Axisymmetric Problem Of Elasticity Theory Research Articles

Analysis of the stress-strain state of a radially inhomogeneous transversely isotropic sphere with a fixed side surface

Introduction. The paper considers an axisymmetric problem of elasticity theory for a radially inhomogeneous transversally isotopic nonclosed sphere containing none of the 0 and 𝜋 poles. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the side surface of the sphere is fixed, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The work objective is an asymptotic analysis of the problem of elasticity theory for a radially inhomogeneous transversally isotropic sphere of small thickness, and a study of a three-dimensional stress-strain state based on this analysis.Materials and Methods. The three-dimensional stress-strain state is investigated on the basis of the equations of elasticity theory by the method of homogeneous solutions and asymptotic analysis.Research Results. After the homogeneous boundary conditions set on the side surfaces of the sphere are met, a characteristic equation is obtained, and its roots are classified with respect to a small parameter characterizing the thickness of the sphere. The corresponding asymptotic solutions depending on the roots of the characteristic equation are constructed. It is shown that the solutions corresponding to a countable set of roots have the character of a boundary layer localized in conic slices. The branching of the roots generates new solutions that are characteristic only for a transversally isotropic radially inhomogeneous sphere. A weakly damping boundary layer solution appears, which can penetrate deep away from the conical sections and change the picture of the stress-strain state.Discussion and Conclusions. Based on the solutions constructed, it is possible to determine the applicability areas of existing applied theories and propose a new more refined applied theory for a radially inhomogeneous transversally isotropic spherical shell.

AXISYMMETRIC PROBLEM OF THE ELASTICITY THEORY FOR THE RADIALLY INHOMOGENEOUS CYLINDER WITH A FIXED LATERAL SURFACE

By the method of the asymptotic integration of the equations of elasticity theory, the axisymmetric problem of elasticity theory is studied for a radially inhomogeneous cylinder of small thickness. It is considered that the elasticity moduli are arbitrary positive continuous functions of the radius of the cylinder. It is also assumed that the lateral surface of the cylinder is fixed, and stresses are imposed at the end faces of the cylinder, which leave the cylinder in equilibrium. The analysis is carried out when the cylinder thickness tends to zero. It is shown that solutions corresponding to the first and second iterative processes that determine the internal stress-strain state of the radially inhomogeneous cylinder with a fixed surface do not exist. The third iterative process defines solutions that have the boundary layer character equivalent to the Saint-Venant end effect on the theory of inhomogeneous plates. The stresses determined by the third iterative process are localized at the ends of the cylinder and decrease exponentially with distance from the ends. The asymptotic integration method is used to study the problem of torsion of the radially inhomogeneous cylinder of small thickness. The nature of the stress-strain state is analyzed.

Asymptotic behavior of the solution of an axisymmetric problem of elasticity theory for a sphere with variable elasticity modules

In the paper the axisymmetric problem of elasticity theory is studied for the radially inhomogeneous sphere of small thickness that does not contain any of the poles 0 and . Here the case is considered when the elasticity modules vary linearly with respect to the radius. It is assumed that the lateral surface of the sphere is free of stresses, and at the ends of the sphere (at the conical sections) the stresses are set, leaving it in equilibrium. A characteristic equation is obtained and, based on its asymptotic analysis, the existence of three groups of roots is established with respect to the small parameter characterizing the thickness of the sphere. The corresponding homogeneous solutions are constructed, depending on the roots of the characteristic equation. It is shown that the penetrating solution corresponds to the first group of roots. The second group of roots corresponds to the solution of the edge effect type, similar to the edge effect in the applied theory of shells. The third group of roots corresponds to the boundary layer type solution localized in the conical sections. The solution corresponding to the first and second groups of roots determines the internal stress–strain state of the sphere. In the first term of the asymptotic, they can be considered as a solution in the applied theory of shells. The question of satisfying the boundary conditions at the ends (on the conical sections) of the sphere is considered using the variational Lagrange principle.

