Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ? J/σ₀) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.     

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.     

Determination of K Id at or below NDTT using instrumented drop-weight testing

Two kinds of surface cracked specimens are numerically analyzed by the three dimensional elastic-plastic FEM. Near tip regions are divided into fine elements, and the stress and displacement fields at the crack tip are compared with HRR solutions.At first, surface cracked specimens subjected to bending are analyzed by changing the aspect ratio and depth/thickness ratio. The effect of the loading condition, crack shape and the crack depth on the stress and displacement fields are discussed. Then the pipe with surface crack subjected to bending is analyzed and the availability of the J-integral concept to the LBB analysis is discussed.In every specimen, it is shown that in the regions very near to the crack tip, the displacement field is similar to HRR solutions of plane strain. In the outer regions, however, the stress and displacement fields depend strongly on the shape, thickness, and loading conditions.     

Dielectric Breakdown Model For An Electrically Impermeable Crack In A Piezoelectric Material

The present work presents a strip Dielectric Breakdown (DB) model for an electrically impermeable crack in a piezoelectric material. In the DB model, the dielectric breakdown region is assumed to be a strip along the crack's front line. Along the DB strip, the electric field strength is equal to the dielectric breakdown strength. The DB model is exactly in analogy with the mechanical Dugdale model. Two energy release rates emerge from the analysis. An applied energy release rate appears when evaluating J-integral along a contour surrounding both the dielectric breakdown strip and the crack tip, whereas a local energy release rate appears when evaluating J-integral along an infinitesimal contour surrounding only the crack tip. Under small yielding conditions, the local energy release rate, if used as a failure criterion, gives a linear relationship between the applied stress intensity factor and the applied electric intensity factor.     

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.     

A theory for the fracture of thin plates subjected to bending and twisting moments

Stress fields near the tip of a through crack in an elastic plate under bending and twisting moments are reviewed assuming both Kirchhoff and Reissner plate theories. The crack tip displacement and rotation fields based on the Reissner theory are calculated here for the first time. These results are used to calculate the J-integral (energy release rate) for both Kirchhoff and Reissner plate theories. Invoking Simmonds and Duva's [16] result that the value of the J-integral based on either theory is the same for thin plates, a universal relationship between the Kirchhoff theory stress intensity factors and the Reissner theory stress intensity factors is obtained for thin plates. Calculation of Kirchhoff theory stress intensity factors from finite elements based on energy release rate is illustrated. A small scale yielding like model of the crack tip fields is discussed, where the Kirchhoff theory fields are considered to be the far field conditions for the Reissner theory fields. It is proposed that, for thin plates, fracture toughness and crack growth rates be correlated with the Kirchhoff theory stress intensity factors.     

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.     

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.     

Structural mechanics modeling of the impact of a double cantilever beam

A Global/Local model is developed for the calculation of dynamic stress intensity history resulting from the impact of a double cantilever beam. The model separates the Global structural dynamics from the Local crack tip zone dominated by singular stresses. The Global model, consisting of connected waveguides, describes the structural dynamics using spectral elements. The Local model, describing the crack tip behavior, is based on an application of the J-integral that converts the dynamic structural resultants directly into strain energy release rate. Previous work used this approach successfully for pure mode II deformation; the present paper extends the modeling to mode I and mixed mode deformation. Of special concern is the local modeling of the elasticity and mass in the vicinity of the crack tip; structural mechanics modeling is combined with Hamilton's principle to properly address this question. The accuracy of this approach is assessed by comparing it to a two dimensional finite element analysis.     

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.     

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.     

HRR fields for damaged materials

The paper presents an investigation of the interaction between a macroscopic crack and distributed damage in an elastic-plastic material based on the HRR field model for virgin materials. This is achieved by describing the mechanical effects of the distributed micro-cracks in terms of the damage variable D on the HRR fields. Damage evolution equation and the constitutive equations coupled with damage are formulated and the resulting boundary value problems are solved numerically. Material constants , n and m ₀are varied to examine their effects on the resulting stress distributions. It is found that the HRR fields for damaged and virgin materials are surprisingly similar although the severity of damage equivalent stress is of several orders of magnitude higher than the conventional plastic equivalent stress without damage consideration. Furthermore, it is shown theoretically and justified numerically that the J-integral loses its path independency for damaged materials, causing the amplitude of the singularity K _D to remain an unknown variable in the asymptotic analysis.     

A novel singular ES-FEM method for simulating singular stress fields near the crack tips for linear fracture problems

This paper presents a novel numerical method for effectively simulating the singular stress field for mode-I fracture problems based on the edge-based smoothed finite element method (ES-FEM). Using the unique feature of the ES-FEM formulation, we need only the assumed displacement values (not the derivatives) on the boundary of the smoothing domains, and hence a new technique to construct singular shape functions is devised for the crack tip elements. Some examples have demonstrated that results of the present singular ES-FEM in terms of strain energy, displacement and J-integral are much more accurate than the finite element method using the same mesh.     

