CC BY
6
УДК 517.958
Solvability of One Nonlinear Boundary-value Problem for a System of Differential Equations of the Theory of Shallow Timoshenko-type Shells
Marat G. Ahmadiev Samat N. Timergaliev* Lilya S. Kharasova^
Naberezhnye Chelny Institute, Kazan Federal University Suumbike, 10A, Naberezhnye Chelny, 423812
Russia
Received 05.09.2015, received in revised form 11.01.2016, accepted 19.02.2016 Solvability of a system of nonlinear second order partial differential equations with given initial conditions is considered in the paper. Reduction of the initial system of equations to one nonlinear operator equation is used to study the problem. The solvability is established with the use of the principle of contracting mappings.
Keywords: system of nonlinear differential equations, equilibrium equations, integral representations, existence theorem.
DOI: 10.17516/1997-1397-2016-9-2-131-143.
1. Problem formulation
Let us introduce in the plane simply connected bounded domain Q and consider a system of nonlinear partial differential equations in the form
+ R = 0, i = 1,2,
T^ + kxTxx + (Tx^w3a, )a, + R3 = 0, (1)
Mi\ - Ti3 + Li = 0, i = 1, 2 under the following conditions at the boundary r:
w1 = =0, (2)
T 12da2/ds - T22 da1 /ds = P2(s), (3)
T 13da2/ds - T23da1/ds + Tnw3aida2/ds -T22w3a2da1/ds+
+T 12(w3a2da2/ds - W3aida1 /ds) = P3(s),
M 12da2/ds - M22da1/ds = N2(s). (5)
From this point on the index ax means differentiation with respect to ax.
1 [email protected] © Siberian Federal University. All rights reserved
In (1)-(5) the following notations are used:
Tij = Tij (a) = Dj^Yl, Mij = Mij (a) = Dfn7L , a = (w1 ,w2, w3, ^ M h
Djn = jh Bijkn(a3)mda3, B1111 = B2222 = E/(1 - j2), B1122 = jE/(1 - j2), (6) 2 B1212 = E/(2(1 + j)), B1313 = B2323 = Ek2/(2(1 + j));
the remainder Bijkn = 0; aj = aj(s)(j = 1, 2) is the equation of the curve r, s is the length of the arc r;
Yjj = Wjaj - kjW3 + wlaj /2 (j = 1, 2), Y12 = w1a2 + W2ai + W3al W3a2,
Yjj = ^jaj (j = 1 2) Yl2 = ^1a.2 + ^2ai , (7)
Y03 = W3aj + tj (j = 1, 2), Y33 = Yfc3 = 0, k =1;^.
The system (1) together with the boundary conditions (2)-(5) describes the state of equilibrium of shallow isotropic elastic homogeneous shell with simply supported edges within the framework of Timoshenko shear model [1]. Here Tij are stresses, Mij are moments; Yij(ij = 1,3, k = 0,1) are components of deformation of the shell middle surface S0 that is homeomorphic to Q; wi(i = 1, 2) and w3 are tangential and normal displacements of the points of So; ti(i = 1, 2) are rotation angles of normal cross-section of So; a is the vector of generalized displacements; Rj(j = 173), Lk(k = 1, 2),N2,P2, P3 are components of the external forces acting on the shell; j = const is the Poisson coefficient, E = const is Young's modulus, k1 ,k2 = const are principal curvatures; k2 = const is the shear coefficient; h = const is the shell width; a1, a2 are the Cartesian coordinates of the points in the domain Q.
We assume summation over repeating Latin indices from 1 to 3 and over Greek indices from 1 to 2 in (1), (6) and in what follows.
System (1) is written in terms of generalized displacements w1 ,w2, w3, t2:
w1a1a1 + ¡1 w1a2a2 + ¡2w2a1a2 = fl, ¡1w2a1 a1 + w2a2 a2 + ¡2w1a1a2 = f2 , k2¡1 (w3a1a1 + w3a2a2 + ^1a1 + 12a2 ) + k3wa + k4w2ly2 - k5w3 + +k3w2a1 /2 + kAw2a2/2 + ¡32 [(TX^w3aX )a, + R3] =0,
^1a1a1 + ¡1^1a2a2 + ¡2^2a1a2 = 9l, ¡1^2a1a1 + ^2a2a2 + ¡2^1a1a2 = 92,
where
(8)
/i = fi(w3) = k3w3ai - w3aiw3aiai - n2w3a2w3aia2 - jiw3aiw3a2a2 - /32R , /2 = /2(w3) = kAw3a2 - w3a2w3a2a2 - n2w3aiw3aia2 - jiw3a2w3aiai - f32R2, (9) 9j = 9j(w3) = ko(w3aj + ) - fiLj,j = 1, 2,
ji = (1 - j)/2, j2 = (1 + j)/2, k3 = ki + ¡jk2, kA = k2 + jki,k5 = k2 + k2 + 2jkik2,
k0 = 6k2(1 - j)/h2, fi = 12(1 - j2)/(h3 E), f2 = (1 - ¡j2)/(Eh).
