Experimental investigation of crack initiation and propagation in high- and gigacycle fatigue in titanium alloys by study of morphology of fracture

Fatigue (high- and gigacycle) crack initiation and its propagation in titanium alloys with coarse and fine grain structure are studied by fractography analysis of fracture surface. Fractured specimens were analyzed by interferometer microscope and electronic microscope to improve methods of monitoring of damage accumulation during fatigue test and verify the models for fatigue crack kinetics. Fatigue strength was estimated for high cycle fatigue (HCF) regime using the Luong method [1] by “in-situ” infrared scanning of the sample surface for the step-wise loading history for different grain size metals. Fine grain alloys demonstrated higher fatigue resistance for both HCF and gigacycle fatigue regimes. Fracture surface analysis for cylindrical samples was carried out using optical and electronic microscopy method. High resolution profilometry (interferometerprofiler New View 5010) data of fracture surface roughness allowed us to estimate scale invariance (the Hurst exponent) and to establish the existence of two characteristic areas of damage localization (different values of the Hurst exponent). Area 1 with diameter ~300 ?m has the pronounced roughness and is associated with damage localization hotspot. Area 2 shows less amplitude roughness, occupies the rest fracture surface and considered as the trace of the fatigue crack path corresponding to the Paris kinetics

MODEL OF GEOMEDIA CONTAINING DEFECTS: COLLECTIVE EFFECTS OF DEFECTS EVOLUTION DURING FORMATION OF POTENTIAL EARTHQUAKE FOCI

This paper describes the statistical thermo-dynamical evolution of an ensemble of defects in the geomedium in the field of externally applied stresses. The authors introduce ‘tensor structural’ variables associated with two specific types of defects, fractures and localized shear faults (Fig. 1). Based on the procedure for averaging of the structural variables by statistical ensembles of defects, a self-consistency equation is developed; it determines the dependence of the macroscopic tensor of defects-induced strain on values of external stresses, the original pattern and interaction of defects. In the dimensionless case, the equation contains only the parameter of structural scaling, i.e. the ratio of specific structural scales, including the size of defects and an average distance between the defects.The self-consistency equation yields three typical responds of the geomedium containing defects to the increasing external stress (Fig. 2). The responses are determined from values of the structural scaling parameter. The concept of non-equilibrium free energy for a medium containing defects, given similar to the Ginzburg-Landau decomposition, allowed to construct evolutionary equations for the introduced parameters of order (deformation due to defects, and the structural scaling parameter) and to explore their solutions (Fig. 3).It is shown that the first response corresponds to stable quasi-plastic deformation of the geomedium, which occurs in regularly located areas characterized by the absence of collective orientation effects. Reducing the structural scaling parameter leads to the second response characterized by the occurrence of an area of meta-stability in the behavior of the medium containing defects, when, at a certain critical stress, the orientation transition takes place in the ensemble of interacting defects, which is accompanied by an abrupt increase of deformation (Fig. 2). Under the given observation/averaging scale, this transition is manifested by localized cataclastic deformation (i.e. a set of weak earthquakes), which migrates in space at a velocity several orders of magnitude lower than the speed of sound, as a ‘slow’ deformation wave (Fig. 3). Further reduction of the structural scaling parameter leads to degeneracy of the orientation meta-stability and formation of localized dissipative defect structures in the medium. Once the critical stress is reached, such structures develop in the blow-up regime, i.e. the mode of avalanche-unstable growth of defects in the localized area that is shrinking eventually. At the scale of observation, this process is manifested as brittle fracturing that causes formation of a deformation zone, which size is proportional to the scale of observation, and corresponds to occurrence of a strong earthquake.On the basis of the proposed model showing the behavior of the geomedium containing defects in the field of external stresses, it is possible to describe main ways of stress relaxation in the rock massives – brittle large-scale destruction and cataclastic deformation as consequences of the collective behavior of defects, which is determined by the structural scaling parameter.Results of this study may prove useful for estimation of critical stresses and assessment of the geomedium status in seismically active regions and be viewed as model representations of the physical hypothesis about the uniform nature of deve­lopment of discontinuities/defects in a wide range of spatial scales

Scaling Analysis of Defect Induced Structure of A6061 Alloy at Dynamic Strain Localization

Plastic strain localization and fracture of dynamically loaded metallic samples, occurred during plug formation, are investigated. These processes are closely related to the instability of plastic flow and can be attributed to structural-scaling transitions in mesodefect ensembles. The multiscale nature of defect structure allows us to use the fractal concept for quantitative analysis of both the fracture surface and the inner structure of a deformed material. The scaling properties of fracture surfaces are established in terms of the roughness index (Hurst exponent) as the characteristics of strain localization and fracture

Absence of Normalizable Time-periodic Solutions for The Dirac Equation in Kerr-Newman-dS Black Hole Background

We consider the Dirac equation on the background of a Kerr-Newman-de Sitter black hole. By performing variable separation, we show that there exists no time-periodic and normalizable solution of the Dirac equation. This conclusion holds true even in the extremal case. With respect to previously considered cases, the novelty is represented by the presence, together with a black hole event horizon, of a cosmological (non degenerate) event horizon, which is at the root of the possibility to draw a conclusion on the aforementioned topic in a straightforward way even in the extremal case.Comment: 12 pages. AMS styl

МОДЕЛЬ ГЕОСРЕДЫ С ДЕФЕКТАМИ: КОЛЛЕКТИВНЫЕ ЭФФЕКТЫ РАЗВИТИЯ НЕСПЛОШНОСТЕЙ ПРИ ФОРМИРОВАНИИ ПОТЕНЦИАЛЬНЫХ ОЧАГОВ ЗЕМЛЕТРЯСЕНИЙ

