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Fracture Mechanics M. Janssen



Description of the CourseLecture, four hours; laboratory, two hours; outside study, four hours. Requisite: course 143A. Engineering and scientific aspects of crack nucleation, slow crack growth, and unstable fracture. Fracture mechanics, dislocation models, fatigue, fracture in reactive environments, alloy development, fracture-safe design. Letter grading.




Fracture Mechanics M. Janssen


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This book provides a thorough introduction to the subject of fracture mechanics, covering the basic concepts of both the linear elastic and elastic-plastic regimes. The development of failure assessment based on elastic-plastic fracture mechanics is comprehensively examined.


A number of chapters are devoted to the fracture mechanics characterisation of crack growth, and special attention is paid to the important topic of the initiation and growth of short fatigue cracks. Sustained load fracture and dynamic crack growth are also discussed, including various test techniques for determining their presence.


Images were taken of the fracture surfaces of each sample using a Zeiss Ultra Plus scanning electron microscope (5 kV; Oberkochen, Germany). Cuticle dimensions (radius and thickness at various locations around the circumference, cut length and cut depth) were measured using Fiji software (an open source image processing package based on ImageJ). Control subjects received no incision but were otherwise treated in the same manner; they were tested at times varying from 3 to 63 days post-moult.


where a is the half-length of the cut measured around the circumference and Q is a factor which depends on a, r and t (given by Takahashi [10]). Individual measurements of radius, thickness and cut length were taken for each sample. These were used to calculate the failure strength for each test, a unique shape factor Q for each injured leg, and hence the fracture toughness for each sample.


Using the experimental values for fracture toughness and nominal strength reported below gives Γ = 0.101 mm, so we used a mesh element size of 0.1 mm throughout the model. Stress intensity (K) values were calculated using a standard approach (described in [12]) in which the length of the crack is extended by a small amount δa and the change in deformation of the sample under load is used to calculate the change in stored strain energy in the body, δW. K can then be found by


SEM photos of repaired and non-repaired cuticle. Panels (a) and (b) show the fracture surface of a tibia that has received an incision (dashed line) and subsequently repaired the area by depositing cuticle (enhanced on the photo by red shading). At high magnification (b), there is a clear difference between the scalpel-cut surface and the fracture surface of the new cuticle. Panels (c) and (d) show the fracture surface of a tibia in which repair did not occur. Panels (e) and (f) are longitudinal sections of tibiae 30 days after receiving incisions. In (e), the incision (dashed line, arrowed) caused a relative displacement of the two sides of the cuticle, which are indicated by white lines. No repair occurred. In (f), there was no such displacement and new cuticle formed (red area) to repair the injury. (Online version in colour.)


To compare mechanical behaviour, we followed procedures established in our previous work [14]. Bending strength was defined as the nominal stress to failure, which is the maximum tensile stress that would occur, at the failure location, in an undamaged circular tube having the same average diameter and thickness as measured for a given sample. We also calculated an apparent fracture toughness. Fracture toughness, Kc, is a measure of the ease with which a crack can propagate in a material. Here, we calculated Kc from the bending strength, assuming the incision to be a sharp crack of the same size in a circular tube [10,14]. As figure 5 shows, both strength and toughness approximately doubled as a result of repair. Uninjured controls of the same age had a strength of 172.4 MPa (s.d. = 31.5 MPa, n = 40). In subjects where repair did not occur, the incision lowered the average strength to 54.3 MPa (s.d. = 19.1 MPa, n = 12) and this was unchanged over time. In subjects where repair occurred, however, strength rose over the first 10 days and then remained approximately constant at an average of 113.7 MPa (s.d. = 16.4 MPa, n = 15), which is 66% of the original, uninjured strength. This increase in the nominal stress to failure was reflected in a similar increase in the bending moment to failure, from 2.50 N mm (s.d. = 1.63 N mm, n = 12) to 4.71 N mm (s.d. = 0.52 mm, n = 15). The tibia will experience high bending moments in vivo while walking, and is loaded almost exclusively in bending during activities such as kicking and jumping [15,16].


