This dissertation presents three studies addressing various modeling and computational aspects of lattice structures. The first study is concerned with characterization of the threshold behavior for very slow (subcritical) crack growth. First, it is shown that this behavior requires the presence of a healing mechanism. Then thermodynamic analysis of brittle fracture specimens near the threshold developed by Rice (1978) is extended to specimens undergoing microstructural changes. This extension gives rise to a generalization of the threshold concept that mirrors the way the resistance R-curve generalizes the fracture toughness. In the absence of experimental data, the resistance curve near the threshold is constructed using a lattice model that includes healing and rupture mechanisms. The second study is concerned with transmission of various boundary conditions through irregular lattices. The boundary conditions are parameterized using trigonometric Fourier series, and it is shown that, under certain conditions, transmission through irregular lattices can be well approximated by that through classical continuum. It is determined that such transmission must involve the wavelength of at least 12 lattice spacings; for smaller wavelength classical continuum approximations become increasingly inaccurate. Also it is shown that this restriction is much more severe than that associated with identifying the minimum size for representative volume elements. The third study is concerned with extending the use of boundary algebraic equations to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as boundary algebraic equations provide seamless bridging between discrete and continuum models. It is shown that, in contrast to regular lattices, boundary algebraic equations for irregular lattices require a statistical rather than deterministic treatment. Furthermore, boundary algebraic equations for irregular lattices contain certain terms that require the same amount of computational effort as the original problem. / text

In systems subjected to long-term loading, subcritical crack growth is the principal mechanism causing the time-dependent deformation and failure of rocks. Subcritical crack growth is environmentally-assisted crack growth, which can allow cracks to grow over a long period of time at stresses far smaller than their failure strength and at tectonic strain rates. The characteristics of subcritical crack growth can be described by a relationship between the stress intensity factor and the crack velocity. This study presents the results of studies conducted to validate the constant stress-rate test for determining subcritical crack growth parameters in Coconino sandstone, compared with the conventional testing method, the double torsion test. The results of the constant stress-rate test are in good agreement with the results of double torsion test. More importantly, the stress-rate tests can determine the parameter A with a much smaller standard deviation than the double torsion test. Thus the constant stress-rate test seems to be both a valid and preferred test method for determining the subcritical crack growth parameters in rocks. We investigated statistical aspects of the constant stress-rate test. The effects of the number of tests conducted on the subcritical crack growth parameters were examined and minimum specimen numbers were determined. The mean and standard deviation of the subcritical crack growth parameters were obtained by randomly selecting subsets from the original strength data. In addition, the distribution form of the subcritical crack growth parameters and the relation between the parameter n and A were determined. We extended the constant stress-rate test technique to modes II and III subcritical crack growth in rocks. The experimental results of the modes I, II and III tests show that the values of the subcritical crack growth parameters are similar to each other. The subcritical crack growth parameter n value for Coconino sandstone has the range of 34 to 38 and the parameter A has the range of 1.02x10⁻² to 6.52x10⁻² m/s. The effect of confining stress, specimen size, and water saturation on subcritical crack growth under mode II loading has also been investigated. Finally strength parameters for Coconino sandstone were determined experimentally, including tensile strength, uniaxial compressive strength, cohesion, internal friction angle, in-plane / our-of-plane shear strength and the fracture toughness under mode I, II, and III loading.

