Crack Tip Stress Intensity Factor Research Articles

The penny crack beneath the surface of a half-space: with application to the blister test

The crack tip stress intensities are found for a penny shaped crack lying beneath the free surface of a half-space, and parallel with it. This is done by employing a novel distributed dislocation approach using axi-symmetric Somigliana dislocations as the kernel of an integral equation, and provides a precise solution at the expense of little computing cost. A comparison is made with the crack tip stress intensity factors for a simple plane crack, and predictions are made for the preferred crack extension direction.

Evaluation of mode I stress intensity factors for edge cracks from 2-D V-notches using composition of constituent SIF weight functions

Stress intensity factor (SIF) weight functions for an edge crack emanating from a single edge V-notch in a finite strip is defined in terms of its constituent SIF weight functions. Solutions are achieved by applying the closed form composition model developed from the composition theory for SIF weight functions [Brennan FP, Teh LS. Determination of crack tip stress intensity factors in complex geometries by composition of weight function solutions. Fatigue Fract Eng Mater Struct 2004;27:1–7]. To overcome the shortage of available constituent reference solutions, finite element analysis (FEA) was employed to evaluate SIFs and the associated stress fields for V-notches. The FEA results were shown to be within 5% of those achieved from the analytical solution. The paper presents weight function coefficients, Cij, for V-notches in semi-infinite bodies which used with the composition model provides extensive SIF solutions for 2-D V-notches under any symmetrical stress system. More importantly, this paper demonstrates the robustness and mathematical simplicity of the composition principle that is independent of the influences of stresses in evaluating new sets of SIF solutions.

Electron probe response function and piezo-spectroscopic behaviour of semiconductor materials in presence of highly graded stress fields

The response function of the electron probe and the stress dependence of cathodoluminescence spectra emitted by selected semiconductor materials have been evaluated by scanning across sharp bi-material interfaces and along highly graded residual stress fields generated at the tip of an equilibrium crack, respectively. These microscopic procedures can be made fully quantitative provided that the crack opening displacement of the investigated crack is preliminarily measured in a scanning electron microscope for an in situ estimate of the crack-tip stress intensity factor. Taking advantage of the high scanning flexibility of the electron probe, capable of nanometric lateral displacements, spectral shifts typical of the K-dominated zone along the axis of crack propagation were recorded as a function of distance from the crack-tip. A plot of equi-biaxial stress versus spectral band shift was then obtained whose slope represents the piezo-spectroscopic coefficient of the selected material band. A theoretical analysis was attempted to analyse the in-plane interaction between sample and electron probe and to put forward suitable conditions for a reliable assessment of highly graded stress fields. Conditions were worked out into generalized plots as a function of spectroscopic and mechanical parameters for GaN, 3C–SiC and GaAs, as paradigm semiconductor materials.

An accumulation damage model for fatigue fracture of ferroelectric ceramics

The accumulated plastic displacement criterion for crack propagation in traditional materials is extended to develop equations to predict the fatigue crack growth of ferroelectric ceramics subjected to combined electromechanical loads. The crack-line is perpendicular to the poling direction of the medium. An electric saturation zone and a stress saturation zone are assumed to develop at the crack tips when the medium is subjected to external electromechanical loads. This assumption makes it possible to obtain the accumulated plastic deformation in closed form. A fatigue crack growth law, which is a fourth-power function of the effective stress intensity factor, similar to the well-known Paris law, is derived. Graphical results for the effect of electric load on the effective crack tip stress intensity factor and crack growth rate are provided.

Micro-mechanical theory of macroscopic stress-corrosion cracking in unidirectional GFRP

A micro-mechanical theory of macroscopic stress-corrosion cracking in a unidirectional glass fibre-reinforced polymer composite is proposed. It is based on the premise that under tensile loading, the time-dependent failure of the composite is controlled by the initiation and growth of a crack from a pre-existing inherent surface flaw in a glass fibre. A physical model is constructed and an equation is derived for the macroscopic crack growth rate as a function of the apparent crack tip stress intensity factor for mode I. Emphasis is placed on the significance of the size of inherent surface flaw and the existence of matrix crack bridging in the crack wake. There exists a threshold value of the stress intensity factor below which matrix cracking does not occur. For the limiting case, where the glass fibre is free of inherent surface flaws and matrix crack bridging is negligible, the relationship between the macroscopic crack growth rate and the apparent crack tip stress intensity factor is given by a simple power law to the power of two.

