A SIMPLE PATH-INDEPENDENT INTEGRAL FOR CALCULATING MIXED-MODE STRESS INTENSITY FACTORS

A path-independent integral is introduced for calculating stress intensity factors. The derivation of the integral is based on the application of the known Bueckner's fundamental field solution for a crack in an infinite body and on the reciprocal theorem. The method was applied to two-dimensional linear elastic mixed-mode crack problems. The key advantage of the present path-independent integral is that the stress intensity factor components for any irregular cracked geometry under any kind of loading can be easily obtained by a contour integral around the crack tip. The method is simple to implement and only the far field displacements and tractions along the contour must be known. The required stress analysis can be made by using any analytical or numerical method, e.g. the finite element method, without special consideration of the modelling of crack tip singularity. The application of this integral is also independent of the crack type, that is, there is no difference between an edge crack and an embedded crack, provided that the crack tip asymptotic behaviour exists.     

A numerical study of the use of path independent integrals in elasto-dynamic crack propagation

The use of the path independent J' integral for dynamic crack propagation, which has the physical meaning of energy release rate is numerically studied by the finite element method. Other path independent integrals are also investigated along with the J' integral. Numerical results show that the combined use of the J' integral and the finite element method is a useful tool to obtain the fracture parameters such as the stress intensity factors and the energy release rates. The use of the several other types of path-independent integrals, despite their lack of a direct interpretation as energy release rates, is also demonstrated. This is so, because the alternate path-independent integrals have been explicitly expressed in terms of the time-dependent K-factors, or the energy release rate, in the present work.     

形状改变比能的J~*积分及应用

本文引进形状改变比能的概念，对平面裂纹起裂扩展问题进行了讨论，给出一个与路径无关的J*积分，通过I型裂纹的应用，其结果与现行公开发表的文献或手册结果一致。     

On The Use Of Vector J-Integral In Crack Growth Criteria For Brittle Solids

In this paper the criterion for crack-growth in solids is investigated on the basis of the concept of potential energy release rate. The expressions for path-independent vector integral J_i (i = 1,2) are derived for brittle crack growth. The relationship is then established between the value of the path-independent vector integral J_i and the potential energy release rate for crack growth in an arbitrary orientation. This allows the prediction of crack re-orientation angles on the basis of the maximum energy release rate (MERR) criterion. The crack growth angle is determined analytically as a function of (). This result is compared with other theoretical formulations of crack growth criteria, as well as with experimental results reported in the literature, and good agreement is found. The formulation provides a rigorous basis for numerical modelling of the processes of crack initiation and propagation.     

Application of the Green and the Rayleigh-Green reciprocal identities to path-independent integrals in two- and three-dimensional elasticity

Summary An elementary but quite general method for the construction of path-independent integrals in plane and three-dimensional elasticity is suggested. This approach consists simply in using the classical Green formula in its reciprocal form for harmonic functions and, further, the more general Rayleigh-Green formula also in its reciprocal form, but for biharmonic functions. A large number of harmonic and biharmonic functions appears in a natural way in the theory of elasticity. Therefore, the construction of path-independent integrals (or, probably better, surface-independent integrals in the three-dimensional case) becomes really a trivial task. An application to the determination of stress intensity factors at crack tips is considered in detail and only the sum of the principal stress components is used in the path-independent integral. Further applications of the method are easily possible.     

A path-independent integral for the characterization of solute concentration and flux at biofilm detachments

A path-independent (conservation) integral is developed for the characterization of solute concentration and flux in a biofilm in the vicinity of a detachment or other flux limiting boundary condition. Steady state conditions of solute diffusion are considered and biofilm kinetics are described by an uptake term which can be expressed in terms of a potential (Michaelis–Menten kinetics). An asymptotic solution for solute concentration at the tip of the detachment is obtained and shown to be analogous to that of antiplane crack problems in linear elasticity. It is shown that the amplitude of the asymptotic solution can be calculated by evaluating a path-independent integral. The special case of a semi-infinite detachment in an infinite strip is considered and the amplitude of the asymptotic field is related to the boundary conditions and problem parameters in closed form for zeroth and first order kinetics and numerically for Michaelis–Menten kinetics.     

Fracture criterion for inelastic solids

The criterion of inelastic crack growth is found on the basis of the principle of the minimum energy dissipation. According to it the variation of dissipation plus the elastic energy rate per unit of crack growth has to reach the critical value. The path-independent integral of the 2nd type about crack-tip zone is given. Some examples are considered.     

