Uniqueness of shear wave modeling in an incompressible, transversely isotropic (ITI) material

Abstract

Five material parameters are required to describe a transversely isotropic (TI) material including two Poisson’s ratios that characterize the compressibility of the material. Both Poisson’s ratios must be specified to model an incompressible, TI (ITI) material. However, a previous analysis of the procedure used to evaluate the incompressible limit in a two-dimensional (2D) space of Poisson’s ratios has shown that elements of the stiffness tensor are not unique in this limit, and that an additional, fourth parameter is required to model these elements for an ITI material. In this study, we extend this analysis to the case of shear wave propagation in an ITI material. Shear wave signals are modeled using analytic Green’s tensor methods to express the signals in terms of the phase velocity and polarization vectors of the shear horizontal (SH) and shear vertical (SV) propagation modes. In contrast to the previous result, the current analysis demonstrates that the phase velocity and polarization vectors are independent of the procedure used to evaluate the 2D limit of Poisson’s ratios without the need to include an additional parameter. Thus, calculated shear wave signals are unique and can be used for comparison with experimental measurements to determine all three model parameters that characterize an ITI material.

1. Introduction

In a recent study, Knight et al (2021) reported observations of shear wave propagation following an acoustic radiation force impulse (ARFI) excitation in vivo in a vastus lateralis muscle using an excitation configuration in which the muscle fibers were tilted with respect to the plane perpendicular to the excitation axis (see figures 1 and 2 of Knight et al (2021)). Because of the tilted excitation, this configuration enabled the observation of both the shear horizontal (SH) and shear vertical (SV) propagation modes with ultrasound, and thereby allowed the measurement of additional material properties that could not be assessed with the commonly used excitation geometry in which the muscle fibers are not tilted.

Knight et al (2021) analyzed their measurements by comparing the measured shear wave speeds to results from model calculations using a linear, elastic, incompressible, transversely isotropic (ITI) material model. Compared to a transversely isotropic (TI) model, the ITI model takes advantage of the fact that biological tissues are nearly incompressible so that the compressional wave speed is much greater than shear wave speeds, and compressional wave motion can be ignored in the model of shear wave propagation. The ITI material model is derived from a TI model by specifying the values of two Poisson’s ratios required to make the model incompressible, and requires only three material parameters compared to five for the TI model. By observing both SH and SV shear wave propagation, Knight et al (2021) were able to measure all three material parameters, and thereby, obtain a full characterization of muscle as a linear, elastic, ITI material.

One difficulty with the use of an ITI material model is that two Poisson’s ratios must be specified to achieve the incompressible limit, and the procedure used to evaluate the limit in a two-dimensional (2D) space of Poisson’s ratios must be considered. In fact, a previous investigation by O’Donnell and Skovoroda (2004) has shown that elements of the stiffness tensor are not uniquely defined in the incompressible limit, and that an additional, fourth material parameter is required to characterize the material. Thus, this observation raises the question of whether or not a three parameter model is sufficient to characterize an ITI material, and if this model can be used to calculate unique shear wave signals for comparison with experimental measurements.

In this technical note, we review the analysis of O’Donnell and Skovoroda (2004) and extend it to the modeling of shear wave propagation in an ITI material. In particular, we consider the analytical calculation of shear wave signals using Green’s tensor methods (Rouze et al 2020) and the result of these calculations in the limit of an ITI material model. The primary result demonstrated here is that shear wave signals are uniquely defined in the limit of a three-parameter, ITI material model, and thus, measurement of these three parameters is sufficient to fully characterize an ITI material.

