SIST EN 14186:2009
(Main)Advanced technical ceramics - Mechanical properties of ceramic composites at room temperature - Determination of elastic properties by an ultrasonic technique
Advanced technical ceramics - Mechanical properties of ceramic composites at room temperature - Determination of elastic properties by an ultrasonic technique
This European Standard specifies an ultrasonic method to determine the components of the elasticity tensor of ceramic matrix composite materials at room temperature. Young's moduli, shear moduli and Poisson coefficients, can be determined from the components of the elasticity tensor.
This European Standard applies to ceramic matrix composites with a continuous fibre reinforcement: unidirectional (1D), bidirectional (2D), and tridirectional (D, with 2 < 3) which have at least orthotropic symmetry, and whose material symmetry axes are known.
This method is applicable only when the ultrasonic wave length used is larger than the thickness of the representative elementary volume, thus imposing an upper limit to the frequency range of the transducers used.
NOTE Properties obtained by this method may not be comparable with moduli obtained by EN 658-1, EN 658-2 and EN 12289.
Hochleistungskeramik - Mechanische Eigenschaften keramischer Verbundwerkstoffe bei Raumtemperatur - Bestimmung von elastischen Eigenschaften mittels Ultraschallwellen
Diese Europäische Norm legt ein Ultraschallverfahren fest, mit dem die Komponenten des Elastizitätstensors von kerami¬schen Verbundwerkstoffen bei Raumtemperatur zu bestimmen sind. Aus den Komponenten des Elastizitäts¬tensors können Youngscher Modul, Schermodul und Querkontraktionszahl bestimmt werden.
Diese Europäische Norm gilt für keramische Verbundstoffe mit unidirektionaler (1D), bidirektionaler (2D) sowie dreidirektio¬naler (xD, mit 2 < × 3) Endlosfaserverstärkung, die mindestens eine orthotrope Symmetrie haben und deren Werkstoff Symmetrieachsen bekannt sind.
Dieses Verfahren ist nur anwendbar, wenn Ultraschall angewendet wird, dessen Wellenlänge größer ist als die Dicke des repräsentativen Volumenelements, wodurch eine obere Grenze für den Frequenzbereich der verwendeten Wandler festgelegt wird.
ANMERKUNG Die nach diesem Verfahren ermittelten Eigenschaften sind möglicherweise nicht mit den Modulen vergleichbar, die nach EN 658 1, EN 658 2 und EN 12289 bestimmt werden.
Céramiques techniques avancées - Propriétés mécaniques des céramiques composites à température ambiante - Détermination des propriétés élastiques par une méthode ultrasonore
La présente Norme européenne spécifie une méthode ultrasonore pour déterminer les composants du tenseur d’élasticité des composites à matrice céramique à température ambiante. Les modules de Young, les modules de cisaillement et les coefficients de Poisson peuvent être déterminés à partir des composants du tenseur d’élasticité.
La présente Norme européenne s'applique aux composites à matrice céramique à renfort fibreux continu unidirectionnels (1D), bidirectionnels (2D) et tridirectionnels (D, avec 2 3), qui ont au minimum une symétrie orthotrope et dont les axes de symétrie sont connus.
Cette méthode est applicable uniquement lorsque la longueur d'onde ultrasonore utilisée est supérieure à l'épaisseur du volume élémentaire représentatif, ce qui impose une limite supérieure à la gamme de fréquences des transducteurs utilisés.
NOTE Les propriétés obtenues au moyen de cette méthode peuvent ne pas être comparables avec les modules obtenus par les méthodes décrites dans l'EN 658-1, l'EN 658-2 et l'EN 12289.
