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.     

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.     

A theoretical note on mode-I crack branching and kinking

An energy-based fracture mode has been derived for the mode-I crack branching and kinking. The classic J_i-integral has been further explored by a new partial integral path and the analytical solution of the energy release rate for crack branching and kinking from a mode-I crack tip has been established. The crack branching/kinking angle has also been analytically derived. It shows that the Griffith’s theorem and conservation law can be applied to both mode-I crack extension and mode-I crack branching and kinking. The branching mechanism for quasi-static mode-I crack has been theoretically investigated. The branching toughness and the K-based criterion for crack branching have been defined. The crack branching phenomena predicted by the present model are in well agreement with the experimental observations reported in the literatures.     

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.     

Frictional contact induced crack initiation in incompressible substrate

Surface crack initiation in an incompressible substrate induced through frictional fretting contact is analyzed using an energy-based fracture mechanics model. A closed-form energy release rate for surface crack initiation at the contact boundary has been derived with the crack growth angle determined by the mixed mode singular stress field at the contact edge. The driving forces in the form of J_i-integral, the critical energy release rate and the critical load for crack initiation from the crack free surface have been formulated. The relations between the friction coefficient and crack initiation angle, critical load have been specified.     

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.     

Fracture analysis of magnetoelectroelastic solids by using path independent integrals

A solution scheme based on the fundamental solution for a generalized edge dislocation in an infinite magnetoelectroelastic solid is presented to analyze problems involving single, multiple and slowly growing impermeable cracks. The fundamental solution for a generalized dislocation is obtained by extending the complex potential function formulation used for anisotropic elasticity. The solution for a continuously distributed dislocation is derived by integrating the solution for an edge dislocation. The problem of a system of cracks subjected to remote mechanical, electric and magnetic loading is formulated in terms of set of singular integral equations by applying the principle of superposition and the solution for a continuously distributed dislocation. The singular integral equation system is solved by using a numerical integration technique based on Chebyshev polynomials. The J_i and M-integrals for single crack and multi-cracks problems are derived and their dependence on the coordinate system is investigated. Selected numerical results for the M-integral, total energy release rate and mechanical energy release rate are presented for single, double and multiple crack problems. The case of a slowly growing crack interacting with a stationary crack is also considered. It is found that M-integral presents a reliable and physically acceptable measure for assessment of fracture behaviour and damage of magnetoelectroelastic materials.     

The energy dissipation rate approach to tearing instability

Stable and unstable tearing in metals is currently analysed by J integral theory, or by the G_R curve approach. This paper explains an alternative analysis route based on energy dissipation rate, D. It is shown that the implication of increasing toughness with crack growth in G_R and J_R curves is misleading. Even in small scale yielding (SSY), it is possible to have stable tearing under increasing G or J whilst at the same time D is constant (or even reducing) with crack growth. New terms: C for crack driving force, D^* for geometry normalised D, D_ssy for D in SSY, and crack stability index are explained. A D based fracture analysis diagram is introduced. Comparisons are made between energy dissipation rate, J integral, and G_R curve instability prediction methods. It is shown that, in most instances, these different approaches are compatible; but that the use of J_R curves derived in fully yielded test pieces to predict failure in SSY has the potential to lead to an unconservative instability prediction. The practical advantage of the energy dissipation rate approach is that it can be applied to all product thicknesses at any extent of crack growth. The major advantage compared to the G_R approach is that toughness measurements can be made on much smaller specimens.     

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.     

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.     

Prediction of mode I crack growth resistance based on a comparative investigation of J-integral and energy dissipation rate concept

An energy dissipation rate concept is employed in conjunction with the J-integral to calculate crack growth resistance of elastic-plastic fracture. Different from Rice’s J-integral, the free energy density is employed in place of the stress working density to define an energy-momentum tensor, which yields that the slightly changed J-integral is path dependent regardless of incremental plasticity and deformational plasticity. The J-integral over the remote contour is split into the plastic influence term and the J _FPZ-integral over the fracture process zone which is an appropriate estimate of the separation work of fracture. Finite element simulations are carried out to predict the plane strain mode I crack growth behavior by an embedded fracture process zone. It can be concluded that J-integral characterization is in essence a stress intensity-based fracture resistance similar to the K criterion of linear elastic fracture, and energy dissipation rate fracture resistance can be taken as an extension of the Griffith criterion to the elastic-plastic fracture.     

An analytical expression for the J2‐integral of an interfacial crack in orthotropic bimaterials

This paper presents a new analytical expression relating the J₂‐integral and stress intensity factors (SIF) in an in‐plane traction‐free crack between two orthotropic elastic solids using the complex function method. The singular oscillatory near tip field of a bimaterial interfacial crack is usually characterized by a pair of SIFs. In linear elastic interfacial fracture mechanics, the majority of numerical and experimental methods rely on the analytical equations relating J_k‐integrals and SIFs. Although an analytical equation relating J₁‐integral or strain energy release rate and SIFs is available, a similar relation for J₂‐integral in debonded anisotropic solids is non‐existent. Using this new analytical expression, in conjunction with the values of J_k, the SIFs can be computed without the need for an auxiliary relation. An example with known analytical solutions for SIFs is presented to show the variation of the J₂‐integral near the crack tip of a bimaterial orthotropic plate. Different bimaterial combinations are considered, and the effect of material mismatch on J_k is demonstrated.     

