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THE FIRST BOUNDARY VALUE PROBLEM FOR A DIFFERENTIAL EQUATION OF ELASTIC EQUILIBRIUM OF A SHALLOW CYLINDRICAL SHELL
S. M. BELONOSOV, M. A. ISMAILOV
I. FORMULATION OF THE PROBLEM
- As is known [1–3], the stressed state of a slightly curved (shallow) cylindrical elastic shell is described by the differential equation
\[ \Delta \Delta W(x,y)-i\lambda^2 \frac{\partial^2 W}{\partial x^2}=0, \tag{1} \]
where
\[ \lambda^2=\frac{\sqrt{12(1-\nu^2)}}{Rh}; \]
\(R\) is the radius of the shell; \(h\) is its height; \(\nu\) is Poisson’s ratio;
\[ \Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}; \]
the coordinates \(x,y\) have the dimension of length and vary respectively along the generator of the circular cylinder and the directing circle of radius \(R\) (when the cylinder is developed onto a plane, the variables \(x,y\) acquire the meaning of ordinary Cartesian coordinates);
\[ W(x,y)=w(x,y)+\frac{iR}{Eh}\lambda^2\varphi(x,y), \tag{2} \]
\(w(x,y)\) is the displacement of a point of the middle surface of the shell in the direction of its normal, and \(\varphi(x,y)\) is the stress function:
\[ T_1=\int_{-h/2}^{h/2}\sigma_x(x,y,z)\,dz=\frac{\partial^2\varphi}{\partial y^2}, \]
\[ S=\int_{-h/2}^{h/2}\tau_{xy}(x,y,z)\,dz=-\frac{\partial^2\varphi}{\partial x\,\partial y}, \tag{3} \]
\[ T_2=\int_{-h/2}^{h/2}\sigma_y(x,y,z)\,dz=\frac{\partial^2\varphi}{\partial x^2}. \]
As \(R\to\infty\), equation (1) in the limit splits into two biharmonic equations for the functions \(w(x,y)\) and \(\varphi(x,y)\).
The problem of finding a biharmonic function from the values of its partial derivatives prescribed on the boundary of a domain is usually called the first boundary-value problem for a biharmonic function.
In this article we study an analogous problem for equation (1).
- We shall consider both the interior and the exterior boundary-value problem for equation (1):
a) Interior problem. One seeks a function \(W(x,y)\) satisfying equation (1) in a finite simply connected domain \(D\) in the \(x,y\)-plane, bounded by a closed, non-self-intersecting smooth contour \(L\). On this contour we prescribe the conditions
\[ \left.\frac{\partial W}{\partial x}\right|_{L}=a(s),\qquad \left.\frac{\partial W}{\partial y}\right|_{L}=b(s), \tag{4} \]
where \(a(s), b(s)\) are prescribed continuous functions of the arc length \(s\).
Subjecting the solution \(W(x,y)\) to the condition of single-valuedness, we must impose on these functions the restriction
\[ \int_L a(s)\,dx+b(s)\,dy=0. \tag{5} \]
b) Exterior problem. Suppose that on the surface of an infinite cylinder there is an opening bounded by a contour \(L\). When the cylinder is developed onto the \(x,y\)-plane, the part of the cylindrical surface exterior to the opening becomes the part of the plane exterior to an infinite row of identical openings arranged along the \(y\)-axis with period \(2\pi R\).
The function \(W(x,y)\), single-valued on the cylinder, should be regarded as periodic in \(y\) with period \(2\pi R\) and satisfying, at each opening in the \(x,y\)-plane, identical conditions of the form (4). At infinity \(W(x,y)\) will be regarded as vanishing together with all its partial derivatives.
In order to shorten the writing of formulas and derivations in the subsequent exposition, we shall consider the exterior problem in an approximate formulation, introducing the simplifying assumption of sufficiently rapid decay of the solution as one moves away from the contour \(L\).
We shall seek a function \(W(x,y)\) satisfying equation (1) in the infinite domain \(D\) (in the \(x,y\)-plane) exterior to a closed, non-self-intersecting smooth contour \(L\), vanishing at infinity and subject on \(L\) to the conditions (4).
