Crack Path Instabilities Research Articles

Three-Dimensional Singular Stress Fields and Interfacial Crack Path Instability in Bicrystalline Superlattices of Orthorhombic/Tetragonal Symmetries

First, a recently developed eigenfunction expansion technique, based in part on the separation of the thickness variable and partly utilizing a modified Frobenius-type series expansion technique in conjunction with the Eshelby–Stroh formalism, is employed to derive three-dimensional singular stress fields in the vicinity of the front of an interfacial crack weakening an infinite bicrystalline superlattice plate, made of orthorhombic (cubic, hexagonal, and tetragonal serving as special cases) phases of finite thickness and subjected to the far-field extension/bending, in-plane shear/twisting, and anti-plane shear loadings, distributed through the thickness. Crack-face boundary and interface contact conditions as well as those that are prescribed on the top and bottom surfaces of the bicrystalline superlattice plate are exactly satisfied. It also extends a recently developed concept of the lattice crack deflection (LCD) barrier to a superlattice, christened superlattice crack deflection (SCD) energy barrier for studying interfacial crack path instability, which can explain crack deflection from a difficult interface to an easier neighboring cleavage system. Additionally, the relationships of the nature (easy/easy, easy/difficult, or difficult/difficult) interfacial cleavage systems based on the present solutions with the structural chemistry aspects of the component phases (such as orthorhombic, tetragonal, hexagonal, as well as FCC (face-centered cubic) transition metals and perovskites) of the superlattice are also investigated. Finally, results pertaining to the through-thickness variations in mode I/II/III stress intensity factors and energy release rates for symmetric hyperbolic sine-distributed loads and their skew-symmetric counterparts that also satisfy the boundary conditions on the top and bottom surfaces of the bicrystalline superlattice plate under investigation also form an important part of the present investigation.

Analysis of crack path instabilities in a quenched glass plate using the phase-field cohesive zone model

Analysis of crack path instabilities in a quenched glass plate using the phase-field cohesive zone model

Numerical analyses of crack path instabilities in quenched plates

Crack path instabilities are observed in rapidly quenched rectangular glass plates whereby wavy crack patterns form as a result of the induced temperature gradients. The peculiar characteristic of these instabilities is that the speed of propagation is several order of magnitudes lower that the Rayleigh wave speed. Experimental studies have shown the dependence of the instabilities on certain geometrical, material, and experimental parameters (e.g. plate width, material toughness, speed of quenching). By perturbing these parameters cracks are observed to propagate along a straight line, oscillate with a periodic sinusoidal or semi-circle like morphology, or propagate in a supercritical manner. Here we formulate the problem of a propagating crack in a brittle thermoelastic material while considering the possibility of the crack undergoing bursts of supercritical crack propagation, by extending the model in Negri and Ortner (2008). We also describe a novel higher order computational framework for its numerical solution centered around Universal Meshes, Mapped Finite Element Methods, and Interaction Integral Functionals. We verify the convergence of the results and compare them against experiments. We reveal crack behaviors not previously observed. Particularly we discuss periods of sudden crack propagation, followed by temporary arrest and crack kinking. We identify various crack morphologies: sinusoidal, asymmetric, semi-circle, kinked and flattened oscillations. We investigate the frequency content of the oscillatory crack paths and study their relation to the dominating problem parameters. Additionally, we identify two new thresholds in phase space corresponding to the transition from oscillatory propagation to rapid propagation and arrest, as well as from permanent crack arrest to temporary crack arrest followed by kinking and branching.

Nature of Crack Path Instabilities in Thin Sheets Cut by Blunt Objects

Cutting a brittle thin sheet with a blunt object leaves an oscillating crack that seemingly violates the principle of local symmetry for fracture. We experimentally find that at a critical value of a well chosen control parameter the straight propagation is unstable and leads to an oscillatory pattern whose amplitude and wavelength grow by increasing the control parameter. We propose a simple model that unifies this instability with a related problem, namely that of a perforated sheet, where through a similar bifurcation a series of radial cracks spontaneously spiral around each other. We argue that both patterns originate from the same instability.

