Exact Three-dimensional Equations Research Articles

The Influence of Imperfect Contact Conditions between Layers of a Hollow Sandwich Sphere on its Natural Frequencies

The influence of imperfect contact conditions between layers of a hollow sandwich sphere on its natural frequencies is studied. It is assumed that the materials of face layers are the same, but different from the core layer material. Each layer is made of a homogeneous isotropic material and has a constant thickness. Investigations are performed using a piecewise homogeneous body model with the use of exact three-dimensional equations and relations of elastodynamics. The solution of the corresponding equations of motion of the hollow sandwich sphere is presented through the Helmholtz potentials in spherical coordinates. Numerical results for the influence of imperfect contact conditions between layers on the natural frequencies of the sphere are presented and discussed. It is established that an imperfect contact between sphere layers can decrease its natural frequencies significantly.

One-dimensional model for the unsteady flow of a generalized third-grade viscoelastic fluid

Specific algorithms for non-Newtonian fluids flow depend on the governing constitutive equation. In this work, we present a constitutive equation for a third-grade fluid in which a specific normal stress coefficient depends on the shear rate. This new three-dimensional model is suitable for studies where phenomena like shear-thinning or shear-thickening occur. Using a function of power-law type, we apply the Cosserat theory to fluid dynamics, reducing the exact three-dimensional equations to a one-dimensional system depending only on time and on a single spatial variable. From this reduced system, we solve the unsteady equations for the wall shear stress and mean pressure gradient depending on the volume flow rate, Womersley number, viscoelastic coefficients, and flow index over a finite section of the tube geometry with constant circular cross section. Attention is focused on numerical simulations of unsteady flows regimes by using a Runge–Kutta method.

Specific Features of Charged Particle Confinement in the T-15 Tokamak in the Presence of Magnetic Perturbations

Charged particle motion in the regions of magnetic islands and near-separatrix ergodicity of magnetic field lines in the Т-15 tokamak is studied numerically. The trajectories are calculated by integration of exact three-dimensional equations of motion for trapped and passing charged particles starting at different pitch angles. It is shown that the presence of resonant magnetic perturbations qualitatively affects the trajectories of passing particles. In the region of magnetic islands, the orbits of passing particles acquire an island structure, while in the region of the near-separatrix ergodicity of magnetic field lines, the orbits are stochastized. It is shown that magnetic perturbations insignificantly affect trapped particle motion. Even in the absence of magnetic surfaces, the trajectories of trapped particles are regular and their cross sections have the shape of a standard banana orbit. The possibility of charged particles to pass between regions with different magnetic topologies is analyzed.

Three-dimensional elasticity solution of layered plates with viscoelastic interlayers

An analytical solution for simply supported layered plates with viscoelastic interlayers under a transverse load is proposed. The deformation of each plate layer is described by the exact three-dimensional elasticity equations. The viscoelastic property of interlayer is simulated by the generalized Maxwell model. The constitutive relation of the interlayer is simplified by the quasi-elastic approximation, which significantly simplifies the analytical process. The solution of stress and displacement fields with undetermined coefficients is derived by solving a group of ordinary differential equations. The undetermined coefficients can be efficiently deduced by using the recursive matrix technique for the plate with any number of layers. The practical convergence is observed during numerical tests. The comparison analysis indicates that the present solution has a close agreement with the finite element solution. However, the solution based on the Mindlin–Reissner hypothesis is significantly different from the present solution for thick plates. Finally, the effect of interlayer thickness on stress and displacement distributions of a five-layer plate is discussed in detail.