System (8) is a system of second order partial differential equations. It is linear with respect to tangential displacements wi, w2, rotation angles ^2 and it is a nonlinear system with respect to deflection w3.
Problem A. Find a solution to system (8) under boundary conditions (2)-(5).
Solvability of the system of nonlinear differential equations that describes shell equilibrium in the framework of the Kirchhoff-Love model has been well studied [2-5]. The questions of the existence of solutions of nonlinear problems in the framework of more general shell models, not based on the Kirchhoff-Love hypotheses, are in the well-known Volovich list of unresolved problems of the mathematical theory of shells [2]. These questions have not been clarified yet. There are a number of works devoted to the solvability of nonlinear problems in the framework of the Timoshenko displacement model [6-10]. The method used in these studies is based on the integral representations of the desired solution of system (8) that contain arbitrary holo-morphic functions. These representations are constructed with the use of general solutions to the inhomogeneous Cauchy-Riemann equation. Holomorphic functions are defined so that the desired solution of system of differential equations (8) satisfies given boundary conditions. At the present time, existence theorems of solutions of nonlinear problems for Timoshenko-type shell with rigidly clamped edges [6,7] and with free edges [8, 9] are obtained. Method developed in [6-9] was applied to system (8) with boundary conditions wl = w3 = =0 that describe the state of equilibrium of Timoshenko-type shell with simply supported edges [10]. The study presented in this paper develops results obtained in [10]. The more complicated case of boundary conditions wl = =0 is considered. These conditions describe elastic bearing against transverse deflection.
Consider boundary-value problem A in a generalized formulation. Let the following conditions hold true:
a) Q is a simply connected domain with the boundary r e Cl (see, for example, [11, p. 23]);
b) external forces Rj(j = 1, 3), Lk(k = 1, 2) e Lp(Q), N2,P2,P3 e (r); in what follows
jp(Q), N~,P2,P- e C/3 ( p> 2, 0 < 1.
(2)
Definition 1. The vector of generalized displacements a = (wl,w2,w3,^l ,^2) e Wp (Q), p> 2, is a generalized solution to the problem A if the vector satisfies almost everywhere the equations of system (8) and it satisfies boundary conditions (2)-(5) in pointwise fashion.
(2)
Here Wf' (Q)
is a Sobolev space. Let us note that due to embedding theorems for Sobolev spaces Wp2)(Q) with p > 2, the generalized solution a belongs to (Q). In what follos a = (P - 2)/p.
2. Solution to problem A with respect to tangential displacements and angles of rotation
Let us consider the first two equations in (8) and initially assume that w3 is fixed. In terms of the complex function w = wlai + w2a2 + i^l(w2oi — wla2) these equations can be represented in the form
w* = f, (10)
where f = (fi + if2)¡2, wz = (wai + iwa2 )/2, z = al + ia2.
Equation (10) is an inhomogeneous Cauchy-Riemann equation. It has general solution [11]:
w(z) = $i(z) + Tf (z), Tf = —1 if pO-didn, Z = e + in, (11)
n J Jq z — z
where $l(z) is an arbitrary holomorphic function that belongs to the space Ca(Q).
It is well-known [11, pp. 41, 53] that Tf is a completely continuous operator which acts in Lp(Q), p > 2, Ck(Q). It maps these spaces into Ca(Q) and Ck+l(Q), respectively. Besides, there exist the generalized derivatives [11, pp. 33-34,53-67]
dTf f dTf_ 1 n f (Z) ded
^ = f, ^ = Sf = — *JJa dedn, (12)
where the integral exists in the principal value sense of Cauchy (almost everywhere when f e Lp(Q),p > 1) and Sf is a linear bounded operator in Lp(Q), Ck(Q).
With the function w0(z) = w2 + iwl, relation (11) can be also rewritten in the form of an inhomogeneous Cauchy-Riemann equation
woz = i(diw + d2w) = id[w], dj = (mi + ( — 1j)/(4ji), j = 1, 2, (13)
The general solution of this equation is
wc(z) = $2(z) + iTd[$i + Tf ](z), (14)
where $2 is an arbitrary holomorphic function of the class C1 (Q).
Thus, for fixed w3 the general solution of the two first equations (8) is of the form (14) and contains two arbitrary holomorphic functions (z),j = 1,2. We define these functions so that tangential displacements wl and w2 will satisfy boundary conditions (2), (3). First, we find $2(z) from the condition wl =0 on r. We have a Rimann-Hilbert problem with the boundary condition Re[i$2(t)] = ReTd[w](t), t e r for the holomorphic function $2(z). Second, we assume that domain Q is the unit disk: \z\ ^ 1. Then the solution of the Riemann-Hilbert problem has the form [12]
*2(z) = — ^¡ReTd [$i + Tf](t) t-±z ^ + co, z e Q, (15)
2n J p t — z t
where co is a arbitrary real constant.
