons in the potential wells of the lattice, there is no tunability of the parameters. In optical lattices, the trap depth proportional to the light intensity is easily varied. This allows a through and nearly complete study of the model Hamiltonian in the laboratory. Since the depth of the potential well is easily tunable, one can study the whole range of phenomena as if the strength of the interaction is varied. In the much discussed example of the phase transition from the superfluid phase to the Mott insulator, the atoms freely tunnel in the superfluid phase when the potential well is shallow; they are collectively coherent, with a common quantum phase. Hence, when the trap is released, matter wave interference results in an interference pattern. As the trap depth is increased, the system goes to the phase transition to an insulator, each atom in individual trap. Then the phase coherence is lost. Atoms released from the trap result in a fully decohered pattern. Several such phenomena, with wider domain of relevance, have been explored with BEC. Two examples of lower dimensional phenomena are the Anderson localization due to weak disorder and the Berezinskii-Kosterlitz-Thouless transition in a Bose gas in layered 2dimensional optical traps.

Metrology

One analogy that is often mentioned is between a laser and a BEC. The analogy is based on the existence of a threshold for the onset of coherence, like a single long wave, and the fact that photons are Bosons. However, a laser is not a Bose-Einstein condensate. Yet, the analogy is helpful in certain applications. A BEC of atoms which is sourced continuously or in pulses from a single source is often called a 'atom laser'. [START_REF] Robins | Atom lasers: Production, properties and prospects for precision inertial measurement[END_REF] At one stage in the peak of these technological developments, BEC was projected as a unique and powerful tool for precision metrology. This was expected from the comparison between the atomic BEC and the immensely useful and universal tool of lasers. The stable wave nature aides in the tremendously precise interferometry, where physical measurements are possible to within a thousandths of the de Broglie wavelength. Concurrently, physical interactions like gravity cause differential phase shift of millions of radians, between two beams separated by a few millimetres! Thus the relative precision could be better than an impressive , in a single run of the experiment. This exceeds the precision achievable in optical metrology in many cases. Laser cooled atoms were already employed in such precision metrology, since the coherence of a BEC is not an essential requirement for such measurements. Atomic interferometry can be performed with just laser cooled atoms, much like optical interferometry without lasers. BEC has certain advantages, like the ability to form a 'interference fringe' directly at the region where the matter waves overlap (fig. 1), but the added experimental difficulty is making the BEC offsets these advantages, except in special metrological situations.

In the important metrological area of atomic clocks, ultra-cold atoms in optical lattices and laser cooled ions have taken the centre stage. BEC with their collective interactions do not figure here because good atomic clocks need to isolate the atoms from even small and subtle interactions. While cold atom interferometry is now commercial technology, BEC based metrology is still confined to the physics laboratory and special situations. I do not foresee a drastic change in this situation in the near future.

Other projected applications

Unlike the decisive technological developments that have been enabled by laser cooled atoms and ions, the technological promise of atomic gas BEC is yet to be realized. Cold atoms have revolutionized atomic clocks, interferometers for navigation and metrology, magnetometry, and improvement of metrological standards and measurement of fundamental constants. In contrast, BEC have proven promise only in some limited areas of inertial and gravitational metrology, till today. Some precision fundamental physics experiments like the test of the equivalence principle, conventionally done using instruments like torsion balances, have been attempted with BEC. The much advertised fundamental difference because of the use of the 'quantum sensor' is a naive interpretation of such tests and need not be given much importance. BEC have been very important in fundamental physics studies of condensed matter and strongly correlated matter, naturally. But I doubt they will assume a comparable stature in applications. The often repeated claim of its potential importance in quantum computing also is unlikely to be realized at a technologically useful scale.

There are some spectacular advances in the technology to produce atomic gas BEC. I mention a recent challenging experiment that produced the BEC of Rubidium atoms in low gravity, in a compact automated laboratory of size 0.5 m diameter and 2.8 m length in space flight on board a sounding rocket. 24 In a parabolic flight with microgravity conditions lasting for about 6 minutes, an international team from Europe successfully produced the BEC on an 'atom chip' magnetic trap several times and conducted a large number of experiments, with the ultimate goal of high precision atom interferometry in space. The production rate was about 10 5 atoms in BEC, in about 1.6 s. This experiment with many technological implications was led by Ernst Rasel, who was also the mainstay in our He* BEC experiment, two decades ago.

A critical element in producing reliable BEC in remotely handled situations, like in space, is what is called the 'atom chip', which has been popular with the cold atom community for two decades. [START_REF] Keil | Fifteen years of cold matter on the atom chip: promise, realizations, and prospects[END_REF] An atom chip is an integrated compact assembly of lithographed current carrying conductors on a reflecting substrate with carefully designed switchable conductor geometries for generating magnetic field configurations suitable for laser cooling and trapping. Usually pre-cooled atoms are loaded close to the surface mirrors of such chips and held in magnetic traps to do further evaporative cooling to a BEC. With careful design, such chip BEC are versatile and reliable. The technological goal is to integrate the entire experimental configuration, including of the laser beams derived from a single laser, on a compact and portable assembly.

S. N. Bose and my research

My scientific trajectory has crossed the terrain named after S. N. Bose a few times by now, transcending the time and generations gap. First was in the Bose-Einstein condensation experiments with Helium and Rubidium.

The next was when I found that that the spin-statistics connection is the consequence of the gravitational coupling of the spin of the particles and the matter-energy of the universe, reaching to the core of the Boson-Fermion distinction. This fundamental feature in physics was revealed in the gravitational paradigm for relativistic physics, called Cosmic Relativity, which I developed. [START_REF] Unnikrishnan | Physics in the Once-Given Universe[END_REF] The spin is the closed current of the charge of gravity, mass-energy, much like the magnetic moment in electromagnetism. Therefore, the phase changes due to the gravitational interaction of an integer spin (Boson) is decisively different from that of an half-integer spin (Fermion).

