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Character of contract pressure distribution between the outside surface of the sealing material and rigid cylinder wall depending on geometrical sizes and mechanical properties of a sealer under its unilateral compression, is defined. The magnitude of the axial load for achieving tightness is determined. The dependence between the magnitude of the axial load necessary for achieving tightness and geometrical sizes is determined. It is shown that with a decrease in the height of the sealing element, the axial load necessary for achieving tightness greatly increases. Threshold height of the sealer, above which contact pressure depends little on the magnitude of the axial load, is defined. The stress-strain state of the sealing element is defined with regard to viscous-elastic properties of its material. It is shown that this greatly influences its sealing ability.

Achieving tightness with the least external load by sealers improves their efficiency and determination of sealing parameters has an important scientific value [

In this paper, based on theoretical investigations, we determine the character of contact pressure distribution between the outside surface of the sealing element and rigid cylinder wall depending on geometrical sizes and mechanical properties of the sealer under its unilateral compression.

The magnitude of the axial load for achieving tightness with regard to differences of strain state before and after contact of outside surface of the sealing element with cylinder wall, is defined. The dependence between the magnitude of the axial load necessary for tightness and geometrical sizes under unilateral compression of the sealing element, is established. It is shown that with decrease in the height of the sealing element the axial load necessary for achieving tightness, greatly increases. Furthermore, the limiting value of the height of the sealing element under unilateral compression, above which contact pressure depends a little on the magnitude of the axial load, is defined.

In this paper, the first section is exposed to research background of this paper’s work and the structure of this paper. Section 2 introduces the elastic statement of problem, and the dependence between the axial load necessary for the first contact of the outer surface of the sealing element with the rigid cylinder wall and its physic-mechanical properties and geometrical sizes is established. In Section 3, the analytic formula allowing to determine the axial load necessary for full contact and tightness of the surface of the sealing element and cylinder wall depending on its physic-mechanical properties and geometrical sizes is found. Section 4 introduces the character of contract pressure distribution between the outside surface of the sealing material and rigid cylinder wall depending on geometrical sizes and mechanical properties of a sealer under its unilateral compression. In Section 5, based on linear laws of heredity the influence of viscous-elastic properties of the cylindrical sealing element on its sealing ability is determined. In Section 6, numerical calculations are conducted under different conditions, and the results of numerical calculations are represented in the form of graphs of contact pressure and external forces necessary achieving sightless and discussed. In Section 7, some conclusions are reached.

First, some parameters are given in

Let us consider a sealing element tightly put on the stock and with a gap

The tightness of the surface of the sealing element and cylinder wall is achieved by unilateral axial compression (

parameter | symbol | parameter | symbol |
---|---|---|---|

deformation of the sealing element in the axial direction | w | instantaneous modulus of elasticity | |

deformation of the sealing element in the radial direction | u | elasticity modulus | |

radial, tangential, axial and shear deformations | shear modulus of the sealing material | G | |

radial, axial and tangential stress | friction coefficient between the washer and the end of the sealer | ||

hydrostatic pressure function | s | dynamical viscosity of the material of the sealing element | |

gap between its outside surface and rigid cylinder wall | bed ratio | ||

height of the sealer | H | axial loads necessary for first and full contact of the outer surface of the sealing element with the cylinder wall | Q, P |

inner and outer radii of the sealer | medium’s pressure | ||

outer radius of the rigid cylinder | Kronecker’s symbol |

Let us consider the first stage. As the material of the sealing element is homogeneous, we accept its deformation as axially-symmetric. Then we can use the hypothesis of plane sections and assuming that the axial deformation of the sealing element depends only on the coordinate z in the axial direction.

