Nonlinear Statement Research Articles

Effect of damping on the postcritical behaviour of autonomous non-conservative systems

Postcritical behavior of non-linear autonomous non-conservative systems is studied with special attention to the effects produced by the damping forces non-uniformly distributed among the natural modes (or the degrees of freedom). The objective is to find out, to what degree the well-known destabilization effect of damping primarily observed in linear systems affects the postcritical behaviour of a system if non-linearities are taken into account. As a bench model, an initially planar elastic panel is considered subjected to the supersonic gas flow. However, the qualitative conclusions may be extrapolated upon a wide class of phenomena in non-linear autonomous non-conservative systems such as airfoils and panel flutter, instabilities induced by the jet thrust or jet pressure, etc. The main conclusion confirms the already formulated statement that, from the rigorous viewpoint, all these phenomena are to be treated taking into account the initial complete positive damping, however small. There are no “paradoxes” neither in the linear nor the non-linear statement of a problem under the condition that the concept of stability is used in the proper sense. On the other hand, a strong effect is demonstrated by the ratio of partial damping factors on the postcritical behaviour. In general, the ratio of these factors influences the postcritical behaviour to a higher degree that their absolute magnitudes, at least if the latter are small or moderate.

Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 2. Geometrically Nonlinear Statement

Based on mixed finite-element approximations, a numerical algorithm is developed for solving linear static problems of prestressed multilayer composite shells subjected to large displacements and arbitrarily large rotations. As the sought-for functions, six displacements and eleven strains of the shell faces are chosen, which allows us to use nonlinear deformation relationships exactly representing arbitrarily large displacements of the shell as a rigid body. The stiffness matrix of a shell element has a proper rank and is calculated based on exact analytical integration. The bilinear element developed does not allow false rigid displacements and is not subjected to the membrane, shear, or Poisson locking phenomenon. The results of solving the well-known test problem on a nonsymmetrically fixed circular arch subjected to a concentrated load and the problem on a locally loaded toroidal multilayer rubber-cord shell are presented.

On Dynamic Behavior of “Self-Stressed” Block Media. Part I. One-Dimensional Mechanico-Mathematical Model

A mechanico-mathematical statement of the problem on dynamic behavior of "self-stressed" block geomedia within a one-dimensional model is given. The structural elements of this model are considered to be absolutely solid bodies of the same dimensions, and the displacements between them are modeled by means of tension and compression of springs, the stiffness of which depends on their elongation along the gravitational force vector. An analytical solution is obtained for the linear case, and its properties are examined. As to the nonlinear statement, a method is proposed for reducing the problem to the set of linear equations.

Bending of a Composite Material Strip with Curved Structure in the Geometrically Nonlinear Statement

To date, bending problems for strips or plates made of composite materials with curved structure have been investigated only in the linear statement. However, in many cases, the necessity arises to investigate the corresponding bending problems in the geometrically nonlinear statement. Therefore, in the present paper, some bending problems for a composite strip with a periodically curved structure is investigated in such a statement using the exact nonlinear equations of elasticity theory in Lagrangian coordinates. The numerical results are obtained by employing the FEM with the use of the Newton-Raphson and the Modified Newton-Raphson algorithms.

Long non-linear waves in a confined basin, induced by transient atmospheric disturbances

In the non-linear statement, the paper studies long waves induced in a confined basin of variable depth by a low atmospheric pressure area, translating at constant speed over the basin. The study is conducted in terms of a non-stationary finite-difference model. Emphasis is made on the behaviour of the wave process after the departure of the disturbance. The dependence of the modal composition of fluid oscillations on the basin parameters and baric disturbances is determined.

Study of the generation of non-stationary non-linear long waves by atmospheric pressure disturbances

In a non-linear statement, this paper considers the generation of long waves by periodic atmospheric pressure disturbances in a confined basin of variable depth filled with homogeneous fluids. The study is conducted using a non-stationary finite-difference model. The contribution of dissipative forces to the development and stabilization of fluid oscillations is analysed. The paper also considers how the growth of the amplitude of atmospheric pressure disturbances may enhance the non-linearity of the wave process. It is shown that as the pressure amplitude becomes larger, the number of significant harmonics with multiple frequencies in the spectrum of sea level elevations increases. For the weakly non-linear case, wave velocities and free surface elevations, with oscillations stabilized, are compared with their counterparts derived through the perturbation technique.

Solution of the problem of oscillations of laminated plates in geometrically nonlinear statement

The article deals with the problem of oscillations of laminated plates with large deflection. It analyzes the fields of application of the refined theory of shear for its solution.

Solution of problems involving the stress state of composite materials with curved layers in the geometrically nonlinear statement

Solution of problems involving the stress state of composite materials with curved layers in the geometrically nonlinear statement

On a certain application of the theory of two-dimensional potential flow to problems on the theory of plates and shells

A new approach based on the use of properties of spherical representation of a surface is proposed for solving a series of problems on the theory of plates and shells. This facilitated another approach to formulation of a resolving equation after having obtained it in a form invariant to the transformation of coordinates. Complex potentials for various flows of the stress-strain state (SSS) are given, i.e., uniform, shear and dipole. Specific applications are presented via a series of results of determination of the basic SSS in plates and shells. Stress concentration factors around holes of complex peripheral form in plates and shells in tension, shear, and torsion are also presented in a physically linear and nonlinear statement. All problems are solved in analytical form and reduced to a quantity.

