Circular Cylindrical Inclusion Research Articles

Scattering of elastic waves by a cluster of cylindrical inclusions: multiple scattering analysis and equivalent scatterer modeling

Abstract The two-dimensional scattering of elastic waves by a cluster of aligned circular cylindrical elastic inclusions embedded in a circular cylindrical domain of an infinitely extended elastic medium is analysed numerically. The analysis is based on a semi-analytical method where the scattered waves are expressed as the eigenfunction expansions of the time-harmonic displacement potentials and the expansion coefficients are determined by a numerical collocation technique. The wave fields within and around the cluster are demonstrated for different spatial arrangements of inclusions in the cluster. It is also examined if the scattering characteristics of the cluster can be simulated by a single equivalent scatterer which has the effective homogenized properties of the matrix/inclusion composite. As a result, it is shown that for sufficiently small inclusion radius-wavelength ratios, the scattering characteristics of the cluster are relatively insensitive to the spatial inclusion arrangements therein, and can be reasonably simulated by those of the equivalent scatterer.

Задача нелинейной теории упругости о растяжении и кручении составного цилиндра с предварительно напряжённым включением, содержащим винтовую дислокацию

The problem of large torsional and tension-compression deformations of a composite circular cylinder of incompressible Bartenev-Khazanovich material is considered. The cylinder contains a central circular cylindrical inclusion in which a concentrated screw dislocation is formed. It is pre-twisted and stretched (or compressed) along the axis and bonded to an unstressed external hollow cylinder. A single reference configuration is used when solving a problem for a composite body. This configuration is natural (unstressed) for the outer hollow cylinder and prestressed for the inner solid cylinder. The large deformation superposition theory is used to write the determining ratios of the material of the inner cylinder. The nonlinear theory of torsion of elastic cylinders containing a screw dislocation is used to solve the problem of the prestressed state of the internal inclusion. This theory is physically correct not for any models of isotropic elastic materials, but only for those for which the screw dislocation in the cylinder has a finite linear energy and creates a longitudinal force of finite value. The Bartenev-Khazanovich model belongs to this class. The problem for a composite cylinder is solved by a semi-inverse method. Using this method, it is reduced to nonlinear ordinary differential equations. The assumption of isotropy and incompressibility of the material makes it possible to find an exact solution to the problem.

Circular inclusion with a refined linearized version of Steigmann–Ogden model

The plane deformation of an infinite elastic matrix enclosing a single circular inclusion incorporating stretching and bending resistance for the inclusion–matrix interface is revisited using a refined linearized version of the Steigmann–Ogden model. This refined version of the Steigmann–Ogden model differs from other linearized counterparts in the literature mainly in that the tangential force of the interface defined in this version depends not only on the stretch of the interface but also on the bending moment and initial curvature of the interface (the corresponding bending moment relies on the change in the real curvature of the interface during deformation). Closed-form results are derived for the full elastic field in inclusion–matrix structure induced by an arbitrary uniform in-plane far-field loading. It is identified that with this refined version of the Steigmann–Ogden model a uniform stress distribution could be achieved inside the inclusion for any non-hydrostatic far-field loading when R = 3 χ int / λ int (where R is the radius of the inclusion, while λ int and χ int are the stretching and bending stiffness of the interface). Explicit expressions are also obtained for the effective transverse properties of composite materials containing a large number of unidirectional circular cylindrical inclusions using, respectively, the dilute and Mori–Tanaka homogenization methods. Numerical examples are presented to illustrate the differences between the refined version and two typical counterparts of the Steigmann–Ogden model in evaluating the stress field around a circular nanosized inclusion and the effective properties of the corresponding homogenized composites.

Numerical Analysis of Chaos in a Phononic Crystal Waveguide with Circular Inclusions of Real Materials

Phononic crystal waveguides (PnCW) have been of great interest due to their properties of manipulating or filtering the acoustic waves with which they interact. Similarly, the presence of the phenomenon of chaos in the classical transport of particles through billiards with analogous geometries has been investigated. With this in consideration, in the present work an acoustic system of a two-dimensional PnCW is modeled, composed of two plane-parallel plates and a periodic arrangement of circular cylindrical inclusions with acoustic surfaces of real materials. In this system, we use the numerical technique of the integral equation, which allows us to obtain the pressure field corresponding to the normal modes in a range of frequencies. In addition, spatial statistical properties of pressure intensity such as the autocorrelation function (ACF) and its standard deviation called correlation length were calculated. The results show that when the correlation length is very small, the system presents disordered patterns of field intensities. Thus under certain conditions, the system under consideration presents a chaotic behavior, similar to the corresponding classical system.

