Perturbation analysis for an imperfect interface crack problem using weight function techniques

The concept of material configurational stress (or called Eshelby stress) is used to describe the material yielding state and to predict the crack propagation in this paper. The equivalent configurational stress \((b_{v})\) is introduced to successfully describe the elastic–plastic boundary near the crack tip. Moreover, the crack propagation initiation is predicted by the total change in potential energy calculated from the magnitude of \(b_{v}\) stored inside the plastic zone reaching a critical value; and the crack propagation is stipulated to take place in the direction ahead of the crack tip along which the distance between the crack tip and the elastic–plastic boundary possesses a stationary minimum value. The validation of the proposed yielding and fracture criteria via configurational stress is addressed by comparing with other theories and experiments.

Fracture of brittle and ductile crystals in the generalized stress state. Strength and strain fracture criteria

When the elastic‐brittle cracked body is subjected to compressive‐shear loading, not only does the phenomenon of wing crack initiation and propagation occur, but there is also a notable incidence of secondary crack initiation and propagation due to shear stress. Moreover, the shear fracture toughness (KIIc) predicted by the fracture criterion in the framework of classical fracture mechanics is usually smaller than the tensile fracture toughness (KIc), which is inconsistent with the actual observations. Therefore, this research firstly establishes a mechanical model of an elastic‐brittle cracked body under biaxial compression considering three kinds of crack parameters and materials' mechanical properties. Then, the stress field characteristics at the crack tip are analyzed. Next, a compression‐shear mixed fracture criterion is proposed. Finally, the effects of the lateral pressure coefficient (λ), critical plastic zone size (rc), friction coefficient (f), and deformation parameters of the crack surface on the failure modes at the crack tip are investigated.

The presence of hydrogen in zirconium alloys causes the hydrides to precipitate when hydrogen solubility limits are exceeded. Hydrides in zirconium alloys can lead to a significant embrittlment effect under certain circumstances, and mechanical properties, especially crack growth resistance, may be affected. As a result, there is a phenomenon of delayed hydride cracking (DHC) of zirconium alloys which is characterized by crack propagation at low values of the stress intensity factor. One of important mechanical properties of zirconium alloys is the threshold stress intensity factor, KIH, below which cracks do not grow. Therefore, the problem of hydrogen embrittlment of zirconium alloys is a very important, particularly for structural components in the nuclear industry. For example, the gradual build-up of deuterium in the Zr-2.5Nb pressure tubes of a pressurized heavy water reactor as a result of corrosion, combined with the possibility of the generation of flaws of sufficient size, makes the tubes susceptible to the initiation of DHC [1]. In this case, the service life of a pressure tube is determined by its capability to resist the crack initiation and propagation. Because of the technological importance of this phenomenon, the crack initiation threshold due to the hydride formation has been studied. A theoretical prediction of the KIH value of zirconium alloys can be based on modelling of the process zone ahead of the crack tip (e.g. [2–4]). The process zone is treated as a zone of fracture process. On the other hand, hydrided zirconium alloy can be interpreted as zirconium-zirconium hydride mixture. In this case, to predict the value of KIH of hydrided zirconium alloy it is necessary to estimate the fracture toughness of the hydride. However, measurements of fracture toughness of δ-zirconium hydrides showed large scatter due to specimen problems because of their extreme brittleness [5]. Therefore, a theoretical estimation of the value of K H IC for bydride is very important for an analysis and optimisation of crack growth resistance in zirconium alloys. This paper presents a theoretical estimation of fracture toughness of zirconium hydrides. The theory is based on a local approach to fracture of zirconium alloys. Recently, a model for the threshold stress intensity factor, KIH, was suggested leading to an analysis of crack growth initiation in zirconium alloys during DHC [4]. Local criterion of brittle fracture and modelling of the fracture process zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. The hydride platelet was assumed to cover the process zone and the crack growth proceeds through fracturing of the hydride. The critical condition for fracture initiation in the hydride is reached when the maximum of the local stress approaches the fracture strength of the hydride at some characteristic (critical) distance, rc, ahead of the crack tip (Fig. 1). It was shown that the theoretical KIH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of KIH for a single hydride platelet at the crack tip characterizes the ultimate lower bound value of crack resistance in hydrided zirconium alloy during DHC, whereas the minimum of the threshold stress intensity factor, (KIH)min, should be considered as the minimal capability of the hydride to resist fracture within the hydride, i.e. the fracture toughness, K H IC, of hydride. The minimum fracture toughness of structural materials can be described by a model called the Kμ-model [6]. The following equation should be valid

