Shear Waves Dispersion in Cylindrically Structured Cancellous Viscoelastic Bones

Abstract

In this chapter we study anti-plane shear waves propagating through a cylindrically structured cancellous bone represented by a two-dimensional mesh of elastic trabeculae filled by a viscous marrow. In the long-wave limit, the original heterogeneous medium can be approximately substituted by a homogeneous one characterized by an effective complex shear modulus. The effect of dispersion is caused by the transmission of mechanical energy to heat due to the viscosity of the marrow (viscoelastic damping). We derive an approximate analytical solution using the asymptotic homogenization method; the cell problem is solved by means of a boundary shape perturbation and a lubrication theory approaches. For short waves, when the wavelength is comparable to the trabeculae size, the effect of dispersion is caused by successive reflections and refractions of local waves at the trabecula-marrow interfaces (Bloch dispersion). Decrease in the wavelength reveals a sequence of pass and stop frequency bands, so the heterogeneous bone can act like a discrete wave filter.

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1 Introduction

Animal and human bones are heterogeneous materials with a complicated hierarchical structure. Bone tissues occur in the two main forms: as a dense solid (cortical or compact bone) and as a porous medium filled by a viscous marrow (trabecular or cancellous bone) [17]. The basic mechanical discrepancy between these two types consists in their relative densities measured by a volume fraction of solids. Both types can be found in the most bones of the body. A classical example of the macroscopic bone structure can be given by the long bones (e.g., humerus, femur, and tibia). They include an outer shell of a dense cortical tissue surrounding an inner core of a porous cancellous tissue.

The microstructure of cancellous bones is often described by two- or three-dimensional mesh of interconnected rods and plates [12, 17, 21]. Despite the obvious simplicity, such idealized models can provide a satisfactory agreement between theoretical predictions and experimental results for the mechanical properties of real bones [10, 11, 15, 19, 25]. In the present paper, we shall deal with a two-dimensional model of cylindrically structured cancellous bones.

Williams and Lewis [27] considered a real 2D section of a trabecular bone and evaluated its elastic constants using the plane-strain finite elements method. The developed approach enables one to predict mechanical properties of cylindrically structured cancellous bones basing on the morphological measurements in the transverse plane.

A challenging problem consists in the detection of the bone structure using non-invasive measurements. The inverse homogenization approach (“dehomogenization procedure”) [7] can help to derive information about the microgeometry of the bone tissue from the magnitudes of static effective moduli. However, the static moduli supply only limited morphological data. Much more information can be obtained studying dynamic response of the bones. Measuring velocities and attenuation of acoustic waves at different frequencies provide us with additional information about the microstructure. Generally, frequency-dependent dynamic properties of the bone may be considered as a kind of “identification portrait”, which is unique for every sample. The larger is the explored frequency range, the more accurate is the “portrait” that can be compiled. This should give a possibility to detect even very small variations of the internal bone texture.

Acoustic waves propagating through cancellous bones undergo dispersion and damping. There are two different physical effects influencing on the dynamic properties of the bone: (i) transmission of mechanical energy to heat due to the viscosity of the marrow (viscoelastic damping and dispersion) [16] and (ii) successive reflections and refractions of local waves at the trabecula-marrow interfaces (Bloch dispersion) [2, 20]. From the theoretical standpoint, both effects are realized simultaneously. However, their intensities are very frequency dependent. For many real materials the effects of viscoelastic damping and Bloch dispersion are observed in rather distant frequency ranges. In such a case they can be analyzed separately.

We study propagation of anti-plane shear waves through a 2D cylindrically structured cancellous bones. In our work, we have shown what kind of dispersion mechanism is dominant in the propagation of viscoelastic waves in the bone. This is an important theoretical result, because it allows choosing an adequate simulation model for the solution of this class of problems. Namely, it is necessary to consider Bloch dispersion and at the same time viscoelastic effects can be neglected, i.e. use an elastic model of the media.

The input dynamic problem is formulated in Sect. 2. In Sect. 3, the long-wave limit is considered. The solution is evaluated by means of the asymptotic homogenization procedure, and the effect of viscoelastic damping is predicted. In Sect. 4, the short-wave case is studied and the effect of Bloch dispersion is analyzed by means of the plane-wave expansion method. Conclusive remarks are presented in Sect. 5.

2 Input Dynamic Problem

We study transverse anti-plane shear waves propagating in the x ₁ x ₂ plane through a regular cancellous structure consisting of a spatially infinite elastic matrix (trabeculae) Ω⁽¹⁾ and viscous inclusions (marrow) Ω⁽²⁾ (Fig. 1). The governing two-dimensional wave equation is

Fig. 1

Cancellous structure under consideration

$$ {\nabla}_x\left( G{\nabla}_x u\right)=\rho \frac{\partial^2 u}{\partial {t}^2}, $$

(1)

where G is the complex shear modulus, ρ is the mass density, u is the longitudinal displacement (in the x ₃ direction), and ∇ _x  = e ₁∂/∂x ₁ + e ₂∂/∂x ₂, e ₁, e ₂ are the unit Cartesian vectors.

