THEORY OF ELASTICITY

M. Z. NARODETSKII

SOLUTION OF PROBLEMS OF THE PLANE THEORY OF ELASTICITY BY MEANS OF SPECIAL FUNCTIONS

(Presented by Academician L. I. Sedov on 26 XII 1956)

In the present article a method is indicated for the effective solution of a broad class of problems of the plane theory of elasticity for doubly connected regions (finite, infinite, and semi-infinite) bounded by circles and subjected to forces applied to the contours of the circles.

Consider an infinite plate (S), weakened by two unequal holes. The origin and the direction of the coordinate axes of the plane (z=x+iy), in which the plate is situated, as well as some notation, are indicated in Fig. 1.

Fig. 1. (z=x+iy;\ \varepsilon=\dfrac{R}{a+b};\ c=\dfrac{r}{R})

To determine the complex stress functions we have the boundary conditions ((^1))

[
\overline{\varphi(t)}+\bar t\,\varphi'(t)+\psi(t)=\overline{f_j(t)}+C_j
\quad \text{on } L_j\ (j=1,2),
\tag{1}
]

where (f_j(t)) are prescribed functions of the complex coordinate (t) of the contour (L_1) or (L_2); (C_j) are certain constants to be determined.

Following D. I. Sherman ((^2)), we compose for (\varphi(z)) the functional equations

[
\frac{1}{2\pi i}\int_L
\frac{\overline{\varphi(t)}+\bar t\,\varphi'(t)}{t-z}\,dt
=
\sum_{m=1}^{2}\frac{1}{2\pi i}\int_{L_m}
\frac{\overline{f_j(t)}}{t-z}\,dt
-C_j
\quad \text{in } \cdot L_j\ (j=1,2).
\tag{2}
]

We introduce special functions (\alpha_n) and (\beta_n), regular, respectively, outside (L_1) and (L_2), and possessing certain convenient properties, which make it possible to reduce the problems to recurrence formulas

[
\alpha_n=
\frac{\xi_{n-1}}{2\pi i}
\int_{L_1}
\frac{\overline{\beta_{n-1}(t)}}{t-z}\,dt,
\qquad
\beta_n=
\frac{\xi^{*}{\,n-1}}{2\pi i}
\int
\frac{\overline{\alpha_{n-1}(t)}}{t-z}\,dt
\quad (n=1,2,\ldots),
]

[
\alpha_0=
-\frac{\varepsilon^{-1}c^{-1}}{2\pi i}
\int_{L_1}
\frac{r}{\bar t+b}\,\frac{dt}{t-z},
\qquad
\beta_0=
\frac{\varepsilon^{-1}}{2\pi i}
\int_{L_2}
\frac{R}{\bar t-a}\,\frac{dt}{t-z},
\tag{3}
]

where (\xi_n) and (\xi_n^*) are certain constants.

The functions (\alpha_n) and (\beta_n), being of the same type in construction, rapidly tend to (\alpha) and (\beta)—the limiting values of the corresponding functions with index (n) as (n\to\infty), even when the boundaries are very close, and are representable, respectively, outside (L_1) and (L_2), in the form of series in powers of (\alpha) and (\beta).

Suppose that the load is continuous, and let us seek (\varphi(z)) in the form

[
\varphi(z)=\chi(z)+a_{01}\alpha_0+b_{01}\beta_0+\sum_{k=1}^{s}\left(A_k\alpha^k+B_k\beta^k\right),
\tag{4}
]

where (a_{01}, b_{01}, A_k), and (B_k) are certain coefficients to be determined, and (s) is a certain fixed number.

We choose the function (\chi(z)) in such a way (also using, for this purpose, the singularities of the functions (\alpha_0) and (\beta_0) on (L)) that, after substituting (\varphi(z)) into (2), the equations do not contain expressions of the form ((r/(z+b))^k) for (j=1) or of the form ((R/(z-a))^k) for (j=2), (k\ne1). Then, expanding the functions (\beta_n) and (\alpha_n) contained in them in powers of (\beta) and (\alpha), and equating to zero in the first equation the expressions at (\beta^k) and (r(z+b)), and in the second those at (\alpha^k) and (R/(z-a)), we obtain recurrence formulas for the indicated coefficients. Comparing the free terms in the same equations, we find the constants (C_1) and (C_2).

