Abstract
Full Text
MATHEMATICS
SUN HSI-SHEN
SOME QUESTIONS ON INFINITESIMAL BENDINGS OF SURFACES
(Presented by Academician I. N. Vekua, 28 V 1958)
In the present paper, which is a further development of the author’s work (¹), a number of questions are considered concerning the rigidity of a surface of positive curvature subject to bushing constraints of the form
[
\vec U \vec \nu = 0 \quad \text{on } L,
\tag{1}
]
where (\vec \nu) is the normal of the bushing; (L) is the aggregate of the contours of the holes of the surface; (\vec U) is the vector of an infinitesimal bending satisfying the equation
[
dU\,dr = 0,
\tag{2}
]
(\mathbf r) is the radius-vector of the surface.
If the bushings are nonrigid, then the bushing constraints have the form
[
\vec U \vec \nu = \gamma \quad \text{on } L.
\tag{3}
]
At each point of (L) we can consider two trihedra (\mathbf s, \mathbf m, \mathbf b) and (\mathbf s, \mathbf l, \mathbf n), where (\mathbf s) is the unit tangent to (L); (\mathbf m) and (\mathbf b) are the unit vectors of the principal normal and binormal of (L); (\mathbf l) is the tangential normal to (L); (\mathbf n) is the external normal of the surface. Let (\tau) be the angle between (\vec \nu) and (\mathbf b), which we assume to be constant along each contour of the surface; (\theta) is the angle between (\mathbf m) and (\mathbf n), measured from (\mathbf m) in the clockwise direction.
The paper also considers the question of rigidity for certain classes of closed nonconvex surfaces.
- Consider a regular surface of positive curvature. Let it be referred to a conjugate isometric system of coordinates. Then equation (2) can be written in the complex form (⁴):
[
\frac{\partial W}{\partial \bar z}+B\overline W=0
\quad
\left(
\frac{\partial}{\partial \bar z}
=
\frac{1}{2}
\left(
\frac{\partial}{\partial x}
+i\frac{\partial}{\partial y}
\right)
\right),
\tag{4}
]
where
[
W=(a\sqrt K)^{-1/2}Ur_{\bar z};\qquad
a=a_{11}a_{22}-a_{12}^{2};\qquad
K \text{ is the Gaussian curvature};
]
[
B=
\frac{1}{4}\left(\Gamma^{1}{22}-\Gamma^{1}}+2\Gamma^{2{12}\right)
+
\frac{i}{4}\left(\Gamma^{2}}-\Gamma^{2{11}-2\Gamma^{1}\right).
]
Equations (3) and (1) reduce to the form
[
\vec U \vec \nu
\equiv
\operatorname{Re}
\left[
\alpha \frac{\partial W}{\partial z}+\beta W
\right]
=
\gamma
\quad \text{on } L;
\tag{5}
]
[
\vec U \vec \nu
\equiv
\operatorname{Re}
\left[
\alpha \frac{\partial W}{\partial z}+\beta W
\right]
=
0
\quad \text{on } L,
\tag{5⁰}
]
where
[
\alpha=
\frac{\sin(\theta-\tau)}{\sqrt[4]{K}},
\qquad
\beta=
\frac{\sin(\theta-\tau)}{K\sqrt a}
\frac{\partial\sqrt{Ka\sqrt K}}{\partial z}
-
\sqrt{a\sqrt K}\cos(\theta-\tau)\frac{d\bar z}{dl}.
\tag{6}
]
We assume that (\sin(\theta-\tau)\ne 0) (on (L)), i.e., the collars are nowhere orthogonal to the surface along the contours. Obviously,
[
\varkappa=\frac{1}{2\pi}[\arg \overline{\alpha}]_L=0 .
\tag{7}
]
Let (l) and (l^) be the numbers of linearly independent solutions of the homogeneous problem (4), ((5^0)) and of the adjoint homogeneous problem for the equation (\partial_{\bar z}V-\overline{B}\,\overline{V}=0). In view of the work ((^2)) and equality (7), we have
[
l-l^*=3(1-m),
\tag{8}
]
where (m+1) is the connectivity of the surface.
Therefore, for a simply connected surface, under the homogeneous condition ((5^0)), equation (4) always has (l\ge 3) solutions, i.e., a surface with one contour under the collar constraint (1) always admits at least three bendings, including motions. (In the works ((^{1,3})) the author proved that in some cases all these bendings are trivial.) Hence the following holds:
Theorem 1. Let (S) be an ovaloid with one plane contour (L). If (L) is not a circle, then under the collar constraint (1), where (\tau\ne 0,\ \pi/2) and (\sin(\theta-\tau)\ne 0) (on (L)), the surface (S) will always admit at least three nontrivial bendings.
This theorem follows from the fact that in this case (S) admits not a single trivial motion.
Remark 1. If (\tau\ne \mathrm{const}) along (L), then (S) admits at least one nontrivial bending.
Let (\vec{\gamma}\equiv \mathbf n); in view of Remark 1, the following holds:
Theorem 2. (S) will be rigid under the collar constraint (\mathbf U\mathbf n=0) (on (L)) if and only if (L) is a circle and (\theta=\mathrm{const}) along (L).
In the case when (L) is a circle and (\theta=\mathrm{const}) along (L), under the additional condition
[
\operatorname{arc\,tg}\frac{R}{h}\le \frac{5\pi}{2}-2\theta
]
(S) will always be rigid ((R) is the radius of the circle (L); (h) is the distance from the point (P) (the point of intersection of the axis of rotation of the circle (L) with the surface) to the plane of the contour (L)).
The second part of this theorem follows directly from the work ((^1)), if one takes into account that in this case (\tau=\theta-\pi/2).
