Dynamic J integral, separated dynamic J integral and component separation method for dynamic interfacial cracks in piezoelectric bimaterials

First, the near-tip stress and electric displacement fields are analytically solved for a dynamically propagating interfacial crack in a piezoelectric bimaterial. Second, from the rate formulation of the energy balance in a piezoelectric material, the path independent dynamic J integral is derived, which has the physical significance of the energy release rate. Using the present near-tip analytical solutions, the relationships between the dynamic J integral and the stress and electric displacement intensity factors are also obtained. It is shown that the path independent dynamic J integral contains the static J integral and the dynamic J integral for elastic materials, and static J integral for piezoelectric materials as special cases. Third, for an interfacial crack in a piezoelectric bimaterial, the path independent separated dynamic J integrals are derived, which have the physical significance of energy flow rates into the propagating interfacial crack tip from the individual material sides or, equivalently, the separated dynamic energy release rates. Fourth, to accurately evaluate mixed-mode stress and electric displacement intensity factors, the component separation method of the dynamic J integral is developed. Finally, the finite element analyses of a static stationary interfacial crack in a piezoelectric bimaterial subject to mechanical, electrical and combined loading are carried out to demonstrate the applicability of the generalized (dynamic) J integral and the separated J integral, and the component separation method.     

Dual path independent integrals in the boundary-value problems of cracks

Path independent integral J in Rice's theory is expressible as the rate of decrease of strain potential energy P with respect to crack length. The dual integral I is path independent, and is expressible as the rate of decrease of potential energy Q for stress field. Approximate upper and lower bounds of integrals are obtained by variational principles.     

Mixed-mode J-integral formulation and implementation using graded elements for fracture analysis of nonhomogeneous orthotropic materials

The path-independent J_k^*-integral, in conjunction with the finite element method (FEM), is presented for mode I and mixed-mode crack problems in orthotropic functionally graded materials (FGMs) considering plane elasticity. A general procedure is presented where the crack is arbitrarily oriented, i.e. it does not need to be aligned with the principal orthotropy directions. Smooth spatial variations of the independent engineering material properties are incorporated into the element stiffness matrix using a “generalized isoparametric formulation”, which is natural to the FEM. Both exponential and linear variations of the material properties are considered. Stress intensity factors and energy release rates for pure mode I and mixed-mode boundary value problems are numerically evaluated by means of the equivalent domain integral especially tailored for orthotropic FGMs. Numerical results are discussed and validated against available theoretical and numerical solutions.     

Computational studies on path independent integrals for non-linear dynamic crack problems

Recently the authors have derived various new types of path independent integrals in which the theoretical limitations of the so-calledJ integral are overcome. First, for elastodynamic crack problems, a path independent integralJ which has the physical meaning of energy release rate was derived. Later, more general forms of path independent integralsT ^* andT were derived, which are valid for any constitutive relation under quasi-static as well as dynamic conditions.This paper presents the theoretical and computational aspects of these integrals, of relevance in non-linear dynamic fracture mechanics. An efficient solution technique is also presented for non-linear dynamic finite element method in which a factorization of the assembled stiffness matrix is done only once throughout the computation for a given mesh pattern. Finite element analyses were carried out for an example problem of a center-cracked plate subject to a uniaxial impact loading. The material behavior was modeled by three different constitutive relations such as linear-elastic, elastic-plastic, elastic-viscoplastic cases. The applicability of theT ^* integral to non-linear dynamic fracture mechanics was shown with the numerical results.This study was supported by the Grant-in-Aid for Scientific Research from the Japanese Ministry of Education, and in part by the Science and Technology Grant from Toray Science Foundation     

Verification of brittle fracture criteria for elements with V-shaped notches

In this paper an analysis of crack initiation in plane elements with V-shaped notches under biaxial loading (mode I and II) was presented. The following fracture criteria were used to evaluate the critical loads and directions of crack initiation: strain energy release rate criterion; strain energy density criterion; modified McClintock's stress criterion; non-local stress criterion.Results of numerical analysis obtained using the boundary element method and path independent H and J integrals were compared with experimental data.     

An extended equivalent domain integral method for mixed mode fracture problems by the p-version of FEM

A new path-independent contour integral formula is presented to estimate the crack-tip integral parameter, J-value, for two-dimensional cracked elastic bodies which may quantify the severity of the crack-tip stress fields. The conventional J-integral method based on a line integral has been converted to an equivalent area or domain integral (EDI) by the divergence theorem. It is noted that the EDI method is very attractive because all the quantities necessary for computation of domain integrals are readily available in a finite element analysis. The details and its implementation are extended to the p-version FE model with hierarchic elements using integrals of Legendre polynomials. By decomposing the displacement field obtained from the p-version finite element analysis into symmetric and antisymmetric displacement fields with respect to the crack line, the Mode-I and Mode-II non-dimensional stress intensity factors can be determined by using the decomposition method. The example problems for validating the proposed techniques are centrally oblique cracked plates under tensile loading. The numerical results associated with the variation of oblique angles show very good agreement with the existing solutions. Also, the selective distribution of polynomial orders and the corner elements for automatic mesh generation are applied to improve the numerical solution in this paper. © 1998 John Wiley & Sons, Ltd.     

