Multiple Circular Inclusions Research Articles

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.

Scattering of Anti-Plane SH-Wave by Multiple Cylindrical Inclusions in Elastic Semi-Space

Multiple circular inclusions exists widely in natural media, engineering materials and modern municipal construction. The scattering field produced by multiple circular inclusions determines the dynamic stress concentration factor around the circular inclusions, and therefore determines whether the material is damaged or not. These problems are complicated, because there are many factors influenced. Researchers solved these problems by analysis and numerical methods. It is hard to obtain analytic solutions except for several simple conditions. In this paper, the solution of displacement field for elastic semi-space with multiple cylindrical inclusions by anti-plane SH-wave is constructed. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the cylindrical inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition around the edge of cylindrical inclusions. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical inclusions. Based on this solution, the problem of interaction of multiple cylindrical inclusions and a linear crack in semi-space can be investigated further.

Scattering of Anti-Plane SH-Wave by the Semi-Cylindrical Gap with Multiple Shallow-Buried Inclusions in Elastic Semi-Space

Semi-cylindrical gap and Multiple circular inclusions exists widely in natural media, composite materials and modern municipal construction. The scattering field produced by semi-cylindrical gap and multiple circular inclusions determines the dynamic stress concentration factor around the gap and circular inclusions, and therefore determines whether the material is damaged or not. These problems are complicated. It is hard to obtain analytic solutions except for several simple conditions. In this paper, the solution of displacement field for elastic semi-space with semi-cylindrical gap and multiple cylindrical inclusions by anti-plane SH-wave is constructed. In complex plane, considering the symmetry of SH-wave scattering , the displacement field aroused by the anti-plane SH-wave and the scattering displacement field impacted by the gap and the cylindrical inclusions comprised of Fourier-Bessel series with undetermined coefficients which satisfies the stress-free condition on the ground surface are constructed. Through applying the method of multi-polar coordinate system, the equations with unknown coefficients can be obtained by using the displacement and stress condition around the edge of the gap and cylindrical inclusions. According to orthogonality condition for trigonometric function, these equations can be reduced to a series of algebraic equations. Then the value of the unknown coefficients can be obtained by solving these algebraic equations. The total wave displacement field is the superposition of the displacement field aroused by the anti-plane SH-wave and the scattering displacement field. By using the expressions, an example is provided to show the effect of the change of relative location of the cylindrical inclusions.

Scattering of flexural wave in a thin plate with multiple circular inclusions by using the multipole method

The multipole method is presented to analytically solve the scattering of flexural wave by multiple circular inclusions in an infinite thin plate. The near-field dynamic moment concentration factor (DMCF) and the far-field scattering pattern are both investigated in this paper. The former has a connection with the fatigue failures and the defects in plate-like structures can be detected by the latter. Owing to the addition theorem, the multipole expansion for the multiple scattering fields can be transformed into one coordinate system centered at one circle where continuity conditions are required. In this way, a coupled infinite linear algebraic system is derived as an analytical model for an infinite thin plate with multiple circular inclusions subject to an incident flexural wave. The convergence analysis is conducted to provide the guideline of usage for the proposed method. The effects of the size and thickness of the flexible inclusion, and the central distance between inclusions on the near-field DMCF and the far-field scattering pattern are investigated in the numerical experiments. It shows that the scattering pattern correlates closely with the size and thickness of weakness, indicating the importance of the scattering pattern to detect the various defects. In addition, the DMCF of two corrosion defects is larger than that of one. Therefore, it is essential to evaluate structural safety when multiple circular defects are very close to each other. The effect of the space between the inclusions on the near-field DMCF is different from that on the far-field scattering pattern.

Interaction of Multiple Circular Inclusions and a Linear Crack by SH-Wave

Multiple circular inclusions exists widely in natural media, engineering materials and modern municipal construction, and defects are usually found around the inclusions. When composite material with multiple circular inclusions and a crack is impacted by dynamic load, the scattering field will be produced. The problem of scattering of SH waves by multiple circular inclusions and a linear crack is one of the important and interesting questions in mechanical engineering and civil engineering for the latest decade. It is hard to obtain analytic solutions except for several simple conditions. In this paper, the method of Green’s function is used to investigate the problem of dynamic stress concentration of multiple circular inclusions and a linear crack for incident SH wave. The train of thoughts for this problem is that: Firstly, a Green’s function is constructed for the problem, which is a fundamental solution of displacement field for an elastic space possessing multiple circular inclusions while bearing out-of-plane harmonic line source force at any point: Secondly, in terms of the solution of SH-wave’s scattering by an elastic space with multiple circular inclusions, anti-plane stresses which are the same in quantity but opposite in direction to those mentioned before, are loaded at the region where the crack is in existent actually; Finally, the expressions of the displacement and stress are given when multiple circular inclusions and a linear crack exist at the same time. Then, by using the expression, an example is provided to show the effect of multiple circular inclusions and crack on the dynamic stress concentration.