Create citation alert 1757-899X/1155/1/012037 Abstract The following thesis will illustrate derivations of crack propagation's formula for investigating brittle and elastic bodies. This work will look at the static and dynamic cases for crack initiation and propagation. Some approaches will be outlined in this work, which are the technics widely used in the field of mechanics for investigating the crack propagation and initiation related problems. At the end of the work crack propagation velocity formula will be derived.


The hazards induced by stratified rock mass creep are still one of the major problems that threaten the safety of underground engineering. This paper takes safe construction of underground roadway in Urumqi mining area as the research background. In this study, we mainly adopted rock mechanics experiments to accomplish the research on creep behavior and crack evolution of stratified structural sandstone. Creep deformation characteristics of stratified structural sandstone under different load were revealed; also, we analyzed the reason why a part of rock samples failed but others were not under the same load. Creep behavior and crack evolution of rock samples without stratified structure have significant randomness. The crack evolution and failure characteristics of stratified structural rock samples were mainly manifested as failure along and cutting through structural plane and their combined forms. Creep strain, creep duration, and creep rate of rock samples with stratified structure had a nonlinear relationship with applied load, such as exponential function or logarithmic function. Understanding the evolutionary relationship between the above parameters and load provides a basis for obtaining the creep behavior of stratified rock mass under different load conditions.


Fabre and Pellet [20] carried out creep experiments on argillaceous rocks under a variety of stress environments and found that the overall mechanical properties of argillaceous rocks deteriorated rapidly when the cracks propagated unsteadily, and the creep of clay particles caused viscoplastic strain. Brantut et al. [21] proposed a micromechanical model that could describe the brittle creep of saturated rock under triaxial stress with time and studied the micromechanics of brittle creep. Davis et al. [22] carried out triaxial compression experiments on dolomites with different particle sizes under variable temperature conditions and revealed the differences of creep mechanism between coarse-grained dolomites and fine-grained dolomites with different grain sizes. Smit et al. [23] studied the structure and microstructure of garnet polycrystals in eclogites and analyzed the creep mechanism of garnet in eclogites by using optical microscopy, element mapping, and electron backscatter diffraction. Rybacki and Dresen [24] carried out creep experiments on plagioclase samples under dry and wet conditions and determined two different creep mechanisms of dry and wet plagioclase. Heap et al. [25] studied the creep mechanism of pore water in sandstone by using microstructure analysis, acoustic emission source location, and macroscopic creep law. Brückl and Parotidis [26] analyzed the deep creep mechanism of slope rock mass with simulation study and pointed out that the main factor controlling the deep creep mechanism was the expansion of subcritical cracks. Bresser [27] obtained the pressure sensitivity and strain rate sensitivity of flow stress through experiments and revealed the creep mechanism of calcite dislocation at high temperature based on the experimental data of microphysical model. Gratier et al. [28] carried out indentation experiments on quartz crystals, which provided characteristic time scales for the transient creep and sealing processes of quartz-rich rocks after earthquakes.


Researches have carried out experiments on rock without primary structures and obtained instructive results [29-31]. However, stratified structural rock mass widely exists in deep engineering, and it is characterized by structural anisotropy. Related studies have found that structural anisotropy has a controlling effect on the creep behavior and crack evolution of rock mass. Therefore, the study on creep behavior and law of stratified structural rock is of guiding significance to discover the failure mechanism of such rock mass. Also, it is an important supplement to the study of rock mechanics. The rock samples used in this study were taken from the surrounding rock of underground roadway in Urumqi mining area. Through systematic creep experiments under different loads, the control effect of structural anisotropy on creep of stratified structural rock samples was studied. And the degree of difference in deformation rate caused by structural anisotropy in rock samples was analyzed to obtain the creep behavior and crack evolution of stratified structural rock mass.


The THMC rheological test system was mainly used in the experiment. And, its calibration curve showed that the relationship between the effective stress σ e and the applied stress σ is σ e=0.244σ -0.511 (Figure 1). Other equipment included strain gauges, SWAES digital multichannel acoustic emission devices, and digital cameras. Axial strain and radial strain were measured by strain gauge and DD1 cantilever strain sensor produced by HBM Company. We used the RMT-150C rock and concrete mechanics test system to conduct the experiment. The size of samples was the same as the ones used in the creep experiments. The number of rock samples was 12.


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