Acoustomechanical constitutive theory for soft materials

Fengxian Xin; Tian Jian Lu

doi:10.1007/s10409-016-0585-z

Acta Mechanica Sinica

October 2016, Volume 32, Issue 5, pp 828–840 | Cite as

Acoustomechanical constitutive theory for soft materials

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Fengxian XinEmail author
Tian Jian LuEmail author

Research Paper

First Online: 18 July 2016

Received: 11 February 2016

Revised: 23 April 2016

Accepted: 28 April 2016

Abstract

Acoustic wave propagation from surrounding medium into a soft material can generate acoustic radiation stress due to acoustic momentum transfer inside the medium and material, as well as at the interface between the two. To analyze acoustic-induced deformation of soft materials, we establish an acoustomechanical constitutive theory by combining the acoustic radiation stress theory and the nonlinear elasticity theory for soft materials. The acoustic radiation stress tensor is formulated by time averaging the momentum equation of particle motion, which is then introduced into the nonlinear elasticity constitutive relation to construct the acoustomechanical constitutive theory for soft materials. Considering a specified case of soft material sheet subjected to two counter-propagating acoustic waves, we demonstrate the nonlinear large deformation of the soft material and analyze the interaction between acoustic waves and material deformation under the conditions of total reflection, acoustic transparency, and acoustic mismatch.

Keywords

Acoustomechanical constitutive theory Acoustic radiation stress Soft material

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Notes

Acknowledgments

The project was supported by the National Natural Science Foundation of China (Grants 51528501, 11532009) and the Fundamental Research Funds for Central Universities (Grant 2014qngz12). F.X. Xin was supported by China Scholarship Council as a visiting scholar to Harvard University. This author appreciates the helpful discussions with Prof. Z.G. Suo on soft material theory.

Appendix: acoustic wave propagation

With reference to Fig. 1, consider a time-harmonic acoustic wave $p( {\varvec{x},t} )=p_{0} \mathrm{e}^{-\mathrm{j}( {\varvec{k}\cdot \varvec{x}-\omega t} )}$ with wavenumber vector $\varvec{k}$ incident on the surface of a soft material sheet from its outside surrounding medium. Generally, the incident acoustic wave will generate reflection and transmission not only on the inlet surface, but also on the outlet surface and, in Eulerian coordinates, wave propagation in the medium is governed by dynamical equation $\nabla \cdot {\varvec{\sigma }} =\rho {\partial ^{2}\varvec{u}}/{\partial t^{2}}$. For ideal fluid-like materials, this equation degrades to the Helmholtz equation $\rho {\partial ^{2}\varvec{u}}/{\partial t^{2}}+\nabla p=\mathbf{0}$. The velocity potential field induced by the incident acoustic wave can be written as

$$\begin{aligned} \phi _1 \left( {\varvec{x},t} \right)= & {} I\mathrm{e}^{-\mathrm{j}\left( {\varvec{k}_1^+ \cdot \varvec{x}-\omega t} \right) }+\beta \mathrm{e}^{-\mathrm{j}\left( {\varvec{k}_1^- \cdot \varvec{x}-\omega t} \right) }, \end{aligned}$$

(A1)

$$\begin{aligned} \phi _2 \left( {\varvec{x},t}\right)= & {} \varepsilon \mathrm{e}^{-\mathrm{j}\left( {\varvec{k}_2^+ \cdot \varvec{x}-\omega t} \right) }+\zeta \mathrm{e}^{-\mathrm{j}\left( {\varvec{k}_2^- \cdot \varvec{x}-\omega t} \right) }, \end{aligned}$$

(A2)

$$\begin{aligned} \phi _3 \left( {\varvec{x},t} \right)= & {} \xi \mathrm{e}^{-\mathrm{j}\left( {\varvec{k}_3^+ \cdot \varvec{x}-\omega t} \right) }, \end{aligned}$$