Wave propagation in a rectangular bar, exposed to the impact tangential forces

In modern technology, there are more and more cases of action of impact loads on work items of buildings, machines and constructions, therefore, the strength calculations of these items under the dynamic effects become important. To solve the problems, arising from the dynamic effects, it is necessary to use the continuum mechanics methods and, in particular in many cases, the methods of the dynamic theory of elasticity. Analytical studies allow to find the exact problem solutions, which is very important, because the exact solutions allow to estimate the main features of the solution in general - the nature and the extent of influence of various set parameters on it. On the other hand, exact solutions are always reference and needed, in particular, for developing numerical methods for more complex cases. This work is a continuation of the works of N. Rassoulova, G. Shamilova, dedicated to studying the propagation of unsteady waves in the prisms of rectangular cross section. Approach to solving this problem differs markedly from all previous issues of the dynamics of rectangular prisms, which mainly investigated their dispersion characteristics. This paper deals with studying the process of propagation of unsteady waves in semi-infinite rectangular bars, exposed to impact shear forces on the face platform. System of exact three-dimensional motion equations of an isotropic elastic body is used. Applying a peculiar integration method, previously developed by the authors of this paper, exact analytic solutions to the posed problem for the final time value are found. The results can be implemented in the production in designing special constructions, where there are impulsive effects on them.

Axis-switching and breakup of low-speed elliptic liquid jets

Theoretical and experimental investigations are conducted to analyze the instability of a low-speed liquid jet emerging from an elliptic nozzle. The complexity of viscous free surface flow analysis for an asymmetric geometry is simplified using an approach based on the Cosserat theory (also called director theory) which reduces the exact three-dimensional equations to a system depending only on time and on a single spatial variable. This work is mainly focused on the spatial instability analysis to examine the key characteristics of an elliptic jet such as jet profile, axis-switching and breakup length. In the experimental part, both natural (free) and excited (forced) breakup behaviors are studied. In the natural breakup, the effects of nozzle’s ellipticity and length to diameter ratio are examined. In the forced breakup case, disturbances are applied to the jet, by modulating the jet exit velocity using a piezoelectric actuator with given sinusoidal perturbations. The spatial evolution of the jet shape is captured with a high speed camera. Liquid jet instability is studied for various nozzle geometries over a specific range of jet velocities and excitation frequencies. Results are compared with conventional circular nozzles which can be considered as a special case of an elliptic jet.

Stress distribution caused by two neighboring out-of-plane locally cophasally curved fibers in a composite material

In the present paper, within the framework of a piecewise homogenous body model, with the use of the exact three-dimensional equations of elasticity theory, a method proposed earlier is developed for investigating the stress distribution caused by two neighboring out-of-plane locally cophasally curved fibers located along two parallel planes in an infinite elastic body. The body is loaded at infinity by uniformly distributed normal forces in the direction of fiber location. The self-equilibrated normal and shear stresses caused by the curved fibers are analyzed, and the influences of interaction between the fibers and of the geometric nonlinearity on the distribution of these stresses are studied. Numerical results for this interaction are obtained.

One-dimensional viscoelastic fluid model where viscosity and normal stress coefficients depend on the shear rate

We study the unsteady motion of a viscoelastic fluid modeled by a second-order fluid where normal stress coefficients and viscosity depend on the shear rate by using a power-law model. To study this problem, we use the one-dimensional nine-director Cosserat theory approach which reduces the exact three-dimensional equations to a system depending only on time and on a single spatial variable. Integrating the equation of conservation of linear momentum over the tube cross-section, with the velocity field approximated by the Cosserat theory, we obtain a one-dimensional system. The velocity field approximation satisfies both the incompressibility condition and the kinematic boundary condition exactly. From this one-dimensional system we obtain the relationship between average pressure and volume flow rate over a finite section of the tube with constant and variable radius. Also, we obtain the correspondent equation for the wall shear stress which enters directly in the formulation as a dependent variable. Attention is focused on some numerical simulation of unsteady/steady flows for average pressure, wall shear stress and on the analysis of perturbed flows.

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.

