Nonlinear Deformation of Flexible Orthotropic Shells of Variable Thickness in an Unsteady Magnetic Field

The analysis of the stress state of a flexible orthotropic shell under the influence of a time-varying mechanical force and a timevarying external electric current is performed, taking into account the mechanical and electromagnetic orthotropy. The effect of thickness on the stress-strain state of the orthotropic shell is investigated. The results obtained indicate the influence of thickness on the deformation of the shell and the need to take this factor into account in the design schemes.


Introduction
The development of research in the theory of magnetoelasticity is associated with the solution of many important problems of modern technology. Such tasks arise in the development of electromagnetic pumps, magnetohydrodynamic accelerators, instrumentation that works with electromagnetic fields, the imposition of magnetic fields in controlling the movement of plasma, in the flow in an elastic shell, the calculation of protective shields, atomic reactors, setting up some physical experiments, etc.
The construction of optimal designs of modern technology operating in magnetic fields is associated with the wide use of structural elements, such as flexible thin-walled shells. The effect of non-stationary fields on metallic thin-walled elements leads to the appearance of bulk electromagnetic forces, capable of causing large deformations of structures under certain field parameters. Recently, the question of determining the stress state of flexible orthotropic shells operating in an alternating magnetic field with regard to orthotropic electrical conductivity has raised considerable interest.

Nonlinear formulation of the problem. Basic equations
Consider the nonlinear behavior of an orthotropic current-carrying conical shell of beryllium of variable thickness, changing in the meridional direction according to the law ℎ = 5 • 10 −4 (1 − ⁄ ) . We believe that the shell is under the influence of mechanical force = 5 • 10 3 2 ⁄ , third-party electric current = −5 • 10 5 2 ⁄ , and external magnetic field 0 = 0.1 л, and that the shell has a finite orthotropic conductivity ( 1 , 2 , 3 ). We assume that the external electric current in the unperturbed state is uniformly distributed over the shell, i.e. External current density does not depend on the coordinates. In this case, the shell is affected by a combined load consisting of the ponderomotive Lorentz force and the mechanical force. Small radius contour = 0 hinged, and the second circuit = free in the meridional direction. Note that in the case under consideration an arbitrary second-order surface has three mutually perpendicular axes of the second order and these axes can be arranged parallel to the second-order crystallographic axes, and the second-order characteristic surface has all the symmetry elements that can be found in the classes of the orthorhombic system [2,3].
Suppose that the geometric and mechanical characteristics of the body are such that to describe the process of deformation, we apply the variant of the geometrically nonlinear theory of thin shells in the quadratic approximation. Also assume that with respect to the electric field strength ⃗ and magnetic field strength ⃗ ⃗ electromagnetic hypotheses are performed [1,4] (1) − shell vector displacement components; , − components of the vectors of the electric and magnetic fields of the shell; − eddy current components; ± − tangential components of the magnetic field on the surfaces of the shell; ℎ − shell thickness.
These assumptions are some electrodynamic analog of the hypothesis of undeformable normals and together with the latter constitute hypotheses of magnetoelasticity of thin bodies. The adoption of these hypotheses allows us to reduce the problem of the deformation of a three-dimensional body to the problem of the deformation of an arbitrarily selected coordinate surface.
The developed methodology for the numerical solution of a new class of related problems of magnetoelasticity of the theory of orthotropic conical shells of revolution with orthotropic electrical conductivity is based on the sequential application of Newmark's finite-difference scheme, the linearization method and discrete orthogonalization [4][5][6].
To effectively use the proposed technique, we assume that when an external magnetic field appears, there are no sharp skin effects along the shell thickness and an electromagnetic process along the coordinate ζ quickly goes to a mode close to steady. This leads to restrictions on the nature of changes in the external magnetic field and on the geometric and electrophysical parameters of the shell.
−characteristic time of the magnetic field. In case of non-fulfillment of this condition, only the equations of shell movement under the action of magnetic pressure should be considered. In this formulation, the system of equations describing the nonlinear oscillations of a flexible current-carrying orthotropic conical shell of variable thickness on the corresponding time layer, according to [5,6], after applying the quasi-linearization method, takes the form The solution of the problem was on the time interval = 0 ÷ 10 −2 , time integration step is equal to ∆ = 1 • 10 −3 .
In the case considered, the anisotropy of the electrical resistivity of beryllium is This is explained by the fact that, according to the boundary conditions, the left end is pivotally fixed, and the right end of the shell is free in the meridional direction.
In addition, the thickness of the shell from the left end to the right end decreases to 2 times when = 0.5. Therefore, the maximum deflection values occur near the right end of the shell.
When taking into account the effect of thickness, the stress of the conical shell was considered as the sum of the mechanical stresses and maxwell stresses, i.e. the general stress state was taken into account.
In fig. 2      The figures show that with increasing parameter value and accordingly with a decrease in thickness ℎ( ), there is an increase in the speed of longitudinal movement along the meridian.
The maximum values of the acceleration of the radial displacement along the meridian occur when = 0.4м, which is associated with boundary conditions and variations in shell thickness.

Conclusion
This article discusses the related problem of magnetoelasticity for a flexible orthotropic conical shell, taking into account the orthotropic electrical conductivity. The effect of thickness on the stress-strain state of an orthotropic shell is investigated. The results obtained indicate the effect of thickness on the deformation of the shell and the need to take this factor into account in the design schemes. As can be seen, the thickness variability has