Numerical modelling of a field soil desiccation test using a cohesive fracture model with Voronoi tessellations

Abstract

Numerical modelling of a field soil desiccation test is performed using a hybrid continuum-discrete element method with a mix-mode cohesive fracture model and Voronoi tessellation grain assemblages. The fracture model considers material strength and contact stiffness degradation in both normal and tangential directions of an interface. It is found that the model can reasonably reproduce the special features of the field soil desiccation, such as curling and sub-horizontal crack. In addition, three significant factors controlling field desiccation cracking, fracture energy, grain heterogeneity and grain size are identified.

Appendix: Mix-mode cohesive fracture model

The cohesive fracture model used here is an extension and application of the elastic–plastic-damage interface constitutive framework originally presented in Ref. [21], and the model has been successfully applied in static [19, 26] and dynamic [20] problem of geomaterials, i.e. rock and soil. To facilitate the readability, the model is described.

The cohesive fracture model takes into account the cohesive effect on both tension and shear. In the model, the fracture interface is idealised with zero thickness; in other words, there is no layer of element embedded at the shared boundary of the two adjacent grains. Figure 2 illustrates the fracture model used in this paper. As shown in Fig. 2b, when the bond is undergoing loading, its displacement can be partitioned into elastic displacement represented by a spring and inelastic displacement represented by a slider and a divider in series. Therefore, the total bond displacement is expressed as

$${\mathbf{u}} = {\mathbf{u}}^{e} + {\mathbf{u}}^{i}$$

(3)

where ${\mathbf{u}}$ is the total displacement tensor of the bond, ${\mathbf{u}}^{e}$ is the elastic displacement tensor and ${\mathbf{u}}^{i}$ is the tensor of the inelastic displacement occurring in the bond. As the inelastic displacement is contributed from the slider and divider in series, it can be further decomposed into plastic displacement described by sliders, which is irreversible and fracture displacement by dividers (reversible) as

$${\mathbf{u}}^{i} = {\mathbf{u}}^{p} + {\mathbf{u}}^{f}$$

(4)

where ${\mathbf{u}}^{p}$ is the plastic displacement tensor from the deformation of the sliders and ${\mathbf{u}}^{f}$ is the fracture displacement tensor measured from the dividers. The norm of the inelastic displacement is computed as

$$u^{\text{ieff}} = ||{\mathbf{u}}^{i} || = ||{\mathbf{u}}^{p} + {\mathbf{u}}^{f} || = \sqrt {(u_{n}^{i2} + u_{s}^{i2} )} = \sqrt {(u_{n}^{p} + u_{n}^{f} )^{2} + (u_{s}^{p} + u_{s}^{f} )^{2} }$$

(5)

where $u_{n}^{i}$ and $u_{s}^{i}$ are the inelastic displacement scalar along the normal and tangential direction of the contact, respectively. In tensile loading, the governing variables are the tensile strength ($\sigma_{t}$) and the norm of the inelastic displacement ($u^{\text{ieff}}$). The tensile strength evolves as a linearly decreasing function of $u^{\text{ieff}}$ as shown in Fig. 2c. It is given as

$$\sigma_{t} \left( {u^{\text{ieff}} } \right) = \left\{ {\begin{array}{*{20}l} {\sigma_{t0} \left( {1 - \frac{{u^{\text{ieff}} }}{{w_{\sigma } }}} \right)} \\ 0 \\ \end{array} \, \begin{array}{c}\quad {u^{\text{ieff}} < w_{\sigma } } \\ \quad{u^{\text{ieff}} \ge w_{\sigma } } \\ \end{array} } \right.$$

(6)

and

$$w_{\sigma } = \frac{{2G_{f}^{I} }}{{\sigma_{t0} }} .$$

(7)

In Eqs. (6) and (7), $w_{\sigma }$, $\sigma_{t0}$ and $G_{f}^{I}$ are the ultimate norm of the inelastic displacement corresponding to zero tensile strength, the initial tensile strength and the mode I fracture energy, respectively. The mode I fracture energy can be obtained through a mode I test. The ultimate norm of the inelastic displacement corresponds to the threshold condition where the fracture is fully developed and the contact is no longer capable of transferring stress. The initial tensile strength is the stress at which the cohesive zone starts to develop and the crack starts to undergo softening.

A micro-damage variable is introduced as the percentage of fracture surface to the overall interface area to degrade the stiffness. This definition can reflect the physical behaviour when the contact is undergoing fracturing. In addition, it also complies with the classical definition of damage parameter in damage mechanics (e.g. [26]). The micro-damage variable can be calculated as

$$D = \frac{{A_{f} }}{{A_{0} }} = 1 - \frac{{k_{ns} }}{{k_{n0} }}$$

(8)

where D is the micro-damage variable and $A_{f}$ and $A_{0}$ are the fracture surface area and the overall contact area, respectively. $k_{ns}$ and $k_{n0}$ are, respectively, the degraded and initial normal stiffness. $k_{ns}$ can be computed as

$$\begin{aligned} k_{ns} & = \frac{{\sigma_{n} }}{{u_{n} - u_{n}^{p} }} = \frac{{\sigma_{t} (u^{\text{ieff}} )}}{{u_{n}^{e} + u_{n}^{p} + u_{n}^{f} - u_{n}^{p} }} \\ & = \frac{{\sigma_{t} (u^{\text{ieff}} )}}{{\sigma_{t} (u^{\text{ieff}} )/k_{n0} + (1 - \eta )u^{\text{ieff}} }} \\ \end{aligned}$$

(9)

where η is the ratio of plastic displacement to the total value of inelastic displacement (i.e. $\eta = u^{p} /u^{i}$ with $u^{i}$ the norm of inelastic displacement before unloading) and it can be determined experimentally using the pure mode I test.

