Axisymmetric Interface Research Articles

Exact formulation of interface stiffness matrix for axisymmetric bodies under non-axisymmetric loading

Based on the concept of the zero-thickness joint element, a new and accurate formulation of the stiffness matrix for an axisymmetric interface element is presented. This element was formulated to model soil-structures interfaces in semi-analytical analysis of axisymmetric solids of revolution subjected to non-axisymmetric loading. The advantages of the closed-form solution of the interface stiffness matrix are manifold. First of all, it is exact and accurate under all conditions. Secondly, it can be easily used in a finite element code. This formulation requires only the angle of the interface element inclination, thus eliminating the need for numerical integration for each orientation of interface element. Thirdly, it is compatible with the commonly used continuum isoparametric elements such as the six-noded triangle or the eight-noded quadrilateral elements. Two example problems involving a rigid footing and a pile subjected to lateral loading are used to show the advantages of using this new exact formulation.

Analysis of stress intensity factors of a ring-shaped interface crack

In this paper, numerical solutions of singular integral equations are discussed in the analysis of axi-symmetric interface cracks under torsion and tension. The problems of a ring-shaped interface crack are formulated in terms of a system of singular integral equations on the basis of the body force method. In the numerical analysis, unknown body force densities are approximated by the products of the fundamental density functions and power series, where the fundamental densities are chosen to express a two-dimensional interface crack exactly. The accuracy of the present analysis is verified by comparing the present results with the results obtained by other researchers for the limiting cases of the geometries. The calculation shows that the present method gives rapidly converging numerical results for those problems as well as for ordinary crack problems in homogeneous material. The stress intensity factors of a ring-shaped interface crack are shown in tables and charts with varying the material combinations and also geometrical conditions.

1118 リング状界面き裂の応力拡大係数の解析

In this paper, numerical solutions of singular integral equations are discussed in the analysis of axi-symmetric interface cracks under torsion and tension. The problems of a ring-shaped interface crack are formulated in terms of a system of singular integral equations on the basis of the body force method. In the numerical analysis, the unknown functions of the body force densities are approximated by the products of the fundamental density functions and power series, where the fundamental densities are chosen to express a two-dimensional interface crack exactly. The accuracy of the present analysis is verified by comparing the present results with the results obtained by other researchers for the limiting cases. The calculation shows that the present method gives rapidly converging numerical results for these problems as well as for ordinary crack problems in homogeneous material. The stress intensity factors of a ring-shaped interface crack are shown in tables and charts with varying in material combination and also geometrical conditions.

Analysis of Stress Intensity Factors of a Ring Shaped Interface Crack

In this paper, numerical solutions of singular integral equations are discussed in the analysis of axi-symmetric interface cracks under torsion and tension. The problems of a ring-shaped interface crack are formulated in terms of a system of singular integral equations on the basis of the body force method. In the numerical analysis, the unknown functions of the body force densities are approximated by the products of the fundamental density functions and power series, where the fundamental densities are chosen to express a two-dimensional interface crack exactly. The accuracy of the present analysis is verified by comparing the present results with the results obtained by other researchers for the limiting cases. The calculation shows that the present method gives rapidly converging numerical results for these problems as well as for ordinary crack problems in homogeneous material. The stress intensity factors of a ring-shaped interface crack are shown in tables and charts with varying in material combination and also geometrical conditions.

Thermocapillary flows in two-fluids liquid bridges

In this paper we consider the flow and interfaces deformation in non-isothermal two-liquids bridges, i.e. in liquid bridges held between two parallel planar disks, posed at different temperature, which are composed by two coaxial immiscible liquid columns, an inner column of liquid ( a) forming an axisymmetric interface with an outer annular column of liquid ( b) bounded by a free surface from the external still gas. The motion in the two bulk phases arises due to surface gradients of the interface tensions, induced by the imposed temperature difference, both at the liquid–liquid interface and at the external free surface; for the interface tensions it is assumed a linear dependence on the temperature. Under the hypothesis of Reynolds, Peclet and capillary numbers very small, the field equations are expanded in a series of successive linear approximations; the zeroth-order approximation is the Stokes problem, that is analytically solved via separation of variables in terms of infinite series of modified first-order Bessel functions of first and second kind and Papkovitch–Fadle bi-orthogonal eigenfunctions. To improve convergence of the series, Cesaro sums are used. The solution is given for the zeroth order temperature, velocity and pressure fields and for the first-order deviation of the interfaces from the hydrostatic shape. Streamlines, velocity profiles and interface shapes are presented and discussed in terms of Marangoni numbers, viscosity ratio of the two bulk phases and aspect ratios of the two liquid columns.

