**4. Test problems**

In this section we will be obtained numerical results of internal and external problems for a homogeneous isotropic body bounded by parabolic curves when normal stress distribution is applied to the parabolic border.

#### **4.1 Internal problem**

We will set and solve the concrete internal boundary value problem in stresses. Let us find the solution of equilibrium equation system (2) of the homogeneous isotropic body in the area Ω<sup>1</sup> ¼ 0<*ξ*< *ξ*<sup>1</sup> f g , 0<*η*<*η*<sup>1</sup> (see **Figure 1a**), which satisfies boundary conditions (7a), (8a), (9a), and (10).

From (14), (8a), and (9a)

$$
\rho\_i = \sum\_{n=1}^{\infty} \rho\_{in}, \quad i = 1, 2,\tag{23}
$$

where *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 :*

By inserting (23) in (11) and (13), we will receive the following expressions for the displacements:

$$\begin{split} \overline{u} &= \sum\_{n=1}^{\infty} \left\{ - \left[ n\eta \xi \cosh \left( n\eta \right) (A\_{1n} + A\_{2n}) + (\kappa - 1) \xi \sinh \left( n\eta \right) A\_{1n} \right] \sin \left( n\xi \right) \right. \right. \\ &\left. + \left[ n\eta \frac{2}{1} \sinh \left( n\eta \right) (A\_{1n} + A\_{2n}) - (\kappa - 1) \eta \cosh \left( n\eta \right) A\_{2n} \right] \cos \left( n\xi \right) \right\}, \\ \overline{\mathbf{v}} &= \sum\_{n=1}^{\infty} \left\{ \left[ n\eta \frac{2}{1} \cosh \left( n\eta \right) (A\_{1n} + A\_{2n}) + (\kappa - 1) \eta \sinh \left( n\eta \right) A\_{1n} \right] \sin \left( n\xi \right) \right. \\ &\left. + \left[ n\eta \xi \sinh \left( n\eta \right) (A\_{1n} + A\_{2n}) - (\kappa - 1) \xi \cosh \left( n\eta \right) A\_{2n} \right] \cos \left( n\xi \right) \right\}, \end{split} \tag{24}$$

but for the stresses the following:

$$\begin{split} \frac{h\_{0}^{2}}{2\mu}\sigma\_{\eta\eta} &= \sum\_{n=1}^{\infty} \left\{ \left[ n^{2} \eta\_{1}^{2} \sinh \left( n\eta \right) (A\_{1n} + A\_{2n}) + n\eta \cosh \left( n\eta \right) \left( \frac{\kappa}{2} A\_{1n} - \frac{\kappa - 2}{2} A\_{2n} \right) \right] \sin \left( n\xi \right) \\ &+ \left[ n^{2} \eta \xi \cosh \left( n\eta \right) (A\_{1n} + A\_{2n}) + n\xi \sinh \left( n\eta \right) \left( \frac{\kappa - 2}{2} A\_{1n} - \frac{\kappa}{2} A\_{2n} \right) \right] \cos \left( n\xi \right) \\ &- \frac{\eta\_{1}^{2} - \eta^{2}}{\xi^{2} + \eta^{2}} [n\eta \cosh \left( n\eta \right) (A\_{1n} + A\_{2n}) \sin \left( n\xi \right) - n\xi \sinh \left( n\eta \right) (A\_{1n} + A\_{2n}) \cos \left( n\xi \right)] \end{split}$$

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

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 determi-

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

**Proposition 2**. The functional series corresponding to (17) and (19) are absolute

In this section we will be obtained numerical results of internal and external problems for a homogeneous isotropic body bounded by parabolic curves when

We will set and solve the concrete internal boundary value problem in stresses. Let us find the solution of equilibrium equation system (2) of the homogeneous isotropic body in the area Ω<sup>1</sup> ¼ 0<*ξ*< *ξ*<sup>1</sup> f g , 0<*η*<*η*<sup>1</sup> (see **Figure 1a**), which sat-

By inserting (23) in (11) and (13), we will receive the following expressions for

f�½ � *nηξ* cosh ð Þ *nη* ð Þþ *A*1*<sup>n</sup>* þ *A*2*<sup>n</sup>* ð Þ *κ* � 1 *ξ* sinh ð Þ *nη A*1*<sup>n</sup>* sin ð Þ *nξ*

� � � �

*<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>*

� � � �

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

þ½ � *nηξ* sinh ð Þ *nη* ð Þ� *A*1*<sup>n</sup>* þ *A*2*<sup>n</sup>* ð Þ *κ* � 1 *ξ* cosh ð Þ *nη A*2*<sup>n</sup>* cosð Þg *nξ* ,

<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>*

*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>2</sup>

*φin*, *i* ¼ 1, 2, (23)

2

2

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

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

(24)

sin ð Þ *nξ*

� ,

cosð Þ *nξ*

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*< ∞ .

nant of block tends to the finite number different from zero. As in the previous subsection, we received the following:

normal stress distribution is applied to the parabolic border.

isfies boundary conditions (7a), (8a), (9a), and (10).