Such an approximate formulation of the exterior problem may be justified if the dimensions of the opening on the cylinder are sufficiently small in comparison with the radius \(R\).
To solve the problem thus posed we shall apply the method of integral equations, following the device of G. Lauricella in the solution of the first boundary-value problem for a biharmonic function [4].
II. TRANSFORMATION OF THE DIFFERENTIAL EQUATION
Let us introduce the functions
\[ u=\frac{\partial W}{\partial x},\qquad v=\frac{\partial W}{\partial y}. \tag{6} \]
These functions satisfy the relation
\[ \frac{\partial u}{\partial y}-\frac{\partial v}{\partial x}=0. \]
Let now
\[ \theta=\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}. \tag{7} \]
Equation (1) can then be written in the form
\[ \frac{\partial}{\partial x}\left(\frac{\partial\theta}{\partial x}-i\lambda^2u\right) = -\frac{\partial}{\partial y}\frac{\partial\theta}{\partial y}, \]
whence it follows that there exists a function \(\chi(x,y)\) such that
\[ \frac{\partial\chi}{\partial y} = \frac{\partial\theta}{\partial x}-i\lambda^2u, \qquad \frac{\partial\chi}{\partial x} = -\frac{\partial\theta}{\partial y}. \tag{8} \]
By virtue of (7) and (8) we arrive at the equations
\[ \left. \begin{aligned} M_1(u,\chi)&\equiv \Delta u-\frac{\partial\chi}{\partial y}-i\lambda^2u=0,\\[4pt] M_2(v,\chi)&\equiv \Delta v+\frac{\partial\chi}{\partial x}=0,\\[4pt] M_3(u,v)&\equiv \frac{\partial u}{\partial y}-\frac{\partial v}{\partial x}=0. \end{aligned} \right\} \tag{9} \]
The system (9) is obviously equivalent to the original equation (1). Indeed, the third of equations (9) ensures the existence of a function \(W(x,y)\) connected with \(u,v\) by relations (6), while elimination of the function \(\chi(x,y)\) from the first two equations (9) leads to equation (1).
III. GREEN’S FORMULAS FOR SYSTEM (9)
THE UNIQUENESS THEOREM FOR THE SOLUTION
- Let \(u,v,\chi\) and \(u',v',\chi'\) be two systems of functions such that the operators \(M_1,M_2,M_3\) of them are continuous in some finite multiply connected domain \(\Omega\), bounded by a smooth contour \(C\).
Then, by the usual method, applying integration by parts, it is not difficult to verify the validity of the relation
\[ \left. \begin{aligned} \iint_{\Omega}\{& u'M_1^*(u,v,\chi)-uM_1^*(u',v',\chi') +v'M_2^*(u,v,\chi)\\ &-vM_2^*(u',v',\chi') +\chi'M_3(u,v)-\chi M_3(u',v')\}\,dxdy\\ &= -\int_C\{u'N_1(u,v,\chi)-uN_1(u',v',\chi')\\ &\qquad\qquad +v'N_2(u,v,\chi)-vN_2(u',v',\chi')\}\,ds, \end{aligned} \right\} \tag{10} \]
where
\[ \left. \begin{aligned} M_1^*&=M_1+\frac{\partial M_3}{\partial y}, \qquad M_2^*=M_2-\frac{\partial M_3}{\partial x},\\[6pt] N_1(u,v,\chi) &= \left(\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right)\cos(n,x) + \left(2\frac{\partial u}{\partial y}-\chi\right)\cos(n,y) \end{aligned} \right\} \tag{11} \]
\[ \left. \begin{aligned} N_2(u, v, \chi) \equiv {}& \left(2\frac{\partial v}{\partial x}+\chi\right)\cos(n,x)+{}\\ &+\left(\frac{\partial v}{\partial y}-\frac{\partial u}{\partial x}\right)\cos(n,y) \end{aligned} \right\} \tag{11} \]
(\(n\) denotes the direction of the outward normal to the contour \(C\)).