The third instance is in my exploration of the unphysical nature of the zero point energy (ZPE) of the electromagnetic vacuum in quantum optics and quantum electrodynamics. The infinite (or at least very large) number of wave modes, each carrying the ZPE , implies a very large energy density, which is in stark conflict with cosmology and the existence of the universe with a history of over 10 billion years. If the ZPE in radiation were true physically, then the rate of expansion of the universe should have been enormously larger, and its dynamical evolution very different from that of the observed universe. This is a serious conflict, which cannot be taken lightly. Further, our recent experiments in quantum optics with a novel homodyne scheme clearly indicates the unphysical nature of the vacuum modes. The particles also have ZPE, but that is finite, proportional to the finite number of particles. It struck me that Bose's derivation that rejected the wave modes avoided the infinity of ZPE completely, though that was not the aim of the derivation. Apparently, Bose realized this at some stage later, as I understand from a conversation with Partha Ghose. There is a good case for going back to Bose's approach and reformulating quantum optics and quantum electrodynamics with fresh foundations.

Recently I completed a reconstruction of quantum mechanics without any of the foundational problems, like the collapse of the wavefunction, the quantum measurement problem, and the nonlocality in entanglement. This required 'calling a particle a particle', as a significant half-step. Much like what Bose had done, trusting and insisting on rational consistency is essential in firmly fixing the correct foundations of physics.

Concluding remarks

It is nearly twenty five years after the first atomic gas Bose-Einstein condensate was created. A subtle prediction by Einstein based on a Bose's radically different way of treating the Planck radiation spectrum has motivated many conceptual and technological advances in the field of atomic physics and atom-light interactions. The fascination with atomic gas BEC and attempts at application to new problems continue. It is expected that this charming ultra-cold quantum state of matter has enough steam to go further in holding the attention of physicists for decades to come.

Figure 1 :

 1 Figure 1: Interference of BEC ( 87 Rb) released from a split-source (double-well) on an 'atom chip'.The fringe spacing decreases as the seperation between the coherent sources increase, similar to the double-slit interference in optics.

Figure 2 :

 2 Figure 2: The contrasting behaviour of Bosons and Fermions at quantum degeneracy is spectacularly shown by two isotopes of ultra-cold Lithium atoms ( 7 Li and 6 Li). Bosons condense and Fermions resist compression (from the lab of R. G. Hulet, www.atomcool.rice.edu)

Figure 3 :

 3 Figure 3: Formation of the BEC in the TIFR experiment. The frame on left shows the cold atoms held in a tight optical trap formed by two crossed CO 2 laser beams. Middle: Ultra-cold atoms slowly expanding in the early stages of evaporative cooling. Right: The dense BEC, when the atoms are cooled to about 200 nano-Kelvin.

Acknowledgements

I thank the several colleagues and research students I worked with in the fascinating field of laser cooling and Bose-Einstein condensates, over two decades. Martine Armand, who witnessed much of my engagement at LKB, Paris, also helped in refining this article.


Paul Leplay
email: paul.leplay@insa-lyon.fr

Julien Réthoré

Sylvain Meille

Marie-Christine Baietto

Damage law identification of a quasi brittle ceramic from a bending test using digital image correlation

Keywords: Mechanical properties (C), Strength (C), Al 2 T iO 5 (D), Digital Image Correlation

Although ceramics are generally considered as to be elastic brittle solids, some of them are quasi brittle. These ceramics show a non linear mechanical behaviour resulting most of the time in a difference between their tensile and compressive stress-strain laws. The characterization of their fracture strengths might be biased if elastic linear formulae are used to analyze classical tests like bending tests. Based on Digital Image Correlation (DIC), an efficient technique to measure full field displacements, a methodology is proposed to characterize and model materials with dissymmetric behaviours between tension and compression. Applying specific basis functions for DIC displacement decompositions for bending, compressive and tensile tests, a stress-strain model and its damage law are identified and then validated for aluminium titanate, a damageable micro cracked ceramic at room temperature. This identification method using DIC can obviously be applied to other quasi brittle materials.

Introduction

Compared to other materials, ceramics are generally considered as to be brittle solids, i.e. with an elastic behaviour followed by an unstable failure. Some of them are quasi brittle and show a non linear mechanical behaviour. Depending on the occurence of the non linearity before or after the maximum load, they are respectively called damageable or strain-softening ceramics. Their quasi brittleness results from the evolution of multiple internal mechanisms. We can mention the most common mechanism and materials, the micro crack propagation for natural rocks [START_REF] Kachanov | Microcrack model of rock inelasticity. part ii: Propagation of microcracks[END_REF], ceramics made of crystals with anisotropic thermal expansion coefficients [START_REF] Swain | Quasi-brittle behaviour of ceramics and its relevance for thermal shock[END_REF][START_REF] Thomas | Aluminium titanate: a literature review. i: Microcracking phenomena. British ceramic[END_REF], refractories [START_REF] Simonin | Thermomechanical behavior of high-alumina refractory castables with synthetic spinel additions[END_REF], concrete [START_REF] Shah | Internal cracking and strain softening response of concrete under uniaxial compression[END_REF] and piezo ceramics [START_REF] Fett | Nonsymmetric deformation behavior of lead zirconate titanate determined in bending tests[END_REF]. We can also mention other mechanisms like the shear yielding of the fibril interfaces for bones [START_REF] Mercer | Mechanisms governing the inelastic deformation of cortical bone and application to trabecular bone[END_REF], the crack bridging and the secondary cracking in the frontal process zone for plasters [START_REF] Meille | Mechanisms of crack propagation in dry plaster[END_REF] and the transformation toughening for some zirconia [START_REF] Rauchs | R-curve behaviour of 9ce-tzp zirconia ceramics[END_REF]. A dissymmetric evolution of these internal mechanisms might appear between tension and compression, leading to differences of both constitutive laws and fracture strengths for tensile and compressive loadings.