Mechanical stresses (i.e., surface tension) inherently exist on a general free surface because its atomic structure differs from its bulk counterpart. The effect of surface tension is amplified at a crack tip because of curvature enhancement. An integral equation describing this effect on the toughness of glass is derived and first‐order approximations using a weight function technique were made. The qualitative results indicate that the geometry‐induced toughening is linearly proportional to surface tension and crack tip curvature and to the square root of crack tip zone size. An illustrative example of a recently observed crack tip in SiO2 glass is given which shows that toughness is enhanced by ∼3 times the intrinsic K1C value.

Directionally solidified superalloy, which has elongated large grains, is used for gas-turbine blade application because of its high creep strength. Since the grain size is not small enough in comparison with the component size or the crack size, the inhomogeneous microstructure strongly affects the crack propagation behavior in such a structure. The authors investigate the effect of microstructural inhomogeneity on crack propagation morphology and crack propagation rate (da/dN) under a creep-fatigue condition. The macroscopic direction of the main crack is perpendicular to the loading axis, though the path is microscopically bumpy due to the effect of inhomogeneous multi-grains. Several subcracks are found around the main crack, and they occasionally coalesce with each other. In order to understand the local fluctuation of (da/dN) due to the inhomogeneous multi-grains, the grain shape and its crystallographic orientation are identified by means of a scanning electron microscope (SEM) and an electron back scattering diffraction (EBSD). The subcracks tend to initiate at the high angle grain boundaries oriented nearly perpendicular to the loading axis. The local fluctuation of da/dN is influenced by two factors. One is the cracking path; the intergranular crack shows higher da/dN. The other factor is the magnitude of Young’s modulus of the grain (in loading direction) at the crack tip. The higher Young’s modulus leads to a lower da/dN in transgranular cracking. The detailed observation of local displacement near the crack tip clarified that the magnitude of cyclic displacement inversely related to the Young’s modulus of the grain in the loading axis, and results in the high strain concentration at the crack tip in the grain with low Young’s modulus. The experiment result suggests that the strain around the crack tip governs the crack propagation rate though it is affected by the inhomogeneous microstructure.

Rock mass behavior is determined not only by the properties of the rock matrix but also mostly by the pre-existing cracks in the rock mass. Before the overall failure of rock, the crack initiation and propagation around the tip of pre-existing cracks (i.e., pre-crack) will occur and contribute to rock failure. In this paper, the deep granite from a gold mine is taken and made to specimens with the pre-crack of 0.3 mm thickness. Uniaxial compression tests are carried out on the pre-cracked specimens. The acoustic emission (AE) sensors and digital image correlation (DIC) system are employed to record the failure characteristics of the specimens. The extended finite element method (XFEM) with the non-local stress field calculation is used to simulate the crack initiation and propagation of pre-cracks. The crack patterns, opening and shearing displacements of the cracked surface, and the crack length development are obtained from numerical simulations. Finally, the effects of friction of crack surface on the crack pattern and crack propagation are investigated and discussed. It has been found that, for pre-cracked specimens, crack initiation and propagation will occur when the stress is much smaller than the rock compressive strength. And in the range of pre-crack angle 30-60°, the larger the pre-crack angle is, the larger the compressive strength is. The crack patterns from numerical simulations have a good agreement with those from experimented DIC results. Moreover, the order of crack propagation speed is consistent with the order of the compressive strength. The crack pattern and crack propagation are affected by the friction coefficient of the cracked surface.

