Array Of Cracks Research Articles

Fe単結晶におけるモードI貫通き裂の局所格子不安定性解析 (周期境界下のき裂配置の影響)

Fe単結晶におけるモードI貫通き裂の局所格子不安定性解析 (周期境界下のき裂配置の影響)

Elastic waves at periodically-structured surfaces and interfaces of solids

This paper presents a simple treatment of elastic wave scattering at periodically structured surfaces and interfaces of solids, and the existence and nature of surface acoustic waves (SAW) and interfacial (IW) waves at such structures. Our treatment is embodied in phenomenological models in which the periodicity resides in the boundary conditions. These yield zone folding and band gaps at the boundary of, and within the Brillouin zone. Above the transverse bulk wave threshold, there occur leaky or pseudo-SAW and pseudo-IW, which are attenuated via radiation into the bulk wave continuum. These have a pronounced effect on the transmission and reflection of bulk waves. We provide examples of pseudo-SAW and pseudo-IW for which the coupling to the bulk wave continuum vanishes at isloated points in the dispersion relation. These supersonic guided waves correspond to embedded discrete eigenvalues within a radiation continuum. We stress the generality of the phenomena that are exhibited at widely different scales of length and frequency, and their relevance to situations as diverse as the guiding of seismic waves in mine stopes, the metrology of periodic metal interconnect structures in the semiconductor industry, and elastic wave scattering by an array of coplanar cracks in a solid.

Dynamic Stresses Due to Time-Harmonic Elastic Wave Incidence on Doubly Periodic Array of Penny-Shaped Cracks

The symmetric frequency-domain problem of the interaction effects in rectangular lattice system of coplanar penny-shaped cracks located in an infinite elastic solid is numerically investigated. The problem is reduced to a boundary integral equation for the crack-opening-displacement in a unit cell by means of 3D periodic elastodynamic Green’s function. This function is adopted for the effective calculation by its representation in the form of exponentially convergent Fourier integrals. A collocation method is used for the solution of the boundary integral equation. Numerical results for the mode-I dynamic stress intensity factor in the crack vicinities are obtained and analyzed depending on the wave number and the lattice size.

Computed tomographic imaging of subchondral fatigue cracks in the distal end of the third metacarpal bone in the thoroughbred racehorse can predict crack micromotion in an ex-vivo model.

Articular stress fracture arising from the distal end of the third metacarpal bone (MC3) is a common serious injury in Thoroughbred racehorses. Currently, there is no method for predicting fracture risk clinically. We describe an ex-vivo biomechanical model in which we measured subchondral crack micromotion under compressive loading that modeled high speed running. Using this model, we determined the relationship between subchondral crack dimensions measured using computed tomography (CT) and crack micromotion. Thoracic limbs from 40 Thoroughbred racehorses that had sustained a catastrophic injury were studied. Limbs were radiographed and examined using CT. Parasagittal subchondral fatigue crack dimensions were measured on CT images using image analysis software. MC3 bones with fatigue cracks were tested using five cycles of compressive loading at -7,500N (38 condyles, 18 horses). Crack motion was recorded using an extensometer. Mechanical testing was validated using bones with 3 mm and 5 mm deep parasagittal subchondral slots that modeled naturally occurring fatigue cracks. After testing, subchondral crack density was determined histologically. Creation of parasagittal subchondral slots induced significant micromotion during loading (p<0.001). In our biomechanical model, we found a significant positive correlation between extensometer micromotion and parasagittal crack area derived from reconstructed CT images (SR = 0.32, p<0.05). Correlations with transverse and frontal plane crack lengths were not significant. Histologic fatigue damage was not significantly correlated with crack dimensions determined by CT or extensometer micromotion. Bones with parasagittal crack area measurements above 30 mm2 may have a high risk of crack propagation and condylar fracture in vivo because of crack micromotion. In conclusion, our results suggest that CT could be used to quantify subchondral fatigue crack dimensions in racing Thoroughbred horses in-vivo to assess risk of condylar fracture. Horses with parasagittal crack arrays that exceed 30 mm2 may have a high risk for development of condylar fracture.

