UDC 539.3

THEORY OF ELASTICITY

Yu. A. DUBINSKII

ON THE SOLVABILITY OF THE SYSTEM OF EQUATIONS OF STRONG BENDING OF PLATES

(Presented by Academician Yu. N. Rabotnov, 2 XI 1966)

In this paper the existence is proved of both a generalized and a smooth solution of system (1), which describes the strong bending of a thin plate; moreover, for proving the existence of a generalized solution only the simplest energy estimate and weak convergence in \(L_2\) are used. We note that the existence of a generalized solution for A. Föppl’s equations, which are a consequence of (1), was obtained earlier by other methods and under stronger assumptions, for example, in papers \((^2,^3)\).

As is known (see, for example, \((^1)\)), the complete system of equations of strong bending of a plate has the form

\[ \frac{h^2E}{12(1-\sigma^2)}\Delta^2\zeta - h\frac{\partial}{\partial x_\beta} \left( \sigma_{\alpha\beta}\frac{\partial\zeta}{\partial x_\alpha} \right) = P(x,y), \qquad \frac{\partial\sigma_{\alpha\beta}}{\partial x_\beta}=0. \tag{1} \]

Here \(h\) is the thickness of the plate; \(E\) is Young’s modulus; \(\sigma\) \((0<\sigma<1)\) is Poisson’s ratio; \(\sigma_{\alpha\beta}\) is the stress tensor, related to the strain tensor \(u_{\alpha\beta}\) by the formulas

\[ \sigma_{\alpha\beta} = E(1-\sigma^2)^{-1} \bigl[(1-\sigma)u_{\alpha\beta}+\sigma\delta_{\alpha\beta}u_{\gamma\gamma}\bigr] \tag{2} \]

(\(\alpha,\beta\), and \(\gamma\) run through the two values \(x\) and \(y\)).

Further,

\[ u_{\alpha\beta} = \frac12 \left( \frac{\partial u_\alpha}{\partial x_\beta} + \frac{\partial u_\beta}{\partial x_\alpha} \right) + \frac12 \frac{\partial\zeta}{\partial x_\alpha} \frac{\partial\zeta}{\partial x_\beta}, \tag{3} \]

where \(\mathbf u=(u_x,u_y)\) is the strain vector; \(\zeta(x,y)\) is the vertical displacement of the point \((x,y)\) of the plate. Thus the unknowns in (1) are the three functions \(u_x(x,y)\), \(u_y(x,y)\), and \(\zeta(x,y)\), defined in the domain \(G\) occupied by the plate.

Let the boundary conditions for system (1) have the form

\[ \zeta(x,y)=u_x(x,y)=u_y(x,y)=0,\qquad \frac{\partial\zeta}{\partial x}(x,y)= \frac{\partial\zeta}{\partial y}(x,y)=0,\qquad \forall(x,y)\in\partial G, \tag{4} \]

which corresponds to the case of a clamped plate.

Definition. A generalized solution of problem (1), (4) is a system of functions \((\zeta,\mathbf u)\) with finite energy integral

\[ E(\zeta,\mathbf u) = \iint_G \bigl[(\Delta\zeta)^2+\sigma_{\alpha\beta}u_{\alpha\beta}\bigr]\,dx\,dy ^*, \]

satisfying the integral identity

\(^*\) The integral \(E(\zeta,\mathbf u)\) is equivalent to the mechanical energy of the plate \(E_m\) in the sense that \(E_m\le c_1E\le c_2E_m\), where \(c_1,c_2>0\) are constants. In addition, we note that from the purely mathematical point of view system (1) is of interest because, besides the principal part, lower-order terms play an essential role in determining the class of generalized solutions.

\[ \frac{h^3E}{12(1-\sigma^2)}\langle \Delta \xi,\Delta \tilde{\xi}\rangle +h\left\langle \sigma_{\alpha\beta}\frac{\partial \xi}{\partial x_\alpha}, \frac{\partial \tilde{\xi}}{\partial x_\beta}\right\rangle =\langle P,\tilde{\xi}\rangle, \]
\[ \left\langle \sigma_{\alpha\beta},\frac{\partial v_\alpha}{\partial x_\beta}\right\rangle=0, \qquad \forall \tilde{\xi}(x,y),\,v_x(x,y),\,v_y(x,y)\in C_0^\infty(G) \tag{3′} \]

\[ \left(\langle u,v\rangle=\iint_G u\cdot v\,dx\,dy; \right. \]
in the last two equations summation is only over \(\beta\)) and satisfying the boundary values (4) in the mean \((^5)\).

Theorem 1. If \(P(x,y)\in W_2^{(-2)}\), then problem (1), (4) has at least one generalized solution.

