Discrete Crack Model Research Articles

Development of discrete cracks in concrete loaded by shock waves

The non-linear behaviour of statically loaded concrete results mainly from the development of cracks. In addition, highly dynamically loaded concrete is influenced by the failure of micro-pores. Several authors use a damage model to consider both effects. The objective of this work is to test the capability of a discrete crack model to simulate shock wave-loaded concrete. Discrete cracks are modelled using the element-free Galerkin method, which allows cracks to be implemented independently of the mesh. The discrete crack model is used in combination with a fracture process zone to simulate the cohesive behaviour of the cracks in concrete. The behaviour of the micro-pores is described by a compression modulus, which depends on hydrostatic strain (Hugoniot curve). The strain rate effect is considered by modifying the failure surface for calculating crack development. Static and dynamic benchmarks with several concrete beams show good agreement with experimental results. Experiments with concrete loaded by contact detonation show the development of cracks below the crater. The simulation model used shows good agreement with the experimental crack pattern.

Towards fully automatic modelling of the fracture process in quasi-brittle and ductile materials: a unified crack growth criterion

Fully automatic finite element (FE) modelling of the fracture process in quasi-brittle materials such as concrete and rocks and ductile materials such as metals and alloys, is of great significance in assessing structural integrity and presents tremendous challenges to the engineering community. One challenge lies in the adoption of an objective and effective crack propagation criterion. This paper proposes a crack propagation criterion based on the principle of energy conservation and the cohesive zone model (CZM). The virtual crack extension technique is used to calculate the differential terms in the criterion. A fully-automatic discrete crack modelling methodology, integrating the developed criterion, the CZM to model the crack, a simple remeshing procedure to accommodate crack propagation, the J2 flow theory implemented within the incremental plasticity framework to model the ductile materials, and a local arc-length solver to the nonlinear equation system, is developed and implemented in an in-house program. Three examples, i.e., a plain concrete beam with a single shear crack, a reinforced concrete (RC) beam with multiple cracks and a compact-tension steel specimen, are simulated. Good agreement between numerical predictions and experimental data is found, which demonstrates the applicability of the criterion to both quasi-brittle and ductile materials.

Energy-based modeling of cohesive and cohesionless cracks via X-FEM

Numerical analyses of structures made of brittle materials such as glass or quasi-brittle materials such as concrete or ceramics require robust models for the opening and propagation of cracks which adequately represent the discontinuous character of the fracture process and consider adequately cohesive forces acting within the fracture process zone. The extended finite element method (X-FEM) is capable to model cracks as propagating discontinuities within the structure independent of the finite element discretization. Nevertheless, the analysis of crack propagation using discrete crack models crucially depends on the crack growth criterion. In this paper, a global energy-based method is proposed for the determination of the crack propagation length as well as for the crack propagation direction. The method is formulated within an X-FEM-based analysis model leading to a variational formulation in terms of displacements, crack lengths and crack angles. Both cohesive and cohesionless cracks are considered. The constitutive model for cohesive cracks considers the transfer of residual stresses parallel as well as orthogonal to the crack faces. For the representation of the displacement field in the vicinity of the tip of cohesive cracks, enriched approximations which do not lead to a singularity of the stresses at the crack tip are proposed in the paper. After linearization, the proposed energy-based finite element crack propagation model results in a coupled system of equations analogous to a multifield problem. The model is verified by means of comparisons with crack growth analyses based on linear elastic fracture mechanics using the maximum circumferential stress criterion for the determination of the crack propagation direction. Furthermore, two examples, characterized by Mode I and Mixed-Mode fracture, respectively, are used to study the performance of the developed energy-based crack propagation model.

鉄筋コンクリートのせん断ひび割れ面における応力伝達機構

FEM analyses have been performed on simply reinforced concrete elements to study the influence of the properties of cracks and the quantity of reinforcing bars upon the shear transfer behaviors in cracks of concrete. The discrete crack models were adopted in the analysis and the simple models that consist of a dilatancy crack and/or a frictional crack in addition to reinforcement were used to make the effects of crack surfaces and reinforcing bars clear. The properties of cracks corresponding to the smooth and rough crack surface were considered by using different aggregate sizes to evaluate interlocking for a dilatancy crack. Furthermore, the coefficient of friction and dilatancy for Coulomb criteria were estimated based on experiments to evaluate sliding behaviors in cracks. The meaningful relations among shear transfer behaviors in cracks, the properties of cracks and the quantity of reinforcing bars have been proven by extensive parametric analyses.

