Crack In Infinite Research Articles

Resonances of Fluid‐Filled Cracks With Complex Geometry and Application to Very Long Period (VLP) Seismic Signals at Mayotte Submarine Volcano

AbstractFluid‐filled cracks sustain a slow guided wave (Krauklis wave or crack wave) whose resonant frequencies are widely used for interpreting long period (LP) and very long period (VLP) seismic signals at active volcanoes. Significant efforts have been made to model this process using analytical developments along an infinite crack or numerical methods on simple crack geometries. In this work, we develop an efficient hybrid numerical method for computing resonant frequencies of complex‐shaped fluid‐filled cracks and networks of cracks and apply it to explain the ratio of spectral peaks in the VLP signals from the Fani Maoré submarine volcano that formed in Mayotte in 2018. By coupling triangular boundary elements and the finite volume method, we successfully handle complex geometries and achieve computational efficiency by discretizing solely the crack surfaces. The resonant frequencies are directly determined through eigenvalue analysis. After proper verification, we systematically analyze the resonant frequencies of rectangular and elliptical cracks, quantifying the effect of aspect ratio and crack stiffness. We then discuss theoretically the contribution of fluid viscosity and seismic radiation to energy dissipation. Finally, we obtain a crack geometry that successfully explains the characteristic ratio between the first two modes of the VLP seismic signals from the Fani Maoré submarine volcano in Mayotte. Our work not only reveals rich eigenmodes in complex‐shaped cracks but also contributes to illuminating the subsurface plumbing system of active volcanoes. The developed model is readily applicable to crack wave resonances in other geological settings, such as glacier hydrology and hydrocarbon reservoirs.

On energy release rates in Peridynamics

This work is concerned with the calculation of fracture energy release rates in Peridynamics. The existing crack growth criterion and the related damage model in Peridynamics, e.g. the critical bond stretch in the prototype micro-elastic brittle (PMB) material model, are based on the energy release rate of an idealized infinite crack model. Therefore, applications of such damage model in Peridynamics to arbitrary fracture patterns, e.g. cracks with finite increment, kinked cracks, and penny-shaped cracks etc., are problematic, because that the idealized crack model assumes an infinitely large crack surface, and the crack growth direction is always assumed being along the direction of the old crack surface.In all engineering applications, we only encounter cracks of finite size. In this work, based on the celebrated Griffith theory that interprets the energy release rate as the surface energy density, we have systematically investigated the fracture energy release rate in Peridynamics for a number of finite crack models: two-dimensional (2D) plane strain finite size horizontal crack, 2D edge crack in a plate, 2D kinked crack, and three-dimensional (3D) penny-shaped crack.The highlights of this work are a set of elegant mathematical solutions: (1) We have proved that for 2D horizontal finite cracks, the energy release rate is exactly the same as that of 2D infinite large crack, i.e. Silling’s original result; (2) We have derived the exact expressions of the energy release rates as well as the related critical bond stretch formulas for 2D finite edge cracks and kinked cracks, and (3) We have found an asymptotic solution of the energy release rate for 3D penny-shaped cracks of finite size in Peridynamics, showing that Silling’s energy release formula for 3D crack model is only the zero-th order approximation of the exact solution.In addition, we have presented some numerical solutions of finite increment crack growths in Peridynamics, and discussed their implications, ramifications, and applications in nonlocal continuum fracture mechanics.

Stress distributions and mechanical properties of laminates [θm/90n]S with closed and open cracks in shear loading

The classical elastic theory of laminates states that the transverse normal stresses in the local axis of the 90° layer of an undamaged laminate [θm/90n]S are positive due to in-plane shear force, the transverse normal stresses will be negative when the shear load is inthe oppositedirection. For the laminate [θm/90n]S with transverse matrix cracks, the cracks in the 90°-layer should be open for the tensile stress and closed for the compressive stress. Therefore, two cases of the transverse cracks in the unbalanced laminates subjected to shear loading are studied such that the cracks are open for one case and the cracks are closed for another case. A stress-based variational model is developed to study the effective in-plane compliance and the stress distribution in symmetrical laminates with two different types of transverse matrix cracks in 90°-ply under an in-plane shear loading. The inter-laminar shear stresses are assumed to be in the form of general functions. A partition coefficient λ is introduced into the normal stresses σx for the closed cracks. The expressions of the stress components for the closed and the open cracks are obtained from the equilibrium equations and the boundary conditions, respectively. The governing equations for the two types of cracks are also derived using a variational approach with the principle of minimum complementary energy. In addition, analysis of strain energy and analysis of deformation on surface are used to estimate the effective in-plane shear properties of the laminates. The coupling coefficients and the shear elements in the compliance matrix are investigated for the two types of cracks. The distributions of the in-planar and the inter-laminar stresses are also presented. Limits of the compliance elements and the stress distributions at the infinite crack density are compared to the results of the classical theory of laminates.

