Modeling crack discontinuities without element‐partitioning in the extended finite element method

In this paper, we model crack discontinuities in two‐dimensional linear elastic continua using the extended finite element method without the need to partition an enriched element into a collection of triangles or quadrilaterals. For crack modeling in the extended finite element, the standard finite element approximation is enriched with a discontinuous function and the near‐tip crack functions. Each element that is fully cut by the crack is decomposed into two simple (convex or nonconvex) polygons, whereas the element that contains the crack tip is treated as a nonconvex polygon. On using Euler's homogeneous function theorem and Stokes's theorem to numerically integrate homogeneous functions on convex and nonconvex polygons, the exact contributions to the stiffness matrix from discontinuous enriched basis functions are computed. For contributions to the stiffness matrix from weakly singular integrals (because of enrichment with asymptotic crack‐tip functions), we only require a one‐dimensional quadrature rule along the edges of a polygon. Hence, neither element‐partitioning on either side of the crack discontinuity nor use of any cubature rule within an enriched element are needed. Structured finite element meshes consisting of rectangular elements, as well as unstructured triangular meshes, are used. We demonstrate the flexibility of the approach and its excellent accuracy in stress intensity factor computations for two‐dimensional crack problems. Copyright © 2016 John Wiley & Sons, Ltd.     

Characteristics of fatigue crack growth in GFRP laminates

Multiple fatigue crack growth behaviour has been studied in quasi-isotropic GFRP laminates under constant amplitude fatigue loading conditions. Characteristics of fatigue crack growth in off-axis plies have been described and comparisons have been made between quasi-static and fatigue crack growth behaviour. Careful monitoring of individual fatigue cracks reveals three distinct stages of crack growth including initiation, steady-state crack growth (SSCG) and crack interaction and saturation. Stress redistribution due to matrix cracking and the associated stiffness reduction have been simulated using finite element models. Strain energy release rates associated with the off-axis matrix cracking have also been obtained and correlated with the measured fatigue crack growth rates.     

A two-dimensional damaged finite element for fracture applications

A novel finite element was developed for fatigue and fracture applications. The new element is two-dimensional with an embedded edge crack. The crack is not physically modeled within the element, but instead, its influence on the local flexibility of the structure is accounted for by the reduction of the element stiffness as a function of the crack length. The components of the stiffness matrix for the cracked element are determined from the Castigliano’s first principle. The element was implemented in the commercial finite element code ABAQUS as a user element (UEL) subroutine. Models using the UEL are shown to produce accurate results when compared with results from traditional models using physical modeling of the crack. The newly developed element is useful for studies focused on the global response of a structure, and does not allow evaluation of the local stress singularity near the crack tip. An advantage of the developed UEL is that the singularity at the crack tip does not need to be captured accurately with a significant number of elements.     

Complete Tangent Stiffness for eXtended Finite Element Method by including crack growth parameters

The eXtended Finite Element Method (XFEM) is a useful tool for modeling the growth of discrete cracks in structures made of concrete and other quasi‐brittle and brittle materials. However, in a standard application of XFEM, the tangent stiffness is not complete. This is a result of not including the crack geometry parameters, such as the crack length and the crack direction directly in the virtual work formulation. For efficiency, it is essential to obtain a complete tangent stiffness. A new method in this work is presented to include an incremental form the crack growth parameters on equal terms with the degrees of freedom in the FEM‐equations. The complete tangential stiffness matrix is based on the virtual work together with the constitutive conditions at the crack tip. Introducing the crack growth parameters as direct unknowns, both equilibrium equations and the crack tip criterion can be handled within the same standard nonlinear iterations. This new solution strategy is believed to provide the modeling capabilities to deal with simultaneous growth of several cracks. A cohesive crack modeling is used. The method is applied to a partly cracked XFEM element of linear strain triangle type with the crack length as the unknown crack growth parameter. In this paper, two examples are given. The first example verifies the theory and the implementation. The second example is the benchmark test three point bending test, where the efficiency of the complete tangential behavior is shown. Copyright © 2013 John Wiley & Sons, Ltd.     

