Mathematics

M. G. Ilyich-Daidovich

INFINITESIMAL BENDINGS OF ONE CLASS OF RIBBED CYLINDROIDS

(Presented by Academician P. S. Aleksandrov, 7 III 1964)

1. We shall agree to call a surface \(C\) a ribbed cylindroid if it consists of \(n\) regular parts of zero Gaussian curvature and if: 1) \(C\) is homeomorphic to a cylindrical belt; 2) the boundary of \(C\) consists of two closed piecewise-smooth curves \(L_1\) and \(L_2\), lying in parallel planes and containing no rectilinear segments; 3) the lines of gluing \(\sigma_i\) \((i=1,2,\ldots,n)\) of the regular parts, the ribs of the surface \(C\), are at the same time generators of this surface.

We shall call a ribbed cylindroid \(C_A\) a ribbed cylindroid of type \(A\) if its tangent planes form, along the ribs \(\sigma_i\), angles \(0<\theta_i<\pi\). Thus the surface \(C_A\) is nonregular, developable in the sense of Lebesgue and, on the whole, nonconvex.

We shall consider infinitesimal bendings of the surface \(C_A\) under which its ribs move as a rigid body; such infinitesimal bendings of the surface \(C_A\) will be called admissible.

For the surface \(C_A\) the following holds.

Theorem. Under an admissible infinitesimal bending of the ribbed cylindroid \(C_A\), on its boundaries \(L_1\) and \(L_2\) there are pairs of points, arbitrarily close to one another, the distances between which increase.

In the proof of this theorem we shall use the lemma (see below) concerning infinitesimal bendings of plane curves.

2. Let a plane developable curve \(L_A\), homeomorphic to a circle, be composed of a finite number of smooth convex arcs \(l_1,l_2,\ldots,l_n\) (the curve \(L\) as a whole being nonconvex), containing no rectilinear segments and having distinct tangents at their common points. For an infinitesimal bending of any arc \(l_i\) \((i=1,2,\ldots,n)\) of such a curve \(L_A\), a theorem of N. V. Efimov holds \((^1)\), and for the curves \(L_A\) defined above the following holds.

Lemma. If, under an infinitesimal bending of the curve \(L_A\), the variation of any of its smooth arcs is equal to zero, and the curvature does not decrease on any of the arcs \(l_i\), then the velocity field of this bending is composed of a trivial summand and a field orthogonal to the plane of the curve \(L_A\).

Proof. Without loss of generality of the conclusions, we shall consider the curve \(L_A\) composed of two arcs \(l_1\) and \(l_2\). The vector equation of the curve \(L_A\) is

\[ \mathbf{x}(s)= \begin{cases} {}^{1}\mathbf{x}, & s_0 \leq s \leq s_1,\\ {}^{2}\mathbf{x}, & s_1 \leq s \leq s_2;\quad s_2 \equiv s_0 \end{cases} \tag{1} \]

(the initial point of the vector \(\mathbf{x}(s)\) is in the plane of the curve \(L_A\); \(s\) is the arc length of the curve \(L_A\)); in this case the end of the vector \(\mathbf{x} = {}^{i}\mathbf{x}(s)\) describes the corresponding arc \(l_i\), and the regular functions \({}^{i}\mathbf{x}(s)\) admit a regular continuation.

Obviously,

\[ {}^{1}\mathbf{x}(s_0) = {}^{2}\mathbf{x}(s_0), \qquad {}^{1}\mathbf{x}(s_1) = {}^{2}\mathbf{x}(s_1), \]

\[ \bigl[{}^{1}\mathbf{x}'(s_0^{+}),\,{}^{2}\mathbf{x}'(s_2^{-})\bigr]\ne 0, \qquad \bigl[{}^{1}\mathbf{x}'(s_1^{-}),\,{}^{2}\mathbf{x}'(s_1^{+})\bigr]\ne 0. \]

If the twice continuously differentiable function \(\mathbf{z}(s)\) is the velocity field of an infinitesimal bending of the curve \(L_A\), then the deformed curve \(L_{A\varepsilon}\) is determined by the vector equation

\[ \mathbf{x} = {}^{i}\mathbf{x}(s) + \varepsilon \mathbf{z}(s) \qquad (s_{i-1}\le s\le s_i;\ i=1,2) \tag{2} \]

(\(\varepsilon\) is the bending parameter, \(\mathbf{z}(s)\) is a periodic function with period equal to the length of the curve \(L_A\)).