Substituting Eq. (9) into Eq. (8), the micro-damage variable $D$ is expressed based on the norm of the inelastic displacement as

$$D = 1 - \frac{{\sigma_{t} (u^{\text{ieff}} )}}{{\sigma_{t} (u^{\text{ieff}} ) + (1 - \eta )u^{\text{ieff}} k_{n0} }}$$

(10)

Accordingly, the normal stress–displacement relationship, i.e. the relationship between $\sigma_{n}$ and $(u_{n} - u_{n}^{p} )$ of the interface, is presented as

$$\sigma_{n} = k_{ns} (u_{n} - u_{n}^{p} ) = \alpha k_{no} (u_{n} - u_{n}^{p} )$$

(11)

where parameter α is the integrity parameter defining the relative active area of the fracture. The integrity parameters is defined as

$$\alpha = 1 - \frac{{\left| {\sigma_{n} } \right| + \sigma_{n} }}{{2\left| {\sigma_{n} } \right|}}D$$

(12)

This integrity parameter is used to simulate the tensile unloading–reloading behaviour in the cohesive fracture model. It can be seen from Eq. (12) that the activation of the micro-damage variable is controlled by the fraction normal stress, which is activated in tension (i.e. $\sigma_{n} > 0$) and deactivated in compression (i.e. $\sigma_{n} < 0$). Thus, the normal stiffness in tension can be degraded, while it is kept unchanged in compression.

The governing variables for the shear loading are cohesion c (the contribution of normal stress to shear strength can be neglected if the friction angle is taken to be zero) and the norm of the inelastic displacement $u^{\text{ieff}}$. As the crack propagates, the cohesion degrades and can be expressed as a linear function of the norm of the inelastic displacement $u^{\text{ieff}}$ as shown in Fig. 2c. It is expressed as

$$c\left( {u^{\text{ieff}} } \right) = \left\{ {\begin{array}{*{20}l} {c_{0} \left( {1 - \frac{{u^{\text{ieff}} }}{{w_{c} }}} \right)} \\ 0 \\ \end{array} \, \begin{array}{c} \quad{u^{\text{ieff}} < w_{c} } \\ \quad{u^{\text{ieff}} \ge w_{c} } \\ \end{array} } \right.$$

(13)

and

$$w_{c} = \frac{{2G_{f}^{\text{IIa}} }}{{c_{0} }}$$

(14)

In Eqs. (13) and (14), $w_{c}$ is the ultimate norm of the inelastic displacement corresponding to zero cohesion, $c_{0}$ is the initial cohesion, and $G_{f}^{\text{IIa}}$ is the mode II fracture energy dissipated during shear at high confining normal stress (i.e. without the influence of the tensile loading regime). The ultimate norm of the inelastic displacement corresponds to zero cohesion. It gives the threshold condition at which the shear crack is fully developed. It also indicates that the material is no longer capable of transferring cohesion. Similar to the softening treatment under tensile loading, the degraded shear stiffness can be written as

$$k_{ss} = \alpha k_{s0}$$

(15)

where $k_{s0}$ and $k_{ss}$ are the initial shear stiffness and the degraded shear stiffness, respectively. The definition of α in Eq. (15) is same as in Eq. (11). The shear stress ($\tau$) is computed by

$$\tau = k_{ss} \left( {u_{s} - u_{s}^{p} } \right) = \alpha k_{so} \left( {u_{s} - u_{s}^{p} } \right)$$

(16)

Now the failure function is described. The normal stress can be either compressive or tensile. As described earlier, with the variation of the norm of the inelastic displacement, the material tensile strength and cohesion are changed. In this paper, the inter-block failure criterion can be described by a failure envelope as shown in Fig. 2d. The failure envelope represented by a solid line is the initial failure envelope. $\sigma_{t0}$ and $c_{0}$ represent the initial tensile strength and cohesion of the fracture, respectively. According to Eqs. (6) and (13), the two parameters decrease with increasing D when the fracture develops. Therefore, the failure envelope shrinks when the fracture is developing, i.e. the square dot line in Fig. 2d. If the fracture is totally destroyed, the failure envelope will become a line which is the conventional Mohr–Coulomb failure envelope with zero cohesion and zero tensile strength, i.e. the long dash line in Fig. 2d. Mathematically, the failure surface function can be expressed as

$$F = \tau^{2} - 2c\tan \left( \varphi \right)\left( {\sigma_{t} - \sigma_{n} } \right) - \tan^{2} \left( \varphi \right)\left( {\sigma_{n}^{2} - \sigma_{t}^{2} } \right) = 0$$

(17)

where $\varphi$ is the friction angle which is kept unchanged during the calculation, while $\sigma_{t}$ and c are evolved as per Eqs. (6) and (13), respectively.