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 modified cohesive zone model for a high‐speed expanding crack

The present work studies a self‐similar high‐speed expanding crack of mode‐I in a ductile material with a modified cohesive zone model. Compared with existing Dugdale models for moving crack, the new features of the present model are that the normal stress parallel to crack faces is included in the yielding condition in the cohesive zone and traction force in the cohesive zone can be non‐uniform. For a ductile material defined by von Mises criterion without hardening, the present model confirms that the normal stress parallel to crack face increases with increasing crack speed and can be even larger than the normal traction in the cohesive zone, which justifies the necessity of including the normal stress parallel to the crack faces in the yielding condition at high crack speed. In addition, strain hardening effect is examined based on a non‐uniform traction distribution in the cohesive zone.     

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.     

Damage Dependent Constitutive Behavior and Energy Release Rate for a Cohesive Zone in a Thermoviscoelastic Solid

In this paper a cohesive zone is introduced ahead of a crack tip in order to avoid the singularity at the crack tip. By applying thermodynamics to the cohesive zone and the surrounding body, a fracture criterion will be established so that the inelastic energy dissipation both in the cohesive zone and the surrounding bulk material can be distinguished from the energy released by fracture, and the propagation of crack can be predicted. In addition, the cohesive zone constitutive equation is constructed utilizing the Helmholtz free energy in the form of a single hereditary integral for a nonlinear viscoelastic material. The resulting constitutive model for the cohesive zone contains an internal state variable which represents the damage state within the cohesive zone. When the cohesive zone opening displacement is known, the energy release rate is thus history dependent, which is expressed in terms of the damage state, the length of separation in the cohesive zone and the geometric configuration of the cohesive zone opening displacement. Example results contained herein demonstrate this effect. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

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 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.     

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.     

Suggestions to the cohesive traction–separation law from atomistic simulations

The cohesive model becomes popular in crack analysis for its clear physical background and flexible implementation. The cohesive traction–separation law, however, is a critical point and will generally be empirically assumed. In the present paper the cohesive traction–separation law is investigated based on constrained three-dimensional atomistic simulations. The computations under mode I conditions show that crack growth even in the nano-scale single-crystal aluminum is in the form of void nucleation, growth and coalescence, which is similar to ductile fracture at meso-scale. The concentrations of the atomic tensile stress and the atomic hydrostatic stress at a certain distance from the crack tip characterize void nucleation and final crack growth. The Mises stress does not play a role in the material failure in the nano-scale. This implies that ductile failure under mode I loading condition is dominated by the normal traction, which agrees with the assumption of the cohesive zone model. Variations of the atomic stresses near the crack tip provide the theoretical background for the cohesive zone model and can be used to identify the cohesive traction–separation law. The traction curve is very sensitive to the distance to the crack tip, which is related with the stress triaxiality. The atomistic simulations show tendentious agreement with the known cohesive traction–separation laws, whereas the scattering of the atomic stress versus separation implies effects of the hydrostatic stress in the traction–separation law. The computation provides important information for constructing the cohesive zone model.     

Comparison of cohesive zone model and linear elastic fracture mechanics for a mode I crack near a compliant/stiff interface

Cohesive zone model has been widely applied to simulate crack growth along interfaces, but its application to crack growth perpendicularly across the interface is rare. In this paper, the cohesive zone model is applied to a crack perpendicularly approaching a compliant/stiff interface in a layered material model. One aim is to understand the differences between the cohesive zone model and linear elastic fracture mechanics in simulating mode I crack growth near a compliant/stiff interface. Another aim is to understand the effects of elastic modulus mismatch and cohesive strength of the stiff layer on the crack behavior near the interface. To simulate crack growth approaching an interface, the cohesive zone model which incorporates both the energy criterion and the strength criterion is an effective method.     

The structure in the vicinity of a crack tip: A general theory based on the cohesive zone model

Consideration is given to the development of a general theory for the determination of the structure in the vicinity of the tip of a crack within an isotropic elastic solid. The basis of the model is the presence of a Barenblatt-type zone, in which cohesive stresses act, thereby allowing non-linear behaviour to he taken into account. The theory enables the stress and displacements within the cohesive zone, the zone size and critical crack extension stress to be determined, for a wide range of stress-displacement laws describing the non-linear behaviour within the cohesive zone, provided its size is small compared with that of the crack.     

