Discrete-orthogonalization Method Research Articles

Stress state modeling of non-circular orthotropic hollow cylinders under different types of loading

Based on a spatial model of the linear theory of elasticity, using an unconventional approach of the reduction of the original three-dimensional boundary value problem described by a system of partial differential equations with variable coefficients to a one-dimensional boundary value problem for a system of ordinary differential equations with constant coefficients, the problem of finding the dimensional stress of hollow elliptic orthotropic cylinders under the influence of various types of loading has been solved under certain boundary conditions at the orientation plane. Reducing the dimensionality of the original problem is carried out using analytical methods of separating variables in two coordinate directions in combination with the method of approximating functions by discrete Fourier series. The one-dimensional boundary value problem is solved by the stable numerical method of discrete orthogonalization.

Stability of cylindrical anisotropic composite shells under torsion in a three-dimensional formulation

The article presents a calculation of the stability of non-thin cylindrical anisotropic layered shells under the action of end torsional moments in a spatial formulation. The anisotropy of the used material is characterized by one plane of elastic symmetry of characteristics. This is caused by the mismatch between the main elastic directions of the composite fibrous orthotropic material and the axes of the curvilinear cylindrical coordinate system.
 A three-dimensional inhomogeneous system of partial differential equations describing the subcritical stress-strain state within the linear theory of elasticity is derived using the Hu-Washizu variational principle. Reducing the dimension of the problem under consideration from three-dimensional to one-dimensional is carried out by taking into account the axial symmetry of the deformation of the cylindrical shell and using the method of straight lines along the generatrix.
 Based on the modified Hu-Washizu variational principle, a three-dimensional system of homogeneous partial differential stability equations is derived within the framework of the spatial theory of elasticity. The reduction of a three-dimensional system to a one-dimensional one is carried out along the generatrix and in the circular direction - by expanding the components of stresses and displacements into double trigonometric series when applying the procedure of the Bubnov-Galorkin method, as well as taking into account the periodicity of the resolving functions.
 An algorithm has been developed, implemented in the form of application software packages for PCs. In it, in a single computational process using the numerical method of discrete orthogonalization in the direction normal to the middle surface of the shell, the establishment of the parameters of the subcritical stress-strain state and the solution on this basis of stability problems for non-thin anisotropic cylindrical shells under the influence of torsion are combined.
 The problem of the influence on the stability of an anisotropic cylindrical non-thin shell of an increase in the number of cross-reinforced layers depending on the angle of rotation of the main directions of elasticity of the material and the direction of application of torque is considered. The obtained results of stability calculations according to the proposed approach were compared with critical torsion loads calculated using an orthotropic model for calculating anisotropic shells. It is shown that for single-layer cylindrical shells the difference between the compared results reaches 69%. An increase in the number of cross-reinforced layers leads to a decrease in this discrepancy, and with seven to eight layers, the difference between the critical loads obtained using the described approach and the orthotropic model is within 5%. This result is consistent with those obtained using classical or refined theories of calculations of both thin and non-thin anisotropic cylindrical shells.

Three-dimensional stability of non-thin anisotropic cylindrical shells under axial pressure

Based on the modified Hu-Washizu variational principle, a three-dimensional system of homogeneous linearized differential equations of stability of non-thin anisotropic cylindrical shells is obtained. The anisotropy type of shells is characterized by one plane of elastic symmetry. Reducing the system to a one-dimensional one is carried out using the Bubnov-Gal'orkin method, which approximates the functions of unknown systems of equations along the generatrix and takes into account their periodicity in the circumferential direction. The numerical solution of the obtained one-dimensional system of equations is carried out using the discrete orthogonalization method. 
 The problem of stability of a non-thin anisotropic cylindrical shell for a different number of cross-laid composite layers against the action of axial pressure is solved. The nature of the dependence of critical forces on the number of layers is analyzed.

On the Equilibrium of Nonthin Cylindrical Shells with a Dent

On the basis of the linear three-dimensional theory of elasticity, we consider the problem of equilibrium of nonthin isotropic cylindrical shells with dents under certain boundary conditions imposed on the end faces. To describe the cross section of the reference surface, we use the Pascal snail equation in polar coordinates. By the method of separation of variables, with the help of the approximation of functions by discrete Fourier series, we reduce the three-dimensional boundary-value problem to a one-dimensional problem, which is solved by the stable numerical method of discrete orthogonalization. We estimate the accuracy of the obtained solutions. The results of solving the problems are presented in the form of plots and tables.

