Stress-strain relations in elastoplastic solids with Dugdale-type cracks

Mechanical behavior of a two-dimensional elastoplastic solid with rectilinear cracks is investigated. Plastic strip model is used to reduce plasticity problem to the equivalent linear elasticity formulation. Two realizations of the mixed mode plastic strip model are considered: in-line plastic strips as proposed by Becker and Gross [Int. J. Fract. 37 (1988) 163], and inclined plastic strips of Panasyuk and Savruk [Appl. Mech. Rev. 47 (1994) 151]. The effective mechanical response predictions are based on the procedure presented in Kachanov et al., [Appl. Mech. Rev. 47 (1994) 151]. Stress-strain relations are obtained for parallel and randomly oriented non-interacting cracks. Results are compared with known elastic solutions.     

Two equal symmetric circular arc cracks in an elastic medium – the Dugdale approach

The Dugdale model for two equal, symmetrically situated coplanar circular arc cracks contained in an infinite elastic perfectly-plastic plate is proposed. Biaxial loads are applied at the infinite boundary of the plate. Consequently, the rims of the cracks open in Mode I and develop a plastic zone ahead of each of the cracks. These plastic zones are then closed by the distribution of uniform normal closing stresses over the rims of the plastic zones. Based on the complex-variable technique and the superposition principle, the solution for the above problem is obtained. Closed-form analytic expressions are obtained for the determination of the sizes of the plastic zones and the crack-opening displacement (COD) at the tip of the crack. Numerical studies are carried out to calculate the load ratio (load applied at infinity/yield point stress applied at the rims of the plastic zones) required for the closure of the plastic zones, for various radii of arc cracks and for various angles subtended by them at the centre. The crack-opening displacement is also investigated with respect to these parameters.     

Size of thermal plastic zones around external cracks

This paper presents a solution of the steady-state thermal stresses and displacements in an infinite solid containing an external circular crack whose surfaces are exposed to an antisymmetric temperature field. The singular stresses round the leading edge of the crack are eliminated by adopting Dugdale's hypothesis concerning the existence of a thin plastic zone at the boundary of the crack. Lengths of plastic zones for various thermal disturbances are calculated and shown in curves.

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Zusammenfassung Diese Abhandlung präsentiert eine Lösung fur die thermalen Drucke im Festzustand and ihre Verschiebungen in einem unendlichen Festkörper, der eine äußere RiBzirkulation zeigt. Die Flächen dieses unendlichen Festkörpers sind einem ungleichmäßigen Temperaturfeld ausgesetzt. Die singularen Drucke, die sich um die Hauptrißkante herum befanden, wurden entfernt. Man nahm Dugdale's Hypothese, die die Existenz einer diinnen, plastischen Zone an der Grenze des Rißes behandelt, an. Die Längen von plastischen Zonen fur verschiedene thermale Störungen wurden berechnet and in Kurven gezeigt.

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Résumé Le mémoire présente une solution a la détermination des contraintes thermiques stationnaires et des déplacements dans un solide infini compbrtant une fissure externe circulaire dont les surfaces sent soumises a l'éffet d'un champ thermique antisymétrique.Les contraintes singulières mises au voisinage du bord directeur de la fissure sont eliminées par l'adoption de l'hypothèse de Dugdale relative à l'existence d'une zone mince de déformations plastiques aux frontières d'une fissure.L'ètendue des zones de déformation plastique est calculèe et exprimèe sous forme de courbes en fonction de divers types de perturbations thermiques.

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This investigation was supported by the National Science Foundation under grant No. GK 3868.     

Ultrasonic detection and sizing of plastic zones surrounding fatigue cracks

The problem of detection and sizing of fatigue cracks that are closed due to the reverse plastic deformation is addressed in this paper. In particular, it is suggested that the zone of plastic deformation can be identified by measuring the changes in the wave speeds in the plastically deformed material. This is supported through experimental results, first by determining the changes in wave speed with monotonic plastic deformation and then by investigating the plastic zones near fatigue cracks. The characteristic changes in the wave speeds are interpreted in terms of the theory of wave propagation in nonlinearly deformed media and elastic plastic fracture mechanics.     

Two cracks with coalesced interior plastic zones — The generalised Dugdale model approach

The problem investigated is of an elastic-perfectly plastic infinite plate containing two equal collinear and symmetrically situated straight cracks. The plate is subjected to loads at infinity inducing mode I type deformations at the rims of the cracks. Consequently, plastic zones are formed ahead of the tips of the cracks. The loads at infinity are increased to a limit such that the plastic zones formed at the neighbouring interior tips of the cracks get coalesced. The plastic zones developed at the tips of the cracks are closed by applying normal cohesive quadratically varying stress distribution over their rims. The opening of the cracks is consequently arrested. Complex variable technique is used in conjugation with Dugdale’s hypothesis to obtain analytical solutions. Closed form analytical expressions are derived for calculating plastic zone size and crack opening displacement. An illustrative numerical example is discussed to study the qualitative behaviour of the loads required to arrest the cracks from opening with respect to parameters viz. crack length, plastic zone length and inter-crack distance. Crack opening displacement at the tip of the crack is also studied against these parameters.     

