Mixed-mode Stress Intensity Factors Research Articles

Fracture Mode Separation for Delamination in Platelike Composite Structures

A computationally efficient approach combining a plate crack-tip element model and a mixed-mode separation technique was developed to calculate mode mixities and stress intensity factors for delamination in platelike composite structures. The crack-tip element was based on classical plate theory, assuming the delamination surfaces to be traction free. Fracture mode separation was achieved by equating the total strain energy release rate from classical plate theory with that from the three-dimensional near-tip solution, along with two supplementary finite element analyses for special loadings. Using this approach, the separation of fracture modes under any other load combinations can easily be carried out. Numerical results showed that the present method gives good evaluations of the mixed-mode stress intensity factors.

Nonlinear Analysis and Experimental Verification of Shell Structure with Axial Multi-Cracks.

Interaction of multiple axial pre-cracks in a pressurized fuselage was investigated using a small scale model of an idealized fuselage. Stain between multiple cracks under rapidly cracks propagation was measured. The recorded crack velocities justified the use of a successive finite element analysis with large shell deformation. The experimental results of a rupturing model fuselage represented the variations in the mixed mode stress intensity factors, KI and KIIand the remote stress components σoxwith cracks extension. The crack kinking location, kinking angles and the off-axis crack trajectories are predicted by numerical analysis, and the validity of the calculated results is discussed in comparison with the experimental data.

Analysis of the driving force for a generally oriented crack in a functionally graded strip sandwiched between two homogeneous half planes

The driving forces for a generally oriented crack problem embedded in a Functionally Graded strip sandwiched between two half plane are analyzed using singular integral equations with Cauchy kernels, and integrated using Lobatto-Chebyshev collocation. Mixed-mode Stress Intensity Factors (SIF) and Strain Energy Release Rates (SERR) are calculated. The Stress Intensity Factors are compared for accuracy with previously published results. Parametric studies are conducted for various non-homogeneity ratios, crack lengths, crack orientation and thickness of the strip. It is shown that the SERR is more complete and should be used for crack propagation analysis.

가중함수법에 의한 볼트 체결부 균열의 임계 경사각 결정에 관한 연구

Mechanical joints such as bolted or riveted joints are widely used in mechanical components. The reliable determination of the stress intensity factors for cracks in bolted joints is needed to evaluate the safety and fatigue life of them. The weight function method is an efficient technique to calculate the stress intensity factors for various loading conditions because only the stress analysis of an uncracked model is required. In this paper the mixed-mode stress intensity factors for cracks in bolted joints are obtained by weight function method, in which the coefficients of weight function are determined by finite element analyses for reference loadings. Critical inclined angle that mode I stress intensity factor becomes maximum is determined and the effects of crack length and the magnitude of clearance on critical inclined angle are investigated.

Estimation of Manifestations of Material Properties and Mechanical Effects on a Macrocrack, Induced by a Cluster of Elliptical Microracks.

A macrocrack in a brittle material containing a cluster of microcracks experiences the stress shielding effect when the macrocrack front locates in the domain of stress relaxation induced by the microcacks. This is called the microcrack toughening effect for the macrocrack in a brittle material such as ceramic. Therefore the existence of microcracks improves the fracture toughness of the macrocracked solid. This paper presents the numerical analyses for the macrocrack in an anisotropically damaged materials. The homogenization method based on the superposition method is used to evaluate the material properties of the damaged materials and the mechanical behavior of the macrocrack in the damaged materials. In order to evaluate the mixed-mode stress intensity factors of a macrocrack in the anisotropically damaged materials, we derive the component separation method based on three dimensional path independent J integral. This paper provides the numerical results for the effective elastic moduli of the damaged materials and for the J integrals and the stress intensity factors of the macrocrack.

332 界面き裂の混合モード応力拡大係数の J 積分成分分離法による評価

332 界面き裂の混合モード応力拡大係数の J 積分成分分離法による評価

Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness

The interface crack problem for a composite layer that consists of a homogeneous substrate, coating and a nonhomogeneous functionally graded interphase was formulated for singular integral equations with Cauchy kernels, which were integrated using the Lobatto–Chebyshev collocation technique. Mixed-Mode Stress Intensity Factors (SIFs) and Strain Energy Release Rates were calculated. The SIFs were compared for accuracy with relevant results previously published. The parametric studies were conducted for the various thickness of each layer and for various nonhomogeneity ratios. Particular application to the Zirconia thermal barrier on steel substrate is demonstrated.

