**2. Problems statement**

#### **2.1 Equilibrium equations and Hooke's law in parabolic coordinates**

It is known that elastic equilibrium of an isotropic homogeneous elastic body free of volume forces is described by the following differential equation [20]:

$$(\lambda + 2\mu) \text{grad } \text{div}\, \vec{U} - \mu \, \text{rot}\, \text{rot}\, \vec{U} = \mathbf{0} \tag{1}$$

where *λ* ¼ *Eν=*½ � ð Þ 1 þ *ν* ð Þ 1 � 2*ν* , *μ* ¼ *E=*½ � 2 1ð Þ � *ν* are elastic Lamé constants; *ν* is !

the Poisson's ratio; *E* is the modulus of elasticity; and *U* is a displacement vector. By projecting Eq. (1) onto the tangent lines of the curves of the parabolic

coordinate system (see Appendix A), we obtain the system of equilibrium equations in the parabolic coordinates.

In the parabolic coordinate system, the equilibrium equations with respect to the function *D*, *K*, *u*, v and Hooke's law can be written as [20–22]:

$$\begin{aligned} \text{(a)}\,D\_{,\xi} - K\_{,\eta} &= \text{0}, \qquad \text{(c)}\,\overline{u}\_{,\xi} + \overline{\nabla}\_{,\eta} = (\kappa - 2) / (\kappa \mu) \cdot h\_0^2 D, \\ \text{(b)}\,D\_{,\eta} + K\_{,\xi} &= \text{0}, \qquad \text{(d)}\,\overline{\nabla}\_{,\xi} - \overline{u}\_{,\eta} = \text{1/}\mu \cdot h\_0^2 K. \end{aligned} \tag{2}$$

$$\begin{aligned} \sigma\_{\eta\eta} &= h\_0^{-1} \Big[ \lambda \overline{u}\_{,\xi} + (\lambda + 2\mu) \overline{\nabla}\_{,\eta} + \Big[ (\lambda + \mu) - \mu h\_0^{-2} \Big] (\xi \overline{u} + \eta \overline{\nu}) \Big], \\ \sigma\_{\xi\xi} &= h\_0^{-1} \Big[ (\lambda + 2\mu) \overline{u}\_{,\xi} + \lambda \overline{\nabla}\_{,\eta} + \Big[ (\lambda + \mu) + \mu h\_0^{-2} \Big] (\xi \overline{u} + \eta \overline{\nu}) \Big], \\ \sigma\_{\xi\eta} &= \mu h\_0^{-1} \Big[ \left( \mathbf{v}\_{,\xi} + \mathbf{u}\_{,\eta} \right) - h\_0^{-2} (\xi \overline{\nu} + \eta \overline{\nu}) \Big], \end{aligned} \tag{3}$$

Boundary conditions that appear in the chapter have the following form:

*Infinite domain (a) Ω<sup>1</sup>* ¼ *0* <*ξ* <*ξ1*, *η* f g *<sup>1</sup>* <*η*< ∞ *bounded by parabolic curve η* ¼ *η<sup>1</sup> and line* y *= 0 and*

<sup>2</sup> ð Þ*<sup>η</sup>* or bð Þ *<sup>u</sup>* <sup>¼</sup> *<sup>G</sup>*<sup>1</sup>

<sup>2</sup> ð Þ*<sup>ξ</sup>* or bð Þ *<sup>u</sup>* <sup>¼</sup> *<sup>H</sup>*<sup>1</sup>

for *ξ* ¼ 0 : ð Þa v ¼ 0, *σξξ* ¼ 0 or bð Þ *u* ¼ 0, *τξη* ¼ 0, (8) for *η* ¼ 0 : ð Þa *u* ¼ 0, *σηη* ¼ 0, or bð Þ v ¼ 0, *τξη* ¼ 0, (9)

for *ξ*<sup>1</sup> ! �∞ : *σηη* ! 0, *τξη* ! 0, *u* ! 0, v ! 0*:* (10)

for *η* ! ∞ : *σηη* ! 0, *τξη* ! 0, *u* ! 0, v ! 0, (10a)

where *Fi*, *Qi* ð Þ *i* ¼ 1, 2 with the first derivative and *Gi*, *Hi* with the first and second derivatives can be decomposed into the trigonometric absolute and uniform

ð Þ*<sup>i</sup>* ð Þ*<sup>η</sup>* , v <sup>¼</sup> *<sup>G</sup>*ð Þ*<sup>i</sup>*

ð Þ*<sup>i</sup>* ð Þ*<sup>ξ</sup>* , v <sup>¼</sup> *<sup>H</sup>*ð Þ*<sup>i</sup>*

<sup>2</sup> ð Þ*η* , (6)

<sup>2</sup> ð Þ*ξ* , (7)

ð Þ*<sup>i</sup>* ð Þ*<sup>η</sup>* , *τξη* <sup>¼</sup> *<sup>F</sup>*ð Þ*<sup>i</sup>*

*(a) D1* ¼ *0* <*ξ*<*ξ1*, *0* <*η* <*η* f g*<sup>1</sup> domain bounded by parabolic curve η* ¼ *η<sup>1</sup> and line* y *= 0 and*

*(b) D* ¼ �*ξ<sup>1</sup>* <*ξ*<*ξ1*, *0* <*η*< *η* f g*<sup>1</sup> domain bounded by parabola η* ¼ *η1.*

*2D Elastostatic Problems in Parabolic Coordinates DOI: http://dx.doi.org/10.5772/intechopen.91057*

*(b) Ω* ¼ �*ξ<sup>1</sup>* <*ξ*<*ξ1*, *η* f g *<sup>1</sup>* <*η* < ∞ *bounded by parabola η* ¼ *η1.*

ð Þ*<sup>i</sup>* ð Þ*<sup>ξ</sup>* , *τξη* <sup>¼</sup> *<sup>Q</sup>*ð Þ*<sup>i</sup>*

for *<sup>ξ</sup>* <sup>¼</sup> *<sup>ξ</sup>*<sup>1</sup> : ð Þ<sup>a</sup> *σξξ* <sup>¼</sup> *<sup>F</sup>*<sup>1</sup>

**Figure 1.**

**Figure 2**.

for *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>1</sup> : ð Þ<sup>a</sup> *σηη* <sup>¼</sup> *<sup>Q</sup>*<sup>1</sup>

convergent Fourier series.

**167**

where *<sup>κ</sup>* <sup>¼</sup> 4 1ð Þ � *<sup>ν</sup>* , *<sup>u</sup>* <sup>¼</sup> *hu=c*2, <sup>v</sup> <sup>¼</sup> *<sup>h</sup>*v*=c*2, *<sup>h</sup>*<sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> <sup>p</sup> , *<sup>h</sup>* <sup>¼</sup> *<sup>h</sup><sup>ξ</sup>* <sup>¼</sup> *<sup>h</sup><sup>η</sup>* <sup>¼</sup> *c* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> <sup>p</sup> are Lamé coefficients (see Appendix A), *<sup>u</sup>*, v are the components of the displacement vector *U* ! along the tangents of *η*, *ξ* curved lines, and *c* is the scale factor (see Appendix A). And in the present paper, we take *c* ¼ 1, ð Þ *κ* � 2 *=*ð Þ� *κμ D* is the divergence of the displacement vector, *K=μ* is the rotor component of the displacement vector; *σξξ*, *σηη* and *τξη* ¼ *τηξ* are normal and tangential stresses; and sub-indexes ðÞ,*<sup>ξ</sup>* and ðÞ,*<sup>η</sup>* denotes partial derivatives with relevant coordinates, for example, *<sup>K</sup>*,*<sup>ξ</sup>* <sup>¼</sup> <sup>∂</sup>*<sup>K</sup>* ∂*ξ* .

#### **2.2 Boundary conditions**

In the parabolic system of coordinates *ξ*, *η* ð Þ �∞ <*ξ*< ∞, 0≤ *η*< ∞ , exact solutions of two-dimensional static boundary value problems of elasticity are constructed for homogeneous isotropic bodies occupying domains bounded by coordinate lines of the parabolic coordinate system (see Appendix A).