Many efforts have been made to analytically model the relationship between the presence of microstructural defects like cracks and the macroscopic dissymmetry of behaviours and fracture strengths [START_REF] Kachanov | Effective elastic properties of cracked solids: critical review of some basic concepts[END_REF][START_REF] Dragon | Anisotropic damage in quasi-brittle solids: modelling, computational issues and applications[END_REF][START_REF] Gambarotta | A microcrack damage model for brittle materials[END_REF]. This dissymmetry between tension and compression is also experimentally difficult to quantify. Indeed, bending tests are often preferred in the literature and in the industry to measure fracture strengths, as uniaxial tensile or compressive tests are challenging and costly to perform on ceramics. Bending tests are certainly simpler to implement and work satisfyingly but they also present some risks and limitations [START_REF] Quinn | Design data for engineering ceramics: a review of the flexure test[END_REF]. For instance, the load-displacement curves of damageable ceramics are usually analyzed in the symmetric elastic linear context [START_REF] Lemaitre | Mécanique des Materiaux Solides[END_REF]. The determination of the fracture strengths might be biased. Therefore extrapolating uniaxial properties from bending test data might lead to significant errors for quasi brittle ceramics.

Obtaining reliable uniaxial results through bending tests is not a new idea [START_REF] Nádai | Plasticity[END_REF] but phenomenological approaches with inverse methods are usually preferred [6, [START_REF] Laws | Derivation of the tensile stress-strain curve from bending data[END_REF][START_REF] Mayville | Uniaxial stress-strain curves from a bending test[END_REF][START_REF] Schajer | Inverse calculation of uniaxial stress-strain curves from bending test data[END_REF]. Numerical inverse methods applied to bending tests certainly provide practical evaluations of uniaxial responses. But the possible combination of strain measurement errors and numerical sensitivity may require some important numerical regularizations, like Tikhonov regularization [START_REF] Schajer | Inverse calculation of uniaxial stress-strain curves from bending test data[END_REF]. Besides, these uniaxial curves are just numerical data but they are not associated to some constitutive laws described by mechanically admissible equations. Thus it is better to directly determine a mechanical behavior law suitable for tension and compression. This last requirement is particularly important to use the results in an ulterior finite element simulation.

Special care must be taken on strain measurement during bending tests because of the dissymmetry between tension and compression for damageable ceramics. The traditional way is to use gauges or extensometers in order to measure strains on the bent beam [START_REF] Simonin | Thermomechanical behavior of high-alumina refractory castables with synthetic spinel additions[END_REF][START_REF] Mercer | Mechanisms governing the inelastic deformation of cortical bone and application to trabecular bone[END_REF][START_REF] Schajer | Inverse calculation of uniaxial stress-strain curves from bending test data[END_REF]. Digital Image Correlation (DIC) is preferred here since it offers the capacity to access the full displacement field on the whole sample surface instead of only an averaged strain at some pre-defined location [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF][START_REF] Chu | Applications of digital-image-correlation techniques to experimental mechanics[END_REF]. This technique based on the optical flow conservation enables the identification of elastic properties [START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF][START_REF] Réthoré | Optimal identification of mechanical properties from digital images[END_REF]. The decomposition adopted for the searched displacement field is a key step for an accurate and reliable displacement measurement. The a piori basis functions choice is primary to include into the numerical problem some mechanical hypotheses [START_REF] Besnard | Finite-element displacement fields analysis from digital images: Application to portevin-le châtelier bands[END_REF]. Other specific decomposition function sets have already been proposed to extract mechanical quantities from a specific mechanical problem like brazilian disk, bending or crack propagation tests [START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF][START_REF] Hild | Displacement measurement technique for beam kinematics[END_REF][START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF][START_REF] Réthoré | On the use of nurbs functions for displacement derivatives measurement by digital image correlation[END_REF].

This paper proposes to develop two new specific function sets for four-point bending and uniaxial tests to enable accurate small strain measurements.

Recently, a similar approach for material identification has been achieved with the virtual field method for elastoplastic metals [START_REF] Pannier | Identification of elasto-plastic constitutive parameters from statically undetermined tests using the virtual fields method[END_REF].

To summarize, the aim of the methodology proposed in this work is to develop new specific decomposition functions for DIC adapted for bending and uniaxial tests (Section 2) in order to identify compressive and tensile constitutive laws and fracture stengths for damageable ceramics (Sections 3 and 4). This methodology is illustrated through an application on aluminium titanate, a micro cracked ceramic whose mechanical and fracture aspects have rather little been investigated in literature to the authors' knowledge.

