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.     

Non - Classical Mixed Interface Problems for Anisotropic Bodies

The paper deals with the three-dimensional mathematical problems of the elasticity theory of anisotropic piece-wise homogeneous bodies. Non - classical mixed type boundary value problems are studied when on one part (S₁) of the interface the rigid contact conditions (jumps of displacement and stress vectors) are given, while conditions of the non - classical interface Problem H or Problem G are imposed on the remaining part (S₂) of the interface, i. e., in both cases jumps of the normal components of displacement and stress vectors are known on S₂ and, in addition, in the first one (Problem H) the tangent components of the displacement vector and in the second one (Problem G) the tangent components of the stress vector are given from the both sides on S₂. The investigation is carried out by the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

Axially symmetric problems of stationary heat conduction and thermoelasticity for a body with thermally active or thermally insulated disk inclusion (crack)

We present solutions of axially symmetric problems of stationary heat conduction and thermoelasticity for a body with a thin thermally active disk inclusion (where the temperature or heat flow is given) and also with a thermally insulated inclusion. The heat conduction problems are reduced to integral equations, and exact solutions are obtained in the case where their right-hand sides are polynomials of arbitrary degree. We determine the components of the stress tensor and displacement vector as well as, in the case of cracks, the stress intensity factors.     

Solving some problems of elasticity theory by the integral equation method for a holomorphic vector

Under study are some problems of elasticity theory with nonclassical boundary value conditions. We assume that the load and displacement vectors are given on a part of the boundary, while on the other parts of the boundary, the load vector or the displacement vector may be given separately, and no conditions are imposed on the remaining part of the surface (of some nonzero measure).We consider the questions of uniqueness for the solutions to these problems. Solving the nonclassical problems is reduced to a system of singular integral equations for a holomorphic vector.     

Asymptotic solutions of coupled dynamic problems of thermoelasticity for isotropic plates

The asymptotic method of solving boundary-value problems of the theory of elasticity for anisotropic strips and plates is used to solve coupled dynamic problems of thermoelasticity for plates, on the faces of which the values of the temperature function and the values of the components of the displacement vector or the conditions of the mixed problem of the theory of elasticity are specified. Recurrence formulae are derived for determining the components of the displacement vector, the stress tensor and for the temperature field variation function of the plate.     

An Algorithm of Linear Combinations: Thermal Conduction

This paper presents computational algorithms that make it possible to overcome some difficulties in the numerical solving boundary value problems of thermal conduction when the solution domain has a complex form or the boundary conditions differ from the standard ones. The boundary contours are assumed to be broken lines (the 2D case) or triangles (the 3D case). The boundary conditions and calculation results are presented as discrete functions whose values or averaged values are given at the geometric centers of the boundary elements. The boundary conditions can be imposed on the heat flows through the boundary elements as well as on the temperature, a linear combination of the temperature and the heat flow intensity both at the boundary of the solution domain and inside it. The solution to the boundary value problem is presented in the form of a linear combination of fundamental solutions of the Laplace equation and their partial derivatives, as well as any solutions of these equations that are regular in the solution domain, and the values of functions which can be calculated at the points of the boundary of the solution domain and at its internal points. If a solution included in the linear combination has a singularity at a boundary element, its average value over this boundary element is considered.     

Irreversible consolidation problem of a saturated porothermoelastic spherical body with a spherical cavity

Analytical solutions have been derived for spherical space with a spherical cavity based on the irreversible coupled governing equations of saturated porous thermoelastic media when subjected to variable mechanical and thermal loading. Two special cases, a solid spherical body and a spherical cavity in an infinite medium, are analyzed. The method proposed in this study takes into account the coupled variable mechanical and thermal loading on inner and outer boundary surfaces, and a so-called “semi-permeability” boundary condition from completely pervious to completely impervious surface as well. Finally, typical examples are presented to illustrate the evolutions of temperature, pore pressure and displacement with time.     

Vector spherical spline interpolation—basic theory and computational aspects

Vector spherical interpolation is discussed from both the theoretical and computational points of view. The theory of vector spherical harmonics is an essential tool. An estimate is given for the absolute error of the interpolation process; an efficient algorithm is developed for the computation of a vector spherical interpoiant. The displacement boundary value problem of determining the elastic field from a finite number of discretely given displacement vectors is solved by the use of vector splines.     

Accurate calculation of spherical and vector spherical harmonic expansions via spectral element grids

We present in this paper a spectrally accurate numerical method for computing the spherical/vector spherical harmonic expansion of a function/vector field with given (elemental) nodal values on a spherical surface. Built upon suitable analytic formulas for dealing with the involved highly oscillatory integrands, the method is robust for high mode expansions. We apply the numerical method to the simulation of three-dimensional acoustic and electromagnetic multiple scattering problems. Various numerical evidences show that the high accuracy can be achieved within reasonable computational time. This also paves the way for spectral-element discretization of 3D scattering problems reduced by spherical transparent boundary conditions based on the Dirichlet-to-Neumann map.     

