Weight Function Method Research Articles

Practical Applications of Numerical Procedures for Structural Integrity Assessment

The paper presents two practical applications in which the weight function method is used to determine the stress intensity factor as the main structural integrity assessment parameter in linear elastic fracture mechanics. In this method, one may obtain the solution for the stress intensity factor for a given crack configuration and a certain loading, provided a complete solution (the Mode I stress intensity factor KIr(a) and the displacements of the crack faces uIr(x,a)) for the same crack problem is known in another loading case called the reference case. Three different cases can be encountered when the weight function method is used: i. The stress intensity factor KIr(a) and the displacement field uIr(x,a) are known; ii. Only the stress intensity factor KIr(a) is known and iii. Both KIr(a) and uIr(x,a) are unknown. In the first case, the method can be applied directly. This paper presents two practical applications for the second and the third case: Stress intensity factors for a strip with a cracked hole subjected to point loads (case 2) and stress intensity factors for an axially cracked thick walled cylinder subjected to internal pressure (case 3). For case 2, the reference stress intensity factors were first calculated using the compounding technique in a reference case (uniform remote traction). Then, an approximate expression of the displacements field found in the literature was used to calculate the weight function. In case 3, the authors propose a solution based on curve fitting of the crack face displacements obtained through finite element analysis. The obtained results were compared with the ones from the literature (where available). In all cases the agreement was very good, showing the reliability of the proposed solutions.

Total Life Approach for Fatigue Life Estimation of Welded Structures

It was shown that estimation of fatigue lives of welded joints can be successfully carried out by considering the fatigue process as a fatigue crack growth from the initial intrinsic crack size of a0=ρ* until the final crack af. Such an approach avoids a somewhat arbitrary division of the fatigue process into the crack initiation and propagation and concentrates on using only one methodology - the fracture mechanics theory. The stress intensity factors can be determined in such cases by the weight function method. The proposed methodology allows estimation of the fatigue life under both constant and variable amplitude loading.

Weight function method for precise determination of top quark mass at Large Hadron Collider

We propose a new method to measure a theoretically well-defined top quark mass at the LHC. This method is based on the “weight function method,” which we proposed in our preceding paper. It requires only lepton energy distribution and is basically independent of the production process of the top quark. We perform a simulation analysis of the top quark mass reconstruction with tt¯ pair production and lepton + jets decay channel at the leading order. The estimated statistical error of the top quark mass is about 0.4 GeV with an integrated luminosity of 100 fb−1 at s=14 TeV. We also estimate some of the major systematic uncertainties and find that they are under good control.

Stress intensity factors for internal surface cracks in autofrettaged functionally graded thick cylinders using weight function method

Stress intensity factors for internal surface cracks in autofrettaged functionally graded cylinders have been studied. It is assumed that the mechanical properties change with a power law in radial direction of cylinder. Effects of autofrettage pressure, volume fraction, crack dimensions and cylinder thickness on the stress intensity factors in deepest and surface points of cracks were determined by using the weight function method. Results show that the changes in volume fractions of ingredients have more effects than the other parameters.

Stress intensity factor calculation for internal circumferential cracks in functionally graded cylinders using the weight function approach

In this work, we describe stress intensity factors obtained for functionally graded cylinders with internal circumferential cracks using the weight function method for different combinations of cylinder geometry, crack depth, and material gradation. A modified form of the J integral is employed to obtain reference stress intensity factors using finite element analysis results. Furthermore, a fitting technique is employed for calculating stress intensity factors not covered in the finite element modeling matrix. The results predicted by the weight function method along with the proposed fitting technique are in good agreement with the results directly obtained from the finite element analysis.

