Formula For Stress Intensity Factor Research Articles

NOTES ON NEW SINGULAR FUNCTION METHOD FOR DOMAIN SINGULARITIES

Recently, a new singular function(NSF) method was posed to get accurate numerical solution on quasi-uniform grids for two-dimensional Poisson and interface problems with domain sin- gularities by the first author and his coworkers. Using the singular function representation of the solution, dual singular functions, and an extraction formula for stress intensity factors, the method poses a weak problem whose solution is in H 2 (›) or H 2 (›i). In this pa- per, we show that the singular functions, which are not in H 2 (›), also satisfy the integration by parts and note that this fact suggests the possibility of dierent choice of the weak formulations. We show that the original choice of weak formulation of NSF method is critical.

A finite element method using singular functions: interface problems

The solution of the interface problem is only in $H^{1+\alpha}(\Omega)$ with $\alpha>0$ possibly close to zero and, hence, it is difficult to be approximated accurately. This paper studies an accurate numerical method on quasi-uniform grids for two-dimensional interface problems. The method makes use of a singular function representation of the solution, dual singular functions, and an extraction formula for stress intensity factors. Using continuous piecewise linear elements on quasi-uniform grids, our finite element approximation is shown to be optimal, $O(h)$, accurate in the $H^1$ norm. This is confirmed by numerical experiments for interface problems with $\alpha < 0.1$. An $O(h^{1+\alpha})$ error bound in the $L^2$ norm is also established by the standard duality argument. For small $\alpha$, this improvement over the $H^1$ error bound is negligible. However, numerical tests presented in this paper indicate that the $L^2$ norm accuracy is much better than the theoretical error bound.

An empirical formula for stress intensity factors of cracks emanating from a circular hole in a rectangular plate in tension

An empirical formula of the SIFs of cracks emanating from a circular hole in a rectangular plate in tension is presented and examined. It is found that the empirical formula is simple, yet accurate for evaluating the SIFs of the crack problem.

Multigrid methods for the computation of singular solutions and stress intensity factors III: interface singularities

It is shown in this paper that two-dimensional interface problems with large jumps in the coefficients can be solved effectively by the piecewise linear finite element method on quasi-uniform grids. This is achieved by combining the full multigrid methodology, the singular function representation of the solution and the extraction formulas for stress intensity factors. The robustness of the algorithm is demonstrated by numerical experiments.

Gradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials—Part I: Crack Perpendicular to the Material Gradation

Anisotropic strain gradient elasticity theory is applied to the solution of a mode III crack in a functionally graded material. The theory possesses two material characteristic lengths, l and l′, which describe the size scale effect resulting from the underlining microstructure, and are associated to volumetric and surface strain energy, respectively. The governing differential equation of the problem is derived assuming that the shear modulus is a function of the Cartesian coordinate y, i.e., G=Gy=G0eγy, where G0 and γ are material constants. The crack boundary value problem is solved by means of Fourier transforms and the hypersingular integrodifferential equation method. The integral equation is discretized using the collocation method and a Chebyshev polynomial expansion. Formulas for stress intensity factors, KIII, are derived, and numerical results of KIII for various combinations of l,l′, and γ are provided. Finally, conclusions are inferred and potential extensions of this work are discussed.

Approximate Green's functions for singular and higher order terms of an edge crack in a finite plate

An edge crack in a finite plate (FSECP) subjected to wedge forces is solved by the superposition of the analytical solution of a semi-infinite crack, and the numerical solution of a FSECP with free crack faces, which is solved by the Williams expansion. The unknown coefficients in the expansion are determined by a continuous least squares method after comparing it with the direct boundary collocation and the point or discrete least squares methods. The results are then used to validate the stress intensity factor (SIF) formula provided by Tada et al. that interpolates the numerical results of Kaya and Erdogan, and an approximate crack face opening displacement formula obtained in this paper by Castigliano's theorem and the SIF formula of Tada et al. These approximate formulae are accurate except for point forces very close to the outer edge, and can be used as Green's functions in the crack-closure based crack growth analysis, as well as in interpreting the size effect of quasi-brittle materials. Green's functions for coefficients relevant to the second to the fifth terms in the crack tip asymptotic field are also provided. Finally, a FSECP with a uniform pressure over a part of the crack faces is solved to illustrate the application of the obtained Green's functions and to further assess their accuracy by comparing with a finite element analysis.

