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## 2D model formulation

The presented model is based on Timoshenko beam elements connecting the grains of material in terms of Voronoi cells. This section provides the enhanced formulation for Timoshenko beam, resulting with embedded discontinuities in local longitudinal direction for mode I failure, and in transversal direction for mode II failure. Two ways for dealing with failure modes are possible. First way is to handle these modes separately, where shear loading invokes mode II failure and at the same time compression force influences the shear failure threshold in terms of Mohr-Coulomb law. Second way is for dealing with tension, where simultaneous softening for both modes is enabled through the criteria presented at the end of this section. The developed model relies on thermodynamics principles. The localized failure is presented by a softening regime in a global macroresponse, where the heterogeneous displacement field is used in order to obtain a meshindependent response. The formulation for fracture process zone with micro-cracks is also presented here through the hardening regime with standard plasticity.

### Enhanced kinematics

The localization implies heterogeneous displacement field which no longer remains regular, even for smooth stress field. Thus, the displacement field ought to be introduced and written as the sum of a sufficiently smooth, regular part and a discontinuous part. Furthermore, the axial and transversal displacement fields need to be calculated independently. A straight finite element with two nodes of length le and cross section A is considered (Figure 2.4). The degrees of freedom at each node i ∈ [1,2] are axial displacement ui, transversal displacement vi and rotation qi. The strain measures for standard Timoshenko element are given e(x) = du(x).

#### Anisotropic model with multisurface criterion

The main difference in anisotropic model with multisurface criterion [Brancherie and Ibrahimbegovic, 2009, Kucerova et al., 2009, Govindjee et al., 1995] is in providing the full stress reduction simultaneously, where only one softening variable exist q. The latter is present in both of the failure surfaces, controlling the evolution of fracture with following failure surfaces.

**Validation test on beam with embedded discontinuities**

The results for a validation test of a beam, fixed at the left end and subjected to tension and shear loading, first separately and then simultaneously, on the right end are presented next. The geometric and material properties of the beam are: l = 1cm (length), b = 1cm, h = 0.8cm (cross-section width and height), E = 1000kN/cm2, n = 0.2, su = 2.2MPa, ty = 0.22MPa, tu = 0.26MPa, K f =2N/m. As stated before, the main mechanisms of rock failure are related to mode I and mode II. Mode I failure takes place when a tension is applied to potential crack which continues to grow perpendicular to the direction of loading. Mode I also occurs as a result of movement on the shear plane, when interlocking of asperities result in the propagation of existing vertical cracks. Dealing with rocks, the most dominant failure occurs under shear movement along a plane that is inclined with respect to principal stress direction. This is a mode II failure. As a final possibility, failure can be a combination of these two modes. The clear insight on crack opening and propagating can be gained in this beam example. The beam is subjected to imposed displacements on the free end and the reactions on the fixed end are monitored.

Figure 2.6.a illustrates a beam response when tension is applied and crack opening in mode I occurs. Shear failure of the beam is presented in Figure 2.6.b. It can be seen from Figure 2.6.b. that for shear load, the fracture process zone starts when the first yielding point is reached. The further stress increase allows to reach the ultimate stress and to start softening phase. Exponential softening drives element to complete failure.

The response of the beams are mesh independent, but initial weakness need to be introduced in one element, so that the plastic deformation localizes inside only weaker element, while the rest of the elements elastically unload.

**Preparation of 2D plain strain rock specimens**

2D plane strain rock specimens are constructed next. The specimens are of dimensions 10×10 cm (with unit thickness) and are meshed with triangles by means of Delaunay algorithm. The mesh generation is carried out with a GMSH [Geuzaine and Remacle, 2009]. The specimen has 253 nodes and 704 elements (Figure 2.8). Timoshenko beam elements are positioned on each edge of every triangle in the specimen. Their geometric properties represent the corresponding part in specimen volume. The main hypothesis in constructing the lattice model is that the cells connected by cohesive links (beams) correspond to the representative part of the specimen which have homogeneous properties, while the heterogeneities are introduced through the cohesive links. Thus, the Voronoi cells are derived from Delaunay triangulation and the beam cross sections are computed from the length of the common size of the neighbouring cells (Figure 2.9). The material parameters are taken the same as in the equivalent standard continuum. This kind of calculating the lattice parameters has already been successfully used by [Ibrahimbegovic and Delaplace, 2003].