Direct determination of cohesive stress transfer during debonding of FRP from concrete

Interface cohesive stress transfer between FRP and concrete during debonding is typically obtained using measured surface strains on the FRP, along the direction of the fibers. The cohesive material law is derived under a set of assumptions which include: (a) the bending stiffness of the FRP laminate is insignificant with respect to that of the concrete test block; (b) the strains in the bulk concrete produced by debonding are negligible, thus concrete substrate can be considered rigid; (c) there is stress transfer between FRP and concrete through the FRP–concrete interface which is of zero thickness; and (d) the axial strain in the FRP composite is uniform across its thickness. In this paper, a test procedure for directly obtaining the through-thickness strains in the FRP and the concrete substrate during cohesive stress transfer associated with debonding is presented. The displacement and strain fields are measured on the side of a direct-shear specimen with the FRP strip attached on the edge. Based on the experimental results, the influence of the assumptions which have been introduced to determine the cohesive law is discussed. Within the stress transfer zone there is a sharp gradient in the shear strain. The location of the interface crack within the stress transfer zone and the cohesive stress transfer during the propagation of the interface crack are determined.     

Crack Growth Across a Strength Mismatched Bimaterial Interface

Crack growth across an interface between materials with different strength is examined by a cohesive zone model. The two materials have identical elastic properties but different fracture process properties, or different yield stresses, which is modeled by different cohesive stresses. The fracture criteria is a critical crack opening displacement. Load is represented by a stress intensity factor defining a remote square root singular stress field. The results show that the ratio between the cohesive stresses of the two materials primarily determines the behavior of the critical stress intensity factor. When the crack approaches a material with a higher cohesive stress the crack tip is shielded, but if the crack approaches a material with smaller critical crack opening displacement the maximum level of shielding is determined by the ratio between the critical crack opening displacements. When a crack approaches a material with a lower cohesive stress it is exposed to an amplified load. This revised version was published online in August 2006 with corrections to the Cover Date.     

J resistance behavior in functionally graded materials using cohesive zone and modified boundary layer models

This paper describes elastic–plastic crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) under mode I loading conditions using cohesive zone and modified boundary layer (MBL) models. For this purpose, we first explore the applicability of two existing, phenomenological cohesive zone models for FGMs. Based on these investigations, we propose a new cohesive zone model. Then, we perform crack growth simulations for TiB/Ti FGM SE(B) and SE(T) specimens using the three cohesive zone models mentioned above. The crack growth resistance of the FGM is characterized by the J-integral. These results show that the two existing cohesive zone models overestimate the actual J value, whereas the model proposed in the present study closely captures the actual fracture and crack growth behaviors of the FGM. Finally, the cohesive zone models are employed in conjunction with the MBL model. The two existing cohesive zone models fail to produce the desired K–T stress field for the MBL model. On the other hand, the proposed cohesive zone model yields the desired K–T stress field for the MBL model, and thus yields J _R curves that match the ones obtained from the SE(B) and SE(T) specimens. These results verify the application of the MBL model to simulate crack growth resistance in FGMs.     

Modelling of crack propagation of gravity dams by scaled boundary polygons and cohesive crack model

Crack propagation in concrete gravity dams is investigated using scaled boundary polygons coupled with interface elements. The concrete bulk is assumed to be linear elastic and is modelled by the scaled boundary polygons. The interface elements model the fracture process zone between the crack faces. The cohesive tractions are modelled as side-face tractions in the scaled boundary polygons. The solution of the stress field around the crack tip is expressed semi-analytically as a power series. It reproduces the singular and higher-order terms in an asymptotic solution, such as the William’s eigenfunction expansion when the cohesive tractions vanish. Accurate results can be obtained without asymptotic enrichment or local mesh refinement. The stress intensity factors are obtained directly from their definition and provide a convenient and accurate means to assess the zero-K condition, which determines the stability of a cohesive crack. The direction of crack propagation is determined from the maximum circumferential stress criterion. To accommodate crack propagation, a local remeshing algorithm that is applicable to any polygon mesh is augmented by inserting cohesive interface elements between the crack surfaces as the cracks propagate. Three numerical benchmarks involving crack propagation in concrete gravity dams are modelled. The results are compared to the experimental and other numerical simulations reported in the literature.     