The primary focus of this research is to evaluate the time-dependent crack growth in rocks using lab tests and numerical modeling and its application to geologic hazard problems. This research utilized Coconino sandstone and Columbia granite as the study materials and produced the subcritical crack growth parameters in both mode I and II loadings using the rock materials. The mode I loading test employs three different types of fracture mechanics tests: the Double Torsion (DT), the Wedge Splitting (WS), and the Double Cantilever Beam (DCB) test. Each test measured the mode I crack velocity. The DT test indirectly measured the crack velocity using the load relaxation method. The WS and DCB tests directly measured the crack velocity by monitoring using a video recording. The different mode I subcritical crack growth parameters obtained from the three tests are discussed. For the mode II loading test, this study developed a new shear fracture toughness test called the modified Punch-Through Shear (MPTS). The MPTS test conducted at different loading rates produced the mode II subcritical crack growth parameters. These fracture mechanics tests were calibrated and simulated using the distinct element method (DEM) and the finite element method (FEM). DEM analysis employed the particle flow code (PFC) to simulate the mixed mode crack growth and to match with the failure strength envelop of the triaxial compressive tests. FEM analysis employed the Phase2 program to analyze the crack tip stress distribution and the FRANC2D program to calculate the modes I and II stress intensity factors. The fracture mechanics tests and numerical modeling showed well the dependency of the mode II subcritical crack growth parameters according to confining pressure, loading rate, and the mode II fracture toughness. Finally, the UDEC modeling based on DEM is utilized in this study to forecast the long-term stability of the Coconino rock slope, as one of geologic hazards. The fracture mechanics approach is implemented in the program using the modes I and II subcritical crack growth parameters obtained from the lab tests and numerical modeling. Considering the progressive failure of rock bridges due to subcritical crack growth, the UDEC results predicted the stable condition of the Coconino rock cliff over 10,000 years. This result was validated by comparing it with the previous planar failure case.

No description available.

A lifetime prediction scheme is proposed based on the assumption that the lifetime (time to failure) of rocks under load is governed by the growth of microstructual defects (microcracks). The numerical approach is based on linear elastic fracture mechanics. The numerical calculation scheme is implemented as a cellular automat, where each cell contains a microcrack with length and orientation following certain distributions. The propagation of the microcrack is controlled by the Charles equation, based on subcritical crack growth. The zone inside the numerical model fails if the microcrack has reached the zone dimension or the stress intensity factor of the crack reached the fracture toughness. Macroscopic fractures are formed by these coalesced propagating microcracks, and finally lead to failure of the model. In the numerical approaches, elasto-plastic stress redistributions take place during the forming of the macroscopic fractures. Distinct microcrack propagation types have been programmed and applied to the proposed numerical models. These numerical models are studied under different loading conditions. Numerical results with excellent agreement with the analytical solutions are obtained with respective to predicted lifetime, important parameters for the microcracks, fracture pattern and damage evolution. Potential applications of the proposed numerical model schemes are investigated in some preliminary studies and simulation results are discussed. Finally, conclusions are drawn and possible improvements to the numerical approaches and extensions of the research work are given. / 本文认为微结构缺陷（微裂纹）的扩展决定了受力岩石的寿命（破坏时间）。基于此假设，提出了岩石寿命预测方法。利用线弹性断裂力学理论，通过FLAC进行了数值模拟。数值模型中每个单元定义一条初始裂纹，其长度与方向服从特定分布。基于亚临界裂纹扩展理论，由Charles方程决定微裂纹的扩展（速度）。如微裂纹发展至单元边界，或应力强度系数到达断裂韧度，则单元破坏。宏观裂纹由微裂纹所联合形成，并最终贯穿模型导致破坏。在形成宏观裂纹的过程中，发生弹塑性应力重分布。在数值模型中，编制了不同类型的微裂纹扩展方式，并在不同的受力条件下加以分析。数值模型的岩石寿命，裂纹形状，破坏方式以及一些重要的参数的数值模拟结果与解析解有较好的一致性。对本文所提出的数值模型的初步实际应用进行了分析，并讨论了计算结果。最后讨论了本文所提出的岩石寿命预测方法的可能改良与发展，并对进一步的研究工作给出建议。

