The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

A computational model and experimental validation of shielding and amplifying effects at a crack tip near perpendicular strength-mismatched interfaces

The stress field around the crack tip near an elastically matched but strength-mismatched interface body in a bimetallic system is influenced when the crack tip yield or cohesive zone spreads to the interface body. The concept of crack tip stress intensity parameter, K _tip, is therefore employed in fracture analysis of the bimetallic body. A computational model to determine K _tip is reviewed in this paper. The model, based upon i) Westergaard??s complex potentials coupled with Kolosov?CMuskhelishvili??s relations between a crack tip stress field and complex potentials and ii) Dugdale??s representation of the cohesive zone clearly indicates shielding or amplifying effects of strength mismatch across the interface, depending upon the direction of the strength gradient, over the crack tip. The model is successfully validated by conducting series of high cycle fatigue tests over Mode I cracks advancing towards various strength-mismatched interfaces in bimetallic compact tension specimens prepared by electron beam welding of elastically identical weak ASTM 4340 alloy and strong MDN 250 maraging steels.     

Effect of residual stress on cleavage fracture toughness by using cohesive zone model

This study presents the effect of residual stresses on cleavage fracture toughness by using the cohesive zone model under mode I, plane stain conditions. Modified boundary layer simulations were performed with the remote boundary conditions governed by the elastic K‐field and T‐stress. The eigenstrain method was used to introduce residual stresses into the finite element model. A layer of cohesive elements was deployed ahead of the crack tip to simulate the fracture process zone. A bilinear traction–separation‐law was used to characterize the behaviour of the cohesive elements. It was assumed that the initiation of the crack occurs when the opening stress drops to zero at the first integration point of the first cohesive element ahead of the crack tip. Results show that tensile residual stresses can decrease the cleavage fracture toughness significantly. The effect of the weld zone size on cleavage fracture toughness was also investigated, and it has been found that the initiation toughness is the linear function of the size of the geometrically similar weld. Results also show that the effect of the residual stress is stronger for negative T‐stress while its effect is relatively smaller for positive T‐stress. The influence of damage parameters and material hardening was also studied.     

Near threshold delayed hydride crack growth in zirconium alloys

Delayed hydride cracking (DHC) in zirconium alloys arises as a consequence of the diffusion of hydrogen atoms to a crack tip, precipitation of hydride platelets and then the fracture of a hydrided region that has formed ahead of the crack tip. This process repeats itself and, consequently, a crack grows in a series of steps. There is a threshold value,K _IH, of the crack tip stress intensity below which DHC crack growth is unable to proceed. The present paper provides a physical picture of the near threshold situation, accounting systematically for the manner in which hydrided material fractures, and consequently obtains an expression forK _IH in terms of the hydrided material's flow and fracture characteristics.     

Three-dimensional modeling of ductile crack growth: Cohesive zone parameters and crack tip triaxiality

For 10 mm thick smooth-sided compact tension specimens made of a pressure vessel steel 20MnMoNi55, the interrelations between the cohesive zone parameters (the cohesive strength, T_max, and the separation energy, Γ) and the crack tip triaxiality are investigated. The slant shear-lip fracture near the side-surfaces is modeled as a normal fracture along the symmetry plane of the specimen. The cohesive zone parameters are determined by fitting the simulated crack extensions to the experimental data of a multi-specimen test. It is found that for constant cohesive zone parameters, the simulated crack extension curves show a strong tunneling effect. For a good fit between simulated and experimental crack growth, both the cohesive strength and the separation energy near the side-surface should be considerably lower than near the midsection. When the same cohesive zone parameters are applied to the 3D model and a plane strain model, the stress triaxiality in the midsection of the 3D model is much lower, the von-Mises equivalent stress is distinctly higher, and the crack growth rate is significantly lower than in the plane strain model. Therefore, the specimen must be considered as a thin specimen. The stress triaxiality varies dramatically during the initial stages of crack growth, but varies only smoothly during the subsequent stable crack growth. In the midsection region, the decrease of the cohesive strength results in a decrease of the stress triaxiality, while the decrease of the separation energy results in an increase of the triaxiality.     

