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15Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 3, pp. 233-250. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd190303
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 74J30
The Study of Wave Propagation in a Shell with Soft Nonlinearity and with a Viscous Liquid Inside
L. I. Mogilevich, S. V. Ivanov
This article is devoted to studying longitudinal deformation waves in physically nonlinear elastic shells with a viscous incompressible fluid inside them. The impact of construction damping on deformation waves in longitudinal and normal directions in a shell, and in the presence of surrounding medium are considered.
The presence of a viscous incompressible fluid inside the shell and the impact of fluid movement inertia on the wave velocity and amplitude are taken into consideration. In the case of a shell filled with a viscous incompressible fluid, it is impossible to study deformation wave models by qualitative analysis methods. This makes it necessary to apply numerical methods. The numerical study of the constructed model is carried out by means of a difference scheme analogous to the Crank-Nickolson scheme for the heat conduction equation. The amplitude and velocity do not change in the absence of surrounding medium impact, construction damping in longitudinal and normal directions, as well as in the absence of fluid impact. The movement occurs in the negative direction, which means that the movement velocity is subsonic. The numerical experiment results coincide with the exact solution, therefore, the difference scheme and the modified Korteweg-de Vries-Burgers equation are adequate.
Keywords: nonlinear waves, elastic cylinder shell, viscous incompressible fluid, Crank-Nick-olson difference scheme
Received April 10, 2019 Accepted July 15, 2019
The work was supported by the RFBR grant No. 19-01-00014a.
Lev I. Mogilevich mogilevichli@gmail.com
Yuri Gagarin State Technical University of Saratov ul. Politechnicheskaya 77, Saratov, 410054 Russia
Sergey V. Ivanov evilgraywolf@gmail. com
Saratov State University
ul. Astrakhanskaya 83, Saratov, 410012 Russia
1. Introduction
The problem of wave propagation in gas dynamics and elastic shells theory is studied using linearized equations. The disturbance propagation velocity is taken to be constant and equal to the sound propagation velocity in an undisturbed medium. However, a number of cases in spite of small values of dependent variables are defined by the dependence of the disturbance propagation velocity on dependent variables values and are studied on the basis of nonlinear equations. These investigations are carried out by perturbation methods.
The study of the wave process in elastic shells is widely applied in various technical fields. The propagation of deformation waves in elastic, viscoelastic, and nonlinear viscoelastic shells and plates was considered in [7-11]. In those papers, the case of interaction of shells and a viscous incompressible fluid is not considered. References [12-14] consider the case of interaction of shells and a viscous incompressible fluid without taking into account wave phenomena; neither were local terms of inertia influence investigated. Various methods are used to solve related and unrelated problems.
When solving unrelated problems, the motion of a fluid interacting with a rigid body is considered. The stress exerted by the fluid on the rigid body, friction and pressure are determined. Thus, it is assumed that there is no influence of the deformation of the shell on the movement of the fluid [15-19]. The parameters obtained are substituted into the equations of the dynamics of an elastic body, then longitudinal and normal (deflection) displacements are found. Thus, the stress-strain state of the elastic structure, as the aim of the uncoupled problem, is determined.
In the case of a related problem, the equations of the dynamics of an elastic body and a fluid are solved simultaneously, taking into account the boundary conditions on impermeable surfaces. This approach has been applied to the study of hydroelastic vibrations [20], as well as in this article to the study of nonlinear deformation waves of elastic shells containing a viscous incompressible fluid, taking into account the inertia of its motion.
It is impossible to investigate models of deformation waves using methods of qualitative analysis in the case of a shell filled with a viscous incompressible fluid [1]. This makes it necessary to apply numerical methods.
This article examines the influence of structural damping in the longitudinal and normal direction, surrounding elastic medium, viscous incompressible fluid inside the shell as well as the inertia of fluid motion on the wave amplitude and velocity.
In the course of investigation the numerical study of the constructed model was carried out by using a difference scheme for an equation similar to the Crank-Nicholson scheme for the heat equation [21].
2. Defining and resolving relations of the physically nonlinear theory of shells
Let us consider an axisymmetric cylindrical shell. Denote: h0 is the thickness of the shell; R is the radius of the medium surface; R\ is the radius of the inner surface; U is the longitudinal elastic displacement; W is deflection directed to the center of curvature.
Longitudinal deformation waves in an infinitely long shell are featured in the long-wave low-frequency model described by the Kirchhoff-Love theory.
