Elastic Cylindrical Inclusion Research Articles

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.

Dynamic responses of a plate with multiple inclusions and depressions subjected to SH waves

Materials and structures in practical engineering inevitably subject to dynamic loads. Surface and internal discontinuities have signification effects on wave propagation and may cause dynamic stress concentration in the vicinity of them. Thus, it is of great significance to determine the dynamic response of materials and structures with surface and internal features under elastic waves. In this article, based on the wave function expansion method, the image method and the complex function method, an analytical solution is presented for the scattering of anti-plane (SH) waves by multiple cylindrical elastic inclusions and semi-cylindrical depressions in an elastic infinite plate. The series expressions of the displacement fields, stress fields, and dynamic stress concentration factors around the inclusion and the depression are derived. Numerical calculations show that the incident wavenumber, shear modulus of the inclusion, plate thickness, and relative position between the inclusion and the depression express great effects on the dynamic stress concentration. The theoretical solutions of this article can provide a reference for the results of numerical methods. And the findings are useful for structural design and nondestructive testing of the plate.

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.

Determination of the Stress State of the Layer with a Cylindrical Elastic Inclusion

A spatial problem of the theory of elasticity for the layer with an infinite round cylindrical inclusion is investigated. At the boundaries of the layer, displacements are given. The cylindrical elastic inclusion is rigidly coupled with the layer and their boundary surfaces do not intersect. The solution to the spatial problem is obtained by the generalized Fourier method, with regard to the Lamé system of equations. The obtained infinite systems of linear algebraic equations are solved by a reduction method. As a result, the values ​​of displacements and stresses in the elastic body are determined. A comparative analysis of the stress state for different geometrical parameters is carried out, and a comparison is made with the stress state in the layer with a cylindrical cavity.

The scattering of steady-state SH waves in a bi-material half space with multiple cylindrical elastic inclusions

The scattering of steady-state SH waves in a bi-material half space with multiple cylindrical elastic inclusions is presented analytically. Mirror method and multi-polar coordinate systems are developed to solve the complex boundary value problem. Considering the displacement and stress continuity conditions, a series of integral equations for unknown coefficients are obtained and solved by truncation. The solution is used to calculate the dynamic stress concentration factor around the edge of the inclusion, and the analysis and numerical results are discussed. The results show that degree of dynamic stress concentration around the circular inclusion is influenced by the incident angle, the incident wave number, and the parameters of medium.

Two Circular Inclusions with Rate-Dependent Interfacial Sliding under Longitudinal Shear

We investigate the time-dependent response of an infinite matrix reinforced by two circularly cylindrical elastic inclusions with viscous sliding interfaces under longitudinal shear. The loadings include remote uniform shear stresses and a screw dislocation fixed in the matrix. The results for remote uniform loading are further used to predict the effective properties of a hybrid composite containing a dilute and random dispersion of perfectly aligned or randomly oriented inclusion pairs. The time-dependent image force acting on the dislocation due to its interaction with the two inclusions is derived, and its far-field asymptotic approximation is obtained. Interestingly, it is found that the asymptotic expression of the image force is intrinsically connected with the expression of the effective shear moduli of the hybrid composite.

Scattering of SH-Wave by an Interface Cylindrical Elastic Inclusion with a Semicircular Debonded above Subsurface Cavity

The Scattering of SH-wave by a cylindrical elastic inclusion on horizontal interface in bi-material space with a semicircular debonded above subsurface circular cavity have been considered using the methods of complex function and Green function. Firstly, we divide the solution domain along the interface and disconnected boundary into two half-spaces, an upper one and a lower one. And Green function was constructed by using the methods of complex function and multi-polar coordinate. Secondly, the bi-material media was connected along the horizontal interface using the idea of interface conjunction, then undetermined anti-plane forces were loaded at the linking sections respectively to satisfy continuity conditions, and a series of Fredholm integral equations of the first kind to determine that the unknown forces could be set up through continuity conditions on surface. Finally, some examples for DSCF around cylindrical elastic inclusion edge are presented and discussed. Numerical results show that subsurface circular cavitys existence notablely influences DSCF of around cylindrical elastic inclusion edge with a semicircular debonded above subsurface circular cavity.

Surface Displacement of a Semi-Cylindrical Hill above a Subsurface Elastic Cylindrical Inclusion under SH-Wave

Scattering of SH-wave by a semi-cylindrical hill above a subsurface elastic cylindrical inclusion in half-space is studied by complex variable function. Firstly, the whole solution domain is divided into two parts, and the solutions that satisfied the boundary conditions are constructed in two parts respectively. Then according to the “conjunction” condition of junction interface and the boundary condition around the subsurface elastic cylindrical inclusion, a set of infinite algebraic equations about the problem can be obtained. Finally the computational results of surface displacement were provided and discussed.

