The application of variational methods to some static contact problems for pliant shells of rotation
We propose a method for finding the finite deformations of hyperelastic domal shells of rotation due to hydrostatic load. The problem is considered under the condition that the deformed surface of the shell enters the domain of perfect contact with coaxial rigid surface. The efficiency of the sugges...
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Інститут математики НАН України
1999
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| Цитувати: | The application of variational methods to some static contact problems for pliant shells of rotation / V.A. Trotsenko // Нелінійні коливання. — 1999. — Т. 2, № 4. — С. 559-573. — Бібліогр.: 6 назв. — англ. |
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Trotsenko, V. A. 2021-02-14T08:05:23Z 2021-02-14T08:05:23Z 1999 The application of variational methods to some static contact problems for pliant shells of rotation / V.A. Trotsenko // Нелінійні коливання. — 1999. — Т. 2, № 4. — С. 559-573. — Бібліогр.: 6 назв. — англ. 1562-3076 https://nasplib.isofts.kiev.ua/handle/123456789/177273 539.3 We propose a method for finding the finite deformations of hyperelastic domal shells of rotation due to hydrostatic load. The problem is considered under the condition that the deformed surface of the shell enters the domain of perfect contact with coaxial rigid surface. The efficiency of the suggested approach is assured by taking into account the singular properties of solutions on the line of transition of shell surface from free domain to the contact one and their asymptotic behavior in vicinity of the shell pole. For a set of examples, which illustate the advantages of the algorithm, we present the basic dependences describing stressly-deformed state of a shell. На основi варiацiйного методу пропонується розв’язок задачi про скiнченнi деформацiї гiперпружних оболонок обертання пiд дiєю гiдростатичного навантаження за умови, що деформована поверхня оболонки вступає в зону iдеального контакту iз спiввiсно розмiщеною жорсткою поверхнею обертання. Ефективнiсть запропонованого пiдходу зумовлена врахуванням властивостей шуканих розв’язкiв на лiнiї переходу поверхнi оболонки вiд вiльної зони до контактної та асимптотичної поведiнки їх в околi полюса оболонки. На конкретних прикладах проiлюстровано можливостi запропонованого алгоритму. The author is grateful to DFG for partial support. en Інститут математики НАН України Нелінійні коливання The application of variational methods to some static contact problems for pliant shells of rotation Застосування варіаційних методів в деяких контактних задачах статики м'яких оболонок обертання Применение вариационных методов в некоторых контактных задачах статики мягких оболочек вращения Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
The application of variational methods to some static contact problems for pliant shells of rotation |
| spellingShingle |
The application of variational methods to some static contact problems for pliant shells of rotation Trotsenko, V. A. |
| title_short |
The application of variational methods to some static contact problems for pliant shells of rotation |
| title_full |
The application of variational methods to some static contact problems for pliant shells of rotation |
| title_fullStr |
The application of variational methods to some static contact problems for pliant shells of rotation |
| title_full_unstemmed |
The application of variational methods to some static contact problems for pliant shells of rotation |
| title_sort |
application of variational methods to some static contact problems for pliant shells of rotation |
| author |
Trotsenko, V. A. |
| author_facet |
Trotsenko, V. A. |
| publishDate |
1999 |
| language |
English |
| container_title |
Нелінійні коливання |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Застосування варіаційних методів в деяких контактних задачах статики м'яких оболонок обертання Применение вариационных методов в некоторых контактных задачах статики мягких оболочек вращения |
| description |
We propose a method for finding the finite deformations of hyperelastic domal shells of rotation due to hydrostatic load. The problem is considered under the condition that the deformed surface of the shell enters the domain of perfect contact with coaxial rigid surface. The efficiency of the suggested approach is assured by taking into account the singular properties of solutions on the line of transition of shell surface from free domain to the contact one and their asymptotic behavior in vicinity of the shell pole.
For a set of examples, which illustate the advantages of the algorithm, we present the basic dependences describing stressly-deformed state of a shell.
На основi варiацiйного методу пропонується розв’язок задачi про скiнченнi деформацiї гiперпружних оболонок обертання пiд дiєю гiдростатичного навантаження за умови, що деформована поверхня оболонки вступає в зону iдеального контакту iз спiввiсно розмiщеною жорсткою поверхнею обертання. Ефективнiсть запропонованого пiдходу зумовлена врахуванням властивостей шуканих розв’язкiв на лiнiї переходу поверхнi оболонки вiд вiльної зони до контактної та асимптотичної поведiнки їх в околi полюса оболонки.
