Relations between structural intensity and J-integral

The present study introduces the concept of structural intensity, which can be interpreted as power flux, into fracture mechanics. It is derived theoretically that the normal component of the structural intensity along crack edge equals J-integral. SI is the power flux or vector representation of the J-integral at the crack tip. Using the finite element method, the structural intensity can be easily calculated. The numerically calculated structural intensity is adopted to visualize the J-integral of a crack tip. Directional power flow path and magnitude at the crack tip is demonstrated schematically with the structural intensity, which facilitates convenient evaluation of the crack propagation status.     

The M-integral for calculating intensity factors of an impermeable crack in a piezoelectric material

In this study, a conservative integral is derived for calculating the intensity factors associated with piezoelectric material for an impermeable crack. This is an extension of the M-integral or interaction energy integral for mode separation in mechanical problems. In addition, the method of displacement extrapolation is extended for this application as a check on results obtained with the conservative integral. Poling is assumed parallel, perpendicular and at an arbitrary angle with respect to the crack plane, as well as parallel to the crack front. In the latter case, a three-dimensional treatment is required for the conservative integral which is beyond the scope of this investigation. The asymptotic fields are obtained; these include stress, electric, displacement and electric flux density fields which are used as auxiliary solutions for the M-integral.Several benchmark problems are examined to demonstrate the accuracy of the methods. Numerical difficulties encountered resulting from multiplication of large and small numbers were solved by normalizing the variables. Since an analytical solution exists, a finite length crack in an infinite body is also considered. Finally, a four point bend specimen subjected to both an applied load and an electric field is presented for a crack parallel, perpendicular and at an angle to the poling direction. It is seen that neglecting the piezoelectric effect in calculating stress intensity factors may lead to errors.     

Methods for calculating stress intensity factors in anisotropic materials: Part II—Arbitrary geometry

The problem of a crack in a general anisotropic material under conditions of linear elastic fracture mechanics (LEFM) is examined. In Part I, three methods were presented for calculating stress intensity factors for various anisotropic materials in which z = 0 is a symmetry plane and the crack front is along the z-axis. These included displacement extrapolation, the M-integral and the separated J-integrals.In this study, general material anisotropy is considered in which the material and crack coordinates may be at arbitrary angles. A three-dimensional treatment is required for this situation in which there may be two or three modes present. A three-dimensional M-integral is extended to obtain stress intensity factors. It is applied to several test problems, in which excellent results are obtained. Results are obtained for a Brazilian disk specimen made of isotropic and cubic material. Two examples for the latter are examined with material coordinates rotated with respect to the crack axes.     

Some new practical equations for rapid calculation of J-integral in plates weakened by U-notches under bending

The paper deals with calculations of the J-integral for a plate weakened by U-notches under Mode I bending loading in the case of a material obeying a linear elastic law. The main aim of the study is to suggest some new equations suitable for calculations of the J-integral under bending loading. Although some closed-form solutions exist in the literature based on elliptic integrals, some expressions, simple to be implemented, are useful as a rapid engineering tool.     

Domain-independent values of the J-integral for cracks in three-dimensional residual stress bearing bodies

This paper describes an approach for computing domain-independent values of the J-integral in the finite element context for three-dimensional bodies containing residual stress. In the analysis of cracked bodies containing residual stress, the usual domain integral formulation results in domain-dependent values of J, and this paper discusses modifications that yield domain independence. Two correction terms are defined. The first of these relates to the spatial gradients of non-mechanical strains in the crack-driving direction, and the second accounts for plastic dissipation included in the material state, but unrelated to fracture. The paper further presents results for two examples recently discussed in the literature. Application of the corrections in these two cases demonstrates the ability of the approach to obtain path-independent domain integral results in residual stress bearing bodies.     

