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.     

Influence of shear parameters on mixed–mode fracture of concrete

The influence of the mode II fracture parameters on the mixed mode fracture experimental tests of quasibrittle materials is studied. The study is based on experimental results and numerical analyses. For the numerical study, a procedure for mixed mode fracture of quasibrittle materials is presented. The numerical procedure is based on the cohesive crack approach, and extends it to mixed mode fracture. Four experimental sets of mixed mode fracture were modelled, one from Arrea and Ingraffea and another from a nonproportional loading by the authors, both with bending concrete beams. Two other sets of experimental fracture were modelled, based on double-edge notched testing; in these tests an important mode II is beforehand expected. The numerical results agree quite well with experimental records. The influence of the main parameters for mode II fracture on the mixed mode fracture is studied for the four experimental set of tests and compared with these results. In all them, large changes in the mode II fracture energy hardly modify the numerical results. The tangential and normal stresses along the crack path during the loading proccess are obtained, also with different values of the mode II fracture energy. For the studied experimental tests it is concluded that the crack is initiated under mixed mode but propagated under predominant mode I. This allows a development of mixed mode fracture models, mainly based on standard properties of the material measured by standard methods, avoiding the problems associated with the measurement of mode II fracture parameters, such as mode II fracture energy and cohesion.     

Modelling the fracture of concrete under mixed loading

A simple and efficient numerical procedure for mixed mode fracture of quasibrittle materials is shown: This technique predicts crack trajectories as well as load-displacement or load-CMOD responses. The model is based on the cohesive crack concept and uses the local mode I approach. Numerical results agree quite well with three experimental sets of mixed mode fracture of concrete beams; one from Arrea and Ingraffea, another from García, Gettu and Carol and from a nonproportional loading by the authors. In constrast to more sophisticated models, this method offers two major advantages: it requires only material properties measured by standardized methods and it can easily be implemented with general multipurpose finite element codes.     

An extended finite element method with analytical enrichment for cohesive crack modeling

A recent approach to fracture modeling has combined the extended finite element method (XFEM) with cohesive zone models. Most studies have used simplified enrichment functions to represent the strong discontinuity but have lacked an analytical basis to represent the displacement gradients in the vicinity of the cohesive crack. In this study enrichment functions based upon an existing analytical investigation of the cohesive crack problem are proposed. These functions have the potential of representing displacement gradients in the vicinity of the cohesive crack and allow the crack to incrementally advance across each element. Key aspects of the corresponding numerical formulation and enrichment functions are discussed. A parameter study for a simple mode I model problem is presented to evaluate if quasi‐static crack propagation can be accurately followed with the proposed formulation. The effects of mesh refinement and mesh orientation are considered. Propagation of the cohesive zone tip and crack tip, time variation of the cohesive zone length, and crack profiles are examined. The analysis results indicate that the analytically based enrichment functions can accurately track the cohesive crack propagation of a mode I crack independent of mesh orientation. A mixed mode example further demonstrates the potential of the formulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Embedded representation of fracture in concrete with mixed finite elements

As an alternative to the smeared and discrete crack representations, an embedded representation of fracture for finite element analysis of concrete structures is presented. The three-field Hu–Washizu variational statement is extended to bodies with internal discontinuities. The extended variational statement is then utilized for formulating elements with a discontinuous displacement field. The new elements are capable of modelling different deformation modes of an internal discontinuity at the element level. The satisfactory performance of the embedded crack representation is verified by several case studies on concrete fracture.     

An adaptive finite element framework for fatigue crack propagation

A new finite element (FE) framework for fatigue crack propagation (FCP) analysis is proposed. This framework combines the simplicity of standard industrial FCP analysis with the generality and accuracy of a full FE analysis and can be implemented on a small computer by combining standard existing computational tools. In this way it constitutes an attractive alternative to existing approaches. The framework is based on linear elastic fracture mechanics and on FE mesh adaptation. Some novel features are introduced in several of its steps in order to make it efficient and at the same time reasonably accurate. Various computational aspects of the scheme are discussed. A few two‐dimensional numerical examples involving FCP in thin sheets under plane‐stress conditions are presented to demonstrate the performance of the framework. Some of the numerical results are compared to those of laboratory experiments. Copyright © 2002 John Wiley & Sons, Ltd.     

