2. Discrete model of the rough surface

materials' properties and microgeometry parameters, there are elastic, viscoelastic, elastic-plastic,

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

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

2. in the sealing joint, all the asperities's contacting is possible, which requires the description

3. when determining the gaps volume (or density), the displacements of the points of the

4. the extrusion of the material into the intercontact space under elastic-plastic contact has

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

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

where Λ is the gaps density in the joint; η is the relative contact area; υ<sup>k</sup> is the probability of a

All the parameters that appear in Eq. (1) depend on the parameters of microgeometry and dimensionless force parameters f <sup>q</sup> or qσ, the determination of which is given in the following

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.

υk

4 1 � <sup>η</sup> <sup>2</sup> , (1)

Cu <sup>¼</sup> <sup>Λ</sup><sup>3</sup>

medium flowing, which depends on the single contact spots fusion.

of the whole bearing profile curve but not only its initial part, as in [2];

asperities surfaces have not been taken into account; and

and rigid-plastic contacts.

4 Contact and Fracture Mechanics

errors, which is explained by the following:

the contacting asperities;

not been taken into account.

material hardening.

sections.

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 Rmax, arithmetic mean deviation of the profile Ra, root-mean-square deviation of the profile Rq, mean height of the profile elements Rp, mean width of the profile elements Sm, bearing profile curve tp, and bearing profile curve on the midline tm. 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:

$$\eta\_u(\varepsilon) = \frac{A\_u}{A\_c} \text{ or } \Phi\_n(u) = \frac{n\_u}{n\_c},\tag{2}$$

where Au is the material cross-sectional area at a relative level ε ¼ h=Rmax; Ac is the contour area; nu is the number of asperities whose peaks are located above the level u; nc ¼ Ac=Aci is the total number of asperities; and Aci 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 tp and the surface ηu(ε), but it is not fulfilled for the peaks and valleys asperities distribution functions of the profile φnl(ul) and the surface φn(u), then the model is based on the bearing profile curve.

Let us assume that the function ηuð Þε 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 ωRmax, and base radius ac <sup>¼</sup> ffiffiffiffiffiffiffiffiffiffiffiffi Aci=<sup>π</sup> <sup>p</sup> . It is necessary to find such a function <sup>φ</sup>nð Þ <sup>u</sup> for which the distribution of the material in the rough layer corresponds to the bearing surface curve.

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

$$A\_{ri} = 2\pi r R\_{\text{max}}(\varepsilon - u),\tag{3}$$

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

$$d\mathfrak{n}\_r = \mathfrak{n}\_c \mathfrak{q}'\_n(\mathfrak{u}) d\mathfrak{u}.\tag{4}$$

Figure 1. The scheme and the bearing curve of a rough surface.

Then, Au ¼ Ar ¼ 2πrRmaxnc Ð ε 0 φ0 <sup>n</sup>ð Þ u ð Þ ε � u du;

$$\eta\_u(\varepsilon) = \frac{A\_r(\varepsilon)}{A\_c} = \mathbb{C} \int\_0^\varepsilon (\varepsilon - u) q\_u'(u) du,\\ \mathbb{C} = \frac{2\pi r R\_{\text{max}} n\_c}{A\_c}. \tag{5}$$

Further, we have

$$\eta(\varepsilon) = \mathbb{C}\left(\varepsilon \left\| \mathbf{q}\_n'(u) du - \int\_0^\varepsilon \mu \mathbf{q}\_n'(u) du\right\| \right) = \mathbb{C}\left(\varepsilon \varrho\_n(\varepsilon) - \mu \varrho\_n(u)\Big|\_0^\varepsilon + \int\_0^\varepsilon \varrho\_n(u) du\right),$$

$$\eta(\varepsilon) = \mathbb{C}\int\_0^\varepsilon \varrho\_n(u) du. \tag{6}$$

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

$$
\mathfrak{n}'(\varepsilon) = \mathsf{Cq}\_{\mathfrak{n}}(\varepsilon), \quad \mathfrak{n}''(\varepsilon) = \mathsf{Cq}'\_{\mathfrak{n}}(\varepsilon); \tag{7}
$$

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

The contact of two rough surfaces zið Þ x; y can be represented as a contact of an equivalent

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]

<sup>η</sup> <sup>¼</sup> <sup>1</sup> � exp � <sup>b</sup> <sup>q</sup>

where b is the coefficient depending on the surface quality, qc is the contour contact pressure,

and E is the elastic modulus. As it follows from Eq. (13), η ! 1 for q ! ∞.

