Mixed Problem Of Elasticity Theory Research Articles

TWO APPROACHES FOR DETERMINING AREAS OF PLASTIC DEFORMATION IN A HOMOGENEOUS BASE FOUNDATION

The paper presents comparison of results of calculations of position, sizes and shapes of plastic deformation areas carried out on the basis of use of Coulomb plasticity condition and solution of elastic-plastic problem and mixed problem of elasticity theory and ground plasticity theory which resulted, among other things, in graphs of dependence of depth of plastic zones development on value of intensity of external influence. Calculations have been executed at three values of coefficient of side pressure of ground about = 0.37; 0.54; 0.72 for uniform strip load (no die), and for absolutely rigid die under condition of full sticking and under condition of no friction on ground-stamp contact. As a result, it was found that the plastic regions corresponding to the "Coulomb" solution differ essentially from similar regions based on elastic-plastic (mixed) solutions. Their sizes, with the same intensity of loading, are larger, their rate of development into the deepening of the base is higher than that of the alternative ones, and when the load reaches the maximum permissible value, they merge under the punch and form a single sickle-shaped area. In the case of "Coulomb" plastic areas, their shape and rate of development into the foundation in the initial stages of loading are rather strongly dependent on the value of soil side pressure coefficient. In the case of "mixed" solution, the "velocity" of this process is practically independent of the value of coefficient o. If we use the condition of closing of plastic regions in the foundation under load as the criterion for determination of the maximum permissible value of intensity of external action, then in "Coulomb" regions this value will be substantially lower than in alternative ones for all considered values of the coefficient of lateral pressure of soil.

Singular solutions of contact problems and block elements

In this work, we consider mixed problems of elasticity theory, in particular, contact problems for cases that are nontraditional. They include mixed problems with discontinuous boundary conditions in which the singularities in the behavior of contact stresses are not studied or the energy of the singularities is unbounded. An example of such mixed problems is contact problems for two rigid stamps approaching each other by rectilinear boundaries up to contact but not merging into one stamp. It has been shown that such problems, which appear in seismology, failure theory, and civil engineering, have singular components with unbounded energy and can be solved by topological methods with pointwise convergence, in particular, by the block element method. Numerical methods that are based on using the energy integral are not applicable to such problems in view of its divergence.

Sharp Constants of the Energy Equivalence Relation in the Method of Conversion of Boundary Conditions for Biharmonic Equation

We propose and justify an iterative algorithm with economic preconditioner for numerical solving the mixed problem of elasticity theory in approximation of plate theory. We obtain unimprovable constants in the energy equivalence relation which are used for optimization of the iterative process. The inversion of the preconditioner is equivalent to the double inversion of the discrete counterpart of the Laplace operator with the Dirichlet condition.

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).

The asymmetrical mixed temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic elastic layer (soft plate) investigated is in ideal contact with an absolutely rigid layer, deformable only by thermal expansion. The generalized plane temperature problem reduces to determining the stress-strain state of the soft anisotropic layer investigated using the equations of the mixed problem of elasticity theory. At the ends of the boundary layer of the soft plate (a thin contact layer), no conditions are imposed. On the remaining part of the ends of the soft plate, the boundary conditions correspond to a free boundary. The problem has a bounded smooth solution. Unlike the approach described earlier [1], it is proposed to seek an accurate solution in the form of ordinary Fourier series with respect to a single longitudinal coordinate. Solutions in polynomials are also used. It is shown that the existence of these solutions in polynomials enables the convergence of the Fourier series to be improved considerably.

The mixed symmetrical temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a symmetrical three-layer plate with absolutely rigid outer layers, deformed solely due to thermal expansion, is solved. The generalized plane temperature problem is reduced to determining the stress-strain state, which is symmetrical with respect to two coordinates, of the inner layer (a soft filler) of transversely isotropic material using the equations of the mixed problem of elasticity theory. The layers are in ideal mutual contact. The conditions at the ends of the filler boundary layer (a thin contact layer) are not specified. On the remaining part of the ends of the filler the boundary conditions correspond to a free boundary. The problem has a finite smooth solution. The construct the exact solution a modification of Mathieu's method [1] is proposed, which consists of the fact that, in addition to ordinary Fourier series, solutions in polynomials are used. It is shown that the presence of these solutions in polynomials enables the convergence of the Fourier series to be accelerated considerably.

