Contact Mechanics of Rough Surfaces in Hermetic Sealing Studies

Abstract

It is indicated that the sealing capacity depends on the contact characteristics—the relative contact area and the gap density in the joint. To determine the contact characteristics, a discrete roughness model is used in the form of a set of spherical segments, the distribution of which in height is related to the bearing curve described by the regularized beta function. The contact of a single asperity is considered with taking into account the influence of the remaining contacting asperities. The equations for determining the relative contact area and gap density in the joint depending on the dimensionless force parameters for elastic and elastic-plastic contacts are provided.

1. Introduction

Tightness is the property of the joints to provide an acceptable leakage value, determined from the conditions of normal operation of various systems and equipment, human safety, and environmental protection. To quantify the tightness, the leakage rate is used, that is, the mass or volume of the medium per unit time per unit length along the SJ’s perimeter. By ‘sealing joint’ (SJ), we mean a set of details that form a structure to ensure tightness.

The SJ’s tightness is provided by loading with a compressive load (the contact sealing pressures), which is largely determined by the stress-strain state in the contact area and depends on the contact interaction of the rough surfaces. The main contact characteristics ensuring SJ’s tightness are the approaching of rough surfaces, the relative contact area, the density of gaps in the joint, and the degree of fusion of contact spots of single asperities. Depending on the materials’ properties and microgeometry parameters, there are elastic, viscoelastic, elastic-plastic, and rigid-plastic contacts.

At present, to solve the tribology problems, we need to use the roughness models and the rough surfaces contacting theory developed by the authors [1, 2] and their followers. However, the use of such models to solve the problems in hermetic sealing studies leads to significant errors, which is explained by the following:

1. the contact pressures of the sealing are approximately 1–2 orders of magnitude higher than for friction and at that, it is necessary to be taken into account the mutual influence of the contacting asperities;

2. in the sealing joint, all the asperities's contacting is possible, which requires the description of the whole bearing profile curve but not only its initial part, as in [2];

3. when determining the gaps volume (or density), the displacements of the points of the asperities surfaces have not been taken into account; and

4. the extrusion of the material into the intercontact space under elastic-plastic contact has not been taken into account.

Therefore, to describe the SJ, a rough surface model is required that adequately describes the real surface and corresponds to the whole bearing curve, and not just its initial part. In addition, in order to improve the accuracy of the calculation of the contact characteristics, the discrete model of a rough surface must be taken into account, the real distribution of dimensions of microasperities and the mutual influence. The criterion of plasticity must take into account the general stress-strain state when contacting of a rough surface and not just of a single asperity. In most cases, the contact of metallic rough surfaces is elastic-plastic, therefore, to determine the contact characteristics, it is necessary to take into account the parameters of material hardening.

To estimate the SJ’s sealing property, in [3, 4], the nondimensional permeability functional is used

$$C_u=\frac{{\mathrm{\Lambda }}^3{\upsilon }_k}{4{\left(1-\mathrm{\eta }\right)}^2},$$E1

where $\mathrm{\Lambda }$ is the gaps density in the joint; $\mathrm{\eta }$ is the relative contact area; ${\upsilon }_k$ is the probability of a medium flowing, which depends on the single contact spots fusion.

All the parameters that appear in Eq. (1) depend on the parameters of microgeometry and dimensionless force parameters $f_q$ or ${\overline{q}}_{\sigma }$, the determination of which is given in the following sections.

The purpose of the given research is to develop methods for calculating the contact characteristics that ensure the given tightness of the immobile joints with taking into account the complex of functional parameters of the sealing surfaces and mutual influence of asperities.

2. Discrete model of the rough surface

We consider that the initial data for the model representation of a rough surface are parameters of roughness according to ISO 4287–1997, ISO 4287/1–1997: maximum roughness depth R_max, arithmetic mean deviation of the profile R_a, root-mean-square deviation of the profile R_q, mean height of the profile elements R_p, mean width of the profile elements S_m, bearing profile curve t_p, and bearing profile curve on the midline t_m. Thus, the standard parameters of the roughness for the developed model must coincide with the corresponding parameters of the real surface.

To describe the entire rough surface, it is required to know one of two functions:

$${\mathrm{\eta }}_u\left(\mathrm{\epsilon }\right)=\frac{A_u}{A_c}\text{ or }{\mathrm{\phi }}_n\left(u\right)=\frac{n_u}{n_c},$$E2

where A_u is the material cross-sectional area at a relative level $\mathrm{\epsilon }=h/R_{\max }$; A_c is the contour area; n_u is the number of asperities whose peaks are located above the level u; $n_c=A{}_c/A_{\mathit{ci}}$ is the total number of asperities; and $A_{\mathit{ci}}$ is the area due to a single asperity.

According to ISO 4287–1997, parameters of roughness are determined from profilograms and the functions describing the distribution for the profile t_p and the surface η_u(ε), but it is not fulfilled for the peaks and valleys asperities distribution functions of the profile φ_nl(u_l) and the surface φ_n(u), then the model is based on the bearing profile curve.

Let us assume that the function ${\mathrm{\eta }}_u\left(\mathrm{\epsilon }\right)$ is monotonic and twice differentiable. A rough surface (Figure 1) is a set of asperities in the form of spherical segments of radius r and height $\mathrm{\omega }{R}_{\max }$, and base radius $a_c=\sqrt{A_{\mathit{ci}}/\pi }$. It is necessary to find such a function ${\mathrm{\phi }}_n\left(u\right)$ for which the distribution of the material in the rough layer corresponds to the bearing surface curve.

Figure 1.

The scheme and the bearing curve of a rough surface.

Figure 2.

The distribution densities of asperities for different values of p and q.

The cross-section of the i-th asperity at the level ε is

$$A_{\mathit{ri}}=2\pi {\mathit{rR}}_{\max }\left(\mathrm{\epsilon }-u\right),$$E3

where u is the relative distance from the peaks level to the peak of the i-th asperity.