Elastic wave propagation and stop-band generation in strongly damaged solids

In this work, we study the propagation of elastic waves in elongated solids with an array of equallyspaced deep transverse cracks, focusing in particular on the determination of stop-bands. We consider solids with different types of boundary conditions and different lengths, and we show that the eigenfrequencies associated with non-localized modes lie within the pass-bands of the corresponding infinite periodic system, provided that the solids are long enough. In the stop-bands, instead, eigenfrequencies relative to localized modes may be found. Furthermore, we use an asymptotic reduced model, whereby the cracked solid is approximated by a beam with elastic connections. This model allows to derive the dynamic properties of damaged solids through analytical methods. By comparing the theoretical dispersion curves yielded by the asymptotic reduced model with the numerical outcomes obtained from finite element computations, we observe that the asymptotic reduced model provides a better fit to the numerical data as the slenderness ratio increases. Finally, we illustrate how the limits of the stop-bands vary with the depth of the cracks.

Dynamic response and localization in strongly damaged waveguides

In this paper, we investigate the formation of band-gaps and localization phenomena in an elastic strip nearly disintegrated by an array of transverse cracks. We analyse the eigenfrequencies of finite, strongly damaged, elongated solids with reference to the propagation bands of an infinite strip with a periodic damage. Subsequently, we determine analytically the band-gaps of the infinite strip by using a lower-dimensional model, represented by a periodically damaged beam in which the small ligaments between cracks are modelled as ‘elastic junctions’. The effective rotational and translational stiffnesses of the elastic junctions are obtained from anad hocasymptotic analysis. We show that, for a finite frequency range, the dispersion curves for the reduced beam model agree with the dispersion data determined numerically for the two-dimensional elastic strip. Exponential localization, boundary layers and standing waves in strongly damaged systems are discussed in detail.

An easy-to-use estimate of the energy-release rate for crack arrays

Formation of crack arrays plays an increasing role in several fields of applied physics. The energy-release rate of the cracks controls the development of the array. Therefore, following the concept of configurational forces, a simplified analytical expression is provided for the energy-release rate, which is based both on numerical studies and on a specially adapted beam model. Comparisons of this easy-to-use estimate of the energy-release rate with established results from the literature as well as detailed numerical results are presented. The provided estimate of the energy-release rate can easily be extended to non-equidistant cracks and an anisotropic material.

Effective antiplane elastic properties of an orthotropic solid weakened by a periodic distribution of cracks

Summary Using the method of asymptotic homogenization in the low-frequency wave propagation regime we derive the effective antiplane elastic properties of a cracked solid which in the absence of cracks would be orthotropic. The solid is a periodic medium defined by a periodic cell containing N cracks of infinite extent in thez direction but with arbitrary shape and orientation in thexy plane. Effective properties are defined in terms of the solution to a so-called cell problem. We propose two convenient schemes by which the cell problem can be solved, one based on a nearly periodic Green’s function, the other based on doubly periodic multipole expansions. Using these methods we compare results with existing approximate expressions for periodic arrays of straight cracks and discuss their regimes of validity by assessing their departure from the exact results obtained by the present method. We go on to consider more complex distributions such as elliptical cavities, non-straight cracks and effects of an orthotropic host phase. In particular we show that specific types of regular arrays of cracks in an orthotropic host medium can induce a macroscopically isotropic response to antiplane shear waves propagating in the xy plane.

Thermal shock resistance behavior of a functionally graded ceramic: Effects of finite cooling rate

This work presents a semi-analytical model to explore the effects of cooling rate on the thermal shock resistance behavior of a functionally graded ceramic (FGC) plate with a periodic array of edge cracks. The FGC is assumed to be a thermally heterogeneous material with constant elastic modulus and Poisson's ratio. The cooling rate applied at the FGC surface is modeled using a linear ramp function. An integral equation method and a closed form asymptotic temperature solution are employed to compute the thermal stress intensity factor (TSIF). The thermal shock residual strength and critical thermal shock of the FGC plate are obtained using the SIF criterion. Thermal shock simulations for an Al2O3/Si3N4 FGC indicate that a finite cooling rate leads to a significantly higher critical thermal shock than that under the sudden cooling condition. The residual strength, however, is relatively insensitive to the cooling rate.