A. A. Ilyushin's deformation theory of plasticity [2, 22] connects the components of the stress tensor ax, a© with the components of the strain tensor ex, e©, and the square of the
strain intensity eu [3, 23]
E
{ex + ß0£@) (l - ; <70 = (e® + ~
Ox =
1 " ßO
1 " ßO
4
£u = 3 (ß1 (£l + t|) - A^ta-t©);
(2.1)
Atl = 3
1 +
ßo
(1 - ßoH
1
№ = 3
1
2ßo
(1 - ßoH
where ^ is Young's modulus, m is a material constant determined from tensile or compression experiments, and fi0 is Poisson's ratio of shell material.
We write down the connection between deformations and elastic displacements in the form of [4]
dU 1 ( dW
dx 2 \ dx
— z-
d2w _
dx2 '
t© = -■
W_
~R'
(2.2)
where x is the longitudinal coordinate along the median surface and z is the normal coordinate in the shell f—— ^ z ^ —
V 2 2
The square of the intensity of deformations is
or
— 2
4
+ ß2
ßl
w_
R
dU_ 1 idW dx 2 I dx
c)U_ 1 fdW dx 2 I dx
d 2 W
— z-
dx2
+
W2_ R2
+
— z-
d2W dx2
2 4 /
£u = 3 ( ßl
dU_ 1 idW\ dx 2 \ dx J
W
+ {r)
W
dU_ l idW dx 2 I dx
2ßl
dU_ 1 idW dx 2 I dx
2
+ ß2
W_
~R
d2W dx2
+ z ßl
fd2WY V dx2 J
(2.3)
(2.4)
Let us define the forces in the middle surface of the shell and the moment according to the following formulas:
h0 2
h0 2
h0 2
Nx = Ox dz; N© = a& dz; Mx =
oxz dz.
(2.5)
where
h0 2
h0 2
h0 2
h0 2
h0 2
h0 2
h0 2
1 - ™r2
dz = hJl-^\
1 - ™r2
W + /i2if
m 4
dZ = ~E 3
dU_ ~dx + 2
ßl
1
dU_ 1 idW dx 2 I dx
+ ^
W R
dwV'
dx J
M 12
+
1
h2 12
dx2 J
+
/ dU _
2M^ + 2
awV
dx /
W ~R
d2w\
dx2 y
(2.6)
1
2
2
2
u
3
2
2
2
2
2
2
2
z
ho 2
h0 2
m
1 - ˣ>
hl m 4
dU 1 ( dW
+
dx 2 V dx
+
W ~R
+
W
dU_ 1 idW\ dx 2 \ dx J
h2° fd2W
Substituting (2.6) into (2.5), we find
Nx =
Ehp 1 "ßo
dU_ 1 ~dx + 2
cW dx
W m 4
dU_ 1 /£W
9a; 2 I 9a;
ßi
dU_ 1
9a; 2 1 9a;
/Wx 2
Ne =
hl ( cPW^ 2 + 12 V dx2
Eho
Í dU 1 ( dW
9^ + 2 19^
/ 9U 1 / 9W Y
+ M 9^ + 2 U^J
W
+ (ß2 - ßlßo)
1 - ß2
ßü
dU_ 1 (dW y
9a; 2 \ dx J
W m 4 ~~R~E 3
, dU
1 /9^yN
+ 2 IJ
ßi
dU . 1
+ № 9^ + 2 1
dU_ 1
9a; 2 1 9a;
+
2
+
~R
+
+
h2 /9 2W
12 V dx2
dU 1 ( dW 3wt0 9^ + 2 9^
+ (ßl - ß2ßü)
"ñ
W
+
(2.7)
Mx = -
ffhg 92iy / m4 12 (1 - jug) 9a;2 \ Atl
dU_ 1
9a; 2 l 9a;
+
+ 2 (ß2 - ßlßü)
dl7 1
9a; 2 \ dx J
W
W
h2
9 2 W
Dynamic equations for shells with structural damping in the longitudinal and normal directions are written as
dNx dx
d2 U
1
E
dU
-poho^+£lTpohoV Po (I-ß2) dt
Qx - W— + U—-dr dx
d 2M,
dW
dx2 dx V dx
x 9
Nx
E
dW
1 , d2W l ,
+ j;N@ = poho-^ + ^Y2Pohod-pQ (1 _ ^ m
+
E
R3 P0 (1 - ßo)
W
qn - w— + ¿7—— 9r 9x
ß
(2.8)
2
2
2
2
2
2
2
2
2
2
x
2
2
x
2
2
2
2
2
2
2
2
2
where t is time, e\ and e2 are the damping coefficients, p0 is the density of shell material, k\ is the coefficient of subgrade reaction, qx and qn are stresses of the fluid inside the circular section, r and x are the cylindrical coordinates, Vr and Vx are velocity vector projections on the axis of the coordinate system, p is the pressure in the fluid, p is the fluid density, v is the kinematic viscosity coefficient, n is the normal to the median shell surface, and nr, n©, i are the basis vectors (r, 0, x) of the cylindrical coordinate system with its center on the geometric axis.