Mixed-mode fracture load prediction in lead-free solder joints

Double cantilever beam (DCB) fracture specimens were made by joining copper bars with both continuous and discrete SAC305 solder layers of different lengths under standard surface mount (SMT) processing conditions. The specimens were then fractured under mode-I and various mixed-mode loading conditions. The loads corresponding to crack initiation in the continuous joints were used to calculate the critical strain energy release rate, J_ci, at the various mode ratios using elastic–plastic finite element analysis (FEA). It was found that the J_ci from the continuous joint DCBs provided a lower bound strength prediction for discrete 2 mm and 5 mm long joints at the various mode ratios. Additionally, these J_ci values calculated from FEA using the measured fracture loads agreed reasonably with J_ci estimated from measured crack opening displacements at crack initiation in both the continuous and discrete joints. Therefore, the critical strain energy release rate as a function of the mode ratio of loading is a promising fracture criterion that can be used to predict the strength of solder joints of arbitrary geometry subject to combined tensile and shear loads.     

Finite element calculation of stress intensity factors for interfacial crack using virtual crack closure integral

This paper presents a successful implementation of the virtual crack closure integral method to calculate the stress intensity factors of an interfacial crack. The present method would compute the mixed-mode stress intensity factors from the mixed-mode energy release rates of the interfacial crack, which are easily obtained from the crack opening displacements and the nodal forces at and ahead of the crack tip, in a finite element model. The simple formulae which relate the stress intensity factors to the energy release rates are given in three separate categories: an isotropic bimaterial continuum, an orthotropic bimaterial continuum, and an anisotropic bimaterial continuum. In the example of a central crack in a bimaterial block under the plane strain condition, comparisons are made with the exact solution to determine the accuracy and efficiency of the numerical method. It was found that the virtual crack closure integral method does lead to very accurate results with a relatively coarse finite element mesh. It has also been shown that for an anisotropic interfacial crack under the generalized plane strain condition, the computed stress intensity factors using the virtual crack closure method compared favorably with the results using the J integral method applied to two interacting crack tip solutions. In order for the stress intensity factors to be used as physical variables, the characteristic length for the stress intensity factors must be properly defined. A study was carried out to determine the effects of the characteristic length on the fracture criterion based the mixed-mode stress intensity factors. It was found that the fracture criterion based on the quadratic mixture of the normalized stress intensity factors is less sensitive to the changes in characteristic length than the fracture criterion based on the total energy release rate along with the phase angle.This work has been supported by ONR, with Dr. Y. Rajapakse as the program official.     

Effect of material plasticity on fatigue crack propagation under complex stress state

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. However, this fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, a modification has been made to G _max criterion to implement the consideration of the plastic strain energy. This criterion is extended to study the fatigue crack growth characteristics of mixed mode cracks in steel pipes. To predict crack propagation due to fatigue loads, a new elasto-plastic energy model is presented. This new model includes the effects of material properties like strain hardening exponent n, yield strength σ_y and fracture toughness and stress intensity factor ranges. The results obtained are compared with those obtained using the commonly employed crack growth law and the experimental data.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis

This paper deals with some basic linear elastic fracture problems for an arbitrary-shaped planar crack in a three-dimensional infinite transversely isotropic piezoelectric media. The finite-part integral concept is used to derive hypersingular integral equations for the crack from the point force and charge solutions with distinct eigenvalues s _i(i=1,2,3) of an infinite transversely isotropic piezoelectric media. Investigations on the singularities and the singular stress fields and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by using the definitions of stress and electric displacement intensity factors and the principle of virtual work, respectively. The hypersingular integral equations under axially symmetric mechanical and electric loadings are solved analytically for the case of a penny-shaped crack.     

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.     

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.     

The crack growth resistance of thin steel sheets under eccentric tension

The stable crack growth in thin steel sheets is the topic of this paper. The crack opening was observed using a videoextensometry system, allowing the crack extension determination. J_R-curve and δ_R-curve were established from obtained data. The ductile tearing properties of different thin sheets of steel were determined, including the impact of the specimen orientation, from test performed on compact tension specimens loaded under two conditions. The effect of the material, the rolling direction, and loading rate on the crack growth resistance of thin steel sheets was analyzed. In addition to the crack growth resistance, J-integral values for crack initiation were also estimated. The relation between J _i and J_0.2 was assessed using the basic mathematical and statistical methods. This relation was described by a linear regression model.