Discrete crack modelling in a new X‐FEM format with emphasis on dynamic response

AbstractThe displacement field in quasi‐brittle material is localized into narrow fracture zones during the process of decohesion. The numerical modelling of the physically propagating displacement discontinuities is considered to be inherently difficult. In this paper, the introduction of continuous discontinuities into the finite element formulation is based on the extended finite element method (X‐FEM). This paper extends the theory by further incorporating arbitrary discontinuities in the approximation and presenting numerical procedures to handle several fields of discontinuities. The discontinuous approximation is accomplished by usage of basis functions of very limited support; only non‐zero in the elements containing the crack. Moreover, the introduction of discontinuities in a continuum is discussed and a robust numerical procedure is proposed. Further, the mechanical behaviour of cracks in a quasi‐brittle material largely governs the overall mechanical response of the material, and the implementation of a cohesive crack model based on anisotropic damage coupled to plasticity is emphasized. Numerical examples for both static and dynamic, transient, loading show that the proposed X‐FEM format in combination with the cohesive crack model has a good performance and leads to an efficient implementation. Finally, the proposed forward method for propagation of cracks and introduction of new ones is robust and stable. Copyright © 2006 John Wiley & Sons, Ltd.

An extended FE strategy for transition from continuum damage to mode I cohesive crack propagation

AbstractAn integrated strategy is proposed for the simulation of damage development and crack propagation in concrete structures. In the initial stage of damage growth, concrete is considered to be macroscopically integer and is modeled by a non‐symmetric isotropic non‐local damage model. The transition to the discrete cohesive crack model depends on the local mesh size and is driven by an analytical estimate of the current bandwidth. When a crack is introduced in the model an extended finite element approach is used to follow the propagation path, independent of the background mesh. The proposed methodology is tested on a notched tension specimen and then applied to the analysis of a wedge splitting test. Copyright © 2006 John Wiley & Sons, Ltd.

Modelling of CFRP–concrete bond using smeared and discrete cracks

This paper presents a finite element study of the bond characteristic between CFRP and concrete. The behaviour of twelve shear-lap specimens was modelled using a combination of smeared and discrete cracks. The smeared crack model is based on Rankine’s failure criterion, whereas the discrete crack model is based on nonlinear fracture mechanics, where both mode I and II fractures are accounted for. The finite element model proved to be able to predict the ultimate loads, crack patterns at failure and CFRP strain distributions reasonably well. The same method was then used to simulate debond failures in retrofitted beams which also showed good correlation with the experimental results.

Discrete crack modelling of ductile fracture driven by non‐local softening plasticity

AbstractA combined approach towards ductile damage and fracture is presented, in the sense that a continuous material degradation is coupled with a discrete crack description for large deformations. Material degradation is modelled by a gradient enhanced damage‐hyperelastoplasticity model. It is assumed that failure occurs solely due to plastic straining, which is particularly relevant for shear dominated problems, where the effect of the hydrostatic stress in triggering failure is less important. The gradient enhancement eliminates pathological localization effects which would normally result from the damage influence. Discrete cracks appear in the final stage of local material failure, when the damage has become critical. The rate and the direction of crack propagation depend on the evolution of the damage field variable, which in turn depends on the type of loading. In a large strain finite element framework, remeshing allows to incorporate the changing crack geometry and prevents severe element distortion. Attention is focused on the robustness of the computations, where the transfer of variables, which is needed after each remeshing, plays a crucial role. Numerical examples are shown and comparisons are made with published experimental results. Copyright © 2005 John Wiley & Sons, Ltd.

Numerical Analysis of Debonding Mechanisms in FRP-Strengthened RC Beams

: Fiber-reinforced polymer (FRP) composites have been increasingly used as externally bonded reinforcement in lieu of their steel counterpart in the rehabilitation and retrofit of existing concrete structures. Without proper understanding of interfacial fracture behavior and failure mechanisms, it is impossible to efficiently develop an effective and rational FRP bonding technique. This article is mainly focused on clarifying the debonding behavior and failure mechanisms caused by different types of flexural crack distributions in FRP-strengthened R/C beams, which has not been solved so far. Using a discrete crack model for concrete crack propagation and a bilinear bond–slip relationship with softening behavior to represent FRP–concrete interfacial behavior, a nonlinear fracture mechanics-based finite-element analysis is performed to investigate the effects of crack spacing and interfacial parameters such as stiffness, local bond strength, and fracture energy on the initiation and propagation of the debonding and the structural performance. It is shown that the debonding behavior and load-carrying capacity are significantly influenced by two important factors: interfacial fracture energy and crack spacing in relation to the effective transfer length of FRP sheets. Based on the numerical results, some suggestions concerning the effect of interfacial properties are made as practical design aids.

