Internal Uniform Hydrostatic Stresses Research Articles

Two circular incompressible liquid inclusions

AbstractWe apply Muskhelishvili's complex variable method to solve the plane strain problem associated with two circular incompressible liquid inclusions embedded in an infinite isotropic elastic matrix subjected to uniform remote in‐plane normal stresses. An analytical solution to the problem is derived primarily with the aid of conformal mapping and analytic continuation. To demonstrate our analytical solution, we present detailed numerical results characterizing the internal uniform hydrostatic stresses within the two liquid inclusions, the nonuniform hoop stresses along the two circular liquid‐solid interfaces on the matrix side and the nonuniform rigid body rotations within the two circular liquid inclusions.

An incompressible liquid slit between dissimilar anisotropic elastic media

We use the Stroh sextic formalism to solve the generalized plane strain problem associated with a finite incompressible liquid slit lying between dissimilar anisotropic elastic half-planes subjected to a system of uniform remote stresses. The liquid slit admits an internal uniform hydrostatic stress field and is perfectly bonded to the surrounding media. By enforcing the interface conditions on the solid-solid and liquid-solid interfaces, a Riemann-Hilbert problem of vector form is obtained and is solved analytically using a decoupling procedure. An application of the residue theorem following the imposition of the incompressibility condition of the liquid slit leads to the determination of the internal uniform hydrostatic tension within the liquid slit. When the elastic bimaterial is orthotropic, the internal uniform hydrostatic tension is equal to the remote normal stress component perpendicular to the surfaces of the liquid slit. The presence of the liquid slit will inhibit crack growth along the anisotropic elastic bimaterial interface.

Interaction between an edge dislocation and a circular incompressible liquid inclusion

We use Muskhelishvili’s complex variable formulation to study the interaction problem associated with a circular incompressible liquid inclusion embedded in an infinite isotropic elastic matrix subjected to the action of an edge dislocation at an arbitrary position. A closed-form solution to the problem is derived largely with the aid of analytic continuation. We obtain, in explicit form, expressions for the internal uniform hydrostatic stresses, nonuniform strains and nonuniform rigid body rotation within the liquid inclusion; the hoop stress along the liquid-solid interface on the matrix side and the image force acting on the edge dislocation. We observe that (1) the internal strains and rigid body rotation within the liquid inclusion are independent of the elastic property of the matrix; (2) the internal hydrostatic stress field within the liquid inclusion is unaffected by Poisson’s ratio of the matrix and is proportional to the shear modulus of the matrix; and (3) an unstable equilibrium position always exists for a climbing dislocation.

A non-circular inhomogeneity near a non-elliptical inhomogeneity designed to admit an internal uniform hydrostatic stress field

We prove that it is possible to maintain an internal uniform hydrostatic stress field within a non-elliptical inhomogeneity in the neighborhood of a non-circular inhomogeneity when the surrounding matrix is subjected to uniform remote in-plane stresses. The non-circular inhomogeneity and the matrix have a common shear modulus but distinct Poisson's ratios. The given non-circular shapes of the inhomogeneity include a Booth's lemniscate, a generalized Booth's lemniscate and a cardioid. Our analysis indicates that the uniform hydrostatic field inside the non-elliptical inhomogeneity is unaffected by the existence of the nearby non-circular inhomogeneity whereas the non-elliptical shape of the inhomogeneity is caused solely by the presence of the non-circular inhomogeneity. Numerical examples are presented to demonstrate the general solutions obtained in each of the three non-circular shapes of the inhomogeneity.

SY3-2. Adipokine/Lipokine and inflammatory process in preeclampsia

We prove that it is possible to maintain an internal uniform hydrostatic stress field within a non-elliptical inhomogeneity in the neighborhood of a non-circular inhomogeneity when the surrounding matrix is subjected to uniform remote in-plane stresses. The non-circular inhomogeneity and the matrix have a common shear modulus but distinct Poisson's ratios. The given non-circular shapes of the inhomogeneity include a Booth's lemniscate, a generalized Booth's lemniscate and a cardioid. Our analysis indicates that the uniform hydrostatic field inside the non-elliptical inhomogeneity is unaffected by the existence of the nearby non-circular inhomogeneity whereas the non-elliptical shape of the inhomogeneity is caused solely by the presence of the non-circular inhomogeneity. Numerical examples are presented to demonstrate the general solutions obtained in each of the three non-circular shapes of the inhomogeneity.