A cohesive element model for mixed mode loading with frictional contact capability

We present a model that combines interface debonding and frictional contact. The onset of fracture is explicitly modeled using the well‐known cohesive approach. Whereas the debonding process is controlled by a new extrinsic traction separation law, which accounts for mode mixity, and yields two separate values for energy dissipation in mode I and mode II loading, the impenetrability condition is enforced with a contact algorithm. We resort to the classical law of unilateral contact and Coulomb friction. The contact algorithm is coupled together to the cohesive approach in order to have a continuous transition from crack nucleation to the pure frictional state after complete decohesion. We validate our model by simulating a shear test on a masonry wallette and by reproducing an experimental test on a masonry wall loaded in compression and shear. Copyright © 2012 John Wiley & Sons, Ltd.     

Bounds for element size in a variable stiffness cohesive finite element model

The cohesive finite element method (CFEM) allows explicit modelling of fracture processes. One form of CFEM models integrates cohesive surfaces along all finite element boundaries, facilitating the explicit resolution of arbitrary fracture paths and fracture patterns. This framework also permits explicit account of arbitrary microstructures with multiple length scales, allowing the effects of material heterogeneity, phase morphology, phase size and phase distribution to be quantified. However, use of this form of CFEM with cohesive traction–separation laws with finite initial stiffness imposes two competing requirements on the finite element size. On one hand, an upper bound is needed to ensure that fields within crack‐tip cohesive zones are accurately described. On the other hand, a lower bound is also required to ensure that the discrete model closely approximates the physical problem at hand. Both issues are analysed in this paper within the context of fracture in multi‐phase composite microstructures and a variable stiffness bilinear cohesive model. The resulting criterion for solution convergence is given for meshes with uniform, cross‐triangle elements. A series of calculations is carried out to illustrate the issues discussed and to verify the criterion given. These simulations concern dynamic crack growth in an Al₂O₃ ceramic and in an Al₂O₃/TiB₂ ceramic composite whose phases are modelled as being hyperelastic in constitutive behaviour. Copyright © 2004 John Wiley & Sons, Ltd.     

Coupled meshfree and fractal finite element method for mixed mode two‐dimensional crack problems

This paper presents a coupling technique for integrating the element‐free Galerkin method (EFGM) with the fractal finite element method (FFEM) for analyzing homogeneous, isotropic, and two‐dimensional linear‐elastic cracked structures subjected to mixed‐mode (modes I and II) loading conditions. FFEM is adopted for discretization of the domain close to the crack tip and EFGM is adopted in the rest of the domain. In the transition region interface elements are employed. The shape functions within interface elements which comprise both the EFG and the finite element (FE) shape functions, satisfies the consistency condition thus ensuring convergence of the proposed coupled EFGM–FFEM. The proposed method combines the best features of EFGM and FFEM, in the sense that no special enriched basis functions or no structured mesh with special FEs are necessary and no post‐processing (employing any path independent integrals) is needed to determine fracture parameters, such as stress‐intensity factors (SIFs) and T‐stress. The numerical results show that SIFs and T‐stress obtained using the proposed method are in excellent agreement with the reference solutions for the structural and crack geometries considered in the present study. Also, a parametric study is carried out to examine the effects of the integration order, the similarity ratio, the number of transformation terms, and the crack length to width ratio on the quality of the numerical solutions. A numerical example on mixed‐mode condition is presented to simulate crack propagation. As in the proposed coupled EFGM–FFEM at each increment during the crack propagation, the FFEM mesh (around the crack tip) is shifted as it is to the new updated position of the crack tip (such that FFEM mesh center coincides with the crack tip) and few meshless nodes are sprinkled in the location where the FFEM mesh was lying previously, crack‐propagation analysis can be dramatically simplified. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite deformation formulation for embedded frictional crack with the extended finite element method

We reformulate an extended finite element (FE) framework for embedded frictional cracks in elastoplastic solids to accommodate finite deformation, including finite stretching and rotation. For the FE representation, we consider a Galerkin approximation in which both the trial and weighting functions adapt to the current contact configuration. Contact and frictional constraints employ two Kuhn–Tucker conditions, a contact/separation constraint nesting over a stick/slip constraint for the case when the crack faces are in frictional sliding mode. We integrate finite deformation bulk plasticity into the formulation using the multiplicative decomposition technique of nonlinear continuum mechanics. We then present plane strain simulations demonstrating various aspects of the extended FE solutions. The mechanisms considered include combined opening and frictional sliding in initially straight, curved, and S‐shaped cracks, with and without bulk plasticity. To gain further insight into the extended FE solutions, we perform mesh convergence studies focusing on both the global and the local responses of structures with cracks, including the distribution of the normal component of traction on the crack faces. Copyright © 2009 John Wiley & Sons, Ltd.     

