Page 1

Journal of Applied Mechanics and Technical Physics, Vol. 43, No. 1, pp. 147–153, 2002

CHARACTERISTICS OF LOCAL COMPLIANCE

OF AN ELASTIC BODY UNDER A SMALL PUNCH INDENTED

INTO THE PLANE PART OF ITS BOUNDARY

UDC 539.3:621.833

I. I. Argatov

An asymptotic solution of the contact problem of an elastic body indented (without friction) by a

circular punch with a flat base is obtained under the assumption of a small relative size of the

contact zone. The resulting formulas involve integral characteristics of the elastic body, which depend

on its shape, dimensions, fixing conditions, Poisson’s ratio, and location of the punch center.These

quantities have the mechanical meaning of the coefficients of local compliance of the elastic body.

Relations that, generally, reduce the number of independent coefficients in the asymptotic expansion

are obtained on the basis of the reciprocal theorem. Some coefficients of local compliance at the center

of an elastic hemisphere are calculated numerically. The asymptotic model of an elastic body loaded

by a point force is discussed.

Introduction. The need for more accurate calculations of contact pressures between machine elements

stimulates the studies of contact problems of the theory of elasticity for bodies different from a half-space. The

existing solutions for an elastic layer [1, 2], elastic spatial wedge [3], and elastic truncated sphere [4, 5] are based

on the explicit representation of the Green function. Het´ enyi [6] and Sheveleva [7] developed a method of specular

reflection for constructing approximately the Green functions for elastic spatial quarter and octant. Tsvetkov and

Chebakov [8] and Aleksandrov and Pozharskii [9] developed a homogeneous-solution method for an elastic plate.

Many numerical algorithms for solving contact problems have been proposed (see, e.g., [10–13]).

Under the assumption of a small contact zone, contact problems can be solved by asymptotic methods [1–3,

5]. Argatov [14] showed that, to construct several first terms of the asymptotic expansion of the contact-pressure

intensity, it suffices to know several coefficients of the asymptotic expansions of regular components of singular

solutions with singularities corresponding to the point force (Green function), point moments, and polymoments.

In the present paper, these coefficients are interpreted as characteristics of local compliance of an elastic body.

1.Formulation of the Contact Problem for a Punch with a Flat Base.

Matrix. We consider an elastic body occupying a three-dimensional domain Ω. At the boundary of the body,

there is a site Σ, which lies in the Ox1x2plane. Let a circular punch whose center coincides with the coordinate

origin be pressed frictionlessly into Σ. We assume that the radius aεof the punch base ω(ε) is small compared to

the characteristic dimension l of the body Ω. For convenience, we set

Local-Compliance

aε= εa∗,

(1.1)

where ε is a small positive parameter and a∗is a quantity comparable with l and independent of ε. As l, we take

the radius of the largest sphere with a center at the point O that can be enclosed in the region Ω. Let the body be

fixed along the part of the boundary Γuand stress-free on Γσand Σ outside the contact region.

We denote the resultant vector and resultant moments of the system of forces acting on the punch by F3and

M1and M2, respectively. As a result of loading, the punch moves translationally for the distance δ0and rotates.

Makarov State Marine Academy, St. Petersburg 199106. Translated from Prikladnaya Mekhanika i Tekhnich-

eskaya Fizika, Vol. 43, No. 1, pp. 177–185, January–February, 2002. Original article submitted March 13, 2001.

0021-8944/02/4301-0147 $27.00 c ? 2002 Plenum Publishing Corporation

147

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The rotation is determined by the angles β1and β2. The contact-pressure intensity under the punch p satisfies the

following integral equation (see, e.g., [1, § 19]):

? ?

ω(ε)

Here G3is the vertical component of the Green vector function with a pole at the boundary point (y1,y2,0).

The unknown quantities δ0, β1, and β2are determined from the equations of equilibrium of the punch

? ?

