Smeared-tip superposition method for cohesive fracture with rate effect and creep

A recent formulation of the smeared-tip superposition method presented by Baant [1], which itself was a generalization and modification of an integral equation formulation with an asymptotic series solution derived by Planas and Elices [2], is further improved, generalized and adapted to an efficient finite difference solution scheme. A crack with bridging stresses is modeled as a superposition of infinitely many LEFM cracks with continuously distributed (smeared) tips having infinitely small intensity factors. Knowledge of the stress intensity factor as a function of the location of the crack tip along the crack path is all that is needed to obtain the load-displacement relation. The solution is reduced to a singular integral equation for a function describing the components of applied load associated with crack tips at various locations. The integral equation is complemented by an arbitrary relation between the bridging stress and the crack opening displacement, which can be rate-independent or rate-dependent. Furthermore, using the creep operator method, the equation is extended to aging linearly viscoelastic behavior in the bulk of the specimen. The previously presented finite difference solution is improved and generalized in a form that leads to a system of nonlinear algebraic equations, which can be solved by an optimization method. Application of the smeared-tip method to the analysis of recent measurements of the size effect in three-point-bend fracture specimens of different sizes is presented and a crack opening law that yields the main qualitative characteristics of the test results, particularly an increase of brittleness with a decreasing loading rate, is presented.     

Cohesive Fracture Model Based on Necking

A linear hardening model together with a linear elastic background material is first used to discuss some aspects of the mathematical and physical limitations and constraints on cohesive laws. Using an integral equation approach together with the cohesive crack assumption, it is found that in order to remove the stress singularity at the tip of the cohesive zone, the cohesive law must have a nonzero traction at the initial zero opening displacement. A cohesive zone model for ductile metals is then derived based on necking in thin cracked sheets. With this model, the cohesive behavior including peak cohesive traction, cohesive energy density and shape of the cohesive traction–separation curve is discussed. The peak cohesive traction is found to vary from 1.15 times the yield stress for perfectly plastic materials to about 2.5 times the yield stress for modest hardening materials (power hardening exponent of 0.2). The cohesive energy density depends on the critical relative plate thickness reduction at the root of the neck at crack initiation, which needs to be determined by experiments. Finally, an elastic background medium with a center crack is employed to re-examine the shape effect of cohesive traction–separation curve, and the relation between the linear elastic fracture mechanics (LEFM) and cohesive zone models by considering the cohesive zone development and crack growth in the infinite elastic medium. It is shown that the shape of the cohesive curve does affect the cohesive zone size and the apparent energy release rate of LEFM for the crack growth in the elastic background material. The apparent energy release rate of LEFM approaches the cohesive energy density when the crack extends significantly longer than the characteristic length of the cohesive zone.     

Crack Extension Resistance and Fracture Properties of Quasi-Brittle Softening Materials like Concrete Based on the Complete Process of Fracture

The crack extension resistance and fracture properties are studied in detail for quasi-brittle materials like concrete with a softening traction-separation law by investigating the complete fracture process. The computed samples are the three-point bending notched beams of concrete with different sizes tested by other researchers. The softening traction-separation law which was proposed by Reinhardt et al. based on direct tension tests for normal concrete materials was chosen in the computations. Different distribution shapes of the cohesive force on the fictitious crack zone were considered for the corresponding loading states. The computations were mainly based on the analytic solutions for this problem using Gauss–Chebyshev quadrature to achieve the integration which is singular at the integral boundary. The crack extension resistance curves in terms of stress intensity (K_R-curves) were determined by combining the crack initiation toughness that is the inherent toughness of the material needed to resist the crack initiation in the case that is in the lack of an extension of the main crack with the contribution due to the cohesive force along the fictitious crack zone during the complete processes of fracture. The situation of crack propagation can be judged by comparing K_R-curves of crack extension resistance with the stress intensity factor curves which were calculated using the lengths of the extending crack and the corresponding loads at each loading states, e.g., when the crack extension resistance curve(K_R-curve) is lower than the stress intensity factor curve, the crack propagation is stable; otherwise, it is unstable. In the computation, the obtained relationship between the crack tip opening displacement CTOD and the amount of crack extension for the complete fracture process is in agreement with the testing results of other researchers. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture Analysis of Concrete Structural Components Accounting for Tension Softening Effect

