Stress intensity factor estimation for unbalanced rotating cracked shafts by artificial neural networks

The knowledge of the stress intensity factor (SIF) values along a crack front is essential to calculate the crack growth rate and the remaining life of a mechanical component. In the case of a rotating shaft, usually it presents disalignments, which modify the SIF data with regard to a balanced one. This paper presents the use of an artificial neural network (ANN) for estimating the SIF at the crack front in an unbalanced shaft under rotating bending, previously, a quasi‐static numerical (finite element) model, which simulates a rotating shaft, has been developed to create the training cases for the ANN. The obtained results allow to study the influence of the unbalance of rotating shafts in the crack breathing mechanism and will allow to predict the influence of this behaviour on the values of the SIF and in the propagation of cracks.     

Experimental and numerical investigation of fatigue crack growth in the cracked gear tooth

The numerical simulation was conducted to analyse the fatigue crack growth in gear with the finite element codes ansys (ANSYS, Inc. Canonsburg, Pennsylvania, USA.) and franc 3d (Fracture Analysis Consultants, Inc. Ithaca, New York, USA.), and the corresponding fatigue test was also carried out. During the simulation, the location of maximal stress induced by the external force was first determined by the code ansys , and then the obtained results were imported into the franc 3d to analyse the crack growth. The analysed results were input into the codes ansys and franc 3d again to compute the stress and the stress intensity factor in the following steps. After several rounds of analysis, the results of the fatigue crack propagation were obtained. The investigations show that the crack mode I is dominant during the crack growth and the stress intensity factor K_I raises with increase of crack growth length and a series of quarter‐elliptical cross sections of the ruptured gear tooth are obtained. The simulation results are in good accordance with experimental findings.     

Assessing the accuracy of fatigue life in surface‐cracked straight pipes

The demonstration of leak before brake (LBB) based on fracture mechanics requires information on the initial size of a defect, initiation of crack growth from the inherent defect and subsequent crack growth rates. In the present paper the prediction methodologies have been tested for three different full scale pipes geometry experimentally tested data. The prediction accuracy of two SIF solutions available in the literature has also been judged. The effect of fatigue crack closure and corrections needed in the numerical prediction methodology using FEM have also been included. The results showed that the FEM could fairly predict the fatigue crack initiation and crack growth life of full‐scale piping components having a constant depth crack profile.     

One‐dimensional analysis of thin‐walled closed beams having general cross‐sections

This paper presents a new one‐dimensional theory for static and dynamic analysis of thin‐walled closed beams with general cross‐sections. Existing one‐dimensional approaches are useful only for beams with special cross‐sections. Coupled deformations of torsion, warping and distortion are considered in the present work and a new approach to determine sectional warping and distortion shapes is proposed. One‐dimensional C⁰ beam elements based on the present theory are employed for numerical analysis. The effectiveness of the present theory is demonstrated in the analysis of thin‐walled beams having pillar sections of automobiles and excavators. Copyright © 2000 John Wiley & Sons, Ltd.     

Transferability of the two‐parameter fracture criterion for 2219 aluminium alloy cracked configurations

Two‐dimensional elastic–plastic finite‐element fracture simulations with the critical crack‐tip‐opening‐angle fracture criterion were used to evaluate the two‐parameter fracture criterion (TPFC). Three different crack configurations under tension and bending loads made of thin‐sheet 2219‐T87 aluminium alloy were analysed. A very wide range of widths (w = 76 to 2440 mm) and initial crack‐length‐to‐width ratios (c_i/w = 0.05 to 0.95) were considered. A relation from the original TPFC was shown to fit the simulated fracture behaviour fairly well for the three different specimen types for net‐section stresses less than the yield stress (σ_y) of the material. Comparisons were also made on measured and simulated fracture tests on middle‐crack‐tension specimens. A relation between the elastic stress‐intensity factor, K_Ie, and net‐section stress, S_n, at failure was found to be linear for S_n < σ_y. The results demonstrated the transferability of the TPFC for different crack configurations for S_n < σ_y, but further study is needed for S_n > σ_y.     