A semi-analytical solution method for problems of cohesive fracture and some of its applications

Cohesive zone models are extensively used for the failure load estimates for structure elements with cracks. This paper focuses on some features of the models associated with the failure load and size of the cohesive zone predictions. For simplicity, considered is a mode I crack in an infinite plane under symmetrical tensile stresses. A traction–separation law is prescribed in the crack process zone. It is assumed by the problem statement that the crack faces close smoothly. This requirement is satisfied numerically by a formulation of the modified boundary conditions. The critical state of a plate with a cohesive crack is analyzed using singular integral equations. A numerical procedure is proposed to solve the obtained systems of integral equations and inequalities. The presented solution is in agreement with other published results for some limiting cases. Thus, an effective methodology is devised to solve crack mechanics problems within the framework of a cohesive zone model. Using this methodology, some problems are solved to illustrate the (i) influence of shape parameters of traction–separation law on the failure load, (ii) ability to account for contact stress for contacting crack faces, (iii) influence of getting rid of stress finiteness condition in the problem statement.     

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.     

The effects of inter-surface cohesive tractions on linear and penny-shaped cracks

A mesoscopic fracture model of equilibrium slit cracks in brittle solids, including inter-surface cohesive tractions acting near the crack tip, is analyzed and the effects of the cohesive tractions on the in-plane stress fields, crack-opening displacement profiles, and crack driving forces examined quantitatively for linear and penny-shaped cracks. The (numerical) analysis method is described in detail, along with results for four different cohesive forces. The assumed distribution of cohesive tractions were found to suppress the in-plane stress field adjacent to cracks in a homogeneous, isotropic medium when uniformly loaded in mode-I, and the suppression was a function of crack length. The crack-opening displacement profile was also perturbed and a new regime identified between the near-field Barenblatt zone and the far-field continuum zone. The extent of this `cohesive zone' was quantified by use of an interpolating function fit to the calculated profiles and found to be independent of crack size for a given cohesive tractions distribution. Furthermore, the crack-opening displacement at the edge of the cohesive zone was found to be independent of crack size, implying that despite significant perturbations to the stress field, the crack driving force at unstable equilibrium remains unchanged with crack size.     

Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials

This work investigates elastic-plastic crack growth in ceramic/metal functionally graded materials (FGMs). The study employs a phenomenological, cohesive zone model proposed by the authors and simulates crack growth by the gradual degradation of cohesive surfaces ahead of the crack front. The cohesive zone model uses 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) to describe the constitutive response of the material in the cohesive zone. A volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model) describes the elastic-plastic response of the bulk background material. The numerical 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. Numerical values of volume fractions for the constituents specified at nodes of the finite element model set the spatial gradation of material properties with isoparametric interpolations inside interface elements and background solid elements to define pointwise material property values. The paper describes applications of the cohesive zone model and the computational scheme to analyze crack growth in a single-edge notch bend, SE(B), specimen made of a TiB/Ti FGM. Cohesive parameters are calibrated using the experimentally measured load versus average crack extension (across the thickness) responses of both Ti metal and TiB/Ti FGM SE(B) specimens. The numerical results show that with the calibrated cohesive gradation parameters for the TiB/Ti system, the load to cause crack extension in the FGM is much smaller than that for the metal. However, the crack initiation load for the TiB/Ti FGM with reduced cohesive gradation parameters (which may be achieved under different manufacturing conditions) could compare to that for the metal. Crack growth responses vary strongly with values of the exponent describing the volume fraction profile for the metal. The investigation also shows significant crack tunneling in the Ti metal SE(B) specimen. For the TiB/Ti FGM system, however, crack tunneling is pronounced only for a metal-rich specimen with relatively smaller cohesive gradation parameter for the metal.     

Application of cohesive zone model in crack propagation analysis in multiphase composite materials

A nonlinear cohesive stress distribution function is employed by relating the cohesive stress to the cohesive zone size (CZS) and the distance from the crack tip to investigate the elastic-plastic fracture behaviors. A crack-inclusion interaction problem is taken as an example to explore the fracture process in the cohesive zone area. The CZS and crack surface opening displacement are evaluated numerically. It is found that for different cohesive parameter combinations, the normalized CZS and crack surface opening displacements change drastically. By reducing the current model to the famous Dugdale model, the results obtained match well with the existing ones.