Magnetoelectroelastic Interaction between a Generalized Screw Dislocation and Multiple Circular Inclusions

AbstractIn this paper, the interaction of a generalized screw dislocation with multiple circular inclusions perfectly bonded to an unbounded matrix under remote magnetoelectromechanical loadings is dealt with. The analytical solutions of electric field, magnetic field and displacement field either in the inclusions or the matrix are obtained by use of the complex variable method. The image force acting on the magnetoelectric screw dislocation is calculated by using the generalized Peach-Koehler formula. Finally, the influence of material combinations on the image force is examined for several practical examples. The obtained solutions can be used as Green's functions for the analysis of the corresponding magnetoelectric crack problem.

Scattering of flexural wave in a thin plate with multiple circular inclusions by using the null-field integral equation approach

The subject of scattering flexural wave in a thin plate with multiple circular inclusions under the incident flexural wave is studied in this paper. A semi-analytical approach is proposed to solve this problem which can be decomposed into several interior circular inclusion problems and an exterior plate problem subject to the incident wave. The scattered field in the associated exterior problem is solved by using the null-field integral formulation in conjunction with degenerate kernels, tensor transformation and Fourier series. All dynamic kernels of plate in the direct formulation are expanded into degenerate forms to avoid the integral singularity and further the rotated degenerate kernels have been derived to consider the general case of multiple circular inclusions. The proposed results for an infinite plate with one circular inclusion are compared with the available analytical solutions to verify the validity of the proposed method. To demonstrate the generality of the proposed method, the cases of multiple inclusions are studied and their quasi-static results are verified by static data of FEM using ABAQUS. Numerical results indicate that the DMCF of two inclusions is apparently larger than that of one when two inclusions are close to each other. Fictitious frequency appearing in the exterior problem can be suppressed by using the more number of Fourier series terms. Numerical results show that the space between scatterers has the opposite effect on the near-field DMCF in comparison with the far-field scattering pattern.

Torsional rigidity of a circular bar with multiple circular inclusions using the null-field integral approach

In this article, a systematic approach is proposed to calculate the torsional rigidity and stress of a circular bar containing multiple circular inclusions. To fully capture the circular geometries, the kernel function is expanded to the degenerate form and the boundary density is expressed into Fourier series. The approach is seen as a semi-analytical manner since error purely attributes to the truncation of Fourier series. By collocating the null-field point exactly on the real boundary and matching the boundary condition, a linear algebraic system is obtained. Convergence study shows that only a few number of Fourier series terms can yield acceptable results. Finally, torsion problems are revisited to check the validity of our method. Not only the torsional rigidities but also the stresses of multiple inclusions are also obtained by using the present approach.

Interaction between screw dislocations and inclusions with imperfect interfaces in fiber-reinforced composites

A three-phase composite cylinder model is utilized to study the elastic interaction between screw dislocations and embedded multiple circular cross-section inclusions (fibers) with imperfect interfaces in composites. By means of complex variable techniques, the explicit solutions of stress and displacement fields are obtained. With the aid of the Peach–Koehler formula, the explicit expressions of image forces exerted on screw dislocations are derived. The equilibrium positions of the appointed screw dislocation near one of the inclusions are discussed for variable parameters (interface imperfection, material mismatch and dislocation position) and the influence of the nearby inclusions and dislocations is also considered. The results show that, if the inclusion is stiffer than the matrix and the magnitude of the degree of interface imperfection reaches the certain value, a new equilibrium position for the screw dislocation in the matrix can always be produced in comparison with the previous solution (the perfect interface). The effect of elastic constants of the inclusion on the image force and the equilibrium position of the appointed screw dislocation is weak when the interface imperfection is strong. It is also seen that the magnitude of the image force exerted on the appointed dislocation caused by multiple inclusions is always smaller than that produced by a single inclusion. The impact of the closer dislocations on the mobility of the appointed dislocation is very significant.