This paper presents methodologies for fracture analysis of concrete structural components with and without considering tension softening effect. Stress intensity factor (SIF) is computed by using analytical approach and finite element analysis. In the analytical approach, SIF accounting for tension softening effect has been obtained as the difference of SIF obtained using linear elastic fracture mechanics (LEFM) principles and SIF due to closing pressure. Superposition principle has been used by accounting for non-linearity in incremental form. SIF due to crack closing force applied on the effective crack face inside the process zone has been computed using Green's function approach. In finite element analysis, the domain integral method has been used for computation of SIF. The domain integral method is used to calculate the strain energy release rate and SIF when a crack grows. Numerical studies have been conducted on notched 3-point bending concrete specimen with and without considering the cohesive stresses. It is observed from the studies that SIF obtained from the finite element analysis with and without considering the cohesive stresses is in good agreement with the corresponding analytical value. The effect of cohesive stress on SIF decreases with increase of crack length. Further, studies have been conducted on geometrically similar structures and observed that (i) the effect of cohesive stress on SIF is significant with increase of load for a particular crack length and (iii) SIF values decreases with increase of tensile strength for a particular crack length and load.     

Residual strength prediction for aircraft panels with Multiple Site Damage,using the “EPFEAM” for stable crack growth analysis

In this paper, a new analytical method for solving stable crack propagation problems in a ductile panel with a row of cracks, is presented. The main purpose of the present study is to estimate the maximum load carrying capacity of such panels accurately. The so called Elastic Plastic Finite Element Alternating Method (Pyo et al. (1994) was extended to account for the propagating cracks. The crack propagation algorithm utilizes the analytic crack solution to release the stresses ahead the crack tip. The T _inf ^sup* integral is employed as the crack extension criterion. This integral parameter accounts for the near tip stress-strain singularity and its critical values for crack propagation can be extracted from the P-a curve of single cracked specimen case. The present method can be applied to the problems of the fuselage skin of aging airplanes, in which a row of cracks develop (MSD; Multiple Site Damage) from rivet holes. The load carrying capacity of such damaged structure reduces by a considerable amount. In order to predict the behavior near the critical load, one must account for plastic deformation, if the material is ductile. Furthermore, the maximum load carried by the structure is often reached after some amount of crack propagation. In this paper, a series of analyses have been conducted and their results compare with the available experimental data.     

Basic issues in the mechanics of high cycle metal fatigue

Mechanics issues related to the formation and growth of cracks ranging from subgrain dimension to up to the order of one mm are considered under high cycle fatigue (HCF) conditions for metallic materials. Further efforts to improve the accuracy of life estimation in the HCF regime must consider various factors that are not presently addressed by traditional linear elastic fracture mechanics (LEFM) approaches, nor by conventional HCF design tools such as the S-N curve, modified Goodman diagram and fatigue limit. A fundamental consideration is that a threshold level for ΔK for small/short cracks may be considerably lower than that for long cracks, leading to non-conservative life predictions using the traditional LEFM approach. Extension of damage tolerance concepts to lower length scales and small cracks relies critically on deeper understanding of (a) small crack behavior including interactions with microstructure, (b) heterogeneity and anisotropy of cyclic slip processes associated with the orientation distribution of grains, and (c) development of reliable small crack monitoring techniques. The basic technology is not yet sufficiently advanced in any of these areas to implement damage tolerant design for HCF. The lack of consistency of existing crack initiation and fracture mechanics approaches for HCF leads to significant reservations concerning application of existing technology to damage tolerant design of aircraft gas turbine engines, for example. The intent of this paper is to focus on various aspects of the propagation of small cracks which merit further research to enhance the accuracy of HCF life prediction. Predominant concern will rest with polycrystalline metals, and most of the issues pertain to wide classes of alloys.     

The role of thermal and residual stresses in linear elastic and post yield fracture mechanics

The general role played by thermal and residual (TR) stresses in fracture mechanics is still unclear. It is sometimes argued (a) that in the linear elastic fracture mechanics (LEFM) regime TR stresses are secondary (rather than primary) i.e. that the overall loading is relaxed (rather than maintained) as well as redistributed as the crack grows, and (b) that because TR stresses do not influence the plastic limit load of a structure they have little influence on the post yield fracture mechanics (PYFM) regime. This paper demonstrates the dangers of these views. Examples are given of TR stresses behaving in either primary or secondary manner in both the LEFM and PYFM regimes. The danger of drawing general conclusions in fracture mechanics from the nearness of a structure to its plastic limit load is demonstrated, and it is shown that local rather than global (limit-analysis) collapse parameters must be used in empirical formulae that interpolate between LEFM and plastic-collapse regimes. In LEFM it is shown that the standard Green's function (weight or influence function) method can be applied to TR stress calculations. The method also applies in the PYFM regime if the Dugdale-type strip yield model is assumed. A general method of analysing fixed-grip loadings in the plastic regime, based on Rice's J contour integral is also given.     

Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity

Summary A modification of the form of the singular integral equation for the problem of a plane crack of arbitrary shape in a three-dimensional isotropic elastic medium is proposed. This modification consists in the incorporation of the Laplace operator into the integrand. The integral must now be interpreted as a finite-part integral. The new singular integral equation is equivalent to the original one, but simpler in form. Moreovet, its form suggests a new approach for its numerical solution, based on quadrature rules for one-dimensional finite part integrals with a singularity of order two. A very simple application to the problem of a penny-shaped crack under constant pressure is also made. Moreover, the case of straight crack problems in plane isotropic elasticity is also considered in detail and the corresponding results for this special case are also derived.With 2 Figures     

Eigenvalue analysis of size effect for cohesive crack model

The paper analyses the effect of structure size on the nominal strength of the structure that is implied by the cohesive (or fictitious) crack model proposed for concrete by Hillerborg et al. A new method to calculate the maximum load of geometrically similar structures of different sizes without calculating the entire load-deflection curves is presented. The problem is reduced to a matrix eigenvalue problem, in which the structure size for which the maximum load occurs at the given (relative) length of the cohesive crack is obtained as the smallest eigenvalue. Subsequently, the maximum load, nominal strength and load-point displacement are calculated from the matrix equilibrium equation. The nonlinearity of the softening stress-displacement law is handled by iteration. For a linear softening law, the eigenvalue problem is linear and independent of the matrix equilibrium equation, and the peak load can then be obtained without solving the equilibrium equation. The effect of the shape of the softening law is studied, and it is found that the size effect curve is not very sensitive to it. The generalized size effect law proposed earlier by Baant, which describes a transition between the horizontal and inclined asymptotes of strength theory and linear elastic fracture mechanics, is found to fit the numerical results very well. Finally some implications for the determination of fracture energy from the size effect tests are discussed. The results are of interest for quasibrittle materials such as concrete, rocks, sea ice and modern tough ceramics.     

On the penny-shaped crack in a non-homogeneous interlayer under torsion

This paper presents a theoretical treatment of a penny-shaped crack in an interfacial zone, along the thickness of which the elastic modulus is assumed as ₂(z) = ( +bz) ^k , wherek represents the distribution parameter independent of material properties and interlayer thicknessh. The theoretical formulations governing the torsion deformation behavior of the material are based on the use of a dislocation density function and integral transform technique. The stress intensity factor is obtained by solving a singular integral equation. Numerical examples are given to show the effects of material properties, interlayer thickness, and especially the distribution parameterk on the stress intensity factor. In the numerical procedure, modified Bessel functions are used, and the rate of convergence depends greatly on the ratio ofh/c, wherec is the crack radius.     

Effect of acid corrosion on crack propagation of concrete beams

The effect of acid corrosion on crack propagation of concrete beams was theoretically studied by the method of crack extension resistance curve. Based on this method, a calculation approach was proposed to determine fracture stress intensity factors in crack propagation of concrete beams. Loop iteration analysis was carried out to calculate maximum bearing capacity load, unstable crack toughness, resistance toughness curve, cohesive toughness curve and load–crack mouth opening displacement. Both bilinear and nonlinear softening traction–separation curves were adopted for each of these calculation parameters. The analysis results of each showed the effect of acid corrosion degrees. The influence of acid corrosion on fracture properties was discussed through the calculated results of cohesive toughness curves. These five kinds of simulated results were basically consistent, before the load attained the maximum value. However, with further crack propagation, cohesive toughness of nonlinear softening model was significantly larger than that of bilinear softening model, and the descending branch of P–CMOD curve by nonlinear law is higher than that by bilinear law. To validate the approach, tests of specimens under six different corrosion periods were experimentally studied, using three-point bending notched concrete beams soaked in sulphuric acid solution. The Double-K fracture parameters were investigated based on the test results, and load–crack mouth opening displacement curves for different acid conditions were obtained using synchronous sampling of a load sensor and clip-gauge. Numerical results by bilinear softening model showed a good correlation with the experimental ones.     

