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 að Þ¼ ri 0:

$$\mathcal{U}\_0 = \mu \mathcal{R}\_{\text{max}} + 2\Theta q\_c (a\_l - a\_c) + 2\omega \mathcal{R}\_{\text{max}} \frac{a\_{ri}^2}{a\_c^2} + + 2\Theta q\_c a\_c \left(1 - \sqrt{1 - \frac{a\_{ri}^2}{a\_c^2}}\right). \tag{37}$$

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

$$
\Delta L\_0 = \varepsilon R\_{\text{max}} + 2\Theta q\_c (a\_l - a\_c). \tag{38}
$$

Since the value of U<sup>0</sup> is constant for all points of the contact regions, it follows from Eqs. (56) and (38) that

$$
\eta\_i + \frac{\Theta q\_c a\_c}{\omega R\_{\text{max}}} \left( 1 - \sqrt{1 - \eta\_i} \right) - \frac{\varepsilon - \mu}{2\omega} = 0. \tag{39}
$$

This equation has a solution

$$\eta\_i = \frac{\varepsilon - u}{2\omega} - f\_q \left( 1 + \frac{f\_q}{2} - \sqrt{\left( 1 + \frac{f\_q}{2} \right)^2 - \frac{\varepsilon - u}{2\omega}} \right) \tag{40}$$

where <sup>f</sup> <sup>q</sup> <sup>¼</sup> <sup>θ</sup>qcac <sup>ω</sup>Rmax :

where a is the radius of the contact area.

materials, the values of k<sup>σ</sup> are given in Ref. [18].

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

4. Contacting rough surfaces

4.1. Elastic contact of rough surfaces

from Eq. (21) under the condition F að Þ¼ ri 0:

U<sup>0</sup> ¼ uRmax þ 2Θqcð Þþ al � ac 2ωRmax

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

η<sup>i</sup> þ

θqcac ωRmax

4.1.1. Relative contact area

and (38) that

This equation has a solution

P E∗

<sup>R</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup> k<sup>σ</sup> � kn

kn <sup>¼</sup> ð Þ <sup>2</sup> <sup>þ</sup> <sup>1</sup>:041<sup>n</sup>

mental data given in Ref. [19], and with the data of FE analysis [20].

ð Þ 1 þ 1:041n

n e � �<sup>n</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

1þ0:5205n

As it was indicated in Ref. [16], the obtained results are in good agreement with the experi-

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],

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

Since the value of U<sup>0</sup> is constant for all points of the contact regions, it follows from Eqs. (56)

ffiffiffiffiffiffiffiffiffiffiffiffi 1 � η<sup>i</sup> � � q � <sup>ε</sup> � <sup>u</sup>

1 �

a2 ri a2 c

þ þ2θqcac 1 �

U<sup>0</sup> ¼ εRmax þ 2θqcð Þ al � ac : (38)

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

<sup>2</sup><sup>ω</sup> <sup>¼</sup> <sup>0</sup>: (39)

: (37)

! s

which was used in solving problems of elastic-plastic contacting of rough surfaces.

<sup>1</sup>þ1:041<sup>n</sup> ð Þ <sup>1</sup>:041<sup>n</sup> <sup>0</sup>:5205<sup>n</sup>

ε<sup>1</sup>�<sup>n</sup> <sup>y</sup> a R � �<sup>2</sup>þ1:041<sup>n</sup>

: (35)

: (36)

Using [16], we have

12 Contact and Fracture Mechanics

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

$$\eta\_c = \frac{N}{A\_c} = \frac{1}{A\_c} \sum\_{i=1}^{n\_r} \eta\_{ci} A\_{ci}; \ \eta = \frac{A\_r}{A\_c} = \frac{1}{A\_c} \sum\_{i=1}^{n\_r} A\_{ci} \eta\_i. \tag{41}$$

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, we represent Eq. (41) in the form.

$$q\_c(\varepsilon) = \int\_0^{\min(\varepsilon, \varepsilon)} q\_{ci} q\_n'(u) du, \ \eta(\varepsilon) = \int\_0^{\min(\varepsilon, \varepsilon)} \eta\_i q\_n'(u) du. \tag{42}$$

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

$$f\_q(\varepsilon) = \frac{\Theta q\_\varepsilon(\varepsilon) a\_\varepsilon}{\omega \aleph\_{\max}} = \frac{\frac{8}{3\pi} \int\_0^{\min(\varepsilon, \varepsilon\_\*)} \eta\_i^{1.5} q\_n'(u) du}{1 - \int\_0^{\min(\varepsilon, \varepsilon\_\*)} \Psi\_\eta(\eta\_i) q\_n'(u) du}, \quad \Psi\_\eta(\eta\_i) = \frac{2}{\pi} \left[ \arcsin \eta\_i^{0.5} - \sqrt{\eta\_i(1 - \eta\_i)} \right]. \tag{43}$$

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

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],

$$V\_i = \begin{cases} V\_{ri} = 2\pi \int\_{\frac{a\_i}{a\_i}}^{a\_c} [z\_{20}(\rho) - z\_{10}(\rho)] \rho d\rho; \\\\ V\_{0i} = 2\pi \int\_{\frac{a\_i}{a\_i}}^{a\_i} [z\_{2r}(\rho) - z\_{1r}(\rho)] \rho d\rho, \end{cases} \tag{44}$$

for contacting asperity z1<sup>r</sup> ¼ z10;

where β ¼ prið Þ0 =pm � 1:

UEci ¼ z20, UEri ¼ ωRmax

Taking into account that <sup>x</sup><sup>2</sup> <sup>¼</sup> <sup>t</sup>, we have

z2<sup>r</sup> ¼

8 ><

>:

f qi x <sup>2</sup>F<sup>1</sup> 1 2 ; 1 2 ; β þ 2;

a point; and cases c and d correspond to the contact under the different loads.

