Circular Inclusion Research Articles

Shaped beam scattering by an object with a uniaxial anisotropic inclusion

The differential scattering characteristics of an object having a uniaxial anisotropic inclusion, for incidence of an arbitrarily shaped beam, are theoretically investigated. The scattering problem is solved in a spherical basis by using the boundary conditions, method of moments technique and Schelkunoff’s equivalence theorem, which leads to a system of linear equations for the expansion coefficients of the scattered and internal fields when the description of the incident shaped beam is known. The theoretical formulation is reviewed, and the numerical problems are discussed. As examples, for a Gaussian beam, zero-order Bessel beam and Hertzian electric dipole radiation striking a spheroid with a uniaxial anisotropic spheroid inclusion and a circular cylinder with a uniaxial anisotropic circular cylinder inclusion, the normalized differential scattering cross sections are calculated, and the scattering properties are analyzed concisely.

Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects

A new model for determining band gaps for wave propagation in two-dimensional (2-D) periodic composite structures is developed using a modified couple stress theory. The general equation of motion and boundary conditions in the elasto-dynamics of the modified couple stress theory are first derived by a variational formulation based on Hamilton’s principle. The in-plane and anti-plane wave equations incorporating microstructure effects are then obtained explicitly from the general equation of motion. The plane wave expansion method and the Bloch theorem for periodic media are used to solve the in-plane and anti-plane wave equations, which are reduced to an eigenvalue problem in each case. The band gaps are determined from solving the characteristic equation and plotting the resulting eigen-frequencies. The new model recovers the classical elasticity-based model when microstructure effects are not considered. To quantitatively illustrate the newly developed model, a parametric study is conducted for 2-D periodic composite structures containing circular and square inclusions. The numerical results reveal that the microstructure effects on the band gaps are significant only when the unit cell size is small for both the composite structures. In addition, it is found that the volume fraction has a significant effect on the band gap size, and the inclusion shape has a large influence on the band gaps.

High-order virtual element method for the homogenization of long fiber nonlinear composites

A high-order virtual element method (VEM) for homogenization of long fiber reinforced composites is presented. In particular, periodic composites are considered studying square or rectangular unit cell arrays and circular inclusions. A suitable displacement representation form is adopted reducing the three-dimensional problem to an equivalent two-dimensional one. Material nonlinearity is taken into account for the matrix which can be plastic or visco-plastic. The formulation is proposed for linear and high-order virtual elements. Numerical applications are performed to assess the accuracy of the VEM formulation in comparison with the classical finite element approach. In particular, convergence investigations on the overall elastic moduli and on the Mises equivalent stress are performed. Elasto-plastic and visco-plastic analyses are carried out exploiting the local mesh refinement features typical of VEM showing efficiency of polygonal discretizations.

Dynamic Green's functions for multiple circular inclusions with imperfect interfaces using the collocation multipole method

This paper presents a semi-analytical approach to solve anti-plane dynamic Green's functions for an elastic infinitely extended isotropic solid (matrix) containing multiple circular inclusions with imperfect interfaces. A linear spring model with vanishing thickness is employed to character the imperfect interface. The multipole expansions of anti-plane displacement of the matrix and inclusion, induced by a time-harmonic anti-plane line force located in the matrix or in the inclusion, are expanded by using Hankel and Bessel functions, respectively. The imperfect interface condition is satisfied by uniformly collocating points along the interface of each inclusion. For the imperfect interface condition, the normal derivative of the anti-plane displacement with respect to a non-local polar coordinate system is developed without any truncation error for multiply-connected domain problems. For the case of one circular inclusion, the proposed quasi-static stress field matches well with the analytical static solution. The proposed quasi-static stress fields containing two and three circular inclusions are critically compared with those calculated by static analysis using the finite element method. Finally, extensive studies are presented to investigate the effects of the frequency of excitation, imperfect interface and separation between inclusions on the dynamic Green's functions.

An elliptical liquid inclusion in an infinite elastic plane

Beyond recent related literature, which focused on spherical incompressible liquid inclusions, the present work studies an elliptical compressible liquid inclusion in an infinite elastic plane under static remote mechanical loading. Here, it is assumed that the change of pressure inside the liquid inclusion is linearly related to the change of inclusion volume with the bulk modulus of the liquid as the proportionality coefficient. Also, the role of the liquid surface tension on the solid–liquid interface is examined especially when the size of the liquid inclusion is comparable to or smaller than the elastocapillary length. Our results show that both the surface tension and the change of liquid pressure have a significant effect on reducing the stress concentration factor at the endpoints of an elliptical liquid inclusion. In addition, the pressure change inside the liquid inclusion is studied when a uniaxial remote stress is applied perpendicular or parallel to the major axis of the elliptical liquid inclusion. In particular, the effective plane-strain Young's modulus of a solid–liquid composite containing circular liquid inclusions predicted by the present model is linearly related to the volume fraction of the liquid inclusions, in reasonable agreement with existing experimental data.