Substituting (2.1), (2.2), (2.5) into (2.8), we find the equations of dynamics in displacements
Ehp 9 / dU 1 idW V 1 — $ dx \ dx 2 \ dx J
W m 4
dU_ 1 idW\ dx 2 \ dx J
2
V o
W ~R
ßl
dU_ 1 (dW\
~dx + 2
2
dx J
W2
f dU 1 / 2\ W
+ ß2 9^ + 2 l-fr) 1T
+
+
h0 (d2 W
12 V dx2
„ / dU 1 (dWs
+2 I 9^"
+ (ß2 - ßlßo)
w
R
E
dU
^d2U 1 I -p0h0^+£llp0h0fß0(l-ß2) dt
EhO
d2 d 2 W
12 (1 - ß2) dx2 \ dx2
1
m4
~E 3
3ßi
urdqx . TTdqx Qx ~ W— + U— dr dx
dU_ 1 idW dx 2 V dx
(2.9)
+ 2 (ß2 - ßlßo)
Eho d
dU_ 1 fdW
dx 2 V dx
^r + (ßl - ß2ßo) ( "77 ) + S^ßi
W ~R
+
dW
1 - ß2 dx \ dx
dU_ 1 idW\ dx 2 \ dx J
2
W R
W m 4
R
+
20'
dU_ 1 idW V dx 2 \ dx J
+
ßo
W_
~R
+
'dU_ 1 fdW\r
14 (d2w\2 12
2
/d2W\2 I dU 1(2\ .
dU 1 fdW
+ ß2 9^ + 2 llfc
W ~R
Eho 1 + -;-— ( ßo
1 - ßO R
dU_ 1 idW\ dx 2 \ dx J
W m 4 ~R~~E 3
/ dU 1 / dW
№ 9^ + 2 l 9^
TU ~R
+
W_
~R
ßl
dU_ 1 /cWs 9t 2 I cte
2
+
W ~R
14 f&w^2
+ 12 v dx2
f dU 1 (dWs 9^ + 2 [W
dU 1 / dWN
+ № +2 V 9^
W
+ (ßl ~ ß2ßo)
TU
"r
+
E
dW
, d2W 1 /
-poho^ + £2T2pohofPo(i-ß2) dt.
+
J.I ho 1 E
R3 Po (1 - ßO)
W
u^dqn t jjdqu
Qn ~ W— + U—-dr dx
R
2
2
2
2
2
2
2
3. The asymptotic method for studying the equations of shells with a fluid
For wave problems, the shell is assumed to be infinite. For longitudinal waves in the shell, dimensionless variables and parameters are introduced. We take the wavelength l for the characteristic length, and um, wm are the characteristic values of the elastic displacements
W = wmu3, U = um ui, x
* - T> t* = r* = 7T-
x
T
l
r R
(3.1)
co
E
P(1 - Po) ho
is the propagation velocity of longitudinal elastic waves in the shell. Set R2
R
= £ < 1,
l2
= 0(£), ^ = 1), ^ = 0(1), ^ = 1), ho £ £
h2 ho
h0 R2
l±nIL = o( i) — = o( i)
I ho [ '' E l I2 R2 I2
(3.2)
£3,
where £ is the small parameter of the problem. In these variables, Eqs. (2.9) take the form
c0 Poho
1 9 [ I umdui 1 (Wm\2 (du3\2\ _
l dx*
l dx*
2 l2 V R ) \dx*
Wm llQ—j^-Uz
m 4
~E 3
umdui 1 )_
I dx* 2 I2 \ R J [dx* At° R U3
Pi
um dui l dx*
+
2
lR^ (^rn\2 \ (Wm
+2Ttf) I.W) I
,'um dui
+ M2 I —+
+
1 R2 iWm\2 f duA2)
2 I2 V R ) \0x* J j R
■Us
+
h2 R
2 wi
12 l4 R2 V dx*2
d2u?
