A DEM investigation of water-bridged granular materials at the critical state

Abstract

The critical state is an important concept for saturated and partially saturated granular materials as the strength and volume become constant and unique under continuous shear. By incorporating the water bridge effect, the mechanical behaviours of wet granular matters can be studied by the discrete element method (DEM). A series of DEM simulations are performed following the conventional triaxial loading path for dry and wet granular materials, and different suction values are applied at various confining stress levels. Unique critical state behaviours have been observed in both macroscopic and microscopic scales. It shows that the confining stress level plays an important role in the critical state behaviour of wet granular materials. The critical stress ratio for a wet material is not a constant value at different stress levels, and it is found that both the critical stress ratio and void ratio in wet granular matters are also much higher with a low confining stress. A framework is proposed by considering both the contact stress and the capillary stress effects to model the critical state lines. At large strain, the coordination number, the mean inter-particle force and fabric anisotropies evolve to constant critical state values for both dry and wet materials. The macro-parameters formulating the critical state stress ratio are found to be associated with the critical state anisotropies in solid skeleton and water phase fabrics, respectively.

Appendix: Calculation of direction tensors

For a granular assembly with $N_{s}$ solid contacts (note that one physical contact point has two contacts), after Oda et al. [23], a moment tensor quantifying the directions of solid contact normals can be expressed as:

$$ N_{ij}^{s} = \frac{1}{{N_{s} }}\sum\limits_{c \in V} {{\mathbf{n}}_{c} \otimes {\mathbf{n}}_{c} } $$

(22)

where $\varvec{n}_{\varvec{c}}$ is the unit vector of the contact normal on the $c$th solid contact. Similarly, a second-rank tensor for water bridge network can also be raised. For a sample with $N_{w}$ particle–water interactions (two times of total water bridge number), the moment tensor can be written as:

$$ N_{ij}^{w} = \frac{1}{{N_{w} }}\sum\limits_{w \in V} {{\mathbf{n}}_{w} \otimes {\mathbf{n}}_{w} } $$

(23)

where $\varvec{n}_{\varvec{w}}$ is the unit vector pointing from water bridge centre to particle centre on the $w$th water–solid interaction. The direction tensors of $D_{ij}^{s}$ and $D_{ij}^{w}$ consider the deviatoric part of the moment tensor being formulated as:

$$ D_{ij}^{s/w} = \frac{15}{2}\left( {N_{ij}^{s/w} - \frac{1}{3}\delta_{ij} } \right) $$

(24)

by taking the corresponding superscript.

For the directional distribution of contact forces, a second-rank moment tensor is also defined. By integrating the tensor product of the average contact force along a particular direction, the moment tensor noted as $K_{ij}^{sf}$ can be expressed in a unit sphere space $\varOmega$ as:

$$ K_{ij}^{sf} = \frac{1}{2\pi }\frac{1}{{N_{s} }}\oint_{\varOmega } {\left. {\left\langle {{\mathbf{f}}_{{{\mathbf{cont}}}} } \right\rangle } \right|_{{{\mathbf{n}}_{c} }} \otimes {\mathbf{n}}_{c} d\varOmega } $$

(25)

where $\left. {{\mathbf{f}}_{{{\mathbf{cont}}}} } \right|_{{{\mathbf{n}}_{{\mathbf{c}}} }}$ is the average value for the contact forces in the $\varvec{n}_{\varvec{c}}$ direction. The direction tensor of contact force, $G_{ij}^{sf}$, is the deviatoric part of the contact force moment tensor in a normalised form:

$$ G_{ij}^{sf} = \frac{{3K_{ij}^{sf} }}{{K_{11}^{sf} + K_{22}^{sf} + K_{33}^{sf} }} - \delta_{ij} $$

(26)

Note that the directional mean contact force is $f_{{{\text{cont}}0}} \approx K_{11}^{sf} + K_{22}^{sf} + K_{33}^{sf}$. Similar to the above procedures, the directional mean capillary force $f_{{{\text{cap}}0}}$ can also be obtained from a moment tensor for the capillary forces which will not be repeated.