Reissner-Sagoci Problem Research Articles

An axisymmetric fundamental solution and the Reissner-Sagoci problem for an internally loaded non-homogeneous transversely isotropic half-space

This paper employs the concept of a ring of sources or forces using the Hankel transform and Fourier cosine transform to derive the axisymmetric fundamental solution for transversely isotropic nonhomogeneous materials in the case of torsional loads. These fundamental solutions determine the kernel functions used in the Boundary Integral Equation method and Body Force Method. Making use of the axisymmetric fundamental solution we consider the classical Reissner-Sagoci problem related to an elastic non-homogeneous transversely isotropic half-space region in which the rotation of the bonded circular disk is induced by a concentrated couple is located along the axis of symmetry and acts at a finite distance from surface. The rotation of bonded rigid disk due to the internally applied concentrated couple is evaluated. Special cases are examined.

The elastic torsion problem for a nonhomogeneous and transversely isotropic half-space

The statical Reissner-Sagoci problem for nonhomogeneous transversely isotropic elastic half-space is investigated. The modulus of rigidity is assumed asCii(r,z)=μiirβf(z),i=4, 6 and β≧0, where μii are constants. The expressions for stresses, displacement and torque are given. The analysis is based on the assumption that the tangential displacement is prescribed within the areaz=0,r≦1 and shearing stress is zero in the outside arear>1,z=0. The integral equation which is obtained is solved in the case whenf(z)=cosh2kz and β≧0, wherek is a constant.

Torsional axisymmetric finite element model for problems in elasticity

This note describes how the displacements and shear stresses of an axisymmetric elastic component, when loaded in torsion, can be computed by modelling the component with torsional axisymmetric finite elements. The model developed represents only minor modifications of the well-known plane stress or plane strain finite element technique.In the analysis, the model is split into a mesh of triangular annuli. Each node of each element has only one degree of freedom, the tangential displacement. The state of strain in each element is represented by a three-term displacement function, one representing a rigid body rotation, the second representing the state of torsion, and the third representing the state of strain in a hollow thin disc.The model has been applied satisfactorily to three torsional problems with known theoretical solutions. The first problem involves the computation of torsional shear stresses of a uniform shaft subjected to pure torsion. In the second problem, the solution is obtained for a conical shaft. In the third problem, known as the Reissner–Sagoci problem, an elastic semi-infinite medium is subjected to a torsional displacement on a small area of the surface.A typical application of the model to the problem of a shrink-fitted assembly subjected to torsion is discussed. Key words: torsion, finite element, elasticity, axisymmetry.

An Elastic Circular Finite Cylinder Under Torsion by a Pair of Annular Rigid Stamps

This paper deals with the so-called Reissner-Sagoci problem of an elastic circular finite solid cylinder or a circular thick plate twisted by a pair of annular rigid stamps. For the problem, a method of approach is developed by applying Green's functions for torsional body force and torsional surface force problems of an elastic thick infinite plate. That is, solutions to the problem of a finite solid cylinder are constructed by Green's functions for problems of an elastic thick infinite plate. Stresses, displacements, torques necessary to twist a stamp and torsional forces on the contact plane under the bottom of a stamp for circular solid cylinders or circular thick plates are shown and compared with the results about the Reissner-Sagoci problems of an elastic half space and an elastic thick infinite plate. A comparision with a simple theory of torsion of a circular cylinder is made.

The impulsive Reissner-Sagoci problem in a composite medium

The small-time Reissner-Sagoci problem in a bi-material elastic half-space under an impulsive twist is investigated. The problem is reduced to an integral equation, which, by asymptotic analysis and application of the Wiener-Hopf technique, is further reduced to a Fredholm integral equation of the second kind. Approximate analytical results, whose accuracy can be improved by retaining higher order terms in the approximations, are obtained for the free surface displacement outside the disc and the reactive couple necessary to keep the disc in the twisted position, for a bi-material medium with small shear wave speed disparity between the two media.

Torsion of a nonhomogeneous transversely isotropic half space

The statical Reissner Sagoci problem for a transversely isotropic, nonhomogeneous elastic solid is investigated. The modulus of rigidity of the medium is assumed to be variable as a power of the radial coordinate in the form r β (β ⩾ 0) . The expressions for stresses, displacement and torque are given.

The statical Reissner-Sagoci problem for an internally loaded transversely isotropic halfspace

The present paper examines the elastostatic problem related to the axisymmetric rotation of a rigid circular punch which is bonded to the surface of a transversely isotropic elastic halfspace region. The rotation of the punch is induced by a concentrated couple which acts at a finite distance from the punch along the axis of symmetry. The resulting rigid rotation of the punch is evaluated in exact closed form.

Torsion of Nonhomogeneous Solid

A solution is given for the Reissner-Sagoci problem for a nonhomogeneous half-space and semi-infinite cylinder in each of which the shear modulus is assumed to vary with the depth. The formulation is reduced to Fredholm integral equations of the second kind. Numerical results have been presented for the physical quantities of interest. Some of these results indicate a marked difference from the corresponding response of a homogeneous isotropic solid. The analysis also reveals that the classical square root singularity in the shear stresses retains this order in the nonhomogeneous solid under consideration in contrast with a material possessing a modulus, in which the singularity is a function of a constant.

On an initial value Reissner-Sagoci problem

A rigid circular cylinder is cemented to the otherwise stress-free surface of a homogeneous isotropic elastic half-space. The cylinder is rotated through a small angle and then released from rest. During the subsequent motion the cylinder remains cemented to the half-space and the problem is the determination of its motion. The problem is solved by setting up an integral equation for the Laplace transform of the angular displacement. Numerical values for the solution of this equation for large and small values of the Laplace parameter s = iω can be obtained from the known results for the steady-state Reissner-Sagoci problem. For intermediate values of the parameter the equation is solved by means of Chebyshev approximation. The inversion of the Laplace transform is carried out by using a Laguerre polynomial approximation technique.

On the Reissner-Sagoci problem at high frequencies

A high frequency asymptotic approximation is obtained for the problem of a rigid circular disc undergoing torsional oscillations on the surface of an elastic half-space. The exact series solution of Reissner and Sagoci is not convenient in this frequency range. Formulas are given for the amplitude of oscillation of the disc and the displacement on the free surface of the half-space.

The Reissner-Sagoci problem

The statical Reissner-Sagoci problem [1, 2, 3] is that of determining the components of stress and displacement in the interior of the semi-infinite homogeneous isotropic elastic solid z ≧ 0 when a circular area (0 ≦ p ≦ a, z = 0) of the boundary surface is forced to rotate through an angle a about an axis which is normal to the undeformed plane surface of the medium. It is easily shown that, if we use cylindrical coordinates (p, φ, z), the displacement vector has only one non-vanishing component uφ (p, z), and the stress tensor has only two non-vanishing components, σρπ(p, z) and σπz(p, z). The stress-strain relations reduce to the two simple equationswhere μ is the shear modulus of the material. From these equations, it follows immediately that the equilibrium equationis satisfied provided that the function uπ(ρ, z) is a solution of the partial differential equationThe boundary conditions can be written in the formwhere, in the case in which we are most interested, f(p) = αρ. We also assume that, as r → ∞, uπ, σρπ and σπz all tend to zero.