Quarter-point Singular Elements Research Articles

Defining the Best Strain Gauge Placement through Numerical Simulations of the Mechanical Behavior of the Modified-WOL Specimen

In structural steels, cracks are an inevitable occurrence, demanding an examination of their ability to resist crack propagation under expected loads and various environmental conditions. The process of experimentally determining fracture toughness comes at a significant expense, involving the use of multiple specimens and extended testing periods, which hinders the swift acquisition of crucial parameters. An alternative approach entails the execution of a constant displacement test using the Modified-WOL specimen, which is equipped with an instrumented bolt for self loading. The primary objective of this method is to provide the Threshold Stress Intensity Factor for Environment-Assisted Cracking, utilizing only a single specimen. The incorporation of electronic instrumentation is essential for obtaining reliable, real-time data. Despite its importance, a consensus is lacking in the literature regarding the best location for sensor installation on the Modified-WOL specimen, potentially influencing its mechanical response. This study aims to explore the mechanical behavior of the three components constituting the Modified-WOL specimen test in accordance with the guidelines outlined by ISO 7539-6. Employing the Finite Element Method with two- and three-dimensional models, this investigation utilizes the Augmented Lagrangian method for formulating the contact regions. Additionally, quarter-point singular elements are employed to accurately represent stress singularity behavior at the crack tip. The numerical computation of fracture parameter (KI) is conducted using the Displacement Correlation Technique. Comparative analyses are then carried out between the numerical and the analytical values of KI determined based on the ISO 7539-6 standard. The findings demonstrate alignment with empirical literature values, ultimately contributing to the identification of the best location for strain gauge installation among the highlighted regions of considerable interest.

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.

Simulation of Quasi-Static Crack Propagation by Adaptive Finite Element Method

The finite element method (FEM) is a widely used technique in research, including but not restricted to the growth of cracks in engineering applications. However, failure to use fine meshes poses problems in modeling the singular stress field around the crack tip in the singular element region. This work aims at using the original source code program by Visual FORTRAN language to predict the crack propagation and fatigue lifetime using the adaptive dens mesh finite element method. This developed program involves the adaptive mesh generator according to the advancing front method as well as both the pre-processing and post-processing for the crack growth simulation under linear elastic fracture mechanics theory. The stress state at a crack tip is characterized by the stress intensity factor associated with the rate of crack growth. The quarter-point singular elements are constructed around the crack tip to accurately represent the singularity of this region. Under linear elastic fracture mechanics (LEFM) with an assumption in various configurations, the Paris law model was employed to evaluate mixed-mode fatigue life for two specimens under constant amplitude loading. The framework includes a progressive analysis of the stress intensity factors (SIFs), the direction of crack growth, and the estimation of fatigue life. The results of the analysis are consistent with other experimental and numerical studies in the literature for the prediction of the fatigue crack growth trajectories as well as the calculation of stress intensity factors.

Nonplanar Crack Growth Simulation of Multiple Cracks Using Finite Element Method

This work deals with a 2D finite element simulation of nonplanar multiple cracks using fracture and crack propagation analysis. This analysis was performed by using the developed source code software written by Visual Fortran Language. This source code includes the adaptive mesh generation utilizing the advanced front method and also the mesh refinement process. In order to correctly represent the field singularity, the quarter‐point singular elements are constructed around the tip of the crack. The crack growth criteria are used to predict the crack growth direction by utilizing the circumferential stress factor in calculating the yielding stress in elastic fracture assumptions. The stress intensity factor determination is one of the most critical procedures as it determines the crack initiation and propagation mechanism. Moreover, the stress intensity factor histories during the crack growth are measured with the use of equivalent domain integral methods. The crack path simulation and stress intensity factor calculations are compared with the literature and revealed that the results are in agreement with research carried in this domain.

An interaction integral and a modified crack closure integral for evaluating piezoelectric crack-tip fracture parameters in BEM

To evaluate the crack-tip field intensity factors of a piezoelectric crack with any inclined angle, the current widely-used interaction integral method (I-integral) is here extended to the boundary element applications under some coordinate transformations. As well, a new modified crack closure integral method (MCCI) is proposed by considering the discontinuous quarter-point singular elements for the crack-face discretization arising from the dual boundary element method (BEM). This dual BEM involves the strongly singular displacement boundary integral equations (BIEs) for the external boundary and the hypersingular traction BIEs for the crack faces. The crack-tip fracture parameters evaluated by the I-integral and MCCI are verified by the existing analytical solutions and meanwhile, compared with those results achieved by the classical displacement extrapolation method and the J-integral. Three examples are presented to show the high accuracy of the interaction integral method and the improvement of MCCI for the piezoelectric crack problems.