Due to the heterogeneity of the medium, the physical properties G and ρ are represented by piecewise continuous functions of co-ordinates:

$$ G\left(\mathbf{x}\right)={G}^{(a)},\kern1em \rho \left(\mathbf{x}\right)={\rho}^{(a)},\mathrm{for}\kern1em \mathbf{x}\in {\varOmega}^{(a)},\kern1em \mathbf{x}={x}_1{\mathbf{e}}_1+{x}_2{\mathbf{e}}_2. $$

(2)

Here and in the sequel the superscript (a) denotes different components of the structure a = 1, 2. In Eq. (2), G ⁽¹⁾ is the real shear modulus of the elastic matrix and G ⁽²⁾ is the frequency-dependent imaginary shear modulus of the marrow. Following the linear theory of viscoelasticity, we can set G ⁽²⁾ = iωη ⁽²⁾, where ω is the frequency of a harmonic wave and η ⁽²⁾ is the viscosity of the marrow.

Equation (1) can be written in the equivalent form:

$$ {G}^{(a)}{\nabla}_{xx}^2{u}^{(a)}={\rho}^{(a)}\frac{\partial^2{u}^{(a)}}{\partial {t}^2}, $$

(3)

$$ {\left.\left\{{u}^{(1)}={u}^{(2)}\right\}\right|}_{\partial \Omega},\kern1em {\left.\left\{{G}^{(1)}\frac{\partial {u}^{(1)}}{\partial \mathbf{n}}={G}^{(2)}\frac{\partial {u}^{(2)}}{\partial \mathbf{n}}\right\}\right|}_{\partial \Omega}, $$

(4)

where ∇ ² _xx  = ∂²/∂x ²₁  + ∂/∂x ²₂ , ∂/∂n is the normal derivative to the contour ∂Ω. From the physical standpoint, Eqs. (4) means the perfect bonding conditions at the trabecula-marrow interface ∂Ω.

3 Long-Wave Approach: Asymptotic Homogenization

3.1 Two-Scale Asymptotic Procedure

We start with the case when the wavelength L is essentially larger than the internal size l of the cancellous structure l < < L. The original heterogeneous bone can be approximately substituted by a homogeneous one with a certain homogenized (effective) complex shear modulus G ₀. Such an approach neglects local reflections and refractions of the waves on microlevel. The effect of dispersion is caused by the transmission of the mechanical energy of the acoustic wave to heat due to the viscosity of the marrow.

Let us study the input boundary value problem (3), (4) by the asymptotic homogenization method [4]. In order to separate macro- and microscale components of the solution we introduce so-called slow x and fast y co-ordinate variables

$$ \mathbf{x}=\mathbf{x},\kern1em \mathbf{y}={\varepsilon}^{-1}\mathbf{x}, $$

(5)

where y = y ₁ e ₁ + y ₂ e ₂, ε = l/L is a natural small parameter, and search the displacement as an asymptotic expansion

$$ {u}^{(a)}={u}_0\left(\mathbf{x}\right)+\varepsilon {u}_1^{(a)}\left(\mathbf{x},\mathbf{y}\right)+{\varepsilon}^2{u}_2^{(a)}\left(\mathbf{x},\mathbf{y}\right)+\dots. $$

(6)

The first term u ₀ of expansion (6) represents the homogenized part of the solution; it changes slowly within the whole sample of the bone and does not depend on the fast co-ordinates (∂u ₀/∂y ₁ = ∂u ₀/∂y ₂ = 0). The next terms, u ^(a) _i , i = 1, 2, 3, …, provide corrections of the orders ε ⁱ and describe local variations of the displacements on microlevel.

The differential operators read

$$ {\nabla}_x={\nabla}_x+{\varepsilon}^{-1}{\nabla}_y,\kern1em {\nabla}_{x x}^2={\nabla}_{x x}^2+2{\varepsilon}^{-1}{\nabla}_{x y}^2+{\varepsilon}^{-2}{\nabla}_{y y}^2, $$

(7)

where ∇ _y  = e ₁∂/∂y ₁ + e ₂∂/∂y ₂, ∇ ² _xy  = ∂²/(∂x ₁∂y ₁) + ∂²/(∂x ₂∂y ₂), ∇ ² _yy  = ∂²/∂y ²₁  + ∂/∂y ²₂ .

