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.     

Fracture of aluminium alloy 2024 under biaxial and triaxial loading

Previous studies on multi-axial fracture of metals have shown that the critical J-integral at fracture may be less than the fracture toughness measured in a standard test. This gives rise to the question: what is the minimum critical J-integral and how can it be obtained? To answer this question a series of uniaxial, biaxial and triaxial tests were carried out. Conducting biaxial and triaxial tests allows the effects of stress state in the fracture of metallic materials to be investigated, particularly when the plasticity is highly constrained. The primary purpose of this paper is to report the experimental findings of the tests performed on specimens fabricated from aluminium alloy 2024. Results of finite element analyses are then used to study further the detailed stress state near the crack tip and to evaluate the intensity of the plastic deformation and relate it to the critical J-integral variation. It was found that indeed high triaxial loading, corresponding to limited plastic deformation prior to the fracture, decreases the critical J-integral even below the values obtained from the biaxial tests, which are already less than the standard uniaxial value.     

Mechanical heterogeneity and validity of J-dominance in welded joints

Fracture criterion of the J-integral finds wide application in the integrity evaluation of welded components, but there exist some confused problems such as the dependence of the fracture toughness on the strength mis-matching and specimen geometry which need to be clarified. It is rough and unsuitable to attribute the variation of J-integral fracture parameter simply to the effect of mechanical heterogeneity. In the present paper, a two-dimensional finite element method is employed to analyze the distribution and variation of crack tip field of welded joints with different strength mis-matching in four kinds of specimen geometry, and then the validity of J-dominance in welded joints is investigated. It is found that the crack tip field of mis-matched joint is different from that of either the weld metal or base metal of which the joint is composed, but it is situated between those of weld metal and base metal. Under the plane strain, there is obvious difference in stress triaxiality for different strength mis-matched joints. The validity of J-dominance in welded joint can not be obtained by comparing whether the stress triaxiality meets that required by the HRR solution because of the existence of mechanical inhomogeneity. By ascertaining if the stress triaxiality of welded joint near the crack tip is dependent of specimen geometry, the conclusion can be arrived at: for plane stress the validity of J-dominance is valid, whilst for plane strain the validity of J-dominance is lost. Based on the above, attempt has been made to point out that the influence of mechanical heterogeneity on the fracture toughness of weldment arises from the variation of constraint intensity-crack tip stress triaxiality. Compared with the effect of mechanical heterogeneity on the stress triaxiality, the losing of validity of J-dominance in mis-matched joint under plane strain may play a more critical role in the variation of J-integral fracture parameter of weldment.     

On the evaluation of the J-integral by a plastic crack tip element

Two crack tip elements are formulated for a stationary, mode I plastic crack in planar structures using hybrid assumed stress approach, based on the secant modulus and the Newton-Raphson schemes, respectively. The stress distribution in the crack tip element is assumed to be the HRR field superimposed by the regular polynomial terms. The formulated (hybrid) crack tip elements are compatible with the isoparametric element so that they can be used conveniently along with the conventional displacement-based finite elements. The intensity of the HRR stress field, the J-integral, is determined directly from the finite element equations together with the nodal displacements. The dominance of the HRR stress field at the crack tip is pertinent to the present approach, which depends on geometry and loading conditions. Since the J-integral is globally path-independent for nonlinear elastic materials (deformation plasticity model), in order to assess the accuracy and efficiency of the methodology as compared to the contour integration approach, numerical studies of common plane-stress cracked configurations are performed for these materials. The results indicate that for a sufficiently small crack tip element size, J from the present approach correlates well, within 6 percent difference, with that from the contour integration for a wide range of material hardening coefficients if the HRR zone exists at the crack tip. These highly accurate results for J from the crack tip stresses could not be achieved without using (newly) modified variational principles and a refined numerical technique. It should be emphasized that the present methodology also can be applied to cracks in J ₂ flow materials under HRR dominance. In such case, the J integral may not be globally path independent, and hence it now must be determined from the stress and strain fields near the crack tip.     

Effect of stress wave propagation phenomenon on the determination of strain energy density theory parameters and dynamic J-integral

Dynamic loading for stationary cracks leads to results that are many times greater in magnitude than their static counterparts. If the dynamic loading is in the form of impact type, stress wave propagation effects become dominant. FRAC3D program comprises enriched element formulation which doesn't require excessive mesh refinement around crack tip for accuracy. Strain energy density (SED) theory parameters and dynamic J-integral are sought in this study to simulate and understand wave propagation phenomenon in detail. Structures under the effect of wave propagations yield more reliable J-integral values by taking the average of the results from multiple domain sizes. Governed by stress waves, space-time variations of minimum energy density locations strongly influence fracture characterization for straight and curved crack fronts. Details given in numerical examples section of this paper make a great contribution to understanding of the response for cracked structures subjected to sudden loading.     

