Partial slip problem in frictional contact of orthotropic elastic half-plane and rigid punch

International Journal of Mechanical Sciences 135 (2018) 168–175

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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci

Partial slip problem in frictional contact of orthotropic elastic half-plane and rigid punch Jing Jin Shen a,∗, Ya Yong Wu a, Jin Xing Lin a, Feng Yu Xu a, Cheng Gang Li b a b

School of Automation, Nanjing University of Posts and Telecommunications Nanjing, Jiangsu (210023), China College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu(210016), China

a r t i c l e

i n f o

Keywords: Frictional contact Orthotropic elasticity Partial slip Linear complementarity

a b s t r a c t Planar partial-slip contact problem of a rigid indenter with flat end and an orthotropic elastic material is analytically and numerically investigated. In the analytical way, the coupled singular equations of this problem are reduced to a Fredholm integral equation with a regular kernel. The analytical solutions are derived in the forms of the Goodman’s and Spence’s approximations. In the numerical way, a linear complementarity formulation is developed by reformulating the governing equations are as coupled Volterra integral equations. And, a Newtonbased optimization algorithm based on smoothing approximation is used to solve the problem. The validness of both the Goodman and Spence approximate solutions is verified by comparing with the numerical results for different orthotropic elastic materials. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Contact between a rigid punch and orthotropic materials can be found in various engineering applications such as instrumented indentation test [1], tactile sensor design [2], metal forming [3]. The knowledge of deformation states during contact is essentially important for better understanding these application processes. Although prior investigations to contact problem have generated amounts of valuable results since the seminal Hertz’s work about one and a half century ago, most of the results are restricted to isotropic elastic solids [4]. And, contact problem of anisotropic materials is still far way from completeness. It is well known that the general solution in anisotropic elasticity can be represented in the Lekhnistskii formulism or the Stroh formulism. As a direct generalization of Muskhelishvili’s approach for planar isotropic elasticity , the Lekhnistskii formulism is developed in terms of elastic compliances. On the other hand, Stroh formulism placing its basis on elastic stiffnesses is easy to generalize to three-dimensional analysis [5]. Based on the two formulisms, many researchers studied contact processes of anisotropic materials with different boundary conditions. By using the Lekhnistskii formulism, [6] simplified the frictionless and the frictional sliding contact problems to be two particular cases of the Hilbert problem and presented corresponding analytical solutions. Based on stress functions corresponding to normal and tangential tractions, [7] presented an iterative scheme for a flat-end punch sliding over an anisotropic half plane, in which the mutual influences

∗

of normal and tangential tractions on each other are neglected in turn. Based on singular equations governing surface tractions and surface displacement gradient, [8] obtained closed-form solutions for indentation of an anisotropic elastic half plane by a flat-end punch under four different boundary conditions. Utilizing the perturbation technique, [9] presented an asymptotic solution for indentation of a parabolic indenter against an orthotropic elastic layer. By using the Stroh formulism, [10] investigated bond contact between a rigid punch with arbitrary profile and an anisotropic elastic half-plane. By splitting the entire contact domain between a rigid indenter and an elastic layer into several regions with Dirichelet or Neumann boundary conditions, [11] presented a technique to find an analytical solution in terms of a series form. Based on the Fourier integral transform, [12] analytically solved the sliding frictional contact of a flat-end or parabolic punch and an orthotropic half plane. Furthermore, there are a lot of research literatures concerning on contact of anisotropic piezoelectric materials [13] and graded solids [14]. However, in the above mentioned literatures, the studied contact problems are either frictionless or sliding frictional. To treat the partial slip contact arisen in normal indentation of an isotropic half-space in presence of finite friction, it needs to treat the boundary conditions imposed on the stick and slip regions. Since the stick/slip boundary appears as an additional unknown parameter, hence the solution procedure becomes much more complex than the frictionless problem. In practical applications, the Goodman’s approximation, in which the influence of the shear stress on the normal stress is neglected, is widely used [15]. For example, based on the Goodman’s approxima-

Corresponding author. E-mail address: wff[email protected] (J.J. Shen).

https://doi.org/10.1016/j.ijmecsci.2017.11.022 Received 1 September 2017; Received in revised form 23 October 2017; Accepted 14 November 2017 Available online 15 November 2017 0020-7403/© 2017 Elsevier Ltd. All rights reserved.

J.J. Shen et al.

International Journal of Mechanical Sciences 135 (2018) 168–175

tion, [16] considered the frictional indentation of a functionally graded coated half-space by a rigid punch, [17] studied the frictional contact between two elastic cylinders. Based on the self-similarity assumption, [18] found that the slip radius is the same for all power-law indenters. Then, [18] compared the results evaluated by the Goodman’s approximation and the coupled integral equations governing the partial slip contact. By using planar bipolar coordinates, [19,20] reduced the problem as a singular equation in terms of the normal stress in the slip region, and presented an analytical solution for the indentation of a rigid cylinder or sphere on an elastic half space. In addition, a few papers paid attention to the partial slip contact in the Cattaneo problem [21,22]. In this paper, the partial contact problem between a rigid punch and an orthotropic elastic half plane is studied. Our primary aims are to investigate the influences of the friction force and the material orthotropy on the splitting boundary between the stick and slip regions, and whether the Goodman’s and Spence’s approximations are valid for the orthotropic elastic solids. The rest of the present paper is organized as follows. In Section 2, the formulation of the problem and the surface Green’s function of the orthotropic elastic half plane are briefly described. Section 3 gives the analytical Goodman’s and Spence’s approximations based on the integral equation governing the partial slip contact of a flat-end punch and a orthotropic elastic half plane. Then, an efficient numerical scheme for determining the contact stresses and the slip/stick boundary is presented in Section 4. Numerical results are provided in section 5 to show the influence of the material anisotropy. Finally, conclusions are drawn in Section 6.

P

punch

x1 -a

orthotropic medium

Fig. 1. Partial slip indentation of flat-ended punch on orthotropic medium.