Development of fracture facets from a crack loaded in mode I+III: Solution and application of a model 2D problem

It is experimentally well-known that a crack loaded in mode I+III propagates through formation of discrete fracture facets inclined at a certain tilt angle on the original crack plane, depending on the ratio of the mode III to mode I initial stress intensity factors. Pollard et al. (1982) have proposed to calculate this angle by considering the tractions on all possible future infinitesimal facets and assuming shear tractions to be zero on that which will actually develop. In this paper we consider the opposite case of well-developed facets; the stress field near the lateral fronts of such facets becomes independent of the initial crack and essentially 2D in a plane perpendicular to the main direction of crack propagation.To determine this stress field, we solve the model 2D problem of an infinite plate containing an infinite periodic array of cracks inclined at some angle on a straight line, and loaded through uniform stresses at infinity. This is done first analytically, for small values of this angle, by combining Muskhelishvili's (1953) formalism and a first-order perturbation procedure. The formulae found for the 2D stress intensity factors are then extended in an approximate way to larger angles by using another reference solution, and finally assessed through comparison with some finite element results.To finally illustrate the possible future application of these formulae to the prediction of the stationary tilt angle, we introduce the tentative assumption that the 2D mode II stress intensity factor is zero on the lateral fronts of the facets. An approximate formula providing the tilt angle as a function of the ratio of the mode III to mode I stress intensity factors of the initial crack is deduced from there. This formula, which slightly depends on the type of loading imposed, predicts somewhat smaller angles than that of Pollard et al. (1982).

Guided fracture of films on soft substrates to create micro/nano-feature arrays with controlled periodicity.

While the formation of cracks is often stochastic and considered undesirable, controlled fracture would enable rapid and low cost manufacture of micro/nanostructures. Here, we report a propagation-controlled technique to guide fracture of thin films supported on soft substrates to create crack arrays with highly controlled periodicity. Precision crack patterns are obtained by the use of strategically positioned stress-focusing V-notch features under conditions of slow application of strain to a degree where the notch features and intrinsic crack spacing match. This simple but robust approach provides a variety of precisely spaced crack arrays on both flat and curved surfaces. The general principles are applicable to a wide variety of multi-layered materials systems because the method does not require the careful control of defects associated with initiation-controlled approaches. There are also no intrinsic limitations on the area over which such patterning can be performed opening the way for large area micro/nano-manufacturing.

An analysis of competing toughening mechanisms in layered and particulate solids

The relative potency of common toughening mechanisms is explored for layered solids and particulate solids, with an emphasis on crack multiplication and plasticity. First, the enhancement in toughness due to a parallel array of cracks in an elastic solid is explored, and the stability of co-operative cracking is quantified. Second, the degree of synergistic toughening is determined for combined crack penetration and crack kinking at the tip of a macroscopic, mode I crack; specifically, the asymptotic problem of self-similar crack advance (penetration mode) versus $$90^{\circ }$$ symmetric kinking is considered for an isotropic, homogeneous solid with weak interfaces. Each interface is treated as a cohesive zone of finite strength and toughness. Third, the degree of toughening associated with crack multiplication is assessed for a particulate solid comprising isotropic elastic grains of hexagonal shape, bonded by cohesive zones of finite strength and toughness. The study concludes with the prediction of R-curves for a mode I crack in a multi-layer stack of elastic and elastic–plastic solids. A detailed comparison of the potency of the above mechanisms and their practical application are given. In broad terms, crack tip kinking can be highly potent, whereas multiple cracking is difficult to activate under quasi-static conditions. Plastic dissipation can give a significant toughening in multi-layers especially at the nanoscale.

The Control of Crack Arrays in Thin Films.

