Circular Inclusion Research Articles

The problem of stationary thermoelasticity for a piecewise homogeneous transversely isotropic space under the influence of a heat flux specified at infinity is considered

An exact solution to the problem of thermoelasticity about interfacial circular inclusion, which is in complete adhesion with different transversely isotropic half-spaces, is constructed. Analytical dependencies are obtained between the parameters of the linear displacement of the inclusion and the characteristics of the force and temperature fields. The order of the singularity of surges and displacements near the inclusion is determined. Expressions are obtained for the stress intensity factor at the inclusion boundary, as well as the numerical dependences of these coefficients on the polar angle, temperature, and load.

A three-dimensional model for rain-wind induced vibration of stay cables in cable-stayed bridges

The most rigorous effective medium approximations for elastic moduli are elaborated for matrix composites made from an isotropic continuous matrix and isotropic inclusions associated with simple shapes such as circles or spheres. In this paper, we focus specially on the effective elastic moduli of the heterogeneous composites with arbitrary inclusion shapes. The main idea of this paper is to replace those inhomogeneities by simple equivalent circular (spherical) isotropic inclusions with modified elastic moduli. Available simple approximations for the equivalent circular (spherical) inclusion media then can be used to estimate the effective properties of the original medium. The data driven technique is employed to estimate the properties of equivalent inclusions and the Extended Finite Element Method is introduced to modeling complex inclusion shapes. Robustness of the proposed approach is demonstrated through numerical examples with arbitrary inclusion shapes.
 Keywords:
 data driven approach; equivalent inclusion, effective elastic moduli; heterogeneous media; artificial neural network.

Two-dimensional Stokes flows in porous medium composed of a large number of circular inclusions

A new approximate approach to solve the two-dimensional boundary-value problems for the Stokes flows in a porous medium composed of a set of circular inclusions is proposed. The idea of the approach, based on the boundary element method (BEM), lies in Fourier series representations of unknown functions on the surfaces of inclusions. It is shown that for small volume concentrations of inclusions, a good accuracy is achieved by taking into account only the first three terms in the Fourier decomposition. Simultaneously, the total number of unknowns, which is equal to the product of the number of inclusions and that of unknowns on every inclusion, is reduced dramatically in comparison with the traditional BEM. The approach is tested by investigating the flow around a circular cylinder in a rectangular periodic cell, the flow in a porous medium with a large number of uniformly or randomly located inclusions. To demonstrate the capacity of the approach, we have calculated the Stokes flows in the porous medium which is modelled by locating 5000 inclusions of two different radii. It is shown that the method can also be used to estimate the properties of the fluid flows in highly porous media formed by non-circular inclusions.

Guiding and Trapping Cracks With Compliant Inclusions for Enhancing Toughness of Brittle Composite Materials

Abstract The problem of toughening heterogeneous materials with a stiff matrix and compliant inclusions is investigated through numerical simulations and experiments. Specifically, the problem of optimizing a combination of effective toughness and effective elastic modulus in the context of a square array of compliant inclusions in a stiff matrix is explored. Crack propagation in the heterogeneous material is simulated using a variational phase-field approach. It is found that the crack can meander between or get attracted to and trapped in the inclusions. Composite specimens with a stiff matrix and compliant circular inclusions were 3D printed, and their fracture toughness was measured using a specially designed loading fixture. The experimental results show agreement with the numerical predictions by demonstrating the attraction and trapping of cracks in the inclusions. This study demonstrates the potential for significant enhancement of toughness through elastic compliance contrast between the matrix and the inclusion without notably compromising the effective elastic modulus of the composite material.