Proof. Let \((\xi^k(x,y),\,v_x^k(x,y),\,v_y^k(x,y))\) be a system of smooth functions finite in the domain \(G\), complete, for example, in \(L_2\). The approximate solution \((\xi^n,u_x^n,u_y^n)\) is sought in the form
\[ (\xi^n,u_x^n,u_y^n) = \sum_{k=1}^{n}(c_{kn}^1\xi^k,\, c_{kn}^2v_x^k,\, c_{kn}^3v_y^k); \]
the constants \(c_{kn}^i,\ i=1,2,3,\) are determined from the system of moment equations
\[ \frac{h^3E}{12(1-\sigma^2)}\langle \Delta \xi^n,\Delta \xi^k\rangle +h\left\langle \sigma_{\alpha\beta}^n\frac{\partial \xi^n}{\partial x_\alpha}, \frac{\partial \xi^k}{\partial x_\beta}\right\rangle =\langle P,\xi^k\rangle, \tag{5} \]
\[ \left\langle \sigma_{\alpha\beta}^n,\frac{\partial v_\alpha^k}{\partial x_\beta}\right\rangle=0, \qquad k=1,\ldots,n, \]
where \(\sigma_{\alpha\beta}^n\) are defined by formulas (2), (3) with \((\xi,\mathbf u)\) replaced by \((\xi^n,\mathbf u^n)\). The solvability of system (5) follows from a lemma of M. I. Vishik \((^4)\); moreover, for the approximate solutions the a priori estimate
\[ E(\xi^n,\mathbf u^n)\le K_1\|P\|_{W_2^{(-2)}}, \tag{6} \]
holds, where \(K_1\) does not depend on \((\xi^n,\mathbf u^n)\). (To obtain estimate (6), it is enough to multiply (5) by \(c_{i n}\), sum over \(k\), then add the resulting equalities and take into account the symmetry of the tensor \(u_{\alpha\beta}\); the right-hand side of (5) is estimated with the aid of Schwarz’s inequality.) From estimate (6) and the known inequalities for the Laplace operator we obtain that \(\xi^n(x,y)\in W_2^{(2)}\), with \(\|\xi^n\|_{W_2^{(2)}}\le K_2\). In addition, it follows from (6) that \(\langle \sigma_{\alpha\beta}^n,u_{\alpha\beta}^n\rangle\le K_3\), or, equivalently,
\[ \frac{E}{1-\sigma^2}\iint_G\bigl[(u_{xx}^n)^2+2\sigma u_{xx}^n u_{yy}^n+(u_{yy}^n)^2\bigr]\,dx\,dy + \frac{2E}{1+\sigma}\iint_G (u_{xy}^n)^2\,dx\,dy \le K_3, \]
after which it follows from formulas (3) that \(\partial u_x^n/\partial x\in L_2\), \(\partial u_y^n/\partial y\in L_2\), \(\partial u_x^n/\partial y+\partial u_y^n/\partial x\in L_2\), and they range over a bounded set in \(L_2\). Hence, it may be assumed (possibly after selecting a subsequence) that \(\xi^n\to\xi\) weakly in \(W_2^{(2)}\); \(\mathbf u^n\to\mathbf u\), \(\partial u_x^n/\partial x\to\partial u_x/\partial x\), \(\partial u_y^n/\partial y\to\partial u_y/\partial y\), and \(\partial u_x^n/\partial y+\partial u_y^n/\partial x\to\partial u_x/\partial y+\partial u_y/\partial x\) weakly in \(L_2\), where \((\xi,\mathbf u)\) is some system of functions with finite energy \(E\) and satisfying the boundary conditions (4) in the mean.

We shall show that \((\xi,\mathbf u)\) is the desired generalized solution. To this end, note that, by virtue of the completeness in \(L_2\) of the system \((\xi^k,v_x^k,v_y^k)\), it is enough to prove that \((\xi,\mathbf u)\) satisfies relations (3′), where \((\tilde{\xi},v_x,v_y)\) are replaced by \((\xi^k,v_x^k,v_y^k)\) for any \(k=1,2,\ldots\).

In other words, it is enough to prove the possibility of passing to the limit in (5) as \(n\to\infty\) with fixed \(k\). In the linear term \(\langle \Delta \xi^n,\Delta \xi^k\rangle\) this is obvious. Further, from the weak convergence \(\xi^n\to\xi\) in \(W_2^{(2)}\) it follows that
\[ \xi^n\to\xi \quad \text{strongly in } W_p^{(1)}\ (p>1\text{ arbitrary}),\qquad \frac{\partial \xi^n}{\partial x_\alpha}\frac{\partial \xi^n}{\partial x_\beta}\to \]

\[ \to \frac{\partial \zeta}{\partial x_\alpha}-\frac{\partial \zeta}{\partial x_\beta} \]
strongly in \(L_2\). Consequently, \(u_{\alpha\beta}^n \to u_{\alpha\beta}\) weakly in \(L_2\), and since \(\sigma_{\alpha\beta}^n\) is expressed linearly in terms of \(u_{\alpha\beta}^n\), it follows that \(\sigma_{\alpha\beta}^n \to \sigma_{\alpha\beta}\) weakly in \(L_2\). This justifies the passage to the limit in the last two equations of system (3).

Lemma. If \(u_n \to u\) strongly in \(L_2\), and \(v_n \to v\) weakly in \(L_2\), then \(u_n v_n \to uv\) weakly in \(L_1\).

For the proof see (6).