На конкретних прикладах проiлюстровано можливостi запропонованого алгоритму.
|
| issn |
1562-3076 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/177273 |
| citation_txt |
The application of variational methods to some static contact problems for pliant shells of rotation / V.A. Trotsenko // Нелінійні коливання. — 1999. — Т. 2, № 4. — С. 559-573. — Бібліогр.: 6 назв. — англ. |
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| fulltext |
т. 2 •№ 4 • 1999
UDC 539 . 3
THE APPLICATION OF VARIATIONAL METHODS
TO SOME STATIC CONTACT PROBLEMS
FOR PLIANT SHELLS OF ROTATION∗
ЗАСТОСУВАННЯ ВАРIАЦIЙНИХ МЕТОДIВ
В ДЕЯКИХ КОНТАКТНИХ ЗАДАЧАХ СТАТИКИ
М’ЯКИХ ОБОЛОНОК ОБЕРТАННЯ
V.A. Trotsenko
Inst. Math. Nat. Acad. Sci. Ukraine,
Ukraine, 252601, Kyiv 4, Tereshchenkivs’ka str., 3
We propose a method for f inding the f inite deformations of hyperelastic domal shells of rotation due to
hydrostatic load. The problem is considered under the condition that the deformed surface of the shell
enters the domain of perfect contact with coaxial rigid surface. The ef f iciency of the suggested approach
is assured by taking into account the singular properties of solutions on the line of transition of shell
surface from free domain to the contact one and their asymptotic behavior in vicinity of the shell pole.
For a set of examples, which illustate the advantages of the algorithm, we present the basic
dependences describing stressly-deformed state of a shell.
На основi варiацiйного методу пропонується розв’язок задачi про скiнченнi деформацiї гiпер-
пружних оболонок обертання пiд дiєю гiдростатичного навантаження за умови, що деформо-
вана поверхня оболонки вступає в зону iдеального контакту iз спiввiсно розмiщеною жорст-
кою поверхнею обертання. Ефективнiсть запропонованого пiдходу зумовлена врахуванням
властивостей шуканих розв’язкiв на лiнiї переходу поверхнi оболонки вiд вiльної зони до кон-
тактної та асимптотичної поведiнки їх в околi полюса оболонки.
На конкретних прикладах проiлюстровано можливостi запропонованого алгоритму.
Consider the pliant domal shell of rotation Σ0 in undeformed state rigidly fixed on a parallel of
radius R0 on walls of an absolutely rigid surface of rotation S so that the axes of symmetry of
Σ0 and S coincide. We suppose that the shell is made of an isotropic incompressible and highly
elastic material, and its thickness h0 is constant and sufficiently small with respect to R0. The
distance of points of the shell generatrix to the axis of symmetry in the initial state is posed by
ξ = ξ(s),
where s is the length of a meridian arc measured from the shell pole. In order to describe the
shape of deformed membrane, we introduce the cylindrical coordinate system (z, η, r) whose
axis Oz coincides with the symmetry axis of the considered mechanical system.
We assume that the shell is under hydrostatic pressure Q. Hence, for some values of
parameters of load, the deformed shell Σ can contact with the rigid surface S, as shown in
* The author is grateful to DFG for partial support.
c© V.A. Trotsenko, 1999 559
Fig. 1. We define the contact domain as the part of the surface of the deformed shell Σ which
is in contact with the surface S. The remaining part of the shell surface will be referred as free
domain.
Fig. 1
Let the generatrix of the surface S have positive (negative) curvature and be described by
the equation
r = ϕ(z). (1)
In what follows we assume that, as the load Q increases, the contact domain grows from the
side of rigid fixing of the shell on the surface S.
By virtue of the assumptions made, the geometrical and physical quantities describing the
deformed state of the shell are the functions of the arc length s only and, hence, we may seek
for the equations describing the meridian of the deformed shell in the following form:
z = z(s), r = r(s). (2)
The equilibrium conditions for the infinitesimal element of the free domain of the deformed
shell enable us to obtain the governing equations [1]
dT1
ds
+
1
r
dr
ds
(T1 − T2) = 0,
T1
R1
+
T2
R2
= Q, Q = C −Dz.
(3)
The internal strains T1 and T2 of the deformed shell in the meridian and parallel directions
are determined from the relations
560 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
Ti = 2h0λ3
(
λ2
i − λ2
3
)(∂W
∂I1
+ λ2
3−i
∂W
∂I2
)
, i = 1, 2,
λ1 =
[(
dz
ds
)2
+
(
dr
ds
)2
]1/2
, λ2 =
r(s)
ξ(s)
, λ3 =
1
λ1λ2
=
h(s)
h0
,
I1 = λ2
1 + λ2
2 + λ2
3, I2 = λ−2
1 + λ−2
2 + λ−2
3 , W = W (I1, I2) .