(A3)

where the subscripts “1, 2, and 3” denote the left medium, the soft material sheet, and the right medium, while the superscripts “+” and “-” correspond to positive- and negative-going waves, respectively. The corresponding velocity field and pressure field can be expressed as

$$\begin{aligned} \varvec{u}_i =-\nabla \phi _i , \quad p_i =\rho _i \frac{\partial \phi _i }{\partial t}, \,i=1,2,3. \end{aligned}$$

(A4)

Continuity of velocity and acoustic pressure requires

$$\begin{aligned}&u_{1z} =\left. {u_{2z} } \right| _{z=0} , \quad u_{2z} =\left. {u_{3z} } \right| _{z=l_3 }, \end{aligned}$$

(A5)

$$\begin{aligned}&p_1 =\left. {p_2 }\right| _{z=0} , \quad p_2 =\left. {p_3 } \right| _{z=l_3 }, \end{aligned}$$

(A6)

which can be rewritten in the form

$$\begin{aligned}&\mathrm{j}k_{1z} I\mathrm{e}^{-\mathrm{j}\left( {k_{1x} x+k_{1y} y-\omega t} \right) }-\mathrm{j}k_{1z} \beta \mathrm{e}^{-\mathrm{j}\left( {k_{1x} x+k_{1y} y-\omega t} \right) }\nonumber \\&\quad =\mathrm{j}k_{2z} \varepsilon \mathrm{e}^{-\mathrm{j}\left( {k_{2x} x+k_{2y} y-\omega t} \right) }-\mathrm{j}k_{2z} \zeta \mathrm{e}^{-\mathrm{j}\left( {k_{2x} x+k_{2y} y-\omega t} \right) }, \end{aligned}$$

(A7)

$$\begin{aligned}&\mathrm{j}k_{2z} \varepsilon \mathrm{e}^{-\mathrm{j}\left( {k_{2x} x+k_{2y} y+k_{2z} l_3 -\omega t} \right) }-\mathrm{j}k_{2z} \zeta \mathrm{e}^{-\mathrm{j}\left( {k_{2x} x+k_{2y} y-k_{2z} l_3 -\omega t} \right) }\nonumber \\&\quad =\mathrm{j}k_{3z} \xi \mathrm{e}^{-\mathrm{j}\left( {k_{3x} x+k_{3y} y+k_{3z} l_3 -\omega t} \right) }, \end{aligned}$$

(A8)

$$\begin{aligned}&\mathrm{j}\omega \rho _1 \left[ {I\mathrm{e}^{-\mathrm{j}\left( {k_{1x} x+k_{1y} y-\omega t} \right) }+\beta \mathrm{e}^{-\mathrm{j}\left( {k_{1x} x+k_{1y} y-\omega t} \right) }} \right] =\mathrm{j}\omega \rho _2 \left[ {\varepsilon \mathrm{e}^{-\mathrm{j}\left( {k_{2x} x+k_{2y} y-\omega t} \right) }+\zeta \mathrm{e}^{-\mathrm{j}\left( {k_{2x} x+k_{2y} y-\omega t} \right) }} \right] ,\qquad \qquad \qquad \qquad \qquad \quad \,\,\,\, \end{aligned}$$

(A9)

$$\begin{aligned}&\mathrm{j}\omega \rho _2 \left[ {\varepsilon \mathrm{e}^{-\mathrm{j}\left( {k_{2x} x+k_{2y} y+k_{2z} l_3 -\omega t} \right) }+\zeta \mathrm{e}^{-\mathrm{j}\left( {k_{2x} x+k_{2y} y-k_{2z} l_3 -\omega t} \right) }} \right] =\mathrm{j}\omega \rho _3 \left[ {\xi \mathrm{e}^{-\mathrm{j}\left( {k_{3x} x+k_{3y} y+k_{3z} l_3 -\omega t} \right) }} \right] . \end{aligned}$$

(A10)