0

<sup>V</sup>0<sup>i</sup> <sup>¼</sup> <sup>π</sup>a<sup>2</sup> c ð 1

Figure 5. The scheme for contacting a single asperity located at level u = 0.5.

z1r, 0 ≤ x < η<sup>0</sup>, <sup>5</sup>

UEri <sup>þ</sup> UEci, <sup>η</sup><sup>0</sup>, <sup>5</sup>

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

<sup>Δ</sup>z0ð Þ<sup>t</sup> dt, Vri <sup>¼</sup> <sup>π</sup>a<sup>2</sup>

i

<sup>i</sup> ≤ x ≤ 1;

i 3π

Contact Mechanics of Rough Surfaces in Hermetic Sealing Studies

þ Ψ η<sup>i</sup>

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

Δzrð Þt dt: (52)

ηi x2 � �, f qi <sup>¼</sup> <sup>8</sup>η<sup>1</sup>,<sup>5</sup>

> c ð 1

> > ηi

(50)

15

� � � <sup>f</sup> <sup>q</sup>, (51)

where z10, z<sup>20</sup> and z1r, z2<sup>r</sup> 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

$$V\_{\mathcal{L}} = \sum\_{i=1}^{n\_r} V\_{ri} + \sum\_{i=1}^{n\_\varepsilon - n\_r} V\_{0i} \tag{45}$$

And the corresponding gap density is equal to

$$\Lambda(\varepsilon) = \frac{V\_{\varepsilon}}{A\_{\varepsilon}R\_{\text{max}}} = \frac{1}{A\_{\varepsilon}R\_{\text{max}}} \left[ \left. \int\_{0}^{\min(\varepsilon,\varepsilon\_{\text{S}})} V\_{n} \boldsymbol{\uprho}\_{n}^{\prime}(u) du + \left. \int\_{\min(\varepsilon,\varepsilon\_{\text{S}})}^{\varepsilon\_{\text{S}}} V\_{0} \boldsymbol{\uprho}\_{n}^{\prime}(u) du \right| \right. \tag{46} \right]$$

Taking into account that Λri ¼ Vri=ð Þ AciRmax и Λ0<sup>i</sup> ¼ V0i=ð Þ AciRmax , it can be represented in the form

$$\Lambda(\varepsilon) = \int\_0^{\min(\varepsilon, \varepsilon\_\S)} \Lambda\_{ri} q\_n'(u) du + \int\_{\min(\varepsilon, \varepsilon\_\S)}^{\varepsilon\_\S} \Lambda\_{0i} q\_n'(u) du. \tag{47}$$

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

$$
\omega\_{10} = \omega R\_{\text{max}} \left[ \frac{\varepsilon - \mu}{\omega} - \mathfrak{x}^2 + 2f\_q(k - 1) \right], \tag{48}
$$

where <sup>x</sup> <sup>¼</sup> <sup>r</sup> ac ; k <sup>¼</sup> al ac ,

$$\omega\_{20} = 2\omega R\_{\text{max}} f\_q \left[ k \cdot\_2 F\_1 \left( -\frac{1}{2}, \frac{1}{2}; 1; \frac{x^2}{k^2} \right) - \_2F\_1 \left( -\frac{1}{2}, \frac{1}{2}; 1; x^2 \right) \right] \tag{49}$$

where <sup>2</sup>F<sup>1</sup> is the Gaussian hypergeometric function,

for contacting asperity z1<sup>r</sup> ¼ z10;

4.1.2. Gaps density of the joint

14 Contact and Fracture Mechanics

To determine the volume of the intercontact space, it is necessary to determine the volumes of

½ � z20ð Þ� r z10ð Þr rdr;

(44)

½ � z2rð Þ� r z1rð Þr rdr,

V0i, (45)

<sup>n</sup>ð Þ u du

3 7

<sup>n</sup>ð Þ u du: (47)

, (48)

, (49)

<sup>5</sup>: (46)

ðac

ari

ð aci

0

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

where z10, z<sup>20</sup> and z1r, z2<sup>r</sup> are the equations describing the surfaces of noncontacting and

Vri þ n <sup>X</sup><sup>c</sup>�nr i¼1

Vriφ<sup>0</sup>

Taking into account that Λri ¼ Vri=ð Þ AciRmax и Λ0<sup>i</sup> ¼ V0i=ð Þ AciRmax , it can be represented in the

<sup>n</sup>ð Þ u du þ

We provide the equations of surfaces of the asperities and the half-space that enter into

ε � u

2 ; 1 2 ; 1; x2 k 2

� �

<sup>n</sup>ð Þ u du þ

εðS

minð Þ ε;ε<sup>S</sup>

<sup>ω</sup> � <sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup><sup>f</sup> <sup>q</sup>ð Þ <sup>k</sup> � <sup>1</sup> h i

� � � �

�2F<sup>1</sup> � <sup>1</sup> 2 ; 1 2 ; 1; x<sup>2</sup>

εðS

V0iφ<sup>0</sup>

minð Þ ε;ε<sup>S</sup>

Λ0iφ<sup>0</sup>

gaps attributable to single contacting and noncontacting asperities [10],

8

>>>>>>>><

>>>>>>>>:

Vri ¼ 2π

V0<sup>i</sup> ¼ 2π

Vc <sup>¼</sup> <sup>X</sup>nr i¼1

> min ð ð Þ ε;ε<sup>S</sup>

Λriφ<sup>0</sup>

2 6 4

0

Vi ¼

contacting asperities and half-spaces, respectively.

