Star-shaped Crack Research Articles

Flow Through Star-Shaped Cracks Filled with a Porous Medium

Flow Through Star-Shaped Cracks Filled with a Porous Medium

Star Crack Formation via Low-Velocity Impact on Glass Windows

The resistance to crack formation induced by gravel impacts on laminated glass windshields is a crucial issue for windshield manufacturers. It is proposed to study the star crack formation on polyvinyl butyral (PVB) laminated glass by designing a new highly-instrumented device that can break the glass under dynamic loading. The set-up is composed of a spring-loaded projectile that mimics a gravel momentum and an input bar equipped with strain gauges, whose end is an indenter initially in contact with the laminated glass sample. A synchronized high-speed camera allows for the observation of the crack growth. A unidimensional model introduces the local non-linear behavior of the indented zone. Crack monitoring synchronized with the Force and Velocity measurements are presented. Live-indentation depth measurements are performed. Experimental results are used in a dynamic simulation to analyze the non-linear damage behavior under the indenter tip. Dynamic and well-instrumented indentation experiments on the laminated glass are performed to investigate the star-shaped crack formation and describe the indented zone. This work is a prelude to the crack length prediction on windshields submitted to gravel pitting.

Analysis of closed branched and intersecting cracks by the boundary element method

The boundary element method (BEM) and a computer code for the analysis of plates with closed branched and intersecting cracks are developed. The BEM enables simple and accurate modelling of cracked plates by using boundary elements. Contact tractions between crack surfaces are computed using an iterative procedure. Stress intensity factors (SIFs) are determined using the path-independent integral. Three numerical examples are studied: a star-shaped crack in a square plate, multiple interacting cracks in an infinite plate and randomly distributed and intersecting cracks in a square plate. The examples demonstrate the simplicity of numerical modelling, the accuracy of the method and the possible applications. The influences of load directions, distances between cracks and the contact of the crack surfaces on SIF are investigated. For the plate with randomly distributed cracks, the effective elastic properties are additionally computed by considering or neglecting contact of crack surfaces. The results show that the importance of the contact procedure depends on how the cracked material is loaded.

A Non-Classical Analytical Approach for Vibration Analysis of Isotropic and Fgm Plate Containing a Star Shaped Crack

ABSTRACTA non-classical analytical model for vibration analysis of thin isotropic and FGM plate containing multiple part-through cracks (star shaped) of arbitrary orientation is proposed. A plate containing four concentric cracks of arbitrary orientation in the form of continuous line is considered for analysis. The proposed governing equation is derived based on classical plate theory and modified couple stress theory. Line spring model is modified to accommodate all the crack terms. The application of Berger’s formulation introduces nonlinearities in the governing equation and then the Galerkin’s method is applied for solving final governing equation. Results for fundamental frequencies for different values of crack length, crack orientation, gradient index and material length scale parameters are presented for two different boundary conditions. Furthermore, to study the phenomenon of bending hardening/softening in a cracked plate, the frequency response curves are plotted for the parameters stated above. Based on the outcomes of this study, it can be concluded that stiffness of the plate is severely affected by the presence of multiple cracks and the stiffness goes on decreasing with increase in number of cracks thereby affecting the fundamental frequency.

Torsion of elastic solids with sparse voids parallel to the twist axis

With the purpose of investigating a linear elastic solid containing a dilute distribution of cylindrical and prismatic holes parallel to the torsion axis, the full-field solution for an infinite elastic plane containing a single void and subject to torsion is derived. The obtained solution is exploited to derive the analytic expressions for the stress concentration factor related to the presence of an elliptical hole, the stress intensity factor for hypocycloidal holes and star-shaped cracks and the notch stress intensity factor for star-shaped polygons. Special sets of the void location are obtained for which peculiar mechanical behaviours are displayed, such as stress annihilation at some points along the boundary of elliptical voids and stress singularity removal at the cusps or points of hypocycloidal or isotoxal star-shaped polygonal voids. By means of finite-element simulations, it is finally shown that the presented closed-form expressions for the stress intensification provide reliable predictions, even for finite-domain realizations; in particular, the infinite-plane solution remains highly accurate when the the smooth and non-smooth external boundaries are larger than twice and five times the void dimension, respectively. Under these geometrical conditions, the derived analytical expressions represent a valid ‘guide tool’ in mechanical design.