If the same medium occupies the left side and the right side, i.e., $\rho _{1} =\rho _{3}$ and $k_{1} =k_{3}$, one has

$$\begin{aligned} \beta= & {} \frac{I\mathrm{e}^{\mathrm{j}k_{2z} l_3 }\left( {\rho _2^2 k_{1z}^2 -\rho _1^2 k_{2z}^2 } \right) +I\mathrm{e}^{-\mathrm{j}k_{2z} l_3 }\left( {\rho _1^2 k_{2z}^2 -\rho _2^2 k_{1z}^2 } \right) }{\mathrm{e}^{\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} +\rho _2 k_{1z} } \right) ^{2}-\mathrm{e}^{-\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} -\rho _2 k_{1z} } \right) ^{2}}, \end{aligned}$$

(A11)

$$\begin{aligned} \varepsilon= & {} \frac{2I\rho _1 k_{1z} \mathrm{e}^{\mathrm{j}k_{2z} l_3 }\left( {\rho _2 k_{1z} +\rho _1 k_{2z} } \right) }{\mathrm{e}^{\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} +\rho _2 k_{1z} } \right) ^{2}-\mathrm{e}^{-\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} -\rho _2 k_{1z} } \right) ^{2}}, \end{aligned}$$

(A12)

$$\begin{aligned} \zeta= & {} \frac{2I\rho _1 k_{1z} \mathrm{e}^{-\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} -\rho _2 k_{1z} } \right) }{\mathrm{e}^{\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} +\rho _2 k_{1z} } \right) ^{2}-\mathrm{e}^{-\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} -\rho _2 k_{1z} } \right) ^{2}}, \end{aligned}$$

(A13)

$$\begin{aligned} \xi= & {} \frac{4I\rho _1 \rho _2 k_{1z} k_{2z} \mathrm{e}^{\mathrm{j}k_{1z} l_3 }}{\mathrm{e}^{\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} +\rho _2 k_{1z} } \right) ^{2}-\mathrm{e}^{-\mathrm{j}k_{2z} l_3 }\left( {\rho _1 k_{2z} -\rho _2 k_{1z} } \right) ^{2}}. \end{aligned}$$

(A14)

When two counter-propagating acoustic waves are normally incident on a soft material sheet (Fig. 2), the acoustic pressure and velocity can be expressed as

$$\begin{aligned}&p_1^\mathrm{{net}}=\mathrm{j}\omega \rho _1 I\mathrm{e}^{\mathrm{j}\omega t}\left[ \mathrm{e}^{-\mathrm{j}k_{1z} z}+\mathrm{e}^{\mathrm{j}k_{1z} z}\frac{\mathrm{j}\sin \left( {k_{2z} l_3 } \right) \left( {\rho _2^2 k_{1z}^2 -\rho _1^2 k_{2z}^2 } \right) +2\rho _1 \rho _2 k_{1z} k_{2z} }{2\rho _1 \rho _2 k_{1z} k_{2z} \cos \left( {k_{2z} l_3 } \right) +\mathrm{j}\left( {\rho _1^2 k_{2z}^2 +\rho _2^2 k_{1z}^2 } \right) \sin \left( {k_{2z} l_3 } \right) } \right] ,\end{aligned}$$

(A15)

$$\begin{aligned}&u_{1z}^\mathrm{{net}}=\mathrm{j}k_{1z} I\mathrm{e}^{\mathrm{j}\omega t}\left[ \mathrm{e}^{-\mathrm{j}k_{1z} z}-\mathrm{e}^{\mathrm{j}k_{1z} z}\frac{\mathrm{j}\sin \left( {k_{2z} l_3 } \right) \left( {\rho _2^2 k_{1z}^2 -\rho _1^2 k_{2z}^2 } \right) +2\rho _1 \rho _2 k_{1z} k_{2z} }{2\rho _1 \rho _2 k_{1z} k_{2z} \cos \left( {k_{2z} l_3 } \right) +\mathrm{j}\left( {\rho _1^2 k_{2z}^2 +\rho _2^2 k_{1z}^2 } \right) \sin \left( {k_{2z} l_3 } \right) } \right] , \end{aligned}$$