Uniform hydrostatic stresses inside a coated non-parabolic inhomogeneity in the vicinity of a concentrated couple

We establish the existence of an internal uniform hydrostatic state of stress inside a coated non-parabolic open elastic inhomogeneity when the surrounding matrix is subjected to a concentrated couple and uniform remote in-plane stresses. The proof of existence is accomplished through analytic continuation and the introduction of a specialized form of the conformal mapping function containing an infinite number of first-order poles. The complex constants appearing in the mapping function can be uniquely determined from a resulting set of non-linear recurrence relations for given material, geometric and loading parameters. The convergence criterion for the series in the mapping function is rigorously established. A simple condition on the remote loading in the matrix is found for given elastic constants of the inhomogeneity and the matrix in order to ensure that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. The analysis indicates that the internal uniform hydrostatic stress field, the constant mean stress in the coating and the constant hoop stress along the inhomogeneity-coating interface on the coating side are all independent of the specific open shapes of the two interfaces and are also independent of the existence of the nearby concentrated couple. However, the non-parabolic open shapes of the two interfaces are significantly influenced by the concentrated couple. We also accomplish the design of an uncoated non-parabolic harmonic elastic inhomogeneity in the presence of an arbitrary number of concentrated couples applied in the matrix.

A circular inhomogeneity interacting with an open inhomogeneity designed to admit internal uniform hydrostatic stresses

We use analytic continuation and conformal mapping techniques to prove the existence of an internal uniform hydrostatic stress field inside a two-dimensional non-parabolic open inhomogeneity interacting with a nearby circular elastic inhomogeneity. The circular inhomogeneity and the surrounding matrix have distinct Poisson's ratios but identical shear moduli. The single complex constant appearing in the conformal mapping function is determined analytically. A simple condition is found that ensures that the internal stresses inside the non-parabolic inhomogeneity are uniform and hydrostatic. We also establish the existence of an internal uniform hydrostatic stress field inside a non-parabolic open inhomogeneity near an arbitrary number of circular elastic inhomogeneities. The circular inhomogeneities and the matrix have distinct Poisson's ratios but a common shear modulus. All of the complex constants in the conformal mapping function are uniquely determined by solving a set of non-linear equations via iterations.

The effect of a nearby circular thermal inclusion on the irregular shape of a coated inhomogeneity with internal uniform hydrostatic thermal stresses

We show that under a uniform temperature change, a uniform hydrostatic residual thermal stress state can still be maintained inside a coated inhomogeneity of irregular shape despite its interaction with a nearby circular thermal inclusion. Our analysis indicates that the internal uniform hydrostatic thermal stress field inside the inhomogeneity, the constant mean stress within the coating and the constant hoop stress along the inhomogeneity-coating interface on the coating side are unaffected by the existence of the nearby circular thermal inclusion. In contrast, the irregular shape of the coated inhomogeneity is attributed solely to the presence of the circular thermal inclusion.

Internal uniform hydrostatic stresses in a three-phase non-elliptical inclusion subjected to a nearby concentrated couple

We apply conformal mapping techniques with analytic continuation to study the existence of a uniform hydrostatic stress field inside a non-elliptical inclusion bonded to an infinite matrix via a finite thickness interphase layer when the matrix is simultaneously subjected to a concentrated couple as well as uniform remote in-plane stresses. We show that the desired internal uniform hydrostatic stress field is possible for given material and geometric parameters provided a certain constraint is placed on the remote loading. Subsequently, when the single loading parameter, five material parameters and three geometric parameters are prescribed, all of the unknown complex coefficients appearing in the series representing the corresponding conformal mapping function can be uniquely determined from a set of nonlinear recurrence relations. We find that the internal uniform hydrostatic stress field, the constant mean stress in the interphase layer and the hoop stress along the inner interface on the interphase layer side are all unaffected by the existence of the concentrated couple whereas the non-elliptical shape of the (three-phase) inclusion is attributed solely to the influence of the nearby concentrated couple.

Two non‐elliptical decagonal quasicrystalline inclusions with internal uniform hydrostatic phonon stresses

AbstractWe use complex variable techniques to prove that the phonon stresses inside two non‐elliptical decagonal quasicrystalline inclusions can be uniform and hydrostatic provided four specific conditions are met. The first condition is a restriction on the phonon elastic constants of the two inclusions, whilst the remaining three conditions can be considered as constraints on the remote phonon and phason stresses for given material and geometric parameters. Once these conditions are satisfied, the hoop phonon stress in the matrix is found to be uniformly distributed along the two inclusion‐matrix interfaces. Moreover, the two inclusions are found to satisfy the harmonic field condition.