l dx*
um dui ,
3Atl <
c^±\2 (f^y ^ ■'......^Wm
2 l2 \ R J \dx*J J
c^poho d2 ui c^po ho dui
-um + £\
l2
dt*
i2 Umdt*
Wm dq
R dr* l dx*
(3.3)
2 ho Hq 1 d2 / R wm d2U3 ( m 4 ^TttPdx** [llld^2 i ~£3
, um dui
I +
1 R fwm\2 (dU3_
2 l2 V R ) \dx
+
um dui , 1 R (Wm\2 (du^W Wm ,, x( Wm\2,
--+ — hr) J J IT3 +(Atl - ^HirJ +
+,i20 /4 I E J ^{dx*2
2 ,1 d / RWm du3 + CoPohojTTT
l dx* \ l R dx*
R
Um dui l dx*
+
+-
(Wm\2 A ^rn
2 l2 \ R ) \dx
+-
2 l^ R ) \dx*)
E3
Pi
Um Qui l dx* Um dug l dx*
+
+
2
2
2
-i
2
-i
+
IE2 2 P
Wm \ 2 / 9U3 R J 1 dx*
+
Wm R
u3
'um dui
+ ß2 I —7T-T +
+
IE2 2 P
Wm R
2 (du,3\ dx* J
Wrr. R
■u.3
+
12 P
Wm\'2 f d2U3
R
dx*2
1R
2 l2 \ R
Wm
du3
dx
w.
(^) ( ) ) - ßißo) ^u3
R
1 dx*
2
3ßi
+
Um du i l dx*
+
2 U1 (Um dui 1 R2 / Wm \2f dU3\2\
Wm
m4
~E 3
ßo
Wrr.
R
■u.3
UmdUl f\ u,m
I dx* 2 P V R ) \dx* J j R ll'3
ßl
Um du 1 l dx*
+
+
IE2 2 ¿2
Wm R
2idu3\2"
\dx* J
+
Wm R
u3
'um dui , + ¿¿2 I —7T-T +
+
1 iL
2 /2
Wm R
2 ( du-3 \ dx* J 2
2
Wn
R
■u,3
+
/?0 R2 w22 f d2u,3 \ ' 12~P~R2 [ßx*2)
l dx*
um dui ,
3ßm 1 Td^+
1 / Wm \ 2 f?U3\ \
2 P V R J \dx*J j
Wm
(ßl ~ ß2ß0) -prU3
R
cO d u3
du3
ho
co
+ £2poho—^ui
Wm dqn um
R dr* l dx*
We introduce independent variables in the form
£ = x* — ct*, t = et*
(3.4)
where t is fast time and c is the dimensionless unknown volume. In these variables (3.4), leaving the terms of orders e and e2 in Eqs. (3.3) and discarding terms with higher degrees, we obtain equations [5]
d_
dl
Um du l I di
' W r- ^ 2
Wm
ßo—rrU3 - — -
R
m4
~E 3
' Um du I v I di
Wm
R
ßi
+
m
~~R
um dui Wm + ß2 — ^r—^-U3
l d£ R
um
~T
du
um
r£lC ae
i
l
Pohoc
2 qx;
um dui Wm m 4 / um dui Wm
" 1fus " E 3 lil0T9[ " 1TU3
+
Wm R
u3
um dui Wm + ß2~--——PTU3
l d£ R
R2 Wm P R
R2 Wm
2 d2 ui
c w " -Y
ßi
2 d2u>3
de
R
Um du■!
I d(
d2ui
d(dr
Um du 1
I d(
d2u-3 d£dr
+
-2ec
+
m l2 du3 , Wm ho ,
P R t2R2° R R lU3 pohoc0
2qn.
(3.5)
2
2
2
2
2
2
2
We represent the dependent variables in the form of an asymptotic expansion.
ui = uio + £uii + ..., us = uso + £u;n + —
(3.6)
We obtain a system of equations by substituting (4.1) into (3.5) and leaving the terms of order £.
d / duio Wml \ 2 d2uio
d£\ d£
Po
Per
du
io
d£
um R
Wml
um, R
uso ) = c
de2
(3.7)
uso = 0.