ω(ε)ω(ε)

Remark 1. The contact pressure under the punch base should be positive. Skewness of the punch may

lead to detachment of the punch edge from the surface of the elastic body. However, it follows from the solution of

the axisymmetric contact problem for an elastic truncated sphere (see [5, § 5.2.1]) that, in the case of translational

indentation of the punch, the condition of full contact holds for reasonably small values of the parameter ε only.

Since the problem is linear, the quantities δ0, β1, and β2should be related to F3, M1, and M2by the linear

relation

Π20

Π21

Π22

The quantities Πkl(ε) characterize the compliance of the elastic body Ω under the action of the punch with a flat

base ω(ε) applied at the point O at the face of the body Σ. For small ε, the displacement of the punch is mainly

due to the local deformation of the body Ω that occurs in the neighborhood of the punch (according to terminology

of [15, § 133], in the local-perturbation region). We call matrix (1.4) the local-compliance matrix. By virtue of the

reciprocal theorem δ??

One of the goals of the present paper is to construct an asymptotic representation of the matrix Π(ε) as

ε → 0.

2. Asymptotic Modeling of Local Contact Interaction between an Elastic Body and a Punch.

We consider the expansion (see, e.g., [1; 16, § 4.14])

G3(y;x) = T3(x1− y1,x2− y2,x3) + g3(y;x),

where g3(y;x) is the projection of the regular component of the Green vector function onto the vertical axis, T3is

the part of the solution of the Boussinesq problem of an elastic half-space loaded by a unit force (see [16, § 5.11]),

where

πE

1 − ν2T3(x1− y1,x2− y2,0) =

Here E and ν are Young’s modulus and Poisson’s ratio of the material of the body Ω, respectively.

We use the following asymptotic formula (notation coincides with that in [14, 17]):

G3(y1,y2;x1,x2,0)p(y1,y2)dy1dy2= δ0+ β1x2− β2x1.

(1.2)

p(y)dy = F3,

? ?

?y2

−y1

?

p(y)dy =

?M1

M2

?

.

(1.3)

Π00

Π10

Π01

Π11

Π02

Π12

F3

M1

M2

=

δ0

β1

β2

.

(1.4)

0F?

3+ β??

1M?

1+ β??

2M?

2= δ?

0F??

3+ β?

1M??

1+ β?

2M??

2, the local-compliance matrix Π is symmetric.

(2.1)

1

?(x1− y1)2+ (x2− y2)2.

(2.2)

(πE/(1 − ν2))g3(y;x1,x2,0) = A0+ B1x1+ B2x2− A(2)

+C11x2

0y1+ A(1)

0y2

1+ 2C12x1x2+ C22x2

2− (B(2)

y2

1x1+ B(2)

2x2)y1

+ (B(1)

1x1+ B(1)

2x2)y2+ (1/2)(A(2,0)

0

1+ 2A(2,1)

0

y1y2+ A(2,2)

0

y2

2) + ... .

(2.3)

Here the dots denote the terms of order O(ε3) [in accordance with (1.1)]. It is noteworthy that, generally, the

coefficients that enter the right side of (2.3) (the method of calculating these coefficients is given in [14, 17]) depend

on the location of the point O (punch center).

Example 1. Let Ω be a layer of thickness h perfectly attached to the rigid base x3= h. In this case, the right

side of (2.3) depends only on the squared distance between the points (y1,y2) and (x1,x2). According to [1], the fol-

lowing representation in the form of an absolutely convergent power series is valid for?(x1− y1)2+ (x2− y2)2< 2h:

πE

1 − ν2g3(y;x1,x2,0) = −1

m=0

h

∞

?

am

h2m[(x1− y1)2+ (x2− y2)2]m.

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Comparing this expansion with (2.3), we infer that the only nonzero coefficients are

A0= −a0/h,C11= C22= −a1/h3,

(2.4)

A(2,0)

0

= A(2,2)

0

= −2a1/h3,B(2)

1

= −2a1/h3,B(1)

2

= 2a1/h3.

The dimensionless coefficients a0and a1as functions of ν are given in [5, Table 1.2] (see also [18]). For example,

a0= 1.3769, and a1= −0.6276 for ν = 0.3.

