Coefficients Of Higher Order Terms Research Articles

Importance of the Higher Order Terms of the Williams series expansion: Experimental Aspects and Finite Element Simulations

The study is devoted to the multi-point Williams series expansion for the crack-tip fields in isotropic linear elastic materials and methods of estimating the coefficients of higher order terms in the power expansion. It is shown that higher order terms are very important in processing of the experimental data obtained from interference – optic methods. It is demonstrated that the accurate interpretation of the experimental data requires keeping, considering and evaluating the higher order terms of the Williams expansion. A large cohort of experiments on cracked specimens from optically sensitive materials using digital photoelasticity method was performed. The interference fringe patterns obtained show that the correct description of the stress fields in the vicinity can be given if we take into account higher order terms in the asymptotic presentation of the near crack tip fields. Two interference-optical methods were used. Digital photoelasticity method and digital holography interferometry technique are applied for determination of the coefficients of the Williams series expansion. It is shown that we can restore theoretically the stress field in the vicinity of the crack tip if at least fifteen terms in the power expansion are hold. The classical over-deterministic method is used for the evaluation of the coefficients.

Dynamic behaviors of general N-solitons for the nonlocal generalized nonlinear Schrödinger equation

The general N-solitons of nonlocal generalized nonlinear Schrodinger equations with third-order, fourth-order and fifth-order dispersion terms and nonlinear terms (NGNLS) are studied. Firstly, the Riemann–Hilbert problem and the general N-soliton solutions of NGNLS equations were given. Then, we study the symmetry relations of the eigenvalues and eigenvectors related to the scattering data which involve the reverse-space, reverse-time and reverse-space-time reductions. Thirdly, some novel solitons and the dynamic behaviors which corresponded to novel eigenvalue configurations and the coefficients of higher-order terms are given. In all the three NGNLS equations, their solutions often collapse periodically, but can remain bounded or nonsingular for wide ranges of soliton parameters as well. In addition, it is found that the higher-order terms of the NGNLS equations not only affect the amplitude variation of the soliton, but also influence the singularity and the motion of the soliton.

Uniform null controllability of a fourth-order parabolic equation with a transport term

In this paper we prove a uniform controllability result for a fourth order parabolic partial differential equation which includes a transport term, when the coefficients of higher order terms vanish. We prove the null controllability of the system with a single boundary control, and also it is obtained that the cost of the control decreases to zero, under the hypothesis of the control time being large enough. Moreover, we prove that, if the control time is small enough, the cost of controllability increases to infinity.

Atomistic Determination of Fracture Mechanics Parameters

Stress intensity factors, T-stresses and higher order coefficients in the Williams series expansion are the fundamental concepts of continuum fracture mechanics for characterizing the stress fields around the crack tip in a homogeneous material in the linear regime. It is well-known that critical values of stress intensity factors stipulate the of materials to growth a crack. Nowadays the modern multi-purpose computational tools such as Simulia Abaqus allow us to calculate T-stresses in cracked specimens and structures along with stress intensity factors. The aim of this study is to understand if one can obtain this fracture parameters of conventional fracture mechanics from atomistic simulations based on molecular dynamics method. The ability to describe fracture processes at atomic scale by means of stress intensity factors, T-stresses and higher-order coefficients will provide the opportunity to take into account many effects such as material microstructures, chemical compositions and concentrations and others. In this study the values of stress intensity factors, T-stress and coefficients of the Williams series expansion are determined using atomistic simulations based on molecular dynamics method with a classical molecular dynamics code Large-scale Atomic/Molecular Massively Parallel Simulator. The over-deterministic method is used to determine SIFs, T-stresses and coefficients of higher-order terms from molecular dynamics modelling of a plate with a central crack. The accuracy of the proposed approach is tested for this rather simple cracked configuration. There is the theoretical analytical solution with all the coefficients of the higher-order terms in the Williams series expansion. The existing theoretical solution allows us to compare the angular distributions of the stress tensor components for a large plane with the central crack. It is shown that results obtained from molecular dynamics simulations and the theoretical analytical solutions coincide qualitatively.