Crack deflection in short fiber reinforced composites: Theory and experiment

AbstractIn the past decades different crack growth and crack deflection criteria have been proposed, e.g. the maximum hoop stress criterion [1], the minimum strain energy density criterion [2] or the maximum energy release rate criterion [3]. These criteria were extended concerning phenomena such as crack path instability [4,5] and the anisotropy of fracture parameters [4]. Another approach for the prediction of crack growth and deflection angle is based on the Jk‐integral [6], assuming the crack to grow into the direction of the J‐vector [7]. The resulting deflection angles coincide well with those from other criteria in the case of small mixed‐mode ratios KII/KI or deflection angles, respectively. One challenge is the accurate calculation of the second coordinate J2 [8] and the appropriateness of this approach has been shown [9]. Nevertheless, in the cases of anisotropic crack resistance parameters, large mode‐II loading or unstable crack paths, the classical J‐vector criterion fails.In this paper, an extended J‐vector criterion is introduced providing the crack growth direction in the case of anisotropic fracture parameters and elastic constants [9,10]. The theory is validated by experimental results for different anisotropy ratios and orientations.

Crack path stability in brittle fracture under pure mode I loading

The crack growth path is expected to be along the initial crack line when the pre-cracked components possess symmetric geometry and loading conditions relative to the crack line. However, it has been previously shown that in some mode I specimens the path of crack growth is not stable and deviates from its initial line after some stages of crack growth. The aim of this paper is to develop an energy-based theoretical model for predicting the instability in the path of crack growth. The theoretical model takes into account both the singular term of stress ahead of the crack tip and the first non-singular term known as the T-stress. The corresponding two-term stresses are replaced in the energy relation around the crack tip and a model is extracted for predicting instability of crack path under mode I loading. The results obtained from the energy-based criterion was then compared to the results obtained from the well-known Generalized Maximum Tangential Stress (GMTS) criterion, which is a two-parameter stress-based fracture criterion that considers the effect of T-stress. To validate the theoretical model, the experimental results published recently from fracture test on several mode I cracked specimens are used. Very good prediction is provided for the path of crack growth in these specimens. It is shown the crack path instability is significantly geometry dependent and can be prevented by modifying the specimen geometry or loading type. The results obtained in this research are important because an appropriate knowledge about the stability of crack path and fracture trajectory can play a key role on the extent of damage that occurs in a cracked structure when it experiences mode I brittle fracture.

Elastic vortices and thermally-driven cracks in brittle materials with peridynamics

Instabilities in thermally-driven crack growth in thin glass plates have been observed in experiments that slowly immerse a hot, pre-notched glass slide into a cold bath. We show that a nonlocal model of thermomechanical brittle fracture with minimal input parameters can predict the entire phase diagram of fracture measured in experiments for the low immersion speed regime. Geometrical restrictions to crack growth commonly found in other approaches are absent here. We discuss a method for determining the appropriate size of the peridynamic horizon based on a data point around a separating line between crack-type zones in the experimental phase diagram. Once the nonlocal size is smaller than the length-scale introduced by the thermal gradient, the computational results show that no fracture criterion is needed beyond Griffith’s criterion to capture the observed instabilities. The combination of thermal gradients and competing contraction forces on the two sides of the crack are behind the observed crack path instabilities. Elastic (velocity) vortices of material points show how and why the cracks develop along the observed paths. Our results demonstrate that thermally-driven fracture in brittle materials can be predicted with accuracy. We anticipate that this model will lead to design protocols for controlled fracture in brittle materials relevant in materials science and advanced manufacturing.

Characteristics of dynamic brittle fracture captured with peridynamics

Using a bond-based peridynamic model, we are able to reproduce various characteristics of dynamic brittle fracture observed in experiments; crack branching, crack-path instability, asymmetries of crack paths, successive branching, secondary cracking at right angles from existing crack surfaces, etc. We analyze the source of asymmetry in the crack path in numerical simulations with an isotropic material and symmetric coordinates about the pre-crack line. Asymmetries in the order of terms in computing the nodal forces lead to different round-off errors for symmetric nodes about the pre-crack line. This induces the observed slight asymmetries in the branched crack paths. A dramatically enhanced crack-path instability and asymmetry of the branching pattern are obtained when we use fracture energy values that change with the local damage. The peridynamic model used here captures well the experimentally observed successive branching events and secondary cracking. Secondary cracks form as a direct consequence of wave propagation and reflection from the boundaries.