E

� �, (13)

<sup>i</sup>¼<sup>1</sup> zið Þ <sup>x</sup>; <sup>y</sup> and a flat surface. The parameters of the microgeometry of

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7

heights are given in work [3, 4].

an equivalent surface are given in [3, 4].

3. Description of contact of a single asperity

Figure 2. The distribution densities of asperities for different values of p and q.

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

rough surface z xð Þ¼ ; <sup>y</sup> <sup>P</sup><sup>2</sup>

$$\!\!\!\!\!\!\!\!\!\!\!\/\(\varepsilon\) = \frac{\eta'(\varepsilon)}{\mathbb{C}}, \quad \!\!\!\!\!\/\left(\varepsilon\right) = \frac{\eta''(\varepsilon)}{\mathbb{C}}.\tag{8}$$

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

$$t\_p(\varepsilon) = \eta(\varepsilon) = I\_\varepsilon(p, q) = \frac{\mathcal{B}\_\varepsilon(p, q)}{\mathcal{B}(p, q)},\tag{9}$$

where

$$p = \left(\frac{R\_p}{R\_q}\right)^2 \left(\frac{R\_{\text{max}} - R\_p}{R\_{\text{max}}}\right) - \frac{R\_p}{R\_{\text{max}}}, \; q = p\left(\frac{R\_{\text{max}}}{R\_p} - 1\right). \tag{10}$$

Вε(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

$$\varphi\_{\mathfrak{n}}(\mu) = \frac{\eta\_{\mathfrak{n}}'(\mu)}{\mathbb{C}} = \frac{\mu^{p-1}(1-\mu)^{q-1}}{\varepsilon\_{\mathfrak{s}}^{p-1}(1-\varepsilon\_{\mathfrak{s}})^{q-1}};\tag{11}$$

$$\mathbf{q}\_{n}^{\prime}(\boldsymbol{u}) = \frac{\eta\_{\boldsymbol{u}}^{\prime}(\boldsymbol{u})}{\mathbb{C}} = \frac{\boldsymbol{u}^{p-2}(1-\boldsymbol{u})^{q-2}[(p-1)(1-\boldsymbol{u}) - (q-1)\boldsymbol{u}]}{\boldsymbol{\varepsilon}\_{\boldsymbol{s}}^{p-1}(1-\boldsymbol{\varepsilon}\_{\boldsymbol{s}})^{q-1}}.\tag{12}$$

The relative height of the spherical asperity is ω ¼ 1 � ε<sup>s</sup> and the radius of spherical asperity is <sup>r</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>c</sup>=ð Þ 2ωRmax :

Contact Mechanics of Rough Surfaces in Hermetic Sealing Studies http://dx.doi.org/10.5772/intechopen.72196 7

Figure 2. The distribution densities of asperities for different values of p and q.

Then, Au ¼ Ar ¼ 2πrRmaxnc

6 Contact and Fracture Mechanics

η εð Þ¼ C ε

Further, we have

where

<sup>r</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup>

asperities, we have

<sup>c</sup>=ð Þ 2ωRmax :

φ0

<sup>n</sup>ð Þ¼ <sup>u</sup> <sup>η</sup><sup>00</sup>

Ð ε 0 φ0

<sup>n</sup>ð Þ u du �

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

<sup>p</sup> <sup>¼</sup> Rp Rq η0

<sup>φ</sup>nð Þ¼ <sup>ε</sup> <sup>η</sup><sup>0</sup>

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

� �<sup>2</sup> <sup>R</sup>max � Rp

<sup>φ</sup>nð Þ¼ <sup>u</sup> <sup>η</sup><sup>0</sup>

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

<sup>u</sup>ð Þ u

Rmax � �

Arð Þε Ac

ðε

0 uφ<sup>0</sup> <sup>n</sup>ð Þ u du

ηuð Þ¼ ε

ðε

0 @

0 φ0 <sup>n</sup>ð Þ u ð Þ ε � u du;

