In this chapter we give the solution of a problem of bending of a transversally-isotropic plate of variable thickness. The asymptotic behavior of the solution for a small thinness parameter is studied.

6.1 Constructing Homogeneous Solutions

Let us consider an axially-symmetric problem of bending of a transtropic plate of thickness \(h = \varepsilon r\).

Suppose that on the end faces of the plate the following boundary conditions are given:

$$\sigma_{\theta } = \left( { - 1} \right)^{n} \sigma \left( r \right),\quad \tau_{r\theta } = T\left( r \right)\quad {\text{at}}\;\;\theta = \frac{\pi }{2} \pm \left( { - 1} \right)^{n} \varepsilon$$
(6.1.1)

On the lateral surface the following stresses are given:

$$\sigma_{r} = f_{1s} \left( \theta \right),\tau_{r\theta } = f_{2s} \left( \theta \right)\quad {\text{for}}\,r = r_{s} \quad \left( {s = 1,2} \right)$$
(6.1.2)

The function \(f_{is} \left( {i = 1,2} \right)\) satisfies the equilibrium conditions

$$\begin{aligned} & 2\pi r_{1}^{2} \int\limits_{\pi /2 - \varepsilon }^{\pi /2 + \varepsilon } {\left[ {f_{11} \sin \varepsilon \eta - f_{21} \left( \eta \right)\cos \varepsilon \eta } \right]} \cos \varepsilon \eta d\eta \\ & \quad = 2\pi r_{2}^{2} \varepsilon \int\limits_{\pi /2 - \varepsilon }^{\pi /2 + \varepsilon } {\left[ {f_{12} \sin \varepsilon \eta - f_{22} \left( \eta \right)\cos \varepsilon \eta } \right]} \cos \varepsilon \eta d\eta = P \\ \end{aligned}$$
(6.1.3)

\(f_{1s} \left( \theta \right)\), \(f_{2s} \left( \theta \right)\) are rather smooth functions. Besides, \(f_{1s} \left( \theta \right)\) are odd, \(f_{2s} \left( \theta \right)\) are even functions with respect to the median plane of the plate.

P is the principle vector of forces acting in the section \(r = const\).

In this chapter we assume that on the end faces of the plate the homogeneous boundary conditions \(\left( {\sigma = \tau = 0} \right)\) are given:

$$\sigma_{\theta } = 0, \quad \tau_{r\theta } = 0\quad {\text{for}}\,\; \theta = \frac{\pi }{2} \pm \varepsilon .$$
(6.1.4)

Using the results of Chap. 5, we represent the solution of the problem in the form:

$$\begin{aligned} & u_{r} = r^{\lambda } \left[ {A_{1} C_{{\gamma_{1} }} F_{{\gamma_{1} }} \left( \theta \right) + A_{2} C_{{\gamma_{2} }} F_{{\gamma_{2} }} \left( \theta \right)} \right] \\ & u_{\theta } = r^{\lambda } b_{0} \left[ {C_{{\gamma_{1} }} F_{{\gamma_{1} }}^{{\prime }} \left( \theta \right) + C_{{\gamma_{2} }} F_{{\gamma_{2} }}^{{\prime }} \left( \theta \right)} \right] \\ & F_{\gamma } \left( \theta \right) = P_{\gamma } \left( {\cos \theta } \right) - P_{\gamma } \left( { - \cos \theta } \right) \\ \end{aligned}$$
(6.1.5)

The remaining denotations are the same as in Chap. 1.

Satisfaction of boundary conditions on the end faces of the plate (6.1.4) gives a homogeneous linear algebraic system of second order with respect to the constants \(C_{{\gamma_{1} }}\), \(C_{{\gamma_{2} }}\).

$$\begin{aligned} & \left\{ {\left[ {\left( {b_{12} \lambda + b_{22} + b_{23} } \right)A_{1} - b_{22} b_{0} \gamma_{1} \left( {\gamma_{1} + 1} \right)} \right]F_{{\gamma_{1} }} \left( {\theta_{1} } \right) + \left( {b_{23} - b_{22} } \right)} \right. \\ & \quad \times \left. {b_{0} tg\varepsilon F^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right)} \right\}C_{{\gamma_{1} }} + \left\{ {\left. {\left[ {\left( {b_{12} \lambda + b_{22} + b_{23} } \right)A_{2} - b_{22} b_{0} \gamma_{2} } \right.\left( {\gamma_{2} + 1} \right)} \right]} \right. \\ & \quad \times F_{{\gamma_{2} }} \left( {\theta_{1} } \right) + \left. {\left( {b_{23} - b_{22} } \right)b_{0} tg\varepsilon F^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right)} \right\}C_{{\gamma_{2} }} = 0, \\ & \left[ {A_{1} + \left( {\lambda - 1} \right)b_{0} } \right]F^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right)C_{{\gamma_{1} }} + \left[ {A_{2} + \left( {\lambda - 1} \right)b_{0} } \right]F^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right) = 0. \\ \end{aligned}$$
(6.1.6)

From the condition of existence of nontrivial solutions of this system, we obtain a characteristic equation for determining the eigen values \(\lambda\):

$$\begin{aligned} \Delta \left( {\lambda ,\theta_{1} } \right) & = C_{11} d_{12} F_{{\gamma_{1} }} \left( {\theta_{1} } \right)F^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right) - C_{12} d_{11} F_{{\gamma_{2} }} \left( {\theta_{1} } \right)F^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right) \\ & \quad + C_{13} \left( {d_{12} - d_{11} } \right)tg\varepsilon T^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right)T^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right) = 0 \\ \end{aligned}$$
(6.1.7)

wherein

$$\begin{aligned} \Delta_{1n} & = C_{{\gamma_{1n} }} = \left[ {A_{2} + \left( {\lambda_{n} - 1} \right)b_{0} } \right]F^{\prime}_{{\gamma_{2n} }} \left( {\theta_{1} } \right), \\ C_{{\gamma_{2} }} & = - \left[ {A_{1} + \left( {\lambda_{n} - 1} \right)b_{0} } \right]F^{\prime}_{{\gamma_{n} }} \left( {\theta_{1} } \right) = \Delta_{2n} \\ \end{aligned}$$
(6.1.8)

The transcendental Eq. (6.1.7) determines the denumerable set \(\lambda_{k}\), with a condensation point at infinity.

Substituting (6.1.8) in (6.1.5), allowing for (5.1.3) and summing over all the roots of \(\lambda_{k}\), we obtain homogeneous solutions of the form:

$$\begin{aligned} u_{r} & = \sum\limits_{n = 1}^{\infty } {C_{n} } r^{{_{{\lambda_{n} }} }} u_{n} \left( \theta \right), \\ u_{\theta } & = \sum\limits_{n = 1}^{\infty } {C_{n} } r^{{_{{\lambda_{n} }} }} w_{n} \left( \theta \right) \\ \sigma_{r} & = G_{1} \sum\limits_{n = 1}^{\infty } {C_{n} } r^{{_{{\lambda_{n} }} }} Q_{rn} \left( \theta \right), \\ \sigma_{\varphi } & = G_{1} r^{ - 1} \sum\limits_{n = 1}^{\infty } {C_{n} } r^{{_{{\lambda_{n} }} }} Q_{\varphi n} \left( \theta \right) \\ \sigma_{\theta } & = G_{1} r^{ - 1} \sum\limits_{n = 1}^{\infty } {C_{n} } r^{{_{{\lambda_{n} }} }} Q_{\theta n} \left( \theta \right), \\ \tau_{r\theta } & = G_{1} r^{ - 1} \sum\limits_{n = 1}^{\infty } {C_{n} } r^{{_{{\lambda_{n} }} }} T_{n} \left( \theta \right) \\ \end{aligned}$$
(6.1.9)

Here \(C_{n}\) are arbitrary constants.

$$\begin{aligned} u_{n} \left( \theta \right) & = A_{1} \Delta_{1n} F_{{\gamma_{1n} }} \left( \theta \right) + A_{2} \Delta_{2n} F_{{\gamma_{2n} }} \left( \theta \right), \\ w_{n} \left( \theta \right) & = b_{0} \left[ {\Delta_{1n} F^{\prime}_{{\gamma_{1n} }} \left( \theta \right) + \Delta_{2n} F^{\prime}_{{\gamma_{2n} }} \left( \theta \right)} \right], \\ Q_{rn} \left( \theta \right) & = \left[ {A_{1} b_{11} \lambda_{n} + 2b_{12} A_{1} - b_{12} b_{0} \gamma_{1n} \left( {\gamma_{1n} + 1} \right)} \right]\Delta_{1n} F_{{\gamma_{1n} }} \left( \theta \right) \\ & \quad \quad + \left[ {A_{2} b_{11} \lambda_{n} + 2b_{12} A_{2} - b_{12} b_{0} \gamma_{2n} \left( {\gamma_{2n} + 1} \right)} \right]\Delta_{2n} F_{{\gamma_{2n} }} \left( \theta \right), \\ \end{aligned}$$
(6.1.10)
$$\begin{aligned} Q_{\varphi n} \left( \theta \right) & = \left\{ {\left[ {\left. {A_{1} b_{12} \lambda_{n} + \left( {b_{22} + b_{23} } \right)A_{1} - b_{23} b_{0} \gamma_{1n} \left( {\gamma_{1n} + 1} \right)} \right]} \right.} \right.F_{{\gamma_{1n} }} \left( \theta \right) \\ & \quad \quad + b_{0} \left. {\left( {b_{22} - b_{23} } \right)ctg\theta F^{\prime}_{{\gamma_{1n} }} \left( \theta \right)} \right\}\Delta_{2n} + \left\{ {\left[ {A_{2} b_{12} \lambda_{n} + \left( {b_{22} + b_{23} } \right)} \right.} \right.A_{2} \\ & \quad \quad - b_{23} b_{0} \gamma_{2n} \left. {\left. {\left( {\gamma_{2n} + 1} \right)} \right]F_{{\gamma_{2n} }} \left( \theta \right) + b_{0} \left( {b_{22} - b_{23} } \right)ctg\theta F^{\prime}_{{\gamma_{2n} }} \left( \theta \right)} \right\}\Delta_{2n} , \\ Q_{\theta n} \left( \theta \right) & = \left\{ {\left[ {\left. {A_{1} b_{12} \lambda_{n} + \left( {b_{22} + b_{23} } \right)A_{1} - b_{22} b_{0} \gamma_{1n} \left( {\gamma_{1n} + 1} \right)} \right]} \right.} \right.F_{{\gamma_{1n} }} \left( \theta \right) \\ & \quad \quad + b_{0} \left. {\left( {b_{23} - b_{22} } \right)ctg\theta F^{\prime}_{{\gamma_{1n} }} \left( \theta \right)} \right\}\Delta_{1n} + \left\{ {\left[ {A_{2} b_{12} \lambda_{n} + \left( {b_{22} + b_{23} } \right)} \right.} \right.A_{2} \\ & \quad \quad - b_{22} b_{0} \gamma_{2n} \left. {\left. {\left( {\gamma_{2n} + 1} \right)} \right]F_{{\gamma_{2n} }} \left( \theta \right) + b_{0} \left( {b_{23} - b_{22} } \right)ctg\theta F^{\prime}_{{\gamma_{2n} }} \left( \theta \right)} \right\}\Delta_{2n} , \\ T_{n} \left( \theta \right) & = \left[ {A_{1} + \left( {\lambda_{n} - 1} \right)b_{0} } \right]F^{\prime}_{{\gamma_{1n} }} \left( \theta \right)\Delta_{1n} + \left[ {A_{2} + \left( {\lambda_{n} - 1} \right)b_{0} } \right]F^{\prime}_{{\gamma_{2n} }} \left( \theta \right)\Delta_{2n} . \\ \end{aligned}$$
(6.1.11)

As in Chap. 5, for effective study of the roots of the characteristic equation, we put

$$\theta = \frac{\pi }{2} + \varepsilon \eta , \quad - 1 \le \eta \le 1$$
(6.1.12)

Substituting (6.1.12) in (6.1.7), we obtain:

$$D\left( {z,\varepsilon } \right) = \Delta \left( {\lambda ,\theta_{1} } \right) = 0$$
(6.1.13)

For the zeros of the function \(D\left( {z,\varepsilon } \right)\) we prove the following statement: the function \(D\left( {z,\varepsilon } \right)\) has two groups of zeros with the following asymptotic properties as \(\varepsilon \to 0\).

The first group consists of four zeros and is characterized by the fact that all of them as \(\varepsilon \to 0\) have finite limit, and two of them are independent of the small parameter \(\varepsilon\).

The second group consists of a denumerable set of zeros that as \(\varepsilon \to 0\) are of order \(O\left( {\varepsilon^{ - 1} } \right)\).

To prove the first statement, we expand the functions \(F_{\nu } \left( \theta \right)\), \(F^{\prime}_{\nu } \left( \theta \right)\) in the vicinity of the plane \(\theta = \frac{\pi }{2}\) in a series with respect to \(\varepsilon\) [1, 2] and get

$$\begin{aligned} F_{\nu } \left( \theta \right) & = \frac{4}{\sqrt \pi }\frac{{{\Gamma } \left( {1 + \frac{\nu }{2}} \right)}}{{{\Gamma } \left( {\frac{\nu + 1}{2}} \right)}}\sin \left( {\nu \frac{\pi }{2}} \right)\varepsilon \eta \left\{ {1 - \frac{1}{3!}\eta^{2} \left[ {\nu \left( {\nu + 1} \right) - 1} \right]\varepsilon^{2} } \right. \\ & \quad + \left. {\frac{1}{5!}\eta^{4} \left[ {\nu^{2} \left( {\nu + 1} \right)^{2} - 4\nu \left( {\nu + 1} \right) + 5} \right]\varepsilon^{4} + \cdots } \right\} \\ \end{aligned}$$
(6.1.14)
$$\begin{aligned} F^{\prime}_{\nu } \left( \theta \right) & = - \frac{4}{\sqrt \pi }\frac{{{\Gamma } \left( {1 + \frac{\nu }{2}} \right)}}{{{\Gamma } \left( {\frac{\nu + 1}{2}} \right)}}\sin \left( {\nu \frac{\pi }{2}} \right)\left\{ {1 - \frac{1}{2}\eta^{2} \left[ {\left( {\nu - 1} \right)\left( {\nu + 2} \right) - 1} \right]\varepsilon^{2} } \right. \\ & \quad + \left. {\frac{1}{4!}\eta^{4} \left[ {\nu^{2} \left( {\nu + 1} \right)^{2} \left( {\nu - 1} \right) - 4\nu \left( {\nu + 1} \right) + 5} \right]\varepsilon^{4} + \cdots } \right\} \\ \end{aligned}$$

Substituting (6.1.14) in (6.1.13), we represent the functions \(D\left( {z,\varepsilon } \right)\) in the form:

$$D\left( {z,\varepsilon } \right) = 3^{ - 1} A\left( {z^{2} - \frac{1}{4}} \right)\varepsilon^{3} \left[ {D_{0} \left( z \right) + \frac{1}{{5\left( {1 - \nu_{1} \nu_{2} } \right)}}D_{1} \left( z \right)\varepsilon^{2} + \cdots } \right],$$
(6.1.15)

where

$$\begin{aligned} A_{kj} & = 16G_{0}^{2} \left[ {\left( {1 + \nu } \right)E_{0} b_{0} \left[ {\gamma_{2} \left( {\gamma_{2} + 1} \right) - \gamma_{1} \left( {\gamma_{1} + 1} \right)} \right]} \right. \\ & \quad \times \sin \left( {\gamma_{1} \frac{\pi }{2}} \right)\sin \left( {\gamma_{2} \frac{\pi }{2}} \right){\Gamma } \left( {1 + \frac{{\gamma_{1} }}{2}} \right){\Gamma } \left( {1 + \frac{{\gamma_{2} }}{2}} \right)\left[ {{\Gamma } \left( {1 + \frac{{\gamma_{1} + 1}}{2}} \right){\Gamma } \left( {1 + \frac{{\gamma_{2} + 1}}{2}} \right)} \right]^{ - 1} \\ \end{aligned}$$
$$\begin{aligned} D_{0} \left( z \right) & = 4z^{2} + 12\nu_{2} - 9 - 4E_{0}^{ - 1} \\ D_{1} \left( z \right) & = - 4\left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)z^{4} + 2\left[ {2\left( {1 - \nu_{1} \nu_{2} } \right)\left( {3 - 2\nu } \right)} \right. \\ & \quad + \left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)\left( {E_{0} + 4E_{0} G_{0} + 2 - 6\nu_{1} } \right) - 2\left( {1 + \nu } \right) \\ & \quad \times \left( {2E_{0} G_{0} - \nu_{1} - 1} \right)\left. {\left( {G_{0} - 1} \right)} \right]z^{2} - \left( {1 - \nu_{1} \nu_{2} } \right)\left( {40E_{0} G_{0} } \right. \\ & \quad - \left. {60\nu_{1} - 2\nu + 23} \right) - \frac{1}{2}\left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)\left( {2E_{0} G_{0} + 2 - 6\nu_{1} + \frac{{E_{0} }}{2}} \right) \\ & \quad + \left( {1 + \nu } \right)\left( {2E_{0} G_{0} - \nu_{1} - 1} \right)\left( {G_{0} - 1} \right) + 4\left[ {\left( {1 + \nu } \right)\left( {\nu_{1} - 1} \right)\left( {2E_{0} G_{0} - 3\nu_{1} + 1} \right)} \right. \\ & \quad + 2\left. {\left( {1 - \nu_{1} \nu_{2} } \right)\left( {3 - 2\nu } \right)E_{0} } \right]\left( {G_{0} - 1} \right) + 8\left( {1 + \nu } \right)\left( {1 - \nu_{1} } \right)E_{0} \left( {G_{0} - 1} \right)^{2} \\ \end{aligned}$$

Here, \({\Gamma } \left( x \right)\) is Euler’s gamma function.

