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.     

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.     

The critical length of the hydride cluster in delayed hydride cracking of Zr-2.5wt% Nb

In delayed hydride cracking (DHC) of Zr-2.5wt% Nb alloy, the hydride cluster at the crack tip has a critical length, which is a function of the stress intensity factor K _I and other parameters. When K _I > K _IH, the threshold stress intensity factor, the hydride cluster must grow to this critical length before it will fracture. On the other hand, when K _I < K_IH, there is a maximum length to which the hydride cluster can grow, and this length is insufficient for fracture i.e. less than the critical length. In this work, the lengths of the hydride cluster were experimentally studied for K _I < K _IH and K _I > K _IH near K _IH. A modified experimental method was used, that permitted the hydride clusters to be formed and fractured individually. The hydride clusters were observed to be wedge-shaped, in agreement with the predictions by Metzger and Sauvé (PVP, vol. 326, ASME, 1996). The lengths of hydride cluster measured in this work are compared with existing theoretical predictions. A good general agreement was obtained, but some differences are discussed.     

A Thermo-mechanical cohesive zone model

In this paper, a cohesive zone formulation that is suitable for the thermo-mechanical analysis of heterogeneous solids and structural systems with contacting/interacting components, is presented. Well established traction-opening relations are adopted and combined with micromechanically motivated heat flux-opening relations reflecting the evolving heat transfer through the interfaces. The finite element approach for a coupled analysis within an operator-split solution framework is presented and demonstrated with an example problem.     

Triaxiality dependent cohesive zone model

A versatile cohesive zone model to predict ductile fracture at different states of stresses is proposed. The formulation developed for mode-I plane strain accounts for triaxiality of the stress-state explicitly by using basic elastic-plastic constitutive relations combined with two stress-state independent new model parameters. Comparison with available predictions of cohesive models based on porous plasticity damage models is used to demonstrate the efficacy of the proposed model. The effect of triaxiality on conventional cohesive parameters is well predicted as peak stresses are shown to increase while the cohesive energy decreases with triaxiality.     

A micromechanical model for a viscoelastic cohesive zone

A micromechanical model for a viscoelastic cohesive zone is formulated herein. Care has been taken in the construction of a physically-based continuum mechanics model of the damaged region ahead of the crack tip. The homogenization of the cohesive forces encountered in this region results in a damage dependent traction-displacement law which is both single integral and internal variable-type. An incrementalized form of this traction-displacement law has been integrated numerically and placed within an implicit finite element program designed to predict crack propagation in viscoelastic media. This research concludes with several example problems on the response of this model for various displacement boundary conditions.     

The prediction of dynamic fracture evolution in PMMA using a cohesive zone model

A cohesive zone model was used in conjunction with the finite volume method to model the dynamic fracture of single edge notched tensile specimens of PMMA under essentially static loading conditions. In this study, the influence of the shape of the cohesive law was investigated, whilst keeping the cohesive strength and separation energy constant. Cohesive cells were adaptively inserted between adjacent continuum cells when the normal traction across that face exceeded the cohesive strength of the material. The cohesive constitutive law was therefore initially rigid, and the effective elasticity of the material was unaltered prior to insertion of the cohesive cells. Notch depths ranging from 2.0 to 0.1 mm were considered. The numerical predictions were compared with experimental observations for each notch depth and excellent qualitative and quantitative agreement was achieved in most cases. Following an initial period of rapid crack tip acceleration up to terminal velocities well below the Rayleigh wave speed, subsequent propagation took place at a constant rate under conditions of increasing energy flux to an expanding process region. In addition, attempted and successful branching was predicted for the shorter notches. It was found that the shape of the cohesive law had a significant influence on the dynamic fracture behaviour. In particular, the value of the initial slope of the softening function was found to be an important parameter. As the slope became steeper, the predicted terminal crack speed increased and the extent of the damage decreased.     

Formulation of a cohesive zone model for a Mode III crack

Cohesive zone models have been proven effective in modeling crack initiation and propagation phenomena. In this work, a possible form for a Mode III cohesive zone model is formulated from elastic stress and displacement fields around a crack with a cohesive zone ahead of the crack tip. A traction-separation relation for the model is derived as a direct consequence of the formulation, which establishes some intrinsic connections between properties of the cohesive zone and those of the bulk material. Interestingly, this model states that the von Mises effective stress in the cohesive zone is constant, which may be related to the bulk material’s yield stress and is consistent with the assumption made in conventional strip-yield elastic-plastic solutions.     

The cohesive zone model: advantages, limitations and challenges

This paper reviews the cohesive process zone model, a general model which can deal with the nonlinear zone ahead of the crack tip--due to plasticity or microcracking--present in many materials. Furthermore, the cohesive zone model is able to adequately predict the behaviour of uncracked structures, including those with blunt notches, and not only the response of bodies with cracks--a usual drawback of most fracture models. The cohesive zone model, originally applied to concrete and cementitious composites, can be used with success for other materials. More powerful computer programs and better knowledge of material properties may widen its potential field of application. In this paper, the cohesive zone model is shown to provide good predictions for concrete and for different notched samples of a glassy polymer (PMMA) and some steels. The paper is structured in two main sections: First, the cohesive model is reviewed and emphasis is on determination of the softening function, an essential ingredient of the cohesive model, by inverse analysis procedures. The second section is devoted to some examples of the predictive capability of the cohesive zone model when applied to different materials; concrete, PMMA and steel.     