3D explicit-BEM fracture analysis for materials with anisotropic multifield coupling

An explicit and unified boundary element formulation for the linear fracture analysis of three-dimensional bodies with general anisotropic multifield coupling is presented. The herein considered multifield media include, as particular cases, the purely elastic, piezo-electric and magneto-electro-elastic material models. The proposed formulation is based on the fundamental solutions for displacements and tractions recently introduced by Buroni and Sáez (2010), which presents the remarkable features of being exact, explicit in terms of the Stroh’s eigenvalues and valid for mathematically degenerate as well as non-degenerate materials. Behaviour around the crack front is modeled with quarter-point elements for the extended displacements (elastic displacements, electric potential and magnetic potential) and singular quarter-point elements for the extended stresses (mechanical stresses, electric displacements and magnetic inductions) which allow for the direct extrapolation of the extended stress intensity factors (the mechanical stress intensity factors in the three classical modes KI, KII and KIII, the electric displacement intensity factor KIV and the magnetic induction intensity factor KV) from the computed nodal values at the singular quarter-point elements. Some benchmark examples are presented and discussed in order to validate this simple and robust approach for the assessment of multifield fracture parameters.

Mixed mode crack growth simulation with/without overloads

In the present paper, mathematical models/procedures are developed in order to investigate the mixed mode crack growth process. The proposed procedures examine the fatigue life estimation and the crack path simulation under either constant amplitude loading or single overload. The fatigue life up to failure is calculated by employing two-parameter driving force model together with the equivalent stress intensity factor. The overload effect is theoretically analyzed by introducing the Wheeler’s retardation parameter. Further, the crack extension under Mode I/II conditions is experimentally investigated for two cracks emanating from two holes in the plate. Then, for the same mixed mode situation with two cracks, the crack growth paths are modeled by using the maximum principal stress criterion and the finite element method. The authors employ analytical and numerical approaches for the stress intensity factor calculations. The numerical investigation is performed by applying quarter-point (Q-P) singular finite elements. The efficiency of proposed procedures is verified through available crack growth data. The different mixed mode calculations are in a good correlation with crack growth observations.

Fatigue crack growth modeling of attachment lugs

In the present paper, computational procedures for the strength analysis of two cracked lug configurations are proposed. The authors consider the lug with single through-the-thickness crack as well as with single quarter-elliptical corner crack. The stress analysis and the fatigue life evaluations are tackled for both configurations. Numerical approaches based on quarter-point (Q-P) singular finite elements are developed for stress intensity factor calculations. Furthermore, the finite element analysis is applied in order to evaluate the polynomial expression for the stress intensity factor calculation of the lug with one through-the-thickness crack. The residual life estimation of cracked lugs is analyzed by employing the crack growth law based on a two-parameter driving force model. Moreover, the crack growth path for the lug with quarter-elliptical crack is simulated. The reliability of the proposed procedures is verified using experimental fatigue crack growth data. Fatigue life calculations and crack path simulations show a very good agreement with experimental observations.

Crack growth modeling via 3D automatic adaptive mesh refinement based on modified-SPR technique

In this paper, the three-dimensional automatic adaptive mesh refinement is presented in modeling the crack propagation based on the modified superconvergent patch recovery technique. The technique is developed for the mixed mode fracture analysis of different fracture specimens. The stress intensity factors are calculated at the crack tip region and the crack propagation is determined by applying a proper crack growth criterion. An automatic adaptive mesh refinement is employed on the basis of modified superconvergent patch recovery (MSPR) technique to simulate the crack growth by applying the asymptotic crack tip solution and using the collapsed quarter-point singular tetrahedral elements at the crack tip region. A-posteriori error estimator is used based on the Zienkiewicz–Zhu method to estimate the error of fracture parameters and predict the crack path pattern. Finally, the efficiency and accuracy of proposed computational algorithm is demonstrated by several numerical examples.

