Patent Publication Number: US-2012036933-A1

Title: Method for the non-destructive and contactless characterization of a substantially spherical multilayered structure and related device

Description:
The invention generally relates to methods for non-destructive and contactless characterization of multilayered structures with a spherical or substantially spherical geometry having at least two layers, such as for example nuclear fuel particles, notably for a high temperature reactor. These particles typically include five layers. Subsequently, the term of particle will designate such multilayer structures. 
     More specifically, the invention according to a first aspect relates to a method for non-destructive and contactless characterization of a multilayered structure with a substantially spherical geometry comprising at least two layers separated by interfaces. 
     BACKGROUND OF THE INVENTION 
     In the case of nuclear fuel particles for a high temperature nuclear reactor, the latter comprise a fissile core coated with layers of dense or porous pyrocarbon, and of ceramic such as silicon carbide or zirconium carbide. The determination of the density, of the thickness, of the Poisson coefficient and of the Young modulus of the core and of each layer making up the fuel particle is essential for qualification of this fuel. 
     The most currently used method for determining the density is a flotation method. Several control particles are sampled in a batch of particles to be characterized. Each particle is cut out and pieces of each layer are separated in order to carry out density measurements. These pieces are placed in turn in a liquid, the density of which strongly varies depending on temperature. The temperature of the liquid is then varied and it is noted at which temperature the pieces are found “in midwater”. The density of the material making up the piece corresponds to the density of the liquid at said temperature. 
     This method has the drawback of using toxic liquids. Moreover, this characterization method is slow and causes destruction of the particles to be characterized. Finally, its application proves to be extremely unwieldy since the pieces of each layer have to be separated and identified one by one. 
     It does not give any information relating to the Poisson coefficient and to the Young modulus. However, it is possible to evaluate these coefficients by methods which have the disadvantage of being destructive (such as for example micro-indentation) and requiring that the measurements be conducted on pieces of each layer, separated and identified one by one. 
     SUMMARY OF THE INVENTION 
     Within this background, the invention is directed to proposing a characterization method which may be applied to particles, which is non-destructive, respectful of the environment, faster to apply, and which allows access to several characteristics in a single measurement. 
     For this purpose, the invention deals with a non-destructive and contactless characterization method for a multilayered structure with a substantially spherical geometry comprising at least two layers, separated by interfaces, the method comprising the following steps: 
     by a laser, locally heating the structure under thermoelastic conditions so that the structure is set into vibration in a non-destructive way; 
     measuring resonance frequencies of the vibration modes of the structure; 
     inferring from resonance frequencies of the structure, at least one characteristic relating to the integrity, or to the geometry or to the mechanical behavior of the structure. 
     The method may also include one or more of the characteristics below, considered individually or according to all technically possible combinations: 
     the measurement of the resonance frequencies is carried out with an optical measurement device, 
     the optical measurement device comprises an interferometric device, 
     the presence or absence of a crack in the structure is inferred from the resonance frequencies, the presence of resonance frequencies in at least one predetermined frequency band being characteristic of the presence of a crack in the structure, and the absence of a resonance frequency in said or each predetermined frequency band being characteristic of the absence of any crack in the structure, 
     at least one sought geometrical or mechanical characteristic of at least one of the layers, selected from density, thickness, Young&#39;s modulus and Poisson&#39;s coefficient, is inferred from the resonance frequencies of the structure, 
     said sought geometrical or mechanical characteristic is inferred by an inverse method by: 
     a) computing theoretical resonance frequencies from respective sets of theoretical or measured values of the geometrical and mechanical characteristics for said or each layer, including first values of said or each sought geometrical or mechanical characteristic, the set of theoretical or measured values comprising for said or each layer the density, the thickness, the Young modulus and the Poisson coefficient; 
     b) computing the difference between the theoretical resonance frequencies and the measured resonance frequencies; 
     c) selecting a new value for said or each sought characteristic from the set of corresponding theoretical or measured values and by iterating steps a), b) and c) until the computed difference in step b) is less than a predetermined limit, 
     the theoretical resonance frequencies are computed in step a) by an analytical vibratory model of the structure, 
     the inverse method is initialized by computing theoretical initial values for the sought characteristics by inverting a linear vibratory model of the structure, from measured resonance frequencies, 
     in step c), the new values of said or each sought characteristic are computed by a linear vibratory model of the structure, from first values of said or each sought geometrical or mechanical characteristic considered in step a) and differences between the theoretical resonance frequencies and the measured resonance frequencies, computed in step b), 
     the structure is a nuclear fuel particle comprising a core and at least two layers surrounding the core, 
     the nuclear fuel particle comprises, from the interior to the exterior, a fissile material core, a layer of porous pyrocarbon, a first layer of dense pyrocarbon, a ceramic layer and a second layer of dense pyrocarbon, the sought geometric or mechanical characteristics comprising at least two of the characteristics selected from Young&#39;s modulus of the porous pyrocarbon layer, Young&#39;s modulus of the first dense pyrocarbon layer, Young&#39;s modulus of the ceramic layer and the density of the porous pyrocarbon layer, 
     the laser is an intensity-modulated laser, for example a pulsed laser delivering energy comprised between 1 μJ and 1 mJ per pulse, each pulse having a duration comprised between 0.5 and 50 nanoseconds, 
     the method comprises the following steps: 
     measuring the period of the echoes resulting from reflections of elastic waves at the interfaces between the layers; 
     inferring from said period at least one characteristic relating to the geometry or to the mechanical behavior of the structure, 
     the propagation velocity of the elastic waves in one of the layers is inferred from the period of the echoes, depending on the thickness of said layer, 
     Young&#39;s modulus of said layer is determined according to the propagation velocity and to the density of said layer. 
     According to a second aspect, the invention relates to an installation for characterizing a multilayered structure adapted for applying the method above, the installation comprising: 
     a laser capable of locally heating the structure under thermoelastic conditions so that the structure is set into vibration in a non-destructive way; 
     a device for measuring resonance frequencies of the vibration modes of the structure; 
     a computer for inferring from the resonance frequencies of the structure, at least one characteristic relating to the integrity, or to the geometry, or to the mechanical behavior of the structure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Other characteristics and advantages of the invention will become apparent from the detailed description which is given of it below, as an indication and by no means as a limitation, with reference to the appended figures, wherein: 
         FIG. 1  is a schematic equatorial sectional view illustrating an exemplary structure of a nuclear fuel particle for a high temperature reactor; 
         FIG. 2  is a schematic view illustrating an installation for applying the characterization method according to invention; 
         FIG. 3  illustrates an experimental signal collected during the application of the method of the invention for measuring the period of the echoes; 
         FIG. 4  illustrates an experimental signal collected during the application of the method of invention for measuring the vibratory signal of the particle; 
         FIG. 5  is a graphic illustration of the vibratory spectrum as inferred from the curve of  FIG. 4 , showing the resonance frequencies of the excited particle; 
         FIG. 6  is a step diagram illustrating the main steps of the method of the invention; and, 
         FIG. 7  is a graphic illustration showing the measured resonance frequencies by means of the installation of  FIG. 2 , for particles with opening cracks, particles with non-opening cracks and particles which are sound. 
     