And the corresponding gap density is equal to

AcRmax

Λ εð Þ¼

<sup>¼</sup> <sup>1</sup> AciRmax

> min ð ð Þ ε;ε<sup>S</sup>

> > 0

z<sup>10</sup> ¼ ωRmax

<sup>z</sup><sup>20</sup> <sup>¼</sup> <sup>2</sup>ωRmax<sup>f</sup> <sup>q</sup> <sup>k</sup>�2F<sup>1</sup> � <sup>1</sup>

where <sup>2</sup>F<sup>1</sup> is the Gaussian hypergeometric function,

Λ εð Þ¼ Vc

form

Eq. (44):

where <sup>x</sup> <sup>¼</sup> <sup>r</sup>

ac ; k <sup>¼</sup> al ac ,

$$z\_{2r} = \begin{cases} z\_{1r\prime} & 0 \le x < \eta\_i^{0.5} \\\\ \mathcal{U}\_{Eri} + \mathcal{U}\_{Eci\prime} & \eta\_i^{0.5} \le x \le 1; \end{cases} \tag{50}$$

$$\mathcal{U}\_{\rm Eci} = \mathfrak{z}\_{20\prime} \cdot \mathcal{U}\_{\rm Eri} = \omega \mathbb{R}\_{\rm max} \frac{f\_{qi}}{\mathbf{x}} \,\_2F\_1\left(\frac{\mathbf{1}}{2}, \frac{\mathbf{1}}{2}; \mathfrak{z} + 2; \frac{\eta\_i}{\mathbf{x}^2}\right) \cdot f\_{qi} = \frac{8\eta\_i^{1.5}}{3\pi} + \Psi(\eta\_i) \cdot f\_{q\prime} \tag{51}$$

where β ¼ prið Þ0 =pm � 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.

Taking into account that <sup>x</sup><sup>2</sup> <sup>¼</sup> <sup>t</sup>, we have

$$V\_{0i} = \pi a\_c^2 \int\_0^1 \Delta z\_0(t)dt,\ V\_{ri} = \pi a\_c^2 \int\_{\eta\_i}^1 \Delta z\_r(t)dt. \tag{52}$$

Figure 5. The scheme for contacting a single asperity located at level u = 0.5.

where Δz<sup>0</sup> ¼ z<sup>20</sup> � z<sup>10</sup> and Δzr ¼ z2<sup>r</sup> � z1r:

Since <sup>Λ</sup><sup>i</sup> <sup>¼</sup> Vi <sup>π</sup>acRmax, after integrating (52), we have

$$\Lambda\_{\rm ei} = \omega \left\{ \frac{1}{2} - \frac{\varepsilon - u}{\omega} - 2f\_q \left[ (k - 1) - k \cdot {}\_2 {}\_1F\_1 \left( -\frac{1}{2}, \frac{1}{2}; 2; \frac{1}{k^2} \right) + {}\_2F\_1 \left( -\frac{1}{2}, \frac{1}{2}; 2; 1 \right) \right] \right\}.\tag{53}$$

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 ηip for distributed at different heights asperities, at which plastic deformation begins. Thus, the value of ηip for any asperity loaded according to Figure 3 can be determined for the

By the Tresca criterion of the maximum tangential stresses, the plastic deformation on the z

The maximum contact pressure is defined as p<sup>0</sup> ¼ Kyσy, where Ky ¼ 1, 613. The mean contact

ari <sup>¼</sup> <sup>3</sup>Pir 4E<sup>∗</sup> � �<sup>1</sup> 3

> c 2ωRmax

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

<sup>8</sup> <sup>β</sup> <sup>þ</sup> <sup>1</sup> � � <sup>f</sup> <sup>y</sup> !<sup>2</sup>

Similarly, we define the criterion of occurrence of plastic deformation at the contact area.

The highest value of the equivalent stress σeqð Þ1 is on the contour of the contact area, where it slightly exceeds σeqð Þ0 in the center of the loading area. It is convenient to represent

<sup>p</sup> <sup>¼</sup> <sup>3</sup>πKy

, a2 ri a2 c ¼ η<sup>i</sup> , σy

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

η∗

2 y

σeqð Þ0 pm

<sup>p</sup> ¼ 1, 605f

According to [23], the equivalent stresses at the center of the area are

depend on loading conditions and this is its advantage.

