Propagation of a cohesive crack crossing a reinforcement layer

This paper analyzes the propagation of a cohesive crack through a reinforcement layer and gives a solution that can be used for any specimen and loading condition. Here it faces the case of a reinforced prismatic beam loaded at three points. Reinforcement is represented by means of a free-slip bar bridging the cracked section, anchored at both sides of the crack at a certain distance that is called the effective slip length. This length is obtained by making the free-slip bar mechanically equivalent to the actual adherent reinforcement. With this model, the crack development depends on three parameters (apart from those that represent the specimen geometry): the size of the specimen, the cover thickness of the layer and the reinforcement strength. The latter depends on the reinforcement ratio and its adherence to the matrix while the reinforcement is in the elastic regime; otherwise, on the reinforcement ratio and its yielding strength. The thickness of the reinforcement cover influences the first stages of the development of the cohesive crack, and thus it also affects the value of the load peak. The computed load-displacement curves display a noticeable size effect, as real cohesive materials do. Finally, the model is able to fit the available experimental results, and accurately reproduces the influence of size, amount of reinforcement and adherence variations in the tests.     

New crack‐tip elements for XFEM and applications to cohesive cracks

An extended finite element method scheme for a static cohesive crack is developed with a new formulation for elements containing crack tips. This method can treat arbitrary cracks independent of the mesh and crack growth without remeshing. All cracked elements are enriched by the sign function so that no blending of the local partition of unity is required. This method is able to treat the entire crack with only one type of enrichment function, including the elements containing the crack tip. This scheme is applied to linear 3‐node triangular elements and quadratic 6‐node triangular elements. To ensure smooth crack closing of the cohesive crack, the stress projection normal to the crack tip is imposed to be equal to the material strength. The equilibrium equation and the traction condition are solved by the Newton–Raphson method to obtain the nodal displacements and the external load simultaneously. The results obtained by the new extended finite element method are compared to reference solutions and show excellent agreement. Copyright © 2003 John Wiley & Sons, Ltd.     

3D dynamic fragmentation with parallel dynamic insertion of cohesive elements

Dynamic fragmentation is a rapid and catastrophic failure of a material. During this process, the nucleation, propagation, branching, and coalescence of cracks results in the formation of fragments. The numerical modeling of this phenomenon is challenging because it requires high‐performance computational capabilities. For this purpose, the finite‐element method with dynamic insertion of cohesive elements was chosen. This paper describes the parallel implementation of its fundamental algorithms in the C++ open‐source library Akantu. Moreover, a numerical instability that can cause the loss of energy conservation and possible solutions to it are illustrated. Finally, the method is applied to the dynamic fragmentation of a hollow sphere subjected to uniform radial expansion. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment

A methodology is developed for switching from a continuum to a discrete discontinuity where the governing partial differential equation loses hyperbolicity. The approach is limited to rate‐independent materials, so that the transition occurs on a set of measure zero. The discrete discontinuity is treated by the extended finite element method (XFEM) whereby arbitrary discontinuities can be incorporated in the model without remeshing. Loss of hyperbolicity is tracked by a hyperbolicity indicator that enables both the crack speed and crack direction to be determined for a given material model. A new method was developed for the case when the discontinuity ends within an element; it facilitates the modelling of crack tips that occur within an element in a dynamic setting. The method is applied to several dynamic crack growth problems including the branching of cracks. Copyright © 2003 John Wiley & Sons, Ltd.     

Stable crack propagation in steel at 1173 K: Experimental investigation and simulation using 3D cohesive elements in large-displacements

This paper deals with the numerical simulation of a stable crack propagation experiment at in a 16MND5 steel. At this temperature, the material is viscoplastic. A cohesive zone model is formulated in order to simulate the rupture of a CT specimen. A large displacement 3D cohesive element with eight nodes is implemented in the finite element code ABAQUS. The associated traction-separation law is of Tvergaard and Hutchinson type, in which an hardening slope has been added. This hardening simulates the material strengthening associated to the increasing strain rate in front of the crack tip when crack tip starts to propagate.We show that in this case the form of the cohesive law has great impact on the simulated propagation velocity.     