Fractography of Dynamic Crack Propagation in Silicon Crystal

The phenomena occurring during rapid crack propagation in brittle single crystals were studied by cleaving silicon specimens on the low energy cleavage planes under tensile and bending. The experiments revealed new phenomena not previously reported, and new crack path instabilities in particular. The well defined boundary conditions of the tested specimens and crack velocity measurements enabled rationalization of the observed phenomena and the velocity-surface instabilities relationship in particular. In contrast to amorphous materials, the observed instabilities are generated at relatively low velocity, while at high velocity the crack path remains stable. No evidences for mirror, mist, and hackle instabilities, typical in amorphous materials, were found.

Global bifurcation criterion for oscillatory crack path instability

A bifurcation criterion for the transition from straight to oscillatory quasistatic crack propagation in an isotropic material is derived from the requirement of pure mode I stress fields at the crack tip (K_{II}=0) on the entire crack path, henceforth called global bifurcation criterion. For a small-amplitude sine-shaped crack path which is observed in experiments at the transition, it is shown to be sufficient to postulate K_{II}=0 only for two phases of the crack path instead, which simplifies calculations. By using the measured temperature fields to solve the thermoelastic problem of dipping a hot thin glass slab into cold water, critical wavelengths of the oscillating crack growth obtained with the derived global bifurcation criterion agree remarkably well with those observed in experiments by Ronsin and Perrin. It is also shown that local bifurcation criteria, which do not take into account K_{II}=0 on the entire crack path, lead to incorrect results for the oscillatory crack path instability.

Dynamic instabilities in {1 1 1} silicon

The phenomena occurring during rapid crack propagation in brittle single crystals was studied by cleaving strip-like silicon specimens along the {1 1 1} low-energy cleavage plane under bending. The experiments reveal phenomena associated with rapid crack propagation in brittle single crystals not previously reported, and new crack path instabilities in particular. In contrast to amorphous materials, the observed instabilities are generated at relatively low velocity, while at high velocity the crack path remains stable. The experiments demonstrate that crack velocity in single crystals can attain the theoretical limit. No evidence for mirror, mist, and hackle instabilities, typical in amorphous materials, was found. The important role played by the atomistic symmetry of the crystals on controlling and generating the surface instabilities is explained; the importance of the velocity and orientation-dependent cleavage energy is discussed. The surface instabilities are generated to satisfy minimum energy dissipation considerations. These findings necessitate a new approach to the fundamentals of dynamic crack propagation in brittle single crystals.

Evaluating crack tip stress field in a thin glass plate under thermal load

The stress field around a propagating crack tip in a quenched thin glass plate is discussed through experimental and theoretical analyses. Instantaneous phase-stepping photoelasticity using a CCD camera equipped with a pixelated micro-retarder array is used for measuring the crack tip stress field. From the successive phase maps of principal direction, the position and the velocity of the crack tip are evaluated. On the other hand, the fracture parameters, that is, the stress intensity factors and the T-stress are determined from the phase maps of the retardation. Experimental results obtained for a straight crack show good agreement with those obtained by theory of elasticity. The results also indicate that the direction of the crack propagation arising in the quenching process is not determined by the direction of the maximum principal stress. Furthermore, the results show that the T-stress criterion is inappropriate to evaluate the crack path instability in a quenched thin glass plate.

Brittle fracture dynamics with arbitrary paths I. Kinking of a dynamic crack in general antiplane loading

We present a new method for determining the elasto-dynamic stress fields associated with the propagation of anti-plane kinked or branched cracks. Our approach allows the exact calculation of the corresponding dynamic stress intensity factors. The latter are very important quantities in dynamic brittle fracture mechanics, since they determine the crack path and eventual branching instabilities. As a first illustration, we consider a semi-infinite anti-plane straight crack, initially propagating at a given time-dependent velocity, that changes instantaneously both its direction and its speed of propagation. We will give the explicit dependence of the stress intensity factor just after kinking as a function of the stress intensity factor just before kinking, the kinking angle and the instantaneous velocity of the crack tip.