We locate the origin of the coordinate system in the center of the lower section of the sealing element, direct the coordinate axis z vertically-upwards, the axis r to the direction of increasing the radius (see

Allowing for above assumptions, we accept the deformation of the sealing element

where

Accepting the sealing element material as incompressible [

where

Allowing for formula (2.1), from expression (2.1) we get

Integrating expression (2.3), we have the equality

where

The boundary condition has the form:

Then, allowing for condition (2.5), from expression (2.4) we get

For the potential energy of the sealing element, after its deformation with regard to axisymmetry, we have the equality [

where H is height;

Then allowing for formulas (2.1), (2.6) and (2.8), from expression (2.7) we get

where G is a shear modulus of the sealing material.

Based on the Euler equation [

where

Integrating differential Equation (2.10) accepting

where

Allowing for boundary conditions (2.12), from the expression (2.11) we get

The radial stress at any point of the sealer, with regard to its compressibility can be defined by the formula [

where s is a hydrostatic pressure function.

s can be determined from the boundary condition

. (2.15)

Then, allowing for expression (2.6) and (2.14), from condition (2.15) we get

We define the axial load Q for repressing the scalar by the formula

on the other hand, we have the equality

Allowing for expressions (2.11), (2.13)-(2.16) and (2.18), from expression (2.17) we get

From expression (2.19) we define q in the explicit form

Now let us define the magnitude of the axial load necessary for complete contact of the outer surface of the sealing element with rigid cylinder wall. We locate the origin of the coordinate system at the center of the lower section of the sealing element and direct the axis z vertically upwards, the axis r to the side increasing of the radius as was shown in

Using the hypothesis of plane sections and assuming that the axial deformation of the sealing element depends only on the coordinate in the axial direction z, we can accept

where

Then allowing for formula (3.1), from the incompressibility condition (2.2) we have

where

The boundary condition has the form

Then allowing for the boundary condition, from expression (3.2) we get

For the potential energy of the sealing element, after its deformation, with regard to axisymmetry of the problem we have the equality [

Substituting expression (3.4) in formula (2.8), the obtained results in expression (3.5) and then integrating it with respect to r , based on the Euler equation [

The solution of differential Equation (3.6) will have the form

where

Allowing for

We define the integration constants

Then allowing for boundary conditions (3.9)-(3.11), from expression (3.8) we have

The axial force, necessary for deformation of the sealing element to its complete contact of its outer surface with cylinder wall, can be determined by the formulas

where

After complete contact of the outer surface of the sealing element with rigid cylinder wall, the boundary condition in the upper section has the form

Then allowing for boundary condition (3.13), from expression (2.14) we get

And allowing for expressions (2.18) and (3.14) the expression (3.12) takes the form

Then allowing for formula (3.8), and expressions of

The contact pressure between the outside surface of the sealing element and cylinder wall after their complete contact can be determined (by the analogy with a beam on an elastic foundation) by the formula

If this time the tightness of the wall is not provided, then we continue to compress the sealing element. To determine the stress-strain state, at the distance z from the lower basis of the sealer we distinguish an annular element of height dz and compose for it the equilibrium equation

where

On the other hand, with regard to incompressibility of the sealing material, we have [

where

Substituting the expression (4.3) in Equation (4.2) and having integrated the obtained expression with regard to the boundary condition

where

The contact pressure distribution between the outer surface of the sealer and rigid cylinder wall can be determined from the expression (4.1) and (4.4) as follows:

Then allowing for formula (4.4), from the expression (4.5) we get

The value of

Accounting of viscous-elastic properties of the material of the sealing element on sealing ability may be realized based on the hypothesis of elastic analogy [

It should be noted that at elastic analogy, all stress components satisfy the dependence between stresses and strain obtained on the basis of the chosen model for a uniaxial stress-strain state.

