# deformation

## Mathematical modelling of critical speed of the multistage core at longitudinal blow

Mathematical modeling of longitudinal elastic central blow of a multistage core about a rigid barrier is carried out, at not keeping communications. Mathematical modeling is carried out by the exact analytical decision of the wave differential equation by a method of Dalamber with the task of necessary initial and boundary conditions. With application of the formula of Euler analytical expression for calculation of critical pretonic speed at which there comes loss of stability of a step core is received.

## Mathematical modelling of loss of stability of system of step and homogeneous cores at blow about the rigid barrier Tymoshenko's method

Mathematical modeling of longitudinal elastic central blow of system of step and homogeneous cores about a rigid barrier is carried out, at not holding communications by a solution of the wave equation by Dalamber's method. On the basis of the law of conservation of energy by Tymoshenko's method the size of critical compressing loading according to which, the size of critical pretonic speed leading to loss of stability of considered rod system further pays off.

## Rotational Invariance of Non-Linear Lagrangians of Type-II Micropolar Thermoelastic Continuum

The paper contains new results related to extension of the field theoretical approach and its formalism to non-linear coupled micropolar thermoelastic media. A mathematical model of micropolar (MP) type-II (GNII) thermoelastic (TE) continuum is considered. A formulation of the least thermoelastic action principle is discussed. Partial differential equations subsequent to the least action principle are derived. The translational symmetries of non-linear Lagrangians are adopted. Those include an additional symmetry: translations of the thermal displacement.

## Mathematical Modeling of Longitudinal Blauofthe System of Homogeneous Rodsabout Rigid Barrir at Increase Long Solids

Mathematical modeling of longitudinal elastic central blow of non-homogeneous rod system about a rigid barrier is carried out, at not-holding connections. Mathematical modeling is carried out by the exact analytical decision of the wave differential equation by method of Dalamber with the setting of necessary initial and boundary conditions. The rod system consists of a step non-homogeneous rod and a homogeneous rod of constant cross section. Connections with a rigid barrier and between rods are not-holding.