Progressive damage through interface microcracking in cementitious composites: A micromechanics based approach

A multiscale model in the framework of micromechanics is developed to study the effect of microcracks present at the interface between coarse aggregates and mortar matrix in cementitious composites. The different stages of damage induced by the propagation of microcracks are analyzed at the mesoscale with the aid of solutions based on interface fracture mechanics. Using the solution of the interaction between an edge dislocation and a circular inclusion with an interface crack, the stress intensity factor of a kinked arc crack is derived numerically by the method of distributed dislocations. The macroscopic response under uniaxial tensile loading is seen to be affected considerably by the presence of microcracks. Various factors such as the inclusion size, inclusion volume fraction, initial size of microcrack and elastic properties of the individual phases are found to influence the overall constitutive behaviour of the composite. The findings of the study can be utilized to provide guidelines to tailor the material properties and enhance the performance of such composites.

Dynamic response of a circular inclusion embedded in inhomogeneous half-space

Dynamic response of a shallow circular inclusion under incident SH wave in radially inhomogeneous half-space is researched by applying complex function theory and multipolar coordinate system. Considering that the mass density of the half-space varies along with the radius direction, the governing equation is expressed as a Helmholtz equation with a variable coefficient. Based on the conformal mapping method, the Helmholtz equation with a variable coefficient is transformed into its normalized form. Then, the expressions of incident wave, reflected wave and scattering wave are obtained, and the standing wave function is deduced by considering the circular inclusion subsequently. According to displacement and stress continuous condition of the inclusion, the undetermined coefficients in scattering wave and standing wave are solved. Finally, dynamic stress concentration factor around the inclusion is calculated and discussed. Numerical results demonstrate the validity of the method and influential factors of dynamic stress concentration factor.

On 3D Anticrack Problem of Thermoelectroelasticity

Abstract A solution is presented for the static problem of thermoelectroelasticity involving a transversely isotropic space with a heat-insulated rigid sheet-like inclusion (anticrack) located in the isotropy plane. It is assumed that far from this defect the body is in a uniform heat flow perpendicular to the inclusion plane. Besides, considered is the case where the electric potential on the anticrack faces is equal to zero. Accurate results are obtained by constructing suitable potential solutions and reducing the thermoelectromechanical problem to its thermomechanical counterpart. The governing boundary integral equation for a planar anticrack of arbitrary shape is obtained in terms of a normal stress discontinuity. As an illustration, a closed-form solution is given and discussed for a circular rigid inclusion.

Uniformity of stresses inside a non-elliptical inhomogeneity interacting with a circular Eshelby inclusion in anti-plane shear

We use conformal mapping techniques to examine the uniformity of stresses inside a non-elliptical inhomogeneity interacting with a circular Eshelby inclusion in an elastic matrix subjected to remote uniform stresses in anti-plane shear. We show that for a prescribed set of two real loading and two complex geometric parameters, it is possible to determine the single unknown complex coefficient in the mapping function and the (unique) shape of the corresponding inhomogeneity enclosing internal uniform stresses. Our results indicate that the shape of the inhomogeneity depends on the circular Eshelby inclusion whereas the uniform stress field inside the inhomogeneity does not. Finally, we note that the influence of the circular Eshelby inclusion in the vicinity of the inhomogeneity allows for the possibility of a sharp corner on the boundary of the inhomogeneity.

Fractured micro-granitoid enclaves: A stress marker

Systematic, parallel, and steeply dipping tensile fractures within rounded to sub-rounded micro-granitoid enclaves are analyzed to decipher the stress condition at Chitradurga granite (Dharwar craton, south India). The study is performed in two steps: 1) Assessing the fracturing condition through thermo-mechanical model as fractures in granite may be related to thermal stresses during its cooling or to tectonics stresses. 2) Performing the plane strain mechanical solution for circular rigid inclusions to fractured enclaves, where rock mechanics concepts of fracturing are integrated with kinematic analysis. We interpret that the parallel, systematic fractures in the micro-granitoid enclaves developed due to the stress amplification inside enclaves when the host pluton was at shallow depth of ∼2.4 km. The derived paleostress orientation fits well with the orientation and sense of movement to the adjacent Chitradurga Shear Zone. The deduced paleostress conditions from fractured micro-granitoid enclaves are also in a good agreement with the previously revealed regional tectonics. We conclude that the fractured micro-granitoid enclaves are reliable stress indicator.

Diffraction of Rayleigh Waves on a Compliant Inclusion in the Elastic Half Space

We consider a three-dimensional dynamical problem of diffraction of Rayleigh plane waves on a circular compliant inclusion in the elastic half space. To solve the problem, we use the method of boundary integral equations. The dynamical stress intensity factors in the vicinities of points of the contour of the inclusion are analyzed.

