The Distinct Element Method (DEM) and Discontinuous Deformation Analysis (DDA) are the two most commonly used discrete element methods in rock mechanics. Discrete element approaches are computationally expensive as they involve the interaction of multiple discrete bodies with continuously changing contacts. Therefore, it is very important to ensure that the method selected for the analysis is computationally efficient. In this research, a general assessment of DDA and DEM is performed from a computational efficiency perspective, and relevant enhancements to DDA are developed. The computational speed of DDA is observed to be considerably slower than DEM. In order to identify reasons affecting the computational efficiency of DDA, fundamental aspects of DDA and DEM are compared which suggests that they mainly differ in the contact mechanics, and the time integration scheme used. An in-depth evaluation of these aspects revealed that the openclose iterative procedure used in DDA which exhibits highly nonlinear behavior is one of the main reasons causing DDA to slow down. In order to improve the computational efficiency of DDA, an alternative approach based on a more realistic rock joint behavior is developed in this research. In this approach, contacts are assumed to be deformable, i.e., interpenetrations of the blocks in contact are permitted. This eliminated the computationally expensive open-close iterative procedure adopted in DDA-Shi and enhanced its speed up to four times. In order to consider deformability of the blocks in DDA, several approaches are reported. The hybrid DDA-FEM approach is one of them, although this approach captures the block deformability quite effectively, it becomes computationally expensive for large-scale problems. An alternative simplified uncoupled DDA-FEM approach is developed in this research. The main idea of this approach is to model rigid body movement and the block internal deformation separately. Efficiency and simplicity of this approach lie in keeping the DDA and the FEM algorithms separate and solving FEM equations individually for each block. Based on a number of numerical examples presented in this dissertation, it is concluded that from a computational efficiency standpoint, the implicit solution scheme may not be appropriate for discrete element modelling. Although for quasi-static problems where inertia effects are insignificant, implicit schemes have been successfully used for linear analyses, they do not prove to be advantageous for contact-type problems even in quasi-static mode due to the highly nonlinear behavior of contacts.

The shear rupture of massive (intact non-jointed) brittle rock in underground high stress mines occurs under a variety of different boundary conditions ranging from constant stress (no resistance to deformation) to constant stiffness (resistance to deformation). While a variety of boundary conditions exist, the shear rupture of massive rock in the brittle field is typically studied under constant stress boundary conditions. According to the theory, the fracturing processes leading to shear rupture zone creation occur at or near peak strength with a shear rupture surface created in the post-peak region of the stress-strain curve. However, there is evidence suggesting that shear rupture zone creation can occur pre-peak. Limited studies of shear rupture in brittle rock indicate pre-peak shear rupture zone creation under constant stiffness boundary conditions. This suggests that the boundary condition influences the shear rupture zone creation characteristics. In this thesis, shear rupture zone creation in brittle rock is investigated in direct shear under constant normal stress and normal stiffness boundary conditions. It is hypothesized that the boundary condition under which a shear rupture zone is created influences its characteristics (i.e., shear rupture zone geometry, load-displacement response, shear rupture zone creation relative to the load-displacement curve, and peak and ultimate strengths). In other words, it is proposed that the characteristics of a shear rupture zone are not only a function of the rock or rock mass properties but the boundary conditions under which the rupture zone is created. The hypothesis is tested and proven through a series of simulations using a two dimensional particle based Distinct Element Method (DEM) and its embedded grain based method. The understanding gained from these simulations is then used in the analysis and re-interpretation of rupture zone creation in two mine pillars. This is completed to show the value and practical application of the improved understanding gained from the simulations. The re-interpretation of these case histories suggests that one pillar ruptured predominately under a constant stress boundary condition while the other ruptured under a boundary condition changing from stiffness to stress control.