Analysis of the Stress State of a Layer with Two Cylindrical Elastic Inclusions and Mixed Boundary Conditions

The spatial problem of the elasticity theory is studied for a layer with two infinite circular solid cylindrical inclusions that are parallel to each other and to the layer boundaries. The physical characteristics of the layer and the inclusions are different from each other and they are homogeneous, isotropic materials. The spatial function of stresses is given at the upper boundary, and the function of displacements is given at the lower layer boundary. Circular cylindrical elastic inclusions are rigidly connected to the layer. It is necessary to determine the stress-strain state of the composite body. The problem solution is based on the generalized Fourier method, which uses special formulas for the transition between the basic solutions of the Lamé equation in different coordinate systems. Thus, the layer is considered in the Cartesian coordinate system, the inclusions – in the local cylindrical ones. Satisfying the boundary and conjugation conditions, systems of infinite integro-algebraic equations were obtained, which were subsequently reduced to linear algebraic ones. The resulting infinite system is solved by the reduction method. After deter-mining the unknowns, it is possible to find the stress values at any point of the elastic composite body. In numerical studies, a comparative analysis of the stress state in the layer and on the surfaces of inclusions at different distances between them is carried out. The analysis showed that when the inclusions approach each other, the stress state in the layer practically does not change. However, its significant change is observed in the bodies of inclusions, so with dense reinforcement ((R1 + R2) / L > 0.5), it is necessary to take into account the distances be-tween the reinforcing fibers. At stress values from 0 to 1 and the order of the system of equations m=10, the accuracy of meeting the boundary conditions was 10-4. With an increase in the system order, the accuracy of meeting the boundary conditions will increase. The given analytical-numerical solution can be used for high-precision determination of the stress-strain state of the given type of problems, and also as a reference for problems based on numerical methods.

Electro-elastic field of a piezoelectric quasicrystal medium containing two cylindrical inclusions

Considering the piezoelectric effect, the electro-elastic field of an infinite one-dimensional quasicrystal medium with two circular cylindrical inclusions is derived under antiplane shear and inplane electric loading. The boundary value problem of the composite material with circular cylindrical inclusions is analytically solved by the use of the conformal mapping technique and analytical continuation theory. The stresses in the phonon and phason fields and the electric displacements are obtained explicitly in the form of a power series both for the matrix and the inclusions. Some typical examples are analyzed to show the effect of the geometric parameters, material properties, and electro-mechanical loading on the electro-elastic fields in the matrix, inclusions, and interfaces. The limiting cases of circular cavities and rigid circular inclusions have also been investigated and discussed.

Eshelby’s circular cylindrical inclusion with polynomial eigenstrains in transverse direction by residue theorem

Cylindrical inclusions with constant cross section in an infinite isotropic matrix are usually treated as plane elasticity problems and solved by complex potential method without considering the longitudinal eigenstrains. This paper provides a closed-form solution for the Eshelby’s circular cylindrical inclusion with eigenstrains which are polynomial in transverse direction and uniform in longitudinal direction. The integrals of Green’s function are decomposed into the sum of customized L-integrals. Two sets of L-integrals for the regions inside and outside the circular cylindrical inclusion are evaluated by using the residue theorem. Further, the stress and strain fields inside and outside the inclusion resulted from the polynomial eigenstrains are obtained. Circular cylinder inclusions with uniform, linear, and quadric eigenstrains are, respectively, used as examples to illustrate the proposed solution. When the cylindrical inclusion only suffers transverse eigenstrains, the solution is appropriate for the circular inclusion with polynomial eigenstrains in plane elasticity. The proposed method has convenient formulae and simplifies the integrals of Green’s function with polynomial eigenstrains.

Stress Concentration Near Stiff Cylindrical Inclusions under Anti-Plane Shear Loading

Analytical solutions for the problems of stress concentration near cylindrical inclusion with circular or elliptical cross section under the anti-plane shear loading are presented. The solutions are obtained in the framework of the isotropic strain gradient elasticity theory with the assumption of high stiffness of the inclusions as compared to the matrix, which corresponds to the typical properties of the fiber-reinforced composite materials. It is shown that near the thin fibers, diameter of which is comparable to the characteristic size of the matrix microstructure, the stress concentration can decrease in comparison with conventional estimates known in the theory of elasticity. For circular cylindrical inclusions, the closed-form solutions are obtained for composites with low volume fraction of inclusions and can be used for the strength prediction of composites under the longitudinal shear and for the identification of additional parameters of the strain gradient theory of elasticity.

