An extended element free Galerkin method for fracture analysis of functionally graded materials

An extended element free Galerkin method (XEFGM) has been adopted for fracture analysis of functionally graded materials (FGMs). Orthotropic enrichments functions are used along with the sub-triangle technique for enhancing the Gauss quadrature accuracy near the crack, and the incompatible interaction integral method is employed to calculate the stress intensity factors. Numerical simulations have proved that XEFGM provides more accurate results by less number of nodes (DOFs) in comparison with the unenriched EFGM and other conventional methods for several FGM problems with different crack locations and loadings. The results have been compared with the reference results, showing the reliability, stability, and efficiency of present XEFGM.

Received 9 June 2014 Accepted 17 September 2014.     

Elasto-plastic element-free Galerkin method

In this paper the element free Galerkin method (EFGM) has been extended to be used in the elastoplastic stress analysis. The developed method has been examined in planar stress analysis around the tip of a crack and in its opening mode of loading. To do this, at the first step by using the incremental relations of plastic deformation a system of elastoplastic EFGM equations has been derived. Since the obtained relations are nonlinear, a nonlinear solution technique has been chosen. To examine the validity of this technique, stress fields in two different plates with and without a crack have been calculated and the results have been compared with other similar analytical works in the literature. In doing so the power law work hardening behavior has been employed and the value of J-integral has been used as a base for comparison of the results.First and second authors wish to express their gratitude to the Office of Research Affairs of Sharif University of Technology for financial support to conduct this research. The second author wishes to appreciate Professor Tom Hyde head of the School of 4 M in the University of Nottingham for his guidance and providing some research facilities.     

Fracture mechanics analysis using the wavelet Galerkin method and extended finite element method

This paper presents fracture mechanics analysis using the wavelet Galerkin method and extended finite element method. The wavelet Galerkin method is a new methodology to solve partial differential equations where scaling/wavelet functions are used as basis functions. In solid/structural analyses, the analysis domain is divided into equally spaced structured cells and scaling functions are periodically placed throughout the domain. To improve accuracy, wavelet functions are superposed on the scaling functions within a region having a high stress concentration, such as near a hole or notch. Thus, the method can be considered a refinement technique in fixed‐grid approaches. However, because the basis functions are assumed to be continuous in applications of the wavelet Galerkin method, there are difficulties in treating displacement discontinuities across the crack surface. In the present research, we introduce enrichment functions in the wavelet Galerkin formulation to take into account the discontinuous displacements and high stress concentration around the crack tip by applying the concept of the extended finite element method. This paper presents the mathematical formulation and numerical implementation of the proposed technique. As numerical examples, stress intensity factor evaluations and crack propagation analyses for two‐dimensional cracks are presented. Copyright © 2012 John Wiley & Sons, Ltd.     

Thermal fracture analysis of orthotropic functionally graded materials using an equivalent domain integral approach

A new computational method based on the equivalent domain integral (EDI) is developed for mode I fracture analysis of orthotropic functionally graded materials (FGMs) subjected to thermal stresses. By using the constitutive relations of plane orthotropic thermoelasticity, generalized definition of the J-integral is converted to an equivalent domain integral to calculate the thermal stress intensity factor. In the formulation of the EDI approach, all the required thermomechanical properties are assumed to have continuous spatial variations through the functionally graded medium. Developed methodology is integrated into a fracture mechanics research finite element code FRAC2D using graded finite elements that possess cubic interpolation. Steady-state and transient temperature distribution profiles in orthotropic FGMs are computed using the finite elements based heat transfer analysis software HEAT2D. EDI method is validated and domain independence is demonstrated by comparing the numerical results obtained using EDI to those calculated by an enriched finite element method and to those available in the literature. Single and periodic edge crack problems in orthotropic FGMs are examined in order to study the influences of principal thermal expansion coefficient and thermal conductivity components, relative crack length and crack periodicity on the thermal stress intensity factors. Numerical results show that among the three principal thermal expansion coefficient components, the in-plane component perpendicular to the crack axis has the most significant influence on the mode I stress intensity factor. Gradation profile of the thermal expansion coefficient parallel to the crack axis is shown to have no effect on the outcome of the fracture analysis.     

