Anisotropic Elastic Body Research Articles

Analytical poro‐elastic solution of deep lined tunnels in anisotropic rock with consideration of tunnel face advance

AbstractThis paper aims at deriving a closed‐form solution for deep lined tunnels within a saturated anisotropic poro‐elastic medium. The derivation of the solution is carried out with the assumption of plane strain conditions along the tunnel axis, an isotropic elastic liner, a steady‐state flow, and perfect contact between the liner and rock. For this purpose, the complex potential function approach pioneered by Lekhnitskii for the anisotropic elastic body was used to develop the mechanical response of the rock mass. A complex hydraulic potential function is also introduced to solve the steady state fluid flow. Then the well‐known Biot theory is chosen to describe the hydro‐mechanical coupling of the anisotropic saturated rock. The stress and displacement solutions of both the liner and the surrounding rock, considering the tunnel face advance with respect to the considered section, are derived based on the principle of the convergence‐confining method. Comparisons with previous closed‐form solutions for isotropic case and finite element solution for anisotropic case were made to show the accuracy of the current closed form solution. Sensitivity analysis is performed based on this analytical solution to highlight the effect of some rock parameters on the stress and displacement of the liner.

Adhesive contact between two-dimensional anisotropic elastic bodies

Adhesion plays a vital role in the design of smart and intelligent high-tech devices such as modern optical, microelectromechanical, and biomedical systems. However, in the literature, adhesive contact is mostly considered for contact of rigid substrates and transversely isotropic and isotropic elastic materials. The composite materials are increasingly used in the mart and intelligent high-tech devices. Since the composite materials are generally anisotropic and contact bodies are all deformable, it is more practical to consider the adhesive contact of two anisotropic elastic materials. In this paper, an adhesive contact model of anisotropic elastic bodies is established, and the closed-form solutions for two-dimensional adhesive contact of two anisotropic elastic bodies are derived. The full-field solutions and the relation for the contact region and applied force are developed using the Stroh complex variable formalism, the analytical continuation method, and concepts of the JKR adhesive model. We will show that the frictionless contact of two anisotropic elastic materials is just a special case of the present contact problem, and its solutions can be obtained by setting the work of adhesion equal to zero. In addition, we also show that our present solutions are valid for the problems of indentation by a rigid punch on an elastic half-space through a proper placement of the contact radius and the corresponding material constant. Numerical results are provided to demonstrate the accuracy, applicability, and versatility of the developed solutions.

Mathematical problems of dynamical interaction of fluids and multiferroic solids

In the paper, we consider a three-dimensional mathematical problem of fluid-solid dynamical interaction, when an anisotropic elastic body occupying a bounded region is immersed in an inviscid fluid occupying an unbounded domain . In the solid region, we consider the generalized Green–Lindsay's model of the thermo-electro-magneto-elasticity theory. In this case, in the domain we have a six-dimensional thermo-electro-magneto-elastic field (the displacement vector with three components, electric potential, magnetic potential, and temperature distribution function), while we have a scalar acoustic pressure field in the unbounded domain . The physical kinematic and dynamical relations are described mathematically by the appropriate initial and boundary-transmission conditions. Using the Laplace transform, the dynamical interaction problem is reduced to the corresponding boundary-transmission problem for elliptic pseudo-oscillation equations containing a complex parameter τ. We derive the appropriate norm estimates with respect to the complex parameter τ and construct the solution of the original dynamical problem by the inverse Laplace transform. As a result, we prove the uniqueness, existence, and regularity theorems for the dynamical interaction problem. Actually, the present investigation is a continuation of the paper [Chkadua G, Natroshvili D. Mathematical aspects of fluid-multiferroic solid interaction problems. Math Meth Appl Sci. 2021;44(12):9727–9745], where the fluid-solid interaction problems for elliptic pseudo-oscillation equations associated with the above mentioned generalized thermo-electro-magneto-elasticity theory are studied by the potential method and the theory of pseudodifferential equations.