Application of Numerical Method to Investigation of Fatigue Crack Behavior Through Friction Stir Welding

Fatigue crack propagation through a friction stir welded (FSW) joint of 2024-T351 Al alloy is investigated numerically. The governing relationships for predicting the crack behavior including incremental crack length, crack growth rate, and crack growth direction are presented. Stress intensity is calculated based on displacement correlation technique, and fatigue crack growth through the FSW joint is investigated under linear elastic fracture mechanics (LEFM) using the Paris model. The concepts of crack closure, residual stress, and stress relaxation are incorporated into the Paris model to support the final results. Maximum circumferential tensile stress method is applied to predict the crack growth direction. Finally, the numerical approaches are employed to the high number of elements in the framework of Fracture Analysis Code (FRANC2D/L) to simulate the fatigue crack propagation through the FSW joint including various zones with different material properties. Fatigue lifetime of the welded joint is predicted by implementing the same procedure for various loading values. The obtained numerical results are validated with the experimental work (Ali et al., Int J Fatigue 30:2030–2043, 2008).

State-of-the-art review on fracture analysis of concrete structural components

This paper presents a critical review of literature on fracture analysis of concrete structural components. Review includes various fracture models, tension softening models, methodologies for crack growth analysis and remaining life prediction. The widely used fracture models which are based on fictitious crack approach and effective elastic crack approach have been explained. Various tension softening models such as linear, bi-linear, tri-linear, etc. have been presented with appropriate expressions. From the critical review of models, it has been observed that some of the models have complex expressions involving many parameters. There is a need to develop some more generalised models. Studies have been conducted on crack growth analysis and remaining life prediction using linear elastic fracture mechanics (LEFM) principles. From the studies, it has been observed that there is significant difference between predicted and experimental observations. The difference in the values is attributed to not considering the tension softening effect in the analysis.     

A new boundary integral equation method for cracked 2-D anisotropic bodies

By using integration by parts to the traditional boundary integral formulation, a traction boundary integral equation for cracked 2-D anisotropic bodies is derived. The new traction integral equation involves only singularity of order 1/r and no hypersingular term appears. The dislocation densities on the crack surface are introduced and the relations between stress intensity factors and dislocation densities near the crack tip are induced to calculate the stress intensity factors. The boundary element method based on the new equation is established and the singular interpolation functions are introduced to model the singularity of the dislocation density (in the order of ) for crack tip elements. The proposed method can be directly used for the 2-D anisotropic body containing cracks of arbitrary geometric shapes. Several numerical examples demonstrate the validity and accuracy of BEM based on the new boundary integral equation.     

Calculation of stress intensity factors for an interfacial crack between dissimilar anisotropic media,using a hybrid element method and the mutual integral

Using Beom and Atluri's complete eigen-function solutions for stresses and displacements near the tip of an interfacial crack between dissimilar anisotropic media, a hybrid crack tip finite-element is developed. This element, as well as a mutual integral method are used to determine the stress intensity factors for an interfacial crack between dissimilar anisotropic media. The hybrid element has, for its Galerkin basis functions, the eigen-function solutions for stresses and displacements embedded within it. The mutual integral approach is based on the application of the path-independent J integral to a linear combination of two solutions: one, the problem to be solved, and the second, an auxiliary solution with a known singular solution. A comparison with exact solutions is made to determine the accuracy and efficiency of both the methods in various mixed mode interfacial crack problems. The size of the hybrid element was found to have very little effect on the accuracy of the solution: an acceptable numerical solution can be obtained with a very coarse mesh by using a larger hybrid element. An equivalent domain integral method is used in the application of the mutual integral instead of the line integral method. It is shown that the calculated mutual integral is domain independent. Therefore, the mutual integral can be evaluated far away from the crack-tip where the finite element solution is more accurate. In addition, numerical examples are given to determine the stress intensity factors for a delamination crack in composite lap joints and at plate-stiffener interfaces.This work was supported by a grant from the NASA Langley Research Center.     

