S. V. Kashtanova

Stability loss in an infinite plate with a circular inclusion under uniaxial tension

Loss of stability under uniaxial tension in an infinite plate with a circular inclusion made of another material is analyzed. The influence exerted by the elastic modulus of the inclusion on the critical load is examined. The minimum eigenvalue corresponding to the first critical load is found by applying the variational principle. The computations are performed in Maple and are compared with results obtained with the finite element method in ANSYS 13.1. The computations show that the instability modes are different when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. As the Young’s modulus of the inclusion approaches that of the plate, the critical load increases substantially. When these moduli coincide, stability loss is not possible.

The Stability of the Plates with Circular Inclusions under Tension

This paper deals with the problem of local buckling caused by uniaxial stretching of an infinite plate with a circular hole or with circular inclusion made of another material. As the Young’s modulus of the inclusion approaches that of the plate, the critical load increases substantially. When these moduli coincide, stability loss is not possible. This paper also shows the difference between them when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. Computational models show that instability modes are different both when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. The case when plate and inclusion have the same modulus of elasticity, but different Poisson’s ratio is investigated too. It is also discussed here the case when a plate with inclusion is under biaxial tension. For each ratio of the modulus of elasticity of plate versus inclusion it’s obtained the range of the load parameters for which the loss of stability is impossible.

Modes of Stability Loss of Materials

Three problems of stability loss are investigated and corresponded buckling modes are discussed. The first one is the stability loss of a compressed transversely isotropic linearly elastic medium. The standard analysis based on the Hadamard condition is conducted to solve this problem. The critical compression could be uniquely defined from the bifurcation equations but not a wave length. So, the buckling mode remains generally indefinite. The second considered problem is the stability loss of a compressed half-space with a free surface. It could be shown that the waviness is localized near the free plane surface but as for an entire space the wave length and the buckling mode are indefinite. These problems are treated in linear and nonlinear statement. In linear approach the pre-buckling deformations are ignored. It is shown that for some values of parameters the linear approach leads not only to the numerical error but also to qualitatively incorrect results. The thisrd problem under investifation is the stability loss of an uniformly compressed plate lying on a soft elastic half-space. In this problem the wave length is uniquely defined. Using the nonlinear post-critical analysis it is shown that the buckling mode could be fully defined and is has a chessboard-like character.

Application of nonclassical models of shell theory to study mechanical parameters of multilayer nanotubes

Stress-strain state of multilayer anisotropic cylindrical shells under a local pressure is studied. Such a problem may model the bending of an asbestos nanotube under the action of a research probe. In earlier works, these authors showed that the application of classical shell theories yields results far from experimental data. More accurate results are obtained by taking into account additional factors, such as the change of the transverse displacement magnitude (according to the Timoshenko-Reissner theory) or the layered structure of asbestos and cylindrical anisotropy (according to the Rodinova-Titaev-Chernykh theory). In the present paper, yet another shell theory, the Palii-Spiro theory, is applied to solve the problem; this theory was developed for shall of average thickness and is based on the following assumptions: (a) the rectilinear fibers of the shell perpendicular to its middle surface before deformation remain rectilinear after deformation; (b) the cosine of the angle between the shell of such fibers and the middle surface of the deformed shell equals the averaged angle of the transverse displacement.Deformation field are studied with the use of nonclassical (the Rodinova-Titaev-Chernykh and Palii-Spiro) shell theories; a comparison with results obtained for three-dimensional models with the use of the Ansys 11 package is performed.

Evaluation of the Mechanical Parameters of Nanotubes by Means of Nonclassical Theories of Shells

In [3] the stiffness of bridges and cantilevers made of natural chrysotile asbestos nanotubes has been studied by means of scanning probe microscopy. The stiffness is defined as a ratio of the value of the local load (applied to the tube) to the value of the displacement. Nanotubes with different fillers are analyzed. Experiments show that the stiffness of the tube depends on the materials for filling. The tubes with water are softer and the tubes filled with mercury are more rigid than tubes without filling materials. It was shown in [3] that the classical theory of bending can not explain the experimental results, but the experimental results well agree with the Timoshenko-Reissner theory (at least qualitatively), when the interlaminar shear modulus of elasticity changes for different filling materials. When additional factors such as lamination of structure and cylindrical anisotropy are taken into account the theory of Rodionova-Titaev-Chernykh (RTC) permits to obtain much more reliable results. In this work the authors also applied another nonclassical shell theory, namely the shell theory of Paliy-Spiro (PS) developed for shells with moderate thickness. The comparison of nonclassical shell theories (RTCh and PS) with experimental data and FEM calculations are presented.