Applying the method of virtual elements to solving contact problems of shells of revolution interacting with surfaces of inconsistent shape

An approach based on the method of virtual elements is used to solve the contact problem for a thin shell of revolution lying on a rigid foundation. In this case, the surface of the base has a shape inconsistent with the surface of the shell. The method makes it possible to determine the contact area and the contact pressure from the contact area unknown in two coordinate directions. Since the contact area is not known in advance, the problem is structurally nonlinear. A thin isotropic shell is described by the classical theory based on the Kirchhoff–Love hypotheses. The base is taken absolutely rigid, but with the presence of an elastic gasket. The shell equations are integrated by S. K. Godunov’s method of discrete orthogonalization. To determine the forces of interaction between the shell and the base, a mixed method of structural mechanics is used. With this aim in view, the maximum possible contact area is discretized by virtual rectangular elements. A constant value of the contact pressure is assumed on each element obtained in this area, and the contact pressure is assumed to be zero on the elements in the area where the shell leaves the base. Based on the assumptions, a system of linear algebraic equations is constructed, which determines the contact pressure and deflection of the shell circumference axis. Since the shell can move away from the base, iterative procedures are applied to search for the real contact area, which depends on the geometric and elastic parameters of the shell and the magnitude of the external load. As an example, the contact interaction of a cylindrical shell (part of the shell of a tank car) lying on a rigid base with a gasket is considered. It is shown how the contact area and contact pressure change depending on the rigidity of the gasket and the difference between the radii of the shell and the base (inconsistency in the shape of the surfaces).

ИССЛЕДОВАНИЕ ЗАКРИТИЧЕСКИХ ДЕФОРМАЦИЙ ПОЛОГИХ СФЕРИЧЕСКИХ ПАНЕЛЕЙ ПОСТОЯННОЙ ТОЛЩИНЫ

An algorithm for studying the stress-strain state of elastic thin-walled shell systems consisting of shells of revolution has been developed. To solve the nonlinear problem of strong bending of a thin isotropic shell of revolution, in which no restrictions are imposed on the angles of rotation of the normal to the original coordinate surface and the relative linear deformation is small compared to unity, we used the Newton-Kantorovich, which reduces the nonlinear boundary value problem to an iterative sequence of linear boundary value problems. A method was applied to reduce the linear boundary problems to several Cauchy problems, which were integrated numerically using the Runge- Kutta method. To ensure the stability of the solution of stiff Cauchy problems, S.K. Godunov’s method of discrete orthogonalization was applied. Based on this algorithm, a computer program was written to determine the parameters of the stress-strain state of shells with a wide range of changes in geometric, physical, and force parameters and boundary conditions. The stress-strain state of sloping spherical panels of constant thickness with pinching on the outer contour under uniform external pressure has been studied. The process of the formation of loops on the deformation curve depending on the height of the shell has been investigated. Changes in the height of the shell with a constant support contour radius simulates the initial irregularity in its manufacture.

Strength of Shell Structure Operating in Hydrogenous Media under Pressure

Estimation of strength and durability of any structure can be carried out after finding components of stress-strain state of all its parts. The present work shows that in order to calculate stresses in a certain element of a structure that works under thermo-mechanical loading and contacts hydrogenous media, one should trace distribution of heat and hydrogen concentration in this element. Based on the results obtained when having solved thermo-diffusion problem of heat and hydrogen distribution in a casing of a diffusion apparatus, its stress state was found. Casing of the diffusion apparatus is represented in a shape of a shell of revolution. It was determined that in a thin shell, homogenous fields of temperature and hydrogen concentration are being set for a period that is significantly less than the design operating life of the apparatus. To solve the problem of stress distribution in a shell, S.K. Godunov’s method of discrete orthogonalization was used. In the structure that does not have any manifested stress raisers, a critical point with maximum value of shear stress intensity was found. Consistent patterns of change of the intensity in the critical point were found under the emergency pressure boost in the apparatus at various hydrogen densities. The paper shows that emergency pressure boost in a shell may cause the appearance of zones of plastic yield; when this boost is accompanied by increase in hydrogen concentration, the fracture pattern changes from viscous to brittle.