From this system we get
Wml duio
;i/3o — Pcr
um R
de
22 C = 1 - Po.
(3.8)
i /2
Thus, uio is an arbitrary function, and the dimensionless volume c = (1 — Po) / , therefore,
the wave velocity equals w—, that is, the velocity of the wave in the rod. Since the shell is of
Po
infinite length,
e =
1
x
-t
We get a system of equations in approximation of order £2
+ (Pi — P2Po)
d / dun duio
Wml m 4
Po--^«31 - —77
um, R
E3
Pi
du
io
de
+
Wml
de \umR
uso
Wml
+ (P2 - PlPo) --¿UiO
um.R
du
io
de
Po-
PiPo duii
wml umR
Wml
uSo
_ _2c9S 92un _ £i^du10
dedr d2e £ de
l2
£um pohoc
2 qx;
de
m
vu3i - — - — Po^rr- H--
umR E 3 V l J \ de umR
Po
du
io
+
wml um. R
2
uto
Wml duio
+ P2-7T ^ Uso
umR de
1 R2 Wm l 2 d2 uso -c
de
Rl
+
e2Wm I ^duso | ^ 1 ho wml ^
£ l2 umR d2e £ R um de £ R umR £umpohoco
2 qn.
(3.9)
By substituting relation (4.3) into Eqs. (5.1), we obtain the system
Po
d2u
ii
92 e
Po
Wml <9-0,31 UmR d(
(tO ^ ~ ^ ^1 + +3 ("ir)
i2
— 2;/1 — Pzo
d uio £2
Wu
io
d 2 u
io
d2 e
d£dr e^!1 ^0 d£ £umpohoc%
qx;
(3.10)
l
s
2
2
s
s
2
2 dun Wml 1 R2 , 2\ <93i/,iO £2 /: 2 ,
At°77 " " el? ( " ßo) № W "7V1" toto^ +
1 duio Rl
£ di £umpohoc20
By multiplying both sides of the second equation by p0 and differentiating with respect to £, we get the following equation:
d2Un Wml du-n _ 1 R2 2 (, 2\ <94-WlO £2 L 2 ^Uya
2 d un wm
Vo—^ö?--
d2C ^ UmR d( £ l2^y £
R 2 /-, 2\ 0 ^10 t2 /1 9 9 , -¡2»0 l1 " to) "7V1" W'O^p +
™ (3-11)
1 ho 2 d2uio Rl dq„
The left-hand side of Eqs. (3.10) and (3.11) coincide. Subtracting, term by term, the first equation of system (3.10) from Eq. (3.11), we obtain the resolving equation
1 R2 2 2\ d4Uio £2 /I n 2<93-»io , l/*o 2 d2u>io tn 10\
+ --¡2 to I1 - to) -g^- - 7 V1 - toto^sf + + (3-12)
R dqr,
ei r ~du, io r
+ — v1 -to-
£ v dC £Umpohoc0
Qx — ßo
l dC
We divide both sides of the equation by 2\Jl — $ and get
d2u,io „ m (Um\2 L 2, 2\ {du10\2 d2uw ,
^-toito + wo+^o)^)
1R21 2 /i 7^10 g2Po93t;-io , Po ly^dVo ,
+ -75-2^0 V:1 - - 7T W + (3-13)
2 1 '2 [ Rdqn
Qx — Po
111 dum Hq _ 1 I
2 £ d£ 2 2^/1 - ^ £Umpoho4
l dC
The left-hand side of the resulting equation corresponds to a modified Korteweg-de Vries-
du
Burgers equation for . In the absence of a fluid, the right-hand side of the equation is
zero, and then a modified Korteweg-de Vries-Burgers equation is obtained. It is necessary to determine the right-hand side by solving the equations of hydrodynamics.
4. The fluid flow stress acting on the shell surface
The stresses of the fluid layer are defined by the formulas [6]
qn = Prr cos (ji, nr j + Prx cos ^n, i^J; qx = — Prx cos (ji, nrj + Pxx cos (ji, ij
dVr (dVx dVr \ dVx
Prr = ~P + 2pv—~] Prx = pv —— + —— ; Pxx = -p + 2 pv-
rx xx -
dr \ or dx ox
(4.1)
If we take down the stress on the unperturbed surface of the shell, we can assume that n = nr and cos (n, nr j = 1, cos ^n, i j = 0.