(4)
Here λ1, λ2 and λ3 are the main degrees of lengthening in the directions of meridian, parallel
and normal to deformed surface, respectively, W is the energy function of deformation for the
shell material, I1, I2 are the invariants of deformations, and h(s) is the thickness of the shell in
the deformed state. The main radii of the curvature of the deformed shell median surface are
given by the formulae
R1 = λ3
1
/(
d2r
ds2
dz
ds
− dr
ds
d2z
ds2
)
, R2 = −(rλ1)
/
dz
ds
.
Let us choose the energy function of deformation in Mooney’s form [2]
W = C1 (I1 − 3) + C2 (I2 − 3) , (5)
where C1, C2 are physical constants, which may be found experimentally.
When assuming that no friction between the shell Σ and the surface S occur, we arrive at
the equilibrium equations (3) for the contact domain
dT1
ds
+
1
r
dr
ds
(T1 − T2) = 0,
T1
R1
+
T2
R2
= Q−QR,
(6)
where QR is the pressure exerted by the surface S on contacting shell.
The account of the relation (2) between the solutions r(s) and z(s) in the contact domain
makes the second equation from (6) auxiliary and enables us to calculate the distribution of
normal pressure QR between the surface S and the shell Σ, using the solution z(s).
Let us introduce the following dimensionless variables
{s∗, r∗, z∗} =
{s, r, z}
R0
, T ∗i = Ti/ (2h0C1) , W ∗ =
W
C1
, Q∗ =
QR0
(2h0C1)
.
In what follows, we use these dimensionless variables but omit the asterisk for the sake of
simplicity.
Let us place the origin of coordinate system Ozηr in the plane of the fixed contour and
write the boundary values of z(s) and r(s) at the ends of integration interval of equilibrium
equations
dz
ds
∣∣∣∣
s=0
= r(0) = z(s0) = 0. (7)
ISSN 1562-3076.Нелiнiйнi коливання, 1999, т. 2, № 4 561
Thus, we have obtained the nonlinear boundary value problem possessing an interesting
feature with respect to the case of free deformation of the shell. Namely, the shell contact
domain is a priori unknown and should be determined while solving the problem.
Let us apply the variational formulation of the considered problem to the construction of
the approximate solution. For the free deformation of the domal shells under the hydrostatic
load, the solution of relevant nonlinear boundary value problem is equivalent to finding the
stationary points of the functional [3]
I =
s0∫
0
Φ(s, r, z, r′, z′)ds, (8)
Φ = [(I1 − 3) + Γ (I1 − 3)] ξ(s) +Qr2dz
ds
, Γ =
C2
C1
.
In the case where the deformed shell contacts with the rigid surface S with generatrix (1),
we must impose on the class of admissible functions both the conditions (7) and an additional
constraint of geometrical nature, which has the form of an inequality:
ψ(z, r) = r(s)− ϕ (z(s)) ≤ 0,
where the equality holds for the contact domain of the shell.
The solution of this problem was first constructed by methods for solving similar problems
in the theory of nonlinear programming [4]. This approach introduces a penalty function so
that the constraint in the form of inequality is reduced to that in the form of equality, and the
relevant problem of finding the conditional extremum of the functional is solved by well-known
methods of calculus of variations. However, this approach turned out to be inefficient because
it does not take into account the differential properties of solutions at the contact line.
Assume that the transition of shell generatrix from contact domain to free one takes place
for s = a. Let us study the properties of solutions r(s) and z(s) in the vicinity of this point. To
do this, let us represent the functional (8) in the form
I =
a∫
0
Φ(s, z, r, z′, r′)ds+
s0∫
a
Φ(s, z, r, z′, r′)r=ϕ(z)ds = I1 + I2. (9)
In order to evaluate the variation of the functional I1, we can use the formula for the
variation of a functional with two unknown functions, where the left boundary point is fixed,
but the right one can move along the line r = ϕ(z). Thus, we obtain [5]
δI1 =
[
Φ− z′Φz′ − r′Φr′
]
s=a
δa+
[(
Φz′ +
∂ϕ
∂z
Φr′
)
δz
]
s=a
+
+
a∫
0
[(
Φz −
d
ds
Φz′
)
δz +
(
Φr −
d
ds
Φr′
)
δr
]
ds. (10)
562 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
In turn, the functional I2 can be represented in the form
I2 =
s0∫
a
Φ
(
s, ϕ(z), z′,
∂ϕ
∂z
z′
)
ds =
s0∫
a
F (s, z, z′)ds,
and its variation will be evaluated as that of a functional with one function with movable left
boundary point. This yields the formula
δI2 = −[F − z′Fz′ ]s=aδa− [Fz′δz]s=a +
s0∫
0
(
Fz −
d
ds
Fz′
)
δzds.
