A Comparison of General Solutions to the Non-Axisymmetric Frictionless Contact Problem with a Circular Area of Contact: When the Symmetry Does Not Matter [original]

Citation: Argatov, I. A Comparison

of General Solutions to the

Non-Axisymmetric Frictionless

Contact Problem with a Circular Area

of Contact: When the Symmetry

Does Not Matter. Symmetry 2022,14,

1083. https://doi.org/10.3390/

sym14061083

Academic Editor: Jan Awrejcewicz

Received: 16 April 2022

Accepted: 17 May 2022

Published: 25 May 2022

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symmetry

Article

A Comparison of General Solutions to the Non-Axisymmetric

Frictionless Contact Problem with a Circular Area of Contact:

When the Symmetry Does Not Matter

Ivan Argatov

Institut für Mechanik, Technische Universität Berlin, 10623 Berlin, Germany; [email protected]

Abstract:

The non-axisymmetric problem of frictionless contact between an isotropic elastic half-

space and a cylindrical punch with an arbitrarily shaped base is considered. The contact problem

is formulated as a two-dimensional Fredholm integral equation of the first type in a fixed circular

domain with the right-hand side being representable in the form of a Fourier series. A number of

general solutions of the contact problem, which were published in the literature, are discussed. Based

on the Galin–Mossakovskii general solution, new formulas are derived for the particular value of

the contact pressure at the contact center and the contact stress-intensity factor at the contour of the

contact area. Since the named general solution does not employ the operation of differentiation of a

double integral with respect to the coordinates that enter it as parameters, the form of the general

solution derived by Mossakovskii as a generalization of Galin’s solution for the special case, when

the contact pressure beneath the indenter is bounded, is recommended for use as the most simple

closed-form general solution of the non-axisymmetric Boussinesq contact problem.

Keywords:

contact problem; non-axisymmetric; circular contact; frictionless contact; general solution;

closed-form solution; series solution; cylindrical punch; contact stress-intensity factor

1. Introduction

The contact mechanics dates back to Hertz (1882), who developed the theory of

unilateral frictionless local contact for two elastic bodies, which in the unloaded state are

shaped as elliptic paraboloids in the vicinity of the point of initial contact, and Boussinesq

(1885), who solved the problem of contact between an elastic half-space and a frictionless

flat-ended cylindrical punch. Since then, particular progress has been made with regard to

solving the axisymmetric frictionless problems with a circular area of contact.

While in the literature, the general solution of the axisymmetric problem is usually

associated with Sneddon’s paper [

] of 1965, different authors give the priority to other

studies. In particular, in his comprehensive review, Borodich [

] highlighted the contribu-

tion made by Galin (1946); Barber [

] in his book referred to the general solution as the

Green and Collins solution; in their historical note [

] (see also [

]), Popova and Popov,

acknowledging the contributions made by Galin and Sneddon, put under a spotlight

the original paper [

] written in German by Schubert in 1942. However, for the sake of

historical truth, it should be underlined that (to the best of the author’s knowledge) the

priority of solving the axisymmetric frictionless contact problem with a circular contact

area belongs to Leonov’s paper [

] published in Russian in 1939. As it was shown by

Argatov and Dmitriev [

], other forms of the general solution follow from Leonov’s results

by the simple change of integration variables. As a compromise, Argatov and Mishuris [

]

suggested to call the general solution of the axisymmetric frictionless contact problem the

Galin–Sneddon solution.

The general solution of the non-axisymmetric contact problem for a cylindrical in-

denter is of great importance in developing the contact stiffness indentation tomography

technique [

]. Another example of the application of the general solution is given by

Symmetry 2022,14, 1083. https://doi.org/10.3390/sym14061083 https://www.mdpi.com/journal/symmetry

Symmetry 2022,14, 1083 2 of 12

the problem of adhesive contact under non-symmetric perturbation of the contact geome-

try [

]. Generally speaking, the general solutions collected below will be useful in solving

the frictionless contact problems with a circular area of contact (e.g., with applications in

geotechnics [13,14]), when the symmetry of the contact geometry does not matter.