Sodobna tehnična keramika - Mehanske lastnosti keramičnih kompozitov pri sobni temperaturi - Določanje upogibnih lastnosti z ultrazvočno tehniko
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2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.HKQLNRHochleistungskeramik - Mechanische Eigenschaften keramischer Verbundwerkstoffe bei Raumtemperatur - Bestimmung von elastischen Eigenschaften mittels UltraschallwellenCéramiques techniques avancées - Propriétés mécaniques des céramiques composites à température ambiante - Détermination des propriétés élastiques par une méthode ultrasonoreAdvanced technical ceramics - Mechanical properties of ceramic composites at room temperature - Determination of elastic properties by an ultrasonic technique81.060.30Sodobna keramikaAdvanced ceramicsICS:Ta slovenski standard je istoveten z:EN 14186:2007SIST EN 14186:2009en01-februar-2009SIST EN 14186:2009SLOVENSKI
STANDARDSIST ENV 14186:20071DGRPHãþD
SIST EN 14186:2009
EUROPEAN STANDARDNORME EUROPÉENNEEUROPÄISCHE NORMEN 14186November 2007ICS 81.060.30Supersedes ENV 14186:2002
English VersionAdvanced technical ceramics - Mechanical properties of ceramiccomposites at room temperature - Determination of elasticproperties by an ultrasonic techniqueCéramiques techniques avancées - Propriétés mécaniquesdes céramiques composites à température ambiante -Détermination des propriétés élastiques par une méthodeultrasonoreHochleistungskeramik - Mechanische Eigenschaftenkeramischer Verbundwerkstoffe bei Raumtemperatur -Bestimmung von elastischen Eigenschaften mittelsUltraschallwellenThis European Standard was approved by CEN on 13 October 2007.CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this EuropeanStandard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such nationalstandards may be obtained on application to the CEN Management Centre or to any CEN member.This European Standard exists in three official versions (English, French, German). A version in any other language made by translationunder the responsibility of a CEN member into its own language and notified to the CEN Management Centre has the same status as theofficial versions.CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland,France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal,Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.EUROPEAN COMMITTEE FOR STANDARDIZATIONCOMITÉ EUROPÉEN DE NORMALISATIONEUROPÄISCHES KOMITEE FÜR NORMUNGManagement Centre: rue de Stassart, 36
B-1050 Brussels© 2007 CENAll rights of exploitation in any form and by any means reservedworldwide for CEN national Members.Ref. No. EN 14186:2007: ESIST EN 14186:2009
EN 14186:2007 (E) 2 Contents Page Foreword.3 1 Scope.4 2 Normative references.4 3 Terms and definitions.4 4 Principle.7 5 Significance and use.10 6 Apparatus.10 6.1 Ultrasonic tank with thermostatic control.10 6.2 Temperature measurement device.10 6.3 Test specimen holder.10 6.4 Transducers.11 6.5 Transducer holders.11 6.6 Pulse generator.11 6.7 Signal recording system.11 7 Test specimens.11 8 Test specimen preparation.11 9 Test procedure.12 9.1 Choice of frequency.12 9.2 Establishment of the test temperature.12 9.3 Reference test without test specimen.12 9.4 Measurement with the specimen.13 10 Calculation.14 10.1 Delay.14 10.2 Calculation of the propagation velocities.14 10.3 Calculation of the refracted angle θθθθr.14 10.4 Identification of the elastic constants, Cij.14 10.5 Back calculation of the phase velocities.18 10.6 Polar plots of the velocity curves.18 10.7 Calculation of the quadratic deviation.18 10.8 Calculation of the engineering constants.18 11 Test validity.19 11.1 Measurements.19 11.2 Criterion of validity for the reliability of the Cij components.19 12 Test report.19 Annex A (informative)
Example of a presentation of the results for a material with orthothropic symmetry.21 A.1 Velocity curves.21 A.2 Stiffness matrix with stiffness components.22 A.3 Engineering constants.23 Bibliography.24
SIST EN 14186:2009
EN 14186:2007 (E) 3 Foreword This document (EN 14186:2007) has been prepared by Technical Committee CEN/TC 184 “Advanced technical ceramics”, the secretariat of which is held by BSI. This European Standard shall be given the status of a national standard, either by publication of an identical text or by endorsement, at the latest by May 2008, and conflicting national standards shall be withdrawn at the latest by May 2008. This document supersedes ENV 14186:2002. According to the CEN/CENELEC Internal Regulations, the national standards organizations of the following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom. SIST EN 14186:2009
