Homogeneous solutions in three-dimensional problems of the theory of elasticity for transtropic plates with restrained ends

Homogeneous solutions of equations of the theory of elasticity for transtropic plates, on the flat faces of which the components of the displacement vector are zero, are obtained. The solution consists of the turbulent and potential states of the plate.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 10–13, 1988.     

Functions of a complex variable in problems in the theory of elasticity with mass forces

The general equations of the theory of elasticity are reduced to an inhomogeneous fourth-order equation assuming that there is a linear dependence of the third component of the displacement vector on the third coordinate and that a mass force potential exists. The solution of this equation is presented, in particular, using two complex Kolosov–Muskhelishvili potentials. A third complex potential is introduced in addition to these. Using the three complex potentials, expressions are obtained for the components of the displacement vector and the stress and strain tensors that take account of mass forces. The application of the three potentials is analysed in problems in the theory of elasticity, and analytical solutions of several plane strain problems are presented.     

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.     

A method of solving three-dimensional problems of elasticity for laminated composite plates

An efficient method of solving 3D elasticity problems for thick and thin laminated composite plates is presented. It is based on a new concept of reference surfaces inside the plate. According to this concept, into each nth layer, In arbitrary reference surfaces parallel to the midsurface are introduced, and the displacement vectors of these surfaces are chosen as unknown functions. Such a choice allows one to represent the governing equations of the high-order theory of plates proposed in a very compact form and to derive strain–displacement relationships correctly describing all rigid-body motions of laminated plates.     

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.     

Solutions of half-space and half-plane contact problems based on surface elasticity

Analytical solutions for the problems of an elastic half-space and an elastic half-plane subjected to a distributed normal force are derived in a unified manner using the general form of the linearized surface elasticity theory of Gurtin and Murdoch. The Papkovitch–Neuber potential functions, Fourier transforms and Bessel functions are utilized in the formulation. The newly obtained solutions are general and reduce to the solutions for the half-space and half-plane contact problems based on classical linear elasticity when the surface effects are not considered. Also, existing solutions for the half-space and half-plane contact problems based on simplified versions of Gurtin and Murdoch’s surface elasticity theory are recovered as special cases of the current solutions. By applying the new solutions directly, Boussinesq’s flat-ended punch problem, Hertz’s spherical punch problem and a conical punch problem are solved, which lead to depth-dependent hardness formulas different from those based on classical elasticity. The numerical results reveal that smoother elastic fields and smaller displacements are predicted by the current solutions than those given by the classical elasticity-based solutions. Also, it is shown that the out-of-plane displacement and stress components strongly depend on the residual surface stress. In addition, it is found that the new solutions based on the surface elasticity theory predict larger values of the indentation hardness than the solutions based on classical elasticity.     

Solution of the Basic Boundary Value Problems of Stationary Thermoelastic Oscillations for Domains Bounded by Spherical Surfaces

The boundary value problems of stationary thermoelastic oscillations are investigated for the entire space with a spherical cavity, when the limit values of a displacement vector and temperature or of a stress vector and heat flow are given on the boundary. Also, consideration is given to the boundary-contact problems when a nonhomogeneous medium fills up the entire space and consists of several homogeneous parts with spherical interface surfaces. Given on an interface surface are differences of the limit values of displacement and stress vectors, also of temperature and heat flow, while given on a free boundary are the limit values of a displacement vector and temperature or of a stress vector and heat flow. Solutions of the considered problems are represented as absolutely and uniformly convergent series.     

正交各向异性体梁弯曲的弹性理论

本文由文献[1]横观各向同性板的弯曲弹性理论关于二维问题的特例,通过比拟,得到了正交各向异性梁弯曲的弹性理论,文中给出了求解正交各向异性梁弯曲问题的一种方法.提出了一种新的深梁理论,并指出了考虑横向剪切变形影响的Reissner理论对于应力分量的近似程度较差.     

Asymptotic solutions of coupled dynamic problems of thermoelasticity for thin bodies of anisotropic inhomogeneous-in-plan materials

Two-dimensional recurrence resolvents for an inhomogeneous thin body (plates of variable thickness and shells) are derived by an asymptotic method based on the three-dimensional equations of the coupled dynamic problem of the thermoelasticity of an anisotropic body, which are solved in the case of anisotropy, having, at each point, one plane of symmetry perpendicular to the transverse axis. Recurrence formulae are derived in a general formulation for determining the components of the stress tensor, the strain vector and the function of the change in the temperature field, when different boundary conditions of dynamic problems of the theory of coupled thermoelasticity and thermal conductivity are given on the end surfaces of a thin body. An algorithm for determining the analytical and numerical (necessary) solutions of these boundary-value problems with an arbitrarily specified accuracy is developed.     

Boundary-value problems of the asymmetric theory of elasticity for thin plates

Boundary-value problems of the three-dimensional asymmetric micropolar, moment theory of elasticity with free rotation are considered for thin plates. It is assumed that the total stress-strain state is the sum of the internal stress-strain state and the boundary layers, which are determined in an approximation using asymptotic analysis. Three different asymptotic forms are constructed for the three-dimensional boundary-value problem posed, depending on the values of dimensionless physical constants of the plate material. The initial approximation for the first asymptotic form leads to a theory of micropolar plates with free rotation, the initial approximation for the second asymptotic form leads to a theory of micropolar plates with constrained rotation, and the initial approximation for the third asymptotic form leads to a theory of micropolar plates with “small shear stiffness.” The corresponding micropolar boundary layers are constructed and studied. The regions of applicability of each of the theories of micropolar plates constructed are indicated.     