Splitting the input problems (3), (4) with respect to ε leads to a recurrent sequence of cell boundary value problems:

$$ {G}^{(a)}\left({\nabla}_{xx}^2{u}_{i-2}^{(a)}+2{\nabla}_{xy}^2{u}_{i-1}^{(a)}+{\nabla}_{yy}^2{u}_i^{(a)}\right)={\rho}^{(a)}\frac{\partial^2{u}_{i-2}^{(a)}}{\partial {t}^2}, $$

(8)

$$ {\left.\left\{{u}_i^{(1)}={u}_i^{(2)}\right\}\right|}_{\partial \Omega}, $$

$$ {\left.\left\{{G}^{(1)}\left(\frac{\partial {u}_{i-1}^{(1)}}{\partial \mathbf{n}}+\frac{\partial {u}_i^{(1)}}{\partial \mathbf{m}}\right)={G}^{(2)}\left(\frac{\partial {u}_{i-1}^{(2)}}{\partial \mathbf{n}}+\frac{\partial {u}_i^{(2)}}{\partial \mathbf{m}}\right)\right\}\right|}_{\partial \Omega}, $$

(9)

where i = 1, 2, 3, …, u ^(a)_− 1  = 0, ∂/∂m is the normal derivative to the interface ∂Ω written in fast variables.

For a spatially periodic medium, the terms u ^(a) _i have to satisfy the conditions of periodicity

$$ {u}_i^{(a)}\left(\mathbf{x},\mathbf{y}\right)={u}_i^{(a)}\left(\mathbf{x},\mathbf{y}+{\mathbf{L}}_p\right), $$

(10)

and normalization

$$ \left\langle {u}_i^{(a)}\right\rangle =0, $$

(11)

where L _p  = ε ^− 1 l _p , l _p  = p ₁ l ₁ + p ₂ l ₂, p ₁, p ₂ = 0, ± 1, ± 2, …, l ₁, l ₂ are the fundamental translation vectors of the cancellous structure (see Fig. 1),

$$ \left\langle \cdot \right\rangle =\frac{1}{S_0}\left({\displaystyle \underset{\Omega_{{}_0}^{(1)}}{\iint}\left(\cdot \right)\mathrm{d}{y}_1\mathrm{d}{y}_2}+{\displaystyle \underset{\Omega_0^{(2)}}{\iint}\left(\cdot \right)\mathrm{d}{y}_1\mathrm{d}{y}_2}\right) $$

(12)

is the homogenizing operator over the unit cell domain Ω₀ = Ω ⁽¹⁾₀  + Ω ⁽²⁾₀ (Fig. 2), and S ₀ = L ² is the area of the unit cell in the fast co-ordinates.

Fig. 2

Periodically repeated unit cell

Conditions (10) and (11) can be approximately replaced by zero boundary conditions for same functions in the center and along the outer contour ∂Ω₀ of the unit cell:

$$ {\left.\left\{{u}_i^{(2)}=0\right\}\right|}_{x, y=0},\kern1em {\left.\left\{{u}_i^{(1)}=0\right\}\right|}_{\partial {\Omega}_0}. $$

(13)

For 1D problems, Eq. (13) appears entirely equivalent to Eqs. (10) and (11). For 2D problems, replacing the periodicity conditions by zero boundary conditions increases stiffness of the system and, thus, provides an upper bound for the effective properties. Analysis of numerical examples has shown that discrepancy between the final solutions in both cases is not essential [1, 3], whereas utilizing approximation (13) leads to a sufficient simplification of the cell problems.

Due to the periodicity of u ^(a) _i (10), Eqs. (8) and (9) can be considered within only one distinguished unit cell. Solution of the cell problems (8), (9), (13) at i = 1 determines the term u ^(a)₁ . In order to find the effective modulus G ₀, the homogenizing operator (12) is applied to Eq. (8) at i = 2. The terms u ^(a)₂ are eliminated by means of Green’s theorem, which together with the boundary conditions (9) and the periodicity relation (10) implies

$$ \left\langle {G}^{(a)}\left({\nabla}_{xy}^2{u}_i^{(a)}+{\nabla}_{yy}^2{u}_{i+1}^{(a)}\right)\right\rangle =0. $$

As a result, the homogenized wave equation of the order ε ⁰ is obtained:

$$ \left\langle {G}^{(a)}\left({\nabla}_{xx}^2{u}_0+{\nabla}_{xy}^2{u}_1^{(a)}\right)\right\rangle =\left\langle {\rho}^{(a)}\right\rangle \frac{\partial^2{u}_0}{\partial {t}^2}. $$

(14)

Substituting here expressions for u ^(a)₁ evaluated below we shall come to a macroscopic wave equation

$$ {G}_0{\nabla}_{xx}^2{u}_0={\rho}_0\frac{\partial^2{u}_0}{\partial {t}^2}, $$

(15)

where ρ ₀ = (1 − c ⁽²⁾)ρ ⁽¹⁾ + c ⁽²⁾ ρ ⁽²⁾ is the effective mass density, c ⁽²⁾ is the volume fraction of the inclusions, c ⁽²⁾ = A/S ₀, A is the size of the inclusion (Fig. 2). The effective modulus G ₀ can be derived after evaluation of the integrals in Eq. (14).

