**C. Finding of** *ξ***<sup>1</sup>**

After the boundary value problem with relevant boundary conditions on *ξ* ¼ *ξ*<sup>1</sup> ¼ *ξ*<sup>11</sup> is solved, the following condition is examined: *F*11*=F*<sup>10</sup> <*ε:*

*ε* is a sufficiently small positive number given in advance (*ε* ¼ 0, 001 � 0, 0001).

$$F\_{11} = \left[\bigcap\_{0}^{\eta\_1} (|\sigma\_{\xi\xi}| + |\sigma\_{\eta\eta}| + |\tau\_{\xi\eta}|) h d\eta\right]\_{\xi = \xi\_1}, \quad F\_{10} = \left[\bigcap\_{0}^{\eta\_1} (|\sigma\_{\xi\xi}| + |\sigma\_{\eta\eta}| + |\tau\_{\xi\eta}|) h d\eta\right]\_{\xi = \eta\_1^2}.$$

*<sup>g</sup>* number will be selected so that on boundary *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*1, point *M g*~*ξ*1, *<sup>η</sup>*<sup>1</sup> � � should correspond to the highest value of expression *σηη g*~*ξ*1, *η*<sup>1</sup> � � � � <sup>2</sup> <sup>þ</sup> *τξη <sup>g</sup>*~*ξ*1, *<sup>η</sup>*<sup>1</sup> � � � � <sup>2</sup> (when stresses are given) or to the highest value of expression *u g*~*ξ*1, *η*<sup>1</sup> � � � � <sup>2</sup> <sup>þ</sup> <sup>v</sup> *<sup>g</sup>*~*ξ*1, *<sup>η</sup>*<sup>1</sup> � � � � <sup>2</sup> (when displacements are given).

If condition *F*11*=F*<sup>10</sup> < *ε* is not valid for*ξ*<sup>1</sup> ¼ *ξ*11, the same problem will be solved at the beginning, but *ξ*<sup>1</sup> ¼ *ξ*<sup>12</sup> will be used instead of *ξ*<sup>1</sup> ¼ *ξ*11. In addition, *ξ*<sup>12</sup> > *ξ*11. Then, if condition *F*12*=F*<sup>10</sup> <*ε* is not still valid, we will continue with the boundary problem, where *ξ*<sup>1</sup> ¼ *ξ*13; besides, *ξ*<sup>13</sup> > *ξ*<sup>12</sup> >*ξ*11, and we will examine condition *F*13*=F*<sup>10</sup> < *ε*. The process will be over at the *k*th stage, if condition *F*1*<sup>k</sup>=F*<sup>10</sup> < *ε* is valid. *2D Elastostatic Problems in Parabolic Coordinates DOI: http://dx.doi.org/10.5772/intechopen.91057*

where *φ*<sup>2</sup> is the harmonic function.

*<sup>ψ</sup>*<sup>1</sup> ¼ �*α*<sup>2</sup> *<sup>ξ</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup>

*<sup>ψ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> *ξηφ*1,*<sup>ξ</sup>* � *<sup>ξ</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup>

Inserting (B8) in (B7c and d), we will get

Inserting (B9) in (B7c and d), we will have

!

Inserting (B10) in (B7c and d), we will get

� �

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

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

After the boundary value problem with relevant boundary conditions on

*ε* is a sufficiently small positive number given in advance (*ε* ¼ 0, 001 � 0, 0001).

, *F*<sup>10</sup> ¼

If condition *F*11*=F*<sup>10</sup> < *ε* is not valid for*ξ*<sup>1</sup> ¼ *ξ*11, the same problem will be solved at the beginning, but *ξ*<sup>1</sup> ¼ *ξ*<sup>12</sup> will be used instead of *ξ*<sup>1</sup> ¼ *ξ*11. In addition, *ξ*<sup>12</sup> > *ξ*11. Then, if condition *F*12*=F*<sup>10</sup> <*ε* is not still valid, we will continue with the boundary problem, where *ξ*<sup>1</sup> ¼ *ξ*13; besides, *ξ*<sup>13</sup> > *ξ*<sup>12</sup> >*ξ*11, and we will examine condition *F*13*=F*<sup>10</sup> < *ε*. The process will be over at the *k*th stage, if condition *F*1*<sup>k</sup>=F*<sup>10</sup> < *ε* is valid.