EXACT SOLUTION OF CIRCULAR INCLUSION PROBLEMS BY A BOUNDARY INTEGRAL METHOD

As an excellent numerical method, boundary element method (BEM) has been widely applied in various scientific and engineering problems. In this paper, a new boundary integral method is obtained based on Somigliana's equation and the properties of Green's function by referring to the idea of boundary element method. It can be used to find the analytic solution of linear elastic problems. The boundary integral method can also be obtained from Betti's reciprocity theorem. By using this new method, the classical problem of elastic circular inclusion under a uniform tensile field at infinity is solved. Firstly, the perfect bonding between inclusion and matrix is set up, and the displacement and stress at interface are expanded according to Fourier series. According to the symmetry of the problem and the orthogonality of trigonometric function, the hypothesis is simplified and the number of undetermined coefficients is reduced. Secondly, the appropriate trial functions are selected (these trial functions satisfy the condition of displacement single value and the control equation of linear elasticity without body force). And the boundary integral method is used to calculate the displacement and stress at the interface. Then the displacement and stress in the domain are solved using similar tricks. The exact analytical solution of the problem is obtained, which is exactly the same with the results in literatures. When the elastic modulus of the inclusion is zero or tends to infinity, it degenerates to the analytical solution of the problem of circular hole or rigid inclusion. The solution process shows that if the problem has boundary conditions at infinity, trial functions should meet the boundary condition at infinity. If the domain of the problem contains the coordinate origin, the displacement and stress of trial functions at the origin should be limited. The results show that the method is effective.

Strength characteristics and fracture evolution of rock with different shapes inclusions based on particle flow code

Natural rock mass contains defects of different shapes, usually filled with inclusions such as clay or gravel. The presence of inclusions affects the failure characteristics and mechanical properties of rock mass. In this study, the strength and failure characteristics of rock with inclusions were studied using the particle flow code under uniaxial compression. The results show that the presence of inclusions not only improves the mechanical properties of rock with defects but also increases the bearing capacity of rock. Circular inclusion has the most obvious effect on improving model strength. The inclusions affect the stress distribution, development of initial crack, change in crack propagation characteristics, and failure mode of rock. In defect models, concentration area of the maximum tensile stress is generated at the top and bottom of defect, and the maximum compressive stress is distributed on the left and right sides of defect. In filled models, the tensile stress and compressive stress are uniformly distributed. Failing mode of defect models is mainly tensile failure, while that of filled models is mainly shear failure.

Interactions of Anisotropic Inclusions on a Fluid Membrane

Biological cells and membranes need to be properly shaped to fulfill many fundamental functions. This shaping is often aided by the aggregation of membrane-bound proteins that both sense the membrane curvature and shape it. Therefore, these protein inclusions interact with each other through the deformation of the membrane that they influence, and a key question is to understand the law that governs their interaction. Whereas the theoretical case of isotropic proteins is well understood, an important feature of many such proteins is their anisotropic interaction with the membrane. Here, we derive an interaction law for rigid circular membrane inclusions which impose an anisotropic contact angle on the surrounding membrane. We include the effects of both membrane bending and tension. Using asymptotic analysis, we identify two distinguished limits corresponding to weak anisotropy/weak tension and strong anisotropy/strong tension, respectively. The resulting laws exhibit a bistability in the equilibrium separation of inclusions. Inclusions with very weak anisotropy equilibrate with large separation, those with very strong anisotropy equilibrate with small separation, while there is a range of anisotropies for which both equilibria are stable. Our results provide a theoretical mechanism for the global aggregation of inclusions seen both in experiments and simulations.

Mathematical modeling of elastic state in a three-component plate containing a crack due to the action of unidirectional tension

Purpose. A two-dimensional mathematical model for the problem of elasticity theory in a three-component plate containing rectilinear crack due to the action of mechanical efforts is examined. As a consequence, the intensity of stresses in the vicinity of tops of the crack increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure. Such a model represents to some extent a mechanism of destruction of the elements of engineering structures with cracks, we determined stress intensity factors (SIFs) at the tops of the crack, which are subsequently used to determine critical values of the tension. Therefore, the aim of present work is to determine the two-dimensional elastic state in plate containing an elastic two-component circular inclusion and crack under conditions of power load in the case of unidirectional tension of the plate perpendicular for the crack line. This makes it possible to determine the critical values of unidirectional tension in order to prevent crack growth, which will not allow the local destruction of the body. Methodology. The methods of studying two-dimensional elastic state body with crack as stress concentrators based on the function of complex variable method by which the problem of elasticity theory is reduced to singular integral equations (SIE) of the first and second kind, the numerical solution by the method of mechanical quadratures was obtained. Findings. In this paper two-dimensional mathematical model in the form of the system of two singular integral equations on closed contour (boundary of inclusion) and unclosed contour (crack) are obtained; numerical solutions of these integral equations were received by the method of mechanical quadratures; stress intensity factors at the tops of a crack are identify and explored to detect the effects of mechanical character. Graphical dependencies of SIFs, which characterize distribution of the intensity of stresses at the tops of a crack as function of elastic properties of inclusion and also as function of the distance between crack and inclusion are obtained. This makes it possible to analyze the intensity of stresses in the vicinity of a crack's tops depending on the geometrical and mechanical factors, as well as to determine the limit of permissible values of unidirectional tension of the plate perpendicular to the crack line at which the crack begins to grow and the body being locally destroyed. It is shown that the proper selection of elastic characteristics of the components of three-component plate can help achieve an improvement in the strength of the body in terms of the mechanics of destruction by reducing SIFs at the crack's tops. Originality. Scientific novelty lies in the fact that the solutions of the new two-dimensional problems of elasticity for a specified region (plate containing an elastic two- component circular inclusion and a rectilinear crack) under the action of unidirectional tension of the plate perpendicular to the crack line are obtained. Practical value. Practical value of the present work lies in the possibility of a more complete accounting of actual stressed-strained state in the piecewise-homogeneous elements of a structure with cracks that work under conditions of different mechanical loads. The results of specific studies that are given in the form of graphs could be used when designing rational operational modes of structural elements. In this case, the possibility for preventing the growth of a crack through the appropriate selection of composite's components with the corresponding mechanical characteristics is obtained.