- The second Green formula (its validity can also be verified by integration by parts) has the form
\[ \left. \begin{aligned} &\iint_{\Omega}\{\overline{u}M_1(u,\chi)+\overline{v}M_2(v,\chi)-\chi M_3(\overline{u},\overline{v})\}\,dxdy +i\lambda^2\iint_{\Omega}|u|^2\,dxdy ={}\\ &= -\iint_{\Omega}\left\{ \left|\frac{\partial u}{\partial x}\right|^2+ \left|\frac{\partial u}{\partial y}\right|^2+ \left|\frac{\partial v}{\partial x}\right|^2+ \left|\frac{\partial v}{\partial y}\right|^2 \right\}\,dxdy -\int_C\{\overline{u}N_1^{*}(u,\chi)+\overline{v}N_2^{*}(v,\chi)\}\,ds, \end{aligned} \right\} \tag{12} \]
where
\[ N_1^{*}(u,\chi)=\frac{\partial u}{\partial n}-\chi\cos(n,y), \]
\[ N_2^{*}(v,\chi)=\frac{\partial u}{\partial n}+\chi\cos(n,x). \tag{13} \]
- If the functions \((u, v, \chi)\) and \((u', v', \chi')\) are solutions of system (9), then formulas (10) and (12) are written as follows:
\[ \int_C\{u'N_1(u,v,\chi)+v'N_2(u,v,\chi)\}\,ds = \int_C\{uN_1(u',v',\chi')+vN_2(u',v',\chi')\}\,ds, \tag{14} \]
\[ \iint_{\Omega}\left\{ \left|\frac{\partial u}{\partial x}\right|^2+ \left|\frac{\partial u}{\partial y}\right|^2+ \left|\frac{\partial v}{\partial x}\right|^2+ \left|\frac{\partial v}{\partial y}\right|^2+ i\lambda^2|u|^2 \right\}\,dxdy = -\int_C\{\overline{u}N_1^{*}(u,\chi)+\overline{v}N_2^{*}(v,\chi)\}\,ds. \tag{15} \]
- From formula (15) there follows directly the uniqueness of the functions \(u(x,y)\), \(v(x,y)\) satisfying the equations under consideration and taking prescribed values on the contour of a finite domain.
If the domain \(\Omega\) is exterior to the contour \(C\) and contains the point at infinity, then formulas (14) and (15) will be valid under the condition of sufficiently rapid decay of the considered solutions of equations (9) as \(r=\sqrt{x^2+y^2}\) increases without bound.
Below we shall prove the validity of formulas (14) and (15) for any solutions of system (9) vanishing at infinity, and thereby the uniqueness theorem will also be established for the exterior boundary-value problem.
- Let \(\Omega\) be a doubly connected domain obtained from the given simply connected domain \(D\) by removing from it a small disk of radius \(\rho\) with center at an interior point \((\xi,\eta)\in D\).
In this case the contour \(C=L+\gamma\), where \(\gamma\) is the circle of radius \(\rho\).
In formula (14) take the following functions:
\[ \left. \begin{aligned} u'&=\frac{\pi}{4i}\frac{\operatorname{sh}\alpha(x-\xi)}{\alpha}\frac{\partial H}{\partial x} +\frac{\pi}{4i}\operatorname{ch}\alpha(x-\xi)H,\\ v'&=\frac{\pi}{4i}\frac{\operatorname{sh}\alpha(x-\xi)}{\alpha}\frac{\partial H}{\partial y},\\ \chi'&=-\frac{\pi}{4i}\operatorname{ch}\alpha(x-\xi)\frac{\partial H}{\partial y} \end{aligned} \right\}, \tag{16'} \]
where \(\alpha=\sqrt{i}\,\dfrac{\lambda}{2}\), \(H=H_0^{(1)}(\alpha r)\) is the Hankel function of the first kind of zero order \(\bigl(r=\sqrt{(x-\xi)^2+(y-\eta)^2}\bigr)\).