Experimental determination and finite element analysis of coefficients of the multi-parameter Williams series expansion in the vicinity of the crack tip in linear elastic materials. Part I

This study aims at obtaining coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate and in the semi-circular notched disk under bending by the use of the digital photoelasticity method. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give a more accurate estimation of the near-crack-tip stress, strain and displacement fields and extend the domain of validity for the Williams power series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. By means of the developed tool, the fringe patterns that contain the whole field stress information in terms of the difference in principal stresses (isochromatics) are captured as a digital image, which is processed for quantitative evaluations. The developed tool allows us to find points that belong to isochromatic fringes with the minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of the stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. The iterative procedure of the over-deterministic method is utilized to find the higher order terms of the Williams series expansion. The procedure is based on the consistent correction of the coefficients of the Williams series expansion. The first fifteen coefficients are obtained. The experimentally obtained coefficients are used for the reconstruction of the isochromatic fringe pattern in the vicinity of the crack tip. The comparison of the theoretically reconstructed and experimental isochromatic fringe patterns shows that the coefficients of the Williams series expansion have a good match.

Deformed breather and rogue waves for the inhomogeneous fourth-order nonlinear Schrödinger equation in alpha-helical proteins

The inhomogeneous fourth-order nonlinear Schrodinger equation is investigated, which models the transport of energy along the inhomogeneous hydrogen bonding spines in alpha-helical proteins. The deformed breather and rogue waves solutions for the equation are derived via Darboux transformation. Thereinto, the modified limit procedure in generalized Darboux transformation is proposed to construct the expressions of higher-order rogue waves. Due to the presence of inhomogeneity, despite their diversity profiles, the shape, amplitude and pulse width of each deformed breather will change with time. When the coefficients of higher-order terms are varied, the propagating trajectories of breathers are changed. Head-on collision, partially coalesced, without interaction between two deformed breathers is found under diverse selected parameters. The temporal-spatial structures of the presented rogue waves exhibit certain valleys around one or several centers, and the peak heights of those rogue waves are more than three times that of the background. Additionally, the distributed area of the rogue wave will tend to be broaden/narrow along the temporal axis, with the variation of the coefficients of inhomogeneity and higher-order terms.

Experimental evaluation of coefficients of multi-parameter asymptotic expansion of the crack-tip stress field using digital photoelasticity

In this study coefficients of the multi-parameter Williams series expansion for the stress field in the vicinity of the central crack in the rectangular plate are obtained by the use of the digital photoelasticity method and finite element analysis. The higher-order terms in the Williams asymptotic expansion are retained. It allows us to give more accurate estimation of stress, strain and displacement fields and to extend the domain of validity for the Williams series expansion. The program is specially developed for the interpretation and processing of experimental data from the phototelasticity experiments. The developed tool allows us to find points that belong to isochromatic fringes with minimal light intensity. The digital image processing with the aid of the developed tool is performed. The points determined with the adopted tool are used further for the calculations of stress intensity factor, T-stresses and coefficients of higher-order terms in the Williams series expansion. Numerical simulations of the same cracked specimen as in the experimental photoelasticity method are performed. The higher-order coefficients are extracted from finite element method calculations implemented in Simulia Abaqus software package and the outcomes are compared to experimental values. The plate with a small central crack has been considered either. This kind of the cracked specimen has been utilized for comparison of coefficients of the Williams series expansion obtained from finite element analysis with the coefficients known from the theoretical solution based on the complex variable theory in plane elasticity. It is shown that the coefficients of the Williams series expansion match with good accuracy.

Perturbative solutions of mathcal{N}={1}^{ast } holography on S4

We apply the recently proposed perturbative technique to solve the supergravity BPS equations of mathcal{N}={1}^{ast } theories put on S4. In particular, we have calculated the coefficients of the leading quartic terms exactly, in the expression of the universal part for the holographic free energy as a function of the mass parameters. We also report on the coefficients of higher order terms up to 10th order, which are computed numerically.

DETERMINATION OF THE WILLIAMS SERIES EXPANSION’S COEFFICIENTS USING DIGITAL PHOTOELASTICITY METHOD AND FINITE ELEMENT METHOD

In the present work, photoelastic and finite element methods have been employed to study the near crack tip fields in isotropic linear elastic cracked bodies under mixed mode loading. The investigated fracture results have been obtained for a series of cracked specimens by testing plates with two parallel cracks, two inclined parallel cracks, three-point bend specimens, four-point bend specimens, inclined edge crack triangular shape specimens subjected to symmetric three point bend loading. The multi-parameter Williams series expansion is used for the crack tip field characterization. Digital photoelasticity method is utilized for determination of the Williams series expansions coefficients. The unknown coefficients in the multi-parameter equation are determined using a linear least squares method in an over-deterministic manner. Together with the experimental determination of the fracture mechanics parameters finite element method is invoked to describe the crack tip stress field. Coefficients of higher order terms are either found numerically by finite element method. A good agreement is found between the numerical and experimental results. The significant advantages using multi-parameter equations in the analysis of the stress field are shown and the errors that a study with a limited number of terms produce is demonstrated. The comparison with finite element analysis highlighted the importance and precision of the photoelastic observation for the evaluation of the fracture mechanics parameters. The experimental SIF, T-stress values and coefficients of higher-order terms estimated using the digital photoelasticity method for the extensive series of cracked specimens are compared with finite element analysis (FEA) results, and are found to be in good agreement.

