Infinite Body Research Articles

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

• Planar crack is analyzed for 2D hexagonal quasicrystals with thermal effects. • Extended displacement discontinuity-boundary integral equation method is proposed. • Green's functions for point EDDs are derived by using the Hankel transform method. • The ordinary 1/2 singularity behavior is obtained at the crack edge. • Extended intensity factors are expressed in terms of EDDs across crack surfaces. An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.

Effect of gravity on generalized thermoelastic diffusion due to laser pulse using dual-phase-lag model

PurposeThe purpose of this paper is to obtain a general solution to the field equations of generalized thermo-diffusion in an infinite thermoelastic body under the effect of gravity in the context of the dual-phase-lag (DPL) model. The half space is considered made of an isotropic homogeneous thermoelastic material. The boundary plane surface is heated by a non-Gaussian laser beam.Design/methodology/approachAn exact solution to the problem is obtained using the normal mode analysis.FindingsThe derived expressions are computed numerically for copper and the results are presented in graphical form.Originality/valueComparisons are made with the results predicted by Lord-Shulman theory and DPL model for different values of time and in the presence and absence of gravity as well as diffusion.

Three-dimensional steady-state general solution for isotropic hygrothermoelastic media

ABSTRACTThe potential theory method was utilized to derive the steady-state general solution for three-dimensional (3D) hygrothermoelastic media. Two displacement functions are introduced to simplify the governing equations, with which the elastic, moisture, and temperature fields are thus simplified. Using the differential operator theory and superposition principle, all the physical quantities can be expressed in terms of two functions, one of which satisfies a harmonic equation and the other satisfies an eight-order partial differential equation. With the aid of generalized Almansi’s theorem, all the physical quantities like displacements, moisture, and temperature are expressed in terms of five quasi-harmonic functions for various cases of material characteristic roots. The obtained general solutions are in simple form and thus they may bring more convenience to certain boundary problems. As an example, the fundamental solutions for a point moisture source combined with heat source in the interior of infinite hygrothermoelastic body are presented by virtue of the obtained general solution. A planar crack of arbitrary shape in an infinite medium subjected to mechanical, moisture, and temperature loads is investigated to illustrate the application of the solution in boundary value problems. Specifically, for a penny-shaped crack under uniform combined loads, the complete, exact solutions are presented.

Improvement of Stress Intensity Factor Analysis Using Overall Defined Basic Density Function on Crack Face

In this study, an improved technique for the evaluation of stress intensity factor (SIF) along the 3D planar crack front is proposed. In the present analysis, a planar triangular element is used to cover the total crack face. The stress field induced by a body force doublet in an infinite body is used for a fundamental solution. In the present analysis, overall defined basic density function of body force doublet is introduced. The crack problem is formulated as hypersingular boundary integral equations and the magnitudes of distributed point force doublets are determined through boundary conditions. The numerical SIF solution obtained using the present approach was compared with the solution obtained using the conventional basic density function. The results indicate that the proposed technique improves the accuracy of SIF. In addition, some numerical examples were examined to verify the effectiveness and the robustness of the proposed technique.

Approximate stress intensity factors for a semi-circular crack in an arbitrary structure under arbitrary mode I loading

Approximate Stress Intensity Factors (SIFs) for a semi-circular surface crack in arbitrary elastic finite and infinite bodies are presented. Utilizing General Point Load Weight Function (GPLWF) concept, an explicit expression is derived to determine SIFs for semi-circular cracks subjected to uniform, linear, and nonlinear loads. The presented formulation not only provides a solution for cracked bodies subjected to arbitrary two-dimensional stress distribution on the crack faces, but also could present SIFs for any point on the crack front. The results of the presented formulation for special cases, i.e. semi-circular cracks in a finite thickness plate subjected to complicated loadings, central and non-central semi-circular cracks in finite-length thick-walled cylinders subjected to a uniform internal pressure, and central semi-circular cracks in an infinite-length thick-walled cylinder subjected to a non-uniform internal pressure are compared with the available results in the literature and those obtained through FEM, indicating a very good accuracy. The proposed procedure will be generalized for semi-elliptical cracks in arbitrary structures and mixed mode loading in future.