Dynamic approachs for system identification applied to deformation study of the dams

To dynamical study of the dams, the prediction of occurred deformation caused by hydrostatic and hydrodynamic pressures during the operation phase as well as the designing phase is necessary. The reservoir water changes also the permanent variations of atmospheric temperature especially in the areas where there are significant seasonal and circadian changes in conditions are important factors to “dynamic system” of the dams. The aim of this paper is to present an applied dynamic method to identify the body of the dam as a dynamic system and to predict its behavior in the variable environmental conditions based on the monitoring observations. Non-parametric methods such as cross-covariance and coherence method are applied for system identification of the dam based on time series of input signal of measured forces and output signal of deformation measurements. The main feature of this paper is using the weighting function method which enables “System Identification” of the dam as a dynamic system, to model the deformation of the dam. The applied weighting function method benefits from measurements of effective factors on system of the dam also the deformation monitoring measurements from different sensors. For practical applicability of the method for a dam, Masjed–Soleiman “earth–dam” in Iran has been chosen. A wide range of instruments has been installed inside the dam to control the deformation; therefore multiple measurements are available for analysis. Since changing atmospheric temperature has not an important role in geotechnical study of an earth-dam singly, the authors decided to investigate the environmental condition such as atmospheric temperature and reservoir water-level changes as input signal to the “dam-sensors-combination” system. Here the authors carried out their investigation using seven epochs of geodetic measurements between 2000 and 2008, moreover the measurements of four anchors inside an installed rod-extensometer between 2007 and 2013, settlement observations on settlement-plates in different levels between 2008 and 2013, and a long-time series of the daily observations of three installed soil-extensometers between 2005 and 2013. A significant correlation between temperature changes and soil-extensometer observations has been determined, besides the coherence method leads to common periods of 351 days between input and output time series. The weighting function method is applied to model the deformation of the dam using deformation measurements of installed soil-extensometer with temperature and water-level changes as effective sources. The RMS of difference between the estimated dam deformations from weighting function model compared to original measurements was calculated at 0.039 mm also from the point of correlation the original deformation measurements and modeled deformations are correlated at 95 %. The RMS of the difference between measurements and the predicted dam deformations using the weighting function model was calculated at 0.057 mm leading to a correlation value of 91 %. If the absolute values are transformed in percentage of the deformation, the maximum error of prediction is 10 %. The results show the success of weighting function method to model the deformation observations of the dam affected by input factors. One of the most important achievements of this paper is to draw a distinction between temperature as a systematic error sources and the other effective factors on the settlement observations of the earth-dam via comparison of the results of non-parametric methods on different sensors.

THEORETICAL ASPECTS AND METHODS OF PARAMETERS IDENTIFICATION OF ELECTRIC TRACTION SYSTEM DEVICES. METHOD OF WEIGHT FUNCTION

Purpose. Development and substantiation of a new method of structural identification of electrical devices of electric traction systems for both DC and AC current. Methodology. To solve this problem the following methods are used: the methods and techniques of the linear electrical engineering, in particular, the Laplace operator method; the numerical method for solving the integral equation, which is based on the representation of the Wiener-Hopf linear equations system (this allows forming the solutions of the problem in a mathematical form of the correlation and weight functions); the factorization method, which provides certain partition of the correlation functions of the stochastic processes. Findings. It was developed the method of weight function of the electrical devices identification, which can be fully used in the systems of electric traction. As the use example of the developed method it was considered a feeder section of DC electric traction with the single power supply. On this section move two electric locomotives of the type DE 1, they have been identified by the weighting functions. The required currents and voltages of electric locomotives are also formulated in the electric traction network in probabilistic and statistical form, that is, the functions of mathematical expectation and the correlation functions are determined. At this, it is taken into account that the correlation function of the sum of random functions is equal to the sum of the correlation functions of additives, and the correlation function of the integral of a random function is defined as the double integral of the correlation function of the output of a random function. Originality. Firstly, originality consists of the adaption of the developed method of structural identification for the devices of electric traction system. Secondly, it lies in the proper development of the new method of weight function. And finally, it lies in the solution of the Wiener-Hopf equation using the correlation method of factorization. Practical value. The developed method and the technique that is based on it, allow predicting electric traction loadings of the traction network, as well as the other devices of the traction energy systems and the electric motive power in the different modes of operation and, in particular, during the high-speed train traffic. The method takes into account the load influence (and its mode of operation) of the adjacent feeder zones on the electromagnetic processes in the studied intersubstation area. Thus, it is important that the required voltages and currents are considered as the random functions.

