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A micromechanics-based elasto-plastic model for granular media combined with Cosserat continuum theory

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Abstract

This paper is devoted to the micromechanical model of granular materials based on the Cosserat continuum theory. The generalised stress–strain relationships of a representative element volume are closely related to the micro elasto-plastic force–displacement and moment–rotation laws. With the help of generalised micro–macro relations, a micromechanical model is presented by aggregating the generalised micro elasto-plastic inter-particle contact laws in each contact orientation. A corresponding implicit multiscale integration algorithm for integrating the micromechanical model is proposed. Furthermore, the micromechanical model and implicit multiscale integration algorithm are simplified to consider a two-dimensional condition, and the Romberg integration of the numerical quadrature formula is also introduced in this condition. Compared with the data of Hostun sand and the results of DEM simulations for different particle shapes, it is demonstrated that the proposed micromechanical model can describe both the macro and micro responses of granular materials relatively well, such as strain softening/hardening, volumetric compaction/dilatancy, particle rotation, and evolution of the coordination number.

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Abbreviations

\(\sigma_{ij}\) :

The stress tensor of the REV

\(\mu_{ij}\) :

The couple stress tensor of the REV

\(\gamma_{ij}\) :

The strain tensor of the REV

\(\kappa_{ij}\) :

The curvature tensor of the REV

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

The inter-particle contact force

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

The inter-particle contact moment

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

The branch vector joining the centres of two contact particles

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

The Cartesian coordinate of inter-particle contact

\(A_{ij}\) :

The fabric tensor

\(P_{ij}\) (\(PP_{ij}\)):

The transform matrix

\(u_{i}^{ce}\) :

The elastic displacement of inter-particle contact

\(\omega_{i}^{ce}\) :

The elastic rotation of inter-particle contact

\(k_{n}\) :

The normal stiffness of inter-particle contact

\(k_{r}\) :

The shear stiffness of inter-particle contact

\(g_{n}\) :

The torsion stiffness of inter-particle contact

\(g_{r}\) :

The bending stiffness of inter-particle contact

\(du_{i}^{cp}\) :

The plastic displacement increment of inter-particle contact

\(d\omega_{i}^{cp}\) :

The plastic rotation increment of inter-particle contact

\({\Delta }_{r}^{cp}\) :

The generalised tangential plastic displacement of inter-particle contact

\(T\) :

The generalised shear force of inter-particle contact

\(x^{c}\) :

The material constant with dimension of length

\(\alpha\) :

The hardening/softening parameter for inter-particle contact

\(\alpha_{m}\) :

The peak inter-particle friction coefficient

\(e_{c}\) :

The void ratio at the critical state

\(e\) :

The void ratio of granular materials

\(\phi_{\mu }^{c}\) :

The inter-particle sliding friction angle

\(\psi^{c}\) :

The material constant in the dilatancy equation

\(\eta\) :

The non-dimensional parameter relating to coefficient of rolling friction

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

The material constant that controls the hardening/softening speed

\(d\lambda\) :

The plastic multiplier

\(H\) :

The plastic hardening modulus

\(Z\) :

The coordination number

\(d_{50}\) :

The mean particle diameter

\(d{\varvec{\sigma}}\) :

The generalised stress increment

\({\varvec{\sigma}}\) :

The generalised stress

\(d{\varvec{\gamma}}\) :

The generalised strain increment

\({\varvec{\gamma}}\) :

The generalised strain

\({\varvec{S}}\) :

The matrix of loading constraint of the generalised stress increment

\({\varvec{E}}\) :

The matrix of loading constraint of the generalised strain increment

\(d{\varvec{X}}\) :

The imposed driven vector

\(d{\varvec{X}}^{{ \star }}\) :

The new imposed driven vector

\({\varvec{R}}\) :

The residual vector

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Acknowledgements

The authors appreciate the financial support from the National Key Research and Development Program of China (Project Nos. 2017YFC1501003; 2018YFC0809605).

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Authors and Affiliations

  1. State Key Laboratory of Hydraulics and Mountain River Engineering, College of Water Resource and Hydropower, Sichuan University, Chengdu, 610065, People’s Republic of China

    Jianqiu Tian & Enlong Liu

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

    Yuanming Lai & Enlong Liu

  3. Ningbo Nottingham New Materials Institute, University of Nottingham Ningbo China, Ningbo, 315100, People’s Republic of China

    Yunming Yang

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  1. Jianqiu Tian
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Correspondence to Enlong Liu.

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Appendices

Appendix A

Constraints matrices and evolution vector (3D)

$${\varvec{S}} = \left[ {\begin{array}{*{20}c} {{\varvec{S}}_{11} } & {{\varvec{S}}_{12} } \\ {{\varvec{S}}_{21} } & {{\varvec{S}}_{22} } \\ \end{array} } \right].$$
(65)
$${\varvec{E}} = \left[ {\begin{array}{*{20}c} {{\varvec{E}}_{11} } & {{\varvec{E}}_{12} } \\ {{\varvec{E}}_{21} } & {{\varvec{E}}_{22} } \\ \end{array} } \right].$$
(66)

Stress-controlled compression:

$${\varvec{S}}_{11} = {\varvec{S}}_{22} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \\ \end{array} } \right].$$
(67)
$${\varvec{S}}_{12} = {\varvec{S}}_{21} = {\varvec{E}}_{11} = {\varvec{E}}_{12} = {\varvec{E}}_{21} = {\varvec{E}}_{22} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ \end{array} } \right].$$
(68)
$${\text{d}}{\varvec{X}} = \left[ {{\text{d}}x_{1} , {\text{d}}x_{2} , {\text{d}}x_{3} , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} \right]^{{\text{T}}} .$$
(69)