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

where *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 :*

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

**4. Test problems**

**4.1 Internal problem**

the displacements:

*<sup>u</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼1

<sup>v</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼1

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

**172**

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

*n*¼1

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

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

*nη*<sup>2</sup>

*n*2 *η*2

�

but for the stresses the following:

From (14), (8a), and (9a)

$$\frac{\mu\_{\xi}^{2}}{2\mu}\varepsilon\_{\xi\eta} = \sum\_{n=1}^{\infty} \left\{ \left[ n^{2} \eta\_{1}^{2} \cosh \left( n\eta \right) (A\_{1n} + A\_{2n}) + n\eta \sinh \left( n\eta \right) \left( \frac{\kappa - 2}{2} A\_{1n} - \frac{\kappa}{2} A\_{2n} \right) \right] \cos \left( n\xi \right) \right. \tag{2.6}$$

$$- \left[ n^{2} \eta \xi \sinh \left( n\eta \right) (A\_{1n} + A\_{2n}) + n\xi \cosh \left( n\eta \right) \left( \frac{\kappa}{2} A\_{1n} - \frac{\kappa - 2}{2} A\_{2n} \right) \right] \sin \left( n\xi \right)$$

$$- \frac{\eta\_{1}^{2} - \eta^{2}}{\xi^{2} + \eta^{2}} \left[ n\xi \cosh \left( n\eta \right) (A\_{1n} + A\_{2n}) \sin \left( n\xi \right) + n\eta \sinh \left( n\eta \right) (A\_{1n} + A\_{2n}) \cos \left( n\xi \right) \right] \right\}, \tag{2.7}$$

$$\frac{\hbar\_{0}^{2}}{2\mu}\sigma\_{\xi\xi} = \sum\_{n=1}^{\infty} \left\{ -\left[ n^{2} \eta\_{1}^{2} \sinh\left(n\eta\right) (A\_{1n} + A\_{2n}) + n\eta \cosh\left(n\eta\right) \left( \frac{\kappa - 4}{2} A\_{1n} - \frac{\kappa + 2}{2} A\_{2n} \right) \right] \sin\left(n\xi\right) \right\}$$

$$-\left[ n^{2} \eta\xi \cosh\left(n\eta\right) (A\_{1n} + A\_{2n}) + n\xi \sinh\left(n\eta\right) \left( \frac{\kappa + 2}{2} A\_{1n} - \frac{\kappa - 4}{2} A\_{2n} \right) \right] \cos\left(n\xi\right)$$

$$+\frac{\eta\_{1}^{2} - \eta^{2}}{\xi^{2} + \eta^{2}} \left[ n\eta \cosh\left(n\eta\right) (A\_{1n} + A\_{2n}) \sin\left(n\xi\right) - n\xi \sinh\left(n\eta\right) (A\_{1n} + A\_{2n}) \cos\left(n\xi\right) \right] \right\}.$$

We have to solve problem (2), (7a), (8a), and (9a) when *Q*1ð Þ¼ *ξ P* and *<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>2</sup>*<sup>μ</sup> σηη* <sup>¼</sup> *<sup>P</sup> h*2 0 is given, but tangent stress is equal to zero. From (16), and (23), we obtain the following equations:

$$\begin{aligned} &\sum\_{n=1}^{\infty} \left[ n^2 \eta\_1 \sinh\left(n\eta\_1\right) (A\_{1n} + A\_{2n}) - n \cosh\left(n\eta\_1\right) \left(\frac{\kappa}{2} A\_{1n} - \frac{\kappa - 2}{2} A\_{2n}\right) \right] \sin\left(n\xi\right) = \frac{P\eta\_1}{\xi^2 + \eta\_1^2}, \\ &\sum\_{n=1}^{\infty} \left[ n^2 \eta\_1 \cosh\left(n\eta\_1\right) (A\_{1n} + A\_{2n}) + n \sinh\left(n\eta\_1\right) \left(\frac{\kappa - 2}{2} A\_{1n} - \frac{\kappa}{2} A\_{2n}\right) \right] \cos\left(n\xi\right) = \frac{P\xi}{\xi^2 + \eta\_1^2}. \end{aligned}$$

From here an infinite system of the linear algebraic equations with unknown *A*1*<sup>n</sup>* and *A*2*<sup>n</sup>* coefficients is obtained:

$$\begin{aligned} \left[ \left( n^2 \eta\_1 \sinh \left( n \eta\_1 \right) - n \frac{\kappa}{2} \cosh \left( n \eta\_1 \right) \right) A\_{1n} \\ + \left( n^2 \eta\_1 \sinh \left( n \eta\_1 \right) + n \frac{\kappa - 2}{2} \cosh \left( n \eta\_1 \right) \right) A\_{2n} \right] &= \bar{F}\_{1n}, \\\ \left[ \left( n^2 \eta\_1 \cosh \left( n \eta\_1 \right) + n \frac{\kappa - 2}{2} \sinh \left( n \eta\_1 \right) \right) A\_{1n} \\ + \left( n^2 \eta\_1 \cosh \left( n \eta\_1 \right) - n \frac{\kappa}{2} \sinh \left( n \eta\_1 \right) \right) A\_{2n} \right] &= \bar{F}\_{2n}, \quad n = 1, 2, \dots \end{aligned} \tag{26}$$

where *F*~1*<sup>n</sup>* and *F*~2*<sup>n</sup>* are the coefficients of expansion into the Fourier series *<sup>f</sup>* <sup>1</sup>ð Þ¼ *<sup>ξ</sup>* <sup>P</sup><sup>∞</sup> *n*¼1 *<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> *n*¼1 *<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>ξ</sup>*2þ*η*<sup>2</sup> 1 and *<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 functions.