The elastic body occupies the following domain (see **Figures 1** and **2**):

$$D\_1(\mathbf{a}) \, D\_1 = \{ \mathbf{0} < \xi < \xi\_1, \ \mathbf{0} < \eta < \eta\_1 \}, \\
\text{(b) } D = \{ -\xi\_1 < \xi < \xi\_1, \ \mathbf{0} < \eta < \eta\_1 \}, \tag{4}$$

$$\mathbf{P}(\mathbf{a})\,\,\mathbf{Q}\_{1} = \{\mathbf{0} < \xi < \xi\_{1}, \,\,\eta\_{1} < \eta < \infty\},\\\mathbf{(b)}\,\,\mathbf{Q} = \{\ -\xi\_{1} \le \xi < \xi\_{1}, \,\eta\_{1} \le \eta < \infty\}.\tag{5}$$

*2D Elastostatic Problems in Parabolic Coordinates DOI: http://dx.doi.org/10.5772/intechopen.91057*

#### **Figure 1.**

Section 4 solves the concrete problems, gains the numerical results, and

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

**2.1 Equilibrium equations and Hooke's law in parabolic coordinates**

ð Þ *λ* þ 2*μ* grad div *U*

function *D*, *K*, *u*, v and Hooke's law can be written as [20–22]:

<sup>0</sup> *λu*,*<sup>ξ</sup>* þ ð Þ *λ* þ 2*μ* v,*<sup>η</sup>*

<sup>0</sup> ð Þ *λ* þ 2*μ u*,*<sup>ξ</sup>* þ *λ*v,*<sup>η</sup>*

� � � *<sup>h</sup>*�<sup>2</sup>

where *<sup>κ</sup>* <sup>¼</sup> 4 1ð Þ � *<sup>ν</sup>* , *<sup>u</sup>* <sup>¼</sup> *hu=c*2, <sup>v</sup> <sup>¼</sup> *<sup>h</sup>*v*=c*2, *<sup>h</sup>*<sup>0</sup> <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>0</sup> v,*<sup>ξ</sup>* þ *u*,*<sup>η</sup>*

the Poisson's ratio; *E* is the modulus of elasticity; and *U*

It is known that elastic equilibrium of an isotropic homogeneous elastic body free of volume forces is described by the following differential equation [20]:

!

By projecting Eq. (1) onto the tangent lines of the curves of the parabolic coordinate system (see Appendix A), we obtain the system of equilibrium equations

ð Þ<sup>a</sup> *<sup>D</sup>*,*<sup>ξ</sup>* � *<sup>K</sup>*,*<sup>η</sup>* <sup>¼</sup> 0, cð Þ *<sup>u</sup>*,*<sup>ξ</sup>* <sup>þ</sup> v,*<sup>η</sup>* <sup>¼</sup> ð Þ *<sup>κ</sup>* � <sup>2</sup> *<sup>=</sup>*ð Þ� *κμ <sup>h</sup>*<sup>2</sup>

� <sup>þ</sup> ð Þ� *<sup>λ</sup>* <sup>þ</sup> *<sup>μ</sup> <sup>μ</sup>h*�<sup>2</sup>

� <sup>þ</sup> ð Þþ *<sup>λ</sup>* <sup>þ</sup> *<sup>μ</sup> <sup>μ</sup>h*�<sup>2</sup>

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> <sup>p</sup> are Lamé coefficients (see Appendix A), *<sup>u</sup>*, v are the components of the

factor (see Appendix A). And in the present paper, we take *c* ¼ 1, ð Þ *κ* � 2 *=*ð Þ� *κμ D* is the divergence of the displacement vector, *K=μ* is the rotor component of the displacement vector; *σξξ*, *σηη* and *τξη* ¼ *τηξ* are normal and tangential stresses; and sub-indexes ðÞ,*<sup>ξ</sup>* and ðÞ,*<sup>η</sup>* denotes partial derivatives with relevant coordinates, for

In the parabolic system of coordinates *ξ*, *η* ð Þ �∞ <*ξ*< ∞, 0≤ *η*< ∞ , exact solu-

ð Þa *D*<sup>1</sup> ¼ 0<*ξ*<*ξ*<sup>1</sup> f g , 0<*η*<*η*<sup>1</sup> , bð Þ *D* ¼ �*ξ*<sup>1</sup> < *ξ*<*ξ*<sup>1</sup> f g , 0< *η*<*η*<sup>1</sup> , (4) ð Þa Ω<sup>1</sup> ¼ 0< *ξ*<*ξ*<sup>1</sup> f g , *η*<sup>1</sup> < *η*< ∞ , bð Þ Ω ¼ �*ξ*<sup>1</sup> ≤ *ξ*<*ξ*<sup>1</sup> f g , *η*<sup>1</sup> ≤ *η*< ∞ *:* (5)

tions of two-dimensional static boundary value problems of elasticity are constructed for homogeneous isotropic bodies occupying domains bounded by

coordinate lines of the parabolic coordinate system (see Appendix A). The elastic body occupies the following domain (see **Figures 1** and **2**):

ð Þ <sup>b</sup> *<sup>D</sup>*,*<sup>η</sup>* <sup>þ</sup> *<sup>K</sup>*,*<sup>ξ</sup>* <sup>¼</sup> 0, dð Þ v,*<sup>ξ</sup>* � *<sup>u</sup>*,*<sup>η</sup>* <sup>¼</sup> <sup>1</sup>*=<sup>μ</sup>* � *<sup>h</sup>*<sup>2</sup>

<sup>0</sup> ð Þ *<sup>ξ</sup>*<sup>v</sup> <sup>þ</sup> *<sup>η</sup><sup>u</sup>* � �,

where *λ* ¼ *Eν=*½ � ð Þ 1 þ *ν* ð Þ 1 � 2*ν* , *μ* ¼ *E=*½ � 2 1ð Þ � *ν* are elastic Lamé constants; *ν* is

In the parabolic coordinate system, the equilibrium equations with respect to the

� *μ* rot rot *U*

!

!

¼ 0 (1)

is a displacement vector.

<sup>0</sup>*D*,

,

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> <sup>p</sup> , *<sup>h</sup>* <sup>¼</sup> *<sup>h</sup><sup>ξ</sup>* <sup>¼</sup> *<sup>h</sup><sup>η</sup>* <sup>¼</sup>

,

(2)

(3)

<sup>0</sup>*K:*

0 � �ð Þ *<sup>ξ</sup><sup>u</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>v</sup> �

along the tangents of *η*, *ξ* curved lines, and *c* is the scale

0 � �ð Þ *<sup>ξ</sup><sup>u</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>v</sup> �

constructs the relevant graphs. Section 5 is a conclusion.

**2. Problems statement**

in the parabolic coordinates.

*σηη* <sup>¼</sup> *<sup>h</sup>*�<sup>1</sup>

*σξξ* <sup>¼</sup> *<sup>h</sup>*�<sup>1</sup>

*τξη* <sup>¼</sup> *<sup>μ</sup>h*�<sup>1</sup>

∂*ξ* .

!

*c* ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

**166**

displacement vector *U*

example, *<sup>K</sup>*,*<sup>ξ</sup>* <sup>¼</sup> <sup>∂</sup>*<sup>K</sup>*

**2.2 Boundary conditions**

*(a) D1* ¼ *0* <*ξ*<*ξ1*, *0* <*η* <*η* f g*<sup>1</sup> domain bounded by parabolic curve η* ¼ *η<sup>1</sup> and line* y *= 0 and (b) D* ¼ �*ξ<sup>1</sup>* <*ξ*<*ξ1*, *0* <*η*<*η* f g*<sup>1</sup> domain bounded by parabola η* ¼ *η1.*

#### **Figure 2**.

*Infinite domain (a) Ω<sup>1</sup>* ¼ *0* <*ξ* <*ξ1*, *η* f g *<sup>1</sup>* <*η*< ∞ *bounded by parabolic curve η* ¼ *η<sup>1</sup> and line* y *= 0 and (b) Ω* ¼ �*ξ<sup>1</sup>* <*ξ*<*ξ1*, *η* f g *<sup>1</sup>* <*η* < ∞ *bounded by parabola η* ¼ *η1.*