Cohesive crack propagation in a random elastic medium

The issue of generating non-Gaussian, multivariate and correlated random fields, while preserving the internal auto-correlation structure of each single-parameter field, is discussed with reference to the problem of cohesive crack propagation. Three different fields are introduced to model the spatial variability of the Young modulus, the tensile strength of the material, and the fracture energy, respectively. Within a finite-element context, the crack-propagation phenomenon is analyzed by coupling a Monte Carlo simulation scheme with an iterative solution algorithm based on a truly-mixed variational formulation which is derived from the Hellinger–Reissner principle. The selected approach presents the advantage of exploiting the finite-element technology without the need to introduce additional modes to model the displacement discontinuity along the crack boundaries. Furthermore, the accuracy of the stress estimate pursued by the truly-mixed approach is highly desirable, the direction of crack propagation being determined on the basis of the principal-stress criterion. The numerical example of a plain concrete beam with initial crack under a three-point bending test is considered. The statistics of the response is analyzed in terms of peak load and load–mid-deflection curves, in order to investigate the effects of the uncertainties on both the carrying capacity and the post-peak behaviour. A sensitivity analysis is preliminarily performed and its results emphasize the negative effects of not accounting for the auto-correlation structure of each random field. A probabilistic method is then applied to enforce the auto-correlation without significantly altering the target marginal distributions. The novelty of the proposed approach with respect to other methods found in the literature consists of not requiring the a priori knowledge of the global correlation structure of the multivariate random field.     

Finite‐deformation irreversible cohesive elements for three‐dimensional crack‐propagation analysis

We develop a three‐dimensional finite‐deformation cohesive element and a class of irreversible cohesive laws which enable the accurate and efficient tracking of dynamically growing cracks. The cohesive element governs the separation of the crack flanks in accordance with an irreversible cohesive law, eventually leading to the formation of free surfaces, and is compatible with a conventional finite element discretization of the bulk material. The versatility and predictive ability of the method is demonstrated through the simulation of a drop‐weight dynamic fracture test similar to those reported by Zehnder and Rosakis. The ability of the method to approximate the experimentally observed crack‐tip trajectory is particularly noteworthy. © 1999 John Wiley & Sons, Ltd.     

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.     

Dynamic steady-state crack propagation in a transversely isotropic viscoelastic body

The problem considered herein is the dynamic, subsonic, steady-state propagation of a semi-infinite, generalized plane strain crack in an infinite, transversely isotropic, linear viscoelastic body. The corresponding boundary value problem is considered initially for a general anisotropic, linear viscoelastic body and reduced via transform methods to a matrix Riemann–Hilbert problem. The general problem does not readily yield explicit closed form solutions, so attention is addressed to the special case of a transversely isotropic viscoelastic body whose principal axis of material symmetry is parallel to the crack edge. For this special case, the out-of-plane shear (Mode III), in-plane shear (Mode II) and in-plane opening (Mode I) modes uncouple. Explicit expressions are then constructed for all three Stress Intensity Factors (SIF). The analysis is valid for quite general forms for the relevant viscoelastic relaxation functions subject only to the thermodynamic restriction that work done in closed cycles be non-negative. As a special case, an analytical solution of the Mode I problem for a general isotropic linear viscoelastic material is obtained without the usual assumption of a constant Poissons ratio or exponential decay of the bulk and shear relaxation functions. The Mode I SIF is then calculated for a generalized standard linear solid with unequal mean relaxation times in bulk and shear leading to a non-constant Poissons ratio. Numerical simulations are performed for both point loading on the crack faces and for a uniform traction applied to a compact portion of the crack faces. In both cases, it is observed that the SIF can vanish for crack speeds well below the glassy Rayleigh wave speed. This phenomenon is not seen for Mode I cracks in elastic material or for Mode III cracks in viscoelastic material.     

