Elliptic Rigid Inclusion Research Articles

A partially debonded rigid elliptical inclusion with a liquid slit inclusion occupying the debonded portion

We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.

Elliptic Rigid Inclusion in Soft Materials of Harmonic Type

Abstract We investigate finite plane deformations of an elliptic rigid inclusion embedded in a soft matrix that is made of a particular class of harmonic-type hyperelastic materials. The inclusion is assumed to be perfectly bonded to the matrix, which is subjected to a constant remote in-plane loading. Utilizing the Cauchy integral techniques associated with conformal mappings, we derive closed-form solutions for the full-field deformation, Piola stress, and Cauchy stress in the entire matrix. Numerical examples are presented to illustrate the current solutions in comparison with those established from linear elasticity theory. We find that in terms of the Cauchy stress around the inclusion, the maximum normal stress component always appears at the endpoints of the major axis of the inclusion, irrespective of the magnitude of the remote loading, while the maximum hoop stress component occurs not exactly at the above-mentioned endpoints when the remote loading exceeds a certain value. In particular, we identify an exact explicit formula for determining the relative rotation of the inclusion during deformation induced by a remote uniaxial loading of arbitrarily given magnitude and direction.

Are the direct and indirect BEM/BIEMs equivalent?

Are the direct and indirect BEM/BIEMs equivalent?

Stress Concentration Due to the Presence of a Rigid Elliptical Inclusion in Porous Elastic Solids Described by a New Class of Constitutive Relations

Stress Concentration Due to the Presence of a Rigid Elliptical Inclusion in Porous Elastic Solids Described by a New Class of Constitutive Relations

Deformation of a plane modelled by John's material with a rigid elliptical inclusion loaded by force and moment

An exact analytical solution is obtained for a non-linear problem of elasticity theory for a plane with a rigid elliptical inclusion. A concentrated force and a moment are applied at the center of inclusion. The elastic properties of the plane are modeled by John’s harmonic material. For this material methods of the theory of functions of a complex variable are using for solving nonlinear plane problems. Nominal stresses and displacements are expressed in terms of two analytical functions of a complex variable, which are determined from the boundary conditions on the contour of inclusion. The problems of the action of a concentrated force and moment on an elliptical core in a plane are considered separately. A comparison with a similar linear problem is made. The influence of the applied force and moment on the magnitude of stresses is studied depending on various parameters of the problem. Calculations of nominal stresses on the contour joining the plane with inclusion are performed.

Interaction of a rigid line inclusion with various discontinuities using experimental and numerical techniques

Using boundary element method (BEM) and photoelasticity, the interaction effects of a rigid line inclusion with various discontinuities embedded in a matrix are investigated. The interaction effect of discontinuities on a rigid line inclusion has not received much attention when compared to the studies on the interaction effect of discontinuities on a crack. The following discontinuities are considered: rigid circular inclusion, rigid elliptical inclusion, circular hole, elliptical hole, and crack when a tensile load is applied along the inclusion length. A circular hole approaching normal to the line inclusion strengthens the strain fields at the rigid line inclusion tip, whereas a hole along the line inclusion alleviates the strain level. However, in the case of rigid circular inclusion, the opposite trends are observed. The influence of the aspect ratio of the elliptical hole on the elastic fields of a rigid line inclusion is also brought out. The influence of a crack near the vicinity of a rigid line inclusion is also studied. The interaction effects of a circular inclusion, a circular hole, and a crack on the rigid line inclusion are further experimentally analyzed for the first time using the photoelastic technique towards validating the boundary element model. The magnitude of singularity at the inclusion tip is quantified using the strain intensity factor. Also, the magnitude of the stress intensity factor (SIF) at the crack tip is reported. Both the strain intensity factor for rigid line inclusion and the SIF for the crack tip are estimated using the domain integral method.