By the condition of the lemma,

\[ d\mathbf{x}\,d\mathbf{z}=0, \tag{3} \]

\[ \mathbf{x}''\mathbf{z}''\ge 0; \tag{4} \]

from (3) it follows that

\[ d\mathbf{z}=[\mathbf{y}\,d\mathbf{x}], \]

so that in the case of a trivial infinitesimal bending,

\[ \mathbf{z}=\mathbf{a}+[\mathbf{c}\mathbf{x}(s)] \]

(\(\mathbf{a},\mathbf{c}\) are constant vectors), while in the case of a nontrivial infinitesimal bending,

\[ \mathbf{z}(s)=\mathbf{a}+[\mathbf{c}\mathbf{x}(s)]+\mathbf{z}^{*}(s). \tag{5} \]

The nontrivial term \(\mathbf{z}^{*}(s)\) satisfies conditions (3) and (4):

\[ \mathbf{x}'\mathbf{z}^{*\,\prime}=0,\qquad \mathbf{x}''\mathbf{z}^{*\,\prime\prime}\ge 0, \]

which we shall write in the form

\[ \mathbf{t}\mathbf{z}^{*\,\prime}=0,\qquad \mathbf{n}\mathbf{z}^{*\,\prime\prime}\ge 0, \tag{6} \]

where, if by \({}^{i}\mathbf{t}\) we denote the unit tangent vector to the arc \(l_i\), then for \(s=s_i\) we shall simultaneously have

\[ {}^{1}\mathbf{t}(s_i)\cdot \mathbf{z}^{*\,\prime}(s_i)=0,\qquad {}^{2}\mathbf{t}(s_i)\cdot \mathbf{z}^{*\,\prime}(s_i)=0 \qquad (i=1,2). \]

Since, by the condition of the lemma, \([{}^{1}\mathbf{t}(s_i),{}^{2}\mathbf{t}(s_i)]\ne 0\), it follows that at the point \(s_i\) \((i=1,2)\) of the curve \(L_A\) the vector \(\mathbf{z}^{*\,\prime}(s)\) is orthogonal to the plane of the curve.

Since

\[ \mathbf{n}\mathbf{z}^{*\,\prime\prime} =(\mathbf{n}\mathbf{z}^{*\,\prime})' -\mathbf{n}'\mathbf{z}^{*\,\prime} =(\mathbf{n}\mathbf{z}^{*\,\prime})' +k\mathbf{t}\mathbf{z}^{*\,\prime} =(\mathbf{n}\mathbf{z}^{*\,\prime})', \]

the second of relations (6) can be written in the form

\[ d(\mathbf{n}\mathbf{z}^{*\,\prime})\ge 0; \]

integrating this total differential along the curve \(l_i\), we obtain

\[ \mathbf{n}\mathbf{z}^{*\,\prime}=0, \]

whence we conclude that also at the interior points of the arcs \(l_i\) the vector \(\mathbf{z}^{*\,\prime}\) is orthogonal to the plane of the curve \(L_A\), i.e.

\[ \mathbf{z}^{*\,\prime}=g(s)\mathbf{b}. \]

(\(\mathbf b\) is a constant unit vector of the binormal, \(g(s)\) is a continuous periodic function). Integrating the last equation, we obtain (5) in the form

\[ \mathbf z(s)=\mathbf a+[\mathbf c\,\mathbf x(s)]+h(s)\mathbf b; \]

this proves the lemma.

1. We shall prove the theorem formulated above for the case \(n=2\), i.e., for the ribbed cylindroid \(C_A\) with two edges \(\sigma_1\) and \(\sigma_2\).

