Numerical study of mixed-mode fracture in concrete

In the present paper, a finite element code based on the microplane model for concrete is used for the analysis of typical mixed-mode geometries: a Single-Edge-Notched beam, a Double-Edge-Notched specimen and a Dowel-Disk specimen. A local smeared fracture finite element analysis is carried out. As a regularization procedure, the crack band method is used. The principal objective of the study was to investigate whether the smeared fracture finite element code is able to predict mixed-mode fracture of concrete with no optimisation of the material model parameters. Comparison between experimental and numerical results shows that the used code predicts structural response and crack patterns realistically for all cases investigated. Moreover, it is shown that for most of the studied geometries a mixed-mode fracture mechanism dominates at crack initiation, however, with increase of the crack length mode-I fracture becomes dominant and the specimens finally failed in mode-I fracture.     

A numerical analysis of mixed-mode delamination in carbon–epoxy prepregs

Several material models can be found in the literature for analysing mixed-mode free edge delamination in laminated carbon–epoxies. Although most of the models predict the laminate behaviour rather well, they are usually not very robust in a numerical analysis. In this paper three material models are assessed. It is shown that a material model based on the formalism of orthotropic damage mechanics performs best from a computational point of view. The model can easily be implemented into a finite element code and shows a robust behaviour in numerical analyses. The delamination growth can be traced over a long range, with relatively large loading steps. The effect of some material parameter variations is studied, to obtain the sensitivity to measurement errors.     

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.     

A geometric nonlinear triangular finite element with an embedded discontinuity for the simulation of quasi-brittle fracture

This paper is aimed to model the appearance and evolution of discrete cracks in quasi-brittle materials using triangular finite elements with an embedded interface in a geometric nonlinear setting. The kinematics for the discontinuous displacement field is presented and the standard variational formulation with respect to the reference configuration is extended to a body with an internal discontinuity. Special attention is paid to the algorithmic treatment. The discontinuity is modeled by additional global degrees of freedom and the continuity of the displacements across the element boundaries is enforced. Finally, representative numerical examples for mode-I and mixed-mode fracture, namely a tension test, different three-point bending tests and a single edge notched beam with structured and unstructured finite element meshes are discussed to study the evolving crack pattern and to show the ability of the model.     

Progressive delamination growth analysis using discontinuous layered element

In this paper, a fracture mechanic approach is used to analyze delamination propagation between layers of composite laminates. A finite element method based on layer-wise theory is extended for the analysis of delamination growth. In this approach, delamination is modeled by jump discontinuity conditions at the interfaces. The layer-wise finite element is developed to calculate the strain energy release rates based on the virtual crack closure technique (VCCT). A procedure is proposed to handle the progressive delamination of laminates. Finally, analyses of the edge delamination propagation for several composite laminates are performed and the corresponding failure stresses are calculated. The predicted results are compared with the available experimental and numerical results. It is shown that the predicted failure stresses using this method are comparable with those obtained using interface elements.     

Mixed mode fracture separation in viscoelastic orthotropic media: numerical and analytical approach by the <Emphasis Type="Italic">M</Emphasis>θ<Emphasis Type="Italic">v</Emphasis> -integral

An analytical and numerical procedure based on an independent integral path and finite element analysis for mixed-mode fracture in viscoelastic orthotropic media is developed. The separated method employs virtual mechanics fields induced by the classical singular analytical forms. The viscoelastic generalization uses a thermodynamic approach by defining an energy release rate only taking into account a perfect uncoupling between free and viscous energies. The implementation of the Mθ-integral in finite element software and its integration into the viscoelastic incremental formulation are presented. As results, the analytical and numerical solutions are compared by the way of the energy release rate in pure mode I, pure mode II and mixed modes. In shows that, the developed model lead to accurate and efficient separated fracture mode in viscoelastic materials.     