Let \(u,v,\chi\) in formula (14) be a solution of system (9) in the domain \(D\), continuous on \(L\) together with \(N_1(u,v,\chi)\), \(N_2(u,v,\chi)\) (and vanishing at infinity if \(D\) is the exterior of the contour \(L\)).
Letting the radius \(\rho\) in formula (14) tend to zero, we obtain an expression for the function \(u(\xi,\eta)\) in terms of the values on the contour \(L\), \(u(s),v(s),N_1[u(s),v(s),\chi(s)]\), \(N_2[u(s),v(s),\chi(s)]\):
\[ \begin{aligned} u(\xi,\eta)=\int_L \{&N_1(u,v,\chi)u' + N_2(u,v,\chi)v'\\ &-uN_1(u',v',\chi')-vN_2(u',v',\chi')\}\,ds. \end{aligned} \tag{17'} \]
Substituting in (10) the functions
\[ \left. \begin{aligned} u''&=\frac{\pi}{4i}\frac{\operatorname{sh}\alpha(x-\xi)}{\alpha}\frac{\partial H}{\partial y},\\ v''&=-\frac{\pi}{4i}\frac{\operatorname{sh}\alpha(x-\xi)}{\alpha}\frac{\partial H}{\partial x} +\frac{\pi}{4i}\operatorname{ch}\alpha(x-\xi)H+1,\\ \chi''&=\frac{\pi}{4i}\operatorname{ch}\alpha(x-\xi)\frac{\partial H}{\partial x} -\frac{\pi}{4i}\alpha\operatorname{sh}\alpha(x-\xi)H \end{aligned} \right\}, \tag{16''} \]
we obtain an analogous integral representation for the function \(v(\xi,\eta)\):
\[ \begin{aligned} v(\xi,\eta)=\int_L \{&N_1(u,v,\chi)u''+N_2(u,v,\chi)v''\\ &-uN_1(u'',v'',\chi'')-vN_2(u'',v'',\chi'')\}\,ds. \end{aligned} \tag{17''} \]
- Formulas \((17')\), \((17'')\) are certainly valid for a finite domain \(D\).
We shall prove their validity for the exterior of the contour \(L\) and, at the same time, give a rigorous proof of the uniqueness theorem for the exterior boundary value problem.
We shall regard the domain \(D\), exterior to the contour \(L\), as the limit of the doubly connected finite domain \(D_R\), bounded by the simply connected contour \(L\) and by a circle of large radius \(R\) with center at the point \((\xi,\eta)\), as \(R\to\infty\).
For the domain \(D_R\), formulas \((17')\), \((17'')\) are valid if in these formulas, instead of the contour \(L\), one takes the contour of integration \(L+\Gamma\), where \(\Gamma\) is a circle of radius \(R\).
Suppose that the functions \(u, v, \chi\), which in the domain \(D\) are a solution of equations (9), vanish at infinity together with \(N_1(u, v, \chi)\), \(N_2(u, v, \chi)\).
On the basis of the asymptotic formula for the Hankel function
\[ H_0^{(1)}(\alpha r)\sim \frac{2}{\sqrt{2\pi\lambda r}}\, e^{-\frac{\lambda r}{2\sqrt{2}}(1-i)-\frac{3\pi}{8}i} \tag{18} \]
it is easy to verify that the integrals in formulas \((17')\), \((17'')\) over the circle \(\Gamma\) tend to zero as \(R\to\infty\).
For example:
\[ I_1=\left|\int_{\Gamma} N_1(u,v,\chi)\,u''\,ds\right| \leq 2\Phi(R)\sqrt{R_0}\,e^{-R_0} \int_0^{\pi/2} e^{R_0\cos\theta}\,d\theta, \]
where
\[ R_0=\frac{\lambda R}{2\sqrt{2}},\qquad \Phi(R)=\max_{\Gamma}|N_1(u,v,\chi)|. \]
By virtue of the formula
\[ J_0(iR)=\frac{2}{\pi}\int_0^{\pi/2}\cos(iR\cos\theta)\,d\theta = \]
\[ =\frac{1}{\pi}\int_0^{\pi/2}(e^{R\cos\theta}+e^{-R\cos\theta})\,d\theta = \frac{e^R}{\sqrt{2\pi R}} \left[1+O\left(\frac{1}{R}\right)\right] \]
we obtain the estimate
\[ I_1\leq \sqrt{\frac{2}{\pi}}\, \Phi(R) \left[1+O\left(\frac{1}{R}\right)\right]. \]
The expression on the right tends to zero together with \(\Phi(R)\) as \(R\to\infty\).