EN 14186:2007 (E) 4 1 Scope This European Standard specifies an ultrasonic method to determine the components of the elasticity tensor of ceramic matrix composite materials at room temperature. Young's moduli, shear moduli and Poisson coefficients, can be determined from the components of the elasticity tensor. This European Standard applies to ceramic matrix composites with a continuous fibre reinforcement: unidirectional (1D), bidirectional (2D), and tridirectional (×D, with 2 < × ≤ 3) which have at least orthotropic symmetry, and whose material symmetry axes are known. This method is applicable only when the ultrasonic wave length used is larger than the thickness of the representative elementary volume, thus imposing an upper limit to the frequency range of the transducers used. NOTE Properties obtained by this method might not be comparable with moduli obtained by EN 658-1, EN 658-2 and EN 12289. 2 Normative references The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies. EN 1389, Advanced technical ceramics — Ceramic composites — Physical properties — Determination of density and apparent porosity CEN/TR 13233:2007, Advanced technical ceramics — Notations and symbols EN ISO/IEC 17025, General requirements for the competence of testing and calibration laboratories (ISO/IEC 17025:2005) ISO 3611, Micrometer callipers for external measurements 3 Terms and definitions For the purposes of this document, the terms and definitions given in CEN/TR 13233:2007 and the following apply. 3.1 stress-strain relations for orthotropic material elastic anisotropic behaviour of a solid homogeneous body described by the elasticity tensor of fourth order Cijkl, represented in the contracted notation by a symmetrical square matrix (6 × 6) NOTE 1 If the material has at least orthotropic symmetry, its elastic behaviour is fully characterised by nine independent stiffness components Cij, of the stiffness matrix (Cij), which relates stresses to strains, or equivalently by nine independent compliance components Sij of the compliance matrix (Sij), which relates strains to stresses. The stiffness and compliance matrices are the inverse of each other. If the reference coordinate system is chosen along the axes of symmetry, the stiffness matrix Cij and the compliance matrix Sij can be written as follows: SIST EN 14186:2009
EN 14186:2007 (E) 5 =654321665544332313232212131211654321000000000000000000000000εεεεεεσσσσσσCCCCCCCCCCCC
=654321665544332313232212131211654321000000000000000000000000σσσσσσεεεεεεSSSSSSSSSSSS NOTE 2 For symmetries of higher level than the orthotropic symmetry, the Cij and Sij matrices have the same form as here above. Only the number of independent components reduces. 3.2 engineering constants compliance matrix components of an orthotropic material which are in terms of engineering constants: []−−−−−−=1213332223111333322211123331222111100000001000000100010001GGEEvEvEvEEvEvEvESij where E11, E22 and E33 are the elastic moduli in directions 1, 2 and 3 respectively; G12, G13 and G23 are the shear moduli in the corresponding planes; ν12, ν13, ν23 are the respective Poisson coefficients 3.3 angle of incidence θθθθi angle between the direction 3 normal to the test specimen front face and the direction ni of the incident wave (see Figure 1 and Figure 2) SIST EN 14186:2009
EN 14186:2007 (E) 6 3.4 refracted angle θθθθi angle between the direction 3 normal to the test specimen front face and the direction n of propagation of the wave inside the test specimen (see Figure 1 and Figure 2) 3.5 azimuthal angle ψψψψ angle between the plane of incidence (3, ni) and plane (2, 3) where ni corresponds to the vector oriented along the incident plane wave and direction 2 corresponds to one of the axes of symmetry of the material (see Figure 1)
Figure 1 — Definition of the angles
Figure 2 — Propagation in the plane of incidence SIST EN 14186:2009
EN 14186:2007 (E) 7
3.6 unit vector n unit vector oriented along the propagation direction of the incident plane wave inside the specimen, with its components nk (k = 1, 2, 3) (see Figure 1 and Figure
2): n1 = sinθr sinψ n2 = sinθr cosψ n3 = cosθr 3.7 propagation velocity V(n) phase velocity of a plane wave inside the specimen in dependence on unit vector n (i.e. in dependence on ψ and θi) NOTE Vo is the propagation velocity in the coupling fluid. 3.8 delay δt(n) difference between the flight time of the wave when the test specimen is in place and the flight time of the wave in the coupling fluid with the test specimen removed under the same configuration of the transducers in dependence on unit vector n 3.9 thickness of the test specimen h thickness of the test specimen 3.10 bulk density ρρρρb bulk density of the specimen 4 Principle The determination of the elastic properties consists of calculating the coefficients of the propagation equation of an elastic plane wave, from a set of properly chosen velocity measurements along known directions. A thin specimen with plane parallel faces is immersed in an acoustically coupling fluid (e.g. water): see Figure 3. The specimen is placed between an emitter (E) and a receiver (R), which are rigidly connected to each other and have two rotational degrees of freedom. Using appropriate signal processing, the propagation velocities of each wave in the specimen are calculated. SIST EN 14186:2009