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 of the body.

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 data: *<sup>ν</sup>* <sup>¼</sup> <sup>0</sup>*:*3, *<sup>E</sup>* <sup>¼</sup> <sup>2</sup> � 106kg*=*cm2, *<sup>P</sup>* ¼ �10 kg*=*cm2, 0*:*1<sup>≤</sup> *<sup>η</sup>*<sup>1</sup> <sup>≤</sup> 3, *<sup>ξ</sup>*<sup>1</sup> <sup>¼</sup> <sup>2</sup> <sup>∗</sup> *<sup>π</sup>*, *<sup>ξ</sup>*<sup>1</sup> <sup>¼</sup> 4 ∗ *π*, and *ξ*<sup>1</sup> ¼ 6 ∗ *π*. Numerical calculations and the visual presentation are made by MATLAB software.

isotropic body in the region Ω<sup>1</sup> ¼ 0<*ξ*<*ξ*<sup>1</sup> f g , *η*<sup>1</sup> <*η*< ∞ , which satisfies the fol-

By inserting (27) in (17) and (19), we will obtain the following expressions for

�*ne*�*n<sup>η</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>*<sup>1</sup> ½ � cosð Þþ *<sup>n</sup><sup>ξ</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* sin ð Þ *<sup>n</sup><sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> <sup>f</sup> ð Þ

*ne*�*n<sup>η</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* cosð Þþ *<sup>n</sup><sup>ξ</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>*<sup>1</sup> ½ � sin ð Þ *<sup>n</sup><sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> <sup>f</sup> ð Þ

�*n<sup>η</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>*<sup>1</sup> ½ � sin ð Þþ *<sup>n</sup><sup>ξ</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ �

*B*1*<sup>n</sup>* þ *B*2*<sup>n</sup>* � �*<sup>ξ</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup>*

2

2

*B*1*<sup>n</sup>* þ 2*B*2*<sup>n</sup>*

Next, we will obtain the numerical results of the following example.

� �,

ð Þ *B*1*<sup>n</sup>* � *B*2*<sup>n</sup>* ½ � *ξ* sin ð Þþ *nξ η* cosð Þ *nξ*

� �*<sup>η</sup>* sin ð Þþ *<sup>n</sup><sup>ξ</sup> <sup>κ</sup>* <sup>þ</sup> <sup>2</sup>

� �

ð Þ *B*1*<sup>n</sup>* � *B*2*<sup>n</sup>* ½ � *η* sin ð Þþ *nξ ξ* cosð Þ *nξ*

�*n<sup>η</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>*<sup>1</sup> ½ � cosð Þ *<sup>n</sup><sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ �

*B*1*<sup>n</sup>* þ *B*2*<sup>n</sup>* � �*<sup>η</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup>*

�*n<sup>η</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>*<sup>1</sup> ½ � sin ð Þþ *<sup>n</sup><sup>ξ</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ �

� �

ð Þ *B*1*<sup>n</sup>* � *B*2*<sup>n</sup>* ½ � *η* sin ð Þþ *nξ ξ* cosð Þ *nξ*

2 *e* �*nη*

2 *e* �*nη*

).

*φin*, *i* ¼ 1, 2, 3, (27)

*B*2*<sup>n</sup>η* cosð Þ *nξ*

*B*2*<sup>n</sup>ξ* cosð Þ *nξ*

� ,

� ,

2

*B*1*<sup>n</sup>ξ* cosð Þ *nξ*

� *:*

<sup>2</sup>*<sup>n</sup> <sup>B</sup>*2*ne*�*n<sup>η</sup>* sin ð Þ *<sup>n</sup><sup>ξ</sup>* .

o ,

(28)

(29)

o ,

lowing boundary conditions: (7a), (8a), (10), and (100

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

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

where *<sup>φ</sup>*1*<sup>n</sup>* <sup>¼</sup> *<sup>B</sup>*1*ne*�*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>*3*<sup>n</sup>* <sup>¼</sup> *<sup>κ</sup>*�<sup>2</sup>

½ � ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>B</sup>*1*<sup>n</sup>* � ð Þ *<sup>κ</sup>* � <sup>2</sup> *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup> <sup>κ</sup>*

½ � ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>B</sup>*1*<sup>n</sup>* � ð Þ *<sup>κ</sup>* � <sup>2</sup> *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup> <sup>κ</sup>*

and for the stresses, we obtain the following formula:

*<sup>B</sup>*1*<sup>n</sup><sup>η</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup> <sup>κ</sup>* � <sup>2</sup>

*<sup>B</sup>*1*<sup>n</sup><sup>ξ</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup> <sup>κ</sup>* � <sup>2</sup>

From (20) and (8a)

displacements:

