Infinite Elastic Research Articles

Solution of the External Pochhammer–Chree Problem and Bending Seismic Vibrations of the Pipeline in Infinite Elastic Continuum

Solution of the External Pochhammer–Chree Problem and Bending Seismic Vibrations of the Pipeline in Infinite Elastic Continuum

Solution of the External Pochhammer-Chree Problem and Bending Seismic Vibrations of the Pipeline in Infinite Elastic Continuu

The possibility of partial splitting of the dynamic equations of the linear theory of elasticity for bulk expansion and the components of the rotation vector of medium particles in cylindrical coordinates in the general case of a nonstationary problem is proved. This result is a generalization of a similar fact established by A. Love with a simple harmonic dependence of the named functions on time and two spatial coordinates. An exact analytical solution of the problem on bending vibrations of an elastic space with a circular cylindrical cavity (external Pochhammer-Chree problem) is given. It is shown that the study published by K. Toki and Sh. Takada on this problem does not provide a solution to the posed problem. On the basis of the solution obtained for the external Pochhammer-Chree problem, bending vibrations of an underground pipeline caused by the action of a seismic wave are studied. The results obtained in this case provide, apparently, the first theoretical substantiation of the engineering theory of rigid jamming of the pipeline in the soil for the case of bending vibrations, widely used in regulatory documents on seismic construction.

Dispersion of Longitudinal Waves by Four Coplanar Mode-I Cracks in an Infinite Elastic Medium

Dispersion of Longitudinal Waves by Four Coplanar Mode-I Cracks in an Infinite Elastic Medium

Misfit Stresses Due to a Cylindrical Dilatational Inclusion of Annular-Sector Cross-Section in an Infinite Elastic Medium

An elastic model for a cylindrical dilatational inclusion of annular-sector cross-section in an infinite elastic medium is considered. The stress fields are found in a closed analytical form and are illustrated by stress maps. Specific features in the stress distribution are revealed and discussed in detail. It is shown that the stress magnitude can be so high that various mechanisms of stress relaxation can be activated.

Stress Solution for an Infinite Elastic Plate Containing Two Arbitrarily Shaped Holes

The problem solved in this paper is an elastic infinite plane containing two arbitrarily shaped holes, each of which is symmetrical about their common axis. Uniform stress at infinity is applied on the plane, and the boundaries of the two holes are subjected to uniform surface traction. The stress solution is derived using the complex variable method. The general form of the mapping function that conformally maps the discussed region into an annulus is proposed. An infinite plane bounded by two round holes or elliptical holes is the particular case of this paper. An example of an infinite plane with two rectangles is investigated using the presented solution. The solution of the ANSYS finite element method is compared with the new solution.

On Sets of Multiple Equally Strong Holes in an Infinite Elastic Plate: Parameterization and Existence

This paper addresses the inverse problem of determining sets of any finite number of “equally strong” holes (with smooth boundaries) in an infinite, homogeneous, isotropic elastic plate that is sub...

Action of Nonaxisymmetric Dynamic Loads on a Circular Hole in the Infinite Elastic Plane

A modified finite-difference method with respect to time and the method of Fourier series with respect to the angular variable are used to solve a plane dynamic problem of the elasticity theory on the action of nonaxisymmetric loads on the boundary of a circular hole in the infinite plane. The distribution of stress is determined. The time dependence of the stress concentration on the boundary of a circular hole is numerically found for various Poisson’s ratios and the load distributed according to a certain law.

Near-Field Representations of the Acoustic Green's Function in a Shallow Ocean with Infinite Elastic Seabed

In this paper, we show how to construct acoustic Green's functions for a water column atop an infinite elastic seabed using a generalization of the method by Ahluwalia and Keller.6Our goal is to obtain a representation of the acoustic Green's function which is valid in the near field. Such a representation is useful in the investigation of the inverse shape problem using the methodology we pioneered for waveguides with reflecting seabeds.