Boundary conditions that appear in the chapter have the following form:

$$\begin{aligned} \text{for} \quad \xi = \xi\_1: \quad \text{(a)} \ \sigma\_{\xi\overline{\xi}} = \mathbf{F}^1\_{(i)}(\eta), \quad \tau\_{\xi\eta} = \mathbf{F}^{(i)}\_2(\eta) \quad \text{or} \quad \text{(b)} \ u = \mathbf{G}^1\_{(i)}(\eta), \quad \mathbf{v} = \mathbf{G}^{(i)}\_2(\eta), \tag{6} \end{aligned} \tag{6}$$

$$\begin{array}{llll}\hline \textbf{for} & \eta = \eta\_1 : & (\textbf{a})\,\sigma\_{\eta\eta} = \textbf{Q}^1\_{(i)}(\xi), \,\,\tau\_{\xi\eta} = \textbf{Q}^{(i)}\_{2}(\xi) \quad \text{or} \quad (\textbf{b})\,\,u = H^1\_{(i)}(\xi), \quad \mathbf{v} = H^{(i)}\_{2}(\xi),\\\\ & & & (7) \end{array}$$

$$\begin{array}{cccc}\text{for} & \xi = \mathbf{0}: & (\mathbf{a})\,\mathbf{v} = \mathbf{0}, & \sigma\_{\xi\xi} = \mathbf{0} \quad \text{or} & (\mathbf{b})\,\boldsymbol{u} = \mathbf{0}, & \tau\_{\xi\eta} = \mathbf{0}, & & \end{array}$$

$$\text{for}\,\eta=\mathbf{0}:\quad \text{(a)}\,u=\mathbf{0},\quad\sigma\_{\eta\eta}=\mathbf{0},\quad\text{or}\quad \text{(b)}\,\mathbf{v}=\mathbf{0},\quad\tau\_{\xi\eta}=\mathbf{0},\tag{9}$$

$$\text{for } \xi\_1 \to \pm \infty: \qquad \sigma\_{\eta \eta} \to 0, \quad \tau\_{\xi \eta} \to 0, \quad u \to 0, \quad \mathbf{v} \to \mathbf{0}. \tag{10}$$

$$\text{for } \eta \to \infty: \quad \sigma\_{\eta \eta} \to \mathbf{0}, \quad \tau\_{\xi \eta} \to \mathbf{0}, \quad u \to \mathbf{0}, \quad \mathbf{v} \to \mathbf{0}, \tag{10a}$$

where *Fi*, *Qi* ð Þ *i* ¼ 1, 2 with the first derivative and *Gi*, *Hi* with the first and second derivatives can be decomposed into the trigonometric absolute and uniform convergent Fourier series.

Boundary conditions on the linear parts *ξ* ¼ 0 and *η* ¼ 0 of the consideration area enable us to continue the solutions continuously (symmetrically or antisymmetrically) in the domain, that is, the mirror reflection of the consideration area in a relationship *y* ¼ 0 line (see **Figures 1b** and **2b**).

From (12) by the separation of variables method, we obtain (see Appendix A)

*φ*1*<sup>n</sup>* ¼¼ *A*1*<sup>n</sup>* cosh ð Þ *nη* cosð Þ *nξ* , *φ*2*<sup>n</sup>* ¼¼ *A*2*<sup>n</sup>* sinh ð Þ *nη* sin ð Þ *nξ*

*φ*1*<sup>n</sup>* ¼¼ *A*1*<sup>n</sup>* sinh ð Þ *nη* sin ð Þ *nξ* , *φ*2*<sup>n</sup>* ¼¼ *A*2*<sup>n</sup>* cosh ð Þ *nη* cosð Þ *nξ :*

For *n* ¼ 0: *φ*<sup>10</sup> ¼ *A*<sup>10</sup> þ *a*02*ξ* þ þ*a*03*η* þ *a*04*ξη*, *φ*<sup>20</sup> ¼ *A*<sup>20</sup> þ *b*02*ξ* þ *b*03*η* þ *b*04*ξη*, where *A*10, *a*02, … , *b*<sup>04</sup> are constant coefficients. When *n* ¼ 0 and 0 <*ξ*<*ξ*1, then the terms *ξ*, *η* and *ξη* will not be contained in *φ*<sup>10</sup> and *φ*20. If the foregoing solutions are presented in expressions of *φ*<sup>10</sup> and *φ*20, then it would be impossible on *ξ* ¼ *ξ*<sup>1</sup> to

2.The boundary conditions given on *η* ¼ *η*1, i.e., stresses or displacements equal

3.When stresses are given on *η* ¼ *η*1, the main vector and main moment equal

*<sup>D</sup>* <sup>¼</sup> *κ σξξ* <sup>þ</sup> *σηη* � �*=*4, *σξξ* <sup>¼</sup> <sup>4</sup>*D=<sup>κ</sup>* � *σηη:*

By ultimately opening expressions *σηη* and *τξη* (in details), we can demonstrate that at point *M*ð Þ 0, 0 , *σηη* and *τξη* (and naturally, *σξξ*, too) are determined, i.e., they

When at *η* ¼ *η*<sup>1</sup> *u* and v are given, then it is expedient to take instead of them as

Considering the homogeneous boundary conditions of the concrete problem, we will insert *φ*<sup>1</sup> and *φ*<sup>2</sup> functions selected from the (14) in the right sides of (15) or

� � � ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*2,

� � <sup>þ</sup> ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*1,

2

*κ* � 2

<sup>2</sup>*<sup>μ</sup> σξη* are given, then instead of them we have to take

*<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>κ</sup>* � <sup>2</sup>

<sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>*

<sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>*,

2 *φ*2,*<sup>η</sup>:*

ð Þ¼ *u* � *η*<sup>1</sup> þ v � *ξ η*<sup>1</sup> *φ*1,*<sup>ξ</sup>* þ *φ*2,*<sup>η</sup>*

*u* � *ξ* � v � *η*<sup>1</sup> ð Þ¼ *η*<sup>1</sup> *φ*1,*<sup>η</sup>* � *φ*2,*<sup>ξ</sup>*

0

<sup>2</sup>*<sup>μ</sup> σηη* � *<sup>η</sup>*<sup>1</sup> � *σξη* � *<sup>ξ</sup>* � � ¼ �*η*<sup>1</sup> *<sup>φ</sup>*1,*ξξ* <sup>þ</sup> *<sup>φ</sup>*2,*ξη* � � � *<sup>κ</sup>*

� � <sup>¼</sup> *<sup>η</sup>*<sup>1</sup> *<sup>φ</sup>*1,*ξη* � *<sup>φ</sup>*2,*ξξ* � � <sup>þ</sup>

*φin*, *i* ¼ 1, 2, (14)

� �*=h i*ð Þ <sup>¼</sup> 1, 2 will not

(15)

(16)

*<sup>φ</sup><sup>i</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼1

*2D Elastostatic Problems in Parabolic Coordinates DOI: http://dx.doi.org/10.5772/intechopen.91057*

satisfy the boundary conditions, and grad *φ<sup>i</sup>*<sup>0</sup> ¼ *φ<sup>i</sup>*0,*<sup>ξ</sup>* þ *φ<sup>i</sup>*0,*<sup>η</sup>*

**Provision.** We are introducing the following assumptions:

1.*ξ*<sup>1</sup> is a sufficiently great positive number (see Appendix C).

be bounded in the point *M*ð Þ 0, 0 .

zero at interval ~*ξ*<sup>1</sup> <*ξ*<*ξ*1.

their equivalent the following expressions:

1 *h*2 0

1 *h*2 0

*h*2 0 <sup>2</sup>*<sup>μ</sup> σηη* and *<sup>h</sup>*<sup>2</sup>

their equivalent following expressions:

<sup>2</sup>*<sup>μ</sup> σηη* � *<sup>ξ</sup>* <sup>þ</sup> *σξη* � *<sup>η</sup>*<sup>1</sup>

zero.

are finite.

It is clear that

and if at *η* ¼ *η*<sup>1</sup>

1

1

**169**

where

or

### **3. Solution of stated boundary value problems**

In this section we will be considered internal and external problems for a homogeneous isotropic body bounded by parabolic curves.

#### **3.1 Interior boundary value problems**

Let us find the solution of problems (2), (3), (4a) (see **Figure 1a**), and (7)–(10) in class *C*<sup>2</sup> ð Þ *D* (for *D* area shown in **Figure 1b**). The solution is presented by two harmonious *φ*<sup>1</sup> and *φ*<sup>2</sup> functions (see Appendix B). From formulas (B11)–(B13), after inserting *α* ¼ *η*<sup>1</sup> and making simple transformations, we will obtain:

$$\begin{split} \overline{u} &= -\left[\eta\left(\rho\_{1,\eta} - \rho\_{2,\xi}\right) + (\kappa - 1)\rho\right]\xi + \left[\frac{\eta\_1^2}{\eta}\left(\rho\_{1,\xi} + \rho\_{2,\eta}\right) - (\kappa - 1)\rho\_2\right]\eta, \\ \overline{\nabla} &= \left[\frac{\eta\_1^2}{\eta}\left(\rho\_{1,\eta} - \rho\_{2,\xi}\right) + (\kappa - 1)\rho\_1\right]\eta + \left[\eta\left(\rho\_{1,\xi} + \rho\_{2,\eta}\right) - (\kappa - 1)\rho\_2\right]\xi; \\ D &= \frac{\kappa\mu}{h\_0^2}\left[\left(\rho\_{1,\eta} - \rho\_{2,\xi}\right)\eta - \left(\rho\_{1,\xi} + \rho\_{2,\eta}\right)\xi\right], \quad K = \frac{\kappa\mu}{h\_0^2}\left[\left(\rho\_{1,\eta} - \rho\_{2,\xi}\right)\xi + \left(\rho\_{1,\xi} + \rho\_{2,\eta}\right)\eta\right], \end{split} \tag{11}$$

where

$$\frac{1}{h^2} (\rho\_{i, \xi\xi} + \rho\_{i, \eta\eta}) = 0, \quad i = 1, 2. \tag{12}$$