The objective of the present research is to build a modeling method for delayed hydride cracking (DHC) of zirconium alloys. DHC tests were carried out on Zircaloy-2 cladding tubes in the chamber of a scanning electron microscope to directly observe the crack propagation and measure the crack velocity in the radial direction. These in situ observations showed that a sharply tipped crack propagated at a relatively high rate, while the velocity decreased when the crack tip was blunted, supporting the occur-rence of intermittent crack propagation that could be expected from the DHC mechanism. V-KI curves or diagrams of crack velocity, V, versus stress intensity factor at a crack tip, KI, were obtained as a function of 0.2 % offset yield stress, hydride orientation, and pre-crack depth. The steady state crack velocity and the threshold stress intensity factor for the onset of the crack propagation tended to increase or decrease, respectively, with an increase in the 0.2 % offset yield stress. Analyses of stress distribution and hydrogen diffusion around a crack tip were made using a finite element computer code. The analyses showed that a strong hydrostatic pressure field was generated concentrically around the crack tip and hydrogen diffused towards the crack tip according to the hydrostatic pressure gradient. The crack velocity was estimated from the calculated hydrogen flux rate assuming the critical hydrogen quantity for the crack propagation. There was good agreement between the experiments and the calculations regarding the crack velocity and its dependency on KI. Calculations showed that the increase in the 0.2 % offset yield stress would accelerate the crack propagation by increasing the hydro-static pressure at the crack tip.

This paper presents an atomistic understanding of the nanoscale processes that govern crack nucleation and propagation in amorphous silica using a combination of theoretical calculations and molecular dynamics simulations with the ReaxFF potential. We show that both crack nucleation and propagation are governed by chainlike nanoscale virial stress-fibers formed by an intricate mixture of Si and O atoms. The stress-fibers are of a few nanometers in length, aligned parallel to the loading direction, and spatially localized at the evolving crack front at the nucleating site or a propagating crack tip. They form and break continuously during the crack nucleation and propagation process and are responsible for localizing stress at the crack nucleation site or propagating crack front. As soon as the stress-fibers reach a critical density the material starts nucleating cracks or leads to the propagation of initial crack. Additionally, the virial stress fields in the domain is highly heterogeneous and species-dependent---and the O and Si atoms play fundamentally distinct roles throughout the deformation process. With increased loading, heterogeneity in virial stress for the Si atoms goes up, whereas for the O atoms it goes down. Furthermore, from the virial and Hardy estimates of atomic stress, it is found that the stress field emanating from the crack tip decays as $1/r$. Also, presence of holes or pores within the interaction distance of a crack tip intensifies the stress state of the stress-fibers near the crack front leading to an improved effective toughness compared to the situation where the pore is far from the crack tip. Nucleation and propagation of cracks are strictly mediated by a complex admixture of localized bond rupture processes across a set of interacting stress-fibers. The details of the atomistic process regulating the underlying mechanisms are undetectable from the macroscopic stress-strain data.

The transient elastodynamic response of a transversely isotropic material containing a semi-infinite crack under uniform impact loading on the faces is examined. The crack lies in a principle plane of the material, but the crack front does not coincide with a principle direction. Rather, the crack front is at an angle to a principle direction and thus the problem becomes more three-dimensional in nature. Three loading modes are considered, i.e., opening, in-plane shear and anti-plane shear. The solutions for the stress intensity factor history around the crack tip are found. Laplace and Fourier transforms together with the Wiener-Hopf technique are employed to solve the equations of motion directly. The asymptotic expression of stress near the crack tip leads to a closed-form solution for the dynamic stress intensity factor for each loading mode. It is found that the stress intensity factors are proportional to the square root of time as expected. Results given here converge to known solutions in transversely isotropic materials with a crack oriented along a principle direction and isotropic materials as special cases. The results of this analysis are used to find approximate strain energy release rates for dynamically loaded penny shaped cracks.

Spontaneous propagation of an anti-plane shear crack is studied using Irwin's fracture criterion. We obtain the numerical solution of crack tip loci and slip motion when the critical stress intensity factor Kc and the dynamic stress drop Δσd are non-uniformly distributed on a fault plane. When rupture prop-agates along a fault plane with a localized high Kc region, the rupture process is divided into three stages. In the first stage, the rupture smoothly propagates with the radiation of elastic waves. In the second stage, when the rupture front reaches the high Kc region, the slip motion is decelerated and the radiated elastic wave is weakened. In the third stage, after the crack tip passes the high Kc region, the slip motion is accelerated again and the intensity of radiated wave increases. This gives a possible interpretation of multiple shock events. In Irwin's criterion, the rupture cannot propagate beyond the high Kc region leaving behind an unbroken area. The rupture process controlled by Griffith's fracture criterion is also studied. The rupture process in Griffith's criterion is similar to that in Irwin's criterion, except that the rupture velocity and the radiated seismic wave energy are somewhat greater than the case of Irwin's criterion.

