Elastic Cylindrical Inclusion Research Articles

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.

Analysis of Intensity of Singular Stress Fields at the End of a Cylindrical Inclusion under Asymmetric Uniaxial Tension.

This paper deals with generalized stress intensity factors at the end of an elastic cylindrical inclusion in an infinite body under asymmetric uniaxial tension. These stress intensity factors control singular stress fields at the end of inclusion. The problem is solved on the superposition of two auxiliary loads;(i)biaxial tension and (ii)plane state of pure shear. The problem is formulated as a system of integral equations with Caushy-type or logarithmic-type singularities, where un-knowns are densities of body force distributed in infinite bodies having the same elastic constants as those of the matrix and inclusion. In the numerical analysis, the unknown functions of the body force densities are expressed as fundamental density functions and weight functions. Fundamental density functions are chosen to express the symmetric stress singularity of the from 1/γ1-λ11/γ1-λ3 and the skew-symmetric stress singularity of the form 1/γ1-λ21/γ1-λ4. 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 cylindrical inclusion under longitudinal tension and the ones of a rectangular inclusion under transverse tension.

Analytical solution for the plane strain inclusion problem of an elastic power-law hardening matrix containing an elastic cylindrical inclusion

An elasto-plastic analytical solution is presented for the plane strain inclusion problem of an elastic power-law hardening matrix containing an elastic cylindrical inclusion and subjected to equi-biaxial tension. Hencky’s deformation theory and von Mises’ yield criterion are used. All stress, strain and displacement components are derived in explicit forms in terms of two constant parameters, which are determined from the boundary conditions using a simple iterative procedure. Three specific solutions of practical interest are obtained as limiting cases. Illustrative numerical results are also provided to demonstrate applications of the solution. It is quantitatively shown that the extent of plastic deformation in the matrix is controlled by the ratio of Young’s moduli of the inclusion and matrix materials, Poisson’s ratio of the inclusion material and the strain-hardening effect of the matrix material, of which the first two factors are dominant. These findings are of practical importance to the design of fiber-filled metal-matrix composites.

Response of an embedded fiber optic ultrasound sensor

In this work the response of an embedded fiber optic Fabry–Perot ultrasound sensor has been investigated. Calculations were performed that model the fiber as a cylindrical elastic inclusion subjected to obliquely incident harmonic plane waves. The phase shift of the light traveling in the fiber was calculated along the principal strain axes as a function of ultrasonic frequency and incident angle. Experiments were performed to validate the calculations. Induced phase shifts were measured for the cases of normally incident ultrasound impinging upon a fiber sensor submerged in water as well as embedded in epoxy. The case of obliquely incident ultrasound at angles up to 4 deg was also investigated for a fiber embedded in epoxy. The experimental measurements were performed over a bandwidth from 2 to 8 MHz.

Analysis of Intensity of Singular Stress at the End of a Cylindrical Inclusion.

This paper deals with numerical solutions of singular integral equations in the problem of an elastic cylindrical inclusion with ends in an infinite body under tension. The problem is formulated as a system of singular integral equations with Cauchy type or logarithmic type singularities, where unknown functions are densities of body forces distributed in infinite bodies having the same elastic constants as those of the matrix and inclusion. In the numerical analysis, the unknown functions of the body force densities are expressed as a linear combination of two types of fundamental density functions and power series, where the fundamental density functions are chosen to express the symmetric stress singularity of the form 1/r1-λ1 and the skew-symmetrics stress singularity of the form 1/r1-λ2. Then, the singular stress fields at one end of a cylindrical inclusion are discussed for various fiber lengths and elastic ratios. The results are also compared with ones for a rectangular inclusion.

Elastic Interphase with Variable Properties in Fibrous Composites

In the present work a theoretical model for composite materials with a Poisson's ratio v = 0.5 is developed in order to evaluate the mechanical stresses around a single cylindrical elastic fibre inclusion. The model considers a gradual and continuous variation of the longitudinal elastic modulus EL in the area between matrix and fibre, the former being under hydrostatic pressure po · EL is expressed as a Fermi-Dirac distribution function of important parameters such as the rate of variation of EL, the extent of the interphase and the fibre volume fraction. Results indicate a strong effect of the above parameters on the stress state developed around the fibre.

On two circular inclusions in harmonic problems

In this paper, we derive the solution for two circular cylindrical elastic inclusions perfectly bonded to an elastic matrix of infinite extent, under anti-plane deformation. The two inclusions have different radii and possess different elastic properties. The matrix is subjected to arbitrary loading. The solution is obtained, via iterations of Möbius transformations, as a rapidly convergent series with an explicit general term involving the complex potential of the corresponding homogeneous problem, i.e., when the inclusions are absent and the matrix material occupies the entire space and is subjected to the same loading. This procedure has been termed “heterogenization."

Diffraction of shear waves by an elastic cylindrical inclusion with two cuts on the phase boundary

A method /1/ similar to that used in the case of one cut /2/ is used to determine the stress and deformation at the boundary of a cylindrical inclusion with two cuts placed on the contact contour. The external perturbation varies sinusoidally and is a plane wave in an isotropic medium. At the boundary of the inclusion the shear wave is reflected as a shear wave.

