Equations Of Neutral Equilibrium Research Articles

Stability of a nonlinear elastic plate under lateral compression

Introduction . Loss of stability and buckling of a round plate may be observed if the plate is loaded on the lateral surface. The solution to this problem is based on a bifurcation approach. In this case, a plate is considered as a nonlinear elastic body. It is important to choose the relation between stresses and deformations in sustainability problems of nonlinear elasticity. Simple laws of state (constitutive equations) were considered in early works devoted to this problem, for example, material of the “harmonic type” (Sensenig). Materials and Methods . Equations of neutral equilibrium for round plates made of Murnaghan and Blatz-Ko materials are obtained. Assuming a uniform initial deformation on the plate, the stability problem is considered. Strict threedimensional neutral equilibrium equations provide exploring related forms of equilibrium taking into account physical and geometric nonlinearity. Derivation of these equations is based on the application of the theory of superposition of small deformation on the final one. Results . Progress in solution to the corresponding secular equation (with non-linear parameter entry) for practically important laws of elasticity of Murnaghan and Blatz-Ko is possible using the numerical methods only. The developed method for calculating bifurcation values of loading parameters makes it possible to analyze the effect of nonlinearity. Discussion and Conclusions. The influence of physical and geometric nonlinearity on the upper critical value of the initial deformation parameter is explored. The results obtained can be used under the assessment of reliability of elastic third-order moduli for various physical materials. Data on these moduli is still scarce. The numerical research has shown that the constants given in some sources should be treated with caution. The use of elasticity moduli in the law of state of Blatz-Ko is also discussed.

Растяжение и раздувание нелинейно-упругого неоднородного полого цилиндра

The influence of the inhomogeneity of material properties on the process of three-dimensional stability loss of a hollow cylinder stretched by axial force and loaded by uniform pressure on the outer or inner side surface is investigated. We used two standard models describing the compressible nonlinearly elastic material's mechanical properties, namely the three-constant Blatz and Ko model, as well as the five-constant Mournaghan model. Usage of the semi-inverse method allows the reduction of a three-dimensional cylinder equilibrium problem to the study of a non-linear boundary-value problem for an ordinary second-order differential equation. Stability analysis was carried out based on the linearization of the equilibrium equations in the vicinity of the constructed solution. The value of a de-formation characteristic for which there were nontrivial solutions of a homogeneous boundary-value problem for the equations of neutral equilibrium obtained in the linearization process was identified with the critical value of the loading parameter, i.e., the value at which the system loses stability. The coefficients of the cylinder's stretching or radial expansion and the dimensionless characteristic of the applied pressure served as such parameters. On the plane of the loading parameters, stability regions are determined. The influence of heterogeneity on the size and shape of these regions is analyzed.

УСТОЙЧИВОСТЬ ЦИЛИНДРА ИЗ МАТЕРИАЛА МУРНАГАНА ПРИ РАСТЯЖЕНИИ, СЖАТИИ И РАЗДУВАНИИ

The problem of equilibrium and stability of a hollow cylinder subjected to simultaneous uniaxial tension/compression and inflation is considered within the framework of the three-dimensional nonlinear theory of elasticity. To describe the mechanical properties of the material of the cylinder five-constant Murnaghan model is used. By the semi-inverse method the three-dimensional problem is reduced to the study of a nonlinear boundary value problem for an ordinary second-order differential equation. For most sets of material parameters known from the literature, the presence of a falling section in the stretching/inflation diagram, indicating the possible existence of instability zones even in the area of tensile stresses, has been found numerically. The stability analysis was carried out using a bifurcation approach based on linearization of the equilibrium equations in the neighborhood of the constructed solution by means of the method of imposing a small strain on a finite one. The value of a particular deformation characteristic, for which non-trivial solutions of a homogeneous boundary-value problem exist for the equations of neutral equilibrium obtained in the linearization process, was identified with the critical value of the loading parameter, i.e. value at which the system loses stability. As a rule, the coefficient of stretching/shortening of the cylinder and the coefficient of increase/decrease of its internal or external radius were chosen as such parameters. On the plane of the above-mentioned deformation characteristics the areas of stability under tension and compression, as well as under compression by external force and inflation by internal pressure, are constructed. The forms of possible of stability loss depending on the type of stress state are constructed, and the effect on the stability of material and geometric parameters is studied.