Simulation of Fatigue Fracture Path Prediction in Materials Containing Holes or Inclusions Using Moving Finite Element Method

It is well known that approximately 70 to 80 percent of the failure accidents that occur in complex structures that have such holes are caused by the fatigue fracture. That is, we need to know fatigue fracture path to evaluate health. In this paper, we carried out fatigue fracture path prediction simulation containing multiple circular holes or inclusions in specimens. Furthermore the effect of their existence on crack growth was examined. The simulations were conducted using the moving finite element method based on Delaunay automatic triangulation, and the numerical result of the specimens containing multiple circular holes was compared with the experimental result. In the fatigue fracture path prediction simulation, Paris's law was used to decide crack propagation length Δa and local symmetry criterion was used to decide crack propagation direction.

Null-Field Approach for the Multi-inclusion Problem Under Antiplane Shears

In this paper, we derive the null-field integral equation for an infinite medium containing circular holes and/or inclusions with arbitrary radii and positions under the remote antiplane shear. To fully capture the circular geometries, separable expressions of fundamental solutions in the polar coordinate for field and source points and Fourier series for boundary densities are adopted to ensure the exponential convergence. By moving the null-field point to the boundary, singular and hypersingular integrals are transformed to series sums after introducing the concept of degenerate kernels. Not only the singularity but also the sense of principle values are novelly avoided. For the calculation of boundary stress, the Hadamard principal value for hypersingularity is not required and can be easily calculated by using series sums. Besides, the boundary-layer effect is eliminated owing to the introduction of degenerate kernels. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. The method is basically a numerical method, and because of its semi-analytical nature, it possesses certain advantages over the conventional boundary element method. The exact solution for a single inclusion is derived using the present formulation and matches well with the Honein et al.’s solution by using the complex-variable formulation (Honein, E., Honein, T., and Hermann, G., 1992, Appl. Math., 50, pp. 479–499). Several problems of two holes, two inclusions, one cavity surrounded by two inclusions and three inclusions are revisited to demonstrate the validity of our method. The convergence test and boundary-layer effect are also addressed. The proposed formulation can be generalized to multiple circular inclusions and cavities in a straightforward way without any difficulty.

On multiple circular inclusions in plane magnetoelasticity

A general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper. By the use of complex variable theory and Laurent series expansion method, the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived. Expanding the definition of the Airy’s stress function of pure elastic field into the magnetoelastic field and applying the superposition method, the general expression then can be reduced to a set of linear algebraic equations and solved in a series form. An approximate closed form solution for the case of two arbitrarily located inclusions is also provided. For illustrating the effect of the pertinent parameters, the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form.

Direct boundary integral procedure for a Boltzmann viscoelastic plane with circular holes and elastic inclusions

A direct boundary integral method in the time domain is presented to solve the problem of an infinite, isotropic Boltzmann viscoelastic plane containing a large number of randomly distributed, non-overlapping circular holes and perfectly bonded elastic inclusions. The holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can, in general, be different. The method is based on a direct boundary integral approach for the problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described by Crouch and Mogilevskaya [1], and a time marching strategy for viscoelastic analysis described by Mesquita and Coda [2–8]. Benchmark problems and numerical examples are included to demonstrate the accuracy and efficiency of the method.

The time-dependent interaction between inclusions and a cracked matrix subjected to antiplane shear

The time-dependent interaction between multiple circular inclusions and a cracked matrix in the antiplane viscoelastic problem is discussed in this paper. The fundamental elastic solution is obtained as a rapidly convergent series in terms of complex potentials via successive iterations of Mobius transformation in order to satisfy continuity conditions on multiple interfaces. Based on the correspondence principle, the Laplace transformed viscoelastic solution is then directly determined from the corresponding elastic one. In association with the singular integral technique, the time-dependent mode-III stress intensity factor of the crack tip can be solved numerically in a straightforward manner. Finally, some typical examples of an arbitrary crack lying in a matrix with various material properties under various loading types are also discussed. The results show that, depending on the relative locations and material properties of inclusions, the evolution of the stress intensity factor (SIF) may increase or decrease with time.