Stress singularity analyses of interface ends in micro-mechanics tests

Singular stress fields exist in the interface ends of micro-mechanics test specimens, which are used to measure the interfacial shear strength (IFSS) in fiber composites. With the aid of asymptotic and variable separation, the stress singularities near the axisymmetric interface ends are studied. The results show that the α-β diagram of the stress singularity exponents for the push-in specimen can be divided into two areas, singular and non-singular. The oscillatory singular area may appear in the diagram of a pull-out specimen. In the micro-droplet specimen, the oscillatory area is dependent not only on Dundurs parameters but also on the angle between the boundaries of the matrix and fiber. The general forms of angular distribution functions are derived in this paper. By the use of axisymmetric BEM and the obtained angular distribution functions, the numerical results of stress distribution near the interface end and the stress intensity factors are obtained. It is concluded that, theoretically, the interfacial shear strength (IFSS) cannot be determined by the existing interfacial micro-mechanics tests.

Modes of Nonaxisymmetry in the Stability of Fixed Contact Line Liquid Bridges and Drops

A method is presented for predicting the onset and stability character of nonaxisymmetric modes in liquid bridges and drops. The analysis applies to any fixed contact line axisymmetric interface in a steady force field. The onset and stability character of nonaxisymmetric equilibria in liquid bridges and drops is determined. Perturbation analysis is used to locate branches to nonaxisymmetry, and the configuration of the branches then gives the stability character. The number of unstable modes to both constant pressure and constant volume disturbances can be determined, so that changes in stability beyond the primary loss of stability may be examined. Although the first nonaxisymmetric mode tends to dominate higher order modes are significant for liquid bridges where length is less than radius and for drops at higher Bond numbers. At Bond numbers significantly greater than unity, the onset of the least unstable nonaxisymmetric modes tend to collapse between the fixed pressure and fixed volume axisymmetric modes of instability. For liquid bridges, two non-singular classes of nonaxisymmetric mode are distinguished: the predominant, classical shift mode; and a previously unreported tilt mode. The range over which the stability character of fixed contact line liquid bridges and drops is understood is significantly extended.

Stress Singularity at the Edge of Fiber/Matrix Interface.

A general model with arbitrary bonding angles for the edge of a fiber/matrix interface is developed. Many practical problems in composite materials can be included in this model. Based on the elastic basic equations for the spatial axisymmetric problem, the stress singularity at the interface edge is investigated. The singular stress field is also deduced theoretically in detail. It is found that the eigenequation determining the stress singular order coincides with that of the corresponding plane strain problem, while the singular stress distribution does not. Moreover, three composite parameters are required to describe the effect of material combinations on the stress field in the present axisymmetric interface problem, but only two are needed in a two-dimensional plane problem.

Singularity at the Interface Edge of Bonded Transversely Isotropic Piezoelectric Dissimilar Materials. 2nd Report. The Solution for an Interface Edge under Axisymmetric Deformation.

The axisymmetric interface edge of bonded transversely isotropic piezoelectric dissimilar materials with arbitrary bonding angles under axisymmetric deformation is analyzed theoretically. The eigenequation determining the stress singular order, and the stress, displacement, electric displacement and electrical potential energy fields near the interface edge are deduced in details. The results show that the stress singularity order under axisymmetric deformation is dependent of the piezoelectric and dielectric constants, as well as the geometry shape of the interface edge and the mechanical properties of bonded materials. The electric displacement near the interface edge also behaves singular and its singularity is the same of stresses, while the electric potential energy does not behave singular. All the stress, displacement, electric displacement and electric potential energy can be described by same parameters. Due to the anisotropic properties, the results of bonded transversely isotropic piezoelectric materials can no be degenerated to piezo/isotropic or isotropic/isotropic dissimilar materials. This paper also gives out the theoretical results of an interface edge in axsymmetric bonded piezo/isotropic materials.

破壊力学 繊維／マトリックス間の摩擦接触をともなう界面き裂の解析

An axisymmetric interface crack with frictional contact is theoretically analyzed. The eigen-equation determining the stress singular order, and the stress and displacement fields near the crack tip are deduced in details. Based on the results obtained by theoretical analysis, the condition under which the contact will occur is discussed. If the condition is not satisfied, the crack can be dealt as a crack with free surface. The results show that the stress singularity order is dependent on the material combination, the frictional coefficient, and also the direction of the relative displacement of the crack-face. When the crack has a contact region, the oscillatory singularity disappears, and the stress field near the crack tip is similar to that of a pure mode II crack, as it does in plane problems. However, to express the effect of material combinations on stress or displacement fields, three composite parameters are needed, while only two is needed in the plane problems.

Order of singularity and singular stress field about an axisymmetric interface corner in three-dimensional isotropic elasticity

The spatial axisymmetric problem of an isotropic, elastic, homogeneous body containing an inclusion of a different material with an interface corner point (along a circular contour) and arbitrary joining angles is considered in this paper. It is found that the order of the stress singularity at the interface corner coincides with that of the corresponding plane strain problem, but the dependence of the singular stress field on the material properties depends on both the Poisson ratios (of the two isotropic materials) as well as on the ratio of their shear moduli. The theoretical results have been confirmed by numerical, finite-element-based results in a special bimaterial case.

Stress Singularity at the Edge of Fiber/Matrix Interface.

Based on the elastic basic equations for the special axisymmetric problem, the stress singularity at the interface edge in a composite is analyzed. It is found that the eigen equation to determine the singular order coincides with that of the corresponding plane strain problem, while the singular stress distribution does not coincide. Three composite parameters are required to describe the stress field in the present axisymmetric interface problem, but only two are needed in two dimensional plane problem.

