Mode III crack growth in linear hardening materials with strain gradient effects

The flow-theory version of couple stress strain gradient plasticity is adopted for investigating the asymptotic fields near a steadily propagating crack-tip, under Mode III loading conditions. By incorporating a material characteristic length, typically of the order of few microns for ductile metals, the adopted constitutive model accounts for the microstructure of the material and can capture the strong size effects arising at small scales. The effects of microstructure result in a substantial increase in the singularities of the skew-symmetric stress and couple stress fields, which occurs also for a small hardening coefficient. The symmetric stress field turns out to be non-singular according to the asymptotic solution for the stationary crack problem in linear elastic couple stress materials. The performed asymptotic analysis can provide useful predictions about the increase of the traction level ahead of the crack-tip due to the sole contribution of the rotation gradient, which has been found relevant and non-negligible at the micron scale.     

A strain energy criterion for mixed mode crack propagation

The strain energy criterion for crack propagation proposed in the paper is based on the principle that the direction of crack propagation takes place along the direction where the distance from the crack tip to a certain contour line of constant distortional strain energy density is minimum, and the crack will begin to propagate when the total strain energy in the region surrounded by the contour line reaches a critical value. In the paper, predicted was compared with the measured results.     

Asymptotic and finite element analyses of mode III dynamic crack growth at a ductile-brittle interface

In this work, dynamic crack growth along a ductile-brittle interface under anti-plane strain conditions is studied. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behavior. Firstly, the asymptotic near-tip stress and velocity fields are derived. These fields are assumed to be variable-separable with a power singularity in the radial coordinate centered at the crack tip. The effects of crack speed, strain hardening of the ductile phase and mismatch in elastic moduli of the two phases on the singularity exponent and the angular functions are studied. Secondly, full-field finite element analyses of the problem under small-scale yielding conditions are performed. The validity of the asymptotic fields and their range of dominance are determined by comparing them with the results of the full-field finite element analyses. Finally, theoretical predictions are made of the variations of the dynamic fracture toughness with crack velocity. The influence of the bi-material parameters on the above variation is investigated.     

Mechanics of mixed mode small fatigue crack growth

An elastic-plastic analysis of a mixed mode crack was conducted under general yielding conditions using a finite element method. Based on the finite element analysis, a formula of the J-integral was developed, and the behavior of the crack opening or sliding displacement was investigated. Crack growth mechanics was discussed to explain mixed mode growth. A model expressed by the J-integral was proposed assuming that the crack growth is determined by the linear summation of relations of pure Mode I and II crack-tip deformation. The crack growth rate obtained using Inconel 718 thin-wall tubular specimens was correlated with the range of ΔJ evaluated from the proposed formula. The crack growth equations in terms of ΔJ for three different biaxial strain ratios were compared with the relations expected from the proposed crack growth model. The difference between the experimental results and the estimation was discussed from the viewpoint of crack closure and the geometry of the surface crack.     

The asymptotic near-tip solution for mode-III crack in steady growth in power hardening media

It is shown that, for mode-III crack in steady growth in power hardening media the near-tip singularity is not of the power function type but is logarithmic. By consideration of the inner boundary layer in the neighborhood of the unloading boundary, the asymptotic near-tip solutions for stress and strain fields are obtained, which satisfy all the necessary relations of elastic-plastic continuum mechanics, including the condition of continuity of the plastic parts of strain-rates across the unloading boundary.     

A hyperelastic-based large strain elasto-plastic constitutive formulation with combined isotropic-kinematic hardening using the logarithmic stress and strain measures

This paper addresses the formulation of a set of constitutive equations for finite deformation metal plasticity. The combined isotropic-kinematic hardening model of the infinitesimal theory of plasticity is extended to the large strain range on the basis of three main assumptions: (i) the formulation is hyperelastic based, (ii) the stress-strain law preserves the elastic constants of the infinitesimal theory but is written in terms of the Hencky strain tensor and its elastic work conjugate stress tensor, and (iii) the multiplicative decomposition of the deformation gradient is adopted. Since no stress rates are present, the formulation is, of course, numerically objective in the time integration. It is shown that the model gives adequate physical behaviour, and comparison is made with an equivalent constitutive model based on the additive decomposition of the strain tensor.     