We define the holomorphic function $l(z) with the use of boundary condition (3). Let us represent this condition in terms of displacements:
¡i(wia2 + w2ai )(t)da2/ds — (¡wiai + w2a2)(t)da1/ds = ^(w3)(t), (16)
t = t(s) = al(s) + ia2(s) e r,
where
¥>(t) = f(ws)(t) = P2(s) + [(¡¡w^ 1 + w3a2)/2 — ¡¡kw — k2w3] da1 /ds— (17)
a 1 w3 a2 da2/ds.
We substitute relations for the tangential displacements w1,w2 from (14) into (16). Taking into account (10), (11) and (14), we obtain
wja3 =Re{$i(z)+ Tf (z)}/2 — ( —1)jIm{$2(z) + iSd[$i + Tf](z)}, j = 1, 2, (18)
wi a2 + w2 a 1 = 2Re{$2(z) + iSd[$i + Tf ](z)}. Hence, boundary conditions (16) take the form
Re{t$2(t)} + Re{itSd[$1] + (t)} — ¡3da1/ds Re$1(t) = y(t)/(1 — ¡¡) + h1f (t), t e r, (19)
where
h1f (t) = Im{tSd[Tf] + (t)} + j3da1/ds ReTf (t), ¡3 = (1 + ¡)/(2(1 — ¡)); (20)
the symbol ^+(t) means the limit of the function ^(z) as z ^ t e r from the interior of the domain Q.
Let us transform relation (19). Representing holomorphic in the domain Q function $1(z) by the Cauchy integral and using (4.7), (8.8a) from [11], we have
Sd[$i] + (t) = dit2[$i(t) — $i(0)], (21)
where constant d1 is defined in (13).
Further, we differentiate relation (15) with respect to z and use (13), (11) for d[$1], Td[$1j. Rearranging the order of integration in the repeated integrals and applying the Cauchy theorem and formula, we have
$2(*) = (-i){d1$1(0) + d2$1(z) + 2Sr(ReTd[Tf])(z)}, z G Q, (22)
S f( ) - dTrf(z) 1 f f(t) d
Sr f (z) = = 2^ (T-z2dT■
In the limit z ^ r taken in (22) from the interior of the domain Q, we find
$2(i) = (-i){d1HM + d2$1(i) + 2(Sr(ReTd[Tf]))+(i)}, t G r, (23)
where constants dk are defined in (13).
Now we substitute (21) and (23) in (19). Then we obtain the Riemann-Hilbert problem for function $1(z) in Q with the boundary condition
Re[it$1(t)] = h[f; v](t), t G r, (24)
where
h[f; v](t) = (j - 1)[hf (t) + 2Re{it(Sr(ReTd[Tf])) + (t)}] - v(t); (25)
operators h1f, Srg are defined in (20), (22).
The index of problem (23) equals -1. Therefore, the solution of this problem is [12]
*1(z) = - - / dt - ; v](z), (26)
n J r t - z t
and the solvability condition
/•fM dt = 0 (27)
Jr t
of problem (24) should be fulfilled.
We substitute expression (26) into (15) and (22) to obtain
$2(z) = -2- i (ReTd[$1 [f; v]](t) + ReTd[Tf](t))^ t + co - [f; v](z) + co,
2n Jr t - z t (28)
$2(t) = (-i){d1 $1[f; v](0) + d2$1[f; v](z) + 2Sr(ReTd[Tf])(z)} - $2[f; v](z), z G Q.
Consider tangential displacements w1 and w2 that satisfy the first two equations (8) and conditions (2), (3). Upon substituting (26), (28) into (14) and assuming that condition (27) is true, we obtain
wo(z) = How3 + co,
(29)
How3 - Ho[f (wa); vK)] = $2[f; v](z) + iTd[$ [f; v] + Tf](z).
Let us obtain integral representations for the derivatives of w1 and w2 (up to second order inclusively). Using (11) and (18), we find
wjaj = Re {$1[f; v] + Tf} /2 - (-1)jIm {$2[f; v] + iSd[$ [f; v] + Tf]} -
- Hjj[f (w3); V(w3)] - HjjW3,
wjak = Re{$2[f; V + iSd [$1[f; v] + Tf]} + (-1)jIm{$1[f; v] + Tf }/(2j1) -
- Hjk[f (w3); v(w3)] - Hjkw3, j,k = 1, 2;
f = f (ws),y = y(ws) are defined in (9), (17).
