Eshelby inclusion of arbitrary shape in an anisotropic plane or half-plane

Summary.  Analytical solution for Eshelby's problem of an anisotropic non-elliptical inclusion remains a challenging problem. In this paper, a simple method is presented to obtain an analytical solution for Eshelby's problem of an inclusion of arbitrary shape within an anisotropic plane or half-plane of the same elastic constants. The method is based on an observation that the interface conditions for arbitrary inclusion-shape can be written in a decoupled form in which three unknown Stroh's functions are decoupled from each other. The solution is given in terms of three auxiliary functions constructed by three conformal mappings which map the exteriors of three image curves of the inclusion boundary, defined by three Stroh's variables, onto the exterior of the unit circle. With aid of these auxiliary functions, techniques of analytical continuation can be applied to the inclusion of any shape. The solution is given in the physical plane rather than in the image plane, and is exact provided that the expansion of every mapping function includes only a finite number of terms. On the other hand, if at least one of the mapping functions includes infinite terms, a truncated polynomial mapping should be used, and thus the method gives an approximate solution. A remarkable feature of the present method is that it gives elementary expressions for the internal stress field within an inclusion in an anisotropic entire plane. Elliptical and polygonal inclusions are used to illustrate the construction of the auxiliary functions and the details of the method. Received February 18, 2002; revised July 22, 2002 Published online: February 10, 2003 Acknowledgement The financial support of the Natural Science and Engineering Research Council of Canada is gratefully acknowledged.     

Theoretical analysis of influence of crack dimensions and geometry on mass transport in corrosion–fatigue cracks

Abstract

Theoretical models are developed to describe the concentration of dissolved species in a corrosion–fatigue crack. The models are based on two separate reaction processes: the cathodic reduction of oxygen on the crack walls, and the anodic dissolution of metal at the crack tip. The concentrations of species in a trapezoidal crack are calculated for a range of crack depths from 0·25 to 20 mm, and for different minimum/maximum load ratios, stress–intensity factors, and frequencies, assuming single–edge–notch geometry. The relative importance of diffusion and fluid flow as functions of these parameters is analysed. The influence the geometry of the specimen has on the composition of the solution is determined, and the use of parallel–sided cracks is discussed. The implications of the conclusions for corrosion–fatigue crack growth are assessed.

MST/73     

Fracture behaviour of 8%Ni 980 MPa high strength steel at various temperatures Part 3 – Four-point notched bend tests

Abstract

Based on the results of four-point notched bend tests together with detailed microscopic observations of fracture surfaces and crack configurations below the unbroken notch roots of double notch specimens, the fracture mechanisms in notched specimens of 8%Ni high strength (980 MPa) steel have been observed to be as follows. A fibrous crack initiates in the bainitic matrix at the notch root and then develops into a cleavage crack at a critical length. The cleavage crack propagates in an unstable manner and causes the final fracture of specimen. The critical event controlling the cleavage fracture is the propagation of the bainitic packet-sized crack, and the local fracture stress is measured as around 1845–2200 MPa.     

Determination and representation of the stress coefficient terms by path-independent integrals in anisotropic cracked solids

Because the elastic T-stress and other coefficients of the higher-order terms play an important role in fracture mechanics such as the stability of crack kinking, crack path, and two-parameter characterization of elastic-plastic crack tip fields, determination of all the coefficients in the crack tip field expansion in an anisotropic linear elastic solid is presented in this paper. Utilizing conservation laws of elasticity and Betti's reciprocal theorem, together with selected auxiliary fields, T-stress and third-order stress coefficients near the crack tip are evaluated first from path-independent line integrals. To determine the T-stress terms using the J-integral and Betti's reciprocal work theorem, auxiliary fields under a concentrated force and moment acting at the crack tip are used respectively. Through the use of Stroh formalism in anisotropic elasticity, analytical expressions for all the coefficients including the stress intensity factors are derived in a compact form that has surprisingly simple structure in terms of one of the Barnett-Lothe tensors, L. The solution forms for degenerated materials, monoclinic, orthotropic, and isotropic materials are also presented.     

Fatigue crack initiation at notches in SA 333 piping steel

Abstract

In this paper are reported two recent investigations into crack initiation at notch roots using techniques developed for remote monitoring of crack growth in high–temperature water environments. In the first, a notched compact–type specimen of a carbon steel, SA 333 Gr. 6, was monitored for crack initiation (defined as a 0·076 mm crack at the notch root) in a variety of water chemistries and testing conditions. The mechanical conditions at the notch root have been analysed using the Neuber–notch method, enabling a direct comparison to be made with the strain–controlled fatigue data curves used for a smooth specimen in the ASME Boiler and Pressure Vessel Code, Section III. In the second investigation a different method was considered for monitoring crack initiation. A modification of the electrical–potential technique, called the reversing dc electrical potential method, was used to obtain quantitative information on the initiation and early growth of small surface cracks in notched bars of a high–strength alloy at elevated temperature. Results obtained by the method are presented and discussed.