    
    
     DETAILED DESCRIPTION OF THE DRAWINGS 
     Opening cracks are cracks opening onto the outer surface of the multilayered structure. Non-opening cracks are cracks, which are not opened at the outer surface of the multilayered structure, the defect then being inside the structure. 
       FIG. 1  schematically illustrates a particle  1  of nuclear fuel for a high or very high temperature reactor (HTR/VHTR). 
     Conventionally, this particle  1  is of a general spherical shape and successively comprises from the interior to exterior; 
     a fissile material core  2 , for example based on UO 2  (these may be other types of fissile material such as UCO, i.e. a mixture of UO 2  and of UC 2  and/or other fissile materials such as compounds based on plutonium, thorium, . . . ), 
     a porous pyrocarbon layer  3 , 
     a first dense pyrocarbon layer  4 , 
     a layer  5  of silicon carbide (or of another ceramic such as zirconium carbide), and 
     a second dense pyrocarbon layer  6 . 
     Upon using such a particle, the porous pyrocarbon is used as a reservoir for fission gases, silicon carbide is used as a barrier against diffusion of fission products, and the dense pyrocarbon ensures mechanical strength of the silicon carbide layer. 
     The core  2  for example has a diameter of about 500 μm, the diameter may vary from 100 μm to 1,000 μm, and the layers  3 ,  4 ,  5  and  6 , have respective thicknesses of 95, 40, 35 and 40 μm, for example. 
     It will be seen that the relative dimensions of the core  2  and of the layers  3 ,  4 ,  5  and  6  have not been observed in  FIG. 1 . 
     These layers, notably the pyrocarbon layers  3 ,  4 ,  6 , are deposited for example by a chemical vapor deposition method applied in an oven with a fluidized bed. 
     The installation illustrated in  FIG. 2  allows: 
     detection of a crack in the particle illustrated in  FIG. 1 ; 
     and/or evaluation of one or more geometrical or mechanical characteristics of the core and/or of one of the layers  3  to  6 . 
     Subsequently in the text, “layer” equally means the core or one of the layers surrounding it. 
     The geometrical or mechanical characteristics which may be evaluated are: the density, the thickness, the Poisson coefficient, the Young modulus. 
     The installation comprises: 
     an optical device  7  for excitation of the particle  1 ; 
     a support  8  on which the particle is placed; 
     a measurement device  9  capable of detecting the vibrations of the particle  1  excited by the device  7 , and of measuring the resonance frequencies of the particle  1  excited by the device  7 ; 
     computing device for detecting a possible crack and/or for evaluating the sought characteristics, from the measured resonance frequencies. 
     The support  8  is provided in order to maintain the particle  1  in position during the measurement, with a minimum contact area between the particle and the support so as not to affect the vibratory behavior of the particle. Preferably, the contact is point-like or according to a circle of small diameter. Preferably, the support  8  includes means for cooling the particle, so that thermal instabilities do not perturb the measurement. 
     The excitation device  7  includes an intensity-modulated laser  11 . The laser  11  delivers pulses of very short duration, comprised between 0.5 ns and 50 ns and for example pulses having a duration of 0.9 ns. 
     The laser  11  delivers at each pulse a power comprised between 1 μJ and 1 mJ, for example 5 μJ. 
     The laser  11  operates at a wavelength comprised between 200 nm and 15,000 nm and having a value of 1047 nm for example. 
     The device  7  includes a set of optomechanical components allowing delivery and shaping of the beam  13  from the laser  11  right up to the particle  1 . 
     The laser  11  is for example of the Nd:YAG type. 
     The laser  11  is adjusted so as to locally heat up the particle  1  so that the latter is excited under thermoelastic conditions. The energy delivered by the laser  11  is deposited with a power density of less than 1 GW/cm 2  in the case of the fuel particle  1  of  FIG. 1 . 
     Indeed, it is known that depending on the irradiation power density, the particle  1  may be excited either under thermoelastic conditions or under material ablation conditions. The limiting power density between both conditions depends on the materials making up the particle  1 . 
     In order to be under thermoelastic conditions, i.e. for non-destructive testing, the power density has to be less than the ablation threshold I S  (in W/cm 2 ) which depends on the thermophysical data hereafter of the material and which is defined by the relationship: 
     