The condition of fatigue crack growth in mixed mode condition

The condition of the initiation of fatigue crack growth in mixed mode conditions has been investigated by using precracked low carbon steel specimens.It is pointed out that, firstly, the critical condition of crack growth should be defined with regard to the modes of fatigue crack growth, i.e. shear mode and tensile mode. Secondly, it is proposed that the critical condition of fatigue crack growth is given by the local tensile stress and shearing stress at the notch tip determined by stress intensity factors $\text{K}{}_{\mathrm{I}}$ and $\text{K}{}_{\mathrm{II}}$, and that this criterion is generally applicable to in-plane-loading conditions, i.e. Mode $\text{I}$, Mode $\text{II}$ and Mixed Mode conditions.     

The mode III full-field solution in elastic materials with strain gradient effects

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical K_III field; stresses have r ^-3/2 singularity near the crack tip, and are significantly larger than the classical K _III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K _III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K _III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral. This revised version was published online in August 2006 with corrections to the Cover Date.     

Singular fields of a mode III interface crack in a power law hardening bimaterial

A high order of asymptotic solution of the singular fields near the tip of a mode III interface crack for pure power law hardening bimaterials is obtained by using the hodograph transformation. It is found that the zero order of the asymptotic solution corresponds to the assumption of a rigid substrate at the interface, and the first order of it is deduced in order to satisfy completely two continuity conditions of the stress and displacement across the interface in the asymptotic sense. The singularities of stress and strain of the zero order asymptotic solutions are –1/(n ₁+1) and –n/(n ₁+1) respectively (n=n ₁, n ₂ is the hardening exponent of the bimaterials). The applicability conditions of the asymptotic solutions are determined for both zero and first orders. It is proved that the Guo-Keer solution [23] is limited in some conditions. The angular functions of the singular fields for this interface crack problem are first expressed by closed form.     

Unsteady growth of mode III crack in elastic perfectly-plastic material

In this paper the assembly of the near-tip fields given by J. R. Rice is completed for the mode III crack growing quasi-statically and unsteadily in elastic perfectly-plastic material. The obtained results provide a particular example for the general theoretical relations between the steady state and unsteady state crack growth. Further, the general expression of the rate of crack opening displacement is obtained, which is similar to one by J.R. Rice and co-workers for mode I crack growing in elastic perfectly-plastic material. The fracture criterion of the critical opening displacement at a prescribed distance behind the crack tip is discussed. As a result, the theoretical J-resistance curves are given.     

A fatigue crack propagation model for strain hardening materials

In this paper a crack propagation model based on Tomkins concept (dl/dN ∝ Δε_p · ω) has been developed using the theoretically developed cyclic plastic zone sizes. The crack propagation rates are found to be functions of stress intensity factor, Elber's effective stress range ratio, cyclic yield strength of material, crack length, specimen width and cyclic strain hardening exponent. Suitably grouped to give the crack growth rate in terms of five constants termed as Loading Constant, Material constant, Crack size constant, specimen Width Constant and Stress Intensity Exponent. The crack growth rates found by theory are compared with the experimental results available in literature and a good agreement is found.     

Path prediction of I + III mixed mode fatigue crack propagation

In this paper, compact tension specimens with tilted cracks under monotonic fatigue loading were tested to investigate I + III mixed mode fatigue crack propagation in the material of No. 45 steel with the emphasis on the propagation rate expression and the path prediction. It is found that during the mode transformation process, the crack propagation rate is still controlled by the mode I stress intensity factor; and Paris equation also holds for the relationship between and ΔK_I . Crack propagation path can be predicted only when both the crack mode transformation rate and propagation rate are available.     