Taking into account the relations
Fz = Φz +
∂ϕ
∂z
Φr +
d
dz
(
∂ϕ
∂z
z′
)
Φz′ , Fz′ = Φz′ +
∂ϕ
∂z
Φr′
and introducing the notation
Φ̃ = Φ(s, z, r, z̃′, r̃′),
where z̃′, r̃′ are the values of the derivatives z′ and r′ at the point s = a when tending to it from
the right, we can represent the variation of the functional I2 in the form
δI2 = −
[
Φ̃− z̃′Φ̃z′ − r̃′Φ̃r′
]
s=a
δa−
[(
Φ̃z′ +
∂ϕ
∂z
Φ̃r′
)
δz
]
s=a
+
+
s0∫
a
[
Φz −
d
ds
Φz′ +
∂ϕ
∂z
(
Φr −
d
ds
Φr′
)]
r=ϕ(z)
δzds. (11)
Since z(s) and r(s) ensure the extremum of the functional I , we have
δI = δI1 + δI2 = 0.
If the functions z(s) and r(s) ensure the extremum of the functional (9) for arbitrary δa
and δz|s=a, then the same obviously remains true for δa = 0 and δx|s=a = 0. Hence the main
lemma of calculus of variations allows us to conclude that the extremum may be achieved only
on integral curves of the Euler system of equations
Φz −
d
ds
Φz′ = 0,
Φr −
d
ds
Φr′ = 0, s ∈ [0, a],
(12)
[
Φz −
d
ds
Φz′ +
∂ϕ
∂z
(
Φr −
d
ds
Φr′
)]
r=ϕ(z)
= 0, s ∈ [a, s0]. (13)
ISSN 1562-3076.Нелiнiйнi коливання, 1999, т. 2, № 4 563
Making simple transformations and keeping the relations (4) in mind, one can show that the
Euler equations (12) are nothing but the equilibrium equations (3) for the shell in free domain,
while equation (13) turns into the first equilibrium equation for the shell in the contact do-
main (6).
Collecting non-integral terms in the variations (10) and (11) and taking into account the
arbitrariness of the variations δa and δz|s=a yields the equations for shell generatrix at the
point of its transition from the domain ψ(z, r) < 0 to the boundary ψ(z, r) = 0:
[
Φ− Φ̃− z′Φz′ + z̃′Φ̃z′ + r̃′Φ̃r′ − r′Φr′
]
s=a
= 0,[
Φz′ − Φ̃z′ +
∂ϕ
∂z
(
Φr′ − Φ̃r′
)]
s=a
= 0.
These equations are satisfied if we set
z′(a) = z̃′(a), r′(a) = r̃′(a).
As follows from the theory of sufficient conditions of extremum, the case where, at the point
of transition from the free domain to the boundary ψ(z, r) = 0, the conditions of continuity of
derivatives are fulfilled is the most important and significant one and takes place in most applied
problems [6].
Let us show that the conditions obtained are sufficient for the actual determination of the
extremal curve, provided that it exists. Let
〈f(s)〉 = lim
s→a−0
f(s)− lim
s→a+0
f(s),
let z = z(s, α1, α2, α3, α4), r = r(s, α1, α2, α3, α4) be the general solution of the Euler equations
(12) in the free domain, and let z = z(s, β1, β2) be the general solution of the equation (13) in
the contact domain. Thus, for determination of the extremal curve we must find four constants
αi, i = 1, 4, and two constants βi, i = 1, 2. The boundary conditions (7) allow us to exclude
three arbitrary constants. The conditions
〈z〉 = 〈r〉 = 〈z′〉 = 0
allow us to obtain additional relations for the other three constants. It seems that we lack one
constant to ensure the continuity of derivative of the function r(s) at the transition through the
point s = a. But since the position of the boundary s = a is unknown, the condition
〈r′〉 = 0
can be considered as a necessary requirement for the determination of the point s = a. This
fact will be used below for the construction of the approximate solution of the problem.
The question of the behavior of higher derivatives of solutions of this problem while passing
through the point s = a is very important, so let us investigate it in detail.
The equilibrium equations for the shell in contact and free domains and the conditions of
continuity of first derivatives of functions z(s) and r(s) yield the relation〈
dT1
ds
〉
= 0,
〈
1
R1
〉
=
(
QR
T1
)
s=a
. (14)
564 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
Furthermore, the continuity of
dT1
ds
implies the continuity of
dλ1
ds
. We can obtain relations
for the second derivatives of functions z(s) and r(s):
d2z
ds2
= − λ1
R1
dr
ds
− r
R2
dλ1
ds
,
d2r
ds2
=
1
λ1
(
dr
ds
dλ1
ds
− rλ3
1
R1R2
)
. (15)
Then, on the basis of formulae (14) and (15), we obtain〈
d2z
ds2
〉
= −
(
λ1
dr
ds
QR
T1
)
s=a
,
〈
d2r
ds2
〉
= −
(
rλ2
1QR
T1R2
)
s=a
. (16)
Using the repeated differentiation of equilibrium equations, elasticity conditions and
expressions (15) and taking into account the relations (16), we can conclude that, except
for some particular cases, the higher order derivatives of the functions z(s) and r(s) have
discontinuities of the first kind on the line of transition from free domain to contact one.