The stress tensor components can be written as

*h*2 0 <sup>2</sup>*<sup>μ</sup> σηη* ¼ � *<sup>η</sup>*<sup>2</sup> 1 *η <sup>φ</sup>*1,*ξξ* <sup>þ</sup> *<sup>φ</sup>*2,*ξη* � *<sup>κ</sup>* 2 *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>κ</sup>* � <sup>2</sup> <sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup> <sup>η</sup>* <sup>þ</sup> *η φ*1,*ξη* � *<sup>φ</sup>*2,*ηη* <sup>þ</sup> *κ* � 2 <sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>* 2 *φ*2,*<sup>η</sup> <sup>ξ</sup>* � *η*2 <sup>1</sup> � *η <sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup> <sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> ξ* , (13) *h*2 0 <sup>2</sup>*<sup>μ</sup> τξη* <sup>¼</sup> *<sup>η</sup>*<sup>2</sup> 1 *η <sup>φ</sup>*1,*ξη* � *<sup>φ</sup>*2,*ξξ* <sup>þ</sup> *κ* � 2 <sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>* 2 *φ*2,*<sup>η</sup> <sup>η</sup>* <sup>þ</sup> *η φ*1,*ξξ* <sup>þ</sup> *<sup>φ</sup>*2,*ξη* � *<sup>κ</sup>* 2 *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>κ</sup>* � <sup>2</sup> <sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup> <sup>ξ</sup>* � *η*2 <sup>1</sup> � *η <sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup> <sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> η* , *h*2 0 <sup>2</sup>*<sup>μ</sup> σξξ* <sup>¼</sup> *<sup>η</sup>*<sup>2</sup> 1 *η <sup>φ</sup>*1,*ξξ* <sup>þ</sup> *<sup>φ</sup>*2,*ξη* � *<sup>κ</sup>* � <sup>4</sup> <sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>κ</sup>* <sup>þ</sup> <sup>2</sup> <sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup> <sup>η</sup>* � *η φ*1,*ξη* � *<sup>φ</sup>*2,*ξξ* <sup>þ</sup> *κ* þ 2 <sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>* � <sup>4</sup> <sup>2</sup> *<sup>φ</sup>*2,*<sup>η</sup> <sup>ξ</sup>* þ *η*2 <sup>1</sup> � *η <sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup> <sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> ξ :*

**168**

From (12) by the separation of variables method, we obtain (see Appendix A)

$$\rho\_i = \sum\_{n=1}^{\infty} \rho\_{in}, \quad i = 1, 2,\tag{14}$$

where

$$\rho\_{1n} = -A\_{1n}\cosh\left(n\eta\right)\cos\left(n\xi\right), \qquad \rho\_{2n} = -A\_{2n}\sinh\left(n\eta\right)\sin\left(n\xi\right).$$

or

Boundary conditions on the linear parts *ξ* ¼ 0 and *η* ¼ 0 of the consideration area enable us to continue the solutions continuously (symmetrically or antisymmetrically) in the domain, that is, the mirror reflection of the consideration area

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

In this section we will be considered internal and external problems for a homo-

Let us find the solution of problems (2), (3), (4a) (see **Figure 1a**), and (7)–(10)

1 *η*

*η* þ *η φ*1,*<sup>ξ</sup>* þ *φ*2,*<sup>η</sup>*

*h*2 0

2

*κ* � 2

*<sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup>*

*κ* � 2

2

 *<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> η* ,

*κ* þ 2

 *<sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> ξ :*

*φ*1,*<sup>ξ</sup>* þ *φ*2,*<sup>η</sup>*

 � ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*<sup>2</sup> 

 � ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*<sup>2</sup> *ξ*;

*φ*1,*<sup>η</sup>* � *φ*2,*<sup>ξ</sup>*

<sup>¼</sup> 0, *<sup>i</sup>* <sup>¼</sup> 1, 2*:* (12)

*<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>κ</sup>* � <sup>2</sup>

<sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>*

<sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>*

*<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>κ</sup>* � <sup>2</sup>

<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>κ</sup>* <sup>þ</sup> <sup>2</sup>

<sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>* � <sup>4</sup>

<sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>*

2 *φ*2,*<sup>η</sup>*

*ξ* , (13)

2 *φ*2,*<sup>η</sup>*

<sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>*

<sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>*

<sup>2</sup> *<sup>φ</sup>*2,*<sup>η</sup>*

*η*

*ξ*

*η*

*ξ*

*η*

*ξ*

 *<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> η* ,

*η*,

(11)

harmonious *φ*<sup>1</sup> and *φ*<sup>2</sup> functions (see Appendix B). From formulas (B11)–(B13), after inserting *α* ¼ *η*<sup>1</sup> and making simple transformations, we will obtain:

ð Þ *D* (for *D* area shown in **Figure 1b**). The solution is presented by two

in a relationship *y* ¼ 0 line (see **Figures 1b** and **2b**).

**3. Solution of stated boundary value problems**

geneous isotropic body bounded by parabolic curves.

<sup>þ</sup> ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup> <sup>ξ</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>2</sup>

*<sup>ξ</sup>* , *<sup>K</sup>* <sup>¼</sup> *κμ*

1

The stress tensor components can be written as

� *η*2 <sup>1</sup> � *η*

> 1 *η*

� *η*2 <sup>1</sup> � *η*

þ *η*2 <sup>1</sup> � *η*

1 *η*

<sup>2</sup>*<sup>μ</sup> σηη* ¼ � *<sup>η</sup>*<sup>2</sup>

*<sup>h</sup>*<sup>2</sup> *<sup>φ</sup><sup>i</sup>*,*ξξ* <sup>þ</sup> *<sup>φ</sup><sup>i</sup>*,*ηη*

þ *η φ*1,*ξη* � *φ*2,*ηη* <sup>þ</sup>

*φ*1,*ξξ* þ *φ*2,*ξη* � *<sup>κ</sup>*

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup>*

� *<sup>κ</sup>*

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup>*

*φ*1,*ξη* � *φ*2,*ξξ* <sup>þ</sup>

þ *η φ*1,*ξξ* þ *φ*2,*ξη*

*φ*1,*ξξ* þ *φ*2,*ξη* � *<sup>κ</sup>* � <sup>4</sup>

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup>*

� *η φ*1,*ξη* � *φ*2,*ξξ* <sup>þ</sup>

 <sup>þ</sup> ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*<sup>1</sup> 

*<sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup>*

**3.1 Interior boundary value problems**

*u* ¼ � *η φ*1,*<sup>η</sup>* � *φ*2,*<sup>ξ</sup>*

*φ*1,*<sup>η</sup>* � *φ*2,*<sup>ξ</sup>*

*φ*1,*<sup>η</sup>* � *φ*2,*<sup>ξ</sup>*

*h*2 0

*h*2 0 <sup>2</sup>*<sup>μ</sup> τξη* <sup>¼</sup> *<sup>η</sup>*<sup>2</sup>

*h*2 0 2*μ*

**168**

*σξξ* <sup>¼</sup> *<sup>η</sup>*<sup>2</sup> 1 *η*

<sup>v</sup> <sup>¼</sup> *<sup>η</sup>*<sup>2</sup> 1 *η*

in class *C*<sup>2</sup>

*<sup>D</sup>* <sup>¼</sup> *κμ h*2 0

where

$$
\rho\_{1n} = -A\_{1n} \sinh\left(n\eta\right) \sin\left(n\xi\right), \qquad \rho\_{2n} = -A\_{2n} \cosh\left(n\eta\right) \cos\left(n\xi\right).
$$

For *n* ¼ 0: *φ*<sup>10</sup> ¼ *A*<sup>10</sup> þ *a*02*ξ* þ þ*a*03*η* þ *a*04*ξη*, *φ*<sup>20</sup> ¼ *A*<sup>20</sup> þ *b*02*ξ* þ *b*03*η* þ *b*04*ξη*, where *A*10, *a*02, … , *b*<sup>04</sup> are constant coefficients. When *n* ¼ 0 and 0 <*ξ*<*ξ*1, then the terms *ξ*, *η* and *ξη* will not be contained in *φ*<sup>10</sup> and *φ*20. If the foregoing solutions are presented in expressions of *φ*<sup>10</sup> and *φ*20, then it would be impossible on *ξ* ¼ *ξ*<sup>1</sup> to satisfy the boundary conditions, and grad *φ<sup>i</sup>*<sup>0</sup> ¼ *φ<sup>i</sup>*0,*<sup>ξ</sup>* þ *φ<sup>i</sup>*0,*<sup>η</sup>* � �*=h i*ð Þ <sup>¼</sup> 1, 2 will not be bounded in the point *M*ð Þ 0, 0 .