Orthotropic enriched element free Galerkin method for fracture analysis of composites

A new approach for modeling discrete cracks in two-dimensional orthotropic media by the element free Galerkin method is described. For increasing the solution accuracy, recently developed orthotropic enrichment functions used in the extended finite element method are adopted along with a sub-triangle technique for enhancing the Gauss quadrature accuracy near the crack. An appropriate scheme for selecting the support domains near a crack is employed to reduce the computational cost. In this study, mixed-mode stress intensity factors are obtained by means of the interaction integral to determine the fracture properties. Several problems are solved to illustrate the effectiveness of the proposed method and the results are compared with available results of other numerical or (semi-) analytical methods.     

Three-dimensional finite element analysis of fracture modes for the pull-off test of a thin film from a stiff substrate

A three-dimensional geometrically nonlinear finite element analysis model is presented to study the interfacial delamination for the pull-off test of a thin film strip debonded from a stiff substrate. The strain energy release rates of all three modes (mode I, mode II, and mode III) along the debond front are considered and calculated to investigate the mixed fracture modes for the entire deformation regime from bending plate to stretching membrane. These results indicate that the individual strain energy release rates and the total energy release rate vary along the width of the debond front and strong three-dimensional edge effects exist near the free edges of the film. Interestingly, residual stress also plays an important role in controlling mixed fracture modes and the variation of the energy release rates. Finally, the three-dimensional finite element model is compared with an analytical solution developed earlier. The three-dimensional finite element model is found to provide additional insights for interfacial delamination for the pull-off test.     

An interaction integral method for analysis of cracks in orthotropic functionally graded materials

This paper presents two new interaction integrals for calculating stress-intensity factors (SIFs) for a stationary crack in two-dimensional orthotropic functionally graded materials of arbitrary geometry. The method involves the finite element discretization, where the material properties are smooth functions of spatial co-ordinates and two newly developed interaction integrals for mixed-mode fracture analysis. These integrals can also be implemented in conjunction with other numerical methods, such as meshless method, boundary element method, and others. Three numerical examples including both mode-I and mixed-mode problems are presented to evaluate the accuracy of SIFs calculated by the proposed interaction integrals. Comparisons have been made between the SIFs predicted by the proposed interaction integrals and available reference solutions in the literature, generated either analytically or by finite element method using various other fracture integrals or analyses. An excellent agreement is obtained between the results of the proposed interaction integrals and the reference solutions. The authors would like to acknowledge the financial support of the U.S. National Science Foundation (NSF) under Award No. CMS-9900196. The NSF program director was Dr. Ken Chong.     

A damage model for simulation of mixed-mode delamination growth

An interface element capable of modelling delamination progression under mixed-mode loading is presented. The kinematics of the element are based on the concept of regularised displacement discontinuity. This concept allows the interfacial constitutive equations to be formulated in terms of the traction vector of the interface and the corresponding displacement discontinuity. The decohesion within the interface, corresponding to delamination progression, is accomplished by assigning a non-associative perfectly plastic material model including isotropic damage to the interface element. All parameters of the model can be determined from experimental material data. Damage initiation is calibrated against the interlaminar fracture stresses whereas the evolution of damage is calibrated against the mixed-mode fracture toughness. The interface element has been implemented in a finite element code and results for simulations of standard fracture toughness tests are shown. The results display the applicability of the proposed model and the calibration procedure.     

Stress intensity factors for cracks in bolted joints

The mixed-mode stress intensity factors (SIFs) of the bolted joint with single and double cracks were examined. Changes in friction, clearance, applied force and crack angle were included in the nonlinear contact finite element analysis. A fine mesh was made between the contact surface and the crack tip in order to obtain an accurate solution. The least-squares method was used to determine the mixed-mode SIFs. Finite element results indicate that reasonable changes in the applied force, frictional coefficient and the clearance will not make significant changes in the normalized SIFs. The pure opening mode for cracked bolted joints does not occur at the horizontal crack but occurs at the crack with the crack angle between 0° and 22.5°. Nevertheless, using the SIF for a horizontal crack as the maximum opening-crack mode is sufficiently reliable. The maximum mode-II crack is approximately at a crack angle of 45° for both isotropic and orthotropic materials; however, at that angle the maximum mode-II SIF is only about one half of the mode-I SIF. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mixed-Mode Delamination – Experimental and Numerical Studies