The Singularity of Stress Field Near the Axisymmetric Interface End of Push-In Test Specimen of Composites

The Singularity of Stress Field Near the Axisymmetric Interface End of Push-In Test Specimen of Composites

Fourier-Finite-element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations -pΔ 3 u = f in three-dimensional axisymmetric domains Ω. Here, p is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H 1 (Ω), if the interface is smooth or if it meets the boundary of Ω at some edge. In general, the solution u contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ω. The rate of convergence of the combined approximation in H 1 (Ω) is proved to be O(h + N -1 ) (h, N : the parameters of the finite-element- and Fourier-approximation, with h →0, N →∞). The theoretical results are confirmed by numerical experiments.

Fourier–Finite‐element Approximation of Elliptic Interface Problems in Axisymmetric Domains

The paper deals with the Fourier-finite-element method (FFEM), which combines the approximate Fourier method with the finite-element method, and its application to Poisson-like equations −p̂Δ3û = f̂ in three-dimensional axisymmetric domains Ωˆ. Here, pˆ is a piecewise constant coefficient having a jump at some axisymmetric interface. Special emphasis is given to estimates of the Fourier-finite-element error in the Sobolev space H1(Ωˆ), if the interface is smooth or if it meets the boundary of Ωˆ at some edge. In general, the solution û contains a singularity at the interface, which is described by a tensor product representation and treated numerically by appropriate mesh grading in the meridian plane of Ωˆ. The rate of convergence of the combined approximation in H1(Ωˆ) is proved to be 𝒪(h+N−1) (h, N: the parameters of the finite-element- and Fourier-approximation, with h→0, N→∞). The theoretical results are confirmed by numerical experiments.

Pressure and Stress Measurement via Image Analysis (P–SIA) of Axisymmetric Drops and Liquid Bridges

The pressure–stress image analysis method for measuring pointwise pressures and stresses from the curvature of liquid interfaces is presented. The weak force of surface tension provides a precise relative measure for other forces, such as pressure, stress, and buoyancy. In particular, the pointwise jump in pressure or normal stress can be determined from the shape of an axisymmetric liquid interface. The resulting stress profile is a more direct measure of the forces on a liquid surface than global measures such as Bond number. In the static case, the jump in normal stress is the pressure jump at the capillary surface. Pointwise precisions of better than 0.01 Pa in pressure are readily attainable for static systems. In such systems, the pressure profile gives the force of surface tension relative to gravity, reproducing the classical method of measuring surface tension from pendent and sessile drop shapes. In flow systems, the stress profile provides information about local flow effects. An application for balancing gravity effects with axial flow is provided as a practical example.

Numerical modeling of cellular/dendritic array growth: spacing and structure predictions

A numerical model of cellular and dendritic growth has been developed that can predict cellular and dendritic spacings, undercoolings, and the transition between structures. Fully self-consistent solutions are produced for axisymmetric interface shapes. An important feature of the model is that the spacing selection mechanism has been treated. A small, stable range of spacings is predicted for both cells and dendrites, and these agree well with experiment at both low and high velocities. By suitable nondimensionalization, relatively simple analytic expressions can be used to fit the numerical results. These expressions provide an insight into the cellular and dendritic growth processes and are useful for comparing theory with experiment.

Domain integrals for axisymmetric interface crack problems

New domain integrals for axisymmetric interface crack problems are derived. Special attention is given to the specification and subsequent treatment of auxiliary fields for the extraction of mixed-mode stress intensity factors using interaction energy integrals. The effect of crack front curvature is shown to play an important role in the derivation of the integrals. Numerical examples include an interface penny-shaped crack and a longitudinal interface crack (fiber pull-out problem), and yield very accurate and consistent results, when compared with analytical solutions from the literature. The method is particularly suited for the analysis of fiber pull-out experiments and the accurate determination of fracture-parameters such as energy release rate G, mixed-mode stressintensity factors KI and KII and phase angle ψ.

Exact Formulation of Axisymmetric‐Interface‐Element Stiffness Matrix

A new and accurate formulation of the stiffness matrix for an axisymmetric interface or joint element is presented. This element is formulated to model rock joints and soil-soil and soil-structure interfaces in the finite element analysis of axisymmetrical solids of revolution. The formulation is done in the local coordinate system and the closed-form solution of the interface stiffness matrix is obtained. The advantages of the new formulation are: (1) It is exact and accurate under all conditions, versus the various approximate forms; (2) it can be easily used in a finite element code; and (3) the planar interface element is a special case of the new interface element. As such, the stiffness matrices of the planar and axisymmetric interface elements can now be expressed in a unified form, thus eliminating the need for two different types of interface elements. An example problem involving a rigid footing is used to show the advantage of using the new formulation in modeling axisymmetric interface behavior.

Evaluation of two-dimensional and three-dimensional axisymmetric fluid interface shapes with boundary conditions

Evaluation of two-dimensional and three-dimensional axisymmetric fluid interface shapes with boundary conditions