EN 14186:2007 (E) 8
Key 1 rotation drive 2 test specimen 3 pulse generator 4 digital oscilloscope 5 micro-computer Figure 3 — Ultrasonic test assembly Depending on the angle of incidence, the pulse sent by the emitter E is refracted within the material in one, two or three bulk waves (one quasi longitudinal wave QL, one quasi transverse wave QT, or two quasi transverse waves QT1, QT2) that propagate in the solid at different velocities and in different directions. The receiver R collects one, two or three pulses, corresponding to each of these waves. The difference in propagation time of each of the waves and the propagation time of the emitted pulse in the coupling fluid without the specimen is measured. The evaluation procedure is based on the measurement of the time difference of the quasi-longitudinal and one or both quasi-transverse waves, and is only valid when the QL and the QT waves are appropriately separated (see Figure 4). SIST EN 14186:2009
EN 14186:2007 (E) 9
Key 1 amplitude 2 incidence angle Figure 4a) — Amplitude of the QL and QT waves as a function of the incidence angle
Key 1 amplitude 2 time Figure 4b) — Temporal waveform of the overlapping QL and QT waves at an incidence angle θθθθi Figure 4 — Overlapping of QL and QT waves at an incidence angle θθθθi From the propagation velocities the components of the elasticity tensor are obtained through a least square regression analysis which minimises the residuals of the wave propagation equations. Young's moduli, shear moduli and Poisson coefficients are determined from these components. SIST EN 14186:2009
EN 14186:2007 (E) 10 5 Significance and use Only two constants (Lamé's coefficients or Young's modulus and Poisson coefficient) are sufficient in order to fully describe the elastic behaviour of an isotropic body. When anisotropy, which is a specific feature of composite materials, shall be taken into account, the use of an elasticity tensor with a larger number of independent coefficients is needed. While conventional mechanical methods allow only a partial identification of the elasticity of anisotropic bodies, ultrasonic techniques allow a more exhaustive evaluation of the elastic properties of these materials particularly transverse elastic moduli and shear moduli for thin specimens. Successful application of the method depends critically on an appropriate selection of the central frequency of the transducers. Frequency shall be sufficiently low for the measurement to be representative of the elementary volume response, but at the same time high enough to achieve a separation between the QL and the QT waves. Contrary to mechanical test methods, the determination of elastic properties by the ultrasonic method described here is not based on the evaluation of the stress-strain response over a given deformation range obtained under quasi-static loading conditions, but is based on a non-destructive dynamic measurement of wave propagation velocities. Therefore the values of Young's moduli, shear moduli and Poisson ratios determined by the two methods might not be comparable, particularly for ceramic matrix composites that exhibit non linear stress-strain behaviour. NOTE Mechanical test methods are based on a measurement performed under isothermal conditions, whereas the ultrasonic method assumes adiabatic conditions. In addition to the ultrasonic method described here, there also exist other non destructive methods to determine the elastic properties, for instance the resonant beam technique and the impulse excitation method. Each of these has its relative merits and disadvantages. The selection of a particular non destructive method shall be considered on a case-by-case basis. 6 Apparatus 6.1 Ultrasonic tank with thermostatic control The ultrasonic tank shall be capable of maintaining the temperature of the coupling fluid constant to within ± 0,1 °C for the full duration of the test. NOTE This requirement is imposed because the wave propagation velocity in the coupling fluid is highly temperature sensitive. 6.2 Temperature measurement device The temperature measurement device shall be capable of measuring the temperature to within 0,1 °C, e.g. as set out in ISO 653. 6.3 Test specimen holder The test specimen holder shall allow rotation of the test specimen around one axis to cover the range of angles of incidence θi between 0 ° and 90 °. Additionally it shall allow for discrete settings of the azimuthal angle Ψ of 0 °, 45 ° and 90 °. The accuracy in the measurement of the angles θi and Ψ shall be better than 0,01 ° and 1 ° respectively. NOTE The accuracy required for the measurement of the angle of incidence θi depends on the nature of the coupling fluid and is higher when using air as the coupling fluid. Commercially available goniometers with automatic positioning are commonly used for this purpose. SIST EN 14186:2009
