Isotropic Body Research Articles

General solutions for elasticity of transversely isotropic materials with thermal and other effects: A review

The well-known general solution for uncoupled thermoelasticity of isotropic bodies was proposed by Goodier, which has been utilized extensively since its birth in 1937. When the steady-state response is considered, the temperature field satisfies Laplace’s equation, and the corresponding elastic field can be expressed in terms of a harmonic function, giving rise to the Williams solution. However, for anisotropic bodies, there are no steady-state thermoelastic general solutions that are expressed in terms of (quasi)harmonic functions until 2000. This article presents a short review of the harmonic general solutions for uncoupled elasticity of transversely isotropic materials with thermal and other effects. These solutions are obtained by simply but forcedly combining the heat conduction equation with other governing equations (e.g., the Navier equations). We will show that the Williams solution as well as some other solutions all can be deduced as the special cases. Two application scenarios of the general solutions are also highlighted to demonstrate their elegance and versatility.

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.

Research of Construction Elements of Structure-inhomogeneous Materials

The paper concentrates on strain-stress state research in construction elements of structurally heterogeneous materials which is of current importance. Classical concepts of solid, homogeneous, isotropic, linearly elastic body do not suit the construction practice, as nearly all materials used in construction and technics are structurally heterogeneous. Present article deals with the most typical structurally heterogeneous materials – polycrystalline materials. Finite element model of structurally heterogeneous body – polycrystal is developed. The problem of determination of body minimum volume, which could be endued with averaged properties by averaging elastic properties, is solved. It allows analyzing structurally heterogeneous body at different volumes. Stress concentration is investigated for small-scale plates with circular or ellipse holes and with various stress states considering microstructural aspects of stress concentration. It is shown, that coefficients of stress concentration depending on anisotropy of elastic properties can have values, which considerably differ from values for isotropic body.

Finite element approximations for near‐incompressible and near‐inextensible transversely isotropic bodies

SummaryThis work comprises a detailed theoretical and computational study of the boundary value problem for transversely isotropic linear elastic bodies. General conditions for well‐posedness are derived in terms of the material parameters. The discrete form of the displacement problem is formulated for conforming finite element approximations. The error estimate reveals that anisotropy can play a role in minimising or even eliminating locking behaviour for moderate values of the ratio of Young's moduli in the fibre and transverse directions. In addition to the standard conforming approximation, an alternative formulation, involving under‐integration of the volumetric and extensional terms in the weak formulation, is considered. The latter is equivalent to either a mixed or a perturbed Lagrangian formulation, analogously to the well‐known situation for the volumetric term. A set of numerical examples confirms the locking‐free behaviour in the near‐incompressible limit of the standard formulation with moderate anisotropy, with locking behaviour being clearly evident in the case of near‐inextensibility. On the other hand, under‐integration of the extensional term leads to extensional locking‐free behaviour, with convergence at superlinear rates.

Diffraction of a Plane Sound Wave on a Thermoelastic Sphere with a Discretely Inhomogeneous Coating

An analytical solution of the problem of the diffraction of a plane monochromatic sound wave on a sphere with a coating of several spherical layers was obtained using the equations of the linear coupled dynamic problem of thermoelasticity of a homogeneous isotropic body. This paper presents the results of calculations of the frequency and angular characteristics of the scattered acoustic field amplitude for a sphere with a multilayer coating and a coating with inhomogeneity continuous across the thickness. It is shown that a continuously inhomogeneous thermoelastic coating can be modeled by a system of homogeneous thermoelastic layers. The effect of the thermoelasticity of the materials of the sphere and its discretely inhomogeneous coating on sound scattering was investigated.

Assessing the mechanical responses for anisotropic multi-layered medium under harmonic moving load by Spectral Element Method (SEM)

• An analytical solution for analyzing the mechanical behavior of multi-layered medium under moving load is proposed. • A more realistic harmonic moving load is applied in the multi-layered medium model. • The impact of the anisotropic property changes with the variety of load properties. • The impact of interlayer condition and top layer thickness are investigated. The article is concerned with the mechanical responses of anisotropic multi-layered medium under harmonic moving load. An analytical solution for two-dimensional anisotropic multi-layered medium subjected to harmonic moving load is devoted via Spectral Element Method (SEM), while the anisotropic property is approximated as transverse isotropy. Starting with the constitutive equations of transversely isotropic body and the governing equations of motion based on the loading properties. The analytical spectral elements in the wavenumber domain are obtained according to the principle of wave superposition and Fourier transformation. Then, the spectral global stiffness matrix of the multi-layered medium is derived by assembling the nodded stiffness matrices of all layers depended on the different interlayer conditions between the adjacent layers, i.e. sliding and bonded. The corresponding analytical solutions are achieved by taking the Fourier series and Inverse Fast Fourier Transform (IFFT) algorithm. Finally, some examples are given to validate the accuracy of the proposed analytical solution, and to demonstrate the impact of both anisotropy, top layer thickness, interlayer conditions, and loading properties (velocity and natural frequency) on the mechanical response of the multi-layered medium.