*<sup>u</sup>* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼0

> �*e* �*nη*

> þ*e* �*nη*

> > *n*¼1

*n*¼1

*h*2 0 2*μ*

**175**

�*n*<sup>2</sup> *e*

�*ne*�*n<sup>η</sup> <sup>κ</sup>* 2

� *<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> *ne*�*n<sup>η</sup>*

> �*n*<sup>2</sup> *e*

�*ne*�*n<sup>η</sup> <sup>κ</sup>* 2

� *<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> *ne*�*n<sup>η</sup>*

*n*¼1

*n*2 *e*

<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> *ne*�*n<sup>η</sup>*

<sup>þ</sup>*ne*�*n<sup>η</sup> <sup>κ</sup>* � <sup>4</sup>

2

*σξξ* <sup>¼</sup> <sup>X</sup><sup>∞</sup>

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

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

<sup>v</sup> <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼1

**Figures 3** and **4** show the distribution of stresses and displacements in the region bounded by curved lines *η* ¼ *η*<sup>1</sup> and *ξ* ¼ *ξ*1*<sup>k</sup>*≔*ξ*<sup>1</sup> (see **Figure 1a**), when (7a), (8a), and (9a) boundary conditions are valid and normal stress is applied to the parabolic boundary. Following conditions (8a) and (9a), at points of the linear parts *ξ* ¼ 0 and *η* ¼ 0 of consideration area *σξξ*ð Þ 0, *η* , *σηη*ð Þ *ξ*, 0 stresses and *u*ð Þ *ξ*, 0 , v 0, ð Þ*η* displacements equal zero which is seen in **Figures 3** and **4**.

## **4.2 External problem**

We will set and solve the concrete external boundary value problem in stresses. Let us find the solution of equilibrium equation system (2) of the homogeneous

**Figure 3.** *Distribution of stresses in the region bounded by curved lines η* ¼ *η<sup>1</sup> and ξ* ¼ *ξ1.*

**Figure 4.** *Distribution of displacements in the region bounded by curved lines η* ¼ *η<sup>1</sup> and ξ* ¼ *ξ1.*

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

isotropic body in the region Ω<sup>1</sup> ¼ 0<*ξ*<*ξ*<sup>1</sup> f g , *η*<sup>1</sup> <*η*< ∞ , which satisfies the following boundary conditions: (7a), (8a), (10), and (100 ).

From (20) and (8a)

data: *<sup>ν</sup>* <sup>¼</sup> <sup>0</sup>*:*3, *<sup>E</sup>* <sup>¼</sup> <sup>2</sup> � 106kg*=*cm2, *<sup>P</sup>* ¼ �10 kg*=*cm2, 0*:*1<sup>≤</sup> *<sup>η</sup>*<sup>1</sup> <sup>≤</sup> 3, *<sup>ξ</sup>*<sup>1</sup> <sup>¼</sup> <sup>2</sup> <sup>∗</sup> *<sup>π</sup>*, *<sup>ξ</sup>*<sup>1</sup> <sup>¼</sup> 4 ∗ *π*, and *ξ*<sup>1</sup> ¼ 6 ∗ *π*. Numerical calculations and the visual presentation are made by

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

displacements equal zero which is seen in **Figures 3** and **4**.

*Distribution of stresses in the region bounded by curved lines η* ¼ *η<sup>1</sup> and ξ* ¼ *ξ1.*

*Distribution of displacements in the region bounded by curved lines η* ¼ *η<sup>1</sup> and ξ* ¼ *ξ1.*

**Figures 3** and **4** show the distribution of stresses and displacements in the region bounded by curved lines *η* ¼ *η*<sup>1</sup> and *ξ* ¼ *ξ*1*<sup>k</sup>*≔*ξ*<sup>1</sup> (see **Figure 1a**), when (7a), (8a), and (9a) boundary conditions are valid and normal stress is applied to the parabolic boundary. Following conditions (8a) and (9a), at points of the linear parts *ξ* ¼ 0 and *η* ¼ 0 of consideration area *σξξ*ð Þ 0, *η* , *σηη*ð Þ *ξ*, 0 stresses and *u*ð Þ *ξ*, 0 , v 0, ð Þ*η*

We will set and solve the concrete external boundary value problem in stresses. Let us find the solution of equilibrium equation system (2) of the homogeneous

MATLAB software.

**4.2 External problem**

**Figure 3.**

**Figure 4.**

**174**

$$
\rho\_i = \sum\_{n=1}^{\infty} \rho\_{in}, \quad i = 1, 2, 3,\tag{27}
$$

where *<sup>φ</sup>*1*<sup>n</sup>* <sup>¼</sup> *<sup>B</sup>*1*ne*�*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>*3*<sup>n</sup>* <sup>¼</sup> *<sup>κ</sup>*�<sup>2</sup> <sup>2</sup>*<sup>n</sup> <sup>B</sup>*2*ne*�*n<sup>η</sup>* sin ð Þ *<sup>n</sup><sup>ξ</sup>* .