Coefficients of the crack tip asymptotic field for a standard compact tension specimen

The coefficients of the crack tip asymptotic field of a standard compact tension (CT) specimen are computed using a hybrid crack element (HCE). It allows the direct calculation (without post-processing) of not only the stress intensity factor (SIF) but also the coefficients of higher order terms of the crack tip asymptotic field. Approximate closed-form expressions for the first five terms for the CT specimen that are accurate for shallow to very deep cracks are obtained by fitting the computed data. The SIF formula proposed by Brown and Srawley (1966) is shown to be accurate when the crack length to depth ratio (α) ranges from 0.35 to 0.75. The formula proposed by Newman (1974) and Srawley (1976) is accurate for α≥0.15. However, the accuracy of available formulas for the second T-term in the literature is quite disappointing. Numerical results also show that, unlike the notched three-point bend beam and the wedge splitting specimen, the second T-term of the CT specimen is always positive.

A crack in a viscoelastic functionally graded material layer embedded between two dissimilar homogeneous viscoelastic layers - antiplane shear analysis

A crack in a viscoelastic functionally graded material (FGM) layer sandwiched between two dissimilar homogeneous viscoelastic layers is studied under antiplane shear conditions. The shear relaxation modulus of the FGM layer follows the power law of viscoelasticity, i.e., μ = μ0exp (βy/h) [t0exp (δy/h) /t]q, where h is a scale length, and μ0,t0,β,δ and q are material constants. Note that the FGM layer has position-dependent modulus and relaxation time. The shear relaxation functions of the two homogeneous viscoelastic layers are μ=μ1(t1/t)q for the bottom layer and μ=μ2(t2/t)q for the top layer, where μ1 and μ2 are material constants, and t1 and t2 are relaxation times. An elastic crack problem of the composite structure is first solved and the `correspondence principle' is used to obtain stress intensity factors (SIFs) for the viscoelastic system. Formulae for SIFs and crack displacement profiles are derived. Several examples are given which include interface cracking between a viscoelastic functionally graded interlayer and a viscoelastic homogeneous material coating. Moreover, a parametric study is conducted considering various material and geometric parameters and loading conditions.

Viscoelastic Functionally Graded Materials Subjected to Antiplane Shear Fracture

In this paper, a crack in a strip of a viscoelastic functionally graded material is studied under antiplane shear conditions. The shear relaxation function of the material is assumed as μ=μ0 expβy/hft, where h is a length scale and f(t) is a nondimensional function of time t having either the form ft=μ∞/μ0+1−μ∞/μ0exp−t/t0 for a linear standard solid, or ft=t0/tq for a power-law material model. We also consider the shear relaxation function μ=μ0 expβy/h[t0 expδy/h/t]q in which the relaxation time depends on the Cartesian coordinate y exponentially. Thus this latter model represents a power-law material with position-dependent relaxation time. In the above expressions, the parameters β, μ0,μ∞,t0; δ, q are material constants. An elastic crack problem is first solved and the correspondence principle (revisited) is used to obtain stress intensity factors for the viscoelastic functionally graded material. Formulas for stress intensity factors and crack displacement profiles are derived. Results for these quantities are discussed considering various material models and loading conditions.

Evaluation of the formula of stress intensity factor K i for semi-elliptical surface crack

The stress-intensity factor (SIF) of the semi-elliptical surface crack in the finite body under extensional stress is sclculated by using the FEM software ANSYS release 5.5. The correction factor M f of SIF at different point along the front of the crack is determined. The relation between M f and the semi-elliptical shape a/c, the relative crack depth a/b, the variation of angle ϑ, the relative crack width 2c/w and the relative height-width ratio h/w are calculated respectively. Finally the application range and the modification of the engineering formula about SIF is proposed.

A new weight function approach using indirect boundary integral method

A new weight function approach to determine SIF (stress intensity factor) using the indirect boundary integral method has been presented. The crack opening displacement field was represented by one boundary integral in the form of a single-layer potential whose kernel was modified from the fundamental solution. The proposed method enables the calculation of SIF using only one SIF formula without any modification of the crack geometries symmetric in a two-dimensional plane, e.g. a center crack in a plate with or without an internal hole, double edge cracks, circumferential crack or radial cracks in a pipe. The application procedure for this variety of crack geometries is very simple and straghtforward with only one SIF formula. The necessary information in the analysis is two reference SIFs. The analysis results, using several examples, verified that the present closed-form solution was in good agreement with those of the literature and applicable to various crack geometries.

