Griffith Crack Research Articles

Thermal shock fracture associated with a unified fractional heat conduction

Recently, a unified fractional heat conduction model is newly proposed, which is a further improvement of the fractional thermoelasticity theory. It would be interesting to know how the newly developed fractional derivative definitions affects the thermal fracture behavior when some defects such as cracks exist in the medium. This work is aimed at analyzing the thermal fracture problem of an elastic half-space and strip with an insulated Griffith crack based on the unified fractional heat conduction model. Fourier and Laplace transforms are employed to solve the mixed boundary problem associated with a time-fractional partial differential equation. Temperature and stress intensity factors are evaluated by solving a system of singular integral equations. Effects of different fractional derivatives and different parameters on the thermoelastic fields are analyzed. A comparison of the temperature and stress intensity factors between the present model and the Fourier model is made. Numerical results show that different fractional definitions have different memory effects on the transient responses, and the effects of fractional definitions on the responses will vary accordingly with the parameter values.

Cracks of a New Type and Models of Some Nano Materials

It is proved in the article that the recently revealed cracks of a new type, which have a different mechanism of destruction of the medium than the Griffith cracks, cannot be obtained as a result of the convergence of the sides of the wedge-shaped cavity in the plate. Thus, cracks of a new type are obtained as a result of the virtual convergence of the lateral sides of a rectangular cavity in the plate. The block element method is used for proof. The same method is used to simulate some types of nanoparticles, materials of mosaic structure and to study the formation of new cracks in such materials.

Fracture analysis of a Griffith crack under thermal-mechanical loading using nonlocal theory

In this paper, the nonlocal theory is utilized to deal with a Griffith crack problem under thermal-mechanical loading for the first time to eliminate stress singularity at the crack tip. A partially, thermally insulated Griffith crack subjected to a remote heat flux and mechanical loading is investigated. Fourier transform is employed to convert the mixed, boundary-value problem to two pairs of dual integral equations, which are then solved using the Schmidt method. The unknown jumps of displacement across crack faces are expressed by a series of Jacobi polynomials. The numerical results show that the stress at the crack tip is finite, and the maximum value occurs at a short distance from the crack tip. The results can be employed to build a maximum stress-based criterion instead of the traditional, stress intensity factor- or energy-based fracture criterion.

Analysis of the Crack Initiation and Growth in Crystalline Materials Using Discrete Dislocations and the Modified Kitagawa–Takahashi Diagram

Crack growth kinetics in crystalline materials is examined both from the point of continuum mechanics and discrete dislocation dynamics. Kinetics ranging from the Griffith crack to continuous elastic-plastic cracks are analyzed. Initiation and propagation of incipient cracks require very high stresses and appropriate stress gradients. These can be obtained either by pre-existing notches, as is done in a typical American Society of Testing and Materials (ASTM) fatigue and fracture tests, or by in situ generated stress concentrations via dislocation pile-ups. Crack growth kinetics are also examined using the modified Kitagawa–Takahashi diagram to show the role of internal stresses and their gradients needed to sustain continuous crack growth. Incipient crack initiation and growth are also examined using discrete dislocation modeling. The analysis is supported by the experimental data available in the literature.

A Semianalytical Solution for a Griffith Crack Nonuniformly Pressurized By Internal Fluid