Whereas axisymmetric contact problems are considered in many publications, includ-

ing textbooks [

] and handbook [

], the situation with the non-axisymmetric contact

problem with a circular area of contact is not so equivocal, even in spite of the fact that this

problem is a direct generalization of the Boussinesq problem for a cylindrical punch with a

non-flat base. This paper aims to bridge this gap by comparing different general solutions

published in the literature.

The main motivation for writing this reviewer paper was to identify in a sense the

simplest closed form of the general solution. Another quite utilitarian motivation was to

collect in one compendium the practically useful results, some of which are not readily

accessible. Herein, we compare only the solutions collected from the literature, and the

discussion of the methods of their derivations falls outside the scope of the present study.

The recent paper [

] on solving Keer’s indentation problem for a cylindrical indenter with

the face in a wedge form can be regarded as a case study for the use of the general solutions.

2. General Solutions of the Frictionless Non-Axisymmetric Contact Problem

2.1. The Boussinesq Contact Problem Formulation

We consider the so-called Boussinesq contact problem for an isotropic elastic half-

space (see Figure 1), which is indented by a frictionless cylindrical punch of radius

with a

non-flat base described by a continuous shape function,

Φ(r

ϕ)

. For the sake of simplicity

we assume that the center of cylindrical coordinates

is taken at the center of the

circular area of contact, and the elastic semi-infinite body occupies the half-space z≥0.

u(r,φ)

(r,φ)

zΦ

zℝ3

Figure 1.

A schematic of the non-axisymmetric Boussinesq contact problem with a circular area

of contact.

Let us assume that the shape function

Φ(r

ϕ)

admits the Fourier series representation

Φ(r,ϕ) = Φ0(r) +

∞

∑

n=1

Φs

n(r)sin nϕ+Φc

n(r)cos nϕ, (1)

where

Φ0(r) = 1

2π

Φ(r,ϕ)dϕ,Φs

n(r)

Φc

n(r)=1

2π

Φ(r,ϕ)sin nϕ

cos nϕdϕ. (2)

Since the function

Φ(r

ϕ)

is assumed to be continuous, it can be shown that

Φs

) =

Φc

n(0) = 0. In addition, we put

Φ0(0) = 0. (3)

Symmetry 2022,14, 1083 3 of 12

Further, let

p(r

ϕ)

denote the contact pressure exerted by the punch under the action

of an external load, F. Then, the condition of static equilibrium implies that

2π

p(r,ϕ)rdrdϕ. (4)

According to the Boussinesq solution of the problem of normal loading of an elastic

half-space, the contact pressure

p(r

ϕ)

produces the following normal surface displace-

ment field:

uz(r,ϕ) = 1

πE∗

2π

p(ρ,ϕ)ρdρdψ

pr2+ρ2−2rρcos(ϕ−ψ). (5)

Here,

E∗=E/(

−ν2)

is the reduced elastic modulus,

and

are Young’s modulus

and Poisson’s ratio of the elastic semi-infinite body,

ϕ)

are coordinates of the point of

observation, and (ρ,ϕ)are coordinates of the point of integration.

Inside the contact area, the normal surface displacements are determined by the shape

of the punch, which under the applied load receives some vertical (normal) displacement,

δ0, such that

uz(r,ϕ) = δ0−Φ(r,ϕ),r∈[0, a],ϕ∈[0, 2π). (6)

Thus, from (5) and (6), it follows that

πE∗

2π

p(ρ,ϕ)ρdρdψ

pr2+ρ2−2rρcos(ϕ−ψ)=δ0−Φ(r,ϕ), (7)

where r∈[0, a]and ϕ∈[0, 2π).

The integral Equation

(7)

is called the governing integral equation of the Boussinesq

contact problem with the circular contact area of radius

. It is to emphasize that in contrast

to the Hertz contact problem, where the contact area is determined from the condition of

vanishing of the contact pressure on the contour of the contact area, in the Boussinesq con-

tact problem, the contact area is assumed to be a priori fixed. Correspondingly, the contact

pressure density

p(r

ϕ)

, which solves Equation

(7)

, may possess a singularity at the contact

contour, as the point of observation (r,ϕ)approaches the contact contour, when r→a.