Blending processes in a silo minimise the fluctuations in the property of bulk solids with the blending performance being strongly influenced by the flow pattern and operating mode among other process parameters such as batch size and type of input fluctuations. An accurate prediction of flow characteristics such as flow channel boundary and velocity profiles is important for understanding and quantifying the blending performance, thereby increasing the scope for new design by minimising the number of expensive pilot scale experiments required. In this thesis, the Discrete Element Method (DEM) is deployed to predict and understand the flow characteristics and blending of cylindrical plastic pellets in a planar flat bottom silo and a multi-flow blender (a silo with an insert and a blending tube). The predictions are validated against high-resolution velocity measurements analysed using Particle Image Velocimetry (PIV) technique. A planar model silo was built to measure the flow of pellets using PIV technique. The existing GeoPIV Matlab module was customised to extract the velocity fields in the Eulerian frame of reference and its accuracy has been verified. The developed tool was then applied to quantitatively investigate the mechanism of evolution of flow in a flat bottom silo and the dependency of the state of developed flow on the depth of the planar silo. It was shown that the development of flow during discharge can be divided into two stages: a rapid upward propagation of plug flow followed by a widening of the flow channel with increasing shearing boundaries. The size of the flow channel was found to be increasing with the depth of the silo. For the 100 mm deep silo, the flow is three dimensional with significant retardation in velocity at the frontal walls, whilst a negligible retardation was found for the 20 and 40 mm deep model silos. The thickness and frontal wall friction in planar silos thus play an important role in the development of flow patterns in model silos. In this thesis, DEM model calibration relating the macro-scale bulk friction and micro- scale particle friction at different rolling friction values was developed from DEM simulations of Jenike direct shear box. During the direct shear simulation, a constant normal force was achieved with the use of a shear lid geometry made with glued spheres thereby eliminating the use of a traditional servo control function. The influence of particle rotations and rolling friction on the limiting bulk friction for different particle sliding friction coefficients was explored. The accuracy of the calibration data was assessed by simulating the flow in a flat bottom silo and comparing the model predictions of flow rate, velocity profiles and flow channel boundary with the experiments. A good quantitative agreement was found between the experiment and simulations. The DEM model predictions were also compared with the kinematic model. Following the validation of the model, it was shown that the frontal friction and rolling friction are the influential parameters in simulating the flow patterns such as semi-mass and internal flow. It was further shown that flow transits from semi-mass flow to internal flow with the increase of frontal wall friction. The drastic influence of frontal wall friction on stress, flow patterns and force chains were analysed highlighting its implications on interpretations in 2D test silos. Finally, the developed DEM and PIV tools are employed to investigate blending in a flat bottom and multi-flow blender silo for different flow patterns. The analysis showed that the blending is more effective with the internal flow when compared to semi-mass flow in a flat bottom silo, in both continuous and discontinuous modes for a variety of process conditions such as batch size, the number of recirculation and frequency of input fluctuations. An algorithm was developed to evaluate the blending performance from the spatially averaged Eulerian velocity fields. The flow in a relatively large-scale multi-flow blender comprising nearly 606,000 particles, thereby fully replicating the test silo, was simulated and the challenges in reproducing the test conditions of continuous and discontinuous modes of operation were discussed. The flow patterns and blending were first analysed from the experiments in different configurations of the insert. Using the same input parameters for the model, it was shown that the model predictions of the velocity profiles along the height of the silo are in good agreement with the experiments. Internal flow, mixed flow and mass flow were predicted for the diverging, straight and converging insert configurations respectively and the blending performance for each of these configurations suggests an optimal configuration of the blender thereby demonstrating the potential of PIV and DEM in design optimisation. The possibility of conducting the DEM simulations under increased gravity in order to reduce the computational time has also been explored.