In-plane fracture of an ice floe: A theoretical study on the splitting failure mode

In practical offshore structure/icebreaker designs, it is critical to capture the salient failure modes of the ice feature. The dominant failure modes are influenced by the structural properties, interaction process, and characteristics of the ice features. For an ice feature of finite size with relatively small lateral confinement, splitting failure has been frequently observed during ice-structure interactions. In literature, the splitting failure of an ice floe has been studied using several different methods, ranging from strength theory to methods based on fracture mechanics. One contribution of this paper is that these various methods are compiled and evaluated collectively. Another contribution of this paper is the construction of an analytical framework to obtain solutions for the splitting process of an ice floe upon serious considerations of relevant fundamental theories. The framework developed for its calculations is presented but not limited these two base cases (i.e., rectangular and circular ice floes). Specifically, because of the complexity of the splitting failure scenario (particularly the ice floe geometry) and our pursuit of analytical solutions, within the presented framework, two different approaches have been proposed: the fracture mechanics approach and plasticity theory approach. All of the employed methods are validated against data from previous field tests or numerical results. Recommendations and discussion were presented for each of these methods’ respective application range. It was found that most of the floe sizes under engineering interests can be accurately described by linear elastic fracture mechanics. The cohesive zone method, which is one type of the nonlinear fracture mechanics based approaches, is found to be potentially applicable in extrapolating laboratory-scale measurements to the field scale. Worth mentioning, the weight function method was effectively used to calculate the fracture properties of an ice floe under various symmetric loading conditions (e.g., boundary confinement). The boundary confinement can easily prevent splitting failure or increase the ice splitting load. As one type of the lateral confinement, the inertia-induced body force, is also studied and compared with previous numerical results. Comparatively, the plasticity theory based approach (i.e., the plastic upper/lower limit theory) enables us to effectively attain a conservative estimation of the ice splitting load in an analytical form. It can serve as a supplementary tool to the fracture mechanics based approach.

Effect of Weld Residual Stress Fitting on Stress Intensity Factor for Circumferential Surface Cracks in Pipe

Recent studies have shown that the crack growth of primary water stress corrosion cracking (PWSCC) is mainly driven by the weld residual stress (WRS) within the dissimilar metal weld. The existing stress intensity factor (K) solutions for surface cracks in pipe typically require a fourth order polynomial stress distribution through the pipe wall thickness. However, it is not always possible to accurately represent the through thickness WRS with a fourth order polynomial fit and it is necessary to investigate the effect of the WRS fitting on the calculated Ks. In this paper, two different methods were used to calculate the K for a semi-elliptical circumferential surface crack in a pipe under a given set of simulated WRS. The first method is the universal weight function method (UWFM) where the through thickness WRS distribution is represented as a piece-wise monotonic cubic fit. In the second method, the through thickness WRS profiles are represented as a fourth order polynomial curve fit (both using the entire wall thickness data and only using data up to the crack-tip). In addition, three-dimensional finite element (FE) analyses (using the simulated weld residual stress) were conducted to provide a reference solution. The results of this study demonstrate the potential sensitivity of Ks to fourth order polynomial fitting artifacts. The piece-wise WRS representations used in the UWFM were not sensitive to these fitting artifacts and the UWFM solutions were in good agreement with the FE results. In addition, in certain cases, it was demonstrated that more accurate crack growth calculations of PWSCC are made when the UWFM is used.

Effect of the model’s geometry in fretting fatigue life prediction

This paper analyses the influence that the type of geometry used to obtain the stress/strain fields in a cylindrical contact has on fretting fatigue life predictions. In addition, this work considers the effect that the fatigue crack shape assumed has on these fretting fatigue life predictions. The strain/stress fields are calculated using a series of finite elements models that consider the following three types of behaviour: plane stress, plane strain (2D geometries) and 3D. Each of these models gives a different crack initiation life and a different evolution of the stress intensity factor (SIF), which are calculated using the weight function method. These models therefore provide different fretting fatigue life predictions. Finally, the lives obtained using the numerical models are compared with experimental lives.