Analytical solutions of some internal boundary value problems of elasticity for domains with hyperbolic boundaries

An analytical solution of two-dimensional problems of elasticity in the region bounded by a hyperbola in elliptic coordinates is constructed using the method of separation of variables. The stress–strain state of a homogenous isotropic hyperbolic body and that with a hyperbolic cut is studied when there are non-homogenous (non-zero) boundary conditions given on the hyperbolic boundary. The graphs for the numerical results of some test problems are presented.

Local uniqueness of the density from partial boundary data for isotropic elastodynamics

We consider an inverse problem in elastodynamics arising in seismic imaging. We prove locally uniqueness of the density of a non-homogeneous, isotropic elastic body from measurements taken on a part of the boundary. We measure the Dirichlet to Neumann map, only on a part of the boundary, corresponding to the isotropic elasticity equation of a 3D object. In earlier works it has been shown that one can determine the sheer and compressional speeds on a neighborhood of the part of the boundary (accessible part) where the measurements have been taken. In this article we show that one can determine the density of the medium as well, on a neighborhood of the accessible part of the boundary.

Uniformly moving screw dislocation in strain gradient elasticity

In the present paper, Mindlin's strain gradient theory of elasticity (form II) is utilized to determine the deformation and strain fields due to a straight screw dislocation that moves uniformly at a subsonic velocity in an infinite isotropic body. The theory employed herein, in addition to the effect of strain gradient, is capable of accounting for that of acceleration gradient through the incorporation of the micro-inertia term into the analysis of the problem. Integral representations are obtained for the displacement and strain fields, and it is shown that the utilization of the strain gradient theory for the continuum description of a moving screw dislocation leads to discarding the discontinuity of the induced displacement field, consequently resulting in the removal of the classical singularities of the associated strain and stress fields.

Stress State Near a Small-Scale Crack at the Corner Point of the Interface of Media

The stress state of a piecewise-homogeneous isotropic elastic body near a small-scale mode I crack at the corner point of the interface of media is analyzed. The exact solution of the corresponding elasticity problem is obtained using the Wiener–Hopf method.

The curse of dimensionality for numerical integration on general domains

We prove the curse of dimensionality in the worst case setting for multivariate numerical integration for various classes of smooth functions. We prove the results when the domains are isotropic convex bodies with small diameter satisfying a universal ψ2-estimate. In particular, we obtain the result for the important class of volume-normalized ℓpd-balls in the complete regime 2≤p≤∞. This extends a result in a work of Hinrichs et al. (2014) to the whole range 2≤p≤∞, and additionally provides a unified approach. The key ingredient in the proof is a deep result from the theory of Asymptotic Geometric Analysis, the thin-shell volume concentration estimate due to O. Guédon and E. Milman. The connection of Asymptotic Geometric Analysis and Information-based Complexity revealed in this work seems promising and is of independent interest.

Spherical inclusion with time-harmonic eigenfields in strain gradient elasticity considering the effect of micro inertia

In the present paper, the complete form of Toupin–Mindlin strain gradient theory of elasticity is utilized to determine the elastic field of a spherical inclusion undergoing a time-harmonic eigenfield in an infinite isotropic body. The theory adopted herein involves three characteristic lengths, two ones of which are present therein due to the incorporation of strain gradients in the strain energy density function of the material under consideration, and the presence of the third one arises from taking into account the role of velocity gradients in its kinetic energy density function, reflecting the effect of micro inertia. After the derivation of an expression for the associated Green’s function of the problem, closed-form solutions are obtained for the interior and exterior elastic fields of a spherical inclusion subjected to a time-harmonic eigenfield whose spatial distribution is expressible as a function of the radial distance of points from the center of the inclusion.