O processo de dano em materiais quasi-frágeis pode ser caracterizado pela perda de isotropia para certos níveis de carga. A localização de deformações, o efeito cooperativo entre regiões danificadas e a avalanche de rupturas são características particulares na medição do dano neste tipo de material. As características mencionadas criam diferentes formas de dissipação de energia, que não são fáceis de representar utilizando métodos baseados na hipótese dos meios contínuos. No presente trabalho uma versão do Método dos Elementos Discretos Formado por Barras é empregado. Neste método a massa do contínuo é concentrada nos nós, os quais são interconectados por barras sem massa. Essas barras possuem uma lei constitutiva bilinear, que é usada para simular a ruptura da estrutura em estudo. A distribuição dos nós permite formar uma treliça tridimensional regular, e a partir dessa discretização espacial é possível chegar a um sistema de equações de movimento, que é resolvido com um esquema explícito de integração numérica (diferenças finitas centrais). Neste método a fratura e a fragmentação são levadas em conta de forma natural, já que as barras que rompem durante o processo são desativadas, respeitando o balanço energético. É possível introduzir heterogeneidade no modelo considerando as propriedades do material como campos espaciais aleatórios com distribuição de probabilidades de Weibull e comprimento de correlação conhecido. Nessa dissertação, é analisado o processo de dano que aparece em estruturas de geometria simples quando solicitadas até o colapso. Diferentes índices são apresentados para realizar a medição do dano. O desempenho desses índices, e a maneira com que eles ajudam na interpretação da evolução do dano, são discutidos nesse trabalho. / The process of damage in quasi-fragile materials is characterized by loss of isotropy for certain load levels. The strain localization, the cooperative effect between damaged regions and the avalanche of ruptures are particular features in measuring the damage in this kind of material. The mentioned features create different forms of energy dissipation, which are not easy to represent with a continuous approach. In the present work a version of the Lattice Discrete Element Method is employed. In this method the mass of the solid is concentrated on node points, which are interconnected by uniaxial elements. These elements have a bilinear constitutive law, which is used to simulate the rupture of the structure under study. The node distribution allows the formation of a regular three-dimensional lattice, and from this spatial discretization it is possible to arrive at a system of equations of motion, which is solved by an explicit numerical integration scheme (central difference). In this method the fracture and fragmentation are taken into account in a natural manner, since the bars that reached their limit strength during the process are disabled of the system, respecting the energy balance. It is possible to introduce heterogeneity in the model considering the material properties as random fields with spatial Weibull probability distribution and known correlation length. In this dissertation, the damage process, which appears in structures of simple geometry, when they are loaded until collapse, is analysed. Different indexes are presented to perform the measurement of the damage. The performance of those indexes, and the way they help in the interpretation of the damage evolution, are discussed in this paper.

A demanda por estruturas mais leves implica num ganho em economia, porém o aumento de esbeltez da estrutura pode tornar ela susceptível a instabilidade frente a tensões compressivas estáticas ou dinâmicas. A instabilidade acontece em várias escalas da estrutura analisada e pode interagir com outras formas de colapso como a propagação instável de fissuras, problema governado pela mecânica da fratura, pela plastificacão do material, ou por uma combinação dos efeitos citados. Neste contexto, no presente trabalho, se explora a capacidade do método dos elementos discretizados por barras (LDEM) na simulação de problemas de instabilidade estática e dinâmica devido as tensões de compressão. Este método permite simular o sólido como um arranjo de barras com rigidez equivalente ao contínuo que se quer representar. Leis constitutivas não lineares permitem modelar ruptura de forma simples. A equação de movimento resultante da discretização permite formular uma equação de movimento desacoplada que pode ser integrada no domínio do tempo com um método explícito (Método das Diferencias Finitas Centrais). O fato das barras serem rotuladas nos seus extremos e a solução do problema ser obtida de forma incremental permite capturar problemas com não linearidade geométrica, entre eles a instabilidade estrutural frente a tensões compressivas. Como último exemplo se realiza a análise de um painel sanduiche por flexão em três pontos, que é composto por um núcleo de poliuretano, com duas lâminas externas de material compósito, neste caso a instabilidade estrutural está associada a flambagem da camada da lâmina comprimida. Finalmente a potencialidade da metodologia de análise utilizada é discutida. / The demand for lighter structures implies a gain in economy, but the increase in slenderness of the structure may make it susceptible to instability against static or dynamic compressive stresses. Instability occurs at various scales of the analyzed structure and may interact with other forms of collapse such as unstable crack propagation, problem governed by fracture mechanics, plastification of the material, or a combination of the cited effects. In this context, in the present work, we explore the ability of the discrete elements methods by bars (LDEM) in the simulation of problems of static and dynamic instability due to the compression stresses. This method allows to simulate the solid as an arrangement of bars with rigidity equivalent to the continuum that one wants to represent. Constitutive non-linear laws allow simple modeling of rupture. The equation of motion resulting from the discretization allows us to formulate a decoupled motion equation that can be integrated in the time domain with an explicit method (Central Finite Differences Method). The fact that the bars are labeled at their ends and the solution of the problem is obtained in an incremental way allows to capture problems with geometric non-linearity, among them the structural instability against compressive tensions. The last example, the analysis of a sandwich panel by three-point bending, which is composed of a polyurethane core, with two external blades of composite material, in this case the structural instability is associated with buckling of the layer of the compressed blade . Finally, the potential of the analysis methodology is discussed.