Below we find approximate solutions of the cell problem (8), (9), (13) and determine the effective shear modulus G ₀ using a boundary shape perturbation and lubrication theory approaches.

3.2 The Case c ⁽²⁾ < < 1: Boundary Shape Perturbation

If the volume fraction c ⁽²⁾ of the marrow inclusions is relatively small, the square shapes of the domains Ω⁽¹⁾, Ω⁽²⁾ can be approximately substituted by the equal circles of radii R ₁, R ₂ so that c ⁽²⁾ = R ²₂ /R ²₁ (Fig. 3). This simplification can be considered as the first approximation of the method of the boundary shape perturbation [1, 14].

Fig. 3

Simplification of the unit cell in the case c ⁽²⁾ < < 1

Let us introduce in the unit cell the polar co-ordinates r ² = y ²₁  + y ²₂ , tan θ = y ₂/y ₁. Equations (8), (9), and (13) at i = 1 read

$$ \frac{\partial^2{u}_1^{(a)}}{\partial {r}^2}+\frac{1}{r}\frac{\partial {u}_1^{(a)}}{\partial r}+\frac{1}{r^2}\frac{\partial^2{u}_1^{(a)}}{\partial {\theta}^2}=0, $$

(16)

$$ \begin{array}{l}{\left.\left\{{u}_1^{(1)}={u}_1^{(2)}\right\}\right|}_{r={R}_2},\\ [10pt] {}{\left.\left\{{G}^{(1)}\left(\frac{\partial {u}_0}{\partial \mathbf{n}}+\frac{\partial {u}_1^{(1)}}{\partial r}\right)={G}^{(2)}\left(\frac{\partial {u}_0}{\partial \mathbf{n}}+\frac{\partial {u}_1^{(2)}}{\partial r}\right)\right\}\right|}_{r={R}_2},\end{array} $$

(17)

$$ {\left.\left\{{u}_1^{(2)}=0\right\}\right|}_{r=0},\kern1em {\left.\left\{{u}_1^{(1)}=0\right\}\right|}_{r={R}_1}, $$

(18)

where ∂/∂n = cos θ∂/∂x ₁ + sin θ∂/∂x ₂.

Solution of the simplified cell problem (16)–(18) is

$$ {u}_1^{(a)}=\left({C}_1^{(a)} r+{C}_2^{(a)}{r}^{-1}\right)\frac{\partial {u}_0}{\partial \mathbf{n}}, $$

(19)

$$ {C}_1^{(1)}=\frac{\left({\lambda}^{(2)}-1\right){c}^{(2)}}{\lambda^{(2)}+1-{c}^{(2)}\left({\lambda}^{(2)}-1\right)},\kern1em {C}_2^{(1)}=-\frac{\left({\lambda}^{(2)}-1\right){R}_2^2}{\lambda^{(2)}+1-{c}^{(2)}\left({\lambda}^{(2)}-1\right)}, $$

$$ {C}_1^{(2)}=-\frac{\left({\lambda}^{(2)}-1\right)\left(1-{c}^{(2)}\right)}{\lambda^{(2)}+1-{c}^{(2)}\left({\lambda}^{(2)}-1\right)},\kern1em {C}_2^{(2)}=0, $$

where λ ⁽²⁾ = G ⁽²⁾/G ⁽¹⁾.

Substituting expressions (19) into the homogenized equation (14), we obtain the effective modulus G ₀ in the closed analytical form:

$$ {\lambda}_0=\frac{\lambda^{(2)}+1+{c}^{(2)}\left({\lambda}^{(2)}-1\right)}{\lambda^{(2)}+1-{c}^{(2)}\left({\lambda}^{(2)}-1\right)}, $$

(20)

where λ ₀ = G ₀/G ⁽¹⁾.

It should be noted that solution (20) is precisely the same as can be obtained by the composite cylinder assemblage model and by the generalized self-consistent scheme [8].