Modelling of the crack growth initiation in viscoelastic media by the Gθv-integral

In this paper, the effects of viscoelastic characteristics on the creep crack growth initiation are studied by developing a new path-independent integral Gθ_v which allows us to compute the energy release rate with the finite element method. The originality of this approach is the perfect uncoupling between the viscous dissipation and the free energy which drives the crack propagation and the crack growth initiation. Coupled with an explicit finite element formulation of the linear viscoelastic behavior, this integral allows us to simulate accurate crack growth initiation.     

Locating a straight crack in an infinite elastic medium by using complex path-independent integrals

Summary The Cauchy integral theorem and the relevant formula (or, equivalently, complex path-independent integrals) have been used in a long series of papers for the determination of zeros and poles of analytic and meromorphic functions. Here this approach is generalized to become applicable to the problem of location of a straight crack inside an infinite plane isotropic elastic medium. The complex path-independent integrals used here contain the first complex potential (z) of Kolosov-Muskhelishvili, which can be obtained experimentally. The present method can be modified to apply to a variety of problems where discontinuity intervals of analytic (or, rather, sectionally analytic) functions are sought.     

Properties of eigenfunction expansion form and relative integral for a crack with completely closed surfaces

The important property of the eigenfunction expansion form found by Chen [Engng Fracture Mech. 22, 673–686 (1985)] is found as the pseudo-orthogonal property. The analysis of this property and relative integral for a semi-infinite crack with completely closed surfaces in homogeneous plane elasticity is studied in detail. It is found that the integral proposed by Bueckner [in Mechanics of Fracture, Vol. 1, pp. 239–314 (1973)] and Chen is no longer path-independent in this circumstance. The differences between the present investigation and that of Chen are discussed.     

Conservation laws, duality and symmetry loss in solid mechanics

The paper deals with conservation laws which are not of the pure divergence type and thus do not provide a path-independent integral for use in Fracture Mechanics. It is shown that Duality is the right tool to re-establish the symmetry between equations and to provide conservation laws of the pure divergence type. The loss of symmetry of some energetic expressions is exploited to derive a new method for solving some inverse problems. In particular, the earthquake inverse problem is solved analytically. Dedicated to George Herrmann.     

The Bueckner work conjugate integral for a permeable crack in piezoelectric materials

Summary The Bueckner work conjugate integral for cracked piezoelectric materials is studied. The analysis is based on the permeable crack model to avoid the unphysically impermeable crack assumption. It is proved that the values of the Bueckner integrals are dependent on the normal electric displacement component on the crack surfaces and the coordinates of the starting point and the end point of the integral contour on the lower and upper crack faces, respectively. This means that, strictly speaking, the Bueckner integral for the permeable crack model is path-dependent unless a special condition is satisfied, i.e., the horizontal coordinates of the starting point and the end points of the integral contour are the same. The present conclusions are quite different from those derived from the impermeable crack model, where the Bueckner integral is strictly path-independent. The universal relation between the Bueckner integral and the J-integral existing in the impermeable crack model is also proved not to be always valid for the present permeable model unless the special integral contour is taken. This implies that the crack electric boundary condition in piezoelectric materials yields some significant influence on the properties of the Bueckner work conjugate integral.     

ANALYSIS OF FREE-EDGE STRESS AND DISPLACEMENT FIELDS IN SCARF JOINTS SUBJECTED TO A UNIFORM CHANGE IN TEMPERATURE

The role of free-edge stresses in controlling the initiation of failure from the interface corner of a scarf joint subjected to a uniform change in temperature is examined. In general, the stress field can be expressed by σ _ij = Hr ^{λ −1} + σ _{ij 0} , where r is the radial distance from the interface corner, λ − 1 is the order of the stress singularity, H is the intensity of the singularity, and σ _{ij 0} is a non-singular constant stress. A combination of the finite element method and a path-independent integral is used to evaluate the magnitude of H for two joint configurations: (i) a scarf joint between two long bi-material strips; and (ii) a scarf joint consisting of a thin elastic layer sandwiched between two substrates. The magnitude of H is linearly dependent on a non-dimensional constant function a; the magnitude of a decreases with increasing level of mismatch in the elastic properties of the bonded materials. A comparison between the values of H evaluated by the path-independent integral method and the commonly used extrapolation method indicate that the extrapolation method could be in error by as much as 25%.     

Life assessment of HK40 reformer tube by fracture mechanics

Creep crack growth measurements have been made at 871°C (design temperature) on HK40 steel which was taken from a reformer tube in service for 81000 h. The path-independent integral C* is a proper parameter correlating creep crack growth rates of different specimen geometries. The residual life of the reformer tube was calculated by the equation da/dt with C* and Webster's model. It has been shown that the results by the two methods are consistent.     