2. Incompressible limit of the stiffness matrix for a TI material

Assuming small displacements and a linear stress-strain relation, an anisotropic material can be modeled using a generalized Hooke’s law to relate the stress tensor σ and strain tensor ϵ through a fourth-order stiffness tensor c with components c_ijkl,

$${\sigma }_{ij}=c_{ijkl}{ϵ}_{kl},$$ (1)

where summation over repeated indices is implied. The stress-strain relation can also be expressed using Voigt notation as a matrix product with a 6 × 6 stiffness matrix C, or compliance matrix S = C⁻¹,

$$\left(\begin{array}{l}{\sigma }_{11}\hfill \\ {\sigma }_{22}\hfill \\ {\sigma }_{33}\hfill \\ {\sigma }_{23}\hfill \\ {\sigma }_{31}\hfill \\ {\sigma }_{12}\hfill \end{array}\right)=C\left(\begin{array}{c}{ϵ}_{11}\\ {ϵ}_{22}\\ {ϵ}_{33}\\ 2{ϵ}_{23}\\ 2{ϵ}_{31}\\ 2{ϵ}_{12}\end{array}\right)\text{and}\left(\begin{array}{c}{ϵ}_{11}\\ {ϵ}_{22}\\ {ϵ}_{33}\\ 2{ϵ}_{23}\\ 2{ϵ}_{31}\\ 2{ϵ}_{12}\end{array}\right)=\text{S}\left(\begin{array}{l}{\sigma }_{11}\hfill \\ {\sigma }_{22}\hfill \\ {\sigma }_{33}\hfill \\ {\sigma }_{23}\hfill \\ {\sigma }_{31}\hfill \\ {\sigma }_{12}\hfill \end{array}\right).$$ (2)

In this note, we consider a TI material characterized by an axis of symmetry, $\widehat{A}$, such that the material properties are unchanged by reflections across any plane parallel to $\widehat{A}$ or rotations about $\widehat{A}$. Skeletal muscle is often modeled as a TI material with the symmetry axis determined by the muscle fibers. Five independent parameters are required to specify the stiffness matrix C, or the compliance matrix S, of a TI material (Lai et al 2010). Using the notation L and T to indicate the longitudinal and transverse directions, these five parameters can be expressed (Lai et al 2010, Rouze et al 2020) in terms of six engineering constants using Young’s moduli E_L and E_T, shear moduli μ_L and μ_T, and Poisson’s ratios ν_LT and ν_TT along with the relation

$${\mu }_T=\frac{E_T}{2\left(1+{\nu }_{TT}\right)}.$$ (3)

In terms of these parameters, independent elements C₁₁, C₁₃, C₃₃, C₄₄, and C₆₆ of the stiffness matrix are given by the relations

$$\begin{gathered} C_{11}=\frac{E_T}{1+{\nu }_{TT}}\frac{1-{\nu }_{LT}^2{E}_T/E_L}{1-{\nu }_{TT}-2{\nu }_{LT}^2{E}_T/E_L},C_{44}={\mu }_L,C_{66}={\mu }_T \\ {C}_{13}=\frac{E_T{\nu }_{LT}}{1-{\nu }_{TT}-2{\nu }_{LT}^2{E}_T/E_L},\text{and}{C}_{33}=\frac{E_L\left(1-{\nu }_{TT}\right)}{1-{\nu }_{TT}-2{\nu }_{LT}^2{E}_T/E_L}. \end{gathered}$$ (4)

The remaining nonzero elements of C are given by symmetry, C_ij = C_ji, and the relations

$$C_{12}=C_{11}-2C_{66},C_{22}=C_{11},C_{23}=C_{13},\text{and}{\text{C}}_{55}={\text{C}}_{44}.$$ (5)

For an ITI material, the dilatation of the material must be zero when subjected to arbitrary stresses. As shown by O’Donnell and Skovoroda (2004) and Rouze et al (2020), this condition requires the two Poisson’s ratios to have specific values,

$${\nu }_{\text{LT},\text{inc}}=\frac{1}{2}\text{and}{\nu }_{\text{TT},\text{inc}}=1-\frac{E_T}{2E_L}.$$ (6)

Thus, three material parameters are required to describe an ITI material. Here, we use the parameters μ_L, μ_T, and the ratio R = E_T/E_L of Young’s moduli as used previously (Rouze et al 2013, Rouze et al 2020). However, with (3) and (6), any three linearly independent combinations of the parameters μ_L, μ_T, E_L, and E_T can be used.

