Estimating stress intensity factors with singular components of the total finite element solution

Displacement formulated singular elements are compared to isoparametric quarter-node elements. The total stress and displacement solution for each element is decomposed into singular and regular components for evaluating stress intensity factors. Specific relationships between nodal displacements and the singular component of stress are presented for the isoparametric element at selected locations within the element. Numerical results for K_I and K_II in the slant crack problem are presented. The singular components are shown to produce superior results when considering crack opening displacements.     

A reappraisal of transition elements in linear elastic fracture mechanics

This article offers a detailed comparison of the transition elements described by P.P. Lynn and A.R. Ingraffea [International Journal for Numerical Methods in Engineering 12,1031–1036] and C. Manu[Engineering Fracture Mechanics 24,509–512]. The source of a numerical phenomenon in using Manu's transitionelement (TE) is explained. The effect of eight-noded TEs with differentquarter-point elements (QPE) on the calculated stress intensity factors (SIFs) isinvestigated. Strain at the crack tip is shown to be singular for any ray emanating from the crack tip within an eight-noded TE, but strain has bothr ^–1/2andr ^–1singularities, withr ^–1/2dominating for large TEs. Semi-transition elements (STEs) are defined and shown to have a marginal effect on the calculated SIFs. Nine-nodedtransition elements are formulated whose strain singularity is shown to be the same as that of eight-noded TEs. Then the effect of eight-noded and nine-noded TEs with collapsed triangular QPEs, and rectangular and nonrectangular quadrilateral eight-noded and nine-noded QPEs, is studied, and nine-noded TEs are shown to behave exactly like eight-noded TEs with rectangular eight-noded and nine-noded QPEs and to behave almost the same with other QPEs. The layered transition elements proposed by V. Murti and S.Valliapan [Engineering Fracture Mechanics 25, 237–258] areformulated correctly. The effect of layered transition elements is shown by two numerical examples.     

Increase of stress intensity near interface edge of elastic-creep Bi-material under a sustained load

The transition from small-scale creep to large-scale creep ahead of a crack tip or an interface edge with strong elastic stress singularity at the loading instant causes stress relaxation and the decrease of stress intensity in general. However, this study shows that the stress near the interface edge of bi-material with no or weak elastic stress singularity increases after the loading instant and brings about the stress concentration during the transition. In addition, the creep strain distribution of this bi-material after the loading instant is different from that occurred in the transition of an interface edge with strong elastic stress singularity or a crack tip (notch root). The criterion for the increase or decrease of stress intensity near the interface edge proved by the finite element method is proposed in this study. The stress intensity near the interface edge increases when the elastic stress singularity is lower than the creep stress singularity (λ^el < λ^cr) and vice versa.     

Isoparametric Hermite elements

Isoparametric Hermite elements are created using Bogner–Fox–Schmit rectangles on a reference domain and mapping these numerically onto the computational domain. The difficulties involved in devising explicit C¹ shape functions for isoparametric elements are thus avoided, and the resulting elements have all the benefits of full C¹ continuity, the simplicity of the Bogner–Fox–Schmit element and the geometrical flexibility expected from higher-order isoparametric elements. The numerical mapping consists in the finite element solution of a linear boundary value problem, which is inexpensive and is carried out as a preprocessing operation—the required derivatives of the mapping then being supplied to the main analysis as data. Some care is required in defining the differential boundary conditions, and guidance on this is provided. Examples are given showing the success of the mapping procedure, and the use of the resulting elements in the solution of some boundary value problems. The numerical results confirm a convergence analysis provided for the new isoparametric Hermite element.     

Mathematical simulation of singularities for isoparametric elements of arbitrary orders

For the general quadrilateral isoparametric elements with 4(n?1) nodes (i.e. n nodes along each side), it is shown that the inverse square root singularity of the strain field at the crack tip can be obtained by a general but simple rule. This amounts to collapsing the quadrilateral elements into triangular elements around the crack tip and placing the (n?2) mid-side nodes at locations γ²/(n?1)² (γ=1,2,…, n?2) times the length of side emanating from the crack tip. These locations are measured from the crack tip. The known results for n = 3 and n = 4 are thus obtained as special cases.     