Coated inhomogeneities of arbitrary shape with internal uniform hydrostatic stress resultants in isotropic laminated plates

We study the internal stress resultants inside an elastic inhomogeneity of arbitrary shape embedded in an infinite Kirchhoff plate (matrix) via an intermediate coating. Each material phase is considered an isotropic laminated plate and the surrounding matrix is subjected to uniform remote membrane stress resultants and bending moments. We find that internal uniform hydrostatic stress resultants within the inhomogeneity can be achieved when (i) the mapping function characterizing the shape of the inhomogeneity is identified and specialized accordingly; (ii) the relative coating thickness should satisfy a higher-order algebraic equation; (iii) several restrictions on the remote loading should be satisfied for given material and geometric parameters of the composite plate. When all these conditions are met, the hoop membrane stress resultant and hoop bending moment on the coating side are uniform along the entire inhomogeneity–coating interface.

A concentrated couple near two non-elliptical inclusions with internal uniform hydrostatic stresses

We employ conformal mapping techniques to study the existence of internal uniform hydrostatic stresses inside two non-elliptical inclusions when the surrounding matrix is simultaneously subjected to a concentrated couple and remote uniform in-plane stresses. The unknown complex coefficients appearing in the corresponding mapping function can be determined analytically for a given pair of loading, one material and three geometric parameters. This allows us to subsequently identify the shapes of the two inclusions. Our analysis further reveals that the shapes of the inclusions depend on the concentrated couple, whereas the corresponding internal uniform hydrostatic stresses do not.

Three-phase elliptical inclusions with internal uniform hydrostatic stress resultants in isotropic laminated plates

Within the framework of the Kirchhoff-Love isotropic and laminated plate theory, we study the internal stress resultants of a three-phase elliptical inclusion which is bonded to an infinite matrix through an interphase layer. The interfaces of the three-phase inclusion are two confocal ellipses. Two conditions are found to ensure that the internal stress resultants within the elliptical inclusion are uniform and hydrostatic. For given material and geometric parameters of the composite system, the two conditions can be interpreted as restrictions on the remote uniform membrane stress resultants and bending moments. When these two conditions are met, the hoop membrane stress resultant and hoop bending moment in the interphase layer are found to be uniform along the entire interphase/inclusion interface.

Uniform hydrostatic thermal stress fields inside two interacting inclusions of irregular shape and related problem

By using the technique of conformal mapping, we construct the irregular shapes of two interacting elastic inclusions with internal uniform hydrostatic thermal stresses due to a uniform change in temperature. It is found that the hoop stresses along the two inclusion/matrix interfaces on the matrix side are constant. The internal uniform hydrostatic thermal stresses within the two inclusions and the hoop stresses along the two interfaces are independent of the specific inclusion shapes. Numerical results are presented to validate and demonstrate the findings. Furthermore, we design the optimum shapes of the two inclusions that permit internal uniform stresses when uniform anti-plane eigenstrains are imposed on the two inclusions of irregular shape.

Coated non-elliptical harmonic inclusions with internal uniform hydrostatic stresses

This paper is concerned with the question of whether a single-coated, non-elliptical inclusion can be designed in such a way as to satisfy the harmonic condition of Bjorkman and Richards (the so-called ‘harmonic inclusion’). The minimization of stress concentration arising from both the internal and external stress fields of the inclusion is accomplished in the sense that the internal stress field is uniform and hydrostatic while the first invariant of the external stress field in the matrix remains constant. The inclusion is taken to be perfectly bonded to the coating at the inclusion-coating interface while the coating–matrix interface is assumed to be inhomogeneously imperfect of the spring-layer type. It is found that only a coated hypotrochoidal inclusion can be made harmonic and only in the case when the coating and the matrix have different Poisson’s ratios and the matrix is subjected to a uniform remote isotropic field. Both exact and approximate solutions are derived.

Three-phase inclusions of arbitrary shape with internal uniform hydrostatic thermal stresses

We investigate the internal thermal stress field of a three-phase inclusion of arbitrary shape which is bonded to an infinite matrix through an interphase layer. The three phases have different thermoelastic constants. It is found that the internal thermal stress field induced by a uniform change in temperature can be uniform and hydrostatic within an inclusion of elliptical or hypotrochoidal shape when the thickness of the interphase layer is properly designed for given material parameters of the three-phase composite. Several examples are presented to demonstrate the solution. The thermal stress analysis of a (Q + 2)-phase inclusion of arbitrary shape with Q ≥ 2 is also carried out under the assumption that all the phases except the internal inclusion share the same elastic constants. It is found that the irregular inclusion shape permitting internal uniform hydrostatic thermal stresses becomes really arbitrary if a sufficiently large number of interphase layers are added between the inclusion and the matrix.