Fig. 2

For loads distributed over separate portions and for concentrated loads, the solution is not essentially changed. In this case it is necessary in (1) to introduce the substitution (\varphi(z)=\varphi_1(z)+\varphi^(z)) and (\psi(z)=\psi_1(z)+\psi^(z)), where (\varphi^(z)) and (\psi^(z)) are the zeroth or the sum of the zeroth and first Schwarz approximations, respectively, for the cases when loads of this kind are applied to one or to both contours.

Let (r=R), and at the point (t=a-R) of the contour (L_1) let a concentrated compressive force (P) be applied (Fig. 2). Taking into account the zeroth Schwarz approximation, instead of (1) we obtain

[
\overline{\varphi_1(t)}+\bar t\,\varphi_1'(t)+\psi_1(t)=\overline{f_j^(t)}+C_j^
\quad \text{on } L_j\ (j=1,2),
\tag{5}
]

where (f_1^*(t)=0) and

[
\overline{f_2^*(t)}
=
\frac{P}{2\pi}
\left(
-\ln\frac{a_0'-(t+a)}{t+a}\,
\frac{2a-R}{t-a+R}
+\frac{R}{t-a}
+\frac{2a-R-a_0'}{t-a+R}
+\mu_0'\frac{R}{t+a}
\right),
\quad
a_0'=R\mu_0',\quad
\mu_0'=\frac{\varepsilon}{1-\varepsilon}.
\tag{6}
]

The functions (\alpha_n) and (\beta_n) ((n=0,1,\ldots)) have the form

[
\alpha_n=\frac{a_n}{a_n+(z-a)},\qquad
\beta_n=\frac{a_n}{a_n-(z+a)},\qquad
a_n=R\mu_n,
]

[
\mu_n=\frac{\varepsilon}{1-\varepsilon\mu_{n-1}},\qquad
\mu_0=\varepsilon,\qquad
\xi_n=\frac{2a-a_n}{a_n},
\tag{7}
]

[
\alpha=\frac{c}{c+(z-a)},\qquad
\beta=\frac{c}{c-(z+a)},\qquad
c=R\mu,\qquad
\mu=\frac{\varepsilon}{1-\varepsilon\mu}<1,\qquad
\xi=\frac{2a-c}{c}.
]

Put in (3) (\alpha_0^*=0) and

[
\beta_0^*=
\frac{\mu_0'^{-1}}{2\pi i}
\int_{L_2}
\frac{R}{\bar t-a+R}\,
\frac{dt}{t-z}.
\tag{8}
]

The expressions for (\alpha_n) and (\beta_n) obtained from the remaining formulas (3) coincide in form with (\alpha_n) and (\beta_n), and their limiting values are also equal, respectively, to (\alpha) and (\beta). We shall agree to attach the sign (') to constants pertaining to the functions (\alpha_n^*) and (\beta_n).

Further, we have

[
\alpha_n^k=\lambda_n^k \sum_{k_1=k}^{\infty} C_{-k}^{k_1-k}\nu_n^{k_1-k}\alpha^{k_1},\quad
\ln \frac{a_n' + (z-a)}{a_m+(z-a)}
=
-\sum_{k=1}^{\infty}\frac{1}{k}\left(\nu_n'^k-\nu_m^k\right)\alpha^k
\quad \text{outside } L_1;
]

[
\beta_n^k=\lambda_n^k \sum_{k_1=k}^{\infty} C_{-k}^{k_1-k}\nu_n^{k_1-k}\beta^{k_1},\quad
\ln \frac{a_n' - (z+a)}{a_m-(z+a)}
=
-\sum_{k=1}^{\infty}\frac{1}{k}\left(\nu_n'^k-\nu_m^k\right)\beta^k
\quad \text{outside } L_2;
\tag{9}
]

[
\lambda_n=\mu_n/\mu,\qquad \nu_n=1-\lambda_n
\qquad (k=1,2,\ldots;\ m,n=0,1,\ldots).
]

Replacing in the first equalities (9) (\mu_n,\lambda_n), and (\nu_n), respectively, by (\mu_n'), (\lambda_n'), and (\nu_n'), we obtain analogous expansions for (\alpha_n^) and (\beta_n^).

