The present work focusses on the development of a method that enables simulation of fragmentation of cohesive frictional materials using the discrete element method without fully resolving the microstructure. This requires modelling of phenomena on different geometric scales. In order to link these scales in a single simulation, coupling of the finite element method (FEM) and the discrete element method (DEM) is suggested which combines the efficiency of the FEM with the accuracy of the DEM. The proposed concept transfers ideas from the quasicontinuum method in the field of atomistics to problems in structural mechanics of cohesive frictional materials. Before describing the method and analysing the results of some numerical examples, different strategies for material modelling are discussed.
On different scales, materials show a different structure. Depending on the scale that the material is looked upon, different methods are applied for modelling. On the macroscopic scale, most materials can be considered continuous. When zooming in, many materials have a discrete microstructure. One example is concrete, where grains, which are also called aggregates, reside in a matrix of cement. On a very fine scale, the atomic scale, all materials are discrete. In order to accurately predict the behaviour of discrete materials until failure, mechanical models are required that are capable of representing the complex processes on the level of the discrete microstructure. Two appropriate methods are presented in this work: the discrete element method (Cundall und Strack 1979), which is often used for simulating granular or cohesive frictional materials, and the molecular dynamics (Alder und Wainwright (1957) and Alder und Wainwright (1959)) for simulations on the atomic scale. The large numerical effort limits the size of computable structures and therefore is a disadvantage of these methods.
On the macroscopic level, methods based on continuum mechanics descriptions like the finite element method (FEM) are efficient and reliable as long as the solution is relatively smooth. A precondition for the application of continuum methods is scale separation. Scale separation means that the dimensions of the global structure are much larger than the dimensions of the microstructure so that homogenisation methods can be applied. Localisation phenomena of the microstructure can only be represented in an average sense and not as detailed as with discrete methods. In addition to the finite element method (Turner u. a. (1956), Argyris (1960) and Zienkiewicz und Cheung (1967)), two other continuum methods are introduced: the smoothed particle hydrodynamics (Lucy (1977) und Gingold und Monaghan (1977)) and the material point method (Sulsky u. a. 1994).
To optimise accuracy and computational efficiency, an obvious idea is to adaptively combine discrete and continuous methods. Local phenomena like crack initiation and evolution require resolution with a discrete model. Therefore, a discrete method is used in areas where localisation appears. In regions where the solution is relatively smooth, a continuum method is sufficiently accurate. This allows for the simulation of large structures for which computations using a fully resolved microstructure exceed the available amount of memory. Coupling the two methods is especially efficient if the regions in which the microstructure needs to be fully resolved are relatively small and identified within an adaptive procedure. Various methods following this strategy may be found in previous publications. Three different concepts for coupling discrete and continuous methods are presented in this dissertation: The bridging domain method (Belytschko und Xiao 2003), the bridging scale method (Wagner und Liu 2003) and the quasicontinuum method (Tadmor u. a. 1996), all of which originate in the field of atomistics.
Based on the quasicontinuum method, a mesh and model adaptive method is developed in order to address issues in structural mechanics. Physical particles build the material of the examined structures on a small scale. Small and large scale of the considered materials differ by only one or two orders of magnitude, but not more. The computations should be started with a coarse finite element discretisation. The transition to the fully resolved discrete model should develop adaptively in the respective areas. The emphasis is not on the development of an advanced discrete element method for certain applications but rather on the adaptive strategy. Therefore, a simple form of the discrete element method is used to model the discrete fine scale, which is basically a lattice method. The nodes of a lattice made up of equilateral triangles represent the centres of round, equally sized and rigid particles. It is assumed this is a good approximation of the microstructure of a material. Hence, the numerical solution obtained with the lattice method represents the material behaviour in a realistic way: The finite element method provides the kinematics of the system and the lattice method drives the constitutive behaviour.
The finite element mesh with triangular elements is refined adaptively in areas of high strain gradients. A specific characteristic of the applied mesh refinement procedure is that new nodes are always placed at the centre of a particle in the microstructure. This ensures that at the end of mesh refinement the microstructure is fully resolved and each particle represents a FE node at the same time. Introducing three different element types, level 1 through level 3, furthermore provides an adaptive transition from the coarse scale (FEM) model to the fully resolved fine scale (DEM) model. The difference between the three kinds of elements is the method of computing the stiffness matrix, at which the microstructure is resolved with different accuracy. In regions with a coarse mesh (level 1) the material is represented by an equivalent continuum obtained from homogenisation. Where the mesh is resolved down to the small scale (level 3), the discrete particles interact according to the small scale model. Level 2 elements serve as transition between the scales.
