Nonaxisymmetric Problem Research Articles

Research on adaptive hydraulic drive optimization control of concrete mixing tank truck for open-pit mine.

The non-axisymmetric problem caused by the fluid sloshing in the tank of a mining concrete mixing tank truck during driving is affected by the excitation of complex road surfaces. The fluid sloshing is coupled with the dynamics of the vehicle body due to the excitation of the complex road surface. The traditional hydraulic drive proportional integral differential (PID) control method is not effective in dealing with such problems, which can easily lead to accidents such as overturning. To improve the accuracy and stability of the hydraulic drive control system, this paper proposes an optimized particle filter PID adaptive control method based on the elastic firefly (FA) algorithm to accelerate the convergence speed of control parameter optimization, and then analyzes its hydraulic drive control characteristics and structural applications, and discusses step steering and double lane change modes are simulated under filling rates of 1.5 and 2.0, respectively. The experimental results show that compared with traditional PID control, the proposed adaptive control method can significantly reduce the average speed error of hydraulic drive control to 0.03km/h and the maximum speed error to 0.17km/h. It also improves the control tracking performance and stability. The practicality of the adaptive hydraulic drive is verified in the filling rate experiments under step steering and double-lane shifting conditions. It has important reference value for the practical application of hydraulic drive control optimization of mining concrete mixing transport tank trucks.

Non-Axisymmetric Coupled Unsteady Thermoelectroelasticity Problem for a Long Piezoceramic Cylinder

Non-Axisymmetric Coupled Unsteady Thermoelectroelasticity Problem for a Long Piezoceramic Cylinder

Thermoelastic response of multilayer cylinders: The direct integration and single-solid approach

An approach for solving the plane nonaxisymmetric thermoelasticity problems in multilayer cylinders is suggested from the single-solid standpoint, which implies considering a multilayer structure as a single unity with a step-wise variation of material properties. The direct integration method was used to the advantage of our approach by applying toward the compatibility equation in terms of strains instead of the one in terms of stresses. As a result, the governing equation is derived without involving generalized derivatives by the radial coordinate and then reduced to an integral equation of the second kind. Using the resolvent-kernel for solving the latter equation, the thermal stresses are computed as the Fourier series whose constituents are explicitly determined by the corresponding constituents of the thermal loading.

Связанная неосесимметричная нестационарная задача термоупругости для длинного цилиндра

Inhomogeneous non-stationary heating of constructions of various purposes induces thermal strains and stresses that should be considered in the comprehensive analysis of elastic systems. The mathematical formulation of the considered linear threedimensional thermoelasticity problems includes coupled non-self-adjoint differential equations of motion and thermal conductivity. Due to their integration difficulty, axisymmetric problems are usually analyzed instead. The purpose of this work is to develop a solution algorithm for the coupled nonaxisymmetric non-stationary problem of the thermoelasticity of a long cylinder. On the internal surface of the hollow anisotropic cylinder, the temperature variation function is known; on the external surface, the convective heat transfer and environmental temperature are given. The rate of the temperature load does not affect the inertial characteristics of the cylinder. Therefore, the equilibrium and heat equations can be added to the initial system of linear equations. The finite Fourier sine and cosine transforms and general biorthogonal transforms are used to study a non-self-adjoint system of differential equations and to develop a closed solution. The obtained solution allows one to determine the temperature field and the stress-strain state of a cylinder with its internal surface affected by non-stationary nonaxisymmetric loading in terms of the temperature variation function.

A General Approximate Solution for the Slightly Non-Axisymmetric Normal Contact Problem of Layered and Graded Elastic Materials

We present a general approximate analytical solution for the normal contact of layered and functionally graded elastic materials for almost axisymmetric contact profiles. The solution only requires knowledge of the corresponding contact solution for indentation using a rigid cylindrical flat punch. It is based on the generalizations of Barber’s maximum normal force principle and Fabrikant’s approximation for the pressure distribution under an arbitrary flat punch in an inhomogeneous case. Executing an asymptotic procedure suggested recently for almost axisymmetric contacts of homogeneous elastic media results in a simple approximate solution to the inhomogeneous problem. The contact of elliptical paraboloids and indentation using a rigid pyramid with a square planform are considered in detail. For these problems, we compare our results to rigorous numerical solutions for a general (bonded or unbonded) single elastic layer based on the boundary element method. All comparisons show the quality and applicability of the suggested approximate solution. Based on our results, any compact axisymmetric or almost axisymmetric contact problem of layered or functionally graded elastic materials can be reduced asymptotically to the problem of indenting the material using a rigid cylindrical flat punch. The procedure can be used for different problems in tribology, e.g., within the framework of indentation testing or as a tool for the analysis of local features on a rough surface.