From (6.1.15) it is directly seen that \(z_{0,1} = \pm \frac{1}{2}\) are the zeros of the function \(D\left( {z,\varepsilon } \right)\). Note that the existence of these zeros follows from the plate equilibrium condition.

To determine the remaining zeros of the first group, we look for them in the form:

$$z_{k} = z_{{k_{0} }} + \varepsilon^{2} z_{{k_{2} }} + \cdots \quad \left( {k = 2,3} \right)$$
(6.1.16)

Substituting (6.1.16) in (6.1.15), we get:

$$\begin{aligned} z_{{k_{0} }} & = \pm \frac{1}{2}\left( {9 + 4E_{0}^{ - 1} - 12\nu_{2} } \right)^{{\frac{1}{2}}} \\ z_{{k_{2} }} & = - \left( {40z_{{k_{0} }} } \right)^{ - 1} D_{2} \left( {z_{{k_{0} }} } \right) \\ \end{aligned}$$

We prove that the remaining zeros of the function \(D\left( {z,\varepsilon } \right)\) unlimitedly increase as \(\varepsilon \to 0\). We proceed from the contrary having assuming that \(z_{k} \to z_{k}^{*} \ne \infty \,\left( {k \ge 4} \right)\) as \(\varepsilon \to 0\). Then the following limit relation \(D\left( {z,\varepsilon } \right) \to \varepsilon^{3} \,D_{0}^{*} \left( {z_{k}^{*} } \right)\) is valid as \(\varepsilon \to 0\). Thus, the limit points of the set \(z_{k} \left( {k \ge 4} \right)\) are determined from the equation \(D_{0}^{*} \left( {z_{k}^{*} } \right) = 0\). In the present case,

$$\begin{aligned} D_{0}^{*} \left( {z_{k}^{*} } \right) & = \left( {z_{k}^{*2} - \frac{1}{4}} \right)\left( {4z_{k}^{*2} - 12\nu_{2} - 9 - 4E_{0}^{ - 1} } \right) \\ & = \left( {z_{k}^{*2} - \frac{1}{4}} \right)D_{0} \left( {z_{k}^{*} } \right) = 0 \\ \end{aligned}$$

From the last equation it follows that these are no other limited zeros except \(z_{0,1}\), \(z_{2,5}\).

So we proved that all remaining zeros of the function \(D\left( {z,\varepsilon } \right)\) tend to zero as \(\varepsilon \to 0\). We can divide them into three groups depending on their behavior as \(\varepsilon \to 0\).

The following limited relations are possible:

(1) \(\varepsilon z_{k} \to 0\); (2) \(\varepsilon z_{k} \to \infty\); (3) \(\varepsilon z_{k} \to const\) as \(\varepsilon \to 0\).

We can prove that cases (1) and (2) are impossible here.

To construct the asymptotics of the zeros of the second group (case 3) we find \(z_{n} \,\left( {n = k - 3,\,\,k \ge 4} \right)\) in the form:

$$z_{n} = \varepsilon^{ - 1} \delta_{n} + O\left( \varepsilon \right)\quad \left( {n = 1,2, \ldots } \right)$$
(6.1.17)

Substituting (1.1.15) in (1.1.17), we have:

$$\begin{aligned} & \tau^{2} - 2q_{1} \delta_{n}^{2} \tau + q_{2} \delta_{n}^{4} = 0, \quad \gamma_{i} = \sqrt {\tau_{i} } , \quad \left( {i = 1,2} \right) \\ & \tau^{2} - \delta_{n}^{22} S_{i}^{2} , \quad S_{i} = \sqrt {q_{1} - \left( { - 1} \right)^{2} \sqrt {q_{1}^{2} - q_{2} } } \\ & 2q_{1} = b_{22}^{ - 1} \left( {b_{11} b_{22} - b_{12}^{2} - 2b_{12} } \right), \quad q_{2} = b_{11} b_{22}^{ - 1} \\ \end{aligned}$$
(6.1.18)

As it was noted in Chap. 1, depending on the characteristics of the material \(\nu ,\nu_{1} ,\nu_{2} ,G_{0}\) the parameters \(q_{1} ,q_{2}\) take different values and this entails different record of the solution by the Legendre function. This in its turn reduces to different asymptotic representations of the Legendre function.

Let us consider the following possible cases:

  1. 1.

    \(q_{1} > 0\), \(q_{1}^{2} - q_{2} > 0\), \(\gamma_{1,2} = \pm S_{1} \delta_{n}\), \(\gamma_{3,4} = \pm S_{2} \delta_{n}\)

    $$\begin{aligned} S_{1,2} & = \sqrt {q_{1} \pm \sqrt {q_{1}^{2} - q_{2} } } , \quad q_{1}^{2} > q_{2} \\ S_{1,2} & = \alpha \pm i\beta \sqrt {q_{1} \pm i\sqrt {q_{2} - q_{1}^{2} } } , \quad q_{1}^{2} < q_{2} \\ \end{aligned}$$
  1. 2.

    The roots of the characteristic Eq. (6.1.18) are multiple

    $$\gamma_{1,2} = \gamma_{3,4} = \pm \delta_{n} p, \quad q_{1} > 0, \; \, q_{1}^{2} - q_{2} = 0,\; \, p = \sqrt {q_{1} }$$
  1. 3.

    \(q_{1} < 0\), \(q_{1}^{2} - q_{2} \ne 0\), \(\gamma_{1,2} = \pm i\delta_{n} S_{1}\), \(\gamma_{3,4} = \pm i\delta_{n} S_{2}\)

    $$\begin{aligned} S_{1,2} & = \sqrt {\left| {q_{1} } \right| \pm i\sqrt {q_{1}^{2} - q_{2} } } , \quad q_{1}^{2} > q_{2} \\ S_{1,2} & = \sqrt {\left| {q_{1} } \right| \pm i\sqrt {q_{2} - q_{1}^{2} } } , \quad q_{1}^{2} < q_{2} \\ \end{aligned}$$
  1. 4.

    \(q_{1} < 0\), \(q_{1}^{2} - q_{2} = 0\), \(\gamma_{1,2} = \gamma_{3,4} = \pm i\delta_{n} p\), \(p = \sqrt {\left| {q_{1} } \right|}\)

In cases 1, 2 after substituting (6.1.17) in (6.1.7) and transforming it by means of asymptotic expansions \(F_{\gamma } \left( \theta \right)\), \(F^{\prime}_{\gamma } \left( \theta \right)\) for \(\delta_{n}\) we get:

$$\left( {S_{2} - S_{1} } \right)\sin \left( {S_{1} + S_{2} } \right)\delta_{n} - \left( {S_{1} + S_{2} } \right)\sin \left( {S_{2} - S_{1} } \right)\delta_{n} = 0$$
(6.1.19)
$$\alpha \sin 2\beta \delta_{n} - \beta sh2\alpha \delta_{n} = 0;$$
(6.1.20)
$$\sin 2p\delta_{n} - 2p\delta_{n} = 0.$$
(6.1.21)

Concerning the cases 3 and 4, the results for them are obtained from cases 1 and 2 formally replacing \(S_{1}\), \(S_{2}\) by \(iS_{1} ,iS_{2}\). These equations coincide with the equations determining the Saint-Venant edge effects indices in the bending problem for a constant thickness plate.

As in the isotropic case [3], we can prove that the function \(D\left( {z,\varepsilon } \right)\) has no other zeros except the above mentioned ones.

6.2 Analysis of Stress-Strain State

We give the characteristics of stresses-strain stresses determined by the above constructed solutions.

Assuming that \(\varepsilon\) is a small parameter, we give asymptotic construction of homogeneous solutions corresponding to different groups of zeros.

For \(z_{0} = \frac{1}{2}\) we obtain the following expressions

$$\begin{aligned} u_{r} & = + C_{0} \sin \varepsilon \eta ,\quad u_{\theta } = + C_{0} \cos \varepsilon \eta \\ \sigma_{r} & = \sigma_{\varphi } = \sigma_{\theta } = \tau_{r\theta } = 0 \\ \end{aligned}$$
(6.2.1)

As can be easily seen, the displacement of the plate as a solid corresponds to this solution.

The solution corresponding to the zero \(z_{1} = - \frac{1}{2}\) has the following asymptotic representation:

$$\begin{aligned} u_{r} & = - \frac{{r_{1} }}{\rho }\eta C_{1} \left\langle {4\left( {1 + \nu_{1} \nu_{2} } \right)} \right. + \frac{1}{3}\varepsilon^{2} \left\{ {2\left( {1 - \nu_{1} \nu_{2} } \right)\left( {\eta^{2} + 3} \right)} \right. \\ & \quad + 2\left( {1 + \nu } \right)\left( {G_{0} - 1} \right)\left( {\eta^{2} - 3} \right) + 2\left[ {G_{0} E_{0} \left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)} \right. \\ & \quad - 2\left( {1 + \nu } \right)\left( {\nu_{1} - 2} \right) - 2\left. {\left( {1 - \nu_{1} \nu_{2} } \right)} \right]\left. {\left. {\left( {G_{0} - 1} \right)\eta^{2} } \right\} + \cdots } \right\rangle \\ u_{\theta } & = \frac{{r_{1} C_{1} }}{\rho }\varepsilon^{ - 1} \left\langle {2\left( {1 - \nu_{1} \nu_{2} } \right)} \right. + \frac{1}{2}\left\{ {\frac{1}{2}\left( {1 + \nu } \right)\left( {2 - \nu_{1} - 3\nu } \right)} \right. \\ & \quad + 2\left( {1 - \nu_{1} \nu_{2} } \right) + \left( {1 - 2\nu_{1} } \right)\left. {\left( {1 + \nu } \right)} \right]\eta^{2} + 4\left( {1 + \nu } \right)\left( {\nu_{1} - 1} \right) \\ & \quad \times \left. {\left. {\left( {G_{0} - 1} \right)} \right\}\varepsilon^{2} + \cdots } \right\rangle \\ \end{aligned}$$
(6.2.2)
$$\begin{aligned} \sigma_{r} & = \frac{{G_{1} C_{1} \eta }}{{\rho^{2} }}\left[ {2\left( {3\nu_{1} - 2} \right) + O\left( {\varepsilon^{2} } \right)} \right] \\ \sigma_{\varphi } & = \frac{{G_{1} C_{1} \eta }}{{\rho^{2} }}\left[ {2\left( {3E_{0} - 2\nu_{1} } \right) + O\left( {\varepsilon^{2} } \right)} \right] \\ \sigma_{\theta } & = O\left( {\varepsilon^{2} } \right), \quad \tau_{r\theta } = O\left( \varepsilon \right).\\ \end{aligned}$$

The remaining zeros of the first group for calculating displacements and stresses, if we represent them in a series with respect to powers of \(\varepsilon\), have the following form:

$$\begin{aligned} u_{r} & = \frac{1}{\sqrt \rho }\eta \sum\limits_{k = 2}^{4} {C_{k} \left\langle {\left( {2z_{{k_{0} }} - 3} \right)\left( {1 - \nu_{1} \nu_{2} } \right)} \right. + \frac{1}{3!}\left\{ {\left( {1 + \nu } \right)\left( {z_{{k_{0} }}^{2} + 2G_{0} - \frac{9}{4}} \right)} \right.} \\ & \quad \times \left( {2\nu_{1} z_{{k_{0} }} + \nu_{1} - 2} \right)\left( {\eta^{2} - 3} \right) + 2\left( {z_{{k_{0} }} - 3} \right)\left( {1 - \nu_{1} \nu_{2} } \right)\left( {\eta^{2} + 3} \right) \\ & \quad - \left( {z_{{k_{0} }} - \frac{3}{2}} \right)\left[ {4G_{0} E_{0} \left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)} \right.z_{{k_{0} }}^{2} - G_{0} E_{0} \left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right) \\ & \quad + 4\left( {1 - \nu_{1} \nu_{2} } \right) + 4\left( {1 + \nu } \right)\left. {\left( {\nu_{1} - 2} \right)} \right]\left( {G_{0} - 1} \right)\eta^{2} \left. { + \left. {2z_{{k_{0} }} \left( {1 - \nu_{1} \nu_{2} } \right)} \right\}\varepsilon^{2} + \cdots } \right\rangle \left( {z_{k} \ln \rho } \right) \\ u_{\theta } & = \frac{1}{\sqrt \rho }\varepsilon^{ - 1} \sum\limits_{k = 2}^{4} {C_{k} \left\langle {2\left( {1 - \nu_{1} \nu_{2} } \right) + \frac{1}{2}\left\{ {2\left( {1 - \nu_{1} \nu_{2} } \right) + \frac{1}{2}\left( {1 + \nu } \right)\left( {2 - 3\nu } \right)} \right.} \right.} \\ & \quad + 2\left( {2\nu_{1} - 1} \right)\left( {1 + \nu } \right)z_{{k_{0} - }} 2\nu_{1} \left. {\left( {1 + \nu } \right)z_{{k_{0} }}^{2} } \right]\eta^{2} + 2\left( {1 + \nu } \right)\left( {2E_{0} G_{0} - \nu_{1} } \right) \\ & \quad \times z_{{k_{0} }}^{2} + 4\left( {1 + \nu } \right)\left( {\nu_{1} - 1} \right)\left( {G_{0} - 1} \right) - \left( {1 - \nu } \right)\left. {\left. {\left( {E_{0} G_{0} - \nu_{1/2} } \right)} \right\}\varepsilon^{2} + \cdots } \right\rangle \exp \left( {z_{k} \ln \rho } \right) \\ \sigma_{r} & = \frac{{G_{1} }}{\rho \sqrt \rho }\eta \sum\limits_{k = 2}^{3} {C_{k} } \left[ {4E_{0} z_{{k_{0} }}^{2} + 4\left( {\nu_{1} - 2E_{0} } \right)z_{{k_{0} }} + 3E_{0} - 2\nu_{1} } \right. \\ & \quad + O\left. {\left( {\varepsilon^{2} } \right)} \right]\exp \left( {z_{k} \ln \rho } \right) \\ \sigma_{\varphi } & = - \frac{{G_{1} }}{\rho \sqrt \rho }\eta \sum\limits_{k = 2}^{3} {C_{k} } \left[ {4\nu_{1} z_{{k_{0} }}^{2} + 4\left( {\nu_{1} - 2\nu_{1} } \right)z_{{k_{0} }} + 3\nu_{1} - 2 + } \right. \\ & \quad + O\left. {\left( {\varepsilon^{2} } \right)} \right]\exp \left( {z_{k} \ln \rho } \right) \\ \sigma_{\theta } & = O\left( {\varepsilon^{2} } \right),\tau_{r\theta } = O\left( \varepsilon \right) \\ \rho & = r_{1}^{ - 1} r. \\ \end{aligned}$$
(6.2.3)