σeq ¼ 2τ1 max ¼ 0, 62p<sup>0</sup> ¼ σy: (55)

Contact Mechanics of Rough Surfaces in Hermetic Sealing Studies

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

17

, (56)

<sup>E</sup><sup>∗</sup> <sup>¼</sup> <sup>ε</sup>y, (57)

, (58)

. Thus, the proposed criterion of plasticity does not

<sup>¼</sup> <sup>0</sup>, 2 1 <sup>þ</sup> <sup>β</sup> � �: (59)

highest asperity at u ¼ 0, qc ¼ 0, and β ¼ 0, 5.

axis corresponds to the equivalent stress [22].

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

Pi <sup>¼</sup> <sup>π</sup>a<sup>2</sup>

pressure is pm <sup>¼</sup> Kyσy<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>β</sup> � �:

and taking into account that.

layer

where <sup>f</sup> <sup>y</sup> <sup>¼</sup> <sup>σ</sup>yac

<sup>E</sup>∗ωRmax :

For the highest asperity η<sup>∗</sup>

$$\begin{split} \Lambda\_{\text{ri}} &= \omega \left\{ \left( 1 - \eta\_{i} \right) \left[ \frac{1 + \eta\_{i}}{2} - \frac{\varepsilon - u}{\omega} - 2f\_{q}(k - 1) \right] + 2f\_{q}k \left[ \,\_{2}F\_{1} \left( -\frac{1}{2}, \frac{1}{2}; 2; \frac{1}{k^{2}} \right) - \\ & \quad - \eta\_{i} \,\_{2}F\_{1} \left( -\frac{1}{2}, \frac{1}{2}; 2; \frac{\eta\_{i}}{k^{2}} \right) \right] - 2f\_{q} \left[ \,\_{2}F\_{1} \left( -\frac{1}{2}, \frac{1}{2}; 2; 1 \right) - \eta\_{i} \,\_{2}F\_{1} \left( -\frac{1}{2}, \frac{1}{2}; 2; \eta\_{i} \right) \right] + \\ & \quad + 2f\_{q} \left[ \,\_{2}F\_{1} \left( -\frac{1}{2}, \frac{1}{2}; \{\theta + 2; \eta\_{i} \right) - \eta\_{i}^{0.5} \,\_{2}F\_{1} \left( -\frac{1}{2}, \frac{1}{2}; \{\theta + 2; 1 \} \right) \right] . \end{split} \tag{54}$$

Substituting the equations obtained in Eq. (47), we determine the joint density Λ εð Þ. To determine the dependence Λ f <sup>q</sup> � �, it is necessary to exclude the parameter <sup>ε</sup> from the dependences f <sup>q</sup>ð Þε and Λ εð Þ.

Figure 6 shows the dependence of the gap density on the complex parameter f <sup>q</sup> 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.

#### 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

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).

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 ηip for distributed at different heights asperities, at which plastic deformation begins. Thus, the value of ηip for any asperity loaded according to Figure 3 can be determined for the highest asperity at u ¼ 0, qc ¼ 0, and β ¼ 0, 5.

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

$$
\sigma\_{c\eta} = 2\tau\_{1\,\max} = 0,\\
62p\_0 = \sigma\_y. \tag{55}
$$

The maximum contact pressure is defined as p<sup>0</sup> ¼ Kyσy, where Ky ¼ 1, 613. The mean contact pressure is pm <sup>¼</sup> Kyσy<sup>=</sup> <sup>1</sup> <sup>þ</sup> <sup>β</sup> � �:

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

$$a\_{ri} = \left(\frac{3P\_ir}{4E^\*}\right)^{\frac{1}{3}},\tag{56}$$

and taking into account that.

where Δz<sup>0</sup> ¼ z<sup>20</sup> � z<sup>10</sup> and Δzr ¼ z2<sup>r</sup> � z1r:

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

2

� �

4.1.3. The criteria for the appearance of plastic deformations

� ��

2 ; 1 2 ; 2; ηi k 2

1 <sup>2</sup> � <sup>ε</sup> � <sup>u</sup>

� �

� <sup>η</sup><sup>i</sup> <sup>2</sup>F<sup>1</sup> � <sup>1</sup>

<sup>þ</sup> <sup>2</sup><sup>f</sup> qi <sup>2</sup>F<sup>1</sup> � <sup>1</sup>

<sup>π</sup>acRmax, after integrating (52), we have

<sup>ω</sup> � <sup>2</sup><sup>f</sup> <sup>q</sup> ð Þ� <sup>k</sup> � <sup>1</sup> <sup>k</sup>�2F<sup>1</sup> � <sup>1</sup>

<sup>ω</sup> � <sup>2</sup><sup>f</sup> <sup>q</sup>ð Þ <sup>k</sup> � <sup>1</sup>

�

� <sup>2</sup><sup>f</sup> <sup>q</sup> <sup>2</sup>F<sup>1</sup> � <sup>1</sup>

� <sup>η</sup><sup>0</sup>, <sup>5</sup>

� � ���

� <sup>ε</sup> � <sup>u</sup>

; β þ 2; η<sup>i</sup> � � 2 ; 1 2 ; 2; 1 k2

� � � � � �

i

2 ; 1 2 ; 2; 1 � �

<sup>i</sup> <sup>2</sup>F<sup>1</sup> � <sup>1</sup>

Substituting the equations obtained in Eq. (47), we determine the joint density Λ εð Þ. To deter-

Figure 6 shows the dependence of the gap density on the complex parameter f <sup>q</sup> when two rough surfaces come into contact. Figure 2 shows that the contact density does not depend on

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

Figure 6. The gap density with/without taking into account the mutual influence of asperities (a) and for different values

the parameters p and q, since the dependences for the different values of p and q.

