THEORY OF ELASTICITY

B. D. ANNIN

ELASTIC–RIGID–PLASTIC TORSION OF A CYLINDRICAL ROD OF OVAL CROSS-SECTION

(Presented by Academician Yu. N. Rabotnov, 25 X 1962)

1. Let the cross-section \(F\) of the rod be bounded by a strictly convex contour \(\Gamma\), having a tangent at each point, and let \(Oxyz\) be a right rectangular Cartesian coordinate system such that the point \(O \in F\), the axis \(Oz\) is directed parallel to the generator of the cylindrical surface, and the tangent to \(\Gamma\) at the point of intersection of \(\Gamma\) with the axis \(Ox\) is perpendicular to the axis \(Ox\). We denote by: \(\alpha > 0\) the relative angle of twist, occurring counterclockwise when viewed from the side of the positive direction of the axis \(Oz\); \(G\) the shear modulus; \(k\) the plasticity constant, \(k(2G\alpha)^{-1} = a\); \(\sqrt{\Omega \pi^{-1}} = b\), where \(\Omega\) is the area of \(F + \Gamma\). Suppose that there is an elastic core—an area \(D\) with boundary \(L\), lying entirely inside \(\Gamma\). Denote by \(B\) the region between \(\Gamma\) and \(L\); in this region the material is in a purely plastic state. The problem of elastic–rigid–plastic torsion (Problem I) is posed as follows \((^{1,2})\):

Problem I. Find a simply connected region \(D\) with boundary \(L\) and a function* \(\psi(x,y)\), defined and continuous in \(F + \Gamma\) and having discontinuous partial derivatives \(\psi_x\) and \(\psi_y\) in \(F + \Gamma\), if:

a) in \(D\)**

\[ \psi_{xx}+\psi_{yy}=-2G\alpha,\qquad \psi_x^2+\psi_y^2<k^2; \]

b) in \(L+B+\Gamma\)

\[ \psi_x^2+\psi_y^2=k^2; \tag{1.1} \]

c) on \(\Gamma\)

\[ \psi|_{\Gamma}=0. \tag{1.2} \]

1. Let \(R\) be an arbitrary point of \(\Gamma\), and let \(x_1Ry_1\) be the moving coordinate system formed by the tangent, directed in the direction of the positive traversal of \(\Gamma\), and the inward normal to \(\Gamma\) at the point \(R\). Let \(\beta\) be the angle between \(Rx_1\) and \(Ox\), \(\pi/2 \le \beta \le \pi/2\). The equation of the contour \(\Gamma\) can be represented in the form \(((^{3}),\) p. 184):

\[ x_{\Gamma}(\beta)=\frac{dM(\beta)}{d\beta}\cos\beta+M(\beta)\sin\beta,\qquad y_{\Gamma}(\beta)=\frac{dM(\beta)}{d\beta}\sin\beta-M(\beta)\cos\beta, \tag{2.1} \]

where \(M(\beta)>0\) is the support function of the contour \(\Gamma\).

Definition. We shall say that a curve \(L\), lying inside \(\Gamma\), has property \(E\) if:

* In fact it is necessary to find \(\psi_x \equiv \psi_x(x,y)=\partial\psi(x,y)/\partial x=-\tau_{yz}\), \(\psi_y=\tau_{xz}\) (\(\tau_{xz}, \tau_{yz}\) are shear stresses); consequently, \(\psi(x,y)\) is sought up to an additive constant.

** \(\psi_{xx}=\psi_{xx}(x,y)=\partial^2\psi(x,y)/\partial x^2\), etc.

1) \(L\) can be represented by the equation

\[ x_L(\beta)=x_\Gamma(\beta)-N(\beta)\sin\beta,\qquad y_L(\beta)=y_\Gamma(\beta)+N(\beta)\cos\beta, \tag{2,2} \]

where \(N(\beta)>0\) is a single-valued, periodic function with period \(2\pi\), continuous in \(\beta\) on \([\pi/2,5\pi/2]\);

2) as \(\beta\) increases, \(L\) is traversed in the positive direction;

3) for any two distinct values of the angle \(\beta\), \(\beta_1,\beta_2\in[\pi/2,5\pi/2]\), the line segments \(R_1Q_1\) and \(R_2Q_2\), where \(R_i\in\Gamma\) corresponds to \(\beta_i\) by (2,1), and \(Q_i\in L\) by (2,2), \(i=1,2\), have no common points*.