Study on the stress intensity factor and the double-degeneracy mechanism in the BEM/BIEM for anti-plane shear problems

Abstract The crack and the rigid-line inclusion problems under the anti-plane shear are the special case of the elliptic hole and elliptic rigid inclusion, respectively. We revisit this kind of problem by using the degenerate kernels in terms of the elliptic coordinates. The stress intensity factor (SIF) is also addressed. Three ways are employed to determine the SIF. One is the extrapolation approach for the boundary or interior displacement near the tip. Another is the extrapolation approach for the boundary stress or interior stress near the tip. The other is the J-integral enclosing the crack tip. It is interesting to find that a rigid-line inclusion case yields a singular influence matrix due to the degenerate scale of length four in the log kernel. However, double-degeneracy including degenerate scale and degenerate boundary may still result in a singular matrix even though the dual BEM/BIEM is employed. The mechanism is well explained thanks to the introduction of degenerate kernel. Without the introduction of degenerate kernel, the mechanism of the double-degeneracy problem in the BIEM can’t be clearly examined. By using the degenerate kernel, the SIF can be easily determined for the crack or the rigid-line inclusion under the anti-plane shear. The path independence of the J-integral can be derived analytically. The reciprocal relation for the SIF between a crack and a rigid-line inclusion with respect to opposite loading is also addressed. In the numerical implementation, the SIF can be determined using the dual BEM. It is worth noting that BEM shows the advantage that the obtained boundary displacement or boundary stress can be directly used to obtain the more accurate SIF for the crack or the rigid inclusion, respectively.

Isotropic laminated plate containing a coated rigid elliptical inclusion subjected to a rotational moment with constant interfacial and hoop stress resultants

We consider the problem of a rigid elliptical inclusion bonded to an infinite Kirchhoff isotropic laminated plate through a coating layer with two confocal elliptical interfaces. The rigid inclusion is subjected to a rotational moment, while the infinite (plate) matrix surrounding the coated inclusion is subjected to uniform remote membrane stress moments and bending moments. We find that the interfacial and hoop stress resultants are uniformly distributed along the inclusion–coating interface provided a set of four conditions on the rotational moment and remote loading are satisfied for specified material and geometric parameters characterizing the three-phase composite plate.

A coated rigid elliptical inclusion loaded by a couple in the presence of uniform interfacial and hoop stresses

We consider a confocally coated rigid elliptical inclusion, loaded by a couple and introduced into a remote uniform stress field. We show that uniform interfacial and hoop stresses along the inclusion–coating interface can be achieved when the two remote normal stresses and the remote shear stress each satisfy certain conditions. Our analysis indicates that: (i) the uniform interfacial tangential stress depends only on the area of the inclusion and the moment of the couple; (ii) the rigid-body rotation of the rigid inclusion depends only on the area of the inclusion, the coating thickness, the shear moduli of the composite and the moment of the couple; (iii) for given remote normal stresses and material parameters, the coating thickness and the aspect ratio of the inclusion are required to satisfy a particular relationship; (iv) for prescribed remote shear stress, moment and given material parameters, the coating thickness, the size and aspect ratio of the inclusion are also related. Finally, a harmonic rigid inclusion emerges as a special case if the coating and the matrix have identical elastic properties.

Elastic Ellipsoidal Inclusion in a Body Under the Action of Constant Temperature on Their Boundary

We obtain the exact solution of a system of three singular integrodifferential equations of a thermoelastic problem for the space containing an elastic heat-conducting ellipsoidal inclusion. It is assumed that the temperature on the interface of the matrix and inclusion is constant. As a result, we deduce the formulas for the stress concentration near the inclusion, stresses inside the inclusion, and the stress intensity factors K I for an elliptic crack and for a perfectly rigid lamellar elliptic inclusion. The influence of the shape of the inclusion on the stress concentration is analyzed in some special cases.

Ellipsoidal Elastic Inclusion in a Body Under the Action of a Constant Heat Flux on Its Surface

We obtain the analytic solution of a system of three singular integrodifferential equations of the thermoelastic problem for the space containing a thin ellipsoidal elastic inclusion. It is assumed that a constant heat flux acts on the surfaces of the inclusion. We deduce the relations for the stress concentration in the vicinity of the inclusion and stresses inside the inclusion. Special cases of the problem for an elliptic crack and a platelike perfectly rigid elliptic inclusion are considered and the corresponding relations for the stress intensity factors are obtained.

Lekhnitskii’s formalism of one-dimensional quasicrystals and its application

By generalizing the complex potential approach developed by Lekhnitskii, plane problems of one-dimensional quasicrystals are solved first by using an octet formalism for which there are four pairs of complex roots. The approach uses a representation of stresses and proceeds by integration of the expressions for deformations and application of the anisotropic constitutive law and the compatibility of displacements. To illustrate its utility, the generalized Lekhnitskii’s formalism is used to analyse the coupled phonon and phason fields in an infinite quasicrystal medium containing an elliptic rigid inclusion.

