Crack Tip Stress Intensity Factor Research Articles

Arbitrarily oriented microcracks near the tip of an interface macrocrack in bonded dissimilar anisotropic materials

It is well known that microcracking in brittle materials results in a reduction of the stress intensity factor (SIF) and energy release rate (ERR). The reduced SIF or ERR represents crack tip shielding which is of significant interest to micromechanics and material science researchers. However, the effect of microcracking on the SIF and ERR is a complicated subject even for isotropic homogeneous materials, and becomes much more formidable in case of interface cracks in bonded dissimilar solids. To unravel the micromechanics of interface crack tip shielding in bonded dissimilar anisotropic solids, an interface crack interacting with arbitrarily oriented subinterface microcracks in bonded dissimilar anisotropic materials is studied. After deducing the fundamental solutions for a subinterface crack under concentrated normal and tangential tractions, the present interaction problem is reduced to a system of integral equations which is then solved numerically. A J-integral analysis is then performed with special attention focused on the J2-integral in a local coordinate system attached to the microcracks. Theoretical and numerical results reassert the conservation law of the J-integral derived for isotropic materials1, 2 also to be valid for bonded dissimilar anisotropic materials. It is further concluded that there is a wastage when the remote J-integral transmits across the microcracking zone from infinity to the interface macrocrack tip. In order to highlight the influence of microstructure on the interfacial crack tip stress field, the crack tip SIF and ERR in several typical cases are presented. It is interesting to note that the Mode I SIF at the interface crack tip is quite different from the ERR in bonded dissimilar anisotropic materials.

Time-dependent craze zone growth at a crack tip in polymer solids

By considering the polymer bulk as a linear viscoelastic body and the craze zone at crack tip as a nonlinear damage zone, the control equation for craze zone growth has been derived. It is shown that for a time-independent craze-zone stress, the craze zone would grow only if the crack-tip stress intensity factor is changed. If the crack-tip stress intensity factor remains constant during loading, the growth rate of the craze zone length will be interrelated to the crack-tip stress, the craze zone length and the rate of change of the craze-zone stress. If both the craze-zone stress and the crack-tip stress intensity factor are time-independent, the craze zone length will be constant during the crack growth, which is the case of self-similar crack growth. Moreover, a new stress distribution model in craze zone is presented based on the constructed damage evolution law, and the lengthening and thickening of the craze zone at the crack tip are also formulated. The numerical calculations from the proposed model agree well with the published experimental data.

Conducting crack growth in ferroelectric ceramics subjected to electric loading

Crack growth with electrically conducting surfaces in ferroelectric ceramics subjected topurely electric loading is investigated. The boundary of the domain switching zone near thetip of a growing crack under small-scale conditions is determined, based on nonlinearelectric theory. The crack tip stress intensity factor induced by domain switching for theconducting crack with growth is numerically obtained. The variation of the electricalfracture toughness obtained from the result of the stress intensity factor is examined for aferroelectric material with electrical nonlinear behaviour. It is shown that theferroelectric material can be either toughened or weakened as the crack grows.Steady state crack growth in nonlinear ferroelectric materials is also discussed.

Implications of interface friction for crack growth in fibre-reinforced metal matrix composites by three-dimensional finite element modelling

A three-dimensional finite element model of a compact tension specimen consisting of a Ti-6Al-4V matrix reinforced with unidirectional, continuous SiC fibres under monotonic and cyclic loading has been developed. This has enabled true Coulomb frictional interface sliding resulting from thermal residual stresses to be modelled. The results, which include the action of individual bridging fibres close to the crack-tip, are compared to results from a two-dimensional weight function method which uses fibre-induced bridging tractions on the crack face based on a constant interface strength. Reasonable agreement was found between the two methods used. An investigation of the fibre stresses showed that together with normal crack bridging tractions a strong bending component is present in the fibres which also affects crack opening and could affect the mode of fibre failure. The influence of processing induced thermal residual stresses and friction at the fibre-matrix interface on the crack growth behaviour during monotonic and cyclic loading has been assessed. It was found that the bridging fibres strongly reduce the crack-tip stress intensity factor. The thermal residual stresses produce a crack-tip opening load in the absence of an external load and have an influence on the crack-tip load ratio. The effect of the crack-tip load ratio on the fatigue threshold has a significant impact on the likelihood of crack arrest.