3.3 The Case c ⁽²⁾ → 1: Lubrication Theory

In the case of densely packed marrow inclusions, when the volume fraction c ⁽²⁾ is close to unit, c ⁽²⁾ → 1, an asymptotic solution of the cell problem can be obtained using as a natural small parameter the non-dimensional width δ = H/L of the trabecula (Fig. 4). Let us suppose δ < < 1. Being restricted by the O(δ ⁰) approximation, for the matrix strips dΩ ₁, dΩ ₂, which separate neighbouring inclusions, one can show

Fig. 4

Unit cell in the case c ⁽²⁾ → 1

$$ \frac{\partial^2{u}_1^{(1)}}{\partial {y}_1^2}>>\frac{\partial^2{u}_1^{(1)}}{\partial {y}_2^2}\kern1.25em \mathrm{for}\kern0.24em \mathbf{y}\in d{\Omega}_1,\kern1.12em \frac{\partial^2{u}_1^{(1)}}{\partial {y}_1^2}<<\frac{\partial^2{u}_1^{(1)}}{\partial {y}_2^2}\kern1em \mathrm{for}\;\mathbf{y}\in d{\Omega}_2. $$

(21)

The physical meaning of estimations (21) is that in the narrow strip dΩ₁ the variation of local stresses in the direction y ₁ is dominant and, hence, the term ∂² u ⁽¹⁾₁ /∂y ²₂ can be neglected in comparison with ∂² u ⁽¹⁾₁ /∂y ²₁ . Vice versa, in the strip dΩ₂, the dominant variation of the local stress field takes place in the direction y ₂, so the term ∂² u ⁽¹⁾₁ /∂y ²₁ can be neglected in comparison with ∂² u ⁽¹⁾₁ /∂y ²₂ . Such a simplification is similar to the basic idea of the well-known lubrication theory, which was used in the theory of composites for many years [8, 9].

Following estimations (21), in the O(δ ⁰) approximation, Eq. (8) reads

$$ {G}^{(1)}\left(\frac{\partial^2{u}_{i-2}^{(1)}}{\partial {x}_s^2}+2\frac{\partial^2{u}_{i-1}^{(1)}}{\partial {x}_s\partial {y}_s}+\frac{\partial^2{u}_i^{(1)}}{\partial {y}_s^2}\right)={\rho}^{(1)}\frac{\partial^2{u}_{i-2}^{(1)}}{\partial {t}^2}, $$

(22)

$$ {G}^{(2)}\left({\nabla}_{xx}^2{u}_{i-2}^{(2)}+2{\nabla}_{xy}^2{u}_{i-1}^{(2)}+{\nabla}_{yy}^2{u}_i^{(2)}\right)={\rho}^{(2)}\frac{\partial^2{u}_{i-2}^{(2)}}{\partial {t}^2} $$

Solution of the simplified cell problems (9), (13), (22) at i = 1 is

$$ {u}_1^{(2)}=-\frac{\left(1-\sqrt{c^{(2)}}\right)\left({\lambda}^{(2)}-1\right)}{\lambda^{(2)}-\sqrt{c^{(2)}}\left({\lambda}^{(2)}-1\right)}{y}_s\frac{\partial {u}_0}{\partial {x}_s}, $$

(23)

$$ {u}_1^{(1)}=-\frac{\sqrt{c^{(2)}}\left({\lambda}^{(2)}-1\right)}{\lambda^{(2)}-\sqrt{c^{(2)}}\left({\lambda}^{(2)}-1\right)}\left(\frac{L}{2}-{y}_s\right)\frac{\partial {u}_0}{\partial {x}_s}\kern1.25em \mathrm{at}\kern1em {y}_s>0, $$

$$ {u}_1^{(1)}=\frac{\sqrt{c^{(2)}}\left({\lambda}^{(2)}-1\right)}{\lambda^{(2)}-\sqrt{c^{(2)}}\left({\lambda}^{(2)}-1\right)}\left(\frac{L}{2}+{y}_s\right)\frac{\partial {u}_0}{\partial {x}_s}\kern1em \mathrm{at}\kern1em {y}_s<0, $$

where s = 1, 2.

For the effective shear modulus we obtain

$$ {\lambda}_0=\frac{\lambda^{(2)}-\sqrt{c^{(2)}}\left(1-\sqrt{c^{(2)}}\right)\left({\lambda}^{(2)}-1\right)}{\lambda^{(2)}-\sqrt{c^{(2)}}\left({\lambda}^{(2)}-1\right)}. $$

(24)

Numerical results, calculated by formulas (20) and (24), are very close (except the case λ ⁽²⁾ < 1, c ⁽²⁾ → 0). This is illustrated at Fig. 5 for real values of λ ⁽²⁾, which correspond to elastic materials. Moreover, in the limit c ⁽²⁾ → 1, the approximate solutions (20) and (24) exhibit the same asymptotic behaviour and give identical expansions for λ ₀ until the order O[(1 − c)²]:

Fig. 5

Effective modulus in the elastic case. Solids—formula (20), dashes—formula (24). (a) λ ⁽²⁾ = 5. (b) λ ⁽²⁾ = 0.2

$$ {\lambda}_0={\lambda}^{(2)}-\frac{1}{2}\left({\lambda}^{(2)}-1\right)\left({\lambda}^{(2)}+1\right)\left(1-{c}^{(2)}\right)+ O\left[{\left(1-{c}^{(2)}\right)}^2\right]\kern1em \mathrm{at}\kern1em {c}^{(2)}\to 1. $$

This fact reveals that for the cancellous structure under consideration, expression (20), originally obtained for the case c ⁽²⁾ < < 1, provides a reasonable approximation in the whole region of the inclusions volume fraction 0 ≤ c ⁽²⁾ ≤ 1.