Upon differentiating relation (13) with respect to z,z, we obtain
^ozz = * {di($i[f; y] + Sf) + dJW)} = Pi[f (ws); y(ws)] = Piw3, (31)
uozz = *{dif (ws) + d2 ($i[f; y] + Sf )} = P2 [f (ws); y(ws)] = Pw. Using formula (8.20) from [11], we obtain
Sd[$i + Tf ](z) = Tr(d[$i + Tf ]/t2)(z) + T (di[$i + Sf ] + df)(z). (32)
Now we differentiate (14) two times with respect to z. Taking into account (32), we have
wozz = Щ [f ; ф + iSr{d[$i [f ; ф] + Tf ]t2} +
(33)
+iS{d1(Ф'1 [f ; ф] + Sf ) + d2f (ws)} = P3[f (ws); фМ] = P3W3. We use the following designations in (31), (33):
Ф'^ ; ф]^) = -1 j f dt; Ф^; ф]^) = (-i){d2Ф'l[f ; ф](г) + 2SfReTd[Tf](z)}, (34)
Sf ReTd[Tf](z) = {dif (z) + d2Sf (z) - S(diSf + d2f)(z) - Sr{T(diSf + d2f)f}(z)-
Jr
One can express derivatives
-Srd[Tf](z)/2 - Sr{Tr(d[Tf]r2)}(z)}/2, Tf (z) = f dt.
2ni J г t - z
wJ-afcafc = Im{ij 1[2^ozz + (-l)fc 1(^0zz + wo^)]}, wjaia2 = Re[ij 1(^ozz - ^0zz)],j,k = 1, 2 in terms of wozz, wozzz, wozz •
Lemma 1. Lei conditions a), b) in Section 1 be fulfilled. Then 1) Pjws(j = 1, 3) are nonlinear
(2)
bounded operators acting from Wp (Q) to Lp(Q), 2 < p < 2/(1 - в); 2) Hjkws(j, к = 1, 2) are
(2)
nonlinear completely continuous operators acting from Wp )(Q) to Lp(Q), 2 <p < 2/(1 - в ) and
Hows is nonlinear bounded operator acting from Wp2) (Q) to Ca(Q), C1 (Q). For any w3s(j =
(2)
1, 2) G Wp )(Q) the following estimates hold
\\pj w3 - pj wз\\Lp(n), \\Hjk wl - Hjk wз\\са(Щ, \\Hows, - Hows Уса (Щ < ^ C(1 + \\w1\w (2) + \\wS\\W (2) )\w3 - wS\\W (2) .
1 p
Proof. Let us note that f (ws) defined in (9) is nonlinear bounded operator acting from WP2)(ft) to Lp(fi), p > 2 and estimate (35) is true. Using formulas (6.10) from [11] and Sokhotski formulas [12], we obtain
SrReTd[Tf](z) = {d[Tf](z) - T(diSf + d2f)(z) - Tr{T(diSf + df )f}(z)-
_ _ (36)
-Trd[Tf](z)/2 - Tr{Tr(d[Tf]r2)](z)}/2.
It is known that if condition b) is fulfilled then Trg and Srg are linear bounded operators acting from (r) to (ft) and to Lp(fi) (2 <p < 2/(1 - ¡3)), respectively [11, pp. 26-27]. Besides, using formula (6.10) from [11], it can be shown that Srg is a linear bounded operator acting from
W(1) (ft) to Lp(ft), p > 2. Taking into account this fact and (32), (34), (36), one can obtain that Sd[Tf](z), SrReTd[Tf](z) e Wp(1)(ft), p > 2. Takingjnto account hif in (20) andjp(w3) in (17), we obtain from (25) that h[f; p](t) = —¡2P2(s) + h(t). Here P2(s) e Cf(r) and h(t) is the boundary value of the function that belongs to the space W(1\ft), p > 2. Therefore, from (34), (28), (26) we have 1) $1[f (w3); p(w3)], $2'[f (w3); p(w3)] are nonlinear bounded operators acting
from Wp(2)(ft) to Lp(ft), 2 <p< 2/(1 - ¡); 2) $i[f (w3); p(w3)], $2 [f (w3); p(w3)] are nonlinear
(2)
completely continuous operators acting from Wp ;(ft) to Lp(ft), 2 < p < 2/(1 — ¡3) and they are nonlinear bounded operators acting from Wp2)(ft) to Ca(ft) These operators satisfy estimates (35) (here min(a,3) = a, when (2 <p < 2/(1 — ¡3))). Lemma 1 follows from (29)-(31) and (33).
Let us consider conditions of solvability (27). With (32) and (36) we find Sd[Tf] + (t), SrReTd[Tf] + (t). Taking into account expressions for operators h1f, Tf, Sf in (20), (11), (12) and holomorphic function Trg(z), we obtain
f f) dt = — -if f2(w3)(z)daida2, f Re{it(SrReTd[Tf])+(t)l dt = 0. (37) Jr t 1 — ^J Jq Jr t
Next using f2(w3) in (9), we have
// f2(w3)(z)da1da2 = {(pw|ai/2 + w|a2/2 — kiw3)dal/ds — j Jq Jr
1 2 2
3a1
Hi Jr
—piw3aiw3a2da2/ds}ds — ¡2 11 R2da1 da2.
(38)
Upon substituting (25), (37), (38) into (27), the condition of solvability take the final form
I P2(s)ds + if R2da1 da2 = 0, (39)
Jr Jq
where P2(s) and R2(a1; a2) are the components of external load.