Axial crack propagation and arrest in a pressurized cylinder: An experimental-numerical analysis

The feasibility of using a previously developed crack-kinking criterion to predict crack arrest at a tear strap in a pressurized fuselage was studied with instrumented axial rupture tests of 21 models of an idealized fuselage. A rapidly propagating axial crack, which was initiated from a precrack, kinked immediately upon extension and propagated diagonally until it turned circumferentially and propagated along the tear straps. An elastodynamic finite element analysis of the rupturing model fuselage yielded the mixed-mode stress intensity factors,KI andKII, and the remote stress component, σOX. This numerical procedure was also used to predict the crack trajectories in full-scale fuselage rupture tests. All numerical results agreed well with their measured counterparts regardless of size.

Fracture propagation of anisotropic plates by the boundary element method

This paper presents a boundary element method combined with the maximum tensile stress criterion for predicting crack initiation angles and simulating fracture propagation paths in cracked anisotropic plates under mixed mode loading. Since the calculated displacements near the crack tip by the BEM have been verified to be very accurate, a decoupling technique can be used to determine the mixed mode stress intensity factors based on the definition of the J‐integral and the relative displacements at the crack tip. Numerical examples are presented for the calculation of stress intensity factors for both isotropic and anisotropic cases. It can be found that our numerical results are in good agreement with those reported by previous authors. Fracture propagation in an anisotropic plate under mixed mode loading is simulated by an incremental crack extension with a piece‐wise linear discretization. A computer program, which can automatically generate a new mesh (required for analyzing sequentially the boundary configuration) has been developed to simulate the fracture propagation process. It was found that the numerical analysis could predict relatively well the direction of crack initiation and the path of fracture propagation.

A dislocation and point force approach to the boundary element method for mixed mode crack analysis of plane anisotropic solids

In this paper we formulate a direct boundary element method (BEM) for plane anisotropic elasticity (i.e., the in‐plane deformation decoupled from the out‐of‐plane deformation) based on distributions of point forces and dislocation dipoles. According to a physical interpretation of Somigliana's identity the displacement field in a finite body R is represented by the continuous distributions of point forces and dislocation dipoles along the imaginary boundary ?R of the finite domain R embedded in an infinite body. We adopt Stroh's complex variable formalism for anisotropic elasticity and represent the point force and the dislocation, their dipoles, and continuous distributions systematically exploiting the duality relations between the point force and the dislocation solutions. Explicit formulas for the displacement and the traction formulations, obtained by analytical integration of the boundary integrals, are given. We apply these formulas to mixed mode crack problems for multiply cracked anisotropic bodies by extending the physical interpretation of Somigliana's identity to cracked bodies and representing the crack by the continuous distribution of dislocation dipoles. With the help of the conservation integrals of anisotropic elasticity, we will demonstrate the capability of the method to determine the mixed mode stress intensity factors (KI and KII ) accurately.

An improved photo-elastic technique for determining the mixed-mode stress-intensity factors for interfacial cracks in a bi-material

Attempts to employ the photo-elastic technique to determine the complex stress-intensity factors of interfacial cracks in a bi-material reveal the weakness of the technique. This weakness is due to the large errors arising from the measurement of the coordinates of data points and fringe orders near the crack tip. In this paper, an improved photo-elastic approach is proposed to reduce the effects of the measurement errors. The results obtained by this modified method are compared with both a traditional photo-elastic solution and a theoretical solution derived from complex potentials. The comparison shows that the modified solution is in better agreement with the theoretical solution as compared with the traditional solution.

Finite element analysis of interface cracking in semiconductor packages

The application of enriched crack tip finite elements for the prediction of interface fracture parameters, e.g., strain energy release rate and mixed mode stress intensity factors, is presented. Of particular interest, is the comparison between fracture results obtained from two-dimensional (2-D) models and related three-dimensional (3-D) (generalized plane strain) calculations. These results show that for thermal cycling problems, one cannot anticipate 3-D fracture results based on 2-D calculations alone, i.e., plane stress, plane strain, and axisymmetric models. On the other hand, it is shown that the 2-D models are quite adequate for modeling interface fracture in the case of pressure loading on the interface, e.g., pressure due to water vapor expansion during solder reflow. The fracture results presented in this paper were obtained using special enriched crack tip elements that contain the analytic asymptotic displacement and stress field. Enriched crack tip elements for 2-D and 3-D elements are shown to provide highly accurate results for simulating debonding in semiconductor packages subjected to thermal cycling and/or moisture absorption.