abstract: In material science, microstructure plays a key role in determining properties, which further determine utility of the material. However, effectively measuring microstructure evolution in real time remains an challenge. To date, a wide range of advanced experimental techniques have been developed and applied to characterize material microstructure and structural evolution on different length and time scales. Most of these methods can only resolve 2D structural features within a narrow range of length scale and for a single or a series of snapshots. The currently available 3D microstructure characterization techniques are usually destructive and require slicing and polishing the samples each time a picture is taken. Simulation methods, on the other hand, are cheap, sample-free and versatile without the special necessity of taking care of the physical limitations, such as extreme temperature or pressure, which are prominent issues for experimental methods. Yet the majority of simulation methods are limited to specific circumstances, for example, first principle computation can only handle several thousands of atoms, molecular dynamics can only efficiently simulate a few seconds of evolution of a system with several millions particles, and finite element method can only be used in continuous medium, etc. Such limitations make these individual methods far from satisfaction to simulate macroscopic processes that a material sample undergoes up to experimental level accuracy. Therefore, it is highly desirable to develop a framework that integrate different simulation schemes from various scales to model complicated microstructure evolution and corresponding properties. Guided by such an objective, we have made our efforts towards incorporating a collection of simulation methods, including finite element method (FEM), cellular automata (CA), kinetic Monte Carlo (kMC), stochastic reconstruction method, Discrete Element Method (DEM), etc, to generate an integrated computational material engineering platform (ICMEP), which could enable us to effectively model microstructure evolution and use the simulated microstructure to do subsequent performance analysis. In this thesis, we will introduce some cases of building coupled modeling schemes and present the preliminary results in solid-state sintering. For example, we use coupled DEM and kinetic Monte Carlo method to simulate solid state sintering, and use coupled FEM and cellular automata method to model microstrucutre evolution during selective laser sintering of titanium alloy. Current results indicate that joining models from different length and time scales is fruitful in terms of understanding and describing microstructure evolution of a macroscopic physical process from various perspectives. / Dissertation/Thesis / Doctoral Dissertation Materials Science and Engineering 2018

A demanda por estruturas mais leves implica num ganho em economia, porém o aumento de esbeltez da estrutura pode tornar ela susceptível a instabilidade frente a tensões compressivas estáticas ou dinâmicas. A instabilidade acontece em várias escalas da estrutura analisada e pode interagir com outras formas de colapso como a propagação instável de fissuras, problema governado pela mecânica da fratura, pela plastificacão do material, ou por uma combinação dos efeitos citados. Neste contexto, no presente trabalho, se explora a capacidade do método dos elementos discretizados por barras (LDEM) na simulação de problemas de instabilidade estática e dinâmica devido as tensões de compressão. Este método permite simular o sólido como um arranjo de barras com rigidez equivalente ao contínuo que se quer representar. Leis constitutivas não lineares permitem modelar ruptura de forma simples. A equação de movimento resultante da discretização permite formular uma equação de movimento desacoplada que pode ser integrada no domínio do tempo com um método explícito (Método das Diferencias Finitas Centrais). O fato das barras serem rotuladas nos seus extremos e a solução do problema ser obtida de forma incremental permite capturar problemas com não linearidade geométrica, entre eles a instabilidade estrutural frente a tensões compressivas. Como último exemplo se realiza a análise de um painel sanduiche por flexão em três pontos, que é composto por um núcleo de poliuretano, com duas lâminas externas de material compósito, neste caso a instabilidade estrutural está associada a flambagem da camada da lâmina comprimida. Finalmente a potencialidade da metodologia de análise utilizada é discutida. / The demand for lighter structures implies a gain in economy, but the increase in slenderness of the structure may make it susceptible to instability against static or dynamic compressive stresses. Instability occurs at various scales of the analyzed structure and may interact with other forms of collapse such as unstable crack propagation, problem governed by fracture mechanics, plastification of the material, or a combination of the cited effects. In this context, in the present work, we explore the ability of the discrete elements methods by bars (LDEM) in the simulation of problems of static and dynamic instability due to the compression stresses. This method allows to simulate the solid as an arrangement of bars with rigidity equivalent to the continuum that one wants to represent. Constitutive non-linear laws allow simple modeling of rupture. The equation of motion resulting from the discretization allows us to formulate a decoupled motion equation that can be integrated in the time domain with an explicit method (Central Finite Differences Method). The fact that the bars are labeled at their ends and the solution of the problem is obtained in an incremental way allows to capture problems with geometric non-linearity, among them the structural instability against compressive tensions. The last example, the analysis of a sandwich panel by three-point bending, which is composed of a polyurethane core, with two external blades of composite material, in this case the structural instability is associated with buckling of the layer of the compressed blade . Finally, the potential of the analysis methodology is discussed.