Substituting (2.1)–(2.3) into (1.2) and integrating, we obtain the equation

1 − ν2

πE

? ?

ω(ε)

p(y1,y2)dy1dy2

?(x1− y1)2+ (x2− y2)2= δ0+ β1x2− β2x1−˜F3A0−˜F3(B1x1+ B2x2)

2

?

−

2

?

M1, and ˜

i=1

˜

MiA(i)

0−˜F3(C11x2

1+ 2C12x1x2+ C22x2

2) −

i=1

˜

Mi(B(i)

1x1+ B(i)

2x2) −

2

?

n=0

˜

M2,nA(2,n)

0

.

(2.5)

Here˜F3and ˜

moments, respectively), which are equal to the quantities calculated from formulas (1.3) multiplied by (πE)−1(1−ν2)

and˜

M2,nare the normalized polymoments of the distributed contact pressures:

M2are the integral characteristics of the contact-pressure intensity (normalized force and

˜

M2,n=1 − ν2

πE

M2,n,Mm,n=1

2Cn

2

? ?

ω(ε)

y2−n

1

yn

2p(y)dy.

(2.6)

Equation (2.5) is a so-called “coupled” integral equation of the contact problem for a finite elastic body [14]

and is a third-approximation equation: the coefficient A0is a first-order correction for the geometry of the elastic

body, the fifth and sixth terms are second-order corrections, and the next terms are third-order corrections.

The method of reducing the integral equation (1.2) of the contact problem for an elastic layer to an ap-

proximate equation by polynomial approximation of the regular component of the integral-operator kernel was

proposed in [19] (see also [1, § 54]). The properties of the solutions of these equations are discussed in [5, § 1.2].

The term “coupled” means that, after a solution (with undetermined coefficients) of the integral equation of the

contact problem for an elastic half-space is constructed, Eq. (2.5) reduces to a system of linear algebraic equations.

Aleksandrov and Shmatkova [18] applied this method to the problem of a parabolic punch pressed into an elastic

layer. Argatov [17] obtained an asymptotic solution of the corresponding nonlinear resulting problem.

3. Solution of the Coupled Equation. Using the results of [20–22], we write the solution of Eq. (2.5) in

the form

p(x1,x2) =

E

π(1 − ν2)

1

√

a2− x2

1− x2

2

?

δ0−˜F3A0−

2

?

i=1

˜

MiA(i)

0−

2

?

n=0

˜

M2,nA(2,n)

0

+˜F3(C11+ C22)a2− 2

?

β2+˜F3B1+

2

?

i=1

˜

MiB(i)

1

?

x1+ 2

?

β1−˜F3B2−

2

?

i=1

˜

MiB(i)

2

?

x2

−2

3

˜F3(5C11+ C22)x2

1−16

3

˜F3C12x1x2−2

3

˜F3(C11+ 5C22)x2

2

?

.

(3.1)

The quantities˜F3,˜

Mi, and˜

M2,nshould be related to δ0, β1, and β2. Integration of (3.1) yields

˜F3= c

?

δ0−˜F3A0−

2

?

i=1

˜

MiA(i)

0−

2

?

n=0

˜

M2,nA(2,n)

0

−1

3(C11+ C22)a2˜F3

?

,

(3.2)

where c = 2π−1a is the translational capacity of the circular punch of base radius a (dependence on the parameter

ε is not indicated).

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Calculating the moments of the contact-pressure intensity (3.1) [see the second formula in (1.3)], we obtain

˜

M1= m

?

β1−˜F3B2−

2

?

i=1

˜

MiB(i)

2

?

,

˜

M2= m

?

β2+˜F3B1+

2

?

i=1

˜

MiB(i)

1

?

,

(3.3)

where m = 4(3π)−1a3is the rotational capacity of the circular punch.

Finally, inserting (3.1) into (2.6), we find that

˜

M2,0=a3

3π

?

δ0+˜F3

?

A0−a2

15(17C11+ C22)

?

