Infinite Elastic Medium Research Articles

Напружений стан у пружному тілі з триланковим жорстким включенням при гармонічних коливаннях поздовжнього зсуву

There is a thin absolutely rigid inclusion that in a cross-section represents three segments broken line in an infinite elastic medium (matrix) that is in the conditions of antiplane strain. The inclusion is under the action of harmonic shear force Pe^{iwt} along the axis Oz. Under the conditions of the antiplane strain the only one different from 0 z-component of displacement vector W (x; y) satisfies the Helmholtz equation. The inclusion is fully couple with the matrix. The tangential stresses are discontinuous on the inclusion with unknown jumps. The method of the solution is based on the representation of displacement W (x; y) by discontinuous solutions of the Helmholtz equation. After the satisfaction of the conditions on the inclusion the system of integral equations relatively unknown jumps is obtained. One of the main results is a numerical method for solving the obtained system, which takes into account the singularity of the solution and is based on the use of the special quadrature formulas for singular integrals.

Helmholtz Decomposition and Boundary Element Method Applied to Dynamic Linear Elastic Problems

The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the longitudinal and transverse fields satisfy scalar Helmholtz equations that can be solved using a desingularized boundary element method (BEM) framework. The curl free longitudinal and divergence free transversal conditions can also be cast as additional scalar Helmholtz equations. When compared to other BEM implementations, the current framework leads to smaller matrix dimensions and a simpler conceptual approach. The numerical implementation of this approach is benchmarked against the 3D elastic wave field generated by a rigid vibrating sphere embedded in an infinite linear elastic medium for which the analytical solution has been derived. Examples of focussed 3D elastic waves generated by a vibrating bowl-shaped rigid object with convex and concave surfaces are also considered. In the static zero frequency limit, the Helmholtz decomposition becomes non-unique, and both the longitudinal and transverse components contain divergent terms that are proportional to the inverse square of the frequency. However, these divergences are equal and opposite so that their sum, that is the displacement field that reflects the physics of the problem, remains finite in the zero frequency limit.

Dynamic kelvinlets

We introduce Dynamic Kelvinlets , a new analytical technique for real-time physically based animation of virtual elastic materials. Our formulation is based on the dynamic response to time-varying force distributions applied to an infinite elastic medium. The resulting displacements provide the plausibility of volumetric elasticity, the dynamics of compressive and shear waves, and the interactivity of closed-form expressions. Our approach builds upon the work of de Goes and James [2017] by presenting an extension of the regularized Kelvinlet solutions from elastostatics to the elastodynamic regime. To finely control our elastic deformations, we also describe the construction of compound solutions that resolve pointwise and keyframe constraints. We demonstrate the versatility and efficiency of our method with a series of examples in a production grade implementation.

P-wave Diffraction by a Crack Under Impact Load

The focus of this paper is centered on diffraction of P-wave by a Griffith crack in an infinite isotropic elastic medium on which impact load is applied. Suddenly applied equal but opposite normal stresses on crack faces causes the separation of the crack faces. Laplace and Fourier transforms have been employed to convert the problem to a Fredholm integral equation of second kind in the Laplace transform plane which has been solved by perturbation method. Stress intensity factor has been computed numerically by Laplace inversion using Zakian’s algorithm and plotted graphically against time for different isotropic materials.

Properties of the Eshelby tensor and existence of the equivalent ellipsoidal inclusion solution

We show that the Eshelby tensor, SE, when written in the 6 × 6 matrix (Voigt) form, is weakly positive definite, i.e., it can be written as a product of two positive definite matrices. All eigenvalues of SE are real and lie between 0 and 1, for an arbitrary anisotropic elastic medium with a positive definite elastic stiffness tensor C. The weakly positive definiteness property leads to a direct proof of the existence of Eshelby’s equivalent inclusion solution for a “transformed” ellipsoidal inhomogeneity in an infinite elastic medium.

Effect of a rigid toroidal inhomogeneity on the elastic properties of a composite

An analytical solution is obtained for the problem of an infinite elastic medium containing a rigid toroidal inhomogeneity under remotely applied uniform strain. The traction on the torus surface is determined as a function of torus parameters and strain components applied at infinity. The results are utilized to calculate components of the stiffness contribution tensor of the rigid toroidal inhomogeneity that is required for calculation of the overall elastic properties of a material containing multiple toroidal inhomogeneities. The analytical results are verified by comparison with finite element model calculations.