To simplify formulas, we introduce the auxiliary notation

f(r,ϕ) = δ0−Φ(r,ϕ)(8)

=f0(r) +

∞

∑

n=1

n(r)sin nϕ+fc

n(r)cos nϕ, (9)

where, in view of (1), we have

f0(r) = δ0−Φ0(r),fs

n(r) = −Φs

n(r),fc

n(r) = −Φc

n(r). (10)

We note that a rigid body, by means of which external loads are transferred to the

surface of an elastic body, is usually called an indenter. Here we prefer the term ‘punch’,

as the term ‘indenter’ (in many cases) assumes unilateral contact, when only positive

contact pressures are allowed inside the contact area.

Finally, we note that, in view of

(3)

, the parameter

δ0

has the exact meaning of the

normal surface displacement at the center of the contact area. If

Φ(r

ϕ)≥

0 for any

r∈[

and

ϕ∈[

0, 2

π)

, then in the unloaded state, the punch touches the surface of the

elastic half-space at the center of the coordinates, and therefore, the parameter

δ0

can be

interpreted as the displacement of the punch under the applied load

. Provided that the

punch shape function

Φ(r

ϕ)

is known, the equilibrium Equation

(4)

establishes a relation

between the contact force Fand the punch displacement δ0.

Symmetry 2022,14, 1083 4 of 12

Remark 1.

It is pertinent to note here that to have an axisymmetric displacement in the Boussinesq

problem, the indenter shape should be axisymmetric. However, an axisymmetric shape of the indenter

in unilateral contact does not necessarily imply that the established axisymmetric contact region

is circular (see, for example, [

]), where indenters of toroidal-type shapes produce an annular

contact region, or [

], where the contact region under a non-convex parametric–homogeneous

punch is composed from a central circular part and a number of concentric annular regions). On the

other hand, the Boussinesq contact problem with a circular area of contact will be non-axisymmetric

if the indenter shape function

Φ(r

ϕ)

essentially depends on the angular coordinate

, that is, if the

Fourier series (9)contains at least one nontrivial term starting from n =1.

2.2. Copson’s Series Solution

In view of (1), the general solution of Equation (7) can be represented in the form

p(r,ϕ) = p0(r) +

∞

∑

n=1

n(r)sin nϕ+pc

n(r)cos nϕ. (11)

According to Copson (1947), the coefficients of the Fourier series

(11)

are determined

by the following formulas [20]:

pn(r) = −E∗

πrn−1d

gn(ρ)ρdρ

pρ2−r2, (12)

where

gn(ρ) = 1

ρ2n

dρ

rn+1fn(r)

pρ2−r2dr. (13)

To be more precise, the coefficients

n(r)

and

n(r)

are given by Formulas

(12) and (13)

upon replacing fn(r)with fs

n(r)and fc

n(r), respectively.

We note that in the case

0, Formulas

(12)

and

(13)

represent the Galin–Sneddon so-

lution of the axisymmetric contact problem. The general solution of the non-axisymmetric

contact problem in the series from

(12)

and

(13)

was also independently derived by

Mossakovskii [21].

2.3. Mossakovskii’s Series Solution

Under the assumption that the functions

f0(r)

n(r)

, and

n(r)

are continuously

differentiable in the interval

(

, Mossakovskii (1953) simplified Formulas

(12)

and

(13)

as follows [21]:

pn(r) = E∗

2(Cnrn

√a2−r2−2rn

x−2ndx

√x2−r2

f00

n(ρ)ρn+1+f0

n(ρ)ρn−n2fn(ρ)ρn−1

px2−ρ2dρ). (14)

Here, the constant Cnis given by the formulas

C0=2

π f0(0) + a

0(ρ)dρ

pa2−ρ2!, (15)

Cn=2

πa1−2n

n(ρ)ρn+n fn(ρ)ρn−1

pa2−ρ2dρ,n=1, 2, . . . (16)

We note that, in light of (3) and (8)–(10), we have f0(0) = δ0.