Predicting fracture swarms — the influence of subcritical crack growth and the crack-tip process zone on joint spacing in rock

Abstract Swarms or clusters represent an exception to the widely accepted idea that fracture spacing in sedimentary rock should be proportional to mechanical layer thickness. Experimental studies and static stress analysis do not provide adequate explanation for fracture swarm occurrence. The problem is re-examined numerically, accounting for the dynamics of pattern development for large populations of layer-confined fractures. Two crucial aspects of this model are: (1) the inclusion of three-dimensional effects in calculating mechanical interaction between simultaneously propagting fractures; and (2) the use of a subcritical crack-propagation rule, where propagation velocity during stable growth scales with the crack-tip stress intensity factor. Three regimes of fracture spacing are identified according to the magnitude of the subcritical index of the fracturing material. For low subcritical index material ( n = 5) numerous fractures propagate simultaneously throughout a body resulting in irregular spacing that is, on average, much less than layer thickness. For intermediate subcritical index ( n = 20) one fracture propagates at a time, fully developing its stress shadow and resulting in a pattern with regular spacing proportional to layer thickness. For high subcritical index cases ( n = 80) fractures propagate in a fashion analogous to a process zone, leaving a fracture pattern consisting of widely spaced fracture clusters.

Electrical fracture toughness for a conducting crack in ferroelectric ceramics

Conducting cracks in ferroelectric ceramics under purely electrical loading are analyzed to investigate the effects of electric fields on fracture behavior. The asymptotic problem of conducting cracks in a mono-domain ferroelectric material as well as in a poly-domain ferroelectric material is considered and the relation between the crack tip stress intensity factor and the applied intensity factor of electric field under small scale conditions is investigated. In order to evaluate the crack tip stress intensity factors due to the domain switching, the shape of the domain switching zone attending the crack tip must be determined. The evaluation of the shape is carried out based on the nonlinear domain switching model. It is shown that the switching zone boundary and the crack tip stress intensity factors due to the switching depend strongly on the angle of the polarization vector and the ratio of the coercive electric field to the yield electric field. The electrical fracture toughness of unpoled poly-domain ferroelectrics is examined. Employing a Reuss type approximation, the crack tip stress intensity factors for an unpoled poly-domain material are obtained. The ratio of the critical electrical energy release rate to the critical mechanical energy release rate is obtained.

Stress intensity factors for an interface crack between a functionally graded coating and a homogeneous substrate

In this paper we consider the problem of a functionally graded coating bonded to a homogeneous substrate with a partially insulated interface crack between the two materials subject to both thermal and mechanical loading. The problem is solved under the assumption of plane strain and generalized plane stress conditions. The heat conduction and the plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the temperature and the displacement fields in the medium as well as the crack tip stress intensity factors. A crack-closure algorithm recently developed by the authors is applied to handle the problem of having negative mode I stress intensity factors. The Finite Element Method was additionally used to model the crack problem and to compute the crack-tip stress intensity factors. The main objective of the paper is to study the effect of the material 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.

Simulation of crack propagation in fiber-reinforced concrete by fracture mechanics

Mode I crack propagation in fiber-reinforced concrete (FRC) is simulated by a fracture mechanics approach. A superposition method is applied to calculate the crack tip stress intensity factor. The model relies on the fracture toughness of hardened cement paste ( K IC) and the crack bridging law, so-called stress–crack width ( σ– δ) relationship of the material, as the fundamental material parameters for model input. As two examples, experimental data from steel FRC beams under three-point bending load are analyzed with the present fracture mechanics model. A good agreement has been found between model predictions and experimental results in terms of flexural stress–crack mouth opening displacement (CMOD) diagrams. These analyses and comparisons confirm that the structural performance of concrete and FRC elements, such as beams in bending, can be predicted by the simple fracture mechanics model as long as the related material properties, K IC and ( σ– δ) relationship, are known.

A partially insulated embedded crack in an infinite functionally graded medium under thermo-mechanical loading

In high-temperature applications, the materials research community has recently been exploring the possibility of using new nonhomogeneous coatings made of functionally graded materials (FGMs) as an alternative to the conventional homogeneous ceramic coatings. In designing components involving FGMs, an important aspect of the problem is the fracture failure. In this paper, we consider an infinite functionally graded medium with a partially insulated crack subjected to a steady-state heat flux away from the crack region as well as mechanical crack surface stresses. The material is modeled as a nonhomogeneous isotropic elastic medium. The problem is solved under the assumption of plane elasticity. The equations of heat conduction and elasticity are converted analytically into singular integral equations which are solved numerically to yield the crack tip stress intensity factors under thermo-mechanical loading. A crack-closure algorithm is developed to handle the problem of having negative mode I stress intensity factors. The main objective of the paper is to study the effect of the material nonhomogeneity parameters, crack-closure and partial crack surface insulation on the crack tip stress intensity factors for the purpose of gaining better understanding of the thermo-mechanical behavior of graded materials.