Low-cost control problems on perforated and non-perforated domains

We study the homogenization of a class of optimal control problems whose state equations are given by second order elliptic boundary value problems with oscillating coefficients posed on perforated and non-perforated domains. We attempt to describe the limit problem when the cost of the control is also of the same order as that describing the oscillations of the coefficients. We study the situations where the control and the state are both defined over the entire domain or when both are defined on the boundary.     

A general procedure for solving boundary-value problems of elastostatics for a spherical geometry based on Love's approach

We develop a general procedure for solving the first and secondfundamental problems of the theory of elasticity for cases whereboundary conditions are prescribed on a spherical surface, usingLove's general solution of the elastostatic equilibrium equationsin terms of three scalar harmonic functions. It is shown thatthis general solution combined with a methodology by Brennerpaves an elegant way to determine the three harmonic functionsin terms of the boundary data. Thus, with this general scheme,solution of any such boundary-value problem is reducible toa routine exercise thereby providing some `economy of effort'.Furthermore, we develop a similar general scheme for thermoelasticproblems for cases when temperature type boundary conditionsare prescribed on a spherical surface. We then illustrate theapplication of the procedure by solving a number of problemsconcerning rigid spherical inclusions and spherical cavities.In particular, apart from furnishing alternative solutions tothe known problems, we demonstrate the use of this general procedurein solving the problem of interaction of a rigid spherical inclusionwith a concentrated moment and that of a concentrated heat sourcesituated at an arbitrary point outside the inclusion. We alsoderive closed-form expressions for the net force and the nettorque acting on a rigid spherical inclusion embedded into aninfinite elastic solid under an ambient displacement field characterizedby an arbitrary-order polynomial in the Cartesian coordinates.To the best of our knowledge, these results are new.     

A fictitious domain model for the Stokes/Brinkman problem with jump embedded boundary conditions

We present and analyze a new fictitious domain model for the Brinkman or Stokes/Brinkman problems in order to handle general jump embedded boundary conditions (J.E.B.C.) on an immersed interface. Our model is based on algebraic transmission conditions combining the stress and velocity jumps on the interface Σ separating two subdomains: they are well chosen to get the coercivity of the operator. It is issued from a generalization to vector elliptic problems of a previous model stated for scalar problems with jump boundary conditions (Angot (2003, 2005) [2], [3]). The proposed model is first proved to be well-posed in the whole fictitious domain and some sub-models are identified. A family of fictitious domain methods can be then derived within the same unified formulation which provides various interface or boundary conditions, e.g. a given stress of Neumann or Fourier type or a velocity Dirichlet condition. In particular, we prove the consistency of the given-traction E.B.C. method including the so-called do nothing outflow boundary condition.     

Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

We consider Cauchy’s equation of motion for hyperelastic materials. The solution of this nonlinear initial-boundary value problem is the vector field which discribes the displacement which a particle of this material perceives when exposed to stress and external forces. This equation is of greatest relevance when investigating the behavior of elastic, anisotropic composites and for the detection of defects in such materials from boundary measurements. This is why results on unique solvability and continuous dependence from the initial values are of large interest in materials’ research and structural health monitoring. In this article we present such a result, provided that reasonable smoothness assumptions for the displacement field and the boundary of the domain are satisfied for a certain class of hyperelastic materials where the first Piola–Kirchhoff tensor is written as a conic combination of finitely many, given tensors.     

SH波对界面含有半圆形脱胶的圆柱形弹性夹杂的散射的近似分析

采用Green函数法、复变函数法研究了SH波对界面附近含有半圆形脱胶的圆柱形弹性夹杂的散射,并给出了动应力集中系数的数值结果．首先,界面将整个空间分成上下两部分．在下半空间,给出在含有半圆形凸起的圆柱形弹性夹杂的弹性半空间中,水平表面上任意一点承受时间谐和的出平面线源荷载作用时的位移函数．其次,取该位移函数作为Green函数．上下空间连接时在界面处满足连续性条件,构造出半圆形脱胶裂纹,进而求出应力和位移的表达式．最后作为算例,给出了动应力集中系数的数值结果,分析了介质参数和入射波参数对动应力集中的影响情况．     

Investigation of a problem of an elastic half space subjected to stochastic temperature distribution at the boundary

The present work investigates the responses of stochastic type temperature distribution applied at the boundary of an elastic medium in the context of thermoelasticity without energy dissipation. We consider an one dimensional problem of half space and assume that the bounding surface of the half space is traction free and is subjected to two types of time dependent temperature distributions which are of stochastic types. In order to compare the results predicted by stochastic temperature distributions with the results of deterministic type temperature distribution, the stochastic type temperature distributions applied at the boundary are taken in such a way that they reduce to the cases of deterministic types as special cases. Integral transform technique along with stochastic calculus is used to solve the problem. The approximated solutions for physical fields like, stress, temperature, displacement etc. are derived for very small values of time where stochastic type boundary conditions are taken to be of white noise type. The problem is further illustrated with graphical representation of numerical solutions of the problem for a particular case. A detailed comparison of the results of stochastic temperature, displacement and stress distributions inside the half space with the corresponding results of deterministic distributions is presented and special features of the effects of stochastic type boundary conditions are highlighted.     