Thus, we may consider the problem about the contact interaction of nonlinearly deformed
pliant shell with rigid surface of rotation as the conjugation problem for the systems of
nonlinear differential equations (3) and (6), where the conjugation point s = a is not known
beforehand.
Let us seek for solutions for functions z(s) and r(s) in the form
z(s) =
2∑
k=1
xkuk(s) +
m∑
k=3
xkuk(s), 0 ≤ s ≤ a;
2∑
k=1
xkuk(s) +
m∑
k=3
xk+2m−2uk(s), a ≤ s ≤ s0,
r(s) =
s
a
ϕ (x(a)) +
m∑
k=1
xk+mvk(s), 0 ≤ s ≤ a;
ϕ (x(s)) , a ≤ s ≤ s0.
(17)
In these expressions, the coordinate functions uk(s) for k = 1, 2 are defined for all s ∈ [0, s0]
and each of them obeys the restriction uk(s0) = 0. The systems of functions uk(s) for k > 2
have different representations on the intervals [0, a] and [a, s0]. Moreover, they must obey some
restrictions at the ends of the intervals.
Namely, for s ∈ [0, a], they must satisfy the boundary conditions uk(a) =
=
duk
ds
∣∣∣∣
s=a
= 0 , while for s ∈ [a, s0], the conditions uk(a) =
duk
ds
∣∣∣∣
s=a
= uk(s0) = 0 must
hold. Similarly, the coordinate functions vk(s), which are defined on the interval 0 ≤ s ≤ a,
must satisfy the condition vk(a) = 0. The constants xk, k = 1, 3m− 2, are to be determined
later. In order to make the reasonable choice of coordinate functions, we expand the required
solutions in Taylor series in a vicinity of the point s = 0 and use the repeated differentiation of
equilibrium equations and geometrical relations describing the shell for the determination of
the coefficients of the expansion.
Consider the undeformed state of the class of domal shells whose generatrices cross the
symmetry axis at the right angles and whose main curvatures in the pole are equal. Taking into
ISSN 1562-3076.Нелiнiйнi коливання, 1999, т. 2, № 4 565
account the symmetry conditions in a pole of deformed shell, which have the form
λ1 = λ2 = λ, T1 = T2 = T,
1
R1
=
1
R2
=
1
R
,
we can show that the solutions bounded for s→ 0 have the structure
z(s) = a1 + a2s
2 + a3s
4 + . . . ,
r(s) = b1s+ b2s
3 + b3s
5 + . . . .
On the other hand, let us choose the systems of coordinate functions with the domain of
definition [a, s0] within the whole class of power functions. In view of the reasoning presented
above, the systems of functions {uk(s)} and {vk(s)} take the form
uk(s) = (s2 − s2
0)(s2 − a2)k−1, k = 1, 2,
vk(s) = (s2 − a2)s2k−1, k = 1,m,
uk(s) =
(s2 − a2)k−1, 0 ≤ s ≤ a,
(s− s0)(s− a)2sk−3, a ≤ s ≤ s0,
k = 3,m.
The direct check shows that the solutions for functions z(s) and r(s) in the form (17) will
trivially satisfy the boundary conditions at the point s = s0 and the conjugation conditions
〈z〉 =
〈
dz
ds
〉
= 〈r〉 = 0.
The finite gap in the higher orders derivatives of solutions is caused by the second terms
in (17).
In fact, the suggested form of solution for function x(s) is a specifically chosen superposition
of two classes of expansions, which describe continuous and singular parts of the solution in
question. We take only two terms in the first sum of expression for z(s), which provides the
continuous transition of the solution and its first derivative through the point s = a, since the
remaining terms of this sum are linearly expressed via the elements of the second sum.
Let us substitute the expressions (17) into functional (9), which as a result, becomes a
function of 3m− 2 variables xi, being the components of unknown vector ~x.
Suppose that the point s = a of transition of the shell generatrix from the contact domain
to free one is known. Then we determine the constants xi from the conditions of stationarity of
functional (9). This yields the system of nonlinear algebraic equations
~g (~x) = 0. (18)
The components of the (3m− 2)-dimensional vector function ~g have the form
566 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
gi+εi =
s0∫
0
[
µ1
dz
ds
dui
ds
−Qλ2
dr
ds
ui
]
ξ(s)ds+
+δik1
a∫
0
[
µ1
dr
ds
+ µ2ξ(s)
]
ξ(s)ds+
s0∫
a
[
µ1
dr
ds
r1i + µ2ri
]
ξ(s)ds,
gi+m =
a∫
0
[
µ1
dr
ds
dvi
ds
+ µ2vi
]
ξ(s)ds, i = 1,m,
(19)
where
µ1 = U(λ1, λ2)/λ1, U(λ1, λ2) =
(
λ1 −
1
λ3
1λ
2
2
)(
1 + Γλ2
2
)
,
µ2 = U(λ2, λ1)/ξ(s) +Qλ2
dz
ds
, k1 =
u1(a)
a
∂ϕ
∂z
∣∣∣∣
z=x1u1(a)
,
ri = ui(s)
∂ϕ
∂z
, r1i =
dri
ds
, δi =
{
1 ∀ i = 1,
0 ∀ i = 2,m.