On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function

In this paper, by the application of methods of weight functions and the use of analytic techniques, a multidimensional more accurate half-discrete Hilbert-type inequality with the kernel of the hyperbolic cotangent function is proved. We show that the constant factor related to the Riemann zeta function is the best possible. Equivalent forms as well as operator expressions are also investigated.

Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains

We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361---366, 1976, Arch. Ration. Mech. Anal. 62:295---301, 1976, Arch. Ration. Mech. Anal. 69:37---52, 1979, Pac. J. Math. 104:77---83, 1980; Knightly in SIAM J. Math. Anal. 3:506---511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361---366, 1976, Arch. Ration. Mech. Anal. 62:295---301, 1976, Arch. Ration. Mech. Anal. 69:37---52, 1979, Pac. J. Math. 104:77---83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633---652, 2012, Quad. Mat. 4:27---68, 1999, Nonlinear Anal. 47:4151---4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159---175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719---732, 2008; Maremonti in Mat. Ves. 61:81---91, 2009, Appl. Anal. 90:125---139, 2011).

Time Delay Signal Processing Research in Sound Source Localization Technology

Time delay estimation is a key technology of array signal processing, which is designed to estimate the time delay of the sound source signal reaches different sensor. Because of differences in the weighting function method, existing methods of generalized cross-correlation time delay estimation have their own anti-noise and anti-reverberation characteristics. In this paper, the existing various delay estimation algorithm comprehensive is compared, which reveals their advantages and disadvantages, and further improved direction of time delay estimation algorithm is given.

A New Optimal Eighth-Order Ostrowski-Type Family of Iterative Methods for Solving Nonlinear Equations

Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2n−1. Thus, we provide a new class which agrees with the conjecture of Kung-Traub for n=4. Numerical comparisons are made to show the performance of the presented methods.

Predicting unstable toughness of concrete based on initial toughness criterion

The fracture processes of concrete were described by a cohesive crack model based on initial toughness criterion. The corresponding analytical method to predict the instability state was proposed. In this model, the initial toughness was adopted as the crack propagation criterion and the weight function method was used to calculate the stress intensity factor and the crack opening displacement caused by the cohesive stress. The unstable toughness can be easily obtained using the proposed method without measuring parameters at the critical state that was necessary in traditional methods. The proposed method was verified by existing experimental data of wedge splitting specimens with different grades of concrete and the sensitivity of the results on the tensile softening curve was discussed. The results demonstrate that the proposed method can well predict the peak load, the critical effective crack length, and the unstable toughness of concrete specimens. Moreover, the calculated unstable toughness is not sensitive to the tensile softening curve.

Residual crack extension resistance of post-fire wedge-splitting normal strength concrete specimen based on the cohesive force function

A comparative study on determining the crack extension resistance curves (KR-curves) for the complete fracture process associated with two functions of the cohesive stress distribution, i.e. Peterson’s softening curve and CEB-FIP Model, 1990 softening curve using an analytical method and the weight function approach is presented in this paper. Fifty wedge-splitting normal strength concrete specimens were prepared to sustain ten different elevated temperatures up to 600°C. The stress intensity factor curves (K-curves) were calculated from the load–displacement curves obtained on these post-fire specimens. At each temperature, the residual fracture toughness KR(Δa) increased with the crack extension length Δa, whereas the KR-curves became lower and flatter with the increasing heating temperatures Tm. It was found that the KR-curve calculated from the weight function method coincided well with the one from the analytical method, whereas different softening curves had observable effects on the KR-curves. The stability of the KR-curve and the influence of temperatures on the crack tip opening displacement–crack extension curves (CTOD–Δa curves) were also analyzed.