The second group of zeros describes the stress-strain state rapidly damping far from the edge of the plate. Expanding the solution of this group in powers of a small parameter \(\varepsilon\), we find the following asymptotic expressions:

$$\begin{aligned} u_{r} & = - \frac{{r_{1} \varepsilon }}{\sqrt \rho }\sum\limits_{n = 1}^{\infty } {B_{n} } \left[ {S_{2} \left( {b_{22} S_{2}^{2} + b_{12}^{2} + b_{12} - b_{11} b_{22} } \right)\cos S_{2} } \right.\delta_{n} \\ & \quad \times \sin S_{1} \delta_{n} \eta - S_{2} \left( {b_{22} S_{2}^{2} + b_{12}^{2} + b_{12} - b_{11} b_{22} } \right)\cos S_{1} \delta_{n} \sin S_{2} \delta_{n} \eta \\ & \quad + \left. {O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ u_{\theta } & = - \frac{{r_{1} \varepsilon S_{1} S_{2} }}{\sqrt \rho }\sum\limits_{n = 1}^{\infty } {B_{n} } \left[ {\left( {b_{22} S_{2}^{2} + b_{12}^{2} } \right)\cos S_{2} \delta_{n} \cos S_{1} \delta_{n} \eta } \right. \\ & \quad - \left( {b_{22} S_{1}^{2} + b_{12} } \right)\cos S_{1} \delta_{n} \cos S_{2} \delta_{n} \eta + \left. {O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ \sigma_{r} & = \frac{{G_{2} }}{\rho \sqrt \rho }\left( {b_{11} b_{22} - b_{12}^{2} } \right)S_{1} S_{2} \sum\limits_{n = 1}^{\infty } {B_{n} } \delta_{n} \left[ {S_{1} \cos S_{2} \delta_{n} \sin S_{1} \delta_{n} \eta } \right. \\ & \quad - \left. {S_{2} \cos S_{1} \delta_{n} \sin S_{2} \delta_{n} \eta + O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ \sigma_{\varphi } & = - \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {B_{n} } \delta_{n} \left[ {S_{2} \left( {b_{11} b_{22} - b_{12}^{2} - 2G_{0} - 2G_{0} b_{12} S_{1}^{2} } \right)} \right. \\ & \quad \times \cos S_{2} \delta_{n} \sin S_{1} \delta_{n} \eta - S_{1} \left( {b_{11} b_{22} - b_{12}^{2} - 2G_{0} b_{12} S_{2}^{2} } \right)\cos S_{1} \delta_{n} \\ & \quad \times \sin S_{2} \delta_{n} \eta \left. { + O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ \sigma_{r} & = - \frac{{G_{1} }}{\rho \sqrt \rho }\left( {b_{11} b_{22} - b_{12}^{2} } \right)\sum\limits_{n = 1}^{\infty } {B_{n} } \delta_{n} \left[ {S_{2} \cos S_{2} \delta_{n} \sin S_{1} \delta_{n} \eta } \right. \\ & \quad - \left. {S_{1} \cos S_{1} \delta_{n} \sin S_{2} \delta_{n} \eta + O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ \tau_{r\theta } & = \frac{{G_{2} }}{\rho \sqrt \rho }\left( {b_{11} b_{22} - b_{12}^{2} } \right)S_{1} S_{2} \sum\limits_{n = 1}^{\infty } {B_{n} } \left[ {\cos S_{2} \delta_{n} \sin S_{1} \delta_{n} \eta } \right. \\ & \quad - \left. {\cos S_{1} \delta_{n} \sin S_{2} \delta_{n} \eta + O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right), \\ \end{aligned}$$
(6.2.4)

where

$$\begin{aligned} & \left( {S_{2} - S_{1} } \right)\sin \left( {S_{2} + S_{1} } \right)\delta_{n} - \left( {S_{2} + S_{1} } \right)\sin \left( {S_{1} - S_{2} } \right)\delta_{n} = 0 \\ & u_{r} = \frac{{r_{1} }}{\sqrt \rho }\varepsilon \sum\limits_{n = 1}^{\infty } {\left[ {F_{1n} \left( \eta \right) + O\left( \varepsilon \right)} \right]} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ & u_{\theta } = \frac{{r_{1} }}{\sqrt \rho }\varepsilon \sum\limits_{n = 1}^{\infty } {\left[ {F_{2n} \left( \eta \right) + O\left( \varepsilon \right)} \right]} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ & \sigma_{r} = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {\left[ {b_{11} \delta_{n} F_{1n} \left( \eta \right) + b_{12} F^{\prime}_{2n} \left( \eta \right) + O\left( \varepsilon \right)} \right]} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ & \sigma_{\varphi } = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {\left[ {b_{12} \delta_{n} F_{1n} \left( \eta \right) + b_{23} F^{\prime}_{2n} \left( \eta \right) + O\left( \varepsilon \right)} \right]} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ & \sigma_{\theta } = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {\left[ {b_{12} \delta_{n} F_{1n} \left( \eta \right) + b_{22} F^{\prime}_{2n} \left( \eta \right) + O\left( \varepsilon \right)} \right]} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ & \tau_{r\theta } = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {\left[ {F^{\prime}_{1n} \left( \eta \right) + \delta_{n} F_{2n} \left( \eta \right) + O\left( \varepsilon \right)} \right]} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right), \\ \end{aligned}$$
(6.2.5)

where

$$\begin{aligned} F_{1n} \left( \eta \right) & = \left( {a_{1} \Delta_{1n} - a_{2} \Delta_{2n} } \right)\cos \beta \delta_{n} \eta sh\alpha \delta_{n} \eta \\ & \quad - \left( {a_{1} \Delta_{2n} - a_{2} \Delta_{1n} } \right)\sin \beta \delta_{n} \eta ch\alpha \delta_{n} \eta . \\ \end{aligned}$$
$$\begin{aligned} F_{2n} \left( \eta \right) & = \left( {b_{12} + 1} \right)\left[ {\left( {\beta \Delta_{2n} - \alpha \Delta_{1n} } \right)} \right.\cos \beta \delta_{n} \eta ch\alpha \delta_{n} \eta \\ & \quad + \left. {\left( {\alpha \Delta_{2n} + \beta \Delta_{1n} } \right)\sin \beta \delta_{n} \eta ch\alpha \delta_{n} \eta } \right]. \\ \end{aligned}$$
$$a_{1} = 1 - b_{22} \left( {\alpha^{2} - \beta^{2} } \right),a_{2} = 2b_{22} \alpha \beta ;$$
$$\begin{aligned} \Delta_{1n} & = D_{n} \left\{ {\alpha \left[ {b_{12} + b_{22} \left( {\alpha^{2} + \beta^{2} } \right)} \right]\sin \beta \delta_{n} sh\alpha \delta_{n} } \right. \\ & \quad + \beta \left. {\left[ {b_{12} + b_{22} \left( {\alpha^{2} + \beta^{2} } \right)} \right]\sin \beta \delta_{n} ch\alpha \delta_{n} } \right\}. \\ \end{aligned}$$
$$\begin{aligned} \Delta_{2n} & = - D_{n} \left\{ {\beta \left[ {b_{12} + b_{22} \left( {\alpha^{2} + \beta^{2} } \right)} \right]\sin \beta \delta_{n} sh\alpha \delta_{n} } \right. \\ & \quad - \beta \left. {\left[ {b_{12} + b_{22} \left( {\alpha^{2} + \beta^{2} } \right)} \right]\cos \beta \delta_{n} ch\alpha \delta_{n} } \right\}. \\ \end{aligned}$$
$$\alpha \sin 2\beta \delta_{n} - \beta sh2\alpha \delta_{n} = 0$$
$$\begin{aligned} u_{r} & = \frac{{\left( {b_{12} + 1} \right)}}{\sqrt \rho }\varepsilon \sum\limits_{n = 1}^{\infty } {E_{n} } \left\{ {\left[ {\sin p\delta_{n} - \left( {b_{12} p^{2} - b_{11} } \right)\left( {b_{12} \rho^{2} + b_{11} } \right)^{ - 1} } \right.} \right. \\ & \quad \times \left. {\frac{{\cos p\delta_{n} }}{{p\delta_{n} }}} \right]\sin p\delta_{n} \eta + \eta \cos p\delta_{n} \cos p\delta_{n} \eta + \left. {O\left( \varepsilon \right)} \right\}\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right). \\ \end{aligned}$$
$$\begin{aligned} u_{\theta } & = - \frac{{\rho^{2} - b_{11} }}{\sqrt \rho }\varepsilon \sum\limits_{n = 1}^{3} {E_{n} } \left\{ {\left[ {\sin p\delta_{n} - 2b_{11} \left( {b_{12} + 1} \right)\rho \left( {b_{11} - \rho^{2} } \right)^{ - 1} } \right.} \right. \\ & \quad \times \left( {b_{12} \rho^{2} + 1} \right)^{ - 1} \left. {\frac{{\cos p\delta_{n} }}{{p\delta_{n} }}} \right]\cos p\delta_{n} \eta + \eta \cos p\delta_{n} \sin p\delta_{n} \eta \\ & \quad + \left. {O\left( \varepsilon \right)} \right\}\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ \end{aligned}$$
$$\begin{aligned} \sigma_{r} & = \frac{{G_{1} \left( {b_{12} \rho^{2} - b_{11} } \right)}}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} } \left[ {\left( {p\delta_{n} \sin p\delta_{n} - \cos p\delta_{n} } \right)\sin p\delta_{n} } \right.\eta \\ & \quad + \left. {\eta p\delta_{n} \cos p\delta_{n} \cos p\delta_{n} \eta + O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) \\ \end{aligned}$$
$$\begin{aligned} \sigma_{\varphi } & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} } \left\langle {\left\{ {\left( {b_{23} \rho^{2} + b_{12} + b_{12} - b_{11} b_{23} } \right)p\delta_{n} \sin p\delta_{n} } \right.} \right. \\ & \quad - \left[ {b_{11} b_{23} + b_{12}^{2} + b_{12} - b_{11} - b_{23} \rho^{2} - 2b_{11} \left( {b_{12} \rho^{2} + 1} \right)} \right. \\ & \quad \left. { \times \left( {b_{23} \rho^{2} + b_{12} } \right)\left. {\left( {b_{12} \rho^{2} + b_{11} } \right)} \right]^{ - 1} \cos p\delta_{n} } \right\}\sin p\delta_{n} \eta \\ & \quad + \left( {b_{23} \rho^{2} + b_{12}^{2} + b_{12} - b_{11} b_{23} } \right)\eta p\delta_{n} \sin p\delta_{n} \sin p\delta_{n} \eta \\ & \quad \left. { + O\left( \varepsilon \right)} \right\rangle \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right). \\ \end{aligned}$$
(6.2.6)
$$\begin{aligned} \sigma_{\theta } & = \frac{{G_{1} \left( {b_{11} b_{22} - b_{12}^{2} } \right)}}{\rho \sqrt \rho }\sum\limits_{n = 1}^{3} {E_{n} } \left[ {\left( {p\delta_{n} \sin p\delta_{n} - \cos p\delta_{n} } \right)} \right.\sin p\delta_{n} \eta \\ & \quad + \left. {\eta p\delta_{n} \cos p\delta_{n} \cos p\delta_{n} \eta + O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right). \\ \end{aligned}$$
$$\begin{aligned} \tau_{r\theta } & = - \frac{{G_{1} \left( {b_{12} \rho^{2} + 1} \right)}}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} \delta_{n} } \left[ {\sin p\delta_{n} \cos p\delta_{n} \eta } \right. \\ & \quad - \left. {\eta \cos p\delta_{n} \sin p\delta_{n} \eta + O\left( \varepsilon \right)} \right]\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right). \\ \end{aligned}$$

where

$$\sin 2p\delta_{n} - 2p\delta_{n} = 0.$$

\(C_{k} ,B_{n} ,D_{n} ,E_{n}\) are arbitrary constants. From the comparison of the solutions of the first and second groups, we can conclude that the first group of solutions determines the main stress state, the second group the edge effect similar to the Saint-Venant edge effect in theory of constant thickness plates.

However, for large \(G_{0}\), some boundary layer solutions damp very weakly and they should be included into penetrating solutions.

Let us now study the stress state described by the homogeneous solutions (6.2.2)–(6.2.6). We consider the relation of homogeneous solutions with the principle stress vector P, acting in the section \(\rho = const\).

$$P = - 2\pi r_{1}^{2} \varepsilon \rho^{2} \int\limits_{ - 1}^{1} {\left( {\sigma_{r} \sin \varepsilon \eta - \tau_{r\theta } \cos \varepsilon \eta } \right)} \cos \varepsilon \eta d\eta$$
(6.2.7)

Assuming \(C_{0} = 0\), we represent the displacements and stresses in the form:

$$\begin{aligned} u_{r} & = u_{1} + \sum\limits_{k = 1}^{\infty } {C_{k} } U_{k} \left( \eta \right)\rho^{{z_{k} - \frac{1}{2}}} , \\ u_{\theta } & = w_{k} + \sum\limits_{k = 1}^{\infty } {C_{k} } W_{k} \left( \eta \right)\rho^{{z_{k} - \frac{1}{2}}} \\ \sigma_{r} & = Q_{{r_{1} }} + \sum\limits_{k = 1}^{\infty } {C_{k} Q_{rk} } \left( \eta \right)\rho^{{z_{k} - \frac{3}{2}}} , \\ \sigma_{\varphi } & = Q_{{\varphi_{1} }} + \sum\limits_{k = 1}^{\infty } {C_{k} Q_{\varphi k} } \left( \eta \right)\rho^{{z_{k} - \frac{3}{2}}} \\ \sigma_{\theta } & = Q_{{\theta_{1} }} + \sum\limits_{k = 1}^{\infty } {C_{k} Q_{\theta k} } \left( \eta \right)\rho^{{z_{k} - \frac{3}{2}}} , \\ \tau_{r\theta } & = T_{1} + \sum\limits_{k = 1}^{\infty } {C_{k} T_{k} } \left( \eta \right)\rho^{{z_{k} - \frac{3}{2}}} \\ \end{aligned}$$
(6.2.8)

In the formulas \(u_{1} , \ldots ,T_{1}\) correspond to eigen values \(z_{1} = - \frac{1}{2}\). The remaining solutions are in the second addend.