Modeling of an incompressible material can be achieved by considering the limit as the Poisson’s ratios ν_LL and ν_LT approach the limiting values (6). As shown in figure 1, this limit can be evaluated in a 2D space of Poisson’s ratios by considering the approach from a position at a distance δ from the limit point along an approach direction specified by an angle α. Using the notation s_α = sin α and c_α = cos α, the Poisson’s ratios can be expressed relative to the incompressible limits as

$${\nu }_{\text{LT}}={\nu }_{\text{LT},\text{inc}}-\delta c_{\alpha }=\frac{1}{2}-\delta c_{\alpha }\text{and}{\nu }_{\text{TT}}={\nu }_{\text{TT},\text{inc}}-\delta s_{\alpha }=1-\frac{E_T}{2E_L}-\delta s_{\alpha }.$$ (7)

Then, the incompressible limit is equivalent to the limit δ → 0.

Figure 1.

Two dimensional space of Poisson’s ratios for a TI material showing the approach to the incompressible limit point (ν_LT,inc, ν_TT,inc) along a trajectory oriented at an angle α relative to the ν_LT axis. Values of the Poisson’s ratios ν_LT and ν_TT can be expressed in terms of α and the distance δ from the limit point using (7).

Elements of the stiffness matrix can be evaluated in the incompressible limit by inserting (7) into (4). From (4), elements C₄₄ and C₆₆ are independent of the Poisson’s ratios and remain constant in the limit. Substituting for E_T from (3) and using the ratio R = E_T/E_L, elements C₁₁, C₁₃, and C₃₃ are given by

$$\begin{gathered} C_{11}=\frac{2{\mu }_T}{D}\left[\left(1-\frac{R}{4}\right)+\delta c_{\alpha }R-{\delta }^2{c}_{\alpha }^{2}R\right], \\ {C}_{13}=\frac{2{\mu }_T}{D}\left\{\left(1-\frac{R}{4}\right)+\delta \left[-\frac{s_{\alpha }}{2}-\left(2-\frac{R}{2}\right)c_{\alpha }\right]+{\delta }^2{s}_{\alpha }{c}_{\alpha }\right\},\text{and} \\ {C}_{33}=\frac{2{\mu }_T}{D}\left[\left(1-\frac{R}{4}\right)+\delta s_{\alpha }\left(\frac{2}{R}-1\right)-{\delta }^2\frac{s_{\alpha }^2}{R}\right] \end{gathered}$$ (8)

where the common factor in the denominators of (4) is given by

$$D=1-{\nu }_{TT}-2{\nu }_{LT}^2{E}_T/E_L=\delta \left(s_{\alpha }+2c_{\alpha }R-2c_{\alpha }^{2}R\delta \right).$$ (9)

Because D is O(δ), elements C₁₁, C₁₃, and C₃₃ diverge in the limit δ → 0. However, the differences between pairs of these elements remain finite,

$$\begin{gathered} C_{11}-C_{13}\underset{\delta \to 0}{\to }\frac{2{\mu }_T}{s_{\alpha }+2c_{\alpha }R}\left[\frac{s_{\alpha }}{2}+c_{\alpha }\left(2+\frac{R}{2}\right)\right], \\ {C}_{11}-C_{33}\underset{\delta \to 0}{\to }\frac{2{\mu }_T}{s_{\alpha }+2c_{\alpha }R}\left[s_{\alpha }\left(1-\frac{2}{R}\right)+c_{\alpha }R\right],\text{and} \\ {C}_{13}-C_{33}\underset{\delta \to 0}{\to }\frac{2{\mu }_T}{s_{\alpha }+2c_{\alpha }R}\left[s_{\alpha }\left(\frac{1}{2}-\frac{2}{R}\right)+c_{\alpha }\left(\frac{R}{2}-2\right)\right]. \end{gathered}$$ (10)

These differences depend on the angle α of approach to the limit point in figure 1. This observation led O’Donnell and Skovoroda (2004) to conclude that three material parameters were not sufficient for a unique description of the stiffness matrix for a ITI material, and that a fourth parameter such as the limit approach angle α was needed to describe the material in the incompressible limit.