−

2

?

i=1

˜

MiA(i)

0−

2

?

n=0

˜

M2,nA(2,n)

0

?

;(3.4)

˜

M2,1= −32a5

45π

˜F3C12; (3.5)

˜

M2,2=a3

3π

?

δ0+˜F3

?

A0−a2

15(C11+ 17C22)

?

−

2

?

i=1

˜

MiA(i)

0−

2

?

n=0

˜

M2,nA(2,n)

0

?

.

(3.6)

With allowance for (3.5), one can express the quantities ˜

determinant of this system is equal to 1+(3π)−1a3(A(2,0)

However, the exact solution is not necessarily required, since formulas (3.1)–(3.6) themselves are approximate.

4. Asymptotic Representation of the Local-Compliance Matrix. We now perform an asymptotic

analysis of relations (3.4)–(3.6) (see also [17]). Let the quantities δ0, β1, and β2 be fixed (independent of the

parameter ε). Then, from (3.2) follows the expansion˜F3= ε˜F0

formula (3.1) is derived by retaining terms of order ε3compared to unity [see (2.5), where√

in accordance with the accuracy of Eq. (2.5), we confine ourselves to the approximation

˜

M2,n?˜

Substitution of (4.1) into (3.2) yields

M2,0 and ˜

) and does not vanish for reasonably small ε = a/l.

M2,2 from Eqs. (3.4) and (3.6). The

0

+A(2,2)

0

3+ ε2˜F1

3+ ... . It should be borne in mind that

x2

1+ x2

2< εa∗]. Thus,

M0

2,n,

˜

M0

2,0=˜

M0

2,2= a3δ0/(3π),

˜

M0

2,1= 0.

(4.1)

?1

c+ A0+a2

3(C11+ C22)

?˜F3+

2

?

i=1

˜

MiA(i)

0 =

?

1 −a3

3π(A(2,0)

0

+ A(2,2)

0

)

?

δ0.

Using similar reasoning to that used in deriving relation (4.1), we replace the last formula by the following one:

?1

Finally, Eq. (3.3) becomes

c+ A0+a2

3(C11+ C22) +

a3

3πc(A(2,0)

0

+ A(2,2)

0

)

?˜F3+ A(1)

0

˜

M1+ A(2)

0

˜

M2= δ0.

(4.2)

B2˜F3+ (1/m + B(1)

2)˜

M1+ B(2)

2

˜

M2= β1,

−B1˜F3− B(1)

1

˜

M1+ (1/m − B(2)

1)˜

M2= β2.

(4.3)

Thus, the force F3and the moments M1and M2are related to the displacements of the punch δ0and its angles

of rotation β1and β2by the approximate equations (4.2) and (4.3). Comparing (4.2) and (4.3) with (1.4), we obtain

the asymptotic formulas for the normalized components of the local-compliance matrix˜Πlk= πE(1 − ν2)−1Πkl:

˜Π00? 1/c + A0+ a2(C11+ C22)/3 + a2(A(2,0)

˜Π10? B2,

˜Π20? −B1,

Formulas (4.4) relate the components of the matrix Π to the capacity characteristics of the punch c and m (which

depend only on the geometry of the punch base) and the coefficients in the asymptotic formula (2.3). The latter

coefficients depend on the shape and size of the elastic body Ω, its fixing conditions, location of the point O, and

Poisson’s ratio.

Since the matrix Π is symmetric, we have

0

+ A(2,2)

0

)/6,

˜Π01? A(1)

˜Π12? B(2)

˜Π22? 1/m − B(2)

0,

˜Π02? A(2)

0,

˜Π11? 1/m + B(1)

˜Π21? −B(1)

2,

2,

(4.4)

1,

1.

A(1)

0

= B2,A(2)

0

= −B1,B(1)

1

= −B(2)

2.