Substituting (6.2.8) in (6.2.7), we obtain:

$$P = C_{1} \gamma_{1} + \rho^{1/2} \sum\limits_{k = 2}^{\infty } {C_{k} } \rho^{{z_{k} - \frac{3}{2}}} \gamma_{k} ,$$
(6.2.9)

where

$$\begin{aligned} \gamma_{1} & = 16G_{1} \pi \left( {\nu_{1} - E_{0} } \right)r_{1}^{2} \varepsilon^{2} + O\left( {\varepsilon^{3} } \right) \\ \gamma_{k} & = - \pi G_{1} \varepsilon \int\limits_{ - 1}^{1} {\left[ {Q_{rk} \left( \eta \right)\sin \varepsilon \eta - T_{k} \left( \eta \right)\cos \varepsilon \eta } \right]} \cos \varepsilon \eta d\eta . \\ \end{aligned}$$

We prove that all \(\gamma_{k} \,\left( {k = 2,3, \ldots } \right)\) equal zero. To this end, we consider the following boundary value problem:

$$\begin{aligned} \sigma_{r} & = \rho_{1}^{{z_{k} - \frac{3}{2}}} Q_{rs} ,\quad \tau_{r\theta } = \rho_{1}^{{z_{k} - \frac{3}{2}}} T_{s} \quad \left( {\rho = \rho_{1} } \right) \\ \sigma_{r} & = \rho_{2}^{{z_{k} - \frac{3}{2}}} Q_{rs} ,\quad \tau_{r\theta } = \rho_{2}^{{z_{k} - \frac{3}{2}}} T_{s} \quad \left( {\rho = \rho_{2} } \right) \\ \end{aligned}$$
(6.2.10)

It is easy to see that the solution of the problem (6.2.10) exists and is obtained from formulas (6.2.8) if we put in them \(C_{k} = \delta_{ks}\), where \(\delta_{ks}\) is the Kronecker symbol.

On the other hand, it is known that the necessary condition of solvability of the first boundary value problem of theory of elasticity is inversion to zero of the principal vector and the principal moment of all external forces.

In the case under consideration, the principal vector of external forces (6.2.10) in the projection on the symmetry axis \(\theta = 0\) has the form:

$$P_{s} = \left( {\rho_{2}^{{z_{k} - \frac{3}{2}}} - \rho_{1}^{{z_{k} - \frac{3}{2}}} } \right)\gamma_{s} = 0$$
(6.2.11)

The last equality is possible only for \(\gamma_{s} = 0\).

For the main vector we finally get:

$$P = C_{1} \gamma_{1}$$
(6.2.12)

Thus, the stress state (6.2.3)–(6.2.6) is self-balanced in every section \(\rho = const\).

Now clarify the picture of stress state corresponding to the zeros \(z_{k} \,\left( {k \ge 2} \right)\). For that for calculate the bending moment in the section \(\rho = const\).

$$\begin{aligned} M & = 2\pi r_{1}^{2} \rho^{2} \varepsilon \int\limits_{ - 1}^{1} {\left[ {\sigma_{r} \sin \varepsilon \eta - \tau_{r\theta } \left( {1 - \cos \varepsilon \eta } \right)} \right]} \cos \varepsilon \eta d\eta \\ & \approx \pi r_{1}^{2} \rho^{2} \varepsilon^{2} \int\limits_{ - 1}^{1} {\eta \sigma_{r} d\eta + O\left( {\varepsilon^{4} } \right)} \\ \end{aligned}$$
(6.2.13)

We calculate the bending moment for stresses (6.2.8) and have

$$\begin{aligned} M_{1} & = \frac{2}{3}\pi r_{1}^{2} G\rho^{1/2} \varepsilon^{2} \sum\limits_{k = 2}^{3} {C_{k} } \left[ {4Ez_{{k_{0} }}^{2} + 4\left( {\nu_{1} - 2E_{0} } \right)z_{{k_{0} }} } \right. \\ & \quad + 3E_{0} - 2\nu_{1} + \left. {O\left( {\varepsilon^{2} } \right)} \right]\exp \left( {z_{k} \ln \rho } \right). \\ \end{aligned}$$
(6.2.14)

Prove that the principle part of the bending moment for stresses, corresponding to the second group of zeros equals zero. Let us consider the solution determined by the formula (6.2.4).

In the same way we consider the other cases.

$$\begin{aligned} M_{2} & = 2\pi r_{1}^{2} \rho^{2} \varepsilon^{2} \int\limits_{ - 1}^{1} {\eta \sigma_{r} } d\eta + O\left( {\varepsilon^{4} } \right) = 2\pi r_{1}^{2} \varepsilon^{2} G_{1} \left( {b_{11} b_{22} - b_{12}^{2} } \right) \\ & \quad \times \rho^{2} \rho^{ - 3/2} \left( {S_{2} \cos S_{2} \delta_{n} \sin S_{1} \delta_{n} - S_{1} \cos S_{1} \delta_{n} \sin S_{2} \delta_{n} } \right) \\ & \quad \times \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right) + O\left( {\varepsilon^{4} } \right) = 2\pi r_{1}^{2} \rho^{2} \varepsilon^{2} \sigma_{\theta } \left( { \pm 1} \right) + O\left( {\varepsilon^{4} } \right) \\ \end{aligned}$$

As \(\sigma_{\theta } \left( { \pm 1} \right) = 0\), we finally obtain \(M_{2} \approx O\left( {\varepsilon^{4} } \right)\).

Thus, the principal parts of the bending moment determine the solution of the first group.

Expanding the bending moment \(M_{k}^{s} \,\left( {k = 2,3} \right)\) acting on the surface \(\rho = \rho_{s}\), in series with respect to \(\varepsilon\)

$$M_{k}^{s} = M_{{k_{0} }}^{s} + M_{{k_{2} }}^{s} \varepsilon^{2} + \cdots$$
(6.2.15)

and finding \(C_{k}\) in the form \(C_{k} = C_{{k_{0} }} + \varepsilon^{2} C_{{k_{2} }} + \cdots\) for determining \(C_{{k_{0} }}\), we obtain the linear system:

$$\begin{aligned} & \frac{3}{2}\pi r_{1}^{2} G_{1} \rho_{s}^{1/2} \sum\limits_{k = 2}^{3} {C_{{k_{0} }} } \left[ {4E_{0} z_{{k_{0} }}^{2} + 4\left( {\nu_{1} - 2E_{0} } \right)z_{{k_{0} }} 3E_{0} - 2\nu_{1} } \right] \\ & \quad \times \exp \left( {z_{{k_{0} }} \ln \rho_{s} } \right) = M_{{k_{0} }}^{s} \quad \left( {s = 1,2} \right) \\ \end{aligned}$$
(6.2.16)

Thus, the constants \(C_{k}\) are determined by the principal parts of bending moments on the lateral surface of the plate.

The first term of the expansion (6.2.2) together with the first term of expansions (6.2.8), can be considered as the solution in applied theory. From (6.2.14), (6.2.16) we obtain that to the first terms of the asymptotics (6.2.4), (6.2.5), (6.2.6) there corresponds the stress state self-balanced in the section \(\rho = const\), and the solution itself has edge effect character equivalent to the Saint-Venant edge effect in theory of constant thickness plates.

We now consider removal of stresses from the lateral surfaces of the plate by means of a class of homogeneous solutions. Let conditions (1.1.2) be given on the lateral surface.

As was shown above, the principal parts of the principal vector and bending moments are determined by the solutions of the first group.

Therefore, below we will assume \(C_{k} = 0\,\left( {k = 1,2,3} \right)\) and consider the case (6.2.4). The other cases are considered similarly.

We will find the solution in the form (6.2.4). To determine the arbitrary constants \(B_{n}\), as in chapter I we use the Lagrange variational principle.

Since homogeneous solutions satisfy equilibrium equations and boundary conditions on the conical surface, the variational principle has the form:

$$r_{1} \varepsilon \sum\limits_{s = 1}^{2} {\rho_{s}^{2} \int\limits_{ - 1}^{1} {\left[ {\left( {\sigma_{r} - f_{1s} } \right)\delta u_{r} + \left( {\tau_{r\theta } - f_{2s} } \right)u_{\theta } } \right]_{{\rho = \rho_{s} }} } } \cos \varepsilon \eta d\eta = 0$$
(6.2.17)

Assuming \(\delta B_{n}\) independent variations, from (6.2.17) we obtain an infinite system of linear algebraic equations:

$$\begin{aligned} & \sum\limits_{k = 1}^{\infty } {M_{kn} } B_{n} = N_{k} \quad \left( {k = 1,2, \ldots } \right), \\ & M_{kn} = \sum\limits_{s = 1}^{2} {\exp \left( {z_{k} + z_{n} } \right)} \ln \rho_{s} \int\limits_{ - 1}^{1} {\left( {Q_{rn} u_{k} + T_{n} w_{k} } \right)} \cos \varepsilon \eta d\eta , \\ & N_{k} = \sum\limits_{s = 1}^{2} {\exp } \left[ {\left( {z_{k} + \frac{3}{2}} \right)\ln \rho_{s} } \right]\int\limits_{ - 1}^{1} {\left( {f_{1s} u_{k} + f_{2s} w_{k} } \right)} \cos \varepsilon \eta d\eta . \\ \end{aligned}$$
(6.2.18)

The solvability and convergence of the reduction method of the systems (6.2.18) follows from [4].

We find the unknown constants \(B_{n}\) in the form:

$$B_{n} = B_{{n_{0} }} + \varepsilon B_{{n_{1} }} + \cdots$$
(6.2.19)

Substituting (6.2.19) in (6.2.18), we obtain the following systems of infinite linear algebraic equations with respect to \(B_{{n_{0} }}\).

$$\sum\limits_{n = 1}^{\infty } {M_{kn} } B_{{n_{0} }} = H_{k} \quad \left( {k = 1,2, \ldots } \right)$$
(6.2.20)
$$\begin{aligned} M_{kn} & = + G\left( {b_{11} b_{22} - b_{12}^{2} } \right)S_{1} S_{2} \sum\limits_{s = 1}^{2} {\exp \left[ {\varepsilon^{ - 1} \left( {\delta_{n} + \delta_{n} } \right)\ln \rho_{s} } \right]} \\ & \quad \times \int\limits_{ - 1}^{1} {\left\{ {\delta_{n} \left( {S_{1} \cos S_{2} \delta_{n} \sin S_{1} \delta_{n} \eta - S_{2} \cos S_{1} \delta_{n} \sin S_{2} \delta_{n} \eta } \right)} \right.} \\ & \quad \times \left[ {S_{2} \left( {b_{22} S_{2}^{2} + b_{12}^{2} + b_{12} - b_{11} b_{22} } \right)\cos S_{2} \delta_{k} \sin S_{1} \delta_{k} \eta } \right] \\ & \quad - \left[ {S_{1} \left( {b_{22} S_{2}^{2} + b_{12}^{2} + b_{12} - b_{11} b_{22} } \right)\cos S_{1} \delta_{k} \sin S_{2} \delta_{k} \eta } \right] \\ & \quad - \left( {\cos S_{2} \delta_{n} \sin S_{1} \delta_{n} \eta - \cos S_{1} \delta_{n} \sin S_{2} \delta_{n} \eta } \right)S_{2} S_{1} \\ & \quad \times \left[ {\left( {b_{22} S_{2}^{2} + b_{12} } \right)\cos S_{2} \delta_{k} \cos S_{1} \delta_{k} \eta - \left( {b_{22} S_{1}^{2} + b_{12} } \right)} \right. \\ & \quad \times \left. {\left. {\cos S_{1} \delta_{k} \cos S_{2} \delta_{k} \eta } \right]} \right\}d\eta ; \\ \end{aligned}$$
(6.2.21)
$$\begin{aligned} H_{k} & = \sum\limits_{s = 1}^{2} {\rho_{s}^{3/2} } \exp \left( {\frac{{\delta_{k} }}{\varepsilon }\ln \rho_{s} } \right)\int\limits_{ - 1}^{1} {\left\{ {f_{1s} \left[ {S_{2} \left( {b_{22} S_{2}^{2} + b_{12}^{2} + b_{12} - b_{11} b_{22} } \right)} \right.} \right.} \\ & \quad \times \left. {\cos S_{2} \delta_{k} \sin S_{1} \delta_{k} \eta } \right] - \left[ {S_{1} \left( {b_{22} S_{1}^{2} + b_{12}^{2} + b_{12} - b_{11} b_{22} } \right)} \right. \\ & \quad \times \left. {\cos S_{1} \delta_{k} \sin S_{2} \delta_{k} \eta } \right] - f_{2k} S_{2} S_{1} \left[ {\left( {b_{22} S_{2}^{2} + b_{12} } \right)\cos S_{2} \delta_{k} } \right. \\ & \quad \times \cos S_{1} \delta_{k} \eta - \left. {\left. {\left( {b_{22} S_{1}^{2} + b_{12} } \right)\cos S_{1} \delta_{k} \cos S_{2} \delta_{k} \eta } \right]} \right\}d\eta. \\ \end{aligned}$$

The matrix of the system (6.2.20) was already encountered in theory of a constant thickness transtropic plate.

The definition of \(B_{{n_{i} }} \,\left( {i = 1,2, \ldots } \right)\) is invariably reduced to the inversion of one and the same matrices that coincide with the matrix (6.2.20).

In conclusion note that for \(G_{0} = 1\) we obtain the results of [3] in the case of bending of a variable thickness isotropic plate.

6.3 Constructing Applied Theories for a Variable Thickness Transtropic Plate

In this section of the chapter we offer a two-dimensional applied theory for removing stresses from the conical surface of a variable thickness transtropic plate.

Homogeneous solutions admitting to remove loads from the spherical part of a variable thickness transtropic plate were constructed above.

In this work, by the methods developed in [3], we construct applied theory for removing stresses from the conical surface of the plate.

We illustrate construction of such theories for determining displacements \(u_{r}\), \(u_{\theta }\), concerning the stresses, they can be determined by means of the generalized Hooke law.

Let the plate be referred to the spherical system of coordinates \(r,\theta ,\varphi\)

$$r_{1} \le r \le r_{2} , \quad \frac{\pi }{2} - \varepsilon \le \theta \le \frac{\pi }{2} + \varepsilon , \quad 0 \le \varphi \le 2\pi$$

Write the equilibrium equations in displacements

$$\begin{aligned} & b_{11} \frac{\partial }{\partial t}\left( {\frac{\partial }{\partial t} + 1} \right)u_{r} + 2\left( {b_{12} - b_{22} - b_{23} } \right)u_{r} + \frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left( {\sin \theta \frac{{\partial u_{r} }}{\partial \theta }} \right) \\ & \quad + \left( {b_{12} + 1} \right)\frac{{\partial^{2} }}{\partial t\partial \theta }\left( {\sin \theta u_{\theta } } \right) + \left( {b_{12} - b_{22} - b_{23} } \right)\frac{1}{\sin \theta }\left( {\sin \theta u_{\theta } } \right) = 0 \\ & \left( {b_{12} + 1} \right)\frac{{\partial^{2} u_{r} }}{\partial t\partial \theta } + \left( {b_{22} + b_{23} + 2} \right)u_{r} + \frac{\partial }{\partial t}\left( {\frac{\partial }{\partial t} + 1} \right)u_{\theta } \\ & \quad + b_{22} \frac{\partial }{\partial \theta }\left( {\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left( {\sin \theta u_{\theta } } \right)} \right) + \left( {b_{22} - b_{23} - 2} \right)u_{\theta } = 0, \\ \end{aligned}$$
(6.3.1)

where \(t = \ln r\).

Assume that on the conical boundaries the conditions \(\sigma_{\theta } = Q_{n} \left( t \right)\), \(\tau_{r\theta } = \tau_{n} \left( t \right)\) are given for

$$\theta = \frac{\pi }{2} + \left( { - 1} \right)^{n} \varepsilon \quad \left( {n = 1,2} \right)$$
(6.3.2)

We shall not clarify the character of boundary conditions on spherical boundaries.