Stress Distribution Near Sharp and Rounded V-Notches In Anisotropic Elastic Body Under Antiplane Strain

Stress Distribution Near Sharp and Rounded V-Notches In Anisotropic Elastic Body Under Antiplane Strain

Multibody contact of two-dimensional anisotropic elastic/piezoelectric/magneto-electro-elastic solids

A unified expression for the generalized two-dimensional linear anisotropic elastic, piezoelectric, and magneto-electro-elastic (MEE) solids is established to simplify the connection between these three different kinds of materials. Here, the two-dimensional assumption includes the most general cases that in-plane and anti-plane deformations are allowed to couple each other. Based upon this unified expression, in this paper a recently developed boundary element method (BEM) for contact of two anisotropic elastic bodies is extended to solve the problems of multibody contact in which the bodies can be any combination of anisotropic elastic, piezoelectric and MEE solids. When a special case of contact between multiple rigid punches and an anisotropic elastic, piezoelectric or MEE half-plane is considered, further reduction can be made by introducing three degrees of freedom (two rigid body translations and one rigid body rotation) for each punch. With this consideration, no meshes are required on the rigid punches. The usual contact modes of separation, stick, and slip all include the situation of conductor and insulator if any one of the multibody is piezoelectric or MEE. In other words, in this paper we successfully develop two BEMs to solve the generalized two-dimensional problems of multibody contact. One is for the general cases whose bodies can be any combination of anisotropic elastic, piezoelectric and MEE solids, and the other is for the special cases of multiple rigid punches indenting on an anisotropic elastic, piezoelectric or MEE half-plane.

Анизотропные твердые цилиндрические волноводы

Object and purpose of research. The article studies the behavior of anisotropic elastic bodies of cylindrical shape (orthotropic shell and transversely isotropic rod). The intention is to find the phase velocities of elastic waves in these bodies using thin shell approximations and a rigorous approach based on the dynamic theory of elasticity using "Debye type" potentials. In previous studies of anisotropic structures, anisotropic media or anisotropic half-spaces were used. Materials and methods. Both an approximate thin shell method and a rigorous approach based on the dynamic theory of elasticity and "Debye type" potentials are used in the paper. Main results. Equations for finding the phase velocities of elastic waves in anisotropic cylindrical bodies are obtained. Phase velocities of longitudinal and bending waves in anisotropic cylindrical rod are calculated. Conclusion. As a result of the conducted research relations for calculating the phase velocities of elastic waves in an orthotropic shell and a transversely isotropic rod were found.

On the response of anisotropic elastic bodies described by implicit constitutive relations

AbstractIn this short note we derive constitutive relations for the response of elastic solids, which are transversely isotropic and those that have two preferred directions of symmetry wherein the material moduli of the body depend on its density, undergoing small deformations. Such constitutive relations have relevance to the response of rocks, concrete, ceramics, bones, and porous metals. The constitutive relations that we develop reduce appropriately to the classical linearized elastic constitutive relation, when appropriate material moduli are set to zero. The new constitutive relations, when restricted to those that are linear in both the stress and the linearized strain, have terms in addition to those for a classical linearized elastic body, and it would be interesting to investigate the consequence of these additional terms with regard to the response of the body.

Optimal Control of Parameters for Elastic Body with Thin Inclusions

In this paper, an equilibrium problem for 2D non-homogeneous anisotropic elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. A connection between the inclusions at a given point is characterized by a junction stiffness parameter. The elastic inclusion is delaminated, thus forming an interfacial crack with the matrix. Inequality-type boundary conditions are imposed at the crack faces to prevent interpenetration. Existence of solutions is proved; different equivalent formulations of the problem are discussed; junction conditions at the connection point are found. A convergence of solutions as the junction stiffness parameter tends to zero and to infinity as well as the rigidity parameter of the elastic inclusion tends to infinity is investigated. An analysis of limit models is provided. An optimal control problem is analyzed with the cost functional equal to the derivative of the energy functional with respect to the crack length. A solution existence of an inverse problem for finding the junction stiffness and rigidity parameters is proved.

