Stress intensity factors for three cracks at the interfaces of a graded layer bonding two different materials

We consider two dissimilar elastic half-planes bonded by a nonhomogeneous elastic layer in which there is one crack at the lower interface between the elastic layer and the lower half-plane and two cracks at the upper interface between the elastic layer and the upper half-plane. The stress intensity factors for these three cracks are solved for when tension is applied perpendicular to the interface cracks. The material properties of the bonding layer vary continuously between those of the lower half-plane and those of the upper half-plane. The differences in the crack surface displacements are expanded in a series of functions that are zero outside the cracks. The unknown coefficients in the series are solved by the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors are calculated numerically for selected crack configurations.     

On one property of solutions of problems of the theory of elasticity for two half-planes or half-spaces

We have shown that the solution of any boundary-value problem for two conjugated half-planes with different elastic constants, in the case where the stresses are persistently continuable across the boundary between half-planes, can be expressed via one common elastic constant if the stresses and external force factors are nondimensionalized by the reduced modulus of elasticity. Owing to this, it is possible to obtain the solution of the problem for two conjugated half-planes directly from the solution of the corresponding problem for one elastic half-plane. This property also holds true for axially symmetric problems formulated for two conjugated half-spaces.     

Contact problems for two anisotropic half-planes with slits

The problem of a stressed state in a nonhomogeneous infinite plane consisting of two different anisotropic half-planes and having slits of finite number on the interface line is investigated. It is assumed that the difference between the displacement and stress vector values is given on the interface line segments; on the edges of the slits we have the following data: boundary values of stress vector (problem of stress) or displacement vector values on one side of the slits, and stress vector values on the other side (mixed problem). Solutions are constructed in quadratures.     

Equivalence Groups for First-Order Balance Equations and Applications to Electromagnetism

We investigate the groups of equivalence transformations for first-order balance equations involving an arbitrary number of dependent and independent variables. We obtain the determining equations and find their explicit solutions. The approach to this problem is based on a geometric method that depends on Cartan's exterior differential forms. The general solutions of the determining equations for equivalence transformations for first-order systems are applied to a class of the Maxwell equations of electrodynamics.     

Analogues of the Kolosov–Muskhelishvili General Representation Formulas and Cauchy–Riemann Conditions in the Theory of Elastic Mixtures

Analogues of the well-known Kolosov–Muskhelishvili formulas of general representations are obtained for nonhomogeneous equations of statics in the case of the theory of elastic mixtures. It is shown that in this theory the displacement and stress vector components, as well as the stress tensor components, are represented through four arbitrary analytic functions.The usual Cauchy–Riemann conditions are generalized for homogeneous equations of statics in the theory of elastic mixtures.     

A circular inhomogeneity with a mixed-type imperfect interface in anti-plane shear

We present a rigorous study of the problem associated with a circular inhomogeneity embedded in an infinite matrix subjected to anti-plane shear deformations. The inhomogeneity and the matrix are each endowed with separate and distinct surface elasticities and are bonded together through a soft spring-type imperfect interphase layer. This combination is referred to in the literature as a ‘mixed-type imperfect interface’ due to the fact that the soft interphase layer (described by the spring model) is bounded by two stiff interfaces arising from the separate surface elasticities of the inhomogeneity and the matrix. The entire composite is subjected to remote shear stresses and we allow for the presence of a screw dislocation in either the inhomogeneity or the matrix. The corresponding boundary value problem is reduced to two coupled second-order differential equations for the two analytic functions defined in the two phases (as well as their analytical continuations) leading to solutions in either series or closed-form. The analysis indicates that the stress field in the composite and the image force acting on the screw dislocation can be described completely in terms of three size-dependent parameters and a size-independent mismatch parameter. Interestingly, in the absence of the screw dislocation, the size-dependent stress field inside the inhomogeneity is uniform. Several numerical examples are presented to demonstrate the solution for a screw dislocation located inside the matrix. The results show that it is permissible for the dislocation to have multiple equilibrium positions.     

Imperfect interface effect for nano-composites accounting for fiber section shape under antiplane shear

This paper investigates the elastic responses of fibrous nano-composites with imperfectly bonded interface under longitudinal shear. The proposed imperfect interface model is the shear lag (or the spring layer) model; the presented nano interfacial stress model is the Gurtin–Murdoch surface/interface model; and the three-phase confocal elliptical cylinder model is the geometry model accounting for the fiber section shape. By virtue of the complex variable method, a generalized self-consistent method is employed to derive the closed from solution of the effective antiplane shear modulus of the fibrous nano-composites with imperfect interface. Five existing solutions can be regarded as the limit form the present analytic expression. The influences of the interface elastic constant, the interfacial imperfection parameter, the size of the elliptic section fiber, the fiber section aspect ratio, the fiber volume fraction and the fiber elastic property on the effective antiplane shear modulus of the nano-composites are discussed. Particularly, numerical results demonstrate that the interfacial elastic imperfection will always cause a significant reduction in the effective antiplane shear modulus; and the fiber interface stress effect on the effective modulus of the fibrous nano-composites will weaken with the interfacial imperfection increases.     