The number of peaks in the layer du and at a distance u is equal to

$${\mathit{dn}}_r=n_c{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}.$$E4

Then, $A_u=A_r=2\pi {\mathit{rR}}_{\max }{n}_c\underset{0}{\overset{\mathrm{\epsilon }}{\int }}{\mathrm{\phi }}_n^{\prime }\left(u\right)\left(\mathrm{\epsilon }-u\right)\mathit{du};$

$${\mathrm{\eta }}_u\left(\mathrm{\epsilon }\right)=\frac{A_r\left(\mathrm{\epsilon }\right)}{A_c}=C\underset{0}{\overset{\mathrm{\epsilon }}{\int }}\left(\mathrm{\epsilon }-u\right){\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du},C=\frac{2\pi {\mathit{rR}}_{\max }n{}_c}{A_c}.$$E5

Further, we have

$$\mathrm{\eta }\left(\mathrm{\epsilon }\right)=C\left(\mathrm{\epsilon }\underset{0}{\overset{\mathrm{\epsilon }}{\int }}{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}-\underset{0}{\overset{\mathrm{\epsilon }}{\int }}u{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}\right)=C\left({\mathit{\epsilon \phi }}_n\left(\mathrm{\epsilon }\right)-u{\mathrm{\phi }}_n\left(u\right)\left|\begin{array}{l}{}^{\mathrm{\epsilon }}\\ {}_0\end{array}\right)+\underset{0}{\overset{\mathrm{\epsilon }}{\int }}{\mathrm{\phi }}_n\left(u\right)\mathit{du}\right),$$

$$\mathrm{\eta }\left(\mathrm{\epsilon }\right)=C\underset{0}{\overset{\mathrm{\epsilon }}{\int }}{\mathrm{\phi }}_n\left(u\right)\mathit{du}.$$E6

Twice differentiating the left and right sides of ε, we have

$${\mathrm{\eta }}^{\prime }\left(\mathrm{\epsilon }\right)=C{\mathrm{\phi }}_n\left(\mathrm{\epsilon }\right),\hspace{1em}{\mathrm{\eta }}^{″}\left(\mathrm{\epsilon }\right)=C{\mathrm{\phi }}_n^{\prime }\left(\mathrm{\epsilon }\right);$$E7

$${\mathrm{\phi }}_n\left(\mathrm{\epsilon }\right)=\frac{{\mathrm{\eta }}^{\prime }\left(\mathrm{\epsilon }\right)}{C},\hspace{1em}{\mathrm{\phi }}_n^{\prime }\left(\mathrm{\epsilon }\right)=\frac{{\mathrm{\eta }}^{″}\left(\mathrm{\epsilon }\right)}{C}.$$E8

To describe the bearing surface curve, we use the regularized beta function:

$$t_p\left(\mathrm{\epsilon }\right)=\mathrm{\eta }\left(\mathrm{\epsilon }\right)=I_{\mathrm{\epsilon }}\left(p,q\right)=\frac{{\mathrm{{\rm B}}}_{\mathrm{\epsilon }}\left(p,q\right)}{\mathrm{{\rm B}}\left(p,q\right)},$$E9

where

$$p={\left(\frac{R_p}{R_q}\right)}^2\left(\frac{R_{\max }-R_p}{R_{\max }}\right)-\frac{R_p}{R_{\max }},\hspace{0.17em}q=p\left(\frac{R_{\max }}{R_p}-1\right).$$E10

В_ε(α,β) и В(α,β) are the incomplete and complete beta-functions.

Double differentiating Eq. (9), from Eq. (8), for the function and the distribution density of the asperities, we have

$${\mathrm{\phi }}_n\left(u\right)=\frac{{\mathrm{\eta }}_u^{\prime }\left(u\right)}{C}=\frac{u^{p-1}{\left(1-u\right)}^{q-1}}{{{\mathrm{\epsilon }}_s}^{p-1}{\left(1-{\mathrm{\epsilon }}_s\right)}^{q-1}};$$E11

$${\mathrm{\phi }}_n^{\prime }\left(u\right)=\frac{{{\mathrm{\eta }}^{″}}_u^{}\left(u\right)}{C}=\frac{u^{p-2}{\left(1-u\right)}^{q-2}\left[\left(p-1\right)\left(1-u\right)-\left(q-1\right)u\right]}{{\mathrm{\epsilon }}_s^{p-1}{\left(1-{\mathrm{\epsilon }}_s\right)}^{q-1}}.$$E12

The relative height of the spherical asperity is $\omega =1-{\mathrm{\epsilon }}_s$ and the radius of spherical asperity is $r=a_c^2/\left(2\omega R_{\max }\right).$

This section describes a model of a rough surface in the form of a set of spherical asperities with constant radii and heights. More complex models with asperities with variable radii and heights are given in work [3, 4].

The contact of two rough surfaces $z_i\left(x,y\right)$ can be represented as a contact of an equivalent rough surface $z\left(x,y\right)={\sum }_{i=1}^2{z}_i\left(x,y\right)$ and a flat surface. The parameters of the microgeometry of an equivalent surface are given in [3, 4].

3. Description of contact of a single asperity

3.1. Contact of a spherical asperity and the low-modulus half-space

Elastic contact occurs when low-modulus materials are used, which are used widely in sealing technology in the form of coatings or individual details [3, 5]. According to the strength criteria, the construction materials belong to the low-modulus materials if the values of the elastic moduli E < 10³ MPa [6]. When contacting metallic rough surfaces, elastic contact is possible for high surface cleanliness classes and large values of the yield strength of the material.

As shown by experiments [7, p. 179] with polymeric interlayers (a coating on one of the conjugate details), loaded by [1] compressive stresses, the real touching area tends to be a constant value, depending on the physico-mechanical properties of the interlayer material.

During elastic contact, the mutual influence of discretely loaded sections leads to the growth retardation of the contact area [3]. It is reflected in the Bartenev-Lavrentyev’s formula [7]

$$\mathrm{\eta }=1-\text{exp}\left(,-b\frac{q}{E}\right),$$E13

where b is the coefficient depending on the surface quality, $q_c$ is the contour contact pressure, and E is the elastic modulus. As it follows from Eq. (13), $\mathrm{\eta }\to 1$ for $q\to \infty$.

The question of the influence of neighboring asperities in the case of elastic contact was considered in [8, 9], where the mutual influence is replaced by the action of equal concentrated forces located at the nodes of the hexagonal lattice.

According to the Saint-Venant’s principle, at a point sufficiently distant from the region of application of the load, the stresses and deformations do not depend on the nature of the load distribution in its application area, in [10, 11]. Using the principle, the influence of the other contacting asperities is replaced by the action of a uniformly distributed load in some circular area. It allows considering the problem posed as an axisymmetric problem.

Let us consider the contact of a single absolutely rigid spherical asperity of radius r, whose peak is located at a distance uR_max from the peaks line of a rough surface with an elastic half-space in the system of cylindrical coordinates z, ρ, and φ with origin at the point О (Figure 3).

Figure 3.

Scheme of contact of a single asperity.

From an analysis of the numerous solutions of contact problems in the theory of elasticity and plasticity, it follows that a change of the distribution of external loads near the contact area under constancy of its average intensity leads to insignificant changes only near the boundary of the contact area.

Then, taking into account, the nature of the mutual location of the individual contact spots, the influence on the contact characteristics of an individual asperity within the circular contact area $W_1\left(\rho =\overline{0,a_{\mathit{ri}}}\right)$ and the circular unloaded area $W\left(\rho =\overline{a_{\mathit{ri}},a_n}\right)$ on the remaining contact spots will be equivalent to the effect of the uniformly distributed load q_cn acting in the circular area $W_2\left(\rho =\overline{a_n,a_l}\right)$, and the assigned problem may be regarded as an axisymmetric (Figure 3). The size of the unloaded area a_n depends on the number of contacting asperities and with increasing applied load, it decreases from a_l to a_c.