Scattering of Sh-Wave by Cylindrical Inclusion Near Interface in Bi-Material Half-Space

ABSTRACTGreen's function and complex function methods are used here to investigate the problem of the scattering of SH-wave by a cylindrical inclusion near interface in bi-material half-space. Firstly, Green's function was constructed which was an essential solution of displacement field for an elastic right-angle space possessing a cylindrical inclusion while bearing out-of-plane harmonic line source load at any point of its vertical boundary. Secondly, the bi-material media was divided into two parts along the vertical interface using the idea of interface “conjunction”, then undetermined anti-plane forces were loaded at the linking sections respectively to satisfy continuity conditions, and a series of Fredholm integral equations of first kind for determining the unknown forces could be set up through continuity conditions on surface. Finally, some examples for dynamic stress concentration factor of the cylindrical elastic inclusion are given. Numerical results show that dynamic stress concentration factor is influenced by interfaces, free boundary and combination of different media parameters.

Dynamic Analysis for a Subsurface Elastic Cylindrical Inclusion with a Semi- Cylindrical Hill Impacted by SH-Wave

An analytic method is developed for dynamic stress concentration of a subsurface elastic cylindrical inclusion below a semi-cylindrical hill under SH-wave. And the dynamic stress concentration factor (DSCF) is given by complex variable function. During the solution, a standing wave and scattered wave displacement functions are constructed in different parts respectively. All of these displacement functions should satisfy the boundary conditions of each part. Employed to the boundary conditions around the elastic cylindrical inclusion, a series of infinite algebraic equations about the problem can be obtained. The calculating results of DSCF around the elastic cylindrical inclusion are plotted to show the effects of some parameters on DSCF.

Solutions to Eshelby’s problems of non-elliptical thermal inclusions and cylindrical elastic inclusions of non-elliptical cross section

Eshelby’s inclusion problem is solved for non-elliptical inclusions in the context of two-dimensional thermal conduction and for cylindrical inclusions of non-elliptical cross section within the framework of generalized plane elasticity. First, we consider a two-dimensional infinite isotropic or anisotropic homogeneous medium with a non-elliptical inclusion subjected to a prescribed uniform heat flux-free temperature gradient. Eshelby’s conduction tensor field and its area average are first expressed compactly in terms of two boundary integrals avoiding the usual singularity and then specified analytically for arbitrary polygonal inclusions and for inclusions characterized by the finite Laurent series. Next, we are interested in a three-dimensional infinite isotropic or transversely isotropic homogeneous medium with a cylindrical inclusion of a non-elliptical cross section that undergoes uniform generalized plane eigenstrains. The solution to this problem is obtained by decomposing a generalized plane eigenstrain tensor into a plane strain part and an anti-plane strain part, exploiting the mathematical similarity between two-dimensional thermal conduction and anti-plane elasticity, and combining the relevant results of Zou et al. (Zou et al. 2010 J. Mech. Phys. Solids 58 , 346–372. ( doi:10.1016/j.jmps.2009.11.008 )) with those derived in the present work for Eshelby’s conduction tensor field and its area average.

Scattering of Plane Sh-Wave From a Partially Debonded Shallow Cylindrical Elastic Inclusion

ABSTRACTThe scattering of plane SH-wave from a partially debonded shallow cylindrical elastic inclusion in half space is investigated in this paper by complex function method and expansion method of wave function. The debonding regions are considered as multiple arc-shaped interface cracks with non-contacting faces. Firstly, in the inclusion district, the standing wave function in the elastic inclusion with unknown coefficients which satisfies the boundary condition is constructed and generated into the Fourier series; in the half space, the stress and displacement boundary condition around the elastic inclusion can be modeled as the same as the standing wave function in the elastic inclusion. Then, a set of infinite algebraic equations can be obtained around the same boundary and the solution of problem can be gained. In the end, numerical examples of the surface displacement are provided and discussed. It is found that the interface cracks can raise the surface displacement amplitudes to a certain degree.

Scattering of SH-wave from interface cylindrical elastic inclusion with a semicircular disconnected curve

Scattering of SH wave from an interface cylindrical elastic inclusion with a semicircular disconnected curve is investigated. The solution of dynamic stress concentration factor is given using the Green’s function and the method of complex variable functions. First, the space is divided into upper and lower parts along the interface. In the lower half space, a suitable Green’s function for the problem is constructed. It is an essential solution of the displacement field for an elastic half space with a semi-cylindrical hill of cylindrical elastic inclusion while bearing out-plane harmonic line source load at the horizontal surface. Thus, the semicircular disconnected curve can be constructed when the two parts are bonded and continuous on the interface loading the undetermined anti-plane forces on the horizontal surfaces. Also, the expressions of displacement and stress fields are obtained in this situation. Finally, examples and results of dynamic stress concentration factor are given. Influences of the cylindrical inclusion and the difference parameters of the two mediators are discussed.

Generalized Stress Intensity Factor at the Corner of Debonded Fiber Ends

This paper deals with intensity of singular stress field at the corner of debonded ends of an elastic cylindrical inclusion in an infinite body under tension. The problem is formulated as a system of singular integral equations, where unknown functions are densities of body forces distributed in infinite bodies having the same elastic constants as those of matrix and inclusion. In the numerical analysis, the unknown function of the body force densities are expressed as a linear combination of two types of fundamental density function and power series, where the fundamental density functions are chosen to express the symmetric stress singularity of the from rλ1-1 and the skew-symmetrics stress singularity of the from rλ2-1. Then, generalized stress intensity factors, which control the singular stress fields at the end of the cylindrical inclusion are discussed with varying the fiber lengths and elastic modulus ratio. The effect of debonding length on the generalized stress intensity factors is also discussed in comparison with the results of a cylindrical cavity. The results are also compared with ones for a rectangular inclusion.