Thin-film fracture can be used as a nano-fabrication technique but, generally, it is a stochastic process that results in non-uniform patterns. Crack spacings depend on the interaction between intrinsic flaw populations and the fracture mechanics of crack channeling. Geometrical features can be used to trigger cracks at specific locations to generate controlled crack patterns. However, while this basic idea is intuitive, it is not so obvious how to realize the concept in practice, nor what the limitations are. The control of crack arrays depends on the nature of the intrinsic flaw population. If there is a relatively large density of long flaws, as commonly assumed in fracture-mechanics analyses, reliable crack patterns can be obtained fairly robustly using relatively blunt geometrical features to initiate cracks, provided the applied strain is carefully matched to the properties of the system and the desired crack spacing. This process is analyzed both for cracks confined to the thickness of a film and for cracks growing into a substrate. The latter analysis is complicated by the fact that increases in strain can either drive cracks deeper into the substrate or generate new cracks at shallower depths. If the intrinsic flaws are all very short, the geometrical features need to be very sharp to achieve the desired patterns. While careful control of the applied strain is not required, the strain needs to be relatively large compared to that which would be required to propagate a large flaw across the film. This results in an approach that is not robust against the introduction of accidental damage or a few large flaws.

Dynamic response of planar cracks in an infinitely long piezoelectric strip

The problem of determining the electro-elastic fields around an array of parallel planar cracks in an infinitely long piezoelectric strip is considered. The cracks, acted upon by dynamic loads, are either electrically impermeable or permeable. A semi-analytic method based on the theory of exponential Fourier transformation is presented for solving the problem in the Laplace transform domain. The Laplace transforms of the jumps in the displacements and electric potential across opposite crack faces are determined by solving a system of hypersingular integral equations. Once these displacement and electric potential jumps are obtained, the displacements and electric potential and other physical quantities of interest, such as the crack tip stress and electric displacement intensity factors, can be recovered with the help of a suitable algorithm for inverting Laplace transforms. The stress and electric displacement intensity factors are computed for some specific cases of the problem.

Propagation of interactive parallel flat elliptical cracks inclined to shear stress

The propagation and interaction of parallel arrays of cracks embedded in rock mass have critical impact on the stability of rock mass subjected to earthquake. To investigate the propagation of these embedded cracks under shear stress, three-dimensional element partition method (3D-EPM) is used to model the pre-existing cracks in conjunction with the augmented virtual internal bond (AVIB) constitutive model to describe the rock matrix. By 3D-EPM, the contact effect of crack faces can be automatically accounted in the original mesh scheme. By AVIB, the failure criterion would be implicitly invoked by the micro fracture mechanism. It is revealed that the propagation pattern of embedded flat cracks to shear stress is related to the inclination of crack relative to the shear force. When the inclination is smaller than 90°, the father crack firstly propagates in wrapping wing pattern. Then, many parallel arrays of descendent cracks, which are vertical to the relative slip of the father crack faces, anti-symmetrically initiate on part of the upper and the lower father crack faces, respectively. With the inclination increasing, the distribution area of the descendent cracks moves from the lower to the upper part of the father crack face. With shear stress increasing, a prior propagation path, vertical to the father crack face, is formed near the middle transect of the father crack. Finally, these prior extended descendent cracks adjacent to different father cracks coalesce together in zigzag at rock bridges. However, when the inclination is bigger than 90°, the father cracks only independently propagate along their minor axis directions. The extended crack is coplanar with the father crack. In all inclination cases, no apparent tensile fracture propagates at the two major axis tips of the original crack. It is also found that the shear strength of the cracked specimen is strongly dependent on the inclination of embedded cracks. When the inclination varies from 0° to 40° or from 180° to 120°, the shear strength remarkably decreases while when the inclination varies from 40° to 120°, the shear strength changes very little. The features of crack propagation obtained and the conclusions drawn in the present paper are significantly valuable for the evaluation of jointed rock slope stability and landslide.