The solution of this problem is given in Ref. [11]. Studies on the effect of the parameter $k_a=a_n/a_c$ on the relative contact area show only 4% increase of last one; therefore, with a margin to tightness ensure, we will give a solution for k_a = 1 or a_n = a_c below.

Let A₁ and A₂ be two points on the surface of the circular contact area W₁. The A₁ and A₂ coming into contact after application of the compressive load. Since the total normal displacement U₀ of the point А₁ is constant for any point in area W₁, we have

$$U_0=U_E+z_1=U_{\mathit{Eri}}+U_{\mathit{Eci}}+z_1,$$E14

where $U_{\mathit{Eri}}$ is the normal contact displacement under the pressure $p_{\mathit{ri}}$ acting in the region W₁; $U_{\mathit{Eci}}$ is the normal displacement under the pressure q_cn; and z₁ is the equation of the surface of a spherical asperity in an unloaded state.

As for the real surfaces, r > > R_max, then

$$z_1=-{\mathit{uR}}_{\max }-\frac{{\rho }^2}{2r}.$$E15

Elementary displacements ${\mathit{dU}}_{\mathit{Eri}}$ and ${\mathit{dU}}_{\mathit{Eci}}$ under pressures q_ri and q_c acting on elementary areas ${\mathit{dw}}_1$ and ${\mathit{dw}}_2$, respectively, are determined by [12]:

$${\mathit{dU}}_{\mathit{Eri}}=\frac{\mathrm{\theta }{q}_{\mathit{ri}}\left({\rho }_1\right)}{\pi R_1}{\mathit{dw}}_1,\hspace{0.17em}{\mathit{dU}}_{\mathit{Eci}}=\frac{\mathrm{\theta }{q}_{\mathit{cn}}}{\pi R_2}{\mathit{dw}}_2;$$E16

where $R_j^2={\mathrm{\rho }}^2+{\mathrm{\rho }}_j^2-2{\mathit{\rho \rho }}_j\text{cos}{\mathrm{\phi }}_j$, j = 1, 2; $\mathrm{\rho }\equiv {\mathrm{\rho }}_i$; $\mathrm{\theta }=\left(1-{\nu }^2\right)/E$, $\nu$ is Poisson’s ratio; ${\mathit{dw}}_1={\mathrm{\rho }}_1\mathit{\text{dρdφ}}$; and ${\mathit{dw}}_2={\mathrm{\rho }}_2\mathit{\text{dρdφ}}$.

After integrating Eq. (16), we have

$$U_{\mathit{Eri}}=\frac{\mathrm{\theta }}{\pi }\underset{W_1}{\int }\frac{p_{\mathit{ri}}\left(\mathrm{\rho }\right){\mathit{dw}}_1}{R_1},$$E17

$$U_{\mathit{Eci}}=\frac{4}{\pi }\mathrm{\theta }{q}_c\left[a_l\mathrm{{\rm E}}\left(\frac{{\mathrm{\rho }}_i}{a_l}\right)-a_{с}\mathrm{{\rm E}}\left(\frac{{\mathrm{\rho }}_i}{a_c}\right)\right],$$E18

where $\mathrm{{\rm E}}\left(x\right)$ is the complete elliptic integral of the second kind.

From Eq. (15), taking into account Eqs. (16)–(18), we have

$$\underset{W_1}{\int }\frac{p_{\mathit{ri}}\left(\mathrm{\rho }\right){\mathit{dw}}_1}{R_1}=f\left({\mathrm{\rho }}_i\right),$$E19

$$f\left({\mathrm{\rho }}_i\right)=\frac{\pi }{\mathrm{\theta }}\left(U_0-{\mathit{uR}}_{\max }-\frac{\omega R_{\max }{\mathrm{\rho }}_i^2}{a_c^2}\right)-2\pi q_c\left[a_l-\frac{2}{\pi }\mathrm{{\rm E}}\left(\frac{{\mathrm{\rho }}_i}{a_c}\right)\right].$$E20

The Eq. (19) is the basic equation of an axisymmetric contact problem. The common decision of Eq. (19) is [13].

$$p_{\mathit{ri}}\left({\mathrm{\rho }}_i\right)=-\frac{1}{2\pi }\underset{{\mathrm{\rho }}_i}{\overset{a_{\mathit{ri}}}{\int }}\frac{F\left(s\right)\mathit{ds}}{\sqrt{s^2-{\mathrm{\rho }}_i^2}},\hspace{0.17em}{P}_i=-\frac{2}{\pi }\underset{0}{\overset{a_{\mathit{ri}}}{\int }}\frac{f^{\hspace{0.17em}}\left(\sigma \right){\sigma }^2\mathit{d\sigma }}{\sqrt{a_r^2-{\sigma }^2}},F\left(s\right)=\frac{2}{\pi }\left[f\left(0\right)+s\underset{0}{\overset{s}{\int }}\frac{f^{\prime }\left(\sigma \right)\mathit{d\sigma }}{\sqrt{s^2-{\sigma }^2}}\right].$$E21

As a result from (21), we have

$$p_{\mathit{ri}}\left({\mathrm{\rho }}_i\right)=\frac{4\omega R_{\max }}{\mathit{\pi \theta }{a}_c^2}\sqrt{a_{\mathit{ri}}^2-{\mathrm{\rho }}_i^2}+\frac{q_c}{\pi }\text{arcsin}\sqrt{\frac{a_{\mathit{ri}}^2-{\mathrm{\rho }}_i^2}{a_c^2-{\mathrm{\rho }}_i^2}},$$E22

$$P_i=\frac{8\omega R_{\max }{a}_{\mathit{ri}}^3}{3\mathrm{\theta }{a}_c^2}+2q_c{a}_c^2\left[\text{arcsin}\frac{a_{\mathit{ri}}}{a_c}-\sqrt{\frac{a_{\mathit{ri}}^2}{a_c^2}\left(1-\frac{a_{\mathit{ri}}^2}{a_c^2}\right)}\right].$$E23

Taking into account that ${\mathrm{\eta }}_i=a_{\mathit{ri}}^2/a_{\mathit{ci}}^2$, $q_{\mathit{ci}}=P_i/\left(\pi a_{\mathit{ci}}^2\right)$, from Eqs. (22) and (23), we have

$$p_{\mathit{ri}}\left({\mathrm{\rho }}_i\right)=\frac{4{\mathrm{\eta }}_i^{0.5}\omega R_{\max }}{\mathit{\pi \theta }{a}_c^2}\sqrt{1-\frac{{\mathrm{\rho }}_i^2}{a_{\mathit{ri}}^2}}+\frac{q_c}{\pi }\text{arcsin}\sqrt{\frac{a_{\mathit{ri}}^2-{\mathrm{\rho }}_i^2}{a_c^2-{\mathrm{\rho }}_i^2}},$$E24

$$q_{\mathit{ci}}=\frac{8\omega R_{\max }{\mathrm{\eta }}_i^{1.5}}{3\mathit{\pi \theta }{a}_c}+\frac{2}{\pi }{q}_c\left[\text{arcsin} {\mathrm{\eta }}_i^{0.5}-\sqrt{{\mathrm{\eta }}_i\left(1-{\mathrm{\eta }}_i\right)}\right].$$E25