Assuming \(\frac{\partial }{\partial t} = \lambda\) and using the results of Chap. 5, we represent the general solution of this problem in the form:

$$\begin{aligned} u_{r} & = \left[ {A_{1} \psi_{{\gamma_{1} }} \left( \theta \right) + A_{2} \psi_{{\gamma_{2} }} \left( \theta \right)} \right]^{{}} \\ u_{\theta } & = b_{0} \left[ {\psi^{\prime}_{{\gamma_{1} }} \left( \theta \right) + \psi^{\prime}_{{\gamma_{2} }} \left( \theta \right)} \right]\,, \\ \end{aligned}$$
(6.3.3)

where

$$\begin{aligned} & \psi_{\gamma } \left( \theta \right) = C_{\gamma } T_{\gamma } \left( \theta \right) + B_{\gamma } F_{\gamma } \left( \theta \right) \\ & T_{\gamma } \left( \theta \right) = P_{\gamma } \left( {\cos \theta } \right) + P_{\gamma } \left( { - \cos \theta } \right) \\ & F_{\gamma } \left( \theta \right) = P_{\gamma } \left( {\cos \theta } \right) - P_{\gamma } \left( { - \cos \theta } \right) \\ & A_{i} = - b_{22} \gamma_{2} \left( {\gamma_{2} + 1} \right) + \lambda \left( {\lambda + 1} \right) + 2\left( {G_{0} - 1} \right) \\ & b_{0} = - \left[ {\left( {b_{12} + 1} \right)\lambda + b_{22} + b_{23} + 2} \right] \\ \end{aligned}$$

\(P_{\gamma } \left( {\cos \theta } \right)\) are the Legendre functions, \(\gamma\) are the roots of the biquadratic equation:

$$\begin{aligned} & b_{22} \gamma^{2} \left( {\gamma + 1} \right)^{2} - \left[ {\left( {b_{11} b_{22} + b_{12}^{2} - 2b_{12} } \right)\lambda \left( {\lambda + 1} \right) + 2b_{22} } \right. \\ & \quad + 2\left( {b_{12} - b_{22} - b_{23} } \right)\left. {\left( {G_{0} - 1} \right)} \right]\gamma \left( {\gamma + 1} \right) + \left[ {\lambda \left( {\lambda + 1} \right) + 2\left( {G_{0} - 1} \right)} \right] \\ & \quad \times \left[ {b_{11} \lambda \left( {\lambda + 1} \right) + 2\left( {b_{12} - b_{22} - b_{23} } \right)} \right] = 0, \\ \end{aligned}$$
(6.3.4)

\(b_{ij}\) are the material constants, \(G_{0} = GG_{1}^{ - 1}\), \(E_{0} = E_{1} E^{ - 1}\).

Such a form admits to partition the general problem for a plate into two independent problems.

Let us consider the quantities

$$\begin{aligned} q_{1} & = \frac{1}{2}\left( {Q_{1} + Q_{2} } \right),\quad q_{2} = \frac{1}{2}\left( {Q_{2} + Q_{1} } \right) \\ S_{1} & = \frac{1}{2}\left( {\tau_{1} + \tau_{2} } \right),\quad S_{2} = \frac{1}{2}\left( {\tau_{2} + \tau_{1} } \right) \\ \end{aligned}$$
(6.3.5)

By their means we partition the general boundary value problem into two problems:

Problem A.

$$\sigma_{\theta } = q_{1} ,\tau_{r \theta } = \left( { - 1} \right)^{\eta } S, \quad {\text{for}}\,\theta = \frac{\pi }{2} + \left( { - 1} \right)^{n} \varepsilon$$
(6.3.6)
$$\sigma_{\theta } = \left( { - 1} \right)^{\eta } q_{2} ,\tau_{r\theta } = S_{2} , \quad {\text{for}}\,\theta = \frac{\pi }{2} + \left( { - 1} \right)^{n} \varepsilon$$
(6.3.7)

By the character of symmetry with respect to the plane \(\theta = \frac{\pi }{2}\) problem A can be called a plate tension-compression problem, B a plate bending problem.

Arbitrary constants \(C_{{\gamma_{1} }}\), \(C_{{\gamma_{2} }}\), \(B_{{\gamma_{1} }}\), \(B_{{\gamma_{2} }}\) are determined from the boundary conditions (6.3.6), (6.3.7)

$$C_{{\gamma_{1} }} = \Delta_{1}^{ - 1} \Delta_{11} , \quad C_{{\gamma_{2} }} = \Delta_{1}^{ - 1} \Delta_{12}$$
(6.3.8)
$$B_{{\gamma_{1} }} = \Delta_{2}^{ - 1} \Delta_{21} , \quad B_{{\gamma_{2} }} = \Delta_{2}^{ - 1} \Delta_{22}$$
(6.3.9)

Substituting (6.3.8), (6.3.9) in (6.3.3), for problem A we obtain

$$\begin{aligned} \Delta_{1} u_{r} & = A_{1} T_{{\gamma_{1} }} \left( \theta \right)\Delta_{11} + A_{2} T_{{\gamma_{2} }} \left( \theta \right)\Delta_{12} \\ \Delta_{1} u_{\theta } & = b_{0} \left[ {T^{\prime}_{{\gamma_{1} }} \left( \theta \right)\Delta_{11} + T^{\prime}_{{\gamma_{2} }} \left( \theta \right)\Delta_{12} } \right] \\ \end{aligned}$$
(6.3.10)

for problem B

$$\begin{aligned} \Delta_{2} u_{r} & = A_{1} F_{{\gamma_{1} }} \left( \theta \right)\Delta_{21} + A_{2} F_{{\gamma_{2} }} \left( \theta \right)\Delta_{22} \\ \Delta_{2} u_{\theta } & = b_{0} \left[ {F^{\prime}_{{\gamma_{1} }} \left( \theta \right)\Delta_{21} + F^{\prime}_{{\gamma_{2} }} \left( \theta \right)\Delta_{22} } \right] \\ \end{aligned}$$
(6.3.11)

where

$$\begin{aligned} \Delta_{1} & = C_{11} d_{11} T_{{\gamma_{1} }} \left( {\theta_{1} } \right)T^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right) - C_{12} d_{11} T_{{\gamma_{2} }} \left( {\theta_{1} } \right)T^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right) \\ & \quad + C_{13} \left( {d_{12} - d_{11} } \right)ctg\theta_{1} T^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right)T^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right) \\ \Delta_{2} & = C_{11} d_{12} F_{{\gamma_{1} }} \left( {\theta_{1} } \right)F^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right) - C_{12} d_{11} F_{{\gamma_{2} }} \left( {\theta_{1} } \right)F^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right) \\ & \quad + C_{13} \left( {d_{12} - d_{11} } \right)ctg\theta_{1} F^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right)F^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right) \\ \end{aligned}$$
$$\begin{aligned} \Delta_{11} & = d_{22} T^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right)q_{1} - \left[ {C_{12} T_{{\gamma_{2} }} \left( {\theta_{1} } \right) + C_{13} ctg\theta_{1} T^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right)} \right]S_{1} \\ \Delta_{12} & = - d_{11} T^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right)q_{1} - \left[ {C_{12} T_{{\gamma_{1} }} \left( {\theta_{1} } \right) + C_{13} ctg\theta_{1} T^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right)} \right]S_{1} \\ \end{aligned}$$
(6.3.12)
$$\begin{aligned} \Delta_{21} & = d_{22} F^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right)q_{2} - \left[ {C_{12} F_{{\gamma_{2} }} \left( {\theta_{1} } \right) + C_{13} ctg\theta_{1} F^{\prime}_{{\gamma_{2} }} \left( {\theta_{1} } \right)} \right]S_{2} \\ \Delta_{22} & = - d_{11} F^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right)q_{2} - \left[ {C_{12} F_{{\gamma_{1} }} \left( {\theta_{1} } \right) + C_{13} ctg\theta_{1} F^{\prime}_{{\gamma_{1} }} \left( {\theta_{1} } \right)} \right]S_{2} \\ C_{1p} & = \left( {b_{12} \lambda + b_{22} + b_{23} } \right)A_{p} - b_{22} b_{0} \gamma_{p} \left( {\gamma_{p} + 1} \right)\,\left( {p = 1,2} \right) \\ C_{13} & = - \left( {b_{22} - b_{23} } \right)b_{0} ,d_{1k} = A_{k} + \left( {\lambda - 1} \right)b_{0} \left( {k = 1,2} \right) \\ \theta & = \frac{\pi }{2} - \varepsilon . \\ \end{aligned}$$
(6.3.13)

Substituting (6.3.12), (6.3.13) in (6.3.10), (6.3.11) and expanding with respect to \(\varepsilon\), we obtain:

$$\begin{aligned} G_{1} \Delta _{1} u_{r} & = 2\left( {1 - \nu _{1} \nu _{2} } \right)\left( {1 + \nu } \right)^{{ - 1}} S_{1} \left( t \right) + \left( {2 - \nu _{1} - 2\nu _{1} z} \right)q_{1} \left( t \right)\varepsilon + \cdots \\G_{1} \Delta_{1} u_{\theta } & = \varepsilon \eta \left\{ {\left( {2\nu_{1} z + 2 - \nu_{1} } \right)S_{1} \left( t \right)} \right. \\ & \quad + \left[ {2E_{0} \left( {1 - \nu } \right)z^{2} - \frac{{\left( {1 - \nu } \right)}}{2}E_{0} - 4\left( {1 - \nu_{1} } \right)} \right]\left. {q_{1} \left( t \right)\varepsilon + \cdots } \right\} \\ \end{aligned}$$
(6.3.14)
$$\begin{aligned} G_{1} \Delta_{2} u_{r} & = \left( {1 - \nu_{1} \nu_{2} } \right)\varepsilon \eta \left[ {\left( {2z - 3} \right)q_{2} \left( t \right) - 2\left( {z^{2} - 9/4} \right)S_{2} \left( t \right)\varepsilon^{2} + \cdots } \right] \\ G_{1} \Delta_{2} u_{r} & = \left( {1 - \nu_{1} \nu_{2} } \right)\left[ { - 2q_{2} \left( t \right) + \left( {2z + 3} \right)S_{2} \left( t \right)\varepsilon + \cdots } \right] \\ \end{aligned}$$
(6.3.15)

where

$$\theta = \frac{\pi }{2} + \varepsilon , \quad - 1 \le \eta \le 1, \quad z = \lambda - \frac{1}{2}.$$
$$\begin{aligned} \Delta_{1} & = \varepsilon G_{0} \left\langle {4E_{0} z^{2} - 4\left( {1 - \nu_{1} } \right) - E_{0}^{2} + 3^{ - 1} \left( {1 - \nu_{1} \nu_{2} } \right)^{ - 1} } \right. \\ & \quad \times \left\{ { - 4\left( {1 + \nu } \right)\left( {E_{0} G_{0} - \nu_{1} } \right)E_{0} z^{4} + \left( {1 + \nu } \right)\left[ {2\left( {E_{0} G_{0} - \nu_{1} } \right)} \right.} \right.E_{0} \\ & \quad + 4\left( {1 + \nu_{1} } \right)\left( {E_{0} G_{0} - \nu_{1} } \right) + \left. {8\left( {1 - \nu } \right)\left( {1 - \nu_{1} \nu_{2} } \right)E_{0} } \right]z^{2} + 4^{ - 1} \left( {1 + \nu } \right) \\ & \quad \times \left( {E_{0} + 4 - 4\nu_{1} } \right)\left( {E_{0} G_{0} - \nu_{1} + 4 + 4G_{0} } \right) + 2\left( {1 - \nu } \right)\left( {1 - \nu_{1} \nu_{2} } \right)E_{0} \\ & \quad + 16\left( {1 - \nu_{1} } \right)\left. {\left. {\left( {1 - \nu_{1} \nu_{2} } \right)} \right\} + \cdots } \right\rangle \\ \Delta_{2} & = G3^{ - 1} \left( {z^{2} - 1/4} \right)\varepsilon^{3} \left\langle {4z^{2} + 2\nu_{2} - 9 - 4E_{0}^{ - 1} } \right. + \frac{1}{{5\left( {1 - \nu_{1} \nu_{2} } \right)}}\varepsilon^{2} \\ & \quad \times \left\{ { - 4\left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)z^{4} + 2\left[ {2\left( {1 - \nu_{1} \nu_{2} \left( {3 - 2\nu } \right)} \right) + \left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)} \right.} \right. \\ & \quad \times \left( {E_{0} + 4E_{0} G_{0} + 2 - 6\nu_{1} } \right) - 2\left( {1 + \nu } \right)\left( {2E_{0} G_{0} - \nu_{1} - 1} \right)\left. {\left( {G_{0} - 1} \right)} \right]z^{2} \\ & \quad - \left( {1 - \nu_{1} \nu_{2} } \right)\left( {40E_{0} G_{0} - 60\nu_{1} - 2\nu + 23} \right) - \frac{1}{2}\left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right) \\ & \quad \times \left( {2E_{0} G_{0} + 2 - 6\nu_{1} + \frac{{E_{0} }}{2}} \right) + \left( {1 + \nu } \right)\left( {2E_{0} G_{0} - \nu_{1} - 1} \right)\left( {G_{0} - 1} \right) \\ & \quad + 4\left[ {\left( {1 + \nu } \right)\left( {\nu_{1} - 1} \right)\left( {2E_{0} G_{0} - 3\nu_{1} + 1} \right) + 2\left( {1 - \nu_{1} \nu_{2} } \right)\left( {3 - 2\nu } \right)E_{0} } \right] \\ & \quad \times \left( {G_{0} - 1} \right) + 8\left( {1 + \nu } \right)\left( {1 - \nu_{1} } \right)E_{0} \left. {\left( {G_{0} - 1} \right)^{2} } \right\}. \\ \end{aligned}$$

As was noted in [3], the relations (6.3.14), (6.3.15) can be used for constructing applied theories for removing stresses from the conical part of the plate. Knowing \(t = \ln r\) and \(z = r\frac{d}{dr} + \frac{1}{2}\), we obtain the following systems of ordinary differential equations:

for problem A

$$\begin{aligned} G\varepsilon D_{1} u_{r} & = 2\left( {1 - \nu_{1} \nu_{2} } \right)\left( {1 + \nu } \right)^{ - 1} S_{1} \left( r \right) + \left( {2 - 2\nu_{1} - 2\nu_{1} r\frac{d}{dr}q_{1} \left( r \right)\varepsilon + \cdots } \right) \\ G\varepsilon D_{1} u_{\theta } & = \varepsilon \eta \left\{ {\left( {2\nu_{1} r\frac{d}{dr} + 2} \right)S_{1} \left( r \right) + \left[ {2E_{0} \left( {1 - \nu } \right)\left( {r^{2} \frac{{d^{2} }}{{dr^{2} }} + 2r\frac{d}{dr}} \right)} \right.} \right. \\ & \quad - \left. {\left. {4\left( {1 - \nu_{1} } \right)} \right]q_{1} \left( r \right)\varepsilon + \cdots } \right] \\ \end{aligned}$$
(6.3.16)