Assessing long-term rock stability is an important aspect in the analysis of slopes, dam and bridge foundations, and other infrastructure. Rock behavior over tens to thousands of years must be anticipated when predicting the performance of, for example, an underground containment facility for nuclear waste. At such long time scales, the time dependence of rock failure, typically ignored in short time scale analyses, has a significant effect and must be included in the analysis. Since time-dependent rock behavior is thought to be caused by the subcritical growth of microcracks, a time-dependent analysis should incorporate a method of simulating subcritical crack growth. In this thesis, a rock bridge damage model was developed using the finite element program Abaqus to simulate subcritical crack growth for all three modes of crack tip displacement in three-dimensional rock masses. Since subcritical crack growth is not among the damage initiation and evolution criteria available in Abaqus, its effect was included in the model through the USDFLD user subroutine. Material properties for the damage model were obtained through laboratory fracture toughness testing of Escabrosa limestone from Kartchner Caverns. Tests included the grooved disk test for mode I, the punch-through shear with confining pressure test for mode II, and the circumferentially-notched cylindrical specimen test for mode III. The subcritical crack growth parameters n and A were calculated for all three modes using the constant stress-rate method. Fracture test results were compared with a previous study by Tae Young Ko at the University of Arizona, which tested Coconino sandstone and determined that the subcritical crack growth parameters were consistent among modes. This thesis expands upon Ko's work by adding the characterization of a second rock material in all three modes; results indicate that for Escabrosa limestone the subcritical crack growth parameters are not consistent among modes. Additionally, the Escabrosa limestone composing the caverns ranges from a more homogeneous, even-grained texture to a more heterogeneous texture consisting of coarse-grained veins and solution cavities set in a fine-grained matrix. To determine if the veined regions are more susceptible to fracturing and act as the nuclei of rock bridge failure, the fracture toughness tests were conducted separately for each texture. Results indicate that the more heterogeneous limestone has a higher fracture strength, fracture toughness, and subcritical crack growth index n than the more homogeneous limestone. This is in agreement with previous studies that determined that a more complex and heterogeneous microstructure produces a larger microcrack process zone and a more tortuous crack path, leading to higher fracture energies and larger values of n. Application of the rock bridge damage model to a simplified Kartchner cave room with a single roof block provided visualization of decreasing rock bridge size and produced time-to-failure estimates of 1,251 to 65,850 years. Multiple models were run to study the effect of (i) using material properties from each of the two textures identified in the Escabrosa limestone and (ii) varying the in-situ stress ratio, K. Both the value of K and the choice of Escabrosa texture had a large effect on the estimated time-to-failure, indicating that for future modeling of Kartchner accurate estimation of the in-situ stress ratio is as important as field identification of homogeneous vs. heterogeneous textures.

O objetivo desta pesquisa foi verificar a influência de diferentes tamanhos de partículas de carga na tenacidade à fratura (KIc), nos parâmetros do crescimento subcrítico de trinca (n e f0) e de Weibull (m e 0), na longevidade estimada pelo diagrama tensão-probabilidade-tempo (SPT) e no limite de fadiga cíclica (LFC) de compósitos resinosos experimentais. Quatro compósitos foram preparados contendo 78% em massa (59% em volume) de conteúdo inorgânico, constituído por 67% de pó de vidro com diferentes tamanhos de partículas (d50 = 0,5; 0,9; 1,2 e 1,9 µm) e 11% de sílica pirogênica. Dados de KIc obtidos pelo método single-edge notched beam (25x5x2,8 mm; n=15) foram submetidos a ANOVA/teste de Tukey (p < 0,05). n e f0 foram determinados através do ensaio de fadiga dinâmica (10-2 a 102 MPa/s) utilizando um dispositivo de flexão biaxial (12x1,2 mm; n=10). Para determinar m e 0, mais 20 espécimes de cada compósito foram testados na taxa de 100 MPa/s. Os diagramas SPT foram obtidos a partir dos dados da fadiga dinâmica e análise de Weibull. No ensaio de fadiga cíclica, um dispositivo de flexão biaxial (12x1,2 mm) foi utilizado para se obter a resistência à flexão