Self-similar Crack Research Articles

Angular distribution of energy release rates and fracture of piezoelectric solids

Applicability of energy release rate criteria to piezoelectric materials is examined in thispaper. The angular distribution of strain and total energy release rates at a crack tip in aplane piezoelectric medium is first formulated by using a branched crack model. Byrelaxing the commonly held assumption of self-similar crack propagation andincorporating in the fracture toughness anisotropy, the criteria of modified strainenergy release rate and modified total energy release rate are applied to discuss thepropagation path of an impermeable crack. Numerical results indicate that thepredictions based on the criterion of modified total energy release rate do not showqualitative agreement with experimental observations, whereas the predictions basedon the criterion of modified strain energy release rate appear more promising.Based on the criterion of modified strain energy release rate, a crack tends tobranch off from a straight path under a positive electric loading, regardless ofcrack orientation. An applied electric field can either promote or retard crackpropagation depending on the direction of electric field, crack orientation andcrack extension direction. The predictions made by the energy-based criteria arecompared with those corresponding to a recently reported stress-based criterion.

Time-dependent craze zone growth at a crack tip in polymer solids

By considering the polymer bulk as a linear viscoelastic body and the craze zone at crack tip as a nonlinear damage zone, the control equation for craze zone growth has been derived. It is shown that for a time-independent craze-zone stress, the craze zone would grow only if the crack-tip stress intensity factor is changed. If the crack-tip stress intensity factor remains constant during loading, the growth rate of the craze zone length will be interrelated to the crack-tip stress, the craze zone length and the rate of change of the craze-zone stress. If both the craze-zone stress and the crack-tip stress intensity factor are time-independent, the craze zone length will be constant during the crack growth, which is the case of self-similar crack growth. Moreover, a new stress distribution model in craze zone is presented based on the constructed damage evolution law, and the lengthening and thickening of the craze zone at the crack tip are also formulated. The numerical calculations from the proposed model agree well with the published experimental data.

On the Self-Healing Fracture Mode

We present the analytical solution for a fundamental fracture mode in the form of a self-similar, self-healing pulse. The existence of such a fracture mode was strongly suggested by recent numerical simulations of seismic ruptures but, to our knowledge, no formal proof of their origin has been proposed yet. We present a two-dimensional, anti-plane solution for fixed rupture and healing speeds that satisfies both the wave equation and crack boundary conditions for a simple Coulomb friction law in the absence of any rate or state dependence. This solution is an alternative to the classic self-similar crack solution by Kostrov. In practice, the self-healing impulsive mode rather than the expanding crack mode is selected depending on details of fracture initiation and is thereafter self-maintained. We discuss stress concentration, fracture energy, and rupture velocity and compare them to the case of a crack. The analytical study is complemented by various numerical examples and comparisons. On more general grounds, we argue that an infinity of marginally stable fracture modes may exist in addition to the crack solution or the impulsive fracture described here. Manuscript received 27 March 2002.

Crack size and speed interaction characteristics at micro-, meso- and macro-scale

The motivation to examine physical events at even smaller size scale arises from the development of use-specific materials where information transfer from one micro- or macro-element to another could be pre-assigned. There is the growing belief that the cumulated macroscopic experiences could be related to those at the lower size scales. Otherwise, there serves little purpose to examine material behavior at the different scale levels. Size scale, however, is intimately associated with time, not to mention temperature. As the size and time scales are shifted, different physical events may be identified. Dislocations with the movements of atoms, shear and rotation of clusters of molecules with inhomogeneity of polycrystals; and yielding/fracture with bulk properties of continuum specimens. Piecemeal results at the different scale levels are vulnerable to the possibility that they may be incompatible. The attention should therefore be focused on a single formulation that has the characteristics of multiscaling in size and time. The fact that the task may be overwhelmingly difficult cannot be used as an excuse for ignoring the fundamental aspects of the problem. Local nonlinearity is smeared into a small zone ahead of the crack. A “restrain stress” is introduced to also account for cracking at the meso-scale.The major emphasis is placed on developing a model that could exhibit the evolution characteristics of change in cracking behavior due to size and speed. Material inhomogeneity is assumed to favor self-similar crack growth although this may not always be the case. For relatively high restrain stress, the possible nucleation of micro-, meso- and macro-crack can be distinguished near the crack tip region. This distinction quickly disappears after a small distance after which scaling is no longer possible. This character prevails for Mode I and II cracking at different speeds. Special efforts are made to confine discussions within the framework of assumed conditions. To be kept in mind are the words of Isaac Newton in the Fourth Regula Philosophandi:

Mesoscopic simulation of dynamic crack propagation in rubber materials

Tear fatigue analyzer (TFA) tests on rubber are simulated, for the first time, with a combination of a mesoscopic model of self-similar crack propagation and a complex viscoelastic rubber model (flow-enhanced linear standard solid). In this way, it is possible to establish interconnections between mechanical fracturing tests (TFA) and intrinsic material properties.

Formation and Modelling of Self-Similar Crack Distributions

Formation and Modelling of Self-Similar Crack Distributions

Experimental Study of Self-Similar Crack Distributions

This paper presents laboratory experiments on polyester resin samples subjected to special regimes of curing under which multiple shrinkage cracks with wide distributions of sizes were produced. Loading these samples produced stress-strain curves that followed a power law within certain stress ranges. The results of the tests were analysed using the mechanics of materials with self-similar crack distributions. The analysis showed that the observed power law in the stress-strain relationship indicated that the crack distribution was approximately self-similar. Thus, the described experiments serve as a verification procedure for the proposed formation mechanism of multiscale self-similar crack distribution in brittle materials. This mechanism is based on: (1) the action of self-equilibrating stress fluctuations as initiators and propagators of cracks and (2) the size discriminating action of crack interaction. The concentration factors for the crack distributions have been retrieved from the experimental data.

Fracture behavior of stitched warp-knit fabric composites

Pilot studies are conducted to characterize the macroscopic fracture resistance behavior using linear elastic fracture mechanics and attempt to quantify the fracture parameters in which may govern the fracture and failure patterns of stitched warp-knit fabric composites. Methods based on the J-integral method and Betti's reciprocal theorem in extracting the fracture parameters, critical stress intensity factors, T-stress, and the second term of σy(r,0) near the crack tip prior to fracture initiation are formulated. Two fracture criteria, [σc,rc] and [ec,rc] are attempted to characterize the failure initiation for the fiber-dominated failure mode and self-similar crack extension in a given thickness of the laminate. Based on linear elastic fracture mechanics principle, these criteria are transformed into crack-driving forces [KQ,T] and [KQ,g32]. The two-parameter fracture criteria, [KQ,T] and [KQ,g32] provide a good correlation for the CCT and SENT specimens, but not for the high constraint CT specimens. With the limited experimental data, the results tend to show that the large tensile T-stress and large magnitude of negative g32 may inhibit the crack extension in the same crack plane and promote crack kinking.

Self-similar analysis of plasticity-induced closure of small fatigue cracks

An analytical model of plasticity-induced crack closure is presented that is consistent with self-similar fatigue crack growth, where the plastic zone size and the plastic-wake thickness increase linearly with increasing crack length. The Dugdale model is used to represent crack tip plasticity, which allows the problem to be solved analytically. Detailed results are presented for the key model variables as a function of stress range and load ratio, in particular for the cyclic crack-tip opening displacement which can serve as a correlating parameter for crack growth rates, both within and beyond the small-scale yielding limit. For cases where the crack growth rate is proportional to the cyclic crack-tip opening displacement, results are presented for the variation of the exponent n characterising the dependence of growth rate on the stress range. It is noted that while n=2 in the small-scale yielding limit, as required for consistency with linear elastic fracture mechanics in that limit, the value of n deviates significantly from the small-scale yielding limit, even for applied stresses just above one-fifth of material yield stress. The present model can also be applied to fatigue crack growth under fully-plastic remote loading, with an appropriate choice for the limit stress and the modulus. This is shown to be particularly relevant for predicting small-crack growth rates at notch roots, in the presence of substantial notch-root plasticity. It is noted that small-crack growth rates predicted by previous models would be significantly unconservative relative to predictions from the present model.

