THEORY OF ELASTICITY

S. M. BELONOSOV

THE PLANE PROBLEM OF THE THEORY OF ELASTICITY FOR A WEDGE WITH GIVEN STRESSES OR DISPLACEMENTS ON THE BOUNDARY

(Presented by Academician S. L. Sobolev, 19 XI 1959)

We consider the problem of finding two functions \(\varphi(z)\) and \(\psi(z)\), regular inside the wedge \(-\alpha\pi/2 < \arg z < \alpha\pi/2\), and satisfying the boundary condition

\[ \varkappa \varphi(z_1)-z_1\overline{\varphi'(z_1)}-\overline{\psi(z_1)}=-f(z_1). \tag{1} \]

Here \(\arg z_1=\pm \alpha\pi/2,\quad 0<|z_1|<\infty,\quad 0<\alpha<2;\ f(z_1)\) is a given function.

The constant \(\varkappa=-1\) in the case of the first fundamental problem and \(\varkappa>1\) for the second fundamental problem of the theory of elasticity (in the terminology of N. I. Muskhelishvili \((^1)\)).

Map the wedge onto the right half-plane \(\operatorname{Re}s>0\) by the function \(z=s^\alpha\), and denote

\[ \Phi(s)=\varphi(s^\alpha),\qquad \Psi(s)=\psi(s^\alpha)+s\,\frac{e^{-i\pi\alpha}}{\alpha}\,\Phi'(s), \]

\[ m=\frac{\sin \pi\alpha}{\pi\alpha},\qquad s=\sigma+i\tau,\qquad f_1(\tau)+i f_2(\tau)=f(\tau^\alpha e^{i\pi\alpha/2}). \tag{2} \]

The functions \(\Phi(s)\) and \(\Psi(s)\), regular in the right half-plane, satisfy on its boundary the condition

\[ \varkappa\Phi(i\tau)-\delta(\tau)\,2\pi m\tau\overline{\Phi'(i\tau)}-\overline{\Psi(i\tau)} =-f_1(\tau)-i f_2(\tau), \tag{3} \]

where \(\delta(\tau)=1\) for \(\tau>0\); \(\delta(\tau)=0\) for \(\tau<0\).

Assume \(\Phi(\infty)=0\). By Harnack’s theorem, equality (3) is equivalent to two functional equations

\[ \varkappa\Phi(s)+m\int_0^\infty \frac{\tau\,\overline{\Phi'(i\tau)}}{i\tau-s}\,d\tau = \frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{f_1(\tau)+i f_2(\tau)}{i\tau-s}\,d\tau \equiv A(s); \tag{4} \]

\[ \Psi(s)=\frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{f_1(\tau)-i f_2(\tau)}{s-i\tau}\,d\tau - m\int_0^\infty \frac{\tau\Phi'(i\tau)}{s-i\tau}\,d\tau. \tag{5} \]

Thus, the problem reduces to solving the functional equation (4) for the function \(\Phi(s)\), after which \(\Psi(s)\) is computed by formula (5).

Applying to equation (4) the one-sided Laplace transform

\[ \Phi(s)=\int_0^\infty u(x)e^{-sx}\,dx,\qquad A(s)=\int_0^\infty F(x)e^{-sx}\,dx, \tag{6} \]

after known transformations \((^2)\) we obtain the integral equation for \(u(x)\):

\[ \varkappa u(x)-m\int_0^\infty \frac{y\,u(y)}{(x+y)^2}\,dy=F(x). \tag{7} \]

The kernel of equation (7) is homogeneous of degree \(-1\); therefore the solution is obtained in closed form by means of the Riemann–Mellin integral transform.

\[ \varkappa u(x)=F(x)+\int_0^\infty tF(tx)M(t)\,dt+\int_0^\infty t\overline{F(tx)}N(t)\,dt . \tag{8} \]

Here

\[ M(t)=\frac{m^2}{\pi^2\varkappa^2 t}\int_0^\infty \frac{\cos\left(\frac{x}{\pi}\ln t\right)} {\left(\frac{\operatorname{sh}x}{x}\right)^2-\frac{m^2}{x^2}}\,dx, \]

\[ N(t)=\frac{m}{\pi^2\varkappa t}\int_0^\infty \frac{\frac{\operatorname{sh}x}{x}\cos\left(\frac{x}{\pi}\ln t\right)} {\left(\frac{\operatorname{sh}x}{x}\right)^2-\frac{m^2}{x^2}}\,dx . \tag{9} \]

By direct integration of formulas (8) and (6) we obtain the following expression for \(\Phi(s)\):

\[ \varkappa\Phi(s)=A(s)+\int_0^1\left[A\left(\frac{s}{t}\right)+A(st)\right]M(t)\,dt +\int_0^1\left[\overline{A\left(\frac{s}{t}\right)}+\overline{A(st)}\right]N(t)\,dt . \tag{10} \]

