Crack growth resistance behavior of a functionally graded material: computational studies

This paper describes crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) using a cohesive zone ahead of the crack front. The plasticity in the background (bulk) material follows J₂ flow theory with the flow properties determined by a volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model). A phenomenological, cohesive zone model with six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) describes the constitutive response of the cohesive zone. Crack growth occurs when the complete separation of the cohesive surfaces takes place. The crack growth resistance of the FGM is characterized by a rising J-integral with crack extension (averaged over the specimen thickness) computed using a domain integral (DI) formulation. The 3-D analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. The paper describes applications of the cohesive zone model and the DI method to compute the J resistance curves for both single-edge notch bend, SE(B), and single-edge notch tension, SE(T), specimens having properties of a TiB/Ti FGM. The numerical results show that the TiB/Ti FGM exhibits significant crack growth resistance behavior when the crack grows from the ceramic-rich region into the metal-rich region. Under these conditions, the J-integral is generally higher than the cohesive energy density at the crack tip even when the background material response remains linearly elastic, which contrasts with the case for homogeneous materials wherein the J-integral equals the cohesive energy density for a quasi-statically growing crack.     

Ductile tearing resistance of metal sheets

The concept of R-curves has been adopted to characterise stable crack extension and predict residual strength of thin-walled structures particularly in the aircraft industry. The present contribution uses results of FE simulations of crack extension in panels by the cohesive model to validate analytical procedures for determining J-integral values at large crack extension from measurable quantities, namely the force vs. displacement records. The numerically determined J-integral is taken as the benchmark for the outcome of the analytical formulas. The geometry dependence of J and CTOD based R-curves is investigated and alternative concepts like CTOA and dissipation rate at crack extension are discussed.     

Residual stresses and fracture mechanics analysis of a crack in welds of high strength steels

This study presented the characteristics of residual stresses in welds of high strength steels (POSTEN60, POSTEN80) whose tensile strengths were 600 MPa and 800 MPa, respectively. Three-dimensional thermal elastic-plastic analyses were conducted to investigate the characteristics of welding residual stresses in welds of high strength steels through the thermal and mechanical properties at high temperatures obtained from the elevated temperature tensile tests. A finite element analysis method which can calculate the J-integral for a crack in a residual stress field was developed to evaluate the J-integral for a centre crack when mechanical stresses were applied in conjunction with residual stresses.The results show that the volumetric changes associated with the austenite to martensite phase transformation during rapid cooling after welding of high strength steels significantly influence on the development of residual stresses in the weld fusion zone and heat-affected zone. For a centre crack in welds of high strength steels where only residual stresses are present, increased tensile strength of the steel, increased the J-integral values. The values of the J-integral for the case when mechanical stresses are applied in conjunction with residual stresses are larger than those for the case when only residual stresses are present.     

A new constraint based fracture criterion for surface cracks

A new methodology for predicting the location of maximum crack extension along a surface crack front in ductile materials is presented. Three-dimensional elastic-plastic finite element analyses were used to determine the variations of a constraint parameter (α_h) based on the average opening stress in the crack tip plastic zone and the J-integral distributions along the crack front for many surface crack configurations. Monotonic tension and bending loads are considered. The crack front constraint parameter is combined with the J-integral to characterize fracture, the critical fracture location being the location for which the product Jα_h is a maximum. The criterion is verified with test results from surface cracked specimens.     