Local and global instabilities associated with continuing crack extension in dissipative solids

The model developed here links microstructural and continuum mechanics aspects of the early stages of the fracture process occurring in dissipative solids. A variable size damage zone, endowed with a structure of its own determined by the microstructural parameters related to the material ductility and grain size, is incorporated into a moving dominant crack. Prior to and during the course of crack extension, the energy dissipation mechanisms of diverse nature are activated within the volume of the localized damage zone, providing a substantial contribution to the effective material toughness. A criterion for quasi-static crack based on self-similarity of the crack-tip region is used to set up a governing equation of motion. The stress-transferring ability of the damage zone depends on the separation distance created between the two opposite boundaries of the fractures zone, and it determines the history of a quasi-static crack development including the attainment of the terminal instability point. Although detailed information regarding the distribution of stress prevailing within the nonlinear zone is lacking, it is shown that certain plausible governing equations may be constructed and employed to define material resistance curve in sufficiently large specimens (within the so-called ssy range), which is then used to predict the onset of catastrophic fracture. Six different specimen configurations are considered and the pertinent macro-mechanical stability analysis is presented in detail. The class of materials susceptible to this type of analysis is not restricted by the usual LEFM or EPFM constraint. This revised version was published online in August 2006 with corrections to the Cover Date.     

A numerical Green's function BEM formulation for crack growth simulation

This paper presents a crack growth prediction analysis based on the numerical Green's function (NGF) procedure and on the minimum strain energy density criterion for crack extension, also known as S-criterion. In the NGF procedure, the hypersingular boundary integral equation is used to numerically obtain the Green's function which automatically includes the crack into the fundamental infinite medium. When solving a linear elastic fracture mechanisms (LEFM) problem, once the NGF is obtained, the classical boundary element method can be used to determine the external boundary unknowns and, consequently, the stress intensity factors needed to predict the direction and increment of crack growth. With the change in crack geometry, another numerical analysis is carried out without need to rebuilding the entire element discretization, since only the crack built in the NGF needs update. Numerical examples, contemplating crack extensions for two-dimensional LEFM problems, are presented to illustrate the procedure.     

Scattering of SH wave by a crack terminating at the interface of a bimaterial

A general method for solving the scattering of plane SH wave by a crack terminating at the interface of a bimaterial is presented. The crack can terminate at the interface in an arbitrary angle. In order to solve the proposed problem, the Greens function for a point harmonic force applied at an arbitrary point of the bimaterial is established by the Fourier transformation method. Using the obtained Greens function and the Betti-Rayleigh reciprocal theorem, the total scattered field of the crack is constructed. The total scattered field of the crack is divided into a regular part and a singular part. The hypersingular integral equation of the crack is obtained in terms of the regular and singular scattered field as well as the free wave field. The stress singularity order and singular stress at the terminating point are analyzed by the hypersingular integral equation and the singular scattered field of the crack. The dynamic stress intensity factor (DSIF) at the terminating point is defined in terms of the singular stresses at the terminating point. Numerical solution of the hypersingular integral equation gives the DSIFs at the crack tips. Comparison of our results with known results confirms the proposed method. Some numerical results and corresponding analysis are given in the paper.Constructive advice from the anonymous reviewers is acknowledged.     

3D elastodynamic crack analysis by a non-hypersingular BIEM

A boundary integral equation method (BIEM) is presented for 3D elastodynamic crack analysis. The method is based on a non-hypersingular BIE formulation, where the unknown quantities are the crack opening dispacements and their derivatives. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square-root crack-tip shape function for elements near the crack front to describe the proper behavior of the unknown quantities at the crack front. A collocation method is applied to convert the non-hypersingular BIEs to a system of linear algebraic equations which are solved numerically. For several geometrical configurations, numerical results are presented for both the elastodynamic stress intensity factors and the scattering cross section. They are in good agreement with those obtained by other authors.     

Eigenvalue method for computing size effect of cohesive cracks with residual stress,with application to kink-bands in composites

The previously developed eigenvalue method for computing the size effect of cohesive crack model is extended to the cohesive crack model with a finite residual stress. In this model, the structure size for which a specified relative length of kink-band corresponds to the maximum load is obtained as an eigenvalue of a homogeneous Fredholm integral equation. This new method is direct and much more efficient than the classical finite element approach in which the entire load-deflection history must be computed to obtain the maximum load. A secondary purpose of the paper is to apply the new method to the effect of structure size on the compressive strength of unidirectional fiber–polymer composites failing by propagation of kink-band with fiber microbuckling. The kink-band is simulated by a cohesive crack model with a linear compressive softening law and a finite residual stress. The simulation shows that the specimens tested have a negative–positive geometry, i.e., the energy release rate of the kink-band for a unit load first decreases but at a certain length of propagation begins to increase. Finally the effect of shape of the softening law of cohesive crack on the size effect curve is studied by using the new eigenvalue method. It is shown that, for a negative–positive geometry, the size effect on the peak load depends on the entire softening curve if the specimens is not too small.