2.2. Indentation of flat punch In a Cartesian coordinates (x1 , y1 ), let a rigid punch with a flat-end profile be brought to contact with an elastic half plane y1 < 0 by normal force P over the contact area −𝑎 ≤ 𝑥1 ≤ 𝑎, as shown in Fig. 1. After the variable changes 𝑥 = 𝑎𝑥1 , 𝑦 = 𝑦1 , the yielded normal and shear stresses at the contact area can be expressed as 𝑃 𝑃 𝑝(𝑥), (𝜎𝑥𝑦 )𝑦=0 = 𝑞(𝑥) (6) 𝑎 𝑎 where p(x) and q(x) are the normal and shear surface tractions for the indentation caused by a punch with flat end −1 < 𝑥 < 1 subjected to the unit normal force, respectively. Due to the effect of the friction force, the total contact area is split to the slip region and the stick region. As a result, the contact boundary conditions are defined as (𝜎𝑦𝑦 )𝑦=0 = −

2.1. Surface Green’s function Considering an orthotropic half plane with its principle axes of orthotropy aligning with the Cartesian coordinates (x, y) , the two dimensional strain-stress relation can be expressed as 𝑠12 𝑠22 0

a

c

y1

2. Problem statement

⎡𝜖𝑥𝑥 ⎤ ⎡𝑠11 ⎢ 𝜖 ⎥ = ⎢𝑠 ⎢ 𝑦𝑦 ⎥ ⎢ 12 ⎣𝜖𝑥𝑦 ⎦ ⎣ 0

-c

0 ⎤ ⎡𝜎𝑥𝑥 ⎤ ⎡𝛿 −2 1⎢ ⎥ ⎢ ⎥ 0 𝜎𝑦𝑦 = −𝜈 ⎥⎢ ⎥ 𝐸⎢ 𝑠66 ⎦ ⎣𝜎𝑥𝑦 ⎦ ⎣ 0

−𝜈 𝛿2 0

0 ⎤ ⎡𝜎𝑥𝑥 ⎤ 0 ⎥ ⎢ 𝜎𝑦𝑦 ⎥ ⎥⎢ ⎥ 𝜅 + 𝜈 ⎦ ⎣𝜎𝑥𝑦 ⎦

(1)

𝑣′0 (𝑥) = 0,

|𝑥| ≤ 1

(7)

where E, 𝜈, 𝛿 and 𝜅 are the effective stiffness, the effective Poisson’s ratio, the stiffness ratio and the shear parameter, respectively. These four parameters have the following relationships with the four engineering constants E11 , E22 , G12 and 𝜈 12

𝑢′0 (𝑥) = 0,

|𝑥| < 𝑐

(8)

𝐸=

√

𝐸11 + 𝐸22 ,

𝜈=

√

𝜈12 𝜈21 ,

𝛿4 =

𝐸11 , 𝐸22

𝜅=

𝑝(𝑥) − 𝑞(𝑥)∕𝜇 = 0,

𝐸 − 𝜈 (2) 2𝐺12

for the plane-stress case, and √ 𝐸 = 𝛿4 =

𝐸11 + 𝐸22 , (1 − 𝜈13 𝜈31 )(1 − 𝜈23 𝜈32 )

𝐸11 1 − 𝜈23 𝜈32 , 𝐸22 1 − 𝜈13 𝜈31

𝜅=

√ 𝜈=

𝐸 −𝜈 2𝐺12

(𝜈12 + 𝜈13 𝜈32 )(𝜈21 + 𝜈23 𝜈31 ) , (1 − 𝜈13 𝜈31 )(1 − 𝜈23 𝜈32 ) (3)

𝑓𝑥 d𝑠 𝑥−𝑠

(4)

𝑎 𝑓 𝑦 d 𝑠 − 𝐴𝑓𝑥 (𝑥) ∫𝑏 𝑥 − 𝑠

(5)

= 𝐴𝑓𝑦 (𝑥) + 𝐵

−𝑣′0 (𝑥) = 𝐶

𝑎

∫𝑏

(9)

𝐶

1 𝑝(𝑡) 𝑑𝑡 − 𝐴𝑞(𝑥) = 0, ∫−1 𝑥 − 𝑡

|𝑥| ≤ 1

(10)

𝐵

1 𝑞(𝑡) 𝑑𝑡 + 𝐴𝑝(𝑥) = 0, ∫−1 𝑥 − 𝑡

|𝑥| < 𝑐

(11)

where u′ is positive. Considering Eqs. (1), (10) and (11), we can see that the effective stiffness E has no influence on the stress distribution in the slip contact area. Taking account of the governing equations for an isotopic half plane

for the plane-strain case. It is well known that the Airy stress function Φ(x, y), which automatically satisfies the equilibrium equations, is particularly suitable for the two-dimensional elasticity. And, Appendix A detailedly presents the elastic responses corresponding to concentrated force. Based on Eqs. (A.19) and (A.20), for the orthotropically elastic half plane subjected to arbitrary distributed loadings fx (x) and fy (x), the surface displacement gradients have the following expressions −𝑢′0 (𝑥)

𝑐 ≤ |𝑥| ≤ 1

where 𝜇 is the friction coefficient, and c represents the extent of the slip region i.e., |x| < c and c ≤ |x| ≤ 1 are the stick and slip regions, respectively. Following the relations in Eqs. (4) and (5), the boundary conditions in Eqs. (7)–(9) can be expressed in terms of p(x) and q(x) as the following Fredholm integral equations

1 𝑝(𝑡) 1 d 𝑡 + ℘𝑞(𝑥) = 0, 𝜋 ∫−1 𝑡 − 𝑥

|𝑥| ≤ 1

(12)

1 𝑞(𝑡) 1 d 𝑡 − ℘𝑝(𝑥) = 0, 𝜋 ∫−1 𝑡 − 𝑥

|𝑥| < 𝑐

(13)

where ℘ = (1 − 2𝜈)∕(2 − 2𝜈) with 𝜈 as the Poisson’s ratio. It is easy to find that when 𝐵 = 𝐶, the orthotropic elasticity reduces to be the isotropic elasticity. And, B/A, C/A are two characteristic parameters for the frictional contact. 169

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International Journal of Mechanical Sciences 135 (2018) 168–175

3. Analytical solution to the indentation of flat punch

where

3.1. The Goodman’s approximate solution

𝜔(𝑥) =

In the treatment of frictional normal contact, the Goodman’s approximation, neglecting the influence of shear tractions on normal displacements, is widely used. In this approximation, Eqs. (10) and (11) are reduced to be

Substituting Eq. (24) into Eq. (23) leads to

(

2𝐶𝑥 2𝐵

1

∫0

𝑝(𝑡) 𝑑𝑡 = 0, 𝑥−𝑡

|𝑥| ≤ 1

1

𝑡𝑞(𝑡) 𝑑𝑡 + 𝐴𝑝(𝑥) = 0, ∫0 𝑥2 − 𝑡2

√

1

𝑐 𝑡𝑞(𝑡) 𝑑𝑡 + 𝜗(𝑥) = 0 ∫0 𝑥2 − 𝑡2

(15)

1

1

𝜆Φ(𝜉) =

∫𝑐

𝑞(𝑥) =

2𝑥 𝜋 2 𝐵 ∫0

(

𝑐2 𝑐2

𝑦2

− − 𝑥2

)1 2

𝜗(𝑦) 𝑑𝑦 − 𝑥2

𝜚=

∫0

𝜁 𝜚−1 𝕂(𝜉, 𝜁 )Φ(𝜁 )𝑑𝜁

lim

(19)

𝜚→0

(20)

1 2

log(1 − 𝛾1 )

log(𝑐) +

log(𝜆) 2𝜚

+

− lim

𝜚→0

𝐵𝜋𝜇 𝐸(𝑐) − = ′ 𝐴 𝐸 (𝑐)

𝜚→0

lim

(21)

where E(c) denotes the complete elliptic integral and √ 𝐸( 1 − 𝑐 2 ).