The equation of motion of a viscous incompressible fluid and the equation of continuity in a cylindrical coordinate system (r, 0, x) in the case of axisymmetric flow are written in the form of [6]
dVr Tr dVr TrdVr 1 dp hd2Vr 1 dVr d2Vr Vr
dt dr dx p dr \ dr2 r dr dx2 r2
dVx TrdVx TdVx 1 dp (d2Vx 1 dVx c)2Vx \ (4 2)
dx p dx \ dr2 r dr dx2 )
dt ' r dr x
dVr Vr dVx
—- + — H--= 0.
dt r dx
The fluid adhesion conditions in Lagrange's approach are satisfied at the boundary with the shell.
dU Tr dVx TTTdVx dW TdVr TJdVr
= + -— = vr + U-^-W-^. (4.3)
dt dx dr dt dx dr
We introduce dimensionless variables and parameters
T/ _ £o _ _ £o_ _ - .. * _
Vr — Wm Vr, Vx — Wm Vx, 7 — , I — I, X — , l K\ K\ l l
p
pvc0lwm Ri ( l\ Wm X D , RlCo
(4.4)
By substituting (4.4) into Eq. (4.2) and the boundary condition (4.3), we obtain the equations and boundary conditions for the dimensionless components of the fluid velocity and pressure. By decomposing pressure and velocity components in powers of a small parameter A
P = P° + AP1 + ..., vx = vx + Avlx + ..., vx = v°r + Avl + ..., (4.5)
for the expansion first terms, we get the equations
dPr „ dvx dPr 1 d ( J)v°x
= 0; Re—-Z- + -— = — — [r* x
dr* dt* dx* r* dr* \ dr* y (4 g)
1 d , ^ rx dv<r
--(r v ) -\--- = 0
r * dr * r dd^o *
and boundary conditions of the form
r du3 r umRl dui *
vr = —k—; vz. =-— —— 11 r = 1;
dt*' Wml dt*
dv dv
r*4r^= 0; r*4r^ = 0 if r* = 0.
dr* dr*
Now we define the stresses of the fluid on the shell in these variables. With an accuracy of order A, W we have
v 2 dvx
; qn = -Po ~ -.--^—poC^P. (4.8)
r* = l w Rlcr
By assuming the harmonic dependence of pressure, fluid velocity and elastic displacements of the shell on time, we find the exact solution of Eq. (4.6)
P
2Re
+ I67
, -, x 1 Um^ d ui
(a - 1) ---~y - a
2 wm dr2
1 um^ d2ui f du3
2 wm dr2
dT
dC
d 2 u3
Ifr2 dC ;
dC
+
dvx dr ^
where
r* = i
n , ^ ,'1 um^ d2ui
d2u3 dr2
+ 87 -
1 um^ dui
2 wm dT
d2u3 dr2
dC
a — 1 =
le qo — 4so
Repo — 4\fReqo + 4so I67
a
Rep0 —
eq0
Repo — 4v Reqo + 4so '
iero
Repo - As/Reqo + 4s0
Here
(4.9)
po = ber2 V~Re- + bei2VRe, qo = berV~Re + bei' V~Re- — ber' VRebeiyÎRe, r° = berVRebei' VRe + bei \f~Re- bei' V~Re, so = ber'2
ber and bei are zero-order Kelvin functions, and the prime denotes a derivative. Note that
4
ct = -, 7 = 1 when Re —> 0,
3
a = 1, a — 1 = \l7 = 2 when Re > 20.
V Re 8
Giving up the assumption of the flow parameters harmony in time in the problem under consideration, we apply the iteration method to the problem (4.6). At the first iteration step, we discard the local inertia term (Re ^ 1) and obtain
P = 16
1 um^ dui
2 wm dT 1 um^ dui
dC) dC,
4
f du3 \
V ; \2 wm dT J dT V
du>3 dr
dus \ + umip du,i dT J Wm dT
(4.10)
In the second iteration step, by substituting the found solution into the local inertia term, we get
P
2 Re
1 um^ d2ui 4 f d2u3
6 wm dT2 3
dC
dvx dr ^
- Rei (lUm1pd2U, 1
r* = 1
3 \ 2 wm dT2
dT2
f d2Uj dT2
+16
1 umip d2ui _ f dus
2 Wm dT2 J dT \
dC;
d2u3
dT2
dC .
(4.11)
Formulas (4.11) coincide with the exact solution (4.9) for the harmonic laws of environmental parameters changes with Re ^ 1, which allows using these formulas for both nonharmonic parameters changes and for nonlinear equations of shell dynamics.