Measurement of plastic zones associated with small fatigue cracks by selected area channeling patterns

The selected area electron channeling pattern method has been used to measure the plasticity associated with “growing” small fatigue cracks in a 7000 series aluminium alloy, allowing study of plastic zone development as well as investigation of the mechanisms by which small fatigue cracks and their plastic zones interact with grain boundaries. It was found that both plastic zone size and shape were dependent on both crack length and growth morphology. Naturally initiated Stage I fatigue cracks were predominantly crystallographic in nature and were accompanied by a relatively long, slender plastic zone shape. However, during subsequent growth over 1–2 grain diameters, this shape evolved, first to a semicircular shape, and finally to the lobed configuration typically found associated with long cracks.     

Assessment of plastic collapse behavior for tubes with collinear cracks

The 40% of wall thickness criterion has been used as a plugging rule of steam generator tubes but it can be applicable just to a single-cracked tubes. In the previous studies preformed by the authors, a total of 10 local failure prediction models were introduced to estimate the coalescence load of two adjacent collinear through-wall cracks existing in thin plates, and the reaction force model and plastic zone contact model were selected as optimum models among them. The objective of this study is to verify the applicability of the proposed optimum local failure prediction models to the tubes with two collinear through-wall cracks. For this, a series of plastic collapse tests and finite element analyses were carried out using the tubes containing two collinear through-wall cracks. It has been shown that the proposed optimum failure models can predict the local failure behavior of two collinear through-wall cracks existing in tubes well. And a coalescence evaluation diagram was developed which can be used to determine whether the adjacent cracks detected by NDE coalesce or not.     

A BOUNDARY ELEMENT MODEL OF PLASTICITY-INDUCED FATIGUE CRACK CLOSURE

This paper describes a plane stress boundary element model of plasticity-induced fatigue crack closure. A simple Dugdale-type strip yield zone is used and quadratic programming techniques are employed to establish crack shape, stress and plastic deformation. The technique is extremely effective and the model can be readily implemented on a personal computer. Predictions of crack closure behaviour are produced for cracks growing under constant amplitude loading, and also following an overload or overload/underload cycle. These results are compared with an empirical R-ratio correction due to Walker and with experimental measurements taken from the literature. The model is found to give good predictions of crack behaviour under constant amplitude loading. Predictions for crack closure levels following an overload cycle give qualitative agreement with experimental results; the differences observed may well be due to the different definition of crack closure in the experiments.     

Constraint effects on plastic crack-tip fields for plane strain mode-I, II and III cracks in non-hardening materials

To explore constraint effects on fully plastic crakc-tip fields, analytical solutions are examined for mode-I, II and III loading in non-hardening materials under plane strain conditions. The results reveal that under mode-II and III loading the crack-tip stress fields are unique, and thus can be characterized by a `single parameter'. Under mode-I loading, however, the crack-tip stress field is non-unique but can be characterized by two sets of solutions or `two parameters'. One set of the solutions is the well-known Prandtl field and the other is a plastic T-stress field. This conclusion corroborates the observation of McClintock (1971) that the slip-line field is non-unique for plane strain tensile cracks. A two-term plastic solution which combines the Prandtl field and the plastic T-stress field with two parameters B ₁ and B ₂ can then characterize the crack-tip stress field of plane strain mode-I crack over the plastic region and quantify the magnitude of crack-tip constraints. These characters are similar to those for hardening materials. Analyses and examples show that the two-term plastic solution can match well with the slip-line field or finite element results over plastic region. Thus the parameters B ₁ and B ₂ can be used to characterize the constraint level for mode-I finite-sized crack specimens in non-hardening materials under plane strain conditions.     

Weight function based Dugdale model for mixed-mode crack problems with arbitrary crack surface tractions

Weight function theory states crack surface displacements can be found for any arbitrary distribution of mode I, or mixed-mode crack face tractions via that geometry’s weight functions. This statement is validated via finite element analysis of the infinite center-cracked plate for various mixed mode loadings. An elastic-perfectly plastic material is considered using a Dugdale approach and compared to elastic-plastic finite element simulations. The weight function method in all cases agrees well with the finite element simulations for small scale yielding at the crack tip. As the maximum traction value approaches one-half the yield strength discrepancies become larger due to violation of small scale yielding.     

An approximation for the cyclic state of stress ahead of cracks and its implications under fatigue crack growth

Numerical methods are mostly used in the field of fatigue to derive the stress intensity factor (SIF) or J-integral solutions to be employed in damage tolerance analysis of cracked components. In this frame, simple assumptions about material properties are taken into account.More refined approaches try to describe the plasticity-induced crack closure in order to account for retardation effects under variable amplitude loading. In these approaches, the cyclic plasticity is used and cyclic finite element analyses are carried out.In the present work, a novel strategy is presented for the calculation of the relevant parameters to the fatigue crack growth, based on the evaluation of local field parameters (J-integral, T-stress) and cyclic material properties. It is demonstrated that, in case of mild steels and under the assumption of a stress ratio R = −1, the global constraint factor α_g widely employed in fatigue crack growth algorithms such as the strip-yield model, can be calculated in a closed-form on the basis of the expression of the crack-tip fields. Moreover, α_g provides a reasonable explanation of the fatigue crack growth behaviour of the A1N steel for different geometrical and loading configurations. Further investigations carried out on different medium and high strength steel grades show that the plastic radius ahead of small and long cracks at their fatigue limits can be considered as a constant for the material.     