Let

\[ \mathbf x(s,v)= \begin{cases} {}^{1}\mathbf x_1(s)+v\mathbf a(s), & s_0\leq s\leq s_1,\\[2mm] {}^{2}\mathbf x_1(s)+v\mathbf a(s), & s_1\leq s\leq s_2, \end{cases} \qquad (0\leq v\leq 1;\ s_2\equiv s_0) \tag{7} \]

be the equation of the surface \(C_A\) (where the origin of the vector \(\mathbf x(s,v)\) is in the plane of the curve \(L_1\)); the regular functions \({}^{i}\mathbf x_1(s)\) admit regular continuations, and \(\mathbf a(s)\) is a periodic function with period equal to the length of the curve \(L_1\). Obviously,

\[ {}^{1}\mathbf x_1(s_i)={}^{2}\mathbf x_1(s_i), \tag{8} \]

and, by the condition of the theorem,
\[ [{}^{1}\mathbf x'_1(s_1^-),{}^{2}\mathbf x'_1(s_1^+)]\ne0,\quad [{}^{2}\mathbf x'_1(s_2),{}^{1}\mathbf x'_1(s_0^+)]\ne0; \]
moreover
\[ \mathbf a'(s)=\lambda(s)\,{}^{i}\mathbf x'_1(s), \]
where the continuous periodic scalar function \(\lambda(s)\) \((1+\lambda(s)>0)\) is such that, for \(s=s_i\),
\[ \mathbf a'(s_i^-)=\lambda(s_i)\,{}^{i}\mathbf x'_1(s_i^-),\quad \mathbf a'(s_i^+)=\lambda(s_i)\,{}^{i}\mathbf x'_1(s_i^+). \]

Let \(\mathbf z=\mathbf z(s,v)\), \(\mathbf y=\mathbf y(s,v)\) be, respectively, the velocity field and the rotation field of an infinitesimal bending of the surface \(C_A\). As is known \((^2)\), on each regular part of the surface \(C_A\), \(\mathbf z(s,v)\) satisfies the equation

\[ d\mathbf x\,d\mathbf z=0, \tag{9} \]

whence follows the condition

\[ d\mathbf z=[{}^{i}\mathbf y,d{}^{i}\mathbf x]\qquad (i=1,2). \tag{10} \]

The vectors \({}^{i}\mathbf y_s,{}^{i}\mathbf y_v\) lie in the tangent plane of the surface \(C_A\); in the relations

\[ \begin{aligned} {}^{i}\mathbf y_s&={}^{i}\alpha\,{}^{i}\mathbf x_s-{}^{i}\beta\,{}^{i}\mathbf x_v,\\ {}^{i}\mathbf y_v&={}^{i}\gamma\,{}^{i}\mathbf x_s-{}^{i}\alpha\,{}^{i}\mathbf x_v \end{aligned} \qquad (i=1,2) \tag{11} \]

the scalar differentiable functions \({}^{i}\alpha(s,v),{}^{i}\beta(s,v),{}^{i}\gamma(s,v)\) are determined from the Gauss–Codazzi equations. In the case under consideration, the latter lead to two systems of equations

\[ \begin{aligned} {}^{i}\alpha_v&=-\frac{2\lambda(s)}{1+v\lambda(s)}\,{}^{i}\alpha,\\ {}^{i}\alpha_s-{}^{i}\beta_v&=0, \end{aligned} \qquad (i=1,2) \tag{12} \]

whose integration leads to the solutions

\[ {}^{i}\mathbf y(s,v)= \int_{s_{i-1}}^{s} \left\{A_i(s)\,{}^{i}\mathbf x'_1(s)-B'_i(s)\mathbf a(s)\right\}\,ds -\frac{vA_i(s)}{1+v\lambda(s)}\,\mathbf a(s)+\mathbf c_i, \tag{13} \]

where \(A_i(s)\) and \(B_i(s)\) are arbitrary scalar functions, and \(\mathbf c_i\) are constant vectors.

The most general form of the function \(\mathbf y(s,v)\) corresponds to infinitesimal bendings under which all chords of the curves \(L_1\) and \(L_2\) decrease or remain stationary; in this case, as follows from the lemma proved above, on the boundaries \(L_1\) and \(L_2\) of the ribbed cylindroid \(C_A\) the nontrivial component of the velocity field of the infinitesimal bending is orthogonal to the planes of the curves \(L_1\) and \(L_2\). Hence we obtain the boundary conditions for the field \(\mathbf y(s,v)\)

\[ \overset{i}{\beta}(s,0)=\overset{i}{\beta}(s,1)=0, \tag{14} \]

on the basis of which the functions \(A_i(s)\) and \(B_i(s)\) in (7) are determined:

\[ A_i(s)=c(1+\lambda(s)),\qquad B_i(s)=0; \]

the fields of rotations of the regular parts

\[ \overset{i}{\mathbf y}(s,v) = c\left\{ \overset{i}{\mathbf x}_1(s)+ \frac{1-v}{1+v\lambda(s)}\,\mathbf a(s) \right\} +\mathbf c_i \qquad (i=1,2) \tag{15} \]

are nontrivial for all \(c\ne 0\).