Modeling Masonry Shear-Compression: Role of Dilatancy Highlighted

In-plane shear and compression are important modes in masonry walls, both in load bearing structures and in framed structures. By these mechanical actions shear forces caused by wind and earthquakes are resisted. Upon shear-slipping along bed joints, brick units in masonry also undergo upward translation, or dilatancy, causing global volume increase. If this dimensional change is prevented, large compressive stresses may build up, increasing the resistance to slip by the Coulomb-friction nature. If this shear-compression interaction is not modeled correctly, large errors may be made. In the extreme case, unlimited shear resistance may be predicted by computational models. A discrete crack modeling approach for masonry, which captures the shear-compression dilatational behavior realistically, is elaborated. Shear-compression experiments on small masonry specimens as well as 1 m×1 m masonry walls are analyzed as validation.

Regularized sequentially linear saw‐tooth softening model

AbstractAfter a brief discussion on crack models, it is demonstrated that cracking is often accompanied by snaps and jumps in the load–displacement response which complicate the analysis. This paper provides a solution by simplifying non‐linear crack models into sequentially linear saw‐tooth models, either saw‐tooth tension‐softening for unreinforced material or saw‐tooth tension‐stiffening for reinforced material. A linear analysis is performed, the most critical element is traced, the stiffness and strength of that element are reduced according to the saw‐tooth curve, and the process is repeated. This approach circumvents the use of incremental–iterative procedures and negative stiffness moduli and is inherently stable. The main part of the paper is devoted to a regularization procedure that provides mesh‐size objectivity of the saw‐tooth model. The procedure is similar to the one commonly used in the smeared crack framework but, in addition, both the initial tensile strength and the ultimate strain are rescaled. In this way, the dissipated fracture energy is invariant with respect not only to the mesh size, but also to the number of saw‐teeth adopted to discretize the softening branch.Finally, the potential of the model for large‐scale fracture analysis is demonstrated. A masonry façade subjected to tunnelling induced settlements is analysed. The very sharp snap‐backs associated with brittle fracture of the façade automatically emerge with sequentially linear analysis, whereas non‐linear analysis of the façade using smeared or discrete crack models shows substantial difficulties despite the use of arc‐length schemes. Copyright © 2004 John Wiley & Sons, Ltd.

Fracture behavior of SFRC slabs on grade

In concrete pavements, steel fibers are particularly suitable both for limiting shrinkage effects and for increasing the bearing capacity and fatigue resistance. In the present paper, the fracture behavior of steel-fiber-reinforced concrete slabs on grade is studied by using Non Linear Fracture Mechanics FE analyses, based on both smeared and discrete crack models. The fracture mechanics models have been validated by results from experiments on full scale slabs on elastic soil subjected to a load in the slab center. The comparison shows that NLFM analyses closely predict crack development, ultimate load and the collapse mechanism of slabs on grade. The smeared crack approach has been used for studying different loading conditions since it does not need an a-priori knowledge of the crack pattern.

Seismic analysis of concrete arch dams by combined discrete crack and non-orthogonal smeared crack technique

There are different techniques to examine the non-linear seismic behavior of concrete arch dams, among which, discrete crack (DC) method is the most popular. A more recent option introduced for this purpose is the non-orthogonal smeared crack (NOSC) approach. In this study, a special finite element program is developed based on the combined discrete crack and non-orthogonal smeared crack (DC–NOSC) technique. The basic concepts of the method are explained initially. Later on, the non-linear dynamic behavior of a typical thin arch dam is considered based on this approach, and the results are compared against other simpler methods mentioned above. Overall, DC–NOSC model is found to be a more rigorous, consistent and realistic approach from different aspects in comparison with both DC and NOSC models alone.

3D nonlinear finite element/fracture mechanics analysis of pressurized nuclear reactor container ring

Nuclear reactor containers, without inner liners, must withstand internal pressurization with minimal leak to the outside through the cracks. This paper reports on a 3D finite element (FE) simulation of a prestressed nuclear reactor container ring subjected to internal pressure. Discrete cracks are used, and the effect of crack surface pressurization on crack openings is investigated. It is shown that the discrete crack model can provide a suitable alternative to the smeared crack one, and that for the selected ring, the effect of additional crack pressurization is minimal.