EN 14186:2007 (E) 11 6.4 Transducers Use piezoelectric broad-band transducers adapted to the coupling fluid and able to generate longitudinal ultrasonic waves. Two identical transducers are used as emitter and receiver. 6.5 Transducer holders The transducer holders shall allow the transducers to be oriented towards each other. The transducers are mounted in such a way that their relative position remains fixed during the test. 6.6 Pulse generator The pulse generator shall be selected in accordance with the characteristics of the transducers. It shall be able to generate short-duration (< 1 µs) pulses of voltage sufficient to provide a mechanical pulse through the transducer. The frequency of the exciting pulse shall be chosen such as described in 9.1. The interval between consecutive pulses shall be long compared with the travel time being recorded, typically greater than 1 ms, so that all signals from the preceding pulse have been dissipated before initiating the next. 6.7 Signal recording system Use any system, for instance: digital oscilloscope or dynamic analogue/digital board, with a minimum sampling frequency of 100 MHz that allows the recording of emitted and received signals. The signal recording system is designed in order to allow to see on the display the generated and the detected pulses on the same time-base and to determine the time-gap separating these two events. 7 Test specimens The choice of test specimen geometry depends on the nature of the material and the reinforcement structure. The thickness shall be large enough to allow separation of the echoes of the quasi longitudinal QL and quasi transverse QT waves, and shall be representative of the material. The largest possible thickness is recommended, at least five times the size of the representative volume element (RVE) in the direction of propagation of the wave. The other dimensions of the test specimen shall be at least twice the diameter of the transducer. A plate with parallel faces is recommended. The plane parallelism of the two faces shall be better than 0,05 mm. 8 Test specimen preparation The material symmetry axes shall be identified. If machining is required, it shall be performed in such a way that the material symmetry axes remain known at all times. Machining procedures that do not cause damage to the test specimens shall be clearly defined and recorded. These procedures shall be followed during machining of the test specimens. NOTE 1 Usually, plate test specimens are cut with their longitudinal axis coinciding with one of the principal directions of the reinforcement. One test specimen is sufficient to perform the test. Multiple measurements can be done on a single test specimen. Care shall be taken to avoid interaction between the coupling fluid and the test specimen (ingress into open porosity, chemical instability, absorption phenomena etc.). NOTE 2 This can for instance be achieved by sealing the test specimen in an evacuated plastic bag, or by applying an appropriate coating. SIST EN 14186:2009
EN 14186:2007 (E) 12
9 Test procedure 9.1 Choice of frequency The selection of the appropriate frequency is critical for the application of the method. The frequency shall be sufficiently low to ensure that the measurement is representative. NOTE 1 An initial selection of dVf2,0<, where d is the size of the RVE in the direction of normal incidence, is proposed (θi = 0). NOTE 2 Because of the inverse relationship between wavelength λ and frequency f (λVf=), this corresponds to a wavelength λ of at least 5d. For the selected frequency the following additional criteria should be met: a) measurable amplitude of the QL wave under normal incidence θi = 0. If the amplitude is too small, the frequency shall be decreased; b) time separation of the waves QL and QT when varying the angle of incidence θi [see Figure 4b)]. This is promoted by increasing the frequency. NOTE 3 A minimum frequency of hV23 is proposed. Because the frequency requirements for meeting the three mentioned criteria may be conflicting, there are cases where the method is not applicable. In these cases the only remaining solution is to increase the specimen thickness beyond the minimum thickness stipulated in clause 7. NOTE 4 For example for a 2D SiC/SiC with a RVE of 0,5 mm (requiring a minimum test specimen thickness of 2,5 mm in accordance with clause 7), the transducer frequency, in order for the measurement to be representative, is lower than 2,25 MHz (corresponding to wave velocities of around 5 000 m/s). On the other hand for obtaining mode separation, the frequency is higher than MHz323=hV. The method can therefore not be applied for the given thickness of 2,5 mm. An increase in thickness to 3,3 mm allows mode separation at a frequency of 2,25 MHz. 9.2 Establishment of the test temperature Switch on the thermostatic control to establish the required temperature of the coupling fluid. Measure the temperature of the coupling fluid at a location between the transducers in the vicinity of the future position of the test specimen. Perform the reference measurement in accordance with 9.3. Mount the test specimen in the test specimen holder in accordance with 9.4.2. Measure the temperature in the vicinity of the test specimen. Make sure that the temperature falls within ± 0,1 °C from that of the reference measurement. Perform the test in accordance with 9.4. 9.3 Reference test without test specimen Record the signals from the emitter and from the receiver versus time without a test specimen mounted. SIST EN 14186:2009