Elastic-wave velocity characterization of gas hydrate-bearing fractured reservoirs in a permafrost area of the Qilian Mountain, Northwest China

There are two types of gas hydrate-bearing reservoirs in the permafrost area of Qilian Mountain. Most of the gas hydrates occur mainly in the fractured mudstone reservoirs and rarely in the pores of the sandstone reservoirs. In this study, for the acoustic velocity characterization of the fractured gas hydrate reservoirs of the Qilian Mountain permafrost area, some mudstone core samples were collected for physical rock experiments, such as the acoustic experiment and the porosity and permeability experiment. An acoustic velocity numerical simulation of gas hydrate reservoirs was performed according to the Biot theory and the differential effective medium theory, with the conditions of multiple gas hydrate occurrence models, including the suspension model, the semi-cementation model and the cementation model, and considering both infinite and penny-shaped cracks. Fracture porosity was added to the core samples that only contain matrix porosity. With fracture porosity ranging from 0.01% to 5%, the variation laws between acoustic velocity with fractured porosity and hydrate saturation are obtained: (1) In the case of an infinite crack, if the fractured porosity is 0.01%–1%, the P-wave velocity decreases rapidly in the case of the three occurrence models. If the fractured porosity is higher than 1%, the acoustic velocity decreases gradually. If the crack shape is a penny-shaped crack, the P-wave velocity decreases almost linearly with increasing fracture porosity. (2) If the hydrate occurrence model is the suspension model, the P-wave velocity increases slightly with increasing hydrate saturation. If the occurrence model is the semi-cementation model or the cementation model, when the gas hydrate saturation of the infinite crack ranges from 0 to 80%, the acoustic velocity increases approximately linearly, whereas when the gas hydrate saturation ranges from 80% to 100%, the velocity increases rapidly. If the crack is a penny-shaped crack, the velocity increases almost linearly with increasing gas hydrate saturation from 0 to 100%. (3) It is found that the fractured gas hydrate reservoirs of the Qilian Mountain permafrost area contain both penny-shaped and infinite cracks, of which the infinite crack is the main crack shape. The gas hydrate occurrence in the Qilian Mountain permafrost area mainly follows the suspension model. This has significance for the seismic exploration and log evaluation of gas hydrate-bearing fractured reservoirs in the permafrost area of the Qilian Mountain in studying the acoustic velocity characterization, the crack shapes and occurrence models of gas hydrate reservoirs in the study area.

On the influence of crack front curvature on the fracture behavior of nanoscale cracks

Atomic-scale fracture processes are traditionally investigated in quasi-2D models of straight, infinite cracks. This approach neglects crack front curvature effects, which might be particularly important for nanoscale crack nuclei in semi-brittle materials. Here we use 3D atomistic simulations to study penny-shaped cracks in body-centered cubic metals. Our results show extensive crack tip plasticity initiated by deformation twinning and followed by emission of screw dislocations which cross-slip along the crack front. Together with the interactions of dislocations and/or twins that are nucleated at differently oriented parts of the crack, these processes determine the fracture behavior of highly curved nanoscale cracks.

Vertical cracks characterization using lock-in thermography: I infinite cracks

Early detection of cracks is a challenging task to prevent failures in working structures. In the last decades the ‘flying spot’ method, based on heating the sample with a moving laser spot and detecting the surface temperature with an infrared detector, has been developed to detect cracks in a fast manner. The aim of this work is to measure the width of an infinite vertical crack using lock-in thermography. An analytical solution for the surface temperature of a sample containing such a crack when the surface is illuminated by a modulated laser beam focused at a fixed spot close to the crack is obtained. Measurements on samples containing calibrated cracks have been performed using an infrared camera. A least square fit of the amplitude and phase of the surface temperature is used to retrieve the thickness of the crack. A very good agreement between the nominal and retrieved thicknesses of fissure is found, even for widths down to 1 µm, confirming the validity of the model.