Abstract: A fracture energy approach for modelling mixed-mode delamination of composite materials and other bonded structures is introduced. The model is incorporated within an explicit finite element (FE) code and ties layered shell elements together via a stiffness condition, a failure criterion and post-failure damage law. The procedure for predictive modelling of delamination using the approach is described and the set of required input parameters is presented. A benchmark test comparing experimental results for a continuous filament random E-glass/polyester composite and explicit FE simulations for standard fracture toughness tests for a range of mode mixities is included.     

Simulation of delamination by means of cohesive elements using an explicit finite element code

This paper presents the formulation of a tri-dimensional cohesive element implemented in a user-written material subroutine for explicit finite element analysis. The cohesive element simulates the onset and propagation of the delamination in advanced composite materials. The delamination model is formulated by using a rigorous thermodynamic framework which takes into account the changes of mixed-mode loading conditions. The model is validated by comparing the finite element predictions with experimental data obtained in interlaminar fracture tests under quasi-static loading conditions.     

Overcoming the cohesive zone limit in composites delamination: modeling with slender structural elements and higher-order adaptive integration

Cohesive element (CE) is a well-established finite element for fracture, widely used for the modeling of delamination in composites. However, an extremely fine mesh is usually needed to resolve the cohesive zone, making CE-based delamination analysis computationally prohibitive for applications beyond the scale of lab coupons. In this work, a new CE-based method of modeling delamination in composites is proposed to overcome this cohesive zone limit on the mesh density. The proposed method makes use of slender structural elements for the plies, a compatible formulation with adaptive higher-order integration for the CEs, and the corotational formulation for geometrically nonlinear analysis. The proposed method is verified and validated on the classical benchmark problems of Mode I, II, mixed-mode delamination, a buckling-induced delamination problem and a double-delamination problem. The results show that elements much larger than the cohesive zone length can be used while retaining accuracy.     

A new M-integral parameter for mixed-mode crack growth in orthotropic viscoelastic material

This paper deals with a new independent path integral which provides the mixed-mode during a creep crack growth process in viscoelastic orthotropic media. The developments are based on an energetic approach using conservative laws. The mixed-mode fracture separation is introduced according to the generalization of the virtual work principle. The fracture algorithm is implemented in a finite element software and coupled with an incremental viscoelastic formulation and an automatic crack growth simulation. This M-integral provides the computation of stress intensity factors and energy release rate for each fracture mode. A numerical validation, in terms of energy release rate and stress intensity factors, is carried out on a CTS specimen under mixed-mode loading for different crack growth speeds.     

Experimental investigation and finite element modeling of mixed-mode delamination in a moisture-exposed carbon/epoxy composite

An investigation of the effects of moisture on mixed-mode I/II delamination growth in a carbon/epoxy composite is presented. Experimental quasi-static and fatigue delamination tests were carried out on composite specimens. The quasi-static fracture test results showed that exposure to moisture led to a decrease in mode II and mixed-mode delamination toughness while mode I toughness was enhanced. The fatigue tests revealed an adverse effect of moisture on delamination growth under mixed-mode loadings. Existing delamination criteria and growth rate models were evaluated to determine which ones best predict delamination toughness and growth, respectively, at any given mixed-mode ratio. Quasi-static and fatigue simulations with a cohesive zone-based finite element model that incorporated the selected mixed-mode delamination models were performed and good agreement between experimental and numerical data was shown for dry and moisture-exposed specimens.     