3.4 Propagation of Long Waves

Let us consider a harmonic wave

$$ {u}_0= U \exp \left(-\mathrm{i}\boldsymbol{\upmu } \cdot \mathbf{x}\right) \exp \left( i\omega t\right), $$

(25)

where U is the amplitude, ω is the frequency, and μ = μ ₁ e ₁ + μ ₂ e ₂ is the wave vector and the direction of propagation is determined by the angle α, tan α = μ ₂/μ ₁.

Separating real μ _R and imaginary μ _I parts of the wave vector μ = μ _R  − iμ _I , expression (25) reads

$$ {u}_0= U \exp \left(-{\boldsymbol{\upmu}}_I\cdot \mathbf{x}\right) \exp \left(-\mathrm{i}{\boldsymbol{\upmu}}_R\cdot \mathbf{x}\right) \exp \left(\mathrm{i}\omega t\right). $$

(26)

Here μ _I  = |μ _I | is the attenuation factor and μ _R  = |μ _R | = 2π/L is the wave number.

For the viscoelastic composite medium, the effective complex modulus G ₀, the attenuation coefficient μ _I and the phase velocity v _p  = ω/μ _R depend on the frequency of the travelling signal. Substituting expression (26) into the macroscopic wave equation (15), we obtain

$$ \left({G}_{0, R}+\mathrm{i}{G}_{0, I}\right){\left({\mu}_I+\mathrm{i}{\mu}_R\right)}^2=-{\rho}_0{\omega}^2, $$

(27)

where G _0,R , G _0,I are, respectively, the real and the imaginary part of G ₀, G ₀ = G _0,R  + iG _0,I . Collecting in Eq. (27) the terms at 1 and i, after routine transformations we derive

$$ {\mu}_I={\mu}_R \tan \left({\varphi}_0/2\right),\kern1em {v}_p^2=\frac{v_0^2}{ \cos {\left({\varphi}_0/2\right)}^2}, $$

(28)

where tan(φ ₀) = G _0,I /G _0,R is the effective loss tangent and ${v}_0=\sqrt{\left|{G}_0\right|/{\rho}_0}$ is the effective velocity in the elastic case.

Adopting for G ₀ the solution (20), we obtain

$$ \begin{array}{l}{G}_0={G}^{(1)}\frac{G^{(1)}\left(1-{c}^{(2)}\right)+\mathrm{i}\omega {\eta}^{(2)}\left(1+{c}^{(2)}\right)}{G^{(1)}\left(1+{c}^{(2)}\right)+\mathrm{i}\omega {\eta}^{(2)}\left(1-{c}^{(2)}\right)},\\[10pt] {} \tan \left({\varphi}_0\right)=\frac{4{G}^{(1)}{c}^{(2)}\omega {\eta}^{(2)}}{\left[1-{\left({c}^{(2)}\right)}^2\right]\left[{\left({G}^{(1)}\right)}^2+{\left(\omega {\eta}^{(2)}\right)}^2\right]}.\end{array} $$

(29)

In the numerical examples presented below we accept some rough estimations of the properties of the components following Bryant et al. [6], Guo [13], and Van Rietbergen and Huiskes [26]. The shear modulus of the trabeculae is G ⁽¹⁾ = 3.85 ⋅ 10⁹ Pa and the viscosity of the marrow is η ⁽²⁾ = 0.15 Pa ⋅ s (at the room temperature of 20 °C) and η ⁽²⁾ = 0.05 Pa ⋅ s (at the body temperature of 37 °C). The trabeculae volume fraction c ⁽¹⁾ = 1 − c ⁽²⁾ can vary from 0.05–0.1 for aged osteoporotic bones to 0.3–0.35 for young normal bones.

Dependencies of the attenuation factor μ _I upon the frequency ω are displayed at Fig. 6 (normal bone, c ⁽¹⁾ = 0.3) and Fig. 7 (osteoporotic bone, c ⁽¹⁾ = 0.1). The dispersion effect vanishes (i) at ω → 0, when the deformation rate is small and the stiffness of the marrow is negligible, and (ii) at ω → ∞, when the deformation rate is high, so the marrow acts like a perfectly stiff medium. Decrease of the trabeculae volume fraction c ⁽¹⁾ leads to the intensifying of the dispersion: the damping frequency region extends and the attenuation factor μ _I grows. Decrease in temperature (i.e. increase in the marrow viscosity η ⁽²⁾) leads to a reduction of the damping frequency. In any case, for physically meaningful values of the bone properties, the effect of viscoelastic damping can be observed starting from the frequencies of the order 100 MHz and higher.