Level 2 elements do not fully resolve the microstructure. However, for computation of the element stiffness matrix, every lattice member inside the element is considered individually. In case of a homogeneous microstructure in which all members of the regular lattice have the same material properties, the displacements of particles inside an element can be obtained from interpolation of the discrete particle movements (Cauchy-Born rule) at the FE nodes. With the particle displacements at hand, the potentials of the lattice members inside the element can be computed in dependence of the nodal displacements. The sum of all these potentials yields the overall potential of the element. The second derivative of the overall potential with respect to the nodal displacements in turn yields the stiffness matrix of the level 2 element.
When applying the Cauchy-Born rule in case of a heterogeneous distribution of material properties among the trusses of the microstructure, the trusses within an element are not in equilibrium. In this case, a different strategy is used. The stiffness matrix is computed from eigenvalues and eigenvectors. The eigenvectors represent a deformation state and the eigenvalues represent the stiffness of the element in the respective state. For determination of the eigenvectors, a homogeneous material law is assumed. The eigenvectors hence correspond to the eigenvectors of the stiffness matrix of a level 1 element. In order to obtain the six eigenvalues, a subproblem on the level of the microstructure is solved for each eigenmode. The eigenvectors are used as displacement boundary conditions for each state. Requiring equivalence of the energies in the fine scale and the coarse scale model provides an equation for computing the eigenvalue of the level 2 element in the applied eigenmode. Since the microstructures of all level 2 elements are independent from each other, the stiffness matrices of the level 2 elements can be computed in parallel. Applicability of the overall concept is demonstrated on four examples. All four examples feature a rectangular structure with a rhombus shaped hole in the centre which is stretched horizontally. In two of these examples, all lattice members share the same material properties (homogeneous case). In the remaining two examples, the material properties are distributed heterogeneously. In one of the two examples per case, a linear elastic material law is used for the lattice members, and in the second example, individual members can damage according to a uniaxial linear softening law. For building the heterogeneous model, an idealised section of concrete, built from round aggregates of different size, is projected on the abovementioned regular triangular lattice. The model becomes heterogeneous due to the fact that the lattice members are assigned different material properties, depending on their position (inside an aggregate, inside the matrix or building the interface). For linear elastic lattice members, stiffness is varied. In case of linear softening, strength is varied.
The linear elastic examples show that the adaptive procedure performs well and that the results of the adaptive and the fine scale simulations are in good agreement. In the homogeneous case, adaptive simulation uses many degrees of freedom less than the reference solution with the fully resolved microstructure. However, during adaptive computation, several steps are necessary for refining the mesh. Nevertheless, the adaptive simulation is three times faster than the fine scale computation (wall clock time). Due to the smaller number of degrees of freedom, less memory is required.
In case of a heterogeneous microstructure fine scale computation is faster than adaptive computation. Since aggregates are distributed in the whole domain, the complete microstructure is heterogeneous. The highest deformation gradients develop at the interface of aggregates and matrix. Hence, the mesh is refined in all these interfaces which leads to a huge number of degrees of freedom- even using the adaptive procedure. In combination with several refinement steps, which are required, computation time is extended. Despite the lack of efficiency, this example demonstrates that the heterogeneous microstructure is recognised and resolved by the refinement procedure and especially the level 2 elements. In this dissertation, parallelisation is not implemented for computing level 2 stiffness matrices.
The materialwise nonlinear examples also show that results obtained with the adaptive simulation are in good agreement with the numerical reference solution. The adaptive procedure resolves the heterogeneous microstructure as expected and a developing crack is directed around the stronger aggregates. In the heterogeneous case computation time can be reduced by about 59 % with the adaptive simulation compared to the fine scale computation. A reduction by 89 % is achieved in case of a homogeneous material.
The method developed is especially powerful in case localisation takes only place in a small region and continuum elements (level 1) can be used in large areas. Compared to the fine scale problem, many degrees of freedoms can be omitted. An efficient computation of large structures is possible without fully resolving the microstructure for a representation of phenomena on a microstructural level.