Deaerator feedwater vessel weld cracking monitored over a service time of around 80,000 hours

This paper describes the fatigue crack growth data which were recorded during a 12-year study on weld cracking in a small deaerator feedwater storage vessel. The sub-critical crack growth mechanism was corrosion fatigue in nature and varied from 5×10 −9 to 5×10 −8 mm/s. Also, crack extension was tunnelled in nature and resulted in a thumbnail shape with the crack depth growth outstripping the surface crack growth rate. A detailed fractographic study indicated that flat fan-shaped facets were common and typified corrosion fatigue in high temperature aqueous conditions. The reported fatigue crack growth data were compared with published results for low alloy steels in high temperature waters in terms of frequency, average crack tip strain rate and stress intensity factor range and favourable agreement was observed in all instances.

Crack–inclusion interaction for mode II crack analyzed by Eshelby equivalent inclusion method

The interaction between an inclusion of arbitrary shape and a crack subjected to a remote mode II stress is investigated. The approximate solutions to predict the variations in crack-tip stress intensity factors of mode II crack induced by the inclusion have been developed both for the cases where the crack tip is near or within the inclusion. The solution technique is based on the transformation toughening theory. The transformation strain induced by the inhomogeneity between matrix and inclusion is estimated based on the Eshelby equivalent inclusion method. As validated by numerical examples, the approximate solutions have satisfactory accuracy for a variety of inclusion shapes and a wide range of the modulus ratios between inclusion and matrix.

PLANE STRAIN AND AXISYMMETRIC SPALLATION OF GRADED COATINGS UNDER THERMAL LOADING

In this study the failure mechanics of thermal barrier coatings under thermally induced in-plane compression is considered. The layered medium studied is assumed to consist of a substrate (the main structural component), a bond coat (the oxygen diffusion barrier), a thin layer of thermally grown oxide that develops during operation, and a homogeneous ceramic or graded metal/ceramic top coat (the primary heat shield). After presenting some results from the plane strain solution, the axisymmetric problem is considered in some detail. Noting that for the geometry and loading studied, namely a layered medium with an interface crack that is subjected to in-plane compression, the small deformation theory would give only a trivial result, and in order to assess the limitation of the plate theory that is often used to study such instability problems, a finite deformation continuum theory is used to solve the instability and the postbuckling problems. A nonlinear finite element method incorporating special enriched and transition elements is used to analyze the problem. The primary objective of the study is the investigation of the effect of property grading parameters, kinematic nonlinearity, plate approximation, and component curvature on the crack-tip stress intensity factors, the strain energy release rate, and the crack-opening displacements.

전기장을 받는 강유전체 세라믹의 분역회전 인성화

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 crack tip stress intensity factor increases or decreases with crack growth, depending on the electrical nonlinear behavior and the direction of an applied electric field. It is shown that the ferroelectric material can be either toughened or weakened as the crack grows. The steady state crack growth in ferroelectric ceramics is also discussed.

The electrical potential difference across cracks in PZT measured by Kelvin Probe Microscopy and the implications for fracture

An indentation crack in a poled PZT ceramic subjected to an electric field is investigated using AFM and KFM to determine the crack opening displacement and the electrical potential difference across the crack. The experimental results are used to calculate the crack tip stress and dielectric displacement intensity factors and the crack tip energy release rate. From the applied electric field and the measured field interior to the crack, the dielectric constant of the crack interior is determined to be 40. The consequences of this permittivity on the crack tip energy release rate are illustrated for a Griffith crack. The theoretically predicted effect of an applied electric field in retarding crack growth decreases significantly with increasing permittivity. In practical situations in terms of crack length, applied load and electric field level, the retardation of crack growth is negligible when the dielectric constant of the crack interior is higher than 20.

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J 1-integral and the path-dependent of J 2-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.