A finite element based interior collocation method for the computation of stress intensity factors and T-stresses

A new, simple and efficient method for simultaneous estimation of the mixed-mode stress intensity factors (SIFs) and T-stresses using finite element computations is proposed in this paper. The current work is based on the formation of overdetermined system of equations using the displacement components near the crack tip. The proposed method can be easily implemented in the existing finite element codes. The results obtained from the present investigation for plane stress problems are validated by comparing with the published results and found to be in very good agreement with them.     

J-integral evaluation for cases involving non-proportional stressing

Calculation of J for cases where the proportional stressing condition cannot be satisfied is investigated. A modified J definition is derived and implemented into an ABAQUS post-processing program for both 2-D and axisymmetric problems. The modified J-integral is path independent for cases of proportional and non-proportional stressing. For cases with proportional stressing, the modified integral gives the same value as does the standard ABAQUS J function. It is also found that the modified J is equivalent to the stress intensity factor for a linear elastic material and provides a measure of the intensity of the crack-tip fields for non-linear elastic and elastic-plastic materials. The modified J formulation is applied to the case of a cylinder with an external circumferential crack under various load conditions.     

Plastic J-integral calculations using the load separation method for the center cracked tension specimen

The load separation method was used to determine the plastic work factor (η_pl) for the center crack tension geometry for power law hardening materials with Ramberg-Osgood hardening exponents n ranging from 2 to 20 and crack sizes a/W ranging from 0.4 to 0.8. The resulting expression for η_pl compares favorably to the analytical work of Rice, Paris, and Merkle and to Sharobeam and Landes. The results were also compared to η_pl calculated from the plastic J results in EPRI Handbook NP-1931. Unlike the EPRI results, η_pl from the load separation method are not a function of crack size.     

Static finite element stress intensity factors for annular cracks

Results of finite element static stress intensity factor calculations for an annular crack around a spherical inclusion (void) are presented and compared with those from approximate analytical methods.     

Fully plastic J-integral solutions for surface cracked plates under biaxial loading

In this paper, surface cracked plates under biaxial tension are studied. Three-dimensional elastic-plastic finite element analyses have been carried out to calculate the J-integral for surface cracked plate for a wide range of geometry, biaxiality and material properties. Fully plastic J-integral solutions along the front of the surface cracks are presented for Ramberg-Osgood power law hardening material of n = 3, 5, 10 and 15. Geometries considered are a/c = 0.2, 1.0 and a/t = 0.2, 0.4, 0.6 and 0.8 and the biaxial ratios of 0, 0.5 and 1. Based on these results, the J-integral along the crack front for general elastic-plastic loading conditions can be estimated using the EPRI scheme. These solutions are suitable for fracture analyses for surface cracked plates under biaxial loading.     

Experimental calculation of mixed-mode notch stress intensity factors for anisotropic materials

This study developed a least-squares method to find the notch stress intensity factors (SIFs) of anisotropic materials using image-correlation experiments without the requirement of smooth procedures. Complex displacement functions are deduced into a least-squares form, and then displacements from the image-correlation experiments are substituted into the least-squares equation to evaluate the notch SIFs for anisotropic materials. Experimental results compared with the H-integral and finite element analyses show that the SIFs evaluated from the current method are acceptably accurate if more than five displacement terms are included. Moreover, the least-squares SIFs are not sensitive to the maximum or minimum radius of the area from which data is included, so experimental data very near the notch tip is not necessary.     