       
         
           
             
               I 
               s 
             
             = 
             
               
                 
                   ( 
                   
                     
                       π 
                        
                       
                           
                       
                        
                       K 
                        
                       
                           
                       
                        
                       ρ 
                        
                       
                           
                       
                        
                       C 
                     
                     
                       4 
                        
                       
                           
                       
                        
                       
                         τ 
                         L 
                       
                     
                   
                   ) 
                 
                 
                   1 
                   / 
                   2 
                 
               
                
               
                 ( 
                 
                   
                     Θ 
                     v 
                   
                   - 
                   
                     Θ 
                     i 
                   
                 
                 ) 
               
             
           
         
       
     
     with K being the heat conductivity; ρ the specific gravity; C the mass heat capacity; Θ v  the vaporization temperature; Θ i  the initial temperature; τ L  the duration of the laser pulse. 
     Under thermoelastic conditions, the material making up the particle  1  at least partly absorbs the energy delivered by the laser beam. The delivered power is variable over time because of the modulation of the laser  11 . This causes a modulation of the heat expansion of the material making up the particle  1 , which in turn causes variation of the mechanical stresses within the material. Consequently, a mechanical vibration occurs within the particle  1 . These vibrations will be detected by the measurement device  9 . 
     When the irradiation power density of the laser  11  exceeds a limiting value, the pulses of the laser beam cause detachment of the material making up the particle  1 . These are then the ablation conditions. 
     The measurement device  9  includes an interferometric device  17  and a computer  19 . The interferometric device  17  includes a laser  21  producing a beam  22 , a splitter  23  dividing the beam  22  into two light waves  24  and  25 , and a detector  27 . The first light wave  24  is the reference wave which is sent towards the detector  27  either directly or indirectly by means of optomechanical components. The optical phase and the polarization of the reference wave  24  may be modified by one of these optomechanical components. 
     The second light wave  25  illuminates the particle  1  directly or indirectly by means of optomechanical components. It illuminates the particle  1  either in one point or in an extended area. The wave  25 , after having been reflected or diffused by the particle, forms a reflected wave  29  directed towards the detector  27  by means of optomechanical components, where it interferes with the reference wave  24 . One of these optomechanical components may modify the optical phase and the polarization of this wave  29 . 
     The vibrations of the surface of the particle  1  modify the optical phase of the wave  25  when the latter is reflected or diffused by the particle  1 . This modification of the phase is expressed by a change in the light intensity which is recorded by the detector  27 . 
     The laser  21  of the interferometric device is a continuous laser having a coherence length comprised between 15 cm and 300 m. It has variable power comprised between 5 mW and 5 W, for example 10 mW. 
     The detector  27  is capable of collecting the vibration of the surface of the particle either in one point or on an extended area of the particle. The collected information is transmitted to the computer  19 . 
     The interferometric device  17  may for example be a stabilized homodyne Michelson interferometer. 
     The signal collected by the detector  27  is illustrated in  FIG. 3 . In this exemplary embodiment, the particle  1  is a particle of nuclear fuel which includes a core and the layers  3  to  5 , but not the second dense pyrocarbon layer  6 . 
     The curve of  FIG. 3  includes several peaks  51  with large amplitudes, regularly spaced out, and a large number of other peaks with smaller amplitudes. 
     The curve of  FIG. 3  illustrates the vibratory response of the particle to a pulse from the laser  11 . The energy deposited by the pulse on the outer surface of the particle is converted by generating elastic waves propagating towards the inside of the latter. Having arrived at the interface between the outermost layer and the underlying layer, a portion is reflected and a portion is transmitted to the underlying layer. The reflected elastic wave, upon arriving at the outer surface of the particle will produce a displacement of the surface which appears as a peak in  FIG. 3 , as well as a reflection of a portion of the wave towards the inside of the layer. These elastic waves will thus accomplish several round trips in the outermost layer of the particle, generating echoes. Every time the elastic waves arrive at the outer surface of the particle, the displacement of the surface which it produces, is detected. Every time the elastic waves arrive at the interface with the underlying layer, a portion of the energy of the wave is transmitted to this underlying layer. 
     The same phenomenon is produced again in the underlying layer and in each of the other layers of the particle. 
     Thus on the curve of  FIG. 3 , a large number of peaks are detected. The four clearly marked peaks  51  correspond to the elastic waves detected after one round trip of the elastic wave in the outermost layer of the particle, two round trips of the elastic wave in said outermost layer of the particle, and three round trips of the elastic wave in the outermost layer of the particle and four round trips of the elastic wave in the outermost layer of the particle respectively. 
     The period separating the peaks  51  from each other therefore corresponds to the duration during which the elastic wave covers twice the thickness of the outermost layer of the particle. The propagation velocity of the elastic wave in the outermost layer of the particle is inferred from this duration further called period of the echoes, if the thickness of the outermost layer is moreover known. From this velocity, Young&#39;s modulus of the relevant layer may be inferred if its density is known or conversely its density if the Young modulus is known. If the velocity of the elastic wave is known, then it is possible to determine the thickness of this layer. The thicknesses and the densities may be known by means of a radiography method as the one described in the patent application of the applicant having the file number FR0606950. 
     The computer  19 , from the time signal detected by the detector  27  ( FIG. 4 ) may compute the spectrum of the resonance frequencies of the vibration modes of the particle  1 . This spectrum is illustrated in  FIG. 5 , and corresponds to the signal of  FIG. 4 . It is obtained by computing the fast Fourier transform of the digitized time signal of  FIG. 4 , performed by the computer  19 . 