Problems of interaction longitudinal shear waves with V-shape tunnels defect

The problem of determining the two-dimensional dynamic stress state near a tunnel defect of V-shaped cross-section is solved. The defect is located in an infinite elastic medium, where harmonic longitudinal shear waves are propagating. The initial problem is reduced to a system of two singular integral or integro-differential equations with fixed singularities. A numerical method for solving these systems with regard to the true asymptotics of the unknown functions is developed.

Efficient numerical solution of the hydraulic fracture problem for planar cracks

An infinite isotropic elastic medium with a planar crack is considered. The crack is subjected to pressure of fluid injected inside the crack at a point of its surface. The fluid is viscous newtonian, incompressible, the medium is impermeable. Description of the crack growth is based on the lubrication equation (local balance of the injected fluid and the crack volume), the elasticity equation for crack opening caused by fluid pressure, Poiseulle equation related the fluid flux with crack opening and the pressure gradient, and the criterion of crack propagation of linear fracture mechanics. The crack growth is simulated by a series of discrete steps. Each step consists of three stages: increasing the crack volume by a constant crack size, crack jump to a new size defined by the fracture criterion, and filling the new crack configuration by the fluid presented in the crack. The problem is ill-posed and requires specific methods for numerical solution. The proposed method is based on an appropriate class of approximating functions for fluid pressure distributions on the crack surface and the theory of solution of ill-posed problems. Example of crack growth in a homogeneous and isotropic elastic medium is considered, influence of the fluid viscosity on the process of crack growth is studied.

Finite element analysis of lung ultrasound surface wave elastography

Interstitial lung disease manifested as fibrotic and stiffened lung tissue, resulting in symptoms such as dyspnea and may lead to respiratory failure. We recently developed a lung ultrasound surface wave elastography for evaluating the elastic properties of superficial lung tissue. In this abstract, we present a finite element analysis to investigate the effects of pleural effusion on the shear wave propagation of superficial lung tissue. The muscle, pleural effusion, and lung tissue were simulated by a 2D planar model of an infinite elastic medium with densities for different materials. A nearly incompressible, linear elastic model was used to model muscle, pleural effusion and lung. The model was excited using a line source on the muscle surface. Harmonic excitations were performed at 100, 150 and 200 Hz with duration of 0.1 s. The boundaries of lung and muscle were attached to an infinite. The mesh of muscle, pleural effusion, and lung were constructed using linear quadrilateral elements. The infinite region was meshed by infinite elements (type CINPE4). The dynamic responses of the tissue model to the excitations were solved by the ABAQUS explicit dynamic solver with automatic step size control. Insignificant difference in shear wave speeds of lung tissue at different excitation frequencies was found in both simulations with and without taking into account the pleural effusion, indicating that pleural effusion does not affect the measurements of shear wave speeds in ultrasound elastography.Interstitial lung disease manifested as fibrotic and stiffened lung tissue, resulting in symptoms such as dyspnea and may lead to respiratory failure. We recently developed a lung ultrasound surface wave elastography for evaluating the elastic properties of superficial lung tissue. In this abstract, we present a finite element analysis to investigate the effects of pleural effusion on the shear wave propagation of superficial lung tissue. The muscle, pleural effusion, and lung tissue were simulated by a 2D planar model of an infinite elastic medium with densities for different materials. A nearly incompressible, linear elastic model was used to model muscle, pleural effusion and lung. The model was excited using a line source on the muscle surface. Harmonic excitations were performed at 100, 150 and 200 Hz with duration of 0.1 s. The boundaries of lung and muscle were attached to an infinite. The mesh of muscle, pleural effusion, and lung were constructed using linear quadrilateral elements. The infinite ...

Revisiting the problem of a 2D infinite elastic isotropic medium with a rigid inclusion or a cavity

The problem of analytically finding the elastic fields inside a 2D infinite elastic isotropic medium containing a rigid inclusion or a cavity and subjected to uniform remote loading is a classical elasticity problem of theoretical and practical interest. In the present work, we revisit the Kolosov–Muskhelishvili potential theory which is a powerful tool for solving the problem in question. In particular, a novel strategy is proposed to deal with the rigid-body displacements that the rigid inclusion or cavity may undergo. When the shape of the rigid inclusion or cavity is described by a Laurent polynomial, a general method is elaborated to solve the problem. The results obtained by using our method include as special cases all the relevant results reported in the literature. In light of our results, some errors in the literature are corrected. Finally, the cases of a rhombus rigid inclusion and a pentagonal cavity saturated with a fluid are studied in detail and some general properties are discussed.