Green's function for a point heat source embedded in an infinite body with two circular elastic inclusions

• The analytical continuation technique and the alternation method are used. • The derived series solution is rapidly convergent. • The interfacial normal stresses become negative with a higher temperature. • Our present proposed method is efficient and general. A general solution for a point heat source interacting with two circular inclusions embedded in an infinite solid is presented in this work. Based on the method of analytical continuation in conjunction with the alternating technique, the solutions to the heat conduction and thermoelastic problems are derived. A rapidly convergent series solution for both the temperature and stress field, which is expressed in terms of an explicit general term of the corresponding homogeneous complex potential, is obtained in an elegant form. As a numerical illustration, the constant contour values of the temperature and stresses are displayed to provide a better understanding of both heat and stress flows for different locations of a point heat source. Numerical results of interfacial stresses are also shown in graphic form and discussed in detail. The solution obtained in this study can be treated as Green's functions which enable us to formulate an integral equation for the crack problem associated with two circular inclusions.

Thermoelastic Interactions in an Infinite Orthotropic Continuum of a Variable Thermal Conductivity with a Cylindrical Hole

In this work, the generalized thermoelasticity theory with phase lags is used to solve a thermoelastic problem for an orthotropic infinite unbounded body containing a cylindrical cavity by approximate techniques. The thermal conductivity of the present body is assumed to vary linearly with the temperature. The surface of the cylinder is traction free and subjected to a uniform step temperature. The general solutions for the temperature, displacement, and thermal stresses are obtained by the method of Laplace transforms. The effects of dual phase lags and the variable thermal conductivity parameter on the studied fields for a cobalt material are discussed.

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part II: Numerical solutions

• 2D hexagonal quasicrystals planar crack with thermal effects analyzed numerically. • Green's functions are derived for uniform triangular and rectangular elements. • The EDD-BEM is proposed for planar crack analysis in infinite 2D hexagonal QC body. • Elliptical crack with different loadings are investigated and presented graphically. The extended displacement discontinuity (EDD) boundary element method is developed to analyze an arbitrarily shaped planar crack in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack face. Green's functions for uniformly distributed EDDs over triangular and rectangular elements for 2D hexagonal QCs are derived. Employing the proposed EDD boundary element method, a rectangular crack is analyzed to verify the Green's functions by discretizing the crack with rectangular and triangular elements. Furthermore, the elliptical crack problem for 2D hexagonal QCs is investigated. Normal, tangential, and thermal loads are applied on the crack face, and the numerical results are presented graphically.

Efficient Numerical Evaluation of Exact Solutions for One-Dimensional and Two-Dimensional Infinite Cylindrical Heat Conduction Problems

Estimation of thermal properties or diffusion properties from transient data requires that a model is available that is physically meaningful and suitably precise. The model must also produce numerical values rapidly enough to accommodate iterative regression, inverse methods, or other estimation procedures during which the model is evaluated again and again. Bodies of infinite extent are a particular challenge from this perspective. Even for exact analytical solutions, because the solution often has the form of an improper integral that must be evaluated numerically, lengthy computer-evaluation time is a challenge. The subject of this paper is improving the computer evaluation time for exact solutions for infinite and semi-infinite bodies in the cylindrical coordinate system. The motivating applications for the present work include the line-source method for obtaining thermal properties, the estimation of thermal properties by the laser-flash method, and the estimation of aquifer properties or petroleum-field properties from well-test measurements. In this paper, the computer evaluation time is improved by replacing the integral-containing solution by a suitable finite-body series solution. The precision of the series solution may be controlled to a high level and the required computer time may be minimized, by a suitable choice of the extent of the finite body. The key finding of this paper is that the resulting series may be accurately evaluated with a fixed number of terms at any value of time, which removes a long-standing difficulty with series solution in general. The method is demonstrated for the one-dimensional case of a large body with a cylindrical hole and is extended to two-dimensional geometries of practical interest. The computer-evaluation time for the finite-body solutions are shown to be hundreds or thousands of time faster than the infinite-body solutions, depending on the geometry.

Fractional order theory to an infinite thermo-viscoelastic body with a cylindrical cavity in the presence of an axial uniform magnetic field

The model of fractional time-derivative of generalized magneto–thermo–viscoelasticity equations for isotropic media in the presence of a constant magnetic field is considered. Some essential theorems on the linear coupled and generalized theories of thermo-viscoelasticity can be easily obtained. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizations, Lord–Shulman theory, and Green–Lindsay theory as well as to the coupled theory.

