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Evolution of meso-structures and mechanical properties of granular materials under triaxial compression state from complex network perspective

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Abstract

The evolution of the network connection of granular materials is investigated by performing a series of numerical simulations in triaxial compression tests with different initial porosities by discrete element method (DEM). Results of evolution characteristics of complex network are reported for both dense and loose assembles. The simulation focuses on the influence of porosity on connectivity evolution, and reveals the correlation between the parameters in macro and mesoscale. Kinds of properties are studied, including degree and its distribution, clustering coefficient, network density and the average shortest path. The results demonstrate the phenomenon of dilataion due to shear deformation are able to be reflected by those mesocope parameters mentioned. Specifically, in the process of the dilatation, the rate of contact disintegration exceeds the rate of contact creation, which means the loss of connectivity, thus the values of some properties decrease, like degree, clustering coefficient and network density, but some increase like the average shortest path. Additionally, the bridging of macro and mesoscope are built regarding the parameters of the Cam-Clay model and complex network. From the results, the parameter M (determined by q = Mp′ at critical state) and the reference parameter \( {\text{T}} \) (\( T_{j}^{s} = L_{j}^{s} \left( {1 - \log_{{D_{j}^{s} }} t} \right) \), calculated according to the average degree \( D_{j}^{s} \) and shortest path \( L_{j}^{s} \) of the critical state) have a positive correlation. And a linear relationship between the slope of isotropic virgin-consolidation λ and the rate of decline of the average shortest path upon loading is represented as well. These achievements are the first step in an ongoing study of establishing the multi-scale constitutive from complex network perspective.

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Abbreviations

\( I \) :

Inertial number

\( \dot{\varepsilon } \) :

Strain rate

\( d \) :

Mean size

\( \rho \) :

Grain density

\( p' \) :

Mean effective stress

\( F_{n} \), \( F_{t} \) :

Normal and tangential forces (with the convention of positive tensile forces), respectively

\( u_{n} \), \( u_{t} \) :

Normal and tangential interparticle displacements, respectively

\( K_{n} \),\( K_{t} \) :

Normal and tangential stiffnesses, respectively

\( \varphi \) :

Interparticle friction angle

\( \varepsilon_{11} \), \( \varepsilon_{22} \), \( \varepsilon_{33} \) :

Strains of the container in the directions of x, y and z

\( \sigma_{11} ,\sigma_{22} ,\sigma_{33} \) :

Stress in the directions of x, y and z, \( \sigma_{11} = \sigma_{33} \)

\( p \) :

Deviator stress, given by \( \left( {\sigma_{22} - \sigma_{11} } \right) \)

\( {\text{R}} \) :

Mean radius

G:

Graph

\( C_{i} \) :

Clustering coefficient at node i

\( k_{i} \) :

Degree of node i

\( t_{i} \) :

Number of triangles attached to the node i

C:

Clustering coefficient

n:

Number of nodes

D:

Density of graph G

m :

Number of edges

L:

Shortest path

V:

Set of nodes in graph G

\( d\left( {s,t} \right) \) :

Shortest path from node s to node t

\( p_{k} \) :

Probability that a vertex chosen uniformly at random has degree k

X :

A discrete random variable

e :

Base of the natural logarithms

k!:

Factorial of k

\( \mu \) :

Shape parameter, equal to the expected value of X and also to its variance

\( {\text{M}} \) :

Critical parameter by q = Mp’

\( T \) :

Reference variable,\( T_{j}^{s} \) for the sample s under confining pressure j

t :

Material parameter, determined by tests to make \( T \) satisfy the conditions

\( \begin{array}{*{20}c} \lambda \\ \end{array} \) :

Slope of isotropic virgin-consolidation line

\( \begin{array}{*{20}c} \beta \\ \end{array} \) :

Slope of the average shortest path

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Acknowledgements

The authors thank the reviewers and Prof. Stefan Luding for their valuable comments, and Mr. Xu Ran for his help when performing the DEM computation. This research was supported by National Key Research and Development of China (Project No. 2017YFC1501003), the 100-Talent Program of the Chinese Academy of Sciences (Granted to Dr. Enlong Liu), and Key Research Program of Frontier Sciences of Chinese Academy of Sciences (QYZDY-SSW-DQC015).

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Authors and Affiliations

  1. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resources and Hydropower, Sichuan University, Chengdu, 610065, China

    Xiaoqiong Jiang, Enlong Liu, Lian Jiang, Jianqiu Tian & Yi Sun

  2. State Key Laboratory of Frozen Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou, 730000, China

    Enlong Liu

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  1. Xiaoqiong Jiang
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Correspondence to Enlong Liu.

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Jiang, X., Liu, E., Jiang, L. et al. Evolution of meso-structures and mechanical properties of granular materials under triaxial compression state from complex network perspective. Granular Matter 20, 54 (2018). https://doi.org/10.1007/s10035-018-0827-9

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