Wrinkling analysis of membranes with elastic–plastic material behavior

This paper concerns membrane wrinkling in conjunction with inelastic effects. A method will be presented, able to treat wrinkling in membranes with elastic–plastic material behavior. Some effort is spent to obtain a suitable initial guess for the internal wrinkling algorithm iteration, which improves the reliability of the method. The wrinkling algorithm is implemented into a FE–program and applied to structural analysis. Numerical and experimental results are discussed.     

Analytical and numerical modeling of R curves for cracks with bridging zones

At the onset of fracture in materials with process zones, the fracture resistance, or R curve, rises as the process zone develops. After process zone development, crack propagation proceeds by steady state growth. By considering J integral contours inside and outside the process zone, the available energy can be partitioned into crack tip energy release rate and process zone energy. To model the rising R curve, however, required assumptions about damage mechanisms in the process zone and partitioning of its energy into released and recoverable energy. By considering process zones that are elastic fiber-bridging zones with softening regions caused by fiber breakage or damage, equations for rising R curves were derived as a function of crack tip toughness and bridging zone mechanics. The new methods were implemented into the Material Point Method for generalized numerical crack propagation simulations with bridging zones. The simulation method includes pure fracture mechanics and pure cohesive zone models as extreme special cases. The most realistic simulations for many materials will likely fall between these two extremes. The results guided comments on interpretation of experimental R curves.     

A crystal plasticity study of cyclic constitutive behaviour, crack-tip deformation and crack-growth path for a polycrystalline nickel-based superalloy

Crystal plasticity has been applied to model the cyclic constitutive behaviour of a polycrystalline nickel-based superalloy at elevated temperature using finite element analyses. A representative volume element, consisting of randomly oriented grains, was considered for the finite element analyses under periodic boundary constraints. Strain-controlled cyclic test data at 650 °C were used to determine the model parameters from a fitting process, where three loading rates were considered. Model simulations are in good agreement with the experimental results for stress–strain loops, cyclic hardening behaviour and stress relaxation behaviour. Stress and strain distributions within the representative volume element are of heterogeneous nature due to the orientation mismatch between neighbouring grains. Stress concentrations tend to occur within “hard” grains while strain concentrations tend to locate within “soft” grains, depending on the orientation of grains with respect to the loading direction. The model was further applied to study the near-tip deformation of a transgranular crack in a compact tension specimen using a submodelling technique. Grain microstructure is shown to have an influence on the von Mises stress distribution near the crack tip, and the gain texture heterogeneity disturbs the well-known butterfly shape obtained from the viscoplasticity analysis at continuum level. The stress–strain response near the crack tip, as well as the accumulated shear deformation along slip system, is influenced by the orientation of the grain at the crack tip, which might dictate the subsequent crack growth through grains. Individual slip systems near the crack tip tend to have different amounts of accumulated shear deformation, which was utilised as a criterion to predict the crack growth path.     

Interactions of fatigue cracks with elastic obstacles

To quantify the growth behaviour of fatigue cracks growing towards microstructural barriers or elastic obstacles, parametric solutions are obtained for crack-tip opening displacement and plasticity-induced crack closure of a mode I fatigue crack growing towards elastic obstacles. Three common bi-material systems are analysed using the finite element method, in which both constituent materials have identical elastic properties but only the phase that contains the crack can deform plastically. It has been found that under monotonic loading the crack-tip opening displacement decreases as the crack-tip approaches the interface boundary, but reaching a non-zero value when the crack-tip terminates at the boundary. For a fatigue crack growing under constant amplitude loading, the crack-closure stress has been found to increase as the crack grows towards the barrier. Based on these results a mechanistic model is proposed to quantify the influence of stress level on the fatigue threshold of microstructurally small fatigue cracks, with predictions being in close agreement with experimental data.     

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 set of interface cracks with contact zones in a combined tension-shear field

Summary. A set of cracks lying along the interface of two dissimilar isotropic materials under a mixed-mode loading is considered. The interface cracks are assumed to be fully open, partially closed with frictionless contact zones and fully closed. The problem is reduced to a homogeneous combined Dirichlet-Riemann boundary value problem, which is solved in closed form. A set of transcendental equations for the determination of the contact zone lengths for an arbitrary number of cracks and the closed-form expressions for the stresses and the displacement jumps on the material interface are obtained. A single crack with one and two contact zones has been considered in details. An explicit set of two transcendental equations for the relative contact zone length and closed-form expressions for the stress intensity factors at the crack tips are obtained for both cases. The contact zone lengths and the stress intensity factors are investigated numerically for different material pairs under different values of the loading, and a comparison of the results for a crack with one and two contact zones is carried out.