The J-integral at Dugdale cracks perpendicular to interfaces of materials with dissimilar yield stresses

The J-integral is applied to a Dugdale crack perpendicular to an interface of materials with equal elastic properties but different yield stresses. It is shown that the integral is path independend with certain limitations to the integration path. Three essentially different paths can be distinguished. The first integration path is totally within the first material, it provides the local crack driving force. Performing the integral around the plastic zone in both materials gives the global crack driving force. An interface force can be defined by evaluating the integral along both sides of the plastically deformed region of the interface. A comparison of these three integrals reveals that the global crack driving force is equal to the sum of the local crack driving force and of the interface force. The derived expression for the J-integral are compared with the crack tip opening displacement published recently. This reveals that the local J describes the plastic deformation at the crack tip. Therefore it represents the crack driving force in bimaterials as it does the conventional J-integral in case of homogeneous materials. The analyses are also extended to cyclic plasticity, where an out-of-phase effect is observed. Finally it is discussed how these results can be used to explain fatigue tests at bimaterial specimens.     

Crack growth characteristics of carbon nanotube-based polymer composites subjected to cyclic loading

This paper presents the characterization of crack growth in carbon nanotube (CNT)-based polymer composites under fatigue loading. Fatigue crack growth tests were performed on single-edge cracked plate specimens of CNT/polycarbonate composites at room temperature and liquid nitrogen temperature (77 K). An elastic–plastic finite element analysis was also conducted to determine the J-integral range. The crack growth rate data were expressed in terms of the J-integral range, and the effect of nanotube addition on the fatigue crack growth behavior was examined. In addition, possible mechanisms of the crack growth in the nanocomposites are discussed based on microscopic observations of the specimen fracture surfaces.     

Computations of energy release rate under monotonic and cyclic loading conditions

In case of an elastic–plastic fracture mechanics analysis, the determination of the energy release rate distribution is a crucial point. In the present paper, three numerical techniques: the virtual crack closure technique (VCCT), J-integral and energy derivative technique (EDT), are used to compute the energy release rate in a middle-crack tension specimen with the combined isotropic/kinematic hardening model. The results obtained by these methods are compared with each other under monotonic and cyclic loading conditions. Finally, it comes out that the difference of the VCCT method to the J-integral is rather insensitive to load increasing, especially when the traction >40% of yield stress, however, the deviation of the VCCT and J-integral results are within 10%, suggesting that one may use the VCCT for plastic cracked specimen analysis. The computations show that the EDT provides the same values for the monotonic as the J-integral if the plastic deformations are not large, but for high plastic loading the EDT overestimates the fracture energy. For cyclic loading case, VCCT method offers closer results as the elastic analytical results, also suggesting that the whole plastic dissipated energy in the loading process should be integrated. While EDT method gives the smaller results than the J-integral because of the energy dissipated in the unloading phase is considered in the loading process.     

Interface crack in sandwich specimen

Fracture of a sandwich specimen loaded with axial forces and bending moments is analyzed in the context of linear elastic fracture mechanics. A closed form expression for the energy release rate for interface cracking of a sandwich specimen with isotropic face sheets is found from analytical evaluation of the J-integral. An approach is applied, whereby the mode mixity for any combination of the loads can be calculated analytically when a load-independent phase angle has been determined. This load-independent phase angle is determined for a broad range of sandwich configurations of practical interest. The load-independent phase angle is determined using a novel finite element based method called the crack surface displacement extrapolation method. The expression for the energy release rate is based on the J-integral and certain stress distributions along the ends of the sandwich specimen. When the stresses from the crack tip interacts with the stresses at the ends, the present analytical calculation of the J-integral becomes inaccurate. The results show that for the analytically J-integral to be accurate the crack tip must be a certain distance away from the uncracked end of the specimen. For a sandwich specimen with face sheet/core stiffness ratio of 100, this distance is in the order 10 times the face sheet thickness. For sandwich structures with face sheet/core stiffness ratio of 1,000, the distance is 30 times the face sheet thickness.     

Residual stresses and fracture mechanics analysis of a crack in welds of high strength steels

This study presented the characteristics of residual stresses in welds of high strength steels (POSTEN60, POSTEN80) whose tensile strengths were 600 MPa and 800 MPa, respectively. Three-dimensional thermal elastic-plastic analyses were conducted to investigate the characteristics of welding residual stresses in welds of high strength steels through the thermal and mechanical properties at high temperatures obtained from the elevated temperature tensile tests. A finite element analysis method which can calculate the J-integral for a crack in a residual stress field was developed to evaluate the J-integral for a centre crack when mechanical stresses were applied in conjunction with residual stresses.The results show that the volumetric changes associated with the austenite to martensite phase transformation during rapid cooling after welding of high strength steels significantly influence on the development of residual stresses in the weld fusion zone and heat-affected zone. For a centre crack in welds of high strength steels where only residual stresses are present, increased tensile strength of the steel, increased the J-integral values. The values of the J-integral for the case when mechanical stresses are applied in conjunction with residual stresses are larger than those for the case when only residual stresses are present.     