inicial (RFI; n=14) e o LFC (n=20). LFC foi obtido pelo método escada após 105 ciclos. Para todos os testes, os espécimes foram armazenados em água destilada a 37oC por 24h. Foi realizada a fractografia dos espécimes fraturados nas taxas 10-2 e 10-1 MPa/s da fadiga dinâmica e nos ensaios para determinação da RFI e LFC. Houve relação direta entre d50 e KIc (C0,5: 1,2±0,1b; C0,9: 1,3±0,1ab; C1,2: 1,3±0,1ab; C1,9: 1,4±0,2a, em MPa.m0,5). C0,5 (31,2±6,2a) e C1,9 (34,7±7,4a) apresentaram valores de n superiores a C0,9 (20,3±3,0b) e C1,2 (17,3±1,8b). C1,2 (166,42±0,01a) apresentou o maior valor de f0 (em MPa), seguido pelo C1,9 (159,82±0,02b), C0,9 (159,59±0,02c) e C0,5 (158,40±0,02d). Não houve diferença estatística entre os valores de m (6,6 a 10,6) e 0 (170,6 a 176,4 MPa) dos compósitos. As reduções na tensão de fratura para uma probabilidade de falha de 5% após 10 anos estimadas pelo diagramas SPT foram de aproximadamente 22% para C0,5 e C1,9 e 36% para C0,9 e C1,2. Não houve diferença estatística entre as médias de RFI (155,4 a 170,7 MPa). C0,5 (93,0±18,6a) apresentou o maior LFC (em MPa), seguido pelo C1,2 (91,8±11,1ab), C1,9 (87,2±3,0b) e C0,9 (82,5±8,0c). Defeitos sub-superficiais e superficiais foram as principais origens de fratura. A trinca se propagou pela matriz polimérica ao redor das partículas (deflexão de trinca) e todas as superfícies apresentaram características de fratura frágil. Como conclusão, compósitos com partículas maiores apresentaram maior KIC, enquanto que partículas menores contribuíram para um maior LFC. Compósitos com distribuição granulométrica mais ampla, independentemente do d50, apresentaram maior resistência ao SCG. Nos demais parâmetros e propriedades avaliados (m, 0 e RFI) não houve influência do tamanho das partículas. / The aim of this study was to verify the influence of different filler sizes in the fracture toughness (KIc), subcritical crack growth (n e f0) and Weibull (m e 0) parameters, longevity estimated by the strength-probability-time (SPT) diagram and cyclic fatigue limit (CFL) of experimental resin composites. Four composites were prepared, each one containing 78 w% (59 vol%) of inorganic content, in which 67 w% were glass powder with different filler sizes (d50 = 0.5; 0.9; 1.2 e 1.9 µm) and 11 w% were pyrogenic silica. KIc data was obtained by the single-edge notched beam test and submitted to ANOVA/Tukey tests (p < 0.05). n and f0 were determined by the dynamic fatigue test (10-2 a 102 MPa/s) using a biaxial flexural device (12x1.2 mm; n=10). 20 specimens of each composite were tested at 100 MPa/s to determine Weibull parameters. SPT diagrams were constructed using the dynamic fatigue and Weibull data. For the cyclic fatigue test, a biaxial flexural device (12x1.2 mm) was used to obtain the initial flexural strength (IFS; n=14) and CFL (n=20). CFL was determined by staircase method after 105 cycles. For all tests, the specimens were stored in distilled water at 37oC for 24h. It was done the fractography of the fractured specimens that was subjected to the 10-2 e 10-1 MPa/s rates of the dynamic fatigue and to the IFS and CFL tests. There was a direct relation between d50 e KIc (C0.5: 1.2±0.1b; C0.9: 1.3±0.1ab; C1.2: 1.3±0.1ab; C1.9: 1.4±0,2a, in MPa.m0,5). C0.5 (31.2±6.2a) and C1.9 (34.7±7.4a) presented higher n values than C0.9 (20.3±3.0b) and C1.2 (17.3±1.8b). C1.2 (166.42±0.01a) showed the highest f0 value (in MPa), followed by C1.9 (159.82±0.02b), C0.9 (159.59±0.02c) and C0.5 (158.40±0.02d). There were no statistical differences among the m (6.6 to 10.6) and 0 (170.6 to 176.4 MPa) values of the composites. The reductions in fracture stress at 5% failure probability for a lifetime of 10 years estimated by the SPT diagrams were approximately 22% for C0.5 and C1.9 and 36% for C0.9 and C1.2. There were no statistical differences among the IFS means (155.4 to 170.7 MPa). C0.5 (93.0±18.6a) showed the highest CFL (in MPa), followed by C1.2 (91.8±11.1ab), C1.9 (87.2±3.0b) and C0.9 (82.5±8.0c). Near-surface and surface flaws were the main fracture origins. The crack propagated by the polymeric matrix around the fillers (crack deflection) and all the fracture surfaces showed brittle fracture features. As conclusion, composites with large fillers presented the highest KIC, while the small fillers contributed to increase the CFL. Composites with broader granulometric size distribution, regardless of d50, showed higher resistance to SCG. There was no influence of the composites filler sizes in the others parameters and properties evaluated (m, 0 and IFS).