Closed form solution for an annular elliptic crack around an elliptic rigid inclusion in an infinite solid

AbstractIn this paper we consider the problem of indentation of an elliptic crack by a rigid elliptic inclusion in anti‐plane shear mode. Making use of integral transforms the solution of the problem is reduced into triple integral equations with cosine kernels and weight functions. Closed form solution of the triple integral equations is obtained and also closed form expressions are obtained for the displacement component and the stress intensity factor.

The three-dimensional problem of a thin inclusion in a composite elastic wedge

The three-dimensional problem of a thin rigid elliptic inclusion in the middle of a composite elastic wedge is investigated. The wedge consists of three connected wedge-shaped layers connected by a sliding clamp, in which the layer containing the inclusion is incompressible. The outer faces of the composite wedge are also under sliding-clamp conditions. The inclusion is completely bonded to the elastic medium in the contact region. Using Fourier and Kontorovich–Lebedev transformations, a system of integral equations of the problems are derived for the shear contact stresses. A regular asymptotic method is used to solve this system. Calculations are carried out. The results can be used for calculations on the strength of rubber-metal articles and structures having a corner line.

Deformable, rigid, and inviscid elliptical inclusions in a homogeneous incompressible anisotropic viscous fluid

Abstract The solution for stress, rate of deformation, and vorticity in an incompressible anisotropic viscous cylindrical inclusion with elliptical cross-section embedded in an incompressible, homogeneous anisotropic viscous medium subjected to a far-field homogeneous rate of deformation is presented. The rate of rotation of a single rigid elliptical inclusion is independent of the ratio of the principal viscosity in “foliation-parallel” shortening or extension to that in foliation-parallel shear, m = η n / η s , and is hence given by the well-known result for the isotropic medium. An analytical expression shows that a thin, very weak elliptical inclusion rotates as though it were a material line in a homogeneous medium [Kocher, T., Mancktelow, N.S., 2005. Dynamic reverse modeling of flanking structures: a source of quantitative kinematic information. Journal of Structural Geology 27, 1346–1354; Kocher, T., Mancktelow, N.S., 2006. Flanking structure development in anisotropic viscous rock. Journal of Structural Geology 28, 1139–1145]. The sense of slip and slip rate across such an inclusion depends on m . The behavior of an isotropic inclusion with viscosity η ∗ in a medium deforming in simple shear parallel to its foliation plane, depends on m and R = η ∗ / η n ; R is the quantity of the same name in Bilby and Kolbuszewski [Bilby, B.A., Kolbuszewski, M.L., 1977. The finite deformation of an inhomogeneity in two-dimensional slow viscous incompressible flow. Proceedings of the Royal Society of London Series A – Mathematical and Physical Sciences 355, 335–353] when the host is isotropic, m = 1. R and m determine ranges of qualitatively different behavior in a finite shearing deformation. For mR = η ∗ / η s mR > 2, depending on initial aspect ratio, a / b , and orientation to the shear plane, ϕ , the inclusions may either undergo periodic motion or asymptotically approach the shear plane as a / b → ∞. In the former case, a stationary point in ϕ , a / b – phase space occurs at ϕ = 0 and ( a / b ) C = ( m [ 1 + R ( m R − 2 ) + 1 ] ) / ( m R − 2 ) . Initial values in the rather broad vicinity of this point undergo periodic motion. For R > R 1 , where m 0.8 R 1 = [ ( η ∗ ) 5 / η n η s 4 ] 1 / 5 ≅ 3.40 , by fit to numerically determined values, all initial pairs of ϕ and a / b lead to periodic motion, which may either be a full rotation about the shear plane or an oscillation.

Anti-plane shear stress problem of two bonded dissimilar half spaces with an elliptical hole or rigid inclusion on the interface

The problem of anti-plane shear stress of two bonded dissimilar half spaces with an elliptical hole or a rigid inclusion at the interface and having interfacial cracks is presented. Uniform anti-plane shear stresses and the stress free or zero displacement boundary conditions on the elliptical hole are considered. The two cases are reduced to Riemann–Hilbert problems and closed form solutions are obtained by use of the complex stress function and the conformal mapping approaches. Stress distributions, as well as stress intensity factor, are shown. When the elliptical hole collapses, the known solutions of the interfacial crack and thin rigid fiber can be obtained. If the coordinates in the Plemelj function are changed, a debonding length can be determined.