From this lemma we obtain that

\[ \left\langle \sigma_{\alpha\beta}^n \frac{\partial \zeta^n}{\partial x_\alpha}, \frac{\partial \zeta^k}{\partial x_\beta} \right\rangle \to \left\langle \sigma_{\alpha\beta} \frac{\partial \zeta}{\partial x_\alpha}, \frac{\partial \zeta^k}{\partial x_\beta} \right\rangle,\qquad k=1,2,\ldots \]

As a result we obtain that \((\zeta,\mathbf u)\) is a solution of problem (1), (4). Theorem 1 is proved.

Remark. The membership \(P(x,y)\in W_2^{(-2)}\) means that \(P(x,y)\) has the form
\[ P(x,y)=\sum_{|\alpha|\le 2} D^\alpha f_\alpha(x,y), \]
where \(f_\alpha(x,y)\in L_2\)
\[ \left(D^\alpha=\frac{\partial^\alpha}{\partial x^{\alpha_1}\partial y^{\alpha_2}},\quad |\alpha|=\alpha_1+\alpha_2\right), \]
i.e. \(P(x,y)\) may be, in particular, a generalized function of the type \(\delta(x,y)\) and its derivatives. This corresponds to loads on the plate concentrated at individual points. For such loads it is impossible to obtain a smoother solution than in Theorem 1.

Theorem 2 (smoothness theorem). Let \(P(x,y)\in L_p\), where \(p<2\) is arbitrary. Then the generalized solution \((\zeta,\mathbf u)\) is such that

\[ \zeta\in W_p^{(4)}\cap \overset{\circ}{W}{}_{2}^{(1)},\qquad \mathbf u\in W_p^{(2)}\cap \overset{\circ}{W}{}_{2}^{(1)}. \]

Proof. Consider the last two equations in (1):

\[ \frac{E}{1-\sigma^2}\frac{\partial}{\partial x} \left(\frac{\partial u_x}{\partial x}+\sigma\frac{\partial u_y}{\partial y}\right) + \frac{E}{2(1+\sigma)}\frac{\partial}{\partial y} \left(\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\right) +f_1(\zeta)=0, \tag{7} \]

\[ \frac{E}{2(1+\sigma)}\frac{\partial}{\partial x} \left(\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\right) + \frac{E}{1-\sigma^2}\frac{\partial}{\partial y} \left(\frac{\partial u_y}{\partial y}+\sigma\frac{\partial u_x}{\partial x}\right) +f_2(\zeta)=0, \]

where
\[ f_1(\zeta)= \frac{E}{2(1-\sigma^2)}\frac{\partial}{\partial x} \left[ \left(\frac{\partial \zeta}{\partial x}\right)^2 +\sigma\left(\frac{\partial \zeta}{\partial y}\right)^2 \right] + \frac{E}{2(1+\sigma)}\frac{\partial}{\partial y} \left( \frac{\partial \zeta}{\partial x}\frac{\partial \zeta}{\partial y} \right), \]
and \(f_2(\zeta)\) coincides with \(f_1(\zeta)\) after interchanging \(x\) and \(y\).

It is easily verified that system (7) is elliptic (indeed, strongly elliptic). Further, since \(\zeta\in \overset{\circ}{W}{}_{2}^{2}\), it follows that \(f_1(\zeta), f_2(\zeta)\in L_p\), where \(p<2\) is arbitrary. Consequently, by the smoothness theorem for linear elliptic systems (see, for example, (7)), \(\mathbf u(x,y)\in W_p^{(2)}\cap \overset{\circ}{W}{}_{2}^{(1)}\).

Further, since \(\partial\sigma_{\alpha\beta}/\partial x_\beta=0\), the first equation in (1) can be written in the form

\[ -\frac{h^3E}{12(1-\sigma^2)}\Delta^2\zeta - h\sigma_{\alpha\beta}\frac{\partial^2\zeta}{\partial x_\alpha\partial x_\beta} = P(x,y), \tag{8} \]

where, since \(\zeta\in W_2^{(2)}\) and \(\mathbf u\in W_p^{(2)}\), \(p<2\) arbitrary, we have
\[ \sigma_{\alpha\beta}\frac{\partial^2\zeta}{\partial x_\alpha\partial x_\beta}\in L_p, \quad p<2 \]
arbitrary. Taking into account that, by assumption, \(P(x,y)\in L_p\), from equation (8) we obtain that the solution \(\zeta(x,y)\in W_p^{(4)}\cap W_2^{(2)}\). Theorem 2 is proved.

Further analogous arguments lead to the following result.

Theorem 3. If \(P(x,y)\in W_p^{(s)}\), \(p>1\), then the solution \((\zeta,\mathbf u)\) is such that \(\zeta(x,y)\in W_p^{(s+4)}\cap \dot W_2^{(1)}\), \(\mathbf u(x,y)\in W_p^{(s+2)}\cap \dot W_2^{(1)}\).

Corollary. If \(P(x,y)\) is an infinitely differentiable function, then the solution \((\zeta,\mathbf u)\) is also infinitely differentiable.

Moscow Power Engineering
Institute

Received
31 X 1966

CITED LITERATURE

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