**5. Calculated investigations of stress state of columnar structure of thermal barrier ceramic coatings with view of influence of centrifugal forces**

The most effective protection of a detail material against a thermal flow occurs in case of use ofelectron beam method for depositing of ceramic coatings ZrO2 (Tamarin & Kachanov, 2008). With the help of the specified method the ceramic coating of column structure on a surface of a metal sublayer (heat resisting coating) of working turbine blade is formed. The specified ceramic barrier coating is generated as columns (Fig. 26), are directed perpendicularly surface on which it is deposited. The columns of the ceramic coating possess low heat conductivity and provides the required durability at thermal cycles. The strength characteristics of ceramics are very low.

Fig. 26. Columnar thermal barrier ceramic coating (EB technique)

Investigations of Thermal Barrier Coatings for Turbine Parts 157

Under action of centrifugal forces the coating columns are exposed to a bend. The low strength of ceramics at a stretching (<sup>В</sup> ≤ 50-200 МПа) leads to breaking off columns during a bend. Therefore the calculation of as much as possible allowable thickness "column" barrier ceramic coating should be carried out in view of operational loadings and also a conFiguration of coating columns in view of a metallized bottom ceramic layer of a coating by thickness 10-15 m. The destruction of the columnar coating at height 10-20 m under influence of operational loadings is shown in Fig. 30 and the spalling of the thermal barrier

Fig. 29. The top part of a columnar coating

Fig. 30. Destruction of the columnar coating

ceramic coating on the turbine blade is presented in Fig. 31.

The calculated investigations of stress state of columnar ceramic coating of GTE blade on view of operated conditions were carried out when the materials of blade and ceramic coating are loaded by centrifugal forces (lepeshkin, 2010). The deformation of a sublayer under action of centrifugal forces together with temperature deformation is accompanied by increase of the distance between legs of columnar coating. Thus the jointed coating surface to crack on blocks and single columns (Fig. 26 and Fig 27).

Fig. 27. Columnar thermal barrier ceramic coating (surface)

Fig. 28. The growth of cracks in a columnar coating leads to occurrence of blocks

The calculated investigations of stress state of columnar ceramic coating of GTE blade on view of operated conditions were carried out when the materials of blade and ceramic coating are loaded by centrifugal forces (lepeshkin, 2010). The deformation of a sublayer under action of centrifugal forces together with temperature deformation is accompanied by increase of the distance between legs of columnar coating. Thus the jointed coating surface

to crack on blocks and single columns (Fig. 26 and Fig 27).

Fig. 27. Columnar thermal barrier ceramic coating (surface)

Fig. 28. The growth of cracks in a columnar coating leads to occurrence of blocks

Fig. 29. The top part of a columnar coating

Under action of centrifugal forces the coating columns are exposed to a bend. The low strength of ceramics at a stretching (<sup>В</sup> ≤ 50-200 МПа) leads to breaking off columns during a bend. Therefore the calculation of as much as possible allowable thickness "column" barrier ceramic coating should be carried out in view of operational loadings and also a conFiguration of coating columns in view of a metallized bottom ceramic layer of a coating by thickness 10-15 m. The destruction of the columnar coating at height 10-20 m under influence of operational loadings is shown in Fig. 30 and the spalling of the thermal barrier ceramic coating on the turbine blade is presented in Fig. 31.

Fig. 30. Destruction of the columnar coating

Investigations of Thermal Barrier Coatings for Turbine Parts 159

The features of calculation of a stress state of coating columns in different cases of fastening at influence of centrifugal forces. The bending moment under action of centrifugal forces in

> <sup>2</sup> () ( - )() *l*

> > 2

2 4

4 64

2 0

*l*

0 () () 0 *l M M rvS v dv <sup>B</sup>* 

> ( ) () *d x S x*

( ) () *d x J x* 

<sup>0</sup> () *d d dx d*

( ) '' ( ) *M x*

2

0 () () 0 *l*

The condition of a fastening of columns in the top part (plane-parallel movement)

*M*

, (3)

. (4)

(5)

(6)

(7)

*<sup>y</sup> EJ x* . (8)

*rvS v dv* . (9)

*B x M x M r v x S v dv* 

The bending moment from action of centrifugal forces in column cantilewer

section of a column at length x:


v – current coordinate.

