Virtual Crack Extension Technique Research Articles

Review and synthesis of stress intensity factor (SIF) solutions for elliptical surface cracks in bolts under tension loading: A Tribute to Juan de la Cierva

This paper presents stress intensity factor (SIF or K) solutions for a semi-elliptical surface crack in a bolt subjected to tension. Solutions were numerically computed by a 3D finite element method (FEM) with quarter-point singular isoparametric elements along the crack line. The SIF was calculated through the stiffness derivative method, by using a virtual crack extension technique to compute the energy release rate. Solutions are compared in the matter of stress intensity values at the center of the crack front and at the external surface of the bolt, and a wide range of crack aspect ratios and relative crack depths are studied, in order to provide the engineer with solutions applicable to different real situations in fracture mechanics or damage tolerance analyses such as static or cyclic (fatigue) loading, environmentally assisted cracking or corrosion-fatigue.In Memoriam: The paper is dedicated to the memory of the prominent Spanish engineer Juan de la Cierva, inventor of “autogiro”.

Review and synthesis of stress intensity factor (SIF) solutions for elliptical surface cracks in bolts under nut loading: A Tribute to Agustín de Betancourt

This paper presents stress intensity factor (SIF or K) solutions for a semi-elliptical surface crack in a bolt subjected to remote tension and nut loading. Solutions were numerically computed by a 3D finite element method (FEM) with quarter-point singular isoparametric elements along the crack line. The SIF was calculated through the stiffness derivative method, by using a virtual crack extension technique to compute the energy release rate. Solutions are compared in the matter of SIF values at the center of the crack front and at the external surface of the bolt, and a wide range of crack aspect ratios and relative crack depths are studied, in order to provide the engineer with solutions applicable to different real situations in fracture mechanics or damage tolerance analyses such as static or cyclic (fatigue) loading, environmentally assisted cracking or corrosion-fatigue. Results of the analysis are relevant to determine the crack path evolution in threaded connections.In Memoriam: The paper is dedicated to the memory of the prominent Spanish engineer Agustín de Betancourt.

Weight function calculation method for analyzing mixed-mode shear cracks in reinforced concrete beams

Based on a virtual crack extension (VCE) technique in finite element method (FEM), this paper proposes a rational method of calculating weight functions for inclined shear cracked RC beams with any geometry. As an inclined shear crack in an RC beam is geometrically determined by the absolute dimensions, shear span to beam depth ratio, crack inclined angle and crack initiation location, the relations between the weight functions and these parameters are investigated by either following theoretical equations or varying the parameters in a certain range. The range is determined to enclose the geometries of inclined critical shear cracks of a large set of RC beams which were tested under a bending load but failed due to the propagation of a critical inclined shear crack. Moreover, a method of calculating the weight functions of a shear cracked RC beam with any geometrical properties is developed numerically using the weight functions calculated and provided in this study. As a result, it is possible that the shear behaviors of RC beams can be even studied conveniently but also comprehensively following the theoretical fracture mechanic approach.

Computation of mode I strain energy release rate of symmetrical and asymmetrical sandwich structures using mixed finite element

The use of composite materials is on the rise in different engineering fields, the main advantage of these materials for the aerospace industry is their low weight for excellent mechanical qualities. The analysis of failure modes, such as delamination, of these materials has received great attention from researchers. This paper proposes a method to evaluate the mode I Strain Energy Release Rate (SERR) of sandwich structures. This method associated a two-dimensional mixed finite element with virtual crack extension technique for the analysis of interfacial delamination of sandwich beams. A symmetrical Double Cantilever Beam (DCB) and asymmetrical Double Cantilever Beam (UDCB) have been analyzed in this study. The comparison of the results obtained by this method and those found in the literature shows efficiency and good precision for the calculation of Strain Energy Release Rate (SERR).

Mixed Finite Element for Kinking Crack Analysis in an Orthotropic Media

Due to the attention brought to the composites materials, it is becoming mandatory to study their mechanical behavior especially in the fracture mechanics cases to determine and follow their conduct in a damaged state. To achieve this, numerical methods have been adapted to simulate the phenomenon of fracture within different materials and in different cases. In this work, an enrichment for modelling kinking cracks in orthotropic media has been added to a newly developed two-dimensional mixed finite element. To evaluate the crack propagation the virtual crack extension technique is used to calculate the energy release rates for this mixed finite element. The element is configured from parent element in a natural (ξ, η) plane to simplify the calculations in each element of the simulation. An example taken from literature with kinking cracks in an orthotropic material is used, the results prove the efficiency of the proposed method.