**Provision.** We are introducing the following assumptions:


It is clear that

$$D = \kappa (\sigma\_{\xi\xi} + \sigma\_{\eta\eta}) / 4,\\ \sigma\_{\xi\xi} = 4D / \kappa - \sigma\_{\eta\eta}.$$

By ultimately opening expressions *σηη* and *τξη* (in details), we can demonstrate that at point *M*ð Þ 0, 0 , *σηη* and *τξη* (and naturally, *σξξ*, too) are determined, i.e., they are finite.

When at *η* ¼ *η*<sup>1</sup> *u* and v are given, then it is expedient to take instead of them as their equivalent the following expressions:

$$\begin{aligned} \frac{1}{h\_0^2} (\overline{u} \cdot \eta\_1 + \overline{v} \cdot \xi) &= \eta\_1 (\rho\_{1,\xi} + \rho\_{2,\eta}) - (\kappa - 1)\rho\_2, \\\frac{1}{h\_0^2} (\overline{u} \cdot \xi - \overline{v} \cdot \eta\_1) &= \eta\_1 (\rho\_{1,\eta} - \rho\_{2,\xi}) + (\kappa - 1)\rho\_1, \end{aligned} \tag{15}$$

and if at *η* ¼ *η*<sup>1</sup> *h*2 0 <sup>2</sup>*<sup>μ</sup> σηη* and *<sup>h</sup>*<sup>2</sup> 0 <sup>2</sup>*<sup>μ</sup> σξη* are given, then instead of them we have to take their equivalent following expressions:

$$\begin{split} \frac{1}{2\mu} \left( \sigma\_{\eta\eta} \cdot \eta\_1 - \sigma\_{\xi\eta} \cdot \xi \right) &= -\eta\_1 \left( \rho\_{1,\xi\xi} + \rho\_{2,\xi\eta} \right) - \frac{\kappa}{2} \rho\_{1,\eta} - \frac{\kappa - 2}{2} \rho\_{2,\xi}, \\ \frac{1}{2\mu} \left( \sigma\_{\eta\eta} \cdot \xi + \sigma\_{\xi\eta} \cdot \eta\_1 \right) &= \eta\_1 \left( \rho\_{1,\xi\eta} - \rho\_{2,\xi\xi} \right) + \frac{\kappa - 2}{2} \rho\_{1,\xi} - \frac{\kappa}{2} \rho\_{2,\eta}. \end{split} \tag{16}$$

Considering the homogeneous boundary conditions of the concrete problem, we will insert *φ*<sup>1</sup> and *φ*<sup>2</sup> functions selected from the (14) in the right sides of (15) or

(16), and we will expand the left sides in the Fourier series. In both sides expressions which are with identical combinations of trigonometric functions will equate to each other and will receive the infinite system of linear algebraic equations to unknown coefficients *A*1*<sup>n</sup>* and *A*2*<sup>n</sup>* of harmonic functions, with its main matrix having a block-diagonal form. The dimension of each block is 2 � 2, and determinant is not equal to zero, but in infinite the determinant of block strives to the finite number different to zero.

*h*2 0 2*μ*

where

or

*φ*1*<sup>n</sup>* ¼ *B*1*ne*

*φ*<sup>10</sup> ¼ *A*10, *φ*<sup>20</sup> ¼ 0.

equal zero.

*h*2 <sup>0</sup>ð Þ *κ* � 1

and if at *η* ¼ *η*<sup>1</sup>

**171**

1

~*ξ*<sup>1</sup> < *ξ*<*ξ*1, will equal zero.

the following expressions as their equivalent:

the following expressions as their equivalent:

1

1

*h*2 0 <sup>2</sup>*<sup>μ</sup> σηη* and *<sup>h</sup>*<sup>2</sup>

*φ*1*<sup>n</sup>* ¼ *B*1*ne*

� *<sup>κ</sup>* � <sup>4</sup>

� *<sup>η</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup> 1 *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup>*

given for *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*1, then *<sup>φ</sup>*<sup>3</sup> <sup>¼</sup> *<sup>κ</sup>*�<sup>2</sup>

<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* <sup>þ</sup>

*2D Elastostatic Problems in Parabolic Coordinates DOI: http://dx.doi.org/10.5772/intechopen.91057*

*σξξ* ¼ � *<sup>φ</sup>*1,*ξξ* <sup>þ</sup> *<sup>φ</sup>*2,*ξη* � �*η*<sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*1,*ξη* � *<sup>φ</sup>*2,*ξξ* � �*<sup>ξ</sup>* � � *<sup>η</sup>* � *<sup>η</sup>*<sup>0</sup> ð Þ

*κ* þ 2

2 Ð *φ*2*dξ*. From (18), by the separation of variables method, we obtain

�*n<sup>η</sup>* sin ð Þ *<sup>n</sup><sup>ξ</sup>* , *<sup>φ</sup>*2*<sup>n</sup>* <sup>¼</sup> *<sup>B</sup>*2*ne*

�*n<sup>η</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup>* , *<sup>φ</sup>*2*<sup>n</sup>* <sup>¼</sup> *<sup>B</sup>*2*ne*

<sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>* � *<sup>φ</sup>*3,*ξξ* � �*<sup>η</sup>* � *<sup>κ</sup>* <sup>þ</sup> <sup>2</sup>

� �*<sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup>* � �*ξ* � �

If *<sup>u</sup>* and v are given for *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*1, then we take *<sup>φ</sup>*<sup>3</sup> <sup>¼</sup> 0, and when *<sup>h</sup>*<sup>2</sup>

*<sup>φ</sup><sup>i</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼1

<sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>* � <sup>2</sup>

*φin*, *i* ¼ 1, 2, 3, (20)

2*n*

2*n*

<sup>0</sup>ð Þ *<sup>κ</sup>* � <sup>1</sup> ð Þ¼ *<sup>u</sup>η*<sup>1</sup> <sup>þ</sup> <sup>v</sup>*<sup>ξ</sup> <sup>φ</sup>*2, (21)

<sup>2</sup>*<sup>μ</sup> σξη* are given, then instead of them we have to take

<sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>φ</sup>*2,*<sup>η</sup>:*

*B*2*ne*

*B*2*ne*

�*n<sup>η</sup>* sin ð Þ *<sup>n</sup><sup>ξ</sup>*

�*n<sup>η</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup> :*

(22)

�*n<sup>η</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup>* , *<sup>φ</sup>*3*<sup>n</sup>* <sup>¼</sup> *<sup>κ</sup>* � <sup>2</sup>

�*n<sup>η</sup>* sin ð Þ *<sup>n</sup><sup>ξ</sup>* , *<sup>φ</sup>*3*<sup>n</sup>* ¼ � *<sup>κ</sup>* � <sup>2</sup>

When *n* ¼ 0, then *φ*<sup>10</sup> ¼ *A*<sup>10</sup> þ *a*02*ξ* þ *a*03*η* þ *a*04*ξη*, *φ*<sup>20</sup> ¼ *A*<sup>20</sup> þ *b*02*ξ* þ *b*03*η* þ *b*04*ξη*, where *A*10, *a*02, … , *b*<sup>04</sup> are constants. From limited of functions *φ<sup>i</sup>*<sup>0</sup> ð Þ *i* ¼ 1, 2 in *η* ! ∞ and satisfying boundary condition for *ξ* ¼ *ξ*1, it implies that *a*<sup>02</sup> ¼ 0, *b*<sup>02</sup> ¼ 0, *a*<sup>03</sup> ¼ 0, *b*<sup>03</sup> ¼ 0, *a*<sup>04</sup> ¼ 0, *b*<sup>04</sup> ¼ 0. Therefore, *φ*<sup>10</sup> ¼ 0, *φ*<sup>20</sup> ¼ *A*<sup>20</sup> or

**Provision.** As in the previous subsection we make the following assumptions:

• At *η* ¼ *η*<sup>1</sup> given boundary conditions, i.e., displacements or stresses on interval

• When stresses are given on *η* ¼ *η*1, the main vector and main moment will

When *u* and v are given at *η* ¼ *η*1, then instead of them, it is expedient to take

*h*2

2 *φ*1,*<sup>η</sup>*,

• *ξ*<sup>1</sup> is a sufficiently large positive number (see Appendix C).