Stability of inhomogeneous micropolar cylindrical tube subject to combined loads

In the present paper, the stability of a nonlinear elastic cylindrical tube made of micropolar material is analyzed. It is assumed that the elastic properties of the tube vary through the wall thickness. The problem is studied for the case of axial compression of the tube under internal and external hydrostatic pressure. Applying linearization the neutral equilibrium equations have been derived, which describe the perturbed state of the tube. By solving these equations numerically the critical curves and corresponding buckling modes have been found, and the stability regions have been constructed in the planes of loading parameters (relative axial compression, relative internal or external pressure). Using these results, the influence of elastic properties, as well as the size of the tube, on the loss of stability is studied. Special attention has been paid to the analysis of how the pattern of change in elastic parameters affects the stability of a cylindrical tube made of micropolar material.

On the forms of loss of stability of a cylindrical shell under an external side pressure

Two statements of the problem of the static loss of stability (through flexural-shearing and non-classical beam flexural modes) of a cylindrical shell made out of an isotropic material under the action of an external pressure on its side surface are considered. The first statement corresponds to the introduction of an assumption concerning the invariability of the direction of the action of the load when the shell changes from the unperturbed equilibrium state to a perturbed equilibrium state and the second statement corresponds to the action of a “tracking” hydrostatic pressure on the shell. The linearized equations of the theory of stability obtained earlier are used with a refinement of the terms in them associated with the action of the “tracking” external pressure and which are written in the linear approximation in the two versions. Exact analytical solutions are obtained for all the formulated problems. The solutions from them describing the beam flexural buckling modes (BMs) correspond to a hinged mounting of the ends of the shell, and they are obtained both on the basis of simplified equations, formulated in the momentless approximation, as well as the unsimplified (moment) equations. It is shown that their realization is only possible in the case of long shells under the action of a “non-tracking” side pressure with a fully explicable loss of stability mechanism. Under the action of a “tracking” pressure, the solutions for beam flexural BMs apparently only exists on account of the asymmetry with respect to the circumferential coordinate of the “loading” terms that occur in the neutral equilibrium equations used when the exact expression for the vector of the unit normal to the deformed middle surface of the shell is replaced by approximate linearized expressions for the vector of the normal that does not have a unit length. The critical values of the pressure corresponding to such beam flexural BMs are considerably greater than the critical pressure corresponding to the “shell” flexural BMs of the shell, and they are therefore of no practical interest whatsoever and only of purely theoretical interest.

The consistent equations of the theory of plane curvilinear rods for finite displacements and linearized problems of stability

Based on a previously constructed, consistent version of the geometrically non-linear equations of elasticity theory, for small deformations and arbitrary displacements, and a Timoshenko-type model taking into account transverse shear and compressive deformations, one-dimensional equations of an improved theory are derived for plane curvilinear rods of arbitrary type for arbitrary displacements and revolutions and with loading of the rods by follower and non-follower external forces. These equations are used to construct linearized equations of neutral equilibrium that enable all possible classical and non-classical forms of loss of stability (FLS) of rods of orthotropic material to be investigated, ignoring parametric deformation terms in the equations. These linearized equations are used to find accurate analytical solutions of the problem of plane classical flexural-shear and non-classical flexural-torsional FLS of a circular ring under the combined and separate action of a uniform external pressure and a compression in the radial direction by forces applied to both faces.

The effect of torsion on the stability of an elastic cylinder under tension

The effect of torsion on instability in the form of the formation of a neck in a stretched rod in the shape of a circular cylinder is investigated. The stability is considered using the three-dimensional equations of neutral equilibrium of an isotropic incompressible solid. The subcritical state is described by the exact solution of the problem of the non-linear theory of elasticity regarding the equilibrium of a twisted and stretched cylinder. After separating the variables, the equations of neutral equilibrium are reduced to a system of ordinary differential equations, which are solved numerically. Numerical results are obtained for a Biderman material and a material with a power dependence of the specific energy on the deformation. A region of stability is constructed in the plane of the loading parameters, which are the longitudinal elongation and the angle of torsion. This region is compared with the region of convexity of the energy per unit length of the twisted and stretched cylinder.