We now turn to functions 0k (k =1,2) in the last two equations (8). These functions should satisfy boundary conditions (2), (5). Taking into account expressions for moments Mjk in (6), we write boundary condition (5) in the form
m(^1a2 + ^2a.i )(t)d,a2/ds — (p^1ai + 02a.2 )(t)da1/ds = p(t), p(t) = ¡1N2(s); (40)
¡1 is defined in (9), N2(s) is the component of the external forces.
Let us note that the structure of left-hand sides in the last two equations (8) coincides with the structure of left-hand sides in boundary conditions (2) and (40). Relations for tangential displacements differ only in the right-hand sides. Therefore at fixed right-hand sides for rotation angles we obtain
0 = 02 + i01 = Ho[g(v); p] + C1, (41)
where
u = U2 + iu1, g(v) = (g1 (u) + ig2(v))/2, (42)
Uj = w3aj + 0j, gj (u) = koVj — ¡1L3 ,j = 1, 2;
c1 an arbitrary real constant, operator H0[g(u); p is defined in (29). As this takes place, the condition of solvability is
dt = 0,
Q
r
where h[g; ¡p](t) is given in (25). These conditions can be reduced to the form
J {N2 + [k1(a1)2 - k2(a2)2]P2/2 - k1a1a2T 1(a) - a2P3}ds + jj' {L2 + [k^a1)2--k2(a2)2R2/2 - k1a1a2R1 - a2R3}da1 da2 + J P2w3ds + Jj R2w3da1da2 = 0,
(43)
where T 1(a) = T 11(a)da2/ds-T 12(a)da1/ds (Tij(a) are defined in (6)); N2,L2,Pk (k = 2, 3), Rj (j = 1, 3) are components of external load.
In a similar way to (30) we obtain the following relations
= Hjk [g; ¡], j, k = 1, 2, (44)
where operators Hjk are defined in (30).
Lemma 2. Let conditions a), b) in Section 1 be fulfilled. Then Hjk[g(u); ¡p] (j,k = 1, 2) and Ho[g(u); ¡p] are linear completely continuous operators with respect to u that act from W(1\Q) to Lp(ft), 2 < p < 2/(1 - 3) and they are linear continuous operators that act from Wp^ft) to Ca(Q) and to Ci (ft), respectively.
Taking into account expressions for g(u) in (42) and indicated above properties of operators Tf,Sf,Trf,Srf, we obtain from (29), (30) that Lemma 2 is true.
Problem A at fixed w3, Uj (j = 1, 2) is solvable with respect to tangential displacements and rotation angles under conditions (39) and (43). Solution of this problem is described in (29) and (41).
In conclusion of Section 2 we represent relationships (41) and (44) in the form convenient for further analysis. First of all we obtain relations for ¡p from (40) and for g(u) from (42):
¡p = + ¡p1, g(u)= g0 + g1(u), = 31 N2(s), ¡P1 = 0, (45)
gk = (g1k + ig2k )/2, k = 0,1, gjo = -31Lj, gj1 = koUj, j = 1, 2.
Let us note that gj (u) are homogenous operators of order j with respect to u.
Now if we substitute (45) into (41) and (44), we arrive at the desired representations for rotation angles and their derivatives
■0 = 0(u) = + 01(u) + C1, ^jak = ^jak (u) = ^joak + ^j1ak (u), (46)
0"(u) = 02n(u) + i01n(u) = Ho[gn (u); p], 0jnak (u) = Hjk [gn(u); ¡pn], j,k = 1,2, n = 0,1. It is easy to see that 0n(u), 0jnak (u) are homogenous operators of order n with respect to u.
3. Reduction of system (8) to a single equation and solvability analysis.
Before considering the third equation in (8) we express the deflection w3 and its derivatives in terms of Uj (j = 1, 2). Taking into account (42) and (46), we obtain
w3aj = w3a.j (u) = w30aj + w31aj (u) - (j - 1)c1, (47)
W30aj = -0jO, w31aj (u) = Uj - 0 j 1 (u) , j = 1, 2.
Using (47), we derive
w3 = w3 (u) = w3o + w31(u) - C1a2 + C2, (48)
/ 1 2\ / 1 2\ n(a ,a ) n(a ,a )
W30 = - 0loda1 + 02oda2, W3i(u) = / [ui - 011(v)]da1 + [u2 - 02i(u)]da2.