The dependence between the stress-strain components for an arbitrary case of loading of a model that describes best the viscoelastic behavior of the material of a sealing element, is of the form [

where

of the sealing element,

Based on elastic analogy we represent

Substituting expression (5.2) in formula (5.1), we get

Integrating expression (4.3) with the initial condition

Introducing the denotation

we can represent the expression (5.4) in the form

For the considered case, when the sealing element at initial moment of deformation

Then allowing for (5.7), from expression (5.5) we get

Allowing for the expression (5.8), from the expression (2.20) we get

Allowing for the expression (5.8), from the expression (3.16) we get

We now consider the case when the sealing element deforms uniformly. Based on the elastic analogy [

where

From formulas (5.5) and (5.12) we get

where

Integrating formula (5.13), we get

Then, allowing for formula (5.14) following from the expression (2.20) we get

From the expression (5.15) we define the magnitude of the axial load necessary for recompressing the sealing element to contact of its outer surface with the cylinder wall.

Substituting formulas (5.14) in expression (3.16) for the axial load necessary for full contact of the outer surface of the sealing element with the cylinder wall with regard to heredity, we get

The axial load necessary for achieving tightness

Thus, based on theoretical investigations, the analytic formulas allowing to determine the contact pressure between the outside surface of the sealing element and rigid cylinder wall and the axial loads necessary for first and full contact of these surfaces depending on its physic-mechanical properties and geometrical sizes is found.

The numerical calculation is made by formulas (2.20), (3.16), (4.6), (5.9), (5.10), (5.15) and (5.16) for the values of parameters, which are showed in

The results of numerical calculations are represented in the form of graphs of contact pressure and external forces necessary achieving sightless (Figures 4-10).

As is seen from

It follows from

The distribution of contact pressure between the outer surface of the sealing element and the rigid cylinder wall depending on the coordinate z was depicted in

Variable | Value |
---|---|

0.073 | |

0.093 | |

0.1 | |

0.003 | |

H, the height of the sealer, (m) | 0.005 |

G, the shear modulus of the sealing material, (Pa) | 1.3·10^{8} |

0.5 | |

0.25 | |

2·10^{7 } | |

6.7·10^{9} | |

0.01 | |

0.1 | |

10, 20, 30, 40, 50, 60 | |

10, 20, 30, 40, 50, 60 |

As is seen from

The same picture is observed when deformation of the sealer happens uniformly. This time axial stress relaxation for different velocities of deformation occurs differently (

In this article, we determine stress-strain state of the sealing element in the form of a hollow cylinder based on theoretical investigations. Influence of viscous-elastic properties of the material of the sealing element on its sealing ability is realized based on the hypothesis of elastic analogy.

1) The dependence between the magnitude of the axial load necessary for tightness and geometrical sizes under unilateral compression of the sealing element, is established. It is shown that, with decrease in the height of the sealing element the axial load necessary for achieving tightness, greatly increases. Furthermore, the limiting value of the height of the sealing element under unilateral compression, above of which contact pressure depends a little on the magnitude of the axial load, is defined.

2) The obtained expression allows to determine the character of contact pressure distribution between the surface of the sealer and the rigid cylinder wall depending on its physical-mechanical characteristics and geometrical sizes.

3) Based on linear laws of heredity the analytic formula allowing to determine the axial load necessary for tightness of the surface of the sealing element and cylinder wall depending on its physic-mechanical properties and geometrical sizes is found. The current work demonstrates that viscous elastic properties of the sealing material greatly influence on its sealing ability and their ignorance may lead to incorrect conclusions.

4) The results of numerical calculations are represented in the form of graphs of external force necessary for achieving sightless. It is shown that, viscous-elastic properties of sealer’s material greatly influence on its sealing ability. Because of heredity of the sealer’s material, the values of external forces in some cases drop about four times.

The authors declare no conflicts of interest regarding the publication of this paper.

Abbasov, S.D.E.M., Rustamova, K.O. and Darishova, A.O. (2020) Contact Pressure between the Outside Surface of the Sealer and Cylinder Wall. Journal of Applied Mathematics and Physics, 8, 349- 366. https://doi.org/10.4236/jamp.2020.82027