The construction of a spatial model for calculating the thermally stressed state of shallow shells on a rigid basis

Modern calculations of layered plates and shells in a three-dimensional formulation are based on a technique where the distribution of the desired functions over the thickness of a structure is sought by the method of discrete orthogonalization. In this article, based on the approaches developed by the authors, the thermally stressed state of layered composite shallow shells with a rigidly fixed lower surface is analyzed. The distribution of the desired functions over the thickness of the structure is found based on the exact analytical solution of the system of differential equations. An approach to studying the thermally stressed state of layered composite shells is also considered, and a spatial model for calculating the thermally stressed state of shallow shells on a rigid basis is constructed. Currently, this is a very urgent task when calculating the pavement of bridges. A feature of this approach is the assignment of the desired functions to the outer surfaces of the layers, which allows one to break the layer into sublayers, reducing the approximation error to almost zero. To build a spatial model, a load option is selected with temperature loads (according to the sine law) and boundary conditions (Navier), which lead to the distribution of the desired functions in terms of a plate with trigonometric harmonics of the Fourier series. A polynomial approximation of the desired functions by thickness is involved. Using the model under consideration, an analysis of flat layered composite shells on a rigid basis under the influence of temperature load was carried out. The considered example showed that the proposed model provides sufficient accuracy in the calculations of layered shallow shells when considering each layer within one sublayer. When dividing each layer into 32, 64, 128 sublayers, almost the same result was obtained. The proposed approach can be used as a reference method for testing applied approaches in calculating the stress states of layered shallow composite shells.

Investigation of the thermally stressed state of shallow shells on a rigid base with a sliding contact layers using analytic solutions of equations оf elasticity theory

Modern calculations of layered plates and shells in a three-dimensional formulation are based on a technique where the distribution of the desired functions over the thickness of a structure is sought by the method of discrete orthogonalization. In this article, based on the approaches developed by the authors, the thermally stressed state of layered composite shallow shells with a rigidly fixed lower surface is analyzed. The distribution of the desired functions over the thickness of the structure is found based on the exact analytical solution of the system of differential equations. An approach to the study of the thermal stress state of shallow composite shells is considered, and an analytical model is constructed for calculating the thermal stress state of shallow shells on a rigid base with a sliding contact of the layers. Currently, this is a very urgent task when calculating the pavement of bridges. A feature of this approach is the assignment of the desired functions to the outer surfaces of the layers, which allows one to break the layer into sublayers, reducing the approximation error to almost zero. Using the model in question, an analysis of flat layered composite shells on a rigid base with a sliding contact of the layers under the influence of temperature loading was carried out. To build a spatial model, a load option is selected with temperature loads (according to the sine law) and boundary conditions (Navier), which lead to the distribution of the desired functions in terms of a plate with trigonometric harmonics of the Fourier series. A polynomial approximation of the desired functions by thickness is involved. Using the model in question, an analysis of flat layered composite shells on a rigid base with a sliding contact of the layers under the influence of temperature loading was carried out. The considered example showed that the proposed model provides sufficient accuracy in the calculations of layered shallow shells when considering each layer within one sublayer. The proposed approach can be used as a reference method for testing applied approaches in calculating various stress states of layered flat composite shells.

Stress-strain state of thick-walled anisotropic cylindrical shells under thermal power load, protected by the functionally graded material

In the article the stress-strain state of thick-walled structurally anisotropic composite cylindrical shells under thermal power load, which are protected by a functionally graded material, are analysed. Based on the interrelations of the spatial theory of elasticity, a system of inhomogeneous differential equations in three-dimensional formulation, which describes the stress-strain state of thick-walled anisotropic cylindrical shells, was obtained. To reduce the dimensionality of this system, the Bubnov-Galerkin analytical method was used. Thus, the obtained one-dimensional system of twelve equations of normal Cauchy form was implemented using the numerical method of discrete orthogonalization. To represent the possibilities of the proposed approach, there were used stress-strain states of two, four and five-layered anisotropic cylindrical shells of fibrous composites, protected from temperature by a layer of transversely isotropic functionally graded material.