Effect of delamination face overlapping on strain energy release rate calculations

The strain energy release rate is often used as a fracture mechanics parameter to describe delamination propagation and onset in composites, and is conveniently evaluated using finite element analysis. A common problem encountered in analysis is overlapping (interpenetration) of the delamination faces if these faces are not constrained. This paper examines the effect of overlapping on strain energy release rate calculated using the virtual crack closure method in conjunction with 3-dimensional anisotropic finite element analysis. A bilinear gap element was used between the delamination faces to prevent these faces from overlapping. Three problems were studied: (1) laminates with an embedded crack and embedded delaminations; (2) laminates with free edge delaminations; and (3) end-notched flexure specimens. The results of this investigation indicated that overlapping has a significant effect on the component values of G_I and G_II as well as on the total strain energy release rate. It was also found that end effects can create non-uniform energy release rates along the crack front in edge delamination problems due to twisting of the sublaminate with unsymmetrical lay-up, so that 3-dimensional finite element analysis is required.     

A THEORETICAL AND EXPERIMENTAL INVESTIGATION OF DYNAMIC DELAMINATION IN COMPOSITES

Abstract A fundamental understanding of dynamic delamination in composites is sought through the application of theoretical and experimental approaches familiar to dynamic fracture mechanics. Analysis of steady-state fracture in an infinite orthotropic strip yields a simple solution which can be used to evaluate numerical procedures and experimental results. The analogous specimen consists of a single edge notched composite strip bonded to stiff steel substrates to enforce the desired displacement boundary conditions. Delamination velocities of the order of 10 to 1000 m/s were measured using a graphite gauge technique. Quasi-static and dynamic finite element methods are applied to investigate the behavior of the specimen and to determine static initiation and dynamic delamination toughness. The experimental observations cannot be explained by linear elastic fracture theory. The absence of a unique G(?) relationship might be rationalized by a simple model relating matrix crack zone size to fiber bridging mechanisms.     

Stress intensity factors as the fracture parameters for delamination crack growth in composite laminates

A “mutual integral” approach is used to calculate the mixed-mode stress intensity factors for a free-edge delamination crack in a laminate under tensile loading conditions. This “mutual integral” approach, for generalized plane strain conditions, is based on the application of the path-independent J integral to a linear combination of three solutions: one, the problem of the laminate to be solved using the quasi 3-D finite element method, the second, an “auxiliary” solution with a known asymptotic singular solution, and the third, the particular solution due to the out-of-plane loading. A comparison with the exact solutions is made to determine the accuracy and efficiency of this numerical method. With this “mutual integral” approach, it was found that the calculated mixed-mode stress intensity factors of the free-edge delamination crack remain relatively constant as the crack propagates into the laminate. It was also found that the fracture criterion based on the mixed-mode stress intensity factors is more consistent with the experimental observations than the criterion based on the total energy release rate, and hence demonstrates the importance of the ability to calculate each individual component of the stress intensity factors. Furthermore, it was found that the fracture toughness measurements from double cantilever beam specimens can be used directly to predict the onset of delamination crack growth between two dissimilar laminae. Using these fracture toughness measurements from the double cantilever beam specimens, some examples are given to show that the fracture criterion based on the mixed-mode stress intensity factors can accurately predict the failure load for various laminates under tensile loading conditions.     

An investigation into the mixed-mode delamination growth in unidirectional T800/924C laminates

This study presents the results of experimental investigations and numerical simulation on mixed-mode I/II delamination growth initiated from an artificial transverse notch. Specimens made of unidirectional carbon fiber epoxy (T800/924C) composite have been tested under three-point-bend condition. A finite element procedure has been introduced to model 3-D stable delamination growth in the specimen to generate numerical growth data including loads, displacements, delamination lengths, and the growing crack front shapes. The simulation method uses strain energy release rate criterion in conjunction with a moving mesh facility. It is shown that very good compatibility exists between experimental and numerical results. A finite element-based data reduction method is then described as an application of the simulation procedure. Based on the obtained results, it is stated that this bending specimen can effectively be used in practice to study the mixed-mode crack growth and to measure interlaminar fracture toughness of unidirectional laminates.