Stability of elastic–plastic conical shells under shear loading

A method of determination of critical loads for thin-walled conical shells loaded by shear forces developed by moment of twist is presented. The three governing equations of neutral equilibrium with respect to basic displacement vector components u, v, and w are used. It is assumed that effective stress in the prebuckling state of stress in the shell can exceed the yield limit of the shell material. The use is made both of physical relations of Nadai–Hencky small elastic–plastic deformation theory of plasticity, and Prandtl–Reuss J 2 incremental plastic flow theory. Also, a bilinear stress–strain material model, material compressibility and Shanley approach will be accepted in the analysis. Galerkin method is applied to solve the problem equations and iterative techniques are accepted in numerical algorithm to determine critical loads for elastic–plastic shells.

Methods of finite element analysis of arbitrary buckling forms of sandwich plates and shells

Two statements of the problem of arbitrary buckling forms (BFs) (including synphasic, antiphasic, mixed flexural, flexural-shear, and shear forms in the tangential directions) of general-form sandwich shells and two schemes of its solution by the FEM are given. The first of the schemes is based on the use of refined linear equations for determination of the precritical stress-strain state and linearized equations of neutral equilibrium with all parametric addends necessary to determine the critical loads and reveal the possible BFs. The second one uses the general geometrically nonlinear relations of elasticity theory for investigation of the whole deformation process up to buckling in terms of a modified incremental (stepwise) statement of the problem. Examples of solution of particular problems are given.

Stability analysis of rectangular plates unidirectionally strengthened by ribs

The exact solution of the equations of neutral equilibrium is derived. The effect of the discrete arrangement of ribs on the behavior of the characteristics of the stress-strain state with distance from the plate edge is analyzed. The conditions of applicability of the theory of structurally orthotropic plates are formulated.

Refined stability theory for sandwich shells with transversally stiff core. 2. Linearized equations of neutral equilibrium

Neutral equilibrium equations of the refined theory of stability for sandwich shells with a transversally stiff core are constructed and used for studying local mixed forms of stability loss (FSL), as well as admitting different variants of simplification, depending on the type of precritical state and realized FSL. The generalized Reissner variational principle used for deriving the stability equations allows us to refine transverse shear stresses in the core as compared to [1]. A method for a highly accurate definition of these stresses is proposed. Namely, after the integration of three-dimensional equilibrium equations over the transverse coordinate, the number of free constants and the number of static conditions to be satisfied are equalized according to the actual stress distribution across the thickness.

Modification in the theory on flexural-torsional buckling of structures

In this paper, the problems existing in the present theory on flexural-torsional buckling of structures are discussed, and the buckling procedure is found to be restricted to certain development order of displacements and rotations by the present theory. A fresh idea is, therefore, proposed for the mathematical description of actual flexural-torsional buckling procedure of structures. New geometric equations are formulated and a set of new potential variational equation and neutral equilibrium equations are got for the flexural-torsional buckling analysis of structures. Examples are given to detect the numerical difference between the modified theory and the present accepted theory.

Stability of the equilibrium of an elastic ring in the presence of radial shear

An exact formulation of the non-linear theory of elasticity is used to study the problem of bifurcation of equilibrium in a circular annulus. It is assumed that the inner boundray is rigidly clamped and tangential stresses act on the outer boundary. The material considered is isotropic and incompressible, with elastic Mooney potential. The subcritical state is determined from the exact solution of the problem of shear under finite deformations. After separating the variables in the equations of neutral equilibrium, the problem is reduced to a system of ordinary differential equations. The critical value of the tangential load intensity is determined by numerical methods.

Stability and postbuckling analysis of nonlinear structures

The complementary energy approach for stability analysis of elastic structures under conservative loading is based on Fraeijs de Veubeke's variational principle. The associate equations of neutral equilibrium and stability criterium are presented for arbitrarily large deformations and rotations. When specialized to the structural behavior of thin plates in moderate rotations this functional yields the von Karman equations. Efficient mixed flat shell finite elements are derived from this functional. If the fundamental path is moderately nonlinear, the buckling load and the initial postbuckling behavior can be obtained by an iterative process of Rayleigh-Ritz type, which is based on Koiter's asymptotic approach. Development of the iterative method within the complementary energy principle allows improvement of computational effectiveness and of the rate of convergence, as shown by means of numerical examples.

On uniqueness conditions in the small for the state of hydrostatic compression of a solid

The governing inequality assuring the stability of any homogeneous anisotropic elastic body subjected to a multilateral follower pressure is formulated. A comparison is given between the uniqueness conditions for the state of hydrostatic compression and the governing inequalities of Hill, Hadamard and Coleman-Noll. The variant when part of the body boundary slides along a smooth solid surface is also considered for an isotropic incompressible body. If the external pressure is a dead load, the condition mentioned does not assure uniqueness of the solutions of the neutral equilibrium equations. The problem of determining the critical pressure for dead loading of an isotropic incompressible body reduces to a problem of minimizing a functional.