Forced Axisymmetric Vibrations of an Electrically Excited Hollow Sphere Made of a Continuously Inhomogeneous Piezoceramic Material*

The forced axisymmetric vibrations of a hollow sphere made of a functionally graded radially polarized piezocelectric material are studied using the spatial theory of electroelasticity. The properties of the material vary continuously as a power function along the radial coordinate. The vibrations are excited by applying a varying electrostatic potential to the sphere surface. Solid and split electrodes are considered. After separation of variables and expression of the components of displacement and electric-flux density, electrostatic potential, and the stress tensor in terms of spherical functions, the original three-dimensional problem is reduced to a boundary-value problem for ordinary differential equations. The problem is solved by the effective method of discrete orthogonalization. The numerical solution is analyzed. Particularly, the distributions of the mechanical and electric parameters in the first modes of forced electroelastic vibrations of the sphere made of homogeneous and inhomogeneous piezoceramic materials are compared and analyzed. The effect of the loading caused by applying a variable electrostatic potential to the sphere surface on the distribution of forced vibration characteristics is studied.

Numerical Analysis of Dynamical Processes in Inhomogeneous Piezoceramic Cylinders (Review)*

The review is devoted to the numerical solution of new problems of electroasticity, namely, determination of the dynamical characteristics of inhomogeneous piezoceramic waveguides of circular cross-section and inhomogeneous piezoceramic cylinders of finite length. To solve these problems, an effective numerical–analytical approach is used. The approach employs various transformations (special functions, Fourier series expansion, and spline-collocation method), which make it possible to reduce the original three-dimensional partial differential equations of electroelasticity to a boundary-value eigenvalue problem for a system of ordinary differential equations. The system is solved by the method of discrete orthogonalization. Using the results obtained, the features of spectral characteristics in an inhomogeneous structure are studied considering the coupled electric field of the piezoceramic layers. The effect of the inhomogeneity and coupled electric field on the dynamical characteristics of the bodies is studied as well. Much attention is paid to the reliability of the numerical results.

Design of Cylindrical Shells with Oblique Cuts in Refined Statement Using Spline-Approximation*

The bending problem for a cylindrical shell with oblique cuts is solved using the equations of a model based on the straight-line hypothesis. The two-dimensional boundary-value problem obtained is reduced by the spline-approximation method to one-dimensional one solved by the method of discrete orthogonalization. The results calculated are compared with those obtained using the classical theory. Shells with different values of the cut angles are analyzed.

Thermal-Magnetic-Elastic Effect Analysis of a Thin Current-Carrying Conical Frustum Shell*

A thermal-magnetic-elastic problem for a thin current-carrying conical frustum shell in a magnetic field is studied. The normal Cauchy form of nonlinear differential equations, which include in total eight basic unknown variables, are obtained by the variable replacement method. Using Newmark’s stable finite equidifferent formulas and the quasi-linearization method, the nonlinear partial differential equations are reduced to a sequence of quasi-linear differential equations, which can be solved by the discrete-orthogonalization method. The temperature field in a thin conical frustum shell and the integral eigenvalues are derived after considering Joule’s heat effect in an electromagnetic field and the thermal equilibrium equation. The change of stresses, displacements, and temperatures in the thin current-carrying conical frustum shell with variation of the electromagnetic parameters is discussed. It is proved that the stresses, strains, and temperatures in thin shells can be controlled by changing the electromagnetic and mechanical parameters by considering a specific example. These results are expected to be a theoretical reference for further analysis of this case.

Stressed state of a steel construction working in hydrogen containing environment

An approach is proposed to solve the problem of estimating the stress state of a shell structure loaded with a thermomechanical load and in contact with a hydrogen-containing medium. The stress state of the steel housing of the diffusion apparatus for the production of highly pure hydrogen was determined. The object of study is presented in the form of a composite shell of rotation, loaded by internal pressure and operating at elevated temperatures. The purpose of the work is to determine the stress state of the shell at normal and elevated pressure, taking into account changes in the mechanical properties of the combined effect of temperature and hydrogen. In the general case, the task of calculating such a structure under given operating conditions is related. In the phenomenological approach, the relationship between thermal diffusion and mechanical problems is manifested in a change in the parameters of the sample deformation diagram with increasing temperature and hydrogen concentration. The integration of differential equations of a boundary value problem for a shell under pressure is performed by the discrete orthogonalization method S.К. Godunov.