Unintentional variations of fluid pressure within a hydraulic fracture will disturb the surrounding stress state, affect the stability of fracture propagation and complicate the fracture intersections during the process of hydraulic fracturing. However, little attention has been paid to the effect of nonuniform fluid pressure inside the hydraulic fracture. This paper presents a semianalytical solution for a Griffith crack nonuniformly pressurized by internal fluid in an impermeable elastic plane. The fluid pressure is hypothetically designated as a general polynomial with respect to the location of the fluid, such that the effect of an arbitrary form of fluid pressure distribution can be explored by polynomial fitting. The semianalytical solution is capable of being degenerated into constant pressure forms which confirms that the solution is an extension of classic constant pressure. In addition, the critical propagation conditions and stress distributions (e.g., σxx) under constant and nonuniform pressures are compared and discussed. The comparison results indicate that the effect of nonuniform fluid pressure accumulated by the number of polynomial terms increases the magnitude of stress (or displacement) but does not change its distribution. Subsequently, the semianalytical solution is validated by comparing the fracture intersection predicted by the semianalytical solution with the laboratory experimental observations and published predictions under constant fluid pressure. Good agreement with the experiments and sufficient advantages over previous predictions are observed. Finally, a sensitivity analysis of existing parameters in the semianalytical solution, including crack length, initial pressure, number of terms and number of subintervals, is conducted to evaluate their influence on surrounding stresses and critical propagation conditions, which further demonstrates the applicability and reliability of the presented semianalytical solution. The new solution enriches hydraulic fracturing theory by considering the nonuniform fluid pressure effect and provides important reference for fracture network design during hydraulic fracturing.

The significant effect of non-recrystallization zone reduction on microstructure and mechanical properties in multi-phase steel from the perspective of crystallographic structure and variant pairing

We elucidate here the significant effect of non-recrystallization zone reduction on microstructure and mechanical properties in multi-phase steel from the perspective of crystallographic structure, particularly, digitization and visualization of variant pairing. The volume fraction of deformation induced ferrite increased with the increase of non-recrystallization zone reduction, accompanied by the increase of acicular ferrite. This behavior is related to suppression of upper bainite by occupying the prior austenite grain boundaries by allotriomorphic ferrite. The increase of non-recrystallization zone reduction led to a transition from upper bainite to acicular ferrite, and consequently influenced the mechanical properties. The multi-phase steel with a microstructure consisting of ultra-fine ferrite, bainite and a large volume fraction of acicular ferrite had an excellent combination of high yield strength (561 MPa), low yield-to-tensile ratio (0.81), high uniform elongation (11.4%) and remarkable cryogenic toughness of 212 J tested at −100 °C. There was a dominance of V1/V8 pairing in upper bainite, resulting in low density of high angle grain boundaries (HAGBs). While the density of HAGBs in acicular ferrite/bainite was significantly higher because of microstructure refinement by acicular ferrite and the transition in variant pairing from V1/V8 to V1/V2. The V1/V2 pair was primarily related to the boundaries between acicular ferrite and bainite. Moreover, the brittle crack propagated straightly in upper bainite. In contrast, the high density of HAGBs in acicular ferrite/bainite impeded and arrested microcracks before they grew to a critical Griffith crack length, and subsequently suppressed the initiation of cleavage crack and decreased the ductile–brittle transition temperature.

SH-waves interaction with crack at orthotropic interface

ABSTRACT The problem of diffraction of SH-waves by two coplanar Griffith cracks situated at the interface of two different orthotropic media has been analyzed. Fourier transform and integral equation method have been used to solve this problem. The expressions for the stress intensity factors (SIFs) around the crack tips are obtained. Numerical values of SIF and the crack opening displacement for several orthotropic materials have been calculated and plotted graphically to show the effect of material orthotropy.

P-Wave Interaction in a Cracked Orthotropic Strip

In this paper diffraction of P-Waves in an infinite orthotropic strip containing two coplanar Griffith cracks is analyzed. Using Fourier transform the problem is reduced to a system of dual integral equation. Abel’s transform technique is used to further reduce the system of dual integral equation to a Fredholm integral equation of second kind and finally the integral equation is solved numerically to find the Stress Intensity Factor (SIF). Graphical representation has been made to show the effect of material orthotropy.