� �

<sup>þ</sup> <sup>2</sup><sup>f</sup> <sup>q</sup><sup>k</sup> <sup>2</sup>F<sup>1</sup> � <sup>1</sup>

<sup>þ</sup> <sup>2</sup>F<sup>1</sup> � <sup>1</sup>

2 ; 1 2 ; 2; 1 k2

� <sup>η</sup><sup>i</sup> <sup>2</sup>F<sup>1</sup> � <sup>1</sup>

; β þ 2; 1

, it is necessary to exclude the parameter ε from the dependences

� � �

2 ; 1 2 ; 2; 1

2 ; 1 2 ; 2; η<sup>i</sup> � ��

:

�

þ

: (53)

(54)

Since <sup>Λ</sup><sup>i</sup> <sup>¼</sup> Vi

16 Contact and Fracture Mechanics

Λoi ¼ ω

Λri ¼ω 1 � η<sup>i</sup>

mine the dependence Λ f <sup>q</sup>

f <sup>q</sup>ð Þε and Λ εð Þ.

of p and q (b).

$$P\_i = \pi a\_{ri}^2 p\_{m'} \ \ r = \frac{a\_c^2}{2\omega \mathcal{R}\_{\text{max}}} \ \ \frac{a\_{ri}^2}{a\_c^2} = \eta\_{i'} \ \ \frac{\sigma\_y}{E^\*} = \varepsilon\_{y'} \tag{57}$$

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

$$
\eta\_p^\* = \left(\frac{3\pi K\_y}{8\left(\beta + 1\right)} f\_y\right)^2,\tag{58}
$$

where <sup>f</sup> <sup>y</sup> <sup>¼</sup> <sup>σ</sup>yac <sup>E</sup>∗ωRmax :

For the highest asperity η<sup>∗</sup> <sup>p</sup> ¼ 1, 605f 2 y . 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

$$\frac{\sigma\_{eq}(0)}{p\_m} = 0, 2\left(1 + \beta\right). \tag{59}$$

The highest value of the equivalent stress σeqð Þ1 is on the contour of the contact area, where it slightly exceeds σeqð Þ0 in the center of the loading area. It is convenient to represent σeqð Þ¼ 1 K<sup>σ</sup> � σeqð Þ0 , where for β ¼ 0, 5 according to the energy theory of shear strains K<sup>σ</sup> ¼ 1, 16, according to the theory of maximal tangential stresses K<sup>σ</sup> ¼ 1, 33.

At the moment of appearance of plastic deformation along the contour of the contact area σeqð Þ¼ 1 σy, and the average contact pressure.

$$p\_m = \frac{5\sigma\_y}{K\_\sigma (1+\beta)}.\tag{60}$$

Ca <sup>¼</sup> Ca <sup>ε</sup>y; <sup>n</sup> � � <sup>¼</sup> <sup>2</sup>

¼ qc σy � σy <sup>E</sup><sup>∗</sup> � ac ωRmax

f <sup>q</sup>ð Þ¼ ε

η εð Þ¼

use the relations.

dependence f <sup>q</sup>ð Þε .

parameter qσ.

(Figure 3).

4.2.2. Gaps density of the joint

<sup>f</sup> <sup>q</sup> <sup>¼</sup> qcac E∗ ωRmax

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

where Cf ¼ Ca � f <sup>y</sup>, η<sup>i</sup> is determined by Eq. (40).

Summing up f qi over all asperities, we have

Similarly, using Eq. (40) and f <sup>q</sup>ð Þε , we have

k<sup>σ</sup> � kn

<sup>q</sup>σ<sup>i</sup> <sup>¼</sup> Ca � <sup>η</sup><sup>1</sup>þ0:52<sup>n</sup>

<sup>f</sup> qi <sup>¼</sup> Ca � <sup>η</sup><sup>1</sup>þ0:52<sup>n</sup>

Cf min Ð ð Þ ε;ε<sup>s</sup>

1 � min Ð ð Þ ε;ε<sup>s</sup> 0

> min ð ð Þ ε;ε<sup>s</sup>

> > 0

Excluding the parameter ε from Eqs. (69) and (70), we obtain the dependence η f <sup>q</sup>

0

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

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

� <sup>2</sup>ωRmax ac

<sup>i</sup> þ q<sup>σ</sup> � Ψη η<sup>i</sup>

<sup>i</sup> þ f <sup>q</sup> � Ψη η<sup>i</sup>

η<sup>1</sup>þ0:52<sup>n</sup> <sup>i</sup> φ<sup>0</sup>

Ψη η<sup>i</sup> � �φ<sup>0</sup>

For a given value ε, we solve the system of transcendental Eqs. (40), (69) and obtain the

η<sup>i</sup> ε; f <sup>q</sup> � �φ<sup>0</sup>

Figures 7 and 8 present the dependencies of the relative contact area on the relative force

The scheme of the action of the loads pr and qc is similar to the scheme for elastic contact

<sup>n</sup>ð Þ u du

<sup>n</sup>ð Þ u du

<sup>¼</sup> <sup>q</sup><sup>σ</sup> � <sup>f</sup> <sup>y</sup>; f <sup>y</sup> <sup>¼</sup> <sup>ε</sup>yac

ωRmax

� �<sup>1</sup>:041<sup>n</sup> n

e � ε<sup>y</sup> � �<sup>n</sup>

Contact Mechanics of Rough Surfaces in Hermetic Sealing Studies

: (65)

19

� �: (66)

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

� �, (68)

: (69)

<sup>n</sup>ð Þ u du: (70)

� � or <sup>η</sup> <sup>q</sup><sup>σ</sup>

� �.