平面弹性裂纹分析的一种有效边界元方法

提出了一种简单而有效的平面弹性裂纹应力强度因子的边界元计算方法．该方法由Crouch与Starfield建立的常位移不连续单元和闫相桥最近提出的裂尖位移不连续单元构成A·D2在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界．算例(如单向拉伸无限大板中心裂纹、单向拉伸无限大板中圆孔与裂纹的作用)说明平面弹性裂纹应力强度因子的边界元计算方法是非常有效的．此外,还对双轴载荷作用下有限大板中方孔分支裂纹进行了分析．这一数值结果说明平面弹性裂纹应力强度因子的边界元计算方法对有限体中复杂裂纹的有效性,可以揭示双轴载荷及裂纹体几何对应力强度因子的影响．     

State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction

This paper is concerned with the determination of the thermoelastic displacement, stress, conductive temperature, and thermodynamic temperature in an infinite isotropic elastic body with a spherical cavity. A general solution to the problem based on the two-temperature generalized thermoelasticity theory (2TT) is introduced. The theory of thermal stresses based on the heat conduction equation with Caputo’s time-fractional derivative of order α is used. Some special cases of coupled thermoelasticity and generalized thermoelasticity with one relaxation time are obtained. The general solution is provided by using Laplace’s transform and state-space techniques. It is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading (thermal shock). Some numerical analyses are carried out using Fourier’s series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically and the effects of two-temperature and fractional-order parameters are discussed.     

Non-classical interface problems for piecewise homogeneous anisotropic elastic bodies

Two non-classical model interface problems for piecewise homogeneous anisotropic bodies are studied. In both problems on the contact surface jumps of the normal components of displacement and stress vectors are given. In addition, in the first problem (Problem H) the tangent components of the displacement vectors are given from both sides of the contact surface, while in the second one (Problem G) the tangent components of the stress vectors are prescribed on the same surface. The existence and uniqueness theorems are proved by means of the boundary integral equation method, and representations of solutions by single layer potentials are established. In the investigation the general approach of regularization of the first kind of integral equations is worked out for the case of two-dimensional closed smooth manifolds. An equivalent global regularizer operator is constructed explicitly in the form of a singular integro-differential operator.     

Heat transfer in a Walter’s liquid B fluid over an impermeable stretching sheet with non-uniform heat source/sink and elastic deformation

In this present article an analysis is carried out to study the boundary layer flow behavior and heat transfer characteristics in Walter’s liquid B fluid flow. The stretching sheet is assumed to be impermeable, the effects of viscous dissipation, non-uniform heat source/sink in the presence and in the absence of elastic deformation (which was escaped from attention of researchers while formulating the viscoelastic boundary layer flow problems)on heat transfer are addressed. The basic boundary layer equations for momentum and heat transfer, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. Analytical solutions are obtained for the resulting boundary value problems. The effects of viscous dissipation, Prandtl number, Eckert number and non-uniform heat source/sink on heat transfer (in the presence and in the absence of elastic deformation) are shown in several plots and discussed. Analytical expressions for the wall frictional drag coefficient, non-dimensional wall temperature gradient and non-dimensional wall temperature are obtained and are tabulated for various values of the governing parameters. The present study reveals that, the presence of work done by deformation in the energy equation yields an augment in the fluid’s temperature.     

The axisymmetric mixed problem of elasticity theory for a cone clamped along its side surface with an attached spherical segment

The axisymmetric mixed problem of the stress state of an elastic cone, with a spherical segment attached to the base, is considered. The side surface of the cone is rigidly clamped, while the surface of the spherical segment is under a load. By using a new integral transformation over the meridial angle the problem is reduced in transformant space to a vector boundary value problem, the solution of which is constructed using the solution of a matrix boundary value problem. The unknown function (the derivative of the displacements), which occurs in the solution, is determined from the approximate solution of a singular integral equation, for which a preliminary investigation is carried out of the nature of the singularity of the function at the ends of the integration interval. Subsequent use of inverse integral transformations leads to the final solution of the initial problem. The values of the stresses obtained are compared with those that arise in the cone for a similar load, when sliding clamping conditions are specified on the side surface of the cone (for this case an exact solution of this problem is constructed, based on the known result).