Fig. 6

Attenuation factor of a normal bone. Solids—η ⁽²⁾ = 0.05 Pa ⋅ s, dashes—η ⁽²⁾ = 0.15 Pa ⋅ s

Fig. 7

Attenuation factor of an osteoporotic bone. Solids—η ⁽²⁾ = 0.05 Pa ⋅ s, dashes—η ⁽²⁾ = 0.15 Pa ⋅ s

It should be noted that in the long-wave limit (l < < L) the cancellous structure under consideration is transversely orthotropic. The obtained solution for anti-plane shear waves is isotropic in the plane x ₁ x ₂, so the parameters G ₀, φ ₀ do not depend on the direction of the wave propagation. The effect of anisotropy is predicted in the case of short waves (see Sect. 4).

4 Floquet-Bloch Approach: Plane-Wave Expansion Method

When the wavelength L is comparable to the internal size l of the cancellous structure, the effect of dispersion is caused by successive reflections and refractions of local waves at the trabecula-marrow interfaces. Decrease in the wavelength reveals a sequence of pass and stop frequency bands (so-called phononic bands) [2, 20]. Thus, a heterogeneous bone can act as a discrete wave filter. If the frequency of the signal falls within a stop band, a stationary wave is excited and the neighbouring trabeculae vibrate in alternate directions. On macrolevel the amplitude of the global wave attenuates exponentially, so no propagation is possible.

In order to explore such a case, let us assume the threshold of the first stop band to be essentially lower than the viscoelastic damping frequencies. The marrow is not involved into the shear deformation, so we can set G ⁽²⁾ = 0, ρ ⁽²⁾ = 0.

Following the Floquet-Bloch theorem [5], a harmonic wave propagating through a periodic cancellous structure is represented in the form

$$ w= F\left(\mathbf{x}\right) \exp \left(\mathrm{i}\boldsymbol{\upmu } \cdot \mathbf{x}\right) \exp \left( i\omega t\right), $$

(30)

where F(x) is a spatially periodic function, F(x) = F(x + l _p ).

We use the plane-wave expansion method [20, 23] and express the function F(x) and the material properties G(x), ρ(x) as infinite Fourier series:

$$ \begin{array}{l} F\left(\mathbf{x}\right)={\displaystyle \sum_{k_1=-\infty}^{\infty }{\displaystyle \sum_{k_2=-\infty}^{\infty }{A}_{k_1{k}_2} \exp \left[\mathrm{i}\frac{2\pi}{l}\left({k}_1{x}_1+{k}_2{x}_2\right)\right]}},\\[6pt] {} G\left(\mathbf{x}\right)={\displaystyle \sum_{k_1=-\infty}^{\infty }{\displaystyle \sum_{k_2=-\infty}^{\infty }{B}_{k_1{k}_2} \exp \left[\mathrm{i}\frac{2\pi}{l}\left({k}_1{x}_1+{k}_2{x}_2\right)\right]}},\\[6pt] {}\rho \left(\mathbf{x}\right)={\displaystyle \sum_{k_1=-\infty}^{\infty }{\displaystyle \sum_{k_2=-\infty}^{\infty }{C}_{k_1{k}_2} \exp \left[\mathrm{i}\frac{2\pi}{l}\left({k}_1{x}_1+{k}_2{x}_2\right)\right]}},\end{array} $$

(31)

where

$$ \begin{array}{l}{B}_{k_1{k}_2}=\frac{1}{l^2}{\displaystyle \underset{\varOmega_0}{\iint } G\left(\mathbf{x}\right) \exp \left[-\mathrm{i}\frac{2\pi}{l}\left({k}_1{x}_1+{k}_2{x}_2\right)\right]\mathrm{d}{x}_1\mathrm{d}{x}_2},\\ {}{C}_{k_1{k}_2}=\frac{1}{l^2}{\displaystyle \underset{\varOmega_0}{\iint}\rho \left(\mathbf{x}\right) \exp \left[-\mathrm{i}\frac{2\pi}{l}\left({k}_1{x}_1+{k}_2{x}_2\right)\right]\mathrm{d}{x}_1\mathrm{d}{x}_2},\end{array} $$

the operator ${\displaystyle {\iint}_{\varOmega_0}\left(\cdot \right)}\mathrm{d}{x}_1\mathrm{d}{x}_2$ denotes integration over a distinguished unit cell Ω₀.