Digital Image Correlation Developments

General Principle : Optical Flow Conservation

DIC is a full field measurement method which enables one to capture both local events and global gradients for the displacement of a structure. It is based on the grey level conservation principle between two images of a same structure. Let us call f the reference image, g the deformed image, u the displacement field, b the noise. This technique consists in comparing two successive images to find the displacement field thanks to the optical flow conservation for each pixel of coordinate x :

f (x) = g(x + u(x)) -b(x) (1)
To determine v, the best approximation possible for u assuming no noise, one can solve this ill-posed inverse problem on chosen regions Ω by minimizing the global error thanks to the functional φ :

φ 2 (x) = Ω [ f (x) -g(x + v(x))] 2 dx (2)
The displacement field is decomposed as a linear combination of basis functions N i :

v(x) = i α i N i (x) (3)
The functional φ 2 becomes a quadratic form of amplitude increments dα i . Using a first order Taylor expansion for dα i , the minimization of φ 2 leads to a linear system written in a matrix form :

Ω (∇g(x + v).N j (x)(∇g(x + v).N i (x)dx dα i = Ω ( f (x) -g(x + v)(∇g(x + v).N j (x))dx (4)
⇔ M ji .dα i = b j (5)
Once the convergence of amplitude increments dα i is achieved, the approximated displacement field v is reconstructed on Ω. The final error between u and v is obtained by comparing the real reference image f to a corrected deformed image calculated from the deformed image g and the displacement approximation v :

error = g(x + v(x)) -f (x) (6)
The displacement field decomposition in equation ( 3) is one of the key tools of DIC. First, it makes feasible to transform an ill-posed inverse problem into a linear system. Second, it enables one to introduce mechanical hypotheses for stress intensity factor measurements [START_REF] Roux | Stress intensity factor measurements from digital image correlation: post-processing and integrated approaches[END_REF], Réthoré et al. for three-point bending kinematics with NURBS functions [START_REF] Réthoré | On the use of nurbs functions for displacement derivatives measurement by digital image correlation[END_REF]. Two new sets of specific functions are now proposed for two particular cases : four-point bending tests with a constant curvature element (Section 2.2) and uniaxial tests with a homogeneous strain element (Section 2.3).

Particular Case : Four-Point Bending Kinematics

The four-point bending test is often used in the scientific literature and the industry to characterize the material strengths, as it requires simple sample geometry and equipment. Compared to the three-point bending test, the volume of loaded material is more important leading to more reliable brittle fracture properties. Assuming the Euler-Bernoulli hypotheses for the four-point bending test, the whole kinematics of the central part can be described with the curvature value and the boundary conditions, because the applied moment is constant.

Some optical approaches have already been used in literature to measure strains during bending tests [START_REF] Meuwissen | Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method[END_REF][START_REF] Eberl | Mechanical characterization of coatings using microbeam bending and digital image correlation techniques[END_REF]. They are nevertheless limited by their incapacity either to detect rigid-body rotations and translations or to capture full fields for large strains.

The bi-linear Q4-DIC approach developed by Besnard et al. [START_REF] Besnard | Finite-element displacement fields analysis from digital images: Application to portevin-le châtelier bands[END_REF] could also be used for the four-point bending test since it is suitable for all mechanical problems with a continuous displacement field. But two drawbacks have to be underlined. First, the noise sensitivity of this general technique may be too important, especially when the displacements are very small like for ceramics (see section 4.1). Second, it is not optimal to average a posteriori the displacement over the beam, since there is no direct access to the curvature due to the vanishing of the second order derivatives of the bi-linear Q4 element.

The basis functions for beam bending developed by Hild et al. [START_REF] Hild | Displacement measurement technique for beam kinematics[END_REF] enables the direct measurement of a beam curvature assumed to be linear over the beam length. This technique is certainly suitable for cantilever beam experiments but not for four-point bending tests.

Therefore developing a specific DIC technique which enables the direct measurement of the constant curvature of a beam is a more efficient approach. This goal can be reached thanks to the a priori choice of basis functions for an element with constant second order spatial derivatives.

[Figure 1 about here.]

For a four-point bending test, the displacement field is expressed according to equation (3) as a sum of five degrees of freedom (dof) within the region of interest Ω :

-Assuming a constant beam curvature, three coefficients (dof : γ 0 ,θ 0 and v x0 ) are required for a two-degree polynomial describing the beam deflection v x :

∂ 2 v x (x, y) ∂y 2 = γ 0 (7) ∂v x (y) ∂y = γ 0 y + θ 0 (8) v x (y) = γ 0 2 y 2 + θ 0 y + v x0 (9)
For an Euler-Bernoulli kinematics, the corresponding horizontal displacement v y is :

v y (x, y) = -x ∂v x (y) ∂y = -x(γ 0 y + θ 0 ) (10)
-Horizontal strain and displacement (dof : v y0 and ǫ yy ) are introduced to accommodate test imperfections :

v y (y) = v y0 + yǫ yy ( 11
)
The vertical position of the neutral axis is not a parameter of this formulation. In fact, equations (7,8,9,10) determine its position at the mid-height of the beam section. Equation ( 11) adds a possible homogeneous strain. A vertical shift of the zero strain axis is thus allowed.

Finally, equation (3) becomes for a four-point bending test kinematics :

              v x v y               =               0.5y 2 y 1 0 0 y 1 0 1 y               .                                               γ 0 θ 0 v x0 v y0 ǫ yy                                               (12)
This specific decomposition being included in matrix system (5), the axial strain becomes directly available with a high accuracy because it is a linear function of the vertical coordinates x. An artificial displacement field is applied in order to test the a priori performance of the proposed basis functions. With translations of amplitudes from 0 to 1 pixel in y direction, the displacement accuracy is 1.2 × 10 -5 pixel and the strain one is 1.1 × 10 -6 % for a mean DIC error of 0.04 %. In x direction, the displacement accuracy is 3.2 × 10 -5 pixel and the strain one is 5.0 × 10 -5 %. The approach developed in this section will henceforth be called 'beam-DIC'.

Particular Case : Uniaxial Tensile and Compressive Kinematics

Even if uniaxial tests are more complicated to implement and run than bending tests, they provide direct information about the material behaviour. To measure displacement and strain fields during such uniaxial tests, the Q4-DIC approach could again be used but the same disadvantages encountered for bending tests appear. As the global formulation of DIC allows displacement decomposition with arbitrary basis functions, it is again better to use specific functions adapted to the test kinematics instead of averaging a posteriori the displacement field in order to extract the searched mechanical quantities.

