Equivalent Domain Integral Method Research Articles

An extended equivalent domain integral method for mixed mode fracture problems by the p ‐version of FEM

A new path-independent contour integral formula is presented to estimate the crack-tip integral parameter, J-value, for two-dimensional cracked elastic bodies which may quantify the severity of the crack-tip stress fields. The conventional J-integral method based on a line integral has been converted to an equivalent area or domain integral (EDI) by the divergence theorem. It is noted that the EDI method is very attractive because all the quantities necessary for computation of domain integrals are readily available in a finite element analysis. The details and its implementation are extended to the p-version FE model with hierarchic elements using integrals of Legendre polynomials. By decomposing the displacement field obtained from the p-version finite element analysis into symmetric and antisymmetric displacement fields with respect to the crack line, the Mode-I and Mode-II non-dimensional stress intensity factors can be determined by using the decomposition method. The example problems for validating the proposed techniques are centrally oblique cracked plates under tensile loading. The numerical results associated with the variation of oblique angles show very good agreement with the existing solutions. Also, the selective distribution of polynomial orders and the corner elements for automatic mesh generation are applied to improve the numerical solution in this paper. © 1998 John Wiley & Sons, Ltd.

An extended equivalent domain integral method for mixed mode fracture problems by thep-version of FEM

An extended equivalent domain integral method for mixed mode fracture problems by thep-version of FEM

Effect of residual stresses on the interfacial fracture behavior of metal-matrix composites

The fiber/matrix interface plays a critical role in the mechanical behavior of the composites. The fiber pushout test is increasingly being used to characterize the interfacial behavior of metal-matrix composites (MMCs). A fracture mechanics approach is used to examine the interfacial debonding process in MMCs and ceramic-matrix composites (CMCs) during a fiber push-out test. The equivalent domain integral (EDI) method is implemented in a finite element code and is used to compute the strain-energy release rates for the interface crack. The cooling process from the composite consolidation temperature, specimen preparation for the push-out test and the actual testing are included in the finite element simulation. A strain-energy-based debonding criterion is used to predict the interfacial behavior. The experimentally observed phenomenon of bottom debonding in MMCs is explained from the energy release rate variation for the loading and support end cracks. It is shown that processing-induced residual stresses significantly affect the initiation and propagation of interface cracks. The advantage of the EDI method over conventional methods for modeling interface crack propagation, by eliminating the need for singular elements and thus remeshing with crack advance is demonstrated through the simulation of the push-out test.

Calculation of stress intensity factors for an interfacial crack between dissimilar anisotropic media, using a hybrid element method and the mutual integral

Using Beom and Atluri's complete eigen-function solutions for stresses and displacements near the tip of an interfacial crack between dissimilar anisotropic media, a hybrid crack tip finite-element is developed. This element, as well as a mutual integral method are used to determine the stress intensity factors for an interfacial crack between dissimilar anisotropic media. The hybrid element has, for its Galerkin basis functions, the eigen-function solutions for stresses and displacements embedded within it. The “mutual integral” approach is based on the application of the path-independent J integral to a linear combination of two solutions: one, the problem to be solved, and the second, an “auxiliary” solution with a known singular solution. A comparison with exact solutions is made to determine the accuracy and efficiency of both the methods in various mixed mode interfacial crack problems. The size of the hybrid element was found to have very little effect on the accuracy of the solution: an acceptable numerical solution can be obtained with a very coarse mesh by using a larger hybrid element. An equivalent domain integral method is used in the application of the “mutual” integral instead of the line integral method. It is shown that the calculated mutual integral is domain independent. Therefore, the mutual integral can be evaluated far away from the crack-tip where the finite element solution is more accurate. In addition, numerical examples are given to determine the stress intensity factors for a delamination crack in composite lap joints and at plate-stiffener interfaces.

Dynamic J Integral and Moving Finite-Element Method in Three-Dimensional Dynamic Fracture Mechanics.

In order to simulate three-dimensional dynamic crack propagation, a three-dimensional moving finite-element method was developed. A new expression was also derived for the dynamic J integral (J') in terms of the equivalent domain integral (EDI) method. The moving finite-element procedure together with the dynamic J integral evaluation procedure made it possible to evaluate the distribution of the dynamic energy release rate along the propagating crack front at each time step. Furthermore, a new formula for converting the dynamic J integral to the stress intensity factor along the crack front was derived by introducing a three-dimensionality parameter. Numerical simulation was carried out for several sample problems.