A comparative study of numerical solutions to non-linear discrete crack modelling of concrete beams involving sharp snap-back

Numerical problems are often encountered in modelling crack propagation in concrete beams using non-linear finite element (FE) analysis, especially when sharp snap-back behaviour in load–displacement relations occurs. This paper firstly identifies 16 arc-length control based numerical strategies based on extensive literature review. They are then used to carefully model the structural behaviour of a four-point single notched shear beam using discrete crack modelling approach in which cracks are represented by interface elements with bilinear softening constitutive laws. Based on extensive FE analyses, detailed comparisons of the merits and demerits of these numerical algorithms are then made. The results indicate that the effectiveness and efficiency of different algorithms may vary considerably from one to another, with the local arc-length based procedures in conjunction with tangential stiffness strategy and reversible unloading model being the most robust.

Modeling of delamination using a discretized cohesive zone and damage formulation

Delamination initiation and growth are analyzed by using a discrete cohesive crack model. The delamination is constrained to grow along a tied interface. The model is derived by postulating the existence of a maximum load surface which limits the adhesive forces in the process zone of the crack. The size of this maximum load surface is made dependent on the amount of dissipated crack opening work, such that the maximum load surface shrinks to zero as a predefined amount of work is consumed. A damage formulation is used to reduce the adhesive forces. Mode I, II and III loading or any combined loading is possible. An analytical solution is obtained for a single mode opening and the implications of this result on the governing equations is discussed. The delamination model is implemented in the finite element code LS-DYNA and simulation results are shown to be in agreement with experimental results.

Parameter identification of a cohesive crack model by Kalman filter

The Kalman filter (KF) methodology is apt to solve parameter identification (inverse) problems in a statistical context, through a sequence of estimations, which starts from an a priori estimation by an “expert” (Bayesian approach) and exploits a time-stepping flow of experimental data until convergence is empirically ascertained. Such methodology is here adopted for the identification of the material parameters, together with their uncertainties, in a mode I cohesive crack model, on the basis of experimental data generated by wedge-splitting tests on concrete specimens. The simulation of the experiments is based on the assumptions of a piecewise-linear cohesive model with four parameters to identify on the crack path, and of linear elasticity elsewhere. In view of regularly progressive fracture processes, the discrete crack model and, consequently, the overall finite element analysis, are formulated as linear complementarity problems. This mathematical construct is exploited to obtain the sensitivity matrix, key ingredient of the KF extended to nonlinear inverse problems, in a computationally convenient closed form. Various issues peculiar of KF identification in the present mechanical context are critically discussed in the light of the numerical solutions achieved.

Predicting failure of composite structures containing cracks

The ability to predict failure loads in composite structures containing through-the-thickness cracks is a requirement in a variety of structural design circumstances. The objective of this study was to develop a practical computational technique that simulates progressive damage growth in the vicinity of the crack tip in a manner that allows the prediction of failure over a wide range of crack sizes. This was accomplished through the use of the discrete crack model with a general purpose finite element program. Calculations were performed for two different laminates with various crack sizes under tensile loading. An analysis was also performed on a section of a composite aircraft fuselage containing a crack and subjected to internal pressure loading. Reasonably good agreement was found between the calculations and the results from tests.

Numerical Model for Minimizing Risk of Damage in Repair System

New cement-based repair overlays are subject to restraint and eigenstresses that can lead to crack formation and delamination. The origin of this complex state of stresses is found in the dimensional instability of the new coating. Drying shrinkage, autogeneous shrinkage, and temperature gradients are the most important loads acting on the system. In most cases the design of such coating systems is carried out in an empirical way. In this paper a new approach based on a numerical model is proposed to determine the relevant key parameters for the design of repair systems. By means of this model time-dependent moisture distributions and crack evolution in a drying concrete repair system can be simulated in a realistic way. Crack formation and delamination is described, with the smeared and discrete crack models, applying nonlinear fracture mechanics. The relevant material parameters influencing shrinkage cracking and delamination are identified by means of a sensitivity analysis. Finally, this approach is a...

Simulation of delamination in fiber composites with a discrete cohesive failure model

Delamination initiation and growth are analyzed by using a discrete cohesive crack model. The model is derived by postulating the existence of a maximum load surface which limits the adhesive forces in the process zone of the crack. The size of the maximum load surface is made dependent on the amount of dissipated crack opening work such that the maximum load surface shrinks to zero as a predefined amount of work is consumed. Mode I, II, III loading or any combined loading is possible. The delamination model is implemented in the explicit finite-element code LS-DYNA and simulation results are found to be in agreement with experimental results.