[Figure 2 about here.]

During tensile or compressive tests, the displacement kinematics can be decomposed with six degrees of freedom :

-Assuming homogeneous strain on the sample surface, three parameters are necessary for vertical and horizontal strains (dof : ǫ xx and ǫ yy ) and one for the shear strain (dof : ǫ xy ) in order to describe the kinematics within the region Ω. One can note that if the material Poisson's ratio was already known, one of the two degrees of freedom ǫ xx or ǫ yy would have been useless.

v x (x) = xǫ xx + y 2 ǫ xy (13) v y (y) = yǫ yy + x 2 ǫ xy (14)
-To accommodate test imperfections, three degrees of freedom are added for the description of the vertical and horizontal translations (dof : v x0 and v y0 ) and for in-plane rotation (dof : θ xy ).

v x (y) = v x0 + yθ xy (15) v y (x) = v y0 -xθ xy ( 16
)
For an uniaxial test kinematics, equation (3) finally becomes :

              v x v y               =               x 0 0.5y y 1 0 0 y 0.5x -x 0 1               .                                                         ǫ xx ǫ yy ǫ xy θ xy v 0x v 0y                                                         (17)
This specific decomposition enables a direct access to the homogeneous strain value for compressive or tensile tests without cumbersome averaging post treatment. With translations of amplitudes from 0 to 1 pixel in x and y directions, the displacement accuracy is 1.1 × 10 -4 pixel and the strain one is 2.3 × 10 -6 % for a mean DIC error of 0.1 %. The approach developed in this section will henceforth be called 'uni-DIC'.

Once specific functions for DIC have been proposed, bending and uniaxial tests can be run to identify mechanical properties of damageable ceramics.

Experimental Procedure

Aluminium titanate

Aluminium titanate Al 2 T iO 5 is a micro cracked ceramic due to the anisotropic thermal expansion coefficient of its crystals [START_REF] Thomas | Aluminium titanate: a literature review. i: Microcracking phenomena. British ceramic[END_REF]. The micro cracks appear at grain boundaries during cooling from the sintering temperature. This is due to the thermal expansion mismatch between crystals. Since closing and healing of these micro cracks occur during later heatings, aluminium titanate has a low thermal expansion exhibiting consequently an excellent thermal shock resistance [START_REF] Kim | Crack healing, reopening and thermal expansion behavior of al2tio5 ceramics at high temperature[END_REF][START_REF] Kim | Excellent thermal shock resistant materials with low thermal expansion coefficients[END_REF]. Secondary phases such as MgO, ZrO 2 , S iO 2 or ZrO 2 are often used to stabilize at high temperature and strengthen pure aluminium titanate [START_REF] Tsetsekou | A comparison study of tialite ceramics doped with various oxide materials and tialite-mullite composites: microstructural, thermal and mechanical properties[END_REF]. This quasi brittle material becomes an excellent potential ceramic for industrial thermostructural applications like, for instance, in the automotive industry with catalyst carriers, thermal insulation liners and gas filters [START_REF] Munz | Ceramics: Mechanical Properties, Failure Behaviour, Materials Selection[END_REF].

The relationship between the sintering process, the microstructure, the secondary silicate phase and the final thermal expansion coefficient has been widely studied [START_REF] Kim | Crack healing, reopening and thermal expansion behavior of al2tio5 ceramics at high temperature[END_REF][START_REF] Kim | Excellent thermal shock resistant materials with low thermal expansion coefficients[END_REF][START_REF] Tsetsekou | A comparison study of tialite ceramics doped with various oxide materials and tialite-mullite composites: microstructural, thermal and mechanical properties[END_REF][START_REF] Giordano | Microstructure and thermal expansion of al2tio5-mgti2o5 solid solutions obtained by reaction sintering[END_REF] but mechanical and fracture aspects are rare in literature [START_REF] Chen | Temperature dependence of mechanical properties of aluminum titanate ceramics[END_REF][START_REF] Hamano | Crack propagation resistance of aluminium titanate ceramics[END_REF][START_REF] Melendez-Martinez | High temperature mechanical behavior of aluminium titanate-mullite composites[END_REF]. Aluminium titanate is however known to have a damageable mechanical behaviour due to the micro crack propagation [START_REF] Hamano | Crack propagation resistance of aluminium titanate ceramics[END_REF].

Samples and Mechanical Tests

The tested material is mainly made of aluminium titanate with a secondary silicate phase. It is a highly porous ceramic with a porosity between 40 and 50 %. The samples were extruded then sintered for a final rectangular section of B×W=5.1×7.3 mm 2 . Five samples were used for each mechanical test at room temperature :

-Four-point bending tests were performed on a hydraulic machine (Instron 8502) with a 5000 N load cell. The lower span was D 1 =60 mm and the upper span D 2 =20 mm. A linear variable differential transducer (LVDT) was used to control the test at a rate of 0.05 mm/min for the deflection at the middle of the beam. Rubber bands have been used to fasten the rollers to the main fixtures (Figure 1).

-Compressive tests were performed on the same hydraulic machine, load cell and crosshead speed. The sample length was about 25 mm.

-Tensile tests were performed on an electromechanical machine (Instron 1195) with a 500 N load cell and a 0.02 mm/min deflection speed. The samples were about 20 mm long and were stuck to the fixtures thanks to a slow-setting glue likely to penetrate into the material pores. A special care for the vertical alignment between the samples and the machine was taken thanks to two cardan joints.