We note that expressions similar to (10) for limiting values of the differences in stiffness matrix elements have previously been given by Rouze et al (2020), but that those expressions differ from (10) as they do not depend on the angle α of approach to the limit point. The reason for this disagreement is that Rouze et al (2020) did not consider the possibility that the limit could depend on the direction of approach to the limit point, but instead, set the Poisson’s ratio ν_TT to its limiting value (6) so that α = 0, and then evaluated the differences (10) in the limit ν_LT → 1/2. In addition, this same procedure was used to evaluate the limiting value for the phase velocity v_SV in expressions (20) of Rouze et al (2013) and (13) of Rouze et al (2020), and also to evaluate the parameters used in the finite element (FE) simulations of wave motion in a nearly ITI (NITI) material in Rouze et al (2013). Thus, results following from those derivations could possibly be affected by the procedure used to evaluate the 2D limit. This possibility is addressed in the following.

3. Modeling shear wave propagation in an ITI material

3.1. Green’s tensor modeling of shear wave propagation in a TI material

In this note, we model shear wave propagation in a linear, elastic, ITI material by calculating shear wave displacements using Green’s tensor methods. As described by Rouze et al (2020), the shear wave displacement $\overrightarrow{u}({\overrightarrow{r}}_{\text{obs}},t)$ at position ${\overrightarrow{r}}_{\text{obs}}$ and time t can be calculated by dividing the excitation source $\overrightarrow{f}\left(r_s,t_s\right)=\overrightarrow{f}\left(r_s\right)W\left(t_s\right)$ into source voxels s and summing the contributions from each voxel. The Green’s tensor $G_{in}({\overrightarrow{r}}_{obs}-{\overrightarrow{r}}_s,t-t_s)$ gives the relative contribution to the signal from each source voxel. Then, in the temporal frequency domain, components $u_i({\overrightarrow{r}}_{obs},\omega )$ of the shear wave displacement can be written as

$$u_i({\overrightarrow{r}}_{obs},\omega )=\sum _s{G}_{in}(\overrightarrow{r},\omega )f_n({\overrightarrow{r}}_s)W(\omega )$$ (11)

where $\overrightarrow{r}={\overrightarrow{r}}_{obs}-{\overrightarrow{r}}_s$ is the relative position between the source and observation positions and summation over the component index n of the force is implied. The Green’s tensor can be calculated as the inverse, 3D Fourier transform of the sum of three terms corresponding to the primary (P) or compressional wave, and the SH, and SV propagation modes, with each term determined by the phase velocity v_N and polarization direction ${\widehat{\mathcal{P}}}^N$ for that mode. For the ITI model considered here, we neglect the P-wave mode so that

$$G_{in}(\overrightarrow{r},\omega )=\sum _{N=SH,SV}\frac{1}{8{\pi }^3}\iiint \frac{{\widehat{\mathcal{P}}}_i^N{\widehat{\mathcal{P}}}_n^N}{\rho k^2{v}_N^2-\rho {\omega }^2}{e}^{i\overrightarrow{k}\cdot \overrightarrow{r}}{d}^3\overrightarrow{k}.$$ (12)

where ρ is the material density.