The mean p_mi and the maximum p_ri(0) stresses at the contact spot are described by equations

$$p_{\mathit{mi}}=\frac{N_i}{A_{\mathit{ri}}}=\frac{q_{\mathit{ci}}}{{\mathrm{\eta }}_i}=\frac{8{\mathrm{\eta }}_i^{0,5}\omega R_{\max }}{3\mathit{\pi \theta }{a}_c}+\frac{2q_c}{{\mathit{\pi \eta }}_i}\left[\text{arcsin} {\mathrm{\eta }}_i^{0.5}-\sqrt{{\mathrm{\eta }}_i\left(1-{\mathrm{\eta }}_i\right)}\right],$$E26

$$p_{\mathit{ri}}\left(0\right)=\frac{4{\mathrm{\eta }}_i^{0,5}\omega R_{\max }}{\mathit{\pi \theta }{a}_c}+\frac{q_c}{\pi }\text{arcsin} {\mathrm{\eta }}_i^{0..5}.$$E27

With sufficient accuracy (with an error of less than 1%), Eq. (24) can be written as.

$$p_r\left({\mathrm{\eta }}_i,{\mathrm{\rho }}_i\right)=p_{r0}\left({\mathrm{\eta }}_i,0\right){\left(1-{\mathrm{\rho }}_i^2/a_r^2\right)}^{\mathrm{\beta }},\hspace{0.17em}\mathrm{\beta }=p_{r0}\left({\mathrm{\eta }}_i,0\right)/p_m\left({\mathrm{\eta }}_i,0\right)-1.$$E28

3.2. The contact of a spherical asperity and the hardenable elastic-plastic half-space

Problems of a spherical asperity elastic-plastic indentation are not studied sufficiently and some suggested solutions are needed for clarification and improvement. One of the important problems is material hardening. The authors’ approach to solve this problem is given in Ref. [14].

In several works [15, 16], the empirical Meyer law linking the spherical indentation load and an indenter diameter was used to allow for material hardening in solving the tribomechanic problems. Let us consider this approach at length.

In describing elastic-plastic characteristics of the hardenable material, the Hollomon’s power law is widely used. According to it, the relation between the true stress S and the true strain ε under uniaxial tension or compression is described by equations

$$S=\left\{\begin{array}{l}\mathit{\epsilon E},\mathrm{\epsilon }\le {\mathrm{\epsilon }}_y;\\ K{\mathrm{\epsilon }}^n,\mathrm{\epsilon }\ge {\mathrm{\epsilon }}_y;\end{array}\right)$$E29

where E is the elastic modulus and n is the strain-hardening exponent.

The constant K is determined from the equality condition for σ at ε_y. Then the second equation in Eq. (29) can be written as.

$$\frac{S}{{\sigma }_y}={\left(\frac{\mathit{E\epsilon }}{{\sigma }_y}\right)}^n={\left(\frac{\mathrm{\epsilon }}{{\mathrm{\epsilon }}_y}\right)}^n,\mathrm{\epsilon }\ge {\mathrm{\epsilon }}_y.$$E30

where ${\sigma }_y\approx S_y$, ${\sigma }_y$ is the yield strength, and ${\mathrm{\epsilon }}_y={\sigma }_y/E$.

Taking into accord that the limiting uniform strain ${\mathrm{\epsilon }}_u=n$, the exponential deformation hardening can be determined according to Ref. [17] from the following equation:

$$n\text{ln}n-n\left(1+\text{ln}{\mathrm{\epsilon }}_y\right)-\text{ln}\frac{\sigma u}{{\sigma }_y}=0,$$E31

where σ_u is the tensile strength.

Meyer was the first who described a material behavior in the elastic-plastic domain. He related the load P to the indentation diameter d as

$$P={\mathit{Ad}}^m.$$E32

The empirical Meyer law is often written as:

$$\frac{4P}{\pi d^2}=\mathit{HM}=A^{\ast }{\left(\frac{d}{D}\right)}^{m-2}.$$E33

where m, A, and A^* are constants. A^* has a dimension of strength.

The equation on the left side is a mean contact area pressure referred to as the Meyer hardness

$$\frac{4P}{\pi d^2}=\frac{P}{\pi a^2}=p_m=\mathit{HM},$$E34

where a is the radius of the contact area.

Using [16], we have

$$\frac{P}{E^{\ast }{R}^2}=\frac{2}{k_{\sigma }}{\left(\frac{n}{e}\right)}^n{\mathrm{\epsilon }}_y^{1-n}{\left(\frac{a}{R}\right)}^{2+1.041n}.$$E35

where $E^{\ast }$ is reduced elastic modulus, $k_{\sigma }=0.333$ for carbon and pearlitic steel, for other materials, the values of $k{}_{\sigma }$ are given in Ref. [18].

$$k_n=\frac{{\left(2+1.041n\right)}^{1+0.5205n}}{{\left(1+1.041n\right)}^{1+1.041n}}{\left(1.041n\right)}^{0.5205n}.$$E36

The limits of using of Eq. (35) are given in Ref. [16].

As it was indicated in Ref. [16], the obtained results are in good agreement with the experimental data given in Ref. [19], and with the data of FE analysis [20].

Thus, the proposed approach suggests an alternative to a more complex method for describing elastic-plastic penetration of a sphere on the basis of the kinetic indentation diagram [14], which was used in solving problems of elastic-plastic contacting of rough surfaces.

