whereεis related to the displacement u εkl= 1
Due to the thermodynamics of reversible deformations and to symmetry of the stain and stress tensors,Chas at most 21 independent coefficients. In a reduced two index notation called Voigt notation, Hooke’s law (1.1) can be expressed as
where pairs of subscripts for the stiffness tensor in Hooke’s law (1.1) 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
If the material istransversely isotropic (one axis of cylindrical symmetry) only five coefficients remains independent, as the following relations hold: C11=C22,C13= C23,C44=C55, andC12=C11−2C44 (for symmetry around axis 3). Finally, if the material is isotropic, only two coefficient remains independent as C11=C22=C33, C44=C55=C66, and C12=C13=C23=C11−2C44.
Hooke’s law (1.1) 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
The basic principle of mechanical testing is to infer an elastic coefficient from a stress-strain curve measured in a low frequency of quasi-static experiment. Usually, a stress field is imposed and the strain measured on the sample using strain gauges or optical methods. If the stress and strain remains sufficiently small, they are linearly related trough Hooke’s law (1.1) and the slope of the curve is equal to the stiffness.
However, due to the tensorial nature of the stress-strain relationship, inferring a par-ticular moduli in a straightforward way requires parpar-ticular stress and strain fields.
For example, measuring the Young’s modulus in a given material direction requires a uniaxial stress field along that direction. Hence, specific specimen shapes are dedicated to the measurement of different coefficients. This usually prevents form assessing completely the anisotropy form a single specimen, although some speci-mens allow the joint measurement of several coefficients (e.g. two Poisson’s ratios can be obtained jointly to the Young’s modulus in uniaxial compression/tension if the lateral strains are measured). This a important limitation of mechanical testing.
Mechanical testing has been employed on cortical bone in numerous studies. For example,Shahar et al.(2007) used compression tests on 6 mm-sized cubes of equine bone in the three orthogonal directions to determine three Young’s moduli and six Poisson’s ratios2. Ohman et al. (2011) used compression tests as well to measure a Young’s modulus on child and adult bones, but on small cylindrical samples (d=2-3 mm, h=14-18 mm). Due to the natural cylindrical beam shape of long bones such as the femur, mechanical tests are sometimes applied on whole bones (Brodt et al., 1999, torsion and bending) or segments of bones (Chattah et al.,2009, compression) instead of samples prepared to a specific regular shape. This is particularly the case in small animal studies for which the preparation of samples can be challenging.
This reduces accuracy, as the equations usually assume regular shapes (van Lenthe et al.,2008). Additionally, the spatial variation of bone tissue properties should also be taken into account to improve accuracy (Kim et al.,2012).
Some of the studies cited above (Chattah et al.,2009;Ohman et al.,2011) used an optical method called electronic speckle pattern interferometry to measure the strain on the tested specimen, instead of using more conventional strain gauges. In addition to being a non-contact method, the main advantage of this increasingly popular optical measurement is that the field strain is measured on a surface of the sample, instead of at a single point.
2Because the Poisson’s ratios and Young’s moduli are not all independent, six independent coefficients were obtained. This seems to be highest number of coefficients that can be measured from mechanical testing on a single sample.
1.3. Assessment of cortical bone elasticity at the millimeter-scale 7
1.3.2 Ultrasonic waves methods
Neglecting body forces and using the stress-strain (1.1) and strain-displacement (1.2) relations, the equation of motion for a continuum media is
ρ∂2ui
∂t2 =Cijkl ∂2uk
∂xj∂xl
, (1.6)
whereρ is the mass density. Assuming a time harmonic plane wave at pulsation ω and phase velocityV propagating in directionn (unit vector)
ui = Uk·exp
the equation of motion (1.6) yields the Christoffel’s equation Γik−ρV2δik
·Uk = 0, (1.8)
with the Christoffel’s matrix Γik =Cijklnjnl and the Kronecker’s delta δik (= 1 if i=k, 0 otherwise).
Christoffel’s equation (1.8) is an eigenproblem for the eigenvaluesρV2and eigen-vectorsU (polarization of the plane wave). For waves propagating in the principal direction of an orthotropic material, i.e. n = (1,0,0), (0,1,0) or (0,0,1), equa-tion (1.8) yields three soluequa-tions. For example, for propagaequa-tion along the first axis the solutions write ρV2 = C11 with polarization vector (1,0,0)3, ρV2 =C66 with polarization vector(0,1,0)4 and ρV2 =C55 with polarization vector(0,0,1)4. De-noting Vij the velocity of a plane wave propagating along direction iwith displace-ment in the direction j, six relations between the diagonal elastic coefficients and velocities are obtained
Cii=ρVii2 (i= 1,2,3), (1.9)
C44=ρV232 =ρV322, (1.10)
C55=ρV132 =ρV312, (1.11)
C66=ρV122 =ρV212. (1.12)
Given a rectangular parallelepipedic sample whose faces are aligned with the material principal directions, the six diagonal coefficients can be obtained from the velocities of waves propagating between a pair of transducers contacting two oppo-site faces, or from a single transducer using the reflection of the waves. Compression transducers must be used for the three first coefficients [Eq. (1.9)] and shear trans-ducers for the three last [Eqs. (1.10), (1.11), and (1.12)]. The off-diagonal coefficients C12,C13, and C23 are related to the velocities of waves propagating in other direc-tions. Their measurement requires additional faces not aligned with the principal direction of elasticity, usually prepared at 45◦ (Fig. 1.3).