¼ C ðε

0

ð Þ ε � u φ<sup>0</sup>

1

ðε

0

ð Þ¼ ε Cφnð Þε , η00ð Þ¼ ε Cφ<sup>0</sup>

ð Þε <sup>C</sup> , <sup>φ</sup><sup>0</sup>

tpð Þ¼ ε η εð Þ¼ Iεð Þ¼ p; q

� Rp Rmax

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

<sup>C</sup> <sup>¼</sup> up�<sup>1</sup>ð Þ <sup>1</sup> � <sup>u</sup> <sup>q</sup>�<sup>1</sup> εs

ε

The relative height of the spherical asperity is ω ¼ 1 � ε<sup>s</sup> and the radius of spherical asperity is

<sup>u</sup>ð Þ u

<sup>C</sup> <sup>¼</sup> up�<sup>2</sup>ð Þ <sup>1</sup> � <sup>u</sup> <sup>q</sup>�<sup>2</sup>

η εð Þ¼ C

<sup>n</sup>ð Þ <sup>u</sup> du, C <sup>¼</sup> <sup>2</sup>πrRmaxnc

A ¼ C εφnð Þ� ε uφnð Þ u

<sup>n</sup>ð Þ¼ <sup>ε</sup> <sup>η</sup>00ð Þ<sup>ε</sup>

Βεð Þ p; q

, q ¼ p

Rmax Rp � 1 � �

½ � ð Þ p � 1 ð Þ� 1 � u ð Þ q � 1 u

0 @ Ac

ε 0 � � � � þ ðε

φnð Þ u du: (6)

0

<sup>n</sup>ð Þε ; (7)

<sup>C</sup> : (8)

<sup>Β</sup>ð Þ <sup>p</sup>; <sup>q</sup> , (9)

<sup>p</sup>�<sup>1</sup>ð Þ <sup>1</sup> � <sup>ε</sup><sup>s</sup> <sup>q</sup>�<sup>1</sup> ; (11)

<sup>p</sup>�<sup>1</sup> <sup>s</sup> ð Þ <sup>1</sup> � <sup>ε</sup><sup>s</sup> <sup>q</sup>�<sup>1</sup> : (12)

: (10)

φnð Þ u du

: (5)

1 A,

> 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 zið Þ x; y can be represented as a contact of an equivalent rough surface z xð Þ¼ ; <sup>y</sup> <sup>P</sup><sup>2</sup> <sup>i</sup>¼<sup>1</sup> zið Þ <sup>x</sup>; <sup>y</sup> 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]

$$\eta = 1 - \exp\left(-b\frac{q}{E}\right),\tag{13}$$

where b is the coefficient depending on the surface quality, qc is the contour contact pressure, and E is the elastic modulus. As it follows from Eq. (13), η ! 1 for q ! ∞.

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.

under constancy of its average intensity leads to insignificant changes only near the boundary

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 <sup>W</sup><sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ari � � and the circular unloaded area <sup>W</sup> <sup>r</sup> <sup>¼</sup> ari ð Þ , an on the remaining contact spots will be equivalent to the effect of the uniformly distributed load qcn acting in the circular area W2ð Þ r ¼ an, al , and the assigned problem may be regarded as an axisymmetric (Figure 3). The size of the unloaded area an depends on the number of contacting asperities and with

The solution of this problem is given in Ref. [11]. Studies on the effect of the parameter ka ¼ an=ac on the relative contact area show only 4% increase of last one; therefore, with a

Let A<sup>1</sup> and A<sup>2</sup> be two points on the surface of the circular contact area W1. The A<sup>1</sup> and A<sup>2</sup> coming into contact after application of the compressive load. Since the total normal displace-

where UEri is the normal contact displacement under the pressure pri acting in the region W1; UEci is the normal displacement under the pressure qcn; and z<sup>1</sup> is the equation of the surface of a

<sup>z</sup><sup>1</sup> ¼ �uRmax � <sup>r</sup><sup>2</sup>

Elementary displacements dUEri and dUEci under pressures qri and q<sup>c</sup> acting on elementary

