Mode Stress Intensity Factors Research Articles

Calculation of mixed mode stress intensity factors for an elliptical subsurface crack under arbitrary normal loading

AbstractNormal loading causes mixed fracture modes in an elliptical subsurface crack because of the nonsymmetrical geometry with respect to the crack face. In this paper, mixed mode weight functions (MMWFs) for elliptical subsurface cracks in an elastic semi‐infinite space under normal loading are derived. Reference mixed mode stress intensity factors (MMSIFs), calculated by finite element analysis, under uniform normal loading are used to derive MMWFs. The cracks have aspect ratios and crack depth to crack length ratios of 0.2–1.0 and 0.05 to infinity, respectively. MMWFs are used to calculate MMSIFs for any point of the crack front under linear and nonlinear two‐dimensional (2D) loadings. So, in order to evaluate the fatigue crack growth phenomenon under complicated 2D stress distributions, MMWFs can be easily used. The comparison between the MMSIFs obtained from the MMWFs and finite element analysis indicates high accuracy.

Thermo-mechanical fracture study of inhomogeneous cracked solids by the extended isogeometric analysis method

Abstract The extended isogeometric analysis method (XIGA) is further developed to study fracture of homogeneous and inhomogeneous materials under mechanical and thermo-mechanical loadings. The domain of the problem is discretized by the knot spans of the isogeometric analysis (IGA) and the same basis functions used for constructing the geometry are employed to discretize the solution. In addition, crack face discontinuity and tip enrichment functions of the extended finite element method (XFEM) are incorporated into the NURBS basis functions of IGA in order to reproduce crack face displacement discontinuity and tip singularity of the stress field. The proper form of the interaction integral method in inhomogeneous materials and thermal conditions is employed to evaluate the mixed mode stress intensity factors. In order to assess the efficacy of the proposed approach, a number of problems with different configurations and loadings are analyzed and the results are verified by comparing with the reference values.

Fatigue strength of CFRP strengthened welded joints with corrugated steel plates

This paper presents an experimental and numerical research on the carbon fibre reinforced polymer (CFRP) strengthened welded joints with corrugated plates. The effectiveness of the strengthening in the improvement of fatigue strength has been examined experimentally on the test joints through varying the number and the layout of the CFRP laminates. The test results show that the joints with transition curvature region reinforcement and single side reinforcement produce slightly lower rigidity but longer fatigue life in contrast to those with full width reinforcement on the double side of the main plate. Furthermore, a simplified two dimensional analytical model which allows for the geometric characteristics of the joint has been proposed to investigate the stress intensity factor of mode I. The proposed analytical model has been simulated by finite element technique and its solution result is compared with previously reported theoretical calculation. Parametric studies have been performed to investigate the effects of the number of CFRP layers and the moduli of carbon fibre & adhesive on the stress intensity factor. The combined influence of the corrugation angle and crack depth has also been considered. It has been found that these effects on the stress intensity factor are more significant for the joints with smaller corrugation angle.

A crack initiation and two debondings development at the interface of a circular rigid inclusion under uniform loading

Competition of a crack and two debondings at the interface of a circular rigid inclusion in an infinite elastic body is analyzed under uniaxial loading in the x and y directions, respectively, and under biaxial uniform loading. It is investigated how these debondings develop along the interface of the inclusion from the initial debondings and where a crack occurs from the tip of debonding. Particularly, when there are both possibilities of the debonding development and of the crack occurrence from the tip of the debonding, it can be decided which phenomenon actually occurs. The angles at which the debondings develop and the crack occurs are determined. The magnitude of the load is also determined. As the criterion for debonding development and crack occurrence, strain energy release rates are used. Moreover, the restricting condition is that the normal stress at the interface ahead of the debonding tip is positive and the stress intensity factor of mode I just after crack occurrence is positive. As for the load, the constant load and the gradually increasing load from zero are considered. The stress analysis is carried out as a mixed boundary value problem with two stress and two displacement boundaries of plane elasticity. The rational mapping function of a sum of fractional expressions and complex stress functions are used for the stress analysis.