Three‐dimensional stress concentrations at circular pin holes in clearance‐fit lugs

The influences of thickness and bonding clearance on stress concentration factors (SCFs) at circular holes in pin‐loaded straight lugs are systematically investigated using the finite element method. The three‐dimensional effect on stress concentration at pin hole is strong when the thickness B of lug is higher than the radius R of pin hole. The maximum tensional SCF K_max normalized by its corresponding plane stress solution K_p–σ increases with increasing B/R when B/R is higher than 2 for all of r/R (the radius of pin to that of lug), and also increases with decreasing r/R for a given B/R. It is also found that the plane stress SCF K_p–σ nearly remains a constant when r/R < 0.98, but is strongly sensitive to r/R and increases by 30% with r/R changing from 0.98 to 1. On the other hand, the friction coefficient, Young's modulus and the load level have also influences on stress concentrations, which should not be neglected in design of structures. An empirical formula of the maximum SCF is obtained for convenience of engineering applications.     

Exact static element stiffness matrices of non‐symmetric thin‐walled curved beams

An evaluation procedure of exact static stiffness matrices for curved beams with non‐symmetric thin‐walled cross section are rigorously presented for the static analysis. Higher‐order differential equations for a uniform curved beam element are first transformed into a set of the first‐order simultaneous ordinary differential equations by introducing 14 displacement parameters where displacement modes corresponding to zero eigenvalues are suitably taken into account. This numerical technique is then accomplished via a generalized linear eigenvalue problem with non‐symmetric matrices. Next, the displacement functions of displacement parameters are exactly calculated by determining general solutions of simultaneous non‐homogeneous differential equations. Finally an exact stiffness matrix is evaluated using force–deformation relationships. In order to demonstrate the validity and effectiveness of this method, displacements and normal stresses of cantilever thin‐walled curved beams subjected to tip loads are evaluated and compared with those by thin‐walled curved beam elements as well as shell elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Neural approach to estimate the stress intensity factor of semi‐elliptical cracks in rotating cracked shafts in bending

In the last decades, neural network approach has often been used to study various and complex engineering problems, such as optimization or prediction. In this paper, a methodology founded on artificial neural networks (ANNs) was used to calculate the stress intensity factor (SIF) in different points of the front of a semi‐elliptical crack present in a rotating shaft, taking into account the shape and depth of the crack, the angle of rotation, and the location of the point in the front. In the event of rotating machines, such as shafts, it is crucial to know the SIF along the crack front because this parameter, according to the Paris Law, is related to the performance of the crack during its propagation. Previously, it was necessary to achieve the data for the ANN training, for this a quasi‐static numerical model was made, which simulates a rotating cracked shaft with a semi‐elliptical crack. The numerical solutions cover a wide range of crack depths and shapes, and rotation angles. The values of the SIF estimated by the ANNs were contrasted with other solutions available in the literature finding a good agreement between them. The proposed neural network methodology is an alternative that offers a very good option for the SIF estimation, because it is efficient and easy to use, does not require high computational costs, and can be used to analyse the propagation of cracks contained in rotating shafts by means of the Paris Law taking into account the nonlinear behaviour of the shaft.     

A super‐element for crack analysis in the time domain

A super‐element for the dynamic analysis of two‐dimensional crack problems is developed based on the scaled boundary finite‐element method. The boundary of the super‐element containing a crack tip is discretized with line elements. The governing partial differential equations formulated in the scaled boundary co‐ordinates are transformed to ordinary differential equations in the frequency domain by applying the Galerkin's weighted residual technique. The displacements in the radial direction from the crack tip to a point on the boundary are solved analytically without any a priori assumption. The scaled boundary finite‐element formulation leads to symmetric static stiffness and mass matrices. The super‐element can be coupled seamlessly with standard finite elements. The transient response is evaluated directly in the time domain using a standard time‐integration scheme. The stress field, including the singularity around the crack tip, is expressed semi‐analytically. The stress intensity factors are evaluated without directly addressing singular functions, as the limit in their definitions is performed analytically. Copyright © 2004 John Wiley & Sons, Ltd.     