𝐸 ′ (𝑐)

=

|𝑥| ≤ 1 |𝑥| < 𝑐

log(𝜆) . 2𝜚

arctan(𝐴𝜇∕(𝐶𝜋)) 𝐵𝜋𝜇 =− log(1 − 𝛾1 ) 𝐴

(22)

If we can define cot (𝜚𝜋) = 𝐶𝜋∕(𝐴𝜇), the solution to Eq. (22) can be expressed as

4.1. Volterra integral equations governing flat-end punch indentation

𝑥 sin(2𝜋𝜚) 𝑐 𝜔(𝑡) 𝜙(𝑡) 𝑑𝑡 ∫0 𝜔 ( 𝑥 ) 𝑥 2 − 𝑡 2 𝜋

(32)

As can be seen, the Spence’s approximation Eq. (34) gives the same friction coefficient 𝜇 corresponding to a given value c for different materials with the same 𝛾 1 . In other words, the Spence’s approximation combines two characteristic parameters C/A and B/A to be one parameter 𝛾 1 . 4. Numerical algorithm to the indentation of flat-end punch

𝑝(𝑥) = sin2 (𝜋𝜚)𝜙(𝑥) −

(31)

Since

(23)

∫0

(30)

Furthermore, substituting Eq. (33) into Eq. (29) and letting limϱ → 0 log(g(ϱ))/ϱ take its first-order accurate term as − log(4) + 𝜚𝜋 2 ∕6, we can get ( ′ )−1 𝜋𝜚 𝐸 (𝑐) 𝜋𝜚 − = − (34) log(1 − 𝛾1 ) 𝐸(𝑐) 3

In common with the analysis for the isotropic medium [18], a new variable, 𝜙(𝑥) = 𝑝(𝑥) − 𝑞 (𝑥)𝑠𝑖𝑔 𝑛(𝑥)∕𝜇 with sign(x) being the signum function, is introduced to reduce the dual integral Eqs. (10) and (11) to a single one. Taking account of the symmetry of p(x) and antisymmetry of q(x), Eqs. (10) and (11) can be reformulated via eliminating q(x) by 𝜙(x) as

2𝑡(𝑝(𝑡) − 𝜙(𝑡)) d𝑡 = 0, 𝑥2 − 𝑡2

1 arctan(𝐴𝜇∕(𝐶𝜋)) 𝜋

the explicit expression of 𝜆∗ can be obtained by combining Eqs. (21), (31) and (32) as ( ) 𝜋 𝐸 ′ (𝑐) 4 − 𝜆∗ = log (33) 𝑐 2 𝐸(𝑐)

3.2. The Spence’s solution

1

𝜚=

arctan(𝐴𝜇∕(𝐶𝜋)) 𝜋 1 = log(1 − 𝛾1 ) 2 log(4) − log(𝑐) − 𝜆∗

where 𝜆∗ = lim𝜚→0

𝐴𝑝(𝑥) + 𝐵𝜇

(29)

log(𝑔(𝜚)) 𝜚

log(𝑔(𝜚)) = − log(4), 𝜚

One sufficient condition satisfying the consistency condition gives the following relation between the friction coefficient, the material constants, and the stick/slip boundary as

1 2𝑥𝑝(𝑡) 𝐶 d𝑡 = 𝜙(𝑥), 𝐴𝜇 ∫0 𝑥2 − 𝑡2

(28)

and taking the limiting value of Eq. (29) yields

𝜗(𝑦)

𝑑𝑦 = 0 √ 𝑐 2 − 𝑦2

𝑝(𝑥) −

(27)

Considering

with the consistency condition as 𝑐

1

∫0

and 𝕂(𝜉, 𝜁 ) is a square integrable kernel that guarantee the existence and uniqueness of c. For the detailed expression about 𝕂(𝜉, 𝜁 ), please refer to [18]. In the following, we will derive the analytical relation of c and ϱ when ϱ → 0. From the definition of ϱ, we can see that 𝜚 = 0 means 𝜇 = 0. Hence, c should be close to 0 as ϱ → 0. From Eq. (28), ϱ can be written as a function of c and 𝜆 in the logarithmic form as

(18)

𝑦2

1

𝜆 = (1 − 𝛾1 )∕(𝑐 2𝜚 𝑔 2 (𝜚))

The solution to Eq. (17) that is non-singular at 𝑥 = 𝑐 is 𝑐

𝐺(𝑥2 , 𝑢2 ) =

𝜚, 𝑥2 )

where

(17)

𝑡𝑝(𝑡) 𝑑𝑡 + 𝐴𝑝(𝑥) 𝑥2 − 𝑡2

1 ,1 + 2

𝑡2 = 𝑐 2 𝜁 , (1 − 𝑥2 ) 2 −𝜚 𝜙(𝑥) = Φ(𝜉) and (1 − 𝑥2 ) 2 −𝜚 𝑝(𝑥) = Ψ(𝜉). After a straightforward but cumbersome process, the following Fredholm integral equation can be obtained

where 𝜗(𝑥) = 2𝐵𝜇

+ 𝜚)∕Γ(1 + 𝜚)Γ( 21 ),

(26)

= 𝐹 (1, with F being the hypergeometric function. To convert Eq. (26) into its canonical form, performing the variable change in Eqs. (24) and (26) as 𝑥2 = 𝑐 2 𝜉,

Notice the normal pressure is independent of the material constants. Substituting Eq. (16) into Eq. (15) and considering 𝑞(𝑥) = 𝜇𝑝(𝑥) for c ≤ |x| ≤ 1 lead to 2𝐵

(25)

𝛾1 = 𝐴2 ∕(𝐵 𝐶𝜋 2 ), 𝑔 (𝜚) = Γ( 21 2 2 [𝑥 𝐴(𝑥 ) − 𝑢2 𝐴(𝑢2 )]∕(𝑥2 − 𝑢2 ) and 𝐴(𝑥2 )

(16)

𝜋 1 − 𝑥2

2

where

It can be seen the orthotropic governing equation system of Eqs. (14) and (15) shares the same mathematic structure as that of the isotropic. For the sake of completeness, we only briefly present the solution process about the problem governed by Eqs. (14) and (15). For the detailed mathematical techniques, readers can refer [15]. Based on the general solution to the Fredholm integral equation of the first kind, the normal pressure can be obtained from Eq. (14) as 𝑝(𝑥) =

) 1 −𝜚

𝑐 ( ) 2 1 − 𝛾1 𝑝(𝑥) + sin(𝜋𝜚)𝑔(𝜚) 𝐺(𝑥2 , 𝑢2 )𝜔(𝑢)𝜙(𝑢)𝑑𝑢 = 0 ∫ 𝜋 0

(14) |𝑥| < 𝑐

1 − 𝑥2 𝑥2

Instead of the Fredholm integral equations, the Volterra integral equations are more suitable for numerical calculation. This section will

(24)

170

J.J. Shen et al.