Theorem 1. There exists no more than one solution of Problem I for which \(L\) has property E.

Suppose that a solution of Problem I for which \(L\) has property E exists. We shall establish certain properties \((1^\circ—5^\circ)\) of this solution, from which Theorem 1 will follow.

Fig. 1

\(1^\circ\). In \(\Gamma+B+L\) the following relations hold, following from the solution by Cauchy’s method of the Cauchy problem for equation (1,1) under condition (1,2), taking (2,1) into account**:

\[ \psi_x=-k\sin\beta,\qquad \psi_y=k\cos\beta; \tag{2,3} \]

\[ -\psi_x y+\psi_y x=k\frac{dM(\beta)}{d\beta}; \tag{2,4} \]

\[ \psi-x\psi_x-y\psi_y=kM(\beta). \tag{2,5} \]

\(2^\circ\). Everywhere in \(D\),

\[ \psi_{xx}\psi_{yy}-\psi_{xy}^{2}\ne 0. \]

Proof. Let at the point \((x_0,y_0)\in D\)

\[ \psi_{xx}\psi_{yy}-\psi_{xy}^{2}=0. \]

Assume further that \(\psi_{xy}(x_0,y_0)=\psi_{xx}(x_0,y_0)=0\). Then ((\(^{5}\), pp. 428—430)) the harmonic function \(\psi_x(x,y)\) takes on \(L\) the value \(\psi_x(x_0,y_0)\) at least at four distinct points, which contradicts (2,3).

\(3^\circ\). The formulas \(\xi=-\psi_x(x,y)/k,\ \eta=-\psi_y(x,y)/k\) realize a homeomorphic mapping of \(D+L\) onto the disk \(K+C:\ \xi^2+\eta^2=1\) of the plane \((\xi,\eta)\). The validity of this assertion follows from known theorems ((\(^{6}\), p. 26; (\(^{7}\), p. 586)).

\(4^\circ\). Everywhere in \(D\),

\[ \psi_{xx}\psi_{yy}-\psi_{xy}^{2}>0. \]

* Obviously, \(L\) is a simple chord-arc curve such that to each value of \(\beta\) there corresponds by (2,2) one and only one point \(Q\in L\); \(N(\beta)\) is the length of this segment in the coordinate system \(x_1Ry_1\), where \(R\) is the point \(\Gamma\) corresponding to the given \(\beta\).

** Taking into account the results of work (\(^{4}\)) and using the equalities (2,3), (2,4), one can show that Problem I, for which \(L\) has property E, can be reduced to the following integral equation for the function \(\gamma(t)\), \(0\le t\le 2\pi\):

\[ \frac{k}{4\pi Ga}\int_0^{2\pi}\cos[\gamma(t)-\gamma(\tau)]\operatorname{ctg}\frac{t-\tau}{2}\,d\tau = f\left(\gamma(t)+\frac{\pi}{2}\right), \]

where \(f(\beta)=k\,dM(\beta)/d\beta,\ \gamma(t)=\beta-\pi/2\).

Proof. Suppose the contrary. Then the continuous vector field \(\mathbf v=(\psi_x,\psi_y)\) in \(D+L\) has in \(D\) a unique singular point (a saddle) and at the same time the index of \(L\) with respect to \(\mathbf v\) is \(+1\) \((^8)\).

\(5^\circ\). Denote
\[ w=-k^{-1}(x\psi_x+y\psi_y-\psi)-\frac{a}{2}k^{-2}(\psi_x^2+\psi_y^2)+\frac{a}{2}. \]
The function\(^*\) \(w=w(\xi,\eta)\) is defined and continuous in \(K+C\); by virtue of (2,5), on the boundary of the circle \(C:\ \xi^2+\eta^2=1\),
\[ w(\xi,\eta)\big|_C=M\left(\theta+\frac{\pi}{2}\right), \tag{2,6} \]
where \(\theta\) is the polar angle in the \((\xi,\eta)\)-plane; it satisfies in \(K+C\) the equation
\[ w_{\xi\xi}w_{\eta\eta}-w_{\xi\eta}^2=a^2 \tag{2,7} \]
and the inequalities
\[ w_{\xi\xi}>0,\qquad w_\eta>0; \tag{2,8} \]
moreover, in \(K\) the formulas \((^9)\) hold:
\[ w_\xi+a\xi=x,\qquad w_\eta+a\eta=y. \tag{2,9} \]