3.2. Phase velocities v_N and polarizations ${\widehat{\mathcal{P}}}^N$ in a TI material

The phase velocities v_N and polarizations ${\widehat{\mathcal{P}}}^N$ in (12) can be determined by considering plane wave propagation with displacement

$$\overrightarrow{u}(\overrightarrow{r},\omega )=\left(\begin{array}{l}{\mathcal{P}}_x\hfill \\ {\mathcal{P}}_y\hfill \\ {\mathcal{P}}_z\hfill \end{array}\right)e^{i(k\widehat{n}\cdot \overrightarrow{r}-\omega t)}$$ (13)

where $\widehat{n}$ is the propagation direction, k is the wavenumber, and v = ω/k is the speed. To describe the polarization and propagation directions relative to the material symmetry axis, we orient the coordinate system so the z-axis is aligned with $\widehat{A}$ and propagation direction is in the x − z plane at an angle θ relative to the z-axis so that $\widehat{n}={\left(s_{\theta }0c_{\theta }\right)}^T$ where s_θ = sin θ and c_θ = cos θ. Inserting (13) into the wave equation leads to an eigenequation for the 3×3 matrix Γ with eigenvectors ρv² and eigenvectors $\widehat{P}$ (Tsvankin (2012), Carcione (2015), Rouze et al (2013)),

$$\left(\begin{array}{ccc}{C}_{11}{s}_{\theta }^2+C_{44}{c}_{\theta }^2& 0& \left(C_{13}+C_{44}\right)s_{\theta }{c}_{\theta }\\ 0& C_{66}{s}_{\theta }^2+C_{44}{c}_{\theta }^2& 0\\ \left(C_{13}+C_{44}\right)s_{\theta }{c}_{\theta }& 0& C_{44}{s}_{\theta }^2+C_{33}{c}_{\theta }^2\end{array}\right)\left(\begin{array}{l}{\mathcal{P}}_x\hfill \\ {\mathcal{P}}_y\hfill \\ {\mathcal{P}}_z\hfill \end{array}\right)=\rho v^2\left(\begin{array}{l}{\mathcal{P}}_x\hfill \\ {\mathcal{P}}_y\hfill \\ {\mathcal{P}}_z\hfill \end{array}\right).$$ (14)

The solution of (14) for the SH propagation mode consists of particle displacements perpendicular to the x − z plane,

$$\rho v_{SH}^2={\mu }_T{s}_{\theta }^2+{\mu }_L{c}_{\theta }^2\text{and}{\widehat{\mathcal{P}}}^{SH}=\left(\begin{array}{l}0\hfill \\ 1\hfill \\ 0\hfill \end{array}\right),$$ (15)

where C₆₆ = μ_T and C₄₄ = μ_L from (4). Note that Rouze et al (2013) refer to this mode as the pure transverse (PT) mode.

Two additional non-trivial solutions of (14) can be found by requiring det (Γ − ρv²I) = 0 where I is the identity matrix. Evaluation of the determinant gives a quadratic equation for the eigenvalues ρv²,

$$\mathcal{A}{\left(\rho v^2\right)}^2+\mathcal{B}\rho v^2+\mathcal{C}=0$$ (16)

where

$$\begin{gathered} \mathcal{A}=1, \\ \mathcal{B}=-\left(C_{11}{s}_{\theta }^2+C_{33}{c}_{\theta }^2+{\mu }_L\right),\text{and} \\ \mathcal{C}=\left(C_{11}{s}_{\theta }^2+C_{33}{c}_{\theta }^2\right){\mu }_L+\left(C_{11}{C}_{33}-C_{13}^2\right)s_{\theta }^2{c}_{\theta }^2-\left(C_{11}+2C_{13}+C_{33}\right){\mu }_L{s}_{\theta }^2{c}_{\theta }^2. \end{gathered}$$ (17)

Then, the phase velocities for the P and SV modes are given by

$$\rho v^2=-\frac{\mathcal{B}}{2}\pm \frac{\mathcal{B}}{2}\sqrt{1-\frac{4\mathcal{C}}{{\mathcal{B}}^2}}.$$ (18)