Effect of biaxial loads on elastic-plastic <Emphasis Type="Italic">J</Emphasis> and crack tip constraint for cracked plates: finite element study

Based on detailed two-dimensional (2-D) and three-dimensional (3-D) finite element (FE) analyses, this paper attempts to quantify in-plane and out-of-plane constraint effects on elastic-plastic J and crack tip stresses for a plate with a through-thickness crack and semi-elliptical surface crack under positive biaxial loading. For the plate with a through-thickness crack, plate thickness and relative crack length are systematically varied, whereas for the plate with a semi-elliptical surface crack, the relative crack depth and aspect ratio of the semi-elliptical crack are systematically varied. It is found that the reference stress based approach for uniaxial loading can be applied to estimate J under biaxial loading, provided that the limit load specific to biaxial loading is used, implying that quantification of the biaxiality effect on the limit load is important. Investigation on the effect of biaxiality on the limit load suggests that for relatively thin plates with small cracks, in particular with semi-elliptical surface cracks, the effect of biaxiality on the limit load can be neglected for positive biaxial loading, and thus elastic-plastic J for a biaxially loaded plate could be estimated, assuming that such plate is subject to uniaxial load. Regarding the effect of biaxiality on crack tip stress triaxiality, it is found that such effect is more pronounced for a thicker plate. For plates with semi-elliptical surface cracks, the crack aspect ratio is found to be more important than the relative crack depth, and the effect of biaxiality on crack tip stress triaxiality is found to be more pronounced near the surface points along the crack front.     

The mechanics of self-similar and self-affine fractal cracks

In this paper we study the mechanical attributes of the fractal nature of fracture surfaces. The structure of stress and strain singularity at the tip of a fractal crack, which can be self-similar or self-affine, is studied. The three classical modes of fracture and the fourth mode of fracture are discussed for fractal cracks in two-dimensional and three- dimensional solid bodies. It is discovered that there are six modes of fracture in fractal fracture mechanics. The J-integral is shown to be path-dependent. It is explained that the proposed modified J-integrals in the literature that are argued to be path-independent are only locally path-independent and have no physical meaning. It is conjectured that a fractal J-integral should be the rate of potential energy release per unit of a fractal measure of crack growth. The powers of stress and strain singularities at the tip of a fractal crack in a strain-hardening material are calculated. It is shown that stresses and strains have weaker singularities at the tip of a fractal crack than they do at the tip of a smooth crack.     

Quasi-static and dynamic fracture behavior of composite sandwich beams with a viscoelastic interface crack

Fracture analysis of sandwich beams with a viscoelastic interface crack under quasi-static and dynamic loading has been studied. Firstly, a three-parameter standard solid material model was employed to describe the viscoelasticity of the adhesive layer. And a novel interfacial fracture analysis model called three material media model was established, in which an interface crack was inserted in the viscoelastic layer. Secondly, a finite element procedure based on Rice J-integral and Kishimoto J-integral theories was used to analyze quasi-static and dynamic interface fracture behavior of the sandwich beam, respectively. Finally, the influence of viscoelastic adhesive layer on the quasi-static J-integral was discussed. In addition, comparison of quasi-static Rice J-integral with Kishimoto J-integral under various loading rates was carried out. The numerical results show that the oscillating characteristic of dynamic J-integral is more evident with shorter loading rise time.     

A method for approximate calculation of the energy integral for notched and cracked bodies

We propose an approximate method for the calculation of the energyJ-integral for bodies with notches (cracks) subjected to elastoplastic deformations based on an analysis of stress and stress concentration at the tip of the notch (crack). The formulas for theJ-integral are obtained in terms of the theoretical stress concentration factor (stress intensity factor), nominal stresses, radius of the notch tip (crack length), and elastoplastic properties of the material. These formulas enable one to representJ-based design curves with account of the effect for material hardening.Blagonravov Institute of Mechanical Engineering, Russian Academy of Sciences, Moscow; Moscow Institute of Engineering Physics, Moscow. Translated from Fiziko-Khimicheskaya Mekhanika Materialov, Vol. 30, No. 3, pp. 82–87, May–June, 1994.     