Instabilities of embedded cylindrical inclusions undergoing isotropic swelling or growth

Infinite circular cylindrical elastic inclusions, or rods, embedded in an unbounded elastic matrix display various modes of instability when they undergo sufficiently large expansion due to either swelling or volumetric growth. In this letter two modes of instability are examined: sinusoidal axisymmetric modes and sinusoidal bending modes. The rod and the matrix are neo-Hookean materials, and the full range of the modulus ratio of rod to matrix is considered. In the primary case examined, deformation is driven by an isotropic volumetric expansion, or transformation, of the rod. A three-dimensional bifurcation analysis of the rod constrained by the matrix reveals the onset of the critical instability mode as dependent on the modulus ratio. Comparisons with related results are discussed, including the compressive buckling of a stiff rod in a compliant matrix and the other limit when the modulus of the rod is very small compared to that of the matrix and behaves effectively as a fluid exerting pressure on the wall of the matrix cavity.11Contributed to celebrate the awarding of the Prager medal to Prof. Horacio Espinosa.

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.

A proof that multiple waves propagate in ensemble-averaged particulate materials.

Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterize wave propagation through an inhomogeneous material, the most crucial parameter is the effective wavenumber. For this reason, there are many published studies on how to calculate a single effective wavenumber. Here, we present a proof that there does not exist a unique effective wavenumber; instead, there are an infinite number of such (complex) wavenumbers. We show that in most parameter regimes only a small number of these effective wavenumbers make a significant contribution to the wave field. However, to accurately calculate the reflection and transmission coefficients, a large number of the (highly attenuating) effective waves is required. For clarity, we present results for scalar (acoustic) waves for a two-dimensional material filled (over a half-space) with randomly distributed circular cylindrical inclusions. We calculate the effective medium by ensemble averaging over all possible inhomogeneities. The proof is based on the application of the Wiener–Hopf technique and makes no assumption on the wavelength, particle boundary conditions/size or volume fraction. This technique provides a simple formula for the reflection coefficient, which can be explicitly evaluated for monopole scatterers. We compare results with an alternative numerical matching method.

Phononic band structure of an acoustic waveguide that behaves as a phononic crystal

Abstract Phononic Crystals (PnCs) are novel synthetic periodic materials for controlling and manipulating the propagation of elastic (or acoustic) waves. In this work, we have considered systems formed by three-dimensional (3D) acoustic rectangular waveguides, which have internal periodic structures. Also, the case of a two-dimensional (2D) rectangular waveguide with a periodic array of circular cylindrical inclusions is exhibited, to carry out a comparative study with the 3D case. Integral numerical methods used to obtain phononic band structures involve a Boundary Element Method, which requires the use of periodic Green’s functions. An interesting detail that resulted from this study was the appearance of band gaps of the proposed structure for different values of the filling fractions. The system considered constitutes itself a PnC whose band structure corresponds in many ways to a conventional one-dimensional PnC. This fact is an alternative to manufacture a PnC that exhibits a double function: acting as a PnC and as an acoustic waveguide. The main result of this work is that discrete modes for some frequency range were obtained, which indicates that Phononic Crystal Waveguides (PnCWs) considered act as unimodal filters when the filling fraction is large. These properties exhibit some interest from a technological point of view.

Surface modes in photonic crystal waveguides coated with a layer of dispersive left-handed material

In this work, we study the presence of surface modes in two different photonic crystal waveguides characterized by the presence of layers of a dispersive left-handed material. The periodicity required to have a photonic crystal waveguide is considered in two distinct ways. One of them is composed of two perfect electrical conducting rippled surfaces, and the other consists of two at infinite surfaces that surround a periodic one-dimensional array of circular cylindrical inclusions of a perfect electrical conductor, coated with a layer of left-handed material. To study these systems, integral numerical methods were applied to determine their band structures and the electromagnetic field distribution, under the TE and TM polarizations. Numerical results show the presence of surface modes for several frequencies with certain geometric configurations. One of these frequencies is known in the literature and corresponds to the frequency of a Surface Plasmon-Polariton.

Application of Mori–Tanaka method in 3–1 porous piezoelectric medium of crystal class 6

We consider a piezoelectric medium of hexagonal crystal system of class 6 containing a uniform distribution of circular cylindrical inclusions aligned with the axis of material symmetry. We determine the piezoelectric Eshelby tensor explicitly. In the case of cylindrical holes, we use the Eshelby tensor together with the multiscale Mori–Tanaka Method to obtain analytical formulae for the effective electroelastic properties of the homogenized medium. These formulae depend upon both the electroelastic properties of the matrix material and the volume fraction of the cylindrical holes. Using electroelastic properties reported in the literature, we obtain graphs of the effective properties of the porous medium versus the volume fraction of the pores and show that these effective electroelastic properties decrease for increasing porous volume fraction, as expected. In particular, for the case of the crystal subclass 6 mm, the results obtained via Mori–Tanaka Method agree well with results obtained via Method of Asymptotic Homogenization and Finite Element Method. This work is important in the evaluation of the effective electroelastic properties of heterogeneous solids with hierarchical structures, such as bones.