(A16)

$$\begin{aligned}&p_2^\mathrm{{net}}=\frac{2\omega I\rho _1 \rho _2 k_{1z} \mathrm{e}^{\mathrm{j}\omega t}\left[ {\rho _2 k_{1z} \sin \left( {k_{2z} \left( {z-l_3 } \right) } \right) -\rho _2 k_{1z} \sin \left( {k_{2z} z} \right) +\mathrm{j}\rho _1 k_{2z} \cos \left( {k_{2z} \left( {z-l_3 } \right) } \right) +\mathrm{j}\rho _1 k_{2z} \cos \left( {k_{2z} z} \right) } \right] }{2\rho _1 \rho _2 k_{1z} k_{2z} \cos \left( {k_{2z} l_3 } \right) +\mathrm{j}\left( {\rho _1^2 k_{2z}^2 +\rho _2^2 k_{1z}^2 } \right) \sin \left( {k_{2z} l_3 } \right) },\end{aligned}$$

(A17)

$$\begin{aligned}&u_{2z}^\mathrm{{net}} =\frac{2I\rho _1 k_{1z} k_{2z} \mathrm{e}^{\mathrm{j}\omega t}\left[ {\rho _1 k_{2z} \sin \left( {k_{2z} \left( {z-l_3 } \right) } \right) +\rho _1 k_{2z} \sin \left( {k_{2z} z} \right) +\mathrm{j}\rho _2 k_{1z} \cos \left( {k_{2z} \left( {z-l_3 } \right) } \right) -\mathrm{j}\rho _2 k_{1z} \cos \left( {k_{2z} z} \right) } \right] }{2\rho _1 \rho _2 k_{1z} k_{2z} \cos \left( {k_{2z} l_3 } \right) +\mathrm{j}\left( {\rho _1^2 k_{2z}^2 +\rho _2^2 k_{1z}^2 } \right) \sin \left( {k_{2z} l_3 } \right) },\end{aligned}$$

(A18)

$$\begin{aligned}&p_3^\mathrm{{net}}=\mathrm{j}\omega \rho _1 I\mathrm{e}^{\mathrm{j}\omega t}\left[ \mathrm{e}^{\mathrm{j}k_{1z} \left( {z-l_3 } \right) }+\mathrm{e}^{-\mathrm{j}k_{1z} \left( {z-l_3 } \right) }\frac{\mathrm{j}\sin \left( {k_{2z} l_3 } \right) \left( {\rho _2^2 k_{1z}^2 -\rho _1^2 k_{2z}^2 } \right) +2\rho _1 \rho _2 k_{1z} k_{2z} }{2\rho _1 \rho _2 k_{1z} k_{2z} \cos \left( {k_{2z} l_3 } \right) +\mathrm{j}\left( {\rho _1^2 k_{2z}^2 +\rho _2^2 k_{1z}^2 } \right) \sin \left( {k_{2z} l_3 } \right) } \right] ,\end{aligned}$$

(A19)

$$\begin{aligned}&u_{3z}^\mathrm{{net}} =\mathrm{j}k_{1z} I\mathrm{e}^{\mathrm{j}\omega t}\left[ -\mathrm{e}^{\mathrm{j}k_{1z} \left( {z-l_3 } \right) } +\mathrm{e}^{-\mathrm{j}k_{1z} \left( {z-l_3 } \right) }\frac{\mathrm{j}\sin \left( {k_{2z} l_3 } \right) \left( {\rho _2^2 k_{1z}^2 -\rho _1^2 k_{2z}^2 } \right) +2I\rho _1 \rho _2 k_{1z} k_{2z} }{2\rho _1 \rho _2 k_{1z} k_{2z} \cos \left( {k_{2z} l_3 } \right) +\mathrm{j}\left( {\rho _1^2 k_{2z}^2 +\rho _2^2 k_{1z}^2 } \right) \sin \left( {k_{2z} l_3 } \right) } \right] . \end{aligned}$$

(A20)

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Authors and Affiliations

Fengxian Xin
- 1
- 2
Email author
Tian Jian Lu
- 1
- 2
Email author

1.State Key Laboratory for Strength and Vibration of Mechanical StructuresXi’an Jiaotong UniversityXi’anChina
2.MOE Key Laboratory for Multifunctional Materials and StructuresXi’an Jiaotong UniversityXi’anChina

Acoustomechanical constitutive theory for soft materials

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