Interface crack in sandwich specimen

Fracture of a sandwich specimen loaded with axial forces and bending moments is analyzed in the context of linear elastic fracture mechanics. A closed form expression for the energy release rate for interface cracking of a sandwich specimen with isotropic face sheets is found from analytical evaluation of the J-integral. An approach is applied, whereby the mode mixity for any combination of the loads can be calculated analytically when a load-independent phase angle has been determined. This load-independent phase angle is determined for a broad range of sandwich configurations of practical interest. The load-independent phase angle is determined using a novel finite element based method called the crack surface displacement extrapolation method. The expression for the energy release rate is based on the J-integral and certain stress distributions along the ends of the sandwich specimen. When the stresses from the crack tip interacts with the stresses at the ends, the present analytical calculation of the J-integral becomes inaccurate. The results show that for the analytically J-integral to be accurate the crack tip must be a certain distance away from the uncracked end of the specimen. For a sandwich specimen with face sheet/core stiffness ratio of 100, this distance is in the order 10 times the face sheet thickness. For sandwich structures with face sheet/core stiffness ratio of 1,000, the distance is 30 times the face sheet thickness.     

A three-point bend specimen with partly tapered cross-section sides

The stress intensity factor and the J-integral have been derived analytically and numerically for a modified three-point bend specimen with partly tapered sides, for various crack lengths, taper and specimen cross-section proportions, in order to allow full-thickness testing of tapered samples, common in older steel structures, to obtain a fair effective fracture toughness value for a through thickness crack in inhomogeneous materials. The stress intensity factor is obtained with the approximate analytical method of Kienzler and Herrmann, based on the concept of material forces. The J-integral is calculated numerically with a 3D finite element model for a linear elastic material and an elastic ideal-plastic material. A simple single specimen fracture toughness evaluation procedure is proposed. It is found that the effect of taper in the range encountered in practice is small, of the order of a few percent.     

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.     

Advantages of the J-integral approach for calculating stress intensity factors when using the commercial finite element software ABAQUS

A predictive method for remaining component lifetime evaluation consists in integrating the crack growth law of the material considered in a finite element step-by-step process. So, as part of a linear elastic fracture mechanics analysis, the determination of the stress intensity factor distribution is a crucial point. The aim of the present work is to test several existing numerical techniques reported in the literature. Both the crack opening displacement extrapolation method and the J-integral approach are applied in 2D and 3D ABAQUS finite element models. The results obtained by these various means on CT specimens and cracked round bars are in good agreement with those found in the literature. Nevertheless, since the knowledge of the field near the crack tip is not required in the energetic method, the J-integral calculations seem to be a good technique to deal with the fatigue growth of general cracks.     

Debonding analysis of fiber-reinforced-polymer strengthened beams: Cohesive zone modeling versus a linear elastic fracture mechanics approach

A linear elastic fracture mechanics (LEFM) approach and a cohesive interface (cohesive zone) modeling approach to the debonding analysis of concrete beams strengthened with externally bonded fiber-reinforced-polymer (FRP) strips are studied and compared. The analytical models that are based on the two approaches are presented and discussed. The cohesive interface model is formulated using a potential function and it takes into account the shear effects, the effect of the peeling stresses, and the coupling of the shear and the peeling effects. This model takes the form of a set on nonlinear differential equations. The LEFM model combines stress analysis using the high order theory and fracture analysis using the concepts of the energy release rate and the J-integral. In addition, an algorithm that converts the results of the LEFM model into the equilibrium path of the debonding process is developed. The main advantages and disadvantages of the two approaches are also discussed. The two approaches are compared in terms of their applicability to quantify and describe the debonding process in various cases that include a single shear test, an edge peeling test, and a beam specimen strengthened with FRP.     

Mechanisms and modeling of low cycle fatigue crack propagation in a pressure vessel steel Q345

The fatigue process near crack is governed by highly concentrated strain and stress in the crack tip region. Based on the theory of elastic–plastic fracture mechanics, we explore the cyclic J-integral as breakthrough point, an analytical model is presented in this paper to determine the CTOD for cracked component subjected to cyclic axial in-plane loading. A simple fracture mechanism based model for fatigue crack growth assumes a linear correlation between the cyclic crack tip opening displacement (ΔCTOD) and the crack growth rate (da/dN). In order to validate the model and to calibrate the model parameters, the low cycle fatigue crack propagation experiment was carried out for CT specimen made of Q345 steel. The effects of stress ratio and crack closure on fatigue crack growth were investigated by elastic–plastic finite element stress–strain analysis of a cracked component. A good comparison has been found between predictions and experimental results, which shows that the crack opening displacement is able to characterize the crack tip state at large scale yielding constant amplitude fatigue crack growth.     