The fourth mode of fracture in fractal fracture mechanics

This paper offers a systematic approach for obtaining the order of stress singularity for different self-similar and self-affine fractal cracks. Mode II and Mode III fractal cracks are studied and are shown to introduce the same order of stress singularity as Mode I fractal cracks do. In addition to these three classical modes, a Mode IV is discovered, which is a consequence of the fractal fracture. It is shown that, for this mode, stress has a weaker singularity than it does in the classical modes of fracture when self-affine fractal cracks are considered, and stress has the same order of singularity when self-similar cracks are considered. Considering this new mode of fracture, some single-mode problems of classical fracture mechanics could be mixed-mode problems in fractal fracture mechanics. By imposing a continuous transition from fractal to classical stress and displacement fields, the complete forms of the stress and displacement fields around the tip of a fractal crack are found. Then a universal relationship between fractal and classical stress intensity factors is derived. It is demonstrated that for a Mode IV fractal crack, only one of the stress components is singular; the other stress components are identically zero. Finally, stress singularity for three-dimensional bodies with self-affine fractal cracks is studied. As in the two-dimensional case, the fourth mode of fracture introduces a weaker stress singularity for self-affine fractal cracks than classical modes of fracture do.

Self-similar crack expansion method for analysis of cracks in an infinite medium or semi-infinite medium

Three-dimensional cracks in an infinite medium or semi-infinite medium are analyzed using the self-similar crack expansion (SSCE) method, in which the stress intensity factors at crack tips are determined by calculating the crack-opening displacements over the crack surface. Calculations show that this method yields very accurate results for stress intensity factors of cracks with various geometries. Based on the numerical calculation by the SSCE method, the effects by Poisson’s ratio of materials on stress intensity factors are discussed for cracks in an infinite medium and semi-infinite medium.

Small scale nonlinear analysis of electrostrictive crack problems

The asymptotic problem of a semi-infinite crack in an electrostrictive ceramic under electric loading is analyzed. A universal relation between the stress intensity factor and the electric field intensity factor under small-scale nonlinear conditions is obtained for a conducting crack as well as for an insulating crack. It is shown that the surfaces of the conducting crack are closed except a small zone near the crack tip, whereas all the surfaces of the insulating crack remain open. Energy release rates due to self-similar crack extension as well as due to dissimilar crack extension are also discussed. The energy release rate for the insulating crack does not always have a negative value, in contrast to a linear dielectric theory.

Self-Similar Crack Expansion Method for Three-Dimensional Cracks Under Mixed-Mode Loading Conditions

Three-dimensional planar cracks under mixed-mode loading conditions are investigated by using the selfsimilar crack expansion method with the boundary integral equation technique. For a planar crack under general loading (tensile and shear) conditions, the normal displacement and tangential displacements on the crack surface exhibit uncoupled characteristics. However, the tangential displacements in the two directions are generally coupled. In this paper, two coupled boundary integral equations for a crack subject to shear loading are solved using the analytically numerical method, where the integrals on elements’ are estimated by using the explicit expression of the close form of the integrals. Combination of the self-similar crack expansion method and the analytically numerical method results in good accuracy, with errors in stress intensity factors of penny-shaped cracks and elliptical cracks less than one percent. This numerical analysis is applicable to the analysis of cracks with arbitrary geometry.