*<sup>u</sup><sup>ξ</sup>* � <sup>v</sup>*η*<sup>1</sup> ð Þ¼ *<sup>φ</sup>*1, � <sup>1</sup>

<sup>2</sup>*<sup>μ</sup> σηη* � *<sup>η</sup>*<sup>1</sup> � *σξη* � *<sup>ξ</sup>* � � <sup>¼</sup> *<sup>κ</sup>*

� � <sup>¼</sup> *<sup>κ</sup>* � <sup>2</sup>

<sup>2</sup>*<sup>μ</sup> σηη* � *<sup>ξ</sup>* <sup>þ</sup> *σξη* � *<sup>η</sup>*<sup>1</sup>

0

<sup>2</sup> *<sup>φ</sup>*2,*<sup>η</sup>* <sup>þ</sup> *<sup>φ</sup>*3,*ξη* � �*<sup>ξ</sup>*

0 <sup>2</sup>*<sup>μ</sup> σηη* and *<sup>h</sup>*<sup>2</sup>

0 <sup>2</sup>*<sup>μ</sup> σξη* is

It is very easy to establish the convergence of (11) and (13) functional series on the area *D* ¼ �*ξ*<sup>1</sup> ≤*ξ*≤ *ξ*<sup>1</sup> f g , 0≤*η*≤*η*<sup>1</sup> by construction of the corresponding uniform convergent numerical majorizing series. So we have the following:

**Proposal 1.** The functional series corresponding to (11) and (13) are absolute and uniform by convergent series on the area *D* ¼ �*ξ*<sup>1</sup> ≤ *ξ*≤*ξ*<sup>1</sup> f g , 0≤ *η*≤ *η*<sup>1</sup> .

#### **3.2 Exterior boundary value problems**

We have to find the solution of problems (2), (3), (5a) (see **Figure 2a**), (7), (8), (10), and (10<sup>0</sup> ), which belongs to the class *C*<sup>2</sup> ð Þ Ω (see region Ω on **Figure 2b**). The solution is constructed using its general representation by harmonic functions *φ*1, *φ*<sup>2</sup> (see Appendix B). From formulas (B11)–(B13), following inserting *α* ¼ *η*<sup>1</sup> and simple transformations, we obtain the following expressions:

$$\begin{split} \mathfrak{T} &= -\Big[ \left( \wp\_{1,\xi} + \wp\_{2,\eta} \right) \eta\_{1} + \left( \wp\_{1,\eta} - \wp\_{2,\xi} \right) \xi \Big] (\eta - \eta\_{1}) - \left[ (\kappa - 1)\wp\_{1} + \wp\_{3,\eta} \right] \xi - \left[ (\kappa - 1)\wp\_{2} - \wp\_{3,\xi} \right] \eta, \\ \mathfrak{T} &= \Big[ \left( \wp\_{1,\xi} + \wp\_{2,\eta} \right) \xi - \left( \wp\_{1,\eta} - \wp\_{2,\xi} \right) \eta\_{1} \Big] (\eta - \eta\_{1}) + \left[ (\kappa - 1)\wp\_{1} + \wp\_{3,\eta} \right] \eta - \left[ (\kappa - 1)\wp\_{2} - \wp\_{3,\xi} \right] \xi, \end{split} \tag{17}$$

$$D = \frac{\kappa \mu}{h\_0^2} \left[ (\rho\_{1,\eta} - \rho\_{2,\xi})\eta - (\rho\_{1,\xi} + \rho\_{2,\eta})\xi \right], \\ K = \frac{\kappa \mu}{h\_0^2} \left[ (\rho\_{1,\eta} - \rho\_{2,\xi})\xi + (\rho\_{1,\xi} + \rho\_{2,\eta})\eta \right],$$

where

$$\frac{1}{h^2} \left( \wp\_{i, \xi \xi} + \wp\_{i, \eta \eta} \right) = \mathbf{0}, \quad i = \mathbf{1}, \mathbf{2}, \mathbf{3}. \tag{18}$$

The stress tensor components can be written as:

*h*2 0 <sup>2</sup>*<sup>μ</sup> σηη* <sup>¼</sup> *<sup>φ</sup>*1,*ξξ* <sup>þ</sup> *<sup>φ</sup>*2,*ξη <sup>η</sup>*<sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*1,*ξη* � *<sup>φ</sup>*2,*ξξ <sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>0</sup> ð Þ þ *κ* 2 *φ*1,*<sup>η</sup>* þ *κ* � 2 <sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>* � *<sup>φ</sup>*3,*ξξ <sup>η</sup>* þ *κ* � 2 <sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>* 2 *<sup>φ</sup>*2,*<sup>η</sup>* <sup>þ</sup> *<sup>φ</sup>*3,*ξη <sup>ξ</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup> 1 *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup> <sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> ξ* , 0 2*μ τξη* <sup>¼</sup> *<sup>φ</sup>*1,*ξξ* <sup>þ</sup> *<sup>φ</sup>*2,*ξη <sup>ξ</sup>* � *<sup>φ</sup>*1,*ξη* � *<sup>φ</sup>*2,*ξξ <sup>η</sup>*'<sup>1</sup> *<sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ� *<sup>κ</sup>* 2 *φ*1,*<sup>η</sup>* þ *κ* � 2 <sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>* � *<sup>φ</sup>*3,*ξξ <sup>ξ</sup>* þ *κ* � 2 <sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>* 2 *<sup>φ</sup>*2,*<sup>η</sup>* <sup>þ</sup> *<sup>φ</sup>*3,*ξη <sup>η</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup> 1 *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup> <sup>ξ</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> η* , (19)

*h*2

*2D Elastostatic Problems in Parabolic Coordinates DOI: http://dx.doi.org/10.5772/intechopen.91057*

$$\begin{split} \frac{h\_{0}^{2}}{2\mu}\sigma\_{\xi\xi} &= -\left[ \left( \rho\_{1,\xi\xi} + \rho\_{2,\xi\eta} \right) \eta\_{1} + \left( \rho\_{1,\xi\eta} - \rho\_{2,\xi\xi} \right) \xi \right] (\eta - \eta\_{0}) \\ & - \left( \frac{\kappa - 4}{2} \rho\_{1,\eta} + \frac{\kappa + 2}{2} \rho\_{2,\xi} - \rho\_{3,\xi\xi} \right) \eta - \left( \frac{\kappa + 2}{2} \rho\_{1,\xi} - \frac{\kappa - 2}{2} \rho\_{2,\eta} + \rho\_{3,\xi\eta} \right) \xi \\ & - \frac{\eta^{2}}{\xi^{2} + \eta^{2}} \left[ \left( \rho\_{1,\eta} - \rho\_{2,\xi} \right) \eta - \left( \rho\_{1,\xi} + \rho\_{2,\eta} \right) \xi \right] \end{split}$$

If *<sup>u</sup>* and v are given for *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*1, then we take *<sup>φ</sup>*<sup>3</sup> <sup>¼</sup> 0, and when *<sup>h</sup>*<sup>2</sup> 0 <sup>2</sup>*<sup>μ</sup> σηη* and *<sup>h</sup>*<sup>2</sup> 0 <sup>2</sup>*<sup>μ</sup> σξη* is given for *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*1, then *<sup>φ</sup>*<sup>3</sup> <sup>¼</sup> *<sup>κ</sup>*�<sup>2</sup> 2 Ð *φ*2*dξ*.