The polarization vectors for these modes are given by Carcione (2015) and Rouze et al (2013),

$$\widehat{\mathcal{P}}=\frac{1}{\mathcal{N}}\left(\begin{array}{c}{\left(C_{33}{c}_{\theta }^2+{\mu }_L{s}_{\theta }^2-\rho v^2\right)}^{1/2}\\ 0\\ {\left(C_{11}{s}_{\theta }^2+{\mu }_L{c}_{\theta }^2-\rho v^2\right)}^{1/2}\end{array}\right)$$ (19)

where $\mathcal{N}={\left(C_{11}{s}_{\theta }^2+C_{33}{c}_{\theta }^2+{\mu }_L-2\rho v^2\right)}^{1/2}$ is a normalization factor. These solutions correspond to the SV propagation mode (upper sign in (18)) with polarization approximately transverse to the propagation direction $\widehat{n}$, and the P propagation mode (lower sign in (18)) with polarization approximately aligned with $\widehat{n}$. Rouze et al (2013) refer to these modes as the quasi-transverse (QT) and quasi-longitudinal (QL) modes, respectively.

3.3. Evaluation of v_N and ${\widehat{\mathcal{P}}}^N$ for an incompressible, TI material

We can evaluate the phase velocities v_N for the P and SV modes in the limit δ → 0 for an incompressible material by expanding the square root in (18),

$$\rho v^2=-\frac{\mathcal{B}}{2}\pm \frac{\mathcal{B}}{2}\left[1-\frac{1}{2}\left(\frac{4\mathcal{C}}{{\mathcal{B}}^2}\right)-\frac{1}{8}{\left(\frac{4\mathcal{C}}{{\mathcal{B}}^2}\right)}^2-\frac{1}{16}{\left(\frac{4\mathcal{C}}{{\mathcal{B}}^2}\right)}^3+\cdots \right].$$ (20)

Using (8) and (9), both $\mathcal{B}$ and $\mathcal{C}$ are O(1/δ) and diverge in the limit δ → 0. Also, terms of the form ${\mathcal{C}}^n/{\mathcal{B}}^m$ are O(δ^(m−n)) and vanish in the limit δ → 0 for m > n. Then the eigenvalues ρv² are given by

$$\rho v^2=-\frac{\mathcal{B}}{2}\pm \frac{\mathcal{B}}{2}\mp \frac{\mathcal{C}}{\mathcal{B}}+O(\delta ).$$ (21)

For the lower sign in (21), $\rho v^2\sim -\mathcal{B}$ and diverges for δ → 0. This solution corresponds to the P propagation mode which describes the compressional wave whose velocity is expected to diverge in an incompressible material.

The upper sign in (21) corresponds to the SV propagation mode so that, using (17),

$$\begin{gathered} \rho v_{\text{SV}}^2=\underset{\delta \to 0}{\lim }\left(-\frac{\mathcal{C}}{\mathcal{B}}\right)=\underset{\delta \to 0}{\lim }[-\frac{\left(C_{11}{s}_{\theta }^2+C_{33}{c}_{\theta }^2\right){\mu }_L}{\mathcal{B}}-\frac{C_{11}{C}_{33}-C_{13}^2}{\mathcal{B}}{s}_{\theta }^2{c}_{\theta }^2 \\ +\frac{C_{11}+2C_{13}+C_{33}}{\mathcal{B}}{\mu }_L{s}_{\theta }^2{c}_{\theta }^2]. \end{gathered}$$ (22)

We consider the terms in (22) separately. First, using (17) and (8), $\mathcal{B}$ is given by

$$\mathcal{B}=-\left(C_{11}{s}_{\theta }^2+C_{33}{c}_{\theta }^2+{\mu }_L\right)=-\frac{2{\mu }_T}{D}\left[\left(1-\frac{R}{4}\right)+O(\delta )\right]-{\mu }_L.$$ (23)

The first term in the limit on the right hand side (RHS) of (22) is given by

$$-\frac{\left(C_{11}{s}_{\theta }^2+C_{33}{c}_{\theta }^2\right){\mu }_L}{\mathcal{B}}=-\frac{\left(-\mathcal{B}-{\mu }_L\right){\mu }_L}{\mathcal{B}}={\mu }_L+\frac{{\mu }_L^2}{\mathcal{B}}\underset{\delta \to 0}{\to }{\mu }_L.$$ (24)

