scholarly journals STABILITY OF A CYLINDER FROM MURNAGHAN MATERIAL UNDER STRETCHING, COMPRESSION AND INFLATION

2019 ◽  
Vol 81 (1) ◽  
pp. 30-39
Author(s):  
M. I. Karyakin ◽  
L. P. Obrezkov

The problem of equilibrium and stability of a hollow cylinder subjected to simultaneous uniaxial tension/compression and inflation is considered within the framework of the three-dimensional nonlinear theory of elasticity. To describe the mechanical properties of the material of the cylinder five-constant Murnaghan model is used. By the semi-inverse method the three-dimensional problem is reduced to the study of a nonlinear boundary value problem for an ordinary second-order differential equation. For most sets of material parameters known from the literature, the presence of a falling section in the stretching/inflation diagram, indicating the possible existence of instability zones even in the area of tensile stresses, has been found numerically. The stability analysis was carried out using a bifurcation approach based on linearization of the equilibrium equations in the neighborhood of the constructed solution by means of the method of imposing a small strain on a finite one. The value of a particular deformation characteristic, for which non-trivial solutions of a homogeneous boundary-value problem exist for the equations of neutral equilibrium obtained in the linearization process, was identified with the critical value of the loading parameter, i.e. value at which the system loses stability. As a rule, the coefficient of stretching/shortening of the cylinder and the coefficient of increase/decrease of its internal or external radius were chosen as such parameters. On the plane of the above-mentioned deformation characteristics the areas of stability under tension and compression, as well as under compression by external force and inflation by internal pressure, are constructed. The forms of possible of stability loss depending on the type of stress state are constructed, and the effect on the stability of material and geometric parameters is studied.

Author(s):  
М.I. Karyakin ◽  
L.P. Obrezkov

The influence of the inhomogeneity of material properties on the process of three-dimensional stability loss of a hollow cylinder stretched by axial force and loaded by uniform pressure on the outer or inner side surface is investigated. We used two standard models describing the compressible nonlinearly elastic material's mechanical properties, namely the three-constant Blatz and Ko model, as well as the five-constant Mournaghan model. Usage of the semi-inverse method allows the reduction of a three-dimensional cylinder equilibrium problem to the study of a non-linear boundary-value problem for an ordinary second-order differential equation. Stability analysis was carried out based on the linearization of the equilibrium equations in the vicinity of the constructed solution. The value of a de-formation characteristic for which there were nontrivial solutions of a homogeneous boundary-value problem for the equations of neutral equilibrium obtained in the linearization process was identified with the critical value of the loading parameter, i.e., the value at which the system loses stability. The coefficients of the cylinder's stretching or radial expansion and the dimensionless characteristic of the applied pressure served as such parameters. On the plane of the loading parameters, stability regions are determined. The influence of heterogeneity on the size and shape of these regions is analyzed.


2003 ◽  
Vol 125 (2) ◽  
pp. 244-245 ◽  
Author(s):  
Igor V. Andrianov ◽  
Jan Awrejcewicz

It is shown by operational method that the boundary value problem of the theory of elasticity related to stresses, which can be reduced to three strains compatibility equations and to three equilibrium equations, in fact is of sixth order. Hence, it is not required to formulate additional boundary conditions.


1993 ◽  
Vol 60 (2) ◽  
pp. 506-513 ◽  
Author(s):  
G. A. Kardomateas

The stability of equilibrium of a transversely isotropic thick cylindrical shell under axial compression is investigated. The problem is treated by making appropriate use of the three-dimensional theory of elasticity. The results are compared with the critical loads furnished by classical shell theories. For the isotropic material cases considered, the elasticity approach predicts a lower critical load than the shell theories, the percentage reduction being larger with increasing thickness. However, both the Flu¨gge and Danielson and Simmonds theories predict critical loads much closer to the elasticity value than the Donnell theory. Moreover, the values of n, m (number of circumferential waves and number of axial half-waves, respectively, at the critical point) for both the elasticity, and the Flu¨gge and the Danielson and Simmonds theories, show perfect agreement, unlike the Donnell shell theory.


2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.


1998 ◽  
Vol 5 (2) ◽  
pp. 121-138
Author(s):  
O. Jokhadze

Abstract Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.


2009 ◽  
Vol 06 (03) ◽  
pp. 577-614 ◽  
Author(s):  
GILLES CARBOU ◽  
BERNARD HANOUZET

The electromagnetic wave propagation in a nonlinear medium is described by the Kerr model in the case of an instantaneous response of the material, or by the Kerr–Debye model if the material exhibits a finite response time. Both models are quasilinear hyperbolic and are endowed with a dissipative entropy. The initial-boundary value problem with a maximal-dissipative impedance boundary condition is considered here. When the response time is fixed, in both the one-dimensional and two-dimensional transverse electric cases, the global existence of smooth solutions for the Kerr–Debye system is established. When the response time tends to zero, the convergence of the Kerr–Debye model to the Kerr model is established in the general case, i.e. the Kerr model is the zero relaxation limit of the Kerr–Debye model.


Author(s):  
Oleksandr Ahafonov ◽  
◽  
Daria Chepiga ◽  
Anton Polozhiy ◽  
Iryna Bessarab ◽  
...  

Purpose. Substantiation of expediency and admissibility of use of the simplified calculation models of a coal seam roof for an estimation of its stability under the action of external loadings. Methods. To achieve this purpose, the studies have been performed using the basic principles of the theory of elasticity and bending of plates, in which the coal seam roof is represented as a model of a rectangular plate or a beam with a symmetrical cross-section with different support conditions. Results. To substantiate and select methods for studying the bending deformations of the roof in the coal massif containing the maingates, the three-dimensional base plate model and the beam model are compared, taking into account the kinematic boundary conditions and the influence of external distributed load. Using the theory of plate bending, the equations for determining the deflections of the coal seam roof in three-dimensional basic models under certain assumptions have a large dimension. After the conditional division of the plate into beams of unit width and symmetrical section, when describing the normal deflections of the middle surface of the studied models, the transition from the partial derivative equation to the usual differential equations is carried out. In this case, the studies of bending deformations of roof rocks are reduced to solving a flat problem in the cross-section of the beam. A comparison of solutions obtained by the methods of the three-dimensional theory of elasticity and strength of materials was performed. For a beam with a symmetrical section, the deflection lies in a plane whose angle of inclination coincides with the direction of the applied load. The calculations did not take into account the difference between the intensity of the surface load applied to the beam. Differences in determining the magnitude of the deflections of the roof in the model of the plate concerning the model of the beam reach 5%, which is acceptable for mining problems. Scientific novelty. To study the bending deformations and determine the magnitude of the roof deflection in models under external uniform distributed load, placed within the simulated plate, a strip of unit width was selected, which has a symmetrical cross-section and is a characteristic component of the plate structure and it is considered as a separate load-bearing element with supports, the cross-sections of this element is remained flat when bending. The deflection of such a linear element is described by the differential equations of the bent axis of the beam without taking into account the integral stiffness of the model, and the vector of its complete displacement coincides with the vector of the force line. Practical significance. In the laboratory, to study the bending deformations and their impact on the stability of the coal seam roof under external loads, it is advisable to use a model of a single width beam with a symmetrical section with supports, the type of which is determined by rock pressure control and secondary support of the maingate at the extraction layout of the coal mine.


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