Thus, formulas \((17')\), \((17'')\) are valid in the domain \(D\), exterior to the contour \(L\), for every solution of system (9) that vanishes at infinity.
From these formulas, taking into account that \(L\) is a closed finite contour, and applying the asymptotic formula (18), it follows directly that every solution of system (9) that vanishes at infinity decreases as \(R\to\infty\) according to the same law as the functions \((16')\), \((16'')\):
\[ \{|u(x,y)|,\ |v(x,y)|\} \underset{R\to\infty}{\sim} \frac{\mathrm{const}\,e^{-R_0(1-|\cos\theta|)}}{\sqrt{R_0}}, \tag{19} \]
where
\[ R_0=\frac{\lambda}{2\sqrt{2}}\sqrt{x^2+y^2},\qquad \theta=\operatorname{arctg}\frac{y}{x}\quad(\lambda>0). \]
On the basis of these asymptotic estimates, one can in an obvious way show the validity of Green’s formulas (14), (15) for any solutions \((u,v,\chi)\), \((u',v',\chi')\) of system (9) that vanish at infinity.
IV. REDUCTION OF THE FIRST BOUNDARY VALUE PROBLEM TO INTEGRAL EQUATIONS
Consider the functions
\[ \left. \begin{aligned} U(\xi,\eta)&=\frac{1}{\pi}\int_L \{N_1(u',v',\chi')\varphi(s)+N_2(u',v',\chi')\psi(s)\}\,ds,\\ V(\xi,\eta)&=\frac{1}{\pi}\int_L \{N_1(u'',v'',\chi'')\varphi(s)+N_2(u'',v'',\chi'')\psi(s)\}\,ds, \end{aligned} \right\} \tag{20} \]
where \(\varphi(s), \psi(s)\) are arbitrary continuous functions.
By direct verification it is not difficult to ascertain the validity of the equalities
\[ \frac{\partial N_1'}{\partial \eta}=\frac{\partial N_1''}{\partial \xi},\qquad \frac{\partial N_2'}{\partial \eta}=\frac{\partial N_2''}{\partial \xi}, \tag{21} \]
from which follows the existence of a function \(W(x,y)\) such that
\[ U=\frac{\partial W}{\partial \xi},\qquad V=\frac{\partial W}{\partial \eta}. \tag{6'} \]
The relations
\[ \left. \begin{aligned} \left[\frac{\partial}{\partial \xi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)-i\lambda^2\frac{\partial}{\partial \xi}\right]N_1' &=-\frac{\partial}{\partial \eta}\left[\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right]N_1'',\\ \left[\frac{\partial}{\partial \xi}\left(\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right)-i\lambda^2\frac{\partial}{\partial \xi}\right]N_2' &=-\frac{\partial}{\partial \eta}\left[\frac{\partial^2}{\partial \xi^2}+\frac{\partial^2}{\partial \eta^2}\right]N_2'' \end{aligned} \right\} \tag{22} \]
also hold; from them it follows that the indicated function \(W(\xi,\eta)\) satisfies equation (1).