, j = 1, 2; r � r<sup>i</sup>

UEri <sup>¼</sup> <sup>θ</sup> π ð

UEci <sup>¼</sup> <sup>4</sup> π W<sup>1</sup>

al � �

<sup>θ</sup>qc al<sup>Ε</sup> <sup>r</sup><sup>i</sup>

2r

dw1, dUEci <sup>¼</sup> <sup>θ</sup>qcn

prið Þr dw<sup>1</sup> R1

> � <sup>a</sup>с<sup>Ε</sup> <sup>r</sup><sup>i</sup> ac

πR<sup>2</sup>

U<sup>0</sup> ¼ UE þ z<sup>1</sup> ¼ UEri þ UEci þ z1, (14)

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: (15)

dw2; (16)

; <sup>θ</sup> <sup>¼</sup> <sup>1</sup> � <sup>ν</sup><sup>2</sup> � �=E, <sup>ν</sup> is Poisson's ratio;

, (17)

� � � � , (18)

margin to tightness ensure, we will give a solution for ka = 1 or an = ac below.

ment U<sup>0</sup> of the point А<sup>1</sup> is constant for any point in area W1, we have

of the contact area.

increasing applied load, it decreases from al to ac.

spherical asperity in an unloaded state. As for the real surfaces, r > > Rmax, then

where R<sup>2</sup>

<sup>j</sup> <sup>¼</sup> <sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup>

dw<sup>1</sup> ¼ r1drdφ; and dw<sup>2</sup> ¼ r2drdφ. After integrating Eq. (16), we have

areas dw<sup>1</sup> and dw2, respectively, are determined by [12]:

<sup>j</sup> � 2rr<sup>j</sup> cosφ<sup>j</sup>

dUEri <sup>¼</sup> <sup>θ</sup>qri <sup>r</sup><sup>1</sup> ð Þ πR<sup>1</sup>

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 uRmax from the peaks line of a rough surface with an elastic halfspace in the system of cylindrical coordinates z, r, and φ with origin at the point О (Figure 3).

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

Figure 3. Scheme of contact of a single asperity.

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 <sup>W</sup><sup>1</sup> <sup>r</sup> <sup>¼</sup> <sup>0</sup>, ari � � and the circular unloaded area <sup>W</sup> <sup>r</sup> <sup>¼</sup> ari ð Þ , an on the remaining contact spots will be equivalent to the effect of the uniformly distributed load qcn acting in the circular area W2ð Þ r ¼ an, al , and the assigned problem may be regarded as an axisymmetric (Figure 3). The size of the unloaded area an depends on the number of contacting asperities and with increasing applied load, it decreases from al to ac.

The solution of this problem is given in Ref. [11]. Studies on the effect of the parameter ka ¼ an=ac 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 ka = 1 or an = ac below.

Let A<sup>1</sup> and A<sup>2</sup> be two points on the surface of the circular contact area W1. The A<sup>1</sup> and A<sup>2</sup> coming into contact after application of the compressive load. Since the total normal displacement U<sup>0</sup> of the point А<sup>1</sup> is constant for any point in area W1, we have

$$\mathcal{U}\_0 = \mathcal{U}\_E + z\_1 = \mathcal{U}\_{Eri} + \mathcal{U}\_{Eci} + z\_{1\prime} \tag{14}$$

where UEri is the normal contact displacement under the pressure pri acting in the region W1; UEci is the normal displacement under the pressure qcn; and z<sup>1</sup> is the equation of the surface of a spherical asperity in an unloaded state.

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

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

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

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

area. It allows considering the problem posed as an axisymmetric problem.

forces located at the nodes of the hexagonal lattice.

8 Contact and Fracture Mechanics

Figure 3. Scheme of contact of a single asperity.

$$z\_1 = -\mu R\_{\text{max}} - \frac{\rho^2}{2r}.\tag{15}$$

Elementary displacements dUEri and dUEci under pressures qri and q<sup>c</sup> acting on elementary areas dw<sup>1</sup> and dw2, respectively, are determined by [12]:

$$d\mathcal{U}\_{Eri} = \frac{\Theta q\_{ri}(\rho\_1)}{\pi \mathcal{R}\_1} dw\_{1\prime} \, d\mathcal{U}\_{Eci} = \frac{\Theta q\_{cn}}{\pi \mathcal{R}\_2} dw\_2 \, \text{s} \tag{16}$$

where R<sup>2</sup> <sup>j</sup> <sup>¼</sup> <sup>r</sup><sup>2</sup> <sup>þ</sup> <sup>r</sup><sup>2</sup> <sup>j</sup> � 2rr<sup>j</sup> cosφ<sup>j</sup> , j = 1, 2; r � r<sup>i</sup> ; <sup>θ</sup> <sup>¼</sup> <sup>1</sup> � <sup>ν</sup><sup>2</sup> � �=E, <sup>ν</sup> is Poisson's ratio; dw<sup>1</sup> ¼ r1drdφ; and dw<sup>2</sup> ¼ r2drdφ.

