Crack Tip Stress Intensity Factor Research Articles

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.

Crack tip stress intensity factors for a crack emanating from a semi-infinite notch with application to the avoidance of fatigue in complete contacts

Crack tip stress intensity factors are found for the problem of a short crack adjacent to the apex of a notch, and lying perpendicular to one of the notch faces. Loading is represented by the two Williams eigensolutions, the ratio between which provides a reference length scale and permits a comprehensive display of the solution. The results are applied to the problem of a crack starting from the edge of a notionally adhered complete contact, and conditions for the avoidance of crack development are found.

Crack tip stress intensity factors for a crack emanating from a sharp notch

Accurate calibrations are provided for the crack tip stress intensity factor for a crack of finite length emanating from the symmetric tip of a sharp notch, of arbitrary angle, in terms of the generalised stress intensity quantifying remote loading of the notch. The solution is applied to example problems and shown to be accurate for cases where the crack is much shorter then the notch depth.

A technique for studying interaction between a screw dislocation dipole or a concentrated load and a mode III crack crossing an interface

A method of potentially wide application is developed for deriving analytical expressions of the elastic interaction between a screw dislocation dipole or a concentrated force and a crack cutting perpendicularly across the interface of a bimaterial. The cross line composed of the interface and the crack is mapped into a line, and then the complex potentials are educed. The Muskhelishvili method is extended by creating a Plemelj function that matches the singularity of the real crack tips, and eliminates the pseudo tips’ singularity induced by the conformal mapping. The stress field is obtained after solving the Riemann–Hilbert boundary value problem. Based on the stress field expressions, crack tip stress intensity factors, dislocation dipole image forces and image torque are formulated. Numerical curves show that both the translation and rotation must be considered in the static equilibrium of the dipole system. The crack tip stress intensity factor induced by the dipole may rise or drop and the crack may attract or reject the dipole. These trends depend not only on the crack length, but also on the dipole location, the length and the angle of the dipole span. Generally, the horizontal image force exerted at the center of the dislocation dipole is much smaller than the vertical one. Whether the dipole subjected to clockwise torque or anticlockwise torque is determined by whether the Burgers vector of the crack-nearby dislocation of the dipole is positive or negative. A concentrated load induces no singularity to crack tip stress fields as the load is located at the crack line. However, as the concentrated force is not located on the crack line but approaches the crack tip, the nearby crack tip stress intensity factor K III u increases steeply to infinity.

Mode-II shielding-curve of Al2O3 from measurement of cone crack angles

Many ceramic materials exhibit the effect of an increasing crack growth resistance during crack extension. Especially in the case of coarse-grained materials, cracks generally follow the grain boundaries, leave the original crack plane and produce a crack-face roughness. In the theoretical analysis forces transmitted by local links between the two crack faces are smoothed and replaced by so-called continuous bridging stresses acting against the crack opening. As the consequence of such bridging stresses, there exists a shielding stress intensity factor term that shields the crack tip partially from the applied loads. This effect as a reason for the occurrence of R-curves is well documented in literature for cracks under pure mode-I loading conditions [1]. In addition it has to be expected that crack-face interactions will also affect crack extension under pure or superimposed mode-II loading as for instance outlined in literature for frictional crack-face interactions [2–4]. In [5] it was outlined that the shear tractions generated under small mode-II load contributions may cause a disappearing effective crack-tip stress intensity factor KII,tip. Also the externally applied mode-II stress intensity factor KII,appl can be reduced by the mode-II shielding so that a disappearing KII,tip can occur. The distribution of the intensity of crack-face interlocking has the same consequences on the bridging tractions normal to the crack face rbr as for the tangentially transferred tractions sbr. At the same distance from the crack tip where the bridging interactions disappear, both the shear and the normal stresses must vanish. Under the assumption of the same traction versus displacement characteristics, the two stresses are proportional sðxÞ 1⁄4 k rbrðxÞ ð1Þ

Experimental Study of the Reduction at Crack-Tip Stress Intensity Factor KI by Bonded Patches

Patch repair of cracked structures has become a rapidly grown technology. The major function of a repair is to reduce the stress intensity factor at the crack-tip. Calculation of stress intensity factor of a repaired crack has been performed by analytical and numerical methods. However, these methods are based on simplifying assumptions regarding material behavior and repair conditions. In the present paper an experimental and an numerical determination of mode-I stress intensity factor (SIF), KI at the tip of an edge crack reinforced with bonded patches is undertaken by using the optical method of caustics and the finite element analysis (FEA). The experimental method of caustics is simple in its application and has successfully been used for the solution of a host of crack problems of engineering importance. The experimental results are compared with the corresponding one obtained by FEA. The program ANSYS 11 was used for the FEA. The cracked used plates were made of Lexan (PCBA) and the patches were made of Plexiglas (PMMA).