Stress intensity factors of a Griffith crack in a porous medium subjected to a time-harmonic stress wave

In many practices, cracks are filled with fluid and their background materials can be modelled within the framework of poroelasticity. Understanding the crack-tip mechanical behaviors of such a cracked medium subjected to dynamic loads is of importance for prediction of fracture failure. In this paper, based on Biot’s theory of poroelastodynamics, solutions of mode-I and mode-II dynamic stress intensity factors (SIFs) of a Griffith crack are derived for the scattering problem of oblique incidence of in-plane time-harmonic P-SV waves. The crack surfaces are assumed to be permeable, and the crack is saturated by fluid. With the aid of Fourier integral transforms, the problem is reduced to a system of coupled Fredholm integral equations with special emphasis placed on finding the near-field solution. Impacts of incident wave type, material permeability and incident angle on the frequency-dependent behaviors of the SIFs have been shown graphically. It is found that the induced fluid pressure inside the crack can significantly change the effective normal stress so that the magnitude of mode-I SIF is much lower than that of a dry impermeable crack. Furthermore, the diffusion-type fluid flow (scattered slow P wave) can greatly lower the magnitude of mode-I SIF at low frequencies (much smaller than the resonance frequency). The characteristic frequency at which the curves of SIFs decline strongest occurs when the characteristic diffusion length is of the same order of the crack size. The fluid flow can also lower mode-II SIF at low frequencies and the effects are strong in the low permeability case. Specifically, in the case of scattering of SV waves, it is shown that the maximum drop of mode-II SIF in peak with respect to that of the corresponding dry crack in an elastic medium is about 10%. The obtained results reveal significant influences of the presence of pore fluid upon the SIFs, both magnitudes and frequency-dependent trends. Such information is useful in predicting the fracture strength of saturated porous materials subjected to oscillating loads.

An Extended Dielectric Crack Model for Fracture Analysis of a Thermopiezoelectric Strip

An extended dielectric crack model is proposed to capture the effects of the physical properties of crack interior on crack-tip thermoelectroelastic fields. The typical crack-face boundary conditions can be retrieved by considering the limiting cases of electrical permeability and thermal conductivity inside a crack. Making use of the Fourier transform technique, the problem of a thermopiezoelectric strip weakened by a Griffith crack is investigated and transformed to solve the system of the second kind Fredholm integral equations with Cauchy kernel. The Lobatto–Chebyshev collocation method is used to form a nonlinear system of algebraic equations, which is solved by elaborating on an algorithm. The crack-tip thermoelectroelastic fields are determined by using the approximate solutions. Numerical simulations are carried out to show the variations of the fracture parameters of concern under applied thermoelectromechanical loads, the physical properties of the dielectric medium inside the crack and the geometry of the cracked thermopiezoelectric strip. Some comparisons with the experimental results are reported to reveal the effectiveness of the extended dielectric crack model.

Scaling Law for Cracking in Shrinkable Granular Packings.

Hydrated granular packings often crack into discrete clusters of grains when dried. Despite its ubiquity, an accurate prediction of cracking remains elusive. Here, we elucidate the previously overlooked role of individual grain shrinkage-a feature common to many materials-in determining crack patterning using both experiments and simulations. By extending classical Griffith crack theory, we obtain a scaling law that quantifies how cluster size depends on the interplay between grain shrinkage, stiffness, and size-applicable to a diverse array of shrinkable granular packings.

Toughening coarse grained heat affected zone of high strength offshore engineering steel by enhancing the completeness of austenite-bainite transformation

Abstract A novel and feasible method of enhancing the low temperature (−40 °C) impact toughness of coarse grained heat affected zone (CGHAZ) is put forward. By slightly increasing the holding time at peak temperature, the completeness of austenite-bainite transformation was enhanced and the toughness of CGHAZ was improved significantly. The product bainite was refined with increased holding time and more crystallographic variants from different Bain groups, namely, more Bain group boundaries were formed, which proved to be effective in preventing crack propagation. Moreover, when the inter-spacing of these boundaries is controlled below the critical Griffith crack length by self-accommodation during transformation, micro-cracks can be arrested by these high angle boundaries, thereby suppressing brittle fracture initiation and increasing fracture properties.