International Journal of Mechanical Sciences 135 (2018) 168–175

derive the governing equations in its Volterra-equation type. Considering Eq. (A.11), it can be found that c1 and c2 in the Airy stress function are even with respect to 𝜔. Meanwhile, for the case of flat punch indentation, the stress 𝜎 yy (x, 0) and 𝑢′0 are even functions of x, while 𝜎 xy (x, 0) and 𝑣′0 being odd. Therefore, in the Fourier domain, 𝜎̄ 𝑦𝑦 (𝜔, 0) and 𝑢̄ ′0 are even, and 𝜎̄ 𝑥𝑦 (𝜔, 0) and 𝑣̄ ′0 odd. Then, the surface stresses and displacement gradients at 𝑦 = 0 can be obtained as 𝜎𝑦𝑦 (𝑥, 0) = 2Re(𝔽𝑐∗ [−𝜔2 (𝑐1 + 𝑐2 ); 𝑥]) = 2𝔽𝑐∗ [−𝜔2 (𝑐1𝑟 + 𝑐2𝑟 ); 𝑥]

−𝑎3

where

√ 𝑃 (𝑋) = 𝜎𝑦𝑦 ( 𝑥),

= 2𝔽𝑐∗ [−𝜔2 [(𝛼 2 − 𝛽 2 + 𝑠12 )(𝑐1𝑟 + 𝑐2𝑟 ) − 2𝛼𝛽(𝑐1𝑖 − 𝑐2𝑖 )]; 𝑥]

𝑄(𝑋) − 𝜇𝑃 (𝑋) = 0,

when

𝑋 ∈ [𝑐 2 , 1]

(47)

𝑄𝑖 = 𝑄(𝑋𝑖 ),

𝑈𝑖 = 𝑈 (𝑋𝑖 ),

with,

𝑋𝑖 =

𝑠𝑖−1 + 𝑠𝑖 , 2 (48)

(49)

𝑎3 𝟏 2

(50)

with

(39)

−𝐔 ≥ 𝟎,

𝐐 − 𝜇𝐏 ≥ 𝟎

(51)

where P, Q and U are N × 1 matrix whose entries are Pi , Qi and Ui , respectively. The coefficients in Eqs. (49) and (50) are explicitly expressed as ( √ ) √ ⎧ 𝑗 + 𝑗 − 𝑖 + 1∕2 ⎪log √ , 𝑖<𝑗 √ ⎪ 𝑗 − 1 + 𝑗 − 𝑖 − 1∕2 ( ) √ √ ⎪ 𝐀𝑈 = ⎨ , 𝑖 + 1∕2 , 𝑖=𝑗 √ ⎪log 𝑖 − 1∕2 ⎪ ⎪0 , 𝑖>𝑗 ⎩

𝑎2 = −2𝛽,

𝑖(< 𝑗 ⎧0 , ) √ ⎪ 𝑖−1 √ ArcTan − + 𝜋∕2, 𝑖=𝑗 ⎪ 1∕2 𝐀𝐿 = ⎨ ( ) √ √ ⎪ 𝑗 ⎪ArcTan − √ 𝑗−1 − ArcTan(− √ ), 𝑖−𝑗+1∕2 𝑖−𝑗−1∕2 ⎩ √ √ ⎧√ || 1 1 || 𝑖<𝑗 ⎪ ℎ|| 𝑗 − 𝑖 + − 𝑗 − 𝑖 − ||, 2 2| ⎪ |√ 𝑈 𝐃𝑖𝑗 = ⎨√ , 1 𝑖=𝑗 ⎪ ℎ 2, ⎪0 , 𝑖>𝑗 ⎩

(40)

(41)

4.2. Discretization strategy

⎛1 ⎜−1 ⎜ ⎜0 1⎜ 𝐅= ⋮ ℎ⎜ ⎜0 ⎜0 ⎜ ⎝

As pointed out in [18], the governing equations expressed in the form of the coupled Volterra integral equations are more suitable to numerical analysis than the form of the singular integral equations. Note that (42)

through the change of variable 𝑠 = 𝑡2 and 𝑋 = 𝑥2 , Eqs. (40) and (41) can be reformulated as

0 1 −1 ⋮ 0 0

0 0 1 ⋱ … …

… … … ⋱ −1 0

0 0 0 ⋮ 1 −1

𝑖>𝑗

0⎞ 0⎟ ⎟ 0⎟ ⋮⎟ ⎟ 0⎟ 1⎟ ⎟ ⎠𝑁×𝑁

𝑈 and 𝐃𝐿 𝑖𝑗 = 𝐃𝑗𝑖

𝑋

1 𝑃 (𝑠) 𝑄(𝑠) 𝑑𝑠 + 𝑎2 𝑑𝑠 √ √ √ √ ∫ 𝑋 2 𝑠 𝑠−𝑋 2 𝑠 𝑋−𝑠 [ ] 𝑋 𝑈 (𝑠) 𝑑 +2 𝑑𝑠 𝑑𝑠 = 0 √ 𝑑𝑋 ∫0 2 𝑋 − 𝑠

𝑈 (𝑋) < 0,

−𝐔(𝐐 − 𝜇𝐏) = 0,

2𝜗(𝜔) = 𝔽𝑠 [𝑣′0 , 𝜔]

1 𝜎 (𝑡) 𝑥 𝑢′0 (𝑡) 𝜎𝑦𝑦 (𝑡) 𝑥𝑦 𝑑𝑡 + 𝑎2 𝑑𝑡 = 𝑑𝑡 √ √ √ ∫0 ∫𝑥 ∫0 𝑥2 − 𝑡2 𝑡2 − 𝑥2 𝑥2 − 𝑡2 ( ) 1 1 𝑡𝜎 (𝑡) 𝑥 𝑡𝜎 (𝑡) 𝑦𝑦 𝑥𝑦 𝑎3 𝜎 (𝑡)𝑑𝑡 − 𝑑𝑡 + 𝑎4 𝑑𝑡 √ √ ∫0 𝑦𝑦 ∫𝑥 ∫0 2 2 𝑡 −𝑥 𝑥2 − 𝑡2 𝑥 𝑡𝑣′0 (𝑡) = 𝑑𝑡 √ ∫0 𝑥2 − 𝑡2

𝑎1

(46)