The influence of biaxial strain ratio and strain range on crack growth mode and crack shape

Axial-torsion tests were used to determine and model the combinations of strain range and crack length at which the crack growth changed from a shear to a tensile mode. Biaxial tests and confocal scanning laser microscopy (CSLM) were used to observe crack shape evolution at various strain ratios. Constant and variable amplitude loading of notched specimens and CSLM were used to determine the dependence of crack shape development on strain range. The results showed that, except for very high strain levels, at strain ratios causing shear mode crack growth multiple cracks initiated and grew until they were semicircular and then linked up to cause failure. For tensile mode crack growth at low strains a single crack grew into and maintained a semicircular shape. At high strain ranges multiple cracks formed and linked up to maintain an elongated crack shape.     

Asymptotic fields ahead a crack for a class of non linear materials under mode III

The paper considers the mechanical fields near the tip of a crack deformed by an anti-plane shear at infinity for a class of non linear elastic materials. For brittle material rupture occurs when a maximal stretch is reached. Taking account of this critical value, the crack is replaced by a totally damaged zone of finite thickness named a quasicrack. Inside this domain, the stress is identically zero and the shape of the boundary between damaged and undamaged body is found analytically.     

Random fatigue crack growth in mixed mode by stochastic collocation method

Fatigue crack growth is uncertain, either for cracking rate or direction. The stochastic models proposed in the literature suffer from limited applicability or lack of physical meaning. In this paper, a new stochastic collocation method is proposed to solve mixed mode fatigue crack growth problems with uncertain parameters. This approach has the advantage of non-intrusive nature methods, such as Monte-Carlo simulations, since it allows us to decouple the stochastic and the mechanical computations. The proposed numerical implementation is very simple, as it requires only repetitive runs of deterministic finite element analysis at some specific points in the random space. The method describes a precise approximation of the mechanical response corresponding to the fatigue life, in order to assess the stochastic properties, namely the statistical moments and the probability density function of fatigue life. The performance of the stochastic collocation method for dealing with this kind of problems has been evaluated through two numerical examples, showing the high performance for practical applications. Moreover, the proposed method is extended in the last example to the failure probability assessment, with respect to the target service life.     

Effect of anisotropic plasticity on mixed mode interface crack growth

Crack growth along an interface between a solid with plastic anisotropy and an elastic substrate is modelled by representing the fracture process in terms of a traction-separation law specified on a crack plane. A phenomenological elastic-viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy. Conditions of small-scale yielding are assumed, and due to the mismatch of elastic properties across the interface the corresponding oscillating stress singularity field is applied as boundary conditions on the outer edge of the region analyzed. Crack growth resistance curves are calculated numerically, and based on these results the dependence of the steady-state fracture toughness on the near-tip mode mixity is determined. Different initial orientations of the principal axes relative to the interface are considered and it is found that the steady-state fracture toughness is quite sensitive to this orientation of the anisotropy.     

Griffith's criterion for mixed mode crack propagation

The fracture criterion of mixed mode crack has been investigated extensively by many people on the basis of Griffith's theory. In this paper, the energy release rate of crack propagation along an arbitrary direction is obtained by the use of Griffith's theory. Then the general expression of the energy release rate criterion has been derived. The results have been compared with the measured data of the fracture testing under combined Mode II–III and Mode I–II–III loading.     

Combination of finite element analysis and analytical crack tip solution for mixed mode fracture

A finite element program was developed which combines the analytical crack tip solution with a conventional finite element analysis and evaluates various crack tip parameters as part of the solution. This program was used to analyze cracked specimens subjected to mixed mode loading. The importance of retaining the second term of the series expansion for local stress, a contribution which is independent of the distance from the crack tip, was demonstrated. It was first shown analytically that the presence of a load applied parallel to the crack reveals itself only through this constant second term, which vanishes only for specific loading conditions. The results of the numerical analysis demonstrate that the stress intensity factor K_I is independent of the load applied parallel to the crack only when this term is included in the analytical crack tip solutions. Failure to include the constant term has the effect that K_I varies with the horizontal load. The parameter K₁₁ is independent of this load in both cases. This indicatesonce again that it is this constant term which accounts solely and entirely for the presence of a load applied parallel to the crack.     