Analytical Solutions and Stress Concentration Factors for Annuli With Inhomogeneous Boundary Conditions

Analytical displacement and stress fields with stress concentration factors (SCFs) are derived for linearly elastic annular regions subject to inhomogeneous boundary conditions: an infinite class of the mth order polynomial antiplane tractions or displacements. The solution of the Laplace equation governing the out-of-plane problem covers both rigid and void circular inclusions forming the core of the annulus. The results show first that the SCF and the loading order are inversely proportional. In particular, the SCF approaches value 2 when either the outer boundary of the annulus tends to infinity or the order of the polynomial loading increases. Second, the number of peculiar points on the inner contour having null stress increases with the increasing loading order. The analytical solution is confirmed and extended to noncircular enclosures via finite element analysis by exploiting the heat-stress analogy. The results show that the closed-form solution for a circular annulus can be used as an accurate approximation for noncircular enclosures. Altogether, the results shown can be exploited for analyzing complex loading conditions and/or multiple rigid or void inclusions for enhancing the design of hollow and reinforced composites materials.

Optimal control of the radius of a rigid circular inclusion in inhomogeneous two-dimensional bodies with cracks

A two-dimensional model describing the equilibrium state of a cracked inhomogeneous body with a rigid circular inclusion is investigated. The body is assumed to have a crack that reaches the boundary of the rigid inclusion. We assume that the Signorini condition, ensuring non-penetration of the crack faces, is satisfied. We analyze the dependence of solutions on the radius of rigid inclusion. The existence of a solution of the optimal control problem is proven. For this problem, a cost functional is defined by an arbitrary continuous functional, with the radius of inclusion chosen as the control parameter.

Atomistic modelling of crack-inclusion interaction in graphene

In continuum fracture mechanics, it is well established that the presence of crack near an inclusion leads to a significant change in the crack-tip stress field. However, it is unclear how atomistic crack-inclusion interaction manifests itself at the nanoscale where the continuum description of matter breaks down. In this work, we conducted molecular dynamics simulations to investigate the interactions of an atomic-scale boron nitride inclusion with an edge crack in a graphene sheet. Numerical simulations of nanoscale tensile tests were obtained for graphene samples containing an edge crack and a circular inclusion. Stress analysis of the samples show the complex nature of the stress state at the crack-tip due to the crack-inclusion interaction. Results reveal that the inclusion results in an increase (amplification) or a decrease (shielding) of the crack-tip stress field depending on the location of the inclusion relative to the crack-tip. Our numerical experiments unveil that inclusions of specific locations could lead to a reduction in the fracture resistance of graphene. Results of the crack-inclusion interaction study were compared with the corresponding results of crack-hole interaction problem. The study also provides an insight into the applicability of well-established continuum crack-microdefect interaction models for the corresponding atomic scale problems.

Near and far field scattering of SH waves by multiple multilayered anisotropic circular inclusions using parallel volume integral equation method

PurposeThe purpose of this paper is to calculate near field and far field scattering of SH waves by multiple multilayered anisotropic circular inclusions using parallel volume integral equation method (PVIEM) quantitatively.Design/methodology/approachThe PVIEM is applied for the analysis of elastic wave scattering problems in an unbounded solid containing multiple multilayered anisotropic circular inclusions. It should be noted that this numerical method does not require the use of the Green’s function for the inclusion – only the Green’s function for the unbounded isotropic matrix is needed. This method can also be applied to solve general elastodynamic problems involving inhomogeneous and/or anisotropic inclusions whose shape and number are arbitrary.FindingsA detailed analysis of the SH wave scattering problem is presented for multiple multilayered orthotropic circular inclusions. Numerical results are presented for the displacement fields at the interfaces and the far field scattering patterns for square and hexagonal packing arrays of multilayered circular inclusions in a broad frequency range of practical interest.Originality/valueTo the best of the authors’ knowledge, the solution for scattering of SH waves by multiple multilayered anisotropic circular inclusions in an unbounded isotropic matrix is not currently available in the literature. However, in this paper, calculation of displacements on interfaces and far field scattering patterns of multiple multilayered anisotropic circular inclusions using PVIEM as a pioneer of numerical modeling enables us to investigate the effects of single/multiple scattering, fiber packing type, fiber volume fraction, single/multiple layer(s), the multilayer’s geometry, isotropy/anisotropy and softness/hardness.