On the evaluation of the J-integral by a plastic crack tip element

Two crack tip elements are formulated for a stationary, mode I plastic crack in planar structures using hybrid assumed stress approach, based on the secant modulus and the Newton-Raphson schemes, respectively. The stress distribution in the crack tip element is assumed to be the HRR field superimposed by the regular polynomial terms. The formulated (hybrid) crack tip elements are compatible with the isoparametric element so that they can be used conveniently along with the conventional displacement-based finite elements. The intensity of the HRR stress field, the J-integral, is determined directly from the finite element equations together with the nodal displacements. The dominance of the HRR stress field at the crack tip is pertinent to the present approach, which depends on geometry and loading conditions. Since the J-integral is globally path-independent for nonlinear elastic materials (deformation plasticity model), in order to assess the accuracy and efficiency of the methodology as compared to the contour integration approach, numerical studies of common plane-stress cracked configurations are performed for these materials. The results indicate that for a sufficiently small crack tip element size, J from the present approach correlates well, within 6 percent difference, with that from the contour integration for a wide range of material hardening coefficients if the HRR zone exists at the crack tip. These highly accurate results for J from the crack tip stresses could not be achieved without using (newly) modified variational principles and a refined numerical technique. It should be emphasized that the present methodology also can be applied to cracks in J ₂ flow materials under HRR dominance. In such case, the J integral may not be globally path independent, and hence it now must be determined from the stress and strain fields near the crack tip.     

Fatigue cracking simulation based on crack closure effects in Al-based sheet materials subjected to biaxial cyclic loads

Fatigue crack growth experiments have been carried out on cruciform specimens in the range of thickness 1.2–10 mm of Al-based alloys, loaded under constant (regular) and variable (irregular) amplitudes of uniaxial and biaxial loads, including sequences of various overloads. Different cases for crack closure effects are considered because of shear lips development, crack-growth direction re-orientation after multiparameters change of cyclic loads, by examining plastic blunting effect at a crack tip during an overload and interaction effects analyzing the crack retardation length and associated parameters together with their relationships. Crack closure effect because of rotation instability of material mesovolumes under biaxial compression–tension has suggested to analyse semi-elliptical cracks. Under biaxial cyclic loads in the range of load ratio-1.4 < λ < +1.5, and R-ratios from 0.05 to 0.8, for frequency variations ?, fatigue striation formation takes place beyond a crack-growth rate close to 4 × 10⁻⁸ m/cycle. The striation spacing and the crack-growth rate increase as the ?-angle of the out-of-phase biaxial loads increases (in the range of ? from 0° to 180°). Cycle loading parameters must be taken into account in order to describe the crack growth period when using a unified method that involves an equivalent stress intensity factor K_e=K_IF(λ,R,?,?). The values of F(λ,R,?,?) are determined. The calculated crack growth period (predicted using F(λ,R,?,?)) in regular and irregular cases of cyclic loads, including material cracking after overloads, is correlated with the experimental data, and the error is of the order of 15%.     

Three-dimensional interaction between a crack front and particles

The three-dimensional interaction of a crack front with particles is investigated under mode-I loading. The J-integral is applied to characterize the crack–inclusion interactions. Numerical examples are presented, using 20 node, isoparametric finite elements, for the compact tension specimen and elastic materials. The J-integral is calculated for various moduli of the particles, distances of the crack from the interface and particle size. The problem of the crack penetrating a cluster of particles is discussed. © 1998 John Wiley & Sons, Ltd.     

A degenerate solid element for linear fracture analysis of plate bending and general shells

A general curved element of arbitrary shape for both thick and thin shells is proposed for the linear fracture analysis of a through crack in a shell or a plate. The element is derived from a degenerate 20-noded solid isoparametric element using reduced integration technique. The 1/√(r) singularity of the strains is obtained by the same procedure proposed earlier for two- and three-dimensional problems,^1,2 viz. by placing the mid-side nodes near the crack at the quarter points. Several illustrated examples ranging from classical solutions to practical problems are given to assess the accuracy of solution attainable.     