Local tangential contact of elastically similar, transversely isotropic elastic bodies

Local tangential contact of elastically similar, transversely isotropic elastic bodies

The mathematic model of and method for solving the Neumann generalized heat-exchange problem for empty isotropic rotary body

The mathematic model of and method for solving the Neumann generalized heat-exchange problem for empty isotropic rotary body

Modeling of Heat-and-Mass Transfer in Fuel Assemblies in Liquid-Metal-Cooled Reactors with Partial Flow-Passage Blockage

The results of modeling the heat-and-mass transfer in the FA of liquid metal cooled reactors with partial flow-passage blockage are presented. The model employed for an isotropic porous body is described, and the obtained results are compared with experimental data and with direct numerical modeling performed with the CONV-3D code. It is shown that the porous-body model developed can be used to analyze threedimensional heat-and-mass transfer processes arising when the flow passage area of FA is partially blocked.

On one method of solving the three-dimensional problem of physically non-linear deformation of transversal-isotropic multi-variable bodies

The method for solving the three-dimensional problem of elastic-plastic deformation of transversely isotropic multiply connected bodies by the finite element method (FEM) is presented in the article. The process of solving the problem consists of: determining the effective parameters of a transversely isotropic medium; construction of the finite element mesh of the body configuration, including the determination by the front method of the local minimum value of the width of the tape of non-zero coefficients of equation systems; constructing the coefficients of the stiffness matrix and the components of the node load vector of the equation of state of an individual finite element according to the theory of small elastic-plastic deformations for a transversely isotropic medium; the formation of a resolving symmetric-tape system of equations by summing the coefficients of the equations of state of all finite elements; solution of the system of symmetric-tape system of equations by means of the square root method; calculation of the elastic-plastic stress-strain state of the body by performing the iterative process of the method of elastic solutions AA Ilyushin. For each stage of solving the problem, effective computational algorithms have been developed that make it possible to reduce the number of computational operations by modifying existing methods of solving and taking into account the structure of the matrix coefficients. As an example, the solution of the problem of deformation of a transversally isotropic body in the form of a rectangle with a circular notch in the center is given.

Distortion-gradient plasticity theory for an isotropic body in finite deformation

This study presents a distortion-gradient model for an isotropic plastically deformed solid within the framework of finite deformation theory. The work aims to provide an alternative form of the Gurtin and Anand (Int J Plast 21:2297–2318, 2005) finite deformation strain-gradient model with a view to relaxing the constraint of irrotationality of plastic flow. The kinematic gradient of deformation is assumed to admit the Kroner–Lee decomposition into elastic and plastic parts. The obtained microforce balance, constitutive relations and plastic flow rule are similar to that obtained by Gurtin and Anand but different in that the present theory used a codirectionality hypothesis to obtain thermodynamically consistent constitutive relations for the dissipative microstresses.

Memory response on thermal wave propagation emanating from a cavity in an unbounded elastic solid

The present paper deals with the thermal memory response of wave propagation in an unbounded, homogeneous, isotropic elastic body, emanating from a spherical cavity. To analyze the memory response, the generalized heat conduction model with the fractional order as well as memory-dependent-derivatives (MDDs) concepts are considered. The solution space is obtained in Laplace transform domain by using the eigenfunction expansion method to the vector-matrix form of the corresponding governing equations. Finally, a comparison study is furnished for thermal displacement, stresses, and temperature changes in the space-time domain and is presented graphically.

Universal spherically symmetric solution of nonlinear dislocation theory for incompressible isotropic elastic medium

The equilibrium problem of a nonlinearly elastic medium with a given dislocation distribution is considered. The system of equations consists of the equilibrium equations for the stresses, the incompatibility equations for the distortion tensor, and the constitutive equations. Deformations are considered to be finite. For a special distribution of screw and edge dislocations, an exact spherically symmetric solution of these equations is found. This solution is universal in the class of isotropic incompressible elastic bodies. With the help of the obtained solution, the eigenstresses in a solid elastic sphere and in an infinite space with a spherical cavity are determined. The interaction of dislocations with an external hydrostatic load was also investigated. We have found the dislocation distribution that causes the spherically symmetric quasi-solid state of an elastic body, which is characterized by zero stresses and a nonuniform elementary volumes rotation field.

The Mathematic Model and Method for Solving the Dirichlet Heat-Exchange Problem for Empty Isotropic Rotary Body

It is the first generalized 3D mathematic model, which is created for calculating temperature fields in the empty isotropic rotary body, which is restricted by end surfaces and lateral surface of rotation and rotates with constant angular velocity around the axis OZ, with taking into account finite velocity of the heat conductivity in the form of the Dirichlet problem. In this work, an integral transformation was formulated for the 2D finite space, with the help of which a temperature field in the empty isotropic rotary body was determined in the form of convergence series by the Fourier functions.