Optical Equipment

A CCD camera has been used to visualize the sample surface during each mechanical test. The maximum image resolution of this camera is 1200×1600 pixels with an 8-bit digitization for grey levels. The acquisition frequency was one image every five seconds. A painted random pattern was spread on the sample surface to enhance the image contrast.

Since the optical magnification of classical lenses depends on the distance to the sample surface, out-of-plane displacements or misalignments might induce artificial magnifications interpreted as in-plane strains by the DIC system.

Therefore a telecentric lens has been used to overcome this possible error since objects have the same magnification at all distances with this type of lenses. Each pixel has a 10.9 µm physical size with a 200 mm telecentric lens.

Results For Aluminium Titanate

Four-Point Bending Test

[Figure 3 about here.]

Figure 3 shows a non linear load-deflection curve. For the five samples, the averaged maximum load P is 22.7±0.4 N for a vertical deflection d of 208±11 µm at the middle of the beam. Since ceramics are known to break at low strains, one can be tempted to use elastic linear formulae [START_REF] Schajer | Inverse calculation of uniaxial stress-strain curves from bending test data[END_REF] and ( 19) to analyze this four-point bending test [START_REF] Timoshenko | Strength of Materials. Part I-Elementary Theory and Problems[END_REF] :

ǫ elastic = 12dW 2D 1 2 + 2D 1 D 2 -D 2 2 ( 18
)
σ elastic = 3P(D 1 -D 2 ) 2BW 2 ( 19
)
where P is the applied load, d the vertical deflection at the beam middle, W the height, B the widht, D 1 and D 2 the lower and upper spans.

These elastic formulae lead to fracture strengths of 5.0±0.1 MPa and failure strains of 0.19±0.1 %. But two problems have to be underlined :

First, formula [START_REF] Schajer | Inverse calculation of uniaxial stress-strain curves from bending test data[END_REF] does not take into account the influence of the shear loading on the global beam deflection d [START_REF] Timoshenko | Strength of Materials. Part I-Elementary Theory and Problems[END_REF].

This might lead to a significant overestimation of the axial strain ǫ (+4 % with our geometrical configuration). Second, these elastic formulae implicitly assume a symmetry between tension and compression -i.e. a neutral axis located at the mid-height of the beam section.

This elastic approach is widely adopted in the literature and in the industry to characterize aluminium titanate at room temperature or at high temperature [START_REF] Chen | Temperature dependence of mechanical properties of aluminum titanate ceramics[END_REF][START_REF] Ohya | Crack healing and bending strength of aluminium titanate ceramics at high temperature[END_REF][START_REF] Oikonomou | Stabilized tialite-mullite composites with low thermal expansion and high strength for catalytic converters[END_REF]. Only Liu and Perera used a different equation [START_REF] Chu | Applications of digital-image-correlation techniques to experimental mechanics[END_REF] to calculate the 'true' fracture strengths of two aluminium titanate ceramics containing magnesium and iron [START_REF] Liu | Long-term thermal stability and mechanical properties of aluminium titanate at 1000-1200˚c[END_REF] :

σ′ true ′ = (D 1 -D 2 ) BW 2 (P + d 2 ∂P ∂d ) (20)
[Figure 4 about here.]

[Figure 5 about here.]

Althougth this formula is better suited for inelastic materials, it is also based on the assumption of a symmetry between tension and compression [START_REF] Nádai | Theory of flow and fracture of solids[END_REF]. In our case, we would have obtained 3.9±0.1 MPa, but this strength value has no physical meaning because of the difference between tensile and compressive mechanical behaviour of the aluminium titanate. This equation has also been used on other damageable ceramics like zirconia [START_REF] Gogotsi | Mechanical behaviour of yttria-and ferric oxide-doped zirconia at different temperatures[END_REF] or synthetic rocks [START_REF] Latella | High-temperature strength behavior of synroc-c[END_REF] without carefully verifying this symmetry hypothesis.

It is thus not possible to quantify the strength of such damageable ceramics from only the deflection and load measurements. Therefore one needs to have access the full strain field during a four-point bending test. The Q4-DIC and beam-DIC approaches are now used to achieve this full field measurement for one of the five samples :

Using the Q4-DIC approach (198 elements / size length = 64 pixels), the measured displacement field amplitudes v y are so small (less than three pixels) that the corresponding strain values ǫ yy are not reliable (Figure 4). Consequently, it is impossible to determine accurately the exact neutral axis position. Using the beam-DIC approach (section 2.2) with one element, the measured displacement field v y and the corresponding axial strain field ǫ yy are more regular and accurate (Figure 5). Even if the zone of interest Ω is wider for the beam-DIC approach (Figure 1), the ratio between inputs (grey level pixels) and outputs (degrees of freedom) per element is higher than for the Q4-DIC approach. Therefore the algorithm convergence is twice faster, the approximated solution is more accurate and the post treatment is simpler.

The axial strain field measured during the bending test reveals a dissymmetric behaviour (Figure 5). Indeed, the neutral axis moves slowly during the test toward the compressive part of the beam. At hte failure load, the beam-DIC approach yields to maximum strains of +0.26 % for the tensile part and of -0.15 % for the compressive one. This neutral axis shift during the test indicates a difference between compressive and tensile behaviours that was previously not taken into account with the elastic linear formulae.

During the bending test, the correlation error calculated with equation ( 6) is homogeneous and the mean error remains between 0.5 and 0.6 % (Figure 5). This low error value confirms that the kinematics hypotheses of the beam-DIC approach are consistent. Besides, this homogeneous error value reveals that there is no visible damage localization until catastrophic failure of the aluminium titanate sample. Since the zone of interest is wider with the beam-DIC approach, the important error at the top corners of the image corresponds merely to the two rubber bands used to fasten the two rollers to the upper fixture.