This article presents the dynamic behavior of a semi-infinite interfacial crack in piezoelectric bimaterials under impact loading. With the help of the transform methods (the Laplace transforms and Fourier transforms), the problem is studied with the Wiener–Hopf technique. This strict proof guarantees the feasibility of this approach. The dynamic stress intensity factor and dynamic electric displacement intensity factor of the interfacial crack propagation characteristics are expressed. Finally, several classic numerical examples are mentioned and discussed to demonstrate that the theoretical deduction is highly accurate for interfacial crack analysis of piezoelectric bimaterials. The results show that the crack propagation is affected by the electromechanical coupling coefficient. In addition, if the velocity of the dynamic crack propagation reaches the generalized Raleigh wave speed, the dynamic stress intensity factor will disappear. Furthermore, for a given time, the ratio of the dynamic stress intensity factor to load increases with the electromechanical coupling coefficient decreasing. Numerical examples are presented to highlight the result.

In this work we develop a complete analytical solution for a double cantilever beam (DCB) where the arms are modelled as Timoshenko beams, and a bi-linear cohesive-zone model (CZM) is embedded at the interface. The solution is given for two types of DCB; one with prescribed rotations (with steady-state crack propagation) and one with prescribed displacement (where the crack propagation is not steady state). Because the CZM is bi-linear, the analytical solutions are given separately in three phases, namely (i) linear-elastic behaviour before crack propagation, (ii) damage growth before crack propagation and (iii) crack propagation. These solutions are then used to derive the solutions for the case when the interface is linear-elastic with brittle failure (i.e. no damage growth before crack propagation) and the case with infinitely stiff interface with brittle failure (corresponding to linear-elastic fracture mechanics (LEFM) solutions). If the DCB arms are shear-deformable, our solution correctly captures the fact that they will rotate at the crack tip and in front of it even if the interface is infinitely stiff. Expressions defining the distribution of contact tractions at the interface, as well as shear forces, bending moments and cross-sectional rotations of the arms, at and in front of the crack tip, are derived for a linear-elastic interface with brittle failure and in the LEFM limit. For a DCB with prescribed displacement in the LEFM limit we also derive a closed-form expression for the critical energy release rate, G_c. This formula, compared to the so-called ‘standard beam theory’ formula based on the assumptions that the DCB arms are clamped at the crack tip (and also used in standards for determining fracture toughness in mode-I delamination), has an additional term which takes into account the rotation at the crack tip. Additionally, we provide all the mentioned analytical solutions for the case when the shear stiffness of the arms is infinitely high, which corresponds to Euler–Bernoulli beam theory. In the numerical examples we compare results for Euler–Beronulli and Timoshenko beam theory and analyse the influence of the CZM parameters.

Nanostructurally small cracks (NSC): A review on atomistic modeling of fatigue

The Griffith concept of energy equilibrium for brittle mode I crack propagation was extended to a combined mode I-II crack extension. First, the maximum energy release rate for an infinitesimally kinked crack extension under combined mode I-II loading, analyzed by other researchers, was successfully expressed with a combined mode stress intensity factor. Second, the fracture energy rate was estimated from the area of the frontal process zone estimated from the area enclosed by the isostress contours of both the maximum principal stress and the maximum shear stress. Third, these considerations led to the estimated KIIC/KIC ratio of 1.20, which was close to the experimental results of 1.1 to 1.3 obtained previously. Lastly, the combined mode I-II fracture criterion was formulated by adopting the same procedure, and the predicted criterion coincided well with Shetty's empirical criterion. The fracture toughness values of mode I and mode II, and the combined mode I-II criterion for polycrystalline alumina, float glass and PMMA (polymethyl methacrylate) were also estimated experimentally using a disk method and a rectangular test. The estimated results of the KIIC/KIC ratio using the disk method ranged from 1.1 to 1.3, which was very close to the predicted value of 1.20, whereas the rectangular test specimens exhibited considerable stable crack growth at the crack tip and yielded a smaller KIIC/KIC ratio, which was related to the mode transformation and stress shielding.