Basic principles

The basic principle of RUS is to use the frequencies of mechanical resonance of a sample of material to infer its stiffness tensor. Indeed, the resonant frequencies of an elastic body depend only on its geometry, mass density and stiffness tensor. If the two first parameters are directly measured, elasticity can then be estimated by matching model-predicted frequencies to the experimental results (inverse problem approach). This requires a setup for the measurement of the resonant frequencies, a method to compute the vibration modes of a sample given its elasticity and finally a procedure to solve the inverse problem. These elements are described in a generic way in the next paragraphs, mostly based on the book byMigliori and Sarrao(1997).

Measurement of the resonant frequencies

The basic setup for a RUS experiment is very simple: it consists of a pair of trans-ducers, a signal generator working at ultrasonic frequencies (usually from about 100 kHz up to a few MHz) and an apparatus to record the output signal (Fig.1.4).

The sample is gently held by points of low symmetry (e.g. corners for a cube) between the transmitting and receiving transducers. This position is expected to

1.4. Resonant Ultrasound Spectroscopy 11

Figure 1.4: Left: the basic setup of a RUS experiment. Right: A portion of a typical spectrum measured for a metallic sample (copper).

maximize the number of measured frequencies because these points are never nodes of the vibration modes (i.e. the displacement in never null). Another reason is that the model assumes stress-free boundary conditions, obtained if the contacting forces on the sample are small and punctual. It is important that the holding forces are as small as possible, as a force of the order of 0.01 N produces a measurable shift of the resonant frequencies (Yoneda,2002). A precise orientation of the sample is however not important, in contrast with wave velocity methods and mechanical testing.

The frequency response of the sample is usually recorded using a stepwise sweep-ing over the frequency band of interest: the sample is excited at a given frequency using a steady signal and the phase and amplitude of the response are recorded be-fore stepping to the next frequency. The main advantage against methods based on the Fourier transform of a temporal signal is that the signal to noise ratio is usually higher. The steady signal can be run for as long as desired and efficient filtering methods such as homodyne or heterodyne detection can be used. In that work, a network analyser was used to perform the frequency sweep as well as the recording of the frequency response (chapters2 and3).

The transducers must be broadband, or at least have no sharp resonances in the frequency band were those of the sample are expected. In the first part of that work (Chapter 2), home made transducers were firstly built following recommendations of Migliori and Sarrao (1997): a small piece of piezoelectric material is suspended to a thin film and put directly in contact with the sample. That way, the system is resonant-free up to the fundamental mode of the piezoelectric piece. Commercial shear transducers were later found out to produce stronger signals (Chapter3). This due to the fact that many resonant modes have a larger displacement in a direction colinear to the surface of the transducers.

Computation of the resonant modes

The natural free-vibration modes of a solid occupying the region Ωbounded by the surface S are the pulsations ω and displacement fields u that solve the harmonic equations of motion⁶

ρω²ui+C_ijkl ∂²u_k

∂x_j∂x_l = 0 inΩ, (1.13)

subject to the stress-free boundary conditions C_ijkl∂u_k

∂x_ln_j = 0 onS. (1.14)

No complete analytical solution exist for that problem except for the isotropic sphere. For some other simple shapes, partial solutions exist. For example, some modes of a rectangular parallelepiped with specific side-length ratios, called the Midlin-Lamé modes, have closed form expressions (Mindlin, 1956). For a general solution, the analysis must use energetic principles and numerical approximations.

According to Hamilton’s principle, the solutions of equations (1.13) and (1.14) must be stationary points of the Lagrangian Lof the body, i.e. they must satisfy

δL=δ

with δ referring to a variation due to a small perturbation δu_i of the displacement field and where the first and second terms of the integrand are respectively the kinetic and potential energy. Indeed, some manipulations⁷ yields

Z Becauseδuiis arbitrary inΩand onS, the two terms in parenthesis in the integrands must vanish independently everywhere over their domain of integration, implying equations (1.13) and (1.14).