; f qi ¼ qσ<sup>i</sup> � f <sup>y</sup>: (67)

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

$$
\eta\_p^{\*\*} = \left(\frac{15\pi}{8K\_\mathcal{O}\left(\beta + 1\right)} f\_y\right)^2. \tag{61}
$$

For the highest asperity η∗∗ <sup>p</sup> <sup>¼</sup> <sup>15</sup>, <sup>42</sup>K�<sup>2</sup> σ f 2 y . According to the theory of maximum tangential stresses η∗∗ <sup>p</sup> <sup>¼</sup> <sup>5</sup>, <sup>405</sup>η<sup>∗</sup> <sup>p</sup>, according to the energy theory of shear deformations η∗∗ <sup>p</sup> <sup>¼</sup> <sup>7</sup>, <sup>105</sup>η<sup>∗</sup> p:

#### 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<sup>c</sup> (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

$$\frac{P\_i}{E^\* R^2} = \frac{2}{k\_{\sigma} \cdot k\_{\pi}} \left(\frac{n}{e}\right)^n \varepsilon\_y^{1-n} \left(\frac{a\_{ri}}{R}\right)^{2+1.041n}.\tag{62}$$

Considering that for the roughness model used <sup>R</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> <sup>c</sup>=ð Þ <sup>2</sup>ωRmax and <sup>η</sup><sup>i</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup> ri=a<sup>2</sup> <sup>c</sup> , from Eq. (62) we have

$$\frac{q\_{ci}}{E^\*} = \frac{P\_i}{E^\* \cdot \pi a\_c^2} = \frac{2}{k\_o \cdot k\_n} \cdot \left(\frac{2\omega R\_{\text{max}}}{a\_c}\right)^{1.041n} \left(\frac{n}{e}\right)^n \varepsilon\_y^{1-n} \eta\_i^{1+0.52n}.\tag{63}$$

For elastic-plastic contact, it is convenient to use the parameter q<sup>σ</sup> ¼ qc=σy, then from Eq. (63) we have

$$\overline{q}\_{oi} = \frac{q\_{ci}}{\sigma\_y} = \mathbb{C}\_a \cdot \eta\_i^{1+0.52n} \text{ .} \tag{64}$$

where

Contact Mechanics of Rough Surfaces in Hermetic Sealing Studies http://dx.doi.org/10.5772/intechopen.72196 19

$$\mathbb{C}\_{\mathfrak{a}} = \mathbb{C}\_{\mathfrak{a}}(\varepsilon\_{\mathfrak{y}}, n) = \frac{2}{k\_{\mathfrak{o}} \cdot k\_{\mathfrak{n}}} \cdot \left(\frac{2\omega R\_{\text{max}}}{a\_{\mathfrak{c}}}\right)^{1.041n} \left(\frac{n}{\mathfrak{c} \cdot \varepsilon\_{\mathfrak{y}}}\right)^{n}. \tag{65}$$

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

$$
\overline{q}\_{\phi i} = \mathbb{C}\_{a} \cdot \eta\_{i}^{1 + 0.52u} + \overline{q}\_{\sigma} \cdot \Psi\_{\eta}(\eta\_{i}).\tag{66}
$$

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

$$f\_q = \frac{q\_\text{c} a\_\text{c}}{E^\ast \omega R\_{\text{max}}} = \frac{q\_\text{c}}{\sigma\_y} \cdot \frac{\sigma\_y}{E^\ast} \cdot \frac{a\_\text{c}}{\omega R\_{\text{max}}} = \overline{q}\_\sigma \cdot f\_y; f\_y = \frac{\varepsilon\_y a\_\text{c}}{\omega R\_{\text{max}}}; f\_{q\text{i}} = \overline{q}\_{\text{oi}} \cdot f\_y. \tag{67}$$

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

σeqð Þ¼ 1 K<sup>σ</sup> � σeqð Þ0 , where for β ¼ 0, 5 according to the energy theory of shear strains

At the moment of appearance of plastic deformation along the contour of the contact area

Then, similarly to the above reasoning, the criterion of the appearance of plastic deformations

<sup>8</sup>K<sup>σ</sup> <sup>β</sup> <sup>þ</sup> <sup>1</sup> � � <sup>f</sup> <sup>y</sup> !<sup>2</sup>

<sup>p</sup>, according to the energy theory of shear deformations η∗∗

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<sup>c</sup>

> n e � �<sup>n</sup>

� <sup>2</sup>ωRmax ac

For elastic-plastic contact, it is convenient to use the parameter q<sup>σ</sup> ¼ qc=σy, then from Eq. (63)