On the C1 continuous discretization of non‐linear gradient elasticity: A comparison of NEM and FEM based on Bernstein–Bézier patches

In gradient elasticity, the appearance of strain gradients in the free energy density leads to the need of C¹ continuous discretization methods. In the present work, the performances of C¹ finite elements and the C¹ Natural Element Method (NEM) are compared. The triangular Argyris and Hsieh–Clough–Tocher finite elements are reparametrized in terms of the Bernstein polynomials. The quadrilateral Bogner–Fox–Schmidt element is used in an isoparametric framework, for which a preprocessing algorithm is presented. Additionally, the C¹‐NEM is applied to non‐linear gradient elasticity. Several numerical examples are analyzed to compare the convergence behavior of the different methods. It will be illustrated that the isoparametric elements and the NEM show a significantly better performance than the triangular elements. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical evaluation of stress intensity factor and J integral in three-point bend specimen

The present paper attempts to evaluate the fracture mechanics parameters, the stress intensity factor (K) and Rice's energy integral (J) in plane strain conditions for three-point bend specimens. Both the parameters have been evaluated by the FEM using higher order isoparametric elements (i.e. quadratic elements). The crack tip elastic singularity (1/√r) has been taken into account by the use of the special crack tip elements of degenerate triangular element type as well as the fine eight-noded isoparametric plane elements. The stress distribution has been compared with the Westergaard solution in the vicinity of the crack. The K and J values have also been-compared with the theoretical results.     

Numerical determination of a class of generalized stress intensity factors

A modification of the collocation method for the numerical solution of Cauchy-type singular integral equations appearing in plane elasticity and, especially, crack problems is proposed. This modification, based on a variable transformation, applies to the case when the unknown function of the singular integral equation behaves like A(x ? c)^α + B(x ? c)^β, where α < 0, 0 < β ? α < 1, near an endpoint c of the integration interval. In plane elasticity such a point is either a crack tip or a corner point of the boundary of the elastic medium. Thus the method seems to be quite efficient for the numerical evaluation of generalized stress intensity factors near such points. A successful application of the method to the classical plane elasticity problem of an antiplane shear crack terminating at a bimaterial interface was also made.     

A numerical study of the deviation of the crack ending at the interface in a multilayer

The study of the deviation of a crack reaching the interface separating two materials in a multilayer in a 4-point deflection is considered. This study is based on the use of special isoparametric finite elements.¹ (The elements surrounding the crack tip have nodes placed at a distance α from the crack tip, and in this way we impose a singularity of the stresses in the neighbourhood of the crack.) The Erdogan² criterion that estimates the deviation angle shows that this angle is a function of the initial angle of the crack, and the properties of each component of the multilayer.     

Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements

Triangular and prismatic quadratic isoparametric elements, formed by collapsing one side and placing the mid-side node near the crack tip at the quarter point, are shown to embody the (1/√r) singularity of elastic fracture mechanics and the (1/r) singularity of perfect plasticity. The procedure of performing the fracture analysis for the case of small scale yielding is discussed, and the finite element results are compared with theoretical results. The proposed elements have wide application in the fracture analysis of structures where ductile fracture is investigated. They permit a determination of the relationship between crack tip field parameters, loading, and geometry. And for a given fracture criterion can be applied to the prediction of fracture in structures such as pressure vessels under in service conditions.     

Transition elements matching the near crack-tip strain field

New transition elements used with the collapsed triangular singular elements are constructed by using the improved isoparametric transformations. Without 1/r singular terms, the new transition elements' strain fields contain only % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac+% cadaGcaaqcaawaaiaadkhaaKqaGfqaaaaa!3B47!\[1/\sqrt r \] singularities and match the Williams solution quite well near the crack tip. It is convenient for the new transition elements to be constructed and introduced in the general purpose finite element programs by adding some modifications. Numerical results show that the transition elements possess good properties and are worth being applied to linear fracture computations.     

Numerical evaluation of the quarter-point crack tip element

This paper attempts to answer two commonly raised questions during the preparation of a finite element mesh, for the linear elastic fracture analysis of cracked structure: how to set up the finite element mesh around the crack tip, and what level of accuracy is to be expected from such a modelling. Two test problems, with known analytical expressions for their stress intensity factors, are analysed by the finite element method using the isoparametric quadratic singular element. The modified parameters were the order of integration, aspect ratio, number of elements surrounding the crack tip, use of transition elements, the singular element length over the total crack length, the symmetry of the mesh around the crack tip. Based on these analyses, a data base is created and various plots produced. The results are interpreted, the accuracy evaluated and recommendations drawn. Contrary to previous reports, it is found that the computed stress intensity factor (SIF) remains within engineering accuracy (10 per cent) throughout a large range of l/a (singular element length over crack length) for problems with a uniform non-singular stress distribution ahead of the crack tip (i.e. double edge notch), and l/a should be less than 0·1 for problems with a non-singular stress gradient (i.e three-point bend). Also, it is found that the best results are achieved by using at least four singular elements around the crack tip, with their internal angles around 45 degrees, and a reduced (2 × 2) numerical integration.     