By inserting (27) in (17) and (19), we will obtain the following expressions for displacements:

$$\begin{split} \overline{u} &= \sum\_{n=0}^{\infty} \left\{-n e^{-n\eta} [(B\_{1n} - B\_{2n})\eta\_{1}\cos\left(n\xi\right) + (B\_{1n} - B\_{2n})\xi\sin\left(n\xi\right)] (\eta - \eta\_{1}) \\ &- e^{-n\eta} [(\kappa - 1)B\_{1n} - (\kappa - 2)B\_{2n}] \xi\sin\left(n\xi\right) - \frac{\kappa}{2} e^{-n\eta} B\_{2n} \eta\cos\left(n\xi\right) \right\}, \\\\ \overline{v} &= \sum\_{n=1}^{\infty} \left\{n e^{-n\eta} [(B\_{1n} - B\_{2n})\xi\cos\left(n\xi\right) + (B\_{1n} - B\_{2n})\eta\_{1}\sin\left(n\xi\right)] (\eta - \eta\_{1}) \\ &+ e^{-n\eta} [(\kappa - 1)B\_{1n} - (\kappa - 2)B\_{2n}] \eta\sin\left(n\xi\right) - \frac{\kappa}{2} e^{-n\eta} B\_{2n} \xi\cos\left(n\xi\right) \right\}, \end{split} \tag{28}$$

and for the stresses, we obtain the following formula:

*h*2 0 <sup>2</sup>*<sup>μ</sup> σηη* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼1 �*n*<sup>2</sup> *e* �*n<sup>η</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>*<sup>1</sup> ½ � sin ð Þþ *<sup>n</sup><sup>ξ</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ � �*ne*�*n<sup>η</sup> <sup>κ</sup>* 2 *<sup>B</sup>*1*<sup>n</sup><sup>η</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup> <sup>κ</sup>* � <sup>2</sup> 2 *B*1*<sup>n</sup>* þ *B*2*<sup>n</sup>* � �*<sup>ξ</sup>* cosð Þ *<sup>n</sup><sup>ξ</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> *ne*�*n<sup>η</sup>* ð Þ *B*1*<sup>n</sup>* � *B*2*<sup>n</sup>* ½ � *η* sin ð Þþ *nξ ξ* cosð Þ *nξ* � , *h*2 0 <sup>2</sup>*<sup>μ</sup> τξη* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼1 �*n*<sup>2</sup> *e* �*n<sup>η</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>*<sup>1</sup> ½ � cosð Þ *<sup>n</sup><sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ � �*ne*�*n<sup>η</sup> <sup>κ</sup>* 2 *<sup>B</sup>*1*<sup>n</sup><sup>ξ</sup>* sin ð Þ� *<sup>n</sup><sup>ξ</sup> <sup>κ</sup>* � <sup>2</sup> 2 *B*1*<sup>n</sup>* þ *B*2*<sup>n</sup>* � �*<sup>η</sup>* cosð Þ *<sup>n</sup><sup>ξ</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> *ne*�*n<sup>η</sup>* ð Þ *B*1*<sup>n</sup>* � *B*2*<sup>n</sup>* ½ � *ξ* sin ð Þþ *nξ η* cosð Þ *nξ* � , (29) *h*2 0 <sup>2</sup>*<sup>μ</sup> σξξ* <sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*¼1 *n*2 *e* �*n<sup>η</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>η</sup>*<sup>1</sup> ½ � sin ð Þþ *<sup>n</sup><sup>ξ</sup>* ð Þ *<sup>B</sup>*1*<sup>n</sup>* � *<sup>B</sup>*2*<sup>n</sup> <sup>ξ</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup> <sup>η</sup>* � *<sup>η</sup>*<sup>1</sup> ð Þ � <sup>þ</sup>*ne*�*n<sup>η</sup> <sup>κ</sup>* � <sup>4</sup> 2 *B*1*<sup>n</sup>* þ 2*B*2*<sup>n</sup>* � �*<sup>η</sup>* sin ð Þþ *<sup>n</sup><sup>ξ</sup> <sup>κ</sup>* <sup>þ</sup> <sup>2</sup> 2 *B*1*<sup>n</sup>ξ* cosð Þ *nξ* � � <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> *ne*�*n<sup>η</sup>* ð Þ *B*1*<sup>n</sup>* � *B*2*<sup>n</sup>* ½ � *η* sin ð Þþ *nξ ξ* cosð Þ *nξ* � *:*

Next, we will obtain the numerical results of the following example.

We have to solve problem (2), (7a), and (8a), when *Q*1ð Þ¼ *ξ P* and *Q*2ð Þ¼ *ξ* 0, i.e., at *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>1</sup> boundary the normal load <sup>1</sup> <sup>2</sup>*<sup>μ</sup> σηη* <sup>¼</sup> *<sup>P</sup> h*2 0 is given, but tangent stress is equal to zero. From (22) and (27), we obtain the following equations:

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

As it can be seen, the main matrix of system (30) has a block-diagonal form, and

, respectively (*f* <sup>1</sup>ð Þ*ξ* , according to sinuses,

*<sup>ξ</sup>*2þ*η*<sup>2</sup> 1

the dimension of each block is 2 � 2. Thus, two equations with two *B*1*<sup>n</sup>* and *B*2*<sup>n</sup>* unknown values will be solved. After solving this system, we find the values of *B*1*<sup>n</sup>* and *B*2*<sup>n</sup>* coefficients and put them into formulas (28) and (29) to get displacements