4. Contacting rough surfaces

4.1. Elastic contact of rough surfaces

4.1.1. Relative contact area

Consider the contact of a rough surface with an elastic-plastic half-space using a roughness model for which the function and the density of the distribution of the asperities are described by Eqs. (15) and (16). The displacement of a rough surface in the general case is determined from Eq. (21) under the condition $F\left(a_{\mathit{ri}}\right)=0$:

$$U_0={\mathit{uR}}_{\max }+2\mathrm{\Theta }{q}_c\left(a_l-a_c\right)+2\omega R_{\max }\frac{a_{\mathit{ri}}^2}{a_c^2}++2\mathrm{\theta }{q}_c{a}_c\left(1-\sqrt{1-\frac{a_{\mathit{ri}}^2}{a_c^2}}\right).$$E37

For an asperity contacting at a point, that is, for a_ri = 0, we have

$$U_0=\mathrm{\epsilon }{R}_{\max }+2\mathrm{\theta }{q}_c\left(a_l-a_c\right).$$E38

Since the value of U₀ is constant for all points of the contact regions, it follows from Eqs. (56) and (38) that

$${\mathrm{\eta }}_i+\frac{\mathrm{\theta }{q}_c{a}_c}{\omega R_{\max }}\left(1-\sqrt{1-{\mathrm{\eta }}_i}\right)-\frac{\mathrm{\epsilon }-u}{2\omega }=0.$$E39

This equation has a solution

$${\mathrm{\eta }}_i=\frac{\mathrm{\epsilon }-u}{2\omega }-f_q\left(1+\frac{f_q}{2}-\sqrt{{\left(1+\frac{f_q}{2}\right)}^2-\frac{\mathrm{\epsilon }-u}{2\omega }}\right),$$E40

where $f_q=\frac{\mathrm{\theta }{q}_c{a}_c}{\omega R_{\max }}.$

Contour pressure in the joint of a rough surface with a half-space and the relative area are described by equations.

$$q_c=\frac{N}{A_c}=\frac{1}{A_c}\sum _{i=1}^{n_r}{q}_{\mathit{ci}}{A}_{\mathit{ci}};\hspace{0.17em}\mathrm{\eta }=\frac{A_r}{A_c}=\frac{1}{A_c}\sum _{i=1}^{n_r}{A}_{\mathit{ci}}{\mathrm{\eta }}_i.$$E41

Considering that for this roughness model $A_{\mathit{ci}}=\text{const}$, $A_c=A_{\mathit{ci}}{n}_c$, and ${\mathit{dn}}_r=n_c{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}$, we represent Eq. (41) in the form.

$$q_c\left(\mathrm{\epsilon }\right)=\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{q}_{\mathit{ci}}{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du},\hspace{0.17em}\mathrm{\eta }\left(\mathrm{\epsilon }\right)=\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{\mathrm{\eta }}_i{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}.$$E42

Taking into account Eq. (25), we have.

$$f_q\left(\mathrm{\epsilon }\right)=\frac{\mathrm{\theta }{q}_c\left(\mathrm{\epsilon }\right)a_c}{\omega R_{\max }}=\frac{\frac{8}{3\pi }\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{\mathrm{\eta }}_i^{1,5}{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}}{1-\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{\mathrm{\Psi }}_{\mathrm{\eta }}\left({\mathrm{\eta }}_i\right){\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}},\hspace{0.17em}{\mathrm{\Psi }}_{\mathrm{\eta }}\left({\mathrm{\eta }}_i\right)=\frac{2}{\pi }\left[\text{arcsin}{\mathrm{\eta }}_i^{0.5}-\sqrt{{\mathrm{\eta }}_i\left(1-{\mathrm{\eta }}_i\right)}\right].$$E43

Figure 4 shows the dependences of the relative contact area on the force elastic-geometric parameter f_q.

Figure 4.

The relative contact area with/without taking into account the mutual influence of asperities (a) and for different values of p and q (b).

4.1.2. Gaps density of the joint

To determine the volume of the intercontact space, it is necessary to determine the volumes of gaps attributable to single contacting and noncontacting asperities [10],

$$V_i=\left\{\begin{array}{l}{V}_{\mathit{ri}}=2\pi \underset{a_{\mathit{ri}}}{\overset{a_c}{\int }}\left[z_{20}\left(\mathrm{\rho }\right)-z_{10}\left(\mathrm{\rho }\right)\right]\mathit{\rho d\rho };\\ V_{0i}=2\pi \underset{0}{\overset{a_{\mathit{ci}}}{\int }}\left[z_{2r}\left(\mathrm{\rho }\right)-z_{1r}\left(\mathrm{\rho }\right)\right]\mathit{\rho d\rho },\end{array}\right)$$E44

where $z_{10},z_{20}$ and $z_{1r},z_{2r}$ are the equations describing the surfaces of noncontacting and contacting asperities and half-spaces, respectively.

Then, the total volume of the intercontact space at the joint is described by the equation

$$V_c=\sum _{i=1}^{n_r}{V}_{\mathit{ri}}+\sum _{i=1}^{n_c-n_r}{V}_{0i},$$E45

And the corresponding gap density is equal to

$$\mathrm{\Lambda }\left(\mathrm{\epsilon }\right)=\frac{V_c}{A_c{R}_{\max }}=\frac{1}{A_{\mathit{ci}}{R}_{\max }}\left[\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_S\right)}{\int }}{V}_{\mathit{ri}}{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}+\underset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_S\right)}{\overset{{\mathrm{\epsilon }}_S}{\int }}{V}_{0i}{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}\right].$$E46

Taking into account that ${\mathrm{\Lambda }}_{\mathit{ri}}=V_{\mathit{ri}}/\left(A_{\mathit{ci}}{R}_{\max }\right)$ и ${\mathrm{\Lambda }}_{0i}=V_{0i}/\left(A_{\mathit{ci}}{R}_{\max }\right)$, it can be represented in the form

$$\mathrm{\Lambda }\left(\mathrm{\epsilon }\right)=\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_S\right)}{\int }}{\mathrm{\Lambda }}_{\mathit{ri}}{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}+\underset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_S\right)}{\overset{{\mathrm{\epsilon }}_S}{\int }}{\mathrm{\Lambda }}_{0i}{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}.$$E47

We provide the equations of surfaces of the asperities and the half-space that enter into Eq. (44):

$$z_{10}=\omega R_{\max }\left[\frac{\mathrm{\epsilon }-u}{\omega }-x^2+2f_q\left(k-1\right)\right],$$E48

where $x=\frac{\mathrm{\rho }}{a_c};k=\frac{a_l}{a_c},$

$$z_{20}=2\omega R_{\max }{f}_q\left[k{\cdot }_2{F}_1\left(-\frac{1}{2},\frac{1}{2};1;\frac{x^2}{k^2}\right){-}_2{F}_1\left(-\frac{1}{2},\frac{1}{2};1;x^2\right)\right],$$E49

where ${}_2{F}_1$ is the Gaussian hypergeometric function,

for contacting asperity $z_{1r}=z_{10};$

$$z_{2r}=\left\{\begin{array}{l}{z}_{1r},0\le x<{\mathrm{\eta }}_i^{0,5}\\ \\ U_{\mathit{Eri}}+U_{\mathit{Eci}},{\mathrm{\eta }}_i^{0,5}\le x\le 1;\end{array}\right)$$E50

$$U_{\mathit{Eci}}=z_{20},\hspace{0.17em}{U}_{\mathit{Eri}}=\omega R_{\max }\frac{f_{\mathit{qi}}}{x}{}_2{F}_1\left(\frac{1}{2},\frac{1}{2};\mathrm{\beta }+2;\frac{{\mathrm{\eta }}_i}{x^2}\right),\hspace{0.17em}{f}_{\mathit{qi}}=\frac{8{\mathrm{\eta }}_i^{1,5}}{3\pi }+\mathrm{\Psi }\left({\mathrm{\eta }}_i\right)\cdot f_q,$$E51

where $\mathrm{\beta }=p_{\mathit{ri}}\left(0\right)/p_m-1.$

Figure 5 shows the different positions of the single asperity in the process of contacting with the rough surface: case a corresponds to original position; case b corresponds to the touching at a point; and cases c and d correspond to the contact under the different loads.

