Unsteady dynamics of a sandwich plate under the influence of a cylindrical wave in an elastic medium

The interaction of a sandwich plate with a damped cylindrical wave in the ground has been investigated. A sandwich plate is considered as a model of a barrier in the ground, described by a system of equations by V. N. Paimushin, placed in the ground dividing it into two parts. The plane problem formulation is considered. The boundary conditions correspond to the hinge attachment of the barrier, and the initial conditions are zero. A cylindrical damped wave is considered as an external influence. To describe the ground movement, the equations of the elasticity theory, the Cauchy relations and the physical principle, or equivalent displacements in potentials and the Lame equations are used. The problem is solved in a related formulation, where the movement of the plate and its surrounding media is considered together. All components of the equations of motion of the plate and media are decomposed into trigonometric series and the Laplace transform is applied to them. As the conditions for the contact of the plate and the ground, the equality of normal displacements at the boundary of the medium and the plate is assumed. It is also assumed that the pressure amplitudes and normal stresses coincide. After determining the constants from the contact conditions, the displacement values and the values of normal and tangential stresses are found, after which their originals are found.

MODELING THE BEHAVIOR OF THE PHYSICAL AND GEOMETRIC NON-LINEAR FUNCTIONAL HETEROGENEOUS MATERIALS

The work is devoted to the problem of modeling the behavior of functionally inhomogeneous materials with the properties of pseudo-elastic-plasticity under complex loads, in particular at large strains (up to 20%), when geometric nonlinearity in Cauchy relations must be taken into account. In previous works of the authors, functionally heterogeneous materials were studied in a geometrically linear formulation, which is true for small deformations (up to 7%). When predicting work with material at large deformations, it is necessary to take into account geometric nonlinearity in Cauchy relations.Studying the behavior of bodies made of functionally heterogeneous materials under unsteady load requires the development of special approaches, methods and algorithms for calculating the stress-strain state. When constructing physical relations, it is assumed that the deformation at the point is represented as the sum of the elastic component, the jump in deformation during the phase transition, plastic deformation and deformation caused by temperature changes.A physical relationship in a nonlinear setting is proposed for modeling the behavior of bodies made of functionally heterogeneous materials. Formulas are obtained that nonlinearly relate strain rates and Formulas are obtained that nonlinearly relate strain rates and displacement rates.Keywords: mathematical modeling, functional heterogeneous materials, geometric nonlinearity, spline functions, pseudo-elastic plasticity, phase transitions

Hooke’s law and elastic constants

Hooke’s law and elastic constants are introduced. The symmetry of the elastic constant tensor follows from the symmetry of stress and strain tensors and the elastic energy density. The maximum number of independent elastic constants is 21 before crystal symmetry is considered, and this leads to the introduction of matrix notation. Neumann’s principle reduces the number of independent elastic constants in different crystal systems. It is proved that in isotropic elasticity there are only two independent elastic constants. The directional dependences of the three independent elastic constants in cubic crystalsare derived. The distinction between isothermal and adiabatic elastic constants is defined thermodynamically and shown to arise from anharmonicity of atomic interactions. Problems set 3involves the derivation of elastic constants atomistically, the numbers of independent elastic constants in non-cubic crystal symmetries, Cauchy relations, Cauchy pressure, invariants of the elastic constant tensorand compatibility stresses.

Theoretical Predication Method of High Order Elastic Constants of Cubic and Tetragonal Crystal Material

The theoretical method of predicating second and third order elastic constants of cubic and tetragonal material are presented by using first-principles total-energy method combined with the means of homogeneous deformation. The predicted results of SrTiO3provide reasonable agreement with the reported experimental data, other theoretical results and Cauchy relations. Since high order elastic constants are very difficult to be measured, the methods presented here provide a valuable guidance for experiments and the investigation of high order elastic properties for cubic and tetragonal materials.

SELF-CONSISTENT THEORY OF ELASTIC PROPERTIES OF STRONGLY ANHARMONIC CRYSTALS II: PROPERTIES OF HEAVY RARE GAS SOLIDS

The method developed in the preceding papers has been used to calculate the full set of equilibrium thermodynamic properties of solid Ar, Kr and Xe at normal pressure. We have used various pairwise interatomic potentials taking into account the three-body Axilrod–Teller forces. For Ar and Kr the multiparameter potentials proposed by Barker et al. provide the best fit to available experimental data, and for Xe the old Lennard–Jones potential does this. Deviations from the Cauchy relations are discussed. Temperature dependencies of the stability coefficients and anisotropy have received much attention. In the unstable point that is about 1.35–1.45 times larger than the experimental melting temperature of each crystal, the isothermal bulk modulus goes to zero (and the thermal expansion coefficient and the isobaric specific heat tend to infinity). Other stability coefficients remain finite and positive.