Uncoupled characteristics of three-dimensional planar cracks

The uncoupled characteristics of three-dimensional planar cracks under arbitrary loading is investigated by boundary integral equation method. From theoretical analysis, the crack displacements under normal loading and shear loading on the crack surface are uncoupled. For a crack under shear loading (mode II and III), the special case of the crack with uncoupled displacements is investigated, whereby the self-similar crack expansion (SSCE) method, proposed in a previous paper by Xu et al. (1996, Journal of Applied Mechanics) is used for the analysis of three-dimensional cracks under shear loading. A number of cracks with arbitrary geometry and mixed mode loading conditions are calculated in this paper. For cracks with uncoupled displacements under various loading conditions, the numerical results by the SSCE method show good accuracy, with errors in stress intensity factors around the penny-shaped crack edge less than 1%. Results of a penny-shaped crack under torsional loading show good applications of uncoupled characteristics to the stress analysis for three-dimensional cracks.

Self-similar crack expansion method for two-dimensional cracks under mixed mode loading conditions

The self-similar crack expansion (SSCE) method is proposed to evaluate stress intensity factors of two-dimensional cracks under mixed-mode loading conditions. With this method, stress intensity factors for the crack in an infinite medium or semi-infinite medium can be determined by calculating the crack-opening displacements over the crack surface. The energy release rate of an extending crack is precisely estimated by taking advantage of the crack self-similarity. In the boundary integral equation technique, the singular integrals near crack tips are evaluated based on closed form expressions. Combination of these two techniques significantly increases the accuracy in the stress intensity factor (SIF) calculations. A variety of SIFs for cracks, such as subsurface cracks, edge cracks and the interactions among them are calculated using various types of crack expansions in the SSCE method. Results show good accuracy for crack analysis, with errors in SIFs of two-dimensional cracks less than 0.5%. This provides an analytical tool for predicting fatigue life of materials. An interesting finding in the relationship between SIFs is presented for a group of aligned cracks under tensile loading.

Self-Similar Crack Expansion Method for Three-Dimensional Crack Analysis

The self-similar crack expansion method is developed to calculate stress intensity factors for three-dimensional cracks in an infinite medium or semi-infinite medium by the boundary integral element technique. With this method, the stress intensity factors at crack tips are determined by calculating the crack-opening displacements over the crack surface, and the crack expansion rate, which is related to the crack energy release rate, is estimated by using a technique based on self-similar (virtual) crack extension. For elements on the crack surface, regular integrals and singular integrals are evaluated based on closed-form expressions, which improves the accuracy. Examples show that this method yields very accurate results for stress intensity factors of penny-shaped cracks and elliptical cracks in the full space, with errors of less than one percent as compared with exact solutions. The stress intensity factors of subsurface cracks are in good agreement with other numerical solutions.

