Internal Uniform Stress Field Research Articles

A non-circular Eshelby inclusion near a non-parabolic inhomogeneity admitting internal uniform stresses

With the aid of conformal mapping and analytic continuation, we prove that within the framework of anti-plane elasticity, a non-parabolic open elastic inhomogeneity can still admit an internal uniform stress field despite the presence of a nearby non-circular Eshelby inclusion undergoing uniform anti-plane eigenstrains when the surrounding elastic matrix is subjected to uniform remote stresses. The non-circular inclusion can take the form of a Booth’s lemniscate inclusion, a generalized Booth’s lemniscate inclusion or a cardioid inclusion. Our analysis indicates that the uniform stress field within the non-parabolic inhomogeneity is independent of the specific open shape of the inhomogeneity and is also unaffected by the existence of the nearby non-circular inclusion. On the other hand, the non-parabolic shape of the inhomogeneity is caused solely by the presence of the non-circular inclusion.

RETRACTED: Thermal mechanical coupling analysis of two-dimensional decagonal quasicrystals with elastic elliptical inclusion

In this paper, the thermal mechanical coupling problem of an infinite two-dimensional decagonal quasicrystal matrix containing elastic elliptic inclusion is studied under remote uniform loading and linear temperature variation. Combining with the theory of the sectional holomorphic function, conformal transformation, singularity analysis, Cauchy-type integral and Riemann boundary value problem, the analytic relations among the sectional functions are obtained, and the problem is transformed into a basic complex potential function equation. The closed form solutions of the temperature field and thermo-elastic field in the matrix and inclusion are obtained. The solutions demonstrate that the uniform temperature and remote uniform stresses will induce an internal uniform stress field. Numerical examples show the effects of the thermal conductivity coefficient ratio, the heat flow direction angle and the elastic modulus on the interface stresses. The results provide a valuable reference for the design and application of reinforced quasicrystal materials.

Uniform stress field inside a non-parabolic open inhomogeneity interacting with a mode III crack

Using conformal mapping techniques, analytic continuation and the theory of Cauchy singular integral equations, we prove that a non-parabolic open inhomogeneity embedded in an elastic matrix subjected to a uniform remote anti-plane stress nevertheless admits an internal uniform stress field despite the presence of a finite mode III crack in its vicinity. Our analysis indicates that: (i) the internal uniform stress field is independent of the specific shape of the inhomogeneity and the presence of the finite crack; (ii) the existence of the finite crack plays a key role in the non-parabolic open shape of the inhomogeneity and in the non-uniform stresses in the surrounding matrix; (iii) the two-term asymptotic expansion at infinity of the stress field in the matrix is independent of the presence of the finite crack. Detailed numerical results are presented to demonstrate the proposed theory.

Need the Uniform Stress Field Inside Multiple Interacting Inclusions Be Hydrostatic?

Since the pioneering work of Eshelby on a single ellipsoidal inclusion embedded in an infinite space, much attention has been devoted in the literature to the question of the uniformity of the stress field inside inclusions surrounded by an elastic matrix. Over the last decade or so, researchers have established the existence of multiple (interacting) inclusions enclosing uniform internal stress distributions when embedded in an infinite elastic matrix subjected to a uniform far-field loading and identified a variety of shapes of such inclusions. In the design of multiple inclusions with uniform internal stresses, it is customary to assume that the uniform stress field inside each inclusion (each with different shear modulus distinct from that of the matrix) is hydrostatic. In this paper, we examine whether this assumption is actually necessary to ensure the required existence of multiple inclusions enclosing uniform stresses. By establishing several theorems in the theory of functions of a complex variable, we prove rigorously that for any collection of multiple inclusions enclosing uniform stresses in an infinite isotropic plane subjected to uniform remote in-plane loading, the internal uniform stress field must indeed be hydrostatic if the corresponding inclusion’s shear modulus is distinct from that of the matrix.

A circular Eshelby inclusion with linear eigenstrains interacting with a coated non-parabolic inhomogeneity with internal uniform anti-plane stresses

We establish criteria which guarantee the uniformity of stresses inside a coated non-parabolic open inhomogeneity located in the vicinity of a circular Eshelby inclusion undergoing linear anti-plane eigenstrains when the surrounding matrix is subjected to uniform remote anti-plane stresses. The associated inverse problem in anti-plane elasticity which identifies the required geometry of the constituents of the composite is successfully solved using conformal mapping techniques and analytic continuation. Three conditions on the remote loading in the matrix, uniform eigenstrains imposed on the non-parabolic inhomogeneity and linear eigenstrains imposed on the circular Eshelby inclusion for given geometric and material parameters of the composite are found in order to ensure the uniformity of internal stresses. The internal uniform stress field is found to be independent of the specific open shapes of the inhomogeneity-coating and coating-matrix interfaces, the shear modulus of the coating and also of the existence of the nearby circular Eshelby inclusion. However, the open shapes of the two interfaces are significantly influenced by the presence of the circular Eshelby inclusion. We also discuss the general case in which arbitrary polynomial eigenstrains are imposed on the circular Eshelby inclusion.

