Determining the intensity factors for stresses,electric-flux density,and electric-field strength in multiply connected electroelastic anisotropic media

Formulas are derived to evaluate the intensity factors of stresses, electric-flux density, and electric-field strength in two-dimensional problems of elasticity, thermoelasticity, electroelasticity, magnetoelasticity, thermoelectroelasticity, and thermomagnetoelasticity for anisotropic bodies with holes and plane (rectilinear) cracks __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 6, pp. 56–62, June 2007.     

Existence of solutions of Kraiko's problem

Three initial-boundary-value problems for the equations of gas dynamics are formulated. Successive solution of these problems yields a solution of Kraiko's problem of the isentropic transition of an ideal gas from a homogeneous state of rest to another state of rest with higher or lower density. Solutions are constructed for plane, cylindrical, and spherical layers of an ideal gas. The existence of locally analytic solutions is proved. Ural State Academy of Service, Ekaterinburg 620034. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 48–55, May–June, 2000.     

Green's function of a point dislocation for the bending of a composite infinite plate with an elliptical hole at interface

Summary  The problem of a hole at bimaterial interface is of practical importance in providing a good understanding of the debonding phenomenon and for determining factors that affect the mechanical properties of composite elements of structures. The problem of a point dislocation in bending bonded dissimilar semi-infinite plates with an elliptical hole at interface is tackled in this paper. Based on the method of analytic continuation and the rational mapping function technique, the problem of obtaining the stress functions in the upper and lower plates is decoupled, and reduced to two Riemann–Hilbert problems. The closed-form solution is obtained. The stress distributions at the bimaterial interface, as well as the debonding at both vertices of the elliptical hole are studied. The stress intensities of debonding are depicted for various parameters. Received 16 March 2000; accepted for publication 12 July 2000     

Plane Strain Bending of Cylindrical Sectors of Admissible Compressible Hyperelastic Materials

In the theory of nonlinear elasticity of rubber-like materials, if a homogeneous isotropic compressible material is described by a strain–energy function that is a homogeneous function of the principal stretches, then the equations of equilibrium for axisymmetric deformations reduce to a separable first-order ordinary differential equation. For a particular class of such strain–energy functions, this property is used to obtain a general parametric solution to the equilibrium equation for plane strain bending of cylindrical sectors. Specification of the arbitrary function that appears in such strain–energy functions yields some parametric solutions. In some cases, the parameter can be eliminated to yield closed-form solutions in implicit or explicit form. Other possible forms for the arbitrary constitutive function that are likely to yield such solutions are also indicated.     

Identification of the shock load on an electroelastic bimorph disk

A numerical–analytic method for the identification of the axisymmetric mechanical shock load on a disk-shaped metal–piezoceramic bimorph transducer is proposed. A problem is formulated based on the theory of thin two-layer plates. The solution is found using the Laplace transform. By recovering the original function analytically, the problem is reduced to a system of Volterra equations, solved numerically using Tikhonov’s regularization algorithm. The finite-element solution of the direct problem is used as input data (potential difference between the electrodes of the piezoceramic layer). The results are analyzed     

One problem of the theory of dimensional electrochemical machining of metals

A method is proposed for determining the shape of the anode-article boundary for a given shape of the cathode-tool in plane problems of the theory of dimensional electrochemical machining of metals. Under the assumptions used, the boundary of the anode-article is divided into the working zone, where metal dissolution occurs, and an adjacent zone, where the treatment (dissolution) is terminated. The initial problem is reduced to a problem of a fictitious plane-parallel potential flow of an ideal fluid with a nonlinear condition on the free surface. The point of separation of the fictitious flow from the solid boundary corresponds to the point separating these two zones of the anode boundary. The Brillouin-Will condition of smooth separation is imposed at the separation point to construct a closed system of equations determining the problem solution. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 214–220, May–June, 2009.     

Anti-plane dynamic hole–crack interaction in a functionally graded piezoelectric media

The anti-plane dynamic problem of a functionally graded piezoelectric plane containing a hole–crack system is treated by a non-hypersingular traction-based boundary integral equation method. The material parameters vary exponentially in the same manner in an arbitrary direction. The system is loaded by an incident SH-type wave, and impermeable boundary conditions are assumed. Using a frequency-dependent fundamental solution of the wave equation, the boundary value problem is transformed into a system of integro-differential equations along the boundary of the hole and on the crack line. Its numerical solution yields the dynamic stress intensity factors and stress concentration factors. A parametric study reveals their dependence on the hole–crack scenario and its geometry, characteristics of the dynamic load and magnitude and direction of material inhomogeneity.     

Kinetics of charge-exchange interaction of dense flows

The nonlinear problem of charge exchange between an ion flow and neutral particles is considered. An exact solution of the equations of charge-exchange interaction in plane geometry is found. Parameters determining the effectiveness of interpenetration of dense flows and the structure of the layer of intense interaction are obtained. Institute of Laser Physics, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 11–19, March–April, 2000.     

Some primal problems of the axisymmetric theory of elasticity for space with a flat slit bounded by a circle

We consider the first and second primal problems of the axisymmetric theory of elasticity for space with a round slit and a mixed problem in which forces are specified on one side of the slit and displacements are specified on the other side. The problems reduce to conjugation problems for generalized analytic functions on rectilinear segments, whose solution is obtained in closed form. Institute of Applied Mechanics, Russian Academy of Sciences, Moscow 117334. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1 pp. 144–151, January–February, 2000.     