Approximate contact solutions for non-axisymmetric homogeneous and power-law graded elastic bodies: A practical tool for design engineers and tribologists

In two recent papers, approximate solutions for compact non-axisymmetric contact problems of homogeneous and power-law graded elastic bodies have been suggested, which provide explicit analytical relations for the force–approach relation, the size and the shape of the contact area, as well as for the pressure distribution therein. These solutions were derived for profiles, which only slightly deviate from the axisymmetric shape. In the present paper, they undergo an extensive testing and validation by comparison of solutions with a great variety of profile shapes with numerical solutions obtained by the fast Fourier transform (FFT)-assisted boundary element method (BEM). Examples are given with quite significant deviations from axial symmetry and show surprisingly good agreement with numerical solutions.

Неосесимметричная нестационарная задача термоэлектроупругости для длинного пьезокерамического цилиндра

A new closed solution to the uncoupled non-axisymmetric problem of thermoelectroelasticity was constructed for a long hollow piezo-ceramic cylinder for the case of non-stationary temperature alteration on its inner surface, taking into account the convective heat transfer between the outer surface and the environment. Cylindrical surfaces were electroded and connected to a measurement device with large input resistance (idle mode), while the internal surface was grounded. The Fourier—Kirchhoff non-stationary heat conduction equation was considered without taking into account the influence of alteration in the body dimensions and the electric field on the temperature field. The closed solution to the heat conduction problem was constructed by the finite integral transformations (FIT) method. The quasi-static coupled electroelasticity problem at a certain temperature field was solved without taking into account the cylinder inertial properties by the FIT method. The calculated relations obtained made it possible to determine the temperature field, the stress-strain state, as well as the electric field in the long piezo-ceramic cylinder under non-stationary non-axisymmetric action in the form of a function of the temperature alteration. Numerical analysis of the results made it possible to determine the cylinder wall thickness and the region of the temperature effect alteration, where deformation could most efficiently transform into the electrical pulse.

Analytical method for elastic-brittle-plastic analysis of a circular tunnel in a non-hydrostatic in-situ stress field considering the unified strength criterion

Based on the unified strength criterion, an analytical solution is proposed to explore the variation law of the impact of tunnel axial stress on the development of failure zone around a circular tunnel. The proposed method aims to deal with the non-axisymmetric problem under a high and complex three-dimensional stress condition. The mechanical behavior of the rock mass is considered to follow an elastic-brittle-plastic model. The conformal transformation is used to deal with the mathematical difficulties caused by the non-circular nature of the boundary. The procedure enables the quantitative representation of the unknown elastoplastic boundary through an undetermined mapping function. The differential-evolution algorithm is used to determine the boundary from the perspective of positive analysis. The presented solution is compared with published analytical solutions. The results show that the solution in this paper covers the first two solutions and is more reasonable than the last one, reflecting the reliability and generality of the new extended method. The parametric study is implemented to discuss the development of failure zone affected by various parameters.

Non-axisymmetric problems of unsteady deformation of cylindrical shells with filler

Circular cylindrical shells, as structural elements, have found wide application in various fields of mechanical engineering. Thin-walled layered shells have also found wide application as components in underwater objects, air and space vehicles and in many other engineering structures. The aim of this work is to study the action of a non-axisymmetric moving wave of normal pressure on a cylindrical shell interacting with an ideal compressible fluid. The problem statement is given, solution methods are developed, and numerical results are obtained for new problems of stationary deformation of infinitely long viscoelastic cylindrical shells on a viscoelastic foundation when a non-axisymmetric normal pressure wave moves along the shell axis with up to resonant velocity. As an example, considered the action of a non-axisymmetric moving wave of normal pressure on a cylindrical shell interacting with an ideal compressible fluid. The solution methods are based on the joint application of the integral Fourier transform along the axial coordinate and the expansion of all given and desired values into Fourier series along the angular coordinate. An efficient algorithm for the joint calculation of integrals and Fourier series has been developed and implemented on a computer. It has been established that paying to attention the viscoelastic properties of the shell material reduces the deflections by 10–15%, and also makes it possible to evaluate the damping capabilities of the system.