Symmetry 2022,14, 1083 5 of 12

2.4. Leonov’s Closed-Form Solution

Let ∆denote the two-dimensional Laplace differential operator, that is

∆u(r,ϕ) = ∂2u

∂r2+1

∂u

∂r+1

∂2u

∂ϕ2. (17)

The general solution to the governing integral Equation

(7)

in a closed form was

first obtained by Leonov (1955). To simplify the writing of his formula, we introduce

the notation

R=qr2+ρ2−2rρcos(ϕ−ψ). (18)

By the definition,

equals the distance between the point of observation

ϕ)

and

the point of integration (ρ,ψ).

So, according to Leonov, the general solution is given by the following formula [22]:

p(r,ϕ) = −E∗

2π2





2∆

2π

f(ρ,ψ)

Rρdρdψ

2π

f(ρ,ψ)"1

R3arctan √a2−r2pa2−ρ2

aR −π

2

R2pa2−ρ2√a2−r2#ρdρdψ





. (19)

We note that arctan(x)−π/2 =−arctan(1/x).

2.5. Mossakovskii’s Form of the General Solution

Starting from the series solution

(11)

–

(13)

, Mossakovskii (1953) derived the following

general solution in the following form [23]:

p(r,ϕ) = −E∗

2π2





∆

2π

f(ρ,ψ)arctan√a2−r2pa2−ρ2

aR ρdρdψ

2π

a f (ρ,ψ)

p(a2−r2)3(a2−ρ2)

(a4−ρ2r2)ρdρdψ

a4−2a2rρcos(ϕ−ψ) + r2ρ2





. (20)

Here the same notation is used as introduced by Formulas (17) and (18).

We note that arctan(x)−π/2 =−arctan(1/x).

2.6. The Galin–Mossakovskii General Solution

We recall that the zeroth term of the Fourier series (9) is defined by the formula

f0(r) = 1

2π

f(r,ψ)dψ. (21)

As a generalization of the general solution obtained by Galin [

], for the special case,

when the contact pressure beneath the indenter is bounded, Mossakovskii (1953) derived

the following formula [23]:

Symmetry 2022,14, 1083 6 of 12

p(r,ϕ) = E∗

f0(0)

√a2−r2

+E∗

2π2

√a2−r2

2π

(a4−ρ2r2)∂f

∂ρ (ρ,ψ) + 2a2rsin(ϕ−ψ)∂f

∂ψ (ρ,ψ)

pa2−ρ2a4−2a2rρcos(ϕ−ψ) + r2ρ2dρdψ

−E∗

2π2

2π

∆f(ρ,ψ)arctan√a2−r2pa2−ρ2

aR ρdρdψ

R.(22)

We note that, in view of

(21)

f0(r)

gives the average value of the function

f(r

ϕ)

a circumference of radius

. That is why, if

f(r

ϕ)

is a continuous function, then

f0(

)

coincides with the limit of f(r,ϕ)as r→0.

2.7. Fabrikant’s General Solutions

Let us introduce the notation

L(k)g(ϕ) = 1

2π

λ(k,ϕ−τ)g(τ)dτ, (23)

where

λ(k,τ) = 1−k2

1+k2−2kcos τ. (24)

We note that Formula

(23)

defines the

-operator [

] that acts on the function

g(ϕ)

defined on a unit circle.

Moreover, we put

η=1

a(a2−r2)1/2(a2−ρ2)1/2. (25)

According to Fabrikant (1986), the general solution to the governing integral equa-

tion of the contact problem under consideration

(7)

, in view of the notation

(8)

, can be

represented in the following closed form [25]:

p(r,ϕ) = −E∗

rL(r)d

xdx

(x2−r2)1/2 L1

x2d

ρdρ

(x2−ρ2)1/2 L(ρ)f(ρ,ϕ). (26)

Another form of the Fabrikant solution is given by the following formula:

p(r,ϕ) = E∗

(a2−r2)1/2

∂

∂a

ρdρ

(a2−ρ2)1/2 Lrρ

a2f(ρ,ϕ)