−𝑎 3 𝐃𝑈 𝐏 + 𝑎 4 𝐃𝐿 𝐐 =

2𝜓(𝜔) = 𝔽𝑠 [𝜎𝑥𝑦 (𝑥, 0); 𝜔],

𝜎𝑦𝑦 (𝑡)𝑑𝑡 = −1

𝑋 ∈ [0, 𝑐 2 )

𝑎1 𝐀𝐿 𝐏 + 𝑎2 𝐀𝑈 𝐐 + 2𝐅𝐃𝐿 𝐔 = 𝟎

𝑥

∫−1

when

Then the governing Eqs. (43) and (44) have the following discrete form

Based on 𝑎1 𝜙ℎ0 (𝑥) + 𝑎2 𝜓ℎ0 (𝑥) = 𝜑ℎ0 (𝑦) and 𝑎3 𝜙ℎ1 (𝑦) + 𝑎4 𝜓ℎ1 (𝑦) = 𝜗ℎ1 (𝑦), the Volterra integral equations governing the indentation from Eqs. (B.1)–(B.6) in Appendix B can be obtained as

1

𝑄(𝑋) − 𝜇𝑃 (𝑋) > 0,

1≤𝑖≤𝑁 +1

where

𝑎1

𝑈 (𝑋) = 0,

𝑃𝑖 = 𝑃 (𝑋𝑖 ),

where 𝔽𝑐∗ denotes the inverse Fourier cosine transform, 𝔽𝑠∗ the inverse Fourier sine transform, cjr and cji are the real part and the imaginary part of cj , respectively. Notice the operation order of Re and 𝔽𝑐∗ (or 𝔽𝑠∗ ) in the above equations is interchangeable. Let 𝜙(𝜔) = −𝜔2 (𝑐1𝑟 + 𝑐2𝑟 ) and 𝜓(𝜔) = −𝜔2 [𝛽(𝑐1𝑟 + 𝑐2𝑟 ) + 𝛼(𝑐1𝑖 − 𝑐2𝑖 )], Eqs. (35)–(38) gives

𝜑 = 𝑎1 𝜙 + 𝑎2 𝜓, 𝜗 = 𝑎3 , 𝑎1 = 𝑠11 (𝛼 2 + 𝛽 2 ) + 𝑠12 , 2𝑠22 𝛽 −𝑠22 𝑎3 = , 𝑎4 = − 𝑠12 𝛼2 − 𝛽 2 𝛼2 − 𝛽 2

(45)

To solve Eqs. (43) and (44), we discretize [0, 1] by 𝑁 − 1 equal intervals with N mesh points. To simplify the calculation, the midpoint rule is employed for each interval, i.e.,

(37)

𝑣′0 = 2Re(𝑖𝔽𝑠∗ [−𝜔2 (𝛾(𝑝1 )𝑐1 + 𝛾(𝑝2 )𝑐2 ); 𝑥]) [ [( ) ( ) ] ] 𝑠22 −𝑠22 = 2𝔽𝑠∗ −𝜔2 − 𝑠12 𝛽(𝑐1𝑟 + 𝑐2𝑟 ) + − 𝑠12 𝛼(𝑐1𝑖 − 𝑐2𝑖 ) ; 𝑥 2 2 2 2 𝛼 −𝛽 𝛼 −𝛽 (38)

2𝜑(𝜔) = −𝔽𝑐 [𝑢′0 , 𝜔],

√ 𝑈 (𝑋) = 𝑢0 ( 𝑥)

(36)

( ) 𝑢′0 = 2Re 𝔽𝑐∗ [−𝜔2 (𝜂(𝑝1 )𝑐1 + 𝜂(𝑝2 )𝑐2 )); 𝑥]

2𝜙(𝜔) = 𝔽𝑐 [𝜎𝑦𝑦 (𝑥, 0); 𝜔],

√ 𝑄(𝑋) = 𝜎𝑥𝑦 ( 𝑥),

(44)

Meanwhile, the complementarity conditions that determines the frictional behavior inside contact area can be written as

(35)

𝜎𝑥𝑦 (𝑥, 0) = 2Re(𝑖𝔽𝑠∗ [𝜔2 (𝑝1 𝑐1 + 𝑝2 𝑐2 ); 𝑥]) = 2𝔽𝑠∗ [−𝜔2 [𝛽(𝑐1𝑟 + 𝑐2𝑟 ) + 𝛼(𝑐1𝑖 − 𝑐2𝑖 )]; 𝑥]

1 𝑋 𝑎 𝑃 (𝑠) 𝑄(𝑠) 𝑑𝑠 + 𝑎4 𝑑𝑠 = 3 √ √ ∫𝑋 2 𝑠 − 𝑋 ∫0 2 𝑋 − 𝑠 2

4.3. Numerical algorithm

∫0

Similar to the analysis of the contact problem in the isotropic elasticity [23], it is easy to see that the anisotropic contact problem described by Eqs. (49)–(51) also is a mixed linear complimentary problem. To

(43)

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International Journal of Mechanical Sciences 135 (2018) 168–175

solve the problem, the complimentary condition in Eq. (51) usually is reformatted in its equivalent form as −𝐔 − max(𝟎, −𝐔 − (𝐐 − 𝜇𝐏)) = 𝟎

input tolerance ε,

(52)

parameter ς

To deal with the nonsmooth max function, its smooth approximation is commonly used in the numerical method. Since the max function is the twice integration of the Dirac function, one of the smooth approximations can be done by appropriately selecting a density function to smooth the Dirac function and its integral. When the sigmoid function of neural networks is selected to be the density function, the max function is smoothly approximated as max(0, 𝑥) ≈ 𝑥 + 𝛽 log(1 + exp(−𝑥∕𝜏)),

𝛽>0

initial guess S0 ,

initialization: set k = 0 and

by eq. (57)

(53)

As 𝜏 → 0, 53 converges to the max function. Therefore, Eq. (52) can be rewritten in its component form as ( ) −𝜇𝑃𝑖 + 𝑄𝑖 − 𝛽 log 1 + exp[(𝑈𝑖 − 𝜇𝑃𝑖 + 𝑄𝑖 )∕𝜏] = 0, 1 ≤ 𝑖 ≤ 𝑁 (54)

R(Sk )

∞

Yes

<ε

End

No ˜ k )) ˜ k )−1 R(S direction dk = −(∇R(S

To simplify the presentation of the solution algorithm, Eqs. (49), (50) and (52) are grouped by the nonlinear equation 𝐑(𝐔, 𝐏, 𝐐) = 𝟎

0

(55)

step size Λk : choose Λk = max(1

and Eqs. (49), (50) and (54) by ̃ (𝐔, 𝐏, 𝐐) = 𝟎 𝐑

ST k+1

(56)

=

ST k

2

, . . .)