Nonlinear and Linear Elastodynamic Transformation Cloaking

We formulate the problems of nonlinear and linear elastodynamics transformation cloaking in a geometric framework. It is noted that a cloaking transformation is neither a spatial nor a referential change of frame (coordinates); a cloaking transformation maps the boundary-value problem of an isotropic and homogeneous elastic body (virtual problem) to that of an anisotropic and inhomogeneous elastic body with a hole surrounded by a cloak that is to be designed (physical problem). The virtual body has a desired mechanical response while the physical body is designed to mimic the same response outside the cloak using a cloaking transformation. We show that nonlinear elastodynamics transformation cloaking is not possible while nonlinear elastostatics cloaking may be possible for special deformations, e.g., radial deformations in a body with either a cylindrical or a spherical cavity. For linear elastodynamics, in line with the previous observations in the literature, we show that the elastic constants in the cloak are not fully symmetric; they do not possess the minor symmetries. We prove that transformation cloaking is not possible regardless of the shape of the hole and the cloak. We next show that linear elastodynamics transformation cloaking cannot be achieved for gradient elastic solids either; similar to classical linear elasticity the balance of angular momentum is the obstruction to transformation cloaking. We finally prove that transformation cloaking is not possible for linear elastic generalized Cosserat solids in 2D for any shape of the hole and the cloak. Particularly, in 2D, transformation cloaking cannot be achieved in linear Cosserat elasticity. We also show that transformation cloaking for a spherical cavity covered by a spherical cloak is not possible in the setting of linear elastic generalized Cosserat solids. We conjecture that this result holds for a cavity of any shape.

A three-dimensional analytical model and parameter determination method of the elastic vibration of a railway vehicle carbody

ABSTRACT Reducing railway vehicles’ three-dimensional (3D) elastic vibration is needed to improve passenger comfort. This paper presents a 3D analytical model and parameter determination method for the elastic vibration of a carbody. The carbody is modelled as a simple box-type structure consisting of an anisotropic 3D elastic body treated as a continuous system. This model can express complicated 3D elastic vibration (such as lateral–vertical coupled vibration) with fewer degrees of freedom and at a lower computational cost than the detailed finite element model. A model parameter determination method using particle swarm optimisation was also proposed. Application to a Shinkansen-type test vehicle showed good agreement between measured and calculated results with a maximum natural frequency differences of the targeted six elastic vibration modes within 0.86%, indicating the effectiveness of the proposed model and parameter determination method.

Analysis of Anisotropic Elastic Body with Three Elliptical Cavities andaCalculation of Stress Intensity Factors

Analysis of Anisotropic Elastic Body with Three Elliptical Cavities andaCalculation of Stress Intensity Factors

Study on contact problem of anisotropic elastic body based on Eshelby-Stroh formalism

Study on contact problem of anisotropic elastic body based on Eshelby-Stroh formalism

Homogenization of Winkler–Steklov spectral conditions in three-dimensional linear elasticity

We consider a homogenization Winkler–Steklov spectral problem that consists of the elasticity equations for a three-dimensional homogeneous anisotropic elastic body which has a plane part of the surface subject to alternating boundary conditions on small regions periodically placed along the plane. These conditions are of the Dirichlet type and of the Winkler–Steklov type, the latter containing the spectral parameter. The rest of the boundary of the body is fixed, and the period and size of the regions, where the spectral parameter arises, are of order $$\varepsilon $$ . For fixed $$\varepsilon $$ , the problem has a discrete spectrum, and we address the asymptotic behavior of the eigenvalues $$\{\beta _k^\varepsilon \}_{k=1}^{\infty }$$ as $$\varepsilon \rightarrow 0$$ . We show that $$\beta _k^\varepsilon =O(\varepsilon ^{-1})$$ for each fixed k, and we observe a common limit point for all the rescaled eigenvalues $$\varepsilon \beta _k^\varepsilon $$ while we make it evident that, although the periodicity of the structure only affects the boundary conditions, a band-gap structure of the spectrum is inherited asymptotically. Also, we provide the asymptotic behavior for certain “groups” of eigenmodes.