Elastic–plastic J and COD estimation schemes for 90° elbow with throughwall circumferential crack at intrados under in-plane opening moment

Leak-before-break (LBB) assessment of primary heat transport piping of nuclear reactors involves detailed fracture assessment of pipes and elbows with postulated throughwall cracks. Fracture assessment requires the calculation of elastic–plastic J-integral and crack opening displacement (COD)¹ for these piping components. Analytical estimation schemes to evaluate elastic–plastic J-integral and COD simplify the calculations. These types of estimation schemes are available for pipes with various crack configurations subjected to different types of loading. However, such schemes for elbow (or pipe bend), which is one of the important components for LBB analyses, is very meager. Recently, elastic–plastic J and COD estimation scheme has been developed for throughwall circumferentially cracked elbow subjected to closing bending moment. However, it is well known that the elbow deformation characteristics are distinctly different for closing and opening bending modes because the ovalisation patterns of elbow cross section are different under these two modes. Development of elastic–plastic J and COD estimation scheme for an elbow with throughwall circumferential crack at intrados subjected to opening bending moment forms the objective of the present paper. Experimental validation of proposed J-estimation scheme has been provided by comparing the crack initiation, unstable ductile tearing loads and crack extension at instability with the test data. The COD estimation scheme has been validated by comparing the COD of test data with the predictions of the proposed scheme.     

On J-integral and potential energy variation

The divergence theorem has been used in a region containing the crack tip to derive the J-integral from the potential energy variation in most fracture mechanics books. Such a derivation is flawed because of the crack tip stress singularity. The present study describes a rigorous and straightforward derivation of the J-integral from the potential energy variation with crack extension by carefully addressing the effect of the crack tip singularity.     

J-integral analysis of the interaction between an interface crack and parallel subinterface cracks in dissimilar anisotropic materials

The J-integral analysis is presented for the interaction problem between a macro-interface crack and subinterface microcracks parallel to the former in the near-tip process zone in dissimilar anisotropic composite materials. Elementary solutions respectively considering an interface crack and a subinterface crack subjected to different loads are given from which the interaction problem is deduced to a system of integral equations with the aid of superimposing technique (i.e., the so called `Pseudo-Traction Method' abbreviated as PTM). After the integral equations are solved numerically, a consistent relation among three kinds of the J-integral values is obtained. They are induced from the macro-interface crack tip, the microcracks, and the remote field, respectively. This consistent relation of the J-integral can be used to confirm the numerical results derived by using whatever kind of technique. With the aid of J-integral analysis, the interaction behaviors between an interface crack and parallel subinterface cracks are investigated in detail, and some special physical phenomena are obtained.     

Interfacial crack-tip constraints and <Emphasis Type="Italic">J</Emphasis>-integral for bi-materials with plastic hardening mismatch

This paper investigates interfacial crack tip stress fields and the J-integral for bi-materials with plastic hardening mismatch via detailed elastic-plastic finite element analyses. For small scale yielding, the modified boundary layer formulation with the elastic T-stress is employed. For fully plastic yielding, plane strain single-edge- cracked specimens under pure bending are considered. Interfacial crack tip stress fields are explained by modified Prandtl slip-line fields. It is found that, for bi-materials consisting of two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and the J-contribution there. The implication of asymmetric J-integral in bi-materials is also discussed.     

Fracture mechanics analysis of a crack in a residual stress field

The standard definition of the J integral leads to a path dependent value in the presence of a residual stress field, and this gives rise to numerical difficulties in numerical modelling of fracture problems when residual stresses are significant. In this work, a path independent J definition for a crack in a residual stress field is obtained. A number of crack geometries containing residual stresses have been analysed using the finite element method and the results demonstrate that the modified J shows good path-independence which is maintained under a combination of residual stress and mechanical loading. It is also shown that the modified J is equivalent to the stress intensity factor, K, under small scale yielding conditions and provides the intensity of the near crack tip stresses under elastic-plastic conditions. The paper also discusses two issues linked to the numerical modelling of residual stress crack problems-the introduction of a residual stress field into a finite element model and the introduction of a crack into a residual stress field.     

A Global/Local approach to lengthwise cracked beams: dynamic analysis

A simple model is developed for the calculation of dynamic stress intensity factors for lengthwise cracked beams subjected to impact or transient loading. The model is based on a Global/Local approach that separates the Global structural dynamics from the Local crack tip zone dominated by singular stresses. The Global model is that of connected waveguides while the Local model is based on a novel application of the J-integral that converts dynamic structural resultants directly into strain energy release rate. The accuracy of this approach is assessed by comparing it to a fully two-dimensional finite element analysis in which the modified crack closure integral is used to calculate the dynamic strain energy release rate. Both mode I and mode II examples are given, and situations with multiple wave reflections are emphasized.