Numerical results are obtained for some characteristic points of the body, in particular, *M*<sup>1</sup> 0, *η*<sup>1</sup> ð Þ, *M*<sup>2</sup> *ξ*1, *η*<sup>1</sup> ð Þ points (see. **Figure 2a**), for the following data: *<sup>ν</sup>* <sup>¼</sup> <sup>0</sup>*:*3, *<sup>E</sup>* <sup>¼</sup> <sup>2</sup> <sup>∗</sup> <sup>10</sup>6kg*=*cm2, *<sup>P</sup>* ¼ �10 kg*=*cm2, 0*:*01≤*η*<sup>1</sup> <sup>≤</sup>3, *<sup>ξ</sup>*<sup>1</sup> <sup>¼</sup> <sup>2</sup> <sup>∗</sup> *<sup>π</sup>*, *<sup>ξ</sup>*<sup>1</sup> <sup>¼</sup> <sup>4</sup> <sup>∗</sup> *<sup>π</sup>*,

The above-presented graphs (see **Figures 5** and **6**) show how displacements and

, max *ut* j j<sup>&</sup>lt; max *un* j j, v*<sup>t</sup>* <sup>¼</sup> <sup>v</sup>*<sup>n</sup>* <sup>¼</sup> <sup>0</sup>*:*

*Tangential stress and normal displacements at points M1 0*, *η<sup>1</sup>* ð Þ *for ξ<sup>1</sup>* ¼ *2* ∗ *π, ξ<sup>1</sup>* ¼ *4* ∗ *π, and ξ<sup>1</sup>* ¼ *6* ∗ *π,*

ð Þ *<sup>j</sup>* <sup>¼</sup> 1, 2, … , 8 , when 0*:*01≤*η*<sup>1</sup> <sup>≤</sup>3 (see **Figure 7**).

stresses change at some characteristic points of body, namely, at points

and *<sup>f</sup>* <sup>2</sup>ð Þ¼ *<sup>ξ</sup> <sup>P</sup><sup>ξ</sup>*

functions *<sup>f</sup>* <sup>1</sup>ð Þ¼� *<sup>ξ</sup> <sup>P</sup>η*<sup>1</sup>

and *ξ*<sup>1</sup> ¼ 6 ∗ *π*.

• At points *M*ð Þ*<sup>j</sup>*

*M*ð Þ*<sup>j</sup>* <sup>1</sup> 0, *η* ð Þ*j* 1 and *<sup>M</sup>*ð Þ*<sup>j</sup>*

**Figure 6.**

**177**

*when 0:01* ≤*η<sup>1</sup>* ≤ *3.*

and *f* <sup>2</sup>ð Þ*ξ* , according to cosines).

and stresses at any points of the body.

<sup>2</sup> *ξ*1, *η*

<sup>1</sup> 0, *η* ð Þ*j* 1

ð Þ*j* 1

From the presented results, we obtain the following:

*<sup>ξ</sup>*2þ*η*<sup>2</sup> 1

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

$$\begin{split} \sum\_{n=1}^{\infty} n e^{-n\eta\_1} \frac{\kappa}{2} B\_{1n} \sin \left( n \xi \right) &= -\frac{P\eta\_1}{\xi^2 + \eta\_1^2}, \quad \sum\_{n=1}^{\infty} n e^{-n\eta\_1} \left( \frac{\kappa - 2}{2} B\_{1n} + B\_{2n} \right) \cos \left( n \xi \right) \\ &= \frac{P\xi}{\xi^2 + \eta\_1^2}. \end{split}$$

Consequently, we obtain the infinite system of the linear algebraic equations with unknown *B*1*<sup>n</sup>* and *B*2*<sup>n</sup>* coefficients:

$$\begin{aligned} \sum\_{n=1}^{\infty} n e^{-n\eta\_1} \frac{\kappa}{2} B\_{1n} \sin \left( n \xi \right) &= -\sum\_{n=1}^{\infty} \tilde{P}\_{1n} \sin \left( n \xi \right), \sum\_{n=1}^{\infty} n e^{-n\eta\_1} \left( \frac{\kappa - 2}{2} B\_{1n} + B\_{2n} \right) \cos \left( n \xi \right), \\\ &= \sum\_{n=1}^{\infty} \tilde{P}\_{2n} \cos \left( n \xi \right), \end{aligned}$$

i.e.,

$$ne^{-n\eta\_1}\frac{\kappa}{2}B\_{1n} = -\tilde{P}\_{1n}, \quad ne^{-n\eta\_1}\left(\frac{\kappa-2}{2}B\_{1n} + B\_{2n}\right) = \tilde{P}\_{2n}, \quad n = 1,2,\ldots. \tag{30}$$

Hence,

$$B\_{1n} = -\frac{2}{\kappa n} e^{n\eta\_1} \tilde{P}\_{1n}, \quad B\_{2n} = \frac{e^{n\eta\_1}}{n} \left( \tilde{P}\_{2n} + \frac{\kappa - 2}{\kappa} \tilde{P}\_{1n} \right),$$