Les travaux de cette thèse concernent le développement d’un modèle par éléments discrets du comportement thermo-mécanique d’un alliage d’aluminium et de son évolution microstructurale lorsqu’il est soumis à des conditions de grandes déformations et/ou de grandes vitesses de déformation (soudage par friction-malaxageou FSW). Le procédé de « soudage par friction malaxage » (Friction Stir Welding,FSW) est un procédé de soudage récent, inventé en 1991 par « The Welding Institute (TWI) ». Ce procédé se distingue des autres par sa capacité à souder la matière à l’état visqueux, sans atteindre le point de fusion. Le principe est relativement simple et comparable à une opération de fraisage, si ce n’est que l’outil coupant est remplacé par un outil malaxant composé d’un pion et d’un épaulement. Ce procédé revêt un fort intérêt pour l’industrie aéronautique, car il offre la possibilité de souder des alliages d’aluminium des séries 2XXX et 7XXX, réputés difficiles à souder par d’autres procédés.Le soudage par friction-malaxage étant un procédé récent, il demeure encore un sujet de recherche actif, pour mieux appréhender certains points comme l’écoulement de la matière, l’influence des paramètres du procédé (vitesse de rotation, forme de l’outil, ...)et la modification de la microstructure. Ce dernier point est particulièrement important pour les alliages à durcissement structural comme ceux des séries 2XXX et 7XXX. Pour ces alliages d’aluminium, les propriétés mécaniques dépendent étroitement de leur état de précipitation, lui-même influencé par l’apport de chaleur généré lors du soudage.L’optimisation des paramètres de soudage par rapport aux propriétés mécaniques passe obligatoirement par la prédiction de l’état de la précipitation le long du joint soudé. Le travail de la thèse vise à mettre en place des modèles à l’échelle de la microstructure permettant de prévoir la modification des propriétés de la matière liées aux paramètres du procédé. / The work of this thesis concerns the development of a model using the discrete element method, for the thermo-mechanical behavior of an aluminum alloy and its microstructural evolution when exposed to conditions of large deformations and/or high deformation rates (Friction StirWelding or FSW). Friction stir welding is a recent welding process invented in 1991 by "The welding Institute" (TWI). This process is different from the others by its ability to weld material in the viscous state, without reaching the melting point. The principle is relatively simple and similar to a milling process except that the cutting tool is replaced by a rotary tool composed of a pin and a shoulder. This process is of great interest in the aircraft industry because it offers the possibility to weld aluminum alloys 2XXX and 7XXX series, known to be difficult to weld by other processes. Since friction stir welding is a recent process, it still remains a subject of active research, to better understand certain issues such as material flow, influence of process parameters (rotation speed, tool shape, ...) and the modification of the microstructure. This last point is particularly important for structural hardening alloys such as those of the 2XXX and 7XXX series. For these aluminum alloys, the mechanical properties are highly dependent on their state of precipitation, which is itself influenced by the heat generated during thewelding. The optimization ofwelding parameters in order to obtain good mechanical properties requires the prediction of precipitation state along thewelded joint. Thework of this thesis is to develop required models at the scale of microstructure allowing to predict the modification of material properties related to the process parameters

O processo de dano em materiais quasi-frágeis pode ser caracterizado pela perda de isotropia para certos níveis de carga. A localização de deformações, o efeito cooperativo entre regiões danificadas e a avalanche de rupturas são características particulares na medição do dano neste tipo de material. As características mencionadas criam diferentes formas de dissipação de energia, que não são fáceis de representar utilizando métodos baseados na hipótese dos meios contínuos. No presente trabalho uma versão do Método dos Elementos Discretos Formado por Barras é empregado. Neste método a massa do contínuo é concentrada nos nós, os quais são interconectados por barras sem massa. Essas barras possuem uma lei constitutiva bilinear, que é usada para simular a ruptura da estrutura em estudo. A distribuição dos nós permite formar uma treliça tridimensional regular, e a partir dessa discretização espacial é possível chegar a um sistema de equações de movimento, que é resolvido com um esquema explícito de integração numérica (diferenças finitas centrais). Neste método a fratura e a fragmentação são levadas em conta de forma natural, já que as barras que rompem durante o processo são desativadas, respeitando o balanço energético. É possível introduzir heterogeneidade no modelo considerando as propriedades do material como campos espaciais aleatórios com distribuição de probabilidades de Weibull e comprimento de correlação conhecido. Nessa dissertação, é analisado o processo de dano que aparece em estruturas de geometria simples quando solicitadas até o colapso. Diferentes índices são apresentados para realizar a medição do dano. O