The Interaction of a Cylindrical Elastic Inclusion with Semicircular Disconnected Curve and Linear Cracks in an Homogeneous Medium by SH-Waves

The scattering of SH waves by a cylindrical elastic inclusion with a semicircular disconnected curve and linear cracks in an homogeneous medium is investigated and the solution of dynamic stress intensity factor is given by Green’s function, complex function method. Firstly, we can divide the space into up-and-down parts along the X axis. In the lower half space, a new suitable Green’s function for the present problem is constructed.In the upper half space, the Green’s function has been given by reference [5]. Thereby the semicircular disconnected curve can be constructed when the two parts are bonded along the interface and the linear cracks can be constructed using the method of crack-division and the integral equations can be obtained by the use of continuity conditions at the X axis. Finally, some examples and results of dynamic stress intensify factor are given and the influence of the parameters is discussed.

Intensity of Singular Stress Fields at the End of a Cylindrical Inclusion

In short fiber reinforced composite it is known that the singular stress at the end of fibers causes crack initiation, propagation, and final failure. The singular stress field is controlled by the generalized stress intensity factors defined at the end of the inclusion. In this study the stress intensity factors are discussed for an elastic cylindrical inclusion in an infinite body under (A) asymmetric uniaxial tension in the x direction, and (B) symmetric uniaxial tension in the z direction. These problems are formulated as a system of integral equations with Cauchy-type or logarithmic-type singularities, where densities of body force distributed in infinite bodies having the same elastic constants as those of the matrix and inclusion are unknown. In the numerical analysis, the unknown body force densities are expressed as fundamental density functions and weight functions. Here, fundamental density functions are chosen to express the symmetric and skew-symmetric stress singularities. Then, the singular stress fields at the end of a cylindrical inclusion are discussed with varying the fiber length and elastic ratio. The results are compared with the ones of a rectangular inclusion under longitudinal and transverse tension.

Interaction of general plane P wave and cylindrical inclusion partially debonded from its viscoelastic matrix

The interaction of a general plane P wave and an elastic cylindrical inclusion of infinite length partially debonded from its surrounding viscoelastic matrix of infinite extension is investigated. The debonded region is modeled as an arc-shaped interface crack between inclusion and matrix with non-contacting faces. With wave functions expansion and singular integral equation technique, the interaction problem is reduced to a set of simultaneous singular integral equations of crack dislocation density function. By analysis of the fundamental solution of the singular integral equation, it is found that dynamic stress field at the crack tip is oscillatory singular, which is related to the frequency of incident wave. The singular integral equations are solved numerically, and the crack open displacement and dynamic stress intensity factor are evaluated for various incident angles and frequencies.

Two circular inclusions with inhomogeneous interfaces interacting with a circular Eshelby inclusion in anti-plane shear

An analytical solution in infinite series form for two circular cylindrical elastic inclusions embedded in an infinite matrix with two circumferentially inhomogeneous imperfect interfaces interacting with a circular Eshelby inclusion in anti-plane shear is derived by employing complex variable techniques. All of those coefficients in the series can be uniquely determined in a simple and transparent way. Numerical examples are given to illustrate the effect of imperfection and circumferential inhomogeneity of the two interfaces as well as the size, location and elastic properties of the two circular inclusions on the stress fields induced within the two circular inclusions and the Eshelby inclusion.

Influence of the cross-section geometry of a cylindrical solid submerged in an acoustic medium on wave propagation

This paper studies wave propagation in the vicinity of a cylindrical solid formation submerged in an acoustic medium generated by point blast loads placed outside the inclusion. The full 3D solution is obtained first in the frequency domain as a discrete summation of responses for 2D problems defined by a spatial Fourier transform. Each 2D solution is computed using the Boundary Element Method, which makes use of two-and-a-half-dimensional Green’s functions. This model is implemented to obtain Fourier spectra responses which make it possible to identify the behavior of both the axisymmetric and non-axisymmetric guided wave modes, when the cross-section of the elastic inclusion changes from circular to smooth oval. When the cylindrical elastic inclusion is submerged in a fluid, thus allowing a dilatational wave velocity greater than the shear wave velocity of the elastic medium (slow formation), our computations show a progressively slower flexural wave and the increased importance of a second flexural mode as the ovality of the inclusion becomes more pronounced. The waves associated with the screw mode become less important as the ovality ratio of the inclusion increases. When the formation is fast (the shear wave velocity of the cylindrical solid inclusion is faster than the pressure wave velocity of the fluid), the responses again indicate a progressively slower flexural wave, as the inclusion becomes more oval. The results computed at the receivers placed on the axis appear to be weakly affected by the ovality ratio of the inclusion.