Longitudinal Shear of Layered Anisotropic Media with Band-Type Inhomogeneities

Using the method of jump functions, we solve the antiplane problem of elasticity theory for a stack of anisotropic strips containing plane band-type inhomogeneities. We model these inclusions by jumps of the stress vector and the derivative of the displacement vector at the middle surfaces. Applying the Fourier integral transformation, we obtain the dependence of the components of the stress tensor and displacement vector on the external load and unknown jump functions. Taking into account the conditions of the interaction between a thin inclusion and an anisotropic medium, we reduce the problem to a system of singular integral equations for the jump functions. A specific example is considered as well.     

关于求解弹性力学平面问题的功的互等定理法

本文推广功的互等定理法于求解具有复杂边界条件矩形板的弹性力学平面问题.首先,我们给出了作为基本系统的四边固定矩形板平面问题的基本解,然后基于在具有复杂边界条件的实际系统与基本系统之间应用功的互等定理,从而求得实际系统的位移表达式.当只存在位移边界条件时,用功的互等定理法求得的位移表达式就是真实的.但是在另一些情况下,当有静力边界条件或混合边界条件时,所求得的位移是容许的.为求得真实位移.必须应用最小势能原理.一些计算表明.对于求解具有复杂边界条件的矩形板弹性力学平面问题,功的互等定理法是一简便通用的方法.显然,这是一个新方法.     

The three-dimensional problem for an anisotropic plate of arbitrary thickness

By expanding the components of the displacement vector in a certain system of functions of the transverse coordinate, we reduce the solution of the three-dimensional problem of the theory of elasticity of an anisotropic body to a series of two-dimensional problems. To determine the displacements we obtain a system of differential equations of infinite order with two independent variables. We show how to pass from the infinite system to a series of finite systems depending on the form of the external forces. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 11–19.     

Asymptotic solution of the first boundary-value problem of the theory of elasticity of the forced vibrations of an isotropic strip

The first boundary-value problem of the theory of elasticity of the forced vibrations of an isotropic strip is solved by an asymptotic method. The asymptotic form of the components of the stress tensor and the displacement vector, which differ in principle from the asymptotic form in the corresponding static problem, is established. All the required quantities in the inner problem are determined and the conditions for resonance to occur are established. The solution in the dynamic boundary layer is constructed and the fundamental (inner) and boundary solutions are matched.     

引用复变量伪应力函数来解幂硬化材料平面应力问题

本文引用复变量伪应力函数将幂硬化材料平面应力问题的协调方程化为双调和方程,从而使此类有强化材料的弹塑性平面应力问题能像线弹性力学平面问题那样采用复变函数法进行求解.本文推导出了幂硬化材料平面应力问题的应力、应变及位移分量的复变函数表达式,可推广应用于满足全量理论的一股弹塑性平面应力问题.作为算例,文中给出了含圆孔幂硬化材料无限大板单向受拉问题的解答,并和有关文献用摄动法获得的同一问题的渐近解进行了比较.     

横观各向同性板的弹性理论和弹性改进理论及一种新的厚板理论

本文从横观各向同性体弹性力学位移形式的基本方程出发,考虑板面承受横向荷载,建立了横观各向同性板弯曲的弹性理论.并由此建立了一个在板的每边能满足三个边界条件的弹性改进理论和一种新的厚板理论.文中求得了周边简支多边形板的弹性改进理论解,数值结果与三维弹性理论精确解的结果非常接近.新的厚板理论和以往的中厚板理论的系统比较表明,我们提出的厚板理论最靠近弹性理论的结果.     

Application of the variational-moment approach to problems of the theory of elasticity of shells

We consider a method of obtaining an approximate system of equations of elasticity theory for shells, based on representing the components of the stress tensor and the displacement vector as a sum of products of moment characteristics depending on the coordinates of the base surface of the shell and functions of the normal coordinate to the base surface.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 56–59.     

Free-edge stress analysis of general composite laminates under extension, torsion and bending

In this study, based on the reduced form of elasticity displacement field for a long laminate, an analytical method is established to exactly obtain the interlaminar stresses near the free edges of generally laminated composite plates subjects to extension, torsion, and bending. The constant parameters being in the displacement field, which describe the global deformation of a laminate, are appropriately calculated by using the improved first-order shear deformation theory. Reddy’s layerwise theory is subsequently employed for analytical and numerical examinations of the boundary layer stresses within arbitrary laminated composite plates. Various numerical results are developed for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates near the free edges. Finally the effects of end conditions of laminates and geometric parameters on the boundary-layer stress are studied.     

A Class of 3D Problems for Layered Plates

Mechanics of Composite Materials - A class of 3D problems of elasticity theory for a layered package of orthotropic plates, which, in particular, models the behavior of the lithospheric plates of...     

Wedge Dislocation in the Geometric Theory of Defects

We consider a wedge dislocation in the framework of elasticity theory and the geometric theory of defects. We show that the geometric theory quantitatively reproduces all the results of elasticity theory in the linear approximation. The coincidence is achieved by introducing a postulate that the vielbein satisfying the Einstein equations must also satisfy the gauge condition, which in the linear approximation leads to the elasticity equations for the displacement vector field. The gauge condition depends on the Poisson ratio, which can be experimentally measured. This indicates the existence of a privileged reference frame, which denies the relativity principle.