Full Text
Reports of the Academy of Sciences of the USSR
1962, Volume 146, No. 1
THEORY OF ELASTICITY
A. L. KRYLOV
JUSTIFICATION OF DIRICHLET’S PRINCIPLE FOR THE FIRST BOUNDARY-VALUE PROBLEM OF THE NONLINEAR THEORY OF ELASTICITY
(Presented by Academician S. L. Sobolev on 28 III 1962)
1. It is known (see, for example, (\(^1,{}^2\))) that the equilibrium equations of an elastoplastic medium can be derived from certain variational principles, consisting in the fact that the solution of the problem gives a minimum to certain positive definite functionals. The fact of the existence of a solution of these variational problems is called Dirichlet’s principle and is widely used in heuristic arguments. Dirichlet’s principle for certain linear equations was proved in a number of works by R. Courant (\(^3\)), S. L. Sobolev (\(^4,{}^8\)); finally, in 1942 it was obtained by K. Friedrichs (\(^5\)) for the first and second boundary-value problems of the theory of elasticity and by D. M. Eidus (\(^6\)) for the third and fourth problems (for more details see (\(^7\))). W. T. Koiter posed the problem of justifying Dirichlet’s principle in nonlinear problems of the theory of elastoplastic media ((\(^1\)), § 7); in the present work this problem is solved for the case when displacements are prescribed on the boundary (the first boundary-value problem).
Variational principles for certain two-dimensional problems of nonlinear elasticity theory were justified by A. Langenbach (\(^{13,14}\)), whose methods are partly analogous to those we use in Sec. 2 (see also (\(^{10}\))). (Langenbach considers, in particular, the same example of the torsion problem, as well as a number of other problems.) Our principal aim, however, was the consideration of the general equations of the theory of elasticity.
2*. By a nonlinear elastic medium we shall mean a medium in which the deviators of the strain tensors \(D_\varepsilon\) (rates of strain) and stresses \(D_\sigma\) are connected by a relation of the form
\[ D_\varepsilon=\psi D_\sigma, \tag{1} \]
where \(\psi=f(T)\), \(T\) is the intensity of the shear stresses (in this case one sometimes speaks of a medium with hardening). For \(\psi=1/2G\) (constant) we have ordinary linear elasticity (Hooke’s law).
The problem that we shall consider is formulated as follows: find the displacement vector
\[ u(x)=\{u_1(x_1,x_2,x_3),\,u_2(x_1,x_2,x_3),\,u_3(x_1,x_2,x_3)\}, \]
defined in a smooth domain \(\Omega\), assuming a prescribed function \(\varphi\) on the boundary of the domain \(D\Omega\), and giving a minimal value to the integral
\[ \Pi(u)=\iiint_{\Omega}\left(\frac{\varepsilon^2}{6k}+\int_{0}^{\Gamma^2}g(\zeta)\,d\zeta\right)d\Omega, \tag{*} \]
where \(k\) is a constant; \(\varepsilon\) and \(\Gamma^2\) are the linear and quadratic invariants of the strain tensor \(\|\varepsilon_{ik}\|\) and of the deviator of the strain tensor \(\|\varepsilon_{ik}-\delta_{ik}\varepsilon\|\), \(g(\Gamma)\) is a function determined from \(f(T^2)\) by the equalities \(f(T^2)=\frac12 T\Gamma\), \(T=2g(\Gamma^2)\Gamma\). The components of the strain tensor \(\varepsilon_{ik}\) are related to the displacement vector \(u_i\) by the formulas
\[ \varepsilon_{ik}=\frac12\,(u_{i,k}+u_{k,i}), \]
where \(u_{ik}=\partial u_i(x_1,x_2,x_3)/\partial x_k\), \(i,k=1,2,3\). The function \(g(\Gamma^2)\) can often be represented in the form \(g(\Gamma^2)=B(\Gamma^2)^{(\beta-2)/2}\), where \(1\ll \beta\ll 2\).
* See (\(^2\)), Chapters I–III (especially §§ 18 and 33). See the notation there as well.
Thus, \(\Pi(u)\) is a functional depending, generally speaking, non-quadratically on linear combinations of the first derivatives of the solution, namely on \(\varepsilon_{ik}=\frac12(u_{i,k}+u_{k,i})\). From the form of the function \(g(\Gamma^2)\) \((^2)\) it follows that \(\Pi(u)\) is a uniformly convex functional of order \(\beta \leqslant 2\) in a sense that will be explained in §§ 3 and 4. For \(\beta=2\) we have the linear case, considered in \((^7)\). The minimum is sought among functions for which the integral is meaningful.