Crack tip and associated domain integrals from momentum and energy balance

A unified derivation of crack tip flux integrals and their associated domain representations is laid out in this paper. Using a general balance statement as the starting point, crack tip integrals and complementary integrals which are valid for general material response and arbitrary crack tip motion are obtained. Our derivation emphasizes the viewpoint that crack tip integrals are direct consequences of momentum balance. Invoking appropriate restrictions on material response and crack tip motion leads directly to integrals which are in use in crack analysis. Additional crack tip integrals which are direct consequences of total energy and momentum balance are obtained in a similar manner. Some results on dual (or complementary) integrals are discussed. The study provides a framework for the derivation of crack tip integrals and allows them to be viewed from a common perspective. In fact, it will be easy to recognize that every crack tip integral under discussion can be obtained immediately from the general result by appropriately identifying the terms in the general flux tensor. The evaluation of crack tip contour integrals in numerical studies is a potential source of inaccuracy. With the help of weighting functions these integrals are recast into finite domain integrals. The latter integrals are naturally compatible with the finite element method and can be shown to be ideally suited for numerical studies of cracked bodies and the accurate calculation of pointwise energy release rates along a curvilinear three-dimensional crack front. The value of the domain integral does not depend on domain size and shape — this property provides an independent check on the consistency and quality of the numerical calculation. The success of the J-based fracture mechanics approach has led to much literature on pathindependent integrals. It will be shown that various so-called path-independent integrals (including path and area integrals) are but alternate forms of the general result referred to above and do not provide any additional information which is not already contained in the general result. Recent attempts to apply these ‘newer’ integrals to crack growth problems are discussed.     

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.     

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.     

Time-dependent crack growth characterization using rate integrals

This paper investigates the utility of the rate forms of Kishimoto et al.'s integral () and Blackburn's integral (J^*) as parameters for correlating time-dependent crack growth rates. These rate integrals are computed from the results of finite element analyses of crack growth in button-head single edge notch specimens of Alloy 718. The specimens are tested under constant strain and constant load conditions at 593°C and 649°C. The crack tip deformation includes large-scale plasticity, primary and secondary creep. The measured crack growth rates are correlated with the computed rate integrals. The results show that both integrals can consolidate the crack growth data very well.     

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.     

Role of elasto-plastic analysis under cyclic loading in fatigue crack growth studies

Linear Elastic Fracture Mechanics (LEFM) has been widely used in the past for fatigue crack growth studies, but this is acceptable only in situations which are within small scale yielding (SSY). In many practical structural components, conditions of SSY could be violated and one has to look for fracture criteria based on elasto-plastic analysis. Crack closure phenomenon, one of the most striking discoveries based on inelastic deformations during crack growth, has significant effect on fatigue crack growth rate. Numerical simulation of this phenomenon is computationally intensive and involved but has been successfully implemented. Stress intensity factors and strain energy release rates lose their meaning,J-integral (or its incremental) values are applicable only in specific situations, whereas alternate path independent integrals have been proposed in the literature for use with elasto-plastic fracture mechanics (EPFM) based criteria. This paper presents certain salient features of two independent finite element (numerical) studies of relevance to fatigue crack growth, where elasto-plastic analysis becomes significant. These problems can only be handled in the current day computational environment, and would have been only a dream just a few years ago. The work presented in this paper is supported by sponsored research projects of the Aeronautics R & D Board, Government of India and their support is acknowledged.     

Formulation and implementation of a high-order 3-D domain integral method for the extraction of energy release rates

This article presents a three dimensional (3-D) formulation and implementation of a high-order domain integral method for the computation of energy release rate. The method is derived using surface and domain formulations of the J-integral and the weighted residual method. The J-integral along 3-D crack fronts is approximated by high-order Legendre polynomials. The proposed implementation is tailored for the Generalized/eXtended Finite Element Method and can handle discontinuities arbitrarily located within a finite element mesh. The domain integral calculations are based on the same integration elements used for the computation of the stiffness matrix. Discontinuities of the integrands across crack surfaces and across computational element boundaries are fully accounted for. The proposed method is able to deliver smooth approximations and to capture the boundary layer behavior of the J-integral using tetrahedral meshes. Numerical simulations of mode-I and mixed mode benchmark fracture mechanics examples verify expected convergence rates for the computed energy release rates. The results are also in good agreement with other numerical solutions available in the literature.     