Corollary 1. A solution of problem I for which \(L\) has property E does not exist if
\[ a<kG^{-1}b^{-1}. \]

3. Theorem 2. Let the oval \(\Gamma\) be symmetric and elongated along the axis \(Ox\), have only four vertices, and let its radius of curvature \(\rho(\beta)\) be an analytic function of \(\beta\) on \([\pi/2,5\pi/2]\); then for
\[ a>kG^{-1}\rho_{\min}^{-1}, \tag{3,1} \]
where \(\rho_{\min}\) is the minimum radius of curvature, there exists a solution of problem I such that the curve \(L\) is symmetric with respect to the axis \(Ox\), the domain \(D\) contains the line \(l\) of discontinuity of tangential stresses in purely plastic torsion*, \(L\) has property E; whereas if \(a<k\cdot 2^{-1}G^{-1}\rho_{\min}^{-1}\), then there is no solution of problem I for which \(L\) has property E.

The proof of Theorem 2 is based on the existence \((^5,\ p. 132)\) of a function \(w(\xi,\eta)\), analytic in \(K+C\), satisfying in \(K+C\) equation (2,7) and inequalities (2,8), and on \(C\) condition (2,6). By formulas (2,9) a homeomorphic mapping of \(K+C\) onto some domain \(D\) with boundary \(L\) of the \((x,y)\)-plane is effected. The equation of \(L\)—the image of \(C\)—can be represented in the form (2,2), where
\[ N(\beta)=M(\beta)-w_r(1,\beta-\pi/2)-a, \]
where
\[ w_r(1,\theta)\equiv \partial w(r,\theta)/\partial r\big|_{r=1},\qquad r=\sqrt{\xi^2+\eta^2},\qquad \beta=\theta+\pi/2. \]
From the a priori estimate of \(w_r(1,\theta)\) and (3,1) it follows that \(L\) lies inside \(\Gamma\).

Corollary 2. Consider the position of the boundary \(L\) for different angles \(\alpha_1,\alpha_2\) \((\alpha_1<\alpha_2)\). Let \(N_1(\beta)\) correspond to \(\alpha_1\), and \(N_2(\beta)\) to \(\alpha_2\); then
\[ N_1(\beta)+kG^{-1}(\alpha_1^{-1}-\alpha_2^{-1}) \ge N_2(\beta)\ge N_1(\beta)+k\cdot2^{-1}G^{-1}(\alpha_1^{-1}-\alpha_2^{-1}). \]

Remark. Theorem 2 and Corollary 2 are also valid for a convex contour \(\Gamma_n\) close to a regular \(n\)-gon****
\[ x=R\,[f\cos t+(n-1)^{-1}f^{1-n}\cos(n-1)t], \]
\[ y=R\,[f\sin t+(n-1)^{-1}f^{1-n}\sin(n-1)t], \]

\[ \overline{\phantom{xxxxxxxx}} \]

\(^*\) If \(w(\xi,\eta)\) is a quadratic polynomial, we obtain the case first considered in work \((^{10})\).

\(^ {**}\) Such contours include, for example, Lamé curves
\[ (xm^{-1})^{2p}+(yn^{-1})^{2p}=1,\quad p=1,2,\ldots, \]
for which
\[ M(\beta)=\left[(m\sin\beta)^{2p/(2p-1)}+(n\cos\beta)^{2p/(2p-1)}\right]^{(2p-1)/2p}. \]

\(^ {***}\) The line \(l\) is the segment of the axis \(Ox\) joining the centers of curvature of the two points of \(\Gamma\) lying on the axis \(Ox\).

\(^ {****}\) See the graph of \(\Gamma_n\) for \(n=3\) and \(f=1.3\) in \((^{11})\), p. 179.

where \(f = \sqrt[n]{n-1} + \delta,\ \delta > 0,\ 0 \leq t \leq 2\pi\). In this case the line \(l\) consists of \(n\) segments drawn from the point of intersection of all axes of symmetry of \(\Gamma_n\) to the centers of curvature of the points of \(\Gamma_n\) corresponding to \(t = \frac{2\pi}{n} i,\ i = 0, 1, \ldots, (n-1)\), while the curve \(L\) is symmetric with respect to the same axes as \(\Gamma_n\).

In conclusion, the author expresses his deep gratitude to Academician Yu. N. Rabotnov for his attention to the present work.

Institute of Hydrodynamics
Siberian Branch of the Academy of Sciences of the USSR

Received
23 X 1962

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