- To make the solution of this problem easier to understand, we shall first examine a simpler example, in which we shall clearly reveal the method of our reasoning. Namely, consider the problem of finding a function \(F(x,y)\) giving a minimum to the functional
\[ I(u)=\iint_{\Omega}\left((F_x^2+F_y^2)^{p/2}-2\omega F\right)\,d\Omega, \tag{2} \]
satisfying the boundary condition
\[ F(x,y)\big|_{\partial\Omega}=0. \tag{3} \]
(This is the problem of torsion of a prism with hardening; here \(F(x,y)\) is the stress function, \(\omega\) is the twist per unit length, and \(p\geqslant 2\) is a constant (\(p=2\) is the linear case), see \((^2)\), § 44.) The minimum is sought in the class of functions having first derivatives (in the Sobolev sense), summable to the power \(p\), and vanishing on the boundary; we shall denote the class of these functions by \(\overset{\circ}{W}{}_{p}^{(1)}\) (see \((^8)\)). In the further arguments we shall follow our note \((^9)\) (for more detail see \((^{10})\)). Using the embedding theorems \((^8)\), let us note that for any \(u(x,y)\in \overset{\circ}{W}{}_{p}^{(1)}\)
\[ \|u\|_{L_q}\leqslant \|u\|_{\overset{\circ}{W}{}_{p}^{(1)}}\qquad (q\geqslant 1\ \text{arbitrary}), \tag{4} \]
where
\[ \|u\|_{L_q}=\left\{\iint_{\Omega}|u|^q\,d\Omega\right\}^{1/q},\qquad \|u\|_{\overset{\circ}{W}{}_{p}^{(1)}}=\left(\iint_{\Omega}(u_x^2+u_y^2)^{p/2}\,d\Omega\right)^{1/p}, \tag{5} \]
and, moreover, for \(p>2\)
\[ \|u\|_{C}\leqslant C\|u\|_{\overset{\circ}{W}{}_{p}^{(1)}}. \tag{6} \]
We at once obtain that \(I(u)\geqslant \mathrm{const}\) on \(\overset{\circ}{W}{}_{p}^{(1)}\), i.e. it is bounded below. Therefore there exists a lower bound \(d\) of \(I(u)\) on \(\overset{\circ}{W}{}_{p}^{(1)}\) and a sequence of functions \(\{F^n\}\in \overset{\circ}{W}{}_{p}^{(1)}\) (called a minimizing sequence) such that
\[ I(F^n)\searrow d. \tag{7} \]
To prove the Dirichlet principle for the given problem means to prove that this sequence converges to some function \(F^*(x,y)\in \overset{\circ}{W}{}_{p}^{(1)}\), and that \(I(F^*)=d\). Our functional \(I(F)\) has the important property of uniform convexity, which consists in the following: if \(I(u)<d+\varepsilon\), \(I(v)<d+\varepsilon\), and \(I\!\left(\frac{u+v}{2}\right)>d\), then \(I(u-v)\to 0\) as \(\varepsilon\to 0\). This property follows from Clarkson’s first inequality \((^8)\)
\[ \left\|\frac{f+g}{2}\right\|_{L_p}^{p} + \left\|\frac{f-g}{2}\right\|_{L_p}^{p} \leqslant \frac12\left(\|f\|_{L_p}^{p}+\|g\|_{L_p}^{p}\right) \qquad (p\geqslant 2). \tag{8} \]
If the convexity criterion is applied to the terms of the minimizing sequence, then we obtain that \(I(F^m-F^n)\to 0\) as \(m,n\to\infty\).
Since the sequence \(\{F^m\}\) is bounded in \(\overset{0}{W}{}^{(1)}_p\), one can choose from it a subsequence such that \(\{F^m\}\) will converge in \(L_q\) (\(q\) arbitrary). Then the term \(\displaystyle \int_\Omega \omega(F^m-F^n)\,d\Omega \to 0\), and we obtain that
\[ \|F^m-F^n\|_{\overset{0}{W}{}^{(1)}_p}\to 0, \]
i.e., the sequence is fundamental in \(\overset{0}{W}{}^{(1)}_p\). By completeness of \(\overset{0}{W}{}^{(1)}_p\) (\(^{8}\))
\[ F^n(x,y)\to F^*(x,y)\in \overset{0}{W}{}^{(1)}_p, \]
with \(I(F^*)=d\). Let us add one more remark on the boundary conditions. For \(p=2\) the problem admits only a one-dimensional boundary, since convergence in this case is weaker than uniform convergence (for details see (\(^{8}\)); whereas in the case \(p>2\) the boundary may consist of a one-dimensional part \(\Gamma_1\) and a zero-dimensional part \(\Gamma_0\) (i.e., individual points). The function \(F^*\) will be discontinuous in such a domain \(\Omega\) (for example, in a disk with the center removed). Of course, the stresses, which are derivatives of the function \(F^*(x,y)\), will have infinite values at these removed points. However, one can consider a sequence of annuli with decreasing inner radius. Then for \(p=2\) the solutions will converge to the solution of the problem on the full disk (without the cut-out), while for \(p>2\) the solution of the limiting problem will differ from the solution of the problem without the cut-out. Thus a linear membrane “breaks” under point fastening, whereas a nonlinear one (for \(p>2\)) withstands it.