Research on Multiple Griffith Cracks at the Interface of Fine-Grained Piezoelectric Coating/Substrate

The behavior of a fine-grained piezoelectric coating/substrate with multiple Griffith interface cracks under electromechanical loads is investigated. In this work, double coupled singular integral equations are proposed to solve the fracture problems. Both the singular integral equation and single-valued conditions are simplified into an algebraic equation and solved by numerical calculation. Thereby, the intensity factors of electric displacement and stress obtained are used to obtain the expression of the energy release rate. Furthermore, numerical results of the energy release rate with material parameters are demonstrated. Based on the obtained results, it could be concluded that the energy release rate is closely related to the size of the interface cracks and the mechanical-electrical loading. For a bimaterial structure, the fine-grained piezoelectric structure exhibited better material performance compared to the large one.

Investigation of fracture behavior of asphalt mixture composite using energy-based approach

This study aims to determine the endurance limit and macro-cracking threshold of a viscoelastic composite material using a fracture mechanics approach. Asphalt mixtures containing different recycled material contents and recycling agent doses were fabricated and long-term oven aged. Dynamic modulus tests were first conducted to obtain linear viscoelastic properties including complex modulus and phase angle, and then damage properties were determined from monotonic semi-circular bending tests. Pseudo displacement was calculated to remove the viscous effect from the total dissipated energy, and the energy balance approach and Griffith crack growth criterion were modified and used to predict the bond energy with the aid of discrete element modeling. With the bond energy computed, the endurance limit was determined to be frequency and mixture composition dependent. The macro-cracking threshold defined as the boundary between healable micro-cracks and non-healable macro-cracks was also identified to be frequency and composition dependent using a damage density concept. Both the endurance limit and the macro-cracking threshold were utilized in the asphalt mixture Black Space diagram, which is capable of illustrating the effects of aging, recycled materials content and recycling agent dose in asphalt mixtures for the prediction of fracture performance.

Thermoelastic analysis of a cracked orthotropic strip under heat conduction effect of crack interior

The problem of a Griffith crack embedded in an orthotropic strip is investigated under a combination of thermal and mechanical loadings. The extended thermal-medium crack model is used to address the effects of thermal conductivity of crack interior on the fracture parameters of concerns. Using the Fourier transform technique, the boundary-value problem is reduced to solving singular integral equations with Cauchy kernel. The closed-form solutions of temperature fields are determined. In solving the elastic fields, the Lobatto–Chebyshev integration formula is applied to discretize the obtained singular integral equations as a system of algebraic equations, which is solved by elaborating an algorithm. Numerical results are reported to illustrate the effects of the thermal conductivity inside crack and the thickness of orthotropic strip on the partial insulation coefficient, the thermal stress intensity factor, and the strain energy density factor. The crack initiation behavior is analyzed according to the normalized strain energy density factor. The observations reveal that the crack-tip thermoelastic fields in an orthotropic strip are dependent on the combined effects of the thermoelastic loadings, the physical and geometrical properties of the material and crack. Some comparisons with the existing works are offered to show the novelty and applicability of the obtained results.

Analytic Solutions to Two-Dimensional Decagonal Quasicrystals with Defects Using Complex Potential Theory

An analytical treatment for two-dimensional point group 10 mm decagonal quasicrystals with defects was suggested based on the complex potential method. On the basis of the assumption of linear elasticity, two new conformal maps were applied to two examples: the first was an arc with an elliptic notch inner surface in a decagonal quasicrystal, where the complex potentials could be exactly obtained; and the second was concerned with a decagonal point group 10 mm quasicrystalline strip weakened by a Griffith crack, which was subjected to a pair of uniform static pressures. Using the basic idea underlying crack theory, the extent of the stress intensity factors was analytically estimated. If the height was allowed to approach infinity, these results can be turned into the known results of an “ordinary” crystal with only phonon elastic parameters when the phason and phonon-phason elastic constants are eliminated.