     Each group of frequencies of the spectrum of  FIG. 5  corresponds to resonance frequencies of a vibration mode of the particle  1 , as measured experimentally. 
     Indeed, a perfectly spherical particle has spheroidal vibration modes noted as nSL, where L is an integer called an orbital number and N is another integer designating the order of occurrence of the spheroidal modes SL. The modes of type nSL have a resonance frequency which is 2L+1 fold degenerate, the mode of type nS 2  having for example a resonance frequency which is degenerate (2×2)+1=5 fold. As these particles have substantially spherical geometry (sphericity defects), group theory provides lifting of degeneracy of the resonance frequencies of a vibration mode of the nSL type. Therefore, these resonance frequencies may be distinct over an interval, as shown in  FIG. 5 . The resonance frequencies of four vibration modes  1 S 1  to  1 S 4  are noted as  1 S 1  to  1 S 4  in  FIG. 5 . 
     The computer  19  will then consider several vibration modes of the particle  1  and will determine by computation, from the resonance frequencies measured experimentally for the relevant vibration modes, one or more geometrical or mechanical characteristics of the particle, according to the procedure illustrated in  FIG. 6  (resolution of the inverse problem). 
     These characteristics are selected from the thickness, the density, the Young modulus and the Poisson coefficient of each of the layers which make up the particle  1 , for example of the core  2  and/or of each of the layers  3  to  6  for the particle of  FIG. 1 . For a particle with N layers, the characteristics are selected from 4N possibilities. 
     For the nuclear fuel particle of  FIG. 1 , the computer will consider certain vibration modes, such as for example the modes  1 S 1  to  1 S 4 . It will determine from the 20 possible characteristics, for example, Young&#39;s moduli E 3  and E 5  of the porous pyrocarbon layer  3  and of the ceramic layer  5 . 
     To do this, the computer  19  will consider for each relevant vibration mode nSL, a so-called experimental resonance frequency. This frequency may for example be selected in the spread interval of the resonance frequencies corresponding to the lifting of degeneracy of the nSL mode. The Monte Carlo method is used for randomly drawing this experimental frequency in a spread interval of the resonance frequencies corresponding to the lifting of degeneracy of the nSL mode. This frequency is retained if it allows convergence of the computing method for the characteristics to be determined The Monte Carlo method also allows evaluation of the uncertainties on the sought characteristics. The experimental frequencies selected for the different relevant vibration modes make up the vector of the experimental frequencies. 
     The computer  19  then computes, from estimated values of the characteristics of the layers of the particle  1 , for example of the core  2  and of the four layers  3  to  6  of the particle of  FIG. 1 , resonance frequencies computed for the relevant vibration modes. 
     The computer  19  starts with the estimated values for the thickness, the density, the Poisson coefficient and Young modulus of each of the layers making up the particle  1 , i.e. for example in the case of nuclear fuel, a total of 20 values. In particular in this case, the computer  19  for Young&#39;s moduli of layers  3  and  5 , the determination of which is sought, considers first values obtained from experimental resonance frequencies, as described later on. The other estimated values are realistic values, having been measured or estimated by computation for particles of nuclear fuels with structures close to the one to be characterized. 
     The most currently used method for directly computing the vibration modes of multilayered objects is the use of simulation software packages by finite elements. This method which has the advantage of simulating objects with geometries close to reality (having defects) has a major drawback which is the computing time which excludes them from real-time characterization in industrial control. 
     In the invention, on the contrary, the resonance frequencies of the vibration modes are computed by means of an analytical vibratory model of the particles. 
     With this analytical vibratory model, it is possible to solve the equation of elastic waves in the case of a multilayered structure with spherical symmetry consisting of N domains and comprising a spherical core and N−1 layers, separated by N−1 spherical interfaces, the N domains are considered as being continuous, elastic, isotropic and homogeneous media. Each numbered (n) domain is characterized by its Young&#39;s modulus E n  its thickness ep n , its specific gravity ρ n  and as well as its Poisson coefficient v n . It is assumed that adhesion is perfect between two adjacent domains. The external layer (n=N) has its outer surface free, forming an interface with the external medium. For example for nuclear fuel particles N=5. 
     The physical parameters required for solving the problem are the specific gravity ρ n , the longitudinal velocity c L,n  and transverse velocity c T,n  of the elastic waves in the different domains which make up the particle (with n=1, 2, . . . , N). 
     The expression of both of these velocities as a function of the mechanical characteristics of each layer is: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         c 
                         
                           L 
                           , 
                           n 
                         
                       
                       = 
                       
                         
                           
                             
                               E 
                               n 
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   v 
                                   n 
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 ρ 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     v 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     v 
                                     n 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         c 
                         
                           T 
                           , 
                           n 
                         
                       
                       = 
                       
                         
                           
                             E 
                             n 
                           
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               
                                 ρ 
                                 n 
                               
                                
                               
                                 ( 
                                 
                                   1 
                                   + 
                                   
                                     v 
                                     n 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
                 
             
           
         
       
     
     The equation of propagation of elastic waves (also called the equation of motion) in each layer n is: 
     
       
         
           
             
               
                 
                   ∂ 
                   2 
                 
                  
                 
                   
                     u 
                     → 
                   
                   n 
                 
               
               
