Cohesive crack analysis of size effect

This paper deals with the analysis of size effect in concrete. An extensive campaign of accurate numerical simulations, based on the cohesive crack model, is performed to compute the size effect curves (CSEC) for typical test configurations. The results are analyzed with reference to the classical Ba?ant’s size effect law (SEL) to investigate the relationship between CSEC and SEL. This analysis shows that as specimen size tends to infinity, the SEL represents the asymptote of the CSEC, and that the SEL parameter known as the effective fracture process zone length is a material property which can be expressed as a function of the cohesive crack law (CCL) parameters. Finally, the practical implications of this study are discussed in relation to the use of the CSEC or the SEL for the identification of the CCL parameters through the size effect method.     

A scale-invariant cohesive crack model for quasi-brittle materials

The fictitious crack model by Hillerborg is the most widely used model to simulate damage and fracture in concrete structures. Its peculiar capability to capture the evolution of the cracking process is accompanied by its simplicity. However, some aspects of the phenomenon are not considered in the model, for instance the size-dependence of the nominal quantities involved in the cohesive law. This affects the predictive capabilities of the model, when it is used to extrapolate results from small laboratory specimens to full-scale structures.In this paper, a scale-independent cohesive law is put forward, which overcomes these drawbacks and permits to obtain a unique constitutive relationship for softening in concrete. By assuming damage occurring in a fractal band inside the specimen, nominal stress, crack opening displacement and nominal fracture energy become scale dependent. Hence they should be substituted by fractal quantities, which are the true material constants. A mutual relation among their fractal physical dimensions puts a strong restriction to disorder. By varying the scaling exponents of the kinematical quantities, a clear transition from discrete to smeared cracking can be obtained. The fractal cohesive law is eventually applied to some tensile test data, showing perfect agreement between theory and experiments.     

An embedded cohesive crack model for finite element analysis of brickwork masonry fracture

This paper presents a numerical procedure for fracture of brickwork masonry based on the strong discontinuity approach. The model is an extension of the cohesive model prepared by the authors for concrete, and takes into account the anisotropy of the material. A simple central-force model is used for the stress versus crack opening curve. The additional degrees of freedom defining the crack opening are determined at the crack level, thus avoiding the need of performing a static condensation at the element level. The need for a tracking algorithm is avoided by using a consistent procedure for the selection of the separated nodes. Such a model is then implemented into a commercial code by means of a user subroutine, consequently being contrasted with experimental results. Fracture properties of masonry are independently measured for two directions on the composed masonry, and then input in the numerical model. This numerical procedure accurately predicts the experimental mixed-mode fracture records for different orientations of the brick layers on masonry panels.     

A cohesive model of fatigue crack growth

We investigate the use of cohesive theories of fracture, in conjunction with the explicit resolution of the near-tip plastic fields and the enforcement of closure as a contact constraint, for the purpose of fatigue-life prediction. An important characteristic of the cohesive laws considered here is that they exhibit unloading-reloading hysteresis. This feature has the important consequence of preventing shakedown and allowing for steady crack growth. Our calculations demonstrate that the theory is capable of a unified treatment of long cracks under constant-amplitude loading, short cracks and the effect of overloads, without ad hoc corrections or tuning.     

Fatigue crack propagation model and size effect in concrete using dimensional analysis

It is well known that fatigue in concrete causes excessive deformations and cracking leading to structural failures. Due to quasi-brittle nature of concrete and formation of a fracture process zone, the rate of fatigue crack growth depends on a number of parameters, such as, the tensile strength, fracture toughness, loading ratio and most importantly the structural size. In this work, an analytical model is proposed for estimating the fatigue crack growth in concrete by using the concepts of dimensional analysis and including the above parameters. Knowing the governed and the governing parameters of the physical problem and by using the concepts of self-similarity, a relationship is obtained between different parameters involved. It is shown that the proposed fatigue law is able to capture the size effect in plain concrete and agrees well with different experimental results. Through a sensitivity analysis, it is shown that the structural size plays a dominant role followed by loading ratio and the initial crack length in fatigue crack propagation.     

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.     

Analysis of a rate-dependent cohesive model for dynamic crack propagation

The effect of including rate-dependence in the cohesive zone modeling of steady-state and transient dynamic crack propagation is analyzed. Spontaneous crack propagation simulations are performed using a spectral form of the elastodynamic boundary integral equations, while the solution to the steady-state problem is obtained by solving the governing Cauchy singular equation on the crack plane. The steady-state analysis shows that the existing techniques for solving the Cauchy singular integral equation are not suitable. A solution technique for the underlying Riemann-Hilbert problem for the chosen rate and damage-dependent cohesive law is presented. Under spontaneous propagation conditions, quasi-steady-state speeds slower than the theoretically predicted shear wave speed are possible. Results also show that, due to the dissipation of energy inside the cohesive zone, the energy required for crack propagation increases with the crack speed.     

Analysis of a rate-dependent cohesive model for dynamic crack propagation

The effect of including rate-dependence in the cohesive zone modeling of steady-state and transient dynamic crack propagation is analyzed. Spontaneous crack propagation simulations are performed using a spectral form of the elastodynamic boundary integral equations, while the solution to the steady-state problem is obtained by solving the governing Cauchy singular equation on the crack plane. The steady-state analysis shows that the existing techniques for solving the Cauchy singular integral equation are not suitable. A solution technique for the underlying Riemann–Hilbert problem for the chosen rate and damage-dependent cohesive law is presented. Under spontaneous propagation conditions, quasi-steady-state speeds slower than the theoretically predicted shear wave speed are possible. Results also show that, due to the dissipation of energy inside the cohesive zone, the energy required for crack propagation increases with the crack speed.     