*ξ* ¼ *ξ*<sup>1</sup> ¼ *ξ*<sup>11</sup> is solved, the following condition is examined: *F*11*=F*<sup>10</sup> <*ε:*

3 5 *ξ*¼*ξ*<sup>1</sup>

*<sup>g</sup>* number will be selected so that on boundary *<sup>η</sup>* <sup>¼</sup> *<sup>η</sup>*1, point *M g*~*ξ*1, *<sup>η</sup>*<sup>1</sup>

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

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

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

� �

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

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

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

ð Þ <sup>b</sup> <sup>v</sup> <sup>¼</sup> *ξηφξ* <sup>þ</sup> ð Þ *<sup>κ</sup>* � <sup>1</sup> *<sup>φ</sup>*1*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*~1,*<sup>ξ</sup>* <sup>þ</sup> *<sup>φ</sup>*2,*<sup>ξ</sup>:* (B11)

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

ð Þa *u* ¼ �*ξηφ*1,*<sup>η</sup>* � ð Þ *κ* � 1 *φ*1*ξ* þ *φ*1,*<sup>η</sup>* þ *φ*~1,*<sup>η</sup>* þ *φ*2,*<sup>η</sup>*,

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

(B10)

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

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

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

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

(B12)

(B13)

3 5 *<sup>ξ</sup>*¼*g*~*ξ*<sup>1</sup>

� � should

<sup>þ</sup> <sup>v</sup> *<sup>g</sup>*~*ξ*1, *<sup>η</sup>*<sup>1</sup> � � � � <sup>2</sup>

� � � � <sup>2</sup> (when

*:*

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

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

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

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

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

ð *η*1

2 4

*σξξ* � � �

� þ *σηη* � � �

� þ *τξη* � � �

<sup>þ</sup> *τξη <sup>g</sup>*~*ξ*1, *<sup>η</sup>*<sup>1</sup>

� � � � <sup>2</sup>

� � �*hd<sup>η</sup>*

0

� � � � <sup>2</sup>

In the third version

ð Þ<sup>a</sup> *<sup>u</sup>* ¼ �*<sup>α</sup> <sup>ξ</sup>*<sup>2</sup> � ð Þ *<sup>η</sup>* � *<sup>α</sup>* <sup>2</sup>

ð Þ<sup>a</sup> *<sup>u</sup>* ¼ �*α*<sup>2</sup> *<sup>ξ</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup>

**C. Finding of** *ξ***<sup>1</sup>**

ð *η*1

2 4

*σξξ* � � �

� þ *σηη* � � �

(when displacements are given).

� þ *τξη* � � �

correspond to the highest value of expression *σηη g*~*ξ*1, *η*<sup>1</sup>

stresses are given) or to the highest value of expression *u g*~*ξ*1, *η*<sup>1</sup>

� � �*hd<sup>η</sup>*

0

*F*<sup>11</sup> ¼

**182**

ð Þ <sup>b</sup> <sup>v</sup> <sup>¼</sup> *<sup>α</sup>*<sup>2</sup> *ξηφ*1,*<sup>ξ</sup>* � *<sup>ξ</sup>*<sup>2</sup> � *<sup>η</sup>*<sup>2</sup>

ð Þ <sup>b</sup> <sup>v</sup> <sup>¼</sup> *α ξηφ*1,*<sup>ξ</sup>* � *<sup>ξ</sup>*<sup>2</sup> � ð Þ *<sup>η</sup>* � *<sup>α</sup>* <sup>2</sup>

*Finding such ξ*<sup>1</sup> ¼ *ξ*1*<sup>k</sup>, for which F*1*k=F*<sup>10</sup> <*ε.*

Distance *<sup>l</sup>* between surfaces *<sup>ξ</sup>* <sup>¼</sup> *<sup>ξ</sup>*<sup>1</sup> and *<sup>ξ</sup>* <sup>¼</sup> <sup>~</sup>*ξ*1, which gives the guarantee for condition *F*1*k=F*<sup>10</sup> <*ε* to be valid in the parabolic coordinate system, will be taken along the axis of the parabola , and the following expression will be obtained:

$$
\xi\_1 = \sqrt{l/c + \tilde{\xi}\_1^2}.
$$

By relying on the known solutions of the relevant plain problems of elasticity, it is purposeful to admit that *l=c* ¼ 4, 5, 6, … , which allows finding *ξ*<sup>1</sup> from the relevant equation. Let us note that when *l=c* ¼ 4, we will denote value *ξ*<sup>1</sup> by *ξ*11, when *l=c* ¼ 5; by *ξ*12, when *l=c* ¼ 6; by *ξ*13, etc. If after selecting *ξ*<sup>1</sup> ¼ *ξ*1*k*, inequality *F*1*k=F*<sup>10</sup> <*ε* is valid; in order to check the righteousness of the selection, it is necessary to once again make sure that, together with condition *F*1*k=F*<sup>10</sup> <*ε*, condition *ε*>*F*1*<sup>k</sup>=F*<sup>10</sup> > *F*1*k*þ<sup>1</sup>*=F*<sup>10</sup> >*F*1*k*þ<sup>2</sup>*=F*<sup>10</sup> > … is valid, too.
