A Critical Review on the Complex Potentials in Linear Elastic Fracture Mechanics

Introducing a crack in an elastic plate is challenging from the mathematical point of view and relevant within an engineering context of evaluating strength and reliability of structures. Accordingly, a multitude of associated works is available to date, emanating from both applied mathematics and mechanics communities. Although considering the same problem, the given complex potentials prove to be different, revealing various inconsistencies in terms of resulting stresses and displacements. Essential information on crack near-tip fields and crack opening displacements is nonetheless available, while intuitive adaption is required to obtain the full-field solutions. Investigating the cause of prevailing deficiencies inevitably leads to a critical review of classical works by Muskhelishvili or Westergaard. Complex potentials of the mixed-mode loaded Griffith crack, sparing restrictive assumptions or limitations of validity, are finally provided, allowing for rigorous mathematical treatment. The entity of stresses and displacements in the whole plate is finally illustrated and the distributions in the crack plane are given explicitly.

A New Stress-Intensity Factor Solution for an External Surface Crack in Spheres

This paper describes a new stress-intensity factor (SIF) solution for an external surface crack in a sphere that expands capabilities previously available for this common pressure vessel geometry. The SIF solution employs the weight function (WF) methodology that enables rapid calculations of SIF values. The WF methodology determines SIF values from the nonlinear stress variations computed for the uncracked geometry, e.g., from service stresses and/or residual stresses. The current approach supports two degrees of freedom that denote the two crack tips located normal to the surface and the surface of the sphere. The geometric formulation of this solution enforces an elliptical crack front, maintains normality of the crack front with the free surface, and supports two degrees of freedom for fatigue crack growth from an internal crack tip and a surface crack tip. The new SIF solution accommodates spherical geometries with an exterior diameter greater than or equal to four times the thickness. This WF SIF solution has been combined with stress variations common for spherical pressure vessels: uniform internal pressure on the interior surface, uniform tension on the crack plane, and uniform bending on the crack plane. This paper provides a complete overview of this solution. We present for the first time the geometric formulation of the crack front that enables the new functionality and set the geometric limits of the solution, e.g., the maximum size and shape of the crack front. The paper discusses the bivariant WF formulation used to define the SIF solution and details the finite element analyses employed to calibrate terms in the WF formulation. A summary of preliminary verification efforts demonstrates the credibility of this solution against independent results from finite element analyses. We also compare results of this new solution against independent SIFs computed by finite element analyses, legacy SIF solutions, API 579, and FITNET. These comparisons indicate that the new WF solution compares favorably with results from finite element analyses. This paper summarizes ongoing efforts to improve and extend this solution, including formal verification and development of an internal surface crack model. Finally, we discuss the capabilities of this solution’s implementation in NASGRO® v10.0.

Secondary Stress Weighting Factor and Plastic Reduction Factor of Surface Cracked Pipes Under Bending

In piping design analysis, the secondary stresses (displacement controlled) may have different design limits than primary stresses (load-controlled stresses). The current design limits for secondary stresses are based on elastic stress analysis. But realistically a flaw in the piping system can cause non-linear behavior due to the plasticity at the crack plane as well as in the adjacent uncracked-piping material. Hence, the actual stresses in a cracked piping system which are elastic-plastic are different than the design stresses which are elastically calculated. To assess margins in the secondary stresses calculated using elastic stress analysis, two parameters are defined in this paper. The first one is the Secondary Stress Weighting Factor (SSWF) on total stress which is defined as the ratio of actual elastic-plastic stresses in a system to the elastic design stress. An alternative approach to applying margins on secondary stresses is to a use a reduction factor only on stresses above the yield stress. This reduced factor is called Plastic Reduction Factor (PRF). In this paper, a methodology developed to determine these factors for circumferential surface-cracked TP304 stainless steel pipes subjected to bending loads at room temperature is described. Four-point-bend tests are conducted on pipes with varying circumferential surface-crack lengths and depths. The moment and rotations needed for the pipe failure for different crack sizes are determined and compared to elastically calculated moments and rotations to establish margins.

