On shells of revolution with the Love-Kirchhoff hypotheses

On the occasion of the 100th anniversary of A.E.H. Love's fundamental paper on thin elastic shell theory, the present article summarizes a line of developments on shells of revolution related to the Love-Kirchhoff hypotheses which form the basis of Love's theory. The summary begins with the Günther-Reissner formulation of the linear theory which is shown to contain the classical first approximation shell theory as a special case. The static-geometric duality is deduced as a natural and immediate consequence of the more general theory. The repeated applications of this duality greatly simplify the solution process for boundary-value problems in shell theory, including the classical reduction of the axisymmetric bending problem and related recent reductions of shell equations for more general loadings to two simultaneous equations for a stress function and a displacement variable. In the nonlinear range, the article confines itself to Reissner's geometrically nonlinear theory of axisymmetric deformation of shells of revolution and Marguerre's shallow shell theory with special emphasis on recent results for elastic membranes, buckling of shells of revolution and applications of asymptotic methods.With fondness and appreciation, the authors dedicate this article to their teacher, collaborator and friend, Professor Eric Reissner, in the year of his seventy-fifth anniversary. Dedication The preparation of this paper has been supported in part by NSF Grant No. DMS-8743445.     

Effect of material heat sensitivity on the deformation of laminar shells of revolution

A method of calculating laminar shells of revolution fashioned from orthotropic heat-sensitive materials is developed within the framework of an accepted variant of finite-displacement theory, which takes into account the nonlinear distribution of transverse-shear strains at the level of hypothesis. The influence exerted by heat sensitivity, type of loading, and boundary conditions on the stress-strain state of laminar cylindrical shells fabricated from orthotropic heat-sensitive materials is analyzed. Translated from Problemy Prochnosti, No. 11, pp. 60–68, November, 1994.     

On the asymptotic behaviour of shells of revolution in free vibration

We consider the free vibration problem of thin shells of revolution of constant type of geometry, focusing on the asymptotic behaviour of the lowest eigenfrequency, as the thickness tends to zero. Numerical experiments are computed using two discretization methods, collocation and finite elements, each corresponding to a different shell model. Our results are in agreement with theoretical results obtained using interpolation theory and cited in literature.     

Cyclically symmetric problems of heat conduction in perforated plates and shells with heat transfer

The problem of determining the steady-state temperature field is analyzed for the case of plates and hollow shells with a circle of holes and boundary conditions of the third kind.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 23, No. 5, pp. 890–897, November, 1972.     

On the phenomenological theory of heat conduction

In this paper the foundations of the phenomenological theory of heat conduction is generalized, then the principle of least dissipation of energy and the Gyarmati variational principle are treated. The second-order differential of Lagrangian of the Gyarmati principle, the general Γ picture and nonlinear cases are investigated. In the last part nonlinear constitutive theories are involved: the method of power series and the method of mollification. Generalizing the mollification method we demonstrate that results obtained by mollifications of different kinds are not equivalent, that is, this method is not unique.     

On finite axi-symmetrical deformations of thin elastic shells of revolution

We derive a generalized version of the known system of two simultaneous second order differential equations for the problem of axi-symmetric torsionless deformations of elastic shells of revolution, for finite deformations and including transverse shear deformations and membrane drilling moments. Our generalization, which involves the introduction of a semicomplementary energy density, comes out in a particularly simple and compact form. We furthermore consider the effect of transverse normal stress deformations and discover the possibility of reducing this problem to a system of three simultaneous second order equations, with the supplementary third equation harmoniously adding itself to the two equations without consideration of transverse normal stress deformation effects.     

On the theory of heat conduction in micromorphic continua

The paper is concerned with the linear theory of heat conduction in micromorphic continua. First, a study of boundary-value problems of the stationary theory is presented. A representation of Galerkin type is established and fundamental solutions are derived. The potentials of single layer and double layer are used to reduce the boundary-value problems to singular integral equations. Existence and uniqueness results are established. The second part of the paper is devoted to non-stationary thermal fields. A continuous dependence result and an existence theorem are presented.     

On asymptotic solutions for nonsymmetric deformations of shallow shells of revolution

The paper presents asymptotic solutions of the differential equations for unsymmetrical deformations of such shallow shells of revolution as behave qualitatively similar to shallow spherical shells. Three distinct classes of asymptotic solutions, two of an interior nature, one of a boundary layer nature, and each having its own amplitude factor, are superimposed. Determination of the relative magnitudes of the amplitude factors follows from a consideration of the relevant boundary conditions, as exemplified for two distinct classes of boundary conditions. Determination of the amplitude factors is equivalent to the determination of the relative orders of magnitude of interior bending and direct stresses and of edge zone bending and direct stresses. The general results are illustrated by the explicit solution for a shallow spherical shell with sinusoidally varying meridional edge load, and with vanishing circumferential edge load and meridional bending moment, the fourth condition being that of vanishing transverse shear force or of vanishing transverse deflection.     