Research of the accuracy of the modified method of lines in the calculation of axisymmetric bodies

This paper shows the application of a modified method of line for determining the stress-strain state of bodies of rotation under the static load. Differential equations of the theory of elasticity recorded in the cylindrical coordinate system are used for this purpose. Given the axial symmetry, the problem is reduced to plane and considered in the coordinate system , circular coordinates of the desired functions do not change. In the first stage of the method, the dimensionality of initial differential equations and boundary conditions is reduced by the Bubnov-Petrov projection method by coordinate, using local basis functions. As a result, the system of equations is reduced to a system of plain first-order differential equations that depend on the coordinate. On the second stage of the method, the reduced system of equations and boundary conditions are solved by S.K. Godunov`s numerical method of discrete orthogonalization in combination with the Runge-Kutta method of Merson. The study revealed all the main features of reducing initial differential equations and boundary conditions. The limits at which the solutions give an approximate result to the analytical value (classical plate theory) are determined. The example of massive bodies shows that its accuracy is not lower than the finite element method. The results obtained in this work are fundamental for further implementation of MML for calculation of thermal elasticity, dynamics problems (in two- and three-dimensional formulation) for objects pertaining to bodies of rotation. The accuracy study was performed on the example of an axisymmetric annular plate of different thicknesses. The results are compared with values of the theory of thin plates and the finite element method.

Stress Analysis of Longitudinally Corrugated Hollow Orthotropic Cylinders

The stress state of longitudinally corrugated hollow cylinders made of orthotropic and isototropic materials is analyzed using a spatial problem statement, analytical variable separation and discrete Fourier series methods and numerical method of discrete orthogonalization. The results are presented as graphs and diagrams of displacement and stress fields and analyzed.

Orthogonal wavelet multiresolution analysis of the turbulent boundary layer measured with two-dimensional time-resolved particle image velocimetry.

The turbulent boundary layer flow measured by two-dimensional time-resolved particle image velocimetry is analyzed using the discrete orthogonal wavelet method. The Reynolds number of the turbulent boundary layer based on the friction velocity is Re_{τ}=235. The flow field is decomposed into a number of wavelet levels which have different characteristic scales. The velocity statistics and coherent structures at different wavelet levels are investigated. It is found that the fluctuation intensities and their peak locations differ for varying scales. The proper orthogonal decomposition (POD) of different wavelet components reveals a cascade of scales of coherent structures, especially the small-scale ones that are usually difficult to be identified in POD modes of the undecomposed flow field. The interactions among the scales are investigated in terms of large-scale amplitude modulations of the small-scale structures. In previous studies the velocity fluctuations are separated into two parts, the large scale and the small scale, divided usually by the boundary layer thickness. In the present study, however, the scales smaller than the boundary layer thickness are further separated. Therefore, the modulation analysis is a refined investigation that differentiates the modulation effects on separated small scales. The results reveal that the modulation effects vary among the small scales.

Phenomenon of embrittlement in titanium shells from hydrogen exposure

Increasing the reliability of equipment used for production, transportation, storage and utilization of hydrogen is directly related to solving the problem of hydrogen embrittlement of metals. Without a fundamental physical theory, it is necessary to predict the bearing capacity of metal structures on the basis of obtained experimental data on the effect which hydrogen have on metal properties. This paper presents a solution (based on the method of discrete orthogonalization proposed by S.K. Godunov) of a physically-nonlinear problem of stress distribution in a titanium shell. Since hydrogen, most notably, reduces plastic properties of metals utilized in structural elements, a critical point was determined where the intensity of shear deformation is maximal. It was found how the intensity changes at a critical point of a shell if the pressure within the device rises to an emergency level. Such a rise of the pressure in the shell could lead to appearance of plastic deformation regions, and hydrogen exposure is manifested in reduced breaking stress and changed fracture pattern.

Influence of Orthotropy on the Stress–Strain State of Quadrangular Plates of Different Shapes

A numerical-analytical approach based on the spline-collocation and discrete-orthogonalization methods is proposed. The approach makes it possible to analyze the static stress–strain state of orthotropic complex-shaped plates. For the plate shape to be taken into account, the coordinate transformation, which reduces the initial domain to a unit square, is applied. Using, as an example, a plate of trapezoidal shape, it is shown how the material properties affect the stress–strain state. It is also studied how the mutual orientation of the plate edges and orthotropy axes influence the distribution of the displacement fields for plates made of materials with essentially dissimilar elastic properties. The dependence of the convergence of the numerical results on mechanical and geometrical parameters of the plates is analyzed.