Effect of electric fields on fracture behavior of ferroelectric ceramics

The asymptotic problem of a semi-infinite crack perpendicular to the poling direction in a ferroelectric ceramic subjected to combined electric and mechanical loading is analyzed to investigate effect of electric fields on fracture behavior. Electromechanical coupling induced by the piezoelectric effect is neglected in this paper. The shape and size of the switching zone is shown to depend strongly on the relative magnitude between the applied electric field and stress field as well as on the ratio of the coercive electric field to the yield electric field. A universal relation between the crack tip stress intensity factor and the applied intensity factors of stress and electric field under small-scale conditions is obtained from the solution of the switching zone. It is found that the ratio of the coercive electric field to the yield electric field plays a significant role in determining the enhancement or reduction of the crack tip stress intensity factor. The fracture toughness variation of ferroelectrics under combined electric and mechanical loading is also discussed.

Secondary elastic crack tip stresses which may influence very slow fatigue crack growth

ABSTRACT Secondary elastic crack tip stress intensity factors are defined in a convenient way consistent with the primary stress intensity factor definition. Methods of evaluation of these secondary stress intensity factors are developed. They are evaluated for example configurations with cracks. It is shown that the first secondary term may be sufficient to evaluate the effects of secondary stresses beyond the usual stress intensity factor and nominal normal stresses parallel to the crack direction.

Finite element modelling of frictional bridging during fatigue crack growth in fibre-reinforced metal matrix composites

In this study, the influence of thermal expansion mismatch, fibre debonding, interface friction and interface degradation on crack-tip shielding due to crack face bridging is investigated for fatigue of a Ti–6–4 matrix reinforced with unidirectional SiC-fibres using a three-dimensional finite element model of a compact tension specimen. The crack-tip stress intensity factors resulting from the residual stresses due to thermal expansion mismatch are calculated. For several levels of applied external loads, the crack-tip stress intensity factors are calculated for cyclic loading to assess the influence of friction on the build-up of crack-tip shielding during crack growth. The results show that fibre bridging, residual stresses and elastic mismatch produce crack-tip stress intensity factors that are widely different from the externally applied stress intensity factor. The numerical results show that an increase in the friction coefficient leads to a reduction of the crack-tip stress intensity factor, whereas thermal expansion mismatch leads to an increase of the crack-tip load. Finally, the degradation of the interface, modelled by a selective reduction of the friction coefficient, leads to a decrease in crack-tip shielding.

Investigating the Effect of Crack Shape on the Interaction Behavior of Noncoplanar Surface Cracks Using Finite Element Analysis

In the last two decades, multiple cracks are often found in aging aerospace and mechanical structures. The interaction and coalescence of multiple cracks may significantly affect the service lives of these aging structures. Knowledge of the behavior of interacting cracks is still limited. The calculation of the crack-tip stress intensity factor, SIF, along the interacting crack fronts is considered a major contribution for the application of any linear fracture mechanics model to investigate the growth life of these cracks. In this paper, a parametric study is presented for two parallel surface cracks in an infinite plate subjected to remote tension or to pure bending loads. This study focuses on constructing a finite element (FE) model that combines the submodeling technique with its ability to generate crack submodels of different lengths and depths, and a mesh generator that can build up a mesh grid based on the size, depth, and orientation of the interacting crack sub-models. The stress intensity factors for these cracks are calculated as a function of the crack front position, depth, shape, and plate thickness. In this paper, the values of the studied crack depth to length ratio, a/c, are 0.33, 0.5, 0.67, and 1.0. Where possible, a comparison of the 3-D with 2-D results is also considered.

Mixed mode asymptotic crack tip fields in orthotropic materials: Derivation and range of dominance

In the first part of this work the asymptotic stress field distribution surrounding a crack in a generally orthotropic solid (i.e. one in which the material and loading axes do not coincide) is derived. The resulting crack tip stress intensity factors are also related to the energy release rate of the cracked solid. In any physical situation, however, the range of dominance of these asymptotic fields will be limited, and will depend on specific geometry and loading parameters. Thus, the second part of this work deals with the range of dominance of the derived stress fields in edge cracked bending loaded fiber reinforced composite plates. The lower and upper limits of the range of dominance of the solution are respectively determined from three-dimensional and two-dimensional full field finite element solutions. The comparison of full field solutions with the asymptotic result provides information on the latter's range of dominance. In all cases the effects of mixed mode loading are considered.