Calculation of K-factor and T-stress for cracks in anisotropic bimaterials

The problem of a crack in a thin layer terminating perpendicular to a layer/substrate interface is analyzed for a general case of elastic anisotropy. The crack is modelled by means of continuous distribution of dislocations, which is assumed to be singular at the crack tip. A system of simultaneous functional equations is obtained that enables to find the singularity exponent λ. The reciprocal theorem (ψ-integral) is used to compute the generalized stress intensity factor (GSIF) through the remote stress and displacement field for a particular specimen geometry and boundary conditions using FEM. The results obtained are compared with the evaluation of GSIF based upon the dislocation arrays technique. Existing semi-analytical solution for singularities in anisotropic trimaterials is applied and its validity for the specimen investigated is checked by FEM. The evaluation of T-stress using the dislocation arrays technique is performed.     

Calculation of the energy J-integral for bodies with notches and cracks

The approximate solutions for calculation of the energy J-integral of a body both with a notch and with a crack under elastic-plastic loading have been obtained. The crack is considered as the limit case of a sharp notch. The method is based on stress concentration analysis near a notch/crack tip and the modified Neuber's approach. The HRR-model and the method based on an equation of equilibrium were also employed to calculate the J-integral. The influence of the strain hardening exponent on the J-integral is discussed. New aspects of the two-parameter J ^* _c-fracture criterion for a body with a short crack are studied. A theoretical investigation of the effect of the applied critical stress (or the crack length) on the strain fields ahead of the crack tip has been carried out.     

A mode I fracture analysis of a center-cracked infinite shape memory alloy plate under plane stress

The problem of a center plane crack in an infinite, thin, pseudoelastic Shape Memory Alloy (SMA) plate subjected to an in-plane uniform tensile stress at infinity is analyzed. The analysis follows closely the Dugdale?CBarenblatt model developed for conventional metals. It is found for low remote stress values??less than a critical value??that the SMA is not fully transformed in the vicinity of a crack tip. Closed form expressions for the size of the partial transformation zone, crack opening displacement and J-integral are given for this case. For remote stress levels above the critical value, the fully-transformed material near a crack tip is assumed to yield plastically. The sizes of the transformed (both partially and fully) and plastic regions are numerically evaluated by solving a system of integral equations and their sensitivity to the transformation characteristics (i.e., maximum transformation strain and temperature) is determined. Moreover, a relationship between the J-integral and the crack-tip opening displacement is derived. The results obtained are important in understanding the effect of stress-induced phase transformation in the fracture behavior of SMAs in the presence of static cracks, and subsequently in formulating conditions for initiation of crack propagation.     

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.     

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.     

Determination of mixed mode cohesive laws

A novel approach is proposed for the determination of mixed mode cohesive laws for large scale crack bridging problems. The approach is based on a plane, two-dimensional analysis utilizing the J integral applied a double cantilever beam specimens loaded with uneven bending moments. The normal and shear stresses of the cohesive laws are obtained from consecutive values of the fracture resistance, the normal and tangential displacements of the end of the cohesive zone. The data analysis involves fitting and determination of partial differentials. This is done by a numerical method using Chebyshev polynomials. The accuracy of the numerical procedure is investigated by the use of synthetic data. It is found that both the shape and peak stress of the cohesive law can be determined with high accuracy, providing that the data possess low noise and a sufficiently high number of datasets are used. The investigation leads to some practical guidelines for experimental use of the proposed approach.     