(4.5)

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The validity of these equalities can be verified by using the Betti formula and coefficients A(1)

in [14, 17] as coefficients in asymptotic formulas of the type (2.3) for certain singular solutions. However, the

reciprocity relations (4.5) and similar relations can easily be obtained directly from (2.3) with allowance for the

following equality implied by the Betty theorem (see, e.g., [23, Chap. 4, § 3.1]):

g3(y1,y2;x1,x2,0) = g3(x1,x2;y1,y2,0).

0,...,B(2)

2

determined

(4.6)

Substituting expansion (2.3) into (4.6), we obtain (4.5) and the equalities

A(2,0)

0

= 2C11,A(2,1)

0

= 2C12,A(2,2)

0

= 2C22.

(4.7)

Thus, in the general case, the reciprocity relations (4.5) and (4.7) reduce the number of different coefficients in the

asymptotic solution constructed above.

Example 2. For a layer (see Example 1), formulas (2.4), (4.2), and (4.3) can be combined to give

?

Expansion of expression (4.8) into a series in powers of the parameter ε = a/h yields

2E

1 − ν2aδ0

4E

3(1 − ν2)a3βi

Formulas (4.9) coincide with formulas (48.2) and (50.2) in [1].

5. Calculation of the Coefficients A0and C11for the Center of an Elastic Hemisphere. Let the

elastic body Ω be shaped like a hemisphere of radius l and fixed over the spherical part of the boundary Γu. We

use the Bubnov–Galerkin method to construct an approximate solution of the problem for a unit force applied to

the center of the cut Σ.

Since the problem of determining the vector function g(0;x) [see (2.1)] is axisymmetric, one can use the

general solution of the Lam´ e equations in cylindrical coordinates r and z, expressed in terms of two harmonic

functions Φ1and Φ2in the Weber form (see [24, § 12]):

gr=1 − ν2

πE ∂r∂z

˜F3=2aδ0

π

1 −2a0

π

ε −8a1

3πε3?−1

,

˜

Mi=4a3βi

3π

?

1 +8a1

3πε3?−1

(i = 1,2).

(4.8)

F3=

?

1 +2a0

π

ε +

?2a0

π

?2

ε2+

??2a0

?

π

?3

+8a1

3π

?

ε3+

??2a0

π

?4

+32a0a1

3π2

?

ε4+ O(ε5)

?

,

(4.9)

Mi=1 −8a1

3πε3+ O(ε5)

?

(i = 1,2).

∂

?

σzz=1

Φ1+ z∂Φ2

+ 2(1 − ν)Φ2

?

,gz=1 − ν2

πE

∂

∂z

?

Φ1+ z∂Φ2

∂z

− 2(1 − ν)Φ2

?

,

(5.1)

r

∂

∂r

?

r∂Φ

∂r

?

,τrz= −∂2Φ

∂r∂z,

Φ = Φ1+ z∂Φ2

∂z.

As stress functions, we use the homogeneous harmonic polynomials

Φn

i= cn

iρnPn(cosθ)(i = 1,2),ρ =

?

r2+ z2,

cosθ = z/

?

1for odd n.

r2+ z2

(5.2)

(Pnis the Legendre polynomial). Substituting expressions (5.1) and (5.2) into the conditions σzz= 0 and τrz= 0

for z = 0, we eliminate one of the coefficients cn

The homogeneous vector polynomial of the nth degree, which satisfies the homogeneous Lam´ e equations in

the half-space z > 0 and the condition that the stresses vanish at the boundary z = 0, has the components

˜Vn

1or cn

2: cn

1= 0 for even n or cn

2= −cn

r= −(1 − α)(1/r)Φn+1+ [2 − (1 + α)(n + 1)](z/r)Φn+ (1 + α)n(z2/r)Φn−1,

˜Vn

(5.3)

z= 2Φn− (1 + α)nzΦn−1,α ≡ ν(1 − ν)−1

for even n and

˜Vn

r= (2/r)Φn+1+ [(1 + α)n − 2](z/r)Φn− (1 + α)n(z2/r)Φn−1,

˜Vn

for odd n. The first three vectors˜ V

(5.4)

z= −(1 − α)Φn+ (1 + α)nzΦn−1

ncalculated by formulas (5.3) and (5.4) have the form

˜ V

0= ez,

˜ V

1= rer− 2αzez,

˜ V

2= −2zrer+ (r2+ 2αz2)ez.