**Figure 5.** *Stresses and displacements at points M2 ξ1*, *η<sup>1</sup>* ð Þ *for ξ<sup>1</sup>* ¼ *2* ∗ *π, ξ<sup>1</sup>* ¼ *4* ∗ *π, and ξ<sup>1</sup>* ¼ *6* ∗ *π, when 0:01* ≤*η<sup>1</sup>* ≤*3.*

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

We have to solve problem (2), (7a), and (8a), when *Q*1ð Þ¼ *ξ P* and *Q*2ð Þ¼ *ξ* 0,

, <sup>X</sup><sup>∞</sup> *n*¼1

Consequently, we obtain the infinite system of the linear algebraic equations

P∞ *n*¼1

*B*1*<sup>n</sup>* þ *B*2*<sup>n</sup>* � �

*n*

*Stresses and displacements at points M2 ξ1*, *η<sup>1</sup>* ð Þ *for ξ<sup>1</sup>* ¼ *2* ∗ *π, ξ<sup>1</sup>* ¼ *4* ∗ *π, and ξ<sup>1</sup>* ¼ *6* ∗ *π, when 0:01* ≤*η<sup>1</sup>* ≤*3.*

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

*κ* � 2 *κ*

� �

*P*~1*n*

,

*<sup>P</sup>*~1*<sup>n</sup>* sin ð Þ *<sup>n</sup><sup>ξ</sup>* ,

*<sup>P</sup>*~2*<sup>n</sup>* cosð Þ *<sup>n</sup><sup>ξ</sup>* ,

2

*<sup>n</sup>η*1*P*~1*<sup>n</sup>*, *<sup>B</sup>*2*<sup>n</sup>* <sup>¼</sup> *en<sup>η</sup>*<sup>1</sup>

*ne*�*nη*<sup>1</sup> *<sup>κ</sup>* � <sup>2</sup> 2

*ne*�*nη*<sup>1</sup> *<sup>κ</sup>*�<sup>2</sup>

equal to zero. From (22) and (27), we obtain the following equations:

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

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

<sup>¼</sup> *<sup>P</sup><sup>ξ</sup> <sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> 1 *:*

*n*¼1

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

*<sup>B</sup>*1*<sup>n</sup>* ¼ �*P*~1*<sup>n</sup>*, *ne*�*nη*<sup>1</sup> *<sup>κ</sup>* � <sup>2</sup>

*κn e*

*<sup>B</sup>*1*<sup>n</sup>* ¼ � <sup>2</sup>

<sup>2</sup>*<sup>μ</sup> σηη* <sup>¼</sup> *<sup>P</sup> h*2 0 is given, but tangent stress is

*B*1*<sup>n</sup>* þ *B*2*<sup>n</sup>* � �

> <sup>2</sup> *B*1*<sup>n</sup>* þ *B*2*<sup>n</sup>* � � cosð Þ *<sup>n</sup><sup>ξ</sup>*

<sup>¼</sup> *<sup>P</sup>*~2*<sup>n</sup>*, *<sup>n</sup>* <sup>¼</sup> 1, 2, … *:* (30)

cosð Þ *nξ*

i.e., at *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*<sup>1</sup> boundary the normal load <sup>1</sup>

with unknown *B*1*<sup>n</sup>* and *B*2*<sup>n</sup>* coefficients:

<sup>2</sup> *<sup>B</sup>*1*<sup>n</sup>* sin ð Þ¼� *<sup>n</sup><sup>ξ</sup>* <sup>P</sup><sup>∞</sup>

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

X∞ *n*¼1

P∞ *n*¼1

i.e.,

*ne*�*nη*<sup>1</sup> *<sup>κ</sup>*

*ne*�*nη*<sup>1</sup> *<sup>κ</sup>* 2

Hence,

**Figure 5.**

**176**

*ne*�*nη*<sup>1</sup> *<sup>κ</sup>* 2

where *P*~1*<sup>n</sup>* and *P*~2*<sup>n</sup>* are the coefficients of expansion into the Fourier series of functions *<sup>f</sup>* <sup>1</sup>ð Þ¼� *<sup>ξ</sup> <sup>P</sup>η*<sup>1</sup> *<sup>ξ</sup>*2þ*η*<sup>2</sup> 1 and *<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 , respectively (*f* <sup>1</sup>ð Þ*ξ* , according to sinuses, and *f* <sup>2</sup>ð Þ*ξ* , according to cosines).

As it can be seen, the main matrix of system (30) has a block-diagonal form, and the dimension of each block is 2 � 2. Thus, two equations with two *B*1*<sup>n</sup>* and *B*2*<sup>n</sup>* unknown values will be solved. After solving this system, we find the values of *B*1*<sup>n</sup>* and *B*2*<sup>n</sup>* coefficients and put them into formulas (28) and (29) to get displacements and stresses at any points of the body.

Numerical results are obtained for some characteristic points of the body, in particular, *M*<sup>1</sup> 0, *η*<sup>1</sup> ð Þ, *M*<sup>2</sup> *ξ*1, *η*<sup>1</sup> ð Þ points (see. **Figure 2a**), for the following data: *<sup>ν</sup>* <sup>¼</sup> <sup>0</sup>*:*3, *<sup>E</sup>* <sup>¼</sup> <sup>2</sup> <sup>∗</sup> <sup>10</sup>6kg*=*cm2, *<sup>P</sup>* ¼ �10 kg*=*cm2, 0*:*01≤*η*<sup>1</sup> <sup>≤</sup>3, *<sup>ξ</sup>*<sup>1</sup> <sup>¼</sup> <sup>2</sup> <sup>∗</sup> *<sup>π</sup>*, *<sup>ξ</sup>*<sup>1</sup> <sup>¼</sup> <sup>4</sup> <sup>∗</sup> *<sup>π</sup>*, and *ξ*<sup>1</sup> ¼ 6 ∗ *π*.