A contact algorithm for cohesive cracks in the extended finite element method

Cracks with quasibrittle behavior are extremely common in engineering structures. The modeling of cohesive cracks involves strong nonlinearity in the contact, material, and complex transition between contact and cohesive forces. In this article, we propose a novel contact algorithm for cohesive cracks in the framework of the extended finite element method. A cohesive-contact constitutive model is introduced to characterize the complex mechanical behavior of the fracture process zone. To avoid the stress oscillations and ill-conditioned system matrix that often occur in the conventional contact approach, the proposed algorithm employs a special dual Lagrange multiplier to impose the contact constraint. This Lagrange multiplier is constructed by means of the area-weighted average and biorthogonality conditions at the element level. The system matrix can be condensed into a positive definite matrix with an unchanged size at a very low computational cost. In addition, we illustrate solving the cohesive crack contact problem using a novel iteration strategy. Several numerical experiments are performed to illustrate the efficiency and high-quality results of our method in contact analysis of cohesive cracks.     

A new cohesive zone model for mixed mode interface fracture in bimaterials

A new two-dimensional cohesive zone model which is suitable for the prediction of mixed mode interface fracture in bimaterials is presented. The model accounts for the well known fact that the interfacial fracture toughness is not a constant, but a function of the mode mixity. Within the framework of this model, the cohesive energy and the cohesive strength are not chosen to be constant, but rather functions of the mode mixity. A polynomial cohesive zone model is derived in light of analytical and experimental observations of interface cracks. The validity of the new cohesive law is examined by analyzing double cantilever beam and Brazilian disk specimens. The methodology to determine the parameters of the model is outlined and a failure criterion for a pair of ceramic clays is suggested.     

Determination of T-stress from finite element analysis for mode I and mixed mode I/II loading

The elastic T-stress has been recognised as a measure of constraint around the tip of a crack in contained yielding problems. A review of the literature indicates that most methods for obtaining T are confined to simple geometry and loading configurations. This paper explores direct use of finite element analysis for calculating T. It is shown that for mode I more reliable results with less mesh refinement can be achieved if crack flank nodal displacements are used. Methods are also suggested for calculating T for any mixed mode I/II loading without having to calculate stress intensity factors. There is good agreement between the results from the proposed methods and analytical results. T-stress is determined for a test configuration designed to investigate brittle and ductile fracture in mixed mode loading. It is shown that in shear loading of a cracked specimen T vanishes only when a truly antisymmetric field of deformation is provided. However this rarely happens in practice and the presence of T in shear is often inevitable. It is shown that for some cases the magnitude of T in shear is much more than that for tension. The effect of crack length is also investigated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Cracking node method for dynamic fracture with finite elements

A new method for modeling discrete cracks based on the extended finite element method is described. In the method, the growth of the actual crack is tracked and approximated with contiguous discrete crack segments that lie on finite element nodes and span only two adjacent elements. The method can deal with complicated fracture patterns because it needs no explicit representation of the topology of the actual crack path. A set of effective rules for injection of crack segments is presented so that fracture behavior beginning from arbitrary crack nucleations to macroscopic crack propagation is seamlessly modeled. The effectiveness of the method is demonstrated with several dynamic fracture problems that involve complicated crack patterns such as fragmentation and crack branching. Copyright © 2008 John Wiley & Sons, Ltd.     