O objetivo desta pesquisa foi verificar a influência de diferentes tamanhos de partículas de carga na tenacidade à fratura (KIc), nos parâmetros do crescimento subcrítico de trinca (n e f0) e de Weibull (m e 0), na longevidade estimada pelo diagrama tensão-probabilidade-tempo (SPT) e no limite de fadiga cíclica (LFC) de compósitos resinosos experimentais. Quatro compósitos foram preparados contendo 78% em massa (59% em volume) de conteúdo inorgânico, constituído por 67% de pó de vidro com diferentes tamanhos de partículas (d50 = 0,5; 0,9; 1,2 e 1,9 µm) e 11% de sílica pirogênica. Dados de KIc obtidos pelo método single-edge notched beam (25x5x2,8 mm; n=15) foram submetidos a ANOVA/teste de Tukey (p < 0,05). n e f0 foram determinados através do ensaio de fadiga dinâmica (10-2 a 102 MPa/s) utilizando um dispositivo de flexão biaxial (12x1,2 mm; n=10). Para determinar m e 0, mais 20 espécimes de cada compósito foram testados na taxa de 100 MPa/s. Os diagramas SPT foram obtidos a partir dos dados da fadiga dinâmica e análise de Weibull. No ensaio de fadiga cíclica, um dispositivo de flexão biaxial (12x1,2 mm) foi utilizado para se obter a resistência à flexão inicial (RFI; n=14) e o LFC (n=20). LFC foi obtido pelo método escada após 105 ciclos. Para todos os testes, os espécimes foram armazenados em água destilada a 37oC por 24h. Foi realizada a fractografia dos espécimes fraturados nas taxas 10-2 e 10-1 MPa/s da fadiga dinâmica e nos ensaios para determinação da RFI e LFC. Houve relação direta entre d50 e KIc (C0,5: 1,2±0,1b; C0,9: 1,3±0,1ab; C1,2: 1,3±0,1ab; C1,9: 1,4±0,2a, em MPa.m0,5). C0,5 (31,2±6,2a) e C1,9 (34,7±7,4a) apresentaram valores de n superiores a C0,9 (20,3±3,0b) e C1,2 (17,3±1,8b). C1,2 (166,42±0,01a) apresentou o maior valor de f0 (em MPa), seguido pelo C1,9 (159,82±0,02b), C0,9 (159,59±0,02c) e C0,5 (158,40±0,02d). Não houve diferença estatística entre os valores de m (6,6 a 10,6) e 0 (170,6 a 176,4 MPa) dos compósitos. As reduções na tensão de fratura para uma probabilidade de falha de 5% após 10 anos estimadas pelo diagramas SPT foram de aproximadamente 22% para C0,5 e C1,9 e 36% para C0,9 e C1,2. Não houve diferença estatística entre as médias de RFI (155,4 a 170,7 MPa). C0,5 (93,0±18,6a) apresentou o maior LFC (em MPa), seguido pelo C1,2 (91,8±11,1ab), C1,9 (87,2±3,0b) e C0,9 (82,5±8,0c). Defeitos sub-superficiais e superficiais foram as principais origens de fratura. A trinca se propagou pela matriz polimérica ao redor das partículas (deflexão de trinca) e todas as superfícies apresentaram características de fratura frágil. Como conclusão, compósitos com partículas maiores apresentaram maior KIC, enquanto que partículas menores contribuíram para um maior LFC. Compósitos com distribuição granulométrica mais ampla, independentemente do d50, apresentaram maior resistência ao SCG. Nos demais parâmetros e propriedades avaliados (m, 0 e RFI) não houve influência do tamanho das partículas. / The aim of this study was to verify the influence of different filler sizes in the fracture toughness (KIc), subcritical crack growth (n e f0) and Weibull (m e 0) parameters, longevity estimated by the strength-probability-time (SPT) diagram and cyclic fatigue limit (CFL) of experimental resin composites. Four composites were prepared, each one containing 78 w% (59 vol%) of inorganic content, in which 67 w% were glass powder with different filler sizes (d50 = 0.5; 0.9; 1.2 e 1.9 µm) and 11 w% were pyrogenic silica. KIc data was obtained by the single-edge notched beam test and submitted to ANOVA/Tukey tests (p < 0.05). n and f0 were determined by the dynamic fatigue test (10-2 a 102 MPa/s) using a biaxial flexural device (12x1.2 mm; n=10). 