The Water Absorption Rule of Vertical Fractured Well

Oil is an important energy and chemical raw materials and strategic materials. Nowadays, the layered water injection test technology become the key factor of oilfield production. According to different types of formation and for the artificial fracturing injection wells, this paper studied the infinite boundary, uniform flow and vertical cracks well, infinite diversion vertical fractures and conductivity vertical fracture wells’ absorbent law. The method to do all of the above work is to solve the equation of dimensionless bottomhole pressure in different formation and boundary conditions. The indicating curve of infinite uniform flow formation, unlimited conduction and limited conduction vertical fractures wells are almost identical, which means that the type of vertical fracture has little effect on the indicate curves of injection wells.

Weight Function Method With Segment-Wise Polynomial Interpolation to Calculate Stress Intensity Factors for Complicated Stress Distributions

Abstract Many solutions of the stress intensity factor have been proposed in recent years. However, most of them take only third or fourth-order polynomial stress distributions into account. For complicated stress distributions which are difficult to be represented as third or fourth-order polynomial equations over the stress distribution area such as residual stress distributions or thermal stress distributions in dissimilar materials, it is important to further improve the accuracy of the stress intensity factor. In this study, a weight function method with segment-wise polynomial interpolation is proposed to calculate solutions of the stress intensity factor for complicated stress distributions. By using this method, solutions of the stress intensity factor can be obtained without employing finite element analysis or difficult calculations. It is therefore easy to use in engineering applications. In this method, the stress distribution area is firstly divided into several segments and the stress distribution in each segment is curve fitted to segment-wise polynomial equation. The stress intensity factor is then calculated based on the weight function method and the fitted stress distribution in each segment. Some example solutions for both infinite length cracks and semi-elliptical cracks are compared with the results from finite element analysis. In conclusion, it is confirmed that this method is applicable with high accuracy to the calculation of the stress intensity factor taking actual complicated stress distributions into consideration.

DYNAMIC PROPAGATION PROBLEMS OF MODE III SEMI-INFINITE CRACK

By means of the theory of complex functions, dynamic propagation problems of mode III semi- infinite crack were researched. The problems studied can be facilely transformed into riemann-hilbert problems and the universal expression of analytical solutions of stress, displacement and dynamic stress intensity factor under the conditions of moving increasing loads Px/t and Pt2/x, respectively, are very facilely obtained using the measures of self-similar functions. In view of corresponding material properties, the mutative rule of dynamic stress intensity factor was illuminated very well.

Fracture dynamics problems of mode I semi-infinite crack for anisotropic orthotropic body

By means of the theory of complex functions, fracture dynamics problems of mode I semi- infinite crack for anisotropic orthotropic body were researched. Analytical solutions of stress, displacement, and dynamic stress intensity factor under the action of moving increasing loads Px3/t3, Pt4/x3, respectively, are very easily obtained utilizing the approaches of self-similar functions. In the light of relevant material’s coefficients, the alterable rule of dynamic stress intensity factor was depicted very well. The correlative closed solutions are attained based on the Riemann–Hilbert problems. After those analytical solutions were applied by the superposition principle, the solutions of discretional complex problems could be attained.

Evaluation of the stress intensity factors and the T-stress in periodic crack problem

This paper investigates the T-stress at crack tips in the periodic crack problem. Remote tension in the y-direction is applied to cracks with an arbitrary inclined angle. The original stress field can be considered a superposition of a uniform stress field and a perturbation stress field. The problem of evaluating the stresses in the perturbation field can be considered a superposition of many single crack problems. A Fredholm integral equation is suggested for the solution of the perturbation stress field. In the equation, the loading on the crack face is chosen as unknown quantity. Once the integral equation is solved, the stress intensity factors and the T-stress at the crack tip can be evaluated immediately. For solving the integral equation and evaluating stresses in the perturbation field, the remainder estimation technique is suggested for evaluating the influences on the central crack from infinite cracks. The technique can considerably improve convergence in computation. Many results for the stress intensity factors and the T-stresses in periodic cracks are presented. It is shown that the interaction is significant for the closer cracks.

Modification of the FRI Crack Growth Model Formulation from a Mathematical Viewpoint

The FRI model of crack growth, which incorporates mechanical properties into the slip oxidation mechanism of crack advance, is an extension of the well-known Ford-Andresen model. When the exponent of the oxidation current decay curve is set close to 1, however, the FRI model gives an infinite crack growth rate. Here, the oxidation decay curve integral is revised to eliminate this divergence, and modified crack growth rate equations are derived. Also presented here is a procedure for determining the oxidation current parameters from the curve-fitting to measurements of crack growth rate. Parameter value determination and crack growth calculations are illustrated for cold-worked Type 316L stainless steel.