A new nonlocal cohesive stress law and its applicable range

In this paper, we attempt to provide a new analytical method to determine the cohesive law in the framework of nonlocal continuum mechanics. Firstly, the equivalence between the cohesive stress and the surface-induced traction (nonlocal surface residual) is established on the basis of the nonlocal stress boundary condition. Then a new cohesive stress law is derived logically from the perspective of rational mechanics, which characterizes the dependence of the cohesive stress on the crack opening displacement (COD) within the cohesive zone. Finally, we apply this new cohesive crack model to two fracture examples with different cohesive zone sizes, and investigate the stress field ahead of the crack tip in detail. The results show that the stress singularity at the crack tip is removed, and the maximum stress occurs within the cohesive zone away from the crack tip. Moreover, the stress in the large-scale cohesive zone drops rapidly to a constant approaching zero, exhibiting a stronger softening behavior.     

Evaluation of Fracture Parameters by Double-G,Double-K Models and Crack Extension Resistance for High Strength and Ultra High Strength Concrete Beams

This paper presents the advanced analytical methodologies such as Double- G and Double - K models for fracture analysis of concrete specimens made up of high strength concrete (HSC, HSC1) and ultra high strength concrete. Brief details about characterization and experimentation of HSC, HSC1 and UHSC have been provided. Double-G model is based on energy concept and couples the Griffith's brittle fracture theory with the bridging softening property of concrete. The double-K fracture model is based on stress intensity factor approach. Various fracture parameters such as cohesive fracture toughness (K_Ic^c), unstable fracture toughness (K_Ic^un) and initiation fracture toughness (K_Icⁱⁿⁱ) have been evaluated based on linear elastic fracture mechanics and nonlinear fracture mechanics principles. Double-G and double-K method uses the secant compliance at the peak point of measured P-CMOD curves for determining the effective crack length. Bi-linear tension softening model has been employed to account for cohesive stresses ahead of the crack tip. From the studies, it is observed that the fracture parameters obtained by using double - G and double - K models are in good agreement with each other. Crack extension resistance has been estimated by using the fracture parameters obtained through double - K model. It is observed that the values of the crack extension resistance at the critical unstable point are almost equal to the values of the unstable fracture toughness K_Ic^un of the materials. The computed fracture parameters will be useful for crack growth study, remaining life and residual strength evaluation of concrete structural components.     

Fatigue and fracture of a 316 stainless steel metal matrix composite reinforced with 25% titanium diboride

Fatigue and fracture mechanisms have been studied in a steel-based metal matrix composite (MMC), comprising a 316L austenitic matrix reinforced with 25 wt.% particulate titanium diboride (TiB₂). The fracture toughness was determined in the as-HIPped condition as being slightly below 30 MPa√m. Fatigue crack growth rates have been determined, and corrected for the effects of crack closure. The fracture surfaces have been studied to determine the mechanisms of damage during crack advance, which are determined as matrix fatigue, reinforcement particle fracture, and ductile rupture of the matrix. We show that the occurrence of damage mechanisms during fatigue of the material is linked to K_max, rather than to ΔK. This is rationalised in terms of a semi-cohesive process zone within the monotonic plastic zone ahead of the crack tip.     

Measuring overload effects during fatigue crack growth in bainitic steel by synchrotron X-ray diffraction

In this work we present the results of in situ synchrotron X-ray diffraction measurements of fatigue crack-tip strain fields following a 100% overload (OL) under plane strain conditions. The study is made on a bainitic steel with a high toughness and fine microstructure. This allowed a very high (60 μm) spatial resolution to be achieved so that fine-scale changes occurring around the crack-tip were captured along the crack plane at the mid-thickness of the specimen. We have followed the crack as it grew through the plastic/residually stressed zone associated with the OL crack location. We observed two effects; one when the enhanced plastic zone is ahead of the crack and one after it has been passed. Regarding the former it was found that the compressive stress at the crack-tip initially falls sharply, presumably due to the increased plastic stretch caused by the OL. This is associated with a concomitant fall in peak tensile stress at K_max, the elastic excursion between K_min and K_max remaining essentially unchanged from before OL. Subsequently discontinuous closure as seen previously for plane stress caused by crack face contact at the OL location limits the elastic strain range experienced by the crack tip and thereby retards crack growth.     