FATIGUE DAMAGE AND SHAKEDOWN IN METAL MATRIX COMPOSITE LAMINATES

SUMMARY

Fatigue failure of metal matrix composite laminates is often preceded by a substantial loss of stiffness associated with cyclic plastic straining and subsequent low-cycle fatigue crack growth in the matrix. Experimental observations suggest that two principal crack patterns are involved; these are related here to the deformation modes predicted by the bimodal plasticity theory of fibrous composites. The relation is utilized in modelling the damage process such that matrix crack growth is regarded as a shakedown mechanism leading to a saturation damage state. For a given program of variable cyclic loading, evaluation of the saturation state is formulated as a non-linear optimization problem, where the total damage in a laminate is minimized subject to non-linear constraints derived from the ply yield criterion, hardening rule, and physically motivated bounds on the damage parameters. Effective elastic stiffness reduction and local stress redistribution predicted by the optimization procedure are compared with experimental measurements on several B/AI laminates. Stress transfer to and overloading of the fibres in certain plies appears to cause final fatigue failure of the laminate.     

Fatigue growth prediction of center slant crack in rectangular plate

This paper presents a numerical prediction model of mixed‐mode crack fatigue growth in a plane elastic plate. It involves a formulations of fatigue growth of multiple crack tips under mixed‐mode loading and a displacement discontinuity method with crack‐tip elements (a boundary element method) proposed recently by Yan is extended to analyse the fatigue growth process of multiple crack tips. Due to an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single‐region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is conveniently modelled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characters of some related elements are adjusted according to the manner in which the boundary element method is implemented. As an example, the present numerical approach is used to analyse the fatigue growth of a centre slant crack in a rectangular plate. The numerical results illustrate the validation of the numerical prediction model and can reveal the effect of the geometry of the cracked plate on the fatigue growth.     

Effect of microcracking on the micromechanics of fatigue crack growth in austempered ductile iron

The effect of microcracking on the mechanics of fatigue crack growth in austempered ductile iron is studied in this paper. The mechanism of fatigue crack growth is modelled using the boundary element method, customized for the accurate evaluation of the interaction effects between cracks and microcracks emanating from graphite nodules. The effects of nodule size and distribution and crack closure are considered, with deviation bounds of computed results estimated through weight-function analyses. A continuum approach is employed as a means of quantifying the shielding effect of microcracking on the dominant propagating crack, due to the reduction of stiffness of the material in the neighbourhood of the crack tip. Although the results obtained may not yield actual numbers for real cases, they are in accordance with experimental observations and demonstrate how the main factors affect the crack growth of the macrocrack.     

Investigating the effects of loading,mechanical properties and layers geometry on fatigue life of asphalt pavements

In this paper, fatigue life, crack growth trajectory, and stress intensity factors of top‐down and bottom‐up cracks have been investigated in a multilayered asphalt pavement using numerical method. Finite element models, Paris law, and maximum tangential stress criteria were used to determine the effects of vehicle position, thickness, and stiffness of layers on fatigue life of pavements. The results show that the minimum fatigue life of pavements with top‐down crack occurs when the wheels are symmetrically located relative to the crack plane. On the other side, in pavements with bottom‐up crack, it occurs when the front tire located on top of the crack. Moreover, the result reveals that the top‐down crack grows slower than bottom‐up crack. Also, the surface layer's stiffness has the most effect on fatigue life of pavement.     

Damage indices for failure of concrete beams under fatigue

In this study, an analytical method is proposed to correlate local damage variables such as relative crack depth and crack tip opening displacement with a newly defined global damage index for a concrete beam under fatigue loading. This global damage index may be used to assess the response of a degraded concrete beam under service loading. The damage is assumed to appear in the form of a major crack that propagates under constant amplitude fatigue loading. The progressive cracking phenomenon is modeled within a finite element framework using a crack beam element, which takes into account the compliance variation due to discrete cracking within the member. The flexural stiffness degradation of the member is computed based on an Eigen analysis of the global stiffness matrix. It is seen that the degree of flexural stiffness degradation due to discrete cracking is the same for geometrically similar specimens when the relative crack depth is used as a local damage parameter. Further, in order to improve the accuracy of the response prediction using the above global damage index, another global damage parameter is defined based on the nature of applied loading.     