Stress intensity factors for circumferential through‐wall cracks in thin‐walled cylindrical shells subjected to tension and torsion

In order to assess the structural integrity of tubular members or pipes containing circumferential through‐wall cracks, their stress intensity factor solutions are required. While stress intensity factors for tension and bending are available, few solutions exist for the case of torsion, even though these components may also be subjected to torque. In this paper, the finite element method is used to compute the stress intensity factors for this geometry under tension and torsion. Shell elements are employed to compute the results for thin shells by the means of the displacement extrapolation technique. The computed results indicate that the available analytical solution for torsional loading, which is based on shallow shell theory, is nonconservative for long cracks in thin shells. Shallow shell theory is in general not applicable to long cracks, and the present work is therefore able to provide solutions for a wider range of crack lengths than what is currently available.     

Stress intensity factors for cracks emanating from a circular hole in an infinite quasi‐orthotropic plane

An infinite quasi‐orthotropic plane with a cracked circular hole under tensile loading at infinity is studied analytically. To this end, complex variable theory of Muskhelishvili is used. In addition, to obtain analytical functions, a new conformal mapping is proposed and expanded to series expressions. Stress intensity factors (SIFs) for two unequal cracks emanating from a circular hole are obtained. To validate the analytical SIFs in a quasi‐orthotropic plane, the results are compared with FEM and the results of isotropic plane. The SIFs for small cracks in a quasi‐orthotropic and an isotropic plane are different, because of difference between stress concentrations in points which cracks emanate from the hole. However, the results of quasi‐orthotropic plane converge to isotropic plane for the large cracks. Therefore, the SIFs of the large cracks in a quasi‐orthotropic plane can be replaced by the results of the center crack with equivalent length in an isotropic plane.     

3D J‐integral evaluation using the computation of line and surface integrals

In the analysis of fracture mechanics of structures using three‐dimensional (3D) J‐integral, an integral evaluation of line and surface is required. However, because surface integral evaluation requires the calculation of the second derivative of displacement field and commercial finite element codes cannot calculate it, then this portion of the integral is neglected in some research. In this paper, a method for computing 3D J‐integral is presented using finite element analysis. In the analysis, the second derivative evaluation of displacement field is employed. The method is implemented in calculating the J‐integral of some 3D cracks and results are compared to well‐known reference values. The results show that the method is reliable and is suitable for applications in engineering. The portion of 3D J‐integral, namely the surface integral value is investigated and it is shown that neglecting this portion can introduce considerable error in the final results.     

The residual stress intensity factors for cold‐worked cracked holes: a technical note

Cold‐working of riveted holes reduces the stress intensity factor associated with cracks that may develop at the hole boundary, by creating a compressive residual stress field. The residual stress field is determined using the finite‐element method and the reduction of the stress intensity factor for different values of the interference is evaluated with the weight function method, in the case of an infinite plate made from an elastic–perfectly plastic material, and having a hole with two symmetrical cracks. Once the weight function of the structure is known, further calculation of the stress intensity factors for different loadings such as a remote uniform stress, or a point load that simulates the action of the rivet can be performed without difficulty.     

An element‐free method for in‐plane notch problems with anisotropic materials

This study developed an element‐free Galerkin method (EFGM) to simulate notched anisotropic plates containing stress singularities at the notch tip. Two‐dimensional theoretical complex displacement functions are first deduced into the moving least‐squares interpolation. The interpolation functions and their derivatives are then determined to calculate the nodal stiffness using the Galerkin method. In the numerical validation, an interface layer of the EFGM is used to combine the mesh between the traditional finite elements and the proposed singular notch EFGM. The H‐integral determined from finite element analyses with a very fine mesh is used to validate the numerical results of the proposed method. The comparisons indicate that the proposed method obtains more accurate results for the displacement, stress, and energy fields than those determined from the standard finite element method. Copyright © 2013 John Wiley & Sons, Ltd.     