Fracture of hollow multiply-twinned particles under chemical etching

Abstract The stress-strain state in a hollow decahedral small particle is modeled as that caused by a positive wedge disclination in an elastic hollow sphere. The stress and dilatation distribution in the particle is analyzed in relation to crack nucleation and the particle's catastrophic fracture during chemical etching. A two-stage process of the fracture is suggested, which includes nucleation of initial cracks along twin boundaries around their quintuple junction and subsequent pattern formation in a fivefold star-shaped crack at the junction. The critical sizes of the initial crack and the decahedral small particle at which this two-stage fracture process occurs, are estimated.

Nanograin formation and reaction-induced fracturing due to decarbonation: Implications for the microstructures of fault mirrors

Principal slip zones often contain highly reflective surfaces referred to as fault mirrors, shown to consist of a nanogranular coating. There is currently no consensus on how the nanograins form, or why they survive weathering on a geological time-scale. To simplify the complex system of a natural fault zone, where slip and heat generation are inherently coupled, we investigated the effect of elevated temperatures on carbonate rock surfaces, as well as their resistance to water exposure. This allows us to isolate the role of the decarbonation process in the formation of nanograins. We used cleaved crystals of Iceland spar calcite, manually polished dolomite protolith, as well as natural dolomite fault mirror surfaces. The samples were heated to 200–800 °C in a ∼5 h heating cycle, followed by slow cooling (∼12 h) to room temperature. Subsequently, we imaged the samples using scanning electron microscopy and atomic force microscopy. Nanograin formation on all sample surfaces was pervasive at and above 600 °C. The Foiana fault mirror samples were initially coated with aligned naturally-formed nanograins, but display a non-directional nanogranular coating after heating. The nanograins that were formed by heating rapidly recrystallized to bladed hydroxides upon exposure to deionized water, whereas the nanograins on unheated fault mirror samples remained unchanged in water. This shows that the nanograins formed by heating alone are different from those formed in fault zones, and calls for a better characterization of nanograins and their formation mechanisms. Furthermore, we find a characteristic star-shaped crack pattern associated with reacted regions of the carbonate surfaces. The existence of this pattern implies that the mechanical stresses set up by the decarbonation reaction can be sufficiently large to drive fracturing in these systems. We propose that this mechanism may contribute to grain size reduction in fault zones.

Modeling complex crack problems using the three-node triangular element fitted to numerical manifold method with continuous nodal stress

A three-node triangular element fitted to numerical manifold method with continuous nodal stress, called Trig3-CNS (NMM) element, was recently proposed for linear elastic continuous problems and linear elastic simple crack problems. The Trig3-CNS (NMM) element can be considered as a development of both the Trig3-CNS element and the numerical manifold method (NMM). Inheriting all the advantages of Trig3-CNS element, calculations using Trig3-CNS (NMM) element can obtain higher accuracy than Trig3 element without extra degrees of freedom (DOFs) and yield continuous nodal stress without stress smoothing. Inheriting all the advantages of NMM, Trig3-CNS (NMM) element can conveniently treat crack problems without deploying conforming mathematical mesh. In this paper, complex problems such as a crucifix crack and a star-shaped crack with many branches are studied to exhibit the advantageous features of the Trig3-CNS (NMM) element. Numerical results show that the Trig3-CNS (NMM) element is prominent in modeling complex crack problems.