Table 1 gives numerical values of the functions \(M(t)\) and \(N(t)\) for various values of the parameter \(m/\varkappa\). These functions are not expressible in elementary form and, for \(t=0\), have a singularity of the form

\[ M(t)=Rt^{\beta-1}+M_0(t), \]

\[ \operatorname{sign}\left(\frac{m}{\varkappa}\right)N(t)=+Rt^{\beta-1}+N_0(t). \tag{11} \]

\(M_0(t)\) and \(N_0(t)\) are continuous functions; \(\beta\) is the smallest positive root of the equation
\[ \frac{\sin\pi\beta}{\pi\beta}=\left|\frac{m}{\varkappa}\right|\quad(0<\beta<1); \qquad R=\frac{|m|\beta}{2(|m|-|\varkappa|\cos\pi\beta)}. \]

Table 1

For practical computations by formula (10), \(M_0(t)\) and \(N_0(t)\) should be approximated by simple piecewise-analytic functions. For qualitative investigations, with an accuracy in determining the stresses of the order of 5–10%, one may take the approximation

\[ M_0(t)\approx a_0=\frac{|m|}{2}\left(\frac{|m|}{\varkappa^2-m^2} -\frac{1}{|m|-|\varkappa|\cos\pi\beta}\right), \tag{12} \]

\[ N_0(t)\approx b_0+b_1(2t-1), \]

where

\[ b_0=\frac{|m|}{2}\left(\frac{|\varkappa|}{\varkappa^2-m^2} -\frac{1}{|m|-|\varkappa|\cos\pi\beta}\right), \]

\[ b_1= \frac{6|m|}{|\varkappa|} \int_0^\infty \frac{\operatorname{sh}x/x\,dx} {\left[(\operatorname{sh}x/x)^2-m^2/x^2\right](\pi^2+x^2)} + \frac{3|m|\beta}{(|m|-|\varkappa|\cos\pi\beta)(\beta+1)} -3b_0 . \]

Numerical values of \(b_1\) are given in Table 1.

Let in what follows \(\alpha<1\). As a concrete example of calculations by the method set forth, we shall investigate the distribution of stresses in a wedge under the action of a concentrated force \(P\), applied to one of the

its faces at a distance \(r_0\) from the vertex. Let the angle formed by the direction of the force with the outward normal be denoted by \(\gamma\). In the present case \(x=-1\), \(\beta=\alpha\);

\[ f(z_1)= \begin{cases} -Pe^{i(\pi\alpha/2+\gamma)}, & \text{for } \operatorname{Im} z_1<r_0\sin\dfrac{\pi\alpha}{2},\\[6pt] 0, & \text{for } \operatorname{Im} z_1>r_0\sin\dfrac{\pi\alpha}{2}, \end{cases} \tag{13} \]

\[ A(s)=\frac{P}{2\pi i}e^{i(\gamma+\pi\alpha/2)}\ln(s-s_0), \qquad \text{where } s_0=ir_0^{1/\alpha}=i\tau_0. \]

From formulas (10) and (13) we find

\[ \begin{aligned} \frac{2\pi\alpha r_0}{P}\, i e^{-i\gamma}\varphi'(z) &=\zeta^{1-\alpha} \biggl\{ \frac{1}{\zeta-1} +\int_0^1 M_0(t) \left[ \frac{1}{\zeta-t}+\frac{t}{t\zeta-1} \right]dt \\ &\quad -e^{-i(\pi\alpha+2\gamma)} \int_0^1 N_0(t) \left[ \frac{1}{\zeta+t}+\frac{t}{t\zeta+1} \right]dt \biggr\} \\ &\quad +\zeta^{1-\alpha}R \biggl\{ I(\zeta,\alpha) -\frac{1}{\zeta}I\!\left(\frac{1}{\zeta},1+\alpha\right) \\ &\quad +e^{-i(\pi\alpha+2\gamma)} \left[ I(-\zeta,\alpha) +\frac{1}{\zeta}I\!\left(-\frac{1}{\zeta},1+\alpha\right) \right] \biggr\}. \end{aligned} \tag{14} \]

Here

\[ \zeta=\frac{s}{s_0} =\left(\frac{r}{r_0}\right)^{1/\alpha} e^{\frac{i}{\alpha}(\arg z-\frac{\pi\alpha}{2})}, \qquad I(\zeta,\alpha)=\int_0^1\frac{t^{\alpha-1}}{\zeta-t}\,dt. \]

For the approximate computation of \(I(\zeta,\alpha)\), the cases \(|\zeta|<1\) and \(|\zeta|>1\) should be separated.