+ Λk dk ,

such that

Then, the solution to Eq. (56) can be found by minimizing the objec̃𝐑 ̃ T ∕2. One of the optimization algorithms proposed tive function 𝑓 = 𝐑 in [24] is presented in Fig. 2 where the parameter 𝜏 is iteratively updated by √ √ 𝑁 ⎧ ‖𝐑(𝐔, 𝐏, 𝐐)‖2 ≤ 𝑁 ‖𝐑 ( 𝐔 , 𝐏 , 𝐐 ) ‖ 2 ⎪ 1 (57) = 𝜛 = ⎨√ √ √ 𝜏 𝑁 ⎪ 𝑁 ‖𝐑 ( 𝐔 , 𝐏 , 𝐐 ) ‖ > 2 ⎩ ‖𝐑(𝐔,𝐏,𝐐)‖

f (Sk ) − f (Sk+1 ) ≥ ιΛk |dT k ∇f (Sk )| 1, 0 < ι < 0.5

with 0

update

∗ k+1

by Sk+1 and eq. (57)

2

5. Results

∗ k+1

To verify the validness of the analytical approximate solutions, we consider four different orthotropic materials whose the material parameters are selected from ranges found in many engineering materials as 𝛿 2 ∈ [0.2, 5], 𝜅 ∈ [0, 1], 𝜈 ∈ [1/7, 5/7] and 𝜇 ∈ [0, 0.9]. Within these ranges, Fig. 3 shows the variations of the characteristic parameters governing the frictional contact, i.e., 𝜅 1 ∈ (0, 0.5), C/A ∈ (0.5, ∞) and B/A ∈ (0, 2). From Fig. 3a, it can be found that 𝛾 1 is independent of 𝛿 2 and C/A is inverse proportional to 𝛿 2 . Considering Eq. (5), when C/A takes a more large value, the influence of the shear traction on the normal traction decrease. Therefore, we can judge that the Goodman’s approximation becomes more accurate as 𝛿 2 tends to zero. To include the effect of the friction force, we consider the characteristic parameter A/(B𝜋𝜇) instead of A/B in the following. Fig. 4 shows the variation of stick region with respect to the ratio A/(B𝜋𝜇), which is estimated by the Goodman’s approximation Eq. (21), the Spence’s approximation Eq. (34), and the numerical algorithm in Section 4. From this figure, it can be seen that the errors of the Goodman’s approximation become large as r1 increases, while the Spence approximations are always close to the numerical results when c < 0.7. For given values of 𝛾 1 and A/(B𝜋𝜇), the shear tractions with different friction coefficients are shown in Fig. 5 where the cusp point of the shear tractions imply the boundary between the stick and slip regions. It can be seen that the stick region keeps the same value while the friction coefficient varies. From our experiences, when 𝛾 1 and A/(B𝜋𝜇) are given, the stick zones for different friction coefficients estimated by the numerical algorithm are indistinguishable for c < 0.95. Although the stick zone is almost the same, the shear tractions gradually increase as the friction coefficient becomes large. Fig. 6 gives the normal tractions for two different materials with different C/A. As expected, lower values of friction lead to smaller stick regions and smaller variation of the normal tractions.

k

Yes

No k+1

=

k

k+1

=

∗ k+1

k=k+1

Fig. 2. Flowchart of the minimization algorithm for the mixed complimentary problem where 𝐒𝑘 = (𝐔, 𝐏, 𝐐).

6. Conclusions The paper considers the normal indentation of a flat-end punch on an orthotropic half plane with finite friction. By utilizing the Fourier transform, we obtain the singular integral equations governing the frictional contact problem. Based on the governing equations, the Goodman’s and the Spence’s analytical approximations for determining the stick/slip contact boundary of orthotropic materials are derived. Meanwhile, on the basis of the governing equations in the form of the Voletrra integral equations, an efficient numerical algorithm for calculating the partial-slip contact problem is presented. For practical engineering materials, the comparisons of the analytical and numerical results show that the Spence’s approximation has enough accuracy for all 𝛾 1 when c < 0.7 and the errors of the Goodman’s approximation becomes large as 𝛾 1 increases. Furthermore, for c < 0.95, the stick/slip contact boundary is mainly determined by 𝛾 1 and A/(B𝜋𝜇), while C/A determines the magnitude of the interfacial tractions. It should note the conclusions 172

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International Journal of Mechanical Sciences 135 (2018) 168–175

Fig. 3. Variations of 𝛾 1 , C/A, B/A with the material parameters.

Fig. 4. Variation of stick region predicted by the analytical approximations and the numerical algorithm.

Fig. 5. Shear tractions during the indentation of flat-ended punch with different A/(B𝜋𝜇).

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International Journal of Mechanical Sciences 135 (2018) 168–175

0.6

1.5 μ = 0.9 μ = 0.6 μ = 0.75

0.5

1

p(x/a) P

p(x/a) P

μ = 0.6 μ = 0.75 μ = 0.9

0.4 0.5

0.3 0

0.2

0.4

0.6

0.8

0

1

0.2

0.4

0.6 x/a

x/a

0.8

1

(b) C/A = 1, γ1 = 0.25

(a) C/A = 4, γ1 = 0.25

Fig. 6. Normal tractions during the indentation of flat-ended punch with different C/A.

can be classified to be two groups, (1) 𝛼1 = −𝛼2 , 𝛽1 = 𝛽2 is positive, (2) 𝛼1 = 𝛼2 = 0, 𝛽 1 and 𝛽 2 are different positive constants. Since both the two groups can be analyzed by using the approach presented in the paper, hence we only consider the first case. At the stage, Eq. (A.6) can be rewritten as

can be extended to the indentation by a rigid punch with a polynomial profile by using the self-similar assumption. Acknowledgments This work was supported by the National Natural Science Fundation of China [51505235, 61473158]; the Natural Science Foundation of Jiangsu Province of China [BK20150844, BK20141414, BK20141430]; the Natural Science Foundation of the Jiangsu Higher Education Institutions of China [15KJB310010]; and the Nanjing University of Posts and Telecommunications Science Foundation [NY214020].