ε<sup>1</sup>�<sup>n</sup> <sup>y</sup> ari R � �<sup>2</sup>þ1:041<sup>n</sup>

� �<sup>1</sup>:041<sup>n</sup> n

<sup>¼</sup> Ca � <sup>η</sup><sup>1</sup>þ0:52<sup>n</sup>

<sup>K</sup><sup>σ</sup> <sup>1</sup> <sup>þ</sup> <sup>β</sup> � � : (60)

. According to the theory of maximum tangential

<sup>c</sup>=ð Þ <sup>2</sup>ωRmax and <sup>η</sup><sup>i</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup>

ε<sup>1</sup>�<sup>n</sup> <sup>y</sup> η<sup>1</sup>þ0:52<sup>n</sup>

<sup>i</sup> , (64)

e � �<sup>n</sup>

: (61)

: (62)

<sup>i</sup> : (63)

<sup>c</sup> , from Eq. (62)

ri=a<sup>2</sup>

<sup>p</sup> <sup>¼</sup> <sup>7</sup>, <sup>105</sup>η<sup>∗</sup>

p:

pm <sup>¼</sup> <sup>5</sup>σ<sup>y</sup>

<sup>p</sup> <sup>¼</sup> <sup>15</sup><sup>π</sup>

σ f 2 y

(Figure 3). We use a discrete roughness model, described by Eqs. (15) and (16).

K<sup>σ</sup> ¼ 1, 16, according to the theory of maximal tangential stresses K<sup>σ</sup> ¼ 1, 33.

η∗∗

<sup>p</sup> <sup>¼</sup> <sup>15</sup>, <sup>42</sup>K�<sup>2</sup>

σeqð Þ¼ 1 σy, and the average contact pressure.

in the contact area is

18 Contact and Fracture Mechanics

stresses η∗∗

we have

we have

where

For the highest asperity η∗∗

4.2.1. Relative contact area

<sup>p</sup> <sup>¼</sup> <sup>5</sup>, <sup>405</sup>η<sup>∗</sup>

4.2. Elastic-plastic contact of rough surfaces

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

Considering that for the roughness model used <sup>R</sup> <sup>¼</sup> <sup>a</sup><sup>2</sup>

qci <sup>E</sup><sup>∗</sup> <sup>¼</sup> Pi <sup>E</sup><sup>∗</sup> � <sup>π</sup>a<sup>2</sup> c

Pi E∗

<sup>R</sup><sup>2</sup> <sup>¼</sup> <sup>2</sup> k<sup>σ</sup> � kn

<sup>¼</sup> <sup>2</sup> k<sup>σ</sup> � kn

> <sup>q</sup>σ<sup>i</sup> <sup>¼</sup> qci σy

$$f\_{qi} = \mathbb{C}\_{\mathfrak{a}} \cdot \mathfrak{n}\_i^{1+0.52n} + f\_q \cdot \Psi\_\eta(\mathfrak{n}\_i) \,\tag{68}$$

where Cf ¼ Ca � f <sup>y</sup>, η<sup>i</sup> is determined by Eq. (40).

Summing up f qi over all asperities, we have

$$f\_q(\varepsilon) = \frac{\mathsf{C}\_f \int\_0^{\min(\varepsilon, \varepsilon\_s)} \mathsf{\eta}\_i^{1+0.52n} q\_n'(u) du}{1 - \int\_0^{\min(\varepsilon, \varepsilon\_s)} \mathsf{\Psi}\_\eta(\eta\_i) q\_n'(u) du}. \tag{69}$$

For a given value ε, we solve the system of transcendental Eqs. (40), (69) and obtain the dependence f <sup>q</sup>ð Þε .

Similarly, using Eq. (40) and f <sup>q</sup>ð Þε , we have

$$\eta(\varepsilon) = \int\_0^{\min(\varepsilon, \varepsilon\_\*)} \eta\_i(\varepsilon, f\_q) \varrho'\_n(u) du. \tag{70}$$

Excluding the parameter ε from Eqs. (69) and (70), we obtain the dependence η f <sup>q</sup> � � or <sup>η</sup> <sup>q</sup><sup>σ</sup> � �.

Figures 7 and 8 present the dependencies of the relative contact area on the relative force parameter qσ.

#### 4.2.2. Gaps density of the joint

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

Total density of gaps with elastic-plastic contact

The total volume of the displaced material

ηi <sup>c</sup><sup>2</sup> � <sup>η</sup>�0, <sup>5</sup>

<sup>i</sup> f qiKβ<sup>0</sup> � ��

Since Λ<sup>p</sup> ¼ Vp=ð Þ AcRmax , we have

0

<sup>c</sup><sup>2</sup> � <sup>η</sup>�0,<sup>5</sup>

Figure 9. Scheme of the unloaded crater.

min ð ð Þ ε;ε<sup>s</sup>

Λ<sup>p</sup> ¼ ω

� <sup>η</sup><sup>i</sup>

plastic displacement of the material into the interfacial space.

determined by Eq. (68) and the parameter β is used in Eq. (72).