Combined effect of M/A constituent and grain boundary on the impact toughness of CGHAZ and ICCGHAZ of E550 grade offshore engineering steel.

The present paper investigated the relationship between low temperature impact toughness and microstructure of bainite in coarse-grained heat affected zone (CGHAZ) and intercritically rehazed CGHAZ (ICCGHAZ) of an offshore engineering steel from both the microstructure morphological and crystallographic aspects. In this work, six groups of samples simulated CGHAZ and ICCGHAZ were designated at three different cooling rates. The Charpy test results showed that the toughness in CGHAZ decreases dramatically with decrease of cooling rate, which was attributed to the microstructural evolution from lath bainite to granular bainite, accompanying with the size increase of Bain zone and the change of M/A morphology from film to block. The increase in hardenability by cooling rate promotes more crystallographic variants from different Bain groups. Meanwhile, the combination with controlled inter-spacing of block boundaries by self-accommodation below the critical Griffith crack length, micro-crack can be arrested by these high angle grain boundaries thereby suppressed brittle fracture initiation and increased fracture properties. However, the variation in toughness of ICCGHAZ is not a concern, since obtaining excellent toughness is scarcely accessible even if the matrix microstructure is analogous to CGHAZ. It was due to the formation of coarse M/A constituents (~2 μm) necklacing at the prior austenite grain boundary. The visualized crystallography suggested that the impact toughness was partially correlated to the configuration manner and the size of Bain zones as well via promoting highly misoriented angle (>45°) boundaries, which in turn effectively deflected or arrested the brittle crack propagation.

The plane strain analysis for one-dimensional hexagonal piezoelectric quasicrystals strip in aperiodical plan

A new stress potential function is introduced, the non periodic plane problem in one-dimensional hexagonal piezoelectric quasicrystals is discussed and the physical equation of the stress-strain relationship in the non periodic plane is constructed. The exact solution of the straight crack in the periodic direction of the one-dimensional hexagonal piezoelectric quasicrystal is obtained. As an application, the problem of straight crack perpendicular to the direction of quasi-periodical in one-dimensional hexagonal piezoelectric quasicrystal with long and narrow body is solved. When the width of the long body becomes infinitely large, the Griffith crack solution is obtained. The results show that the stress at the crack tip remains singularity, which is basically consistent with the crack problem that penetrates along the quasi periodic direction. When the phonon field and the phase field get to zero, the above analytical solution degenerates into the fracture problem of isotropic piezoelectric materials, the results are in agreement with the existing results.

Transient response of an infinite row of collinear Griffith cracks in a saturated porous medium

The transient stress distribution is obtained for a saturated porous medium containing an array of equally distributed collinear cracks when each crack is opened by suddenly applied internal pressure. The integral and Laplace transforms are employed to reduce the mixed boundary-value problem to solving the Fredholm integral equation. The influence of pore fluid and the size of the crack with respect to the distance between the neighbouring cracks on the dynamic stress intensity factor are investigated. Numerical results are presented and compared with the corresponding values of the dry medium.

Uniformity of stresses inside a non-elliptical inhomogeneity interacting with a mode III crack

Using conformal mapping techniques and the theory of Cauchy singular integral equations, we prove that it is possible to maintain a uniform internal stress field inside a non-elliptical elastic inhomogeneity embedded in an infinite matrix subjected to uniform remote stress despite the fact that the inhomogeneity interacts with a finite mode III crack. The crack can be modelled either as a Griffith crack or as a Zener–Stroh crack. Our analysis further indicates that the existence of the crack plays a key role in influencing the shape of the corresponding inhomogeneity but not the internal uniform stress field inside the inhomogeneity. Numerical examples are presented to demonstrate the solution.