T-spline based XIGA for fracture analysis of orthotropic media

Fracture analysis of orthotropic cracked media is investigated by applying the recently developed extended isogeometric analysis (XIGA) (Ghorashi et al., 2012) using the T-spline basis functions. The signed distance function and orthotropic crack tip enrichment functions are adopted for extrinsically enriching the conventional isogeometric analysis approximation for representation of strong discontinuity and reproducing the stress singular field around a crack tip, respectively. Moreover, by applying the T-spline basis functions, XIGA is further developed to make the local refinement feasible. For increasing the integration accuracy, the ’sub-triangle’ and ’almost polar’ techniques are adopted for the cut and crack tip elements, respectively. The interaction integral technique developed by Kim and Paulino (2003) is applied for computing the mixed mode stress intensity factors (SIFs). Finally, the proposed approach is applied for analysis of some cracked orthotropic problems and the mixed mode SIFs are compared with those of other methods available in the literature.

Mixed-mode fracture & non-planar fatigue analyses of cracked I-beams, using a 3D SGBEM–FEM Alternating Method

In the present paper, computations of mixed mode stress intensity factor (SIF) variations along the crack front, and fatigue-crack-growth simulations, in cracked I-beams, considering different load cases and initial crack configurations, are carried out by employing the three-dimensional SGBEM (Symmetric Galerkin Boundary Element Method)–FEM (Finite Element Method) Alternating Method. For mode-I cracks in the I-beam, the computed SIFs by using the SGBEM–FEM Alternating Method are in very good agreement with available empirical solutions. The predicted fatigue life of cracked I-beams agrees well with experimental observations in the open literature. For mixed-mode cracks in the web or in the flange of the I-beams, no analytical or empirical solutions are available in the literature. Thus mixed-mode SIFs for mixed-mode web and flange cracks are presented, and non-planar fatigue growth simulations are given, as benchmark examples for future studies. Moreover, because very minimal efforts of preprocessing and very small computational burden are needed, the current SGBEM–FEM Alternating Method is very suitable for fracture and fatigue analyses of 3D structures such as I-beams.

A general methodology for calculating mixed mode stress intensity factors and fracture toughness of solder joints with interfacial cracks

Solder joints in electronic packages undergo thermo-mechanical cycling, resulting in nucleation of micro-cracks, especially at the solder/bond-pad interface, which may lead to fracture of the joints. The fracture toughness of a solder joint depends on material properties, process conditions and service history, as well as strain rate and mode-mixity. This paper reports on a methodology for determining the mixed-mode fracture toughness of solder joints with an interfacial starter-crack, using a modified compact mixed mode (CMM) specimen containing an adhesive joint. Expressions for stress intensity factor (K) and strain energy release rate (G) are developed, using a combination of experiments and finite element (FE) analysis. In this methodology, crack length dependent geometry factors to convert for the modified CMM sample are first obtained via the crack-tip opening displacement (CTOD)-based linear extrapolation method to calculate the under far-field mode I and II conditions (f1a and f2a), (ii) generation of a master-plot to determine ac, and (iii) computation of K and G to analyze the fracture behavior of joints. The developed methodology was verified using J-integral calculations, and was also used to calculate experimental fracture toughness values of a few lead-free solder-Cu joints.

Investigating the effects of micro-defects on the dynamic properties of rock using Numerical Manifold method

In this study, the Numerical Manifold method (NMM) is extended to investigate the effects of micro-defects on the dynamic mechanical properties of rock under different strain rates. The displacement decomposition technique is incorporated in the NMM to estimate the mixed mode stress intensity factors at the crack tip. A dynamic crack growth criterion is also incorporated in the NMM for crack growth analysis. The developed NMM is first validated by a simple sliding crack model. The developed model is then applied to investigate the effects of the micro-cracking properties such as initial micro-crack length, initial micro-crack inclination angle and initial micro-crack separation distance on the dynamic mechanical properties of a granite under different strain rates ranging from 10−4/s to 100/s. The effect of confining stress on the granite dynamic strength is also investigated. Simulation results illustrated that the initial micro-crack length and the confining stress have a significant effect on the dynamic strength. The effect of micro-crack separation distance, on the other hand, is heavily dependent on the ratio of separation distance to the initial micro-crack length.

Evaluation of the SIF for the Multiple Crack Problems

In elastic fracture mechanics the evaluation of the stress intensity factor (SIF) for multiple cracks problems is an important issue. In the paper the scaled boundary finite element method (SBFEM) is used to solve the SIF of mode I of multiple crack problems. The solving domain is partitioned into several sub-domains according to the number of cracks. Every sub-domain has its own scaling center. The characteristics of the SBFEM are preserved in every sub-domain. Numerical examples show that the SBFEM is effective with high accuracy in evaluating the multiple cracks fracture problems. It can be applied to treat the anisotropic materials conveniently. The stress intensity factors of unequal double-edged cracks in orthotropic material are provided.