Figure 5.

The scheme for contacting a single asperity located at level u = 0.5.

Taking into account that $x^2=t$, we have

$$V_{0i}=\pi a_c^2\underset{0}{\overset{1}{\int }}\mathrm{\Delta }{z}_0\left(t\right)\mathit{dt},\hspace{0.17em}{V}_{\mathit{ri}}=\pi a_c^2\underset{{\mathrm{\eta }}_i}{\overset{1}{\int }}\mathrm{\Delta }{z}_r\left(t\right)\mathit{dt}.$$E52

where $\mathrm{\Delta }{z}_0=z_{20}-z_{10}$ and $\mathrm{\Delta }{z}_r=z_{2r}-z_{1r}.$

Since ${\mathrm{\Lambda }}_i=\frac{V_i}{\pi a_c{R}_{\max }}$, after integrating (52), we have

$${\mathrm{\Lambda }}_{\mathit{oi}}=\omega \left\{\frac{1}{2}-\frac{\mathrm{\epsilon }-u}{\omega }-2f_q\left[\left(k-1\right)-k{\cdot }_2{F}_1\left(-\frac{1}{2},\frac{1}{2};2;\frac{1}{k^2}\right)+{}_2{F}_1\left(-\frac{1}{2},\frac{1}{2};2;1\right)\right]\right\}.$$E53

$$\begin{array}{ll}{\mathrm{\Lambda }}_{\mathit{ri}}=& \omega \left\{\left(1-{\mathrm{\eta }}_i\right)\left[\frac{1+{\mathrm{\eta }}_i}{2}\right)\right)-\left.\frac{\mathrm{\epsilon }-u}{\omega }-2f_q\left(k-1\right)\right]+2f_{q}k\left[{}_2{F}_1\left(-\frac{1}{2},\frac{1}{2};2;\frac{1}{k^2}\right)\right)-\\ & -\left.{\mathrm{\eta }}_i {}_2{F}_1\left(-\frac{1}{2},\frac{1}{2};2;\frac{{\mathrm{\eta }}_i}{k^2}\right)\right]-2f_q\left[{}_2{F}_1\left(-\frac{1}{2},\frac{1}{2};2;1\right)-\left.{\mathrm{\eta }}_i {}_2{F}_1\left(-\frac{1}{2},\frac{1}{2};2;{\mathrm{\eta }}_i\right)\right]+\right)\\ & +2f_{\mathit{qi}}\left[{}_2{F}_1\left(-\frac{1}{2},\frac{1}{2};\mathrm{\beta }+2;{\mathrm{\eta }}_i\right)-\left.\left.{\mathrm{\eta }}_i^{0,5}{}_2{F}_1\left(-\frac{1}{2},\frac{1}{2};\mathrm{\beta }+2;1\right)\right]\right\}\right).\end{array}$$E54

Substituting the equations obtained in Eq. (47), we determine the joint density $\mathrm{\Lambda }\left(\mathrm{\epsilon }\right)$. To determine the dependence $\mathrm{\Lambda }\left(f_q\right)$, it is necessary to exclude the parameter $\mathrm{\epsilon }$ from the dependences $f_q\left(\mathrm{\epsilon }\right)$ and $\mathrm{\Lambda }\left(\mathrm{\epsilon }\right)$.

Figure 6 shows the dependence of the gap density on the complex parameter $f_q$ when two rough surfaces come into contact. Figure 2 shows that the contact density does not depend on the parameters p and q, since the dependences for the different values of p and q.

Figure 6.

The gap density with/without taking into account the mutual influence of asperities (a) and for different values of p and q (b).

4.1.3. The criteria for the appearance of plastic deformations

To determine the limits of using the above equations for metal surfaces, it is necessary to have a reliable criterion of plasticity. The closest coincidence with the experimental data on the indentation into elastic-plastic media was shown by the energy Mises’ theory of shear strain and the theory of the maximum tangential stresses of Tresca. The difference between the two criteria is small; therefore, it is advisable to use the Tresca criterion because of its algebraic simplicity. The problem of determining the plasticity criterion for the considered loading scheme for a single asperity (Figure 3) was considered in [21]. In this case, the data of the effect of an axisymmetric load of the form Eq. (28) on the stress-strain state were taken into account. An important conclusion of [21] is the statement of stability of the values of the relative contact area ${\mathrm{\eta }}_{\mathit{ip}}$ for distributed at different heights asperities, at which plastic deformation begins. Thus, the value of ${\mathrm{\eta }}_{\mathit{ip}}$ for any asperity loaded according to Figure 3 can be determined for the highest asperity at $u=0$, $q_c=0$, and $\mathrm{\beta }=0,5$.

By the Tresca criterion of the maximum tangential stresses, the plastic deformation on the z axis corresponds to the equivalent stress [22].

$${\sigma }_{\mathit{eq}}=2{\tau }_{1\text{max}}=0,62p_0={\sigma }_y.$$E55

The maximum contact pressure is defined as $p_0=K_y{\sigma }_y$, where $K_y=1,613$. The mean contact pressure is $p_m=K_y{\sigma }_y/\left(1+\mathrm{\beta }\right).$

Using Hertz’s expressions for the radius of the contact area.

$$a_{\mathit{ri}}={\left(\frac{3P_{i}r}{4E^{\ast }}\right)}^{\frac{1}{3}},$$E56

and taking into account that.

$$P_i=\pi a_{\mathit{ri}}^2{p}_m,\hspace{0.17em}r=\frac{a_c^2}{2\omega R_{\max }},\hspace{0.17em}\frac{a_{\mathit{ri}}^2}{a_c^2}={\mathrm{\eta }}_i,\hspace{0.17em}\frac{{\sigma }_y}{E^{\ast }}={\mathrm{\epsilon }}_y,$$E57

We obtain the value of the criterion for the appearance of plastic strains in the near-surface layer

$${\mathrm{\eta }}_p^{\ast }={\left(\frac{3\pi K_y}{8\left(\mathrm{\beta }+1\right)}{f}_y\right)}^2,$$E58

where $f_y=\frac{{\sigma }_y{a}_c}{E^{\ast }\omega R_{\max }}.$

For the highest asperity ${\mathrm{\eta }}_p^{\ast }=1,605f_y^2$. Thus, the proposed criterion of plasticity does not depend on loading conditions and this is its advantage.