The Rayleigh-Ritz method expands the displacement field as a finite sum of known basis functions φ_λ with yet unknown coefficients a_i,λ

ui =X

λ

ai,λφλ. (1.17)

The Lagrangian can then be expressed in a matrix form L= 1

2ω²a^tMa−1

2a^tKa (1.18)

6A rectangular coordinate system is assumed here for simplicity. A cylindrical or spherical coordinate system can be more adapted to some shapes. See Chapter3for the cylindrical case.

7Integration by part followed by the use of the divergence theorem, and discarding the second and higher order terms inδui.

1.4. Resonant Ultrasound Spectroscopy 13

where the vector a contains the coefficients of the expansion (1.17), and M and Kare the mass and stiffness matrices, related respectively to kinetic and potential energy. Requiring δL= 0 now impose ∂L/∂ai,λ = 0, which leads to

ω²Ma−Ka= 0, (1.19)

an eigenproblem for the eigenvalues ω² and eigenvectors a. Expressions for the elements of the matrices are

For simple geometries and polynomial basis functions, the integrals have simple analytical expressions and the eigenproblem (1.19) can then be numerically solved.

The number of basis function usually required to get satisfactory accuracy is of the order of 1000. An eigenproblem of this size is manageable for even modest desktop computers, particularly when considering the splitting of the eigenproblem in smaller independent sub-problems allowed for specific shapes and elastic symmetries (see Chapter3 and references therein).

The eigenvectors a allow representing the shapes of the normal modes approxi-mated on the polynomial basis. Fig. 1.5represents some of the first normal modes of a 2 mm-sized steel cube, computed from the Rayleigh-Ritz method with Legendre polynomials up to order 10 (basis functions are described in Chapter 3).

Inverse estimation of the stiffness

The objective of the inverse problem is to find the set of elastic coefficients giving the best match between the measurement and the model. This is usually done through the minimization of a quadratic cost function F

F =

whereN is the number of measured resonant frequencies. The Levenberg-Marquardt algorithm is the usual method employed for the minimization. It is an iterative algorithm for non-linear optimization, which mixes search in the direction of the gradient and Gauss-Newton steps (Press et al.,1992). The partial derivatives of the frequencies with respect to the elastic coefficients, required in the inversion process, are obtained after solving the eigenproblem from the derivative of equation (1.19).

The main difficulty of the inverse problem is however not the minimization in itself, but the identification, orpairing, of the resonant modes. Indeed, the measured

f=704.8 kHz f=949.0 kHz f=965.2 kHz f=1096.3 kHz

f=1128.4 kHz f=1182.8 kHz f=1338.9 kHz f=1360.9 kHz

Figure 1.5: The eight first modes of a steel cube (l = 2mm, E=200 GPa,ν = 0.3, ρ = 8000 kg·m⁻³). Colors map the norm of the displacement vector (red: large, blue: small). The fourth mode is a Midlin-Lamé mode whose exact frequency is f = 1/(√

2l)×p

C44/ρ. The approximation is accurate to ten digits for that mode.

resonant frequencies are not informative of which predicted vibration mode they cor-respond to, and this causes ambiguities in the definition of the cost function (1.22).

Because the frequencies computed for the initial guess of the elasticity may appear in a different order than for the "true" elasticity, and because some predicted mode are potentially not observed in the measurement, there is no straightforward way to correctly index the frequencies, i.e. finding which predicted mode corresponds to the n-th experimental frequency in equation (1.22). Solving this so-called pairing problem was an important part of this work.

It is usually admitted that a precise estimation of the elastic coefficients requires about 10 to 20 measured resonances for isotropic materials, and at least four to five times the number of independent coefficients for anisotropic materials (Migliori and Maynard,2005). However, the present work will show that an accurate and precise RUS measurement can be done with a smaller number of measured frequencies.