Calculation of a nonhomogeneous cylinder with a rigid inclusion

A method for the approximate solution of mixed problems of elasticity theory for nonhomogeneous bodies of revolution af finite size is proposed. Based on this method the stress-formed state of a cylinder wtih a rigid inclusion is determined.

On possibilities of the method of homogeneous solutions in the mixed problem of elasticity theory for a half-strip

The mixed plane boundary value problem for an elastic half-strip under kinematic loading of its endface and lateral sides free of force loads is examined. An asymptotic is obtained for the coefficients of the series expansion of the displacement vector in homogeneous solutions. Regularization of the series in homogeneous solutions is proposed which would permit computation of the stress field on the half-strip enface.

Exact solution of the antiplane contact problem for finite canonical domains

The exact solution of a mixed problem of elasticity theory concerning pure shear by a stamp (in general, deformable) of a cylindrical body that occupies a domain bounded in section by coordinate lines of an orthogonal curvilinear coordinate system in a plane whose Lamé coefficients satisfy certain conditions, is obtained by constructing the closed solutions of integral equations of the first kind that contain Jacobi elliptic functions as kernels. Analogous problems were studied in /1, 2/, etc., in the special case of a strip and a ring. A scheme is proposed /1/ for constructing the exact solution of these problems by conformal mapping of the strip into a finite domain.

On the asymptotic method of “large λ”

In developing the results in /1/, where an easily realizable method is presented for constructing any number of terms of the series expansion of the solution for one class of mixed axisymmetric problems of elasticity theory by using the method of large λ, a method is described for also constructing any number of terms of such an expansion for another extensive class of integral equations of mixed problems of elasticity theory and mathematical physics. The algorithm results in simple arithmetic recursion relations which enables the domain of application of the large λ method to be extended to its theoretical limits and enables the solution to be constructed with any degree of accuracy. Two problems on the interaction of a stamp with a rectangle are considered as examples, for which certain new results are obtained. The large-λ method was proposed and developed in /2–5/, etc.

On variations in the solutions of integro-differential equations of mixed problems of elasticity theory for domain variations

The behaviour of the solution of the boundary value problem for a pseudodifferential equation (PDE), Green's function of this problem, and also some of their local and global characteristics, during variation of the domain is investigated. Formulas are proposed that enable the solution of a broad class of PDE in a domain to be expressed in terms of the solution in the near domain. Local characteristics of the solution are expressed in terms of the local characteristics of the solution in the near domain. A double asymptotic form of Green's function for both arguments tending to the domain boundary occurs in the variation formula. The variation of this double asymptotic form as the domain varies is expressed in terms of this same asymptotic form. The system of variation formulas obtained is closed. It enables the PDE solution in the domain to be reduced to the solution of an ordinary differential equation in functional space. The local characteristics of the solution can also be found by this method without calculating the solution itself. If there is sufficient symmetry in the initial operator, then conservation laws in the Noether sense are obtained for its Green's function and its asymptotic form. The behaviour of the quantities under investigation is studied under inversion. The investigation of variations of the solutions of problems for the variation of the domain occurs in the paper by Hadamard /1/, who studied the variation in conformal mapping and obtained a formula similar to (1.4). The formula for the variation of the solution of the boundary value problem for an elliptic differential equation is obtained in /2/. Variation formulas for the case of the operator of the problem about a crack and a circular domain are obtained in /3, 4/. The Irwin formula /5/ is obtained from formulas (1.4) and (1.21) by substitution.

Certain variational problems with a small parameter in the theory of elasticity

For the mixed problem of elasticity theory on the deformation of a transversely isotropic cylinder, it is proved than in the selection of a specific small parameter the zeroth approximation equations agree with the bending equations for an elastic isotropic plate based on the Kirchhoff-Love hypotheses /1/. Certain singularly perturbed contact problems of the Signorini type are considered.