−E∗

2π2

2π

Rarctanη

R∆f(ρ,ψ)ρdρdψ. (27)

Yet, another form of the Fabrikant solution is given

p(r,ϕ) = E∗

πa

√a2−r2

ρdρ

pa2−ρ2

dρρLrρ

a2f(ρ,ϕ)

−E∗

2π2

2π

dψ

Rarctan√a2−r2pa2−ρ2

aR ∆f(ρ,ψ)ρdρ. (28)

Symmetry 2022,14, 1083 7 of 12

It is pertinent to note here that in terms of the

operator, the Mossakovskii solu-

tion (20) can be represented as follows [26]:

p(r,ϕ) = −E∗

π



−∆

(x2−r2)1/2

ρdρ

(x2−ρ2)1/2 Lrρ

x2f(ρ,ϕ)

(a2−r2)3/2

ρdρ

(a2−ρ2)1/2 Lrρ

a2f(ρ,ψ)





. (29)

We also note that, in view of

(25)

, the last terms on the right-hand sides of Formulas

(22)

and (27) coincide.

3. Contact Pressure at the Center of Circular Contact

For the Fourier series representation (21), it follows that

lim

r→0p(r,ϕ) = p0(0). (30)

When comparing the series solutions due to Copson

(12)

(13)

and Mossakovskii

(14)

(15)

it is readily seen that only the Mossakovskii series solution allows to evaluate directly the

right-hand side of Equation (30).

From Equation (14), it follows that

p0(r) = E∗

2(C0

√a2−r2−2

√x2−r2

f00

0(ρ)ρ+f0

0(ρ)

px2−ρ2dρ),

so that

p0(0) = E∗C0

2a−E∗

f00

0(ρ)ρ+f0

0(ρ)

px2−ρ2dρ). (31)

By changing the order of integration, we easily transform Formula (31) as follows:

p0(0) = E∗C0

2a−E∗

0π

2−arcsin ρ

a∆f0(ρ)dρ. (32)

Here,

∆f0(ρ) = f00

0(ρ) + (

/ρ)f0

0(ρ)

, and

is given by

(15)

. We also note that the

integrand in (32) can be further transformed, using the trigonometric formulas

π/2 −arcsin x=arccos x=arctan(p1−x2/x).

Finally, we recall that the function f0(r)is defined by Formula (21).

Now, when comparing the closed-form solutions due to Leonov

(19)

, Mossakovskii

(20)

and Fabrikant

(26)

with the Galin–Mossakovskii solution

(22)

and the Fabrikant solu-

tions

(27)

and

(28)

, we conclude that only the latter three formulas allow to evaluate directly

the contact pressure at the contact center.

By setting r=0 in the Galin–Mossakovskii formula (22), we readily obtain

pr=0=E∗f0(0)

πa+E∗

2π2

2π

∂f

∂ρ (ρ,ψ)dρdψ

pa2−ρ2

−E∗

2π2

2π

∆f(ρ,ψ)arctanpa2−ρ2

ρdρdψ. (33)

Symmetry 2022,14, 1083 8 of 12

By taking into account

(15)

and

(21)

, it can be easily verified that Formulas

(32)

and

(33)

are in complete agreement, and they can be rewritten as

pr=0=E∗

π





f0(0)

0(ρ)dρ

pa2−ρ2−

arccosρ

a∆f0(ρ)dρ





, (34)

where f0(r)is defined by Formula (21).

Now, by setting r=0 in the Fabrikant solution (27), we obtain

pr=0=E∗

2π2

∂

∂a

ρdρ

pa2−ρ2

2π

f(ρ,ψ)dψ

−E∗

2π2

2π

arctanpa2−ρ2

ρ∆f(ρ,ψ)dρdψ. (35)

Here the following formula is used (see Equations (23) and (24)):

L(0)f(ρ,ϕ) = 1

2π

f(ρ,τ)dτ. (36)

The first term on the right-hand side of Equation (35) can be simplified as follows:

2π

∂

∂a

2π

f(ρ,ψ)ρdρdψ

pa2−ρ2=∂

∂a

f0(ρ)ρdρ

pa2−ρ2

=∂

∂a

a f0(0) +

0(ρ)qa2−ρ2dρ

(37)

=f0(0) + a

0(ρ)dρ

pa2−ρ2.