Hypocycloidal Inclusions in Nonuniform Out-of-Plane Elasticity: Stress Singularity vs. Stress Reduction

Stress field solutions and Stress Intensity Factors (SIFs) are found for $n$-cusped hypocycloidal shaped voids and rigid inclusions in an infinite linear elastic plane subject to nonuniform remote antiplane loading, using complex potential and conformal mapping. It is shown that a void with hypocycloidal shape can lead to a higher SIF than that induced by a corresponding star-shaped crack; this is counter intuitive as the latter usually produces a more severe stress field in the material. Moreover, it is observed that when the order $m$ of the polynomial governing the remote loading grows, the stress fields generated by the hypocycloidal-shaped void and the star-shaped crack tend to coincide, so that they become equivalent from the point of view of a failure analysis. Finally, special geometries and loading conditions are discovered for which there is no stress singularity at the inclusion cusps and where the stress is even reduced with respect to the case of the absence of the inclusion. The concept of Stress Reduction Factor (SRF) in the presence of a sharp wedge is therefore introduced, contrasting with the well-known definition of Stress Concentration Factor (SCF) in the presence of inclusions with smooth boundary. The results presented in this paper provide criteria that will help in the design of ultra strong composite materials, where stress singularities always promote failure. Furthermore, they will facilitate finding the special conditions where resistance can be optimized in the presence of inclusions with non-smooth boundary.

Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields. Part II: Singularities, annihilation and invisibility

Notch stress intensity factors and stress intensity factors are obtained analytically for isotoxal star-shaped polygonal voids and rigid inclusions (and also for the corresponding limit cases of star-shaped cracks and stiffeners), when loaded through remote inhomogeneous (self-equilibrated, polynomial) antiplane shear stress in an infinite linear elastic matrix. Usually these solutions show stress singularities at the inclusion corners. It is shown that an infinite set of geometries and loading conditions exist for which not only the singularity is absent, but the stress vanishes ('annihilates') at the corners. Thus the material, which even without the inclusion corners would have a finite stress, remains unstressed at these points in spite of the applied remote load. Moreover, similar conditions are determined in which a star-shaped crack or stiffener leaves the ambient stress completely unperturbed, thus reaching a condition of 'quasi-static invisibility'. Stress annihilation and invisibility define optimal loading modes for the overall strength of a composite and are useful for designing ultra-resistant materials.

Isotoxal star-shaped polygonal voids and rigid inclusions in nonuniform antiplane shear fields. Part I: Formulation and full-field solution

An infinite class of nonuniform antiplane shear fields is considered for a linear elastic isotropic space and (non-intersecting) isotoxal star-shaped polygonal voids and rigid inclusions perturbing these fields are solved. Through the use of the complex potential technique together with the generalized binomial and the multinomial theorems, full-field closed-form solutions are obtained in the conformal plane. The particular (and important) cases of star-shaped cracks and rigid-line inclusions (stiffeners) are also derived. Except for special cases (addressed in Part II), the obtained solutions show singularities at the inclusion corners and at the crack and stiffener ends, where the stress blows-up to infinity, and is therefore detrimental to strength. It is for this reason that the closed-form determination of the stress field near a sharp inclusion or void is crucial for the design of ultra-resistant composites.

An Exact Analytical Solution for Star-Shaped Cracks

Using the method of complex analysis and by constructing conformal mapping, the study investigates the plane elasticity problem of star-shaped cracks and provides an analytical solution for the stress intensity factor (SIF) of crack-tip type I and II. Problems of the classic Griffith crack, the cross-shaped crack, concurrent uniformly distributed three-cracks and symmetrical eight-cracks are also simulated.

A partition of unity formulation referring to the NMM for multiple intersecting crack analysis

A partition of unity method (PUM) is presented for multiple intersecting crack analysis. In the present PUM, the influence domain at each node is defined by a polygon composed of all the elements connected to the node. The nodal influence domain is partitioned by the material discontinuities, e.g. cracks into several sub-domains where the local displacement functions are constructed using the partition of unity (PU). The proposed PUM unifies the continuum analyses and discontinuum analyses within the same framework, which has a straightforward formulation to implement. The developed PUM is capable for modeling complex cracks such as star-shaped cracks and branched cracks. A number of numerical examples indicate the accuracy and robust of the present method.