Some auxiliary parameters:

the area of section of a column at length x

moment of inertia of section at length x

diameter of circular section at length x

Then

If the top part is free:

Here - density of a material (ceramic) of a column,

r – distance from a column up to an axis of rotation, MB - moment of action of external forces in top section,

Fig. 31. Spalling of the thermal barrier ceramic coating on the turbine blade

The analytical calculations of stress state of columns of a ceramic coаting of turbine blade were carried out under following conditions: frequency of rotation - 10000 r.p.m, radius - 400 mm from an axis of rotation, density of a coating - 4450 kg/m3, parameters of columns: d1 = 0,5 m - diameter of the basis of a column, d2 = 0,5÷5,0 m - diameter of the top part of a column, *l* height of a column (thickness of a coating). Two calculated cases were considered (Lepeshkin & Vaganov, 2010). In the first case the stress state of a single column was considered with fastening his leg in cantilewer. In the second case the calculation of stress state of a column in the block was carried out at fastening his leg in the basis of the block and his top part in a continuous surface of the block in view of a hypothesis of plane-parallel movement. The positions of the given hypothesis consist of the following. The top part of the block of a coating is formed by connection of the top parts of columns and represents a continuous surface. The roof of coating block starts to move in parallel the basis of the block under influence of centrifugal forces on the block. In view of the specified conditions the stress state of a column in the block in a field of action of centrifugal forces is calculated. The calculated circuits for determination of the stress state of columns of a ceramic coating are resulted in Fig. 32. From Fig. 26 follows that columns have the cone form that also is shown in circuits in Fig. 32.

Fig. 32. Calculated schemes of determination of a stress state: 1 - single columns; 2 - columns in blocks; *d*1 - diameter of the basis of a column, *d2* - diameter of the top part of a column,

The features of calculation of a stress state of coating columns in different cases of fastening at influence of centrifugal forces. The bending moment under action of centrifugal forces in section of a column at length x:

$$M(\mathbf{x}) = M\_B + \int\_{\mathbf{x}}^{l} \rho \rho o^2 r(\mathbf{v} \cdot \mathbf{x}) S(\mathbf{v}) d\mathbf{v} \,\,\,\,\,\tag{3}$$

Here - density of a material (ceramic) of a column,


r – distance from a column up to an axis of rotation,

MB - moment of action of external forces in top section,

v – current coordinate.

158 Ceramic Coatings – Applications in Engineering

Fig. 31. Spalling of the thermal barrier ceramic coating on the turbine blade

The analytical calculations of stress state of columns of a ceramic coаting of turbine blade were carried out under following conditions: frequency of rotation - 10000 r.p.m, radius - 400 mm from an axis of rotation, density of a coating - 4450 kg/m3, parameters of columns: d1 = 0,5 m - diameter of the basis of a column, d2 = 0,5÷5,0 m - diameter of the top part of a column, *l* height of a column (thickness of a coating). Two calculated cases were considered (Lepeshkin & Vaganov, 2010). In the first case the stress state of a single column was considered with fastening his leg in cantilewer. In the second case the calculation of stress state of a column in the block was carried out at fastening his leg in the basis of the block and his top part in a continuous surface of the block in view of a hypothesis of plane-parallel movement. The positions of the given hypothesis consist of the following. The top part of the block of a coating is formed by connection of the top parts of columns and represents a continuous surface. The roof of coating block starts to move in parallel the basis of the block under influence of centrifugal forces on the block. In view of the specified conditions the stress state of a column in the block in a field of action of centrifugal forces is calculated. The calculated circuits for determination of the stress state of columns of a ceramic coating are resulted in Fig. 32. From

Fig. 26 follows that columns have the cone form that also is shown in circuits in Fig. 32.