On the rib \(\sigma_i\), for example on \(\sigma_0\) \((s=s_0=s_2)\), we have

\[ \overset{1}{\mathbf y}(s_0,v) = c\left\{ \overset{1}{\mathbf x}_1(s_0)+ \frac{1-v}{1+v\lambda(s)}\,\mathbf a(s_0) \right\} +\mathbf c_1, \]

\[ \overset{2}{\mathbf y}(s_0,v) = c\left\{ \overset{2}{\mathbf x}_1(s_0)+ \frac{1-v}{1+v\lambda(s)}\,\mathbf a(s_0) \right\} +\mathbf c_2. \tag{16} \]

The velocity field \(\mathbf z(s,v)\) is single-valued on the whole surface \(C_A\) if along \(\sigma_i\)

\[ [\overset{2}{\mathbf y}\,d\overset{2}{\mathbf x}] = [\overset{1}{\mathbf y}\,d\overset{1}{\mathbf x}], \]

where \(\overset{i}{\mathbf x}\) is the vector \(\mathbf x(s,v)\) describing the corresponding regular part of the surface \(C_A\). Since along \(\sigma_i\)

\[ d\overset{2}{\mathbf x}=d\overset{1}{\mathbf x}=\overset{i}{\mathbf x}_v\,dv=\mathbf a(s_i)\,dv, \]

the last relation may be written in the form

\[ [\overset{2}{\mathbf y}-\overset{1}{\mathbf y},\,\mathbf a(s)]=0 \qquad \text{(along \(\sigma_i\))} \]

or

\[ [\mathbf y,\mathbf a(s)]=0 \qquad \text{(along \(\sigma_i\))} \tag{17} \]

\((\mathbf y=\overset{2}{\mathbf y}-\overset{1}{\mathbf y};\) thus the vector \(\mathbf y\) along \(\sigma_i\) is collinear with the vector \(\mathbf a(s_i)\)). On the other hand, taking (1) into account, from (16) we derive:

\[ \mathbf y=\overset{2}{\mathbf y}-\overset{1}{\mathbf y}=\mathbf c_2-\mathbf c_1=\mathbf C; \tag{18} \]

in the case where the vectors \(\mathbf a(s_i)\) are not collinear, it follows from (17) and (18) that along \(\sigma_i\), \(\mathbf y(s,v)=0\).

The nontrivial, for \(c\ne 0\), field of rotations

\[ \mathbf y(s,v)= \begin{cases} c\left\{\overset{i}{\mathbf x}_1(s)+\dfrac{1-v}{1+v\lambda(s)}\,\mathbf a(s)\right\}+\mathbf C, & s\ne s_i,\\[6pt] \mathbf C, & s=s_i \end{cases} \qquad (\mathbf C=\mathrm{const}\ge 0). \]

ensures the integrability of the equation \(d\mathbf z=[\mathbf y\,d\mathbf x]\).

Applying directly the arguments of N. V. Efimov \((^2)\), we derive that under the boundary conditions (14) the field \(\mathbf y\) is single-valued on the surface \(C_A\) only in the case when \(c=0\) and \(\mathbf y=\mathbf C\;(\ge 0)\). The theorem is proved.

1. As is known, the fields \(\mathbf z\) and \(\mathbf y\) are invariant under projective transformations, so that the theorem proved is also valid for ribbed cylindroids with nonparallel bases that are projectively transformable into ribbed cylindroids with parallel bases (the boundary conditions being preserved).

Received
29 II 1964

REFERENCES

\(^1\) N. V. Efimov, Matem. sborn., 20, 1 (1947). \(^2\) N. V. Efimov, UMN, 3, 2 (1948).