Using (8), the numerator in the second term on the RHS of (22) is given by

$$-\left(C_{11}{C}_{33}-C_{13}^2\right)=-{\left(\frac{2{\mu }_T}{D}\right)}^2\left[\delta \left(1-\frac{R}{4}\right)\left(\frac{2s_{\alpha }}{R}+4c_{\alpha }\right)+O\left({\delta }^2\right)\right]$$ (25)

so that, with (9) and (23), the second term on the RHS of (22) is given by

$$-\frac{C_{11}{C}_{33}-C_{13}^2}{\mathcal{B}}{s}_{\theta }^2{c}_{\theta }^2\underset{\delta \to 0}{\to }\frac{4{\mu }_T}{R}{s}_{\theta }^2{c}_{\theta }^2.$$ (26)

Finally, the last term in (22) can be evaluated using (8) and (23),

$$\frac{\left(C_{11}+2C_{13}+C_{33}\right){\mu }_L}{\mathcal{B}}{s}_{\theta }^2{c}_{\theta }^2=\frac{1}{\mathcal{B}}\frac{2{\mu }_T}{D}\left[4\left(1-\frac{R}{4}\right)+O(\delta )\right]{\mu }_L{s}_{\theta }^2{c}_{\theta }^2\underset{\delta \to 0}{\to }-4{\mu }_L{s}_{\theta }^2{c}_{\theta }^2.$$ (27)

Then, combining (22), (24), (26), and (27) gives

$$\rho v_{\text{SV}}^2\underset{\delta \to 0}{\to }\rho v_{\text{SV},\text{inc}}^2={\mu }_L+4\left(\frac{{\mu }_T}{R}-{\mu }_L\right)s_{\theta }^2{c}_{\theta }^2.$$ (28)

Thus, this result is independent of the limit approach angle α and agrees with the previous result from Rouze et al (2013) which was evaluated without consideration of the procedure used to evaluate the 2D limit of Poisson’s ratios.

We note that for the special case R = 4, the terms (1 − R/4) vanish in (23), (25), and (27), and the evaluation of $\rho v_{\mathit{SV}}^2$ will require higher order terms in δ that lead to dependencies on the approach angle α. From a practical standpoint, this issue does not affect the result found above for two reasons. First, it is unlikely the ratio R = E_T/E_L = 4 will occur in muscle because we expect E_T < E_L and R < 1. For example, the measurements of Knight et al (2021) reported the tensile anisotropy in the vastus lateralis muscle to be ${\chi }_E=\left(E_L-E_T\right)/E_T=4.67$ with the corresponding value R = 0.18. Second, even if calculations of shear wave displacements with R = 4 are required, they can be performed using a value of R that is arbitrarily close to 4. In effect, this approach is equivalent to saying that we can evaluate the incompressible limit to obtain (28) with R ≠ 4, and then apply the result with R = 4.

Polarization vectors for the P and SV modes can be determined by defining a polarization angle ϕ relative to the z-axis, so that, using (19),

$$\tan \varphi =\frac{{\mathcal{P}}_x}{{\mathcal{P}}_z}=\frac{{\left(C_{33}{c}_{\theta }^2+{\mu }_L{s}_{\theta }^2-\rho v^2\right)}^{1/2}}{{\left(C_{11}{s}_{\theta }^2+{\mu }_L{c}_{\theta }^2-\rho v^2\right)}^{1/2}}$$ (29)

From (28) and (8), $\rho v_{\mathit{SV}}^2$ remains finite in the incompressible limit while C₃₃ and C₁₁ both diverge, and

$${\varphi }_{\text{SV}}\underset{\delta \to 0}{\to }{\varphi }_{\text{SV},\text{inc}}={\tan }^{-1}\left(\frac{c_{\theta }}{s_{\theta }}\right)={\tan }^{-1}\left(\frac{\sin (\theta -\pi /2)}{\cos (\theta -\pi /2)}\right)=\theta -\pi /2.$$ (30)