Similarly, we define the criterion of occurrence of plastic deformation at the contact area. According to [23], the equivalent stresses at the center of the area are

$$\frac{{\sigma }_{\mathit{eq}}\left(0\right)}{p_m}=0,2\left(1+\mathrm{\beta }\right).$$E59

The highest value of the equivalent stress ${\sigma }_{\mathit{eq}}\left(1\right)$ is on the contour of the contact area, where it slightly exceeds ${\sigma }_{\mathit{eq}}\left(0\right)$ in the center of the loading area. It is convenient to represent ${\sigma }_{\mathit{eq}}\left(1\right)=K_{\sigma }\cdot {\sigma }_{\mathit{eq}}\left(0\right)$, where for $\mathrm{\beta }=0,5$ according to the energy theory of shear strains $K_{\sigma }=1,16$, according to the theory of maximal tangential stresses $K_{\sigma }=1,33$.

At the moment of appearance of plastic deformation along the contour of the contact area ${\sigma }_{\mathit{eq}}\left(1\right)={\sigma }_y$, and the average contact pressure.

$$p_m=\frac{5{\sigma }_y}{K_{\sigma }\left(1+\mathrm{\beta }\right)}.$$E60

Then, similarly to the above reasoning, the criterion of the appearance of plastic deformations in the contact area is

$${\mathrm{\eta }}_p^{\ast \ast }={\left(\frac{15\pi }{8K_{\sigma }\left(\mathrm{\beta }+1\right)}{f}_y\right)}^2.$$E61

For the highest asperity ${\mathrm{\eta }}_p^{\ast \ast }=15,42K_{\sigma }^{-2}{f}_y^2$. According to the theory of maximum tangential stresses ${\mathrm{\eta }}_p^{\ast \ast }=5,405{\mathrm{\eta }}_p^{\ast },$ according to the energy theory of shear deformations ${\mathrm{\eta }}_p^{\ast \ast }=7,105{\mathrm{\eta }}_p^{\ast }.$

4.2. Elastic-plastic contact of rough surfaces

Contact characteristics for elastic-plastic contact will be considered taking into account the mutual influence of the contacting asperities. By analogy with the elastic contact, we assume that the mutual influence of the asperities is equivalent to the action of the additional load q_c (Figure 3). We use a discrete roughness model, described by Eqs. (15) and (16).

4.2.1. Relative contact area

According to Eq. (33), the load applied to a single asperity

$$\frac{P_i}{E^{\ast }{R}^2}=\frac{2}{k_{\sigma }}{\left(\frac{n}{e}\right)}^n{\mathrm{\epsilon }}_y^{1-n}{\left(\frac{a_{\mathit{ri}}}{R}\right)}^{2+1.041n}.$$E62

Considering that for the roughness model used $R=a_c^2/\left(2\omega R_{\max }\right)$ and ${\mathrm{\eta }}_i=a_{\mathit{ri}}^2/a_c^2,$ from Eq. (62) we have

$$\frac{q_{\mathit{ci}}}{E^{\ast }}=\frac{P_i}{E^{\ast }\cdot \pi a_c^2}=\frac{2}{k_{\sigma }}\cdot {\left(\frac{2\omega R_{\max }}{a_c}\right)}^{1.041n}{\left(\frac{n}{e}\right)}^n{\mathrm{\epsilon }}_y^{1-n}{\mathrm{\eta }}_i^{1+0.52n}.$$E63

For elastic-plastic contact, it is convenient to use the parameter ${\overline{q}}_{\sigma }=q_c/{\sigma }_y$, then from Eq. (63) we have

$${\overline{q}}_{\mathit{\sigma i}}=\frac{q_{\mathit{ci}}}{{\sigma }_y}=C_a\cdot {\mathrm{\eta }}_i^{1+0.52n},$$E64

where

$$C_a=C_a\left({\mathrm{\epsilon }}_y,n\right)=\frac{2}{k_{\sigma }}\cdot {\left(\frac{2\omega R_{\max }}{a_c}\right)}^{1.041n}{\left(\frac{n}{e\cdot {\mathrm{\epsilon }}_y}\right)}^n.$$E65

By analogy with Eq. (25), taking into account Eq. (64), for an elastic-plastic contact, we have

$${\overline{q}}_{\mathit{\sigma i}}=C_a\cdot {\mathrm{\eta }}_i^{1+0.52n}+{\overline{q}}_{\sigma }\cdot {\mathrm{\Psi }}_{\mathrm{\eta }}\left({\mathrm{\eta }}_i\right).$$E66

In order to preserve the acceptability of the equations for elastic and elastic-plastic contacts, we use the relations.

$$f_q=\frac{q_c{a}_c}{E^{\ast }\omega R_{\max }}=\frac{q_c}{{\sigma }_y}\cdot \frac{{\sigma }_y}{E^{\ast }}\cdot \frac{a_c}{\mathit{\omega R}{}_{\text{max}}}={\overline{q}}_{\sigma }\cdot f_y;\hspace{0.17em}{f}_y=\frac{{\mathrm{\epsilon }}_y{a}_c}{\omega R_{\max }};\hspace{0.17em}{f}_{\mathit{qi}}={\overline{q}}_{\mathit{\sigma i}}\cdot f_y.$$E67

Then Eq. (66) can be represented in the form

$$f_{\mathit{qi}}=C_a\cdot {\mathrm{\eta }}_i^{1+0.52n}+f_q\cdot {\mathrm{\Psi }}_{\mathrm{\eta }}\left({\mathrm{\eta }}_i\right),$$E68

where $C_f=C_a\cdot f_y$, ${\mathrm{\eta }}_i$ is determined by Eq. (40).

Summing up $f_{\mathit{qi}}$ over all asperities, we have

$$f_q\left(\mathrm{\epsilon }\right)=\frac{C_f\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{\mathrm{\eta }}_i^{1+0.52n}{\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}}{1-\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{\mathrm{\Psi }}_{\mathrm{\eta }}\left({\mathrm{\eta }}_i\right){\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}}.$$E69

For a given value $\mathrm{\epsilon }$, we solve the system of transcendental Eqs. (40), (69) and obtain the dependence $f_q\left(\mathrm{\epsilon }\right)$.

Similarly, using Eq. (40) and $f_q\left(\mathrm{\epsilon }\right)$, we have

$$\mathrm{\eta }\left(\mathrm{\epsilon }\right)=\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{\mathrm{\eta }}_i\left(\mathrm{\epsilon },f_q\right){\mathrm{\phi }}_n^{\prime }\left(u\right)\mathit{du}.$$E70

Excluding the parameter $\mathrm{\epsilon }$ from Eqs. (69) and (70), we obtain the dependence $\mathrm{\eta }\left(f_q\right)$ or $\mathrm{\eta }\left({\overline{q}}_{\sigma }\right)$.