The above-presented graphs (see **Figures 5** and **6**) show how displacements and stresses change at some characteristic points of body, namely, at points ð Þ*j* and *<sup>M</sup>*ð Þ*<sup>j</sup>* ð Þ*j*

*M*ð Þ*<sup>j</sup>* <sup>1</sup> 0, *η* 1 <sup>2</sup> *ξ*1, *η* 1 ð Þ *<sup>j</sup>* <sup>¼</sup> 1, 2, … , 8 , when 0*:*01≤*η*<sup>1</sup> <sup>≤</sup>3 (see **Figure 7**). From the presented results, we obtain the following:

• At points *M*ð Þ*<sup>j</sup>* <sup>1</sup> 0, *η* ð Þ*j* 1 , max *ut* j j<sup>&</sup>lt; max *un* j j, v*<sup>t</sup>* <sup>¼</sup> <sup>v</sup>*<sup>n</sup>* <sup>¼</sup> <sup>0</sup>*:*

**Figure 6.**

*Tangential stress and normal displacements at points M1 0*, *η<sup>1</sup>* ð Þ *for ξ<sup>1</sup>* ¼ *2* ∗ *π, ξ<sup>1</sup>* ¼ *4* ∗ *π, and ξ<sup>1</sup>* ¼ *6* ∗ *π, when 0:01* ≤*η<sup>1</sup>* ≤*3.*

• Two concrete internal and external boundary value problems in stresses are

The bodies bounded by the parabola are common in practice, for example, in

building, mechanical engineering, biology, medicine, etc., the study of the deformed state of such bodies is topical, and consequently, in my opinion, setting the problems considered in the chapter and the method of their solution is interest-

set and solved.

ing in a practical view.

*x*, *y* Cartesian coordinates *ξ*, *η* parabolic coordinates

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

*λ*, *μ* elastic Lamé constants

ð Þ *<sup>u</sup>*, v displacement vector

*h<sup>ξ</sup>* ¼ *h<sup>η</sup>* ¼ *h* ¼ *c*

The coordinate axes are parabolas

<sup>0</sup> *<sup>x</sup>* � *<sup>c</sup>ξ*<sup>2</sup>

scale coefficient, *x*, *y* are the Cartesian coordinates.

*σξξ*, *σηη*, *τξη* ¼ *τηξ* normal and tangential stresses

**A. Some basic formulas in parabolic coordinates**

q

In orthogonal parabolic coordinate system *ξ*, *η*(�∞ < *ξ*< ∞, 0≤*η*< ∞, see

where *hξ*, *h<sup>η</sup>* are Lame's coefficients of the system of parabolic coordinates, *c* is a

Laplace's equation Δ*f* ¼ 0, where *f* ¼ *f*ð Þ *ξ*, *η* , in the parabolic coordinates has the

*f* ¼ *X*ð Þ� *ξ E*ð Þ*η* ,

*X*} *X* þ *E*0 *E* � �

*X*} þ *mX* ¼ 0, *E*} � *mE* ¼ 0,

<sup>2</sup> *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup> � � <sup>¼</sup> <sup>0</sup>*:*

¼ 0*:*

, *<sup>x</sup>* <sup>¼</sup> *<sup>c</sup> <sup>ξ</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup> � �*=*2, *<sup>y</sup>* <sup>¼</sup> *<sup>c</sup>ξη*,

<sup>0</sup> *<sup>x</sup>* <sup>þ</sup> *<sup>c</sup>η*<sup>2</sup>

<sup>0</sup>*=*<sup>2</sup> � �, *<sup>η</sup>*<sup>0</sup> <sup>¼</sup> const*:*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ξ</sup>*<sup>2</sup> <sup>þ</sup> *<sup>η</sup>*<sup>2</sup>

<sup>0</sup>*=*<sup>2</sup> � �, *<sup>ξ</sup>*<sup>0</sup> <sup>¼</sup> const, *<sup>y</sup>*<sup>2</sup> ¼ �2*cη*<sup>2</sup>

*<sup>f</sup>*,*ξξ* <sup>þ</sup> *<sup>f</sup>*,*ηη* � �*=<sup>c</sup>*

We have to find solution of the equation in following form

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

and then by separation of variables, we will receive

*E* and *v* modulus of elasticity and Poisson's ratio

**Notations**

**Appendix**

**Figure A1**) [23, 24]; we have

*<sup>y</sup>*<sup>2</sup> ¼ �2*cξ*<sup>2</sup>

From here

**179**

form

*U* !

#### **Figure 7.**

*Infinite region bounded by parabola marked with points, when obtaining the above-presented numerical results.*


Here superscript *t* and *n* denote the tangential and normal displacement or the stress, respectively.