J(0,0) J(0,0)
Upon substituting expressions (47), (48) into (9), (17) and then the resulting expression into (29) and (30), we obtain the decomposition of tangential displacements and their derivatives into linear and nonlinear operators
^0 = ^0 (u) = W0i(u) + ^02(u) + ^0, (49)
Wjak = Wjak (u) = Wjiak (u) + Wj2ak (u) + Wjak, j = 1, 2,
where
W0j(u) = W2j(u) + iwij(u) = H0[fj(u); <jP(u)], wjnak (u) = Hjk[fn(u); ^n(u)], (50) ^0 = w2 + iw'i = H)[f •; + C0, wjak = Hjk [f ■; ym], j,k,n = 1, 2, fj (u) = [fij(u) + if2j (u)]/2, fji(u) = k2+jw3iaj (u), fj2 (u) = k2+jW30aj - --W3aj (u)w3aj aj (u) - (J,2W3cc3-j (u)w3a1 a2 (u) - HiW3aj (u)w3a3-j a3-j (u) , j = 1, 2, f ■ = -iciki/2, ^i(u) = -kiW3i(u)da1/ds, ^2(u) = ^2 P2 (s) + {-k4W30 + m(w30«i + +W3iai )2/2 + (W30a2 + W3ia2 )2/2 - ci (w30a2 + W3ia2 )}dai/ds - ni{w3ai (w30a2 + +W3ia2) - ci(w30ai + W3ia2)}da2/ds, = (c2/2 + cik4a2 - k4C2)dai/ds,
operators H0[f; g], Hjk[f; g], (j, k =1, 2) are defined in (29), (30).
After some cumbersome mathematical treatment one can derive the explicit expression
^0 = -cik4(a2)2/2 + (c2k4 - c2/2)a2 + cik4/4 + c0. (51)
Now we turn to the third equation in (8). Replacing generalized displacements by relations (46)-(49), we reduce the third equation to the equivalent system with respect to u = u2 + iui:
du/dz =[02a.i (u) - 0ia2 (u) + if3(u)]/2 = f0(u), (52)
f3(u) = f3(W3(u)) = -{k3Wiai (u) + k4W2a2 (u) - k5W3(u) + k3wlai (u)/2 + k4wla2 (u)/2+ +&[T^(u)w3«>(u)]a, + ^2^3}/(k2Mi), TA^(u) = Tx»(a(u)) (A, p = 1, 2). Boundary condition (4) is transformed to
uida2/ds - u2dai/ds = y>0(u)(t), t £ r, (53)
Mu)(t) = Mw3(u))(t) = &[P3(s) - Tii(u)w3ai (u)da2/ds + T22(u)w3a2 (u)dai/ds-
-Ti2(u)(w3a2 (u)da2/ds - w3ai (u)dai/ds)], = 2(1 + ¡j)/(k2Eh).
So, problem A is now to find solution to equation (52) under boundary condition (53). Equivalent form of equation (52) is
u = $(z) + Tf0(u)(z), (54)
where $(z) is an arbitrary holomorphic function of the class Ca(Q) and operator Tf is defined in (11).
We define the holomorphic function $(z) so that the function u from (54) satisfy (53). We assume for the time being that y0(u), f0(u) in the right-hand sides of (53), (54) are fixed. Substituting (54) into (53), we obtain the Riemann-Hilbert problem for $(z) in the unit disk.
The boundary condition for this problem is Re[(—i)t$(t)] = l(v)(t), t G r. The solution of this problem is
*(z) - *[l(v)](z) = ~f ^ J, (55)
n J r t — z t
1 f l(v)(t) dt n J r t — z t
where l(v)(t) should satisfy the condition f l(v)(t)
h t
-dt = 0, l(v)(t) = po(v)(t) + Re[itTfo(v)(t)].
This condition can be represented in the form
I (k1a1Tx(a) + k2a2P2 + P3)ds + I I (k1a1R1 + k2a2 R2 + R3)da1da2 = 0, (56) JF J JQ
where T 1(a) is defined in (43), Pk(k = 1, 2) and Rj(j = 1, 3) are components of external load. Substituting (55) into (54), we obtain the following equation for v G W(1\ p > 2
v — $[l(u)j — Tfo(v) = 0. (57)
Now we represent equation (57) in a slightly different form. Taking into account relations (46), (48), (49), (51), we obtain for f3(v), fo(v), l(v) the decompositions into linear and nonlinear terms:
fs(v) = fsi(v) + fs2(v), fo(u) = foi(u) + fo2(u), l(v) = li(u) + h(v), (58)
where
f31 (u) = —[k^wiia! (v) + kiW21a2 (v) — k5 W31 (v)]/(k2^i ), f32(u) = —[k3Wi2ai (v) + kAW22o2 (v) + k2+\(w3oax + w3iax (v))2/2 — k5w3o — —kici(w3o a2 + w31a2 (v))+32 (T ^ (v)w3ax (v))a^ + 32R + k2(1 ^ )(c1a c2)]/(k fti^ (59) foi (v) = [021a.i (v) — 011a2 (v) + if31(v)]/2, fo2(v) = [02oai (v) — ^Wa.2 (v) + if32(v)]/2,
li(u) = Re[itTfoi(u)], l2(v) = ^o(u) + Re[itT fo2(v)], t G T. Let us introduce the following operators
Kv = $[li(u)] + Tfoi(u), Gv = $[l2(u)] + Tfo2(u). (60)
Then equation (57) takes the form
u — Kv — Gv = 0. (61)
Let us consider the solvability of equation (61) in the space wP^(Q), p > 2.