Properties of the nonequilibrium high-velocity “Tails” of the molecular distribution function in plane couette flow of a compressible gas

The results of an analytic and numerical investigation of the properties of the high-velocity “tails” of the distribution function are given for the solution of the BGK model of the kinetic Boltzmann equation for plane Couette flow of a compressible gas. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 183–190, July–August, 1998. The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-00573; grant in support of leading science schools No. 96-15-9603).     

Plane waves in a heavy two-layer liquid excited by a cylinder moving at an angle to the horizontal

A solution of an initial-boundary-value problem for a system of integrodifferential equations which describes the plane waves excited in an initially stationary heavy two-layer ideal fluid by a cylinder moving at an angle to the horizontal is investigated. The homogeneous fluid fractions of different densities are assumed to be separated by an evolving fluid interface (horizontal plane, if the liquid is at rest). An approximate solution of two problems for the waves excited by a cylinder moving with a constant acceleration and an oscillating cylinder is constructed analytically. Nizhnii Novgorod. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–152, July–August, 1998.     

Problem of filling of a spherical cavity in a Kelvin-Voigt medium

This paper is concerned with mathematical modeling and solution of the problem of the collapse of a spherical cavity in a viscoelastic medium under the action of constant pressure at infinity. A differential equation of motion for the cavity boundary is constructed and solved numerically. The existence of three modes of motion of the boundary is established, and a map of these modes in the plane of the determining parameters is constructed. Asymptotic forms of the solutions of the problem for all modes are constructed. The problem of cavity collapse with capillary forces taken into account is formulated and solved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 93–101, September–October, 2008.     

Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.     

Deformation of flexible noncircular cylindrical shells under concurrent loads of two types

The paper presents an exact analytic solution to a nonlinear boundary-value problem for a long noncircular cylindrical shell with variable cross-sectional curvature under two concurrent loads: a surface force and bending moments. The behavior of the shell under these loads is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 58–65, July 2007.     

Mixed problem of crack theory for antiplane deformation

The elastic equilibrium of an isotropic plane with one linear defect under conditions of longitudinal shear is considered. The strain field is constructed by the solution of a twodimensional boundary-value Riemann problem with variable coefficients. A special method that reduce the general two-dimensional problem to two one-dimensional problems is proposed. The strain field is described by three types of asymptotic relations: for the tups of the defect, for the tips of the reinforcing edge, and also at a distance from the closely spaced tips of the defect and the rib. The general form of asymptotic relations for strains with finite energy is deduced from analysis of the variational symmetries of the equations of longitudinal shear. A paradox of the primal mixed boundary-value problem for cracks is formulated and a method of solving the problem is proposed. Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 3, pp. 163–172, May–June, 1998.     

一维正方准晶椭圆孔口平面弹性问题的解析解

利用复变方法,引入广义保角映射,研究了一维正方准晶中具有椭圆孔口的平面弹性问题,给出了各应力分量的复变表示,并在特殊情况下转化为Griffith裂纹,得到该裂纹尖端处的应力强度因子的解析解.当准晶体的对称性增加时,正方准晶椭圆孔口平面弹性问题退化为一维四方准晶中具有椭圆孔口的平面弹性问题,同样在特殊情况下转化为Griffith裂纹,得到裂纹尖端处的应力强度因子的解析解.     

Experimental study of the strength of composite plates in bending

The strength of composite plates in bending and the effect of the angular parameters at the tips of the joint of the constituents on the strength are studied. An approximate experimental finite-stress curve (λ=1) that separates the low-stress zone from the stress-concentration zone near the angular point of the joint is obtained in the plane of the angles (α, β) by analyzing the photoelastic patterns (isochromatic curves), fracture lines, and rupture moments. A comparison with the theoretical curve obtained from the solution of the corresponding problem is performed. An experimental relation between the strength of the plate and the angular parameters of the constituents is obtained for the stress-concentration and low-stress states near the joint tips. Institute of Mechanics, Armenia National Academy of Sciences, Erevan 375019. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 211–215, July–August, 2000.     

Stress–strain state of nonthin orthotropic spherical shells of variable thickness

The stress–strain state of an orthotropic spherical shell with thickness varying in two coordinate directions is analyzed. Different boundary conditions are considered, and a refined problem statement is used. A numerical analytic method based on spline-approximation and discrete orthogonalization is developed. The stress–strain state of spherical orthotropic shells with variable thickness is studied     

Modeling the effect of diffusion of the ambient medium on the long-term strength of a hollow cylinder under uniaxial tension

The problem of the long-term strength of an extended thick-walled tube containing a corrosive medium in the internal cavity is solved. The diffusion of this medium into the tube material is analyzed. The diffusion equation is solved approximately by introducing the diffusion front, and the error of the solution is estimated. The dependence of the time of fracture of the tube on the variable tensile stress and the concentration of the medium filling the cavity is obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 88–93, July–August, 2007.     

Numerical simulation of shock wave propagation in microchannels using continuum and kinetic approaches

Numerical simulations of shock wave propagation in microchannels and microtubes (viscous shock tube problem) have been performed using three different approaches: the Navier–Stokes equations with the velocity slip and temperature jump boundary conditions, the statistical Direct Simulation Monte Carlo method for the Boltzmann equation, and the model kinetic Bhatnagar–Gross–Krook equation with the Shakhov equilibrium distribution function. Effects of flow rarefaction and dissipation are investigated and the results obtained with different approaches are compared. A parametric study of the problem for different Knudsen numbers and initial shock strengths is carried out using the Navier–Stokes computations.