Quantitative Evaluation of the Flaking Strength of Rolling Bearings with Small Defects: (Part 1: FEM Analyses of the Stress Intensity Factor, KII, under Rolling Contact)

It has been demonstrated that the rolling contact fatigue (RCF) test, using a specimen with a small drilled hole, is a useful means of evaluating the influence of a minor defect on the flaking strength of steels. In this study, RCF tests were conducted on rolling bearings with small drilled holes. Flaking failure was determined to be caused by the shear-mode fatigue cracks that emanated from the small defects. As a first step to quantifying the crack-growth threshold according to fracture mechanics principles, using the finite element method (FEM), it was necessary to analyze the Mode II stress intensity factor (SIF) range, ΔKII, of a ring-shaped crack, as emanated around the edge of a drilled hole after the passage of a rolling element. Subsequently, the derived values were correlated with the ΔKII values of penny-shaped cracks in an infinite body under uniform shear via a correlation factor, fdrill. The SIF of the ring-shaped crack was also uniformly correlated with that of the penny-shaped crack, using the single factor, fdrill, irrespective of the hole diameter, d, the depth of the hole-edge, h′, and the maximum contact pressure, qmax, within the following ranges: d = 0.05 ~ 0.2 mm, h′ = 0.05 ~ 0.345 mm and qmax = 2.0 ~ 3.0 GPa. The obtained results were later applied towards the quantification of the RCF test results, as detailed in Part 2 of the report on this research.

転がり軸受のはく離強度に及ぼす微小欠陥寸法の影響（第1報:転がり接触下のモードII応力拡大係数のFEM解析）

It has been shown that rolling contact fatigue (RCF) test using specimen having a small drilled hole is a useful method for evaluating the effect of small defect on flaking strength of steels. In this study, RCF tests of rolling bearings having a small drilled hole were carried out. The flaking failure was considered as a problem of shear-mode fatigue crack emanating from the small defect. As a first step to quantify the crack-growth threshold on the basis of fracture mechanics, mode II stress intensity factor range, ΔKII, of a ring-shaped crack emanated around the edge of a drilled hole under the passing of a rolling element was analyzed by using finite element method. And then, the obtained values were correlated with the ΔKII values of penny-shaped cracks in an infinite body under uniform shear through the intermediary of a correlation factor, fdrill. The stress intensity factor of the ring-shaped crack was uniformly correlated with that of the penny-shaped crack by the single factor fdrill irrespective of hole diameter, d, depth of hole edge, h’, and maximum contact pressure, qmax, within the ranges: d = 0.05 ~ 0.2 mm, h’ = 0.05 ~ 0.345 mm and qmax = 2.0 ~ 3.0 GPa. The obtained results will be applied for the quantification of RCF test results shown in the subsequent paper.

On the accuracy of higher order displacement discontinuity method (HODDM) in the solution of linear elastic fracture mechanics problems

The higher order displacement discontinuity method (HODDM) utilizing special crack tip elements has been used in the solution of linear elastic fracture mechanics (LEFM) problems. The paper has selected several example problems from the fracture mechanics literature (with available analytical solutions) including center slant crack in an infinite and finite body, single and double edge cracks, cracks emanating from a circular hole. The numerical values of Mode I and Mode II SIFs for these problems using HODDM are in excellent agreement with analytical results (reaching up to 0.001% deviation from their analytical results). The HODDM is also compared with the XFEM and a modified XFEM results. The results show that the HODDM needs a considerably lower computational effort (with less than 400 nodes) than the XFEM and the modified XFEM (which needs more than 10000 nodes) to reach a much higher accuracy. The proposed HODDM offers higher accuracy and lower computation effort for a wide range of problems in LEFM.

Two approaches to solving the problem of diffraction by a cylindrical body with a coordinate-dependent refractive index

An algorithm based on solving a two-dimensional integral equation for the problem of diffraction by an infinite cylindrical body with a coordinate-dependent refractive index is proposed. For the diffraction by a circular cylinder, rigorous approach reduced to solving the Helmholtz equation by expanding the field in a combined basis of splines and trigonometric functions is used. The comparison of two methods for the case when the squared wavenumber depends parabolically on the x-coordinate is performed. The method on the basis of the integral equation was tested on a problem of plane-wave scattering by a cylinder with an elliptic or a rectangular cross section under the assumption that the wavenumber of the medium inside the cylinder may be constant. A special case of the diffraction by an inhomogeneous circular cylinder from a metamaterial is also considered.

Analysis of rectangular cracks in elastic bodies

The solution of Volterra dislocation is derived in an infinite elastic body. Stress components are Cauchy singular at dislocation location. The stress field is utilized to construct integral equations for rectangular cracks with arbitrary arrangement in the body. The solution to integral equations is used to determine stress intensity factors on the crack edges. Numerical results are presented for a rectangular crack under various loads. Furthermore, interaction between two cracks, having a line of symmetry, is studied.