- Let us now consider the problem of the minimum of the integral (*). Since \(\Pi(u)\geq 0\), there exists a minimizing sequence \(u^n\in \overset{\varphi}{W}{}^{(1)}_\beta\) of functions for which
\[ \Pi(u^n)\searrow d. \]
From the properties of uniform convexity of \(\Pi(u)\), which for \(\beta\leq 2\) follows from the second Clarkson inequality (\(^{8}\)), analogously to item 3 we obtain
\[ \Pi(u^n-u^m)\to 0\quad \text{as } m,n\to\infty. \]
Unlike the preceding considerations, it is impossible from this to draw directly a conclusion about the convergence of \(\{u^n\}\) in the norm \(W^{(1)}_\beta\), since the functional \(\Pi(v)\) contains powers not of the derivatives \(v_{i,k}\) themselves, but of their linear combinations \(\varepsilon_{ik}=\frac12(v_{i,k}+v_{k,i})\). An analogous situation also occurs in the linear case, where it is overcome by proving Korn’s inequality (\(^{5,7}\))
\[ \Pi(v)\geq \|v\|_{\overset{0}{W}{}^{(1)}_\beta} \tag{9} \]
(of course, for the case \(\beta=2\)). We shall prove inequality (9) for all \(\beta>1\).
It is easy to show that from \(\Pi(u)\to 0\) it follows that
\[ \int_\Omega \left[\sum_{i,k=1}^{3}\varepsilon_{ik}^{2}\right]^{\beta/2}\,d\Omega \to 0; \]
thus, in order to prove (9), it is sufficient to show that
\[ \|v\|_{\overset{0}{W}{}^{(1)}_\beta} \leq C\int_\Omega \left[\sum \varepsilon_{ik}^{2}\right]^{\beta/2}\,d\Omega \tag{10} \]
(where \(\displaystyle \|v\|_{\overset{0}{W}{}^{(1)}_\beta}=\sum_{i=1}^{3}\|v_i\|_{\overset{0}{W}{}^{(1)}_\beta}\)).
Thus, we have to estimate the norms of \(v_i\) in \(\overset{0}{W}{}_{\beta}^{(1)}\) in terms of the norms of
\(\varepsilon_{ik}=v_{i,k}+v_{k,i}\) in \(L_\beta\). Consider, for example, \(v_1\). We have
\[ \varepsilon_{11}=v_{1,1}; \qquad \varepsilon_{12}=\tfrac12\left(v_{1,2}+v_{2,1}\right); \qquad \varepsilon_{13}=\tfrac12\left(v_{1,3}+v_{3,1}\right), \]
and so on. Hence it is easy to obtain that \(v_1\) satisfies the Poisson equation
\[ \Delta v_1=h,\qquad v_1|_{\gamma\Omega}=0 \]
with right-hand side belonging to \(W_{\beta}^{(-1)}\), i.e., being a generalized function (the derivative of a function from \(L_\beta\)). Such a problem has a solution \((^{11})\), obtained from the work of Calderon and Zygmund \((^{12})\), for which the estimate
\[ \|v\|_{\overset{0}{W}{}_{\beta}^{(1)}}\leq C\left\|\sum_{i,k=1}^{3}|\varepsilon_{ik}|\right\|_{L_p}. \]
is valid. Hence follows the validity of (10), and therefore of Korn’s generalized inequality (9). After this we obtain the convergence of \(\{u^n\}\) in \(\overset{\varphi}{W}{}_{\beta}^{(1)}\), and the Dirichlet principle for (*) is proved analogously to § 3.
Apparently, the proposed method can also be applied to the case of other boundary-value problems of the nonlinear theory of elasticity.
Moscow State University
named after M. V. Lomonosov
Received
15 III 1962
REFERENCES
\(^{1}\) V. T. Koiter, General Theorems of the Theory of Elastic-Plastic Media, IL, 1961.
\(^{2}\) L. M. Kachanov, Mechanics of Plastic Media, 1948.
\(^{3}\) R. Courant, D. Hilbert, Methods of Mathematical Physics, 2, 1945.
\(^{4}\) S. L. Sobolev, Mat. sborn., 2, No. 3 (1937).
\(^{5}\) K. O. Friedrichs, Ann. of Math., 48, No. 2 (1947).
\(^{6}\) D. M. Eidus, DAN, 76, No. 2 (1951).
\(^{7}\) S. G. Mikhlin, The Minimum Problem of a Quadratic Functional, 1952.
\(^{8}\) S. L. Sobolev, Certain Applications of Functional Analysis in Mathematical Physics, L., 1950.
\(^{9}\) A. L. Krylov, DAN, 115, No. 1 (1957).
\(^{10}\) A. L. Krylov, Dissertation, Moscow State University, 1958.
\(^{11}\) A. L. Krylov, DAN, 119, No. 5 (1958).
\(^{12}\) A. P. Calderon, A. Zygmund, Acta Math., 88, 1–2, 85 (1952).
\(^{13}\) A. Langenbach, Vestn. LGU, ser. mathematics and mechanics, No. 1, issue 1 (1961).
\(^{14}\) A. Langenbach, DAN, 121, No. 2 (1958).