Dynamic behaviour of rigid inclusions with circumferential damage embedded in an elastic matrix

This work addresses the evaluation of the displacement field in an elastic matrix due to the presence of an embedded homogeneous inclusion under time-harmonic SH-wave loads. We consider the bond between the circular cylindrical inclusion and the surrounding matrix of infinite extent to be partly damaged in the circumferential direction. The material of both inclusion and matrix is modelled as homogeneous, isotropic, and linearly elastic. An analytical approach is introduced here for describing the scattering of SH-waves by inclusions with partly damaged bond. Subsequently, a numerical example serves to illustrate the methodology, which can be extended to the scattering of SV/P-waves in a straightforward manner.

Formation of a Prismatic Dislocation Loop in the Interface of a Circular Cylindrical Inclusion Embedded in a Thin Slab

The introduction of a prismatic dislocation loop in the interface of an axisymmetric precipitate embedded in a thin slab of infinite lateral extension has been theoretically investigated. The critical misfit strain resulting from the lattice mismatch between the inclusion and the slab has been characterized for the loop formation versus the thickness of the slab and the radius of the inclusion. The case where the precipitate is embedded in a semi-infinite matrix is also discussed and a stability diagram of the structure is displayed with respect to the loop introduction versus the geometric and misfit parameters.

Analytical mean Green operators/Eshelby tensors for patterns of coaxial finite long or flat cylinders in isotropic matrices

Recently, an analytical solution has been provided for mean and axial Green operators (GO) of circular cylindrical inclusions with finite length in 3D isotropic (elastic-like and dielectric-like) materials, from long rods to truly penny-shaped ones. This result provides a situation for examining the commonly used “shape spheroidal approximation” of same aspect ratio as well as possibly more appropriate alternatives. For cases when such a spheroidal approximation would remain needed, we propose and compare an “operator spheroidal approximation” in terms of best matching GO, which is exactly operator-identical for dielectric-like properties and nearly identical for elastic-like ones. Next, a simple analytical solution is explicated for the mean isotropic GO of all patterns of coaxial circular long or flat cylinders with same radii, obtained from solving the coaxial cylinder pair interaction problem. These patterns can be seen as either fragmented cylinders or as cylinder alignments. The obtained GO solution form is further shown to formally hold for similar patterns of domains having more general shapes. The established general property of such patterns is that the interaction part of their mean GO can be expressed as a linear combination of the mean GO of all the pattern sub-domains. The mean GO solution for one such pattern is further proved to be valid as well for the domains – of the double inclusion type – obtained in connecting the pattern elements by their outer envelop of zero thickness, thanks to the GO blindness to connectivity differences. This GO equivalency is finally used to define a global operator spheroidal approximation in terms of best matching GO for any such coaxial cylinder pattern. Effective properties for an isotropic matrix reinforced by congruent cylinders or patterns are estimated within a mean field approximation context.

Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites Part II: Higher-order constitutive properties and application cases

Starting from a Cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first-order by a certain discrepancy tensor (see part I of the present article), it is shown that the equivalent second-gradient Mindlin elastic solid: (i) is positive definite only when the discrepancy tensor is negative defined; (ii) the non-local material symmetries are the same of the discrepancy tensor, and (iii) the non-local effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response. Furthermore, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in the particular cases of: (a) circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix, (b) n-polygonal cylindrical voids in an isotropic matrix, and (c) circular cylindrical voids in an orthotropic matrix.

Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing an anti-plane strain inclusion of arbitrary cross-sectional shape prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for anti-plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite anti-plane strain elastic body. The solution reduces to that of the infinite-domain anti-plane strain inclusion problem when the boundary effect is not considered. The problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti-plane strain deformations is analytically solved by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical linear elasticity-based Eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.

Strain gradient solution for the Eshelby-type anti-plane strain inclusion problem

The solution for the Eshelby-type inclusion problem of an infinite elastic body containing an anti-plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The Green’s function based on the SSGET for an infinite three-dimensional elastic body undergoing anti-plane strain deformations is first obtained by employing Fourier transforms. The Eshelby tensor is then analytically derived in a general form for an anti-plane strain inclusion of arbitrary cross-sectional shape using the Green’s function method. By applying this general form, the Eshelby tensor for a circular cylindrical inclusion is obtained explicitly, which is separated into a classical part and a gradient part. The former does not contain any classical elastic constant, while the latter includes the material length scale parameter, thereby enabling the interpretation of the particle size effect. The components of the new Eshelby tensor vary with both the position and the inclusion size, unlike their counterparts based on classical elasticity. For homogenization applications, the average of this Eshelby tensor over the circular cross-sectional area of the inclusion is obtained in a closed form. Numerical results reveal that when the inclusion radius is small, the contribution of the gradient part is significantly large and should not be ignored. Also, it is found that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. These components approach from below the values of their counterparts based on classical elasticity when the inclusion size becomes sufficiently large.