Extending service life of rails in the case of a rail head defect

Rails are subjected to the processes of wear, corrosion and contact and bending fatigue during their lifecycle. As a result of these processes, various types of damage and defects are formed in rails. The residual life of rails depends on the size, position, and orientation of defects. Maximum permissible crack-size values are calculated in this paper using the finite element method. The crack plane orientation relative to the contact surface plane is analysed. The dependence of the stress intensity factor on the crack area is established. This allows continued use of defective rails and safe operation on low-activity railways.

Thermal Activation and Ductile vs. Brittle Behavior of Microcracks in 3D BCC Iron Crystals under Biaxial Loading by Atomistic Simulations

We present the results of free 3D molecular dynamics (MD) simulations, focused on the influence of temperature on the ductile-brittle behavior of a pre-existing central Griffith through microcrack (1¯10)[110] (crack plane/crack front) under biaxial loading σA and σB in tension mode I. At temperatures of 300 K and 600 K, the MD results provide new information on the threshold values of the stress intensity factor K and the energy release rate G, needed for the emission of <111>{112} blunting dislocations that support crack stability. A simple procedure for the evaluation of thermal activation from MD results is proposed in the paper. 3D atomistic results are compared with continuum predictions on thermal activation of the crack induced dislocation generation. At elevated temperature T and biaxiality ratios σB/σA ≤ 0.8 dislocation emission in MD is observed, supported by thermal activation energy of about ~30 kBT. With increasing temperature, the ductile-brittle transition moves to a higher biaxiality ratios in comparison with the situation at temperature of ~0 K. Near the transition, dislocation emission occurs at lower loadings than expected by continuum predictions. For the ratios σB/σA ≥ 1, the elevated temperature facilitates (surprisingly) the microcrack growth below Griffith level.

Dynamic Crack Propagation and Its Interaction With Micro-Cracks in an Impact Problem

The dynamic fracture behavior of brittle materials that contain micro-level cracks should be examined when material subjected to impact loading. We investigated the effect of micro-cracks on the propagation of macro-cracks that initiate from notch tips in the Kalthoff–Winkler experiment, a classical impact problem. To define predefined micro-cracks in three-dimensional space, we proposed a two-dimensional micro-crack plane definition in the bond-based peridynamics (PD) that is a non-local form of classical continuum theory. Randomly distributed micro-cracks with different number densities in a constant area and number in expending area models were examined to monitor the toughening of the material. The velocities of macro-crack propagation and the time required for completing fractures were considered in several predefined micro-cracks cases. It has been observed that toughening mechanism is only initiated by exceeding a certain number of micro-cracks; therefore, there is a positive correlation between the density of predefined micro-cracks and macro-crack propagation rate and, also, toughening mechanism.

Tensile-dominant fractures observed in hydraulic fracturing laboratory experiment using eagle ford shale

SUMMARY Hydraulic fracturing plays a vital role in the development of unconventional energy resources, such as shale gas/oil and enhanced geothermal systems to increase the permeability of tight rocks. In this study, we conducted hydraulic fracturing experiments in a laboratory using carbonate-rich outcrop samples of Eagle Ford shale from the United States. We used a thermosetting acrylic resin containing a fluorescent compound as a fracturing fluid. Immediately after fracturing, the liquid resin penetrated in the fractured blocks was hardened by applying heat. Then, the crack was viewed under UV irradiation, where the fluorescent resin allowed the induced fracture to be clearly observed, indicating the formation of simple, thin bi-wing planar fractures. We observed the detailed structure of the fractures from microscopy of thin cross-sections, and found that their complexity and width varied with the distance from the wellbore. This likely reflects the change in the stress state around the tip of the growing fracture. The interaction between fractures and constituent grains/other inclusions (e.g. organic substances) seemed to increase the complexity of the fractures, which may contribute to the efficient production of shale gas/oil via hydraulic fracturing. We first detected acoustic emission (AE) signals several seconds before the peak fluid pressure was observed, and the active region gradually migrated along the microscopically observed fracture with increasing magnitude. Immediately after the peak pressure was observed, the fluid pressure dropped suddenly (breakdown) with large seismic waves that were probably radiated by dynamic propagation of the fracture; thereafter, the AE activity stopped. We applied moment tensor inversion for the obtained AE events by carefully correcting the AE sensor characteristics. Almost all of the solutions corresponded to tensile events that had a crack plane along the maximum compression axis, as would be expected based on the conventional theory of hydraulic fracturing. Such domination of tensile events has not been reported in previous studies based on laboratory/in situ experiments, where shear events were often dominant. The extreme domination of the tensile events in the present study is possibly a result of the use of rock samples without any significant pre-existing cracks. Our experiments revealed the fracturing behaviour and accompanying seismic activities of very tight rocks in detail, which will be helpful to our understanding of fracturing behaviour in shale gas/oil resource production.