The quantities εi in (19) are equal to 2(m− 1), if i > 2 and s > 0, and zero otherwise.
We solve the algebraic system (18) by using the Newton iteration procedure
~x(k+1) = ~x(k) −H−1
(
~x(k)
)
~g
(
~x(k)
)
, (20)
where H (~x) is the Jacobi matrix of the system of functions g1, g2, . . . , g3m−2 with respect to the
variables x1, x2, . . . , x3m−2.
The upper symmetric part of the nonzero elements hij of the matrix H is given by
hi+εi,j+εj =
s0∫
0
(
α1
dui
ds
duj
ds
+ α2uiuj
)
ξ(s)ds+
+δiδjk2
a∫
0
(
µ1
dr
ds
+ µ2ξ(s)
)
ξ(s)ds+ k1
a∫
0
[
(γ1 + ξ(s)γ2)
(
δi
duj
ds
+ δj
dui
ds
)
+
+γ3ξ(s) (δiuj + δjui) + δiδjk1
(
β1 + 2ξ(s)β2 + ξ2(s)β3
)]
ξ(s)ds+
+
s0∫
a
[
γ1
(
dui
ds
r1j +
duj
ds
r1i
)
+ γ2
(
dui
ds
rj +
duj
ds
ri
)
+ γ3 (uirj + ujri)
]
ξ(s)ds+
+
s0∫
0
[
µ1
dr
ds
(
∂2ϕ
∂z2
d
ds
(uiuj) +
∂3ϕ
∂z3
dz
ds
uiuj
)
+ µ2
∂2ϕ
∂z2
uiuj
]
ξ(s)ds+
ISSN 1562-3076.Нелiнiйнi коливання, 1999, т. 2, № 4 567
+
s0∫
a
[
β1r1ir1j + β2
d
ds
(rirj) + β3rirj
]
ξ(s)ds,
hi+m,j+m =
a∫
0
[
β1
dvi
ds
dvj
ds
+ β2
d
ds
(vivj) + β3vivj
]
ξ(s)ds, i = 1,m; j = 1,m, (21)
hi,j+m =
a∫
0
[
γ1
dui
ds
dvj
ds
+ γ2vj
dui
ds
+ γ3vjui +
+ δik1
(
β1
dvj
ds
+ β2
d
ds
(ξ(s)vj) + β3ξ(s)vj
)]
ξ(s)ds, i, j = 1,m.
In (21), we have used the following notation:
k2 =
u2
1(a)
a
∂2ϕ
∂z2
∣∣∣∣
z=x1u1(a)
, α1 =
z1(λ1, λ2)
λ2
1
(
dz
ds
)2
+ µ1
(
dr
ds
)2 1
λ2
1
,
α2 = −∂Q
∂z
λ2
dz
ds
, β1 =
z1(λ1, λ2)
λ2
1
(
dr
ds
)2
+
µ1
λ2
1
(
dz
ds
)2
, β2 =
z2(λ1, λ2)
ξ(s)λ1
dr
ds
,
β3 =
1
ξ(s)
[
z1(λ2, λ1)
ξ(s)
+Q
dz
ds
]
, γ1 =
[
z1(λ1, λ2)
λ1
− µ1
λ1
]
dz
ds
dr
ds
,
γ2 =
z2(λ1, λ2)
ξ(s)λ1
dz
ds
+Qλ2, γ3 = λ2
dz
ds
∂Q
∂z
,
z1(λ1, λ2) =
(
1 +
3
λ4
1λ
2
2
)(
1 + Γλ2
2
)
, z2(λ1, λ2) =
2
λ3
1λ
3
2
(
1 + Γλ4
1λ
4
2
)
.
To determine the point s = a, we use the continuity condition for the first derivative of the
function r(s) at this point. This yields one more nonlinear algebraic equation for the parameter
a:
ϕ(z)
2a
∣∣∣∣
s=a
− a
[
x1 + (a2 − 1)x2
] ∂ϕ
∂z
∣∣∣∣
z=a
+
m∑
k=1
xk+ma
2k = 0. (22)
We seek a solution of the algebraic system (18) and the equation (22) by the method of
successive approximations. For the chosen zero approximation ~x(0), we solve equation (22) with
respect to the parameter a by the chord method. Then we precise the vector ~x(0) by iterations
(20) and then return to solving equation (22). We repeat this process as many times as necessary
to provide the required accuracy of the solutions of the equations (18) and (22).