A virtual crack extension method for thermoelastic fracture using a complex-variable finite element method

A virtual crack extension (VCE) technique using the complex-variable finite element method (ZFEM) has been developed and demonstrated for thermoelastic fracture problems. This VCE approach provides an accurate computation of the energy release rate (G) as a byproduct of the complex-variable finite element analysis and obviates the use of energy conservation integrals such as the J-integral. The crack extension is induced using imaginary coordinates and a single global estimate of G is obtained as a sum of the element contributions. Numerical examples are provided for 2D, including a bimaterial cracked plate, axisymmetric, and 3D structures. The results indicate that the complex-variable VCE approach is of the same accuracy as the J-integral formulation.

Weight function determinations for shear cracks in reinforced concrete beams based on finite element method

An efficient approach of evaluating the weight functions for the generally load applied faces of shear cracked RC beams with varying shear span/beam depth ratios is presented using the corresponding weight functions for the RC beams with two shear span/beam depth ratios. The shear cracked RC beam under applied load can be considered as an oblique edge crack elastic geometry under mixed load conditions if the nonlinear bridging force is evaluated according to a crack bridging model separately. With pure bend load conditions, both Mode I and II weight functions for the oblique edge crack geometries with three shear span/beam depth ratios for different crack length/beam depth ratios are evaluated through applying the Virtual Crack Extension technique with symmetric mesh around the crack-tip into finite element analysis, where all weigh function components exhibit orderly trends with respect to the changing shear span/beam depth. As a result, the weight functions for the shear crack RC beams with the other shear span/beam depth ratios can be calculated following the orderly trends. Since the crack-tip singular behaviors of fracture problems is observed only in the primary crack-face nodal weight functions, to facilitate application, the primary crack-face weight functions are formulated accurately as a function of crack length and distance away from the crack-tip employing the fitting and interpolation methods.

A decomposition technique of generalized degrees of freedom for mixedmode crack problems

SummaryThe numerical manifold method (NMM) builds up a unified framework that is used to describe continuous and discontinuous problems; it is an attractive method for simulating a cracking phenomenon. Taking into account the differences between the generalized degrees of freedom of the physical patch and nodal displacement of the element in the NMM, a decomposition technique of generalized degrees of freedom is deduced for mixed mode crack problems. An analytic expression of the energy release rate, which is caused by a virtual crack extension technique, is proposed. The necessity of using a symmetric mesh is demonstrated in detail by analysing an additional error that had previously been overlooked. Because of this necessity, the local mathematical cover refinement is further applied. Finally, four comparison tests are given to illustrate the validity and practicality of the proposed method. The aforementioned aspects are all implemented in the high‐order NMM, so this study can be regarded as the development of the virtual crack extension technique and can also be seen as a prelude to an h‐version high‐order NMM. Copyright © 2017 John Wiley & Sons, Ltd.

Fatigue Debonding Three-dimensional Simulation with Cohesive Zone

The cohesive zone model (CZM) has found a wide acceptance as a tool for the simulation of debonding in adhesive joints. Recently, fatigue-devoted implementations of CZM have been proposed. In earlier works, the authors have developed a model of the cohesive zone able to correctly simulate the propagation of fatigue of two-dimensional (2D) defects in joints. The procedure has been implemented in the finite-element software (Abaqus®) using software-embedded subroutines. That model was then extended to 3D cracks with quasi-straight crack front, where G could be evaluated by contour-integral on slices along the crack front.Inspiring to the approach used by Xie and Biggers in the case of virtual crack extension technique, the aim of this work is to extend the fatigue crack growth evaluation to 3D cracks of general shape. The new procedure has been compared with the previously developed one in the case of planar, 3D cracks showing a good agreement. The evolution of a corner crack is also shown as a qualitative example of the descriptive potential of the new procedure.