Dynamic crack propagation in elastoplastic thin‐walled structures: Modelling and validation

In this paper, a method to analyse and predict crack propagation in thin‐walled structures subjected to large plastic deformations when loaded at high strain rates—such as impact and/or blast—has been proposed. To represent the crack propagation independently of the finite element discretisation, an extended finite element method based shell formulation has been employed. More precisely, an underlying 7‐parameter shell model formulation with extensible directors has been extended by locally introducing an additional displacement field, representing the displacement discontinuity independently of the mesh. Of special concern in the paper has been to find a proper balance between, level of detail and accuracy when representing the physics of the problem and, on the other hand, computational efficiency and robustness. To promote computational efficiency, an explicit time step scheme has been employed, which however has been discovered to generate unphysical oscillations in the response upon crack propagation. Therefore, special focus has been placed to investigate these oscillations as well as to find proper remedies. The paper is concluded with three numerical examples to verify and validate the proposed model.Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of the integration of cohesive elements in regard to utilization of coarse mesh in laminated composite materials

An analysis of the solver convergence difficulties and erroneous results when large cohesive elements are utilized in delamination propagation simulations in laminated composites is presented. Special focus is given to the numerical integration of the cohesive element force vector and stiffness matrix. The magnitude and variation of the integration error are analyzed, and the results show that contrary to statements found elsewhere in the literature, the 2 × 2 point Newton‐Cotes quadrature, commonly used in commercial software, results in large errors and is one of the limiting factors for using larger elements. By reducing the integration error in the damage process zone of the interface, more accurate results can be obtained, and larger elements can be utilized with less iterations, thereby decreasing the computational cost. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamic crack response to a localized shear pulse perturbation in brittle amorphous materials: on crack surface roughening

Linear Elastic Fracture Mechanics (LEFM) provides a coherent framework to evaluate quantitatively the energy flux released at the tip of a growing crack. However, the way in which the crack chooses its path in response to this energy flux remains far from completely understood: the growing crack creates a structure on its own as conveyed by crack surface roughening even in brittle amorphous materials such as glass. We report here experiments designed to uncover the primary cause of surface roughening in brittle amorphous materials. Therefore, we investigate the response of a growing crack to local perturbation introduced as a shear wave pulse of controlled duration, amplitude, frequency and polarization. This pulse is shown to induce a local mode III perturbation in the loading of the crack front, which makes it twist locally, without fragmenting. This response is linear both in amplitude and frequency with respect to the perturbation, and disappears with it. We also show that shear wave pulses are emitted when the propagating crack encounters the heterogeneity. Implications of these observations for possible sources of roughening are finally discussed.     

Dynamic stress intensity factor due to concentrated loads on a propagating semi-infinite crack in orthotropic materials

The elastodynamic response of an infinite orthotropic material with a semi-infinite crack propagating at constant speed under the action of concentrated loads on the crack faces is examined. Solution for the stress intensity factor history around the crack tip is found for the loading modes I and II. Laplace and Fourier transforms along with the Wiener-Hopf technique are employed to solve the equations of motion. The asymptotic expression for the stress near the crack tip is analyzed which lead to a closed-form solution of the dynamic stress intensity factor. It is found that the stress intensity factor for the propagating crack is proportional to the stress intensity factor for a stationary crack by a factor similar to the universal function k(v) from the isotropic case. Results are presented for orthotropic materials as well as for the isotropic case.     

A damage to crack transition model accounting for stress triaxiality formulated in a hybrid nonlocal implicit discontinuous Galerkin‐cohesive band model framework

Modelling the entire ductile fracture process remains a challenge. On the one hand, continuous damage models succeed in capturing the initial diffuse damage stage but are not able to represent discontinuities or cracks. On the other hand, discontinuous methods, as the cohesive zones, which model the crack propagation behaviour, are suited to represent the localised damaging process. However, they are unable to represent diffuse damage. Moreover, most of the cohesive models do not capture triaxiality effect. In this paper, the advantages of the two approaches are combined in a single damage to crack transition framework. In a small deformation setting, a nonlocal elastic damage model is associated with a cohesive model in a discontinuous Galerkin finite element framework. A cohesive band model is used to naturally introduce a triaxiality‐dependent behaviour inside the cohesive law. Practically, a numerical thickness is introduced to recover a 3D state, mandatory to incorporate the in‐plane stretch effects. This thickness is evaluated to ensure the energy consistency of the method and is not a new numerical parameter. The traction‐separation law is then built from the underlying damage model. The method is numerically shown to capture the stress triaxiality effect on the crack initiation and propagation.     