A boundary integral method for multiple circular inclusions in an elastic half-plane

This paper presents a numerical method for solving the problem of an isotropic elastic half-plane containing a large number of randomly distributed, non-overlapping, perfectly bonded circular elastic inclusions. The boundary of the half-plane is assumed to be traction free and a uniform far-field stress acts parallel to that boundary. In general, the inclusions may have different elastic properties and sizes. The analysis is based on a numerical solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. The resulting numerical method allows one to calculate the elastic fields everywhere in the matrix and inside the inclusions. Numerical examples are included to demonstrate the effectiveness of the approach.

A numerical procedure for multiple circular holes and elastic inclusions in a finite domain with a circular boundary

This paper describes a numerical procedure for solving two-dimensional elastostatics problems with multiple circular holes and elastic inclusions in a finite domain with a circular boundary. The inclusions may have arbitrary elastic properties, different from those of the matrix, and the holes may be traction free or loaded with uniform normal pressure. The loading can be applied on all or part of the finite external boundary. Complex potentials are expressed in the form of integrals of the tractions and displacements on the boundaries. The unknown boundary tractions and displacements are approximated by truncated complex Fourier series. A linear algebraic system is obtained by using Taylor series expansion without boundary discretization. The matrix of the linear system has diagonal submatrices on its diagonal, which allows the system to be effectively solved by using a block Gauss-Seidel iterative algorithm.

A Galerkin boundary integral method for multiple circular elastic inclusions with uniform interphase layers

The paper presents a numerical method for solving the problem of an infinite, isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with uniform interphase layers. The bonds between the inclusions and the interphases as well as between the interphases and the matrix are assumed to be perfect. In general, the inclusions may have different elastic properties and sizes; the thicknesses of the interphases and their elastic properties are arbitrary. The analysis is based on a numerical solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex Fourier series. The Galerkin technique is used to obtain a system of linear algebraic equations. The resulting numerical method allows one to calculate the elastic fields everywhere in the matrix and inside the inclusions and the interphases. Using the assumption of macro-isotropic behavior in a plane section one can find the effective elastic moduli for an equivalent homogeneous material. The method can be viewed as an extension of our previous work (Int. J. Solids Struct. 39 (2002) 4723) where simpler spring-like interface conditions were modeled. The problem of overlapping of the fibers and matrix inherent to spring type interface is discussed in the context of the present model. Numerical examples are included to demonstrate the effectiveness of the new approach.

A Galerkin boundary integral method for multiple circular elastic inclusions with homogeneously imperfect interfaces

A Galerkin boundary integral method is presented to solve the problem of an infinite, isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with homogeneously imperfect interfaces. Problems of interest might involve thousands of inclusions with no restrictions on their locations (except that the inclusions may not overlap), sizes, and elastic properties. The tractions are assumed to be continuous across the interfaces and proportional to the corresponding displacement discontinuities. The analysis is based on a numerical solution of a complex hypersingular integral equation with the unknown tractions and displacement discontinuities at each circular boundary approximated by truncated complex Fourier series. The method allows one to calculate the stress and displacement fields everywhere in the matrix and inside the inclusions. Numerical examples are included to demonstrate the effectiveness of the approach.

A Galerkin boundary integral method for multiple circular elastic inclusions

AbstractThe problem of an infinite, isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered. The analysis procedure is based on the use of a complex singular integral equation. The unknown tractions at each circular boundary are approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using the classical Galerkin method and the Gauss–Seidel algorithm is used to solve the system. Several numerical examples are considered to demonstrate the effectiveness of the approach. Copyright © 2001 John Wiley & Sons, Ltd.

On the general treatment of multiple inclusions in antiplane elastostatics

This article provides an explicit general solution of an infinitely extended plate containing any number of circular inclusions under antiplane deformation. The derived solution of the present heterogeneous problem associated with multiple inclusions is obtained in terms of the corresponding homogeneous solution by a simple algebraic substitution. This is accomplished by the technique of analytical continuation and the method of successive approximations. A rapidly convergent series solution either in the matrix or in the inclusions, which is expressed in terms of an explicit general term involving the complex potential of the corresponding homogeneous problem, is derived in an elegant form. The present derived solution can also be applied to the inclusion problem with straight boundaries. Numerical examples of three circular inclusions embedded in an infinite matrix, in a half-plane matrix, and in a strip are discussed in detail and displayed in graphic form. Interaction of a crack with multiple circular inclusions is also considered.