The plasticity-corrected stress intensity factor for plane stress mode I and mode II cracks

An approximate analytical solution for plasticity-corrected stress intensity factor (PC–SIF) is developed on the basis of Eshelby equivalent inclusion theory and transformation toughening theory both for plane stress mode I and mode II cracks. As validated by detailed finite element analysis the approximate solution has good accuracy. The proposed PC–SIF provides a new mechanical parameter to describe the fatigue and fracture behaviors associated with crack tip plastic zone for mode I and mode II cracks under plane stress loading conditions.

Six degrees of freedom coupled dynamic response of rotor with a transverse breathing crack

More coupling mechanisms between the six degrees of freedom (DOFs) are introduced by considering the contribution of the general transverse forces to stress intensity factor of mode I crack in predicting the crack additional flexibility matrix of the cracked rotor. And the obtained flexibility elements show a good agreement with the experiment result for a wide range of the crack depth ratio. Six DOFs coupled dynamic equations for cracked rotor are formulated by introducing three rotational DOFs. A response-dependent non-linear breathing crack model is applied to simulate the breathing behavior during operation in this paper. Numerical investigations are carried out to simulate the various parametric conditions on the dynamic characteristics of cracked rotor, including the crack depth, shaft slenderness ratio, and rotating speed ratio. A perturbation frequency component and its combinations with harmonic components are observed in the dynamic response obtained by six DOFs coupled models (SDCM). Some differences in evolution of whirling orbit obtained by SDCM and three DOFs coupled models are found, when the cracked rotor passes through the sub-critical speeds.

A coupled FEM–Bernstein approach for computing the J k integrals

The stress intensity factors of mixed mode I–II crack loading are often evaluated numerically via FEM implementation combined with the equivalent domain integral method to derive the J k integrals. Mesh refinement or singular elements are classic techniques for reproducing the sharp stress gradient due to material fracture. In this context, high-order Bernstein shape functions are proposed for a model that couples a radial spectral expansion for the stress field around crack tip with a finite element mesh in the outer regions. Lagrange multipliers are used to enforce the coupling in displacements along the FEM–Bernstein approximations interface, in which non-matching nodal patterns are allowed.

Mixed Mode Stress Intensity Factor Determination for Single and Multiple Cracks in an Aircraft Wing

Structural failure of aircraft wings due to nucleation and propagation of cracks is one of the main reasons for failure of aged aircrafts. Reported studies on aircraft failures indicate that the main cause of wing failure is due to fatigue cracks which nucleate from the wing root region. Thus, determination of residual life of the cracked wing structure using fracture mechanics approach becomes important. In the present work an attempt has been made to estimate the SIF of single and multiple cracks in an aircraft wing subjected to lift force. Crack depth ratios ranging between 0.1 and 0.4 and aspect ratios of 0.6 and 1.0 have been considered. Single and multiple cracks are introduced at the wing rib region and the lift force is applied at the bottom surface of the wing. Geometric correction factor (Y) is estimated with the additional consideration of mode II and mode III fracture. The effect of crack depth ratio and number of cracks on SIF is determined. Non symmetric SIF distribution is observed with increase of crack depth ratio. It is also noted that SIF values are always higher at the crack surface region compared to crack middle region irrespective of crack depth ratio.

J-Integral Computation in Mixed Mode Ι and ΙΙ

Computation of a path independent contour integration but it yields directly the independent values of the opening mode and shear mode stress intensity factors. The calculations are not sensitive to the choice of contour. To the J-integral calculation, the present scheme calculates separately the factors KI and KII. A simple method to determine the SIFs for the three-dimensional crack is proposed based on the conservation integral.