̃ = Φ

Since the stresses are bounded as |y| → ∞, hence, it easy to find that c3 and c4 should be zero for 𝜔 ≥ 0 and c1 and c2 zero for 𝜔 ≤ 0 by inspecting Eq. (A.8). To calculate the strain/stress filed distribution, the formula expressing the stresses and displacements in terms of the Airy stress function are needed. Based on Eq. (A.1), in the Fourier domain, these formula are summarized as the following,

For two-dimensional elasticity, the in-plane stresses can be expressed by Φ(x, y) as 𝜕2 Φ , 𝜕𝑦2

𝜎𝑦𝑦 =

𝜕2 Φ , 𝜕𝑥2

𝜎𝑥𝑦 = −

𝜕2 Φ 𝜕 𝑥𝜕 𝑦

𝜎̃ 𝑥𝑥 =

(A.1)

Substituting Eq. (A.1) into the compatible equation 2𝜖𝑥𝑦,𝑥𝑦 = 𝜖𝑥𝑥,𝑦𝑦 + 𝜖𝑦𝑦,𝑥𝑥 , we can get 𝑠22

𝜕4 Φ 𝜕𝑥4

+ (2𝑠12 + 𝑠66 )

𝜕4 Φ 𝜕𝑥2 𝑦2

+ 𝑠11

𝜕4 Φ 𝜕𝑦4

=0

𝑢̃ =

(A.2)

̃ ̃ 𝜕2 Φ 𝜕4 Φ + 𝑠11 =0 𝜕𝑦2 𝜕𝑦4

[

(A.3)

where the symbol tilde denotes a variable in the Fourier domain. And, the Fourier transform pairs are defined as ̃ 𝜔, 𝑦) =𝔽 (Φ(𝑥, 𝑦), 𝜔) = Φ(

∞

∫−∞

̃ 𝜔, 𝑦), 𝑥) = Φ(𝑥, 𝑦) =𝔽 −1 (Φ(

Φ(𝑥, 𝑦)𝑒−𝑖𝜔𝑥 𝑑𝑥

4 ∑ 𝑗=1

𝑐𝑗 𝑒𝑖𝜔𝑝𝑗 𝑦

) ( ̃ 1 𝑑2Φ ̃ , 𝑠11 − 𝑠12 𝜔2 Φ 2 𝑖𝜔 𝑑𝑦

𝑣̃ = 𝑠12

] [ 𝑓𝑦 −1 = 𝜔2 𝑓𝑥 𝑝1

−1 𝑝2

][ ] 𝑐1 , 𝑐2

(A.5)

𝑠11 𝑝 + (2𝑠12 + 𝑠66 )𝑝 + 𝑠22 = 0

(A.9)

̃ 𝑑Φ ̃ Φ𝑑𝑦 − 𝑠22 𝜔2 ∫ 𝑑𝑦

(A.10)

(A.11)

𝜔 ≥ 0;

with

(A.12)

Similarly, the values of c3 and c4 can be determined as (A.6)

2

̃ 𝑑Φ 𝑑𝑦

𝜔≥0

with

𝑓𝑥 + 𝑝2 𝑓𝑦 𝑓𝑥 + 𝑝1 𝑓𝑦 , 𝑐2 = − , (𝑝2 − 𝑝1 )𝜔2 (𝑝2 − 𝑝1 )𝜔2 𝑐1 = 𝑐2 = 0, with 𝜔<0

𝑐3 = 𝑐4 = 0,

where cj are coefficients only depending on 𝜔, pj are the two pairs of complex conjugate roots of the corresponding characteristic equation 4

𝜎̃ 𝑥𝑦 = 𝑖𝜔

𝑐1 =

A general solution to the ordinary differential Eq. (A.3) is ̃ = Φ

̃ 𝜎̃ 𝑦𝑦 = −𝜔2 Φ,

Obviously, c1 and c2 can be determined as

(A.4)

∞

1 ̃ 𝜔, 𝑦)𝑒𝑖𝜔𝑥 𝑑𝜔 Φ( 2𝜋 ∫−∞

̃ 𝑑2Φ , 𝑑𝑦2

For the concentrated normal and tangential forces, fy and fx , acting on the 𝑦 = 0 surface of the half plane, the following relations can be obtained from Eqs. (A.8) and (A.9)

To get a general form of Φ, applying the Fourier transform for Eq. (A.2) over the coordinate x leads to ̃ + (2𝑠12 − 𝑠66 )𝜔2 𝑠22 𝜔4 Φ

(A.8)

𝑗=1

Appendix A. Elastic responses due to concentrated forces

𝜎𝑥𝑥 =

2 ∑ (𝑐𝑗 𝑒−𝜔𝛽𝑗 𝑦 + 𝑐𝑗+2 𝑒𝜔𝛽𝑗 𝑦 )𝑒𝑖|𝜔|𝛼𝑗 𝑦

𝑐4 = −

(A.7)

To simplify the notation, these two pairs of roots are expressed as 𝑝𝑘 = 𝛼𝑘 + 𝑖𝛽𝑘 and 𝑝𝑘+1 = 𝑝̄𝑘 , 𝛼 k and 𝛽 k are real, 𝑘 = 1, 2. For orthotropy, pk

with

𝑓𝑥 + 𝑝3 𝑓𝑦 (𝑝4 − 𝑝3 )𝜔2

,

𝜔 > 0; with

𝑐3 = 𝜔≤0

𝑓𝑥 + 𝑝4 𝑓𝑦 (𝑝4 − 𝑝3 )𝜔2

, (A.13)

Then, the stresses fields due to the concentrated force can be obtained by transforming Eq. (A.9) into the physical domain, 174

J.J. Shen et al.

[

International Journal of Mechanical Sciences 135 (2018) 168–175 𝑓 +𝑝 𝑓 −𝑖 ⎤⎡ 𝑥 2 𝑦 ⎤ 𝑥+𝑝2 𝑦 ⎥⎢ (𝑝2 −𝑝1 ) ⎥ + 𝑖𝑝2 ⎥⎢ 𝑓𝑥 +𝑝1 𝑓𝑦 ⎥ 𝑥+𝑝2 𝑦 ⎦⎣− (𝑝2 −𝑝1 ) ⎦

] ⎛⎡ − 𝑖 𝜎𝑦𝑦 1 ⎜⎢ 𝑥 + 𝑝 1 𝑦 = 𝜎𝑥𝑦 2𝜋 ⎜⎢ 𝑖𝑝1 ⎝⎣ 𝑥 + 𝑝 1 𝑦

⎡ −𝑖 ⎢ 𝑥+𝑝3 𝑦 ⎢ 𝑖𝑝1 ⎣ 𝑥+𝑝3 𝑦

𝑓 +𝑝 𝑓 −𝑖 ⎤⎡ 𝑥 4 𝑦 ⎤⎞ 𝑥+𝑝4 𝑦 ⎥⎢ (𝑝4 −𝑝3 ) ⎥⎟ 𝑖𝑝4 ⎥⎢ 𝑓𝑥 +𝑝3 𝑓𝑦 ⎥⎟ 𝑥+𝑝4 𝑦 ⎦⎣− (𝑝4 −𝑝3 ) ⎦⎠

𝜓ℎ1 (𝑥) = =

(A.14) Expanding Eq. (A.14) leads to [ 2 ] 2𝛽𝑦2 1 (𝛽 + 𝛼 2 )𝑦𝑓𝑦 + 𝑥𝑓𝑥 𝜋 𝑥4 + 2(𝛽 2 − 𝛼 2 )𝑥2 𝑦2 + (𝛼 2 + 𝛽 2 )𝑦4