Λ ¼ Λ<sup>e</sup> � Λ<sup>p</sup> ¼ Λe<sup>0</sup> þ Λer � Λp, (73)

Contact Mechanics of Rough Surfaces in Hermetic Sealing Studies

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

21

� �: (74)

<sup>n</sup>ð Þ u du: (75)

� ð Þ <sup>ω</sup>Rmax <sup>2</sup> 3a<sup>2</sup> c

�

(76)

where Λ<sup>e</sup> is the density of gaps due to the elastic punching of the half-space, which accounted for single contacting and noncontacting asperities; Λ<sup>p</sup> is reduction of the gap density due to the

The value of Λ<sup>e</sup> is determined, similarly to the elastic contact, by Eq. (47). In this case, f qi is

Let us determine the volume of the displaced material for a single contacting asperity (Figure 9). Let us assume that the unloaded crater has a constant radius Rfi and the unloaded depth from the level of the initial surface hfi. The volume of plastically displaced material falling on a single

> fi Rfi � hfi 3

> > Vpiφ<sup>=</sup>

<sup>i</sup> f qi Kβ<sup>0</sup> � Kβ<sup>c</sup> � � h i�<sup>1</sup>

crater is equal to the volume of a spherical segment of height hf and radius Rfi:

Vp ¼ nc

(

<sup>i</sup> f qiKβ<sup>0</sup> � �<sup>2</sup>

> φ= <sup>n</sup>ð Þ u du:

Vpi <sup>¼</sup> <sup>π</sup>h<sup>2</sup>

min ð ð Þ ε;ε<sup>s</sup>

0

<sup>0</sup>; 5 1 � <sup>η</sup>�1,<sup>5</sup>

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 ε<sup>y</sup> and n.

For an elastic-plastic contact

$$\overline{P}\_i = \frac{P\_i}{E^\* R^2} \propto \left(\frac{h\_i}{R}\right)^{0.5205n+1} \text{ \AA} \tag{71}$$

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

$$p(r) = p\_0 \left(1 - \frac{r^2}{a^2}\right)^{\beta},\tag{72}$$

where <sup>p</sup><sup>0</sup> <sup>¼</sup> pm <sup>1</sup> <sup>þ</sup> <sup>β</sup> is pressure at <sup>r</sup> <sup>¼</sup> 0, pm is the mean pressure on contact area and β ¼ 0, 5205n:

Total density of gaps with elastic-plastic contact

$$
\Lambda = \Lambda\_{\epsilon} - \Lambda\_p = \Lambda\_{\epsilon 0} + \Lambda\_{\epsilon r} - \Lambda\_{p\prime} \tag{73}
$$

where Λ<sup>e</sup> is the density of gaps due to the elastic punching of the half-space, which accounted for single contacting and noncontacting asperities; Λ<sup>p</sup> is reduction of the gap density due to the plastic displacement of the material into the interfacial space.

The value of Λ<sup>e</sup> is determined, similarly to the elastic contact, by Eq. (47). In this case, f qi is determined by Eq. (68) and the parameter β is used in Eq. (72).

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

Let us assume that the unloaded crater has a constant radius Rfi and the unloaded depth from the level of the initial surface hfi. The volume of plastically displaced material falling on a single crater is equal to the volume of a spherical segment of height hf and radius Rfi:

$$V\_{pi} = \pi h\_{\hat{f}}^2 \left( R\_{\hat{f}} - \frac{h\_{\hat{f}}}{3} \right). \tag{74}$$

The total volume of the displaced material

$$V\_p = n\_c \int\_0^{\min(\varepsilon, \varepsilon\_s)} V\_{pi} \mathbf{q}\_n^{\prime}(u) du. \tag{75}$$

Since Λ<sup>p</sup> ¼ Vp=ð Þ AcRmax , we have

$$\begin{split} \Lambda\_{\mathbb{P}} &= \omega \int\_{0}^{\min(\varepsilon, \mathfrak{t}\_{i})} \left( \frac{\eta\_{l}}{c^{2}} - \eta\_{i}^{-0.5} f\_{qi} K\_{\mathbb{P}0} \right)^{2} \left\{ 0, \mathbb{E} \left[ 1 - \eta\_{l}^{-1.5} f\_{qi} (K\_{\mathbb{P}0} - K\_{\mathbb{P}c}) \right]^{-1} - \frac{\left( \omega R\_{\max} \right)^{2}}{3a\_{c}^{2}} \times \\ & \quad \times \left( \frac{\eta\_{l}}{c^{2}} - \eta\_{i}^{-0.5} f\_{qi} K\_{\mathbb{P}0} \right) \right\} \Phi\_{n}^{\prime}(u) du. \end{split} \tag{76}$$

Figure 9. Scheme of the unloaded crater.

For an elastic-plastic contact

Figure 8. The relative contact area for different values of ε<sup>y</sup> and n.

different values of p and q (b).

20 Contact and Fracture Mechanics

β ¼ 0, 5205n:

Pi <sup>¼</sup> Pi E∗

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

<sup>R</sup><sup>2</sup> <sup>∝</sup> hi R

Figure 7. The relative contact area with/without taking into account the mutual influence of asperities (a) and for

p rð Þ¼ <sup>p</sup><sup>0</sup> <sup>1</sup> � <sup>r</sup><sup>2</sup>

where <sup>p</sup><sup>0</sup> <sup>¼</sup> pm <sup>1</sup> <sup>þ</sup> <sup>β</sup> is pressure at <sup>r</sup> <sup>¼</sup> 0, pm is the mean pressure on contact area and

<sup>0</sup>, <sup>5205</sup>nþ<sup>1</sup>

a2 <sup>β</sup> , (71)

, (72)

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).

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 qσ.