Use of J-integral to predict static failures in sharp V-notches and rounded U-notches

In the present paper, the physical meaning of J_V (namely, the classic J-integral applied to either sharp V-notch) is discussed. Consider a Cartesian reference frame having the x-axis parallel to the notch bisector, each mode of J_V, for a given circular path, is proportional to the correspondent mode of the classic J-integral of a virtual crack having length equal to the path radius and emanating from the tip of the V-notch. Analytical and numerical results have been performed for linear elastic materials. Additionally, in order to verify the formulations of J_V, experimental result of embedded cracks of sharp V-notch was considered.Then, by introducing a characteristic path radius ρ^∗, assumed to be dependent only on the material properties, the J_V parameter was used for the estimation of the static failure load of sharp V-notches specimens under mode I loading.Furthermore, the J_Vρ parameter (namely, the classic J-integral applied to U-rounded notches) was used to analyze the static failure of two new series of specimens with double U-notches made of brittle material (PMMA and PVC glass) subjected to tensile loading. This method allowed us to prove that when the ratio between the notch tip radius and ρ^∗ is small the approach agrees with the classic J-integral, whereas when ρ^∗ becomes small with respect to the notch tip radius, the J_Vρ method agrees with the classic peak stress approach.     

Application of finite element method for determination of stress intensity factor and plastic zone geometry in an aluminium alloy sheet under uniaxial tension

High strength materials have gained prominence in the fields of aero-structures, space missiles, ship-building, pressure vessels etc. However, high strength materials are often characterised by low values of crack resistance or fracture toughness. Knowledge of stress intensity factor (SIF) is essential to predict their fracture toughness. SIF values can be obtained both theoretically and experimentally. Theoretical methods include analytical techniques as well as the finite element method (FEM). The former is used for simpler geometries and the latter for complicated geometries of engineering structures. The SIF as a function of crack size in an aluminium alloy 2024-T₃ (Al-4·5% Cu, 1·5% Mg, 0·6% Mn) sheet was determined by a computer method. These values were obtained directly from the stresses as well as indirectly from strain energy release rateG andJ integral. The results agree well with the normalised values obtained from an ASTM formula. The size and shape of the plastic zone at the crack tip have been determined as a function of nominal stress for a fixed crack length. The plastic zone has the form of two ellipsoids with their maximum spreads oriented around 69° to the crack axis.     

J-integral evaluation of austenitic-martensitic functionally graded steel in plates weakened by U-notches

In this paper, a numerical method for J-integral evaluation of plates weakened by U-notches for brittle or quasi-brittle functionally graded steel (FGS) has been proposed. The material contains austenite phase in addition to martensite layer produced by electroslag remelting (ESR). The Young’s modulus and the Poisson’s ratio have been assumed to be constant, while other mechanical properties vary exponentially along the specimen width. The effect of notch depth on the J-integral and the critical fracture load has been studied. A comparison of the J-integral between functionally graded and homogeneous steels was made, where the notch tip in the functionally graded steel is situated in a layer with same mechanical properties as the homogeneous steel.     

Dislocation-based semi-analytical method for calculating stress intensity factors of cracks: Two-dimensional cases

Determination of the stress intensity factors of cracks is a fundamental issue for assessing the performance safety and predicting the service lifetime of engineering structures. In the present paper, a dislocation-based semi-analytical method is presented by integrating the continuous dislocation model with the finite element method together. Using the superposition principle, a two-dimensional crack problem in a finite elastic body is reduced to the solution of a set of coupled singular integral equations and the calculation of the stress fields of a body which has the same shape as the original one but has no crack. It can easily solve crack problems of structures with arbitrary shape, and the calculated stress intensity factors show almost no dependence upon the finite element mesh. Some representative examples are given to illustrate the efficacy and accuracy of this novel numerical method. Only two-dimensional cases are addressed here, but this method can be extended to three-dimensional problems.     

Calculation of stress intensity factors by the force method

The force method is a simple and accurate technique for calculating stress intensity factors (SIFs) from finite element (FE) models, but it has been scarcely used. This paper shows three important advantages of the force method, which make it particularly attractive for designers and researchers. First, it can be employed without special singular quadratic finite elements at the crack tip. Actually, linear reduced integration elements may be used. Second, the force method can be applied to highly anisotropic materials without requiring knowledge of complicated elasticity relations for the stress field around the crack tip. Third, it can handle mixed-mode fracture problems.     