Energy release rate for kinking crack using mixed finite element

A numerical method, using a special mixed finite element associated with the virtual crack extension technique, has been developed to evaluate the energy release rate for kinking cracks. The element is two dimensional 7-node mixed finite element with 5 displacement nodes and 2 stress nodes. The mixed finite element ensures the continuity of stress and displacement vectors on the coherent part and the free edge effect. This element has been formulated starting from a parent element in a natural plane with the aim to model different types of cracks with various orientations. Example problems with kinking cracks in a homogeneous material and bimaterial are presented to assess the computational accuracies.

A continuous/discontinuous deformation analysis (CDDA) method based on deformable blocks for fracture modeling

In the framework of finite element meshes, a novel continuous/discontinuous deformation analysis (CDDA) method is proposed in this paper for modeling of crack problems. In the present CDDA, simple polynomial interpolations are defined at the deformable block elements, and a link element is employed to connect the adjacent block elements. The CDDA is particularly suitable for modeling the fracture propagation because the switch from continuous deformation analysis to discontinuous deformation analysis is natural and convenient without additional procedures. The SIFs (stress intensity factors) for various types of cracks, such as kinked cracks or curved cracks, can be easily computed in the CDDA by using the virtual crack extension technique (VCET). Both the formulation and implementation of the VCET in CDDA are simple and straightforward. Numerical examples indicate that the present CDDA can obtain high accuracy in SIF results with simple polynomial interpolations and insensitive to mesh sizes, and can automatically simulate the crack propagation without degrading accuracy.

Calculation of stress intensity factors for functionally graded materials by using the weight functions derived by the virtual crack extension technique

The weight function method provides a powerful approach for calculating the stress intensity factors for a homogeneous cracked body subjected to mechanical loadings. In this paper, the basic equations of weight function method for mode I and mixed mode crack problems in a two-dimensional functionally graded crack system are derived based on the Betti’s reciprocal theorem. The weight functions derived by the virtual crack extension technique are further used to calculate the stress intensity factors of functionally graded materials (FGMs). The practicability and accuracy of this proposed method has been confirmed by the comparison with theoretical or numerical solutions available in the literatures. On account that the repeated extractions of the distributions of normal stress and shear stress in the uncracked component along the prospective crack line under different loadings can be avoided using the method presented in this paper, this method can be potentially used for optimal design for FGMs under multiple-load cases.

Computational Modeling of Surface Fracture of Polyethylene Acetabular Cup in Total Artificial Hip Replacement

In this paper, a crack analysis model based on finite element method and virtual crack extension technique was proposed to investigate the occurrence of surface fracture of polyethylene acetabular cup under gait loadings. To this, a simplified hip joint model was created for facture analysis. The stress intensity factor (SIF) at crack site was estimated and used to evaluate the propagation of the surface crack. Current results show that under normal gait loading, the SIF at crack tip within polyethylene cup was predicted to be lower than the fatigue threshold of polyethylene material. However, under the heel strike instant, the crack tip SIF exceeds the fracture strength of polyethylene subject to gamma radiation, which may drive the crack to propagate to final fracture. Overall, the presented analysis model has demonstrated the probability of severe surface damage occurring in polyethylene cup under impact walking conditions. This provides a valuable reference to the improvement of the mechanical properties or design of bearing materials in clinical orthopedic application.

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

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

Continuum shape sensitivity analysis of mixed-mode fracture using fractal finite element method

This paper presents a new fractal finite element based method for continuum-based shape sensitivity analysis for a crack in a homogeneous, isotropic, and two-dimensional linear-elastic body subject to mixed-mode (modes I and II) loading conditions. The method is based on the material derivative concept of continuum mechanics, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed in the proposed method to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations predicts the first-order sensitivity of J-integral or mode-I and mode-II stress-intensity factors, K I and K II, more efficiently and accurately than the finite-difference methods. Unlike the integral based methods such as J-integral or M-integral no special finite elements and post-processing are needed to determine the first-order sensitivity of J-integral or K I and K II. Also a parametric study is carried out to examine the effects of the similarity ratio, the number of transformation terms, and the integration order on the quality of the numerical solutions. Four numerical examples which include both mode-I and mixed-mode problems, are presented to calculate the first-order derivative of the J-integral or stress-intensity factors. The results show that first-order sensitivities of J-integral or stress-intensity factors obtained using the proposed method are in excellent agreement with the reference solutions obtained using the finite-difference method for the structural and crack geometries considered in this study.

