Bulk and Surface Stability of Materials

Abstract

Buckling modes of a linearly elastic compressed medium are discussed. Stability of a plate lying on a soft elastic foundation is investigated. To the plate surface a membrane with initial stresses is attached. The stresses in the membrane simulate difference of equilibrium distances between atoms of crystal lattices of a surface layer (membrane) and of a plate. Also the influence of force and thermal stresses is studied. The chessboard-like stable buckling mode appears on the membrane surface at the stability loss. The ways of control of the wave length of buckling mode are discussed.

Bending Stiffness of a Multilayered Plate

“…Calculating the integrals in Eqs. (15) and 17we find the coordinate z = a of the neutral layer, the bending stiffness D according to the KL model, and the coefficients A g and A ν in Eqs. (17) as follows:…”

“…The present paper is concerned with a thin plate of constant thickness made of a linearly elastic material that is transversally isotropic and heterogeneous in the thickness direction. For a transversally isotropic material the accepted simplification [15], for which a 3D system of sixth order of the theory of elasticity splits into systems of second and fourth orders, is possible. Asymptotic expansions in powers of the small thickness parameter µ are constructed, the bending equation of second-order accuracy (the SA model) is obtained.…”

Modes of Stability Loss of Materials

“…The elastic potential energy density for transversely isotropic material is (4) where (5) are displacements, and denote elsticity modulus sutisfying (for transversally isotropic material) to the following relations (6) In further examples we will use the modules (6) in the form [6,7] (7) Formulae ( 7) contains two dimensionless parameters: the Poisson ratio and the measure of anisotropy . At the material is isotropic, and at it is strongly anisotropic (…”

“…In addition to (12) let assume that the elastic modulus ( 6) may depend on z. We will try to seek the solution in the form (3) and instead and we will introduce the new unknown variables U, V by relations [6][7][8] (14) Then the bifurcation system divides into two parts (15) (16 ) Eq. ( 15) describes the shear deformations and displacements just in the horizontal plane.…”

“…In the case 3 the sign of the value depends on the elastic modules. For the modules (7) the division of the plane into the parts with type 1 and type 2 of buckling and then with and was demonstrated.…”