After integrating Eq. (16), we have

$$
\mathcal{U}\_{Eri} = \frac{\Theta}{\pi} \int\_{W\_1} \frac{p\_{ri}(\rho) dw\_1}{R\_1},\tag{17}
$$

$$\mathcal{U}L\_{Eci} = \frac{4}{\pi} \Theta q\_c \left[ a\_l \mathcal{E} \left( \frac{\rho\_i}{a\_l} \right) - a\_c \mathcal{E} \left( \frac{\rho\_i}{a\_c} \right) \right],\tag{18}$$

where Εð Þx is the complete elliptic integral of the second kind.

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

$$\int\_{W\_1} \frac{p\_{ri}(\rho)dw\_1}{R\_1} = f\left(\rho\_i\right),\tag{19}$$

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

problems. Let us consider this approach at length.

in Eq. (29) can be written as.

where σ<sup>u</sup> is the tensile strength.

the load P to the indentation diameter d as

The empirical Meyer law is often written as:

under uniaxial tension or compression is described by equations

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

S σy

where σ<sup>y</sup> ≈ Sy, σ<sup>y</sup> is the yield strength, and ε<sup>y</sup> ¼ σy=E.

<sup>¼</sup> <sup>E</sup><sup>ε</sup> σy <sup>n</sup>

ening can be determined according to Ref. [17] from the following equation:

4P

4P <sup>π</sup>d<sup>2</sup> <sup>¼</sup> <sup>P</sup>

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

nln n � n 1 þ ln ε<sup>y</sup>

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

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

> <sup>S</sup> <sup>¼</sup> <sup>ε</sup>E, <sup>ε</sup> <sup>≤</sup> <sup>ε</sup>y; Kεn, ε ≥ εy;

The constant K is determined from the equality condition for σ at εy. Then the second equation

<sup>¼</sup> <sup>ε</sup> εy <sup>n</sup>

Taking into accord that the limiting uniform strain ε<sup>u</sup> ¼ n, the exponential deformation hard-

Meyer was the first who described a material behavior in the elastic-plastic domain. He related

<sup>π</sup>d<sup>2</sup> <sup>¼</sup> HM <sup>¼</sup> <sup>A</sup><sup>∗</sup> <sup>d</sup>

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

D <sup>m</sup>�<sup>2</sup>

� ln <sup>σ</sup><sup>u</sup>

σy

(29)

11

, ε ≥ εy: (30)

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¼ 0, (31)

: (33)

<sup>P</sup> <sup>¼</sup> Ad<sup>m</sup>: (32)

<sup>π</sup>a<sup>2</sup> <sup>¼</sup> pm <sup>¼</sup> HM, (34)

$$f(\boldsymbol{\rho}\_{i}) = \frac{\pi}{\Theta} \left( \boldsymbol{\mathcal{U}}\_{0} - \boldsymbol{\mu} \mathbf{R}\_{\text{max}} - \frac{\boldsymbol{\omega} \mathbf{R}\_{\text{max}} \boldsymbol{\rho}\_{i}^{2}}{\boldsymbol{a}\_{c}^{2}} \right) - 2\pi q\_{c} \left[ \boldsymbol{a}\_{l} - \frac{2}{\pi} \boldsymbol{\mathcal{E}} \left( \frac{\boldsymbol{\rho}\_{i}}{\boldsymbol{a}\_{c}} \right) \right]. \tag{20}$$

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

$$p\_{\pi}(\rho\_{i}) = -\frac{1}{2\pi} \int\_{\rho\_{i}}^{a\_{\pi}} \frac{F(s)ds}{\sqrt{s^{2} - \rho\_{i}^{2}}},\\P\_{i} = -\frac{2}{\pi} \left[ \frac{f'(\sigma)\sigma^{2}d\sigma}{\sqrt{a\_{r}^{2} - \sigma^{2}}},\\F(s) = \frac{2}{\pi} \left[ f(0) + s \left\{ \frac{f'(\sigma)d\sigma}{\sqrt{s^{2} - \sigma^{2}}} \right\} \right. \tag{21}$$