Fine elastic circular inclusion in the area of harmonic vibrations of an unlimited body under smooth contact

Fine elastic circular inclusion in the area of harmonic vibrations of an unlimited body under smooth contact The problem of the interaction of harmonic waves with a thin elastic circular inclusion, which is located in an elastic isotropic body (matrix), is solved. On both sides of the inclusion between it and the body (matrix), the conditions of smooth contact are realized. The solution method is based on representing the displacements in the matrix through discontinuous solutions of the Lamé equations for harmonic vibrations. This made it possible to reduce the problem to Fredholm integral equations of the second kind with respect to functions associated with jumps in normal stress and radial displacement to included ones. After the realization of the boundary conditions on the sides of the inclusion, a system of singular integral equations is obtained to determine these jumps. Keywords: elastic inclusions, cylindrical waves, matrix, stress intensity factor.

Own Weight Effect on Static Analysis of Pre-Stretched Plate-Strip Containing Twin Circular Inclusions Under Bending

Bu çalışma, eğilme etkisi altında içerisinde aynı malzemeden yapılmış ve merkezleri serbest yüzeye paralel bir doğrultuda olan ikiz dairesel şekilli dolgular içeren öngerilmeli bir kompozit şerit-plağın statik analizine kendi ağırlığının etkisini ele almaktadır. Dolgular içeren şerit-plak üzerindeki hacimsel kuvvetlerin (kendi ağırlığı) ve yüzeysel kuvvetlerin (öngerilme yüklemesi) etkisi birlikte öngerilmeler olarak düşünülmüştür. Bu ön​​gerilmelerinin, ek bir eğilme yükü altında bir şerit-plak içindeki dolgular etrafındaki gerilme dağılımlarının analizleri üzerindeki etkileri, düzlem şekil değiştirme durumu altında Lineerize Edilmiş Üç Boyutlu Elastisite Teorisi (LEÜBET) ile araştırılmaktadır. Ayrıca, ele alınan sınır değer problemlerinin çözümleri Sonlu Elemanlar Yöntemi kullanılarak gerçekleştirilmiştir. Sonuçlara göre, eğilme etkisi altındaki ikiz dairesel dolgular içeren şerit-plağın kendi ağırlığının plağın statik analizi üzerinde önemli bir etkisi olduğu ortaya çıkmıştır.

A circular Eshelby inclusion interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

Using conformal mapping techniques and analytic continuation, we prove that when subjected to anti-plane elastic deformations, a non-parabolic open inhomogeneity continues to admit an internal uniform stress field when a circular Eshelby inclusion is placed in its vicinity and the surrounding matrix is subjected to uniform remote stresses. Explicit expressions for the non-uniform stress distributions in the matrix and in the circular Eshelby inclusion are obtained. The internal uniform stress field is independent of the shape of the inhomogeneity and the presence of the circular Eshelby inclusion, whereas the existence of the circular Eshelby inclusion exerts a significant influence on the shape of the non-parabolic open inhomogeneity as well as on the non-uniform stress distributions in the matrix and in the circular Eshelby inclusion itself.

