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Elastic constants obtained analytically from microscopic features for regularly arranged elliptical particle assembly

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Abstract

By the micro-structural mechanics approach, this study establishes the quantitative stress–strain relationships at the elastic stage with respect to the microscopic features of the particles for anisotropic granular materials that are composed of regularly arranged elliptical particles with the same size. Firstly, the elliptical particle assembly is equated with a lattice network described by beam elements attached to the center of particles. Then, the elastic stress–strain relationships, which exactly show the features of orthotropic micropolar continuum, are established through analyzing the triangular and hexagonal cells based on the principle of energy balance. Finally, the analytical expressions of eight independent parameters in stress–strain relationships are obtained as the functions of particle shape, size, and microscopic contact stiffnesses. Further, the relationship between microscopic parameters and macroscopic elastic constants for anisotropic granular materials is proposed. The analytical expressions are verified by comparison between theoretical and discrete element method (DEM) results for elliptical particle assembly, and the influences of microscopic parameters on the macroscopic elastic constants are investigated in detail according to the proposed analytical expressions. Our work provides some useful insights for the microscopic explanation and the influences of microscopic parameters on the macroscopic mechanical behavior of granular materials.

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Abbreviations

\(a\), \(b\) :

The major and minor axes of ellipse

\(e_{ijk}\) :

Permutation tensor

\(E_{i} (i = 1,2)\) :

Elastic modulus along i direction

\(f_{i}^{c}\) :

Contact force

\(G\) :

Shear modulus

\(K_{ij}^{c}\) :

Contact stiffness tensor

\(l_{k}^{c}\) :

Length vector corresponding to the beam c in the triangle cell

\(m\) :

Major-minor axis ratio of elliptical particles

\(M_{i}^{c}\) :

Contact moment

\(R_{E}\), \(R_{B}\) :

Elastic and bending modulus ratios, respectively

\(R_{G}\) :

Shear modulus ratio

\(T_{ij}^{c}\) :

Rotation stiffness tensor

\(u_{i,k}^{o}\) :

Displacement gradient

\(V\) :

The area of the unit cell

α :

The angle between tangential direction and the x-axis at contact points A2 (A3)

\(\beta\) :

Base angle of the isosceles triangle

γ :

Contact anisotropy ratio

\(\delta_{i}^{c}\) :

Relative displacement

\(\varepsilon_{ki}^{o}\) :

Strain tensor

\(\zeta\) :

Shape parameter

\(\eta_{p}^{c}\) :

Position vector from centroid of the triangle cell to the contact point corresponding to the beam c

\(\theta_{i}^{c}\) :

Relative rotation

\(\kappa_{i} (i = 1,2)\) :

Bending modulus along i direction

\(\lambda\) :

Tangential-normal contact stiffness ratio

\(\mu_{pj}\) :

Couple stress tensor

\(\nu_{ij} (i \ne j)\) :

Poisson’s ratio that deformation along j direction due to applied stress along i direction

\(\xi\) :

Rolling-normal contact stiffness ratio

\(\rho\) :

The local rigid rotation ratio

\(\sigma_{mq}\) :

Cauchy stress tensor

\(\omega_{j,p}^{o}\) :

Rotation gradient

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Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant Nos. 11872281, 51639008, and 51890911), and the State Key Laboratory of Disaster Reduction in Civil Engineering (Grant No. SLDRCE19-A-06).

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Authors and Affiliations

  1. College of Civil Engineering, Tongji University, Shanghai, 200092, China

    Z. H. Zhou

  2. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    H. N. Wang

  3. Department of Civil Engineering, Tian** University, Tian**, 300072, China

    M. J. Jiang

  4. State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai, 200092, China

    H. N. Wang & M. J. Jiang

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  1. Z. H. Zhou
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Correspondence to M. J. Jiang.

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Zhou, Z.H., Wang, H.N. & Jiang, M.J. Elastic constants obtained analytically from microscopic features for regularly arranged elliptical particle assembly. Granular Matter 23, 29 (2021). https://doi.org/10.1007/s10035-021-01088-4

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