This paper describes the statistical thermo-dynamical evolution of an ensemble of defects in the geomedium in the field of externally applied stresses. The authors introduce ‘tensor structural’ variables associated with two specific types of defects, fractures and localized shear faults (Fig. 1). Based on the procedure for averaging of the structural variables by statistical ensembles of defects, a self-consistency equation is developed; it determines the dependence of the macroscopic tensor of defects-induced strain on values of external stresses, the original pattern and interaction of defects. In the dimensionless case, the equation contains only the parameter of structural scaling, i.e. the ratio of specific structural scales, including the size of defects and an average distance between the defects.The self-consistency equation yields three typical responds of the geomedium containing defects to the increasing external stress (Fig. 2). The responses are determined from values of the structural scaling parameter. The concept of non-equilibrium free energy for a medium containing defects, given similar to the Ginzburg-Landau decomposition, allowed to construct evolutionary equations for the introduced parameters of order (deformation due to defects, and the structural scaling parameter) and to explore their solutions (Fig. 3).It is shown that the first response corresponds to stable quasi-plastic deformation of the geomedium, which occurs in regularly located areas characterized by the absence of collective orientation effects. Reducing the structural scaling parameter leads to the second response characterized by the occurrence of an area of meta-stability in the behavior of the medium containing defects, when, at a certain critical stress, the orientation transition takes place in the ensemble of interacting defects, which is accompanied by an abrupt increase of deformation (Fig. 2). Under the given observation/averaging scale, this transition is manifested by localized cataclastic deformation (i.e. a set of weak earthquakes), which migrates in space at a velocity several orders of magnitude lower than the speed of sound, as a ‘slow’ deformation wave (Fig. 3). Further reduction of the structural scaling parameter leads to degeneracy of the orientation meta-stability and formation of localized dissipative defect structures in the medium. Once the critical stress is reached, such structures develop in the blow-up regime, i.e. the mode of avalanche-unstable growth of defects in the localized area that is shrinking eventually. At the scale of observation, this process is manifested as brittle fracturing that causes formation of a deformation zone, which size is proportional to the scale of observation, and corresponds to occurrence of a strong earthquake.On the basis of the proposed model showing the behavior of the geomedium containing defects in the field of external stresses, it is possible to describe main ways of stress relaxation in the rock massives – brittle large-scale destruction and cataclastic deformation as consequences of the collective behavior of defects, which is determined by the structural scaling parameter.Results of this study may prove useful for estimation of critical stresses and assessment of the geomedium status in seismically active regions and be viewed as model representations of the physical hypothesis about the uniform nature of deve­lopment of discontinuities/defects in a wide range of spatial scales. В работе описана статистико-термодинамическая эволюция ансамбля дефектов в геосреде в поле внешнего приложенного напряжения. Авторами вводятся тензорные  структурные переменные, ассоциированные с двумя характерными типами дефектов: трещинами и локализованными сдвигами (рис. 1). Процедура осреднения структурных переменных по статистическому ансамблю дефектов позволила получить уравнение самосогласования, определяющее зависимость макроскопического тензора деформации, индуцированной дефектами, от величины внешних напряжений, исходной структуры и взаимодействия дефектов, которое в безразмерном случае содержит только один параметр – параметр структурного скейлинга. Параметр структурного скейлинга определяется отношением характерных структурных масштабов: размером дефектов и средним расстоянием между дефектами.В результате решения уравнения самосогласования получено три характерных реакции геосреды с дефектами на рост внешнего напряжения (рис. 2), которые определяются величиной параметра структурного скейлинга. Формулировка неравновесной свободной энергии для среды с дефектами в форме, аналогичной разложению Гинзбурга-Ландау, позволила записать эволюционные уравнения для введенных параметров порядка (деформации, обусловленной дефектами, и параметра структурного скейлинга) и исследовать их собственные  решения (рис. 3).Показано, что первая реакция соответствует устойчивому квазипластическому деформированию среды, локализованному в регулярно расположенных пространственных областях, характеризующихся отсутствием коллективных ориентационных эффектов. Уменьшение параметра структурного скейлинга приводит ко второй реакции, которая характеризуется появлением области метастабильности в поведении среды с дефектами, когда при некотором критическом напряжении происходит ориентационный переход в ансамбле взаимодействующих дефектов, сопровождающийся резким скачком деформации (рис. 2). При этом на масштабе наблюдения (осреднения) этот переход проявляется в виде локализованной катакластической деформации (множества слабых землетрясений), мигрирующей по пространству со скоростью, на порядки меньшей скорости звука – «медленной» деформационной волны (рис. 3). Дальнейшее уменьшение параметра структурного скейлинга приводит к вырождению ориентационной метастабильности и формированию в среде локализованных диссипативных дефектных структур, которые при достижении критического напряжения развиваются в режиме с обострением – режиме лавинно-неустойчивого роста дефектов в локализованной пространственной области, уменьшающейся с течением времени. На масштабе наблюдения этот процесс проявляется в виде хрупкого разрушения с формированием зоны разрушения, соизмеримой с самим масштабом наблюдения, и соответствует появлению сильного землетрясения.Таким образом, построенная модель поведения геосреды с дефектами в поле внешних напряжений позволяет описать основные способы релаксации напряжений массивами горных пород: хрупкое крупномасштабное разрушение и катакластическое деформирование, которые являются следствиями коллективного поведения дефектов, определяемого величиной параметра структурного скейлинга.Полученные результаты могут быть полезны для оценки критических напряжений и состояний геосреды в сейсмоактивных районах, а также могут рассматриваться как модельные представления физической гипотезы о единстве природы развития несплошностей (дефектов) на широком спектре пространственных масштабов