Assessment of viscoelastic crack bridging toughening in refractory materials

Viscoelastic bridges can be formed in refractory ceramics while cooling from high temperatures. Such bridges can shield crack tips, thus reducing the effective crack tip stress intensity factors leading to higher resistance to creep and thermal shock. The extent to which the crack tip stress intensity is reduced can be estimated from fracture mechanics models that include experimental measurement of crack bridging and microstructural parameters. In this paper a novel approach is proposed for the assessment of the effective crack bridging toughening from combining destructive and non-destructive test methods. Fracture toughness values were determined applying chevron notched specimen technique and surface damage of the specimen was monitored by image analysis. Different cordierite–mullite compositions characterized by different microstructure morphologies and crack propagation behaviour were investigated. A brief discussion about the correlation between thermo-mechanical properties, microstructure, crack propagation behaviour and thermal shock resistance is presented. Moreover, an empirical model able to determine the presence and effectiveness of the viscoelastic crack bridging ligaments acting in the microstructure under thermal shock conditions and their degradation with increasing thermal shock cycles from parameters measured at room temperature is presented.

Piezospectroscopic measurement of the stress field around an indentation crack tip in ruby using SEM cathodoluminescence

Piezospectroscopy using stress-sensitive lines in the cathodoluminescence (CL) spectrum generated in response to excitation by the electron beam of an SEM has recently been shown to be a promising technique for submicron resolution stress measurements in alumina and other ceramics. This paper develops and applies the technique by mapping the wavelength shifts of the R 1 CL line around the tip of an indentation crack in a ruby single crystal. Accounting for crystallographic anisotropy, the shifts observed were quantitatively consistent with the classical crack tip stress field for all polar angles ahead of the crack tip and indicated a crack tip stress intensity factor of 1.0 MPa m 1/2. This is significantly lower than the fracture toughness of the crack plane (4.5 MPa m 1/2), and indicates the post-indentation development of lateral cracks and slow crack growth. The spatial resolution of the stress measurements was measured as 550 nm and the effects of specimen heating by the electron beam were shown to be negligible.

Fatigue Crack Propagation Under Complex Loading

The paper describes the recent progress of the understanding of fatigue crack propagation under complex loading. The behavior of fatigue crack propagation is dependent on the macroscopic mode of crack growth : tensile mode and shear mode growth. Tensile mode growth takes place on the plane perpendicular to the maximum tangential stress range, satisfying the local symmetry. Shear mode growth takes place on the plane of the maximum shear stress when a precrack is subjected to high cyclic shear stress. When the shear stress is low, a bent or branched crack is formed at the precrack tip, showing tensile crack growth. Under an intermediate stress, a crack first shows shear mode growth and turns to tensile mode after some extension. This transition is caused by the decrease in the mode II crack-tip stress intensity factor due to increasing crack surface contact. The criterion of the maximum crack growth rate may control the mode of fatigue crack growth. In notched components under complex loading, the life of crack initiation at the notch root is calculated from the critical plane model using the stress and strain at a critical distance from the notch root. The propagation of a fatigue crack from a notch is dependent on the geometry of notched component. The resistance-curve method is applicable for the prediction of fatigue thresholds of thin walled tubular specimens subjected to in-phase and out-of-phase combination of cyclic torsion and tension-compression.

A Surface Crack in a Graded Coating Bonded to a Homogeneous Substrate under Thermal Loading

The elastostatic problem of a surface crack in a graded coating bonded to a homogeneous substrate under steady-state heat flux is considered. The coating is graded along the thickness direction and modeled as a nonhomogeneous medium with an isotropic stress-strain law. The problem is solved under the assumption of plane strain or generalized plane stress conditions. The resulting crack problem is of mode I because the orientations of the crack axis, the material gradient and the heat-flux are all parallel. The equivalent crack surface tractions are first obtained and substituted in the plane elasticity equations which are then converted analytically into a singular integral equation. The resulting equation is solved numerically using orthogonal Jacobi polynomials to yield the Mode I stress intensity factor. The main objective of the article is to study the effect of the layer thickness and nonhomogeneity parameters on the crack tip stress intensity factor for the purpose of gaining better understanding on the behavior of graded coatings under thermal loading.