Determination of higher order stress terms in cracked Brazilian disc specimen under mode I loading using digital image correlation technique

The digital image correlation (DIC) technique is used to calculate the coefficients of higher order terms of the Williams' expansion in the centrally-cracked Brazilian disc specimens of different crack lengths under pure mode I loading. The specimens are subjected to diametral-compression loading and the displacement field is obtained for the cracked Brazilian disc by a correlation between the undeformed and deformed images captured before and after loading. The rigid body motion and rotation of each specimen are detected and eliminated from the displacement field by using a code developed based on the Williams' series solution. Then, by employing an over-deterministic system of equations, the coefficients of higher order terms of the Williams expansion are calculated. The same specimens are then simulated using finite element method. It is shown that there is good agreement between the DIC and the finite element results. Therefore, the DIC technique can be proposed as a reliable method to experimentally obtain the mode I stress intensity factor KI, the T-stress and the coefficients of higher order terms in the Williams' series expansion.

Taylor Series Expansion for Zeros of Sampled-Data Systems

This paper introduces key properties that are useful for the computation of a Taylor expansion of discretization zeros or single intrinsic zeros of sampled-data systems. Furthermore, regularity is shown to exist in the suffixes of the coefficients of the expansion expressions, which implies that the sampling zeros or every coefficient in the numerator of the transfer function of the sampled-data systems is dominated by the coefficients of higher order terms in the numerator and denominator of the continuous-time counterpart. Moreover, it is shown that the regularity can reduce the calculation effort for expansions of higher order systems.

Digital image correlation method for calculating coefficients of Williams expansion in compact tension specimen

The digital image correlation (DIC) method is used to obtain the coefficients of higher-order terms in the Williams expansion in a compact tension (CT) specimens made of polymethyl methacrylate (PMMA). The displacement field is determined by the correlation between reference image (i.e., before deformation) and deformed image. The part of displacements resulting from rigid body motion and rotation is eliminated from the displacement field. For a large number of points in the vicinity of the crack tip, an over-determined set of simultaneous linear equations is collected, and by using the fundamental concepts of the least-squares method, the coefficients of the Williams expansion are calculated for pure mode I conditions. The experimental results are then compared with the numerical results calculated by finite element method (FEM). Very good agreement is shown to exist between the DIC and FE results confirming the effectiveness of the DIC technique in obtaining the coefficients of higher order terms of Williams series expansion from the displacement field around the crack tip.

Analytical Approximation of Crack-Tip Stress Field: Study on Efficient Determination of Coefficients of Higher Order Terms of Williams Power Expansion

The paper presents a study on accuracy of the multi-parameter approximation of the stress field in a cracked body using Williams power series calibrated according to results of FEM computation. Main attention is paid to a detailed analysis of suitable selection of FE nodes, whose results serve as inputs to the over-deterministic method (ODM) employed for determination of the coefficients of terms of the Williams expansion describing the crack-tip field. Two different ways of FE nodal selection are compared – nodes selected from the crack tip vicinity versus nodes selected from a specific part of the test specimen body in a specific way. Comparison is made with the usage of the authors’ own developed procedures enabling both the determination of the coefficients of the analytical approximation of stress field based on the FE results and the backward reconstruction of the field from those determined terms’ coefficients/functions. The wedge-splitting test (WST) specimen with a crack is taken as example for the study.

Evaluation of Crack Tip Stress Field Using Combination of Advanced Implementation of Over-Deterministic Method and FE Analysis

Creation of an automatic utility to determine the values of coefficients of higher order terms of Williams power series by usage of over-deterministic method applied to results of finite element analysis is a main goal of this research. The developed procedure based on the support of Java programming language considerably simplifies analyses on optimization of selection of FE nodal results for improvement of accuracy of the near-crack-tip fields’ approximation using Williams series.

Evaluation of the Higher Order Terms of the Wedge-Splitting Specimen Based on the SBFEM

The scaled boundary finite element method (abbr. SBFEM) is a semi-analytical method developed by Wolf and Song. The analytical advantage of the solution in the radial direction allows SBFEM converge to the Williams expansion. The coefficients of the Williams expansion, including the stress intensity factor, the T-stress, and higher order terms can be calculated directly without further processing. In the paper the coefficients of higher order terms of the crack tip asymptotic field of typical wedge splitting specimens with two different loading arrangements are evaluated using SBFEM. Numerical results show the method has high accuracy and effectiveness. The results have certain significance on determining crack stability of the wedge-splitting specimen.