Figures 7 and 8 present the dependencies of the relative contact area on the relative force parameter ${\overline{q}}_{\sigma }$.

Figure 7.

The relative contact area with/without taking into account the mutual influence of asperities (a) and for different values of p and q (b).

Figure 8.

The relative contact area for different values of ${\epsilon }_y$ and $n$.

4.2.2. Gaps density of the joint

The scheme of the action of the loads p_r and q_c is similar to the scheme for elastic contact (Figure 3).

For an elastic-plastic contact

$${\overline{P}}_i=\frac{P_i}{E^{\ast }{R}^2}\propto {\left(\frac{h_i}{R}\right)}^{0,5205n+1},$$E71

therefore, the pressure distribution in the contact area described by [4]

$$p\left(r\right)=p_0{\left(1-\frac{r^2}{a^2}\right)}^{\mathrm{\beta }},$$E72

where $p_0=p_m\left(1+\mathrm{\beta }\right)$ is pressure at $r=0$, p_m is the mean pressure on contact area and $\mathrm{\beta }=0,5205n.$

Total density of gaps with elastic-plastic contact

$$\mathrm{\Lambda }={\mathrm{\Lambda }}_e-{\mathrm{\Lambda }}_p={\mathrm{\Lambda }}_{e0}+{\mathrm{\Lambda }}_{\mathit{er}}-{\mathrm{\Lambda }}_p,$$E73

where ${\mathrm{\Lambda }}_e$ is the density of gaps due to the elastic punching of the half-space, which accounted for single contacting and noncontacting asperities; Λ_p is reduction of the gap density due to the plastic displacement of the material into the interfacial space.

The value of ${\mathrm{\Lambda }}_e$ is determined, similarly to the elastic contact, by Eq. (47). In this case, $f_{\mathit{qi}}$ is determined by Eq. (68) and the parameter $\mathrm{\beta }$ is used in Eq. (72).

Let us determine the volume of the displaced material for a single contacting asperity (Figure 9).

Figure 9.

Scheme of the unloaded crater.

Let us assume that the unloaded crater has a constant radius $R_{\mathit{fi}}$ and the unloaded depth from the level of the initial surface $h_{\mathit{fi}}$. The volume of plastically displaced material falling on a single crater is equal to the volume of a spherical segment of height h_f and radius $R_{\mathit{fi}}$:

$$V_{\mathit{pi}}=\pi h_{\mathit{fi}}^2\left(R_{\mathit{fi}}-\frac{h_{\mathit{fi}}}{3}\right).$$E74

The total volume of the displaced material

$$V_p=n_c\underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{V}_{\mathit{pi}}{\mathrm{\phi }}_n^{/}\left(u\right)\mathit{du}.$$E75

Since ${\mathrm{\Lambda }}_p=V_p/\left(A_c{R}_{\max }\right),$ we have

$$\begin{array}{ll}{\mathrm{\Lambda }}_p& =\omega \underset{0}{\overset{\text{min}\left(\mathrm{\epsilon },{\mathrm{\epsilon }}_s\right)}{\int }}{\left(\frac{{\mathrm{\eta }}_i}{c^2}-{\mathrm{\eta }}_i^{-0,5}{f}_{\mathit{qi}}{K}_{\mathrm{\beta }0}\right)}^2\left\{0,5,{\left[1-{\mathrm{\eta }}_i^{-1,5}{f}_{\mathit{qi}}\left(K_{\mathrm{\beta }0}-K_{\mathit{\beta c}}\right)\right]}^{-1},-,\frac{{\left(\omega R_{\max }\right)}^2}{3a_c^2}\right)\times \\ & \times \left(\left(\frac{{\mathrm{\eta }}_i}{c^2}-{\mathrm{\eta }}_i^{-0,5}{f}_{\mathit{qi}}{K}_{\mathrm{\beta }0}\right)\right\}{\mathrm{\phi }}_n^{/}\left(u\right)\mathit{du}.\end{array}$$E76

Substituting Eq. (76) into Eq. (73), we find the total gap density for elastic-plastic contact.

Figure 10 presents the dependencies of the gap density on the relative force parameter ${\overline{q}}_{\sigma }$.

Figure 10.

The gap density with/without taking into account the mutual influence of asperities (a) and for different values of p and q (b).

5. Ensuring specified tightness

Ensuring specified tightness or leakage rate is related to the determination of the force parameters $f_q$ or ${\overline{q}}_{\sigma }$. The sealing capacity of the SJ is evaluated by the permeability functional by Eq. (1). The contact characteristics—the relative contact area η and the gap density Λ, included in Eq. (1), are defined in the previous section. Included in Eq. (1), the probability $v_k$ of the medium flowing through the SJ is determined by the fusion of contact spots and is given in Ref. [3]. Two adjacent asperities will merge if ${\mathrm{\eta }}_i>0.5$ for each asperity.

Figure 11 shows the dependences for the elastic and elastic-plastic contacts.

The required permeability functional is determined by [3]

$$C_u^{\ast }=\frac{2l\mu G_{{}^l}^{*}}{R_{\max }^3\mathrm{\rho }\mathrm{\Delta }\mathrm{p}},$$E77

where $G_{{}^l}^{*}$ is the specified tightness; $\mathrm{\rho }$ is the density of the sealed medium; $p_1$ and $p_2$ are the inlet and outlet pressures; $\mu$ is the dynamic viscosity; $\mathrm{\Delta }p=p_1-p_2$; and $l$ is the compacting band width.

Figure 11.

The dependences of the permeability functional for the elastic (a) and elastic-plastic (b) contacts.

The force parameters $f_q$ or ${\overline{q}}_{\sigma }$, that providing a given level $C_u^{\ast }$ are determined from the $C_u\left(f_q\right)$ or $C_u\left({\overline{q}}_{\sigma }\right)$ (Figure 11).

6. Conclusion

Using the proposed model of roughness as a result of the studies, methods for determining the contact characteristics and the conditions for ensuring a specified tightness of the joints were developed and established:

1. Contact characteristics and the permeability functional are determined depending on the introduced dimensionless power parameters f_q for the elastic and ${\overline{q}}_{\sigma }$ for elastic-plastic contacts.

2. The relative contact area and the gap density for elastic contact do not depend on the values of the parameters of the bearing curve p and q. To a large extent, the mutual influence of asperities affects, and at f_q > 0.47, the determining factor affecting the permeability functional is the probability $v_k$ of the medium flowing (Figure 11).

3. To describe the elastic-plastic contact, Mayer’s law and the relation between the hardening exponent n and the Mayer index m were used.

4. In the case of elastic-plastic contact, the exponent of hardening n has a greater effect on the contact characteristics and to a lesser extent, the parameter ε_y and the mutual influence of the asperities. For the considered range of the parameter ${\overline{q}}_{\sigma }$, the fusion of the contact spots is insignificant.