Combination of the material force concept and the extended finite element method for mixed mode crack growth simulations

A novel approach to simulate crack growth within an extended finite element framework is presented. The introduced approach combines the material force concept and the extended finite element method (xFEM) that is not straight forward and faces the major problem that a crack tip node, which is required for the evaluation of the material force, is not available within an xFEM framework. The introduced concept enables an efficient single step evaluation of the crack state and the crack growth direction based on a continuum mechanics approach and represents an alternative to the common procedure of using the stress intensity factor solution within a stress or energy‐based empirical formulation for the determination of the crack growth direction. Two different approaches are introduced that evaluate the crack tip material force within the xFEM based on a domain or contour approach, both providing equivalent results. After an evaluation of the method, a major focus is set on crack growth investigations with increased complexity, including mixed mode loading and crack interaction with other discontinuities. The influence of different evaluation parameters is studied by comparing the results with empirical, experimental and alternative numerical solutions and confirms the applicability and capability of the proposed combination of both concepts. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Modelling cohesive crack growth using a two-step finite element-scaled boundary finite element coupled method

A two-step method, coupling the finite element method (FEM) and the scaled boundary finite element method (SBFEM), is developed in this paper for modelling cohesive crack growth in quasi-brittle normal-sized structures such as concrete beams. In the first step, the crack trajectory is fully automatically predicted by a recently-developed simple remeshing procedure using the SBFEM based on the linear elastic fracture mechanics theory. In the second step, interfacial finite elements with tension-softening constitutive laws are inserted into the crack path to model gradual energy dissipation in the fracture process zone, while the elastic bulk material is modelled by the SBFEM. The resultant nonlinear equation system is solved by a local arc-length controlled solver. Two concrete beams subjected to mode-I and mixed-mode fracture respectively are modelled to validate the proposed method. The numerical results demonstrate that this two-step SBFEM-FEM coupled method can predict both satisfactory crack trajectories and accurate load-displacement relations with a small number of degrees of freedom, even for crack growth problems with strong snap-back phenomenon. The effects of the tensile strength, the mode-I and mode-II fracture energies on the predicted load-displacement relations are also discussed.     

Extended isogeometric analysis for cohesive fracture

The objective of this study is to present an extended isogeometric formulation for cohesive fracture. The approach exploits the higher order interelement continuity property of nonuniform rational B-splines (NURBS), in particular the higher accuracy that results for the stress prediction, which yields an improved estimate for the direction of crack propagation compared to customary Lagrangian interpolations. Shifting is used to ensure compatibility with the surrounding discretization, where, different from extended finite element methods, the affected elements stretch over several rows perpendicular to the crack path. To avoid fine meshes around the crack tip in case of cohesive fracture, a blending function is used in the extension direction of the crack path. To comply with standard finite element data structures, Bézier extraction is used. The absence of the Kronecker-delta property in the higher order interpolations of isogeometric analysis impedes the enrichment scheme and compatibility enforcement. These issues are studied comprehensively at the hand of several examples, while crack propagation analyses show the viability of the approach.     

A material model to reproduce mixed‐mode fracture in concrete

This paper presents a material model to reproduce crack propagation in cement‐based material specimens under mixed‐mode loading. Its numerical formulation is based on the cohesive crack model, proposed by Hillerborg, and extended for the mixed‐mode case. This model is inspired by former works by Gálvez et al but implemented for its use in a finite element code at a material level, that is to say, at an integration point level. Among its main features, the model is able to predict the crack orientation and can reproduce the fracture behaviour under mixed‐mode fracture loading. In addition, several experimental results found in the literature are properly reproduced by the model.     

Finite element modelling of multiple cohesive discrete crack propagation in reinforced concrete beams

This paper presents a finite element (FE) model for fully automatic simulation of multiple discrete crack propagation in reinforced concrete (RC) beams. The discrete cracks are modelled based on the cohesive/fictitious crack concept using nonlinear interface elements with a bilinear tensile softening constitutive law. The model comprises an energy-based crack propagation criterion, a simple remeshing procedure to accommodate crack propagations, two state variable mapping methods to transfer structural responses from one FE mesh to another, and a local arc-length algorithm to solve system equations characterised by material softening. The bond-slip behaviour between reinforcing bars and surrounding concrete is modelled by a tension-softening element. An example RC beam with well-documented test data is simulated. The model is found capable of automatically modelling multiple crack propagation. The predicted cracking process and distributed crack pattern are in close agreement with experimental observations. The load-deflection relations are accurately predicted up to a point when compressive cracking becomes dominant. The effects of bond-slip modelling and the efficiency and effectiveness of the numerical algorithms, together with the limitations of the current model, are also discussed.