Optimization of vertical axisymmetric thermal displacements in a round plate using heat sources

We consider the problem of using heat sources for optimal control of the distribution of the vertical axisymmetric thermal displacements of a thin round plate with edge clamped at an angle of revolution. On the basis of the method of the inverse thermoelasticity problem we construct a solution of the control problem. For specific cases of heating the plate we carry out a numerical analysis of the behavior of the optimal control.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Poly a, Issue 36, 1992, pp. 105–110.     

Optimization of axisymmetric thermal displacements in a given section of an unbounded layer

We study the problem of optimal control of the distribution of axisymmetric vertical or radial thermal displacements in a given section of an unbounded layer. Using the method of the inverse problem of thermoelasticity we construct the solution of the control problem. For specific cases of heating of the layer we give a numerical analysis of the behavior of the optimal control.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 55–61.     

A counterexample to Hencky plasticity in the case of a thin plate under vertical load

It is shown that the variational problem describing the bend of a perfect plastic plate under vertical load, in general, has no solution. Bibliography: 5 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 192–198.     

Note on the large deflection of a circular plate under a concentrated load

Zusammenfassung Es werden die grossen Deformationen einer Kreisplatte unter zentrischer Einzellast nach der Methode vonBerger ermittelt.     

Inverse heat conduction problem in a thin circular plate and its thermal deflection

An inverse problem of transient heat conduction in a thin finite circular plate with the given temperature distribution on the interior surface of a thin circular plate being a function of both time and position has been solved with the help of integral transform technique and also determine the thermal deflection on the outer curved surface of a thin circular plate defined as 0 ? r ? a, 0 ? z ? h. The results, obtained in the series form in terms of Bessel’s functions, are illustrated numerically.     

Bifurcation and chaos of thin circular functionally graded plate in thermal environment

A ceramic/metal functionally graded circular plate under one-term and two-term transversal excitations in the thermal environment is investigated, respectively. The effects of geometric nonlinearity and temperature-dependent material properties are both taken into account. The material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power law distribution of the volume fraction of the constituents. Using the principle of virtual work, the nonlinear partial differential equations of FGM plate subjected to transverse harmonic forcing excitation and thermal load are derived. For the circular plate with clamped immovable edge, the Duffing nonlinear forced vibration equation is deduced using Galerkin method. The criteria for existence of chaos under one-term and two-term periodic perturbations are given with Melnikov method. Numerical simulations are carried out to plot the bifurcation curves for the homolinic orbits. Effects of the material volume fraction index and temperature on the criterions are discussed and the existences of chaos are validated by plotting phase portraits, Poincare maps. Also, the bifurcation diagrams and corresponding maximum Lyapunov exponents are plotted. It was found that periodic, multiple periodic solutions and chaotic motions exist for the FGM plate under certain conditions.     

Quasi-static thermal stresses in a thick circular plate

The present paper deals with the determination of a quasi-static thermal stresses in a thick circular plate subjected to arbitrary initial temperature on the upper face with lower face at zero temperature and the fixed circular edge thermally insulated. The results are obtained in series form in terms of Bessel’s functions and they are illustrated numerically.     

Wave propagation and transient response of a functionally graded material plate under a point impact load in thermal environments

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load in thermal environments are studied. The thermal effects and temperature-dependent material properties are taken into account. The temperature field considered is assumed to be a uniform distribution over the plate surface and varies in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Considering the effects of transverse shear deformation and rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived from Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Using the dispersion relation and integral transforms, exact integral solutions of the functionally graded plate under a point impact load in thermal environments are obtained. The influences of the volume fraction distributions and temperature field on the wave propagation and transient response of functionally graded plates are discussed in detail. The results carried out can be used in the ultrasonic inspection techniques and provide a theoretical basis for engineering applications.     

Thermal deflection of an inverse thermoelastic problem in a thin isotropic circular plate

In this paper an attempt has been made to solve the inverse problem of thermoelasticity in a thin isotropic circular plate by determining the unknown temperature gradient, temperature distribution and the thermal deflection on the edge of the circular plate. The results are obtained in terms of series of Bessel’s function and illustrated numerically.     

Elastic state of a plate with a circular plug and a rectilinear thin elastic inclusion

There is considered the problem of the state of stress of an infinite elastic plane with a bonded circular plug and an arbitrarily located thin elastic inclusion under biaxial tension. Conditions of ideal mechanical contact are satisfied on the line separating the materials. By using the complex Kolosov — Muskhelishvili potentials, the problem is reduced to a system of integro-differential equations which is solved numerically by utilization of a mechanical quadrature method. A numerical analysis is given for the solution of the problem of the elastic equilibrium of a plane with a circular hole and an arbitrarily located thin inclusion.     

Plastic flow of an axisymmetric layer under an end load

A solution is constructed for the problem of the stress state of a thin plastic axisymmetric layer subjected to an end load and compression between rigid slabs. The unknown parameters in the solution are determined from two integral equations. A numerical study is made of the range of compressive loads for which plastic flow occurs. The components of the stresses in the layer are also determined.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 106–110, 1988.     