Heat transfer analysis of composite slabs using meshless element Free Galerkin method

This paper deals with three dimensional heat transfer analysis of composite slabs using meshless element free Galerkin method. The element free Galerkin method (EFG) method utilizes moving least square (MLS) approximants to approximate the unknown function of temperature Tx). These approximants are constructed by using a weight function, a basis function and a set of coefficients that depends on position. Penalty and Lagrange multiplier techniques have been used to enforce the essential boundary conditions. MATLAB codes have been developed to obtain the EFG results. Two new basis functions namely trigonometric and polynomial have been proposed. A comparison has been made among the results obtained using existing (linear) and proposed (trigonometric and polynomial) basis functions for three dimensional heat transfer in composite slabs. The effect of penalty parameter on EFG results has also been discussed. The results obtained by EFG method are compared with those obtained by finite element method     

Dynamic analysis of fixed cracks in composites by the extended finite element method

This paper is dedicated to simulation of dynamic analysis of fixed cracks in orthotropic media using an extended finite element method. This work is in fact an extension to dynamic problems of the recently developed orthotropic extended finite element method for fracture analysis of composites. In this method, the Heaviside and near-tip enrichment functions are used in the framework of the partition of unity for modeling crack discontinuity and crack-tip singularities within the classical finite element method. In this procedure, elements that include a crack are not required to conform to crack edges. Therefore, mesh generation can be performed without any need to comply to crack edges and the method is capable of modeling the crack propagation without any remeshing. To determine the fracture properties, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated by means of domain separation integral (J^′-integral) method. Results of the proposed method are compared with other available analytical and computational results.     

Interaction integrals for fracture analysis of functionally graded magnetoelectroelastic materials

This paper presents the domain form of interaction integrals based on three independent formulations for computation of stress intensity factors, electric displacement intensity factors and magnetic induction intensity factors for cracks in functionally graded magnetoelectroelastic materials. Conservation integrals of J-type are derived based on the governing equations for magnetoelectroelastic media and the crack tip asymptotic fields of homogeneous magnetoelectroelastic medium as auxiliary fields. Each of the formulations differs in the way auxiliary fields are imposed in the evaluation of interaction integrals and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded magnetoelectroelastic medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.     

Dual boundary element method for three-dimensional thermoelastic crack problems

This paper describes the formulation and numerical implementation of the three-dimensional dual boundary element method (DBEM) for the thermoelastic analysis of mixed-mode crack problems in linear elastic fracture mechanics. The DBEM incorporates two pairs of independent boundary integral equations; namely the temperature and displacement, and the flux and traction equations. In this technique, one pair is applied on one of the crack faces and the other pair on the opposite one. On non-crack boundaries, the temperature and displacement equations are applied. This revised version was published online in July 2006 with corrections to the Cover Date.     

A hybrid boundary element method for three-dimensional fracture analysis

At first, a hybrid boundary element method used for three-dimensional linear elastic fracture analysis is established by introducing the relative displacement fundamental function into the first and the second kind of boundary integral equations. Then the numerical approaches are presented in detail. Finally, several numerical examples are given out to check the proposed method. The numerical results show that the hybrid boundary element method has a very high accuracy for analysis of a three-dimensional stress intensity factor.     

On fracture analysis using an element overlay technique

In this paper, an element overlay technique (s-FEM [Comput. Struct. 43 (1992) 539]) is applied to various two dimensional linear fracture problems. When s-FEM is adopted, local finite element model concerning cracks can be built independently from the global finite element mesh for modeling overall structure. The local model is superposed on the global one. Therefore, it is tractable to introduce cracks in an existing finite element model. The accuracy of s-FEM is critically examined and it is found that the size of local mesh region needs to be larger than or roughly equal to that of an element in the global mesh.     

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.     

An indirect boundary element method for three-dimensional dynamic fracture mechanics

Indirect boundary element methods (fictitious load and displacement discontinuity) have been developed for the analysis of three-dimensional elastostatic and elastodynamic fracture mechanics problems. A set of boundary integral equations for fictitious loads and displacement discontinuities have been derived. The stress intensity factors were obtained by the stress equivalent method for static loading. For dynamic loading the problem was studied in Laplace transform space where the numerical calculation procedure, for the stress intensity factor K_I(p), is the same: as that for the static problem. The Durbin inversion method for Laplace transforms was used to obtain the stress intensity factors in the time domain K_I(t). Results of this analysis are presented for a square bar, with either a rectangular or a circular crack, under static and dynamic loads.     

