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As discussed in Section 2.1, the macroscopic behaviour of porous geomaterials is the emergent result of processes that occur at lower scales. Understanding the influence of these lower scale processes on the overall material response is valuable when designing for many engineering applications involving geomaterials. In the present thesis, the focus is on numerical microscale modelling.

vast majority of boundary value problems for engineering applications are modelled using the macroscopic approach. The main reason for this choice is the high computational cost of modelling large boundary value problems with a microscale approach. The large length scale of the engineering applications of interest would encompass an enormous number of individual particles, pore bodies and throats. Therefore, the computational information needed to simulate the whole geotechnical structure would result in impractically time consuming computations. However, this impracticality has been tackled by other numerical approaches that incorporate microscale approaches within the modelling of boundary value problems.

Since the boundary value problems cannot be solved by microscale modelling, the investi- gation of influence of the microstructure on the overall material response can be analysed by applying proper boundary conditions to a numerical cell with volume much smaller than specimens used in laboratory experiments as presented in Section 2.3.2. These conditions are chosen based on the assumption that the numerical cell is part of a larger region of microstructure that consists of identical cells that all behave similarly. Then, the overall cell behaviour is evaluated based on a so called computational homogenisation scheme. Hence, the link between the two scales can be analysed at the material point level (Thornton, 2000; Athanasiadis et al., 2016).

To connect the microscale and boundary value problem in a numerical framework in an effi- cient manner, a scheme known as a multiscale approach has been proposed (Kouznetsova et al., 2002). This approach is based on the discretisation of the material, at the macroscopic level, into elements that contain one or more integration points. Then, an individual microscale problem is assigned to each discrete macroscopic constituent. For the approach to be effi- cient, the macroscopic element must be much larger than the microscale problem. When the element chosen for the microscale problem presents a representative response (its behaviour is invariant to its random generation and to its size increase) then only one cell is assigned to each macroscopic element. Otherwise, a number of microscopic problems are assigned to each element and their average response determines the macroscopic behaviour. Hence, the boundary value problems can be solved by analysing a series of microstructural prob- lems instead of using phenomenological constitutive laws (Miehe et al., 2010; Wang and Sun, 2016).

The present thesis focuses on relating the individual microscopic phenomena to the overall material behaviour. This involves development of a new framework for a hydro-mechanical network model for porous geomaterials describing fluid retention, conductivity and mechanical response at the capillary pore scale. The network model can be used in the future to inform macroscopic constitutive hydro-mechanical relations.

Chapter 3

Model definition

The modelling approach used in this thesis is introduced. The domain discretisation, the mechanical, transport and hydro-mechanical models along with the model parameters used are presented. Furthermore, a new technique is proposed to model Periodic Boundary Conditions (PBCs).

3.1

Point Generation

The domain is discretised by means of an irregular dual Voronoi/Delaunay tessellation. For this purpose, points are initially placed randomly, based on a trial and error approach that ensures a minimum distance dminbetween the points. A trial point is generated in a cuboid domain (cell) of dimensions a, b and c. When a trial point contravenes the minimum distance requirement to a previously placed point, the trial point is discarded and a new trial point is generated. This point placement is performed until a specified number of consecutive trial points are rejected for contravening the minimum distance criterion. The cell is then considered to be saturated with points down to the specified minimum spacing dmin.

To generate periodic cells, new points, called from now on image points, must be generated outside the cell. When a trial point with coordinate vector x = (x, y, z)T is accepted, the vector is used to generate 26 image points located outside the rectangular domain. The image points are generated based on the translation rule

where x0 is the coordinate vector of one of the image points, M is the translation matrix defined as M = diag[1 + kxa, 1 + kyb, 1 + kzc] where kx, ky, kz ∈ {−1, 0, 1}. The kx, ky and kz coefficients define the direction of the shift from the original point to the image point. The coordinates of the 26 image points are the result of the coordinate translations described by (3.1) for all kx, ky, kzcombinations except for the case where kx= ky = kz= 0.

A 2D schematic representation of the translation rule is presented in Figure 3.1 for the case of kx= −1 and ky= 1. The bold lines denote the borders of the cell within which the initial point x is generated and the dashed lines denote the two linearly independent translations of the coordinates of x to generate x0.

Figure 3.1: Point generation: 2D schematic representation of the generation of point x0 as the result of the translation of point x according to the translation rule (3.1) for kx= −1 and ky= 1.