Let us consider a shell which has, in the undeformed state, the shape of a circular membrane
as the first example of calculation of chracteristics of pliant shells under contact and static
loading. As a rigid surface of rotation S, we take the conic surface of angle 90◦. We assume
that while entering the contact interaction with the considered surface the membrane is under
568 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
constant pressure (D = 0). In all the numerical results given below the parameter Γ describing
the ratio of constants in the elastic potential (5) is supposed to be equal to 0,1.
The results of numerical computation of values of the functions under study and their first
two derivatives at the point s = 0,3 (free domain) in the function of the number m of iterations
in expansions (17) for C = 1, 6 are given in Table 1 below.
Table 1
m z −z′ z′′ r r′ −r′′
1 0,44501 0,29341 0,9780 0,34300 1,1062 0,37144
2 0,52587 0,45520 1,3743 0,35753 1,1278 0,62024
3 0,53301 0,43626 1,4503 0,35964 1,1391 0,60854
4 0,53251 0,43967 1,4384 0,35944 1,1377 0,61108
5 0,53251 0,43966 1,4383 0,35944 1,1377 0,61100
6 0,53251 0,43966 1,4383 0,35944 1,1377 0,61099
The similar convergence takes place for the solution z(s) which was evaluated in the domain
of contact of deformed shell with cone. The relative error
ε =
∣∣∣∣ T1
R1
+
T2
R2
−Q
∣∣∣∣ / |Q| , (23)
to within which the constructed approximate solutions satisfy the second equilibrium equation
for all values of the parameter s, 0 ≤ s ≤ 1, when one keeps six terms in expansions (17), does
not exceed 10−4.
Figure 2 displays the behavior of solutions z(s) and r(s) and their first derivatives on the
whole interval of integration of initial equations (D = 0, C = 1, 6). While the functions z(s)
and r(s) are smooth, their first derivatives possess an obvious fracture at the point s = a.
The presented results show that the suggested variant of the Ritz method for solution of the
problem of contact interaction of pliant shells of rotation with axially symmetric rigid surfaces
allows us to obtain the uniform convergence of solutions and their first derivatives within
their domain of definition. The main reason for this is taking into account the asymptotics of
solutions for s → 0, as well as their differential properties on the line of transition of the shell
from contact domain to free one.
It should also be mentioned that one may use the representations of solutions of conjugation
problem for systems of nonlinear differential equations with unknown point of conjugation,
which are different from (17). But the advantage of the algorithm is that, while the realization
of iterative procedure, described above, we usually have no problems with choosing initial
approximations which ensure its convergence.
The profiles of the deformed membrane for different values of the loading parameter C
with taking into account the contact interaction with the conic surface in absence of restrictions
for displacements (dashed curve) are shown at Fig. 3. The vertical dashed lines indicate the
points of transition of the generatrix of deformed shell from the contact domain to free one.
ISSN 1562-3076.Нелiнiйнi коливання, 1999, т. 2, № 4 569
Fig. 2 Fig. 3
The main relative lengthening and stresses for the median surface of deformed shell in
function of the parameter s (Γ = 0,1, D = 0, C = 1,6) are presented at Fig. 4. The dashed
lines refer to the case of free deformation of membrane. As one may observe, the presence
of restricting surface S implies the considerable decrease in efforts and deformations of the
median surface of the shell with respect to the case where the contact is absent.
Let us consider the example where the membrane is rigidly fixed in unstressed state on the
walls of circular cylinder orthogonally to its symmetry axis. To solve this contact problem, we
apply the above approach with the only difference that we take into account higher smoothness
of solutions. For example, it can be easily seen from relations (16) that the second derivative of
the function z(s) at the point s = a is continuous. At the same time the derivatives of higher
orders of z(s) remain discontinuous.
In this connection, let us represent the solutions for functions z(s) and r(s) in the form
z(s) =
3∑
n=1
xnun(s) +
m∑
n=4
xnun(s), 0 ≤ s ≤ a;
3∑
n=1
xnun(s) +
m∑
n=4
xn+2m−3un(s), a ≤ s ≤ 1,
r(s) =
s
a
+
m∑
n=1
xn+mvn(s), 0 ≤ s ≤ a;
1, a ≤ s ≤ 1,
(24)
where
un(s) =
(
s2 − 1
) (
s2 − a2
)n−1
, n = 1, 2, 3,
un(s) =
(
s2 − a2
)3
s2n−8, 0 ≤ s ≤ a;(
s2 − 1
) (
s2 − a2
)n−1
, a ≤ s ≤ 1,
n = 4,m,
vn(s) =
(
s2 − a2
)
s2n−1, n = 1,m.