MST/72     

Antiplane interaction of an anisotropic elliptic inclusion with an arbitrarily oriented crack

The antiplane interaction problem for an anisotropic elastic inclusion embedded in an anisotropic elastic matrix with an arbitrarily oriented crack, located either in the matrix or in the inclusion, is considered in this paper. The proposed analysis is based upon the use of conformal mapping, analytical continuation and Laurent series expansion of the corresponding complex potentials. By applying the existing solutions for dislocation functions, the integral equations for a line crack are formulated and the mode-III stress intensity factors are obtained numerically. Several numerical examples are given to demonstrate the effects of geometrical parameters and material property combinations on the strength of the antiplane stress singularity.     

Mode-III stress intensity factors for two circular inclusions subject to a remote uniform shear load

ABSTRACT

An analytical solution to the antiplane elasticity problem associated with two circular inclusions interacting with a line crack is provided in this article. A series solution for the stress field is derived in an elegant form by using complex variable theory in conjunction with the alternation method. Based on the superposition method, a singular integral equation (SIE) is established from the traction-free condition along the crack surface. After solving the SIE, the mode-III stress intensity factors (SIFs) can be obtained to quantify the singular behavior of the stress field ahead of the crack tips. Numerical results of the SIFs, when a crack is embedded either in the inclusion or in the matrix, are discussed in detail and displayed in graphic form.     

Prediction models of creep crack initiation for different specimen geometry

Abstract

In this article, the prediction models of the creep crack initiation for the specimen geometry was quantified by six different types of cracked specimens [including C-ring in tension CS(T), compact tension C(T), single notch tension SEN(T), single notch bend SEN(B), middle tension M(T), and double edge notch bend tension DEN(T)]. Load-independence constraint parameter Q* was introduced to quantify the in-plane constraint. The specimen order of Q* and the creep damage accumulation rate of the different specimen geometries from high to low was C(T), CS(T), SEN(B), SEN(T), DEN(T), and M(T), which generally represented the distinctions of in-plane constraint level in these specimens. For a specific load level, C(T) and CS(T) specimens showed the highest crack damage accumulation rate or the shortest creep crack initiation time, whereas the lowest rates or the longest CCI time existed in M(T). Moreover, the relationship between the CCI times and specimen thickness and crack depths was obtained, and a series of empirical equations were fitted. Finally, the power law relation between the CCI times and constraint parameter Q* was extrapolated.     

Ermittlung des Bruchwiderstands an Oxidkeramik und Hartmetallen mit verschiedenen Methoden

Fracture Toughness Determination of Alumina and Cemented Carbide with Different Testing Methods Fracture toughness of a sintered alumina and two tungsten carbidecobalt materials was determined using four-point-bend specimens with straight through and chevron notches and with the short rod specimen. With the specimens with a straight through notch a lower K_Ic was measured for Al₂O₃ and a higher for WC-Co compared to the chevron-notched specimens. This behavior was explained by the different shapes of the crack growth resistance curves and the different critical notch radii. For Al₂O₃ a steeply rising crack growth resistance curve was measured in a controlled fracture test, for WC-Co a flatter curve was obtained. The effect of the shape of the crack growth resistance curve and of notch width on the evaluated toughness is discussed.     

Stress intensity factor analysis of an interface crack between dissimilar anisotropic materials under thermal stress using the finite element analysis

New numerical methods were presented for stress intensity factor analyses of two-dimensional interfacial crack between dissimilar anisotropic materials subjected to thermal stress. The virtual crack extension method and the thermal M-integral method for a crack along the interface between two different materials were applied to the thermoelastic interfacial crack in anisotropic bimaterials. The moving least-squares approximation was used to calculate the value of the thermal M-integral. The thermal M-integral in conjunction with the moving least-squares approximation can calculate the stress intensity factors from only nodal displacements obtained by the finite element analysis. The stress intensity factors analyses of double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load were demonstrated. Excellent agreement was achieved between the numerical results obtained by the present methods and the exact solution. In addition, the stress intensity factors of double edge cracks in jointed dissimilar anisotropic semi-infinite plates subjected to thermal loads were analyzed. Their results appear reasonable.     