Micro-mechanical modelling of mode III fatigue crack growth in rotor steels

A study has been made of fatigue crack propagation in mode III (anti-plane shear) for A469 and A470 commercial rotor steels (tensile strength 621 and 764 MN/m² respectively) using torsionally-loaded circumferentially-notched cylindrical specimens. For crack growth under both small-and large-scale yielding conditions, radial mode III crack propagation rates are observed to be similar in both steels and to be uniquely related to the plastic intensity range 163-1 per cycle, provided friction, abrasion and interlocking between sliding crack faces is minimized by the application of a small tensile mean load. Over the range studied (i.e., 10^-6 to 10^-1 mm/cycle), mode III growth rates (dc/dN)_III are found to be independent of load ratio (for R=–1.0 and –0.5) and to be a power law function of 163-1 or the mode III cyclic crack tip displacement. When compared to mode I crack growth at equivalent cyclic crack tip displacements, however, crack propagation rates in mode III are seen to be two orders of magnitude smaller than in mode I. Based on fractographic evidence of elongated voids, parallel to the crack front, at the tip of the fatigue crack, several models for mode III crack growth are proposed utilizing the concept that mode III crack advance occurs by the initiation and coalescence of voids formed at inclusions directly ahead of the crack tip. By considering the linkage of these voids to take place by mode II shear parallel to the main crack front, expressions for the mode III crack propagation rate are developed based either on considerations of the local mode II crack tip displacementsor the mode II accumulated crack tip strain (computed from the Coffin-Manson damage relationship). Whereas both types of models predict mode III growth rates to be a small fraction of the cyclic crack tip displacements per cycle, the damage accumulation model in particular is found to provide excellent agreement with experimentally measured growth rates in the present rotor steels.

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Résumé On a procédé à une étude de la propagation en mode III (cisaillement antiplanaire) des fissures de fatigue dans les aciers commerciaux pour rotors des nuances A 469 (UTS 621 MN/m²) et A 47 à (UTS 764 MN/m²) à l'aide d'éprouvettes cylindriques à entaille circonférentielle soumises à tension.Pour des croissances de fissures demeurant sous les conditions d'écoulement plastique à petite ou à grande échelle, on observe que les vitesses de propagation d'une fissure radiale en mode III sont similaires dans les deux aciers. Si on minimise les frottements, l'abrasion et les couplages mécaniques entre les faces de la fissure en appliquant une légère contrainte de traction, on observe également que ces vitesses sont en seule relation avec l'étendue de la déformation plastique III par cycle. Dans la gamme étudiée (F de 10^-6 à 10^-1 mm/cycle) on trouve que la vitesse de croissance de la fissure en mode III est indépendante de R (pour R=–1 et –0,5) et est une fonction parabolique de III ou du déplacement d'extrémité de la fissure. Toutefois, si on compare les vitesses de croissance en mode III à celles obtenues en mode I sous les mêmes déplacements d'extrémités de fissure, on constate que les premières sont inférieures aux secondes de deux ordres de grandeur. On propose divers modèles pour la croissance de fissure en mode III, en se basant sur l'observation fractographique de la présence de lacunes allongées parallèles sur front de fissuration.Ces modèles recourent au concept de préfissuration par coalescences des lacunes formées sur des inclusions en amont de la fissure. En considérant que ces lacunes se réunissent par un cisaillement de mode II parallèle au front de fissuration principale, on a développé des expressions pour la vitesse de propagation suivant le mode III, qui reposent soit sur les déplacements locaux de l'extrémité de la fissure suivant un mode II, soit sur la déformation de mode II accumulée à l'extrémité de la fissure et calculée par la relation de dommages cumulés de Coffin-Manson.Si les deux types de modèle prédisent que les taux de croissance selon le mode II ne sont qu'une faible fraction des déplacements par cycle, le modèle recourant au concept de dommages cummulés se montre quant à lui en excellent accord avec les vitesses de propagation mesurées expérimentalement sur lesdits aciers pour rotors.