                 ∂ 
                 
                   t 
                   2 
                 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       c 
                       
                         L 
                         , 
                         n 
                       
                       2 
                     
                     - 
                     
                       c 
                       
                         T 
                         , 
                         n 
                       
                       2 
                     
                   
                   ) 
                 
                  
                 
                   
                     ∇ 
                     → 
                   
                    
                   
                     ( 
                     
                       
                         ∇ 
                         → 
                       
                        
                       
                         · 
                         
                           
                             u 
                             → 
                           
                           n 
                         
                       
                     
                     ) 
                   
                 
               
               + 
               
                 
                   c 
                   
                     T 
                     , 
                     n 
                   
                   2 
                 
                  
                 Δ 
                  
                 
                     
                 
                  
                 
                   
                     u 
                     → 
                   
                   n 
                 
               
             
           
         
       
     
     Wherein {right arrow over (u)} n ({right arrow over (r)},t) represents the displacement field in the domain n. 
     In a domain n, the displacement field {right arrow over (u)} n  which solves the wave equation for objects of spherical symmetry (r,θ,φ) is expressed as a function of a scalar potential ψ 1,n  and of vector potentials {right arrow over (ψ)} 2,n  et {right arrow over (ψ)} 3,n  such as: 
         {right arrow over (u)} ( {right arrow over (r)},t )={right arrow over (∇)}ψ 1,n +{right arrow over (∇)} ({right arrow over (∇)} {right arrow over (ψ)} 2,n )+{right arrow over (∇)} {right arrow over (ψ)} 3,n  
 
     The vector potentials are radial, being expressed as {right arrow over (ψ)} 2,n =rψ 2,n {right arrow over (e)} r  and {right arrow over (ψ)} 3,n =rψ 3,n {right arrow over (e)} r  with {right arrow over (e)} r  the unit vector along the radial direction. 
     The potentials ψ j,n  verify d&#39;Alembert&#39;s equation with different velocities 
     
       
         
           
             
               
                 
                   
                     ∂ 
                     2 
                   
                    
                   
                     Ψ 
                     
                       j 
                       , 
                       n 
                     
                   
                 
                 
                   ∂ 
                   
                     t 
                     2 
                   
                 
               
               - 
               
                 
                   c 
                   
                     j 
                     , 
                     n 
                   
                   2 
                 
                  
                 
                   
                     ∇ 
                     2 
                   
                    
                   
                     Ψ 
                     
                       j 
                       , 
                       n 
                     
                   
                 
               
             
             = 
             0 
           
         
       
     
     with j=1, 2, 3. 
     The expression of the velocities is expressed by 
     
       
         
           
             
               c 
               
                 j 
                 , 
                 n 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         
                           c 
                           
                             L 
                             , 
                             n 
                           
                         
                          
                         si 
                          
                         
                             
                         
                          
                         j 
                       
                       = 
                       1 
                     
                   
                 
                 
                   
                     
                       
                         
                           
                             c 
                             
                               T 
                               , 
                               n 
                             
                           
                            
                           si 
                            
                           
                               
                           
                            
                           j 
                         
                         = 
                         2 
                       
                       , 
                       3 
                     
                   
                 
               
             
           
         
       
     
     A particular solution of d&#39;Alembert&#39;s equation for spherical geometry in each domain is of the form (in spherical coordinates) for a sinusoidal vibration of angular frequency ω, written as: 
       ψ j,n ( r,θ,φ,t )=[ A   j,n   L,m   j   L ( k   j,n   r )+ B   j,n   L,m   n   L ( k   j,n   r )] Y   L   m (θ,φ)×exp(− iωt )
 
     wherein ω=k j,n ×c j,n  is the angular velocity Y L   m (θ,φ)=Y L   m,s (θ,φ) is the function of spherical harmonics of harmonic degree L≧0 and of azimuthal order m (−L≦m≦L). A j,n   L,m  and B j,n   L,m  are constants. 
     The angular portion Y L   m,c (θ,φ) corresponds to the real part of non-normalized spherical harmonics defined by: 
         Y   L   m,c (θ,φ)= Re[Y   L   m (θ,φ)]= Re[P   L   |m| (cos θ)×exp( im φ)]= P   L   |m| (cos θ)×cos( m φ)
 
     The angular portion Y L   m,s (θ,φ) corresponds to the imaginary part of non-normalized spherical harmonics defined by: 
         Y   L   m,s (θ,φ)= Im[Y   L   m (θ,φ)]= Im[P   L   |m| (cos θ)×exp( im φ)]= P   L   |m| (cos θ)×sin( m φ)
 
     The radial portion is expressed from spherical Bessel functions of the first kind 
     
       
         
           
             
               
                 j 
                 L 
               
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 
                   
                     x 
                     L 
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           1 
                           x 
                         
                       
                        
                       
                          
                         
                            
                           x 
                         
                       
                     
                     ) 
                   
                 
                 L 
               
                
               
                 ( 
                 
                   
                     sin 
                      
                     
                         
                     
                      
                     x 
                   
                   x 
                 
                 ) 
               
             
           
         
       
     
     and of the second kind 
     
       
         
           
             
               
                 n 
                 L 
               
                
               
                 ( 
                 x 
                 ) 
               
             
             = 
             
               
                 
                   
                     x 
                     L 
                   
                    
                   
                     ( 
                     
                       
                         - 
                         
                           1 
                           x 
                         
                       
                        
                       
                          
                         
                            
                           x 
                         
                       
                     
                     ) 
                   
                 
                 L 
               
                
               