Cracks in three dimensions: A dynamic dual boundary element analysis

In this paper the dual boundary integral equation for three-dimensional dynamic problems in Laplace space is presented for the first time. The dual boundary element method is used to calculate dynamic stress intensity factors for three-dimensional cracked structures. The application of Laplace transforms to the time-dependent equations of elasticity reduce the problem to a static one in Laplace space. The displacement and traction boundary integral equations for the transformed variables are established by Somigliana's identity. By applying the displacement integral equation on one of the crack surfaces and the traction boundary integral equation on the other, a general mixed-mode crack problem can be solved in a single region, in the same way as a static problem. The transformed mixed-mode stress intensity factors are calculated from the transformed displacement discontinuities near the crack tip. The dynamic stress intensity factors in the time domain are obtained by the use of Durbin's inversion method. The accuracy of this method is demonstrated by its application to embedded penny-shaped and elliptical cracks, and edge cracks.

A variational technique for boundary element analysis of 3D fracture mechanics weight functions: static

The dual boundary element method coupled with the weight function technique is developed for the analysis of three-dimensional elastostatic fracture mechanics mixed-mode problems. The weight functions used to calculate the stress intensity factors are defined by the derivatives of traction and displacement for a reference problem. A knowledge of the weight functions allows the stress intensity factors for any loading on the boundary to be calculated by means of a simple boundary integration without singularities. Values of mixed-mode stress intensity factors are presented for an edge crack in a rectangular bar and a slant circular crack embedded in a cylindrical bar, for both uniform tensile and pure bending loads applied to the ends of the bars. © 1998 John Wiley & Sons, Ltd.

A finite-element simulation method for a system of growing cracks in a heterogeneous material

In this paper a numerical simulation method is proposed for the growth of interacting non-collinear cracks in a finite, heterogeneous, and two-dimensional elastic solid. The stress analysis is based on a step-by-step finite element method using the method of superposition of analytical and finite-element solutions accompanied by the quadtree automatic mesh generation algorithm. An incremental crack growth pattern is predicted considering the interaction of all growing cracks by the Schwartz–Neumann alternating procedure using a first order perturbation solution of slightly kinked and curved extension of all growing cracks, which analytically gives the mixed mode stress intensity factors at extending crack tips. The local symmetry criterion is used as the crack path criterion at the extending crack tips. Several numerical examples, including the avoidance of collinear edge cracks and cracks initiated at the ends of fibers show the validity of the present simulation method.

A Finite Element Analysis of Crack Initiation and Propagation in a Notched Disk Submitted to Rolling Contact Fatigue

A finite element model has been developed to predict crack initiation and propagation in a notched disk submitted to rolling contact fatigue. The aim of this study is to validate the model with experimental results obtained by tests carried out on a two-disk machine. First, a three-dimensional finite element analysis is performed. A unidimensional equivalent damage stress is calculated by applying a plastic criterion in an attempt to estimate the damage location and the time necessary to initiate cracks from the notches. Then a two-dimensional calculation based on linear fracture mechanics is conducted to determine mixed mode stress intensity factors at the tip of a crack initiated from the notch. Several crack growth criteria are used to evaluate crack growth direction and rate. Numerical results are in good agreement with experimental ones.

Mixed-mode weight functions in three-dimensional fracture mechanics: static

In this paper, the indirect boundary element method, coupled with the weight function concept, is developed for the analysis of three-dimensional elastostatic fracture mechanics mixed-mode problems. A set of boundary integral equations for the fictitious loads and the displacement discontinuities are derived. The weight functions used to calculate the stress intensity factors are determined by incrementing the crack front over a small region. The stress intensity factors were also evaluated by the equivalent stress method to compare with those obtained from the weight functions. Results of mixed-mode stress intensity factors are presented for a square bar, with either a rectangular or a circular crack, under static load.

Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks

New domain integrals for extracting mixed-mode stress intensity factors along curved, three-dimensional bimaterial interface cracks are derived. In the derivation, the asymptotic auxiliary fields for the plane problem of a bimaterial interface crack are imposed along a curved crack front. The general crack-tip interaction integral (a contour integral surrounding a point on the crack front which is evaluated in the limit as the contour is shrunk onto the crack tip) is then expressed in domain form which is more suitable for numerical computation. A consequence of imposing the auxiliary fields along a curved crack front is that the auxiliary stress fields do not satisfy equilibrium, and the auxiliary strain fields do not satisfy compatibility. The terms which arise due to the lack of equilibrium and compatibility are not sufficiently singular to contribute to the crack-tip interaction integral or to affect its path independence; however, these terms become important when domain integral representations are introduced, because they involve fields which are not asymptotically close to the crack tip. In order to compute the pointwise stress intensity factors along the crack front, the domain integrals are evaluated as a post-processing step in the finite element method. In the numerical results, it is demonstrated that it is crucial to incorporate the terms that arise due to lack of equilibrium and compatibility of the auxiliary fields, especially in regions where the local crack front curvature is high. In the paper, we present two numerical examples. As a benchmark, we first consider the problem of a penny-shaped interface crack embedded in a cylinder. The results for the complex stress intensity factor and phase angle are found to be in excellent agreement with the analytical solution. The problem of an elliptical crack embedded between two dissimilar isotropic materials is also considered, and the results are discussed.

Identifying failure mechanisms of composite structures under compressive load

Various failure mechanisms involving both local and global deformation mechanisms of layered structures, consisting of differently oriented orthotropic laminae, are investigated with a large deformation finite element analysis. Under large compressive loads, fiber-reinforced composite structures can fail by buckling, kink band formation and/or delamination growth. Our aim is to identify the dominant mode which leads to the structural failure under a given boundary condition and geometrical shape. It is assumed that these structures contain initial interlaminar flaws represented by embedded delaminations. Such flaws can play a significant role in defining overall structural integrity and deformation mechanisms. Our study considers two structures with distinguished shapes, both consisting of four layer laminae. The first structure is a flat panel under compressive load and the second structure is a cylindrical shell subjected to external pressure. In both cases, the energy release rate and mixed-mode stress intensity factors are computed to quantify the crack driving force and used to measure likelihood of delamination growth. In the flat panel model, two buckling modes, a global and a local, are observed under quasi-static loading conditions. Depending upon the delamination length and laminate stacking orientations, the interaction of these two modes can produce an unstable post-buckling behavior. It is found that the energy release rate exceeds experimentally estimated fracture toughness values only after buckling occurs. In the cylindrical shell study, lower critical buckling loads are observed for models with longer interlaminar delamination as in the flat panel model. However, unlike the flat panel case, the energy release rate surpasses the critical toughness well before the applied pressure reaches the buckling load of the flawed cylindrical shell. This behavior implies that a shell containing an embedded defect along an interface can fail by delamination growth and therefore has a failure load lower than its critical buckling load. Also, for thicker cylindrical shells, the compressive internal stress exceeds the predicted stress required for kink band formation prior to structural buckling. The competition among these failure modes, buckling, delamination growth and compressive kinking in fiber-reinforced composite structures, is studied and a design diagram is introduced here. This diagram predicts a likely dominant failure mode for given structural dimensions and material properties. In addition, unstable delamination growth along the lamina interface under dynamic loading conditions is simulated using an iterative computational procedure. The results predict that the crack tip velocity can reach a fraction of the material wave speed leading to total structural failure in a very short time. For the cylindrical shell, the inertial effect on the pressure-deformation relation is investigated by comparing it with the static results.

Stress intensity factors as the fracture parameters for delamination crack growth in composite laminates

A “mutual integral” approach is used to calculate the mixed-mode stress intensity factors for a free-edge delamination crack in a laminate under tensile loading conditions. This “mutual integral” approach, for generalized plane strain conditions, is based on the application of the path-independent J integral to a linear combination of three solutions: one, the problem of the laminate to be solved using the quasi 3-D finite element method, the second, an “auxiliary” solution with a known asymptotic singular solution, and the third, the particular solution due to the out-of-plane loading. A comparison with the exact solutions is made to determine the accuracy and efficiency of this numerical method. With this “mutual integral” approach, it was found that the calculated mixed-mode stress intensity factors of the free-edge delamination crack remain relatively constant as the crack propagates into the laminate. It was also found that the fracture criterion based on the mixed-mode stress intensity factors is more consistent with the experimental observations than the criterion based on the total energy release rate, and hence demonstrates the importance of the ability to calculate each individual component of the stress intensity factors. Furthermore, it was found that the fracture toughness measurements from double cantilever beam specimens can be used directly to predict the onset of delamination crack growth between two dissimilar laminae. Using these fracture toughness measurements from the double cantilever beam specimens, some examples are given to show that the fracture criterion based on the mixed-mode stress intensity factors can accurately predict the failure load for various laminates under tensile loading conditions.