Evaluation of polymer fracture parameters by the boundary element method

The boundary element method (BEM) for two-dimensional linear viscoelasticity is applied to polymer fracture. The time-dependence of stress intensity factors is assessed for various viscoelastic models as well as loading and support conditions. Various representations of the energy release rate under isothermal conditions are adopted. Additional boundary integral equations for the displacement gradient in the domain are linked to algorithms for the evaluation of path-independent J-integrals. The consistency of BEM predictions and their good agreement with other analytical results confirms BEM as a valid modelling tool for viscoelastic fracture characterisation and failure assessment under complex geometric and loading conditions.     

Boundary element analysis of ductile crack growth using the traction-free fundamental solution

A new boundary element formulation in two-dimensional rate-independent plasticity is given. This new formulation uses a so-called traction-free fundamental solution so that the resulting boundary integral equation converges in the normal sense, and more important, a formal differentiation of the boundary integral equation leads to a valid integral representation for the in-plane stress component on the boundary. No finite difference approximation is needed to construct the stress recovery routine. The new boundary element method is then used to solve the problem of quasi-static ductile crack growth. Numerical simulations based on a set of experimental data have been carried out to evaluate a new path-independent integral,T* _M . TheT* _M ,-integral is a modified version of Atluri'sT*-integral. This modified version has an advantage of having a less singular domain integral near the crack flank so that it is numericaly preferable toT*.     

A path-independent integral for 2-dimensional cracks in homogeneous isotropic conductive plate

A path-independent integral which is denoted by j_e, is introduced for the 2-dimensional crack problems in the homogeneous isotropic conductor in which the steady current flows. By utilizing the j_e-integral the distributions of the electric potential, current density and the Joule heating rate near the crack tip are derived. It is shown that the j_e-integral provides a parameter which dominates the distributions of the electric potential and current density near the crack tip as the square of the amplitude.     

Path-independent H integrals for three-dimensional fracture mechanics

In fracture mechanics, a number of real applications have intrinsically three-dimensional crack geometries, thereby requiring a means of extracting stress intensity factors under such circumstances. Two approaches to this end are examined here: one, a three-dimensional J-integral; the other, three-dimensional H integrals for each mode. The first integral is well accepted by the fracture mechanics community; the second integrals are newly developed herein. The two are compared on three-dimensional test problems with closed-form solutions that are constructed for this purpose. Analysis is via quarter-point elements on two successively refined grids for each test problem. The results demonstrate that both types of path-independent integral can furnish estimates of stress intensity factors which converge to good levels of accuracy in return for reasonable levels of computational effort. This revised version was published online in July 2006 with corrections to the Cover Date.     

Calculation of stress intensities at sharp notches in anisotropic media

We present a procedure to calculate mode I and II notch stress intensities in anisotropic media using the path-independent H-integral. The method is based on coupling the analysis of asymptotic stress and displacement fields near a sharp notch with a path independent integral that results from the application of Betti's reciprocal theorem to the notched solid. The approach is demonstrated for two loading/geometry combinations that arise naturally in etched single crystal silicon: mode I loading of a 70.53° notch and mixed mode I and II loading of a T-structure with a 90° notch. Results agree well with those obtained by correlating computed notch-flank displacements with the asymptotic solution.     

Finite element methodology for path integrals in fracture mechanics

Based on the energy foundation of the path-independent integral in non-linear fracture mechanics, I* integral as the dual form of Rice's J is presented, it is also path-independent and is equivalent to J in value but it relates to the complementary energy. It is proved that, in numerical implementation, the path independence of J and I* can be ensured by using the assumed displacement finite elements and the assumed stress finite elements, respectively. Regarding the bounds of crack parameters, it is demonstrated that the lower bound of J can be estimated by the displacement compatible elements, and the upper bound of I* can be estimated by the stress equilibrium elements. In view of the difficulties in formulating stress equilibrium model, instead of it, a quasi-equilibrium model is proposed, which makes hybrid stress elements be able to estimate the bound of I*, and do not lose the characteristics of stiffness formulation. Two four-node plane elements are suggested; of them, the incompatible one can be used in incompressible/fully plastic fracture analysis, and the penalty-equilibrium one can be implemented to estimate the bound of I*. Furthermore, an incremental formulation is developed for I*, and can be extended into the calculations of ductile fracture under monotonic loading. For attestation, quite a number of numerical experiments is carried out, and some significant results are offered. © 1998 John Wiley & Sons, Ltd.