1
v
x
It should be noted that the convergence of the iteration method was proved in [6]. Taking into account that variables (3.4) are introduced, and using the relation c = \/l — $, we obtain, with an accuracy of order e,
^v^K2/"^-1^"'»)
1 D f0 umRi duio
- -Re \ 8«30--j-uio—-—
3 V wm l d£
\A " ^ o
(4.12)
where
dvx dr *
r* = 1
= V1 — Joi 4 u30 —
umR\ du\o
wml
. uml
As-— «30
wm Ri
duio
, we get
umRi d2uw
wml d(2
(4.13)
Qx ~ jo ■
R dqn I
R1 2 1 2\ um
1 — 2 jo
R_
Ri
R
1-2/zo-^-Ri
R
du
io
+ M 2 jo
Ri
d£ 2 d2 u
io
2
(4.14)
Therefore, we have the equation
d2u10 1 R2 Ho V1 ~ tlo <94-wio
d£dT e l2
2
d£4
-2-
m
\JI - nl ( jl + j2 jo + MlMo)
+ k
1 ^o
j 2o
d2uio
1 I f v
+
um
~T
1 ei duio
2 ( du
io
d2u
io
d£ ) d£2 1 e2 d3uio 2 _
+
2 e d£ 2 e d£3
-Vo
2
epoho | Eic0
p4
1 — 2 jo
Ri
du
io
(4.15)
Rl 1 A-2
Tp6 V At°
R 2 R 2
for
The resulting equation generalizes the modified Korteweg-de Vries-Burgers equation
d-o-io
With an accuracy of order e (4.15), we can put Ri = R.
Assuming
du
io
= n = c^, t = c2t, we obtain the generalized modified Korteweg-
de Vries-Burgers (MKdV-B) equation
dt dn dn3
d20 , w n 037^2 + + ^ =
(4.16)
2
2
2
2
2
where
we assume that
ci
2 m fUm\2 I'2 (pi + ß'2ßo + ßißl)
3 Ë \TJ Ë2 J4
l J R2
lm A 2 ( , , 2\ fUm\' c2 = ci V1 - K (ßi + toPo + ßlßoj J
2
pl
(71 = - (1 - 2Ai0)2-r^—'
C2 epoho Rico
ci 1 £2 2
^ =
f 2 Pl = — — \/l -Ato"
^2
ci
Po
^ 1 ho
C2 vT^I 2 e E
£i J_
c2 12
pohoe l
_Ll£I
0-4 c2 2 t ' (1 — 2po)2 + 3(2po)2
(4.17)
(4.18)
Provided that (a4 + cti) = 0, in the absence of longitudinal structural damping (a4 = 0) and ¿to = ^ f°r the incompressible material (<ti = 0), the MKdV-B equation
ËÉ. + (a.2 - a5 - QA2) 9<^ | 9 V dt drj dr/3
has an exact solution in the form of a kink-antikink
&3
d2(j> dn2
(4.19)
(f) = ±-(73 ± k t.anh <! k 6
The phase velocity is The wave velocity is
rj + ( 2k2 - ((72 - (75) + -<r'l)t
w l2 1 2
(4.20)
C =
_C2 (2fc2 + \oj + (75 - (72) Po \ ci v/1 - ß20
If the numerator of the fraction 2k2 + ^aè — (a2 — (75) > 0, ( (a2 — (75) < 2k2 + 7703 Y
6 \ 6 /
then the speed is subsonic. If the numerator of the fraction 2k2 + ^aè — (a2 — (75) < 0,
6
^((72 — (75) > 2fc2 + ^c2^ > then the speed is supersonic. The effect of subgrade reaction a2
increases the wave velocity. At the same time, the fluid motion inertia of t5 reduces the wave speed. Thus, they act in different directions.
The constructing damping in the longitudinal direction t4 > 0 and the influence of a viscous fluid t\ > 0 affect the amplitude of the wave. Thus, they act in one direction. This effect is investigated by numerically solving the generalized modified Korteweg-de Vries-Burgers equation for (t4 + t\) > 0.
The wave number k in (4.20) is arbitrary.
5. Computing experiment
For a numerical study of the wave motion model of a physically nonlinear elastic shell with structural damping (energy dissipation) interacting with the surrounding elastic medium under the influence of the fluid, we write Eq. (4.16) in an integral form
fidn —
((T2 - (T5) (f) - 2(f)3 -<73TT" +
dfi d dn dn2
dt + / / (04 + 01) ^dtdn = 0
(5.1)
for any area Q. To transfer a discrete formulation, let us compare un = 0 (tn, nj) and select the basic contour shown in Figure 1.