Let us study the form of the functions \(N_1, N_2\) when the point \((\xi,\eta)\) approaches a fixed point \((x_0,y_0)\) of the contour \(L\), i.e., when
\[ r=\sqrt{(x-\xi)^2+(y-\eta)^2}\to 0. \]
For small \(r\), the Hankel function \(H_0^{(1)}(\alpha r)\) has the form
\[ H_0^{(1)}(\alpha r)=\frac{2i}{\pi}\ln r\{1+O(\alpha^2r^2)\}+Q(r^2), \]
where \(Q(r^2)\) is an entire function. Hence, for small \(r\), we find
\[ \left. \begin{aligned} N_1(u',v',\chi')&=2\left(\frac{\partial r}{\partial y}\right)^2\frac{\partial \ln r}{\partial n} +\lambda^2 k_{11}(x,y;x_0,y_0,\lambda^2),\\ N_2(u',v',\chi')&=-2\frac{\partial r}{\partial x}\frac{\partial r}{\partial y}\frac{\partial \ln r}{\partial n} +\lambda^2 k_{12}(x,y;x_0,y_0,\lambda^2),\\ N_1(u'',v'',\chi'')&=-2\frac{\partial r}{\partial x}\frac{\partial r}{\partial y}\frac{\partial \ln r}{\partial n} +\lambda^2 k_{21}(x,y;x_0,y_0,\lambda^2),\\ N_2(u'',v'',\chi'')&=2\left(\frac{\partial r}{\partial x}\right)^2\frac{\partial \ln r}{\partial n} +\lambda^2 k_{22}(x,y;x_0,y_0,\lambda^2), \end{aligned} \right\} \tag{23} \]
where \(k_{i,j}(x,y;x_0,y_0,\lambda^2)\) are continuous functions, and differentiation with respect to the normal to the contour \(L\) is performed at the point of integration \((x,y)\in L\).
Thus, the kernels of the integral operators (20) have a weak singularity of the same type as the kernels in double-layer potentials.
We shall seek the solution of the boundary-value problem under consideration for the domain \(D\) with the conditions
\[ u\big|_L = a(s), \quad v\big|_L = b(s) \tag{4'} \]
in the form of the integrals (20). Letting the point \((\xi,\eta)\) tend to the point \(\xi_0=x(s_0), \eta_0=y(s_0)\) of the contour \(L\), we obtain, for the unknown functions \(\varphi(s)\), \(\psi(s)\), Fredholm integral equations of the second kind
\[ \left. \begin{aligned} \varphi(s_0) - \frac{1}{\pi}\int_L \Bigg\{& \left[2\left(\frac{\partial r}{\partial y}\right)^2 \frac{\partial \ln r}{\partial n} +\lambda^2 k_{11}(s,s_0,\lambda^2)\right]\varphi(s) \\ &- \left[2\left(\frac{\partial r}{\partial x}\right) \left(\frac{\partial r}{\partial y}\right) \frac{\partial \ln r}{\partial n} -\lambda^2 k_{12}(s,s_0,\lambda^2)\right]\psi(s) \Bigg\}\,ds = a(s_0), \\[6pt] \psi(s_0) - \frac{1}{\pi}\int_L \Bigg\{& \left[-2\left(\frac{\partial r}{\partial x}\right) \left(\frac{\partial r}{\partial y}\right) \frac{\partial \ln r}{\partial n} +\lambda^2 k_{21}(s,s_0,\lambda^2)\right]\varphi(s) \\ &- \left[2\left(\frac{\partial r}{\partial x}\right)^2 \frac{\partial \ln r}{\partial n} +\lambda^2 k_{22}(s,s_0,\lambda^2)\right]\psi(s) \Bigg\}\,ds = b(s_0). \end{aligned} \right\} \tag{24} \]
For \(\lambda=0\), equations (24) were studied by G. Lauricella [4], who proved their solvability under condition (5).
By the contraction mapping principle, equations (24) are, evidently, solvable for all sufficiently small \(\lambda\).
References
- Vekua I. N. New Methods for Solving Elliptic Equations. OGIZ, Gostekhizdat, 1948.
- Vlasov V. Z. PMM, VIII, 1944, pp. 109–140.
- Lur’e A. I. Statics of Thin-Walled Elastic Shells. Gostekhizdat, 1947.
- Lauricella G. Sur l’intégration relative à l’équilibre des plaques élastiques encastrées. Acta Math., 32, 1909, pp. 201–256.
Received by the editors
December 3, 1964.
Institute of Mathematics, Academy of Sciences of the BSSR