Green’s functions for a bi-material problem with interfacial elliptical rigid inclusion and applications to crack and thin rigid line problems

The Green’s functions for a point force and dislocation interacting with interfacial elliptical rigid inclusion in a bonded bi-material system are obtained by applying complex variable method and conformal mapping technique. The problem of an internal crack or thin rigid line interacting with the interfacial inclusion is then examined. For mapping the half plane with a semi-elliptic notch a rational mapping function is used. This helps in evaluating certain contour integrals quite easily. The Green’s function solutions are then used to simulate internal cracks or thin rigid lines to study their behavior in the presence of interfacial inclusion. Some interesting observations pertaining to the interaction between rigid inclusion and crack as well as between rigid inclusion and thin rigid line are discussed. In particular, stress intensity factors (SIF) at the tips of internal crack or stress singularity coefficients (SSC) at the tips of thin rigid line exhibit markedly different behavior depending on loading direction and distance between interfacial inclusion and crack (thin rigid line).

Laminated anisotropic thin plate with an elliptic inhomogeneity

This work is concerned with a through-thickness elliptic elastic inhomogeneity in a laminated anisotropic elastic thin plate within the context of the Kirchhoff theory. By means of the octet formalism recently established by the authors, an exact closed-form solution is obtained, for the first time, for coupled stretching and bending deformations of the plate subjected to remote uniform membrane stress resultants and bending moments. The stress resultants inside the elastic elliptic inhomogeneity are uniform, which is consistent with the uniformity property of the Eshelby inclusion solution in three-dimensional elasticity. In two special limit cases where the elliptic inhomogeneity becomes an elliptic rigid inclusion or hole, the corresponding solutions are also obtained. A relation between the rotational moment and the rotation is given for the elliptic rigid inclusion problem. The concentration factors of hoop membrane stress resultants and hoop bending moments are given for the elliptic hole problem. The intensity factors of membrane stress resultants and moments are obtained for a Griffith crack. Displacements, slopes, membrane stress resultants and bending moments along the elliptic boundary for the elliptic inhomogeneity, rigid inclusion and hole problems are all presented in a real form.

Modes of detachment at the inclusion–matrix interface

Abstract With the help of a 2D theoretical model, this paper analyses the modes of matrix detachment around a circular–elliptical rigid inclusion under pure and simple shear bulk deformations. Three modes of matrix detachment are possible: Mode 1 —the matrix is displaced normal to the inclusion boundary, forming fissures at the interface; Mode 2 —the matrix slips along the inclusion boundary without any separation; Mode 3 —the detachment occurs by a combination of Modes 1 and 2. In order to determine the onset of detachment at any point on the coherent inclusion–matrix interface, the tensile and shear stresses were derived at that point, and compared with imposed critical values. Numerical simulations based on the theoretical model reveal that the three modes occur systematically along the inclusion–matrix interface, the geometrical dispositions of which depend on the aspect ratio ( R ) and orientation ( φ ) of the inclusion. In the case of circular inclusions, Mode 1 and Mode 2 domains are separated by a Mode 3 domain and the disposition shows an internal symmetry. On the other hand, it is asymmetrical when the inclusions are elliptical and oriented oblique to the bulk extension direction in pure shear and to the shear direction in simple shear. In simple shear, Mode 2 detachment with synthetic slip occurs dominantly when φ is less than 45° and greater than 135°. The results obtained from numerical models are complemented with observations in physical experiments. The paper also determines theoretically the critical stresses for detachment to occur as a function of R for different φ values, revealing that for a given mechanical strength of the inclusion–matrix interface, a particular mode of detachment can take place only when the aspect ratio crosses a threshold value.

A Rigid Elliptic Inclusion in an Elastic Medium With a Slipping Interface

ABSTRACTWhen an elastic medium containing an elliptic inclusion with a sliding interface is subjected to a remote pure shear, it was found that the inclusion behaves like a cavity. Since a circle is a special case of an ellipse, the solution should be applicable to a circular inclusion as well. However, it breaks down when the ellipse degenerates into a circle. This implies that the solution is questionable. In this paper the problem is examined by considering a rigid elliptic inclusion in an elastic medium with sliding interface between them. By taking account of a large rotation of the inclusion instead of a small rotation, we obtain a uniformly valid solution applicable to a circular inclusion as well as to an elliptic inclusion. The solution reveals a remarkable snapping behavior of the inclusion under a critical load. A simple condition for its occurrence is derived.