EN 14186:2007 (E) 13 9.4 Measurement with the specimen 9.4.1 Measurement of the bulk density and of the thickness 9.4.1.1 Measurement of the bulk density Measure the bulk density in accordance with EN 1389. 9.4.1.2 Measurement of the thickness Measure the thickness in three positions on the test specimen with a micrometer with an accuracy of 0,01 mm in accordance with ISO 3611. 9.4.2 Mounting of the specimen The specimen shall be oriented perpendicularly to the incoming beam. The accuracy of the perpendicularity between the beam and the specimen shall be 0,1 °. The test specimen shall be mounted in such a way that one of the symmetry axes coincides with Ψ = 0 ° to within 1 °. 9.4.3 Acquisition of different angles of incidence Set acquisition plane by selecting azimuthal angle Ψ = 0 °, 45 ° and 90 °. For each acquisition plane measurements are made of the QL and QT signals at given values of the angle of incidence θi. The incidence angle θi varies from 0 ° up to a maximum defined by a decrease of the amplitude of the QL wave to approximately one third of its maximum. The number of incidence angles shall be selected to optimise coverage over the range in which both the QL and QT waves appear. NOTE 1 Over the total range of θi usually a minimum of 20 measurements is performed. In the angular range where QL and QT overlap the step in angle θi shall be reduced. NOTE 2 This range of θi can be defined as ± 5° from the incidence angle where QL and QT have the same amplitude. In this range the step is set at 0,5 °. NOTE 3 Maximum θi is also configuration limited. Only signals recorded at values of θi and Ψ meeting the following conditions can be used for subsequent calculation and evaluation of Cij: a) the bulk waves are clearly identified (i.e. they can unambiguously be separated from other propagating waves); b) the longitudinal QL and the transverse QT waves are clearly separated in time, making it possible to clearly separate QL from QT. NOTE 4 This is usually verified by representing the experimental results by the velocity curves as shown in Annex A. NOTE 5 Signal stability should be secured by repeating the experiment under given conditions (Ψ, θi) at different time periods. SIST EN 14186:2009
EN 14186:2007 (E) 14 10 Calculation 10.1 Delay For each value of Ψ and θi, the delay δt(n) on the QL and the QT waves is determined by comparing the signal received in the coupling fluid alone (reference signal), and the signal received when the specimen is in the coupling fluid. NOTE The delay δt(n) is usually obtained by computer assisted signal processing techniques. 10.2 Calculation of the propagation velocities For each measurement of δt(n) the associated propagation velocity V(n) is determined by the following equation: −+=icos)()()(θδδ21000hntVhntVVnV (1) where V(n) is the propagation velocity in the material, in metres per second (m⋅s-1); Vo is the propagation velocity in the coupling fluid, in metres per second (m⋅s-1); h is the specimen thickness, in metres (m); δt(n) is the delay, in seconds (s); θi
is the angle of incidence, in degrees (°). 10.3 Calculation of the refracted angle θθθθr θ×=θoirVsinVn(arcsin (2) where θr is the refracted angle, in degrees (°). 10.4 Identification of the elastic constants, Cij 10.4.1 Basic considerations The phase velocities of the three propagating waves QL, QT1, QT2 are given by the eigenvalues of the propagation tensor Γij according to the following equation: Det(Γij – pb V2(n) δij) = 0 (3) and the polarization directions are the corresponding eigenvectors, with δij Kronecker´s symbol and pb the bulk density. The wave propagation tensor in the case of an anisotropic material has the following general form: Γij = Cijkl nk nl (4) SIST EN 14186:2009
EN 14186:2007 (E) 15 where Cijkl are the components of the stiffness matrix (in the contracted notation: Cij); nk and nl (k, l = 1, 2, 3) are the components of the propagation direction vector n = (n1, n2, n3); n1, n2, n3
are the direction cosines (n1 = sinθr sinψ, n2 = sinθr cosψ, n3= cosθr). In the case of an orthotropic material the components of the propagation tensor Γij have the following
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