Fracture behaviour of concrete containing limestone fines

The use of limestone fines as partial replacement of cement can lower the cost of concrete, protect the environment by reducing carbon dioxide emissions and thus contribute towards sustainable development. The effect of incorporating limestone fines on the mechanical and durability properties of mortar and concrete has been thoroughly investigated. However, little research has been done so far on the fracture behaviour of mortar and concrete beams containing limestone fines. In the present study, the effect of limestone as partial replacement of cement on strength and fracture behaviour of mortar and concrete was investigated. This was carried out experimentally by determining the load–deflection curves of concrete or mortar using the three-point flexural tests on notched samples so that the material characteristics at rupture, such as the fracture energy GF and fracture toughness, KI (stress intensity factor) for mode I of fracture could be deduced. The results showed that strength and fracture parameters were reduced with the increase of limestone fines in cement for both notched mortar and concrete beams.

A New Comparative Method To Evaluate the Fracture Properties of Laminated Composite

In this paper a new method to determine the fracture properties and strain energy release rate for Carbon- Polyester composite has been introduced. Fracture characteristics such as critical stress intensity factor and critical strain energy release rate for mode I, mode II and mixed mode loading were determined using Arcan type specimen. 130 layers of carbon fiber polyester woven composite with each of 0.2mm thickness were put on each other. Theoretical studies to determine strain energy release rate were done using three methods: Corrected Beam Theory (CBT), Compliance Calibration Method (CCM), Virtual Crack Closure Technique (VCCT), and results were recorded and were compared with the results from experimental and numerical attempts. Critical loads were recorded with experimental attempts then applied to the finite element software. Results were recorded and compared with each other to determine the best method. Results show that compliance calibration method and virtual crack closure technique determine strain energy release rate value closer to J-integral in comparison with corrected beam theory. Finally, the fracture surfaces were examined by scanning electron microscope to gain insight the failure responses that shows the fracture surface for mode II is rougher than the fracture surface for mode I and mixed mode.

Influence of crack offset distance on interaction of multiple collinear and offset edge cracks in a rectangular plate

In the present work, photoelastic and finite element method have been employed to study the interaction between two collinear cracks and two offset edge cracks. CY 230 resin and hardener HY 951 are used for preparation of photoelastic material. Mode I and mode II stress intensity factors were determined by using finite element method and compared with the experimental results. A good agreement is found between the numerical and experimental results. Second degree polynomial functions are presented to evaluate the mixed mode stress intensity factors from mode I stress intensity factor and crack offset distance H. The limits of crack offset distance for significant crack interaction have been discussed.

Crack Growth Threshold Criterion Based on Calculation of Dynamic Stress Intensity Factors in the Mixed Mode (Complex Functions Theory Approach)

In this paper, an analytical method is presented for crack growth threshold criterion based on calculation of dynamic stress intensity factors (DSIF) in the mixed mode. The basis of the method is comparison of DSIF and fracture toughness. The analytical method is grounded on solving two-dimensional wave equations in the frequency domain, using complex functions and mapping technique. Using this method, the J-integrals are calculated and based on relation between the J-integrals and DSIF, these factors are derived. The proposed criterion for crack growth threshold is defined as a comparing calculated DSIF (KI, KII) with fracture toughness (Kc), using complex functions and mapping technique. The mapping technique helps to convert the complicated boundaries geometry into a simple one (unit circle). Finally, since using complex functions theory is an analytical method, the results have a high precision compared with numerical methods.

FGM Gradation Direction Effects on Mixed-Mode Crack Initiation Angle by MLPG Method

In this article, the meshless local Petrov–Galerkin method is used to analyze the cracked plate made of functionally graded material. The stress intensity factor of Mode I & II, maximum energy release rate and crack initiation angle are determined under the influence of various nonhomogeneity ratios, crack length, and material gradation angle. To solve discrete dynamic equations, a new effective method is extended and utilized. To find crack initiation angle, a simple and easy to use formula is suggested here. The edge-cracked functionally graded plate is considered under the uniform membrane and fixed grip conditions; also, the center-cracked functionally graded plate is analyzed under the uniform membrane and impact loads.

The mixed-mode fracture mechanics analysis of an embedded arbitrary oriented crack in a two-dimensional functionally graded material plate

Mixed-mode fracture mechanics analysis of an embedded arbitrarily oriented crack in a two-dimensional functionally graded material using plane elasticity theory is considered. The material properties are assumed to vary exponentially in two planar directions. Then, employing Fourier integral transforms with singular integral equation technique, the problem is solved. The stress intensity factors (SIFs) at the crack tips are calculated under in-plane mechanical loads. Finally, the effects of crack orientation, material non-homogeneity, and other parameters are discussed on the value of SIF in mode I and mode II fracture.