desempenho desses índices, e a maneira com que eles ajudam na interpretação da evolução do dano, são discutidos nesse trabalho. / The process of damage in quasi-fragile materials is characterized by loss of isotropy for certain load levels. The strain localization, the cooperative effect between damaged regions and the avalanche of ruptures are particular features in measuring the damage in this kind of material. The mentioned features create different forms of energy dissipation, which are not easy to represent with a continuous approach. In the present work a version of the Lattice Discrete Element Method is employed. In this method the mass of the solid is concentrated on node points, which are interconnected by uniaxial elements. These elements have a bilinear constitutive law, which is used to simulate the rupture of the structure under study. The node distribution allows the formation of a regular three-dimensional lattice, and from this spatial discretization it is possible to arrive at a system of equations of motion, which is solved by an explicit numerical integration scheme (central difference). In this method the fracture and fragmentation are taken into account in a natural manner, since the bars that reached their limit strength during the process are disabled of the system, respecting the energy balance. It is possible to introduce heterogeneity in the model considering the material properties as random fields with spatial Weibull probability distribution and known correlation length. In this dissertation, the damage process, which appears in structures of simple geometry, when they are loaded until collapse, is analysed. Different indexes are presented to perform the measurement of the damage. The performance of those indexes, and the way they help in the interpretation of the damage evolution, are discussed in this paper.

O processo de dano em materiais quasi-frágeis pode ser caracterizado pela perda de isotropia para certos níveis de carga. A localização de deformações, o efeito cooperativo entre regiões danificadas e a avalanche de rupturas são características particulares na medição do dano neste tipo de material. As características mencionadas criam diferentes formas de dissipação de energia, que não são fáceis de representar utilizando métodos baseados na hipótese dos meios contínuos. No presente trabalho uma versão do Método dos Elementos Discretos Formado por Barras é empregado. Neste método a massa do contínuo é concentrada nos nós, os quais são interconectados por barras sem massa. Essas barras possuem uma lei constitutiva bilinear, que é usada para simular a ruptura da estrutura em estudo. A distribuição dos nós permite formar uma treliça tridimensional regular, e a partir dessa discretização espacial é possível chegar a um sistema de equações de movimento, que é resolvido com um esquema explícito de integração numérica (diferenças finitas centrais). Neste método a fratura e a fragmentação são levadas em conta de forma natural, já que as barras que rompem durante o processo são desativadas, respeitando o balanço energético. É possível introduzir heterogeneidade no modelo considerando as propriedades do material como campos espaciais aleatórios com distribuição de probabilidades de Weibull e comprimento de correlação conhecido. Nessa dissertação, é analisado o processo de dano que aparece em estruturas de geometria simples quando solicitadas até o colapso. Diferentes índices são apresentados para realizar a medição do dano. O desempenho desses índices, e a maneira com que eles ajudam na interpretação da evolução do dano, são discutidos nesse trabalho. / The process of damage in quasi-fragile materials is characterized by loss of isotropy for certain load levels. The strain localization, the cooperative effect between damaged regions and the avalanche of ruptures are particular features in measuring the damage in this kind of material. The mentioned features create different forms of energy dissipation, which are not easy to represent with a continuous approach. In the present work a version of the Lattice Discrete Element Method is employed. In this method the mass of the solid is concentrated on node points, which are interconnected by uniaxial elements. These elements have a bilinear constitutive law, which is used to simulate the rupture of the structure under study. The node distribution allows the formation of a regular three-dimensional lattice, and from this spatial discretization it is possible to arrive at a system of equations of motion, which is solved by an explicit numerical integration scheme (central difference). In this method the fracture and fragmentation are taken into account in a natural manner, since the bars that reached their limit strength during the process are disabled of the system, respecting the energy balance. It is possible to introduce heterogeneity in the model considering the material properties as random fields with spatial Weibull probability distribution and known correlation length. In this dissertation, the damage process, which appears in structures of simple geometry, when they are loaded until collapse, is analysed. Different indexes are presented to perform the measurement of the damage. The performance of those indexes, and the way they help in the interpretation of the damage evolution, are discussed in this paper.