As a result from (21), we have

$$p\_{ri}(\rho\_i) = \frac{4\omega R\_{\text{max}}}{\pi \theta a\_c^2} \sqrt{a\_{ri}^2 - \rho\_i^2} + \frac{q\_c}{\pi} \arcsin\sqrt{\frac{a\_{ri}^2 - \rho\_i^2}{a\_c^2 - \rho\_i^2}},\tag{22}$$

$$P\_i = \frac{8\omega R\_{\text{max}} a\_{ri}^3}{3\Theta a\_c^2} + 2q\_c a\_c^2 \left[ \arcsin\frac{a\_{ri}}{a\_c} - \sqrt{\frac{a\_{ri}^2}{a\_c^2} \left(1 - \frac{a\_{ri}^2}{a\_c^2}\right)} \right]. \tag{23}$$

Taking into account that <sup>η</sup><sup>i</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> ri=a<sup>2</sup> ci, qci <sup>¼</sup> Pi<sup>=</sup> <sup>π</sup>a<sup>2</sup> ci � �, from Eqs. (22) and (23), we have

$$p\_{ri}(\rho\_i) = \frac{4\eta\_i^{0.5}\omega R\_{\text{max}}}{\pi\theta a\_c^2} \sqrt{1 - \frac{\rho\_i^2}{a\_{ri}^2}} + \frac{q\_c}{\pi} \arcsin\sqrt{\frac{a\_{ri}^2 - \rho\_i^2}{a\_c^2 - \rho\_i^2}}\tag{24}$$

$$q\_{ci} = \frac{8\omega R\_{\text{max}} \eta\_i^{1.5}}{3\pi\epsilon\theta a\_c} + \frac{2}{\pi} q\_c \left[ \arcsin \eta\_i^{0.5} - \sqrt{\eta\_i (1 - \eta\_i)} \right]. \tag{25}$$

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

$$p\_{mi} = \frac{N\_i}{A\_{ri}} = \frac{q\_{ci}}{\eta\_i} = \frac{8\eta\_i^{0.5}\omega R\_{\text{max}}}{3\pi\theta a\_c} + \frac{2q\_c}{\pi\eta\_i} \left[\arccos\,\eta\_i^{0.5} - \sqrt{\eta\_i(1-\eta\_i)}\right] \tag{26}$$

$$p\_{ri}(0) = \frac{4\eta\_i^{0.5}\omega R\_{\text{max}}}{\pi\Theta a\_c} + \frac{q\_c}{\pi}\arcsin\eta\_i^{0.5}.\tag{27}$$

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

$$p\_r(\eta\_i, \rho\_i) = p\_{r0}(\eta\_i, 0) \left(1 - \rho\_i^2 / a\_r^2\right)^{\beta}, \beta = p\_{r0}(\eta\_i, 0) / p\_m(\eta\_i, 0) - 1. \tag{28}$$

#### 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

$$S = \begin{cases} \varepsilon E, & \varepsilon \le \varepsilon\_y; \\ K \varepsilon^n, & \varepsilon \ge \varepsilon\_y; \end{cases} \tag{29}$$

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 εy. Then the second equation in Eq. (29) can be written as.

$$\frac{S}{\sigma\_y} = \left(\frac{E\varepsilon}{\sigma\_y}\right)^n = \left(\frac{\varepsilon}{\varepsilon\_y}\right)^n, \varepsilon \ge \varepsilon\_y. \tag{30}$$

where σ<sup>y</sup> ≈ Sy, σ<sup>y</sup> is the yield strength, and ε<sup>y</sup> ¼ σy=E.

Taking into accord that the limiting uniform strain ε<sup>u</sup> ¼ n, the exponential deformation hardening can be determined according to Ref. [17] from the following equation:

$$n\ln n - n\left(1 + \ln \varepsilon\_y\right) - \ln \frac{\sigma u}{\sigma\_y} = 0,\tag{31}$$

where σ<sup>u</sup> is the tensile strength.

where Εð Þx is the complete elliptic integral of the second kind.