Determination of masonry crack evolution due to differential displacements: a numerical study*

Brick walls of ceramics without any mortar covering or paint are used extensively in building façades in Spain. One of the most used masonry wall systems is based on non‐bearing panels partially supported, about two‐thirds of the brick width, over the edge beams of the structural skeleton. The edge beam is veneered with special thinner bricks to achieve the visual continuity of the façade. A considerable number of these walls show cracking. In a previous work, finite element simulations were performed in order to gain insight into the causes of cracking. A special finite element, based on the strong discontinuity analysis and the cohesive crack theory, is used in the numerical simulations. The results agree with the overall cracking patterns observed but if an imposed displacement is applied in the range allowed by the standards, extensive cracking occurs. This implies that the design displacements are not the actual ones. In this work, an elastic study using the principle of superposition is used to determine the effective deflections under service loading. Then, these deflections are applied to the structure and the evolution of cracking is studied. This study shows that the masonry panels of the first and last store have the major probability of cracking. Another parametric study is carried out changing the elastic and tensile properties of the masonry. This study shows that although the cracking of the masonry panels starts at different loads for different tensile properties, the crack patterns are similar for a given panel geometry and loading. This numerical study provides a method of design to determine the crack width for different geometries, loadings and fracture properties.     

A refined diffuse cohesive approach for the failure analysis in quasibrittle materials—part I: Theoretical formulation and numerical calibration

In this work, a refined interelement diffuse fracture theoretical model, based on a cohesive finite element approach, is proposed for concrete and other quasibrittle materials. This model takes advantage of a novel micromechanics‐based calibration technique for reducing the artificial compliance associated with the adopted intrinsic formulation. By means of this technique, the required values for the elastic stiffness parameters to obtain nearly invisible cohesive interfaces are provided. Furthermore, the mesh‐induced toughening effect, essentially related to the artificial crack tortuosity caused by the different orientations of the interelement cohesive interfaces, is numerically investigated by performing comparisons with an additional fracture model, newly introduced for the purpose of numerical validation. These comparisons are presented to assess the reliability and the numerical accuracy of the proposed fracture approach.     

An enriched FEM technique for modeling hydraulically driven cohesive fracture propagation in impermeable media with frictional natural faults: Numerical and experimental investigations

In this paper, an enriched finite element technique is presented to simulate the mechanism of interaction between the hydraulic fracturing and frictional natural fault in impermeable media. The technique allows modeling the discontinuities independent of the finite element mesh by introducing additional DOFs. The coupled equilibrium and flow continuity equations are solved using a staggered Newton solution strategy, and an algorithm is proposed on the basis of fixed‐point iteration concept to impose the flow condition at the hydro‐fracture mouth. The cohesive crack model is employed to introduce the nonlinear fracturing process occurring ahead of the hydro‐fracture tip. Frictional contact is modeled along the natural fault using the penalty method within the framework of plasticity theory of friction. Moreover, an experimental investigation is carried out to perform the hydraulic fracturing experimental test in fractured media under plane strain condition. The results of several numerical and experimental simulations are presented to verify the accuracy and robustness of the proposed computational algorithm as well as to investigate the mechanisms of interaction between the hydraulically driven fracture and frictional natural fault. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamic stress intensity factors due to scattering of harmonic waves by a crack in an infinite medium

An analytical method for calculating dynamic stress intensity factors in the mixed mode (combination of opening and sliding modes) using complex functions theory is presented. The crack is in infinite medium and subjected to the plane harmonic waves. The basis of the method is grounded on solving the two‐dimensional wave equations in the frequency domain and complex plane using mapping technique. In this domain, solution of the resulting partial differential equations is found in the series of the Hankel functions with unknown coefficients. Applying the boundary conditions of the crack, these coefficients are calculated. After solving the wave equations, the stress and displacement fields, also the J‐integrals are obtained. Finally using the J‐integrals, dynamic stress intensity factors are calculated. Numerical results including the values of dynamic stress intensity factors for a crack in an infinite medium subjected to the dilatation and shear harmonic waves are presented.