Wake length and loading history effects on crack closure of through-thickness long and short cracks in 304L: Part I – Experiments

A detailed study of the propagation of bidimensional (2D) through-thickness cracks has been conducted on a 304L austenitic stainless steel used in nuclear industry in view of providing a detailed description of the effective driving force to be introduced in numerical simulation of thermal fatigue crack arrays in some critical components. To answer this problem, crack closure is precisely determined for long cracks and 2D short cracks artificially obtained in CT specimens, for different loading conditions. The present results will be used to validate numerical simulations of crack closure presented in a second part as a sister paper.

Radially aligned microchannels prepared from ordered arrays of cracks on colloidal films

Arrayed microchannels with dimensions of tens to hundreds of micrometres show great potential for application in microfluidics, microreactor devices and other areas. In this work, we report the formation of ordered arrays of cracks on solution-cast colloidal films, and the preparation of radially aligned microchannels. The polymer film coating enables the colloidal film to be detached from the substrate, retaining the crack patterns on the bottom side. The subsequent chemical corrosion converts the cracks into microchannels. Crack-patterned colloidal films are also used as structured substrates for the preparation of honeycomb films by the breath figure method. Hierarchical honeycomb structures are obtained, giving rise to improved support for colloidal films.

Interaction between periodic cracks and periodic rigid-line inclusions in piezoelectric materials

Piezoelectric materials with a doubly periodic array of cracks and rigid-line inclusions under far-field antiplane mechanical load and inplane electric load are investigated. By employing the conformal mapping technique and the elliptical function theory, an exact solution of the whole-field stress and electrical displacement is obtained. The closed-form formulae for the stress and electrical displacement intensity factors at the tips of cracks and rigid-line inclusions and the effective electroelastic moduli of the composites are presented. Some existing solutions can be regarded as degenerated cases of the present results. Numerical examples are provided to show the interesting electroelastic interaction between multiple cracks and multiple rigid-line inclusions.

The stress intensity factors for a periodic array of interacting coplanar penny-shaped cracks

The effect of crack interactions on stress intensity factors is examined for a periodic array of coplanar penny-shaped cracks. Kachanov’s approximate method for crack interactions [Kachanov, M., 1987. Elastic solids with many cracks: a simple method of analysis. International Journal of Solids and Structures 23 (1), 23–43] is employed to analyze both hexagonal and square crack configurations. In approximating crack interactions, the solution converges when the total truncation number of the cracks is 109. As expected, due to high density packing crack interaction in the hexagonal configuration is stronger than that in the square configuration. Based on the numerical results, convenient fitting equations for quick evaluation of the mode I stress intensity factors are obtained as a function of crack density and angle around the crack edge for both crack configurations. Numerical results for the mode II and III stress intensity factors are presented in the form of contour lines for the case of Poisson’s ratio ν=0.3. Possible errors for these problems due to Kachanov’s approximate method are estimated. Good agreement is observed with the limited number of results available in the literature and obtained by different methods.

Interactions of 3D embedded parallel vertically inclined cracks subjected to uniaxial compression

Sets of embedded parallel joints are frequently encountered in geotechnical engineering. The interactions between these embedded parallel joints have significant influence on failure process of rock mass. To investigate the interactions of joints, the present paper firstly develops a simple constitutive model based on the augmented virtual internal bond (AVIB) method, by which no separate fracture criterion is needed in simulating fracture propagation. To simulate the pre-existing cracks, the 3D element partition method is employed, which avoids the problems of specially setting up joint element and modification of mesh scheme. The simulation results suggest that when the parallel cracks are vertically aligned, they propagate independently in wrapping wing pattern, but there is neither prior propagation nor coalescence path. In contrast to the aligned vertically parallel crack cases, the nonaligned vertically parallel crack case has prior propagation and coalescence path. The crack always propagates toward and coalesces with its adjacent crack by which a wing crack array is formed. During the wing-array cracks’ propagation and coalescence, the other crack’s growth is restrained to a certain extent due to the release of stress concentration. The drawn conclusions provide meaningful reference for the analysis of landslide of jointed rock slope.