Strain-controlled drained triaxial:

$${\varvec{S}}_{11} = {\varvec{S}}_{22} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} } \\ \end{array} } \right].$$
(70)
$${\varvec{S}}_{12} = {\varvec{S}}_{21} = {\varvec{E}}_{12} = {\varvec{E}}_{21} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ \end{array} } \right].$$
(71)
$${\varvec{E}}_{11} = {\varvec{E}}_{22} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ \end{array} } \right].$$
(72)
$${\text{d}}{\varvec{X}} = \left[ {{\text{d}}x_{1} , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0} \right]^{{\text{T}}} .$$
(73)

Constraints matrices and evolution vector (2D)

Stress-controlled compression:

$${\varvec{S}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} } \\ \end{array} } \right].$$
(74)
$${\varvec{E}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ \end{array} } \right].$$
(75)
$${\text{d}}{\varvec{X}} = \left[ {{\text{d}}x_{1} ,{\text{d}}x_{2} , 0, 0, 0, 0} \right]^{{\text{T}}} .$$
(76)

Strain-controlled drained biaxial:

$${\varvec{S}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} } \\ \end{array} } \right].$$
(77)
$${\varvec{E}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \\ \end{array} } \\ \end{array} } \right].$$
(78)
$${\text{d}}{\varvec{X}} = \left[ {{\text{d}}x_{1} ,0, 0, 0, 0, 0} \right]^{{\text{T}}} .$$
(79)

Appendix B

The stress–strain relationships are closely related to the plastic part of the inter-particle contact and macro parameters (i.e. the reference pressure \(p_{{{\text{ref}}}}\), the reference void ratio \(e_{{{\text{ref}}}}\), and the slope of the critical state line \(\lambda\)) of the plastic part for inter-particle contact are determined directly from the critical state of the Hostun sand. So the micro parameters (i.e. material constant \(k_{p}^{c}\), the inter-particle sliding friction angle \(\phi_{\mu }^{c}\), and material constant \(\psi^{c}\)) of the plastic part for inter-particle contact are discussed here. Based on parameters for Hostun sand, the confining pressure 800 kPa and initial void ratio 0.72, \(k_{p}^{c}\), \(\phi_{\mu }^{c}\), and \(\psi^{c}\) are discussed.

The effects of micro parameters on the stress–strain relationships are illustrated in Fig. 

Fig. 15
figure 15

The parametric study for \(k_{p}^{c}\), \(\phi_{\mu }^{c}\), and \(\psi^{c}\)

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15. Figure 15a depicts effects of \(k_{p}^{c}\) on the stress–strain relationships. The speed of hardening is faster and volumetric compaction is smaller for larger values of \(k_{p}^{c}\). According to Coulomb type yield function (Eq. 16) and definition of the hardening/softening parameter (Eq. 18), the contact force and moment increase faster with larger \(k_{p}^{c}\). After relationship between macro and micro variables (Sect. 2.1) and homogenisation method (Sect. 2.5) are applied, we can see that the deviatoric stress increases faster for larger values of \(k_{p}^{c}\). According to the dilatancy equation (Eq. 20), the direction of plastic contact displacement increment is closer to the direction of \(d\Delta_{r}^{cp}\) for larger values of \(k_{p}^{c}\). After relationship between macro and micro variables (Sect. 2.1) and homogenisation method (Sect. 2.5) are applied, we can see that the volumetric compaction is smaller for larger values of \(k_{p}^{c}\). Figure 15b depicts effects of \(\phi_{\mu }^{c}\) on the stress–strain relationships. The deviatoric stress is larger and volumetric compaction is smaller for larger values of \(\phi_{\mu }^{c}\). Figure 15c depicts effects of \(\psi^{c}\) on the stress–strain relationships. The deviatoric stress is larger and volumetric compaction is larger for larger \(\psi^{c}\). These phenomena in Fig. 15b, c can be also analysed like discussions for Fig. 15a.

Appendix C

Based on the parameters of square particles, the micromechanical model proposed here is used to predict the stress–strain relations of square particles with relative density 40% (i.e. initial void ratio 0.206) [55]. In Fig. 

Fig. 16
figure 16

The stress–strain curves of medium dense sand predicted by micromechanical model

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16, under a low confining pressure the strain softening and volumetric dilatancy after volumetric compaction are observed, and under a high confining pressure the strain hardening and a light volumetric dilatancy after volumetric compaction are observed. These characteristics belong to medium dense sand.

The representative volume element for simple shear is shown in Fig. 

Fig. 17
figure 17

The schematic of specimen and stress–strain curves of simple shear

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17a. Initially, an isotropic compressive stress (200 kPa, 400 kPa, and 600 kPa) is applied on the representative volume element, and with shear strain increasing, the vertical and horizontal stresses (i.e. \(\sigma_{xx}\) and \(\sigma_{yy}\)) remain constant. Based on the parameters of the elongated grains (Table 4), the stress–strain relations are predicted by the micromechanical model (Fig. 17b). Figure 17b depicts the evolution of shear stress versus shear strain. The strain softening is observed and the shear stress is higher for higher isotropic compressive stresses. Figure 17b also depicts the evolution of volumetric strain versus shear strain. After an initial stage of volumetric compaction, volumetric dilatancy occurs during a large range of shear strain until the critical state is reached. The volumetric compaction is larger and dilatancy is lower for higher isotropic compressive stresses. These stress–strain relations of simple shear predicted by the micromechanical model conform to the rule of dense granular materials sheared under simple shear condition.

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Tian, J., Lai, Y., Liu, E. et al. A micromechanics-based elasto-plastic model for granular media combined with Cosserat continuum theory. Acta Geotech. 17, 2259–2285 (2022). https://doi.org/10.1007/s11440-021-01336-0

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