Thus, in view of

(21)

and

(38)

, Formula

(35)

also completely agrees with Formula

(34)

Further, from the Fabrikant solution (28), in view of (21) and (36), it follows that

pr=0=E∗

πa2

pa2−ρ2d

dρρf0(ρ)dρ

−E∗

arctanpa2−ρ2

ρ∆f0(ρ)dρ, (38)

and, taking into account the identity

pa2−ρ2d

dρρf0(ρ)dρ=a f0(0) + a2

0(ρ)dρ

pa2−ρ2,

it is readily seen that Formula (38) is equivalent to Formula (34).

4. Contact Stress Intensity Factor

We define the stress-intensity factor (SIF) of the contact stresses as follows:

KI(ϕ) = −lim

r→aq2π(a−r)p(r,ϕ). (39)

Symmetry 2022,14, 1083 9 of 12

It is to note that the normal stress produced by the punch on the surface points inside

the contact area is equal to

−p(r

ϕ)

. It also is worth noting that the contact SIF analysis

under a circular punch was considered recently in [16].

4.1. Borodachev’s Formula for the Contact SIF

By using the Fabrikant solution

(28)

, Borodachev (1991) derived the following closed-

form result [27]:

KI(ϕ) = E∗a1/2

2π3/2

2π

dψ

0ρf(ρ,ψ)−a f (a,ϕ)dρ

pa2−ρ2a2−2aρcos(ϕ−ψ) + ρ2. (40)

It is warned that different normalizations can be used in the definition of the SIF.

4.2. Fabrikant’s Formula for the Contact SIF

By utilizing his general solution (26) and the general property

lim

r→a √a−rd

g(x)dx

√x2−r2!=−g(a)

√2a,

Fabrikant (1998) derived the following formula [28]:

KI(ϕ) = E∗

√πaL1

a∂

∂a

ρdρ

pa2−ρ2L(ρ)f(ρ,ϕ), (41)

where the L-operator is defined by (23).

4.3. A New Formula for the Contact SIF

Observe that the general solutions

(19)

and

(20)

given by Leonov and Mossakovskii

contain the application of the Laplace differential operator

∆

to the double integral,

and therefore, the direct usage of Formula

(39)

for evaluating the contact SIF is impossi-

ble. On the other hand, the Galin–Mossakovskii solution

(22)

employs the operation of

differentiation only under the integral sign.

Let us assume that the function

f(r

ϕ)

is twice continuously differentiable over the

closed circle 0

≤r≤a

, 0

≤ϕ<

. Then, it can be shown (see Appendix A) that the third

term on the right-hand side of Equation

(22)

is not singular at the contact contour. Hence,

from Equations (22) and (39), it follows that

KI(ϕ) = −E∗f0(0)

√πa−E∗√a

2π3/2

2π

(a2−ρ2)∂f

∂ρ (ρ,ψ) + 2asin(ϕ−ψ)∂f

∂ψ (ρ,ψ)

pa2−ρ2a2−2aρcos(ϕ−ψ) + ρ2dρdψ. (42)

Using integration by parts, it can be shown that Formula

(40)

follows from

(42)

, that

is—in other words—Formulas (40) and (42) are equivalent.

5. Discussion

First, we observe that all general solutions considered above make use of both the

operations of integration and differentiation. From the computational point of view, it

is preferable to avoid differentiating integrals with respect to parameters. That is why,

Mossakovskii’s series solution

(14)

–

(16)

is, in this sense, better than Copson’s series solu-

tion (12) and (13).

Among the closed-form general solutions, a special interest represents the Galin–

Mossakovskii solution

(22)

, since all differentiations are performed under the integral

sign. Moreover, in Formula

(22)

, all operations of differentiation are applied directly to

Symmetry 2022,14, 1083 10 of 12

the function

f(r

ϕ)

, and thus, are equivalent to differentiating the shape function of the

punch Φ(r,ϕ).