Analysis of power type singularities using finite elements

A finite element formulation is described for problems with solution functions known to have local rλ variation (s), 0<λ<1, and thus singular gradients. Special 3-node triangular elements encircle the singularity and focus to share a common node at the singular point. The shape function of each triangle has the appropriate r λ mode and a smooth angular mode expressed in element natural co-ordinates. As with standard elements, the unknowns are the nodal values of the function. Even if the precise angular form of the asymptotic solution is known, the formulation makes no attempt to embed it, but instead piecewise approximates it. This allows assembly of the element coefficient matrix using standard procedures without nodeless variables and bandwidth complications. The conditions of continuity, low order solution capability, and accurate numerical integration of the singularity element are discussed with a view towards establishing the general range of applicability of the formulation. Numerical applications to the elastic fracture mechanics problems of composite bondline cracking and crack branching are discussed.     

Moving crack at the interface between two dissimilar magnetoelectroelastic materials

Summary Analytical solutions for an anti-plane Griffith crack moving at the interface between two dissimilar magnetoelectroelastic media under the conditions of permeable crack faces are formulated using the integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic materials. Expressions for stresses, electric displacements, and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities, and it is found that the dynamic stress intensity factor (DSIF), the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF) are independent of the remote electromagnetic loads. The moving speed of the crack has influence on the DEDIF and the DMIIF. When the crack is moving at lower speeds 0 ≤ M≤ M_c1 or higher speeds M_c2 < M < 1, the crack will propagate along its original plane, while in the range of M_c1 < M < M_c2 , the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.     

FRACTOGRAPHIC ANALYSES OF FATIGUE CRACK GROWTH IN D16T ALLOY SUBJECTED TO BIAXIAL CYCLIC LOADS AT VARIOUS R-RATIOS

Abstract Fractographic peculiarities of fatigue crack development are studied in cruciform specimens of D16T aluminium alloy under biaxial tension and tension-compression. In the range of the biaxial load ratios λ from - 1 to +1.5, and in the range of R-ratios 0.05 to 0.8, fatigue striation formation took place over a crack growth rate near to 4×10^?8 m/cycle. The striation spacing and the crack growth rate decrease as the ratios λ and R increase. The ratio between the increment of crack growth, da/dN, and the striation spacing, δ, is approximately 1:1 when da/dN is greater than 4×10-^?8 m/cycle. The relationship between the number of cycles from the beginning of a test up to the growth rate of 2.14×10^?7 m/cycle (N_d), and the crack growth period, _Np, from when the crack initiates up to the instant when that growth rate is reached, was determined for different λ and R-ratios. The value of N_d increases as the stress ratio, λ, is increased. Cycle loading parameters must be taken into account in order to describe the crack growth period when using a unified method involving an equivalent stress intensity factor K_e, =K₁,F(λ, R_s). The values of F(λ, R) for the growth rate (F(λ, R)_s) and for the striation spacing (F(λ, R_s) were determined and compared. The fatigue crack growth period, N^t_p, applicable to the stage of fatigue striation formation, (predicted by using both of the F(λ, R) values) is correlated with the experimental data and the error is of the order of 15%.     

Fatigue behavior of aluminum alloys under biaxial loading

The biaxiality effect, especially the effect of non-singular stress cycling, on the fatigue behavior was studied, employing cruciform specimens of aluminum alloys 1100-H14 and 7075-T651. The specimens, containing a transverse or a 45^o inclined center notch, were subjected to in-phase (IP) or 100% out-of-phase (hereinafter referred to as “out-of-phase or OP”) loading of stress ratio 0.1 in air. The biaxiality ratio λ ranged from 0 to 1.5, and 3 levels of stress were applied. It was observed that: (1) at a given λ, a lower longitudinal stress induced a longer fatigue life under IP and OP loading, and the fatigue life was longer under IP loading, (2) the fatigue crack path profile was influenced by λ, phase angle (0^o or 180^o), and initial center notch (transverse or 45^o inclined); (3) the fatigue crack path profiles, predicted analytically and determined experimentally, had similar features for the specimens with a transverse center notch under IP loading; and (4) the fatigue crack growth rate was lower and the fatigue life longer for a greater λ under IP loading, whereas it changed little with change in λ under OP loading. These results demonstrate that non-singular stress cycling affects the biaxial fatigue behavior of aluminum alloys 1100-H14 and 7065-T651under IP and OP loading.