Analysis of the crack growth propagation process under mixed-mode loading

In the present paper, a computational model for crack growth analysis under Mode I/II conditions is formulated. The focus is on two issues – crack path simulation and fatigue life estimation. The finite element method is used together with the maximum principal stress criterion and the crack growth rate equation based on the equivalent stress intensity factor. To determine the mixed-mode stress intensity factors, quarter-point (Q-P) singular finite elements are employed. For verification purposes, a plate with crack emanating from the edge of a hole is examined. The crack path of the plate made of 2024 T3 Al Alloy is investigated experimentally and simulated by using the finite element method with the maximum tangential stress criterion. Then, the validation of the procedure is illustrated by applying the numerical evaluation of the curvilinear crack propagation in the polymethyl methacrylate (PMMA) beam and the Arcan specimen made of Al Alloy for which experimental results are available in the literature. In order to estimate fatigue life up to failure of the plate with crack emanating from the edge of a hole, the polynomial expression is evaluated for the equivalent stress intensity factor using values of stress intensity factors obtained from the finite element analysis. Additionally, the fatigue life up to failure of the Arcan specimen is analyzed for different loading angles and compared with experimental data. Excellent correlations between the computed and experimental results are obtained.

Finite Element Analysis of the Crack Propagation for Solid Materials

Problem statement: The use of fracture mechanics techniques in the assessment of performance and reliability of structure is on increase and the prediction of crack propagation in structure play important part. The finite element method is widely used for the evaluation of SIF for various types of crack configurations. Source code program of two-dimensional finite element model had been developed, to demonstrate the capability and its limitations, in predicting the crack propagation trajectory and the SIF values under linear elastic fracture analysis. Approach: Two different geometries were used on this finite element model in order, to analyze the reliability of this program on the crack propagation in linear and nonlinear elastic fracture mechanics. These geometries were namely; a rectangular plate with crack emanating from square-hole and Double Edge Notched Plate (DENT). Where, both geometries are in tensile loading and under mode I conditions. In addition, the source code program of this model was written by FORTRAN language. Therefore, a Displacement Extrapolation Technique (DET) was employed particularly, to predict the crack propagations directions and to, calculate the Stress Intensity Factors (SIFs). Furthermore, the mesh for the finite elements was the unstructured type; generated using the advancing front method. And, the global h-type adaptive mesh was adopted based on the norm stress error estimator. While, the quarter-point singular elements were uniformly generated around the crack tip in the form of a rosette. Moreover, make a comparison between this current study with other relevant and published research study. Results: The application of the source code program of 2-D finite element model showed a significant result on linear elastic fracture mechanics. Based on the findings of the two different geometries from the current study, the result showed a good agreement. And, it seems like very close compare to the other published results. Conclusion: A developed a source program of finite element model showed that is capable of demonstrating the SIF evaluation and the crack path direction satisfactorily. Therefore, the numerical finite element analysis with displacement extrapolation method, had been successfully employed for linear-elastic fracture mechanics problems.

Numerical Computation of Stress Intensity Factors for Bolt-hole Corner Crack in Mechanical Joints

The three-dimensional finite element method is used to solve the problem of the quarter-elliptical corner crack of the bolt-hole in mechanical joints being subjected to remote tension. The square-root stress singularity around the corner crack front is simulated using the collapsed 20-node quarter point singular elements. The contact interaction between the bolt and the hole boundary is considered in the finite element analysis. The stress intensity factors (SIFs) along the crack front are evaluated by using the displacement correlation technique. The effects of the amount of clearance between the hole and the bolt on the SIFs are investigated. The numerical results indicate that the SIF for mode I decrease with the decreases in clearance, and in the cases of clearance being present, the corner crack is in a mix-mode, even if mode I loading is dominant.

An Effective Method for Calculation of Corner Stresses With Applications to Plastic IC Packages

Plastic integrated circuit (IC) packages are usually composite components made of multiple bonded materials with different mechanical and thermal properties. High stresses often occur at corners between different components within the packages where discontinuities of geometry or material properties are present. Delaminations usually initiate from these corners when packages undergo adverse thermal or moisture environments. Hence, in order to prevent delaminations from occurring and to improve package reliability performance, it is crucial to accurately and efficiently evaluate the stresses at corners within the package. However, with conventional finite element methods, it is always a challenge to give an accurate description of the stresses at the corners since these corners represent stress singularity points. In this paper, an effective method is developed to precisely evaluate the stresses in the vicinity of internal corners within the packages. A new variable-order singular boundary element is constructed with a built-in accurate description of the stresses at the corner. This method is versatile for solving general corner problems involving wedges, two-material and three-material corners and interfacial cracks that are common in the IC packages. This method is verified by solving a bimaterial interface crack problem with known solution. Comparisons are made with other conventional methods such as displacement-based quarter-point singular elements and normal quadratic elements, on the calculation of interfacial stress intensity factors. Results show that the new method has significant advantages in giving more accurate results with much less computational resources needed. The new method is applied to a typical plastic IC package with multiple internal corners and interface cracks. The possible failure sites and failure modes within the package are predicted and the results agree well with package evaluations done in the industry.

