This paper presents a micromechanics-based Cosserat continuum model for microstructured granular materials. By utilizing this model, the macroscopic constitutive parameters of granular materials with different microstructures are expressed as sums of microstructural information. The microstructures under consideration can be classified into three categories: a medium-dense microstructure, a dense microstructure consisting of one-sized particles, and a dense microstructure consisting of two-sized particles. Subsequently, the Cosserat elastoplastic model, along with its finite element formulation, is derived using the extended Drucker-Prager yield criteria. To investigate failure behaviors, numerical simulations of granular materials with different microstructures are conducted using the ABAQUS User Element (UEL) interface. It demonstrates the capacity of the proposed model to simulate the phenomena of strain-softening and strain localization. The study investigates the influence of microscopic parameters, including contact stiffness parameters and characteristic length, on the failure behaviors of granular materials with microstructures. Additionally, the study examines the mesh independence of the presented model and establishes its relationship with the characteristic length. A comparison is made between finite element simulations and discrete element simulations for a medium-dense microstructure, revealing a good agreement in results during the elastic stage. Some macroscopic parameters describing plasticity are shown to be partially related to microscopic factors such as confining pressure and size of the representative volume element.

Granular materials are composed of solid particles and inter-particle voids, which exhibit nonlinearity, heterogeneity, and other microscopic characteristics that usually cause complex failure behavior, i.e., localized failure phenomena for granular materials [

Granular materials have been studied using the Cosserat continuum theory [

This study uses a micromechanics-based Cosserat constitutive model, in which the macroscopic constitutive modulus tensors are obtained by microstructural information. Different microstructures of granular materials are determined by particle’s arrangements, sizes, void ratios and coordination numbers, and it can categorize microstructures into a medium dense one and two dense ones. Using this micromechanics-based Cosserat model, macroscopic constitutive modulus tensors are determined for granular materials with different microstructures. In addition, a micromechanics-based Cosserat elastoplastic model is proposed using an extended Drucker-Prager yield criterion, and this model provides the finite element formulation. Numerical simulations are performed using the User Element (UEL) interface in ABAQUS to investigate the capacity of the model and the influences of microscopic parameters on modeling of failure behaviors of granular materials with different microstructures. Mesh independence is analyzed for the presented model. The results are compared with those based on DEM to associate some macroscopic parameters with microscopic ones.

According to the Cosserat continuum theory, material points are treated as infinitesimal solids with characteristic lengths. There are three translational DOFs and three rotational DOFs, namely the displacement vector

1) Kinematic equations

where

2) Elastic constitutive relationships

where

where

3) Balance equations and boundary conditions

where

in which

The micromechanics-based Cosserat model can identify the constitutive modulus tensors in

where

(A) Isotropic directional density distribution function of contacts

There is usually a difference in contact stiffness parameters and internal lengths among contacts within a volume element. The material is assumed to be isotropic to simplify the analysis, and we consider an isotropic directional density distribution function of contacts

where

where

(B) Specific microstructures

To include more microstructural information, we use three different three-dimensional microstructures defined by different ordered particle arrangements as shown in our previous study [

Microstructures | Medium dense | dense_onesized | dense_twosized |
---|---|---|---|

Coordination number | 8 | 12 | 14 |

Void volume ratio (%) | 39.5 | 26.0 | 27.2 |

Then, the constitutive modulus tensors of these granular materials with different microstructures can be obtained by directly solving the discrete summations as shown in

(1) Medium dense microstructure

(2) Dense microstructure in one-sized particles

(3) Dense microstructure in two-sized particles

According to Forest et al. [

where

where

The Cosserat continuum is used to describe plastic behavior of granular materials by using an extended Drucker-Prager yield criterion:

where

Based on the piecewise linear hardening/softening assumption, the cohesion is written as follow:

where

The non-associated flow rule is assumed in granular materials, which implies the plastic potential function

in which

where

The numerical examples focus on the following issues: (1) the capacity of the micromechanics-based Cosserat model on modeling strain localization and softening of granular materials; (2) effects of different microstructures on failure behaviors; (3) effects of microscopic parameters including contact stiffness parameters and characteristic length on failure behaviors; (4) the mesh independence of the presented model; (5) comparisons of simulations between FEM and DEM.

The numerical simulations concern a rectangular panel with the size of 1 m × 2 m (

Parameter | Value | Parameter | Value |
---|---|---|---|

First,

Furthermore,

The Cosserat model can describe the rotational DOF for granular materials.

This part investigates the effects of contact stiffness parameters on failure behaviors of granular materials with microstructures for the micromechanics-based Cosserat model. For the sake of simplicity, GM medium is used as a representative.