Solution of heat conduction problems for a body of revolution made of inhomogeneous material

Solutions of the heat conduction equation are obtained which permit methods of solving plane boundary problems for analytic functions to be applied when the boundary conditions on a surface of revolution are satisfied.     

On the modelling of heat conduction problem in laminated bodies

Summary The paper deals with the heat conduction modelling problem for micro-periodic composites. The models proposed there take into account certain micromorphic effects resulting from the fine periodic structure of the body. The governing equations describing heatfluxes and temperature distributions in these composites in the nonlinear as well as linear cases have been derived. Two examples of the application of the presented models are detailed.With 4 Figures     

Axial wave propagation in membrane shells of revolution

The dynamic finite element technique, which is referred to in the literature as ‘computer-analysis’, is applied to wave propagation problems occurring in finite and semi-infinite linearly elastic membranes of revolution. Both semi-infinite and finite versions of cylindrical and conical membranes are considered, and a finite membrane having a meridional curve which is parabolic is solved. The source of excitation is generally a constant velocity motion of one end of the membranes, but the results for a stress-pulse input at one end of a semi-infinite cylindrical membrane are also given. The results for the finite membranes are new, and the results for the semi-infinite problems are discussed with respect to previously published results. The two-dimensional state of stress in the membrane requires careful ordering of the calculations, and the boundary conditions for finite membranes are shown to follow logically from this ordering of the calculations. The difference in the solutions resulting from prescribing an axial or a tangential velocity excitation at the end of a conical membrane is presented, and the mesh size necessary for convergence to the solution is indicated. The graphs of the results clearly indicate regions in space and time where the membrane model should be replaced by the shell formulation to represent a realistic structure. The technique is shown to be self-contained and independent of any formal method such as the method of characteristics.     

On the domain of attraction for steady states in heat conduction

Energy methods are employed to obtain information on the domain of attraction of steady state solutions to a class of heat conduction problems with nonlinear heat generation. Essential use is made of an inequality of Sobolev type, which may be of interest beyond the present context.     

Stress waves in the coupling zones of composite shells of revolution

The problem of the axisymmetric pulsed deformation of a composite shell of revolution and complex geometric shape is solved in the two-dimensional statement. The problem was solved numerically using the Wilkins explicit finite-difference scheme. The results obtained from the one-dimensional and two-dimensional theories, on the basis of which it is concluded that for ultrashort-term loadings with a duration of approximately 10^–1–10^–2 sec, the influence of stress risers is manifested only in the immediate proximity of points of abrupt change in the body's geometry (corner points), are compared. Here, the stress calculation must be performed in accordance with two-dimensional equations, since the onedimensional equations yield values on the low side in certain cases. In sections that are 1 mm and more distant from the corner points, it is possible to use the one-dimensional theory, since the error in stresses, which is obtained in this case, goes into the factor of safety.Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR, Kiev. Moscow Institute of Thermotechnics. Translated from Problemy Prochnosti, No. 10, pp. 68–71, October 1989.     

Flows of the hele-shaw type between coaxial shells of revolution

Flows are considered in narrow gaps, whose mean surfaces are being formed by two planar parts of a sphere, a right circular cone, an ellipsoid, one- and two-sheet ellipsoids, a paraboloid, and a torus.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 52, No. 6, pp. 940–948, June, 1987.     

On the effect of a chemical reaction on heat conduction and dynamic pressure

Summary When a chemical reaction occurs in a mixture of gases it facilitates heat conduction and creates a non-equilibrium pressure. Both phenomena are explicitly related to the heat of reaction in this paper.Dedicated to Prof. Franz Ziegler on the occasion of his 60^th birthday     

周期性非稳态传热对低温容器的影响机理及实验

对低温容器分别在环境温度下和有外热源条件下进行热响应实验,发现低温容器静态蒸发率呈周期性变化,但变化周期仅为环境温度变化周期的1/2,环境温度的周期性变化并非是使高真空多层绝热容器蒸发率测量时产生周期变化的主要因素。在进行外热源加热容器外壁实验时发现,外壁温度变化并没有对低温容器的漏热量产生明显的影响,低温容器的绝热层对低温容器外壁面的温度波动有较强的衰减及延迟作用,热量在传入的过程中发生了剧烈的衰减,使容器内壁近似形成一个"等温层",从而使高真空多层绝热低温容器的漏热量几乎不受外界环境温度变化的影响。     

On the stability of difference schemes for nonlinear heat conduction problems

A nonlinear heat conduction problem with nonlinear boundary conditions is solved by the finite-difference method. The stability of the difference scheme is investigated.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.19, No.2, pp.283–289, August, 1970.     

On the numerical solution of phase change problems in transient non-linear heat conduction

A finite difference solution for the transient non-linear heat conduction with phase change in a finite slab is proposed. A two-time level implicit method is used while Taylor's forward projection method is employed for taking into account the non-linearities. The stability due to the boundary conditions at the moving front is verified. The numerical solution is compared with the analytical solutions and found to be in close agreement.