(A.15)

[ 2 ] 2𝛽𝑥𝑦 1 =− (𝛽 + 𝛼 2 )𝑦𝑓𝑦 + 𝑥𝑓𝑥 𝜋 𝑥4 + 2(𝛽 2 − 𝛼 2 )𝑥2 𝑦2 + (𝛼 2 + 𝛽 2 )𝑦4

(A.16)

𝜎𝑦𝑦 = − 𝜎𝑥𝑦

𝜑ℎ0 (𝑥) = = 𝜗ℎ1 (𝑥) = =

(A.18)

𝑓𝑥 𝑥

(A.19)

−𝑣′0 (𝑥) =𝐶

𝑓𝑦 𝑥

− 𝐴𝑓𝑥 𝛿(𝑥)

(A.20)

where 𝐴 = 𝑠12 + 𝑠11 (𝛼 2 + 𝛽 2 ),

𝐵=

2𝑠11 𝛽 , 𝜋

𝐶 = 𝐵 (𝛼 2 + 𝛽 2 )

(A.21)

Appendix B. Hankel transforms of relations in Eq. (39) Applying the Hankel transform of order 0 and 1 on the relations in Eq. (39) leads to ) ∞ ∞( ∞ 1 𝜙ℎ0 (𝑥) = 𝜙(𝜔)𝐽0 (𝜔𝑥)𝑑 𝜔 = 𝜎𝑦𝑦 (𝑡) cos(𝜔𝑡)𝑑 𝑡 𝐽0 (𝜔𝑥)𝑑 𝜔 ∫0 ∫0 2 ∫0 ( ∞ ) ∞ 1 = 𝜎𝑦𝑦 (𝑡) 𝐽0 (𝜔𝑥) cos(𝜔𝑡)𝑑𝜔 𝑑𝑡 ∫0 2 ∫0 𝑥 𝜎 (𝑡) 𝑦𝑦 1 = 𝑑𝑡 (B.1) √ 2 ∫0 𝑥2 − 𝑡2 ) ∞ ∞( ∞ 1 𝜙ℎ1 (𝑥) = 𝜙(𝜔)𝐽1 (𝜔𝑥)𝑑 𝜔 = 𝜎𝑦𝑦 (𝑡) cos(𝜔𝑡)𝑑 𝑡 𝐽1 (𝜔𝑥)𝑑 𝜔 ∫0 ∫0 2 ∫0 ( ∞ ) ∞ 1 = 𝜎𝑦𝑦 (𝑡) 𝐽1 (𝜔𝑥) cos(𝜔𝑡)𝑑𝜔 𝑑𝑡 ∫0 2 ∫0 ( ) ∞ ∞ 𝑡𝜎 (𝑡) 𝑦𝑦 1 = 𝜎𝑦𝑦 (𝑡)𝑑𝑡 − 𝑑𝑡 (B.2) √ ∫𝑥 2𝑥 ∫0 𝑡2 − 𝑥2 ) ∞ ∞( ∞ 1 𝜓ℎ0 (𝑥) = 𝜓(𝜔)𝐽0 (𝜔𝑥)𝑑 𝜔 = 𝜎𝑥𝑦 (𝑡) sin(𝜔𝑡)𝑑 𝑡 𝐽0 (𝜔𝑥)𝑑 𝜔 ∫0 ∫0 2 ∫0 ( ∞ ) ∞ 1 = 𝜎𝑥𝑦 (𝑡) 𝐽0 (𝜔𝑥) sin(𝜔𝑡)𝑑𝜔 𝑑𝑡 ∫0 2 ∫0 =

1 2 ∫𝑥

∞

𝜎𝑥𝑦 (𝑡) 𝑑𝑡 √ 𝑡2 − 𝑥2

∞

∫0

∞

1 2 ∫0

∞

∫0

1 2𝑥 ∫0

(

∞

∫0

𝑢′0 (𝑡)

𝜗(𝜔)𝐽1 (𝜔𝑥)𝑑𝜔 = 𝑥

𝑡𝑣′0 (𝑡) 𝑑𝑡 √ 𝑥2 − 𝑡2

) 𝜎𝑥𝑦 (𝑡) sin(𝜔𝑡)𝑑 𝑡 𝐽1 (𝜔𝑥)𝑑 𝜔 (B.4)

1 𝑑𝑡 √ 2 ∫0 𝑥2 − 𝑡2 ∞

∞

∫0

𝑡𝜎𝑥𝑦 (𝑡) 𝑑𝑡 √ 𝑥2 − 𝑡2

𝜑(𝜔)𝐽0 (𝜔𝑥)𝑑𝜔 =

(

) 𝑢′0 (𝑡) cos(𝜔𝑡)𝑑𝑡 𝐽0 (𝜔𝑥)𝑑𝜔 (B.5)

1 2 ∫0

∞

( ∫0

∞

) 𝑣′0 (𝑡) sin(𝜔𝑡)𝑑𝑡 𝐽1 (𝜔𝑥)𝑑𝜔 (B.6)

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where 𝛿(x) is the Dirac delta function and i is the imaginary unit. Applying Eq. (A.18) to Eq. (A.17), the surface displacement gradients in the physical domain can be obtained as −𝑢′0 (𝑥) =𝐴𝑓𝑦 𝛿(𝑥) + 𝐵

1 2𝑥 ∫0

𝑥

1 2 ∫0

References

where 𝑢′0 = 𝑑 𝑢(𝑥, 0)∕𝑑 𝑥, 𝑣′0 = 𝑑 𝑣(𝑥, 0)∕𝑑 𝑥, 𝜂(𝑝) = 𝑠11 𝑝2 + 𝑠12 , 𝛾(𝑝) = 𝑠12 𝑝 + 𝑠22 ∕𝑝. It is well known that the inverse Fourier transform of the Heaviside function H(𝜔) is [25] 1 𝑖𝑥

∫0

𝜓(𝜔)𝐽1 (𝜔𝑥)𝑑 𝜔 =

𝑥

Based on Eq. (A.10), the relations between the surface displacement gradients on 𝑦 = 0 and the surface stresses, which are essential for the contact analysis, can be expressed in the Fourier domain as [ ′] ([ ][ ] [ ][ ]) 𝑢̃ 0 𝜂(𝑝1 ) 𝜂(𝑝2 ) 𝑐1 𝜂(𝑝3 ) 𝜂(𝑝4 ) 𝑐3 2 = − 𝜔 + (A.17) 𝑣̃′0 𝛾(𝑝1 ) 𝛾(𝑝2 ) 𝑐2 𝛾(𝑝3 ) 𝛾(𝑝4 ) 𝑐4

𝔽 −1 (𝐻(𝜔); 𝑥) = 𝜋𝛿(𝑥) −

∞

(B.3)

175