First, to consider the effects of microscopic parameters, we should compare failure behaviors simulated by the micromechanics-based Cosserat model to those by the classical Cosserat one without microscopic parameters. The elastic constants of the classical Cosserat model can be identified by

Then, the effect of the contact stiffness parameter related to displacement

The effects of contact stiffness parameters related to rotation

Similarly, only GMmedium is discussed in this part. The characteristic length can reflect the size of the microstructure in granular materials, which is an important parameter in the Cosserat model and other microstructural models. Effects of the characteristic length

It is noted that the micromechanics-based Cosserat model with infinitesimal

The Cosseat model can provide a regularization mechanism by introducing a characteristic length to reduce the pathological mesh dependence for the localization problem in the classical continuum model, and our previous study [

Therefore, we further investigate the mesh dependence for situations with larger characteristic length

It is noted that the non-orthogonal flow behavior [

The micromechanics-based Cosserat model can provide the effects of microscopic information on failures of granular materials, therefore, it is meaningful to compare simulations of failures by this model with those by the discrete element model. In this part, GMmedium is investigated. Then, there is an issue that how to compare results of simulations between FEM and DEM. Usually, in the computational homogenization method for granular materials, the integration point is considered as a representative volume element (RVE) of the particle assembly. In this study, an RVE is considered by a hexagonal close-packed particle assembly as shown in

As mentioned above, the choice of RVE’s size in DEM is open. Then, we compare the stress-strain relationships simulated by FEM with those simulated by DEM with different sizes of RVEs based on

To ensure the generality of the macroscopic mechanical responses, the comparative study between DEM and FEM is needed under the same microstructure of granular materials when different microscopic parameters such as the contact stiffness are arranged.

The effects of microstructures on failure behaviors in granular materials can be analyzed by the micromechanics-based Cosserat model. First, we consider granular materials with different microstructures consisting of different particle arrangements and sizes, void ratios, and coordination numbers. Using the extended Drucker-Prager yield criterion, a Cosserat elastoplastic model is proposed for granular materials with microstructures, and a user element program coded by the ABAQUS UEL subroutine interface is implemented. Then, the presented model is used to investigate failure behaviors of different granular materials and the main conclusions are given as follows:

(1) Strain localization and softening behaviors are obtained for granular materials respectively with medium dense microstructure (GMmedium), dense microstructure in one-sized particles (GMdense_onesized) and dense microstructure in two-sized particles (GMdense_twosized), which are compared with those for granular materials based on isotropic contact density distribution (GMiso). Similar and greater degrees of strain localization and strain softening for GMdense_onesized and GMdense_twosized, followed by GMmedium, and finally, GMiso.

(2) The contact stiffness parameters related to displacement have obvious positive correlations with degrees of rotation, strain localization, and softening of the GMmedium. The contact stiffness parameters related to rotation have no effect on the elastic and hardening stages, but have an obvious effect on the softening stage of granular materials.

(3) The characteristic length has no effect on the elastic stage. After the elastic stage, the degrees of strain localization and softening are negatively related to the characteristic length when the characteristic length is greater than

(4) The mesh independence is related to the characteristic length. The mesh dependence is prominent when the characteristic length is below

(5) The FEM simulations for GMmedium by the micromechanics-based Cosserat model can agree well with the DEM ones in the elastic stage, but the hardening and softening stages show some differences because of the arrangements of particles in DEM due to discreteness, which is absent in FEM. Macroscopic parameters

The authors would like to thank Dr. Jiao Wang of Southwest Jiaotong University for suggestions on DEM simulations, and thank the editors and the anonymous referees for helpful guidance.

The authors are pleased to acknowledge supports of this work by the National Natural Science Foundation of China through Contract/Grant Numbers 12002245, 12172263 and 11772237, and by Chongqing Jiaotong University through Contract/Grant Number F1220038.

The authors confirm contribution to the paper as follows: study conception and design: Chenxi Xiu, Xihua Chu; data collection: Chenxi Xiu; analysis and interpretation of results: Chenxi Xiu, Xihua Chu; draft manuscript preparation: Chenxi Xiu. All authors reviewed the results and approved the final version of the manuscript.

Data are available from the corresponding author on reasonable request.

The authors declare that they have no conflicts of interest to report regarding the present study.

For a plane strain problem, the displacement vector in the Cosserat continuum is given by

According to the displacement fields, the strain vector is written by

The relation between displacement components and strain components are rewritten in a matrix form by

in which

The stress vector is given by

Then, the constitutive equation is obtained by

where the subscript p represents plasticity, and

The matrix form of the equilibrium equation is as follows:

For a plane strain problem, we consider an eight-node iso-parametric element with four integration points (CPER8), and the node displacement of the element is written by

in which the superscript

where

where

When the element displacement is obtained after interpolation of node displacement, the strain vector can be obtained as follows:

where

And the partitioned matrix

Then, the element stress can be obtained by the constitutive equation:

where

And the partitioned matrix

Based on the principle of virtual work, the matrix form of the incremental total potential energy for an element is obained by

where

where the element stiffness matrix

Finally, the FEM equation for a whole structure can be derived by

For a rate-independent elastoplastic constitutive model, the mathematical equations describing the stress-strain relationship are generally defined by a set of ordinary differential equations with constraints as follows:

where

An important step in the secondary development of UEL in ABAQUS is to update the stress, also known as constitutive iteration, and then perform equilibrium iteration to achieve the purpose of numerical calculation. Scholars are engaged in the development of stress integration algorithms. For instance, Lu et al. [

The updated stress is

where

Here, the superscript

The consistent elastoplastic tangent modulus matrix is presented here.

where

^{ d}_{e}^{ m}_{e}