An engineering model for propagation of small cracks in fatigue

The propagation of small cracks in fatigue has received considerable attention over the past decade. Microstructurally and even mechanically small cracks have been shown to consistently exhibit higher crack growth rates than predicted using standard threshold and Paris growth law concepts, based on linear elastic fracture mechanics (LEFM) applied to mechanically long cracks. This has been commonly attributed to several factors, including the influence of microstructure, the breakdown of LEFM parameters for representation of the crack tip field, and the transient development of plasticity-induced closure towards some steady-state value associated with long crack behavior. Other mechanisms related to microstructural effects such as roughness-induced closure and crack face bridging/interference are also potential contributors. Quantitative attempts to explain the fatigue propagation of small cracks in terms of plasticity-induced closure, along with adoption of an additional component of the driving force (e.g. crack tip opening displacement) to reflect the contribution of cyclic plastic strain have met with some success in correlation of the so-called “anomalous” propagation behavior, including crack deceleration and acceleration transients. However, these models rely on the adoption of highly idealized assumptions regarding the self-similarity of crack growth, neglect of local anisotropy and heterogeneity associated with microstructure, etc.; in spite of these compromises, they still involve a considerable degree of complexity. Here, we adopt the viewpoint that multiple, microstructure interactions and closure effects may simultaneously influence the propagation of small cracks; moreover, driving force parameters based on self-similar crack growth arguments of elastic-plastic fracture mechanics (EPFM) for mechanically long cracks, such as the cyclic J-integral or crack tip opening displacement, may apply in principle, but not rigorously, as the driving force for small cracks. As an engineering approach, we consider a recent extension of the multiaxial microcrack propagation model first proposed by McDowell and Berard[1,2] for the growth of microstructurally small and mechanically small fatigue cracks[3] in multiaxial fatigue. Integrated between initial and final crack lengths, the model is fully consistent with standard strain-life laws of fatigue crack initiation mechanics under various states of stress, [1,2] and therefore bridges the mechanics of classical initiation and LEFM/EPFM to some extent. The existence of a fatigue limit (nonpropagating crack limit) is neglected in this particular work. It is shown for uniaxial loading of both 1045 steel and Inconel 718 that the model is able to describe, to first order, the anomalous high propagation rates of small cracks and convergence with long crack d a/ d N - ΔK data as the crack transitions from small to mechanically long scales. The limits of validity of engineering schemes based on decomposition of total fatigue life into “initiation” and propagation phases that rely on strainlife and long crack propagation laws are discussed[4]. Moreover, it is shown that the model essentially reflects a closure transient in the context of a cyclic J-integral approach, similar to the EPFM plasticity-induced closure modeling concepts set forth by Newman[5] and McClung et al. [6] for small cracks in fatigue. However, the present model implicitly reflects multiple forms of crack tip shielding effects, not just plasticity-induced closure. Finally, the model is shown to provide realistic treatment of cumulative damage in two level loading sequences, as reflected by comparison with the damage curve approach of Manson and Halford[7].

Transonic extension of a self-similar interface crack

The transonic self similar propagation of a mode III interface crack is examined. The analysis employs the moving dislocation distribution approach

Stiffness Evaluation for Solids Containing Dilute Distributions of Inclusions and Microcracks

Materials, such as ceramics, intermetallics, and rocks, contain varying amounts of inhomogeneities, and the matrix material is vulnerable to microcracking in the neighborhood around these inhomogeneities. In an attempt to model the micromechanical aspects of this type of material, a solid containing dilute inclusions surrounded by cracks is investigated in this paper. The dilute-inclusion assumption neglects any interactions among different inclusion-crack clusters, but local inclusion-crack and crack-crack interactions are taken into account fully. It is shown that additional strain due to microcracking in a solid containing inclusions can be represented by an integral of crack opening displacements weighted by a nonuniform stress field induced by inclusions alone (in the absence of microcracking). An effective numerical approach is then developed to evaluate the effective moduli and additional macroscopic strain due to microcracking in composites. It is found that an increase in the number of hard inclusions may not always lead to expected strengthening of the materials, if the matrix material is vulnerable to microcracking around inclusions and a relatively large microcracking zone develops. The limited calculations show that a quasi-static crack-growing process can lead to an actively growing crack being arrested or to a stationary crack starting to grow. This suggests that self-similar crack growth may not be enough to describe the behavior of microcracked composites.

Elastic-plastic analysis of crack propagation with branching

A numerical procedure is described for the elastic-plastic finite element analysis of crack propagation with branching. The constraint equations are used to model the closing and opening of the crack. The constraint conditions are imposed by using the penalty method for self-similar crack propagation and the elimination method for off-axis propagation. Multiple constraints are used to determine the contact mode at the crack branching point. Frictional sliding on the contacting crack surfaces under cyclic loads is also considered in the numerical procedure. The method is then applied to (i) self-similar crack growth in a single edge notch specimen, (ii) self-similar propagation followed by interfacial splitting in a center-cracked 0° composite plate and (iii) bifurcation of a crack in a compact tension specimen.