for problem B

$$\begin{aligned} GD_{2} u_{r} & = \varepsilon \eta \left( {1 - \nu_{1} \nu_{2} } \right)\left[ {\left( {2r\frac{d}{dr} - 2} \right)q_{2} \left( r \right)} \right. \\ & \quad - \left. {2\left( {r^{2} \frac{{d^{2} }}{{dr^{2} }} + 2r\frac{d}{dr} - 2} \right)S_{2} \left( r \right)\varepsilon + \cdots } \right] \\ GD_{2} u_{\theta } & = \left( {1 - \nu_{1} \nu_{2} } \right)\left[ { - 2q_{2} \left( r \right) + 2\left( {r\frac{d}{dr} + 2} \right)S_{2} \left( r \right)\varepsilon + \cdots } \right] \\ D_{1} & = 4E_{0} \left( {r^{2} \frac{{d^{2} }}{{dr^{2} }} + 2r\frac{d}{dr}} \right) - 4\left( {1 - \nu_{1} } \right) + 3^{ - 1} \left( {1 - \nu_{1} \nu_{2} } \right)^{ - 1} \\ & \quad \times \left\{ { - 4\left( {1 + \nu } \right)\left( {E_{0} G_{0} - \nu_{1} } \right)E_{0} d_{1}^{2} + \left( {1 + \nu } \right)\left[ {2\left( {E_{0} G_{0} - \nu_{1} } \right)E_{0} } \right.} \right. \\ & \quad + 4\left( {1 - \nu_{1} } \right)\left( {G_{0} - 1} \right)E_{0} + 4\left( {1 - \nu_{1} } \right)\left( {E_{0} G_{0} - \nu_{1} } \right) + 8\left( {1 - \nu } \right) \\ & \quad \times \left. {\left( {1 - \nu_{1} \nu_{2} } \right)E_{0} } \right]d_{1} + 4^{ - 1} \left( {1 + \nu } \right)\left( {E_{0} + 4 - 4\nu_{1} } \right)\left( {E_{0} G_{0} - \nu_{1} } \right. \\ & \quad + \left. {4 - 4G_{0} } \right) + 2\left( {1 - \nu } \right)\left( {1 - \nu_{1} \nu_{2} } \right)E_{0} + 16\left( {1 - \nu_{1} } \right)\left. {\left( {1 - \nu_{1} \nu_{2} } \right)} \right\}\varepsilon^{2} + \cdots \\ D_{2} & = G3^{ - 1} \left( {r^{2} \frac{{d^{2} }}{{dr^{2} }} + 2r\frac{d}{dr}} \right)\left\langle {4\left( {r^{2} \frac{{d^{2} }}{{dr^{2} }} + 2r\frac{d}{dr}} \right) + 2\nu_{1} } \right. - 8 \\ & \quad - 4E_{0}^{ - 1} + 5^{ - 1} \left( {1 - \nu_{1} \nu_{2} } \right)^{ - 1} \left\{ { - 4\left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)d_{1}^{2} + 2\left[ {2\left( {1 - \nu_{1} \nu_{2} } \right)} \right.} \right. \\ & \quad \times \left( {3 - 2\nu } \right) + \left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)\left( {E_{0} + 4E_{0} + 2 - 6\nu_{1} } \right) - 2\left( {1 + \nu } \right) \\ & \quad \times \left( {2E_{0} G_{0} - \nu_{1} - 1} \right)\left. {\left( {G_{0} - 1} \right)} \right]d_{1} - \left( {1 - \nu_{1} \nu_{2} } \right)\left( {40E_{0} G_{0} - 60\nu_{1} } \right. \\ & \quad - \left. {2\nu + 23} \right) - \frac{1}{2}\left( {1 + \nu } \right)\left( {G_{0} - \nu_{2} } \right)\left( {2E_{0} G_{0} + 2 - 6\nu_{1} + \frac{{E_{0} }}{2}} \right) + \left( {1 + \nu } \right) \\ & \quad \times \left( {2E_{0} G_{0} - \nu_{1} - 1} \right)\left( {G_{0} - 1} \right) + 4\left[ {\left( {1 - \nu } \right)\left( {\nu_{1} - 1} \right)\left( {2E_{0} G_{0} - 3\nu_{1} + 1} \right)} \right. \\ & \quad + 2\left. {\left( {1 + \nu_{1} \nu_{2} } \right)\left( {3 - 2\nu } \right)E_{0} } \right]\left( {G_{0} - 1} \right) + 8\left( {1 + \nu } \right)\left( {1 - \nu_{1} } \right)\left. {E_{0} \left( {G_{0} - 1} \right)^{2} } \right\}\varepsilon^{2} + \cdots \\ d_{1} & = r^{2} \frac{{d^{2} }}{{dr^{2} }} + 2r\frac{d}{dr} + \frac{1}{4}. \\ \end{aligned}$$
(6.3.17)

Rejecting from the left and right the terms of highest order smallness than \(\varepsilon^{n}\), we obtain applied theory of a variable thickness transtropic plate. Thus, we obtain a number of applied theories with preassigned accuracy with respect to \(\varepsilon\).

Retaining in (6.3.16), (6.3.17) more number of terms, we will obtain more exact theories.

Note that the given applied theories, generally speaking, are only for removal of stresses from the conical part of the boundary, as removal of stresses from the spherical part of the boundary is performed by homogeneous solutions.

In conclusion note that for \(G_{0} = 1\) we get the results of [3] in the isotropic case as well.

6.4 Constructing Homogeneous Solutions to a not Axially-Symmetric Problem of Elasticity Theory for a Variable Thickness Transtropic Plate

The plate is referred to the spherical system of coordinates \(r,\theta ,\varphi\) varying within:

$$r_{1} \le r \le r_{2} , \quad \pi /2 - \varepsilon \le \theta \le \pi /2 + \varepsilon , \quad 0 \le \varphi \le 2\pi$$

Assume that on the end faces of the plate the following homogeneous boundary conditions are given

$$\sigma_{\theta } = 0, \quad \tau_{r\theta } = 0, \quad\tau_{\theta \varphi } = 0 \quad{\text{for}}\;\,\theta = \theta_{n} \quad \left( {n = 1,2} \right)$$
(6.4.1)

From the lateral surfaces, the following load acts on the plate

$$\sigma_{r} = q_{r}^{\left( s \right)} \left( {\theta ,\varphi } \right), \quad \tau_{r\theta } = q_{r\theta }^{\left( s \right)} \left( {\theta ,\varphi } \right), \quad\tau_{r\varphi } = q_{r\varphi }^{\left( s \right)} \left( {\theta ,\varphi } \right),$$
(6.4.2)

for \(r = r_{s} \,\left( {s = 1,2} \right)\).

Using the results of Chap. 2 of Sect. 2.7, we look for the solution of the system (2.7.16–2.7.19) in the form:

$$\begin{aligned} u_{r} & = r^{\lambda } u\left( \theta \right)e^{im\varphi } \\ \phi & = r^{\lambda - 1} \upsilon \left( \theta \right)e^{im\varphi } \\ \psi & = ir^{\lambda - 1} w\left( \theta \right)e^{im\varphi } \\ i & = \sqrt { - 1} \\ \end{aligned}$$
(6.4.3)

Substituting (6.4.3) in (2.7.16)–(2.7.19), (6.4.1), after separation of variables we obtain

$$\begin{aligned} & \left[ {b_{11} \lambda \left( {\lambda + 1} \right) + 2\left( {b_{12} - b_{22} - b_{23} } \right)} \right]\,u + u^{\prime\prime} + ctg\theta u^{\prime} - \frac{{m^{2} }}{{\sin^{2} \theta }}u \\ & \quad + \left[ {\left( {b_{12} + 1} \right)\lambda + b_{12} - b_{22} - b_{23} } \right]\,\left( {\upsilon^{\prime\prime} + ctg\theta \upsilon^{\prime} - \frac{{m^{2} }}{{\sin^{2} \theta }}\upsilon } \right) = 0 \\ & \left[ {\left( {b_{12} + 1} \right)\lambda + b_{22} + b_{23} + 2} \right]u + \left[ {\lambda \left( {\lambda + 1} \right) + 2\left( {G_{0} - 1} \right)} \right]\upsilon + b_{22} \\ & \quad \times \left( {\upsilon^{\prime\prime} + ctg\upsilon^{\prime} - \frac{{m^{2} }}{{\sin^{2} \theta }}\upsilon } \right) = 0 \\ \end{aligned}$$
(6.4.4)
$$\begin{aligned} & \left[ {\lambda \left( {\lambda + 1} \right)2\left( {G_{0} - 1} \right)} \right]w + G_{0} \left( {w^{\prime\prime} + ctg\theta w^{\prime} - \frac{{m^{2} }}{{\sin^{2} \theta }}w} \right) = 0 \\ & \left[ {\left( {b_{12} \lambda + b_{22} + b_{23} } \right)u + b_{22} \upsilon^{\prime\prime} + b_{23} ctg\theta \upsilon^{\prime} - \frac{{b_{23} }}{{\sin^{2} \theta }}} \right.m^{2} \upsilon \\ & \quad - 2G_{0} m\left. {\left( {\frac{w}{\sin \theta }} \right)} \right]_{{\theta = \theta_{n} }} = 0 \\ & \left[ {u^{\prime} + \left( {\lambda - 1} \right)\upsilon^{\prime} - \frac{{\left( {\lambda - 1} \right)}}{{\sin^{2} \theta }}w} \right]_{{\theta = \theta_{n} }} = 0 \\ & \left[ {2m\left( {\frac{1}{{\sin^{2} \theta }}\upsilon } \right)^{\prime } - \sin \theta \left( {\frac{{w^{\prime}}}{{\sin^{2} \theta }}} \right)^{\prime } - \frac{{m^{2} }}{{\sin^{2} \theta }}w} \right]_{{\theta = \theta_{n} }} = 0 \\ \end{aligned}$$
(6.4.5)

Not going to the details, we give the final solution of Eq. (6.4.4)

$$\begin{aligned} u\left( \theta \right) & = A_{1} Z_{{\gamma_{1} }} \left( \theta \right) + A_{2} Z_{{\gamma_{2} }} \left( \theta \right) \\ \upsilon \left( \theta \right) & = b_{0} \left[ {Z_{{\gamma_{1} }} \left( \theta \right) + Z_{{\gamma_{2} }} \left( \theta \right)} \right] - \frac{m}{\sin \theta }F_{{\gamma_{2} }} \left( \theta \right) \\ w\left( \theta \right) & = Z_{{\gamma_{3} }} \left( \theta \right) \\ \end{aligned}$$
(6.4.6)

Here

$$\begin{aligned} Z_{\gamma } \left( \theta \right) & = C_{\gamma } T_{\gamma } \left( \theta \right) + B_{\gamma } F_{\gamma } \left( \theta \right) \\ T_{\gamma } \left( \theta \right) & = P_{\gamma }^{\left( m \right)} \left( {\cos \theta } \right) + P_{\gamma }^{\left( m \right)} \left( { - \cos \theta } \right) \\ & = P_{\gamma }^{\left( m \right)} \left( {\sin \varepsilon \eta } \right) + P_{\gamma }^{\left( m \right)} \left( { - \sin \varepsilon \eta } \right) \\ F_{\gamma } \left( \theta \right) & = P_{\gamma }^{\left( m \right)} \left( {\cos \theta } \right) - P_{\gamma }^{\left( m \right)} \left( { - \cos \theta } \right) \\ & = - \left[ {P_{\gamma }^{\left( m \right)} \left( {\sin \varepsilon \eta } \right) - P_{\gamma }^{\left( m \right)} \left( { - \sin \varepsilon \eta } \right)} \right] \\ \theta & = \pi /2 + \varepsilon \eta , - 1 \le \eta \le 1 \\ \end{aligned}$$

\(\gamma_{1} ,\gamma_{2}\) are the roots of the biquadratic equation

$$\begin{aligned} & b_{22} \gamma^{2} \left( {\gamma + 1} \right)^{2} - \left[ {\left( {b_{11} b_{22} - b_{12}^{2} - 2b_{12} } \right)\left( {z^{2} - \frac{1}{4}} \right)} \right. \\ & \quad + \left. {2b_{22} + 2\left( {b_{12} - b_{22} - b_{23} } \right)\left( {G_{0} - 1} \right)} \right]\gamma \left( {\gamma + 1} \right) \\ & \quad + \left[ {b_{11} \left( {z^{2} - \frac{1}{4}} \right) + 2\left( {b_{12} - b_{22} - b_{23} } \right)} \right]\left[ {z^{2} - \frac{1}{4} + 2\left( {G_{0} - 1} \right)} \right] = 0 \\ & A_{k} = b_{22} \gamma_{k} \left( {\gamma_{k} + 1} \right) + z^{2} - \frac{1}{4} + 2\left( {G_{0} - 1} \right) \\ & b_{0} = - \left[ {\left( {b_{12} + 1} \right)\left( {z - \frac{1}{2}} \right) + b_{22} + b_{23} + 2} \right] \\ & z = \lambda + \frac{1}{2};\gamma_{3} = \frac{1}{{G_{0} }}\left[ {z^{2} - \frac{1}{4} + 2\left( {G_{0} - 1} \right)} \right] \\ \end{aligned}$$
(6.4.7)

\(C_{\gamma }\), \(B_{\gamma }\) are arbitrary constants.

\(P_{\gamma }^{\left( m \right)} \left( {\sin \varepsilon \eta } \right)\) is the Legendre associated function.

The chosen form of solutions admits to divide the general problem for a plate into two independent ones:

A plate tension-compression problem and plate bending problem.

In the first case in (6.4.3) we should put \(C_{{\gamma_{3} }} = B_{{\gamma_{1} }} = B_{{\gamma_{2} }} = 0\), in the second case \(C_{{\gamma_{1} }} = C_{{\gamma_{2} }} = B_{{\gamma_{3} }} = 0\). In this case, we can represent the displacement vector components in the form:

$$\begin{aligned} u_{r} & = r^{\lambda } \left[ {A_{1} T_{{\gamma_{1} }} \left( \theta \right) + A_{2} T_{{\gamma_{2} }} \left( \theta \right)} \right]e^{i\omega \varphi } \\ u_{\theta } & = r^{\lambda } \left\{ {b_{0} \left[ {\frac{{dT_{{\gamma_{1} }} \left( \theta \right)}}{d\theta } + \frac{{dT_{{\gamma_{2} }} \left( \theta \right)}}{d\theta }} \right] - \frac{m}{\sin \theta }F_{{\gamma_{3} }} \left( \theta \right)} \right\}e^{i\omega t} \\ u_{\varphi } & = ir^{\lambda } \left\{ {\frac{{mb_{0} }}{\sin \theta }\left[ {T_{{\gamma_{1} }} \left( \theta \right) + T_{{\gamma_{2} }} \left( \theta \right)} \right] - \frac{{dF_{{\gamma_{3} }} \left( \theta \right)}}{d\theta }} \right\}e^{im\varphi } \\ \end{aligned}$$
(6.4.8.)