Micro-mechanical analysis and TEM study of crack initiation in dislocation free zone

Dislocation emission, dislocation-free zone (DFZ) formation and crack initiation in the DFZ and/or at the crack tip were analyzed by micromechanics. The results show that a DFZ is formed after dislocation emission under constant load. The DFZ size decreases with increase in the applied stress intensity factor K _Ia or lattice friction stress _f. There are two stress peaks ahead of the crack tip. The first one is located at the blunt crack tip and the second one in the DFZ. With increasing in the applied stress intensity factor K _Ia, the peak stress at the crack tip may decrease while the peak stress in the DFZ increases monotonically. Microcrack will initiate when the peak stress is equal to the cohesive strength. In situ tensile tests in a transmission electron microscope (TEM) show that microcrack initiates in DFZ or/and at a blunt crack tip after dislocation emitting and DFZ formation.     

On the Cohesive Zone Model for a Finite Crack

The paper concentrates on the development of the crack tip model with the cohesive zone in an infinite plate with a finite crack of mode I. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration according to Westergaard's theory with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. The calculation results of the crack tip opening displacement are compared with the Dugdale solution for the plane stress. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical investigation on stable crack growth in plane stress

Large deformation finite element analysis has been carried out to investigate the stress-strain fields ahead of a growing crack for compact tension (a/W=0.5) and three-point bend (a/W=0.1 and 0.5) specimens under plane stress condition. The crack growth is controlled by the experimental J-integral resistance curves measured by Sun et al. The results indicate that the distributions of opening stress, equivalent stress and equivalent strain ahead of a growing crack are not sensitive to specimen geometry. For both stationary and growing cracks, similar distributions of opening stress and triaxiality can be found along the ligament. During stable crack growth, the crack- tip opening displacement (CTOD) resistance curve and the cohesive fracture energy in the fracture process zone are independent of specimen geometry and may be suitable criteria for characterizing stable crack growth in plane stress. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Weight function approach for determining crack extension resistance based on the cohesive stress distribution in concrete

The use of universal form of weight functions for determining the K_R-curves associated with the cohesive stress distribution for complete fracture process of three-point bending notched concrete beam is reported in the paper. Closed form expressions for the cohesion toughness with linear and bilinear distribution of cohesive stress in the fictitious fracture zone during monotonic loading of structures are obtained. Comparison with existing analytical method shows that the weight function method yields results without any appreciable error with improved computational efficiency. The stability analysis and the size-effect study using K_R-curves of crack propagation are also described.     

Stress wave emission and plastic work of notched specimens

The stress wave energy released from notched specimens of structural steel was measured in order to compare it with the recently proposedJ-integral which takes account of the effect of large plastic deformation around the crack tip in ductile materials. Very close agreement was observed between theJ-integral and the differential stress wave energy released. This suggests that the increment of the stress wave energy released is proportional to the decrement of the work done on the specimen during tensile testing under the plane stress condition. This result, combined with information obtained from linear elastic fracture mechanics, leads to a relationship between the differential stress wave energy released and the stress intensity factorK, [Δ(SWER)/Δa] ∝K ². It was also found that in the region before general yielding, the stress wave energy release was proportional to the development of plastic zone size. A larger portion of the accumulated stress wave energy released was generated after general yielding due to void formation and coalescence. The accumulated stress wave energy released at the catastrophic crack growth point reached virtually the same value for each specimen, independent of the initial crack length. This implies that void formation and coalescence are not influenced by the initial crack length, but by the geometry of the crack tip.     

A model for the static fracture toughness of ductile structural steel

Fracture of ductile structural steels generally occurs after void initiation, void growth and void coalescence. In order for ductile fracture of structural steels to occur, energy must be spent to induce void initiation and void growth. Therefore, fracture toughness for ductile fracture should be contributed from void initiation and void growth. On the basis of this suggestion static fracture toughness (K_IC) of ductile structural steels is decomposed into two parts: void nucleation-induced fracture toughness (denoted as K_IC.n) and void growth-induced fracture toughness (K_IC.g). K_IC.n, defined as the stress intensity factor at which voids ahead of a crack begins to form, is calculated from crack tip strain distribution and void nucleation strain distribution. In contrast, K_IC.g is determined by the void growth from the beginning of void nucleation to void coalescence. Therefore, K_IC.g relates to the void sizes and void distribution. In this paper, the expression for K_IC.g is given from the void sizes directly from fracture surfaces. The relationship between K_IC.n, K_IC.g and K_IC is expressed in the form (K_IC)²=(K_IC.n)²+(K_IC.g)². The newly developed model was applied to the fracture toughness evaluation of three structural steels (SN490, X65 and SA440), and the theoretical calculation agrees with the experimental results.     