Notch fatigue behaviour of low-temperature gaseous carburised 316L austenitic stainless steel

ABSTRACT

The influence of low-temperature gaseous carburisation on notch fatigue behaviour of 316L steel under cyclic axial loading was investigated. After carburisation, the carburised case was well distributed at the surface region and was not influenced by the notch geometry. Low-temperature carburisation considerably enhanced the notch fatigue performance, which led to 32% and 44% increase in the endurance limits for the specimens with stress concentration factors K _t?=?1.91 and 3.91, respectively. The notch sensitivity of 316L steel reduced after carburisation. Irrespective of the applied stress amplitude, the fatigue crack nucleation sites were always at the notch root surface for the untreated specimens. For the carburised specimens, fatigue cracks nucleation changed from surface at high-level stress to subsurface at low-level stress.     

The growth of short fatigue cracks ahead of a notch in high strength steel

This paper is concerned with the short crack growth behaviour in notched specimens of high strength steels mainly used for automotive parts. Attention is focused on the appropriate estimation of the stress intensity range ΔK, when the short cracks grow in the stress field of a notch and attempts are made to investigate whether this is appropriate or whether further allowance may have to be made for short crack effects at low ΔK.The stress distribution for the notch in bending was taken into account for the estimation of ΔK. This enabled us to produce a more precise evaluation of short crack growth in the stress field of the notch. Our findings are that the short cracks, which propagate in the notch field, grow faster at low ΔK when their lengths are extremely short compared with the notch root radius. The short crack effect at low ΔK in the notch stress field is analysed by expressing the crack growth data in terms of the parameter, ΔK_s, in which stress gradient ahead of the notch is taken into account.     

Crack propagation from a pre-existing flaw at a notch root – II: Detailed form of the stress intensity factors at the initial crack tip and conclusion.

This paper pursues the study of crack kinking from a pre-existing crack emanating from some notch root. It was shown in Part I that the stress intensity factors at the tip of the small initial crack are given by universal (that is, applicable in all situations, whatever the geometry of the body and the loading) formulae; they depend only on the `stress intensity factor of the notch' (the multiplicative coefficient of the singular stress field near the apex of the notch in the absence of the crack), the length of the crack, the aperture angle of the notch and the angle between its bisecting line and the direction of the crack. Here we identify the universal functions of the two angles just mentioned which appear in these formulae, by considering the model problem of an infinite body endowed with a notch with straight boundaries and a straight crack of unit length. The treatment uses Muskhelishvili's complex potentials formalism combined with some conformal mapping. The solution is expressed in the form of an infinite series involving an integral operator, which is evaluated numerically. Application of Goldstein and Salganik's principle of local symmetry then leads to prediction of the kink angle of the crack extension. It is found that although the direction of the crack is closer to that of the bisecting line of the notch after kinking than before it, the kink angle is not large enough for the crack tip to get closer to this line after kinking, except perhaps in some special situations.     

Calculation of the energy J-integral for bodies with notches and cracks

The approximate solutions for calculation of the energy J-integral of a body both with a notch and with a crack under elastic-plastic loading have been obtained. The crack is considered as the limit case of a sharp notch. The method is based on stress concentration analysis near a notch/crack tip and the modified Neuber's approach. The HRR-model and the method based on an equation of equilibrium were also employed to calculate the J-integral. The influence of the strain hardening exponent on the J-integral is discussed. New aspects of the two-parameter J ^* _c-fracture criterion for a body with a short crack are studied. A theoretical investigation of the effect of the applied critical stress (or the crack length) on the strain fields ahead of the crack tip has been carried out.     

Cracks at rounded V-notch tips: an analytical expression for the stress intensity factor

An analytical expression for the stress intensity factor related to a crack stemming from a blunted V-notch tip is put forward. The analysis is limited to mode I loading conditions, as long as the crack length is sufficiently small with respect to the notch depth. The proposed formula improves significantly the predictions of a recently introduced relationship, by considering a notch amplitude dependent parameter. Its values are estimated through a finite element analysis: different notch amplitudes, ranging from $0^{\circ }$ to $180^{\circ }$ , and different crack length to root radius ratios, ranging from 0 to 10, are taken into account. The evaluation of the apparent generalized fracture toughness according to equivalent linear elastic fracture mechanics concludes the paper.     