Eigenvalue Approach on Thermoelastic Interactions in an Infinite Elastic Solid with Voids

The linear theory of generalized thermoelastic materials has been applied to determine the change in volume fraction field, the distribution of temperature, and deformation in an infinite elastic solid with voids. The non-dimensional forms of the solutions in transformed domains have been obtained analytically by applying eigenvalue approach methodology. Numerical results are displayed graphically to illustrate the solution, and discussion of the results are made with parameters for a magnesium crystal-like material.

Effect of Couple-Stresses on the Transient Dynamic Stress Intensity Factors for a Crack in an Infinite Elastic Medium Under an Impact Stress Wave

This study applies linearized couple-stress theory to evaluate the dynamic stresses around a crack in an infinite elastic medium that is subjected to an incoming shock stress wave impinging normal to the crack. The boundary conditions with respect to the crack are reduced to dual integral equations using a Fourier transform in the Laplace domain. To solve these equations, the differences in the displacement and rotation at the crack are expanded by a series of functions that are zero-valued outside the crack in the Laplace domain.

Effect of Couple Stresses on the Stress Intensity Factors for Two Parallel Cracks in an Infinite Elastic Medium under Tension

Stresses around two parallel cracks of equal length in an infinite elastic medium are evaluated based on the linearized couple‐stress theory under uniform tension normal to the cracks. Fourier transformations are used to reduce the boundary conditions with respect to the upper crack to dual integral equations. In order to solve these equations, the differences in the displacements and in the rotation at the upper crack are expanded through a series of functions that are zero valued outside the crack. The unknown coefficients in each series are solved in order to satisfy the boundary conditions inside the crack using the Schmidt method. The stresses are expressed in terms of infinite integrals, and the stress intensity factors can be determined using the characteristics of the integrands for an infinite value of the variable of integration. Numerical calculations are carried out for selected crack configurations, and the effect of the couple stresses on the stress intensity factors is revealed.

Progressive Frictional Delamination of an Infinite Elastic Film on a Rigid Substrate Due to In-Plane Point Loading

The paper presents a solution to a delamination problem of an infinite elastic film resting on a rigid substrate and loaded by a monotonically increasing in-plane point force. A rigid-slip contact is assumed between the film and the substrate, leading to the development of two regions at the interface: a damaged zone with a relative slip between the materials, and a region where the interface remains intact. Both film natural and essential boundary conditions are zero on the boundary between these two interfacial zones with the shape of the boundary being a part of the solution. Problem's self-similarity enables us to obtain an approximate distribution of interfacial traction within the delaminated zone and a shape of the zone itself. For film's Poisson's ratio ν =− 1 the approximate solution becomes exact. It is argued that this can be treated as a special case of a rigid film sliding on a rigid substrate. The presented approach can be used to obtain approximate closed-form solutions to similar delamination problems.

Hole Crack Problems in Infinite Elastic Plate in Tension

This paper deals with crack(s) emanating from a hole in infinite elastic plate in tension. Such a crack problem is called a hole crack problem for short. By extending Buckner’s principle suited for a crack to a hole crack problem in infinite plate in tension, here, the original problem (the hole crack problem in infinite plate in tension) is divided into a homogeneous problem (the one without hole crack) subjected to remote loads and a hole crack problem in an unloaded body with applied tractions on the surfaces of the hole and crack. Thus, the results in terms of the stress intensity factors (SIFs) can be obtained by considering the latter problem, which is analyzed easily by means of a hybrid displacement discontinuity (a boundary element method) proposed recently by Yan. Numerical examples are included to illustrate that the numerical approach is very simple and effective for analyzing the hole crack problem in infinite plate in tension. By using the proposed approach, three hole crack problems (i.e., a pair of cracks emanating from an elliptical hole, a pair of cracks emanating from a rhombus hole, and a crack emanating from a triangular hole in infinite plate in tension) are analyzed in detail. By changing the hole geometry form and the hole geometry parameters and by comparing the SIFs of the hole crack problem with those of the center crack problem, the effect of the hole geometry form and the hole geometry parameters on the SIFs is revealed. It is found that a hole has a shielding and an amplifying effect on the SIFs of crack(s) emanating from the hole. The shielding and amplifying effects varied with the hole geometry form and the hole geometry parameters.