                 ( 
                 
                   - 
                   
                     
                       cos 
                        
                       
                           
                       
                        
                       x 
                     
                     x 
                   
                 
                 ) 
               
             
           
         
       
     
     The 2L+1 fold degeneracy of the resonance frequency of the nSL mode gives the possibility of only considering the mode corresponding to m=0 for computing the resonance frequency. The modes m=0 (an axisymmetrical mode) do not depend on φ like: 
       ψ j,n ( r,θ,t )=[ A   j,n   L,0   j   L ( k   j,n   r )+ B   j,n   L,0   n   L ( k   j,n   r )] Y   L   0,c (θ)×cos(ω t )
 
     because Y L   0,s (θ,φ)=P L   |0| (cos θ)×sin(0×φ)=0. 
     The solution in the central domain (n=1), corresponding to the core should be limited in its centre whence B j,n   L,0 =0 since n L (0)=∞ is of the following form: 
       ψ j,n ( r,θt )= A   j,n   L,m   j   L ( k   j,n   r ) Y   L   m,c (θ)×cos(ω t )
 
     In order to find the vibration eigenmodes, we should consider boundary conditions at the interfaces and on the free surface: therefore, the wave equation should be solved inside the multilayered sphere, meeting the continuity conditions of displacement and stresses at the interfaces (perfect adhesion), except for the free surface, where the stress is cancelled. 
     Given the spherical symmetry of the object, the normal at any point of the interface is in a radial direction i.e. along {right arrow over (n)}=(n r ,n θ ,n φ )=(1,0,0). The components of the stress in spherical coordinates are calculated with: 
     
       
         
           
             
               
                 ( 
                 
                   
                     
                       
                         σ 
                         rr 
                       
                     
                     
                       
                         σ 
                         
                           r 
                            
                           
                               
                           
                            
                           θ 
                         
                       
                     
                     
                       
                         σ 
                         
                           r 
                            
                           
                               
                           
                            
                           ϕ 
                         
                       
                     
                   
                   
                     
                       
                         σ 
                         
                           r 
                            
                           
                               
                           
                            
                           θ 
                         
                       
                     
                     
                       
                         σ 
                         
                           θ 
                            
                           
                               
                           
                            
                           θ 
                         
                       
                     
                     
                       
                         σ 
                         
                           θ 
                            
                           
                               
                           
                            
                           ϕ 
                         
                       
                     
                   
                   
                     
                       
                         σ 
                         
                           r 
                            
                           
                               
                           
                            
                           ϕ 
                         
                       
                     
                     
                       
                         σ 
                         
                           θ 
                            
                           
                               
                           
                            
                           ϕ 
                         
                       
                     
                     
                       
                         σ 
                         
                           ϕ 
                            
                           
                               
                           
                            
                           ϕ 
                         
                       
                     
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     
                       1 
                     
                   
                   
                     
                       0 
                     
                   
                   
                     
                       0 
                     
                   
                 
                 ) 
               
             
             = 
             
               ( 
               
                 
                   
                     
                       σ 
                       rr 
                     
                   
                 
                 
                   
                     
                       σ 
                       
                         r 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                   
                 
                 
                   
                     
                       σ 
                       
                         r 
                          
                         
                             
                         
                          
                         ϕ 
                       
                     
                   
                 
               
               ) 
             
           
         
       
     
     The displacement components are: 
     
       
         
           
             
               u 
               → 
             
             = 
             
               ( 
               
                 
                   
                     
                       u 
                       r 
                     
                   
                 
                 
                   
                     
                       u 
                       θ 
                     
                   
                 
                 
                   
                     
                       u 
                       ϕ 
                     
                   
                 
               
               ) 
             
           
         
       
     
     The sphere comprises N domains and therefore N−1 interfaces. 
     At the interfaces and at the core (domain 1) 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         
                           
                             u 
                             → 
                           
                           i 
                         
                          
                         
                           ( 
                           
                             R 
                             i 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           
                             u 
                             → 
                           
                           
                             i 
                             + 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             R 
                             i 
                           
                           ) 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         
                           σ 
                           
                             rr 
                             
                               
                                 r 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               
                                 r 
                                  
                                 
                                     
                                 
                                  
                                 ϕ 
                               
                             
                           
                           i 
                         
                          
                         
                           ( 
                           
                             R 
                             i 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           σ 
                           
                             rr 
                             
                               
                                 r 
                                  
                                 
                                     
                                 
                                  
                                 θ 
                               
                               
                                 r 
                                  
                                 
                                     
                                 
                                  
                                 ϕ 
                               
                             
                           
                           
                             i 
                             + 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             R 
                             i 
                           
                           ) 
                         
                       
                     
                   
                 
               
                 
             
           
         
       
     
     with i=1, . . . , N−1 being the number of the layer and R i  its outer radius. 
     At the free surface 
     
       
         
           
             
               
                 σ 
                 
                   rr 
                   
                     
                       r 
                        
                       
                           
                       
                        
                       θ 
                     
                     
                       r 
                        
                       
                           
                       
                        
                       ϕ 
                     
                   
                 
                 N 
               
                
               
                 ( 
                 
                   R 
                   N 
                 
                 ) 
               
             
             = 
             0 
           
         
       