Three-dimensional BEM for piezoelectric fracture analysis

A boundary element approach with quadratic isoparametric elements, quarter-point elements and singular quarter-point elements for three-dimensional crack problems in piezoelectric solids under mechanical and electrical loading conditions, is presented in this paper for the first time. The procedure is based on Deeg's fundamental solution for anisotropic piezoelectric materials, and the classical extended displacement boundary integral equation. Stress and electric displacement intensity factors are directly evaluated as system unknowns, and also as functions of the computed nodal displacements and electric potentials at crack faces. Special attention is paid to the fundamental solution evaluation. Several three-dimensional crack problems in transversely isotropic bodies under mechanical and electrical loading conditions are analysed. Numerical solutions computed for prismatic cracked 3D plate problems with a plane strain behaviour are in very good agreement with their corresponding 2D BE solutions. Results for a penny shape crack in a piezoelectric cylinder are presented for the first time. The proposed approach is shown to be a simple, robust and useful tool for stress and electric displacement intensity factors evaluation in piezoelectric media.

Interfacial steady‐state and transient thermal fracture of dissimilar media using the boundary element contact analysis

AbstractA boundary element procedure is formulated to treat frictional contact problems of thermally stressed structures. The aim is the computation of steady‐state and time‐dependent thermal stress intensity factors of interfacial cracks in two‐dimensional bimaterial structures when crack closure conditions are present. The effect of friction between the crack faces is taken into consideration. The problems considered are formulated in an incremental and iterative fashion because of their non‐linear nature. Near crack tip singularities of temperature and displacement fields are modelled through appropriate quarter‐point singular elements. Fracture parameters are evaluated from nodal displacements of singular elements utilizing proper formulas. Numerical results are compared with available solutions from the literature, where possible. Good agreement between them can be found. Copyright © 2005 John Wiley & Sons, Ltd.

Time-domain BEM for three-dimensional fracture mechanics

The present paper deals with time-domain analysis of three-dimensional transient dynamic crack problems. The time-domain formulation of the boundary element method for 3-D elastodynamic problems is used. Quarter-point and singular quarter-point elements represent displacements and tractions, respectively, near the crack front. Special attention is paid to integration and algorithms to preserve stability. Cracks in finite and unbounded regions under single and mixed mode dynamic loading conditions are studied. To the authors’ knowledge, no previous BE approach for 3-D elastodynamic crack problems based on the time-domain displacement representation exists.

Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems

In this paper the dual reciprocity boundary element method in the Laplace domain for anisotropic dynamic fracture mechanic problems is presented. Crack problems are analyzed using the subregion technique. The dynamic stress intensity factors are computed using traction singular quarter-point elements positioned at the tip of the crack. Numerical inversion from the Laplace domain to the time domain is achieved by the Durbin method. Numerical examples of dynamic stress intensity factor evaluation are considered for symmetric and non-symmetric problems. The influence of the number of Laplace parameters and internal points in the solution is investigated.

시간적분형 운동방정식을 바탕으로 한 동적 응력확대계수의 계산

In this paper, the dynamic stress intensity factors of fracture mechanics are numerically computed in time domain using the FEM. For which the finite element formulations are derived applying the Galerkin method to the equations of motion in convolution integral as has been presented in the previous paper. To assure the strain fields of r-½ singularity near the crack tip, the triangular quarter-point singular elements are imbedded in the finite element mesh discretized by the isoparametric quadratic quadrilateral elements. Two-dimensional problems of the elastodynamic fracture mechanics under the impact load are solved and compared with the existing numerical and analytical solutions, being shown that numerical results of good accuracy are obtained by the presented method.

The boundary element method applied to time dependent problems in anisotropic materials

In this paper a multi-domain boundary element formulation is developed for the solution of two-dimensional dynamic anisotropic problems with or without cracks. The inertia effects are treated as body forces and the dual reciprocity formulation is used to transform the domain integral into a boundary integral. In fracture problems, traction singular quarter point elements are used in order to give a correct representation of displacement and traction fields around the crack tip. The dynamic stress intensity factors are computed directly from the tractions at the crack tip node. The results obtained demonstrate good agreement with other reported results and show strong dependence on the material anisotropy.