The size of the cohesive zone at a crack tip

Although it is markedly dependent on both the stress at the cohesive zone tip and the displacement at the crack tip, the cohesive zone size at the critical stage of crack extension is shown to be relatively insensitive to the detailed form of the force law describing the non-linearity of material behaviour.     

Algorithmic implementation of a generalized cohesive crack model

Smeared fictitious crack models can be regarded as generalized cohesive crack models. The classic fictitious crack models, i.e. the fixed crack, multiple fixed crack, rotating crack and microplane model, are based on different assumptions for the orientation of developing cracks. A smooth transition between the extreme cases, the fixed crack and the rotating crack model, is provided by the adaptive fixed crack model. In this approach, the critical direction of failure is uniquely identified based on Mohr's hypothesis. Thus, the critical direction depends on the character of the failure criterion and the type of loading. The numeric implementation of the adaptive fixed crack model has given rise to some subtle questions. It is shown that even for a classical fixed crack concept, the algorithmic tangent stiffness may have to include components of crack rotation, depending on the imposed strategy for the global equilibrium iteration scheme.     

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.     

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.     

The cohesive frictional crack model applied to the analysis of the dam-foundation joint

The mechanical behaviour of dam-foundation joints plays a key role in concrete dam engineering since it is the weakest part of the structure and therefore the evolutionary crack process occurring along this joint determines the global load-bearing capacity. The reference volume involved in the above mentioned process is so large that it cannot be tested in a laboratory: structural analysis has to be carried on by numerical modelling. The use of the asymptotic expansions proposed by Karihaloo and Xiao [13] at the tip of a crack with normal cohesion and Coulomb friction can overcome the numerical difficulties that appear in large scale problems when the Newton-Raphson procedure is applied to a set of equilibrium equations based on ordinary shape functions (Standard Finite Element Method). In this way it is possible to analyze problems with friction and crack propagation under the constant load induced by hydro-mechanical coupling. For each position of the fictitious crack tip, the condition K₁=K₂=0 allows us to obtain the external load level and the tangential stress at the tip. If the joint tangential strength is larger than the value obtained, the solution is acceptable, because the tensile strength is assumed negligible and the condition K₁=0 is sufficient to cause the crack growth. Otherwise, the load level obtained can be considered as an overestimation of the critical value and a special form of contact problem has to be solved along the fictitious process zone. For the boundary condition analyzed (ICOLD benchmark on gravity dam model), after an initial increasing phase, the water lag remains almost constant and the maximum value of load carrying capacity is achieved when the water lag reaches its constant value.     

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.     

An embedded cohesive crack model for finite element analysis of mixed mode fracture of concrete*

An embedded cohesive crack model is proposed for the analysis of the mixed mode fracture of concrete in the framework of the Finite Element Method. Different models, based on the strong discontinuity approach, have been proposed in the last decade to simulate the fracture of concrete and other quasi‐brittle materials. This paper presents a simple embedded crack model based on the cohesive crack approach. The predominant local mode I crack growth of the cohesive materials is utilized and the cohesive softening curve (stress vs. crack opening) is implemented by means of a central force traction vector. The model only requires the elastic constants and the mode I softening curve. The need for a tracking algorithm is avoided using a consistent procedure for the selection of the separated nodes. Numerical simulations of well‐known experiments are presented to show the ability of the proposed model to simulate the mixed mode fracture of concrete.     

A cohesive crack model coupled with damage for interface fatigue problems

An semi-analytical formulation based on the cohesive crack model is proposed to describe the phenomenon of fatigue crack growth along an interface. Since the process of material separation under cyclic loading is physically governed by cumulative damage, the material deterioration due to fatigue is taken into account in terms of interfacial cohesive properties degradation. More specifically, the damage increment is determined by the current separation and a history variable. The damage variable is introduced into the constitutive cohesive crack law in order to capture the history-dependent property of fatigue. Parametric studies are presented to understand the influences of the two parameters entering the damage evolution law. An application to a pre-cracked double-cantilever beam is discussed. The model is validated by experimental data. Finally, the effect of using different shapes of the cohesive crack law is illustrated     

An irreversible cohesive zone model for interface fatigue crack growth simulation

Fatigue crack growth (FCG) along an interface is studied. Instead of using the Paris equation, the actual process of material separation during FCG is described by the use of an irreversible constitutive equation for the cyclic interface traction-separation behavior within the cohesive zone model (CZM) approach. In contrast to past development of CZMs, the traction-separation behavior does not follows a predefined path. The model definition, its predicted cyclic material separation behavior and application to a numerical study of interface FCG in double-cantilever beam, end-loaded split and mixed-mode beam specimens are reported.     

Numerical techniques for the analysis of crack propagation in cohesive materials

The aim of the paper is the development, assessment and use of suitable numerical procedures for the analysis of the crack evolution in cohesive materials. In particular, homogeneous as well as heterogeneous materials, obtained by embedding short stiff fibres in a cohesive matrix, are considered. Two‐dimensional Mode I fracture problems are investigated. The cohesive constitutive law is adopted to model the process zone occurring at the crack tip. An elasto‐plastic constitutive relationship, able to take into account the processes of fibre debonding and pull‐out, is introduced to model the mechanical response of the short fibres. Two numerical procedures, based on the stress and on the energy approach, are developed to investigate the crack propagation in cohesive as well as fibre‐reinforced materials, characterized by a periodic crack distribution. The results obtained using the stress and energy approaches are compared in order to evaluate the effectiveness of the procedures. Investigations on the size effect for microcracked periodic cohesive materials, and on the beneficial effects of the fibres in improving the composite material response, are developed. Copyright © 2003 John Wiley & Sons, Ltd.