Numerical modeling of an underwater explosion in an air cavity

An efficient numerical algorithm based on Godunov method is proposed that permits qualitative and quantitative calculations of the hydrodynamic flows resulting from the detonation of explosive charges in an air cavity. Calculations are performed by a difference scheme using moving difference grids in which the moving boundaries are the contact surfaces between the detonation products and air and between air and water, and the shock-wave front. The reliability of the calculations is confirmed by experimental data. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev 252057. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 33–37, July–August, 1998.     

On one iteration method for solving problems on the stress-strain state of thin shells

Decomposition methods are used in solving problems on the stress-starin state of thin shells. The problem is reduced to the iterative solution of a system of equations that have a simpler structure and permit constructing a computing parallel-integration algorithm. The stress-strain problem is solved by an example of a shallow spherical shell. The effect of the curvature of the shell on the number of iterations necessary for reaching the accuracy prescribed is studied. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 98–103, June, 2000.     

Analysis of the thermoelastoplastic nonaxisymmetric stress-strain state of laminated circular cylindrical shells

A technique for analysis of the nonaxisymmetric thermoelastoplastic stress-strain state of laminated circular cylindrical shells is developed. It is assumed that the layers in a laminated package do not slip and separate relative to each other. The problem is solved using the geometrically linear theory of shells that is based on the Kirchhoff-Love hypotheses. The equations of thermoplasticity are written in the form of the method of additional strains. The order of the obtained system of partial differential equations is reduced with the help of trigonometric series in the cyclic coordinate. The systems of ordinary differential equations thus obtained are solved by Godunov's method of discrete orthogonalization. As an example, the nonaxisymmetric thermoelastoplastic stress-strain state of a two-layer cylindrical shell is considered. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 105–110, February, 2000.     

Hurling of shells by hollow charges

Results of the numerical solution of the problem of one-dimensional hurling of shells by hollow explosive charges are elucidated. The results of the numerical solution are compared with asymptotic formulas. Numerous domestic and foreign papers have been devoted to the question of hurling shells by explosive charges. A numerical solution of the problem of convergence of a ring to the center under the effect of detonation products is presented in [1–3]. The problem of hurling a shell by a hollow explosive charge with an internal lining is considered in [4]; the solution of the problem of hurling a shell by a hollow explosive charge without the cavity lining is presented in [5] on the basis of the energy-balance equations; however, the complete picture of the processes occurring in the detonation products is not considered.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 161–166, May–June, 1976.     

Longitudinal impact on the end of a rigid cylindrical shell with an elastic filler

The nonstationary problem on the axisymmetric antiplane deformation of a hollow cylinder due to a longitudinal impact on the end of the rigid cylindrical shell is considered. The solution of the boundary-value problem is reduced to a system of Volterra integral equations of the second kind. A numerical analysis is carried out and its results are plotted. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 98–102, May, 2000     

Solving problems on the stress state of thin shells of revolution under local loads

An approach to the solution of problems on the deformation of locally loaded orthotropic shells of revolution is proposed. Some analytical transformations are carried out to reduce the initial resolving system of partial differential equations to the form where the constant terms representing the surface loads are continuous functions of the circumferential coordinate. This allows us to accelerate the convergence of the trigonometric series that represent the desired solution. One-dimensional boundary-value problems are solved by a stable numerical method. Examples of solving problems on the stress state of a cylindrical shell are presented. The scientific results of the present work were obtained during implementation of Project No. 14 of the Ukrainian Scientific and Technological Center. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 106–113, April, 2000.     

On the representation of the solutions of the equilibrium equations of a piezoelectric transversally-isotropic spherical shell

A method is proposed for constructing equilibrium equations for thickness-polarized transversally isotropic piezoceramic shells. The method is based on Fourier expanding the required functions in Legendre polynomials. The appropriate system of differential equations is formed for the expansion coefficients as functions of two independent variables. The equilibrium equations are given in particular for transversally isotropic spherical shells. A method is given for constructing the general solution in the first approximation. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 7, pp. 59–68, July, 1999.     

Shock loading of a cylindrical shell filled with viscous compressible liquid

The effect of a uniformly distributed shock load on a thin-walled cylindrical shell filled with viscous compressible liquid is considered. An analytical method of inverse Laplace transformation of the general solution of the linearized Navier-Stokes equations is proposed. A numerical example is presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 55–61, November, 1999.     

Numerical simulation of gas-dynamic processes in a pipe open at one end

The problem of sudden opening of one end of a circular pipe containing a pressurized gas is considered. A new form of the boundary condition at the open end of the pipe is proposed that takes into account the local hydrodynamic drag due to the nonlinearity of the real physical problem. The system of gas-dynamic equations is integrated by the Godunov numerical method of discontinuity decay. The procedure of numerical realization of the nonlinear boundary condition at the open end of the pipe is described in detail. Comparison of the graphs obtained in the calculations with experimental data indicates that the proposed technique is appropriate. Deceased. State Scientific Engineering Center of Systems of Control and Emergency Response, State Atomic Energy Committee of Ukraine, Kiev 252011. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 74–79, January–February, 1999.     

Stability of motion of a two-mass system with nonlinear friction

This paper concerns the motion of different elastically coupled masses. One of the masses is subjected to a motive force, while the other mass is acted upon by friction. The motive force decreases linearly, while friction changes nonlinearly. The differential equations of motion are derived and are reduced to the standard form (after Bogolyubov). The averaging method is used to find steady-state solutions, one of which agrees with the exact steady-state solution of the initial system of equations. It is found that the actual conditions of stability of the steady-state solution are differ greatly from the conditions calculated on the basis of avaraged equations. These differences are due to the difference in the degrees of the characteristic Rouse-Hurwitz polynomials calculated on the basis of the initial and averaged equations. The analytical results are illustrated by modeling on a microcomputer. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev; Regional Scientific Research and Experimental Design Institute, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 94–100, August, 1999.     