Satisfying homogeneous boundary conditions (6.4.1), we obtain a characteristic equation with regard to eigen values z:

$$\begin{aligned} D\left( {z,\theta_{1} } \right) & = \left[ {d_{12} D_{11} \left( {\theta_{1} } \right)\frac{{dT_{{\gamma_{2} }} }}{d\theta }\left( {\theta_{1} } \right) - d_{11} D_{12} \left( {\theta_{1} } \right)\frac{{dT_{{\gamma_{1} }} }}{d\theta }\left( {\theta_{1} } \right)} \right]L\left( {\theta_{1} } \right) \\ & \quad + 2m^{2} \left( {z - 3/2} \right)b_{0} \left( {\sin \theta_{1} } \right)^{ - 2} F_{{\gamma_{3} }} \left( {\theta_{1} } \right)\left[ {l_{2} \left( {\theta_{1} } \right)D_{11} \left( {\theta_{1} } \right)} \right. - \left. {l_{1} \left( {\theta_{1} } \right)D_{12} \left( {\theta_{1} } \right)} \right] \\ & \quad - 2m^{2} b_{0} G_{0} \left( {\sin \theta_{1} } \right)^{ - 2} \left[ {\frac{{dT_{{\gamma_{2} }} }}{d\theta }\left( {\theta_{1} } \right) - ctg\theta_{1} F_{{\gamma_{3} }} \left( {\theta_{1} } \right)} \right]H\left( {\theta_{1} } \right) = 0 \\ \end{aligned}$$
(6.4.9)

where

$$\begin{aligned} D_{1k} \left( \theta \right) & = \left( {C_{1k} + \frac{{2b_{0} G_{0} m^{2} }}{{\sin^{2} \theta }}} \right)T_{{\gamma_{k} }} - 2b_{0} G_{0} ctg\theta \frac{{dT_{{\gamma_{k} }} }}{d\theta }\left( \theta \right)\,\left( {k = 1,2} \right) \\ L\left( \theta \right) & = 2ctg\theta \frac{{dF_{{\gamma_{3} }} \left( \theta \right)}}{d\theta } + \left[ {\gamma_{3} \left( {\gamma_{3} + 1} \right) - \frac{{2m^{2} }}{{\sin^{2} \theta }}} \right]F_{{\gamma_{3} }} \left( \theta \right) \\ l_{k} \left( \theta \right) & = \frac{{dT_{k} \left( \theta \right)}}{d\theta } - ctg\theta T_{{\gamma_{k} }} \left( \theta \right) \\ H\left( \theta \right) & = \left( {d_{11} - d_{12} } \right)\frac{{dT_{{\gamma_{1} }} }}{d\theta }\left( \theta \right) + ctg\theta \\ & \quad \times \left[ {d_{12} T_{{\gamma_{1} }} \left( \theta \right)\frac{{dT_{{\gamma_{2} }} \left( \theta \right)}}{d\theta } - d_{11} T_{{\gamma_{2} }} \left( \theta \right)\frac{{dT_{{\gamma_{1} }} }}{d\theta }\left( \theta \right)} \right] \\ C_{1k} & = \left[ {b_{12} \left( {z - \frac{1}{2}} \right) + b_{22} + b_{23} } \right]A_{k} - b_{22} b_{0} \gamma_{k} \left( {\gamma_{k} + 1} \right) \\ d_{1k} & = A_{k} + \left( {z - \frac{3}{2}} \right)b_{0} ,C_{13} = - G_{0} b_{0} \\ \end{aligned}$$

The transcendental function (6.4.9) as an entire function of the parameter \(z_{1}\) determines a denumerable set \(z_{n}\) with a condensation point at infinity. Summing over all the roots, we obtain the homogeneous solutions of the following form:

$$\begin{aligned} u_{r} & = \frac{1}{\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} u_{1n} \left( \theta \right)e^{im\varphi } \\ u_{\theta } & = \frac{1}{\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} u_{2n} \left( \theta \right)e^{im\varphi } \\ u_{\varphi } & = \frac{1}{\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} u_{3n} \left( \theta \right)e^{im\varphi } \\ \sigma_{r} & = \frac{{G_{1} }}{r\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} Q_{1n} \left( \theta \right)e^{im\varphi } \\ \sigma_{\varphi } & = \frac{{G_{1} }}{r\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} Q_{2n} \left( \theta \right)e^{im\varphi } \\ \sigma_{\theta } & = \frac{{G_{1} }}{r\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} Q_{3n} \left( \theta \right)e^{im\varphi } \\ T_{r\theta } & = \frac{{G_{1} }}{r\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} T_{1n} \left( \theta \right)e^{im\varphi } \\ T_{r\varphi } & = \frac{{G_{1} }}{r\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} T_{2n} \left( \theta \right)e^{im\varphi } \\ T_{\theta \varphi } & = \frac{{G_{1} }}{r\sqrt r }\sum\limits_{n = 1}^{\infty } {C_{n} } r^{{z_{n} }} T_{3n} \left( \theta \right)e^{im\varphi } , \\ \end{aligned}$$
(6.4.10)

where

$$\begin{aligned} u_{1n} \left( \theta \right) & = A_{1} \Delta_{1n} T_{{\gamma_{1n} }} \left( \theta \right) - A_{2} \Delta_{2n} T_{{\gamma_{2n} }} \left( \theta \right) \\ u_{2n} \left( \theta \right) & = b_{0} \left[ {\Delta_{1n} \frac{{dT_{{\gamma_{1n} }} \left( \theta \right)}}{d\theta } - \Delta_{2n} \frac{{dT_{{\gamma_{2n} }} \left( \theta \right)}}{d\theta }} \right] - \frac{m}{\sin \theta }\Delta_{3n} F_{{\gamma_{3n} }} \left( \theta \right) \\ u_{3n} \left( \theta \right) & = \frac{{mb_{0} }}{\sin \theta }\left[ {\Delta_{1n} T_{{\gamma_{1n} }} \left( \theta \right) - \Delta_{2n} T_{{\gamma_{2n} }} \left( \theta \right)} \right] - \Delta_{3n} \frac{{dF_{{\gamma_{3n} }} \left( \theta \right)}}{d\theta } \\ \end{aligned}$$
$$\begin{aligned} Q_{1n} \left( \theta \right) & = \left\{ {\left[ {b_{12} \left( {z_{n} - \frac{1}{2}} \right) + 2b_{12} } \right]A_{1} - b_{0} b_{12} \gamma_{1n} \left( {\gamma_{1n} + 1} \right)} \right\}\Delta_{1n} T_{{\gamma_{1n} }} \left( \theta \right) \\ & \quad - \left\{ {\left[ {b_{12} \left( {z_{n} - \frac{1}{2}} \right) + 2b_{12} } \right]A_{2} - b_{0} b_{12} \gamma_{2n} \left( {\gamma_{2n} + 1} \right)} \right\}\Delta_{2n} T_{{\gamma_{2n} }} \left( \theta \right) \\ \end{aligned}$$
$$\begin{aligned} Q_{2n} \left( \theta \right) & = \left\langle {\left\{ {\left[ {b_{12} \left( {z_{n} - \frac{1}{2}} \right) + b_{22} + b_{23} } \right]A_{1} - b_{13} b_{0} \gamma_{1n} } \right.\left( {\gamma_{1n} + 1} \right)} \right. \\ & \quad - \left. {\frac{{2m^{2} G_{0} b_{0} }}{{\sin^{2} \theta }}} \right\}T_{{\gamma_{1n} }} \left( \theta \right) - 2G_{0} b_{0} ctg\theta \left. {\frac{{dT_{{\gamma_{1n} }} \left( \theta \right)}}{d\theta }} \right\rangle \Delta_{1n} - \left\langle {\left\{ {\left[ {b_{12} \left( {z_{n} - 1/2} \right)} \right.} \right.} \right. \\ & \quad \left. { + b_{22} + b_{23} } \right]A_{2} - b_{0} b_{13} \gamma_{2n} \left( {\gamma_{2n} + 1} \right) - \left. {\frac{{2m^{2} G_{0} b_{0} }}{{\sin^{2} \theta }}} \right\}T_{{\gamma_{2n} }} \left( \theta \right) - 2G_{0} b_{0} ctg\theta \\ & \quad \times \left. {\frac{{dT_{{\gamma_{2n} }} \left( \theta \right)}}{d\theta }} \right\rangle \Delta_{2n} - \frac{{2G_{0} m}}{\sin \theta }\Delta_{3n} \left[ {\frac{{dF_{{\gamma_{3n} }} \left( \theta \right)}}{d\theta } - ctg\theta F_{{\gamma_{3n} }} \left( \theta \right)} \right] \\ \end{aligned}$$
$$\begin{aligned} Q_{3n} \left( \theta \right) & = D_{11} \left( \theta \right)\Delta_{1n} - D_{12} \left( \theta \right)\Delta_{2n} - \frac{{2G_{0} m}}{\sin \theta } \\ & \quad \times \Delta_{3n} \left[ {\frac{{dF_{{\gamma_{3n} }} \left( \theta \right)}}{d\theta } - ctg\theta F_{{\gamma_{3n} }} \left( \theta \right)} \right] \\ \end{aligned}$$
$$\begin{aligned} T_{1n} \left( \theta \right) & = d_{11} \Delta_{1n} \frac{{dT_{{\gamma_{1n} }} \left( \theta \right)}}{d\theta } - d_{12} \Delta_{2n} \frac{{dT_{{\gamma_{2n} }} \left( \theta \right)}}{d\theta } - \frac{{m\left( {z_{n} - \frac{3}{2}} \right)}}{\sin \theta }\Delta_{3n} F_{{\gamma_{3n} }} \left( \theta \right) \\ T_{2n} \left( \theta \right) & = \frac{m}{\sin \theta }\left[ {d_{11} \Delta_{1n} T_{{\gamma_{1n} }} \left( \theta \right) - d_{12} \Delta_{2n} T_{{\gamma_{2n} }} \left( \theta \right)} \right] - \left( {z_{n} - \frac{3}{2}} \right)\Delta_{3n} \frac{{F_{{\gamma_{3n} }} \left( \theta \right)}}{d\theta } \\ T_{3n} \left( \theta \right) & = \frac{{2mb_{0} }}{\sin \theta }\left[ {l_{1} \left( \theta \right)\Delta_{1n} - l_{2} \left( \theta \right)\Delta_{2n} } \right] - L\left( \theta \right)\Delta_{3n} \\ \end{aligned}$$
$$\begin{aligned} \Delta_{1n} & = d_{12} \frac{{dT_{{\gamma_{2n} }} \left( {\theta_{1} } \right)}}{d\theta }L\left( {\theta_{1} } \right) + \frac{{2m^{2} \left( {z_{n} - 3/2} \right)b_{0} }}{{\sin \theta_{1} }}F_{{\gamma_{3n} }} \left( {\theta_{1} } \right)l_{2} \left( {\theta_{1} } \right) \\ \Delta_{2n} & = d_{11} \frac{{dT_{{\gamma_{1n} }} \left( {\theta_{1} } \right)}}{d\theta }L\left( {\theta_{1} } \right) + \frac{{2m^{2} \left( {z_{n} - 3/2} \right)b_{0} }}{{\sin \theta_{1} }}F_{{\gamma_{3n} }} \left( {\theta_{1} } \right)l_{1} \left( {\theta_{1} } \right) \\ \Delta_{3n} & = \frac{{2mb_{0} }}{{\sin^{2} \theta_{1} }}H\left( {\theta_{1} } \right). \\ \end{aligned}$$

The solution of the bending problem is obtained from (6.4.10) by the substitution \(T_{\gamma } \leftrightarrow F_{\gamma }\). In the case of thinness of the plate, from the above-mentioned formulas we can obtain simple asymptotic formulas admitting to calculate the stress-strain state of the plate. Illustrate this on an example of a plate tension-compression problem when the ends of the plate are rigidly built-in:

$$u_{r} = 0, \quad u_{\theta } = 0, \quad u_{\varphi } = 0\quad {\text{for}}\;\,\theta = \pi /2 \mp \varepsilon$$
(6.4.11)

In this case, the characteristic equation has the form:

$$\begin{aligned} D\left( {z,\theta_{1} } \right) & = \frac{{dF_{{\gamma_{3} }} \left( {\theta_{1} } \right)}}{d\theta }\left[ {A_{1} T_{{\gamma_{2} }} \left( {\theta_{1} } \right)\frac{{dT_{{\gamma_{1} }} \left( {\theta_{1} } \right)}}{d\theta } - A_{2} T_{{\gamma_{1} }} \left( {\theta_{1} } \right)\frac{{dT_{{\gamma_{2} }} \left( {\theta_{1} } \right)}}{d\theta }} \right] \\ & \quad + \frac{{m^{2} }}{{\sin^{2} \theta_{1} }}\left( {A_{1} - A_{2} } \right)T_{{\gamma_{1} }} \left( {\theta_{1} } \right)T_{{\gamma_{2} }} \left( {\theta_{1} } \right)F_{{\gamma_{3} }} \left( {\theta_{1} } \right) = 0 \\ \end{aligned}$$
(6.4.12)

Prove that the function \(D\left( {z,\varepsilon } \right)\) as \(\varepsilon \to 0\) has no limited zeros. To this end, assuming \(\theta = \theta_{0} + \varepsilon \eta\), \(- 1 \le \eta \le 1\), and expanding the functions \(T_{\gamma } \left( \theta \right)\), \(\frac{{dT_{\gamma } }}{d\theta }\left( \theta \right)\), \(F_{\gamma } \left( \theta \right)\), \(\frac{{dF_{\gamma } }}{d\theta }\left( \theta \right)\) in the vicinity of the plane \(\pi /2\) in a series with respect to \(\varepsilon\), we obtain

$$\begin{aligned} T_{\gamma } \left( \theta \right) & = \frac{{2^{m + 1} }}{\sqrt \pi }\frac{{{\Gamma } \left( {\frac{1 + \gamma + m}{2}} \right)}}{{{\Gamma } \left( {1 + \frac{\gamma - m}{2}} \right)}}\cos \pi /2\left( {\gamma + m} \right) \\ & \quad \times \left\{ {1 - \frac{1}{2}\eta^{2} \varepsilon^{2} \left[ {\gamma \left( {\gamma + 1} \right) - m^{2} } \right] + \cdots } \right\} \\ \frac{{dT_{\gamma } \left( \theta \right)}}{d\theta } & = \frac{{2^{m + 1} }}{\sqrt \pi }\frac{{{\Gamma } \left( {\frac{1 + \gamma + m}{2}} \right)}}{{{\Gamma } \left( {1 + \frac{\gamma - m}{2}} \right)}}\cos \pi /2\left( {\gamma + m} \right)\varepsilon \eta \\ & \quad \times \left\langle {\gamma \left( {\gamma + 1} \right) - m^{2} \frac{1}{3}\left\{ {\left[ {\gamma \left( {\gamma + 1} \right) - m^{2} } \right]\left[ {\gamma \left( {\gamma + 1} \right) - 2 - m^{2} } \right] + 2m^{2} } \right\}\eta^{2} \varepsilon^{2} + \cdots } \right\rangle \\ \end{aligned}$$
(6.4.13)
$$\begin{aligned} F_{\gamma } \left( \theta \right) & = \frac{{2^{m + 1} }}{\sqrt \pi }\frac{{{\Gamma } \left( {\frac{1 + \gamma + m}{2}} \right)}}{{{\Gamma } \left( {1 + \frac{\gamma - m}{2}} \right)}}\sin \pi /2\left( {\gamma + m} \right) \\ & \quad \times \eta \varepsilon \left\{ {1 - \frac{1}{3!}\left[ {\gamma \left( {\gamma + 1} \right) - m^{2} - 1} \right]\eta^{2} \varepsilon^{2} + \cdots } \right\} \\ \frac{{dF_{\gamma } \left( \theta \right)}}{d\theta } & = \frac{{2^{m + 1} }}{\sqrt \pi }\frac{{{\Gamma } \left( {\frac{1 + \gamma + m}{2}} \right)}}{{{\Gamma } \left( {1 + \frac{\gamma - m}{2}} \right)}}\sin \pi /2\left( {\gamma + m} \right) \\ & \quad \times \left\{ {1 - \frac{1}{2}\left[ {\gamma \left( {\gamma + 1} \right) - m^{2} - 1} \right]\eta^{2} \varepsilon^{2} + \cdots } \right\} \\ \end{aligned}$$
(6.4.14)

Substituting (6.4.13), (6.4.14) in (6.4.12), we obtain

$$\begin{aligned} D\left( {z,\varepsilon } \right) & = \frac{{2^{3m + 3} }}{\pi \sqrt \pi }\frac{{{\Gamma } \left( {\frac{{\gamma_{1} + m + 1}}{2}} \right)}}{{{\Gamma } \left( {1 + \frac{{\gamma_{1} - m}}{2}} \right)}}\frac{{{\Gamma } \left( {\frac{{\gamma_{2} + m + 1}}{2}} \right)}}{{{\Gamma } \left( {1 + \frac{{\gamma_{2} - m}}{2}} \right)}}\frac{{{\Gamma } \left( {\frac{{\gamma_{3} + m + 1}}{2}} \right)}}{{{\Gamma } \left( {1 + \frac{{\gamma_{3} - m}}{2}} \right)}} \\ & \quad \times \cos \pi /2\left( {\gamma_{1} + m} \right)\cos \pi /2\left( {\gamma_{2} + m} \right)\sin \pi /2\left( {\gamma_{3} + m} \right) \\ & \quad \times \left[ {\gamma_{2} \left( {\gamma_{2} + 1} \right) - \gamma_{1} \left( {\gamma_{1} + 1} \right)} \right]\varepsilon \left[ {z^{2} - \frac{1}{4} + 2\left( {G_{0} - 1} \right) + O\left( {\varepsilon^{2} } \right)} \right] \\ \end{aligned}$$
(6.4.15)

\({\Gamma } \left( x \right)\) is Euler’s gamma function.

From (6.4.15) it is seen that \(z = \pm \sqrt {\frac{9}{4} - 2G_{0} }\) are the roots of the characteristic equation.

