Three-dimensional contact problem for a two-layered extra loaded elastic base

Introduction. Bodies with coverings represent a widespread class of materials, so their study has the great theoretical and practical significance. L. A. Galin was probably the first who has considered the contact problem for a half-space with an additional concentrated force applied outside the contact area [1]. A similar contact problem for a two-layered elastic base was investigated earlier [2] without additional force. The Galanov’s method used below to take the additional concentrated force into account allows us to estimate the influence of the extra force onto the contact pressure as well as onto the force applied of the punch. This problem is of interest for the contact mechanics of the bodies with coverings. Statement of problem. Consider the contact problem of the indentation of a punch into an elastic layer of the thickness h completely attached to an elastic half-space. The punch is acted on by a force P. The shape of the punch foot is described by the function f(x, y). The elastic characteristics of the layer and half-space are G1, ν1 and G2, ν2, respectively. For simplicity, we assume that the punch has the form of an elliptic paraboloid, i. e.

^f ( x , У ) =

x2

— +.

2R12

The force P is applied to the punch so that the punch penetrates without rotation by a depth δ. Assume that an additional normal concentrated force Q is applied at the point x = c , y = 0.

By means of the methods of the operational calculus [2], one can derive the following integral equation with respect to the normal contact pressure q(x, y ):

JJ q ( ^,q ) K | R | dd = 2п Л 9 1 ( 5 - f ( x , y ) ) - QK | R 4 ( x , y )_e Q,              (1)

я           V h ⁾                                  к h )

”

K ( t ) = J N ( u ) J 0 ( ut ) du , R = ^( x - £ ) ² + ( y - n ) 2 , R 1 =V ( X - c ) 2 + y 2 ,              (2)

N           M + 4u exp ( - 2u ) - L exp ( - 4u )

^{( U} ) = M - ( 1 + 4 u ² + LM ) exp ( - 2 u ) + L exp ( - 4 u ) ,

The research was supported by the Russian Foundation for Basic Research (12-01-00065).

Physical and mathematical sciences gк + g , gк, - Gк „    Gn       , „   , и

M =    --- 4 L =    --- 2^, o„ = ——, K„ = 3 - 4v„ ( n = 1,2).

G ₁ - G ₂ ,       G ₁k₂ + G 2      ⁿ 1 - v _n    ^{n n}

Here, Ω is an unknown contact area.

To determine the connection between the force P and the settlement δ one can use the integral condition of equilibrium of the punch

JJ q ( x , y ) dxdy = P .

Ω

The Galanov’s method. To solve the contact problem and equation (1) we use the method of nonlinear boundary integral equations based on the theorems discussed in [2, 3]. Let the contact area be a priori included into the rectangle

5 ={| x | <  a, |y | <  b } , b >  a .

We introduce the following dimensionless notation:

b 2 R 1

xyR b=x- b=y; b=R;

= a b = в qty

2 R ₂        2πθ₁

R 1

b " R¹ ,

δ ha

T = ⁵ , T = ^A, T = ^£ , bbb

c b c,

= q ( ^x' y > , 2Пё 1 Ь 2 = ^P'

Q

2πθ₁ b ²

= Q', etc.

We will omit the primes. The dimensionless parameter λ characterizes the relative thickness of the elastic layer.

To calculate the function K(t) in formula (2) one should extract its principal term by using the integral

”                    1

J J 0 ( ut ) du = -.

0 t

The elastic materials chosen for calculations and their elastic parameters (Young's modulus E and Pois son's ratio v) are presented in Table 1. The shear modulus is given by formula G = —E—-.

² ( ^{1 + v} )

Table 1

Elastic parameters

Modulus Material	Е хЮ-4 (MPa)	ν
Carbon steel	20	0.28
Cold-drawn brass	9	0.35
Concrete	2	0.17

The two following cases were taken for the calculations: steel on brass (case A) and steel on concrete (case B). The values of the contact pressure q_Q = q(0, 0) and of the force P applied to the punch are tabulated in Table 2 for δ = A = B = ε = 1.

Table 2

Values of pressure q 0 and force P

λ	Q	c	q 0	P	q 0	P
			Case A		Case B
1	0.1	1.5	0.229	0.163	0.127	0.0322
0.5	0.1	1.5	0.191	0.137	0.0631	0.0167
1	0.05	1.5	0.236	0.183	0.159	0.0547
0.5	0.05	1.5	0.194	0.154	0.112	0.0343
1	0.1	1.2	0.226	0.156	0.111	0.0267
0.5	0.1	1.2	0.188	0.129	0.0145	0.0102
1	0.05	1.2	0.235	0.179	0.155	0.0542
0.5	0.05	1.2	0.193	0.149	0.104	0.0304

As one can see from Table 2, for case A the force P and the pressure q 0 are greater than those in case B. As the value of λ decreases, the force P and the pressure drop because the top steel layer becomes thinner. As the force Q increases and also as the value of c decreases (the force Q goes to the contact area), the force P and the pressure will drop because of the force Q input becomes significant. Conclusion. The obtained results have a clear mechanical sense helping us to estimate the extra force influence onto the typical contact characteristics. The Galanov’s method is quite effective for the threedimensional elastic contact analysis in the presence of an additional concentrated force applied outside the contact area.