Fig. 32. Calculated schemes of determination of a stress state: 1 - single columns; 2 - columns in blocks; *d*1 - diameter of the basis of a column, *d2* - diameter of the top part of a column,

The bending moment from action of centrifugal forces in column cantilewer

$$M(0) = M\_{\mathcal{B}} + \int\_0^l \rho o^2 r v S(v) dv \,. \tag{4}$$

Some auxiliary parameters:

the area of section of a column at length x

$$S(\mathbf{x}) = \,\_2\pi \frac{d(\mathbf{x})^2}{4} \tag{5}$$

moment of inertia of section at length x

$$J(\mathbf{x}) = \, \_\pi J(\mathbf{x}) \, ^4 \tag{6}$$

diameter of circular section at length x

$$d(\mathbf{x}) = \,\_0d\_0 + \frac{d\_2 - d\_0}{l} \tag{7}$$

Then

$$\mathbf{y''} = \frac{M(\mathbf{x})}{E(\mathbf{x})}.\tag{8}$$

If the top part is free:

$$M(0) = \int\_0^l \rho \rho o^2 r v S(v) dv \,. \tag{9}$$

The condition of a fastening of columns in the top part (plane-parallel movement)

Investigations of Thermal Barrier Coatings for Turbine Parts 161

column in 100 m with the cross-section sizes d1 = 0,5 m, d2 = 2,0 m in Fig. 36 shows that the stress in the basis of a column in the block is less than in the basis of a single column in 7 times. For comparison of stresses in the basis of a column in different cases the curves 1 and 2 with ratios d2/d1 from 1 up to 2 for a single column and curves 3, 4 and 5 with ratios d2/d1 from 1 up to 10 columns in the block at increase in length of a column are shown on Fig. 37. From the analysis of Fig. 37 follows that there are following restrictions on length of a column in view of stresses in the basis: more than length 120 m at d2/d1 = 10 and 100 m at d2/d1 = 4 and 80 m at d2/d1 = 1 for length of a column in the block and 80 m at d2/d1 = 10

Fig. 33. Distribution of the bending moment in a column by length 100 m (*d*1 = 0,5 m, *d*2 =

Fig. 34. Distribution of stresses in a single column by length 100 m, the column of the different cross-section sizes: 1 - *d*1 = 0,5 m, *d*2 = 1,0 m; 2 - *d*1 = 0,5 m, *d*2 = 0,5 m

and 40 m at d2/d1 = 4 for a single columns.

2,0 m): 1 - single column, 2 - column in the block

$$d(\mathbf{x}) = \,\_0d\_0 + \frac{d\_2 - d\_0}{l} \,. \tag{10}$$

$$y'(l) = \begin{array}{c} 0 \end{array} \tag{11}$$

As y'(0) = 0 (rigid fastening in the basis), then

$$\int\_{0}^{l} y''(v)dv = y'(l) - y'(0) = 0\tag{12}$$

Hence, the condition for a moment

0 ( ) <sup>0</sup> ( ) *<sup>l</sup> M v dv EJ v* . (13)

Substituting in (10) the formula (1) we receive the moment of action of external forces in the top section of a column:

$$M\_B = -\frac{\rho o^2 r \int\_0^l \frac{l}{v} (u-v)S^2(u) du}{\int\_0^l \frac{d(v)^4}{d(v)^4}},\tag{14}$$

Thus, it is known M(x):

$$\mathbf{M}(\mathbf{x}) = -\frac{\rho \rho^2 \mathbf{r} \left\| \frac{\mathbf{v}}{\rho} \frac{\mathbf{v}}{d(\mathbf{v})^4} d\mathbf{v} \right\|}{\int \frac{\mathbf{d} \mathbf{v}}{d(\mathbf{v})^4}} d\mathbf{v} + \frac{1}{\int \rho \rho^2 \mathbf{r} (\mathbf{v} \cdot \mathbf{x}) \mathbf{S}(\mathbf{v}) d\mathbf{v}}.\tag{15}$$

