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A micromechanics-based micromorphic model for granular materials and prediction on dispersion behaviors

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Abstract

One of the purposes in this study is to develop a micromorphic continuum model for granular materials based on a micromechanics approach. A symmetric curvature tensor is proposed in this model, and a symmetric couple stress tensor conjugated with the symmetric curvature tensor is derived. In addition, a symmetric stress tensor is obtained conjugating a symmetric strain tensor. The presented model provides a complete deformation pattern for granular materials by considering the decomposition for motions (displacement and rotation) of particles. Consequently, the macroscopic elastic constitutive relationships and constitutive moduli are derived in expressions of the microstructural information. Furthermore, the balance equations and boundary conditions are obtained for the presented micromorphic model. The other purpose in this study is to predict the dispersion behaviors of granular materials using the micromechanics-based micromorphic model. Five wave modes are predicted based on the presented model, including coupled transverse–rotational transverse, longitudinal, rotational longitudinal, transverse shear and rotational transverse waves. Investigating the propagations of these waves in the elastic granular media, the dispersion behaviors are predicted for coupled transverse–rotational transverse, longitudinal, rotational longitudinal waves, and the corresponding frequency band gaps are obtained.

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Acknowledgements

The authors are pleased to acknowledge the support of this work by the National Natural Science Foundation of China through contract/Grant Nos. 11772237 and 11472196, and to acknowledge the open funds of the State Key Laboratory of Water Resources and Hydropower Engineering Science (Wuhan University) through contract/grant number 2015SGG03.

Author information

Authors and Affiliations

  1. School of Civil Engineering, Wuhan University, Wuhan, 430072, China

    Chenxi Xiu, Xihua Chu, Jiao Wang & Wenping Wu

  2. State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, 430072, China

    Xihua Chu, Wenping Wu & Qinglin Duan

  3. State Key Laboratory for Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Qinglin Duan

Authors
  1. Chenxi Xiu
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  2. Xihua Chu
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  3. Jiao Wang
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  4. Wenping Wu
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Correspondence to Xihua Chu or Wenping Wu.

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Appendix: Wave equations

Appendix: Wave equations

Considering the propagation of this plane wave along \(x_{1}\) axis, we suppose the kinematic measures as

$$\left\{ {\begin{array}{*{20}l} {\bar{u}_{i} = \bar{u}_{i} \left( {x_{1} , t} \right)} \hfill \\ {\varGamma_{i} = \varGamma_{i} \left( {x_{1} , t} \right)} \hfill \\ {\psi_{ij} = \psi_{ij} \left( {x_{1} , t} \right)} \hfill \\ {\varphi_{ij} = \varphi_{ij} \left( {x_{1} , t} \right)} \hfill \\ \end{array} } \right.$$
(A1)

Substituting Eq. (A1) into Eq. (51), wave equations are obtained as follows:

$$\begin{aligned} \rho \ddot{\bar{u}}_{1} & = \left( {C_{1111} + A_{1111} } \right)\bar{u}_{1,11} - \left( {A_{1111} \psi_{11,1} + A_{1122} \psi_{22,1} + A_{1133} \psi_{33,1} } \right) \\ \rho \ddot{\bar{u}}_{2} & = \left( {C_{2121} + A_{2121} } \right)\bar{u}_{2,11} - \left( {A_{2112} \psi_{12,1} + A_{2121} \psi_{21,1} } \right) - \left( {D_{3131} + B_{3131} } \right)\bar{u}_{2,1111} \\ & \quad + \left( {D_{3131} + B_{3131} } \right)\varGamma_{3,111} - \left( {B_{3113} \varphi_{13,11} + B_{3131} \varphi_{31,11} } \right) \\ \rho \ddot{\bar{u}}_{3} & = \left( {C_{3131} + A_{3131} } \right)\bar{u}_{3,11} - \left( {A_{3113} \psi_{13,1} + A_{3131} \psi_{31,1} } \right) - \left( {D_{2121} + B_{2121} } \right)\bar{u}_{3,1111} \\ & \quad - \left( {D_{2121} + B_{2121} } \right)\varGamma_{2,111} + \left( {B_{2112} \varphi_{12,11} + B_{2121} \varphi_{21,11} } \right) \\ \end{aligned}$$
(A2)
$$\begin{aligned} \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{11} & = A_{1111} \bar{u}_{1,1} - \left( {A_{1111} \psi_{11} + A_{1122} \psi_{22} + A_{1133} \psi_{33} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{22} & = A_{2211} \bar{u}_{1,1} - \left( {A_{2211} \psi_{11} + A_{2222} \psi_{22} + A_{2233} \psi_{33} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{33} & = A_{3311} \bar{u}_{1,1} - \left( {A_{3311} \psi_{11} + A_{3322} \psi_{22} + A_{3333} \psi_{33} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{12} & = A_{1221} \bar{u}_{2,1} - \left( {A_{1212} \psi_{12} + A_{1221} \psi_{21} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{13} & = A_{1331} \bar{u}_{3,1} - \left( {A_{1313} \psi_{13} + A_{1331} \psi_{31} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{21} & = A_{2121} \bar{u}_{2,1} - \left( {A_{2112} \psi_{12} + A_{2121} \psi_{21} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{23} & = - \left( {A_{2323} \psi_{23} + A_{2332} \psi_{32} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{31} & = A_{3131} \bar{u}_{3,1} - \left( {A_{3113} \psi_{13} + A_{3131} \psi_{31} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{32} & = - \left( {A_{3223} \psi_{23} + A_{3232} \psi_{32} } \right) \\ \end{aligned}$$
(A3)
$$\begin{aligned} \rho I\ddot{\varGamma }_{1} & = \left( {D_{1111} + B_{1111} } \right)\varGamma_{1,11} - \left( {B_{1111} \varphi_{11,1} + B_{1122} \varphi_{22,1} + B_{1133} \varphi_{33,1} } \right) \\ \rho I\ddot{\varGamma }_{2} & = \left( {D_{2121} + B_{2121} } \right)\varGamma_{2,11} - \left( {B_{2112} \varphi_{12,1} + B_{2121} \varphi_{21,1} } \right) + \left( {D_{2121} + B_{2121} } \right)\bar{u}_{3,111} \\ \rho I\ddot{\varGamma }_{3} & = \left( {D_{3131} + B_{3131} } \right)\varGamma_{3,11} - \left( {B_{3113} \varphi_{13,1} + B_{3131} \varphi_{31,1} } \right) - \left( {D_{3131} + B_{3131} } \right)\bar{u}_{2,111} \\ \end{aligned}$$
(A4)
$$\begin{aligned} \rho^{\prime } J^{\prime } \ddot{\varphi }_{11} & = B_{1111} \varGamma_{1,1} - \left( {B_{1111} \varphi_{11} + B_{1122} \varphi_{22} + B_{1133} \varphi_{33} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{22} & = B_{2211} \varGamma_{1,1} - \left( {B_{2211} \varphi_{11} + B_{2222} \varphi_{22} + B_{2233} \varphi_{33} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{33} & = B_{3311} \varGamma_{1,1} - \left( {B_{3311} \varphi_{11} + B_{3322} \varphi_{22} + B_{3333} \varphi_{33} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{12} & = B_{1221} \varGamma_{2,1} - \left( {B_{1212} \varphi_{12} + B_{1221} \varphi_{21} } \right) + B_{1221} \bar{u}_{3,11} \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{13} & = B_{1331} \varGamma_{3,1} - \left( {B_{1313} \varphi_{13} + B_{1331} \varphi_{31} } \right) - B_{1331} \bar{u}_{2,11} \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{21} & = B_{2121} \varGamma_{2,1} - \left( {B_{2112} \varphi_{12} + B_{2121} \varphi_{21} } \right) + B_{2121} \bar{u}_{3,11} \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{23} & = - \left( {B_{2323} \varphi_{23} + B_{2332} \varphi_{32} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{31} & = B_{3131} \varGamma_{3,1} - \left( {B_{3113} \varphi_{13} + B_{3131} \varphi_{31} } \right) - B_{3131} \bar{u}_{2,11} \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{32} & = - \left( {B_{3223} \varphi_{23} + B_{3232} \varphi_{32} } \right) \\ \end{aligned}$$
(A5)