Analysis of axisymmetric problem from the theory of elasticity for an isotropic cylinder of small thickness with alternating elasticity modules

Asymptotic methods play an important role in solving three-dimensional elasticity problems. The method of asymptotic integration of three-dimensional equations of elasticity theory takes an important place in solving the problems of the limited transition from three-dimensional problems to two-dimensional for elastic membranes. Based on the method of asymptotic integration of equations of the elasticity theory, the axisymmetric problem of elasticity theory for radially non-homogeneous cylinder of small thickness is explored. The case when elasticity modules change by the radius according to the linear law is considered. It is expected that the lateral part of the cylinder is free from stresses and boundary conditions, leaving a cylinder in equilibrium, are assigned at the ends of a cylinder. The stated boundary-value problem is reduced to the spectral problem. The behavior of solutions to the spectral problem both in the inner part of a cylinder, and near the ends of a cylinder if the parameter of thinness of cylinder’s walls tends to zero, is studied. Three groups of solutions were obtained and the nature of the constructed homogeneous solutions was explained. The solution corresponding to the first iterative process determines the penetrating stressed-strained state of a cylinder. The solution corresponding to the second iterative process represents edge effects in the applied theory of shells. The third iterative process determines the solution which has the character of a boundary layer. The solution corresponding to the first and second iterative processes determines the internal stressed-strained state of the cylinder. In the first term of asymptotics, they can be regarded as a solution on the applied theory of shells. It was shown that the stressed-strained state, similar to the case of a homogeneous cylinder of small thickness, consists of three types: penetrating stressed state, simple edge effect and a boundary layer. The problem of meeting the boundary conditions on the ends of a radially non-homogeneous cylinder using the Lagrangian variation principle was considered.

Application of the geometric immersion method based on the Castigliano variational principle for the axisymmetric problems of elasticity theory

The fundamentals of the geometric immersion stress method for axisymmetric problems of elasticity theory have been stated. The geometric immersion method involves reduction of the initial problem for Clapeyron’s free form elastic body to an iteration sequence of elasticity theory problems on some canonical domain. The iteration procedure for the variational equation of the geometric immersion method has been stated, as well as its discrete analog construction technique suggested with the finite-element stress method for the axisymmetric problem of the elasticity theory in the cylindrical coordinates. A practical application of the method has been demonstrated by a test problem. A reasonably good fit between the stress fields definition and the numerical solution by the traditional finite-element displacement method has been obtained.

Modeling of Temperature Field in Flexible Pavement

The nonstationary temperature field is modeled by the finite element method. Results of the numerical analysis conform fully with the experimental data. By means of the developed finite element model, temperature variation has been obtained in points of asphalt concrete layers of pavement within one decade of the hottest period in the southern region of Kazakhstan. Using the solution of axisymmetric problem of elasticity theory for the layered media, stresses and deformations in specific points of flexible pavement under loading have been calculated at night period (when asphalt concrete layers temperature is minimum) and at day period (when asphalt concrete layers temperature is maximum). The obtained results showed that depending on temperature stresses and deformations in specific points of pavement change significantly. Therefore correct modeling of temperature field in flexible pavement is important in their designing.

Exact solution of the mixed axisymmetric problem of elasticity theory for an infinite elastic layer weakened by cylindrical cavity

Exact solution of the mixed axisymmetric problem of elasticity theory for an infinite elastic layer weakened by cylindrical cavity

Nonsteady-state damage accumulation processes in composite flanges under cyclic loads

We consider two typical flanges: for a powerplant casing and for the casing of suspensions with a sound-absorbing duct. We model the behavior of flanges under the action of cyclic external loads based on solution of the nonlinear axisymmetric problem of elasticity theory. In order to take into account fatigue fracture of the layers of the structure, we use maximum stress criteria, a tensor damage function, the rule of linear summation of damage, and reduction schemes for the deformation characteristics of the layers, describing different fracture mechanisms. We indicate the possible nature of the development of fracture zones in composite flanges. We have established the spare load-bearing capacity of the flanges after fracture begins.

Mathematical modeling of axisymmetric problems of elasticity theory for piecewise-homogeneous bodies of revolution

Mathematical modeling of axisymmetric problems of elasticity theory for piecewise-homogeneous bodies of revolution

Contact problem for elastic half-space with an ellipsoidal cavity

We consider the axisymmetric problem of elasticity theory for a space with an elongated ellipsoidal cavity with mixed boundary conditions of smooth contact on its surface. The method of p-analytical functions is applied to reduce the solution of the problem to an infinite quasi-completely regular system of linear algebraic equations with upper bounded free terms that tend to zero as the index increases. The behavior of the normal stress near the contact line of the different boundary conditions is analyzed.