Knowing M(x) is possible to find the maximal stretching stress in section of a column:

$$
\sigma\_{\text{max}}(\mathbf{x}) = \frac{M(\mathbf{x})d(\mathbf{x})}{2J(\mathbf{x})} \tag{16}
$$

The results of calculated investigations have been conducted. The stress distributions on length of single columns and the columns which are taking place in blocks of a ceramic coating in height 100 microns at influence of centrifugal forces (as shown in Fig. 33-36) are received. The stresses (curves) in the basis of a column depending on length are submitted on Fig. 37. The taper of a columns is determined by a ratio d2/d1. From Fig. 35 follows that at increase of a ratio d2/d1 from 1 up to 10 the stresses in the basis of the column which is taking place in the block are reduced twice. The analysis of stress distribution on length of a

''( ) '( ) '( ) 0 0

( ) <sup>0</sup> ( ) *<sup>l</sup> M v dv*

Substituting in (10) the formula (1) we receive the moment of action of external forces in the

*l l v*

 

2

*r*

*B l*

*M*

l l 2 v

 

r

0

2

*u v S u du*

( ) ()

*d v*

4

0 2

( ) M(x) - r(v-x)S(v)dv. dv

4

Knowing M(x) is possible to find the maximal stretching stress in section of a column:

2 max ()() ( ) ( ) *M xdx*

The results of calculated investigations have been conducted. The stress distributions on length of single columns and the columns which are taking place in blocks of a ceramic coating in height 100 microns at influence of centrifugal forces (as shown in Fig. 33-36) are received. The stresses (curves) in the basis of a column depending on length are submitted on Fig. 37. The taper of a columns is determined by a ratio d2/d1. From Fig. 35 follows that at increase of a ratio d2/d1 from 1 up to 10 the stresses in the basis of the column which is taking place in the block are reduced twice. The analysis of stress distribution on length of a

*x*

( )

*d v*

0

l

0

( ) - '

2

*u v S u du*

( ) ()

*d v*

4

l

x

(16)

4

( )

*dv*

*J x*

*dv d v*

<sup>0</sup> () *d d dx d*

As y'(0) = 0 (rigid fastening in the basis), then

Hence, the condition for a moment

top section of a column:

Thus, it is known M(x):

0

0

*l*

2 0

. (10)

*y l* '( ) 0 (11)

*y v dv y l y* (12)

*EJ v* . (13)

(14)

(15)

*l*

column in 100 m with the cross-section sizes d1 = 0,5 m, d2 = 2,0 m in Fig. 36 shows that the stress in the basis of a column in the block is less than in the basis of a single column in 7 times. For comparison of stresses in the basis of a column in different cases the curves 1 and 2 with ratios d2/d1 from 1 up to 2 for a single column and curves 3, 4 and 5 with ratios d2/d1 from 1 up to 10 columns in the block at increase in length of a column are shown on Fig. 37.

From the analysis of Fig. 37 follows that there are following restrictions on length of a column in view of stresses in the basis: more than length 120 m at d2/d1 = 10 and 100 m at d2/d1 = 4 and 80 m at d2/d1 = 1 for length of a column in the block and 80 m at d2/d1 = 10 and 40 m at d2/d1 = 4 for a single columns.

Fig. 33. Distribution of the bending moment in a column by length 100 m (*d*1 = 0,5 m, *d*2 = 2,0 m): 1 - single column, 2 - column in the block

Fig. 34. Distribution of stresses in a single column by length 100 m, the column of the different cross-section sizes: 1 - *d*1 = 0,5 m, *d*2 = 1,0 m; 2 - *d*1 = 0,5 m, *d*2 = 0,5 m

Investigations of Thermal Barrier Coatings for Turbine Parts 163

Fig. 37. Stresses in the basis of a column depending on length: Single column: 1 - *d*1 = 0,5 m, *d*2 = 2,0 m; 2 - *d*1 = 0,5 m, *d*2 = 0,5 m; Column in the block: 3 - *d*1 = 0,5 m, *d*2 = 0,5 m; 4 - *d*<sup>1</sup>