Direct and inverse elastic scattering from anisotropic media

Assume that a time-harmonic elastic wave is incident onto a penetrable anisotropic elastic body embedded in a homogeneous isotropic background medium. The scattering problem is reduced to a truncated domain. Uniqueness and existence of weak solutions are proved by applying the Fredholm alternative and using properties of the Dirichlet-to-Neumann map in both two and three dimensions. The Fréchet derivative of the near-field solution operator with respect to the boundary of the scatterer is derived. As an application, a descent algorithm is designed for recovering the interface from the near-field data of one or several incident directions and frequencies. Numerical examples in 2D are presented to show the validity and accuracy of the algorithm.

Analysis for Anisotropic Elastic Body with Two Elliptical Holes Subjected to In-Plane and Out-of-Plane Shear Loadings

Analysis for Anisotropic Elastic Body with Two Elliptical Holes Subjected to In-Plane and Out-of-Plane Shear Loadings

An initial boundary value problem for modeling a piezoelectric dipolar body

This study deals with the first initial boundary value problem in elasticity of piezoelectric dipolar bodies. We consider the most general case of an anisotropic and inhomogeneous elastic body having a dipolar structure. For two different types of restrictions imposed on the problem data, we prove two results regarding the uniqueness of solution, by using a different but accessible method. Then, the mixed problem is transformed in a temporally evolutionary equation on a Hilbert space, conveniently constructed based on the problem data. With the help of a known result from the theory of semigroups of operators, the existence and uniqueness of the weak solution for this equation are proved.

A Three-Dimensional BEM for Dynamic Analysis of Anisotropic Elastic Multi-Connected Bodies

In this paper, the direct boundary element method in the Laplace domain is applied for the solution of three-dimensional transient dynamic problems of anisotropic elasticity in multi-connected domains. The formulation is based upon the integral representations of anisotropic dynamic fundamental solutions. As numerical example the problem of an anisotropic elastic prismatic solid with cubic cavity is investigated.

Green’s-function-based-finite element analysis of fully plane anisotropic elastic bodies

In the paper, an anisotropic Green’s function based hybrid finite element was developed for solving fully plane anisotropic elastic materials. In the present hybrid element, the interior displacement and stress fields were approximated by the linear combination of anisotropic Green’s functions derived by Lekhnitskii formulation, the element frame fields were constructed by the interpolation of general shape functions widely used in the conventional finite element, and then they are linked by a new double-variable hybrid functional. Because the approximated interior fields exactly satisfied the governing equations related to anisotropic elasticity, all integrals in the present hybrid functional were performed along the element boundary and theoretically arbitrary hybrid polygonal element can be constructed. Finally, the present hybrid polygonal element with four edges was verified by making comparison of numerical results and exact solutions in a cantilever composite beam made with angled lamina.

Energy release rates for interfacial cracks in elastic bodies with thin semirigid inclusions

In this paper, we present some rigorous results for an equilibrium problem arising from the study of fiber-reinforced composites. We consider a two-dimensional homogeneous anisotropic linear elastic body containing a thin semirigid inclusion. The semirigid inclusion is an anisotropic thin structure that stretches along one direction and moves like a rigid body possessing both rotational and translatory motion along the perpendicular direction. A pre-existing interfacial crack is subject to nonlinear conditions that do not allow the opposite crack faces to penetrate each other. We focus on a variational approach to modelling the physical phenomenon of equilibrium and to demonstrate that the energy release rate associated with perturbation of the crack along the interface is well defined. A higher regularity result for the displacement field is formulated and proved. Then, taking into account this result, we deduce representations for the energy release rates associated with local translation and self-similar expansion of the crack by means of path-independent energy integrals along smooth contour surrounding one or both crack tips. Finally, some relations between the integrals obtained are discussed briefly.

Numerical study of composite bulkhead partition strength with in-situ X-ray monitoring

The aim of this work is the stress-strain analysis of bulkhead partition made of carbon fiber reinforced plastic (CFRP) at mechanical testing. The problem was solved in a general 3-D statement for an anisotropic elastic body. The strength prediction of the construction was done by stress analysis in each ply of composite laminate. The numerical simulation was obtained with ANSYS Workbench software. The detailed ply-by-ply stress analysis with special attention to areas of plies curvature was done. The results of numerical simulation were compared with the test data for verification of the proposed numerical algorithm.