Eigenfunction expansion method to analyze thermal shock behavior in magneto-thermoelastic orthotropic medium under three theories

ABSTRACTThis article is concerned with a study of thermal shock response in infinite thermoelastic medium under the purview of Lord–Shulman model, Green–Naghdi theory III, and three-phase-lag model of generalized thermoelasticity. The medium under consideration is assumed to be homogeneous, orthotropic, and thermally conducting. The fundamental equations of the two-dimensional problem of generalized thermoelasticity with three-phase-lag model in an infinite elastic medium under the influence of magnetic field are obtained as a vector–matrix differential equation form using normal mode analysis which is then solved by the Eigenfunction expansion method. Numerical results for the temperature, displacements and thermal stress distribution are presented graphically.

Closed-form elastic solution for irregular frozen wall of inclined shaft considering the interaction with ground

An analytical solution is derived that provides a closed-form formulation for stresses and displacements of a deep inclined shaft with irregular frozen wall liner in an infinite elastic medium, subjected to a non-uniform stresses. This solution is based on: (i) the conformal mapping of a doubly connected domain of prescribed shape onto a circular ring by an appropriate numerical optimization scheme which is achieved by MATLAB program, (j) the complex variable method. For a particular case, the distributions of hoop stress and radial displacement inside frozen wall are presented with analytical solution. The accuracy of stress functions is verified by comparison of different number of coefficients in stress functions and it show that the number of coefficients k≥15 is advised. The validity of analytical solution is verified by a comparison between its result and that obtained from the finite element program ANSYS. Noting that the shaft excavation can be regarded as relaxation of initial ground stresses around shaft excavation boundary, this paper presents the mechanical problem as two separate problems: (i) initial stresses and displacements model where stresses keep constant and displacements equal zero around frozen wall and ground, (j) stresses relaxation model with relaxed normal and shear stresses acted on excavated boundary. The solution is worked out by the principle of superposition in elastic medium. The Young's modulus ratio (β) of the frozen wall to the ground reflects the influence of ground to frozen wall, and the influence decreases with the growth of β. The proposed solution can provide analytical basis for frozen wall design and be used as a quick-solver for back-analysis of in situ stresses and displacements.

Effect of surface tension and geometry on cavitation in soft solids

Unbounded growth of cavity in soft solids subjected to internal pressure is a commonly observed phenomenon. Such phenomenon has been harnessed by developing cavitation rheology technique (CR) to investigate the local mechanical properties of many complex soft materials. The elasticity of the material, surface energy, and geometric factors in combination can dictate the cavitation behavior in a complex manner. We report a finite element based framework capturing the coupled effect of elastic strain energy and surface tension on the cavitation phenomenon. We show results for a spherical cavity in an infinite elastic media and for the CR geometry. The surface tension is shown to increase the critical pressure for cavitation. The energy release rate also depends on the surface tension. In CR geometry, by varying the distance between the needle and the immovable sample boundaries, we have captured the conditions leading to confinement. Our results provide new understanding regarding the effects of geometry and surface tension on the cavitation phenomenon in soft solids.

Discrete shear-transformation-zone plasticity modeling of notched bars

Plane strain tension analyses of un-notched and notched bars are carried out using discrete shear transformation zone plasticity. In this framework, the carriers of plastic deformation are shear transformation zones (STZs) which are modeled as Eshelby inclusions. Superposition is used to represent a boundary value problem solution in terms of discretely modeled Eshelby inclusions, given analytically for an infinite elastic medium, and an image solution that enforces the prescribed boundary conditions. The image problem is a standard linear elastic boundary value problem that is solved by the finite element method. Potential STZ activation sites are randomly distributed in the bars and constitutive relations are specified for their evolution. Results are presented for un-notched bars, for bars with blunt notches and for bars with sharp notches. The computed stress–strain curves are serrated with the magnitude of the associated stress-drops depending on bar size, notch acuity and STZ evolution. Cooperative deformation bands (shear bands) emerge upon straining and, in some cases, high stress levels occur within the bands. Effects of specimen geometry and size on the stress-strain curves are explored. Depending on STZ kinetics, notch strengthening, notch insensitivity or notch weakening are obtained. The analyses provide a rationale for some conflicting findings regarding notch effects on the mechanical response of metallic glasses.