Consider an ovaloid (S) with (m+1) circular holes, on which nowhere orthogonal nonhomogeneous collar constraints (3) are imposed. In view of equality (8), the following will hold:
Theorem 3. If, under homogeneous collar constraints (1), (S) is rigid, then under nonhomogeneous collar constraints (3), (S) admits nontrivial infinitesimal bendings if and only if the function (\gamma) satisfies (l^) integral conditions, where: 1) (l^=3m-3), if (m>1); 2) (l^=2), if (m=1,\ \tau\equiv\pi/2) and the planes of the contours are parallel; 3) (l^=1), if (m=1,\ \tau\equiv\pi/2) and the planes of the contours are parallel, or (m=0,\ \tau=\pi/2).
In the case when (m=0,\ \tau\ne\pi/2) or (m=1) and the planes of the contours are not parallel, the ovaloid always admits nontrivial infinitesimal bendings under nonhomogeneous collar constraints (3) with any right-hand side.
2. Consider closed nonconvex surfaces of revolution. Let (S) be a closed surface of revolution consisting of two spherical segments (S_1, S_2) and a cylinder (T), whose meridians are expressed, respectively:
* In the work ((^2)), the formulation of Corollary 4 for the boundary-value problem (5) must read: (l-l^=\varkappa-\varkappa^=2\varkappa-3(m-1)). This was pointed out by I. I. Danilyuk.
[
\begin{aligned}
\rho_1(z)&=\sqrt{R_1^2-(z-z_1)^2}, && z_0 \leqslant z \leqslant R_1+z_1;\
\rho_2(z)&=\sqrt{R_2^2-(z+z_2)^2}, && -(R_2+z_2)\leqslant z\leqslant -z_0;\
\rho_0&=\sqrt{R_1^2-h_1^2}=\sqrt{R_2^2-h_2^2}, && -z_0\leqslant z\leqslant z_0\
&&(h_1=z_1-z_0,\quad h_2=z_2-z_0,\quad z_0,z_1,z_2\geqslant 0).
\end{aligned}
]
(S_1) and (T) are glued along the circle (L_1) with center (z=z_0) (on the (z)-axis) and radius (\rho_1(z_0)=\rho_0), and (T) and (S_2) are glued along the circle (L_2) with center (z=-z_0) (on the (z)-axis) and radius (\rho_2(-z_0)=\rho_0).
Theorem 4. (S(=S_1+T+S_2)) admits nontrivial bendings if and only if one of the following conditions is satisfied:
1) [
h_1=\frac{R_1}{k},\qquad h_2=\frac{R_2}{k}\quad (k=2,3,\ldots);
]
2) [
z_0=\frac{1}{k}\,\frac{R_1^2-h_1^2}{2}
\left(\frac{1}{kh_1-R_1}+\frac{1}{kh_2-R_2}\right)
\quad \text{for } \quad
k\geqslant \max\left(\left[\frac{R_1}{h_1}\right],\left[\frac{R_2}{h_2}\right]\right)+1
]
or
[
\left[\frac{R_1+R_2}{h_1+h_2}\right]\geqslant k\geqslant
\min\left(\left[\frac{R_1}{h_1}\right],\left[\frac{R_2}{h_2}\right]\right)+1;
]
3) [
z_0\equiv 0,\qquad \frac{h_1+h_2}{R_1+R_2}=\frac{1}{k}\quad (k=2,3,\ldots).
]
In the case when (\rho_1(z)\leqslant \rho_0) and (\rho_2(z)\leqslant \rho_0), obviously, (S) will always be rigid. If (S_1) and (S_2) are concave into the cylinder (T), then the closed surface obtained is likewise always rigid.
Let us consider one more class of closed surfaces of revolution ({S}); (S) consists of (S_1) and (S_2), whose meridians are expressed respectively by
(\rho_1(z)=c_1(a-z)^\alpha) ((0\leqslant z\leqslant a,\ 0<\alpha<1)),
(\rho_2(z)=c_2\sqrt{R^2-(z+h)^2}) ((-R-h\leqslant z\leqslant 0)).
(S_1) and (S_2) are glued along the circle (L) with center (z=0) (on the (z)-axis) and radius (\rho_1(0)=c_1a^\alpha), where (c_1a^\alpha=c_2\sqrt{R^2-h^2}).
Theorem 5. (S(=S_1+S_2)) admits nontrivial bendings if and only if
[
h=\frac{c_1^2}{c_2k^2}\,a^{2\alpha-1}
\left[
\frac{1}{2}
+\frac{1}{2}\sqrt{4\alpha(1-\alpha)(k^2-1)+1}
+(k^2-1)\alpha
\right]
+\frac{1}{k}.
]
In particular, for (\alpha=1/2) we shall have
[
h=\frac{c_1^2}{2c_2}\left(1+\frac{1}{k}\right)+\frac{1}{k}.
]
Hence, obviously, in the case when (h\leqslant \dfrac{c_1^2}{2c_2}) or (h>\dfrac{3c_1^2+2c_2}{4c_2}), (S) will always be rigid.
Remark 2. Theorems 4 and 5 show that, in the case of gluing, non-rigidity is a comparatively rare case.
In conclusion I express my gratitude to Acad. I. N. Vekua, under whose supervision this work was carried out.
Moscow State University
named after M. V. Lomonosov
Received
25 V 1958
CITED LITERATURE
- Sun He-shen, DAN, 116, No. 5 (1957).
- B. Boyarsky, DAN, 102, No. 2 (1955).
- Sun He-shen, DAN, 122, No. 5 (1958).
- I. N. Vekua, Czechoslovak Mathematical Journal, 6 (81), 143 (1956).