Approximate J estimates for tension-loaded plates with semi-elliptical surface cracks

This paper provides a simplified engineering J estimation method for semi-elliptical surface cracked plates in tension, based on the reference stress approach. Note that the essential element of the reference stress approach is the plastic limit load in the definition of the reference stress. However, for surface cracks, the definition of the limit load is ambiguous (“local” or “global” limit load), and thus the most relevant limit load (and thus reference stress) for the J estimation should be determined. In the present work, such limit load solution is found by comparing reference stress based J results with those from extensive 3-D finite element (FE) analyses. Based on the present FE results, the global limit load solution proposed by Goodall for surface cracked plates in combined bending and tension was modified, in the case of tension loading only, to account for a weak dependence on w/c and was defined as the reference normalizing load. Validation of the proposed equation against FE J results based on actual experimental tensile data of a 304 stainless steel shows excellent agreements not only for the J values at the deepest point but also for those at an arbitrary point along the crack front, including at the surface point. Thus the present results provide a good engineering tool for elastic-plastic fracture analyses of surface cracked plates in tension.     

Numerical methods for determination of stress intensity factors of singular stress field

The asymptotic solution of the singular stress field near a singular point is generally comprised of one or more singular terms in the form of Kr^λ-1f_ij(θ). Based on the asymptotic solution of the singular stress field and the common numerical solution (stresses or displacements) obtained by an ordinary tool such as the finite element method or boundary element method, a simple and effective numerical method is developed to calculate stress intensity factors for one and two singularities. Three examples show that the stress intensity factors evaluated using the method proposed in this paper are very accurate.     

J-T characterized stress fields of surface-cracked metallic liners bonded to a structural backing - I. Uniaxial tension

Surface crack-tip stress fields in a tensile loaded metallic liner bonded to a structural backing are developed using a two-parameter J-T characterization and elastic-plastic modified boundary layer (MBL) finite element solutions. The Ramberg-Osgood power law hardening material model with deformation plasticity theory is implemented for the metallic liner. In addition to an elastic plate backed surface crack liner model, elastic-plastic homogeneous surface crack models of various thicknesses were tested. The constraint effects that arise from the elastic backing on the thin metallic liner and the extent to which J-T two parameter solutions characterize the crack-tip fields are explored in detail. The increased elastic constraint imposed by the backing on the liner results in an enhanced range of validity of J-T characterization. The higher accuracy of MBL solutions in predicting the surface crack-tip fields in the bonded model is partially attributed to an increase in crack-tip triaxiality and a consequent increase in the effective liner thickness from a fracture standpoint. After isolating the effects of thickness, the constraint imposed by the continued elastic linearity of the backing significantly enhanced stress field characterization. In fact, J and T along with MBL solutions predicted stresses with remarkable accuracy for loads beyond full yielding. The effects of backing stiffness variation were also investigated and results indicate that the backing to liner modulus ratio does not significantly influence the crack tip constraint. Indeed, the most significant effect of the backing is its ability to impose an elastic constraint on the liner. Results from this study will facilitate the implementation of geometric limits in testing standards for surface cracked tension specimens bonded to a structural backing.     

Stress intensity factor analysis of a three-dimensional interface crack between dissimilar anisotropic materials

A new numerical method to calculate the stress intensity factors (SIFs) of a three-dimensional interface crack between dissimilar anisotropic materials was developed. In this study, the M-integral method was employed for mode separation of the SIFs. The moving least-square method was utilized to calculate the M-integral. Using the M-integral with the moving least-square method, SIFs can be automatically calculated with only the nodal displacements from the finite element method (FEM). Here, SIFs analyses of some typical three-dimensional problems are demonstrated. Excellent agreement was achieved between the numerical results obtained by the present method and the corresponding results proposed by other researchers. In addition, the SIFs of a single-edge crack, a through crack, and a semi-circular crack between two anisotropic solids in three-dimensional structures were analyzed.