Propagation of mechanical waves through a stochastic medium with spherical symmetry

We theoretically analyze the propagation of outgoing mechanical waves through an infinite isotropic elastic medium possessing spherical symmetry whose Lamé coefficients and density are spatial random functions characterized by well-defined statistical parameters. We derive the differential equation that governs the average displacement for a system whose properties depend on the radial coordinate. We show that such an equation is an extended version of the well-known Bessel differential equation whose perturbative additional terms contain coefficients that depend directly on the squared noise intensities and the autocorrelation lengths in an exponential decay fashion. We numerically solve the second order differential equation for several values of noise intensities and autocorrelation lengths and compare the corresponding displacement profiles with that of the exact analytic solution for the case of absent inhomogeneities.

A complex function method of SH wave scattering in inhomogeneous medium

The complex function method is applied in the solution of the scattering problem for an irregularly shaped boundary in an infinite inhomogeneous elastic medium, which is deduced from the scattering problem in a homogeneous one. The potential function of the scattering wave which is generated by the irregularly boundary is obtained by applying the complex function method in the inhomogeneous medium. The reduced Helmholtz equation with variable coefficients is solved by separation of variables. Then, the potential function is expressed as the complex domain functions series. By truncating a set of infinite algebraic equations, the coefficient of the series are determined. In order to verify the validity of this method, the wave equation in a inhomogeneous medium is degenerated to the equation with constant coefficients. The domain function is discussed. The dynamic stress concentration factor around an elliptical cavity is calculated in an exponentially inhomogeneous medium.

Forced vibration of the elastic system consisting of the hollow cylinder and surrounding elastic medium under perfect and imperfect contact

The bi-material elastic system consisting of the circular hollow cylinder and the infinite elastic medium surrounding this cylinder is considered and it is assumed that on the inner free face of the cylinder a point-located axisymmetric time harmonic force, with respect to the cylinder\'s axis and which is uniformly distributed in the circumferential direction, acts. The shear-spring type imperfect contact conditions on the interface between the constituents are satisfied. The mathematical formulation of the problem is made within the scope of the exact equations of linear elastodynamics. The focus is on the frequency-response of the interface normal and shear stresses and the influence of the problem parameters, such as the ratio of modulus of elasticity, the ratio of the cylinder thickness to the cylinder radius, and the shear-spring type parameter which characterizes the degree of the contact imperfectness, on these responses. Corresponding numerical results are presented and discussed. In particular, it is established that the character of the influence of the contact imperfection on the frequency response of the interface stresses depends on the values of the vibration frequency of the external forces.

Study of thermo‐elastic cruciform crack with unequal arms in an orthotropic elastic plane

The objective of this article is concerned with a study of a static cruciform crack problem in an infinite orthotropic elastic medium opened by prescribed normal pressure distributions subjected to steady state temperature field. The problem is reduced to Fredholm singular integral equations of first kind, which are solved using Chebyshev polynomials. Analytical expressions of the stress intensity factors and crack energies are found and the numerical computations for the orthotropic elastic material Boron‐epoxy composite for different particular cases are depicted through figures. The striking feature of the article is the graphical presentation of possibility of crack arrest and variations of crack energies with the variations of the arms of the cruciform crack.

A Crack in the Form of a Three-Link Broken Line Under The Action of Longitudinal Shear Waves

We solve the problem of determination of the dynamic stress intensity factors for a crack in the form of a three-link broken line. The crack is located in an infinite elastic medium with propagating harmonic longitudinal shear waves. The initial problem is reduced to a system of three singular integrodifferential equations with fixed singularities. A numerical method is proposed for the solution of this system with regard for the true asymptotics of the unknown functions.

Penny-shaped crack in elastic medium with surface energy effects

In this paper, a penny-shaped crack in an infinite elastic medium subjected to vertical pressure loading at the crack surface under the influence of surface stress is considered. The Gurtin–Murdoch continuum theory of elastic material surfaces is adopted, and the Hankel integral transform is employed to solve this axisymmetric boundary value problem. A set of simultaneous dual integral equations is solved by employing an appropriate numerical solution scheme. Selected numerical results are presented to portray the influence of the surface stress on the elastic field. Numerical results reveal that the surface stress has a significant influence on both stress and displacement fields. It is also found that the material becomes tougher with the presence of surface stress. In addition, the elastic field also shows size-dependent behavior deviating from the classical crack solution. The solutions presented in this study provide fundamental understanding of the influence of surface stress on fracture mechanics problems. It can also be used as a benchmark solution for the development of numerical techniques, such as FEM and BEM, for analysis of more complicated problems associated with cracks under the influence of surface stress.