Three-Phase Inclusions of Arbitrary Shape With Internal Uniform Hydrostatic Stresses in Finite Elasticity

We study the internal stress field of a three-phase two-dimensional inclusion of arbitrary shape bonded to an unbounded matrix through an intermediate interphase layer when the matrix is subjected to remote uniform in-plane stresses. The elastic materials occupying all three phases belong to a particular class of compressible hyperelastic harmonic materials. Our analysis indicates that the internal stress field can be uniform and hydrostatic for some nonelliptical shapes of the inclusion, and all of the possible shapes of the inclusion permitting internal uniform hydrostatic stresses are identified. Three conditions are derived that ensure an internal uniform hydrostatic stress state. Our rigorous analysis indicates that for the given material and geometrical parameters of the three-phase inclusion of a nonelliptical shape, at most, eight different sets of remote uniform Piola stresses can be found, leading to internal uniform hydrostatic stresses. Finally, the analytical results are illustrated through an example.

Neutrality in the case of N-phase elliptical inclusions with internal uniform hydrostatic stresses

A novel method is proposed to design neutral N-phase ( N ⩾ 3) elliptical inclusions with internal uniform hydrostatic stresses. We focus on the study of the internal and external stress states of an N-phase elliptical inclusion which is bonded to an infinite matrix through ( N − 2) interphase layers. The interfaces of the N-phase elliptical inclusion are ( N − 1) confocal ellipses. The design of the resulting overall composite material consists of four stages: (i) an inner perfectly bonded interphase/inclusion interface which is necessary to make the internal uniform stress state hydrostatic; (ii) outer imperfect interphase layers properly designed to make the coated inclusion harmonic (i.e., the uniform mean stress of the original field within the matrix is unperturbed); (iii) the aspect ratio of the elliptic inclusion uniquely chosen for a given material and thickness parameters to make the resulting coated inclusion neutral (i.e., the prescribed uniform stress field in the matrix remains undisturbed); and finally (iv) the derivation of a simple condition relating the remote uniform stresses and the thickness parameters of the ( N − 2) interphase layers for given material parameters which lead to internal uniform hydrostatic stresses. We note that another interesting feature of the present results is that the mean stress is found to be constant within each interphase layer, and the hoop stress in the innermost interphase layer is uniform along the entire interphase/inclusion interface.

Diffusionally modified elastic interaction of edge dislocations with circular inclusions

This research contains two relevant parts. In the first part, we investigate the problem of an edge dislocation interacting with a thickly coated circular inclusion. Both interface slip and diffusion occur on the inclusion/coating and coating/matrix interfaces. We further assume that all these stress relaxation processes on the two interfaces have come to an end. A closed-form solution to the problem is derived. It is observed that: (i) the dislocation will induce an internal uniform hydrostatic stress field within the three-phase circular inclusion; (ii) the dislocation will induce the same uniform rigid body rotation within the inclusion and within the coating layer; and (iii) when a compliant coating layer is thick enough, there exists no climb repulsion region. In the second part, an exact solution to the problem of an edge dislocation inside a circular inclusion with a fully relaxed interface is obtained. We observe that the climb force on the dislocation tends to keep it at the center of the inclusion.

Three-phase elliptical inclusions with internal uniform hydrostatic stresses in finite plane elastostatics

We consider the internal stress field of a three-phase elliptical inclusion bonded to an infinite matrix through an interphase layer when the matrix is subjected to remote uniform stresses. The elastic materials comprising all the three phases belong to a particular class of harmonic materials, and the formed interfaces are two confocal ellipses. A condition leading to internal uniform hydrostatic stresses is derived. This condition relates the two remote principal stresses with the geometric parameters (the thickness of the interphase layer and the aspect ratio of the elliptical inclusion) of the three-phase elliptical inclusion. When this condition is met, the hoop stress in the interphase layer along the entire interphase/inclusion interface is also uniform. Five special situations of practical importance are discussed in considerable detail to demonstrate the unique phenomena inherent in harmonic materials. Our discussions indicate that when this condition is met, it is permissible for the two remote principal stresses to have opposite signs and that for given geometric and material parameters, the remote loading ratio is no longer constant and multiple external loading states exist leading to internal uniform hydrostatic stresses. It is found that this condition can be written into a hyperbola for the two remote principal stresses when the interphase layer is extremely compliant or relatively stiff or when the inclusion is almost rigid. When the magnitudes of the remote stresses are sufficiently large, this condition becomes a very simple one relating the remote loading ratio with the geometric parameters of the composite. Interestingly, it is clearly observed from the simple condition that for given geometric parameters of the three-phase elliptical inclusion, there exist two different values of the remote loading ratio, both of which lead to an internal uniform hydrostatic stress state.