Error estimation of numerical stress intensity factors for semi-elliptical cracks

Error estimates of numerical results and empirical formulae for stress intensity factors are usually based on author's experience and on comparison of some test results with closed analytical solutions (e.g., circular and elliptical cracks in an infinite solid). Such evaluation of K~-data in three-dimensional crack problems may become insufficient, as it was demonstrated for some part-through crack configurations [1]. Another way to check the quality of K~-results [2,3] is based on the evaluation of average, or integral, errors through the calculation of energy integrals associated with a cracked body. This procedure has been developed for the error estimation of some known stress intensity factor equations for part-elliptical cracks in plates [1,3]. It can also be a useful tool for the verification of numerical results on K~-factor provided the latter are derived for a wide range of configuration parameters of a cracked body and different locations of points on the crack front. The present paper deals with the evaluation of accuracy of boundary element data on K~-distdbution for circumferential and axial part-elliptical cracks in a hollow cylinder. Average errors are estimated at different kinds of polynomial loading applied to crack faces. It is concluded that the evaluation procedure can be applied for an approximate error analysis of local K~-estimates at some characteristic points of a crack front.

Some asymptotic behavior and formulae of stress intensity factors for collinear and parallel cracks under various loadings

In this paper, the relations between stress intensity factors and N are given in a more refined manner with theoretical proofs than previously. They apply to cracks, holes and inclusions of arbitrary shapes. Based on these relations, reliable formulae of the stress intensity factors are proposed for collinear and parallel cracks under various loadings.

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W2(y) and W3(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.

Analogies between crack tip and rigid line tip stresses and displacements

The analogies between crack tip and rigid line tip stresses and displacements are stated in respect of loading modes with singular stresses, loading modes with non-singular stresses, distribution of singular stresses, ligament related limit value formulae for stress intensity factors, notch stress related limit value formulae for stress intensity factors and elementary explicit formulae for stress intensity factors.

Equally Spaced Equal Collinear and Parallel Cracks under Various Loads. Some Asymptotic Behavior and Formulae of Stress Intensity Factors.

Previously, Isida analysed a row of equal cracks distributed with equal spacings subjected to tension, in-plane shear, longitudinal shear and classical bending. In all the problems, the stress intensity factors were found to be nearly linear with 1/N for fixed values of λ, where N and λ are the number of cracks and the length-spacing ratio, respectively. In this paper, the relationships between the stress intensity factors and N are given in a more refined manner with theoretical proofs. They apply not only to cracks, but also to holes and inclusions of arbitrary shapes. Based on these relationships, reliable formulae of the stress intensity factors are proposed for collinear and parallel cracks under various loads.

Simplified formulae for stress intensity factors of spot welds

The procedure for calculating the stress intensity factors at the edge of spot welds on the basis of the structural stresses is further simplified. The results are compared with a recent solution from literature.

Stress intensity factors of two parallel 3d surface cracks

Two parallel semicircular surface cracks in a finite plate subjected to remote tensile loading have been studied. This paper presents an empirical stress intensity factor formula for this crack configuration and the stress intensity factors used to develop the formula were obtained from finite element analysis. The formula covers a wide range of configuration parameters and is within 6% of the finite element results. The comparison with a bi-dimensional estimation is also considered.

Formulae of stress intensity factors at tips of kinked cracks under various loadings

A number of papers have been published on kinked cracks by using various analytical methods, but most of them are concerned with special crack geometries or special loading conditions, and often give unreliable values for cracks with a short kink or with a large kink angle. The purpose of this paper is to give reliable formulae of the stress intensity factor at the tips of kinked cracks in a wide plate under various loading conditions at infinity. The analysis is based on the body force method combined with a perturbation procedure. Calculations are performed for three fundamental loads at infinity, and for various values of the kink angle β. The numerical results of the stress intensity factors are fitted to a power series of β and b a , the ratio of the lengths of the kink and the main crack. The stress intensity factors of arbitrarily kinked cracks under arbitrary uniform loads at infinity are given by linear combinations of the above obtained formulae for the three fundamental loading cases.

Simple procedure for ditermination of stress intensity pactor by boundary element mtdhod.

The method that the present authors have recently proposed on using the finite element method has been incorporated into a boundary element method to analyse twodimensional elastostatic stress intensity factors. Namely, the present method is based on the expectation that the ratio of the resultant force around the crack tip in the theoretical value and the numerical one is constant. Numerical-emprical formulas for stress intensity factors of the mode I and II deformations are established from the results of stress analysis for the strips with central crack and the rectangular plates with contral inclined crack, respectively. The linear strain element has been used to the calculations of stress for the cracked plates. The usefulness of the method has been tested with a number of plates with crack, and the results have turned out to be fairly satisfactory.