Solution of a mixed nonaxisymmetric problem of the theory of elasticity for anisotropic bodies of revolution

The paper developed a technique for solving mixed nonaxisymmetric problems of the theory of elasticity for bounded bodies of revolution made of a transversely isotropic material under the action of surface forces specified according to a cyclic law. The technique involves the development of the energy method of boundary states, which is based on the concepts of spaces of internal and boundary states, conjugated by isomorphism, which makes it possible to establish a one-to-one correspondence between the elements of these spaces. The internal state includes the components of the tensor of stresses, deformations, and the displacement vector. The boundary state includes efforts and displacements at the boundary of the body. The isomorphism of the state spaces is proved, which allows finding the internal state to be reduced to the study of the boundary state isomorphic to it. The basis is formed on the basis of the general solution of the boundary value problem of elastostatics for a transversely isotropic body of revolution. Orthogonalization of state spaces is carried out, where the internal energy of elastic deformation is used as scalar products in the space of internal states; in the space of boundary states, the work of external forces is used. Finally, finding the desired state is reduced to solving an infinite system of algebraic equations for the Fourier coefficients. The solution of the problem with mixed boundary conditions for a circular in plan cylinder of transversely isotropic coarse dark gray siltstone with anisotropy axis coinciding with the geometric axis of symmetry is presented. The solution is analytical and the characteristics of the stress-strain state have a polynomial form. Explicit and indirect signs of convergence of problem solutions and graphical visualization of the results are presented.

On the complete solution of the three-dimensional solid space problems based on a novel curvilinear elasticity representation

A solution scheme is presented for the three-dimensional elasticity theory in the curvilinear cylindrical coordinate, without introducing any new physically dubious potential function. Exact analytical solutions for the essential 3-D solid full- and half-space problems are developed based on the introduced elasticity approach. The Kelvin's classical problem of concentrated force acting in the interior of an infinite solid space, as well as the Boussinesq's half-space problem of concentrated force normal to the free surface of a semi-infinite 3-D solid are treated. Furthermore, the three-dimensional non-axisymmetric problem of semi-infinite solid space under concentrated tangential force on its surface (the so-called Cerruti's problem) is revisited. It is demonstrated that unlike most of the conventional solutions in the literature for the aforementioned problems, the present approach is capable of obtaining the complete solution without needing to combine with other approaches. Evidently, the completeness issue associated with any new method of potentials is not present for the established direct elasticity approach. At the end, an extension of the approach to a nonhomogeneous elastic media is presented and, as an application, the Boussinesq's problem for a nonhomogeneous half-space is solved analytically. The developed representation may serve as an efficient replacement of the original three-dimensional Navier's elasticity equations for solving different 3-D elastic problems in the future.

Resistivity contribution tensor for nonconductive sphere doublets

• The resistivity contribution tensor for nonconductive sphere doublets is analytically evaluated. • Making reference to tangent sphere coordinates, the temperature fields are expressed by means of convergent integrals. • Based on a stream function, the axial component of the resistivity contribution tensor is found in closed form. • The transverse component of the resistivity contribution tensor is numerically assessed based on an Euler shooting method. • Comparison with respect equivalent spheroids is provided too. The distribution of the temperature and heat flux fields around a couple of unequal nonconductive tangent spherical inhomogeneities (or pores) embedded in an infinite medium under a steady-state and remotely applied heat flux is addressed in the present work. Owing to the 3D geometrical layout of the inhomogeneity, use is made of the tangent sphere coordinate system. A corrective temperature field expressed in terms of convergent integrals is superposed to the fundamental one to fulfill the BCs at the surfaces of the spheres. When the heat flux is aligned to the symmetry axis (axisymmetric problem), the solution can be found straightforwardly by introducing a stream function, which allows for transforming the Neumann BCs into a Dirichlet boundary value problem. Conversely, for the transversal heat flux (non-axisymmetric problem), the problem is formulated in terms of temperature, thus leading to a system of two ODEs which is handled numerically through a Euler shooting method, after preliminary asymptotic expansions. Once the temperature fields are known, the components of the resistivity contribution tensor are assessed varying the aspect ratio of the two spheres. It is found that the extrema of the thermal resistivity are achieved for spheres of equal size. The study allows assessing the effective thermal conductivity of a wide range of smart composites involving insulating inhomogeneities resembling sphere doublets.

