# Three-Dimensional Network Model for Coupling of Fracture and Mass Transport in Quasi-Brittle Geomaterials

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## Abstract

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## 1. Introduction

## 2. Network Approach

#### 2.1. Discretisation

#### 2.2. Structural Network Model

#### 2.2.1. Structural Element

#### 2.2.2. Structural Material

#### 2.3. Transport Model

#### 2.3.1. Transport Element

#### 2.3.2. Transport Material

## 3. Analyses

#### 3.1. Steady-State Potential Flow

#### 3.2. Nonstationary Transport Analysis

#### 3.3. Coupled Structural-Transport Benchmark

## 4. Conclusions

- The network of structural elements, defined by the Delaunay edges, provides element geometry and size independent load-displacement curves, as demonstrated through cohesive fracture simulations of double cantilever beams. The traction free condition is approached without stress locking. Local deviations of the fracture path due to random network generation has very little influence on the load-displacement curves.
- The network of transport elements, defined by the Voronoi edges, provides results for non-stationary transport which are in very good agreement with analytical solutions, and are independent of element geometry and size. The proposed discretisation scheme for the transport network facilitates the enforcement of boundary conditions. Local to a domain boundary, transport elements have one node on the boundary and are directed perpendicular to the boundary.
- The proposed method for coupling the effect of crack opening, determined by the structural network, with transport properties of the transport network yields objective results with respect to element geometry and size. This dual network approach facilitates the simulation of transport along crack paths and from crack faces into the bulk material.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Network models for coupled problems: (

**a**) common approach in which the structural and transport network nodes are coincident. Both structural and transport elements are on the Delaunay edges; (

**b**) simulated crack in structural network; and (

**c**) improved approach in which transport elements are on the Voronoi edges and therefore aligned with potential cracks.

**Figure 2.**Spatial arrangement of structural and transport elements of the 3D transport-structural network approach: (

**a**) geometrical relationship between Delaunay and Voronoi tessellations; (

**b**) structural element with cross-section defined by the associated Voronoi facet; and (

**c**) transport element with cross-section defined by the associated Delaunay facet.

**Figure 3.**Discretisation of domain boundaries: (

**a**) Voronoi facet of Delaunay edge i–j located on the surface of the domain after initial tessellation; and (

**b**) modified arrangement used for definition of transport nodes and elements.

**Figure 5.**Steady-state simulation of potential flow: (

**a**) Voronoi tessellation of domain; (

**b**) conventional network solution; and (

**c**) proposed network solution.

**Figure 10.**The coarse dual networks (${d}_{\mathrm{min}}/L=0.06$) for (

**a**) structural and (

**b**) transport analysis.

**Figure 12.**Crack patterns for (

**a**) coarse; (

**b**) medium; and (

**c**) fine network for a load-point-displacement of $\delta =0.15$ mm in Figure 11. The shaded polygons represent the mid-cross-sections of elements with $\tilde{w}>10$ $\mathsf{\mu}$m.

**Figure 13.**Influence of element size on the cumulative volume of inflow normalised by the domain volume.

**Figure 14.**Contour plots of capillary suction ${P}_{\mathrm{c}}$ at 3.33 h for the (

**a**) x–z plane at $y=0.125$ m and (

**b**) y–z plane at $x=0.3$ m.

Symbol | (Units) | Definition |
---|---|---|

${A}_{\mathrm{t}}$ | (m${}^{2}$) | cross-sectional area of the tetrahedron face |

a | (Pa) | parameter in van Genuchten model |

${\mathbf{B}}_{1}$, ${\mathbf{B}}_{2}$ | matrices expressing rigid body kinematics | |

c | (s${}^{2}$/m${}^{2}$) | capacity of the material |

${\mathbf{C}}_{\mathrm{e}}$ | (m s${}^{2}$) | element capacity matrix |

C | (m) | centroid of mid-cross-section |

${c}_{\mathrm{s}}$ | ratio of compressive and tensile strength | |

${\mathbf{D}}_{\mathrm{e}}$ | (Pa) | material stiffness matrix |

${d}_{\mathrm{min}}$ | (m) | minimum distance between nodes |

${e}_{\mathrm{p}}$, ${e}_{\mathrm{q}}$ | (m) | eccentricities between the midpoint of the network element and the centroid C |

E | (Pa) | Young’s modulus |

${\mathbf{f}}_{\mathrm{s}}$ | (N) | acting structural forces |

${f}_{\mathrm{d}}$ | loading function | |

${f}_{\mathrm{t}}$ | (Pa) | tensile strength |

${f}_{\mathrm{q}}$ | (Pa) | shear strength |

${f}_{\mathrm{c}}$ | (Pa) | compressive strength |

f | (kg/m${}^{2}$) | outward flux normal to the boundary |

${G}_{\mathrm{F}}$ | (J/m${}^{2}$) | fracture energy |

h | (m) | length of structural element |

${h}_{\mathrm{t}}$ | (m) | length of transport element |

$\mathbf{I}$ | unity matrix | |

${I}_{\mathrm{p}}$ | (m${}^{4}$) | polar moment of area |

${I}_{1}$ and ${I}_{2}$ | (m${}^{4}$) | two principal second moments of area of the cross-section |

