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The evaluated uncertainties on the elastic constants are approximately 0.4%, 3% and 5% for shear, longitudinal and off-diagonal constants respectively. This sorting of the uncertainties associated with each type of constants is typical in RUS (Migliori and Sarrao, 1997), and originates in the fact that (1) many of the first vibrational modes of a weakly anisotropic solid with small aspect ratios are shear modes; (2) the relative weight of off-diagonal stiffness constants in the measured frequencies is small. Since the shear moduli and the shear stiffness constant are equal in the orthotropic elastic symmetry (Bower,2009), their uncertainty are equal (0.4%). However, uncertainties on the Young’s moduli and Poisson’s ratios are respectively about0.6%and2.5%, which is significantly less than the uncertainties on the longitudinal and off-diagonal stiffness constants. As a consequence of the physics of waves propagation involved in vibrational motion, resonant frequencies are more related to ratios of stiffness constants than to the constants themselves.

Accordingly RUS can perform more precise measurements of these ratios. Since the expressions of Young’s moduli and Poisson’s ratios mainly involve ratios of stiffness constants (Bower,2009), these are determined with a lower uncertainty. However, because we measured only one specimen and only six spectra, our uncertainty values have to be confirmed with repeated measurements on several specimens.

In its current development stage, our implementation of the RUS method re-quires several repositioning of the measured specimen to limit the number of missed modes and to obtain a satisfactory measurement precision. This repositioning could be automated or avoided by the use of different transducers types. Using an au-tomated identification procedure, combined with a random testing of many sets of initial values, it should be possible to improve the robustness of the method.

In this paper we have presented results of the measurement of a unique specimen for the purpose of illustrating the method. Another bone specimen and a sample of bone-mimicking material have also been successfully measured (results not shown).

2.5 Conclusion

This study demonstrates that RUS is suitable for an accurate measurement of cor-tical bone anisotropic elasticity despite the strong viscoelastic damping. RUS lends itself to a precise and automated measurement of all the stiffness constants of a bone specimen. We foresee that the the availability of such method will stimulate investi-gations of normal and pathological bone material properties. Since in principle RUS does not require a minimum size for the tested specimens, it may be appropriate for the characterization of small bone specimens (∼1 mm3) and for bone phenotyping in small animal studies.

Chapter 3

RUS for viscoelastic characterization of anisotropic attenuative solid materials

This chapter is a research article published under the tile Resonant ultrasound spec-troscopy for viscoelastic characterization of anisotropic attenuative solid materials in The Journal of the Acoustical Society of America in 2014, and referred to in this manuscript as Bernard et al. (2014). The full text of the article is reproduced here with no addition and no modifications except in the form.

This work presents results obtained on two materials whose damping is close to that of bone: PMMA, an isotropic polymer and a bone-mimicking composite mate-rial. Both rectangular and cylindrical specimens are used and the viscoelastic damp-ing of the materials is estimated from the width of the resonant peaks. Descriptions of the technical aspects of RUS for the application to attenuative materials are more detailed than in Chapter 2. In addition, a first Bayesian framework is introduced for the resolution of the inverse problem. The proposed algorithm for the pairing of resonant frequencies can be seen as an automated version of the procedure proposed in Chapter 2.

3.1 Introduction

Resonant ultrasound spectroscopy (RUS) is an accurate and efficient method to characterize the elastic properties of isotropic or anisotropic solid materials (May-nard,1992; Migliori et al., 1993; Migliori and Sarrao,1997;Migliori and Maynard, 2005). In a RUS experiment, the mechanical resonant frequencies of a freely vi-brating sample are measured, and the material elastic properties are adjusted until model-predicted frequencies match the measured frequencies (inverse problem ap-proach). Likewise, viscoelastic damping of the material can be inferred from the width of the resonances (Ogi et al.,2003;Leisure et al.,2004). In the present work, we are interested in the application of RUS to strongly attenuative viscoelastic ma-terials.

The main advantage of RUS over concurrent elasticity measurement methods is that the full anisotropic elastic tensor can be assessed from a single sample in a single experimental configuration. Quasi-static mechanical tests usually require a dedicated sample for each elastic modulus and wave velocity-based methods requires longitudinal and shear waves propagation in several directions. Another advantage is that RUS is well-adapted to small samples (a few millimeters or less). Velocity-based methods require samples much larger than the wavelength, and this is not always

possible to achieve for heterogeneous materials by scaling down the wavelength.

Indeed, such materials require that the wavelength remains much larger than the inhomogeneities to avoid strong dispersion effects. Finally, RUS is reputed as being very accurate. Since the method minimizes the coupling between the sample and the apparatus it avoids errors due to inaccurate modeling of such problems (Maynard, 1992). Moreover, RUS takes into account the complex vibrations of a finite sample, instead of assuming idealized states of stress and strain.

RUS has been successfully applied to many low damping materials, such as single crystals (Isaak and Ohno, 2003), quasi crystals (Spoor et al., 1995), or metallic composites (Ledbetter et al.,1995), most often on rectangular parallelepipeds, but also on cylinders (Heyliger et al.,1993;Jaglinski and Lakes,2011). In contrast, its application to attenuative materials has not received much attention. This is due to the difficulty caused by the broadening of the resonant peaks in the presence of mechanical damping. Indeed, as damping increases, the quality factorQ=f /∆f of the resonant peaks decreases, peaks overlap, and it becomes complicated to retrieve the resonant frequencies. In geophysics, RUS has been applied to rock samples with a Qabout 150 by Ulrich et al.(2002). Lebedev(2002) proposed the use of a signal processing method to retrieve resonant frequencies with Q=50. In the context of bio-materials Lee et al.(2002) applied RUS to cortical bone but could not measure all the elastic moduli, due to the low Qfactor about 30. Recently, we applied RUS to a cortical bone sample (Bernard et al., 2013) with a comparable Q, using the processing method proposed by Lebedev.

Although this last study was successful in measuring bone elastic properties, it reported that several predicted resonant modes could not be observed, despite signal processing. Indeed, due to the width of the resonant peaks, weakly excited resonant modes were masked by strongly excited ones (Bernard et al., 2013). This was a source of trouble in the inverse problem, and will be as well for all highly attenuative materials. Missing resonant frequencies means that there is less data available to identify the mechanical properties and that we don’t know which of the overnumerous predicted frequency compare to each measured frequency. For these reasons, the inverse problem could benefit from a probabilistic formulation, in a Bayesian framework, which allows one to conveniently introduce available prior information (Tarantola,2005) and, as we will see, to define a quantitative criterion for the pairing of each measured frequency to its corresponding predicted one.

Applicability of RUS to attenuative materials could benefit several fields. In the context of biological materials (bone, tooth material) there is a lack of accurate methods for anisotropic viscoelasticity complete characterization. Design of appro-priate materials for prosthesis and implants also requires characterization methods.

Many composites used in industrial applications are anisotropic and viscoelastic,

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