     
     The expression of the boundary conditions leads to an eigenvalue equation. Solving the eigenvalue equation gives the eigenfrequency of each mode. 
     With the analytical vibratory model it is possible to determine for the relevant vibration modes, the vector of the computed resonance frequencies F calc , corresponding to the vector of the experimental resonance frequency F exp . For example in the case of the nuclear fuel particle of  FIG. 1 , at least four vibration modes may be identified in  FIG. 5 , and the vectors of the experimental and computed frequencies each have four components. 
     For solving the inverse problem, the computer  19  evaluates whether the quadratic distance between the experimental resonance frequencies and the computed resonance frequencies is less than a predetermined limit L. For this, the computer uses the following formula: 
       ∥ F   exp   −F   calc ∥ 2   &lt;L  
 
     wherein L is the predetermined limit, ∥ ∥ designates the Euclidean norm of a vector. Both frequency vectors have the same number of components which is the number of relevant vibration modes. 
     If the quadratic distance is less than L, then the computer considers that the first values of the sought characteristics (for example Young&#39;s moduli E3 and E5), used for evaluating the computed resonance frequencies, are satisfactory and retains them as final values. 
     On the contrary, if the quadratic distance is greater than the limit L, the computer  19  performs an additional iteration. 
     For this purpose, the computer  19  generates new values of the sought characteristics (for example Young&#39;s moduli E3 and E5). These new values of the characteristics are generated by means of standard error minimization routines which exist in different computing software packages. 
     These standard routines generally require several tens of iterations which requires excessive computing time. 
     The invention uses a linear function which computes in an approximate way the resonance frequencies and which will subsequently be called a linear vibratory model. 
     This linear vibratory model allows rapid computation of the resonance frequencies of the particle for the relevant vibration modes. 
     In this model, the vector F of the resonance frequencies is written as a sum of a linear function F 0 +S·X and of an error vector E i.e. F=F 0 +S·X+E, 
     with F=(f i ) 1=1, . . . , m , a vector with m components of resonance frequencies of the vibration modes obtained experimentally or obtained with the analytical vibration model (target frequencies); F 0  is a constant vector with m components; X=(x α ) a=1, . . . , n  is a vector with n components of the sought characteristics; S is the sensitivity matrix of dimensions n×m; E is an error vector with m components. 
     This formalization allows acceleration of the convergence of the quadratic deviation minimization function. 
     The use of the least squares method (multilinear regression) on the relationship F exp =F 0 +S·X+E leads to the first values (estimations) of the sought characteristics X 1 . 
         X   1 =( t   S·S ) −1 · t   S ·( F   exp   −F   0 )
 
     wherein F exp  corresponds to the vector of the experimentally determined resonance frequencies. 
     From the value of X 1 , the vector of the resonance frequencies is computed by means of the analytical vibratory model. These computed frequencies F 1  are expressed in the following way by means of the linear model: 
     
       
      
       F 
       1 
       =F 
       0 
       +S·X 
       1 
       +E 
       1  
      
     
     The deviation between the experimental frequencies and the computed frequencies is expressed in the following way: 
       ( F   exp   −F   1 )= S ·( X−X   1 )+( E−E   1 )
 
     At each iteration, the error vector E′ 1 =E−E 1  decreases. 
     The second estimation of the sought characteristics is computed with 
         X   2   =X   1   +ΔX=X   1 +( t   S·S ) −1 · t   S ·( F   exp   −F   1 )
 
     At each estimation or iteration, the deviation between the experimental and computed frequencies decreases. 
     In the same way, the value of the vector X considered at iteration i+1 (X i+1 ) is inferred from the value of the vector X used at iteration i (X i ) by using the following equation (method of least squares): 
         X   i+1   =X   i +( t   S·S ) −1 · t   S ·( F   exp   −F   i )
 