Lemma 3. Let conditions a), b) in Section 1 be fulfilled. Then 1) Kv are linear completely continuous operators in p > 2; 2) Gv are nonlinear bounded operators in W(1)(Q), 2 <
p < 2/(1 — ¡3) and for any vj G W(1\Q) (j = 1, 2) which belong to the ball ||v||W(i)(Q) < r, the following estimate takes place
IIGvl — ^Wpq < c[qo + (1 + |w3(0)||wP>(Q) + r)(||w3(0)||w,2>(n) + r)]|v1 — v^wiD^,
22 qo = £ ||TV(0)||C(Q) Uk2+Xw3a> (0)+ RXhp(Q), T^(0) - T^(a(0)), \,v=1 X=1
a(0) = (wi(0),w2(0),w3(0),^i(0),^2(0)), wj(0) (j =1,3), w3ax (0), (0) (A = 1, 2) are defined in (49), (48), (47), (46) at v = 0.
Lemma 3 follows from (60) and (59), in view of Lemmas 1, 2 and properties of operators Tnf, Sf, Trf and Srf.
Consider the homogenous equation
u - Ku = 0. (62)
Let u £ Wp(1)(ft), 2 <p < 2/(1 - 3) be nonzero solution of equation (62). In view of (46), (48), (50), this solution is associated with the generalized displacements wji(u) (j = 1,3), ^ji(u) (j = 1, 2) which satisfy the system of linear homogenous equations
Wiaiai + MiWia2a2 + ¡J2W2aia2 - k3W3ai =0,
Miw2aiai + W2a2 a2 + M2Wiai a2 - k4W3a2 = 0, (63)
k2^i (W3aiai + W3a2a2 + 0iai + 02a2 ) + k3Wiai + k4W2a2 - k5W3 = 0, Wiaiai + Mi0ia2a2 + ¡¡202aia2 - k0 (w3ai + 0i) = 0, Mi02aiai + 02a2a2 + ¡¡20iai a2 - W3a2 + 02 ) = 0
and homogenous static boundary conditions (2) and (16) with y>(t) = 0, boundary conditions (40) with p(t) = 0 and boundary conditions (53) with ^>0(t) = 0. We multiply equalities (63) by wii, w2i, w3i, 0ii,02i, integrate the resulting relations over the domain ft, and add up the result of integration. Then upon integrating by parts the resulting relation and taking into account boundary conditions, we obtain uj =0, j = 1,2, i.e., u = 0 in ft. Therefore, equation (62) has only zero solution in Wi(i)(ft), 2 < p < 2/(1 -3). Thus, there exists the inverse operator (I - K)-i bounded in W(1\ft), 2 < p < 2/(1 - 3). It reduces equation (61) to the equivalent form
u - G*u = 0, G*u = (I - K)-iGu. (64)
It follows from the established above properties of the operator Gu that G*u is a nonlinear bounded operator in Wp^(ft), 2 <p < 2/(1 -3). For any uj £ W?(i)(ft) (j = 1,2) which belong to the ball ||u||w.(i) < r, in view of Lemma 3, the following estimate holds
\\G*ul -G*u2||w(l)(n) ^ q*Wul -u2Ww(D(n),
where q* = c||(I - K)-i||Wu>(fi)[90 + (1 + llW3(0)llWjp)(fi) + r)(llW3(0)llW!(2)(fi) + r)].
Let us assume that the radius r of the ball and the external forces exerted on the shell are such that the following conditions hold
q* < 1, |G*(0)|w(D(„) < (1 - q*)r, (65)
where G*(0) is given by relations that follow from (53), (59), (60) at u = 0.
Let us note that to fulfill conditions (65) it is enough, for example, to require that the external load and the radius of the ball are sufficiently small.
Under these conditions we can apply the principle of contracting mappings to equation (64) [13]. According this principle equation (64) has the unique solution u £ Wi(i)(ft), 2 < p < 2/(1 - 3) in the ball ||u||w(i) < r. This solution can be represented in the form u = "RG*(0), where R is the resolvent operator G* (u) - G* (0).
Using u = RG* (0), (46), (48) and (49), we obtain the generalized displacements Wj £ w(2)(ft) (j = 173), 0j £ W(2)(ft) (j = 1,2), 2 < p < 2/(1 - 3). Finally we obtain the generalized solution a = (wi,w2,w3,0i,02) of problem A. It can be represented in the form a = a0 + a", where a" = (0, w2, -cia2 + c2,0, ci) (w'2 is defined in (51)); a0 is the vector with
components wj1(v) + wj2(u) (j = 1, 2), w3o + w31(u), 0jo + 0ji{v) (j = 1, 2), that are defined in (50), (48), (46).