The wave equations can be divided into 8 groups as followings:

Group A:

$$\begin{aligned} \rho \ddot{\bar{u}}_{2} & = \left( {C_{2121} + A_{2121} } \right)\bar{u}_{2,11} - \left( {A_{2112} \psi_{12,1} + A_{2121} \psi_{21,1} } \right) - \left( {D_{3131} + B_{3131} } \right)\bar{u}_{2,1111} \\ & \quad + \left( {D_{3131} + B_{3131} } \right)\varGamma_{3,111} - \left( {B_{3113} \varphi_{13,11} + B_{3131} \varphi_{31,11} } \right) \\ \rho I\ddot{\varGamma }_{3} & = \left( {D_{3131} + B_{3131} } \right)\varGamma_{3,11} - \left( {B_{3113} \varphi_{13,1} + B_{3131} \varphi_{31,1} } \right) - \left( {D_{3131} + B_{3131} } \right)\bar{u}_{2,111} \\ \rho^{\prime } I^{\prime } \ddot{\psi }_{12} & = A_{1221} \bar{u}_{2,1} - \left( {A_{1212} \psi_{12} + A_{1221} \psi_{21} } \right) \\ \rho^{\prime } I^{\prime } \ddot{\psi }_{21} & = A_{2121} \bar{u}_{2,1} - \left( {A_{2112} \psi_{12} + A_{2121} \psi_{21} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{13} & = B_{1331} \varGamma_{3,1} - \left( {B_{1313} \varphi_{13} + B_{1331} \varphi_{31} } \right) - B_{1331} \bar{u}_{2,11} \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{31} & = B_{3131} \varGamma_{3,1} - \left( {B_{3113} \varphi_{13} + B_{3131} \varphi_{31} } \right) - B_{3131} \bar{u}_{2,11} \\ \end{aligned}$$
(A6)

Group B:

$$\begin{aligned} \rho \ddot{\bar{u}}_{3} & = \left( {C_{3131} + A_{3131} } \right)\bar{u}_{3,11} - \left( {A_{3113} \psi_{13,1} + A_{3131} \psi_{31,1} } \right) - \left( {D_{2121} + B_{2121} } \right)\bar{u}_{3,1111} \\ & \quad - \left( {D_{2121} + B_{2121} } \right)\varGamma_{2,111} + \left( {B_{2112} \varphi_{12,11} + B_{2121} \varphi_{21,11} } \right) \\ \rho I\ddot{\varGamma }_{2} & = \left( {D_{2121} + B_{2121} } \right)\varGamma_{2,11} - \left( {B_{2112} \varphi_{12,1} + B_{2121} \varphi_{21,1} } \right) + \left( {D_{2121} + B_{2121} } \right)\bar{u}_{3,111} \\ \rho^{\prime } I^{\prime } \ddot{\psi }_{13} & = A_{1331} \bar{u}_{3,1} - \left( {A_{1313} \psi_{13} + A_{1331} \psi_{31} } \right) \\ \rho^{\prime } I^{\prime } \ddot{\psi }_{31} & = A_{3131} \bar{u}_{3,1} - \left( {A_{3113} \psi_{13} + A_{3131} \psi_{31} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{12} & = B_{1221} \varGamma_{2,1} - \left( {B_{1212} \varphi_{12} + B_{1221} \varphi_{21} } \right) + B_{1221} \bar{u}_{3,11} \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{21} & = B_{2121} \varGamma_{2,1} - \left( {B_{2112} \varphi_{12} + B_{2121} \varphi_{21} } \right) + B_{2121} \bar{u}_{3,11} \\ \end{aligned}$$
(A7)