Analysis of wellbore stability for overbalanced drilling in marine methane hydrate-bearing sediments under non-hydrostatic stresses

New challenges have emerged in drilling marine methane hydrate-bearing sediments (MHBSs); these challenges are closely related to methane hydrate (MH) dissociation and complex multi-physics coupling. This study investigates wellbore stability in overbalanced drilling by proposing exact solutions for multi-physics fields, taking into account the more general non-hydrostatic far-field stresses, partial multi-field coupling and the influences of hydrate dissociation, i.e., the water/heat production/absorption and the change in the physical properties of reservoirs due to hydrate dissociation.Wellbore drilling is simplified as a non-axisymmetric plane strain problem for the mechanical field and an axisymmetric problem for the pressure/temperature/salt concentration fields, in which the drilling domain includes four annular regions with different physical properties. The closed-form solutions for the temperature, pore pressure and salt concentration fields are obtained by considering the influence of water/heat production/absorption in the compatibility conditions of the steady-state governing equations. The solutions can explain the secondary formation of MHs around the dissociation front. The stress and drilling-induced displacement are then derived for the multi-region elastic problem subjected to non-hydrostatic far-field stresses considering seepage flow, with which the initial failure zone can be addressed by considering a failure criterion. The analytical solutions are verified by the good agreement between analytical and numerical results based on the same assumptions. Furthermore, the comparison between the analytical solutions and the results from complex numerical models suggests the applicability of the proposed solutions. Based on the conditions in marine hydrate field trials in the Shenhu area of the South China Sea, the influence of key parameters on wellbore stability in drilling in MHBSs is investigated in detail by the proposed solutions. Some useful dimensionless charts are provided.

A Comparison of General Solutions to the Non-Axisymmetric Frictionless Contact Problem with a Circular Area of Contact: When the Symmetry Does Not Matter

The non-axisymmetric problem of frictionless contact between an isotropic elastic half-space and a cylindrical punch with an arbitrarily shaped base is considered. The contact problem is formulated as a two-dimensional Fredholm integral equation of the first type in a fixed circular domain with the right-hand side being representable in the form of a Fourier series. A number of general solutions of the contact problem, which were published in the literature, are discussed. Based on the Galin–Mossakovskii general solution, new formulas are derived for the particular value of the contact pressure at the contact center and the contact stress-intensity factor at the contour of the contact area. Since the named general solution does not employ the operation of differentiation of a double integral with respect to the coordinates that enter it as parameters, the form of the general solution derived by Mossakovskii as a generalization of Galin’s solution for the special case, when the contact pressure beneath the indenter is bounded, is recommended for use as the most simple closed-form general solution of the non-axisymmetric Boussinesq contact problem.

Analytical layer element analysis for non-axisymmetric problem of multilayered thermoelastic media

This paper obtained the analytical layer element solution for the non-axisymmetric problem of multilayered thermoelastic media. Firstly, starting from the governing equations, the relationship between the generalized stress and displacement vectors on the surface and bottom of a single-layered medium is obtained by applying Hankel transform and Fourier series expansion. Then, this relationship is extended to multilayered media by introducing the interlaminar continuity condition, and the total stiffness matrix equation is obtained. By combining the boundary conditions at both ends, the solution of the non-axisymmetric problem in the transformed domain can be obtained. The solution in physical domain is achieved by applying the Hankel transform inversion. Finally, the theory and program are verified by several examples, and the sensitivity analysis (SA) is taken on the parameters of the media. The proposed method has the advantages of high efficiency and stability in calculation.

A singularity analysis in Keer’s elastic indentation problem

A non-axisymmetric three-dimensional frictionless contact problem for an elastic half-space and a cylindrical indenter with the face in a wedge form is considered under the assumption of a circular region of contact. Explicit formulas are derived for the critical indenter displacement and contact force that are needed to ensure full contact over the entire contact region. It is shown that the contact pressure beneath the indenter maintains a logarithmic singularity, which is produced by the wedge edge and is similar to the singularity of the contact pressure in the corresponding two-dimensional contact problem for a wedge rigid indenter.