3Pure longitudinal wave, propagation and displacement directions are colinear.
4Pure shear wave, propagation and displacement directions are orthogonal.
Figure 1.3: Cubic specimen with additional 45◦-oriented faces for the measurement of the off-diagonal elastic constants. FromRho(1996)
.
The above equations were derived for propagation in an infinite medium (bulk wave propagation), but still hold if the wavelength is sufficiently smaller than the cross-section of the sample. If not, waves are perturbed by the boundaries of the domain and the stiffness coefficients are no longer simply related to the velocities. In the opposite case (wavelength much larger that the cross section), waves propagate in a bar wave regime, in which the propagation is guided by the sample shape. In that case, longitudinal wave velocities are directly related to the Young’s moduli of the material (e.g. ρV112 =E1).
Lang (1969) was the first to introduce ultrasound bulk wave velocity measure-ments in the context of bone research as an alternative to mechanical tests. He measured five independent elastic coefficients (transversely isotropic symmetry) on bone samples from the time-of-flight of ultrasonic pulse waves. The method was subsequently used by other researchers to study bone elastic anisotropy. Yoon and Katz (1976a,b) argued for the transversely isotropic symmetry of human cortical bone and measured the five elastic coefficients on dried specimens. Van Buskirk et al. (1981) find significant deviations from transverse isotropy for bovine corti-cal bone and argued for orthotropy. On human tibial corticorti-cal bone, Rho (1996) also observed slightly orthotropic material properties. Espinoza Orias et al. (2008) studied the variation in magnitude and anisotropy of the elastic coefficients along the human femur and found that, while the transverse isotropy assumption was ac-curate for the mid-diaphysis, tissue at extreme (distal and proximal) positions of the diaphysis showed moderate orthotropy. Applying the same method to human femoral bone samples (mid-diaphysis),Granke et al.(2011) also observed transverse isotropy. It is noteworthy that only the diagonal elastic coefficients were measured in this two last studies, due to the difficulty of cutting samples with 45◦-oriented faces.
In consequence, the Young’s moduli and Poisson’s ratios could not be evaluated.
Ashman et al.(1984) introduced a slightly different method, based on the use of continuous waves instead of pulses. They also discussed the sample size constraints to respect for bulk wave propagation. They showed that the transition from bulk
1.3. Assessment of cortical bone elasticity at the millimeter-scale 9
waves to bar waves occurs for a product wavenumber×cross-section inferior to 10.
For the wavelength of 2.25 MHz compression wave in bone5, the minimal sample size is 2.5 mm. This size is comparable or even larger than the cortical thickness in most of the bones of the human body, and obviously much larger that samples from small animals. This constraint cannot be simply overcome by scaling up the frequency because the wavelength must remain much larger that bone heterogeneities. At 2.25 MHz the wavelength is about 1.6 mm, close to 1.5 mm, the minimal length over which human cortical bone can be consider homogeneous with respect to elasticity (Grimal et al.,2011). Moreover, the attenuation of ultrasound waves increases with frequency. For these reasons, wave velocity measurements have been limited to large human bones (tibiae and femur) and bones from large animals (mostly bovines and equines).
Granke et al. (2011) estimated the repeatability of the measured elastic coeffi-cients to be 3.4%and 4.9%for longitudinal and shear coefficients, respectively. The lack of repeatability in velocity measurements is in part caused by the limited preci-sion of the time-of-flight measurements. Indeed, due disperpreci-sion of ultrasound waves in bone, the shape of the transmitted signal changes during propagation. Moreover, the signal is also very sensitive to the transducers delicate positioning and coupling with the sample (Maynard,1992;Grimal et al.,2009).
Mechanical tests vs ultrasound velocity
The equations of linear elasticity introduced for both methods hold only for small strains. When stress and strain increase above a certain threshold, they are no longer linearly related as the material enters regions of non-linear elasticity and plasticity (permanent deformation) before eventually fracturing. Mechanical prop-erties beyond the elastic region can be measured using mechanical testing, but not using ultrasonic methods. This is a considerable advantage of mechanical methods.
Due to the difficulties linked to sample geometry for both mechanical testing and wave velocity measurement, very few studies compared the results of the two meth-ods on the same specimens. However, two recent studies reach similar conclusions (Grimal et al., 2009; Berteau et al., 2014): 1) elastic properties measured by me-chanical tests and ultrasonic wave velocities are only moderately correlated (R=0.63 - 0.76) and 2) the mechanical Young’s modulus is significantly lower than the ultra-sound modulus (e.g. for Grimal et al.: Emech. = 17.3±1.3 and Eac. = 22.1±0.9).
Although Grimal et al. have shown that the lack of correlation in their results may be entirely explained by the uncertainties in both methods, the measured bias seems to be physically relevant. A possible cause is bone viscoelaticity, which entails a variation of elastic modulus with frequency, but this is still an open question.
5Assuming a velocity of 3600 m.s−1