By direct verification we can establish that the trivial solutions correspond to the roots \(z = \pm \sqrt {\frac{9}{4} - 2G_{0} }\). As in the axially-symmetric case, we can prove that all remaining zeros of the function \(D\left( {z,\varepsilon } \right)\) unlimitedly grow as \(\varepsilon \to 0\) and only the case \(\varepsilon z_{n} \to const\) as \(\varepsilon \to 0\) is possible here.

To construct the asymptotics of the second group of zeros, we find them in the form

$$z_{n} = \frac{{\delta_{n} }}{\varepsilon } + O\left( \varepsilon \right).$$
(6.4.16)

In this case, the characteristic Eq. (6.4.7) takes the form:

$$\begin{aligned} & \tau^{2} - 2q_{1} \tau + q_{2} = 0\quad \gamma_{n} = \sqrt {\tau_{n} } \\ & 2q_{1} = \frac{1}{{b_{22} }}\left( {b_{11} b_{22} - b_{12}^{2} - 2b_{12} } \right)\delta_{n}^{2} ,q_{2} = b_{11} b_{22}^{ - 1} \delta_{n}^{4} \\ \end{aligned}$$
(6.4.17)

Let us consider the following cases:

  1. 1.

    \(q_{1} > 0\), \(q_{1}^{2} - q_{2} > 0\), \(\gamma_{1,2} = \pm s_{1} \delta_{n}\), \(\gamma_{3,4} = \pm s_{2} \delta_{n}\),

    $$\begin{aligned} s_{1,2} & = \sqrt {q_{1} \pm \sqrt {q_{1}^{2} - q_{2} } } , \quad q_{1}^{2} > q_{2} \\ s_{1,2} & = \alpha + i\beta = \sqrt {q_{1} \pm i\sqrt {q_{2} - q_{1}^{2} } } , \quad q_{1}^{2} < q_{2} . \\ \end{aligned}$$
  1. 2.

    The roots of the characteristic Eq. (6.4.17) are multiple.

    $$\gamma_{1,2} = \pm p\delta_{n} , \quad q_{1} > 0, \;\, q_{1}^{2} - q_{2} = 0\;\, p = \sqrt {q_{1} } .$$
  1. 3.

    \(q_{1} < 0\), \(q_{1}^{2} - q_{2} \ne 0\), \(\gamma_{1,2} = \pm i\delta_{n} s_{1}\), \(\gamma_{3,4} = \pm i\delta_{n} s_{2}\)

    $$\begin{aligned} s_{1,2} & = \sqrt {\left| {q_{1} } \right| \pm \sqrt {q_{1}^{2} - q_{2} } } , \quad q_{1}^{2} > q_{2} \\ s_{3,4} & = \sqrt {\left| {q_{1} } \right| \pm i\sqrt {q_{1} - q_{1}^{2} } } , \quad q_{1}^{2} < q_{2} \\ \end{aligned}$$
  1. 4.

    \(q_{1} < 0\), \(q_{1}^{2} - q_{2} = 0\), \(\gamma_{1,2} = \gamma_{3,4} = \pm ip\delta_{n}\), \(p = \sqrt {\left| {q_{1} } \right|}\).

In cases 1, 2 substituting (6.4.17) in (6.4.12) and transforming it by means of the asymptotic expansion \(T_{\gamma } \left( \theta \right)\), \(\frac{{dT_{\gamma } }}{d\theta }\left( \theta \right)\), \(F_{\gamma } \left( \theta \right)\), \(\frac{{dF_{\gamma } }}{d\theta }\left( \theta \right)\) for \(\delta_{n}\), we obtain. For a vortex problem

$$\cos \frac{{\delta_{n} }}{{\sqrt {b_{22} } }} = 0.$$
(6.4.18)

For a potential problem

$$\frac{{1 + b_{22} s_{1} s_{2} }}{{1 - b_{22} s_{1} s_{2} }}\left( {s_{2} - s_{1} } \right)\sin \left( {s_{2} + s_{1} } \right)\delta_{n} + \left( {s_{2} + s_{1} } \right)\sin \left( {s_{2} - s_{1} } \right)\delta_{n} = 0$$
(6.4.19)
$$\frac{{p^{2} + b_{11} }}{{p^{2} - b_{11} }}\sin 2p\delta_{n} + 2p\delta_{n} = 0$$
(6.4.20)
$$\begin{aligned} & \beta \left( {1 + b_{22} \beta^{2} - 3b_{22} \alpha^{2} } \right)\sin \alpha \delta_{n} \\ & \quad + \alpha \left( {1 - b_{22} \alpha^{2} - 3b_{22} \beta^{2} } \right)\sin 2\beta \delta_{n} = 0 \\ \end{aligned}$$
(6.4.21)

Concerning 3 and 4, the results for them are obtained from cases 1 and 2 by formal replacement of \(s_{1} ,s_{2}\), p by \(is_{1} ,is_{2} ,ip\).

These equations have a denumerable set of roots and in fact coincide with the characteristic equations of the similar problem for a transversally-isotropic elastic layer.

We give asymptotic construction of homogeneous solutions corresponding to different roots of the characteristic equation.

Group 1.

$$\begin{aligned} u_{r}^{\left( 1 \right)} & = \frac{{r_{1} \varepsilon }}{\sqrt \rho }\sum\limits_{n = 1}^{\infty } {C_{n} } U_{n}^{\left( 1 \right)} \left( \eta \right)\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ u_{\theta }^{\left( 1 \right)} & = \frac{{r_{1} \varepsilon }}{\sqrt \rho }\sum\limits_{n = 1}^{\infty } {C_{n} } U_{n}^{\left( 1 \right)} \left( \eta \right)\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ u_{\varphi }^{\left( 1 \right)} & = 0 \\ \sigma_{r}^{\left( 1 \right)} & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {C_{n} Q_{rn}^{\left( 1 \right)} } \left( \eta \right)\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ \sigma_{\varphi }^{\left( 1 \right)} & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {C_{n} Q_{\varphi n}^{\left( 1 \right)} } \left( \eta \right)\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ \sigma_{\theta }^{\left( 1 \right)} & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {C_{n} Q_{\theta n}^{\left( 1 \right)} } \left( \eta \right)\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ \tau_{r\theta }^{\left( 1 \right)} & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {C_{n} T_{n}^{\left( 1 \right)} } \left( \eta \right)\exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ \tau_{r\varphi }^{\left( 1 \right)} & \approx 0,\tau_{\theta \varphi }^{\left( 1 \right)} \approx 0 \\ \end{aligned}$$
(6.4.22)

where

$$\begin{aligned} \rho & = rr_{1}^{ - 1} \\ u_{n}^{\left( 1 \right)} \eta & = s_{2} \left( {1 - b_{22} s_{1}^{2} } \right)\sin s_{2} \delta_{n} \cos s_{1} \delta_{n} \eta \\ & \quad - s_{1} \left( {1 - b_{22} s_{2}^{2} } \right)\sin s_{1} \delta_{n} \cos s_{2} \delta_{n} \eta \\ \upsilon_{n}^{\left( 1 \right)} \left( \eta \right) & = \left( {b_{12} + 1} \right)s_{1} s_{2} \left( {\sin s_{2} \delta_{n} \sin s_{1} \delta_{n} \eta - \sin s_{1} \delta_{n} \sin s_{2} \delta_{n} \eta } \right) \\ Q_{rn}^{\left( 1 \right)} \left( \eta \right) & = \delta_{n} \left[ {b_{11} U_{n}^{{{\prime }\left( 1 \right)}} - \delta_{n} b_{12} \upsilon_{n}^{\left( 1 \right)} } \right] \\ Q_{\varphi n}^{\left( 1 \right)} \left( \eta \right) & = \delta_{n} \left[ {b_{11} U_{n}^{{{\prime }\left( 1 \right)}} - \delta_{n} b_{22} \upsilon_{n}^{\left( 1 \right)} } \right] \\ Q_{\theta n}^{\left( 1 \right)} \left( \eta \right) & = \delta_{n} \left[ {b_{11} U_{n}^{{{\prime }\left( 1 \right)}} - \delta_{n} b_{23} \upsilon_{n}^{\left( 1 \right)} } \right] \\ Q_{n}^{\left( 1 \right)} \left( \eta \right) & = \left[ {U_{n}^{{{\prime }\left( 1 \right)}} - \delta_{n} \upsilon_{n}^{\left( 1 \right)} } \right] \\ \end{aligned}$$

Group 2.

$$\begin{aligned} u_{r}^{\left( 2 \right)} \left( \eta \right) & = \frac{{r_{1} \varepsilon }}{\sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} } U_{n}^{\left( 2 \right)} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ u_{\theta }^{\left( 2 \right)} \left( \eta \right) & = \frac{{r_{1} \varepsilon }}{\sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} } \upsilon_{n}^{\left( 2 \right)} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ \sigma_{r}^{\left( 2 \right)} \left( \eta \right) & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} } Q_{rn}^{\left( 2 \right)} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ \sigma_{\varphi }^{\left( 2 \right)} \left( \eta \right) & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} } Q_{\varphi n}^{\left( 2 \right)} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ \sigma_{\theta }^{\left( 2 \right)} \left( \eta \right) & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} } Q_{\theta n}^{\left( 2 \right)} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ T_{r\theta }^{\left( 2 \right)} \left( \eta \right) & = \frac{{G_{1} }}{\rho \sqrt \rho }\sum\limits_{n = 1}^{\infty } {E_{n} } T_{n}^{\left( 2 \right)} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ \tau_{r\varphi }^{\left( 2 \right)} & \approx 0,\tau_{\theta \varphi }^{\left( 2 \right)} \approx 0,u_{\varphi }^{\left( 2 \right)} \approx 0; \\ u_{n}^{\left( 2 \right)} & = \left( {b_{12} + 1} \right)\left\{ {\left[ {\left( {p^{2} - b_{11} } \right)\cos p\delta_{n} + \left( {p\delta_{n} } \right)^{ - 1} \left( {p^{2} + b_{11} } \right)\sin p\delta_{n} } \right]} \right. \\ & \quad \times \left. {\cos p\delta_{n} \eta + \eta \left( {p^{2} - b_{11} } \right)\sin p\delta_{n} \sin p\delta_{n} \eta } \right\} \\ \upsilon_{n}^{\left( 2 \right)} \left( \eta \right) & = \left( {b_{11} - p^{2} } \right)\left( {\cos p\delta_{n} \sin p\delta_{n} \eta - \eta \sin p\delta_{n} \cos p\delta_{n} \eta } \right). \\ \end{aligned}$$
(6.4.23)

The expressions for \(Q_{rn}^{\left( 2 \right)}\), \(Q_{\varphi n}^{\left( 2 \right)}\),…, \(T_{n}^{\left( 2 \right)}\) are obtained from (6.4.22) by simple replacement of \(u_{n}^{\left( 1 \right)}\), \(\upsilon_{n}^{\left( 1 \right)}\) by \(u_{n}^{\left( 2 \right)}\), \(\upsilon_{n}^{\left( 2 \right)}\), respectively.

Group 3.

$$\begin{aligned} u_{r}^{\left( 3 \right)} & = \frac{{r_{1} \varepsilon }}{\sqrt \rho }\sum\limits_{n = 1}^{\infty } {D_{n} } u_{n}^{\left( 3 \right)} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ u_{\theta }^{\left( 3 \right)} & = \frac{{r_{1} \varepsilon }}{\sqrt \rho }\sum\limits_{n = 1}^{\infty } {D_{n} } \upsilon_{n}^{\left( 3 \right)} \exp \left( {\frac{{\delta_{n} }}{\varepsilon }\ln \rho } \right)e^{im\varphi } \\ u_{\varphi }^{\left( 3 \right)} & \approx 0, \\ u_{n}^{\left( 3 \right)} & = \left( {a_{0} \cos \beta \delta_{n} \eta ch\alpha \delta_{n} \eta - b_{0} \sin \beta \delta_{n} \eta sh\alpha \delta_{n} \eta } \right)\Delta_{1n} \\ & \quad + \left( {b_{0} \cos \beta \delta_{n} \eta ch\alpha \delta_{n} \eta + a_{0} \sin \beta \delta_{n} \eta sh\alpha \delta_{n} \eta } \right)\Delta_{2n} \\ \upsilon_{n}^{\left( 3 \right)} & = \left( {b_{12} + 1} \right)\left[ {\left( {\alpha \cos \beta \delta_{n} \eta sh\alpha \delta_{n} \eta + \beta \sin \delta_{n} \eta ch\alpha \delta_{n} \eta } \right)} \right.\Delta_{1n} \\ & \quad - \left. {\left( {\beta \cos \beta \delta_{n} \eta sh\alpha \delta_{n} \eta + \alpha \sin \delta_{n} \eta ch\alpha \delta_{n} \eta } \right)\Delta_{2n} } \right] \\ a_{0} & = 1 - b_{22} \left( {\alpha^{2} - \beta^{2} } \right),b_{0} = 2b_{22} \alpha \beta \\ \Delta_{1n} & = - \alpha \cos \beta \delta_{n} sh\alpha \delta_{n} + \beta \sin \beta \delta_{n} ch\alpha \delta_{n} \\ \Delta_{2n} & = \beta \cos \beta \delta_{n} sh\alpha \delta_{n} + \alpha \sin \beta \delta_{n} ch\alpha \delta_{n} \\ \end{aligned}$$
(6.4.24)

The expressions for \(Q_{rn}^{\left( 3 \right)} , \ldots ,T_{n}^{\left( 3 \right)}\) are obtained from (6.4.22) replacing \(u_{n}^{\left( 1 \right)}\) by \(u_{n}^{\left( 3 \right)}\), \(\upsilon_{n}^{\left( 1 \right)}\) by \(\upsilon_{n}^{\left( 3 \right)}\), respectively. It is important to note that (6.4.24) is typical only for anisotropic bodies, for \(G_{0} = 1\) it completely disappears.

Concerning the solutions (6.4.21), (6.4.22), for \(G_{0} = 1\) they emerge one another and completely coincide with Saint-Venant’s edge effect in theory of isotropic plates.

\(C_{n} ,E_{n} ,D_{n}\) are arbitrary constants.

Similarly, for the vortex problem we obtain

$$u_{r} \approx 0,u_{\theta } \approx 0,\sigma_{r} \approx 0,\sigma_{\theta } \approx 0,\tau_{r\theta } \approx 0.$$
$$\begin{aligned} u_{\varphi } & = \frac{{r_{1} i\varepsilon }}{\sqrt \rho }\sum\limits_{k = 1}^{\infty } {B_{k} } \cos \frac{{\delta_{k} }}{{\sqrt {b_{22} } }}\eta \exp \left( {\frac{{\delta_{k} }}{\varepsilon }l_{n} \rho } \right)e^{im\varphi } \\ \tau_{r\varphi } & = \frac{{G_{1} i}}{\rho \sqrt \rho }\sum\limits_{k = 1}^{\infty } {\delta_{k} B_{k} } \cos \frac{{\delta_{k} }}{{\sqrt {b_{22} } }}\eta \exp \left( {\frac{{\delta_{k} }}{\varepsilon }l_{n} \rho } \right)e^{im\varphi } \\ \tau_{\theta \varphi } & = \frac{{G_{1} i}}{\rho \sqrt \rho }\sum\limits_{k = 1}^{\infty } {B_{k} } \frac{{\delta_{k} }}{{\sqrt {b_{22} } }}\sin \frac{{\delta_{k} }}{{\sqrt {b_{22} } }}\eta \exp \left( {\frac{{\delta_{k} }}{\varepsilon }l_{n} \rho } \right)e^{im\varphi } \\ \end{aligned}$$
(6.4.25)

In the general case of loading, the arbitrary constants \(C_{n} ,E_{n} ,D_{n} ,B_{k}\) may be determined by means of the Lagrange variational principle [5, 6]. Under special plate edge support conditions they are exactly determined by means of the generalized orthogonality condition.