We shall seek the function (\varphi_1(z)) in the form

[
\varphi_1(z)=\frac{P}{2\pi}\left(
a_{01}\alpha_0+b_{01}\beta_0+
\ln \frac{a_1' + (z-a)}{a_0+(z-a)}
\frac{a_0'-(z+a)}{z+a}
-
\right.
]

[
\left.
{}-\varepsilon^2\alpha_0^2+(1-\mu_0'^2)\beta_0^*
+\sum_{k=1}^{s}\left(A_k\alpha^k+B_k\beta^k\right)
\right).
\tag{10}
]

The functional equations (2), after substituting into them (6), (9), and (10), and comparing in both sides of these equations the expressions in (\beta) and (\alpha) in equal powers, respectively for (j=1) and (j=2), lead to recurrence formulas for the coefficients (A_k) and (B_k). These formulas, with the aid of the notation

[
a_{01}'=a_{01}+b_{01},\qquad a_{01}''=a_{01}-b_{01},
]

[
A_k'=A_k+B_k,\qquad A_k''=A_k-B_k
\qquad (k=1,2,\ldots,s)
\tag{11}
]

are conveniently written in the form

[
\sum_{k_1=k}^{s} A_{k_1}'\left(
C_{k_1}^{k}\left(1-(-1)^{k_1+k}\xi^{-(k_1+k)}\right)
-
k_1\left(\xi^{-k+1}-(-1)^{k_1+k}\xi^{-(k_1+1)}\right)
\right)
=
]

[
=\xi^{-k}\left(\delta_k^{+}+\Delta_k^{+}\right);
\tag{12}
]

[
\sum_{k_1=k}^{s} A_{k_1}''\left(
C_{k_1}^{k}\left(1-(-1)^{k_1+k}\xi^{-(k_1+k)}\right)
+
k_1\left(\xi^{-k+1}-(-1)^{k_1+k}\xi^{-(k_1+1)}\right)
\right)
=
]

[
=\xi^{-k}\left(\delta_k^{-}+\Delta_k^{-}\right),
\tag{13}
]

where the notation has been introduced

[
\delta_k^{+}
=
a_{01}'\left(
\lambda_0\left(\varepsilon^{-2}(1-\xi_0^{-2})+\lambda_0(1-\xi_0^{-1})\right)\nu_0^{-1}C_{-2}^{k-2}\nu_0^{k-1}
+
\xi_0^{-1}\lambda_1\nu_1^{k-1}
\right);
\tag{14}
]

[
\delta_k^{-}
=
-a_{01}''\left(
\lambda_0\left(\varepsilon^{-2}(1-\xi_0^{-2})+\lambda_0(1-\xi_0^{-1})\right)\nu_0^{-1}C_{-2}^{k-2}\nu_0^{k-1}
-
\xi_0^{-1}\lambda_1\nu_1^{k-1}
\right);
\tag{15}
]

[
\Delta_k^{+}
=
\lambda_0'\left(
-1+\mu_0'^{-1}\mu_2'^{-1}-\mu_1'^2+\xi_0'^{-2}
-
\right.
]

[
\left.
{}-(\mu_0'^{-1}-\mu_0')(1-\mu_1')\lambda_0'\nu_0'^{-1}C_{-2}^{k-2}\nu_0'^{k-1}
-\frac{1}{k}\left(\nu_2'^k-\nu_1^k\right)
+
\right.
]