Intergranular Fracture Prediction and Microstructure Design

A model based on discrete unit events coupled with a graph search algorithm is developed to predict intergranular fracture. The model is based on two hypotheses: (i) the key unit event associated with intergranular crack propagation is the interaction of a grain boundary crack with a grain boundary segment located at an angle with the initial crack plane; and (ii) for a given crack path, the overall crack growth resistance can be calculated using the crack growth resistance of a collection of unit events. Next, using a directed graph containing the connectivity of grain boundary junctions and the distances between them, and crack deflection versus crack growth resistance data, a directed graph in the J-resistance space is created. This graph contains information on the crack growth resistance for all possible crack paths in a given grain microstructure. Various crack growth resistance curves are then calculated including those corresponding to: (i) a local resistance minimum; (ii) a global minimum; and (iii) for verification, a path specified by microstructure-based finite element calculations. The results show that the proposed method based on discrete unit events and graph search can predict the crack path and the crack growth resistance for cracks that propagate from one grain boundary junction to another. The proposed computationally inexpensive model can be used to design material microstructures with improved intergranular fracture resistance, and/or to assess the overall crack growth resistance of materials with a known distribution of grain morphology.

Failure analysis of a motor vehicle suspension helical spring

The purpose of this work is to reveal the cause of the failure of the motor vehicle rear suspension barrel-shaped spring with the progressive elasticity characteristic and predict measures to increase the lifetime of springs of this type. The fracture of the spring occurred on the middle coil, which operates under conditions of more severe stress in comparison with other coils. The chemical composition of the spring material, determined by X-ray fluorescence spectral and microstructural analyzes, corresponded to chromium-silicon steel 54SiCr6. In terms of structure and mechanical properties, the spring material met the standards. No traces of decarburization were detected, and no crack initiation, caused by non-metallic inclusions, was found in the material of the fractured spring. Macroscopic examination of the spring surface did not reveal any cracks, scratches, dents, traces of blows with stones and marks of spring coiling tool. Instead, extensive areas of exfoliation of the protective coating were found. The metallographic analysis revealed selective corrosion in the form of pitting damage in places of exfoliation of the protective coating. The fatigue crack propagates from the certain deep pit with the reorientation of the crack plane along the spiral surface to the central axis of the coil wire. After depletion of the safety margin, the spring broke down quickly. The fast fracture zone contains steps of the river pattern formed due to the spiral reorientation of the fracture surface. The research can be used to understand the importance of adhesive strength and wear resistance of protective coatings on the spring surface. Their local exfoliation causes subsequent corrosion damage to the spring, which stimulates its fatigue fracture.

Tracking capsule activation and crack healing in a microcapsule-based self-healing polymer

Structural polymeric materials incorporating a microencapsulated liquid healing agent demonstrate the ability to autonomously heal cracks. Understanding how an advancing crack interacts with the microcapsules is critical to optimizing performance through tailoring the size, distribution and density of these capsules. For the first time, time-lapse synchrotron X-ray phase contrast computed tomography (CT) has been used to observe in three-dimensions (3D) the dynamic process of crack growth, microcapsule rupture and progressive release of solvent into a crack as it propagates and widens, providing unique insights into the activation and repair process. In this epoxy self-healing material, 150 µm diameter microcapsules within 400 µm of the crack plane are found to rupture and contribute to the healing process, their discharge quantified as a function of crack propagation and distance from the crack plane. Significantly, continued release of solvent takes place to repair the crack as it grows and progressively widens.