An Embedded Crack in a Graded Coating Bonded to a Homogeneous Substrate Under Thermo-Mechanical Loading

ABSTRACT The problem of an embedded partially insulated crack in a graded coating bonded to a homogeneous substrate under thermal and mechanical loading is considered. The heat conduction and the plane elasticity equations are converted into singular integral equations which are solved to yield the temperature and the displacement fields in the medium as well as the crack tip stress intensity factors. A crack-closure algorithm is applied to avoid interpenetration. The main objective of the paper is to study the effect of the coating nonhomogeneity parameters, partial insulation of the crack surfaces and crack-closure on the crack tip stress intensity factors for the purpose of gaining better understanding of the thermo-mechanical behavior of graded coatings.

Evaluation of Transformation Zone Around Propagating Cracks in Zirconia Biomaterials Using Raman Microprobe Spectroscopy

Structural reliability, biocompatibility and bioinertness are fundamental prerequisites for bioceramics used in artificial hip and knee joints. Among structural properties, superior fracture toughness is necessary for guaranteeing high reliability during implantation lifetime. Bioinert ceramics employed in artificial joints are mainly limited to alumina and zirconia materials. In this paper, the critical crack-tip stress intensity factor, KI0, and the tetragonal-to-monoclinic phase-transformation behavior of a 3 mol % Y2O3-doped tetragonal ZrO2 polycrystals (3Y-TZP) were studied as a function of grain size. 3Y-TZP’s with four different grain sizes were prepared and the size and morphology of the monoclinic transformation zone generated around the tip of an indentation crack were analyzed by quantitative Raman microprobe spectroscopy. The stress intensity factor, KI0, was evaluated by the crack opening displacement (COD) method using a recently proposed equation for calculating the compliance of an indentation crack.

Analysis of Thermoelastic Crack Problems Using Green–Lindsay Theory

A boundary element method using the Laplace transform in time domain is presented for the analysis of fracture mechanics under thermal shock using the Green and Lindsay (GL) theory of thermoelasticity. The dynamic thermoelastic model of Green and Lindsay is selected to show the effect of thermal wave propagation at finite speed on crack tip stress intensity factor evaluation. The singular behavior of the temperature and displacement fields in the vicinity of the crack tip is modeled by the quarter-point elements. Thermal dynamic stress intensity factor for mode I is evaluated from computed nodal values, using the well-known displacement and traction formulas. The accuracy of the method is investigated through comparison of the results with the available data in literature. Condition where the inertia term plays important role is discussed and variations of dynamic stress intensity factor is investigated. Different relaxation times are chosen to briefly show their effects on stress intensity factor in the Green and Lindsay theory.

The edge dislocation in a three-quarter plane. Part II: Application to an edge crack

Abstract This paper presents the solution for the crack tip stress intensity factors for a short crack emanating from the corner of a three-quarter plane. The solution employs the influence functions for a dislocation along one of the projection lines of a three-quarter plane together with the Williams semi-infinite wedge solutions, and the result is valid for any remote loading configuration. The sole proviso is that the crack is short enough to be considered well within the region in which the asymptotes govern the stress state. This general solution is then applied to a specific geometry and is compared with a calibration for the stress intensity factors for a similar geometry.

Multiple axial cracks in a coated hollow cylinder due to thermal shock

The transient thermal stress problem of an inner-surface-coated hollow cylinder with multiple pre-existing surface cracks contained in the coating is considered. The transient temperature, induced thermal stress, and the crack tip stress intensity factor (SIF) are calculated for the cylinder via finite element method (FEM), which is exposed to convective cooling from the inner surface. As an example, the material pair of a chromium coating and an underlying steel substrate 30CrNi2MoVA is particularly evaluated. Numerical results are obtained for the stress intensity factors as a function of normalized quantities such as time, crack length, convection severity, material constants and crack spacing.

Mode I stress intensity factors for edge cracks emanating from 2-D U-notches using composition of SIF weight functions

The recently developed Composition Principle for SIF Weight Functions [Brennan FP, Teh LS. Determination of crack tip stress intensity factors in complex geometries by composition of weight function solutions. Fatigue Fract Eng Mater Struct 2004; 27:1–7.] has been applied to generate mode I stress intensity factor (SIF) solutions for an edge crack emanating from a single U-notched plate. While weight function coefficients for other constituents were well presented [Teh LS, Brennan FP. Stress intensity factors for 2-D semicircular notches using the superposition of SIF weight functions. Fatigue Fract Eng Mater Struct superposition 2005;28:423–435.], SIF solutions and the associated stress distributions for U-notches in semi-infinite strips are required for the composition model. These were obtained by finite element analysis (FEA) and details of the FEA modelling are presented. The FEA results were verified giving good agreement with analytical solutions. The weight function for cracks from U-notches in semi-infinite bodies presented here can also be used in any composition model requiring this geometry as one of its constituents. This paper also demonstrates the mathematical efficiency and reliability of the SIF weight function composition concept.