From (18), by the separation of variables method, we obtain

$$
\rho\_i = \sum\_{n=1}^{\infty} \rho\_{in}, \quad i = 1, 2, 3,\tag{20}
$$

where

(16), and we will expand the left sides in the Fourier series. In both sides expressions which are with identical combinations of trigonometric functions will equate to each other and will receive the infinite system of linear algebraic equations to unknown coefficients *A*1*<sup>n</sup>* and *A*2*<sup>n</sup>* of harmonic functions, with its main matrix having a block-diagonal form. The dimension of each block is 2 � 2, and determinant is not equal to zero, but in infinite the determinant of block strives to the finite

*Solid State Physics - Metastable, Spintronics Materials and Mechanics of Deformable…*

It is very easy to establish the convergence of (11) and (13) functional series on the area *D* ¼ �*ξ*<sup>1</sup> ≤*ξ*≤ *ξ*<sup>1</sup> f g , 0≤*η*≤*η*<sup>1</sup> by construction of the corresponding uniform

**Proposal 1.** The functional series corresponding to (11) and (13) are absolute

We have to find the solution of problems (2), (3), (5a) (see **Figure 2a**), (7), (8),

*h*2 0

*η*

*ξ*

*<sup>η</sup>* <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup>

1 *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup>*

solution is constructed using its general representation by harmonic functions *φ*1, *φ*<sup>2</sup> (see Appendix B). From formulas (B11)–(B13), following inserting *α* ¼ *η*<sup>1</sup> and

ð Þ Ω (see region Ω on **Figure 2b**). The

*<sup>ξ</sup>* � ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*<sup>2</sup> � *<sup>φ</sup>*3,*<sup>ξ</sup>*

*<sup>η</sup>* � ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*<sup>2</sup> � *<sup>φ</sup>*3,*<sup>ξ</sup>*

 *<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> η* ,

*φ*1,*<sup>η</sup>* � *φ*2,*<sup>ξ</sup>*

<sup>¼</sup> 0, *<sup>i</sup>* <sup>¼</sup> 1, 2, 3*:* (18)

2 *φ*1,*<sup>η</sup>* þ *κ* � 2

 *<sup>ξ</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> η* ,

<sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>* � *<sup>φ</sup>*3,*ξξ*

*ξ*

(19)

*η*,

(17)

*ξ*,

and uniform by convergent series on the area *D* ¼ �*ξ*<sup>1</sup> ≤ *ξ*≤*ξ*<sup>1</sup> f g , 0≤ *η*≤ *η*<sup>1</sup> .

convergent numerical majorizing series. So we have the following:

), which belongs to the class *C*<sup>2</sup>

simple transformations, we obtain the following expressions:

*η*<sup>1</sup>

*<sup>ξ</sup>* , *<sup>K</sup>* <sup>¼</sup> *κμ*

1

The stress tensor components can be written as:

<sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>*

*<sup>ξ</sup>* � *<sup>φ</sup>*1,*ξη* � *<sup>φ</sup>*2,*ξξ*

2

*<sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ� ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*3,*<sup>η</sup>*

*<sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þþ ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*<sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*3,*<sup>η</sup>*

*<sup>h</sup>*<sup>2</sup> *<sup>φ</sup><sup>i</sup>*,*ξξ* <sup>þ</sup> *<sup>φ</sup><sup>i</sup>*,*ηη*

*<sup>η</sup>*<sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*1,*ξη* � *<sup>φ</sup>*2,*ξξ*

2

*φ*2,*<sup>η</sup>* þ *φ*3,*ξη*

*κ* � 2

*<sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>0</sup> ð Þ

<sup>2</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>* � *<sup>φ</sup>*3,*ξξ*

 *<sup>η</sup>*'<sup>1</sup> *<sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ� *<sup>κ</sup>*

*φ*2,*<sup>η</sup>* þ *φ*3,*ξη*

 *<sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup> ξ* ,

number different to zero.

(10), and (10<sup>0</sup>

*u* ¼ � *φ*1,*<sup>ξ</sup>* þ *φ*2,*<sup>η</sup>*

v ¼ *φ*1,*<sup>ξ</sup>* þ *φ*2,*<sup>η</sup>*

*<sup>D</sup>* <sup>¼</sup> *κμ h*2 0

where

*h*2 0

*h*2 0 2*μ*

**170**

**3.2 Exterior boundary value problems**

*<sup>η</sup>*<sup>1</sup> <sup>þ</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup>*

*<sup>η</sup>* � *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>η</sup>*

*<sup>ξ</sup>* � *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup>*

<sup>2</sup>*<sup>μ</sup> σηη* <sup>¼</sup> *<sup>φ</sup>*1,*ξξ* <sup>þ</sup> *<sup>φ</sup>*2,*ξη*

þ *κ* 2 *φ*1,*<sup>η</sup>* þ

þ

*τξη* ¼ *φ*1,*ξξ* þ *φ*2,*ξη*

*κ* � 2

þ

*κ* � 2

<sup>þ</sup> *<sup>η</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup> 1 *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> *<sup>φ</sup>*1,*<sup>η</sup>* � *<sup>φ</sup>*2,*<sup>ξ</sup>*

<sup>2</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* � *<sup>κ</sup>*

*φ*1,*<sup>η</sup>* � *φ*2,*<sup>ξ</sup>*

$$\rho\_{1n} = B\_{1n} e^{-n\eta} \sin \left( n\xi \right), \qquad \rho\_{2n} = B\_{2n} e^{-n\eta} \cos \left( n\xi \right), \quad \rho\_{3n} = \frac{\kappa - 2}{2n} B\_{2n} e^{-n\eta} \sin \left( n\xi \right)$$

or

$$\rho\_{1\mathfrak{n}} = B\_{1\mathfrak{n}} e^{-n\eta} \cos \left(n\mathfrak{k}\right), \qquad \rho\_{2\mathfrak{n}} = B\_{2\mathfrak{n}} e^{-n\eta} \sin \left(n\mathfrak{k}\right), \qquad \rho\_{3\mathfrak{n}} = -\frac{\kappa - 2}{2\mathfrak{n}} B\_{2\mathfrak{n}} e^{-n\eta} \cos \left(n\mathfrak{k}\right).$$

When *n* ¼ 0, then *φ*<sup>10</sup> ¼ *A*<sup>10</sup> þ *a*02*ξ* þ *a*03*η* þ *a*04*ξη*, *φ*<sup>20</sup> ¼ *A*<sup>20</sup> þ *b*02*ξ* þ *b*03*η* þ *b*04*ξη*, where *A*10, *a*02, … , *b*<sup>04</sup> are constants. From limited of functions *φ<sup>i</sup>*<sup>0</sup> ð Þ *i* ¼ 1, 2 in *η* ! ∞ and satisfying boundary condition for *ξ* ¼ *ξ*1, it implies that *a*<sup>02</sup> ¼ 0, *b*<sup>02</sup> ¼ 0, *a*<sup>03</sup> ¼ 0, *b*<sup>03</sup> ¼ 0, *a*<sup>04</sup> ¼ 0, *b*<sup>04</sup> ¼ 0. Therefore, *φ*<sup>10</sup> ¼ 0, *φ*<sup>20</sup> ¼ *A*<sup>20</sup> or *φ*<sup>10</sup> ¼ *A*10, *φ*<sup>20</sup> ¼ 0.

**Provision.** As in the previous subsection we make the following assumptions:


When *u* and v are given at *η* ¼ *η*1, then instead of them, it is expedient to take the following expressions as their equivalent:

$$\frac{1}{h\_0^2(\kappa - 1)}(\overline{u}\xi - \overline{v}\eta\_1) = \rho\_1, \qquad -\frac{1}{h\_0^2(\kappa - 1)}(\overline{u}\eta\_1 + \overline{v}\xi) = \rho\_2,\tag{21}$$

and if at *η* ¼ *η*<sup>1</sup> *h*2 0 <sup>2</sup>*<sup>μ</sup> σηη* and *<sup>h</sup>*<sup>2</sup> 0 <sup>2</sup>*<sup>μ</sup> σξη* are given, then instead of them we have to take the following expressions as their equivalent:

$$\begin{split} \frac{1}{2\mu} \left( \sigma\_{\eta\eta} \cdot \eta\_1 - \sigma\_{\xi\eta} \cdot \xi \right) &= \frac{\kappa}{2} \rho\_{1,\eta}, \\ \frac{1}{2\mu} \left( \sigma\_{\eta\eta} \cdot \xi + \sigma\_{\xi\eta} \cdot \eta\_1 \right) &= \frac{\kappa - 2}{2} \rho\_{1,\xi} - \rho\_{2,\eta}. \end{split} \tag{22}$$

Just like that in the previous subsection, considering the homogeneous boundary conditions of the concrete problem, we will insert *φ*<sup>1</sup> and *φ*<sup>2</sup> functions selected from (20) in Eq. (21) or (22), and we will expand the left sides in the Fourier series. Both sides of the expressions, which show the identical combinations of trigonometric functions, will equate to each other and will receive the infinite system of linear algebraic equations to unknown coefficients *A*1*<sup>n</sup>* and *A*2*<sup>n</sup>* of harmonic functions, with its main matrix having a block-diagonal form. The dimension of each block is 2 � 2, and the determinant does not equate to zero, but in the infinity, the determinant of block tends to the finite number different from zero.

*h*2 0 <sup>2</sup>*<sup>μ</sup> τξη* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

*h*2 0 <sup>2</sup>*<sup>μ</sup> σξξ* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

X∞ *n*¼1

X∞ *n*¼1

*n*2

*n*2

*n*2

*n*2

<sup>þ</sup> *<sup>n</sup>*<sup>2</sup>

<sup>þ</sup> *<sup>n</sup>*<sup>2</sup>

h

�

*<sup>f</sup>* <sup>1</sup>ð Þ¼ *<sup>ξ</sup>* <sup>P</sup><sup>∞</sup>

*<sup>f</sup>* <sup>2</sup>ð Þ¼ *<sup>ξ</sup> <sup>P</sup><sup>ξ</sup> <sup>ξ</sup>*2þ*η*<sup>2</sup> 1

of the body.