Fracture resistance via topology optimization

The fracture resistance of structures is optimized using the level-set method. Fracture resistance is assumed to be related to the elastic energy released by a crack propagating in a normal direction from parts of the boundary that are in tension, and is calculated using the virtual crack extension technique. The shape derivative of the fracture-resistance objective function is derived. Two illustrative two-dimensional case studies are presented: a hole in a plate subjected to biaxial strain; and a bridge fixed at both ends subjected to a single load in which the compliance and fracture resistance are jointly optimized. The structures obtained have rounded corners and more material at places where they are in tension. Based on the results, we propose that fracture resistance may be modeled more easily but less directly by including a term proportional to surface area in the objective function, in conjunction with nonlinear elasticity where the Young’s modulus in tension is lower than in compression.

Spar Corner Radius Integrity for the A400M Wing

The paper focuses on the structural integrity of the corner radius of the carbon fibre composite, ‘C’-section spar for the Airbus A400M wing. The corner radius is subject to opening moments generated by internal wing box fuel pressures. The low inter-lamina strength of composites makes de-lamination of the corner of prime concern. The paper describes initial development of analytical techniques to calculate the through-thickness tensile stresses and inter-lamina shear stresses developed in a corner radius under applied bending moments and transverse shear forces. A test programme is also described, aimed at the determination of the failure moment of curved laminates under pure bending moments. Using the analytical expressions developed, a through-thickness failure stress is calculated from the failure moments. A variation of the failure stress with specimen thickness is indicated, showing that thicker specimens fail at higher inter-lamina stresses – a characteristic that must be exploited in the design of the spar. Using finite element analysis of the test configuration, in conjunction with virtual crack extension techniques, it is demonstrated that, at the failure load, a constant rate of strain energy release accompanies inter-lamina crack growth in the different test specimens. A critical energy release rate for uncontrolled crack growth is thus established, which is used, in conjunction with further finite element analysis, to predict the failure stress of specimens with different values of thickness and corner radius. It is concluded that this fracture mechanics approach to integrity can be applied to the A400M spar corner and to similar aircraft structures. Recommendations for further testing and correlation with analysis are proposed to strengthen the theoretical basis for such integrity assessments.

Continuum shape sensitivity and reliability analyses of nonlinear cracked structures

A new method is proposed for shape sensitivity analysis of a crack in a homogeneous, isotropic, and nonlinearly elastic body subject to mode I loading conditions. The method involves the material derivative concept of continuum mechanics, domain integral representation of the J-integral, and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is required in the proposed method. Since the governing variational equation is differentiated before the process of discretization, the resulting sensitivity equations are independent of any approximate numerical techniques. Based on the continuum sensitivities, the first-order reliability method was employed to perform probabilistic analysis. Numerical examples are presented to illustrate both the sensitivity and reliability analyses. The maximum difference between the sensitivity of stress-intensity factors calculated using the proposed method and the finite-difference method is less than four percent. Since all gradients are calculated analytically, the reliability analysis of cracks can be performed efficiently.

Continuum Shape Sensitivity analysis of a mode-I fracture in functionally graded materials

This paper presents a new method for conducting a continuum shape sensitivity analysis of a crack in an isotropic, linear-elastic, functionally graded material. This method involves the material derivative concept from continuum mechanics, domain integral representation of the J-integral and direct differentiation. Unlike virtual crack extension techniques, no mesh perturbation is needed to calculate the sensitivity of stress-intensity factors. Since the governing variational equation is differentiated prior to the process of discretization, the resulting sensitivity equations are independent of approximate numerical techniques, such as the meshless method, finite element method, boundary element method, or others. In addition, since the J-integral is represented by domain integration, only the first-order sensitivity of the displacement field is needed. Several numerical examples are presented to calculate the first-order derivative of the J-integral, using the proposed method. Numerical results obtained using the proposed method are compared with the reference solutions obtained from finite-difference methods for the structural and crack geometries considered in this study.

An energy-based crack growth criterion for modelling elastic–plastic ductile fracture

A crack growth criterion is derived based on the Griffith energy concept and the cohesive zone model for modelling fracture in elastic–plastic ductile materials. The criterion is implemented in the finite element context by a virtual crack extension technique. An automatic modelling of the ductile fracture process is realised by combining a local remeshing procedure and the criterion. The validity of the derived criterion is examined by modelling a compact tension specimen.