On the computation of mixed-mode K-factors for a dynamically propagating crack,using path-independent integrals J'k

The path-independent integral $\text{J'}{}_{\mathrm{k}}$, which has the meaning of energy release rate in elastodynamic crack-propagation, is used to numerically obtain the mixed-mode dynamic stressintensity factors for a crack propagating in a prescribed direction with a prescribed velocity. Moving isoparametric (non-singular) elements are used to model crack propagation. Even though J' is a vector integral and hence is coordinate invariant, the desirability of using specific coordinate systems to improve the accuracies of the numerical solutions for $\text{K-}\text{factors}$ is pointed out. Two procedures for extracting the mixed-mode $\text{K-}\text{factors}$ from the J' integral for a propagating crack are given. It is found that the component of J' along the crack-axis, i.e. $\text{J'}{}_1{}^0$, is always equal to or greater than the product of a crack-velocity-function and the component normal to the crack-axis, $\text{J'}{}_2{}^0$. Several examples of a slanted crack are presented to demonstrate the practical utility of the J' integral. A discussion is also presented concerning the velocity factors for dynamic $\text{K-}\text{factors}$, and energy release rate, in a finite body.     

Dynamic stress intensity factors using Ĵ-integral and finite element method

A formula is derived for determining the stress intensity factors from the path independent $\text{J}\text{?}\text{-}\text{integral}$ which has been formulated in the previous paper as the energy release rate by taking the effect of inertia into account. Both pure and mixed mode problems of a suddenly loaded crack can be analyzed by making use of the formula together with the conventional finite element method. Several computational examples have been given to demonstrate the accuracy of this method.     

Fracture toughness parameters and elasticplastic analysis of non-moderate fracture conditions using finite element methods

Non-moderate fracture conditions of load and geometry, in which plastic zones remain neither constant in size nor uniform in shape as load or crack progress, were simulated using two dimensional elasticplastic finite element calculations. Comparison of conventional crack extension energy rate G, as determined by elastic fracture mechanics and by a compliance method suitable for non-linear deformation, is made with Rice's ‘J integral’, with crack opening displacement, and also with plastic strain energy. The relationship of crack opening displacement to crack extension force can be linearised by selecting values of crack opening displacement at locations which nominally follow the calculated location of the elastic-plastic boundary on the crack face. The relatively large plastic zones associated with high loading are shown, and the field determined in the usual manner from incremental load increases is compared with the field resulting when only two steps are used to reach the same load. Some insight is gained into the use of finite element calculations for determining, from among the various fracture indicators, a single parameter capable of describing nonmoderate fracture conditions.     

Numerical assessment of T-stress computation using a p-version finite element method

Two path independent integrals for T-stress computations, one based on the Betti-Rayleigh reciprocal theorem and the other based on Eshelby's energy momentum tensor are studied. Analytical as well as numerical equivalence between the two integrals is found. To quantify and assess the accuracy of computed values, error analysis for the proposed numerical computation of the T-stress is presented. Specifically, it is found that the error of the computed T-stress is proportional to the ratio of the stress intensity factor divided by the square root of the characteristic dimension of the integration domain where the path independent integral is evaluated. Using a highly accurate hierarchical p-version finite element method, the convergence and accuracy of computed values are easily monitored, and it is shown for numerical examples that the error of the computed T-stress complies with the described error analysis. We conclude that path independent integrals, in conjunction with hierarchical p-version finite element methods, provide a powerful and robust tool to obtain highly accurate numerical results for the T-stress.     

Calculation of mixed-mode stress intensity factors for a crack normal to a bimaterial interface using contour integrals

A pair of contour integrals J_kε are proposed in this paper. The integrals are shown to be path-independent in a modified sense and so they can be accurately evaluated without using any particular singular finite elements. Also, the relationship between J_kε and the generalized stress intensity factors (SIFs) is analytically derived and expressed as functions of the bimaterial mechanical constants. Once the J_kε-integrals are accurately computed, the generalized SIFs and, consequently, the asymptotic mixed-mode stress field can then be properly determined. Numerical results in this study show that the contribution from mode II stress component appears to be more dominant when the uncracked material is relatively stiffer, and vice versa.     

Tubular channel growth in piezoelectric ceramics under electric fields

Tubular channel growth in a piezoelectric material with a conductive channel is investigated. Breakdown tests are performed on PZT807 samples with cylindrical bar shapes under purely electrical loading. It is experimentally observed that dielectric breakdown occurs via the formation of tubular channels, and the new tubular channel propagates in a straight direction through the specimen. The three-dimensional J integral for a tubular channel is used as a criterion for dielectric breakdown failure. The J integral at the onset of breakdown is calculated numerically through finite element analysis. The critical J integrals at the onset of breakdown are obtained.