Numerical Technique for Surface Opening Displacement of an External Crack in an Infinite Elastic Space

This paper presents a numerical technique for determination of surface opening displacements of an external circular crack embedded in an infinite elastic space. Under the applied constant displacement at infinity, a Barenblatt – Dugdale crack whose crack tip plasticity governed by Tresca and von Mises yield criteria is numerically investigated. The proposed technique of a discretizing approach is qualitatively developed. It is then employed for the purpose of predicting the opening displacement of such crack with von Mises yielding criterion whose corresponding analytical expressions are not available. By superposition of Dugdale model's displacement functions, the influence function which is the essential basis of numerical implementation is obtained. The accuracy and convergence of the proposed technique has been judged by the comparison studies between Tresca analytic crack opening displacement and the one obtained from these numerical computations. Accordingly, this technique provides the sufficiently accurate results and shows a good agreement with the exact value and always converges to a stable solution. This leads to a conclusion that the problem can be estimated accurately by the simple numerical technique proposed within. Finally, the surface opening displacement of the cohesive external crack is compared for the three different yield conditions namely Dugdale, Tresca and von Mises.

G0301-6-5 調和振動荷重を受ける無限弾性板の正方形孔による応力集中(材料力学部門一般(6))

G0301-6-5 調和振動荷重を受ける無限弾性板の正方形孔による応力集中(材料力学部門一般(6))

Ultrasonic Reflection Tomography vs. Canonical Body Approximation: Experimental Assessment of an Infinite Elastic Cylindrical Tube

Comparisons were made between the results obtained using two quantitative ultrasound imaging methods on the solid cross section of a cylindrical tube that is infinite in the axial direction. The first method tested was the classical reflection tomography method based on the first-order Born approximation, which can only be used under conditions to obtain limited reconstruction of the external boundaries of the high contrast scatterer. The results were compared with those obtained using another inversion scheme based on the Intercepting Canonical Body Approximation (ICBA) in a large frequency range, which gives accurate complete geometrical information about the tube (thickness measurements). The numerical and experimental results obtained show the feasibility of the latter approach.

Boundary Perturbation Solution for Nearly Circular Holes and Rigid Inclusions in an Infinite Elastic Medium

The boundary perturbation method is used to solve the problem of a nearly circular rigid inclusion in a two-dimensional elastic medium subjected to hydrostatic stress at infinity. The solution is taken to the fourth order in the small parameter epsilon that quantifies the magnitude of the variation of the radius of the inclusion. This result is then used to find the effective bulk modulus of a body that contains a dilute concentration of such inclusions. The corresponding results for a cavity are obtained by setting the Muskhelishvili coefficient κ equal to −1, as specified by the Dundurs correspondence principle. The results for nearly circular pores can be expressed in terms of the pore compressibility. The pore compressibilities given by the perturbation solution are tested against numerical values obtained using the boundary element method, and are shown to have good accuracy over a substantial range of roughness values.

P012 部分的に埋め込まれた剛体丸棒によりねじりを受ける無限弾性厚板の応力特異性(フェロー賞表彰対象ポスターセッション)

P012 部分的に埋め込まれた剛体丸棒によりねじりを受ける無限弾性厚板の応力特異性(フェロー賞表彰対象ポスターセッション)

Shape Derivative of Energy Functional in an Infinite Elastic Strip with a Semi-Infinite Crack

In this paper we study linear elasticity equations in an infinite elastic strip with a semi-infinite crack. We find the derivative of the energy functional as the crack shifts with an angle. Then we obtain the formula given by surface force and the angle.

Dynamic T-Stress for a Mode-I Crack in an Infinite Elastic Plane

An integral equation method is presented to determine dynamic elastic T-stress. Special attention is paid to a single crack in an infinite elastic plane subjected to impact loading. By using the Laplace and Fourier transforms, the associated initial-boundary value problem is transformed to a Fredholm integral equation. The dynamic T-stress in the Laplace transform domain can be expressed in terms of its solution. Moreover, an explicit expression for initial T-stress is derived in closed form. Numerically solving the resulting equation and performing the inverse Laplace transform, the transient response of T-stress is determined in the time space, and the response history of the T-stress is shown graphically. Results indicate that T-stress exhibits apparent transient characteristic.