Complete theoretical analysis for higher order asymptotic terms and the HRR zone at a crack tip for Mode I and Mode II loading of a hardening material

Summary A complete development for the first two terms of the crack tip fields for both Mode I and Mode II loading of a hardening material in either plane stress or plane strain is performed, including the elastic deformation in the analysis. It is shown that the determination of the order of the second term depends on bothn and whether plane stress or plane strain is considered. In addition, regions of HRR dominance at a crack tip for the field variables are estimated. Comparison of the analytic predictions with finite element results indicates that the analytic results for the zone of HRR dominance are in agreement with numerical predictions.     

Effect of thermomechanical loads on the propagation of crack near the interface brittle/ductile

The purpose of this paper is to understand the combined effect of thermal and mechanical loading on the initiation and behaviour of sub-interface crack in the ceramic. In this study a 2D finite element model has been used to simulated mixed mode crack propagation near the bimaterial interface. The assembly ceramometalic is subjected simultaneously to thermomechanical stress field. The extent of a plastic zone deformation in the vicinity of the crack-tip has a significant influence on the rate of its propagation. The crack growth at the joint specimen under four-point bending (4PB) loading and the influence of residual stresses was also evaluated by the maximum tensile stress criterion. The J-integral at the crack tip is generally expressed by the thermomechanical local stresses. The results obtained show the effect of the temperature gradient ΔT, the size of the crack and the applied stresses on the J-integral.     

Analytical and numerical study of cohesive zones for a crack in an adhesive layer between identical isotropic materials

A crack in a thin adhesive elastic-perfectly plastic layer between two identical isotropic elastic half-spaces is considered. Uniformly distributed normal stress is applied to the substrates at infinity. First, stress distribution in the cohesive zones and the J-integral values are defined numerically by the finite element method (FEM). Further, a mathematical formulation of the problem is given and its analytical solution is proposed. It is assumed that, at the crack continuations, there exist cohesive zones. The interlayer thickness is neglected since it is much smaller than the crack length. The distribution of the normal stress, which was obtained by means of the FEM, is now approximated by a piecewise-constant function and assumed to be applied at the faces of the cohesive zones. The formulated problem is solved analytically and an equation for determination of the cohesive zone lengths is derived. Also, closed expressions for the crack tip opening displacement and for the J-integral are obtained in an analytical form. These parameters are found with respect to the values of the normal stress applied at infinity. Finally, a universal approximating function, which describes the stress distribution in the cohesive zones, is constructed. This function depends on the ratio between the interlayer thickness and the crack length and on the ratio between the normal stress applied at infinity and the yield limit of the interlayer’s material. Once again, the problem is solved analytically, but this time for the stress distribution prescribed by the universal approximating function. The cohesive zone lengths, the values of the crack tip opening displacement and of the J-integral are calculated. A comparative analysis of the obtained results is carried out. A good agreement of the J-integral values calculated by means of the developed analytical models and by the associated finite element analysis is demonstrated.     

Advantages of the J-integral approach for calculating stress intensity factors when using the commercial finite element software ABAQUS

A predictive method for remaining component lifetime evaluation consists in integrating the crack growth law of the material considered in a finite element step-by-step process. So, as part of a linear elastic fracture mechanics analysis, the determination of the stress intensity factor distribution is a crucial point. The aim of the present work is to test several existing numerical techniques reported in the literature. Both the crack opening displacement extrapolation method and the J-integral approach are applied in 2D and 3D ABAQUS finite element models. The results obtained by these various means on CT specimens and cracked round bars are in good agreement with those found in the literature. Nevertheless, since the knowledge of the field near the crack tip is not required in the energetic method, the J-integral calculations seem to be a good technique to deal with the fatigue growth of general cracks.     

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.     

A new constraint based fracture criterion for surface cracks

A new methodology for predicting the location of maximum crack extension along a surface crack front in ductile materials is presented. Three-dimensional elastic-plastic finite element analyses were used to determine the variations of a constraint parameter (α_h) based on the average opening stress in the crack tip plastic zone and the J-integral distributions along the crack front for many surface crack configurations. Monotonic tension and bending loads are considered. The crack front constraint parameter is combined with the J-integral to characterize fracture, the critical fracture location being the location for which the product Jα_h is a maximum. The criterion is verified with test results from surface cracked specimens.     

Two-Parameter (J-M) Description of Crack Tip Stress-Fields for an idealized weldment in small scale yielding

The crack tip stress-field in a trimaterial finite element model has been examined. The model represents an idealised steel weldment with a crack located at the fusion line. The model was loaded with a K _I displacement field to simulate small scale yielding conditions. The effect of changing the weld metal plastic properties and the HAZ layer thickness on the crack tip stress-field was studied, keeping the material properties of the HAZ and base metal constant. The results show that the calculated J-integral remains path independent in the trimaterial model. It is confirmed that the crack tip stress-fields can be normalised by the J-integral. The mismatch constraint can be characterised by a difference field, which is independent of the normalised distance from the crack tip. The results show that changes of HAZ thickness only have a small effect on the stress-fields close to the crack tip. The hardenability of the weld metal influences on the slope of the crack tip stress distribution, but for small changes in hardenability, this effect can be neglected. The results indicate that the difference fields show some radial dependence when a homogeneous reference field is used, but the radial dependence was removed by introducing an inhomogeneous reference field. The effect of changes in the weld metal yield strength has been described with a two parameter (J-M) formulation using the inhomogeneous reference field.