CC BY
0DEFORMATION OF FLEXIBLE SHELLS IN A MAGNETIC
FIELD
Kholjigitov S.M.
Teacher at Jizzakh State Pedagogical University https://doi.org/10.5281/zenodo.12677649
Abstract. The work mathematically models the deformation of shells under the influence of magnetic forces and mechanical loads. Numerical results were obtained and the results were analyzed.
Keywords: shell, deformation, stress, electromagnetic field, magnetoelasticity.
Introduction. An important place in the mechanics of conjugate fields is occupied by the study of the motion of a continuous medium, taking into account electromagnetic effects. To calculate the magneto mechanical deformation of a magnetoelastic medium, it is necessary to use the equations of elasticity theory (equations of motion and boundary conditions), Maxwell's equations of electrodynamics and know the equations of state for a magnetic elastic medium. In order to take into account the presence of eddy currents in a dynamic mode in metallic media, it is necessary to solve the equations of the theory of elasticity together with the equations of electrodynamics. Since in dynamic mode resonance phenomena can also occur, in practical calculations of magneto mechanical deformation of current-carrying bodies, the magnetic
induction vector B is chosen as the quantity that determines the magnetic state of the magnetic elastic medium. This is due to the fact that the occurrence of an electromotive force in a current-carrying body is determined by a change in the magnetic induction flux.
Formulation of the problem. Basic equations. We will consider non-ferromagnetic flexible shells of variable thickness along the meridian, which are under the influence of non-stationary
electromagnetic E and mechanical fields H . Neglecting the influence of polarization and magnetization processes, we assume that an alternating electric current is supplied to the end of the shell from an external source. It is assumed that the external electric current in an unperturbed state is uniformly distributed over the body (the current density does not depend on the coordinates).
We also assume that the electromagnetic hypotheses are satisfied regarding the electric field strength and magnetic field strength [1,2]. These assumptions are some electrodynamic analogues of the hypothesis of non-deformable normal and, together with the latter, constitute the hypotheses of magnetoelasticity of thin bodies. Accepting these hypotheses allows us to reduce the problem of deformation of a three-dimensional body to the problem of deformation of an arbitrarily chosen coordinate surface [1].
In the mechanics of deformable bodies, there are two ways to specify the motion of body points. The Lagrange method is based on the fact that each point of the body is assigned coordinates (m = 1,2,3), which do not change during the deformation process and are called Lagrange variables, or material variables [2]. It is known that in mechanics two Lagrangian coordinate systems are used: initial and actual. When moving from the initial to the current system, the equations in the components change their form. This is due to the fact that the formulas for the transition from the components of the itensor vectors in the initial Lagrangian coordinate system
to the corresponding components in the actual one do not coincide with the usual formulas for the transformation of tensor components when passing from one coordinate system to another in the same space.
The spaces of the initial and actual states with the same coordinates of the points ^ ^3must be considered as different spaces with different metrics due to deformations. The initial Lagrangian coordinate system has advantages over the actual one, since when using the actual system, to completely solve the problem, it is also necessary to determine the position of the coordinate system relative to the reference system. When specifying the movement of body points using the Euler method, the coordinates xm (m = 1,2, 3)are used as variables, which determine the location of the points in the body, which changes during the deformation process. The variables xm are called Eulerian (spatial), and the description of the motion is called a description in spatial variables.
If point Po had coordinates (^1, before deformation, then after deformation it will
take position P, and its coordinates relative to the same reference system will be (xx, x2, x3). The law of motion of body points can be written in the form
xi = t) = xi(|,t), (1)
where t varies from t to T. Thus, the motion of a deformable body is considered known if three functions from (1) are known.
Knowing dependencies (1), you can set other motion characteristics: displacement field £ = t) = x(|, t) - (2)
velocity field
acceleration field
• ^ N du dx
a = v = v(^,t)= = (3)
» _ dV _ a2u _ a2x
u = at = at2 = at2, (4)
and also calculate, in accordance with physical laws that describe the properties of the material of the body at point x 7 the change in other fields that accompany the movement of the body Q, for example, a change in the density of the medium
p=p(x :t)=p(x 7(£ :t),t) . (5)
It is assumed that dependencies (1) are continuously differentiable and at any given time. It follows, firstly, that
>0, v te[tc,n (7) = 1, and secondly, that dependence (1) can be resolved with respect to^:
det
since at t = o det
^ = ^^(x1,x2,x3,t) = ^'(x,t) (8)
Let the motion be given by the Lagrange method; the transition to the Euler variables x is carried out, as noted, using relations (8), inverse (1).