     wherein F i  is a vector with m components corresponding to the resonance frequencies obtained with the analytical vibratory model starting from the characteristics X i . 
     This recurrence relationship leads to two convergent series: one convergent series X i  converging towards the limit X calc  of the sought characteristics, and one convergent series F i  of the frequencies converging towards a limit F calc  such that ∥F exp −F calc ∥ 2 &lt;L. The values forming X calc  are then retained by the computer as final values. 
     In the example of the nuclear fuel particle of  FIG. 3 , m has the value four (number of experimentally identified eigenmodes) and n has the value two (number of characteristics to be sought). 
     The computer  19  carries out several iterations, by considering at each iteration new values of the two sought Young moduli E 3  and E 5 , estimated by means of the inversion of the linear model, until the quadratic deviation between the computed frequencies and the experimental frequencies is less than the predetermined limit. 
     In practice, consideration of four experimental resonance frequencies is sufficient for determining at least two characteristics from the twenty characteristics mentioned above. 
     If five experimental resonance frequencies are considered, it is possible to determine at least three or four of the twenty characteristics. 
     If an even larger number of resonance frequencies is considered, it is possible to determine more than four of the twenty characteristics. 
     In order to determine whether the particle  1  includes cracks, the computer  19  considers the spectrum of the resonance frequencies of the particle  1 , and determines whether it includes resonance frequencies in certain predetermined frequency intervals. 
     Experimental results corresponding to measurements carried out on different types of particles are gathered in  FIG. 7 . Each horizontal line corresponds to the spectrum of a particle. These spectra were obtained with an installation like the one illustrated in  FIG. 2 . In each line, the symbols (circle, cross, plus sign, etc.) are placed at each of the main resonance frequencies of the vibration modes, as measured experimentally. 
     The upper line corresponds to a particle having a crack opening out onto the outer surface of the particle. 
     The intermediate line corresponds to a bead which is sound, i.e. not having any cracks. 
     The line at the bottom corresponds to a particle having a non-opening defect, i.e. a crack which is not open at the outer surface of the particle. 
     Moreover, frequency ranges in which resonance frequencies of sound beads are found, are materialized by vertically elongated rectangles. Between these frequency ranges are found other ranges referenced as BI 1  to BI 5  (forbidden band) in  FIG. 7 , in which a resonance frequency for particles which are sound, is never found. 
     Thus, in order to determine whether the particle includes cracks or not, the computer determines whether some of the resonance frequencies of the experimentally measured vibration modes for the particle are found in one of the intervals BI 1  to BI 5 . The intervals BI 1  to BI 5  are predetermined intervals, depending on the type of beads, on the nature of the layers, on the thickness of the layers, etc. These intervals are experimentally determined, by considering a large number of particles including defects and also considering a large number of sound particles. 
     The method described above is not limited to the detection of opening or non-opening cracks in the particle  1 . According to this same principle, it is possible to detect decohesions between layers, abnormal porosities in certain layers, sphericity flaws. 
     By decohesion is meant areas where two contiguous layers do not have proper adhesion to each other at their mutual interface. By porosity is meant an area of a layer where the material is abnormally porous because of the existence of micro-cavities within the material. 
     The method described above has multiple advantages. 
     By locally heating the multilayered structure to be characterized under thermoelastic conditions, by means of a laser and by inferring at least one characteristic related to the integrity or to the geometry or to the mechanical behavior of the structure from the resonance frequencies of the vibration modes of the structure, it is possible to characterize this contactless structure, in a non-destructive, rapid way. With the method it is possible to access certain characteristics such as Young&#39;s modulus or the density of one or more of the layers of the structure or of the core, which is extremely difficult with other methods. 
     The presence of cracks in the structure may be detected in a simple and rapid way, by seeking whether the vibratory spectrum of the particle includes resonance frequencies in one or more predetermined frequency bands. With the method it is possible to detect both opening and non-opening cracks. This method is simple, rapid, reliable and contactless. 
     The use of an analytical vibratory model for calculating resonance frequencies of a particle contributes to reducing the required computing time for determining the sought characteristics of the particle by the method of least squares. Indeed, computing the resonance frequencies of the particle with such an analytical model is much faster than computing said resonance frequencies with a finite element model. 
     Also, by using the linear vibratory model described above for determining the new values to be taken into account at the following iteration, it is possible to considerably shorten the computing time and accelerate convergence. 
     The method may also be used for determining from the period of the echoes resulting from the reflections of elastic waves at the interfaces between the layers, the velocity of elastic waves in at least one of the layers and/or Young&#39;s modulus of said layer. 
     The method described above may have multiple alternatives. 
     In the case of a nuclear fuel particle, the number of geometrical or mechanical characteristics to be sought may be two, three or more than three depending on the number of relevant experimental resonance frequencies. The number of experimental resonance frequencies which may be used depends on the quality of the signal detected by the interferometric device. Thus, by considering five resonance frequencies, it is possible to determine with good accuracy the combinations of four characteristics from the thicknesses, the Young moduli, the densities and the Poisson coefficients. 
     It is possible not to use any linear vibratory model for determining the values of the sought characteristics to be taken into account during the following iteration. In this case, standard routines (Nelder-Mead, quasi-Newton, conjugate gradients, etc.) for minimization which exist in most computing software packages may perform this operation. 
     Also, it is possible to use a model different from the analytical vibratory model described above for determining the resonance frequencies such as finite element models used in most simulation software packages. 
     If the number of experimentally obtained resonance frequencies is less than the number of parameters of the object as soon as the number of layers N  2 , the resolution of the inverse problem becomes subdetermined. In order that the inverse problem admits a solution, the parameters which may be determined in a robust way and those which may be known a priori, have to be selected. This selection depends on the structure of the multilayered object. 
     In this context, the invention proposes a solution to the inverse problem which uses the sensitivity (or effect) of each parameter on each of the resonance frequencies. 
     
       
         
           
             
               S 
               
                 i 
                  
                 
                     
                 
                  
                 α 
               
             
             = 
             
               
                 ∂ 
                 
                   f 
                   i 
                 
               
               
                 ∂ 
                 
                   x 
                   α 
                 
               
             
           
         
       
     
     is a component of the matrix of sensitivities 
     
       
         
           
             
               S 
               = 
               
                 
                   ( 
                   
                     S 
                     
                       i 
                        
                       
                           
                       
                        
                       α 
                     
                   
                   ) 
                 
                 
                   
                     
                       i 
                       = 
                       1 
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     m 
                   
                   
                     
                       α 
                       = 
                       1 
                     
                     , 
                     … 
                      
                     
                         
                     
                     , 
                     n 
                   
                 
               
             
             ; 
           
         
       
     
     wherein f i  is a resonance frequency which is a component of the vector of frequencies F=(f i ) i=1, . . . , m  and x α  is one of the 4N characteristics of the object which is a component of the vector of the characteristics to be sought X=(x α ) α=1, . . . , n . The use of the correlation matrix (degree of similarity) between the vectors of the effects S iα  of a parameter x α  on the frequency f i  allows optimum selection of the parameters which may be determined in a robust way, for example the Young moduli E 3  and E 5  of the particle of  FIG. 1 . By using the methodology of the experimental schemes, the sensitivities S iα  may be computed and an approximate linear function may subsequently be provided which will be used for solving the inverse problem. 
     It should be noted that for determining the periods of the echoes, the beam  13  of the laser  11  and the second optical wave  25  produced by the laser  21  are applied to the same point of the particle  1 . This is not necessarily the case for determining the spectrum of the resonance frequencies of the vibration modes of the particle, the application points of the beam  13  and of the optical wave  25  may be different.