$\mathbf{K}$ | element stiffness matrix | |

${\mathbf{K}}_{\mathrm{r}}$ | rotational stiffness at point C | |

${l}_{\mathrm{c}}$ | (m) | crack length |

L | (m) | length of specimen |

m | parameter in van Genuchten model | |

n, p, q | (m) | coordinates of mid-cross-sections |

${P}_{\mathrm{c}}$ | (Pa) | capillary suction (tension positive) |

${P}_{\mathrm{w}}$ | (Pa) | pressure in the wetting fluid |

${P}_{\mathrm{d}}$ | (Pa) | pressure in the non-wetting fluid |

${q}_{\mathrm{s}}$ | ratio of shear and tensile strength | |

S | degree of saturation | |

t | (s) | time |

${u}_{\mathrm{n}}$, ${u}_{\mathrm{p}}$, ${u}_{\mathrm{q}}$ | (m) | displacement discontinuities |

${u}_{\mathrm{x}}$, ${u}_{\mathrm{y}}$, ${u}_{\mathrm{z}}$ | (m) | translational degrees of freedom |

${\mathbf{u}}_{\mathrm{e}}$ | vector of degrees of freedom of structural element | |

${\mathbf{u}}_{\mathrm{t}}$ | (m) | vector of translational part of degrees of freedom |

${\mathbf{u}}_{\mathrm{r}}$ | vector of rotational part of degrees of freedom | |

${\mathbf{u}}_{\mathrm{C}}$ | (m) | vector of displacement discontinuities |

V | (m${}^{3}$) | volume |

${V}_{\mathrm{avail}}$ | (m${}^{3}$) | available volume to be filled |

${V}_{\mathrm{tot}}$ | (m${}^{3}$) | total volume of the specimen |

${w}_{\mathrm{n}}$, ${w}_{\mathrm{p}}$ and ${w}_{\mathrm{q}}$ | (m) | crack opening components |

${w}_{\mathrm{f}}$ | (m) | displacement threshold which determines the initial slope of the softening curve |

$\tilde{w}$ | (m) | equivalent crack opening |

x, y, z | (m) | Cartesian coordinates |

${\alpha}_{0}$ | (s) | initial conductivity of the undamaged material |

${\alpha}_{\mathrm{c}}$ | (s) | change of the conductivity due to fracture |

α | (s) | conductivity |

${\alpha}_{\mathrm{e}}$ | conductivity matrix | |

γ | input parameter, which controls Poisson’s ratio of the structural network | |

${\mathsf{\Gamma}}_{1}$, ${\mathsf{\Gamma}}_{2}$ | boundary segments | |

δ | (m) | load-point-displacement |

ε | strain vector | |

${\epsilon}_{\mathrm{n}}$, ${\epsilon}_{\mathrm{p}}$, ${\epsilon}_{\mathrm{q}}$ | strain components | |

${\epsilon}_{0}$ | strain threshold | |

θ | (kg/m${}^{3}$) | moisture content |

${\theta}_{\mathrm{r}}$ | (kg/m${}^{3}$) | residual moisture content |

${\theta}_{\mathrm{s}}$ | (kg/m${}^{3}$) | saturated moisture content |

κ | (m${}^{2}$) | intrinsic permeability |

${\kappa}_{\mathrm{d}}$ | history variable in damage model | |

${\kappa}_{\mathrm{r}}$ | relative permeability | |

μ | (Pa s) | dynamic (absolute) viscosity |

ξ | tortuosity factor | |

ρ | (kg/m${}^{3}$) | density of the fluid |

${\sigma}^{\mathrm{c}}$ | (Pa) | continuum stress |

σ | (Pa) | stress vector |

${\sigma}_{\mathrm{n}}$, ${\sigma}_{\mathrm{p}}$, ${\sigma}_{\mathrm{q}}$ | (Pa) | stress components |

${\varphi}_{\mathrm{x}}$, ${\varphi}_{\mathrm{y}}$, ${\varphi}_{\mathrm{z}}$ | rotational degrees of freedom | |

ω | damage variable | |

∇ | divergence operator |

Network Type | Node Definition | Element Definition | Nodal Count * | Element Count * |
---|---|---|---|---|

Conventional | Delaunay vertex | Delaunay edge | 330 | 1800 |

Proposed | Voronoi vertex | Voronoi edge | 2880 | 5440 |

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**MDPI and ACS Style**

Grassl, P.; Bolander, J. Three-Dimensional Network Model for Coupling of Fracture and Mass Transport in Quasi-Brittle Geomaterials. *Materials* **2016**, *9*, 782.
https://doi.org/10.3390/ma9090782

**AMA Style**

Grassl P, Bolander J. Three-Dimensional Network Model for Coupling of Fracture and Mass Transport in Quasi-Brittle Geomaterials. *Materials*. 2016; 9(9):782.
https://doi.org/10.3390/ma9090782

**Chicago/Turabian Style**

Grassl, Peter, and John Bolander. 2016. "Three-Dimensional Network Model for Coupling of Fracture and Mass Transport in Quasi-Brittle Geomaterials" *Materials* 9, no. 9: 782.
https://doi.org/10.3390/ma9090782