Crack growth in ferroelectric ceramics under electric loading

A crack with growth in ferroelectric ceramics under purely electric loading is analyzed. The crack tip stress intensity factor for the growing crack under small scale conditions is evaluated by employing the model of nonlinear domain switching. The electrical fracture toughness is obtained from the result of the stress intensity factor. It is shown that the ferroelectric material can be either toughened or weakened as the crack grows. Fatigue crack growth in a ferroelectric material under cyclic electric loading is also examined. The incremental fatigue crack growth under cyclic electric loading is obtained numerically. The fatigue crack growth rate is affected strongly by the electrical nonlinear behavior. It is found that the curve of fatigue crack growth rate versus electric field intensity factor is linear on the log-log plot at intermediate values of the electric field intensity factor.

Thermomechanical analysis of thermal shock fracture in the brittle/ductile transition zone—Part II: Numerical calculations and interpretation of the test results

The integrity of PWR pressure vessels is assured by keeping the crack tip stress intensity factor below the toughness of the material under monotonic isothermal loading. To study the effects of sudden cooling associated with a thermal gradient, a specially modified compact specimen has been developed. This has been used to carry out tests in the transition zone with different loading-temperature sequences liable to call the conventional criteria into question. The test is described in detail in Part I of this article [Chapuliot S, et al. Thermomechanical analysis of thermal shock fracture in the brittle/ductile transition zone. Part I: Description of the tests. Engng Fract Mech, 72, 2005, 661–73]. The second part describes numerical investigations to estimate the local mechanical fields at the crack tip and the overall parameters of the fracture mechanics. Finite element thermomechanical calculations are used to interpret the results of these new thermal shock tests using the master curve concept [ASTM E 1921–1997. Standard test method for determination of reference temperature To for ferritic steels in the transition range, 1997] and the Beremin statistical model [Beremin FM. A local criterion for cleavage fracture of a nuclear pressure vessel steel. Metall Trans A, 14A, November 1983, 2287–777].

An analytical model for the effect of elastic modulus mismatch on laminate threshold strength

We present an analytical model for predicting the crack tip stress intensity factor in a laminated composite material with alternating zones of residual biaxial compressive and tensile stresses. Such an arrangement of residual stress zones has been shown to be useful for enhancing the damage tolerance properties of the structure, which are manifested as a ‘threshold’ strength below which the material cannot fail catastrophically. This model improves upon a previous scheme for predicting crack behaviour in such a material, which assumes homogeneous material properties throughout the laminate. To make the problem tractable, this scheme models the system as a homogeneous anisotropic material when calculating the stress fields at the crack tip arising from tractions applied far away from the crack tip. A comparison of the predictions from both schemes with those from a finite element model of such a system shows that our assumptions yield an improved representation of the material's behaviour. We also present results for optimization of geometrical parameters of the system to obtain the highest possible threshold strength under the limitation of a given set of material properties. We conclude that while this scheme improves upon the homogeneous model, more work is required to predict the stresses at the tip as the crack approaches a material interface before a complete analytical model can be obtained.

Transient Elastodynamic Crack Growth in Functionally Graded Materials

A generalized elastic solution for an arbitrarily propagating transient crack in functionally graded materials (FGMs) is obtained through an asymptotic analysis. The shear modulus and mass density of the FGM are assumed to vary exponentially along the gradation direction. The out-of-plane displacement field and its gradients about the crack tip were obtained in powers of radial coordinates, with the coefficients depending on the time rates of change of crack tip speed and stress intensity factor. The effects of the transient terms on the contours of constant out-of-plane displacement are duly discussed.

A new photoelastic model for studying fatigue crack closure

The photoelastic analysis of crack tip stress intensity factors has been historically developed for use on sharp notches in brittle materials that idealize the cracked structure. This approach, while useful, is not applicable to cases where residual effects of fatigue crack development (e.g., plasticity, surface roughness) affect the applied stress intensity range. A photoelastic model of these fatigue processes has been developed using polycarbonate, which is sufficiently ductile to allow the growth of a fatigue crack. The resultant stress field has been modeled mathematically using the stress potential function approach of Muskhelishvili to predict the stresses near a loaded but closed crack in an elastic body. The model was fitted to full-field photoelastic data using a combination of a generic algorithm and the downhill simplex method. The technique offers a significant advance in the ability to characterize the behavior of fatigue cracks with plasticity-induced closure, and hence to gain new insights into the associated mechanisms.