It has been verified on some images that the kinematics obtained using the beam-DIC approach leads to the same deflection than the value measured using the LVDT at the middle of the beam. In addition to the low mean correlation error, it confirms once again the quality of the kinematics measurement.

Finally, some tensile and compressive tests must be run to go further in the material characterization because of the dissymmetry observed during the four-point bending test. Actually, a phenomenological approach with an inverse method could have been used to apprehend uniaxial behaviours but it has been preferred to search directly mechanically admissible models from experimental data of compressive test (Section 4.2) and tensile test (Section 4.3).

Uniaxial Compressive Test

The measured maximum stress is 27.5±1.1 MPa. Even if the Q4-DIC approach (204 elements / size length = 64 pixels) allows one to identify the axial and shear displacement fields on the sample surface, it is impossible to obtain a reliable value for the axial strain (Figure 6). On the contrary, the uni-DIC approach (Section 2.3) leads directly to a homogeneous strain value (Figure 7) : the axial strain at failure is -0.74±0.03 %.

The stress-strain curve (Figure 8) shows that the mechanical behaviour is elastic linear until a strain of -0.50±0.05

% is reached. The secant modulus remains almost equal to 4.6 GPa until this strain limit. Oscillations of the secant modulus curve are observed for strains ranging from 0.0 to -0.03 % due to a lack of accuracy in both the strain and load measurements. When strains vary from -0.50 to -0.74 %, the progressive increase of the DIC error indicates that the strain is no more homogeneous everywhere on the sample surface. In fact, a strain localization begins at one of the beam corners, so one knows where the unstable crack will appear before the sample failure. For aluminium titanate tested in compression, the failure shows a very limited strain-softening behaviour.

[Figure 6 about here.]

[Figure 7 about here.]

Strains have been quantified according to the beam-DIC approach during the four-point bending test in section 4.1 :

the value of -0.15 % was reached in the compressive part of the beam just before the catastrophic failure. It means that the mechanical behaviour for the compressive part of the bent beam can be considered as elastic linear with a Young modulus of 4.6 GPa.

[Figure 8 about here.]

Uniaxial Tensile Test

The measured maximum stress is equal to 3.0±0.3 MPa for an axial strain of +0.11±0.05 % using the uni-DIC approach (Section 2.3). The stress-strain curve is not linear and the secant modulus descreases progressively from 4.7 to 2.7 GPa when the axial strain increases (Figure 9). As for the compressive test, oscillations of the secant modulus curve are observed when for strains ranging to 0.00 to +0.01 % because of the insufficient accuracy of strain and load measurements. The initial modulus is almost equal to the one measured in compression. This damageable behaviour is linked to the progressive propagation of the grain boundary micro cracks of the aluminium titanate. Since the mean error is low and almost constant, it means that the strain field is really homogeneous without visible damage localization in the sample surface until catastrophic failure.

[Figure 9 about here.]

Failure occurred always close to the glue interface at one of the sample extremities. But it has to be highlighted that the fracture surfaces were clean (i.e. without glue) so that the obtained results do characterize the porous ceramic.

The maximum strain is +0.11 % in the tensile test whereas values of +0.26 % have been measured during the fourpoint bending test. Smaller strains in tension than in bending tests have already been observed in literature [START_REF] Laws | Derivation of the tensile stress-strain curve from bending data[END_REF][START_REF] Bazant | Modulus of rupture: size effect due to fracture initiation in boundary layer[END_REF].

From the bending tests, the Weibull modulus have been roughly estimated to be higher than 40. A difference of sample size cannot totally explain this difference of failure strains. Two hypotheses can be put forth to explain why smaller strain values are reached during an uniaxial test rather than during the previous bending one. First, the rigidity increases at the interface due to the glue, leading surely to a stress concentration in this region. Higher strains might be reached for a tensile test with a better boundary condition control. Second, a structural effect may be taken into account between two different tests [START_REF] Bazant | Modulus of rupture: size effect due to fracture initiation in boundary layer[END_REF] : whereas the crack propagation is quickly unstable during a tensile test, the cracking in the outer layer of a bent beam is forced to remain distributed because of the constraint provided by the uncracked compressed zone : higher strains can thus be reached before failure. The transition from stability to instability depends on the whole stress field of the considered structure. Such structural size effects in bending tests have also been highligthed for mechanisms of crack propagation for other highly porous materials like plaster [START_REF] Meille | Mechanisms of crack propagation in dry plaster[END_REF].

Prediction of the Bending Curve through a Mechanical Model

The aim of this last section is to identify a stress-strain model which can be used henceforth, for instance, in a finite element simulation. The model validation is achieved by predicting the experimental load during a four-point bending test at room temperature.

It has been shown that the mechanical behaviour is elastic for strains from 0.0 to -0.5 % (Section 4.2). The lowest compressive strains measured using the beam-DIC approach being -0.15 %, an elastic law is well suited to describe the compressive behaviour during a bending test :

σ compression (ǫ) = E 0 .ǫ (21)
It has been shown that the mechanical behaviour is damageable in tension (Section 4.3). The fracture strain measured using the uni-DIC approach was +0.11 % during the tensile test, whereas strains of +0.26 % obtained using the beam-DIC approach were reached during the bending test. To overcome this lack of experimental data between +0.11 and +0.25 %, a damage law is proposed to describe the tensile behaviour. The micro crack propagation law is represented by an exponential function D(ǫ) with two new parameters a and b. Such exponential damage laws have already been observed on other quasi brittle ceramics [START_REF] Swain | R-curve behavior and thermal shock resistance of ceramics[END_REF]. The same modulus E 0 is used in order to impose a modulus continuity between tensile and compressive laws. This modulus continuity has been experimentally observed in uniaxial tests (4.6 GPa in compression and 4.7 GPa in tension).