Stress intensity factors for cracked triangular cross‐section thin‐walled tubes

For one kind of finite‐boundary crack problems, the cracked equilateral triangular cross‐section tube, an analytical and very simple method to determine the stress intensity factors has been proposed based on a new concept of crack surface widening energy release rate and the principle of virtual work. Different from the classical crack extension energy release rate, the crack surface widening energy release rate can be defined by the G*‐integral theory and expressed by stress intensity factors. This energy release rate can also be defined easily by the elementary strength theory for slender structures and expressed by axial strains and loads. These two forms of crack surface widening energy release rate constitute the basis of a new analysis method for cracked tubes. From present discussions, a series of stress intensity factors are derived for cracked equilateral triangular cross‐section tubes. Actually, the present method can also be applied to cracked polygonal tubes.     

Strain rate concentration and dynamic stress concentration for double‐edge‐notched specimens subjected to high‐speed tensile loads

Engineering plastics provide superior performance to ordinary plastics for wide range of the use. For polymer materials, dynamic stress and strain rate may be major factors to be considered when the strength is evaluated. Recently, high‐speed tensile test is being recognized as a standard testing method to confirm the strength under dynamic loads. In this study, therefore, high‐speed tensile test is analysed by the finite element method; then, the maximum dynamic stress and strain rate are discussed with varying the tensile speed and maximum forced displacement. The maximum strain rate increases with increasing the tensile speed u/t, but the strain rate concentration factor is found to be constant independent of tensile speed, which is defined as the maximum strain rate appearing at the notch root over the average nominal strain rate at the minimum section . It is found that the strain rate at the notch root depends on the dynamic stress rate at the notch root and independent of the notch root radius ρ. It is found that the difference between the static and dynamic maximum stress concentration (σ_yA,max ? σ_yA,st) at the notch root is proportional to the tensile speed when u/t = 5000 mm/s. Strain rate concentration factors are also discussed with varying the notch depth and specimen length. Based on the elastic strain rate concentration factor, the master curve is obtained useful for understanding the impact fracture of polycarbonate for the wide range of temperature and impact speed.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Extended finite element method coupled with face‐based strain smoothing technique for three‐dimensional fracture problems

In this work, the extended finite element method (XFEM) is for the first time coupled with face‐based strain‐smoothing technique to solve three‐dimensional fracture problems. This proposed method, which is called face‐based smoothed XFEM here, is expected to combine both the advantages of XFEM and strain‐smoothing technique. In XFEM, arbitrary crack geometry can be modeled and crack advance can be simulated without remeshing. Strain‐smoothing technique can eliminate the integration of singular term over the volume around the crack front, thanks to the transformation of volume integration into area integration. Special smoothing scheme is implemented in the crack front smoothing domain. Three examples are presented to test the accuracy, efficiency, and convergence rate of the face‐based smoothed XFEM. From the results, it is clear that smoothing technique can improve the performance of XFEM for three‐dimensional fracture problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Effects of CFRP bond locations on the Mode I stress intensity factor of centre‐cracked tensile steel plates

The fatigue life of cracked steel members can be greatly extended by externally attached carbon fibre reinforced plastics (CFRP), which reduces the stress intensity factors (SIFs) at the crack tip. Access to cracks is sometimes limited and the CFRP has to be attached away from the cracks. There is a lack of knowledge on SIFs for such strengthening scheme. This paper presents the effects of CFRP bond locations on the Mode I SIF of centre‐cracked tensile (CCT) steel plate. The Mode I SIF at the crack tip is calculated using the finite element (FE) models. A correction factor is introduced as a function of CFRP bond location and crack length. The FE results are compared and agree well with experimental tests conducted by the authors. By combining with another two factors (one considering CFRP mechanical properties and the other considering CFRP bond width) derived previously by the authors, SIF formulae are proposed for CFRP reinforced CCT steel plates.