No abstract

It is noted in [1] that a characteristic feature of the behavior of many structures under the action of compressive loads is periodicity of the initial buckling mode in the direction of compression upon loss of stability. However, the f'mal buckling mode of such structures often has the form of a clearly defined single buckle or a small number of buckles. Buckling localization is due to the existence of a bifurcation point when the load reaches the maximal value or thereafter, at which the initial buckling mode bifurcates [1]; after bifurcation the periodic buckling mode is replaced by localized buckling, and this often takes place instantaneously. It is noted in [2] that the mechanism of buckling localization may consist of nonlinear interaction of the buckling modes with similar wavelengths. Study of the localization of buckling of models of real structures shows that the basic laws governing the process are described by the one-dimensional model of the behavior of a bar [1]. It is noted in [3] that the mechanism of buckling in a composite structure, in spite of specific peculiarities, is analogous to the mechanism of the buckling of a bar in an elastic medium.In the present work we study experimentally the buckling of a flexible bar resting on an elastic base. We should point out the difficulties that arise in this problem. Thus, the validity of the Euler formula for the critical load of an axially compressed bar (1744) was finally confirmed 150 years later by the experiments of Bauschinger (1889), Consider6 (1889), Tetmaier (1903), and yon Karman (1910), in which much attention was devoted to the hinged support, central application of the compressive load, and the satisfaction of other conditions that are anticipated by the theory. Thanks to the adoption of these precautions, the experimental results approached the Euler load with accuracy to 1.5 % [4]. The theoretical study of the problem of the buckling of bars resting on several elastic supports was first (1902) performed by Yasinskii [4, 5]. However, in spite of the fact that the theory of beams and plates resting on an elastic base is at the present time a very highly developed branch of mechanics, the existing computational methods are still far from perfect and do not answer many of the questions advanced by practical experience. It is noted in [6] that many of these methods are too complex for practical calculations; the hypotheses which are taken as the basis for the formulation of the mathematical models also can not be considered to be without fault.In the present work we establish experimentally the possibility of unstable behavior of a bar resting on an elastic base, which agrees with the theoretical arguments. In the case of repeated loadings there is noted a reconfiguring of the buckling modes, which is associated with the instability of the realization of the buckling process because of the high density of the critical load spectrum.Experimental Setup and Bar Specimens. Figure 1 shows the specimen loading scheme. The bars 1 were polished steel...

The differential equation of equilibrium of a bent bar axis [I-3] or the integral expression of the system's potential energy [1,[4][5][6][7] are conventionally used as the starting expression in analyzing the stability of a bar on elastic foundation.Equal values of the critical loads for the buckling of the system are obtained in both cases. With the advent of the catastrophe theory, these results were elucidated from a new, more common viewpoint providing a clear description of the influence of initial imperfections on the behavior of the system. Nevertheless, postbuckling behavior of the bar-foundation system has not been sufficiently studied. In the present paper the modes of buckling and postbuckling behavior are studied by the perturbation theory method within the framework of three mathematical models, two of which are classical. It appeared that all three models provide dissimilar description of the postbuckling behavior of the system. A distinctive feature of the problems under consideration is the fact that several possible forms of bar buckling correspond to certain values of rigidity of an elastic foundation, i.e., the appropriate eigenfunction and eigermumber problems have multiple eigenvalues.1. Statement of the Problem. Consider a hinge-supported bar of length L lying on elastic foundation and loaded by axial compression P whose value and direction remain invariant upon deformation of the bar (Fig. 1). The length L of the bar axis is assumed to be invariable. Denote the distance between the bar ends by l. Assume that the bar axis may bend in the plane (x, y) only. Let us study the buckling modes and postbuckling behavior of the bar-foundation system with the use of different models ( Figs. 1 and 2) describing the behavior of the system. 2. Classical Model of an Elastic Foundation. Suppose that, in bending, the reaction forces of an elastic foundation at every point of the bar are invariably directed upright to the Ox axis ( Fig. 1) and proportional to the bar deflection. In this case the expression for the total potential energy of the bar is written [5] L

SHEAR STRAIN EFFECTS ON THE THEORETICAL STRENGTH OF AN ATOMIC LATTICEN. S. Astapov and V. M. Kornev UDC 539.3 A four-atomic unit cell corresponding to a close-packed layer of atoms is considered. It is shown that with occurrence of a shear the system prematurely loses stability. It is concluded that in Novozhilov-type integral criteria for brittle strength, it is reasonable to take into account shear strains.Modified Novozhilov's discrete criteria of brittle strength [1] use the theoretical strength of crystals, which is commonly estimated ignoring shear strains. The shear strains occurring when an ideal crystal is stretched along the symmetry axis of the atomic lattice results in a premature shear deformation [2, 3]. Macmillan and Kelly [4] performed a statistical analysis of the stability of an ideal crystal using the Newtonian approach and described the interatomic interaction by semiempirical potentials such as the Lennard-Jones and Born-Mayer potentials. In this case, in studies of the effects of external conservative forces on the mechanical behavior of crystals, the sum of the force potential and effective energy of interatomic interactions is chosen as a function of the total potential energy of the system. As noted in [3], such an approach often provides appropriate information for description of the macroscopic mechanical properties of a solid.In the present work, a four-atomic unit cell corresponding to a close-packed layer of atoms is considered. The shear strain effects on the stability of a rhombic four-atomic cell stretched along the diagonal under homogeneous deformation is studied within the framework of the approach described in [3, 5]. Interatomic interaction is taken into account by the Morse potential [6, 7]. Although for almost all metals, the interatomic forces are not central even approximately, most of the energy change due to changes in the atomic configuration at constant atomic volume can be described in terms of the central interaction [6][7][8]. Therefore, even if noncentral interactions make-a substantial contribution to the energy of the atomic lattice, it is still possible to obtain satisfactory estimates of some properties using a simple model of pair central interactions [1, 7, 9].The system studied exhibits unstable supercritical behavior. It is found that with occurrence of a shear, the system prematurely loses stability and the critical point on the strain line is determined by the parameters of the Morse potential function. Therefore, in Novozhilov's integral criteria of brittle strength, it is reasonable to use a refined estimate of the theoretical strength of crystals taking into account shear strains.Formulation of the Problem. Let us assume that the potential energy of interaction between any two atoms is a function w(s), which is a spherically symmetric two-body interaction potential for which the force of interaction is directed along the line connecting their centers (s is the distance between the centers). Below, we confine ourselves to considering the interactio...

Delamination of bimaterial composed of two structured materials is considered. A crack is located at the interface between two media. Under tension applied at infinity, I mode fracture is implemented. The improved Leonov-Panasyuk-Dugdale model (LPD model) is proposed to be applied in combination with the Neuber-Novozhilov approach. The case when elastic material characteristics are identical and strength ones essentially differ is analyzed in detail. Analytical description of plotting the fracture diagram of quasi-brittle bimaterial for the plane stress state is given. Numerical modeling of the plasticity zone in bimaterial under quasi-static loading has been performed. The updated Lagrange formulation of solid-state mechanics equations is used in a numerical model. This formulation is most preferable for modeling of bodies made from elasto-plastic material subjected to large strain. Using the finite element method, a plastic zone in the vicinity of a crack tip has been described. It is shown that the shape of the plastic zone in bimetal essentially differs from that in a homogenous medium. Numerical experiments are in good agreement with the proposed analytical model of the pre-fracture zone in the weakest material.