Behavior of solution of the elasticity problem for a radial inhomogeneous cylinder with small thickness

A non-axisymmetric problem of the theory of elasticity for a radial inhomogeneous cylinder of small thickness is studied. It is assumed that the elastic moduli are arbitrary positive piecewise continuous functions of a variable along the radius. Using the method of asymptotic integration of the equations of the theory of elasticity, based on three iterative processes, a qualitative analysis of the stress-strain state of a radial inhomogeneous cylinder is carried out. On the basis of the first iterative process of the method of asymptotic integration of the equations of the theory of elasticity, particular solutions of the equilibrium equations are constructed in the case when a smooth load is specified on the lateral surface of the cylinder. An algorithm for constructing partial solutions of the equilibrium equations for special types of loads, the lateral surface of which is loaded by forces polynomially dependent on the axial coordinate, is carried out. Homogeneous solutions are constructed, i.e., any solutions of the equilibrium equations that satisfy the condition of the absence of stresses on the lateral surfaces. It is shown that homogeneous solutions are composed of three types: penetrating solutions, solutions of the simple edge effect type, and boundary layer solutions. The nature of the stress-strain state is established. It is found that the penetrating solution and solutions having the character of the edge effect determine the internal stress-strain state of a radial inhomogeneous cylinder. Solutions that have the character of a boundary layer are localized at the ends of the cylinder and exponentially decrease with distance from the ends. These solutions are absent in applied shell theories. Based on the obtained asymptotic expansions of homogeneous solutions, it is possible to carry out estimates to determine the range of applicability of existing applied theories for cylindrical shells. Based on the constructed solutions, it is possible to propose a new refined applied theory.

New approach to interface crack problems in transversely isotropic materials

The novelty of the approach contains several aspects: the method of derivation of the governing equations, the form and number of equations, the optimal choice of elastic constants. While majority of published articles derives the governing equations from the Green’s function for the compound space and the use of the reciprocal theorem, we use the combination of Green’s functions for 2 different half-spaces and Fourier transform. The usual approach leads to 3 hypersingular integral equations, we arrive at 2 integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic elastic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications. The new form of the governing equations allows us to obtain an exact closed form solution for an axisymmetric interface crack problem by elementary means. The same method can be applied to obtain the exact solution for a non-axisymmetric problem. A comparison is made with the existing exact solutions and some are shown to be incorrect.

On the Electro-Mechanical Stability of Elastomeric Coaxial Fibers

Abstract The stability of cylindrical coaxial fibers made from soft elastomeric materials is studied for electro-static loadings. The general configuration considered is a three-component axisymmetric fiber having a conducting core bonded to a dielectric annulus in turn bonded to an outer conducting annular sheath. A voltage difference between the conducting components is imposed. The stresses and actuated elongation in the perfectly concentric fiber are analyzed, and the critical voltage at which stability of the concentric configuration is lost is determined via solution of the non-axisymmetric bifurcation problem. The role of the geometry and moduli contrasts among the components is revealed, and the sub-class of two-component fibers is also analyzed. The idealized problem of a planar layer with conducting surfaces that is bonded to a stiff substrate on one surface and free on the other exposes the importance of short wavelength surface instability modes.

Determination of non-axisymmetric stresses in the bodies of revolution based on regulized intergral equations

A solution to the 3D elasticity theory problem in the form of a Fourier series expansion by an angular coordinate, the coefficients of which are determined from the system of one-dimensional integral equations, is constructed. The relation for determining the kernels of these equations for harmonics of arbitrary order is written in analytical form. The regularization of the obtained equations is carried out. For this purpose, the kernels of the equations are represented as the sum of regular functions and singular components, which are determined through the kernels of the plane problem of the elasticity theory in the Cartesian coordinate system. In the stress tensor, the components that contain logarithmic functions as multiplier, are additionally highlighted. The test calculations are performed, which confirm the reliability of the analytically obtained relations and integral equations built on their basis. Testing of the developed numerical algorithm for solving integral equations is carried out and examples of calculation of non-axisymmetric stress concentration near cavities in bodies of various shapes are resulted. The obtained results allow to expand the application of boundary elements method to study of practically important non-axisymmetric problems for the bodies of rotation.