Crack path instabilities in DCDC experiments in the low speed regime

We studied the low speed fracture regime (10 −4–10 −9 m s −1) in different glassy materials (soda-lime glass, glass ceramics) with variable but controlled length scale of heterogeneity. The chosen mechanical system enabled us to work in pure mode I (tensile) and at a fixed load on double cleavage drilled compression specimen. The internal residual stresses of studied samples were carefully relaxed by appropriate thermal treatment. By means of optical and atomic force microscopy techniques fracture surfaces have been examined. We have shown for the first time that the crack front line underwent an out-of-plane oscillating behavior as a result of a reproducible sequence of instabilities. The wavelength of such a phenomenon is in the micrometer range and its amplitude in the nanometer range. These features were observed for different glassy materials providing that a typical length scale characterizing internal heterogeneities was lower than a threshold limit estimated to few nanometers. This effect is the first clear experimental evidence of crack path instabilities in the low speed regime in a uniaxial loading experiment. This phenomenon has been interpreted by referring to the stability criterion for a straight crack propagation as presented by Adda-Bedia et al. [Phys. Rev. Lett. 76 (1996) 1497].

Process zone size effects on naturally curving cracks

In a fuselage crack turning application, a crack may grow under stable tearing conditions for some length, achieving a somewhat steady-state condition, then encounter a region of high tensile T-stress as it nears a stiffener, causing the crack to turn. While a T-stress related crack path instability has been described by Cotterell and Rice [Int J Fract 16 (1980) 155], recent experimental evidence indicates that crack turning is also influenced by the size of the process zone. In order to study this problem, the perturbed crack path solution of Cotterell and Rice is extended to include the effects of a process zone represented by cohesive tractions on the crack flanks trailing the crack tip. The characteristic length, r c, over which the tractions are prescribed, represents a strain-localization zone which precedes the physical crack tip, and through which the crack presumably must pass. The strain-localization zone is assumed to be smaller than common measures of plastic zone size, and other than the strain-localization zone, plasticity is not explicitly modeled. The solution, which uses cohesive tractions analogous to the Dugdale–Barenblatt crack tip model [J Mech Phys Solids 8 (1960) 100; Advances in Applied Mechanics, vol. VII, Academic Press, 1962. p. 55], is accurate to first-order deviations from a straight crack path in an infinite medium for small strain-localization zones. A correction factor is provided to the process zone parameter to approximate the solution for larger process zones, but loses accuracy as the strain-localization zone approaches the plastic zone size. Increased process zone size is shown to result in an increase in the perturbation sensitivity of a crack in a positive T-stress environment, causing more rapid turning of the crack, as has also been experimentally observed.

Crack path instabilities in a quenched glass plate

A crack driven by steadily quenching a hot glass plate into a water bath exhibits fascinating path instabilities that depend on the severity of the temperature jump involved in the quenching process. An experimental and analytical investigation of these crack path instabilities is presented in this paper. First, real-time observations of the growth of the crack at different temperature jumps are presented. With increasing temperature jump, the crack path changes from a straight path to a periodic oscillatory path, then to a chaotic oscillatory path, to an unstable (dynamic) straight crack path, and finally to multiple crack paths. The experimental observations show that the crack growth is not a steady process even though the quenching occurs at a constant speed. The experimental observations provide crucial insight and the basis for the analysis of the problem. The problem is then examined analytically by solving the underlying thermoelastic problem, within the framework of linear elastic fracture mechanics. The results of the numerical simulations in conjunction with the experimental observations are used to show that (i) the T-stress criterion commonly used for evaluating crack path instability is inapplicable to this problem, (ii) the crack path stability can be determined by using the maximum tangential stress criterion for crack advance and examining the divergence of adjacent crack paths, (iii) the initiation of a structurally unstable crack can be determined, and (iv) the complete oscillatory crack path can be obtained through an incremental solution of the thermoelastic problem.