Hyperbolic boundary layers in compound cylindrical shells

A survey of papers dealing with the mathematical modeling of transient wave propagation in compound and stiffened shell constructions by asymptotic methods based on exact three-dimensional elasticity equations can be found in [1]. Longitudinal actions of tangential and bending types are considered. An asymptotic model of wave propagation in a semi-infinite shell of rotation is used, which employs the two-dimensional Kirchhoff-Love (tangential and flexural) components, the solutions of the quasiplane problem of elasticity, the parabolic boundary layer near the quasifront, and the hyperbolic boundary layer near the expansion wave front. The boundary value contact problems for the incident, reflected, and transmitted waves are posed and solved on the basis of approximate theories for the corresponding components. If the boundary conditions are satisfied at the shell faces in the exact three-dimensional form in small neighborhoods of the shell end contact region, then the construction of the quasistatic boundary layer satisfying the boundary conditions together with the two-dimensional component is completed. Methods proposed for solving boundary value problems by using integral transforms and asymptotic approaches to their inversion are described. But the methods developed for constructing hyperbolic boundary layers near the wave fronts are not described in [1].

Magnetoelectric effects in laminated multiferroic shells

Two-dimensional equations for laminated multiferroic shells of piezoelectric and piezomagnetic layers are obtained from the exact three-dimensional equations. Mechanically, the two-dimensional equations represent the so-called first-order theory that describes coupled extensional, flexural and thickness-shear motions. The electric and magnetic fields are described layer by layer. The two-dimensional equations obtained are used to study magnetoelectric coupling in a laminated multiferroic circular cylindrical shell. Both static coupling and the frequency dependence of the coupling in time-harmonic motions are examined.

Axisymmetric Motion Of A Second Order ViscousFluid In A Circular Straight Tube Under PressureGradients Varying Exponentially With Time

The aim of this paper is to analyze the axisymmetric unsteady flow of a nonNewtonian incompressible second order fluid in a straight rigid and impermeable tube with circular cross-section of constant radius. To study this problem, we use the one dimensional (1D) nine-directors Cosserat theory approach which reduces the exact three-dimensional equations to a system depending only on time and on a single spatial variable. From this system we obtain the relationship between mean pressure gradient and volume flow rate over a finite section of the tube. Assuming that the pressure gradient rises and falls exponentially with time, the 3D exact solution for unsteady volume flow rate is compared with the corresponding 1D solution obtained by the Cosserat theory using nine directors.

Stress distribution in a composite materialwith a row of antiphase periodically curved fibers

The stress distribution in an infinite matrix containing a row of antiphase periodically curved fibers is studied using the piecewise-homogeneous body model and the exact three-dimensional equations of elasticity theory for anisotropic bodies. It is assumed that the midlines of the fibers are located in the same plane and the materials of the fibers are the same. Numerical results are presented for the case where the materials of the fibers and matrix are isotropic and homogeneous

Local instability of horizontal tunnels of polygonal shape in viscoelastoplastic masses

The local instability of a horizontal mine tunnel with a regular polygonal cross section in a viscoelastoplastic rock mass is studied within the framework of the exact three-dimensional stability equations. The effect of rock-mass parameters on the critical pressure is estimated.

Modeling the mechanical behavior of metal-reinforced plastic pipes under internal pressure

Natural and forced vibrations of a thick rectangular plate fabricated from a composite material with a spatially locally curved structure are investigated with the use of exact three-dimensional equations of motion of the theory of elastic anisotropic bodies. The investigations are carried out within the framework of the continuum approach developed by Akbarov and Guz’. It is supposed that the plate is clamped at all its edges and is loaded on the upper face with uniformly distributed normal forces periodically changing with time. The influence of the parameters of local curving on the fundamental frequency of the plate and on the distribution of the normal stress acting in the thickness direction under forced vibration is studied. The corresponding boundary-value problems are solved numerically by employing the three-dimensional FEM modelling.