In order to validate the lattice model parameters, the tension and shear tests are conducted in the linear elastic regime on the proposed homogeneous specimen (shown in Figure 2.8) in two versions: lattice model and equivalent standard continuum model with triangular solid elements. The material parameters are the same for each test version: E = 1000kN/cm2, n = 0.2. The results are presented in Figures 2.10.a and 2.10.b. The equivalent standard continuum model operate only in linear elastic regime and its response matches with linear elastic regime of lattice models before the failure phase, showing that the proposed model is capable of reproducing classical linear elastic continuum with such computed lattice parameters.

**Numerical tests on heterogeneous specimens**

For further tests, heterogeneous specimens are introduced, where the phase II elements are initially weaker and have a lower value of modulus of elasticity, representing preexisting micro-cracks and other defects. The distribution of phase II is spread across the specimen by a random distribution. The initiation of crack propagation starts when the loading is applied.

**Table of contents :**

Contents

List of Figures

List of Tables

**1 Introduction **

1 Motivation

2 Overview of the numerical methods applied to rock mechanics

2.1 Finite Difference and Finite Volume Method (FDM, FVM)

2.2 Finite Element Method (FEM)

2.3 Meshless methods

2.4 Boundary Element Method (BEM)

2.5 Discrete Element Method (DEM)

2.6 Discrete Fracture Network (DFN)

2.7 Hybrid methods

3 Aims, scopes and methodology

4 Outline

**2 2D rock mechanics model **

1 Model description

2 2D model formulation

2.1 Enhanced kinematics

2.2 Equilibrium equations

2.3 Constitutive model

2.4 Computational procedure

2.5 Anisotropic model with multisurface criterion

3 Numerical results

3.1 Validation test on beam with embedded discontinuities

3.2 Preparation of 2D plain strain rock specimens

3.3 Numerical tests on heterogeneous specimens

4 Final comments on the presented 2D rock mechanics model

Rock mechanics and failure phenomena

**3 3D rock mechanics model **

1 Model description

2 3D model formulation

2.1 Kinematics of strong disontinuity

2.2 The finite element approximation

2.3 Virtual work

2.4 Constitutive model

2.5 Computational procedure

2.6 The global solution procedure

2.7 The failure criteria

3 Numerical examples

3.1 Embedded discontinuity beam test

3.2 The construction of rock specimens

3.3 Rock specimen under uniaxial (unconfined) compression and tension test

3.4 Influence of pre-existing defects

4 Final comments on the presented 3D rock mechanics model

**4 Influence of specimen shape deviations on uniaxial compressive strength **

1 Research motivation

2 Preparation of specimens with targeted shape deviations

3 Numerical model

4 Numerical results

5 Final comments on research of shape deviations influence to the UCS

**5 2D model for failure of fluid-saturated rock medium **

1 Model introduction

2 The porous media formulation

3 The discrete lattice model description

3.1 The discrete lattice mechanical and fluid flow formulations

3.2 The strong discontinuities in poro-elastoplastic solid

4 The enhanced finite element formulation

4.1 The finite element interpolations

4.2 The enhanced weak form

4.3 The finite element equations of a coupled poroplastic problem

4.4 The operator split algorithm

5 Numerical simulations

5.1 Uncoupled fluid flow across the lattice

5.2 Uniaxial coupled poroelastic problem

5.3 Drained compression test of the poro-plastic sample with the localized failure

**6 Final comments on the presented failure model of fluid-saturated rock medium **

6 Conclusions and future perspectives

1 Conclusions

2 Future perspectives

**Bibliography **