For the P mode, the lower sign in (21) gives $\rho v^2\sim -\mathcal{B}$ so that, with (17) and (8),

$${\varphi }_{\text{P}}\sim {\tan }^{-1}\left[\frac{{\left(C_{33}{c}_{\theta }^2+{\mu }_L{s}_{\theta }^2+\mathcal{B}\right)}^{1/2}}{{\left(C_{11}{s}_{\theta }^2+{\mu }_L{c}_{\theta }^2+\mathcal{B}\right)}^{1/2}}\right]\underset{\delta \to 0}{\to }{\tan }^{-1}\left(\frac{s_{\theta }}{c_{\theta }}\right)=\theta .$$ (31)

Thus, in the incompressible limit, the polarization vectors of the SV and P modes are perpendicular and parallel to the propagation direction, respectively. This result agrees with the result of Rouze et al (2013) which was evaluated without consideration of the approach direction α in the evaluation of the 2D limit of Poisson’s ratios.

4. Discussion and conclusion

The key result of the analysis presented in section 3 is that, in the limit of an ITI material, the phase velocity v_SV and polarization ${\widehat{\mathcal{P}}}^{SV}$ for the SV propagation mode are independent of the procedure used to evaluate the incompressible limit. Thus, these quantities are unique, and shear wave signals calculated using Green’s tensor methods are also unique and can be used for comparison with experimental measurements to characterize the material using a linear, elastic, ITI material model. In particular, this result demonstrates that previous observations of non-unique elements of the stiffness tensor by O’Donnell and Skovoroda (2004) do not extend to the modeling of shear wave propagation in an ITI material.

In fact, it might be argued that models of shear wave propagation in an ITI material should not depend on the procedure used to evaluate the incompressible limit because compressional wave propagation, which depends on the compressibility of the material through the Poisson’s ratios ν_LT and ν_TT, is excluded in the model. Instead, the ITI model considers only shear wave motion which is an isovolumetric process and is not dependent on the material compressibility or Poisson’s ratios.

On the other hand, nearly ITI (NITI) material models (Pitre et al (2020), Rouze et al 2013) include compressional wave propagation using a five-parameter, TI material model and set the Poisson’s ratios ν_LT and ν_TT near to the values required in the incompressible limit. For example, in their FE modeling of shear wave propagation in a NITI material, Rouze et al (2013) set the Poisson’s ratios to the values ν_TT = ν_TT,inc and ν_LT = 0.499 so that δ = 0.001 and α = 0 in (7). At the same time, μ_L, μ_T, and the ratio R = E_T/E_L were set to the specific values to be modeled, and individual values of E_L, E_T needed for the FE simulations were calculated using (3) and (6). An alternative approach with, for example, a different approach angle α, would be expected to give essentially identical results for the shear wave signals, with similar, but different, results for the compressional wave signals.

In conclusion, this note addresses the question raised by O’Donnell and Skovoroda (2004) regarding the number of parameters required to characterize the stiffness tensor in an incompressible, transversely isotropic (ITI) material, and the extension of this question to the case of shear wave propagation in these materials. Beginning with a five-parameter, TI material model, O’Donnell and Skovoroda (2004) found that, even with two Poisson’s ratios specified by values consistent with the incompressible limit, three material parameters were not sufficient to describe the stiffness tensor of an ITI material, and instead, a fourth parameter was required to describe the approach to the incompressible limit in a 2D space of Poisson’s ratios. In contrast, the analysis presented in this note demonstrates that three material parameters are sufficient to uniquely characterize shear wave propagation in an ITI material. Thus, analysis of experimentally observed shear wave signals by comparison with calculated signals based on an ITI model are sufficient to determine all three parameters necessary to characterize the material using a linear, elastic, ITI model.