Evaluation of the T-Stress and the Higher Order Terms of the Elastic Crack Based on the SBFEM

The Scaled Boundary Finite Element Method (abbr. SBFEM) developed by Wolf and Song is a numerical method which has a half analytical nature. In the paper, the asymptotic fields of central crack tip and single edge crack tip of the plane elastic plates are evaluated based on the SBFEM. Numerical examples are provided to demonstrate its high accuracy and effectiveness, and the numerical results show that SBFEM can calculate the SIFs, T-stress and the coefficients of higher order terms with higher efficiency and accuracy. The singular fields of crack-tip with complex configuration can be evaluated combining the sub-structuring technique (or super-element).

Size effects in mode II brittle fracture of rocks

In this paper, a stress-based criterion is suggested for predicting the mode II fracture resistance of rock materials by considering the effects of specimen size. This criterion is a modified form of the maximum tangential stress criterion which takes into account the second and third terms of Williams series expansion in addition to the singular terms. In order to assess the proposed criterion, a number of modes I and II fracture tests are conducted on the semi-circular bend specimens manufactured from marble. For using the proposed criterion, the constant coefficients of higher order terms are determined for the semi-circular bend specimen with different crack length ratios and various loading span to radius ratios. Finally, it is shown that the proposed criterion can provide good estimates for the mode II fracture resistance of marble from its mode I fracture results while the effects of specimen size are taken into account.

Experimental determination of stress field parameters in bi-material notches using photoelasticity

The experimental technique of photoelasticity has been utilized for calculating bi-material notch stress intensities as well as the coefficients of higher order terms. Employing the equations of multi-parameter stress field allows data collection from a larger zone from the notch tip and makes the data collection from experiments more convenient. Moreover, the effects of higher order terms in the region near the notch tip are taken into account. For the photoelasticity experiments, a laboratory specimen known as the Brazilian disk with a central notch, consisting of Aluminum and Polycarbonate, has been utilized. Using this specimen, different mode mixities could be easily produced by changing the loading angle. The bi-material notch stress intensities and the first non-singular stress term (called I-stress) were then calculated for different test configurations. In order to utilize the advantages of whole-field photoelasticity and minimize the experimental errors, a large number of data points were substituted in the multi-parameter stress field equations. Then the resulting system of nonlinear equations was solved by employing an over-deterministic least squares method coupled with the Newton–Raphson algorithm. It has been shown that considering the I-stress term improves, to a large extent, the accuracy of the stress intensities calculated through the photoelasticity technique. Moreover, by reconstructing the isochromatic fringes, the effects of the I-stress term on the shape and size of these fringes around the notch tip were investigated for a 30° notch. Finally, the experimental photoelasticity results were compared with the corresponding values obtained from finite element analysis and a good correlation was observed.

An over-deterministic method for calculation of coefficients of crack tip asymptotic field from finite element analysis

An over-deterministic method has been employed for calculating the stress intensity factors (SIFs) as well as the coefficients of the higher-order terms in the Williams series expansions in cracked bodies, using the conventional finite element analysis. For a large number of nodes around the crack tip, an over-determined set of simultaneous linear equations is obtained, and using the fundamental concepts of the least-squares method, the coefficients of the Williams expansion can be calculated for pure mode I, pure mode II and mixed mode I/II conditions. A convergence study has been conducted to examine the effects of the number of nodes used, the number of terms in Williams expansion and the distance of the selected nodes from the crack tip, on the accuracy of the results. It is shown that the simple method presented in this paper, yields accurate results even for coarse finite element meshes or in the absence of singular elements. The accuracy of SIFs and the coefficients of higher-order terms are validated by using the available results in the literature.

Determination of NSIFs and coefficients of higher order terms for sharp notches using finite element method

An overdeterministic method was used for calculating the Notch Stress Intensity Factors (NSIFs) as well as the coefficients of the higher order terms for structures containing sharp notches. The series solution of displacement fields around the notch tip was fitted to a large number of nodal displacements obtained from finite element analysis. An over-determined set of simultaneous linear equations was then derived and the nodal displacements reduced to a small set of unknown coefficients by employing the concept of the least-squares method. The efficiency of the proposed method was assessed through analyzing several notched specimens under pure mode I, pure mode II and mixed modes I/II loading. The accuracy of the NSIFs and the coefficients of higher order terms was evaluated by comparing them with the results available in the literature, or the results obtained by the boundary collocation method. While the presented method is simple, it yields very good results.