Analysis of cracked plates using an iterative hybrid technique of boundary element method and distributed dislocation method

An iterative hybrid technique of boundary element method (BEM) and distributed dislocation method (DDM) is introduced for solving two dimensional crack problems. The technique decomposes the problem into (n + 1) subsidiary problems where n is the number of crack branches. The required solution will be the sum of these (n + 1) solutions. The first subsidiary problem is to find the stress distribution induced in the plate in the absence of the crack using BEM. All of the remaining subsidiary problems, are stress disturbance ones that will be solved using DDM. The results will be added and compared with the boundary conditions of the original problem. Iteration will be performed between the plate boundaries and crack faces until all of the boundary conditions are satisfied.     

A sub‒domain smoothed Galerkin method for solid mechanics problems

A sub?domain smoothed Galerkin method is proposed to integrate the advantages of mesh?free Galerkin method and FEM. Arbitrarily shaped sub?domains are predefined in problems domain with mesh?free nodes. In each sub?domain, based on mesh?free Galerkin weak formulation, the local discrete equation can be obtained by using the moving Kriging interpolation, which is similar to the discretization of the high?order finite elements. Strain smoothing technique is subsequently applied to the nodal integration of sub?domain by dividing the sub?domain into several smoothing cells. Moreover, condensation of DOF can also be introduced into the local discrete equations to improve the computational efficiency. The global governing equations of present method are obtained on the basis of the scheme of FEM by assembling all local discrete equations of the sub?domains. The mesh?free properties of Galerkin method are retained in each sub?domain. Several 2D elastic problems have been solved on the basis of this newly proposed method to validate its computational performance. These numerical examples proved that the newly proposed sub?domain smoothed Galerkin method is a robust technique to solve solid mechanics problems based on its characteristics of high computational efficiency, good accuracy, and convergence. Copyright © 2014 John Wiley & Sons, Ltd.     

Mixed-mode thermoelastic fracture analysis of orthtropic composites

This study demonstrates ability to determine the in-plane stress intensity factors, K _I and K _II, simultaneously under mixed-mode conditions in orthotropic composites by the combined use of least-squares, stress representations which are valid away from the crack and distant measured temperatures. Recognizing the stresses near a crack-tip are dominated by the stress intensity factors, it has not been uncommon to neglect the higher-order stress terms when evaluating these factors. However, and among other considerations, it is typically difficult to obtain accurate temperature information very near the crack-tip. It can therefore be advantageous to employ measured data which originate away from the crack and to retain six to eight terms in the stress functions when evaluating the stress intensity factors. On the other hand, errors in K _I and/or K _II can be appreciable if only the r^–1/2 terms are employed with distant input information.     

Dual boundary element method modelling of aircraft structural joints with multiple site damage

The computation of crack growth from a bolt or rivet hole in a structural joint practically requires that the geometry be approximated to some degree. In this paper a simplified quasi-2D stress analysis method, using the boundary element method is presented, where the load transfer rate and the contact stresses at the hole edge for the full 3D geometry are fairly well approximated. Coupled with a dual boundary element formulation for the crack propagation problem, this model is used to evaluate stress intensity factors for through cracks emanating from holes in several double shear lap joint configurations. As the calculated stress intensity factors compare well with experimental data, this procedure is considered to approximate satisfactorily the load transfer rate and the contact stresses at the hole edge of the full 3D geometry, when secondary bending is not a factor.     

Complex variable method of computing Jk for bi-material interface cracks

A method using functions of a complex variable is developed for evaluation of J₁ and a modified J₂ integrals for bi-material interface cracks. This method, used in conjunction with the finite element method, would be useful in the prediction of stress intensity factors for cracks lying between the interface of two dissimilar materials. Since the direct evaluation of J₂ poses difficulties in modeling the singular behavior in the near vicinity around the crack tip for bi-material crack problems, it is modified by evaluating it around a contour path of small radius from the crack tip within the singularity dominated zone. It is shown that the stress intensity factors for a bi-material interface crack can be accurately evaluated using these j_k integrals.     

Parametric quadratic programming method for elastic contact fracture analysis

A solution procedure for elastic contact fracture mechanics has been proposed in this paper. The procedure is based on the quadratic programming and finite element method (FEM). In this paper, parametric quadratic programming method for two-dimensional contact mechanics analysis is applied to the crack problems involving the crack surfaces in frictional contact. Based on a linear complementary contact condition, the parametric variational principle and FEM, a linear complementary method is extended to analyze contact fracture mechanics. The near-tip fields are properly modeled in the analysis using special crack tip elements with quarter-point nodes. Stress intensity factor solutions are presented for some frictional contact fracture problems and are compared with known results where available.