A methodology for computing nonlinear fracture parameters for a bulging crack in a pressurised aircraft fuselage

A computational methodology for obtaining nonlinear fracture parameters which account for the effects of plasticity at the tips of a bulging crack in a pressurised aircraft fuselage is developed. The methodology involves a hierarchical three stage analysis (global, intermediate, and local) of the cracked fuselage, with the crack incorporated into the model at each stage. The global analysis is performed using a linear elastic shell finite element model in which the stiffeners are treated as beam elements. The geometrically nonlinear nature of the bulging phenomenon is emulated in the intermediate analysis using a geometrically nonlinear shell finite element model. The local analysis is a three-dimensional solid finite element model of the cracked skin using a hypoelastic-plastic rate formulation. Kinematic boundary conditions for each stage are obtained from the preceding stage in the hierarchy using a general mesh independent mechanism. The T*integral, which accounts for both large deformations and plasticity, is taken to be the fracture parameter characterising the severity of the conditions at the crack tip, and is evaluated from the local analysis using the Equivalent Domain Integral (EDI) method. The implementation of the EDI technique for finite deformations in shell space is also outlined. The methodology is applied to a number of example problems for which correction factors relating the nonlinear T* values to those obtained from a linear elastic stiffened shell analysis are computed. The issue of flapping is addressed by investigating the behaviour of the longitudinal stress parallel to the crack for various cases.

An analytical-numerical alternating method for elastic-plastic analysis of cracks

A new algorithm based on the Schwartz-Neumann alternating technique is developed for the solution of elastic-plastic fracture mechanics problems. An analytical solution for an elastic crack, with arbitrary crack-face loading, is used inside an initial stress iterative procedure as an addition to the finite element solution for the uncracked body. Iteration processes of the alternating method and of the initial stress method are performed simultaneously. Numerical examples show that the proposed elastic-plastic alternating method in conjunction with the equivalent domain integral method provides reasonable values of the J-integral.

An equivalent domain integral method for three-dimensional mixed-mode fracture problems

A general formulation of the equivalent domain integral (EDI) method for mixed-mode fracture problems in cracked solids is presented. The method is discussed in the context of a 3-D finite-element analysis. The J- integral consists of two parts: the volume integral of the crack front potential over a torus enclosing the crack front and the crack surface integral due to the crack front potential plus the crack-face loading. In mixed-mode crack problems the total J- integral is split into J I , J II , and J III , representing the severity of the crack front in three modes of deformation. The direct and decomposition methods are used to separate the modes. These two methods were applied to several mixed-mode fracture problems in isotropic materials. Several pure and mixed-mode fracture problems were analyzed and results found to agree well with those available in the literature. The method lends itself to be used as a post-processing subroutine in a general purpose finite-element program.

An equivalent domain integral method in the two-dimensional analysis of mixed mode crack problems

An equivalent domain integral (EDI) method for calculating J- integrals for two-dimensional cracked elastic bodies is presented. The details of the method and its implementation are presented for isoparametric elements. The total and product integrals consist of the sum of an area or domain integral and line integrals on the crack faces. The line integrals vanish only when the crack faces are traction free and the loading is either pure mode I or pure mode II or a combination of both with only the square-root singular term in the stress field. The EDI method gave accurate values of the J- integrals for two mode I and two mixed mode problems. Numerical studies showed that domains consisting of one layer of elements are sufficient to obtain accurate J- integral values. Two procedures for separating the individual modes from the domain integrals are presented. The procedure that uses the symmetric and antisymmetric components of the stress and displacement fields to calculate the individual modes gave accurate values of the integrals for all problems analysed. The EDI method when applied to a problem of an interface crack in two different materials showed that the mode I and mode II components are domain dependent while the total integral is not. This behavior is caused by the presence of the oscillatory part of the singularity in bimaterial crack problems. The EDI method, thus, shows behavior similar to the virtual crack closure method for bimaterial problems.

Comparison of different methods of evaluation of weight functions for 2-D mixed-mode fracture analyses

In this paper, two different techniques for mixed-mode type engineering fracture analysis, by using the weight-function method, are presented. These two methods are (a) the virtual crack extension technique (VCE), and (b) the equivalent domain integral method (EDI). For mixed-mode weight-function evaluation, a technique of using two reference load states, both of which cannot be either pure mode I or pure mode II, is combined with the above mentioned numerical schemes. The numerical results show that the weight functions calculated by these two methods coincide with each other well. With the proposed techniques, it is very straightforward to perform fracture analysis of isotropic, orthotropic and composite materials.

Calculation of fracture mechanics parameters for an arbitrary three‐dimensional crack, by the ‘equivalent domain integral’ method

AbstractIn this paper, an equivalent domain integral (EDI) method and the attendant numerical algorithms are presented for the computation of a near‐crack‐tip field parameter, the vector Jϵ‐integral, and its variation along the front of an arbitrary three‐dimensional crack in a structural component. Account is taken of possible non‐elastic strains present in the structure; in this case the near‐tip Jϵ‐values may be significantly different from the far‐field values Jf, especially under non‐proportional loading.