Measurement of Cortical Bone Elasticity Tensor with Resonant Ultrasound Spectroscopy

Abstract

Resonant Ultrasound Spectroscopy estimates the stiffness coefficients of a material from the free resonant frequencies of a single specimen. It is particularly suitable for complete stiffness characterization of anisotropic materials available only as small samples (typically a few mm), and it does not suffer from some limitations associated to quasi-static mechanical test and ultrasound wave velocity measurements. RUS has been used for decades on geological samples and single crystals, but was until recently not applied to mineralized tissues such as bone. The reason is the significant mechanical damping presents in these materials, which causes the resonant peaks to overlap and prevent a direct measurement of the resonant frequencies. This chapter describes the use of RUS for the elastic characterization of mineralized tissues, cortical bone in particular. All steps are described, from sample preparation and measurement setup to signal processing and data analysis, including the developments and adaptions necessary to overcome the difficulties linked to damping. Viscoelastic characterization, from the width of the resonant peaks, is also presented. Mostly technical aspects are developed in this chapter, while the data obtained from RUS on several collections of mineralized tissues specimens are presented and discussed in Chap. 13.

Appendix: Anisotropic Elasticity

Hooke’s law for continuum media is a tensorial relation between the stress tensor σ, the linear strain tensor 𝜖, and the fourth-order stiffness tensor $\mathbb {C}$

$$\displaystyle \begin{aligned} \sigma_{ij}=C_{ijkl}\epsilon_{kl}, {} \end{aligned} $$

(12.23)

where 𝜖 is related to the displacement u, in the Cartesian frame (x ₁, x ₂, x ₃),

$$\displaystyle \begin{aligned} \epsilon_{kl}=\frac{1}{2}\left(\frac{\partial u_k}{\partial x_l}+ \frac{\partial u_l}{\partial x_k}\right). {} \end{aligned} $$

(12.24)

Due to the thermodynamics of reversible deformations and to symmetry of the stain and stress tensors, $\mathbb {C}$ has at most 21 independent coefficients (Bower, 2009). In a reduced two index notation called Voigt notation, Hooke’s law (12.23) can be expressed as

(12.25)

where pairs of subscripts for the stiffness tensor in Hooke’s law (12.23) have been mapped to a single subscript: 1 ⇔ 11; 2 ⇔ 22; 3 ⇔ 33; 4 ⇔ 23; 5 ⇔ 13; 6 ⇔ 12.

If the material possesses symmetries, this relation further simplifies. For an orthotropic material (three orthogonal mirror symmetry planes) the stiffness tensor has only nine independent non-zero coefficients in the coordinate system defined by the symmetry directions

(12.26)

If the material is transversely isotropic (one axis of cylindrical symmetry) only five coefficients remains independent, as the following relations hold: C ₁₁ = C ₂₂, C ₁₃ = C ₂₃, C ₄₄ = C ₅₅, and C ₁₂ = C ₁₁ − 2C ₄₄ (for symmetry around axis 3, which for bone is chosen to be aligned with the axial direction of long bones, and hence aligned with the main orientation of the pores). Finally, if the material is isotropic, only two coefficients remain independent as C ₁₁ = C ₂₂ = C ₃₃, C ₄₄ = C ₅₅ = C ₆₆, and C ₁₂ = C ₁₃ = C ₂₃ = C ₁₁ − 2C ₄₄.

Hooke’s law (12.23) can be inverted to yield 𝜖 = C ⁻¹ : σ, where the inverse of the stiffness tensor is called the compliance tensor, and can be expressed using the engineering moduli (Young’s moduli E, shear moduli G, and Poisson’s ratios ν). For orthotropic materials (Bower, 2009)

$$\displaystyle \begin{aligned}{}[C_{ij}]^{-1}= \begin{bmatrix} \frac{1}{E_1}&-\frac{\nu_{12}}{E_2}&-\frac{\nu_{13}}{E_3}&0&0&0\\ -\frac{\nu_{21}}{E_1}&\frac{1}{E_2}&-\frac{\nu_{23}}{E_3}&0&0&0\\ -\frac{\nu_{31}}{E_1}&-\frac{\nu_{32}}{E_2}&\frac{1}{E_3}&0&0&0\\ 0&0&0&\frac{1}{G_{23}}&0&0\\ 0&0&0&0&\frac{1}{G_{13}}&0\\ 0&0&0&0&0&\frac{1}{G_{12}}\\ \end{bmatrix}. \end{aligned} $$

(12.27)