A direct method of integrating the equations of two-dimensional problems of elasticity and thermoelasticity for orthotropic materials

For a plane, a half-plane, and a strip, we propose a direct method of integrating the differential equations of equilibrium and continuity with respect to the stresses in the case of two-dimensional problems of elasticity and thermoelasticity for orthotropic materials. We find the relations between the components of the stress tensor, the key integro-differential equation and the equation of continuity equivalent to it for determining one of the components of the normal stresses. Translated fromMatematichni Metodi i Fiziko-Mekhanichni Polya, Vol. 40, No. 1, pp. 24–29.     

Fundamental solution of a plane problem of elasticity theory for a composite anisotropic medium

The plane problem on the action of an arbitrarily oriented concentrated force, applied at some point of an elastic plane, composed of two different anisotropic half-planes, is considered. By a special choice of a particular solution the problem reduces to a well-known differential equation of the anisotropic theory of elasticity with discontinuous coefficients. The latter reduces, by the method of the integral Fourier transform, to the Riemann boundary value problem. Expressions for the stresses and displacement derivatives at an arbitrary point of the plane are obtained. The application of the obtained results is illustrated on the example of a problem on an elastic linear inclusion (strap).Translated from Dinamicheskie Sistemy, No. 4, pp. 40–45, 1985.     

On the existence of vanishing at infinity symmetric solutions to the plane stationary exterior Navier–Stokes problem

We study a nonhomogeneous boundary-value problem for the steady-state Navier–Stokes equations in a two-dimensional exterior domain with two orthogonal symmetry axes. The existence of a solution which tends to zero uniformly at infinity is proved under suitable parity conditions on the data of the problem. The result is obtained for arbitrary values of the flux of the boundary datum.     

The stressed state of a transversally isotropic medium with a spheroidal inclusion under nonideal mechanical contact

We study the problem of the stressed state of a transversally isotropic medium containing a foreign inclusion in the form of a prolate spheroid under an arbitrary homogeneous stress field at infinity. On the “medium-inclusion” interface there is slipping without flaking. The stressed state is constructed in the medium and in the inclusion using the exterior and interior problems for a prolate spheroid on the basis of potential functions. The solution of the problem is reduced to studying infinite systems of linear algebraic equations. The results of numerical studies are shown as graphs that describe the stress distribution in both the transversally isotropic medium and in the inclusion under various boundary conditions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 15–26.     

Hamilton体系下层合板弱粘接模型的应用

层合板的层间粘接模型对整个层合板的结构有重要影响．运用Hamilton正则方程对层合板层间的不同类型的粘接模型进行了分析．结合弹性材料修正后的Hellinger Reissner变分原理和插值函数,构建了直角坐标系下8节点层合板的每一层的线性方程;考虑到脱层板的连接界面处应力和位移的关系,改进了现有的常用弱粘接模型,建立不同粘接模型的控制方程;最后通过求解整个板的控制方程,得到层合板的层间应力和位移．数值算例验证了该模型的正确性,并研究了层间界面为线性和非线性时的问题．结果表明:应用改进后的弱粘接模型,能够更好地模拟层合板的弱界面失效过程．     

Drained analyses of cylindrical cavity expansion in sand incorporating a bounding-surface model with state-dependent dilatancy

This paper presents a semi-analytical drained solution for cylindrical cavity expansion in sand. By introducing an auxiliary variable, defined as the ratio of the original position to the current position of a material particle, the governing differential equations of the cylindrical cavity expansion problem can be transformed into a group of first-order ordinary differential equations. These equations are solved as an initial value problem by incorporating a bounding-surface model with state-dependent dilatancy. This approach does not require the division of the material around the cavity into an elastic zone and a plastic zone. The state-dependent dilatancy model employed in this study allows the investigation of the effects of the initial relative density and mean normal stress of the sand, whereas the rigorous definitions of the invariant stresses permit the examination of the effect of the initial ratio of the horizontal stress to the vertical stress. Moreover, the model parameters that are of paramount importance for the cylindrical cavity expansion analyses are determined via comprehensive parametric studies.     

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.     