20 specimens of each composite were tested at 100 MPa/s to determine Weibull parameters. SPT diagrams were constructed using the dynamic fatigue and Weibull data. For the cyclic fatigue test, a biaxial flexural device (12x1.2 mm) was used to obtain the initial flexural strength (IFS; n=14) and CFL (n=20). CFL was determined by staircase method after 105 cycles. For all tests, the specimens were stored in distilled water at 37oC for 24h. It was done the fractography of the fractured specimens that was subjected to the 10-2 e 10-1 MPa/s rates of the dynamic fatigue and to the IFS and CFL tests. There was a direct relation between d50 e KIc (C0.5: 1.2±0.1b; C0.9: 1.3±0.1ab; C1.2: 1.3±0.1ab; C1.9: 1.4±0,2a, in MPa.m0,5). C0.5 (31.2±6.2a) and C1.9 (34.7±7.4a) presented higher n values than C0.9 (20.3±3.0b) and C1.2 (17.3±1.8b). C1.2 (166.42±0.01a) showed the highest f0 value (in MPa), followed by C1.9 (159.82±0.02b), C0.9 (159.59±0.02c) and C0.5 (158.40±0.02d). There were no statistical differences among the m (6.6 to 10.6) and 0 (170.6 to 176.4 MPa) values of the composites. The reductions in fracture stress at 5% failure probability for a lifetime of 10 years estimated by the SPT diagrams were approximately 22% for C0.5 and C1.9 and 36% for C0.9 and C1.2. There were no statistical differences among the IFS means (155.4 to 170.7 MPa). C0.5 (93.0±18.6a) showed the highest CFL (in MPa), followed by C1.2 (91.8±11.1ab), C1.9 (87.2±3.0b) and C0.9 (82.5±8.0c). Near-surface and surface flaws were the main fracture origins. The crack propagated by the polymeric matrix around the fillers (crack deflection) and all the fracture surfaces showed brittle fracture features. As conclusion, composites with large fillers presented the highest KIC, while the small fillers contributed to increase the CFL. Composites with broader granulometric size distribution, regardless of d50, showed higher resistance to SCG. There was no influence of the composites filler sizes in the others parameters and properties evaluated (m, 0 and IFS).

Fracture mechanics is an effective approach for characterizing material resistance to interfacial failure and for making interface reliability predictions. Because interfacial bond integrity is a major concern for performance and reliability, the need to evaluate the fracture and delamination resistance of an interface under different environmental conditions is very important. This study investigates the effects of temperature, solution chemistry and environmental preconditioning, in several solutions on the durability of silicon/epoxy and glass/epoxy systems. A series of experiments was conducted using wedge test specimens to investigate the adhesion performance of the systems subjected to a range of environmental conditions. Both silicon and glass systems were relatively insensitive to temperature over a range of 22-60°C, but strongly accelerated by temperatures above 60°C, depending on the environmental chemistry and nature of the adhesive system used. Silicon/commercial epoxy specimens were subjected to preconditioning in deionized (DI) water and more aggressive solution mixtures prior to wedge insertion to study the effect of prior environmental exposure time on the system. The wedge test data from preconditioned specimens were compared with standard wedge test results and the system was insensitive to preconditioning in DI water but was affected significantly by preconditioning in aggressive environments. Plots describing - G (crack velocity versus applied strain energy release rate) characteristics for a particular set of environmental conditions are presented and a comparison is made for different environmental conditions to quantify the subcritical debonding behavior of systems studied. A kinetic model to characterize subcritical debonding of adhesives for microelectronic applications is also proposed based on molecular interactions between epoxy and a silane coupling agent at the interface and linear elastic fracture mechanics, which could help predict long-term deterioration of interfacial adhesion. / Master of Science

Subcritical crack growth in composite materials plays quite an important role in determination of the service life and leads to the defects growth below the fracture toughness. The thesis presents search of the several scientific works dealing with the determination of the crack growth direction in composite materials. The aim of this diploma thesis was to create 2D and 3D finite element models to analyse crack growth in Low Temperature Co-fired Ceramics (LTCC). These models were compared based on the time calculation of the crack propagation under the subcritical growth conditions.