Influence of physical and geometrical nonlinearity on the value of the upper critical pressure in buckling of a hollow sphere

The framework of the Murnaghan model of a nonlinearly elastic body is used to investigate the problem of stability of a closed sphere acted upon by hydrostatic pressure. The theory of small deformations of an elastic body superimposed on a finite deformation is used [1]. The initial stress-strain state of equilibrium of the sphere is assumed to be centrally symmetric, and the neighboring state of equilibrium to be axisymmetric. The conditions of bifurcation of equilibrium lead to an eigenvalue problem with a nonlinear entry of the parameter. A solution of this problem is obtained by numerical methods for spheres of varying thickness, with various constants appearing in the equation of state taken into account. Several variants of the equations of neutral equilibrium are compared, depending on the accuracy of solution of the initial problem. All this makes possible the investigation of the influence of the physical and geometrical nonlinearity on the value of the critical pressure to be carried out.

Equations of bifurcation of equilibrium of an elastic isotropic body in terms of rates of change of lagrangean coordinates: PMM vol. 38, n≗4, 1974, pp. 693–702

A new version of constructing the three-dimensional theory of elastic stability is advanced. Bifurcation is considered to be an interchange of material particles at a fixed point in space. As a kinematic variable we take the rate of change of Lagrangean particle coordinates. On the basis of obtained exact solutions, an approximate method is developed, valid in the case of small precritical deformations and rotations. Whenever the usual Lagrangean presentation is used for the motion of a continuous medium [1, 2], the equations which determine the changes in the stress tensor necessarily contain the rotations of material particles. As a result, the linearized deformation equations of equilibrium in the general case contain the sought critical stresses [1]. It is of interest to study that version of a boundary value stability problem for which the parameters enter essentially only into the boundary conditions. One of these versions was suggested by Leibenzon [3], and then independently by Ishlinskii [4], However, it cannot be obtained using the Lagrangean presentation while linearizing the original equations of the nonlinear theory of elasticity. In the present paper, nonlinear equations of the theory of elasticity in Eulerian representation are obtained using as a kinematic variable the rates of change in Lagrangean coordinates. On the basis of these equations, bifurcation of equilibrium of an isotropic elastic body is considered. The advantage of the suggested version is that the changes in the Cauchy stress tensor are related only to the deformation tensor of introduced velocities. Therefore, the differential equations of equilibrium contain only those parameters of the prebifurcation state which are related to the change in physical properties of the body during deformation. The components of the rotation tensor of these velocities enter only into the boundary conditions in connection with the change of shape of the body at the instant of bifurcation. The parameters which enter as factors of the components of the rotation tensor in the boundary conditions are the most essential part of parameters of the precritical state which enter into the structure of the obtained boundary value problem for neutral equilibrium. For an isotropic elastic body which is only slightly deformed in the precritical state, the indicated circumstance permits to suggest a simple approximate version of the equations of neutral equilibrium in which the sought parameter of the critical loading enters only into the boundary conditions. If the physical content of the sought functions which enter the differential equations and the boundary conditions is moved into the background, then it appears that the approximate version of the boundary value problem is close to the one used by Leiben zon.

Theory of stability of an elastic cosserat surface

The theory of elastic stability of a Cosserat surface subjected to conservative, nongyroscopic surface and edge loading is presented. The equations of neutral equilibrium and a study of stability at the critical load is included. For the special case in which the deformation is such that the director may be identified with the unit normal to the surface, the results are shown to reduce to the generally accepted theory of stability of thin shells within the Kirchoff-Love hypothesis.

Accurate buckling equations for arbitrary and cylindrical elastic shells

Accurate equations for the neutral equilibrium of arbitrary elastic shells are derived. These equations involve no displacement components and thus complement alternate sets of neutral equilibrium equations proposed by Koiter and Budiansky. The equations are specialized to a uniformly loaded circular cylindrical shell and reduced to a single partial differential equation. For a simply supported cylindrical shell, expressions for the buckling loads are obtained which are as simple as those predicted by the Donnell equations, yet as accurate as those predicted by the more elaborate Flügge equations.