Green's function in plane anisotropic bimaterials with imperfect interface

^** Email: lsudak{at}ucalgary.ca The fundamental solutions or Green's functions for 2D or 3Danisotropic media with imperfect interface remain a challengingproblem. In this paper, a general method is presented for therigorous solution for the 2D Green's function in an anisotropicelastic bimaterial subject to a line force or a line dislocation.Most significant is the fact that the bonding along the bimaterialinterface is considered to be homogeneous imperfect. Specifically,the tractions are continuous but the displacements are discontinuousand proportional, in terms of interface stiffness parameters,to their respective traction components. Using complex variabletechniques, the basic boundary-value problem for two analyticvector functions is reduced to a coupled linear first-orderdifferential equation for a single analytic vector functiondefined in the lower half space. The coupled linear differentialequation for the single analytic vector function can be subsequentlydecoupled into three independent linear first-order differentialequations for three newly defined analytic functions. Closed-formsolutions for the 2D Green's function are derived in terms ofthe exponential integral. Unlike previous works which involvesome sort of inverse transform method to obtain the physicalquantities from the transform domain, the key feature of thepresent method is that the physical quantities can be readilycalculated without the need to perform any inverse transformoperations.     

Numerical modeling of turbulence-induced interfacial instability in two-phase flow with moving interface

The present paper introduces a new interfacial marker-level set method (IMLS) which is coupled with the Reynolds averaged Navier–Stokes (RANS) equations to predict the turbulence-induced interfacial instability of two-phase flow with moving interface. The governing RANS equations for time-dependent, axisymmetric and incompressible two-phase flow are described in both phases and solved separately using the control volume approach on structured cell-centered collocated grids. The transition from one phase to another is performed through a consistent balance of kinematic and dynamic conditions on the interface separating the two phases. The topological changes of the interface are predicted by applying the level set approach. By fitting a number of interfacial markers on the intersection points of the computational grids with the interface, the interfacial stresses and consequently, the interfacial driving forces are easily estimated. Moreover, the normal interface velocity, calculated at the interfacial markers positions, can be extended to the higher dimensional level set function and used for the interface advection process. The performance of linear and non-linear two-equation k–ε turbulence models is investigated in the context of the considered two-phase flow impinging problem, where a turbulent gas jet impinging on a free liquid surface. The numerical results obtained are evaluated through the comparison with the available experimental and analytical data. The nonlinear turbulence model showed superiority in predicting the interface deformation resulting from turbulent normal stresses. However, both linear and nonlinear turbulence models showed a similar behavior in predicting the interface deformation due to turbulent tangential stresses. In general, the developed IMLS numerical method showed a remarkable capability in predicting the dynamics of the considered two-phase immiscible flow problems and therefore it can be applied to quite a number of interface stability problems.     

The stress state of planar surface of a nanometer-sized elastic body under periodic loading

The paper is concerned with the model of an elastic body in the form of a half-plane whose boundary is subjected to periodic loading. It is assumed that there exists an additional surface stress, which is characteristic of nanometer-sized bodies and which obeys the laws of surface elasticity theory. With the use of the boundary properties of analytical functions and the Goursat-Kolosov complex potentials, the boundary value problem in its general setting with an arbitrary load is reduced to a hypersingular integral equation with respect to the derivative of the surface stress. For a periodic load, the solution of this equation is obtained in the form of a Fourier series. The effect of the surface stress upon the stress state of the boundary of the half-plane is examined with independent action of periodically distributed tangential and normal loads. In particular, the size effect was discovered, which is manifested in the dependence of stresses versus the period of loading within several dozens of nanometers. Normal loads are shown to be responsible for tangential stresses on the boundary, which are zero in the classical solution.     

An approximate solution of the problem on quasistatic thermal stresses in a thin plate of a certain noncanonical shape

Making use of L. V. Kantorovich's variational approach and of the methods of the analytic theory of differential equations, one determines the stresses in a thin plate, bounded by two arbitrary curves and two parallel straight lines.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 36–40, 1987.     

无穷远处Ⅲ型均布荷载与集中荷载作用下应变加强层连接唇形裂纹的解析解

通过对保角映射及解析延拓的应用,获得了无穷远处Ⅲ型均布荷载和集中荷载作用下,连接在唇形裂纹面上的内埋应变加强层的级数形式应力解析解．分析了材料匹配、界面连接及几何特征对界面应力的影响,研究发现对于不同的荷载形式,合理的材料匹配、连接及几何特征能够有效地减少应力集中和界面应力．     

On Nonhomogeneous Slip Boundary Conditions for 2D Incompressible Exterior Fluid Flows

The paper examines the issue of existence of solutions to the steady Navier-Stokes equations in an exterior domain in ℝ². The system is studied with nonhomogeneous slip boundary conditions. The main results proves the existence of weak solutions for arbitrary data.