Group C:

$$\begin{aligned} \rho \ddot{\bar{u}}_{1} & = \left( {C_{1111} + A_{1111} } \right)\bar{u}_{1,11} - \left( {A_{1111} \psi_{11,1} + A_{1122} \psi_{22,1} + A_{1133} \psi_{33,1} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{11} & = A_{1111} \bar{u}_{1,1} - \left( {A_{1111} \psi_{11} + A_{1122} \psi_{22} + A_{1133} \psi_{33} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{22} & = A_{2211} \bar{u}_{1,1} - \left( {A_{2211} \psi_{11} + A_{2222} \psi_{22} + A_{2233} \psi_{33} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{33} & = A_{3311} \bar{u}_{1,1} - \left( {A_{3311} \psi_{11} + A_{3322} \psi_{22} + A_{3333} \psi_{33} } \right) \\ \end{aligned}$$
(A8)

Group D:

$$\begin{aligned} \rho I\ddot{\varGamma }_{1} & = \left( {D_{1111} + B_{1111} } \right)\varGamma_{1,11} - \left( {B_{1111} \varphi_{11,1} + B_{1122} \varphi_{22,1} + B_{1133} \varphi_{33,1} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{11} & = B_{1111} \varGamma_{1,1} - \left( {B_{1111} \varphi_{11} + B_{1122} \varphi_{22} + B_{1133} \varphi_{33} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{22} & = B_{2211} \varGamma_{1,1} - \left( {B_{2211} \varphi_{11} + B_{2222} \varphi_{22} + B_{2233} \varphi_{33} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{33} & = B_{3311} \varGamma_{1,1} - \left( {B_{3311} \varphi_{11} + B_{3322} \varphi_{22} + B_{3333} \varphi_{33} } \right) \\ \end{aligned}$$
(A9)

Group E:

$$\begin{aligned} \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{22} & = A_{2211} \bar{u}_{1,1} - \left( {A_{2211} \psi_{11} + A_{2222} \psi_{22} + A_{2233} \psi_{33} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{33} & = A_{3311} \bar{u}_{1,1} - \left( {A_{3311} \psi_{11} + A_{3322} \psi_{22} + A_{3333} \psi_{33} } \right) \\ \end{aligned}$$
(A10)

Group F:

$$\begin{aligned} \rho^{\prime } J^{\prime } \ddot{\varphi }_{22} & = B_{2211} \varGamma_{1,1} - \left( {B_{2211} \varphi_{11} + B_{2222} \varphi_{22} + B_{2233} \varphi_{33} } \right) \\ \rho^{\prime } J^{\prime } \ddot{\varphi }_{33} & = B_{3311} \varGamma_{1,1} - \left( {B_{3311} \varphi_{11} + B_{3322} \varphi_{22} + B_{3333} \varphi_{33} } \right) \\ \end{aligned}$$
(A11)

Group G:

$$\begin{aligned} \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{23} & = - \left( {A_{2323} \psi_{23} + A_{2332} \psi_{32} } \right) \\ \rho^{{\prime }} I^{{\prime }} \ddot{\psi }_{32} & = - \left( {A_{3223} \psi_{23} + A_{3232} \psi_{32} } \right) \\ \end{aligned}$$
(A12)

Group H:

$$\begin{aligned} \rho^{{\prime }} J^{{\prime }} \ddot{\varphi }_{23} & = - \left( {B_{2323} \varphi_{23} + B_{2332} \varphi_{32} } \right) \\ \rho^{{\prime }} J^{{\prime }} \ddot{\varphi }_{32} & = - \left( {B_{3223} \varphi_{23} + B_{3232} \varphi_{32} } \right) \\ \end{aligned}$$
(A13)

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Xiu, C., Chu, X., Wang, J. et al. A micromechanics-based micromorphic model for granular materials and prediction on dispersion behaviors. Granular Matter 22, 74 (2020). https://doi.org/10.1007/s10035-020-01044-8

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