**173**

*n*¼1

*n*¼1

� *<sup>n</sup>*<sup>2</sup>

� *η*2 <sup>1</sup> � *<sup>η</sup>*<sup>2</sup>

*n*¼1

� *<sup>n</sup>*<sup>2</sup>

þ *η*2 <sup>1</sup> � *<sup>η</sup>*<sup>2</sup>

�

*n*2 *η*2

*2D Elastostatic Problems in Parabolic Coordinates DOI: http://dx.doi.org/10.5772/intechopen.91057*

> � *<sup>n</sup>*<sup>2</sup> *η*2

and *A*2*<sup>n</sup>* coefficients is obtained:

*η*<sup>1</sup> sinh *nη*<sup>1</sup> ð Þ� *n*

*η*<sup>1</sup> cosh *nη*<sup>1</sup> ð Þþ *n*

functions.

�

<sup>1</sup> cosh ð Þ *<sup>n</sup><sup>η</sup>* ð Þþ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup> <sup>n</sup><sup>η</sup>* sinh ð Þ *<sup>n</sup><sup>η</sup> <sup>κ</sup>* � <sup>2</sup>

<sup>1</sup> sinh ð Þ *<sup>n</sup><sup>η</sup>* ð Þþ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup> <sup>n</sup><sup>η</sup>* cosh ð Þ *<sup>n</sup><sup>η</sup> <sup>κ</sup>* � <sup>4</sup>

We have to solve problem (2), (7a), (8a), and (9a) when *Q*1ð Þ¼ *ξ P* and

stress is equal to zero. From (16), and (23), we obtain the following equations:

� � � � sin ð Þ¼ *<sup>n</sup><sup>ξ</sup> <sup>P</sup>η*<sup>1</sup>

� � � � cosð Þ¼ *<sup>n</sup><sup>ξ</sup> <sup>P</sup><sup>ξ</sup>*

From here an infinite system of the linear algebraic equations with unknown *A*1*<sup>n</sup>*

*ηξ* sinh ð Þ *<sup>n</sup><sup>η</sup>* ð Þþ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup> <sup>n</sup><sup>ξ</sup>* cosh ð Þ *<sup>n</sup><sup>η</sup> <sup>κ</sup>*

*ηξ* cosh ð Þ *<sup>n</sup><sup>η</sup>* ð Þþ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup> <sup>n</sup><sup>ξ</sup>* sinh ð Þ *<sup>n</sup><sup>η</sup> <sup>κ</sup>* <sup>þ</sup> <sup>2</sup>

*<sup>Q</sup>*2ð Þ¼ *<sup>ξ</sup>* 0, i.e., at *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>1</sup> boundary the normal load <sup>1</sup>

*<sup>η</sup>*<sup>1</sup> sinh *<sup>n</sup>η*<sup>1</sup> ð Þð Þ� *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup> <sup>n</sup>* cosh *<sup>n</sup>η*<sup>1</sup> ð Þ *<sup>κ</sup>*

*<sup>η</sup>*<sup>1</sup> cosh *<sup>n</sup>η*<sup>1</sup> ð Þð Þþ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup> <sup>n</sup>* sinh *<sup>n</sup>η*<sup>1</sup> ð Þ *<sup>κ</sup>* � <sup>2</sup>

*κ* <sup>2</sup> cosh *<sup>n</sup>η*<sup>1</sup> ð Þ � �*A*1*<sup>n</sup>*

*κ* � 2 2 sinh *<sup>n</sup>η*<sup>1</sup> ð Þ � �*A*1*<sup>n</sup>*

> *κ* <sup>2</sup> sinh *<sup>n</sup>η*<sup>1</sup> ð Þ � �*A*2*<sup>n</sup>*

where *F*~1*<sup>n</sup>* and *F*~2*<sup>n</sup>* are the coefficients of expansion into the Fourier series

As seen, the main matrix of system (26) has a block-diagonal form, dimension of each block is 2 � 2. Thus, two equations with two *A*1*<sup>n</sup>* and *A*2*<sup>n</sup>* unknown values will be solved. After solving this system, we find *A*1*<sup>n</sup>* and *A*2*<sup>n</sup>* coefficients, and in putting them into formulas (24) and (25), we get displacements and stresses at any points

Numerical values of displacements and stresses are obtained at the points of the finite size region bounded by curved lines *η* ¼ *η*<sup>1</sup> and *ξ* ¼ *ξ*<sup>1</sup> (see **Figure 1a**), and relevant 3D graphs are drafted. The numerical results are obtained for the following

*n*¼1

*κ* � 2 2 cosh *<sup>n</sup>η*<sup>1</sup> ð Þ � �*A*2*<sup>n</sup>*

*η*<sup>1</sup> sinh *nη*<sup>1</sup> ð Þþ *n*

*η*<sup>1</sup> cosh *nη*<sup>1</sup> ð Þ� *n*

*<sup>F</sup>*~1*<sup>n</sup>* sin ð Þ *<sup>n</sup><sup>ξ</sup>* and *<sup>f</sup>* <sup>2</sup>ð Þ¼ *<sup>ξ</sup>* <sup>P</sup><sup>∞</sup>

� � � � cosð Þ *<sup>n</sup><sup>ξ</sup>*

� � � � sin ð Þ *<sup>n</sup><sup>ξ</sup>*

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> ½ � *<sup>n</sup><sup>ξ</sup>* cosh ð Þ *<sup>n</sup><sup>η</sup>* ð Þ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup>* sin ð Þþ *<sup>n</sup><sup>ξ</sup> <sup>n</sup><sup>η</sup>* sinh ð Þ *<sup>n</sup><sup>η</sup>* ð Þ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup>*

� � � � cosð Þ *<sup>n</sup><sup>ξ</sup>*

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> ½ � *<sup>n</sup><sup>η</sup>* cosh ð Þ *<sup>n</sup><sup>η</sup>* ð Þ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup> <sup>n</sup><sup>ξ</sup>* sinh ð Þ *<sup>n</sup><sup>η</sup>* ð Þ *<sup>A</sup>*1*<sup>n</sup>* <sup>þ</sup> *<sup>A</sup>*2*<sup>n</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup>*

2

2

� � � � sin ð Þ *<sup>n</sup><sup>ξ</sup>*

2

2

<sup>2</sup>*<sup>μ</sup> σηη* <sup>¼</sup> *<sup>P</sup> h*2 0

*<sup>A</sup>*1*<sup>n</sup>* � *<sup>κ</sup>* � <sup>4</sup> 2 *A*2*<sup>n</sup>*

*A*2*<sup>n</sup>*

*<sup>A</sup>*1*<sup>n</sup>* � *<sup>κ</sup>* � <sup>2</sup> 2 *A*2*<sup>n</sup>*

2

2

*<sup>A</sup>*1*<sup>n</sup>* � *<sup>κ</sup>* � <sup>2</sup> 2

> *<sup>A</sup>*1*<sup>n</sup>* � *<sup>κ</sup>* 2 *A*2*<sup>n</sup>*

> > �

i

<sup>¼</sup> *<sup>F</sup>*~1*<sup>n</sup>*,

<sup>¼</sup> *<sup>F</sup>*~2*<sup>n</sup>*, *<sup>n</sup>* <sup>¼</sup> 1, 2, …

*<sup>F</sup>*~2*<sup>n</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup>* , respectively, *<sup>f</sup>* <sup>1</sup>ð Þ¼ *<sup>ξ</sup> <sup>P</sup>η*<sup>1</sup>

*<sup>A</sup>*1*<sup>n</sup>* � *<sup>κ</sup>* 2 *A*2*<sup>n</sup>*

*<sup>A</sup>*1*<sup>n</sup>* � *<sup>κ</sup>* <sup>þ</sup> <sup>2</sup> 2 *A*2*<sup>n</sup>* � ,

(25)

� *:*

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> 1 ,

*<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> 1 *:*

(26)

*<sup>ξ</sup>*2þ*η*<sup>2</sup> 1 and

is given, but tangent

As in the previous subsection, we received the following:

**Proposition 2**. The functional series corresponding to (17) and (19) are absolute and a uniformly convergent series on region Ω ¼ �*ξ*<sup>1</sup> ≤*ξ*≤*ξ*<sup>1</sup> f g , *η*<sup>1</sup> ≤*η*< ∞ .