Analysis of the axisymmetric thermoelastoplastic state of branched laminated transversally isotropic shells

A technique is developed for determination of the axisymmetric thermoelastoplastic stress-strain state of branched laminated transversally isotropic shells of revolution under loads that cause the meridional stress state and torsion. The method is based on the rectilinear-element hypotheses for the whole package of layers. To describe the processes of active elastoplastic deformation of a transversally isotropic material, deformation-type equations, which are constructed without recourse to the plastic-potential existence condition, are used. The scalar functions in the constitutive equations depend on the shear-strain rate and temperature. The solution of the problem is reduced to numerical integration of systems of differential equations. An example of determination of the elastoplastic state of a two-layer cylindrical shell stiffened with a rigid ring support is presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 125–131, April, 2000.     

On solution of problems of dynamics of plates and shells with local structure heterogeneities

A general approach to the solution of nonstationary problems of the theory of shells with local structure heterogeneities is proposed. The approach is based on representation of solutions for the systems of differential motion equations of shell theory in the form of expansions in terms of free forms of vibrations (FFV). Two implementations of the approach, involving FFV expansions of certain members and FFV expansions of the system as a whole, are considered. The capabilities of the approach are illustrated by results for specific problems. In the first implementation, we determine the dynamic stress-strain state of a square plate with three rod supports. A round plate connected to a coaxial circular cylindrical shell is considered as an example of FFV expansion of the system as a whole. Special Design and Engineering Office, S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 46–53, October, 1999     

Magnetoelastic nonlinear deformation of a conical shell of variable stiffness

A complete system of equations of nonlinear magnetoelasticity of shells is presented. A resolving system of equations of a conical shell of variable thickness is constructed for the axisymmetric case. A numerical example of a flexible shell illustrates the possibility of using the proposed technique to solve this class of problems. T. G. Shevchenko National University, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 34–39, November, 1999.     

Motion of the detonation products of a point-ignited charge in a cylindrical shell

A numerical solution is obtained for the two-dimensional nonsteady problem of the motion of detonation products from a cylindrical high-explosive charge enclosed in a shell with the initiation of detonation at a central point in the end of the charge. The detonation products propagate in vacuum. The strength of the shell is not considered. A three-term equation of state is used for the detonation products.     

Intensification of a detonation wave in the zone of secondary reactions in a two-phase medium

A method is proposed for the numerical calculation of one-dimensional nonsteady-state flows of a mixture of a gas with particles, based on the separation of a system of differential equations for a two-phase medium into two subsystems. The problem is solved concerning the propagation of a plane detonation wave in a mixture of a detonating gas with particles, behind the front of which secondary chemical reactions are taking place between the vapors of the particle material and the detonation products. The velocity profiles of the gas and of the thermodynamic functions behind the detonation wave front are determined, and also the dependence of the detonation velocity on the distance to the point of initiation. The conditions for intensification of the detonation wave are obtained in the zone of secondary reactions.Leningrad. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 92–96, September–October, 1972.     

Experimental investigation of the dynamics of shells of revolution (review)

The results of experimental investigations carried out at the S. P. Timoshenko Institute of Mechanics of the National Academy of Sciences of Ukraine are analyzed in this review. The analysis suggests that experimental investigations of shell dynamics must be contained and developed if the method used to calculate the stressed-strained state and stability of dynamically loaded shells is to be developed further and made more precise. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, 35, No. 7, pp. 3–11, March, 1999.     

Method for direct integration of the equations of an axisymmetric problem on thermoelasticity in stresses for unbounded regions

We consider a method for direct integration of equilibrium and consistency equations for axisymmetric quasistatic thermoelasticity problems in regions with unbounded radial coordinates. Relations between stress tensor components are obtained. We derive a system of two integro-differential equations in stresses and an equivalent system of differential equations. We present solutions for the thermoelasticity problem and integral equilibrium conditions for the first stress invariant for a space, half-space, and unbounded layer. Ya. S. Podstrigach Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine, L'vov, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 49–56, March, 1999.     

Calculation of spatial vibrations of plates of complex shape

We consider problem of the spatial vibrations of plates with complex shape within Timoshenko's theory of plates. We write the constitutive equations in a local oblique coordinate system where all the bounding contours coincide with coordinate lines. We have demonstrated the effectiveness of the proposed method using as an example the vibration of a rectangular plate weakened by a regular hexagonal hole under the action of an impulsive surface load. We analyze the effect of nonlinear terms on the frequency and amplitude of normal displacements of the shell as a function of the load amplitude. S. P. Timoshenko Institute of Mechanics. National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 8, pp. 44–53, August, 1999.     

Numerical solution of the two-dimensional nonstationary problem of the motion of a shell under the action of detonation products

The problem of the motion of an incompressible cylindrical shell with an explosive charge is solved numerically for the propagation of a plane detonation wave from the end of the charge. The strength of the shell is not taken into account. A three-term equation of state [1] is assumed for the detonation products. A comparison is made with the one-dimensional case.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 76–79, July–August, 1972.The authors thank G. S. Roslyakov and V. M. Paskonov for assistance in the work and for helpful advice.     

Constructing boundary layers for a shallow shell by the energy-asymptotic method

The energy-asymptotic method is used to construct the boundary layers and the internal stress-strain state of a shallow shell. The basic equations are represented in the form of the displacement method. The efficiency of the method is shown. Pridneprovskaya Academy of Building and Architecture, Dnepropetrovsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 99–104, February, 2000