D tension (ǫ) = a.(1 -exp(-ǫ/b)) (22)
σ tension (ǫ) = E 0 .(1 -D(ǫ)).ǫ (23)
[Figure 10 about here.]

To identify the numerical values of the three parameters (E 0 , a, b), the load applied during the four-point bending test is quantified according to the previous constitutive laws. For every pixel of coordinate x, the strain is measured using the beam-DIC approach and the corresponding stress is calculated using equations [START_REF] Hild | Digital image correlation: from displacement measurement to identification of elastic properties-a review[END_REF][START_REF] Réthoré | Optimal identification of mechanical properties from digital images[END_REF][START_REF] Besnard | Finite-element displacement fields analysis from digital images: Application to portevin-le châtelier bands[END_REF]. There are some pixels which correspond to the beam but are just above or below the DIC zone of interest. Knowing the real height W of the section, the strain field has been linearly extrapolated for these few pixels not included in the zone of interest. Then, the load P is calculated by integrating the moment on the beam section :

P = 4B D 1 -D 2 +W/2 -W/2 σ(ǫ(x)).(x - W 2 )dx (24)
[Figure 11 from the load measurements. Figure 10 presents a load-deflection curve with the parameter values : E 0 =4.67 GPa, a=0.874 and b=0.00136 (Figure 11). The load values come from either the load cell or the identified constitutive laws applied to the strain fields measured using the beam-DIC approach. The deflection values come here from the LVDT.

The simulated load curve fits well the experimental data.

It has been checked that the E 0 value is consistent with uniaxial test data and with an ultrasonic measurement which led to an Young modulus of 4.69±0.15 GPa. Finally, it has been verified that these constitutive laws lead for a bending test to the same neutral axis position than the beam-DIC approach on figure 5.

These elastic and damageable laws for aluminium titanate are mechanically admissible [START_REF] Lemaitre | Mécanique des Materiaux Solides[END_REF]. Thus they can directly be used in a finite-element numerical simulation.

[Figure 12 about here.]

Whereas elastic linear formulae gave fracture strengths of ±5.0 MPa in tension and in compression, the identified constitutive law leads to a strength of +3.5 MPa for tension and a stress of -7.1 MPa for compression at failure (Figure 12). Relative errors of 30-40 % were thus done with the elastic formulae. Furthermore, the damageable behaviour of aluminium titanate observed during the bending test is only dictated by the micro crack propagation in the tensile part. Finally, the catastrophic failure of the bent beam without visible damage localization results also from the tensile behaviour.

Conclusion

Elastic linear formulae must not be used to analyze mechanical tests on quasi brittle ceramics, otherwise significant errors might be made concerning the identification of fracture strengths. Therefore a methodology based on Digital Image Correlation (DIC) has been proposed to characterize damageable ceramics which have different mechanical behaviours under tensile and compressive loadings.

DIC is an efficient optical technique for full field strain measurements, especially when associated with a specific displacement decomposition. It enables one to introduce mechanical hypotheses into a numerical problem. Two specific basis function sets have been developed : a constant curvature element for four-point bending tests (beam-DIC approac) and a homogeneous strain element for uniaxial tests (uni-DIC approach). Two main advantages about using specific decompositions have to be reminded. First, the searched mechanical quantities are easily extractable since they are directly among the degrees of freedom of the system. Second, the obtained results are more accurate, so very small strains can be measured. This last point is very important for the characterization of ceramics.

The specific tools developed for DIC have been applied to aluminium titanate at room temperature. During a fourpoint bending test, the non centered neutral axis position indicated a difference between tension and compression.

Uniaxial tests revealed that the mechanical behaviour can be considered as elastic in compression and as damageable in tension. The non linear behaviour and the catastrophic failure of bending tests result exclusively from the micro crack propagation inside the tensile part of the bent beam. Then a stress-strain model and its damage law have been identified and validated by successfully predicting the load curve of the four-point bending test. A difference of 30-40 % was observed for fracture strengths between the identified constitutive laws and the elastic linear formulae. Finally, this model is mechanically admissible so it can be directly used in a finite-element simulation.

This methodology brings a new alternative to characterize quasi brittle ceramics instead of using unadapted linear formulae. In addition to its simplicity, it allows a direct identification of uniaxial stress-strain models from a classical bending test. This methodology has been applied in this paper to aluminium titanate. Obviously, it can be applied to

  into a numerical problem leading to a regularization of the displacement solution v. Following this concept, specific basis function sets have already been proposed in the literature : Besnard et al. for arbitrary mechanical problem with bi-linear Q4 elements [23], Hild et al. for cantilever beam kinematics with a linear curvature element [24], Roux et al.

  about here.] A Levenberg-Marquardt algorithm has been used to solve this inverse problem to identify the values (E 0 ,a,b)

Figure 3 :Figure 4 : 24 200Figure 5 :

 34245 Figure 3: Experimental load-deflection curve for one of the four-point bending tests on aluminium titanate at room temperature

Figure 10 :

 10 Figure 10: Comparison between the identified and experimental applied loads for the bending test

Figure 11 :

 11 Figure 11: Damage law accounting for micro crack propagation for aluminium titanate under a tensile loading

Figure 12 :

 12 Figure 12: Measured and identified stress-strain curves for aluminium titanate

Acknowledgements

The authors wish to thank Fabiano Rodrigues and his colleagues at the CREE (Saint-Gobain group) for providing the samples and financially supporting this research project.

other quasi brittle materials among biomaterials, cements, plasters, concrete, composites, rocks and porous ceramics.

List of Figures y(pixels