71 + 1
Tl
j j +1 j + 2 Fig. 1. Basic contour for equation (5.1)
Let us add the integral relations
mj+i
J uvdn = u (t,nj+i) — u (t,nj),
mj+i
J uvvdn = uv (t, nj+i) — uv (t,nj).
(5.2)
We use a trapezoid formula for integration in time and even derivatives in n and an average value formula for odd derivatives in n. We assume (tn+i — tn) = t, (nj+i — tj) = h. We substitute all these expressions into formulas (5.1), (5.2) and, using the Grobner basis method, we obtain the following difference scheme for Eq. (4.16) similar to the Crank-Nickolson scheme for the heat equation.
u^1 — un
+ (02 — 05)
ul+1 — uj-+i) + [ul+1 — ul~1
2
(vï+11 — ^ + (■
4h
uj+1 uj-1
4h
+
+
(ujj+2 — 2u>j+1 + 2u>j-1 — uj-2^ + {uj+2 — 2uj+1 + 2uj-1 — ujj—2^
(5.3)
4h3
03
j — j + u^h) + j — 2u + u—)
4h2
+ (04 + 01 )
-
2
0
with the initial condition in the form of an exact solution (4.20) at t = 0, choosing the plus
sign, k = ~.
6
j
6. The results and conclusions
In the absence of environmental influences, structural damping in the longitudinal, normal directions and the influence of the fluid, the velocity and amplitude of the wave do not change. The movement occurs in the negative direction (Fig. 2). This means that the speed of movement is subsonic. The result of the computational experiment coincides with the exact solution, therefore, the difference scheme and the KdV-B equation are adequate. The dimensionless deformation 0, the dimensionless phase variable n, and dimensionless fast time t are presented in Figs. 2-6.
In the absence of fluid influence, the influence of the surrounding elastic medium (a2) leads to an increase in the wave velocity, up to a supersonic one. The presence of damping in the normal direction (0-3) changes the amplitude by constant volume (leads to the structure of the stretch shock wave 0 > 0) and reduces the velocity of the wave. The presence of structural damping in the longitudinal direction (04) leads to a drop in the amplitude of the wave (Fig. 3).
Fig. 2. The absence of environmental influence (02 = 0), structural damping in both the longitudinal (04 = 0) and normal (o3 = 0) direction as well as the absence of fluid influence (04 = 0,o5 = 0).
Fig. 3. The absence of fluid effect (04 = 0,o5 =0). o2 = 1, o3 = 1, o4 = 1.
Fig. 4. The absence of environmental influence (t2 = 0), structural damping both in the longitudinal (t4 = 0) and normal (t3 = 0) direction. ti = 1, t5 = 1.
Fig. 5. All coefficients ti , t2 , t3 , t4 , t5 are nonzero and equal to 1, the stress from the fluid side is greater than the environmental influence (t2 < t5) .
Fig. 6. All coefficients ti , t2, t3, t4, t5 are nonzero, the stress from the fluid side is less than the environmental influence (t2 > t5 )
The effect of viscous fluid stress on the shell (ai) leads to a drop in the wave amplitude, and the inertia of fluid motion (a5) leads to a decrease in the wave velocity (Fig. 4).
The presence of the medium (a2) leads to an increase in the wave velocity, while the construction damping in the longitudinal direction (a4) leads to a decrease in the wave amplitude. If the inertia of fluid motion (a5) is greater than the influence of the medium (a2 < a5), the wave velocity decreases. The influence of fluid stress leads to a greater decrease in the wave amplitude (Fig. 5).
If the inertia of fluid motion (a5) is less than the influence of the medium (a5 < a2), the increase in the wave velocity due to the presence of the medium (a2) reduces (Fig. 6).
In the absence of the fluid influence, the influence of the elastic medium leads to an increase in the velocity of wave motion up to a supersonic one. The presence of construction damping in the normal direction changes the wave amplitude by constant volume (leads to the structure of the stretch shock wave) and reduces the velocity of wave motion. The presence of construction damping in the longitudinal direction reduces the wave amplitude.
The influence of the inertia of fluid movement reduces the deformation wave velocity, while fluid viscous stress on the shell reduces the wave amplitude.
The authors are grateful to the anonymous referees for a careful checking of the details and for helpful comments and suggestions that have improved this paper.
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