Of note, Fabrikant’s solutions

(26)

(27)

and

(28)

are given by in terms of the

operator, and their effective application requires the knowledge of its properties (for

instance, L(k1)L(k2) = L(k1k2)).

Another important point to note is the apparent singularity of the Leonov and

Mossakovskii general solutions, as each of the terms on the right-hand sides of

Equations (19)

and

(20)

, generally speaking, has a singularity of the order

(a−r)−3/2

r→a

. At the

same time, the sought-for solution of Equation

(7)

, generally speaking, should possess the

square root singularity, that is, the singularity of the order

(a−r)−1/2

r→a

. This means

that the higher-order singularity terms should cancel each other.

It is necessary to note here that the general solutions outlined above hold true in a

more general case of a transversely isotropic elastic half-space, provided that the plane of

isotropy is parallel to the half-space surface (see, for example, [

]). The potential for further

generalization and development of the general solutions presented above relies on the fact

that the problem of elastic contact is a core issue in similar contact problems with a circular

contact region for functionally graded [

], viscoelastic [

], thermoelastic [

poroelastic [

], magneto-electro-elastic [

], multiferroic [

] semi-infinite media

as well for elastic semi-infinite media with surface effects [41,42].

The main results of the present paper are given by Formulas

(34)

and

(42)

, which in

view of (8), can be rewritten as follows:

pr=0=E∗

π





δ0

a−

Φ0

0(ρ)dρ

pa2−ρ2+

arccosρ

a∆Φ0(ρ)dρ





, (43)

KI(ϕ) = −E∗δ0

√πa+E∗√a

2π3/2

2π

0(qa2−ρ2∂Φ

∂ρ (ρ,ψ)

+2asin(ϕ−ψ)

pa2−ρ2

∂Φ

∂ψ (ρ,ψ))dρdψ

a2−2aρcos(ϕ−ψ) + ρ2. (44)

Here, Φ0

0(r)is the angle-averaged shape function, i.e.,

Φ0(r) = 1

2π

Φ(r,ψ)dψ.

It is of interest to note that, in contrast to formulas due to Borodachev

(40)

and

Fabrikant

(41)

, Formula

(44)

separates the contributions to the contact SIF from the punch

displacement δ0and the punch shape function Φ(r,ϕ).

To the best of the author’s knowledge, Formulas

(43)

and

(44)

have been reported in

the literature for the first time.

The Galin–Mossakovskii general solution is recommended for use as the most simple

closed-form general solution of the non-axisymmetric Boussinesq contact problem.

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: Not applicable.

Acknowledgments:

The financial support from the Ba-Yu Scholar program of Chongqing City (China)

is gratefully acknowledged. We acknowledge support by the German Research Foundation and the

Open Access Publication Fund of TU Berlin.

Symmetry 2022,14, 1083 11 of 12

Conflicts of Interest: The author declares no conflict of interest.

Appendix A. Derivation of the Contact SIF from the Galin–Mossakovskii

General Solution

By setting r=a(1−ε)in Equation (22), we can rewrite

pa(1−ε),ϕ=E∗

πa

f0(0)

pε(2−ε)

+E∗

2π2

pε(2−ε)

2π

a2−ρ2(1−ε)2∂f

∂ρ (ρ,ψ) + 2a(1−ε)sin(ϕ−ψ)∂f

∂ψ (ρ,ψ)

pa2−ρ2a2−2a(1−ε)ρcos(ϕ−ψ) + (1−ε)2ρ2dρdψ

−E∗

2π2

2π

∆f(ρ,ψ)arctanpε(2−ε)pa2−ρ2

Rερdρdψ

Rε,

(A1)

where Rε=pa2(1−ε)2+ρ2−2a(1−ε)ρcos(ϕ−ψ).

Further, by the definition (39), we have

KI(ϕ) = −lim

ε→0+√2πaεpa(1−ε),ϕ.(A2)

Hence, by substituting (A1) into Formula (A2) and letting

tend to zero, we immedi-

ately arrive at Formula

(42)

, since the third term on the right-hand side of Equation (A1) is

not singular as ε→0.

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