Efficient Analysis of 3D Mixed-Mode Cracks of a Pressure Vessel Based on Schwartz-Neuman Alternating Method

The Schwartz-Neuman alternating method is employed to analyze 3D cracks in structural components with complicated geometries.The SIFs of Mode I, II and III for the mixed-mode cracks under complicated stress state are obtained from the alternating computing scheme between finite element method solution for the uncracked body and the symmetric Galerkin boundary element method solution for the crack in an infinite body. The SIFs of the different surface cracks postulated at a regular pipe and elliptical surface cracks at the nozzle-cylinder intersection of a typical reactor pressure vessel are investigated by using the Schwartz-Neuman alternating method. The comparison with other analysis models and results reported in the literature shows that the Schwartz-Neuman alternating method can efficiently and accurately evaluate SIFs of surface cracks of different shapes and depths with much smaller computational models, which indicates that the Schwartz-Neuman alternating method is an efficient method in the evaluation of the SIFs of 3D cracks.

Exergy Analysis of Monoethylene glycol recovery processes for hydrate inhibition in offshore natural gas fields

Gas hydrate formation is an issue in natural gas production. In offshore deepwater scenarios the situation is aggravated by the inaccessibility, salinity, low temperatures and high pressures. Monoethylene glycol (MEG) injection in well-heads is one of the used technologies for flow assurance in gas flowlines. Rich MEG is processed in MEG Recovery Units (MRU) to be recovered as Lean MEG returned to well-heads. In offshore rigs, besides determining energy requirements, the assessment of energy degradation is also important, which can be done by Exergy Analysis. In this work, an exergy formulation is developed and three technologies of offshore MRUs are assessed. Since exergy is a property that depends on the definition of a reference environment reservoir (RER), Exergy Analysis is performed via two RER approaches, both presenting consistent results. Approach #1 prescribes the usual standard atmosphere, with the addend that it is saturated by equilibrium with an infinite body of liquid water and MEG is in chemical equilibrium with air species. Since MEG is spontaneously oxidable, this entails very high exergy flows for MRU streams with MEG. As MEG is practically conserved, high exergy efficiencies result for all MRUs hindering their discrimination. Approach #2 prescribes also the standard atmosphere in equilibrium with liquid water containing MEG at infinite dilution, but not in chemical equilibrium with air. The MEG condition in MRU streams is now more akin with the MEG state in this RER, resulting that MEG streams have lower magnitude of exergy flow, leading to realistic exergy efficiencies that allow MRU discrimination. The underlying reason is that the input exergy flows now have magnitude comparable with exergy losses.

Generalized thermoelastic problem of an infinite body with a spherical cavity under dual-phase-lags

The aim of the present contribution is the determination of the thermoelastic temperatures, stress, displacement, and strain in an infinite isotropic elastic body with a spherical cavity in the context of the mechanism of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord–Shulman (2TLS) model and two-temperature dual-phase-lag (2TDP) model of thermoelasticity are combined into a unified formulation with unified parameters. The medium is assumed to be initially quiescent. The basic equations are written in the form of a vector matrix differential equation in the Laplace transform domain, which is then solved by the state-space approach. The expressions for the conductive temperature and elongation are obtained at small times. The numerical inversion of the transformed solutions is carried out by using the Fourier-series expansion technique. A comparative study is performed for the thermoelastic stresses, conductive temperature, thermodynamic temperature, displacement, and elongation computed by using the Lord–Shulman and dual-phase-lag models.

2D steady-state general solution and fundamental solution for fluid-saturated, orthotropic, poroelastic materials

The 2D steady-state solutions regarding the expressions of stress and strain for fluid-saturated, orthotropic, poroelastic plane are derived in this paper. For this object, the general solutions of the corresponding governing equation are first obtained and expressed in harmonic functions. Based on these compact general solutions, the suitable harmonic functions with undetermined constants for line fluid source in the interior of infinite poroelastic body and a line fluid source on the surface of semi-infinite poroelastic body are presented, respectively. The fundamental solutions can be obtained by substituting these functions into the general solution, and the undetermined constants can be obtained by the continuous conditions, equilibrium conditions and boundary conditions.

Research on stress intensity factors of elliptical hole in infinite body

Research on stress intensity factors of elliptical hole in infinite body