Analytical Solution of the Star-Shaped Crack in One-Dimensional Hexagonal Quasicrystals

With the use of complex analysis, and by introducing adequate conformal mapping, the anti-shear problem of the star-shaped crack in One-dimensional Hexagonal Quasicrystals was studied. An analytical solution to the crack tip stress intensity factor is found.

An Analytical Theory for Radial Crack Propagation: Application to Spherical Indentation

A simple analytical theory is proposed for estimating the number of radial cracks which will propagate in brittle materials subjected to axisymmetric transverse surface loads. First, an expression is obtained for the stress intensity factor of a traction-free star-shaped crack in an infinite elastic membrane subjected to axisymmetric transverse loads. Combining this relation with the critical stress intensity factor criterion for fracture, an implicit expression is obtained which defines the number of cracks as a function of the applied loading, initial flaw size, and fracture toughness. Based on the form of this expression, we argue that if the initial flaw size is sufficiently small compared to the length scale associated with the loading, then the number of cracks can be determined approximately in closed-form from the analysis of a traction-free star-shaped crack in a thin body subjected to uniform equibiaxial in-plane tension. In an attempt to validate the theory, comparisons are made with spherical micro-indentation experiments of silicon carbide (Wereszczak and Johanns, 2008, “Spherical Indentation of SiC,” Advances in Ceramic Armor II, Wiley, NY, Chap. 4) and good agreement is obtained for the number of radial cracks as a function of indentation load.

The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials

Using the complex variable method and a new conformal mapping, the fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials is considered under remotely uniform in-plane electric and anti-plane mechanical loadings. The analytic solutions of the field intensity factors and the energy release rate are presented by taking the effect of dielectric permittivity into consideration. Known results can be derived as special cases from the general solutions. The results are illustrated with plots, showing that the piezoelectric material containing three radial cracks spaced equally at 120° apart originating from a circular hole is the easiest to fail for all cases of cracks originating from a circular hole under in-plane electric and anti-plane mechanical loadings. Moreover, if there exist multiple cracks (n ≥ 3) originating from a circular hole in piezoelectric materials, an increase in the number of cracks can enhance the reliability of these materials. The star-shaped cracks possess similar characteristics.

The Propagation of Star-Shaped Brittle Cracks

The paper studies the dynamical propagation of star-shaped cracks symmetrically arranged in an elastic thin plate, subjected to the action of instantly applied, comprehensively (uniformly) stretching stresses, which implies a self-similar problem with homogeneous stresses and velocities of particles. Occurrence of such motion patterns is established through experiments. By using the Smirnov–Sobolev functional-invariant solutions method and a careful choice of mappings, the problem is reduced to some boundary value problem of the theory of complex variable functions, and exact analytic solution of the original problem, including a closed-form solution for important stress intensity coefficient near the end of the crack, is derived. We also establish a fundamental theoretical limit imposed on the number of cracks—there has to be at least three cracks.

First Report of Acidovorax avenae subsp. citrulli Infecting Edible Seed Watermelon (Citrullus lanatus var. lanatus) in China.