Note that in the case of using Lagrange variables £ ^the rate of change of any parameter is determined by the partial derivative of this parameter with respect to time.
Let us now know the task of motion according to the Euler method. Let's move on to Lagrange variables. To do this, first of all, consider a material particle located at a given moment of time t at a point in space x 7 it has speed (V) ~(x 71) and at time t+At we will have coordinates
x + dx = X + Vdt. (9)
From dependence (9) follows the system of equations
= Vi(x1,x2,x3,t), (10)
which is a system of ordinary (nonlinear) differential equations with respect to the coordinates of points of the body Q. As it is known, the general solution of system (10)
xl = xi(C1,C2,C3,t) (11)
depends on three arbitrary constants, which are determined from the condition xi(C1,C2,C3,t) EH. (12)
Having resolved the system of algebraic equations (12) with respect to the constants C1 and substituting the solution into (11), we obtain
xl = xi(Z1,Z2,Z3,t) (13)
(the form ofjunctions1 will change in this case). This completes the transition from Euler variables to Lagrange variables, because if the density field is known
p = p(x,t), (14)
then by substituting (13) into (14) we move on to the Lagrange variables.
Methodology for solving a related problem. After some transformations [3], we obtain a complete system of nonlinear differential equations of magnetoelasticity in the Cauchy form, which describes the stressed state of an electrically conductive flexible shell under the influence of mechanical and magnetic fields.
a u_ i— vv0,N v cos^
a s
a w as
eh
u-
v0 sin^
i
r
r
w — 02 ; 2 2
= —0
D0
S
S 5
as
12(1 — vVn ) , , k 0
v s — Mv--0
cos^
r
V
a s
— Ps + hJecTB^ — <yx h
/ \ eB
vs— — 1
e
S
eh -
NS + e0h
v0 cos^
r
0
S
r cosrn sinrn ^ -— u +--— w
y
V
r
r
y
Ee B^ + 0.5 ^W B,(B+ + B-) —
a t
au ~dt
B2
+ ph
a2 u
at2
aQs a s
cos^
r
e0 sin^
sin^ ( cos^
Qs + V ^ ^^ N + 1 ^^ u + w I — Pr —
e
r
r
sin^
r
r
— 0.5 h Jec T (b+ + h
0.5 Ee(B++ Bs) — 0.25 aW (B++ B—)2
1 a w 12 a t
(b+— bsj + 0.5 aubc(b++ Bs) +
h a0
at
DMs _ cos^
a s r
v
sin^
0—1
2s y
•3
12 a t
e0h cos^
BC(B+ + Bs )] + ph
a2 w
M +0
12 r
+ Qs + Ns0s —
e0h cos^
s e s 12 r
^. h a 0
0s +--s;
s 12 at2
r
=
Ö s
ÖE0 _ cos^
+ BS — BS h
E,
Q s Q t r 0
In relations (1) the notations generally accepted in the theory of shells and the theory of electromagnetoelasticity are used. The technique for solving the nonlinear problem of magnetoelasticity of current-carrying bodies is based on the sequential use of the Newmark scheme, the quasi-linearization method and the discrete orthogonalization method [1,2,3,4,5,6,7].
Analysis of the results obtained.
Let us study the behavior of a conductive shell of variable thickness in a magnetic field. Figure 1 shows the distribution of electric field Ee strength as a function of time s = 0.04 m for options: 1 - isotropic aluminum cone; 2 - orthotropic beryllium cone.
Fig.1 Electric field strength distribution
As can be seen from the graphs, in the cases considered, the maximum values E at the left end of the shell coincide, and as they move away from the ends, their differences are observed.
Conclusion. The work analyzes the stressed state of a flexible shell under the influence of a time-varying mechanical force and a time-varying external electric current, taking into account geometric nonlinearity.
The magnetoelastic nonlinear problem for a shell is considered in a coupled form. Numerical results were obtained and the stress-strain state was analyzed.
The presented results make it possible to evaluate the influence of nonlinearity on the stress-strain state of the shell.
REFERENCES
A. Ambartsumyan, G.E. Bagdasaryan, and M.V. Belubekyan, Magnetoelasticity of Thin Shells and Plates [in Russian], Nauka, Moscow (1977).
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