Two cracks with coalesced interior plastic zones — The generalised Dugdale model approach

The problem investigated is of an elastic-perfectly plastic infinite plate containing two equal collinear and symmetrically situated straight cracks. The plate is subjected to loads at infinity inducing mode I type deformations at the rims of the cracks. Consequently, plastic zones are formed ahead of the tips of the cracks. The loads at infinity are increased to a limit such that the plastic zones formed at the neighbouring interior tips of the cracks get coalesced. The plastic zones developed at the tips of the cracks are closed by applying normal cohesive quadratically varying stress distribution over their rims. The opening of the cracks is consequently arrested. Complex variable technique is used in conjugation with Dugdale’s hypothesis to obtain analytical solutions. Closed form analytical expressions are derived for calculating plastic zone size and crack opening displacement. An illustrative numerical example is discussed to study the qualitative behaviour of the loads required to arrest the cracks from opening with respect to parameters viz. crack length, plastic zone length and inter-crack distance. Crack opening displacement at the tip of the crack is also studied against these parameters.     

Surface effects on mode-I crack tip fields: A numerical study

Surface energy often significantly influences the deformation and failure behavior of materials and devices at the nanoscale. However, how it alters the local deformation around a crack tip remains unclear. In the present paper, we investigate the surface effects on the near-tip fields of a mode-I blunt crack (or notch). The theory of surface elasticity is incorporated into the finite element method. It is found that when the curvature radius of the crack root shrinks to nanometers, surface effects considerably affect the local stress distributions near the crack tip. We also calculate the J-integral, which is almost independent of surface effects except when the integral path approaches the crack tip. This demonstrates that surface effects are localized in a small zone around the crack tip, where the classical fracture mechanics solutions neglecting surface effects should be modified.     

Creep-fatigue crack propagation in lead-free solder under cyclic loading with various waveforms

Crack propagation tests of lead-free solder were conducted using center-notched plate specimens under cyclic tension-compression of three load waveforms: pp waveform having fast loading and unloading, cp-h waveform having a hold time under tension, and cc-h waveform having a hold time under tension and compression. In the case of fatigue loading, i.e. pp waveform, the path of crack propagation was macroscopically straight and perpendicular to the maximum principal stress direction, showing tensile-mode crack propagation. The introduction of the creep components by hold time in cc-h and cp-h waveforms promoted shear-mode crack propagation. For fatigue loading of pp wave, the crack propagation rate was expressed as a power function of the fatigue J integral and the relation was identical for load-controlled and displacement-controlled conditions. The creep component due to the hold time greatly accelerates the crack propagation rate when compared at the same values of the fatigue J integral or the total J integral (the sum of fatigue J and creep J integrals). The creep crack propagation rate was expressed as a power function of the creep J integral for each case of cp-h and cc-h waveforms. The crack propagation rate for cp-h waveform is higher than that for cc-h waveform. The predominant feature of fracture surfaces was striations for pp waveform and grain boundary fracture for cp-h waveform. Grain fragmentation was abundantly observed on the fracture surface made under cc-h waveform.     

Cohesive-bridging zone model of FRP-concrete interface debonding

External bonding of FRP plates or sheets has emerged as a popular method for strengthening reinforced concrete structures. Debonding along the FPR-concrete interface can lead to premature failure of the structures. In this study, a combined cohesive/bridging zone model is presented to simulate the debonding procedure between the FRP and concrete interface. In this model, the crack processing zone of the interface is modeled by a cohesive zone model and the particle interlocking zone of the interface is modeled by a bridging zone model. Two different linearly softening bond stress-slip laws are used to describe these two different zones. Closed-form solutions of interfacial stress, FRP stress and ultimate load are obtained for a typical single-lap specimen and verified with experimental results. The pulling force applied to the FRP plate is found to be proportional to the square root of the energy release rate at the debonding tip for this model. Such a relationship is then extended to any general shapes of bond stress-slip law through J-integral method. A new approach to experimentally determine the bond stress-slip law is also proposed.