Analysis of a finite matrix with an inhomogeneous circular inclusion subjected to a non-uniform eigenstrain

The mechanical model of eigenstrains could not be always taken as uniform distributions in engineering applications when performing micromechanics analysis of the inclusion-matrix system. In the framework of plane strain, this paper presents the analytical solution to an inhomogeneous circular inclusion with a non-uniform eigenstrain concentrically embedded in a finite matrix. First, the equivalent eigenstrain equation is extended to satisfy the condition of the finite matrix through the equivalent eigenstrain principle. The modified equation is used to transform the inhomogeneous inclusion in a finite matrix into the corresponding homogeneous inclusion. Then, the model of the inhomogeneous circular inclusion is accordingly formulated, and the stress and strain distributions are found. Finally, the stresses for the case of the polynomial series distribution of eigenstrains are obtained. The effects of non-uniformity of eigenstrains, the material mismatch and the inclusion size on stress distributions are shown graphically. The results indicate the stiffer inclusion induces the larger stress under the specific eigenstrain distribution. The analytical solutions obtained here also help to predict failure and optimize the designs of composite structures.

Cloaking of a circular cylindrical elastic inclusion from antiplane elastic waves and resonance effects

Cloaking of a circular cylindrical elastic inclusion embedded in a homogeneous linear isotropic elastic medium from antiplane elastic waves is studied. The transformation or change-of-variables method is used to determine the material properties of the cloak and the homogenization theory of composites is used to construct a multilayered cloak consisting of many bi-material cells. The large system of algebraic equations associated with this problem is solved by using the concept of multiple scattering with wave expansion coefficient matrices. Numerical results for cloaking of an elastic inclusion and a rigid inclusion are compared with the case of a cavity. It is found that while the cloaking patterns for the three cases are similar, the major difference is that standing waves are generated in the elastic inclusion and the multilayered cloak cannot prevent the motion inside the elastic inclusion, even though the cloak seems nearly perfect. Waves can penetrate into and cause vibrations inside the elastic inclusion, where the amplitude of standing waves depend on the material properties of the inclusion but are very much reduced when compared to the case when there is no cloak. For a prescribed mass density, the displacements inside the elastic cylinder decrease as the shear modulus increases. Moreover, the cloaking of the elastic inclusion over a range of wavenumbers is also investigated. There is significant low frequency scattering even if the cloak consists of a large number of layers. When the wavenumber increases, the multilayered cloak is not effective if the cloak consists of an insufficient number of layers. Resonance effects that occur in cloaking of elastic inclusions are also discussed.

The implementation of B-splines to Hashin and Shtrikman variational principle based FFT method for the homogenization of composite

Fast Fourier transform (FFT) has been successfully used to estimate the effective properties of composites with meso periodic structure for more than two decades. Numerous improvements have been done to make it adequate for infinite contrast, bound elastic energy and composite voxels. The Hashin and Shtrikman variational principle based FFT method handles these three problems simultaneously and is very attractive. In this paper, B-splines were adopted to discretize the polarized field. The approximate polarized field was substituted into the Hashin and Shtrikman variational principle to obtain the discretized system. The constitutive matrix of a composite voxel was calculated by both the energetically consistent way and the laminate mixing rule. Numerical example of a square matrix with a circular inclusion shows that the results are more accurate than the remaining cases when the first and second order B-splines, as well as the laminate mixing constitutive matrix, are used. For higher order B-spline, the summation to compute the discrete Green operator for periodic boundary conditions converges faster while the discretized polarized field needs more iterations to converge. Only when the polarized field is sufficiently smooth, high order B-spline improves the numerical accuracy and efficiency.

‘Explicit’ and ‘implicit’ non-local continuous descriptions for a plate with circular inclusion in tension

Increasing application of composite structures in engineering field inherently speed up the studies focusing on the investigation of non-homogeneous bodies. Due to their capability on capturing the size effects, and offering solutions independent of spatial discretization, enriched non-classical continuum theories are often more preferable with respect to the classical ones. In the present study, the sample problem of a plate with a circular inclusion subjected to a uniform tensile stress is investigated in terms of both ‘implicit’/‘weak’ and ‘explicit’/‘strong’ non-local descriptions: Cosserat (micropolar) and Eringen theories, by employing the finite element method. The material parameters of ‘implicit’ model is assumed to be known, while the nonlocality of ‘explicit’ model is optimized according to stress concentration factors reported for infinite Cosserat plates. The advantages/disadvantages, and correspondence/non-correspondence between both non-local models are highlighted and discussed apparently for the first time, by comparing the stress field provided for reference benchmark problem under various scale ratios, and material parameter combinations for matrix-inclusion pair. The results reveal the analogous character of both non-local models in case of geometric singularities, which may pave the way for further studies considering problems with noticeable scale effects and load singularities.