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Page 6

TABLE 1

νA0

C11= C22

0.2

0.25

0.3

0.35

0.4

−1.5442

−1.6027

−1.6899

−1.8214

−2.0236

0.5943

0.6642

0.7585

0.8891

1.0763

Thus, the cylindrical components of the regular part of the Green vector function G(0;x) satisfy the boundary

conditions

πE

1 − ν2gr

πE

1 − ν2gz

for ρ = l and 0 ? θ ? π/2. We use the following approximation of the vector function g(0;x):

?

In this case, the unknown quantities are calculated by the formulas

???

Γu= −

1

2(1 − ν)

???

1

l

?

sinθcosθ − (1 − 2ν)

?

sinθ

1 + cosθ

?

?

,

(5.5)

Γu= −1

l

1 +

1

2(1 − ν)cos2θ

vN(r,z) =1 − ν2

πE

N

n=0

cn

ln+1˜ V

n(r,z).

(5.6)

A0= c0/l,C11= C22= c2/l3.

To determine the coefficients c0, c1, ..., and cN, we obtain a system of N + 1 linear algebraic equations using the

condition that the discrepancy in the boundary conditions (5.5) due to approximation (5.6) is orthogonal to each

vector˜ V

were performed for N = 2–17. It should be noted that the relative error in determining A0is as small as 2% even

for N = 2. To verify the results obtained, calculations by the method of boundary collocation for equidistant nodes

were also performed.

Conclusions. To refine the asymptotic formula (2.3), it is necessary to use additional parameters that

characterize the geometry of the elastic body. All local-compliance coefficients can be obtained by the complete

asymptotic expansion.Practically, it is possible to obtain explicitly only a few first terms of the asymptotic

representation of the contact pressure (see, e.g., [1, 5]).

By virtue of the reciprocity relations (4.5) and (4.7), expansion (2.3) is simplified:

0,˜ V

1, ..., and˜ V

Nover the hemisphere. The calculation results are listed in Table 1. The calculations

(πE/(1 − ν2))g3(y;x1,x2,0) = A0+ Bixi+ Biyi+ Cijxixj+ bijxiyj+ Cijyiyj+ ... .

Here b11= −B(2)

that the linear and quadratic forms appearing in this expression are invariant with respect to rotation of the

coordinate axes implies that, in passing to a new coordinate system, the quantities Biand Cij and bij should be

transformed as a vector and tensors, respectively.

If the body Ω and the parts of the boundary Γuand Γσare symmetric, then b12= 0 and b11= b22at the

point O. However, the last coefficients are determined by solving the corresponding nonaxisymmetric problem.

Formulas (1.4) and (4.4) can be considered as an asymptotic model of an elastic body under point loads

(forces and moments). It is well known (see, e.g., [25; 10, Chap. 10, § 1]) that the singular solution G(0,x) is

meaningless in the neighborhood of the point at which the force is applied. In particular, the displacement at this

point is unbounded, whereas formula (1.4) relates the force F3e3to the generalized displacement δ0e3(compare

with Example 1 in [26, § 8.9]).

It is also known (see [27, Chap. 7, § 21]) that the real distribution law of local loads is difficult to determine,

whereas the resultant force vector is usually known with a high degree of accuracy. A priori knowledge of the

distribution law of local loads is necessary (see [28, p. 301]) if, in calculations, they are reduced to point forces by

passing to the limit (see [29] and [28, Chap. 3, § 6]). At the same time, to construct an asymptotic model of a

point force, it is necessary to solve the coupled integral equation of the contact problem for an elastic body of finite

dimensions and thereby to determine approximately the pressure under the base of a small punch.

The author is grateful to S. A. Nazarov for useful discussions.

152

1, b12= b21= B(1)

1, and b22= B(1)

2; summation is performed over repeated indices. The condition

Page 7

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