Edible seed watermelon, Citrullus lanatus var. lanatus, is a type of watermelon from which only the seeds are consumed. Between 2003 and 2005, a bacterial disease was discovered on this type of watermelon in several regions of Xinjiang Province in western China, with a diseased area of 40,000 ha. The average disease incidence on fruit and seedlings was approximately 30 and 20%, respectively. Symptoms were most noticeable on fruit. Lesions on the fruit rind were first noticeable as small pinpoint water-soaked areas. As the lesions rapidly enlarged, black, starshaped cracks formed in the rind, and the rind along with the flesh around the spots hardened. If no secondary invaders or saprophytes entered through the cracks, the inside flesh often disintegrated into dry stiff cavities. The fruit with necrotic spots became markedly malformed. The pathogen also infected seedlings. First symptoms on the seedling appeared as dark, water-soaked lesions on the underside and edge of the cotyledons. As the cotyledon expanded, the lesions become necrotic and often extended along the length of the midrib and eventually merged into large, black, withered areas. To determine the causal agent of the disease, 54 bacterial strains were isolated from cotyledons, fruits, or seeds of edible seed watermelon. By spray inoculation, these strains were inoculated onto edible seed watermelon cv. Xinzigua1 at the two-true-leaf stage with bacterial concentrations at 107 CFU/ml. All strains were strongly virulent to cv. Xinzigua1, which is planted in large areas in Xinjiang, by forming many water-soaked spots on cotyledons and true leaves. There were no symptoms in the control that was sprayed with sterile deionized water. Through biological and biochemical characteristics, including Gram reaction and tests for catalase, oxidase, and oxygen requirement, all strains were identified as Acidovorax avenae subsp. citrulli. To verify pathogen identification, specific polymerase chain reaction was also carried out. One set of 16S primers, WFB1/WFB2 for A. avenae subsp. citrulli, was used (1). A single unique band of approximately 360 bp was amplified for all strains tested. To our knowledge, this is the first report of A. avenae subsp. citrulli infecting edible seed watermelon. Reference: (1) R. R. Walcott and R. D. Gitaitis. Plant Dis.84:470, 2000.

Superposition method for calculating singular stress fields at kinks, branches and tips in multiple crack arrays

A method is developed for calculating stresses and displacements around arrays of kinked and branched cracks having straight segments in a linearly elastic solid loaded in plane stress or plain strain. The key idea is to decompose the cracks into straight material cuts we call `cracklets', and to model the overall opening displacements of the cracks using a weighted superposition of special basis functions, describing cracklet opening displacement profiles. These basis functions are specifically tailored to induce the proper singular stresses and local deformation in wedges at crack kinks and branches, an aspect that has been neglected in the literature. The basis functions are expressed in terms of dislocation density distributions that are treatable analytically in the Cauchy singular integrals, yielding classical functions for their induced stress fields; that is, no numerical integration is involved. After superposition, nonphysical singularities cancel out leaving net tractions along the crack faces that are very smooth, yet retaining the appropriate singular stresses in the material at crack tips, kinks and branches. The weighting coefficients are calculated from a least squares fit of the net tractions to those prescribed from the applied loading, allowing accuracy assessment in terms of the root-mean-square error. Convergence is very rapid in the number of basis terms used. The method yields the full stress and displacement fields expressed as weighted sums of the basis fields. Stress intensity factors for the crack tips and generalized stress intensity factors for the wedges at kinks and branches are easily retrieved from the weighting coefficients. As examples we treat cracks with one and two kinks and a star-shaped crack with equal arms. The method can be extended to problems of finite domain such as polygon-shaped plates with prescribed tractions around the boundary.

Crack initiation, propagation and branching in a disk of brittle material under axisymmetric tension

The most stable crack nucleation and crack branching patterns in an infinite plate under axisymmetric tension were analysed by using Griffith theory. The analytical results were compared with the experimental results of flexural tests on glass, ceramic, and acryl disks. The strain energy released by nucleation of a star-shaped crack, U, was approximately given by U = (2πσ 2a 2/E)(1 − 1/n) , and the strain energy release rate, G n , by G n = (47πσ 2a/E)(1/n)(1 − 1/n) , were σ is nominal tensile stress, a is crack length, n is crack number, and E is elastic modulus. The energy consideration on the basis of Griffith theory indicated that the most stable crack nucleation shape in a homogeneous brittle material is a line-shaped crack ( n = 2). In addition, the most stable crack length after the ith branching has occurred, a i , is given as: a i = {(2 2i + 1)/(2 i + 1 − 1)}{Eγ/πσ i 2])} , where σ i is the nominal stress at the ith crack branching and γ is the surface energy per unit area. The experimental results of flexural tests on the disk specimens of brittle materials corresponded well to the analytical results.