Threshold intensity factors as lower boundaries for crack propagation in ceramics

BackgroundSlow crack growth can be described in a v (crack velocity) versus KI (stress intensity factor) diagram. Slow crack growth in ceramics is attributed to corrosion assisted stress at the crack tip or at any pre-existing defect in the ceramic. The combined effect of high stresses at the crack tip and the presence of water or body fluid molecules (reducing surface energy at the crack tip) induces crack propagation, which eventually may result in fatigue. The presence of a threshold in the stress intensity factor, below which no crack propagation occurs, has been the subject of important research in the last years. The higher this threshold, the higher the reliability of the ceramic, and consequently the longer its lifetime.MethodsWe utilize the Irwin K-field displacement relation to deduce crack tip stress intensity factors from the near crack tip profile. Cracks are initiated by indentation impressions. The threshold stress intensity factor is determined as the time limit of the tip stress intensity when the residual stresses have (nearly) disappeared.ResultsWe determined the threshold stress intensity factors for most of the all ceramic materials presently important for dental restorations in Europe. Of special significance is the finding that alumina ceramic has a threshold limit nearly identical with that of zirconia.ConclusionThe intention of the present paper is to stress the point that the threshold stress intensity factor represents a more intrinsic property for a given ceramic material than the widely used toughness (bend strength or fracture toughness), which refers only to fast crack growth. Considering two ceramics with identical threshold limits, although with different critical stress intensity limits, means that both ceramics have identical starting points for slow crack growth. Fast catastrophic crack growth leading to spontaneous fatigue, however, is different. This growth starts later in those ceramic materials that have larger critical stress intensity factors.

Macrocrack–microcrack interaction problem in magnetoelectroelastic solids

Aiming at investigating the macrocrack–microcrack interactions in magnetoelectroelastic solid, the elementary solutions for a semi-infinite macrocrack and a microcrack subjected to various loading conditions are derived first. Then by using the pseudo-traction–electric-displacement–magnetic-induction method that is motivated by the well-known pseudo-traction method [J. Mech. Phys. Solids 35 (1987) 601], the present interaction problem is reduced to a system of integral equations. After solving these equations numerically by using the Chebyshev numerical integration technique, the crack tip stress intensity factors (SIFs), the electric displacement intensity factor (EDIF), the magnetic induction intensity factor (MIIF) as well as the mechanical strain energy release rate (MSERR) are evaluated subsequently. The J-integral for magnetoelectroelastic medium is further deduced, and it is employed to examine the correctness of the calculation results derived. Finally, take both the stress intensity factors and the mechanical strain energy release rate as measures to investigate the influences of various loading conditions, the crack location and orientation as well as the number of microcracks on the macrocrack–microcrack interactions. It is found that both the electric and magnetic loadings have a significant influence on the fracture toughness of magnetoelectroelastic solid.

An embedded crack in a functionally graded coating bonded to a homogeneous substrate under frictional Hertzian contact

In an attempt to simulate nonuniform coating spalling, the elastostatic problem of an embedded crack in a functionally graded coating bonded to a homogeneous substrate subject to normal and tangential surface loading is considered. The coating is graded in the thickness direction and is orthogonal to the crack direction which is parallel with the free surface. The graded coating is modeled as a nonhomogeneous medium with an isotropic stress–strain law. Using integral transforms, the governing equations are converted into singular integral equations, which are solved numerically to yield the displacement field as well as the crack-tip stress intensity factors. Since the loading is compressive, a crack-closure algorithm is developed to avoid interpenetration assuming frictionless contact between the crack faces. The main objective of the paper is to study the influence of material nonhomogeneity on the ability of the coating to maintain its integrity with and without using the crack-closure algorithm.

The interaction of mode I crack with multi-inclusions in a three-phase model

An approximately close form solution has been developed for mode I crack interacting with multi-inclusions in composite materials. The crack-tip stress intensity factor is evaluated in a three-phase model, which combines the present knowledge that the inclusions only in the immediate neighborhood of the crack-tip have strong effect on the stress intensity factor and that the far inclusions have an overall effects which can be estimated by effective properties of the composites. As validated by numerical examples, the solution has good accuracy for a wide range of the modulus ratios between the inclusion and matrix material.