Mechanical response of a capacitive microsensor under thermal load

A considerable fraction of commercial sensors are electrostatically actuated. Many sensor diaphragms are operated in different thermal environments that affect their performance. Because the interplay between the thermal and electrostatic loadings is of interest to designers, in this work we investigate such an interplay. We start with the coupled heat conduction equation and the Saint–Venant plate model. We use nondimensional analysis to show that the dissipation and the elastic coupling vary on a slow scale and hence they can be neglected. Consequently, the heat equation is uncoupled from the plate equation. We consider the case in which the temperature at the boundary is kept at a constant value above the ambient temperature. Substituting the resulting temperature distribution into the plate equation yields an equation with an equivalent compressive load and an electrostatic load due to a DC voltage. Then, a reduced-order model is used to investigate the influence of the dual loading on the plate deflection and their interplay.     

The nonstationary problem of thermoviscoelasticity for a plane-parallel layer under pulsed thermal action

We solve a problem of linear thermoviscoelasticity for a layer on whose boundary a thermal pulse is given. Using the example of the Maxwell model we study the influence of the finite speed of propagation of heat and the size of the connectivity coefficient of the temperature and deformation fields on the nature of the wave process.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 45–50.     

Strongly nonlinear internal soliton load on a small vertical circular cylinder in two-layer fluids

The solutions of MCC theory are used to investigate larger-amplitude strongly nonlinear internal soliton load on a small surface-piercing circular cylinder in two-layer fluids. By comparing the wave profiles and instantaneous horizontal velocities calculated by MCC theory with those of KdV theory and experimental data, we verify the validity of MCC theory for larger-amplitude strongly nonlinear internal soliton. The accelerations are computed, and then force and torque on a small cylinder are estimated based on Morison’s formula for both MCC and KdV theories. Computed results show that the internal soliton force and torque become more and more large and wide with the increase of amplitude for MCC theory. The location of torque crest calculated by MCC theory departs from origin (moving to the right) as the amplitude grows and whenever the inertial term is included or not, the wave forces computed based on the two theories both have small discrepancies for the same amplitude, but when the inertial term is included, the torque obtained by MCC theory will be much larger and the torque obtained by KdV still have a small discrepancy. The reasons are presented in detail. The internal wave force will be underestimated if the traditional KdV theory is used. Therefore, ocean engineers should consider the large-amplitude strongly nonlinear internal soliton load on marine construct carefully.     

Bending of a rectangular orthotropic glass-reinforced plastic plate under a transverse load

The Bubnov-Galerkin method is used to solve the problem of the bending of a clamped flexible orthotropic plate. An equation relating the deflection and the load is obtained. An experiment on a plate made of polyester plastic reinforced with satin-weave glass fabric is described. It is shown that within certain limits the deflections at the center of the plate and the combined stresses along the clamped edges are in good agreement with the theoretical values. The limits of applicability of the first-approximation formulas are investigated.All-Union Scientific-Research Institute of Railroad Transport, Ural Division, Sverlovsk. Translated from Mekhanika Polimerov, No. 4, pp. 674–679, July–August, 1969.     

Unsteady axisymmetric stagnation flow on a circular cylinder

Unsteady, axisymmetric stagnation flow about a circular cylinderis examined when the far-field flow is a periodic function oftime with a fixed time average and an oscillatory part of prescribedamplitude and frequency. Solutions are computed for arbitraryvalues of the Reynolds number, quantifying the effects of surfacecurvature, and a frequency parameter based on the period ofthe far-field flow. It is found that solutions remain regularand periodic provided that the far-field amplitude lies belowa critical value. Above this value, solutions terminate in afinite-time singularity. The blow-up time is delayed by increasingthe curvature of the surface. These results are corroboratedby asymptotic predictions valid in the limits of small and largeamplitude and frequency. For large Reynolds number, the problemreduces to the two-dimensional stagnation-point flow againsta plane wall studied by previous authors.     

Bending of a thin cantilevered orthotropic plate with a concentrated load at the free edge

The problem of a thin rectangular cantilevered plate of constant thickness with a concentrated load in the center of the free edge is considered. The plate is assumed to be orthotropic, the fixed edge coinciding with the principal direction of elasticity. An equation is obtained for the normal stresses at an arbitrary point on the plate. The theoretical results are compared with experiment.A. A. Zhdanov Gor'kii Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 739–741, July–August, 1968.     

An axisymmetric steady-state thermoelastic problem of an external circular crack in an isotropic thick plate

A steady state thermoelastic mixed boundary value problem for an isotropic thick plate is considered in this paper. The faces of an external circular crack situated in the mid-plane of the plate are opened up by the application of temperature while the bounding surface of the plate are maintained at a constant zero temperature. Solution valid for large values of the ratio of the plate thickness to the diameter of the crack has been obtained. Expressions for various quantities of physical interest are derived by finding iterative solutions of the equations and the results are shown graphically.