570 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
Fig. 4 Fig. 5
Expressions (24) take into account a priori the asymptotic behavior of solutions in vicinity
of the pole of deformed membrane and satisfy the conjugation conditions
〈z〉 =
〈
dz
ds
〉
=
〈
d2z
ds2
〉
= 〈r〉 = 0
which imply the existence of discontinuities of the first kind in the derivatives of higher order.
The components of the (3m− 3)-dimensional vector function ~g in the case considered have
the form
gi+εi =
1∫
0
[
µ1
dz
ds
dui
ds
−Qλ2
dr
ds
ui
]
sds,
gi+m =
a∫
0
[
µ1
dr
ds
dvi
ds
+ µ2vi
]
sds, i = 1,m.
(25)
Here, the quantities εi are equal to 2m − 3 if s > a and i > 3, and to zero if at least one of
these conditions is not fulfilled.
The condition of continuity of the first derivative of the function r(s) at the point s = a
yields one more nonlinear equation for the parameter a
1 + 2
m∑
n=1
xn+ma
2n+1 = 0. (26)
Since the function z(s) has a discontinuity of the first kind in its third derivative, we can
construct a simpler algorithm for solving the problems, in which the computation of the second
derivatives of the functions z(s) and r(s) at the points of integration of initial equations is
not required. Namely, we can seek the solutions for functions z(s) in the class of continuous
functions. In this case, we can seek a solution in the form
ISSN 1562-3076.Нелiнiйнi коливання, 1999, т. 2, № 4 571
Fig. 6 Fig. 7
z(s) =
p∑
n=1
xnun(s), s ∈ [0, 1],
r(s) =
s
a
+
p∑
n=1
xn+pvn(s), s ∈ [0, a],
1, s ∈ [a, 1],
(27)
where
un(s) =
(
s2 − 1
)
s2n−2, vn(s) =
(
s2 − a2
)
s2n−1.
We must also assume that εi = 0 in expressions (25) and thus reduce the problem to the
simultaneous solution of 2m nonlinear equations (18) and equation (26).
As in the previous example, the parameter Γ in the energy function is taken to be equal to
0,11.
The numerical computation of the values of required functions and their derivatives and
the quantity ε (23) at the point s = 0,6 in the function of the number m of approximations in
expansions (24) are listed in Table 2. The parameters of hydrostatic loading are C = −3,5, D =
= 1. We have chosen values of m such that the expansions (24) take into account the
discontinuity of the function z(s).
Table 2
m −z z′ r r′ ε
4 0,62642 1,5852 0,94180 0,70409 3 · 10−3
5 0,62617 1,5869 0,94186 0,70312 2 · 10−4
6 0,62617 1,5869 0,94186 0,70313 3 · 10−5
572 ISSN 1562-3076. Нелiнiйнi коливання, 1999, т. 2, № 4
Figure 5 shows the behavior of the second derivative of the function z(s) calculated from the
fifth (sixth) approximation, when the solutions were represented in the form (24) (solid line)
and in the form (27) (dashed line). The results obtained demonstrate the uniform convergence
of solutions and their first two derivatives, when one takes into account the presence of
discontinuities in the solutions.
The profiles of deformed shell for different values of the loading parameter C with account
of contact interaction with cylindrical surface of unit radius are shown at Fig. 6. Dashed
lines refer to the case of absence of restrictions on displacements. Figure 7 demonstrates
the dependences of main relative lengthenings and efforts in the shell in the function of the
parameter s (C = −3, 5, D = 1). T1 in the contact domain is constant, which is in agreement
which the first equilibrium equation.
1. Green A., Adkins J. Large elastic deformations and nonlinear mechanics of continuum media. — Moscow:
Mir, 1965. — 465 p. (in Russian: Translation from English).
2. Oden J. Finite elements of nonlinear continua. — Moscow: Mir, 1976. — 464 p. (in Russian).
3. Trotsenko V.A. Axially symmetric problem on equilibrium of a circular membrane under the hydrostatic
load // Phys. and Techn. Appl. of Boundary Problems. — Kiev: Naukova dumka, 1978. — P. 126 — 140 (in
Russian).
4. Himmelblau D. Applied nonlinear programming. — Moscow: Mir, 1975. — 536 p. (in Russian).
5. Gel’fand I.M., Fomin S.V. Calculus of variations. — Moscow: Fizmatgiz, 1961. — 228 p. (in Russian).
6. Lavrent’ev M. and Lyusternik L. Foundations of calculus of variations. — Moscow: ONTI, 1935. — Vol. 1,
Pt 2. — 400 p. (in Russian).
Received 09.03.99
ISSN 1562-3076.Нелiнiйнi коливання, 1999, т. 2, № 4 573
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