Analytical solution in nonlinear statement of the problem of calculating a circular flexible plate with rigid center subjected to a concentrated bending moment or a concentrated axial force

Within the framework of the nonlinear theory of plates the article suggests a method by which analytical dependences are obtained for calculating the torsional and axial rigidities, and also the state of stress of a circular plate with a rigid center, constrained along the outer circumference, which can be loaded either by an axial force or by a concentrated bending moment applied at the center.

Numeric analysis of the accuracy of models of a stiffened spherical dome subjected to pulsed loading

The stress-strain state (SSS) of a stiffened spherical shell is investigated in the nonlinear statement. The feasibility of replacing a ribbed shell by a smooth one is confirmed from investigation of its deformed state.

The entry of a wedge into an incompressible fluid

The similarity problem of the entry of a rigid wedge into an ideal weightless incompressible fluid occupying a half-space is studied. The difficulty is that a non-linear boundary conditon has to be satisfied on the free surface of the fluid, whose position is unknown and has to be found during the solution. Three types of fluid motion are considered: flow past the wedge without break-away, the case when one wedge face is not wetted (a semi-infinite plate), and the intermediate case, when a cavity forms on one face. The problem amounts to solving a non-linear system of integral equations. A method of solving this system is given for the flow without break-away and the plate case. Examples of calculations are given. The results for thin and thick wedges are compared with approximate data. The penetration of a wedge into a fluid was first studied in /1/. In /2/ the linear problem of normal collision with a water surface was solved. An approximate solution can be found e.g., in /3–5/. In /6/ a solution was obtained for the special case the entry of a wedge into a fluid. In /7/ the problem of normal wedge entry was solved in the exact non-linear statement, and the same problem was considered in /8/. The method below is based on that of /7/.

Comparative analysis of two approaches to more accurate calculation of laminate shells made of composite materials

1. We worked out an approach based on the theory of multilayered anisotropic shells which makes it possible to investigate the state of stress and strain of structures made of composite materials and taking into account the local effect in the geometrically nonlinear statement. 2. We worked out an approach for calculating laminate anisotropic shells of revolution on the basis of the geometrically nonlinear theory of elasticity with numerical realization by the finite element method. 3. For the first time all the stress tensors calculated with a view to the geometrically nonlinear deformation of laminate shells of composite materials were compared with each other.

A difference scheme on a non-uniform mesh for the equations of one-dimensional magnetic gas dynamics

A new difference scheme for the equations of the one-dimensional motion of a viscous heat-conducting gas in a magnetic field is studied. In the non-linear statement of the problem, the existence, uniqueness, and stability of the solution of the scheme are proved and error estimates are obtained in terms of the data. All the results are obtained in the large with respect to time.

Stability of shells of revolution formed from fibrous composites in the geometrically nonlinear finite-element statement

Stability of shells of revolution formed from fibrous composites in the geometrically nonlinear finite-element statement

Unsteady-state extraction from a falling droplet with nonlinear dependence of distribution coefficient on concentration

An asymptotic analysis of the problem of unsteady-state heat and mass transfer of a droplet at commensurable phase resistances in the case of high Peclet numbers is carried out. The problem is considered in nonlinear statement when in the boundary condition of phase equilibrium on the droplet surface the concentration of a dispersed phase is arbitrarily dependent on continuous phase concentration. The dynamics of heat and mass transfer is shown to be qualitatively very different over three characteristic time intervals, the boundaries of which depend on Peclet number. The first time interval, 0 ⩽ t < O(ln Pe) ( t is the dimensionless time), is characterized by the formation of unsteady-state diffusional boundary layers on both sides of the droplet surface (which are as yet qualitatively identical), with the internal boundary layer generating a diffusional wake near the flow axis. Over the second time interval, O( ln Pe) ⩽ t < O(Pe), the developed internal diffusional wake starts to interact with the boundary layer and ‘smears’ it severely (here, the boundary layers outside and inside the droplet are already substantially different), as a result of which the thickness of the internal boundary layer increases considerably in a gradual way. Over the last time interval, 0( Pe) ⩽ t, a further rearrangement of the concentration field is observed so that the boundary layers are practically no longer present; the concentration outside the droplet becomes constant and equal to the nonperturbed concentration at infinity, while inside the droplet a substantially unsteady-state process occurs when concentration on each fixed streamline has already fully become equalized (at the expense of repeated liquid circulation along the closed streamlines), while mass is transferred by molecular diffusion in the direction normal to the streamlines.

Jet flow around an elastic shell

In [1] an investigation was made of jet flow around an elastic plate. Below, in an exact nonlinear statement, a study is made of the problem of jet flow around an elastic cylindrical shell, fastened at one end and having the second end free. With certain limitations on the form of the shell, the single-valued solvability of the problem is demonstrated, and a method for its solution is proposed. Some results of calculations are given. A statement and a solution of the inverse problem of static hydroelasticity are also given.

An algorithm for the numerical solution by the mesh method of the cylindrical blast problem, allowing for back pressure

AN algorithm is given for the numerical solution of the cylindrical problem by the mesh method in Lagrange variables. An implicit computational scheme is employed. Two types of scheme are used in computations of ideal gas flows involving jumps: schemes with artificial dissipative terms simulating the properties of the actual physical medium, where the jumps are manifested automatically, and schemes in which the isolation of singularities has to be provided for in advance. Both types of scheme have been used to solve the spherical blast problem in its non-linear statement [1–3]. The method of integral relationships differs from the above [4]; it has been used for solving both the spherical and the cylindrical problems [5, 6]. The present paper describes an algorithm for the numerical solution of the cylindrical problem by the mesh method in Lagrange variables. An implicit computational scheme is employed.