[
+ \varepsilon^2 \xi_0^{-2}(2-\lambda_1\nu_1^{-1}C_{-2}^{k-2})\lambda_1\nu_1^{k-1}
+{\lambda_0(1-\xi_0^{-1}+2\xi_0^{-3}+2(1-\xi_0^{-2})\lambda_0\nu_0^{-1}C_{-2}^{k-2}
]
[
+2\xi_0^{-1}\lambda_0^2\nu_0^{-2}C_{-3}^{k-3})\nu_0^{k-1}
-\lambda'_1(\mu_1'^{-1}(\mu'_0-\mu'_2)
]
[
+(1-\mu_0'^2)\xi_0'^{-1})\nu_1'^{\,k-1}
+\frac{1}{k}(\nu_1'^k-\nu_0'^k)}.
\tag{16}
]

Comparing in the same equations the expressions for (R/(z+a)) and (R/(z-a)), respectively for (j=1) and (j=2), as well as the free terms, we obtain

[
\varepsilon^{-1}a'{01}(2-\xi_0^{-2})
+\mu^{-1}\sum\bigr)=}^{s} kA'_k\bigl(1+(-1)^k\xi^{-(k+1)
]
[
=\varepsilon(4+2\xi_0^{-3})-\mu_1+\mu'_1+\mu'_2-(\mu_0'^{-1}-\mu'_0)(1-\xi_0'^{-2});
\tag{17}
]

[
\varepsilon^{-1}\xi_0^{-2}a''{01}
+\mu^{-1}\sum\bigr)}^{s} kA''_k\bigl(1-(-1)^k\xi^{-(k+1)
=
]
[
=1-2\varepsilon(1+\xi_0^{-3})-3\mu'_0+\mu_1+\mu_1'^{-1}+\mu_3'^{-1}+(\mu_0'^{-1}-\mu'_0)\xi_0'^{-2};
\tag{18}
]

[
C'1=C_1^+C_2^
=a'\bigr)-}(1-\xi_0^{-1})+\sum_{k=1}^{s} A'_k\bigl(1+(-1)^k\xi^{-k
]
[
-\varepsilon^2(1+\xi_0^{-2})+(1-\mu_0'^2)(1-\xi_0'^{-1})+\ln\frac{\mu_1}{\mu'_2};
\tag{19}
]

[
C''1=C_1^-C_2^
=a''\bigr)-}(1+\xi_0^{-1})+\sum_{k=1}^{s} A''_k\bigl(1-(-1)^k\xi^{-k
]
[
-\varepsilon^2(1-\xi_0^{-2})-(1-\mu_0'^2)(1+\xi_0'^{-1})-\ln\frac{\mu_1}{\mu'_2}.
\tag{20}
]

The coefficients (A'k) ((k=s,\ s-1,\ldots,2)) and (A''_k) ((k=s,\ s-1,\ldots,1)) are determined from the recurrence formulas (12) and (13), where the former contain the as yet unknown coefficient (a'), and the latter (a''{01}). The coefficient (a') is found from formula (12) for (k=1), and (a''{01}) from (18). After this, from formulas (17), (19), and (20), we successively find the coefficient (A'_1) and the constants (C'_1) and (C''_1). Finally, returning to formulas (11), (19), and (20), we find the coefficients (A_k) and (B_k) ((k=1,2,\ldots,s)), (a), as well as the constants (C_1^}) and (b_{01) and (C_2^).

Figure 2 gives the diagram of (\sigma_\theta) ((\theta) is the polar angle, measured from the polar axis (r') with origin at the point (z=-a)) at various points of (L_2) for (R/a=0.9) and (s=11). We note that the boundary condition (5) on (L_1) is satisfied here exactly for any fixed (s), and on (L_2), for the chosen (s), the greatest deviation from the boundary condition does not exceed (0.002\%).

The numerical results obtained can be extended to a plate of finite width if the width of the plate considerably exceeds the diameter of the holes.

Unlike the known methods for solving analogous problems, the present method leads to rapidly convergent processes for any closeness of the boundaries and for arbitrary loads.

Received
25 IV 1956

REFERENCES CITED

(^{1}) N. I. Muskhelishvili, Some Problems of the Theory of Elasticity, 1935.
(^{2}) D. I. Sherman, Applied Mathematics and Mechanics, vol. 6 (1951).