ð

prið Þr dw<sup>1</sup> R1

¼ f r<sup>i</sup>

i a2 c

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

ð Þ <sup>σ</sup> <sup>σ</sup><sup>2</sup>d<sup>σ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2

<sup>r</sup> � <sup>σ</sup><sup>2</sup> <sup>p</sup> ,F sð Þ¼ <sup>2</sup>

þ qc π arcsin

> ac �

þ qc π arcsin

<sup>i</sup> �

arcsin η<sup>0</sup>:<sup>5</sup>

arcsin η<sup>0</sup>::<sup>5</sup>

þ qc π

, β ¼ pr<sup>0</sup> η<sup>i</sup>

<sup>i</sup> �

<sup>q</sup> � � � �

; <sup>0</sup> � �=pm <sup>η</sup><sup>i</sup>

<sup>q</sup> � � � �

� <sup>2</sup>πqc al � <sup>2</sup>

π <sup>Ε</sup> <sup>r</sup><sup>i</sup> ac

<sup>π</sup> <sup>f</sup>ð Þþ <sup>0</sup> <sup>s</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ri � <sup>r</sup><sup>2</sup> i

a2 <sup>c</sup> � <sup>r</sup><sup>2</sup> i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �, from Eqs. (22) and (23), we have

s

a2 <sup>c</sup> � <sup>r</sup><sup>2</sup> i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η<sup>i</sup> 1 � η<sup>i</sup>

> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η<sup>i</sup> 1 � η<sup>i</sup>

<sup>i</sup> : (27)

; <sup>0</sup> � � � <sup>1</sup>: (28)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ri � <sup>r</sup><sup>2</sup> i

<sup>1</sup> � <sup>a</sup><sup>2</sup> ri a2 c

s

a2 ri a2 c

" # <sup>s</sup> � �

2 4

� � � �

� �, (19)

ðs

f 0 ð Þ <sup>σ</sup> <sup>d</sup><sup>σ</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>s</sup><sup>2</sup> � <sup>σ</sup><sup>2</sup> <sup>p</sup>

0

: (20)

3

, (22)

: (23)

, (24)

: (25)

, (26)

5: (21)

W<sup>1</sup>

<sup>θ</sup> <sup>U</sup><sup>0</sup> � uRmax � <sup>ω</sup>Rmaxr<sup>2</sup>

� �

π ð ari

0

q

<sup>þ</sup> <sup>2</sup>qca<sup>2</sup>

ci, qci <sup>¼</sup> Pi<sup>=</sup> <sup>π</sup>a<sup>2</sup>

s

<sup>i</sup> ωRmax πθa<sup>2</sup> c

i 3πθac

<sup>¼</sup> <sup>8</sup>η<sup>0</sup>, <sup>5</sup>

prið Þ¼ 0

þ 2

<sup>i</sup> ωRmax 3πθac

> 4η<sup>0</sup>,<sup>5</sup> <sup>i</sup> ωRmax πθac

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

; <sup>0</sup> � � <sup>1</sup> � <sup>r</sup><sup>2</sup>

f 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 ri � <sup>r</sup><sup>2</sup> i

<sup>c</sup> arcsin ari

ci

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � <sup>r</sup><sup>2</sup> i a2 ri

<sup>π</sup> qc arcsin <sup>η</sup><sup>0</sup>:<sup>5</sup>

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

þ 2qc πη<sup>i</sup>

<sup>i</sup> =a<sup>2</sup> r � �<sup>β</sup>

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

f r<sup>i</sup> � � <sup>¼</sup> <sup>π</sup>

Eq. (19) is [13].

10 Contact and Fracture Mechanics

pri r<sup>i</sup>

� � ¼ � <sup>1</sup>

As a result from (21), we have

Taking into account that <sup>η</sup><sup>i</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup>