A model for the static fracture toughness of ductile structural steel

Fracture of ductile structural steels generally occurs after void initiation, void growth and void coalescence. In order for ductile fracture of structural steels to occur, energy must be spent to induce void initiation and void growth. Therefore, fracture toughness for ductile fracture should be contributed from void initiation and void growth. On the basis of this suggestion static fracture toughness (K_IC) of ductile structural steels is decomposed into two parts: void nucleation-induced fracture toughness (denoted as K_IC.n) and void growth-induced fracture toughness (K_IC.g). K_IC.n, defined as the stress intensity factor at which voids ahead of a crack begins to form, is calculated from crack tip strain distribution and void nucleation strain distribution. In contrast, K_IC.g is determined by the void growth from the beginning of void nucleation to void coalescence. Therefore, K_IC.g relates to the void sizes and void distribution. In this paper, the expression for K_IC.g is given from the void sizes directly from fracture surfaces. The relationship between K_IC.n, K_IC.g and K_IC is expressed in the form (K_IC)²=(K_IC.n)²+(K_IC.g)². The newly developed model was applied to the fracture toughness evaluation of three structural steels (SN490, X65 and SA440), and the theoretical calculation agrees with the experimental results.     

Study on cleavage fracture criteria of the quasi-brittle and micro-inhomogeneous materials

On the bases of recent achievements on the micro-mechanism of cleavage this paper analyses the inherent deficiencies of the stress intensity factor K _I which is used to evaluate the fracture toughness of quasi-brittle and micro-inhomogeneous materials. The K _I parameter can uniquely determine the field intensity ahead of a crack tip in the condition of elastic and small scale yielding (SSY). However, the K _I cannot uniquely determine the critical condition triggering the cleavage fracture in a quasi-brittle and inhomogeneous steel where the cleavage fracture process is not a direct extension of the precrack but is initiated at a variable distance from the precrack tip. The variable distances of cleavage initiations invoke varied critical values of K _I. On the bases of authors' experiments, this paper analyses the physical meaning of the local fracture stress _f, its stability and the feasibility to be used as an engineering parameter for assessing the fracture toughness.     

Crack tip shielding by asperity contact as determined by acoustic measurements

Asperity contact along the fracture surface of a crack is one of the mechanisms of crack closure. This contact shields the crack tip, in part, from the externally applied driving force. We have now succeeded in using information from acoustic transmission and diffraction experiments, obtained under plane strain conditions, to determine the size and density of the contacting asperities in the closure region. We have also succeeded in estimating values for the static stress across a partially closed crack as well as the stress intensity factor,K_I (local), which shields the crack tip below the stress intensity factor K_Iclosure at which the first contact during unloading occurs. It is suggested that when crack closure has an important influence on crack propagation, the shielding stress intensity factor provides information that can be used to estimate the fatigue crack propagation rate.     

Modeling of crack tip high inertia zone in dynamic brittle fracture

A phenomenological modification is proposed to the existing cohesive constitutive law of Roy and Dodds to model the crack tip high inertia region proposed by Gao. The modification involves addition of a term which is attributed to fracture mechanisms that result in high energy dissipation around the crack tip. This term is assumed to be a function of external energy per unit volume input into the system. Finite element analysis is performed on PMMA with constant velocity boundary conditions and mesh discretizations based on the work of Xu and Needleman. The cohesive model with the proposed dissipative term is only applied in the high inertia zone and the traditional Roy and Dodds model is applied on cohesive elements in the rest of the domain. The results show that crack propagates in three phases with a speed of 0.35c_R before branching, confirming experimental observations. The modeling of high inertia zone is one of the key aspects to understanding brittle fracture.