A stabilization technique for the regularization of nearly singular extended finite elements

In this contribution a simple, robust and efficient stabilization technique for extended finite element (XFEM) simulations is presented. It is useful for arbitrary crack geometries in two or three dimensions that may lead to very bad condition numbers of the global stiffness matrix or even ill-conditioning of the equation system. The method is based on an eigenvalue decomposition of the element stiffness matrix of elements that only possess enriched nodes. Physically meaningful zero eigenmodes as well as enrichment scheme dependent numerically reasonable zero eigenmodes are filtered out. The remaining subspace is stabilized depending on the magnitude of the respective eigenvalues. One of the main advantages is the fact that neither the equation solvers need to be changed nor the solution method is restricted. The efficiency and robustness of the method is demonstrated in numerous examples for 2D and 3D fracture mechanics.     

A methodology for predicting creep/fatigue crack growth rates in Ti 6246

A methodology has been developed which is capable of predicting creep/fatigue crack growth rates at ambient and elevated temperatures in Ti 6246. Predictions are based on finite element analysis and strain-control testing of plain specimens. The prediction of fatigue crack growth rates for a given crack configuration and cyclic plastic zone size is assumed to be consistent with the processes leading to crack initiation in plain specimens. Such an assumption leads to the conclusion that a similar stress–strain profile will lead to similar lives in both the plain specimens and in the cyclic plastic zone ahead of a crack in a notched specimen. Therefore, fatigue crack growth results from the accumulation of damage in the cyclic plastic zone ahead of the crack tip. Once the damage accumulated in this element of material becomes critical, the crack propagates through the damaged region into a new region of virgin material where the process of damage accumulation begins again. The creep/fatigue model is described and assessed with reference to measured fatigue crack growth rate data for Ti 6246 at 20 °C and 500 °C.     

Adherend thickness influence on fatigue behavior and fatigue failure prediction of adhesively bonded joints

The effect of the adherend thickness, h, on mode-I fatigue behavior of a toughened epoxy adhesive system was examined in terms of the substrate global stiffness and curing residual stress. It was found that a change in adherend thickness from 1.6 mm to 12.7 mm caused a reduction in the fatigue performance; i.e. the threshold energy release rate decreased and the crack growth rate increased for a given applied energy release rate. Finite element modeling showed that the fatigue results could be explained in terms of an increase in the crack tip stresses and an enlarged plastic zone due to the greater global stiffness of thicker joints. No difference in fatigue behavior was observed for mixed-mode loading at relatively small phase angles; however, it is expected that at higher phase angles the adverse effect of h would be observed.     

Extended finite element method and fast marching method for three-dimensional fatigue crack propagation

A numerical technique for planar three-dimensional fatigue crack growth simulations is proposed. The new technique couples the extended finite element method (X-FEM) to the fast marching method (FMM). In the X-FEM, a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modeled by finite elements with no explicit meshing of the crack surfaces. The initial crack geometry is represented by level set functions, and subsequently signed distance functions are used to compute the enrichment functions that appear in the displacement-based finite element approximation. The FMM in conjunction with the Paris crack growth law is used to advance the crack front. Stress intensity factors for planar three-dimensional cracks are computed, and fatigue crack growth simulations for planar cracks are presented. Good agreement between the numerical results and theory is realized.     

An enriched element‐failure method (REFM) for delamination analysis of composite structures

This paper develops an enriched element‐failure method for delamination analysis of composite structures. This method combines discontinuous enrichments in the extended finite element method and element‐failure concepts in the element‐failure method within the finite element framework. An improved discontinuous enrichment function is presented to effectively model the kinked discontinuities; and, based on fracture mechanics, a general near‐tip enrichment function is also derived from the asymptotic displacement fields to represent the discontinuity and local stress intensification around the crack‐tip. The delamination is treated as a crack problem that is represented by the discontinuous enrichment functions and then the enrichments are transformed to external nodal forces applied to nodes around the crack. The crack and its propagation are modeled by the ‘failed elements’ that are applied to the external nodal forces. Delamination and crack kinking problems can be solved simultaneously without remeshing the model or re‐assembling the stiffness matrix with this method. Examples are used to demonstrate the application of the proposed method to delamination analysis. The validity of the proposed method is verified and the simulation results show that both interlaminar delamination and crack kinking (intralaminar crack) occur in the cross‐ply laminated plate, which is observed in the experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

A mixed augmented Lagrangian-extended finite element method for modelling elastic–plastic fatigue crack growth with unilateral contact

The complete modelling of fatigue crack growth is still an industrial challenging issue for numerical methods. A new technique for the finite element modelling of elastic–plastic fatigue crack growth with unilateral contact on the crack faces is presented. The extended finite element method (X-FEM) is used to discretize the equations, allowing for the modelling of arbitrary cracks whose geometries are independent of the finite element mesh. This paper presents an augmented Lagrangian formulation in the X-FEM framework that is able to deal with elastic–plastic crack growth with treatment of contact. An original formulation, which takes advantages of two powerful numerical methods, is presented. Next the numerical issues such as contact treatment and numerical integration are addressed, and finally numerical examples are shown to validate the method. Copyright © 2007 John Wiley & Sons, Ltd.     