2π ð ari

ri

F sð Þds ffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>s</sup><sup>2</sup> � <sup>r</sup><sup>2</sup> i <sup>q</sup> , Pi ¼ � <sup>2</sup>

pri r<sup>i</sup>

pri r<sup>i</sup>

pmi <sup>¼</sup> Ni Ari

pr η<sup>i</sup> ; ri � � <sup>¼</sup> pr<sup>0</sup> <sup>η</sup><sup>i</sup>

Pi <sup>¼</sup> <sup>8</sup>ωRmaxa<sup>3</sup>

3θa<sup>2</sup> c

� � <sup>¼</sup> <sup>4</sup>η<sup>0</sup>:<sup>5</sup>

qci <sup>¼</sup> <sup>8</sup>ωRmaxη<sup>1</sup>:<sup>5</sup>

¼ qci ηi

ri=a<sup>2</sup>

� � <sup>¼</sup> <sup>4</sup>ωRmax πθa<sup>2</sup> c

ri

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

$$P = Ad^m.\tag{32}$$

The empirical Meyer law is often written as:

$$\frac{4P}{\pi d^2} = HM = A^\* \left(\frac{d}{D}\right)^{m-2}.\tag{33}$$

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

$$\frac{4P}{\pi d^2} = \frac{P}{\pi a^2} = p\_m = HM\_r \tag{34}$$

where a is the radius of the contact area.

Using [16], we have

$$\frac{P}{E^\*R^2} = \frac{2}{k\_{\sigma} \cdot k\_n} \left(\frac{n}{e}\right)^n \varepsilon\_y^{1-n} \left(\frac{a}{R}\right)^{2+1.041n}.\tag{35}$$

<sup>η</sup><sup>i</sup> <sup>¼</sup> <sup>ε</sup> � <sup>u</sup>

qc <sup>¼</sup> <sup>N</sup> Ac ¼ 1 Ac Xnr i¼1

> min ð ð Þ ε;ε<sup>s</sup>

> > 0

η<sup>1</sup>, <sup>5</sup> <sup>i</sup> φ<sup>0</sup>

Ψη η<sup>i</sup> � �φ<sup>0</sup>

qcð Þ¼ ε

8 3π min Ð ð Þ ε;ε<sup>s</sup> 0

0

1 � min Ð ð Þ ε;ε<sup>s</sup>

where <sup>f</sup> <sup>q</sup> <sup>¼</sup> <sup>θ</sup>qcac

f <sup>q</sup>ð Þ¼ ε

parameter fq.

different values of p and q (b).

<sup>ω</sup>Rmax :

represent Eq. (41) in the form.

θqcð Þε ac ωRmax

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

¼

described by equations.

<sup>2</sup><sup>ω</sup> � <sup>f</sup> <sup>q</sup> <sup>1</sup> <sup>þ</sup>

0 B@

f q 2 � vuut

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

qciAci; <sup>η</sup> <sup>¼</sup> Ar

<sup>n</sup>ð Þ u du, η εð Þ¼

, Ψη η<sup>i</sup>

� � <sup>¼</sup> <sup>2</sup> π

Considering that for this roughness model Aci ¼ const, Ac ¼ Acinc, and dnr ¼ ncφ<sup>0</sup>

<sup>n</sup>ð Þ u du

<sup>n</sup>ð Þ u du

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

Figure 4. The relative contact area with/without taking into account the mutual influence of asperities (a) and for

qciφ<sup>0</sup>

Ac ¼ 1 Ac Xnr i¼1

> min ð ð Þ ε;ε<sup>s</sup>

> > 0

ηi φ0

arcsinη<sup>0</sup>:<sup>5</sup>

<sup>i</sup> �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� <sup>ε</sup> � <sup>u</sup> 2ω

Aciη<sup>i</sup>

1

Contact Mechanics of Rough Surfaces in Hermetic Sealing Studies

CA, (40)

http://dx.doi.org/10.5772/intechopen.72196

: (41)

<sup>n</sup>ð Þ u du: (42)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi η<sup>i</sup> 1 � η<sup>i</sup> <sup>q</sup> � � � �: (43)

<sup>n</sup>ð Þ u du, we

13

1 þ f q 2 !<sup>2</sup>

where <sup>E</sup><sup>∗</sup> is reduced elastic modulus, <sup>k</sup><sup>σ</sup> <sup>¼</sup> <sup>0</sup>:333 for carbon and pearlitic steel, for other materials, the values of k<sup>σ</sup> are given in Ref. [18].

$$k\_n = \frac{(2 + 1.041n)^{1 + 0.5205n}}{(1 + 1.041n)^{1 + 1.041n}} (1.041n)^{0.5205n}.\tag{36}$$

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.