Assessment of thermal fatigue damage caused by local fluid temperature fluctuation (part II: crack growth under thermal stress)

In order to assess the crack growth due to thermal fatigue caused by a local temperature fluctuation, it is important to consider the stress gradient in the thickness direction. Even if the stress fluctuation on the surface is sufficiently large enough to initiate fatigue cracks, the initiated cracks may stop growing due to the stress gradient in the thickness direction. In this study, not only the stress on the surface but also the stress in the body was investigated in order to evaluate the crack growth behavior. A trapezoidal temperature fluctuation and a sinusoidal temperature fluctuation were analyzed. It was revealed that the stress range on the surface caused by the former fluctuation could be larger than that caused by the latter. Based on the evaluated stress distribution the stress intensity factor (SIF) was derived by the weight function method. It was shown that, in a crack growth analysis under thermal stress fluctuation, the crack could penetrate the wall thickness when the threshold SIF ΔKth for crack arrest was not assumed. If the ΔKth was considered, cracks could stop growing under the sinusoidal temperature fluctuation, although it was still difficult for cracks to stop growing under the trapezoidal temperature fluctuation. The reason for the arrest of crack growth at the mixing tee and the crack penetration at the branched elbow was reasonably explained from the analyzed results.

Weight Function Method With Segment-Wise Polynomial Interpolation to Calculate Stress Intensity Factors for Complicated Stress Distributions

Abstract Many solutions of the stress intensity factor have been proposed in recent years. However, most of them take only third or fourth-order polynomial stress distributions into account. For complicated stress distributions which are difficult to be represented as third or fourth-order polynomial equations over the stress distribution area such as residual stress distributions or thermal stress distributions in dissimilar materials, it is important to further improve the accuracy of the stress intensity factor. In this study, a weight function method with segment-wise polynomial interpolation is proposed to calculate solutions of the stress intensity factor for complicated stress distributions. By using this method, solutions of the stress intensity factor can be obtained without employing finite element analysis or difficult calculations. It is therefore easy to use in engineering applications. In this method, the stress distribution area is firstly divided into several segments and the stress distribution in each segment is curve fitted to segment-wise polynomial equation. The stress intensity factor is then calculated based on the weight function method and the fitted stress distribution in each segment. Some example solutions for both infinite length cracks and semi-elliptical cracks are compared with the results from finite element analysis. In conclusion, it is confirmed that this method is applicable with high accuracy to the calculation of the stress intensity factor taking actual complicated stress distributions into consideration.

A mode III interfacial crack under nonuniform ferro-elastic domain switching

This paper deals with a mode III interfacial crack while considering the effect of domain switching. The analysis will focus on the variation of stress intensity factor caused by domain switching. Due to the mismatch of material properties, an asymmetric nonuniform domain switching zone around the interfacial crack is achieved under the nonuniform domain switching criterion. Employing the weight function method, we obtain analytic forms of mono-domain and multi-domain toughening effects about stationary and quasi-static steady-state growing cracks, respectively. Three coupling parameters are proposed to evaluate the influences of elastic and ferroelectric bimaterial properties. In conclusion, domain switching of mode III quasi-static steady-state growing interfacial crack can toughen the material. The research will give us ideas on material and initial poling orientation selection to maximize the ferro-elastic toughening effect.

Local thermal stress intensity factors for an axial semi-elliptical crack in a hollow cylinder using the finite element method

METHODS: In this paper, based on the three-dimensional finite element method, the local thermal stress intensity factor is computed using ANSYS software. The weight function method is also used to determine closed-form stress intensity factors at the deepest point of the crack. The effects of crack relative depth, aspect ratio and heat transfer coefficient on the stress intensity factor along the crack front are studied. The results show to be in accordance with those cited in the literature in the special case of loading. CONCLUSIONS: The obtained results show that the small inner surface crack in a cylinder due to thermo-mechanical loading sometimes is more critical than that for the deeper one. It is also shown that the maximum of thermal stress intensity factors are always in the case of conductive heat transfer instead of the convective one.