Substituting Ansatz (30) and expansions (31) into the wave equation (1) and collecting the terms exp[i2πl ^− 1(j ₁ x ₁ + j ₂ x ₂)], j ₁, j ₂ = 0, ± 1, ± 2, …, we come to an infinite system of linear algebraic equations for the unknown coefficients ${A}_{k_1{k}_2}$:

$$ \begin{array}{l}{\displaystyle \sum_{k_1=-\infty}^{\infty }{\displaystyle \sum_{k_2=-\infty}^{\infty }{A}_{k_1{k}_2}}}\left\{{B}_{\begin{array}{l}{j}_1-{k}_1,\\ {}{j}_2-{k}_2\end{array}}\right.\left[\left(\frac{2\pi}{l}{k}_1+{\mu}_1\right)\left(\frac{2\pi}{l}{j}_1+{\mu}_1\right)+\right.\\ {}\kern1em +\left.\left(\frac{2\pi}{l}{k}_2+{\mu}_2\right)\left(\frac{2\pi}{l}{j}_2+{\mu}_2\right)\right]\left.-{C}_{\begin{array}{l}{j}_1-{k}_1,\\ {}{j}_2-{k}_2\end{array}}{\omega}^2\right\}=0.\end{array} $$

(32)

System (32) has a nontrivial solution if and only if the determinant of the matrix of the coefficients is zero. Equating the determinant to zero, we derive a dispersion relation for ω and μ. It should be noted that the plane-wave expansion method does not use explicitly the bonding conditions (4), whereas they are “embedded” implicitly into Eq. (1) and expansions (31).

To illustrate the appearance of phononic band gaps, let us rewrite Ansatz (30) separating real μ _R and imaginary μ _I parts of the wave vector μ = μ _R  − iμ _I :

$$ w= F\left(\mathbf{x}\right) \exp \left(-{\boldsymbol{\upmu}}_I\cdot \mathbf{x}\right) \exp \left(\mathrm{i}{\boldsymbol{\upmu}}_R\cdot \mathbf{x}\right) \exp \left( i\omega t\right). $$

(33)

The imaginary part μ _I  ≡ |μ _I | represents the attenuation factor. Frequency regions where μ _I  ≠ 0 correspond to stop bands (signal (30) attenuates exponentially), while regions where μ _I  = 0 correspond to pass bands.

In numerical examples the dispersion relations are calculated approximately by the truncation of the infinite system (32) supposing − j _max ≤ j _s  ≤ j _max. The number of the kept equations is (2j _max + 1)². We expect that increase in j _max shall improve the accuracy of the solution. From the physical point of view, such a truncation means cutting off the higher frequencies.

Figure 8 displays dispersion curves for a normal bone with the following properties of the trabecular tissue: G ⁽¹⁾ = 3.85 ⋅ 10⁹ Pa, ρ ⁽¹⁾ = 1900 kg/m³, c ⁽¹⁾ = 0.3. Calculations are performed at j _max = 3. The diagram consists of two (left and right) parts separated by a vertical dash line. The right part displays a solution for the orthogonal direction (α = 0) and the left part for the diagonal direction (α = π/4) of the wave propagation. The results for the frequency ω are normalized to ω ₀ = v ₀ μ _R  = 2πv ₀/L. We can observe that in the long-wave case (ω → 0, l/L → 0) the solution is isotropic. However, with the increase in ω and decrease in L, the cancellous structure exhibits an anisotropic behaviour.

Fig. 8

Dispersion curves of a normal bone

Shaded areas at Fig. 8 indicate the threshold of the first stop bands. Let us estimate the corresponding values ω _s of the frequency. We obtain ω _s l/(ω ₀ L) ≈ 0.29 at α = 0 and ω _s l/(ω ₀ L) ≈ 0.41 at α = π/4. The typical length of trabeculae is about l ≈ 10^− 3m. Taking into account ω ₀ = 2πv ₀/L, ${v}_0=\sqrt{G_0/{\rho}_0}$, we derive ω _s  ≈ 2.0 MHz at α = 0 and ω _s  ≈ 2.8 MHz at α = π/4.

5 Conclusions

For anti-plane shear waves the effect of Bloch dispersion, caused by the heterogeneity of cancellous bones, appears at essentially lower frequencies than the effect of viscoelastic damping, caused by the viscosity of the marrow. Bloch dispersion is expected to play the primary role in the processes of ultrasonic diagnostic, which usually deals with acoustic waves in the regions 1–10 MHz. The viscoelastic damping can be neglected until the frequency of about 100 MHz. Obtained results may be used for the development of new methods of non-invasive testing and diagnostic.

In the present paper, a perfectly regular arrangement of marrow inclusions is investigated. It is clear that the microstructure of real bones is not regular. At the same time, it has been shown in a number of studies [18, 22, 24] that regular structures exhibit the narrowest stop bands in comparing to disordered systems. Thus, the obtained solutions may be treated as theoretical bounds for the stop band thresholds that appear in randomly disordered bone tissues.