The formed single columns are exposed to a bend under influence of centrifugal forces under operating conditions at cracks in coatings and at their length more than 40-100 m can break. In the educated blocks the columns are loaded in a field of action of centrifugal forces and at their length no more than 100-140 m can be kept without destruction. Thus the probability of destruction of columns in the block decreases at increase taper - ratio d2/d1. It is shown that at designing columnar ceramic coverings their allowable thickness should make no more than 100-140 microns in conditions of influence of the centrifugal

One of the directions of future investigations are improvement of the designs and rise of durability of ceramic coatings and the development of advanced technologies for their depositing. The blades and turbine components of gas turbine engines have a ceramic coating of uniform thickness. The disadvantages of the part design are the increased thickness of the ceramic coating that maintains or increases the unevenness of temperature distribution and thermal stresses in the metal blade. In addition, in the increased ceramic coating mass of the turbine blades will arise the increased stresses under the influence of centrifugal forces. The factors can lead to lower life of a coating and blade. On a turbine parts is possible to put ceramic coatings of variable thickness. The coatings of variable thickness allow to raise the durability of a turbine part due to increase of strength of a covering and reduction thermal stresses in part metal. The durability raises due to variable thickness of a coating and due to uniformity of temperature distribution in a junction of a coating with metal of a part and due to performance of a coating of the maximal thickness in zones of the maximal temperatures and the minimal thickness in zones of the minimal temperatures on a surface of a coating. In result the temperature drops and thermal stresseson a profile and height of a part (blade) are reduced. Also the probably of occurrence of defects, cracks and chipping of ceramics in the areas of stress concentration decreases and the durability of a coating and blade increases in view of influence of centrifugal forces and

= 0,5 m, *d*2 = 2,0 m; 5 - *d*1 = 0,5 m, *d*2 = 5,0 m

forces on the basis of carried out investigations.

**6. Thermal barrier ceramic coatings of variable thickness** 

Fig. 35. Distribution of stresses in a column by length 100 m (in the block), the column of the different cross-section sizes: 1 - *d*1 = 0,5 m, *d*2 = 0,5 m; 2 - *d*1 = 0,5 m, *d*2 = 1,0 m; 3 - *d*<sup>1</sup> = 0,5 m, *d*2 = 2,0 m; 4 - *d*1 = 0,5 m, *d*2 = 5,0 m

Fig. 36. Distribution of stresses in a column by length 100 m (*d*1 = 0,5 m, *d*2 = 2,0 m): 1 single column, 2 - column in the block

Fig. 35. Distribution of stresses in a column by length 100 m (in the block), the column of the different cross-section sizes: 1 - *d*1 = 0,5 m, *d*2 = 0,5 m; 2 - *d*1 = 0,5 m, *d*2 = 1,0 m; 3 - *d*<sup>1</sup>

Fig. 36. Distribution of stresses in a column by length 100 m (*d*1 = 0,5 m, *d*2 = 2,0 m): 1 -

= 0,5 m, *d*2 = 2,0 m; 4 - *d*1 = 0,5 m, *d*2 = 5,0 m

single column, 2 - column in the block

Fig. 37. Stresses in the basis of a column depending on length: Single column: 1 - *d*1 = 0,5 m, *d*2 = 2,0 m; 2 - *d*1 = 0,5 m, *d*2 = 0,5 m; Column in the block: 3 - *d*1 = 0,5 m, *d*2 = 0,5 m; 4 - *d*<sup>1</sup> = 0,5 m, *d*2 = 2,0 m; 5 - *d*1 = 0,5 m, *d*2 = 5,0 m

The formed single columns are exposed to a bend under influence of centrifugal forces under operating conditions at cracks in coatings and at their length more than 40-100 m can break. In the educated blocks the columns are loaded in a field of action of centrifugal forces and at their length no more than 100-140 m can be kept without destruction. Thus the probability of destruction of columns in the block decreases at increase taper - ratio d2/d1. It is shown that at designing columnar ceramic coverings their allowable thickness should make no more than 100-140 microns in conditions of influence of the centrifugal forces on the basis of carried out investigations.