A Theoretical Model of Conductive Cracks in Piezoelectric Materials under Electromechanical Cycling

This article develops an analytical model for the conductive cracks in piezoelectric ceramics under electromechanical cycling. An electrical yielding strip and a mechanical yielding strip are assumed to develop at the crack tips when the medium is subjected to external electromechanical loads. The yielding strip is assumed to have constant stress and constant electric field. Based on these assumptions and extending the accumulated plastic displacement criterion for crack propagation in traditional structural materials to piezoelectric ceramics, a fatigue crack growth law is derived. The law, similar to the well known Paris law, is a fourth-power function of the effective stress intensity factor. Graphical results for the effect of electric load on the effective crack tip stress intensity factor and crack growth rate are provided. The law is given in closed-form so that it might be used in the future for the fatigue and reliability analysis and design of piezoelectric materials.

Determination of the effective mode-I toughness of a sinusoidal interface between two elastic solids

A finite element model of crack propagation along a sinusoidal interface with amplitude A and wavelength λ between identical elastic materials is presented. Interface decohesion is modeled with the Xu and Needleman (J Mech Phys Solid 42(9):1397, 1994) cohesive traction–separation law. Ancillary calculations using linear elastic fracture mechanics theory were used to explain some aspects of stable and unstable crack growth that could not be directly attained from the cohesive model. For small aspect ratios of the sinusoidal interface (A/λ ≤ 0.25), we have used the analytical Cotterell–Rice (Intl J Fract 16:155–169, 1980) approximation leading to a closed-form expression of the effective toughness, K Ic , given by $$K_{Ic}\sqrt{(1-\nu^{2})/E\phi_n}=2/\left(1+\left[1+4\pi^{2}(A/\lambda)^{2}\right]^{-1/2}\right),$$ where $$\phi_n$$ is the work of separation, E is Young’s modulus, and ν is Poisson’s ratio. For A/λ > 0.25, both the cohesive zone model and numerical J-integral estimates of crack tip stress intensity factors suggest the following linear relationship: $$K_{Ic} \sqrt{(1-\nu^{2})/E\phi_n}=0.81+1.89(A/\lambda).$$ Parametric studies show that the length of the cohesive zone does not significantly influence K Ic , although it strongly influences the behavior of the crack between the initiation of stable crack growth and the onset of unstable fracture.

Effect of stress-induced martensitic transformation on the crack tip stress-intensity factor in Ni–Mn–Ga shape memory alloy

The Ni–Mn–Ga shape memory ferromagnetic shape memory alloys (FSMAs) are prone to fracture during thermal cycling. The present research shows that the primary reason for the thermally induced fracture of Ni–Mn–Ga FSMAs is the increase in the crack tip stress-intensity factor (SIF) due to stress redistribution around the crack tip as a result of stress-induced martensitic (SIM) transformation. On lowering the temperature to M s, the crack tip SIF of Ni–Mn–Ga FSMA increases significantly, being very different from that of Ni–Ti and Cu–Al–Ni SMAs. The sensitive temperature dependence of crack tip SIF in Ni–Mn–Ga is responsible for its brittleness under thermal cycling. The temperature dependence of crack tip SIF is strongly related to the yield stress, the temperature dependence of the critical stress for SIM transformation and the transformation interval. Temperature rate also plays an important role in the fatigue behavior of Ni–Mn–Ga FSMAs under thermal cycling.

Accumulation Damage Mode for Ferroelectric Ceramics Subjected to Mode III Fatigue Loading Conditions

Based on the accumulated plastic displacement criterion, equations to predict the fatigue crack growth of piezoelectric ceramics are developed. The medium is subjected to anti-plane mechanical and in-plane electrical loads. Under such electromechanical loading conditions, it is assumed that an electric saturation zone and an elastic yielding zone near the crack tips are developed. The accumulated plastic deformation is obtained in a closed-form, which is expressed in terms of the applied loads and material constants. A fatigue crack growth law of a fourth-power function of the effective stress intensity factor is derived. The effect of electric field load on the effective crack-tip stress intensity factor and crack growth rate are shown graphically.