Scattering of SH guided wave by a circular inclusion and semicircular depression in an infinite piezoelectric material strip

ABSTRACT By means of the guided wave theory, the scattering of SH guided wave by a circular inclusion and semicircular depression in an infinite piezoelectric material strip is investigated. The multi-polar coordinate change method is used to deal with semicircular sag at the boundary position. By using the method of multiple images, the theoretical expressions of scattering waves, satisfying the stress-free and electrical insulation conditions at the upper and lower boundaries of the band domain, are constructed. Based on the boundary conditions, the integral equations are established. The analytical expressions of the dynamic stress concentration coefficient and electric field intensity concentration coefficient are obtained. By analyzing the order of the guided wave, the physical parameters of the medium, the distance between the center of a circular inclusion, the location of a semicircular depression and a circular inclusion and the radius of the semicircular depression, the dynamic stress concentration factor and electric field intensity concentration factor are analyzed and compared with the existing literature in calculating example.

Thermal stress analysis of current-carrying media containing an inclusion with arbitrarily-given shape

The presence of inclusions in metal-based composites subjected to an electric current or a heat flux induces thermal stresses. Inclusion geometry is one of the important parameters in the stress distribution. In this study, the plane problem of an arbitrarily-shaped inclusion embedded in an infinite conductive medium is investigated based on the complex variable method. The shape of the inclusion is defined approximately by a polynomial conformal mapping function. Faber series and Fourier expansion techniques are used to solve the corresponding boundary value problems. The obtained results show that the shape, bluntness and rotation angle of the inclusion have a significant effect on the stress concentration around the inclusion induced by the far-field electric current. In addition, for the considered inclusion-matrix system under given electric loading, a lower amount of the Von Mises stress concentration than that around a circular inclusion could be achieved by appropriate selection of the inclusion shape and orientation.

Nonlinear thermal conductivity of periodic composites

Composites have been widely used to realize various functions in thermal metamaterials, thus becoming important to predict heat transport properties according to geometric structures and component materials. Based on a first-principles approach, namely, the Rayleigh method, here we develop an analytical way to calculate temperature-dependent (i.e., nonlinear) thermal conductivities of a composite with circular inclusions arranged in a periodic rectangular array. We focus on both weak and strong nonlinearity. As a result, we find that the temperature-dependence (nonlinearity) coefficient of the whole periodic composite can be larger than that of the nonlinear component inside this composite. Simulation results from finite element analysis show that the Rayleigh method can be also more accurate than the Maxwell-Garnett or Bruggeman effective medium approximations. As a model application, we further tailor the nonlinearity to design a thermal diode, for which heat flux along one direction is much larger than that along the opposite. This work provides a different theory for handling periodic structure with thermally responsive thermal conductivities, and it could be useful for designing thermal metamaterials with diverse properties including rectification.

Solution of the Problem of Heat Conduction for the Transversely Isotropic Piecewise-Homogeneous Space with Two Circular Inclusions

Solution of the Problem of Heat Conduction for the Transversely Isotropic Piecewise-Homogeneous Space with Two Circular Inclusions

The Effect of Own Weight on Dynamic Analysis of a Pre-Stretched Composite Plate-Strip Containing Twin Circular Inclusions Under Bending Using Finite Element Method

This manuscript describes how the weight of a simply supported composite plate-strip (containing twin circular inclusions) affects its dynamic behaviors when exposed to bending load as analyzed via the finite element method (FEM). The centers of the twin circular inclusions are on a line parallel to the free surface, and the materials of both inclusions are the same. First, the effects of body forces (the plate-strip’s own weight) and surface forces (pre-stretching load) on the plate-strip (both are considered to be initial stresses) are identified using the classical linear theory of elasticity. Next, the consequences of these stressors (identified in the first step) under additional time harmonic bending load on the forced vibration around the inclusions are determined using the three-dimensional linearized theory of elasticity (TDLTE) under the plane-strain state. The data herein suggest that the plate-strip’s weight can significantly effect dynamic characteristics of the considered plate-strip.