A hole of irregular shape interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

We use conformal mapping techniques together with analytic continuation to show that a non-parabolic open elastic inhomogeneity continues to admit a state of uniform internal stress when a hole with closed curvilinear traction-free boundary is placed in its vicinity and the surrounding matrix is subjected to uniform remote anti-plane stresses. The internal uniform stress field inside the inhomogeneity is found to be independent of the existence of the nearby hole and the specific non-parabolic shape of the inhomogeneity. In contrast, the non-parabolic shape of the inhomogeneity is influenced solely by the existence of the nearby hole.

A circular Eshelby inclusion interacting with a non-parabolic open inhomogeneity with internal uniform anti-plane stresses

Using conformal mapping techniques and analytic continuation, we prove that when subjected to anti-plane elastic deformations, a non-parabolic open inhomogeneity continues to admit an internal uniform stress field when a circular Eshelby inclusion is placed in its vicinity and the surrounding matrix is subjected to uniform remote stresses. Explicit expressions for the non-uniform stress distributions in the matrix and in the circular Eshelby inclusion are obtained. The internal uniform stress field is independent of the shape of the inhomogeneity and the presence of the circular Eshelby inclusion, whereas the existence of the circular Eshelby inclusion exerts a significant influence on the shape of the non-parabolic open inhomogeneity as well as on the non-uniform stress distributions in the matrix and in the circular Eshelby inclusion itself.

A screw dislocation near a coated non-elliptical inhomogeneity with internal uniform stresses

We consider the anti-plane shear deformation of a three-phase inhomogeneity-coating-matrix composite containing a coated non-elliptical inhomogeneity whose surrounding matrix is subjected to the action of a screw dislocation and uniform remote anti-plane shear stresses. Our objective is to establish conditions under which the inhomogeneity maintains an internal uniform stress field. Our analysis, which is based on a carefully chosen conformal mapping function, clearly indicates that such an internal uniform stress distribution can be achieved independently of the action of the screw dislocation, which influences the shape of the inhomogeneity depending on its proximity to the dislocation. In fact, we find that when the screw dislocation is located far from the coated inhomogeneity, the corresponding material interfaces become two confocal ellipses as reported previously in the literature. A simple criterion for the convergence of the series in the conformal mapping function is established.

A circular Eshelby inclusion interacting with a coated non-elliptical inhomogeneity with internal uniform stresses in anti-plane shear

We consider a coated non-elliptical inhomogeneity interacting with a nearby circular Eshelby inclusion inside an infinite elastic matrix subjected to anti-plane shear deformations and uniform remote stresses. Using conformal mapping techniques, we prove that despite the presence of the Eshelby inclusion, it is possible to design the system to achieve a uniform stress distribution inside the inhomogeneity. The conformal mapping function used in the analysis is constructed to give rise to an infinite number of first-order poles inside the unit circle in the image plane in order to satisfy all of the conditions required by the complex potential in the matrix. Our analysis indicates that the inhomogeneity's internal uniform stress field is unaffected by the Eshelby inclusion whereas the non-elliptical shape of the coated inhomogeneity is attributed solely to the nearby Eshelby inclusion.

Achieving a uniform stress field in a coated non-elliptical inhomogeneity in the presence of a mode III crack

We establish design criteria which guarantee uniformity of stresses inside a coated non-elliptical inhomogeneity influenced by the presence of a finite mode III crack in a matrix subjected to uniform remote anti-plane shear stresses. We employ a particular conformal mapping function containing an unknown real density function which is obtained via the numerical solution of an associated Cauchy singular integral equation with the aid of the Gauss–Chebyshev integration formula. Interestingly, in contrast to the (non-elliptical) shape of the coated inhomogeneity which is influenced solely by the presence of the nearby crack, the resulting internal uniform stress field remains unaffected by the crack.

Uniformity of stresses inside a non-elliptical inhomogeneity interacting with a mode III crack

Using conformal mapping techniques and the theory of Cauchy singular integral equations, we prove that it is possible to maintain a uniform internal stress field inside a non-elliptical elastic inhomogeneity embedded in an infinite matrix subjected to uniform remote stress despite the fact that the inhomogeneity interacts with a finite mode III crack. The crack can be modelled either as a Griffith crack or as a Zener–Stroh crack. Our analysis further indicates that the existence of the crack plays a key role in influencing the shape of the corresponding inhomogeneity but not the internal uniform stress field inside the inhomogeneity. Numerical examples are presented to demonstrate the solution.