Modification of the FRI Crack Growth Model Formulation from a Mathematical Viewpoint

The FRI model of crack growth, which incorporates mechanical properties into the slip oxidation mechanism of crack advance, is an extension of the well-known Ford-Andresen model. When the exponent of the oxidation current decay curve is set close to 1, however, the FRI model gives an infinite crack growth rate. Here, the oxidation decay curve integral is revised to eliminate this divergence, and modified crack growth rate equations are derived. Also presented here is a procedure for determining the oxidation current parameters from the curve-fitting to measurements of crack growth rate. Parameter value determination and crack growth calculations are illustrated for cold-worked Type 316L stainless steel.

Analysis of Slant Surface Cracks with Step Notches Subject to Contact Loadings by a Variational Boundary Integral Method

Modes I and II stress intensity factors are analyzed by means of a variational boundary integral method (VBIM) for slant surface-breaking cracks in a half-plane with surface steps subject to contact loadings. This method represents the crack as a continuous distribution of dislocation loops. The crack opening displacements, which are related to the geometry of loops and their Burgers vectors, can be determined by minimizing the elastic potential energy, obtained from the known expressions of the interaction energy of a pair of dislocation loops, of the solid. In contrast to other methods, this approach finally reduces to a symmetric system of equations with milder singularities of the type 1/R, which facilitate the numerical treatments. By modeling the surface boundary of the half-plane as half part of an infinite crack breaking through an infinite solid, this paper demonstrates that the VBIM can be well extended to solve the fracture problems of inclined surface-breaking cracks in a half-plane with curve or step notches subject to combined contact loadings, and presents results of stress intensity factors for a variety of loadings, cracks and step surface configurations. Numerical results of test examples are in good agreement with the existing results in the literature.

Fracture of a finite piezoelectric strip with a crack vertical to its borders – An exact analysis and its applications

In this paper, we developed a closed-form solution for a crack in a piezoelectric strip of finite width, subjected to in-plane mechanical and electrical loads. Explicit expressions for the stresses and the electric fields, together with their intensity factors are obtained for the two extreme cases of impermeable crack and permeable crack. Solutions for some special cases, such as a center crack, an edge crack, a semi-infinite medium, an infinite medium, two symmetric and collinear cracks, and a row of collinear cracks, are also obtained. All solutions are given in closed-form.

The shielding effects of a screw dislocation near an elliptically blunted crack tip

An explicit solution for describing the shielding effects arising from a screw dislocation near an elliptically blunted crack is presented. This is a general solution which may degenerate to the solutions of a finite crack and an infinite crack, and may apply for a dislocation either emitted from the crack surface or originating elsewhere and for a dislocation located anywhere around the crack tip. Based on the solution, the effects of the morphology of the blunt crack and the position of the dislocation on the shielding are examined.

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.

Surface displacement characteristics of line crack trespassing the Rayleigh wave speed barrier as influenced by velocity and applied stress dependent local zone of restrain

Surface displacement characteristics of line crack trespassing the Rayleigh wave speed barrier as influenced by velocity and applied stress dependent local zone of restrain

Analysis on the cohesive stress at half infinite crack tip

The nonlinear fracture behavior of quasi-brittle materials is closely related with the cohesive force distribution of fracture process zone at crack tip. Based on fracture character of quasi-brittle materials, a mechanical analysis model of half infinite crack with cohesive stress is presented. A pair of integral equations is established according to the superposition principle of crack opening displacement in solids, and the fictitious adhesive stress is unknown function. The properties of integral equations are analyzed, and the series function expression of cohesive stress is certified. By means of the data of actual crack opening displacement, two approaches to gain the cohesive stress distribution are proposed through resolving algebra equation. They are the integral transformation method for continuous displacement of actual crack opening, and the least square method for the discrete data of crack opening displacement. The calculation examples of two approaches and associated discussions are given.

On semi-infinite crack problems in elastic–plastic bodies; uniqueness and examples

First, the conditions for uniqueness of the stress boundary value problem for infinite cracks in elastic–plastic bodies are discussed on the basis of the Laurent series representation of the plastic boundary in an elastic perfectly plastic body in anti-plane strain (mode III). Next, for two cases, exact closed-form solutions of the shape of the elastic–plastic boundary are found in terms of elementary functions. A crack under shear stress acting on its surfaces, and a crack under constant remote shear stress σ ∞ 13=− p, are considered. In the first case, also the complete stress distribution is obtained, for the second the physical coordinates as functions of stresses are found. The new elastic–plastic solutions are compared with ones predicted by linear elastic fracture mechanics.