First and second fundamental axisymmetric problems of elasticity theory for doubly-connected domains bounded by the surfaces of a sphere and a spheroid

The method of constructing solutions of the fundamental boundary-value problems for a homogeneous Lamé equation of multiconnected domains bounded by canonical surfaces of cylindrical and spheroidal coordinate systems described in /1/ is extended to domains with other geometry. The problems under consideration reduce to infinite systems of linear algebraic equations of the second kind with completely continuous opertors. A solution in the form of expansions in a small parameter for the problem of the hydrostatic pressure of a sphere with a centrally located spheroidal cavity is presented as an example.

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 the torsion of a finite cylindrical rod by a flat annular stamp

The mixed axisymmetric problem of elasticity theory on the torsion of a finite cylindrical rod with various cross-section by a stamp is considered. The stamp is fixed rigidly to one plane face of the rod, the other plane face of the rod is fixed, and conditions for displacement or stresses are given on the side surface of the rod. The problem is investigated by the direct boundary element method using the fundamental solutions of a circumferential ring load in the infinite elast medium. Numerical results are presented.

Optimality conditions in axisymmetric problems of elasticity theory

The result /1/ of the optimality of equi-stressed contours in a loaded plane is extended to the case of a space with cavities that is axisymmetric. Boundary conditions are found to determine the shape of the optimal cavities, the inverse axisymmetric problem of elasticity theory, and its analytic solution is obtained in the case of a single cavity.

Method of homogeneous solutions in the mixed problem for a finite circular cylinder: PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented. A solution of the problem for the case of a large value of the ratio ( R − a) / h and small values of the ratio λ = h / a is obtained in [2].

On further development of the “method of large λ” in the theory of mixed prOBLEMS: PMM vol. 40, n≗ 3, 1976, pp. 561–565

The method of large λ [1], when the solution of the integral equations is represented as an asymptotic expansion in negative powers of some dimensionless parameter λ is used extensively, among the asymptotic methods of investigating the integral equations of the theory of mixed problems. As a rule only several terms of such an asymptotic expansion are constructed successfully. Certain types of integral equations of the second kind, for which a method is proposed for the construction of all terms of the asymptotic expansion, are investigated below by the method of large λ. The coefficients and expansions of the required solution in negative powers of λ are represented as polynomials in the main argument and recursion formulas are obtained for the coefficients of these polynomials. Considered as examples are the axisymmetric mixed nonstationary problem of heat conduction for a homogeneous half-space and the axisymmetric problem of elasticity theory for the torsion of a truncated sphere by a press.

Dynamics of a hollow symmetrically loaded circular elastic cylinder: PMM vol. 38, n≗ 3, 1974, pp. 561–564

A solution is presented for the dynamic axisymmetric problem of elasticity theory for a hollow elastic circular cylinder which permits determination of both the displacements and the stresses at any point of the cylinder during symmetric loading of its side surface. The problem of the natural vibrations of a hollow elastic thickwalled cylinder has been considered in [1, 2]. A transcendental equation has been derived in [1] to obtain the natural frequencies in the case of the axisymmetric vibrations of the cylinder, and the propagation of elastic waves in a hollow circular cylinder has been investigated in [2] on the basis of the general elasticity equations. A numerical method for solving problems of such a class has been elucidated in [3].

On the state of stress and strain in a finite cylinder subjected to dynamic loads: PMM vol. 37, n≗4, 1973, pp. 724–730

A solution is presented of the dynamical axisymmetric problem of elasticity theory for a cylinder of arbitrary length with given displacements on its curved and planar surfaces. The initial non-self-adjoint equations are converted into equivalent first order equations for an extended eigenvector by introducing certain auxiliary functions. Arbitrary displacements given on the flat endface of the cylinder are expanded in series of eigensolutions of the problem by using these eigenvectors. Final formulas are obtained for the expansion coefficients. As a particular case, the solution of the statics problem of a cylinder [1] follows for ω → 0. An analogous problem has been examined in [2] where it was reduced to solving an infinite system of equations. The numerical method for solving problems of such a class has been elucidated in [3].

On the approximate equations of the axisymmetric problem of elasticity theory for a transversally isotropic base

On the approximate equations of the axisymmetric problem of elasticity theory for a transversally isotropic base

A variational-difference method for solving the axisymmetric problem of elasticity theory in terms of stress functions

A variational-difference method for solving the axisymmetric problem of elasticity theory in terms of stress functions