Investigation of 3D crack propagation problems via fast BEM formulations

Abstact The simulation of 3D fatigue crack growth is investigated and in combination with suitable experiments it is aimed to identify a reliable crack growth criterion. To perform the crack growth simulation as effectively as possible the boundary element method (BEM) in terms of the 3D Dual BEM is utilized. The relevant boundary integral equations are evaluated in the framework of a collocation procedure. To compress the system matrix and to speed-up the solution procedure the adaptive cross approximation (ACA) method is applied. It is an algebraic technique acting purely on the system matrix itself. The efficiency of this procedure with respect to crack problems will be shown on both a standard fracture specimen and an industrial application.     

Fatigue crack growth in thin notched woven glass composites under tensile loading. Part II: Modelling

Fatigue propagation of a through-the-thickness crack in thin woven glass laminates is difficult to model when using homogeneous material assumption. Crack growth depends on both the fatigue behaviour of the fibres and of the matrix, these two phenomena occurring at different time and space scales. The developed finite element model is based on the architecture of the fabric and on the fatigue behaviours of the matrix and the fibre, even if the pure resin and fibre behaviours are not used. That thus limits the physical meaning of this model. Basically, the objective of this simulation is to illustrate and to confirm proposed crack growth mechanism. The fatigue damage matrix is introduced with user spring elements that link the two fibre directions of the fabric. Fibre fatigue behaviour is based on the S-N curves. Numerical results are compared to experimental crack growth rates and observed damage in the crack tip. Relatively good agreement between predictions and experiments was found.     

Probabilistic fatigue crack growth analysis of spot‐welded joints

This paper presents a probabilistic fatigue crack growth life prediction methodology for spot‐welded joints under variable amplitude loading history. The loading is multi‐axial and is obtained from transient response analysis of a vehicle model using finite‐element analysis. A three‐dimensional (3D) finite element model of a simplified joint with four spot welds is developed, and the static stress analysis of this joint is performed. Then the fatigue crack inside the base material sheet is modelled as a surface crack. Probabilistic crack growth model is combined with the stress analysis result to develop a probabilistic fatigue crack growth life prediction methodology for spot welds. This new method is implemented with MSC/NASTRAN and MSC/FATIGUE and is useful for the reliability assessment of spot‐welded joints against fatigue crack growth.     

Experimental and numerical study of fatigue crack growth of aluminum alloy 2024-T3 single lap simple bolted and hybrid (adhesive/bolted) joints

In this paper, the effects of tightening torque (clamping force) on the fatigue crack growth rate and stress intensity factors in cracked single lap simple bolted and hybrid (adhesive/bolted) joints have been studied experimentally and numerically. To do so, series of fatigue crack growth tests for two different amounts of tightening torque in Aluminum alloy 2024-T3 pre-cracked joints have been carried out to record the fatigue crack growth and also the fatigue life of specimens. In the numerical part, finite element method was employed to obtain the stress intensity factors and also the effective stress intensity factor ranges for different crack lengths to explain the behavior of fatigue crack propagation. It was found that the hybrid joint has longer fatigue crack growth life compared to the simple bolted joint at a given bolt tightening torque. The results also showed that a higher bolt tightening torque provides improved fatigue crack growth life for both types of the joints.     

Fatigue crack propagation between holes and particles

This study investigates the behaviour of a fatigue crack, propagating between holes or particles. These inhomogeneities are found to play a significant role on the fatigue propagation behaviour. The study is performed by simulations, both experimentally and numerically. The experiments show that holes or partly debonded particles on the average accelerate the crack growth, even though they may exert a decelerating effect, due to shielding, during some phase of a crack tip passage between the holes or particles. Besides debonding, a high material stiffness ratio between matrix and particles also tends to accelerate fatigue crack growth. The numerical simulations are able to explain the essential experimental results quite satisfactorily, even though a few seemingly anomalous experimental results are still not convincingly explained.