For \(|\zeta|<1\),

\[ I(\zeta,\alpha)= \int_1^\infty \frac{t^{\alpha-1}\,dt}{\zeta-t} -\frac{e^{i\pi\alpha}\pi}{\sin\pi\alpha}\,\zeta^{\alpha-1} = \zeta\int_0^1\frac{t^{1-\alpha}\,dt}{t\zeta-1} -\frac{e^{i\pi\alpha}\pi}{\sin\pi\alpha}\,\zeta^{\alpha-1} -\frac{1}{1-\alpha}. \]

For \(|\zeta|>1\),

\[ I(\zeta,\alpha)=\frac{1}{\alpha\zeta} +\frac{1}{\zeta}\int_0^1\frac{t^\alpha}{\zeta-t}\,dt. \]

In the last two integrals one may use the approximate formula (3)

\[ t^\alpha \simeq t\,\frac{(2-\alpha)+\alpha t}{\alpha+(2-\alpha)t}. \]

Let us compute the normal stresses \(\sigma_r\) on the faces of the wedge in the case \(\gamma=0\):

\[ \sigma_r=4\operatorname{Re}\varphi'(z)=\frac{2P}{\pi\alpha r_0}\,\beta(\zeta). \tag{15} \]

For \(\zeta>0\),

\[ \beta(\zeta)= \zeta^{1-\alpha} \left\{ \pi M(\sigma) +\sin\pi\alpha\int_0^1 N(t) \left[ \frac{1}{t+\zeta}+\frac{t}{t\zeta+1} \right]dt \right\}. \tag{16} \]

For \(\zeta=-x<0\),

\[ \beta(-x)= x^{1-\alpha}\sin\pi\alpha \left\{ \frac{1}{1+x} +\int_0^1 M(t) \left[ \frac{1}{t+x}+\frac{t}{tx+1} \right]dt +\pi x^{1-\alpha}N(x) \right\}. \]

Table 2 gives the values of \(\beta(\zeta)\) and \(\beta(-x)\) for a right-angle wedge, when \(\alpha=1/2\). In compiling this table, the above approximation of the functions \(M_0(t)\) and \(N_0(t)\) was used. To estimate the error of the table, we note that for \(\zeta=x=1\) the integrals (16) are easily computed exactly: \(\beta(1)=-0.234\); \(\beta(-1)=0.183\).

We note that in paper (4) an erroneous solution of this problem is given. Passing in formula (14) to the limit as \(r_0 \to 0\), we obtain Michell’s well-known solution (5) of the problem of the deformation of a wedge by a force applied at its vertex:

\[ e^{-i(\pi\alpha/2+\gamma)}\frac{2\pi\alpha}{P}\,z\varphi'(z) = 1+2\int_0^1 M(t)\,dt - 2e^{-i(\pi\alpha+2\gamma)}\int_0^1 N(t)\,dt = \]

\[ = \frac{1}{1-m^2} + e^{-i(\pi\alpha+2\gamma)}\frac{m}{1-m^2}. \]

With the above-indicated approximation of the functions \(M_0(t)\) and \(N_0(t)\), in the case \(\alpha=\tfrac12\) the following approximate solution of Michell’s problem is obtained:

\[ \frac{\pi}{P}e^{i(\pi/4-\gamma)}z\varphi'(z)\approx 1.6814-1.0705e^{2i(\pi/4-\gamma)}. \]

The exact value of the right-hand side is: \(1.6815-1.0705e^{2i(\pi/4-\gamma)}\). For the solution of the problem in the case of an arbitrary load on the upper face of the wedge, knowing the function \(\beta(\zeta)\), one can compute \(4\operatorname{Re}\varphi'(z)\) by integrating the right-hand side of expression (15) with respect to \(r_0\) over the interval \((0,\infty)\). In particular, for \(\tau(r_0)\equiv 0\), \(\rho(r_0)=-p=\mathrm{const}\), in this way we obtain (with the aid of the indicated approximation of \(M_0(t)\) and \(N_0(t)\)) the following approximate solution:

\[ 4\varphi'(z)\approx -1.011\,p=\mathrm{const}. \]

Table 2

The exact solution of the last problem, found by M. Lévy, is determined by the formula

\[ 4\varphi' z = -p. \]

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

Received
16 XI 1959

CITED LITERATURE

1. N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Publishing House of the Academy of Sciences of the USSR, 1949.
2. S. M. Belonosov, Trudy Voronezh. Gos. Univ., 27, 30 (1954).
3. A. N. Khovanskii, The Application of Continued Fractions and Their Generalizations to Questions of Approximate Analysis, Moscow, 1956.
4. A. I. Lur’e, B. Z. Brachkovskii, Trudy Leningrad. Politekhn. Inst., No. 3 (1941).
5. P. F. Papkovich, Theory of Elasticity, 1939.