Stresses and stress intensities at notches: ‘anomalous crack growth’ revisited

In this paper, a new notch stress estimation scheme is presented within the context of conventional finite element solutions. The estimation method is based on a separation of an actual notch stress state into two parts. One part is a far-field stress in the form of membrane and bending components that satisfy far-field equilibrium conditions, and the other is a self-equilibrating part that provides an effective measure of the notch stress state. The self-equilibrating part can be directly related to notch geometry and captured using a rather coarse finite element model. The singular stress behavior at an arbitrary notch is then described by a closed form solution formulated using the self-equilibrating part of the stress state for a small crack emanating from a notch. The corresponding stress intensity factor solutions are then presented by considering both the far-field stress (also called structural stress) and self-equilibrating notch stress. The stress intensity solutions are formulated using existing solutions for typical simple crack geometry. One important feature of the notch stress and stress intensity solutions is that the current solutions not only capture the singular characteristics as the notch tip is asymptotically approached, but also recover the far-field (or nominal) stress state.Finally, the effectiveness of the present notch stress estimation scheme is demonstrated by using a series of well-known short crack growth data exhibiting ‘anomalous growth’. It has been found that, for instance, the anomalous growth discussed in, for example, Fat Eng Mat Struct 6 (1983) 315; Eng Fract Mech 29 (1988) 301; as well as Surface Crack Growth: Models, Experiments, and Structures (1990) 333, can be unified with long crack data as straight lines, without resorting crack closure considerations. As a result, a two-stage crack growth model is proposed within the context of K-dominant crack growth. The first stage is dominated by the notch-induced self-equilibrating part of the stress state and the second stage is dominated by the equivalent far-field stress state or structural stresses that satisfy equilibrium conditions and can be effectively computed in a mesh-insensitive manner. The implications of these findings on fatigue growth rate data generation and fatigue life predictions will also be discussed.     

Analysis of a crack sited at a notch in an elastic-perfectly plastic strip subjected to longitudinal shear

An exact linear elastic-perfectly plastic solution is given for a problem of a strip of finite width, which contains a crack sited at the apex of a sharp notch. The strip is subjected to uniform anti-plane loading far from the notch. The solution is obtained by an extension of a conformal mapping procedure employed by J. R. Rice and is expressed in terms of definite integrals. Particular quantities that are of interest in relation to fracture criteria, namely the J-integral and the maximum length of the plastic zone, are presented graphically for a 45° notch, for a variety of crack lengths and all possible applied loads. The results indicate that the stress ahead of the crack is approximated well by the stress ahead of an appropriately chosen "equivalent" crack in an un-notched strip, even when the crack is much smaller than the notch depth and the loads are high enough to induce extensive yielding.

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Résumé On fournit une solution exacte en système d'élasticité linéaire et de plasticité parfaite au problème d'un feuillard de largeur finie comportant une fissure localisée à la racine d'une entaille aigüe. Le feuillard est soumis à une sollicitation uniforme antiplanaire appliquée à grande distance de l'entaille. La solution résulte d'une extension d'une procédure de représentation conforme utilisée par J. R. Rice et exprimée sous forme d'intégrales définies. Les quantités intéressantes à faire intervenir dans les critères de rupture, à savoir l'intégrale J et la longueur maximum de la zône plastique, sont présentées sous forme graphique dans le cas d'une entaille à 45°, d'un échelonnement de longueurs de fissure, et de toutes les configurations de charges possibles.Les résultats montrent que la contrainte en avant de la fissure peut être donnée avec une bonne approximation par la contrainte qui se trouverait en avant d'une fissure "équivalente" dans un feuillard sans entaille, même lorsque la fissure réelle est bien plus petite que la profondeur de l'entaille, et que les charges sont suffisamment élevées pour entraîner un écoulement plastique significatif.

     

Initiation and growth of corrosion-fatigue cracks near stress concentrators in V95pchT2 aluminum alloy

We investigate the kinetics of small fatigue cracks starting from notches in samples of V95pchT2 aluminum alloy in air and in a 3.5% NaCl solution. The influence of salt water is manifested in the initiation of pits on the notch surface and subsequent accelerated growth of small cracks (2–3 times faster than in the laboratory air). The size of an initial macrocracka _i does not depend on the level of stress or strain, notch radius, or environment and is equal to the characteristic distanced ^*=100 μm of the prefailure zone which is a material constant. The effect of crack closure in the presence of a corrosive medium is observed at the crack lengtha>100 μm. Within the frameworks of the approach proposed earlier for the study of initiation of a macrocrack near a notch, the basic dependences of values of local stresses or strains on the duration of the period of initiation of a macrocrack with lengtha _i =d ^* are established, and the standard diagrams of cyclic crack resistance are constructed for V95pchT2 aluminum alloy at the stage of macrocrack growth in air and in a 3.5% NaCl solution. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 35, No. 1, pp. 7–15, January–February, 1999.     

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 materials 1 , 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.     

The problem of two plastic and heterogeneous inclusions in an anisotropic medium

We demonstrate an integral equation for the total local strain ε^T in an anisotropic heterogeneous medium with incompatible strain ε^p and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Horn's approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.