Then we substitute the solution u = u2 + iu1 = RG* (0) of Eq. (64) into (58), (43). Taking into account relations
T 1(a) = T1 (ao)(u) + T 1(a■) + ci/o(u),
lo(u) = {(w3oai + w3iai (u))da1/ds — /^(w3oa2 + w3ia2 (u))da2/dsj/fo
and calculating integrals that contain T 1(a"), we transform the solvability conditions (58), (43) into the form
/ (k1a1T 1(ao) + k2a2P2 + P3)ds + (k1a1R1 + k2a2R2 + R3)da1da2+ Jr J JQ
+C1 j k1a1 lo(u)(s)ds + nc2k2(p? — 1)/@2 = 0,
J {N2 + [k1 (a1)2 — k2(a2)2]P2/2 — k1a1a2T 1(ao) — a2P3}ds + JQ {L2 + [k1 (a1)2— (66)
—k2(a2)2R2/2 — k1a1a2R1 — a2R3}da1da2 + P 2w3ds + R2w3da1da2 —
Jr J JQ
—k1c1 J a1a2lo(u)(s)ds — nc1k2i(1 — p2)/(2^2)=0.
Let us note that relation (66) is the system of equations with respect to arbitrary constants c1 and c2. Thus, the solvability conditions (58), (43) depend on constants c1, c2. Note that at zero external load c1 = c2 =0.
Therefore, we obtain the generalized solution of problem A, where components w1,w3, 01, 02 are defined uniquely and component w2 depends on constant co.
Condition (39) is not only sufficient but also necessary for the solvability of problem A. Indeed, if a = (w1, w2, w3,01,02) is a generalized solution of problem A then, upon integrating by parts second equality in (1) over the domain Q and taking into account condition (2), we come to condition (39).
Thus we have proved the following basic theorem.
Theorem 1. Let conditions a), b) in Section 1 be fulfilled and inequality (65) holds. Then geometrically nonlinear boundary value problem for elastic shallow Timoshenko-type shell with
simply supported edge is solvable if and only if condition (39) is satisfied. Then the problem has
(2)
generalized solution a = (w1, w2, w3, 01,02) e Wp (Q), 2 <p< 2/(1 — ¡3). Components w1, w3, 01, 02 are uniquely defined and component w2 depends on constant co.
References
[1] K.Z.Galimov, Principles of the Nonlinear Theory of Thin Shells, Kazan Univ. Press, Kazan, 1975 (in Russian).
[2] I.I.Vorovich, Mathematical Problems of Nonlinear Theory of Shallow Shells, Nauka, Moscow, 1989 (in Russian).
[3] N.F.Morozov, Selected two-dimensional problems of elasticity theory, LGU, Leningrad, 1978 (in Russian).
[4] M.M.Karchevskii, Solvability of Variational Problems of the Nonlinear Theory of Shallow Shells, Differents. Uravneniya, 27(1991), no. 7, 1196-1203 (in Russian).
[5] S.N.Timergaliev, Existence Theorems in Nonlinear Theory of Thin Elastic Shells, Kazan Univ. Press, Kazan, 2011.
[6] S.N.Timergaliev, Solvability of geometrically nonlinear boundary-value problems for the Timoshenko-type anisotropic shells with rigidly clamped edges, Russian Mathematics, 55(2011), no. 8, 47-58.
[7] S.N.Timergaliev, Proof of the Solvability of a System of Partial Differential Equations in the Nonlinear Theory of Shallow Shells of Timoshenko Type, Differential Equations, 48(2012), no. 3, 458-463.
[8] S.N.Timergaliev, On existence of solutions to geometrically nonlinear problems for shallow shells of the Timoshenko type with free edges, Russian Mathematics, 58(2014), no. 3, 31-46.
[9] S.N.Timergaliev, On the Existence of Solutions of a Nonlinear Boundary-Value Problem for the System of Partial Differential Equations of the Theory of Timoshenko Type Shallow Shells with Free Edges, Differents. Uravneniya, 51(2015), no. 3, 373-386 (in Russian).
[10] S.N.Timergaliev, A.N.Uglov, L.S.Kharasova, Solvability of geometrically nonlinear boundary-value problems for shallow shells of Timoshenko type with pivotally supported edges, Russian Mathematics, 59(2015), no. 5, 41-51.
[11] I.N.Vekua, Generalized Analytic Function, Nauka, Moscow, 1988 (in Russian).
[12] F.D.Gakhov, Boundary-Value Problems, Fizmatgiz, Moscow, 1963 (in Russian).
[13] M.A.Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Gostekhizdat, Moscow, 1956 (in Russian).
Исследование разрешимости одной нелинейной краевой задачи для системы дифференциальных уравнений теории пологих оболочек типа Тимошенко
Марат Г. Ахмадиев Самат Н. Тимергалиев Лилия С. Харасова
Институт Набережных Челнов Казанского федерального университета
Сююмбике, 10A, Набережные Челны, 423812
Россия
Работа посвящена исследованию 'разрешимости системы нелинейных дифференциальных уравнений с частными производными второго порядка при заданных граничных условиях. Метод исследования заключается в сведении исходной системы уравнений к одному нелинейному операторному уравнению, разрешимость которого устанавливается с помощью принципа сжатых отображений.
Ключевые слова: система нелинейных дифференциальных уравнений, уравнения равновесия, интегральные представления, теорема существования.