Mechanics of two-stage crack growth in fretting fatigue

Motivated by experimental observations, we carry out a numerical analysis of the two-stage crack growth under fretting fatigue by using an efficient and accurate boundary element method. To start with, the variation of stress field during a loading cycle is analyzed. Various values of friction coefficient in the contact zone are considered, which is shown to considerably affect the stress field. Then, by assuming crack initiation to occur in the shear mode, a surface-breaking crack is introduced to the specimen at the location of highest shear-stress amplitude. The crack-tip stress intensity factors (SIFs) are calculated for various crack lengths and at various crack angles ranging from 25° to 45° about the contact surface. It is shown that, for a loading ratio of 0.5, the cyclic mode-II SIF amplitude decreases with increasing crack length, whilst its mean value increases. It suggests that the (first-stage) shear crack would sooner or later become dormant, or switch to another mode that can provide continuous support of growth. Then, the first-stage shear crack is manually kinked into a second-stage opening crack, and the follow-on driving force is analyzed. It is shown that the kinking event is only favored after the first-stage crack has grown to a certain length. The present study thus provides insights in the mechanics of two-stage crack growth that has been frequently observed in a typical dovetail joint under fretting fatigue. It also suggests an improved experimental setup to quantitatively investigate the fretting fatigue in dovetail joints.

Fretting fatigue test analysis of contact

Fretting fatigue tests conducted by Nowell and Szolwinski are analysed accurately by introducing several modifications needed to the classical formulation. With the total state of stress established, the crack-tip stress intensity factor for a crack growing inward from the trailing edge of the contact is determined by the distributed dislocation technique. Finally, the results are correlated with local solutions for the contact stress field that enable an estimate of the crack nucleation life, and hence a characteristic material property quantifying initiation, to be found.

Spatially resolved crack-tip stress analysis in semiconductor by cathodoluminescence piezospectroscopy

A spatially resolved cathodoluminescence piezospectroscopic analysis is attempted for the high-resolution evaluation of the stress field developed ahead of the tip of an equilibrium crack propagating in a semiconductor. GaN was selected for this assessment as a paradigm semiconductor material. Quantitative measurements of in-plane luminescence probe response function (PRF) were preliminarily performed at different acceleration voltages upon scanning across a straight and atomically sharp interface between GaN and gold metal. Then, based on the knowledge of PRF, the convoluting effect due to the finite size of the electron probe could be corrected and an improved plot of the crack-tip stress field could be retrieved by a computer-aided data restoration procedure. The crack-tip stress intensity factor KI obtained by the cathodoluminescence piezospectroscopic method was compared with that obtained on the same crack path according to high-resolution measurements of crack-tip opening displacement. This study not only shows that a nanometer-scale spatial resolution can be experimentally achieved in a stress analysis of semiconductor materials but also that a stress data treatment based on experimental PRF is a viable approach to greatly reduce the error involved with the finite size of the electron probe.

Field intensity factors of a penny-shaped crack in a magnetoelectroelastic layer

This paper considers the coupled field of a horizontal penny-shaped crack in a transversely isotropic magnetoelectroelastic layer subjected to mechanical, electric and magnetic loading. Hankel integral transform techniques and integral equation methods are used to formulate the boundary-value problem corresponding to a crack. The integral equation system is solved by using numerical quadrature. Solutions for the crack-tip stress intensity factors, electric displacement intensity factor and magnetic induction intensity factor are derived and computed to examine the behaviour of a penny-shaped crack for different layer heights, crack depths and loading. It is found that both positive electric and negative magnetic loading enhances crack propagation. Stress intensity factors show substantial dependence on crack depth whereas electric displacement and magnetic induction intensity factors are relatively insensitive to the depth of a crack.

Reflections on identifying the real Δ Keffective in the threshold region and beyond

The use of the crack tip stress intensity factor, K, has survived almost 50 years as the key parameter correlating fatigue crack growth. As time past the range of the stress intensity, Δ K, was recognized as causing alternating plasticity at the crack tip. The threshold level for Δ K was discovered. Further, the occurrence of crack closure was noted which affected the Δ K for different load ratios, R of cyclic loading. The ASTM method of counting the linear part of the load displacement for determining Δ K open was found to understate the Δ K effective, which correlates data for different load ratios. One approach to adjust for this problem is the “Partial Closure Model”, where the closure only occurs away from the crack tip. Here it will be discussed that such a model leads to a universal growth law. Moreover, this law shows application in estimating the order of magnitude of crack growth life (<10 7 cycles) for example with very high cycle fatigue (>10 9 cycles). Some advances in this application will also be cited.

Stress intensity factor for a crack within a strip under arbitrary loading

The solution for an edge dislocation in a strip is found in a very straightforward manner, using only the elementary solution for a dislocation in a half-plane. This is then employed to find the crack tip stress intensity factor (SIF) for an edge crack subject to an example loading condition, and the answers are compared with known solutions in the literature. The distributed dislocation method gives results in very close agreement with previously published results. The technique described has wide application to the solution of crack problems in strips when the nominal stress gradient is severe.