Acoustic waveguiding in a silicon carbide phononic crystals at microwave frequencies

Two dimensional SiC–air phononic crystals have been modeled, fabricated, and tested with a measured bandgap ranging from 665 to 693 MHz. Snowflake air inclusions on a hexagonal lattice were used for the phononic crystal. By manipulating the phononic crystal lattice and inserting circular inclusions, a waveguide was created at 680 MHz. The combined insertion loss and propagation loss for the waveguide is 8.2 dB, i.e., 39% of the energy is guided due to the high level of the confinement afforded by the phononic crystal. The SiC–air phononic crystals and waveguides were fabricated using a CMOS-compatible process, which allows for seamless integration of these devices into wireless communication systems operating at microwave frequencies.

In silico simulation and in vitro evaluation of an elastomeric scaffold using ultrasonic shear wave imaging

Biodegradable elastomeric scaffolds for soft tissue repair represent a growing area of biomaterials research. Mechanical strength is one of the key factors to consider in the evaluation of candidate materials and the designs for tissue scaffolds. It is desirable to develop non-invasive evaluation methods of the mechanical property of scaffolds which would provide options for monitoring temporal mechanical property changes in situ. In this paper, we conduct in silico simulation and in vitro evaluation of an elastomeric scaffold using a novel ultrasonic shear wave imaging (USWI). The scaffold is fabricated from a biodegradable elastomer, poly(carbonate urethane) urea using salt leaching method. A numerical simulation is performed to test the robustness of the developed inversion algorithm for the elasticity map reconstruction which will be implemented in the phantom experiment. The generation and propagation of shear waves in a homogeneous tissue-mimicking medium with a circular scaffold inclusion is simulated and the elasticity map is well reconstructed. A PVA phantom experiment is performed to test the ability of USWI combined with the inversion algorithm to non-invasively characterize the mechanical property of a porous, biodegradable elastomeric scaffold. The elastic properties of the tested scaffold can be easily differentiated from the surrounding medium in the reconstructed image. The ability of the developed method to identify the edge of the scaffold and characterize the elasticity distribution is demonstrated. Preliminary results in this pilot study support the idea of applying the USWI based method for non-invasive elasticity characterization of tissue scaffolds.

A concentrated couple inside or outside a circular inclusion in finite plane elastostatics

We consider a circular elastic inclusion embedded in an infinite matrix each from a particular class of compressible hyperelastic materials of harmonic type. A concentrated couple is applied either inside the circular inclusion or in the matrix. Closed-form solutions of the corresponding boundary value problems are obtained using complex variable methods, in particular the principle of analytic continuation. Our analysis reveals several interesting conclusions including: the sum σ11+σ22 of the normal stresses inside the inclusion remains constant when the concentrated couple is located in the surrounding matrix; the sum of the normal stresses is zero everywhere in the matrix when the concentrated couple is located inside the inclusion.

A simplified model of fracking based on the topological derivative concept

This paper deals with a simplified hydraulic fracture model based on the concept of topological derivatives. It means that we consider a two dimensional idealization in which the rock is assumed to be impermeable, while the fracturing process is activated by a given pressure acting within the existing geological faults. The basic idea consists in adapting the Francfort–Marigo damage model to the context of hydraulic fracture. The Francfort–Marigo damage model is a variational approach to describe the behavior of brittle materials under the quasi-static loading assumption, focusing on the evolution of damaged regions under an irreversibility constraint. In our model problem, the loading comes out from a pressurized damaged region embedded into the rock, which is used to trigger the hydraulic fracturing process. In particular, a shape functional given by the sum of the total potential energy of the system with a Griffith-type dissipation energy term is minimized with respect to a set of ball-shaped pressurized inclusions by using the topological derivative concept. Thus, the topological asymptotic expansion of the shape functional with respect to the nucleation of a circular inclusion endowed with non-homogeneous transmission condition on its boundary is obtained. The associated topological derivative, which corroborates with the famous Eshelby theorem, is used to devise a simple topology optimization algorithm specifically designed to simulate the whole nucleation and propagation process of hydraulic fracturing. To assess our model, some numerical examples are presented, showing typical features of hydraulic fracture phenomenon, including the characterization of the fault-activation pressure and specific crack path growth, allowing for kinking and bifurcations.

Scattering of SH-wave by an elliptical inclusion with partial debonding curve in half-space

ABSTRACTFor the purpose of revealing the dynamic properties of elliptical inclusion in the half-space, solving the problem of scattered SH-wave, the method of ‘conformal mapping’ is used to map an elliptical inclusion into a circular inclusion. The displacement field and the stress field of the elliptical inclusion while bearing out-plane line source load are obtained by Green’s function method. Then, infinite system of linear equations is established by displacement and stress continuous boundary conditions, to solve unknown coefficients of wave function. Finally, the ‘partial debonding curve model’ is constructed, and the equal stress but opposite in direction is applied in the partial debonding curve. It is obtained the total displacement field of the elliptical inclusion with a partially debonding curve in the half-space. Numerical results demonstrate that dynamic stress concentration factor is influenced in the incident angle, the frequency of incident wave, the depth of inclusions and the partial debonding curve angle.