Excitation of elastic waves in a half-space containing a horizontal elastic cylindrical inclusion

We present a method for the rigorous analysis of wave fields excited in an elastic halfspace containing a horizontal elastic cylindrical inclusion by a distributed harmonic surface load. For simplicity we illustrate the application of the method by investigating a model problem of antiplane oscillations of the composite elastic medium. We derive asymptotic representations of the solutions when the inclusion is rigidly bonded, which permit a rather simple analysis of the wave field in the medium. The proposed method can be applied without changes to similar problems in two and three dimensions. In doing this only the awkwardness of the derived relations is increased significantly.

The Born approximation for the scattering theory of elastic waves by two-dimensional flaws

First, we present formal aspects of the scattering of plane elastic waves by a single elastic cylindrical inclusion or a cavity of an arbitrary cross section embedded in an isotropic homogeneous elastic medium of different properties. An integral equation is used to derive expressions for the displacement field valid in the entire domain. Thereafter the Born approximation for the solution is derived. Formulas for the scattered amplitudes and the scattering cross sections are presented. The validity of the approximation is examined by comparing the results with those of the simple bodies for which the exact solutions are available in the literature. Some new results for the corresponding three-dimensional case are also presented.

The influence of porosity on the elastic response of homogeneous and structural composite media

Continuum theories of composites are employed to analyze the influence of inclusions and porosity on the elastic response of both homogeneous and laminated composite media. The general model analyzed consists of a periodic array of two perfectly bonded laminates; one of which consists of an elastic homogeneous material while the other is made up of a periodic array of cylindrical elastic inclusions that are distributed in another elastic matrix material. Several specific models are deduced as special cases. In all cases, porosity is simulated in the limit as the properties of the inclusions identically vanish. It is demonstrated that porosity plays a major role in the geometric dispersion of such media; in particular, it increases the arrival and rise times (spreading) of a propagating transient pulse. For the special case of elastic inclusions in a homogeneous matrix media, the present results correlate very well with existing experimental data and other approximate analyses.

Wave front analysis of a plane compressional pulse scattered by a cylindrical elastic inclusion

The plane-strain problem of a stress pulse striking an elastic circular cylindrical inclusion embedded in an infinite elastic medium is treated. The method used determines dominant stress singularities that arise at wave fronts from the focusing of waves refracted into the interior. It is found that a necessary and sufficient condition for the existence of a propagating stress singularity is that the incident pulse has a step discontinuity at its front. The asymptotic wave front behavior of the first few P and SV waves to focus are determined explicitly and it is shown that the contribution from other waves are less important. In the exterior, it is found that in most composite materials the reflected waves have a singularity at their wave front which depends on the angle of reflection. Also the wave front behavior of the first few singular transmitted waves is given explicitly.The analysis is based on the use of a Watson-type lemma, developed here, and Friedlander's method [5]. The lemma relates the asymptotic behavior of the solution at the wave front to the asymptotic behavior of its Fourier transform on time for large values of the transform parameter. Friedlander's method is used to represent the solution in terms of angularly propagating wave forms. This method employs integral transforms on both time and θ, the circumferential coordinate. The θ inversion integral is asymptotically evaluated for large values of the time transform parameter by use of appropriate asymptotics for Bessel and Hankel functions and the method of stationary phase. The Watson-type lemma is then used to determine the behavior of the solution at singular wave fronts.The Watson-type lemma is generally applicable to problems which involve singular loadings or focusing in which wave front behavior is important. It yields the behavior of singular wave fronts whether or not the singular wave is the first to arrive. This application extends Friedlander's method to an interior region and physically interprets the resulting representation in terms of ray theory.

Generalised plane strain problem for a circular inclusion in anisotropic elasticity

This paper is concerned with a generalised plane deformation problem in the linear theory of anisotropic elasticity. As is well known, the generalised plane deformation is the deformation of a body of infinite length bounded by a cylindrical surface, when all the stress and strain components exist but they are functions of two co-ordinates x 1, and x 2 only. It may be shown that if u 3 = 0, it is impossible to satisfy all the three equations of equilibrium of anisotropic elastic body. One has to choose u 3 as a non-zero function of x 1, x 2 for satisfying equations of equilibrium. In isotropic elasticity, u 3 = 0, makes the third equation of equilibrium identically equal to zero. The problem in this paper concerns an elastic circular cylindrical inclusion embedded in a matrix of different anisotropic material. The matrix and the inclusion are perfectly bonded at the interface. Each of the two materials possesses anisotropy of a general form with all the 21 elastic constants. The matrix is subjected to a uniform stress at infinity. The equations of elasticity theory demand that the rotation component ω 3 must also be prescribed at infinity. The complex variable technique is used and exact analytical expressions are derived for the elastic field in both the regions.

A numerical method for computing the stresses around an axisymmetrical inclusion

A numerical method is described for finding stresses in an elastic solid containing an axisymmetrical inclusion subjected to uniform tension. The method is based on the classical three-function approach in the theory of elasticity in conjunction with an improved point-matching technique. Three problems, namely the rigid spherical inclusion, the rigid cylindrical inclusion of finite length and the elastic cylindrical inclusion of finite length, have been studied. Results are presented in graphical form showing the variations of bond stresses with aspect ratio, corner radius and modulus ratio.