Comment on ‘A combined SIF and temperature model of delayed hydride cracking in zirconium materials’, by A.A. Shmakov et al.

Given the experimental facts related to delayed hydride cracking in zirconium alloys and the DHC models proposed so far, critical comments on “A combined SIF and temperature model of delayed hydride cracking in zirconium materials” by A.A. Shmakov were made, demonstrating that the authors’ DHC model is irrational.     

Calibration of a probabilistic cohesive zone model for generating a fracture nomograph

In the recent years, Cohesive Zone Models (CZMs) have gained increasing popularity for modelling the fracture process [ 1 ] and also in other applications like composite de‐lamination [ 2 ] solder failures [ 3 ] in circuits, etc. This can be attributed to the ability of the CZM to adapt to the nonlinearities in the process it represents by adjusting the model parameters. These parameters that are selected to represent the material behaviour in the vicinity of the crack or a damage zone are non‐deterministic in nature resulting in random fracture strength estimates. Currently, there are no standardized tests for measuring the CZM parameters and their random scatter. Numerous researchers in the literature suggest values for the CZM parameters based on their experience from limited test data. Traditionally, fracture toughness is determined through coupon tests for any material system that is being analysed using Linear Elastic Fracture Mechanics to determine the fracture strength of a specimen. Since data for fracture toughness are available, this research is aimed at determining the probability density functions (PDFs) for the cohesive zone parameters that would give the same scatter in fracture strength as that obtained from the test statistics. Correlations between the model parameters were introduced to improve the accuracy of the fracture strength PDF. A finite width cracked plate was selected as a test case to demonstrate the process. This paper also presents evidence that material scatter can be isolated from the geometric effects to determine a normalized PDF of fracture strength for a given material. This normalized PDF can then be scaled, using mean fracture strength, to any crack configuration to develop a nomograph that can be used to rapidly assess risk without the need for a probabilistic fracture analysis.     

Simulation of fatigue crack growth with a cyclic cohesive zone model

Fatigue crack growth is simulated for an elastic solid with a cyclic cohesive zone model (CZM). Material degradation and thus separation follows from the current damage state, which represents the amount of maximum transferable traction across the cohesive zone. The traction–separation relation proposed in the cyclic CZM includes non-linear paths for both un- and reloading. This allows a smooth transition from reversible to damaged state. The exponential traction–separation envelope is controlled by two shape parameters. Moreover, a lower bound for damage evolution is introduced by a local damage dependent endurance limit, which enters the damage evolution equation. The cyclic CZM is applied to mode I fatigue crack growth under \(K_{\mathrm{I}}\) -controlled external loading conditions. The influences of the model parameters with respect to static failure load \(K_{\mathrm{0}}\) , threshold load \(\varDelta K_{\mathrm{th}}\) and Paris parameters \(m, C\) are investigated. The study reveals that the proposed endurance limit formulation is well suited to control the ratio \(\varDelta K_{\mathrm{th}}/K_{\mathrm{0}}\) independent of \(m\) and \(C\) . An identification procedure is suggested to identify the cohesive parameters with the help of Wöhler diagrams and fatigue crack growth rate curves.     

A computationally efficient cohesive zone model for fatigue

A cohesive zone model has been developed for the simulation of both high and low cycle fatigue crack growth. The developed model provides an alternative approach that reflects the computational efficiency of the well‐established envelop‐load damage model yet can deliver the accuracy of the equally well‐established loading‐unloading hysteresis damage model. A feature included in the new cohesive zone model is a damage mechanism that accumulates as a result of cyclic plastic separation and material deterioration to capture a finite fatigue life. The accumulation of damage is reflected in the loading‐unloading hysteresis curve, but additionally, the model incorporates a fast‐track feature. This is achieved by “freezing in” a particular damage state for one loading cycle over a predefined number of cycles. The new model is used to simulate mode I fatigue crack growth in austenitic stainless steel 304 at significant reduction in the computational cost.     

An analysis of crack growth in thin-sheet metal via a cohesive zone model

A cohesive zone model (CZM) is applied to crack growth in thin sheet metal. CZM parameters are determined from results of global measurements and micromechanical damage models. Crack propagation in constrained center-cracked panels is analyzed to verify the choice of CZM parameters. Special attention is paid to the interaction between buckling and crack growth and to crack link-up in multi-site damaged specimens. The good agreement found between the predicted and experimental data demonstrates that the approach is attractive in investigation of structural integrity of thin-walled structures and does not require assumptions regarding the geometry and size dependence of crack growth parameters.     

Modelling the quasi-static behaviour of bituminous material using a cohesive zone model

This paper investigates the applicability of a cohesive zone model for simulating the performance of bituminous material subjected to quasi-static loading. The Dugdale traction law was implemented within a finite volume code in order to simulate the binder course mortar material response when subjected to indirect tensile loading. A uniaxial tensile test and a three-point bend test were employed to determine initial stress-strain curves at different test rates and the cohesive zone parameters (specifically, fracture energy and cohesive strength). Numerical results agree well with the experimental data up to the peak load and onset of fracture, demonstrating the value of the cohesive zone modelling technique in successfully predicting fracture initiation and maximum material strength.