A special relationship in spheroidal wave functions and its application to contact problems

A spectral and kindred relationship are set up by methods of the theory of the generalized potential /1/ for an integral operator generated by a symmetric difference kernel in the form of a Macdonald function in two identical semi-infinite intervals {(− ∞, − α), (α, ∞)} that contain spheroidal wave functions. The formula for the expansion of an arbitrary function in these functions is also set up by a well-known method /2/. On the basis of the results obtained, a solution is then constructed for the integral equation of the contact problem of the impression of two identical stamps with half-plane bases into a half-space being deformed in a power-law form in the formulation of /3/. This contact problem can be described by the same integral equation when the elastic modulus of a linearly elastic half-space changes with depth according to a power law /1/. The spectral relationships in classical orthogonal polynomials for extensive classes of integral operators in mathematical physics are presented in /4,5/, where the method of orthogonal polynomials based on them is also elucidated, and numerous applications of this method are given to contact and mixed problems of elasticity theory. We also mention /6–9/ which are related directly to the investigation presented here.

Application of the boundary integral equations method to solve mixed problems of elasticity theory

Application of the boundary integral equations method to solve mixed problems of elasticity theory

On quasiperiodic boundary value problems and their applications in the theory of elasticity

The fundamental boundary value problems of analytic function theory are considered on certain systems of contours possessing translational symmetry: the Riemann problem on an oblique lattice of arbitrary contours, the Hubert problem and the mixed problem for a half-plane, the Dirichlet problem for a plane with a periodic system of slits on a line. In contrast to /1/, where the listed problems are solved under the condition of periodicity of their coefficients and free terms, this condition is here imposed only on the coefficients. By applying a discrete Fourier transform and periodicity of the boundary conditions for an elementary cell, the formulated quasiperiodic problems are reduced to periodic problems and are solved in closed form. The results obtained are used to solve (in quadratures) new mixed problems of elasticity theory in translationally symmetric domains with nonperiodic boundary conditions.

A method for solving a mixed problem of elasticity theory for a strip

A method for solving a mixed problem of elasticity theory for a strip

Method of orthogonal polynomials in plane antisymmetric mixed problems of elasticity theory with two contact sections

The possibility is shown of applying the method of orthogonal polynomials to solve some-integral equations of a special kind if the eigenfunctions of the integral operator corresponding to the principal (singular) part of the kernel are unknown. Use of the classical scheme [1–3] is impossible in this case. However, by using modified Chebyshev polynomials, an integral equation of the form ∫k1 ε(ξ) ln |ξ+xξ−x| dξ = πf(x) − ∫k1 ε(ξ)G (ξ, x, λ)dξG (ξ, z, λ) = ξxG∗ (ξ, x, λ), k ⩽ x ⩽ 1, λ ∈ (0, ∞), k ∈ (0, 1) is successfully reduced to an infinite algebraic system of the first kind convenient for approximate solution. Here λ, k are dimensionless parameters, G∗ is a continuous, even, and symmetric function in ξ,x. Plane antisymmetric mixed problems of elasticity theory with two contact sections, odd in x. reduce to equations of the type (0.1). The odd function f (x) describes the shape of the boundary layer on the contact section k ⩽ | x | ⩽ 1 altered under the effect of stamps. Considered as an illustration is the problem of impressing two flat stamps into a strip.

Periodic convolution equations on segments which occur in elasticity theory and mathematical physics: PMM, vol. 35, n≗ 5, 1971, pp. 861–868

Integral convolution equations given on a system of segments are considered; the kernel of the integral equation is a periodic function with period T The unique solvability of the equations is established for kernels of a general kind, an approximate factorization is given a foundation, and a method for construction of the approximate solution is also indicated. Integral equations of the kind mentioned occur in mixed problems of elasticity theory and mathematical physics, posed for finite bodies [1, 2], as well as for infinite bodies with periodic boundary conditions [3]. The result herein, just as in [4], is necessary for the correct formulation and solution of dynamic contact problems.

An approach to the solution of a mixed problem of elasticity theory

An approach to the solution of a mixed problem of elasticity theory

On the method of orthogonal polynomials in plane mixed problems of elasticity theory: PMM vol. 34. n≗4, 1970, pp. 643–652

On the method of orthogonal polynomials in plane mixed problems of elasticity theory: PMM vol. 34. n≗4, 1970, pp. 643–652