Stress Distribution in an Elastic Body with a Periodically Curved Row of Fibers

Within the framework of a piecewise homogeneous body model, with the use of exact three-dimensional equations of elasticity theory for anisotropic bodies, a method is developed for investigating the stress distribution in an infinite elastic matrix containing a periodically curved row of cophasal fibers. It is assumed that fiber materials are the same and fiber midlines lie in the same plane. The self-balanced stresses arising in the interphase in uniaxial loading the composite along the fibers are investigated. The influences of problem parameters on these stresses are analyzed. The corresponding numerical results are presented.

Influence of the interaction between two neighboring periodically curved fibers on the stress distribution in a composite material

A method is developed for a stress analysis in an infinite elastic body containing two neighboring periodically cophasaly curved fibers located along two parallel lines. The stress distribution is studied when the body is loaded at infinity by uniformly distributed normal forces in the fiber direction. The investigation is carried out within the framework of a piecewise homogeneous body model with the use of exact three-dimensional equations of elasticity theory. Numerical results related to the stress distribution considered and the influence of interaction between the fibers on this distribution are given.

Stress Analysis for a Thick Rectangular Plate of a Composite Material with a Spatially Periodically Curved Structure under Forced Vibration

Natural and forced vibrations of a thick rectangular plate fabricated from a composite material with a spatially periodically curved structure are investigated with the use of exact three-dimensional equations of motion of the theory of elastic anisotropic bodies. The investigations are carried out within the framework of the continuum approach developed by Akbarov and Guz'. It is supposed that the plate is clamped at all its edges and is loaded on the upper face with uniformly distributed normal forces periodically changing with time. The influence of curving parameters on the fundamental frequency of the plate and on the distribution of the normal stress acting in the thickness direction under forced vibration is studied. The corresponding boundary-value problems are solved numerically by employing the three- dimensional FEM modeling.

A PARAMETRIC STUDY ON THE AXISYMMETRIC MODES OF VIBRATION OF MULTI-LAYERED SPHERICAL SHELLS WITH LIQUID CORES OF RELEVANCE TO HEAD IMPACT MODELLING

The free vibration of spheres composed of inviscid compressible liquid cores surrounded by spherical layers of linear elastic, homogeneous and isotropic materials are studied using three-dimensional elasticity equations. The exact three-dimensional equations are first derived for an N -layered sphere with a liquid core and an extensive parametric study is then presented for the first few natural frequencies of the spheroidal modes of vibration. Non-dimensional frequency parameters are compared with values obtained using lower order membrane and shell theories. It is shown that for a remarkably wide range of geometric and material parameters, which encompasses values typical for the human head, the first ovalling mode of a fluid-filled shell behaves like a membrane filled with incompressible fluid and a simple closed-form expression is derived which closely approximates the natural frequencies obtained using the exact three-dimensional equations.

Asymptotically approximate model equations for nonlinear dispersive waves in incompressible elastic rods

In literature, nonlinear waves in elastic rods have been studied by many authors. Usually, the Navier-Bernoulli hypothesis (the assumption that plane cross-sections remain planar and normal to the rod axis) is used. Intuitively, one would expect that this would be a good approximation when one is mainly interested in longitudinal waves. However, there are no rigorous theoretical justifications available. Also, a defect of this assumption is that comparing with the exact three-dimensional theory the boundary conditions on the lateral surface can never be satisfied. Recently, three papers have been published to overcome this defect, but they contain some algebraic errors (which implies that the approach adopted there cannot be used to overcome this defect). So, this problem remains open. In this paper, we present our recent research results for this problem, and we have managed to establish asymptotically valid one-dimensional rod equations which are consistent with the lateral boundary conditions. Further, their dispersion relation can match with that of the exact three-dimensional field equations to any asymptotic order in the long-wave limit. For solitary waves in the far field, we derive to the leading order the KdV equation. Comparing its solitary-wave solution with that of the KdV equation obtained through the Navier-Bernoulli hypothesis, we find that the difference is very small. This provides some evidence to the validity of the assumption that plane cross sections remain planar and normal to the rod axis.