Uniform stress field inside an anisotropic non-elliptical inhomogeneity interacting with a screw dislocation

Abstract We find that an anisotropic non-elliptical inhomogeneity interacting with a screw dislocation in a matrix subjected to remote uniform stresses and anti-plane shear deformations may still admit an internal uniform stress field. Our analysis indicates that the screw dislocation does not affect the uniform stresses inside the inhomogeneity but does play a crucial role in determining the shape of the inhomogeneity. In fact, we find that the influence of the screw dislocation gives rise to the possibility that the boundary of the inhomogeneity may have up to three sharp corners. Our discussion extends to the case when multiple screw dislocations interact with the non-elliptical inhomogeneity leading to a conjecture concerning the maximum allowable number of sharp corners in an inhomogeneity with Eshelby's uniformity property.

Solution of multiple confocally elliptical layers with dissimilar properties in antiplane elasticity with eigenstrains and remote loading

This paper provides a general solution for the problem of multiple confocally elliptical layers in antiplane elasticity. In the problem, the elastic medium is composed of many confocally elliptical layers with different elastic properties, and it is assumed to be isotropic. The eigenstrain is placed in the inclusion. The complex potentials for the inclusion and many layers are assumed in a particular form with two undetermined coefficients. The continuity conditions for the traction and the displacement along the interfaces are formulated exactly. From those conditions, the relation between two sets for two undetermined coefficients in two adjacent layers can be evaluated. By using the above-mentioned relation, a definite relation among (1) the remote loading, (2) the stress in inclusion and (3) the eigenstrian in inclusion can be evaluated. Within the three components, two of them are independent. The proposed condition of the uniform remote stress and the eigenstrain admits an internal uniform stress field. Many computed results are provided, which give a relation among the remote loading, the eigenstrain and stress in inclusion.

Decagonal quasicrystalline elliptical inclusions under thermomechanical loading

In this work, an elegant method is proposed to derive the thermoelastic field induced by thermomechanical loadings in a decagonal quasicrystalline composite composed of an infinite matrix reinforced by an elliptical inclusion. The thermomechanical loadings include a uniform temperature change, remote uniform in-plane heat fluxes and remote uniform in-plane stresses. The corresponding boundary value problem is ultimately reduced to the solution of two independent sets of four coupled linear algebraic equations, each of which involves four complex constants characterizing the internal stress field. The solution demonstrates that a uniform temperature change and remote uniform stresses will induce an internal uniform stress field, and that uniform heat fluxes will result in a linearly distributed internal stress field within the elliptical inclusion. The induced uniform rigid body rotation within the inclusion is given explicitly.

Three-phase anisotropic elliptical inclusions with internal uniform in-plane and anti-plane stresses

We study the stress field of a three-phase composite in which an internal elliptical inclusion is bonded to the surrounding matrix through an interphase layer. The linearly elastic materials occupying both the inclusion and the matrix are generally anisotropic, whereas the interphase layer is made of an isotropic elastic material. The two interfaces of the three-phase composite are confocal ellipses. Two conditions are found that ensure that the internal in-plane and anti-plane stress field is uniform. When these conditions are met, the mean stress within the isotropic interphase layer is also uniform. A real form expression of the internal uniform stress field inside the inclusion is derived. Several examples are presented to demonstrate and validate the obtained results.

Multi-coating an inclusion of arbitrary shape to achieve uniformity of interior stresses

The idea of a multi-coating has been used extensively in the design of invisibility cloaks in which the objective is to achieve near-invisibility of a particular structure. To this end, this paper is concerned with the internal stress field of an elastic inclusion of arbitrary shape bonded to the surrounding matrix through multiple coatings when the matrix is subjected to remote uniform anti-plane stresses. For any number of intermediate coatings, we identify non-elliptical shapes of the inclusion leading to an internal uniform stress field. The problem is solved analytically for an inclusion of arbitrary shape with two coatings, and iteratively for an inclusion of arbitrary shape with three or more coatings. Our results indicate that the shape of an inclusion permitting an internal uniform stress field can essentially be arbitrary when the number of the coatings is sufficiently large.

Finite plane deformations of a three-phase circular inhomogeneity-matrix system

We consider finite plane deformations of a three-phase circular inhomogeneity-matrix system in which the inhomogeneity, the interphase layer and the matrix belong to the same class of compressible hyperelastic materials of harmonic-type but with each phase possessing its own distinct material properties. We obtain the complete solution when the system is subjected to general classes of remote (Piola) stress, specifically, remote stress distributions characterized by stress functions described by general polynomials of order n ⩾ 1 in the corresponding complex variable z used to describe the matrix. As a particular case of the aforementioned analysis, we establish an Eshelby-type result namely that, for this class of harmonic materials, a three-phase circular inhomogeneity under uniform remote stress and eigenstrain, admits an internal uniform stress field when subjected to plane deformations.

Uniformity of stresses within a three-phase elliptic inclusion in anti-plane shear

In this paper, we show that a three-phase elliptic inclusion under uniform remote stress and eigenstrain in anti-plane shear admits an internal uniform stress field provided that the interfaces are two confocal ellipses. The exact closed-form solution is used to quantify the effect of the interphase layer on the residual stresses within the inclusion and the dependency of this effect on the aspect ratio of the elliptic inclusion.